J . Phys. Chem. 1987, 91, 6403-6409
6403
Generalized Lattice-Model Treatment of Nematic-Isotropic Phase Equilibrium in Uniaxial Binary Mixtures of Hard Molecules Daniel E. Martire* and Samir Ghodbane Department of Chemistry, Georgetown University, Washington, D.C. 20057 (Received: March 9, 1987; In Final Form: June 29, 1987)
Addition of a nonmesomorphic (perturbing) solute to a nematogenic liquid-crystalline solvent depresses the nematic-isotropic transition temperature (TNI)and causes the formation of a two-phase region (Figure 1). The moduli of the initial slopes, 0,”and &“,of the nematic and isotropic phase boundary lines in reduced transition temperature (T* = T/TNI) vs solute mole fraction (x2) diagrams, which reflect the ability of a solute impurity to destabilize the nematic phase, are determined theoretically by using a simple cubic lattice model. Previous lattice-model treatments of uniaxial binary mixtures of hard molecules (anisotropic repulsive forces only) are consolidated, extended, and simplified. Examined are the effects of solute molecular structure (volume, m2,shape and flexibility, E , , / k ) and solvent molecular structure (length-to-breadth ratio of the rigid-rod core, r l , and length,f,, and flexibility, E b , / k ,of the end chain(s)) on nematic phase stability. New numerical results are presented for cubic (model analogue of spherical), rigid-rod, and semiflexible-chain solutes, varying m2 and (for the chains) E , , / k , in several solvents, varying r l , f , ,and E b , / k . These results (6” values) and their trends are discussed and interpreted in terms of the structure and calculated orientational order parameters of the solute and solvent molecules, and the calculated values of the pure-solvent and dilute-solution contributions to @-, viz., the transition entropy and the ratio of solute activity coefficients in the two phases, respectively.
Introduction With few exceptions (vide infra), solute impurities that are not sufficiently rodlike and rigid in their molecular structure depress the nematic to isotropic liquid transition temperature ( TNI)of liquid-crystalline solvents. Moreover, consistent with classical thermodynamics and the first-order nature of the phase transition, and as was clearly demonstrated in an earlier study,’ the presence of such impurities leads to a two-phase region. Shown in Figure 1 is a typical phase diagram for a nonmesomorphic solute and nematogenic solvent mixture at low solute mole fraction ( x 2 5 0.05),where P = T/TN, is the reduced temperature. Incremental addition of solute at a fixed temperature P’ to an originally pure nematic material results in initial appearance of the isotropic phase at X2.N and final appearance of the nematic .’ and in the manner of our more recent phase at x ~ , ~Alternatively, experiment^,^-^ when a binary nematic (isotropic) system of composition x i is heated (cooled), the isotropic (nematic) phase begins to appear at TN* (TI*) and the nematic (isotropic) phase completely disappears at TI* ( TN*). Of particular interest is the negative of the slope of the lower phase-boundary line, PN = - ( d F / d ~ ~ which ) ~ , has been used as a measure of the ability of a solute (impurity) to destabilize (positive ON) or, as the case may be, even stabilize (negative PN)the orientationally ordered nematic fluid.’-5 Note that, for sufficiently small x2, the slope of the upper phaseboundary line, PI = -(dTC/du,)], bears a simple thermodynamic relationship to PN (vide infra).2-5 Useful information on the effects of molecular size, shape, and flexibility on the orientational order and stability of conventional (i.e., nonpolymeric, nondiscotic, and nonreentrant) nematic me(1) Peterson, H. T.; Martire, D. E. Mol. Cryst. Liq. Cryst. 1974, 25, 89. (2) Martire, D. E.; Oweimreen, G. A.; Agren, G. I.; Ryan, S . G.; Peterson, H. T. J . Chem. Phys. 1976,64, 1456. (3) Martire, D. E. In Molecular Physics of Liquid Crystals; Luckhurst, G. R., Gray, G. W., Eds.; Academic: London, 1979; Chapter 10 and references cited therein. (4) Oweimreen, G. A.; Martire, D. E. J . Chem. Phys. 1980, 72, 2500 and references cited therein. ( 5 ) Ghodbane, S.; Martire, D. E. J . Phys. Chem., following article in this issue.
(6) Kronberg, B.; Gilson, D. F.;Patterson, D. J . Chem. SOC.,Faraday Trans. 2 1976, 72, 1673. (7) Gidley, M. A.; Stubley, D. J . Chem. Thermodyn. 1982, 14, 785. (8) Oweimreen, G. A,; Hasan, M. Mol. Cryst. Liq. Cryst. 1983, 100, 357. (9) Kronberg, B.; Bassignana, I.; Patterson, D. J . Phys. Chem. 1978, 82, 1714.
0022-3654/87/2091-6403$01.50/0
sophases has been gained through systematic thermodynamicl-10 studies of nematieisotropic phase and statistical-mechani~al~*~*’~-~~ equilibrium in binary mixtures. Experiments have been conducted ~.~ on mixtures of quasi-spherical,’-8 hai in like,',^" r ~ d l i k e , and platelikelo solutes dissolved in nematogenic solvents and the results have been compared in some detail with the predictions of various theoretical models. While the experiments’O and t h e ~ r yon ~~,~~ platelike solutes are cited for completeness and the unfolding of such interesting phase behavior as the possibilities of elevation second-order phase transitions, and biaxial states, their of TNI,24 detailed consideration is beyond the scope of the present work which focuses on low solute concentrations, first-order transitions, and uniaxial systems. The statistical-mechanical approaches which have been developed to treat these binary mixtures may be summarized as follows: (a) lattice models of (model analogue of spheres), (rigid rectangular parallelepipeds), and flexible or semiflexible chains4~12-’5~16 as the solute molecules, and solvent molecules that are completely rigid rods2,4*11-16 or have rodlike “cores” and semiflexible “tails”4~’2*15~17 (hard, anisotropic repulsive
(10) Sigaud, G.; Achard, M. F.; Hardouin, F.; Gasparoux, H. Chem. Phys. Lett. 1977, 48, 122. (11) Agren, G. I.; Martire, D. E. J . Phys. (Paris) 1975, 36, C1-141. (12) Martire, D. E. In Molecular Physics ofLiquid Crystals; Luckhurst, G. R., Gray, G. W., Eds.; Academic: London, 1979; Chapter 11 and references cited therein. (13) Peterson, H. T.; Martire, D. E.; Cotter, M. A. J . Chem. Phys. 1974, 61, 3547. (14) Cotter, M. A. Mol. Cryst. Liq. Cryst. 1976, 35, 33. (15) Dowell, F. J . Chem. Phys. 1978,69, 4012. (16) Dowell, F.; Martire, D. E. J . Chem. Phys. 1978, 69, 2332. (17) Martire, D. E.; Dowell, F. J . Chem. Phys. 1979, 70, 5914. (18) Agren, G. I. Phys. Rev. A 1975, 11, 1040. (19) Humphries, R. L.; Luckhurst, G. R. Proc. R . SOC.London Ser. A 1976, 41, 352. (20) Palffy-Muhoray, P.; Dunmur, D. A,; Price, A. Chem. Phys. Lett. 1982, 93, 572. (21) Cotter, M. A.; Wacker, D. C. Phys. Rev.A 1979, 18, 2669, 2676. (22) Alben, R. J . Chem. Phys. 1973, 59, 4299. (23) Rabin, Y.; McMullen, W. E.; Gelbart, W. M. Mol. Cryst. Liq. Cryst. 1982, 89, 61. (24) Note that other solutes which are not particularly ‘rodlike” have also been found to elevate T N I :Park, J. W.; Bak, C. S . ; Labes, M. M. J . Am. Chem. SOC.1975, 97, 4398. Apparently, the stabilization (negative &) arises from solute-induced molecular association.
0 1987 American Chemical Society
6404
The Journal of Physical Chemistry, Vol. 91, No. 25, 1987 I
I
Figure 1. Typical reduced temperature, P,vs solute mole fraction, x2. phase diagram for nonmesomorphic solute + nematogenic solvent mixtures at low x2.
interactions only, except for two studies14J5 where anisotropic attractive interactions are also included). (b) virial-expansion treatment of rigid spheres and rigid spherocylindersI* (hard, anisotropic repulsion only). (c) mean-field (Maier-Saupe type) treatments of rigid spheror rodlike'2,20solute molecules and rigid rodlike solvent ica12,4-'2q'9 molecules (anisotropic attractive interactions only; anisotropic repulsion neglected). (d) van der Waals model of rigid spheres and rigid spherocylinders, where the hard cores are treated by scaled-particle the0ry~9~l (balanced model, incorporating anisotropic repulsion and attraction). (e) lattice-mode122and excluded-~olume~~ treatments of binary mixtures of rigid plates and rigid rods (hard, anisotropic repulsion only). Of the above theories, the lattice models, which emphasize the important role of anisotropic repulsive interactions or molecular packing effects, have proved to be especially successful and versatile. Indeed, with the possible exception of the Maier-Saupe type treatment^,^^,^^ none of the other approaches can be readily extended to the stage of tractability for binary mixtures consisting of molecules other than completely rigid ones, thus hampering their use in examining chainlike solutes and the effects of solvent end-chain flexibility and length. We cite this drawback and possible exception in the light of previous evidence that nemat~ * ~bi-~ - ~ ~ ic-isotropic phase equilibria in s i n g l e - c ~ m p o n e n t ~and nary4.1115-17 systems are very sensitive to both the length-to-breadth ratio of the core and the length and intrinsic flexibility of the tail(s) of the solvent molecules. It is particularly noteworthy that, in a lattice model of hard molecules, inclusion of some end-chain flexibility leads to predictions for rigid-rod solutes which are in semiquantitative agreement with experiment and yield the observed maximum in PN vs. m2 (solute length-to-breadth ratio).I7 Note also that lattice-model studies4*'2~'4J5~29 indicate that, while the explicit incorporation of attractive interactions (in lieu of fixing a high external pressure to achieve realistic densities) may lead to moderately improved agreement with experiment, the anisotropy of attractive interactions appears to be of minor consequence. Accordingly, in the present article we consolidate, extend, and simplify previous lattice-model treatments of nematic-isotropic equilibria in uniaxial binary mixtures of hard molecules at low solute concentrations. Based on specific forms derived else~ h e r e , " - ' ~ ~we ' ~ obtain J~ a generalized expression for the configurational Helmholtz free energy of binary mixtures where the solvent molecule consists of a rigid rodlike core and semiflexible (25) Marcelja, S. J. Chem. Phys. 1974, 60, 3599. (26) Janik, B.; Samulski, E. T.;Toriumi, H. J. Phys. Chem. 1987, 92, 1842 and references cited therein. (27) Martire, D. E. Mol. Cryst. Liq. 1974, 28, 63 and references . Cryst. . cited therein. (28) Agren, G. I.; Martire, D. E. J. Chem. Phys. 1974, 61, 3959. (29) Dowell, F.; Martire, D. E. J. Chem. Phys. (a) 1978, 68, 1088; (b) 1978, 68, 1094.
Martire and Ghodbane tail(s), and where the solute molecule may be a cube, rigid rod, or semiflexible chain. Moreover, guided by classical thermodynamics, we develop a considerably simplified procedure for determining PN and PI at low solute concentration (Henry's law region). This generalized approach represents an extension for cubic solutes (previously, only rigid-rod solvents were treated") and provides new numerical results. It permits the systematic prediction and interpretation of the effects of solute and solvent molecular structure and geometry on (uniaxial) nematic phase stability. Its utility and efficacy are further demonstrated in the following a r t i ~ l ewhere ,~ salient theoretical results are compared with experimental results for quasi-spherical, chainlike, and rodlike solutes uniformly obtained with a common nematogenic solvent. Thermodynamic Background Referring to Figure 1, from classical thermodynamics it can
-
be shown'q2 that, at very low solute concentration (formallv x , 0), the negatives of the limiting slopes of the lower or nematic phase boundary line (P,") and the upper or isotropic phase boundary line (PI") are given by Pn"
= -(dT*/dxAn =
[(72,nm/72,1m)
- Il[R/mniI
(1)
Pi" = -(dT*/dx2)1 = [ 1 - ( y 2 , l m / ~ 2 , n " ) 1 [R/mnil (2) where we now employ lower-case subscripts to denote the nematic and isotropic liquid phases, respectively, and where T* and x2 have already been defined. In eq 1 and 2, R is the gas constant, ASnl is the nematic-to-isotropic transition entropy of the pure nemaand y2,,"are the infitogenic solvent (component l), and y2,nm nite-dilution (Henry's law region) solute activity coefficients in the nematic and isotropic phases, respectively, each extrapolated to T,, (P 1) and based on the Raoult's law convention (y2 1 as x2 l).30*31Note that y2"is proportional to the Henry's law constant, K2. The smaller the value of y2"or K 2 ,the more soluble is the solute and the greater is its compatibility with the solvent. From eq 1 and 2 it follows that
- --
(Pn"/Pi")
=
P'(see Figure - (Pn")-'I [1 - T*'I
and, since y 2 , n ~ 2=, ny2,1x2,1 at fixed (x2,1 - ~ 2 , n )=
[(PI")-'
(3)
(y2,nm/y2,l")
1)
(4a)
= (ASnl/R)(l - T*') (4b) Equation 4b clearly shows that if A& > 0, then a two-phase region must exist and its size at a given T*', x2,, - x ~ ,depends ~ , only on the magnitude of AS,, (independent of the s o l ~ t e ) . ' - ~ ~ ~ ~ We see from eq 1 and 2 that P" is equal to the product of a pure solvent property (R/AS,,) and a dilute-solution property (y2,nm/y2,1"). Both properties influence the magnitude of P", but the latter property alone determines its sign. The smaller the value of AS,,/R, the larger is P". Thus, solvents exhibiting more weakly first-order nematic-isotropic transitions are more easily perturbed by solute impurities. With respect to the solution property, it is appears in eq 1 important to note that just a ratio, y2,nm/y2,1m, and 2. This means that p" depends only on the relative preference of the solute for the two phases. When ~ 2 , ~ " / y ~>, ,1"( e l ) , the solute is more compatible with the isotropic (nematic) phase, p" is positive (negative) and the solute depresses (elevates) the nematic-isotropic transition. The above equations and analysis have proved to be useful in interpreting both experimental and theoretical results. It has been demonstrated4 that P, and PI values obtained by using the "visual" meth~d,~ which - ~ utilizes solute mole fractions in the range 0.01 < x2 < 0.06, are virtually the same (&lo%)as the true limiting values, &" and Pi", determined by using an alternative (but more (30) Chow, L. C.; Martire, D. E. J. Phys. Chem. 1971, 75, 2005. (31) Oweimreen, G. A,; Lin, G. C.; Martire, D. E. J . Phys. Chem. 1979, 83, 2111. (32) Note that eq 1-4 may also be applied toprst-order smectic(s)-nematic(n) phase transitionsSby simply replacing the subscripts n and i by s and n, respectively.
The Journal of Physical Chemistry, Vol. 91, No. 25, 1987 6405
Uniaxial Binary Mixtures of Hard Molecules restrictive) m e t h ~ d based ~ ? ~ on eq 1 and 2. Therefore, with ; are relevant to confidence that theoretical results for 0", and @ the visual-method results reported in the following a r t i ~ l eeq ,~1 and 2 will be employed here to develop a much simplified procedure for calculating these limiting slopes.
of component j about the preferred direction is (axial symmetry assumed)
Theory In this section we describe a generalized and economical theoretical approach for evaluating @", and @,".For those details of the derivation not presented here, the reader is referred to previous lattice-model s t u d i e ~ . ~ I - ~ ~ , ~ ~ The model is a steric, mean-field, simple cubic lattice model for a binary mixture of hard solute molecules (cubes, rigid rods, and semiflexible chains) and hard solvent molecules composed of rigid, rodlike cores, and semiflexible pendant chains (tails). Letting uo be the volume of a single lattice site, and M , V = Muo, and N be the total number of lattice sites, volume, and number of molecules, respectively, the number density, p , is given by p = N / M = NUO/V (5)
where s, is the fraction of molecules of component j with cores lying in a nonpreferred direction ( b or c ) . The core order parameter, v, is then 7, = 1 - 33, (12)
Also, each molecule of typej (j= 1 for solvent a n d j = 2 for solute) consists of m, segments, where each segment occupies a single lattice site and has a molecular volume of uJ = mpo. Letting N 1 and N2 be the number of solvent and solute molecules, respectively, and letting xJ = N J / N ( N = C;=,NJ)be the mole fraction of componentj, the number density, pJ, and occupied volume fraction, O,, of component j are given by PJ = xJP = N J / M
(6)
0, = xJmJp= N J m J / M= N,uJ/V
(7)
where OJ is proportional to pJ, and where
'
L
0 = pCxjmj = C N j m j / M = j= 1
j= 1
C N ~ ~ ~ / V(8) j=1
is the total occupied volume fraction. Each solute molecule contains m 2segments, where m2 = D23 for cubes (D2is the side dimension), m2 = r2 for rigid rods, and m2 = 2 fifor semiflexible chains; r2 is the number of rigid, collinear segments (for semiflexible chains, r2 = 229b)and& is the number of semiflexible segments. Each solvent molecule is comprised of m l = r l + f l segments, where r 1 is the number of rigid core segments and fl is the number of semiflexible tail segments. Therefore, except for the cubic solute molecules (vide infra), each molecule consists of rJ rigid core segments (and r, 1 "bonds") a n d & = mJ - rJ semiflexible pendant segments U; "bonds"). In this restricted-orientation model, the r, - 1 collinear bonds of the rodlike core of the molecules may lie in any one of three mutually perpendicular directions. A positive energy, Eb, is required for one of the& semiflexible bonds to bend in a noncore ("nontrans") direction, whereas no energy is required for one to lie in a a r e ("trans") d i r e ~ t i o n . ' ~This , ~ ~measure ~ of the intrinsic flexibility of these bonds is then entirely an internal property. The fraction, uJ,of the& semiflexible bonds in moleculej lying in a noncore direction is governed by'5,29b,33
+
-
= NJ(b)/NJ = NJ(C)/NJ
(10)
1 - 2sj = Nj(a)/Nj
(11)
where 7, = 0 (s, = for an isotropic distribution of the cores and 7, > 0 (sJ < for an anisotropic one. The fraction of semiflexible bonds of molecules of type j in a nonpreferred core direction, t,, is15*29b tJ = SJ UJ(l - 3 S J ) (13)
+
and is 1 - 2tJ in the preferred core direction. The corresponding order parameter, T,, for these semiflexible bonds is then T, = 1 - 3tJ (14) where, with any value of uJ,we have sJ = fc = and rJ = 0 for an isotropic distribution, and, with u, < 13,we have sJ < ]I3,t, < and T, > 0 for an anisotropic one. We also define an average order parameter for molecules of type j , w,: wJ
= [(rJ- ')?J + f , T J l / [ m J -
'1
(153)
If we apply eq 12-14 and recall that mJ = rJ +&, eq 15a becomes wJ
= 7J[l - ( 3 u f , / ( m J- ')I1
(133)
Finally, since cubic solute molecules lack a unique axis, they are always isotropically distributed; thus, s2 = 'I3and q 2 = 0. In this purely steric model (no attractive interactions) and with E b treated as an internal property, the configurational partition function, Q,and the configurational Helmholtz free energy, A , are simply related:11-16,29 -A/kT = In 0
(16)
DowellI5 had derived an expression for In Q of rigid-rod and semiflexible chain solute molecules mixed with rigid-rod core semiflexible tail solvent molecules, which we may simplify here by neglecting the attractive interaction energy terms. Extending, by straightforward induction, the Agren-Martire result" for hard-cube solute molecules mixed with hard rigid-rod solvent molecules to solvent molecules also having semiflexible tails, and melding it with Dowell's result for In Q,we obtain the following generalized expression -A/NkT = (1 / N ) In 0 = (p)-'[Qa In Qa 2Qb In Qb - Qc In Qc - QdI - Qe (17)
+
where 2
Qa = 1 - p c x j [ g j ( l - 2sj) - 2uifj(l - 3sj)]
(18a)
Qc = 1 - p Cxj6j
(18c)
j=1
-
where k is the Boltzmann constant and T i s the absolute temas Eb, 0 (completely flexible limit) perature. Note that uj 0 as E,, m (completely rigid limit). and uj Let directions a be the preferred direction, and b and c be the directions perpendicular to a. The distribution of the rigid cores
-
'J
j=1
-+
2
Qd =
[ (gj j=1
+ 1 - 6j)
/5j]
[( 1 - P X j c j ) In (1 - pxjcj)]
2
(33) In the model system the inherent molecular flexibility is determined by Eb/kT,where the parameter Eb is referred to as a 'bond-bending energy" and is assumed to be independent of pressure, density, and all other molecular properties of the solute and solvent. Note that Ebis not a bending energy in the usual sense, but rather an energy associated with having a tail bond in a nonwre direction. Also, in applications of the theory a tail bond is estimated to represent three or four methylene groups,29whereas each such bond in the model is assumed to have only three possible orientations with respect to the preceding bond.
Q, = E x j [ (1 - 2sj) In (1- 2sj) j=1
+ 2sj In sj + In pxjhj]
(1 8d) (1 8e)
where the molecular structural parameters C,, g,, and h, are defined and listed in Table I. Note that, except for cubic solute molecules, gj + 1 - 6,is zero in eq 18d. Therefore, with Qd = 0, eq 17 and 18 become Dowell's expression, as applied to hard molecules. Also, whenf, = 0, eq 17 and 18 reduce to the Agren-Martire result,
6406
The Journal of Physical Chemistry, Vol. 91, No. 25, 1987 Y2 In [Qa(l)/Qb(l)I
TABLE I: Molecular Structural Parameters Used in the Lattice Model
9"
molecule
m solvent rigid-rod solute m semiflexible chain solute m cube solute D'
5; (2m+ 1)/3 (2m+ 1)/3 (2m 1 ) / 3 D2
+
Y; h', m- 1 5m-1 1 m- 1 m- 1 1 m-1 5m - 1 1 3 ( D 3 - D2) 0 3 g/'
" E J = u,/uo, where u, is the volume occupied by a single molecule and uo is the volume of a single lattice site or cell. b5J = a / a , where uJ is J .o the surface area of a single molecule and a. ( = 6 units) IS the surface area of a single lattice site or cell. ' g J = 3(.!jJ - 4 ) . dyJ = gJ - 3 u f , = 3(5J - 5, - US,),where u, a n d 4 are defined in the text, and where yJ = 0 for cube solutes (see eq 2 5 ) . eSymmetry number.
where s2 = for cubic solutes. The configurational chemical potential of component I , p/,is obtained from eq 17 and 18"-16 p / / k T = [ ~ ( A / ~ T ) / ~ N / I T , M=, NgA(1 , - 2sd In Qa + 2s/ In QbI + (1 - 33/)(2uh)[In Qb - In Qal 6, In Qc - (g! 1 - DI) In (1 - px,i;/) + In (px,hl) + (1 - 2sl) ln(1 - 2sJ + 2sl In sI (19)
+
where 1 = 1 or 2 and N-I refers to the other component (2 or 1). For the pure solvent ( I = 1, x l = l ) , eq 19 yields ~ l ' / k T = gl[(1 - 2s1O) In Qa(1) + 231'1n Qb(l)I + + (1 - 3s1')(2u~f!f,)[lnQb(l) - ln Qa(1)1 - 61 In (plohl) (1 - 2sI0) In (1 - 2sI0) 2.9,' In sIo (20)
+
+
where the superscript 0 refers to the pure-substance value, and where, from eq 18a-18c Qa(1)
= 1 - plo[gl(l - 2~1') - 2 ~ d 1 ( -1 3~1')l
(21a)
= 1 - P1'k1si0 + u d i ( 1 - 3slo)1
(2 1b)
= 1 - PI'&
(21c)
Q,(I)
-
For the solute ( I = 2) at very low x2 (formally, as x2 0 and x1 1; Henry's law region, denoted by the superscript a), eq 19 yields -+
(&"/kT) - In X2 = g2[(1 - 2s2-1 In Qa(1) 2SzmIn Qb(l)I (1 - 3s2")(24f~)[lnQb(1) - In Qa(1)I - 62 In Qc(11 + In (p10h2)+ (1 - 2sZm)In (1 - 2sZm) 2s2mIn sZm (22)
+
for where (p2"/kT) - In x 2 remains finite and where s2" = cubic solutes, independent of the solvent environment." The dimensionless pressure-to-temperature ratio, @, is obtained from eq 17 and 18"-16 @ E Pvo/kT = -[6(A/kT)/6M]T,, = In Q,
+ 2 In Qb - In Q, - 2 [(g, + 1 - ~,)/E,IIn [ 1 - px,~,] J=
1
(23) where P is the pressure. For the pure solvent (gl + 1 = C1), eq 23 becomes *IO
= In
e a ( , ) -I-2
In
- In
(24)
Qc(I)
To determine the equilibrium orientational order of the system (as marked by sl and s2),we also require a minimization condition. Applying eq 17 and 18 we get [6(A/NkT)/6S/],,,s., = 0 = Y / In [Qa/QbI + In [(I - 23/)/s/I (25) where I = 1 or 2, s-, refers t o the other component (2 or 1) and where y l is defined and listed in Table I. Note that for cubic solutes (s2 = always) the general solution to eq 25 requires that y 2 = 0. For the pure solvent ( I = 1, x1 = l ) , eq 25 becomes Y I In [Qa(l)/Qb(l)I + In [(I - ~~IO)/SIOI =0
and for the solute ( I = 2) at very low x2 (x2
-.
0, xi
Martire and Ghodbane
(26)
-
1)
+ In [(I - ~ S Z " ) / ~ Z =" ] 0
(27)
where Qa(l)and Qb(l)are given by eq 21. The possible solutions to eq 26 and 27 are (i) sIo= and s2" = which corresponds to an isotropic state; (ii) sl0< and s2" I which corresponds to an anisotropic or "nematic" state."-I6 The equilibrium between the nematic (n) and isotropic (i) states of the pure solvent at a given is governed by the following equations pl,,O/kTni = pl,lO/kTni = @ I ,I
'
Y I In [Qa(~),n/Qb(i),nI+ In [(I - 2Sl,n?/Ji,,OI = 0
(28) (29) (30)
where pl,:/kT,, and are given by eq 20 and 24, respectively, and where Qbfl),",and Qc(l),nare given by eq 21, with sl0 and p I o = pl,,". Also, from eq 21, with sIo= sl,: = sl,: < = and p I o = pl,: Qa(l),i
=
Q~(I),I
Qc(l).l
= 1 - (Pl,,Og1/3)
= 1 - PI,?~I
(31a) (31b)
which may be substituted into eq 20 and 24, to obtain expressions for pl,:/kT,, and (PI PmChalns > (following the same trend in y2,nm/y2,1"), in each of the solvents. Since cubes cannot order (q2" = 0) and semiflexible chains are less ordered than rigid rods (as measured by q2-) in the anisotropic solvent, as can be seen in Table VII, this trend in the ability of a solute type to induce the anisotropic ("nematic")-isotropic transition also follows the trend of its inability to order in and be accommodated by the anisotropic solvent. Considering the cubic solutes (where 1 ID,I2) in more detail, y2,,"/y2,,"and both p," and PI" increase with increasing Dz, primarily reflecting increasing incompatibility between the bulky solute and the ordered solvent. Comparing the cubic solute results in the solvents having the same ml value but different EbJk values (Tables 111-V, where ml = 5 , f l = 0 may be regarded as ml = 5 , f , = 1 with E b , / k =. a), we see that y2,,"/y2,,"decreases as & , / k and the orientational order (both ql0 and wIo) of the anisotropic solvent decrease. That is, relative to the disordered solvent, the solute becomes less incompatible with the ordered solvent as the flexibility of the solvent molecular tail increases. Since ASJR of the pure solvent increases in going from E b , / k = m to E b , / k = 600 K, this effect amplifies the solution effect (see eq 1 and 2), leading to a pronounced decrease in both p," and p," (see curves a and c' in Figure 2). However, AS,,/R decreases in going from Eb,/k = 600 K to Eb,/k = 400 K, resulting in slightly higher p," and p," values for the more flexible solvent molecule (see curves c' and c in Figure 2). Examining the cubic-solute results for the two solvents having &,/k = 400 K and rl = 4, but differentf, values (Tables V and VI), we first note that, consistent with e ~ p e r i m e n t AS,,/R ,~~ increases and the orientational order of the anisotropic solvent (qIoand wl0) decreases a s h increases. Following the latter trend, y2,nm/y2,1m also decreases as the number of tail segments in the solvent molecule increases. This reinforces the trend in AS,,/R, leading to another pronounced drop in p," and PI" (see curves c and b in Figure 2). The experimental results for quasi-spherical solute^^^^ bear out this theoretical prediction. With increasing m2 of the rodlike solutes, p," and p," (both initially positive) increase, go through a maximum, and then decrease dramatically and become negative at about m2 = 5 (see Figure 3), where the solute molecules become more ordered than the solvent molecules in the anisotropic phase. Again, this follows the behavior of y2sl"/y2sm, i.e., the relative preference of the solute for the anisotropic and isotropic phases. In contrast to cubic solutes, where an increase in m2 leads to progressive destabilization of the anisotropic phase by lowering the orientational order per unit volume of solution, the rodlike solutes, whose orientational order increases with m2 (see Table VII), initially destabilize the
Uniaxial Binary Mixtures of Hard Molecules
The Journal of Physical Chemistry, Vol. 91, No. 25, 1987 6409
anisotropic phase (because of insufficient ordering) but eventually stabilize it as m2 increases (hence, the observed m a ~ i m a ~ , ~ , " ) . Comparing the rodlike solute results in the solvents having the same ml value but different Eb,/k values (Tables 111-V), it is seen that 72,nm/72,im decreases as E b , / k and the orientational order of the anisotropic solvent decrease. Since ASni/Ris the smallest for & , / k = m, we find once more that the 6" values are the highest for the stiffest solvent tail (curve a, Figure 3). However, compensation between the pure-solvent term (favoring lower 0"'s for & , / k = 600 K) and the solution term (favoring lower p"s for &,/k = 400 K) leads to rather similar /3" values for the two more flexible solvent species in the region 1 Im2 5 3, but with the more stabilizing solution effect eventually dominating at larger m2 for E b , / k = 400 K (curves c' and c, Figure 3). For the rodlike solutes in the solvents ml = 5, f l = 1 and ml = 6 , f 2 = 2 , both with E b , / k = 400 K (Tables V and VI), there is a complex interplay between the solution and pure-solvent is smaller contributions to @".Initially (smaller m2) y2,,"/y2,im in the longer and less ordered solvent molecules, but eventually (larger m2) becomes smaller in the shorter and more ordered solvent molecules, reflecting the match between solute and solvent orientational order parameters in the anisotropic phase (Tables I1 and VII). Therefore, since ASni/R is larger for f l = 2 than forfi = 1, the solution and pure-solvent effects reinforce each other at smaller m2,leading to lower p" values forfi = 2, whereas they oppose each other at larger m2,with the former (more stabilizing) effect dominating, leading to the observed crossover and lower /3" values for f l = 1 (curves c and b, Figure 3). Moreover, the maximum is predicted to occur at somewhat larger m2 and with a lower @", for the solvent molecule with the longer tail. These predictions are, for the most part, borne out by experimentas As expected, the semiflexible chain solutes fall between the cubic and rodlike solutes in their @" values and general behavior. In each of the solvents and in the range 1 Im2 I6 , initially (smaller m2)B increases with increasing m2at a rate intermediate between the rates for cubic and rodlike solutes, and then (larger m2) goes through a maximum, followed by a decrease not as pronounced as the decrease for rodlike solutes (see Figure 4). In two cases (curves a and d), the maximum is quite shallow and there is a region of m2 where @" is fairly constant (which also holds (approximately) for curve b). This general behavior follows the trend (here, always > l ) , the values of which also fall in 72,nm/y2,im between those of cubic and rodlike solutes.
Comparing the results in Tables I11 and IV, and, in order of descending p" and ascending ASni/R values, curves d, d', and c in Figure 4, it is apparent that the more similar the semiflexible-chain solute molecule is to the solvent molecule in its inherent and the flexibility ( E b / k ) ,the smaller is the value of 72,nm/72,im smaller is the m2 value at which the maximum occurs (marking the shift toward reduced destabilization). Considering curve c ( E b / k = 600 K) and curve b ( E b / k = 400 K) in Figure 4, and the results in Tables IV and V (ml = 5,f1 = 1 in both cases), note that (a) ASni/R is larger for the E , , / k = 600 K solvent is smaller for the more (favoring lower p"s), and (b) 72,nm/72,im flexible molecules at lower m2 and smaller for the less flexible ones at higher m2. Therefore, the pure-solvent term always leads to lower /3" values for the less flexible molecules and more so at higher m2,where it is reinforced by the solution term (eq 1 and 2). Comparing the chainlike solutes in the two solvents differing only in the length of their molecular end chains (Tables V and VI; curves b and a in Figure 4), we see that both the pure-solvent and solution effects promote lower p" values for the longer solvent molecule, a prediction supported by e ~ p e r i m e n t .Also ~ note that the theory predicts that, for chainlike solutes at very low concentrations in the anisotropic solvent near Tni,the average order parameters (w2") should range from 0.2 to 0.4 and increase with increasing solute chain length. These predictions are consistent with deuterium N M R data (quadrupolar splittings) for n-alkane solutes dissolved in nematogenic solvent^.^^,^^ Finally, Tables V and VI summarize the 8" values for the three types of solutes in the solvents ml = 5,f1 = 1 and m2 = 6 ,f1 = 2 , both with E b , / k = 400 K (physically, the most realistic "bond-bending energy" considered2'). These theoretical results, based on hard molecules, reveal the expected magnitude of the @" values and the distinctly different behavior of cubic (model analogue of spherical), rodlike, and chainlike solutes as a function of solute molecular volume and the length of the solvent molecular end chain. They will be subjected to a definitive experimental test in the following a r t i ~ l e . ~
Acknowledgment. This material is based upon work supported by the National Science Foundation under Grant CHE-8305045. We are also grateful to F. Dowel1 for helpful discussions. (34) Samulski, E. T.Ferroelectrics 1980, 30, 8 3
