J. Phys. Chem. 1985, 89, 2058-2064
2058
Icm-’I Figure 7. IR spectra of PCP complexes with TBNO and triethylamine (TEA): solid line, phenol to base ratio 1:l; dotted line, phenol to base ratio 5:l; dashed line, differential spectrum.
The Structure of ( A w 2 B Adducts. The analysis of steric conditions might show that the arrangement of two 2,6-dichlorophenol molecules at one oxygen atom seems to be rather impossible. One could expect the structure of (AH)2B adducts to be of type IV or V. For a confirmation of such an anticipation the IR spectra have been studied of 2:l adducts in which two phenol molecules are evidently nonequivalent. In Figure 6 are shown the absorption spectra of TBNO-phenol systems in the 1600-3 500-cm-’ region. The differential spectra obtained when a solution of stoichiometry CAo/CBo= 5 has been put into the test beam and that of CAo/CBo = 1 into the reference beam indicate the contribution of at least two different types of hydrogen bonds. There are two
alternative interpretations of the observed picture. Either the equilibrium of I and 111 complexes is present or a contribution of IV adducts takes place, in which the second molecule of phenol is attached to the oxygen atom of complexed phenol. To solve the question, all the spectra of the PCP-TBNO system for Cmo/CBoequal to 1:l and 5:l were analyzed. The PCP-TBNO system in CC14 belongs to the so-called inversion region where the proton transfer equilibrium HB * PT appeared to be extremely sensitive to the environment effects. The spectrum of 1:l species is typical for nearly pseudosymmetric O.-H.-O bonds with the gravity center of low-frequency protonic background absorption a t about 1000 cm-’. For the systems with CAHo/CBo > 1 one observes a drop in the absorption in the 400-1000-cm-’ region and an increase in the absorption between 1000 and 1600 cm-I. This means that a change in the potential energy curve takes place, favoring the proton transfer ionic form. Thus, an association of the second phenol molecule must take place via the lone electron pair of the hydroxyl group in the AHB complex, causing an increase of its proton donor ability. In the case of structure I11 one should expect the opposite effect, Le. a decrease of the proton transfer contribution, because of lowering the proton acceptor affinity of lone electron pairs. More evidence of such an association could be the comparison of the results obtained for the PCP-TEA (triethylamine) system in which there is only one possible way of bonding additional phenol molecules to the AHB complex, namely via lone electron pairs of hydroxyl groups. In Figure 7 we compared the evolution of IR spectra for PCP-TEA and PCP-TEA and PCP-TBNO systems at an excess of PCP under the same conditions. Acknowledgment. We thank Mr. Jerzy Jafiski for his assistance in the calculations. Reniptry NO. DPSO, 945-51-7; DBESO, 621-08-9; TPPO, 791-28-6; TOPO, 78-50-2; TPASO, 1153-05-5; DCNP, 618-80-4;PCP, 87-86-5; TCP, 88-06-2;DCP, 87-65-0;DPSEO, 7304-91-8;DCMP, 2432- 12-4; DPEPO, 1733-57-9; DPOPO, 29701-85-7;TBNO, 7529-21-7.
The Equal 0 Analysis. A Comprehensive Thermodynamics Treatment for the Calculation of Liquid Crystaltine Phase Diagrams Gerald R. Van Hecke Department of Chemistry, Harvey Mudd College, Claremont, California 91 711 (Received: June 19, 1984: In Final Form: February 1 , 1985)
Equilibrium biphasic regions in isobaric binary phase diagrams are usually calculated on the basis of setting equal the chemical potentials of each component in each phase. However, setting equal the total Gibbs energies of the two phases in equilibrium will define a composition that as a function of temperature will also describe the phase coexistence region. This procedure, called the equal G analysis, is applied to isobaric liquid crystalline phase diagrams and shown to offer ease of computation and wide applicability. Further, liquid crystalline phase diagrams are classified as ideal, nonideal, or reentrant depending on the values of the thermodynamic properties of heat capacity and excess Gibbs energy. To illustrate the technique, two phase diagrams are calculated: the binary system of 2-[4-n-heptylphenyl]-5-[4-(n-heptyloxy)phenyl]pyrimidineand 2-[4n-nonylphenyl]-5-[4-(n-nonyloxy)phenyl]pyrimidine,which exhibits miscible isotropic, smectic A, C, F, and G phases, and the binary system of 4-(n-octyloxy)-4’-cyanobiphenyl and 4-n-heptyl-4’-cyanobiphenyl, which is reentrant with a minimum temperature-composition point in the reentrant region.
Introduction
Mixtures, especially binary mixtures of liquid crystals, have been, and continue to be, extensively studied. Very often such study involves the determination of the phase diagram exhibited by a particular pair of mesogens. While a few efforts to quantify such diagrms have been reported using thermodynamic’ or sta(1) (a) Domon, M.; Billard, J. Pramana, Suppl. No. I , 131. (b) Cox, R. J.; Johnson, J. F. Zbm J . Res. Develop. 1978, 22, 51. (c) Van Hecke, G. R. 3. Phys. Chem. 1979,83, 2344.
tistical thermodynamic techniques,2 in the main binary phase diagrams have not been calculated or fit to any quantifiable theory. In this paper is presented a simple thermodynamic approach for the quantification of liquid crystal phase diagrams. The approach (2) (a) Humphries, R. L.; Luckhurst, G. R. Proc. R. SOC.London, Ser. A 1976, 352,41. (b) Phaovibul, 0.; Tang, I-M. 3. Sci. SOC.Thailand 1980, 6, 112. (c) Palffy-Muhoray, P.; Dunmar, D. A.; Miller, W. H.; Balzarini, D. A. “Liquid Crystals and Ordered Fluids”, Vol. 4, Griffin, A. C., Johnson, J . F., Eds.; Plenum Press: New York, 1984; p 615.
0022-3654/85/2089-2058$01 .50/0 0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 10, 1985 2059
Calculation of Liquid Crystalline Phase Diagrams
Figure 1. Qualitative isobaric plots of total Gibbs energy as a function
of temperature and composition for two phases in equilibrium. The compositions of the phases in equilibrium are determined by the double tangent construction. The composition for which the total Gibbs energies are equal, xg, is also shown. is called the equal G analysis and is based on setting equal the total Gibbs energies of the phases in equilibrium. In the literature of materials science, the resultant temperature-composition, T-x, curve is called the equiconcentration or allotropic curve. Here we will call the resultant T-x curve the equal G curve following Oonk who detailed the equal G analysis followed here in his m~nograph.~ Perhaps the reluctance to calculate liquid crystal phase diagrams has been because of concern for the computational labor involved. Application of the equal G analysis to liquid crystalline diagrams so reduces the computational effort that now systematic, quantifiable studies of liquid crystal phase diagrams can begin. Moreover properties of mixtures of liquid crystals (indeed liquids) should be better understood when the excess energies @, HE,and TSEdescribing the mixtures are understood. Knowledge of the experimental T-x diagram alone will allow excess energies to be estimated and quantitatively compared system to system, particularly for selected homologous series, hopefully to provide significant new insights. It is the purpose of this paper to illustrate the computational techniques with various examples.
General Theory For any phase under constant pressure the total Gibbs energy of that phase can be written as a sum of the partial molar Gibbs energies, that is, the chemical potentials of each component i in the phase of interest. Thus G(TJ) = CXiPi(T,X) i
(1)
where xi is the customary mole fraction of component i. Note also at constant pressure that G( T,x) and pi( T,x) are only functions of temperature and composition. Substituting the usual form for pi(T,x) gives the total Gibbs energy in the phase as
where p*i( T ) is a reference potential at the pressure of interest and is only a function of T. In the event the system is nonideal (see below) it is customary to add the terms p E i , the excess chemical potentials which are functions of temperature and pressure. The last term is just the excess G i b b energy of the phase GE. The logarithmic term, often abbreviated as R T L N ( x ) , is just the ideal Gibbs energy of mixing. For two components 1 and
Figure 2. Qualitative, possible shapes for ideal binary phase diagrams
(after van Laar ref sa). The equal G temperaturecomposition curve is the dashed line in each diagram. 2, just considering the a phase, we can write the Gibbs energy as
where the appropriate equation for a second, say /3, phase is obtained by replacing a by /3 in eq 3. In Figure 1, G, and G, are plotted in a qualitative manner but at a temperature chosen so that the two curves cross at least once. The traditional criterion of phase equilibrium, that the chemical potentials of each component be equal in each phase, is satisfied by the double tangent line and the intersection points along the temperature axes for pure 1 and 2. Suppose, however, that, instead of analysis of the phase equilibrium and hence phase diagrams in terms of the chemical potentials and the equilibrium phase compositions x, and x,, analysis is carried out using the composition determined by the intersection of G, and G,, that is, where G, equals G,. One composition, xgrdefines this point at the temperature for which G, and G, were drawn. This equal G composition will always be such that x, Ix8 Ix, where the equal signs apply to those cases where a maximum or minimum is observed in the diagram (see Nonideal Systems). Setting eq 3 for a phase equal to its parallel for /3 phase will give the equal G equation (1 - ~ ) A f i * i ( T ) xAp*Z(T) + AGE = 0 (4) where the composition x is taken to be the mole fraction of component 2 and Ap*i(T) is the chemical potential difference for species i between a and B phases. Here we consider a to be the higher temperature phase. Solutions to eq 4 give a temperature-composition phase diagram consisting of a single line describing the equilibrium between the phases LY and p. The situation is illustrated in Figure 2, where the traditional diagram is drawn in solid lines and the equal G curve drawn in dashed lines. Since many liquid crystal phase diagrams exhibit very narrow two-phase regions, the equal G analysis in a practical sense calculates such diagrams “exactly”. To proceed further with the analysis, expressions are required for &*,( T ) and AGE. Consider &*i( T ) first, noting that &*i( Tj depends only on the properties of pure component i. At a transition temperature Ti, P * ~ , ( T ) equals P*~,( T ) and this can be used as a reference point. Also, since (ap/aT), equals -S and (as/”), equals Cp/T then
+
2060 The Journal of Physical Chemistry, Vol. 89, No. 10, 1985
which gives for the (3 to a transition h ~ * i ( T ) = p * j , ( T ) - p*jp(T) = -AS*j(T- Ti) - Z(AC*pj) ( 6 ) where AS*i is a constant, the transition entropy at the transition temperature Ti,and Z(ACpj) is shorthand for the integral obtained from eq 5 for a and (3 phases. Perhaps it is worth pointing out here that, if the transition a to (3 were second order, AS*i equals 0 and A P * ~ T ( ) is entirely determined by the heat capacity term Z( A C*pj ) . For AGE, any functional form of temperature and composition can be used providing that the excess chemical potentials derived therefrom satisfy the Gibbs-Duhem equation. A common form for GE is the Redlich-Kister expansion4 GE = X ( 1 - x ) C A j ( 1 - 2x)i-I
(7)
I
where the Aj are usually taken to be constants or perhaps functions of temperature in the form H, - TSj. Using, for example, only the first term and considering the equal G analysis for two equilibrium phases (Y and (3, one finds that AGE becomes AGE = GE,
- GEp = ( A , - A ~ ) x1 (- X ) = AAx( 1
-X)
J
The above equal G equation given the thermodynamic parameters, the AS*/s, T/s, and AAis, can estimate any binary phase diagram with actually a minimum of computational difficulty. The remaining presentation in this paper will illustrate the application of this analysis to three cases which we will call ideal, nonideal, and reentrant. We will only briefly introduce the analysis for reentrant systems and defer detailed discussions to a future communication. We will show that ideal cases are described by AAj equal zero and A P p j zero or about zero, while nonideal cases require AAj to be nonzero and again AC*pj zero or about zero. Reentrant phenomena will be described by A P p i nonzero and hAj zero or about zero. Before taking up specific cases, it is worthwhile to develop an equation for the slope of the equal G curve, since analysis of the slope, particularly at initial concentrations, turning points,maxima, or minima, provides useful relationships between the required thermodynamic parameters. In general, the derivative of the equal G curve with respect to composition will yield an expression for the slope (dT/dx). Thus differentiating eq 4 after AGE has been replaced by eq 7 gives
- = - A H * ~ ( T+) Ar*z(T) AX
+
C A A j ( a [ X ( l- x ) ( l - 2 x y - q / a x ) / X
[
-(1 - x)
dAP* 1( T )
aT
aAAj aAP**(T) - x( 1 - x ) C - ( 1 -2 x p aT aT
excess Gibbs energies equal to zero. If it is also assumed that heat capacity effects are small compared to the entropies, substituting eq 6 into eq 4 and solving for temperature as a function of composition gives T=
( 1 -x)AS*,T, ( 1 - x)AS*,
+xAS*~T~ + xAS*2
(11)
It might be noted that this equation has the form AHav/ASav. Several comments on ideality should be made at this time. First, the above equation does not predict a linear relationship between T and x unless AS*,equals AS*2. If a linear relationship is observed, it is a special case of ideal behavior. Of course, if AS*, were to a good approximation equal to AS*2, then the T-x diagram would appear linear. As pointed out by van Laar in 1908, any of the phase diagrams shown in Figure 2 are ideal and can, therefore, be described by eq 11 given different values of entropies and temperatures (types iv and v need Cp data also h o w e ~ e r ) . ~ The analysis of the initial slopes of a phase diagram can give a quick test for the suitability of the ideal G analysis. Since for an ideal system AAj = 0, the slope eq 10 becomes
(8)
where recall x is the equal G composition in terms of mole fraction of component 2. This simple form for ACE is equivalent to the regular solution approximation. Using the Redlich-Kister expansion for AGE and the previous expressions for Ap*i( 7'),one can write the general q u a l G equation as ( 1 - X)[-AS*i(T- T i ) - Z(AC*pi)] + x[-AS*Z(T- Tz) Z(AC*,,)] + ~ ( -l x ) C A A j ( l - 2 x y ' = 0 ( 9 )
aT
Van Hecke
1
(10)
Equation 10 is the slope of the equal G curve at any temperature and composition. The use of this relationship will be illustrated below with specific examples. Ideal Systems A system exhibiting an ideal phase diagram will be one that can be described by the equal G curve obtained by setting the (3) Oonk, H. A. J. "Phase Theory"; Elsevier: New York, 1981. (4) Redlich, 0.;Kister, A. T. Ind. Eng. Chem. 1948, 40, 345.
Considering now an initial slope analysis where x = 0, T = T I , and thus A P * ~ ( T ~=) 0, the slope equation becomes
where s is the entropy ratio LL!?*~/AS*,.If the transition entropies (or enthalpies) are known, comparing the initial slope of the experimental phase diagram to the value calculated by eq 13 provides a sensitive test for system's ideality. Whether or not the transition entropies are known, the initial slope analysis can still test for ideality. Combining the initial slope equation obtained at x = 1 and T = T2 with eq 13 gives
Thus, if the product of the initial slopes does not satisfy eq 14, the diagram is not ideal (at least to the neglect of heat capacity effects). If the phase diagram is shown to be ideal by initial slope analysis but the entropies are not known, the phase diagram can still be calculated since eq 14 can be reduced to yield T=
+XST~ ( 1 - x ) + xs
( 1 - X ) TI
where the entropy ratio s can be used as a parameter to fit the experimental diagram. As an example of the technique, consider the smectic A-isotropic equilibrium shown in Figure 3 for the binary system 2[4-n-heptylphenyl] -5- [4-(n-heptyloxy)phenyl] pyrimidine (707) and 2-[4-n-nonylphenyl]-5-[4-(n-nonyloxy)phenyl]pyrimidine (909). Figure 3 is redrawn from Figure 17 of ref 6a. Enthalpy data are available for this binary mixture.6b-C However, the following discussion will take the approach that such data are not available. Such an approach represents the more practical case since the number of just phase diagrams is much larger than the number of phase diagrams with enthalpy data. Thus the analysis will be carried out using only the phase diagram and the results of the analysis will then be compared to the literature enthalpy ( 5 ) (a) van Laar, J. J. Z.Phys. Chem. 1908,63,216. 1908,64, 257. (b) See also ref la. ( 6 ) (a) Biering, A,; Demus, D.; Richter, L.; Sackmann, H.; Weigeleben, A.; Zaschke, H. Mol. Cryst. Liq. Cryst. 1980, 62, 1. (b) Wiegeleben, A.; Marzotko, D.; Demus, D. Cryst. Res. Technol.1981, 16, 1205. (c) Wiegeleben, A.; Richter, L.; Deresch, J. Demus, D. Mol. Cryst. Liq. Cryst. 1980, 59, 329.
Calculation of Liquid Crystalline Phase Diagrams
of the entropies and hAj/As*,are required to calculate a phase diagram. Estimates for the excess parameters, the AA], can be obtained from initial slope analyses and/or the presence of maxima or minima. The equation for the initial slope with x = 0 and T = TI and thus A P * ~ ( T J= 0 is
4 50
i T ‘
The Journal of Physical Chemistry, Vol. 89, No. 10, 1985 2061
I
f-
f 410
C
iaxcl
=
A
-AS*~(T, - T ~ + ) c u / ( a [ X ( i - ~ ) ( -i z ~ y l l / a ~ ) , , S * 1
i 370
1
(17)
For a case where only the first term of the expansion j = 1 is sufficient, the initial slope becomes
s,
909
707 9 z
Figure 3. The isobaric phase diagrams for the binary system of 2-[4-nheptylphenyl]-5-[4-(n-heptyloxy)phenyl]pyrimidine(707) and 2-[4-nnonylphenyl]-5-[4-(n-nonyloxy)phenyl]pyrimidine(909). The solid lines are the calculated equal G curves based on AGE = hAx(1 - x). The X’s are the experimental points redrawn from ref 6 .
values. Since the diagram might be ideal, use of the initial slopes and eq 15 is suggested. Measurement of the initial slopes gives values (aT/ax)o = -16.3 K and (aT/ax)l = -6.4 K which are not consistent with the ideal hypothesis eq 14. The disagreement is not terrible, however, and a quite reasonable phase diagram based on eq 15 with s = 1.39 (an average value derived from the initial slopes) can be calculated. Just looking a t Figure 3, which is the complete phase diagram for the above binary system and shows the existence of several mesophases, one would have to infer from the smectic A-isotropic portion of the diagram that equilibrium could only be considered ideal if the nonideality of the isotropic phase exactly matched the obvious nonideality of the smectic A phase. (The smectic A is obviously nonideal because of the minimum in the spectic A-smectic C portion of the diagram-see below.) The smectic A-isotropic portion of the phase diagram is interesting because it is nearly ideal yet the lower temperature phase is definitely not ideal, implying then that the isotropic phase is nearly as nonideal as the smectic A phase in this case. Nonideal Systems The smectic A-isotropic portion of the phase diagram mentioned above cannot be exactly described by the ideal G equation. Furthermore, a slope analysis of the ideal equal G, eq 11, will show that such a treatment cannot describe a maximum or minimum in the temperature-composition diagram, since eq 11 will have a zero slope only for the rare case of TI equal to T2. Thus, all portions of the 7071909 phase diagram are nonideal and some nonzero expression for AGE must be used. To a first approximation, heat capacity effects can be neglected so Z(AC*pl) Z(AC*,2) is about zero and the equal G eq 9 can be solved for temperatures as a function of composition to give
+
(1 - ~ ) A s * l T l + ~ A s * 2 T 2 ~ ( -x)CAAj(l l -2~Y-l
It is well to mention here some properties of this analysis. First, if AAj equals zero, then A@ is zero, and the ideal equal G equation is obtained. Second, while not obvious from eq 16, the nonideal equal G curve must go exactly through any maximum or minimum in the phase diagram. That this is true is most easily seen from Figure 1, for if the points of tangency and intersection become the same, then at that point x, = xg = x,. Third, again only ratios
Provided the entropies are known, the initial slope analysis gives a means to estimate the excess Gibbs energy parameter AA. In the event that a maxima or minima is exhibited by the system, the fact that the slope is zero a t such a point provides another equation for the excess parameters and the entropies. Again, in the event that only t h e j = 1 term is sufficient
where T,, and x, are the temperature and composition of the observed maximalminima in the diagram. If the entropies are unknown, eq 19 can be reduced to give
where the entropy ratio can be determined from
which is obtained by eliminating AA‘between eq 16 and 20. Even if AS*I and As*2are known separately, eq 21 still provides a test of experimental consistency. Furthermore, eq 21 is quite useful for analysis of liquid crystal phase diagrams obtained by the contact method, since, in such cases, often the entropies of pure materials are known which then combined with an experimentally determined T, allow estimation of x,. Should the phase diagram or, more exactly, the excess Gibbs energy, not be well fit using just the simple LUX( 1 - x ) term, some options of increasing complexity are A G= ~ Mlx(1 -x)
+ AA2x(1 - x)(l - 2x)
AGE = u(T - T&(l
AGE = U(T - r0)x(i - X)
- X)
(22a) (22b)
+ u Z x ( i - ~ ) ( -i 2 4
(22c)
A G E = u ( T - T o ) x ( l - x ) + b ( T - T ‘ o ) x ( l -x)(l -2x) (224
where in eq 22c,d AAl has been given a temperature dependence of a (T - To) while also in eq 22d, AA2 has been given a temperature dependence of b(T - Tb) as well. It is important to point out that the excess Gibbs energies can be found from the experimental phase diagram without any “fitting” procedure at all. Rewriting eq 16 which solves the equal G equation for T, we can obtain an equation directly for AGE AGE=
E
T-
(1 - x)AS*IT, + x A S * ~ T ~ (1 - x ) A s * , + xAs*2 [(l - x)AS*l
1.
+ ~ A s * 2 ](23)
Here T is the “experimentally observed equal G temperature” for composition x , and thus, the first bracketed term is just the
2062
The Journal of Physical Chemistry, Vol. 89, No. 10, 1985
Van Hecke
TABLE I: Reduced Entropies, Initial Slopes, and Excess Cibbs Energy Parameters Derived for the Isotropic and Smectic, A, C, F, and C Phases (707) and Exhibited in the Binary Mixture 2-[4-o-Heptylp~ayl]-5-[4-(a-heptyloxy)phenyllpyrimidiw
2-14-n-Nonylpbenyl)-5-l4-(n -nonyloxy)phenyllpyriimidine (909)
dT/dx, K
U*,(a@)/
phase I
x=o
s*707(a@)
x=l
4 a ) - A(@)/ AS*,,,(a@), K
obsd
exptO
obsd
calcd
obsd
calcd
-3.6
1.15
1.13
-16.3
-16.3'
-6.4
-6.4
-42
2.23
2.21
-8.3
-22.9'
12.5
22.5
0.68
-30
2.24
2.83
-8.0
-16.3'
11.2
15.8
2.21
-100
2.80
?(b)
-37.8
-76.2'
35.1
38.8
0.048
s.4
A(a)/
U*107(a@)/
J/(mol K)"
J/mole 1756
18.6 1823 1851
SC
SF
1919
1926
SG
ODSC results from ref 6b text. bExperimentallyDSC was unable to detect this transition (ref 6b). This analysis estimates AHtw(SGSF)to be 48 J/mol which should be large enough to observe by DSC. 'The agreement is exact since slope data was used to calculate parameters. dThe minimum point data was judged more reliable than the slopes and the parameters were choosen to better fit the minimum point. CCalculatedfrom A ( a ) = A ( @ )+ u * 7 0 7 ( ( Y @ ) u ' ( a @ ) . fA(SG) value derived from 909 K-SO equilibrium. The ASSw(KSo) value was found to be 94 6 J/mol which compares favorably to the DSC observed 105 J/mol (ref 6b), especially since the 909 compound exhibits solid-state polymorphism and the reported transition enthalpy may be a partly composite value. difference between the experimental temperature and that expected for an ideal diagram. Again, even if the entropies are not known, the eq 23 can be reduced to give AG& = AGE U*l
[
T-
+ XSTZ (1 - x) + xs
(1 -x)TI
1
[(l - x)
+ xs]
M(s8F)/s*707(SCsF),
m * W S -
and s * W S It should be kept in mind that, even in the presence of a minimum or maximum, initial slopes via eq 18 can be used to determine the reduced entropies and excess Gibbs energy parameters. The estimates obtained for s and M'values by the two methods should of course be the same. However, in practice inconsistencies may appear due to uncertainties in determining initial slopes and/or the minimim/maximum compositions and temperatures. In the treatment below, both initial slope and minimum point data were used to arrive at the s and AA' values reported in Table I (see below). Moreover, since there is an equilibrium between SG and crystalline 707 and 909, it is possible to determine A(SG) uniquely by using a modified Schroeder-van Laar equation. Knowledge of A(SG),combined with the M'values will allow determination of each excess parameter A (using the experimental entropies-see below and Table (sCsF)/s*707(SCsF), (s&)/M*707(S$G).
0
(24)
where s is the same as before and, in the case of a maximum or minimum diagram, can be found from eq 21. Thus, in a case where it is not desired to fit the excess Gibbs energy analytically, it can still be easily determined and used for comparing one system to another. The above procedure is what Oonk calls 'reading AGE from the phase diagram".' The 707/909 system mentioned previously will serve to illustrate reading A@ from the diagram as well as the ease of an analytical fit using M x ( 1 - x) for AGE. Moreover, it is possible to make suggestions as to the relative nonidealities of the isotropic and smectic A, C, F, and G phases exhibited by this 707/909 system. To quantify the nonidealities of these phases, the excess parameters are required. As mentioned above, the form PAX(l x) will be used for AGE, and, here also the approach will be to assume no enthalpy data are available, do the calculations in terms of reduced variables, and them compare results with literature enthalpy data. For the systems exhibiting minima, eq 20 and 21 can be used to obtain the appropriate r e d u d entropies and excess G i b b energy parameters which here are h A ( s A s C ) / m * 7 , ( s A s & s*909(SASC)/~*707(SASC)r
20
M(s$G)/s*707(s$G),
1).
The appropriately modified Schroeder-van Laar equation is obtained as follows. Writing the most general equations describing (7) Oonk, H. A. J. "Phase Theory";Elsevier: New York, 1981; p 125.
'0E
0
-A G
c c
0 707
909 xsoi
Figure 4. The isobaric excess Gibbs energies for the phase equilibria exhibited by the system 2-[4-n-heptylphenyl]-5-[4-(n-heptyloxy)-
phenyllpyrimidine (707) and 2-[4-n-nonylphenyl]-5-[4-(n-nonyloxy)phenyllpyrimidine (909). The excess Gibbs energies have been reduced by the transition entropies for pure 707. Thus AG"(afl) = AGE(a@)/ h S * 7 0 7 ( ( Y @ ) . The open circles are the calculations using AGE = AAx( 1 - x) and the X's using AGE = AAlx(l - x) + AA2x(1- x ) (1 - 2 x ) . phase behavior in terms of chemical potentials and equilibrium phase compositions we haveIb
RT In
(Xl,/Xl&
+ [A,(1 -
- A,(1 - xlg)21 = U * , , , ( T - TI) (25)
RT In [(I - xIa/(l
- X1g)l +
[AA1a2 - AgXI,Zl =
s*Z,,(T-
T2) (26)
Now here let a refer to mesophase and j3 to solid and consider the phase equilibrium between mesophase and pure solid component 1. By setting xIBequal to one for the pure solid, and setting A A g equal to zero for the pure solid by assuming the solid phase to be ideal, we obtain from eq 25 R T In xl,
+ A,(1
- x1J2 = AS*,,,(T
- TI)
(27)
If we specifically consider smectic G-solid equilibrium for 909, ( S G - K ~ )eq , 27 allows the determination of the excess energy parameter, A&), and the transition entropy, hs*909(s&p& at T = 88.5 K from the experimental temperature and composition
Calculation of Liquid Crystalline Phase Diagrams
The Journal of Physical Chemistry, Vol. 89, No. 10, 1985 2063
data provided by the phase diagram The results are summarized in Figures 3 and 4 and Table I. The solid lines in Figure 3 are the equal G calculations for the liquid phase equilibria based on U x ( 1 - x) for AGE. The X’s are the experimental points. In Table I are the reduced excess parameter differences, AA’,and the entropy ratios, AS*909(a/3)/AS*7w(a/3), used to calculate the phase diagrams in Figure 3 and to estimate the excess Gibbs energies presented in Figure 4. Also in Table I are estimates for the individual A values. The solid lines in Figure 4 are the reduced excess Gibbs energies calculated by using eq 24. The open circles are estimates from the use of AA’x(1 - x) (Table I data), and the x’s are estimates obtained by using a reduced eq 22a. The entropy ratios and reduced excess parameters found in Table I are all that are necessary to obtain the results given in Figures 3 and 4. The values of A(&+) and A S * m ( S G b ) , found by use of eq 27, are also in Table I. Since the approach here is one assuming that enthalpy data are not known, other than A(SG), all the actual numerical values for the individual excess parameters depend on the various liquid phase transition entropies of 707. At this point, however, since in fact the enthalpy values are known, the reduced entropy values derived from the equal G analysis can be compared to the experimental ratios.6b.c Also the absolute values of the excess Gibbs energy parameters can be calculated. The results are all given in Table I. The agreement between derived (called obsd in Table I) and experimental entropy ratios is very good. Moreover, the analysis allows an estimate of the enthalpy and entropy for the SGS, transition in pure 909, a transition not detected by DSC, though given its magnitude it should be. Looking at the excess Gibbs energy parameters shows clearly that phase nonidealities increases in the sequenbe I < SA < Sc < SF < So. Both the I and SAphases are considerably nonideal even though their phase diagram does not appear very nonideal. Both the SA and Sc and the SFand So phases seem to be quite similar thermodynamically in that the entropies and nonidealities of these pairs of phases are very similar. Here we take the SG phase to be the most nonideal on the basis that it has the greatest excess Gibbs energy. This trend is borne out in Figure 4, where it is obvious that the nonidealities involving the SFand SG phases are the greatest. As mentioned above the solid lines in Figure 4 are the ‘experimental” values obtained by reading AGE from the phase diagrams. The open circles are the estimates based on AA’x( 1 - x) and the x’s are based on a two-term expansion eq 22a as mentioned before. The simple AA’x( 1 - x) function is symmetric about x = 0.5, and so while AGR(IS)A and ACE(SgG) are reasonably symmetric about the composition midpoint, the considerable asymmetry of the SA& and the ScSFexcess energies can never be well fit by use of only the quadratic function x ( l - x). An excess Gibbs energy based on eq 22a clearly fits well and can actually be used conveniently, since using the fact that the slope aAGE ax is zero at the maximum at any of the curves, analytical expressions can be developed for U land PA2 which only require knowledge of AG“max and x,,,. While obviously a two-parameter A@ equation better calculates the “experimental” AG“ values, the end result in terms of the phase diagrams in Figure 3 is not as sensitive, and, at first look, the single term ( U ’ x ( 1 - x) equation calculates the phase diagrams reasonably well. Of course, to further study this 707/909 system, other forms for AGE could be employed and better agreement between experimental and calculation be achieved. However, such improvement in agreement comes at the cost of losing the chance to find the correlation with molecular properties based on a single parameter which is easily comparable from system to system. As an aside, if eq 22c were used for AG“, it would be possible to develop analytical expressions to determine s, a, T,,,and AA ’by using experimental initial slope and maximum or minimum point data. The fact that eq 22a can also be analytically fit to AGE data has already been measured. These methods will not be further illustrated here. As a last comment on the form GE= Ax( 1 - x), not all A values are possible. For a phase to be stable, for it not to phase separate
a
Figure 5. Chemical potential vs. temperature and temperature vs. com-
position diagrams for a binary system that exhibits reentrant behavior with only the first component doing so in the pure state. The possibility of a phase transition between LY and @ phases at a temperature lower than T I is suggested by the dashed lines in (c). and show, in the case of liquids, liquid-liquid immiscibility, the second derivative of the total Gibbs energy with respect to composition must be greater than zero, thus [(RT/x(l - x ) ) - 2 4 > 0. If A > 0, phase stability requires 2A < [ R T / x (1 - x)]; if A < 0, the phase can never separate. Normally the equal G analysis will yield only AA values unless some phase equilibria occur that can be considered ideal as in the case of the mesophase-solid above. Generally A values are taken to be positive but they should be checked to assure that they do not predict phase separation. None of the values in Table I predict phase separation. It is not our purpose here to offer any detailed interpretation of each such curves and excess energy parameters a t this time, but mainly point out that they can be conveniently determined and quantified by analytical expressions for liquid crystalline systems. Since GE = HE - TSE,the expansion for AGE could be written AGE = ~ ( -lx)C(AHj - TASJ)(l - 2xY-l J
(28)
where it is noticed that the AA, have k e n replaced by iw,- T U p If AAj has no temperature dependence, which would mean AS, equals zero, than AA, describes the excess heat of mixing (which in this approximation is also temperature independent). Thus considering only the first term AGE = AHE = ~ ( -lx ) A H ~
(29)
and, therefore, it is possible to estimate excess enthalpy of mixing from the equal G analysis of the phase diagram. Conversely, as it is often done in “normal” liquid mixtures, could be measured to estimate AGE. To date, no reports of heats of mixing measurements on binary mixtures of mesogens have appeared in the literature. Since it is the purpose of this paper to only introduce the equal G technique for the analysis of liquid crystal phase diagrams, further examples of nonideal systems will be discussed in future communications.
Reentrant Systems Reentrant phase behavior such as illustrated in Figure 5b is “new” only to liquid crystals. Pure iron, for example, exhibits reentrant behavior as do several of its alloys. For a pure material to exhibit reentrant behavior, the chemical potentials of the phases involved must have the characteristics shown in Figure 5. Clearly the slopes of the chemical potentials are quite different and the origin of the difference is the temperature dependence of the entropies, which in turn, are determined by the heat capacities.
2064
The Journal of Physical Chemistry, Vol. 89, No. 10, 1985
Thus the expression for r ) must now include the A C P i term. If in the first approximation ACpiis taken to be a nonzero constant, Ap*i(r) frome eq 5 is AM*^( r ) = -AS*i(T - T i ) + AC*J T - ‘Pi- T In ( T / pi)) (30) where the higher transition temperature ‘Pihas been used as the reference temperature. This equation does have two roots, one where A P * ~ ( = ~ )0 for T = ‘Pi,and the other for T = Ti. The ratio Ti/‘Pi is entirely determined by the ratio AC*pi/AS*i. It should be kept in mind that all of these properties are characteristic of pure material i, and this point have no dependence on the equal G assumptions. It should also be noted that here AC*piis taken to be the first-order difference in the heat capacities of the phases above and below the transition temperature ‘P,. To apply the equal G analysis to reentrant diagrams we need to consider two cases: (i) only one component exhibits reentrant behavior in the temperature range studied; (ii) both components exhibit reentrant behavior in the temperature range of interest. The first case, the only one considered here, is illustrated in Figure 5 . In this case, for the second component, the a phase is always more stable than the /3 phase in the temperature range studied. This is illustrated in Figure 5c. The possibility of a lower temperature /3 to a phase transition is described by the dashed line in Figure 5c. Case i, which is the apparently more often experimentally observed one, is described by an equal G equation of the form (1 - x)[-AS*I(T- ‘ P I ) + AC*pl(T- Ttl T In ( T / P l ) ) ]+ xAp*,
+ AGE = 0
(31)
where ApLZcan be taken as a constant or as a function of temperature with negligible heat capacity effects. In figure 5c the assumption of Ap** constant would mean the entropies of the two phases a and /3 are about equal, that is, the chemical potentials have the same slope. Whatever the exact form used for Ap*, is, it must, for case i, satisfy the condition that no transitions from the a to fl phase be allowed for the second component in the temperature range TI < T < pl.It is worth noting here that eq 3 1 with AGE zero or nonzero predicts reentrant behavior for large enough heat capacity effects. Thus, the mixing of a nonreentrant component 2 cannot affect the reentrant behavior of pure component 1, though properties of 2 obviously affect the binary region of reentrant behavior. Figure 6 shows an example of the use of the equal G analysis for an experimental system which not only is reentrant but also exhibits a minimume8 The equal G equation used for this case involved a temperature dependence for both Ap** and the excess Gibbs energy in the manner shown below
+
(1 - x ) [ - ~ S * ~ ( T -‘PI) A C * p l ( T - ‘PI- T l n ( T / ‘ P , ) ) ] + x [ - h S * z ( T - Tz)] u ( T - T,)x(l - x ) = 0 (32)
+
This four-parameter equation, in which each parameter has a thermodynamic meaning, can be eaily applied to any case i type of reentrant system. By use of initial slopes and the turning point, the four parameters, AS*2/AS*1, a/AS*l, T,, and To,can be found (8) Gobl-Wunsch, A.; Heppke, G . ;Hopf, R. Z . Narurforsch. A 1981,36, 213. The author thanks Dr.W. Wagner for making available his numerical results before publication.
Van Hecke
340h 300
28 0
IL
SCBP
I
x7cs
C
7CBP
Figure 6. The calculated isobaric phase diagram for the reentrant system 4-(n-octyloxy)-4’-cyanobiphenyl (8CBP) and 4-n-heptyl-4’-cyanobiphenyl (7CBP) where only SCBP shows reentrant behavior in the pure state. The solid line was calculated by using eq 39 in the text with reduced parameter values (AC,~cBP/AS*~cBP(SAN,340 K) = 10.8, AS‘*7cep(SAN,340K) = -1.36694, T2= 611.22 K, To = 316.64 K. The X’s are redrawn experimental points from ref 8.
analytically. This particularly example well illustrates the power and convenience of the equal G analysis since no other analysis technique that has been put forward can treat a reentrant system that has a minimum (or a m a x i m ~ m ) . ~It, will ~ ~ be noted, that T2 is indeed outside the T I - PIrange and suggests that 4-nheptyl-4‘-cyanobiphenyl would undergo a virtual smectic A to nematic transition at 611 K. The application of the equal G technique will be further illustrated in a forthcoming communication treating several homologous series.” Acknowledgment. These studies were initiated in the laboratories of Dr. H. D. Koswig of the Central Institute for Electron Physics of the Academy of Sciences of the German Democratic Republic while the author was on an exchange visit under the auspices of the Academics of Sciences of the United States and the German Democratic Republic. The Author thanks Dr. Koswig and Dr. W. Wagner for the many discussions and hospitalities extended to him during the visit. Also, he thanks Professors H. Sackmann and D. Demus for their hospitality and encouragement during his visit to Halle and the Martin Luther University. Registry No. 8CBP, 52364-73-5; 7CBP, 52364-72-4; 2-[4-n-heptylphenyl]-5- [4-(n-heptyloxy)phenyl]pyrimidine 2- [4-n-nonylphenyl]-5[4-(n-nonyloxy)phenyI]pyrimidine, 6561 5-80-7. (9) Heppke, G.; Schneider, F. ‘Liquid Crystals”, Chandrasekhar, S., Ed.; Heyden and Son: Philadelphia, 1980; p 147. (10) Billard, J. “Liquid Crystals”, Chandrasekhar, S., Ed.; Heyden and Son: Philadelphia, 1980; p 155. ( 1 1 ) Wagner, W.; Van Hecke, G . R. ‘Proceedings of the Tenth International Liquid Crystal Conference”, York, England, 1984, submitted for publication.
