Theory of athermal lyotropic liquid crystal systems

Polymer Science U.S.S.R. Vol. 30, No. 2, pp. 316-324, 1988 Printed in Poland

0032-3950/88 $10.00+ .00 O 1989 r ~ m o n Pren plc

THEORY OF ATHERMAL LYOTROPIC LIQUID CRYSTAL SYSTEMS* T. M. BIRSHTEIN,B. I. KOLEGOV and V. A. PRYAMITSYN Institute of High Molecular Compounds, U.S.S.R. Academy of Sciences

(Received 19 July 1986) The liquid crystal ordering of solutions of homogeneous and heterogeneous rod-like molecules and macromoleeules consisting of freely connected rods is described within the framework of the Flory lattice model and a virial expansion (Onsager approximation). The qualitative agreement and the quantitative differencesin the results for these two models are indicated. Phase separation into two anisotropic and one isotropic phases occurs with solutions of mixtures of strongly heterogeneous rods of two different lengths.

As Is known, Onsager [1] and Fiery [2] (with refinements by Fiery and Rouca [3]) were the first to study theoretically the thermodynamics of solutions of rod-like, essentially polymeric molecules, i.e. the formation of polymeric lyotropic liquid crystals. The approximate methods of calculating the statistical properties of such systems proposed by Onsager and Fiery [1, 2] are still important and continue to be ever more widely used, especially in recent years, because of the growing interest in low molecular weight liquid crystals and liquid crystal polymer systems [4, 5]. In the case of the above mentioned simplest systems the Onsager and Fiery methods agree qualitatively, but the results differ quantitatively: in the case of athermal solutions of rod-like macromolecules having a given d~gree of elongation the values of the solution concentrations corresponding to the transition from the disordered into the ordered state differ by a factor of more than two. Intuitively it would seem natural to prefer the results given by the Onsager method, which makes use of a virial expansion, and which is asymptotically precise for long rod-like particles of finite diameter in a continuous complex medium. On the other hand, the Fiery method is based on a discrete lattice model with an artificially introduced continuous distribution in the orientations of the rod-like macromolecules (in the more systematic DiMarzio model [6], the distribution of the orientations is discrete, so that it cannot adequately describe the liquid crystal phase with a high, differing from unity, order parameter). On the other hand, much experimental data on solutions of rigid chain polymers [7, 8] is in good agreement with the quantitative results of the Flory method. Furthermore, the lattice model system, which does not make use of a virial expansion, also * V y s o k o m o l . soyed. A30: N o . 2, 348-354, 1988.

316

Athermal lyotropic llqBid crystal systems

317

has undoubted advantages, e.g. it enables liquid crystal polymer systems to be considered at high concentrations, even when there is no solvent. A comparative analysis of the results of the Onsager and Flory methods would thus be useful, and in this paper athermal solutions of monodisperse rod-like molecules (rods), mixtures of rods of different lengths, and polymer chains made up of freely connected segments of different lengths will be considered. Solutions of monodisperserod-likeparticles. These systems were considered in the basic works of Onsager [1] and Flory [2, 3]. The reasons for the differences in the results of the two methods will be discussed in this paper. In the Onsager method the free energy of a solution of rod-like molecules is represented in the form (virial expansion)

NkTF-f----lnc+I f(n)ln(4nf(n))d~+lc f f d~df~'B2(~)f (n)f (n')'

(1)

where N is the number of rods in solution, c is the rod concentration, f (n) is an angular function of the rod distribution, B2(?) is the second virial coefficient of rod interaction, the direction vectors of which n and n' made an angle ? between each other, Ba(?) =2P2S31sin ~] (P=L/D, where L is the rod length and D the rod diameter), k is the Boltzmann constartt, and T is the temperature. The Onsager problem is solved by a variational method, using a test function

cosh(~cos0) f(n)=4n

sinh~

'

(2)

where g is a variational parameter indicating rod ordering, 0 is the angle between discrete nematie order and the rod axis, and

1

--~(2sin(O/2)) 2

(3)

In the Flory lattice model [2, 3]

F =ln c + 1- ccP In Nk---T

- h In (1 -

1- c

c (P - y))

+(y-1)-21n(y/P)+const,

(4)

where y/P= [sin 0 I. Expanding the logarithmic terms in (4) into a series with respect to concentration (assuming the solution is dilute) and considering only the first term of the expansion, with due allowance for the ordering characteristics of the solution y and ~, y[P--~2-/~, equations (1) and (4) can be reduced to a single form. The appropriate equations for an i~otropie solution I [5, 6] and for an anisotropie solution [7, 8] are Fz =In c + P2c + const

NkT

(5~

318

T.M.

BIRSHTBIN

et aL

Fl = In c + p2c/2 + const NkT = l n c + l n e + p 2 c 4_ + c o n s t - Fa NkT x/ Ir~

--Fa----=In c +2 In NkT

+ P e~-+const,

(6)

(7)

(8)

where eqns (5) and (7)correspond to the Onsager method and (6) and (8) to the Flory method. As can be seen, the equations for the free energy of the isotropic phase differ only in the value of the coefficient of "c", i.e. by the value of the second virial coefficient. In the Flory model this is less than half that of the Onsager model. In the anisotropic phase tb.is difference is retained with only small variations of the ratio of the coefficients, which is equal to 4/v/2-~"-, 1.6. This difference also results in an approximately twofold diff.~rence in the concentrations, corresponding to the boundaries of the two-phase region in the 1-.A phase transition. According to Onsager's results, the critical volume fractions of the rods in solution at the phase interface are equal to ~ = 3.3]P;

Oa = 4.5/P

Or= 8"0/P;

Ba= 12"5/P

(9)

and for the Flory data (10)

It is important to note that fundamentally this difference is associated only with the introduction of a discrete volume element (cell lattice), which is occupied by part of the dissolved substance or by an isotropic solvent molecule. In fact, the results of another two models of a solution of rods will be compared, in which only a discrete orientation of the particles is permitted along three mutually perpendicular directions. In the work of Zwansig [9] and Onsager [1] is is assumed that the medium is continuous (also using a virial expansion), and in the work of DiMarzio [6] a lattice model is considered. A twofold concentration difference is also obtained fol the I ~ A transition: 0t-~l.4/P; #a--- 1-7/P in [9] and ,91"~2.5/P; ,ga~3.1/P in [6]. Furthermore, Sokolova and Tumanyan [10] considered a solution of rods, of crosssection equal to the values of W for the lattice cells, within the framework of a lattice model with. discrete orientation. The critical volume fraction of the rods obtained was a diminishing function W, varying from 2.8/P at W= 1 to 1.6[P at W ~ oo, i.e. on passing to a continuous medium model. This confirms Flory's assertion [3] that the difference between the values of eqns (9) and (10) is associated mainly with effective allowance for the molecule dimensions in the lattice model, where they are assumed equal to the transverse dimensions of the rods. This conclusion correlated completely with known [11, 12] data on the relation between the binary distribution function in a system of solid rods and the concentration

319

Athermal lyotrop~cliquid crystal systems

of the system. At zero concentration, the binary function determines the second wrial coefficient for the interaction of solid spheres in a vacuum (or a continuous medium), and at high concentrations the second virlai coefficient for the interaction of two specific spheres in a solution of spheres similar to the two selected spheres. In this case the binary distribution function is non-oteady, and the second virial coefficient decreases because of the appearance of eff:ctwe attraction. Since in real system~ the solvent molecules are not negligibly small compared with the transverse dimensions of the dissolved rodlike molecules, the b,tter agreement of the experimental data [7, 8] with eqn. (9) rather than with eqn. (10) has a physical basis. Solutions of rod-like molecule mixtures. Flory and his coworkers [13, 14] generahzed a Iattice model scheme for the case of a solution of a mixture of molecules of different length and calculated phase diagrams for solutions of mixtures of rod-like molecules of two types with the asymmetry parameters PL and P2
F NkT

Inc+(l-x)ln(l-x)+xlnx+(I

x)jfl(nj)ln(47rf1(n))dQ

+~x~q2

d~d~2,1sinT(n, n,)if2(n)]2(n, )

.

(ll)

where Nj and Nz are the number of rod~ of the first and second type, and 7g

2

V = N I + N 2 ; b=--4Pl D3 ; q=P1/P2; x=N2/N. Birshtem et aL [16] used a function of type (2) as a test function, and Lekkerkerker and Coulon [17] used a precision numer;cal method for finding the angular distribution functions. For systems wzth a P1/P2 ratio of 2 the calculation results [16, 17] agree quahtatively with those of Abe and Flory [14]. Figure 1 shows some phase diagrams for the ternary ~ystem with P1 =40 and P2=20, obtained within the framework of a lattice model

320

T.M. BIRSHT'dlNet aL

and a virial expansion [16]. A constant factor is introduced to obtain convergence of the points ~q2~=0. In the case of rods with PI/P2 =5 Lekkerkerker and Coulon [17] failed to observe two-phase A~ +A2 and three-phase [+A~ +,42 regions, and assumed these to be a lattice model artifact. On the basis of the method proposed by these authors [17], accurate

o/ ,, \

\o o I

I1

FIQ. 1. Phase diagram for a solution of rod mixtures: I-Pxffi40 and Pz=20, as calculated by the Onsager (full line) and Flory (broken line) methods. The volume fractions a~ and ~2, as calculated by the Flory method, multiplied by a constant factor of about 0"4 for indicating the extreme point ~92~ffi0;I I - P , =50 and P2= 10, as calculated by the Onsager method, the triangle xyz indicating the three-phase region; the region between the lines yz and the broken line corresponds to the region where two anisotropic phases coexist. solutions of the integral-differential equations were obtained by analyzing ternary systems with P1/P2>>.3, selecting G=oala/(~q,A+~q2A) as the independent variable, (but not G=~qtlA/(,91,t+-~92a), as did the above authors [17],. The net result was i \

-

,~

/

that a three-phase region is also observed (Fig. 1) in the case of strongly heterogeneous systems within the framework of the virial expansion. As found by Abe and Flory [14], the stability boundary of the isotropic phase has a metastable loop. It should be noted that the boundary between the A and A, +Az regions (as given by Abe and Flory [14]) provisional, since it is extremely difficult to calculate accurately. Accordingly, Onsager and Flory provide very close relations for such systems, which differ only in the values of a single numerical factor. As shown by Birshtein et aL [16], this also concerns the order parameter along the boundary of the anisotropic phase. It is important to note that in using the wrial expansion method for systems of appreciable heterogeneity with PI[P2 >12, test functions of the same type as eqn. (2) are unsuccessful for the comparatively low values obtained for the orientation parameter ~ for short rods. Correct results can be obtained by using the method given in [17] for accurate numerical solution of the equations. The original version of the Flory method [2] is also tmsuecessful when the orientation of one of the lattice model components is comparatively weak, and it is necessary to use a more accurate approach [3] (for further details see [18]).

321

Athermal lyotropie liquid crystal systems

As a further example of a system with intermolecular heterogeneity, Fig. 2 shows phase diagrams for solutions of mixtures of which one component is a rod-like molecule, and the other is a polymer chain consisting of ra freely joined segments of given length.

¥

I

//

Fro. 2. Phase diagram for rod mixtures with t'2=40, 48, and a freely joined chain of rods: I-P1 =P2--48, m2--1, ml-~l (1), 100 (2), m (3); II-Pt--48, P~=40, m2=l, ml--10 (1), 101)(2), ao (3). The broken hnes correspond to nodes connecting the corresponding states.

Systems with P1=48 (m~ = 1, 10, 100) and P2=40 and 48 (m2= 1) were considered by Birshtein et aL [16] on the basis of a virial expansion. It can be seen that increase in the degree of polymerization of one of the components results in an effective increase in its degree of mesogenicity. Even when P1 =P2, the components are redistributed between the phases: the isotropic phase is enriched in the monomer component (mz = 1), and the anisotropic phase in the polymer component. The effect is strengthened when the inequality P, > P2 applies; a noticeable widening of the two-phase region is observed (Fig. 2). Figure 2 is constructed from calculation data based on an eqn. (2) test function. It ~hows tile inadequacy of the approach, since the boundary of the anisotropic region on the right hand side of the diagram is not shown. Solutions of chains consistin# of different length segments. Cham~ made up of freely joined segments of different lengths are the simplest model of macromolecules containing sections of different rigidity and capable of liquid crystal ordering. A theoretical analysis of the behaviour of macromolecule solutions consisting of two types of segments is given in previous papers [18-21] within the framework of the Flory model. Analysis of typical phase diagrams of such systems in the form of ternary phase diagrams (see papers [18] and [20] for further details) shows that two limiting types of phase diagram are possible. In the first case, when the difference between Pt and Pz is not too large an I---,At transition is observed, where A1 i~ an extremely ordered phase, and the width of the two-phase region remains approximately constant. In the second case, the I--,A2 transition involves a sudden contraction of the two-phase region. The second type of diagram is obtained when the rods differ significantly in length, and the short rods

322

T.M. BIRSHTEINet al.

exhibit hardly any ord:ring at the b3undary of the amsotropic region, thus acting as "quasi-isotropically" bond:d solvent. Calculations for similar systems based on a virial expansion confirm this result. The mixed phase diagram shown in Fig. 3 ha~ the most complex structure. It can be seen that both. methods give qualitatively similar results. On changing the proportion of short segments in the backbone from 0 to about 90 ~ (left hand part of the diagram in Fig. 3) the two-phase region of coexistence of the liquid crystal phase and the isotropic solution is steadily contracted. Thus, m the case of the diagram of Fig. 3 (I)

/

\ \

7/ 0.£

o-% 0.5

\ \0.4

o./

o.,

o e~ I

'" 11

FIG. 3. Phase diagram for an endless freelyjoined chain with the components PI = 50 and P2 = 10, as calculated by the Onsager (II) method, and a chain wtlth the components P1=80 and P2= 10, as calculated by the Flory method (I).

the relative width (~9a - oat)13a of the two-phase region varies from about 0.3 to 0.01, and at the centre part of the diagram of Fig. 3 the transition boundary is almost a straight line. Figure 4 shows the ordering characteristics at tile boundary of the anisotropic phase as a function of the volume fraction of the short segments of the chain. Two characteristics are shown, i.e. the parameter y determining the mean projection of a segment on a plane perpendicular to the nematic order dlrcctor, and the conventional order parameter S, as calculated on the assumption of a lattice-free model. As can be seen from Fig. 4, in the left hand part of Fig. 3 there is a 1 - ~ A 2 transition, and the degree of orientation of the short segments at the boundary of the anisotropic phase is extremely small. When the content of short segments is high (right hand part of Fig. 3) the width of the two-phase region is the same as for chains from segment,~ of the same (approximately average) length, and both the long and short segments acquire appreciable ordering (Fig. 4), i.e. there is an I - * A 1 transition. Birshtein and Merkureva [18] and Birshtein and Kolegov [20] concluded from analysis of a lattice model that there is a continuous change in the anisotropic phase from A2 to At. In this work this section of the "ternary diagram" is analyzed more

Athermal lyotropie liquid crystal systems

323

carefully and in greater detail. As can be seen from diagram I of Fig. 3, at the g~ven values o f P~ and P2 and oa2A/(oalA--92,t)~0"9 theory predicts that two phase transitions occur in succession when the solution concentration is increased, i.e. an I--*A2 transition and an A2--*A~ transition (the two-phase regions I+A2 and A2+A~ respectively correspond to each phase transition). When ~2A,/(81a + ~92A)-~0"938 a ternary

Y 2O

II

10

5 0-7 -

I I

8.3 0.I

I o5

I 0.9 G

I-to 4. Order parameters S and disorientation y as a funcuon of 6 = 9z;'(ont+ 82) on the hne of the

I--,A transition for freely 3olned chains with P~=50 and Pz= 10 (I, 2, 1', 2") (Onsager method) and P~ =80 and P2= 10 (3, 4) (Flory method). 1, 1', and 3' correspond to P~ segments, and 2, 2', and 4 to P2 segments

zone can appear, indicating a tllree-phase equthbnum, i.e. of the ~sotroplc phase I and the two amsotrop~c phases A1 and A2, which have different degrees of ordering. The coexistence of nematlc lyotroplc liquid crystal phases in macromolecule solutions becomes possible only when the system has extreme intramolecular heterogeneity. When PI/P2 is decreased, up to three transitions acquire I-~A2 character, and an A2--+A~ transttion is not observed. In conclusion it is noted that the complete agreement between the results based on the Ftory lattice model and the Onsager continuous model on the one hand confirm the rehability of these results, and on the other hand indicate the possibility of using each method to d,~scrlbe the polymer systems. The contraction of thc t~vo-phase region, ~.e. the convergenc~ of the properties of the isotroplc and anisotroplc phases when there ~s apprecmble intramolecular herterog~,neity and the fact that a phase transition can occur between two amsoptropic phase can be added to the important results obtained in thl, work.

Translated by N. S'rAND~N

324

T . M . BXRSMTEZNe t

at.

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17. 18. 19. 20. 21.

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