Development of helical cholesteric structure in a nematic liquid crystal due to the dipole-dipole interaction

1. Phys. Chem. Solids. 1975. Vol 36. pp. IO5FlC41

Pergamon Press.

Printed in Great Britain

DEVELOPMENT OF HELICAL CHOLESTERIC STRUCTURE IN A NEMATIC LIQUID CRYSTAL DUE TO THE DIPOLE-DIPOLE INTERACTION A. G. KHACHATURYAN Institute of Crystallography, Academy of Sciences of USSR, Leninsky prospect 59, Moscow, USSR (Received 14 Ocfober 1974)

Abstract-A nematic liquid crystalline phase is considered whose rod-like non-centrosymmetric molecules possess a permanent dipole moment. This phase is a “liquid ferroelectric” if all the molecules are oriented along the same “preferred” direction. It is shown that a liquid ferroelectric can not exist in a homogeneous nematic state. It is transformed into a more stable helical structure (the vector of the spontaneous polarization of such a structure rotates aroung the helical axis). There is a variety of domain structures for the specific case when the anisotropy coefficient of the polarization is equal to zero. Since each elementary dipole moment is rigidly bound to its molecule, the “preferred” alignment direction of the rod-like molecules as well as the polarization vector rotates with respect to the same axis in a helical manner. Therefore a nematic phase with a nonzero spontaneous polarization has a cbolesteric structure. Its helical pitch is determined by the geometric size of the sample, the absolute value of the spontaneous polarization, and the elastic moduli. Apparently, we can consider some cholesteric phases to be liquid ferroelectrics with helical domain structure.

1. INTRODUCTION If a liquid crystal consists of rod-like noncentrosymmetric molecules, it is convenient to introduce a direction vector u&)-the unit vector characterizing the “preferred” direction of the molecules within the unit volume at point r. The structure of a nematic phase is characterized by an overall parallel alignment of the long molecular axes, i.e. the direction vector s(r) for nematic phase is constant. On the other hand the direction vector no(r) of the cholesteric phase slowly rotates as one proceeds in a direction h perpendicular to n&)(hlao(r)). The result is a helical structure (Fig. 1) with the helical axis directed along the vector h. As a rule, the molecules of a liquid crystal possess a permanent dipole moment of about 1 to 2 D. Permanent dipole-dipole interaction makes a small contribution to the total interaction energy. The major part of the energy is due to the dispersion forces. However, the dipole-dipole interaction must be taken into account because, in contrast to the dispersion forces, it is of long-range character. It will be shown below that a homogeneous state of the nematic phase is unstable with respect to a slight permanent dipole-dipole interaction. This results in destruction of the nematic phase and the development of helical cholesteric structure. It is easy to see that a homogeneous nematic liquid crystalline phase is a “liquid ferroelectric” if each molecule has an elementary permanent dipole moment rigidly bound to it (we consider the case in which all tSuch a case can arise if En - Eti < 0 and [Et, - Etl( - XT, where X is the Boltzmann constant, T is absolute temperature, and En and Etl are typical interaction energies of a pair of molecules with the same and opposite directions, respectively. A nematic state in which polarization is zero arises if 1En - EU/ Q XT, since in this case the statistical distribution that includes equal average numbers of molecules in some given direction and in the opposite direction is realized.

Fig. 1. Helical cholesteric structure of the liquid crystal phase: a is the helical pitch; the arrows show the “preferred” orientation direction of the molecules.

molecules and, consequently, all elementary dipole moments in a nematic phase are aligned along the same direction).? A liquid ferroelectric, like a solid one (see for example[l]), can not be in a homogeneous state, since both are unstable with respect to the formation of ferroelectric domains. The only difference is that anisotropy of the alignment orientation (spontaneous polarization orientation) is absent in a liquid ferroelectric, but this

1055

1056

A. G.

KHACHATURYAN

causes the domain structure of a liquid ferroelectric to be substantially different from that of a solid crystal ferroelectric. 2.

FREE

WERGY OF THE IumRoGENEous NEMATICPHASE

Let us describe the orientation of each noncentrosymmetric rod-like molecule of a liquid crystalline phase by a unit vector n directed along the direction of the elementary permanent dipole momentum of the molecule. The thermodynamic average (n) depends on the temperature and characterizes the preferred direction of the molecular alignment (((n)/l(n>])= ILo(r If (n) = const (nematic phase), there is a nonzero spontaneous polarization P (the average dipole moment of the unit volume). The quantity P is determined by the equation P = &(n)c = d( T)c

Since we are considering the particular case of the heterogeneous distribution in which IP( = P = cons& the equality (P(r))*= P* holds. This means that we can transform eqn (6) to the form (7) Therefore eqn (7) shows that the second term in eqn (4) does not depend on the character of the spatial distribution P(r) and consequently can be included in the free energy FO. Thus the free energy change connected with a spatial heterogeneity of the polarization distribution is

(1)

where do is the value of the elementary dipole moment of the molecule, d(T) = do(n) is the average dipole moment of a single molecule, and c is the number of molecules per unit volume. The free energy of the homogeneous nematic phase in which the dipole-dipole interaction is not taken into account is

(2) where f(P/doc, T) is the specific free energy, and V is the liquid crystal volume. The equilibrium value of the spontaneous polarization connected with the long-range order parameter (n) is determined by the minimum condition

Since the summable function in (8) is positive, one has Afi 3 0,

i.e., the minimum possible value which AFd can take on is zero. The main contribution to the free energy change connected with heterogeneities in the liquid crystalline phase arises from the short-range interactions caused by the dispersion forces. This change in the free energy connected with the smooth heterogeneities is [2,3] AF.=;/

(V)

’ 110)’ {KI ,(drv + K~(no curl 110)’ t K&to, curl n$} d V

The free energy (2) is degenerate with respect to the direction of the polarization vector P. This means that arbitrary heterogeneous distributions P(r) in which only the directions of vectors P(r) are changed (but lP(r)] = const) provide the same value for the free energy Fo. Taking into account the dipole-dipole interaction as well as the interaction associated with the heterogeneities of the liquid crystalline phase eliminates this degeneracy. The energy of the dipole-dipole interaction in krepresentation can be expressed in the form

where i’(k) =

dV P(r)eeikr

(5)

(9)

(10)

where Ku, Kz2, Ks are the positive elastic constants characteristic of the nematic phase. The positivity of the elastic constants Kit, Kz2, Ka leads to stability of the homogeneous nematic phase without the dipole-dipole interaction. In (10) we chose as a direction vector the unit vector in the polarization direction Ilo(r)= F.

(11)

3. DEl’ERWNATION OF THE OPTIMAL SPATIAL. DB’IXIBUTION OF THE IJOLAWZATlON The optimal distribution of the vectors P(r) or (which is the same) preferred directions Ilo of the molecular alignments is determined by the free energy minimum condition. Since the free energy (2) does not change if one varies the polarization distribution within the function set /P(r)1= P(T) = const, we are to minimize the sum

AF = AFd + AF. in the Fourier-transform of the polarization P(r), and the summation is taken over all points of quasi-continuum connected with the periodic boundary conditions. The second term in eqn (4) can be simplified:

which is a small addition to the free energy Fo. We can see from (8) that the minimum value (Afi),,,,” = 0 is attained only if klP(k) for all values of k

Developmentof helical cholesteric structure vectors for which P(k) # 0. This situation is realized if: (a) P(k) takes on nonzero values within a long and thin rod in k-space directed along a unit vector h, (b) P(k)lh, and (c) P(k)20 if k-0. It follows from (c) that p(O)= kv, d V P(r) = 0.

(12)

Let us choose the Cartesian coordinate axes so that the z-axis is directed along vector h, and x and y axes lie in the plane perpendicular to the vector h. Then it follows from condition (b) that P(r) = Z%(r)

1057

of a along the z-axis, and the function e”(r)at distances of the order of Ax - Ay - L, one can see that choice of the function P(r) in the form (16) automatically provides realization of the inequalities (14) and (15). Therefore, if the function P(r) comes from the set of functions (16), i.e. if the director Ilo in (13b) depends on all three coordinates r = (x, y, z), it yields an increase in the free energy AFd as well as the free energy AF.. One can easily prove this by substituting (13b) into (10). The result is

AK=:iv,{Kll(cos

Q$-sin

Q$$’

(13a)

where no(r) = (cos Q(r), sin O(r), 0)

UW

and Q(r) is a function of the spatial coordinates r. Choice of the director so(r) in the form (13b) simultaneously provides fullfilment of the identity IP(r))= P and of condition (b). If condition (b) is satisfied, one can see from eqn (8) that the free energy AFd tends to its minimum value (zero) as the length-to-thickness ratio of the rod in k-space where P(k) # 0. Therefore the condition of the asymptotic smallness of AFd is reduced to the inequalities (14) where A&: is the length and Ak, -A/c, the thickness of the rod in k-space. Since Ak, - (2?r/a), Ak, -Ak, -(2r/L) (a is the characteristic length of the heterogeneities along the z-axis, and L is the length along the x and y axes), one can rewrite the inequality (14) in the form +1.

(15)

The inequality (15) means that the free energy AFd tends to zero if the heterogeneity dimensions along the z-axis are minimum while the dimensions along the x and y axes are maximum. Taking into account that the maximum possible dimension along x and y is the size of the liquid crystal sample along these directions, one can write the expression (13) for P(r) providing the minimum of the free energy AFd: P(r) = Ps(z)B(r)

(16a)

no(z) = (cm Q(z), sin Q(z), 0)

(16b)

where

is the direction vector, a unit vector a&g the polarization direction (see eqn (13b)), Q(z) is some function dependent on the coordinate z, J(r) is a function of the shape of the liquid crystal sample that is unity inside the sample and zero outside it. Since n&r) varies substantially at distances of the order JPCSVol 36.No IC-.C

The dependence of the function Q(r) and, consequently, of the function n&) on the coordinates x, y leads to the additional positive term in the free energy AF. (the first term in eqn 17). Thus we conclude that the minimum value of the total free energy AF = AFd + AF, is provided by some function PO(r)which is like the function (16). The problem of the complete determination of PO(r) is reduced to the determination of the function Q(z) in (16b). Let the function no(z) be a periodic function: no(z t a) = n,,(z), where a is the period. The last assumption does not limit the generality of the problem, since a nonperiodic distribution is a limiting case of a periodic one, with the period a tending to infinity. The general form of Q(z) providing the periodicity of no(z) in (16b) is Q(z) = koz+ Hz)

(18)

where k. = (2a/a) is a wave vector, and I/J(Z)is some periodic function (#(z t a)= q(z)). We shall show further that the optimal distribution PO(r)corresponds to the case 4(z) = const, since in this case the free energy AF has its minimum value. The periodic function n,(z) can be represented in a form of the Fourier series n,,(z) =

2

A,e’sm* Ill=--m

(19)

where the prime to the summation sign means that there is no zeroth term in the sum (this is a consequence of eqn 12). A,,, = Oa$na(z) exp (- ikomz). I

(20)

The Fourier coefficients A, satisfy the normalization condition

Substituting (19) into (16a) and performing the Fourier-

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A. G. KHACHATURYAN

transformation, one obtains W, @k)=P

AFd = ; L,cP’

c A,B(k-km) mzmrn

(29)

(22)

where C is the perimeter length, and /3,r= (l/C) #r p,pj dl is a second-rank tensor in the (x, y)-plane. For a circular cross section, PI, = f&, where S, is the Kroneker symbol. Substituting pz, = is,, into (28), one obtains

where &r)e-“’ d V

O(k) =

2 r -!- &An,,A2, m=--”ko(m(

-cs

(30)

Substituting eqn (18) into (17), one has with k = (T, k,), 7 = (&, k,), and r = (p, z), p = (x, y). The vector ko, whose modulus is equal to 27r/a, is parallel to the z-axis. Let us assume that the volume of the liquid crystalline phase is a cylinder whose axis is directed along the z-axis. Then the Fourier-transform of the shape function is 8(k) = Odkz)&(T)

(24)

8,(k,) = 2 sin (%LJ k: ’

@a)

It follows from eqn (30) and (31) that the total change in the free energy is

where

t%(T)=

II

d2pe-‘“.

(25b) It is shown in the Appendix that the helical distribution

0)

The integration in (25b) is carried out over the area 5’ of the cylinder section defined by the (x, y)-plane. Substituting (25a, b) into (22), one obtains ‘ti(k) = P c A,B,(k, -km)&(~). mm-cc

no(z) = (cos kc, sin koz,0)

(33)

provides the minimum value of the

(26)

(34)

Using eqn (26) in (8), one can represent the expression for

of the free energy (32) with respect to the rest of all periodic distributions n&z) with the same period a. The optimal helical pitch a = uo can be found from the minimum condition for the free energy (34) with respect to ko = (2~/n):

A.Fd within the accuracy of (a/L) < 1:

dAFm,n_ 0. dko

In (27) we used the equalities:

(35)

It follows from (35) that 8K, V w = 27r(ro2R)“3 >

a, = 2ir p2L,c

(

as well as the identity. The integral in (27) has been evaluated previously (see the Appendix in [4]). Using the calculations[4] in (27), one can obtain

*fi =;p2LCz_&$r

IA,p(‘dl.

(28)

The integration in (28) is carried out over the perimeter I of the section of the cylinder defined by the plane (x, y), and p is the unit vector in the plane (x, y) perpendicular to the linear element d\ of the closed contour I. The expression (28) can be presented also in the form

(36)

where a0 is the optimal helical pitch, V = nRTL,, C = 2~rR, R is the cylinder radius, and r. = 2~(Kzz/P2) is a constant with the dimension of length. This result is changed slightly if one takes into account the Debye screening. In this case the helical pitch is a. = [(rozR)2’3- Xa*]1’2 where SC, = (2r/r9), and ra is the Debye radius. If (r02R)“3s 3&, then the helical structure does not arise at all and a nematic phase is stable. It is interesting that the sample dimension L, consists of an integral number m of the helical pitch: L, = mao. Otherwise eqn (12) is violated,

1059

Development of helical cholesteric structure and this results in an increase in the free energy above its

minimum value. The physical reason for the increase in free energy is the development of nonzero total polarization when (L/so) # m. This results in the formation of non-compensated charge on the sample surfaces. 4. DISCUSSION It

is necessary to bear in mind that two kinds of nematic phase can exist. The first includes nematic phases that become liquid ferroelectrics if the molecules have a permanent dipole moment. In these phases all molecules are oriented along one preferred direction (see Fig. 2a). It was shown above that such a homogeneous nematic phase is unstable and must be transformed into the helical cholesteric structure shown in Fig. 1. This is a variety of the domain structure.

“3

- (PAna- Pe(l -

nn))2’3. (39)

Equation (39) determines the dependence of the helix wave vector ko on the molecular fraction n,+ It is seen from eqn (39) that there is a composition n., = nX which provides the zeroth value of the polarization,

Then t--f 0 and, consequently, a0+ m, i.e. the cholesteric phase becomes a nematic phase. The dependence (39) is different from the linear dependence ko=PAnA-PB(1-rzA)

(4)

proposed in[6] for the cholesteric solutions. However, the concentration dependencies (39) and (40) are sufficiently close (see Fig. 3).

(a)

(b)

Fig. 2. Orientation of the molecules of the nematic phase. The arrows show the orientation of the molecules as well as the orientation of the elementary dipole moment of the molecules. (a) Orientation of molecules associated with “liquid ferroelectric” state. (b) Orientation of moleculesassociatedwith “liquidantifer-

roelectric”state.

The second kind includes the nematic phases whose molecules are aligned along one direction but have opposite orientations. In this case the sum of dipole moments per unit volume is zero and spontaneous polarization does not arise. The corresponding liquid crystalline phase is analogous to the antiferroelectric crystal (see Fig. 2b). Nematic phases of the second kind are stable in the homogeneous state. It follows from the above consideration that the free energy AF is degenerate with respect to both right-handed and the left-handed helical structures. This degeneracy can be lifted by short-range intermolecular interactions if the molecular shape is not planar[5] (the left-handed or right-handed structure can provide better packing in accordance with the shape of the molecules). Consider the case in which the liquid crystal phase is a mixture of two kinds of molecules, A and B, aligned along the same axis but in opposite directions. Then the polarization per unit volume is

where P.., and PB are the polarizations per unit volume of the corresponding single-component system, and nA is the molecular fraction of the A molecules. In this case the equation for the helix wave vector can be obtained from eqns (38) and (36):

Fig. 3. Dependence of helix wave vector k, on liquid crystalline phase composition. The solid line corresponds to the dependence (39).The dashed line corresponds to the linear dependence (41).

Finally, we consider a question: may the liquid ferroelectrics with helical domain structure be identified with the cholesteric phase! It is known that the dispersion forces contribute more than 90% of the total interaction energy of the liquid crystalline phase. However, the dispersion forces cannot provide stability to a cholesteric phase whose helical pitch is of the order of 10’ to 10’. The fact is that a helical structure can be stable only if the characteristic scale of the intermolecular interactions is determined by a macroscopic length of the same order as the helical pitch. The mechanism considered above based on the long-range dipole-dipole interaction, in principle, allows one to overcome this shortcoming, to explain some properties of the cholesteric phase, and consequently, to assume that in some cases the cholesteric phase is a liquid ferroelectric with the helical structure. However, there is a finding which, at first sight, con-

A. G. KHACHATURYAN

1060

tradicts the last statement. Experimental studies seem to show that the helical pitch in large samples of the cholesteric phases depends only slightly or not at all on sample size. This is in contradiction to eqn (36) which relates the helical pitch with the sample dimension R. However, an analogous difficulty arises in the analysis of the domain structure of large samples of crystalline ferroelectrics and magnetics. Their domain sizes, also, are not determined by sample dimensions. This problem is usually solved by means of the assumption that domain sizes are related to distances between crystal lattice defects. Apparently, there are no reasons that would not allow one to assume an analogous situation in liquid ferroelectries with sufficiently large dimensions. The helical pitch in liquid ferroelectrics as well as the domain size in crystalline ones may be supposed to be related to the characteristic distance between defects (for example, alloying atoms and molecules, their clusters, molecules in other conformations, and so on). In that case, the characteristic distance between defects plays the role of the parameter R in eqn (36), and the value of the helical pitch will not be dependent on sample size and, consequently, will be an inherent property of the cholesteric phase. REFERENCES 1.

2. 3. 4. 5. 6.

KittelC., Introduction to Solid State Physics. Wiley, New York (1956). Oseen C. W., Trans. Faraday Sot. 29, 883 (1933). Frank C. F., Disc. Faraday Sot, 25, 19 (1958). Khachaturyan A. G. and Hairapetyan V. N., Phys. Stat. Sol. (b) 57, 801 (1973). Grossens W. J. A., Physics Letters 31A, 413 (1970). Baessler H. and Labes hf. M., J. Chem. Phys. 52,631(1970).

into account (A3a), we have SF=AF,+AF.~~K,,Vkd+g~~K,Vk~‘tg,

t&P’L,C~~~e’*“‘/1.

AF, 3 &

0

P2L,C(A,/*

(Al)

where IA,r=IA,rr+IA,yr,A,=

AT,

k(B. + B,),

AI, =

cos (kox t t)(z)) =

A,, =

sin(k,zt$(z))=~(B.-B,).

(A2a)

(‘46)

The quantity & is a minimum value of the functional (b. The function $ = &(.z) providing the minimum value I#J,,of the functional 4 can be found from the Lagrange equation - K,Vs

d’

I&+&- P*L,C[-

0

yI sin J10t yz cos I&] = 0

(A7) which was obtained from the condition &#I= 0 where SI$is a first variation of 4 with respect to function JI(z). Here y, = .fE(dzla) cos & and y2 = .fo”(dzla) sin &(.z) are constants. Since &(z) is a periodic function, the boundary conditions of eqn (A7) are (2),_.

= (Z),_.

= 0, Jl(0) = $(a).

(A8)

The solutions of eqn (A7) correspond to the maxima, saddle points, and minima of the functional do.It is necessary to analyse the sign of the second variation of Q:

APPENDIX

Since all terms in (30) are positive, one can obtain an inequality omitting all terms except those corresponding to m = 1 and m=-I:

(AS)

where

t

2

oa:cos #o&b (I >I

to determine what solutions of eqn (A7)provide the minima of $. The function r&(z) is connected with the minima of I$ if the second variation 8’4 is of a positive definite quadric form. AU solutions of the nonlinear integro-diierential equation (A7) are unlikely to be.found in an explicit form. However, there is no necessity to do this. It will be shown below that all solutions of eqn (A7) whiih depend on the coordinate z correspond to saddle points of the functional 4 and, consequently, they are not the subject of interest. There is a single solution of eqn (A7) &a(z)= $ = const.

(A2b) (A3a)

which provides the minimum of 4. In order to prove this, let us take the derivative of eqn (A7)with respect to z. The result is

Bz=

Substituting &?a, b) into (Al), we obtain

Using eqn (AlO) one can easily prove that the variation S$ = eo(dtiO/dz)(e. is an infinitesimal constant) makes the functional (A9) equal to zero. Taking

AFaay;(/B.p+IB,i’)

3 j& P2LCIB,r. 0

(A4)

Combining the inequality (A4) with the equality (28) and taking

s*=c

%+e(z) ’ dz

(All)

where e(t) is an infinitesimal function, and using the boundary conditions (A8), we have

1061

Development of he&J cholesteric structure

x [y, cos& + y2

sinI&

I

value &of the functional ek The function &(z) = E = coast. is the single solution of eqn (A7) providing the minimum of the fuactional 4, since all other solutions correspond to saddie points of the functional 6. Substituting the solution &,(z) = $ = coast. into eqa (A6), we have

One can essay prove by explicit subs~tu~o~ that the rn~~um v&e of the free energy bF if (d&dr) $80 sad le P eF we caa obtaia &“I#CO by simpIy changing the sign of 4, since the second term in (AJ2) is quadratic and the first one is linear with respect to E(Z).This means that the minimum of Cpcan not be related to the coordinate dependent salution ~&r) of eqa (A7). However, there is one other solution of eqn (A7)which does not depend on coordinates:

can be obtained for the distribution of the polarization P(r) = P&!(z) where so(z) = (cos(k0r + j), sin(koz+ j;), 0).

(A161

@O(Z) = $ = coast. Actually, it follows from (Ala) that Substituting this solution iato eqa (A9), we obtain

The functioa~ (Al3) is of a positive detbtite quad& form. Therefore the extreme function I&= $ provides tJte minimum

Ibsen #rG= $ into eqn (31) and (A J7) into eqa (3CJ)* one has dF= dF, +BF, = AF,,, where AFmF,,is deterasiaed by eqa (AS).