TK ASsr.TkG A

TK ASsr.TkG

KFKI- 1985 -ilO

N, ÉBER A , J Á K L I

A CONTINUUM THEORY OF CHIRAL SMECTICS C*

PART I.

ELECTRO- AND THERMODYNAMICS OF POLARIZED MEDIA

'Hungarian 'Academy o f Sciences

CENTRAL RESEARCH

INSTITUTE F O R PHYSICS

BUDAPEST

■J

I

;

A CONTINUUM THE O R Y OF CHIRAL SMECTICS C*

PART I.

ELECTRO- AND THERM O D Y N A M I C S OF POLARIZED MEDIA

N. ÉBER, A. JÁKLI

Central Research Institute for Physics H-1525 Budapest 114, P.O.B.49, Hungary

HU ISSN 0368 5330

A continuum theory of polarized media is presented. The electromagnetic field is incorporated into the balance equations and an expression for the energy dissipation of the medium is obtained. Reversible and irreversible phenomena are separated. The final equations serve as a basis for setting up the constitutive equations for chiral smectic C* liquid crystals.

АННОТАЦИЯ

Представляется континуальная теория поляризованных сред. Электромагнит­

ное поле подставляется в уравнения сохранения и выражается диссипитивная функция среды. Различаются обратимые и необратимые процессы. Конечные уравне­

ния служат основой написания материальных уравнений спиральных смектических С* жидких кристаллов.

KIVONAT

Polarizált közegek kontinuum elméletét ismertetjük. Az elektromágneses teret beépítjük a mérlegegyenletekbe, és kifejezzük a közeg energiadisszipá­

cióját. Szétválasztjuk a reverzibilis és az irreverzibilis folyamatokat.

Egyenleteink alapul szolgálnak a csavart szmektikus

c*

Since the discovery of liquid c rystals a lot of work has been

devoted to the study of macroscopic, "bulk" properties of these materials.'*'-^

Parallel with expe r i m e nt s several c o nt inuum theories have been developed to explain e x pe r i m e n t a l data. 1-23

Perhaps the best known of them is the powerful E r ic k sen-Leslie theory of nematic and cholesteric liquid c r y s t a l s .6-11 Though it has been c r iticized from some points of v i ew and other theories for these phases also

12-21

exist, the Ericks e n -L e s l ie theory can give an a c count on many reversible phenomena (e.g. e l a s t i c deformations, Free d e r i ck s t r a n s i t i on s 2-4) as well as

8 _ni some i r re v e r s ib l e o n es (e.g. v i s c o u s flow, t h er m o m e ch a n i ca l coupling ).

Un fo r t u n at e l y it ca nn o t be g e n e r a l i ze d easily to describe smectic phases, while o t he r theories of smectics usually deal with some special aspects o n l y .2-3, 5 , 2 2

An alte r n a t iv e approach to the p r ob l e m is the unified hydrodynamic theory of Martin, Parodi and P e r s h a n . ^ Wi th i n the f r am ework of this theory a rigorous f o rm u l a t io n of the r e ve r s i b l e d ynamics of v a ri o u s liquid crystal- Tine p h as e s has been worked out, ’ ’ but much less attention has-been paid to irreversible phenomena. 19 However these theories are primarily devoted to d e sc r i b e fluc t u a t io n s and Tight s c attering in the a b se n c e of external el ec t r o m ag n e t ic fields, thus u n f o r t u n a t e l y are less a p p l i c a b l e to explain effects due to the external fields. Irreversible phenomena connected with e l e c t r o m ag n e t ic fields, e.g. d i el e c t r i c relaxation, are a priori .neglected arguing that rela x a t i on of p o la r i z a t i o n takes place on a micr o s c o pi c time scale w h i l e hy dr o d y n am i c s is v alid only for chara c t e ri s t i c times much longer than the time between molecular collisions. Though this a rgument holds for most cases, there are exceptions since in smectic liquid crystals especially in fe rr o e l e ct r i c c h i r a l smectics C* there are relaxational phenomena at very Tow frequencies^'’" ^ too.

A c o mb i n a t io n of e l e c t r o - and t h er m o dynamics of polarized media can be f o und in two books of de G r o o t . ' ^ ’'5* However their equations are valid for s y stems without internal d e g r e e s of freedom but are not for liquid c r y s ­ tals. A g e ne r a l i z a t i o n for n e matic and c h olesteric phases has been done

13-14

wi th i n the f r am ework of m i c r o p o l a r c o nt i n u u m theories , but to our k n o w ­ ledge no such theory exists for the smectic liquid crystals.

Very recently we have found an expe r i m e nt a l evidence 27 that i r re v e r s ib l e p h en omena due to e l e c t r o m ag n e t ic fields may play an important role in the f e rr o e l e ct r i c ch ir a l smectic C* liquid crystals. This moved us to try to combine hydro-, elec t r o - and t h er m o d y na m i c s of chiral smectics C in a two-part p aper e s t a b l i s h i n g theoretically our observation.

In Part I. we focus on the basic equations of the c o nt inuum theory I n c o r p o r at i n g e l ec t r o d y n a m i c s into hydro d y n am i c s we follow the c o nception of de Groot"5® but we use a d i ff e r e n t r e pr e s entation and the SI system of units.

We d e r i v e e q ua t i o n s valid for any i n homogeneous a n is o t r o pi c polarized media having internal degrees of. freedom.

In the subsequent Part I I . w e construct the constitutive e q u a t i o n s for chiral and a c hi r a l smectic C liquid crys t a l s covering r e v e r ­ sible and i r re versible p h en o m e n a as well and discuss the relationship

b e t w e e n chirality and the e x is t e n c e of new c r o s s - e f f e c t s in these materials.

2. STATE V A RI A B L E S OF THE M E DI U M

A c o n t i n u u m is c h ar a c t e r i z e d by its motion and its internal thermo d y n a m i c state. We pretend that, as it is usual in n o n- e q u i l i b r i u m t h er m o ­ dynamics, our m e d i u m is in local equilibrium. The m o t i o n can be described by the velocity field v_(r) but the usual t h e r m o d y na m i c state variables (internal energy p u, entropy p s, density p , t e m p e r a t ur e T, pressure p) alone do not give a comp l e t e d e s c r i p t i o n of the t h er m o d y n a m i c state of the medium.

The e l e c t r o m a g n e t i c field, wh en interacting with the medium, modifies its inte r n a l state, cons e q u e nt l y one has to i ncorporate into thermodynamics some e l ec t r o m a g n e t i c state v a ri a b l e s too (e.g. p o la r i z a t i o n and m a gnetization or e l e c t r i c field and magnetic induction). F urthermore liquid crystals or any other ordered systems have further internal degrees of freedom which have to be taken into account (e.g. director for nematics or displacement of layers for smec t i c s e.t.c.).

As it is usual in field theories, two frames of reference will be О О _ T n

used. The laboratory frame serves for the d e s c r i p t i o n of the e l e c t r o ­ m a g n e t i c field and the m o ti o n of the medium. However the thermodynamic q u a n t i t i e s c h a r a c t e ri s i n g the internal state of the m e d i u m will be given in the mate r i a l frame, i.e. in the frame co-moving with the medium. This choice makes possible an easy f o rm u l a t io n of a Galileian invariant electro- and thermodynamics. The e l e c t r o m ag n e t ic fields detected in these two frames are different, since the frames are moving relatively to each other."5®

To ma ke a distinction, dashed quantities will be used to denote e l e c t r o ­

m a g n e t i c variables in the material frame and these dashed quan t i t i es will appear in thermodynamics. We pretend t h roughout this paper that the v elocity of the medium is small enough to remain in the n o n- r e l a t i v i s t i c approximation.

The rules of trans f o r ma t i o n between frames for the e l e c t r o m ag n e t ic quan t i t i es are given by Eq.(A.l.) in Appendix 1.

3. CONSERVATION LAWS

Since the e l ec t romagnetic field i n te racts with the medium, n e ither of them alone can be regarded as a closed system. The c o ns e rvation laws can be w r it t e n only for the system being comp o s e d of the medium and the e l e c t r o ­ m a g n e t i c field. Though it could be done in both frames, the labo r a t o ry frame is p r ef erred because of the presence of e l e c t r o m a g n e t i c terms. However the t r an s f ormation rules of the fluxes are taken into account, i.e. the conv e c t i ve terms are separated in the balance equations. In the non- r e l a ti v i s ti c a p p r o x i ­ ma ti o n the electro m ag n e t ic mass is n e gl e c t e d thus we have c o ns e r v a ti o n laws for the mass, the total linear momentum and the t o tal energy. The integral form of these e q ua tions in the laboratory frame are

jj^fpdV = - ф Pvd n (3.1)

J ( Р У + s f i e l d )dV = -<j> ( a - p v.v - T)d n (3.2)

(тур v 2 + P u + ef l e l d )dV = - $ Jed p (3.3)

where д / * e ^ d and are the linear mome n t u m and energy of the e l e c t r o ­ m a g n e t i c field respectively, _£ and J are the m e c h a n i c a l and the Maxwell stress tensors respectively. The total energy flux is not s p ec i f i e d at this point, it will be given later via a c o n s t i t u t i v e equation.

The above c o ns e rvation laws have to be supplemented wi-th the entropy balance e quation

^ j p s dV = -<j) (у д + p s v ) d P + j у dV (3.4) where д is the heat current and R is the energy dissipation.

4. BALANCE E Q UA T I O N S FOR THE E L E C T R O M A G N E T I C FIELD

The e l e c t r o m ag n e t ic field can be described in the laboratory frame by the Maxwell e q u a t i o n s given in SI.32

7 0 = pe 7 B = 0 V x E

э в a t

(4.1) B D

7 x H = J + -- 9 t

where pg is the c h ar g e density, £ is the c u r r e n t density and o = e0£ + P

H = - В - M (4.2)

- u„~

defines the p o l a r i z a t i o n £ and m a g n e t i z a t i o n M of the medium.

These e q ua t i o n s can be r e wr itten into the form of a m o m e n t u m and energy balance as s h ow n in Appendix 2 and 3.

- gfield = V .T .. - F. (4.3)

at 1 J 1J 1

A efield = 7 jfield + r field (4.4)

a t J J

where £ is the force exerted on the medium by the field, ^he

e l e c t r o m ag n e t ic ener g y flux, rlleld is the rate of t r an s f o r ma t i o n from field energy into kinetic or internal one, and the summation conv e n t i on on repeated indeces has been used. However the d e f i n i t i o n of the e l e c t r o m a g n e t i c momentum and energy is not unique, the m e d i u m and the e l e c t r o m ag n e t ic field cannot be s e pa rated u n am b i g u o u s l y because of their interaction."51 We have chosen the representation"51 w here

field

a ^{e0 E x B (4.3)}

and field

e 1 1 R2

2 й Д BM (4.6)

since it has led to a consequent G a li leian invariant t r ea tment of the ther m o d y na m i c s of p o la rized media in the n o n - r e l a t i v i s t i c approximation.

For further details including the d e f i n i t i o n of the other quantities of Eqs. (4.3) and (4.4) we refer to Appendix 1-3.

5. BALANCE EQUATIONS FDR THE ME DI U M

Using Eqs.(4.3) and (4.4) the c o ns e r v a ti o n laws E q s .(3.1)-(3.3) can be c o nv erted into balance e q u a t i o n s for the mass, linear momentum, kinetic energy and internal energy of the m e d i u m . ® In the material frame they read

г г ° - - p v j vj

dt

P V, — V. o. . - n v . V.

J 1J v. V. v- + F.1 J

(5.1) (5.2)

± . I p V 2 = . Vj(v. o ^ ) - | P v 2 V.v. + FiVi + 0 i j V. Vi and

3 t » u “ Л И ' VJ ° U } - * " V i -

- °ij V i * v i ield - r“ Bld - ' л

d 3

where + _v J7 is the mate r i a l time derivative.

From Eq.(3.4) the entropy balance is

(5.3)

(5.4)

dt pS = “ (T qj} ~ pS V j + T (5.5)

6. C O NS E R V A T I O N OF ANGULAR M O M E N T U M

The only c o ns erved quantity, which we have not yet paid attention to, is the angular momentum. E m pl o y i n g Egs.(4.3) and (5.2) the balance e q ua tions for the angular mome n t u m of the field and the m e d i u m are r e s p e c ­ tively

ж ' « “ »i ■ V i j k V k i ’ - ' ^ i - ‘ i j k V i

Ht (£*pv)i = - ( Ei.k rj 0 k l ) + (rxF). + eijk a k .

Adding these two e q ua tions and c o mp a r i n g with Eq.(3.2) it follows i m m e d i a ­ tely that the cons e r v a ti o n of the total angular mome n t u m requires

(6.1)

(6.2)

s j k c k j * c i j k Tk j * V ü k £ k j i <6- , ) where is an arbitrary an ti s y m m et r i c tensor. In our r e p r e s e n t a ­ tion the M a xw e l l stress tensor is not symmetric c o ns e q u e nt l y the mechanical stress tensor must have an a n t i s y m m et r i c part too. In the absence of e l e c t r o ­ m a g n e t i c field Eq.(6.3) reproduces the usual argu m e n t that a symmetric stress tensor automa t i ca l l y m eets the requ i r e m en t of c o n s e r v a t i o n of angular m o m e n ­ t u m . 1 6 -20

In E q s .(3.2),(5.2) and (6.2) we p r et e n d e d that the medium had no extra internal linear or angular momentum. This means no restrictions in case of liquid crystals, since on c ontrary to the E r i c k s e n - l e s l i e 1 11 or the

m i c r o p o l a r 1 2 - 1 ^ theories which have to introduce such quantities, hydro-

1 6 - 2 0 2 3 - 2 4 3

d y namic t h e o r i e s ’ can d e s c r i b e the same phenomena without the need for such extra momehtums.

7. G E N E R A L I Z E D FREE ENERGY

The internal energy is the th er m o d y na m i c potential belonging to I I

the set of independent state v a ri a b l e s j s , £ , M However it is more

p r ac tical to use for independent variables the t e mp e r a t ur e instead of entropy, the electric field instead of p o la r i z a ti o n and the m agnetic induction instead of magn e t i z at i o n . This transition in variables c o rr e s p o nd s to the Legendre- - t r a n s f o r m a t i o n

p f * ( T , E V ) = p u(p s,p!m' )

I t I f

Tps - P E - M В (7.1)

w here the gene r a l i ze d free energy pf is the new t h er m o d y n a m i c potential for

I I .

the new set of independent state v a ri ables {T,E_,B_, . . .§ . Though either of the above two r e p r e s e n ta t i o ns could be applied to desc r i b e the same phenomena, we prefer the latter one since it has many a d va n t a g es when setting up the c o n s t i t u t i v e equations for chiral s mectics in Part I I . ^

I I

Besides T,JE and E3 a m e d i u m has some other independent state

variables too. These are the density, temperature gradient, velocity gradient and the i nternal degrees of freedom denoted by X a ( a = l ,2,...). These latter q u a n t i t i e s •should be s p ec ified sepa r a t e ly for each media.

With the above arguments the general form of pf* is

Pf * = pf * ( T , E ' , B_' , p , V T . V o v . X “ ) (7.2)

This e x p r e s s i on is Galileian invariant since it contains only quantities g i v ­ en in the m aterial frame but not the velocity of medium.

0. THE ENERGY D ISSIPATION

The basic equation of irre v e r s ib l e phenomena is the expression for the energy dissipation.

Using Eqs . (5.4), (5.5),(7.1) and ( A . 21) one gets for the energy dissipation

R - - г О » - - pf“ - Ц у * o'Ej' - p13 V3vt -

d pf dT , dE. , dB.

-- - p s — - P • — A - M . — -*A

dt dt ^ dt ^ dt

(8.1)

where the nota t i o n -.X e

- V vi 0 ij Vj { ^ PV/2 + p f * + PTS + P i^-X- >i} - (В.2) was introduced.

With indirect derivation of E q .(7.2) one gets

Эрf , I I

R = - V (Jx - q ) - ( pf* - p --- ) V v - A q H + J E -

J J J 0p J J j J J J J

ij 3 1

3pf* dE. 9Pf*

- (P. + ' , -) — 1- - (M. + -- r- 3

Э Е з dt ³

9 B j 3pf* dT 9pf* dV.T

- ( p s + --- ) - - - _ J__

ЭТ dt 9V.T dt

_ # a ЭрГ dX

3 i

(8.3)

which has to be s u pp l emented by the c o n s t i t u t i v e equations desc r i b i ng the 16 18

time e v ol ution of the internal d e gr e e s of freedom. ’

Xa = - Za

at a = 1,2, (8.4)

9. S E PARATION OF REVERSIBLE AND IRRE V E R S IB L E PHENOMENA

In general reversible and irre v e r s ib l e processes coexist in a medium. Their d e sc r i p t io n requires d i ff erent tools so one has to separate them. This s e p a r a t i on can be done on the basis, that reve r s i b le processes are invariant under time reversal, while irreversible ones are not. N e v e r t h e ­ less this i n variance concerns the equa t i o n describing the p r ocess and not the individual physical quantities. In general any physical quan t i t y can be

splitted into an equilibrium, reversible and a non-equilibrium, irreversible

in an opposite way. It is quite natural to regard the independent state

v a ri a b l e s as purely r e v e r s i b le ones. Moreover the generalized free energy and e n tr o p y desc r i b e e q u i l i b r i u m systems c o ns e q u e nt l y they are also r e ve rsible as well as their partial derivaties. The r e ve rsible parts of the other q u a n t i ­ ties are d e te r m i n ed by the requirement, that in equilibrium, where all i r r e v ­ e r si b l e terms vanish, the balance e q ua tions ( 5 . 1 ) — (5.5) must be invariant u n der time reversal.

To summarize, the purely reversible variables are T ,e' V , p ,v,x“ ,pf*, ps

as well as their time d e r i v a t i ve s and gradients, while others split into two parts

R =-- Rr + Rlr ; ^It

ж -,ХГ -,Х1Г

2 + Д ; а = аг + а 1г

1

P

1

= P r + p'ir

; м' = м'Г + м'1Г ; 1а = z“ r + z“ ir 1

3 = 0 -,'ir

+ 2 ; g _ = 0 + ^ 1Г

w h er e the latter two, name l y the electric and heat currents have only i r r e v ­ e r si b l e parts.

We i l lustrate the above m e nt ioned m e t h o d of separation on the energy d i s s i p a t i o n term. The e n tropy is r e versible and is invariant under time r e v e r ­ sal. Owing to the d e r i v a t i o n with respect to time the left-hand side of

E q.(5.5) c h anges its sign if time is reversed. In equilibrium this entropy b a la n c e equa t i o n is i n va riant under time reversal, consequently the r e versible part R r of the energy d i s s i p a t io n has to change its sign, while the irrev- e r s i b l e part R i г has to be invariant if time is reversed. Similar s p ec ulations can be followed for the o t her quantities in Eq.(9.1). For chiral and achiral smec t i c s C the resulting trans f o r ma t i o n rules are listed- in Table 1. of Part II.33

After s e p a r a t i n g reversible and irreversible phenomena one gets f rom E q .(8.3)

7.J*r 9pf* 9pfM dE.

- ( pf* - p ) V.v. - о г . V .v . - (f%r + - - г-) — ^

J J 3 P J J ij J 1 J ЭЕ dt

9pf* dB'. 3Pf*

Z“r - ( P s +

3Pf* dT (M r + - - Г— ) — j— -

E

J Зв. dt a ЭТ dt

dpi* d ^ T 9Pf* d7.vi dy^T dt ay.Vi dt

-ДГ V. (Jxir - q .) - — q • 7 -T +

1 i

J J j ^J j j J.E.J J

i dEi dß! Spf^

Pj

1 Г J

-- „j — L * E dt a

dt э х “

,air

(9.2)

(9.3)

10. SECOND LAW OF T HERMODYNAMICS

Second law of thermodynamics introduces one more dist i n c t io n b e ­ tween reversible and irreversible phenomena. It states that the energy d i s s i ­ pation has to be zero in all r e v e r s i b le processes while in irreversible ones energy d i ssipation is always positive.

R r = 0 and R ir > 0 (10.1)

Since the material time d e ri v a t i ve s of the independent variables can be adjusted a r bi t r a r il y and in de p e n d en t l y from any other quantity, Eqs.(9.2) and (10.1) yield

8pf*

---- = 0 3V г 8Pf *

--- = - ps ЭТ

9 Pf*

3V.v.

J 1 3pf*

= 0

3pf*

(10.2)

i.e. the generalized free energy has to be independent of temperature gradient and velocity gradient. Thus in g e neral the infinitesimal change of the

p f * = p f * ( T ,e‘ ,b’ , p ,Xa ) (10.3) generalized free energy can be written as

* 'г 1 -г 1 9pf Э р Г я

d p f* = - psdT - P . r dE. - M.rdB. + --- dp + £ - - - - dX01

1 1 1 1 3p „ 9Xa

(10.4)

and there is a c o nstraint between reversible quantities

V * r V i - (pf* - p — -) V v + f T - 0

J J 1 J J 1 э р J J „ a x “

8pf

(10.5)

Up till now the medium under c o ns i d e r a t i o n has not been specified at all thus the equations derived above are valid for any polarized c o n t i ­ nuous media, i.e. for liquids, crystals or liquid crystals as well. However these general equations do not give a c omplete d escription of the behaviour of the materials, one still has to set up a series of constitutive equations giving the depe n d e n ce of physical q u antities listed in Eq.(9.1) on the independent state variables. The cons t r u c ti o n of these constitutive e q u a ­ tions for chiral and achiral smectic C liquid c rystals is described in the subsequent Part I I . ^ of our paper.

U . SUMMARY

The c o nt inuum theory of polarized media described in this paper is 16 18

a g eneralization of former hydrodynamic theories. ’ We incorporated the el ec t r o m ag n e t ic field into the c o ns ervation laws which has led to the m o d i ­ fications listed below.

a^, There is an e l ec t romagnetic force in the equation of motion of the medium /Eqs . ( 5 .2),(A .8) and ( A . 11)/

b, The conservation of total angular momentum requires the m e chanical stress tensor to be asymmetric /Eqs . (6.3),(A .7) and ( A . 10)/.

£, In the presence of an e l ec t r o m a g n e t i c field the adequate thermodynamic potential, d e scribing reversible phenomena in the medium is the g eneralized free energy, which contains electr o m ag n e t ic c o ntributions too. / E q s .(7.1),(10.3) and

(10.4).

jj, After separating reversible and irreversible processes three irreversible electromagnetic terms remain in the expression of energy dissipation, which are related to the Joule-heat,

dielectric and m a gn e t i c relaxations. /Eq.(9.3)/

E q s .(8.4),(9.3) and (10.3)-(10.3) stand for the starting point in c o ns t r u c ­ tion of the constitutive equations for different media.

5

A p p e ndix 1. TRANSFORMATION RULES OF ELECTR O M AG N E T IC FIELDS

The transformation rules of electr o m ag n e t ic field between moving frames can be determined from the fact that the Maxwell equations (4.1) are Lorentz invariant. In the non-relativistic a p p r o x i m at i o n neglecting terms prop o r t i on a l to ^ - « 1 one gets the t ransformation rules'1^ in the SI system of units

e p e ^{|C-J II} J + P V

e—

в = в ' LU| II E - vxB

D = o ' ^II

1

H + V X D

p = p ' M =

1

M - vx£

1 = 1 ^II

(ОЙ

d l ' &

(A . 1)

where the dashed q u an tities are the ones meas u r e d in the material frame, m o v ­ ing with the velocity v^ relatively to the laboratory frame.

With this transformation rules the Maxwell equations (4.1) can be rewritten in the material frame as

« д = pe

v в’ = 0

I dJB t ,

5 xE = --- + (B 9)v - В (Vv) dt

, . dD , ,

9xH = J + --- ( 0 У )v - 0 (V v ) dt

(A.2)

I

which shows that in the non-r e l a ti v i s ti c a p pr o x i m at i o n the Maxwell equations become Galileian invariant /The extra terms cont a i n i ng velocity gradients disappear in a Galilei transformation, where v^ = constant/.

A ppendix 2 . BALANCE OF E L EC T R O M A G N E T I C MOMENTUM

Deriving the balance equations we follow the method of de Groot and Mazur"50 but we use SI and define the e l ec t r o m a g n e t i c mome n t u m as

a field = 6o(e x bi . 31 Its time derivative

— flfield = — ee (ExB) = — (DxB) - — (PxB)

at at at at

( A . 3 )

From the Maxwell equations (4.1) follows, that

a a o ЭВ

— (DxB) = — xB + Ox ^— at 9t " " at

= v |b oH + fl»E - K i . B2

(A.4)

I n t r o d u c in g £ = and using the d e fi nition of the material time derivative

— (PxB) = — p (£xB) = p — — (£xf3) - i [ v o ( P x B ) ]

at at d t

( A . 5)

Thus we get the balance equa t i o n of the e l e c t r o m ag n e t ic mome n t u m in the form

g field = У.T. • - F.

at 1 J lj 1 ^(A.6)

where

T lj ■ OjE, * Ej«, * »j - »ij ( i . Bk Bk . i [.Ek Ek - Bk Mk ) ( A . 7) and

F, = peEl * O x B ) , . Pj 5 lEj . Mj ii6j . pf/fixB) ( A . 8)

The Maxwell stress tensor and the e l ec t r o m ag n e t ic force can be e x p r e s s e d with the dashed q u antities too. Using the t r an s formation rules (A.l), neglecting terms of the order ^ - « 1 and using the identity

v^(P )^ + B ^ v x P )^ + P (B xv)A = 6 ^ v(P xB ) (A.9)

one gets

I I I I I I I I

Tij ■ V i * V i - 6ij < к вЛ * Ь Ek Ek - Bk Mk >

and

(A.10)

I l i i I I

Fi • pe Ei * У -I >i * Pj 'iEj * Mj 5 iBj * V j ( A . 11)

A ppendix 3 . BALANCE OF E L E C T R O M AG N E T IC ENERGY

The formal balance equation for the energy of the field is 3 field Г7 -.field „field

3t E = - V J + г (A.12)

From the Maxwell equation (4.1) one can get easily the Poynting theorem

3D ЭВ

E -- + H — + V(ExH) + JE = 0

3t 3t

(A.13)

We define the energy flux of the field as

jfieid = ExH = S (A.14)

w h ic h is the Poynting vector.

Using the trans f o r ma t i o n rules (A.l) Eq.(A.13) can be rewritten into the form

(A.15)

,30 , 3D ,3B , ЭВ

E --- (vxB ) — -t- H — + (vxD ) —

3t 3t 3t at

- J (vxB ) = 0

N e gl e c t i ng terms of the order 1 it can be t r ansformed further

, 1 I о J. 19 I t \ ion a I I

{ - e0 E + -- В - В M l + V S_ + E— + В — + v— (P xB ) +

1 2 2u0 ’ 3t 9t Bt

(A.16) I I

+ Д E + Pe)(E - Д (vxjl ) = 0

C o mp a r i n g Eq.(A.16) with Eq.(A.12) now we can define the field energy efield

and the energy supply term r field as

field 1 '2 x 1 „'2 ' ' _ 1. c2 . 1 0 2 я м '

e = 2 e° — + ' I * - 2 e° £ + T iТ0 й - I M ( A . 17)

and

„field ^{, ЭР , ЭМ}

E ---- В — 3t

“ V. — (P. xE3 ) - 2 £ “ P„vE + 2 (v.xJL )

at at e

(A.18)

The balance equation of the internal energy of the medium (5.4) c o n t a i n s the el ec t r o m ag n e t ic terms

infield _ rfield _ ^ > which can now be expressed with the field v a r i a b l e s .

From Eq. ( A . 11) one gets easily

V i • к в Ei*i * “i * ’ i {»ipjEj * * vivj cr ' * s '5 j } * vi “а^н.* s'

■[Ei Pi * "Л * vj C- ' xB' ^ i vi - Ej vl ViPj - 6j vi ' i Mj - Y i ' i <P xa' )i (A. 19)

and finally

V i ■ - e V i - * »Í { vt [P^Ej ♦ m!b: * vj(p'*B}]}- (E/; * .

, эр! , эм' 3 . I , dp! , dM*

+ E . — J- + B. — J- + v. — (P xB ). - E.-1 - B. — 1 (A.20)

з at ^ at ^ at ** ^ dt ^ dt

With the d efinitions (A.14) and (A.IB) one gets

v f eld - rfleld - V i = ' i { «*!<>! - » i ( E / j * B/ y } - (e’p: . в :м:)чл . , , , dP, , dM.

* ¥ i * Ei ^ * pi - (A.21)

which yields a simple, Galileian invariant expression for the energy d i ss i p a t io n of the medium. /Eq.(B.l)/.

ACKNOWLEDGEMENT

We would like to thank Dr.L.Bata, D r . Ä . Buka and Dr.O.Verhás for the critical reading of the manuscript and the s timulating discussions which helped us to get this final from of the theory.

REFERENCES

1. H. Kelker, R. Hatz. Handbook of Liquid C rystals (Verlag Chemie:

Weinheim).1980.

2. P.G. de Gennes. The Physics of Liquid Crystals ( C l a r e n d o n : O x f o r d ).1974.

3. M.J. Stephen, J.P. Straley, R e v .M o d .P h y s . 4^, 617 (1974)

4. L.M. Blinov. E lectro- and M a g n e t o o pt i c s of Liquid Crystals (Nauka:

M o s c o w ) .1978.

5. S.A. Pikin. S t ru c t u r e Transitions in Liquid Crystals (Nauka: Moscow).

1981.

6. F.C. Frank, Disc. Faraday Soc. 2Jj, 19 ( 1958) 7. J.L. Ericksen, Trans . Soc . R h e o l . J5 , 23 ( 1961 )

8. J.L. Ericksen, in Liquid Crystals 2 . Part I. Ed. G.H. Brown (Gordon and Breach: L o n d o n ) .1969, p.177.

9. F.M. Leslie, P r o c .Roy . S o c . А Ж 7 , 359 ( 1968)

10. F.M. Leslie, in Liquid Crystals 2. Part II. Ed. G.H. Brown (Gordon and Breach: London). 1969, p.473.

11. F.M. Leslie, S y m p .C h e m .S o c .Faraday Div. 5, 33 (1971)

12. J.D. Lee, A.C. Eringen, in Liquid Crystals and Ordered Fluids 2. Eds.

J.F. Johnson, R.S. Porter (Plenum: New Y o r k ) . 1974, p.315 and p.383.

13. A.C. Eringen, J.Math.Phys. 20, 2671 ( 1979)

14. A.C. Eringen, M o l . C r y s t . Liq . C r y s t . 5J^, 21 ( 1979) 15. T.C. Lubensky, Phys.Rev. A 6, 452 ( 1972)

16. P.C. Martin, 0. Parodi, P.S. Pershan, Phys.Rev. A 6^, 2401 (1972)

17. D. Forster. Hydr o d y n am i c Fluctuations, Broken Symmetry and C o rrelation Functions (Benjamin: Reading). 1975.

18. H. Brand, H. Pleiner, J.Physique 41 , 553 ( 1980) 19. H. Pleiner, H. Brand, Phys.Rev. A 25^, 995 ( 19B2)

2 0 . ' H. Pleiner, H. Brand, J.Physique Lett. <44, L-23 (1983) 21. J. Verhás, Acta Phys.Hung. 55 , 275 ( 1984)

22. S.A. Pikin, V.L. Indenbom, Usp. F i z . Na u k (USSR), 125, 251 (1978) 23. H. Brand, H. Pleiner, Phys.Rev. A 24., 2777 ( 1981)

24. H. Brand, H. Pleiner, J.Physique 4_5, 563 ( 1984)

25. A. Levstik, B. Zeks, I. Levstik, R. Blinc, C. Filipic, M o l .C r y s t .L i q . Cryst. Lett. 56, 145 (1980)

26. Á. Buka, L. Bata, K. Pintér, M o l .C r y s t .L i q .C r y s t . 108, 211 (1984) 27. A. Jákli, L. Bata, Á. Buka, N. Éber, I. Jánossy, submitted to

J.Physique Lett.

28. C. Truesdell, R.A. Toupin, in Ency c l o p ed i a of Physics. Vol.III/1.

Ed.S. Flügge. (Springer: Berlin)1960, p.226.

29. H.3. K r e u z e r . N o ne q u ilibrium T h er m o d y na m i c s and its Statistical Foun d a t i on s (Clarendon: O x f o r d ) .1981.

30. S.R. de Groot,, P. Mazur. N o n - e q u i l i b r i u m Thermodynamics (North - H o il a n d A m s t e r d a m ) . 1962.

31. 5.R. de Groot,, L.G. Suttorp. Foun d a t i on s of Electrodynamics ( N o r t h - H o l l a n d : Amsterdam). 1972.

32. В.В. Laud. Elec t r o m ag n e t ic s (Wiley Eastern: New Delhi). 1983.

33. N. Éber, A. Jákli, Report K F K I - 1 9 8 5 - 4 1 .

(

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I

Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Kroó Norbert

Szakmai lektor: Bata Lajos Nyelvi lektor: Buka Ágnes

Példányszám: 230 Törzsszám: 85-241 Készült a KFKI sokszorosító üzemében Felelős vezető: Töreki Béláné

Budapest, 1985. április hó