Reentrant transitions in liquid crystals: Theoretical predictions of new phenomena

Volume 115, number 6 PHYSICS LETTERS A 21 April 1986

REENTRANT TRANSITIONS IN LIQUID CRYSTALS: THEORETICAL PREDICTIONS OF NEW PHENOMENA

James S. WALKER and Jonathan L. MACE

Department of Physics, Washington State University, Pullman, WA 99164, USA

Received 30 December 1985; accepted for publication 18 February 1986

A new mechanism is proposed for the reentrant nematic-smectic A-nematic transitions seen in several liquid crystals. The

mechanism applies in particular to systems with strong dipolar head-group interactions, and involves a subtle interplay

between these forces and the Lennard-Jones interactions of the tails. A simple model which embodies the mechanism is

developed, and solved exactly. In addition to providing an understanding of the experiments to date, our model makes definite

predictions for new types of reentrant phase diagrams.

Reentrant phase transitions have been observed in a variety of systems, such as amorphous metal hy- drides [ 11, adsorbed monolayers [2], liquid mixtures [3] and liquid crystals [4,5]. In each case a transition occurs from a “more ordered” to a ‘less ordered” phase as temperature is lowered; thus the less ordered phase is reentered. A transition of this type seems to violate thermodynamic stability, hence reentrance serves as a challenge to our microscopic understand-

ing of these systems. In systems such as liquid mixtures the underlying causes of the reentrance are well understood [6,7], though in others, such as liquid crystals, the situation is less settled.

In this letter a new microscopic mechanism for the reentrance observed in certain liquid crystals is

proposed, and its consequences are explored. We car- ry out an exact solution of a simple model based on this mechanism, and find good qualitative agreement with experiments. Other mechanisms, involving flop- py tails [B], optimum densities [9] and frustration [ 10,111, also compare well with experiment, though none goes beyond current observations to predict new effects that would further test the theories. The mechanism described here, however, differs in that it predicts, as a natural consequence, phase diagrams that have not yet been seen experimentally, but which may indeed be physically realizable.

We focus our attention on those liquid crystals

which exhibit the reentrant sequence of phases, nematic-bilayer-smectic-A-nematic, as temperature is lowered [4]. In these systems there are strong dipolar head-group interactions which, by favoring anti- alignment of neighboring molecules, play a dominant role in producing the bilayer smectic phase. Each molecule also has a hydrocarbon chain extending in one direction from the head group. Thus, molecules interact with one another not only through dipole forces, but by LennardJones interactions between the hydrocarbon tails as well. It should be noted, too, that the tails experience steric hindrance due to hard-core repulsions with one another [B] .

Given these elements it is easy to see how reentrance can come about. First, dipole interactions alone would simply drive a transition from nematic to bilayer smectic, with neighboring molecules point- ing in opposite directions. The tails, on the other hand, can be seen to favor either alignment or anti- alignment, depending on whether the system is at high or low temperatures, respectively. For example, alignment of neighboring molecules is favorable in terms of the LennardJones attraction between the tails, but unfavorable in that many tail configurations are disallowed due to steric hindrance. That is, energy is lowered in such a case, but so too is entropy. When molecules are anti-aligned the favorable Lennard-Jones attraction is no longer present, but

281


neither are there as many constraints on the tail configurations; thus energy considerations are unfavor-

represented by Uii = 1, or none interact at all, Ujj = 0. The partition function for the full model is then

able, whereas entropy is increased. defined as Therefore, at high temperatures, where entropy

is of most importance, the tails favor anti-alignment, and thus assist the dipole interactions in forming the bilayer smectic. At lower temperatures the tails favor alignment of the molecules, to take advantage of the Lennard-Jones interactions, and thus the system

can be driven back into the nematic phase, and ulti- mately, at still lower temperatures, into a monolayer smectic. Thus, the mechanism proposed here depends crucially on the ratio of the dipole and Lennard-Jones interaction energies, as well as on the entropy associated with tail configurations.

2 = Tr Tr

As a final comment, our calculations, presented below, show that the mechanism just outlined is suf- ficient to produce reentrance. In real systems, however, it is probable that a variety of mechanisms act in concert. Here, we study our newly proposed mechanism in isolation, and reserve for the discussion a consideration of the likely results of including additional effects such as frustration.

The hamiltonian we study deals with interactions

between molecules in a single layer only, as in our previous work [lo], and identities long-range in-plane antiferroelectric order with formation of the bilayer smectic phase. For a given set of nearest neighbor molecules, i and j, the reduced bond hamiltonian is

+ a2uij)(1 + sisj)/21 exp[-PH(i,i)l , (2) where a(uii) is the degeneracy associated with the various tail interactions. In particular, we let 52(l) = 4,

where by 4 we denote the number of configurations appropriate to a single tail, and we assume that if neighboring tails are in the same configuration they can interact fully. When neighboring tails are in different configurations we take there to be no interaction at all. If a given tail is in one of 4 possible contigura- tions, its neighbor can be in a different configuration in one of q - 1 ways. But not all of these configurations will be allowed, due to steric hindrance. Thus we let Cl(O) = q(Aq - 1), where A < 1 is the acceptance fraction, in this way incorporating, at least crude ly, the effects of hardcore repulsions. When the molecules are anti-aligned there are no constraints on the

tail configurations, thus oii = 0 and Uii = 1 both con- tribute a degeneracy of q2/2.

-/W(i,j) = K,SiSi t nK2 Uii(l t SiSi)/2 , (1)

where 0 = 1 /kB T, with k, the Boltzmann constant and T the absolute temperature. The variables Si = +1 are Ising spins, and refer to the molecular orientations; namely, whether a molecule points up or down with- in the horizontal layer. Thus, K, , which represents the dipole-dipole interaction, is taken to be negative so that anti-alignment of neighbors is preferred.

Before proceeding, it is of interest to note that we have carried out calculations in which tail configurations are explicitly counted on a suitable lattice. The results of this complex procedure, however, are in complete qualitative agreement with those obtained from the simple model used here, These calculations are useful, though, in determining reasonable values for the parameters 4 and A to be used in our model.

Since the focus here is on the qualitative aspects of these systems, we present only the least involved calculations.

The parameter K2, on the other hand, is the attrac- tive (K2 > 0) interaction between each of the n units

of the tail, and thus, comes into play only when neighboring molecules have the same orientation. In order for the tails to interact, however, it is necessary not only that the molecules be aligned, but that the tails assume configurations which bring them near one another. In our simplified model we assume an all or none situation in this regard; thus, for a given coniig- uration either all n units of the tails interact together,

To study the system described by eq. (2) we per- form a trace over the uii variables, a calculation that can be carried out exactly. The physical motivation behind this procedure is that the tail degrees of free- dom , represented by the uV , do not themselves under- go a phase transition. Rather, the transition involves antiferroelectric ordering of the molecules, that is, the si variables. The tails, then, serve merely to renor- malize the bare dipole-dipole interactions, and in this way cause the reentrance. This type of “trace- out” calculation has been used with a great deal of success in studies of reentrant liquid mixtures [7].

282


When the trace in eq. (2) is completed, one is left with a partition function involving only the Ising variables, Si. The result is

(3)

where l/IK,I 2 -

-oHI (i, j) = K, SiSi •t K, . (4)

The effective Ising coupling, K,, and the energy per bond, K,, are determined from the following two

conditions:

exp(KI +Ko)

= 4 exp(K1 + nK2) + 4644 - 1) exp(K1) ,

l-

0 I I I I I I I t I

0 1 2 3

Si = Sj 3 (5)

exp(-K, t Ko) =q2 exp(-K1), Si f~i. (6)

3.2 I I

- (b) N

3.1 -

Thus we find,

KI=K1t~ln{[exp(nK2)-l]/q+A},

and

(7) 3.0 -

T/T, _

K. = f ln{q3 [exp(nK2) t Aq - 11) . (8)

Note that this transformation is exact since the uV of each bond are completely independent of one another.

We are now in a position to use the known proper- ties of the two-dimensional Ising model [ 121. In particular, we choose to consider a square lattice since this w-ill eliminate frustration as a factor, and since

the precise choice of lattice should not affect the qualitative results.‘Thus, the critical points are defined by K, = *ln(l + fi)/2, where the plus or minus sign corresponds to the transition into the antiferroelectric phase (bilayer smectic) or the ferroelectric phase (monolayer smectic), respectively.

A convenient and useful way to present our results is with a dimensionless temperature, T = l/IKy,I, and

a ratio of the IerinardJones to dipole interactions, R = K,/IK, I. A typical phase diagram, for given values ofA and 4, is shown in fig. la. Note the reentrant character of the bilayer smectic A phase (labeled SA), and the appearance at lower temperatures of a monolayer smectic phase (SAI). For different values ofA and ~7 a family of curves is produced, all with the same features as in fig. 1 a. For example, when 9 is increased

Fig. 1. Phase diagrams resulting from the model defined by eq. (2). (a) A typical phase diagram in reduced temperature (l/]Kr I) versus the ratio of the Lennard-Jones to dipole- dipole interaction energies (R = Ka /I K1 I). The physical range of interest for most systems will be a small region near the nose of the SA phase. (b) Phase diagram obtained from (a) by incorporating the dependence of the interaction energies on intermolecular separation, r, as described in eqs. (9) and (11). At atmospheric pressure the intermolecular separation is represented by ro. Note the reappearance at smaller values of r/ro, i.e. at higher pressures, of another reentrant smectic phase. In all figures the number of tail configurations is taken to be q =A [2(d - 1) ] “, where d is the dimensionality (d = 3) and n is the number of hydrocarbon units in each tail (n = 8). The factor [2(d - l)]” results from counting the possible configurations of an uninhibited single tail on a lattice, and A, the acceptance fraction, represents steric hindrance and has the value 0.8 in all three figures.

R=K,/ IK,l

RN

b/r0 I $,

0.98 0.9% 0.94

r/r0

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Volume 115, number 6 PHYSICS LETTERS A

there is more entropic advantage for anti-alignment, thus the stability of the bilayer smectic is increased. Similarly, a decrease in A means more steric hindrance,

21 April 1986

which again favors anti-aligned molecules, and increases the range of the bilayer phase.

Clearly the qualitative features of the experimental systems are reproduced in this model, but it is possible to go further in this respect by explicitly using

the separation dependence of the dipole and Lennard- Jones interactions. This can be done as follows; first,

note that at atmospheric pressure the system is char- acterized by a given intermolecular separation, ro, which in turn implies a certain dipole (Kl) and LennardJones (K2) interaction, and thus a given value of R. If this value of R is less than Rmax, shown

in fig. 1 a, the bilayer smectic phase will be observed

over a range of values of l/lK, 1. As pressure is increased one can expect the separation r to decrease approximately linearly. Since K, is simply the ratio of the dipole interaction energy to the thermal energy, and since the dipole interaction increases as r is made smaller, it follows that the thermal energy must increase correspondingly. Thus, the temperature as a function of r is

T/To = [Ed @)/Ed (%)I lKI 1-l ? (9)

where the normalization temperature is defined by

TO = E, (r#‘R . In addition, since the dipole and Lennard-Jones

interactions obey different r dependences, an increase in pressure will affect the value of R. To illu- strate this, fig. 2a presents the LennardJones (ELJ) and dipole (Ed) interaction energies, normalized by

IEd (ro) I, plotted versus the separation, which we normalize by ro. In particular,

Ed (r)/IKd (‘0) 1 = -(r/ro)-3 , (10)

and

EL J (r) --=-m[(,),, -($--1, (11) IEd@&

where o is a measure of the relative strengths of the two interactions, and n sets the length scale for E,J.

Note that Ed is given for the case of anti-aligned molecules, whereas ELJ applies to molecules that are aligned. The tendency for alignment versus anti-alignment is quantified in the ratio R = K2/IK, I. Since Kl

= &Ed(r) < 0 and K2 = +!?L J (r) > 0, one obtains

E/ IE,(r,) I -2.0

-3.0

-A C-l .._ 1.0 0.95 0.90 0.05 0.60

r/r0

2.75

R=Kp/lK,l

2.70

2.65 1 .oo 0.95

r/r0

0.90

Fig. 2. Dependence of the Lennard-Jones and dipole-dipole interaction energies, EL, and Ed, and their ratio, R, on the reduced average molecuti separation, r/re. The energies are reduced by Ed(&. (a) Interaction energies which generate fig. lb. Pointsrr/ro and rz/re in this figure and fig. lb corresponds to R = Rmax, the maximum value of R for which the SA phase exists. The vertical dashed line represents the range of r/r0 in (b) and in figs. 3 and lb. The minimum of the Lennard-Jones potential occurs at r_Jre. (b) Typical re- . ..a..

sults for R versus r/i-o. The ratio of ELJ (rmin) to Ed 0-O) for a single hydrocarbon tail unit, w, has values -1.756 in (a) and kg. lb, and -1.763, -1.755; and -1.712 in(i), (ii); and (iii) of(b) respectively (see eq. (10)). Since there are two values of r/r0 corresponding to each vahre of R, i.e. one at low pressure and another at high pressure, both low pressure and high pressure reentrant SA phases are possible. Curve (i) represents phase diagrams of the type shown in FT. lb whereas curves (ii) and (iii) correspond exactly to the phase diagrams of figs. 3a and 3b.

284


R = r~o(r/r~)~ [igy,,, -(Yg]* (12)

The results for R, corresponding to different values of w (q = 0.8 throughout), are shown in fig. 2b. Re-

call that R,,, is the largest value of R for which the

bilayer smectic is possible. Of particular interest is the generic form of R ver-

sus r/r,-,; namely, it first increases, then decreases, as

r/r0 is steadily decreased. For example, if a system displays the bilayer smectic phase at atmospheric pressure, an increase in pressure may cause it to pinch off, then reappear again at a higher pressure, as in fig.

lb, This is the behavior one expects when R has the

form of curve (i) in fig. 2b. If R is given by the curve (ii), the smectic phase pinches off to a point, then immediately opens again, as shown in fig. 3a. Finally,

curve (iii) in fig. 2b results in the phase diagram shown in fig. 3b, where the smectic region pinches in slightly, then widens out again as pressure is increased further.

These phase diagrams, figs. lb, 3a and 3b, are all obtained by making only small changes in w, the ratio of the LennardJones to dipole strengths, which indicates that slight modifications of the molecules might suffice to sweep through this global family of curves. In fact, some systems which appear not to be reentrant at all, may actually be like the system in fig. 3b, in which case a modest increase the tail interactions, or decrease in the dipole forces, could cause the smectic to pinch off as in figs. 3a and lb. Also, some of these new phase diagrams may occur in systems already studied, though experiments carried out to higher pressure would be needed to see them. For example, the low pressure (r/r0 - 1) SA phase in fig. lb is almost identical to experimental temper-

ature-versus-pressure phase diagrams in 80CB [4], which would indicate that the high pressure SA

phase may reappear at pressures on the order of 3 000 to 4 000 atmospheres.

Thus we see that this simple mechanism provides a good understanding of the present experimental

situation, and in addition predicts new effects which, if found, would yield additional insight into these systems. Finally, frustration has not been included in these considerations, though it may be of consid- erable importance in real systems. If frustration is present, and becomes important at relatively low pressures, then the high pressure effects predicted

r/r0

r/r0

Fig. 3. Reentrant bilayer smectic phase diagrams corresponding to ewes (ii) and (iii) of fig. 2b. The parameter w is ad- justed so that the maximum value of R is&,x (see eq. (11) and (ii) of fig. 2b). (b) The value of w in (a) is decreased slightly so that the maximum of R is less than R,,, (see @ii of fig. 2b).

here may not be seen. If in other systems, however, frustration becomes important only at fairly high pressures, then a high pressure SA phase may occur, only to vanish again at still higher pressures as a result of frustration. These and other possibilities will be considered in detail in a future article.

The authors wish to thank the Department of Physics, Washington State University, for its generous support.

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Volume 115, number 6

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PHYSICS LETTERS A 21 April 1986

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Reentrant transitions in liquid crystals: Theoretical predictions of new phenomena