Modeling elongational viscosity and brittle fracture of polystyrene solutions [original]

ORIGINAL CONTRIBUTION

Modeling elongational viscosity and brittle fracture

of polystyrene solutions

Manfred H. Wagner

&Esmaeil Narimissa

2,3

&Leslie Poh

2,3

&Taisir Shahid

4,5

Received: 14 February 2021 /Revised: 13 April 2021 /Accepted: 29 April 2021

#The Author(s) 2021

Abstract

Elongational viscosity data of well-characterized solutions of 3–50% weight fraction of monodisperse polystyrene PS-820k

(molar mass of 820,000 g/mol) dissolved in oligomeric styrene OS8.8 (molar mass of 8800 g/mol) as reported by André et al.

(Macromolecules 54:2797–2810, 2021) are analyzed by the Extended Interchain Pressure (EIP) model including the effects of

finite chain extensibility. Excellent agreement between experimental data and model predictions is obtained, based exclusively

on the linear-viscoelastic characterization of the polymer solutions. The data were obtained by a filament stretching rheometer,

and at high strain rates and lower polymer concentrations, the stretched filaments fail by rupture before reaching the steady-state

elongational viscosity. Filament rupture is predicted by a criterion for brittle fracture of entangled polymer liquids, which assumes

that fracture is caused by scission of primary C-C bonds of polymer chains when the strain energy reaches the bond-dissociation

energy of the covalent bond (Wagner et al., J. Rheology 65:311–324, 2021).

Keywords Polymer melt .Polymer solution .Fracture .Failure .Chain scission .Elongation .EIP model .Interchain pressure .

Finite extensibility

Introduction

Substantial progress in measuring the elongational viscosity

of polymer melts and solutions up to high Hencky strains was

made by the use of the filament stretching rheometer with

locally controlled deformation and deformation rate

developed by Hassager and coworkers (Huang et al. 2016a).

By measuring the local diameter of the polymer sample during

elongation, the true Hencky strain and strain rate can be de-

termined and controlled, while by elongational rheometers

prescribing the global deformation of the filament only nom-

inal values of strain and strain rate can be obtained.

Elongational viscosity measurements with filament stretching

rheometers have revealed surprising differences between the

elongational rheology of polymer melts and solutions (Huang

et al. 2013a,2013b), which in turn has sparked different the-

oretical explanations as discussed, e.g., in Narimissa et al.

(2020a,2021) and Ianniruberto et al. (2020).

Progress in understanding failure of polymer samples in

elongational flow was hampered by the fact that failure of

stretched filaments includes the phenomena of ductile failure

(“necking”) and cohesive failure (“rupture”or “brittle frac-

ture”), and until recently, the experimental separation of these

two fundamentally different failure modes has been difficult

or even impossible. A further achievement of the filament

stretching rheometer as shown by Huang et al. (2016b),

Huang and Hassager (2017) and Huang (2019) was that when

the true Hencky strain rate is controlled rather than nominal

Hencky rate, the four failure zones of the so-called Malkin

plot (Malkin and Petrie 1997) (purely viscous zone,

*Manfred H. Wagner

manfred.wagner@tu-berlin.de

*Esmaeil Narimissa

esmaeiln@technion.ac.il

Polymer Engineering/Polymer Physics, Berlin Institute of

Technology (TU Berlin), Ernst-Reuter-Platz 1,

10587 Berlin, Germany

Department of Chemical Engineering, Technion–Israel Institute of

Technology (IIT), Technion City, 32 000 Haifa, Israel

Department of Chemical Engineering, Guangdong Technion–Israel

Institute of Technology (GTIIT), Shantou 515063, China

Bio and Soft Matter, Institute on Condensed Matter and

Nano-science, Université catholique de Louvain,

Louvain-la-Neuve, Belgium

DSM Materials Science Center, P.O. Box 18, NL-6160

MD Geleen, The Netherlands

https://doi.org/10.1007/s00397-021-01277-1

/ Published online: 14 June 2021

Rheologica Acta (2021) 60:385–396

viscoelastic zone with failure by necking, rubbery zone,

glassy zone) are reduced to just two possible states: liquid or

solid, and a clear distinction exists between liquid-like behavior

(unlimited steady-state elongation) and solid-like behavior (brit-

tle fracture). A quantitative criterion for brittle fracture of

entangled polymer liquids (Wagner et al. 2018) was recently

extended by taking finite chain extensibility and polymer fraction

of the solutions into account (Wagner et al. 2021). Filament

rupture follows from scission of primary C-C bonds, when the

strain energy of an entanglement segment reaches the bond-

dissociation energy of the covalent bond. Thermal fluctuations

lead to short-time concentration of the strain energy on one C-C

bond of the entanglement segment, and the chain ruptures.

In the present paper, we analyze the elongational viscosity

data of well-characterized solutions of 3–50% mass fraction of

monodisperse polystyrene (PS-820k with a molar mass of

820,000 g/mol) dissolved in oligomeric styrene (OS8.8 with

a molar mass of 8800 g/mol) as reported by Shahid (2018)and

André et al. (2021). The polymer solutions show increasing

strain hardening behavior with decreasing polymer concentra-

tion, which is associated with an increasing tendency of fila-

ment rupture at higher elongation rates. Experimental data of

elongational stress growth coefficient and steady-state

elongational viscosity as well as stress and strain at fracture

are compared to the prediction of the Extended Interchain

Pressure (EIP) model (Narimissa et al. 2020a,2021) including

the effects of finite chain extensibility and a recently devel-

oped fracture criterion for brittle fracture of polymer melts and

solutions (Wagner et al. 2021). Modeling is based exclusively

on linear-viscoelastic characterization of the solutions and the

ratio of carbon-carbon bond energy to thermal energy, and

does not require any fit parameter. The EIP model with the

fracture criterion is the only model presently available for

quantitative modeling of time-dependent nonlinear-viscoelas-

tic flows including brittle fracture of polymer systems.

The paper is organized as follows: We first give a short

report of the experimental data and the linear-viscoelastic

characterization of the polymer systems considered, followed

by a short account of the EIP model and the fracture criterion.

The main focus of the paper is on comparison of experimental

data and model predictions, and on the conclusions that can be

drawn from this comparison.

Experimental data and linear-viscoelastic

characterization

The polymer PS-820k with a molar mass of 820 kg/mol and

polydispersity index of 1.02 was obtained from Polymer

Source, Inc. (Montreal, Canada). Preparation of solutions is

described in detail by Shahid (2018) and André et al. (2021),

and the molecular characteristics of the polystyrene solutions

are summarized in Table 1. The samples are named in the

form of PS-820k/8.8k-X, where 820k characterizes the molar

mass of the polystyrene, 8.8k the molar mass of the oligomeric

styrene solvent with 8.8 kg/mol and with polydispersity index

of 1.1, and X the percentage of mass and (assuming equal

density of PS melt and solution) volume fraction of the PS

in the solution.

Details of mechanical spectroscopy and elongational vis-

cosity measurements are presented in Shahid (2018).

Elongational measurements using a VADER 1000 (Huang

et al. 2016a) were performed at T = 130 °C. Storage (G′)and

loss modulus (G″) were measured at iso-T

temperatures T

i.e., temperatures with equal distance to the glass transition

temperature T

with T

+23.4K,andshiftedtoT=

130 °C by time-temperature shifting (TTS) according to the

WLF equation with shift factor a

log10aTg¼−c0

1T−T0

ðÞ

2T−T0

ðÞ ð1Þ

and coefficients c0

1¼8:99 and c0

2¼81:53K (Wagner 2014).

From the mastercurves of G′and G′, parsimonious relaxation

spectra were obtained,

GtðÞ¼∑

giexp −t=τi

ðÞ ð2Þ

for characterization of the linear viscoelasticity (LVE) in the

experimentally accessible window of all polymer systems

considered here. The partial moduli g

and relaxation times

as determined by the IRIS software (Winter and Mours

2006) result in excellent agreement with the linear-

viscoelastic data of G′and G″(see Fig. S1and Table S1of

the Support Information).

From the plateau modulus G

GN¼ρRT

Með3Þ

the entanglement molar mass M

is obtained, and with M

being the molar mass of the polymer, the number of entangle-

ments per chain Zis given by

Z¼M

Með4Þ

The relation between the entanglement molar mass of the

polymer in solution, M

, and entanglement molar mass in the

melt, M

,isgivenby

Me¼Memφ−αð5Þ

with φbeing the polymer volume fraction in the solution. To

be consistent with earlier works (Huang et al. 2013a,2013b,

2016b;Wagner2014; Narimissa et al. 2020a,2021), we take

=2.5⋅10

Pa and M

= 13,300 g/mol for polystyrene in the

melt state. (The index mcharacterizes the melt state in the

386 Rheol Acta (2021) 60:385–396

following.) The value of the dilution exponent αis model

dependent (1 < α< 4/3), and a value of α= 1 is taken here.

The number of Kuhn segments or “monomers”between en-

tanglements is N

, where M

= 610 g/mol is the molar

mass of the Kuhn monomer assumed to be independent of

dilution.

According to the Doi-Edwards model, the Rouse time τ

the disengagement (or reptation) time τ

, and the zero-shear

viscosity η

are given by (Dealy et al. 2018)

τR¼Z2τeð6Þ

τd¼3ZτRð7Þ

η0¼π

2GNτdð8Þ

is the entanglement segment equilibration time. We

identify here τ

with the mean quadratic average of the relax-

ation times of the discrete relaxation spectrum and calculate η

from the discrete relaxation spectrum:

τd¼

∑

giτ2

∑

giτið9Þ

η0¼∑

giτið10Þ

We use Osaki’s approach (Osaki et al. 1982; Takahashi

et al. 1993; Isaki et al. 2003; Menezes and Graessley 1982)

for the quantification of the Rouse time τ

, which extrapolates

the Rouse time of unentangled polymer systems to the Rouse

time of entangled polymer melts and solutions, and takes into

account the power of 3.4 scaling of the zero-shear viscosity

with molar mass Mand polymer fraction φ(Wagner 2014).

This leads to the relation

τR¼12Mη0

π2ρRTφ

Mcm

Mφ



2:4

ð11Þ

for the Rouse stretch relaxation time. M

denotes the crit-

ical molar mass in the melt state, when the entanglement

effect becomes apparent by a change of the power of 1 to

power of 3.4 scaling of the zero-shear viscosity as a func-

tion of molar mass. For monodisperse polystyrene, we take

the well-documented value of M

= 35kg/mol (Wagner

2014), and Eq. (11) has been used successfully for the

modeling of the transient and steady-state elongational

and shear viscosities of PS melts and well-entangled poly-

mer solutions (Narimissa et al. 2020a,2020b,2021;

Wagner et al. 2021).

For the solutions of PS-820k in OS8.8k, the power of

3.4 scaling of the zero-shear viscosity of polymer solutions

with polymer fraction φat iso-T

temperatures of T

+ 23.4 K is shown in Fig. 1. The solutions with polymer

fractions of 50, 40, 30, and 20% (full symbols in Fig. 1)

follow nicely the relation η

=η

3.4

with a value of

=1.07⋅10

Pas for the zero-shear viscosity of the melt,

while for solutions with polymer fractions of 5 and 3%

(open symbols), the zero-shear viscosity is higher than ex-

pected, possibly affected by the viscosity of the solvent

OS8.8k, which has a value of η

=1.0⋅10

Pas at 130 °C.

Thus, the solutions with polymer fractions φ≤0.1 deviate

fromthescalingpresumedbyEq.(11) and cannot be de-

termined from this relation. To be consistent with previous

investigations (Narimissa et al. 2020a;Wagneretal.2021),

we calculate τ

by using Eq. (11) for melt 820k and its

solutions with polymer fractions of 20–50%, while at low-

er polymer concentration, the Rouse time of the solution is

taken as the time-temperature-shifted value of the Rouse

time of the melt.

Table 1shows glass transition temperature (T

), shift factor

(aTg), entanglement molar mass (M

), number of entangle-

ments per chain (Z), number of Kuhn monomers (N

) per

entanglement, zero-shear viscosity (η

), disengagement/

reptation time (τ

), and Rouse stretch relaxation time (τ

)at

T = 130 °C of the monodisperse polymer samples considered

in this study.

Table 1 Sample characterization at T = 130 °C

PS-820k

melt

PS-820k/

8.8k-50

PS-820k/

8.8k-40

PS-820k/

8.8k-30

PS-820k/

8.8k-20

PS-820k/

8.8k-10

PS-820k/

8.8k-05

PS-820k/

8.8k-03

[°C] 106.6 102.3 101.3 100.7 99.8 98.6 98.4 98.1

[−] 1 0.35 0.28 0.25 0.20 0.16 0.15 0.14

[kg/mol] 13.3 26.6 33.3 44.3 66.5 133 266 443

Ζ[−] 61.7 30.8 24.7 18.5 12.3 6.2 3.1 1.8

21.8 43.6 54.5 72.7 109 218 436 727

[Pa s] 1.07Ε+10 3.87Ε+8 1.46Ε+8 4.01Ε+7 8.01Ε+6 6.01Ε+5 1.02Ε+5 3.01Ε+4

[s] 429,918 21,461 12,634 6631 4369 1591 906 114

[s] 1644 628 505 369 293 259 248 233

387Rheol Acta (2021) 60:385–396

The Extended Interchain Pressure (EIP) model

The molecular stress function (MSF) model is a generalized

tube segment model with strain-dependent tube diameter

(Wagner 1990; Wagner and Schaeffer 1992; Wagner et al.

2001,2003,2005; Narimissa et al. 2021b). The extra stress

tensor σ(t) of the MSF model with consideration of finite

chain extensibility effects (Rolon-Garrido et al. 2006)isgiven

by a history integral of the form

σtðÞ¼ ∫

−∞

∂Gt−t0

ðÞ

∂t0fλSIA

DEt;t0

ðÞdt0ð12Þ

tis the time of observation when the stress is measured, and

t′indicates the time when a tube segment was created by

reptation. The strain measure SIA

DErepresents the contribution

to the extra stress tensor originating from the affine rotation of

the tube segments assuming “Independent Alignment (IA)”

(Doi and Edwards 1978,1979), and is given by

SIA

DE t;t0

ðÞ≡5u0u0

u02



o¼5St;t0

ðÞ ð13Þ

with S(t,t′) is the relative second-order orientation tensor.

u'u' is the dyad of a deformed unit vector u0¼u0t;t0

ðÞ,

u0¼F−1

t⋅uð14Þ

F−1

t¼F−1

tt;t0

ðÞis the relative deformation gradient tensor,

and u' is the length of u'. The orientation average is indicated

by <…>

…

o≡1

4π∯…½sinθodθodϕoð15Þ

i.e., an average over an isotropic distribution of unit vectors

λ=λ(t,t')represents the inverse of the relative tube diame-

ter a/a

, and at the same time the relative length of a deformed

tube segment,

λt;t0

ðÞ¼a0

at;t0

ðÞ

¼lt;t0

ðÞ

l0ð16Þ

At time t=t′, the tube segment was created with equilibri-

um tube diameter a

and equilibrium length l

In the Gaussian limit, the molecular stress function f,i.e.,

the relative tension in the chain, is equal to the tube stretch λ.

However, this is valid only as long as λ<0.5λ

max

(Bird et al.

1987), where λmax≅ﬃﬃﬃﬃﬃﬃ

prepresents the maximum stretch (i.e.,

a fully extended chain), and N

the number of Kuhn mono-

mers in an entanglement segment. Outside the Gaussian re-

gime, tension in the chain can be described by the inverse

Langevin function, or, due to its mathematical complexity,

approximations like the Padé approximations (Cohen 1991).

Therefore, the nonlinear elasticity caused by finite extensibil-

ity (FENE) is implemented in the EIP theory in the following

way:

f¼cλ

ðÞ

λð17Þ

cis a nonlinear spring coefficient, representing a relative

Padé inverse Langevin function with

c¼

3−λ2

λ2

max

⋅1−1

λ2

max

3−1

λ2

max

⋅1−λ2

λ2

max

! ð18Þ

Maximal stretch λ

max

is defined as

λ2

max ¼Ne¼Nemφ−1ð19Þ

with N

given in Table 1.

While SIA

DE is determined directly by the deformation his-

tory according to Eq. (13), λis found as a solution of an

evolution equation considering affine tube segment deforma-

tion balanced by Rouse relaxation and interchain pressure

(Wagner et al. 2021). We modified the evolution equation of

Narimissa et al. (2020a) by including the effect of finite ex-

tensibility into the interchain pressure term in the same way as

explained in detail by Rolon-Garrido et al. (2006),

∂λ

∂t¼λκ:SðÞ−λ−1

τR

1−2

3φ4



−2φ4

9τR

λ2λf2−1



ð20Þ

-2

-1

[-]

0[Pa s]

Fig. 1 Zero-shear viscosity of solutions of PS-820k in OS8.8 with

polymer fractions of 50, 40, 30, and 20% (full symbols), and 5 and 3%

(open symbols) at iso-T

temperatures T

+ 23.4 K. Line indicates

the relation of η

=η

3.4

with η

=1.07⋅10

Pas

388 Rheol Acta (2021) 60:385–396

In Eq. (20), the first term on the right-hand side describes

an affine deformation, the second term Rouse relaxation, and

the third term represents the interchain pressure contribution.

Equation (20) reduces to the evolution equation of the EIP

model of Narimissa et al. (2020a) in the Gaussian limit, i.e.,

when c=1andf=λ.Equations(12)and(20) represent the

EIP model with finite chain extensibility and are solved

numerically.

From Eq. (20) follows at high Weissenberg numbers Wi

¼˙"τR(with elongational strain rate ˙") and large deforma-

tions, when the equilibrium stretch is reached and ∂λ/∂t=0

that the product of molecular stress fand stretchλis propor-

tional to the square root of Wi and inverse proportional to the

square of the polymer fraction,

fλ¼cλ2¼3

2φ−2ﬃﬃﬃﬃﬃﬃﬃﬃ

2Wi

pð21Þ

In this asymptotic limit and neglecting the glass transition,

the tensile stress is expected to reach a value of

σ¼5GNfλ¼15

2GNφ−2ﬃﬃﬃﬃﬃﬃﬃﬃ

2Wi

pð22Þ

From Eq. (22) and considering that G

, the uni-

versal relation for the high Wi tensile stress of melts and

solutions of Narimissa et al. (2020a)isrecovered,

σ¼15

2GNm ﬃﬃﬃﬃﬃﬃﬃﬃ

2Wi

pð23Þ

with G

being the plateau modulus of the melt.

The fracture criterion

The thermal energy w

at a temperature Tof 403 K (130 °C)

weq ¼3kT ¼1:67⋅10−20 Jð24Þ

with Boltzmann’sconstantk=1.38⋅10

−23

J/K. On the other

hand, the bond-dissociation energy of a single carbon-carbon

bond in hydrocarbons is (Wagner et al. 2018)

U¼348kJ

NA¼5:78⋅10−19 J≅35weq ð25Þ

with Avogadro’snumberN

=6.02⋅10

.Thus,thebonden-

ergy Uis about 35 times larger than the thermal energy w

130 °C, which is why the polymer chain will not break due to

Brownian motion at equilibrium.

As explained by Wagner et al. (2021), the strain energy of a

chain segment is taken as

ðÞ¼3kTfλφ ð26Þ

with N

being the number of Kuhn monomers of an en-

tanglement segment in the melt. When the strain energy of the

segment reaches the critical energy

wc¼3kTf cλcφ¼Uð27Þ

the total strain energy of the chain segment will be concentrat-

ed on one C-C bond by thermal fluctuations, and this bond

then ruptures. We recall that stretch and tension are relative

quantities, and therefore, the strain energy w=w(t,t') is also a

relative quantity. Chain segments with long relaxation times,

i.e., those preferably in the middle of the chain, will be the first

to reach the critical energy w

and will fracture. Chain seg-

ments closer to the ends of the macromolecule, which due to

reptation processes have shorter relaxation times and see less

stretch and tension within the time interval of t′(creation) and t

(observation), will not reach w

and are less likely to fracture.

We assume that as soon as the strain energy w=w(t,t'=0)

accumulated between the start-up of deformation at time t′=0

and time t=t

reaches the critical energy w

=w(t=t

,t' = 0),

a sufficient concentration of locally ruptured chains is reached

and crack initiation will occur. Crack initiation is followed by

crack growth, which leads within a very short time (about

200 ms according to Huang et al. (2016b)) to brittle fracture

of the sample. At time t=t

, the critical Hencky strain at frac-

ture, εc¼˙"tc, is reached and the critical tensile stress at frac-

ture, σ

=σ(t

), is given by the stress equation (Eq. (12)).

Chain fracture preferably in the middle of the polymer chain

is in agreement with earlier findings of Ballauf and Wolf

(1984): They studied the degradation of solutions of 4.9–

20 wt% of polystyrene in trans-decalin by use of a shear cell

at shear rates of 5:0⋅103<

γ<104s−1, and showed that only

a Gaussian breakage probability of C-C bonds with the center

of the probability distribution at the midpoint of the chain can

reproduce the experimentally observed changes in the molar

mass distribution.

From the fracture hypothesis defined by Eq. (27), the max-

imum achievable product of critical molecular stress f

and

critical stretch λ

is obtained,

fcλcφ¼cλ2

cφ¼U

3kT ≅35 ð28Þ

We called this fracture mode “entropic fracture”(Wagner

et al. 2018), as it is caused by thermal fluctuations, in contrast

to the “enthalpic fracture”hypothesis of Lake and Thomas

(1967)asmodifiedbyMazichandSamus(1990). These au-

thors assumed that all bonds are fully stretched at fracture and

when a chain with NC-C bonds between two entanglement

points ruptures, the strain energy w

=NU corresponding to

thebondenergyofall N C-C bonds in the entangled chain

segment is dissipated.

Combining the fracture criterion of Eq. (28) with the as-

ymptotic tensile stress at high Wi and large stretch according

389Rheol Acta (2021) 60:385–396

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