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Electrohydrodynamic flow in nematic liquid crystals
PHYSICS
ELECTROHYDRODYNAMIC
LETTERS
FLOW
IN
1 June
NEMATIC
LIQUID
1970
CRYSTALS
H. KOELMANS and A. M. van BOXTEL Pllilips
Rcsc~avcb
Labovutorics.
h’. I’. PIIilips
GlocilanlpE,i!~bric,krr,.
Eindlzolwz Tlzc NrIl~~~vluwds
Kcm:rtic liquid cr!-st:lIs are :I convenient tool in the study of clectrohydrodynamics. lizht scatterinrr connected \vith turbulent flo\v is found to be proportional to viscosity pO’rtion:rI to thf suure of the clcctric field.
On increasing the voltage on a thin (lo-100 pm) layer of a nematic liquid crystal sandwiched between planar electrodes, two distinctly different phenomena occur. Above about 5 V a stationary domain-like pattern is observed ]I]. When studying the behaviour of tiny solid particles suspended in the liquid we observed that the fluid within the domains is not at rest but shows a cellular laminar flow. Similar observations were recently made by Durand et al. [2]. When the voltage is further increased the laminar flow pattern gradually changes into a regime of turbulent flow in which the liquid crystal strongly scatters light ]3]. We again studied the motion of solid particles and noticed no dramatic change in velocity when the sample was passed from the nematic to the isotropic range. In the nematic phase particle motion closely corresponds to the motion of neighbouring nematic threads, which proves that the particles are dragged by the liquid and do not themselves move with respect to the liquid. Contrary to what is assumed in a model recently proposed by Helfrich [4] the anisotropy in liquid crystals apparently contributes little to its electrodynamic flow propoerties. We investigated the rise time t, of turbulent electrohydrodynamic flow by measuring the rise in intensity of the concomitant light scattering. Most experiments were done with p-methoxy benzilidene p-n butyl-aniline, which is nematic between 2o°C and 41°C. Measured rise times ranged between 15 and 600 msec. It was found that t, varies with the electric field as E-2. Furthermore tr is proportional to the viscosity n *. The latter was con*?I wns obtained tal through 1 mm.
32
from the efflux time of the liquid crysa cylindrical tube with a diameter of about
The rise time of and inversely pro-
eluded from a comparison of rise times in nematics of different viscosity and from the temperature dependence of t,, which coincides with the temperature dependence of 77. These results combined with dimensional analysis suggest the equation
t, =
cp E2
where 6 is the dielectric constant and C is a numerical constant, which was found in our case to be about I02. Recently a similar equation was reported for electrohydrodynamic flow in isotropic liquids [5]. Due to injection from the electrodes a low conductivity isotropic liquid at high electric field usually contains a net ionic space-charge. By exchange of momentum the force exerted on this net space-charge is transferred to the liquid. Analogous to the thermal case (the Benard problem) hydrodynamic instability may occur; the liquid is set in motion. Eq. (1) may be derived when unipolar injection giving rise to a spacecharge limited current is assumed [5-71. The rise time t, may be interpreted as the sum of th and t,, where th is the time to develop hydrodynamic instability and t, is the time to speed up the liquid to maximum velocity. Both th and t, obey eq. (1) [5-V]. Filippini and co-workers [5] estimate th >> tv and thus t, x th. If this were true one would expect a rise curve characterised by a silent period followed by a steep rise to maximum response. Experimentally, however, we find a rather gradual increase in light scattering over most of the rise period, which therfore suggests t, = t,.
Volume
32A.
number
PHYSICS
1
References 111 R. Williams, J. Chem. Phvs. i2j G. Durand et al.. C. R. Acad. [3] G. H. Heilmeier. L.A. Zanoni Proc. IEEE 56 (1968) 1162. [4] W.Helfrich, J. Chem. Phys.
1 June
LETTERS
1970
[5] J. C. Filippini et al.. Comptes. Rend. Acad. SC. 269B (1969) 736. [6] N.Felici. Rev. Gen. Elec. 78 (1969) 717. [7] P. K. Watson, J. M. Schneider and T. R. Till, Intern. Symp. on Electrohydrodvnamics. Cambridge.
39 (1965) 384. SC. 270B (1970) 97. and L. A. Barton.
USA, April 1969.
51 (1969) 1092.
*****
THE
ANISOTROPIC
(2s
+ 1)
LEVEL
ISING
FERROMAGNET
G. B. TAGGART Department
of Physics,
Temple
Pennsylvania,
University, 19122.
Philadelphia,
USA
Received 20 April 1970
By generalizing the method of Lagrange multipliers for the simple two-level tion and free energy per ion can be calculated in a mean field approximation for tern with uniaxial anisotropy.
system. the magnetizaa (2.8 + 1) level Ising sys-
The properties of higher level Ising systems, i. e. , S > $, have received some attention in the literature recently [l-3]. Most of these treatments have used the method of Green functions [e. g. 31 or of the Bragg-Williams approximation [e. g. 11. We wish to point out here the advantage of using the method of Lagrange multipliers [4] in order to determine the system magnetization and free energy within a mean field approximation. In particular, consider the Ising ferromagnet in an external field, I_LH, having an exchange interaction Z(gf) > 0 and uniaxial crystal anisotropy, D, i. e. Q=
- p HcSz-
Dc(Sz)2
ff
-cZ(sf)SzSz
ff
67c
The free energy of the system F=(Be>
- kTln W(oZ)
gf
(1)
.
can be written as,
,
(2)
where (. . . ) represents the usual thermal average, and lnW(oZ) is a function of 2S independent variables which represents the system entropy. The aZ are defined by, UZ 9 ((+)Z>
) I = 1,2 )....)
and W(oZ) is determined
2s
,
(3)
by the condition,
w(uZ) = Tr’ 1 ,
(4)
where Tr’ is a sum over all states with fixed uZ. Using Lagrange multipliers, X Z, and eq. (3), we can rewrite w(q)
=
3;
~le~p{-\I(Nu~
-
eq. (4) in the form, (5)
states 01,
33
ELECTROHYDRODYNAMIC
LETTERS
FLOW
IN
1 June
NEMATIC
LIQUID
1970
CRYSTALS
H. KOELMANS and A. M. van BOXTEL Pllilips
Rcsc~avcb
Labovutorics.
h’. I’. PIIilips
GlocilanlpE,i!~bric,krr,.
Eindlzolwz Tlzc NrIl~~~vluwds
Kcm:rtic liquid cr!-st:lIs are :I convenient tool in the study of clectrohydrodynamics. lizht scatterinrr connected \vith turbulent flo\v is found to be proportional to viscosity pO’rtion:rI to thf suure of the clcctric field.
On increasing the voltage on a thin (lo-100 pm) layer of a nematic liquid crystal sandwiched between planar electrodes, two distinctly different phenomena occur. Above about 5 V a stationary domain-like pattern is observed ]I]. When studying the behaviour of tiny solid particles suspended in the liquid we observed that the fluid within the domains is not at rest but shows a cellular laminar flow. Similar observations were recently made by Durand et al. [2]. When the voltage is further increased the laminar flow pattern gradually changes into a regime of turbulent flow in which the liquid crystal strongly scatters light ]3]. We again studied the motion of solid particles and noticed no dramatic change in velocity when the sample was passed from the nematic to the isotropic range. In the nematic phase particle motion closely corresponds to the motion of neighbouring nematic threads, which proves that the particles are dragged by the liquid and do not themselves move with respect to the liquid. Contrary to what is assumed in a model recently proposed by Helfrich [4] the anisotropy in liquid crystals apparently contributes little to its electrodynamic flow propoerties. We investigated the rise time t, of turbulent electrohydrodynamic flow by measuring the rise in intensity of the concomitant light scattering. Most experiments were done with p-methoxy benzilidene p-n butyl-aniline, which is nematic between 2o°C and 41°C. Measured rise times ranged between 15 and 600 msec. It was found that t, varies with the electric field as E-2. Furthermore tr is proportional to the viscosity n *. The latter was con*?I wns obtained tal through 1 mm.
32
from the efflux time of the liquid crysa cylindrical tube with a diameter of about
The rise time of and inversely pro-
eluded from a comparison of rise times in nematics of different viscosity and from the temperature dependence of t,, which coincides with the temperature dependence of 77. These results combined with dimensional analysis suggest the equation
t, =
cp E2
where 6 is the dielectric constant and C is a numerical constant, which was found in our case to be about I02. Recently a similar equation was reported for electrohydrodynamic flow in isotropic liquids [5]. Due to injection from the electrodes a low conductivity isotropic liquid at high electric field usually contains a net ionic space-charge. By exchange of momentum the force exerted on this net space-charge is transferred to the liquid. Analogous to the thermal case (the Benard problem) hydrodynamic instability may occur; the liquid is set in motion. Eq. (1) may be derived when unipolar injection giving rise to a spacecharge limited current is assumed [5-71. The rise time t, may be interpreted as the sum of th and t,, where th is the time to develop hydrodynamic instability and t, is the time to speed up the liquid to maximum velocity. Both th and t, obey eq. (1) [5-V]. Filippini and co-workers [5] estimate th >> tv and thus t, x th. If this were true one would expect a rise curve characterised by a silent period followed by a steep rise to maximum response. Experimentally, however, we find a rather gradual increase in light scattering over most of the rise period, which therfore suggests t, = t,.
Volume
32A.
number
PHYSICS
1
References 111 R. Williams, J. Chem. Phvs. i2j G. Durand et al.. C. R. Acad. [3] G. H. Heilmeier. L.A. Zanoni Proc. IEEE 56 (1968) 1162. [4] W.Helfrich, J. Chem. Phys.
1 June
LETTERS
1970
[5] J. C. Filippini et al.. Comptes. Rend. Acad. SC. 269B (1969) 736. [6] N.Felici. Rev. Gen. Elec. 78 (1969) 717. [7] P. K. Watson, J. M. Schneider and T. R. Till, Intern. Symp. on Electrohydrodvnamics. Cambridge.
39 (1965) 384. SC. 270B (1970) 97. and L. A. Barton.
USA, April 1969.
51 (1969) 1092.
*****
THE
ANISOTROPIC
(2s
+ 1)
LEVEL
ISING
FERROMAGNET
G. B. TAGGART Department
of Physics,
Temple
Pennsylvania,
University, 19122.
Philadelphia,
USA
Received 20 April 1970
By generalizing the method of Lagrange multipliers for the simple two-level tion and free energy per ion can be calculated in a mean field approximation for tern with uniaxial anisotropy.
system. the magnetizaa (2.8 + 1) level Ising sys-
The properties of higher level Ising systems, i. e. , S > $, have received some attention in the literature recently [l-3]. Most of these treatments have used the method of Green functions [e. g. 31 or of the Bragg-Williams approximation [e. g. 11. We wish to point out here the advantage of using the method of Lagrange multipliers [4] in order to determine the system magnetization and free energy within a mean field approximation. In particular, consider the Ising ferromagnet in an external field, I_LH, having an exchange interaction Z(gf) > 0 and uniaxial crystal anisotropy, D, i. e. Q=
- p HcSz-
Dc(Sz)2
ff
-cZ(sf)SzSz
ff
67c
The free energy of the system F=(Be>
- kTln W(oZ)
gf
(1)
.
can be written as,
,
(2)
where (. . . ) represents the usual thermal average, and lnW(oZ) is a function of 2S independent variables which represents the system entropy. The aZ are defined by, UZ 9 ((+)Z>
) I = 1,2 )....)
and W(oZ) is determined
2s
,
(3)
by the condition,
w(uZ) = Tr’ 1 ,
(4)
where Tr’ is a sum over all states with fixed uZ. Using Lagrange multipliers, X Z, and eq. (3), we can rewrite w(q)
=
3;
~le~p{-\I(Nu~
-
eq. (4) in the form, (5)
states 01,
33