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The interaction between a disclination in a nematic liquid crystal and a rubbed surface
Solid State Communications,
Vol. 12, pp.585—588, 1973.
Pergamon Press.
Printed in Great Britain
THE INTERACTION BETWEEN A D1SCLINATION IN A NEMATIC LIQUID CRYSTAL AND A RUBBED SURFACE Robert B. Meyer Division of Engineering and Applied Physics, Harvard University, Cambridge, Massachusetts 02138, U.S.A. (Received 17 November 1972 by P. G. de Gennes)
For the case of a rubbed surface with a finite anisotropic surface tension, a model is developed for the structure and energy of a surface disclination and of an edge disclination near the surface. It is found that there is an energy barrier against detaching a disclination from the surface, and also that there is a critical distance within which a disclination is attracted, and beyond which it is repelled. by the surface. These properties may provide a measurement of the anisotropy of the surface tension.
INTRODUCTION
= 0 plane is Or. The anisotropic part of the surface tension ‘y is modeled by the simplest function with the right symmetry,
THE HOMOGENEOUS alignment of nematic liquid crystals at solid surfaces which have been rubbed or otherwise given a direction of preferred orientation is an effect of practical importance and the subject of recent theoretical investigation.’ ~ One of the common defects at such surfaces is a line disclination. Besides
2 (0 Or). = ~ A sin Consider a disclination parallel to the ii axis at = 0. Assuming the splay and bend elastic constants equal (k 11 = k,, =1~ coordinates ak.~b). nd z =and transforming the total energy per unit to x = of (k,,/k~b) length the discination can be written as:5 —
~,
being a nuisance. theseallowing lines may provide a probe the surface structure, measurement of theof anisotropic part of the surface tension, in this cornmunication. a model is developed to calculate the structure and energy of a surface disclination. as well as the interaction of an edge disclination in the liquid
I
00 + 00
E
=
~(ksb/k22 )~ ô
—~o
{k, 2(VO)’ +A6(z)sin’(O Or)}dXdZ. from which one derives the Euler equation ~(z)(~ A sin 2(0 Or) k,~80’ k 20 = 0. V +: That is, we must solve Laplace’s equation 22 in the half plane, with the boundary condition at z = 0 given by setting the coefficient of ~(z) equal to zero. Furthermore, for the lowest strength disclination. the solution far from the origin must approach 0 ±0, with 0 being the polar angle about the —17 axis. measured form the x axis. One can look for two kinds —
crystal with the anisotropic surface. Williams. Vitek and Kléman have proposed an approximate solution 4 De Gennes for the structure of a surface disclination. has made some general observations on the nature of anisotropic surfaces, and given the image solution for a disclination near a strongly anisotropic surface.’
—
—
—
GENERAL MODEL The simplest problem to consider, which occurs in the case of the twist elastic constant. k 22. being the smallest, is that of a planar sample. in cartesian coordinates (~, ~,~)consider the nematic in the +~ half space, with the director n parallel to the ~ plane, and n~ = cos 0. The rubbing direction in the
of solution: (a) Surface disclinatfon (Continuous solution) Qualitatively, the surface disclination can be described by saying that the core of the disclination has 585
586
DISCLINATION NEAR A RUBBED SURFACE
Vol. 12, No.7
moved out of the upper half plane and combined with its image at a distance s below the surface. The solution is then 0
=
Or+tan~ [(z+s)/x]
=Or+
~,1i, ~R
in which i~iis the polar angle in coordinates centered atx = 0,z = —s. Atz=O, @0/az) = x/(x2 + ~2) = 1s111 2 (8 so that s
—
Or)]
/(2s),
_
________________________ _____
—d FIG. 1. The geometry, the variables, and the path of —
__________
integration for the elastic energy of a disclination at a distance, d, above the surface.
= k
22 IA. The width lineorigin. is 0r as of s/xthefarsurface from the ‘—2s, with 8 approaching Using the unit of energy E 0 = lr(k.bk22 )f/4 the total energy per unit length of this structure is = 2E 0 {ln(R/2s) + I).
The first term is the elastic energy and the second is the surface energy. R is a large radius cutoff.
O~gives the correct behavior near the core and far 0im function = °r at from the origin, while 01 is a continuous approaching zero at large radii. Since z = 0, the boundary condition on 01 at z = 0 becomes O’/s =
ao’ /8z + x/(x’ + d’).
The general form for 0’ is
(b) Disclination near a surface
0’ (x, z)
=
(irr’
f dk fiIk)e~sinkx, 0
Consider the lowest strength disclination, fl = 1 (ir rotation of the director in one Buerger circuit),
in which f(k) is obtained from the boundary condition, which gives
at x = 0,and z =0+d. Ifd>>Or.s,Inthen 80/8z << at the surface, (zwith = 0)a~second the 1/s simple image solution, nthis = limit, 1 disclination at z = —d is valid (see Fig. l)
f(/ç)
0
= ~
= Or +
~ ~i + ~ 02.
+00
exponential integrals, 1(w) = f (ë~/u)du,which are tabulated.6 With w = (d + z ~ ix)/s.
0~and 02 are polar angles measured around the two 01 =
disclinations. The elastic energy can be converted to a line in. tegral, ~m
~
-~-~i
xsinkx dx = 7re~’1/(k + (x2 + d2) s/ -00 81 can therefore be expressed in terms of complex k + — S /
=
Im [ewI(w)]
Along z = 0, the integrand of the elastic energy line integral for this solution is k
2. 220(80/82)k220r(8018z)4
(k
22ksb)~#0V0Cdl, in which c is the outward normal unit vector, around the path shown in Fig. 1. The radius e of the small circular path around the core is the short range cutoff for linear elasticity theory. For e <
non-linear boundary condition is difficult. However, O 0,. remains small at the surface, then the linearized boundary condition (80/8z) = (0 Or)/S is sufficiently accurate. The solution for the linearized problem may be written as 0 = + 01 in which —
—
A(O’)
The first term is odd in x, and the second term cancels the surface energy density, so once again the total energy is obtained from the integral along the branch cut. With e << (d or s) <
Vol. 12, No.7 -~
I
DISCLINATION NEAR A RUBBED SURFACE I
The minimum value of is approximately a molecular dimension at low temperatures. It may become larger near the clearing temperature, T~.
I
0 ‘U
‘U
_______
o
E
/ 0
I
I
I
0
I
_________________________________________ r
FIG. 3. Surface orientation of the director, ~ 0,., for an edge disclination at about the critical distance. dm, from the surface. —
ofpositive lines ofnet strength n to the surface.
E(core) can be estimated by setting the energy density within the core equal to the elastic energy density at e. This makes E(core) = ~ E0, independent ofe.
For comparison of cases (a) and (b), E8 can be rewritten as E8
.0
FIG. 2. Energy, Ed —E1,in units of E0, and force per unit length, in units of E0/s, of an edge disclination as a function of distance from the surface.
C0
=
587
+2. and is attracted
To check the validity of the linearized problem as a calculation of dm, the value of 0 at the surface is plotted in Fig. 3 for d/s = 0.3. Its maximum value is < 0.54 so the approximation is good. The nonlinear problem would give a slightly larger value of dm. DISCUSSION The important parameters for evaluating these results are s, e, and E (core). The length s is the same as the ‘extrapolation length’ found by de Gennes for the kind of anisotropic surface discussed here. sinusoidal For the 1”3 with groove model of the rubbed grooves of wavelength X and surface, depth ~ = (1/ir3)(k 2 X, which is approximately temperature 22/ kgb) (X/6) independent, far from a nematic—smectic phase change.
= E1 +
[2E0—E(core)] —Eoln(4s/e).
The minimum value of s/c for which the calculation in case (b) is valid is about 10. Combining this with the estimate of E(core), E8 E1 —2.2E0, still about E0 below the maximum of Ed, so there is a large activation energy for removing the disclination from the surface. Near 7~,s/c may approach I, so that this activation energy disappears. Disclinations are observed to detach themselves from rubbed surfaces near T~. —
Another useful comparison to make is between E8 and the minimum energy of the n = I disclination in a finite planar sample of thickness t. The two sample surfaces are rubbed at right angles to one another. For the surface line, in this case, R = t/ir. The energy of the disclination is a 7mimimum and is Ed when = E it is in the mid. plane of the sample. 0 ln[t!2ire] + E (core). Comparing E8 and Ed: E~ Ed = [2E’0 E(core)] E0 in [2irs’/(et)] For4cm a largeEd, eitherbut E~orEd in practice, may be with larger,lOdepending on the value of s. —
—
—
Finally, if it is large enough. s can be measured directly by examining surface disclinations with polarized light. For smaller values of s, less direct experiments which measure either E 8 or dm may be practical.
Acknowledgements The author is most grateful to the work Alfredthrough P. Sloana Sloan Foundation for their supportThis of his Research research was also supported by the Fellowship. National Science —
Foundation throughof grants GH-33576 GH-3440l and by the Division Engineering andand Applied Physics, Harvard University.
588
DISCUNATION NEAR A RUBBED SURFACE
Vol. 12, No.7
REFERENCES 1. 2.
DE GENNES P.G., Lecture Notes (1970). BERR.EMAN D.W., Phys. Rev. Len. 28, 1683 (1972), and Proc. Fourth mt. Liq. Oyst. Conf (1972).
3.
CREAGH L.T. and KMETZ A.R., Proc. Fourth mt. Liq. Cryst. Conf (1972).
4.
WILLIAMS C., VITEK V. and KLEMAN M., Proc. Fourth mt. Liq. Cryst. Conf (1972).
5.
The divergence terms in the free energy discussed by FRANK F.C., Disc. Faraday Soc. 25, 19 (1958) and NEHRJNG J. and SAUPE A., J. Chem. Phys. 54, 337 (1971), both integrate to zero in this problem. Tables of Sine, Cosine, and Exponential Integrals Vol. II, National Bureau of Standards, 1940, and Tables of the Exponential Integral for Complex Arguments, Applied Math Series 51, National Bureau of Standards, 1958. MEYER R.B. and ARGUIMBAU N., oral presentation at Pont-a-Mousson, 1971; paper in preparation. See also SHEFFER T.J., .Phys. Rev. AS, 1327, (1972).
6. 7.
Dans Ic cas d’une surface frottée, avec une energie anisotrope finie, on calcule Ia structure et l’energie d’une disclination a Ia surface, et d’une disclination coin prés de la surface. On trouve une barrière d’energie pour Ic détachement d’une disclination de la surface, et aussi une distance critique pour l’attraction de la disclmation a la surface. Par delà cette distance, Ia disclination est repoussée de la surface. Ces propriétés doivent fournir une mesure de l’anisotropie de I’energie de Ia surface.
Vol. 12, pp.585—588, 1973.
Pergamon Press.
Printed in Great Britain
THE INTERACTION BETWEEN A D1SCLINATION IN A NEMATIC LIQUID CRYSTAL AND A RUBBED SURFACE Robert B. Meyer Division of Engineering and Applied Physics, Harvard University, Cambridge, Massachusetts 02138, U.S.A. (Received 17 November 1972 by P. G. de Gennes)
For the case of a rubbed surface with a finite anisotropic surface tension, a model is developed for the structure and energy of a surface disclination and of an edge disclination near the surface. It is found that there is an energy barrier against detaching a disclination from the surface, and also that there is a critical distance within which a disclination is attracted, and beyond which it is repelled. by the surface. These properties may provide a measurement of the anisotropy of the surface tension.
INTRODUCTION
= 0 plane is Or. The anisotropic part of the surface tension ‘y is modeled by the simplest function with the right symmetry,
THE HOMOGENEOUS alignment of nematic liquid crystals at solid surfaces which have been rubbed or otherwise given a direction of preferred orientation is an effect of practical importance and the subject of recent theoretical investigation.’ ~ One of the common defects at such surfaces is a line disclination. Besides
2 (0 Or). = ~ A sin Consider a disclination parallel to the ii axis at = 0. Assuming the splay and bend elastic constants equal (k 11 = k,, =1~ coordinates ak.~b). nd z =and transforming the total energy per unit to x = of (k,,/k~b) length the discination can be written as:5 —
~,
being a nuisance. theseallowing lines may provide a probe the surface structure, measurement of theof anisotropic part of the surface tension, in this cornmunication. a model is developed to calculate the structure and energy of a surface disclination. as well as the interaction of an edge disclination in the liquid
I
00 + 00
E
=
~(ksb/k22 )~ ô
—~o
{k, 2(VO)’ +A6(z)sin’(O Or)}dXdZ. from which one derives the Euler equation ~(z)(~ A sin 2(0 Or) k,~80’ k 20 = 0. V +: That is, we must solve Laplace’s equation 22 in the half plane, with the boundary condition at z = 0 given by setting the coefficient of ~(z) equal to zero. Furthermore, for the lowest strength disclination. the solution far from the origin must approach 0 ±0, with 0 being the polar angle about the —17 axis. measured form the x axis. One can look for two kinds —
crystal with the anisotropic surface. Williams. Vitek and Kléman have proposed an approximate solution 4 De Gennes for the structure of a surface disclination. has made some general observations on the nature of anisotropic surfaces, and given the image solution for a disclination near a strongly anisotropic surface.’
—
—
—
GENERAL MODEL The simplest problem to consider, which occurs in the case of the twist elastic constant. k 22. being the smallest, is that of a planar sample. in cartesian coordinates (~, ~,~)consider the nematic in the +~ half space, with the director n parallel to the ~ plane, and n~ = cos 0. The rubbing direction in the
of solution: (a) Surface disclinatfon (Continuous solution) Qualitatively, the surface disclination can be described by saying that the core of the disclination has 585
586
DISCLINATION NEAR A RUBBED SURFACE
Vol. 12, No.7
moved out of the upper half plane and combined with its image at a distance s below the surface. The solution is then 0
=
Or+tan~ [(z+s)/x]
=Or+
~,1i, ~R
in which i~iis the polar angle in coordinates centered atx = 0,z = —s. Atz=O, @0/az) = x/(x2 + ~2) = 1s111 2 (8 so that s
—
Or)]
/(2s),
_
________________________ _____
—d FIG. 1. The geometry, the variables, and the path of —
__________
integration for the elastic energy of a disclination at a distance, d, above the surface.
= k
22 IA. The width lineorigin. is 0r as of s/xthefarsurface from the ‘—2s, with 8 approaching Using the unit of energy E 0 = lr(k.bk22 )f/4 the total energy per unit length of this structure is = 2E 0 {ln(R/2s) + I).
The first term is the elastic energy and the second is the surface energy. R is a large radius cutoff.
O~gives the correct behavior near the core and far 0im function = °r at from the origin, while 01 is a continuous approaching zero at large radii. Since z = 0, the boundary condition on 01 at z = 0 becomes O’/s =
ao’ /8z + x/(x’ + d’).
The general form for 0’ is
(b) Disclination near a surface
0’ (x, z)
=
(irr’
f dk fiIk)e~sinkx, 0
Consider the lowest strength disclination, fl = 1 (ir rotation of the director in one Buerger circuit),
in which f(k) is obtained from the boundary condition, which gives
at x = 0,and z =0+d. Ifd>>Or.s,Inthen 80/8z << at the surface, (zwith = 0)a~second the 1/s simple image solution, nthis = limit, 1 disclination at z = —d is valid (see Fig. l)
f(/ç)
0
= ~
= Or +
~ ~i + ~ 02.
+00
exponential integrals, 1(w) = f (ë~/u)du,which are tabulated.6 With w = (d + z ~ ix)/s.
0~and 02 are polar angles measured around the two 01 =
disclinations. The elastic energy can be converted to a line in. tegral, ~m
~
-~-~i
xsinkx dx = 7re~’1/(k + (x2 + d2) s/ -00 81 can therefore be expressed in terms of complex k + — S /
=
Im [ewI(w)]
Along z = 0, the integrand of the elastic energy line integral for this solution is k
2. 220(80/82)k220r(8018z)4
(k
22ksb)~#0V0Cdl, in which c is the outward normal unit vector, around the path shown in Fig. 1. The radius e of the small circular path around the core is the short range cutoff for linear elasticity theory. For e <
non-linear boundary condition is difficult. However, O 0,. remains small at the surface, then the linearized boundary condition (80/8z) = (0 Or)/S is sufficiently accurate. The solution for the linearized problem may be written as 0 = + 01 in which —
—
A(O’)
The first term is odd in x, and the second term cancels the surface energy density, so once again the total energy is obtained from the integral along the branch cut. With e << (d or s) <
Vol. 12, No.7 -~
I
DISCLINATION NEAR A RUBBED SURFACE I
The minimum value of is approximately a molecular dimension at low temperatures. It may become larger near the clearing temperature, T~.
I
0 ‘U
‘U
_______
o
E
/ 0
I
I
I
0
I
_________________________________________ r
FIG. 3. Surface orientation of the director, ~ 0,., for an edge disclination at about the critical distance. dm, from the surface. —
ofpositive lines ofnet strength n to the surface.
E(core) can be estimated by setting the energy density within the core equal to the elastic energy density at e. This makes E(core) = ~ E0, independent ofe.
For comparison of cases (a) and (b), E8 can be rewritten as E8
.0
FIG. 2. Energy, Ed —E1,in units of E0, and force per unit length, in units of E0/s, of an edge disclination as a function of distance from the surface.
C0
=
587
+2. and is attracted
To check the validity of the linearized problem as a calculation of dm, the value of 0 at the surface is plotted in Fig. 3 for d/s = 0.3. Its maximum value is < 0.54 so the approximation is good. The nonlinear problem would give a slightly larger value of dm. DISCUSSION The important parameters for evaluating these results are s, e, and E (core). The length s is the same as the ‘extrapolation length’ found by de Gennes for the kind of anisotropic surface discussed here. sinusoidal For the 1”3 with groove model of the rubbed grooves of wavelength X and surface, depth ~ = (1/ir3)(k 2 X, which is approximately temperature 22/ kgb) (X/6) independent, far from a nematic—smectic phase change.
= E1 +
[2E0—E(core)] —Eoln(4s/e).
The minimum value of s/c for which the calculation in case (b) is valid is about 10. Combining this with the estimate of E(core), E8 E1 —2.2E0, still about E0 below the maximum of Ed, so there is a large activation energy for removing the disclination from the surface. Near 7~,s/c may approach I, so that this activation energy disappears. Disclinations are observed to detach themselves from rubbed surfaces near T~. —
Another useful comparison to make is between E8 and the minimum energy of the n = I disclination in a finite planar sample of thickness t. The two sample surfaces are rubbed at right angles to one another. For the surface line, in this case, R = t/ir. The energy of the disclination is a 7mimimum and is Ed when = E it is in the mid. plane of the sample. 0 ln[t!2ire] + E (core). Comparing E8 and Ed: E~ Ed = [2E’0 E(core)] E0 in [2irs’/(et)] For4cm a large
—
—
Finally, if it is large enough. s can be measured directly by examining surface disclinations with polarized light. For smaller values of s, less direct experiments which measure either E 8 or dm may be practical.
Acknowledgements The author is most grateful to the work Alfredthrough P. Sloana Sloan Foundation for their supportThis of his Research research was also supported by the Fellowship. National Science —
Foundation throughof grants GH-33576 GH-3440l and by the Division Engineering andand Applied Physics, Harvard University.
588
DISCUNATION NEAR A RUBBED SURFACE
Vol. 12, No.7
REFERENCES 1. 2.
DE GENNES P.G., Lecture Notes (1970). BERR.EMAN D.W., Phys. Rev. Len. 28, 1683 (1972), and Proc. Fourth mt. Liq. Oyst. Conf (1972).
3.
CREAGH L.T. and KMETZ A.R., Proc. Fourth mt. Liq. Cryst. Conf (1972).
4.
WILLIAMS C., VITEK V. and KLEMAN M., Proc. Fourth mt. Liq. Cryst. Conf (1972).
5.
The divergence terms in the free energy discussed by FRANK F.C., Disc. Faraday Soc. 25, 19 (1958) and NEHRJNG J. and SAUPE A., J. Chem. Phys. 54, 337 (1971), both integrate to zero in this problem. Tables of Sine, Cosine, and Exponential Integrals Vol. II, National Bureau of Standards, 1940, and Tables of the Exponential Integral for Complex Arguments, Applied Math Series 51, National Bureau of Standards, 1958. MEYER R.B. and ARGUIMBAU N., oral presentation at Pont-a-Mousson, 1971; paper in preparation. See also SHEFFER T.J., .Phys. Rev. AS, 1327, (1972).
6. 7.
Dans Ic cas d’une surface frottée, avec une energie anisotrope finie, on calcule Ia structure et l’energie d’une disclination a Ia surface, et d’une disclination coin prés de la surface. On trouve une barrière d’energie pour Ic détachement d’une disclination de la surface, et aussi une distance critique pour l’attraction de la disclmation a la surface. Par delà cette distance, Ia disclination est repoussée de la surface. Ces propriétés doivent fournir une mesure de l’anisotropie de I’energie de Ia surface.