Simulation and theory of fluid demixing and interfacial tension of mixtures of colloids and non-ideal polymers

0.24 MB·English

by  R. L. C. Vink

Simulation and theory of fluid demixing and interfacial tension of mixtures of colloids
and non-ideal polymers
R. L. C. Vink
Institut fu¨r Physik, Johannes-Gutenberg-Universit¨at, Staudinger Weg 7, D-55099 Mainz, Germany.
Matthias Schmidt∗
Soft Condensed Matter, Debye Institute, Utrecht University,
5 Princetonplein 5, 3584 CC Utrecht, The Netherlands.
0 (Dated: 27 December 2004)
0
2 An extension of theAsakura-Oosawa-Vrijmodel of hard spherecolloids and non-adsorbingpoly-
mers, that takes polymer non-ideality into account through a repulsive stepfunction pair potential
n
between polymers, is studied with grand canonical MonteCarlo simulations and density functional
a
J theory. Simulationresultsvalidateprevioustheoreticalfindingsfortheshiftofthebulkfluiddemix-
ingbinodaluponincreasingstrengthofpolymer-polymerrepulsion,promotingthetendencytomix.
4
For increasing strength of the polymer-polymer repulsion, simulation and theory consistently pre-

dicttheinterfacial tensionofthefreecolloidal liquid-gasinterfacetodecreasesignificantlyforfixed
]
t colloiddensitydifferenceinthecoexistingphases,andtoincreaseforfixedpolymerreservoirpacking
f
o fraction.
s
. PACSnumbers: 82.70.Dd,61.20.Ja,64.75.+g,61.20.Gy
t
a
m
I. INTRODUCTION potential between polymers in Ref. 22. The colloid-
-
d polymer interaction was kept as that of hard spheres.
n Thisintroducesoneadditionalmodelparameter,namely
Various different levels of description have been em-
o the ratio of stepheight andthermal energy,interpolating
c ployed in order to study mixtures of colloidal parti- between the AOV case when this quantity vanishes, and
[ cles and non-adsorbing globular polymers [1, 2]. While
theadditivebinaryhardspheremixturewhenitbecomes
colloid-colloid interactions are reasonably well described
1 infinite. Following the hard sphere [23] and AOV cases
v by those of hard spheres, the effective interactions be- [12, 13], the well-defined particle shapes of this model
7 tweencolloids and polymers,as wellas that between the allowed to obtain a geometry-based density functional
3 centers of two polymers, are more delicate. The model
theory (DFT) [24, 25] specifically tailoredfor this model
0 due to Asakura and Oosawa[3] and Vrij [4] (AOV), tak-
[22]. Thetheorycan,inprinciple,beappliedtoarbitrary
1
ing the colloids to be hard spheres and the polymer-
0 inhomogeneous situations, which constitutes an a poste-
polymerinteractionstovanish,isarguablythemostsim-
5 riorijustificationforusingthesemodelinteractions. The
ple description of real colloid-polymer mixtures. Its ap-
0 trendsfoundforthefluid-fluiddemixingtransitionintoa
/ pealclearlyliesinitssimplicity,ratherthaninveryclose colloidalliquidphase (thatis richin colloidsandpoorin
t
a resemblanceofexperimentalcolloid-polymersystems,en- polymers) and a colloidal gas phase (that is poor in col-
m abling detailed simulation [5, 6, 7, 8, 9] and theoretical
loidsandrichinpolymers),uponincreasingthepolymer-
[6,10,11,12,13]studiesofbulk[5,6,7,8]andinterfacial
- polymerinteractionstrength,demonstratedimprovedac-
d [9, 14, 15, 16, 17, 18, 19] properties. More realistic ef-
cordance with the experimental findings of Ref. 26, as
n fectiveinteractionsbetweentheconstituentparticlescan
comparedto the AOV case [17, 22], where the DFT pre-
o
be systematically obtained by starting at the polymer
c dicts the same phase diagram as the free volume theory
segment level, and integrating out the microscopic de-
: [11]. Furthermore the extended AOV model was used in
v grees of freedom of the polymers [8, 20]. The resulting
thestudyofthe“floatingliquidphase”thatwasfoundin
i
X colloid-polymer interaction is a smoothly varying func- sedimentation-diffusion equilibrium [27]. Further discus-
tion of distance, and excluded volume between polymer
r sionofthemodelanditsrelationtotheAOVprescription
a segments leads to a soft Gaussian-like polymer-polymer
is given in Ref. 22.
repulsion. Such effective interactions have been used to
calculate bulk phase behavior [8, 21]. Theaimofthepresentpaperistwofold. First,wewant
to assessin detail the accuracyof the DFT of Ref. 22 by
In order to retain most of the simplicity of the AOV
comparing to results from direct simulations of the ex-
model,butstilltocapturepolymernon-ideality,theAOV
tended AOV model. While phase-coexistence is often
model was extended using a repulsive stepfunction pair
studied in the Gibbs ensemble [28], we instead choose
to take advantage of recently developed methods [7, 29]
that rely on the grand canonical ensemble. Benefits are
∗Present address: Institut fu¨r Theoretische Physik II, Heinrich- accurate estimates of the interfacial tension [7, 30] and
access to the critical region [31]. The binodal we ob-
Heine-Universita¨t Du¨sseldorf, Universita¨tsstraße 1, D-40225
Du¨sseldorf,Germany. tain in the simulations is compared to that from DFT,
2
and good agreement between both is found. This result center distance r between two particles, given as
strongly supports the original claim [22] that a straight-
forward perturbation theory, taking the AOV system as ∞ r <σ
c
the reference state and adding polymer-polymer interac- Vcc(r) = (1)
(0 otherwise,
tions in a mean-field like manner, will fail, as the bulk
fluid-fluid binodal from this approach differs markedly ∞ r <(σ +σ )/2
c p
V (r) = (2)
fromthatofthegeometry-basedDFT,asdiscussedinde- cp
(0 otherwise,
tailinRef.22. Instead,eventolowestorderinstrengthof
the polymer-polymer interaction, a non-trivial coupling ǫ r <σ
p
V (r) = (3)
to the colloid density field needs to be taken into ac- pp
(0 otherwise.
count; the DFT of Ref. 22 does this intrinsically. Our
second aim is to study the interfacial tension between
Particle numbers are denoted by N , and as bulk ther-
i
demixed colloidal gas and liquid phases, which is known
modynamicparametersweusethepackingfractionsη =
i
from experiments [32, 33, 34, 35], theory [14, 16, 17, 36]
πσ3N /(6V),whereV isthesystemvolume. Asanalter-
and simulation [7, 37] to be orders of magnitude smaller i i
native to η , we use the packing fraction ηr = πσ3ρr/6
than that of atomic substances. While much work has p p p p
in a reservoirof pure polymers, interacting via V (r) as
pp
been devoted to the case of non-interacting polymers
givenabove,thatisinchemicalequilibriumwiththesys-
[7, 14, 16, 17, 18, 19, 36, 37], only quite recent studies
tem, with ρr being the polymer number density in the
addressed the effect of polymer non-ideality [38, 39, 40]. p
reservoir.
Aarts et al. [38] use a square gradient approach based
The model is characterized by two dimensionless con-
on the free energy of a free volume theory, to include
trolparameters,namely the polymer-to-colloidsize ratio
polymer interactions [41], and the mean spherical ap-
q = σ /σ , and the scaled strength of polymer-polymer
proximationforthedirectcorrelationfunction. Theyfind p c
repulsion βǫ, with β =1/(k T), k the Boltzmann con-
thegas-liquidinterfacialtensiontodecreaseascompared B B
stant,andT theabsolutetemperature. Aslimitingcases,
to the case of non-interacting polymers, when plotted
the present model possesses the AOV model for βǫ = 0,
as a function of the density difference in the coexisting
wherepolymer-polymerinteractionsareideal,andthebi-
phases. Similar findings were reportedby Moncho-Jord´a
nary additive hard sphere mixture in the limit βǫ →∞.
et al.[39],whoalsousedasquaregradientapproach,but
One can use the parameterβǫ to match to a realsystem
based on an effective colloid-colloid depletion potential
at a given thermodynamic statepoint, polymer type and
that reproduces simulation results accurately [42]. Our
solvent, by imposing equality of the second (polymer-
present study goes beyond Refs. 38, 39, and also beyond
polymer) virial coefficients. See Ref. 22 for an in-depth
the veryrecentRef.40, asweemployanon-perturbative
discussion of this procedure.
DFT treatingthe full two-componentmixture ofcolloids
andinteractingpolymers. Wecompareresultsforthein-
terfacialtensiontoourdatafromdirectsimulationofthe
III. METHODS
binary mixture. Besides its intrinsic interest, the inter-
facial tension is moreover relevant for the occurrence of
capillarycondensationinconfinedsystems[43,44,45]as A. Density functional theory
is apparent from a treatment based on the Kelvin equa-
tion [46]. To investigate bulk and interfacial properties of the
present model, we use the geometrically-based DFT of
The paper is organizedas follows. In Sec. II we define
Ref.22thathasitsrootsingeneralizationsofRosenfeld’s
the extended AOV model taking into account polymer-
fundamental-measure theory for additive hard sphere
polymer repulsion. In Sec. III we briefly sketch the the-
mixtures [23], namely the treatment of the “penetrable
oreticaland simulationtechniques. InSec. IV results for
sphere model” [47] (equivalent to the present polymer
fluid-fluid phase behavior and the interfacial tension are
particles)andtheDFTgenuinelydevelopedfortheAOV
presented, and we conclude in Sec. V.
model [12]. The minimization of the grand potential is
carried out with a simple iteration technique. We refer
the reader directly to Ref. 22 for all details about the
(approximate) Helmholtz free energy functional.
II. MODEL
B. Simulation method
We consider a mixture of hard sphere colloids (species The simulations are carried out in the grand canoni-
c) of diameter σ , and effective polymer spheres (species cal ensemble, where the fugacities z and z , of colloids
c c p
p) of diameter σ , that interact via pairwise potentials and polymers, respectively, and the total volume V are
p
V (r) with (i,j) ∈ (c,p), as function of the center-to- fixed, while the numbers of particles, N and N , are
ij c p
3
allowedtofluctuate. Weusearectangularboxofdimen- Close to the critical point the simulation moves back
sionsL×L×D,withperiodicboundaryconditionsinall andfortheasilybetweenthe gasandliquidphases,while
threedirections,andsimulatethe full mixtureasdefined further away from the critical point, i.e. at higher poly-
by the pair potentials given in Eqs. (1)-(3), i.e. the posi- mer fugacity, the free energy barrier between the two
tional degrees of freedom of both colloids and polymers phasesincreases. Hencetransitionsfromonetotheother
are explicitly taken into account. Note that asymmetric phasebecomelesslikely,andthesimulationspendsmost
binary mixtures are in general difficult to simulate, and of the time in only one of the two phases. A crucial
proneto long equilibrationtimes. To alleviatethis prob- ingredient in our simulation is therefore the use of a
lem,werelyonarecentlydevelopedclustermove[7,48], biased sampling technique. We employ successive um-
that has already been applied successfully to the stan- brellasampling,aswasrecentlydevelopedbyVirnauand
dard AOV model [7, 31, 48, 49]. Here we perform the Mu¨ller [29], to enable accurate sampling in regions of η
c
generalization to βǫ>0 which is straightforward. where P(η ), due to the free energy barrier separating
c
During the simulation, we measure the probability the phases, is very small. In this approach, states (or
P(η ) of finding a certain colloid packing fraction η . windows) are sampled successively. In the first window,
c c
Atphasecoexistence,the distributionP(η )becomesbi- the number of colloids is allowed to vary between 0 and
c
modal,withtwopeaksofequalarea,onelocatedatsmall 1, in the second window between 1 and 2, and so forth.
values of η corresponding to the colloidal gas phase, The number of polymers is allowed to fluctuate freely in
c
and one located at high values of η corresponding to each window. Our sampling scheme is thus strictly one-
c
the colloidal liquid phase. Typical coexistence distri- dimensional: the bias is put on the colloid density only.
butions for the standard AOV model can be found in Note that this becomes problematic for very large sys-
Ref. 7, and our present results display similar behavior. tems because of droplet formation. An additional free
The equal area rule [50] implies that hηciP(η )dη = energy barrier, which grows with the size of the simu-
∞ 0 c c lation box, must be crossed before the transition from
P(η )dη , with hη i the average of the full distri-
hηci c c c R the droplet state, to the slab geometry occurs (which is
∞
Rbution hηci = 0 ηcP(ηc)dηc, where we assume that required if the interfacial tension is to be determined).
∞
P(ηc) has beenRnormalized to unity 0 P(ηc)dηc = 1. As pointed out in Refs. 51, 52, this additional barrier
The packing fraction of the colloidal gas ηg now fol- is not one in colloid density, but rather in the energy-
c
R
lows trivially from the average of P(ηc) in first peak like parameter, which for our system would be the poly-
ηg = 2 hηciη P(η )dη , and similarly for the colloidal mer density. Hence, for very large systems, a naive one-
c 0 c c c
liquid ηl = 2 ∞ η P(η )dη , where the factors of 2 are dimensional biasing scheme such as described above, is
cR hηci c c c pronetoseveresamplingdifficulties(moreappropriatein
a consequence of the normalization of P(η ).
R c thiscasewouldbeatwo-dimensionalsamplingschemein
The interfacial tension γ is extracted from the loga-
both the colloid and the polymer density). For the sys-
rithm of the probability distribution W ≡k T lnP(η ).
B c tem size used by us, however, no problems in obtaining
Note that −W correspondsto the free energy ofthe sys-
the slab geometry were encountered.
tem. Therefore, the height F of the peaks in W, mea-
L In this work, we simulate up to a colloid packing frac-
suredwithrespecttotheminimuminbetweenthepeaks,
tionofη ≈0.45,correspondingtoamaximumofaround
equalsthefreeenergybarrierseparatingthecolloidalgas c
1000 colloidal particles. The maximum number of poly-
from the colloidal liquid [30]. F may be related to the
L mers is obtained at low η . While the precise number
interfacial tension γ by noting that, at colloid packing c
depends on ηr and βǫ, a value of 3000 is typical. In
fractions between the peaks ηg ≪ η ≪ ηl, the system p
c c c each window, O(107) grand canonical cluster moves are
consists of a colloidal liquid in coexistence with its va-
attempted, of which O(105) are accepted at low η , and
por. A snapshot of the system in this regime, would c
O(103)athighη (thegrandcanonicalclustermovethus
reveal a so-called slab geometry, with one region dense c
becomeslessefficientwithincreasingcolloidpackingfrac-
in colloids, and one regionpoor in colloids, separated by
tion). The typical CPU time investment to obtain a sin-
an interface (because of periodic boundary conditions,
gle distribution P(η ) is 48 hours on a moderate com-
twosuchinterfacesareactuallypresent). Ifanelongated c
puter.
simulation box with D > L is used, rather than a cubic
box with D = L, the interfaces will be oriented perpen-
dicular to the elongated direction, since this minimizes
the interfacialarea,andhence the freeenergyofthe sys- IV. RESULTS
tem. The totalinterfacialareain the systemthus equals
2L2 and,followingRef.30,γ =F /(2L2). Anadditional InFig. 1 we plotresults for the bulk fluid-fluid demix-
L
advantage of using an elongated simulation box is that ing binodal as obtained from theory and simulation in
interactions between the interfaces are suppressed. This the (η ,η )-plane, i.e. the “system representation”. For
c p
enhances a flat region in W between the peaks, which the AOV case (recovered for βǫ = 0), the DFT predicts
is required for an accurate estimate of the interfacial the same bulk free energy for fluid states, and hence the
tension. In this work, an elongated box of dimensions same demixing binodal, as free volume theory [11]. The
D/σ =16.7 and L/σ =8.3 is used. result is known to compare overall reasonably well with
c c
4
simulationresultsforavarietyofsizeratiosq,butto de- sion with increasing βǫ is confirmed up to βǫ = 0.25,
viateclosetothe criticalpoint[5,7,8,9]. Forincreasing but not for βǫ = 0.5. The simulation data for the lat-
strength of the polymer-polymer repulsion, the theoret- ter case, however, should be treated with some care,
ical critical point shifts toward higher values of η , and as in grand canonical simulations, the quantity ηr does
c p
very slightly to lower values of η . The accompanying not play the role of a direct control parameter, which
p
shift of the binodal leads to a growth of the one-fluid ratherthepolymerfugacity,z ,does. Consequently,con-
p
region in the phase diagram, hence polymer-polymer re- version to the reservoir representation requires an addi-
pulsiontends to promotemixing. The simulationresults tional step, namely the determination of ηr as a func-
p
indicatethesametrend,butshowaquantitativelylarger tion of z . With the exception of βǫ = 0, in which case
p
shift of the binodal toward higher values of η upon in- ηr = πσ3z /6 holds trivially, this conversion introduces
c p p p
creasing βǫ. Note also the pronounced finite-size devia- an additional statistical uncertainty. Fig. 2 shows that,
tions of the simulation data in the vicinity of the critical for βǫ = 0.5, the binodal has become very flat, so even
point. Asaresult,theslightdecreaseinthecriticalvalue small uncertainties in ηr imply rather large uncertain-
p
ofη withincreasingβǫ,aspredictedbyDFT,cannotbe ties in ∆η . Hence, for very flat binodals, the reservoir
p c
identified in the simulation data. To better access the representationis not convenientin simulations (more re-
critical region, a finite size scaling analysis [53, 54, 55] liableinthiscaseisthesystemrepresentation,seeFig.3,
wouldbe required. While suchaninvestigationhasbeen which, as a benefit, is experimentally more relevant). A
carried out for the standard AOV model [31], it would second point is that the Monte Carlo cluster move be-
requireextensiveadditionalsimulationsforthe extended comeslessefficientwithincreasingη andβǫ (seeRef.7)
c
model, which is beyond the scope of the present work. andthis willalsoadverselyaffectthe data(especiallyfor
When plotted in the (ηc,ηpr)-plane or “reservoirrepre- βǫ = 0.5, since then both βǫ and ηcl are substantial).
sentation”, the theoretical results display a similar shift Therefore, we conclude that the shift of the simulation
of the binodal toward larger values of η , and consid- data for βǫ = 0.5 in Fig. 4 most likely reflects a simula-
c
erable broadening of the coexistence region, see Fig. 2. tion artifact.
Thisimpliesthatatagivenvalueofηr abovethecritical
p
point, increasingthe polymer-polymerrepulsionleads to
alargerdensitydifferencebetweenthecoexistingphases. V. CONCLUSIONS
The broadening of the coexistence region is strikingly
confirmed by the simulation data. We again observe an In conclusion, we have investigated the effect of
overall shift of the binodals toward higher values of ηc. polymer-polymer repulsion on the fluid-fluid demixing
InaccordancewithfindingsforthestandardAOVmodel phase behavior and on the (colloidal) liquid-gas inter-
[5, 7, 8, 9], the critical value of ηr obtained by DFT un- faceofamodelcolloid-polymermixture. Wehaveuseda
p
derestimates the simulation value. This is related to the simplisticpairpotentialbetweenpolymers,givenbyare-
universalityclassofthe AOVmodel,whichisthatofthe pulsivestepfunction,toextendthestandardAOVmodel
3D Ising model [31], and hence, close to criticality, devi- tocasesofinteractingpolymers. GrandcanonicalMonte
ations froma mean-fieldtreatment like the presentDFT Carlosimulationsofthefullmixturedemonstratetherea-
are to be expected. sonableaccuracyofthetheory,withatendency toquan-
In Fig. 3, we show results for the gas-liquid interfa- titatively underestimate the shifts in the bulk fluid-fluid
cial tension, γ, plotted as a function of the difference demixing binodal, and the gas-liquid interfacial tension,
between the colloid packing fraction in the coexisting upon increasing strength of the polymer-polymer repul-
phases, ∆η = ηl −ηg. Both DFT and simulation indi- sion. The present study demonstrates the usefulness of
c c c
catethattheinterfacialtensiondecreasesmarkedlyupon the modelassuch,asitclearlydisplayspreviouslyfound
increasing βǫ, in accordance with square-gradient treat- featuresdue to polymer non-ideality. This offers waysto
ments [38, 39]. Note that this change is not due to in- study further interesting inhomogeneous situations, like
terfacial contributions alone, but also to the pronounced the wetting properties at substrates. Such investigations
changesinthe locationofthe bulkdemixing binodaldis- could test the robustness of the results obtained for the
cussedabove. Thedecreasemightseemreasonablebased surface phase behavior at a hard wall using ideal poly-
on the profound increase in the density of the coexisting mers [9, 16, 17, 18, 19].
liquid upon increasing βǫ, see again the phase diagram
in reservoir representation (Fig. 2). As is apparent from
Fig. 2, a given value of ∆η corresponds to a reduction Acknowledgments
c
of ηr at coexistence upon increasing βǫ. We believe that
p
this decrease induces the observed decrease in γ. We thank Ju¨rgen Horbach, Peter Virnau, Marcus
Alternatively, comparing the interfacial tension as a Mu¨ller,KurtBinder,MarjoleinDijkstraandAndreaFor-
function of ηr thus reveals an increase with increasing tini for many useful and inspiring discussions. The work
p
βǫ, see the upper panel of Fig. 4 which shows the DFT of MS is part of the research program of the Sticht-
result. The lower panel of Fig. 4 shows the correspond- ing voor Fundamenteel Onderzoek der Materie (FOM),
ing simulation data. The increase in the interfacial ten- which is financially supported by the Nederlandse Or-
5
ganisatievoorWetenschappelijk Onderzoek(NWO).Sup-
port is acknowledged by the SFB-TR6 “Physics of col-
loidal dispersions in external fields” of the Deutsche
Forschungsgemeinschaft (DFG). RLC also acknowledges
generous allocation of computer time on the JUMP at
the ForschungszentrumJu¨lich GmbH.
6
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7
1.4
(a) βε=0
1.2 0.1
0.25
1 0.5
0.8
p
η
0.6
0.4
0.2
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
η
c
1.4
(b) βε=0
1.2 0.1
0.25
1 0.5
0.8
p
η
0.6
0.4
0.2
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
η
c
FIG. 1: Bulk fluid-fluid demixing phase diagram of the ex-
tended AOV model as a function of colloid packing frac-
tion, ηc, and polymer packing fraction, ηp, for size ratio
q=0.8andincreasingstrengthofthepolymer-polymerrepul-
sion βǫ=0,0.1,0.25,0.5 as indicated. a) The binodal (lines)
andcriticalpoint(symbols) asobtainedfrom DFT.Thecase
βǫ=0 corresponds to the result from free volume theory for
theAOVmodel. b)Thebinodalasobtainedfromsimulations
(symbols indicate data points; lines are guides to theeye).
8
1.4
(a) βε=0
1.2 0.1
0.25
1 0.5
0.8
rp
η
0.6
0.4
0.2
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
η
c
1.4
(b) βε=0
1.2 0.1
0.25
1 0.5
0.8
rp
η
0.6
0.4
0.2
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
η
c
FIG. 2: The analogue of Fig. 1, but as a function of colloid
packing fraction, ηc, and polymer reservoir packing fraction
ηr in a reservoir of interacting polymers.
p
9
0.7
βε=0
0.6 0.1
0.25
0.5 0.5
2 0.4
σc
βγ 0.3
0.2
(a)
0.1
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
∆η
c
0.7
βε=0
0.6 0.1
0.25
0.5 0.5
2 0.4
σc
βγ 0.3
0.2
(b)
0.1
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
∆η
c
FIG. 3: Dimensionless interfacial tension βγσ2, of the free
c
colloidalgas-liquidinterface,asafunctionofthedifferencein
colloid packing fractions of the coexisting states, ∆ηc =ηcl −
ηg. Thesizeratioisq=0.8,andthestrengthofthepolymer-
c
polymerrepulsionequalsβǫ=0,0.1,0.25,0.5asindicated. a)
Results from DFT; b) results from simulation.
10
0.7
βε=0
0.6 0.1
0.25
0.5 0.5
2 0.4
σc
βγ 0.3
0.2
(a)
0.1
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
η r
p
0.7
βε=0
0.6 0.1
0.25
0.5 0.5
2 0.4
σc
βγ 0.3
0.2
(b)
0.1
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
η r
p
FIG. 4: The analogue of Fig. 3, but as a function of the
polymer reservoir packing fraction ηr. a) Results from DFT;
p
b) results from simulation.