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1982 a simple molecular statistical treatment for cholesterics
1982 a simple molecular statistical treatment for cholesterics
1982 a simple molecular statistical treatment for cholesterics
1982 a simple molecular statistical treatment for cholesterics
1982 a simple molecular statistical treatment for cholesterics
1982 a simple molecular statistical treatment for cholesterics
1982 a simple molecular statistical treatment for cholesterics
1982 a simple molecular statistical treatment for cholesterics
1982 a simple molecular statistical treatment for cholesterics
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1982 a simple molecular statistical treatment for cholesterics

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  • 1. Physica l13A (1982) 587-595 North-Holland Publishing Co. A SIMPLE MOLECULAR STATISTICAL TREATMENT OF A M O D E L FOR CHOLESTERICS P.M.L.O. SCHOLTE Solid State Physics Laboratory, Melkweg 1, University of Groningen, 9718 EP Groningen, The Netherlands and G. VERTOGEN Institute for Theoretical Physics, Toernooiveld, Catholic University of Nijmegen, 6525 ED Nijmegen, The Netherlands Received 18 January 1982 The cholesteric state is discussed in terms of a model consisting of the Maier-Saupe and a twist interaction. The model can be easily solved in the mean field approximation. In the practical case of a large pitch analytical expressions are obtained for relevant quantities like the order parameters and the free energy. Then it appears that the pitch is temperature independent, which is a general result, and does not influence the discontinuity of the main (nematic) order parameter at the isotropic-cholesteric transition. Interactions giving rise to a temperature-dependent pitch are discussed briefly. 1. Introduction The cholesteric phase is described as a twisted nematic phase. The appear- ing helix structure might be attributed to a combination of the dipole-dipole interaction and the dipole-quadrupole interaction as proposed by Goossenst). This type of interaction can be considerably simplified because of the rapid rotations of the molecules around their long molecular axes. This molecular behaviour suggests strongly the absence of long-range correlations between the rotations around the long molecular axes. Consequently the molecules may be assumed to rotate uncorrelated in a first approximation. Then the relevant interaction between two molecules 1 and 2 boils down essentially to - K12(al" a2)(al ^ a2" ul2), (1.1) as shown by van der Meer et alfl). Here the unit vectors a~ and a2 denote the orientations of the long molecular axes of the molecules 1 and 2 respectively, while ut2 is a unit vector pointing from the center of mass of molecule 1 towards the center of mass of molecule 2. The coupling constant KI2 denotes the interaction strength and depends on the distance rl: between the centers of mass of both molecules. According to Goossens 1) K~2 is proportional to r~7. 0378-4371/82/0000-00001502.75 © 1982 North-Holland
  • 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.

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