Gas Sorption and Swelling in Glassy Polymers
Combining Experiment, Phenomenological Models and
Detailed Atomistic Molecular Modeling
von der
Fakultät III Prozesswissenschaften
der Technischen Universität Berlin
genehmigte
DISSERTATION
zur Erlangung des akademischen Grades
doctor rerum naturalium
(Dr. rer. nat.)
vorgelegt von
Herrn Dipl.-Phys. Ole Hölck
geboren am 06.06.1972 in Kiel
Promotionsausschuss:
Vorsitzender: Prof. Dr. Bernhard Senge
1. Gutachter: Prof. Dr.-Ing. Manfred H. Wagner
2. Gutachter: Prof. Dr. Dieter Hofmann
Tag der wissenschaftlichen Aussprache: 22.01.2008
Berlin 2008
D 83
Zusammenfassung
Durchgeführte Arbeiten: In der vorliegenden Arbeit werden grundle-
gende Fragen zur Gas-Sorption und der damit verbundenen Quellung von gla-
sigen Polymeren untersucht. Dabei wurde eine kombinierte Analyse aus ex-
perimenteller Charakterisierung, detailliert atomistischer Modellierung und
phänomenologisch-theoretischer Betrachtung angewandt.
Drei unterschiedliche Polymere, ein Polysulfon (PSU), ein Polyimid (PI4) und
ein neuartiges Polymermaterial (PIM-1), wurden experimentell bezüglich ih-
rer Sorptions- und Quellungseigenschaften unter CO2- und CH4-Atmosphären
bis zu 50 bar charakterisiert. Die Kinetik der Prozesse der experimentell ge-
messenen Gasaufnahme und Volumendilatation wurde analysiert, wobei zwei
Anteile unterschieden werden konnten: Ein diffusiv/elastischer Anteil und
ein relaxiver Anteil, der bei längerer Messzeit und höheren Drücken signifi-
kant wird. Zusätzlich zu dieser Ermittlung von Konzentrations-Druck- bzw.
Dilatations-Druck-Isothermen (differentielle Messung) wurden sogenannte in-
tegrale Sorptions- und Dilatationsmessungen (‘Ein-Schritt’-Messungen) durch-
geführt, die als Referenz für eine entsprechende molekulardynamische (MD)
Simulation elastischer Dilatationseffekte dienten.
Für die detailliert atomistischen MD Simulationen wurden equilibrierte Pa-
ckungsmodelle der (ungequollenen) Polymere erstellt. Für jeweils einen weite-
ren Referenzzustand, charakterisiert durch Druck, aufgenommene Gasmenge
und Quellung, wurden CO2- und CH4-beladene (gequollene) Packungsmo-
delle erstellt und durch NpT-MD Simulation equilibriert. Die Packungsmo-
delle der reinen (ungequollenen) Polymere und der CO2-gequollene Zustand
wurden jeweils bezüglich ihres freien Volumens quantitativ charakterisiert.
Die gefundenen Unterschiede konnten weiterhin durch eine detaillierte 3D-
Visualisierung des Freien Volumens veranschaulicht werden.
Für alle Packungsmodelle wurden großkanonische Monte Carlo (gcmc) Si-
mulationen durchgeführt, die jeweils zu Konzentrations-Druck-Isothermen
führten. Experimentelle Daten und Simulationsergebnisse wurden in Bezug
auf drei theoretische Modelle (Dual Mode Sorption Model (dm), Site Distri-
bution Model (sd), und Non-Equilibrium Thermodynamics of Glassy Poly-
mers (net-gp)) ausgewertet und diskutiert.
Außerdem wurden die ungequollenen Packungsmodelle nach experimentel-
ler Spezifikation aus integralen Sorptionsmessungen mit dem jeweiligen Gas
beladen und die elastische Dilatation während der folgenden NpT-MD Equi-
librierung beobachtet.
Ergebnisse: Die konsistente Anwendung der Ergebnisse der kinetischen
Analyse führt zu einer verbesserten Übereinstimmung theoretischer Modelle
mit dem Experiment und deutlich zuverlässigerer Bestimmung von Modell-
parametern. Deren physikalische Bedeutung ermöglicht erste Ansätze für ei-
ne universelle Beschreibung grundlegender Stofftransporteigenschaften. Eine
mögliche Limitierung des sd Modells bezüglich der Anwendbarkeit auf hoch-
freivolumige Polymere, wie sie im Rahmen dieser Arbeit festgestellt wurde,
konnte dahingehend erklärt werden, daß die Gas-Matrix Wechselwirkung eine
andere Besetzungsreihenfolge von Sorptionsplätzen bedingt, als bisher vom
sd Modell angenommen.
Die simulierten Isothermen konnten durch die Einführung eines linearen
Übergangs zwischen der jeweils für ungequollenes und gequollenes Packungs-
modell berechneten gcmc-Isothermen in gute Übereinstimmung mit dem Ex-
periment gebracht werden. Die Verwandschaft der Methode mit dem net-gp
Modell wird erörtert und die Ergebnisse mit der Vorhersage der Sorption des
Modells verglichen.
Die Freie-Volumen-Analyse zeigt deutliche Unterschiede zwischen den Poly-
meren PSU und PI4 einerseits und PIM-1 andererseits. In PIM-1 liegt ein
großer Teil des Freien Volumens in einer Art losem Verbund einzelner ‘Lö-
cher’ vor, der in dieser Arbeit als ‘Lochphase’ (‘void phase’) diskutiert wird.
Im Gegensatz zu den gequollenen Packungen von PSU zeigen diejenigen von
PI4 bereits Ansätze zur Bildung einer solchen Lochphase. Diese ist in ge-
quollenen PIM-1 Packungen stärker ausgeprägt. Die gefundenen Größenver-
teilungen Freier Volumen-Elemente in den ungequollenen Packungsmodellen
zeigen eine gute Übereinstimmung mit den aus der sd Analyse erhaltenen
Gaußverteilungen.
Außerdem konnte die elastische Dilatation, die in integralen Messungen ex-
perimentell beobachtet wird, erfolgreich in entsprechenden MD-Simulationen
nachempfunden werden. Ein Zusammenhang zwischen den Abweichungen der
Absolutwerte und möglichen anelastischen Reaktionen der Polymeratrix wird
diskutiert.
Die gute Übereinstimmung experimenteller Ergebnisse mit den Resultaten
aus der Simulation sorgfältig ausgewählter Aspekte der Gas-Sorption und
Quellung in glasigen Polymeren bestätigt sowohl die Qualität der erstell-
ten Packungsmodelle als auch die Vorteile der generellen Herangehensweise.
Die Ergebnisse dieser Arbeit zeigen, dass die Diskrepanz in Zeitskalen von
Experiment und Simulation und die daraus resultierende scheinbare Inkom-
patibilität mit Hilfe phänomenologischer Modelle überbrückt werden kann,
und damit wertvolle Erkenntnisse über die zugrunde liegenden Phänomene
gewonnen werden.
Contents
1 Introduction 1
2 Background 6
2.1 Glassy Polymers and Free Volume . . . . . . . . . . . . . . . . 6
2.2 Permeability, Solubility and Diffusion . . . . . . . . . . . . . . 8
2.3 Partial Molar Volume . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Stress-strain Relationships . . . . . . . . . . . . . . . . . . . . 12
3 Theoretical Models 15
3.1 Dual Mode Sorption Model . . . . . . . . . . . . . . . . . . . 15
3.2 NET-GPModel.......................... 17
3.3 Site Distribution Model . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Summary ............................. 25
4 Experimental 26
4.1 Materials ............................. 26
4.1.1 Gases ........................... 26
4.1.2 Polymers.......................... 26
4.2 Sorption Measurements . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Dilation Measurements . . . . . . . . . . . . . . . . . . . . . . 30
4.4 Kinetic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.5 Experimental Results and Discussion . . . . . . . . . . . . . . 35
4.5.1 Sorption and Dilation Isotherms . . . . . . . . . . . . . 35
4.5.2 Integral Sorption and Dilation . . . . . . . . . . . . . . 46
i
5 Modeling 53
5.1 Forcefield based Molecular Modeling . . . . . . . . . . . . . . 54
5.1.1 The Forcefield . . . . . . . . . . . . . . . . . . . . . . . 55
5.1.2 Molecular Dynamics (MD) . . . . . . . . . . . . . . . . 56
5.1.3 The Concept of Ensembles . . . . . . . . . . . . . . . . 57
5.1.4 Periodic Boundary Conditions . . . . . . . . . . . . . . 58
5.1.5 Packing Procedure . . . . . . . . . . . . . . . . . . . . 59
5.2 Modeling Techniques . . . . . . . . . . . . . . . . . . . . . . . 62
5.2.1 Free Volume Analysis . . . . . . . . . . . . . . . . . . . 62
5.2.2 Grand Canonical Monte Carlo Calculations . . . . . . . 64
5.2.3 Integral Dilation Simulation . . . . . . . . . . . . . . . 65
5.3 Modeling Results and Discussion . . . . . . . . . . . . . . . . 67
5.3.1 Detailed Atomistic Molecular Packing Models . . . . . 67
5.3.2 Free Volume Analysis................... 70
5.3.3 Sorption Isotherms . . . . . . . . . . . . . . . . . . . . 78
5.3.4 Integral Dilation Simulation . . . . . . . . . . . . . . . 84
6 Combined Discussion 90
6.1 Sorption Isotherms . . . . . . . . . . . . . . . . . . . . . . . . 90
6.2 Free Volume Distributions .................... 97
6.3 Integral Dilation . . . . . . . . . . . . . . . . . . . . . . . . . 105
7 Synopsis 110
Bibliography 114
A Appendix 125
A.1 Abbreviations...........................125
A.2 Selected Notations . . . . . . . . . . . . . . . . . . . . . . . . 126
A.3 Slices of Packing Models . . . . . . . . . . . . . . . . . . . . . 127
A.4 Rotation Analysis of Bond Angles . . . . . . . . . . . . . . . . 130
ii
Preliminary Remarks
This work was prepared in the framework of a joint project between the Federal
Institute for Materials Research and Testing (Bundesanstalt für Materialforschung
und -prüfung, BAM) and the GKSS-Forschungszentrum Geesthacht (GKSS Re-
search Center). The project was financially supported by the Deutsche Forschungs-
gemeinschaft (DFG, German Research Foundation). All experimental measure-
ments were conducted at the BAM, which also hosted the author of this work.
All simulation efforts were performed at the Institut für Polymerforschung of the
GKSS in Teltow (Institute for Polymer Research). Data analysis was carried out
at both project partners’ facilities. Experiments and data analysis of both, sim-
ulated and experimental results were predominantly conducted by the author of
this work. However, operation of the simulations, in particular the procedure of
packing detailed atomistic molecular models, strongly depended on the expertise
of the project partners at the GKSS Research Center in Teltow, where the author
could resort to the basic procedures of packing detailed models, as well as exist-
ing analyzing scripts. The packing and equilibration of the models of PSU and
PSU80,PIM ,PIM95 and PIM206, as well as the bond angle analysis on PSU
and PI4 were realized by the modeling group of Prof. D. Hofmann of the GKSS
Research Center in Teltow, as part of the joint research project. The packing of
the models PSU35,PI4,PI76 and PI156 was operated under the supervision of
Dr. M. Heuchel (gkss) by the author of this thesis.
Besides the publication of two abstracts for the Conference Proceedings of the Eu-
romembrane 2006 (Heuchel et al., Desal. 199(1):443, 2006 and Hölck et al., Desal.
200(1):166, 2006), some of the results of this work have already been published or
have been accepted for publication in peer reviewed scientific journals:
[1]: M. Heuchel, M. Böhning, O. Hölck, M. R. Siegert, and D. Hofmann. J. Polym. Sci.
B: Polym. Phys., 44(13):1874-1897, 2006.
[2]: O. Hölck, M. R. Siegert, M. Heuchel, and M. Böhning. Macromolecules, 39(26):9590-
9604, 2006.
[3] O. Hölck, M. Heuchel, M. Böhning, and D. Hofmann. J. Polym. Sci. B: Polym. Phys.,
(in press), 2007.
The concept of preswollen packing models was introduced in [1] where also sorption
isotherms were calculated using the gcmc technique. In the second publication
[2], the experimental sorption and dilation data were kinetically and phenomeno-
logically analyzed and results were compared to nonswollen and swollen packing
models of PSU with respect to CO2-sorption, free volume and integral dilation.
The experimental and simulated integral dilation results of the polymers PSU and
PI4, induced by CO2and CH4sorption and insertion, respectively, were published
in reference [3]. However, these aspects are presented and discussed in full context
for the first time in this work.
iii
1 Introduction
The behavior of amorphous polymers in contact with gas atmospheres is
still an area of both fundamental scientific and applied industrial research.
Applications range from the use as barrier materials or protective coatings
to active layers in sensor applications (‘artificial nose’) and the large field
of gas separation membranes. In all these applications, high concentrations
of small penetrant molecules may lead to a plasticization of the polymer.
This effect is utilized in processing applications, where supercritical carbon
dioxide (CO2) can be used as a plasticizer.4The phenomenon of penetrant
induced plasticization of glassy polymers is also observed in gas separation
membranes.5In the process of natural gas sweetening, the CO2content of the
gas mixture is reduced by separation of the CO2from the fuel gas methane
(CH4) to avoid corrosion of pipelines and to enhance the fuel value. Solubility
and diffusivity of the respective gas determine the separation performance
of the membrane material, i.e., the permselectivity. Both parameters are
connected to the internal structure of the polymer and its free volume. To
achieve high throughputs, e.g. to enhance cost-effectiveness, it is desirable
to increase the CO2solubility and mobility. However, the observed plas-
ticization and the associated relaxations in the polymer matrix change its
structure and free volume, and thereby affect the selectivity of the material.6
In addition, other properties of the polymer are influenced, e.g. a reduction
of glass transition temperature,7yield stress8and creep compliance9have
been observed. The origin and mechanism of these structural relaxations are
poorly understood, as are the factors that influence solubility and mobility of
the plasticizing penetrant. This lack of knowledge leads to a development of
new or optimized materials, which is in part determined by trial and error. A
deeper understanding of the phenomena that accompany gas sorption on the
molecular level is therefore needed to control material properties and enable
a targeted design of functional materials. Therefore, in this work, laboratory
experiments are combined with detailed atomistic molecular simulations.
1
1. Introduction
Modeling. In detailed atomistic molecular modeling, the interactions of
an assembly of atoms, e.g. a polymer molecule, are calculated according to
known physical laws. Several established analysis methods allow an indirect
determination of certain properties of such assemblies, others can even be
directly calculated.10 However, CPU-power limits both the size and the sim-
ulation time of such assemblies. The size of the simulated packing models
used in this work (∼5000 atoms) ranges among the larger models found in
the literature. Forcefield based Molecular Dynamics (MD) simulations are
calculated in femtosecond steps, but reliable results are usually not obtained
until a nanosecond of net simulation time has been performed. Millions of
interactions need to be calculated, making the time effort for these ‘virtual
experiments’ comparable to laboratory experiments. However, increasing
speed of single processors and the possibility of parallel processing will fur-
ther reduce the evaluation times for such simulations in the future. The goal
of computer simulations is therefore to establish reliable methods to predict
material properties. Properties of new materials could then be assessed by
simulations first and only the most promising materials need to be synthe-
sized for further testing, reducing the expense of trial and error.
Although some methods already exist to predict polymer/gas properties from
simulations, which show well agreeing results in ideal circumstances, they fre-
quently fail when applied to less moderate conditions, e.g., high penetrant
concentrations, long time scales, large penetrants etc. The aforementioned
gas induced plasticization of polymers presents such a case where the gap
of time scales between experiment and available simulation time amounts to
several orders of magnitude. The time scale of simulations is limited to a few
nanoseconds and therefore it is not possible to directly simulate relaxations of
the glassy matrix as they are observed experimentally. Experiments, on the
other hand, yield results of the real macroscopic system, and though molec-
ular details cannot be observed individually, the accumulated effects permit
the analysis through models on a statistical or phenomenological basis. It
is the aim of this work to survey new approaches of a combined analysis of
experimental and modeling results and to establish, where possible, a conver-
gence of boundary conditions or, alternatively, an identification and isolation
of comparable aspects of these seemingly incompatible methods of research.
To this effect, phenomenological models are utilized as a means of interpreta-
tion of experimental data as well as to construe modeling results and thereby
putting the assumptions and implications of these models to the test.
Phenomenological Models. The well known Dual Mode Sorption Model
(dm model),11, 12 is the most widely used model to describe gas sorption
2
1. Introduction
in glassy polymers, due to its easy applicability and the ability to success-
fully describe sorption in a wide variety of polymer/gas systems. Though
the initial assumption of two distinct penetrant populations could not un-
ambiguously be confirmed,13 the concept has proven to be flexible, and the
parameters are beyond their physical interpretation a valuable means to in-
terpolate or exchange data.
Several other authors successfully developed phenomenological models to de-
scribe or predict penetrant concentration in glassy polymers.5Most of these
models approach sorption from the thermodynamic point of view,13–16 often
treating the system of polymer matrix, free volume and penetrants as a lat-
tice of partly occupied sites,17–19 extending earlier models of lattice fluids20
by introducing order parameters that describe the non-equilibrium nature of
the glassy state.
This is done in the Non-Equilibrium Thermodynamics of Glassy Polymers
(net-gp) model, by considering the density as a parameter indicating the
deviation from equilibrium. Using an equation of state, applying suitable
mixing rules for the two components, polymer and penetrant, and assuming
the density of the matrix to decrease linearly with penetrant concentration,
a quasi-equilibrium concentration can be calculated for each pressure (chem-
ical potential) of the penetrant gas. The calculation of Grand Canonical
Monte Carlo (gcmc) isotherms in conjunction with the transition between
discrete swollen states which will be introduced later in this work, correspond
exactly to the net-gp procedure and the respective results will be compared
accordingly.
Free Volume. It is agreed upon in the literature, that the free volume
plays a major role in penetrant solubility, transport and matrix mobility.21 A
better understanding of its structure, volume fraction and distribution will
also contribute to the understanding of sorption and swelling phenomena.
The Site Distribution (sd) model explicitly constitutes a conception of the
free volume. Though this model does not incorporate relaxational swelling of
the matrix, its main feature comprises the penetrant-induced elastic stresses,
which have been recognized by Newns.22 Newns suggested that the second-
stage-sorption of vapors into glassy polymers is controlled by the rates of
relaxations in the polymer. The rapid initial stage of sorption, following
Fickian diffusion kinetics until a quasi-equilibrium is reached, was thought
to induce stresses within the polymer matrix. Limited mobility of the matrix
would lead to a relaxation of the stresses and hence a decrease in chemical
potential of sorbed penetrant molecules, relatively to the gas or vapor phase.
This again would lead to a self sustaining cycle of sorption-induced stresses
3
1. Introduction
and relaxations which is only limited by the ability of the matrix to relax
stresses and thus by the (concentration dependent) plasticizing ability of the
penetrant molecule. Following this interpretation, in order to understand
relaxational swelling behavior, the forces that lead to the softening and sub-
sequent relaxation need to be understood. In contrast to the thermodynamic
approach, the sd model relates the distribution of sorption site volumes to
a distribution of corresponding site energies. In this mechanical view, the
partial molar volume connects the solubility of penetrant molecules to the
structure of the free volume of the polymer matrix. It thus provides a favor-
able basis of comparison to detailed atomistic packing models, which can be
analysed with respect to the free volume by the insertion of a small probe
sphere, scanning the static polymer matrix following a grid pattern to detect
non-occupied space.
Laboratory Experiments. Gas separation with polymeric membranes is
a rather complex process, involving a gas mixture and a steady-state concen-
tration gradient through the membrane. Properties of the membrane mate-
rial and its changes as well as transport properties of the gases are therefore
not easily attributed to individual phenomena. The fundamental processes of
penetrant sorption, transport (diffusion) and relaxation of the matrix which
are underlying the gas separation are, with regard to the combined analysis
of experiment, phenomenology and simulation, better investigated utilizing
sorption and dilation isotherms of single gases. Here, the step by step pro-
cedure of increasing the pressure allows a thorough kinetic analysis and the
determination of (quasi-) equilibrium conditions. In addition to the com-
parison of sorption isotherms and size distributions, this thesis presents an
attempt to further investigate the experimentally observed dilation induced
by integral gas sorption. The volume change of specially prepared pack-
ing models of binary mixtures under molecular dynamics simulations (NpT)
show satisfying agreement to the experimental results and opens promising
possibilities to investigate the elastic nature of the dilation process in its ini-
tial state, as well as the partial molar volumes of the penetrant gases in the
polymers investigated in this work.
Investigated Systems. Six polymer/gas systems were selected for inves-
tigation in this work. CO2and CH4were selected as penetrant gases because
of their vital roles in industrial applications, as mentioned in the begin-
ning. Polysulfone (PSU) is a widely investigated conventional glassy poly-
mer with regard to CO2sorption,23–29 and can therefore be used to validate
the experimental procedures used in this work. Polymers of the class of
4
1. Introduction
6FDA-polyimides, generally exhibiting a larger free volume than PSU, are
known to show excellent transport and solubility characteristics with respect
to gas separation applications. They furthermore tend to be susceptible to
plasticization,30 making the 6FDA-TrMPD (PI4) an ideal choice for the desired
investigations. Still larger free volume is expected to be present in Polymers
of Intrinsic Microporosity (PIMs). This new class of polymers, with struc-
tures of varying degree of order, shows promising features which have yet
to be fully investigated.31 The selection of PIM1 (see Section 4.1.2) as an
amorphous, membrane forming polymer allows for this work to contribute to
a very recent field of research.
Outline. In this thesis, three different polymers are investigated with re-
spect to the sorption of two gases and the induced dilation effects, involving
three phenomenological models to analyze the experimental data and several
simulation techniques and analysis methods regarding the modeling data.
To provide some background to the general subject, the following chapter
briefly summarizes some general features of polymer/gas systems and gas
separation. Chapter 3 addresses the phenomenological models and the rele-
vant formulae are established. In Chapter 4, the investigated polymers and
experimental methods are introduced and, to retain lucidity, the immediate
experimental results are presented at the end of this chapter, along with
the related results of phenomenological model analyses. Correspondingly,
Chapter 5 introduces to detailed atomistic molecular simulations, the applied
techniques and analysis methods, followed by the presentation of the results.
The discussion of Chapter 6 is reserved to the combined interpretation and
comparison of experiment, simulation and phenomenological models. If con-
venient, results that were presented in the preceding chapters may be shown
again, in some cases introducing enhancements of the presentation or evalu-
ation of data. This work is summarized and concluded in Chapter 7, where
a short outlook will be given as well.
5
2 Background
The polymer/gas applications named in the Introduction highly depend on
the sensitivity of the material to particular gases. Especially gas separa-
tion membranes are required to meet high standards regarding the selectivity
while at the same time the permeability of the target gas species should be
high enough to allow effective separation on an industrial scale. Glassy dense,
i.e., non-porous, polymeric materials are ‘frozen’ into a non-equilibrium state.
Between the disordered and in their mobility quite limited polymer chains,
they contain some excess free volume which provides the room (‘sorption
sites’) to accommodate penetrants and to allow movement within the poly-
mer matrix (‘diffusion paths’). Differing size and interaction with the matrix
allows the separation of small penetrants in a solution-diffusion process, the
basics of which will be shortly recapitulated below.
2.1 Glassy Polymers and Free Volume
In the thermodynamic description of materials, it is generally distinguished
between first and second order transitions.32 The first order transitions are
marked by a continuous free energy function of state variables (e.g. pressure
por temperature T) which is discontinuous in the first partial derivatives
with respect to the relevant state variables. At a melting point, a typical
first order transition, the Gibbs Free energy Gis continuous, but there is a
discontinuity in entropy S, volume Vand enthalpy H:
∂G
∂T !p
=−S ∂G
∂p !T
=V ∂(G/T)
∂(1/T)!p
=H(2.1)
Second order transitions are classically defined33 by discontinuities in the
second partial derivatives of the free energy function while the function itself
as well as the first partial derivatives S,Vor Hare continuous, leading to
6
2.1 Glassy Polymers and Free Volume 2. Background
discontinuities in the heat capacity Cp, compressibility κand the thermal
expansion coefficient γ:
∂S
∂T !p
=Cp
T ∂V
∂p !T
=−κV ∂H
∂T !p
=Cp ∂V
∂T !p
=−γV
(2.2)
The nature of the glass transition is still the subject of discussions. Though
it exhibits features of a second order transition, there is still disagreement
whether it is purely kinetic or if it is a kinetic manifestation of an underly-
ing thermodynamic transition.32 However, when a liquid is cooled to form a
glassy solid, it transforms from an equilibrium state (liquid) to a nonequilib-
rium state (glass), and its appearance, i.e., the glass transition temperature
Tg, is dependent on the rate or time scale of the experiment (see Figure 2.1),
contradicting the definition of thermodynamic phase transitions. A way to
determine the glass transition temperature Tgis provided by differential scan-
ning calorimetry (DSC). Here, the heat flow of the sample to a reference is
monitored and at the glass transition temperature Tga step is observed that
is caused by the discontinuity of the heat capacity Cp.
V
vdW
V
c
matrix volume
v.d.W aals volume
hypothetical cristal
glass
s
T
T
g
V
T
g
'
l
liquid
V
sp
Figure 2.1:
Schematic diagram of vol-
ume definitions in polymers
and temperature dependen-
cies. The glass transition Tg
depends on the cooling rate
of the liquid state.
Above the glass transition temperature, glass forming amorphous polymers
are in a liquid-like or rubbery state. At these temperatures, the enhanced
molecular motion permits assemblies of polymer chain-segments to move in
a coordinated manner, and hence allowing the material to flow,34 the degree
of cooperative movement depending on temperature and molecular weight.
The enhanced molecular mobility mutually depends on the presence of free
7
2.2 Permeability, Solubility and Diffusion 2. Background
volume which provides the space that is required for the rearrangement of
the polymer segments to take place.
With decreasing temperature, the mobility of the polymer segments decreases
and hence the specific volume Vsp of the polymer decreases according to the
thermal expansion coefficient of the liquid state γl, see Fig. 2.1. At the
glass transition temperature, long range cooperative movement of polymer
segments ceases, and while short-ranged rearrangements of individual mobile
units of the polymer chain may still be possible, the restricted coordinated
movement of the entangled macromolecular chains below the glass transition
temperature turns the polymer into a solid glass, preserving the amorphous
structure of the liquid or rubbery state.
Volume changes now follow the thermal expansion coefficient of the solid
γs. The transition temperature Tgdepends on the rate of cooling, as does
the amount of frozen-in free volume. At slower rates, the glass transition
temperature decreases (T0
g) and less free volume is conserved. Figure 2.1 also
serves to illustrate different concepts of free volume. Starting from the specific
volume of the polymer Vsp different occupied volumes may be subtracted to
give a measure of the free volume:
i) Vsp −VvdW , the van der Waals volume of the polymer chains, gives the
free volume at 0 K.
ii) Vsp −Vc, the volume of a hypothetical, close packed crystal, gives the
excess free volume
iii) Vsp −Vl, the extrapolated volume of an undercooled liquid, gives the
amount of unrelaxed free volume
Commonly glassy polymers are characterized with respect to the free volume
by calculation of the fractional free volume using the method of Bondi:35
ΦF V =Vfree
Vsp
= 1 −1.3VvdW
Vsp
(2.3)
2.2 Permeability, Solubility and Diffusion
Gas separation with membranes made from dense amorphous polymers under
a concentration gradient is based on a solution-diffusion mechanism,5that is
the sorption and transport of small molecules (penetrants), which essentially
depends on the free volume and its size distribution. The solution-diffusion
8
2.2 Permeability, Solubility and Diffusion 2. Background
mechanism comprises the three steps of (i) sorption of the gas molecules at
the so called ‘feed’ or ‘upstream’ side of the membrane, (ii) diffusion of the
molecules through the dense matrix and (iii) desorption of the gas molecules
at the ‘permeate’ or ‘downstream’ side. Mass transport of a penetrant species
ithrough a membrane of thickness dis given by the flux Ji:36
Ji=Pi
pup −pdown
d(2.4)
where pup and pdown denote the feed and permeate pressure and Piis the
coefficient of permeability of species i. Characteristic for the separating
performance of a membrane with respect to a gas pair i, j is the ratio of
permeabilities, called the ideal selectivity αij:
αij =Pi
Pj
(2.5)
The ideal selectivity αij is determined by separate measurements of the per-
meabilities for the two gases i, j; for real gas mixtures different partial pres-
sures and competitive sorption have to be taken into account.37, 38
Driving force of the permeation process is the pressure difference between
up- and downstream side of the membrane ∆p=pup −pdown, which leads to
a difference of the concentration ∆Cbetween the gas/polymer interfaces
∆C=S(pup −pdown) (2.6)
where Sdenotes the solubility of the gas in the polymer. The flux within the
polymer membrane is, according to Fick’s first law for an isotropic material
J=−D ∂C
∂x !(2.7)
where xis the direction perpendicular to the membrane plane. The constant
Dis called the diffusion coefficient]and gives a measure of the penetrant
mobility in the polymer matrix. Equation 2.7 is valid only in the steady
state, when in the process of permeation a constant concentration gradient
is established. In this case, the time dependency formulated in Fick’s second
law ∂C
∂t != ∂2C
∂x2!(2.8)
]Throughout this work, by ‘diffusion coefficient’ the coefficient of mutual diffusion is
meant. In contrast to the tracer diffusion, which describes the statistical motion of a single
particle, the driving force for mutual diffusion is a concentration gradient.
9
2.2 Permeability, Solubility and Diffusion 2. Background
vanishes and, assuming a linear concentration gradient (Dindependent of
concentration), the flux is
J=−D dC
dx !=−D∆C
∆x=DCup −Cdown
d(2.9)
At constant upstream pressure pup and small downstream pressure pdown ≈0,
as is usually the case in permeation setups, equations 2.4, 2.6 and 2.9 yield
the permeability coefficient
Pi=SiDi(2.10)
In a typical permeation measurement, the pressure increase with time in a
small downstream volume is monitored at nearly constant upstream pressure.
Assuming the steady state condition, the permeability Pis then retrieved
from the slope of the curve, whereas the diffusion coefficient can be calculated
from the time lag (zero pressure extrapolation of the steady state regime).
However, the aforementioned conditions are not necessarily valid in longtime
experiments and at high concentrations. The phenomenon of swelling alters
the structure of the free volume making the direct observation of the applica-
tion relevant property Pdifficult. Equation 2.10 emphasizes that solubility
and diffusion are the fundamental processes which determine the permeabil-
ity. Therefore, in this work, experiments were performed which allow more
reliable assertions of the fundamental processes in question. In the sorption
measurements, which will be discussed in more detail in Section 4.2, the phe-
nomenon of mass uptake (sorption) is observed directly and time resolved,
allowing the determination of the diffusion coefficient D. For a plane sheet
geometry, Fick’s second law has been solved by Crank:39
M(t)
M∞
= 1 −
∞
X
n=0
8
(2n+ 1)2π2exp(−D(2n+ 1)2π2t/d2) (2.11)
where M(t) and M∞denote the total mass uptake at time tand at infinite
time, respectively. By variation of M∞and the diffusion coefficient D, equa-
tion 2.11 can be fitted to sorption data, yielding the information on solubility
(using eq. 2.6) and diffusion for a polymer/gas system (see also Section 4.4).
10
2.3 Partial Molar Volume 2. Background
2.3 Partial Molar Volume
The molar volume Vm,i of a pure species imay be defined as the ratio between
its volume Viand its quantity ni, measured in moles, at a given temperature
and pressure:
Vm,i =Vi
niT,p
(2.12)
Formally, the partial molar volume (pmv)˜
Vm,i of a species iin a mixed system
of volume Vis defined as the variation of the volume with the amount of
substance niof species iand constant amount of other species nj6=i
˜
Vm,i = ∂V
∂ni!T,p,nj6=i
(2.13)
In practice, it may be measured as the ratio of the change in the volume of
the system ∆V, and in the number of molecules ∆niof species i:
˜
Vm,i =∆V
∆ni
(2.14)
In liquids or rubbery polymers, the mobility of the matrix is sufficient to
compensate the insertion of a penetrant molecule into a site of free volume
such that the latter is kept nearly constant. In this case the relaxation
around a penetrant molecule is complete and the pmv ˜
Vm,i may be regarded
as a ‘dynamic volume’ Vg,i, and thus as a property of the penetrant phase of
species i:˜
Vm,i =Vg,i (2.15)
In glassy polymers the situation is different. The mobility of the matrix is
limited, and therefore occupation of free volume cannot be fully compen-
sated within reasonable time scales. The consumption of free volume by the
penetrant molecules leads to a smaller partial molar volume which must now
be regarded as a property of the matrix/penetrant system. A mechanical
interpretation of this phenomenon is provided by Eshelby:40
The volume change ∆Vof an elastic continuum containing a spherical hole
of volume Vh, upon occupation of this hole with an elastic sphere of volume
Vg≥Vhis given by
∆V= const(Vg−Vh) = ˜
Vi(2.16)
where const = 1 if penetrant and matrix have the same elastic properties.
The partial molar volume ˜
Vitherefore reveals information about the structure
of the free volume, provided that the penetrant phase volume Vg,i of species
11
2.4 Stress-strain Relationships 2. Background
iis known and the matrix can be regarded as an elastic continuum. For
convenience, the ‘tilde’, indicating partial molar volumes will be omitted in
the following. Vgwill be used to indicate the molar ‘dynamic volume’ of the
penetrants (CO2and CH4) and Vp≡˜
Vm,i for their partial molar volumes.
Vhdenotes the molar volume of sorption sites (see Section 3.3). On rare
occasions, for the sake of a simplified discussion, the denominations are used
to describe single holes or penetrants, their meaning being easily recognized
out of the context. In addition, Appendix A.2 provides a list of selected
notations used in this thesis.
2.4 Stress-strain Relationships
Polymers below their glass transition temperature Tgexhibit complex behav-
ior when subjected to an external stress. While chemical bonds and physical
entanglements of the polymer chain prevent long range cooperative motion
and ensure a certain form stability that gives rise to an elastic reaction of the
polymer matrix, localized rearrangements of smaller subunits may still take
place below the glass transition and lead to a limited relaxation. This behav-
ior is called viscoelastic. For viscoelastic materials, the mechanical behavior,
that is, the response to a (uniaxial) stress with a strain closely resembles
that of models built from discrete elastic and viscous elements.41 If a mate-
rial is linear-elastic, the relationship between the stress σand the strain is
described by Hooke’s law
σ=E (2.17)
st ra in
time t
stress
Figure 2.2: The stress-strain relationship of a spring module is propor-
tional and instantaneous.
12
2.4 Stress-strain Relationships 2. Background
where Eis the Young’s modulus. The reaction of a purely elastic material is
therefore instantaneous, proportional to the stress. The element representing
elastic behavior is a spring. When the applied stress is removed, the strain
recovers in full as depicted in Figure 2.2.
In materials exhibiting viscous flow, on the other hand, the stress is propor-
tional to the strain rate, according to Newton’s law
σ=ηd
dt (2.18)
where ηis the viscosity.34 Viscous flow, represented by a dashpot, corresponds
to a plastic deformation of the polymer matrix and is not recovered upon
removal of the stress (Figure 2.3).
st ra in
time t
stress
Figure 2.3: The strain of a dashpot module is proportional to the stress
rate and linear in time.
The two models may be connected in various ways, depending on the na-
ture of viscoelasticity specific to the material. A common combination to
describe a single retardational process of a glassy polymer, is the ‘Kelvin’
model (sometimes called the ‘Voigt’ model), where spring and dashpot are
connected in parallel (Figure 2.4). The stress-strain relationship becomes34
σ=ηd
dt +E (2.19)
and under conditions of constant stress, equation 2.19 can be integrated to
=σ
E(1 −exp(−t/τ)), τ =η
E(2.20)
13
2.4 Stress-strain Relationships 2. Background
where τdenotes the retardation time. Generally, deformations in glassy
polymers are not recovered upon removal of the stress. However, the Kelvin
model implies a reversibility of the strain. This apparent contradiction can
be solved, if different time constants are assumed for strain and recovery.
st ra in
time t
stress
Figure 2.4: The Kelvin module is a combination of spring and dashpot.
Its response to a stress is reversible but nonlinear in time.
In this work, the term ‘relaxational’ will be used in a more general sense
as the generic term for retardational (viscoelastic behaviour under constant
stress) as well as relaxational behaviour (constant strain), as is common in
the literature regarding sorption induced swelling of glassy polymers. Since
most often relaxational behavior cannot be attributed to one single process
but a spectrum of relaxation times has to be assumed in addition to an elastic
response of the polymer matrix, a superposition of an infinite sum of Kelvin
models seems more appropriate to completely describe the complex behavior
of viscoelastic polymers. However, in most cases one or two Kelvin models
in series with a spring are sufficient to describe experimental data42, 43 as is
the case in this work, where a spring and two Kelvin models are connected
in series (eq. 4.3) to kinetically analyse experimental sorption and dilation
data (see Section 4.4).
The Kelvin model (Fig. 2.4) may also serve to describe anelastic behavior.
Anelastic materials, e.g. metals,44 are known to exhibit a time dependent
but completely reversible contribution to the strain that is usually observed
on short time scales. With respect to polymers, anelastic behavior has been
discussed in the context of the secondary relaxation processes by Boyd et
al.45, and in the context of internal friction caused by small molecules (Snoek
effect) by Böhm et al.46. In this work, possibly anelastic effects will be
considered in Section 6.3.
14
3 Theoretical Models
3.1 Dual Mode Sorption Model
Of the several phenomenological models that have been developed to describe
sorption of small molecules in glassy polymers and the induced swelling be-
havior, the Dual Mode Sorption Model (dm model) is the one most commonly
used. It combines the two independent processes of Henry solution, as it is
observed in liquids or rubbery polymers, with the Langmuir adsorption pro-
cess on inner surfaces of microporous materials. In the framework of this
model, a penetrant molecule is either sorbed into a Henry-type sorption site,
which is believed to be in the intermediate space of neighboring polymer seg-
ments, or adsorbing onto the surface of Langmuir type sites, microcavities
which considerably contribute to the excess Free Volume of a glassy polymer.
For solution according to Henry’s law, the solubility coefficient kDconnects
the penetrant concentration CDin the polymer linearly with the pressure p:
CD(p) = kDp(3.1)
The linear relationship between concentration and pressure of Henry mode
sorption is depicted by a dotted line in Figure 3.1. The Langmuir isotherm
CHcan be viewed as a sort of ‘hole filling’ of the microcavities with a sat-
uration capacity C0
Hand the quotient of ad- and desorption rate bis called
the affinity constant:
CH(p) = C0
Hbp
1 + bp (3.2)
In total, the pressure dependent concentration C(p) amounts to
C(p) = CD+CH=kDp+C0
Hbp
1 + bp (3.3)
The total concentration, i.e., the resulting sorption isotherm depicted by a
compact line in Figure 3.1, is dominated by the steep increase of Langmuir
15
3.1 Dual Mode Sorption Model 3. Theoretical Models
mode sorption in the low pressure regime (dashed line), while at higher pres-
sures it reaches saturation and the influence of Henry mode sorption is more
pronounced. Originally, the dm model was only intended to describe sorp-
tion of small penetrants in glassy polymers. Property changes of the matrix
like plasticisation were neglected. There were several attempts to modify
the dm model or incorporate other models,47 since the implied existence of
two distinct populations of sorbed molecules could not unambiguously be
validated,13 nor could the parameters be ascribed to physical parameters in
a straightforward way. However, due to its easy applicability and frequent
use, the dm model of the form 3.3 remains the most important model to
interpolate and exchange sorption data. If the Langmuir adsorption eq. 3.2
is interpreted as a mechanism of filling up microcavities or rather covering
their surfaces, dilation should only be caused by Henry type sorption 3.1.
This linear volume change is then related to the concentration of penetrants
via their partial molar volume Vp:
∆V
V0
=Vp
Vid
CD=Vp
Vid
kDp(3.4)
where V0is the volume of the dry polymer and Vid the volume of an ideal gas
at standard conditions (stp). To be able to also describe dilation isotherms
that are concave to the pressure axis, Kamiya et al.48 introduced the factor
0≤f≤1, to capture the contribution of penetrants sorbed in microcavities
to the dilation:
∆V
V0
=Vp
Vid
CD+fVp
Vid
CH=Vp
Vid
kDp+fVp
Vid
C0
Hbp
(1 + bp)(3.5)
0 10 20 30 40 50
0
10
20
30
40
C / cm
3
(S T P )/ c m
3
p / bar
Henry mode
Langmuir mode
Dual mode
Figure 3.1:
According to the dm
model, the total concen-
tration (solid line) is a
superposition of Henry
mode and Langmuir
mode sorption.
16
3.2 NET-GP Model 3. Theoretical Models
In other words, the two populations of sorbed penetrants exhibit different
partial molar volumes (p.m.v.):
The p.m.v. of the Henry population Vp=VDas also observed in liquid or
rubbery polymers, and the smaller p.m.v. fVp=VHof the penetrants sorbed
in microcavities.
3.2 NET-GP Model
The Non Equilibrium Thermodynamics of Glassy Polymers model (net-
gp)19, 49–51 is based on the description of liquid polymer/gas systems as lat-
tice fluids. It presents a modification of the ‘Lattice Fluid Theory’ (LF) of
Sanchez and Lacombe:20, 52
The polymer/gas system is treated as a partially occu-
pied lattice for which the equilibrium chemical potential
of the components can be calculated from parameters
of the lattice, statistical considerations of possible con-
figurations of gas and polymer, and their interaction
energies. Sanchez and Lacombe consider a polymer of
N r-mers, each mer occupying a lattice site of molar volume]v∗, and the
number of N0unoccupied lattice sites. The total number Nrof lattice sites
and the total volume Vof the system are then given by
Nr=N0+rN V = (N0+rN)v∗=V∗Nr
rN (3.6)
Here, V∗is the volume of the ideally packed polymer ‘crystal’ (i.e., no vacant
sites in the lattice), which is related to the characteristic density ρ∗
V∗=N(rv∗)rv∗=M/ρ∗(3.7)
where Mis the molar mass of the pure polymer. For nmole of a pure
polymeric fluid at temperature T, pressure pand density ρ, the expression
for the Gibb´s free energy is20
G=rnRT∗(−˜ρ+˜p
˜ρ+˜
T
˜ρ(1 −˜ρ) ln(1 −˜ρ+˜ρ
rln(˜ρ))(3.8)
]The denominations of the net-gp model somewhat differ from the notations used in
this thesis. However, since the quantities are discussed only once outside this section in
an unambiguous context, the notation of the authors of the model is used to conveniently
stay in compliance with the literature.
17
3.2 NET-GP Model 3. Theoretical Models
where ris the number of lattice sites occupied by a molecule and Ris the
universal gas constant. The reduced quantities of density ˜ρ, temperature ˜
T
and pressure ˜pare defined by
˜ρ=ρ/ρ∗˜p=p/p∗˜
T=T/T∗(3.9)
where p∗and T∗are the characteristic pressure and temperature, respectively.
The molar volume v∗of the closely packed polymer is assumed independent
of temperature and pressure
v∗=RT∗
p∗(3.10)
In the case of thermodynamic equilibrium (EQ)
∂G
∂ρ !T,p,n
= 0 (at EQ) (3.11)
holds. To fulfill the condition for equilibrium 3.11 at a given temperature
and pressure, the density can be varied.
For a two-component system, the LF theory can be expanded by combining
the model parameters ρ∗, p∗, T∗and v∗using appropriate mixing rules20:
(1) The reduced molar volume becomes
v∗=φ0
1v∗
1+φ0
2v∗
2(3.12)
where the concentration variable φ0
1is defined by the number of moles niof
component i(i= 1,2)
φ0
1=r0
1n1
r0
1n1+r0
2n2
(3.13)
(2) For the ratio of the number of sites occupied by a molecule of species 1
in the mixture (r1) to the number of sites occupied by the same molecule in
the pure state (r0
1), the following assumption is made:
r1
r0
1
=v∗
1
v∗(3.14)
(3) For the characteristic pressure p∗
p∗=φ1p∗
1+φ2p∗
2−φ1φ2∆p∗(3.15)
where φ1is the volume fraction of the penetrant species which is defined by
φ1=φ0
1
v∗
1
v∗(3.16)
18
3.2 NET-GP Model 3. Theoretical Models
The mixing rules result in a relation of the characteristic density ρ∗and
the composition of the mixture which is expressed by the mass fraction ωi
(i= 1,2) of the components
1
ρ∗=ω1
ρ∗
1
+ω2
ρ∗
2
(3.17)
The expression for the Gibbs free energy of a mixture containing n1moles
(penetrant) and n2moles of the polymer then becomes
G=RT∗(r1n1+r2n2)(−˜ρ+˜
P
˜ρ+˜
T
˜ρ"1(1 −˜ρ) ln(1 −˜ρ) + φ1
r1
˜ρln(φ1˜ρ)#)
(3.18)
For a given temperature, pressure and composition, the equilibrium value of
the density minimizes the Gibbs energy:
∂G
∂ρ !T,p,n1,n2
= 0 (at EQ) (3.19)
However, since the molecular mobility is limited, glassy polymers are not at
equilibrium. On the contrary, the density may be utilized to describe the
nonequilibrium character of the system. Doghieri and Sarti19 introduced the
density of the polymer matrix ρ2as an internal state variable which is related
to the reduced density ˜ρof the system as
˜ρ=ρ2
ω2
1
ρ∗(3.20)
The chemical potential of the penetrant in the sorbed phase
µS
1= ∂G
∂n1!T,p,n2,ρ2
(3.21)
can now be expressed as a function of temperature, pressure, concentration
and density of the matrix using 3.18:
µS
1
RT = ln(˜ρφ1)−[r0
1+(r1−r0
1)/˜ρ] ln(1−˜ρ)−r1−˜ρ[r0
1v∗
1(p∗
1+p∗−φ2
2∆p∗)]/(RT)
(3.22)
Equation 3.22 provides the means to calculate the solubility of a penetrant
phase at a given temperature, matrix density and pressure (chemical poten-
tial of the gas phase of the penetrant). Only one binary parameter, ∆p∗, and
19
3.2 NET-GP Model 3. Theoretical Models
the variation of matrix density with pressure is needed to predict a concen-
tration/pressure isotherm from otherwise pure-component parameters only.
The former may be approximated by the arithmetic mean value19
∆p∗= (qp∗
1−qp∗
2)2(3.23)
For the density change of the matrix with pressure, i.e., the dilation isotherm
corresponding to the concentration isotherm, a linear relationship is assumed:
ρ∞
2(p) = ρ0
2(1 −ksp)(3.24)
where ρ∞
2(p)is the pseudo-equilibrium matrix density at pressure p,ρ0
2the
initial density and ksthe swelling coefficient, which can be used as the single
free parameter to fit sorption data. Moreover, the resulting swelling coeffi-
cient ksmay be used to entirely predict swelling behavior or vice versa, pre-
dict sorption from known swelling behavior. Only the latter will be done in
this work, since this way the procedure is more closely related to the method
of calculating sorption isotherms using modeling techniques presented later
in this work. The net-gp calculations were performed using the Microsoft
Excel sheet xlEOS.xls and the library EOS.dll that are freely available for
download in the internet as Nonequilibrium package.53 The necessary pure-
component data of the gases CO2and CH4are contained within this package
and were published in reference [54]. The pure component data of the poly-
mers were calculated using equation 3.22 from experimental solubility data
of three gases54 and are listed in Table 3.1.55
Table 3.1: Pure component parameters used for net-gp calculations.
polymer ρ0p∗T∗ρ∗
g/cm3MPa K g/cm3
PSU 1.240 503 899 1.310
PI4 1.320 421 864 1.603
PIM 1.124 624 728 1.554
20
3.3 Site Distribution Model 3. Theoretical Models
(a) Volume element (b) Potential landscape
Figure 3.2: a): Sketch of a volume element Voccupied by polymer seg-
ments (cube) and the enclosed free volume site Vh(hatched). b): Sketch
of a potential landscape as seen by penetrant molecules, caused by a dis-
tribution of site volumes Vh.
3.3 Site Distribution Model
The site distribution model of Kirchheim56, 57 considers the localized free vol-
ume of an amorphous polymer (‘holes’) as possible sorption sites for pene-
trants. It is the size distribution of these holes which, by way of an additional
energy term caused by elastic deformation upon occupation of a penetrant,
leads to a distribution of sorption energies as well. Once the parameters of
this energy distribution have been obtained from sorption data, dilation data
can be used to obtain the parameters of the size distribution of ‘holes’ and
hence information of the structure of the free volume is gained. The shape of
the distribution is deduced from above Tgconsiderations following the treat-
ment of Bueche:58 A number of polymer segments in a polymer matrix occupy
a certain volume V(see Figure 3.2(a)). Above Tg, thermal fluctuations lead
to the probability to find this volume in an interval [V, dV ]
P(V)dV ∝exp −G0(V)
kT !dV (3.25)
where the Gibbs free energy is G0=E0−T0S+p0Vat constant pressure
p0and temperature T0. The most probable state is when G0=Gmin is
at a minimum. For small volume fluctuations ∆V=V−Vmin,G0can be
expanded to yield
∆G=G0−Gmin = ∂G0
∂V !T·∆V+1
2 ∂2G0
∂V 2!T·∆V2+··· (3.26)
21
3.3 Site Distribution Model 3. Theoretical Models
Dropping higher order terms and taking into account the conditions for the
minimum ∂G0
∂V !T
= 0 and ∂2G0
∂V 2!T
=− ∂p
∂V !T≥0(3.27)
it follows, that
G0(V) = Gmin + (∆V)2
2V0K!(3.28)
and hence the probability for the volume to be in the interval [V, dV ]becomes
P(V)dV =Bexp −(V−V0)2
2kT0V0K!dV (3.29)
where K=−1/V (∂V/∂p)Tis the compressibility, kthe Boltzman constant
and Bis a factor to be determined from the normalization condition. If the
probability 3.29 holds true for a number of segments occupying a volume
V, it should also be valid for the ‘hole’ volume Vhwhich is enclosed by the
segments, because the volume of the segments does not vary (Fig. 3.2(a)).
The distribution of ‘holes’, n(Vh), enclosed by polymer segments, becomes
n(Vh) = N0
√πσV
exp (Vh−Vh0)2
σ2
V!(3.30)
with the number of ‘holes’ per unit volume N0. The width of the distribution
σVis deduced from the normalization condition
Z∞
−∞
n(V)
N0
dV = 1 →σV=q2kT0Vh0K(3.31)
Below the glass transition temperature Tg, the segmental mobility of the
polymer is limited and the distribution of free volumes is quenched-in, pro-
viding the space to accommodate penetrants (sorption sites). To account for
the size effect of the sorption sites on the Gibbs free energy of sorption G,
it is considered as the sum of an elastic contribution Gel and the total of all
other interactions Gr(‘rest’), the latter is assumed to be independent of site
volume Vhand equal for all sites:
G=Gel +Gr(3.32)
To assess the elastic contribution, Kirchheim simplifies by idealizing the poly-
mer matrix as an elastic continuum, containing the free volume in the form of
spherical holes. If an elastic sphere (penetrant) of volume Vg> Vhis inserted
22
3.3 Site Distribution Model 3. Theoretical Models
into a hole of volume Vh, the elastic energy of the sphere Egcan be derived
from the distortion ∆Vgof the sphere40
Eg=Kg(∆Vg)2
2Vg
(3.33)
where Kgis the compressibility of the sphere. The energy Emstored in the
matrix is derived from the distortion field surrounding the deformed hole Vh
Em=µm(∆Vh)2
3Vh
(3.34)
with the shear modulus µmof the matrix. Comparison of the pressures
acting on sphere and hole, 3.33 and 3.34 lead to a relation between the
misfit (Vg−Vh)and the macroscopic volume change, i.e., the partial molar
volume Vp
Vp=γ
γ0(Vg−Vh)(3.35)
where γand γ0contain the elastic properties of sphere and matrix
γ= 1 + 4µm
3Km
γ0= 1 + 4µm
3Kg
(3.36)
The total energy stored in the polymer/penetrant system by occupation of
a sorption site adds up to
Gel =Eg+Em=2µm
3γ0
(Vg−Vh)2
Vh
(3.37)
Inserting 3.37 into 3.32 and expanding around Vh0and dropping higher order
terms leads to
G(V) = Gr+Gel(V)(3.38)
=Gr+GV h0+2µm
3γ0
(V2
g−V2
h0)
V2
h0(Vh0−Vh) + ··· (3.39)
≈G0+const.(Vh0−Vh)(3.40)
GV h0denotes the elastic energy needed to occupy a sorption site of volume
Vh0, and G0the total sorption energy of a sorption site of average volume Vh0
G0=Gr+GV h0(3.41)
The expression 3.40 can now be inserted into the distribution of site volumes
3.30, which leads to a distribution of sorption energies
n(G) = N0
√πσG
exp (G−G0)2
σ2
G!(3.42)
23
3.3 Site Distribution Model 3. Theoretical Models
The width σGof the distribution satisfies the normalization condition and is
connected to σV(cf. Equation 3.31) as
σG=σV
2µm(V2
g−V2
h0)
3γV 2
h0
(3.43)
The distribution of site volumes of Gaussian form 3.30 obviously leads to a
similar distribution of site energies 3.42 as well. For a penetrant, the polymer
matrix presents itself as a potential landscape (see Figure 3.2(b)) of sorption
sites (energy minima) and ‘bottlenecks’, energy barriers that might be con-
sidered as diffusion paths. Assuming thermal occupancy of the sorption sites
following Fermi-Dirac statistics, the concentration of the penetrant is given
by
C(µ) = Z∞
−∞ n(G)dG/(1 + exp[(G−µ)/kT]) (3.44)
where µis the chemical potential of the gas phase of the penetrant, which is
related to the pressure pvia
µ=µ0+kT ln p
p0!(3.45)
with µ0being the standard value for p=p0= 1 atm. The parameters G0and
σGare varied to fit the sorption isotherms, their values result in a Gaussian
distribution of site energies. In order to calculate the elastic volume dilation,
equation 3.32 and 3.41 are subtracted and equation 3.37 inserted:
G=G0+2
3µs (Vg−Vh)2
Vh−(Vg−Vh0)2
Vh0!(3.46)
The latter equation can be solved for Vh, yielding57
Vh(G) = Vg+3(G−G0)
4µs
+(Vg−Vh0)2
2Vh0−v
u
u
t 3(G−G0)
4µs
+(Vg−Vh0)2
2Vh0!2
−V2
g
(3.47)
The macroscopic volume change ∆Vcan now be calculated combining equa-
tion 3.35 and 3.44, considering the contribution of all molecules occupying
sorption sites of free enthalpy Gby integrating over all sites
∆V
V0
=Z∞
−∞ Vp(G)C(G)dG
=Z∞
−∞
γ[Vg−Vh(G)]
γ0
n(G)
(1 + exp[(G−µ)/kT])dG (3.48)
24
3.4 Summary 3. Theoretical Models
Equation 3.48 contains only one unknown parameter, provided that the pa-
rameters of the energy distribution are known from a fit of eqn. 3.44 to
sorption data and reasonable values for the volume of the penetrants Vg, the
elastic properties γand γ0as well as the number of sorption sites N0can be
assumed. Following Kirchheim,56 for the volume Vgof the penetrants their
partial molar volumes in liquids or rubbery polymers, which may be viewed
as dynamic or relaxed partial molar volume of the penetrants, are used in this
work. Gotthardt et al.59 proposed that the bulk moduli of liquid gases and
glassy polymers are about equal, yielding γ/γ0≈1and simplifying equations
3.35 and 3.48. The number of sorption sites per unit volume N0has been
estimated to be in the range of 3·1021 to 6·1021 cm3/mol, and a value of
6.7·1021 cm3/mol was chosen for easy unit conversion.56 In this thesis how-
ever, an analysis of detailed atomistic molecular packing models is utilized
to determine the value of N0for each polymer individually. This will be dis-
cussed in more detail in Section 6.2. The fit of equation 3.44 to sorption and
of equation 3.48 to dilation data was performed using the numerical integra-
tion algorithm of the package ‘Statistics_NonlinearFit’ of the Mathematica
5.2 software of Wolfram Research Inc., Champaign (usa).
3.4 Summary
In the above sections, the details of three theoretical models were provided
which will be used later to analyse the experimental data of this work (see
Section 4.5). Selected aspects of the results will be compared to simulated
data in Section 5.3. All three models approach the phenomenon of gas sorp-
tion and sorption induced dilation in glassy polymers from different view-
points. The dm model combines, in a more or less empirical way, the ob-
servation of sorption in liquid or rubbery polymers (Henry part) with the
adsorption process on inner surfaces (Langmuir part); while this intuitive
combination seemed to fit sorption data rather well, the attempt to connect
the parameters of the model to independently determinable physical param-
eters was only partly successful.60 In contrast, the net-gp approach and the
site distribution model arise from physical considerations, enabling a view on
the phenomena of a more substantiated nature. However, both models show
limitations regarding their applicability which are inherent in the model as-
sumptions and can therefore not be accounted as ‘general’, as will be shown
later in this work.
25
4 Experimental
4.1 Materials
4.1.1 Gases
Carbon dioxide (CO2) and methane (CH4) of purity >99.5%were used as re-
ceived from Air Liquide Deutschland GmbH. As ‘dynamic volume’ Vgin terms
of a relaxed partial molar volume of the gas dissolved in liquids or rubbery
polymers, the values of 46.2cm3/mol and 52 cm3/mol were used that were re-
ported by Pope et al.61 for CO2and CH4in silicone rubbers, respectively. For
unit conversion, the value of Vid = 22413.6cm3/mol was used as the volume
of an ideal gas at standard conditions (stp:T= 273.15 K,p= 1.013 bar).
4.1.2 Polymers
Polysulfone
Poly(sulfone) is an amorphous high-performance thermoplast used in a va-
riety of applications and therefore readily available from a number of dif-
ferent suppliers. In this work, Ultrason S was obtained from BASF AG,
Germany, as a melt extruded film of d= 100 µmthickness. The repeat
unit (see Figure 4.1) has a molecular weight of 469.8g/mol and the sup-
plier reports an average molecular weight of 400 kg/mol. Measurements in a
differential-scanning-calorimeter (DSC), heating rate 10 K/min showed non-
crystallinity and a glass transition at 462 K(190 ◦C). The density was de-
termined in a density-gradient column (DGC,Ca(NO3)2-solution) at 295 K
(23 ◦C) to ρ= 1.240 g/cm3, agreeing well with the value given by the sup-
plier (1.24 g/cm3). From the density a value of Φfree = 13.5%results for the
fraction of free volume, calculated by the Synthia software of Accelrys Inc.,62
using the Bondi method.35
26
4.1 Materials 4. Experimental
Polyimide 4 (6FDA-TrMPD)
The polyimide 6FDA-TrMPD (named PI4 as in ref. [63]) was synthesized64 at
the GKSS research center, resulting in chains of an average molecular weight
of about 300 kg/mol. The chemical structure of the repeat unit, which has
a molecular weight of 558.4g/mol, is shown in Figure 4.2. Films were cast
from a dichloromethane solution onto a glass plate and spread by a coating
knife. After slow solvent evaporation in a chamber of decreasing solvent vapor
atmosphere, the films were shortly immersed in water to detach them from
the glass plate. The thickness of the resulting dry and transparent films was
determined to d= 120 µm.dsc measurements showed some residual water
contents in the film and a glass transition of 650 K(380 ◦C). After several
days of degassing of the sample inside the pressure cell (prior to sorption
and dilation measurements), no significant influence of the residual water on
volume or weight of the sample could be detected. The density of the cast
film was determined to be ρ= 1.352 g/cm3and the free volume fraction to
be Φfree = 18.7%.
PIM-1
The Polymer of Intrinsic Microporosity (PIM-1), a recent development of
Budd et al.,31, 65–67 was synthesized at the GKSS research center, resulting in
an average molecular weight of 150 kg/mol.68 The chemical structure of the
repeat unit with 460.5g/mol is shown in Fig. 4.3. Films were cast from a
chloroform solution into a petri dish situated inside a chamber of decreasing
solvent vapor atmosphere for slow evaporation. Immediately after detaching
the films in a water bath for some hours, they were immersed in ethanol
for 8hto reduce the water content and then dried in vacuum and 552 K
(150 ◦C) for 12 h. The transparent film of yellow color had a thickness of
d= 150 µm and a density of ρ= 1.124 g/cm3. As reported in ref. [65], no
glass transition could be observed by DSC below the degradation temperature
of Tdeg = 722 K(350 ◦C). From the density, the free volume fraction was
calculated to Φfree = 19.2%.
27
4.1 Materials 4. Experimental
Figure 4.1: Chemical structure of PSU and representation of the repeat
unit by Van-der-Waals spheres.
Figure 4.2: Chemical structure of PI4 and representation of the repeat
unit by Van-der-Waals spheres.
Figure 4.3: Chemical structure of PIM-1 and representation of the repeat
unit by Van-der-Waals spheres.
28
4.2 Sorption Measurements 4. Experimental
100 1000 10000 100000
0.25
0.50
0.75
1.00
PSU/CO
2
buoyancy
effect
m / m g
t / s
Figure 4.4:
Typical mass uptake curve on
logarithmic time scale following
a gas pressure step. The buoy-
ancy effect is easily identified.
Data points are shown with a fit
according to the kinetic analysis
(cf. Section 4.4).
4.2 Sorption Measurements
Gravimetric Sortpion Balance
Gravimetric sorption measurements were carried out using an electronic mi-
crobalance Sartorius M25D-P of Sartorius GmbH, Göttingen. Details of the
setup may be found in reference [69]. The balance is situated in a pressure
cell designed to bear pressures well over 50 bar. The temperature of the
setup is held constant by an air-bath at 35 ±0.1◦C. Deviating from ref.
[69], in this work data acquisition was computer aided, recording the signal
of the microbalance automatically at a rate of 0.5s−1. The film sample of
uniform thickness is cut into several pieces (∼10x10x0.1 mm3) and put onto
the balance pan inside the cell, which is then evacuated at p < 10−5mbar un-
til any significant weight change has ceased. After thorough degassing of the
sample, the gas-pressure is increased in a series of stepwise increments and
the weight gain ∆mof the sample is observed for at least 24 hours at each
step (Figure 4.4); in the literature, this procedure is referred to as differential
sorption measurement. For the so called integral sorption measurements, the
gas-pressure is increased in a single integral step to (in this work) 10 bar and
the weight gain of the sample is observed for at most 4 hours to ensure that a
quasi-equilibrium is reached but no significant relaxations take place. After-
wards, the pressure cell is evacuated and the desorption process is observed.
The effect of buoyancy is either eliminated from the data or estimated using
Archimedes law. The weight gain is converted into the commonly used units
29
4.3 Dilation Measurements 4. Experimental
of cm3(stp)/cm3(polymer)]using
∆C=Vid∆m
Mgas ·ρ0
m0
(4.1)
where Vid is the molar volume of an ideal gas at standard conditions (stp),
Mgas the molar mass of the penetrant gas and ρ0and m0the density and
mass of the polymer prior to any measurement.
4.3 Dilation Measurements
Gas pressure Dilatometer
A number of different techniques to measure sorptive dilation of glassy poly-
mers are reported in the literature. While earlier methods were based on
visual readouts,70, 71 more recently induction gauges,59, 72, 73 ellipsometry30
or image analysis74 are utilized to monitor the swelling of polymer/gas sys-
tems. In this work, the volume changes in the sample due to sorption of gas
molecules were investigated using a gas-pressure-dilatometer, whose principle
is shown in Figure 4.5 and which has been described in more detail in refer-
ences [69] and [29]. It is based on a capacitive distance measuring system:
Figure 4.5:
Principle of the capaci-
tive dilatometer.
A strip (∼20x10x0.1 mm3) of the sample film is
clamped at both ends. The upper clamp is fixed
to the cell-wall, while at the lower end a small
metal disk is attached perpendicular to the film
plane. Thus the disk is suspended freely above
the capacitance-sensor, serving as its counter plate.
Distance changes between the two plates of this ca-
pacitor can be measured linearly within a range
of 1000 µmand with an accuracy of 0.25 µm. The
weight of lower clamp and metal disk (<10 g) is not
expected to significantly influence the measurement
as the exerted stress is almost three orders of mag-
nitude below the yield stress of the polymer speci-
men. Similar to the sorption setup, the dilatometer
is placed in a pressure cell which is held at constant
temperature by an air bath. In contrast to ref. [69],
a second sensor has been added to the chamber with
]Throughout this work, for better readability the unit cm3(STP)/cm3(polymer) will be
expressed by cm3/cm3.
30
4.4 Kinetic Analysis 4. Experimental
a well defined distance to its fixed counter plate; its signal is used to eliminate
the effects due to dielectric changes of the gas atmosphere resulting from its
variations in the density upon each pressure step. The Labview-based soft-
ware allows data-acquisition rates of up to 1s−1, limited by the instruments
and interfaces. Following the same procedure as in the sorption measure-
ments, after thorough degassing, the length change of the sample is recorded
for a series of pressure increments, and at least 24 hours observation time per
step were scheduled for the differential dilation isotherms, and accordingly
a pressure step to 10 bar for several hours and subsequent degassing for 24
hours following the integral dilation procedure. Assuming isotropic swelling,
the length change ∆lof the sample can be easily converted to volume change
∆Vusing
∆V/V0= (1 + ∆l/l0)3−1(4.2)
where l0and V0denote the initial length and volume of the sample.
100 1000 10000 100000
0
5
10
15
20
25
30
PSU/CO
2
l / µm
t / s
oscillatory displacement
during pressurization
Figure 4.6:
Typical length change curve on
logarithmic time scale following
a gas pressure step. At times
larger than 104s, the sample
rate is decreased and the data
is smoothed by an averaging al-
gorithm. Data points are shown
with a fit according to the ki-
netic analysis (cf. Section 4.4).
4.4 Kinetic Analysis
According to Crank,39 three cases of diffusion of penetrants in glassy polymers
are distinguished:
i Case I or Fickian diffusion in which the rate of diffusion is much less
than the rate of relaxations;
ii Case II diffusion in which diffusion is very rapid compared with the
relaxation processes;
31
4.4 Kinetic Analysis 4. Experimental
iii Non-Fickian or anomalous diffusion which occurs when diffusion and
relaxation rates are at a comparable level.
For the obtained data of all investigated polymer/gas systems at 308 K, the
shape of the mass uptake curve implies anomalous diffusion with a two-
stage sorption process, where the initial rapid stage is controlled by Fickian
diffusion, whereas the second stage is dominated by relaxational processes of
the polymer matrix. A typical example for the PSU/CO2system is shown in
Figure 4.4. The same two-stage behavior is observed for the dilation curve
(Figure 4.6). Thus, for a detailed analysis of the kinetics of sorption and
dilation, the following model function has been fitted to the sorption and
dilation data, respectively:
∆X(t) = ∆Xf·f(t, D) + ∆Xg1·g1(t, τ∗
1) + ∆Xg2·g2(t, τ∗
2)(4.3)
where
f(t, D) = 1 −
∞
X
n=0
8
(2n+ 1)2π2exp(−D(2n+ 1)2π2t/d2)(4.4)
and
gi(t, τ∗
i) = 1 −exp(−t/(d2τ∗
i)) (4.5)
∆X(t)denotes the time dependent change of mass or length, respectively.
The first term on the right side of eq.4.3 reflects Fickian diffusion kinetics
into a plane sheet of thickness d/2as derived by Crank39 (eq.4.4 ≡eq.2.11).
It contains the diffusion coefficient Dand the diffusive fraction of the mass
uptake ∆Xfas fit-parameters. Sufficient accuracy was achieved using the
first ten terms in the summation of eq.4.4 for calculations while higher order
terms were dropped for practical reasons (n≤9). To account for non-Fickian
diffusion, two exponential relaxation functions g1and g2are implemented
with the relaxational fractions ∆Xgi and the thickness normalized relaxation
times]τ∗
ias fit-parameters. This approach was proposed by Berens and
Hopfenberg,42 based on the treatment of solvent-vapor sorption in glassy
polymers. In their model, the kinetics of vapor sorption are considered to
be a linear superposition of independent contributions from Fickian diffusion
and sorption controlled by relaxational processes of the polymer matrix. In
general the relaxational contribution may be written as an infinite sum of
exponential functions representing first order relaxations. However, one or
]Strictly speaking, the τ∗
iare reciprocal (effective) diffusion coefficients; however, this
stage of sorption is kinetically controlled by relaxations of the polymer matrix, character-
ized by a relaxation time τi= (d2τ∗
i). The normalization of the relaxation times to the
film thickness is done for convenient comparison to the diffusion coefficient and should not
imply a thickness dependence.
32
4.4 Kinetic Analysis 4. Experimental
two relaxational terms have proven to successfully describe even complex
sorption data.42, 43 Wessling et al.43 applied this method to CO2-sorption in
glassy polyimides, and suggested to use the same kinetic model to fit the
corresponding CO2-induced dilation.
5 6 7 8 9 10
0
1
2
3
PSU/CO
2
relaxational
fractions
dilation
sorption
extrapolation
X (t )/ X
f
log(t/d
2
)
elastic/diffusive fraction
Figure 4.7:
Thickness normalized ki-
netics of mass uptake and
dilation on logarithmic
time scale, normalized
to the elastic/diffusional
fraction and extrapolated
to large times (hatched
area).
As a typical example for the behavior in the PSU/CO2-system, Figure 4.7
shows mass uptake and dilation of the pressure step 25 to 30 bar, that were
shown in Figures 4.4 and 4.6, normalized to the diffusive/elastic fraction
and on a thickness-normalized logarithmic time scale. As can be easily seen,
Figure 4.8:
Spring-dashpot model to
describe viscoelastic be-
havior.
there is a clear relation between the time scales of
the different processes involved in sorption and di-
lation, respectively. This leads to the viscoelastic
interpretation by Newns22 as discussed in the intro-
duction: Equation 4.3 is the mathematical repre-
sentation of a spring and dashpot model (cf. Sec-
tion 2.4) as shown in Figure 4.8. A spring accounts
for the elastic stresses imposed during the initial,
rapid stage of sorption of penetrant molecules. Be-
cause this reaction of the matrix is nearly instanta-
neous, the slower kinetics of diffusion into the ma-
trix is controlling the process and eq.4.4 applies to
this elastic fraction of dilation. Two Kelvin mod-
ules (see also Section 2.4) represent the viscoelastic
relaxations of the polymer matrix. Their time con-
stants τ∗
i, normalized to the film-thickness d, are of
33
4.4 Kinetic Analysis 4. Experimental
the order τ∗
i≈108. . . 109s/cm2, whereas the reciprocal diffusion coefficient
D−1is about 107s/cm2for CO2in PSU at 308 Kat a CO2pressure of 30 bar
(see Figure 4.7). In the following, the first part, with kinetics controlled by
Fickian diffusion, will be referred to as the diffusive fraction of sorption and
elastic fraction of dilation, respectively. As there is no explicit relation of the
relaxation functions gito specific molecular motions or processes assumed
here, both relaxational contributions will be discussed in terms of one relax-
ational fraction of sorption or dilation, meaning the sum of both relaxational
functions at the end of an experimental measurement cycle (t=te= 24 h):
∆Xr(te) = ∆Xg1(te) + ∆Xg2(te)(4.6)
The respective contributions to sorption (∆Cf(te)and ∆Cr(te)) and dilation
(∆Vf(te)and ∆Vr(te)) for each pressure step may now be added up resulting
in separate isotherms of the diffusive/elastic and the relaxational fraction,
which may be further analyzed:
Cdiff =X
steps
∆Cf(te)(4.7)
Crelax =X
steps
∆Cr(te)(4.8)
Ctotal =Cdiff +Crelax (4.9)
∆Velast =X
steps
∆Vf(te)(4.10)
∆Vrelax =X
steps
∆Vr(te)(4.11)
∆Vtotal = ∆Velast + ∆Vrelax (4.12)
34
4.5 Experimental Results and Discussion 4. Experimental
4.5 Experimental Results and Discussion
4.5.1 Sorption and Dilation Isotherms
The application of the kinetic analysis (eq. 4.3) that was described in the
previous section, yields the fractions of mass uptake (sorption) and volume
change (dilation) that follow diffusive/elastic and relaxational kinetics for
each experimental step, respectively. The resulting values for each pressure
step are added, resulting in two separate contributions (diffusive/elastic and
relaxational) to the total isotherm. Summation of the two parts gives the
total sorption or dilation isotherm, respectively, as it is usually presented.
In the following, the data acquired for the PSU/CO2system will be thor-
oughly discussed exemplarily in terms of the Dual Mode (dm) and the site
distribution (sd) model. The quality of the fits and the reliability of the
fitparameters will be addressed and the kinetic analysis will be consistently
implemented with regard to the origin of the sd model.
Polysulfone - CO2
A number of authors have published data on CO2sorption in polysulfone.24–29
Erb and Paul23 examined polysulfone samples that were aged (free vol-
ume reduction by sub-Tgannealing) or conditioned (free volume increase
by high pressure CO2treatment). The resulting isotherms, emphasized by
dm-fitcurves in Figure 4.9 (solid lines), approximately mark the lower and
upper boundaries of sorption data at a temperature of 35 ◦C. Aside from a
variety of different experimental setups, the sample prehistory, which is not
always known, and the experimental time scale seem to be the main causes of
the observed differences. As can be seen from Figure 4.9, the primary sorp-
tion data gathered in this work (displayed in red) are well within the limits
of the variation of reported literature data.2For the CO2-induced dilation
there is no literature data available to a similar extent. However, Wang and
Kamiya25 arrived at a dilation of 0.5%at a pressure of 2.3bar and Böhning
and Springer69 measured a dilation of 5%and 8.5%for short- and long-term
measurements, respectively, in an isotherm up to 50 bar CO2pressure using
the same experimental setup as in this work.
35
4.5 Experimental Results and Discussion 4. Experimental
0 10 20 30 40 50
0
10
20
30
40
50
this work
Erb and Paul: conditioned
Erb and Paul: aged
other experimental data
C / cm
3
(S T P ) /c m
3
CO
2
pressure / bar
Figure 4.9: Sorption data of CO2in PSU at 35 ◦Cof this work (red)
and literature data ( ) from refs.24–29 The data of Erb and Paul23 on
conditioned ( ) and aged ( ) samples mark the upper and lower boundary
of the variation found in the literature.
Figure 4.10 shows the experimental results of sorption and dilation in the
PSU/CO2system obtained in this thesis. In Figure 4.10(a) the increase of
concentration with pressure is shown. The total mass uptake is shown in
full circles ( ), at each pressure step indicating the content of CO2within
the polymer sample. The second isotherm that is displayed shows the diffu-
sive/elastic part of the sorption ( ). At each pressure step, only the fraction
of sorbed CO2following Fickian diffusion kinetics is added to the isotherm.
It should be noted that this ‘diffusive isotherm’ indicates the amount of pen-
etrant gas that was sorbed at short time scales following Fickian diffusion
kinetics; however, the classification as diffusive (and relaxational) isotherm
should not imply different populations of sorbed CO2.\At low pressures
both, the total and the diffusive isotherm coincide, because no significant
relaxations take place within the time scale of the experiment. At pressures
above 10 bar, the diffusive isotherm deviates from the total isotherm and
the relaxational fraction ( ) can be obtained by subtraction of the diffusive
part. It should be noted that the relaxational fraction and hence the total
isotherm result from relaxational processes that are still in progress at the
end of a measurement step (te= 24 h).
\To illustrate this point, imagine a room with a door and a window. It is possible to
count how many people entered through the door, and how many entered through the
window; but once inside it is impossible to tell who entered through the door.
36
4.5 Experimental Results and Discussion 4. Experimental
0 10 20 30 40 5 0
0
10
20
30
40
50
PSU/CO
2
C / cm
3
(S T P )/ c m
3
p / bar
(a) Sorption
0 10 20 30 4 0 50
0
2
4
6
PSU/CO
2
V /V
0
/ %
p / bar
(b) Dilation
Figure 4.10: Isotherms of CO2in PSU. The solid lines represent a best
fit of the dm model through the total isotherms ( ). The dashed line
through the diffusive/elastic fraction ( ) represents a best fit of the sd
model. The relaxational fraction ( ) of the dilation is fitted linearly (cf.
text).
The concentration, total or relaxational, at each pressure is therefore not at
a quasi-equilibrium, as is the diffusive fraction of sorption.
In Figure 4.10(b) the corresponding dilation isotherms are shown, using the
same symbols for the total ( ), elastic ( ) and relaxational ( ) isotherm.
Comparison of the concentration and dilation isotherm (Figure 4.10) and of
the time dependent sorption and dilation of individual pressure steps (Fig-
ure 4.7) shows a clear correlation between the diffusive sorption and elastic
dilation and the respective relaxational fractions, which may be interpreted
in the following way:2, 22, 43
Following the pressure step, the chemical potential of the CO2in the gas
phase and within the polymer matrix is balanced by a mass flux, following
Fickian diffusion kinetics (eq. 4.4), until a (pseudo-) equilibrium concentra-
tion is reached in the polymer. The elastic dilation that is caused by the
misfit between gas molecule and sorption site (hole) in the matrix (cf. Sec-
tion 2.3) appears instantaneously upon occupation of a sorption site, which
is the reason why the elastic/diffusive process is kinetically controlled by
the slower sorption kinetics (Fickian diffusion). Since in the glassy polymer
PSU (at a low concentration level) the mobility of the matrix is quite lim-
ited, any sorption induced stresses are conserved and the system remains in
a metastable state. At higher pressures (concentration levels) of CO2the
PSU-matrix is plasticized, the mobility of the polymer chains is enhanced
and sorption induced stresses are partly reduced, resulting in relaxational
37
4.5 Experimental Results and Discussion 4. Experimental
dilation. The reduction of sorption induced stresses lowers the chemical po-
tential of the sorbed gas molecules, which again leads to a diffusive mass
flux into the matrix. The characteristic times of the relaxational processes
are larger than that of diffusion (τ∗
i> D−1) and therefore the relaxational
dilation dominates the sorption kinetics.
Dual Mode Sorption Model
In Chapter 3, three different phenomenological descriptions of sorption and
concomitant swelling behavior of polymer/gas systems were introduced. The
Dual Mode Sorption Model (dm model) describes sorption as two superposing
processes, one describing a ‘hole filling’ of microcavities within the polymer
matrix (Langmuir mode) and the other the occupation of interstitial space
(Henry mode) (see Section 3.1). Equation 3.3 was fitted to the the total
concentration pressure isotherm. The resulting fit parameters are listed in
Table 4.1 on page 40, along with the results of the extended dm model to fit
the gas induced dilation (Section 3.1 eq. 3.5). The fit curves of the dm model
are in excellent agreement with the experimental data (Fig. 4.10, solid lines).
However, the dual mode sorption parameters show a clear dependence on the
maximum pressure.75
0 10 20 30 40 50
1
10
10000
20000
dual mode:
G
0
/ J/mol
G
/ J/mol
G
0,f
/ J/mol
G,f
/ J/mol
k
D
/ cm
3
(STP)/cm
3
/bar
C
H
'
/ cm
3
(STP)/cm
3
b
/ bar
-1
pa r am et er v al u e
maximum pressure / bar
site distribu tion:
Figure 4.11: Model parameters in dependence on the maximum pressure
included in a fit to sorption data of CO2/PSU. The strong dependency
of the dual mode parameters (kD, C0
H, b) levels off at higher pressures.
Reported literature data23, 25–28 (open symbols) fit well into this scheme.
Site distribution parameters from the total sorption ( , ) lose reliability
with the onset of relaxational swelling, while parameters obtained from
diffusive fraction ( , ) are stable over the whole pressure range.
38
4.5 Experimental Results and Discussion 4. Experimental
Figure 4.11 shows the variation of model parameters with pressure. On the
y-axis, the parameter value is plotted against the maximum pressure (Fig.
4.10) of which the data pair [C, p] was included in the fit (as if the mea-
surement was finished at this pressure)]. The dependence of the dm model
parameters (kD, C0
H, b) on this maximum pressure is quite strong at pres-
sures below 20 bar and levels off at higher pressures. Speaking in terms of
the model, Langmuir-type sorption sites are saturated at this pressure and
the Henry parameter kDmay be determined with reasonable certainty. Any-
how, due to the uncertainty of the exact pressure of Langmuir site saturation
and the physical ambiguity of the parameters it is recommended to use dm
parameters for interpolation only.
0 10 20 30 40 50
0
10
20
30
40
50
PSU/CO
2
C / cm
3
(S T P ) /c m
3
p / bar
Figure 4.12: dm fit (——) to the experimental sorption data of
CO2/PSU. Obviously, there is no connection between the diffusive sorp-
tion ( ) and the Langmuir mode (---) or the relaxational fraction ( )
and the Henry mode (····).
At first glance, a comparison of Figure 3.1 and the three sorption isotherms
of Figure 4.10 suggest a correlation between Henry mode sorption and the
relaxational fraction of sorption on the one hand and between Langmuir
mode and diffusional sorption on the other. However, the separate display
of Henry mode sorption (dotted line) and Langmuir mode sorption (dashed
line) in Figure 4.12, as obtained by a best fit of eq. 3.3, shows that the
observed similarity with the kinetic separation of diffusive and relaxational
fractions of sorption is misleading, and a connection of the observed kinetic
]Note that the DM-parameters at the highest pressure correspond to the DM-fit in Fig.
4.10, i.e., all datapoints were included in the fit.
39
4.5 Experimental Results and Discussion 4. Experimental
processes and the two sorption modes of the model cannot be made in a
straightforward way.
Table 4.1: Dual Mode Sorption model parameters (cf. eqs. 3.3 and 3.5).
polymer/ kD/ C0
H/ b/ Vp/ f
gas cm3/cm3bar−1cm3/cm3bar−1cm3/mol
PSU /CO20.805 12.8 0.384 30.1 0.79
PSU /CH40.381 3.35 0.206 10.7 1.6
PI4 /CO21.56 54.6 0.544 75.2 0.16
PI4 /CH40.482 31.5 0.153 29.5 0.13
PIM-1/CO21.64 78.6 0.480 37.6 0.29
PIM-1/CH40.116 76.4 0.109 312 0.00
Site Distribution model
The site distribution model is derived using the assumption of a continuous,
elastic polymer matrix (see Section 3.3). The concentration of penetrants
within the matrix is calculated, assuming that elastic energy, resulting from
the deformation of a sorption site, is stored in the matrix. Ideally, the SD
model then yields parameters of a size distribution of sorption sites of the
pure polymer matrix. Relaxational swelling processes are not considered by
the model, which is why the author of the model disregards any sorption
and dilation data beyond the pressure at which relaxational swelling typi-
cally sets in (20 bar).56 But the omission of data points always involves a
loss of reliability or accuracy, as Figure 4.11 has already shown for the dm
parameters. Figure 4.11 also contains the pressure dependent parameters of
the SD model, obtained in the same fashion as the dm model parameters.
Obviously, the pressure dependence of the SD model is not as pronounced.
However, if all data of the total isotherm are included in the fit, the parame-
ters of the SD-model decrease slightly with pressure. This result is consistent
with the model: if relaxational swelling takes place, the average site volume
should rise and therefore the elastic energy which is necessary to occupy the
site decreases; hence additional sorption occurs. Also, as plasticization is
progressing, i.e. the mobility of the matrix is significantly increased, stresses
induced in the matrix upon gas sorption are easier to be relaxed and, as the
differences in site energy lessen, the distribution of Gaussian shape decays
40
4.5 Experimental Results and Discussion 4. Experimental
to a Dirac-delta function (one site energy for all sites). As more and more
data points are included in the fit, both effects, swelling and plasticization,
change the structure of the amorphous polymer matrix, and the result of the
SD model fit that should give information about the original structure of the
matrix is altered.
Therefore it seems reasonable to exclusively use the diffusive/elastic fraction
of sorption and dilation over the whole pressure range and thus abiding by the
supposed origin of the sd model (i.e., elastic deformation).2It must be noted
that, even if only the diffusive fraction is taken into account, the structural
changes of the matrix are taking place nonetheless, increasing the width of the
size distribution of sorption sites. Yet, at the regarded levels of concentration,
this change does not lead to significant differences in the diffusive fraction of
sorption. Therefore it is justified to utilize the complete diffusive fraction of
the sorption isotherm to obtain structural information of the original matrix.
The dashed lines in Figure 4.10 show the best fit of eqs. 3.44 and 3.48 to the
sorption and dilation data, respectively. The sorption data (left) was fitted
resulting in best-fit-parameters of the energy distribution G0and σG(center
and width of the Gaussian distribution). These values are used as input for
eq. 3.48, where only the average volume of a sorption site Vh0is varied to give
a best fit to the dilation data. The fit curves agree well with the experimental
data. The resulting parameters of Gaussian size and energy distributions are
compiled in Table 6.2 on page 98 and will be discussed in the context of free
volume distributions (Sec. 6.2).
NET-GP
The third model that was discussed in Chapter 3 is called Non-Equilibrium
Thermodynamics of Glassy Polymers (net-gp) or Non-Equilibrium Lattice
Fluids (nelf). As will be argued in Section 6.1, the model only takes into ac-
count irreversible volume changes. Since the elastic dilation is not considered
by this model, besides using the total dilation, in Section 6.1 the exclusive
use of the relaxational dilation isotherms will also be discussed to determine
the swelling factor ksneeded for the prediction of sorption isotherms. As
can be seen in Figure 4.10, the relaxational dilation resembles far better a
linear relationship with pressure, which is assumed by the model, than the
diffusive or total isotherms do. Figure 4.10 shows the best fitting straight
line (through zero, dotted line). It results in a swelling coefficient ksr that
was used to predict sorption isotherms according to equation 3.22 (cf. Table
3.1). Owing to the close relationship of the procedures, the results of the
sorption prediction by the nelf model will be presented in Section 6.1 in the
41
4.5 Experimental Results and Discussion 4. Experimental
context of gcmc isotherms (see Section 5.2.2), including a more thorough
discussion of the exclusive use of relaxational dilation.
0 10 20 30 4 0 50
0
5
10
15
20
25
C / cm
3
(S T P )/ c m
3
p / bar
PSU/CH
4
(a) Sorption
0 10 20 30 4 0 50
0.0
0.2
0.4
0.6
0.8
1.0
1.2
PSU/CH
4
V /V
0
/ %
p / bar
(b) Dilation
Figure 4.13: Sorption and dilation isotherms of CH4in PSU. (——): dm
model through the total isotherms ( ). (- - -): sd model fit through the
diffusive/elastic fraction ( ). The relaxational fraction ( ) of the dilation
is fitted linearly (····)(cf. text).
Polysulfone - CH4
Figure 4.13 shows the sorption and dilation isotherms of CH4in polysulfone.
Expectedly, the concentration level reached is much lower than that for CO2.
This is expressed by the lower values of the dm model parameters (Tab. 4.1
on page 40), which represent a well agreeing fit through the (total) sorption
data. The diffusive/elastic fraction of sorption and dilation was fitted by the
site distribution model, in compliance with the arguments discussed in the
previous paragraph. The curves fit the data reasonably well and the resulting
size distribution of sorption sites will be discussed in Section 6.2.
PI4 -CO2
Figure 4.14 shows the sorption and dilation isotherms of CO2in 6FDA-TrMPD
polyimide (PI4). Due to technical reasons, the measurement had to be
aborted at a pressure of 35 bar. The CO2solubility is extraordinarily high,
as is the accompanying dilation. It is worth mentioning that at the maximum
pressure of 35 bar, the relaxational fraction of dilation reaches the level of
elastic dilation, indicating the strong plasticizing ability of CO2. The dm pa-
rameters reflect this sorption and dilation behavior and again yield excellent
42
4.5 Experimental Results and Discussion 4. Experimental
fitcurves. While the diffusive sorption is well represented by the SD model,
the dilation, where only one parameter (Vh0) is adjusted, overestimates the
dilation at low pressures and underestimates at high pressures.
0 10 20 30 4 0 50
0
20
40
60
80
100
PI4/CO
2
C / cm
3
(S T P )/ c m
3
p / bar
(a) Sorption
0 10 20 30 4 0 50
0
5
10
15
20
PI4/CO
2
V /V
0
/ %
p / bar
(b) Dilation
Figure 4.14: Sorption and dilation isotherms of CO2in PI4. (——): dm
model through the total isotherms ( ). (- - -): sd model fit through the
diffusive/elastic fraction ( ). The relaxational fraction ( ) of the dilation
is fitted linearly (····)(cf. text).
However, the agreement between fitcurve and data is still acceptable. The
unexpectedly low value for the average site volume Vh0, when compared to
polysulfone, will be discussed in Section 6.2.
PI4 -CH4
In Figure 4.15 the sorption and dilation isotherms of the PI4/CH4system
are presented. Again, a considerably lower concentration and dilation level
is observed for methane in comparison to CO2, but the solubility is higher in
PI4 than in PSU. The dm model fits the data quite well. The total dilation
seems, in contrast to both dilation isotherms in PSU, to be almost linear,
which is, same as for CO2, reflected by the small value of f, denoting the
fraction of Langmuir mode sorption contributing to the dilation. The site
distribution model yields excellent fits to the diffusive fraction of sorption
and the elastic fraction of dilation, respectively.
43
4.5 Experimental Results and Discussion 4. Experimental
0 10 20 30 4 0 50
0
10
20
30
40
50
PI4/CH
4
C / cm
3
(S T P )/ c m
3
p / bar
(a) Sorption
0 10 20 30 4 0 50
0
1
2
3
4
PI4/CH
4
V /V
0
/ %
p / bar
(b) Dilation
Figure 4.15: Sorption and dilation isotherms of CH4in PI4. (——): dm
model through the total isotherms ( ). (- - -): sd model fit through the
diffusive/elastic fraction ( ). The relaxational fraction ( ) of the dilation
is fitted linearly (····)(cf. text).
PIM-1 -CO2
Sorption and dilation isotherms of the PIM-1/CO2system are presented in
Figure 4.16. The solubility of CO2in PIM-1 exceeds even that of CO2in PI4,
while at the same time exhibiting less pronounced dilation. Also the com-
paratively low relaxational fraction suggests a high fractional free volume
(only little dilation necessary to accommodate penetrants) as well as a rela-
tively rigid matrix (poor plasticisation ability), as is expected for a ‘polymer
of intrinsic microporosity’. The dm model yields a nice fit and the highest
value for the Langmuir capacity C0
H. The sd model again fits the sorption
very well, while the dilation fit is of poorer quality (see also the discussion
in Section 6.2).
PIM-1 -CH4
Figure 4.17 shows the sorption and dilation isotherms of the PIM-1/CH4
system. Similar to the methane isotherms in PI4, a high level of concentration
is reached. The low value of the Henry parameter kDentails an unusually high
Langmuir capacity C0
H. In combination with the linear dilation, unreasonable
values of the partial molar volume Vpand fVpresult. The sd model fit
curve does not represent the experimental data as well as in all other cases.
However, the agreement can still be called fairly well.
44
4.5 Experimental Results and Discussion 4. Experimental
0 10 20 30 4 0 50
0
25
50
75
100
125
150
PIM/CO
2
C / cm
3
(S T P )/ c m
3
p / bar
(a) Sorption
0 10 20 30 4 0 50
0
5
10
15
PIM/CO
2
V /V
0
/ %
p / bar
(b) Dilation
Figure 4.16: Sorption and dilation isotherms of CO2in PIM-1. (——):
dm model through the total isotherms ( ). (- - -): sd model fit through
the diffusive/elastic fraction ( ). The relaxational fraction ( ) of the
dilation is fitted linearly (····)(cf. text).
0 10 20 30 4 0 50
0
20
40
60
80
PIM/CH
4
C / cm
3
(S T P )/ c m
3
p / bar
(a) Sorption
0 10 20 30 4 0 50
0
2
4
6
8
PIM/CH
4
V /V
0
/ %
p / bar
(b) Dilation
Figure 4.17: Sorption and dilation isotherms of CH4in PIM-1. (——):
dm model through the total isotherms ( ). (- - -): sd model fit through
the diffusive/elastic fraction ( ). The relaxational fraction ( ) of the
dilation is fitted linearly (····)(cf. text).
45
4.5 Experimental Results and Discussion 4. Experimental
Despite the fact that all experimental data may be well fitted by the discussed
models, the following observations can be made:
E1) Overall sorption is represented very well by the dm model. The Lang-
muir capacity C0
Has well as the Henry solubility kDarrange in the
order PSU <PIM-1 <PI4, giving a measure of the concentration level
reached by CO2and CH4. The only exception presents the PIM-1/CH4
sorption. A reliability analysis of dm parameters of data of this work
and literature data suggests their use for interpolation only.
E2) By way of kinetic analysis, the sorption and dilation process can be
successfully divided into a fraction following Fickian diffusion kinetics
and a relaxational fraction. Dilation which is induced in the former,
more rapid stage appears instantly upon sorption of a penetrant and
is therefore regarded as elastic. The slower process is dominated by
viscoelastic or plastic deformation kinetics of the polymer matrix and
therefore regarded as irreversible (on a similar time scale).
E3) The sd model, which is based on the assumption of an elastic defor-
mation, fits the diffusive/elastic fraction of sorption and dilation fairly
well, allowing to make use of the full pressure range of the data. The
reliability analysis of the PSU/CO2justifies this approach.
E4) The relaxational fraction can be fitted linearly to derive a swelling
parameter to be used for net-gp sorption predictions (see Section 6.1).
4.5.2 Integral Sorption and Dilation
In contrast to differential sorption and dilation measurements, where a se-
ries of relatively small pressure increments at regular time intervals lead to
sorption and dilation isotherms as described in the previous section, inte-
gral sorption and dilation refers to the experimental procedure of a single,
larger pressure step.42 In this work, a step to 10 bar is reversed after a given
time to be repeated following the desired pattern. This procedure bears
the advantage of a relatively quick assertion of the concentration and dila-
tion level for one individual pressure, although a point by point scan of a
complete isotherm would take considerably more time, due to the necessity
of relatively long degassing times in-between steps. Anyway, it should be
noted that the latter method of measuring a sorption or dilation isotherm
not necessarily leads to the same result as a differentially measured isotherm,
because of the difference in sample history and the difference in the rate of
46
4.5 Experimental Results and Discussion 4. Experimental
stress application. In this work, the integral sorption and dilation procedure
was utilized to assess the diffusive concentration level of the penetrant gases
at 10 bar and use this information as input for ‘integrally loaded’ packing
models in preparation to integral dilation simulation (see Section 5.2.3). The
observed experimental dilation is then compared to the volume dilation of
thusly prepared packing models in NpT-MD simulations.
Figure 4.18 shows the integral sorption and dilation of polysulfone (PSU)
under 10 bar CO2pressure. At the start of the measurement, following
thorough degassing, the pressure is raised quickly in one single step. The
increase in concentration is following Fickian diffusion kinetics, reaching a
(quasi-) equilibrium after a few hours. The pressure step is then reversed and
desorption is observed. Three consecutive measurements (No. 2 and 3 shown
as lines; note the slightly different starting point of the desorption) show the
reproducibility of the measurement. The desorption of the penetrant gas
is complete but somewhat slower than the sorption. This is due to the
concentration dependency of the diffusion coefficient caused by penetrant
induced plasticization, decreasing the rate of desorption in the process. The
dilation is following the same kinetics as the sorption and is, upon desorption,
completely recovered. Qualitatively, these are the two major results of the
experimental measurements of integral sorption and dilation: The congruent
kinetics of sorption and dilation suggest that the dilation appears instantly
(within experimental resolution) upon sorption of a penetrant. Furthermore,
the gas induced dilation is reversible on the same time scale.
0 1 2 3 4 5
0
5
10
15
PSU/CO
2
10 bar CO
2
0 bar
C / cm
3
(S T P )/ c m
3
t / h
(a) Sorption
0 1 2 3 4 5
0.0
0.5
1.0
1.5
10 bar CO
2
0 bar
PSU/CO
2
V /V
0
/ %
t / h
CO
2
sorption
CO
2
desorption
(b) Dilation
Figure 4.18: (a) Integral sorption of CO2at 10 bar gas pressure in PSU
() and subsequent desorption at 0 bar ( ). (b) Corresponding dilation
() and contraction ( ). Lines denote second and third measurement in
each diagram.
47
4.5 Experimental Results and Discussion 4. Experimental
Both results confirm the interpretation as elastic deformation (=instant and
reversible, see Section 3.3), induced by penetrant sorption in the rapid stage,
and justify the splitting of the isotherms (see previous section) into diffu-
sive/elastic and relaxational parts. The mass uptake and dilation were fitted
using equation 4.3, yielding the (quasi-) equilibrium values of CO2in PSU
concentration C= 15.6cm3/cm3at 10 bar and the induced elastic dilation
∆V/V0= 1.62 %as well as the diffusion coefficient D= 2.6·10−8cm2s−1.
The ratio of sorption and dilation may be regarded as a measure of the
partial molar volume Vp(eq. 2.14), which will be further discussed in Section
6.3. All values are the averaged result of three subsequent measurements and
are compiled in Table 4.2 on page 51, along with the results of the following
integral measurements.
Figure 4.19 shows the results of integral sorption and dilation in the PSU/CH4
system. Only one representative measurement of each, sorption and dilation,
is shown in the graph for clarity. The concentration level reached by CH4at
10 bar pressure is significantly lower than that for CO2, and the partial molar
volume Vpreveals a smaller dilation effect per CH4molecule than for CO2.
The diffusion coefficient Dis lower by an order of magnitude. This relatively
slow diffusion of CH4in PSU gave rise to the longest sorption and desorp-
tion times needed to ensure the determination of the (quasi) equilibrium
concentration and dilation, compared to the other investigated polymer gas
systems.
0 2 4 6 8 10
0
1
2
3
4
5
PSU/CH
4
C / cm
3
(S T P )/ c m
3
t / h
(a) Sorption
0 2 4 6 8 10 12
0.0
0.1
0.2
0.3
PSU/CH
4
V /V
0
/ %
t / h
(b) Dilation
Figure 4.19: (a) Integral sorption of CH4at 10 bar gas pressure in PSU
( ) and subsequent desorption at 0 bar ( ). (b) Corresponding dilation
() and contraction ( ). Lines for the second and third measurement are
omitted here and in subsequent diagrams.
48
4.5 Experimental Results and Discussion 4. Experimental
0 1 2 3 4 5
0
20
40
60
80
PI4/CO
2
C / cm
3
(S T P )/ c m
3
t / h
(a) Sorption
0 1 2 3 4 5
0
1
2
3
4
5
6
PI4/CO
2
V /V
0
/ %
t / h
(b) Dilation
Figure 4.20: (a) Integral sorption of CO2in PI4 ( ) and subsequent
desorption ( ). (b) Corresponding dilation ( ) and contraction ( ).
0 1 2 3 4 5
0
5
10
15
20
PI4/CH
4
C / cm
3
(S T P )/ c m
3
t / h
(a) Sorption
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
PI4/CH
4
V /V
0
/ %
t / h
(b) Dilation
Figure 4.21: (a) Integral sorption of CH4in PI4 ( ) and subsequent
desorption ( ). (b) Corresponding dilation ( ) and contraction ( ).
Figure 4.20 shows the integral sorption and dilation of CO2in PI4. The high
level of concentration is reached very fast, due to a large value of the diffusion
coefficient. The low partial molar volume Vpindicates a considerably higher
amount of free volume in PI4 than in PSU. This is confirmed by the integral
sorption an dilation measurement of PI4/CH4shown in Figure 4.21. Again,
diffusion of CH4is slower than CO2in PI4 by an order of magnitude, and
the concentration level reaches only about a quarter of that of CO2. The low
partial molar volume compared to CH4in PSU suggests a higher free volume
in PI4, as stated above. It is interesting to note that a small fraction of the
49
4.5 Experimental Results and Discussion 4. Experimental
0 1 2 3 4 5
0
20
40
60
80
100
PIM/CO
2
C / cm
3
(S T P )/ c m
3
t / h
(a) Sorption
0 1 2 3 4 5
0
2
4
6
8
PIM/CO
2
V /V
0
/ %
t / h
(b) Dilation
Figure 4.22: (a) Integral sorption of CO2in PIM-1 ( ) and subsequent
desorption ( ). (b) Corresponding dilation ( ) and contraction ( ).
CO2induced dilation seems to recover following kinetics that differ from the
(faster) desorption kinetics. This could be due to anelastic effects that are
discussed in more detail in Section 6.3. The dilation effect of methane is
obviously too small to induce such mechanisms and appears therefore purely
elastic.
Figure 4.22 shows the results of integral CO2sorption and dilation in PIM-1.
The concentration level exceeds that of CO2in PI4. The dilation]effect is
rather large and the larger value of the partial molar volume in comparison
with PI4/CO2suggests a smaller fractional free volume in PIM-1 than in PI4.
This unexpected result will be discussed in the context of the size distribution
of the free volume in Section 6.2. Again, the differing kinetics of desorption
(faster) and contraction (slower) reveal the existence of some anelastic effects,
which in the case of PIM-1 also occur in the CH4measurements. However,
in all cases (PI4/CO2,PIM-1/CO2,PIM-1/CH4) the dilation is completely
recovered within several hours. Diffusion of CH4in PIM-1 is quite fast, the
determined value of the diffusion coefficient at 10 bar is even slightly higher
than that for CO2.
]The poor quality of the signal of the volume dilation of the PIM-1 sample is due to a
resonance vibration of the freely suspended metal disc, which does not lead to significant
errors due to its regular sinusoidal form.
50
4.5 Experimental Results and Discussion 4. Experimental
0 1 2 3 4 5
0
10
20
30
40
PIM/CH
4
10 bar CH
4
0 bar
C / cm
3
(STP)/ cm
3
t / h
(a) Sorption
0 1 2 3 4 5
0.0
0.5
1.0
1.5
2.0
PIM/CH
4
V /V
0
/ %
t / h
(b) Dilation
Figure 4.23: (a) Integral sorption of CH4in PIM-1 ( ) and subsequent
desorption ( ). (b) Corresponding dilation ( ) and contraction ( ).
Table 4.2: Results of integral sorption and dilation.
polymer/ C/ ∆V/V0/ Vp/ D/
gas cm3/cm3% cm3/mol 10−8cm2/s
PSU /CO215.6 1.62 23.29 2.6
PSU /CH45.6 0.35 14.18 0.44
PI4 /CO281.4 6.13 16.89 32.2
PI4 /CH420.8 0.89 9.60 5.6
PIM-1/CO299.5 7.75 17.45 34.6
PIM-1/CH440.4 1.94 10.79 46.3
In continuation of experimental results, measured integral sorption and dila-
tion can be summarized as follows:
E5) In all cases, the dilation follows the kinetics of sorption, implying an
instantaneous reaction to sorption of penetrants.
E6) In all cases, the dilation is completely recovered upon desorption, im-
plying a reversible dilation effect, although in some cases it is not in-
stantaneously reversible.
E7) The findings E5) and E6) suggest the gas induced dilation to be of
elastic nature. Better time resolution or slower processes (as in some
cases the contraction) might reveal an anelastic nature of the dilation.
51
4.5 Experimental Results and Discussion 4. Experimental
E8) As may be expected from the calculated free volume (Bondi35, cf. Sec-
tion 4.1) of the different polymers, the concentration level of CO2and
CH4at 10 bar gas pressure order as follows: PSU <PI4 <PIM-1.
E9) The same order holds true for the dilation.
E10) Unexpectedly, the partial molar volume of the gases, which might be
expected in reversed order, arrange as PSU >PIM-1 &PI4.
E11) Equally unexpected is the fast diffusion of CH4in PIM-1; while for the
other polymers the ratio of the diffusion coefficients is DCO2/DCH41,
for PIM-1 it is DCO2/DCH4<1.
52
5 Modeling
Atomistic molecular modeling is a field of growing importance in the inves-
tigation of molecular phenomena and development of innovative materials.
Its potential lies within the detailed description of the behavior of ensembles
under a set of given conditions. The elaborate representation of molecular
systems allows a detailed insight into individual, small scale processes which
can otherwise only be grasped collectively as a statistical and thermody-
namic accumulation of individual effects. Although the physical description
cannot be called ‘complete’, relying on approximations and simplifications
to reduce complexity, it is nevertheless able to predict basic properties with
reasonable certainty and, by analysis and visualization of individual aspects,
adverts ideas of interpretation and conception of the investigated phenom-
ena. Once a procedure is validated, the performance of ‘virtual experiments’
may allow the characterization of new materials beforehand, saving a great
deal of effort and expenses associated with the synthesis and experimental
characterization.
It will become clear in the following paragraphs, that processes with large
time constants are impossible to simulate for large systems and with the
available CPU-power and -time. It was a goal of this work to establish ways
to work around the alleged incompatibility of experimentally observed long
term processes and the simulation of large systems. As was presented in
Section 4.5.1 as a result of the kinetic analysis (section 4.4), relaxational
processes on large time scales take place during sorption. In fact, sorption
that is kinetically controlled by Fickian diffusion must be called a long term
process with regard to simulation times, too, as the following estimation76
shows:
A typical value for a single diffusional step, as e.g. obtained from MD-
simulations77) is a≈5·10−8cm (5Å), a typical small molecule diffusion
coefficient for conventional glassy polymers is D≈10−8cm2s−1and a
common MD-simulation covers a simulation time of t≈10−9s(1ns).
Random walk of a particle with this diffusion coefficient Din a cubic
53
5.1 Forcefield based Molecular Modeling 5. Modeling
lattice of grid spacing aleads, via the Einstein equation,78 to a number
of observable penetrant jumps n= 6Dt/a21.
Obviously, sorption processes on these time scales cannot be directly simu-
lated nor is it possible to simulate relaxational processes on even larger time
scales. However, in the course of this work, the concept of preswollen pack-
ing models was developed,1proposing to construct and analyse two separate
states of a polymer/gas system with respect to the volumetric dilation of the
matrix and penetrant concentration. In combination with Grand Canonical
Monte Carlo simulations (section 5.2.2), sorption isotherms on static packing
models of two reference states can be calculated and combined2and thus be
compared to experimental data.
The same argument as above impedes the direct simulation of integral sorp-
tion. However, as was shown in Section 4.5.2, the dilation which is accom-
panied by the sorption of penetrants, seems to be of elastic nature, i.e., it
is (nearly) instantaneous and reversible with regard to the experimental res-
olution. The method of setting the penetrant concentration according to
experimental results within a static packing model and subsequently simu-
lating the volumetric dilation3is described in Section 5.2.3. A short intro-
duction to the background of the applied molecular modeling techniques will
be given first, followed by a simple analysis technique to characterize the free
volume of the packing models. For the molecular simulations described in
detail in the following paragraphs, the Insight II (4.0.0p),Cerius2, as well as
the Materials Studio (3.2) software of Accelrys, Inc. (San Diego, CA), were
used. To prepare the three-dimensional visualizations, the DS Viewer Pro 6.0
of Accelrys Inc. was used.
5.1 Forcefield based Molecular Modeling
The interaction of particles, bonded or nonbonded, is subject to a quantum
mechanical description. In principle, for the physically most accurate de-
scription available, the Schrödinger-equation for each particle (electrons and
nuclei) must be devised, leading to a complex set of equations. Its solution
is, though possible, beyond reasonable effort for the size of the systems and
the CPU-power available. However, the properties which are under investi-
gation in this work, i.e., static, thermodynamic and dynamic (transport and
relaxational) properties of non-reactive organic polymers, are well described
using forcefield based molecular mechanics (MM), molecular dynamics (MD)
and classical Monte Carlo (MC) simulations,79, 80 methods based on classical
54
5.1 Forcefield based Molecular Modeling 5. Modeling
mechanics of multi-particle systems. Needless to say, some quantum mechan-
ical information and experimental data are needed to establish the forcefield
in the first place. The information is obtained for relatively small units and
is, by extrapolation, assumed to be valid for larger systems of equal classes.
5.1.1 The Forcefield
Forcefields allow the calculation of the potential energy Uof an ensemble of
Natoms, as a function of their coordinates (r1. . . rN). It is composed of indi-
vidual contributions which describe the interactions of bonded atoms (bond-
lengths, -angles, conformation angles) and nonbonded interactions (van der
Waals, electrostatic).
U(r1,r2, . . . , rN) = X
bonds
bond-length-deformations
+X
angles
bond-angle-deformations
+X
conf.angles
torsional-deformation
+X
atom−pairs
nonbonded-interactions (5.1)
The contributions of bonded interactions are represented by anharmonic os-
cillators of the form81
Uij(rij) = k1(rij −r0
ij)2+k2(rij −r0
ij)3+k3(rij −r0
ij)4(stretch)
Uijk(θijk) = k1(θijk −θ0
ijk)2+k2(θijk −θ0
ijk)3+k3(θijk −θ0
ijk)4(angle)
Uijkl(Φi−l) = k1[1 −cos(Φi−l−Φ0
1)]2+k2[1 −cos(Φi−l−Φ0
2)]3(torsion)
+k3[1 −cos(Φi−l−Φ0
3)]4(5.2)
The force constants kiand the equilibrium positions r0
ij,θ0
ijk and Φ0
i−lare
based e.g. on results of quantum mechanics and constitute the integral part
of the forcefield. The nonbonded interactions are expressed in the applied
forcefield by a van der Waals term with a 9,6-potential and a coulomb-term
Uij(rij, qi, qj) = Aij
r9
ij −Bij
r6
ij !+qiqj
0rij
(nonbonded) (5.3)
where Aij and Bij are parameters describing the strength of the repulsive and
attractive force, qiand qjare the partial charges of the interacting atoms and
0is the vacuum permittivity.
55
5.1 Forcefield based Molecular Modeling 5. Modeling
(bond length) (bond angle) (torsion)
Figure 5.1: Illustration of (bonded) forcefield contributions.
The forcefield is defined by the functional form (eqns. 5.2, 5.3) and a set
of parameters ki, r0
ij,... which are specific to types of atoms, i.e., account
for different bonded states of the atoms. For a given molecular structure,
the forcefield results in a potential energy surface, which can be evaluated
with respect to local energy minima. These methods are known as molecular
mechanics (MM).81 In the course of optimization, geometrically reasonable
(static) structures can be obtained from the initially guessed geometry by
varying the atom positions and minimizing the potential energy of the system.
5.1.2 Molecular Dynamics (MD)
To perform molecular dynamics (MD) simulations, the forces that result from
the forcefield and which act on each atom are applied to the system of fi-
nite temperature, i.e., finite kinetic energy Ekin. Each of the Nparticles is
assigned a random (Boltzmann) velocity vector ˙
riso that the total kinetic
energy of the system corresponds to the desired temperature T
Ekin =
N
X
i=1
1
2mi˙
r2
i=(3N−6)
2kBT(5.4)
Here (3N−6) is the number of degrees of freedom and kBis the Boltz-
mann constant. In the course of a simulation, integration of the Newtonian
equations of motion
Fi=−∇riUi(r1, ..., rN) = mi¨
ri(5.5)
leads to a new velocity of each particle which can be extrapolated over
the time step ∆t= 1 fs of the simulation to determine the new coordi-
nates ri. The force Fi, acting on a particle iof mass mi, results from
the gradient of the potential energy Ui(see eq. 5.5) determined by the
forcefield eq. 5.1. The choice of the time step ∆tresults from the con-
sideration that the fastest vibration of the system should be sufficiently re-
solved. In a typical IR-spectrum, the C-H-bond shows a characteristic peak
56
5.1 Forcefield based Molecular Modeling 5. Modeling
at λC-H ≈3000cm−1, which corresponds to an oscillation period of τC-H ≈10fs.
The MD-simulations that were performed in this thesis using the Discover en-
gine with the commercial COMPASS force field82, 83 of Accelrys Inc.
5.1.3 The Concept of Ensembles
In statistical thermodynamics, details of individual particles are usually not
of great importance. On the contrary, for a realistic representation of ther-
modynamic behavior, the expectancy of observable properties are regarded.
This may be achieved by taking the mean value with respect to time or
as the average of a number of configurations. In molecular dynamics, the
evolution of a system with respect to time is observed. Depending on the
property under investigation, different ensembles are evaluated, i.e., different
state-variables are held constant to observe the behavior of others.
In the microcanonical ensemble the number of particles N, the total energy E
and the volume of the simulation cell Vare held constant. While keeping N
and Econstant is quite straightforward and needs no further explanation, the
volume Vof a system may be kept constant by periodic boundary conditions
(see 5.1.4), forcing a particle that leaves the virtual simulation cell to enter
on the opposite side by assigning the appropriate coordinates.
Acanonical ensemble is referred to if N,Vand the temperature Tare kept
constant. The easiest way to control the temperature is to directly scale
the particle velocities ˙
ri, whenever the system temperature Tsys leaves a
predefined temperature window T0±∆T:
˙
ri,new =˙
ri,old T0
Tsys !1/2
(5.6)
More refined methods like the Berendsen thermostat84 are more commonly
used because a temperature change per simulation time step leads to a more
smooth progression of the temperature. In this work, molecular dynamics
are applied to canonical ensembles as part of the equilibration steps in the
packing procedure (paragraph 5.1.5) and referred to as NV T-MD (according
to the state variables N,Vand Tthat are kept constant).
If, at constant Nand T, the pressure pis held constant instead of the volume
Vthe ensemble is called isothermal-isobaric. The pressure is evaluated using
the virial Ξand kinetic energy Ekin ∝NkBTbased on centers of mass:84
Ξ(t) = 1
2X
i<j
rij(t)Fij(t)(5.7)
57
5.1 Forcefield based Molecular Modeling 5. Modeling
It is controlled by changing the volume Vof the simulation cell according to
the relation
pV =NkBT+2
3hΞi(5.8)
Molecular dynamics simulations of isothermal-isobaric ensembles (NpT-MD)
are performed as final equilibration procedure of the packing models (see
Section 5.1.5) and they constitute the core part of the integral dilation sim-
ulation which is described in detail in Section 5.2.3.
An ensemble of constant pressure, volume, temperature and chemical poten-
tial µis called grand canonical. Here, the number of particles Nis allowed
to fluctuate. This is achieved by randomly inserting or deleting molecules.
The chemical potential of the molecules within the matrix is balanced with
a reference chemical potential, e.g. that of a surrounding gas phase at corre-
sponding pressure and temperature. Using a Monte Carlo algorithm, a new
configuration is energetically evaluated and accepted if more favorable than
the previous and otherwise rejected, allowing for thermal fluctuations using
a temperature dependent Boltzmann factor (see also Section 5.2.2). Grand
Canonical Monte Carlo (gcmc-) simulations are used in this work to calculate
sorption equilibria on static packing models for several pressures (chemical
potentials) at constant volume and temperature to determine concentration-
pressure isotherms which are referred to as gcmc-isotherms (see Section
5.3.3). The gcmc-simulations were realized with the ‘Fixed Pressure’ MC
algorithm of the Solid_Sorption module of the Cerius2package of Accelrys.
The same procedure was also used to insert a specified number of penetrants
for the integral dilation simulations by adjusting the chemical potential ac-
cordingly (see Section 5.2.3).
5.1.4 Periodic Boundary Conditions
An ensemble of some thousand atoms is large when regarded with respect to
computing effort, but the resulting size of the simulation cell does not allow
to calculate ‘bulk’ properties. The partial absence of nearest neighbors and
the corresponding forces for atoms at and near the surface of the simulation
cell falsifies results with regard to bulk properties at this
surface to volume ratio. The common way to avoid this
problem is the application of three-dimensional periodic
boundary conditions (3D-PBC). To this end, the cubic
simulation cell is surrounded by identical copies of the
original cell. Atoms at or near the surface of the original
cell now interact with surface atoms of the surrounding
58
5.1 Forcefield based Molecular Modeling 5. Modeling
replica. In the course of a simulation, if a molecule or atom leaves the
simulation cell through one wall, one of its replications enters the original
cell at the opposite side of the wall. It should be noted that the boundary
of the periodic box (wall) has no special significance other than defining the
shape and orientation of the primitive cells of a periodic lattice. To save
computing time and to avoid artifacts resulting from this forced periodicity,
e.g. interaction of an atom with its own images or more than one image of
other atoms in the replicated cells, the nonbonded interactions need to be
cut off at a reasonable distance, which should not exceed one half of the
edge-length of the simulation cell. Since the van der Waals interaction (eq.
5.3) decrease with the power of six of the distance, the error due to the
cut-off distance can be neglected. For electrostatic interactions, groups of
atoms, e.g. a methyl group (–CH3), are collected into charge-groups which
are assumed to be neutral to the outside and the interactions are calculated
only within this group. Since in this thesis no electric charges (ions) are part
of the simulations, a cut-off distance of 15 Åwas used for the packing models
whose edge-lengths are in all cases larger than 35 Å.
5.1.5 Packing Procedure
The detailed atomistic packing models that were investigated in this thesis
were constructed following the basic packing and equilibration procedure
that is based on the Theodorou-Suter method85, 86 and described in detail in
reference [10]. Basis for the construction of packing models is an equilibrated
repeat unit as depicted in Figures 4.1, 4.2 and 4.3. A head and a tail of the
repeat unit are defined and a template chain is grown adding repeat units
step by step and, as e.g. in the case of a CH2–CH2- bond, choosing one of
the favorable bond angles (cis, trans, gauche) at random, until a specified
number of repeat units is connected to the polymer chain (see e.g. Fig.5.2).
The ‘open’ bonds at each end of the chain are terminated by a hydrogen
atom. The single template chain is the subjected to energy minimizing using
molecular mechanics (MM).
To obtain a packing model, a cubic simulation cell is defined such that the
template chain confined to the volume of the simulation cell would yield a
density of about 10 %of the experimental value. 400 small molecules (CO2
or CH4) are distributed at random positions within this simulation cell as
obstacles. The low density and the obstacles within the cell prevent packing
artifacts such as ring-catenations or -spearings in the following packing step.
Using a random position within the simulation cell as starting point, the
template chain is packed, atom by atom along the backbone of the chain,
59
5.1 Forcefield based Molecular Modeling 5. Modeling
Figure 5.2: Template chain of PI4, containing 80 repeat units.
Figure 5.3: PI4 polymer chain in simulation cell of very low density with
3D-PBC applied (obstacles removed). Indicated in light red, the original
chain (grey) leaves the simulation cell and the replica (red/dark grey)
enters on the opposite side.
with 3D-PBC applied, i.e., if the chain ‘grows out’ of the simulation cell at
one side, it appears at the opposite side (growing in). If an obstacle is hit
during this procedure, a second choice of the available favorable angles is
tried. If all of the available angles fail, the angle of the previous repeat unit
is changed and the accretion reattempted. As a result, a packing model is
obtained containing a number of obstacles and a polymer chain of a very low
density which fulfills the periodic boundary conditions explained in Section
5.1.4. Figure 5.3 shows such a low density packing model with all obstacles
60
5.1 Forcefield based Molecular Modeling 5. Modeling
Figure 5.4: Equilibrated swollen packing model of 6FDA-TrMPD con-
taining 156 molecules of CO2(PI156). Triatomic CO2molecules are col-
ored in black. The polymer is represented as one chain (left) and as one
cell (3D-PBC applied) (right).
removed. The light grey chain shows the complete chain. In dark grey, all
atoms contained in the original simulation cell are displayed, demonstrating
how replica of the light-grey chain enter and leave the simulation cell. At
one position in the chain such behavior is marked in light-red (original chain)
and red (replica).
To obtain a swollen model at an experimentally determined concentration
of penetrants, most of the obstacles are deleted at random, leaving only the
specified number which are then replaced by the desired penetrant species.
The density is adjusted by scaling of the coordinates of all atoms and re-
sizing the cell dimensions. It is obvious, that this compression is achieved
by brute force: Therefore the nonbonded and torsional contributions to the
forcefield (see eqns. 5.1) are turned on at a level of 1% and increased one by
one to 100 %in several steps, performing minimizations (MM) and dynamics
simulations (NV T-MD) in between.
The packing models are further equilibrated by stimulated annealing, several
NV T-MD simulations at decreasing temperatures, typically starting above
experimental glass transition. Finally an NpT-MD simulation of at least
1.3ns is performed, allowing the volume to fluctuate. If the volume of the
packing model is stable within thermal fluctuations, an adequate level of
equilibration is reached and further analyses may be performed. Figure 5.4
shows an equilibrated swollen packing model of PI4 at the correct density.
On the left hand side, the polymer chain is represented as a continuous chain,
while on the right side 3D-PBC are applied to show the space filling of the
model. CO2molecules are depicted as black triatomic molecules. To obtain
61
5.2 Modeling Techniques 5. Modeling
nonswollen models of the pure polymer, all penetrants are removed from
the swollen model and the compression/ annealing/ equilibration treatment
is performed as described above. The amorphous polymer packing models
were constructed using the Theodorou-Suter method85, 86 as implemented in
the Amorphous_Cell module.87
5.2 Modeling Techniques
5.2.1 Free Volume Analysis
Several definitions of free volume in glassy polymers are employed in the
literature,1depending on the method of evaluation or the subject under in-
vestigation. In this work, the accessible free volume based on the insertion
of a test particle is used. To estimate the size distributions of free volume
elements (fves), a recently developed computer program88 was applied to
the packing models. The free volume is derived by the superimposition of a
fine grid over the cubic packing model. At every grid point a hard sphere
is inserted as test particle. If the test particle overlaps with any atoms of
the polymer matrix, which are also represented by hard spheres of van der
Waals radii, the grid point is classified as ‘occupied’ (see red circle in Figure
5.5(a). If there is no overlap (large green circle), the grid point is considered
as ‘free’ and contributes to the free volume (green dots). Neighboring free
grid points are collected into groups which represent individual holes.
(a) Vcon (b) Rmax
Figure 5.5: 2D-illustration of the free volume analysis: (a) The Vcon-
method groups free gridpoints, where the test-sphere (green) does not
overlap the matrix (grey), according to next neighborship (green dots).
(b) The Rmax-method further subdivides these groups by detecting con-
strictions (‘bottlenecks’, cf. text).
62
5.2 Modeling Techniques 5. Modeling
The grouping is done in two ways.88, 89 In the first approach (named Vcon, con
for connected), affiliation to a group is defined through next neighborship:
every point of a group has at least one next neighbor which is also member
of this group. This approach identifies holes, which may be of complex shape
and of large volume. In Figure 5.5(a) all gridpoints marked green contribute
to one fve according to the Vcon method. In a second approach (named
Rmax) for every free grid point the distance to the nearest matrix atom is
determined. By calculation of the gradient, the grid points are assigned to
the nearest local maximum in this distance. The Rmax-approach may divide
larger free volume regions of elongated or highly complex shape into smaller,
more compact regions (as illustrated in Figure 5.5(b)).
Figure 5.6 further illustrates the method for three dimensions: Atoms of the
polymer matrix are colored grey. Only free gridpoints are shown, depicted
as a small ball. The size of the test-particle is indicated by the transparent
surface (green) around the marked free gridpoint (black) in the lower left
corner. All free gridpoints that are shown belong to a single free volume ele-
ment in the sense of the Vcon method. Different coloring of the free gridpoints
indicate connectivity according to the Rmax method of grouping.
Figure 5.6: Visualisation of the free volume analysis in three dimen-
sions. One Vcon-fve is shown: Matrix atoms are grey, only free grid-
points are displayed and colored according to the Rmax-method. The
half-transparent test-sphere (green) is centered on the black grid-point.
63
5.2 Modeling Techniques 5. Modeling
Originally introduced to match better the situation in pals?experiments,
where the positronium probe can obviously not completely sample very large
holes of complex topology,88, 89 the second approach also seems more ade-
quate to depict the environment of a sorbed molecule:1, 2 An oblong hole
that is constricted at some point would be regarded as a single hole by the
Vcon method (see e.g. Fig. 5.13). However, a penetrant molecule would have
to jump an energy-barrier to pass this ‘bottleneck’ and therefore ‘see’ two
separate sorption sites. By defining a hole according to the Rmax method
this separation would be recognized (Fig. 5.13). With the help of a small
computer program, the gridpoints of a free volume element are assigned the
volume of a cube of edge length according to the grid spacing; corrections
are made for grid points that are at the surface of the fve. The volumes of
the grid points add up to the total volume of a free volume element. Results
of the free volume analysis and a visualization of the Free Volume will be
presented in Section 5.3.2.
5.2.2 Grand Canonical Monte Carlo Calculations
The atomistic packing models of the swollen and nonswollen state were used
to calculate CO2- and CH4-sorption isotherms, assuming a static polymer
matrix and therefore exclusively using the free volume elements of the ma-
trix as sorption sites. This was carried out by Monte Carlo simulations of
a Grand Canonical ensemble (gcmc). This well documented technique per-
mits the calculation of phase equilibria between gas- and sorbate phase. The
properties of the sorbed CO2(CH4) molecules in the fves of the polymer
matrix are calculated by statistic sampling of molecular configurations that
are consistent with temperature and chemical potential of the penetrants.
Assuming phase equilibrium, the chemical potential is calculated from the
gas phase at the specified temperature and pressure using an adequate equa-
tion of state (here: Peng Robinson93). For several pressures up to 50 bar,
it is tried to add single penetrant molecules at random positions within the
polymer matrix (overlaps with matrix atoms are not allowed), and consistent
with the chemical potential of the (virtual) gas phase. The new configura-
tion is energetically evaluated, compared to the previous configuration and
accepted or rejected according to a temperature dependent fluctuation factor.
In the same manner, a penetrant molecule is deleted from the configuration
or translated or rotated within the configuration. This way the number of
?Positron Anihilation Lifetime Spectroscopy, for a more detailed discussion in this
context see ref. [1], for pals data on PSU,PI4 and PIM-1 see refs. [90, 91], [64] and [66],
respectively, and for a study relating pals with the site distribution model see ref. [92].
64
5.2 Modeling Techniques 5. Modeling
penetrants may fluctuate and after some million iterations the system reaches
an equilibrium concentration of penetrants within the static polymer matrix
which corresponds to the pressure of the penetrant gas phase. As a result,
concentration pressure isotherms are obtained which are presented in Section
5.3.3. The insertion was realized with the ‘Fixed Pressure’ MC Algorithm of
the Solid_Sorption module of the Cerius2package of Accelrys.
5.2.3 Integral Dilation Simulation
In Section 4.5.2, the experimental results of integral sorption and dilation
were presented. As already mentioned at the beginning of this chapter,
the diffusion process of penetrants into the polymer matrix is too slow to
be directly simulated with our modeling method. Therefore, our approach,
also presented in ref. [2], is focused on the dilation only, bypassing the slow
diffusion process by establishing the experimentally observed concentration
within nonswollen static packing models by insertion of penetrants at ade-
quate sites.
To this end, the general scheme to establish the experimentally determined
concentration of penetrants within the packing models and to observe the
induced dilation, comprises the following steps of insertion/dilation:
1. Insertion of as many of the specified number of penetrants as possible,
using the Solid_Sorption tool described above.
2. Short NV T-MD run (20 ps) to obtain a ‘realistic’ velocity distribution
at T=308 K.
3. Longer NpT-MD run (300 ps) allowing volume fluctuations (‘dilation’)
consisting of two steps: (i) The first 20 ps are calculated with timesteps
of 0.1 fs for better resolution and (ii) the following 280 ps with timesteps
of 1.0 fs to save CPU time and to reduce the amount of data.
4. Deletion of all penetrants from the packing models.
5. Retry of insertion of the specified number of penetrants (steps 1.-3.).
The insertion was carried out by the ‘Fixed Loading’ MC algorithm of the
Solid_Sorption module (cf. Appendix A). Due to differences in the concen-
tration levels of the different gases and the initial free volume characteristics
of the polymer matrix, the target gas concentration may be reached after a
single insertion cycle (step 1.-3.), or multiple insertion cycles following the
65
5.2 Modeling Techniques 5. Modeling
general scheme (step 1.-5.), as for example for PI4/CO2. In this case, the
dilation result is the sum of the observed dilation during the 300 ps NpT -
MD run (step 3.), following each insertion step needed to reach the target
concentration. To check the reversibility of the observed dilation, after the
300 ps of NpT-MD, (step 3.), the penetrants were deleted from a copy of the
packing model and subsequently an NpT-MD simulation allowing volume
fluctuations was carried out to see the ‘contraction’ behavior of the dilated
packing models. All (dilation and contraction) NpT -MD-simulations were
continued up to a total simulation time of 1.3 ns.
66
5.3 Modeling Results and Discussion 5. Modeling
5.3 Modeling Results and Discussion
5.3.1 Detailed Atomistic Molecular Packing Models
In Section 5.1.5, the general packing and equilibration procedures that were
applied in this work have been introduced. Using the input parameters that
are compiled in Table 5.1 (on page 70), of each polymer and reference state
(characterized by density and penetrant concentration) three]packing models
were obtained that showed reasonable agreement in characteristics with ex-
perimental specifications.
Polysulfone
The packing models of pure polysulfone,1named PSU,\contain a single chain
consisting of 94 repeat units (r.u.) as depicted in Figure 4.1, amounting to
5048 atoms within the cubic simulation cell of average edge length 38.6Å.
This corresponds to an average density ρpure = 1.200 g/cm3of the PSU pack-
ing models, deviating from the experimentally specified value by about −3%.
Figure 5.7 shows a representative slice of 5.5Åthickness cut off one pack-
ing model for convenient visualization. All slices of one packing model per
polymer and CO2-reference state may be viewed in Appendix A.3, Figures
A.1-A.6. Since the swelling effect of CH4is small, no additional visualisations
of CH4-swollen packing models are shown in this thesis.
For the CO2-swollen packing models of PSU at a pressure of 50 bar, the
experimentally reached concentration level of CCO2= 52.6cm3/cm3corre-
sponds to the number of 80 CO2molecules per packing model (mpp) and
were therefore named PSU80.2The experimentally obtained volume dilation
of ∆V/V0= 6.5%corresponds to a density of ρexp = 1.262 g/cm3of the bi-
nary system. On average, the packing models PSU80 reached a density of
ρsim = 1.233 g/cm3, i.e., 39.3Åedge length of the simulation cell, agreeing
quite well with the experimental specifications (−1.8%). A slice of 5.6Å
thickness of a PSU80mpacking model with CO2penetrants removed (index
m)[is depicted in Figure 5.8.
]With the exception of PSU80 and PSU35, the swollen models of PSU/CO2and
PSU/CH4, respectively, where four packing models were used for all analyses.
\Note that the packing models of the polymers are typeset in italic, whereas the ‘real’
material is abbreviated in normal font.
[The swollen packing models with removed penetrant molecules are marked by the
index mas in PSU80mto indicate that only the polymer matrix has been analysed.
67
5.3 Modeling Results and Discussion 5. Modeling
Experimental results specify a number of 35 CH4molecules per simula-
tion cell of 38.9Å(ρexp = 1.242 g/cm3) for the swollen PSU/CH4system
(22.4cm3/cm3). Packing and equilibration efforts resulted in PSU35 models
of average density ρsim = 1.189 g/cm3(−4.2%). It should be noted, that
the packing models of pure polysulfone (PSU ) and CH4-swollen polysulfone
(PSU35) were obtained by deleting CO2molecules from the PSU80 models
(and in the case of PSU35 sufficient random replacement by CH4, see Section
5.1.5) and subsequent densification; therefore identical specifications regard-
ing the number of chains and matrix atoms apply to all polysulfone packing
models.
PSU PI4 PIM
Figure 5.7: Representative slices of nonswollen packing models of PSU,
PI4 and PIM . For more slices confer to Figures A.1, A.3 and A.5 in
Appendix A.3.
PSU80mPI156mPIM206m
Figure 5.8: Representative slices of nonswollen packing models of
PSU80m,PI156mand PIM206m. For more slices confer to Figures A.2,
A.4 and A.6 in Appendix A.3.
6FDA-TrMPD Polyimide (PI4)
The packing models of the pure polyimide 6FDA-TrMPD, named PI4, each
contain a single chain of 80 repeat units (see Figure 4.2). The simulation cells
of 38.8Åaverage edge length consist of 4482 atoms, corresponding to an av-
erage density of ρsim = 1.273 g/cm3, showing a relatively high deviation from
68
5.3 Modeling Results and Discussion 5. Modeling
the target density (−5.8%, see also Section 5.3.2). A slice of the nonswollen
packing model PI4 of 5.5Åthickness is depicted in Figure 5.7.
The reference state for the CO2-swollen PI4 packing models was taken at
35 bar CO2pressure (see Section 4.5.1). The concentration of 105.6cm3/cm3
corresponds to 156 molecules of CO2within the simulation cell at a dila-
tion of ∆V/V0= 21.4%. The obtained packing models were stable under
NpT-equilibration at an average density of ρsim = 1.359 g/cm3(edge length
39.8Å), deviating by −5.8%from experimental specifications. Figure 5.8
shows a slice of the CO2-depleted packing model PI156 (5.7Å).
CH4-swollen models were constructed meeting the experimentally determined
concentration (69.4cm3/cm3) and dilation (3.5%) state at 50 bar CH4pres-
sure. This corresponds to 76 CH4molecules in the PI76 packing models,
which reached an average packing density of ρsim = 1.276 g/cm3(39.1Åedge
length), deviating by −4.9%from the target density.
PIM-1
The property of intrinsic microporosity of the PIM-1 investigated in this work
is a consequence of the rigid and contorted repeat unit (see Figure 4.3).65
However, this rigidity also poses difficulties for the packing algorithm (section
5.1.5), slowing the process of finding a valid configuration of a single chain
packed inside a simulation cell. It has proven to be more successful to pack
aPIM-1 model as a number of smaller chains.94 The packing models of the
pure PIM-1 (named PIM) therefore contain 5 chains of 15 repeat units, i.e.,
4145 atoms, within the simulation cell of average density ρsim = 1.073 g/cm3
(37.7Åedge length), a deviation of −4.5%to the experimental density. The
slice of the PIM packing model presented in Figure 5.7 has a thickness of
5.4Å.
Experimental sorption results at 45 bar CO2pressure correspond to 206 CO2
molecules per packing model (150.1cm3/cm3) for the swollen reference state
of the PIM-1/CO2system.95 The PIM206 packing models reached an average
density of ρsim = 1.229 g/cm3(39.3Åedge length), agreeing well with experi-
mental specifications (−2.8%). Figure 5.8 shows a 5.6Åslice of a PIM206m
packing model.
The packing models of the swollen PIM-1/CH4reference state at 45 bar con-
tain 95 molecules of CH4(69.4cm3/cm3).95 The density of the packing models
ρsim = 1.047 g/cm3deviates from the experimentally determined value (vol-
ume dilation of 7.5%) by 4%.
69
5.3 Modeling Results and Discussion 5. Modeling
Table 5.1: Construction parameters of packing models.
polymer/gas name r.u.achains atoms pref ρtbρocdev.
bar g/cm3g/cm3%
PSU (pure) PSU 94 1 5078 0 1.240 1.200 −3.1
PSU /CO2PSU80 +240 50 1.262 1.233 −1.8
PSU /CH4PSU35 +175 50 1.242 1.189 −4.2
PI4 (pure) PI4 80 1 4482 0 1.352 1.273 −5.8
PI4 /CO2PI156 +468 35 1.284 1.359 −5.8
PI4 /CH4PI76 +380 50 1.342 1.276 −4.9
PIM-1 (pure) PIM 15 5 4145 0 1.124 1.073 −4.5
PIM-1 /CO2PIM206 +618 45 1.229 1.194 −2.8
PIM-1 /CH4PIM95 +475 45 1.091 1.047 −4.0
aNumber of repeat units.
bTarget density.
cObtained density.
5.3.2 Free Volume Analysis
In Section 5.2.1 the method of analyzing the free volume of a packing model
by insertion of a testparticle was introduced. In the following, the results
of this insertion method will be presented. All packing models of the pure
polymers were analysed using a positronium sized test particle (1.1Åradius)
and a grid spacing of 0.5Å. Two different approaches of group affiliation, Vcon
and Rmax, were applied. That way, for each polymer, two size distributions
of free volume elements (fves) were obtained. It is common in the literature
to assign each volume Vthe radius of a volume-equivalent sphere (r.e.s.)
r=3
qV3/4π. The size distributions are displayed as the fractional free
volume (ffv)
ffvi=Ni(4/3πr3
i)
V0
(5.9)
where Niis the number of fves found by the respective method within
an interval [ri±0.25 Å], and V0is the analyzed volume of the nonswollen
polymer. It should be noted that, strictly speaking it is a distribution of
equivalent sphere radii, rather than a distribution of fve volumes. In the
same manner, the CO2-swollen packing models were analyzed after removing
all CO2molecules. The volume of the nonswollen polymer V0is also used for
the ffv of the swollen models to emphasize on the increase in free volume.
The swelling effect of the polymer/CH4systems was too small for the analysis
70
5.3 Modeling Results and Discussion 5. Modeling
to contrast the nonswollen models and therefore in this thesis, the focus of
the free volume analysis will be on pure and CO2-swollen packing models. In
the following, for each polymer, swelling state and method of group affiliation
the distributions are presented.
PSU and PSU80m
Figure 5.9(a) shows the analysis results for the nonswollen and swollen poly-
sulfone packing models PSU and PSU80m. The distribution of the fractional
free volume of the nonswollen packing models PSU shows that the free vol-
ume is distributed rather evenly among all sizes of fves found in the analysis.
However, it should be kept in mind that for small sizes a larger number of
individual fves is needed for the same fraction of the free volume. This will
become more apparent when displayed as a number distribution of sorption
site volumes as it is done in Section 6.2. No exceptionally large free volume
elements have been detected in the packing models of pure polysulfone.
For the swollen models PSU80mthe shift to larger fves is apparent. The
difference in area of the distributions may be viewed as a measure of the
increase in free volume in PSU and PSU80m. However, it should be kept in
mind, that both distributions are based on the volume V0of the nonswollen
packing models PSU, for better comparison. Besides the general shift to
higher radii of equivalent spheres the tendency of smaller fves merging to
larger elements of free volume can be observed.
0 2 4 6 8 10
0
1
2
PSU / PSU80
FF V / %
radius of equivalent sphere / Å
(a) Vcon
0 1 2 3 4 5 6 7
0
1
2
3
4
5
6
FF V / %
radius of equivalent sphere / Å
PSU / PSU80
(b) Rmax
Figure 5.9: Fractional free Volume in nonswollen (hatched) and swollen
(blue) models of PSU, as detected by a positronium sized sphere on a
0.5Ågrid. (a) ffv distribution according to the Vcon-method and (b)
after subdivision by the Rmax-method.
71
5.3 Modeling Results and Discussion 5. Modeling
In contrast to the PSU packing models, the swollen packing models also
contain few rather large fves at around 10 Åradius, within the analysed
volume. It should be noted that only two of these voids were detected in
the three packing models of PSU80m; however, at this size they contribute a
rather large fraction to the free volume of the polymer. Figure 5.9(b) shows
the result of the free volume analysis using the Rmax-method. According
to this method, group affiliation is not determined by a mere connectivity
test, but also constrictions or ‘bottlenecks’ are detected (cf. Section 5.2.1).
As a result, large volumes of complex shape, which would appear as one
fve according to the Vcon-method, are divided into several fves of a more
spherical shape. Consequently, more fves are detected and no particularly
large ones are observed. Furthermore, the distribution of the fractional free
volume now shows a distinct peak at around 2År.e.s. for the PSU models.
This peak is shifted to about 3Åfor the swollen PSU80mpacking models in
addition to a gain in area, again indicating the increase in the amount of free
volume.
PI4 and PI156m
Figure 5.10(a) shows the results of the Vcon-method applied to the nonswollen
and swollen packing models of the polyimide PI4. For small radii of equivalent
spheres, i.e., up to 6Å, the size distribution of fves of the nonswollen PI4
packing models appears similar to that of the PSU packing models: The
fractional free volume is distributed evenly in that range and no peak is
observed (see inlet of Figure 5.10).
2 4
0
1
0 5 10 15 20
0
2
4
6
8
10
12
14
16
PI4 / PI156
FF V / %
radius of equivalent sphere / Å
(a) Vcon
0 1 2 3 4 5 6 7
0
1
2
3
4
5
6
PI4 / PI156
FF V / %
radius of equivalent sphere / Å
(b) Rmax
Figure 5.10: Fractional free Volume in nonswollen (hatched) and swollen
(blue) models of PI4. (a) ffv distribution according to the Vcon-method
and (b) after subdivision by the Rmax-method.
72
5.3 Modeling Results and Discussion 5. Modeling
However, each individual packing model shows one rather large fve in the
range 11 Åto 14 Å, that contributes (due to its size) the main fraction to
the free volume. Upon swelling (PI156mpacking models), the shift to larger
fves is even more apparent than in PSU. Compared to the nonswollen PI4,
only few percent of the ffv is organized in the small r.e.s. range, while the
three large fves appear to make up for all the swelling of the polymer.
Again, the Rmax analysis divides the large fves into smaller units (Figure
5.10(b)). Both, PI4 and PI156mpacking models show a distinct peak struc-
ture in the ffv distribution (2.5Åand 3Å), and while the shift between PI4
and PI156mis not as pronounced as for the packing models of polysulfone,
the broader shoulder to higher r.e.s. is more developed upon swelling, and
area and larger radii attest a large free volume. It should be noted that this
swelling state of PI156 serves as reference state at only 35 bar CO2pressure,
in contrast to the packing models PSU80 and PIM206 (50 bar).
PIM and PIM206m
The results of the free volume analysis (Vcon) on the PIM packing models
are shown in Figure 5.11(a). Only a small fraction of the free volume of the
PIM packing models is present as more or less evenly distributed fves of
small radii. On the contrary, most of the free volume (of each of the three
individual PIM packing models) is located in one extraordinary large fve
with an equivalent sphere radius above 15 Å. As observed in PI4, the analysis
of the PIM206mpacking models shows that the swelling (gain of free volume)
takes place in this large fve and at the cost of smaller fves. It should be
noted that the shift of the large fves to about 20 År.e.s. contributes to
volume gain in the third power.
If ‘bottlenecks’ and constrictions are considered, as in the Rmax analysis, the
large fves of complex shape are divided into smaller units and a distribution
of the ffv is obtained that show peaks at 2.6Åand 3Åfor the nonswollen and
swollen models, respectively. Similar to PI4/PI156, the peak of the swollen
packing models PIM206mshows a broadened shoulder towards higher r.e.s.
and an enlarged area, indicating the increase in free volume.
73
5.3 Modeling Results and Discussion 5. Modeling
2 4 6
0
1
0 5 10 15 20
0
5
10
15
20
PIM / PIM206
FF V / %
radius of equivalent sphere / Å
(a) Vcon
0 1 2 3 4 5 6 7
0
2
4
6
PIM / PIM206
FF V / %
radius of equivalent sphere / Å
(b) Rmax
Figure 5.11: Fractional free Volume in nonswollen (hatched) and swollen
(blue) models of PIM-1. (a) ffv distribution according to the Vcon-
method and (b) after subdivision by the Rmax-method.
Free Volume Visualization
In the free volume analysis method of test particle insertion discussed above,
a three-dimensional grid is superimposed on the simulation cell containing the
packing model. At the coordinates of the overlap-free gridpoints, a sphere
of testparticle-size may be inserted to visualize the free volume.However,
looking at the packing model PI156 displayed in Figure 5.4 makes clear that
it does not suffice to display the simulation cell as a whole, because the matrix
molecules obstruct the view and the great number of holes interfere with each
other if displayed together. To get a better impression, the simulation cell is
cut into slices of about 5Åthickness. For even better viewing, matrix atoms
are displayed in ‘stick-style’ and are colored grey. It should be noted that
due to the slicing process and to periodic boundary conditions some matrix
atoms as well as fves appear fragmented or continue on the opposite side.
Figure 5.12 shows representative slices, cut out of one packing model of each
polymer and reference state (nonswollen and CO2-swollen). In the top row,
the fves of the PSU and PSU80 slices are visualized. Next to a number of
small individual fves (orange\), only few larger fves contribute to the free
volume of the PSU -slice, and all fves are evenly distributed between the
matrix atoms. The slice of the PSU80 packing model is shown on the right
side.
\If necessary, the spheres are colored differently according to connectivity, to distinguish
between individual fves. However, the colors green, red and purple always refer to one
single fve according to the Vcon-method.
74
5.3 Modeling Results and Discussion 5. Modeling
PSU
PI4
PIM
PSU80
PI156
PIM206
Figure 5.12: Slices of nonswollen (left) and swollen (right) packing
models of PSU, PI4 and PIM-1. Green, red or purple color indicates one
single fve (associated free grid-points according to Vcon), whereas other
colors are used to mark individual fves. Matrix atoms are represented as
grey ‘sticks’.
75
5.3 Modeling Results and Discussion 5. Modeling
The fves are represented by semi-transparent surfaces, to permit the view
on the CO2penetrant molecules, which are colored black. The shift to fves
of larger size is easily noticeable. Obviously the larger fves contain more
than one CO2molecule, supporting the employment of the Rmax-method
when considering sorption sites. Figure 5.13 shows the purple fve of the
PSU80 slice. On the left hand side, the Vcon-fve is shown in a wire-mesh
representation, containing three CO2molecules. On the right side, the Rmax-
fves are shown as solid surface representations. These are more compact
and of a more spherical size than the Vcon-fves and generally provide space
for only one CO2molecule. It has to be noted that in this visualization
the Rmax-fves overlap because the grid size is smaller than the testparticle
radius.
The second row of Figure 5.12 shows the slices of PI4 and PI156. For the
nonswollen packing model the number of small fves is reduced in comparison
to PSU . Large fves dominate the visualization. The largest fve (colored
green) spans the entire height of the simulation cell and continues through
neighboring slices.]This becomes even more pronounced in the slice of the
swollen packing model PI156. As the quantitative analysis has shown (cf.
Figure 5.10), most of the free volume of a single packing model is present in
one individual fve. The high complexity of the shape suggests the formation
of a dendritic mesh of connected ‘holes’ in the polymer matrix, which could be
thought of as a void-phase, as opposed to singular, disconnected and smaller
fves as they are found in PSU (and PSU80). The majority of the CO2
penetrants are contained within this void-phase, emphasizing its importance
for penetrant sorption and transport.
In the bottom row of Figure 5.12, the visualization of fves of the PIM and
PIM206 slices is shown. Of the three nonswollen polymers, the PIM packing
models contain the largest single fves, exhibiting equivalent sphere radii
larger than 15 Å, which is not much less than those of the swollen PI156
packing models. Accordingly, the PIM-slice shows a void-phase for the PIM
packing models even in the nonswollen state. The corresponding fve (colored
green) expands wide over the plane of the slice and its fragmentation points
at the three dimensional character of the expansion.]Upon swelling to the
PIM206 reference state, this void-phase grows at the cost of smaller fves
and dominates the appearance of the free volume, while loosing none of its
complexity in shape. As in PI156, the penetrant CO2are mainly located in
the void-phase.
]In this method of visualization, discontinuities within one fve point to the continuance
within a neighboring slice and reentry at another position into the shown slice. Likewise,
‘flat’ surfaces at the edge of the slice point to a fve reaching across slices.
76
5.3 Modeling Results and Discussion 5. Modeling
(a) Vcon (b) Rmax
Figure 5.13: (a) Single fve as analyzed by the Vcon-method in PSU80.
(b) Same fve subdivided by the Rmax-method.
02468
0
2
4
non-sw ol len models
PSU PI4 PIM
FF V / %
radius of equivalent sphere / Å
(a) non-swollen packing models
0 2 4 6 8
0
2
4
6
sw ollen models
PSU PI4 PIM
FF V / %
radius of equivalent sphere / Å
(b) swollen packing models
Figure 5.14: Fractional free volume distributions in nonswollen pack-
ing models of PSU, PI4 and PIM-1 and in the swollen packing models
PSU80m,PI156mand PIM206m, according to the Rmax-method.
The results of the free volume visualization and analysis of the polymer pack-
ing models can be summarized as follows:
M1) The free volume was successfully probed using a test particle insertion
method. Two different approaches of group affiliation (Vcon and Rmax)
yield diverse size distributions which allow the perception of free vol-
ume from different perspectives: While the Vcon-method yields more
information about the connectivity of free volume, the Rmax-method
resembles better the surrounding volume of sorbed penetrants.
M2) The fractional free volume (ffv) detected by the Vcon-method in non-
swollen packing models is distributed rather evenly in size up to an
77
5.3 Modeling Results and Discussion 5. Modeling
equivalent sphere radius (r.e.s.) of about 8Å. However, the PI4 pack-
ing models each contain one extraordinary large free volume element
(fve) and in the PIM models one fve of highly complex shape contains
the major fraction of the free volume (void-phase).
M3) Upon swelling, all polymers tend to develop larger fves at the cost of
smaller ones. While the PSU80mpacking models show fves of only
moderate size, the PI156mpacking models exhibit a void-phase similar
to all PIM /PIM206 packing models.
M4) The Rmax-method breaks up larger fves by considering ‘bottlenecks’.
The size distributions of the ffv of all packing models exhibit a similar
peak structure slightly increasing in peak position and width in the
order PSU <PI4 .PIM , reflecting the expected amount of free volume
present in each polymer, respectively (cf. Fig. 5.14(a)).
M5) The swollen models of each polymer show a shift of the peak position
towards larger fves. However, the shift between polymers is not as
pronounced as for the nonswollen models (cf. Fig. 5.14(b)).
M6) The Rmax-method divides the exceptionally large fves found by the
Vcon-method, including the void-phase, into smaller units that better
represent ‘sorption sites’ for penetrant molecules.
5.3.3 Sorption Isotherms
In Section 5.2.2 the possibility to calculate concentration-pressure isotherms
using Grand Canonical Monte Carlo (gcmc) simulations on static packing
models was introduced. To this end, penetrants are inserted at adequate po-
sitions into the static matrix until an equilibrium is reached that is consistent
with the chemical potential of a virtual gas phase at a given pressure and
temperature (here: 308 K). Repeated calculation at several pressures leads to
the sorption isotherms on static packing models. In the following, the results
of the calculations are presented that were conducted for CO2and CH4on the
nonswollen packing models and on the swollen packing models with removed
penetrants. For each polymer, gas and swelling state, the gcmc sorption
isotherm was calculated for all packing models. The results are presented as
an average of the respective packing models with error bars indicating the
standard deviation. Dual mode sorption fits were used to interpolate the
simulated sorption data and fitparameters are compiled in Table 5.2.
78
5.3 Modeling Results and Discussion 5. Modeling
Table 5.2: Dual Mode Sorption model parameters for gcmc-simulated
sorption isotherms (see eq. 3.3).
Model kD/ C0
H/ b/
/gas cm3/cm3bar−1cm3/cm3bar−1
PSU /CO20.066 14.0 0.384
PSU80m/CO20.113 45.2 0.815
PSU /CH40.076 9.40 0.176
PSU35m/CH40.073 22.7 0.247
PI4 /CO20.243 51.6 0.931
PI156m/CO20.254 96.4 0.980
PI4 /CH40.187 40.4 0.179
PI76m/CH40.229 54.4 0.205
PIM /CO20.269 72.6 1.165
PIM206m/CO20.321 121.9 0.815
PIM /CH40.195 53.3 0.183
PIM95m/CH40.270 74.5 0.183
Simulated CO2and CH4sorption in Polysulfone
Figure 5.15(a) shows the calculated sorption isotherms for CO2on the packing
models PSU () and PSU80m( ) along with the experimentally measured
sorption isotherm ( , cp. Fig. 4.10 on page 37). Dual mode sorption fits
(represented as broken lines) were used to interpolate the data, and a cir-
cle indicates the experimental reference data used to construct the swollen
packing models. Neither simulated gcmc-isotherm is able to represent the
experimentally measured data over the full pressure range. The nonswollen
PSU sorption isotherm compares well to experimental data at low pressures;
however, after the initial increase, the calculated isotherm levels off rapidly
at a pressure of about 5bar, fairly underestimating experimental data at
intermediate and high pressures. In contrast, the PSU80msorption isotherm
increases rapidly at low pressures, overestimating the experimental data over
the whole pressure range but agreeing well at the highest pressure. The
flattening of the curve occurs at about the same pressure as for the PSU
isotherm.
79
5.3 Modeling Results and Discussion 5. Modeling
0 10 20 30 40 50
0
10
20
30
40
50
PSU/CO
2
C / cm
3
(S T P )/ c m
3
p / bar
(a) CO2sorption
0 10 20 30 40 50
0
5
10
15
20
25
PSU/CH
4
C / cm
3
(S T P )/ c m
3
p / bar
(b) CH4sorption
Figure 5.15: CO2(a) and CH4(b) gcmc isotherms calculated for non-
swollen ( ) and swollen ( ) models of PSU. The high-pressure reference
data taken from experimental measurement ( ) are marked by a red circle.
This is an expected result. The gcmc calculations are performed with static
packing models. No rearrangements of the polymer matrix may take place
to better accommodate the CO2penetrants within the matrix; any swelling,
elastic or relaxational, as it is observed in sorption induced dilation experi-
ments (Section 4.5.2) or in the integral dilation simulations (Section 5.3.4),
is excluded. Solely the ‘movement’, i.e., rotation or relocation of the pene-
trants by the gcmc insertion algorithm, may lead to a reduction of chemical
potential of the ensemble and hence to further ‘sorption’. The simulated
sorption may thus be regarded as a ‘hole filling’ process. The nonswollen
packing models provide enough energetically favorable sorption sites in the
beginning, agreeing with the experimental data rather well until at moder-
ate concentrations most of the sorption sites are occupied and the sorption
isotherm levels off. In contrast, the swollen packing models PSU80mare at an
artificially low density because they were constructed to provide space for the
experimentally determined concentration at 50 bar CO2pressure, hence the
overestimation. However, the agreement of the simulated sorption isotherms
of the nonswollen packing models PSU and the fact that after removal of the
CO2penetrants from the swollen packing models PSU80, the experimental
concentration at the highest pressure is reached by way of Grand Canonical
Monte Carlo simulations on the PSU80m, can be regarded as a good result
and a validation of the gcmc method in this context. A way to further
improve this result is introduced in Section 6.1.
Figure 5.15(b) shows the gcmc-sorption isotherms for CH4on the nonswollen
PSU packing models and the swollen PSU35mpacking models in the same
80
5.3 Modeling Results and Discussion 5. Modeling
fashion and using the same symbols for swollen, nonswollen and experimen-
tal data as Figure 5.15(a) for CO2. Both isotherms show the same main
features as already observed for CO2sorption: The isotherm of the non-
swollen packing models agrees well with experimental data at low pressures
and underestimates at high pressures. The isotherm of the swollen models
PSU35moverestimates the experimental sorption isotherm but for the highest
pressure, where experimental and simulated concentration agree well.
Simulated CO2and CH4sorption in 6FDA-TrMPD
Figure 5.16(a) shows the simulated CO2-sorption isotherms in the nonswollen
PI4 and swollen PI156mpacking models. Here, the reference pressure of
35 bar was selected from the experimental data due to technical reasons
(cf. Section 4.5.1). All the same, the gcmc calculations were performed
up to a pressure of 50 bar. This leads to the peculiarity that the sorption
isotherm of the PI156mpacking models seems to underestimate extrapolated]
experimental data ( ) in the high pressures range above the reference point.
However, the agreement of experiment and model is excellent at the reference
pressure itself, as should be expected. It is questionable if the extrapolation
of Dual Mode sorption model results is feasible, however, the method was
used here to illustrate the reasonably expected underestimation by the gcmc
sorption isotherm of the static PI156mpacking models above the reference
point.
0 10 20 30 40 50
0
20
40
60
80
100
120
PI4/CO
2
C / cm
3
(S T P )/ c m
3
p / bar
(a) CO2sorption
0 10 20 30 40 50
0
10
20
30
40
50
60
PI4/CH
4
C / cm
3
(S T P )/ c m
3
p / bar
(b) CH4sorption
Figure 5.16: CO2(a) and CH4(b) gcmc isotherms calculated for non-
swollen ( ) and swollen ( ) models of PI4.
]By calculation of the concentration at 40 and 50 bar, using the dm parameters listed
in Table 4.1 on page 40.
81
5.3 Modeling Results and Discussion 5. Modeling
The CH4-sorption isotherms for the nonswollen and swollen packing models
PI4 and PI35mare presented in Figure 5.16(b). Compared to the previ-
ous isotherms, the isotherm of the nonswollen packing model PI4 follows the
experimental data over a wide pressure range up to about 40 bar. The agree-
ment between the experimental reference point at about 50 bar is satisfying,
considering the noticeably smaller difference between isotherms for swollen
and nonswollen packing models (by comparison with the previous isotherms).
Simulated CO2and CH4sorption in PIM-1
In Figure 5.17(a) the CO2gcmc-sorption isotherms for the packing models
of the polymer of intrinsic microporosity (PIM-1) are shown. The isotherm of
the nonswollen PIM represents the low pressure range well, as expected. The
PIM206 packing models, on the other hand, do not seem to be able to take
up as many penetrant molecules at 50 bar as the models were constructed
for, understating the experimental data at the reference point. It should be
noted that at 35 bar, where experimental and gcmc isotherms cross, the
agreement is merely coincidental and should not be compared with that of
the CO2-PI4 isotherm in Figure 5.16, where a reference state at 35 bar was
chosen.
The CH4-PIM-1 isotherms are presented in Figure 5.17(b). Again, the iso-
therms of the nonswollen packing models PIM represent the experimental
data in the low pressure range up to 5bar quite well and underestimate at
higher pressures.
0 10 20 30 40 50
0
20
40
60
80
100
120
140
PIM/CO
2
C / cm
3
(S T P )/ c m
3
p / bar
(a) CO2sorption
0 10 20 30 40 50
0
20
40
60
80
PIM/CH
4
C / cm
3
(S T P )/ c m
3
p / bar
(b) CH4sorption
Figure 5.17: CO2(a) and CH4(b) gcmc isotherms calculated for non-
swollen ( ) and swollen ( ) models of PIM-1.
82
5.3 Modeling Results and Discussion 5. Modeling
As was the case for PI4-CH4(Figure 5.16(b)), the swollen packing models
PIM95mtake up slightly more penetrants than the model was built for. How-
ever, the deviation is within acceptable limits, considering the concentration
difference between swollen and nonswollen packing models at the reference
pressure. It is obvious from Figures 5.15 to 5.17), that the CH4isotherms
in all three polymers do not level off as rapidly as do the CO2-sorption iso-
therms. This could be due to the fact, that none of the CH4isotherms reach a
concentration level at which the respective CO2-isotherm levels off, implying
that the polymer matrix still provides enough space in the free volume for fur-
ther sorption of penetrants without the need for rearrangement or relaxation
of the polymer chain (dilation).
Continuing the summary on page 77, the modeling results of the concentra-
tion-pressure isotherms calculated by gcmc simulations at 35 ◦Con non-
swollen and swollen packing models for CO2and CH4can be resumed as
follows:
M7) The gcmc sorption isotherms that were calculated for nonswollen and
swollen packing models of three polymers and two gases are in good
agreement to the experimental data within the pressure range of the re-
spective reference state, validating the method of calculating isotherms
using gcmc simulations.
M8) Outside the respective pressure ranges, the observed underestimation of
experimental data for the nonswollen packing models can be explained
by the impossibility of the polymer matrix to relax stresses which may
be induced by the insertion of penetrants.
M9) The sorption isotherms of swollen packing models overestimate the ex-
perimental data in low and intermediate pressure ranges. This may be
understood using the simple picture of a hole filling process in low den-
sity packing models that provide enough free volume to accommodate
large concentrations of penetrants.
M10) In general, the calculated CH4-isotherms seem to level off less rapidly
with increasing pressure than do the CO2-isotherms, suggesting that
not all possible ‘sorption sites’ are occupied at the highest simulated
pressure.
83
5.3 Modeling Results and Discussion 5. Modeling
0 250 500 750 1000 1250
0.0
0.5
1.0
1.5
2.0
2.5
3.0
PSU/CO
2
model 1
model 2
model 3
V /V
0
/ %
t / ps
CO
2
insertion
10 bar NpT, C=15.6 cm
3
(STP)/cm
3
(a) individual packing models
0 250 500 750 1000 1250
0.0
0.5
1.0
1.5
2.0
2.5
10
-1
10
0
10
1
10
2
10
3
0
1
2
PSU/CO
2
10 bar NpT, C=15.6 cm
3
(STP)/cm
3
V /V
0
/ %
t / ps
CO
2
insertion
(b) average
Figure 5.18: Simulated CO2induced dilation of PSU . (a) Dilation is
shown for all 3 packing models. (b) The average is shown ( ) with result-
ing standard deviation (grey bars), the mean standard deviation (blue)
and an arbitrary fit (——) to emphasize the kinetics (inlet on logarithmic
time scale).
5.3.4 Integral Dilation Simulation
In Section 4.5.2 the results of the experimental integral-sorption measure-
ments were presented. The experimental values for the (quasi-) equilibrium
concentration at 10 bar gas pressure that are listed in Table 4.2 (p. 51), were
converted to the number of the respective penetrant molecules per packing
model to construct the loaded packing models and perform the simulations
of integral dilation as described in Section 5.2.3. After sufficient repetition
of the insertion steps (see p. 65) and a short NV T -MD run (=at fixed vol-
ume), the packing models were equilibrated in an NpT-MD run at 10 bar
and 35 ◦C, allowing the volume of the initially nonswollen packing models
PSU ,PI4 and PIM to adjust to the respective penetrant load.
The result of the volume dilation for the three individual packing models of
PSU /CO2is shown in Figure 5.18(a). Even though there are considerable
fluctuations in the volume of a simulation cell and some discrepancies be-
tween individual packing models, a distinct volume dilation can be clearly
recognized. For better clarity, the average of the three packing models is
taken and depicted in Figure 5.18(b). The grey shadow at each timestep
denotes the standard deviation. The mean of the standard deviation with
respect to time, taking all data points into account, is displayed as a blue
error bar, which is set arbitrarily to 750 ps and which is positioned on the
grey line. This grey line is the result of a fit to an arbitrarily chosen expo-
84
5.3 Modeling Results and Discussion 5. Modeling
nential function. The inlet of Figure 5.18(b) shows the averaged data and
the fit curve on a logarithmic time scale to emphasize the fact that the dila-
tion is following some kinetics rather than appearing instantly. However, the
absolute parameter values of the fit and the exact nature of the fit function
will not be discussed beyond this general statement.
In the following, the resulting volume dilation of each polymer/gas-system is
presented as the average of the three respective packing models versus simu-
lation time, along with the contraction after removal of the penetrants. The
error bar indicates the mean standard deviation of the averaging procedure
as detailed above, to provide a measure of the deviation while preserving the
lucidity of the Figures.
Simulated dilation of PSU
Figure 5.19(a) shows the dilation ( ) and contraction ( ) of the PSU packing
models after insertion/removal of CO2penetrants. The simulation cell ex-
pands rapidly upon insertion (i.e., upon starting the NpT -MD simulation),
levels off after a few picoseconds and settles, within thermal fluctuations, at
a volume dilation of ∆V≈1.6%(data listed in Table 5.3 on page 86). To
check the reversibility of the observed dilation effect, the penetrants were re-
moved from a copy of the dilated mixed ensemble at 300 ps. For this depleted
packing model, the NpT-MD simulation was continued at 1bar.
0 250 500 750 1000 1250
0.0
0.5
1.0
1.5
2.0
2.5
PSU/CO
2
10 bar NpT, C=15.6 cm
3
(STP)/cm
3
1 bar NpT
V /V
0
/ %
t / ps
CO
2
insertion
CO
2
removal
(a) CO2dilation
0 250 500 750 1000 1250
0.0
0.2
0.4
0.6
0.8
1.0
1.2
PSU/CH
4
10 bar NpT, C=5.6 cm
3
(STP)/cm
3
1 bar NpT
V/V
0
/ %
t / ps
(b) CH4dilation
Figure 5.19: Simulated gas induced dilation ( ) and contraction ( )
after insertion/removal of penetrants into the PSU packing models (aver-
age) and subsequent NpT -MD simulation. The mean standard deviation
is indicated by a black error bar centered on the fit-curve.
85
5.3 Modeling Results and Discussion 5. Modeling
The depleted model shows nearly immediate contraction down to a level off
about 25 %of the dilation effect. Over the remaining simulation time a slight
tendency of further contraction can be ascertained, resulting in a recovery
of the dilation effect of 88 %. It should be noted that the individual pack-
ing models behave different, two recovering to the original volume while one
of the models only marginally contracts. The three packing models are ex-
pected to show slightly different behavior within the range of usual thermal
fluctuations. However, deviations in the fraction of the free volume as well as
the local free volume distribution will lead to further discrepancies between
the packing models that are loaded with the same number of penetrant mol-
ecules.
Figure 5.19(b) shows the average dilation and contraction of the CH4loaded
and depleted PSU packing models. Because the concentration level of CH4
is quite low at a pressure of 10 bar, the dilation effect is also rather small.
The contraction of the packing models upon removal of the penetrants is even
less pronounced. Again the packing models behave differently, one recovering
to an even smaller volume than the original while two do not show a clear
contraction trend. It is worth to note that the dilation effect of PSU /CH4
reaches only the magnitude to which the PSU/CO2packing models contract
rapidly and then show very slow trends of contraction. In view of this,
the observed indistinct response of the matrix to the removal of the CH4
molecules might be expected.
Table 5.3: Data for simulated integral dilation.
polymer/ C/∆V/V0/ recovery Vp/
gas mpp % % cm3/mol
PSU /CO225 1.63 88 23.5
PSU /CH49 0.55 43 21.9
PI4 /CO2120 3.80 104 10.5
PI4 /CH431 0.57 96 6.0
PIM /CO2130 2.94 79 6.6
PIM /CH455 0.57 32 3.0
86
5.3 Modeling Results and Discussion 5. Modeling
0 250 500 750 1000 1250
-1
0
1
2
3
4
5
6
7
PI4/CO
2
10 bar NpT, C=73.6 cm
3
(STP)/cm
3
10 bar NpT, C=81.4 cm
3
(STP)/cm
3
1 bar NpT
V /V
0
/ %
t / ps
(a) CO2dilation
0 250 500 750 1000 1250
-0.5
0.0
0.5
1.0
PI4/CH
4
V /V
0
/ %
t / ps
(b) CH4dilation
Figure 5.20: Simulated gas induced dilation ( ) and contraction ( ) of
PI4. Note that a second insertion step ( ) is needed to reach the specified
concentration of CO2(a).
Simulated dilation of PI4
In the PSU /CO2and PSU /CH4systems presented above, the insertion algo-
rithm of the Solid_Sorption module was able to find non-overlapping sorp-
tion sites within the polymer matrix for all penetrant molecules needed to
match the experimental concentration, i.e., only one insertion-cycle (step 1.
to 3.) was needed. In the case of PI4/CO2, step 1. yielded a concentration
of C= 73.6cm3/cm3, that is, 109 CO2molecules per packing model (mpp),
and a second insertion cycle (step 4. and 5.) was needed to reach the ex-
perimentally determined concentration of C= 81.4cm3/cm3(120 mpp). The
result of the volume dilation of both steps is shown in Figure 5.20(a). After
a first rapid step of volume dilation, a smaller second rapid dilation step is
observed following the second insertion step. To check the stability of the
first step, the initial MD-simulation was continued to 1.3ns containing the
initial amount of penetrants (not shown). Both the first and the second
CO2-induced dilation are completely reversible on a short time scale, as the
contraction behavior of the depleted packing models after removal of the
penetrants shows (not shown for the first dilation step in Fig. 5.20(a)).
In the case of PI4/CH4again only one insertion cycle was needed to load
the packing models with the experimentally specified number of molecules
of 31 mpp. The dilation result of the subsequent NpT-MD simulation is
shown in Figure 5.20(b). Similarly to the PSU /CH4system, the dilation
effect of methane in PI4 is not very pronounced. However, after removal of
the CH4molecules, the volume dilation completely recovers to the original
87
5.3 Modeling Results and Discussion 5. Modeling
volume within very short time, in contrast to the results for polysulfone but
in accordance to the CO2induced dilation in PI4. Within the range of the
usual volume fluctuations, all three packing models show this behavior.
Simulated dilation of PIM
Figure 5.21(a) shows the dilation and contraction of the PIM packing model
after insertion and removal of the CO2molecules, respectively. Despite the
rather large number of 130 mpp, only one insertion step was necessary to
prepare the mixed ensembles according to the experimentally determined
specification (95 cm3/cm3). The PIM packing models dilate at a relatively
slow rate, when compared to PSU or PI4, and although the CO2content
of PIM is the largest, the volume dilation does not exceed the dilation of
the first insertion step of PI4. This might be expected considering that only
one insertion step was needed, which suggests the presence of a sufficient
number of possible sorption sites within the free volume. The contraction
upon removal of the CO2molecules is incomplete (79 %recovered). However,
as the large error bar indicates, the individual models behave different. As
in the PSU /CO2contraction, two of the packing models virtually recover
to their original volume, while one packing model does not show a distinct
contraction.
CH4induced dilation of the PIM packing models is shown in Figure 5.21(b).
In contrast to PIM /CO2, the dilation occurs as fast as for all previously
shown packing models. Interestingly, the volume dilation that is induced by
CH4is nearly the same for all three polymers (∼0.5%).
0 250 500 750 1000 1250
0
1
2
3
4
PIM/CO
2
V /V
0
/ %
t / ps
(a) CO2dilation
0 250 500 750 1000 12 50
0.0
0.2
0.4
0.6
0.8
PIM/CH
4
V /V
0
/ %
t / ps
(b) CH4dilation
Figure 5.21: Simulated gas induced dilation ( ) and contraction ( ) of
PIM .
88
5.3 Modeling Results and Discussion 5. Modeling
Upon removal of the 55 CH4molecules only minor contraction is observed.
By way of fitting the data, it can still be determined that 32 %of the dilation
is recovered, although a large error needs to be considered.
Table 5.3 on page 86 lists the partial molar volumes (pmv)Vpthat were
calculated from the ratio of induced dilation and CO2and CH4concentra-
tion in the polymer packing models, respectively, using equation 2.14. For
both penetrants an increasing pmv is observed with decreasing amount of
fractional free volume. The partial molar volumes will be discussed in more
detail in Section 6.3.
The results of the simulated integral dilation of nonswollen packing models by
insertion of penetrants and subsequent NpT equilibration can be summarized
as follows:
M11) It was shown that a distinct volume dilation induced by penetrants
may be simulated using the presented approach.
M12) In all cases the dilation appeared very rapidly but clearly following
some kinetics on a time scale of a few picoseconds. Only PIM /CO2
showed a timescale of a few hundred picoseconds.
M13) The dilation was shown to be mainly recoverable on the same timescale.
However, only the PI4 packing models contracted to the original volume
and especially the CH4loaded models exhibit large relative errors.
M14) While results M11) and M12) fit into the concept of elastic wave propa-
gation as suggested in the beginning of this chapter, result M13) implies
relaxational processes which are also part of the dilation.
M15) The partial molar volumes of CO2and CH4order as PSU>PI4>PIM,
as would be expected from the calculated fractional free volume (cf.
Section 4.1).
In the following chapter, the results of the experimental chapter and those
of this chapter are subject to a combined analysis and comparison, including
a further discussion of the phenomenological models and further insights to
aspects of the free volume.
89
6 Combined Discussion
In Sections 4.5 and 5.3 the results of the experimentation and simulation ef-
forts were presented, respectively. In some cases, where input of the other was
needed, the connection between ‘real’ and ‘virtual’ experiment has already
been established and some of the results from phenomenological analysis have
been shown along with the experimental results to allow a direct comparison
with the data. However, primarily the immediate results have been shown
and were discussed independent of each other. To comply with the objective
of this work, that is, to combine experiment, phenomenological model and
detailed atomistic molecular modeling, in this chapter the results will be in-
terpreted and discussed in direct comparison. For convenience, some of the
results that have been presented in the preceding chapters are shown again,
introducing varying ways of presentation.
6.1 Sorption Isotherms
In Section 5.3.3 the results of the Grand Canonical Monte Carlo (gcmc) sim-
ulations for two gases and nonswollen and swollen packing models of three
polymers were presented in the form of sorption isotherms. For all systems
investigated, the simulated data compared well with the experimentally ob-
tained data, provided that only the pressure range of the reference state is
regarded. However, the simulated data of the intermediate pressure range
is not in agreement with experimental data. As stated in Section 5.3.3,
this result is anticipated because of the resemblance of the gcmc calcula-
tions on static packing models with a ‘hole-filling’ process. To be able to
describe the experimental data in the intermediate pressure range, the con-
tinuous swelling of the polymer matrix must be taken into account. The
concept of preswollen packing models was especially designed to avoid time
and resource consuming simulation methods. To approximate the sorption
behavior at intermediate pressures, it is therefore proposed to implement a
90
6.1 Sorption Isotherms 6. Combined Discussion
transition between the sorption isotherm of the nonswollen and the swollen
packing model. The most obvious difference between both reference states
is the density of the polymer matrix (depleted of penetrants). In the pheno-
menological model ‘Non-Equilibrium Thermodynamics of Glassy Polymers’
(net-gp19, see Section 3.2), the density change of the matrix is assumed to
be linear with pressure (equation 3.24 on page 20). Adopting this supposition
and further assuming a linear relationship between density and concentra-
tion as well, a simple transition between the isotherms of the nonswollen and
swollen packing models can be achieved mathematically:
C(p) = (1 −p
psw
)·Cno(p) + p
psw ·Csw(p)(6.1)
Here, Cno(p)indicates the calculated sorption isotherms of the nonswollen
(index no) packing model, and Csw(p)indicates the isotherm of the swollen
model (index sw), as they are represented by the respective Dual Mode
sorption fits (cf. Table 5.2). Using the pressure of the swollen reference
state psw as basis, equation 6.1 introduces the concentration C(p)as a linear
weighted average of nonswollen and swollen isotherm, giving the former full
weight at zero pressure while at psw only the latter contributes.
Figure 6.1(a) shows the result of the transition (solid line) from the non-
swollen to the swollen gcmc isotherm that were calculated for CO2in poly-
sulfone (broken lines). The experimental data ( ) is represented excellently
over the whole pressure range.
0 10 20 30 40 50
0
10
20
30
40
50
PSU/CO
2
C / cm
3
(S T P )/ c m
3
p / bar
(a) CO2sorption
0 10 20 30 40 50
0
5
10
15
20
25
PSU/CH
4
C / cm
3
(S T P )/ c m
3
p / bar
(b) CH4sorption
Figure 6.1: Model representation of experimental sorption data ( ) of
CO2(a) and CH4(b) in PSU by linear transition (——) of gcmc isotherms
for swollen and nonswollen packing models (- - -). A red circle indicates
the reference data for the swollen model.
91
6.1 Sorption Isotherms 6. Combined Discussion
Figure 6.1(b) confirms the usefulness of the method for methane in PSU.
While the transition overestimates the experimental data a little, the shape
of the isotherm could be improved. Also note the difference in scales of
Figures 6.1 (a) and (b), enhancing the deviation in the latter Figure.
Figure 6.2 shows the transition of CO2and CH4sorption isotherms calculated
for PI4. Again, the experimental data, especially for the CO2isotherm, are
represented very well. Sorption isotherms in glassy polymers are known to
show an upward curvature at high pressures in some cases,96 which is why
the agreement to the extrapolated experimental data (cp. Fig. 5.16) beyond
the reference state should not be overrated. However, it also shows that the
method is capable to afford reasonable first approximation predictions. The
CH4transition isotherm exceeds experimental data, but the shape could be
improved as before.
Figure 6.3 shows the transition isotherms for CO2and CH4in PIM-1. As
already stated in Section 5.3.3, the gcmc isotherm of the swollen packing
models PIM206mdoes not reach the concentration of the experimental ref-
erence state. Consequently, the transition isotherm underestimates the ex-
perimental data. However, the agreement is still very satisfactory, as is the
agreement of the transition of the methane isotherms from PIM to PIM95m.
0 10 20 30 40 50
0
20
40
60
80
100
120
PI4/CO
2
C / cm
3
(S T P )/ c m
3
p / bar
(a) CO2sorption
0 10 20 30 40 50
0
10
20
30
40
50
60
PI4/CH
4
C / cm
3
(S T P )/ c m
3
p / bar
(b) CH4sorption
Figure 6.2: Model representation of experimental sorption data ( ) of
CO2(a) and CH4(b) in PI4 by linear superposition (——) of gcmc
isotherms for swollen and nonswollen packing models (- - -). The linear
transition covers also the extrapolation of experimental data ( ) beyond
the reference state (red mark) of CO2/PI4 (cf. text).
92
6.1 Sorption Isotherms 6. Combined Discussion
0 10 20 30 40 50
0
25
50
75
100
125
150
PIM/CO
2
C / cm
3
(S T P )/ c m
3
p / bar
(a) CO2sorption
0 10 20 30 40 50
0
20
40
60
80
PIM/CH
4
C / cm
3
(S T P )/ c m
3
p / bar
(b) CH4sorption
Figure 6.3: Model representation of experimental sorption data ( ) of
CO2(a) and CH4(b) in PIM-1 by linear superposition (——) of gcmc
isotherms for swollen and nonswollen packing models (- - -).
Comparison with NET-GP
The gcmc isotherm transitions that were presented in the previous para-
graph have been achieved by building a packing model for a nonswollen and
a swollen reference state, respectively, relying on experimental density data.
In principle, packing models could be built for several pressure steps and the
linear transition performed between each step, refining the method. This cor-
responds exactly to the procedure of the net-gp phenomenology presented
shortly in Section 3.2. Based on experimental density data, a linear swelling
factor ksdescribes the density change of the polymer matrix (eq. 3.24 on
page 20). For each pressure step, i.e., chemical potential of the gas phase,
the equilibrium chemical potential of penetrant gas can be calculated from
pure component parameters, statistical considerations of possible configura-
tions of the components on the lattice and parameters of the latter, including
the density as an order parameter.
Figure 6.4(a) shows the prediction of CO2and CH4sorption in PSU as cal-
culated by the net-gp model using a swelling factor of kst = 1.461 %/MPa−1
calculated from the total dilation (cf. Figure 4.14(b) and Table 3.1) and the
pure component data of reference [54] and Table 3.1. For CO2(solid line) the
experimental data at larger pressures is overestimated, while net-gp yields
an excellent prediction for CH4sorption (broken line).
93
6.1 Sorption Isotherms 6. Combined Discussion
0 10 20 30 40 50
0
10
20
30
40
50
60
PSU/CO
2
PSU/CH
4
C / c m
3
(S T P )/ cm
3
p / bar
(a) kst from total dilation
0 10 20 30 40 50
0
10
20
30
40
50
NET-GP
GCMC
PSU/CO
2
PSU/CH
4
C / c m
3
(S T P )/ cm
3
p / bar
(b) ksr from relaxational dilation
Figure 6.4: net-gp sorption predictions for CO2(——) and CH4(- - -)
in PSU. (a) Swelling coefficient calculated from total dilation data (kst)
and (b) from relaxational fraction (ksr), including the linear transitions of
the gcmc isotherms (blue).
0 10 20 30 40 50
0
50
100
150
200
PI4/CO
2
PI4/CH
4
C / c m
3
(S T P )/ cm
3
p / bar
(a) kst from total dilation
0 10 20 30 40 50
0
20
40
60
80
100
NET-GP
GCMC
PI4/CO
2
PI4/CH
4
C / c m
3
(S T P )/ cm
3
p / bar
(b) ksr from relaxational dilation
Figure 6.5: net-gp sorption predictions for CO2and CH4in PI4.
0 10 20 30 40 50
0
50
100
150
200
PIM/CO
2
PIM/CH
4
C / c m
3
(S T P )/ cm
3
p / bar
(a) kst from total dilation
0 10 20 30 40 50
0
20
40
60
80
100
120
140
NET-GP
GCMC
PIM/CO
2
PIM/CH
4
C / c m
3
(S T P )/ cm
3
p / bar
(b) ksr from relaxational dilation
Figure 6.6: net-gp sorption predictions for CO2and CH4in PIM-1.
94
6.1 Sorption Isotherms 6. Combined Discussion
Looking at the prediction for CO2in PI4 (Fig. 6.5(a)), the overestimation of
experimental data is even larger, while the prediction for CH4in PI4 is still
acceptable. The predictions for CO2and CH4in PIM-1, however, are not
satisfying, both isotherms being largely overestimated (Figure 6.6(a)).
The single one parameter emanating from this work that affects the calcu-
lation is the swelling factor ks, fitted from the dilation data. It is indeed
possible to obtain excellent fits of the sorption data by adjusting ks. How-
ever, if done that way, the predicted dilation exceeds the experimental dila-
tion data in all cases equally unsatisfying. To improve the result, it is useful
to review how the polymer density influences the behavior of the net-gp
model: As stated in Section 3.2, a polymer of N r-mers is considered, ‘each
mer occupying a lattice site of molar volume v∗’. Since the volume of a ‘mer’,
meaning one mobile unit of the polymer, is a constant, the size of a lattice
site remains fixed as well. Thus the model implies that volume changes of
the polymer are effected by an increase in the number of lattice sites (Nr, cf.
equation 3.6) rather than their volume v∗.
From the kinetic analysis of the experimental data (section 4.4) two different
types of dilation processes can be distinguished and separately quantified.
The elastic fraction stems from the picture of a misfit between penetrant
and sorption site, dilating the surrounding matrix reversibly. Clearly, no
generation of new ‘lattice sites’, i.e. increase of free volume would be expected
by this process.
That leaves the relaxational fraction of dilation. Here, the polymer matrix
relaxes the stresses induced by penetrant sorption by rearranging mobile
subunits of the polymer chain. This process can certainly be connected to
the generation of free volume and is in fact known as conditioning.97 Based on
this in some respect admittedly speculative notion, the swelling factors ksof
the polymer/gas systems were obtained by linearly fitting the experimental
data of the relaxational fraction of dilation only, resulting in swelling factors
ksr < kst (Table 3.1, p. 20).
If these swelling factors are applied, predictions for sorption of CO2and CH4
in PSU result, that are displayed in Figure 6.4(b). For convenient comparison
to the simulated gcmc calculations the respective isotherms are included as
solid blue lines. Using the swelling coefficient ksr resulting from relaxational
dilation data, the sorption capacity of PSU is predicted to be considerably
lower, now underestimating the experimental data. The representation of
experimental data in the intermediate pressure range (10 to 20 bar) could
be remarkably improved. On the other hand, the prediction of the CH4
sorption isotherm by the net-gp model is somewhat impaired. An obvi-
95
6.1 Sorption Isotherms 6. Combined Discussion
ous improvement of the prediction is achieved for the CO2isotherm in PI4,
shown in Figure 6.5(b). The net-gp model relying on relaxational dilation
input excellently matches experimental sorption data. No worsening of the
CH4sorption prediction is observed; on the contrary, the lower relaxational
swelling coefficient ksr even provides for a slight improvement. The grand
overestimation of CO2and CH4sorption in PIM-1 is corrected by the relax-
ational swelling coefficient (Figure 6.6(b)). However, a stronger curvature
results in an unsatisfactory representation of the data in the intermediate
pressure region for CO2, and some residual overestimation is still observed
for CH4at large pressures.
The transition isotherms of the gcmc simulations are included for easy com-
parison in all Figures 6.4(b) to 6.6(b). In general, the transition isotherms
give an equally good or better representation of the experimental data than
do the isotherms calculated by the net-gp model. But while the simula-
tions depend on density data as well as on concentration data for construc-
tion of the packing models at the reference state, the net-gp phenomeno-
logy relies on input on swelling (and pure component data) only, to yield
a concentration-pressure isotherm that is purely predictive in nature. All
things considered, both methods provide rather good descriptions of experi-
mental sorption data for the gases and polymers investigated in this work.
Table 6.1: Swelling coefficients used for net-gp sorption prediction.
polymer PSU PI4 PIM-1
gas CO2CH4CO2CH4CO2CH4
kst [%/MPa]a1.461 0.249 6.371 0.758 3.961 1.761
ksr [%/MPa]b0.584 0.042 2.378 0.210 0.490 0.519
aObtained from linear fit through total dilation data.
bObtained from linear fit through relaxational fraction of dilation data.
96
6.2 Free Volume Distributions 6. Combined Discussion
6.2 Free Volume Distributions
In Section 3.3 the phenomenology of the site distribution (sd) model was
introduced, which provides the means to determine a size distribution of
Gaussian form by way of analyzing the sorption and dilation data of gases
in glassy polymers. The curves of the fitting procedure have already been
presented along with the experimental data on two gases and three polymers
in Section 4.5. In this section the Gaussian size distribution of sorption sites
are to be discussed and compared with the size distributions obtained by
the free volume analysis of detailed atomistic molecular packing models that
were presented in Section 5.3.2, albeit in a different exposition.
Performing the least square fit of eqs. 3.44 and 3.48 (see page 24) to the ex-
perimental sorption and dilation data (diffusive/elastic fraction), ultimately
leads to the parameters Vh0and σv, i.e., center and width, that characterize
the Gaussian size distribution of the free volume of the investigated poly-
mers, respectively. The parameter values are listed in Table 6.2. To obtain
reasonable values for the number of sorption sites N0, the free volume anal-
ysis of the nonswollen packing models was consulted. In Section 5.2.1 it
was already argued, that in view of the high geometric complexity of large
free volume elements (fves) as determined by the Vcon-method, the Rmax-
method would be preferred to obtain a size distribution of fves that pass as
sorption sites in terms of the sd model. One further adjustment was made,
namely the exclusion of fves with volumes smaller than the positronium-
sized probe-sphere. It is inherent in the method of analysis to exhibit the
greatest uncertainties for small volumes, where the dimensions of the volume
are comparable to the grid spacing. Furthermore, it is an artifact of the
calculation algorithm to yield undersized volumes if the number of free grid
points is small. Therefore, for the determination of the number of sorption
sites N0, all fves of volumes not larger than that of the probe-sphere were
assumed to be ‘bottlenecks’ (potential diffusion paths), rather than sorption
sites, and excluded. This procedure may appear somewhat arbitrary, how-
ever, the resulting values listed in Table 6.2 are well within the range of the
estimates made by Kirchheim56 and therefore it seems an adequate approach
to determine the number of sorption sites individually.
97
6.2 Free Volume Distributions 6. Combined Discussion
Table 6.2: Site distribution model parameters.
polymer Vg/aG0/ σG/ Vh0/ σV/ N0/
/ gas cm3/mol kJ/mol kJ/mol cm3/mol cm3/mol 1021/cm3
PSU /CO246.1 17.5 10.5 21.0 6.4 4.9
PSU /CH452.2 20.7 8.6 27.7 7.8 4.9
PI4 /CO246.1 14.7 14.4 13.8 3.7 6.6
PI4 /CH452.2 18.8 11.6 26.3 10.4 6.6
PIM-1/CO246.1 13.1 13.5 14.5 8.8 7.8
PIM-1/CH452.2 18.4 13.8 18.5 11.8 7.8
a‘Dynamic volume’ of penetrant,61 cf. eq. 2.15 on page 11.
0 20 40 60 80 100
0
1
2
3
4
5
0.0 12.0 24.1 36.1 48.2 60.2
PSU
V
site
/ cm
3
/mol
N (V
site
) / 1 0
20
/c m
3
V
site
/ Å
3
Figure 6.7: Number distribution of fve volumes (Rmax) in PSU packing
models and Gaussian distribution of sorption sites obtained from sd fit to
CO2(——) and CH4(- - -) sorption and dilation data.
PSU
Figure 6.7 shows the results of the sd analysis of sorption and dilation data of
CO2and CH4in PSU, that is, the Gaussian size distributions of free volume
in PSU, along with the Rmax distribution of fves of the nonswollen pack-
ing models PSU. In principle, it is the same distribution as in Figure 5.9(b)
98
6.2 Free Volume Distributions 6. Combined Discussion
(hatched columns), yet with the aforementioned exclusion of the smallest
fves and this time the distribution is displayed as a number-distribution of
fve-volumes, as opposed to the fractional free volume distribution of radii
of volume equivalent spheres. Noticeably, by this last point, the rather sym-
metrical shape of the distribution in Figure 5.9(b) is stretched to the large
volume range. In fact, for convenient comparison with the Gaussian size
distributions, the Rmax-volume distribution is only shown up to a volume
of 100 Å; several larger fves exist only in small numbers and hence do not
contribute to the shape of the distribution.]The size distribution of fves
detected by the Rmax-method in nonswollen PSU packing models does not
resemble a Gaussian shape. The number distribution is increasing from large
volume tail until a sharp cutoff that is due to the exclusion of smallest vol-
umes. Since the detection of smaller and smaller fves depends on the probe-
as well as the grid-size, it has to be presumed that this sharp peak could be
pushed to smaller volumes, while increasing in intensity. Likewise, increas-
ing the probe-size would shift the peak to larger volumes at the cost of the
number of fves detected. The choice of a positronium-sized test-sphere is
a reasonable compromise, meeting the detection limit of pals experiments
while maintaining the ability to map geometries of the fves.
The Gaussian size distribution of the free volume in PSU, as established
by the sd model using CO2and CH4sorption and dilation data, are dis-
played in Figure 6.7 as a solid and broken line, respectively. In theory, both
distributions should coincide. Instead, the CH4data yields a distribution
that is shifted to larger volumes. The message of this outcome is that ob-
viously CH4induces less dilation in PSU than would be expected from its
‘dynamic volume’\Vg= 52 cm3/mol, given that the assumptions made by
the sd-model hold and the CO2data yield the ‘correct’ distribution. In fact,
this was assumed by Gotthardt et al.,59 who used the dynamic volume of
CH4as fit-parameter to adjust equation 3.48 to experimental dilation data
induced by methane in Bisphenol-A-Polycarbonate (BPA-PC). A value of
Vg= 71.7Å3(43.2cm3/mol) resulted using the size distribution in BPA-PC
obtained from CO2sorption and dilation data and its dynamic volume of
76.7Å3(46.2cm3/mol). This qualitatively corresponds to the shift in the
distribution observed in Figure 6.7: In terms of the mechanical point of view
assumed by the sd model, dilation is caused by the misfit between penetrant
and ‘hole’ volume (Vg−Vh, eq. 3.35, p. 23). However, to stay consistent within
]Although it has to be mentioned that these contribute considerably to the fractional
free volume, due to their large size.
\That is, the partial molar volume of the penetrant in rubbery polymers or liquids, cf.
Section 2.14.
99
6.2 Free Volume Distributions 6. Combined Discussion
this work, the ‘dynamic volumes’ of only one reference for both gases were
used,61 and the procedure of fitting the data was not modified. In addition
to the uncertainties regarding the dynamic volumes of the penetrants, the
sd model assumes a spherical geometry of the penetrants and concentration-
independent interaction other than that of elastic nature. Especially in the
case of CO2, this may not be the case at higher concentrations, despite the
exclusive usage of diffusive/elastic fraction of sorption and dilation data.
In view of these simplifications, the agreement between both Gaussian size
distributions within a few Ångstrom is still acceptable.
For all the differences in shape, center and width of the Gaussian distri-
butions shown in Figure 6.7 with the Rmax-distribution of fves of detailed
atomistic packing models, the accordance within the same order of magni-
tude can be counted as a success. For their comparison, it has to be kept in
mind that, while the sd model is derived from considerations of the above-Tg
dynamics (cf. eq. 3.25, p. 21), the volume analysis is performed on the static
atomistic model of polysulfone. Moreover, continuum mechanics and ideal-
ized spherical shapes are assumed in the sd model, whereas the volumetric
analysis does not distinguish between geometrical shapes. But large volumes
detected by model analysis may well be narrow in one or two dimensions and
not necessarily accommodate penetrant molecules as well as would spherical
holes of the same volume.
100
6.2 Free Volume Distributions 6. Combined Discussion
0 20 40 60 80 100
0
1
2
3
4
5
6
0.0 12.0 24.1 36.1 48.2 60.2
PI4
V
site
/ cm
3
/mol
N (V
site
) / 1 0
20
/c m
3
V
site
/ Å
3
Figure 6.8: Size distribution (Rmax) of fves in PI4 packing models and
Gaussian distribution of sorption sites obtained from sd fit to CO2(——)
and CH4(- - -) sorption and dilation data.
PI4
Similar observations as for the size distributions in PSU can be made for PI4,
as shown in Figure 6.8. Using the methane data, the Gaussian distribution is
shifted and considerably broadened. However, recalling the quality of the fits
to the data (Figure 4.14 on page 43), one is lead to believe that in this case
the distribution derived from CO2data is less reliable. The unexpectedly low
center (Vh0(PI4)< Vh0(PSU), cf. Table 6.2 on page 98) adds to that suspicion.
In PI4, a larger fraction of free volume (18.7%) than in PSU (13.5%) is present
(see Section 4.1). If, as the result of the Rmax-analysis on PI4 suggests, this
is at least in part due to the presence of larger sorption sites, there should
be a smaller misfit and hence less dilation. The opposite is observed, and
for this reason the sd model yields a low value for Vh0. The main features
of the fve distribution agree with those of PSU as could be expected from
Figure 5.10(b) (p. 72). Again, sorption sites in the same order of magnitude
resulted from both, molecular simulation and sd analysis.
101
6.2 Free Volume Distributions 6. Combined Discussion
0 20 40 60 80 100
0
1
2
3
4
5
6
0.0 12.0 24.1 36.1 48.2 60.2
PIM
V
site
/ cm
3
/mol
N (V
site
) / 1 0
20
/c m
3
V
site
/ Å
3
Figure 6.9: Size distribution (Rmax) of fves in PIM packing models and
Gaussian distribution of sorption sites obtained from sd fit to CO2(——)
and CH4(- - -) sorption and dilation data.
PIM-1
Figure 6.9 shows the results of the free volume analysis of the Rmax-method
and sd analysis in PIM-1. The mismatch between the Gaussian distributions
obtained from CO2and CH4data is smaller than for PSU and PI4. The distri-
butions are rather broad, and on first sight the agreement to the distribution
of fve sizes seems better than for the other polymers. However, like in PI4,
the distribution of fves in PIM exhibits a long tail towards large volumes
which seems not to be detected by the sd model, where the size distribution
is determined making use of the ratio of experimental sorption and dilation.
Possible Limits of the Model
Although the agreement of the size distributions as determined by packing
model analysis and by sd analysis of polymer/gas-systems is satisfying, being
within the same order of magnitude, the question remains why the trend of
the size distributions of fves towards a higher number of large volumes for
PI4 and PIM in comparison to PSU (cf. Fig. 5.14(a)) is not reflected by the
position of the centers of the Gaussian size distributions, displayed together
in Figure 6.10.
102
6.2 Free Volume Distributions 6. Combined Discussion
0 20 40 60
0
1
2
3
4
5
6
0.0 12.0 24.1 36.1
V
site
/ cm
3
/mol
N (V
site
) / 1 0
20
/c m
3
V
site
/ Å
3
PSU
PI4
PIM
Figure 6.10: Gaussian distributions of sorption sites obtained from sd
fit to CO2sorption and dilation data in PSU (——), PI4 (- - -) and
PIM-1 (····).
Since experimental pals data suggest the same trend (equivalent sphere radii
of 2.9Åfor PSU,91 4.0Åfor PI4,64 and 4.8Åfor PIM-1,98) as the free volume
analysis of packing models, a closer look at the sd model seems appropriate
to resolve the question. The center of the Gaussian size distribution Vh0is
the result of a fit of equation 3.48 to experimental dilation data:
∆V/V0=Z∞
−∞[Vg−Vh(G)]C(G)dG
As a result of equation 3.47, which states that large sorption sites have the
lowest sorption energy G, and the fact that integration in eq. 3.48 starts at
low energies, the mathematics of the sd model implies that penetrants of
volume Vgoccupy the largest volumes Vhfirst, dilation setting-in once the
misfit [Vg−Vh(G)] is positive. This assumption may not be correct. The
step by step insertion of penetrants that was performed to calculate sorption
isotherms in the nonswollen polymer packing models, provides the means to
actually analyse which of the fves present in a packing model are occupied
by the ‘first’ 10 CO2molecules. To account for complex shapes even the
Rmax-method does not cancel out, the fves were characterized by the radius
of the largest sphere that would fit into an fve without overlap. As it
turns out, the fves that are occupied first in the gcmc procedure exhibit an
equivalent sphere radius of 1.9±0.3Åin all PSU packing models, 2.2±0.3Å
in PI4 and 2.4±0.3Åin PIM . However, the largest volumes present in the
polymer packing models range from 2.4Åto 1.9Åin PSU ,3.9Åto 2.9Åin
103
6.2 Free Volume Distributions 6. Combined Discussion
PI4 and 3.2Åto 2.7Åin PIM . While in the PSU packing models the fves
that are occupied first range among the largest that are available, clearly for
PI4 and PIM this is not the case.
This train of thought is supported by the behavior of a slit-pore potential.
Here, the potential Φsp of a molecule between two rigid walls is examined
in one dimension. The commonly used 10–4 potential is sketched out in
Figure 6.11 for two distances.99 Without going into further detail, Fig. 6.11
illustrates that the potential exhibits a local minimum close to the walls
of the slit-pore. This minimum is optimized when the walls are at a finite
distance.
Related to the matter at hand, this means that, deviant from the mathe-
matics of the sd model, where large sorption sites are occupied first because
of the little or absent elastic energy contribution, in reality penetrants may
occupy sorption sites first, where such a minimum distance to the polymer
matrix in all directions is given, i.e., where a maximum of ‘constructive’ inter-
action between penetrant molecule and matrix can be achieved. In the case
of PSU this would not effect the sd analysis much, since the ‘preferred size’
of the fves coincides more or less with the largest available fves. However,
if, as the analysis of the packing models of PI4 and PIM suggests, the largest
sites present in these two polymers are left unoccupied in favor of smaller
sorption sites, at low concentration levels of penetrants a dilation would be
observed that is similar to polysulfone. This dilation, in addition to a high
number of sorption sites N0would then be ‘mathematically’ misinterpreted
by the sd model, yielding undersized Gaussian distributions. Incidentally,
the Gaussian distributions obtained for PI4 and PIM center in the same vol-
ume range, lending more support to the hypothesis that the sd model is
‘blind’ to oversized volumes.
0
w all
en er g y [a .u .]
position [a.u.]
w all
Figure 6.11:
Sketch of the adsorption poten-
tial of a molecule between the
two walls of a slitpore (——). If
the distance between the walls
is reduced (····) to a finite dis-
tance, the potential minimum is
optimized (- - -).
104
6.3 Integral Dilation 6. Combined Discussion
PSU PI4 PIM PSU PI4 PIM
0
1
2
3
4
5
6
7
CH
4
V /V
0
/ %
CO
2
Figure 6.12: Gas induced volume dilation in polymers (black) and cor-
responding dilation in polymer packing models (blue).
6.3 Integral Dilation
In Section 4.5.2 the results of the experimentally observed volume dilation
induced by integral gas sorption were presented. Using the experimental
sorption data, nonswollen packing models of each polymer were prepared
containing the specified number of penetrant molecules. The resulting vol-
ume dilation of the simulation cell in a subsequent NpT-MD simulation were
presented in Section 5.3.4. Figure 6.12 summarizes the data that were listed
in Tables 4.2 and 5.3 (see pages 51 and 86).
In all six cases, the polymer matrix responds to the sorption and accordingly
to the insertion of penetrants with a volume dilation. This volume dilation,
which seems to be instantaneous upon sorption of a penetrant in the cor-
responding experiments, appears to follow a kinetics in the corresponding
simulation experiments that is not possible to resolve in the experimental
setups used in this work. The absolute value of the dilation effect found by
detailed atomistic molecular dynamics simulation deviates from the experi-
mentally obtained values by not more than a factor of three. Upon removal
of the penetrants, all polymer/gas systems were shown to contract. But
while in the experiments the contraction is complete, showing only a slower
diffusion kinetics as is expected for desorption, all packing models except for
PI4 show a residual volume dilation after equilibrating the depleted packing
models.
105
6.3 Integral Dilation 6. Combined Discussion
PSU PI4 PIM PSU PI4 PIM
0
5
10
15
20
25
CH
4
p.m .v. / cm
3
/m o l
CO
2
Figure 6.13:
Partial molar volume of
CO2and CH4in polymers
(black) and corresponding
pmv in polymer packing
models (blue).
Considering the limited size of the packing models and the difference in time
scales of experiment (minutes) and modeling (picoseconds) of more than
10 orders of magnitude, these results can be regarded as being in rather
satisfying agreement to the experimental data.
All polymer/gas systems were investigated at the same pressure of the re-
spective penetrant gas. Since the concentration levels are quite different, it
is instructive to relate concentration and dilation effects. In Section 2.3 the
concept of the partial molar volume (pmv) was outlined and equation 2.14
was offered as a practical means to calculate the pmv Vpof a penetrant from
experimental concentration (change in penetrant number ∆ni) and dilation
data:
Vp=∆V
∆ni
In Figure 6.13 the pmvs of CO2and CH4in the three polymers are dis-
played as they were calculated from experimental integral sorption and di-
lation data and calculated from the simulated integral dilation of polymer
packing models (see Tables 4.2 and 5.3 on pages 51 and 86).
It was already noted in Section 4.5.2, that the experimental results order as
PSU >PIM-1 &PI4, despite the expectation that the calculated free volume
(Bondi)35 suggests PSU >PI4 >PIM-1 for the concentration levels reached in
the experiment. The latter order is actually found in the simulated integral
dilation. However, both simulated dilation effects of PI4 and PIM are consid-
erably lower (about 60 %and 40 %, respectively) than in the corresponding
experiments. Two aspects have to be considered in regard to this difference:
106
6.3 Integral Dilation 6. Combined Discussion
(1) The densities of the PI4 and PIM packing models deviate from the experi-
mental densities by −6%and −5%, while those of the PSU packing models
deviate only by −3%. The negative sign of the deviation suggests a larger
fractional free volume in the packing models than in the ‘real’ samples and
hence a smaller dilation effect should be observed. However, a rough es-
timate shows that the difference in fractional free volume between packing
model and ‘real’ sample amounts to +34 %for PI4 and +25 %for PIM . Due
to the smaller fractional free volume in polysulfone, the relatively smaller
deviation of the obtained density still amounts to +25 %for the PSU packing
models. These values actually suggest for the simulated dilation of PSU to
show some deviations as well; however, no such effects can be observed in this
case. Also, in the discussion of the site distribution model in the previous
section evidence was presented that large sorption sites are not occupied first
in the simulation and it is therefore not clear that the surplus of free vol-
ume necessarily leads to the observed deviation in dilation behavior. In any
case, it should be noted, that an improvement of the density of the PI4 and
PIM packing models would probably lead to an improvement of the dilation
results as well, qualitatively emphasizing the simulated dilation effect.
Although it cannot be ruled out for the density deviations to contribute to
the deviations in dilation, another approach is possible in order to explain the
‘incomplete’ dilation of the PI4 and PIM packing models which was presented
in reference [3]:
(2) The larger deviation of the dilation values of PI4 and PIM in compar-
ison to the very good results for PSU may be explainable in terms of the
stiffness of the respective repeat units. As was explicated in reference [3],
polysulfone has a higher density of flexible, i.e., more or less rotatable bonds
along the polymer chain than the polyimide PI4 does (cf. Fig. 4.1). There-
fore e.g. rearrangements of phenylene rings are more easily possible than in
PI4. In PI4, bulky side groups, like the -CF3groups, or the methyl-groups
in ortho-position to the imide group, represent additional hindrances to lo-
cal reorientation of groups (cf. Fig. 4.2). Besides the considerably higher
glass transition temperature of PI4, compared to PSU, this reasoning was
supported by an analysis of the rotational mobilities of selected bond angles
in PSU and PI4 packing models (see Appendix A.4)3as well as the mean
square displacement of backbone atoms of (PSU>PI4).1, 63 The absence of
a glass transition at temperatures well above that of PI4 and a supposedly
even more rigid polymer matrix insinuates that the same reasoning should
apply to PIM-1 as well, even if no such thorough investigations for PIM-1
were committed in the course of this work.
Assuming the second explanation to be the main reason for the deviations
107
6.3 Integral Dilation 6. Combined Discussion
between experiment and modeling, the incomplete dilation of PI4 and PIM
packing models could be interpreted as the result of an anelastic contribution
to the dilation on a time scale that is well below the resolution limit of the
experimental setups used in this work, while being fast enough to be detected
within the nanoseconds of simulation time only in the case of PSU. Anelastic
relaxation behavior in glassy polymers was investigated by Boyd et al.45 and
related to secondary relaxation processes (β-relaxation) that may well match
the intermediate time scales between experiment and modeling. However,
extensive investigations of relaxation spectra as well as further simulations
would be necessary to unambiguously correlate these processes.
Partial Molar Volume and Hole Volumes
The partial molar volumes can be further used to gain information about the
localized free volume. Equation 3.35, which was introduced in the context of
the site distribution model, connects the partial molar volume Vpof individual
penetrants via their ‘dynamic volume’ Vgwith the volume of the occupied
‘hole’ Vh:
Vp=const(Vg−Vh)
Assuming const = 1 and the ‘dynamic volumes’ Vg= 46.2cm3/mol and
Vg= 52 cm3/mol for CO2and CH4, respectively, as was done in the sd analy-
sis, the average site volumes of the occupied sorption sites result from the
partial molar volumes shown in Figure 6.13 that can be expressed as the radii
of volume equivalent spheres. Figure 6.14 shows the resulting values for the
three investigated polymers.
PSU PI4 PIM PSU PI4 PIM
0.0
0.5
1.0
1.5
2.0
2.5
r.e.s. / Å
CO
2
CH
4
Figure 6.14:
Average equivalent sphere
radii (r.e.s) of sorption sites
occupied by CO2and CH4
in polymers (black) and
polymer packing models
(blue).
108
6.3 Integral Dilation 6. Combined Discussion
A slight trend may be discerned towards the occupation of larger hole volumes
in the case of CH4compared to CO2, and larger sites seem to be occupied
in the different polymers, taking on the order PSU <PI4 <PIM-1 for both
penetrants, however, the differences are rather small. Recalling the discussion
of the sd model in the previous section, the values of 1.9Å3,2.2Å3, and 2.4Å3
were evaluated for the fves occupied by the first 10 CO2molecules in PSU ,
PI4 and PIM packing models, respectively. Given the higher concentration
level of penetrants in the integral dilation experiments and simulations, the
observed increase is expected, because increasing elastic contribution to the
sorption energy upon occupation of smaller and smaller sites will eventually
make the occupation of larger-than-optimum sites more ‘favorable’.
109
7 Synopsis
Summary Sorption and dilation characteristic of three polymers (PSU,
PI4 and PIM-1) in contact with two gases (CO2and CH4) have been inves-
tigated utilizing experimental characterization methods, techniques of de-
tailed atomistic molecular modeling and theoretical data analysis. Differ-
ential sorption and dilation isotherms were recorded up to gas pressures of
50 bar and were kinetically analyzed. The resulting diffusive/elastic fractions
of the isotherms were subjected to an analysis by means of the site distri-
bution (sd) model, yielding well agreeing fits and resulting in parameters
of Gaussian size distributions of the localized free volume of each polymer.
The density and concentration data at the minimum
and maximum pressure of each sorption and dilation
isotherm were chosen as reference states to construct
detailed molecular packing models of the nonswollen
and swollen state of each polymer/gas system. Grand
Canonical Monte Carlo simulations (gcmc) were performed to obtain simu-
lated sorption isotherms for the nonswollen and swollen state, respectively,
and a linear transition with pressure was proposed that agrees excellently
with the experimental data. The close resemblance to the Non Equilibrium
Thermodynamics of Glassy Polymers (net-gp) model was identified and re-
sults compared to the sorption predictions of the model, using both, the total
and the relaxational fraction of the experimental dilation isotherms to obtain
the swelling factor needed for this purpose.
A thorough analysis and visualization of the free volume
was conducted on the static packing models of the pure
polymers, revealing separated free volume elements (fves)
for PSU and PI4, whereas PIM exhibits large fves of
highly complex shape that were denominated as a void
phase, taking up the main portion of the fractional free volume (ffv). The
same analysis on the CO2-swollen packing models showed that PI4 developed
a similar void phase upon swelling (PI156) while no such change could be
110
7. Synopsis
observed in swollen PSU models (PSU80) at the investigated dilation levels.
The size distributions of fves according to the Rmax-method in nonswollen
models were found to be of the same order of magnitude as Gaussian size
distributions obtained by sd analysis of experimental data. Differences in
shape were discussed to be a matter of the dynamic/static viewpoint of the
experimental and modeling technique, respectively. Possible limits of the sd
model regarding the occupation sequence of sorption sites have been pointed
out for PI4 and PIM-1.
Integral sorption and dilation measurements were car-
ried out at 10 bar pressure for all polymer/gas systems.
The dilation was recognized as an elastic process on
the time scale of the experiments. The sorption data
were used as input to prepare nonswollen yet penetrant-
loaded packing models which showed reasonably agreeing dilation in subse-
quent NpT-MD simulations. The deviation towards lower dilation in PI4 and
PIM as well as the incomplete recovery of the dilated PSU and PIM packing
models were attributed to anelastic relaxations in the time range interme-
diate to experiment and simulation. The partial molar volumes of the two
penetrant gases within the respective polymers were related to the volume
of occupied sorption sites using the ‘dynamic volume’ of the penetrants.
Conclusions The experimental investigation of the phenomena associated
with gas sorption and dilation in glassy polymers demands a considerable
effort of time and resources. The prospect of assessing the properties of
polymer/gas systems by examination of detailed atomistic molecular packing
models, selecting only the most promising materials for further investigation,
seems a reasonable presumption, given the progress in computing power and
availability of CPU time is sustained. However, the large gap of several orders
of magnitude in system size and time scales between experiment and simula-
tion will not be bridged by capacity and speed alone. Methods are called for
that combine experiment and simulation and make possible a deeper under-
standing of the mechanisms that govern sorption and dilation in polymer/gas
systems.
In this work, several approaches were presented that address this apparent
incompatibility of experiment and simulation. Since the relaxational swelling
of the polymer matrix that is observed at elevated concentration levels of pen-
etrants, is orders of magnitudes too long to directly simulate the respective
molecular dynamics in reasonable time and effort, representative reference
states were evaluated experimentally and packing models constructed ac-
cording to the specifications. Even though this method is still dependent on
111
7. Synopsis
experimental input, the successful description of experimental sorption data
using a transition between gcmc isotherms of nonswollen and swollen model
validates both, the quality of the molecular models and the general approach,
which is further backed up by its resemblance to the well established net-
gp model. In addition, the free volume analysis of nonswollen and swollen
packing models revealed fundamental differences in the structure of the free
volume of the investigated polymers in their initial and swollen states. In the
recently developed new class of Polymers of Intrinsic Microporosity (PIM),
the existence of a void phase could be detected and quantified. The visual-
ization showed the void phase as a large fraction of the free volume of highly
complex shape that is loosely connected and penetrates far reaches of the
simulation cell. This sort of structure is present in PI4 only in the swollen
state and not at all in the investigated PSU reference states. Certainly, this
void phase plays an important role in sorption and transport characteristics
of penetrant gases that have yet to be studied.
Some disagreement was encountered concerning the size distributions of fves
and the Gaussian distributions obtained via the sd model. Although gener-
ally of the same order of magnitude, which already can be clearly counted as
a success given the disparate conditions of the two approaches, the Gaussian
distributions suggested a different trend concerning the center of the distri-
bution. Here, the tool of detailed molecular modeling provided the means
to put model assumptions to the test, exposing possible limits or weaknesses
of the model with respect to polymers that exhibit sorption sites of large
volume. However, the elastic response of the matrix to the insertion of pen-
etrants could successfully be simulated. Furthermore the sd model is able
to describe the experimental sorption and dilation data quite well, if the
kinetic analysis and its results are consistently implemented, that is, if the
diffusive/elastic fraction is used exclusively. To a certain extent, this lends
some credibility to the perception of the processes involved as established by
the sd model.
In experimental measurements of gas induced dilation as they were studied
in this work, sorption is necessarily associated with the diffusion of the pen-
etrant gas. The method of inserting penetrant molecules in a nonswollen
packing model to observe the dilation effect in NpT-MD simulations avoids
the ‘slow’ process of diffusion with respect to simulation time and thus al-
lows the comparison to the experiment, with quite satisfying agreement. Al-
though some deviation remains, the results obtained in this work point at
the existence of possible short-time relaxations or anelastic reactions of the
polymer matrix that are specific to each polymer. However, the successful
secondary insertion of CO2molecules into the dilated PI4 matrix and the
112
7. Synopsis
fact that the isotherms of the nonswollen models show good agreement to
the experimental isotherms in the low pressure region, shows that in princi-
ple, no experimental input may be needed to record concentration/dilation
isotherms that compare to the elastic fraction of experimental dilation.
Outlook In this work methods of analysis of the sorption and dilation phe-
nomena in polymer/gas systems were presented that combine experiment,
theory, and simulations. Even though promising results could already be
achieved, further efforts are necessary to add extra confirmation and to an-
swer the questions that were raised in the course of this work. In the context
of the integral dilation simulation anelastic processes were surmised that ac-
count for the ambiguous behavior of the packing models. The rough analysis
of bond angle rotations for PSU and PI4 packing models that was conducted
as a scouting experiment in the course of this work (see Section 6.3 and Ap-
pendix A.4), should be refined and extended to PIM packing models, along
with the analysis of the dynamic mechanical or dielectric spectra of the ‘real’
counterparts, to confirm or disprove the conjecture. As already mentioned in
the previous paragraph, the notion of simulating sorption and dilation by al-
ternately loading (gcmc) and equilibrating (NpT-MD) a nonswollen packing
model provides an intriguing way to record diffusive/elastic concentration-
and dilation-pressure isotherms independent of experimental input.
A follow-up on these encouraging new ways of comparison of laboratory ex-
periments and molecular modeling, to further converge the respective bound-
ary conditions and to reconcile them with the assumptions made by pheno-
menological models, is recommended to improve the compatibility of the
results and thereby ensure targeted material design in the future.
113
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A Appendix
A.1 Abbreviations
3D-PBC 3-dimensional boundary conditions
CPU Central Processing Unit
dm Dual Mode
DGC Density Gradient Column
DSC Differential Scanning Calorimetry
ffv Fractional Free Volume
ffe Free Volume Element
gcmc Grand Canonical Monte Carlo
LF Lattice Fluid Theory
MC Monte Carlo
MM Molecular Mechanics
MD Molecular Dynamics
net-gp Non-Equilibrium Thermodynamics of Glassy Polymers
pals Positron Annihilation Lifetime Spectroscopy
PI4 6FDA-TrMPD, Polyimide 4
PIM Polymer of Intrinsic Microporosity
pmv Partial Molar Volume
PSU Polysulfone
r.e.s. Radius of Equivalent Sphere
sd Site Distribution
stp Standard Temperature and Pressure
125
A.2 Selected Notations A. Appendix
A.2 Selected Notations
symbol meaning unit
baffinity constant (dm)cm3/mol
Cconcentration cm3(stp)/cm3
C0
HLangmuir capacity (dm)bar−1
Ddiffusion coefficient cm2/s
fscaling factor for Langmuir dilation (dm)
G0center of sorption-energy distribution (sd)kJ/mol
kDHenry solubility (dm)cm3/cm3bar−1
kst swelling coeff. of total dilation (net-gp)MPa−1
ksr swel. coeff. of relax. dilation (net-gp)MPa−1
N0molar number of sorption sites (sd)mol−1
p∗characteristic pressure (net-gp)MPa
ρ∗characteristic density (net-gp)g/cm3
ρ0density of pure polymer g/cm3
ρoobtained density g/cm3
ρttarget density g/cm3
σGwidth of sorption-energy distribution (sd)kJ/mol
σVwidth of size distribution (sd)cm3/mol
T∗characteristic temperature (net-gp)K
V0initial volume of polymer cm3/mol
∆Vvolume change of polymer cm3/mol
Vg‘dynamic volume’ of penetrant cm3/mol
Vhvolume of sorption site (sd)cm3/mol
Vh0center of size distribution (sd)cm3/mol
Vppartial molar volume of penetrant cm3/mol
Vid volume of ideal gas (stp)cm3/mol
126
A.3 Slices of Packing Models A. Appendix
A.3 Slices of Packing Models
Figure A.1: Slices of one packing model PSU , thickness 5.5Åand edge
length 38.6Å.
Figure A.2: Slices of one packing model PSU80, thickness 5.6Åand
edge length 39.3Å.
127
A.3 Slices of Packing Models A. Appendix
Figure A.3: Slices of one packing model PI4, thickness 5.5Åand edge
length 38.8Å.
Figure A.4: Slices of one packing model PI156, thickness 5.7Åand edge
length 39.8Å.
128
A.3 Slices of Packing Models A. Appendix
Figure A.5: Slices of one packing model PIM , thickness 5.4Åand edge
length 37.7Å.
Figure A.6: Slices of one packing model PIM206, thickness 5.6Åand
edge length 39.3Å.
129
A.4 Rotation Analysis of Bond Angles A. Appendix
A.4 Rotation Analysis of Bond Angles
To support the qualitative difference of backbone flexibility of the two poly-
mers a rough analysis of dihedral angle rotations was carried out as a ‘scout-
ing’ experiment (the analyzed bonds are indicated by arrows in Figure A.8
(a) and (b)).3For one packing model of PI4 Figure A.7(a) shows individually
for all 80 repeat units (y-axis) of a PI4 backbone chain the change of the
dihedral angle for one of the two most ‘flexible’ bonds –C–C(CF3)2–, during
an NpT-MD run of 1000 ps (x-axis) of the pure matrix at 1 bar bulk pres-
sure. Each dot in the graph represents a state where the dihedral angle has
changed more than 60◦with respect to its initial value at t= 0. The value of
60◦was chosen to distinguish significant changes from thermal fluctuations.
200 400 600 800 1000
0
20
40
60
80
PI4
t / ps
re p ea t un it n u m b er
'stable' rotation events
(a) –C–C(CF3)2– bond rotations
200 400 600 800 1000
0
20
40
60
80
rotation reversed
t / ps
re p ea t un it n u m b er
PSU
rotation events
(b) –C–C(CH3)2– bond rotations
Figure A.7: Visualization charts of dihedral angle rotation events in PI4
(a) and PSU (b). The most flexible angle of each repeat unit (y-axis) was
chosen for PI4, and the least flexible one (capable of rotation) for PSU
(see Fig. A.8). Changes in dihedral angle (>60◦) in comparison to the
value at t= 0 that are ‘stable’ with respect to simulation time (x-axis),
appear as lines parallel to the x-axis. To guide the eye, lines were added
to mark some of the rotation events that are considered ‘stable’ (red). In
PSU, some of the bonds rotate back to the initial bond angle (marked
green) within the simulation time.
Four lines parallel to the x-axis can be observed, which represent ‘stable’
changes in dihedral angle which then last over the remaining simulation time.
Figure A.7(b) shows a similar plot for a bond –C–C(CH3)2– in each of the
94 repeat units (y-axis) of a PSU backbone chain. In comparison to PI4,
two observations can be made. There is a larger number of ‘stable’ changes
of dihedral angle, and the ‘life time’ of these is in average shorter (reverse-
130
A.4 Rotation Analysis of Bond Angles A. Appendix
rotations to the initial angle are marked in green). The PSU repeat unit
contains eight bonds which are capable of rotation. Of these, the one chosen
for the analysis presented in Figure A.7(b), is the least flexible. The repeat
unit of PI4 contains four bonds capable of rotation, of which the one indicated
in Figure A.8(a) is one of the two more flexible (analysis presented in Fig.
A.7(a)). These two bonds were chosen for the comparison to emphasize the
difference in mobility of the two polymer backbones. The findings support
the statement that the backbone of PSU is more flexible in comparison to
PI4. See also reference [3].
(a) PI4 (b) PSU
Figure A.8: Chemical structure of (a) 6FDA-TrMPD polyimide (PI4)
and (b) poly(sulfone) (PSU). Arrows indicate the analyzed dihedral
bonds.
131
Danksagung
Diese Arbeit ist das Ergebnis eines gemeinsamen Projektes, und ich bin ei-
ner Reihe von Leuten zu Dank verpflichtet, die mich bei der Fertigstellung
unterstützt haben.
Ich danke Herrn Prof. Dr.-Ing. M. H. Wagner, der sich bereit erklärt hat,
diese Arbeit seitens der TU Berlin zu betreuen und zu begutachten.
Prof. Dr. Dieter Hofmann hat sich ebenfalls zur Begutachtung der Arbeit
bereit erklärt und stand mir mit seiner Modelling Gruppe des Instituts für
Polymerforschung (GKSS, Teltow) in allen Fragen des Simulierens hilfreich
zur Seite. Insbesondere die von ihm zur Verfügung gestellten PIM-1 Packungs-
modelle haben die Abrundung der Untersuchungen ermöglicht.
Besonderer Dank gilt meinen beiden fachlichen Betreuern Dr. Martin Böh-
ning (BAM) und Dr. Matthias Heuchel (GKSS), die das Projekt bei der Deut-
schen Forschungsgemeinschaft (DFG) beantragt haben und sich viel Zeit ge-
nommen haben, die einzelnen Schritte und Ergebnisse meiner Arbeit mit mir
zu diskutieren. Ihre Erfahrungen und Vorarbeiten sowohl hinsichtlich detail-
liert atomistischer MD-Simulation als auch experimenteller Charakterisierung
glasiger Polymere haben wesentlich zum Gelingen dieser Arbeit beigetragen.
Ich danke der DFG für die finanzielle Unterstützung im Rahmen der gemein-
samen Projekte [He2108/2-1] und [Bo1921/1-1], sowie der Bundesanstalt für
Materialforschung und -prüfung (BAM), die mich auch im Rahmen ihres
Doktorandenprogramms finanziert hat. Ich danke auch Herrn Prof. Dr. J.
F. Friedrich (BAM VI.5) und Herrn Dr.-Ing. W. Mielke (BAM VI.3), die sich
diesbezüglich für mich eingesetzt haben.
In der Arbeitsgruppe von Priv. Doz. Dr. Andreas Schönhals habe ich mich
sowohl fachlich als auch persönlich sehr gut aufgehoben gefühlt. Meine Kolle-
gen Dr. Hao Ning, Diana Labahn und Dr. Harald Goering haben viel zu der
angenehmen Arbeitsatmosphäre beigetragen. Auch habe ich mich bei mei-
nen häufigen Besuchen in Teltow in der Kaffeerunde der Modelling Gruppe
immer willkommen gefühlt.
Ich danke Dr. Jörg Frahn (GKSS) für die Synthese des Polyimids PI4 und
Kathleen Heinrich und Dr. Detlev Fritsch (GKSS) für die Herstellung von
PIM-1. Dr. Martin R. Siegert (GKSS) verdanke ich die Ergebnisse der Bin-
dungswinkelanalyse.
Dr. Maria Grazia De Angelis vom Chemical Engineering-Department der
Universität von Bologna hat mir beim Verständnis des net-gp Modells ent-
scheidend weitergeholfen und mir die nötigen Parameter zur Verfügung ge-
stellt.
Mit Andrea Pfitzner konnte ich über die Vorzüge von L
A
TEX schwärmen und
die Probleme des Layouts lösen.
Auf Annettes Rückhalt und insbesondere ihre Geduld während des Endspurts
konnte ich jederzeit vertrauen. Für die Ablenkung zur richtigen Zeit waren
unsere Kinder Jorid und Piet zuständig, und meine Mutter hat geholfen
die Ablenkung zur unrechten Zeit abzuwenden. Meinen Spaß an der Physik
verdanke ich meinem Vater, Physiker und Pädagoge, und ich wünschte, er
hätte das noch erleben können.
Lebenslauf
Name: Ole Hölck
Geboren: 06.06.1972 in Kiel
Staatsangehörigkeit: deutsch
Familienstand: verheiratet, 2 Kinder
1978–1979: Hardenberg-Schule in Kiel
1980–1982: Claus-Rixen-Schule in Altenholz
1982–1992: Kieler-Gelehrten-Schule in Kiel
1988–1989: Prosser Highschool, Washington State, USA
Mai 1992: Abitur
Sept.1992–Nov. 1993: Zivildienst
Okt. 1994–Nov. 2000: Physikstudium,
Georg-August-Universität Göttingen
Okt. 1996 Vordiplom Physik
Okt. 1998–Sept.2000: Diplomarbeit am Institut für Materialphysik:
Aufbau einer Vibrating Reed Anlage zur Unter-
suchung der Inneren Reibung kleiner Moleküle in
Polymeren Gläsern
Betreuer: Prof. Dr. R. Kirchheim
Nov. 2000: Diplom im Studiengang Physik
Jan. 2001–Sept.2003: Wissenschaftl. Hilfskraft, Inst. f. Materialphysik
Feb. 2004–Nov. 2007: Wissenschaftlicher Mitarbeiter der Bundesan-
stalt für Materialforschung und -prüfung
Feb. 2004–Nov. 2007: Dissertation:
Gas Sorption and Swelling in Glassy Polymers
Eidesstattliche Versicherung
Ich versichere an Eides statt, dass ich die von mir vorgelegte Dissertation
selbstständig angefertigt habe und alle benutzten Quellen und Hilfsmittel
vollständig angegeben habe. Die Zusammenarbeit mit anderen Wissenschaft-
lern habe ich kenntlich gemacht. Diese Personen haben alle bereits ihr Pro-
motionsverfahren abgeschlossen.
Teile dieser Arbeit sind bereits veröffentlicht:
[1]: M. Heuchel, M. Böhning, O. Hölck, M. R. Siegert, and D. Hofmann.
J. of Polym. Sci. B: Polym. Phys., 44(13):1874-1897, 2006.
[2]: O. Hölck, M. R. Siegert, M. Heuchel, and M. Böhning.
Macromolecules, 39(26):9590-9604, 2006.
[3]: O. Hölck, M. Heuchel, M. Böhning, and D. Hofmann.
J. of Polym. Sci. B: Polym. Phys., (accepted), 2007.
sowie zwei Abstracts für Conference Proceedings:
M. Heuchel, M. Böhning, O. Hölck, M. R. Siegert, and D. Hofmann.
Desal. 199(1):443, 2006.
O. Hölck, M. Böhning, M. Heuchel, M. R. Siegert
Desal. 200(1):166, 2006.
Eine Anmeldung der Promotionsabsicht habe ich an keiner anderen Fakultät
oder Hochschule beantragt.
