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Molecular association and theisotropic-nematic transitionSadhana R. Sharma aa Department of Physics , University of Guelph , Guelph,Ontario, N1G 2W1, CanadaPublished online: 22 Aug 2006.

To cite this article: Sadhana R. Sharma (1993) Molecular association and the isotropic-nematic transition, Molecular Physics: An International Journal at the Interface BetweenChemistry and Physics, 78:3, 733-746, DOI: 10.1080/00268979300100481

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MOLECULAR PHYSICS, 1993, VOL. 78, NO. 3, 733-746

Molecular association and the isotropic-nematic transition

By SADHANA R. SHARMA

Department of Physics, University of Guelph, Guelph, Ontario, N1G 2W 1, Canada

(Received lOth June 1991; accepted 26th June 1992)

A mean field theory for polar nematic liquid crystals is presented. The liquid crystalline system is considered to be a mixture of anti-parallel pairs and unpaired molecules. An attempt is made to understand the behaviour of such polar nematogens using an extended version of the generalized Maier-Saupe theory of nematic mixtures which incorporates a version of the dielectric theory of Maier and Meier extended to mixtures. The theory is applied to the case of nCBs (n-cyanobiphenyls). The effect of including the electronic dielectric con- stant is also considered. Reasonable agreement is obtained with experimental data for the behaviour of the dielectric permittivity as well as the change in the average dielectric permittivity at the transition. As expected, we also obtained a jump in the mole fraction of the 'associated pairs' at the I -N transition.

1. Introduction

There is a considerable body of evidence which shows that liquid crystals with polar end groups behave differently from the more classic liquid crystalline materials. This difference is most apparent in the behaviour of the dielectric permittivity. In the higher temperature isotropic phase the molecules are distributed randomly with no directional or spatial order; while in the lower temperature nematic phase, though their centres of mass are still randomly distributed there is directional order in the system, i.e. there is a preferred axis for the system (along the director h). Due to this directional order, in the nematic phase the dielectric permittivity has different values parallel (ell) and perpendicular (e• to the director, while in the isotropic phase it has one directionally independent value (q) . While in the case of non-polar liquid crystals the values of ell and e• are not much different from q , for polar liquid crystals ell and e• change substantially from the value q in the isotropic phase.

The average dielectric permittivity ~ is defined as

g _- ell + 2e• 3 (1)

At the isotropic-nematic ( I - N ) transition the change in the average permittivity 6~NI (tS~NI = ~ N - ei) can be positive, negative or zero depending on the liquid crystalline material [1, 2]. Also, as the temperature decreases further in the nematic phase, ~ can increase, decrease or remain constant. In the case of nCBs [3-5] which we shall study here, the dipole lies along the long molecular axis, 6~NI is negative and in the nematic phase ~ decreases slowly with decreasing temperature. In the present work we find that, if the dipolar molecules are assumed to interact with each other only through the Maier-Saupe pseudo-potential [6], the components of the dielectric permittivity are overestimated in the nematic phase and the average dielectric per- mittivity is continuous at the I - N transition. We also find that the inclusion of the

0026-8976/93 $10.00 �9 1993 Taylor & Francis Ltd.

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dipole-dipole interactions through a reaction field energy term in the total pseudo- potential does not substantially change the behaviour of the dielectric permittivity. Hence it appears that to understand the behaviour of the dielectric permittivity in the case of n CBs, the effective dipole moment # must decrease at the I -N transition as well as decrease further with decreasing temperature in the nematic phase. A way to account for this behaviour is to consider some sort of 'association' between the molecules [7]. Such a model for polar liquid crystals with 'association' was first suggested by Cladis et al. [8] and used by Longa and de Jeu [9] to explain the smectic-re-entrant nematic transition. Here we shall extend this model to the isotropic-nematic transition. In our model the polar nematic liquid crystalline material is taken to be a mixture of anti-parallel pairs with a zero dipole moment and unpaired molecules with a dipole moment # [8-10]. This mixture is in a state of dynamical equilibrium, with anti-parallel pairs forming and unforming; also, the concentration of pairs is a function of the temperature of the system.

As a first step towards understanding the behaviour of polar nematogens we start with the dielectric theory of Maier and Meier [11], which is an extension of Onsager's theory [12] to anisotropic systems. In other work [13] we have discussed, in a preliminary manner, the extension of the dielectric theory of Maier and Meier to nematic mixtures and used it to examine the thermodynamics of the system. In the case of polar liquid crystals, this mixture is made up of unpaired polar molecules with a dipole moment # and anti-parallel pairs with a zero dipole moment. In that work we used this extended dielectric theory to study the latent heat at the I -N transition and the average dielectric permittivity in the isotropic phase. In the present work we shall use the extension of the dielectric theory of Maier and Meier to polar and non-polar mixtures to obtain the reaction field energy of the unpaired molecules. An estimate is made for the pairing energy of the anti-parallel pairs [14]. The reaction field energy and the pairing energy are then included with the Maier-Saupe pseudo- potential for nematic mixtures to give us the total pseudo-potential for polar nema- togens. This total pseudo-potential is then used to evaluate the I -N transition temperature, the mole fraction of the paired molecules, the latent heat of the transi- tion as well as ell, e• and g.

Until now, most of the applications of the 'association' model have been in the study of the nematic-smectic-re-entrant nematic transitions [9, 15, 16]. There have also been a few studies of the nematic phase in mixtures of nematogens and solutes [17] as well as attempts [1] at understanding the behaviour of the dielectric permit- tivity in the isotropic and nematic regions using the Kirkwood-Fr6hlich theory [18, 19]. The 'association' model has also been applied to isotropic polar liquids by Monaschi and Chierico [20] to understand the behaviour of their dielectric properties. But these papers do not try to explain the behaviour of the components of the dielectric permittivity in the region of the isotropic-nematic transition as well as the change in the average permittivity at that transition. The present work is a first attempt at understanding the behaviour of the permittivity of polar nematogens at the isotropic-nematic transition using the 'association' model and including the reaction field.

The rest of the paper is arranged as follows. In Section 2 we consider a system of unpaired dipolar molecules which interact with each other through a combination of the reaction field energy obtained from the dielectric theory of Maier and Meier and the simple Maier-Saupe pseudo-potential. We examine the effect this combined pseudo-potential has on the interaction parameters needed at the I -N transition.

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Molecular association and I-N transition 735

In Section 3 we extend the dielectric theory of Maier and Meier [11] to mixtures of anti-parallel pairs and unpaired molecules fi la Cladis [8] or Longa and de Jeu [9]. This is then incorporated into an extended version of the generalized Maier-Saupe theory of nematic mixtures [21] and applied to the case of 5CB and 7CB. We also consider the effect of the electronic dielectric constant. Finally, the concluding remarks are given in Section 4.

2. Single component polar nematic

If a polar liquid crystalline material is made up only of 'unassociated' dipolar molecules, the components of the dielectric permittivity parallel and perpendicular to the director are given by

47r - - ~ h F a+ 3--KaT 1 - ( 1 - 3 c ~ ' (2a)

according to the dielectric theory of Maier and Meier [I 1]. Using equation (1) the average dielectric permittivity is then given by

g - 1 ( F#2 ~ (2c) 4re -- ~ P hF # ~ + ~ j .

Here NA is Avogadro's number, p the mass density, M the molecular weight, ~ the average polarizability, A a the polarizability anisotropy, r/the angle between the dipole and the long molecular axis and # the dipole moment of the molecule. In equation (2),

Q=(q)=(3c~ ) (3a)

is the order parameter and 0 the angle between the long axis of the molecule and the director. The other parameters in equation (2) are

3g h = (3b)

2 ~ + 1 '

the cavity field factor and

1 F = - - (3c)

1 - ~ f '

where the reaction field factor f is given by

1 2 ( g - 1) (3d) f = a 3 (2g+ 1)'

with a the radius of the spherical cavity around each dipolar molecule [22, 23]. Here we shall only consider nematogens whose dipoles lie along the long mol-

ecular axis, i.e. r /= 0, which is the case for n CBs. Due to the surrounding medium the reaction field at the dipole is R = fFI.t. The

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736 S.R. Sharma

polarization due to the reaction field is

p e - 1 N A P- - ~ R, (4)

where ~ is the tensorial dielectric permittivity. Using the parallel and perpendicular components of this tensor along with the components of the reaction field we obtain the various components of the polarization vector P. The energy of orientation due to the reaction field is then given by

ua = h P . R

~..[ fF~ 2

+2[fAa+fF#2 ] 1 }fF#2. "-~fBTJ Q(q - sQ )

Now the single component Maier-Saupe pseudo-potential is

E= _UQ(q_ lQ) ,

(5)

(6)

with u the anisotropic interaction parameter and v = V/N the volume per molecule. Since the dipoles interact with each other through a combination of the Maier- Saupe and the reaction field pseudo-potentials, the total pseudo-potential can be written as

A -- -/~(E + U.)

= (am -- aa)Q(q _IQ) +Fa, (7)

= a o Q ( q - l Q ) + F a .

The interaction parameters a m and a a arise from the Maier-Saupe pseudo-potential and the reaction field term respectively and can be easily obtained from equations (5) and (6). am, aa and F a are given by

U

am - - vKa T' (8 a)

2 2F 2 fF# 2 ( f A a +fF#2"~ O~a='3 h ~ B T ~ ~BT: ' (8b)

I 2 2 ( +fF#2) fF#2 (8c) F a =-'~h F fa 3-'-KBT) KBT"

Since the quantity F a is independent of the order parameter Q and contains only the isotropic (orientationally independent) part of the total pseudo-potential, it is an irrelevant parameter as far as the isotropic-nematic transition is concerned.

The nematic order parameter Q is the self-consistent solution of the equation

.[ q exp(A) dO Q = fexp(A) dS2 (9a)

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Molecular association and I -N transition 737

and the free energy density is defined as

G = -KBT ln ( I exp( A) dI2 ) �9 (9b)

The transition from the isotropic to the nematic phase will take place when the free energy of the nematic phase is lower than that of the isotropic phase. Fa has no effect on the order parameter Q as it cancels out from the numerator and denominator on the right-hand side of equation (9 a), just adds a constant additive term to the free energy in equation (9 b), and hence is of no importance to the I - N transition. The values of the parameter a o and the nematic order parameter Q at the I - N transition are the same universal values as in the classic Maier-Saupe theory, but due to the reaction field term the new Maier-Saupe parameter am(TNi ) will be larger than t~ o, the classic Maier-Saupe parameter, with

Otm(TNi ) = O~ o --1- OLa(TNI ). (10)

The average dielectric permittivity ~ is evaluated iteratively at the temperature TNI in the isotropic phase using equations (2 c) and (3). This vlaue of ~ is then used in calculating ~(TNI) from equation (8 b). The behaviour of aa(TNi) as a function of TNI for various values of ~ 3 is shown in figure 1 in the case of 5CB, where the electronic dielectric constant n 2 has been taken into account as explained in the next section. All the calculations were done for a case where the dipole lies along the long molecular axis, i.e. ~ = 0.

Hence there is no longer a universal constant for the Maier-Saupe pseudo- potential at the I - N transition, as in the classic Maier-Saupe case. Now, we find from the above arguments and from equations (7), (8) and (10), that for polar nematogens the new Maier-Saupe interaction parameter t~,,(TNI) depends on the

0

1.50

1.00

0 .50

0 .00 , �9 250 .00

. . . . . . . . . . . . . j , .

- - " A

, L

. . . . I ~ , i i i . . . . i �9 ~ ~ - i .

265 .00 2 8 0 . 0 0 2 9 5 . 0 0 ,:310.00

TNI/K Figure 1. c~ a versus TNt in the case of 5CB. Electronic dielectric constant n 2 has been taken

into account. ( )a 3 = 5-0b 3, ( - - ) a 3 -- 4.0b 3, (...)a 3 = 3-0b 3 and (- - -)a 3 = 2'0b 3. A represents the (C~a, TNI ) points used in the calculation. (47r/3)b 3 is the volume per molecule.

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738 S.R. Sharma

reaction field for the particular liquid crystal and TNI, the isotropic-nematic transi- tion temperature, as well as a 3, a parameter of the order of the volume of the cavity around each dipole. Thus we find that even through the classic Maier-Saupe inter- action parameter ao maintains its universal value, due to the presence of the reaction field interaction the new effective Maier-Saupe interaction parameter am(TNI) has a material-dependent value which is not universal.

The order parameter Q and the average dielectric permittivity ~ are evaluated self-consistently as follows. We notice from equation (2 c) that ~ is independent of Q, hence first the average dielectric permittivity can be evaluated iteratively at a temperature T and then using ~ the order parameter Q can be calculated. Starting with an initial value for the average dielectric permittivity ~ at a certain tempera- ture T in the nematic phase and using equations (2 c) and (3) a new ~ is evaluated and this calculation is repeated until ~ converges. The order parameter Q is then evaluated self-consistently using equation (9 a) at the same temperature T in the nematic phase as follows. Using the above value of the average dielectric permit- tivity, an initial value for the order parameter Q, and equations (3), (5) and (6), the reaction field energy and the Maier-Saupe pseudo-potential are calculated. This gives us the total pseudo-potential equation (7), which is used in equation (9 a) to evaluate a new value for the order parameter Q. This process is repeated till Q converges. Once ~ and Q have been evaluated, the free energy G and the components of the dielectric permittivity can be obtained easily from equations (9b) and (2).

We find that the components of the dielectric permittivity in the nematic phase, calculated using this combination of the Maier-Saupe pseudo-potential and the reaction field energy term, overestimate the experimentally obtained values for nCBs. The average dielectric permittivity ~ is also found to be continuous across the I -N transition, while experimentally it is discontinuous with ~N being slightly smaller than ~I. The behaviour of the dielectric permittivity is basically the same, even if only the Maier-Saupe pseudo-potential is taken into consideration and the reaction field is neglected. Thus we find that assuming the polar nematic liquid crystal to be made up of simple unassociated dipoles does not explain the experi- mental behaviour of the dielectric permittivity. We find that some form of 'association' is needed to explain the decrease in the average dielectric permittivity at the I -N transition.

3. Mixture of polar and non-polar nematics

Following Cladis [8] and Longa and de Jeu [9] the polar liquid crystalline system is now taken to be a mixture of unpaired dipolar molecules with a dipole moment # and 'associated' anti-parallel non-polar pairs with a zero dipole moment. The indices 1 and 2 indicate the unpaired molecules and the associated pairs, respectively. The volume per molecule of species i is v i so that the total volume of the mixture is

2

V = ~ _ N i v i , (11) i=1

where Ni is the number of molecules of species i. The Maier and Meier expressions for the parallel and perpendicular components

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Molecular association and 1-N transition 739

of the dielectric permittivity are now generalized to

41r -- hF xldq +Xza2 +~[xlAalQ, +xzAa2Q2]

F#z 2QI)}, +x, 3- .T (1+

(12a)

4re - - h F xl(~ l q- x2(~ 2 - 1 [ x 1 A o q Q I q_ x2Ao~2Q2 ]

(lZb) FU 2

(1 - a , ) } -~- X1 3---KB T

and using equation (1) the average dielectric permittivity g is given by

~ - I __ NAP. F ( _ F# 2 "~ 47r -~ n ~xlal + x2~2 + xi 3---KsTj, (12c)

where xi, the mole fraction of species i, is given by

Ni w i t h x 1 + x 2 = 1. (13) xi = N1 + N2

The definitions of the other parameters are the same as in the last section. Since each 'associated' pair is made up of two single polar molecules, we shall hence- forth assume that t~ 2 ---2~1 and zXa2 = 2Aal. Since the separation between the unpaired molecules increases as the concentration of the paired molecules increases we put

Xl q'- X2V ' / ' 02 (14) a 3 = ~

x1

where t~ is a parameter of the order of the volume of the cavity around each dipole. Using the same procedure as in the last section the reaction field energy can now be written as

UI =h2F2 ~ -2 + 3--KBT)

(15) 2 "Aa. fv#2 ~Q1) } fF# l . + g [ f j + K B T ] Q j ( q _ 1 2

According to the generalized Maier-Saupe theory or nematic mixtures [21] the single component pseudo-potential for a particle of species i is

Ei(h) = - ~ Z P j ")'ij + 2uijQj(q- Qi) , (16) j = l

where pj = Nj /V is the number density of species j, 3'ij and uij are the isotropic and anisotropic interaction parameters, respectively. The total pseudo-potential for this mixture is then given by

Di = Ei + Vi, (17)

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740 S .R . Sharma

where the reaction field energy term contributes only to the pseudo-potential of the unpaired molecules and U2, the energy of pairing for the associated pairs, is esti- mated to be of the order of KBTl(r=3oo) [14]. The nematic order parameter Qi for each species i is the self-consistent solution of

ai : ~ q exp (- f lDi) dO (18) f exp (--/~Di) dO "

The free energy of the system is then given by

G = - K B T ~ - - ~ l n 1 e x p ( - 3 D j ) d ~ (19) j = l

apart from irrelevant constants. The anisotropic interaction parameters between the like species Un and u22 can be estimated from the I - N transition temperatures, the molar volumes and in the case of polar molecules, by also taking into account the reaction field term as shown in the last section. The anisotropic interaction par- ameter between the unlike species is taken to be the geometric mean of that between the like species, i.e. ul2 = x/(UllUE2). There is no such prescription for the isotropic interaction parameters. The isotropic interaction parameters, the 7ij, do not have any effect on the order parameters but are important as far as the free energy is concerned. Through the free energy, they have a pronounced effect on the I - N transition temperature of the system. Since there is a change in the mole fraction of the associated pairs at the I - N transition, the latent heat of the transition is evaluated through the change in the entropy

L = TSI - TSN where S = - ( 0 G / 0 T )v. (20)

In this model, as there is no change in the total volume of the system at the isotropic-nematic transition and as two dipolar molecules make up one non-polar 'associated' pair, the ratio of the volumes of the two components can be taken to be Vl/V2 = 0"50. Since the measured change in density at the I - N transition is quite small this is not a bad assumption. Taking the long thin fiat cigar shapes of the dipolar liquid crystalline molecules into consideration the arrangement of dipolar molecules with the lowest energy is that of anti-parallel pairs. Due to the shape of the molecules this configuration is even lower in energy than the configuration of dipoles lying end on end. Hence the I - N transition temperature of the pure 'associated' system should be higher than that of the purely dipolar molecules, i.e. TNI ~ / TNI2 < 1.

As a test of the theory we have applied it to the case of nCBs where all the parameters needed for the reaction field part of the calculation except n have been determined experimentally by Dunmur and Miller [1] and are given in table 1. Also,

Table 1. Experimentally measured parameters needed for the calculation of the reaction field of 5CB and 7CB and obtained from [1] and [26]; with # the dipole moment [1], a the polarizability [1], Aa the polarizability anisotropy [1], n the optical refractive index [1] and PI and PN the densities in the isotropic and nematic phases [26].

# a Aa Pl PN esu - cm x 10-23cm 3 x 10 -23 cm 3 n 2 g/cm 3 g/cm 3

5CB 4.77 3"37 1.76 2.547 1"005 1.013 7CB 4"92 3"81 1"58 2.484 0-982 0.991

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Molecular association and I - N transition 741

as this is a constant-volume theory we neglect the change in the mass density at the I -N transition and the mass density in the isotropic phase PI is used in all the calculations. The dielectric permittivity of n CBs has been measured experimentally by various groups as a function of temperature in the isotropic and nematic phases [3, 4, 5].

Initially we pick values for the parameter ~r which are of the order of the size of the cavity around each molecule. The latter is determined by the volume available to each molecule [22]. Then for each value of n the average dielectric permittivity ~ is calculated self-consistently at a temperature T in the isotropic phase using equations (3) and (12 c) for a range of mole fractions of the associated pairs. The value of ~ is known experimentally at this particular temperature T in the isotropic phase, and we then determine the mole fraction of associated pairs needed at that particular n from ~'(T)exp ~--- ~(T)cal c. Then TNII, TNI l/TNI2, 711,722 and 712 are picked for each n such that the isotropic-nematic transition for the total system occurs at the experimen- tally determined temperature TNI and the average dielectric permittivity in the isotropic phase at a temperature T > TNI falls at ~(T)ex p for the same mole fraction of associated pairs as obtained in the previous calculation. Once the value of TNI~ and ~ are known, the new Maier-Saupe parameters am can be calculated easily, as shown in Section 2 for the dipolar system. The value for the energy of association U2 is estimated from the observations of Schad [14] as

U2 = -mKB T ](r=3oo). (21)

Initially all the calculations were done for m = 1. Now as all the necessary parameters are known, the components of the dielectric

permittivity in the nematic phase are calculated as follows. For each temperature T the average dielectric permittivity ~ and the order parameters (Q1, Q2) are deter- mined self-consistently for a range of mole fractions of the associated pairs using equations (12)-(18). As can be seen from equation (12 c), ~ is not a function of the order parameters (Q1, Q2), and its value can be evaluated independently of the order parameters. Hence, first ~ is evaluated iteratively and then the order parameters are evaluated self-consistently as follows. For each mole fraction of the associated molecules x2, we start with an initial value for the average dielectric permittivity and using equations (3) and (12 c) obtain a new value for ~. This process is repeated until the value for the average dielectric permittivity converges. Then initial values are picked for the order parameters (Q1, Q2). Using these along with the converged value for the dielectric permittivity ~, the total pseudo-potential is calculated using equations (15)-(17). Then using this pseudo-potential the new order parameters (Q1, Q2) are calculated from equation (18). These order parameters are then used to re-calculate the total pseudo-potential and to repeat the calculations until (Q1, Q2) converge. Using the calculated values of ~ and (Q1, Q2), the free energy G, the entropy S, as well as ell and e• can be evaluated from equations (19), (20) and (12). Then at each temperature the minimum in the free energy gives us the most stable state of the system. Hence we obtain the mole fraction of the associated pairs, the order parameters, the components of the dielectric permittivity and the entropy of the system at a given temperature. Using these we can determine the mole fraction of the associated pairs and the components of the dielectric permittivity on both sides of the I -N transition, as well as the latent heat at the transition.

The components of the dielectric permittivity are plotted as a function of temperature in figure 2 for 5CB, and in figure 3 for 7CB. The parameters used in the

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742 S . R . S h a r m a

1 8 . 0 0

14.00

1 0 . 0 0

1~ . . . . . . . . . . . . O . .

. . . . . _ 2 : " - ~ ' . - . . . . . . . . . . . . . . . .

�9 �9 �9 �9 �9 �9 �9 �9 �9 �9 �9 0 �9 �9 ,0,

~11 �9 �9

EL AA . . . . . . . . . . . . . . . . . . . . . . , . . . . . A �9 . � 9 - . A . . �9

6 . 0 0 r - - - - - . . . . . . . . . . . . . . . . . . . . . . . . 300.00 302.00 304.00 306.00 308.00 310.00

Temperature/K

Figure 2. Dielectric permittivity versus temperature for 5CB. A are the experimental data points from Dunmur et al. [3], T~a = 308.00K, o from Dunmur [17] and O from R a y n e s [18]. ( - - - ) ~ = 5.0b 3, n 2 = 1'0, U2 = --UD; (...)~r = 5"0b 3, n 2 ~ 1"0, U2 = --Up; (--)to = 2"0b 3, n 2 ~ 1"0, U2 = - U p ; (- - -)to = 2"0b 3, n 2 ~ 1"0, U 2 =

--I'5UD. (4zr/3)b 3 is the volume per molecule and Up = KaTI(r=aoo) [15].

ca lcula t ions and ob t a ined f rom exper imenta l results [1, 26] for 5CB and 7CB are given in table 1, the pa rame te r s eva lua ted ini t ia l ly as shown above as well as the ca lcu la ted quant i t ies are given in tables 2 and 3 for 5CB and 7CB, respectively. W e find tha t even t h rough 6g is negat ive in bo th cases and there is an increase in the mole f rac t ion o f the assoc ia ted pairs , the values o f ~fl are still too large. Reduc ing the size o f the cavi ty makes the con t r ibu t ion o f the reac t ion field t e rm too large and unphy-

Table 2. Parameters evaluated for 5CB and the quantities calculated from the present theory (47r/3)b 3 is the volume per molecule, UD = KBT I(r=300) [14], the latent heat at constant volume is measured to be 650-4-50J/mol [25] and n 2 r 1 implies that the electronic dielectric constant was taken into account.

TNII 6X2m L Electronic dielectric

K tc/b 3 [U2/Un[ 6~Nt ~ ~ X~ J/mot constant

283"80 5"0 1"00 -0"20 0"02 895"7 n 2 = 1 0.01 --* 0"03

304"00 5"0 1'00 0"003 0"00 952"2 n 2 ~ 1 0"05 ~ 0"05

301"50 4"0 1"00 -0 '12 0"01 905"0 n 2 ~ 1 0"06 ~ 0"07

296"00 3"0 1"00 -0"08 0"01 889-2 n 2 ~ 1 0"07 ~ 0"08

267"00 2"0 1.00 -0"17 0"02 886"9 n 2 ~ 1 0.10--~ 0.12

266.00 2.0 1.50 -0.25 0.03 889"0 n 2 5 1 0.10---~ 0.13

TNI 1 / TN12 = 0"92, vl/v2 = 0"50.

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Molecular association and 1 - N transition 743

17.00

13.00

9.00

5.00 �9 303.00

E

%

' A �9 �9 �9 �9

. . . . i . . . . i . . . . i . . . . = .

306.00 309.00 312.00 315.00 Temperature/K

Figure 3. Dielectric permittivity versus temperature for 7CB. A are the experimental data points from Dunmur et al. [3], TNI = 315.60K. ( - - - ) n = 5"0b 3, n 2 = 1"0, U2 = --UD; ('2" .)t~ = 5-0b 3, n2# 1-0, U 2 = - U p ; (--)t~ = 2.063, n2# 1'0, U2 = --UD;(- - -)t~ = 2-0b 3, n # 1'0, U2 = --I'5UD. (47r/3)b 3 is the volume per molecule and UD = KBT[(r=30o) [15].

sical. Also , the a m o u n t o f assoc ia t ion remains near ly cons t an t t h r o u g h o u t the nemat ic region. Hence af ter decreas ing ini t ia l ly at the I - N t rans i t ion , the average dielectr ic permi t t iv i ty remains near ly cons t an t in the nemat ic phase. Exper imen ta l ly the average dielectr ic pe rmi t t iv i ty decreases s lowly as the system goes deeper into the nemat ic phase.

The electronic con t r i bu t ion to the dielectr ic permi t t iv i ty has been neglected up to now and it wou ld be interes t ing to see the effect o f separa t ing the electronic con-

Table 3. Parameters evaluated for 7CB and the quantities calculated from the present theory. 3 (47r/3)b is the volume per molecule, UD = KB T [(r=300) [14], the latent heat at constant

volume is measured to be 8 801 100J/mol [25] and n 2 # 1 implies that the electronic dielectric constant was taken into account.

TNI ~ 6x2m L Electronic dielectric

K t~/b 3 [ U2 / UD[ 6ENI X I 4 X N J/tool constant

264"00 5"0 1"00 -0"25 0"03 999"9 n 2 = 1 0"07 4 0" 10

306"00 5"0 1"00 -0-19 0"03 798"8 n 2 ~ 1 0"14---~ 0"17

304"00 4"0 1"00 -0"19 0"03 814-6 n 2 ~ 1 0 .1440 .17

297-50 3'0 1"00 -0"20 0"03 815.3 n2 ~ 1 0 .1640 .19

275"00 2"0 1-00 -0"14 0"02 864"8 n2 ~ 1 0"18 ~ 0"20

274.40 2"0 1"50 -0"29 0"04 850"6 n 2 ~ 1 0.18 ~ 0"22

TIvIz/T~ 6 = 0"92, vl/v2 = 0"50.

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744 S .R . Sharma

tribution n 2, (where n is the optical refractive index) from ~. Hence we generalize the model so that a dipole with dipole moment # is embedded in a spherical cavity of dielectric permittivity n 2, which is surrounded by a medium of average dielectric permittivity ~ [18]. The electronic contribution to the dielectric permittivity changes the equations slightly with (Ell- 1)/47r and (E• 1)/47r being replaced by (ell - n2)/47r and (E• - n 2 )/47r in equation (12). The expressions for the cavity field factor and the reaction field factor are now given by

3~ h - ( 2 ~ + n 2) and (22a)

1 2 ~ - n 2 f = a 3 n 2 2~+ n 2" (22b)

Using these expressions the reaction field term is re-calculated. Using the same procedure as in the previous case, the average dielectric permittivity ~ and (QI, Q2) are re-evaluated self-consistently and the I - N transition temperature and the com- ponents of the dielectric permittivity are re-calculated. We now find that there is better agreement between the experimental and theoretical values for ell and E• as can be seen from figures 2 and 3. We see from the values given in tables 2 and 3 that there is also a negative jump in the average dielectric permittivity at the I - N transi- tion as well as an increase in the mole fraction of the 'associated' pairs at the transition. Also, for the smaller values of ~;, the parameter of the size of the cavity around each dipole, the amount of association does increase a little as we go deeper into the nematic phase. Hence, the average dielectric permittivity decreases at the I - N transition but still remains nearly constant as the temperature decreases further in the nematic phase.

Finally, we consider the effect of changing the value of the energy of association U2. Schad [14] estimated the energy of formation of the anti-parallel pairs, the energy of association, to be of the order of KBT[(r=3oo). The calculations until now were done with U 2 = --KB T [(T=300). Increasing the magnitude of the energy of association from this value makes the associated pairs more stable against dissociation into single polar molecules. Changing the energy of association s l ight ly to U 2 = - 1 . 5 KBTI(T=300) does not change the behaviour of the system drastically. The average dielectric permittivity decreases slightly more at the I - N transition, the jump in the amount of association is slightly larger, and as the temperature decreases further the amount of association increases a little but g still remains nearly constant. Changing the energy of association to U2 =--2KBTI(r=300) does not alter the behaviour much from the above, but values of U2 = -2.5KBT [(T=300) or larger have a very dramatic effect on the behaviour of the system. The magnitude of the energy of association is now too large and it is energetically favourable for the system to be nearly completely associated. So, although there is a local minimum of the free energy at a smaller amount of association (x2 = 0.03 at 308 K for 5CB) the actual global minimum is at a much larger value (x2 = 0.95) and such a large degree of association is clearly unphysical. Hence, the energy of association can be changed within very narrow limits to produce slight changes in the behaviour of the system, but changing the energy of association beyond these limits leads to very unphysical behaviour.

The dipoles in nCBs are not at the centres of the molecules, but are placed eccentrically, and this could have an effect on the behaviour of the system, although

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Molecular association and I - N transition 745

according to Dekker [24] the average reaction field due to an eccentrically placed dipole is the same as the reaction field of a dipole placed at the centre of a spherical cavity. Thus the dipoles of the unpaired molecules can still be treated as dipoles placed at the centre of a cavity with dielectric permittivity n 2 immersed in a material of dielectric permittivity ~.

4. Conclusions

We have treated a polar nematic liquid crystalline material as a mixture of unpaired molecules with a finite dipole moment and anti-parallel pairs with a zero dipole moment. The molecules interact with each other through a combination of the generalized Maier-Saupe pseudo-potential for nematic mixtures and a reaction field energy term calculated from an extension of the dielectric theory of Maier and Meier. This simple theory gives reasonable agreement with the experimental data for the behaviour of the components of the dielectric permittivity of n CBs through the isotropic-nematic transition. We also find that the average dielectric permittivity decreases at the I -N transition in agreement with experimental results.

Even though this mean field theory gives reasonable agreement with the overall behaviour of the dielectric permittivity in the nematic phase, the behaviour of ~ is not the same as that observed experimentally. We find that after decreasing initially at the I -N transition, g remains nearly constant through the rest of the nematic region. Experimentally, ~ decreases with decreasing temperature over a much larger range of temperature in the nematic phase. This could be due to the fact that we have neglected the interactions with the nearest neighbours and have treated all the molecules outside the cavity around the dipole as being part of a continuum. A way to improve this would be through a Kirkwood-Fr6hlich [18, 19] type of theory and by taking into account the hindered rotation of the molecules.

At the I - N transition there is also a small change in the density of the system which has been neglected here as this is a constant-volume model. Including the effects of excluded volume through a van der Waals type of theory would account for the change in density at the first order I -N transition.

Some of the behaviour of the dielectric permittivity close to the I -N transition is due to pre- and post-transitional behaviour which cannot be explained through a mean field theory and would need to be tackled with much more sophisticated techniques.

We have applied this theory only to the case of nCBs, where the dipole lies along the long molecular axis. It would be an important test of the theory to apply it to materials where the dipole makes a non-zero angle with the long molecular axis and ell could be less than e• We plan to extend the theory to such cases. We should be able to predict the behaviour of the dielectric permittivity as well as its dependence on the dipolar angle and the concentration of the 'associated' pairs. Even though there have been some measurements of the dielectric permittivity for such materials [11], the other parameters needed for the calculation have so far not been determined experimentally.

We have benefited from discussions with B. Bergersen, B. G. Nickel, S. Goldman and D. E. Sullivan. We would also like to thank D. A. Dunmur and E. P. Raynes for the use of their unpublished experimental data. The financial support of the National Sciences and Engineering Research Council of Canada is acknowledged.

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746 S .R . Sharma

References

[1] DUNMUR, D. A. and MILLER, W. H., 1980, Molec. Crystals liq. Crystals, 60, 281. [2] DE JEU, W. H., 1978, Solid-St. Phys. SuppL, 14, 109. [3] DtmMUR, D. A., MANTERFIELD, M. R., MILLER, W. H., and DUNLEAW, J. K., 1978,

Molec. Crystals liq. Crystals, 45, 127. [4] DtrNMUR, D. A., 1981, unpublished. [5] RAYNES, E. P., 1985, unpublished. [6] MAmR, W., and SAUPE, A., 1958, Z. Naturf. A, 13, 564; 1959, ibid. 14, 882; 1960, ibid. A,

15, 287. [7] DRUON, C., WACRENmR, J.-M., 1978, Ann. de Phys., 3, 199. [8] CLADIS, P. E., BOGARDUS, R. K., and AADSEN, D., 1978, Phys. Rev. A, 18, 2292. [9] LONGA, L., and DE JEU, W. H., 1982, Phys. Rev. A, 26, 1632.

[10] MAOHUSUDANA, N. V., and CHANDRASEKHAR, S., 1973, Proc. International Liquid Crystal Conference, Bangalore, Pramana Supplement I, p. 57; LEADBEYrER, A. J., RICHARDSON, R. M., and COLLING, C. N., 1975, J. de Phys. (Paris) Colloq, 36, 37; TORIVAMA, K., and DtmMUg, D. A., 1985, Molec. Phys., 56, 479; DtmMUR, D. A., and TORIVAMA, K., 1986, Liq. Crystals, 1, 169.

[11] MA~ER, W., and MEIER, G., 196l, Z. Naturf. 16a, 262; 1961, ibid., 16a, 470; 1961, ibid., 16a, 1200.

[12] ONSAGER, L., 1936, J. Am. Chem. Soc., 58, 1486. [13] SHARMA, S. R., and BERGERSEN, B., 1992, Liq. Crystals, to be published. [14] SCHAD, HP., and OSMAN, M. A. J., J. chem. Phys., 1981, 75, 880. [15] BERKER, A. N., and WALKER, J. S., 1981, Phys. Rev. Lett., 47, 1469; INDEKEC, J. O., and

BERKER, A. N., 1986, Physica, 140A, 368; 1986, Phys. Rev. A, 33, 1158; 1988, J. Phys. France 49, 353; INDEKEU, J. O., 1988, Phys. Rev. A, 37, 288.

[16] DE JEW, W. H., 1982, Solid-St. Comm., 41, 529. [17] TORIYAMA, K., and DUNMUR, D. A., 1985, Molec. Phys., 56, 479; 1986, Molec. Crystals

liq. Crystals, 139, 123; DUNMUR, D. A., and TORIYAMA, K., 1986, Liq. Crystals, 1, 169. [18] FROLICH, H., 1948, Trans. Faraday Soc., 44, 238; 1958, Theory of Dielectrics (Oxford

University Press). [19] KIRKWOOD, J. G., 1939, J. chem. Phys., 7, 911. [20] MOGNASCHI, E. R., and CHmRICO, A., 1989, Molec. Phys., 68, 241. [21] SHARMA, S. R., PALFFY-MuHoRAV, P., BERGERSEN, B., and DUNMUR, D. A., 1985, Phys.

Rev. A, 32, 3752. [22] B6YrCHER, C. J. F., 1973, Theory of Electric Polarization Volume I, (Elsevier). [23] CHANDRASEKHAR, S., 1977, Liquid Crystals (Cambridge University Press). [24] DEKKER, A. J., 1946, Physica A, 12, 209. [25] ORWOLL, R. A., SULLIVAN, V. J., and CAMPBELL, G. C., 1987, Molec. Crystals liq. Crystals,

149, 121. [26] BEGUIN, A., BILLARD, J., BONAMV, F., BUISINE, J. M., CUVEILIER, P., DuBoIs, J. C., and LE

BARMY, P., 1984, Molec. Crystals liq. Crystals, 115, 276.

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