J . Phys. Chem. 1986, 90, 5426-5430
5426
during cooling when slowly cooled, but it remains unfrozen during cooling when rapidly cooled and crystallizes during subsequent warming. A conceivable reason is that the extent of continuity between the compartments increases with the water content, as mentioned before. As a result of the increase in the extent of continuity, compartmentalized water crystallizes spontaneously during cooling when slowly cooled. The extent of continuity between the outside of the gel beads and the compartments may also increase as the fraction of water outside the gel beads increases with water content. Then, compartmentalized water is easy to be ice-seeded from the outside. When rapidly cooled, the network structure may change too fast for most of compartmentalized water to crystallize, and it remains unfrozen even if the water content is high. In the case of a Sephadex G-15 gel, each compartment can accommodate less than 100 water molecules considering the exclusion limit in molecular weight.34 Then, most of the water in the compartment may exist in the vicinity of gel matrices.23 Therefore, it can hardly be compartmentalized as "bulk" water. Water in the compartment, in this case, crystallizes during cooling when cooled slowly. By rapid cooling, compartmentalized water remains unfrozen and crystallizes during subsequent warming in the same way as a G-25 gel. In the case of a G-25 gel, most compartments can accommodate about a few hundred water molecules34 and a substantial part of the water is to be compartmentalized as almost "bulk" water. So, it remains unfrozen when cooled slowly as well as when cooled rapidly if the water content is about 50 wt %. The amount of crystallizable water during warming in the case of a G-25 gel was larger when rapidly cooled than that when slowly cooled previously. It means that some part of the water in the vicinity of gel matrices remains unfrozen as a part of compartmentalized water when rapidly cooled and crystallizes during subsequent warming as was explained in the case of a G-15 gel. The size of the compartment of a G-100 gel in a fully swollen state is so large that compartmentalized water may easily crystallize spontaneously when cooled. However, it may also be true that compartmentalized water in a G-100 gel crystallizes as the extent of continuity between the adjacent compartments is high, as explained before. Then, it is important to indicate that the size of the compartment and the
extent of continuity between the adjacent compartments may not be independent in the case of these cross-linked polymer gels as the extent of continuity is related to the density of the cross-links. Similar results dependent on the size of the compartment have been obtained by studies using cross-linked polyacrylamides (Bio-Gels) (Bio-Rad Laboratories, Richmond, CA). A Bio-Gel P-6 gel, whose compartment size is known to be almost the same as that of a Sephadex G-25 gel, shows the most remarkable heat evolution during warming among Bio-Gels.6,11 The size of the compartment of a G-25 gel in a fully swollen state calculated from its accommodations for water molecules is estimated to be about 20-25 A in diameter, assuming that compartments are spherica1.23J9 It is very interesting to know that this size coincides with the size reported to be necessary for small molecules or ions to maintain unrestricted fast Remarkably, the ice crystallization during warming has not been observed, thus far, with gels made from polymers without covalent cross-linkage, although it has been observed with some biomaterials containing ~ a t e r . ~ One ~ ' ~of~the ' ~reasons why it is scarcely observed with these gels may be the difficulty in preparing compartments of appropriate size uniformly. The behavior of ice crystallization during warming is dependent on the nature of cross-linked polymers; Le., its behavior observed with a Sephadex G-25 gel is slightly different from that observed with a Bio-Gel P-6 gel.IoJ1 It may be due to the difference in the hydration properties of the material forming the gel. The mechanism speculated in this paper is still incomplete; however, it may become complete by the more precise studies using other gelling systems. Studies of gels made from biopolymers and compartmentalized water by the gel's network are encouraged from the standpoint of biological freezing. Registry No. H,O, 7732-18-5; dextran, 9004-54-0. (39) Accommodations of the compartment of Sephadex G-25 for water molecules were estimated by simply dividing the exclusion limit of the compartment (SO00 Da for G-25) by the molecular weight of water, 18. The number of water molecules thus obtained, 2SCb300 molecules, is considered to be taken in the spherical compartment. Assuming that the density of water is 1 g/cm', the diameter of the spherical compartment is estimated to be 20-25 8, (ref 23).
Phase Transitions In Nematic Liquid Crystals: A Mean-Field Theory of the Isotropic, Uniaxial, and Biaxial Phases D. K. Remler and A. D. J. Haymet* Department of Chemistry, University of California, Berkeley, California 94720 (Received: September 17, 1985; In Final Form: March 26, 1986)
The model of nematic liquid crystals proposed by Freiser is solved within the mean-field approximation (with no further approximations), over the full range of biaxiality. Both uniaxial and biaxial ordered phases are found. By comparison with a computer simulation of the same model we find that, in general, those aspects of the uniaxial transition predicted poorly by mean-field theory continue as the degree of biaxiality is increased. In addition, we examine the effect of an external magnetic field on biaxial liquid crystals. The mean-field estimate for the critical field strength, above which the "paranematic"-nematic (uniaxial) transition is suppressed, decreases as the degree of biaxiality is increased.
1. Introduction
Liquids crystals continue to be a subject of intense experimental, theoretical, and industrial interest.' In this paper we focus on an approximate model of nematic liquid crystals proposed by F r e i ~ e r . ~This , ~ model, which allows for both uniaxial and biaxial phases, is solved within mean-field theory with no further physical * NSF Presidential Young
Investigator 1985-1990.
0022-3654/86/2090-5426$01.50/0
or mathematical approximations. Luckhurst et aL4 examined the same model 10 years ago and studied the influence of increasing degree of biaxiality on the isotropic to uniaxial nematic transition. (1) Gray, G. W. Proc. R . SOC.London Ser. A 1985, 402, 1. (2) Freiser, M. L. Mol. Crysr. Liq. Cryst. 1971, 14, 165. (3) Freiser, M. L. Pkys. Rev. Lett. 1970, 24, 1041. (4) Luckhurst, G . R.; Zannoni, C.; Nordio, P. L.; Segre, U. Mol. Phys. 1975, 30, 1345.
0 1986 American Chemical Society
Phase Transitions in Nematic Liquid Crystals Building on the earlier ~ o r k ,this ~ , study ~ focuses instead on the three possible phase transitions. Our reexamination of Freiser’s model of a single-component nematogen is motivated by formal (mathematical) similarities with recently studied models of ordering in amorphous and the recent experimental,8 t h e o r e t i ~ a l , and ~ ~ ’ ~computational” studies of biaxial phases in mixtures of liquid crystals. In section 2 we summarize briefly the unresolved problem concerning first-order phase transitions in nematic liquid crystals. The model Hamiltonian of Freiser2 is reviewed in section 3 in the context of the present calculation, and our results are presented in section 4. The influence of an external magnetic field on the isotropic uniaxial nematic transition has been examined by computer simulationsI2 and mean-field theory.l3*I4 These studies indicate that the mean-field theory estimate for the critical magnetic field strength yc (above which the nematic-paranematic transition vanishes) is too high to observe the field-induced critical point in real systems. In section 5 we examine approximately the effect of an external magnetic field on biaxial liquid crystals and find that the critical magneic field strength is reduced rapidly as the degree of biaxiality is increased.
2. First-Order Phase Transitions in Nematic Liquid Crystals One unsolved problem in nematic liquid crystals may be stated simply. Liquids composed of anisotropic molecules, such as p azoxyanisole (PAA), undergo a phase transition from the isotropic liquid to the nematic liquid-crystal phase, which has long-range orientational order. Despite much recent effort15J6this first-order phase transition is not completely understood: it is a “small” first-order phase transition, and yet it has features more often associated with second-order transitions. At the theoretical level, the properties of real isotropic-nematic liquid-crystal phase transitions (as well as computer simulation “experiments” on simplified model systems) disagree with the predictions of mean-field theory15 and have not yet been completely understood in the context of the renormalization group (RG) theory.” As emphasized by Gelbart,15 simple molecular theories of nematic liquids fall into three categories, based upon the particular interaction deemed to be most important: (i) attractive forces (e.g. Maier-Saupe,ls F r e i ~ e r ,Straley19); ~.~ (ii) repulsive forces (e.g. Onsager,20Zwanzig,21Flory22);or (iii) van der Waals theory (e.g. Alben,23g24Gelbart and B a r b ~ y ~ ~These ) . theories predict remarkably similar behavior for liquid crystals, although the microscopic details included in the theories are obviously very different. This fact emphasizes the importance of symmetry in (5) Haymet, A. D. J. Phys. Rev. B 1983, 27, 1725. (6) Haymet, A. D. J. J . Non-Cryst. Solids 1985, 75, 79. (7) Remler, D. K.; Haymet, A. D. J. Phys. Rev. B Condens. Matter, to be published. (8) Yu, L. J.; Saupe, A. Phys. Rev. 1980, 45, 1000. (9) Alben, R. J . Chem. Phys. 1973, 59, 4299. (10) (a) Rabin, Y.; McMullen, W. E.; Gelbart, W. M. Mol. Cryst. Liq. Cryst. 1982, 89, 67. (b) Chen, 2.-Y.; Deutch, J. M. J . Chem. Phys. 1984, 80.2151. (cl Stroobants. A,: Lekkerkerker. H. N. W. J . Phvs. Chem. 1984. 88; 3669. id, Palffy-Muhoray, P.; de Bruyn, J. R.; Dunmur: D. A. J . Chem: Phys. 1985, 82, 5294. (11) Hashim, R.; Luckhurst, G. R.; Romano, S. Mol. Phys. 1984, 53, 1535. (12) Luckhurst, G. R.; Simpson, P. Chem. Phys. Lett. 1983, 95, 159. (13) Wojtowicz, P. J.; Sheng, P. Phys. Lett. 1974, 48A, 235. Our definition of the external field includes an extra factor of 6 (for the coordination number) compared to this reference. (14) Palffy-Muhoray, P.; Dunmur, D. A. Phys. Left. 1982, 91A, 121. (15) Gelbart, W. M. J . Phys. Chem. 1982, 86, 4298. (16) The Molecular Physics of Liquid Crystals, Luckhurst, G. R., Gray, G . W.. Ed.: Academic: London. 1979. (17) Priest, R. G. Phys. Rev. 2 1981, 23, 3279. Priest, R. G.; Lubensky, T. C. Phys. Rev. B 1976, 13, 4159. (18) Maier, W.; Saupe, A. 2.Naturforsch. A 1958, 13, 564. 1959, 14, 882. 1960, 15, 287. (19) Straley, J. P. Phys. Reu. A 1974, 10, 1881. (20) Onsager, L. Ann. N.Y. Acad. Sci. 1949, 51, 627. (21) Zwanzig. R. W. J . Chem. Phvs. 1963, 39. 1714. (22) Flory, F. J. Proc. R . Soc. London, Ser. A 1956, 234, 73. (23) Alben, R. Mol. Cryst. Liq. Cryst. 1971, 13, 193. (24) Alben, R. Phys. Reu. Lett. 1973, 30, 778. (25) Gelbart, W. M.; Barboy, B. Acc. Chem. Res. 1980, 13, 290.
The Journal of Physical Chemistry, Vol. 90, No. 21, 1986 5421 the theory of nematic liquids. The alternative to molecular theories is, of course, the highly developed continuum elastic Analysis of the phase transition properties of these theories does not lead to good agreement with the experimental results from real liquid crystals such as PAA. At least three important deficiencies have been found in the assumptions of the theories: (a) breakdown of mean-field theory, (b) the effect of b i ~ x i a l i t y ? ~ , ~ ~ and (c) the effect of f [ e ~ i b i l i t y . ~ ~ . ~ l This paper uses mean-field theory to study the role of biaxiality. Real liquid-crystal molecules, such as PAA, are obviously not perfectly uniaxial, and a slightly more realistic model (although still far from perfect) is to allow a certain degree of asymmetry in the plane perpendicular to the “long axis”. A model Hamiltonian which allows for this effect was developed by Freiser,2 who analyzed the model by series expansion techniques. In this contribution we do not attempt a comprehensive review of all the recent studies of biaxial phases. We cite the work which directly influenced our calculations and apologize for any accidental omissions. A number of closely related studies have appeared in the past 10 years, including the work of Alben; Straley,19 and the similar, but not identical, SU(3) model of Mulder and R ~ i j g r o k .Luckhurst ~~ et aL4 have examined many aspects of Freiser’s model, especially the dependence of the orientational properties of the uniaxial mesophase on the deviation from cylindrical symmetry in nematogens, and the implications for the interpretation of experimental results. The Landau theory of biaxial nematics has been developed by Allender, Lee and H a f i ~ , ~ ~ and Szumlin and M i l ~ z a r e k . ~In~ addition, the hydrodynamic of biaxial nematics p r o p e r t i e and ~ ~ ~ccntinuum ~~~ elastic have been developed recently. In this work we define a quantity which measures the degree of biaxiality (denoted r below) and solve the Freiser model exactly within mean-field theory over the full range of biaxiality, from perfectly uniaxial to perfectly biaxial. This involves solving for two independent order parameters, which can be viewed as measuring uniaxial and biaxial order, respectively. The results are qualitatively similar to the series expansion results of Freiser’ and agree with the numerical calculations of Luckhurst et aL4 where the two calculations overlap. An important conclusion of the present work is obtained by comparison with a single Monte Carlo computer simulation by Luckhurst and Romano39for the same model at one particdar biaxiality. (Unfortunately, we know of no other computer studies of this Hamiltonian.) We amplify the comparison between simulation and mean-field made in this important paper and find that the predictions of mean-field theory for biaxial nematogens have the same weaknesses as for the purely uniaxial case. In particular the mean-field estimate for the entropy change at the transition is much too large. At this point it may be worthwhile to review the properties of the “Lebwohl-Lasher” perfectly uniaxial lattice model.40 In this model, nearest-neighbor nematogens on a fixed cubic lattice interact with a potential energy proportional to the second Legendre (26) Chandrasekhar, S. Liquid Crystals; Cambridge University Press: Cambridge, 1977. (27) de Gennes, P. G. The Physics of Liquid Crystals; Clarendon: Oxford, 1975. (28) Faber, T. E. Phil. Trans. R . SOC.London, Ser. A 1983, 309, 115. (29) Mulder, B. M.; Ruijgrok, Th. W. Physica 1982, 113A, 145. (30) Dowell, F.; Martire, D. E. J . Chem. Phys. 1978, 68, 1088, 1094. (31) Emsley, J. W.; Luckhurst, G. R.; Stockley, C. P. Proc. R . SOC.London, Ser. A 1982, 381, 117. (32) Allender, D. W.; Lee, M. A.; Hafiz, N. Mol. Cryst. Liq. Cryst. 1985, 124, 45. See also: Allender, D. W.; Lee M. A. Mol. Cryst. Liq. Cryst. 1984, 110, 331. (33) Szumilin, K.; Milczarek, J. J. Mol. Cryst. Liq. Cryst. 1985, 127, 289. (34) Saupe, A. J. Chem. Phys. 1981, 75, 5118. (35) Liu, M. Phys. Reu. A 1981, 24, 2720. (36) Brand, H.; Pleiner, H. Phys. Reu. A 1981, 24, 2777. 1982, 26, 1783. (37) Trebin, H. R. J . Phys. (Paris) 1981, 42, 1573. (38) Grovers, E.; Vertogen, G. Phys. Rev. A 1984, 30, 1998. erratum 1985, 31, 1957. Physica A (Amsterdam) 1985, 133, 337. (39) Luckhurst, G. R.; Romano, S. Mol. Phys. 1980, 40, 129. (40) Lebwohl, P. A.; Lasher, G. Phys. Rev. A 1972, 6,426; erratum. 1973, 7, 2222.
5428
The Journal of Physical Chemistry, Vol. 90, No. 21, 1986
polynomial, P,(cos e), where 0 is the relative angle between the adjacent molecules. This simple model has been studied in great detail by “Monte Carlo” simulation t e c h n i q ~ e s . ~ The , ~ ’ properties discussed below indicate a striking disagreement between “experiment” and mean-field theory, due to Maier and Saupe. l8 A further major discrepancy is the magnitude of the difference between the transition temperatue, To,and the ”instability” temwhich is defined (by extrapolation of experimental perature, P, results) as the temperature at which the order parameter would vanish. Typical experiments show this difference to be less than 1 “C: mean-field theory predicts 30 0C.17327 In view of these results for the perfectly uniaxial case, it is important to discriminate between (i) the deficiencies of mean-field theory and (ii) the deficiencies of perfectly uniaxial models. We find below that, for moderately biaxial systems in Freiser’s model, mean-field theory is in no better agreement with the simulations of Luckhurst and Romano. 3. The Model Hamiltonian The interaction potential between two liquid crystal molecules can be expanded in the spherical harmonics, as shown by Freiser.2 Consider two molecules, centered at positions ri and rj, whose orientations with respect to fixed external axes are completely determined by the Euler angles Qi (cui,Pi,yi)and Qj, respectively. Then the potential energy Vi, between two molecules may be written in standard form42
Remler and Haymet assumed to be identical, ellipsoidal particles. Since these particles have inversion symmetry, all e f , with 1 odd are zero. The final approximation in the potential energy is to truncate the expansion (eq 2) after the first angle-dependent term, namely the term with 1 = 2. Higher order terms, such as the 1 = 4 term, have been investigatedM6 and apparently play an important but secondary role. The internal axes can be chosen2 so that the coefficients Q f ] vanish and Qi2) = Q!;). Hence the symmetry of the molecules is completely determined by two numbers, Qi2)and QQ). If Q$*)= 0, we recover the well-known Lebwohl-Lasher40 Pz(cos PI,) interaction between purely uniaxial nematogens. The final, approximate form of the potential energy between each pair of molecules in the liquid is 2
VI = a2 m=-2 C
Without further loss of generality we assume that the potential is attractive and define a coupling strength J = -a2. Hence the model is defined by three quantities J , QS), and Qf). We now solve the model defined by the potential energy (eq 4) within the mean-field approximation. Each molecule is assumed to interact with a constant number z of nearest neighbors. To be definite, we assume from now on that z = 6. Each molecule is assumed to interact with the average “field” due to its neighbors. The Hamiltonian H in the mean-field approximation may be written
-PH = PJ
C C (Qrn(Q,))Q*m(QL) (?I)
where ai,,,,,(rij)are coefficients which depend on the particular molecules under study, the symbol in parentheses in the usual “3j symbol”, Q‘,(Qi) is the (1,m)th multipole moment of molecule “i”, Y,, is the spherical harmonic, rij = ITj - rjl is the distance from the center of molecule “i” to the center of molecule “j”, and 0 and 4 are the spherical polar angles of the vector rii in the fixed external axes. Similar expansions have been used to study orientational ordering in supercooled liquids and amorphous The first approximation of the theory is to average the potential energy (eq 1) over all possible directions of rij. This means that the interaction energy is assumed to depend only on the relative orientation between the molecules. This is certainly an approximation of the interactions found in nature, but it has proved useful in earlier studies of liquid crystals2 The average potential energy may be written /
Jj‘i
= Cai(rij) /
C m=-l
QL(Qj)
QI(Qi)
(2)
The second approximation is to replace the distance-dependent coefficient al(rij)with an “average” value a, which is independent of distance. For dense liquid systems, where the average nearest-neighbor distance cannot vary widely, this is thought to be a useful first approximation. The multipole moments Q’,(Qi) of a given molecule “i”,with orientation ai, in the external coordinate system may be related to multipole moments QL (with no argument) in a given internal coordinate system (i.e. a coordinate system fixed in the molecule) by the equation
(4)
QkYQJ Qg)*(Q,)
(5)
m
where 8’= k T , the sum is over pairs of nearest neighbors, the angle brackets denote equilibrium average, and the superscripts 1 = 2 have been dropped. By defining the external coordinate system to be that of the average orientation, we can set
AmQm = ( Q m ( Q ) ) , m = - 2 , O , 2 (6) where {Am]is a set of order parameters to be calculated from the theory. Hence the Hamiltonian becomes
-PH= PJzC I
C
m=O.f2
AmQm
C D*,m ( Q i ) Q n
n=O,f2
(7)
At zero temperature all the molecules are perfectly aligned, ) Q, and A , = 1. At high temperatures, in the and so ( Q m ( Q ) = isotropic state, the molecules are randomly oriented, and so ( Q m ( Q )= ) 0 and Am = 0. The goal of the theory is to calculate how the two independent order parameters A , and A2 vary with temperature. There are three possible states of the system: (i) isotropic ( A , = 0 and A 2 = 0); (ii) uniaxial ( A , # 0 and A2 = 0, or equivaiently, with a relabeling of axes, A , = 0 and A , # 0); and (iii) biaxial ( A , # 0 and A2 # 0). The partition function Z(P) is given by Z(P) =
J dQ exp(-PH)
(8)
where JdQ
= (8r2)-’J2“ da
STd p sin /3
J2“
0
dy
(9)
The order parameters A . and A2 must be determined from the coupled implicit equations for the average orientation
(3)
AoQo = Z-’ I d 0 W,(Q)exp(-PH)
(10)
where (D;,J Q i ) ) define the three-dimensional rotation group. Throughout this paper the conventions of E d m ~ n d are s ~ ~used. The use of internal coordinates is convenient because, with judicious choice of axes, the symmetry of the molecules causes many of the Q f ,to vanish. In the liquid under study, the molecules are
A2Q2= Z - l S d Q W2(Q)exp(-PH)
(11)
QL(Qi)
=
xGz,m(Qi)Qf, n
(41) Luckhurst, G. R.; Simpson, P. Mol. Phys. 1982, 47, 251. (42) Gray, C. G.; Gubbins, K. E. Theory of Molecular Fluids; Clarendon: Oxford, 1984; Section 2.3. (43) Edmonds, A. R. Angular Momentum in Quantum Mechanics; Princeton University Press: Princeton, 1957.
where wn(fi) = CDm,n(Q)Qm m
(12)
(44) Chandrasekhar, S.; Madhusudana, N. V. Acta Crystallogr. 1971, 27, 303.
(45) Humphries, R. L.; James, P. G.; Luckhurst, F. G. J . Chem. SOC., Faraday Trans. 2 1972, 68, 1031. (46) Gelbart, W. M.; Barboy, B. Mol. Crysr. Lip. Crysr. 1979, 55, 209.
The Journal of Physical Chemistry, Vol. 90, No. 21, 1986 5429
Phase Transitions in Nematic Liquid Crystals
I .o
I .5 Isotropic
*
L
W
1.0
c
E)
k
Fo
2!3
a
c
E
L
W
W
a
E
0.5
z 0
0.5
F
0.0 0.0
0.0 0.1
0.2
0.3
0.4
Figure 1. Phase diagram calculated from mean-field theory for the model Hamiltonian described in the text.
since the mean-field Hamiltonian (-pH) depends parametrically on the order parameters. The (dimensionless) free energy of a phase with order parameters A. and A2, relative to the isotropic phase at the same temperature, is given by m
- In Z(P)
1.5
Temperature T *
Degree of Biaxiality
P A F / N = ‘/2zJP&Im2Q*,Qm
1.0
0.5
(
(13)
The integrals 8, 10, and 11 have been evaluated numerically by using 24-point Gauss-Legendre quadrature, with techniques developed earlier.s A Newton-Raphson method was used to find the solution of the implicit equations 10 and 11. The degree of biaxiality r can be defined by where we have normalized the I = 2 multipole moments with respect to the internal axes so that
For a perfectly uniaxial system Q2 = 0 and hence r = 0. As shown by Freiser,2 a “perfectly biaxial” system has r = 6-’12 i= 0.408. Systems with r > 6-’12 (“plates”) can be related to systems with r C 6-’12 (“rods”) by a relabeling of axes, and so we consider the full range of biaxiality by finding numerical solutions for 0 < r < 6-1/2. Within mean-field theory the properties of the system are completely determined by two numbers, the degree of biaxiality r, and the dimensionless product PJz.
4. Results: The Effect of Biaxiality The phase diagram of this model biaxial system, calculated by using mean-field theory, is shown in Figure 1, which displays transition temperatures T* = (PJ)-’as a function of biaxiality r. For purely uniaxial nematogens ( r = 0), the transition is the familiar l 8 first-order isotropic to uniaxial transition. For the perfectly biaxial system ( r = 6-’/2) there is a single second-order transition to a biaxially ordered state at a slightly lower temperature. For intermediate degrees of biaxiality, there are two transitions, a first-order isotropic to uniaxial transition at high temperature, and a second-order uniaxial to biaxial transition at a lower temperature. The quantitative behavior of the system is summarized in Table 11. As the degree of biaxiality is increased, the “jump” in the order parameter A. at the isotropic uniaxial transition decreases, as does the accompanying discontinuity in entropy PS. Also, the temperature at which the biaxial transition occurs increases, until at r = 6-*12the two transition coalesce into the single second-order, isotropic to biaxial transition. The behavior of the
Figure 2. Order parameters as a function of (dimensionless) temperature T“ for three degrees of biaxiality: 1, A . for the purely uniaxial case ( r = 0); 2, A . for the intermediate case ( r = 0.2); 3, A. and A2 for the perfectly biaxial case ( r = 6lI2); 4, A 2 for the intermediate case ( r = 0.2). TABLE I: Comparison of Computer Simulations and Mean-Field Theorv uniaxial, r = 0
biaxial, r = 0.2 ASlNk A A n” ToASINk AAn 1.12 0.098 0.33 1.22 0.08 i 0.01 0.31
Tn
Monte Carlo‘ mean-field theory 1.32
0.42
0.43 1.27 0.27
0.35
’References 39 and 41.
two independent order parameters is displayed in Figure 2, for three biaxialities r = 0, 0.2, and 6-1/2. Where comparison is possible, our calculations of the order parameters appear to be in agreement with the earlier calculation of Luckhurst et aL4 The qualitative features of the phase diagram (Figure 1) have been obtained from a variety of microscopic models.25 Mulder and R ~ i j g r o obtained k~~ similar results from their SU(3) model. Rabin et al.Ioa obtain the phase diagram from a mathematical limit in their theory of mixtures of rodlike and platelike nematogens, which is “essentially equivalent to a one-component liquid of biaxial molecules”.’08 Chen and Deutchlobobtain corresponding results from their lattice van der Waals model of mixtures. As pointed out Allender et a qualitatively different phase diagram, which contains a fine of isotropic to biaxial transitions, is possible both in Landau theory and within Alben’s model.24 Most models, including the one discussed here and the model of rectangular plates with hard-core repulsive potential^,^^ lead to the same qualitative feature, namely that increasing the degree of biaxiality decreases the jump in order parameter at the isotropic uniaxial transition. Hence, biaxiality is one factor which is often cited in the remarkable disagreement in the “size” of the first-order transition between the Maier-Saupe model and real nematogens. However, the Lebwohl-Lasher simulation calculation shows that much of the problem in the uniaxial limit is due to mean-field theory. When the degree of biaxiality is increased, it has apparently not been appreciated that mean-field theory is also in poor agreement with the available Monte Carlo “experiment”. Luckhurst and Romano39 have performed a simulation of an fcc lattice model (12 neighbors per spin) with biaxiality r = 0.2, and some of the results are collected in Table I. Note that the measurement of order parameters in the simulation suffers from so that the usual limitation connected with director fl~ctuations$~ identification of AAo in Table I is not unambiguous. We have ~
~
~~
(47) Shih, C. S.; Alben, R. J . Chem. Phys. 1972, 57, 3057. (48) Keyes, P. H.; Shane, J. R. Phys. Reu. Letr. 1979, 42, 722. (49) Masters, A. J. Mol. Phys. 1985, 56,887.
5430 The Journal of Physical Chemistry, Vol. 90, No. 21, 1986
Remler and Haymet 0.50
TABLE 11: Isotropic-Uniaxial (and Isotropic-Biaxial Phase Transition in the Freiser Model of Liauid Crystals
I
\
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.375 0.39 6-1 /2
1.321 1.317 I .306 1.288 1.267 1.243 1.222 1.207 1.202 1.20 1.20
0.43 0.43 0.41 0.39 0.35 0.3 1 0.24 0.14 0.08 0.06 0
0.42 0.42 0.38 0.34 0.27 0.20 0.12 0.04 0.01 0.008 0
0
a L
e,
z
c
2
y’o
Q L
e,
?
0
performed our own comparison of mean-field with the simulations, and added the relevant numbers to Table I. A useful quantity for the purpose of comparison is the entropy change ASINk at the isotropic uniaxial transition. In the uniaxial limit ( r = 0), Luckhurst and Simpson4’ find from a refined Lebwohl-Lasher simulation that a / N k = 0.098, whereas Maier and Saupe found 0.42 in mean-field theory, more than a factor of four error. By comparison, for biaxial nematogens the simulation measures an entropy jump of 0.08 f 0.01, whereas mean-field theory predicts S / N k = 0.27, a slightly smaller error. However, Table I shows that, for the biaxial nematogens, mean-field predictions for the transition temperature and change in order parameter are in better agreement with the simulation. The results of our calculation confirm that mean-field theory does not provide an adequate description of the phase transition even in (slightly) more realistic models which include the correct degree of biaxiality found in actual nematogens. Certainly, fluctuations play a dominant role in the phase transitions found in Hamiltonian of the form of eq 4, and not just in the purely uniaxial limit.
5. Nematogens in an External Magnetic Field In weak external magnetic fields, uniaxial systems undergo a first-order transition from the (relatively disordered) “paranematic” phase to the (more ordered) nematic phase as the temperature is lowered. Above a certain critical magneic field strength, this line of first-order transitions terminates in a field-induced critical point, above which there is no phase transition. To our knowledge this field-induced critical point has not been observed experimentally, although attempts have been made.48 The uniaxial Maier-Saupe theory has been generalized to include the effect of an external magnetic field H by Wojtowicz and Sheng.13 Mathematically, this modification consists of adding a term to the Hamiltonian - Y C P ~ ( C OPSi )
0.25
0
(16)
I
where the external field is along the director axis and y is proportional to the molecular diamagnetic anisotropy13 and to the external field squared. The mean-field estimate for the critical field strength is, in our units, yc = 0.0626 which is reportedi2J3 to imply an external field considerably greater than that produced by dc magnets. The effect of an external field on the Lebwohl-Lasher model has been investigated in Monte Carlo simulations by Luckhurst and Simpson.’* They find at the extremely high field y = 0.2 that mean-field theory is in good agreement with the computer simulations. At such very high field strength, more than three times the mean-field estimate for the critical field strength, this good agreement is expected, since high fields damp any possible fluctuation effects. Unfortunately, from our point of view, there are no simulations in the interesting range of external field strengths.
0.00
I
1.3
4
Temperature T Figure 3. The order parameter A. as a function of temperature for a system with biaxiality r = 0.2, for various external fields y . From left to right, the curves correspond to field strengths y equal to 0, 0.012, 0.018, 0.025, 0.030, and 0.048.
Here we analyze the effect of an external magnetic field on biaxial nematogens. In this first approximation we include only the dominant uniaxial contribution (eq 16) to the field-molecule interaction. However, our first comment concerns the reliability of the mean-field estimate for the critical external field yc for purely uniaxial systems. Despite the high-field simulation result, we believe that mean-field theory significantly overestimates yc, possibility by as much as a factor of two, just as mean-field theory overestimates the change of order parameter and the entropy discontinuity in zero field. Given the continuing advances in “warm bore” superconducting magnets, and our results below, it is important to establish a reliable estimate for yc. We now turn to biaxial molecules in an external field. Note that the extra piece of the Hamiltonian (eq 16) couples directly to one order parameter (Ao,by definition) but not the other. For perfectly biaxial systems ( r = 6-’/*), this external field simply depresses the transition temperature (from T+= 1.2 at zero field to T* = 1.04 at the high field y = 0.04) without qualitatively affecting the transition. This behavior is also seen at the low temperature, uniaxial to biaxial transition in systems of intermediate biaxiality. The most interesting behavior is seen in the isotropic (paranematic)-uniaxial (nematic) transition. As the degree of biaxiality is increased, the critical field strength drops rapidly. For example, the critical field strength for nematogens with degree of biaxiality r = 0.2 is yc = 0.025, just 40% of the mean-field estimate for perfectly uniaxial systems. The order parameter A , for biaxial molecules ( r = 0.2) is shown as a function of temperature for a number of external fields in Figure 3. The results of our calculations indicate that the field-induced critical point is most likely to be seen experimentally in systems which possess not only a large, positive molecular diamagnetic anisotropy but also biaxiality. We emphasize that the reliability of mean-field estimates for the critical field strength has not been established, even for purely uniaxial systems. Acknowledgment. We acknowledge the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research, and a grant from the (US.) National Science Foundation under the Presidential Young Investigator Program. We thank the referees for helpful comments on an earlier version of this manuscript.
