Nematic Ordering in Mixtures of Polymers
and Liquid Crystals
Akihiko Matsuyama∗(Department of Chemistry for Materials)(Received August 30, 2002)

Abstract
We review our recent theoretical works on the phase ordering in mixtures ofpolymers and liquid crystals. A mean field theory is introduced to describe nematicisotropic phase transitions and phase separations in binary mixtures of a low-molecularweight liquid crystal molecule and a polymer chain which has various degrees of flexibility. We calculate the phase diagrams on the temperature-concentration plane,
depending on the stiffness of polymer chains and the strength of anisotropic interactions between rigid molecules. We also study spinodal decomposition (SD) in polymer/liquid crystal mixtures by solving linearized time-dependent Landau-Ginzburg
equations for concentration (conserved order parameter) and orientation (nonconserved order parameter). We find a new SD process driven by instability of theorientational order parameter. The simulations in two dimension are consistent withthe analytical results.
Keywords: polymer dispersed liquid crystals, phase separations, spinodal decompositions, nematicisotropic phase transitions

1

Introduction

Mesomorphic mixtures comprising of polymers and liquid crystal molecules (so called polymer
dispersed liquid crystals: PDLC) are of interest because of their important technological applications in high modulus fibers, nonlinear optics, and electro-optical devises. The performancesof these systems are closely related to a chain extension in a liquid crystal phase and phaseseparations[1]. Low-molecular weight liquid crystals are modeled as rigid rodlike molecules. On
the other hand, polymer chains have the variety of their stiffness and so when the polymer chainsare mixed with the liquid crystal molecules we can expect various types of phase diagrams depending on the stiffness of the polymer chain.
In mixtures of a flexible polymer and a liquid crystal, broad biphasic regions between an
isotropic phase and a nematic phase appear below the nematic-isotopic transition (NIT) temperature of the pure liquid crystal molecule. The liquid crystalline phases are destroyed as increasing∗

email: matuyama@chem.mie-u.ac.jp

9Res. Rep. Fac. Eng. Mie Univ., Vol. 27, pp.9-22(2002)10

the polymer concentration. These nematic-isotropic phase separations have been investigatedboth experimentally[2-8] and theoretically[8-14]. In contrast, mixtures of a liquid crystalline
polymer and a liquid crystal have good miscibility even at a liquid crystalline phase[15]. In thesesystems it is important to consider the co-occurrences of orientational ordering of the polymer
chain and phase separations[16-19].
In this review we introduce a mean field theory to describe phase behaviors in binary mixturesof a liquid crystal molecule and a polymer chain which has various degrees of flexibility. We
calculate the phase diagrams on the temperature-concentration plane. We also study the phaseseparation dynamics (spinodal decompositions) by solving linearized time-dependent LandauGinzburg equations for concentration (conserved order parameter) and orientation (nonconservedorder parameter) and show simulations in two dimensions.

2Free Energy for Mixtures of a Polymer and a Liquid
CrystalConsider binary mixtures of a non-nematic polymer and a liquid crystal (nematogen). The
polymers and liquid crystals interact through orientational dependent van der waals interactions
and excluded volume interactions. We here assume that the two neighboring bonds on thepolymer chains have either bent or straightened conformations[20, 21] and the conformationalenergy of the straightened bond is |0 | less than that of the bent bonds. The straightened stateof the bonds on the chains is energetically favored and, however, it is entropically unfavorable.
The nematic behaviors of the straightened bonds as well as liquid crystals can be induced by theanisotropic interactions. To describe the nematic-isotropic phase transition and phase separations,
we consider thermodynamics of our systems. Hereafter we refer the segments in straightened
bonds as ”rigid” segments.
Let Np and N` be the number of the polymers and the liquid crystal molecules, respectively.
Let np be the number of segments on the polymer and n` be that of the liquid crystal. The totallattice sites of our systems is given by Nt = n` N` + np Np and φ = np Np /Nt is the volume fractionof the polymer chains. The volume fraction of the liquid crystals is given by φ` = 1 − φ. The freeenergy of our systems can be given by
F = Fbent + Fmix + Fnem .

(1)

The first term shows the free energy change needed to straighten bent bonds on the polymers
and is given by[21, 22]hi

βFbent = Np nr (βf0 ) − Scomb /kB − ∆Sconf /kB ,(2)where β ≡ 1/kB T , T is the absolute temperature, kB is the Boltzmann constant, f0 is the
local free energy difference between the bent and straightened conformations, and nr shows thenumber of the rigid segments on the polymer chain. We here assume that each chain has thesame conformation. The second term in Eq. (2) is the combinatorial entropy related to thenumber of ways to select nr rigid segments out of the np segments on the polymer and is given
Scomb /kB = −np [x ln x + (1 − x) ln(1 − x)], where x(≡ nr /np ) shows the fraction of the rigid
segments on the polymer chain. The volume fraction φr of the rigid (straightened) segments on
the polymer chain is given by φr = nr Np /Nt = xφ. The third term in Eq. (2) shows the change
in conformational entropy to bring a chain from a crystalline (straightened) state to a flexible

Res. Rep. Fac. Eng. Mie Univ., Vol. 27, pp.9-22(2002)

11

amorphous state. According to the Flory’s lattice theory, the conformational entropy Sconf (n)of the polymer chain with n flexible segments is given by entropy of disorientation[23]. Whenthe nr segments out of the np segments on a polymer chain are replaced by rigid segments, theconformational entropy change ∆Sconf is given by ∆Sconf = Sconf (np − nr ) − Sconf (np ).
The second term in Eq. (1) is the free energy of the isotropic mixing for the polymers andliquid crystals. According to the Flory theory, the free energy is given by[23]βFmix /Nt =φ1−φln(1 − φ) +ln φ + χφ(1 − φ),
n`
np

(3)

where χ(≡ U0 /kB T ) is the Flory-Huggins interaction parameter related to the isotropic van derwaals interactions between unlike molecular species.[23]The third term in Eq. (1) shows the free energy for the nematic ordering. On the basis ofboth the Maier-Saupe model[24, 25] for orientational dependent-attractive interactions and theOnsager model[26] for excluded volume interactions, the free energy of the nematic ordering isgiven byβFnem /Nt =

φr
1−φ
1
f` (θ) ln 4πf` (θ)dΩ +fr (θ) ln 4πfr (θ)dΩ − ν`` S`2 (1 − φ)2
n`
nr
2
2
−ν`r S` Sr (1 − φ)φr + (ρ`` − 1)(1 − φ) + 2(ρ`r − 1)(1 − φ)φr ,
(4)
Z

Z

where dΩ ≡ 2π sin θdθ, θ is the angle between the rigid segments and the director of the orienting
field. The ν`` shows the orientational dependent (Maier-Saupe) interactions between the liquid
crystals, ν`r is that between the liquid crystal and the rigid segment on the polymer chains.
We here focus on the nonnematic polymer and so we can neglect the terms of the anisotropicinteraction νrr and the excluded volume interaction ρrr between rigid segments on the polymer
chain in Eq. (4). The f` (θ) and fr (θ) show the orientational distribution functions of the liquid
crystals and that of the rigid segments on the polymers, respectively. The scalar orientational
order parameter S` of the liquid crystals and that Sr of the rigid segments on the polymer isgiven byZ
Si =P2 (cos θ)fi (θ)dΩ,(5)i = `, r. where P2 (cos θ) ≡ 3(cos2 θ −1/3)/2. The last two terms in Eq.
(4) show the excluded volRR
ume interactions and the function ρij (i, j = `, r) is given by ρij = π4 sin γ(θ, θ 0 )fi (θ)fj (θ 0 )dΩdΩ0 .
In the isotropic phase, we have fi (θ) = 1/(4π) and ρij = 1[26].
The orientational distribution function f` (θ) of the liquid crystals and fr (θ) of the rigid segments on the polymer are determined by the free energy (4)with respect to these functions:
R(∂Fnem /∂fi (θ))x,fj = 0, under the normalization conditions fi (θ)dΩ = 1. In the framework of
our mean field approximations[21], we obtain the distribution function:fi (θ) =1
exp[ηi P2 (cos θ)],
Zi

(6)

h
i
5
5
η` ≡ n` (ν`` + )S` (1 − φ) + (ν`r + )Sr xφ ,
(7)
4
4
5
ηr ≡ np x(ν`r + )S` (1 − φ),(8)4where the constants Zi (i = `, r) are determined by the normalization condition as Zi = 2πI0 [ηi ],where the function I0 [ηi ] is defined as

Im [ηi ] ≡

Z 1h

im

P2 (cos θ)

0

h

i

exp ηi P2 (cos θ) d(cos θ),(9)Res. Rep. Fac. Eng. Mie Univ., Vol. 27, pp.9-22(2002)

12

m = 0, 1, 2, · · · . Substituting Eq. (6) into (5), we obtain two self-consistency equations for thetwo order parameters S` and Sr :Si = I1 [ηi ]/I0 [ηi ],(10)
and the average value of the order parameters is given by S = S` (1 − φ) + Sr φr .
The fraction x of the rigid segments on the polymer chain is determined by minimizing the freeenergy (1) with respect to x: (∂F/∂x)S` ,Sr = 0. This yields
5
Sr = D(x)/[(ν`r + )S` (1 − φ)],
4
"

(11)
#

i
1
x
+
,
D(x) ≡ ln(1 − x)λnp (1 − x)h(12)and λ ≡ (e/(z−1)) exp(−βf0 ), where z is the coordination number of a quasilattice[21]. By solving
the coupled equations (10) and (11), we can obtain the values of the two order parameters S` ,Sr , and the fraction x of the rigid segments on the polymer chain as a function of temperatureand concentration.
In our numerical calculations, we further split the local free energy difference f0 in Eq. (2)into two parts: f0 = 0 − T s0 , where s0 (= kB ln ω0 ) is the local entropy loss which is given byω0 = 1/(z − 2) [20] and 0 (< 0) is the energy change needed to straighten a bent bond. Then we
obtain λ = ω exp(−β0 ), where ω ≡ e/[(z − 1)(z − 2)]. The stiffness of the polymer is controlled
by the 0 . The larger values of 0 correspond to the stiffer polymers. The most flexible polymer
chain is realized when 0 = 0. The coexistence curve (binodal) and the spinodal line of the phase
diagram can be calculated by the chemical potentials[21]. The anisotropic interaction parameterν`` between liquid crystals is given to be inversely proportional to temperature: ν`` = Ua /kB T [25].
We also define the dimensionless nematic interaction parameter α ≡ ν`` /χ = Ua /U0 , the stiffness
parameter  ≡ |0 |/Ua of a polymer chain and c ≡ ν`r /ν`` .

3Phase Diagrams

3.1

Mixtures of a flexible polymer and a liquid crystals

When the polymer chain is sufficiently flexible, we can assume that Sr = 0,  = 0, and ν`r = 0.
The theory results in that for mixtures of a flexible polymer and a liquid crystals[14].
Figure 1(a) shows the binodal curve for α = 2.5, nr = 2.The temperature is normalized by
the NIT temperature TNLI of the pure nematogen. The number np of segments on a polymer
chain is changed. For low molecular weight polymers, the biphasic region between the nematicand isotropic phases (N+I) appears at lower temperatures. The stable nematic phase appears
in a dilute region of the polymer concentration. The nematic phase consists almost of purenematogens and the coexisting isotropic phase consists of nematogens and flexible polymers. For
larger values of np , we have the isotopic-isotropic (I+I) phase separation with an upper criticalsolution temperature (UCST) such as observed for polystylene with 7CB[5]. We also find thethree phase equilibrium (triple point: TP) where two isotropic phases and a nematic phase cansimultaneously coexist. At high temperatures, the system is in isotropic phase (I) because theentropy of mixing is dominant. As decreasing temperature, the I+I phase separation takes place
where χ parameter becomes dominant. Further decreasing temperature, the attractive interaction
ν`` between nematogens dominates and the N+I phase separation occurs at the lower temperatureside of the TP.

Res. Rep. Fac. Eng. Mie Univ., Vol. 27, pp.9-22(2002)

I
1.1

100

(b)

1000
I

L

T / TNIL(I + I)1.0α=2.5n l =2
T / TNI

(a)

13

50

50
n p=10

0.95

30

1.0

(N+I+I)N

20(N + I)

0.9

0

α=5
n l =2(N + I)n p=10

0.2

0.4

φ

0.6

0.9

0

0.1

0.2

0.3

0.4

0.5

φ

Fig. 1: (a) The binodal curves for α = 2.5, nr = 2. The temperature is normalized by the NIT
temperature TNLI of a pure liquid crystal. The number np of segments on a polymer chain ischanged. (b) The binodal curves for α = 5, nr = 2. When the nematic interaction parameter α isincreased, the isotropic-isotropic phase separation disappears and we have the nematic-isotropic
phase separation

Figure 1(b) shows the phase diagram for α = 5 and nr = 2. The number np of segments ona polymer is changed. When the nematic interaction parameter α is increased, the isotropicisotropic phase separation disappears and we only have the nematic-isotropic phase separation.
The width of the two phase region increases with increasing np . The type of phase diagram ischaracterized by the nematic interaction parameter α[14, 27].

3.2

Induced-nematic phase

In the following calculations we use α = 5,  = 3, ω = 0.025 and c = 1 for a typical example[21].
Figure 2 shows the phase diagram on the temperature-concentration plane for n` = 2. The
number np of segments on the polymer is changed from (a) to (b): (a) np = 10; (b) np = 50. The
temperature is normalized by the NIT temperature TNLI of the pure liquid crystal molecule. The
solid curve is referred to the binodal and the dashed and dotted line shows the spinodal. The
short-dashed lines show the NIT line. In the phase diagram, we have two different metastableregions: an isotropic metastable (Im); a nematic metastable (Nm). We also have a nematic
unstable region (Nu). As shown in Fig. 2 (a), at higher temperatures we have the two-phasecoexistence (N1+I) region between the nematic and the isotropic phase, which was observed in
the mixtures of polystyrene (PS) with (p-ethoxybenzylidene)-p-n-butylaniline (EBBA)[2, 3]. The
nematic phase (N1) almost consists of the pure liquid crystals and is stabilized by orientational
ordering of the liquid crystals. As decreasing temperature, we find a triple point (three phaseequilibrium) where the two nematic phases and an isotropic phase can simultaneously coexist(N1+N2+I). At the lower temperature side of the triple point, a new stable nematic phase (N2)
is induced by the anisotropic coupling between polymers and liquid crystals. This induced nematic
phase (N2) is stabilized by the orientational ordering of both polymer chains and liquid crystals.
We also find the (N1+N2) and (N2+I) coexistence regions. As increasing the polymer length np ,Res. Rep. Fac. Eng. Mie Univ., Vol. 27, pp.9-22(2002)

1.10
n p = 10

(a)

0.95

ε=3

L

T / TNI

Im

N1
0.85

(N1+I)(I+N2)

1.00
N1

0.80

(N1+N2+I)(N1+N2)

Nm

0.70
0

0.2

N2
Nm

0.4

N2

Im

0.95

Nu

0.75

nl =2

1.05

I

0.90

(b) n p = 50

I

nl =2

L

T / TNI

1.00

14(N1+N2)(N2+I)

0.6

0.8

Nm

0.90

φf

0

0.2

0.4

0.6

0.8

φ

Fig. 2: Phase diagram on the temperature-concentration plane for n` = 2. The number np ofsegments on the polymer is changed from (a) to (b): (a) np = 10; (b) np = 50.

the induced nematic phase (N2) shifts to higher temperatures due to the correlation between thetwo neighboring bonds along the polymer chain. In Fig. 2(b), we also find the azeotrope point at
which two equilibrium nematic and isotropic phases have the same composition. We predict thatthe partial orientational ordering of polymers by liquid crystal solvents can give rise to a newstable nematic phase (induced-nematic phase), depending on the polymer length and the stiffnessof the polymer chain.

4Phase Separation Dynamics

In this section, we study spinodal decompositions in mixtures of a flexible polymer and liquid
crystals. In these mixtures, as shown in Fig.1, biphasic regions between an isotropic and a nematic
phase appear below the NIT temperature of the pure liquid crystal molecule. When the systemis thermally quenched from a stable isotropic phase into an unstable part of the biphasic region,
the fluctuations of concentration and of orientation take place and isotropic or nematic droplets
appear with time[28-30]. The instability of these systems is driven by the competition between
phase separation and nematic ordering.
To elucidate the time evolution of the concentration and orientation fluctuations during the
spinodal decomposition, we calculate structure factor for concentration and for orientation using
time-dependent Landau-Ginzburg (TDLG) equations for concentration and orientational orderparameters[31]. We also show simulations in two dimension (but without linearization) to further
understand our analytical results for the structure factors.

4.1

Kinetic equations

We consider mixtures of a flexible polymer and liquid crystal (nematogen) described by one
conserved order parameter (volume fraction φ of nematogen) and one nonconserved order pa-

Res. Rep. Fac. Eng. Mie Univ., Vol. 27, pp.9-22(2002)15

rameter (orientational order parameter Sij of nematogens)[31-36]. Since the orientational orderparameter is a traceless symmetric tensor, its components can be expressed as[25]
"

#

1
d
Sij = S(r) ni (r)nj (r) − δij ,
2
d

(13)

where i, j = x, y, z denote the components along three orthogonal coordinate axes, d is thespace dimension, n(r) is a local director, δij is the Kronecker δ function, and S(r) is the scalarorientational order parameter referred to previously. The dynamics of the mixture is describedby the coupled time-dependent Ginzburg-Landau equations[33-36] for the two order parameters.
In the inhomogeneous system under nonequilibrium conditions, spatial variations occur in thetwo order parameters. The total free energy (F ) can be expressed in terms of a local bulk freeenergy density f (φ, Sij ) and the gradients of the two order parameters[31]:F [φ, Sij ] =Z"

dr f (φ, Sij ) +K0(∇φ)2 + L0 (∂i φ)(∂j Sij )
2
#

L1
L2
+ (∂k Sij )2 +(∂i Sik )(∂j Sjk ) ,
2
2

(14)

where the free energy F and f are dimensionless quantities (divided by kB T ), T is the absolutetemperature, kB is the Boltzmann constant and K0 , L0 , L1 , L2 are phenomenological coefficients
derived from a mean field theory[33, 34]. In this paper we take these coefficients as constant. The
bulk free energy density is given by[31]f (φ, Sij ) =φ1−φln(1 − φ) + ln φ + χφ(1 − φ)
np
nl
1
1
1
+ A(φ)Sij Sji − B(φ)Sij Sjk Ski + C(φ)Sij Sjk Skl Sli ,
2
3
4

(15)

where the coefficients A(φ), B(φ), and C(φ) are given as a function of temparature and concentration[31].
In considering the nonequilibrium equations of motion for our system, we adopt a thermodynamic point of view. The phenomenological equation of motion for the concentration φ, which
ensures local conservation of material, is given by[37]:∂φ(r, t)∂t

= Γφ ∇2 δF 

δφ

#

"

∂f
− K0 ∇2 φ − L0 ∂i ∂j Sij ,= Γφ ∇2∂φ(16)

where the thermodynamic force which drives the flux is given by the gradient of the chemical
potential µ = δF/δφ and Γφ is the mobility, assumed constant. On the other hand, for thenonconserved order parameter Sij , we take the local rate of change to be linearly proportional to
the local thermodynamic force ∂F/∂Sij [37]. The equation of motion for Sij is then given by∂Sij (r, t)
∂t

= −ΓS

 δF

δSij

= −ΓS

"

+ Λ(r, t)δij

∂f− L0 ∂i ∂j φ − L1 ∇2 Sij
∂Sij
#

L2
− (∂i ∂k Skj + ∂j ∂k Ski ) + Λ(r, t)δij ,2(17)

Res. Rep. Fac. Eng. Mie Univ., Vol. 27, pp.9-22(2002)16

where the transport coefficients Γφ and ΓS are taken as constant. The kinetic equations could in
principle be made more general, by writing the Onsager coefficient Γ as a matrix. This would allow
one order parameter to be driven by gradients in the chemical potential of the other[33]. However,as this is not the phenomenon under investigation, we set the off-diagonal matrix elements to zerofor simplicity. The orientational order parameter evolves in such a way as to lower the free energy,
but it must do so subject to the constraint that it remains traceless. The Lagrange multiplier Λ
in Eq.(17) will be chosen to ensure conservation of the trace of Sij .
By linearizing Eqs. (16) and (17)[31], we can calculate the structure factor for concentration:Sφ (q, t) ≡ h|δφ(q, t)|2 i,(18)

and that for the component Szz of the orientational order parameter: given by
SS (q, t) ≡ h|δSzz (q, t)|2 i,(19)

where q is the wave number in the Fourier space and we define a z axis parallel to the director.

4.2

Spinodal decompositions
(I)

(II)

1
I
0.9
L

T / TNI

Im

N

0.8
Iu

0.7

Nu

Nm

0.6
(A)

0.5
0.4

(C)

(B)
0.6

0.8

1

φ (volume fraction of liquid crystals)

Fig. 3: Phase diagram of a polymer/liquid crystal mixture with np = nl = 2 and ν/χ = 1.4. The
solid curve refers to the binodal and the dotted line shows the first-order NIT line. The dashdotted line shows the spinodal. Closed circles indicate temperature quenches from the stableisotropic phase into the isotropic unstable (Iu; A) and nematic unstable (Nu;B, C) regions.

A typical phase diagram on the temperature-concentration plane is shown in Fig. 3 which is
calculated with np = nl = 2 and ν/χ = 1.4 in Eq. (15)[14]. The reduced temperature τ (≡ T /TNLI )is normalized by the nematic-isotropic transition (NIT) temperature TNLI of the pure liquid crystal(at φ = 1). The critical solution point in the isotropic phase is at φ = 0.5 and τ = 0.95. The
solid curve refers to the binodal and the dotted line shows the first-order NIT of a hypotheticalhomogeneous phase. The dash-dotted line shows the spinodal. Note that the origin is suppressedon the φ-axis. When τ = 0.831, we have a triple point where two isotropic liquid phases (L1 + L2 )Res. Rep. Fac. Eng. Mie Univ., Vol. 27, pp.9-22(2002)17

and a nematic phase (N ) can simultaneously coexist. Below the triple point, we have the twophase coexistence between an isotropic and a nematic phase. Such phase diagrams are observedin mixtures of n-tetracosane and nematic liquid crystal (PAA)[4]. In the biphasic region between
the nematic and the isotropic phases, we have two different metastable regions: an isotropic
metastable (Im) and a nematic metastable (Nm), and two unstable regions: an isotropic unstable(Iu; A) and a nematic unstable (Nu; B,C)[8, 13, 14]. On increasing the molecular weight np of
the polymer, the critical solution point shifts to higher temperatures and higher concentrations of
mesogens and the Nu and Iu regions also shift to higher temperatures and higher concentrations
with increasing np [8, 13, 14].
Filled circles indicate temperature quenches from the stable isotropic phase into the isotropic
unstable (Iu;A) and nematic unstable (Nu;B, C) regions. The region (A), lying below the isotropic
spinodal curve and above the NIT line, corresponds to a system which is initially unstable withrespect to concentration fluctuations, but metastable to orientation fluctuations. The region (B),between the isotropic spinodal curve and the NIT line, is initially unstable to both concentrationand orientation fluctuations. In the region (C) between the isotropic spinodal curve and thenematic spinodal curve, the system is initially unstable with respect to orientation fluctuations,but metastable to concentration fluctuations. Thus if we thermally quench from an isotropic
phase to these different regions, we can expect a variety of SD processes even in the early stages.

qqFig. 4: Temporal evolution of the compositional structure factor (a) Sφ and of the orientational
structure factor (b) SS , for the temperature quench into the Iu region A (τ = 0.6, φ = 0.55) inFig. 3Figure 4 shows the temporal evolution of the compositional structure factor Sφ and of theorientational structure factor SS , respectively, for the temperature quench into the Iu region(τ = 0.6, φ = 0.55) in Fig. 3. The structure factor for concentration has a maximum at qm
which corresponds to the peak wave number of the growth rate[31]. With time the corresponding
mode grows exponentially and the peak position qm is invariant. The time evolution of thestructure factor Sφ is the same as that of the Cahn theory for isotropic SD[38, 37]. The structure
factor SS decreases with increasing q at very early times. The amplitude of the peak at q = 0
decreases with time because fSS ≡ (∂ 2 f /∂S 2 ) > 0. With time another peak appears in the SS
curve, corresponding to the peak at qm , and the orientation fluctuation grows exponentially. In
this quench, the concentration fluctuation initially induces the SD and the orientational ordering
within the domains subsequently takes place due to the coupling between the two order parameters

Res. Rep. Fac. Eng. Mie Univ., Vol. 27, pp.9-22(2002)
as time progresses.

18

Further increasing the initial concentration of mesogen, the orientationalqqFig. 5: Temporal evolution of the compositional structure factor (a) Sφ and of the orientational
structure factor (b) SS , for the temperature quench into the Nu region C (τ = 0.6, φ = 0.85) inFig. 3φ =0.55

0.10

0.72

log(q m )

0.00

0.77

-0.10

0.78

-0.20

0.79

-1/3
0.8

-0.30

0.85
τ=0.6

-0.40
-0.50

-1/2
0.5

1.0

1.5

2.0

log(t)

Fig. 6: Temporal evolutions of the scattering wave number qm at which the compositional structure factor has a maximum. The initial concentration is varied at fixed τ .

fluctuation becomes dominant. Figure 5 shows the temporal evolutions of the structure factors
for a temperature quench into the Nu region (C) (τ = 0.6, φ = 0.85), where the system is
initially unstable with respect to orientational order parameter (fSS < 0) and metastable withrespect to concentration (fφφ ≡ (∂ 2 f /∂φ2 ) > 0). In the very early stages, the concentration
fluctuation becomes weak with time because fφφ > 0. However the orientational fluctuations
grow exponentially with time because fSS < 0. Further increasing time, a peak in Sφ appears
and shifts to lower values of the wave number. There is no longer any time stage in which thepeak position in Sφ is invariant, which was predicted by Cahn’s linearized theory for isotropic SD
in the early stages[37, 38]. The instability of the orientational ordering initially induces the SD
and the concentration fluctuation is induced by the coupling between the two order parameters.

Res. Rep. Fac. Eng. Mie Univ., Vol. 27, pp.9-22(2002)

19

To summarize these results of our linearized analysis, we show in Fig. 6 the values of qm for thedensity structure factor Sφ for various initial concentrations φ0 . In the Iu region A, the peak wave
number qm is invariant during the early stages of the SD. On increasing the initial concentration,
we find that the scattering peak shifts to the lower values with time. When φ0 = 0.85 (region C),
the scattering peak for Sφ changes as t−1/3 (the average domain size increases as t1/3 even in theearly stages). The instability of the orientational ordering induces the concentration fluctuation
through the coupling between two order parameters.

4.3

Simulations in two dimensions
t=6.0

10.0

15.0

50.0

φ

S

Fig. 7: Time evolution of the compositional (upper) and orientational (lower) order parameters
for the temperature quench (τ = 0.6, φ0 = 0.35) in Fig. 3.

To further understand our analytical results, nonlinear coupled differential equations (16,17)
were simulated in two spatial dimensions (x, y) with periodic boundary conditions. In the twodimensions, we have, with the constraints Tr Sii = 0 and Sij = Sji , a set of two coupled nonlinear
partial derivative equations describing the spatiotemporal evolutions of the tensor components
Sxx
q
2 .
2
and Sxy in Eq. (17) and the scalar orientational order parameter is given by S = 2 Sxx + Sxy
The time step and grid spacing are ∆t = 0.0001 and ∆h = 0.3 respectively and 128 grid points
were used. The initial conditions for the concentration φ(x, y) and the tensor order parameters
Sij (x, y) at each lattice point are given by random numbers distributed uniformly in φ(x, y) =φ0 ± 0.02 and Sij (x, y) = ±0.02, respectively. Initially, the system is in an isotropic phase. We
neglect the random noise terms in the kinetic equations and believe no significant effects on thephase separation dynamics. Figures 7, 8, and 9 show the results of simulations for temperature
quenches from the isotropic state to τ = 0.6 with φ0 = 0.35, φ0 = 0.55, and φ0 = 0.85, respectively.
The upper(lower) figures show the temporal evolution of the concentration (scalar orientation
order parameter). The darkness is proportional to the value of φ and S. In region A, at φ0 = 0.33
[Fig. 7] and φ0 = 0.55 [Fig. 8] , we first observe the concentration fluctuation in the earlystage and the nematogen-rich isotropic droplets appear in a polymer-rich matrix. With time theconcentration fluctuation and the orientational fluctuation within the droplets grow and form
nematic droplets [Fig. 7] or bicontinuous networks of interconnected domains[Fig. 8].
In the case of φ0 = 0.85 [region C, Fig. 9], we first observe the orientation fluctuations inRes. Rep. Fac. Eng. Mie Univ., Vol. 27, pp.9-22(2002)

30.0

10.0

6.0

t=4.0

20

φ

S

Fig. 8: Time evolution of the compositional (upper) and orientational (lower) order parameters
for the temperature quench (τ = 0.6, φ0 = 0.55) in Fig. 3.

the very early stage (t = 0.2), but the system is still in an isotropic state. As time increases,
the orientation fluctuations become large and induce concentration fluctuations. We find thepolymer-rich isotropic region (white areas) form a fibrillar network morphology in a nematogenrich nematic matrix (t ∼ 1.0). Further increasing time, these networks break up (t = 1.4) andform polymer-rich isotropic droplets (t = 12.4). At late stages (t = 12.4), the droplets becomes
noncircular in the surrounding nematogen-rich nematic phase. The nematic ordering can significantly influence domain morphology. These simulations show how the phase separation dynamics
in polymer/liquid crystal mixtures is driven by the competition between phase separation andnematic ordering. On increasing the concentration of liquid crystal, the instability of the orientational ordering becomes dominant and the mechanism (or morphology) of the SD is changed
from concentration fluctuation-induced SD to orientation fluctuation-induced SD. The cross term
between gradients plays a significant role in the early stage SD.
t=0.2

1.0

1.4

12.4

50.0

φ

S

Fig. 9: Time evolution of the compositional (upper) and orientational (lower) order parameters
for the temperature quench (τ = 0.6, φ0 = 0.85) in Fig. 3.

Res. Rep. Fac. Eng. Mie Univ., Vol. 27, pp.9-22(2002)

5

21

Summary

In this review, we have theoretically studied the phase behaviors and phase separation dynamics in mixtures of polymers and liquid crystals. The theory can qualitatively describe theobserved phase diagrams. We predict that the partial orientational ordering of polymers by liquid
crystal solvents can give rise to a new stable nematic phase (induced nematic phase), dependingon the polymer length and the stiffness of the polymer chain. These phase behaviors will be usefulfor further studies for the polymer dispersed liquid crystals and the investigations of intermolecular interactions, association phenomena, and dipole-dipole correlations between polymers andliquid crystals. Though we have performed a linear analysis of the spinodal decompositions inpolymer/liquid crystal mixtures, these results will also be an useful to understand other nematic
systems including semiflexible polymer, liquid crystalline polymers, and rodlike colloids[39-42].
The nematic ordering can significantly influence domain morphology.
Finally, volume phase transitions of a liquid crystalline gel immersed in a nematogen [43, 44]
are also closely related to the phase separations stated in this review.

References[1] Liquid Crystals in Complex Geometries, edited by G. P. Crawford and S. Zumer (Taylor
Francis, London, 1996).
[2] B. Kronberg, I. Bassignana, and D. Patterson, J. Phys. Chem. 82, 1714 (1978).
[3] A. Dubaut, C. Casagrande, M. Veyssie, and B. Deloche, Phys. Rev. Lett. 45, 1645 (1980).
[4] H. Orendi and M. Ballauff, Liq. Cryst. 6, 497 (1989).
[5] W. Ahn, C. Y. Kim, H. Kim, and S. C. Kim, Macromolecules 25, 5002 (1992).
[6] G. W. Smith, Phys. Rev. Lett. 70, 198 (1993).
[7] F. Roussel, J. M. Buisine, U. Maschke, X. Coqueret, and F. Benmouna, Phys. Rev. E 62,
2310 (2000).
[8] C. Shen and T. Kyu, J. Chem. Phys. 102, 556 (1995).
[9] F. Brochard, J. Jouffroy, and P. Levinson, J. Phys. (France) 45, 1125 (1984).
[10] M. Ballauff, Mol. Cryst. Liq. Cryst. 136, 175 (1986).
[11] P. Maissa and P. Sixou, Liq. Cryst. 5, 1861 (1989).
[12] R. Holyst and M. Schick, J. Chem. Phys. 96, 721 (1992).
[13] T. Kyu and H. W. Chiu, Phys. Rev. E 53, 3618 (1996).
[14] A. Matsuyama and T. Kato, J. Chem. Phys. 105, 1654 (1996); Progr. Colloid Polym. Sci.
106, 175 (1997).
[15] Liquid Crystalline and Mesomorphic Polymers, edited by V. P. Shibaev and L. Lam(Springer-Verlag, New York, 1993).
[16] P. J. Flory, Proc. R. Soc. London, Ser. A 234, 60 (1956); 234, 73 (1956).

Res. Rep. Fac. Eng. Mie Univ., Vol. 27, pp.9-22(2002)22[17] M. Warner and P. J. Flory, J. Chem. Phys. 73, 6327 (1980).
[18] A. ten Bosch, P. Maissa, and P. Sixou, J. Chem. Phys. 79, 3462 (1983).
[19] T. Sato and A. Teramoto, Adv. Polym. Sci. 126, 85 (1996).
[20] A. Matsuyama, Y. Sumikawa, and T. Kato, J. Chem. Phys. 107, 4711 (1997).
[21] A. Matsuyama and T. Kato, Phys. Rev. E 59, 763 (1999); Phys. Rev. E 58, 585 (1998); J.
Chem. Phys. 109, 2023 (1998).
[22] A. Matsuyama and T. Kato, J. Chem. Phys. 112, 1046 (2000).
[23] P. J. Flory, Principles of Polymer Chemistry (Cornell University, Ithaca, 1953).
[24] W. Maier and A. Saupe, Z. Naturforsch, A 14a, 882 (1959).
[25] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford. Sci., London,1993).
[26] L. Onsager, Ann. N. Y. Acad. Sci. 51, 626 (1949).
[27] A. Matsuyama and T. Kato, J. Chem. Phys. 108, 2067 (1998).
[28] C. Casagrande, M. Veyssie, and C. M. Knobler, Phys. Rev. Lett. 58, 2079 (1987).
[29] W. K. Kim and T. Kyu, Mol. Cryst. Liq. Cryst. 250, 131 (1994).
[30] A. Nakai, T. Shiwaku, W. Wang, H. Hasegawa, and T. Hashimoto, Macromolecules 29, 5990
(1996).
[31] A. Matsuyama, R. M. L. Evans, and M. E. Cates, Phys. Rev. E 61, 2977 (2000); Mol. Cryst.
Liq. Cryst. 367, 455 (2001); A. Matsuyama and T. Kato, J. Chem. Phys. 113, 9300 (2000).
[32] J. R. Dorgan and D. Yan, Macromolecules 31, 193 (1998).
[33] A. J. Liu and G. H. Fredrickson, Macromolecules 29, 8000 (1996).
[34] J. Fukuda, Phys. Rev. E 58, R6939 (1998); 59, 3275 (1999).
[35] H. W. Chiu and T. Kyu, J. Chem. Phys.110, 5998 (1999).
[36] A. M. Lapena, S. C. Glotzer, A. S. Langer, and A. J. Liu, Phys. Rev. E 60, R29 (1999).
[37] Solids far from equilibrium, J. S. Langer, Chapter 13 (Cambridge University Press, New
York, 1992).
[38] J. W. Cahn, Trans. Mat. Soc. AIME 242, 166 (1968).
[39] S. P. Meeker, W. C. K. Poon, J. Crain, and E. M. Terentjev, Phys. Rev. E 61, R6083 (2000).
[40] J. Yamamoto and H. Tanaka, Nature 409, 321 (2001).
[41] M. Adams, Z. Dogic, S. L. Keller, and S. Fraden, Nature 393, 349 (1998).
[42] A. Matsuyama and T. Kato, Eur. Phys. J. E 6, 15 (2001).
[43] K. Urayama, Y. Okuno, T. Kawamura, and S. Kohjiya, Macromolecules 35, 4567 (2002).
[44] A. Matsuyama and T. Kato, J. Chem. Phys. 114, 3817 (2001); Phys. Rev. E 64, 010701R(2001); J. Chem. Phys. 116, 8175 (2002).