A Model of Capillary Rise of Nematic Liquid Crystals - Langmuir (ACS

Langmuir 2003, 19, 3677-3685

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A Model of Capillary Rise of Nematic Liquid Crystals Alejandro D. Rey* Department of Chemical Engineering, McGill University, 3610 University Street, Montreal, Quebec, Canada H3A 2B2 Received August 29, 2002. In Final Form: December 4, 2002 A general model for the capillary rise for uniaxial nematic liquid crystals has been derived using fundamental principles and classical liquid crystal physics and partially validated using existing experimental data. A rigorous formulation of the contributions of surface and bulk nematic elasticity was implemented. The surface contribution is a function of the surface anchoring strength at the liquid crystalcapillary wall. The exact bulk elasticity contribution is a function of the director field in the meniscus. The specific form of the capillary rise equation for four typical nematic textures was developed and analyzed. It is found that capillary rise depends on the presence of bulk disclinations and on the orientation field close to the contact line. It is found that orientation gradients at the contact line are the most significant nematic contribution to capillary rise. The model explains unusual features in experimental capillary rise measurements, including why parallel nematic orientation at the capillary wall exhibits a higher capillary rise than orthogonal orientation.

Introduction The capillary rise method is a standard and popular procedure to measure surface tension of liquids.1 In this method the height h of a liquid column in a capillary above a reference level in a large reservoir is measured. For isotropic liquids the height of the liquid is

h)

γL cos θ ∆FgA

(1)

where A is the cross section, L is the wetted perimeter, γ is the liquid surface tension, θ is the static contact angle, and ∆F is the density difference between the liquid and the ambient gas. Depending on the contact angle, the height can take any value in the interval -L/∆FgA < h < +L/∆FgA. The capillary rise method is one of the most accurate means to measure surface tension. Details of the method can be found in the literature.1 The only material property that enters in the equation used to measure surface tension using the capillary rise method is the surface tension itself. For soft matter, such as complex fluids, the equation is inapplicable because longand short-range bulk elastic modes are not included in eq 1. This paper presents an analysis of capillary rise for a typical single-component single-phase complex fluid, a low molar mass nematic liquid crystal,2 in the absence of transitional phenomena. Nematic liquid crystals are anisotropic viscoelastic liquid crystals,2 whose orientational order is described by a unit vector n, known as the director. Spatial director gradients (∇n * 0) store longrange elastic energy, known as Frank elasticity.2 Since nematic liquid crystals are orientationally ordered materials, they can exhibit defects; linear defects are known as disclinations2 and arise due to incompatibilities due to geometry or orienting fields. In the capillary rise of nematic liquid crystals, defects arise due to the presence on orienting surfaces. The surface physics of liquid crystals * E-mail: [email protected]. Tel: (514) 398-4196. Fax: (514) 398-6678. (1) Hiemenz, P. C. Principles of Colloid and Surface Chemistry, 2nd ed.; Marcel Dekker: New York, 1986. (2) de Gennes, P. G.; Prost, J. The Physics of Liquid Crystals; Clarendon Press: Oxford, 1993.

has been recently reviewed.3-5 Surface tension measurements and theories for low molar mass thermotropic nematic liquid crystals using the Wilhelmy method are available.6-8 The importance, magnitude, and mathematical description of the nematic ordering contributions have not been discussed in the literature but are certainly critical given the central importance of the capillary rise method in surface science and the need of interpreting experimental surface tension data of a nematic liquid crystal (NLC).9 A unique model equation of the capillary rise method for NLC such as eq 1 does not exist because the measurements involve the selection of one out of several bulk distortion modes, depending among other things on the NLC-capillary surface properties. Thus, distortion mode selection is at the core of the problem. The orientation at the free surface of NLC can also be tangential, planar, or tilted, and thus the magnitude and even sign of the orientation-dependent part of the surface tension, known as anchoring energy,3-5 will depend on the nature of the NLC in question. The approach taken in this work is to derive the governing balance equation for the capillary rise and then apply it to a number of realistic particular cases. No attempts at modeling the capillary rise method are currently available, but they are certainly necessary to interpret and use existing experimental data.9 Capillary rise measurements of several low molar mass rodlike nematic liquid crystals have been performed, using several carefully controlled physicochemical treatments of the capillary walls containing the liquid crystals.9 The physicochemical treatments produce fixed orientation of the NLC, known as strong anchoring condition. The carefully conducted experiments show that when the (3) Je´roˆme, B. Rep. Prog. Phys. 1991, 54, 391. (4) Sonin, A. A. The Surface Physics of Liquid Crystals; Gordon and Breach: Amsterdam, 1995. (5) Yokoyama, H. In Handbook of Liquid Crystal Research; Collins, P. J., Patel, J. S., Eds.; Oxford University Press: New York, 1997; Chapter 6, p 179. (6) Gannon, M. G. J.; Faber, T. E. Philos. Mag. 1978, 37, 117. (7) Chandrasekhar, S. Liquid Crystals, 2nd ed.; Cambridge University Press: New York, 1992. (8) Rey, A. D. Langmuir 2000, 16, 845. (9) Tsvetkov, V. A.; Tsvetkov; O. V.; Balandin; V. A. Mol. Cryst. Liq. Cryst. 1999, 329, 305.

10.1021/la020750h CCC: $25.00 © 2003 American Chemical Society Published on Web 03/27/2003

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Figure 1. Schematic of the capillary rise surface tension measurement method and effect of nematic orientation. For low molar mass thermotropic nematic liquid crystals, it is experimentally found that the capillary rise height when the orientation is parallel to the capillary wall (h||) is significantly higher than when it is perpendicular (h⊥).

director orientation at the capillary bounding surface was parallel, the capillary rise h| was significantly higher than when it was orthogonal h⊥ (parallel orientation is known as planar, and orthogonal orientation is known as homeotropic in the liquid crystal literature).9 A schematic of the capillary rise experimental setup and nematic orientation is given in Figure 1. For example, using a capillary of thickness 100 µm, the highest capillary rise of a typical NLC, MBBA (4-methoxybenzylidene-4′-butylaniline), for planar wall orientation was 75 mm, while the lowest for homeotropic wall orientation was 65 mm. The unusual finding (i.e., h|| > h⊥) was confirmed using different surface treatment methods and different NLCs. The experimental results are thus inconsistent with eq 1, since no information on orientation boundary conditions can be included in any of the quantities appearing there. The objectives of this paper are (1) to present a general model equation for the capillary rise surface tension method for uniaxial nematic liquid crystals at static, isothermal conditions, (2) to characterize the nematic contributions to the measured capillary rise for the most likely nematic textures, (3) to identify the material systems, experimental conditions, and geometric factors that enhance the impact of nematic ordering on the capillary rise surface tension measurements, and (4) to explain the experimental findings9 on why planar wall orientation results in a higher capillary rise than homeotropic orientation. Nonequilibrium phenomena are beyond the scope of this work. Model Equations In this section we present the derivation of the general balance equation for the capillary rise measurement method for a uniaxial nematic liquid crystal. The geometry and schematic of the capillary rise method are given in Figures 1 and 2. The coordinate system is rectangular {x,y,z}, with z along the vertical direction. The respective unit vectors are {δˆ x, δˆ y, δˆ z}. The meniscus of height h forms in a rectangular capillary of width Lx and Ly. The static contact angle forms at the intersection of the solid surface (S), the nematic liquid crystal free surface (NLC), and the vapor (V), and is denoted by θ. The intersection of these three interphases is the contact (triple) line, Ctl. The lateral bounding surface area of the rectangular capillary is Sω.

Rey

Figure 2. Schematic of the capillary rise geometry. The coordinate system is rectangular {x,y,z}, with z along the vertical direction. The respective unit vectors are {δˆ x, δˆ y, δˆ z}. The meniscus of height h forms in a rectangular capillary of width Lx and Ly. The static contact angle forms at the intersection of the solid surface (S), the nematic liquid crystal free surface (N), and the vapor (V) and is denoted by θ. The intersection of these three interphases is the contact (triple) line, Ctl. The lateral bounding surface area of the rectangular capillary is Sω, and the length of triple line is Ctl. The unit normal to Sω is η, and the unit normal to Ctl is µ.

The cross sectional area is A ) LxLy, and the wetted perimeter is P ) 2(Lx + Ly). After the meniscus formation, the static integral balance of forces along the vertical direction δˆ z is

where g is the acceleration of gravity, η is the unit normal to Sω, tb is the 3 × 3 bulk elastic stress tensor in the NLC phase, µ is the unit normal to Ctl and tangent to the nematic-vapor (N/V) interface, and tsNA is the 2 × 3 elastic surface stress tensor of the N/V surface at the contact line. Equation 2 is the bulk, surface, and lineal force balance acting on the system and governs the capillary rise h. For simple fluids10 the bulk elastic stress contribution tb is zero, but for NLCs, bulk elastic stresses arise due to orientation gradients. For simple fluids the elastic surface stress contribution ts is isotropic, but for NLCs, elastic surface stresses depend on surface director orientation.11-14 In eq 2 and everywhere below we neglect the density of the vapor phase. The static force balance equation that determines the contact angle θ is given by the junction sum of surface stress vectors and the junction integral of the nematic bulk stress vector

ν‚Ts + Ijun (k‚tb) dl ) 0 ∑ jun

(3)

where the junction sum of surface stress vectors is defined by

ν‚Ts ) νNV‚tsNV + νVS‚tsVS + νSN‚tsSN|C ∑ jun

tl

(4)

and the junction integral of the NLC bulk stress vector is defined by

Ijun k‚tb dl ) lim δfrc

∫D

δ

k‚tb dl

(5)

(10) Slattery, J. C. Interfacial Transport Phenomena; SpringerVerlag: New York, 1990.

Capillary Rise of Nematic Liquid Crystals

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Figure 4. Schematic of the geometry and integration path used in eq 5. In this paper the contact line Ctl is also a nematic defect since it is assumed that the director n has a fixed and constant orientation at the N/V and S/N interfaces. Since these two surfaces are nonparallel, a defect arises at the contact line. To compute the effect of long-range elastic energy on the defect, we evaluate the limit of the integral of the stress vector along an arc of radius δ toward rc, where rc is the defect core size. The core size has molecular dimensions and is not shown. The unit vector k is normal to the arc.

Figure 3. Schematic of interfaces, unit tangent vectors, and unit normal vectors in the vicinity of the contact line Ctl. (a) The contact (triple) line Ctl forms at the intersection of three interfaces: vapor-solid interface (V/S), solid-nematic interface (S/N), and vapor-solid interface (V/S). (b) Tangent vectors pointing away from Ctl: νVS along the vapor-solid interface (V/S), νNV along the nematic-vapor interface (N/V), and νSN along the solid-nematic interface (S/N). (c) Anticlockwise orthogonal set of unit normals: kVS to the vapor-solid interface (V/S), kNV to the nematic-vapor interface (N/V), and tSN to the solid-nematic interface (S/N).

Here k is the outward unit normal to the arc Dδ that spans the nematic phase and rc is a cutoff radius. Figure 3 shows a schematic of all the unit vectors appearing in eq 4 and defines the nomenclature as follows: (a) unit tangent vectors away from Ctl, νVS is the tangent along the vaporsolid interface (V/S), νNV is the tangent along the nematicvapor interface (N/V), νSN is the tangent along the solidnematic interface (S/N); (b) anticlockwise unit normals, kVS is the normal to the vapor-solid interface (V/S), kNV is the normal to the nematic-vapor interface (N/V), and tSN is the normal to the solid-nematic interface (S/N). Figure 4 shows a schematic of the geometry and integration path used in eq 5. In this paper the contact line Ctl is also a nematic defect since it is assumed that the director n has a fixed and constant orientation at the N/V and S/N interfaces. Since these two surfaces are nonparallel, a defect arises at the contact line. (Examples of free-surface defects are discussed in the classical liquid crystal textbook.2) The arc of radius δ is shrunk toward rc, where rc is the defect core size, which is not shown in the figure since it has molecular dimensions. Equation 3 is the force balance equation acting at each point on the triple line Ctl. For isotropic fluids no long-range energy appears in eq 3.10 To complete the model equations, constitutive equations for the bulk stress tensor tb in the nematic phase, and (11) Cheong A. G.; Rey, A. D. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2002, 66, 021704. (12) Rey, A. D.; Denn, M. M. Annu. Rev. Fluid Mech. 2002, 34, 233. (13) Rey, A. D. J. Chem. Phys. 2000, 113, 10820. (14) Rey, A. D. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2000, 61, 1540.

surface stress tensors (tsNV, tsVS, tsSN) for the three interphases are required. The bulk elastic stress tensor in the NLC phase is given by2

tb ) fbI -

∂fb ‚(∇n)T ∂∇n

(6)

where I is the unit tensor and fb is the bulk (long range) Frank elastic energy density. Assuming that the system is elastically isotropic, the expression for the Frank elastic energy density fb simplifies to2

fb(∇n) )

K ∇n:(∇n)T 2

(7)

and the bulk elastic stress tensor tb becomes

tb ) - fbI -

[

)]

(

∂fb ∂fb ‚(∇n)T :(∇n)T I ∂∇n ∂∇n

(8)

The elastic surface stress tensor tsVS for the V/S interface is11-14

tsVS ) γVSIsVS

(9)

IsVS ) I - kVSkVS where γVS is the surface tension and IsVS is the surface unit tensor for the V/S surface. The dimension of the tsVS tensor is 2 × 2, and it represents tension (normal) stresses. The elastic surface stress tensor tsNV for the N/V interface is given by11-15

(

)

∂γNV tsNV ) γNVIsNV - IsNV‚ NV kNV ∂k γ

NV

)

γisoNV

(10)

γanNV NV NV 2 (n ‚k ) + 2

where γisoNV is the isotropic surface tension and γanNV is the anchoring energy.3-5,15 For NLC interfaces, if γan > 0, the (15) Rapini, A.; Papoular, M. J. Phys., Colloq. 1969, 30, C4 54.

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director tends to orient parallel to the surface, and if γan < 0, the director tends to orient normal to the surface. The term γanNV(nNV‚kNV)2/2 is the anisotropic surface tension and is a function of the director n orientation with the respect to the unit normal k to the N/V interface. The dimension of the elastic surface stress tsNV is 2 × 3, and it contains normal tnNV and bending tbNV contributions

the bounding surfaces (S/N and N/V) are nonparallel. Thus the capillary rise and static contact angle depend on the director field as follows

tnNV ) γNVIsNV

where nSN,nNV denote the director orientation at the S/N and N/V bounding surfaces. The director gradients ∇n close to the N/V surface and the contact line also play an important role. In the orientation gradient regime the model equations (2-13) become

(

(11a)

)

∂γNV tbNV ) - IsNV‚ NV kNV ∂k

(11b)

In liquid crystalline surfaces, the normal stresses are associated with changes in surface area while bending stresses are associated with changes in surface orientation.11-14 The elastic surface stress tensor tsSN for the S/N interface is given by

(

)

∂γSN tsSN ) γSNIsSN - IsSN‚ SN kSN ∂k γSN ) γisoSN +

γanSN SN SN 2 (n ‚k ) 2

(12a)

h ) h(nSN,nNV,∇n)

(15a)

θ ) θ(nSN,nNV,∇n)

(15b)

FgAh - δˆ z‚

∫S

w

η‚tb dA - γNV L cos θ ) 0 (16a)

γNV cos θ - δˆ z‚Ijun (k‚tb) dl + γSN - γSV ) 0 (16b) The capillary rise h of a nematic liquid crystal is thus given by the sum of four distinct contributions

(12b)

where γisoSN is the isotropic surface tension and γanSN is the anchoring energy. The term γanSN(nSN‚kSN)2/2 is the anisotropic surface tension and is a function of the director n orientation with the respect to the unit normal k to the S/N interface. The elastic surface stress tsSN is a 2 × 3 tensor that contains normal tnSN and bending tbSN contributions

tnSN ) γSNIsSN

(

(13a)

)

∂γSN tbSN ) - IsSN‚ SN kSN ∂k

(13b)

The model for capillary rise h and contact angle θ of a NLC in a rectangular capillary is given by eqs 2-13. The length scales of the model are the external characteristic length scale of the capillary size Le and the internal length scale Li defined by

Li )

γanSN

2K + γanNV

(14)

Comparing these two length scales, two regimes arise: (a) Le . Li, gradient orientation regime with a spatially inhomogeneous director field, (b) Le ≈ Li, homogeneous orientation regime with constant director field. The former describes a system with fixed director orientation at the S/N and N/V interfaces, in which long-range elastic energy is less costly than anchoring energy, while the latter represents a system in which anchoring energy is less costly than long-range elasticity. Real systems will be well approximated by the former case, since the length scales over which director gradients are likely to occur are large enough and hence the cost of long-range elasticity will be sufficiently low. The Four Modes in the Gradient Orientation Regime In the gradient orientation regime the director field is fixed at the bounding surfaces. This condition is known as the strong anchoring condition.2 Close to the N/V surface of the capillary, the director field is space dependent since

and is a function of the director gradients close to the N/V surface (first term), the director gradients at the contact line (second term), the anisotropic interfacial tension at the S/N interface (third term), and the classical isotropic interfacial tension term (fourth term). Performing an order of magnitude analysis on h (eq 17) gives

( ) ()

FgA K K h≈O +O + O(γanSN) + O(γSV - γisoSN) L Ly rc (18) where O denotes the order of magnitude of the enumerated terms presented in eq 17. Typical values of thermotropic rodlike nomadic liquid crystals2-5 are K ) 10-7erg/cm, γanSN ) 1-10 - 2erg‚cm-2, γisoSN ≈ γSV ) 10-30 erg‚cm-2, rc ≈ 10 nm, while a typical value for the length scale used in the experiments9 is Ly ) 100 µm. Hence we can expect that the main nematic contribution comes from the longrange elastic gradient at the contact line. Furthermore this effect is of the same order of magnitude as the isotropic surface tension. More precise predictions require more detailed calculations of the orientation field close to the meniscus. Next we identify and compute the capillary rise for four characteristic cases. (a) Planar-Planar Director Surface Orientations. Figure 5a shows a schematic of the director field for the planar-planar mode. The director field n is parallel to both interfaces

n ) δˆ z on S/N; n ) lNV on N/V

(19)


Langmuir, Vol. 19, No. 9, 2003 3681

In the present geometry, the order of magnitude of the y and z director gradients are

2 Ly

(21a)

1 2 cos θ||,|| ) lz Ly 1 - sin θ||,||

(21b)

∇yno ≈ ∇zno ≈

The scale in the z direction is taken to be equal to height of the meniscus. The nematic surface energy densities and long-range force for this case are

δˆ z‚

∫S

w

η‚tb dA )

∫S

γ||,||NV ) γisoNV

(22a)

γ||,||SN ) γisoSN

(22b)

w

- K(∇yno∇zno) (-2) dx dz ) 4C||,||K

Lx (22c) Ly

where C||,|| is a factor of order one. Replacing eqs 22 into the integral force balance (2) gives

h||,|| )

4KC||,||

+

FgLy2

γisoNVL cos θ||,|| FgA

(23)

showing that long-range forces contribute to the capillary rise. The contact angle computed from eq 3 gives

γ||,||NV cos θ||,|| - δˆ z‚Ijun(k‚tb) dl + γ||,||SN - γSV ) 0 (24) Figure 5. (a) Schematic of the director field for the planarplanar mode (||,||). In this mode the director is parallel to the S/N and V/S interfaces. Since the S/N and V/S interfaces are nonparallel, a defect nucleates at (Ly/2, lz), where lz is measured from the N/V interface. In addition, the contact line is itself a defect since θ||,|| is not zero. (b) Schematic of the director field for the homeotropic-homeotropic mode (||,⊥). For this director field, no defect nucleates in the bulk of the NLC, because the effect of the nonparallel N/V and S/N surfaces is compatibilized by director gradients. The director field n is parallel to the S/N interface and normal to the N/V interface. (c) Schematic of the director field for the homeotropic-homeotropic mode (⊥,⊥). In this mode the director is normal to the S/N and V/S interfaces. Since the S/N and V/S interfaces are nonparallel, a defect nucleates at (Ly/2, lz), where lz is measured from the N/V interface. In addition, the contact line is itself a defect since θ||,|| is not zero. (d) Schematic of the director field for the homeotropic-planar mode (⊥,||). For this director field, no defect nucleates in the bulk of the NLC, because the effect of the nonparallel N/V and S/N surfaces is compatibilized by director gradients. The director field n is normal to the S/N interface and parallel to the N/V interface.

where lNV is the unit tangent to the N/V interface. Since the S/N and N/V interfaces are nonparallel, a defect nucleates at (Ly/2, lz), where lz is measured from the N/V interface. In addition, the contact line is itself a defect since θ||,|| is not zero. Let the capillary rise and contact angle be denoted by

h||,|| ) h(δz;lNV;∇n) θ||,|| ) θ(δz;l

NV

;∇n)

To compute the long-range contribution (second term above), we need the expression for the bulk long-range elastic energy fb and the director field next to the contact line. Using a local polar coordinate system (φ,r) centered at the contact line, assuming a planar director field (nx ) 0), the director angle ψ, measured with respect to the y axis, that satisfies the Laplace equation 2

∇

ψ)0

(25)

and the boundary conditions for the planar-planar case

ϑ ) π/2, ψ ) π/2; ϑ ) 3π/2 - θ||,||, ψ ) π/2 - θ||,|| (26) is

ψ)

θ||,|| θ||,|| π π + ϑ 2 2 (π - θ||,||) (π - θ||,||)

(27)

The director field close to the contact line is purely radial (splay2 distortion) and at a distance rc from the center of the contact line the bulk range elastic energy density is

fb||,|| )

K ∂ψ 2 K ) E 2 2 ∂ϑ 2 ||,|| 2rc 2rc

( )

(28a)

(20a) (20b)

E||,|| )

θ||,|| (π - θ||,||)

(28b)

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where E||,|| is the director gradient coefficient. The longrange contribution to the contact angle becomes

where C||,⊥ is a factor of order one. Replacing eqs 36 and37 into the integral force balance (2) gives

δˆ z‚Ijun(k‚tb) dl ) Ijun δˆ zk:tb dl ) -Ijun δˆ zk:I dl ) K -Ijun δˆ zk:I dl ) E 2 sin θ||,|| (29) 2rc ||,|| and the contact angle satisfies

γ||,||NV cos θ||,|| -

K E 2 sin θ||,|| + γ||,||SN - γSV ) 0 2rc ||,|| (30)

showing that long range decreases the contact angle. Thus the capillary rise h||,|| is

h||,|| )

(

)

K 4C||,|| L L (-γisoSN + + E 2 sin θ||,|| + Fg L 2 2Arc ||,|| FgA y γSV) (31)

Equations 30 and 31 show that for this case long-range and isotropic nematic interfacial tension contribute to the capillary rise and contact angle

h||,||(K,γisoSN,γisoNV)

(32a)

θ||,|| ) θ||,||(K,γisoSN,γisoNV)

(32b)

h||,|| )

n ) δz on S/N, n ) k

on N/V

h||,⊥ ) h(δz;kNV;∇n)

(34a)

θ||,⊥ ) θ(δz;kNV;∇n)

(34b)

The order of magnitude of the y and z director gradients are

1 Ly

(35b)

The surface energy densities and long-range force for this case are

γ||,⊥

)

γisoNV

SN

γ||,⊥ δˆ z‚

∫S η‚tb dA ) ∫S w

w

)

γanNV + 2

(36a)

γisoSN

(36b)

- K(∇yno∇zno) (-2) dx dz ) Lx (37) 2C||,⊥K Ly

γisoNVL cos θ||,⊥ γanNVL cos θ||,⊥ + FgA 2FgA (38)

showing that long-range forces and anisotropic surface tension contribute to the capillary rise. The contact angle computed from eq 2 gives

γ||,⊥NV cos θ||,⊥ - δˆ z‚Ijun(k‚tb) dl + γ||,⊥SN - γSV ) 0 (39) To compute the long-range contribution (second term above), we follow the same process as above (see eqs 2529). The director angle ψ, measured with respect to the y axis, that satisfies the Laplace equation and the boundary conditions for the planar-planar case

ϑ ) π/2, θ ) π/2; ϑ ) 3π/2 - θ||,⊥, θ ) π - θ||,⊥ (40) is

(

) (

)

π/2 - θ||,⊥ π π π/2 - θ||,⊥ + ϑ 2 2 π - θ||,⊥ π - θ||,⊥

(41)

The director field close to the contact line has bend-splay2 distortions, and at a distance rc from the center of the contact line the bulk range elastic energy density is

fb,||,⊥ )

K ∂ψ 2 K ) E 2 2 ∂ϑ 2 ||,⊥ 2rc 2rc

E||,⊥ )

( )

(

)

π/2 - θ||,⊥ π - θ||,⊥

(42a)

(42b)

and the long-range contribution to the contact angle becomes

δˆ z‚Ijun(k‚tb) dl ) -Ijunfb δˆ zk:I dl )

K E 2 sin θ||,⊥ 2rc ||,⊥ (43)

and the contact angle satisfies

γ||,⊥NV cos θ||,⊥ -

(35a)

2 cos θ||,⊥ 1 ∇zno ≈ ) lz Ly 1 - sin θ||,⊥

NV

FgLy2

+

(33)

Let the capillary rise and contact angle be denoted by

∇yno ≈

2KC||,⊥

ψ)

(b) Planar-Homeotropic (||,⊥) Director Surface Orientations. Figure 5b shows an schematic of the director field for the planar-homeotropic mode. For this director field no defect nucleates in the bulk of the NLC, because the effect of the nonparallel N/V and S/N surfaces is compatibilized by director gradients. The director field n is parallel to the S/N interface and normal to the N/V interface NV

h||,⊥ )

K E 2 sin θ||,⊥ + γ||,⊥SN - γSV ) 0 2rc ||,⊥ (44)

showing that long-range effect increases the contact angle due to the fact that long-range energy decreases with larger contact angles. Thus the capillary rise h|,⊥ is

h||,⊥ )

(

)

K 2C||,⊥ L L (-γisoSN + + E 2 sin θ||,⊥ + Fg L 2 2Arc ||,⊥ FgA y γSV) (45)

Equations 44 and 45 show that for this case long-range and isotropic and anisotropic nematic interfacial tensions contribute to the capillary rise and contact angle

h||,⊥ ) h||,⊥(K,γisoSN,γisoNV,γanNV)

(46a)

θ||,⊥ ) θ||,⊥ (K,γisoSN,γisoNV,γanNV)

(46b)


Langmuir, Vol. 19, No. 9, 2003 3683

(c) Homeotropic-Homeotropic (⊥,⊥) Director Surface Orientations. This mode is analogous to the planarplanar mode. Figure 5c shows a schematic of the director field for the homeotropic-homeotropic mode. For this director field a defect nucleates in the bulk of the NLC, because of the effect of the nonparallel N/V and S/N. The director field n is parallel to both interfaces

n ) kSN on S/N, n ) kNV on N/V

h⊥,⊥ ) h(kSN;kNV;∇n)

(48a)

θ⊥,⊥ ) θ(kSN;kNV;∇n)

(48b)

The order of magnitude of the y and z director gradients are given by eqs 21. The surface energy densities and long-range force for this case are

γ⊥,⊥NV ) γisoNV +

γanNV 2

(50a)

γ⊥,⊥SN ) γisoSN +

γanSN 2

(50b)

∫S η‚tb dA ) ∫S w

w

- K(∇yno∇zno) (-2) dx dz ) 4C⊥,⊥K

Lx (51) Ly

where C⊥,⊥ is a factor of order one. Replacing eqs 50 and 51 into the integral force balance (2) gives

h⊥,⊥ )

4KC⊥,⊥ FgLy2

+

fb,⊥,⊥ )

γisoNVL cos θ⊥,⊥ γanNVL cos θ⊥,⊥ + FgA 2FgA (52)

γ⊥,⊥NV cos θ⊥,⊥ - δˆ z‚Ijun(k‚tb) dl + γ⊥,⊥SN - γSV ) 0 (53) To compute the long-range contribution (second term above), we follow the same process as above (see eqs 2529). The director angle ψ, measured with respect to the y axis, that satisfies the Laplace equation and the boundary conditions for the homeotropic-homeotropic case

ϑ ) π/2, ψ ) 0

(54a)

ϑ ) 3π/2 - θ⊥,⊥, ψ ) -θ⊥,⊥

(54b)

(56a)

-θ⊥,⊥ π - θ⊥,⊥

(56b)

( )


δˆ z‚Ijun(k‚tb) dl ) - Ijun(k‚tb)δˆ zk:I dl ) K E 2 sin θ⊥,⊥ (57) 2rc ⊥,⊥ and the contact angle satisfies

γ⊥,⊥NV cos θ||,|| -

K E 2 sin θ⊥,⊥ + γ⊥,⊥SN - γSV ) 0 2rc ⊥,⊥ (58)

showing that long range decreases the contact angle. Thus the capillary rise h⊥,⊥ is

h⊥,⊥ )

(

)

K 4C⊥,⊥ L + E 2 sin θ⊥,⊥ + Fg L 2 2Arc ⊥,⊥ y

(

SN

γan L -γisoSN FgA 2

)

+ γSV (59)


h⊥,⊥ ) h⊥,⊥(K,γisoSN,γanSN,γisoNV,γanNV)

(60a)

θ⊥,⊥(K,γisoSN,γanSN,γisoNV,γanNV)

(60b)

θ⊥,⊥ )

The contact angle computed from eq 3 gives

K ∂θ 2 K ) E⊥,⊥2 2 ∂ϑ 2rc 2rc2

E⊥,⊥ )

(47)

Let the capillary rise and contact angle be denoted by

δˆ z‚

at a distance rc from the center of the contact line the bulk range elastic energy density is

(d) Homeotropic-Planar (⊥,||) Director Surface Orientations. This case is analogous to the planarhomeotropic case. Figure 5d shows an schematic of the director field for the homeotropic-planar mode. For this director field no defect nucleates in the bulk of the NLC, because the effect of the nonparallel N/V and S/N surfaces is compatibilized by director gradients. The director field n has hybrid surface anchoring

n ) kSN on S/N, n ) lNV on N/V

(61)

where lNV is a unit tangent vector. Let the capillary rise and contact angle be denoted by

is

(

) (

)

-θ⊥,⊥ π -θ⊥,⊥ + ϑ ψ)2 π - θ⊥,⊥ π - θ⊥,⊥

(55)

The director field close to the contact line is distorted, and

h⊥,|| ) h(kSN;lNV;∇n)

(62a)

θ⊥,|| ) θ(kSN;lNV;∇n)

(62b)

The orders of magnitude of the y and z director gradients

3684


Rey

are given by eqs 35. The surface energy densities and long-range force for this case are

γ⊥,||NV ) γisoNV γ⊥,||SN ) γisoSN + δˆ z‚

∫S

w

η‚tb dA )

∫S

w

(63a)

γanSN 2

(63b)

where C⊥,|| is a factor of order one. Replacing eqs 63 and 64 into the integral force balance (2) gives

2KC⊥,| FgLy2

γisoNVL cos θ⊥,|| + FgA

To compute the long-range contribution (second term above), we follow the same process as above (see eqs 25 and 29). The director angle ψ, measured with respect to the y axis, that satisfies the Laplace equation and the boundary conditions for the planar-planar case

ϑ ) π/2, θ ) 0

(67a)

ϑ ) 3π/2 - θ⊥,||, θ ) π/2 - θ⊥,||

(67b)

is

(

) (

)

π/2 - θ⊥,|| π π/2 - θ⊥,|| + ϑ 2 π - θ⊥,|| π - θ⊥,||

(68)

The director field close to the contact line has bend-splay distortions,2 and at a distance rc from the center of the contact line, the bulk range elastic energy density is

K ∂ψ 2 K ) E⊥,||2 2 ∂ϑ 2rc 2rc2

(69a)

π/2 - θ⊥,|| π - θ⊥,||

(69b)

( )

E⊥,|| )


δˆ z‚Ijun (k‚tb) dl ) - Ijun δˆ zk:I dl )

)

(

SN

γan L -γisoSN FgA 2

)

+ γSV (72)


h⊥,|| ) h⊥,||(K,γisoSN,γanSN,γisoNV)

(73a)

θ⊥,|| ) θ⊥,||(K,γisoSN,γanSN,γisoNV)

(73b)

Application

γ⊥,||NV cos θ⊥,|| - δˆ z‚Ijun (k‚tb) dl + γ⊥,||SN - γSV ) 0 (66)

fb,⊥,|| )

(

K 2C⊥,|| L + E 2 sin θ⊥,|| + Fg L 2 2Arc ⊥,|| y

(65)

showing that long-range forces and anisotropic surface tension contribute to the capillary rise. The contact angle computed from eq 3 gives

ψ)-

h⊥,|| )

- K(∇yno∇zno)(-2) dx dz ) Lx (64) 2C⊥,||K Ly

h⊥,| )

angle due to the fact that long-range energy decreases with larger contact angles. Thus the capillary rise h⊥,|| is

K E 2 sin θ⊥,|| 2rc ⊥,|| (70)

In the previous sections we established that the most significant nematic contribution to capillary rise is the long-range elasticity effect at the contact line. Since this effect is present whenever strong anchoring conditions prevail, it remains to establish which boundary conditions are most likely to result in the highest capillary rise. The experimental results show that planar orientation at the S/N interface results in greater capillary rise than with homeotropic orientation: h|| > h⊥ (see Figure 1 and ref 9). Since there is no experimental indication of the director field in the vicinity of the N/V interface, the model predictions cannot be fully validated due to the lack of data. Nevertheless the model prediction can be evaluated for the two likely cases: (a) planar orientation at the N/V surface, and (b) homeotropic orientation at the N/V surface. It is worth noting that oblique orientation at the N/V interface may be present, and therefore the present predictions are only bounds. (a) Planar Orientation at the Nematic-Vapor Interface. For this case we need to evaluate the capillary rise difference (∆h)|| ) h||,|| - h⊥,|| . Using eqs 31 and 65, we find

(∆h)|| ) h||,|| - h⊥,|| )

2KC KL + (E||,||2 sin θ||,|| FgLy2 2FgArc E⊥,||2 sin θ⊥,||) -

L|γanSN| (74) 2FgA

where we assumed C ) C|| ) C⊥,|| and where we used the fact that for homeotropic surface orientation γanSN < 0. The first term on the right-hand side is due to differences in long-range energies at the bulk, the second to longrange energies at the contact line, and the third and last to the anisotropic surface tension. Long-range elastic effects in the bulk promote a positive (∆h)||, anisotropic surface tension promotes a negative (∆h)||, while the nature of the dominant long-range effects at the contact line depend on the magnitude of the contact angle. The capillary rise difference (∆h)|| is positive if SN

and the contact angle satisfies

γ⊥,||NV cos θ||,⊥ -

K E 2 sin θ⊥,|| + γ⊥,||SN - γSV ) 0 2rc ⊥,|| (71)

showing that the long-range effect increases the contact

L|γan | L 2C 2 2 > (E sin θ E sin θ ) + ⊥,|| ⊥,|| ||,|| ||,|| 2AK Ly2 2Arc (75) By use of the order of magnitude estimates presented above (eq 18), the model predicts a positive capillary rise difference only if the dominant term is negative: (E⊥,||2


Langmuir, Vol. 19, No. 9, 2003 3685

sin θ⊥,|| - E||,||2 sin θ||,||) < 0. Using eqs 28b and 69b, we find that this occurs when

θ||,|| g θ⊥,|| > π/4

θ||,⊥ < θ⊥,⊥ < π/4

(76)

The inequality (76) establishes the principle of capillary rise in nematics: the higher the long-range effect at the contact line, the higher the capillary rise. In partial summary, a necessary condition of the present model to replicate the experimental data is found when the contact angle is greater than π/4. (b) Homeotropic Orientation at the NematicVapor Interface. For this case we need to evaluate the capillary rise difference (∆h)⊥ ) h⊥,|| - h⊥,⊥. Using eqs 45 and 59, we find

(∆h)⊥ ) h||,⊥ - h⊥,⊥ ) -

difference only if (E||,⊥2 sin θ||,⊥ - E⊥,⊥2 sin θ⊥,⊥) > 0. Using eqs 42b and 56b, we find that this occurs when

KL 2KC + 2 2FgAr FgLy c

(E||,⊥2 sin θ||,⊥ - E⊥,⊥2 sin θ⊥,⊥) -

L|γanSN| (77) 2FgA

where we assumed C ) C||,⊥ ) C⊥,⊥ and where we used the fact that for homeotropic surface orientation γanSN < 0. The first term on the right-hand side is due to differences in long-range energies at the bulk, the second to longrange energies at the contact line, and the third and last to the anisotropic surface tension. Long-range elastic effects in the bulk promote a negative (∆h)||, anisotropic surface tension promotes a negative (∆h)||, while the nature of the long-range effects at the contact line depends on the magnitude of the contact angle. The capillary rise difference (∆h)|| is positive only if SN 2C L|γan | L > (E||,⊥2 sin θ||,⊥ - E⊥,⊥2 sin θ⊥,⊥) > 2 + 2Arc 2AK L

(79)

The inequality (79) predicts, consistently with the previous case, that the higher the long-range effect at the contact line, the higher the capillary rise. In partial summary, a necessary condition of the present model to replicate the experimental data is found when the contact angle is less than π/4. Full validation of the model presented here can only be established by a determination of the director field close to the free surface and in the immediate neighborhood of the contact line. Conclusions A general model for the capillary rise of nematic liquid crystals has been derived using fundamental principles and classical liquid crystal physics. It is shown that when compared to the equation valid for isotropic liquids, the equation for nematic liquid crystals contains three additional contributions arising from anisotropic surface tension and from the nematic bulk elasticity. The surface contribution is a function of the surface anchoring strength of the nematic/solid interface. The long-range elasticity contributes through bulk gradients and gradients at the contact line. The most significant contributions to capillary rise from nematic ordering is through long-range elasticity at the contact line. The model was used to analyze and explain the salient features of available experimental data such as why capillary rise depends on surface orientation at the capillary wall, among other things. The necessary conditions under which the experimental observation that parallel wall orientation leads to higher capillary rise than homeotropic orientation are identified. This work has shown that a mathematical framework based on liquid crystal surface physics and nematic elasticity is needed to understand and use measurements of capillary rise for nematic liquid crystals, since classical equations are insufficient.

y

0 (78) By use of the order of magnitude estimates presented above (see eq 18), the model predicts a positive capillary rise

Acknowledgment. This work is supported by a grant from the donors of the Petroleum Research Fund (PRF), administered by the American Chemical Society. LA020750H

A Model of Capillary Rise of Nematic Liquid Crystals - Langmuir (ACS

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