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The Journal of Physical Chemistry, Vol. 83, No. 18, 7979
Gerald R. Van Hecke
polarization of all the other molecules in region k: = Ek'(X)
Ep(X)
+ @k(h)
(33)
The type of electrostatic potential energy contribution of interest here, Ukk(X), results from interaction of the moment of molecule :k occupying the region containing (r,O,$),with the field Ek(X) from all the other molecules in region k. For simplicity in notation, the prime on one k in Ukk(X) has been dropped. In terms of the volume element, du, of region k, the differential contribution, f%.&k(X), to this energy is t?Ukk(X)
=
-Ek(X)'P(X) do
(34)
where P(X) is the polarization, or the electric moment per unit volume. P(h) a t (r,6',$) is given byI5 t-1P(X) = ---E(X) (35) 4n
P(X) = 4 3(t ~ ( 2-+ t') 1)(-)Z(r,O,$) 1 - ag The field &,(A)
(36)
is15
With the definition of g from eq 3, the result given in eq 24 is obtained:
References and Notes (1) L. Onsager, J . Am. Chem. SOC., 58, 1486 (1936). (2)J. G. Kirkwood, J. Chem. Phys., 7, 911 (1939). (3) B. Linder, Adv. Chem. Phys., 12, 225 (1967). (4) S. Nir, Biophys. J., 16, 59 (1976). (5) J. H. van Vleck, J. Chem. Phys., 5, 556 (1937). (6) R. Fowler and E. A. Guggenheim, "Statistical Thermodynamics", Cambridge University Press, London, 1952,Chapter 14. (7) K. Shinoda, J. Phys. Chem., 81, 1300 (1977). (8) A. Ben-Naim, "Water and Aqueous Solutions. Introduction to a Molecular Theory", Plenum Press, New York, 1974,Chapters 4,6-8. (9) H. S. Frank and M. W. Evans, J . Chem. Phys., 13, 507 (1945). (10) W. Kauzmann, Adv. Protein Chem., 14, 1 (1959). (11) G. Nemethyand H. A. Scheraga, J . Chem. Phys., 36, 3401 (1962). (12) C. Coulson and D. Eisenberg, Proc. R. SOC.London, Ser. A , 291, 454 (1966). (13)J. M. Deutch. Annu. Rev. Phvs. Chem., 24, 301 (1973) (14i J. G. Kirkwood. Chem. Rev..-l9. 275 11936). (l5j L. Page, "Introduction to Theoretical Physics", Van Nostrand, Princeton, N.J., 1952,Chapter 10. (16) A. Ben-Naim, J. Wilf, and M. Yaacobi, J. Phys. Chem., 77, 95 (1973). (17) M. Yaacobi and A. Ben-Naim, J . Phvs. Chem., 78, 175 (1974). (18) I. M. Barclay and J. A. V. Butler, Trans. Faraday SOC.,34, 1445 (1938). (19) A. Ben-Naim, J . Phys. Chem., 82, 792 (1978). (20) R. C. Weast, Ed., "Handbook of Chemistry and Physics", 55th ed, CRC Press, Cleveland, 1974. (21)J. T. Edsall and J. Wyman, "Biophysical Chemistry", Academic Press, New York, 1958,Chapters 2 and 6. (22) 0. Sinanoglu and S. Abdulnur, Fed. Proc., 24, S12 (1965). (23) R. A. Pierotti, J. Phys. Chem., 69, 281 (1965). (24) A. Ben-Naim, J . Phys. Chem., 69, 1922 (1965). (25) A. Ben-Naim, J . Chem. Phys., 54, 1387 (1971). (26) R. 6. Hermann, J . Phys. Chem., 75, 363 (1971). (27) M. Lucas, J . Phys. Chem., 76, 4030 (1972). (28) M. H. Klapper, Prog. Bioorg. Chem., 11, 55 (1973). (29) H. DeVoe, J . Am. Chem. SOC.,98, 1724(1976). (30) C. Tanford, "The Hydrophobic Effect", Wiley, New York, 1973. ~
(37) Therefore, from eq 28 and eq 32-37
Then, from eq 34 Ukk(h) = JBUkk(h)
=
-J[Ek(r,6',4,x).P(r,e,$,X)]du
(39) Substituting from eq 36 and 38, with dv = r2 dr sin 0 d8 d$, one finds that the integration, over r from a to 03, over 6' from 0 to n, and over 4 from 0 to 27r, gives
Use of Regular Solution Theory for Calculating Binary Mesogenic Phase Diagrams Exhibiting Azeotrope-Like Behavior for Liquid Two-Phase Regions. 1. Simple Minimum Forming Systems Gerald R. Van Hecke Department of Chemistry, Harvey Mudd College, Claremont, California 9 1711 (Received September 29, 1978, Revised Manuscript Received June 11, 1979) Publication costs assisted by Petroleum Research Fund and Harvey Mudd College
Many binary phase diagrams of mesomorphic components exhibit minimum azeotrope-like phase behavior. Regular solution theory is successfully applied to calculate such phase diagrams. In addition, an empirical correlation between the regular solution parameters and molar volume ratio of the components comprising a mixture is proposed and shown to have considerable utility.
Azeotrope-Like Behavior for Liquid Two-Phase Regions
of ideal solution behavior, but can be conveniently described in terms of regular solution theory. We will briefly introduce the regular solution approximation, and then show the usefulness of the theory by calculating the phase diagrams reported for some homologous pairs of symmetric azoxybenzenes. Further, we will discuss some simple schemes for estimating regular solution parameters that would then allow at least preliminary predictions of the liquid two-phase regions exhibited by these systems.
Ideal Solutions For a binary system of components 1 and 2 in equilibrium between a phase CY (nematic, smectic, e.g.) and a /3 phase (isotropic, nematic, e.g.), standard thermodynamic treatments yield two equations that describe the equilibrium phase compositions as
The Journal of Physical Chemistty, Vol. 83, No. 78, 7979 2345
coefficient. Typically activity coefficients are measured for vapor-liquid systems, but here, for liquid crystalline systems, they are experimentally not measurable.12 Thus it is as useful to discuss the nonideality in terms of A parameters as it is in terms of activity coefficients. Usually for vapor-liquid systems when the vapor pressure is low, the vapor phase is typically treated as an ideal mixture, leaving whatever nonideality is present to be described by a single parameter for the liquid phase. With liquid crystalline azeotropes, best results appear to be obtained by using a parameter for each phase. Substituting regular solution parameters for the excess chemical potentials into the appropriate equations similar to eq 3 and to that which yielded eq l a and l b , we obtain In ( x l p / x l , )
+ (Aaxzg2
In
+ ( A p Q - A,x1,2)/RT
(x20/xza)
-
A,xZa2)/RT= H I
(5b)
While eq l a and l b can be solved explicitly for the composition as a function of temperature, eq 5a and 5b must be solved by some iterative procedure. Since xl, xza = 1 and xlb xZp= 1,iterative solution of (5a) and (5b) is possible for xla and xlp once given T , the properties of the pure components, and some estimates for A , and APi3 For vapor-liquid systems the A parameters are functions of the enthalpies of vaporization of the two components comprising the mixture. However, little or no heat of vaporization data are available for liquid crystalline compounds, mainly because of the decomposition occurring before boiling. Thus at present no practical means affords estimates for the parameters. However, it is anticipated that empirical methods (vide supra) will prove useful. In the case of the appearance of a minimum or maximum point in the phase diagram, eq 5a and 5b simplify a great deal, since at the azeotrope point T = T,, xl, = x l p = xlm, and xz, = x24 = x2,, which results in
+
+
where AHoiapis the CY + /3 transition enthalpy a t Ti, assumed independent of temperature. Equations l a and l b are just the Schroeder-van Laar equations and allow calculation of ideal solution phase diagrams. The various types of ideal two-phase spindles that can exist depend on the magnitudes of the enthalpies and temperatures for the pure materials and were noted some time ago by van Laar and more recently quite thoroughly discussed by Reisman and Domon and Billard.'~~However, the azeotrope-like behavior of liquid crystal systems cannot be treated ideally as a brief calculation will demonstrate. Suppose Hl and Hz were small enough to expand the exponential as exp(Hi) N 1 + Hi, then eq l a and l b could be solved for the composition of x,,, say, as
= H2
(54
( A , - A , ) x ~ ,=~RTmH1
(64
(Ap - A,)xlm2 = RT,H2 Since 0 Ixla 5 1,the permissible values of Tare such that, for T z > T,, T must be less than T z or greater than T I , otherwise xla > 1, which is not possible. Thus an ideal two-phase region must have one of the spindle shapes described originally by van Laar and be bounded by the pure component transition temperatures T1and T2. Thus a phase diagram that exhibits any temperature greater or lesser than maximum or minimum T2or T,, respectively, violates the T criteria above and must therefore be discussed by the introduction of nonideality.
Nonideal Solutions Nonideality, as it has been found in vapor-liquid and alloy systems, is usually treated in terms of excess functions. Thus an excess chemical potential term is added to the normal expression for the chemical potential of component 1 in a phase CY giving = poia + RT 1n xia + piaE (3) where klaEis the excess chemical potential of (Y phase for component 1. Regular solution theory substitutes for the excess chemical potential of component 1 in CY phase pi,
2 Ao,xza (4) where A , is a positive parameter with units of free energy, independent of temperature and composition, and dependent only on the characteristics of the p h a ~ e . ~ - l l Alternatively plaE could be written in terms of an activity
piaE =
Moreover, a relationship for xlm, depending only on T,, H,, and H,, can be found by taking the ratio of (6a) to (6b) and again noting that xlm + xz, = 1: XIm = [l + (H,/H2)'/2]-1 (7) Further, the difference in the regular solution parameters can also be found from knowledge of only T , by
( A , - A,) = RTmH2[1+ ( H ~ / H 2 ) 1 ~ 2 ] 2 (8) Therefore, only the pure component transition temperatures and enthalpies and knowledge of T , are required to calculate the composition of the azeotrope point and the regular solution parameters which, in turn, allow estimation of the phase diagram. It is convenient that so much can be accomplished with only knowledge of T,, since the commonly used contact method for determining binary phase diagrams of mesogenic mixtures will provide only temperature information.
Applications/Results To illustrate the method, we will study the binary phase diagrams of homologous 4,4'-dialkoxyazoxybenzenes as reported by Hsu and Johnson?6 and Demus et al.14 Demus et al. present phase diagrams for all the 15 possible binary combinations of the homologues in the series from 4,4'dimethoxy- (PAA) to 4,4'-dihexyloxyazoxybenzene (mixtures will be designated 1-6, e.g., meaning a methoxy-1 and hexyl-6 mixture). Of the 15 possibilities, 6 show
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The Journal of Physical Chemistry, Vol. 83,
No. 18, 1979
Gerald R. Van Hecke
TABLE I: Observed Temperatures and Compositions of the Minima Observed in Various Binary Mixtures of 4,4'-Di-n-alkyloxyazoxybenzenes as well as the Regular Solution Parameters Derived from Minimum Temperature regular s oh parameters system 1-3' 1-4 1-5 1-6 3-5 3-6
Xi,(obsd)
K
Tm,
b 395.2 403.7 389.5 397.4 394.7 395.7
A N / R ,K A I / R , dK
C
b
c
395.3 403.2
X,, = 0.74 X,, = 0.40
0.68 0.47
Xsm=0.62 394.5
X,, = 0.40 X , , = 0.55 Xsm=0.28
0.50
6.433 1.744 13.717 10.263 2.406 5.484
2.145 2.582 4.572 3.421 0.802 1.828
pred
A N / R ,K A I / R , eK
Xi,
0.72 0.40 0.59 0.53 0.52 0.25
pred
Xi,
6.286 8.441
2.096 2.814
0.73 0.40
14.429
4.810
0.50
a 1-3 denotes the methoxy ( 1 ) and propoxy (3) mixture. Demus et al., ref 14. Hsu and Johnson, ref 4 . Calculated from only T, observed by Demus et al. via eq 7-9. e Calculated from only T, observed by Hsu and Johnson via eq 7-9.
. +
4001
/. -L
6
L.
T K
+
c
t
d
X1-
+
I+
A
-1410
4-
4
P 1---
b
1 ; x: ; I ~ : x : : : ~ : : : : ~ : : : ~ I ; : x: : [ ; ; ; ~ l IC
4'
Xt-
X 1-
XT
6'
440
4-5
'
06
400
MOLE
FRACTION
Figure 1. Calculated and observed nematic-isotropic phase diagrams for binary mixtures of homologous 4,4'-di-n-alkyloxyazoxybenzenes. The solid line is calculated by regular solution approximation. A denotes points calculated assuming ideal solution. (E denotes experimentally observed point. a-k have the same temperature scale (410-390) as do I, m and n (440-400).5-6 means 4,4'-di-n-pentyloxyazoxybenzene in a mixture with 4,4'-di-n-hexyloxyazoxybenzene. Xi is mole fraction of ith component.
minimum azeotrope behavior. The observed minimum temperature and composition data (as estimated from the published figures), the derived regular solution parameters, AN and AI (where LY = N, the lower temperature phase, and 6 = I, the higher temperature phase), as well as the composition of the minimum predicted by eq 7 are given in Table I for the six minimum forming systems. The temperature and enthalpy data for the pure compounds used in the calculations were obtained from Arnold.lG As seen in Table I, the lower temperature phase parameter, that is, here the excess free energy of the nematic phase, is greater than that of the isotropic phase. Maximum diagrams would require the reverse to be true. Figure 1 (a-d, f, and g) shows the calculated ideal and regular solution phase diagrams as well as the observed points for the six minimum forming systems. The other phase diagrams in Figure 1 were calculated from parameters estimated by the empirical method discussed below. For the minimum systems, the agreement between the calculated and observed is excellent. The width of the two phase region deserves comment. While the calculated width is small ( N 0.003) it does exist. The observation of a very
narrow two-phase region is in keeping with the calculations of Cox and Johnson on systems not exhibiting minima or maxima and accounts for the frequently encountered difficulty of experimentally observing the two phase region.15 What, if any, empirical rules can be derived from correlation of the parameters in Table I with the molecular structure and/or physical properties of the pure materials? First it should be noted that since no actual two-phase region data exist, only the differences in the parameters AN and AI can be calculated from the T,. Some other means must be used to obtain values of the individual parameters. What has been used here is to choose arbitrarily values equally spaced around the difference, that is AN = -1.5(AI - AN) AI = -0.5(A1- AN) (9) This effectively reduces the parameters from two to only one but the formulism is still convenient, especially for other types of systems. Other choices of AN and AI make little difference in the calculated results as long as the difference is kept at the determined value. It might be
The Journal of Physical Chemistry, VoL 83, No. 18, 1979
Azeotrope-Like Behavior for Liquid Two-Phase Regions
2347
TABLE 11: Regular Solution Parameters Observed and Derived for Binary Mixtures of 4,4'-Di-n-alkyloxyazoxybenzenes
1-3' 1-4 1-5 1-6 3-5 3-6
VI I vz 0.76 0.68 0.62 0.57 0.81 0.74
1-2 2-3 2-4 2-5 2-6 3-4 4-5 4-6 5-6
0.87 0.88 0.79 0.71 0.65 0.89 0.90 0.83 0.91
1-7 3-7 5-7
0.52 0.68 0.84
system
(AA/R )ob&, K
(AA/R)aH,'
d
e
K
-4.289 -5.163 -9.144 -6.842 -1.604 -3.656
-4.191 -5.627
-0.48 -8.08 -1.06 -8.52 -0.11 -4.94
-9.619
(AA/R)(V~,V,),~ AHz(T,K T , ) / R T , ,K
-20.50 -14.69 -2.84 -12.24 -2.59 -4.61 -3.29 -0.01 -3.57
-3.52 -6.15 -8.12 -9.76 -1.89 -4.18 0.08 0.41 -2.54 -5.17 -7.13 0.74 1.07 -1.23 1.40
- 1.98 -0.24 -2.04 -1.05 -0.08 -1.72 -12.0 -16.4 -11.5 -16.6 -14.4 -4.0 -4.1 -2.3 -1.9
-11.40 -6.15 -0.9
-1.9 -0.1 -0.3
1-3 denotes the ' (AA/R)AH= -[AH,,, t A H z ~ -l 2 ( A H l ~ ~ A H z ~ ~ ) 1 ' Z(AAIR) ] / R . v /v,) = 28.46 - 32.81( Vl/V,). From data of Demus et al., rek1'4. e From data of Hsu and Johnson, ref 4. methoxy (1)and propoxy ( 3 ) mixture.
noted, however, that for systems with one common component, 1-3, 1-4, 1-5, 1-6, and 3-5, 3-6, there is an increase in the AN, AI, and the AI - AN with the increase in molecular weight (or molar volume) of the second component. In fact, data for the mixtures involving methoxy are essentially linear. It has been mentioned above that perhaps some empirical scheme can be used to estimate the parameters which then could be used to estimate phase diagrams for other similar or homologous systems. For vapor-liquid systems AIiq has been estimated by9 Aliq =
[(mlvap)"'
- (4E~vap)~''I'
(10)
where it has been assumed in writing eq 10 that the molar volumes are approximately equal. If it is further assumed that Hivap N AEivap, that is, the pV term is small, then for the isotropic phase we write a similar equation AI = [(AH1Ivap)"'
- ( ~ Z ~ v a p ) l / ~ I ~ (11)
and for the nematic phase we take into account the nematic transition by I M 1 1 v a p ) 1 ' 2 - (AH2NI + AH21vap)1/212 AN = [ ( A H ~ N + (12)
the difference AI - AN can be found in terms of only AHlNI and A H Z N I by expanding and simplifying. Assuming A H i N I / A H i I v a p
