Field effects and the critical end point in polymeric nematics

Volume 119, number 4

PHYSICS LETTERS A

15 December 1986

FIELD EFFECTS AND THE CRITICAL END POINT IN POLYMERIC NEMATICS X.-J.WANG Department ofPhysics, Tsinghua University, Beijing, PR China

and M. WARNER RutherfordAppleton Laboratory, Chilton, Didcot, Oxon, OXIJ OQX, UK Received 25 May 1986; revised manuscript received 10 October 1986; accepted for publication 10 October 1986

We calculate the effect ofan external field on nematic polymer liquid crystals with polarizabledipoles along the chain backbone. In close analogy to conventional nematics we find a line of first-order transitions from nematic to paranematic as the field is increased, culminating in a critical end pointas the critical field is attained. This critical point is related to those found by other authors. The results hold for electric and magnetic fields, the latter coupling to diamagnetic anisotropy. The electric case will be complicated ifpermanent dipoles are present. We demonstrate the equivalence of these results to the I..andau approach in a field. Shortcomings in previous work on nematic polymers in fields are analysed.

Polymeric nematics, like their monomeric counterparts, are highly anisotropic in their molecular shape and other properties. Monomeric systems are well known [1,2] to respond to externally applied fields which couple either to, say, the positive diamagnetic anisotropy of molecules or their anisotropy ofmolecular polarizabiity in the electric case. A close analogy to magnetic and fluid systems exists. There is a first-order transition from nematic to isotropic fluid as temperature is raised, persisting until a critical value ofthe applied field, whereupon the discontinuity of the order parameter S= vanishes. Here P2(x) 2 —1)12 andis second denotesLegendre averaging polyover nomial (3x the angles 0 that rod axes make with the ordering direction. Estimates, for monomeric systems of the field at this critical end point disagree [1,2], but the qualitative picture given above is in close accord with conventional critical systems. We shall discuss backbone nematic polymers where the shape (stiffness) and other anisotropic properties are arranged along the main chain (in contrast to comb systems). Also we consider only polarizable, 0375-9601/861$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

rather than permanent dipole systems, for which the part of the field energy associated with the anisotropic phase is, for a chain of length L: Uf=

—

~0E2 ~

$

L

ds P 2(cos 0(s))

(1)

where L~ais the anisotropy in molecular polarizability per unit length of backbone, E is the applied electric field and 0(s) is the angle the backbone at arc length point s makes with E, considered to be in the ordering direction for the to fluid. magnetic formula is entirely analogous (1). The Permanent dipoles represent a system ofgreat complexity and are treated elsewhere [3]. This is because their field energy depends on P 1 rather than P2 in (1) and in (2)between below. This difference in symmetry, distinguishing up and down relates to hair pins [4] and other nonperturbative aspects. For high degrees of ordering, any perturbative approach is doomed to failure. areThe the other bend contributors energy Ub: to the energy of a molecule

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PHYSICS LETFERS A

Ub=~Jds ~2(s)

(2)

~

.3L~

and the nematic mean field U~:

15 December 1986

.002’

L

Unzz~aS$dsP2(cos0(s)),

(3) .1~-(SN_Sp)~

where ~ is the bend modulus for the main chain consideredasaworm [5—7],u(s) is the (unit) tangent vector ofthe chain at s and a is the nematic coupling constant (independent of temperature in MS theory [1], or with a temperature component if steric interactions are considered The partition function7]). is derived by integrating the Boltzmann factor associated with (1) —(3) functionally over chain configurations:

phase?~ change until the field a critical value ofcritical y~=0.0111 with =0.39958 and an reaches order parameter at this end point of S~=0.1782. In this mean field theory the coexistence

z=

curve is parabolic and the critical isochamp is cubic around S~,

J

öu(s)

xexp{(— Ub[u]

—

U~[u]

—

Uf[u])IkBT}.

(4)

.38

39

Fig. 1. Order parameters in the nematic (SN) and paranematic (Sr) phases as a function of reduced temperature 2’ forTK various there reduced external fields, y. At the clearing temperature is a discontinuity oforderparameter characteristic of a first-order

influence, it is interesting to compare the field

This allows a complete description of all chain properties and involves [7] a coupling constant, in the E= 0 case, 42 = 3acSI (kBT)2 This dimensionless measure of the strength of the nematic field shows immediately how to define the reduced temperature ?fora chain:

behaviour ofpolymeric with that ofsimple nematics. In fact the results below show that they are overwhelmingly and surprisingly similar. Adding Uf to U~means that the coupling constant, in analogy with ref. [7], becomes 423(5+y)/?2 (6)

7= kB TI~J~,

(5)

where y = ~(i~aI3a)~E2. For long chains the calcu-

revealing the first difference from the conventional MS case where 7= kBTIa’ and a’ is an energy rather than an energy per unit length. In (5) the stiffness ~ representing the internal degrees of freedom of the chain, plays a role. Indeed the internal entropy of a chain (the many complexions in (4)) usually dominates polymer properties. Polymer nematics are interesting because this tendency competes with the nematic requirements of stiffness and conformity to the molecular field. We have shown [81that the consequence is that polymeric nematics are much more strongly first order in their transition than monomeric systems the chain soaks up a lot of conformational entropy on entering the isotropic phase. Experimentally this is so, see our discussion after (10). For reasons of scaling, (5), and the large entropic

lation ofthe free energy ofthe nematic and isotropic phases, their equality locating the transition and the self-consistency condition determining S, is straightforward, see eqs. (12)— (16) of ref. [8]. Our results are closely analogous to those of Wojtowicz and Sheng [1]. Fig. 1 shows the variation of order of order parameter S with reduced temperature ? for a variety of fixed values of y. For ~ = 0.0111 there is a jump in S from the paranematic value S~to the nematic value SN. For y =0, S~,= 0, which is the conventional first-order situation [9]. As y increases the jump SN S~,decreases while the temperature at which the jump occurs, ?K( ~), increases. In fact the coexistence region is parabolic: (SNS)2l067(??K) , (7)

—

—

182

,

—

where

~ is the critical end point when y = y~and

Y [El Fig. 2. Reduced clearing temperature FK as a function of the effective force y (broken line). It is a straight line given by &( y) = Fk( 0) + 1.079 y. Since y scales as E* with E the applied field, Fk( E), the full line, is parabolic. These curves represent the sequence of first-order transitions, nematic to paranematic, as the field is increased until the critical end point is reached.

the jump vanishes. The critical end-point temperature is F; = 0.39958. Mean field theory and the parabolic nature of (7) are unlikely to be applicable there. The law of rectilinear diameters, (S, +S,)/2 =S” is also obeyed [ 1 ] with the order parameter at the critical point, S”=O.1782. The critical isochamp ( y = yc) is seen to have the cubic form expected of mean field theory. Fig. 2 shows the rise in Fk(y) with y which is linear: &( j> = Fk( 0) + 1.079y )

15 December 1986

PHYSICS LETTERS A

Volume 119, number 4

(8)

where the zero field value Tk(O) =0.3878 [ 5,8,9]. Since y aE2 the curve F,(E) is quadratic with E, shown also in fig. 2. Both curves represent a line of first-order transitions culminating at a critical end point. A critical point has been found in the original introduction of the worm model for nematic environments [ 51, that is, lipid bilayers. Such layers impose an external field of dipolar symmetry on the chains so the result of ref. [ 51 is not comparable with ours. Critical points have been found [ lo] for fields applied to solutions of worm chains, the solvent effects (phase separation) giving results somewhat different from ours. We now contrast our and other related [ 5, lo] results with those of Lee [ 111. He purports to con-

sider permanent dipoles, see his fig. 5 and the comments that follow (note that the molecule he discusses has both permanent and polarizable dipoles). His field energy, his eq. (3)) is however the same as our eq. (1) and is appropriate only to polarizable dipoles and not to permanent dipoles. The latter have an energy N -E cos 0 which changes sign on reversing E. Permanent dipoles only contribute in a form analogous to Lee’s (3) in a Landau theory where E represents part of the macroscopic dielectric tensor and not a molecular quantity as required in a microscopic theory of this type. Taking Lee’s eq. (3) to represent a molecular polarizable dipole energy instead, we note several erroneous aspects of his results. In his fig. 2 the zero field S-T curve displays a discontinuity with a nonzero value of S at temperatures above the discontinuity. This is in distinction to other work on nematic polymers [5,7-lo] and also indicates an apparent second-order vanishing of the order parameter at some higher temperature. Second-order behaviour in the absence of a field is in discord with usual results for systems with quadrupolar (P2) symmetry. Such symmetry imposes a first-order character on nematits. With finite fields the discontinuity gets larger in Lee’s fig. 2. We assume that this precludes the existence of a critical end point in Lee’s analysis. Below we demonstrate that the Landau theory for nematic polymers in a field is identical to that for conventional nematics [ 21. This Landau theory has been demonstrated [ 21 to have a critical end point. It remains to make contact with the Landau approach of Homreich [ 21. The Landau theory of Rusakov and Shliomis [9] for liquid crystalline polymers in zero field predicts a first-order nematic to isotropic phase change. We can quickly recover their results by noting [ 81 that the free energy per unit length of chain scaled by a is FIaL=$(~210+S2)’

(9) where lo = fd2 +A0 and no is the lowest eigenvalue of the spheroidal wave equation [ 71. Using the small A2 expansion of A0 and writing p = Sl F ’ one obtains the conventional Landau expansion: FIaLF’ = 4 ( T2 -&)p2

-4

&yp’+!

334xx51,3x7P4...

.

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Volume 119, number 4

PHYSICS LETTERS A

We recall [9] that F is only the actual free energy at the absolute minimum, at which point p gives the actual order parameter and self-consistency is attained. This limit to the stability of the isotropic phase on supercooling into the nematic region is = where the p2 term vanishes, i.e. at ~/~= 0.3652, whereupon this fictitious second-order transition is at a reduced temperature (?K D * )/?K = 0.0584, about half the Maier—Saupe value for monomeric nematics. Roughly an order of magnitude separates theory and experiments in that case. For polymeric nematics, with their large latent entropy, one might expect first-orderbehaviour ofthe Landau theory to be more nearly obeyed. This is indeed so; the reduced fictitious second-order transition for the polymer “DDA 9-L” has been observed [12] to be 0.0662 providing encouraging agreement with ref. [7]. With an applied field present (6) gives 42 to be put into (9) and yields additional terms to those in (10): —

—

(2/1 5?2)yp _(2/3X5X7?4)(y2p+yp2?2)

.

tinuity in order parameter as field increases, terminate in a critical end point. A qualitative similarity is expected since both simple and polymeric systems have quadrupolar symmetry. However the quantitative similarity is surprising because the connectivity of polymers, with its attendant entropy, yields strong first-order behaviour in zero field in distinction to the marginal behaviour of conventional nematics. Additionally the interplay of entropy, bend and nematic field yields a scaling behavious quite different from Maier—Saupe theory. We conclude on a note of caution about comparing with experiment if permanent dipoles are present. Because the system is first order (a finite order exists below TK) and because of collective effects in the coupling to the external field [3], the response can be non-linear and one does not necessarily expect TK E2 to hold when permanent dipoles are present [5]. We thank Dr. J.M.F. Gunn for helpful discussions. W.X.-J. thanks the Rutherford Appleton Laboratory for hospitality during the execution ofthis work.

(11)

The first term is the conventional —SE2 term, the second and third are normally neglected in Landau theories of nematics. Insertion ofthe numerical values of 5, y and 7 shows that the second term is 16% of the first at the critical end point and its neglect may have quantitatively consequences. Eq. (10) augmented by the first term of (l1)is the same as Hornreich’s starting point and his analysis follows. Because our results are similar to his, it follows that the neglect of the last two terms of (11) has no qualitative consequences for the phase behaviour. Since we have derived the Landau free energy from our microscopic starting point the coefficients are related to each other, see (10) and (11), in a simple way. We could instead make the coefficIents arbitrary and fit them as Hornreich did to get a more accurate prediction of the end point location. In summary, we find that nematic polymers with polarizable dipoles along their lengthexhibit a strong similarity with conventional nematics when an external field is applied. First-order transitions from nematic to paranematic, with a diminishing discon-

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15 December 1986

References [11P.J. Wojtowicz and P. Sheng,

Phys. Lett. A 48 (1974) 235. [2] R.M. Hornreich, Phys. Lett. A 109 (1985) 232. [3] ratory J.M.F. Report Gunn and 86. M. Warner, Rutherford Appleton Labo[4] P.G. de Gennes, in: Polymer liquid crystals, eds. A. Cifferri, W.R. Krigbaum and R.B. Meyer (Academic Press, New York, 1982). [5] F. Jhhnig,(1981) J. Chem. Phys. 70 (1979) 3279; Mol. Cryst. Liq. Cryst63 157. [6] A.ten Bosch, P. Maissa and P. Sixou, J. Phys. Lett. 44(1983) 105. [7] M. Warner, J.M.F. Gunn and A. Baumgärtner, J. Phys. A 18(1985)3007. [8] X.-J. Wang and M. Warner, J. Phys. A 19 (1986), to be published. [9] V.V. Rusakov and M.. Shliomis, J. Phys. (Paris) Lett. 46 (1985) L935. [10] A.R. Khokhlov and A.N. Semenov, Macromolecules 15 (1982) 1272. [11] A. Lee, Phys. Lett. A 113 (1986) 391. [12] Blumstein, G. Maret, F.Proc. Volino, A.F. Martins (Strasand A. 27thR.B. mt. Blumstein, Symp. Macromolecules bourg, 1981).