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J. Phys. Chem. B 2008, 112, 14198–14205
Thermodynamic Assessment of Smectic A-Nematic Tricritical Points through the Equal Gibbs Energy Analysis D. O. Lo´pez,*,† P. Cusmin,† J. Salud,† S. Diez-Berart,†,‡ M. R. de la Fuente,§ and M. A. Pe´rez-Jubindo§ Laboratori de Caracteritzacio´ de Materials (LCM-GPFM), Departament de Fı´sica i Enginyeria Nuclear, E.T.S.E.I.B UniVersitat Polite`cnica de Catalunya, Diagonal, 647, 08028 Barcelona, Spain, Liquid Crystal Institute, Kent State UniVersity, Kent, Ohio 44242-0001, and Departamento de Fı´sica Aplicada II, Facultad de Ciencia y Tecnologı´a, UniVersidad del Paı´s Vasco, Apartado 644, E-48080 Bilbao, Spain ReceiVed: July 24, 2008; ReVised Manuscript ReceiVed: September 05, 2008
A complete thermodynamic analysis on four two-component phase diagrams between liquid crystals belonging to the nCB and nOCB series of compounds, the so-called cyanobiphenyls, has been performed through the Oonk’s equal Gibbs energy method. The binary systems dealt with in this paper show as a common feature the existence of a tricritical point at the SmA-to-N phase transition, all of which reported elsewhere. As a singular finding of the work proposed in this paper, the Oonk’s method is extended in order to account for second-order phase transitions. Likewise, this extension allows performing calculations of the tricritical temperature. 1. Introduction During the past 30 years, there has been certain interest in the order of the mesophase transitions in liquid crystals and also in their binary mixtures. In particular, the nature of the smecticA (SmA)-to-nematic (N) phase transition has been the subject of numerous theoretical and experimental studies. From the theoretical point of view, in the 1970s, McMillan1 and Kobayashi,2 working independently on the basis of the mean-field model, developed a molecular theory, according to which the SmA-to-N phase transitions could be second order or first order in nature, depending on the nematic range. Nearly at the same time, de Gennes3 suggested that this transition could be described by a Landau-Ginzburg functional and anticipated that the SmA-to-N phase transition should belong to the 3DXY Universality class. However, owing to a coupling between smectic and nematic order parameters, strongly influenced by the extension in temperature of the nematic range, a crossover or intermediate behavior from the strict 3D-XY second-order transition until up to the tricritical point (TCP) is expected to be observed. The TCP could be defined as the limit point at which the SmA-to-N second-order phase transition becomes a first-order one. As a parameter to control the nematic range, and so that coupling, McMillan proposed the ratio TAN/TNI (TAN and TNI are the SmA-to-N and the N-to-isotropic (I) transition temperatures, respectively), called the McMillan’s ratio, which should be 0.87 from the TCP. However, experimental evidence point out values somewhat higher, ranging from 0.942 to 0.994, displaying a nonuniversal value, strongly dependent on the polarity or nonpolarity of the material molecules. Experimental evidence for the TCP have been reported on temperature vs pressure phase diagrams,4,5 but the most widely studied twocomponent systems, in which TCP is readily attainable, are those built up at ordinary pressure.6-14 * To whom correspondence should be addressed. E-mail: david.
[email protected]. † Universitat Polite ` cnica de Catalunya. ‡ Kent State University. § Universidad del Paı´s Vasco.
Contrarily, Halperin, Lubensky, and Ma15,16 published, in the 1970s as well, the so-called HLM theory on the basis of the analogy with superconductors, suggested by de Gennes.3 According to this theory, the SmA-to-N phase transition should always be weakly first order in nature, and so conventional tricritical behavior is ruled out. Obviously, some assumptions are required to extract such a conclusion, and they have not yet been rigorously proved. Experimental studies in support of this theory were undertaken in the 1990s and continue currently.17-23 From the above-mentioned discussion, it follows that the problem of the SmA-to-N TCP may be rather delicate because it is very sensitive to various approaches.24 In fact, the most important problem is how to explain the experimental observations. Relatively few efforts have been made in order to quantify the experimental liquid crystal phase diagrams by means of pure thermodynamic procedures14,25-30 or statistical thermodynamic techniques.31-34 In addition, to our knowledge, nearly no thermodynamic calculations have been reported in order to obtain the tricritical temperature in binary liquid crystal mixtures with the exception of a previous work published by some authors of the present paper.14 The work envisaged in this paper is devoted to the presentation of a simple thermodynamic approach in order to estimate the TCP temperature. In addition, we have carried out a complete thermodynamic analysis on several binary phase diagrams for which experimental TCP temperatures on the SmAto-N phase transition have been experimentally proved to exist.11-13 This thermodynamic analysis has been performed on the basis of the so-called Oonk’s equal Gibbs energy method,35 the general trends of which will be exposed further on. The paper is organized as follows. In section 2, we describe the general theory of the Oonk’s equal Gibbs method to be applied in order to perform the thermodynamic analysis as well as the TCP temperature prediction. In section 3, the results addressed to the calculated SmA-to-N and N-to-I phase transitions of the chosen two-component liquid crystal systems are
10.1021/jp806544j CCC: $40.75 2008 American Chemical Society Published on Web 10/22/2008
Equal Gibbs Energy Approach for a TCP Prediction
Figure 1. Qualitative two-component system A + B in which pure component A displays a second-order β-to-R phase transition and pure component B displays a first-order β-to-R phase transition (a). Molar Gibbs energy curves for pure component A (µA*,R, µA*,β) (b) and for pure *,β component B (µ*,R B , µB ) (c) vs temperature, in both cases in a qualitative mode. The EGC denotes the equal Gibbs composition curve. FEGC corresponds to the first EGC composition curve. XTCP and TTCP stand for the composition and temperature of the tricritical point, respectively.
displayed. TCP temperature calculations are performed and compared with experimental results. The discussion is driven according to the McMillan’s ratios and the excess Gibbs energy functions. Finally, in section 4, a summary of the main conclusions is made. 2. General Theory Under isobaric conditions, the thermodynamic properties of a binary mixture of a two-component system A + B can be determined if, for each possible phase of this mixture, the molar Gibbs energy as a function of temperature and composition is known. The Gibbs energy function of a phase R is described by the following expression in terms of X moles of B and (1 X) moles of A (i.e., the mixture A1-XBX)
GR(X, T) ) (1 - X)µA*,R(T) + XµB*,R(T) + RT[(1 - X) ln(1 - X) + X ln X] + GE,R(X, T) (1) where GE,R(X,T) is the excess Gibbs energy, µA*,R and µ*,R B are the molar Gibbs energies of pure components which are temperature dependent, R is the gas constant, and T is the thermodynamic temperature. In Figure 1a, an arbitrary two-component system A + B is shown in a qualitative manner. Two phases R and β are present, but for pure compound A the β-to-R phase transition is of second order whereas the same transition is of first order for pure compound B. In Figure 1b,c, the molar Gibbs energy of pure compounds A and B are plotted against temperature also in a qualitative way. In Figure 1b, for temperatures lower than TA, the Gibbs energies of both phases are similar and the slopes
J. Phys. Chem. B, Vol. 112, No. 45, 2008 14199
Figure 2. Qualitative representation of the Gibbs energy functions corresponding to R and β phases, at the temperature T1 according to Figure 1, vs mole fraction (a). The same representation of the Gibbs energy functions modified by the subtraction of the term RT[(1 - X) ln(1 - X) + X ln X] (b).
are equal up to TA. This means that there are no differences between both phases. For temperatures higher than TA, both molar Gibbs energy curves are diverging, being the lower value for phase R (µ*,R A ) which is the stable phase at these temperatures. There are no changes in molar entropy between R and β (no changes in the slopes) at TA, but they exist in their curvature due to changes in specific heat. On the contrary, in Figure 1c we have the typical plot corresponding to a first-order phase transition in which both molar Gibbs energy curves cross one to another at the transition temperature TB. There exists a change in the slope at TB for which a change in molar entropy is present. In Figure 2a, the Gibbs energy functions of the R and β phases (GR and Gβ), as described by eq 1, are plotted in a qualitative manner but at a temperature T1 chosen so that the two curves intersect. At this temperature takes place a first-order phase transition (see Figure 1a), and so a two-phase equilibrium region between R and β occurs. The traditional criterion of such an equilibrium is satisfied by the double tangent line, which gives rise to the equilibrium compositions (XR and Xβ). In this picture, both compositions are very close, and a very narrow two-phase region has to be expected. The inset represents this situation in a zoom window. In addition, these compositions are very close to the intersection of both Gibbs energy functions, named equal Gibbs composition (EGC). This EGC changes with temperature giving rise to the equal Gibbs composition curve (also named EGCC),35 which in a practical sense provides the representative line of the very narrow two-phase equilibrium in the T-X plot (Figure 1a). In Figure 2b, both Gibbs energy functions, for which the term RT[(1 - X) ln(1 - X) + X ln X] has been subtracted, denoted as G′′, are shown. This picture enables us to better observe the small differences when both G curves are very similar, taking into account that the EGC remains unchanged. The situation of Figure 2b occurs when the transition entropies of the pure compounds are very small or zero, being usual for many liquid crystal mesophase transitions. In addition, both XR
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Lo´pez et al. other) changes to a set of EGC points (the Gibbs energies are coincident during a composition range). To quantify the situations arising from Figures 2 and 3 for a real two-component system similar to that of Figure 1, some expressions must be defined. First of all, setting eq 1 for R-phase and its parallel for β-phase, the equilibrium compositions (XR and Xβ, as it can be seen in Figure 2) and the EGC could be *,β *,β calculated. However, in practice, µA*,R, µ*,R B , µA , and µB are unknown, and Oonk proposed to add the same and adequate linear contribution function F(X,T) to both Gibbs energy equations, to overlook this inconvenience.35 The new Gibbs energy functions, denoted as G′, lead to the same equilibrium compositions (XR and Xβ) and EGC that those raised for the previous Gibbs functions. The F(X,T) chosen has been
F(X, T) ) -(1 - X)µA*,β(T) - XµB*,β(X)
(2)
In such a way that *,β (GR)′(X, T) ) GR(X, T) - (1 - X)µ*,β A (T) - XµB (X) (3a)
Figure 3. Qualitative representation of the Gibbs energy functions corresponding to R and β phases, at the temperature T2 according to Figure 1, vs mole fraction (a). The same representation of the Gibbs energy functions modified by the subtraction of the term RT[(1 - X) ln(1 - X) + X ln X] (b).
and Xβ can be substituted by the EGC, and Figure 2b may account for the phase stability. So, on the two-component systems, a single line describing the two-phase equilibrium is drawn, and this corresponds to a very good approximation with the EGC curve. In Figure 3a, the Gibbs energy functions of the R and β phases (GR and Gβ), as described by eq 1, are plotted in a qualitative mode against composition at a temperature T2 for which a second-order transition is expected at a certain composition (X ≈ 0.2), as can be seen in Figure 1a. From Figure 3a, it cannot be observed any crossing between both Gibbs energy functions. In Figure 3b, G′′R and G′′β, for which the term RT[(1 - X) ln(1 - X) + X ln X] has been subtracted from GR and Gβ, are plotted against composition in order to better observe the details. This picture clearly evidence three facts: (i) both modified Gibbs functions (G′′R and G′′β) are coincident from pure component A up to X ≈ 0.2, which supposes that both GR and Gβ are coincident in the same composition range as well; (ii) phase β is doubtless the more stable phase from X ≈ 0.2 up to pure component B, because Gβ is lower than GR in such a composition range and (iii) between pure component A and X ≈ 0.2, there is more than one equal Gibbs composition (EGC) at this temperature. All these facts provide a coherent thermodynamic explanation for a second-order transition in binary mixtures. In addition, from a quantitatively point of view, a method to calculate second-order transition points is obtained. According to this method, the second-order transition points are identified as the first EGC point (FEGC) at which both Gibbs energy functions are coincident when the composition at each temperature is moving from pure component B (the compound which experiments first-order phase transition) to pure component A (the compound which experiments second-order phase transition). The TCP temperature could be defined as the temperature at which an EGC point (the Gibbs energies are crossing each
(Gβ)′(X, T) ) Gβ(X, T) - (1 - X)µA*,β(T) - XµB*,β(X) (3b) To simplify both eqs 3a and 3b, the logarithmic term (RT[(1 X) ln(1 - X) + X ln X]) in both Gibbs functions can also be subtracted, leading to new Gibbs functions denoted as G′′:
(GR)′′(X, T) ) (1 - X)∆µA*(T) + X∆µB*(T) + GE,R(X, T) (4a) (Gβ)′′(X, T) ) GE,β(X, T)
(4b)
where ∆µ*i (T) ) µ*,R - µ*,β i i (i ) A, B). These Gibbs functions lead to the same equal Gibbs curve equation which corresponds to the solution of
GR(X, T) - Gβ(X, T) ) (GR)′′(X, T) - (Gβ)′′(X, T) ) 0 (5) Following eq 5 by combining both eqs 4a and 4b, we obtain
(1 - X)∆µA*(T) + X∆µB* (T) + ∆GE(X, T) ) 0
(6)
To proceed further, expressions for ∆µ*i (T) and ∆GE(X,T) are required. For ∆µ*i (T), using the transition temperature as a reference point, we have as a good approximation * ∆µi* ≈ -∆Si*(T - Ti) + ∆Cpi (T - Ti - T ln(T/Ti)) (7)
where ∆S*i is the transition molar entropy at the transition temperature Ti and ∆C*pi is the change in specific heat also at the transition temperature. It should be noticed that if the transition β-to-R were second order, ∆S*i would equal zero and ∆µ*i would be entirely determined by the change in specific heat. The change ∆C*pi can be, in turn, finite, infinite or zero depending on the type of second-order transition.
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Figure 4. Schematic representation of the four two-component phase diagrams, at the same temperature scale, for which the thermodynamic analysis through the EGC method, has been performed. The TTCPEXP stands for the experimental tricritical temperature at the SmA-to-N phase transition.
TABLE 1: Thermodynamic Properties of Phase Transitions for Pure Liquid Crystal Compounds phase transition SmA-N (stable) compound 8CBa 8OCBb 9OCBc
∆S (J mol-1 K-1)
T (K) 306.43 340.37 351.08
N-I (stable) ∆Cp (J mol-1 K-1)
0e
0e
0e 0.456
0e 2.67
T (K)
∆S (J mol-1 K-1)
313.75 353.39 353.22
2.008 1.783 2.378
phase transition SmA-I (stable) compound
T (K)
10CBa 10OCBd
324.40 357.02
-1
∆S (J mol
-1
K )
8.724 7.044
a Data taken from ref 12. b Data taken from ref 11. second-order 3D-XY Universality class.
c
N-I (metastable) ∆Cp (J · mol · K )
T (K)
∆S (J mol-1 K-1)
-
322.25 355.95
6.415 4.629
-1
-1
Data taken from refs 11, 14, and 36.
d
Data taken from ref 13.
e
Assumed to be
TABLE 2: Excess Gibbs Energy Differences Referred to Redlich-Kister Coefficients (According to Eq. 8) along the Equal Gibbs Energy Curves Corresponding to the N-to-I and SmA-to-N Phase Transitions N-to-I phase transition
a
SmA-to-N phase transition
binary system
∆A1 (J mol-1)
∆A2 (J mol-1)
∆A1 (J mol-1)
∆A2 (J mol-1)
8CB + 9OCB 8OCB + 9OCB 8CB + 10CB 8OCB + 10OCB 7OCB + 9OCBa
-0.3 8.9 19.0 13.0 8.8
-0.1 -1.5 13.0 -4.0 -0.2
3.5 24.5 61.0 40.5 10.4
1.1 -2.1 -16.0 -3.0 1.4
∆A3 (J mol-1)
10 7.0
According to ref 14.
For ∆GE(X,T) a two-parameter of Redlich-Kister expansion is commonly used
∆GE(X, T) ) X(1 - X)
∑ [∆Ai(T)(1 - 2X)i-1]
(8)
i
where the ∆Ai(T) are usually taken to be constants or as functions of temperature in the form ∆HEi - T∆SEi . Usually two coefficients are proved to be enough. The EGC method requires some experimental data related to the phase transition of the mixtures in order to perform an iterative procedure in which one obtains a reasonable EGC curve as well as the ∆GE(X,T) along this curve. This procedure can be automatically executed by means of the WINIFIT 2.2 software (based upon the old version WINIFIT 2.0,36 which has been slightly modified to include the possibility of strictly
zero entropy change for pure compounds). To calculate TCP temperatures, WINIFIT numerically compares the Gibbs functions of both phases at each temperature (every 0.05 K) over all temperatures for which the calculations want to be performed. At each temperature, the numerical coincidence between the Gibbs functions over a composition range is tested by means of a parameter, denoted as TOL, which represents the numerical discrepancy. Values of TOL ranging from 0.1 to 0.01 J mol-1 have been considered to be enough to determine the FEGC. The first temperature at which FEGC is numerically reported is considered as the TCP temperature. 3. Results and Discussion In previous works11-13 made by some of the authors of the present paper, several two-component systems built up with cyanobiphenyl liquid crystals have been experimentally determined. The four chosen examples, schematically shown in
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Figure 5. Experimental (full circles) and calculated (dotted line, first order; empty dotted line, second order) two-component phase diagrams, 8OCB + 9OCB (a) and 8CB + 9OCB (b). Modified Gibbs energy functions for the SmA and N-phases ((GSmA)′′ and (GN)′′) computed at the tricritical temperature for the 8OCB + 9OCB (c) and 8CB + 9OCB (d) phase diagrams.
TABLE 3: Experimental and Calculated McMillan’s Ratios at the Tricritical Temperature (TTCP) McMillan’s ratio experimental
EGC analysis
binary system
TTCP/TNI
ref
TTCP/TNIa
8CB + 9OCB 8OCB + 9OCB 8CB + 10CB
0.987 0.986 0.989 0.985 0.981 0.976c
11 11 7 12 13 14
0.983 0.973 0.981
0.985 0.977 0.983
0.971 0.954c,d
0.976 0.959c
8OCB + 10OCB 7OCB + 9OCB
TTCP/TNIb
a According to a TOL ) 0.01 (J mol-1). b According to a TOL ) 0.1 (J mol-1). c Only for the SmA-to-N TCP temperature. d According to ref 14.
Figure 4, at the same temperature scale, display a tricritical point in their SmA-to-N phase transition. The cyanobiphenyl liquid crystal pure components can be classified in two series: the nCB and the nOCB series (n being the number of carbons in the alkyl or alkoxy chain, respectively). From the nCB series, only the octylcyanobiphenyl (8CB) and decylcyanobiphenyl (10CB) are considered, whereas from the nOCB series the octyloxycyanobiphenyl (8OCB), nonyloxycyanobiphenyl (9OCB), and decyloxycyanobiphenyl (10OCB) are taken into account. In Table 1, all the thermal information required for pure cyanobiphenyls to perform the thermodynamic analysis on the several two-component systems is collected. In the presentation of our results about the four twocomponent systems, two different groups have been taken into account. The first one corresponds to the systems 8OCB + 9OCB and 8CB + 9OCB, the experimental results of which have been recently published jointly11 by some authors of the present paper, and no indications of other studies, neither
experimental nor theoretical, are known by us. Likewise, the methodology to perform their thermodynamic analysis is the same. The second group consists of the 8CB + 10CB and 8OCB + 10OCB binary phase diagrams, which share the same thermodynamic calculation procedure. Concerning the 8OCB + 10OCB phase diagram, as far as we know, there exists only one experimental work13 published nearly 3 years ago, but regarding the 8CB + 10CB phase diagram, several experimental studies have been published up to now. As early as 1985, Marynissen et al.7 built up this two-component system from very precise adiabatic scanning measurements and as one of the most noticeable findings it was the existence of a TCP for the SmA-to-N phase transition. However, Oweimreen and Hwang22 republished later the same two-component system concluding that the SmA-to-N phase transition is of first order whatever the composition is. Soon after, some of the authors of the present paper have published two pieces of the experimental evidence for the existence of a TCP in such a binary system.12 3.1. The 8OCB + 9OCB and 8CB + 9OCB TwoComponent Systems. The first step in the thermodynamic analysis of both sets of mixtures is concerning to the [N + I] two-phase equilibrium. According to eq 1, one Gibbs energy function is needed for the I phase and another for the N phase. The required input for the thermodynamic analysis includes the experimental data points read from Figure 5 (full circles, taken from Sied et al.11), together with the experimental thermal properties of the pure 8OCB, 9OCB, and 8CB, corresponding to the first-order N-to-I phase transition read from Table 1. The EGC method provides the excess Gibbs energy difference between N and I phases along the EGC curve (given by an expression like eq 8) with the coefficients, ∆Ai (taken as temperature-independent), shown in Table 2. As one can
Equal Gibbs Energy Approach for a TCP Prediction
J. Phys. Chem. B, Vol. 112, No. 45, 2008 14203
Figure 6. Experimental N-to-I transition temperature vs mole fraction for the two-component 8CB + 10CB (a) and 8OCB + 10OCB (b) phase diagrams. The square root of the SmA-to-N latent heat vs mole fraction for the two-component 8CB + 10CB (c) and 8OCB + 10OCB (d) phase diagrams. TN-I(m) and ∆HSmA-N(m) stand for metastable temperature and latent heat, respectively.
Figure 7. Experimental (full circles) and calculated (dotted line, first order; empty dotted line, second order) two-component phase diagrams, 8CB + 10CB (a) and 8OCB + 10OCB (b). Modified Gibbs energy functions for the SmA and N phases ((GSmA)′′ and (GN)′′) computed at the tricritical temperature for the 8CB + 10CB (c) and 8OCB + 10OCB (d) phase diagrams.
observe, in the set of 8OCB + 9OCB and 8CB + 9OCB mixtures, both N and I phases show comparable deviations with regard to the ideal mixing behavior, though they are stronger for the former. The optimized EGC curve representative of the calculated [N + I] two-phase equilibrium for both set of mixtures has been drawn in Figure 5a,b. The second step in the thermodynamic analysis is addressed to the SmA-to-N phase transition. In such a case, the EGC method through eq 6 is applied with Gibbs energy functions built up with the linear contribution F(X,T), referred to as the
I-phase, in such a way that the generic β-phase should be changed by I-phase in eq 2. The modified (GSmA)′′ and (GN)′′ are numerically computed and iteratively compared via eq 6 using as input data the experimental SmA-to-N phase transition points for the mixtures (see Figure 5a,b) as well as transition temperatures, molar entropy, and specific heat changes at the SmA-to-N phase transition for pure components (see Table 1) and the excess Gibbs energy difference between N and I phases. The output data from the thermodynamic calculation are the excess Gibbs energy difference between SmA and N phases
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(see Table 2) and the optimized EGC curve representative of the SmA-to-N phase transition. The calculation is performed every 0.05 K for all the composition range, from the pure component which the SmA-to-N phase transition is of first order in nature toward the component in which this phase transition is second order in nature. The first temperature at which no crossing between both Gibbs functions, (GSmA)′′ and (GN)′′, is observed, i.e., a continuous set of EGC compositions is computed because they are coincident, is denoted as the TCP temperature. At temperatures below the TCP temperature, there is no crossing between both (GSmA)′′ and (GN)′′. Figure 5c,d shows this situation for both set of 8OCB + 9OCB and 8CB + 9OCB mixtures. It is important to realize that in Figure 5d, although the Gibbs function (GI)′′ for the I-phase would be needed to account for all the stability domains at this temperature, it is not shown for sake of clarity. A crossing between (GI)′′ and both (GSmA)′′ and (GN)′′ is expected to be at the composition at which the N-phase changes to the I-phase (the gray-hatched part at the right side of the Figure 5d). In Table 3, the experimental McMillan’s ratios for both set of mixtures according to Sied et al.11 along with those calculated from the EGC method are shown. It has been clearly stated that the tricritical temperature is slightly sensible to the TOL value, with tricritical McMillan’s ratio of about 0.98. The experimental value is very close to the calculated one in the case of 8CB + 9OCB mixtures, and it seems to be fairly independent over the used TOL value. However, in the case of 8OCB + 9OCB mixtures, it seems to exist more discrepancy between the experimental and calculated tricritical McMillan’s ratios, and even in this case, the value is heavily dependent on the TOL value. 3.2. The 8CB + 10CB and 8OCB + 10OCB TwoComponent Systems. As we can observe from Figure 4, both two-component systems present a crossing between the [N + I] two-phase equilibrium and the SmA-to-N phase transition at the so-called INA point, defined as the point at which the nematic range becomes zero. At intermediate nematic ranges between the INA point and pure 8CB or 8OCB the experimental TCPs in both sets of mixtures are observed. In Table 3, the experimental tricritical McMillan’s ratios for both binary phase diagrams are collected, showing similar values ranging between 0.981 and 0.989. In such a case, given that the [N + I] two-phase equilibrium does not exist over the whole composition range (it only exists between 8CB or 8OCB up to the INA point), the N-to-I phase transition in both pure 10CB and 10OCB does not exist as a stable transition. Therefore, the first step in our thermodynamic analysis will consist in determining the thermal data (temperature and molar entropy change) corresponding to the metastable N-to-I phase transition in both pure compounds. As for the N-to-I transition temperature, from Figure 6a,b, it can be easily extrapolated from experimental data taken from Lafouresse et al.12 for the 8CB + 10CB mixtures and from Sied et al.13 for the 8OCB + 10OCB mixtures. As far as the entropy change of the N-to-I phase transition is concerned, first of all, we proceed to calculate the associated latent heat (∆HN-I) following the expression
∆HSmA-I ) ∆HSmA-N + ∆HN-I
(9)
for which the latent heat associated with the SmA-I (∆HSmA-I) is directly obtained in an ordinary experiment for both 10CB and 10OCB because this phase transition is the only stable one. The SmA-to-N latent heat (∆HSmA-N) can be estimated by
extrapolation according to the procedure, initially evidenced by Thoen and co-workers,6 whereby [∆HSmA-N]0.5 varies linearly with the composition, as is shown in Figure 6c,d. These experimental data have been carefully measured and reported elsewhere.12,13 The linear extrapolation leads to the value of the latent heat at the SmA-to N phase transition for both pure 10CB (Figure 6c) and pure 10OCB (Figure 6d). These values through eq 9 give rise to the latent heats associated with the virtual N-to-I phase transition which allow us to calculate the entropy changes at this transition, dividing the latent heat by the transition temperature, as it has been consigned in Table 1. The next step in our thermodynamic analysis consists in building up the Gibbs energy functions corresponding to I, N, and SmA phases. To that end, an expression as eq 1 is considered for each one of the mentioned phases, for which the linear contribution F(X,T), referred to the I-phase, is subtracted. The three Gibbs energy functions are combined among them, following eq 5 representative of the EGC curve, leading to the EGC curve expressions for the [N + I] and [SmA + I] twophase equilibria:
(1 - X)∆µA*,N-I(T) + X∆µB*,N-I(T) + ∆GE,N-I(X, T) ) 0 (10) (1 - X)∆µA*,SmA-I(T) + X∆µB*,SmA-I(T) + ∆GE,SmA-I(X, T) ) 0 (11) and also for the SmA-to-N phase transition:
(1 - X)∆µA*,SmA-N(T) + X∆µB*,SmA-N(T) + ∆GE,SmA-N(X, T) ) 0 (12) The subscripts A and B of eqs 10-12 correspond to 8CB and 10CB in one set of mixtures and 8OCB and 10OCB in the other one. The terms ∆µ*i (T) (i ) A or B) are evaluated through eq 7 from the data consigned in Table 1. Experimental data of the different transitions in each set of mixtures, either 8CB + 10CB or 8OCB + 10OCB, are used as input data in order to get, in a simultaneous way, the three optimized EGC curves provided by eqs 10-12 as well as the excess Gibbs energy differences along the EGC curves. Among the latter, only ∆GE,N-I and ∆GE,SmA-N are given in Table 2. Parts a and b of Figure 7 show the experimental data, taken from Lafouresse et al.12 and Sied et al.,13 respectively, together with the optimized EGC curves calculated by means of WINIFIT software, by computing at every temperature eqs 10-12. It is important to realize that in reference to the SmA-to-N phase transition for the mixtures, at temperatures very close to the INA point, both (GSmA)′′ and (GN)′′ are crossing each other, pointing out that the SmA-to-N phase transition is of first order in nature. To lower temperatures, the first temperature at which both Gibbs functions do not cross each other, in such a way that they coexist over a composition range is defined as the TCP temperature. Parts c and d of Figure 7 show the Gibbs energy functions, computed by means of WINIFIT software, at the corresponding SmA-to-N TCP temperatures of 8CB + 10CB and 8OCB + 10OCB mixtures, respectively. In Table 3, the calculated McMillan’s ratios for both sets of mixtures are presented. In the case of 8CB + 10CB mixtures, the calculated value is very close to the experimental one
Equal Gibbs Energy Approach for a TCP Prediction extracted from Lafouresse et al.12 In addition, the calculated value seems to be very little sensitive to the TOL value. As for the 8OCB + 10OCB mixtures, the calculated McMillan’s ratio is slightly more sensitive to the TOL value. Even so, there exists a more than acceptable agreement with the experimental value. 3.3. Overall Discussion. Table 2 contains the excess Gibbs energy changes along the EGC curves representative of the calculated phase transitions corresponding to the four twocomponent systems given in Figure 5. At a first glance, these differences are relatively small, much more for the [N + I] twophase equilibria than for the SmA-to-N phase transitions. This means that the N mixtures are much more ideal solutions than the SmA ones for each binary phase diagram. In a more quantitative way, the magnitude of the excess Gibbs energy for both phases, N and SmA, are in a ratio 1 to 3 with the exception of the system 8CB + 9OCB, the only system in which the pure components belong to different series. Let us consider the SmA-to-N phase transition. The asymmetry of the excess Gibbs energy difference between the SmA and N-phases, quantified by the ratio between the second and first coefficients of the Redlich-Kister expansion (eq 8), is the same in absolute value for the two-component systems 8CB + 10CB and 8CB + 9OCB, of about 0.3. However, the other two binary systems (8OCB + 10OCB and 8OCB + 9OCB) display excess Gibbs energy differences much more symmetric, with an asymmetry ratio, in absolute value, of about 0.1 or even lesser. Curiously, both binary systems show calculated TCP temperatures more sensitive to the TOL value than the other. In order to know whether the symmetry of the excess Gibbs energy difference between the SmA and N phases could be the key parameter, let us consider the two-component system 7OCB + 9OCB, published elsewhere,14 and which has been the subject of the first EGC prediction of the TCP at the SmA-to-N phase transition. Its asymmetry ratio, according to Table 2, is about 0.1 and the calculated McMillan‘s ratio, according to Table 3, ranges between 0.954 and 0.959, which implies a relative variation of about 0.5% with regards to the TOL value. This relative variation is comparable to what is shown for the systems 8OCB + 10OCB and 8OCB + 9OCB. 4. Concluding Remarks The general details of the equal Gibbs energy approach for the TCP temperature prediction in two-component systems are given. To test the viability of this approach, four experimental examples, for which a documented TCP on the SmA-to-N phase transition exists, are considered. As a whole, the EGC method is able to determine TCP temperatures provided that experimental information that allows the elaboration of a complete thermodynamic analysis of binary phase diagrams is available. In general, at least for the examples considered in the present work, calculated McMillan’s ratios are in quite good agreement with the experimental ones. In a few cases, there seems to exist a certain dependence of the calculated TCP temperature with the TOL value. As discussed previously, it seems that this dependence could be caused by the symmetry of the excess functions at the SmA-to-N-phase transition. Obviously, more experimental two-component systems displaying TCP at the SmA-to-N phase transitions would be needed to extract reliable conclusions about this topic. Finally, it would be important to observe that the experimental McMillan’s ratios as well as the calculated ones are clearly higher than the theoretical value of 0.87, at least for the four examples considered in this paper and even for the 7OCB +
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