Numerical simulation of thermally induced phase separation in polymerdispersed liquidcrystalsP. I. C. Teixeira and B. M. Mulder Citation: The Journal of Chemical Physics 105, 10145 (1996); doi: 10.1063/1.472842 View online: http://dx.doi.org/10.1063/1.472842 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/105/22?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Kinetic theory of phase separation induced by nonuniform photopolymerization J. Appl. Phys. 80, 3285 (1996); 10.1063/1.363237 Annealing and memory effects in polymerdispersed chiral liquid crystals J. Appl. Phys. 80, 2586 (1996); 10.1063/1.363173 Fine droplets of liquid crystals in a transparent polymer and their response to an electric field Appl. Phys. Lett. 69, 1044 (1996); 10.1063/1.116925 Electricfield controlled color effect in cholesteric liquid crystals and polymerdispersed cholesteric liquid crystals J. Appl. Phys. 80, 1907 (1996); 10.1063/1.362984 Theory of domain boundary effects in a phaseseparated mixture of polymer and liquid crystal J. Chem. Phys. 104, 2725 (1996); 10.1063/1.471011

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Numerical simulation of thermally induced phase separationin polymer-dispersed liquid crystals

P. I. C. Teixeira and B. M. MulderFOM Institute for Atomic and Molecular Physics, Kruislaan 407, NL-1098 SJ Amsterdam, The Netherlands

~Received 25 March 1996; accepted 10 July 1996!

We have developed a model of polymer-dispersed liquid crystal~PDLC! formation by thermallyinduced phase separation. Spinodal decomposition in the thermoplastic-LC mixture is modelled bythe cell dynamical systems method of Oono and Puri, suitably modified to describe a continuoustemperature quench. Numerical calculations performed on a two-dimensional system for acomposition of 30% LC170% thermoplastic reveal that the final morphology depends strongly onthe quench rate: If cooling is much faster than phase separation, then complete decomposition isprecluded, whereas for slower quenches we recover the usual LC-rich droplet pattern ofconstant-temperature studies. The droplet size distributions are quite broad, and the mean dropletsize at a given temperature decreases as a power of the quench rate, consistently with what isobserved in real PDLCs. ©1996 American Institute of Physics.@S0021-9606~96!50639-2#

I. INTRODUCTION

Dispersions of liquid-crystal rich droplets in a polymermatrix, or polymer-dispersed liquid crystals~PDLCs!, havearoused a great deal of interest recently.1,2 Their relevancestems not only from a multitude of possible technologicalapplications,3,4 but also from the fundamental questions theyraise in connection with the dynamics of phase separationand ordering, structure selection and growth, and the behav-ior of LCs in confined geometries.

The main advantage of PDLC devices is their simplicityof operation. By matching the refractive index of the poly-mer and the ordinary refractive index of the LC, PDLC filmscan be switched from a translucent ‘‘off’’ to a transparent‘‘on’’ state by application of an electric field.5 Unlike moreconventional LC displays, they require neither complex sur-face treatments, nor polarisers.4

Techniques for preparing PDLCs all involve the~macro-scopically! incomplete phase separation~‘‘microphase sepa-ration’’! of an initially homogeneous mixture~see Ref. 6 fora review!. In the simplest of these, known as thermally in-duced phase separation~TIPS!, the LC is dissolved in thepolymer ~thermoplastic! melt, followed by cooling. Eventu-ally the glass transition temperature of the polymer isreached, below which phase separation virtually ceases.TIPS is the method of choice for fundamental studies: It isreversible, cheap, and—unlike polymerization-induced phaseseparation~PIPS!—easily controllable~the relevant controlparameter being the temperature rather than the progress of achemical reaction!. Moreover, it produces no noxious fumes,which are one of the drawbacks of solvent-induced phaseseparation~SIPS!.7

Because the optical properties of these materials dependcrucially on their morphology,8–10 it is important to gain amore detailed understanding of the determining factorsthereof. Of particular relevance is the distribution of dropletsizes, which affects both switching times and threshold volt-ages. Experimentally, the mean droplet size can be con-trolled by varying the cooling rate~TIPS!, the intensity of uv

light ~PIPS, photopolymerisation!, the temperature or theconcentration of chemical accelerators~PIPS, thermal cur-ing!, or the rate of solvent evaporation~SIPS!.4 However,much less effort has been devoted to a unified theoreticaldescription of these effects. One line of approach consists inassuming a droplet structure as a starting point, and thenusing empirical arguments to extract information on PDLCproperties.11–14Alternatively, one may choose to concentrateon the phase separation aspects of the problem,15–19 as de-scribed by Cahn-Hilliard~CH! theory.20–23 To our knowl-edge, only three theoretical studies of spinodal decomposi-tion at non-constant temperature have been performed.24–26

The first of these24 was restricted to the linear regime, whilethe second, although allowing for heat exchange through theboundary, only considered the case of constant external tem-perature. Finally, Jin and co-workers have reported MonteCarlo simulations of both TIPS27 and PIPS,28 also includinggravity effects.29 They did not, however, implement continu-ous cooling.

Recently we proposed a phase-field model30 for the dy-namics of formation of PDLCs by PIPS31 that employs thecell dynamical systems~CDS! method of Oono andPuri.32–34In this paper we show how the same formalism canbe extended quite naturally to study TIPS. In Section II weintroduce our model, which is based upon a modification ofthe basic CDS algorithm to describe phase separation duringcontinuous cooling. In Section III results are presented forthe LC/thermoplastic concentration patterns corresponding todifferent cooling rates, as well as for the circularly averagedpolymer structure factor, LC droplet size distribution andmean droplet size, as a function of temperature. The depen-dence of the relevant length scale in the system on the rate ofcooling is also investigated, and our data are shown to be inqualitative agreement with observations on real PDLCs. Fi-nally, in Section IV we conclude by summarizing our results.

II. THEORY

Consider a two-component system consisting of a nem-atic LC and polymer~thermoplastic!. As in previous work,31

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we shall neglect the nematic character of the LC component.This should not affect our results substantially, since in thistype of theory ordering only occurs above a minimum~local!LC concentration, which is only achieved after a dropletstructure has already formed.

We need to model, first, phase separation; and second,the glass transition of the polymer. Start by assuming that theincompressibility condition is satisfied, viz.

cLC1c51, ~1!

wherecLC andc are the concentrations of LC and polymer,respectively. Note that this implies that there is only oneindependent variable; in what follows we shall usec. Weemphasize that Eq.~1! need not be true of a real pre-PDLCmixture, which in principle requires a more sophisticatedtreatment. One possibility would be to add ‘‘voids’’ as anadditional, non-interacting, component, the chemical poten-tial of which would be proportional to the pressure.36 As afirst approach, however, we restrict ourselves to the simplestcase.

Let us concentrate on phase separation dynamics first.Rather than using the conventional formulation in terms ofpartial differential equations, we opted for the computation-ally efficient cell dynamics systems~CDS! method of Oonoand Puri,32–34where space is assumed discrete from the out-set and divided into ‘‘cells.’’ Modelling then consists of twosteps:~i! modelling of each cell; and~ii ! connecting cells.The first step is essentially the calculation of the thermody-namic force that drives the order parameter within that cell tothe value pertaining to either of the phases in equilibrium,and away from the single unstable state. Here the need todefine a non-equilibrium free energy is circumvented by in-

troducing a map~i.e., an injection of the set of real numbers,where the order parameter takes values! with stable fixedpoints corresponding to the two coexisting phases, and anunstable fixed point corresponding to the unstable one-phasestate. Next, cells are connected to take cooperative interac-tions into account: The non-local driving force for order pa-rameter change in a given cell is taken to be proportional tothe difference between the value of the order parameter inthat cell and a suitably defined average of its values in neigh-boring cells. In addition, in the conserved-order-parametercase which concerns us here, a correction must be applied toensure conservation. For details of the method we refer thereader to the original publications.32–34

We consider a two-dimensional~2D! system for simplic-ity, since we do not expect any significantly new physics toemerge in 3D in this type of model.35 The evolution ofcabove the glass transition is given by

c~x,t11!5F @c~x,t !#2^^F @c~x,t !#2c~x,t !&&1j~x,t !,~2!

with

F @c~x,t !#5 f ~c~x,t !!1D@^^c~x!&&2c~x,t !#. ~3!

In Eq. ~3!, D is a constant which sets the time scale of phaseseparation, and * &&2* is the isotropized discrete ‘‘Laplac-ian’’ on the 2d square lattice,33

^^* ~x,t !&&51

6 (nn

* ~x,t !11

12(nnn

* ~x,t !, ~4!

wherenn and nnn denote nearest and next-nearest neigh-bors, respectively.37 In Eq. ~2!, j(x,t) is a noise term whichwe take to be of the form34

j~x,t !5e¹•h~x,t !.ehx~ t,x1h,y!2hx~ t,x,y!1hy~ t,x,y1h!2hy~ t,x,y!

h, ~5!

whereh is the lattice spacing, taken as the unit of length,e isthe noise amplitude, andh5(hx ,hy)(x,t) consists of tworandom numbers, uniformly distributed in the interval@0,1#, assigned at each timet to each lattice sitex. A noiseterm thus defined satisfies the fluctuation-dissipationrelation38

^j~x,t !j~x8,t8!&522e2¹2d~x2x8!d~ t2t8!. ~6!

For f (x) we choose a simple generalization of the tanhmap, with stable fixed pointsa andb, unstable fixed point(a1b)/2, viz.

f ~x!51

2 H a1b1~b2a!

3A tanhF tanh21S 1AD 2x2~a1b!

b2a G J , ~7!

with A51.3 as in previous work.32,33 To model continuouscooling, we now allow the fixed points to be time-dependent.In common with usual experimental practice, we shall as-sume that the rate of cooling is uniform, i.e., that the tem-perature is a linear function of time, viz.

T~ t !5Tbin~c0!2t/tcool, ~8!

whereTbin is the temperature at which a mixture of polymervolume fractionc0 crosses the binodal line, andtcool51/Q isthe inverse cooling rate. It is implicitly assumed that coolingis always slow enough that it is meaningful to define an‘‘instantaneous temperature,’’ i.e., that the system can be‘‘instantaneously’’ in thermal equilibrium. To make closercontact with the usual CH approach in terms of a quartic freeenergy, further assume that the phase diagram is parabolic in

10146 P. I. C. Teixeira and B. M. Mulder: Phase separation in polymer-dispersed liquid crystals

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the temperature and exhibits an upper critical solution tem-peratureTc ;

39 here we takeTc51 ~see Fig. 1!

a~ t !5 12 ~12A12T~ t ! !, ~9!

b~ t !512a~ t !, ~10!

which yields Tbin512(2c021)2. As t→tmax5tcoolTbin(c0), a→0 andb→1. So we start right on thecoexistence curve and gradually plunge into the two-phaseregion, which justifies the inclusion of the noise term in Eq.~2!: Any inhomogeneity introduced through the initial con-dition only, as in our previous work,31 would die out beforephase separation could start.

Below T5Tglass the system is in the glassy state, wherethe dynamics gets very slow, i.e., the time scale for phaseseparation changes. In general, the glass transition tempera-ture Tglass is determined by the previous history of the sys-tem, in particular by the cooling rate. However, in the ab-sence of more detailed information on the functionaldependence ofTglass~which appears to be weak in practice

7!,we took it to be a constant for a givenc0.

Our modelling of PDLCs now proceeds as follows: Acalculation is started by assigningc i , j (t50)5c0 to all lat-tice sites, where we have definedc i , j (t)[c(xi ,yj ,t), withxi5( i21)h, yj5( j21)h ( i51,2,...,N; j51,2,...,N!. Eqs.~2! and ~3! are then advanced in time up tot5tglass5INT( tcool(Tbin2Tglass)), where INT stands for ‘‘in-teger equivalent of.’’40 Thus for T,Tglass, c(x,t) is as-sumed to be completely frozen. This is the simplest possibleidealization of the glass transition.41 Note that there is noconsensus among specialists on whether the microphase-separated structure of real PDLCs is an equilibrium one, al-though samples are reported to remain stable for months oreven years~see Ref. 42 for a discussion of ageing in thesesystems!.

III. RESULTS

We have performed our calculations on a square latticeof sizeN2564364, with periodic boundary conditions, forc050.7, A51.3, D50.5, e50.01, and several sets of(Tcool,Tglass). For this compositionTbin50.84.c0 is chosenso as to be in the droplet-forming region of the phase dia-gram of a binary mixture atT50; it lies above the site per-colation threshold for the square lattice ('0.5927543! andcorresponds toC52c2150.4, whereC P (21,1) is themore usual order parameter of spinodal decomposition stud-ies. This is consistent with the experimental requirement thatLC concentrations should be below;53% volume, abovewhich an inverted phase of polymer droplets in a LC matrix~‘‘polymer ball’’ ! is found.12

As an illustration we present snapshots of the system atdifferent temperatures~5times!, each of which may be seenas ‘‘a’’ Tglass. In all contour plots, the depth of shading isproportional to the value ofc. Figure 2 shows a fast quench(tcool5102), Fig. 3 one that is twenty times as slow(tcool52 3 103), and Fig. 4 the slowest quench considered(tcool5104). The fastest quench produces no droplets; in ei-ther of the two slower ones, the dynamics visibly slowsdown belowT50.44. The final size of the droplets is clearlyan increasing function oftcool, i.e., the slower the quench,the bigger the droplets at a given temperature, in agreementwith the experimentally observed trend. This will be mademore quantitative below.

All results presented henceforth are averages over tenruns, corresponding to different initializations of the noiseterm. In order to perform a droplet size analysis we startedby ‘‘hardening’’ the pattern at timet by assigning to eachlattice site (i , j ), an ‘‘Ising spin variable’’ si , j such thatsi , j51 if c i , j,0.5, andsi , j50 otherwise. This allowed us toidentify LC droplets as clusters of 1’s. These clusters werethen labelled, and their size distributions found. In Fig. 5 weplot the time evolution of the mean droplet size, for a num-ber of quenches. The faster the quench, the later the systementers the droplet regime, characterized by a power-law de-pendence of the droplet size~quenches faster than thoseshown do not produce droplets at any temperature!. We havefitted the data for the two slowest quenches shown to a

FIG. 1. Phase diagram~binodal line! of the LC-polymer mixture intemperature-composition space (T2c); here the upper critical solution tem-peratureTc51. The arrow is the path followed by the system in allquenches considered in this paper (c050.7).

FIG. 2. Polymer concentration fortcool5102 ~the fastest quench considered!,at ~a! T50.64 and~b! T50. No droplet structure has had time to form.

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power law of the form Ata, with a50.9960.02~tcool57.53103!; and a51.0160.06 ~tcool5104!. ~Note thatby ‘‘cluster size’’ we mean a measure of the cluster area.!Unlike in our previous model of PIPS,31 there is quite con-siderable polydispersity, which seems to increase withtcool. This is clear from the droplet size frequency histo-grams of Figs. 6 (tcool523103) and 7 (tcool5104), forT50.34,T50.29,T50.44, andT50. ~See also Tables I andII.! Sizes are given as no. spins/droplet, and the class widthis 4. Except at very early times when most droplets are verysmall, no class ever contains more than;26% of all drop-lets. Such spread is a consequence of the broadening of therange of wavelengths receiving amplification owing to con-tinuous cooling.24 ~Note also that our study did not extend tolate enough times to probe the asymptotic regime, where asingle length scale is expected to be dominant, at least forinstantaneous quenches.44,45!

We also computed the 2d polymer structure factor, de-fined by

S~k,t !5K 1

N2 U(i , j

@c i , j~ t !2c0#eik•~ i , j !U2L , ~11!

where k5(2p/N)(me11ne2), m,n52(N/2)11,...,N/2,and the angular brackets denote averaging over thermalnoise. The circularly averaged structure factor is then

S~k,t !5(kS~k,t !Y (

k1, ~12!

with k52pn/N, n50,1,2,...,N, and the sum(k is over acircular shell defined byn2 1

2<ukuN/(2p)<n1 12. Figures 8

and 9 showS(k,t) vs k for tcool52 3 103 andtcool5104, re-spectively. At early times~hence for the whole duration of

FIG. 3. Polymer concentration fortcool52 3 103, at ~a! T50.34; ~b!T50.29; ~c! T50.24; and~d! T50.

FIG. 4. Polymer concentration fortcool5104 ~the slowest quench consid-ered!, at ~a! T50.34; ~b! T50.29; ~c! T50.14; and~d! T50.

FIG. 5. Log-log plot of the mean droplet size~no. sites/droplet! vst5t/tcool , starting at tcool51000, for a number of quenches. Triangles:tcool5103; filled left triangles: tcool51.53103; inverted triangles:tcool52.53103; filled circles: tcool543103; 3: tcool553103; crosses:tcool57.53103; stars:tcool5104. The solid lines are power-law fits to the lasttwo sets of data; see the text for details.

10148 P. I. C. Teixeira and B. M. Mulder: Phase separation in polymer-dispersed liquid crystals

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the faster quenches! the peaks increase in height ratherquickly, due to polymer-LC segregation, and slowly shift tosmaller values ofk as the characteristic length scale in-creases. Furthermore, the height of the peaks at constanttincreases with increasingtcool, as this is equivalent to goingto laterabsolutetimes.

Because of the glass transition leading to effectivelyarrested growth, our studies are restricted to the early/

intermediate stages of phase separation. This precludesan investigation of the asymptotic behavior of the character-istic length scale, or of the form of the structure factor,which are the focus of much work on the dynamics offirst-order phase transitions.44,45 It is, however, of interest tocharacterize the functional dependence of the droplet size onthe cooling rate. To this end we plot the mean droplet sizes~areas! as determined from the cluster analysis vstcool inFig. 10. At sufficiently low temperatures, there exists aregime where the mean droplet size scales astcool

b . Fittingthe large tcool data at the three lowest temperatures con-sidered yieldsb50.5360.04 ~for tcool>2500 atT50.04);and b50.5460.02 ~for tcool>2000 at T50). Note thatpower-law behavior is only obtained for sufficientlyslow quenches, where droplets have had time to form~cf.Fig. 5!.

An alternative measure of~linear! size is provided by thereciprocal of the first moment of the circularly averagedstructure factor, defined as

k1~ t !5(k50N kS~k,t !

(k50N S~k,t !

. ~13!

This is insensitive to whether a droplet structure has formedand so allows us to probe a wider range of quench rates.1/k1 at as a function oftcool is shown in Fig. 11: Again, thisexhibits a power-law dependence of the formtcool

c which isvery weak at the higher temperatures~at which no structurehas yet emerged!, while at later times it agrees with thatderived from Fig. 10; exponentsc for several temperaturesare collected in Table III.

The above result is at variance with Cahn and co-workers’ theoretical prediction of a crossover between expo-nents 1

6 ~slow quenches! and zero~fast quenches!.24 Note,however, that this prediction was made on the basis of thelinearized CH equation for a system with temperature-

FIG. 6. Droplet size frequency histograms for the faster quench (tcool523 103). From left to right:T50.14 andT50. The corresponding meandroplet sizes and standard deviations are given in Table I.

FIG. 7. Droplet size frequency histograms for the slower quench(tcool5104). From left to right:T50.34,T50.29,T50.14 andT50. Thecorresponding mean droplet sizes and standard deviations are given inTable II.

TABLE I. Mean droplet sizes and standard deviations fortcool52 3 103.

t Mean droplet size Standard deviation

0.65 7.1 5.60.70 11.5 8.10.75 16.8 12.20.80 24.6 15.90.84 29.5 17.5

TABLE II. Mean droplet sizes and standard deviations fortcool5104.

t Mean droplet size Standard deviation

0.50 18.4 11.70.55 32.6 25.60.60 49.0 32.60.65 54.9 35.00.70 59.3 40.30.75 63.7 42.90.80 65.8 46.90.84 69.7 52.4

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dependent mobility and undergoing an exponential, ratherthan linear, temperature quench. To check the sensitivity ofour results to the details of the cooling mechanism, we re-peated our calculations for an exponential quench, where in-stead of Eq.~8! we have

T~ t !5Tbine2t/tcool. ~14!

In Fig. 12 we plot 1/k1 at as a function oftcool in thiscase, forT50.64 ~a very early time! and T50.01 ~a rela-tively late time!. The data are well fitted by power lawswith exponentsc850.07860.002 ~T50.64! and c850.23860.004 (T50.01!, over the whole range oftcool considered.We find no evidence of deviation from single power-lawbehavior, even for the fastest quenches. Yet we have notallowed the mobility to be a function of the temperature as inRef. 24, and in addition the nature of our model restricts usto tcool>1. It may thus be the case that Cahnet al.’s predic-

tion does hold if either, or both, of these restrictions areremoved.

IV. CONCLUSIONS

We have developed what is perhaps the simplest modelof PDLC formation by TIPS. Our approach is based on theCDS method and is thus less computationally expensive thandirect integration of nonlinear partial differential equations~typically the computational cost was less than 0.3 s perwhole lattice update on a Silicon Graphics Indigo 3000workstation!. This is all the more important when thermalnoise is included, calculation of which is very time consum-

FIG. 8. Circularly-averaged polymer structure factor for the faster quench(tcool52 3 103) at ~a! T50.34 ~solid line! andT50.29 ~dashed line!; ~b!T50.14 ~dotted line! andT50 ~dot-dashed line!.

FIG. 9. Circularly-averaged polymer structure factor for the slower quench(tcool5104) at T50.34 ~solid line!; T50.29 ~dashed line!; T50.14 ~dottedline!; andT50 ~dot-dashed line!.

FIG. 10. Mean droplet size vstcool , the inverse quench rate~log-log plot!.Stars:T50; crosses:T50.043: T50.09; inverted triangles:T50.14; filledtriangles:T50.19; diamonds:T50.24; filled squares:T50.29; solid lines:power-law fits. See the text for details.

10150 P. I. C. Teixeira and B. M. Mulder: Phase separation in polymer-dispersed liquid crystals

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ing. Results obtained, in particular for the dependance of themean droplet size on the rate of cooling, are in accord withqualitative experimental trends.

We believe that our model captures most of the essentialphysics of the phenomena under study: in spite of the factthat it contains several oversimplifications, it does reproduce,qualitatively at least, the experimentally observed behaviors.Some possible improvements would be:

~1! Inclusion of the nematic order parameter is obviouslyessential if we want to model the optical properties of PDLCmaterials. Moreover, it would be interesting to study the in-terplay of phase separation and nematic ordering in this con-centration range and at non-constant temperature, extendingthe work of Lansacet al.46

~2! A more realistic treatment of the glass transition,including its dependence on composition and the detailedcooling mechanism. In a CH formulation the obvious solu-tion would be to allow the mobility to be a function of thelocal concentration~and of temperature!; it is not so clearhow the same can be done within the CDS formalism.

ACKNOWLEDGMENTS

We thank Yeong-Sik Kim, Wonsool Ahn and John Westfor the discussions that formed the impetus for this work;Henk Boots and Hans Kloosterboer for sharing their knowl-edge on real PDLCs with us; Peter Bladon for writing thecluster labelling routine; Maarten Hagen for help with thefigures; Thierry Biben for advice on Fourier transforms; andChristopher Lowe for a critical reading of the manuscript.The work of the FOM Institute is part of the research pro-gramme of FOM and is supported by the Nederlandse Or-ganisatie voor Wetenschappelijk Onderzoek~NWO!. Wegratefully acknowledge the hospitality of the Liquid CrystalInstitute at Kent State University, where part of this workwas performed.

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FIG. 11. Inverse first moment ofS(k,t) vs tcool , the inverse quench rate~log-log plot!. Symbols as in Fig. 10, and circles:T50.34; filled invertedtriangles: T50.44; squares:T50.54; filled diamonds:T50.64; left tri-angles:T50.74. See the text and Table III for details.

FIG. 12. Inverse first moment ofS(k,t) vs tcool for the exponential quench,~log-log plot!, at T50.64 ~filled diamonds! andT50.01 ~stars!. The solidlines are power-law fits.

TABLE III. Exponents c of power-law fits to data in Fig. 11.T is thetemperature.

T c

0.74 0.04260.0030.64 0.04960.0030.54 0.07360.0020.44 0.11060.0030.09 0.28260.0080.04 0.28060.0060.0 0.27760.006

10151P. I. C. Teixeira and B. M. Mulder: Phase separation in polymer-dispersed liquid crystals

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