On the tensor vs. the vector character of the director field in a nematic liquid crystal

Physica A 150 (1988) 299-309 North-Holland, Amsterdam

ON THE TENSOR VS. THE VECTOR CHARACTER OF THE DIRECTOR FIELD IN A NEMATIC LIQUID CRYSTAL G. VAN DER

ZWAN

Department of Physical and Theoretical Chemistry, Vrije Universiteit, de Boelelaan 1083, 1081 HV Amsterdam, The Netherlands

Received

8 January

1988

The description of the director field of the nematic phase of a liquid crystalline material is investigated. It is shown that utilizing a tensor field to represent the director has exactly the same physical consequences as the more commonly used vector field. Furthermore it is proven that a nematic tensor field can always be replaced by the square of a vector field for which the Frank-Oseen expression holds.

1. Introduction It is sometimes suggested that the description of the director field in a nematic liquid crystal by a tensor field is more natural or fundamental than the commonly used vector field’). In a recent paperlh) it was even claimed that such a description has different physical consequences for measurable quantities. It seems therefore worthwhile to devote a paper to the subject in order to point out the equivalency of both descriptions, and to show that the use of either one is merely a question of taste. The problem can be stated as follows: Molecules have tensorial properties related to their structure, such as a moment of inertia tensor or a magnetic susceptibility tensor. For the molecules constituting a liquid crystal these (symmetric) tensors have at least two different eigenvalues, three if the molecule has no axial symmetry. In order to arrive at the director field in its usual form we can designate the molecular axis with the smallest eigenvalue the molecular z-axis, and average over the phase space of all molecules in which case we arrive for the nematic phase at a nonzero quantity: the director field2). It is assumed, and so far there is no experimental evidence to the contrary, that the averaging process leads to axial symmetry even if the molecules themselves are not axially symmetric. On the other hand it is of course also possible to average the tensor property immediately, without first taking one of the axes as 037%4371/88/$03.50 0 (North-Holland Physics

Elsevier Science Publishers Publishing Division)

B.V

G. VAN DER ZWAN

300

a preferred one, thus arriving process evens the differences molecular tensor certain properties

at a tensorial director field. The averaging between the two largest eigenvalues of the

property chosen and consequently the related to the resulting axial symmetry.

These two processes any difference whether

average

tensor

has

ought to lead to the same results. It should not make we choose a molecular z-axis as the microscopic vector,

average, and arrive at the director field; or first average, get an average tensor, and subsequently choose the axis connected with the exceptional eigenvalue as the vectorial form of the director field. At first the number of components of the tensor and the vector field may seem to be different but there are restrictions on the tensor-, as well as on the vector field. For the vector field the length of the vector is fixed at each point in space, one habitually takes it equal to one, which makes the number of varying components equal to two; these in turn can be related to the two angles (polar and azimuth) the director vector makes with a suitably chosen coordinate system. For the tensor field the eigenvalues of the tensor have to be constants. This gives rise to three relations between the components. Furthermore. after averaging, there has to be axial symmetry, which is an additional condition. We thus arrive at again possibly two independently variable quantities, which can now be related to the angles the principal axis makes with the external coordinate frame. The

aim

of this

paper

is twofold.

First

we wish

to clarify

the

above

statements and show rigourously that the two processes indicated above do indeed lead to fields that have the same consequences for the physically measurable quantities depending on them. Secondly, it allows us to show that a consistent application of either one of the formalisms does not lead to contradictions or convergence problems3). The organization of this paper is as follows. In the next section we review the Frank-Oseen expression for the free energy of the nematic liquid crystal. In section 3 we introduce a similar expression for the director field in tensorial form, which is analyzed in section 4. In the final section we show that a tensor field with axial symmetry can always be written as the square of a vector field. For this derivative vector field the Frank-Oseen expression holds.

2. Free energy of the direct or (vector) field In order to compare with the usual vector mode structure of the point in this section is

the results to be derived from a tensor field approach field expressions we briefly review in this section the conventional free energy for a nematic. Our starting the Frank-Oseen expression for the deformational free

DIRECTOR

energy

of a nematic

9 = $,, + i

I

FIELD IN NEMATIC LIQUID CRYSTAL

liquid

crystal.

dr { K,(div

It is given

n(r))’

301

by4) - curl n(r))’

+ K,(n(r)

V +

K,(n(r)

-

.

grad n(r))“}

(2.1)

In this equation n(r) denotes the (conventional) director field; K,, K2 and K3 are the elasticity constants for splay, twist and bend, respectively. The integration is over the volume of the system and 9(, is the free energy of the system in the absence of deformations. When the deformations are small, it is convenient to write the fields n(r) as IZ” + &z(r), where the average field H,, is constant over a macroscopic volume and &z(r) are small deviations from the average value. In the remainder of this section n(r) denotes the fluctuations of the director field around the value n,,. Up to lowest order in the fluctuations the free energy becomes

$kF”+I

2

I

dr { K,(div

n(r))” + K,(n,

- curl n(r))’ + K,(n,, *grad n(r))*},

V (2.2)

Strictly speaking the integration is no longer over the total volume of the system but over a volume in which n, is a constant; since both volumes are macroscopic, we neglect the difference. In order to preserve the length of the vector n, + n(r) we have the auxiliary condition n, - n(r) = 0 on the field n(r). We introduce the spatial fourier transform of a field and its inverse as n(k) =

dr n(r) eik’r

and

n(r) = i

c

Substitution

9-

of eq. (2.3)

9(,=

n(k) eik’? .

(2.3)

k

V

$

c

into eq. (2.2)

{K,jk-n(k))2+

gives

K,ln,-(kx

n(k))12+

K,~n,+z(k)(*}.

k

(2.4) Since fluctuations take place in a plane perpendicular to n, it is advantageous to expand them in a basis in that plane. We make the following choice for the two basis vectors: a, =

%’

(It”’ k, k,

and

a = 2

aoxk

(2.5)

G. VAN DER ZWAN

302

where k: = k2 - kf The unit basis.

vectors

and

k,, = n,, - k .

a, and a,,

The fluctuations

together

(2.6) with n,,, form

can be written

a complete

orthonormal

as

n(k) = n,(kb, + n,(k)a2

(2.7)

Upon introduction of eqs. (2.5) and (2.7), we get after calculation the following expression for the free energy:

a straightforward

(2.8) or for every value of k the ncr(k) are Expression (2.8) is invariant under the exponential of the free energy fluctuation eq. (2.8) can be used director field fluctuationsJ.5).

3. Free energy of the director

the two independent modes of the system. rotations around n,, as it should be. Since is directly related to the probability of a to find the correlation functions of the

(tensor) field

All physical expressions related to the director field depend on it in tensorial form. The reason is the invariance under the inversion n-+ -n, which appears to hold for all nematics. We will use as an example the potential of mean force between a (probe) molecule and its nematic surroundings as introduced by Polnaszek and Freed”‘), Ii = -Q: 1212,where Q is some tensorial property of the guest molecule. We might equally well consider the quantity relevant for light scattering, the dielectric susceptibility tensor, of which the anisotropic part is also proportional to nn ‘). Measurable quantities are related to corwhere the averaging is over the fluctuarelationfunctions of the type (r/U), tions of the directorfield’.‘). If one takes the more “fundamental” point of view the above potential could be written as U = -Q: N, where N is a tensor field describing the director. This necessitates a fluctuation theory for the tensor field N. The field 1211is not obviously the same as N. Fluctuations in 1212 can be derived from those in 12 directly: we can write s(nn) = n,,Sn + i%zn,,. which is consistent with the free energy expansion given in the previous section. However, the fields n and N are averaged quantities and it is by no means evident that the product of two averages is the average of the product.

DIRECTOR

Therefore

one should

FIELD IN NEMATIC

start

at a microscopic

303

LIQUID CRYSTAL

level as indicated

in the introduc-

tion. Consequently, the free energy for deformations cannot be derived from the free energy given in section 2, but one should again start by searching for scalar

quantities

in terms

of the derivatives

of the tensor

field.

First we note that since after averaging the system has axial symmetry, can always choose our tensor N with the following properties’): N2=

N

and

Tr(N)=l.

we

(3.1)

Although the tensor nn has these properties it does not a priori mean that N can always be written as nn; that is only the case when N can be considered what in quantum statistical mechanics is called a “pure state” “). The immediate consequences of eq. (3.1) are that N has two eigenvalues zero and one eigenvalue equal to one “). Such a tensor can always be constructed starting from, for instance, the moment of inertia tensor, and may be taken to represent the director field since we are not interested in changes in the eigenvalues but in singling out the main principal axis. The next step is to write down the free energy for deformations based on the derivatives of the tensor field introduced above. A systematic search for scalars using the methods given in refs. 4 and 5 gives the following result for the “tensor” free energy ST: ST - SC0=

J

dr { K,(Div

- N(r) - (Div N(r))

N(r))

V

+ i&(Curl + K,(Div

where

the divergence

(Div N)cX = apNPu

N(r)): N(r).

(Curl N(r))

(7 - N(r)) - (Div N(r))}

and the curl of tensors and

(Curl

N),,

are defined

= eafivaKLNVP.

,

(3.2) as (3.3)

In eq. (3.3) ~~~~ is the Levi-Civita tensor, and the Einstein summation convention is followed. In the derivation of eq. (3.2) we used the properties (3.1), surface terms were neglected, and we used a few relations given in the appendix for completeness. The elasticity constants can be found in several ways. One way which is, strictly speaking, not correct, since possible coherence effects are neglected, is to substitute N = nn. The Frank-Oseen expression is then recovered. An other, independent way is to substitute the tensors belonging to splay, twist and bend deformations and show that these give just the respective terms in eq. (3.2).

304

G. VAN DER ZWAN

Expression (3.3) is equivalent to the tensorial free energy expression given in ref. lb. We now follow the same procedure as in the previous section for the fluctuations. lowest order performed 9’

First we write N = N,, + SN(r) and expand the free energy in the fluctuations. Subsequently a spatial Fourier transform

to is

to obtain - q, = $

{K, Tr[ N(k) - kk * N*(k) - NJ

c k

+ i K, Tr[(k x N(k)) - (k x N*(k))] + K,Tr[N(k)*kk.

N”(k)*(l

- N)]}

(3.4)

The 6 is again deleted and N stands for the fluctuations around the average value N,. The symbol Tr stands for the trace of a tensor and the definitions of k. N and k X N can be derived from eq. (3.3). As in the previous section we have conditions on the fluctuations N. Eq. (3.1) has to be satisfied by the tensor N,, as well as by N,, + N. From this we derive, by expansion to lowest order, the following relations for the fluctuating fields: Tr[ N] = 0

and

N,, - N + N - N,, = N .

(3.5)

These relations are essential to our analysis, since they make certain that at every point the director field has a constant set of eigenvalues and axial symmetry. In the next section we will elucidate the mode structure of the above free energy in the small deviation approximation.

4. The normal modes of the tensor field The set of symmetric tensors can define an inner product as (A, B) = Tr[A* B]

of rank

two form a vector

space

in which we

(4.1)

Therefore one possible way to proceed is to construct a complete orthonormal basis in this space, expand the fluctuations in this basis and attempt to diagonalize expression (3.4). Although it seems artificial, and it is indeed not necessary, we let ourselves be guided by the results of the previous section to obtain such a basis. It is always possible to write N,, as no+,, even though this does not (yet) mean that n,, is the conventional director field. We can always

DIRECTOR

find a local coordinate

FIELD

system

IN NEMATIC

LIQUID

in which for instance

CRYSTAL

(No)lZ is nonzero,

305

whereas

all the other elements are zero. Since an n,, can be identified, we can use the vectors u, , u2 together with n, to construct the desired basis. It is given by the following set of six tensors:

N,, = n,jn,,,

A, = a,~,,

A, = ~2~2,

A, = $

(w,

+ urn,,) 3 (4.2)

A, = L

(~2

v3

+

and

a2d

A, = -&

(~,a,

+ a2u,).

One can easily verify that this set is orthonormal under the trace inner product. Since according to the second of the relations (3.5) the fluctuations N have to be normal to N,, in the above inner product sense, we can write

N(k) =

2

N;(k)A; ,

(4.3)

,=I

which automatically lets N satisfy the condition Tr[ N - No]= 0: the fluctuations are perpendicular to N,,. From eq. (4.2) we get Tr[A,] = Tr[A,] = 1, whereas the trace of the other tensors is zero. The first of the conditions (3.5) thus implies N, + N, = 0. Substitution of eq. (4.3) into the second relation of (3.5) gives in fact N, = N2 = Ns = 0. As expected we find only two normal modes. It remains to be shown that these two modes have the same eigenvalues as the ones found earlier. To that end we substitute eq. (4.3) into eq. (3.4). After a somewhat lenghty but again straightforward calculation we obtain B”-

So = &

c k

c

{K,mZk;

+ K,k;}IN,(k)l*.

u=3.4

(4.4)

As an application we calculate the correlation function (UU) for a probe molecule at the origin of the coordinate system using both the formalisms. From

tensor

formalism

we get (4.5)

whereas

the vector

(UU)=

c

0=1,2

formalism

gives

(n,.Q-aU+n~.Q.n,)‘~(In,(k)12).

(4.6)

Using the definitions of the A,, eqs. (4.2), together with the values of the averages, which can be derived from eqs. (4.4) and (2.8) (note the 1/4V in eq.

306

G. VAN DER ZWAN

(4.4) and the 112V in eq. (2.8)),

(UU>

=

C (n,,-Q*uCx)‘T
This completes

our treatment

5. Large deviations,

remarks

we finally

arrive

.‘,~~~~~z (Y .

in both cases at the result”‘)

(4.7) 3 II

of the fluctuation

theory

in both cases.

and conclusions

We have shown in the previous two sections that, at least for small fluctuations, both the tensor formalism and the vector formalism can be used with equal success and with the same results. It remains to be shown that it is also true in general. The results for the fluctuation theory indicate the propriety, after all, to replace N by nn, provided we take for the fluctuations in nn only the terms linear in ih, in order to be consistent with relations (3.5). It is allowed to replace N by nn as a consequence of the conditions on the tensor field, given by eq. (3.1). Deviations from the average value N,, can only be the result of rotations. Eq. (3.1) implies we can always find a local coordinate system in which N,, has the form

NC,=!:

i

!]=[e]l”

o

l]=n,,n,,.

Deviations from this situation can only the above coordinate frame. Arbitrary be obtained by decomposing N(, into their transformation properties under

N =

be the result of rotations with respect to rotations of this tensor can for instance its spherical components and consider a general rotation”). The result is

sin% COS’C$ sin% sin C$cos 4 sin% sin 4 cos C$ sin% sin’+ sin H cos H cos 4 sin 0 cos H sin 4

It was derived using that the trace of a tensor spherical components of a traceless symmetric

N,:, = i k-

D;,,,(fl)N,

.

(5.1)

sin 0 cos 0 cos 4 sin H cos H sin C$ co& transforms tensor as

as D11,,(0),

(5.2)

and the

(5.3)

2

where fl = ($. H. 4) is the set of Euler angles over which the rotation takes are the Wigner rotation matrices. Rotations around the place and Df,,(R)

307

DIRECTOR FIELD IN NEMATIC LIQUID CRYSTAL

principal because rotation

axis (which

n=(sin8cos+ In other

have

magnitude

(cl) do not

enter

into

the expressions

of the axial symmetry. It is obvious from eq. (5.2) that we can still introduce a vector n such that N = nn, namely sin8sin4

words:

c%l,n,%!

a “pure

state”

after

cos8). remains

-’ = (%!n,)(%!n,) =

the

(5.4) a “pure

state”

after a rotation

2:

(5.5)

nn .

It is therefore a consequence of the relations (3.1) that we are allowed to replace N by nn for arbitrary deviations from the average field N,. These relations are themselves a reflection of the axial symmetry existing after the director field is obtained by averaging some microscopic field. The above considerations are sufficient to show the suitability of letting a vector field represent the director in the nematic phase. In addition we ought to make certain that for this vector field the Frank-Oseen expression holds. This can be done by inserting N = nn into eq. (3.2); eq. (2.1) is then recovered. The quantity n may thus for all intents and purposes be considered to represent the conventionally introduced vectorial director field, and the Frank-Oseen expression is the correct free energy of deformation. We have therefore shown that the use of a vector field to describe uniaxial nematics is fully justified for both small fluctuations around an average directorfield as well as for large deformations. Care has to be taken in comparing the fluctuation theories. Replacement of N by nn in expressions like the ordering potential, or in the dielectric susceptibility, does not mean that fluctuations in N include a term Sn8n. Such a term may lead to divergences in the calculation of the standard deviation of the potential. Indeed, inclusion implicates the necessity of expanding the (vector) free energy to a higher order. The differences found in ref. lb in the consequences of both field theories may well be attributable to a nonconsistent application of either one of them. A vector field to describe a nematic has one advantage over a tensor field: one is not tempted to include in the fluctuations terms of the type &z&z. This seems a trivial point, but the treatment of fluctuations is obscured by the fact that sometimes one considers the Euler angles as the fluctuating quantitie?‘.‘“) in which case it is hard to keep track of where the terms like 13’ come from.

Acknowledgment The author

would

like to thank

Dr. D. Bedeaux

for valuable

comments.

308

G. VAN DER ZWAN

Appendix In this appendix

we give a few relations

eq. (3.2). To show the general term

not

relevant

procedure

for

followed

nematics,

but

which are useful in the derivation in the derivation

which

is present

we first derive in cholesteric

of the

liquid

crystals.

We look for scalars containing first derivatives of the tensor field: The only scalar one can find using just this field is a contraction with “&a. the Levi-Civita tensor but it vanishes since the latter tensor is antisymmetric in is symmetric. Next we look for combinations of NC,, all indices whereas NcYa and a,N,, . Using the properties (3.1) there remains only one possible term. given by

E‘~,~,,N,,~I?~N~~ = N: Curl N .

(A.11

Upon introduction of N = nn (and using that IZ. n = 1) this term indeed reduces to II * curl n ‘.‘). Terms quadratic in the derivatives can be treated in a similar way. Surface terms are neglected so that we can write for example

“a&3 a&Jrrg = In eq. (A.2)

aPN,,8N,,

the difference

= (Div N).(Div

between

the right

N).

and left-hand

This term occurs under the integral sign in eq. converts it to a surface integral. In addition we used the following relations:

(JKN+)(a,N,,,)

(A.21

= (Curl N): (Curl N) +2(Div

(3.2)

and

N).(Div

side is given

Gauss’

N)

by

theorem

(A.4)

and

(Curl Wap By a tedious

(Curl N)pa = FUpy(Div N),

systematic

procedure

we arrived

(A.5) at eq. (3.2).

DIRECTOR

FIELD

IN NEMATIC

LIQUID

CRYSTAL

309

References 1) (a) R. Blinc, in: NMR Basic Principles and Progress, vol. 13, Introductory Essays, M.M. Pintar, ed. (Springer, Berlin, 1976), pp. 97-111; and references therein. (b) E. Govers and G. Vertogen, Physica A 141 (1987) 625. 2) L. Onsager, Ann. N.Y. Acad. Sci. 51 (1949) 627. 3) The convergence problem referred to is one occurring in molecular ordering. If one takes as the ordering potential of mean force U = Q: nn, where n is the director field, then fluctuations in this potential should not include the term 6n8n. See e.g. (a) CF. Polnaszek and J.H. Freed, J. Chem. Phys. 79 (1975) 2283, and (b) L. Plomp, M. Schreurs and J. Bulthuis, J. Chem. Phys. March (1988), esp. note 19, to be published. 4) P.G. de Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1974), chap. 3. 5) L.D. Landau and E.M. Lifshitz, Statistical Physics, part 1, Course of Theoretical Physics, vol. 5, 3rd ed. (Pergamon. Oxford, 1980), chap. 12, and 9140, $141. 6) J.H. Freed, J. Chem. Phys. 66 (1977) 4183. 7) Although they do not state so explicitly, the authors of ref. lb do make use of these relations, otherwise their free energy expression is not complete. 8) R.P. Feynman, Statistical Mechanics, a Set of Lectures (BenjaminiCummins, London, 1982), chap. 2. 9) The eigenvalue equation can be written as A’ - A” Tr[N] + h((Tr[N])* - Tr[N’])/2 case reduces to A3 ((Tr[N])3 + 2 Tr[N’] - 3 Tr[N] Tr[N’])I6 = 0, which for the present A?=O. 10) For a more extensive treatment of this molecular ordering problem, see G. van der Zwan and L. Plomp, Liquid Crystals, submitted. II) M.E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957). chaps. 4 and 5.