- Email: [email protected]
Relation between frank constants and viscosity coefficients in an ellipsoid model for nematic liquid crystals
Volume 57A, number 1
PHYSICS Lf';TFERS
17 May 1976
R E L A T I O N B E T W E E N F R A N K C O N S T A N T S A N D V I S C O S I T Y C O E F F I C I E N T S IN AN E L L I P S O I D M O D E L F O R N E M A T I C L I Q U I D C R Y S T A L S *~ Y.R. LIN-LIU, Y.M. SHIH and C.-W. WOO
Department of Physics,Northwestern Unil,ersity,Evanston,Illinois 60201, USA Received 22 March 1976 The Frank constants and viscosity coefficients for a nematic liquid crystal are calculated for a microscopic model: 1 1 a one-parameter ellipsoid model. A simple relation is established: K33[ ~-(K11 + K22) = 2-(r/a + r/c)/r~b,and appears well confirmed by experiment.
Frank's theory [1 ] expresses the deformation energy in a nematic liquid crystal as:
~v=2flKll(V.a)2 +K22(n.v × a)2 + K33Qi×
V X ti) 2] dr,
(1)
where ti(r) is.the director at r, a macroscopically local unit vector characterizing the orientation of the fluid particle, The three terms represent three kinds of orientational deformation, as permitted by symmetry considerations applied to a nematic: respectively splay, twist, and bend. It is experimentally observed that the Frank constants K l l and K22 are of similar magnitude, while K33 is about twice as large. The Frank constants have been calculated from microscopic models by Nehring and Saupe [2], Priest [3], and Straley [4]. Our calculation here will be similar to that of Priest in physical content, but is rather different in presentation. We begin with a nematic of density p and an intermolecular potential of the form:
v(1,2) = Vo(r12) +
v2(r12)e2(f~l, ~z2).
(2)
The internal energy of the system is given by U = l f p ( 2 ) ( r 1, ~1, r2, ~22) V(1,2) dr 1 d(21
dr 2 d ~ 2,
(3)
where p(2)(rl, O1 r2, ~ 2 2 ) = N ( N - 1 ) f e x p '
Z
[- i
(4)
-P(~)(r 1 , al)P(1)(r2, a z ) g ( 1 , 2 ) ~ P(1)(r I , al)P(l)(r2, a2)g(rl2), ~
N
P(l'(rl' ~ l ) - - z - f e x p
I- ~
i
V(i,])/kT..I1 dr 2 d ~ 2 ... drN d~N =Pf(~l'n(rl))'
(s)'
and
z =fexp I- i
(6)
On account of the simple form of V(t, 2), U can be rewritten in the following form after twice applying the addition theorem: Work supported in part by the National ScienceFoundation through Grant No. DMR73-07659-A02. 43
Volume 57A, number I
PHYSICS LETTERS
17 May 1976
-' ' U--~Np70 +-fNP72 o 2 + ~p202 ffg(r12)V2(r12)iP2(ci(rl).li(r2) ) where
7i
=fg(r) v~(r) dr,
o
1] dr 1 dr2,
.(7)
1
=ff(0)P2(cos 0) d(2.
The latter defines the usual orientational order parameter• At this point, a macroscopic distortion is introduced such that a IV~(r)l ~ 1, where a measures the intermolecular spacing. For Irl >a,g(r) V2(r ) -~ 0. For Irl ~ a , a Taylor expansion can be carried out for ~(rl). Substituting the expansion into P2(it(rl)'it(r2) ) and keeping only the leading term we obtain from eq. (7) the deformation energy
6U = - ~p2o2h&r f V¢nc~(r) VTn~(r ) dr, with
(Einstein summation used)
(8)
h~7 = fg(r) V2(r)r¢r v dr.
By assuming cylindrical symmetry about the z-directi0n, (thus blurring the distinction between splay and twist) hll = h22 4= h33 , and hi3.r = 0 for ~ 4: 7. The equilibrium director becomes ho = ~ = (0, 0, 1), and the deformed n(r) can be written in first order as:
it(r) = ito + 6it(r),
5h(r) = ( 6nx, 5ny, 0).
Eq. (8) then reduces to the same form as eq. (1), yielding Kll =K22 = - 2_a^2~2u ~ ' v "11,
K33 = _ ~ p 2 o 2 h 3 3
Let us now take a specific model: one in which the intermolecular potential observes a particularly simple ellipsoidal symmetry:
Vi(r ) = Vi(R, z) = Vi(N/R 2 + z2/a2),
(9)
where R ~ x / ~ x2 +y2. Then g(r) V2(r ) ~ V(R, Z)' K l l =K22 ~ - ~ p1 2 o 2 a j V/ ~( ~'~") ~ 2
One finds
d~,
K33
~
- - ~1 p 2 o 2 a 3 j/ V' ~ (~~" ) ~ 2
d~,
and consequently, 1
K33/~(Kll +K22 ) =a 2.
(10)
We now turn to Kirkwood's theory of Brownian motion [5, 6]. The frictional coefficient u for an isotropic liquid is given in terms of the correlation function (F(0)F(s)> O, thus:
u = ~-~
(F(O)F(s)> 0 ds,
(11)
_oo
where F(t) denotes the force exerted on molecule [1] at time t: 3/4 • ~H F~(t) = ~ (t) = e x p 0 L o t ) ~-7--(0) = exp(iLot ) F~(0), t,, a vrce
where
Lo = i =
Vrl + FI" VPI '
i =1 zm
(12) i
The average ( )o is taken over an equilibrium distribution. By introducing a mean field approximation on the molecular orientations, we show in a recent paper [7] that an "effective pair correlations" theory can be constructed which treats the spatial correlations by first averaging the potential V(1,2) over the orientations (21 and (22 to yield an effective potential: 44
Volume 57A, number 1
PHYSICS LETTERS
17 May 1976
Table 1 Frank constants and viscosity coefficients for PAA and MBBA Viscosity coefficients (10 -2 poise) Frank constants [9, 10] (10 -7 dyne)
Set I [11,121
PAA
Kll
K22
K33
K33 l ~(Kll + K22)
at 125°C
4.5
2.9
9.5
2.6
MMBBA at 25°C
6
4
7.5
1
r/a 3.4
r/b
r/c
2.4
9.2
16.1 16.3
25.2
1.5
Ve(1, 2) = V0(rI2 ) +
Set II [13,141 7(r~a + rTc) nb 2.6
r/a 4.1 41
1.3
V2(r12)o2.
r/b 1.5 24
~-(r/a + ~c) nb
r/c
8.6 4.2 103 3.0 (13)
Thus, in the ellipsoid model, N (rli)c~ d Fc~(O) = ~ Ve(~'li; o), "= a~f li d~'li where {'li =- (x/R
A.__2+ z~i/a2),
as =
(14) 0, a = 1,2, a, Or=3.
The evaluation of the different components o f / / : / / 1 1 , / / 2 2 and//33, is then straightforward. We find//11 =//22 =a2//33 . Now, assuming that Stoke's Law still holds, these three frictional coefficients can be related to the three viscosity coefficients r/a, r/b, and r/c as defined conventionally [8] for a nematic. Since 1
//11 =//22 cc 2-(r/a + r/c), •
//33 cc r/b,
1
we obtain 7 ( % + %)/r/b = a2, and finally with eq. (10), the relation 1
K33/2(Kll + K22)
1
= ~(r/a + r/c)/r/b"
(I 5)
Table 1 lists experimental data on the Frank constants and viscosity coefficients, for PAA at 125°C and MBBA at 25°C. The ratii shown in eq. (15) are given for comparison. We note that in each case one set of the viscosity measurements confirm our predicted relation almost perfectly, while the other set is off by a factor o f 2. For a model as simple as ours, the agreement should be regarded as quite acceptable. The need for more reliable viscosity data is clearly indicated.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
F.C. Frank, Discuss. Faraday Soc. 25 (1958) 19. J. Henring and S. Saupe, J. Chem. Phys. 56 (1972) 5527. R.G. Priest, Mol. Cryst. Liq. Cryst. 17 (1972) 129; Phys. Rev. A7 (1973) 720. J.P. Straley, Phys. Rev. A8 (1973) 2181. J.G. Kirkwood, J. Chem. Phys. 14 (1946) 180. F. Reif, Fundamental of statistical and thermal physics (McGraw-Hill, 1965) 15.8. Y.M. Shih, Y.R. Lin-Liu and C.-W. Woo, Phys. Rev. Lett., to be published. P.G. DeGennes, The physics of liquid crystals (Oxford, 1974) p. 164. Orsay Liquid Crystal Group, Phys. Rev. Lett. 22 (1969) 1361. I. Haller, J. Chem. Phys. 57 (1972) 1400. M. Miesowicz, Nature (Lond.) 158 (1946) 27. O. Langevin, J. Phys. (Paris) 33 (1972) 249. Orsay Liquid Crystal Group, Mol. Cryst. Liq. Cryst. 13 (1971) 187. C. G~ihwiller, Phys. Lett. A36 (1971) 311. 45
PHYSICS Lf';TFERS
17 May 1976
R E L A T I O N B E T W E E N F R A N K C O N S T A N T S A N D V I S C O S I T Y C O E F F I C I E N T S IN AN E L L I P S O I D M O D E L F O R N E M A T I C L I Q U I D C R Y S T A L S *~ Y.R. LIN-LIU, Y.M. SHIH and C.-W. WOO
Department of Physics,Northwestern Unil,ersity,Evanston,Illinois 60201, USA Received 22 March 1976 The Frank constants and viscosity coefficients for a nematic liquid crystal are calculated for a microscopic model: 1 1 a one-parameter ellipsoid model. A simple relation is established: K33[ ~-(K11 + K22) = 2-(r/a + r/c)/r~b,and appears well confirmed by experiment.
Frank's theory [1 ] expresses the deformation energy in a nematic liquid crystal as:
~v=2flKll(V.a)2 +K22(n.v × a)2 + K33Qi×
V X ti) 2] dr,
(1)
where ti(r) is.the director at r, a macroscopically local unit vector characterizing the orientation of the fluid particle, The three terms represent three kinds of orientational deformation, as permitted by symmetry considerations applied to a nematic: respectively splay, twist, and bend. It is experimentally observed that the Frank constants K l l and K22 are of similar magnitude, while K33 is about twice as large. The Frank constants have been calculated from microscopic models by Nehring and Saupe [2], Priest [3], and Straley [4]. Our calculation here will be similar to that of Priest in physical content, but is rather different in presentation. We begin with a nematic of density p and an intermolecular potential of the form:
v(1,2) = Vo(r12) +
v2(r12)e2(f~l, ~z2).
(2)
The internal energy of the system is given by U = l f p ( 2 ) ( r 1, ~1, r2, ~22) V(1,2) dr 1 d(21
dr 2 d ~ 2,
(3)
where p(2)(rl, O1 r2, ~ 2 2 ) = N ( N - 1 ) f e x p '
Z
[- i
(4)
-P(~)(r 1 , al)P(1)(r2, a z ) g ( 1 , 2 ) ~ P(1)(r I , al)P(l)(r2, a2)g(rl2), ~
N
P(l'(rl' ~ l ) - - z - f e x p
I- ~
i
V(i,])/kT..I1 dr 2 d ~ 2 ... drN d~N =Pf(~l'n(rl))'
(s)'
and
z =fexp I- i
(6)
On account of the simple form of V(t, 2), U can be rewritten in the following form after twice applying the addition theorem: Work supported in part by the National ScienceFoundation through Grant No. DMR73-07659-A02. 43
Volume 57A, number I
PHYSICS LETTERS
17 May 1976
-' ' U--~Np70 +-fNP72 o 2 + ~p202 ffg(r12)V2(r12)iP2(ci(rl).li(r2) ) where
7i
=fg(r) v~(r) dr,
o
1] dr 1 dr2,
.(7)
1
=ff(0)P2(cos 0) d(2.
The latter defines the usual orientational order parameter• At this point, a macroscopic distortion is introduced such that a IV~(r)l ~ 1, where a measures the intermolecular spacing. For Irl >a,g(r) V2(r ) -~ 0. For Irl ~ a , a Taylor expansion can be carried out for ~(rl). Substituting the expansion into P2(it(rl)'it(r2) ) and keeping only the leading term we obtain from eq. (7) the deformation energy
6U = - ~p2o2h&r f V¢nc~(r) VTn~(r ) dr, with
(Einstein summation used)
(8)
h~7 = fg(r) V2(r)r¢r v dr.
By assuming cylindrical symmetry about the z-directi0n, (thus blurring the distinction between splay and twist) hll = h22 4= h33 , and hi3.r = 0 for ~ 4: 7. The equilibrium director becomes ho = ~ = (0, 0, 1), and the deformed n(r) can be written in first order as:
it(r) = ito + 6it(r),
5h(r) = ( 6nx, 5ny, 0).
Eq. (8) then reduces to the same form as eq. (1), yielding Kll =K22 = - 2_a^2~2u ~ ' v "11,
K33 = _ ~ p 2 o 2 h 3 3
Let us now take a specific model: one in which the intermolecular potential observes a particularly simple ellipsoidal symmetry:
Vi(r ) = Vi(R, z) = Vi(N/R 2 + z2/a2),
(9)
where R ~ x / ~ x2 +y2. Then g(r) V2(r ) ~ V(R, Z)' K l l =K22 ~ - ~ p1 2 o 2 a j V/ ~( ~'~") ~ 2
One finds
d~,
K33
~
- - ~1 p 2 o 2 a 3 j/ V' ~ (~~" ) ~ 2
d~,
and consequently, 1
K33/~(Kll +K22 ) =a 2.
(10)
We now turn to Kirkwood's theory of Brownian motion [5, 6]. The frictional coefficient u for an isotropic liquid is given in terms of the correlation function (F(0)F(s)> O, thus:
u = ~-~
(F(O)F(s)> 0 ds,
(11)
_oo
where F(t) denotes the force exerted on molecule [1] at time t: 3/4 • ~H F~(t) = ~ (t) = e x p 0 L o t ) ~-7--(0) = exp(iLot ) F~(0), t,, a vrce
where
Lo = i =
Vrl + FI" VPI '
i =1 zm
(12) i
The average ( )o is taken over an equilibrium distribution. By introducing a mean field approximation on the molecular orientations, we show in a recent paper [7] that an "effective pair correlations" theory can be constructed which treats the spatial correlations by first averaging the potential V(1,2) over the orientations (21 and (22 to yield an effective potential: 44
Volume 57A, number 1
PHYSICS LETTERS
17 May 1976
Table 1 Frank constants and viscosity coefficients for PAA and MBBA Viscosity coefficients (10 -2 poise) Frank constants [9, 10] (10 -7 dyne)
Set I [11,121
PAA
Kll
K22
K33
K33 l ~(Kll + K22)
at 125°C
4.5
2.9
9.5
2.6
MMBBA at 25°C
6
4
7.5
1
r/a 3.4
r/b
r/c
2.4
9.2
16.1 16.3
25.2
1.5
Ve(1, 2) = V0(rI2 ) +
Set II [13,141 7(r~a + rTc) nb 2.6
r/a 4.1 41
1.3
V2(r12)o2.
r/b 1.5 24
~-(r/a + ~c) nb
r/c
8.6 4.2 103 3.0 (13)
Thus, in the ellipsoid model, N (rli)c~ d Fc~(O) = ~ Ve(~'li; o), "= a~f li d~'li where {'li =- (x/R
A.__2+ z~i/a2),
as =
(14) 0, a = 1,2, a, Or=3.
The evaluation of the different components o f / / : / / 1 1 , / / 2 2 and//33, is then straightforward. We find//11 =//22 =a2//33 . Now, assuming that Stoke's Law still holds, these three frictional coefficients can be related to the three viscosity coefficients r/a, r/b, and r/c as defined conventionally [8] for a nematic. Since 1
//11 =//22 cc 2-(r/a + r/c), •
//33 cc r/b,
1
we obtain 7 ( % + %)/r/b = a2, and finally with eq. (10), the relation 1
K33/2(Kll + K22)
1
= ~(r/a + r/c)/r/b"
(I 5)
Table 1 lists experimental data on the Frank constants and viscosity coefficients, for PAA at 125°C and MBBA at 25°C. The ratii shown in eq. (15) are given for comparison. We note that in each case one set of the viscosity measurements confirm our predicted relation almost perfectly, while the other set is off by a factor o f 2. For a model as simple as ours, the agreement should be regarded as quite acceptable. The need for more reliable viscosity data is clearly indicated.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
F.C. Frank, Discuss. Faraday Soc. 25 (1958) 19. J. Henring and S. Saupe, J. Chem. Phys. 56 (1972) 5527. R.G. Priest, Mol. Cryst. Liq. Cryst. 17 (1972) 129; Phys. Rev. A7 (1973) 720. J.P. Straley, Phys. Rev. A8 (1973) 2181. J.G. Kirkwood, J. Chem. Phys. 14 (1946) 180. F. Reif, Fundamental of statistical and thermal physics (McGraw-Hill, 1965) 15.8. Y.M. Shih, Y.R. Lin-Liu and C.-W. Woo, Phys. Rev. Lett., to be published. P.G. DeGennes, The physics of liquid crystals (Oxford, 1974) p. 164. Orsay Liquid Crystal Group, Phys. Rev. Lett. 22 (1969) 1361. I. Haller, J. Chem. Phys. 57 (1972) 1400. M. Miesowicz, Nature (Lond.) 158 (1946) 27. O. Langevin, J. Phys. (Paris) 33 (1972) 249. Orsay Liquid Crystal Group, Mol. Cryst. Liq. Cryst. 13 (1971) 187. C. G~ihwiller, Phys. Lett. A36 (1971) 311. 45