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Adsorption of Acicular Particles at Liquid-Fluid Interfaces and the Influence of the Line Tension Lichun Dong and Duane T. Johnson* Department of Chemical and Biological Engineering, University of Alabama, Tuscaloosa, Alabama 35487-0203 Received August 29, 2004. In Final Form: January 4, 2005 In this paper, the adsorption energy of an acicular (prolate and cylindrical) particle onto a liquid-fluid interface and the effect of the line tension are investigated. The results show that, without line tension, acicular particles always prefer to lie flat in the plane of the interface. However, line tension plays a significant role in determining the adsorption of an acicular particle. First, the line tension creates an energy barrier for the adsorption of particles onto an interface. The planar configuration has a larger energy barrier due to the longer contact line. Therefore, the particles prefer to enter the interface in a homeotropic configuration and then rearrange to a planar configuration or an oblique configuration with a small tilt angle. Second, for prolate particles, an energy maximum occurs at some tilt angles when the line tension is large. Therefore, once the prolate particle is adsorbed on the interface in a homeotropic configuration or with a larger tilt angle, it must conquer an energy barrier to rearrange to a planar configuration. For cylindrical particles, when the line tension is higher, the planar configuration will not be the most energy-favorable configuration. The cylindrical particles prefer to stay in the interface with a small tilt angle.
Introduction The possibility of using particles as stabilizers in liquid films, foams, and emulsions is linked to their adsorption to the relevant interfaces, which act in many ways as surfactant molecules.1,2 Therefore, it is important to examine the state of the particle on the interface including the attachment energy and geometric configuration. There have been several investigations of the adsorption of spherical particles at the water-air or water-oil interface. The results show that the adsorption of spherical particles to the interface is an energy-favorable process and the particles are strongly attached to the interface. Levine and Bowen3 calculated the adsorption free energy of a particle to a spherically curved oil-water interface. Their expression for the free energy of adsorption, ∆E, of a single particle (with contact angle θ < 90°) onto a curved droplet interface is given as a truncated series
{
∆E ) πR2γRβ(1 - cos θ)2 1 -
2R(2 + cos θ) + 3Rs
3R2(1 + cos θ)2 + ... 4Rs
}
(1)
where R is the radius of the solid particle, Rs is the radius of the curved interface before the particles are adsorbed, and γRβ is the interfacial tension between two fluids. For a flat liquid interface (1/Rs ) 0), eq 1 simplifies to eq 2 (for θ < 90°)
∆E ) πR2γRβ(1 - cos θ)2
(2)
Another approach by Aveyard et al. has led to an explicit expression for the adsorption free energy as1
∆E ) 2πγRβ{R2(1 - cos R) cos θ - R2cur(1 - cos β)} (3) in which R ) sin-1(r/R) and β ) sin-1(r/Rcur), r being the
radius of the three-phase contact line around the particle at the interface. Rcur is the radius of the curved interface after the particles are adsorbed. Values of ∆E by eq 1 and eq 3 are in close agreement. The calculations show that for small particles (for example, R ) 0.1 µm, γ ) 25 mN/m, and θ ) 90°) the energy required to move the particle from the interface into one of the bulk phases is on the order of 106 kBT. This means that particles with an appropriate contact angle would be so strongly adsorbed to the interface that the thermal energy would not be enough to remove them. The previous derivation ignored line tension, which might be significant when particles are small, e.g., nanoparticles. Line tension arises as a result of the excess free energy associated with a unit length of a three-phase contact line.4-7 Theoretically, line tension is more complicated than surface tension because only two bulk phases can meet at a surface, whereas several bulk and also surface phases meet simultaneously at a line. Experimentally, line tension is typically small and hence much more difficult to measure than surface tension. Experimental values in the literature ranging from 10-11 to 10-5 N are reported with both positive and negative signs.7-11 In theoretical studies, most of the estimates for the magnitude of line tension are near the lower limit of the (1) Aveyard, R. Adv. Coll. Int. Sci. 2003, 100-102, 503. (2) Binks, B. P. Curr. Opin. Colloid Interface Sci. 2002, 7, 21. (3) Levine, S.; Bowen, B. D. Colloids Surf., A 1991, 59, 377. (4) Gibbs, J. W. The scientific papers of JW Gibbs; Dover: New York, 1961; Vol. 1, p 288. (5) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Oxford Science Publications: Oxford, 1984. (6) Boruvka, L.; Neumann, A. W. J. Phys. Chem. B 1977, 66, 5464. (7) Amirfazli, A.; Neumann, A. W. Adv. Colloid Interface Sci. 2004, 110, 121. (8) Gu, Y. Colloids Surf., A 2001, 181, 215. (9) Aveyard, R.; Clint, J. H. J. Chem. Soc., Faraday Trans. 1995, 91, 175. (10) Toshev, B. V.; Platikanov, D.; Scheludko, A. Langmuir 1988, 4, 74. (11) Amirfazli, A.; Hanig, S.; Muller, A.; Neumann, A. W. Langmuir 2000, 16, 2024.
10.1021/la047851v CCC: $30.25 © 2005 American Chemical Society Published on Web 03/18/2005
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From eq 8, the critical dimensionless line tension, τ/C, which corresponds to ∆E ) 0, can be calculated as
1 τ/C ) (1 - cos θ) sin θ 2 Figure 1. The force balance at the contact line of a spherical particle adsorbed onto a flat interface.
experimental range.7,10,12 In practice, line tensions of 10-6 N would have quite a drastic effect when the size of particles is less than 20 nm. For a particle at the interface between two fluids, the mechanical equilibrium condition at the three-phase line will be the modified Young’s equation.6,10,13 If the system has a constant line tension and is restricted to moderate curvatures, the modified Young’s equation can be written as follows
γsR - γsβ + τκgs cos θ ) γRβ cos θ
(4)
where τ is the line tension, γsR is the interfacial energy between fluid R and the particle, γsβ is the interfacial energy between fluid β and the particle, and κgs is the geodesic curvature of the three phase line. For a small spherical particle where gravity can be ignored, the thermodynamically stable position for the particle at the interface is shown in Figure 1, in which the interface is flat. Because the three-phase line is circular, the geodesic curvature will be equal to the reciprocal of the three-phase circle, (κgs ) 1/(R sin θ)). After substituting κgs into eq 4, we obtain eq 5.
γsβ + γRβ cos θ ) γsR +
τ cos(θ) R sin(θ)
(5)
Inspection of eq 5 shows that the influence of the line tension on the contact angle depends on the particle size. The size of the particles used as stabilizers in foams and emulsions are usually less than 1 µm. For these conditions, the influence of the line tension on the adsorption of a particle into the interface could be significant. When the line tension can be ignored, the classical Young’s equation is recovered
(6)
where θ0 is the original contact angle when line tension is zero. Substitution of eq 6 into eq 5 gives the expression,
τ* ≡
[
]
cos θ0 τ ) sin θ 1 , γRβR cos θ
(7)
where τ* is the dimensionless line tension. Equation 7 shows that, for a given γRβ and τ, θ depends on R. This equation has been used to obtain curves of θ against dimensionless line tension for various values of θ0.9,13-16 When the affect of line tension is included, the adsorption energy, ∆E, of a particle with a flat interface between fluid R and fluid β is (θ < 90°)1
1 - cos θ ∆E ) πR (1 - cos θ) γRβ - 2πRτ sin θ 2
2
When τ* ) τ/C, the minimum value of the adsorption energy is zero. When τ* < τ/C, the minimum value is less than zero, indicating the adsorption of a particle to an interface is an energy-favorable process. When τ* > τ/C, the minimum value becomes greater than zero and the adsorption energy becomes negative, indicating that the adsorption of a particle to an interface is not an energyfavorable process. If the original contact angle, θ0, is known, eqs 6, 7, and 9 can be used to calculate the critical line tension. For example, when θ0 ) 80°, the calculated critical dimensionless line tension is 0.35. If a particle at the interface can obtain sufficient energy, it will leave the interface irreversible. To obtain the energy required for this, imagine a spherical particle that is originally at the equilibrium state and then is pushed into the β phase (for θ < 90). Assume the line tension remains unchanged. The free energy of the particle at any immersion depth, h, is
Eh ) 2πR(2R - h)γsβ + 2πRhγsR - πr2γRβ + 2πrτ (10) The first term on the right side of eq 10 represents the interfacial energy due to the area of the particle exposed to the lower fluid, β, which is a function of how far the particle has moved into the top fluid, R. The second term represents the interfacial energy between the particle and the top fluid. The third term represents the change in interfacial energy caused by the removal of an area between fluid R and fluid β, whose area is circumscribed by the contact line. The fourth term represents the energy needed to create the contact line. With reference to the free energy for a particle totally in β phase, E1 ) 4πR2γsβ, and substituting Young’s equation, the change of the free energy at any immersion depth, h, can be determined
∆E hh )
γsR - γsβ cos θ0 ) γRβ
(
)
(8)
(12) Babak, V. G. Rev. Chem. Eng. 1999, 15, 157. (13) Aveyard, R.; Clint, J. H. J. Chem. Soc., Faraday Trans. 1996, 92, 85.
(9)
∆Eh 4πγRβR2
) -h h (1 - cos θ0 - h h ) + τ*xh h (1 - h h) (11)
where h h ≡ h/2R is the dimensionless height, ∆Eh is the change of the free energy, which is the activation energy required for a particle to leave the interface irreversibly and become completely wetted, and ∆E h h is the change of the dimensionless free energy. A similar result has been derived by Aveyard and Clint.9,13 In their deviation, they assumed the interface is flat. However, our results are identical. Before proceeding, we must first look at the range of applicability of these equations. For example, when the radius of curvature is very small, Young’s equation becomes invalid. This can be seen by subtracting the classical Young’s equation (eq 6) with the modified Young’s equation (eq 4) to get,
cos θ - cos θ0 )
τ , γRβR
(12)
(14) Drelich, J. Colloids Surf., A 1996, 116, 43. (15) Clint, J. H.; Taylor, S. E. Colloids Surf., A 1992, 65, 61. (16) Aveyard, R.; Clint, J. H.; Nees, D. Colloid Polym. Sci. 2000, 278, 155.
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Figure 2. Change in the free energy with respect to the dimensionless height of a spherical particle with different line tensions. At h ) 0, the particle is completely in the lower (R) phase. At h ) 2R, the particle is completely in the upper (β) phase. The positive values of ∆E for small h and small τ* give an energy barrier to the adsorption of a particle onto an interface.
where R is the radius of curvature. As the left-hand side of the equation can never be greater than 2 (or less than -2), this puts an upper bound on the radius of curvature for which Young’s equation is valid. For example, using the interfacial tension of water-air and a moderate and positive line tension of 10-9 N/m, the smallest the radius of curvature can be is approximately 10 nm. If Young’s equation, which is a static force balance, is not valid, then one could postulate that a particle with high curvature (e.g., small radius or sharp corners) and positive line tension would never be in mechanical equilibrium in the interface. Another way of looking at this is to take the derivative of the dimensionless adsorption energy for a sphere (eq 11) with respect to the dimensionless height to get the force it would take to move the sphere through an interface.
F)
∂E h 1 - 2h ) cos θ0 + 2h - 1 + τ ∂h h xh(1 - h)
(13)
The above equation diverges as h f 0 or h f 1. When h is near 0 or 1, the spherical particle is just penetrating the interface and the radius of curvature of the contact line approaches zero. One must therefore conclude that these calculations are invalid for any particle in an interface where the radius of curvature of the contact line is very small. However, evidence exists that the line tension is a function of the line length and/or the contact angle when the radius of curvature is small.17,18 This phenomenon could be explained by adding an additional term to the modified Young’s equation. Because the nature of these derivative terms is even less-known than the value of the line tension, we will neglect their contribution to our overall energy calculations. In other words, we will assume that the modification of the line tension when the radii of curvature are very small exactly balances the divergence of the force and allows us to assume Young’s equation is valid. Figure 2 shows the change of the free energy with the dimensionless height for a spherical particle with different line tensions (θ ) 80°). The figure shows that the increase in line tension moves the curve to higher energies. When the dimensionless line tension is not very high (τ* < 0.3),
Figure 3. The forces acting at the contact line between two fluids and a solid prolate particle in the (a) homeotropic, (b) planar, and (c) oblique configurations. Because the curvature is not constant for the planar and oblique configurations, the contact angles are also not constant. The two images in the planar configuration are the side and front view, and the two images in the oblique configuration are the side view and an image of the particle rotated so its major axis is in the vertical direction.
the free energy has two maximum values near h h ) 0 and h h ) 1 and one minimum value. The minimum value represents the equilibrium position of the particle at the interface. The opposite of this value is called the free energy of the adsorption (adsorption energy). On further increasing the line tension, the minima will disappear and no stable or metastable configurations are possible. Particle adsorption will not occur. The maximum values near h h) 0 correspond to an energy barrier that the particle must conquer before adsorbing onto the interface. To estimate the influence of the line tension, assume a conservative value of τ ) 10-9 N, γRβ ) 50 mN/m, and R ) 0.1 µm. This gives a dimensionless line tension of τ* ) 0.2. For these values, the energy barrier for adsorbing a particle from β phase to the interface is ∆E h ) 0.012, or about 104 kBT. This energy barrier is quite significant and will inhibit the adsorption of particles simply due to thermal fluctuations. Conversely, a particle adsorbed onto the interface will become “stuck”. Figure 2 also shows that negative line tensions move the curves to lower energies and increases the adsorption energy. However, the negative line tension does not change the profile of the curve when the line tension is zero. Therefore, negative line tensions will always favor the adsorption of particles to an interface. In this paper, we extend the previous work to prolate and cylindrical particles. The adsorption energy of the acicular particles was calculated for different orientations with respect to a planar interface and various particle aspect ratios. In all of the cases, line tension is considered. As previously discussed, the negative line tension always benefits the adsorption of a particle to an interface and does not create an adsorption barrier as does a positive line tension; therefore, only the influence of positive line tension is discussed. Prolate Particles
(17) Ivanov, I. B.; Kralchevsky, P. A.; Nikolov, A. D. J. Colloid Interface Sci. 1986, 112, 97. (18) Marmur, A. J. Colloid Interface Sci. 1997, 186, 462
A prolate particle is an ellipsoid particle with two equal axes. There are three configurations for a prolate particle
Adsoprtion of Acicular Particles at Liquid-Fluid Interfaces
at the interface: homeotropic (Figure 3a), planar (Figure 3b), and oblique (Figure 3c). We will discuss each configuration separately and ignore the influence of gravity. Homeotropic Position. Consider a smooth thermodynamically stable prolate particle (s) at a planar interface between fluid R and fluid β (Figure 3a). The interfaces sR, sβ, and Rβ have interfacial tensions γsR, γsβ, and γRβ, respectively. Assume for simplicity that the particle has uniform wetting properties over its entire surface, i.e., θ0 is independent of position. Furthermore, we shall neglect the influence of contact angle hysteresis. Balancing the forces along the tangent to the solid surface at the contact line gives eq 14.
(14)
where r is the radius of the cross sectional-area created by the intersection between the particle and the fluidfluid interface. After substituting Young’s equation into eq 14, we get eq 15.
(
τ γRβr
)
(15)
According to the geometry, when θ < 90, r can be calculated as,
a2 tan θ
r)
xb2 + a2 tan2 θ
)
ax2bh - h2 b
(16)
a2(2bh - h2) a4 tan2 θ ) π ARβ ) π 2 b + a2 tan2 θ b2
(17)
The surface area of the particle in fluid R is
πab2
(
xb2 - a2
πa2b2xtan2 θ + 1 + b2 + a2 tan2 θ
sin-1
πa2 -
xb2 - a2 b
- sin-1
)
x
b2 - a2 ) b2 + a2 tan2 θ
πa(b - h)xa2(b - h)2 + b2h(2b - h)
πab2
(
xb2 - a2
sin-1
b2
xb2 - a2 b
- sin-1
+
)
(b - h)xb2 - a2 b2
(18) The free energy of the particle at the interface is
E2 ) γsRAsR + γsβ(Atotal - AsR) - γRβARβ + 2πrτ
(19)
where Atotal is the outside area of the whole particle. Because the energy of the particle totally in fluid β is E1 ) γsβAtotal, the adsorption energy from the β phase to the interface is
∆E ) (γsβ - γsR)AsR + γRβARβ - 2πrτ Substituting eq 14 into eq 20 gives eq 21.
(20)
(
AsR cos θ - 2πr r
)
(21)
After substituting the equations for the areas, eq 21 is made dimensionless.
∆E hh )
∆Eh 4πγRβa2
)h h2 - h h+
[
h )x(1 - 2h h )2 + 4m2h h (1 - h h) + 0.25 1 - (1 - 2h
(
sin-1
xm2 - 1 xm2 - 1
h) sin-1(1 - 2h
m
xm2 - 1
)]
m
-
cos θ0 + τ*xh h-h h 2 (22)
Here h h ≡ h/2b is the dimensionless immersion depth, ∆E hh is the change of the dimensionless free energy, τ* ≡ τ/γRβa is the dimensionless line tension for a prolate particle, and m ) b/a is the aspect ratio. The critical line tension, where the minimum value of ∆E is zero, is given in eq 23.
r (ARβ - AsR cos θ) a ) τ/C ) γRβa 2ARβ - AsR cos θ τ
where a is the minor axis, b is the major axis of the particle, and h is the height of the particle in the upper fluid. The area inscribed by the contact line is
AsR ) πa2 -
∆E ) γRβ(ARβ - AsR cos θ) + τ
m2
τ cosθ γsβ + γRβ cos θ ) γsR + r
cos θ0 ) cos θ 1 -
Langmuir, Vol. 21, No. 9, 2005 3841
(23)
The critical line tension for a prolate particle can be calculated using eqs 15-18 and 23, which is a function of the aspect ratio and the original contact angle, θ0. For a particle with θ0 ) 80° and m ) 2, 4, 6, and 8, the critical dimensionless line tension is 0.28, 0.20, 0.15, and 0.12, respectively. Figure 4 shows the change of the free energy of prolate particles of different length ratios at different immersion depths when the particle is adsorbed in the interface homeotropically. As was the case for spherical particles, an increase in the line tension moves the curves to higher energies. Therefore, the adsorption energy decreases with an increase in the line tension. When τ* > τ/C, the minimum value of the change of the free energy (the negative of the adsorption energy) becomes positive and the adsorption of a particle to an interface is not an energyfavorable process. The existence of the line tension also creates two maxima. Our results show the energy barriers the particles must conquer are similar for particles with the same minor axis. Planar Position. For a prolate particle lying in the interface in a planar configuration, the contact line is an ellipse. Because the curvature along the ellipse changes (eq 4), the contribution of the line tension to the force balance is different along the contact line. This will cause different contact angles along the contact line, and the result is the deformation of the interface. Figure 3b shows the forces acting along two axes of the elliptical contact line. For the prolate particle, we cannot assume the interface is flat. However, it is still a good assumption that the contact line is flat and horizontal. Because the curvature and the contact angle are different along the contact line for the prolate particle lying in the plane of the interface, we cannot get a specific relationship between the actual contact angle and the original contact angle as eq 7 for spherical particles and eq 15 for prolate particles lying homeotropically in the interface. Therefore, we cannot calculate the critical line
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Figure 4. Change of the free energy of prolate particles of different aspect ratios at different immersion depths when the particles lie in the interface in a homeotropic configuration. The figures show that shorter particles are more favorably absorbed than longer particles that have the same minor axis. The figures also show that shorter particles penetrate further into the top fluid.
tension directly. However, we still can calculate the change of the free energy of particles at different immersion depths. For a prolate particle lying in the plane of the interface (Figure 3b) between fluid R and fluid β, the area in fluid R at any height, h, is
mh h )
∫ ∫ 2π
0
1
0
) 2πa2 +
x
(m2 - 1) (h h - mh h 2)r2 cos2 ψ m rdrdψ 1 - 4m(h h - mh h 2)r2
1-4
2πm2a2
xm2 - 1
(h h < 1/m) sin-1
x1 - m2 -
4a2m2(h h - mh h 2)
x
∫02 x1 - (1 - m2) sin2 ψ dψ
lRβ,h ) 8axm(h h - mh h 2)
π
(26)
Therefore, the change of the free energy at an immersion height, h, is
ARβ ) 4a2m2(h h-
2
The length of the contact line is
m
∫ ∫ 2π
0
1
0
(m2 - 1) (h h - mh h 2)r2 cos2 ψ m rdrdψ 1 - 4m(h h - mh h 2)r2
1-4
(h h > 1/m) (24)
The area of the elliptical cross-sectional area is
b ARβ,h ) π (2ah - h2) ) 4πa2m2(h h - mh h 2) a
(25)
∆Eh ) (γsR - γsβ)AsR,h - γRβARβ,h + lRβ,hτ
(27)
Substituting Young’s equation into eq 27 gives eq 28.
∆Eh ) γRβ(AsR,h cos θ0 - ARβ,h) + lRβ,hτ
(28)
Figure 5 shows the change of the free energy of prolate particles of different length ratios at different immersion depths when the particle is adsorbed at the interface in a planar configuration. As was the case for particles in the interface homeotropically, the increase in line tension moves the curve to higher energies. The existence of the line tension causes two maximum values and a minimum value of the free energy. Oblique Position. When a prolate particle at the interface lies at an angle with respect to the interface (Figure 3c), the contact angle is different along the contact line because of the different curvatures. Therefore, a direct expression between the actual contact angle with the original angle and the critical line tension cannot be derived. However, we can still calculate the change of the free energy of particles at different immersion depths.
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Langmuir, Vol. 21, No. 9, 2005 3843
Figure 5. Change of the free energy of a prolate particle that is lying in the plane of the interface versus the immersion depth for aspect ratios, m ) 2, 4, 8, and 12. The figures show that longer particles have a larger adsorption energy (minimum of the change of free energy). The figures also show that longer particles have a higher adsorption energy barrier (positive values of ∆E near h ) 0.
For a prolate particle at an oblique angle to the interface (Figure 3c), the coordinates (x, y) of the two points A and B are
The surface area of the particle in the R phase is
ARs )
xA )
∫yy
B
A
(
-2a2(b - h)/tan φ - 2abxa2/tan2 φ - (h2 - 2bh) 2(b2 + a2/tan2 φ) (29) yA )
2
2
2
2(b2 + a2/tan2 φ)
(30)
xB ) -2a (b - h)/tan φ + 2abxa /tan φ - (h - 2bh) 2
2
2
2
2(b2 + a2/tan2 φ) (31) yB )
2(b2 + a2/tan2 φ)
(32)
where h is the height shown in Figure 3c, which ranges from b - xb +a /tan φ (point C) to b + (point D). 2
2
2
xb +a /tan φ 2
2
2
)
πayBxa2xA2 + b2(b2 - yB2)
ARs ) πa2 -
b2
(
πab2
xb2 - a2
∫yy
B
A
2b2(b - h) + 2ab/tan φxa2/tan2 φ - (h2 - 2bh)
1+
(b - xb2 + a2/tan2 φ < h < 0) (33)
2b (b - h) - 2ab/tan φxa /tan φ - (h - 2bh) 2
x
a2 y2 cos-1 2 2 2 b b -y -(y - b + h) tan φ dy a 2 xb - y2 b
a 2 xb - y2 b
2
sin-1
xb2 - a2 b
x
(
)
yBxb2 - a2 b2
+
a2 y2 cos-1 b2 b2 - y2 -(y - b + h) tan φ dy a 2 xb - y2 b
a 2 xb - y2 b
2
- sin-1
+
1+
)
(0 < h < b + xb2 + a2/tan2 φ) (34) The length of the short axis of the cross sectional
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Figure 6. Change of the free energy of a prolate particle (m ) 4) versus immersion depth for different tilt angles.
area is
across )
x
(
)
2 2 yA + yB - 2(b - h) a2 (yA + yB) tan φ a - 2 4 2 b (35) 2
The length of the long axis of the cross sectional area is
bcross ) x(xB - xA)2 - (yB - yA)2
(36)
The cross sectional area is
ARβ ) πacrossbcross
(37)
and the length of the contact line is
∫
lRβ ) 4across
π 2 0
x
1-
across2 - bcross2 across2
sin ψ dψ
(38)
As before, the energy of a particle at the interface at any immersion depth, h, is
∆Eh ) γRβ(AsR cos θ0 - ARβ) + lRβτ
(39)
Figure 6 is the change of the free energy of a prolate particle versus immersion depth for different tilt angles (m ) b/a ) 12). When the tilt angle is zero, the particle is planar. When the tilt is 90 °C, the particle is homeotropic. As before, the free energy has a minimum value, which is less than zero when the line tension is less than the critical line tension. The existence of the line tension also causes two maximum values, indicating that particles
must conquer an energy barrier before adsorbing onto the interface. Figure 7 is the minimum free energy of a prolate particle at an interface versus tilt angle. When the line tension is zero, the minimum free energy monotonically increases as the tilt angle increases. However, the story is different for the nonzero line tensions. Figure 7 shows that, at large line tensions, although the planar configuration is still the most energy-favorable configuration, the free energy has a maximum value at some tilt angle. Therefore, once a particle is adsorbed at the interface in a homeotropic configuration or with a larger tilt angle, it must conquer an energy barrier before it rearranges to the low-energy planar configuration. Cylindrical Particles As was the case for a prolate particle, there are three configurations for a cylindrical particle at the interface: homeotropic (Figure 8a), planar (Figure 8b), and oblique (Figure 8c). Homeotropic Position. Consider a smooth cylindrical particle at the interface in a homeotropic configuration (Figure 8a). The interface is not flat except when θ ) 90°. However, when the particle is uniform, the contact line around the particle is flat. Therefore, the free energy of the particle at the interface is given as
E2 ) 2πRHγsβ + 2πRh(γsR - γsβ) + πR2(γsR + γsβ - γRβ) + 2πRτ (40) The free energy of the particle totally in the β phase (Figure 7a) is E1 ) 2πRγsβ(H + R). Therefore, the change of the free energy of the particle at any
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Langmuir, Vol. 21, No. 9, 2005 3845
Figure 7. The minimum free energy of a prolate particle at an interface versus the tilt angle for different aspect ratios and line tensions.
h ) 0 when θ < 90, i.e., γsR > γsβ, which is
∆Emin ) πR2(γsR - γsβ - γRβ) + 2πRτ
(42)
The forces at the cylindrical particle surface are
γsβ + γRβ cos θ ) γsR +
τ cos θ R
(43)
Substituting eq 43 into eq 42 gives
(
∆Emin ) πR2 γRβ cos θ -
τ cos θ - πR2γRβ + 2πRτ R (44)
)
From eq 44, we can calculate the critical dimensionless line tension as Figure 8. The forces acting at the contact line between two fluids and a cylindrical particle in the (a) homeotropic, (b) planar, and (c) oblique configurations. The two images in the planar configuration are the side and front view, and the two images in the oblique configuration are the same particle at different immersion depths.
immersion depth h is
∆E ) 2πRh(γsR - γsβ) - πR2(γsβ - γsR + γRβ) + 2πRτ (41) Inspection of eq 41 shows that ∆E increases monotonically with an increase in h from 0 to H and has a minimum at
τ/c )
(cos θ - 1) τ ) RγRβ (cos θ - 2)
(45)
eq 43 can be arranged into
(1 - τ*) cos θ ) cos θ0
(46)
Using eq 45 and eq 46, the critical dimensionless line tension can be calculated.
τ/c )
1 - cos θ0 2
(47)
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Dong and Johnson
Figure 9. Change of the free energy versus immersion depth for a cylindrical particle of different aspect ratios lying in a planar configuration.
When a cylindrical particle lies in the plane of the interface, the surface area in the R phase is
( )
AsR ) 2R cos-1 Figure 10. The seven different configurations of a cylindrical particle lying at an angle in the interface between fluid β and fluid R.
(
R-h H+ R
2R2 cos-1
(R - h)x2Rh - h R-h R R2
(
)
)
2
(48)
and the cross sectional area is When the line tension is larger than the critical line tension, the particle will stay in one of the bulk phases. When the line tension is less than the critical line tension, the preferred homeotropic position of a cylindrical particle will stay in one of the bulk phases with the top or bottom just touching the interface. Planar Position. When a smooth cylindrical particle lies in the plane of the interface (Figure 8b), the contact line is rectangular. The curvature of the contact line is zero on all sides. However, at the corners of the rectangular contact line, the radius of curvature is zero. As mentioned above, this problem was ignored and its effect needs to be clarified in the future. Because of the lack of curvature, the line tension does not change the contact angle. The actual contact angle along the contact line is always equal to the original contact angle, θ0. The force balance for this situation is given by Young’s equation. The interface for this situation is not flat; however, we still can assume the contact line is flat.
ARβ ) 2Hx2Rh - h2
(49)
The length of the contact line is given in eq 50
lRβ ) 2(H + 2x2Rh - h2)
(50)
The energy of the particle at the interface is
E2 ) AsRγsR + (Atotal - AsR)γsβ - ARβγRβ + lRβτ
(51)
and the free energy of the particle totally in the β phase is E1 ) Atotalγsβ; therefore, the change of the free energy is
∆E ) AsR(γsR - γsβ) - ARβγRβ + lRβτ Substituting Young’s equation into eq 52 gives
(52)
Adsoprtion of Acicular Particles at Liquid-Fluid Interfaces
Langmuir, Vol. 21, No. 9, 2005 3847
Figure 11. Change of the free energy of a cylindrical particle versus immersion depth for different tilt angles and line tensions.
∆E ) γRβ(AsR cos θ0 - Aab) + lRβτ
(53)
m ∆E ) - xh h-h h2 + ∆E h ) 2 π 4πγRβR 1 (m + 1) cos θ0 cos-1(1 - 2h h) 2π 1 1 (1 - 2h h )xh h-h h 2 cos θ0 + τ*(m + 4xh h-h h 2) (54) π 2π where m ) H/R is the aspect ratio and h h ) h/2R is the dimensionless immersion depth. Figure 9 shows the change of the free energy for a cylindrical particle lying in the plane of the interface at different immersion depths. For this situation, once the particle touches the interface, the contact line will be 2H. Figure 9 shows two large maxima, or adsobtion barriers, h ) 1-. The figure also shows that the longer at h h ) 0+ and h particles have larger adsorption energies. Because of the large energy barrier, it is very difficult for a cylindrical particle to be adsorbed onto the interface directly in a planar configuration. Oblique Configuration. When a smooth cylindrical particle stays at the interface in an oblique configuration, there are five possible positions (positions 2-6 in Figure 10). Let us first consider position 4 in Figure 10. With reference to Figure 8c, the surface energy of the particle is
[(
E4 ) πR2 1 +
]
γRβ H-h 2h γ + 1+2 γsβ R sR R cos φ (55)
)
(
(
))
Because 2h ) x + y, eq 55 can be arranged to eq 56,
E4 )
[(
πR2 1 +
)
]
γRβ (x + y) 2H - x - y γsR + 1 + γsβ R R cos φ (56)
( (
))
and,
∂E4 ) πR(γsR - γsβ) ∂x
(57)
When γsR > γsβ, ∂E4/∂x > 0, and E4 has a minimum value when x ) 0. Therefore, when a cylindrical particle lays in the interface at an oblique configuration, it takes position 2 or position 6 in Figure 10; position 4 is not an energyfavorable position. When γsR > γsβ, it takes position 2. When γsR < γsβ, it takes position 6. When the particle is at position 2 of Figure 10, the surface area in the R phase is
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AsR )
Dong and Johnson
R-z 2 z R 1+ + (z - R)x2Rz - z2 R R tan φ z < H (58) tan φ
(
cos-1
) (
)
(
)
∆E h )
cosθ0 ∆E [cos-1(1 - 2h ) h) + 2 4π 4πγRβR 2(2h h - 1)xh h-h h 2 + m(cos-1(1 - 2h h) + h + m tan φ))] cos-1(1 - 2h
R-z 2 AsR ) cos R (1 + m) + (z - R)x2Rz - z2 + R R - z + H tan φ 2 R (1 + m) + cos-1 R
(
)
-1
(
h h - 0.5
∫2hhsin- mφ tan φ - 1 x1 - 4x2 sin2 φdx +
1 π
)
2 sin φ
(z - H tan φ - R)x2R(z - H tan φ) - (z - H tan φ)2 z > H (59) tan φ
(
)
The cross section is a part of an ellipse, whose area is
ARβ ) 2
ARβ ) 2
z-R sin φ -R sin φ
∫
z H tan φ
(
)
(61)
The length of the contact line is
lRβ ) 2x2Rz - z2 + z-R
sin φ ∫-R
2
sin φ
x
1+
x2 sin2 φ dx R - x2 sin2 φ 2
(tanz φ < H) (62)
lRβ ) 2x2Rz - z2 + 2x2R(z - H tan φ) - (z - H tan φ)2 + z-R
2
sin φ ∫-R sin φ
x
1+
x2 sin2 φ dx R2 - x2 sin2 φ
τ* [2xh h-h h2 + 2π
(tanz φ < H) (63)
x2(2hh - m tan φ) - (2hh - m tan φ)2] + τ* π
h h - 0.5 sin φ 2h h - m tan φ - 1 2 sin φ
∫
x
1+
4x2 sin2 φ dx 1 - 4x2 sin2 φ
(tanz φ > H) (66) Figure 11 shows the change of the free energy of a cylindrical particle at the interface versus immersion depths for different tilt angles (m ) H/R ) 8). When the tilt angle is zero, the particle is planar. When the tilt angle is 90 °C, the particle is homeotropic. The plot shows that with an increase in tilt angle, the minimum value increases, which means that the adsorption energy decreases. Therefore, particles prefer to be adsorbed at the interface with smaller tilt angles. Figure 12 is the minimum of the free energy of a cylindrical particle at an interface versus tilt angle. When the line tension is zero, the minimum free energy monotonically increases with an increase in tilt angle. When the line tension is not zero, a local minimum occurs at a nonzero tilt angle. Therefore, when the line tension is zero or smaller, the planar configuration is the energy-favorable configuration for a cylindrical particle at the interface. However, when the line tension is higher, the cylindrical particles prefer to stay at the interface with a nonzero tilt angle. Summary
And the change of the free energy is
∆E ) γRβ(AsR cos θ0 - ARβ) + lRβτ
(64)
which can be arranged into eq 65 or eq 66.
∆E h )
[
1 2h h ∆E ) cos-1(1 - 2h h) 1 + + tan φ 4πγRβR2 4π
]
(
)
h-h h 2 cos θ0 2(2h h - 1)xh h h - 0.5 sin φ -1 2 sin φ
∫
[
1 τ* π
1 π
x1 - 4x2 sin2 φdx + h h - 0.5
xhh - hh 2 + ∫ sin φ -1 sin φ
x
1+
]
4x2 sin2 φ dx 1 - 4x2 sin2 φ
(
)
z < H (65) tan φ
In this paper, we investigated the adsorption of acicular (prolate and cylindrical) particles onto a liquid-fluid interface and the affect of the line tension. The results showed that the planar configuration is always the most energy-favorable position when line tension is zero. However, the existence of line tensions significantly affects the adsorption and configuration of acicular particles in the interface. First, when a particle enters the interface, an energy barrier will occur due to the line tension. The longer the contact line, the bigger the energy barrier. The planar configuration has a larger energy barrier due to the longer contact line. Therefore, the particles prefer to enter the interface in a homeotropic configuration and then rearrange to a planar configuration or an oblique configuration with a small tilt angle. Second, for prolate particles, an energy maximum occurs at some tilt angles when the line tension is large. Therefore, if the prolate particle adsorbs onto the interface in a homeotropic configuration, it must conquer an energy barrier to rearrange to a planar configuration. For cylindrical particles, when the line tension is larger, the planar configuration will not be the most energy-favorable configuration and the cylindrical particles prefer to stay in the interface with a nonzero angle.
Adsoprtion of Acicular Particles at Liquid-Fluid Interfaces
Langmuir, Vol. 21, No. 9, 2005 3849
Figure 12. The minimum free energy of a cylindrical particle in an interface versus the tilt angle for different aspect ratios and line tensions.
Previous research showed that the surface tension of a liquid crystal has a component that depends on the surface particle orientation, which arises from the deviation of the particle orientation from the preferred direction (easy axis).18-20 Our results show that surface particles always prefer to align at the surface in a planar configuration except when the line tension is very high. Therefore, a change of the bulk particle orientation will change the surface easy axis and affect the deviation of the particle orientation from its easy axis, which will change the surface tension.
Acknowledgment. This work was supported by the Materials for Information Technology center, the Alabama Space Grant Consortium (Grant No. SUB2002-036) and the University of Alabama Albert Simmons Endowment Fund. Supporting Information Available: Additional equations and figures. This material is available free of charge via the Internet at http://pubs.acs.org. LA047851V
(19) Barbero, B.; Durand, G. Liquid Crystals in Complex Geometries; Crawford, G. P., Zumer, S., Eds.; Taylor and Francis: London, 1996.
(20) Rey, A. D. Phys. Rev. E 1999, 61, 1540.
