HAL Id: jpa-00215894
https://hal.archives-ouvertes.fr/jpa-00215894
Submitted on 1 Jan 1975
HAL
is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire
HAL, estdestinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
SELF-DIFFUSION IN NEMATIC LIQUID CRYSTALS
K.-S. Chu, D. Moroi
To cite this version:
K.-S. Chu, D. Moroi. SELF-DIFFUSION IN NEMATIC LIQUID CRYSTALS. Journal de Physique
Colloques, 1975, 36 (C1), pp.C1-99-C1-101. �10.1051/jphyscol:1975117�. �jpa-00215894�
JOURNAL DE PHYSIQUE Colloque C1, supplkment au no 3, Tome 36, Mars 1975, page C1-99
Classification Physics Abstracts
7.130
-
7.610SELF-DIFFUSION IN NEMATIC LIQUID CRYSTALS
(*) K.-S. C H U (**) and D. S. MOROIDepartment of Physics and Liquid Crystal Institute, Kent State University Kent, Ohio 44242, U. S. A.
Rksum6. - En utilisant une forme fonctionnelle, aux parametres approprib, de la fonction d'autocorr6lation du moment, nous avons calcule sous une forme analytique fermee le coefficient moyen de diffusion D des cristaux liquides nematiques en fonction de paramktres molCculaires tels que les distances intermoleculaires, la force de couplage et les dimensions mol6culaires. A partir de l'hypothkse de (( cluster N et des proprietes de transformation du tenseur de diffusion, nous avons obtenu les Bquations qui couplent le coefficient moyen de diffusion a m composantes anisotropes du tenseur de diffusion par I'intermediaire du parametre d'ordre orientationnel. Les coefficients de diffusion anisotrope ont BtB obtenus en resolvant ces equations. Pour le PAA B 122 OC, nous trou- vons D = 3,8
x
10-6 crnzls, D l , = 5,9 x 10-6 cmz/s et D, = 2,8 x 10-6 cmz/s, ce qui est en assez bon accord avec les resultats experimentaux rBcents de Topler et al.Abstract. - Using a properly parameterized functional form of the momentum autocorrelation function, we have calculated, in a closed analytical form, the average diffusion coefficient D of nematic liquid crystals in terms of the molecular parameters such as intermolecular distances, coupling strength and molecular dimensions. Using the cluster assumption and the transformation properties of the diffusion tensor, we have derived the equations which couple the average diffusion coefficient and the anisotropic components of the diffusion tensor through the orientational order parameter. Thus the anisotropic diffusion coefficients are obtained by solving these equations.
For PAA at 122 OC, we have found that D = 3.8 x 10-6 cmz/s, Dll = 5.9 x 10-6cm2/s, and D, = 2.8 x 10-6 cmz/s, which are in fairly good agreement with the very recent experimental results of Topler et al.
1. Introduction. - The self-diffusion coefficients for PAA liquid crystals have been measured by the qua- sielastic neutron scattering experiments [ I ] , the radio- active tracer techniques [2] and the proton-spin echo experiments [ 3 ] . There are some discrepancies among these experimental results.
Using the modified Kirkwood theory, Franklin [4]
has calculated the diffusion coefficients for PAA.
The purpose of the present paper is t o develop a general diffusion theory for ordered fluids. The appli- cation of this theory to the liquid crystals other than nematic liquid crystals will be published elsewhere.
The results of the present calculation are compared with the above experimental results.
2. Theory.
-
Based on the properties of the time correlation function, Douglas [5] has suggested a momentum autocorrelation function in the form of damped oscillation to calculate the diffusion constants of normal fluids. The results of these calculations [6]agree qualitatively with the Rahman's computer experiments 171.
(*) Supported in part by the National Science Foundation under Grant No. GH-34164 X.
(**) Based in part on a dissertation submitted to the Depart- ment of Physics at Kent State University in partial fulfillment of the Ph. D. degree in 1974.
T o calculate the average diffusion coefficient of a nematic liquid crystal, we use this functional depen- dence of the momentum autocorrelation function
= cos (ant) sech (at)
.
Here
< ... >
indicates the equilibrium ensemble average and the parameters a and I are determined by comparing the coefficients of a power series in t of the two expressions in eq. (1). Thus we haveThe equipartition theorem gives
< p 2 > = 3 k B T , (3) where k , is the Boltzmann constant.
Since the $ ( t ) is an even function o f t , the coefficient of the first power in t yields
Taking the time derivative of eq. (4), we have
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1975117
C1-100 K.-S. CHU AND D. S. MOROI Using the equations of motion for a diffusing molecule,
eq. ( 5 ) becomes
< i 2 >
= ~ , T < V : V ~ > , (6) where Vl is a one-body potential experienced by the diffusing molecule. If we use a pairwise potentialeq. ( 6 ) reduces to
where N is the number of molecules in the system.
Similarly, we have
(8) where three-body and four-body interactions which are involved in the above calculation have been neglected.
It is widely accepted that the dispersion force and repulsive force are dominating interactions between the molecules of nematogenic compounds, in parti- cular, the long conjugated double bond or aromatic molecules. Considering this fact and the softening anharmonic effects within a cluster region, we assume that each liquid crystal molecule is of the rod-like shape of diameter d and length I and has ( 2 n
+
1 ) centers of interaction, where 2 n+
1 r IJd. Thus we derive a two-body potential in the relative cylindrical coor- dinates [8].x' cosh [2(m
+
1) bdz]I . (9) Here a is an energy parameter, Ro the equilibrium separation of the molecules, and b = bo R0 ', where bo is the steepness of the potential.
Substituting eq. ( 9 ) into eqs. (7) and ( 8 ) and using a pair correlation function g(2)(r,2) z exp[- V12/k, TI, we have
where po is the equilibrium density, 8, the maximum attractive energy at p = Ro and z = 0, r, the radius of the cluster, and h, is the height of the cluster.
The average diffusion coefficient is given by D = (k, TJm)
jw
$(f) df0 (12)
= (nkg T / 2 mol) sech ( 7 4 2 )
.
It is reiated to the Dll and D,, the diffusion coefficients parallel and perpendicular to the preferred direction of molecular alignment by
Since the average diffusion coefficient is a scalar quantity, the same relation as eq. (13) holds in the cluster ; i. e.,
Here
I>;
andD:
are the diffusion coefficients parallel and perpendicular to the preferred direction in the cluster region in which the orientational order para- meter S equals to unity. Transforming the diffusion tensor from the cluster frame to the laboratory frame and taking the orientational ensemble average over the angles involved, we haveIn order to compute the anisotropy ratio of the diffusion coefficients in a cluster region, let us consider two planes, one parallel and the other perpendicular to the preferred direction. The areas All and A , of a rod-like molecule projected onto these planes are
71.
All = dl and A , = - d2, respectively. Since the 4
diffusion coefficient is proportional to the number of molecules which can go through the unoccupied space in a unit area perpendicular to the direction of diffusion, the anisotropy ratio of the diffusion coeffi- cients in the cluster region is
where o is the number of the molecules per unit area.
With the help of eq. (13') and (1 5), eq. (14) becomes
where D and y are given by eqs. (12) and (15).
SELF-DIFFUSION I N NEMATIC LIQUID CRYSTALS C1-101
3. Comparison with the experimental results.
- < 2 >
= 1.33 xlo-''
erg2/cm2,
For PAA at 122 OC, we have
<? >
= 2.99 x 1012 erg2/cm2. s2,
.e0 =2.18 x 10-l4 erg, po= 5.52 x loZ0 molecules/cm3,
c = 1.19,1= 4.45, and a = 9.52 x 1 0 ~ ~ s "
.
b0=3.32, r C = 2 x cm, hC=14.4x cm,
With the help of these values, we have obtained the
<
p2>
=7.04x gmlerg,
results which are listed in the following table :Diffusion Coefficient in
cm2/s
-
D Dll DL0101
D,,/D,
Experimental Results by
Present Topler Yun and Blinc Janik
theory et al. Fredrickson et al. et
al.
-
-- -
3.8 4.1 3.4 7.5 14
5.9 3.9 19
2.8 3.4 3.1 10
1.3 1.2 1.1 1.4
2.1 1.2 1.9
Here we note that the experimental values given by for the pair correlation function. More accurate Janik et al. are measured in the temperature range of results would be obtained if the exact pair correlation
123-125 OC. function for the nematic phase were known. This
Our calculations are based on a crude assumption work will be one of our future investigations.
References
[I] TOPLER, J., ALEFELD, B. and SPRINGER, T., presented at the Vth International Liquid Crystal Conference in Stock- holm, Sweden (June 17-21, 1974) ; JANIK, J. A., JANIK, J. M., OTNES, K. and RISTE, T., Mol. Crysf and Liquid Cryst., 15 (1971) 189 ; OTNES, K., PYNN, R., JANIK, J. A. and JANIK, J. M., Phys. Lett. 38A (1972) 335 ; BLINC, R. and DIMIC, V., Phys. Lett. 31A (1970) 531.
[2] YUN, C. K. and FREDRICKSON, A. G., Mol. Cryst., Liqu.
Cryst. 12 (1970) 73.
[3] BLINC, R., HOGENBOOM, D. L., O'REILLY, D. E. and PETER- SON, E. M., Phys. Rev. Lett. 23 (1969) 969 ; GHOSH,
S. K. and TETTAMANTI, E., Phys. Lett. 43A (1973) 361.
[4] FRANKLIN, W., Mol. Cryst., Liqu. Cryst. 14 (1971) 227.
[S] DOUGLAS, D. C., J. Chem. Phys. 33 (1960) 1376.
[6] ISBISTER, D. J. and MCQUARRIE, D. A., J. Chem. Phys. 56 (1972) 736.
[7] RAHMAN, A., Phys. Rev. 136 (1964) 405.
[8] CHU, K. S., Ph. D. Dissertation, Kent State University (June, 1974).
[9] GRAY, G. W., Molecular Structure and the Properties of Liquid Crystals (Academic Press Inc., New York, N. Y.) 1962.
