THERMODYNAMICS OF THE RE-ENTRANT NEMATIC-BILAYER SMECTIC A TRANSITION

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THERMODYNAMICS OF THE RE-ENTRANT NEMATIC-BILAYER SMECTIC A TRANSITION

N. Clark

To cite this version:

N. Clark. THERMODYNAMICS OF THE RE-ENTRANT NEMATIC-BILAYER SMEC- TIC A TRANSITION. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-345-C3-349.

�10.1051/jphyscol:1979367�. �jpa-00218763�

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THERMODYNAMICS OF THE RE-ENTRANT NEMATIC-BILAYER SMECTlC A TRANSITION

(*)

N. A. CLARK

Dept. of Physics and Astrophysics, University of Colorado, Boulder, CO 80309, U.S.A.

Rksum6. - On analyse des mesures ricentes sur la transition ntmatique-smectique A dans des cristaux liquides smectiques A formant des bicouches. On presente et discute un modele thermodyna- mique qui rend compte de la forme elliptique de la ligne de transition et qui facilite la determination des parametres dtterminant la ligne de transition.

Abstract. - Recent measurements of nematic-smectic A pressure-temperature phase boundaries obtained in bilayer forming smectic A liquid crystals are analyzed. A thermodynamic model which accounts for the elliptical shape of the phase boundaries and which facilitates determination of the various parameters determining the phase boundary is presented and discussed.

1. Introduction. - In the recent letter [l]

High pressure in~lestigation of the re-entrant nematic bilayer srnectic A transition by Cladis, Bogardus, Daniels, and Taylor, henceforth referred to as [l], measurements of P-T phase diagrams of a group of liquid crystalline one and two component systems which form bilayer smectic phases were presented.

The observed nematic-smectic A (N-SA) phase boun- daries were found to be reentrant, i.e. they curl around in the P-T plane so that, at sufficiently high pressure, a nematic to smectic A phase transition is obtained upon both heating and cooling, up to some maximum pressure, P,, above which only the nematic is observ- ed. Figure l shows measurements (obtained from figure 1 of [l]) of the nematic-smectic A P-T phase boundary of COOB (4 cyano-4' octyloxy biphenyl), which clearly demonstrate this effect.

This behavior represents an exciting departure from heretofore observed N-SA phase boundaries and presents an excellent opportunity to connect moIecuiar structure and macroscopic phenomena. The purpose of this comment is to present a thermodynamic model that accounts for the data of [l] and a statement of the conclusions concerning the N-SA transition which thermodynamics allows.

2. Thermodynamics of the nematic-smectic A phase transition. - We begin with a qualitative analysis of the phase diagram. Along the phase boundary the molar Cibbs free energy difference between the nematic and smectic A phases, AG = G, - G,, is

(*) Work supported by NSF MRL contract DMR 76-01 111, NSF contract DMR 76-22452 and the Joint Services Electronics Program.

FIG. 1. - Measured nematic-smectic A phase boundary of COOB from Ref. [l]. Solid curve is the best fit of an ellipse to the data.

zero and the Clausius-Clapyron equation may be applied :

Here S and V denote molar entropy and volume respectively. At atmospheric pressure it is generally observed that A S > 0 at the N-SA transition so that

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979367

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C3-346 N. A. CLARK

A S > 0 at P = 1 atm is a reasonable assumption.

Since, from figure 1 (6P/6T)A,,o is positive in this region we must have that AV > 0 a t P = l atm.

As the pressure is increased (6P/6 T),;,',

,

approaches zero and then changes sign implying that A V decreases with increasing P, passing through zero (AV = 0 at P = 1.6 kbar and T = 83 OC) and then going negative. Defining the compressability of a phase as

we may therefore conclude that

is negative and since

bN

and

p,

are both negative that

l PN l

>

I P, l.

Continuing around the phase boundary (GPIST),,,, approaches zero and changes sign implying that A S decreases passing through zero (AS = 0 at P ⁼P, = 1.84kbar, T = 77OC) and going negative. In the reentrant portion of the phase boundary then both AS and A V are negative. Defining specific heat as C, = T(BS/aT), the fact that AS decreases with decreasing temperature implies that the specific heat difference AC, = C,, - C,, is positive for the transition i.e. C,, > C,,.

The shape of the phase boundary may be quantita- tively related to the thermodynamic parameters of the two phases by obtaining an expression for the P, T dependence of the free energy difference AG(P, T).

In the vicinity of some reference point P,, To AG may be expressed in a Taylor's series as AG = AGO

+

AVo(P - P,) - AS,(T - T,)

+

where the following notation is used [2] :

AGO = AG(Po, To) AS, = AS(T,, P,) AVo = A V(Po, To)

The neglect of higher order terms in applying eq. (2) is equivalent to assuming that AB, Acr, and AC, are

sufficiently independent of P and T over the range of application. As we shall see this assumption appears to be reasonable in these systems. The N-SA phase boundary in the P-T plane is obtained from eq. (2) by setting AG to zero and solving for P(T). The result- ing equation is of the form

which is the general equation for a conic section. Thus P - T phase boundaries are at least locally expressable as conic sections and in systems, apparently like the ones under consideration, where AC,, Acr, AB do not depend strongly on P and T, boundaries will have the shape of conic sections over a large area in the P - T plane. The shape of the N-SA phase boundary obtained in [l] suggested an ellipse to be the appropriate conic section. Elliptical phase boundaries will be obtained from eq. (4) when

1

Acr2 T,/AP AC,

I

< 1.

The fit of the data of figure 1 to an ellipse was therefore attempted. At first glance this fit involves varying 8 independent parameters : AP, P,, Aa, To, QC,, AV,, AS,, and AGO. However since the choice of the reference point is arbitrary we may put it at the center of the fitted ellipse. At this point AGO is maximum and the first derivatives of AG, namely AV, and AS, are zero, effectively eliminating four parameters : P,, To, AV, and AS,. The fitted ellipse will then be characterized by three quantities, the lengths of its major and minor axes a and b respectively and an orientation angle do of the major axis relative to the T axis. We will thus be left with three equations and four unknowns (AP, Aa, AC,, and AGO), allowing the determination of ratios among any of these four quantities. The fit of an ellipse to the data of [l] on COOB is shown in figure 1 and is excellent, verifying our assumption that Aa, AB, and AC, are reasonably constant over the area covered by the phase boundary in the P-T plane. The fit yields the following ratios :

with the center at P, = 4.0 atm,

P (kbar) To = 42.85 OC , do = tan-' 2

= 49.40 T("C> 1

p -

50 and

( Acr2 To/AP ACp ( = 0.51

.

The following units have been used

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S (cal/mole OC)

,

u(cc/mole O C ) ,

P

((~c)~/rnole cal), and

C, (cal/mole OC)

.

The center of the ellipse falls in the smectic A phase so that AGO is positive. Hence as expected from the qualitative analysis we have AC, > 0 and AP < 0.

The entropy and volume differences between the nematic and smectic A phases for any P, T'may be obtained from eq. (2) and are as follows :

T - To AS(P, T ) = - Aa(P - P,)

+

AC,-

To (6) Av(p, T) = Ap(P - P,)

+

^Au(T- To)

.

The lines of zero entropy and volume difference may be obtained from these equations and are indicated in figure 1. From the fitted ellipse the ratios AS(P, T)/AG, and A V(P, T)/AG, may be calculated at any point on the phase boundary. Eq. (6) shows that A S decreases continuously along the phase boundary as the A S = 0 line ( P = P,,,) is approached. This is in accord with the observation made in [l] that a smaller Pm corre- lates with a smaller latent heat at P = 1 atm. Eqs. (2) and (6) may be combined to give Pm and Tm, the maximum pressure and temperature at which the smectic A will be stable. The results are :

where again P,, To give the ellipse center.

Without further data eq. (5) to (7) represent the total quantitative information which may be obtained from a phase diagram like that of figure 1. The addition of a single measurement of either AS or A V at any place on the phase boundary would allow using eq. (6) the determination of absolute numbers for AP, Aa, AC, and AGO. Hence augmenting such P-T measurements with A S and/or AV measurements would be extremely useful, allowing determination of differences of thermodynamic parameters as well as independent checks of this model. Alternatively determi- nations of the P-T diagram such as presented in [l], could be used to establish relative values of AS, permitting calorimetric measurements of A S to be checked.

We now present some additional qualitative arguments concerning the nature of the P-T phase boundary for various values of the thermodynamic parameters which may be made on the basis of this model.

It is clear from the preceding discussion that for a reentrant N-SA phase transition to occur the smectic phase must have a smaller bulk isothermal compres-

sability magnitude and a smaller isobaric heat capa- city than the nematic, i.e. that

I PS l

<

l pN I

and

C,, < C,,. Furthermore, eq. (7) shows that the

larger AB is the lower will be the pressure required to produce reentrant behaviour. However the proper location of the ellipse center is also necessary to observable reentrant behaviour and is an additional key property of bilayer smectic A materials.

An elliptical P-T boundary may be characterized by the location of its center (P,, To), its size (major axis length, a), its shape (E = bla), and its orientation 8,. The shape and orientation of the ellipse depend only on relative values of Ap, AC,, and Aa. The orientation angle depends primarily on the ratio

generally decreasing with increasing A. For a particu- lar elliptical P-T boundary P and T coordinates can be chosen to make 80 = 450. In this coordinate system the eccentricity of the ellipse is given by

Thus the extension of the ellipse along a line of positive slope with the P-T plane such as for COOB in figure 1 indicates that E < 1 (E = 0.297 for COOB) and therefore that AalAC, is positive. Increasing Au will further increase the eccentricity stretching the ellipse out a P-T diagonal. The influence of Aa in the ellipse shape is also evident in the A S = 0 lines in figure 1. For Au = 0 the AS = 0 line is parallel to the P axis and the A V = 0 line is parallel to the T axis. As Aa increases from zero these lines approach each other becoming nearly parallel to the ellipse major axis for large AM.

The size of the ellipse depends on the ratios AuIAG,, APIAG,, and AC,/AGo, i.e. with fixed relative values of Aa, Ap, and AC,, the ellipse will maintain a constant shape but its dimensions will scale as a, b K

m.

The position (P,, To) of the ellipse center is that point for which the entropy difference, AS, and volume difference, AV, between the two phases are zero. The ellipse center will be shifted in the P-T plane by any change which produces entropy or volume differences which are independent of P, T.

It is easy to show that a change in A S will shift the ellipse center along the AV = 0 line and a change in A V will shift it along the A S ⁼0 line. Changes in A S and AV will in general alter AGO, i.e. AG at the ellipse center and therefore the ellipse size will change as it moves.

We may now apply these considerations to the P-T boundary data presented in [I] on the CBNA-CBHA mixture. For a mixture differences in thermodynamic parameters obtained are those for each species having equal chemical potential in both the nematic and

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C3-348 N. A. CLARK

smectic A phases. Figure 3a of [l] presents P-TN-SA boundaries for various concentrations of CBHA (N-p-cyanobenzylidene-p-heptylaniline) in CBNA (N- p-cyanobenzylidene-p-nonylaniline), showing a sys- tematic shift of the P-T boundary toward lower P and T with increasing CBHA concentration. Elliptical P-T phase boundaries were fit to these measurements.

For concentrations where a sizable part of the ellipse was determined experimentally (0.17 c 0.51) unambiguous values of the parameters could be obtained and the fitted ellipses agree well with the data. As defined in [l]

satisfactory fits were obtained by assuming that the shape of the phase boundary was not changing ( A M , AB, AC, independent of c ) but that it was only translating (AV, AS changing). Hence the c = 0.51 ellipse was shifted in the P, T plane without altering other parameters until a fit was obtained and the new origin (P,, To) noted for each c 2 0.51. For c = 0 a reliable fit could not be obtained. Figure 2 shows the data of reference [l] along with the fitted ellipses for each curve. The parameters derived from the fits are given in table I. The open circles with dots

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indicate the origin P,, To of a given ellipse. For c r weight CBHAlweight (CBHA

+

CBNA)

.

c 2 0.32 these points lie nearly on a line of slope 6Po/6To = 4.6 kbar/oC where 6P0 and 6T0 are the For concentrations c > 0.51 data sufficient for an

accurate fit are not available. For these concentrations

FIG. 2. -Data from ref. [l] showing nematic-smectic A phase boundaries obtained in CBNA-CBHA mixtures of various concentrations. Solid curves are best fit ellipses to measured phase boundaries. The dotted open circles indicate the ellipse origin PO, To for a particular concentration. Paths of zero entropy (AS = 0 ) and zero volume (AV = 0) differences are indicated for several concen-

trations. c = weight CBHA/weight (CBNA

+

^CBHA).

shifts in ellipse center produced by a weight ratio change 6c. The AS = 0 and A V = 0 lines for several of the best fitted ellipses are also shown in figure 2.

It is evident that the line along which P,, To moves with increasing c is nearly parallel to the AS = 0, AV = 0 lines. Using the preceding discussion this is the behaviour expected if 6P0 and 6T0 are caused by changes 6 AS and 6 AV in the entropy and volume differences respectively with increasing c. Quantita- tively 6P0, and ST, are related to 6 AS and 6 AV by

X ( - AD6 AS - Aa6 A V ) .

For c 2 0.32 6Po/6c and 6T0/6c are both negative and (P,, To) moves on a line that lies between the AV = 0 and AS = 0 lines. For this region then 6 AS/& > 0 and 6 AV/& < 0. Hence increasing the CBNA weight fraction either decreases the volume difference AV, - AV, or increases the entropy diffe- rence ASN - AS, between the nematic and smectic A phases or both for c > 0.32.

The changes in parameters with increasing c suggest

Thermodvnamic parameters obtainedfiom fits of ellipses to measured nematic-smectic A P-Tphase boundaries in a CBHA-CBNA mixture. Uaits are deJnedfollowing Eq. ( 5 ) in tlze text

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two regimes of behaviour : for c ,< 0.3 and c 2 0.3.

As c increases from zero to 0.3 AGO decreases by roughly an order of magnitude, A V decreases, A S decreases, and Ap increases by about a factor of 2.

For c > 0.32, AGO, Aa, AB, and AC, cease to change significantly but an increasing A S and/or decreasing A V continue to move (P,, To) to lower temperatures and pressures with increasing c.

3. Conclusion. - We have presented a thermody- namic model of the P - T behaviour of the nematic- smectic A phase transition. The necessary conditions for the occurrence of reentrant phase boundaries as observed in [l] have been shown to be

IP, I

>

I P, 1,-

C,, > C,,, and a, > a,. None of these conditions

appear to be particularly unusual so that most N-SA transitions may be expected to be reentrant at some pressure P,. The thermodynamic arguments presented here show that P, will be lower in materials having (i) a larger

I

^Ap

I,

(ii) lower A V(P, T), (iii) larger AS(P, T ) , and (iv) smaller AGO. The CBNA-CBHA mixture data of [l] showed that the addition of CBHA pro- duced relatively little significant change in Aa, AC,, or AP and yet dramatically reduced P,,, by reducing A V(P, T) and AGO and increasing AS(P, T). This suggests that it is (ii) to (iv) which are most important in affecting the low value of P, in the bilayer smectic A materials. The bilayer smectic A structure must therefore allow a smectic A phase with a relatively small volume and energy difference andtor large entropy difference from the nematic at a given P, T.

References [l] CLADIS, P. E., BOGARJUS, R. K., DANIELS, W. B and TAYLOR,

G. N., ((High Pressure Investigation of the Re-Entrant Nematic Bilaver Smectic A Transition D.

[2] In addition to P, T, V, and S two additional thermodynamic variables are reauired to thermodvnamicallv characterize the smectic A phase, D, the smectic layer spacing and @, the conjugate force. If pi, is the smectic A stress tensor and z the uniaxial direction then

P = (c,,

+

^c,,

+

^ffr,)/3

and @ = oz. - cXX. Six independent thermodynamic derivatives relate S, V , and D to P, T, and Q 3 . Bulk P-T measurements are carried out at constant @ ( @ = 0)

so that thermodynamic derivatives in all of the suc- ceeding discussion are all carried out at constant Q, e.g.

p, = (aV/dP),,,. The variation of D in the P, T plane is given by

These derivatives are independent of those determining the P-T phase boundary, namely (aS/aT),.,, (aVIo'T),.,, (av/ap)T',@, (aG/aT)~,,, ( a G / a P ) ~ , ~ .

[3] MARTIN, P. C., PARODI, 0. and PERSHAN, P. S., Phys. Rev.

A 6 (1972) 2401.