A new theory of the nematic-smectic A-smectic C transition in liquid crystals

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A new theory of the nematic-smectic A-smectic C transition in liquid crystals

B.S. Andereck, B.R. Patton

To cite this version:

B.S. Andereck, B.R. Patton. A new theory of the nematic-smectic A-smectic C transition in liquid

crystals. Journal de Physique, 1987, 48 (8), pp.1241-1245. �10.1051/jphys:019870048080124100�. �jpa-

00210548�

1241

A new theory of the nematic-smectic A-smectic C transition in liquid crystals

B.S. Andereck and B.R. Patton+

Department ^of Physics ^and Astronomy, ^Ohio Wesleyan University, Delaware, ^Ohio 43015, ^U.S.A.

+Department ^of Physics, ^{The Ohio State} University, Columbus, ^Ohio 43210, ^U.S.A.

(Recu ^{Ie S9 janvier 1987, rivisi Ie 16 juin, accepti le 16 juin} 1987)

Résumé.- Une nouvelle énergie ^libre pour la transition nématique-smectique A-smectique ^{C est} introduite et utilisée pour calculer le facteur de structure liquide près de la transition nématique-smectique ^C. Quand ^la température ^est réduite, ^les pics de l’intensité de diffusion s’écartent de l’axe des ^z et simultanément changent ^{de forme} passant d’une _queue quadratique à ^une queue quartique. ^Des expériences ^récentes sont correctement décrites _par les résultats

théoriques.

Abstract.- A ^new free _energy for the nematic-smectic A-smectic C transition is introduced and used to calculate the

liquid ^{structure factor} just above the nematic-smectic C transition. As the temperature ^is lowered, ^the peaks ^{in the} scattering intensity ^move off the z-axis and simultaneously change ⁱⁿ shape ^from tails characterized by quadratic ^to

those with quartic behaviour. Recent experimental data is well described by the theoretical results.

LE JOURNAL ^DE PHYSIQUE

J. Physique ⁴⁸ (1987) ^1241-1245 AOÛT 1987, ¹

Classification

z

Physics ^Abstracts 64.70M

Recent experimental work has revealed an in-

triguing ^structure associated with transitions in li-

quid-crystal ^mixtures which exhibit nematic (N),

smectic (A), and smectic-C (C) phases. Experiments

have seen direct evidence of the crossover from A to C type fluctuations : scattering ^data [1,21 ^{well above}

the NC transition show peaks centred about ±go) the wavevector characterizing the smectic-A density

wave. As T --> TNO, ^the scattering regions ^broaden

into rings ⁱⁿ momentum space with _{qz m} ±qo, q i m

sin(w).qo, the direction being ^defined by ^{the aver-}

age molecular direction no. ^In addition, ^{the scatter-} ing develops striking quartic ^{tails for} q in the plane ^of

the ring, ^a foreshadowing ^{of the} transverse instabil-

ity of the smectic phases pointed ^out by ^Landau [3].

Previous theories [4,6] ^{describe some of the qualita-}

tive features of the experimental results, ^{but lead to} quantitative ^{fits with} unacceptable properties.

We suggest ^{that the} biaxiality exhibited in liq-

uid crystals ^{is due to} staggered alignment ^of neigh- bouring liquid crystal molecules. In the N phase ^this stagger ^is short-range ; ^{in the C} phase stagger ^{is com-} monly ^{referred to as} tilt and becomes long-range. ^We

combine a Ginzburg-Landau theory ^for stagger (the

order parameter ^c below) ^{with a} generalized ^{form of}

deGennes’ NA free _energy [4] ^{to produce a new uni-}

fied theory for the NAC system. ^{The smectic order} is characterized by ^the parameter y, ^{which is related}

to the density p by

p(r) = Po {I + 1tJ1(r) I cos (qo.r - e)}, (1)

where 0 is the phase, 21r/qo ^the wavelength ^{of the} density wave, and _qo is in the _average direction of the

density ^wave. We introduce ^{a new} order parameter

c to describe the local stagger ^of adjacent ^molecules.

Although ^{it is} possible ^to give ^a precise definition of

c [7], ^{here we just note that, like in} the hexatic phase

of two dimensional liquids, ^{it is} possible ^{to have} long-

range orientational order in the directions of neigh- boring molecules without having positional ^{order of}

the centres of masses. However, ^{in the} presence of the

layered order of the smectic, ^the average stagger, co,

gives the inclination or tilt of the density ^{wave from}

the _average molecular direction no :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048080124100

1242

where (A ^{is the} wavelength ^{of the} density ^{wave in the}

A phase. ^From figure 1, ^the magnitude of the tilt is

given by leol = ^{tan W, where cv} is the tilt angle, ^while

qo ⁼ 2x/(AAcos w) , indicating ^{that the} wavelength

of the density ^wave decreases with increasing ^tilt [8].

Fig.l.- The relation between the density ^{wave vector} qo, the averge molecular direction no, ^{and the} average tilt _{vector co.}

In the _presence of thermal fluctuation in both the director n(r) ^{and the} tit c (r), ^the density ^wave

contribution to our free _energy is

where a ⁼ ao. (T - TNA) I TNA, ^{b is a} positive ^cons- tant, rii ^is the inverse mass tensor, and Q ⁼ 2x(f ⁺ czar. The value of the tilt order parameter ^is gov- erned by ^{the tilt free} energy

where a ⁼ ao. (T - TAC) /TAc, ^and {3, 11, ’Y2) and 73 ^are positive constants. We expect ^on physical grounds ^that 73 ^« Tfi, ’y2 due to the weakness of the end-to-end molecular interactions. There is an

attractive coupling ^{between the smectic and tilt or-}

der parameters ^{of the form}

In addition, ^we include the usual Frank elastic _energy

where K10, Kg, Kg ^are the unrenormalized splay, twist, and bend elastic constants in the N phase. ^Ours

free _energy is consistent with deGenne’s in appropri-

ate limits. When a is large ^and positive, ^{the value of}

c is zero and equations (3)-(8) ^{reduce to deGennes’}

free _energy [4] for the NA transition (if r is deGennes’

inverse mass tensor). ^{When a is} large ^and negative, ^a

static density ^wave forms and the equations ^are equiv-

alent to deGennes’ [9] description of the AC transi- tion. In the N phase ^{above the A or C} phases, ^our theory differs from that of Chen and Lubensky [5]

by ^the presence of an extra degree of freedom corre-

sponding ^to the direction of c. This degree ^{of freedom}

is essential in characterizing the broken symmetry ^of the C phase and enables our theory ^{to describe the}

AC transition in a unified fashion. Finally, although

our free _energy has a stagger ^order parameter ^similar

to that of Chu and McMillan [6], the form of our free energy differs from reference [6] ⁱⁿ that, 1) ^{our c free}

energy is not proportional ^to l"p 12 , ^and 2) ^{the struc-}

ture of our gradient term, ^unlike that of reference

[6], ^{has a} gauge-coupled form like that of deGennes’

smectic A free _energy [4].

Stagger order leads to correlations that are

stronger in two-dimensional sheets containing ^{the mo-}

lecules than in the perpendicular ^direction [10]. ^As

a result of this 2D nature, fluctuations may suppress the transition temperature considerably ^{below the}

mean-field transition temperature TMF where short- range stagger correlations develop. In the mixtures

typically studied which exhibit an NAC point, ^this suppression eliminates _any long-range ^{biaxial nematic}

(stagger) ^order [11]. ^{Finally, as seen below, our re-}

sults _very close to the smectic transition are also con-

sistent with the scaling ^relation recently ^{found be-}

tween the coherence length exponents : vll ⁼ 2v_L

[12].

In this _paper ^we report ^{the structure factor} S(q)

predicted by ^{our free} energy in the nematic region ^the

C phase ^{in the} presence of short-range stagger ^corre- lations. The quantity of interest is the density-density

correlation function g(r) = Tro,,ft,. [e-FlleT p(r)p(O)],

where the trace is over the degrees of freedom y, n, ^{and c in our free} energy, F the ^sum of (3)-(6), G(r) is modified above the smectic transition due ^to the gauge-like coupling between the density ^fluctua-

tions of the incipient ^smectic layers and the director

(see Eq.(3)). This is the same coupling that leads to the renormalization of the elastic constants Kio ^{of the}

Frank _energy (6) ^{above this} transition. In calculating

the effect of 0 ^{and c in} renormalizing ^{the structure}

factor of the N phase, ^{we focus on the} region ^{not too}

close to the NAC point. ^{In this} region ^{the smectic}

fluctuations are weak and _may be treated in lowest

order. Well below TMF, strong local correlations in c

imply ^{that the} magnitude ^{of c is fixed near its mean}

field value, while fluctuations in the direction ^{of c}

wash out long range stagger (c) order. We neglect ^the

effect of short-range angular correlations and perform

a powder average. Then G is given ^{in terms of the} self-energy E by G(q) = Tr,, [a ⁺ a2 Iql. - ql.Ol2 + ae 2 (q - ^{qllo )2 -} E (q) I-’. ^The leading one-loop ^con-

tributions ^{to the} self-energy ^{E are}

where _qlo = 90 $in (w ) ^{and Z =} e2 qo2kBT/e//Kl7r.

The important feature of (7) ^is the fact that the term

quadratic ⁱⁿ (ql - qlo) ^is positive and increases mo-

notonically ^{as the} transition is approached. ^This

decreases the quadratic ^{term in} Go ^{and leads to the}

dominance of the quartic contribution. The (qx -

q//0)2 contribution in (7) ^has the opposite sign ^and

has no significant effect ^{on G.} Performing ^{the trace}

over p and defining R ⁼ (1- 41Z/30)1)2 ^{/ (226Z/}

105)1/4, ^{a measure of the crossover from} quadratic ^to quartic behaviour, ^{we obtain} (with ^{qz =} qllo)

where A2 == 1- R4, B:I: == R2 + e21 (q_L ± q_Lo)2 (113Z/840) ^{, and a’ = a} -E(O). Far above the NC transition, ^for Z 1, S(ql.) ^is Lorentzian and similar to the result of Chu and McMillan [6], which,

of the previous theories, gave the best fit to the ex-

perimental data in this region. As the transition is

approached, the coefficient of (ql - ql.O 12 ^{in Go de-}

creases while that of the [q i - ql.O 14 ^{term increases,}

leading ^{to enhanced} quartic ^{tails to} S(q) ^{and a pro-}

nounced squaring ^{of the} shape ^{of the structure} factor,

a form which is closer to that of Chen and Luben-

sky [5]. ^{As the} transition is approached further, ^the quadratic (ql - qlo) ^{term in} G(q) --> ^{0, requiring}

that Z -; constant. Since Ki ^{is not} renormalized,

this would imply ^that 1 Qf.II’ ^{or that the relation}

2v1 ⁼ ₁₁₁₁ ^{hold as the} second order instability ^tem- perature ^is asymptotically approached [12]. ^{An in-}

tervening first order transition, however, ^is expected

to make the second order transition inaccesible [13].

This _appears to be the case for the system [2,14] ^con-

sidered below, for, although ^Z changes significantly ⁱⁿ

the temperature ^range involved, ^{it does not} approach

its limiting ^value.

We _compare our result with experimental ^data

on mixtures of 7S5/8S5 [2,15]. The critical concentra- tion _XNAC ⁼ 42.2 mol % , ^while X-ray scattering ^mea-

surements precisely ^{fix c=0.200 at 70% and} T-TNC ⁼

0.015° C [2]. For other concentrations and tempera-

tures we approximate the local value of lei by ^mean

field behaviour ; ^{for T s5} TNO ^{c varies as the} square root of z - XNAC, while for fixed _x, ^c varies as the square root of TMF - T. This reduces the number of

independent parameters ^{used to fit the} data, ^{and the} resulting values for c appear ^to be in good agreement with the data on ’785/885. It is known experimen- tally that the N phase above the NC line in 785/8S5

is characterized by weak fluctuations and mean field behaviour [15]. We therefore assume that the coher-

ence lengths ^{scale as} t-1/2 (as does Z in this region),

where the reduced temperature t ⁼ (T - T*)/T* ^is

measured from the second-order instability tempera-

ture T*, which lies below the (first-order) ^transition

temperature TNC. Using these constraints to fit the data of Safinya ^{et al.} [2] ^{for both 50 and 70%} produces

the curves shown in figure 2, ^with TNC - ^{T* ~ 20C}

for both concentrations. The latter values is in good agreement ^with TN C - ^{T* =} 2.7° C for 100% ^obtained by Witanachi et al. [14]. ^{Table I compares the fitted}

values of the coherence length j_ ^{and the crossover} parameter ^{R to} the theoretical values ; ^the agreement is within 5% in general. ^A discrepancy ^{occurs for the}

T - TNC ^{= 7.12° C} 50% data because our mean-field

approximation ^{for c} underestimates the degree ^{of lo-}

cal correlations in tilt above TMF - Correction to finite

experimental resolution were neglected ^since they ^are

small [16] ; ^we estimate their inclusion should change

the values in table I by ^{at most a few} percent. ^Fi-

nally, using ^our results and the data for the specific

heat jump [15], ^{the Ginzburg} criteria for the size of the critical regions ^{is found to} be less than 10-2oC for both concentrations, ^which places the critical re-

gion will below TNC, consistent with the mean-field behaviour observed above TNC.

Previous fits of the X-ray scattering data, using

the Chen-Lubensky ^model [5] give reasonable fits in the region ^close enough ^{to the C} phase ^{for the} quartic

tails to be prominent. However, ^a systematic ^{fit for}

all temperatures and concentrations does not seem

possible. Specifically, ^the quartic ^{tails must be ar-} tificially reduced close to the critical point [2], ^good

fits are not possible ^{in the crossover} region [2], ^and

an unphysical variation of the coherence length ^with

temperature is obtained [16]. ^Other attempts ^{to fit} the data using the theories of Chu and McMillan [6]

or deGennes [9] give ^{even less} satisfactory ^{results. ;}

1244

Table I.-Comparison of fits of equation (8) ^{to the data} of reference [21 for different concentrations and temperatures. The values of ^the perpendicular ^coherence length f iex ^{and the crossover} function Rex ^were

determined from ^the fits ^{to the data in} figure 2. ^The theoretical values were calculated f rom _L,// ^{at-1/2 and}

the definition of ^R following equation (7), ^{using the data} for ^{the lowest} temperature for ^each concentration.

Fig.2.- ^A comparison between the theoretical structure factor S (q.L) ^from equation (8) (solid line), ^and experimental ^{data on} 7S5/gS5, from reference [2]. ^{Each plot offset 5} counts/s ^from

the one below it.

a.z= 50% ; ^{T -} TNC ⁼ 0.005, 1.17, 4.78°C for top, middle,

bottom

b.z=70% , ^{T -} TNO ⁼ 0.015, 4.62, 7.12°C for top, middle, bottom.

In _summary, we have proposed ^{a new free} energy which describes the NA, ^{AC and NC} liquid crystal

transitions by including ^{an order} parameter ^charac- terizing ^{the local stagger or} misalignment ^of neigh- bouring molecules. We have applied ^this theory ^to

the calculation of the liquid ^{structure factor near the}

NC transition. The basic result of ^our calculation, ^the

crossover from quadratic ^to quartic ^behaviour repre-

senting ^the tendency towards the Landau instability

of the smectic phase, ^is clearly ^{seen in} the data. In ad-

dition, ^{in a} region ^{which is} expected ^to be mean-field

like, ^{we are able to} give ^a quantitative ^{account with}

a coherence length ^which diverges ^{with the} expected exponent ^{at a} temperature consistent with known val-

ues. Recent experiments [ 17] ^{on other systems} having

the NAC multicritical point have indicated the uni- versal nature of the phase diagram. ^In these mixtures it _appears that the NC transition is pushed ^{to lower}

temperature, leading ^{to the} speculation ^{that the win-}

dow for local biaxial order might ^be larger. ^{If the} length scale for the biaxial correlations becomes large enough, light scattering experiments ^{on these} systems could provide additional information on the nature of

ordering ^{in these} very interesting ^materials.

Acknowledgments

This work ^was supported by the National Science Foundation under Grant N°. DMR8114842. We wish to thank D.L. Johnson, ^C.R. Safinya, ^{and R.J.} Birge-

neau for useful discussions and S. Kumar for pointing

out reference lOb.

References

[1] ^McMILLAN, W.L., Phys. ^{Rev. A8} (1973) ^328.

[2] SAFINYA, C.R., BIRGENEAU, R.J., LITSTER, J.D., ^and NEUBERT, M.E., Phys. Rev. Lett. 47

(1981) ^{668 ;}

SAFINYA, C.R., MARTINEZ-MIRANDA, L.J., ^KA- PLAN, M., LITSTER, J.D., ^and

BIRGENEAU, R.J., Phys. Rev. Lett. 50 (1983)

56.

[3] LANDAU, ^{L.D. and} LIFSHITZ, E.M., Statistical

Physics (Pergamon) ^{1980, sec. 137.}

[4] DEGENNES, P.G., Solid State Commun. 10

(1972) ^753.

[5] CHEN, ^{J.L. and} LUBENSKY, T.C., Phys. ^Rev.

A14 (1976) ^1202.

[6] ^{CHU, K.C. and} McMILLAN, W.L., Phys. ^Rev.

A15 (1977) ^1181.

[7] ^{One way} in which such an order parameter may be defined in the absence of layers is R//R> / R2>, ^{where R is the} distance between the cen-

tres of mass of nearest neighbour molecules,

R// = ^{R.n, and R = R -} R//n. ^{The pres-}

ence of this correlation selects, ^{a direction in the} plane perpendicular ^to n, and adds a biaxiality

to the nematic phase.

[8] TAYLOR, T.R., ARORA, S.L., ^and FERGASON,

J.L., Phys. Rev. Lett. 25 (1970) ^{722 ;}

SAFINYA, C.R., KAPLAN, M., ALS-NIELSEN, J., BIRGENEAU, R.J., DAVIDOV, D., LITSTER, J.D., JOHNSON, ^{D.L. and} NEUBERT, M., Phys. ^Rev.

B21 (1980) ^4149.

[9] DEGENNES, P.G., ^Mol. Crys. Liquid Crys. ²¹ (1973) ^49.

[10] a) SAUPE, A., ^Mol. Crys. Liquid Crys. 7 (1969)

59 ;

b) AZAROFF, L.V., Proc. Natl. Acad. Sci. 77

(1980) ^1252.

[11] GRINSTEIN, ^{G. and} TONER, J., Phys. ^{Rev. Lett.}

51 (1983) ^2386.

[12] NELSON, D.R., ^and TONER, J., Phys. ^{Rev. B24}

(1981) ^{363 ;}

LUBENSKY, T.C., ^and MCKANE, A.J., ^J. Physi-

que Lett. 43 (1982) ^L-217.

[13] BRAZOVSKY, S.A., ^Zh. Eksp. Teor. Fiz. 68

(1975) ¹⁷⁵ [Sov. Phys.-JETP ⁴¹ (1975) 85] ;

SWIFT, J., Phys. ^{Rev. A14} (1976) ^2274.

[14] WITANACHI, S., HUANG, ^{J. and} Ho, J.T., Phys.

Rev. Lett. 50 (1983) ^594.

[15] DEHOFF, R., BIGGERS, R., BRISBIN, ^{D. and} JOHNSON, D.L., Phys. ^{Rev. A25} (1982) ^472.

[16] SAFINYA, C.R., private communication.

[17] BRISBIN, D., JOHNSON, D.L., FELLNER, H., ^and

NEUBERT M.E., Phys. Rev. Lett. 50 (1983) ^178.