Layer fluctuations and hexatic order in liquid crystals

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Layer fluctuations and hexatic order in liquid crystals

Jonathan V. Selinger

To cite this version:

Jonathan V. Selinger. Layer fluctuations and hexatic order in liquid crystals. Journal de Physique,

1988, 49 (8), pp.1387-1396. �10.1051/jphys:019880049080138700�. �jpa-00210819�

E 1387

Layer fluctuations and hexatic order in liquid crystals

Jonathan V. Selinger

Department ^of Physics, ^Harvard University, Cambridge, ^{MA 02138, U.S.A.}

(Requ le 14 octobre 1987, accepté ^sous forme définitive le 25 avril 1988)

Résumé.

Dans les phases smectique-A ^et hexatique-B les couches ne sont pas planes ^mais présentent ^des

fluctuations de grande amplitude. Afin de déterminer l’effet de ces fluctuations sur la transition smectique-A- hexatique-B ^nous construisons un hamiltonien de Ginzburg-Landau qui dépend ^du paramètre ^d’ordre hexatique 03C8 ^et des fluctuations u. Le couplage ^des champs 03C8 ^{et u est} dû à la frustration d’origine géométrique

introduite par la courbure des couches. Nous intégrons les fluctuations u pour obtenir ^un hamiltonien effectif

en fonction de 03C8. ^Cette intégration produit ^une renormalisation finie des coefficients de l’hamiltonien effectif.

Par ce mécanisme, la fluctuation des couches peut transformer la transition smectique-A-hexatique-B ^{en une}

transition du premier ordre si les constantes de rigidité hexatiques ^{sont assez} grandes. ^{Cet effet est} peut-être apparenté ^aux anomalies de la chaleur spécifique ^observ6es expérimentalement à ^cette transition.

Abstract.

In the smectic-A and hexatic-B phases, ^the layers ^{are not} planar ^but undergo large fluctuations. In order to find the effect of these fluctuations on the smectic-A-hexatic-B transition, ^{we construct a} Ginzburg-

Landau Hamiltonian in terms of the hexatic order parameter 03C8 ^{and the} layer fluctuations u. The fields 03C8 ^{and u are} coupled because of the geometrical frustration introduced by ^{the curvature of the} layers. ^We integrate ^{out the u field to obtain an} effective Hamiltonian in terms of 03C8 alone. This integration ^{leads to a finite}

renormalization of the coefficients in the effective Hamiltonian. By ^this mechanism, layer fluctuations may drive the smectic-A-hexatic-B transition to be first-order if the hexatic stiffness constants are sufficiently large.

This effect may be related ^to observed anomalies in the specific ^{heat at} that transition.

J. Phys. ^{France 49} (1988) ^{1387-1396 AOÙT} 1988,

Classification

Physics ^Abstracts

64.70M - 61.30

64.60C

1. Introduction.

Three-dimensional (3D) layered liquid crystal sys- tems can exhibit several phases ^{with different} types of in-plane ^{2D order. The} least-ordered layered phase is the smectic-A phase, ^{which has} short-range

bond-orientational and positional ^{order within each} layer. ^{At lower} temperature ^the system ^{can form the} hexatic-B phase, which has long-range ^bond-

orientational order but only short-range positional

order within each layer [1-3]. ^{At even lower tempera-}

ture the system freezes into a crystalline phase, ^with long-range bond-orientational and positional ^order.

In the bulk smectic-A and hexatic-B phases, ^the

molecules do not lie in static, planar layers ; rather,

the layers ^are constantly fluctuating. In any layered system ^without long-range in-plane positional order,

the magnitude ^{of the} Landau-Peierls [4, 5] ^fluctu-

ations of the layers diverges logarithmically ^{with the}

system ^size [6, 7]. ^The smectic-A-hexatic-B ordering

transition therefore takes place ^{in a} system ^of

strongly-fluctuating layers. ^{It is} interesting ^{to con-}

sider the effects of these strong fluctuations on the smectic-A-hexatic-B transition.

In a recent paper Nelson and Peliti [8] derived the elastic Hamiltonian of a single curved hexatic mem-

brane. They ^showed that the Hamiltonian must

involve the covariant derivative of the bond-angle

field 0 (x ) to account for the frustration due to the membrane curvature. In this paper ^we extend their

theory ^to the stacked system ^{of curved} layers ^{in a}

smectic-A or hexatic-B liquid crystal. ^We thereby

obtain a coupling ^{between the} coarse-grained ^hexatic

order parameter .p (x) = e6 i 8 (x» ^{and the layer}

fluctuations u (x). ^This Hamiltonian involves a vector

potential ^{term which is similar to the} coupling

between the superconductor ^order parameter ^and the electromagnetic field in the normal-supercon-

ductor transition, ^{and to the} coupling ^{between the}

smectic order parameter and nematic director fluctu- ations in the nematic-smectic transition. Halperin, Lubensky, ^{and Ma} [9, 10] ^{have studied how fluctu-}

ations in the electromagnetic ^{field or} the nematic director field affect the latter two transitions. By

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049080138700

integrating ^{out these} fluctuations, they ^{show that}

these transitions are driven weakly first-order. In this paper ^we perform ^the analogous calculation for the smectic-A-hexatic-B transition : we integrate ^out

the u field to obtain an effective Hamiltonian in terms of qi alone. In this case, integration ^{of the}

u field leads only ^{to a finite} renormalization of the coefficients of the terms in the effective Hamiltonian.

Coupling ^with layer fluctuations may or may ^not

cause the smectic-A-hexatic-B transition to be first-

order, depending ^on the size of the coefficients. In

particular, the transition will be driven first-order if the hexatic stiffness constants are sufficiently large.

This coupling between hexatic order and layer

fluctuations _may help ^to explain ^{observed anomalies}

in the specific ^{heat at the} smectic-A-hexatic-B tran-

sition. Experiments have studied this transition in several pure and binary liquid crystal systems [11- 17]. ^{In some the} transition was found to have a large specific ^heat exponent ^{a = 0.48 to 0.67. In others} the transition showed some thermal hysteresis ^and

was judged ^{to be} weakly first-order. The large ^value

of ^a is close to the value ^{a =} 0.5 that one would expect ^{from a} nearby tricritical point. ^For systems with herringbone packing ^order, Bruinsma and Aep- pli [18] have attributed the tricritical point ^to coupl- ing between the hexatic and herringbone ^order

parameters. ^For systems ^without herringbone order, Aharony ^{et al.} [19] ^have argued ^{that a} tricritical

point should exist on the smectic-A-hexatic-B line

near the smectic-hexatic-crystal triple point ^because

of coupling between the hexatic and crystalline ^order parameters. ^The coupling between hexatic order and

layer fluctuations discussed in this paper provides

an additional mechanism for the transition to be driven first-order. If the tricritical point hypothesis ^is

correct, ^then this coupling ^{should move} the tricritical

point ^{farther from the} triple point ^{than one would} expect ^{from the} argument ^of Aharony ^{et al.}

In principle ^the theory in this paper should also be relevant to the smectic-A-smectic-C transition. That transition is described by ^the complex order par- ameter 0 ⁼ w e where w is the angle of the tilt away from the layer ^normal and 0 ^{is the azimuthal} angle [20]. By ^the argument of Nelson and Peliti, ^the Hamiltonian should involve covariant derivatives of 03A6 , ^{and the} analysis of this paper should go through exactly ^{as in the hexatic case. However,} experiments

have shown that the smectic-A-smectic-C transition is well described by ^a simple ^{mean field} theory ⁱⁿ

03A6 with an unusually large sixth-order term [21-23].

To the best of our knowledge, ^no experiments ^have

seen a weakly first-order transition or an anomalous

peak ^{in the} specific heat that could be due to

coupling between the tilt order parameter ^and layer

fluctuations. This coupling ^{must be too weak to}

drive the smectic-A-smectic-C transition first-order, probably ^because smectic-C stiffness constants are

typically much weaker than the corresponding ^hexa-

tic stiffness constants [24].

The plan of this paper is ^as follows. In section 2 we

extend the Nelson-Peliti theory ^{of a} single ^hexatic

membrane to a 3D stacked hexatic liquid crystal ^and

obtain expressions ^{for the} intralayer ^and interlayer coupling. ^{In section 3 we} integrate ^out layer ^fluctu-

ations to obtain an effective Hamiltonian for the hexatic order parameter t/1, ^{and we} show that the

coupling ^with layer fluctuations can cause the smec-

tic-A-hexatic-B transition to be first-order if the hexatic stiffness constants are large enough.

2. Coupling between hexatic order and layer ^fluctu-

ations.

In any layered liquid crystal system, ^each molecule is surrounded by (on average) ^{six nearest} neighbours

in the same layer. ^{In the hexatic-B} phase, ^{there are} long-range correlations in the orientations of the

nearest-neighbour bonds. To describe this phase,

the Hamiltonian must contain a term that tends to

align the bond orientations at different points ^{in the} system. ^{To construct such a term, one must define} the local bond angle field, ^{and one must find a}

method for comparing ^{the bond} orientations at different points. ^{If the} layers ^are planar, ^{then one}

can measure the angles of all the bonds relative to a

single fixed axis (e.g. ^the x-axis) ^{in the} layer planes,

and thereby ^{obtain a} bond-angle ^field 0 (x) ^defined

modulo 2 7r/6. ^{One can} compare the bond orien- tations at x and x’ by simply taking the difference

0 (x’) - 0 (x). ^{The hexatic} elastic Hamiltonian is therefore

where ^z is the coordinate normal to the layers. ^In general the elastic coefficients KA ^and KN ^for

variations within and normal to the layers may be different.

If the layers ^{are not} planar, then it is much more

difficult to construct the hexatic elastic Hamiltonian.

The problem is illustrated in figure 1. At any given point ^{on a} layer, ^{the six} nearest-neighbour ^{bonds lie}

in the local tangent plane ^{to the} layer, ^not in the xy-

plane. Because the tangent plane varies from point

to point, ^{one cannot measure the bond} angles ^{in a} single plane ^{relative to a} single ^fixed axis, ^{and one}

must find another way ^to compare the bond orien- tations at different points. Nelson and Peliti [8] ^have

studied this problem ^{for a} single 2D hexatic mem-

brane. They ^use parallel transport ^to compare the bond vectors at different points ^{on the same mem-} brane, ^and they ^{arrive at a} hexatic elastic Hamilto- nian involving the covariant derivative of the bond-

angle field. This parallel-transport ^{method is not}

adequate ^{for a} 3D stacked hexatic system ^because

1389

Fig. ^1.

Comparison of the bond orientations in the tangent planes ^{at x} and x’. The vector v lies in both tangent planes.

one cannot parallel transport ^{bond vectors from one} layer ^{to another.} In this section we develop ^a general

method for comparing ^{the bond} orientations at

different points ^{in the same or different} layers, ^and

we use it to construct the elastic Hamiltonian of a

stacked hexatic system. ^{In the} appendix ^{we show}

that our method is equivalent ^to parallel transport ⁱⁿ the case of a single hexatic membrane.

Consider a layered liquid crystal system ^{with the} layers specified by

where n is an integer, ^{d is the} equilibrium interlayer spacing, ^and u (x) ^{is the} layer displacement ^{field. At}

any point ^on any of the layers, ^{there is a normal}

vector and a tangent plane. The normal vector is

and a natural choice of basis for the tangent plane ^is

This basis is not orthonormal because

In this paper ^we will use the convention that Greek indices run over the values 1 and 2, ^{and Latin indices}

run over the values 1, 2, ^and 3, unless otherwise noted. We can define an orthonormal basis for the tangent plane by

where c(x) ^and è(x) ^{are chosen so that}

Consistency ^of (2.6a) ^and (2.6b) implies

and (2.7) implies

It is straightforward ^{to solve} (2.8) ^and (2.9) ^{to obtain}

exact expressions ^for c(x) ^and c (x ), valid in any coordinate system. ^{At a later} stage ^of this calcu-

lation, it will be convenient to work in a coordinate system ⁱⁿ which z is the average direction normal to the layers and Vu is small. At lowest order in

Vu,

This approximation ^for c(x) ^and c (x ) gives e03B1 ^to

second order in Vu.

If we neglect disclinations, then each molecule in each layer ^{has six} nearest-neighbour bonds in the six tangent directions given by

The field 0 (x) specifies ^{the bond} angles in the local tangent plane ^{relative to the} locally-defined ^axis ê1. Now consider the bond configurations ^{at two} nearby points ^{x and x’,} which may be in the ^{same or} different layers. ^The geometry ^{is shown in} figure ^1.

We cannot compare the bond configurations by simply subtracting 0 (x’) - 0 (x), ^because those ang- les are defined in different tangent planes ^{relative to}

different axes. To compare them, ^{we must use a}

common reference direction in both tangent planes.

To find a reference direction, ^{note that the two} tangent planes ^{have a common} tangent ^{vector v,} which is unique up ^to scalar multiplication. ^The

vector v can be expressed ^as

in the orthonormal bases at x and x’. We can use v as

a benchmark for comparing ^the bond orientations at

x and x’. The angle 0 (x) - 0 gives ^{the bond}

orientation relative to v in the tangent plane ^{at x,}

and 0 (x’) - .0’ gives ^{the same at x’. If}

0 (x) - 0 ^{= 0} (x’ ) - 0’, ^{then we can} map the bond

configuration ^{at x onto the bond} configuration ^at

x’ by ^a single rotation about _v, and the bond

configurations ^{are as} closely aligned ^as they ^{can be.}

We hypothesize that there is an energy penalty ^for disalignment ^of

for d0 = O(x’)-O(x) ^{and d0 =0’-O both}

small. Note that 0 ^{is not} just ^{a function of x ; rather,} cp ^and -0’ ^are both derived from v, which is derived from the intersection of the tangent planes ^{at x and}

x’. In the appendix ^{we show that} comparison ^{of the}

bond orientations with respect ^{to v is} equivalent ^to comparison by parallel transport ^{if x and x’ are in the}

same layer.

Because of the So term, ^{the hexatic elastic} Hamiltonian couples ^the bond-angle ^field 0 (x) ^and

the layer fluctuations u (x ). ^{To see this} coupling explicitly, ^{we must derive} expressions ^{for v and} 5 0 ^{in terms of u. We can let v = n x} n’, ^{where n}

and n’ are the normal vectors at x and x’ respectively.

In terms of the natural bases for the tangent planes

at x and x’ we have

where ððiU ⁼ aju (x’) - aju (x) ^and Ea{3 is the anti-

symmetric symbol ^with E12 =1. In the limit of small ðx ⁼ x’ - x, 5 0 ^is determined by

At this point ^{we can} greatly simplify the calculations

by choosing ^a coordinate system ^{in which z is the} average direction normal to the layers ^{and Vu is}

small. If we combine equations (2.15), (2.14), (2.6)

and (2.10), ^and expand ^all quantities ^to lowest order in Vu, ^{we obtain}

For small 6x, ^{we can write}

where

or in standard 3D notation

A is a vector potential ^{for the} bond-angle ^{field in this} system. Because 0 ^{is not} just ^a function of _x, A is not

necessarily ^the gradient ^{of a scalar function.}

We have argued ^that the energy penalty ^for

variations in 0 (x) ^{between x} and x’ should be

proportional ^to [ (V 0 + A ) . SX]2 . ^The coefficient may be different for intralayer ^and interlayer ^vari-

ations. We can therefore use the Hamiltonian

We have shown that A is of order (Vu )2. ^{If we}

assume that VO is also of order (ou )2 (we ^{will check}

this assumption self-consistently ^{in the next} section),

then we can neglect ^the difference between n and z to obtain the Hamiltonian

valid to order (Vu )4. ^{If we} rescale distances in the z-

direction by ^{a factor of} eo = .J KN/ KA, ^{the Hamil-}

tonian simplifies yet ^{further to}

where KA ^{has been} implicitly ^rescaled.

So far we have considered systems ^{that are well} inside the hexatic phase ^{and hence} have well-estab- lished hexatic order. In these systems ^{the coarse-}

grained hexatic order parameter t/J (x) = e 6 i 0 (x))

has constant amplitude. ^At higher temperatures, close to the smectic-A-hexatic-B transition, ^{we must}

consider variations in the amplitude ^{as well as the} phase ^of 03C8. ^{In that case the} Hamiltonian becomes

The constants KA ^and KA ^{are related} by

KA ⁼ KÁ 1 t/J 2 ^{at low} temperature.

1391

3. Integration ^of layer fluctuations.

We have derived a geometrical coupling (2.22)

between the hexatic order parameter w ^{and a vector} potential ^{A due to} layer fluctuations. This coupling ^is analogous ^{to the} coupling between the superconduc-

tor order parameter 4/ ^{and the} electromagnetic

vector potential ^{A in the} normal-superconducting

transition. It is also analogous ^{to a} coupling ^{term in}

the nematic-smectic-A transition, where w ^corres- ponds ^{to the} amplitude ^and phase of the smectic

density ^{wave and A to} the nematic director field.

Halperin, Lubensky, ^{and Ma} [9] [10] ^{have shown}

that the critical properties ^{of the} normal-supercon- ducting and nematic-smectic-A transitions are ident- ical : in each case, fluctuations in the A field drive the transition from w ^{= 0} to qi :0 ^{0 to be} weakly first-order, ^at least in 4 - E dimensions. Although

the smectic-A-hexatic-B transition is similar to the other two transitions, ^{it is not} exactly equivalent ^to

them because in this transition the vector potential ^A

is really ^a composite ^of derivatives of the u field. It is therefore interesting ^{to ask what effect} fluctuations in A have on the smectic-A-hexatic-B transition. In this section we answer that question by integrating

out fluctuations in A to obtain an effective Hamilto- nian in terms of the hexatic order parameter 03C8 ^alone.

As a first step ^{in this} calculation, ^we must construct the full Hamiltonian for the hexatic order parameter and layer fluctuations. In addition to the coupling

term (2.22) developed in the last section, ^{the Hamil-}

tonian must contain (a) ^a Ginzburg-Landau expan- sion in It/J 2 ^and (b) ^the energy of layer fluctuations

u (x ). ^The Ginzburg-Landau expansion ^can easily ^be

written as

Grinstein and Pelcovits [25] have shown that the lowest-order rotationally-invariant form for the en-

ergy of layer fluctuations is

where distances in the x, y, and z-directions ^are measured in the same units. If we work in a

coordinate system ^{in which z} is the average direction normal to the layers, ^{then the} cubic and quartic

terms in Vu will be small. Grinstein and Pelcovits have shown that these terms give only ^a logarithmic

renormalization of the elastic constants B and

Kl ^at long wavelengths. We will therefore neglect

these terms for the purpose of assessing ^the signifi-

cance of the coupling ^{between qi and u. Further-}

more, ^{we can} neglect ^the a2U ^{terms in} comparison

with the a,u ^{term at} long wavelength. ^{If we now}

rescale distances in the z-direction by ^{a factor of}

= J KN/ KA, ^and implicitly rescale all coef- ficients in (3.1) ^and (3.2) appropriately, ^{we obtain}

the total Hamiltonian

For any specific realization of the layer ^fluctu-

ations u (x), ^{the state} qi (x) = ^{constant does not}

minimize the Hamiltonian. This fact is not surpris- ing : ^{if the} layers ^are curved, ^{there is no reason to} expect ^{the state in} which all bond angles ^are aligned

with the locally-defined ^axis ê1 ^{to be the} ground

state. Suppose ^the amplitude ^of hexatic order

141 (x) I ^{is constant} in space. To find the ground ^state

of the bond-angle ^{field, we write the vector} potential

as

where P j ⁼ 6ij - °iOj/V2 is the 3D transverse pro-

jection operator. ^The ground ^{state occurs when the} phase of qi (x) satisfies 0 (x) ⁺ eo L (x ) ^{= constant.}

We therefore make a gauge transformation [10] by defining

Variations in 0 (x ), ^the phase ^of IV (x), give ^the

fluctuations in the bond-angle field away from the

ground ^state for any specific realization of u (x). ^In

terms of 03C8 (x), ^the Hamiltonian becomes

The transverse part ^{of A} gives ^{an extra} energy that cannot be reduced by any configuration of the bond-

angle ^{field. It} gives ^a geometrical frustration in the hexatic phase.

Because the state 11 (x) ^{= constant} minimizes the

Hamiltonian, we use mean field theory ⁱⁿ 1/1’ (x) ^and neglect fluctuations in 0(x). ^{In this mean field} theory ^VO is of order A, ^which is of order

(Vu )2. ^In physical ^{terms, this mean field} theory

assumes that the correlation length ^{for the} amplitude

of the hexatic order parameter ^{is much} greater ^than

the correlation length ^for layer fluctuations. In the

analogous superconductor problem, ^this assumption corresponds ^{to the} assumption ^{that the coherence} length 6 ^{is much} greater ^than the London pen- etration depth A, ^{or that the} superconductor ^is type ^I. We will discuss the validity ^{of this} approxi-

mation below.

To obtain an effective Hamiltonian as a function

of W alone, ^{we must} integrate ^out the fluctuations in

u by writing

With the neglect ^of spatial fluctuations in 1/1’ (x), equation (3.7) implies

where fi is the system ^{volume. We must now} calculate the expectation ^value (f d3x ^{PT A 2 as a function}

of 1/1’. In contrast with the superconductor problem, ^{we cannot} evaluate this expectation ^value exactly

because of the quartic ^{term in u that is} implicit in 1 p T A 12. ^{Rather, we must use} perturbation theory ⁱⁿ

Kg e)[ W [ 03C8. ^Let Ho represent ^{the first two terms in} equation (3.6), and write the coupling ^{term as}

r

.1 f

The expectation value is then

where (. ) o denotes the average with respect ^to the Gaussian weighting function e - f3Ho.

We can express equation (3.10) ^{as a} graphical perturbation ^{series. Let the vertex in} figure ² represent

Fig. ^2.

^Vertex corresponding ^{to the} operator

Each slash over a leg represents ^{a factor of} momentum, and the dashed line indicates how to contract the momenta with the transverse projection operators. ^At zero-th order in KA e 0 21 T 2, ^the only non-vanishing

contributions to

(f d3x I p T A 12) ^{V, are the graphs in figure 3, which represent}

1393

Fig. ^3.

Contribution to

( d’x I pTA 12 ) ^{w at} zero-th order in Kg e) [ W [.2

This term is IR convergent ⁱⁿ 3D, ^{and it can} be made finite by ^{the use of a UV} cutoff. When inserted into

equation (3.8), ^{this term} gives ^{a finite} correction to the coefficient a in Heff (03C8).

The contribution to

( f d 3X I pT A12 ) ^{W at} first order in KÁ eJI1Ji’ 12 ^is

which is explicitly negative. ^This contribution is

represented by ^the graphs ^{with the} loop ^structures

shown in figure 4. Because of the transverse projec-

tion operator, ^{none of these} graphs factor into

independent integrals. By power counting, ^{all of}

these graphs ^{are IR} convergent in 3D. When inserted into equation (3.8), ^this contribution to

(f d3x I pT A 12) ^{gives a finite, negative correction}

to the coefficient b in Heff (AP ).

Fig. ^4.

Loop structures of the contributions to

f d’x I pT A 12) ^at first order in K, A e21 0 lp 2.

In the normal-superconducting ^and nematic-smec- tic-A problems, integration ^of fluctuations in the vector potential generates ^a non-analytic ^{term in} Heff( 1/1’). ^{This term} goes as - IT14-, ^{in 4 - e}

dimensions, ^{or as} + I T I I In 11/1’ I in 4 dimensions.

This non-analytic ^term is what forces those tran-

sitions to be weakly first-order. If such a term were

present ⁱⁿ Heff (T) ^{in the} smectic-A-hexatic-B ^tran-

sition, ^then a2 H eft/ a ( 11/1’ 12)2 ^would diverge ^at

1/1’ ⁼ 0, and there would be an infinite, negative

correction to the coefficient b. Because this correc-

tion is only finite, ^no non-analytic ^term of this form is present, ^and Heft ( 1/1’) ^{can be written as}

where the subscript ^R indicates renormalization of a and b. The Halperin-Lubensky-Ma ^mechanism

therefore does not force the smectic-A-hexatic-B transition to be first-order.

As an aside, ^we could also calculate how the

coupling ^term in the Hamiltonian (3.6) renormalizes the elastic constants B and Kl ^{in the hexatic} phase, just ^{as Grinstein} and Pelcovits [25] calculated how the (ozu) (Vu )2 ^and (VU)4 ^terms renormalize B and

Kl. ^{In the case of a} single ^{2D hexatic} membrane, Nelson and Peliti [8] have shown that coupling ^with

hexatic order gives ^a logarithmically divergent, wavevector-dependent correction to the rigidity

K. In a 3D stacked hexatic, however, ^the corrections to B and Kl ^are convergent. Coupling ^{with hexatic}

order in 3D gives only ^{a finite} renormalization of these elastic constants.

Although ^the coupling ^{between hexatic order and} layer fluctuations does not generate divergent ^correc-

tions to b, B, ^or Ki, it may still have important

effects on the smectic-A-hexatic-B transition. If the hexatic stiffness constants KA ^and KN ^are sufficiently large, ^{then the} coupling may drive bR negative. ^In

that _case, Heff(T) ^{must contain a} positive ^{term of}

3CR I T I ’. ^for thermodynamic stability. ^{We would}

then get ^a first-order transition from the smectic-A

phase ^to the hexatic-B phase. ^{As usual we assume}

that aR OC (T - TO), ^where To ^is constant, ^and

bR and CR ^are independent ^of temperature. ^{In mean} field theory ⁱⁿ 1/1’, ^the first-order transition occurs at

and the value of the hexatic order parameter just

below the transition is

We can use the Ginzburg ^criterion [26] ^{to deter-}

mine whether fluctuations in 1/1’ invalidate mean

field theory. According ^{to the} Ginzburg criterion,

fluctuations have a significant ^effect only ^{inside a}

critical region I aR I ^- aGe If the hexatic stiffness constants are sufficiently large ^that bR ^{0 and}

3 bR/16 CR > aG, then ^mean field theory ^{is self-}

consistent. In that case the second-order transition at aR ⁼ 0 is preempted by ^the first-order transition, and aR ^{= 0 is} only ^{the limit of} metastability ^{of the}

smectic-A phase. ^{However, if 3} bR/16 CR aG then

mean field theory ^{is not} self-consistent.

It is difficult to make a quantitative estimate of how large the hexatic stiffness constants must be to induce a first-order smectic-A-hexatic-B transition for two reasons. First, ^{all the} integrals ^{that contribute}

to the renormalization of b are quite ^UV divergent.

Their numerical values are therefore extremely

sensitive to the choice of the UV cutoffs on

qz and q, . Second, ^{we cannot} determine the renor-

malized coefficient bR ^{unless we} know the bare coefficient b. This bare coefficient is unobservable.

However, ^{we can make some} general comparisons

between different systems. ^{As mentioned in the} introduction, ^the theory ^{in this} paper should apply ^to

the smectic-A-smectic-C transition as well as to the smectic-A-hexatic-B transition. By ^the argument ^of section 2, the hexatic-B and smectic-C phases ^should

have the same coupling ^between orientational order and layer fluctuations, ^{where the} bond-angle ^{field in}

the hexatic-B phase corresponds ^to the azimuthal

angle ^{of the} tilt in the smectic-C phase. ^{Because of} packing constraints, ^{we would} expect the hexatic-B stiffness constants to be larger ^{than the} correspond- ing ^{stiffness constants in the smectic-C} phase. ^Recent experiments ^{on thin} liquid crystal films confirm that hexatic stiffness constants are much greater ^than smectic-C stiffness constants [24]. ^{For that reason,}

the coupling ⁱⁿ this paper is ^more likely ^{to affect the}

smectic-A-hexatic-B transition than to affect the smectic-A-smectic-C transition.

It is interesting ^to compare the results of ^our calculations with the observed anomalies in the smectic-A-hexatic-B transition. As noted in sec-

tion 1, this transition is observed to be second-order with a specific-heat exponent ^{a = 0.48 to 0.67 in}

some systems, ^and weakly first-order in other sys- tems. The large ^value of a in the former case could arise from the influence of a nearby tricritical point,

which Bruinsma and Aeppli ^{attribute to} coupling

between hexatic and herringbone ^order parameters and Aharony et ^{al. to} coupling between hexatic and

crystalline ^order parameters. This tricritical point

must exist in the temperature-concentration phase diagram ^{of the} 650BC-40.8 binary system, ^which shows a second-order transition at low concentration and a first-order transition at higher concentration of 40.8 [13]. However, ^no tricritical point ^{has been}

seen in the temperature-chain length phase diagram

of the homologous ^series nmOBC, ^which always

shows a second-order smectic-A-hexatic-B transition with a large ^{value of a} [17]. ^{If that} system ^{has a} tricritical point, ^{it must be} unobservably ^{close to the} smectic-hexatic-crystal triple point. ^{It is} surprising

but possible ^{that such a} tricritical point ^{could affect}

a for a wide range of chain length. Coupling

between hexatic order and layer fluctuations _pro- vides an additional reason for the smectic-A-hexatic- B transition to be driven first-order in a part ^{of the} phase diagram ^{where the} hexatic stiffness constants are large. ^{It therefore} provides ^{an additional}

mechanism to generate ^a tricritical point, ^which

could account for a large ^{value of a.}

Acknowledgments.

I am grateful ^{to D. R.} Nelson for suggesting ^this problem and for many helpful discussions about it,

and to R. G. Petschek for drawing my attention ^to

errors in a previous version of this paper. This work

was supported by the National Science Foundation

through ^{Grant No. DMR 85-14638 and} through ^the

Harvard Materials Research Laboratory.

Appendix.

In section 2 we develop ^a general method for

comparing ^the bond orientations at two nearby points ^x and x’ in the same or different layers. ^{If the} points ^{are in the same} layer, ^{then we} could also compare the bond orientations by ^{the more standard}

method of parallel transport: ^{we could} parallel- transport ^{the bond vectors} ba along ^a geodesic ^from

the tangent plane ^{at x to the} tangent plane ^at

x’ and then compare them with the bond ^{vectors at} x’. In the limit of x’ - ^{x -+} 0, ^{tlfis method} gives ^the

covariant derivative of the bond-vector field in the

given layer. Nelson and Peliti [8] ^use this covariant

derivative to construct the elastic Hamiltonian of a

1395

single hexatic membrane. In this appendix ^{we show}

that comparing bond orientations by the method of section 2 gives ^{the same result as} comparing ^bond

orientations by parallel transport ^{if x and x’ are in}

the same layer. Our Hamiltonian (2.21) is therefore

a generalization ^of the Hamiltonian of Nelson and Peliti.

To see that the two methods of comparing ^bond

orientations are equivalent, ^{consider a} single layer ⁱⁿ

the Monge parametrization x(o-) = (o- , 0’

u (o,- 1, Q 2)). The natural basis for the tangent plane

at any point ^is ta ⁼ a a x, ^where a,, =- d/do-a. ^{In the} Monge parametrization, ^{the metric tensor is}

and the connection coefficients are

As in section 2, ^{let v be a vector} that is in the tangent planes ^{at both x and x’.} By ^the arguments ^of section 2, ^{it can be} expressed ^{as v = v "} ta ^{= v’"} ta,

where

in the tangent planes ^{at both x and x’.}

Now consider the two methods of comparing ^bond

orientations geometrically. In the method of sec-

tion 2, ^{we use the common} tangent ^{vector v as a} reference direction to compare the bond angles ⁱⁿ

the tangent planes ^{at x} and x’. In the parallel- transport method, ^{we draw a} geodesic ^{from x to}

x’ and use the geodesic ^{as a} reference direction in the two tangent planes. ^{Those methods are consist-}

ent if the angle ^{between v and the} geodesic ^{at x} equals ^the angle ^{between v and the} geodesic ^at

x’. Suppose ^{that the} geodesic ^has tangent ^{vector G at}

x ⁼ x(o-) ^and tangent ^{vector G’ at x’ =} x (cr ⁺ Su).

We then obtain

For small 8 cr , ^the angle ^{between v and the} geodesic

varies only ^as (8 u )2. Comparison ^with respect ^{to v}

and comparison by parallel transport ^{are therefore,} equivalent ways of comparing bond orientations at

nearby points.

We can also show explicitly that the hexatic Hamiltonian of Nelson and Peliti is consistent with the Hamiltonian developed in section 2. We define a

local orthonormal basis ê(IA-)( U ) ⁼ e (IA-) ^{ta for the}

tangent plane ^{at (1, and let} b = cos 0 e (1) ^{+ sin} 6ê(2) } be any of the six bond-vector fields. If we compare

b at nearby points by parallel transport, ^{we obtain} the elastic Hamiltonian

where d2s is the element of area and Vab13 ^{is the}

covariant derivative of b. Nelson and Peliti show that this Hamiltonian reduces to

where Aa is defined by

They ^calculate Eaf3 ðaAf3 and hence the transverse

component of the 2D vector potential ^{A for a} general ^basis ê(JL )(0’). ^{We can} calculate both compo- nents of A with a specific ^choice of ê (JL ) (0’ ). ^Specializ-

ing (2.6a) ^{to a} single ^2D layer, ^{we choose} e(JL) ⁼ SJLa - c(O’) ðJLu ðau. ^{In a} coordinate system ⁱⁿ which z is the average direction normal ^to the layers

and au is small, ^{we have} c -- 2 . ^We then obtain

explicitly

and hence

at lowest order in au. The vector potential (A.9) ^{is a}

2D special ^{case of the vector} potential (2.18). ^The

3D hexatic Hamiltonian derived in section 2 is therefore a generalization of the Hamiltonian (A.5)

derived from covariant differentiation.

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