Phase Transitions in Quasi-Two-Dimensional Molecular Solids. A

Phase Transitions in Quasi-Two-Dimensional Molecular Solids: Tilt-Induced Elastic Multipoles and Their Interaction and Ferroelasticity in Langmuir Mon...
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J. Phys. Chem. 1995,99, 8872-8887

8872

Phase Transitions in Quasi-Two-Dimensional Molecular Solids. A Microscopic Theory of Tilt and Structural Instabilities in Langmuir Monolayers Tadeusz Luty*tt and Craig J. Eckhardt” Department of Chemistry, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0304 Received: January 12, 1995@

A microscopic theory of orientational and structural instabilities associated with Langmuir monolayer phases is developed based upon methods of solid-state analysis. Surface harmonics are used to describe molecular orientational fluctuations providing a concise relation to the symmetries of the net. An explicit treatment of the translational-rotational coupling permits calculation of indirect molecular rotational coupling that is shown to lead to orientational and structural instabilities that are related to the nature of the intermolecular interactions and manifested by net symmetry, tilt, etc. Transitions from the highest symmetry “superliquid” phase are shown to be driven by an elastic instability which induces lattice strain. A mechanism for coupling the lattice strain to molecular tilt is developed. These results permit a consistent explanation of the formation of the Langmuir monolayer phases of n-alkanoic acids. It is demonstrated that some of the phase transitions may be viewed as ferroelastic. The thermoelastic properties of Langmuir monolayers are calculated and the free energy is derived as a function of orientational order parameters.

1. Introduction The phase behavior and textures of molecular monolayers at the air-water interface, Langmuir films, are rich and rival those of three-dimensional systems.’ These phenomena have not been investigated as extensively for the Langmuir-Blodgett films where the Langmuir film is transferred to a solid substrate. It is a matter of interest to inquire into the nature of the intemolecular forces that lead to such fascinating thermodynamical processes. The detailed determinati~n~-~ of the phase behavior of these systems may be attributed to the advent of new techniques for their study: synchrotron X-ray diffra~tion,~,~ atomic force microscopy: second harmonic generation,* and Brewster angle9 and polarized fluorescence2 microscopies. These methods have significantly augmented the more traditional surface pressure @)-area (A) isotherms.Io In particular, recent X-ray diffraction experiments4.’ have shown that Langmuir monolayer phases exhibit a variety of structures. Generally, monolayer phases may be divided into two groups: (1) mesophases (hexatics) that display long-range orientational and short-range translationalorder and are observed in higher temperature regimes, and ( 2 ) crystalline phases that have long-range translational order and are seen at low temperatures. Grazing incidence X-ray diffraction experiments’ have shown the monolayers to be crystalline in both compressed and uncompressed states. These crystalline films, essentially 2-D “powders”, are weak X-ray scatterers,’I and distinction between the “powder” and mesophases is neither easy nor definite. The analogy of monolayer phases to smectic liquidcrystalline phases is the current paradigmI4 with significantly less attention given to the early ob~ervation’~ of great similarities to crystalline phases. The current view is that the phase behavior of a monolayer displays mesomorphic and solid states and that the subtle and almost continuous changes between these phases even admit amorphous states. This is based on monolayer phase behavior of “classical” fatty acid amphiphiles which are said to reveal most subtleties in Langmuir film phase b e h a ~ i o r . ~

’ Permanent address: Institute of Physical and Theoretical Chemistry, The Technical University of Wroclaw, Wroclaw, Poland. @Abstractpublished in Advance ACS Abstracts, May 1, 1995.

Temperature (T) Figure 1. Schematic phase diagram for n-alkanoic acids. The heavy phase equilibrium lines represent determinations by X-ray, optical, and surface isotherm measurements. The light phase equilibrium lines have been located by X-ray and optical methods. The broken phase equilibrium lines have been found only by X-ray measurement. Italic letters denote those phases that require further confirmation but which are consistent with the proposed theoretical development.

Although there are many factors that may lead to detailed structure in the phase diagram arising from differences in translational and orientational order, predominant are the chemical nature, shape, and flexibility of the amphiphiles. This is exemplified by the striking difference in crystallinity at room temperature between amphiphiles comprised of alkane chains and those that are perflu~rinated.~~.’~ This has been attributed to higher rigidity and interchain van der Waals interaction for the latter.I8 Moreover, for the homologous series of n-alkanoic acid molecules, the extent of two-dimensional (2-D) crystalline order is larger for systems with more attractive lattice energies.I8 Thus, crystallinity is favored by the longer molecules. A schematic phase diagram for n-alkanoic acids is presented in Figure 1. It represents a rather accepting c~mpilation’~ based on a variety of measurements. Some phases, denoted by italics, require confirmation but are included because most are consis-

0022-365419512099-8872$09.00/0 0 1995 American Chemical Society

J. Phys. Chem., Vol. 99, No. 21, 1995 8873

Phase Transitions in Quasi-Two-Dimensional Solids

TABLE 1: Characterization of n-Alkanoic Langmuir Monolayer Phases of Figure '1 phase

lattice

backbone order (related to T scale)

LS

hexagonal rect/hex" rect/hexa rect/hexa rectangular rectangular rectangular rectlobliqued rectlobliqued rectlobl iqued

rotational disorder rotational disorder rotational disorder rotational disorder static disorder" parallel order' parallel ordef herringbone order herringbone order herringbone order

ov LZd

LI'

S LZ* LZh

cs S' LF

tilt order (related to n scale)

vertical NNN NN NNANNN

vertical NNN NN

vertical NNN NN

" Molecules form a hexagonal, close-packed lattice in the plane perpendicular to their long axes. Molecular backbones are in two equivalent orientations in every lattice site. c Molecular backbones are ordered parallel to each other. It is not certain if the herringbone order of molecular backbones is realized on a rectangular or oblique lattice. e Italics indicate unconfirmed phases that do not appear in the phase diagram of ref 19.

n

Figure 2. Representation of the axes, angles, and directors used in description of the orientation of rigid rod model n-alkanoic acid molecules. x, y , and z define the Cartesian coordinate system with polar angles 8 and 4. u is the angle formed by a crystallographic hexagonal "bond axis with the x-axis.

tent with the theory developed herein. The structural information for the represented phases is displayed in Table l. This phase diagram can be further partitioned according to the method of measurement employed in the determination of each phase. Phases LS, L2, L2', L2", and CS are found by surface-isotherm studies. All these phases and two others, Ov and S, have been observed by optical methods such as Brewster angle microscopy and polarized fluorescence mi~roscopy.'~In addition, the three phases, LI', L2&and L2h within the L2 phase have been observed by X-ray scattering? According to ref 1, the phases S' and LQ* within the L2' phase have also been seen in X-ray diffraction experiments. To better characterize these phases, and for a clearer distinction of theoretical models relevant to the ensuing discussion, introduction of angles and directors specifying the orientation of molecules in the monolayer is desirable. These are defined in Figure 2 where the molecular chain is modeled as a rigid rod. In addition to two polar angles (e, #), there is the angle v formed by one crystallographic hexagonal axis (a socalled "bond") with the x-axis of the coordinate system. The latter has been introduced to characterize hexatic order, and it plays an important role in the textures of mesophases*O and

monolayers.2' It may be neglected in characterizing the aforementioned phases, i.e., by assuming hexatic rigidity, v = 0. In the LS phase the molecules are, on average, not tilted ((e) = 0), and indirect evidence suggests that the molecules are almost free rotors.' I Mesophases (differing in the orientation of the c director (see Figure 2)) have (e) 0 and are characterized by the azimuthal angle 4. In the L1' phase: (4) = fa m(&/6), with a an angle intermediate between 0 (molecules tilted in the direction toward nearest neighbors (NN), e.g., L2h and L2d phases) and n/6 (molecules tilted toward next nearest neighbors (NNN),e.g., Ov and &* phases). A distinction between subphases within the L2' and L2 phases is based on the order of molecular backbones (see Table 1). Although not conclusive, transitions between LS and L2 phases are believed to be continuous.' In the low-temperature regime, phases possess a higher degree of crystallinity. The S phase contains vertical molecules, thought to be orientationally disordered, on a distorted hexagonal lattice which is, more precisely, a centered rectangular net.' I The CS phase shows long-range translational and orientational order of the spines of the molecule^,^ but the pattern of the molecular backbones is not uniquely determined. It is believed that in the low-temperature regime the molecules are packed in the herringbone pattern analogous to structures in 3-D solids. Thus, lowering the temperature has the effect of reducing the symmetry of the high-pressure phase from hexagonal (LS) to centered rectangular (S) and finally to the possibly oblique superstructure (CS). This symmetry reduction is, no doubt, related to translational-rotational coupling since the molecules become hindered rotors with decreasing temperature. The most crystalline phase (CS) may even be compared to the 3-D crystalline structure of alkanes well below their melting points where their molecular backbones are locked into a herringbone pattern." The order-disorder transition from the S to the CS phase is completely analogous to the orientational transitions in molecular solids. The generalized phase diagram presented in Figure 1 can be viewed as resulting from two tendencies in the thermodynamic behavior of the system. First, with decreasing temperature there is an increasing order of molecular backbones and the system passes from a rotationally disordered state to herringbone order through a centered rectangular lattice. Second, the effect of increasing 2-D pressure is to suppress tilting of molecules and the system evolves from NN tilting to the vertical structure by passing through a state with N" tilting. The two tendencies in conjunction with the principle of continuity provide a conceptual basis for understanding formation of all phases, with the exception of LI', represented by the phase diagram. Molecular dyr~amics~~-~O and Monte C a r l ~ ~ simulations I-~~ have been performed for Langmuir monolayers. These have involved a variety of approximations, ranging from conceptualization of the monolayer as a system of rods grafted to a surface, through the inclusion of different interactions such as chain-chain and chain-surface, tb different modelings of intermolecular potentials ranging fkom the idealized to the realistic. Most of these numerical experiments concentrated on hexatic phases with results supportive of the collective tilt measurement of experiment but with less attention being paid to the hexagonal lattice distortion. It yet remains for studies to focus on the microscopic aspects of the stabilities, structures, and transitions of the Langmuir monolayer phases. Computer simulations using realistic potentials offer much more accurate descriptions of monolayer ordering for specific systems than do microscopic and mean-field theories, but they

*

+

8874 J. Phys. Chem., Vol. 99, No. 21, 1995

Luty and Eckhardt

cal Landau free-energy expansion using three order parameters cannot yield the powerful analytical solutions available from to describe the order and symmetries of the phases represented the latter and thus fail to provide the broad conceptualizations in the fatty acid phase diagram.I9 An order parameter involving and trends desired for understanding the phase behavior of 9 (eq 1.2) was used to describe the collective tilt of molecules Langmuir monolayers. Moreover, such theories are essential along with two others that describe the so-called onefor better interpretation of both physical and computer experidimensional “weak crystallization”. In this context, we shall ments. show that spherical harmonics offer a natural and logical set of In an extensive study comparing intermolecular forces of variables to define vector and higher-rank tensor order paramLangmuir monolayers of perhydro and perfluoro amphiphiles, eters for Langmuir monolayer phases. Their Fourier transforms Cai and R i ~ e developed ~ ~ - ~ a~molecular theory based on density may be exploited to describe the superlattice ordering of the functional formalism to describe transitions between untilted monolayer crystallinephases instead of the “weak (LS) and tilted hexatic phases. Numerical calculation^^^^^^ using crystallization” order parameters. realistic Lennard-Jones potentials for reasonably strong moleculeThis study is aimed at a further development of the surface interactions have shown that the hexatic phase with microscopic theory of Langmuir monolayer phases and their vertical molecules is the most stable. Thus, a delicate balance transitions by viewing them as essentially solid-state phenomena between the potential and chain-chain interaction dictates the and describing them by exploiting their full symmetry. The tilting characteristics for the hexatic phases. The nature of the purpose is to develop an elaborated microscopic theory that transitions depends critically on the intermolecular interactions. provides a consistent framework for understanding Langmuir For perfluorinated amphiphiles, only a first-order phase transiThree strategies are used to achieve this. First, tion between an ordered and disordered dilute phase is f o ~ n d ’ ~ , ~monolayers. ~ we consistently use surface harmonics to describe fluctuations with no evidence of a continuous tilt transition. However, a in molecular orientations. This produces a quite compact continuous tilt transition is commonly found for perhydro treatment of the rotational degrees of freedom, allows description systems.’ The crystallinity of Langmuir films of molecules of orientational fluctuations of any symmetry or magnitude, and without polar groups38 may be attributed to the assumed gives a consistent and logical set of order parameters for the increased rigidity of perfluro chains over that of perhydro chains. phase transitions. Second, we explicitly treat the translationalThe model of grafted rods, where molecules are approximated rotational coupling, calculate effective rotational interactions, by rigid, rodlike particles attached to a planar, impenetrable and analyze possible orientational instab es from generalized surface has been most extensively s t ~ d i e d . ~The ~ - ~discrete ~ rotational susceptibilities. Third, we calculate thermoelastic version of the model, spin- 1 Ising-variable Hamiltonian, has properties and find the free energy of the system in terms of been solved by mean-field39 and renormalization-group meththe orientational order parameters which are the average values o d ~ The . ~ results ~ deal with competing roles of interparticle of the surface harmonics. Contrary to all current Langmuir and particle-surface interactions but are limited to transitions monolayer theories which are formulated from a liquid state between LS and isotropic liquid phases. This model, augmented viewpoint, our approach may be identified with solid-state by continuous orientational variables, has been proposed by methods. We exploit physical concepts employed in describing Somoza and DesaLU translational-rotational coupling in molecular crystals with Consideration of the problem of order parameters used in orientational d i ~ o r d e r . ~This ~ - ~approach ~ may, on first conthe theoretical studies is crucial, and it is useful, at this point, sideration, appear to contradict the idea of making comparisons to emphasize their importance to extant theories and to that to mesophases. However, upon recognizing that all Langmuir subsequently developed here. As previously noted,# and despite monolayer phase transitions are related to orientational fluctuaa large number of models and theories, orientational order in tions, the need of a powerful method to describe these must monolayers is usually studied by imposing azimuthal symmetry first be identified. It is this that generates the similarity to and examining the nematic order parameter, which, following orientational disorder in solids. When we approach the hexatic the standard formulation$5 has the form, phases and their transformations with this point of view, we concentrate on similarities between the mesophases and the 2-D solid state rather than their differences. We begin by defining the Langmuir monolayer as a system In the context of Langmuir monolayers, the Q 3 3 component of close-packed rigid rods of hexagonal symmetry. The ((3 cos2 8 - I)), is not the symmetry-breaking parameter and potential energy is then partitioned explicitly for a singlethe 2-D version of eq 1.1 is used to define the symmetrymolecule orientational potential and intermolecular couplings: breaking order p a ~ a m e t e r . ~We ~ - ~shall ~ show that this limitarotational-rotational, translational-rotational, and translationaltion has no justification in microscopic considerations. Models translational. Every part of the potential is derived for hexagonal which have made the orientational degrees of freedom dissymmetry using the five lowest surface harmonics to describe rete^^.^^ have used the nematic order parameter and sought the the orientational fluctuations. The coupling matrices are so-called biaxial order associated with 2-D nematics where the expressed as Fourier transforms thus forming a dynamical matrix tilt order of molecules was neglected. Thus, Somoza and for the system. With the expression of the translationalDesaiU focused on tilt order but neglected the biaxial order. translational part in terms of elastic constants, the effective They introduced the tilt order parameter,44 orientational potential is found. With this potential we calculate the rotational susceptibility matrix and analyze possible orienq = (sin 8 cos 4) (1.2) tational instabili ed to different orientational fluctuations. This arises naturally from employing spherical harmonics for Initially, the ins for no translational symmetry breaking the description of a molecular orientational probability distribuare discussed and expected symmetry changes are predicted. tion function.44 We too shall develop the description in terms Subsequently, we consider orientational instab of spherical harmonics and show their convenience and power lational symmetry breaking and conclude with phase transitions for both microscopic theory and the free-energy expansion. to a superlattice. We show how the predicted orientational instabilities may drive phase transitions between the different Kaganer and c o - w ~ r k e r s ~ have ~ - ~developed ~ a phenomenologi-

Phase Transitions in Quasi-Two-Dimensional Solids phases found in the experimental phase diagram. Next, we calculate the thermoelastic properties of the monolayer system and show how the structural (elastic) instabilities are driven by the orientational ones. The temperature-dependent elastic constants are calculated. Then we derive the expression for the free energy of the monolayer in terms of primary order parameters-the averages of the surface harmonics. Finally, we show how the phenomenological free-energy expansion follows from microscopic considerations. The conclusion stresses how this paper contributes to the general theory and understanding of Langmuir monolayers. 2. Model Building

We shall derive a model which will take into account several features of the Langmuir monolayers relevant to the phase diagram of the system. The following points are employed in the model-building procedures. (i) The phase transitions result from instabilities which follow from a competition between intermolecular interactions. We shall identify the competing interactions and, in particular, we shall focus on direct and indirect (lattice mediated) rotational interactions. (ii) Rotational degrees of freedom are important variables in the system, and we choose to represent them in terms of surface harmonics. (iii) The sites of the 2-D lattice are noncentrosymmetric by the very nature of the Langmuir monolayer, and various translationalrotational couplings appear. We suggest that such coupling is a driving force for orientational and structural instabilities in the system. The Langmuir monolayer model system is defined as a hexagonal planar lattice with amphiphiles perpendicular to the net. The molecules are attached to the impenetrable lattice at the sites. Because we want to keep our model reasonably general, the detailed structure and chemical nature of the molecules will be neglected. In principle, the cross section of the rigid rod molecules mimic reasonably well the structure and chemical nature of a particular m~lecule.~’ The close-packing structure of such a Langmuir monolayer has c6v point group symmetry. This symmetry reflects the local environment around a molecular site which is the symmetry of the hexatic (LS) phase (see the Appendix for details) with one molecule in a primitive unit cell of 2-D space group p6m.55,56The crystalline model of the hexatic phase assumes hexatic rigidity (the “bond” angle LJ = 0, Figure 1) being mindful that it is only a reference structure, a network. The crystallinity of a phase depends on the correlation length which is determined by intermolecular couplings and temperature. At nonzero temperature, fluctuations destroy the long-range order and regions of uniform crystallinity will extend only over a finite correlation length. The lengths, L, and 4,are, in general, different and vary from tenths to tens of thousands the size of the hexagonal lattice constant.’’ The correlation lengths can be used to define a discrete set of wave vectors, q, in reciprocal space. The final size of the crystallinity of the hexatic phase, which serves as the reference structure, will then give a particular meaning for the special points of the Brillouin zone defined for the lattice (see the Appendix). Let us denote by X ( k ) the position of the kth molecule on the 2-D lattice. This is where the head group of a molecule is grafted to the impenetrable subphase lattice. Momentary deviations, u(k), are described relative to R(k), a well-defined equilibrium position,

X ( k ) = R(k)

+ u(k)

The orientation of a rigid-rod molecule is given by Q(k) which contains the polar angles, 8 and 4 (see Figure 2).

J. Phys. Chem., Vol. 99, No. 21, 1995 8875 The potential energy of the system is

2k

k‘

and is approximated by the molecule-molecule pair potential,

(2.3) This potential can be modeled by any kind of (semi)empirical atom-atom potential, where details of the molecules are taken into account, or by nonrealistic potentials, where molecules are represented by some geometrical objects, as is often done in computer experiments. We expand the potential in (2.2) to second order in translational displacements. This harmonic approximation for the displacements is incorporated into the potential,

+

+

v = VR VTR vT (2.4) written as the sum of a purely rotational part, VR, a translationalrotational part, VTR (first-order in the displacement, u), and a translational part, as the second-order term. We shall derive these three contributions to the total energy. A. Rotational Potential. The rotational potential corresponds to the zeroth-order term in the expansion in (2.4) with respect to translational displacements, u. Therefore, for the rotational potential we have,

v,

which describes the interaction of molecules in orientations, Q ( k ) and Q(K), that are at equilibrium positions on the lattice. The potential is further decomposed into the single-molecule orienting potential, (the term with k = k‘ in the summation, eq 2.5), and the rotational interaction,

6

where k

The single-molecule orientational potential is the sum of orienting potentials experienced by a molecule at site k when all surrounding molecules are kept in their equilibrium positions, R(k’), and with orientations, Q(k’) = (8=0, @O). This potential contains a contribution from the subphase of the monolayer. The single-molecule potential, eq 2.7, possesses full hexagonal symmetry ( c 6 v ) . The most convenient way to specify the potential, #, is to expand it in terms of spherical harmonics57 which transform according to the totally symmetric representation of the c6v point group. In principle, one has to expand the single-molecule orientational potential in terms of symmetry-adapted rotator functions. (For the theory of symmetry-adapted rotator functions that were originally introduced for solid methane?* see refs 59-61.) The symmetry-adapted rotator functions transform according to irreducible representations of the product of the site group and point group of a molecule.60 This becomes important for nonlinear molecules when one wants to take into account, explicitly, the symmetry of a molecule. It has been shown60 that for the case of a linear rod molecule (Cmv symmetry), the symmetry-adapted functions depend on the spherical coordinates of the long molecular axis and, therefore, they are just surface harmonics57 relative to the site group.

Luty and Eckhardt

8876 J. Phys. Chem., Vol. 99, No. 21, 1995 For molecules in a Langmuir monolayer, the vector of a molecular axis spans only the upper half space ( z > 0) above the surface. As pointed out by Somoza and DesaiU an expansion of the rotational potential with the full set of spherical harmonics, Yl,m(8,4), leads to a problem of redundancy or overcompleteness. Thus, following their suggestion, we restrict the set of spherical harmonics in problems of Langmuir monolayers to those for which

+

e.g., limited to harmonics with I m = even. For the site symmetry, c6v, the single-molecule orientational potential is written as

e

+

where = (Yl,6 ~l,-6)/2/2and al,P I are the coefficients of the expansion. More explicitly, the orientational potential (2.9) is written in the form

(E,)

G(Q)= a. + C a, sin2, e + n = l , ...

(

trostatic, induction, dispersion, and repulsion interactions. For long molecules it is convenient to partition the potential into attractive and repulsive parts and relate the latter to the excluded area,U A[Q(k),Q(k’)] (the area excluded by molecule k in orientation 8 ( k ) as seen by another molecule k’ in orientation R(k’)). Consequently, the rotational-rotational coupling constants will be a sum of the corresponding contributions. At this point we specify the set of surface harmonics describing the rigid rod’s orientational fluctuations. Since there is no symmetry constraint imposed on the set, the number of these variables depends on how much rotational freedom one expects for the molecules, i.e., on how strong the singlemolecule orientational potential is. For Langmuir monolayers we do not expect a very strong single-molecule orientational potential and we would like to keep the formalism reasonably clear, so we limit consideration to surface harmonics up to 1 = 2. Following the constraint, eq 2.8, we take the following surface harmonics to represent variables of the orientational fluctuations of molecules,

C b, in^+^^ 6) cos 6 4 + ... (2.10)

Y, = (Y1J

+ YI,-,)/2/2 = cx

(2.15)

(E,)

y2 = -i(Yl,l - ~,,-,)/2/2= cy

(2.16)

(A,)

Y3 = Y2,0 = c’(2z2 - x 2 - y 2)

(2.17)

(E2)

Y4 = (Y2,2 Y2,-2)/2/2 = c”(x2 - y2)

n = l , ...

The orientational probability distribution, P(Q), for a molecule at the c 6 v symmetry site is

P ( Q )=

exp[-P$(Q)I

(2.11)

+

(2.18)

where P = ( k ~ Z ‘ - l and the single-molecule rotational partition function is where c = (3/47~)”~, c’ = (5/167~)’/~, c” = (15/16n)’/*, irreducible representations of the c 6 v point group are given, and the Cartesian coordinates are x = sin 8 cos 4, y = sin 8 sin 4, For a strong orientational potential, eq 2.9, molecules are and z = cos 8. This set of surface harmonics is representative localized in quasidiscrete states (pocket states62)which often for the problem and contains the lowest surface harmonics are further approximated by discrete states. The spin-1 model Hamiltonian considered for the Langmuir m o n ~ l a y e rcor~ ~ . ~ ~ important to the description of different symmetries of the orientational fluctuations. The functions, Y1 and Y2, belong to responds to such an approximation. By keeping the orientational the doubly degenerate EI irreducible representation of the c6v potential expressed in terms of spherical harmonics, we allow point group and transform as the x and y components of a vector. a continuous change in molecular orientation. The function, Y3, belongs to the totally symmetric representation The rotational-rotational coupling term, the second in eq 2.6, A1 and transforms as the symmetric components of a seconddescribes direct coupling between orientational fluctuations of rank tensor. Finally, the functions, Y4 and Y5, belong to the different molecules ( k t k’). Expressing the fluctuations in doubly degenerate E2 representation and transform as the terms of surface harmonic^,^' we write the interaction in the nonsymmetric (deviatoric) part of a second-rank tensor. The form symmetry properties of the harmonics are important because they permit relation of every observable property, vectorial or (2.13) tensorial, to statistical averages of the corresponding surface ktk‘ harmonics. We shall refer to this when discussing order where summation over repeated indices is assumed. The vector, parameters. Y1 and Y2 are also used to represent the angular Y(k){Y,[Q(k)]}, represents a set of surface harmonics that distribution of atomic orbitals, px and py, respectively, while describe orientational fluctuations of the kth molecule. J(kK), Y3, Y4, and Y5 are similarly associated with the d,z, dX2+ and with elements, Jap(kk’), represents the matrix of rotationaldq orbitals, respectively. This analogy will be helpful as the rotational coupling constants which couple the ath harmonic discussion proceeds. of the kth molecule to the Pth harmonic of the Kth molecule. With the set of surface harmonics, eqs 2.15-2.19, we shall Equation 2.13 is a condensed notation for a more general form be able to define orientational order parameters of A1 (nonof the interaction potential between molecule^.^^^^^ The coupling symmetry-breaking) and E1 and E2 (symmetry-breaking) symconstants, Jap(k,K), can be calculated from an assumed intermetries. Still, for a more complete set of surface harmonics, molecular potential between two molecules, k and K , and, using and the order parameters corresponding to them, the surface the orthogonality properties of the surface harmonics, harmonics, Y6 = (Y3,3 f Y3,-3)/J2 Of Bl symmetry, Y7 = -i(Y3,3 Jd(k,k’) = JdQ(k)JdQ(k’)V[R(k) - R(k’);Q(k),Q(k’)] x - Y3,-3)/2/2 Of B2 Symmetry, and Yg = -i(Y6,6 - Y6,-6)/2/2 Of A2 symmetry, need to be included. The surface harmonic, Y8, Y,[Q(~)lYp[Q(k’)l (2.14) is a natural choice for the chiral order parameter. In this paper, The coefficients, Jap(kk’), consist of contributions from elecwe limit ourselves to the set given by eqs 2.15-2.19.

J. Phys. Chem., Vol. 99, No. 21, 1995 8877

Phase Transitions in Quasi-Two-Dimensional Solids

nearest neighbors, and denoting the constants between molecules located at (0,O) and (a, 0) as Via, we construct the V(q) matrix (eq 2.301,

We introduce the Fourier transforms,

and k’

where q is the wave vector for the simple hexagonal planar lattice. There is a finite set of wave vectors determined by the extent of crystallinity of the reference phase. For such a set, the Fourier transform in eq 2.20 is equivalent to the “weak crystallization”~arameter.“~-~* The direct rotational-rotational part of the energy in reciprocal space is (2.22) The matrix, J(q), can be calculated for six nearest neighbors by symmetry arguments (see the Appendix) and the elements of the matrix are specified in the Appendix. For q =. 0 the rotational-rotational matrix is diagonal as required by c 6 v pointgroup symmetry. B. Translational-Rotational Coupling. The translationalrotational coupling term, is the first term in the expansion of the total potential with respect to translational displacement, u, eq 2.4. It is

vR,

VTR= C C e [ R ( k ) - R(K);Q(k)]ui(K) k

+

+

wherefl(q) = cos 2 a 2 cos a cos /3 - 3, f z ( q ) = i(sin 2 a sin a cos /3), andf3(q) = i(sin a sin /3), with a = (1/2)q# and /3 = ( J 3 / 2 ) q ~ . The character a is the hexagonal lattice constant, and qx and qy are components of the wave vector in the orthogonal axis system (see the Appendix). In the limit q 0, there is no translational-rotational coupling for harmonics Y1 and Y2. This is due to the translational invariance of the system potential which requires &vja(kk’) = 0 for every (ia) c0mponent.6~ At q 0, the system has c6v symmetry and the orientational fluctuations of type E, (YI and Y2 harmonics) can couple bilinearly to a displacement vector, u, which transforms also as the El representation. This coupling, a force, is compensated when the system is at equilibrium. At q t 0, the system has lower symmetry than c 6 v and fluctuations of type E2 and A1 can couple bilinearly to the molecular displacements. In the limit q +. 0, the translationalrotational coupling matrix is approximated as -E)

+

(2.23)

k’

q,

where stands for the ith component of a force acting between the k and K molecules at the equilibrium distance, R(k) - R(K), when molecule K is in the equilibrium orientation, Q(K),and molecule k is in the orientation Q(k). The force is calculated as

Better insight into the physical meaning of the translationalrotational matrix in the limit q 0 can be gained from elasticity theory and the elastic dipole concept and will be the subject of a separate contribution. C. Translational Potential. The purely translational part of the system potential is conveniently expressed in terms of a translational-translational dynamical matrix, M(q), =+

and represents an angular distribution. Therefore, we express the force in terms of orientational fluctuations of the kth molecule, e.g., in terms of the surface harmonics,49

0). This is a general result that is independent of the assumption of the nature of the intermolecular potential. At To(E1) the system becomes simultaneously unstable against both fluctuations described by Yl and Y2. We define the order parameter for the transition driven by this instability as

+

V(EJ = a,(Y,) + a2(Y2)

(3.21)

e.g., a linear combination of parameters (Yl) and (Y2). For the general case, al,a2 f 0, ~ ( E Imeasures ) the collective tilt of molecules in an arbitrary direction, as seen from the more explicit form of the order parameter: ~ ( E I=) &in O)(al(cos 4) &in 4)), see eqs 2.15 and 2.16. Since the instability, eq 3.20, is due to competition between the single-molecule orientational potential and direct rotational-rotational interaction, translational degrees of freedom do not contribute and there is no lattice deformation involved. The phase which would result from this instability will contain tilted molecules on the 0, the hexagonal, undeformed lattice. When al and a2 symmetry change at the transition is c 6 v * cl,with the resulting phase reminiscent of L1' on the phase diagram, Figure 1. For al = 1, a2 = 0, and ~ ( E I=) (Yl), molecules in the corresponding phase are tilted toward NN in an angular distribution analogous to a px orbital. For al = 0, a2 = 1, and ~ ( E I=) (Y2) molecules are tilted toward NNN. The symmetry change in both cases is c 6 v * c, with the phase suggestive of L2d ( ( Y I ) )and o v ((Y2)) in the schematic phase diagram, Figure 1. This designation of phases is not final since the observed tilt phases are all on a deformed lattice and a coupling with a lattice deformation will make the assignment more specific. Phenomenologically, one can understand a coupling of El type tilting with a deformation as a breaking of the hexagonal symmetry. The susceptibility matrix, eq 3.19, then contains offdiagonal terms proportional to the symmetry-breaking defonnation e, and (exx - cry) components of the strain tensor. Consequently, the degeneracy of the tilting instabilities of El type is lifted and discrimination between tilts toward NN and NNN is obtained. The deformation of the lattice will decide which of the instabilities will occur first; this can be analyzed from the 2 x 2 susceptibility matrix. The conclusion is that the elasticity of the system will determine which of the instabilities will take place. Therefore, the interaction of the molecules with the surface can be important.

+

*

where 6 = 3m(CPl)-'. For the orthogonal direction, the matrix is

Introducing these matrices into eq 3.14 and calculating the inverse of the orientational susceptibility, X-'(q+O), we find that the matrix is diagonal in Y I ,Y2 (E1 symmetry), and Y5 (E2 symmetry) and well-defined for q =+ 0. There is a 2 x 2 matrix on the diagonal which corresponds to a coupling (a hybridization) of Y3 (AI symmetry) and Y4 (E2 symmetry) surface harmonics, and this matrix depends on the direction by which the q * 0 point is approached (see eqs 3.17 and 3.18). Therefore, there are three possibilities for which our model of the Langmuir monolayer will become unstable without translational symmetry breaking: (i) orientational fluctuations of El symmetry, (ii) orientational fluctuations of E2 symmetry, and (iii) orientational fluctuations of hybridized AI and E2 symmetries.

(3.19)

Luty and Eckhardt

8880 J. Phys. Chem., Vol. 99, No. 21, 1995 Next, we discuss the instability of the Langmuir monolayer model system with respect to the orientational fluctuation of E2 symmetry without translational symmetry breaking. This instability is expected when (X(q*O))i; = 0, and its temperature is calculated as

corresponding to this is

(xi; G;) x-l

=

X-'(q+;A,,E,)

33

x-l

34

(3.24)

where

This instability arises from a competition between the singlemolecule orientational potential which determines x 4 and the effective rotational-rotational interaction. The indirect, latticemediated coupling between orientational fluctuations, Y5, helps the system to gain energy when molecules are collectively reoriented in the pattern described by the surface harmonic, Y5. The energy gain measured by the first term in the square bracket of eq 3.22 is inversely proportional to the system elasticity (GI) and critically depends on the strength of the translationalrotational coupling. The indirect rotational-rotational interaction should be very important in the Langmuir monolayer systems and be responsible for the orientational instabilities. For systems where direct interactions are described by quadrupolar-like interactions, more specifically by the xy component of a quadrupole, (J44 J55) -= 0, the indirect interaction increases the temperature of the instability. If the instability of the E2 type takes place, the new phase is characterized by the order parameter,

+

= -6v13v14

x;6(qy*)

(3.28)

where we note the difference in the off-diagonal terms for approaching the q 0 (T-point) from the x and y directions. The instability condition, eq 3.16, -+

(3.29) determines the instability temperature, TO. At this temperature, the lowest eigenvalue of the matrix, eq 3.24, becomes zero and the system is unstable against orientational fluctuationsdescribed by the corresponding eigenvector, = Y3 cos v,

+ Y4 sin v,

(3.30)

when approaching the r-point from the x direction and and the symmetry change expected at the transition is c6v * C2L2.In this case, molecules are tilted collectively at every site with equal probability in the four directions, [ l , 11, [ l , -11, [-1, 11, and [-1, -13 (in the orthogonal axial system), and with the hexagonal lattice deformed according to the xy strain component. The deformation of the hexagonal lattice is involved in the E2 type instability because the lattice-mediated indirect interaction contributes to the instability, eq 3.22. For symmetry reasons, such a phase can be associated with the S region of the phase diagram, Figure 1. The pattern of orientational disorder assumed for phase S, due to two mutually orthogonal orientations of a molecular backbone, is completely similar to the pattern of tilt reorientations described by Y5. The harmonics, Y3, Y4, and Y5, transform as components of a second-rank tensor; therefore, every observable tensorial property will be described by its thermal averages. Quantitative relations for macroscopic strain components are derived at the end of the section. Here, we relate these averages to a mean cross section of a molecule. The cross section of a rigid rod which models a molecule can be represented as a 2-D second rank tensor similar to the nematic order parameter, eq 1.1 Therefore, the statistical averages of the surface harmonics, (Y& (Y& and (Y5), can be a measure of an average cross section of the rigid rod. Furthermore, this cross section would correspond to an average orientation of a molecular backbone. Physically, this means that a rigid rod with circular cross section represents a molecule rotating around the long axis, and the rotation becomes hindered by a hexagonal lattice deformation and/or tilt of the molecule. This is reflected in nonzero orientational order parameters with the tensorial ones interpreted as indicating the angular distribution of the molecular backbone. Finally, for instabilities without translational symmetry changes, we shall discuss an instability due to coupled, Y3 (AI) and Y4 (E& harmonics. The inverse susceptibility submatrix

tj = -Y3

sin v,

+ Y4 cos v,

(3.31)

for the y direction, where tan 2 p = 21X; l(XTi - Xi1)-'. The eigenvectors express hybridization of the orientational fluctuations described by the Y3 and Y4 surface harmonics. The order parameters corresponding to the instability are defined as statistical averages of the eigenvectors, VX(A,,E,) = (E?;

$(AI&> =

(tY)

(3.32)

Observe that the order parameters are orthogonal. It is the indirect, lattice-mediated,rotational interaction which makes the difference in the ordering of molecules, depending on the direction of approaching the r-point (see eqs 3.27 and 3.28). This is a well-known problem in solids with magnetic or electrical dipolar interactions. Here, the lattice-mediated, indirect, interaction described by the matrix L(q) represents the interaction of elastic dipoles.50 Therefore, we can expect a shape dependence for the interaction. Because a real system breaks up into domains, no shape dependence can be observed for the system as a whole. However, the domains would then have most favorable shapes, because in this way a system can lower its free energy. Detailed analysis of the shape dependence problem would require lengthy considerations and derivations, but some insight into the preferred shapes of domains can be developed. Clearly, eq 3.32 defines order parameters for two domains of the same phase. The domain with the qx order parameter is formed with relative dimensions 4, >> L,, and conversely, the domain characterized by the order parameter, q y , has relative dimensions L, >> 4%. In other words, the phase with the order parameter that is a hybrid between (Y3) and (Y4) will split into two kinds of domains with mutually perpendicular stripes as their favored shapes. Indeed, it has been found that monolayer crystallites are stripes elongated perpendicular to the molecular tilt direction." Every domain will be characterized by C2v symmetry

J. Phys. Chem., Vol. 99, No. 21, 1995 8881

Phase Transitions in Quasi-Two-Dimensional Solids since the (Y4) component is responsible for the symmetry breaking. Due to the lattice-mediated interactions which take part in the instability mechanism, the phase transition would be accompanied by a lattice distortion measured by a linear combination of (exx e,) (AI symmetry) and (en - e,) (E2 symmetry) strain tensor components. The discussion of orientational instabilities concludes with the case where translational symmetry is broken. Consider the direction C(0, qr) with the end point M(0,2x/43a) at the Brillouin zone boundary (see the Appendix). For this direction, the J(q)matrix, as well as the matrix of indirect couplings, L(q), is specified in the Appendix. It follows from the symmetry of the matrix, E(q), in the 2 direction that there is a coupling between YI and Y5 and between Y2, Y3, and Y4. Consequently, the inverse susceptibility matrix, X-I(C), is decomposed into 2 x 2 and 3 x 3 submatrices. For practical reasons, we analyze the 2 x 2 matrix corresponding to hybridization between YI and Y5 in the C direction. A related orientational instability will be determined by the lowest eigenvalue of the matrix,

+

1 A(X) = -{XI; 2

+ 5;- [(X,’

dependent) elastic constants. We define the translational susceptibility matrix,

The thermal averages of the displacement variables are calculated with the total Hamiltonian of the system, eq 2.37. On introducing eqs 3.1 and 3.2 into the above equation, we get

M-’(q) = P(w(q>w(-q)) +

1

If (3J22 - J11) < 0, then the instability at the M-point will be preferred only when 486V:, > 1(3& - J I I ) ~In . this case, the instability at the M-point is a result of successful competition between the indirect rotational interaction and the direct one. It would be favored for a sufficiently soft lattice which gives a large 6 coefficient. In the case (3522 - J ~ I>) 0, the instability related to the Y I orientational fluctuation will be preferred at the M-point, and the new phase will be characterized by the order parameter,

@IM)

= (Y,(M))

(3.38)

(see eq 2.15). This phase is characterized by the tilt of molecules in the direction of NN (x), with a modulation described by exp[iq(m)R(k)]. The modulation term may be interpreted as equivalent to the “weak crystallization” order ~arameter:~-~~ The transition to this phase results in a doubling of the unit cell and creation of a herringbone pattern of molecular backbones. The new phase can be assigned to L2” of the schematic phase diagram, Figure 1. The symmetry change would be p6m pg.56.57 C. Thermoelasticity and Structural Instabilities. Thermoelasticity implies a temperature dependence of the elastic properties of a system, and thus, we calculate effective (T +

+ c12)%qy

1 $cI

+ c12)qxqy

1

1

2

Tc66qx

+ cllq:

I

(3.42)

In principle, the effective elastic constants should be calculated from the exact equation, eq 3.40. However, this is tedious, and instead, we use an approximate equation which gives insight into the renormalization of the elasticity due to the translationalrotational coupling. We use, as an approximation to (3.40),

D(q-0) One then finds that the minimum eigenvalue is obtained for the wave vector, qy = 2nI43a, e.g., for the M-point which is the end of Brillouin zone in the C direction. At this point, Yl and Y5 are decoupled. Moreover, the indirect interactions are nonzero only for the YI and Y2 fluctuations. Since the interaction is larger for the Yl harmonic, the instability temperature is found from

1

1 2 I d + zc664y

T(cll

where the elements of the inverse susceptibility submatrix are

(3.34)

(3.41)

In the limit q 0, eq 3.40 gives a link between the rotational susceptibility, X(q), and the elastic constants in the presence of the translational-rotational coupling, V(q). Matrix D(q) has exactly the same form as matrix M(q) with bare elastic constants replaced by the effective ones. Without the central forces assumption, the matrix takes the form,

D(q-0)=

+ Jll(X)

(3.40)

where we have used

- x;;)2 + 4(x;;)211’2} (3.33)

- L,,(X)

+ Br(q)x(q)Br(-q)M-l(-q)l

D-’(q) = M-“

c1

x,’ = p-’x;’

(3.39)

D-’(q) = P[(u(q>u(q>)- (u(q))(u(q))I

2 M(q-0)

- v(q”o)x(q=o)v(q’o)

(3.43)

The rotational susceptibility, X(q=O), is determined by direct and self-interactions only and is diagonal as required by the symmetry of the c 6 v point group. The result of calculations for the effective elastic constants is

c,,= cy, - 9(X33V:, + x44v:4,

(3.44)

C6, = Cg6 - 1 8X,,V:,

(3.45)

C,,

+ C,, = (Cyl + C;12)- 18X3,V:,

(3.46)

where the rotational susceptibilities are

The susceptibilities increase with decreasing temperature consequently making the effective elastic constants smaller. The thermoelasticity makes the system softer with decreasing temperature, an effect totally due to the translational-rotational coupling. An elastic instability occurs when one of the eigenvalues of the effective dynamical matrix, D(q), eq 3.42, becomes zero. The diagonalization of the matrix for q = (qx, 0) and q = (0, qy)yields the same eigenvalues, and the lowest one corresponds to the c 6 6 elastic constant. Thus, the elastic instability in the system will be determined by the condition, c 6 6 0. This means that the system becomes unstable against shear strain of the 2-D lattice, e.g., the eq = e6 component of the strain tensor. For the hexagonal lattice, the relation, c 6 6 = (1/2)(c11- CIZ), implies that the hexagonal lattice also becomes unstable with respect to the (e, - e?)) = (el - e2) strain. Both strains +

8882 J. Phys. Chem., Vol. 99, No. 21, 1995

Luty and Eckhardt

transform according to the E2 irreducible representation and will cause a corresponding lattice symmetry change: c 6 v C2,,. The elastic instability will take place simultaneously with the orientational instability causing E2 type symmetry changes as discussed above. This is due to the translational-rotational coupling effect, and as a result, phases characterized by the orientational order parameters, as defined in eqs 3.23 and 3.32, will show lattice deformations measured by e6 and (el - e2) strains, respectively. They are equivalent to the orientational order parameters of E2 symmetry. D. Free Energy. From the total Hamiltonian of the system, eq 2.37, the free energy can be written as a sum,

F = F,

+ PR+ F~~ + F~

Finally, we derive the free-energy contribution in the mesoscopic limit (q -40). The corresponding parts of the free energy are (3.54)

F T -- 5 l[1 $Cyl

+ Cy2)(e, +

1 $Cy1 - C;,)(e, -

(3.49)

Here, FO is the single-molecule orientational free energy per molecule,

where the partition function, ZR,is calculated classically with the effective single-molecule orientational potential given by eq 3.7. The other parts of the free energy can be obtained by replacing the variables, u(q) and Y(q), in eqs 2.40-2.42 by their instantaneous thermal expectation values.61 The result is

Recall that Xo is the single-molecule orientational susceptibility and J(q) stands for the direct rotational-rotational coupling matrix. The average, (Y(q)), represents a vector of orientational order parameters. For a given orientational configuration, (Y(q)), we minimize the free energy with respect to the displacements, (u(q)), and obtain

Substituting back into eq 3.51 yields

+ + CE6e:]

(3.56)

The minimization of the free energy, given by the above equations with respect to the strain components, yields macroscopic strain as a response to the orientational order parameters,

+ e2) = -6(Cy, + Cy2)-1V13(Y3)

(3.57)

(el - e,) = -6(Cp, - Cy2)-'Vl4(Y4>

(3.58)

(el

(3.59) Note that eqs 3.57 and 3.58 are coupled as the averages; (Y3) and (Yd), are coupled in the "hybridized" order parameter, eq 3.32. Decoupling leads to relations el = V(A,,E2) and e2 = ~;IY(AI,E~). Moreover, eq 3.57 gives important information on the change of the area of the 2-D unit cell, AA = Ao(e~+ez), where A0 is the area of unit cell in the high-symmetry (f&) phase. In analogy to eqs 3.57-3.59, one can draw qualitative relations between the effective cross section of a rigid rod and the orientational order parameters of a second-rank tensor. This analogy is used in representing the effective molecular shapes in the phase diagram, Figure 3. On introducing eqs 3.57-3.59 into 3.55 and 3.56, we obtain the free energy in terms of the orientational order parameters, (Y), which characterize phases of the system without translational symmetry change. In the presence of an extemal 2-D pressure, E , we consider the Gibbs free energy per molecule,

G = F[e,(Y)]

+ x(el+e,)

(3.60)

where the first term represents the free energy expressed by eq 3.53. The equilibrium condition, eq 3.57, changes to where %(q) is the coupled orientational susceptibility matrix for the system defined by eqs 3.10-3.14. Equation 3.52 and 3.53 give the most general result: the free energy of the system is expressed in terms of primary orientational order parameters, (Y(q)). These order parameters are more general than those used so far. In particular, the statistical average of the Fourier transform of the spherical harmonics is equivalent to a product of the tilt order parameter and the so-called "weak crystallization" order ~ a r a m e t e r . ~This ~ - ~is~ clear when one remembers that the set of q vectors is determined by the extent of crystallinity defined by the correlation lengths (see the Appendix). Equation 3.52 gives translational displacements (as a measure of lattice distortion) which accompany the change in orientation of molecules, e.g., it gives the elastic response of the lattice to the primary order parameter. Instabilities were analyzed using the condition det(X-'(q)l = 0, since the microscopic nature of intermolecular interactions and mutual competition is contained in this quantity.

x = -(Cy1

+ CY2)(e, + e2) - 6V13(Y3)

(3.61)

This serves as an equation of state for the system. It reflects the physical behavior of the system in that the extemally applied pressure (a stress) is compressed by macroscopic stress due to the bare elasticity of the lattice (the first term on the right-hand side) and local stress due to reorientation of the molecules. The second term forms the diagonal, isotropic part of the local stress tensor, conveniently expressed as an elastic dipole tensor. The equation of state shows that only totally symmetric reorientations of the molecules, measured by (Y3), are 2-D pressure dependent. Therefore, only transitions to phases characterized by (Y3) 0, e.g., due to instability with respect to hybridized (AI, E2) orientational fluctuations, will depend on the pressure. The conclusion is that phase transitions driven by El type and E2 type tilt instabilities are weakly pressure dependent. When a nonuniform pressure is applied, it couples to el and e2 separately, and the local stress is due to reorientations expressed by the

*

J. Phys. Chem., Vol. 99, No. 21, 1995 8883

Phase Transitions in Quasi-Two-Dimensional Solids

Temperature (T) Figure 3. Schematic phase diagram of n-alkanoic acids with pictorial description of the orientational order parameters and symmetries of the phases. Vectorial (dipolar) order parameters are represented by arrows showing the tilt direction, and tensorial (quadrupolar) order parameters are designated by the effective molecular cross section.

order parameters AI,E2) and yV(Al,E$, respectively. More extensive discussion of this aspect will be provided in a forthcoming paper. 4. Discussion

The aim of this study has been to approach Langmuir monolayers from a solid-state point of view in order to understand the microscopic mechanisms which cause the richness of phase behavior of these systems. We have attempted to provide a uniform description of different phases through a logical choice of order parameters. The key variables in our theory are symmetry-adapted surface harmonics which appear to offer a guideline to yet unresolved questions. (i) What are the intermolecular interactions related to a possible phase (instability) transition? (ii) How should the phases of Langmuir monolayer systems be characterized? Starting from the highest symmetry phase, the LS phase of c6v point group symmetry, we have analyzed orientational instabilities which can cause different phase transitions. The instabilities are possible responses of the model system of the assumed symmetry. We have shown that the instabilities taking place depend, however, on the details of intermolecular interaction. Before we summarize the results, we introduce the following terminology: dipolar phases (instabilities) are those which are characterized by the order parameters, (YI) and (Y2), and which transform as vector components; quadrupolar phases (instabilities) are those which are characterized by order parameters related to (Y3), (Y4), and (Ys) and which transform as secondrank tensor components. The order parameters of quadrupolar tilt phases correspond to the 2-D nematic order parameter tensor components, eq 1.1.

Dipolar instabilities without translational symmetry breaking have been analyzed. The instabilities are caused by competition between the single-molecule orientational potential, which includes the molecule-surface interaction, and the direct rotational-rotational coupling. No lattice-mediated interaction is involved. Therefore, the instability does not cause any deformation of the hexagonal lattice. The dipolar tilt on the hexagonal lattice requires strong, attractive dipolar-like intermolecular interactions. This can be understood as the interaction of elastic counterparts of an electric dipole, e.g., as forces created by the tilted molecules. Thus, the dipolar tilt instability would be preferred for longer molecules. The proper order parameter for the transition is the linear combination of the expectation values of surface harmonics, (YI) and (Y2), which transform as the x and y components of a vector, respectively. For symmetry reasons, the E1 type, dipolar instability is isotropic in the hexagonal plane of the monolayer and there is no preferred direction for a molecular tilting where a precession of molecules is expected. The experimentally observed transitions from the LS phase to the tilted, deformed phases, therefore, must be initiated by quadrupolar instabilities causing the lattice deformations. Quadrupolar instabilities are of two types and are driven by both direct and indirect rotational interaction. The E2 type instability is driven by a coupling between the xy components of molecular quadrupoles. The instability involves hexagonal lattice deformation measured by the strain component, e6, and is due to the indirect interaction. The hybridized (AI, E2) type instability is driven by the coupling between diagonal, xx and yy, components of molecular quadrupoles. The instability involves hexagonal lattice deformations, (el e2) and (el -

+

Luty and Eckhardt

8884 J. Phys. Chem., Vol. 99, No. 21, 1995 e2). In fact, the deformations, e6 and (el - ez), can be considered as equivalent to primary order parameters for the quadrupolar phases. We propose the following mechanism for the tilt transitions from the LS phase. Phases Ov, L1’ and L2d (Figure 1) are assigned as dipolar phases. Phase L1’ may result from an instability with respect to Yl and Y2 orientational fluctuations. The order parameter for the phase is given by eq 3.21, and molecules are tilted on the hexagonal lattice. Because there is no preferred tilting direction, the molecules may show random tilt directions. Phases Ov and L2d have the molecules tilted towards N” and NN, respectively, and they are characterized by order parameters (Y2) and (Yl), respectively. The rotating molecules in all the dipolar phases have circular effective cross sections which allow retention of hexagonal packing in the plane perpendicular to the molecular long axes. L2h and L2* are deformed hexagonal phases. We suggest they are driven by a structural (elastic) instability, c 6 6 * 0 (see eq 3.48), causing e6 and (el - e2) deformations. Because the transition to these phases seem to be pressure dependent, we expect that a quadrupolar instability of the hybridized (AI,E2) type drives the transitions. The deformation, (el - e2) 0, a result of the instability, lowers the system symmetry and causes coupling to the dipolar order parameters. Molecular backbones are ordered on a centered rectangular lattice, and the molecules are tilted. For (el - e2) > 0, which represents contraction of the lattice in the direction NN, the dipolar interaction between molecules tilted in that orientation would become more attractive and the system would gain more energy by tilting the molecules in this direction. As a result, the phase L2h is established and characterized by the order parameters (el - e2) > 0 and (YI) # 0. The dipolar interaction between tilted molecules would favor the direction of lattice contraction along the NNN direction corresponding to the (el - e2) < 0 deformation. This generates phase L2* which is characterized by (el - e2) 0 and (Y2) # 0. We posit that the L2h and L2* phases appear as a result of ferroelastic domain formation and the transitions from the LS phase to the L2* and the L2h phases are a ferroelastic phase transitions. This assignment of lattice strains to these phases is particularly well-supported by recent studies68of structures and phase diagrams of Langmuir monolayers composed of mixtures of heneicosanic acid and heneicosanol. We take the “swiveling transition” observed as a 2-D pressure-driven68 transformation to be a “transition” between ferroelastic domains. From this perspective, it is interesting that fatty acids show this phenomenon while corresponding alcohols prefer to retain a NNN tilt direction and (el - e2) < 0 deformation. We attribute this to details of the molecule-surface interaction and thus to the 2-D elasticity of the lattice. The mechanism of the tilt transitions can, therefore, be understood by representing the tilted molecules by force vectors (zero-order elastic multipoles) in the system’s plane. The force is proportional to (Yl) in the x direction (NN) and to (Y2) in the y direction (NNN). The forces directed along the vector joining interacting molecules attract each other in the same way as electric dipoles with, however, a different distance dependence. In contrast, forces directed perpendicular to the joining vector repel each other. In the hexagonal, undeformed, lattice, the tilting is possible only when there are strong, dipolar-like interactions which overcome all other interactions with the molecules subsequently undergoing precession. Lattice deformations suppress the precession, and the direction of lattice contraction determines the direction of collective tilt. Thus, ferroelastic domains are formed. There is strong correlation between the lattice deformation and the ordering of molecular

*

TABLE 2: Characterization of Phases of the Phase Diagram, Figure 3, Using Primary Orientational Order Parametersb Dhase

order Darameter(s)

instability

symmetry‘

~

ov L2d Ll’ L2h

L2*

S S’

LT cs

cm cm Pl cm cm cmm Pg Pg Pmg

a Symmetry of the phases is given in symbols of 2-D space groups.56 For the phases appearing as a result of quadrupolar instabilities the lattice strain and effective cross section of molecules serve as equivalent order parameters (see text).

backbones so that every deformation results in molecular ordering with tilting occurring in the direction perpendicular to the backbone located on the centered rectangular lattice. The phase transition LS * S can be driven by a quadrupolar instability of the E2 type which involves the lattice distortion, e6 and (Ys) # 0. This instability is temperature driven and almost 2-D pressure independent. What is believed to be the phase with molecules in an orientationally disordered state’ is mirrored here by the average value of the orientational fluctuations described by the Y5 surface harmonic. The order parameter (Y5) describes the angular distribution of the orientational fluctuationsat every site of the rectangular (deformed hexagonal) lattice. Finally, the phases believed to possess a herringbone pattem of molecular backbones, L2”, S’, and CS, are consequences of the corresponding instabilities at the M-point of Brillouin zone. The order parameters suggested in this study are more general than any used to date. In particular, the order parameters defined as statistical averages of Fourier transforms of the surface harmonics contain information about both the orientational fluctuations and crystallinity, since the wave vector q has its discrete values determined by the size of a crystalline sample. Thus, what has been suggested as an order parameter for “weak c r y ~ t a l l i z a t i o n ” ~is~naturally -~~ included in our formalism. Figure 3 presents a pictorial assignment of phases to the schematic diagram of Figure 1 according to our considerations. Table 2 gives our description of the phases in terms of the orientational order parameters and symmetries. There is a simple convention used to represent the order parameters. The vectorial parameters, (Yl) and (Y2), are represented by a vector (tilt direction), and the tensorial parameters, (Y$, (Y4), and (Ys), are designated by the effective cross section of molecules. The cross sections are, of course, related to the macroscopic strain. 5. Conclusion

We have formulated a new approach to understanding Langmuir monolayers from the viewpoint of the solid state and have used methods commonly associated with solid-state analysis. We have exploited the use of surface harmonics for the description of orientational fluctuations of molecules in the context of earlier work on monolayers by Somoza and D e ~ a i . ~ ~ This has provided a compact treatment of rotational degrees of freedom and allowed description of orientational fluctuations of El as well as E2 symmetry. We have included vector and second-rank tensor order parameters that are consistently defined by the average values of the surface harmonics. By explicitly treating translational-rotational coupling, we have been able to calculate indirect rotational coupling between the molecules

J. Phys. Chem., Vol. 99, No. 21, I995 8885

Phase Transitions in Quasi-Two-Dimensional Solids

terms of the Cartesian coordinates as gI = ( k / u ) ( - 1 , l/d3), g2 = (k/u)(O, 2/43). Figure 4b shows the vectors in the reciprocal lattice and the corresponding Brillouin zone. Highsymmetry points, (r,C, and M), discussed in the paper are indicated. In terms of the reciprocal lattice vectors, the coordinates of the points are r(O,O),Z(0,0