Liquid crystalline polymers at interfaces: Deformations and phase transitions

Prog. Polym.Sci., Vol. 21, 1089-1113, 1996

~

Copyright © 1996 ElsevierScienceLtd Printed in Great Britain. All rights reserved. 0079-6700/96 $32.00

Pergamon PII: S0079-6700(96)00011-1

LIQUID CRYSTALLINE POLYMERS AT INTERFACES: DEFORMATIONS AND PHASE TRANSITIONS D. R. M. W I L L I A M S a and A. H A L P E R I N 'b

alnstitute of Advanced Studies, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, Australia bCentre de Recherches sur la Physico-Chimie des Surfaces Solides, 24, Avenue du PrOsident Kennedy, 68200 Mulhouse, France

CONTENTS 1. Introduction 2. Hairpins and configurations of LCPs 2.1. Statics 2.2. Dynamics 2.3. Thermodynamics 3. Chain elasticity 3.1. Other effects of hairpins 4. Grafted LCPs: nematic polymer brushes 4.1. Isotropic brushes 4.2. Anchoring 4.3. Nematic brushes and valves 5. Mechanically induced Frederiks transitions 6. Buckling of flexible rods: a pressure induced Frederiks transition 6.1. Electric field effects 6.2. Different buckling modes 6.3. Electro-mechanical effects 7. LCPs in nematic solvents: on tilting phase transitions at interfaces 8. Nematic-isotropic diblock copolymers 8.1. Triblock copolymers and nematic mesogels 9. Concluding remarks Acknowledgements References

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1. I N T R O D U C T I O N

Thermotropic liquid crystals of low molecular weight are utilized as working media in display devices. Their polymeric counterparts are much less successful in this respect because of their slow response times. A potential application of thermotropic liquid crystalline polymers (LCPs) in which the response times is not critical is optical data storage systems. This has been thoroughly explored but commercial utilization is still a distant prospect. In this article we reconsider the potential of thermotropic LCPs for use in display devices. Our theoretical considerations suggest that LCPs may prove useful in such applications if they were used as surface modifiers rather than as bulk working media. From a fundamental point of view our considerations concern both interfacial and deformation 1089

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(b) (c)

(e)

Fig. 1. Molecular architectures of liquid crystalline polymers. (a) Main-chain LCPs, with the

nematic mesogens(rectangles)built into the chain backbone. The mesogensare joined by short, flexible, spacer chains (dark line). (b) Side-on-fixedLCPs. As in (a) the mesogensare essentially parallel to the chain backbone. (c) Side-chainLCP with the mesogensperpendicularto the chain backbone. (d) Rod-like LCP. (e) Rod-coil LCP diblock. (f) LCP-isotropic diblock copolymer formedby joining an isotropic chain to a semiflexible,main-chainLCP. behaviour and, in particular, their ramifications for a system comprised of block copolymers and for solutions of LCPs in nematic solvents. The materials we will consider are semiflexible, nematic LCPs. 1,2 In particular, we are interested in LCPs where the mesogenic monomers are aligned parallel to the polymer backbone. 3-7 In such materials there is a strong coupling between the nematic order and the configurations of the chains. This situation can be achieved in two ways. The first is in "main-chain" architecture, where the mesogenic monomers are incorporated directly into the backbone [Fig. l(a)]. The chain then consists of mesogens joined head-to-tail by flexible spacers. The synthesis of long main-chain LCPs, typically via polycondensation reactions, produces comparatively short chains of high polydispersity. The second architecture, so-called "side-on-fixed" LCPs, may allow these difficulties to be overcome. Here the mesogens are attached to a flexible backbone by pendant spacer chains while locally maintaining parallel orientation to the backbone [Fig. l(b)]. Such polymers may be produced by anionic polymerization, which affords the possibility of obtaining long chains of low polydispersity. For the purposes of this article these two kind of polymers will be treated as if they are identical. These polymers combine the orientational order of monomeric nematics with the flexibility and randomness inherent in polymers. In these architectures the two ingredients are strongly coupled. These two related types of LCP should be distinguished from side-chain varieties [Fig. 1(c)] where the pendant mesogenic monomers are largely decoupled from the backbone. These, as well as rigid, rod-like polymers such as the Tobacco Mosaic Virus [Fig. l(d)], will not be considered in the following. We will, however, treat certain block copolymers comprising an LCP block and an isotropic block [Fig. l(f) and (e)]. Finally, much of our discussion concerns solutions of LCPs in nematic solvents of low molecular weight. The properties of such solutions and of the block copolymers have received comparatively little attention until recently. The appearance of an oriented, nematic phase can be controlled either by temperature or, as in lyotropic systems, s-l° by a change of concentration. Here we discuss the former, "thermotropic", case where the onset of orientational order occurs upon lowering the temperature T. Below a characteristic temperature, Tc, the polymer chains change their

LIQUID CRYSTALLINE POLYMERS AT INTERFACES

(a)

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q

(b) _

) i"1

Fig. 2. Hairpins. (a) A single hairpin on a liquid crystalline chain. The nematic mesogens are represented by the rectangles, the remainder of the chain consists of flexible spacers. The direction of the nematic field imposed by surrounding chains is given by ft. (b) The mode of motion of a chain with a single hairpin. Length is removed from one arm and placed on the other arm. (c) A typical configuration for a chain with many hairpins. Here the thermal fluctuations of the chain are depicted. For clarity the mesogens in (b) and (c) are omitted from the picture.

configuration and orientation. Above Tc both are random, while below Tc the chains are locally aligned along a particular direction, the director ft. Some of the effects we discuss have counterparts in lyotropic systems. However, many of our results depend on the coupling of LCPs to the liquid crystalline order of nematic solvents. This is not possible for lyotropic systems which are only soluble in isotropic solvents. The effects we shall be considering arise from a number of distinctive properties of main-chain LCPs which are due, primarily, to their anisotropy. Unlike isotropic polymers, LCPs are characterized by two radii of gyration, one parallel to the director, Rllo, and one perpendicular to it, R ±o. In general RHo > R ±o. As a result, the overall chain shape is that of a prolate ellipsoid aligned with its major axis parallel to the local director. This anisotropy has important consequences for the elastic behaviour of the LCPs. In particular, it is much easier to deform an LCP along its major axis than along its minor axis. 11'12 A second consequence of the anisotropic elasticity is that, in comparison to their isotropic counterparts, LCPs are much easier to deform. The second trait gives rise to drastic modifications of the phase diagram of block copolymers incorporating LCPs. The combination of the first and the third traits leads to a variety of phase transitions at interfaces. 13-~5As we shall see, these transitions may have applications in nematic displays because they are predicted to allow for an arbitrary reduction of the critical voltage for the Frederiks transition. As noted above, much of our discussion rests on the anisotropy of flexible nematic polymers. Essentially there are two extreme states. In the isotropic state, i.e. at high temperatures, a long LCP chain is totally isotropic and forms a random coil. At low temperatures, deep in the nematic phase, a single chain will behave as a bendable rod which is aligned with the nematic director. 4 At intermediate temperatures the chain can be viewed as a series of well-aligned rod-like segments separated by sharp bends known as "hairpins" [Fig. 2(a)]. These were predicted independently by de Gennes 5 and by Khokhlov and Semenov 9 more than a decade ago. We shall mainly focus on the regime where hairpins dominate the chain configurations. As we shall see, their presence has important effects. Among the affected properties are the dimensions 3-5'1°-12 and the elasticity of the

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chains, 11'12 the elastic behaviour of the bulk nematic 16 and its dielectric response. 3'4 For example, they lead to a novel "Ising" elasticity parallel to the chain direction. One should note that hairpins in LCPs are superficially reminiscent of similar configurations in proteins and in homopolymers which undergo fold crystallization. These similarities are, however, misleading because hairpins in these systems are permanent structures. In marked contrast, hairpin defects in LCPs are mobile topological excitations that are created and annihilated continuously. 2. H A I R P I N S

AND CONFIGURATIONS 2.1.

OF LCPS

Statics

For long enough LCPs the chain configurations are governed largely by the presence of hairpins (Fig. 2). The simplest model of an LCP considers a single chain in a nematic field due to the surrounding chains. The chemical sequence is smeared out and the chain is modelled as a uniform inextensible string of length L which is characterized by a bending constant e and a coupling constant an. 3-7 an measures the strength of the interaction between the chain and the molecular field due to the orientational order of the medium. Because of the rotational symmetry with respect to fi it is sufficient to consider the two-dimensional behaviour of the chain. In this case the orientation of the tangent to a chain is specified by the angle O(s), where s is the arc-length along the chain. The energy of a single chain consists then of two terms: 3-7

U=~J~ds[e(~s)2+3ansin20(s)l

(1)

The first term arises from the resistance of the chain to bending: the chain has a lowenergy state when it is fully extended and dO/ds -- 0. The second term reflects the nematic contribution. The chain prefers to align along fi, where sin O(s) = 0. Minimization of the energy (1) leads to the trivial result O(s) -- O, i.e. the chain is aligned along fi and is perfectly straight. In this configuration both the bending and the nematic terms attain their lowest possible values. However, this state is associated with a very small configurational entropy. One way in which the chain can increase its entropy is by introducing small thermal undulations in its trajectory. A second way was proposed by de Gennes 5 and, independently, by Khokhlov and Semenov. 9,10This is to form hairpin bends in the chain [Fig. 2(a)]. The structure of a hairpin can be understood as a balance between two contributions: the bending term which promotes a very gradual bend, and the nematic term which favours an abrupt change in orientation so that the chain is aligned, for most of its length, with ft. This balance specifies the characteristic length scale, X (elan)1/2,and the energy, Uh 2(3can) 1/2, of the hairpin. Mathematically the defect can be understood as an extremum of the energy U from eqn (1), i.e. as a solution of the Euler-Lagrange equation derived from U. This problem is analogous to a pendulum undergoing oscillations in a gravitational field. The energy U corresponds to the action of such a pendulum if we replace the arc-length by time and the tangent angle 0 by q~/2, where 4) is the angle made by the pendulum with the vertical. The --

--

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E

Fig. 3. A single roller on chain of dipoles. The direction of the dipoles is represented by arrows. This kind of defect is analogous to a hairpin.

small undulations made by a chain then correspond to the pendulum undergoing small oscillations. A hairpin, on the other hand, is the counterpart of a pendulum "looping-theloop", with the tip of the hairpin corresponding to the maximum height of the pendulum. A hairpin represents a soliton-like disturbance in the chain trajectory. All the curvature and all the associated energy are localized around a small region of length X. However, the similarity to a soliton is incomplete. For instance, the dynamics of hairpins are diffusive rather than inertial. 6,7 It is possible to have more than one hairpin on a single chain. In fact, a detailed calculation shows that the maximum number, in mechanical equilibrium, is nmax = L~ (71"X).6'7 At such high hairpin densities the hairpins begin to interact very strongly, and the chain is aligned almost perpendicular to the director along much of its length. At smaller densities the interactions between hairpins are usually weak. These interactions can be divided into two kinds. The first are communicated along the backbone of the chain 6'7 and can be calculated directly from the energy functional (1). These attractive interactions have the form typical of localized defects in one dimension, i.e. it is Ui,t--Uhexp(- s/X), where s is the arc-length distance between two hairpins. Thus an LCP can lower its energy by placing all the hairpins in one region. This is of course prevented by the large decrease in entropy. The second type of interactions are more complex and are mediated by the surrounding nematic medium. The hairpin bend distorts the nematic environment and causes a long-range interaction 5 between hairpins which has a strong angular dependence. 17 The full details of this interaction are yet to be fully calculated. In general, all models up to the present time have assumed that hairpin-hairpin interactions s are negligible. Many of the properties of hairpins depend strongly on the symmetry of the nematic field, which does not distinguish between fi and -ft. This is in marked contrast to a dipolar field such as an electric or magnetic field. In chains with permanent dipoles a hairpin-like defect can also occur. 18 In these the chain incorporates permanent dipoles rather than mesogens in its backbone. In an electric field an energy similar to (1) may be written, but with sin 2 0 term replaced by cos 0, i.e. in an electric field the chain can distinguish " u p " from " d o w n " . This has drastic consequences for the type of chain defects which can be generated. In this case hairpins do not occur, but are replaced by a tight circular loop, christened a "roller" (Fig. 3). Although mathematically similar, the physical differences between hairpins and rollers should be very marked. For instance a single hairpin on a chain roughly halves the length of the chain, whereas a single roller has negligible effects on the chain dimensions. 2.2.

Dynamics

The extremization of the energy U, eqn (1), describes a chain with a single hairpin. The solution is not a local minimum of the energy but a saddle point. 6,7 There is one unstable

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D . R . M . WILLIAMS and A. HALPERIN

eigenmode corresponding to transfer of length from one arm of the hairpin to the other [Fig. 2(b)]. This process barely changes the shape of the bend, where all the energy is localized. The lowest eigenvalue, though negative, is very small, decreasing as ~-exp(L/X). This implies that a hairpin is only barely unstable. The stability of the hairpins is not relevant for the static properties and statistical mechanics of LCPs. Their average number is constant and their removal is balanced by creation due to thermal excitation. However, the unstable mode of a single hairpin does play an important role in the dynamics. 6,7 This involves an exchange of length between the arms separating the various hairpins. In turn, this implies that the dynamics are "reptative", i.e. the monomers move along the length of the chain in a tube created by the surrounding nematic environment. As an example we can calculate the time needed to remove a single hairpin from a chain with only one hairpin. As in reptation, 19 the characteristic time scales as p£3 where/z is a friction constant. One factor of L arises from the fact that the frictional force is proportional to the chain length. The remaining factor of L 2 arises from the necessity to perform a one-dimensional random walk of length L. It is important to note that these remarks on dynamics apply only to a continuum model of the chain. If the chain is represented as a series of discrete monomers the hairpins can be stabilized by potential barriers and an activation energy is needed to maintain motion. 6,7 2.3.

Thermodynamics

The above discussion was limited to the mechanical equilibrium of the chain at zero temperature. Accordingly, the problem was reduced to extremization of the energy functional (1). At finite temperature it is necessary to allow for the role of entropy. Finite temperature affects the chain trajectory in two respects. First is the creation of hairpins by thermal excitation. As a result, the chain trajectory performs a one-dimensional random walk along the director. Second, the chain also develops a two-dimensional random walk component in the direction perpendicular to 6. The statistical mechanics of hairpins in an unperturbed chain are analogous to those of a one-dimensional ideal gas. 3-5 Each hairpin is represented as a point particle on a line. To a good approximation the hairpins are noninteracting thermal excitations. 5-7 Their number per chain is thus given by no = (L//)exp(Uh/kT), where l is a microscopic " d e Broglie" wavelength associated with the localization of the hairpin along the chain. The inter-hairpin distance along a chain, lexp(Uh/kT), is an effective "monomer" size 20 which depends strongly on temperature. This strong T dependence has important consequences. The most obvious is the spatial extent of the chain. Because of the nematic field imposed by surrounding LCPs the chain is anisotropic. It is thus characterized by two dimensions, RLIo and R±o, parallel and perpendicular to the director, respectively. The perpendicular span results from a two-dimensional random walk caused by the thermal undulations and is given by R 2±o-Ll. 34' The parallel dimension is due to a one-dimensional random walk caused by the presence of hairpin defects. Each time the chain encounters such a defect it reverses its direction. It thus traverses no+l steps of size L~ no. Accordingly, the parallel dimension is R~lo~(no+l)(L/no)Z-Llexp(Uh/kT) << R2o . The hairpins control Rllo, thus endowing it with an exponential T dependence. This is in marked contrast to the familiar, isotropic polymers whose radius of gyration is only weakly dependent on the temperature. The strong effect of hairpins on the dimensions of LCPs provides the most direct route for proving their existence. In a recent, careful study

LIQUID CRYSTALLINE POLYMERS AT INTERFACES

ttttttt° tttttt 51

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/

® ~ -)

/

4F

r.

/

/

/

lr, 0

0.2

0.4

RIt/L

0.6

0.8

1.0

Fig. 4. A plot of the scaled force on a chain, y = fb/kT, vs scaled extension R]t/L in the parallel direction. Here f i s the force, b a monomer size and L the total chain length. The solid line depicts the Ising elasticity for a nematic chain. 11J2 The dashed line is for a freely-jointed chain in an isotropic solvent. 24 The inset shows the correspondence between the Ising spin model in a magnetic field and the stretching of an LCP.

Li et al.21 managed to overcome difficulties due to transesterification and provided clear experimental evidence for the existence of hairpins in main-chain LCPs. Results of several other experiments have also been interpreted in terms of hairpin effects. 22,23 3. C H A I N E L A S T I C I T Y The deformation behaviour of polymers has been the subject of much interest over the past five decades. The behaviour of flexible isotropic polymers under deformation is reasonably well understood. 19 The expressions for the elastic free energy are universal, i.e. independent of the chemical structure of the particular polymer involved. Thus for a chain of N monomers with monomer size b, the unperturbed radius in a melt or theta solvent is Ro-bN m. The extension free energy penalty when the end-to-end distance is R, such that Ro << R << Nb, is kTR2/Nb 2. Similarly, under compression, the free energy penalty is kTNbZ/R 2. The behaviour under strong deformations is often well described by a freely-jointed chain model, 24 which gives a "Langevin" elasticity. These results are universal because they depend only on the configurational entropy of the chain. Similar elastic behaviour is also expected in weakly deformed LCPs. There are, however, three distinct differences between the elasticity of isotropic and nematic polymers. First, the anisotropy of the chain induces an anisotropic elastic response. Thus for weak 2 2 extensions of the major axis the free energy penalty is kTRI]/RII o while for the minor axis the penalty is kTR2/R2o . For equal extensions in both directions R± = RII, the elastic penalty of the major axis is much weaker. A similar trend is found for compression. The second point is closely related. A nematic polymer in the nematic state should be much easier to deform along the parallel direction than the equivalent isotropic polymer, since RlIo >> Ro. The third difference occurs under strong extension and is due to the nature of the hairpins. RIIo is determined by a one-dimensional random walk due to the hairpins. However, since the hairpins are thermal excitations, their number nh may vary with the applied tension. This situation is qualitatively different from that encountered in isotropic chains where the number of monomers is a constant set by the polymerization degree. A new framework is needed to the describe this situation. This framework is supplied by an

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D.R.M. WILLIAMS and A. HALPERIN

analogy between an LCP and a one-dimensional Ising chain in a magnetic field. 11,12 Hairpins are analogous to domain boundaries separating domains of spin +1/2 from domains of spin -1/2 (Fig. 4). The applied tension plays the role of the magnetic field and the magnetization is the counterpart of the extension, RII. This correspondence allows us to obtain the Ising elasticity H'12 associated with strong extension of Rtl, when nh decreases with an increase in tension. The restoring force is (Fig. 4): f = krRIlexp( - U h / k r ) / l ( L 2 -R]0-1/2

(2)

and the number of hairpins on the chain is: nh =no(1

-R~I/L2) 1/2

(3)

where no is the number of hairpins prior to extension. The anisotropic elasticity of LCPs and its distinctive temperature dependence are reflected in swelling of dilute LPCs in good nematic solvents. H'12 In a good solvent the monomers of the chain repel one another, thus favouring chain swelling. This swelling is however opposed by the elastic response of the chain. Within the framework of the Flory theory of swelling, the corresponding free energy is: F / k T = b3N2 /(R 2Rll) + R 2 / (L1) - no(1 - R~/L2)l/2

(4)

where the first term allows for the excluded volume interactions between monomers, the second term reflects the swelling of the chain in the perpendicular direction, and the final term allows for the swelling in the parallel direction. Here Rtl and R± are the radii of gyration in the swollen state. Expanding for weak swelling and minimizing over Rlt and R± yields the equilibrium radii of gyration in the swollen state: Rlleq = (l/b)4/5exp[2Uh/5kT]RF

R±eq = (l/b)7/l°exp[ - Uh/IOkT]RF

(5)

where RF -- bN 3/5 is the radius of the swollen isotropic chain. Both of the equilibrium radii exhibit the familiar Flory exponent of 3/5. However, the temperature behaviour is very distinctive. In particular, the parallel radius increases with decreasing temperature whereas the perpendicular radius decreases. This temperature dependence is fairly strong, i.e. exponential, in marked contrast to that of isotropic polymers in good solvents. The latter exhibit a very weak temperature dependence. 19 3.1. Other effects o f hairpins The splay elastic constant is also strongly affected by hairpins. In a nematic melt of rods the resistance to splay ~6 depends on packing of the rods. Rod ends are necessary in order to satisfy the constant density constraint. For long rods this packing becomes extremely difficult because of the scarcity of such ends. The situation is different for LCPs with hairpins since each hairpin acts like a rod end. The effective rod size thus scales as the inter-hairpin distance and the splay constant is accordingly much smaller. Furthermore, as a result, the splay elastic constant is also expected to exhibit an exponential T dependence. A very different example concerns the cyclization of LCPs and their dynamics. In the absence of hairpins the LCP behaves as an undulating rod and the two ends do not encounter each

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other. Cyclization is only possible in the presence of hairpins. This leads to a novel dependence of the cyclization probability on L and T.25

4. G R A F T E D

LCPS: NEMATIC

POLYMER

BRUSHES

4.1. Isotropic brushes The past decade has seen an intense interest in the grafting of polymers, i.e. terminal anchoring of the chains to a surface. 26-28 In the case of isotropic, flexible polymers grafted to a flat surface immersed in a good solvent, one may distinguish two regimes. The regimes depend on the grafting density a. Below the overlap threshold the chains form individual "mushrooms" comparable in size to the free polymer in dilute solution. At higher grafting densities the chains crowd each other and stretch away from the surface thus forming a "brush". In the brush regime the height of the layer can be calculated using a simple Flory theory. As in the case of polymer swelling, we balance the deformation penalty, which opposes swelling, against an excluded volume term promoting swelling. Thus, the free energy per chain is:

F/kT=b3N2a/H +H2/R 2

(6)

Minimizing with respect to the height H of the brush yields:

H = a l / 3 b ( N R o ) 2/3

(7)

4.2. Anchoring The alignment of the nematic director at properly treated surfaces is known as anchoring. One distinguishes between two types of anchoring, weak and strong. 29-31 For weak anchoring there is a preferred direction for the director at the surface. However, the director can assume other orientations at the price of an energy penalty. We consider here only the case of strong anchoring, where the energy penalty is infinite. We further limit the discussion to homeotropic and homogeneous anchoring where the director at the interface is respectively perpendicular and parallel to the surface. This kind of anchoring is imposed on the (a)

(b)

l:I ll,l,II,l:lI,l,l,l l: l:l ll'Ill'l:It 'l

Fig. 5. A nematic brush-valve made from LCP chains grafted to a slit in a nematic solvent. In the isotropic state (a) the height of the brush is small and solventcan flow easily through the slit. In the nematic state (b) the chains expand to fill the slit. The flow of solvent is then severely restricted.

1098

D.R.M. WILLIAMSand A. HALPERIN (a)

~,'fl' lfl/]l ',/1' (b)

Fig. 6. A tilting transition of an LCP chain in a nematic solvent (shown as rectangles), under compression. (a) For weak compression the chain is aligned along the direction imposed by the surface anchoring, in this case, normal to the plates. (b) For stronger compressions the chain tilts in order to escape the confinement. This tilt is at the expense of nematic bend distortion of the bulk. The resulting tilting transition is analogous to the ordinary Frederiks transition used in nematic displays. In this case, however, the driving field is mechanical rather than electrical. molecules directly in touch with the surface. Although we impose strong anchoring we shall see that nematic brushes, can, under appropriate circumstances, impose effective weak anchoring on the bulk. Grafted layers thus provide a way of varying the nature of the anchoring conditions at a surface. 4.3. Nematic brushes and valves There have been numerous studies of isotropic brushes and their properties. In marked contrast, nematic brushes received much less attention. 12-15,20,32 They are of interest because of their rich and distinctive phenomenology and their potential for applications. In the following we shall study several surface phase transitions of nematic brushes. Before we proceed, we consider the swelling of LCP brushes for the two anchoring conditions. In particular we discuss a brush in a good nematic solvent which undergoes an isotropic-tonematic transition. We first assume perpendicular anchoring conditions at the surface. In the isotropic state the height of the brush is given by eqn (7). In the nematic state the same expression applies but Ro is replaced by Rllo = (I/b)exp(U6kT) >> Ro. The height thus undergoes a large increase, by a factor of (l/b)2/3exp(2Uh/3kT), over a small temperature range. This suggests the design of a "nematic brush-valve". We consider two brushes lining a slit (Fig. 5). In the isotropic state the brushes barely penetrate into the slit and the flow of fluid through it is unrestricted. In the nematic state the brushes expand into the slit. Even though the polymer volume fraction is low the flow is effectively screened in the (a)

(b)

SSSS

SSS

SS

SS

Fig. 7. The anchoring transition predicted for grafted layers subject to parallel, homogeneous, anchoring. Each chain is represented as an ellipse and the nematic solvent is represented by the small rectangles. (a) At low grafting densities the chains do not overlap and there is no tilt. (b) At higher grafting densities the chains overlap, repel each other, stretch away from the surface and tilt. This system may provide a way of lowering the switching voltage for the Frederiks transition in display devices.

LIQUID CRYSTALLINE POLYMERS AT INTERFACES

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brush. Thus, the flow through the slit is severely restricted. This system acts thus as a temperature-sensitive valve. One way of creating such a valve involves porous materials containing terminally adsorbed LCPs. Each pore acts as a tiny microvalve, and the whole assembly acts as a macroscopic valve. Similar microvalves have been proposed before using flow-induced switching 33 in isotropic solvents. Note that similar arguments yield the thickness of a brush subject to parallel anchoring. In this case Ro is replaced by R • and the brush thickness, b a l / 3 N l 2/3, is thus significantly smaller. 5. M E C H A N I C A L L Y

INDUCED

FREDERIKS

TRANSITIONS

Most studies of thermotropic LCPs focus on their melt behaviour. However, LCPs are soluble in monomeric nematic solvents 22'34 and their solution behaviour is of interest. Of special interest is a family of second-order tilting phase transitions predicted to occur upon constraining long LCPs with hairpins to the solid-nematic interface. The scenarios studied thus far involve confined LCPs !2-15 (Fig. 6), as well as adsorbed 2° and grafted L C P s 13-15 (Fig. 7). Two crucial ingredients are involved. One is the imposition of anchoring conditions at the interface i.e. alignment of fi in a perpendicular or parallel direction with respect to the surface. The second is the anisotropy, in shape and elasticity, of the LCPs. The LCPs at the interface are oriented by virtue of the anchoring conditions. For the appropriate choice of anchoring this results in an increase of free energy which is relieved by tilting the major axis of the LCPs. For example, perpendicularly oriented LCPs will tilt in order to avoid confinement [Fig. 6(b)]. The flat, rigid surfaces play a double role. First, they serve to constrain and deform the LCPs. Second, the surfaces also provide nematic anchoring. This family of second-order phase transitions afford novel possibilities to improve the performance of liquid crystalline displays by using "active alignment". 13-15,20 The critical voltage for the Frederiks transition, 29'3° Vc, is a material constant of order 2 V. To reduce the price of the driving electronics it is desirable to lower Vc. Doing so by chemical modification of the nematogens is a difficult undertaking with (. a)

', ,, ,, . . . . ,'"" .... i ;:flrl ,

rg", ', (c) ,',' ~,,,,

~:,',

ii

(d)

,t',

,'I

i

,',,:

Fig. 8. The states of a system of rod-like, semiflexible LCPs and a nematic solvent, confined between two slits imposing perpendicular, homeotropic, anchoring. (a) The unperturbed system of rods in a nematic solvent. The rods are grafted to the bottom plate only. (b) Buckling under pressure in the high-density regime occurs while lateral uniformity is maintained. (c) Buckling under pressure in the low-density regimes involves formation of columns of width ~ L (shaded area). (d) The Frederiks transition in an electric field at low rod density. The nematic fluid is distorted (shaded area) except in regions of size of the nematic correlation length, (, around each rod. In both (c) and (d) the unshaded regions only show weak distortion of the nematic director.

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D . R . M . WILLIAMS and A. HALPERIN

limited prospects of major reduction in Vc. Vc can be lowered significantly by modification of the alignment layer of the device, in particular by grafting LCPs with subcritical density, i.e. at the threshold of the second-order tilting phase transition. Since the susceptibility of the system diverges at the vicinity of the critical point, V¢ can be made arbitrarily small. In the next section we study several tilting transitions involving LCPs with hairpins. However, some insight may be gained by first considering a simpler system exhibiting similar phenomena; in particular, a brush comprising short, semiflexible LCPs that are short enough or stiff enough so that no hairpins are present. 6. B U C K L I N G

OF FLEXIBLE FREDERIKS

RODS: A PRESSURE TRANSITION

INDUCED

The semiflexible LCPs, of length L, are modelled as bendable rods. A unit area of the surface carries a LCPs (Fig. 8). The unperturbed rods are straight. Upon bending, the chains incur an energy penalty of 1/2e(dO/ds) 2 per unit length, e is a bending constant for the rod, 0 is the angle of the tangent to the chain and s is the arc-length. In our discussion homeotropic anchoring conditions prevail at both surfaces of the slit, i.e. the nematic director at the interface is perpendicular to the surface. Strong coupling between the trajectories of the chains and the nematic fluid is assumed: the chain follows the local director field and vice versa. If the distance between the plates is H > L the chains are perpendicular to the surface and are undeformed. However, when H < L the chains must buckle. This buckling is opposed not only by the elasticity of the rods but also by the distortion of the nematic solvent. Initially, we shall assume that the nematic distortion is laterally uniform, i.e. it is a function of the altitude z only and there is no variation in the horizontal x direction. This assumption will be relaxed below. Thus if O(z) is the angle between the director and the surface normal, the distortion free energy per unit area is: 1 L Fdist = ~(ea+K') [ ds(dO/ds) 2 (8) J0 This arises from the rod deformation contribution, 1/2ea(dO/ds) 2, and a bend term due to the nematic fluid, 1/2K'(dO/ds) 2. Here K is the nematic bend constant of the fluid, K' -K(1-ab 2) and the factor (1-ab 2) allows for the volume fraction occupied by rods, each of volume Lb 2. For most practical grafting densities K' ~ K. Note that this expression does not involve H. This is because of the strong coupling assumption and the requirement that the free ends of the confined rods are at the upper surface. The onset of the distortion is characterized by a critical pressure, Pc, which must be applied to the plates. To obtain Pc it is necessary to consider the total free energy per unit area, F. F Fdist+PAVand since H = J~ dscosO(s) --~L - ½fL dsO2(s), we obtain: =

ds(dO/ds) 2- ~P

F-- ~[ea+K ] 0

dsOZ(s)

(9)

0

Since the nematic fluid in the reservoir is undistorted it does not contribute to the total free energy of the system. Minimizing (9) with respect to O(s) leads to: d20 ds 2

- X20

(10)

LIQUID CRYSTALLINE POLYMERS AT INTERFACES

1101

where the characteristic length is Xc [P/(eg+K'] -1/2. Equation (10) has a non-trivial solution 0 -- Oosin(res/L), which satisfies the boundary conditions 0 -- 0. Substituting this solution into (10) yields the condition kc -- L/re, i.e. the transition occurs when the characteristic length is comparable to the chain length. The corresponding critical pressure is thus: ---

Pc° = re2L-2(ea + K ' )

(11)

This transition is similar to that expected upon confinement of long, main-chain LCPs. In both cases the effect is similar to the familiar field-induced Frederiks transition. All three systems exhibit features of a second-order phase transition. However, in the polymeric systems the transition is mechanically induced and the applied pressure plays the role of the electric (magnetic) field. Distinctively, in the present case the chain elasticity is due to the rigidity of the chains rather than to their configurational entropy. Also, in this system there is a close relationship to the buckling of an array of rigid rods.35 However, the effective modulus of our system also includes a contribution due to the distortion of the nematic order. This contribution depends on the grafting density of the rods. If we assume K = kT/b, where b is a mesogen size, then a for which the contributions due to the nematic solvent and rods are equal occurs at across ~ K'/e ~ kT/(be), e/kT, which can be identified as the persistence length of the chain, is at least L. Altogether Crcross ~ (Lb)- 1(l+b/L)- 1 ~ 1~Lb. 0 2 2 Thus, the criticaloPressure can be readily estimated at ~ = across where Pc = re 2L- K. For L = 400 A, b = 5 A and K = kT/b, the critical pressure is roughly i atmosphere. However, as we show below, depending on o one may distinguish three different regimes and the critical pressure can vary significantly between them. In the above discussion the nematic distortion was uniform in the direction parallel to the plates. However, a different scenario is expected at low grafting densities. The nematic fluid distorts in " t u b e s " of radius R around each rod [Fig. 8(c)] and the remainder of the fluid is unaffected. It is convenient to estimate R in the case of a single rod in an infinite nematic layer of thickness H. In this case the only relevant length scale is H, which is essentially equivalent to L. Thus a rod is surrounded by a tube of radius R ~ L where the distortion closely follows the rod contour. Beyond this radius the distortion relaxes exponentially with a decay length L. Accordingly in the low-density regime, L2a < 1, only pillars of radius R ~ L support the applied pressure. The critical pressure is then weaker by a factor of ~x~rL2, where oe is a numerical constant, in comparison to Pc as given by eqn (11). In this regime we automatically have ~ < ~c~ossso that the elastic contribution of the rods is negligible. The existence of this columnar transition is a characteristic feature of the polymeric system. In marked distinction, in the familiar, field-driven, Frederiks transition the system is always uniform apart from the presence of walls and defects. Altogether, the three regimes identified are summarized as follows:

p0 =

ocre2Ko

if

re2L 2K

if L - 2

reeL-2ea

if cr > (Lb) -1.

-

a < L -2, < (7

< (Lb) -

1,

(12)

In the low- and high-density regimes we find a linear dependence on o. In the former case this is because each rod couples the pressure to the nematic medium. In the latter regime the

1102

D.R. M, WILLIAMS and A. HALPERIN

pressure is opposed only by the rigidity of the rods, thus giving rise to a linear a dependence. In the intermediate regime the rods provide negligible elastic contribution. Their only role is to couple the external pressure to the nematic distortion of the solvent. This indirect role is reflected in the absence of a dependence. Another point of interest concerns the H or L dependence of the critical pressure. In the familiar Frederiks transition the critical electric field scales as H -1. In the pressure-induced transition we find a markedly different L dependence. A stronger, L -z, dependence is found in the high-density regime. In the low-density regime the critical pressure is, to a good approximation, independent of the rod length. This last feature can be understood as follows. The nematic distortion free energy density is roughly Accordingly, the distortion free energy per column of volume L 3 scales as KL. Since the distortion term per unit area is aKL, the corresponding pressure is

K/L2.

aK.

6.1.

Electricfield effects

For a Frederiks transition induced by an electric field, the critical field is proportional to Accordingly, the critical voltage is independent of H. It is a material constant which may be modified in a limited range by altering the molecular structure of the mesogens. The grafting of semiflexible LCPs provides an alternative method for the control of the critical voltage. This method involves a surface treatment rather than chemical modification of the working medium. Suppose we apply an electric field E perpendicular to the plates for a sample which has a negative dielectric anisotropy A~ -- ell-el < 0. For simplicity we consider the case when the field is applied to the slit as well as to the reservoir. If we assume a laterally uniform director field, the free energy (9) may then be written as:

1/H.

1

_IK,

L

H

L

H

~edIOds(dO/ds)2+ 2 J~odz(dO/dz)2+P Io dscos0(s)-~IAeIE 2 Jo dzc°s20(z)

(13)

The reservoir contributes a constant term to the total free energy of the system, a term that is ignored. Noting that dz -- dscos 0 we obtain as the free energy:

ds[

+½1';'(cosO)(dO/ds)2 +PcosO(s)-lzX,lE2cos30(s)]

(14)

I(dO/ds)2-(P+ ~IAelE2)O2(s)]

(15)

or

+K']I F = ~[ea

[~ J0 ds

The onset of the distortion can be characterized by the method used above, yielding the condition P + (3/2)IAeLE2 > 7rZL-2(ecr+K'). The critical voltage for the onset of the nematic distortion can be thus written as: 0

V

LIQUID CRYSTALLINE POLYMERS AT INTERFACES

1103

where Vc° ~ a - V @ lAd is the critical voltage in the absence of any rods. Here Pc° is given by (11). II - 7r2K/L2 is half the critical pressure at a ffi across. Expression (16) exhibits two interesting features. The most important is that Vc can be made arbitrarily small if the applied pressure is set close enough to the critical pressure, p0. When P ~ p0 yet P < p0, an arbitrarily weak field can drive the system to buckle. The second feature is more subtle• In our system the contour length of the rods, L, is fixed, but the distance between the plates, H = f~ dscosO(s), can change. In marked contrast, the familiar Frederiks transition is typically studied under the constraint of constant H. Accordingly, in the present case some of the nematic fluid escapes under the transition thus giving rise to a factor of cos 3 0 in the free energy (14). This replaces the cos 2 0 factor which appears in the constant H case, thus leading to a critical voltage which is lower by 20%. Note that for a Frederiks transition involving free plates and a neat monomeric nematic fluid all the fluid escapes to the reservoir. 36 Earlier in this section we have assumed that there is no x variation of the nematic director, i.e. lateral uniformity. We now relax this assumption and focus on the case of zero applied pressure. In this case, an applied field can induce a novel form of non-uniform Frederiks transition. Suppose the grafting density is low while the field exceeds E °. In the absence of LCPs, E > Ec° induces a Frederiks transition. This distortion is opposed in our system by the presence of the rods. As a result the system is expected to undergo a nonuniform transition. While E < Ec the rods are unbuckled and perpendicularly oriented. In this regime the rods are surrounded by pillars of radius R ~ ( where ~ ffiE-a(K/IAe [)1/2 is the electric coherence length 29 of the nematic [Fig. 8(d)]. Within each pillar the nematic field is only weakly distorted, and at the rod surface it is not distorted at all. Throughout the remainder of the system the field is distorted as it would be for an ordinary Frederiks transition. No transition can occur for high grafting density or weak fields when ~ is larger than the inter-rod distance. However, when the field is higher than a critical value, Epin~, we have a < ~-2 and small regions of the system undergo a distortion. The condition a ~ (- 2 yields Epillar =/3 a ~ e l where /3 is a numerical constant. Naturally we also require E > Ec° ffi7rL-1 ~ , otherwise no Frederiks transition can take place. The expected sequence is thus as follows. A "pillared" Frederiks transition takes place in parts of the sample at a critical field of E = max(Epmar, E°). The regions undergoing this transition will grow with the applied field and finally, at a field E = Ec (16), the rods buckle and the distortion becomes laterally uniform. This discussion assumes perfectly uniform grafting density. In reality the rods will probably be grafted randomly with occasional patches of flee space separating regions of high grafting density. As the field is increased the large patches distort first followed by the smaller patches. •

6.2. Different buckling modes It is of interest to note that different buckling modes may be obtained in certain circumstances. When the rods are grafted to both plates so as to produce bridges one may distinguish between two cases. When lateral displacement of the two plates is allowed, the preceding discussion is applicable. However, when the system is constrained so as to prevent such shear strain, the lowest buckling mode, 0 = OosinOrs/L), is disallowed. The next solution which satisfies the boundary conditions 0 = 0 at each plate and induces no relative displacement is

1104

D . R . M . WILLIAMS and A. HALPERIN

O(s) = Oosin(27rs/L). This solution induces a novel "high wavenumber" Frederiks transition, and increases the critical voltage (16) and critical pressure (11) by a factor of 4. 6.3. Electro-mechanical effects Thus far we have considered how the Frederiks transition is affected by compression due to an imposed pressure. However, the inverse effect can also o c c u r , 37 i . e . an applied electric field can result in compression of the slit. The field couples to the nematic fluid and via the fluid to the rods. This in turn causes a mechanical deformation of the system. This deformation can take various forms, depending on how the chains are attached to the walls. In the system described above, where the chains are attached to one wall only, the mechanical effect is simple compression, i.e. a vertical relative displacement of the plates. In the case where the polymers are attached to both plates a shear strain is expected, i.e. deformation parallel to the plates can take place. To investigate these distortions it is necessary to expand the free energy (14) to higher order terms in the distortion amplitude. In doing this we assume a distortion of the form 0 -- Oosin(Trs/L). This kind of distortion will be approximately true close to the transition but will only be qualitatively correct at very high distortions. Expansion of (14) with respect to 00 yields:

F/L=~OZI2(P°-P)+3II(~c)2]+ ] ~1 0 4o I2P + 4II + 21II (if---6)21

(17)

Minimization with respect to 00 gives: 2

02

02= 24II(EZ-EZ)/(2E ° P + 21E2II +4E c I-I)

(18)

This result takes a simpler form in the limit of zero applied pressure and sufficiently low grafting density, ab 2 < 1, so that K' --~ K and hence E c - - E ° v / ~ V / 1 + ea/K. In this case 02 = 24(E 2 -E2)/(21E 2 +4E °2)

(19)

In the limit a ~ across,where the contribution of the rods and the nematic medium are comparable, the transition occurs at E c = 2/x/~E °. In this case if we choose a critical field E 2Ec we find 00 ~ 1. Accordingly, a field of a few volts is sufficient to induce a substantial deformation. The vertical displacement of the chain ends is given by 6z= ~[~dscos0(s)L ~ - 1/2 f0L dsO2(s)= - 1/4L02. Thus, near the critical field, E=Ec(1 +6), the vertical 2 displacement scales as E 2 -Ec2 ~Ec6, i.e. above the critical field, the displacement increases linearly with E-Ec. The horizontal displacement of the chain ends is given by 62:= f~ dssin0(s)--2L00/Tr. When the chains are grafted to both surfaces then displacement of the chain ends induces a shear strain between the two surfaces. However, in this case the relative displacement scales a s 61/2 . For near critical fields the horizontal displacement can thus be substantially larger than the vertical displacement. 7. L C P S IN N E M A T I C S O L V E N T S : ON T I L T I N G TRANSITIONS

PHASE

AT INTERFACES

The buckling transitions discussed in the previous section are the simplest examples of

LIQUID CRYSTALLINE POLYMERS AT INTERFACES

1105

distortional phase transitions involving LCPs at interfaces. Similar scenarios are expected for long LCPs which support hairpins. However, in this second system a number of extra ingredients are present. 12-15,20 The buckling transitions involve only the bending rigidity of the rods. In the case of long LCPs it is necessary to allow for the entropic, Gaussian elasticity of both the major and the minor axes of the ellipsoidal LCPs. Also, in the examples considered below repulsive monomer-monomer interactions can play an important role. For long LCPs with hairpins a number of "tilting" transitions are predicted. These involve a variety of driving forces: the elasticity of the deformed chains or the interaction penalty due to monomer-monomer interactions. These penalty terms are relieved by tilting the major axis of the chain at the price of an induced distortion of the nematic director of the solvent. Two situations are considered in some detail. (i) Doubly grafted LCPs in a slit. In this case the LCPs bridge the two surfaces since each end group is grafted to a different surface. Our discussion concerns the compression and extension in a theta nematic solvent. The shear behaviour has been considered elsewhere. 37 (ii) The uniform adsorption of LCPs in a good nematic solvent onto a flat, impenetrable plate. In this situation all monomers are equally attracted to the surface. In each of these cases the initially undeformed LCPs are oriented via the alignment imposed on the nematic by the anchoring conditions. As the free energy penalties of the LCPs grow, it becomes beneficial to induce a nematic distortion so as to alleviate them. It is important to note that such scenarios are only possible for thermotropic LCPs in nematic solvents. It is impossible to realize such effects for lyotropic LCPs since these are only soluble in isotropic solvents which cannot be aligned. In the following we focus on tilting transitions as they occur in laterally uniform layers. However, the columnar, non-uniform transitions considered in the previous section are expected to occur in these systems as well, provided the area fraction of the LCPs is low enough. The essential features of the tilting phase transitions are introduced in the discussion of doubly grafted LCPs. We consider two plates bridged by LCPs and swollen by a theta nematic solvent, i.e. monomer-monomer interactions do not come into play. The area per chain is a and the width of the slit is H. a is chosen at the overlap threshold so as to ensure lateral uniformity while retaining the single-chain behaviour. The width of the slit at equilibrium, Ho, corresponds to the unperturbed size of the LCPs. In the case of parallel anchoring Ho ~ R±o while for perpendicular anchoring Ho ~ Rilo. H can be adjusted by pumping nematic fluid into or out of the slit. The behaviour of the system for the two anchoring conditions is markedly different. When parallel anchoring is imposed, the confinement results in simple Gaussian compression of the coils. On the other hand, a tilting phase transition is expected when the slit is expanded and the LCPs are stretched. The opposite trend is encountered for perpendicular anchoring (Fig. 6). A tilting phase transition is expected upon confinement while simple Gaussian behaviour is predicted when the slit is expanded. The familiar Gaussian behaviour obtains when tilting Rimwith respect to its orientation at Ho does not relieve the deformation of the LCPs. The following, somewhat oversimplified, argument explains these tilting phase transitions. The behaviour of the LCPs may be described at the level of a Flory-type approximation, assigning uniforfli tilt to Ril. 13-15The confinement of LCPs subject to perpendicular anchoring results in the compression of Ril. Tilt is beneficial since it enables Ril to be larger than H. Roughly speaking R, ~ H/cos 0 where 0 is the tilt angle of Ril with respect to the normal.

1106

D . R . M . WILLIAMS and A. HALPERIN

the elastic penalty is reduced from Fd(RO/kT~R~Io/H 2 to (R~Io/H2)cos20. Similar reasoning may be applied to the extension of LCPs

Consequently,

Fe~(RII)/kT

-~

subject to parallel anchoring. In this case R< is stretched. Tilt is favourable since it allows R± to be smaller than H. In particular, R± --~ Hcos 0 where 0 is now the angle between RnI and the surface. The elastic penalty is thus reduced from Fel(R±)/kT-~HZ/R2 o to Fel(R±)/kT . - ~ (H2/R2o)COS20. The elastic benefits are obtained at the price of a nematic distortion. This is described by means of the continuum theory of nematics. 29.3o In the case of confinement subject to perpendicular anchoring, the angle between li and the normal is approximated by O(z) = QsinQrz/H). Here z is the altitude with respect to the lower plate and Q is a variational parameter 0 -< Q << 1. For the extension of LCPs subject to parallel anchoring conditions, the angle between fi and the surface is taken as O(z) = Qsin (Trz/H). Both choices allow for continuous variation of O(z) subject to the anchoring conditions at the boundaries. The nematic distortion free energy, Fdist , is: Fdist = (crK/2) ]oH (d0/dz)2dz = (TrZ/4)(crK/H)Q2

(20)

where K is the elastic constant of the nematic. To relate the two descriptions we identify 0 as the average O(z), i.e.

0 =H- 1

O(z)dz = 2Q/Tr

(21)

In reality, the trajectory of the LCP between the hairpins, locally, on average, follows ft. However, at the level of the Flory approximation our interest is limited to overall chain dimensions and this detailed description may be avoided. We now expand the total free energy per chain, FT = Fel+Fdist, to the lowest order in Q. In view of the rough approximations used, we adopt a scaling approach and discard all numerical prefactors. For the expansion case we find:

FT -~ H2 /R2o + [(crK/ kTH) - HZ /RZo]Q2

(22)

The coefficient of Q2 vanishes for H --~ (aKRZo/kT) V3. This is the signature of a secondorder tilting phase transition between an untilted state, Q -- 0, and a tilted one, Q > 0. It occurs for H-R±o, when the LCPs are weakly stretched beyond their equilibrium span. For the compression case we similarly obtain:

FT/kT ~ R~o/H 2 + [(aK/kTH) -R{o/H2]Q 2

(23)

and the second-order tilting phase transition is expected to occur upon confinement below H ~ Riio(RltokT/aK)--R,o. Note that although this transition bears some resemblance to the buckling of stiff chains in nematic solvents there is one major difference. Here the chain deforms significantly before it tilts. It is possible to use similar arguments to analyse the expected adsorption behaviour of LCPs in good nematic solvents. LCPs, like most polymers, 38-4° should exhibit strong propensity to adsorb onto solid surfaces. This trait occurs because the polymerization lowers the translational entropy per monomer by a factor of 1/N while hardly affecting the interaction energies. However, in the case of adsorption of LCPs, a richer

LIQUID CRYSTALLINE POLYMERS AT INTERFACES

1107

phenomenology is predicted because of the interplay between chain anisotropy and the anchoring conditions imposed by the surface. In particular, the adsorption of LCPs onto a flat impenetrable surface is expected to involve a second-order tilting phase transition. 2° This effect is of special interest since it should be particularly easy to observe optically, using cross polarizers. At this stage we overlook the detailed structure of the layer. 4° Following de Gennes, 41 we assume a step-like concentration profile. The monomer volume fraction, q~, within a layer of thickness D a n d 3`/b2 monomers per unit area is q~ -- 7b/D • Accordingly, the fraction of monomers at the surface is b/D. An adsorption energy of 6kT with 6 << 1 is assigned to a monomer at the surface. The adsorption free energy is thus FadskT ~ -6N(b/D). For low surface coverages, the adsorption results in confinement and D is smaller than the unperturbed chain size. As 3' increases, repulsive monomer-monomer interactions grow in importance• Within our picture, the associated interaction free energy per chain is Fint ~ 3,N(b/D). This term causes the layer thickness to increase. Eventually, D of the saturated layer is comparable to the Flory radius of the swollen chain. Thus far, our considerations apply to isotropic as well as LCP chains. For LCP chains we should also allow for two possible tilting phase transitions. In the case of perpendicular anchoring, tilt is favoured in the low 3` regime when it relieves the confinement of Rb The tilt is expected to disappear when 3' increases and the associated swelling leads to D > RIIo.In the following we focus on a second, simpler, scenario involving parallel anchoring. In this case the tilting phase transition is predicted for large 3`, when D > Rio, and tilt helps alleviate the stretching penalty of R±. Our analysis focuses on the laterally uniform adsorption layer, when the adsorbed LCPs are already overlapping. It is thus sufficient to allow only for the stretching free energy. In the absence of tilt D = R± and this term is

Fel/kT "~ D 2/R2o

(24)

The free energy of the layer may be reduced by tilting R, by an angle of 0 with respect to the surface. Within the approximation scheme introduced earlier we have R ± --~D cos 0 and R[I ~ D sin 0. The stretching penalty is accordingly reduced to: Fel IkT -~ (D IR ±o)2cos20

(25)

thus allowing for stronger swelling with concomitant reduction in 4~ and, as a result, in Fin t. The tilt is however associated with a penalty due to the distortion of the nematic environment. As before this term is of the form: Fdist ~ (K N b / 3`D )O2 ~ ( k TNb / 3"D )O2

(26)

where the nematic elastic constant was approximated as K .-~ kT/b. Note that on this occasion we express Fdist in terms of 0 rather than Q. Altogether, the free energy per chain, in the limit of 0 << 1, is: Fchain/k T

(3bN 3`bN + D2 (1 _ 02"~ + 02 Nb - + - D D Nb 2" " 3`D

(27)

where the translational entropy term has been neglected. Minimization with respect to D yields the equilibrium thickness:

Deq/N2/3b = (3` - 6) 1/3 + (1 + 32 _ 63")3"-l(,y

_ 6) - 2/302

(28)

1108

D . R . M . WILLIAMS and A. HALPERIN

Note that this expression is only valid for 3, > 6 when D > R±o. At lower coverages the chains are compressed rather than stretched and Fchain used above is inappropriate. Upon substitution of Deq into Fchai, we obtain the free energy per chain at equilibrium: Fsurr/N1/3kT -- (3' - 6) 2/3 + 3'- 1(3' _ 6)- 1/3(2 - 3'2 + 3'6)02

(29)

The coefficient of 02 can change sign, thus signalling the occurrence of a second-order tilting phase transition. The sign of the coefficient is governed by the factorA -- 2-3,2+3'& For A > 0 there is no tilt. A second-order phase transition occurs atA -- 0 and forA < 0 the chains are tilted. Since it was assumed that 6 << 1 the transition takes place, 3'--3'c--x/2 + (1/2)6 ~ 1. This corresponds to an accessible coverage of (1/Nb 2) chains per unit area. Note that 3' increases linearly with 6. This is because, within our approximation scheme, tilt lowers the fraction of monomers at the surface. The adsorption isotherm is obtained by equating the chemical potential of the free chains, /~b~k ~ kTln(c/N), with that of the adsorbed polymers, IZad/kT = ln(3"/N)+O3"Fsurf/03"./-~bulkis only due to the translational entropy of the dilute LCPs, whose concentration is c/Nb 3. gad is due to an Fchaincontribution supplemented by the two-dimensional translational entropy of the chains at the surface. In particular: 1

I.~surf/kT=N1/3(3"-6)-l/3(~'y-@

+ ln(3,/N)

(30)

The plateau of the adsorption isotherm is then given by 3' -- N-m[ln (c)] 3/2 while the critical bulk concentration Cc, corresponding to 3"¢, is: Cc--3'cexp r[_ 51N 1/2.(rc _ 6)-1/3(36-3'c)1] ~ exp[_N1/3(1

6) 1/3]

(31)

Similar phenomena are expected in a number of related systems: (i) solutions of LCPs confined to a slit imposing perpendicular anchoring; 13,14(ii) confined brushes of LCPs grafted onto a surface providing perpendicular anchoring; 15 (iii) a tilting phase transition can also occur on a single plate in the presence of excluded-volume interactions. In this case the system incorporates LCPs grafted onto a surface imposing parallel alignment and immersed in a good nematic solvent. The osmotic pressure gives rise to swelling as discussed in the adsorption case. The onset of the transition is controlled by the grafting density. This phenomenology is of interest from a number of view points. From the polymer science perspective, the tilting phase transitions are distinctive features of LCPs. Isotropic polymers do not exhibit related effects. Similar effects are known for monomeric liquid crystals. Yet, the LCP phenomenology exhibits distinctive polymeric features. Thus, the tilting transitions in the slit configuration are reminiscent of the Frederiks transition. However, in the LCP case, the driving force is due to the deformation penalty of the LCPs rather than to the effect of an electric field. Similarly, the tilting phase transitions involving single plates are related to the anchoring phase transitions reported for monomeric nematics. 31 Finally, the tilting phase transitions are of technological potential for liquid crystalline displays. 8. N E M A T I C - I S O T R O P I C D I B L O C K C O P O L Y M E R S

Diblock copolymers consisting of A and B polymer chains joined end-to-end have been

LIQUID CRYSTALLINE POLYMERS AT INTERFACES

1109

C~

0 R

14

..,......,0.-

I

f

12

1o

L

/ S

!J

8~

L 4

H

0.2

0.4

0.6

0.8

if

Fig. 9. An example of the phase diagram for a nematic-isotropic diblock copolymer,sl The inverse temperature 0 = 1/T is plotted against the length fraction f of the nematic block. The isotropic-nematic transition, shown by the dotted line, occurs at 0 = 7.9. The dashed line shows where the nematic block effectivelybecomes a rod. HereH denotes homogeneous,S spherical, C cylindrical, L lamellar and R rod-coil. The shaded regions are also cylindrical. Along the line marked ~xthe following re-entrant sequence should be seen: H, S, C, L, C, S, C, L, R. extensively studied during the last decade. 42 These studies have concentrated mainly on the tendency of such diblocks to form microphases at low temperatures. Microphase separation is driven by the enthalpic penalty for AB contacts. For A and B homopolymers this penalty leads to a macrophase separation. For diblocks this is not possible and results in microphases with periods of order 100-1000 A. At low enough temperatures the AB interfaces are sharp and the system is, in effect, an array of polymer brushes. It is the competition between the chain stretching energy in the brush and the AB surface tension which then determines the size of the microphases. Various microphases are possible: lamellae, cylinders, spheres and more exotic bicontinuous phases. The particular mesophase obtained depends on the relative sizes of the A and B blocks. Most of the attention has focused on the case where A and B are both isotropic, flexible polymers. There has been a certain degree of theoretical work on the case where B is a rod andA is an isotropic, flexible p o l y m e r - - s o called "rod-coils" [Fig. 1(e)].43-47 These systems have been recently synthesized and the phase diagrams investigated.48-so However, the case where A is isotropic and B is a semiflexible nematic polymer has been almost totally ignored until very recently. 51-s3 Such nematic-isotropic diblocks should exhibit a drastically different phase diagram compared with their isotropic counterparts. Here we discuss this phase diagram. Consider a diblock where A is an isotropic, flexible chain and B is an LCP which undergoes an isotropic-to-nematic transition at some temperature T~. We focus on the case where TNr < TOp, where Top is the order-to-disorder transition temperature, i.e. the microphase separation occurs before the isotropic-nematic transition. We also assume that the B block is long enough to support several hairpins and that the glass transition

1110

D.R.M. WILLIAMS and A. HALPERIN

temperature of the two domains is well below TM. The diblock is described by an asymmetry parameter f - NB/(NA+NZ), where Ni is the degree of polymerization of the ith block. The predicted phase diagram is depicted in Fig. 9. The domain boundaries are shown in the 1/T vs f plane. Since possible defect structures in the continuous LCP phases are ignored, the !/2 -< f--- i region is not fully reliable. Upon lowering T below TM, the phase diagram undergoes a marked change. The symmetry with respect to f = 1/2 is lost and the domain boundaries are roughly oriented along the horizontal rather than the vertical direction. When the system is cooled down along the line denoted by a, it undergoes a sequence of re-entrant phase transitions, in particular, lamellar to spherical to cylindrical and back to lamellar. The physical origin of the predicted behaviour is simple. The phase diagram in the strong segregation limit is obtained by comparing the free energies of the possible mesophases. The free energy per chain incorporates three terms, Fchain= Fel(A)+Fsurface+Fel(B). Fd(A) and Fel(B) allow for the Gaussian stretching penalties of the A and B blocks. Fsuaaceis the free energy associated with the sharp interface between the A and B domains. Fsurfa~e favours the increase of the number of chains incorporated into the elementary units (micelles, cylinders or lamellae) of the mesophase. The surface free energy per chain decreases with the number of diblocks per unit area of the boundary. The growth of the domains is arrested by the stretching penalties Fel(A) and Fel(B). Since the AB junctions are constrained to the sharp AB boundary, both blocks are, in effect, grafted. This, together with the constant density requirement, enforces chain stretching. Lamellar mesophases are obtained for nearly symmetric blocks, f ~ 1/2. For more asymmetric blocks cylindrical and spherical mesophases are obtained. In these geometries the volume available to each block increases with the radius, r. It is thus possible to relieve the elastic deformation of the longer blocks by placing them in the exterior, large r, regions where the stretching is weaker. These general considerations apply both above and below T~. The dominant new features found for T < TM are traceable to the role of hairpins, requiring the replace2 2 2 ment of Fel(B ) b y Fel(Rll). In the isotropic state NB determines Fe1(B) ~ R / R o, since Ro NBb2, and the volume per chain is NBb3. The volume per chain does not vary when T < TM. In marked contrast, RIIo in the nematic state is determined by the number of hairpins, no, rather than NB. As a result, the elasticity of the B blocks is abruptly reduced. Furthermore, while NB is independent of the temperature, no~exp(- Uh/kT) exhibits an exponential T dependence. To clarify the point further, we follow the development of the system along the oe line. Above T>a the diblocks are nearly symmetric with NA ~ NB giving rise to R2(A) NAb2 and R 2 ~ Neb 2. When the temperature is lowered below TM there is no change in

Fig. 10. A section of a lamellar mesogel. The two exterior A layers are glassy. The interior B layer consists of LCPs swollen by a nematic solvent. Some of the LCPs bridge two different A domains. The A layers provide mechanical integrity for the system.

LIQUID CRYSTALLINE POLYMERS AT INTERFACES

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the behaviour of A. However, the number of steps of the random walk determining RILois reduced from N8 to no and the average step length increases from B to L/no. The elastic free energy of the B block is thus lowered to Fel (RO/kT ~ (R~ ~if)no. Accordingly, the two blocks are no longer symmetric since the elasticity of the B block is now much softer. Consequently, the system favours the spherical mesophase with B cores. As the temperature is lowered further, no decreases and the LCPs approach rod-like configurations. In this limit the lamellar phase is again favoured. 8.1. Triblock copolymers and nematic mesogels The phase diagram of AnBm diblock copolymers and of A,,B2mAn triblock copolymers is typically indistinguishable. The mesophases formed by the triblock copolymers exhibit however bridging of A domains by B blocks, thus giving rise to thermotropic elastomers. 54-56 When the B block is a thermotropic LCP, the bridging fraction can be significantly different. In particular, the equilibrium fraction of bridging chains for T < T• is enhanced by the nematic field.57 These systems also afford interesting opportunities for the design of "smart materials". A possible design involves lamellar mesogels incorporating LCP B blocks and swollen by nematic solvents. Such materials should be obtainable by shear treatment of lamellar melt followed by a quench to below the Tg of the A blocks. The resulting "lamellar mesonetwork" consists of a stack of doubly grafted LCP B blocks bridging slits formed by the glassy A layers.54 A nematic mesogel is then obtained by swelling the sample with a selective nematic solvent 57 (Fig. 10). The anchoring conditions imposed are determined by proper selection of the A blocks. In the following we briefly remark on the expected behaviour in the case of perpendicular anchoring. A variety of mechano-optic, electro-mechanic and electro-optic effects are expected. Among them: 37 (i) Mechanical shear of the lamellar mesogel should induce a nematic distortion. This is due to the coupling between the nematic solvent and the bridging chains which are rigidly anchored to the sheared A layers. (ii) An electric field inducing a Frederiks transition in the nematic layers is expected to cause shear strain. This quasi-piezoelectric effect arises because the field-induced distortion couples to the A layers via the bridging B blocks. (iii) The shear modulus of a nematic mesogel can be made vanishingly small by subjecting it to a suitable subcritical field. This reflects the diverging susceptibility of the system as it approaches the critical voltage of the Frederiks transition. 9. C O N C L U D I N G

REMARKS

The study of main-chain LCPs has focused, thus far, on the bulk properties of melts. This orientation was apparently motivated by the interest in structural polymers of this family. When viewed from the perspective of polymer science, a number of topics are conspicuous in their neglect: solutions of LCPs in nematic solvents, interfacial behaviour, the properties of block copolymers incorporating LCP blocks. All of these systems exhibit distinctive features compared with the familiar scenarios obtained with isotropic, flexible polymers. Among the examples of such features are the Ising extension elasticity and the interfacial phase transitions. When compared with the behaviour of monomeric liquid crystals, the situation is somewhat less dramatic. Many of the polymeric effects do have monomeric

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counterparts. Yet the LCP effects do have uniquely polymeric features. Thus, a polymeric system can undergo a Frederiks transition which is triggered by pressure rather than by an electric or magnetic field. This rich behaviour is not only of academic interest. The phenomenology predicted in this review suggests a novel, rational strategy to improve the performance of liquid crystalline displays. In particular, it suggests a strategy for the reduction of the switching voltage. It also suggests a rotational design for smart materials exhibiting a variety of electro-optic, electro-mechanic and mechano-optic effects. Very little experimental work exists in this area. However, this is not due to fundamental difficulties. The materials are available: the existence of LCPs soluble in nematic solvents is well established. There is strong evidence for the existence of hairpins. Recent synthetic developments suggest that anionic synthesis of long main-chain LCPs of low polydispersity is feasible. A suitable experimental technique is also available using the Surface Force Apparatus (SFA). This has already been used to study monomeric liquid crystals. 58 The technology for applying fields within the SFA is well established. 59 The mica surfaces used in the SFA are known to impose homeotropic anchoring conditions. It is also known how to modify this anchoring. Finally, the SFA does allow for optical characterization of the confined material. Computer simulation is also lacking in this field. However, as in the experimental domain, this is not due to fundamental difficulties. It is hoped that this review may contribute to the intensification of the activity in this area. ACKNOWLEDGEMENTS

D.R.M.W. was funded by a QEII Fellowship. This contribution was written while he was a visitor at the CRPCSS/CNRS in Mulhouse. He thanks the members of the laboratory for their hospitality during his visit. REFERENCES

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