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Wettability of Nematic Liquid Crystals
Journal of Colloid and Interface Science 237, 145–146 (2001) doi:10.1006/jcis.2001.7421, available online at http://www.idealibrary.com on
NOTE Wettability of Nematic Liquid Crystals Organic liquids such as nematic liquid crystals should wet solids with high surface energies like mica, yet they generally do not. In the model proposed here, the affinity to wet the solid in form of Hamaker forces is opposed by elastic effects due to nematic order. Results predict correctly that such liquids show small contact angles and the formation of ultrathin liquid films ahead of the bulk drops. ° C 2001
For the wedge-shaped contact line region shown in Fig. 1, the director has components only in the tangential (z) and the normal (y) directions. When the contact angle of the wedge is small, the variations in the z direction can be ignored in the presence of variations in the y direction. The Oseen–Z¨ocher– Frank equation for the free energy per unit volume (8) becomes F=
1 02 fθ , 2
[2]
Academic Press
Key Words: liquid crystals; nematic; wettability; contact angle.
where f = k11 cos2 θ + k33 sin2 θ ; that is, there is only splay (modulus k11 ) and bend (k33 ), no twist, θ 0 = dθ/dy, and θ is the angle of the director. The hydrostatic equilibrium deals only with the torsion and
Nematic liquid crystals are anisotropic and show special optical properties. Thin films of such optically active materials are of interest in displays and in a growing number of applications (1). At the molecular level they show a medium range order, which in continuum formalism is represented through the director vector n denoting the local orientation. The persistence of order is also being pursued for identification by using thin films, as they amplify various features on the surface (2). As a result there has been some attempt at studying the spreading of drops of liquid crystals on a solid surface (3) as a means of generating these thin films. One observation has been that some nematic liquid crystals do not appear to wet solid surfaces such as those of mica (3). The equilibrium contact angles are small, below 16◦ . This is also observed in the study of thin films between two parallel plates (1), where the contact angle can be related to the excess potential, that is, the disjoining pressure using an overall balance. The lack of wetting is surprising because the values of the surface tensions of organic liquids cannot be so large that they exceed the surface tension of mica, which would be necessary to explain this lack of wetting (4). It is likely that the elastic stresses due to the deformation of the nematic order have an important role, and the object here is to obtain a quantitative description. The shapes of the profiles of the liquid films as the contact line is approached are investigated theoretically and fall into two groups. In the first group, the liquid has a greater affinity (as described by dispersion forces, for instance) for itself than for the solid and forms a wedge-shaped profile (5). In the second group, the wedge thickness never falls to zero but steadies to a constant value as shown in Fig. 1. This case treated by Derjaguin (6) and Frumkin (7) is possible only if there are both an attractive force and a repulsive force between the two surfaces. This is particularly appropriate here because we expect that the dispersion forces are attractive and the elastic forces are repulsive (1). Thus, in the absence of the elastic forces, the liquid would be wetting. The repulsive force, or the increase in potential, is due to two different anchorings at the two surfaces (see Fig. 1) separated by a small distance and hence a strong elastic deformation. The Derjaguin–Frumkin equation is
2 f θ 00 + f θ θ 02 = 0,
λ2 = −
2 γ
[3]
where f θ = ∂ f /∂θ. One integration of Eq. [3] leads to f =
a , θ 02
[4]
where a is a constant of integration. Substituting into Eq. [2], one has F = a/2, a constant, but this constant can be a function of the film thickness h. A second integration of Eq. [4] leads to p p √ ± k11 [E(θh , α) − E(θ0 , α)] = ± k11 1E = ah,
[5]
where E(θ, α) is the elliptic integral of the second kind and the modulus α = (k11 − k33 )/k11 . θh and θ0 are the angles that the director makes at y = h and y = 0, respectively, that is, the anchorings. Substituting a = k11 [1E]2 / h 2 into F and then into the expression for disjoining pressure, one has 5=
AH k11 [1E]2 − , 12π h 3 2h 2
[6]
where the first term on the right-hand side represents the dispersion potential as appropriate for a wetting liquid and the second term is the elastic contribution, which has been derived earlier (9) in about the same form. Note that the excess potential per unit volume is the negative of the disjoining pressure (10). For the particular case where k11 = k33 (they are in general close), 1E = 1θ , where 1θ = θh − θ0 . Since there is no net force in the bulk, there should be no net force in the precursor film, leading to 5 = 0 there and hence from Eq. [6] h0 =
Z∞
AH . 6π k11 (1E)2
[7]
Finally, the Derjaguin–Frumkin equation, Eq. [1], is used to get 5 dh,
[1]
·
h0
where γ is the surface tension of the liquid, h 0 is the thickness of the thin film, and λ is the equilibrium contact angle, which is assumed to be small. 5 is the disjoining pressure.
λ = k11 (1E)2
3π γ AH
¸1 2
.
[8]
Which shows that the contact angle does go up as the anchoring mismatch 1E or 1θ goes up but it is unaffected by the sign of the mismatch. 145
0021-9797/01 $35.00
C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.
146
NOTE feature. However, MBBA [n-(4-methoxybenzylidene)-4-butylaniline] does not. A key feature that is missing is that we do know that the anchoring, which is a thermodynamic phenomenon, also has a transition temperature (1). It is also not known whether the bulk transition temperature comes before the surface transition temperature, etc., and what it means from the point of view of the free energies. In addition, the role of short-range forces such as acid–base forces has not been considered. Nevertheless, the significance of the elastic effects on the contact angles is clear.
REFERENCES
FIG. 1. Coordinates, the local film thickness h, and the dissimilar anchoring at the two interfaces are shown. The bulk liquid is in the form of a wedge. Ahead of the drop is a film of constant thickness h 0 , too small to be observed at this scale. The details of the profile near the junction can be complex.
The extremum in the disjoining pressure occurs at hm =
AH 4πk11 [1E]2
[9]
k11 [1E]2 . 6h 2m
[10]
and is 5min =
Equation [9] was reported earlier by Vandenbrouck et al. (11), who, to quantify thickness of the thin film ahead of the liquid they observed, minimized the energy of a flat film. Since the wedge is not accounted for, they could not calculate the contact angle, and the Derjaguin–Frumkin approach accounts for and provides us with this key quantity. These investigators verify quantitatively that thin films lie ahead of the bulk liquid, one of the few such reports in this area. For k11 = 5 × 10−7 dynes (12), 1E = π/4, A H = 10−13 ergs, and γ = 27 dynes/cm, one has h m = 8.1 × 10−8 cm, 5max = 7.8 × 106 ergs/cm3 , and λ = 0.6◦ . 1E is not equal to 1θ , but it is close. Also, half of the largest value of 1θ has been used. The results indicate not only that there is a lack of wetting but also that the contact angles are low, in keeping with the experimental observations (in fact, in one system they are too small to be measured with a goniometer). At sufficiently high temperatures, a nematic to isotropic transition occurs. The nematic order and the lack of wetting should disappear above this temperature. EBBA [n-(4-ethoxybenzylidene)-4-butylaniline] on mica does show this
1. Yokoyama, H., in “Handbook of Liquid Crystal Research” (Collins, P. J., and J. S., Patel, Eds.), p. 179, Oxford University Press, New York, 1997. 2. Gupta, V. K., Skaife, J. J., Dubrovosky, T. B., and Abbott, N. L., Science 279, 2077 (1998). 3. Lin, C.-M., Neogi, P., and Ybarra, R. M., Chem. Eng. Sci. 55, 37 (2000). 4. Zisman, W. A., Advances in Chemistry Series V. 43, p. 1, American Chemical Society, Washington, D.C., 1964. 5. Miller, C. A., and Ruckenstein, E., J. Colloid Interface Sci. 48, 368 (1974). 6. Derjaguin, B. V., Zh. Fiz. Khim. 14, 137 (1940). 7. Frumkin, A., Zh. Fiz. Khim. 12, 337 (1938). 8. Chandrasekhar, S., in “Liquid Crystals,” Second ed., p. 98, Cambridge University Press, 1992. 9. Perez, E., Proust, J. E., and Ter-Minassian-Saraga, L., in “Thin Liquid Films” (Ivanov, I. B., Ed.), p. 891, Marcel Dekker, New York, 1988. 10. Dyzlonshinskii, I. E., Lifshitz, E. M., and Pitaeveskii, L. P., Soviet Phys. JETP 10, 161 (1960). 11. Vandenbrouck, F., Valignat, M. P., and Cazabat, A. M., Phys. Rev. Letts. 82, 2693 (1999). 12. de Gennes, P. G., and Prost, J., “The Physics of Liquid Crystals,” Second ed., p. 105, Clarendon Press, 1993. P. Neogi1 Xianzhong Zhang Department of Chemical Engineering University of Missouri-Rolla Rolla, Missouri 65409-1230 Received September 1, 2000; accepted December 29, 2000
1 To
whom correspondence should be addressed.
NOTE Wettability of Nematic Liquid Crystals Organic liquids such as nematic liquid crystals should wet solids with high surface energies like mica, yet they generally do not. In the model proposed here, the affinity to wet the solid in form of Hamaker forces is opposed by elastic effects due to nematic order. Results predict correctly that such liquids show small contact angles and the formation of ultrathin liquid films ahead of the bulk drops. ° C 2001
For the wedge-shaped contact line region shown in Fig. 1, the director has components only in the tangential (z) and the normal (y) directions. When the contact angle of the wedge is small, the variations in the z direction can be ignored in the presence of variations in the y direction. The Oseen–Z¨ocher– Frank equation for the free energy per unit volume (8) becomes F=
1 02 fθ , 2
[2]
Academic Press
Key Words: liquid crystals; nematic; wettability; contact angle.
where f = k11 cos2 θ + k33 sin2 θ ; that is, there is only splay (modulus k11 ) and bend (k33 ), no twist, θ 0 = dθ/dy, and θ is the angle of the director. The hydrostatic equilibrium deals only with the torsion and
Nematic liquid crystals are anisotropic and show special optical properties. Thin films of such optically active materials are of interest in displays and in a growing number of applications (1). At the molecular level they show a medium range order, which in continuum formalism is represented through the director vector n denoting the local orientation. The persistence of order is also being pursued for identification by using thin films, as they amplify various features on the surface (2). As a result there has been some attempt at studying the spreading of drops of liquid crystals on a solid surface (3) as a means of generating these thin films. One observation has been that some nematic liquid crystals do not appear to wet solid surfaces such as those of mica (3). The equilibrium contact angles are small, below 16◦ . This is also observed in the study of thin films between two parallel plates (1), where the contact angle can be related to the excess potential, that is, the disjoining pressure using an overall balance. The lack of wetting is surprising because the values of the surface tensions of organic liquids cannot be so large that they exceed the surface tension of mica, which would be necessary to explain this lack of wetting (4). It is likely that the elastic stresses due to the deformation of the nematic order have an important role, and the object here is to obtain a quantitative description. The shapes of the profiles of the liquid films as the contact line is approached are investigated theoretically and fall into two groups. In the first group, the liquid has a greater affinity (as described by dispersion forces, for instance) for itself than for the solid and forms a wedge-shaped profile (5). In the second group, the wedge thickness never falls to zero but steadies to a constant value as shown in Fig. 1. This case treated by Derjaguin (6) and Frumkin (7) is possible only if there are both an attractive force and a repulsive force between the two surfaces. This is particularly appropriate here because we expect that the dispersion forces are attractive and the elastic forces are repulsive (1). Thus, in the absence of the elastic forces, the liquid would be wetting. The repulsive force, or the increase in potential, is due to two different anchorings at the two surfaces (see Fig. 1) separated by a small distance and hence a strong elastic deformation. The Derjaguin–Frumkin equation is
2 f θ 00 + f θ θ 02 = 0,
λ2 = −
2 γ
[3]
where f θ = ∂ f /∂θ. One integration of Eq. [3] leads to f =
a , θ 02
[4]
where a is a constant of integration. Substituting into Eq. [2], one has F = a/2, a constant, but this constant can be a function of the film thickness h. A second integration of Eq. [4] leads to p p √ ± k11 [E(θh , α) − E(θ0 , α)] = ± k11 1E = ah,
[5]
where E(θ, α) is the elliptic integral of the second kind and the modulus α = (k11 − k33 )/k11 . θh and θ0 are the angles that the director makes at y = h and y = 0, respectively, that is, the anchorings. Substituting a = k11 [1E]2 / h 2 into F and then into the expression for disjoining pressure, one has 5=
AH k11 [1E]2 − , 12π h 3 2h 2
[6]
where the first term on the right-hand side represents the dispersion potential as appropriate for a wetting liquid and the second term is the elastic contribution, which has been derived earlier (9) in about the same form. Note that the excess potential per unit volume is the negative of the disjoining pressure (10). For the particular case where k11 = k33 (they are in general close), 1E = 1θ , where 1θ = θh − θ0 . Since there is no net force in the bulk, there should be no net force in the precursor film, leading to 5 = 0 there and hence from Eq. [6] h0 =
Z∞
AH . 6π k11 (1E)2
[7]
Finally, the Derjaguin–Frumkin equation, Eq. [1], is used to get 5 dh,
[1]
·
h0
where γ is the surface tension of the liquid, h 0 is the thickness of the thin film, and λ is the equilibrium contact angle, which is assumed to be small. 5 is the disjoining pressure.
λ = k11 (1E)2
3π γ AH
¸1 2
.
[8]
Which shows that the contact angle does go up as the anchoring mismatch 1E or 1θ goes up but it is unaffected by the sign of the mismatch. 145
0021-9797/01 $35.00
C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.
146
NOTE feature. However, MBBA [n-(4-methoxybenzylidene)-4-butylaniline] does not. A key feature that is missing is that we do know that the anchoring, which is a thermodynamic phenomenon, also has a transition temperature (1). It is also not known whether the bulk transition temperature comes before the surface transition temperature, etc., and what it means from the point of view of the free energies. In addition, the role of short-range forces such as acid–base forces has not been considered. Nevertheless, the significance of the elastic effects on the contact angles is clear.
REFERENCES
FIG. 1. Coordinates, the local film thickness h, and the dissimilar anchoring at the two interfaces are shown. The bulk liquid is in the form of a wedge. Ahead of the drop is a film of constant thickness h 0 , too small to be observed at this scale. The details of the profile near the junction can be complex.
The extremum in the disjoining pressure occurs at hm =
AH 4πk11 [1E]2
[9]
k11 [1E]2 . 6h 2m
[10]
and is 5min =
Equation [9] was reported earlier by Vandenbrouck et al. (11), who, to quantify thickness of the thin film ahead of the liquid they observed, minimized the energy of a flat film. Since the wedge is not accounted for, they could not calculate the contact angle, and the Derjaguin–Frumkin approach accounts for and provides us with this key quantity. These investigators verify quantitatively that thin films lie ahead of the bulk liquid, one of the few such reports in this area. For k11 = 5 × 10−7 dynes (12), 1E = π/4, A H = 10−13 ergs, and γ = 27 dynes/cm, one has h m = 8.1 × 10−8 cm, 5max = 7.8 × 106 ergs/cm3 , and λ = 0.6◦ . 1E is not equal to 1θ , but it is close. Also, half of the largest value of 1θ has been used. The results indicate not only that there is a lack of wetting but also that the contact angles are low, in keeping with the experimental observations (in fact, in one system they are too small to be measured with a goniometer). At sufficiently high temperatures, a nematic to isotropic transition occurs. The nematic order and the lack of wetting should disappear above this temperature. EBBA [n-(4-ethoxybenzylidene)-4-butylaniline] on mica does show this
1. Yokoyama, H., in “Handbook of Liquid Crystal Research” (Collins, P. J., and J. S., Patel, Eds.), p. 179, Oxford University Press, New York, 1997. 2. Gupta, V. K., Skaife, J. J., Dubrovosky, T. B., and Abbott, N. L., Science 279, 2077 (1998). 3. Lin, C.-M., Neogi, P., and Ybarra, R. M., Chem. Eng. Sci. 55, 37 (2000). 4. Zisman, W. A., Advances in Chemistry Series V. 43, p. 1, American Chemical Society, Washington, D.C., 1964. 5. Miller, C. A., and Ruckenstein, E., J. Colloid Interface Sci. 48, 368 (1974). 6. Derjaguin, B. V., Zh. Fiz. Khim. 14, 137 (1940). 7. Frumkin, A., Zh. Fiz. Khim. 12, 337 (1938). 8. Chandrasekhar, S., in “Liquid Crystals,” Second ed., p. 98, Cambridge University Press, 1992. 9. Perez, E., Proust, J. E., and Ter-Minassian-Saraga, L., in “Thin Liquid Films” (Ivanov, I. B., Ed.), p. 891, Marcel Dekker, New York, 1988. 10. Dyzlonshinskii, I. E., Lifshitz, E. M., and Pitaeveskii, L. P., Soviet Phys. JETP 10, 161 (1960). 11. Vandenbrouck, F., Valignat, M. P., and Cazabat, A. M., Phys. Rev. Letts. 82, 2693 (1999). 12. de Gennes, P. G., and Prost, J., “The Physics of Liquid Crystals,” Second ed., p. 105, Clarendon Press, 1993. P. Neogi1 Xianzhong Zhang Department of Chemical Engineering University of Missouri-Rolla Rolla, Missouri 65409-1230 Received September 1, 2000; accepted December 29, 2000
1 To
whom correspondence should be addressed.