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LATTICE MODEL FOR A BINARY MIXTURE OF HARD RODS AND HARD CUBES. APPLICATION
TO SOLUTE INDUCED NEMATIC → ISOTROPIC TRANSITIONS
G. Ågren, D. Martire
To cite this version:
G. Ågren, D. Martire. LATTICE MODEL FOR A BINARY MIXTURE OF HARD RODS AND HARD CUBES. APPLICATION TO SOLUTE INDUCED NEMATIC → ISOTROPIC TRANSI- TIONS. Journal de Physique Colloques, 1975, 36 (C1), pp.C1-141-C1-145. �10.1051/jphyscol:1975127�.
�jpa-00215904�
Abstract. — A statistical mechanical treatment for a two component mixture of hard rods of dimensions L% lx 1 (L = 5.10) and hard cubes of dimensions D* Dx D (1 =s D =S 2), placed on a simple cubic lattice, is described. The dimensionless pressure-to-temperature ratio 0 = Pvo/kT (where vo is the volume of a lattice site) is chosen so that the system is anisotropic when only rods are present. At constant <2> the partially aligned anisotropic mixture can be induced to undergo a first-order transition to the isotropic phase by increasing the concentration x of the cubes. A small two phase region is found. The dependence of this transition on 0, x, L, and D is described. Recent experimental results for mixtures of nematics with CCU are cited and compared with the findings of the lattice calculation. The model successfully predicts the existence, the general position and the extent of the observed two phase region, as well as the correct magnitude of the solute induced nematic — isotropic transition temperature depression. In agreement with experiments, the transi- tion order parameter of the rods is found to be independent of the concentration or size of the cubes.
The role of repulsive forces and the limitations of this and other mean field treatments of nematic mixtures are discussed.
1. Introduction. — Several different theoretical the Maier-Saupe theory of the nematic mesophase to approaches are available to aid in understanding, on a multicomponent systems. Starting with a general form molecular level, the nematic-isotropic transition in of the interaction between two molecules, they ave- pure liquid crystals. Much less attention has been paid raged over all coordinates of one of the molecules, to the theory of mixtures of liquid crystals [1, 2] and thereby obtaining for each component an anisotropic mixtures of liquid crystalline materials with non- pseudo-potential for one molecule in the form of a mesomorphic ones [1-4]. A study of the latter problem sum of Legendre polynomials. Evaluation of the should provide additional information concerning the orientational molar Helmholtz function, in this mole- forces stabilizing the nematic phase. For example, to cular field treatment, led to the result that for a what extent can a theoretical model employing only mixture of rods and spheres :
repulsive forces successfully account for the observed _ ( 1 )
transition behavior ? It should also aid in assessing the ~~ *• — X2' N _ I *• ' sensitivity of the transition temperature to possible o r
solute impurities.
Humphries, James, and Luckhurst [1] have extended l ai _ £z = _ 1 M M
- I N _ J C1X2 OX2
(*) Permanent Address : Institute of Theoretical Physics,
FACK, S-402 20 Goteborg, Sweden. where x2 is the mole fraction of the spherical solute, Classification
Physics Abstracts 7.130
LATTICE MODEL FOR A BINARY MIXTURE OF HARD RODS AND HARD CUBES. APPLICATION TO SOLUTE INDUCED
NEMATIC -> ISOTROPIC TRANSITIONS
G. I. AGREN (*) and D. E. MARTIRE Georgetown University, Washington, D. C. 20057, U. S. A.
JOURNAL DE'PHYSIQUE Colloque C l , supplément au n° 3, Tome 36, Mars 1975, page Cl-141
Résumé. — Par un traitement de mécanique statistique, on étudie les propriétés d'un mélange binaire composé de bâtons durs Lx lx 1 (L = 5,10) et de cubes durs Dx Dx D (1 =S D =S 2) placés sur un réseau cubique simple. Le nombre sans dimension 0 = Pvo/kT, où vo est le volume d'une maille du réseau, est choisi pour que le système soit anisotrope lorsque les bâtons durs sont seuls présents. A 0 constant, on peut induire une transition du premier ordre du mélange partiellement anisotrope vers une phase isotrope en augmentant la concentration x des cubes. On trouve une petite zone où deux phases coexistent. On étudie cette transition en fonction de 0, x, L et D et on compare nos prédictions théoriques aux résultats expérimentaux récents obtenus pour les mélanges de CCU dans une phase nématique. Notre modèle prévoit avec succès l'existence, la position et la largeur de la région à deux phases, et donne l'ordre de grandeur observé pour la dépression de tempéra- ture à la transition nématique-isotrope induite par un soluté. On trouve que le paramètre d'ordre des bâtons à la transition est indépendant de x et D, résultat vérifié expérimentalement. On discute le rôle des forces répulsives et les limitations d'une approximation de champ moyen pour traiter les mélanges nématiques.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1975127
G. I. AGREN A N D D . E. MARTIRE
TABLE I
Comparison of experimental results with lattice model predictions
Quantity
- dT*/dvz
- dT*/dx2 ( ~ 2 ) (a)
( X ; - X,) (a)
Experimental
A
DHAB (b) MBPA ('j C--
Lattice model -.
D = 1 D = l
L = l 0 L = 5
(a) At T* = 0.97.
(b) Reference [g] ; DHAB is p,pl-dihexyloxyazoxybenzene ; solute is CCI,.
(? Reference [S] ; MBPA is p-methoxybenzylidene-p'-n-propylaniline ; solute is CC],.
TA'?~ is the nematic-isotropic transition temperature of the pure liquid crystal,rods, and T and T* (= T / T ~ ? ~ ) are, respectively, the nematic-isotropic transition temperature and reduced transition temperature of the mixture. Thus, dT*/dx, is predicted to be independent of the nature of either the liquid crystal solvent or the solute, a prediction which is not borne out by experiment (see Table I). Also, no mention is made of a temperature range of phase separation, even though such behavior is required by the laws of thermodynamics for a first order transition with dT*/dx, # 0 [2, 4, 51.
Possible reasons for the inadequacy of the Hum- phries, James, Luckhurst [l] mean field treatment of mixtures might be examined. Certain approximations which are made concerning the pair distribution function are, at the very least, debatable. The predicted independence of dT*/dx, on the solute can be directly inferred from their eq. (2.13), where they make the approximation n?2)(r) = X, d2)(r). n$)(r) is the radial pair distribution function for species 1 and 2, and n(')(r) is a pair distribution function independent of the nature of the solute and solvent. Another convenient (and implicit) approximation which is made in order to perform their averaging is to factorize the pair distribution function (which now is not only a function of r, but also of the orientations of the molecules and the intermolecular axis) into one factor depending only on v, one factor depending only upon the orien- tation of one of the molecules, one corresponding factor for the other molecule, and one factor depending only on the orientation of the intermolecular axis.
Without access to the real pair distribution function it is impossible to assess how severe these approxima- tions are. However, it is felt that any attempt to draw conclusions from the resulting equations about the intermolecular forces should be tempered in light of the above. Finally, one might note that their results could be deduced in a direct, but less elegant, manner.
Their resulting eq. (3.10) (our eq. (la)) and (3.15)
(for a binary mixture of liquid crystals) are obtainable phenomenologically by simply taking the nematic- isotropic transition temperature of the mixture as the mole fraction average of the transition temperatures of the two pure components (with that for spheres being taken as zero Kelvin). Such corresponding states arguments are not without precedence [6]. Fur- ther, the use of an adjustable mixed interaction para- meter [l] to modify either of these equations might be acceptable in a phenomenological treatment, but is not a meaningful test of the proposed theory when such a parameter is required to generate agreement between expeiiment and theory.
Recently, Peterson et al. [4] employed a lattice model to study a binary mixture of hard rods of different lengths. They considered solvent rods of size L,, 1,l and solute rods of size L,, 1, 1, where L, = 5,10 and 1 ,< L, ,< L, - 1. For the chosen values of @* (equi- valent to T*, see later), the pure solvent rods pos- sessed a stable anisotropic phase, while the pure solute rods did not. With @* < 1 and held constant, the addition of a sufficient number of solute rods led to the onset of a two phase region, with an anisotropic phase of solute mole fraction X, in equilibrium with an iso- tropic phase of solute mole fraction X& At mole fractions less than x2 the system was anisotropic, while at values greater than x i the system was isotro- pic. Comparison with experiment [5] indicated that the lattice model correctly predicts the existence, the general position and the extent of the observed two phase region. Also, the predicted values of I d@*/dv, I,
where v, is the volume fraction of occupied sites occupied by the solute, follow the observed solute trend within a given solvent, i. e., I dT*/dv, I decreasing with increasing L,. In further agreement with experi- ment [7], the degree of alignment of the system on the anisotropic side of the transition is independent of either @* or L,. However, the model predicts that
I d@*/dv, I should increase with increasing L, for a given solute, while the reverse trend is observed [8].
LATTICE MODEL FOR SOLUTE INDUCED N + I TRANSITIONS Cl-143 In the present paper we extend the lattice calcula-
tioni to mixtures of hard rods of size L, 1 , 1 ( L = 5,10) and hard cubes of length D (1 D 2 ) . The purpose of this study is to further investigate the role of particle dimensions in solute induced anisotropic- isotropic transitions where the solute particle under- goes no preferential alignment in the anisotropic state. These calculations supplement those already carried out on mixtures of plates and rods [2] and rods and rods [4] and will provide a theoretical basis for comparison with future experimental studies on mixtures of liquid crystalline solvents and quasi- spherical solutes.
2. The lattice model. - In a system of elongated particles the Helmholtz free energy depends upon the order parameter y in addition to the usual variables T and V. In the lattice model as developed by DiMarzio [9], the molecules are divided into cubic blocks which are then placed on the lattice sites in such a way as to reconstitute the original molecule.
Rodlike molecules are treated as hard, rigid particles and are restricted to lie in one of three orthogonal directions. The procedure permits us to calculate, in a mean field approximation, the configurational parti- tion function as a function of T, V, and y. Since the
counting procedure has been extensively described elsewhere [2, 4, 91, we merely quote the result for the configurational partition function of our system, which consists of m cubes with side D and ni rods in direction i ( i = 1 , 2, 3 ; n, + n2 + n3 = n) with length-to- breadth ratio L packed in a simple cubic lattice of M sites :
--
[ M !IT
Q C ( M ¶ m ¶ { ni 1) = 1 + 2 D 3 - 3 D 2 *
[ ( M - m o 3 ) ! ] - D3
where the Z,'s are chosen to maximize Q,. Singling out direction 3 as the preferred direction and making the following changes in notation
we obtain the following expression for the configura- tional Helmholtz free energy (A,) of the mixture
+ 2 ( M - m ( 0 3 - 0 2 ) - sn(L - 1) ) ( In [ M - m ( 0 3 - 0 2 ) - sn(L - l ) ] - 1 )
+ ( ~ - m ( ~ ~ - ~ ~ ) - ( 1 - 2 s ) n ( L - l ) } ( l n [ M - m ( ~ ~ - D 2 ) - ( 1 - 2 s ) n ( L - l ) ] - 1 ) - 2 ns ( In (ns) - 1 ) - n(l - 2 S ) ( ln [n(l - 2 S ) ] - 1 )
- ( M - m ~ ~ - n ~ ) ( l n ( ~ - m ~ ~ - n L ) - 1 ) - m ( l n m - 1 ) . (4)
The order parameter y is related to s through cannot be done directly ; but, it can readily be shown (5) that
y = 1 - 3 s .
In experimental systems the usual thermodynamic
(2)
.,p,m,.
=(3
.,M,.,. . (6)variables are temperature, pressure and composition. . A
Hence, the equilibrium position is determined by the Thus, at that volume corresponding to a given pressure minimum in the configurational Gibbs free energy G,. one can determine the s value that minimizes G, by However, since G, cannot be expressed here in closed finding the extrema of eq. (4). This gives the following form as a function of P and T, this minimization set of transcendental equations :
+ 2 In [V - x2(D3 - 0 2 ) - X , s(L - l ) ] + In [V - x , ( D ~ - D') - x l ( l - 2 S) ( L - l ) ] (7b)
Cl-144 G. I. AGREN AND D. E. MARTIRE where X, = n/(n + m), x2 = m/(n + m), V= M/(n + m)
and v, is the volume of a unit cell. One also has the chemical potentials for the rods and cubes given by, respectively,
In a system of hard particles, the temperature and pressure are not independent variables but only their ratio, as can be seen in eq. (7b). A variation of @ in our calculation will be interpreted as a variation of temperature at constant pressure [4].
3. Results. - We express our results concerning the solute induced transition temperature depression at constant pressure using the reduced variable @*, where
for X; (isotropic phase mole fraction of cubes) and X,
(nematic or anisotropic phase mole fraction of cubes).
In the region investigated 1 > @* > 0.93, @* is
virtually a linear function of X, or v,, where
Given in figure 2 are the results for the slopes d@*/dx, (comparable to dT*/dx,) and d@*/dv, (comparable to dT*/dv,). In contrast to the prediction of Humphries et al. [l], i. e., eq. (lb), both d@*/dx, and d@*/dv,
where G,-, is the transition point for the pure rod system. Two different rod lengths (L = 5,lO) and six different cube sizes (D = 1 .O, 1.2, . . ., 2.0) were treated.
The general features of our results are the same as those reported previously [4] and are depicted in figure 1 for L = 5 with D = 1.0 and D = 2.0. The
0 . 9 0 ~ I I I
0.05 0.1Q
X 2
FIG. 1. - Reduced transition temperature of the mixture @*
as a function of the mole fraction of cubes x2 for rods of L = 5.
Lines marked A denote the boundary of the anisotropic phase, and those marked I the boundary of the isotropic phase.
transitions are first order, and the anisotropic and iso- tropic phase coexist over a mole fraction range at constant @* for Qi* < 1. The position and extension of the two phase region were determined by graphical solution of the equations
#"(X;) = p;em(x2) (10a)
&"(X;) = prm(x2) (lob)
FIG. 2. - d@*/dq as a function of the solute cube dimension D, where d@*/dq is the slope of the linear plot of @* (reduced transi- tion temperature of the mixture) against either xz (mole fraction of. cubes) or v 2 (occupied volume fraction of cubes) on the anisotropic side of the transition.
depend strongly on both solute and solvent size. The extension of the two phase region is interesting in that it follows a simple corresponding states expression :
where K = 0.30 for L = 5 and IC = 0.46 for L = 10, independent of the size of the solute. Another way of viewing the extent of the two phase region is to consider the temperature range (A@*) between the onset of the anisotropic phase and the disappearance
LATTICE MODEL FOR SOLUTE INDUCED N + I TRANSITIONS Cl-145 of the isotropic phase upon cooling a system of total
composition 5,. For example, for 3, = 0.02, A@* = 0.04 with L = 5 and D = 2 (see Fig. l), and A@*=0.005
with L = 10 and D = 1.
Our model predicts that the order parameter y at the transition is essentially independent of solute concen- tration and solute size, in agreement with experi- ment [7]. For cubes with D = 1 the y values are 0.799 (L = 5) and 0.864 (L = 10), the same (within the precision of the calculation, + 0.001) as those found with pure rods (at @* = 1). For cubes with D = 2, the transition values vary slightly, reaching 0.818 (L = 5) and 0.873 (L = 10) at @* = 0.93.
Another quantity we find to be constant at the onset of the two phase region (anisotropic side) is the volume fraction of sites occupied by the rods, i. e.,
X, LIV. For rods of length 5 it is 0.711, while for rods of length 10 it is 0.359.
4. Discussion. - The predictions of the lattice model are compared in Table I with experimental results for mixtures of CCI, (a quasi-spherical mole- cule) with MBPA [5] and DHAB [8]. The hard sphere diameter of CCI, is roughly equal to the breadth of the two liquid crystal solvents ; thus, D w 1. The lengths of MBPA and DHAB were estimated from molecular models, assuming the pendant hydrocarbon portions to be fixed in the completely extended conformation. This assumption is probably not valid for DHAB [3], and flexibility of the hexyloxy groups could account for the discrepancies between experi- ment and our calculations, which are based on rigid
rods. In particular, note that at T* = 0.97 'the mole fraction of solute necessary to induce the onset of the isotropic phase is observed to be much larger with DHAB than with MBPA or in the lattice model. If the alkoxy groups of DHAB in the nematic state have some fluidity, one would expect that this would favor the accommodation of CCl,, as is observed. Neverthe- less, with the exception of dT*/du,, the experimental quantities listed in Table I follow the same trend with decreasing L as given by the lattice model. Further- more, the quantitative agreement with the experimen- tal results for MBPA, for which L w 3 [10], is quite decent, and the existence and extension (xi - xz) of the two phase region are satisfactorily predicted for both solvents.
Among the drawbacks of the lattice model are the large free volume fractions [(V - X, L - X , D3)/V]
and transition order parameters encountered with increasing L. This behavior is common to all mean field treatments of hard, rigid particles [ll], and could account (in part, at least) for some of the discrepancies in Table I. Nevertheless, the reasonable success of this model further demonstrates the impor- tant role of repulsive forces, i. e., particle size and shape, in governing nematic phase stability, and suggests that additional experiments involving larger quasi-spherical solutes (D > 1) with MBPA might be illuminating.
Acknowledgment. - This work was supported through a basic research grant from the U. S. Army Research Office, Durham, N. C.
References
[l] HUMPHRIES, R. L., JAMES, P. G. and LUCKHURST, G. R., Symp. Faraday Soc. 5 (1971) 107.
[2] ALBEN, R., J. Chem. Phys. 59 (1973) 4299.
[3] MARTIRE, D. E., Mol. Cryst. Liqu. Cryst., in press.
[4] PETERSON, H. T., MARTIRE, D. E. and COTTER, M. A., J. Chem. Phys., 61 (1974) 3547.
[5] PETERSON, H. T. and MARTIRE, D. E., Mol. C w t . Liqu.
Cryst. 25 (1974) 89.
[6] HUMANS, J. and HOLLEMAN, Th., Advan. Chem. Phys. 16 (1969) 223.
[7] CHEN, D. H. and LUCKHURST, G. R., Trans. Faraday Soc.
65 (1969) 656.
[g] PETERSON, H. T. and MARTIRE, D. E., J. Phys. Chem., submitted.
[9] DIMARZIO, E. A., J. Chem. Phys. 35 (1961) 658.
[l01 Direct comparison with a lattice calculation for L = 3 is not possible. No anisotropic phase exists at any density with L < 3.65, an unfortunate artifact of the lattice model.
See, ALBEN, R., Mo1. Cryst. Liqu. Cryst. 13 (1971) 193.
[l11 See, e. g., COTTER, M. A. and MARTIRE, D. E., J. Chem.
Phys. 53 (1970) 4500.
