J. Phys. Chem. 1988, 92, 1406-1419
1406
of very dominant transition moments associated with only a few of the transition-metal cluster states. The combination of a high transition moment coupled with a large cluster concentration would appear to afford an appealing means to overcome the loss mechanisms which must plague not only transition-metal but also
other metal and metalloid cluster quantal probes. We intend to use intense cluster beams sources in an attempt to develop quantum level probes of the ground states of the transition-metal trimers. Registry
No. Ni, 7440-02-0; Ni3, 63055-48-1.
FEATURE ARTICLE Effect of Etecttlc and Magnetic Flelds on Orlentatlonal Disorder In Liquid Crystals David A. Dunmur* Department of Chemistry, The University, Sheffield, S3 7HF. U.K.
and Peter Palffy-Muhorayt Department of Physics, University of British Columbia, Vancouver, British Columbia, Canada V6T 2A6 (Received: September 1 1 , 1987; In Final Form: December 15, 1987)
Liquid crystals are fluid states of matter in which molecular orientations are partially ordered. The extent of order and resulting broken symmetry determine the liquid crystal properties and phase type. Because the molecular order is partial, dynamic disorder is also a feature of liquid crystals and results in fluctuations, which are manifested in various experiments. Application of external electric and magnetic fields can influence the fluctuations and hence disorder in liquid crystals, and lead to measurable changes in physical properties, new phase types, changes in transition temperatures, and a variety of field-induced critical phenomena. In this article, the disorder in liquid crystals is considered on the basis of a scale separation into (i) local or microscopic disorder resulting from fluctuations in molecular orientationsand (ii) macroscopic disorder associated with fluctuations in the symmetry axes of the local order. Molecular statistical theories are used to describe the effects of external fields on the microscopic order, while the quenching of macroscopic fluctuations by external fields is best explained in terms of the continuum theory of torsional elasticity. Predictionsof these theories are compared with the results of experiments, and it is concluded that continuum theory provides the best interpretation of existing measurements of field-induced order. Conceptual problems concerning scale separation, and the marrying of theories of local and macroscopic disorder, are also discussed.
I. Introduction: Symmetry and Order in Liquid Crystals Liquid crystals are intermediate states of matter that are formed between the crystalline and isotropic liquid states by some compounds or mixtures.' In a crystal, molecular species are orientationally ordered and positionally ordered on a regular lattice. With increasing temperature this order is lost a t phase transitions to more symmetric phases, but the material may not become completely disordered at a single phase transition. Translational and orientational order may be lost progressively through a series of phase transitions linking the crystal state with the isotropic liquid state, and some of these intermediate states (mesophases) may be liquid crystals. Essential features of liquid crystalline mesophases are fluidity and anisotropy. This definition excludes the special case of the cubic smectic D phase,2 which is probably anisotropic on a microscopic scale. The fluidity implies that there must be translational disorder, while the anisotropy results from partial orientational order of constituent molecules. Smectic liquid crystals are phases in which there is some translational order present, and they give rise to distinct X-ray diffraction patterns. These mesophases are layered structure^,^ and in the ordered smectic phases (hexatic B, I, and F) there is positional order within layers. If the layers become correlated, then the phases are better described as crystalline, although they may still be highly disordered. In the nematic phase all trans'Present address: Department of Physics, Kent State University, Kent, O H 44242.
0022-365418812092-1406$01.50/0
lational order has been lost, apart from liquidlike short-range correlations, and the anisotropic properties result solely from the remaining molecular orientational order. The symmetry of translational and orientational order can be characterized by appropriate space groups and point groups, and in Figure 1 the symmetry types of the main liquid crystalline (as opposed to crystalline) mesophases are given. Most thermotropic low molecular weight nematic liquid crystals have uniaxial symmetry Dmh, but very recently single component biaxial (DZh)nematic phases (CmJnematics have so have been d i ~ c o v e r e d :no ~ ~ferroelectric ~ far been identified. In principle there could be biaxial nematic phases of lower orientational symmetry, e.&, Dnh, where n = 3 or 4, and it has been suggested6 that liquid crystals with point groups including C, and C, elements, which are excluded for crystalline lattices, are also possible. The limits of stability of liquid crystalline mesophases are determined by symmetry-breaking phase transitions. These may (1) For recent reviews covering materials and applications see Gray, G. W. Proc. R . SOC.London, A 1985, 402, 1, and Gray, G. W., Ed. Thermotropic Liquid Crystals; CRAC 22; Wiley: Chichester, 1987. (2) Etherington, G.; Leadbetter, A. J.; Wang, X. J.; Gray, G. W.; Tajbakhsh, A. Uq.Crpr. 1986, I, 209. (3) Gray, G. W.; Goodby, J. W. Smectic Liquid Crystals; Leonard Hill: Glasgow, 1984. (4) Malthete, J.; Tinh, N. H.; Levelut, A. M. J . Chem. SOC.,Chem. Commun. 1986, 1548. ( 5 ) Malthete, J.; Liebert, L.; Levelut, A. M.; Galerne, Y. C. R. Acod. Sci., Ser. 2 1986, 303, 1073. (6) Kats, E. I. Sou. Phys. Usp. 1984, 27, 42.
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 6, 1988 1407
Feature Article
Ordering tensors of odd rank are absent because of the assumption of a centre of symmetry for the phase. Anisotropy in the physical properties of liquid crystal phases results from the orientatiodal ordering of molecular axes. Since this article is about field effects, the important physical properties will be electric and magnetic susceptibilities K , ~ : both of these are second-rank tensor quantities, and they can be related to the orientational order parameters. It is convenient to define the traceless susceptibility tensor such that
DieCotic Cylindricd
K:B = Kcla - ( 1 / 3 ) K , , L B (3) In a principal axis system the susceptibility tensors are diagonal, so that for a uniaxial phase 0 1
Cholcsterrc D,
A
x
TO)
4
Uniaxial Smectic
-
A,
Doh
x T(2)
Figure 1. Liquid crystal phase symmetries. T(N) indicates translational
degrees of freedom.
be first order (e.g., nematic/isotropic or certain smectic/nematic) or second order such as some smectic C/smectic A transitions. Absolute thermodynamic stability is not always necessary for the observation of liquid crystal phases, and it is sometimes possible to obtain metastable liquid crystals a t temperatures below the normal crystal melting temperature. Such phases are obtained by supercooling from the liquid state and are known as monotropic phases (i.e., they form only on cooling); phases that form on heating and cooling are enantiotropic. Metastable liquid crystal behavior can be investigated by using mixtures. Materials that do not form liquid crystal mesophases in the pure state may become liquid crystalline in a mixture with a suitable solvent. Extrapolation of measurements on the liquid crystal mixtures to zero solvent concentration can often yield information on the liquid crystal tendencies of compounds. Liquid crystal transition temperatures obtained in this way are known as "virtual transitions"; i.e., they do not actually occur in the pure material. In molecular fluids order and symmetry are closely connected: the higher the symmetry, the lower the ordering of molecules, and for each symmetry type there will be a number of order parameters that can be used to describe the extent of orientational7 or translational order.* It is not the purpose of this article to explore the relationship between order and symmetry in liquid crystals, and fortunately for the simpler phases order parameters can be defined without recourse to group theory. In a uniaxial nematic phase (Dmh)the orientational distribution functionf(0) for molecular axes can be written as' f ( 0 ) = (1 / S d ) [ 1
+ 5SDan,na+ higher terms]
(1)
where n, are the direction cosines of the nematic symmetry axis with respect to a molecule-fixed axis system. Sa,is the lowest rank orientational ordering tensor defined by
(7) Zannoni, C.In NMR of Liquid Crystals; Emsley, J. W., Ed.; Reidel: Dordrecht, 1985; p 1. (8) Prost, J.; Barois, P. J . Chim. Phys. 1983, 80, 65.
The anisotropy in this case is AK = ( K , , - K ~ and ) it vanishes when the orientational order is zero: the subscripts 11 and I refer to directions parallel or perpendicular to a symmetry axis. In this paper we define the local director axis system as follows: we consider a volume V(r) at position r with linear dimensions equal to a few molecular lengths. The ensemble-averaged susceptibility tensor for this volume may be diagonalized, and the corresponding principal axes define the local director axis system. If the sample is locally uniaxial, the symmetry axis is the local director; samples lacking cylindrical symmetry are biaxial, and for these at least two unique directions need to be specified. In each volume element orientational disorder has been attributed to randomness of molecular orientations, but there is another source of disorder that influences macroscopic liquid crystal properties. Liquid crystals are fluid; their symmetry axes may not be fixed in space and may fluctuate in a laboratory fKed frame. These director fluctuations are observable through light scattering, and they introduce two further complicating features. One is that the measured value of any bulk property defined with respect to the laboratory frame will include contributions from director fluctuations? and secondly the local symmetry itself may be broken during a fluctuation.'O Thus a locally uniaxial liquid crystal may be subject to biaxial fluctuations, and particular experimental probes may be sensitive to these."J2 To take account of director fluctuations, it is convenient to consider averaging molecular orientations over two distance scales: the validity and significance of doing this will be discussed later. Thus the local traceless susceptibility tensor of the liquid crystal fluid defined with respect to the director axis system is related to the molecular susceptibility by
= (2/3)s$$34m,6"'
(K",)$Q,
(5)
where S$"'6 is a generalized ordering matrix defined in the Appendix. If the local director symmetry is uniaxial, then eq 5 can be simplified to (AK)$((,,= S$K$') (6) where S:t = ( 1/2(3n,n, - 6&)) is the ordering matrix of the local director axis with respect to a molecular frame. n, is the component of the unit-vector n along the molecular axes. We can sim lify eq 6, by choosing the molecular frame such that siK !$ diagonal. It is then natural to define two susceptibility anisotropies: KI
=
[Kz,
K2
-
YZ(KXX
+ KyJl
(7)
= [K, - K y y l
and eq 6 further reduces to ( AK)$((r)= K I V ; d
+ (1 /2)Kz(S;id
(8)
- S$d)
(9)
(9) Warner, M. Mol. Phys. 1984, 52, 677. (IO) Yu Val'kov, A.; Romanov, V. P. Sou. Phys. JETP (Engl. Trans/.) 1977, 56, 1028. (1 1) Pokrovsku, V. L.; Kats, E. I. Sou. Phys. JETP (Engl. Troml.) 1977, 46, 405. (12) Lacerda Santos, M. B.; Durand, G . J . Phys. (Les. Ulis, Fr.) 1986, 47, 529.
1408 The Journal of Physical Chemistry, Vol. 92, No. 6, 1988
f
tuations in the isotropic state of liquid crystalline materials, and the effect of external fields on this state is also described. In section 111 the various theoretical approaches and predictions will be discussed, and an attempt will be made to rationalize some of the apparent theoretical disagreements. The test of theoretical controversy must lie in experiment, and various experimental investigations of field effects on liquid crystalline order will be reviewed in section IV. Consequences of these results for the theories will be discussed in section V, and an attempt will be made to summarize the present situation.
Figure 2. Polar angles.
The anisotropy of the susceptibility in a laboratory frame will be obtained by averaging eq 6 over the sample to give
It has been assumed that the averaging over the two axis systems can be carried out independently, and the generalized order paand s$& describe the macroscopic order and local rameters Sb;ds,ra order, respectively. A number of special cases can now be identified, and these are given in detail in the Appendix. The most usual situation for field-free nematics is biaxial molecules which order locally and macroscopically to give a uniaxial phase, and in these circumstances eq 10 becomes
The simplest situation is uniaxial molecules ordered uniaxially on a local and macroscopic scale so that
All of the above ordering matrices can be expressed in terms of averages over appropriate rotation matrices, but usually only the diagonal components of the ordering matrices are relevant, and it is convenient to write S,, in diagonal form as sop=
0
-'/z(Q
-
Dunmur and Palffy-Muhoray
0 O
'1
t P) 0
(13)
Q
where Q is the uniaxial order parameter, Q = (1/2(3 cos2 0 - l)), and P is the biaxial order parameter P = ( 3 / 2 sin2 0 cos 24); 0 and 4 define the orientation of a uniaxial molecule as indicated in Figure 2. The two important aspects of liquid crystals identified above are symmetry and order, and fields can influence both of these. We will restrict attention to uniform applied electric or magnetic fields, although inhomogeneous fields can have interesting effects.13 Because of the intrinsic anisotropy of liquid crystals, static distortions in the director field can be induced by external fields; this is the basis of liquid crystal display devices.' Although obviously of great importance, this aspect of the interaction of liquid crystals with fields will not be considered here. Macroscopic realignment may couple with viscous flow, and many electrohydrodynamic effects have been identified in liquid crystals;I4 this area is also outside the scope of this article. A plan of the paper is as follows: in the next section we show how fields may be included in alternative static theoretical descriptions of nematic liquid crystals. The predicted effects of fields on liquid crystal properties will then be considered, including the influence of fields on mesophase transition temperatures, changes in order parameters, and the induction of new phases of different symmetry. Because of the weak first-order nature of the nematic/isotropic phase transition, there are strong nematic fluc(13) Marcerou, J. P.; Prost, J. Mol. Cryst. Liq. Cryst. 1980, 58, 259. (14) Blinov, L. M. Electro-Optical and Magneto-Optical Properties of Liquid Crystals; Wiley: Chichester, 1983: p 50.
11. Theoretical Background: Including External Fields A complete theory of liquid crystals would correctly predict the symmetry, transition temperatures, and properties of all the mesophases. For nematic phases we would like to predict the nematic/isotropic transition temperature and the order parameter as a function of temperature, pressure, and external field strength. Molecular theories including only attractive forces (Maier-Saupe) provide a reasonable estimate of the degree of arder, but using realistic potential parameters give very low estimates of the transition temperature. Repulsive interactions can be included through van der Waals theories, and these lead to higher transition temperatures, but also to higher order parameters not in agreement with experiment. These theories have been reviewed by Gelbart,15 who considers also the importance of additional aspects such as biaxiality, molecular flexibility, and multiple-site interactions. The phenomenological Landau theory of liquid crystals16 makes assumptions about the phase symmetry and nature of the order parameters. It is then possible to make qualitative predictions about physical properties and phase transitions, or the theory can be used to fit experimental results to obtain quantitative predictions of other properties. The macroscopic behavior of liquid crystals is most conveniently described by continuum mechanics, and the Frank-Oseen theory of torsional elasticity provides the basis for explaining many of the field realignment phenomena that are exploited in devices." This theory also provides a way of including director fluctuations into the description of physical phenomena.'* II.1 .I. Molecular Statistical and Landau Theories. In this section we wish to describe the influence of electric and magnetic fields on the local orientational order of a liquid crystal. Disorder due to fluctuations of the director and thermally excited changes in local symmetry are ignored, although it will be shown later that the equilibrium symmetry of liquid crystal phases can be altered by external fields. Two theoretical descriptions of field effects on order have been proposed; both are mean field theories, but we shall refer to them as the molecular statistical theory and Landau-de Gennes theory. The latter is an extension of Landau's theory of phase transitions to liquid crystals: it is essentially empirical, and the particular aspects of orientational order to be considered are included in the initial Landau expansion for the free energy. The molecular statistical theory models the molecular interactions and makes appropriate approximations to allow simple solutions to be obtained. The mean field theory of liquid crystals proposed by Maier and Saupe replaces the interactions of one molecule with its neighbors by a pseudopotential, which is a measure of the single molecule energy in an averaged local environment. Depending on how the mean field theory is applied, it can in principle (see later) include both short-range and longrange interactions. However, the usual formulation of molecular statistical theory does not include director axis fluctuations. Our interest is in the effect of external fields on liquid crystal order, and we will consider this within both the molecular statistical and Landau-de Gennes theories. In general the molecules constituting liquid crystals lack cylindrical symmetry. Effective single-particle potentials taking this into account have been constructed based on either or attractive dispersion force^.^^-^' The effect of external fields in these models has not (15) Gelbart, W. M. J . Phys. Chem. 1982, 86, 4298. (16) For a recent review see Gramsbergen, E. F.; Longa, L.; de Jeu, W. H. Phys. Rep. 1986, 135, 196. (17) See, e.g., ref 14. (18) de Gennes, P. G. Mol. Cryst. Liq. Cryst. 1969, 7 , 325.
The Journal of Physical Chemistry, Vol. 92, No. 6, 1988 1409
Feature Article yet been fully investigated. In this paper, therefore, we will concentrate on cylindrically symmetric particles. II.1 .la. Molecular Statistical Mean Field Theory. ( i ) The Nematicllsotropic Equilibrium. For a fluid of cylindrically symmetric rigid molecules, the rotationally invariant singleparticle pseudopotential (e,) for a nematic can be written asI9 t,
=
to,
- 2/3puS$(a,p
- j/2S$) - Y~AK(~)F,FBu~, (14)
where
S$$ = (Y2Waip - Sag)) =
(gag)
F is the external field, p is the number density, u is an energy parameter, and tonis the isotropic liquid potential. Equation 14 can be written in terms of the biaxial and uniaxial order parameters defined in eq 13 and becomes t,
=
to,
C
+
- 1) - XpuP sin2 0 cos 24 Y2pu(Q2 )/3Pz) - KpAdm)F,Fg~g, (1 5)
- Y2puQ(3 cos2 8
+
Within the mean field theory, self-consistent equations can be written for the order parameters Q and P in terms of an average over the pseudopotential given in eq 15: Q = '172(3 Z
1
P = -]y2 Z
COS2
8 - 1) eXp(-t/kgT) dQ
(16)
sin2 8 cos 24 exp(-t/kgT) dQ
(17)
and the equilibrium state in the presence of a field is that for which the self-consistent order parameters minimize 9. For small values of the order parameters, the free energy density can be expanded as a power series in Q and PI8 and to lowest order is quadratic in the order parameter and field strength: Qp') +
+ &'P'
9 4-
where Z = exp(-c/kgT) dSZ is the orientational part of the configurational partition function. In the absence of a field eq 16 and 17 have three degenerate solutions corresponding to P = 0, Q = QMS, P = *3Q, and Q = -lI2QMs. The solution QMS is the Maier-Saupe order parameter, which is obtained if the field is zero and the director is chosen to be along the z axis. The other two degenerate solutions correspond to the director along the y and x axes. In the presence of a field, the degeneracy of the solutions may be lifted, and the possibility of field induced biaxiality emerges. The free energy density is obtained from 9 = -pkgT In ( Z / ~ T ) (18)
f(Q4
4-
+ $) + p?;
[
l/z
(Q - P) if F = F, ( Q + P) if F = F y ifF=F,
1
t
terms in F Q ' , PP', ... (19)
where a = p(5kgT- U), b = U2p/7k~T,and C = U3p/35(k~T)'. In this representation of the effect of an external field, three separate terms are necessary to describe the field along different axes. Only negative field-energy terms in eq 19 have any physical significance, since if the free energy is increased by application of a field, the liquid crystal will undergo a macroscopic deformation (e.g., reorientation of the director) to a lower free energy state. Such a deformation is not taken into account in the molecular statistical theory, which assumes a uniform director distribution. Thus an equilibrium field deformation is only possible with F along x or y axes if Adm) is negative, since P = 0 for F = 0. It has recently been shownB that the interaction of a field with a material of negative susceptibility anisotropy can be conveniently represented as the interaction of a material of positive susceptibility anisotropy with two fields along perpendicular axes. (19) Palffy-Muhoray, P.; Dunmur, D. A. Mol. Cryst. Liq. Cryst. 1983, 97, 337. (20) Frisken, B. J.; Bergersen, B.; Palffy-Muhoray, P. Mol. Cryst. Liq. Cryst., in press.
/ , b
4-
I
,
0
n
08
09
C
/I,, 10
11
2
T, Figure 3. Molecular statistical calculation of Q, P and lo' F / p u for different values of field applied perpendicular to the director of a material of negative susceptibility anisotropy. Results for a positive material with F along the director are similar to those given for Q but with the induced biaxiality P equal to zero. Adm)P/3u = 0.0 (a); -0.015 (b); -0.05 (c).
The first calculations of the order induced in materials of positive susceptibility anisotropy by fields using mean field theory were presented by Hams2' and Wojtowicz and Sheng.22 We have also carried out similar calculations including biaxial ~ r d e r i n g , ~ ~ , ~ ~ and results for order parameters Q and P and the free energy for different field strengths are given in Figure 3. These calculations were performed numerically by using the self-consistent equations for the order parameters, eq 16 and 17. The points marked as Tc,TNIand TKon Figure 3 are as follows. T" is the second-order phase-transition temperature for the change isotropic or paranematic to nematic. (In the presence of a field the transition is between a state of low orientational order (paranematic) and a state of high orientational order (nematic). TNIis the first-order transition temperature between the states and always occurs at a higher temperature than T",while T K is the temperature for which dQ/dT = and marks the highest temperature for metastability of the nematic phase. In fact, T" and TK are spinodal points for which d29/aQ2 = 0. (ii) SmecticlNematic Equilibrium. We shall restrict attention to the SA phase, as field effects in other smectic phases have scarcely received any attention from a molecular point of view. For the smectic A/nematic transition, an approach similar to that given above for the nematic/isotropic transition has been reported. Neglecting the possibility of induced biaxiality, the effect of a field on smectic ordering can be modeledz4by adding a field term to the McMillan pseudopotential: €s = to, - YzpUQ(3 COS2 8 - 1) Y2pWR(3 cos2 0 - 1) cos qoz - '/,pAdm)F2(3cos2 8 - 1) (20) where w = 2u exp[-(~r,,/d)~] is the smectic ordering energy. In (21) Hanus, J. Phys. Rev. 1969, 178,420. (22) Wojtowicz, P. J.; Sheng, P. Phys. Lett. A 1974, 48A, 235. (23) Waterworth, T. F. Thesis, University of Sheffield, 1985. (24) Hama, H. J . Phys. SOC.Jpn. 1985, 54, 2204.
Dunmur and Palffy-Muhoray
1410 The Journal of Physical Chemistry, Vol. 92, No. 6, 1988 r
1
-----0.5
,
\
"
\
"
\
0.3
0.1
1
I
I / I
I
I
I
I
1
I
I
this equation the director and field are assumed to be along the z axis (normal to the smectic layers), d = 2a/q0 is the layer spacing, and ro is a parameter of the theory identified as the length of the rigid part of the molecule. The quantity R = ( 1/2(3 cos2 6 - 1) cos 402) is an order parameter describing the coupling of translational and orientational order and is characteristic of the smectic phase. Numerical solution of the self-consistent equations for Q and R (eq 21), together with an evaluation of the free energy,
R = $$y2(3
cos2 0 - 1) cos qoz exp(-t/kBT) dz dO
(21)
gives the effect of an external field on smectic ordering. Results for the order parameters Q and R taken from the paper by Hama24 are given in Figure 4. The effect of a field on a material of positive susceptibility anisotropy is to increase both Q and R , and for a particular critical field the first-order transition smectic A/nematic becomes second order. II.1. l b . Landau-de Gennes Theory. The Landau-de Gennes approach to the description of field effects in liquid crystals has received much a t t e n t i ~ n . ' ~ , ~ It ~ -starts ~ ' from an empirical expansion of the free energy density as a Taylor series in powers of the order parameters: field effects are taken into account by adding a suitable field-energy term. A quadratic dependence of the free energy on order parameter leads to a second-order nematic/isotropic transition; however, inclusion of a cubic term ensures that the Landau expansion correctly models the first-order character of the transition. Biaxial ordering can be easily included in the expansion, and smectic order and coupling between different types of smectic/nematic order is readily modeleds by adding terms of suitable symmetry to the free energy expansion. The difficulty with the Landau-de Gennes theory is that it is empirical and can only model those features of liquid crystal order that are included at the outset. Furthermore, the interpretation of the coefficients in the free energy expansion in terms of physical parameters is difficult. Some help can be provided by comparing molecular statistical expressions with the Landau-de Gennes free energy, but this also exposes some inconsistencies in the latter. The Landau-de Gennes free energy expansion is formally the same as the low-field expansion, eq 19, obtained from molecular statistical theory, except that the higher order cross terms between the field and the order parameters are absent. Also, the coefficients a, b, and c are empirical parameters. To find the equilibrium state of a system, the free energy is minimized with respect to the order parameters Q and P, and inclusion of the field term allows field-dependent order parameters to be calculated. In order to carry out the minimization, certain assumptions have to be made about the dependence of the ex( 2 5 ) Fan, C. P.; Stephen, M. J. Phys. Rev. Left. 1970, 25, 500. (26) Priest, R. G. Phys. Lett. A 1975, 47A, 474. (27) Keyes. P. H. Phys. Lett. A 1978, 67.4, 132.
pansion coefficients on external variables. It is usual to assume that b and c are independent of temperature, pressure, and external field strength, while a is given a Landau-type functionality of a = a. ( T - T*), where T* is the second-order transition temperature. Comparison of the Landau expansion with expansions derived from the molecular statistical theory show that these assumptions are not strictly valid. C a l c ~ l a t i o n sof~ the ~ Landau free energy and order parameters Q and P give curves similar to those obtained from the molecular statistical theory (Figure 3). Although the shapes of the curves are similar, the Landau-de Gennes theory has four material parameters (including the susceptibility anisotropy), while molecular statistical theory has only two independent parameters. Thus the Landau expansion can be readily fitted to experimental results but does not contain any fundamental description of the molecular interactions that lead to field-induced order. The Landau-de Gennes expansion has been applied to the description of field-induced effects in SA phase^.^^,^* Interaction between the field and the translational order in smectic phases only occurs through the coupling terms linking translational and orientational order. The effects of external fields on the order and phase behavior in smectic liquid crystals have yet to be fully explored. II.l.2. Continuum Theory. In this section we concentrate on the disorder that results from orientational fluctuations in the local director axis system. Generally the local microscopic order will be biaxial; therefore, the free energy associated with a distortion will depend on torsional elastic constants for a biaxial medium. As the theory for this has not k e n fully worked out, we will restrict attention to microscopically uniaxial systems. As before, the ordering matrix Si$describes the macroscopic order of the director axes with respect to a laboratory fixed axis system:
In the diagonal representation (eq 21), this becomes
z is assumed to be the average direction for the director. If the director is not uniformly aligned in the sample (as implied by eq 22), then there will be a torsional elastic contribution to the free energy density, which in the presence of an external field is
3 = - 1$ 2v
[kl,(div n)2 + kz2(n.curl n)2 + k3,(n X curl n)2 -
y3) - i i p p ] d3r ( 2 5 )
A ~ ~ p F ? ( ( n . f-) ~
where k l l , kZ2,and k33are the torsional elastic constants for splay, twist, and bend, respectively. p is the molecular number density, and f is a unit vector along the field direction: for simplicity we have written ( A K ~ ) ~=~A, ,K ~ . The spatial variation of the director n(r) can be expressed in terms of Fourier components:
V
n J r ) = -Jn,(q) (2*)3
~~
exp[-iq-r] d3q
~
(28) Rosenblatt, C. J . Phys., Lett. 1981, 42, L9.
(27)
The Journal of Physical Chemistry, Vol. 92, No. 6, 1988 1411
Feature Article For biaxial nematics we would have to consider five normal modes; however, for a uniaxial nematic, eq 28 can be expressed in terms of two normal modes ni(q) and nz(q):
3' + '/zA~dpF,Z = '/zC[X+n12(q) + X-nz2(q)] 4
~
0.9
(29) 0.8
where A+ and X- are the eigenvalues of the transformation matrix.I0' It is assumed that the z axis is along the average director. The two normal modes appearing in eq 29 can be represented in the absence of an external field as splay-bend and twist-bend deformations. The effect of a field on these modes depends on the orientation of the field with respect to the z axis and on the sign of the susceptibility anisotropy. For a material of positive susceptibility anisotropy, with the field along z, the splay-bend and twist-bend modes remain uncoupled but reduce in amplitude. In any other field configuration, the modes can no longer be represented as splay-bend and twist-bend, and the mean square amplitudes depend on all three elastic constants. For simplicity we will assume that k , , = k22 = k33 = k, and under these circumstances
1,d szz G. 7
0. 6
0. 5
0. 4
and a. o
I
1
I
I
AXF' The mean square amplitudes can be obtained by applying the equipartition principle to eq 29:
(n12(q))= pkBT[X+(q)l-'
(32)
I
I
I
1
I
+
Figure 5. A plot of Si: = 1 - const[kq2 AXPI-' against field strength to illustrate the effect of q and F on director fluctuations. The results have been scaled so that kq2 = 0.1 corresponds to complete director disorder. Curves are labeled a (q = 3.16 X IOs m-l), b ( 4 = 2 X lo5d), c (q = 1.41 X lo5 m-l), d (q = 1.22 X lo5 m-I), and e (q = l o 5 d).
The effect is strongly dependent on q or more particularly AKdp/kq2and vanishes for large q; i.e., long-wavelength fluctuations are the most important: this behavior is illustrated in Figure Two sample configurations are of particular interest and give compact results. (i) Field parallel to z axis and AKpositive: the order parameters eq 23 and 24 become Q l" d = 1 - -
'( -1' 2 2n
k T
1
4n2q2dq (34) kq2+ A ~ ~ p p
qm
4mn
5.
In these expressions long- and short-wavelength cutoffs are of crucial importance. We imagine that qminis fixed by the sample dimension D, so that qmin= 2 n / D . Since the sample is usually anisotropic, Le., a thin film, then anisotropic boundary conditions for qminhave to be considered, and it has been shownI0' that these can have significant quantitative effects on the calculated induced order. The short-wavelength cutoff qmaxis related to the distance scale that defines the local order. An alternative approach is to fix the cutoff on the basis of a fixed total number of fluctuation modes. A uniaxial molecule requires two angles to specify its orientation, while a biaxial molecule requires three. Thus the total number of modes should be 2N or 3N respectively, where N is the number of molecules. If these modes can be represented by then the short-wavelength cutoff a spherical q space of radius qmman, is given by29
(ii) Field perpendicular to z axis and AK negative nlP =
Thus the effect of the field is to increase both the uniaxial and biaxial order parameters. For a particular q the field-dependent part of the order parameter is proportional to [kq2+ AxdpP]-', and for small field strengths the induced order varies as the square of the field strength; at higher field strengths saturation occurs.
For a typical liquid crystal this gives 2n/qmax= 1 = 1 nm, compared with an average intermolecular distance of 0.6 nm. In our description of short- and long-range disorder it would be necessary to apportion modes as short range or long range, and it would be difficult to establish a proper physical criterion for doing this. On the other hand, fitting experimental results for field-induced order to the theory30provides a value for 1 in the range of 1-2 nm, which indicates that a volume encompassing 10-100 molecules is sufficient to define the local ordering matrix S$'. 11.2.1. Predictions of Field Effects Using Molecular Field and Landau Theories. Provided that similar order parameters are included in the molecular statistical and Landau-de Gennes (29) Faber, T. E. Proc. R . SOC.London, A 1977, 353, 247. (30) Dunmur,D. A,; Waterworth, T. F.; Palffy-Muhoray, P. Mol. Cryst. Liq. Cryst. 1985, 124, 73.
1412 The Journal of Physical Chemistry, Vol. 92, No. 6, 1988
Dunmur and Palffy-Muhoray rl
c c < -
006
-OM
-004
-003
0 02
0 01
om
001
I I2
+Ax F’
Figure 6. Reduced temperature/field phase diagram calculated from molecular statistical theory. Dotted lines represent T* and TK.
theories, the predicted field-dependent behavior is similar for both theories. Results of calculations of order parameters for uniaxial and biaxial nematics and smectic A phases have been illustrated in Figures 3 and 4. There are multiple solutions for the order parameters from both the self-consistent equations of molecular statistical theory and from the Landau-de Gennes power series expansion: the equilibrium solutions are those that minimize the corresponding free energies. Although we have only given results for cylindrical particles, it is worth pointing out that for biaxial particles the susceptibility anisotropy (eq 9) contains two material parameters. It follows, therefore, that for such systems the measured values of electric and magnetic susceptibilities,or electric susceptibilities at different frequencies, do not in general have the same temperature dependence. This discrepancy has been discussed by Bunning, Crellin, and Faber.98 It is found that for materials of positive susceptibility anisotropy, the effect of a field is to induce orientational order in both the nematic and isotropic (paranematic) phases. The transition temperature increases with field strength, and at a critical field the transition from nematic to paranematic becomes continuous; Le., there is a critical end point. For materials of negative susceptibility anisotropy, an external field induces biaxiality in the nematic phase, while the high-temperature phase is uniaxial. The phase-transition temperature increases with increasing field strength, and the transition changes from first order to second order at a field-induced tricritical point. This field-dependent behavior is illustrated in Figure 6. Calculated values of the field strength corresponding to critical end points and tricritical points depend on assumed values for material parameters. As an example,*O parameters equivalent to the nematogen 5CB (4-npentyl-4’-cyanobiphenyl) give critical field strengths equal to 7.8 X lo7 V m-l or 847 T, while for a material of equal but opposite (negative) susceptibility anisotropy, the tricritical field strength would be 1.62 X lo8 V m-l or 1765 T. These values were obtained from a molecular statistical calculation, but using Landau-de Gennes theory with experimentally adjusted parameters gives somewhat lower results for the critical field strengths: e.g., Hornreich31 has suggested a typical critical field strength of 0.92 x 106 V m-l. The behavior of nematics with two fields applied simultaneously has been investigated,20and the corresponding 3-D phase diagram is illustrated in Figure 7. The surface represents first-order transition points, and has spokes lying in the planes defined by -E,T, -H,T, and (E/E+H/H),T. There are also three vertical planes containing second-order transition lines. These are equivalent to the negative part of Figure 6 and correspond to transitions a t field strengths above the tricritical point between a biaxial nematic and a uniaxial paranematic. It has already been explained that fields will only influence smectic ordering through the coupled translationallorientational (31) Hornreich, R. M. Phys. Leu. A 1985, 109A. 232.
e
3
Figure 7. Phase diagram for simultaneous response to two perpendicular fields (h, e), each equal to A K ( ” P / ~ U .
biaxial
uniaxial
0
1.o
O
Y,
0
Figure 8. Phase diagram showing phase separation in the presence of a field in a mixture of nematogens of positive (component 1) and negative susceptibility anisotropy. The composition axis is volume fraction of component 1.
order parameters. Application of molecular mean field theory to the SA isotropic transition with an applied field predicts a field-induced nematic phase.32 Molecular ~ t a t i s t i c a land ~~ Landau-de Gennes2*calculations of the effect of an external field on the nematic/SA transition show a field-induced tricritical point on the nematic/SA phase boundary. Estimates of the tricritical field strength are given as 500 T or around 5 X lo7 V m-I, and it is clear that the effect of accessible fields on the transition is likely to be rather small. Significantly larger effects of external fields on transition temperatures have been reported33 for chiral smectic C materials, where there is a linear interaction between an electric field and the spontaneous polarization of the material. The theory of field-induced effects in tilted and chiral smectic phases remains to be investigated. (32) Rosenblatt, C. Phys. Left. A 1981, 83A, 221. (33) Heppke, G. In Proceedings of the 5th European Winter Liquid Crysfal Conference;Borovets, Bulgaria, March 1987; Mol. Cryst. Liq. Cryst. 1987, 151, 69,
Feature Article
The Journal of Physical Chemistry, Vol. 92, No. 6, 1988 1413
Another area of study in terms of field effects is with mixtures." A molecular statistical theory of nematic mixtures has been devel0ped,3~which leads to some interesting predictions; for example, nematic/nematic phase separation has already been observed, as predicted.wJOOIf a field term is included in the p~eudopotential,~~ then, in a mixture of mesogens of opposite susceptibility anisotropies, an external field can cause phase separation into uniaxial and biaxial nematic phases of different compositions. This is illustrated in Figure 8. With increasing field strength the uniaxial (positive) nematic/paranematic transition becomes continuous at a critical end point, while the biaxial nematic/uniaxial (negative) paranematic transition becomes second order. 11.2.2. Predictions of Field Effects Using Continuum Theory. The main result of continuum theory is eq 32 and 33, which give the mean square amplitude for director fluctuations of wave vector q. Individual q modes can be probed by light scattering, and the corresponding intensity is proportional3' to (n12(q)+ jnz2(q)), where j is a polarization factor dependent on the orientation of incident and scattered light polarization vectors. According to eq 35 the quenching of each mode is proportional to FZ at low field strengths and inversely proportional to q2. Integrating over all modes gives the effect of fluctuation quenching on the order parameters. For small values of the field and thin samples there is a quadratic dependence of the induced order on field strength, which becomes linear, or more strictly modular, at higher field strengths and for thicker samples. We have shown that for particular sample configurations an external field can induce biaxiality in the sample through differential quenching of director fluctuation^.^^ The field dependence of this effect is similar to the uniaxial result. We have only discussed the result for equal elastic constants; however, if this assumption is removed, anisotropic boundary conditions, even in the absence of a field, can impose biaxiality on a sample. It is clear from the results of this section that fluctuations contribute to the macroscopic disorder and will reduce the apparent anisotropy of physical properties. Fluctuations will also contribute to transition temperatures and transition entropies but are neglected in the usual molecular field and Landau theories of liquid crystal phase transitions. A recent paper39 reports a calculation based on a lattice model which includes fluctuations in a molecular field theory of the nematic/isotropic transition. In the context of continuum theory, elastic strain increases the free energy of a nematic and so lowers the transition temperature to the isotropic phase. For a field-free fluctuating nematic, the free energy is partitioned between the fluctuating modes: quenching of these will reduce the free energy and hence increase the nematic/isotropic transition temperature. The magnitude of this effect has yet to be determined. 11.3. Field Effects in the Isotropic State of a Liquid Crystal. In the isotropic liquid state just above a liquid crystal phase transition, molecular correlations are still fairly long range and the liquid exhibits many properties characteristic of an ordered fluid. This pretransitional region has been extensively studied by optical, electrooptical, magnetooptical, and NMR technique^,^' (34) Sinha, K. P.; Subburam, R.; Khetrapal, C. L. Chem. Phys. Lett. 1985, 96, 472. (35) Palffy-Muhoray, P.; de Bruyn, J. J.; Dunmur, D. A. Mol. Cryst. Liq. Crvst. 1985. 127. 301. '(36) Sharma,'S. R.; Palffy-Muhoray, P.; Bergersen, B.; Dunmur, D. A. Phys. Rev. A 1985, 32, 3152. (37) Chandrasekhar, S. Liquid Crystals; Cambridge University Press: Cambridge, 1977; p 155. (38) Dunmur, D. A.; Szumilin, K.; Waterworth, T. F. Mol. Cryst. Liq. Cryst. 1987, 149, 385. (39) Matsui, T.; Hofsass, T.; Kleinert, H . Phys. Rev. A 1986, 33, 660. (40) For a review up to 1980 see Dunmur, D. A. Molecular Electrooprics; Krause, S.,Ed.; Plenum: New York, 1981; p 435. (41) Pouligny, B.; Marcerou, J. P.; Lalanne, J. R.; Coles, H. J. Mol. Phys. 1983, 49, 583. (42) Dunmur, D. A,; Tomes, A. E. Mol. Cryst. Liq. Cryst. 1981, 76, 231. (43) Rosenblatt, C. Phys. Rev. A 1985, 32, 1924; Phys. Rev. A 1986, 34, 3551. (44) Tsvetkov, V. N.; Ryumtsev, E. I. Mol. Cryst. Liq.Cryst. 1986, 133, 125.
and it is appropriate to review the background to the interpretation of such studies. The molecular field and Landau theories of nematic liquid crystals outlined above predict that at sufficiently high field strengths the nematic/isotropic transition becomes second order for a material of negative susceptibility anisotropy and continuous if the anisotropy is positive. For both positive and negative materials the high-temperature phase is uniaxial so P = 0, and neglecting higher order terms, the induced uniaxial order is (39) In the Landau theory T* and a. are parameters, but the results of molecular field theory show that a. = 5kp and T* = u/5k, and Q becomes
-
Equation 39 shows that the induced order diverges as T T* and is proportional to the field strength squared. Molecular field theory predicts a T * / T N 1that is much smaller than is observed experimentally, but in contrast to the Landau result it is field and temperature dependent. For a noninteracting molecular system u = 0 and Q = h('")P/ 15kT, which is the result for a dilute gas.@ In the absence of a field the average order parameter in the isotropic phase is zero, but it will fluctuate about zero with an amplitude determined by appropriate elastic constants. De G e n x ~ eintroduced s~~ two elastic constants (Ll,L2)for the isotropic phase which determine the contribution to the free energy (AS) arising from fluctuations in the magnitude of the local order parameter: they may be defined by
Considering only uniaxial fluctuations, the mean square order parameter corresponding to a deformation associated with wave vector q becomes
The isotropic phase elastic constants can be related to longitudinal (tl) and transverse (Et) coherence lengths, which measure the spatial persistence of an ordered region parallel and perpendicular to the axis of local order. These coherence lengths are given by (43) (44) We might inquire what the effect of a field would be on the fluctuations in the isotropic phase. The situation is now more complicated because a field destroys the symmetry of the phase. Since the induced anisotropy now depends on the field strength squared, the quenching of mean square fluctuations of the order parameter in the isotropic phase is predicted to depend on the fourth power of the field strength. 111. Comments on the Theories of Field-Induced Order In this section we wish to examine critically the theoretical predictions concerning field-induced order. Two theoretical ap(45) Nicastro, A. J.; Keyes, P. H . Phys. Rev. A 1984, 30, 3156. (46) Attard, G. S.;Beckmann, P. A.; Emsley, J. W.; Luckhurst, G. R.; Turner, D. L. Mol. Phys. 1982, 45, 1125. (47) Ruessink, B. H.; Barnhoorn, J. B. S.; MacLean, C. Mol. Phys. 1984, 52, 939. (48) Buckingham, A. D. Discuss. Faraday SOC.1967, 43, 205. (49) de Gennes, P. G. Mol. Cryst. Liq. Cryst. 1971, 12, 193.
1414 The Journal of Physical Chemistry, Vol. 92, No. 6,1988
proaches are outlined above, and it is clear that the predictions of these differ qualitatively and quantitatively from each other. To some extent the two theories address different aspects of the problem, and experiments have been performed that provide data to test both approaches. However, it should ultimately be possible to unify the different theoretical treatments and provide a complete description of the interaction of anisotropic fluids with external fields. III.1. Microscopic and Macroscopic Order. Our starting point for the discussion of field effects on liquid crystal order was that the orientational order of molecules could be separated on the basis of two distance scales into a local or microscopic order referring to molecular axes and a macroscopic order that described the degree of order of the local symmetry axes. An additional source of disorder is the internal flexibility of a molecule,50 which has not been considered here, but which could be characterized in terms of a third distance. We must ask how valid it is to consider different types of order in this way. The representation of a molecule as an average over different conformations is commonly done, but different segments of a molecule may have different degrees of order. The separation of orientational order into microscopic and macroscopic order is equivalent to the separate consideration of internal and external degrees of freedom. Its validity depends on there being a large difference in energy between the different interactions responsible for internal and external motion. In the context of this paper, we argue that effects due to strong intermolecular forces acting on a molecular scale can be separated from the long-range behavior due to weak torsional elastic forces. We will assume that it is possible to define axes within a molecule, e.g., average principal inertial axes, and order parameters can be assigned to describe the orientational order of these axes with respect to any other appropriate axis system. In a series of p a p e r ~ ~ sFaber ~ l - ~developed ~ a model for disorder in nematic liquid crystals based on the idea that all orientational disorder was due to the excitation of director fluctuation modes. This is equivalent to setting 5$"' in eq 12 equal to 1; biaxial ordering was not considered by Faber. Field effects based on this model should be qualitatively the same as the continuum theory predictions with the accompanying problems of the mode cutoff and number of modes. Another question raised by Faber52 is the q dependence of the elastic constants. In the Debye theory of translational disorder in crystals, upon which Faber's theory is based, force constants do depend on the wave vectors of the normal modes. It is to be expected that torsional elastic constants of liquid crystals should also depend on q, but no such dependence has been measured and is usually implicitly ignored. The separation of long-range and short-range influences on orientational order in liquid crystals was first considered by de G e n n e ~ . ' ~He, ~proposedIE ~ that the extent of short-range order was determined by coherence lengths El and tt (eq 43 and 44), which are likely to be on the order of 10 times the molecular length, and the correlation function for the order parameter has the Ornstein-Zernike form: (45) Faber has also applied his model to calculate short-range correlations between director orientation^.^'.^^ No simple analytical ) is provided, but the limiting long-range form for (Szz(0)Szz(r) behavior for the correlation function is established. Calculations using this model agree with the results of computer simulation for a 10 X 10 X 10 lattice of points.53 The effect of a strong external field on director fluctuations using this model have also been ~ a l c u l a t e d and , ~ ~ these results agree with a computer-sim~~
~
(50) don, A (51) (52) (53) (54) (55)
Emsley, J. W.; Luckhurst, G. R.; Stockley, C. P. Proc. R . SOC.Lon-
Dunmur and Palffy-Muhoray ulation study of a similar The long-range picture of orientational order introduces a field-dependent coherence length t(F)= (k/AKdpp)'/2.Orientation imposed by a field dominates when {(F) is smaller than the sample dimensions. For small deformations from n, = 1 , correlations between director orientations at different points in a liquid crystal are given by1*
For a particular fluctuation mode q, the corresponding mean square fluctuations in the director are (47)
In an important paper Masterss7 has shown that even if the direct correlation function for molecular orientation is only of short range, the indirect correlation function can be long ranged and so lead to macroscopic fluctuations in orientational order. The corresponding expression for the mean square fluctuation is identical with eq 47. Masters claims a consequence of this result is that a complete mean field theory should include long-range fluctuations and that mean field order parameters will include disorder due to director fluctuations. We have already seen that long-range effects depend on sample dimensions and boundary conditions, but mean field theories as usually constituted do not include any such dependence. Approximate mean field theories such as the Maier-Saupe (M-S) theory do not include long-range disorder. The coefficient of the M-S single-particle pseudopotential is independent of temperature. However, if this pseudopotential is to include contributions from long-range fluctuations, which are more readily excited thermally than those of short range, it should have an explicit temperature dependence. To avoid double counting of disordered complexions in the calculation of the order parameter it is important that the averaging volumes for longrange and short-range order are carefully defined and that the local order parameter refers to a microscopic volume with linear dimensions less than k B T / k k . The problem of separating molecular motion and cooperative reorientation in liquid crystals was addressed by Zientara and Freed,58 who investigated the rotational diffusion of rods in a coupled linear array subjected to a field. They found that for small wave vectors the model gave the same results as continuum theory (eq 47) for the mean square amplitude of fluctuations, but for larger wave vectors (small distances) there was a modified dependence on q. This result would seem to contradict the conclusions of Masters, who showed that the expression for the mean square amplitude of fluctuations is the same for all q, although the elastic constants may depend on q. 111.2. Molecular Statistical and Landau-de Gennes Mean Field Theories. The molecular statistical mean field theory assumes a form for the single-particle potential and, depending on the number and type of terms included, will predict the values of various order parameters together with the free energy and entropy of the system. Order parameters are defined self-consistently, and there is no analytical expression for the order parameters or the dependence of thermodynamic properties on the order parameters. By contrast, the Landau-de Gennes free energy is an empirical function of the order parameters, the choice, and number of terms depending on the effects to be modeled. The coefficients of the order parameters in the Landau-de Gennes free energy are parameters that may be adjusted to fit experimental results. A problem with the mean field theory is that the predicted value of (TNI- P)/ TNIis approximately an order of magnitude larger than is observed experimentally. As a result, calculated fieldinduced effects are considerably smaller than observed. One
1982, 381, 117.
Faber, T. E. Proc. R . SOC.London, A Faber, T. E.Proc. R. SOC.London, A Faber, T. E. Proc. R. SOC.London, A Faber, T. E.Proc. R. SOC.London,A Faber, T. E. Proc. R. Soc. London, A
1977, 353, 1977, 353, 1980, 370, 1981, 375, 1984, 396,
261. 247. 509. 579. 351.
(56) Luckhurst, G. R.; Simpson, P.;Zannoni, C . Chem. Phys. Lett. 1981, 78, 429. (57) Masters, A. J. Mol. Phys. 1985, 56, 887. (58) Zientara, G. P.; Freed, J. H. J . Chem. Phys. 1983, 79, 3077.
Feature Article remedy may be to include the particle excluded volume in the pseudopotential through a van der Waals theory. Recent calcul a t i o n ~using ~ ~ such an approach suggest that the predicted ( T N I - T * ) / T N may I be smaller than is obtained with the excluded volume neglected. Work to include field-induced behavior in the van der Waals description of nematics is in progress. In Landau theory, T* is a parameter to be determined by experiment. In such circumstances experimental data for field-induced order can be readily fitted to the Landau-de Gennes theory, and predicted effects are much larger than is obtained from the molecular statistical theory. The Landau free energy is a power series in the order parameters. This expansion is one particular nonequilibrium free energy functional and may span thermodynamically inaccessible states. On the other hand, the nonanalytic dependence of the molecular statistical free energy on order parameters is constrained by the requirement that the order parameters are self-consistent within the mean field approximation. A detailed analysis of the free energy expansions in the Landau-de Gennes and molecular statistical approaches@’has been given by Katriel et a1.61 They derive a result that is independent of the assumed form of the pseudopotential. It differs slightly from our result@’because of our neglect of higher order parameters in the pseudopotential. One conclusion of this work is that the L a n d a u 4 e Gennes expansion coefficients should not be regarded as arbitrary parameters to be fixed by experiment: furthermore, we have also shown that these may be field dependent. Examination of the convergence properties of the Landau expansion raises questions61 about the validity of the Landau-de Gennes treatment of transitions from the nematic to more ordered smectic phases. 111.3. Characteristic Lengths and Cutoffs for Fluctuation Modes. The continuum model for orientational disorder introduces a short-wavelength cutoff (qmax)for the director fluctuation mode spectrum, and its value determines the quantitative predictions of the model. For example, the disorder due to director fluctuations in the absence of an external field is predicted by simple application of the theory to be9
The Journal of Physical Chemistry, Vol. 92, No. 6,1988 1415 conceptual identification of macroscopic and microscopic order requires that the distance scales for the respective order parameters are properly defined. It is important that fluctuation modes are not counted twice, but the number of modes may not be fixed by a simple Debye relationship because internal molecular motion can contribute to the orientational disorder. It is well-known that static liquid crystal properties are considerably affected by surface. interactions and boundary effects; the fluctuation mode spectrum will also be restricted by the sample size and shape. Theoretical results for infinite samples set qminto zero, but it is apparent from the results of section 11.1.2 and from recent experiments3’ on field-induced birefringence that the long-wavelength cutoff is important for thin samples. Further complications may also arise for different shaped samples.
IV. Experiments: Observation of Field-Induced Order The theoretical predictions of the effect of external fields on orientational order were outlined in section 11. It was shown that external fields can increase the microscopic and macroscopic order in fluids, with an accompanying reduction in orientational disorder or fluctuations. Thus any physical property that depends on the order can be used to probe field effects, or equivalently the quenching of fluctuations (disorder) will also reflect the fieldinduced order. In reviewing the experiments that have been camed out, it is convenient to consider these two approaches separately. ZV.1. Light Scattering Measurements of Orientational Disorder. In both isotropic and anisotropic fluids, fluctuations of the birefringence results in depolarized light scattering. For isotropic fluids the fluctuations are about a mean birefringence of zero. This scattering may be molecular in origin from the free rotation of anisotropically polarizable molecules, but angular correlation greatly enhances the scattering and is responsible for the turbid appearance of liquid crystals. As a liquid crystal transition is approached from the isotropic phase, the depolarized intensity ZVH diverges; it is proportional to the mean square order parameter and from eq 42 and 43 becomes IVH
If the cutoff is on the order of molecular dimensions (10 X lo-’’ m), qmx is 6 X lo9 m-l and the order parameter predicted by eq 48 is 0.6. This is similar to the molecular order parameter, and we conclude that macroscopic and molecular disorder have been accounted for by eq 48. The Debye model would predict a value for q- of 5 X lo9m-l, and similar values for qmx were proposed by Berreman.62 Analysis of N M R measurements by Doane, Tam, and Nickersod3 gave a cutoff of 4 X lo9 m-l, corresponding to a director order parameter of 0.74. Fitting depolarized light scattering intensities in the pretransitional region to a perturbed Gaussian modelI6 suggests that qmaxshould be 0.25 X lo9 m-l. This result would require that the order parameter describing director fluctuations was 0.98; Le., the disorder due to thermally excited fluctuations in the director orientation was very small. Measurements of fluctuation quenching through field-induced birefringence can also give information on qmax(see next section). Comparison of the results of these experiments with theory3’ gave values for qmx ranging from 6.3 X lo9 to 3.0 X lo9 m-l. These were obtained from measurements on a number of samples of different thicknesses. Thus there appears to be no clear guidance from theory or experiment as to an appropriate value for qmax. Presumably different experiments probe different interaction volumes in the liquid crystal, and so qmaxmay be determined to some extent by the technique used to probe it. However, the (59) Palffy-Muhoray, P.; Bergersen, B. Phys. Rev. A 1987, 35, 2704. (60) Palffy-Muhoray, P.; Dunmur, D. A. Phys. Letf. A 1982, 91A, 121. (61) Katriel, J.; Kventsel, G.F.; Luckhunt, G.R.;Sluckh, T. J. Liq. Cryst. 1986, 1 , 337. (62) Berremann, D. W. J . Chem. Phys. 1975, 62, 776. (63) Doane, J. W.; Tarr, C. E.; Nickerson, M. A. Phys. Rev. Lett. 1974, 33, 620.
=
const (Acr)’kBT 11 + t12q21-1 X4Vao(T- P)
(49)
Aa is the molecular optical anisotropy, and Vis the scattering volume. There have been many experiments on pretransitional light scattering from liquid c r y s t a l ~ , 6 ~and ” ~ these have been deinterpreted through eq 49. Deviations from the ( T - P)-’ pendence are observed close to the transition6’ but have not been satisfactorily explained. Recent light scattering experiments and their interpretation have been reviewed by Gramsbergen et a1.16 Angular correlation also affects the dynamics of light scattering, and the inverse line widths (relaxation times) of the light scattered from the isotropic phase also diverge as the temperature approaches the liquid crystal transition.68 Application of external fields is expected to influence the pretransitional isotropic/liquid crystal light scattering, but no experiments have been reported. We have measured the depolarized scattered intensity as a function of temperature and applied electric field strength for 44’-n-pentylcyanobiphenyl,and obtained an estimate of the field dependence of P. Our result69 for K V2m2 is in fair agreement with a mean d P / d E 2 = 6.5 X K V2m2. Unfortunately these field c a l c ~ l a t i o nof~ 3.5 ~ X measurements were not precise enough to obtain quantitative information on the field quenching of fluctuations in the isotropic phase. (64) Stinson, T. W.; Litster, J. D.; Clark, N. A. J . Phys. (Les Ulis, Fr.) 1972, 33, C1-69. (65) Coles,H. J.; Strazielle, C. Mol. Cryst. Liq. Cryst. 1979, 55, 237. (66) Zink, H.; de Jeu, W. H. Mol. Cryst. Liq. Cryst. 1985, 124, 287. (67) Grohin, A,; Destrade, D.: Gasparoux, H.; Prost, J. J. Phys. (Les Ulis, Fr.) 1983, 44, 427. (68) Litster, J. D.; Stinson, T. W. J . Appl. Phys. 1970, 41, 996. (69) Dunmur, D. A,; Tomes, A. E., unpublished results. Tomes, A. E. Thesis, University of Sheffield, 1984. (70) Waterworth, T. F. Thesis, University of Sheffield, 1985.
,
1416 The Journal of Physical Chemistry, Vol. 92, No. 6, 1988 Light scattering from liquid crystal phases is used to probe both their elastic and viscoelastic properties. The depolarized intensity is proportional to the mean square fluctuations in the optical anisotropy, and depending on the scattering geometry, various fluctuation modes can be probed.37 Recently this technique was used to detect for the first time biaxial fluctuations in the orientational order of the biaxial lyotropic nematic phase of potassium lauryl sulfate, decanol, and Dz0.71 The intensity of depolarized light scattering from the nematic phase in the presence of an external field is72
IVH = ( c o n ~ t ) k ~ T C [ k+~ki,qL2 ~q~+ ~ ~p A ~ ( ~ ) p ] -(50) l
Dunmur and Palffy-Muhoray , 2:
,
LID
B I Tesla
3 0 10'6An 2 0
1 0
i
In this equation qli and q L are the components of the scattered wave vector parallel and perpendicular to the director. The corresponding equation for the reciprocal line width of the scattered light is73,74
r-' = Cvi(q)[k33ql12+ kiiqLz +
pAdd)P]-'
(51)
I
where vi(q) is a viscoelastic coefficient that depends on the scattering vector q. Various workers have confirmed the validity of eq 50 and 51, and measurements on the scattered light (usually the line width) have provided values for elastic and viscoelastic coefficients of a number of material^.^^,^^ IV.2. Field-Induced Birefringence Measurements of Orientational Order. The most sensitive experimental technique available to study changes in orientational order is field-induced birefringence. Changes in optical path length (6Anl), where 1 is the sample length and 6An is an induced change in birefringence, can be measured directly by using a suitable compensator or through calibrated modulation techniques. Estimated sensitivities for 6Anl as reported for liquid crystal samples cover the range 2 X 1O-lO-5 X 1V8m for changes induced by magnetic field^$^,^^ 3X m for static electric fields$5 and 2 X m for modulated electric fields.30 Translating these sensitivities into detectable changes in induced order depends on the sample path length: increased path length resulting in increased sensitivity to induced order. However, thick samples give rise to problems due to multiple scattering and depolarization. For a 50-pm cell existing techniques can detect field-induced orientational order parameters in liquid crystals in the range 10-5-10-8. Most experiments have been carried out in the pretransitional (isotropic/nematic) region, where the induced order varies as the square of the field strength and is for the most part adequately described by eq 39. The relationship between the induced birefringence 6An and the induced order Q depends on the effective optical anisotropy A a of the liquid crystal molecules interacting with their environment. If problems associated with anisotropic internal fields are neglected, then
nz + 2 P ~ A ~ A K ( ~ ) P 6An = (52) 18cOn ao(T - T * ) Deviations from a squared field dependence have been observed for strong magnetic fields, and measurements of the temperature dependence of dz6An/d(F2)2 have been used to establish the socalled gap exponent A. This is predicted by Landau theory78to have the value of 2 for critical behavior (as opposed to tricritical behavior), which is in reasonable agreement with recent meas u r e m e n t ~on~ a~ lyotropic system. As with the divergence of the (71) Reference 12. (72) Reference 14, p 95. (73) Martinand, J. L.; Durand, G. Solid State Commun. 1972, 10, 815. (74) van der Meulen, J. P.; Zijlstra, R. J. J. J . Phys. (Les Ulis, Fr.) 1984, 45, 1627. (75) Ivanov, S.; Yu Vetrov, V. Sou. Phys. Crystallogr. (Engl. Transl.) 1982, 27, 609. (76) Leslie, F. M.; Waters, C. M. Mol. Cryst. Liq. Cryst. 1985, 123, 101. (77) Seppen, A.; Maret, G.; Jansen, A. G. M.;Wyder, P.; Janssen, J. J. M.; de Jeu, W. H. Springer Proc. Phys. 1986, 11, 18. (78) Keyes, P. H.; Shane, J. R. Phys. Rev. Lett. 1979, 42, 722.
10
20
30
50
LO
E 110% m-1
F v 9. Experimental results for field-induced birefringence in nematics of positive susceptibility anisotropy. Dashed line: magnetic birefringence in 7CB;82 sample thickness = 200 pm. Full lines: electric birefringence in 5CB;30 sample thickness = 111 (0),55 (X), and 15 pm (+). The electric and magnetic field axes have been scaled according to (Axei,/ AXUlap.
depolarized light scattering, deviations from a (T - P)-l dependence of the field-induced order have been measured close to the nematic/isotropic transition for both electric and magnetic birefringen~e.~~.~~ The effect of an external electric or magnetic field on the birefringence of a liquid crystal phase has been measured by a number of g r o ~ p s Initial . ~ ~experiments81,82 ~ ~ ~ ~ ~ involved ~ ~ ~the ~ application of a field along the director axis of a nematic material of positive susceptibility anisotropy and the measurement of the consequent change in birefringence as a function of field strength. Equation 35 gives the order induced through fluctuation quenching for this configuration. Making the assumption that qmin= 0 and qmax>> which is valid for normal laboratory fields, gives 6An =
87rtonk
For thick samples at moderately high field strengths a linear dependence of induced birefringence and hence order parameter on electric and magnetic field strength has been observed, consistent with eq 53. Results have been reported for the magnetic field induced birefringence in the nematic phase of 7CB,828CB,82 lyotropic C S P F O , ~the ~ electric field induced birefringence in nematic 5CB:O and the electric field induced birefringence in the smectic phase of 8CB.30 Some experimental results are given in Figure 9. The predicted dependence of induced order on the modulus of the field strength has been confirmed by an experiment30 using alternating electric fields. For an applied field of frequency w, it was found that the induced birefringence was modulated at 2w and 4w in the proportion of 5:l as required by the Fourier expansion of IEI: (Eosin wtl =
"( 7r
2 2 1 - - cos 2wt - - cos 4wt 15
+ .-.>
(54)
Measurements on thin samples showed a deviation from a linear field dependence at low field strengths (see Figure 9). This is consistent with the behavior predicted by eq 35 when the approximation qmin= 0 cannot be used. In fact, the measured results over the full range of field strengths and cell thicknesses can be accurately fitted by eq 35. Although the continuum theory outlined in section I1 gives a good qualitative description of the birefringence induced in nematic liquid crystals by external fields, there appear to be some quantitative differences between theory ~~
~
(79) Kumar, S.; Litster, J. D.; Rosenblatt, C. Phys. Rev. A 1984,29, 1010. (80) Ratna, E.R.; Viyaya, M. S.; Shashidhar, R.; Sadashiva, B. K. Pramana, Suppl. 1973, 1, 69. (81) Poggi, Y . ;Filippini, J. C. Phys. Rev. Lett. 1977, 39, 150. (82) Malraison, E.,Poggi, Y. and Guyon, E. Phys. Rev. A 1980, 21, 1012.
The Journal of Physical Chemistry, Vol. 92, No. 6,1988 1417
Feature Article B I Tesla
1.o
5.0
3.0
10
05
15
1O-6E I Vm-’
Figure 10. Experimental results for field-induced biaxiality in nematics of negative susceptibility anisotropy. (a) Magnetic biaxiality in mixture of p-cyano@’-alkylcyclohexyl)cyclohexanes;77 sample thickness = 200 pm, (b) Electric biaxiality in cis-(4’-cyano-4,4’-n-pentyl-n-pentyl)bic y ~ l o h e x a n e ;sample ~~ thickness = 20 pm. (c) Electric biaxiality in 4n-pentyl-2-fluorophenyl-4’-n-pentyl c y c l ~ h e x a n o a t e ;sample ~~ thickness = 25 pm.
and experiment. Fitting the electric field induced birefringence results for thin samples to the theory requires the introduction of an effective cell thickness which is considerably less than the actual cell d i m e n ~ i o n s .This ~ ~ suggests that the director fluctuations close to the cell walls are already quenched by surface interactions, and the full quenching effect of an external field is observed only in thick samples. Macroscopic field-induced biaxiality has been observed in nematic liquid crystals of negative susceptibility anisotropy. For such materials a field perpendicular to the director does not induce macroscopic realignment, and differential quenching of fluctuations leads to induced biaxiality. Measurements have been reported of this effect for both magneti~’~ and electric fieldsS3*The induced biaxiality is found to be proportional to the modulus of the applied field strength, and the theory of section 11.1.2 would seem to be appropriate. In Figure 10 we reproduce the reported experimental results for biaxiality induced by both electric and magnetic fields. The measurements scale roughly according to the square root of the susceptibility anisotropies of the materials, although the magnitudes are about of the corresponding uniaxial birefringence. Field-induced biaxiality has been observeds3in the cholesteric blue phase I, though not in blue phase 11. The effect was detected by using optical microscopy, but no quantitative measurements of the field induced biaxiality were reported. It is usually claimed that smectic phases will not support bend or twist elastic deformations since k22and kj3 are infinite. This means that there can be no field-induced birefringence from fluctuation quenching. However, measurements on the electric field birefringence in the smectic phase of 8CB have been reported.30 It was found that the effect was about lOOX smaller than in the nematic phase and proportional to the square of the electric field strength. From eq 35 it is apparent that the effect of increased elastic constants is similar to a reduced dell thickness, at least insofar as they affect the low field dependence of the induced order. We have already seen that for thin samples of nematics the field-induced birefringence is proportional to the field strength squared, so the observed similar field dependence in smectics could be attributed to large but finite values of k22and k33 in the smectic A phase. An alternative expl’anation is that macroscopic director fluctuations are indeed totally quenched in smectics, and the field-induced order arises from the field dependence of the microscopic ordering matrix s$. Further studies of induced order in smectics are clearly called for and may help (83) Porsch, F.,Stegemeyer, M. and Hiptrop, K. Z. Naturforsch.,A 1984, 39A, 475.
to resolve the conceptual problems associated with the partitioning of fluctuations into microscopic and macroscopic disorder. Magnetic birefringence measurements have been made in the region of an SA/SF phase t r a n ~ i t i o n . ~Since ~ this transition involves the development of molecular tilt and hence a biaxial phase, the pretransitional tilt susceptibility can lead to magnetic field induced biaxiality detectable through the induced birefringence. ZV.3. Other Experiments. As explained in the introduction to this section, any physical property that depends directly or indirectly on the order parameter can be used to probe field-induced order. However, there are very few reports of experiments using techniques other than light scattering or induced birefringence. Nonlinear dielectric polarization should be observable in liquid crystals, but no experiments have been reported. Laser field induced birefringence has been measured in a variety of liquid crystals,85~86 and various other nonlinear optical effects have been observed in nematics, e.g., self-focussing, harmonic generation, and parametric mixing. These phenomena all depend on the refractive indices being field dependent. The refractive indices of liquid crystals depend on both the molecular properties and the degree of order, so that nonlinear optical responses can arise from a field dependence of molecular properties and/or the orientational order. For the experiments performed so far no attempts have been made to separate the relative contributions of these two influences. Magnetic field induced order in the pretranslational isotropic phase of a nematic has been detected with magnetic res0nance,8~~~~ but no measurements in the nematic phase have been reported.
V. Discussion and Conclusions The molecular statistical and continuum approaches to the description of liquid crystal properties can each be characterized by appropriate lengths. We assume that for the molecular theory the scale is the distance over which director fluctuations can be discounted, while the continuum theory takes account of director fluctuations. Contact between these approaches can be made through statistical calculations of elastic constants.89~90Because of the relatively weak anisotropic forces in liquid crystals, fluctuations in orientational order are large and experimentally accessible, but their influence on equilibrium liquid crystal properties is still a matter for d i s c u s s i ~ n . ~ ~ ~ ~ In this article we have considered the effects of external fields on the order and disorder in liquid crystals; some of the predicted phenomena have already been observed, while others await experimental verification. The description of field effects in liquid crystals has been approached conventionally, so that some phenomena are interpreted in terms of molecular or Landau-de Gennes theory, while continuum theory is used to explain other effects. All the predicted behavior of liquid crystals in external fields can be attributed to the effects of the field on the orientational ordering matrix, through changes in the magnitude of the components and/or trace, changes in symmetry giving biaxial phases, or changes in the orientation of the principal axes. The effect of fields on thermodynamic properties such as transition temperaturesg2and transition entropies is a consequence of the ~
~~
~
~~
~
~~
~
(84) Rosenblatt, C.; Ho, J. T. J . Phys. (Les Ulis, Fr.) 1983, 44, 383. (85) Inoue, K.; Shen, Y. R. Mol. Crysf. Liq. Crysf. 1979, 51, 179. (86) Coles, H. J.; Jennings, B. R. Mol. Phys. 1978, 36, 1661. (87) Reference 46. (88) Ruessink, B. H.; Barnhoorn, J. B. S.;MacLean, C. Mol. Phys. 1984, 52, 939. (89) Poniewierski, A.; Stecki, J. Mol. Phys. 1979, 38, 1931. (90) Gelbart, W. M.; Ben-Shaul, A. J . Chem. Phys. 1983, 77, 916. (91) Clark, M. G. Mol. Cryst. Liq. Cryst. 1985, 127, 1. (92) Rosenblatt, C. Phys. Reu. A 1981, 24, 2236. (93) Straley, J. P. Phys. Reu. A 1974, 10, 1881. (94) Mulder, B. Liq. Crysf.1986, I , 539. (95) Freiser, M. J. Phys. Reu. Lett. 1970, 24, 1041. (96) Luckhurst, G. R.; Romano, S.Mol. Phys. 1980, 40, 129. (97) Bergersen, B.; Palffy-Muhoray, P.; Dunmur, D. A. Liq.Cryst. 1987, 2, 381. (98) Bunning, J. D.; Crellin, D. A.; Faber, T. E. Liq. Cryst. 1986, I , 37.
1418 The Journal of Physical Chemistry, Vol. 92, No. 6, 1988
dependence of the free energy on order parameter and hence field strength. Over a microscopic volume containing perhaps 100-1000 molecules, corresponding to a characteristic distance of 2-5 nm, the average orientational order will not fluctuate to any significant extent as the free energy associated with such fluctuations will be large compared to kT. An external field will influence the local order through interaction with the molecular susceptibility anisotropy, and the induced effects will be of the same order of magnitude as those observed in simple liquids. Expressions for the free energy and entropy derived by using molecular statistical theory can be expanded as power series in the order parameter, and comparison with the empirical Landau-de Gennes expression for the free energy provides some basis for using the Landau theory to explain field-induced phenomena. The advantage of the empirical Landau-de Gennes theory, which predicts essentially the same phenomena as the molecular statistical theory, is that it can be parametrized in terms of experimental quantities. Thus the field effects predicted by Landau theory are considerably larger than those predicted by the molecular statistical theory. In the isotropic phase of liquid crystals close to the liquid crystal transition, the measured field-induced order is much larger than for ordinary liquids, and so it is to be expected that the molecular statistical theory will considerably underestimate the magnitude of field effects in liquid crystals. The orientational order induced by external fields in the isotropic phase above a liquid crystal transition diverges as the temperature approaches the transition temperature. Qualitatively the experimental results are in accord with Landau-de Gennes and molecular statistical theories, apart from a discrepancy close to the transition. As well as affecting the order, an external field will also increase the transition temperature, but again the magnitude of the effect calculated by molecular mean field theory is too small. The predicted quadratic dependence of induced order on field strength is confirmed experimentally. No attempts have been made to apply continuum theory to the description of the effects of field on phase transitions. Since the elastic constants depend on the microscopic order parameter, and the latter is field dependent, it is possible in principle to determine the field dependence of the elastic free energy and hence the field dependence of the transition temperature. A more important influence on the transition temperatures is likely to be fluctuations, but the consequences of field quenching of fluctuations on liquid crystal phase transitions has yet to be considered. The most direct manifestation of the effect of external fields on orientational order is the birefringence induced in thermotropic and lyotropic nematic phases by electric and magnetic fields. Both continuum and mean field theories can be used to interpret the experimental results, but as before the molecular statistical theory predicts induced order that is much smaller than observed. However, there is a more serious conflict between the two approaches, since continuum theory predicts a modular dependence of induced order on the field strength, instead of the quadratic dependence that comes from Landau and molecular statistical descriptions. Measurements reported on nematic phases all exhibit a linear dependence of induced birefringence on field strength, and the modular functionality has also been experimentally confirmed. In smectic phases the induced birefringence is much smaller and appears to be quadratic in the field. This result suggests that materials with large elastic constants, in which long-range fluctuations are restricted, may conform more closely to the mean field or Landau descriptions. At low field strengths, and for thin samples, nematics also show a field-induced birefringence that is quadratic in the field strength. We have explained this in terms of the low-field limit of the continuum expression, eq 3 5 , but we note that low fields correspond to small values of which also result from large elastic constants; i.e., there is some correspondence between the continuum theory and mean field
rl,
Dunmur and Palffy-Muhoray theories for materials with large elastic constants. Recent measurements of the biaxial birefringence induced in materials of negative susceptibility anisotropy by both electric and magnetic fields confirm the linear dependence of induced order on field strength predicted by continuum theory. We have shown in section 11.1 that both Landau and molecular statistical theories predict a field-squared induced biaxiality in materials of negative susceptibility anisotropy, but the measured effects are considerably larger than predicted. We conclude that mean field theory provides a reasonable description of the field dependence of the transition temperature and phase behavior but appears to be inadequate in describing induced order. The continuum model on the other hand adequately describes birefringence measurements of field-induced order and light scattering experiments on fluctuation quenching in nematic liquid crystals. These complementary approaches can be combined through the scale separation scheme discussed in this paper. Under these circumstances a measured property depends on both the local and macroscopic disorder. Masters has correctly pointed out a weakness in this approach, since a complete mean field theory, in which the pseudopotential is obtained by averaging over short-range and long-range complexions of the system, must include short-range and long-range disorders. It is significant that the director correlation function derived by Masters from mean field theory is the same as that used in the continuum theoryfluctuation model. From a fundamental point of view the “joining” of microscopic and continuum theories is unsatisfactory, particularly in view of the fact that the short-wavelength cutoff for the fluctuation mode spectrum becomes a parameter of the continuum theory. In a more complete formalism, as pointed out by Mast e r ~the , ~ long-wavelength ~ fluctuations would be taken into account in the single-particle pseudopotential: this formalism has yet to be developed.
Acknowledgment. Our contributions to the research reviewed above have been supported by the United Kingdom Science and Engineering Research Council and Ministry of Defense, and the National Science and Engineering Research Council of Canada. Valuable discussions and correspondence with Dr. T. E. Faber, Dr. A. J. Masters, and Dr. M. Warner are gratefully acknowledged. Appendix The general rotation matrix R$“ in terms of Euler angles 8, 9, and x relates the susceptibility in the molecular axes (m) to the local director axes (d): it can be written as
bo
=
1
(cos e cos @ c o sx - sin Q sin x ) (-cos 8 cos @ sin x - sin 0 cos x) sin 0 cos 0 (cos e sin 0 cos x + cos 0 sin X ) (-cos e sin 0 sin x + cos Q cos x) sin e sin 0 -sin e cos x sin e sin x cos e
[
(Ai)
and the relationship between the susceptibility tensors becomes
2 = -Sdm
am)
3 o;e,raKrb
(‘42)
If the molecular susceptibility is defined in a principal axis system, then
For uniaxial molecular symmetry written as
K*
is zero, and eq A 3 can be
(99) Pratibha, R.; Madhusudana, N. V. Mol. Cryst. Liq. Cryst. Lett. 1985, 1, 111.
(100) Veyssie, M.; Casagrande, C. Phys. Rev. Lett. 1987, 58, 2079. (101) Dunmur, D. A,; Szumilin, K., to be published.
If on the other hand the molecule is biaxial but the local nematic symmetry is uniaxial, then
J. Phys. Chem. 1988,92, 1419-1425
1419
(ii) Local biaxial ordering of uniaxial molecules giving a macroscopically uniaxial phase: = S:tK$)
where S$ = (1/2(3n,nb- tiab))is the ordering matrix of the local director axis with respect to the molecular frame. In order to write down the measured susceptibility anisotropy with respect to a laboratory fixed axis system, eq A2 has to be averaged over the orientations of the local director, giving =
(KtS)&pIe
!!.(3RldRld - 6
1 a8ya),(3R$?R:km
(
d.7 Ba - a7&))KP ('46) 9 2 Assuming the random phase approximation, averaging over the director and molecular axes may be done separately so eq A6 becomes
(Ktp)$kple
= (4/9)st;d,,yas$6&")
( b ~ ) k k= ~ l(2/3)s$$,d,.gmA~(~) ~
645)
(A7)
(iii) Biaxial molecules giving a locally uniaxial nematic, which is macroscopically biaxial-this could represent the orientational ordering in a tilted smectic phase: (Kap)kpIe
Id m d am)
(2/3)Sd.gS7d Ky6
(A 10)
(iv) Biaxial molecules which order locally and macroscopically to give a uniaxial phase: ( AK)
kkPle= S > % ; ~ K $ ~ )
('41 1)
In the above the second-rank ordering matrices can be written as simple averages,over appropriate rotation matrices; for example, S$? is given by Sd.m =
A number of special cases can now be identified: (i) Uniaxial molecules with local biaxial order which generates a macroscopic biaxiality:
( ~ t ~ ) =z (4/9)st;d,,y,s$'A~(m) k~~~
('49)
e cos' m - 1 c o s 0 sin cos e cos @
sin' e c o s sin @ 3 sin' e sin' m - 1 sin e c o s e sin 0
('48)
sin e c o s e c o s sin e c o s e sin @ 3 cos' e - 1
I)
(-412)
ARTICLES Optical Second Harmonic Generation Studies of Molecular Adsorption on Pt( 111) and Ni(111) S. G.Grubb,*t A. M. DeSantolo,t and R. B. Hall* Exxon Research and Engineering Company, Annandale, New Jersey 08801 (Received: December 2, 1986; In Final Form: October 13, 1987) The nonlinear optical process of second harmonic generation (SHG) is, from symmetry considerations, inherently surface specific and has the sensitivity to detect submonolayer coverages of molecules. We have used SHG to monitor the adsorption of CO, 02,and H2 on Ni( 11 1) and Pt( 111) surfaces. In the case of Ni( 11l), hydrogen adsorption follows Langmuir kinetics while the adsorption of both CO and O2 is characteristic of adsorption from a precursor state. The SHG response from these surfaces is dependent on the relative orientation of the crystalline axes and input polarization vector. Under specific wavelength and polarization conditions the SHG response is seen to correlate with work function changes upon adsorption. In the case of Pt( 11l), a striking frequency dependence of the SHG response is observed, suggesting the participation of surface states or resonant effects associated with interband transitions.
Introduction The utility of optical second harmonic generation (SHG) as a surface probe. has now been well demonstrated in a wide variety of Second-order nonlinear optical processes such as S H G are forbidden, within the electric dipole approximation, in media with inversion symmetry. Thus, the SHG process exhibits an intrinsic sensitivity to a symmetry-breaking surface or interface. In addition, S H G is sufficiently sensitive to detect submonolayer coverages of adsorbates on surfaces. The SHG technique has been used to monitor the orientation and structural symmetry of monolayers of adsorbates on a variety of dielectric surfaces.*' In these cases, the nonlinear susceptibility t Present address: Ammo Research Center, Physical Technology Division, P.O. Box 400, Naperville, IL 60566. *Present address: AT&T Bell Laboratories, 600 Mountain Ave, Murray Hill. NJ 07974.
of the adsorbates and, hence, the S H G signal are greater than or comparable to that of the substrate. However, in the case of r ~ ~ ~ ~ the adsorption on smooth metal8 and s e m i c o n d u c t ~ surfaces, ( 1 ) Shen, Y. R. In Chemistry and Structure at Interfaces, New Laser and Optical Investigations; Hall, R. B., Ellis, A. B., Eds.; VCH: Publishers Deerfield Beach, FL, 1986; and references therein. (2) Shen, Y.R. J . Vac. Sci. Technol., B 1985, 3, 1464. (3) Heinz, T. F.; Chen, C. K.; Ricard, D.; Shen, Y. R. Phys. Rev. Lett. 1982, 48, 478. (4) Heinz, T. F.;Tom, H. W. K.; Shen, Y. R. Phys. Rev. A 1983,28, 1883. (5) Tom, H. W. K.; Heinz, T. F.; Shen, Y. R. Phys. Rev. Lett. 1983,51, 1983. ( 6 ) Rasing, Th.; Shen, Y. R.; Kim, M. W.; Valint, P., Jr.; Bock, J. Phys. Rev. A 1985, 31, 537. (7) Rasing, Th.; Shen, Y. R.; Kim, M. W.; Grubb, S . G. Phys. Rev. Lett. 1985, 55, 2963. (8) Tom, H. W.K.; Mate, C. M.; Zhu, X. D.; Crowell, J. E.; Heinz, T. F.;Somorjai, G.; Shen, Y. R. Phys. Rev. Lett. 1984, 52, 348.
0022-3654/88/2092-1419$01 .50/0 0 1988 American Chemical Society
