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Pantoja, Christian F.; Muñoz-Muñoz, Y. Mauricio; Guastar, Lorraine; Vrabec, Jadran; Wist, Julien (2018).
Composition dependent transport diffusion in non-ideal mixtures from spatially resolved nuclear magnetic
resonance spectroscopy. Physical Chemistry Chemical Physics, 20(44), 28185–28192. https://doi.
org/10.1039/c8cp05539d
Christian F. Pantoja, Y. Mauricio Muñoz-Muñoz, Lorraine Guastar, Jadran
Vrabec, Julien Wist
Composition dependent transport diffusion
in non-ideal mixtures from spatially
resolved nuclear magnetic resonance
spectroscopy
Accepted manuscript (Postprint) Journal article |

ARTICLE
Composition De pendent Transport Dif fusion in Non - Ideal
Mixtures from S patially Resolve d Nuclear Magnetic
Resonan ce Spectr oscopy

Christian F. Pantoja, a Y. Mauricio Mu ñoz- Mu ñoz b , Lorraine Guastar a , Jadran Vra bec b , and Julien
Wist *a
Nuclear mag netic resonan ce (NMR) is a well - establ ished technique for the measuremen t of intra - diffusion co efficients.
Recently, such information has bee n used as a basis of p redictive mod els to extrap olate to the Fick diffusion coefficient o f
liquid mixtures . T he present work presents a new approach to directl y access the Fi ck diff usion coeff icient by spatially
resolved NMR ex periments. The Fi ck diffusi on coeffi cient o f t he binary mixture TE A/H2O was d etermin ed at two
temperatures , 283.2 K and 275.2 K. The results ar e consisten t wit h values previ ously rep orted eit her from optica l
experiments or predict ive Darken - ty pe models develo ped for thi s system. The propo sed methodol ogy adds hi gh - resolution
NMR to the toolbox for the study of transp ort diffusion of multicompone nt mixtures. It is, however , still limited to mixture s
with liquid - liquid eq uilibrium phase separation .
Introductio n
Transport diffusi on is wide ly studie d because of it s importance to
understand natural phenomena and improve industr ial processes 1 2
3 . Different experi mental techniques are being employ ed for that
task 4 , typically providing diffusi on coefficient data. B ased on the
nonetheless still surprisingly small experi mental database 5 , a range
of numerical models wa s proposed for their p redictio n 6 7 .
Transport diffusion can be descr ibed by means of two formalis ms ,
which can be transformed i nto each other if sufficient
thermodynamic information on the sy stem is avail able. The Maxwell -
Stefan approach descri bes diffusion from a physical ly sound
perspecti ve and postulates chemical po tentia l gradients as driving
forces for m ass flux 8 9 . It is being us ed e.g . for stud ie s of ca talytic
performance in react ors 10 , pervaporatio n membranes 11 or for
predicting transpor t diffusion coeffici ents by equilibrium mol ecular
dynamics simulation 12 . Al ternativ ely, Fick’s “l aw” assumes
concentrati on gradients as driving force s for mass f lux, which is
beneficial from a prac tical standpoint . The related diff usion
coefficients are usuall y measured by Taylor dispersi on 13 or
interferometry t echniques 14 15 .
However, the precise measurem ent of diffusio n coeffici ents entails
significant exper imental effort and i s only possibl e for mixtures
consisting of a small number of components (typically ≤3). Also , mo st
numer ical pred ictive mo dels e xplicitly focus on such systems only,
mainly because of the lac k of appropri ate data. Consequently, the
uncertainty associ ated with transport diffusion in m ulti - component
mixture s is hig h 16 .
Diffusion c oefficients o f liquid mixtures tend t o be strong ly
dependent on co mposition. In co ntrast to common as sumptions, thi s
holds for both Fick and Maxwell- Stefan diffusi on coeffic ients.
Experiments thus ha ve to be carried out in a r epetitive manner
varying th e mixture compositi on. An approach first proposed in the
1970s 17 has recent ly been revitalize d by Bardow et al. 18 . It allows for
the determination of the Fi ck diffusion coefficient of a l iquid mixture
in the entire composit ion range with a single experi mental run if a
varying spatial distr ibution of composition can be m easured over
time with a resol ution that is sufficient for fitting a diffusion model
to these data. A simil ar strategy has successfull y been employed to
gain insight about absor ption 19 as well as t ransport in ani onic gels 20
and polyelectrol ytes 21 . Moreover, the corresponding data can be
used to discriminate model s, such as in the incremental- model
approach proposed by B ardow and cowork ers. This model- free
methodology was validated on the basis o f Raman spectr oscopy
data 18 22 23 . Because Raman spec troscopy is cap able to sample th e
individual contribution of al l components constit uting a mixture, it
may pave the way t o the study of truly mul ti-component systems.
Nuclear Magnetic Resonance (NMR) spect roscopy offers comparabl e
advantages as Raman spectrosco py. Indeed, the mola r fraction of all
components in a mixt ure can be measured with a good temporal and
spatial resolution borrowing co nce pts from Magnetic Resonan ce
Imaging (MRI) 24 . Moreover, NMR spectro scopy is suitabl e for the
concurrent measureme nt of intra - diffusion coeff icients , which
describe the random motio n of molecular species in a mixture 25 . It
has also been used to i mprove model- based schemes, which rely on
Raman spectrosc opy measurements, e specially for mo lar fractio n
regimes where the presence of small concentration gr adients
prohibits good estimati ons of transport diffusion coefficient s 26 .
NMR spectrosco py was used in preceding work of our group to
observe the temporal and spati al evolution of all molar fractions of a
multi- component mi xture during its m ixing proce ss, s tarting f rom a
liquid - liq uid eq uilib rium (LLE) s tate point 27 . In the prese nt work, we
propose a sc heme for m easuring t he Fick diffusion coeffi cient of
binary liquid mixtures by co mbining that experimental tec hnique
with a numerical solution of partial differential equations (P DE) for
the mixing process inside a cyli ndr ical tube . The b inary mixture
Triethyl amine (TEA) / Wate r (H 2 O) was chosen to v alidate this
approach. Due to i ts highly non-ideal thermody namic behavior , it
was necessary to employ a c oordinate transformatio n as shown by
Bardow et al. 23 . TEA / H 2 O has also recently been used to validate a
predictive model by D ’Agostino et al . 28 , which very ac curately
performs for its Fic k diffusion c oefficient over a wi de compositi on
range. Moreo ver, the pr esent results ar e compared to experimental
literature val ues an d other recen t predict ive models 29 30 .

Experimen tal
Sample prepar atio n
To evaluate the proposed metho dology, four liquid TEA/H 2 O mixture
samples were prepared toget her with Deuterium oxi de (D 2 O)
purchased from Sigma - Aldric h. D 2 O was added for technica l reaso ns
only, i.e . , to lock the resonance frequency during multipl e scan
acquisitions, and its amount was kept as low as possible (30.5 μL).
The composition of these samples is reported in Table 1.
Table 1
Experimental condi tions employed in th e present w ork. T stand s for
the temperature and 𝑥𝑥 TEA , 𝑥𝑥 H 2 O , 𝑥𝑥 D 2 O are the mol ar fractions of TEA,
water and deuteri um oxide, respectively. Numb ers in parentheses
stand for the uncert ainties .

T / K

(±0.1)
𝑥𝑥 TEA /

mol.mol -1
(±0.0001)

𝑥𝑥 H 2 O /

mol.mol -1
(±0.0001)

𝑥𝑥 D 2 O /

mol.mol -1
(±0.0001)

283.2

0.1149

0.7799

0.1052

283.2

0.1127

0.7881

0.0992

278.2

0.1168

0.7825

0.1007

278.2

0.1167

0.7802

0.1031

The influence of the isot opic substitution of H 2 O with D 2 O has
previously been studied for this mixture 31 . Its low er criti cal so lut ion
temperature (L CST) is reduced by 3.8 0 K in c ase of complete
substitution. On the oth er hand, it was found that the tran sport
properti es, such as the Fic k diffusion c oefficient , are independent of
the deg ree of deuteratio n 32 .
The amount of eac h component was car efully chosen to place the
interface betwe en the two phas es und er LLE as c lose as possibl e to
the center of the NMR coil ( z cc ), i.e. approximately 2 cm above the
bottom of the NM R tube, cf. Fig. 1A. Onc e prepared, the sam ples
were allowed to rest for a t least 12 h ours at ambie nt tempe rature
and pressure.

S patially selective sampling
The experiment s were carried o ut with a 400 MH z Bruker Avance II
NMR spectrometer , equipped with a double channel 5 mm probe
(BBO) and triple axi s gradients. A r obust temperature contro l (±0.1
K) was achieved w ith a BCU1 unit (Br uker, Rheinstetten) and hi gh
quality 5 mm NMR t ube s were use d as sa mpling ce ll.
A Double Pulsed Field Gradi ent Selective Echo (DPFGSE) puls e
sequence was used to measur e the compositi on distribution along
the 𝑧𝑧 coordi nate because it is suit able for inho mo geneous system s 33 ,
cf. Fig 1 A. The excit ation pulse frequency was set to 22.2 kH z. A
Gaussian - shaped pulse of 1 ms and an encoding gradient of 10 .61
G/cm were used in the gradient echo. The offs et Ω 𝑖𝑖 of the s elective
pulse was vari ed between - 20 kHz to 24 kHz , whic h se lects horizo ntal
slices o r isochromats loc ated at 1.53 cm and 2.53 c m. The thick ness
of the isochromat i s defined by the bandwidth o f the pulse and was
determined to be 0.5 mm.
Once the mixing process was established in the NMR t ube, spatially
selective sampli ng was performed to map the conc entration profil e
along the z directi on. Each mapping consisted of measuring 23 sli ces
along z in random order , an operation t hat was achieved in 24 4 s. For
each sample ac cording to Table 1, a total of 83 mappings w as
recorded every 5 mi nutes for a total duration of 7 hours. The 23
isochromats were chosen to c over a region of 1.1 cm lo cated inside
the NMR co il active regio n. The acquis ition time of the free i nduction
decay (FID) was set to 1 s , and the width of th e spectr al window was
set to 15 ppm. The resulting 32k complex points were stored in a
matrix of dimension 83x( 23x32k). Further data pro cessing, such as
apodization (3 Hz) and baseline correction, was performed using
Topspin 2.5 (Bruker R heinstetten), whil e integration of the signals
was done with in-house sc ripts writt en for Scilab 5.5.
Calibra tion of the spati al co ordinat e and det erminat ion of t he bul k
temperat ure
To precisel y estimate t he loc ation 𝑧𝑧 of the select ed isoc hromats, i t is
essential to determine t he strengt h of the puls ed field gra dient 𝐺𝐺 𝑧𝑧
with a good accu racy , cf. supplem entary material . This was achieved
by calibration with solvents of well - known self - diffusion coeffici ents.
For that purpose, D 2 O with a purity of 99.9% (Sigma - Aldrich) was
used. Transport phenomena in liqui d systems are strongly
temperature dependent, thus the experimental set up must contro l
this property. Deuter ated Methanol (met hanol-d4) with a purity of
99.8 % ( Sigma- Aldrich ) was used to accurately cali brate the bulk
temperature measureme nt, cf. supplementary material .
Extraction of molar fractions
The intensity 𝐼𝐼 𝑗𝑗 of the signal wa s obtained by inte grating the sign al
area and is related to the num ber of molecules 𝑛𝑛 𝑗𝑗 presen t in the
sample by 34
𝐼𝐼 𝑗𝑗 = 𝐾𝐾 𝐴𝐴 𝑗𝑗 𝑛𝑛 𝑗𝑗 .

(1)
The rein, 𝐾𝐾 is a constant that depends on the experimental setu p
and 𝐴𝐴 𝑗𝑗 is the number of pro tons involved . T he integral of the signal
was preferred o ver the signal height because it is less depende nt on
the individual spin relax ation rates, cf. supplementary materia l . It
is possible to demonstrate tha t the molar fr action 𝑥𝑥 𝑗𝑗 is readily
obtained from the intensiti es by
𝑥𝑥 𝑗𝑗 = 𝐴𝐴 𝑗𝑗 −1 𝐼𝐼 𝑗𝑗
∑ 𝐴𝐴 𝑘𝑘
−1 𝐼𝐼 𝑘𝑘
𝑛𝑛
𝑘𝑘 =1 .

(2)
In the present experiments, the relaxation rat es depend on time a nd
spatial location. Therefor e, a longer rel axation time of 4 s was c hosen
to ensure that all spins had relaxed. Fi nally, the ac cumulation of only
two acquisitions was necessary to obt ain a high signal- to - noise ratio.

Figure 1 . A) Double Pulsed Field G radie nt Sele ctive Ech o (DPFG SE)
pulse sequence 33 . NMR signal s can be spatially encoded by applying
a pulse field gradient simul taneously wit h a narrowband rf- pulse 35 .
B) The field gradient shifted the Larmor fr equency linearly i n the 𝑧𝑧
coordi nate, while the narrow band pulse 36 was used to selec t the
vertica l region of i nterest by t uning it s resonance fr equency of fset
Ω 𝑖𝑖 . The resonance frequency at thi s 𝑧𝑧 c oordinate is thus defined a s
shown in panel B), where 𝜔𝜔 𝑧𝑧 ( 𝑧𝑧 ) represents t he resona nce freque ncy
of the nuclei wit h gyromagnetic c onstant 𝛾𝛾 , 𝐵𝐵 0 is the magni tude of
the external ma gnetic field and 𝐺𝐺 𝑧𝑧 the strength of th e gradient at
coordinate 𝑧𝑧 . The t hickness of t he select ed slice is direct ly related to
the bandwidth of pulse. Repeating the exper iment by varying the
frequency o ffset of th e select ive pulse al lows for the o bs ervation of
the composition the di fferent coordi nates 𝑧𝑧 .
Measuring a gradient in the NMR tub e
TEA/H 2 O exhibits a LLE region with a LCST, i .e. , starting f rom a two -
phase state point, t he temperature has to be decreased to enter t he
homogeneous region wher e mixi ng occurs, cf. Fig. 2B (insert) .
Therefore, to establi sh and measure a gradient of concentration in
the NMR tube, t he system i s first set to eq uilibrat e a temperat ure
where both phase c oexist ( 𝑇𝑇 𝑖𝑖𝑛𝑛𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 300 K ). Subsequently, this system
was cooled belo w the mixing temperat ure ( 𝑇𝑇 𝑚𝑚𝑖𝑖𝑚𝑚𝑖𝑖𝑛𝑛𝑚𝑚 =293.7 K ) to a
final temperature ( 𝑇𝑇 𝑜𝑜𝑜𝑜𝑜𝑜 = 278,2 𝐾𝐾 𝑜𝑜𝑜𝑜 283.2 𝐾𝐾 ), as descri bed in
Figure 2B (insert ). While reac hing its new equilibrium, sampling of
the concentrati on gradient were recor ded as described above.
Prior to the experim ent , the concentration profile was measure d to
locate t he position of t he inter face sepa rating th e two phases under
LLE, which w as near the center of the NMR co il, cf. Fig ure 2 A. B efore
acquiring data, a time delay 𝒕𝒕 𝒐𝒐𝒐𝒐𝒐𝒐 (F igu re 2B) was necessary to avoid
the sampling of contr ibutions from convecti ve phenomen a p resen ts
in step 1 and 2. The measurements takes place during step 3 w hen
the remaining flux 𝑵𝑵 𝒊𝒊 in the tube was only caused by diffusive
contributions 𝑱𝑱 𝒊𝒊 . Thi s process was re peated for all samples li sted in
Table 1.

Figure 2 . T he e xperimental set up to obser ve the time t evolution of
the molar fractions 𝑥𝑥 𝑖𝑖 ( 𝑧𝑧 , 𝑡𝑡 ) during the mixing process, where z is the
relevant spatial coordinate A) T he sampled sections wer e chosen to
be near the center of the observat ion window (red vertical line),
defined by the height of the coi l in the NMR setup (not depict ed).
Experimentally, this window is determined as the largest portion o f
the coil for w hich the response is homogeneous, while the intensity
of the signal is reduced at the e dges of the coil. The preparati on of
the sample ensur ed that the LLE int erface was at the c enter of the
observation window, i.e. , about 2 cm above the bottom of the NMR
tube in the pre sent setup. B) C ooling curve and pha se diagram
(insert) of the TEA /H 2 O system 37 . The measure ment cycle co nsisted
of preparing the sam ple at a temperature where two phase s
coexist 𝑇𝑇 𝑖𝑖𝑛𝑛𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 , cool it d own below 𝑇𝑇 𝑚𝑚𝑖𝑖𝑚𝑚𝑖𝑖𝑛𝑛𝑚𝑚 to a targ et temp erature
𝑇𝑇 𝑜𝑜𝑜𝑜𝑜𝑜 in the homogeneous regi on , and sample how this new
equilibrium is attained. The mixi ng temperature is defined as the
crossin g poin t in th e liquid - liquid coexistence curve (red diamonds)
for the composition of our system ( 𝑋𝑋 𝑇𝑇𝑇𝑇𝑇𝑇 = 0.1149 ). During cool ing,
diffusive 𝐽𝐽 𝑖𝑖 and convective contributions to the flux ar e present
(step s 1 and 2). Measureme nts started after a t emporal delay that
ensured that contribut ions from c onvective pr ocesses ha d subsided
(step 3 ).

Modelin g the mixin g process i n a n NMR tube
The assumption o f constant volu me is not valid for the binary mixtur e
TEA/H 2 O, whi ch e xhibits a sign ificant ex cess v olume 𝑣𝑣 𝑇𝑇 ( 𝑇𝑇 , 𝑝𝑝 , 𝑥𝑥 1 ) at
283.15 K 38 and 278.15 K 39 . An appropri ate mass transfer model was
proposed by B ardow et al. 23 employing a coo rdinate transf ormation
proposed in Ref. 40 . The continui ty equation is solved i n a molar
reference fra me because th e total number of mole s is constant
during the diff usion proc ess. However , this referenc e frame is
defined using a t ransformed concent ration 𝜉𝜉 𝑖𝑖 𝑀𝑀 = 𝑥𝑥 𝑖𝑖 𝑣𝑣 𝑛𝑛
0
⁄ , where 𝑣𝑣 𝑛𝑛
0 is
the molar volume of a pur e reference component. In t his new
reference frame, t he continuity equation of component 1 in the
binary mixture can be r ecast as
𝜕𝜕 𝜉𝜉 1
𝑀𝑀
𝜕𝜕𝑡𝑡 = 𝜕𝜕
𝜕𝜕 𝑧𝑧  � 𝐷𝐷 12 ( 𝑐𝑐 𝑖𝑖 𝑣𝑣 1
0 ) 2 𝜕𝜕 𝜉𝜉 1
𝑀𝑀
𝜕𝜕 𝑧𝑧  � .

(3)
Therei n, 𝑣𝑣 1
0 ( 𝑇𝑇 , 𝑝𝑝 ) is the molar vo lume of TEA , 𝐷𝐷 12 th e Fick d iffu sion
coefficie nt, 𝑧𝑧 the laboratory framewor k coordinate and 𝑧𝑧 represents
a fiction al dista nce w hich is re late d to z throug h 23
𝑧𝑧 = ∫ 𝑐𝑐 𝑖𝑖 ( 𝑡𝑡 , 𝑧𝑧′ ) 𝑣𝑣 𝑛𝑛
0 𝑑𝑑𝑧𝑧′
𝑧𝑧
0

. (4)
Furthermore, the t otal concentratio n is given by 1 𝑐𝑐 𝑖𝑖
⁄ =
∑ 𝑥𝑥 𝑖𝑖 𝑣𝑣 𝑖𝑖 0 ( 𝑇𝑇 , 𝑝𝑝 ) + 𝑣𝑣 𝑇𝑇 ( 𝑇𝑇 , 𝑝𝑝 , 𝑥𝑥 1 )
𝑖𝑖 . For the excess volume, a Redlich - Kister
type correlat ion was fitted to experiment al data
𝑣𝑣 𝑇𝑇 ( 𝑇𝑇 , 𝑝𝑝 , 𝑥𝑥 1 ) = 𝑥𝑥 1 ( 1 − 𝑥𝑥 1 ) [ 𝐴𝐴 + 𝐵𝐵 ( 2 𝑥𝑥 1 − 1 ) +
𝐶𝐶 ( 2 𝑥𝑥 1 − 1 ) 2 + 𝐷𝐷 ( 2 𝑥𝑥 1 − 1 ) 3 ]

.
(5)
w hich excellently descr ibes the experimental l iterature data 38 39 , cf.
Fig. 3C , D . Coefficients f or both targ et tempe ratures are re ported in
Table 1S of the supplem entary material . The effect of D 2 O as a third
component on the excess pro perties has been well studied and is
expected to be small for compositi ons where its molar frac tion is
below 0 .9 99 41 .
Results and discus sion
To illustrat e how the present approach may be used to access the
composition dependenc e of the Fick diffusio n coefficient of a bin ary
mixture, TEA/ H 2 O systems wer e prepared as described abov e. A
given sample was c ooled down below t he temperatur e of mixing
and , after a temporal delay that allowed t he convective flux to
vanish, the conc entration profil e along the z coo rdinate was
measured over time. Fig. 3 show s the first sampled concentr ation
profile that was use d as initial condit ion for TEA in the new
coordinate s ystem. A strategy to s olve the resulting PDE is ou tlined
in the f ollowing .
Initial condition
Although the mixi ng process occurs inside the N MR magnet, it is not
possible to start the measure ment immediately , and a temporal
delay is necessary to ensure t hat contributions from convect ive
processes ha ve termi nated, i.e. , the temperature had to att ain a
constant value. Here, this c ondition was chosen t o init iate the first
mapping at 𝑡𝑡 𝑜𝑜𝑜𝑜𝑜𝑜 , cf. F igure 2 B. This issue is not s pecific to t he present
method and the effects of such a choice have been discuss ed for
other experimental techniques 42 . An error func tion was used t o fit
the initial condition in the new referenc e frame at 𝑡𝑡 = 𝑡𝑡 𝑜𝑜𝑜𝑜𝑜𝑜 .
𝜉𝜉 ( 𝑧𝑧 , 𝑡𝑡 = 𝑡𝑡 𝑜𝑜𝑜𝑜𝑜𝑜 ) = 𝑈𝑈 + 𝑉𝑉 erf [ 𝑌𝑌 ( 𝑧𝑧 + 𝑊𝑊 ) ] .

(6)
Therein, 𝑼𝑼 , 𝑽𝑽 , 𝒀𝒀 , and 𝑾𝑾 are fitting p arameters as re ported in Ta b le
1S of the supplementary mat erial . Because the observation window
was limited by the lengt h of the act ive volume of t he coil, the
concentrati on profile of the mixture beyon d these limit s was
extrapolated. Fig. 3 shows th e initial condit ion in the new reference
frame for both target temperat ures. It allow s for the defini tion of
boundary conditions that are required to sol ve Eq. ( 3 ).

Figure 3. Initial con dition 𝜉𝜉 ( 𝑧𝑧 , 0 ) for solving Eq. ( 3 ) for te mperatures
278.2 K and 283.2 K as wel l as excess vo lume 𝑣𝑣 𝑇𝑇 ( 𝑇𝑇 , 𝑝𝑝 , 𝑥𝑥 1 ) . The initial
condition was built using the ficti onal distanc e 𝑧𝑧 obtained by means
of Eq. ( 6 ) and a new composit ion variable calculated using mo lar
volume 𝑣𝑣 1
0 ( 𝑇𝑇 , 𝑝𝑝 ) for TEA. The exp erimental dat a (A) and (B) can be
approximated wi th the error func tion. The parameters cal culated by
least squares ar e presented for each temperat ure. Note that the
separation between the points i s not regularly spaced becaus e the
fictiona l distanc e is prop ortional to 𝑐𝑐 𝑖𝑖 at specif ic loca tions. The f our
parameter Redlic h- Kister model was fitted to experimental excess
volume data (C) and (D) from t he literature 38 39 . Negati ve excess
volume values reflect c on traction of the liquid d uring mixing.
Numerical solution
The PDE , Eq. (3), was solved numerically for TEA/H 2 O with the
method of the lines 43 as implemented in the Ma thematica softwar e.
The fict ional distanc e 𝑧𝑧  was chosen as the variable to be discretized.
The composit ion dependence of Fick diffusi on coeffic ient ca n b e
represented by a pol ynomial 44
𝐷𝐷 12 ( 𝑇𝑇 , 𝑝𝑝 , 𝑥𝑥 1 ) = � 𝜈𝜈 𝑘𝑘 𝑥𝑥 1

𝑘𝑘− 1
𝑁𝑁
𝑘𝑘= 1 .

(7)

Therein, x 1 is the molar fractio n of component 1 and 𝜈𝜈 𝑘𝑘 are f itting
parameters. In c ase of the mixt ure Toluene/Cyclohexane, a
combination of low order pol ynomials was sufficient to describe the
Fick diff usion coef ficient in the e ntire com posit ion range 45 . Here , t he
following functionality w as deduced by adjustment to present
experimental data
𝐷𝐷 12 ( 𝑇𝑇 , 𝑝𝑝 , 𝑥𝑥 1 ) = ν 4 𝑥𝑥 1
3 + ν 2 ( 𝑥𝑥 1 − 𝑥𝑥 1
2 ) .

(8)
This expressio n was tra nsformed in t erms of the new composi tion
variable 𝜉𝜉 and insert ed into Eq. (3) . Once the PD E was sol ved in th e
new coordinate space ( 𝜉𝜉 , 𝑧𝑧 ), the results were converted ba ck to the
laborato ry reference fr ame by rever sing the exp ression 𝜉𝜉 𝑖𝑖 𝑀𝑀 = 𝑥𝑥 𝑖𝑖 𝑣𝑣 𝑛𝑛
0
⁄
𝑧𝑧 = 1

𝑣𝑣 1
0 ( 𝑇𝑇 , 𝑝𝑝 )
� [ 𝑥𝑥

1
𝑣𝑣

1
0
( 𝑇𝑇 , 𝑝𝑝 ) + ( 1 − 𝑥𝑥

1
) 𝑣𝑣

2
0
( 𝑇𝑇 , 𝑝𝑝 )
𝑧𝑧 

0
+ 𝑣𝑣 𝑇𝑇 ( 𝑇𝑇 , 𝑝𝑝 , 𝑥𝑥 1 ) ] 𝑑𝑑 𝑧𝑧  .

(9)
Solutions for spec ific combinations of th e coeffici ents 𝜈𝜈 2 and 𝜈𝜈 4 were
parametrized to maxi mize the correl ation coefficient 𝑅𝑅 2 . The resu lts
obtained for the target temper atures 278.15 K and 283.15 K are
shown in Fig. 4 and report ed numericall y in Table 2S of th e
supplementary material .
Compa rison to othe r data
Predic tive models often assu me a relati onship between the
propagation of mol ecular species quanti fied by the intra - di ffusion
coeffici ents 𝐷𝐷 𝑖𝑖 ∗ and the Fick diffusion coeffi cient 𝐷𝐷 12 . The Darken
equation 46 is particular ly stra igh t forward
𝐷𝐷 12 = ( 𝑥𝑥 2 𝐷𝐷 1
∗ + 𝑥𝑥 1 𝐷𝐷 2
∗ )

. (10 )
This expression can be meani ngful if the involved components
exhibit similar intermol ecular interactions between like and unlik e
species, as i s the case for some metal al loys 46 or ideal mi xtures 47 .
However, once the behavior o f mixtures is asso ciated with no n -
ideality, it is necessary to c onsider the thermodynamic facto r [ 1 +
𝜕𝜕 ln γ 1 𝜕𝜕 𝑥𝑥 1
⁄ ] 48
𝐷𝐷 12 = ( 𝑥𝑥 2 𝐷𝐷 1
∗ + 𝑥𝑥 1 𝐷𝐷 2
∗ ) � 1 + 𝜕𝜕 ln γ 1
𝜕𝜕 𝑥𝑥 1 � .

(11)
The thermodynamic factor is usually extr acted from vapor- liquid
equilibrium (VLE) data. D' Agostino et al. 28 calculated this factor for
the TEA/H 2 O mixture using experimental VLE dat a reported by
Counsell 49 . Modifi cations to Eq. ( 11 ) wer e recently present ed for
binary mixtures that have a consulate p oint, such a s the
Hexane/Nitrobenzene 50 or for non i deal mixtures 51 . Given i ts success
in pred icting 𝐷𝐷 12 over a wide compositi on range, a similar
modification was propos ed for TE A/H 2 O in the vicinity of its
consulate point 28
𝐷𝐷 12 = ( 𝑥𝑥 2 𝐷𝐷 1
∗ + 2 𝑥𝑥 1 𝐷𝐷 2
∗ ) � 1 + 𝜕𝜕 ln γ 1
𝜕𝜕 𝑥𝑥 1 � 𝛼𝛼 .

(12)
Theoretical c onsiderations have been present ed to motivate the
exponent 𝛼𝛼 52 . However, it can also be seen as a fitting paramet er to
alleviate shortc omings of the Dar ken equation whic h is more closely
related to th e Maxwell - Stefan diffusion coeffic ient.
The present result s obtained by solving the PDE are co ntrasted in Fig.
5 with numerical models and experiment al literature data 29 30 . For
the sake of compari son, we applied our methodolo gy at
temperatures for w hich Fick diffusion coeffic ient data for TEA/H 2 O
are available in the li terature.

Figure 4. Expe rimental data measured at 278.2 K (A) and 283.2 K (C)
are contrasted with the numeric al solution of the P DE for each c ase.
The spatial variable was di scretized by 500 points between 1.4 cm
and 2.8 cm. Par ameters 𝜈𝜈 2 an d 𝜈𝜈 4 were fitte d to maximize th e
correlat ion coe fficien t 𝑅𝑅 2 (B and D).
Both predictive mod els achieve a better match wi th the
experimental liter ature data tha n the present ap proach, in part icular
for low TEA mola r fractions ( 𝑥𝑥 1 < 0.2 mol/mol ). However, when the
entire co mposition r ange is consi dered, the present r esults a re
equall y consistent wi th the experimental literature data. For a fair
assessment , the follo wing issues have to be consider ed:
1. The predictive m odels are more a ccurate, but they r equire
a large amount of experimental tr ansport data as an input
because they rest on intra - diffusion c oefficient s of both
components over the enti re composition r ange.
2. The predictive models also rely o n the thermodynamic
factor that may great ly vary depending on the ch oice of the
activity coeff icien t mode l 53 and requires informatio n on
the VLE.
I t must be returned to t he central i dea of the method first propose d
by Gupta and Cooper 17 t o understand the discrepanci es obs erved f or
diluted states, cf . Supplementary material . Instead of carrying out
several exp eriments vary ing the m olar fra ction, i ts tim e evolu tion is
followed by a single experiment. In this approac h, data about low (or
high) TEA molar fractions may on ly be captured d uring the very sharp
gradients present immediately after the mixing occurs ( or after
infinite time). Because of conv ective phenomena pr esent at the
beginning of the mixing process it is not p ossib le to rec ord a n
infinitely sharp gradient, nor it is possible to continue the experiment
until any gradient at all. . For instance , our experiment s tarts from a
composition dictat ed by the LLE of t he system ( 𝑥𝑥 1 = 0.75 mol/mol)
and stops befo re perfect equili brium is attained . Given t hese
condition s , som e degree of uncertainty i s unavoidable in dilu te
regimes. This pr oblem has already been observed in the
impleme ntation of an incremental model for Ethyl
acetate/Cyclohexane 26 and its explanatio n is consistent with our

results; devi ations obs erved bet ween the tw o repeti tions are l arger
for molar frac tions near infinite d ilution. The use of an a priori model
when solving t he PDE has been discussed elsew here 19 20 . Ideally, one
would like to dis cover an appropriate model from the experime ntal
data only .

Figure 5. Fick diffusion coefficient o btained by solving the PD E for
TEA/H 2 O in com paris on to expe rimental l iteratu re data by Dudley
and Tyrell at 278.15 K 29 A) and 283.15 K 30 B) , interpolat ing the data
for the latter. The predict ive models of Darken and D'Agostino et al.
were applied by using int ra - diffusion co efficients and
thermodynamic facto r data report ed in Ref . 28 . Th e green lines
represe nt t he Fick diffusion coefficient obtained by NMR. The da rk
and light green lines represent t he first and second re plicate,
following the order in Tabl e 1.
Indeed, such an approach has been successful ly applied in recent
work using Raman spectroscopy 54 . It is assumed that the same
method, translated to o ur work, may deli ver appropriat e models and
accurate diffusion c oefficients, provided that it is support ed by a
suitable experimental design. For instance, a b etter understandi ng of
the imp act of t he in itial coo lin g perio d is cru cial b ecaus e it af fects
several factors, such as the posi tion of the center and s hape of the
first measured gradient. Further experiments in this di rection are
currently under developme nt.
Conclus ions
A complementary strat egy is presented for the deter mination of the
Fick diffusion coeffici ent in a wide composition r ange for non- ideal
systems. This a pproach is nev ertheless limit ed to systems t hat
poss ess an LCST or an upper critical sol ution temperature (UC ST) , as
a mean to establish a reproducible gradient inside of an NMR tube .
Even with th e strong res trictions impos ed by the NMR tube, d ifferent
experimental design s may , in the future, extend this method to other
systems without c onsolute point . T he formulated PDE for the binary
mixture TEA/Water was sol ved numerically, adopting a reduced
functionality to estimate the binar y Fick diffusion co efficient 𝐷𝐷 12
through fitting to NMR data. The results a re consistent with
experimental literature data measured with other techniques.
Predic tive models de veloped f or the TEA/H 2 O system were als o
compared with the funct ionality obtained for the Fick diffusion
coefficie nt.
Despite the r estric tions discussed above, the reducti on in
experimental effort is notable, which may co mpensate for a larger
uncertainty in di lute regimes. Con sider ing the spectr oscopic ben efits
to be comparable to those provided by Raman, the NMR setup allows
for the mea surement at differ ent temperatu res out -of- the - box ,
which is e ssent ial for a broader study of mas s tran sport phenomena.
Conflict s of interest
Th ere are no c onflicts to d eclare .
Acknowl edgements
The Paderborn chair gratefully acknowledges financial support by
Deutsche Forschungsgemein schaft (grant VR6/1 1).

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