Theory of phase separation kinetics in polymer–liquid ...weinan/s/theory phase.pdf ·...

JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 11 15 MARCH 2002

Theory of phase separation kinetics in polymer–liquid crystal systemsC. B. Muratova)

Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102

Weinan Eb)

Department of Mathematics, Princeton University, Princeton, New Jersey 08544

~Received 20 February 2001; accepted 17 October 2001!

We introduce a kinetic model describing the phase separation in the mixture of long rod-likemolecules and long chain-like molecules. The model uses the angular distribution function for theorientations of the rods as a dynamical variable. The energetics is based on the nonlocal Onsagertheory for the rods combined with a nonlocal extension of the Flory–Huggins theory. The kineticsexplicitly takes into account the preferential diffusion along the rods. We computed the phasediagrams in this model and found a number of transitions leading to phase separation. We alsoperformed numerical simulations of the phase separation kinetics and studied the resultingmorphologies. ©2002 American Institute of Physics.@DOI: 10.1063/1.1426411#

ea,;icathh

ericei-

neo

danom

a

tauthpin-orbela

b

ac-le.that

beach,on.dede-

ro-efu-idtalsool-the

heiona

ol-en-d

hendy-n

ayoth

eruid

s-m-ges-like

an

I. INTRODUCTION

Polymer–liquid crystal mixtures have attracted a grdeal of attention lately.1 From the viewpoint of applicationspolymer matrix is used to stabilize liquid crystal displays2

polymeric composite materials have interesting mechanand electrical properties which make them good candidfor a variety of applications.1,3,4 Phase separation of sucsystems can be induced either through a thermal quenc3,4

or through polymerization.2

In order to better control the properties of these matals, it is crucial to understand the morphology and dynamof the phase separation process. Theoretical work on thproblems is relatively scarce. Following earlier work of Shmada, Doi, and Okano,5 Liu and Fredrickson derived TDGL~time-dependent Ginzburg–Landau!-type equations using adynamical mean field theory.6 Their work illustrated the in-terplay between orientational order of the liquid crystals athe phase separation process.6,7 But it also shares the samunpleasant feature that for a complex system such as theconsidered here, a Ginzburg–Landau approach is bounresult in a large number of terms. This not only makes qutitative comparisons difficult but also seems to deviate frthe original spirit of a Landau expansion.

Our approach is slightly more microscopic. We workthe level of kinetic theories such as Doi’s equations.8 Weextend Doi’s equation to the case of polymer–liquid crysmixtures. The advantage of this approach is that we can bin directly the microscopic interaction potential betweenmolecules, which in the present paper is taken to be emcal, but in principle can be calculated directly from first priciples. The disadvantage is that it is computationally mdemanding, particularly since the interaction potentialtween rods involves a multifold integral over both transtional ~position! and orientational~angle! degrees of free-dom. This problem in the present paper is overcome

a!Electronic mail: [email protected]!Electronic mail: [email protected]

4720021-9606/2002/116(11)/4723/12/$19.00

Downloaded 14 Dec 2006 to 128.112.16.126. Redistribution subject to AI

t

ales

,

i-sse

d

neto-

t

lilderi-

e--

y

resorting to a gradient expansion which localizes the intertion with respect to position, but retains nonlocality in angSuch a gradient expansion is considerably simpler thanof Ref. 6.

The main physical assumptions of our model cansummarized as follows. We are using a mean-field approso any fluctuation effects are not taken into consideratiWe use simple phenomenological models of the excluvolume interaction. Specifically, to model the interaction btween liquid crystal molecules we use the potential intduced by Priest and Straley.9–11 For the kinetics, we assumthat it is dominated by the translational and rotational difsion of the liquid crystal molecules, thus ignoring all flueffects. We neglect transversal diffusion of the liquid crysmolecules compared to their longitudinal diffusion. We alassume that the mixture of polymer and liquid crystal mecules is incompressible. In choosing the parameters ofmodel, we follow Doi.8

First we derive the free energy functional that gives tfree energy of the system in terms of the angular distributfunction of the liquid crystal molecules. Next, we introducekinetic model that relates the fluxes of the liquid crystal mecules to the gradients of the liquid crystal chemical pottial. The mobilities of the liquid crystal molecules obtainein this way are density- and orientation-dependent. We tintroduce the gradient approximation that captures thenamics of sufficiently slowly varying spatial distributiofunctions and use it in our simulations.

Our simulations show the importance of the interplbetween liquid crystal ordering and phase separation, benergetically and kinetically. We find that at high polymvolume fractions, the system phase separates into liqcrystal-rich droplets followed by ordering of the liquid crytal inside the droplets and leading to the formation of asymetric droplets. At the same time, in the intermediate staof this process the system evolves into connected beadstructures similar to the ones observed in experiments.12 Wefind that this morphology is a result of the kinetic rather th

3 © 2002 American Institute of Physics

P license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

emoea

ly-ly

sr

mieee

edy-rsibstr

a

li

naonhe-e

ituaan

mliq

thth

negtatsthe

gym

alr

efreen a

e,n,d

ed

en-is

the

tri-nstial

theeeirheore

-nu-

n

-

ol-althe

el-

sf a

4724 J. Chem. Phys., Vol. 116, No. 11, 15 March 2002 C. B. Muratov and W. E

energetic effects. At equal volume fractions, the systevolves through the formation of a transient bead-like mphology into a stripe-like bicontinuous morphology, with thliquid crystal oriented along the stripes and the defectstached to the polymer–liquid crystal interfaces. At low pomer volume fractions we get distorted droplets of the pomer in the ordered liquid crystal matrix.

This paper is organized as follows. In Sec. II we discuthe energetic and kinetic aspects of our model. We will wowith the orientation distribution function as the order paraeter. We then discuss the phase diagrams in Sec. III. Gradexpansion for the interaction potential is discussed in SIV. Our simulation results are discussed in Sec. V, followby conclusions.

II. MODEL

In contrast to most previous works, we will be interestin a detailed description of the interaction of flexible polmers with the nematic liquid crystal molecules. As was firealized by Onsager, the fundamental quantity that descrthe relevant degrees of freedom of the nematic is the dibution function f n̂ of the nematic orientationn̂.8,13,14 On-sager used this approach to explain the isotropic–nemphase transition for lyotropic liquid crystals.13 Priest andStraley used this approach to calculate the elastic moduthe nematic solutions.9–11 Doi used the distribution functionf n̂ to derive the kinetic model which describes the rotatiodiffusion of the solutions of rod-like molecules. The commfeature of all these works is that they focus only on tdistribution of theorientationof the nematic while neglecting the variation in its overall density. This is justified for thhomogeneous solutions of the nematics. However, the stion becomes different when phase separation occursthere exists an interplay between the nematic orderingspatial inhomogeneity.

This interplay leads to a number of interesting phenoena. First, the free energy of the interaction between theuid crystal and the polymer is going to be affected byinhomogeneity of the polymer concentration and lead toalignment between the orientationn̂ of the nematic mol-ecules and the concentration gradients. Second, the orietional order should strongly affect the kinetics of phase sregation in a strongly entangled polymer–liquid crysmixture. In the following, we will incorporate these effecinto a model which describes both the energetics andkinetics of the polymer–liquid crystal mixtures whose dscription is given in terms of the distribution functionf n̂ .

A. Energetics

We start by writing the phenomenological free enerfunctionalF of the polymer–liquid crystal system as a suof three terms:

F5F lc1Fpoly1F int , ~1!

whereF lc is the free energy functional for the liquid crystsubsystem,Fpoly is the free energy of the flexible polymesubsystem, andF int is the interaction free energy.


r-

t-

-

sk-nt

c.d

tesi-

tic

of

l

a-ndd

--

ee

ta--

l

e-

Let us look at the liquid crystal part first. We will use thnonlocal extension of the Onsager theory to describe theenergy due to the alignment of the nematic molecules ispatially inhomogeneous system:

F lc5E ddx dn̂S f n̂ ln f n̂11

2 E ddx8 dn̂8Bn̂n̂8~x2x8!

3 f n̂~x! f n̂8~x8! D , ~2!

whered is the dimensionality of space,n̂ is the unit vectorspecifying the orientation of the liquid crystal moleculf n̂(x) is the distribution function of the nematic orientatioand Bn̂n̂8(x2x8) is the pair interaction between the liquicrystal molecules~rods! with orientationn̂ and n̂8 located atx and x8, respectively. If one considers only the excludvolume interaction, one can takeB to be equal to 1 if therods are intersecting and 0 otherwise.9–11,14 In Eq. ~2! thefirst term gives both the translational and the rotationaltropy of the liquid crystal molecules, and the second termthe excluded volume interaction. Note that we choseunits in whichkBT51.

To describe the free energy of the inhomogeneous disbution of the polymer density, we use the Flory–Huggitheory with the square gradient term that penalizes spainhomogeneities

Fpoly5E ddxS b2~¹f!2

21

f

Nln f1xpolyf

2D , ~3!

wheref is the volume fraction of the polymer,b is the co-herence length,N is the polymerization index, andxpoly isthe Flory–Huggins parameter for the interaction betweenpolymer segments. In Eq.~3!, the first term represents thconformational entropy of the chains, the second is thtranslational entropy, and the third is the interaction of tsegments with each other. Note that one could use a mappropriate Lifshitz–DeGennes termb2(¹f)2/f instead ofthe gradient square term in Eq.~3!.15,16 This leads to essentially the same results, but complicates somewhat themerical simulations.

Finally, we will model the contact interactions betweethe polymer and the liquid crystal by

F int5x intE fr ddx, ~4!

wherex int is the polymer–liquid crystal interaction parameter,

r~x!5E dn̂E2L/2

L/2

dl f n̂~x1n̂l ! ~5!

is the segment number density of the liquid crystal mecules, andL is the number of segments in a liquid crystmolecule. Let us point out here the distinction betweendistribution functionf n̂(x), which gives the number of liquidcrystal molecules with the center of mass in the volumeementddx and oriented within the solid angledn̂, from thesegment densityr(x). For long slender rods the latter igiven by Eq.~5!, since in this case the segment density o


n

es

onha

tete

ionglnlaufon-

le

ho-ass

heule

inin

ssesuidthel-

aeranrl

-id

x

ehe

eesuseshe

us

idt

heally-e

t-

reen-tede-

4725J. Chem. Phys., Vol. 116, No. 11, 15 March 2002 Theory of phase separation kinetics

single liquid crystal molecule centered atx8 and orientedalongn̂ can be taken as*2L/2

L/2 d(x2x81n̂l )dl, whered(x) isthe Dirac delta-function.

One should not forget the incompressibility conditiowhich relates the values off(x) and f n̂(x) in space, so infact f n̂(x) is the only relevant variable. In our case it takthe following form:

f~x!1r~x!51. ~6!

This equation plays the role of the normalization conditifor f n̂(x) in the spatially inhomogeneous system. Note tthis incompressibility condition isnonlocal in the variablef n̂(x).

The above-mentioned set of equations completely demines the free energy of the spatially inhomogeneous sysin terms of the distribution functionf n̂(x) of the nematicorientations and density. The physically relevant situatcorresponds tod53. In this case the distribution functiof n̂(x) depends on three spatial coordinates and two ancoming from the unit vectorn̂. Furthermore, the interactioBn̂n̂8(x2x8) is nonlocal in both the positions and the anguvariables. Thus, using this distribution function in the simlations is a formidable challenge. For this reason, in thelowing we will work with systems constrained to two dimesions (d52). In this caseBn̂n̂8(x2x8) can be writtenexplicitly as

Bn̂n̂8~x2x8!5H 1, x2x8,PABCD

0, x2x8,¹ABCD,~7!

where ABCD is the rhombus with the sides of lengthL,whereL is the length of the liquid crystal molecules, paralto n̂ and n̂8, respectively, whose center is at the origin~seeFig. 1!. The distribution functionf n̂(x), in turn, dependsonly on two spatial and one angular variable. Let us empsize, however, that we are not trying to model twdimensional nematics here. It is well known that the fluctution effects destroy nematic ordering in two dimensions,our mean-field approach is not appropriate for thesystems.17 Nevertheless, we believe that on the level of tmean-field theory the restriction to two dimensions shocapture well some qualitative features of the physically revant three-dimensional system.

FIG. 1. Interaction potential geometry.


t

r-m

n

es

r-l-

l

a-

-oe

dl-

B. Kinetics

Let us now turn to the kinetics of the phase separationpolymer–liquid crystal systems. Since we are interestedphase separation, we need to study slow relaxation procethat lead to the segregation of phases. In polymer–liqcrystal systems these should be governed primarily bytranslational diffusion of polymer and liquid crystal moecules and the rotational diffusion of the liquid crystal.8

Let us look at the translational diffusion first. Instrongly entangled mixture of liquid crystal and polymmolecules the long rod-like liquid crystal molecules cmove only in the direction of their orientation. In othewords, only the longitudinal diffusion of the liquid crystawith the diffusion coefficientD i is allowed while the trans-versal diffusion is almost completely suppressed (D'50).8

If we were dealing with the liquid crystal in a polymer matrix whose kinetics were driven by the gradient of the liqucrystal part of the chemical potential, the fluxj n̂ of the liquidcrystal molecules with the orientation vectorn̂ would be

j n̂52D i f n̂~ n̂^ n̂!¹m lc , ~8!

wherem lc5dF/d f n̂ , and we took into account that the fluis proportional to the densityf n̂ . Observe that in Eq.~8! theflux is strongly coupled to the orientation distribution of thliquid crystal molecules. Similarly, if we were dealing witthe polymer in a liquid crystal matrix, we would write thpolymer flux jf as

jf52Dff¹mf , ~9!

wheremf5dF/df with f n̂ fixed, and the dependence of thmobility on f is taken into account. Note that if one assumRouse dynamics for flexible polymers and simple viscodrag for the liquid crystal, one gets the following estimatfor the dependence of the diffusion coefficients on tparameters:8

Df;1, D i;L21, ~10!

where we measure time in the units in which the viscodrag coefficients of a single polymer segment~monomer! isof order 1.

In an incompressible mixture of the polymer and liqucrystal the fluxesj n̂ and jf are no longer independent, busatisfy:

¹"S jf1E dn̂E2L/2

L/2

j n̂~x1n̂l !dl D 50. ~11!

In writing this equation we again took into account that tmass flux of the liquid crystal segments depends nonlocon the liquid crystal fluxj n̂(x) and neglected the contributions from the rotational diffusion, which are small in thentangled mixture.

A proper way to introduce incompressibility to the kineics would be to introduce a Lagrange multiplier~a scalarfield! into the free energy and solve for one of the fluxes.18 Inthe context of polymer–liquid crystal systems, in which theis a significant dependence of the mobilities on the conctrations of the species, this turns out to lead to a complicaelliptic problem. To overcome this difficulty, we will assumthatD i!Df /L, so the kinetics is limited by the liquid crys


ial

owec

enl

nn-offi-c

too

is

noeda

rte

esodal

mslow


tal component. In this case the fluxj n̂ must satisfy Eq.~8! inwhich instead ofm lc one should take the chemical potentm n̂(x) of the system with the incompressibility constraint:

m n̂~x!5E dn̂8E2L/2

L/2

dlE2L/2

L/2

dl8$2b2D f n̂8~x1n̂8l 82n̂l !

22x f n̂8~x1n̂8l 82n̂l !%21

N E2L/2

L/2

dl lnS 1

2E dn̂8E2L/2

L/2

dl8 f n̂8~x1n̂8 l 82n̂l !D 1 ln f n̂~x!

1E dn̂8 dx8Bn̂n̂8~x2x8! f n̂8~x8!, ~12!

where

x5x int2xpoly . ~13!

This equation, however, does not take into account the sling down of the dynamics at low volume fractions of thpolymer. To account for it, we will use a standard approaand introduce a density-dependent diffusion coefficiD(f) instead ofD i .16 Thus, the flux of the liquid crystamolecules in our model has the form

j n̂52D~f! f n̂~ n̂^ n̂!¹m n̂ , D~f!5D iDff

LD i1Dff. ~14!

So far, we were dealing with the translational diffusioof the polymer and the liquid crystal, and the effect of icompressibility. Let us now turn to the rotational diffusionthe liquid crystal. In a highly entangled mixture the coefcient of the rotational diffusion has the following dependenon the parameters:8

Dr;a2

L5 , ~15!

wherea is the entanglement length which we will assumebe of order 1, that is, comparable to the monomer size, inmodel. The rotational diffusion produces an extra fluxj n̂

r inthe configuration space off n̂ . In our model, we will ignorethe effects of the tube dilations and will simply write thflux as

j n̂r 52Dr f n̂¹n̂m n̂ . ~16!

Here¹n̂ is the gradient in the space of orientations.Summarizing all the above-presented results, we are

able to write the continuity equation which gives a closevolution equation forf n̂ in the course of the phase separtion

] f n̂

]t1¹"j n̂1¹n̂"j n̂

r 50. ~17!

III. PHASE DIAGRAMS

We calculated the phase diagrams for our system foset of values ofN andL. These phase diagrams are presen


-

ht

e

ur

w

-

ad

in Figs. 2–7. In all these figures the solid lines are the linof phase coexistence, and the dashed lines are the spinand the nematic transition lines.

There are a number of regions in the phase diagraseparated by these lines which can be distinguished. Be

FIG. 2. Phase diagram forL56 andN51.



nsroi-m

e of

blee--richle-


the coexistence lines and to the right of the nematic tration the system in equilibrium is in the homogeneous isotpic state~HI!. To the left from the isotropic–nematic transtion line and below the coexistence line the equilibriuphase is homogeneous nematic~HN!. Above the coexistence




i--line and below the spinodal lines the homogeneous statthe system is isotropic metastable~MI ! to the right of theisotropic–nematic transition line or nematic metasta~MN! to the left of this line. At long times these homogneous metastable states will phase separate into polymerisotropic and liquid crystal-rich nematic droplets via nuc





FIG. 8. Order parameters as a function of the polymer concentrationf for L56 ~a! andL510 ~b!.

biti

you

-

rgins

eiof

u

theons.

pear.the

ic–

al

to

ity

sses

ation. Above the spinodal lines the homogeneous statecomes unstable and results in the spinodal decomposand phase separation~PS!.

A. Nematic transition

We first look at the energetics of the homogeneous stem. The free energy per unit volume of the homogenesystem is given by

Fhom

V5

1

N S 12LE dn̂ f n̂D lnS 12LE dn̂ f n̂D1E dn̂ f n̂ ln f n̂1

1

2 E dn̂ dn̂8b~ n̂,n̂8! f n̂f n̂8

1xLE dn̂ f n̂S 12LE dn̂8 f n̂8D , ~18!

where

b~ n̂,n̂8!5L2un̂Ãn̂8u ~19!

is the second virial coefficient which is obtained from Eq.~7!by integration overx andx8 ~it is just the area of the rhombusABCD in Fig. 1!.

The free energy in Eq.~18! is then minimized numeri-cally by discretizing the dependence off n̂ on n̂ for a givenvalue off subject to the condition in Eq.~6!. Depending onthe value off, the equilibrium distribution function is eithea constant corresponding to the isotropic phase, or a sinpeaked function corresponding to the nematic phase. Sour system is two-dimensional, the isotropic–nematic trantion is second order~more precisely, the bifurcation of thfree energy minimizers at the isotropic–nematic transitpoint is a supercritical pitchfork!. Figure 8 shows the plot othe order parameters defined in two dimensions as

s52^cos2&u21, ~20!

where averaging is with respect to the equilibrium distribtion function f n̂

eq, as functions off for different values ofL.The nematic transition point is located atf5f IN with

f IN5121

Lub2u, ~21!


e-on

s-s

le-cei-

n

-

whereb2520.212~see the following!. Note that in our sys-tem the isotropic–nematic transition exists only forL*5.

B. Phase coexistence and spinodal curves

If we substitute the obtained equilibrium distributionf n̂eq

in Eq. ~18!, we can calculateFhom as a function off. Thephase coexistence lines are then obtained fromFhom

eq (f) us-ing the double tangent construction.

Usually, there are only two coexisting phases whensystem is in the metastable or phase separation regiHowever, for some values ofL andN there are regions of theparameter space where several coexisting phases can apThe reason for this can be seen from Fig. 9, which showsplot of the chemical potentialm as a function off for aparticular set of parameters. Because of the isotropnematic transition the dependencem(f) has a jump in thederivativedm/df at f5f IN . As a result, there are severvalues ofm and the corresponding values off for which thecondition of phase equilibrium can be satisfied. This leadsa peculiarity in the energetics of the system forL.6.

The spinodal line is obtained by requiring non-negativof the second variationd2Fhom. Let us first concentrate onthe isotropic homogeneous state. Since this state posse

FIG. 9. Chemical potentialm as a function of the polymer densityf for L56, N56, andx50.68.


n

n

tro

e

ta

on

rr

a

us

.he

ticc-

y

rs

ose

-0.

itheviorison.themslo-


rotational invariance, the normal modes of the fluctuatiowill be d f n̂5eimu, whereu is the azimuthal angle andm isan even integer@recall thatf n̂(x) does not depend on the sigof n̂#. It is then not difficult to show from Eq.~18! that thesecond variation of the free energy withd f n̂5(m d f meimu

relative to the homogeneous isotropic state is given by

d2Fhom52p2L2V(m

H 1

L~12f!1dm,0S 1

Nf22x D

1bmJ ud f mu2, ~22!

wheredm,0 is the Kronecker delta,

bm51

p E0

p

eimu sinu du52

p~12m2!. ~23!

Note thatb0.0 andbm,0 for m>2.From Eq.~22! one can see that the homogeneous iso

pic state becomes unstable with respect to them50 modewhen

x>x051

2 S 1

L~12f!1

1

Nf1b0D , ~24!

which corresponds to the spinodal. Equation~22! also showsthat at f5f IN given by Eq. ~21! the homogeneous statbecomes unstable with respect to them52 mode. Numeri-cally, we haveb050.637 andb2520.212.

When the system is in the homogeneous nematic sone has to calculate the second variationd2Fhom for the de-viations d f n̂ of f n̂ from the equilibrium distributionf n̂

eq ob-tained in Sec. III A. This leads to the following expressifor d2Fhom:

d2Fhom5V

2 H L2S 1

fN22x D E d f n̂d f n̂8dn̂ dn̂8

1E ~d f n̂!2

f n̂eq dn̂1E b~ n̂,n̂8!d f n̂d f n̂8dn̂ dn̂8J .

~25!

This second variation defines a quadratic form and a cosponding generating Hermitian integral operator:

d2Fhom5pL2VE d f n̂L̂homd f n̂dn̂, ~26!

with

L̂homd f n̂51

2p H d f n̂

L2f n̂eq1E S 1

Nf22x

1un̂Ãn̂8u D d f n̂8 dn̂8J . ~27!

Discretizingd f n̂ and diagonalizingL̂hom numerically, weobtained the spinodal curves for the homogeneous nemphase. They correspond to the values ofx as functions off,f IN for which the lowest eigenvalue of the operatorL̂hom

becomes negative.


s

-

te,

e-

tic

C. Inhomogeneous fluctuations

Let us now consider the stability of the homogeneostate of the system with respect to arbitrary fluctuations~notnecessarily homogeneous!. In this case, according to Eq~12!, the second variation of the free energy will have tform

d2F51

2 E d2xdn̂ dn̂8E2L/2

L/2

dlE2L/2

L/2

dl8H S 22x

11

Nf D d f n̂~x1n̂l !d f n̂8~x1n̂8l 8!1b2¹d f n̂~x

1n̂l !"¹d f n̂8~x1n̂8l 8!12pL

12fd f n̂~x!d f n̂8~x!d~ n̂

2n̂8!1E dx8 B~x2x8;n̂,n̂8!d f n̂~x!d f n̂8~x8!J .

~28!

Because of the translational symmetry, the quadraform in this equation can be partially diagonalized. Introduing d f n̂(x)5(q d f qn̂e

iqx, after a straightforward but lengthcalculation, we get from Eq.~28!

d2F5pL2V(qE dn̂ d f qn̂

* L̂qd f qn̂ , ~29!

where the operatorL̂q is defined as

L̂qd f qn̂51

L~12f!d f qn̂1

1

2pL2 E dn̂8S 1

Nf22x

1b2q21un̂Ãn̂8u D

3

4 sinFL

2~q"n̂!GsinFL

2~q"n̂8!G

~q"n̂!~q"n̂8!d f qn̂8 . ~30!

By discretizingd f qn̂ in n̂ and diagonalizing the operatoL̂q from Eq. ~30! numerically, we found that the instabilitiealways occur first with respect to the wave vectorq50, sothe only phase transitions that occur in our model are thconsidered previously. An example of the spectrumlq of theoperatorL̂q as a function ofq for the set of parameters corresponding to the simulation in Fig. 14 is shown in Fig. 1

IV. GRADIENT EXPANSION

The fully nonlocal chemical potential given by Eq.~12!is very difficult to use in the simulations. For one thing,gives very detailed information about the behavior of tsystem at short distances. If we are interested in the behaof relatively long wavelength modes, we can simplify thfree energy functional by introducing a gradient expansiThis approach should be justified near the spinodal orisotropic–nematic transition lines in the phase diagrasince the dominant instability modes in these cases arecated atq50 ~see Sec. III!.



FIG. 10. The spectrum of the operatorL̂q from Eq.~30!as a function of the wave vectorq at N56, L56,b52, f50.5, andx51.25.

ia

la

cu

lo

at

ubin

thlo

no

er,e

-

d

op-o-ofcan

mn

anyhat

cut-

a-esh

The gradient approximation for the chemical potentm n̂ is obtained by assuming thatf n̂ is a slowly varying func-tion of space and expanding the expressions likef n̂(x1n̂8l 82n̂l ) in the Taylor series inl and l 8. Using theseapproximations, after a lengthy but straightforward calcution we can write the expression form n̂ from Eq. ~12! ~up toa constant as!

m n̂5 ln f n̂2L

Nln f12xLf1Lb2Df

1L3

24Nf2 ~ n̂•“f!21L2E un̂Ãn̂8u f n̂8 dn̂8

1L4

24E S 1

Nf22x1un̂Ãn̂8u D“"@~ n̂^ n̂1n̂8

^ n̂8!¹ f n̂8#dn̂8, ~31!

where to the leading order in the gradient expansion

f~x!512LE f n̂~x!dn̂. ~32!

Note a term proportional to (n̂"“f)2 in Eq. ~31! which re-sults in anchoring of the liquid crystal molecules perpendilarly to the direction of“f. The evaluation ofm n̂ involvesonly the calculation off n̂ itself and the derivatives up tosecond off n̂ . Thus, the problem associated with the noncality in space of the original interaction kernelBn̂n̂8(x2x8) has been eliminated. Equation~31! together with Eqs.~14!, ~16!, ~32!, and ~17! are a closed set of equations thdetermine the evolution of the distribution functionf n̂ fromgiven initial conditions.

One still has to make sure that the obtained set of eqtions is well-posed, that is, there is no blowup of the artrarily short wavelength modes. In the case of the fully nolocal theory this should clearly be the case sincecontributions ofBn̂n̂8(x2x8) and the polymer–liquid crystainteraction vanish for these modes. As a result, at shwavelengths the system should behave as a mixture of


l

-

-

-

a---e

rtn-

interacting polymer and liquid crystal molecules. Howevwe will see in the following that this is in general not thcase for the reduced theory@Eq. ~31!#.

Let us see how the inhomogeneous fluctuationsd f qn̂with the wave vectorq affect the free energy for small deviations from the stable homogeneous state. For smalld f qn̂the variation of the free energyd2F will be given by Eq.~29!, whereL̂q has to be replaced byL̂q

g with

L̂qgd f qn̂5L̂homd f qn̂1

b2q2

2p E d f qn̂8dn̂82L2

48p E S 1

Nf

22x1un̂Ãn̂8u DqT~ n̂^ n̂1n̂8^ n̂8!qd f qn̂8dn̂8,

~33!

and L̂hom given by Eq.~27!.We still have to verify that the problem is well-pose

since there is a possibility that the operatorL̂qg has large

negative eigenvalues for large enoughq. This is in fact gen-erally the case. In order to analyze the spectrum of theeratorLq

g for largeq, we discretized this operator and diagnalized it numerically. The spectrum for a particular setthe parameters is presented in Fig. 11. From Fig. 11 onesee that for large enoughq there is a branch of the spectruwith lq,0, signifying an instability. On the other hand, ithe region of interest in whichq;1 the spectrum of theoperatorL̂q

g approximates well the spectrum ofL̂q . There-fore, the negative sign oflq for large enoughq is an artifactof the gradient expansion and is not associated withphysical behavior. It is simply a consequence of the fact tthe gradient expansion breaks down for large enoughq. Toavoid this problem, one needs to introduce a short-rangeoff q5qmax, so thatlq.0 for all q,qmax. In practice, thecutoff is naturally introduced in the numerical implementtions of the equations by choosing sufficiently coarse msizeDx, so thatqmax>p/Dx.



FIG. 11. The spectrumlq of the operatorL̂qg from Eq.

~33! for b52, L56, N56, x51.25,f50.5.

kitiothb

io

is

eiceympunly-ti-m

ewtheta--

henotefor

enfreethe-

rethe

s insitytheap-

V. SIMULATIONS OF THE KINETICS

We have performed the numerical simulations of ournetic model based on the gradient expansion approxima~Sec. IV!. The system was quenched into the regions ofphase diagram in which the homogeneous isotropic statecomes thermodynamically unstable. The governing equatwere discretized both in time and the angular variablen̂,with periodic boundary conditions. We found that the dcretization stepsDx51.5, Dt50.1, andDu50.26 ~for theazimuthal angle! were adequate for the parameters we usThe initial conditions were taken in the form of the isotropuniform mixture with small random fluctuations. The modwas then run up to sufficiently long times at which the dnamics of the system becomes very slow. Note that the silations of our model require a substantial amount of comtational power since they involve the discretization not oin space, but also inn̂, with the couplings between all possible n̂’s @see Eq.~31!#. For this reason, the code was opmized for running on an R10000 SGI Origin2000 supercoputer, also allowing for parallelization.


-n

ee-

ns

-

d.

l-u--

-

Before reporting our simulation results, let us say a fwords about the choice of the parameters. In order forsimulations not to run into the regime in which the oriention distribution functionf n̂ is sharply peaked in the orientation space, we chose a modest value ofL56. Since the sys-tem more easily undergoes the spinodal decomposition wL.N, in the si\mulations we chose them to be equal. Nthat the phase diagram of our system is most complexthese values ofN andL ~see Fig. 3!. The ratio between thetranslational and rotational diffusion coefficients was chosso as to reflect the effect of the entanglement. Thus theparameters in our simulations are the volume fraction ofpolymerf and the polymer–liquid crystal interaction parameterx.

In the first example we chose a polymer-rich mixtu(f50.7). Figure 12 shows a sequence of snapshots ofstate of the system at different times. The upper paneleach snapshot show the distribution of the polymer denwith gray corresponding to the pure polymer and black toliquid crystal, respectively. The lower panels in each sn

FIG. 12. Numerical simulations of the model forL56, N56, f50.7, b52, x51.25, LD i51, Df50.1,Dr58.331024. The system is 1503150.


re

th

drh

uite

ti

tetiolele

liqhet

exhi

meigh

e-

pa-alen-heid

idtsr-al

g-g in

etu-tal

rm.theing

ofle,ro-ain

turetal-or-

ke

ctgeliq-ticinges.talthees

ninge inmi-not

as(14.sseds.heer,

therop-usirdhehe

the

or

tria

f


shot show the magnitude of the order parameter at diffepoints in space.

One can see three qualitatively different regimes. Inearly stages of the phase separation~the first five snapshotsin Fig. 12! a bead-like morphology is formed. After a perioof linear growth of about 1000 time units the systems stato form an interconnected network of liquid crystal-ricbeads. For these times the rotational diffusion of the liqcrystal molecules is negligible, so the transport is dominaby the translational diffusion.

The formation of the beads is accompanied by a kineordering of the liquid crystal in the polymer-rich regions~seethe lower panels in Fig. 12!. We would like to emphasize thathis ordering is not related with the energetic nematic ording which occurs in the later stages of the phase separaWhat happens is, as the minority liquid crystal molecumove together into localized regions, the fastest molecuare those oriented perpendicularly to the gradient of theuid crystal concentration. If the localized region in which texcess liquid crystal molecules accumulate is elongated,molecules that are orientedperpendicularlyto the axis of thiselongated region move into this region first, leaving ancess of the molecules oriented in parallel with the axis of tregion behind in the polymer-rich region.

This effect can be illustrated by Fig. 13. Figures 13~a!and 13~b! show the director field scaled by the order paraeter, and the polymer density, respectively, at the momwhen the kinetic ordering occurs. One can see from F13~a! that at the time of the picture the liquid crystal-ricregions@dark regions in Fig. 13~b!# are not ordered, whilethe polymer-rich regions@light regions in Fig. 13~b!# arestrongly ordered.

This kinetic ordering in the polymer-rich regions b

FIG. 13. The kinetic and energetic ordering of the liquid crystal: the disbution of the director scaled by the magnitude of the order parametert51200 ~a!, the distribution of the polymer density att51200 with lighterregions corresponding to the polymer-rich regions~b!, the distribution of thedirector scaled by the magnitude of the order parameter att52000 ~c!, thedistribution of the polymer density att52000 with lighter regions corre-sponding to the polymer-rich regions~d!. In all four panels the parameters othe simulation are those of Fig. 12. The system is 75375.


nt

e

ts

dd

c

r-n.

ss-

he

-s

-nt.

comes quite substantial, with the magnitude of the orderrameter reaching;0.5. Note that because the liquid crystmolecules in the polymer-rich regions are oriented perpdicularly to the gradient of the polymer concentration, ttransport of these liquid crystal molecules into the liqucrystal-rich regions becomessuppressed. Indeed, since therotational diffusion is negligible at this time scale, a liqucrystal molecule can only diffuse in the direction of i~fixed! orientation. Therefore, if the molecule is oriented pependicularly to the direction of the gradient of the chemicpotential, no motion will occur. This effect produces lonrange correlations in the dynamics of the system, resultinthe formation of the network of beads.

After a time of order 1500 time units the second regimstarts. The growth of the liquid crystal-rich regions is sarated and the droplets made of almost pure liquid crysphase in the matrix of the almost pure polymer phase foAfter that the characteristic time scale of the variations ofconcentration dramatically decreases, substantially slowdown the dynamics. At this point the rotational diffusionthe liquid crystal molecules, which so far was negligibbecomes important and destroys the kinetic ordering pduced in the first stage of the phase separation. The mdriving mechanism for the dynamics becomes the curvaof the interfaces between the polymer and the liquid crysrich phases. Since the constraint imposed by the kineticdering of the liquid crystal now disappears, the network-listructure of the liquid crystal-rich phase breaks down.

At times of the order of 2000 time units the compaliquid crystal-rich droplets form, and the third and final staof the dynamics begins. As the droplets are formed, theuid crystal molecules inside start to order into the nemaphase, making the droplets a little elongated and producpoint defects at the most curved portions of the interfacThe anchoring effects produce a layer of the liquid crysmolecules oriented perpendicularly to the interfaces onoutside of the droplets and in parallel with the interfacinside them. This can be seen in Figs. 13~c! and 13~d!. Thelate stage dynamics is evaporation-condensation coarseof these droplets. The defects do not seem to play any rolthe dynamics here. Since the liquid crystal phase is thenority here, the defects attached to different droplets dointeract.

In the second example the polymer volume fraction wchosen to be the same as that of the liquid crystalf50.5). The results of the simulation are presented in Fig.One can see that the three stages of the dynamics discupreviously are still present, but with certain modificationAs in the previous simulation, the initial stage leads to tformation of an interconnected network of beads. Howevbecause of the greater liquid crystal volume fraction, insecond stage the morphology does not break up into a dlet morphology, but rather develops into a bicontinuostripe-like morphology. As the dynamics enters the thstage, the ordering of the liquid crystal in the direction of tstripes produces a global anisotropy in the system. Tstripes then coarsen, but the overall global direction ofnematic ordering remains.

During coarsening, the stripes break up into droplets

-t



FIG. 14. Numerical simulations of the model forL56, N56, f50.5, b52, x51.25, LD i51, Df50.1,Dr58.331024. The system is 1503150.

nfhintechein

se5

orii

asioro

et

thee dorentto-

ion-pletcess

hehisas

tionurthe

coalesce. The point defects~dark spots in the sea of white ithe order parameter pictures! that form in the second stage othe phase segregation remain attached to the points ofcurvature of the interfaces. We do not see any substamotion of the defects along the interfaces once the defare attached. The annihilation of the defects occurs wportions of the interfaces containing them merge durcoarsening.

In the third example, the polymer is the minority pha(f50.2). The results of the simulation are shown in Fig. 1In this case we do not see the formation of the netwstructure or the kinetic ordering. As the droplets of the mnority polymer-rich phase form, a nematic phase transitionthe liquid crystal matrix occurs, with the droplets workingthe initiators. The nematic ordering results in the formatof the point defects that quickly move to the droplet intefaces. Once again, we do not typically see the annihilationthe defects attached to the same droplet. Because of thefects, the interfaces become distorted, producing asymm


ghialtsn

g

.k-n

n-f

de-ric

droplets. These droplets once again coarsen viaevaporation-condensation mechanism. Here, however, wsee interaction between the defects attached to diffedroplets. These defects attract, pulling the droplets closergether. When the droplets come closer, the evaporatcondensation mechanism intensifies, so the smaller drodisappears before the droplets come in contact. This proincreases somewhat the speed of coarsening.

VI. CONCLUSION

In conclusion, we developed a model describing tphase separation in polymer–liquid crystal mixtures. Tmodel uses the angular orientation distribution functionthe dynamical variable and thus gives a detailed descripof the diffusion of the liquid crystal molecules. Thus, oapproach can be considered more microscopic thanTDGL approach of Refs. 6 and 7.

FIG. 15. Numerical simulations of the model forL56, N56, f50.2, b52, x51.25, D iL51, Df50.1,Dr58.331024. The system is 1503150.


bitothtdore

do

ists

thti

leth

uebea

wiprmm

ss

o.

s

s

la-


We found that the anisotropic transport due to the inaity of the liquid crystal molecules to move perpendicularlytheir axis produces kinetic ordering of these molecules inearly stages of the phase separation which is contrary toordering of the liquid crystal in the nematic phase. This leato the formation of a transient connected bead-like morphogy for low volume fractions of the liquid crystal. Similamorphologies are observed in experiments on polymstabilized liquid crystals.12

The late stages of the phase separation are dominatethe dynamics of the interfaces and the coarsening of the mphology due to the evaporation-condensation mechanThe nematic ordering of the liquid crystal in the dropleleads to their distortions and significant deviations fromcircular shape. Also, point defects forming in the nemaattach to the interfaces. On the time scales we were abrun our simulations we did not see a significant effect ofdefects on the motion of the interfaces.

The next step in our approach would be to perform simlations of the fully nonlocal model without resorting to thgradient expansion. Computationally, this is challengingcause it would require evaluating multiple convolutionseach moment in time. We hope that this can be achievedthe use of the spectral methods in space. The extra comtational cost associated with using fast Fourier transfomay be compensated by the ability of choosing larger tisteps. This should also allow one to use the methodLagrange multipliers to properly account for the incompreibility of the mixture.


l-

ehesl-

r-

byr-

m.

ectoe

-

-tthu-

eof-

ACKNOWLEDGMENT

This work is supported in part by an Air Force Grant NF 49620-98-1-0256.

1National Research Council Report,Liquid Crystalline Polymers~NationalAcademy Press, Washington, DC, 1990!.

2G. P. Crawford and S. Zumer,Liquid Crystals in Complex GeometrieFormed by Polymer and Porous Networks~Taylor and Francis, London,1996!.

3J. H. Aubert and R. L. Clough, Polymer26, 2047~1985!.4C. L. Jackson and M. T. Shaw, Polymer31, 1071~1990!.5T. Shimada, M. Doi, and K. Okano, J. Chem. Phys.88, 7181~1988!.6A. Liu and G. F. Fredrickson, Macromolecules29, 8000~1996!.7A. M. Lapena, S. Glotzer, S. A. Langer, and A. Liu, Phys. Rev. E60, R29~1999!.

8M. Doi and S. F. Edwards,The Theory of Polymer Dynamics~Clarendon,New York, 1986!.

9J. P. Straley, Mol. Cryst. Liq. Cryst.22, 333 ~1973!.10J. P. Straley, Mol. Cryst. Liq. Cryst.24, 7 ~1973!.11R. G. Priest, Phys. Rev. A7, 720 ~1973!.12C. V. Rajaram, S. D. Hudson, and L. C. Chien, Chem. Mater.8, 2451

~1996!.13L. Onsager, Ann. N.Y. Acad. Sci.51, 627 ~1949!.14A. Y. Grosberg and A. R. Khokhlov,Statistical Physics of Macromolecule

~AIP Press, New York, 1994!.15I. M. Lifshitz, A. Y. Grosberg, and A. R. Khokhlov, Rev. Mod. Phys.50,

683 ~1978!.16P. G. de Gennes, J. Chem. Phys.72, 4756~1980!.17D. Forster,Hydrodynamic Fluctuations, Broken Symmetry, and Corre

tion Functions~Addison–Wesley, Redwood City, CA, 1990!.18W. E and P. Palffy-Muhoray, Phys. Rev. E55, R3844~1997!.


Date post:	26-Mar-2018
Category:	Documents
Upload:	hoangduong
View:	213 times
Download:	0 times

Theory of phase separation kinetics in polymer–liquid ...weinan/s/theory phase.pdf ·...

Documents