New critical points in frustrated smectics

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New critical points in frustrated smectics

P. Barois, Jacques Prost, T.C. Lubensky

To cite this version:

P. Barois, Jacques Prost, T.C. Lubensky. New critical points in frustrated smectics. Journal de

Physique, 1985, 46 (3), pp.391-399. �10.1051/jphys:01985004603039100�. �jpa-00209978�

New critical points in frustrated smectics

P. Barois, ^{J. Prost}

Centre Paul Pascal, ^Domaine Universitaire, ³³⁴⁰⁵ Talence, ^France

and T. C. Lubensky

University ^of Pennsylvania, Department ^of Physics, Philadelphia, ^PA 19104-3859, U.S.A.

(Reçu ^{le 23} juillet ^1984, accepté le 5 novembre 1984)

Résumé.

Nous étudions un modèle phénoménologique de Landau de systèmes smectiques frustrés dans lequel

sont en compétition ^{deux vecteurs d’onde de} modulation k1 ^et k2 incommensurables. Outre les phases nématique

et smectique monocouche (SA1), ^les diagrammes ^de phase calculés montrent l’existence de deux phases ^bicouches

distinctes (SA2) séparées par ^une ligne de transition du premier ^{ordre se terminant en un} point critique. ^{Les inten-}

sités de diffusion des rayons X ^sont également calculées : elles conduisent à identifier l’une des phases SA2, présen-

tant une très faible modulation harmonique ^{au vecteur d’onde} (0, 0, ² qo), ^{à la} phase partiellement bicouche SAd

connue expérimentalement ^{Un nouveau} point bicritique ^{N-SA1-SA2 est mis en évidence en cas} de forte incom- mensurabilité.

Abstract.

A phenomenological model for frustrated smectics with competition ^{for order at} incommensurate wavevectors k1 ^and k2 ^{is used to} study phase diagrams and x-ray scattering intensity ^{within mean field} theory.

It is shown that in addition to the nematic (N) ^and monolayer (SA1) phases, ^the phase diagram may involve two distinct bilayer (SA2) phases separated by ^{a first order line} terminating ^{in a critical} point One of the SA2 phases

exhibits very little harmonic modulation and can be identified with the experimentally characterized partial bilayer

SAd phase. ^{In addition for} larger incommensurability, ^{an unusual} bicritical point ^where N, SA1 and SA2 phases

meet is identified.

Classification Physics ^Abstracts

61.30E - 64. 70M

1. Introduction.

In 1978 G. Sigaud and coworkers [1] challenged ^the

traditional classification of liquid crystals ^{with the} discovery ^{of a} transition between two phases ^macro- scopically classified as smectic A phases. ^{Since then}

no less than five distinct smectic A phases ^{have been}

identified in polar smectogenic compounds [2]. ^These phases ^{are most} easily distinguished by their x-ray

scattering intensities (Fig. 1). The classical mono-

layer ^{smectic A exhibits a} Bragg peak ^{at 2} qo =(2 nj l) ⁿ

where I is the molecular length ^{and n} the nematic director. In addition, ^there may be diffuse scattering

at wavevectors I q I 2 qo I. ^This phase ^{is the}

SA 1 phase. ^The bilayer ^SA2 phase ^is characterized

by ^two Bragg peaks ^at qo ⁼ (2 x/2 1) ^{n and 2} qo.

Finally ^the experimental signature [3] of the SAd

phase ^{is a} Bragg peak ^at qo ⁼ (2 x/1’) ^{n where}

Fig. ^1.

x-ray scattering intensities for the (a) nematic, (b) SAI, (c) ^{SA2 and} (d) ^SAd phases. The dark dots represent

quasi-Bragg peaks ^{and the dotted} ellipses ^diffuse scattering.

The vertical axis is along qo.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004603039100

392

1 l’ 2 1, generally ^{with diffuse} scattering ^for q I > qa 1. ^{In addition, there are} anti-phases [2, 4]

(A ^and C), ^{which will not concern us here, with} Bragg peaks ^{off of the n axis.}

Most of the above smectic polymorphism ^{has been} successfully explained ^{in terms of a} phenomeno- logical model introduced by ^Prost [5-7]. ^{The nature}

of the SAd phase and its relation to the SAI and SA2 phases ^have not, ^until now, been understood

completely. ^{In this} paper, ^we will use the Prost model and mean field theory ^to address this problem. ^Our

results can be summarized as follows. The SA1 phase

is characterized by ^the non-vanishing amplitude I l I ^{of the} Bragg peak ^{at 2} qo whereas the SA2

phase ^is characterized by ^{the two non-zero} ampli- tudes, I ql2 and I t/J 1 I, ^{of the} Bragg peaks ^at qo and 2 qo. For certain _ranges of model parameters, ^{we find} that there can be two bilayer phases, ^{which we label}

SA2 and SA2’ characterized by different values of the ratio i Ijl t/J 2 1 ^{and of} qo. In the SA2 phase, 102 1/1 4(l I ^{is not small, and} qo is of order 2 x/2 ¹

whereas in the SA2’ phase, I t/J 2 1/101 I ^{is small and}

qo is of order (2 x/1’) > (2 n/2l). ^We calculate the x-ray intensities for the two phases and find that the

Bragg peak ^at 2 qo in the SA2’ phase ^{can be} comple- tely ^masked by ^diffuse scattering. Thus, ^we identify

the SA2’ phase ^{with the} experimentally characterized SAd phase. ^The symmetries of the SA2 and SA2’

phases ^{are identical so there cannot be a second}

order phase transition between them. We find that there is a first order phase boundary along ^which

the SA2 and SA2’ phases ^can coexist that terminates in a critical point ^{C similar to the} liquid gas critical

point. ^The prediction ^{of this critical} point ^is new, and it would be of some interest to find a physical

system in which it is exhibited. The _cyano series of reference [8] ^{seem to be} good candidates for revealing

its existence. We note, however, ^{that other} phases

such as the A and C phases may intervene in real systems and eliminate the critical point ^{C. In addi-}

tion we find a mean field bicritical point ^{where the} N, SA1 and SA2’ phases ^{meet in} systems ^with large enough mismatch between I and I’. This bicritical

point ^{is unusual in that} it arises from a model in which there is a trilinear coupling (of ^{the form} I t/J 1 12

I t/J 2 I) between the two order parameters. ^{Such a} coupling usually ^{does not allow} t/J 1 and 02 ^{to be} simultaneously critical but rather causes a first order transition to occur before criticality.

The outline of this _paper is as follows. In section 2,

we define the model and discuss mean field theory.

In section 3, ^we discuss the phase diagrams predicted by ^{mean field} theory. ^{In section} 4, ^{we calculate} x-ray

scattering intensities for the various phases ^and

show that the predicted intensities for the SA2’ phase

are in agreement ^{with the} experimentally ^observed

intensities in the SAd phase. ^{In section} 5, ^{we review}

our results and make ^some comments about critical corrections to mean field theory. Finally ^{in an} appen-

dix we consider in more detail the bicritical point

where both qll and tf¡ 2 ^{are critical.}

2. Landau theory.

As discussed in reference [7], ^{two order} parameters

are needed to describe the properties of frustrated smectics. The first p(r) ^{is the centre of mass} density

of the constituent molecules. The second Pz(r) ^des-

cribes long range head-to-tail correlations of polar

molecules along ^{the z-axis} (normal ^to the smectic

layers). ^In the absence of coupling ^between p and Pz,

p would develop spatial modulations along ^{the z-axis}

at wavenumber k2 ^{= 2} 7r/l where I is of the order of

a molecular length ^whereas Pz(r) ^would develop

modulations at wavenumber k1 ^{= 2} 7r//’ ^{where l’} > 1 is a length associated with two molecules. To describe the appearance of modulated order, ^{we write}

where qll(r) ^and V12(r) ^are complex ^{fields :}

In terms of these fields, ^the Landau-Ginzburg ^free

energy of the Prost model in d-dimensions is

where rl = al(T - T1c) and r2 = a2(T - T2c) ^mea-

sure the temperature ^{from the} noninteracting ^mean

field transition temperatures T lc ^and T 2c ^{of the fields} 01 ^and t/12’ Ol ^{is a} derivative in the plane perpendi-

cular to the z-axis, parallel ^{to n.} The I (d ⁺ k2)01 12 and (d ⁺ ki) t/12 12 ⁱⁿ equation (2.3) ^favour qp=kl and qp ^{= k2 whereas the I V _Lol 12 and} 1 V.L t/12 12

terms favour qp ^and qp parallel ^to the z-axis. The Re 02 t/1! ^term favours lockin at qp ^{= 2} qp.

To study ^{the mean field} phase diagram ^of linearly

modulated phases, ^{we choose} _qp ^and qp parallel ^to

n ⁼ êz ^{and seek} spatially independent ^fields

which minimize ð.Fs’ ^We expect ^the following phases :

1) the nematic phase (N) with I t/J 1 I = I t/J2 ^{1 = 0,}

2) ^the monolayer ^smectic phase (SAl) with I t/J 1 ^{I = 0,}

and

3) ^the bilayer anti-ferroelectric smectic phase (SA2)

with and

Note that an anti-ferroelectric smectic A phase

with no long range ^mass density modulation (I fj 1 I =1= 0, I fj 2 ^{I =} 0) ^{is never stable} (see ^Ref. [5]) ^{because a non-} zero I fj 1 1 always generates ^a non-zero I fj 2 1 ^{via the}

third order term in ð.Fs’ ^In addition" ^{there are anti-} phases, which will ^not concern us here, ^{in which} qp is not collinear with qp [2, 7].

The mean field free _energy density ^{of the N} phase

is zero :

The free _energy density of the SAI phase is obtained

by minimizing

with respect to 11/12 I to yield

The free _energy density ^{of the} bilayer ^SA2 phase ^is

obtained by minimizing

with respect to I t/J 1 1, I t/J 2 1 and qo. The minimization with respect ^to qo is straightforward, ^{and we obtain}

To proceed ^{with the} algebraically complex ^minimiza-

tion with respect to [ gli [ and I t/J 2 I, ^{we introduce}

rescaled variables

so that

x, therefore, ^{measures the} degree ^of order and 0 the relative amplitude ^of t/J 1 and ql2 with small 0 cor-

responding ^to larger t/J l’ ^{In terms of these} variables,

the free energy density ^becomes

where

yl and _Y2 are the temperature variables, ^{and z2 is}

the incommensurability parameter measuring ^the degree of mismatch between k1 ^and (1/2) k2. ^The

usual stability requirements ^{on fourth order terms} require ðUl,2 > - 1. We now write

Minimizing f ^with respect ^{to x, we obtain}

For a given 0, f ^{is a minimum either at}

(The ^{smaller root x =} [1/(2 a)] (- ^b corresponds ^{to a} local maximum of the free energy). We, therefore, ^obtain

Finally, ^we performed ^a numerical minimization of

equation (2.17b) ^with respect ^{to 0 to} locate the lowest free energy solutions. Plots of f (x(9), 0) ^{vs. 0 for}

different values of _y, at constant y2 ^are shown in

figure ^{2. The various} phase diagrams ^{we obtained are}

shown in figure ³ and will be discussed in the next section.

3. Phase diagrams.

For small incommensurability parameter Z2 ^and

symmetric elastic and fourth order terms (bul =ðU2)’

the phase diagram (Fig. 3a) looks like the theoretical N-SA1-SA2 diagram of reference [5]. ^{There is a}

second order N-SA 1 line terminating ^{at a mean field}

394

Fig. ^2.

Free energy ^{as a} function of sin 0, (a) ^{in the} strongly

first order region ^and (b) ^{in the} vicinity ^{of the critical} point.

In each figure, ^{there are three curves} at constant y2, but

differing Y1’ In figure 2(a) the absolute minimum of curve 1

corresponds ^{to the SA2’} phase and that of curve 3 to the SA2 phase. Curve 2 with two energetically equivalent ^minima corresponds ^{to the} phase boundary where the SA2 and SA2’ phases co-exist. In figure ^2a, y1 ^{= -} 0.200, and Y2 ⁼

-

0.569 ^x 10-2, - 1.069 ^x 10-2 and - 1.569 x 10-2 in

curves 1, 2 and 3. In figure 2b, Y1 ^{= -} 0.375, and Y2 ⁼ + 1.520 ^x 10- 3, - 3.480 ^x 10- 3 and - 8.480 x 10- 3 for

curves 1, ^{2 and 3.}

critical end point [9] Q ^{where the} N, ^{SA 1 and SA2} phases ^{meet. There is a} second order N-SA2 line

terminating ^{at a} tricritical point ^{P. The line} QP ^is

a first order N-SA2 line which continues into the smectic region ^{as a} first order SAI-SA2 line termi-

nating ^{at a} tricritical point ^R. Beyond R, ^{there is a} second order SAI-SA2 line that is expected ^{to be in}

the Ising universality ^class [5, 10]. ^The wavevector qo

depends on yl and Y2’

If the elastic constants D1 ^and D2 ^are different and favour an easy compression ^of t/11 (i.e., bul ^increases

and bu2 decreases), then, ^{at small values of} Z2 , ^{a new}

first order phase boundary separating ^{two SA2} phases appears. The two SA2 phases ^are distinguished by different values of 0 and from equation (2.11)

different values of _{qo. For} 0 small, qo is of order kl,

Fig. ^3.

^Phase diagrams in the Y 1 - Y 2 plane for different values of the incommensurability parameter z2 ^{and the}

potentials ðU1 ^and bu2. ^Solid (dashed) ^lines correspond ^to

first (second) order transitions.

a) ^{z’ =} 0.25, ðU1 ⁼ 0, 6U2 ^{= 0. This} diagram ^{shows N,}

SAl 1 and SA2 phases and is almost identical to the phase diagram ^{for z2 = 0} calculated in reference [5].

b) ^{z2 =} 0.25, bul ⁼ 6, du2 ^{= -} 0.85 +. There are now two SA2 phases, characterized by different values of qo and

I t/J 1 II I t/J 2 1, separated by the first order co-existence line PC.

The point ^{C is a critical} point ^{similar to the} liquid gas critical

point ^{and the} point ^{P is a mean} field critical end point.

c) ^{z2 =} 0.45, 6ui ^{= 6,} 6U2 ^{= - 0.85 +. A mean} field bicri- tical point ^{B and a} triple point ^T appear in this diagram.

The first order SA2-SA2’ co-existence line and critical point

C remain.

and t/J 1 ^{is much} larger ^than t/J 2’ ^{Thus a small 0 or}

SA2’ phase ^can be identified with the experimentally

observed [3] ^SAd phase ^even though t/J 2 ^is strictly speaking ^{non-zero. We shall} strengthen this identi- fication in the next section when we consider _x-ray

scattering. ^{For 0} intermediate between 0 and 7r/2,

the amplitudes ^of t/J 1 and 4/2 ^are comparable leading

to the traditional SA2 phase. ^We will label the tra- ditional bilayer phase ^{SA2 and the small 0, SAd-like} phase SA2’. The dicontinuities in W02 ^{and sin2 0 across}

the new phase boundary ^are related from _equa- tion (2 .10) ^via

The new phase boundary ^is tangent ^to the first order N-SA2 line at P (Fig. 3b) ^and terminates at a critical

point ^{C where} Oqo goes ^{to zero.} Critical fluctuations in the vicinity ^{of C similar to} those encountered at the

liquid gas critical point ^are expected. ^The presence of the critical point ^{C indicates as} expected ^{that there is no}

difference in symmetry between the SA2 and SA2’

phase ^with different relative amplitudes ^of 0, and t/J 2’

The experimental phase diagram [11] ^{for the} binary

mixture DB7-TBBA is _very similar to the central part of figure ^3b provided ^one identifies the small 0 SA2’

phase ^{with the} experimental ^SAd phase. ^As just discussed, ^{this is a} reasonable identification that will be further justified ^{in the next section. In addition}

the experimental phase diagrams ^{for the DBn and}

DBnO series [12] ^{are similar to the} region ^{around P}

in figure ^3b provided ^{one assumes that the} points Q

and C are outside the _range of experimental ^obser-

vation.

When the incommensurability parameter Z2 ^is further increased or bul increased and bu2 decreased,

a new first order SAl-SAd line (Fig. 3c) appears

terminating ^{at a mean} field bicritical point ^{B where}

the N, SAI and SAd phases ^{meet, and a} triple point ^T

where the SAI, SA2 and SAd phases co-exist The critical point ^C marking ^the end of the line distin-

guishing the SA2 and SAd phases ^{continues to exists}

The existence of the mean field bicritical point ^{B is} surprising ^{in that it} arises from a theory ^{with a trilinear}

t/Ji t/Ji coupling ^that normally ^{leads to a} first order transition as in figure ^{3b. In} appendix A, ^we give ^an

alternate derivation of this result This phase ^dia-

gram in the vicinity ^{of B is in} good agreement ^{with the} experimental phase diagram [ 1 3] ^{of the} bindery ^mixture DB7-NO2-DB8NO2. Note, however, ^{that the} experi-

mental diagram ^{does not show an} SAd-SA2 transition.

Rather there is an SAd-C transition, ^the possibility ^of

which we do not consider in this _paper. This is an

indication, ^{that the new critical} point C may be _pre-

empted by non-collinear anti-phases. Finally, ^{we note}

that there is a small re-entrant nematic region ^{in the} experimental phase diagram ^{which is} probably ^due

to fluctuations beyond ^{mean field} theory ^{not included}

in the present theory [6, 7, 14].

Fig. ^4.

qo ^{as a} function of y2 ^for different yi. Note the

similarity between this diagram and the PV liquid-vapour diagram.

In figure 4, ^we plot qo ^versus the temperature variable _y2 for different _yi in the vicinity ^{of C. Note}

the close resemblance of the behaviour of this variable to that of the liquid-gas density difference in the

vicinity ^{of the} liquid gas critical point.

4. _x-ray scattering.

The most definitive experimental probe ^{of the structure}

of smectic liquid crystals ^is x-ray scattering. ^{In this}

section we will calculate the _x-ray structure factors for the SA2 and SA2’ phases discussed in the previous

section by considering Gaussian fluctuations about

mean field theory.

The _x-ray scattering intensity ^at wavenumber q ^is proportional ^to

where pel(r) is the electronic density ^{which can be} expressed ^{as a} linear combination of the smectic order parameters :

Thus, ^{we have}

with

for A, ^{B =} Pz, ^{p. To} discuss correlation functions of Pz(r) ^and p(r), ^{we write}

where t/11 ) = I t/11 ^I and t/12) ⁼ I t/12 I ^{are the}

mean field order parameters calculated in section 2.

396

The qJ-fields represent deviations of t/1I and t/12 ^from

their equilibrium values and by definition have

expectation ^value zero, (i.e. qJ Ix ) ^{= 0, etc.). After}

some simple algebra, ^{we obtain}

where

and

Equation (4. 6a) ^{follows from the} fact that fluctuations in Pz ^{and p occur in different Brillouin zones.}

The maximum intensities of the two Bragg peaks ^at wavevectors qo ^{and 2} qo arising ^{from the first terms in} equation (4. 7) ^and (4. 8) ^are proportional respectively ^to ai V2 I t/11) 12 ^and CX2 2 V21 V’2 > t2 corresponding

to long range anti-ferroelectric and mass density ordering. ^Thermal averages of the diffuse scattering ^{terms can}

be calculated in the harmonic approximation ^{as follows. Let} 45. ⁼ ((Plxl (Ply, (P2x, (P2y) ^{be the four component}

vector describing fluctuations out of equilibrium. ^{The free} energy AF, is second order in ø (X ^{is then}

where ðFMF is the volume integral of the free energy density ^of equation (2.8). ^{There is no linear term in ø since} øa ) ^{= 0} and ( VØa) ^{= 0. The} stability ^matrix Maa(q) ^is

Thermal _averages of 0 .. (q) Øp( - ^{q) are obtained by}

inverting ^{the matrix} Map(q) :

can now be obtained from equations (4.6)-(4.8) ^and equation (4.12). ^Different profiles ^for I(q_L ^{= 0,} qz) ^are

shown in figure ⁵ for different points of interest in the

phase diagram. The finite scattering ^volume V,

as well as the coefficients _al and _a2 (whiqh ^{are taken}

to be equal ^for simplicity) ^are maintained constant in all profiles ^so that relative intensity domparisons

can be made _among the different curves.

We note the following behaviour shown in figures ^5a

to 5d.

The nematic phase ^{exhibits two diffuse} spots

(Fig. 5a) ^{centred at the} incommensurate wavenumbers

k1 ^and k2.

A strong quasi-Bragg peak ^condenses at wavevector

k2 ⁼ 2 qo in ^{the SA1} phase ^{while the} intensity ^{of the}

diffuse scattering is lowered. The diffuse spot ^{at wave}

vector ki is still incommensurate with 2 qo ^as observed

experimentally [3, 15]. ^{Note that} figure ^{5b is in}

agreement with other computations ^of x-ray diffrac- tion patterns [16].

A second quasi-Bragg peak appears in the bilayer

SA2 phase at wavevector qo(lj2) k2 qo kl) ^com-

Fig. ^5.

Typical x-ray intensities /(0, 0, qz) (q2 scans) ^for

different temperatures. ^{The other} parameters ^{are the same}

as in figure 3c. Reduced co-ordinates are used as explained

in the text (Eq. (2.13)). Arbitrary ^{units are} used in the figures

but relative intensities can be compared provided ^{one mul-} tiplies intensities by factors of 10 and 5 in figures ^{5b and 5c.}

a) ^N phase, the maxima of the diffuse spots ^{are at} the natural

(incommensurate) wavenumbers k1 ^and k2.

b) ^SA1 phase, ^{there is a} single quasi-Bragg peak ^at k2 ^and

a diffuse maximum at k 1.

c) ^SA2 phase ^near the SA1-SA2 phase boundary ^for large 0,

there are two commensurate quasi-Bragg peaks ^at qo and 2 qo with qo k1 and 2 qo very slightly greater than k2

so that kl ¹ > qo > k2/2 qo.

d) ^SA2 phase ^{for small} 0, ^{there are two} quasi-Bragg peaks

at qo - k1 and 2 qo and a large ^diffuse scattering peak ^at

qz ’" k2. Note that the Bragg peak ^at 2 qo is almost entirely

hidden by the diffuse scattering peak. ^The intensity ^{of the} Bragg peak grows ^as V2 and should always ^dominate

diffuse peaks ^for large enough scattering volume V. V is, however, always ^bounded by mosaicity ^{and other} experi-

mental limitations.

mensurate with that of the monolayer ^{order of} 2 qo

(Fig. 5c). ^The intensity ^{of this new} peak ^at (0, 0, qo)

grows continuously ^{from zero} away from the SA 1- SA2 phase boundary ^{when this} transition is second order [10] whereas the intensity ^{of the} (0, 0, ² qo) peak ^{does not} change significantly. Deep ^{in the SA2} phase ^the intensities of the (0, 0, qo) ^and (0, 0, ² qo) peaks ^are comparable. ^Note that there remains a

small diffuse scattering amplitude (Fig. 5c) ^centred

around the wavevector k2 incommensurate with the established order at qo and _{2 qo.}

Although ^the scattering profiles of the SA2 and

SA2’ (SAd ^{or small} angle SA2) phases ^are qualitatively equivalent, they ^are quantitatively ^{distinct. As} already discussed, ^the intensity ^{of the} (0, 0, qo) peak ^{in the}

SA2’ phase ^is greater ^than that of the (0, 0, ² qo) peak

whereas the converse is generally ^true of the SA2

phase. ^This difference is particularly ^{visible as the}

first order SA2-SA2’ line is crossed in the vicinity ^of

the triple ^or pseudo-triple points ^{T and P in} figures ^3b

and 3c. In fact, ^the intensity ^{into the} (0, 0, ² qo) peak

goes ^{to zero as} the second order N-SA2’ phase ^boun- dary ^is approached ^{in these} figures. ^{In the SA2} phase,

the intensity ^{of the} (0, 0, ² qo) peak ^is significantly

greater than that of the (0, 0, qo) peak (Fig. 5c). ^{In the}

SA2’ phase ^at the order side of the boundary (Fig. 5d)

the intensity ^{of the} (0, 0, ² qo) peak ^is strongly ^lowered

and almost completely overshadowed by ^{an intense}

diffuse spot centred around (0, 0, k2) and the first order peak ^at (0, 0, qo) ^is dominant with the lockin wavevector qo close ^to k1 (Fig. 5d). ^{If the} (0, 0, ² qo) peak ^were completely ^hidden by the diffuse spot ^at (0, 0, k2), ^{in an} experiment of standard resolution,

then the profile ^of figure 5d would be identical to the

experimental ^SAd profile of reference [15]. ^The signa-

ture of the SA2-SA2’ (SA2-SAd) transition is thus

a discontinuous jump ^{from an} x-ray pattern ^with

a dominant Bragg peaks ^at wavevector (0, 0, _{qo -} k2/2) ^and (0, 0, 2 qo) with diffuse scattering ^at k1

to one with a Bragg peak ^at (0, 0, qo - kl) ^with

diffuse scattering ^at k2 ^{as is} commonly ^observed [15].

Recently ^{the weak} (0, 0, ² qo) ^Bragg peak ^{has been}

observed in high resolution _x-ray experiments ^on

DB6 [17] ^{in excellent} agreement ^{with the} present

theory. ^{As the critical} point ^{C is} approached, ^the

intensities of the two Bragg peaks ^{in each} phase

become more nearly equal. Finally ^{at the} point C, the distinction between the SA2 and SA2’ phases disappears. ^Stated differently, the classical SAd phase changes continuously ^{into the} bilayer ^SA2 phase ^{as C}

is approached. The difference between the lockin wavevector qo of the SA2 and SA2’ phases ^also approaches ^{zero as C is} approached.

5. Discussion.

In this _paper, we have used the Prost model to study phase transitions in frustrated smectics in mean field

theory. ^We find nematics (N), monolayer (SA1) ^and

two bilayer phases (SA2) ^and (SA2’). ^The general types of mean field phase diagrams ^{are shown in} figure ^3.

Of particular ^{interest are} figures ^{3b nd 3c where}

there _appears a new critical point te inating ^{a first}

order phase boundary separating the A2 and SA2’

phases. ^We identify ^{the SA2’} phase ^{with the} experi-

mental SAd phase. With this identification, ^the

calculated phase diagrams ^of figure ^{3 and are in} good agreement ^{with the} experimental phase diagrams ^of

references [ 11-13].

This _paper was devoted exclusively ^{to mean field} theory. There remain the questions of whether critical fluctuations will lead to qualitative changes ⁱⁿ any of the mean field phase diagrams ^{and what} universality

class any surviving ^critical points will fall into. The

critical point ^{C would} appear ^to have Ising symmetry

in analogy ^{with the} liquid-gas transition. Coupling

398

of the order parameter ^{to elastic} degrees ^{of freedom} could, however, ^{lead to a different} universality ^class

and is currently being investigated. The tricritical

point ^{R and} critical end points Q ^{and P} probably ^will

survive the inclusion of fluctuation. The situation for the bicritical point ^B is, however, less clear because of the third order coupling between 01 ^and 02, ^which

would normally ^{drive a} transition first order. It seems

plausible, however, ^{that the} incommensurability ^which

makes the mean field bicritical point possible ^will also, make the bicritical point possible ^{in a} complete theory. However, ^even if the bicritical point survives,

its universality ^{class is} uncertain. The second order N-SA 1 and N-SA2’ lines are believed to be in the inverted _xy universality ^class [18] ^with negative specific ^heat exponent ^a. Normally ^{when two second}

order lines with negative ^a meet, ^the resulting ^multi-

critical point is tetracritical [9] rather than bicri- tical. Thus, either the mean field bicritical point

becomes a tetracritical point when fluctuations are

included, ^or it is in a new universality class. These

points ^{are under current} investigation.

Appendix.

In this appendix, ^we will review why a t/Jî t/J! coupling usually prevents t/J 1 and 02 ^from being simultaneously

critical and show how the incommensurability ^{of the} present model allows them to be. Consider first fSA2

in equation (2. 8) with k21 = qo and k2 ⁼ 4 q2

When r 1 r2l I t/J 1 1 is critical before I t/J 2 1, ^{and we}

can seek an effective free energy in ^terms of I t/J 1 I only.

Minimizing fsA2 ^with respect to 1 t/J 2 I, ^{we find}

Inserting ^{this into} equation (A. 1) ^{we find}

The coefficient ueff = u1 - W2/r2 of I ol 14 ^becomes negative ^for small r2 leading ^{to a} first order transition.

Thus, ^a second order transition with rl 1 = r2 is

impossible. The first order lines QP ⁱⁿ figures ^3a

and 3b are a result of this effect.

Now consider the full free _energy equations (2.14)

and (2.15) ^{in terms} of the reduced variables. This free energy has already been minimized with respect ^to qo. Here ^we minimize first with respect ^{to 0 rather}

than with respect ^{to x.} Seeting q ^{= sin} 0, ^{we find}

where

Thus f is ^a minimum either for cos 0 ⁼ 0 (i.e. t/J ^{1 =} 0)

or for g(q) ⁼ 0. The former ^case corresponds ^{to the}

N-SAI transition with ordered phase free energy

g(n) ^{is a cubic} polynomial in q ^{so there are three}

solutions to g(n) ^{= 0. When x =} 0, ^{these are} trivially

For x small, ^{there are} corrections to the above solu- tions that can be expressed ^{as a} power series in ^x.

The first solution (Eq. ^A. 7a) ^becomes

Leading ^{to an} effective free _energy

This predicts ^a second order transition at yl = 0 if 1 + bul > (z’ ⁺ y2)-1 with ordered phase ^free energy

The SA2’ subscript indicates that the ordered phase

is an SA2’ phase with 1 2 1 0 1 1 0 ^{0’ We have}

thus established that there ^are two second order extrema leading ^to N-SA 1 and N-SA2’ transitions

meeting ^{at a} multi-critical point ^at yl ⁼ Y2 ^{= 0.} If this multi-critical point ^{is not} eliminated by ^the

alternate solution to f(~) ^{= 0} (Eq. A. 6b), ^{then there}

will be a first order transition for _Yl 0, y2 0 between the SAI and SA2’ phases ^when fSAl = ,fSA2’

or along ^{the line}

for small I y, I and y2 I. ^This gives ^a topology ⁱⁿ

agreement ^with that in the vicinity ^{of B in} figure ^3c.

We now consider the alternate solution to f( q). ^It

leads to the existence of ^a first order N-SA2 transition line, ^{which can be} calculated in closed and exact form

by using the fact that the f(x, 0) ^{= 0} equation ^{has a}

double root in x other than zero. The parametric representation ^{of this} phase boundary ^{reads :}

Varying ^{0 in the} [0, n/2] interval defines the curve, which has a physical meaning only ^as long ^as y, and Y2 ^are positive. ^{Let us} first remark that for vanishingly

small incommensurability (Z2 _ 0), yl is always positive ^so that this first order N-SA2 transition

occurs before the second order N-SA2 and N-SA2’

transitions can be reached Increasing ^Z2 shifts the

curve toward negative values of both y1 ^and Y2’

It will cross the bicritical point ^when equation (A. .12),

admits _{the yl} ⁼ _{Y2 = 0} solution, ^{in a} physically acceptable domain. This requires :

for z’ zf, the situation is that of figures ^3a and ^3b,

as just discussed. For Z2 > Z2c, the first order line no

longer passes through ^{the first} quadrant; the alternate solutions to f(~) ^{take over} leading ^to the second order N-SA2 and N-SA2’ and first order SA2-SA2’ transi- tions meeting ^{in a} bicritical point ^{at B shown in} figure 3c. The existence of this point requires ^further 6U2 > - 1 (stability of the fourth order term) ^and bul > 1 (cos2 ^e 1).

Acknowledgments.

One of us (T.C.L.) ^received partial support ^{for this} work from the National Science Foundation under grant ^no. DMR82-19216. J. P. is also grateful ^{to NSF}

under the above contract for partial support ^{for a visit}

to the University ^of Pennsylvania where this work

was initiated. P. B. acknowledges the assistance of P.

Richetti in computer calculations.

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