Langmuir 1989,5, 1358-1363
1358
tion into larger aggregates that might not exist in the solutions used for the spectroscopic measurements.6C Evaporation at a higher temperature than those used in the precipitation of the dispersion might also lead to growth of the CdS particles. To investigate these possibilities, samples were prepared on quartz slides in the same way as those for microscopic examination, as described in the Experimental Section, a t different temperatures, and the absorption spectra were examined. The absorption spectra (Figure 2) still showed essentially the same blue shift of the absorption edge as the solution samples prepared a t lower temperature. This result indicates that the colloids on the substrate are of comparable size to those in solution. The TEM images of CdS colloids prepared a t 25 and -42 OC are shown in Figure 3. For the higher temperature preparation, more aggregation of the tiny crystallites occurred; however, individual particles still can be clearly seen. With the lower temperature preparation, the aggregation was less pronounced, but the size of the individual particles was essentially the same. The contrast in these micrographs is poor and does not permitthe resolution of the fine structure of the size distribution. Figure 4a shows an STM image of CdS particles on HOPG prepared at -42 "C. The image, part of a larger area image (1000 A X 1000 A), shows many isolated particles of size ca. 15 A, as well as some clusters of larger
size. It also shows several isolated larger particles of size ca. 45 A. In the middle section of this image, there are some particles that appear to be undergoing stepwise displacement along the scan direction of the tip. Such motion could be induced by the tip motion and might be associated with the poor conductivity of the particles and their weak interaction with the substrate. Unfortunately, such instability often reduces the reproducibility of the STM image from scan to scan. Figure 4b is a magnification of part of Figure 4a, which clearly reveals that the large clusters are mainly composed of aggregates of particles of size ranging from 15 to 40 A. As suggested in Figure 4c, which shows an STM image of CdS particles prepared at room temperature, the extent of aggregation of the particles seems to be greater at higher temperature, although the size of individual particles still ranges from about 15 to 40 A. It is still not clear whether the observed clustering is induced by the sample preparation technique, as mentioned previously, or is an intrinsic property of CdS particles in MeCN solution.
Acknowledgment. The support of this research by the National Science Foundation (CHE8805865)and the Texas Advanced Research Program is gratefully acknowledged. We thank Dr. M. Schmerling for his help with the TEM. We also appreciate the assistance of Dr. J. Kwak with the computer interfacing to the STM. Registry No. CdS, 1306-23-6;MeCN, 75-05-8.
Orientational Ordering in a Monolayer of Hard Oblate Spheroids Jodi Wesemann,+ Lihong &in,$ and Paul Siders' Department of Chemistry, University of Minnesota-Duluth, Duluth, Minnesota 55812 Received December 16, 1988. In Final Form: June 12, 1989 The Onsager theory of orientational ordering is applied to a monolayer of hard oblate spheroids. The spheroids are free to rotate, but their centers are confined to a plane. Order parameters and configurational free energy are calculated, within the Onsager theory. It is found that orientational ordering occurs in two steps, as density is increased from zero. First the spheroids' symmetry axes gradually tip into the plane. Then, in a second-order transition to a nematic phase, the spheroids' major axes align. The second transition shifts to lower density as spheroid eccentricity increases.
Introduction We present a study of the orientational behavior of a monolayer of hard oblate spheroids. Oblate spheroids of various eccentricities represent shapes from thin disks to spheres. Spheroids are inherently orientable, so an assembly of spheroids may exhibit cooperative orientational ordering. We have studied spheroids with their centers confined to a plane. The system studied crudely represents the shape-induced orientational behavior that may be expected in a monolayer of disklike molecules Address after September 1989 Department of Chemistry, Indiana University, Bloomington, IN 47401. Present address: Department of Physics, University of Minne-
*
sota, Minneapolis, MN 55455.
(e.g., porphyrins at an air-water interface). Liquid-crystallinephases of disklike molecules have been observed experimentally in three-dimensional systems.'" Strongly first-order orientational transitions were observed.'~~The orientational properties of disklike molecules, and of chlorophylls in particular, dissolved in liquid crystals have been studied. Effects of chlorophyll (1) Chandrasekhar, S.; Sadashiva, B. K.;Suresh, K. A. Prarnana 1977,5,471-480. (2) Billiard, J.; Dubois, J. C.; Tinh, N. H.; Zann, A. N o w . J. Chirn. 1978. ..-, 2. -, 52.5-540. - - - - .-. (3) Destrade, C.; Mondon-Bernaud, M. C.; Tinh, N. H. Mol. Cryst. Liq. Cryst. 1979, 49, 169-174. (4) Chandrasekhar, S.; Sadashiva, B. K.; Suresh, K. A.; Madhusudana, N. V.; Kumar, s.; Shaahidhar, R.; Venkatash, G. J. Phys. (Les Ulis, Fr.) 1979,40(C3), 120-124. (5) Levelut, A. M. J. Phys. (Les Ulis, Fr.) 1979,40, L81-L84.
0143-7463/89/2405-1358$01.50/0 0 1989 American Chemical Society
Monolayers of Hard Oblate Spheroids
Langmuir, Vol. 5, No. 6, 1989 1359
orientation on polarized absorptions8 and energy-transfer fluorescence quenchingg*10have been observed. The energy-transfer rate has been reported to be sensitive to orientational ordering of chlorophyll disks, in a monolayer solution of chlorophyll in hexadecane." These experimental systems are more complex than the model problem we have studied. Nonetheless, the present study is sufficient to predict qualitatively the nature and extent of orientational ordering due simply to the geometry of disks in a monolayer. There are several recent theoretical studies of the phase diagrams of hard spheroids. Frenkel et al. found orientational phase transitions in three-dimensional assemblies of hard They reported an isotropic to nematic transition, when the ratio of major to minor axes exceeds 512. They found the transition weakly first order, with strong precursor fluctuations. Their Monte Carlo study of the limiting case of hard disks in three dimensions also clearly shows a discontinuous isotropicnematic transition.16.17 Other studies of orientable model particles in three dimensions also show weakly discontinuous orientational phase transition^.'^'^ The phase behavior of strictly two-dimensional systems is less clear. Vieillard-Baron's early Monte Carlo cal~ulations'~ on hard ellipses in a plane show an orderdisorder transition, apparently of first order. Frenkel and Eppenga studied thin hard rods (of finite length) in two dimension^.^^ They found an apparently continuous transition from isotropic to uniaxial order. They also repeated some of the hard-ellipse calculations and concluded that system's transition is also continuous but with a strong peak in compressibility located at slightly lower density than the true phase transition. The Percus-Yevick equation also yields a continuous growth of orientational order for hard ellipses in two dimensions.26 The monolayer system we have studied is two dimensional with respect to translations, but rotations are fully three dimensional. The three- and two-dimensional results discussed above suggest that an orientational phase transition will occur, at least for sufficiently eccentric spheroids. A similar system, hard spherocylinders with midpoints in a plane, was studied by Coldwell et al.27 They
z
(6) Journeaux, R.; Viovy, R. Photochem. Photobiol. 1978,28,243248. (7) Frackowiak, D.; Bauman, D.; Stillman, M. J. Biochim. Biophys. Acta 1982,681,273-285. (8) Frackowiak, D.;Hatchandani, S.; Leblanc, R. M. Photobiochem. Photobiophys. 1983,6,339-350. (9) Bauman. D.:Wrobel. D. BioDhvs. Chem. 1980.12.83-91. (10) Frackowiak, D.;SzurkowsG, i.;Hotchandani, S.';LeBlanc, R. M. Mol. Cryst. Liq. Cryst. 1984,111,359-372. (11) Gonen, 0.; Levanon, H.; Patterson, L. K. Isr. J. Chem. 1981, 21,271-276. (12) Frenkel, D.Molec. Phys. 1987,60,1-20, (13) Frenkel. D.: Mulder. B. M.: McTarme. J. P. Phvs. Reo. Lett. 1984,52,287-296. . (14) Frenkel, D.;Mulder, B. M.; McTague, J. P. Mol. Cryst. Liq. Cryst. 1985,123,119-128. (15) Frenkel, D.;Mulder, B. M. Molec. Phys. 1985,55,1171-1192. (16) Frenkel, D.;Eppenga, R. Phys. Reu. Lett. 1982,49,1089-1092. (17) Eppenga, R.; Frenkel, D. Molec. Phys. 1984,52,1303-1334. (18) Boublik, T.; Nezbeda, I.; Trnka, 0. Czech. J. Phys. 1976,26, 1081-1087. (19) Gelbart, W.M.;Barboy, B. Acc. Chem. Res. 1980,13,290-296. (20) Decoster, D.;Constant, E.; Constant, M. Mol. Cryst. Liq. Crvst. 1983.97.263-276. "(21) Pe;ra&J. W.;Wertheim, M. S.; Lebowitz, J. L.; Williams, G. 0. Chem. Phys. Lett. 1984,105,277-280. (22) Allen, M. P.; Frenkel, D. Phys. Reu. Lett. 1987,58,1748-1750. (23) Frenkel, D.J.Phys. Chem. 1988,92,3280-3284. (24) Vieillard-Baron, J. J. Chem. Phys. 1972,56,4729-4744. (25) Frenkel, D.; Eppenga, R. Phys. Reu. 1985,31,1776-1787. (26) Ward, D.A.; Lado, F. Molec. Phys. 1988,63,623-638. (27) Coldwell, R. L.; Henry, T. P.; Woo, C.-W. Phys. Reo. A 1974, 10,897-902. I
.
h Y
L
a
'
X' Figure 1. Oblate spheroid model particle. Euler angles, axes, and shape parameters are shown. approximately located an isotropic-nematic transition, for spherocylinders with length 5 times width. The transition appeared to be discontinuous, with a 10% density change! The approximate symmetry between disks and rods observed in three dimensions12suggests that a monolayer of disks may also show a discontinuous orientational transition. On the other hand, rods in a monola. 9r respond to pressure by aligning their symmetry axes, ',,de disks need not, so it may be that in two dimensions disks will not share the phase behavior of rods. The rotational freedom in the present model is to simulate the rotational motions of large oblate molecules adsorbed on surfaces. This model treats the interaction of adsorbed molecules with the surface exceedingly simply, as an orientation-independent &function potential. The out-ofplane reorientation of real molecules involves significant potential energy changes, which may modify adsorbate orientational ordering considerably. We have studied the monolayer of hard spheroids within the Onsager theory of orientational ordering.28 This wellknown theory is based on second-order virial expansion of the configurational free energy. Cotter2' has given a particularly clear exposition and critique of Onsager's method. The nonlinear integral equation that results from Onsager's theory has been analyzed as a bifurcation problem, by Kayser and Ra~eche.~'We make use of their bifurcation analysis in understanding our results. Because the Onsager method is based on a low-order expansion, it cannot be quantitatively accurate at high densities. That is not of great concern in our study, as we are looking for qualitative understanding of orientational ordering. Zwanzig31 extended the Onsager theory to include virial coefficients through B,, for a three-dimensional gas of long thin rods. He found an orientational transition at all orders, qualitatively, as predicted at the B2 level. Lasher32 compared scaled-particle and Onsager theories for hard rods in three dimensions. Both theories predicted a first-order isotropic-nematic transition, with scaled-particle theory yielding the smaller discontinuity. (28) Onsager, L. Ar ,. N. Y. Acad. Sci. 1949,51,627-659. (29) Cotter, M. In 2 he Molecular Physics of Liquid Crystals;Luckhurst, G. R., Gray, G. W., Eds.; Academic Press: New York, 1979;Chapter 7,pp 169-180. (30) Kayser, R. F.; Raveche, H. J. Phys. Reu. A 1978,17, 2067On""
LU I L.
(31) Zwanzig, R.J. Chem. Phys. 1963,39,1714-1721. (32) Lasher, G. J. Chen. Phys. 1970,53,4141-4146.
1360 Langmuir, Vol. 5, No. 6, 1989
Wesemann et al.
Method Cotter has clearly laid out the Onsager theory for hard particles.29 We give the most important equations here, both to be definite about our methods and because the monolayer problem differs slightly from the more familiar, higher symmetry, three-dimensional problem. The geometry of the hard particles we have considered is oblate spheroidal. An oblate spheroid can be generated by rotating an ellipse about its minor axis. We denote the semiminor axis b and the semimajor axis a. The centers of the spheroids lie in the (x,y) plane of the laboratory coordinate system. Orientations of an oblate spheroid are described by Euler angles: a, P, and y. We have used Arflen's convention3' for the definition of these angles. The third Euler angle, y, is not used to describe particle orientations, because oblate spheroids are symmetric with respect to rotation about their principal axes. The angles, axes, and parameters are depicted in Figure 1. The pressure of this system of oblate spheroids can be expanded in a virial series. Truncation at B,, the second virial coefficient, yields
P/ (pkr ) = 1 + For hard particles on a plane, B, is given by &J
(1)
B, = (1/2)SSf(3) f(Q') A(Q,0')dQ dQ' (2) The differential solid angle dQ = sin pdpda, with 0 Ia < 2a and 0 5 C a. In eq 2, f ( 3 ) is the single-particle orientational distribution function, normalized to unity: 1 = Sf(Q)d 0 (3) The function A(Q,3')is the area in the (x,y)plane excluded to the center of a spheroid having orientation Q' by a second spheroid having orientation 3. Excluded area A is calculated numerically, as follows: Locate a spheroid, with orientation Q', at the origin. Then the location of a second spheroid, with orientation 3, is given in plane polar coordinates (r,8). Let R(8) be the least r for which the two spheroids do not overlap, with 8 fixed. Then, A = ' / J R 2 ( 8 ) d8, with the integration from 0 to 2a. We use Vieillard-Baron's overlap criterionz4to calculate R(8)and then calculate A by numerical integration. Both Gauss-Legendre and rectangular quadrature work well for the integration. The excluded areas used for the results reported here were obtained with 256-point rectangular quadrature. The configurational free energy per particle, A,, also truncated at the B, level, is given in eq 4. The usual approach (and our approach) to calculating f is to solve for the f ( 3 ) that minimizes A J k T . That is, we solve eq 5. The parameter u in eq 5 is the Lagrange multiplier for the constraint eq 3.
+ J f ( Q )In [ 4 ~ f ( Q )dQ ] + pB2 0 = In [4af(Q)]+ 1 - u + pJf(Q') A(3,O') dQ'
A J k T = In
(p) - 1
(4)
(5)
We solve eq 5 iteratively, with the convergence criterion JIlf, +l - f,lz d 3 C lo-''. The largest eigenvalue of the Frechet derivative of the iteration operator F f ,
-
(33) Arfken, G . Mathematical Methods for Physicists, Academic Press: New York, 1970; pp 178-180. A spheroid is oriented as follows: Define a local ( r y , z )coordinate system on the spheroid, with the z axis being the spheroid's principal axis. Initially, the local coordinate system is aligned with the laboratory system. Rotate the local system through an angle CY about the laboratory z axis. Then rotate through an angle ¶, about the (newly rotated) local y axis. Finally, rotate through y about the local z axis. All rotations are in the right-hand sense.
f,,, is estimated n~merically'~ during the iteration process. The corresponding eigenfunction, also estimated numerically, is used to speed convergence. The function f ( 3 ) is represented as a square N X N matrix. The results reported in this paper are for N = 16. Dependence on N , discussed in the Results, appears not to be a problem. Integrals in eq 2-5 are evaluated by rectangular quadrature, using the N X N points at which f ( 3 ) is calculated. We find that at sufficiently high density the thermodynamically stable solution of eq 5 is ordered in both a and p. We refer to the phase represented by that solution as "nematic". In the nematic phase, we use f ( 3 ) to locate a nematic director and to calculate three orientational order parameters: llfz0ll, Ilfzzll, and S. We use the Q-matrix to locate the director
QE
[
(Ux2)
(UPy)
(UXU,)
I
(u:) (uyu,) -(1/3)1 (6) ( U P , ) (u,uy) ( U 2 ) In eq 6, I is the identity matrix and ii = (u,,u,,u,) is the unit vector along a spheroid's principal axis. Notice that Q is closely related to the covariance matrix for the components of 6. Because of the symmetry of our model system, when a director exists it lies in the (x,y) plane. For this reason, we need diagonalize only the (x,y) block of Q. The usual nematic order parameter S is 312 times the largest eigenvalue of Q, and the nematic director is the associated eigenvector. We calculate also two orientational order parameters defined by Zann~ni,'~ given in eq 7. In eq 7 a,,, is the value of a along which the nematic director lies. In this paper, we report only norms of fzz and fzo. The parameter llfzoll reflects only ordering in and is zero in the absence of (3 ordering. The parameter llfzzII reflects both P and a ordering and is zero unless both a and /3 are ordered (uYu,)
where d 2 = (312) cos2 @ - '1, and d 2 = (3/8)l/' sin2 p.
Results The monolayer of oblate spheroids shows a transition to a nematic phase for axis ratio a l b > 6.25. The orientational phase transition is clear in Figure 2, where the nematic order parameter is graphed uersus pressure, for a / b = 10 and 20. The order parameter increases sharply at the transition point. The order parameter increases rapidly, but not discontinuously, above the transition point. The high-pressure, nematic solution grows smoothly from the low-pressure solution. The low-pressure, less ordered orientational distribution function can still be calculated for some pressures above the transition pressure. The configurational chemical potential of the low-pressure solution is greater than that of the nematic solution, over the pressure range for which both can be calculated. The chemical potentials of the two branches are tangent at the bifurcation point, so the transition is continuous. There is no evidence of a van der Waals' loop in the pressure, further supporting the continuous nature of the transition. (34) Jones, G. L.; Lee, E. K.; Kozak, J. J. J. Chem. Phys. 1983, 79, 459-468. (35) Vieillard-Baron, J. Molec. Phys. 1974, 28, 809-818. (36) Zannoni, C. The Molecular Physics of Liquid Crystals; Luckhunt, G. R., Gray, G. W., Eds.; Academic Press: New York, 1979; Chapter 3, pp 51-83.
Langmuir, Vol. 5, No. 6, 1989 1361
Monolayers of Hard Oblate Spheroids
1
a/b=lO
i
P/(P,kT)
Figure 2. Nematic order parameter S versus pressure for a / b = 10 and 20. Points representing the thermodynamically stable phase are connected by lines. Unconnected points represent the unstable continuation of the low-pressure phase. The close-packing density is po. 0.4
'r'
'
i i 0.31
0.1
'
i
'
'
'
1
i
11 0.2
'
Figure 4. Orientationalorder parameters for a / b = 10. Points representing the thermodynamically stable phase are connected by lines. Unconnected points represent the unstable continuation of the low-pressure phase.
'.'b=lo
0.6
0.4
I 0.8
Figure 5. Orientational distribution function f(a,cosj3) in the low-pressure phase, at p / p o = 0.30, for a / b = 10.
PIPo
Figure 3. Norm of the bifurcating nematic solution. It is difficult to distinguish tangency of the chemical potentials from a slight slope discontinuity, so we havecalculated the norm of the difference between the highand low-pressure distribution functions, near the bifurcation point. Figure 3 shows llhll versus PIP,,, near the transition points for a / b = 10 and 20. The function llhll is defined to be JJlf,., ,(Q) - fio,(Q)12 dQ. For p > p*, is the thermodynamically unstable solution and is less ordered orientationally than the nematic solution fnem. According to Kayser and Raveche's analysis3' of bifurcation in Onsager's theory, a first-order transition ("hard" bifurcation) occurs if dllhll/dp < 0 at the bifurcation point, while dllhll/dp > 0 signals a continuous transition ("softn bifurcation). For the present monolayer of oblate spheroids, it is clear that dllhll/dp is positive, and the transition is continuous. It appears that dllhll/dp increases as a / b increases. The behavior of the limiting hard-disk geometry may be quite interesting. The orientational ordering in this system is qualitatively unlike the ordering of spheroids in three dimensions. At zero density, the system is isotropic. At all non-zero densities, however, the orientational distribution function depends on /3: the system is orientationally ordered. As pressure is increased from zero, first @ orders. The spheroids tip up, with their symmetry axes tending to lie in the plane. This ordering continues smoothly as the pressure increases. The order parameter llfmll, graphed
Figure 6. Orientational distribution function f(a,cos8) in the nematic phase, at p / p o = 0.60, for a / b = 10. in Figure 4, shows this ordering. The transition to a nematic phase reflects ordering in a. The already tilted disks swing into alignment. The ordering in a is shown by l fz211, graphed in Figure 4. The a ordering is rapid but continuous. The maximum possible values of l f2011 and Ilfz2ll are 0.199 and 0.244, respectively. Figures 5 and 6 show graphs of the orientational distribution function f(a,cos P), for a / b = 10, in the low-pressure and nematic phases. Isotropic orientation distribution would correspond to f(a,cos P) = 1 / ( 4 ~ ) . It is clear from eq 5 that ~ ( C Y , C O SP) must show P ordering at non-zero densities. The excluded area A can be written as a function of only three angles: A(Q,Q')= A(a'a,cos p',cos 0).The integral JJA(a'-a,cos @,c'os 0) f(a'a,cos p') sin p' dpl d(a'-a) in the right-hand side of eq 5
1362 Langmuir, Vol. 5, No. 6,1989 is a function of cos 0.I t follows that f ( Q ) must be a function of cos P if p > 0. Of course, this argument is based on a low-order virial expansion. The expansion is valid as p 0, so the prediction of low-density orientational ordering is real. Monte Carlo calculations qualitatively support this Onsager picture of the orientational ordering.37 At the nematic transition, P is already substantially ordered. If the ordering were complete, all the spheroids would have P = a/2-they would be tipped up. Then the present model of a monolayer of oblate spheroids would reduce to the planar system of hard ellipses. Because of that similarity, the nematic transition in this system may be much like the transition of hard ellipses. VieillardBaronz4concluded on the basis of Monte Carlo calculations that the hard ellipse transition is first order, unlike the continuous transition observed here. Later calculations16sz6 suggest, however, that the transition may in fact be continuous. The similarity between ellipses and &ordered spheroids is intriguing but need not, of course, imply identical phase behavior. Lebowitz et al.38 recently studied an analogous connection one dimension lower and found that hard ellipses on a line behave differently than hard rods, even in the limit of orientational order imposed by high pressure or eccentricity. The Onsager theory is based on second-order virial expansion and so must err quantitatively at high density. At low density, B,p = [ P / ( p k T )- 11. We have calculated that quantity from eq 2 and from a 128-particle NPT Monte Carlo ~ i m u l a t i o n .The ~ ~ results are in Figure 7. The Monte Carlo calculation is not based on a virial expansion, so the Monte Carlo results in Figure 7 represent virial coefficients to all orders, at least for densities at which a virial expansion would converge. The Onsager pressure lies below the Monte Carlo pressure, as expected. The transition to a nematic phase appears at p N po/3 in the Monte Carlo calculation and is preceded by nearly complete p ordering. This is qualitatively the same sequence of orientational ordering (although at lower densities) as predicted by the Onsager theory. The slope of the virial pressure changes at the nematic transition. This change of slope implies discontinuity in the isothermal compressibility, K,, and hence implies that the transition is "second order". The compressibility K , = ( l / p ) ( a p/aP)., so the relationship between K , and the slope in Figure 7 is
Wesemann et al. -
-
Fitting the slope to Figure 7 , near the transition density p*/po = 0.583, yields (pok7')KT = 0.19 below the transition and 0.56 in the nematic phase: a large increase in compressibility. The Monte Carlo results also show an increase in compressibility, though not nearly so dramatic. The dependence of transition density and pressure (p* and P)on eccentricity is illustrated in Figure 8. The orientational transition moves to unphysically large densities p* > po for a / b < 6.25. The large-eccentricitybehavior is intriguing. We have not made calculations with a l b > 20. The size of the matrix used to represent f(a,cos /3) does affect results. In fact, with some small matrices we observed an apparent van der Waals' loop in pressure versus density. All the results published here used a 16 (37) Siders, P. Unpublished work. (38) Lebowitz, J. L.; Percus, J. K.; Talbot, J. J. Stat. Phys. 1987, 49,1221-1234.
a-
'
I
I ' MontkCarlo
o
1
I
.
'
Onsager
a/b=lO
6-
i
I
1
0 , 0.0
2
1 0.2
8
,
,
I
0.4
,
0.6
I
,
1 .o
0.8
P/P,
Figure 7. Pressure-density behavior from the Onsager theory
and from 128-particle constant-NPT Monte Carlo simulation. 1 .G
Q"
\ Q
0.5
0
. 0.
1.
0
2.
1 3.
4.
1 5.
P* pokT
Figure 8. Dependence of the transition point on axis ratio a / b. Points on the graph are labeled with corresponding values of a / b. P* and p* are the pressure and density of the transi-
tion to nematic phase. The close-packing density is po.
Table I. Location of the Nematic Transition for a / b = 10 with Aa.cos 6)bmesented by an N X N Matrix ~
N
P*IPO
P/(POm
4 6 8 12 16 20
0.525 0.578 0.583 0.583 0.583 0.583
1.865 2.460 2.497
2.501 2.501
2.501
X 16 matrix. Table I shows the effect of matrix size on transition point, for a / b = 10. The model system studied here can represent a monolayer of disklike molecules, such as porphyrins at an airwater interface. The orientation of such molecules and their dipole moments have been studied experimentally by linear d i ~ h r o i s m . ~Optical , ~ ~ ~properties ~~ can be predicted by using the orientational distribution functions we have calculated. We may locate a transition dipole moment lit in each spheroid. In order to represent a dipole moment that lies in the molecular plane, we align the dipole moment with the semimajor axis of the spheroid. The absorbencies of light polarized parallel and perpendicular to the (x,y) plane are sensitive to the orientation
(39) Tweet, A. G.; Bellamy, W. D.; Gaines, G. L., Jr. J.Chem. Phys. 1964,41,2068-2077. (40) Cherry, R. J.; " I , K.; Chapman, D. Biochim. Biophys. Acta 1972,267,512-522.
Langmuir, Vol. 5, No. 6, 1989 1363
Monolayers of Hard Oblate Spheroids of the dipole moment. A measure of this sensitivity is the dichroic ratio4' defined
D = (m;)/(my2)
(9)
In eq 9, ma and my are the z and y components of the transition dipole moment riz. The angle brackets represent ensemble averages, which are accomplished by averaging over the orientational distribution f(a,cos0,~). The third Euler angle y is the angle by which riz is rotated about the spheroid's principal axis. In our hard-spheroid model, the interparticle interaction is independent of y, so our calculated distribution functions give no information about ordering in y. For lack of any detailed information, we have chosen a random in-plane rotation riz about the symmetry axis. That is, we set f(a,cos 0,y) = f(a,cos 0)/27r. We can then use our calculated distribution functions f(a,cos 0)to calculate the dichroic ratio. The necessary integrals are given in eq 10. Calculated dichroic ratios, for a / b = 10 and 20, are given in Figure 9.
(m:) = x ' ~ : f ( a , c o s0)(1- cos'
0:o
0.2
Oj4
016
0.8
1 .o
P/P,
Figure 9. Dichroic ratio calculated by using orientational distribution functions from the Onsager theory at a / b = 10 and 20. For this calculation, the transition dipole moment was assumed to lie in the symmetry plane of the spheroid and rotated randomly about the symmetry axis.
0)d(cos 0)d a
(m;) = J*$_:f(a,cos @)[I+cos2 P + sin20cos 2a]d(cos 0)d a (10)
Summary The orientational ordering of oblate spheroids in a monolayer occurs in two steps. First, as density is increased from zero, the spheroids' short axes gradually tip into the plane. Second, the spheroids' long axes align in a continuous transition to a nematic phase. Over the range studied, the transition density decreases as spheroid eccentricity increases. The behavior of disks, the limiting geometry, is not yet known. The Onsager theory may err on the order and strength of the nematic transition but is correct in predicting orientational order at low density. The monolayer of hard oblate spheroids has no isotropic phase.
The calculated orientational distribution functions can be applied to calculate properties of monolayers of disklike molecules. Orientation-sensitive properties may change dramatically at the nematic phase transition. The dichroic ratio of a monolayer of spheroids with embedded dipole momenta is calculated and shows a strong density dependence in the nematic phase.
Acknowledgment. It is a pleasure to acknowledge partial support of this research by a Northwest Area Foundation Grant of Research Corp. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. Additional support was provided by the Graduate School of the University of Minnesota. We are especially grateful for J. Wesemann's support as an ACS-PRF Summer Research Fellow during her tenure at the University of Minnesota in Duluth.
