2. 588 P.M.L.O, S C H O L T E A N D G. V E R T O G E N
Putting Kj2 = 0 the nematic phase is obtained. This means that additional
intermolecular interactions should be added to the twist interaction (1.1). In
order to keep the mathematics as simple as possible the nematic phase is
described here in the usual way, namely in terms of the Maier-Saupe model 3)
-Jl2(al • a2)2, (1.2)
where the coupling constant only depends on the intermolecular distance rl2.
The resulting model was solved by van der Meer et al. 2) in the molecular
field approximation under the further assumption that the distribution of the
possible orientations of the long molecular axis a around the local director is
uniaxial, i.e. the system is locally nematic. The purpose of this paper is to
present a simplified version of the model composed of the interactions (1.1)
and (1.2). The proposed model is closely related to an earlier published
simplified version of the Maier-Saupe model4). The main advantage of this
simplified model for a cholesteric is that it can be solved analytically in the
molecular field approximation. This means that a clear picture of the original
model and the accompanying assumption of local uniaxiality can be obtained
without hardly any numerical effort.
This paper is organized in the following way. In section 2 the model is
described. Section 3 deals with the molecular field approximation. The solu-
tions of the relevant equations are given in section 4. Finally the results are
discussed in section 5.
2. The model
Starting point is the model composed of the interactions (1.1) and (1.2) but
with one important modification. The vector a~ is no longer a unit vector but
needs only to satisfy the socalled spherical constraint
N
Y~ a{ = N, (2.1)
i=l
where N denotes the number of molecules. As pointed out earlier 4) this
modification boils down, as far as thermodynamics is concerned, to a
weakening of the constraint a 2= 1 to the constraint (a~}= 1, i.e. the ther-
modynamic expectation value of its length squared must be one.
In order to determine the thermodynamic properties of the system the
partition function Z has to be calculated. The spherical constraint can be
easily taken into account by means of the method of Lagrange multipliers.
3. STATISTICAL TREATMENTOF A MODEL FOR CHOLESTERICS 589
The partition function reads
Z = ; d3al " " f d 3 a N e x p [ ~ ~ (JijSo~+ K,j~o~su,jDa,~ait3aj~aj~
N
where /3 = 1/kBT, ks is Boltzmann's constant and T the temperature; a,/3, 7,
= x, y, z and the Einstein summation convention is used for repeated Greek
indices. The tensors 60v and E0v, denote the Kronecker and Levi-Civita tensor
respectively. The Lagrange multiplier ~, is determined by
0_j_/= 1 0 In Z = O, (2.3)
0X /3N 0X
where f denotes the free energy per molecule. Obviously the exact calculation
of Z is an extremely hard problem. In the next section the partition function
will be evaluated in an approximate way using mean field theory.
3. The mean field approximation
In order to apply the mean field approximation the following identity is
used
ai~ai~ajaaj, = ( aioai~ - ( a~ai~))( aj~aj, - ( aj~aj~)) + ( ai~ai~)aj~aj,
+ (aj~aj.>a~ai~ - (a~a~)<aj~aj.). (3.1)
The mean field approximation then neglects the coupling between the local
fluctuations. Consequently the partition function Z reads in this ap-
proximation
Z = f d3at . . - f d3aN exp[/3 ~ (J0,~. + K,,,~.,u,,,,(a,.a,.,(ai.ai~-~(a,.ai~,,
N
-,h(i~=tai, a,o- N ) ] . (3.2,
The normal cholesteric phase is characterized by a helix-axis, which is
identified here with the macroscopic x-axis. Then the second rank tensor glj
with elements
Qi, o = ( a~,,aj~) (3.3)
only depends on x in this phase. It holds
Oj = R(qr,jx)O,R-t(qr,~), (3.4)
4. 590 P.M.L.O. SCHOLTE AND G. VERTOGEN
with
o
R(qriix) = ( i cos(qrijx) -sin(qrijx , (3.5)
sin(qrij~) cos(qrij~) /
w h e r e q d e n o t e s the helix w a v e n u m b e r and r~ix is the p r o j e c t i o n of inter-
m o l e c u l a r distance v e c t o r r~j = rj - ri on the x-axis. The matrix R describes a
rotation a r o u n d the x-axis o v e r an angle qrijx. C o n s e q u e n t l y the partition
f u n c t i o n can be written like
N
Z -- exp[/3AN] l'-[ Z,, (3.6)
i=1
where
Zi = d3 ai exp[~Li~o(ai~aio - ,;(ai~ait~)) - flAai~ai,], (3.7)
f ,
with
Li~ = ~ ( Jij6~ + Kijet3~uijs)R~. ( qri~x)Qi.~R -.~(qrgx).
I
(3.8)
i
It is clear that Zi does not d e p e n d on i. T h e r e f o r e the integral (3.7) only n e e d s
to be e v a l u a t e d for an arbitrary molecule d e s c r i b e d b y the v e c t o r a. The
calculation p r o c e e d s as follows. B e c a u s e of the t w o - f o l d s y m m e t r y of the
helix axis the following elements of the t e n s o r Q are zero,
(axay) = (axa~) = 0. (3.9)
F u r t h e r the m a c r o s c o p i c c o o r d i n a t e s y s t e m can be c h o s e n such that
(aya..) = 0. (3.10)
T h e r e f o r e it holds
L ~ -- J(a~), (3.1 l a )
1 2
Lyy = J[~((ay) + (a~)) + ~A(q)((a 2 - (a 2))],
' y) (3. l i b )
L.. = J[~((a 2 + (a ~.))- ' A(q)((a y) - (a ~))],
1
~> (3.1 lc)
L~y = Lr.~ = L~, = Lzx = Lyz = Lzy = 0, (3.1 ld)
where
J = ~, Jij, (3.12a)
i
JA(q) = y_, [Jij cos(2qrijx) + Kiiui~x sin(2qri~)]. (3.12b)
i
5. STATISTICAL TREATMENT OF A MODEL FOR CHOLESTERICS 591
Then
Z = exp[/3AN - ~/3NLx~(a 2 - ~[3NLyr(a 2 - ~/3NLzz(a 2
1 x) 1 y) ! z)]
× [f dax f day f da~ exp[/3(Lxx-h)a2x+/3(Ly,-h)a2y
"1N
+ /3(L~ - A)a2)]] . (3.13)
This means that the free energy per molecule, f, is given by
1 2 1 2 1 2 ~ "iT
f=~Lxx(ax)+~Lyy(ay)+~L~(a~)-h- - , - In/3(A ~Lxx )
2/3 In/3(h - L , ) 2/3 In/3(h - Lz~)" (3.14)
The Lagrange multiplier h is determined using relation (2.3), i.e.
1 1 1
~- -~ - 1. (3.15)
2/3(h - L~x) 2/3(h - Lyy) 2/3(h - Lz~)
The order parameters (a~), i = x, y, z, are determined selfconsistently or,
equivalently, follow from OflO(a 2) = 0;
(a2) = 2/3(h 1- Lii) , i = x, y , z . (3.16)
Finally the helix wave number q0 of the cholesteric is obtained by solving
oy/Oq = 0, or
( (a2)_ (a2z) 1 + _1 )((aZr)- (aa))~OA
/3(X-Lyy) /3(X L~) 0,=0. (3.17)
In order to obtain the thermodynamic properties of the system the order
parameters and q0 must be determined by solving (3.16) and (3.17). The
solution, that gives rise to the lowest free energy, is the thermodynamically
stable one.
4. T h e s o l u t i o n
According to (3.15) two independent order parameters exist. Instead of (a2),
i = x, y, z, it is advantageous to use the order parameters R and S defined by
(a 2) = ~(1 - 2S), (4.1a)
(a2y) = -~(1+ S - R), (4.1b)
6. 592 P.M.L.O. SCHOLTE AND G. VERTOGEN
(a~) = l(1 + S + R). (4.1c)
This means, using (3.17), that the helix wave number qo is determined by
R2 3A = 0, (4.2)
After substituting (4.1) into (3.16) and thereupon eliminating h from the three
resulting equations the following equations for the order parameters R and S
are obtained:
9R
R - - 2[3JA(qo)[(1 + S) 2- R2] ' (4.3a)
313S(1 + S) - R 2]
S = 2/3J[1 - 2S][(1 + S) ~- R2]" (4.3b)
Clearly the isotropic solution R = S - - 0 fulfils these equations. The cor-
responding free energy per molecule, f~, reads
3 1 3 2~-
/3f~- 2 6/3J-~ln=~-" (4.4)
At low temperature the equations (4.3) allow more solutions. The correct
solution minimizes the free energy, or equivalently the difference Af between
the free energy per molecule, f, and the free energy of the isotropic state. This
difference Af is given by
~Af= ¢U(3S2+A(qo)R 2) 1--2S ~- / 3 J S - ~ l n ( I - 2 S )
- ~ ln[(1 + S):- R21. (4.5)
As for the anisotropic solutions equation (4.3a) implies
(a) R = 0, (4.6a)
(b) (1 + S) 2 - R 2 - 9
2[3JA(qo)" (4.6b)
(a) Substitution of R = 0 in (4.3b) gives, besides S = 0, the solutions
4 ]1[2
So,_~= - - ~1 + 3 1 1 -- ~ j ' j ,
-- (4.7)
provided that the temperature T is lower than J[4kB. The solution (4.7) gives
rise to a difference in free energy
~Af,,± = - ~,¢ J S , . + ~In 4 /32J2(1_
11 soq .
, ,48)
7. STATISTICAL TREATMENT OF A MODEL FOR CHOLESTERICS 593
Note that this type of solution does not determine the helix wave number qo.
The free energy belonging to the Sn- solution equals the free energy of a
nematic ordered along the x-axis, where the nematic is described by the
Maier-Saupe model (1.2).
(b) It follows directly from (4.6b) and (4.3b) that
1/2
3-A(qo) ÷3(l+A(qo))[1 - 4(3+A(qo)) ]
, (4.9a)
S~,. = 4(3 + A(qo)) - 4(-3 + A(qo))L /3J(l + A(qo))2J
R~ = A~q0) [A(qo) - 1 + (A(qo) + 2)Sc,_+]Sc,., (4.9b)
whereas the corresponding difference in free energy is given by
1 3S~,.
~3Arc,+_= -~ /3J[(A(qo) + 3)S~,_++ (A(qo) + 1)So,_+] I - 2S¢,_+
-2
'In[ 2/319 (qo) (1 - 2Sc'-+)]" (4.10)
The helix wave number q0 is determined by the relation OA(q)/Oq = O.
In order to determine the solution corresponding with the lowest free
energy use is made of a perturbation expansion in qo. Such a procedure can be
followed safely, because qo is small in practice. The coupling parameters J~j
and K~j decrease rapidly with the distance r~j. T h e r e f o r e A(q) may be ap-
proximated by
A(q) = 1 - 2(qro)2 + 2Kqro, (4.11)
with
1 z 1
r~=-f ~,j Ji,rii~, K =-f~o~f/ Ki, ui,xri,x. (4.12)
Then it follows directly from OAlOq = 0 that
K
qo = 2r---~' (4.13)
i.e.
A(qo) = 1 + 2(qoro)2. (4.14)
It appears to be advantageous to introduce a new set of order parameters Sz
and R~ defined by
Sz = -~(S + R), Rz = ½(R - 3S) (4.15)
8. 594 P.M.L.O. SCHOLTE AND G. VERTOGEN
i.e.
(a2x) = 1 ~(1- Sz + R~),
<(/2,,>=~(l _ & _ R=),
<(/~> -- 3(1 + 2S=).
l
Then lim,0_~)R= = 0 , i.e, in the limit q 0 ~ 0 the nematic state, that is ordered
along the z-direction, is obtained. The order parameters S= and R: and the
difference in free energy appear to read up to order (qoro) 2
27
S=+ = S.:,~ q 4 ~ J ( 4 S . : + - 1) (qoro)', (4.16)
R:+ = ~(1 - Snz,_+)(qoro)2, (4.17)
I
[3Afc~ = ~ A f .... - ~[3JS .... (q0ro)', (4.18)
where
3 [1 - 4 ]1/2
S ~_ _l -t- ~
q ~J • (4.19)
/3Af. . . . =-~[3JS2.=,~+~ln[4~2j2(l+ 2S .... )]. (4.20)
The quantities So~,+and [3Af .... refer to the uniaxial phase already discussed in
ref. 4, where it is shown that the thermodynamically stable state is described
by S.=,+ and [3Afo~,+. As for the cholesteric phase it is verified easily that the
thermodynamically stable solution is given by S .... R~,+ and ~Af ....
The isotropic-cholesteric phase transition occurs as soon as [3Af~,+ = 0. Up
to order (qoro) 2 the transition temperature T~ and the values of the order
parameter Sz. and R~,+ at the transition temperature T~ are found to be
T~ = 0.247(1 + ~(q0ro) : ) k--BB'
3 J (4.21a)
S~,+ = 0.335, (4.21b)
R=,+ = 0.333(q0r0) 2, (4.21 c)
i.e. the jump in the order parameter &,+ is unaffected up to order (qoro) 2.
5. Discussion
It is clear that a description of the cholesteric state must be based upon
more than one order parameter. In the underlying model two order
9. STATISTICAL TREATMENT OF A MODEL FOR CHOLESTERICS 595
parameters appear. For large values of the pitch, however, one order
parameter suffices, because the deviation of local uniaxiality is of the order of
(qoro) 2. In the underlying model calculation this deviation reads
(a2x) - (a~) = 0.222(q0r0) 2. (5.1)
This means that in practice, where Iqorol ~- l0 -2, the deviation of the uniaxial
symmetry is of the order of 10-5 and consequently negligible, i.e. the assump-
tion of van der Meer et al. 2) concerning local uniaxiality is fully justified.
Further it should be mentioned that the model gives rise to a pitch, which
does not influence the jump of the main (nematic) order parameter up to order
(q0r0)2 and does not depend on temperature.
In order to obtain a temperature-dependent pitch, as experimentally obser-
ved, the model has to be extended. The first way is to change whether the
nematic component of the model by incorporating higher order terms like
P4(ai • aj) or the twist component by adding terms like (a~ ¢lj)a(ai A aj Uij) or
• °
both components2'5"6). The second way is through introducing the biaxiality of
the molecules7'S). Because of the shape of the molecules the rotations around
the molecular principal axes will be hindered. Assuming that this hindered
rotation gives rise to long range correlations, i.e. the introduction of order
parameters is justified, still two options are open. The first option is to
attribute the temperature dependence of the pitch mainly to the order
parameter that describes the tendencies of the transverse molecular axes to
be oriented along the average director7). Then a locally uniaxial (nematic)
state can be maintained. The second option accepts the relevance of all order
parameters describing the hindered rotationsS). Such a point of view of
ascribing the temperature-dependence of the pitch to local biaxiality is
obviously equivalent with a rejection of the starting-point of the macroscopic
continuum theory of cholesterics. Up to the present the problem concerning
the origin of the temperature dependence of the pitch is still largely unsolved,
i.e. the relative relevance of the possible interactions is unknown.
References
1) W.J.A. Goossens, Mol. Cryst. Liq. Cryst. 12 (1971) 237.
2) B.W. van der Meet, G. Vertogen, A.J. Dekker and J.G.J. Ypma, J. Chem, Phys. 65 (1976) 3935.
3) W. Maier and A. Saupe, Z. Naturforsch. 14a (1959) 882; 15a (1960) 287.
4) G. Vertogen and B.W. van der Meer, Physica 99A (1979) 237.
5) B.W. van der Meer and G. Vertogen, Phys. Lett. 71A (1979) 486.
6) H. Kimura, M. Hosino and H. Nakano, J. Physique Colloq. 40 (1979) C3-174.
7) B. W. van der Meer and G. Vertogen, Phys. Lett. 59A (1976) 279.
8) W.J.A. Goossens, J. Physique Colloq. 40 (1979) C3-158.
