Structure of Liquid Crystalline Polymers

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Theories of Liquid Crystalline Polymers 65 The con tribut ion fro m Rus sia n sci entis ts is wo rth y of special men tio n. We will introduce their theory later. It should be pointed out that to meet the second virial approximation, molecules must have a large L/D so that at the transition the solution is dilute. For mo lecules of axia l ratio less than 10 the theory does not work well. In addition, the Onsager value of the density dierence at the nematic isotro pic transiti on is great er than the experimenta l data. Introduci ng higher vir ial terms may extend the Onsage r the ory to concentrated solutions (Khokhlov & Semenov, 1981). The Flory theory discussed in the next section is another important theory on rigid liquid crystalline polymers. Because of its clear picture of the lattice model and the incorporation of the Onsager theory, it has become a basic method for the theoretical study of liquid crystalline polymers. As a result of the constant eorts of Flory and his co-workers, the theory has been applied to binary and poly-disperse systems and also includes the “soft” interactions. 2.2. FLOR Y THEOR Y FOR RIGI D ROD LIQUID CRYSTALLINE POLYMERS 2.2.1. Partition funct ion of a rigid rod solution Flory (19 56, 198 4) adopte d the lattic e mode l. The Flory the ory start s with the partition function of systems consisting of rigid rods and solvent molecules. Assume the long axis of the rigid rods makes an angle ψ with respect to the director of the system and the director is along one principal axis of the cubic lattice. Divide each rod into x basic units of equal width. Each basic unit occupies one cell in the lattice. x is actually the axial ratio of the rods. For simplicity, suppose that the dimension of a solvent molecule is compat ible to the size of a cel l lattice. In this section we adopt the same assignations as Flory. These may be dierent from those used in the precedi ng section by Onsag er. In order to put a rod into the lattic e, a postulate is made, whic h suggests that each rigid rod be divided into y sub-particles as shown in Figure 2.4b y = x sin θ. (2.28) Each sub-pa rticle has x/y basic unit and its long axis is along the director. If a particle is perfectly aligned along the director, y is zero. As a

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Theories of Liquid Crystalline Polymers 65

The contribution from Russian scientists is worthy of special mention.We will introduce their theory later.

It should be pointed out that to meet the second virial approximation,molecules must have a large L/D so that at the transition the solutionis dilute. For molecules of axial ratio less than 10 the theory does notwork well. In addition, the Onsager value of the density difference at thenematic — isotropic transition is greater than the experimental data.

Introducing higher virial terms may extend the Onsager theory toconcentrated solutions (Khokhlov & Semenov, 1981).

The Flory theory discussed in the next section is another importanttheory on rigid liquid crystalline polymers. Because of its clear picture of the lattice model and the incorporation of the Onsager theory, it has becomea basic method for the theoretical study of liquid crystalline polymers. As aresult of the constant efforts of Flory and his co-workers, the theory hasbeen applied to binary and poly-disperse systems and also includes the“soft” interactions.

2.2. FLORY THEORY FOR RIGID — ROD LIQUID

CRYSTALLINE POLYMERS

2.2.1. Partition function of a rigid rod solution

Flory (1956, 1984) adopted the lattice model. The Flory theory startswith the partition function of systems consisting of rigid rods and solventmolecules.

Assume the long axis of the rigid rods makes an angle ψ with respectto the director of the system and the director is along one principal axis of the cubic lattice. Divide each rod into x basic units of equal width. Eachbasic unit occupies one cell in the lattice. x is actually the axial ratio of the rods. For simplicity, suppose that the dimension of a solvent moleculeis compatible to the size of a cell lattice. In this section we adopt thesame assignations as Flory. These may be different from those used in thepreceding section by Onsager.

In order to put a rod into the lattice, a postulate is made, which suggeststhat each rigid rod be divided into y sub-particles as shown in Figure 2.4b

y = x sin θ. (2.28)

Each sub-particle has x/y basic unit and its long axis is along thedirector. If a particle is perfectly aligned along the director, y is zero. As a

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66 Liquid Crystalline Polymers

Ψ

(a) (b)

 x x/y x

Figure 2.4. A rigid rod in the Flory lattice. (a) a rod making an angle Ψ to the director;(b) the rod divided into y  sub-particles.

particle is aligned off the director the value of y increases. Therefore, y canbe regarded as a measure of a rod’s deviation off the director. y is calledthe off-orientation degree or disorder degree.

Assume that the total number of cells in the system is n0 and ( j − 1)rods have been placed in the lattice. They have occupied x( j − 1) latticecells and hence n0 − x( j − 1) lattice cells remain unoccupied. In this case,there are ν j ways to put the j-th rod into the lattice

ν j = [n− x( j − 1)]N (x−yj)j P 

(yj−1)j , (2.29)

where yj is the number of the sub-particles of the rod; the first termrepresents the number of ways of putting the first basic unit of the first

sub-particle into the lattice which is the number of unoccupied cells. P j isthe ways of putting the first unit of remaining (y − 1) sub-particles; N j isthe number of ways of placing the remaining (x − yj) units entering intothe lattice.

First we will work out P  j . Once the first sub-particle’s position is deter-mined each of other sub-particles must be the closest neighbor to thepreceding sub-particle shown in Figure 2.4. In addition, the first unit mustbe immediately next to the last unit of the preceding sub-particle. Theprobability of such an arrangement is the volume fraction of unoccupiedcells in the system, so that

P j =

[n− x( j − 1)]

n . (2.30)

All units of each sub-particle must be in same row of the cell. Once thefirst unit has been put into the lattice (the cell must be unoccupied and isallowed to put in) each of remaining units must be positioned immediatelynext to preceding unit (the cell must be unoccupied). There are two pos-sibilities: the cell may be unoccupied and is allowed to enter in; the otherpossibility is that it has been occupied by the first unit of a sub-particle

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Theories of Liquid Crystalline Polymers 67

of another rod and is not allowed to enter in. The possibility N j of empti-ness for the cell is the fraction of empty cells in both empty cells and totalnumber of sub-particles. Thus N  j is

N  j =[n− x( j − 1)]

[n− x( j − 1) − y( j − 1)]

=[n− x( j − 1)]

[n− (x− y)( j − 1)],

(2.31)

where y is the average of  y of the ( j − 1) rods already in the system,y( j − 1) is the total number of sub-particles already in the lattice, i.e., thetotal number of the first unit of all the sub-particles.

Substitute P j and N j to Equation 2.29 one obtains

ν j = [n− x( j − 1)]N (x−yj)j P 

(yj−1)j

=[n− x( j − 1)]x

{[n− (x− y)( j − 1)](x−yj)n(yj−1)}

≈[n− x( j − 1)]![n− (x− y) j]!

(n− xj)![n− (x− y)( j − 1)]!n(yj−1).

(2.32)

Assume there are n p identical rigid rods in the system and the con-tribution of the n p rods to the partition function of the system can bewritten as

Z comb =1

n p!

npj=1

ν j , (2.33)

where the factor of (1/n p!) is introduced to avoid repeatedly countingidentical rods. Substituting Equation 2.32 into Equation 2.33 gives

Z comb =(ns + yn p)!

ns!n p!nnp(y−1), (2.34)

where ns = n− xn p is the number of empty cells left in the system whichare occupied by solvent molecules.

It is shown that both the numerator and denominator in Equation 2.34increases as the off-orientation degree y increases, but the denominatorincreases more rapidly. As a result, Z comb decreases with increasing y.It illustrates that if the disorder of configuration of rods in the systemincreases, the excluded volume occupied by each rod increases and the space

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68 Liquid Crystalline Polymers

in which rods can move freely decreases. Therefore, the collision betweenparticles increases and the entropy decreases, reducing the stability of thesystem. Inversely, if y decreases, the degree of orientation along the direc-tor becomes high, hence Z comb becomes great and the contribution to theenergy reduction accordingly becomes important.

In fact, the above-mentioned equation is valid only for the perfectlyordered case, i.e., the rods are all aligned in parallel. This illustrates thatthe Flory theory works well for concentrated solutions.

Another contribution to the total partition function of the system arisesfrom the orientation, i.e., Z orient

Z orient =y

ωyn pn py

npy, (2.35)

where n py is the number of rods with off-orientation degree y, ωy is thesolid angle fraction associated with y, and n py/n p represents the orientationdistribution function. The average of  y is given by

y =y

yn pyn p

(2.36)

It is illustrated from Equation 2.35 that if the system is in a perfectlyordered state, y = 1; thus n py = n p, Z orient  becomes very small. Otherwise,the system is in disorder (y = x) then ωy = n py/n p and Z orient = 1.

According to the Flory’s (1956) approximation, when the orientationalorder is high, n py/n p is important only in the range θ ≪ θ′. Assume thatn py/n p is uniform within the range. When θ > θ′, n py/n p is zero. In therange θ ≤ θ′ the solid angle becomes approximately (y/x)2. Therefore,

Z orient ≈ (y/x)2np . (2.37)

The fact that the orientational partition function Z orient increases as yincreases can be understood. Suppose the next neighbor of each cell in a

lattice is six. If the orientation is random each basic unit of a particle hasfive ways and hence the particle of  x units has 5x ways to enter into thelattice and thus the contribution to the entropy of the system is kBx ln5.In the perfectly ordered state, after the first unit is put into the lattice theremaining units enter the lattice via the same direction. The contributionto the system entropy is about zero, and thus is not favored, taking onlythe orientational entropy into account. Therefore, the fundamental reason

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Theories of Liquid Crystalline Polymers 69

for rigid rods to form a liquid crystal phase must be attributed to the stericrepulsion effect between the rods. Consequently, the partition function isgiven by

Z = Z combZ orient  =(ns + yn p)!(y/x)

2np

ns!n p!nnp(y−1). (2.38)

The equation describes the dependence of the partition function on n p (thenumber or concentration of rods), x (the axial ratio or degree of poly-merization of rods), and y (the averaged off-orientation degree). From thepartition function one can find the critical concentrations of phase sepa-

ration of the underlying system as a function of the axial ratio and otherquantities which are of interest.

2.2.2. Formation of the liquid crystal phase

From the partition function in Equation 2.38 the free energy can beobtained as

kBT = − lnZ = ns ln(1 − φ) + n p ln

φ

x

+ n p(y − 1)

− (ns + yn p) ln

1 − φ1 − y

x

− 2n p ln yx. (2.39)

For the isotropic system y = x and Equation 2.39 becomes

kBT = ns ln(1 − φ) + n p ln

φ

x

+ n p(y − 1), (2.40)

where

φ =xn pn

1 − φ =nsn

are the volume fractions of solvents and rods, respectively.The relation of the partition function and off-orientation degree yin Equation 2.39 is depicted in Figure 2.5 where x= 100. Each curvecorresponds to a different volume fraction.

As shown in Figure 2.5, for the volume fraction of rods less than criticalvalue φ∗ (here φ∗ = 0.0784) e.g., φ = 0.060, Z  increases monotonously withincreasing y. In other words, the more disorder the higher Z .

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70 Liquid Crystalline Polymers

Figure 2.5. The partition function vs. off-orientation degree of a rod system of axialratio x = 100. (Modified from Flory, 1956.)

As φ > φ∗ there is a maximum Z max  with y. If Z max  is greater than Z 0,the value for y = x, then Z max  gives the equilibrium state; otherwise, itis the meta-equilibrium state.

Another conclusion from Figure 2.5 is that the y value at Z  = Z max 

is much less than x/2, even for φ = φ∗. As φ increases, y at Z  = Z max 

decreases. It is illustrated that when the system transforms from the dis-ordered state (y = x) to the liquid crystal state, y changes abruptlyto x/2.

If  φ and x are great enough, there are two extremes in Z . The maximalvalue is associated with the stable (or meta-stable) states, in which y is lessthan that at the minimum Z . Differentiate F  in Equation 2.39 with respectto y and make d(− ln Z )/dy equal to zero. The y for the two extremes are

given by

φ =

x

(x − y)

1 − exp

−2

y

. (2.41)

There are two solutions for y. The smaller solution corresponds to the stableor meta-stable state in which Z  is at a maximum, as shown in Figure 2.5.The dependence of  φ on y can be obtained numerically.

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Theories of Liquid Crystalline Polymers 71

To obtain the critical value of φ∗ when the system starts to appear in themeta-stable state, i.e., the minimum value of  φ for the existence of liquidcrystal state, let

dy= 0

and φ∗ is given implicitly by

x = y∗ + y∗2 [exp(2/y∗) − 1]

2, (2.42)

where y∗ is the value at φ = φ∗, or

φ∗ = 1 −

1 − 2y∗

exp

−2y∗

(2.43)

Substituting 2.42 into 2.41 Flory et al ., in their 1956 approximation,obtained the critical φ∗ as the function of the axial ratio x

φ∗

8

x

1−

2

x

. (2.44)

If  x > 10, the error of the approximation in the above equation is lessthan 2%. It is concluded from Equation 2.44 that the larger the axial ratioof rods x, the less is φ∗.

Equation 2.44 is the well-known Flory formula which is widely used in

the study of liquid crystalline polymers. It should be pointed out that φ∗

isonly the minimal solution of Equation 2.41 at which the partition functionfirst shows a maximum. At this volume fraction φ∗, Z max  is actually lessthan the Z  at the disordered state (y = x). The system is at a meta-stablestate only when the volume fraction further increases to a greater value inwhich the system is indeed at a stable state.

It is shown in Equation 2.43 that as φ∗ increases up to φ∗ = 1,i.e., the neat polymer system, y∗ = 2. Substituting y∗ = 2 into Equation2.42 the axial ratio is

x = 2e = 5.44,

which is approximately the minimum axial ratio of rods that the systems

are able to show a liquid crystal phase.

2.2.3. Two phase equilibrium

According to the above analyses, when the concentration of rods in the

solvent increases beyond φ∗, phase separation occurs, and the biphasic

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72 Liquid Crystalline Polymers

coexistence of liquid crystal and isotropic phases appears. The chemicalpotentials of each component must be equal at the coexistence of twophases, i.e.,

µs = µ′

s

µ p = µ′

 p,(2.45)

where the subscript “s” is designated for the solvent while p is for the rigidrods, and µ and µ′ are the chemical potentials at the isotropic and liquidcrystal phase, respectively.

According to Equation 2.38, the derivative of (− lnZ ) with respect to

the volume fraction ns of the solvent gives

(µ′

s − µ0s)

RT = ln(1 − φ′) +

φ′(y − 1)

x− ln

1 − φ′

1 −

y

x

. (2.46)

At equilibrium d(− lnZ )/dy = 0. Replace the last term by 2/y fromEquation 2.41 and thus Equation 2.46 becomes

(µ′

s − µ0s)

RT = ln(1 − φ′) +

φ′(y − 1)

x+

2

y. (2.47)

The chemical potential for solvents in the isotropic phase is

(µs − µ0s)RT 

= ln(1 − φ) + φ

1 − 1x

. (2.48)

Similarly, the chemical potential of rods in the liquid crystal and in theisotropic phase are given respectively by

(µ′

 p − µ0 p)

RT = ln

φ′

x

+ φ′(y − 1) + 2 − 2 ln

yx

(2.49)

and(µ p − µ0

 p)

RT = ln

φ

x

+ φ(x− 1). (2.50)

The equilibrium state of coexistence of two phases is governed by thefollowing set of equations

ln(1 − φ′) +φ′(y − 1)

x+

2

y= ln(1 − φ) + φ

1 −

1

x

ln

φ′

x

+ φ′(y − 1) + 2 − 2 ln

yx

= ln

φ

x

+ φ(x− 1).

(2.51)

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Theories of Liquid Crystalline Polymers 73

Figure 2.6. Phase diagrams for various rod axial ratios. (From Flory & Ronca, 1979a.)

and Equation 2.43, where φ′ and φ are the volume fraction of rods in the

liquid crystal and isotropic phase, respectively.For various axial ratios x of the rods the numerical solutions of the above

set of equations are summarized in Figure 2.6.The following important conclusions can be obtained from Figure 2.6:

(1) Those rigid molecules capable of showing a stable liquid crystal phasemust have the axial ratio greater than x = 6.7. This value is some-what greater than the estimated value of  x = 5.44. We have emphasizedthat the estimate of the minimum axial ratio for forming a liquid crystalphase (x = 5.44) is that at which the partition function starts to take amaximum.

(2) At the equilibrium state, the volume fraction of rods in the two phases

decreases as the axial ratio x increases. The volume fraction of theliquid crystal phase is slightly greater than that of the isotropic phase,the ratio between these two critical volume fractions increases withincreasing x, but is always less than 1.56.

For enough large x, the critical volume fractions are respectively

φ =8

x, φ∗ =

12.5

x.

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74 Liquid Crystalline Polymers

Figure 2.7. Critical volume fraction vs. axial ratio. (Modified from Flory, 1961.Reproduced by permission of John Wiley & Sons, Inc.)

(3) y/x increases smoothly as x increases, however, the variation is notgreat. The value of y/x is very small which illustrates that the orderingin the liquid crystal phase is very high.

An experiment was carried out for PBLG to verify the theory. PBLGmolecules in solution adopt an extended α-helical conformation so thatthey are rigid rods. In Figure 2.7, the critical volume fractions of PBLG insolvents (Flory, 1961) were depicted as a function of the axial ratio. Thecurve A is for φ at which the liquid crystal phase starts to appear whilethe curve B is for φ′ at which the isotropic phase completely disappearsand the system becomes entirely a liquid crystal. The solid curve is thetheoretical expectation while the dashed line is the experimental result.Both are in good agreement.

2.2.4. Effect of “soft” interaction between molecules

Later Flory further took the two “soft” interactions between the moleculesinto account. The anisotropic interaction is associated with molecular ori-entations while the isotropic one is irrelevant of the molecular orientation.In fact, the anisotropic interaction was the basis of another well-knowntheory in liquid crystals — Maier–Saupe theory (Maier & Saupe, 1959).Flory successfully captured the essence of the theory.

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Theories of Liquid Crystalline Polymers 75

First we introduce the isotropic interaction which results from themixing of rods and solvents. The previous theory, without the mixing con-tribution to entropy, is only applicable to an athermal system. If thereis a mixing entropy contribution, the free energy in Equation 2.39 isimplemented by a term χφns, where χ is the Flory–Huggins interactionparameter. Flory called the mixing term the isotropic “soft” interaction todistinguish it from the steric interaction of rods.

At the coexistence of the two phases the chemical potentials of the rodsand solvents in both the liquid crystals phase and isotropic phase must beequal and thus the following set of equations for φ, φ′ and y holds

ln(1 − φ′) +φ′(y − 1)

x+

2

y+ χφ′

2

= ln(1 − φ) + φ

1 −

1

x

+ χφ2

ln

φ′

x

+ φ′(y − 1) + 2 − 2 ln

yx

+ χx(1 − φ′)2

= ln

φ

x

+ φ(x− 1) + χx(1 − φ)2. (2.52)

Figure 2.8 shows the numerical result of the rod/solvent system for the

rod axial ratio x = 100.The ordinate is the Flory–Huggins parameter χ and the abscissa is

the volume fraction of rigid rods. For negative χ, the two phase equilib-rium is basically independent of  χ. A positive χ has a significant effecton the two-phase equilibrium. The diagram can be divided into threeregions: in Region I the concentration is small and the system is in a singleisotropic phase; in Region II the system is in the liquid crystal phase; andin Region III both the liquid crystal and isotropic phases coexist. For χ lessthan, say, 0.07, the region of biphasic coexistence is narrow, φ∗∗/φ∗ = 1.5.For χ greater than 0.07 the volume fraction difference between the twophases becomes larger.

Figure 2.9 shows the experimental results of the PBLG/dimethylformamide(DMF)-methanol system (Nakajama et al ., 1968). The axialratios were 150 and 350, respectively, and the χ value was controlled byvarying the concentration of methanol. The more methanol the greater is χ.It was shown that for small concentrations of methanol, the biphasic rangeis narrow. As the volume fraction of methanol increases (up to 0.10–0.12)the biphasic range becomes wider.

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76 Liquid Crystalline Polymers

0

0.2

0.1

0

–0.1

–0.2

0.2 0.4

I + L CЉ

LC

LCЉLCЈ + LCЉ      I     +      L      C 

                            Ј

LCЈ

III

III

0.6 0.8 1.0

Figure 2.8. Phase diagram for rod axial ratio of 100. (From Flory, 1956.)

Figure 2.9. Experimental phase diagram of PBLG/dimethyl formamide(DMF)-methanol system. (Modified from Nakajama et al ., 1968.)

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Theories of Liquid Crystalline Polymers 77

Figure 2.10. The phase equilibrium vs. temperature. (From Miller et al ., 1974.)

The value of χ varies with the temperature, i.e., χ decreases as the tem-

perature increases. Wee & Miller (1971) examined the phase equilibriumas a function of temperature. The results are shown in Figure 2.10. Fortemperatures below 35 ◦C the phase diagram is basically the same as theFlory theory; while at higher temperatures, the curve deflects to a high con-centration regime. This phenomenon was observed in a system of cellulosederivatives (Navard et al., 1981).

Warner and Flory (1980) found that the introduction of the anisotropicattractive force predicts this effect. This anisotropic interaction associatedwith molecular orientation is expressed by

1

2

xn pφS 

2T ∗

, (2.53)

where S  is the order parameter and T ∗ is the characteristic temperaturewhich is a function of the anisotropy of the longitudinal and transverseelectric susceptibilities

kBT ∗

∝ r−6(∆α)2,

where r is the distance of neighboring rods.