Landau Theory of the Reentrant Nematic-Smectic A Phase Transition

Résumé. - Nous


-64.70E
Cladis and coworkers have recently demonstrated that for some liquid crystals, the boundary between nematic and smectic A phases in the P, T plane is reentrant in that at high enough pressures the nematic phase exists for temperatures both lower and higher than for the smectic phase [1]. In some materials that have a smectic-A to nematic phase transition which is not reentrant, reentrant behaviour can be induced by mixing it with a second liquid crystal that only has a nematic phase [2]. Cladis also observed that reentrant behaviour appears only in materials for which the smectic phase is of the bilayer type and suggested that at densities above some optimum value the interactions stabilizing smectic order would be suppressed by steric effects. The purpose of this article is to demonstrate that if one assumes the existence of this optimum density most of the observed properties of the reentrant transition are predicted by the Landau theory of the nematic to smectic A phase transition [3,4].
Reentrant behaviour has also been observed at the normal to superconducting phase boundary in materials containing magnetic impurities [5] and in one of the 3He phase transitions [6]. In view of the often used analogy between smectics and superfluids [3,4] it is important to note that reentrant behaviour is simply a consequence of two competing interactions whose sum can be optimized through control of another variable. Its appearance in superconductors, 3He and liquid crystals does not necessarily imply the microscopic interactions responsible for the effects, are analogous.
In the usual form of the Landau theory the difference between the free energy per unit mass of the smectic and nematic phases is expanded as a power series in the smectic order parameter 1 t/J I where Aa(T -T*). If B &#x3E; 0 the transition is second order and occurs at T = T*; if B 0 a first order transition occurs at T = T* + 3 B2/16 aC. The Landau theory has been widely applied to a number of different phase transitions and examples where the order parameter couples to some other variable are also common [4,7]. In the present case we want to describe coupling of 1 tjJ to both the density p and the relative concentration x of a binary mixture.
The simplest assumption is to add a term to eq. (1) of the form g(p, x) ~ ~ 12, expand g as a power series in (ppo, x -xo) and keep only the leading terms. If we consider only the density term first, and choose po to be the optimum density referred to above, the T* appearing in eq. (1) can be replaced by where t2 0 insures that p =1= po suppresses the phase transition.
The general procedure [8] for obtaining the nematic to smectic phase boundary is to first obtain expressions for the pressure P = p2(aF/ap)x,T, and the chemical potentials and . ~,_ .
--.,.for the two components and for both phases. Setting /~ //, and P for the smectic phase equal to those of the nematic phase defines the phase boundary. 0, implying a first order transition, or B &#x3E; 0 implying one of second order. On the other hand, since coupling between 1 tf¡ 12 and either p or x makes a negative contribution to B it also increases the tendency to first order behaviour. Also, since for dilute solutions (a,u/ax)-1 oc x one can reasonably expect that the effect of adding a second component to a pure material that exhibits a second order s -N transition may be to induce a tricritical point at some finite x.
Neglecting the terms described by eq. (5) the transition temperature can be obtained from eqs. (1) and (2) by setting T -T*(XN, pN) equal to either zero for a second order transition or 3 B 2/ 1 b aC if it is first order. Taking where aN and #N are respectively the thermal expansion and compressibility for the nematic phase. The phase boundary surface is predicted to have the form In figure 1 we show the fit of eq. (6) to data reported by Cladis et al. for a sample of pure 4-cyano-4'octyloxy biphenyl (80CB) [1]. In terms of the parameters of eq. (6) the maximum temperature, is indi- Although Tl, T2 and T3 are completely adjustablẽ N/~N can be estimated since the thermal expansion coefficient r:J..N f8tt.I 10-3 K-1 for almost all condensed fluids and also for nematic liquid crystals [9,10]. Also PN can be obtained from sound speeds in either the isotropic or nematic phases [11]. Typical values for similar materials obtain ~iN' ~ 4 or 5 x 1010 dyne . cm-2 ~ 40 or 50 kbar obtaining very good agreement between the expected value of ~iN/aN ~ 20 to 25 K/kbar, and the value that fits the data. This is the strongest evidence sup-porting the premise that the phase boundary curvature can be interpreted in terms of an optimum density for smectic order. Cladis already noted that in binary mixtures of N-p-cyanobenzylidene-p-nonylaniline (CBNA) and N-p-cyanobenzylidene-p-heptylaniline (CBHA) the maximum pressure PI is a linear function of the ratio of CBNA to CBHA [12]. This follows from the above considerations if we take To(x) to be a linear function of this ratio, e.g., where Since T1 (x) -To(x) is a constant PI is also linear in xJ(1 -x). Figure 2 contains one possible fit of eq. (6) to Cladis's data for this mixture using Although the linear dependence of T* on x(1 -X)-l is ad hoc it is interesting that it is sufficient to qualitatively describe all of the concentration effects. Furthermore, the ratio aN/#N is in reasonable agreement with the values expected from an optimum density model. More generally, one would certainly expect po to depend on x and there is no reason why the other parameters shouldn't also. Although inclusion of these effects would undoubtably improve the fit it does not seem worthwhile without some specific microscopic model. If we take P = 0 in eq. (6) (i.e., atmospheric pressure) the nematic-smectic A phase boundary is predicted to have the same form, e.g., that Cladis [2] used to describe her observations on binary mixtures of p[p-hexyloxy-benzylidene]-aminobenzonitrile (HBAB) and N-p-cyanobenzylidene-p-noctyloxyaniline (CBOOA). Expressions for TNs and YNs in terms of the parameters in eq. (6) are easily obtained. Mixtures of 4-cyano-4'-hexyloxybiphenyl (60CB) and (80CB) behave similarly [12].
According to the Landau theory there is no essential difference between the high and low temperature nematic phases and experiments support this. For example Cladis's measurements of the bend elastic constant K3 in HBAB-CBOOA mixtures behaved similarly on both sides of the smectic phase. Furthermore at concentrations high enough that a smectic phase does not appear (i.e., x &#x3E; xo) the standard Landau theory [3] predicts that both the bend and twist elastic constants, K3 and K2 should diverge as the correlation length. For x &#x3E; xo this will have the form [(7~ -T)2 + (~~')2, -1/2, Figure 3 demonstrates the fit between this expression and Cladis's data [2]. Schaetzing et al. [13] have done light scattering studies on the nematic phase of 60CB-80CB mixtures as the smectic is approached from both higher and lower temperatures. Although for x xo the tran- sitions appear weakly first order and are not well described by a Landau type of mean field theory the critical exponents in both cases are identical (i.e. y ~ 0.67 ± 0.05); again indicating the similarity of the high and low temperature nematic.
In summary we have demonstrated that the principle experimentally observed properties of reentrant nematic phases can be understood simply in terms of an optimum density for smectic ordering. The Landau theory expresses this qualitative idea in quantitative forms through the demonstration that the ratio o~/r equired to fit the data is the value expected from independent measurements. Although Clark previously [14] argued that the phase boundary should be elliptical, rather than parabolic, in the P-T plane, the extra terms required to change the parabolic form into the elliptical are small in comparison with the quadratic term in eq. (2).