ARTICLE pubs.acs.org/Langmuir
Interactions between Spheroidal Colloidal Particles P. Schiller, S. Kr€uger, M. Wahab,* and H.-J. M€ogel Department of Physical Chemistry, TU Bergakademie Freiberg, Leipziger Strasse 29, 09599 Freiberg, Germany ABSTRACT: Using Derjaguin’s approximation, we have evaluated the interaction energy associated with van der Waals, electrostatic, depletion, and capillary forces between colloidal spheroids. If the interaction range between spheroids is distinctly smaller than the lengths of their principal axes, then simple pair potentials that depend on particle distance and orientation can be derived. Attractive interactions between adjacent spheroids favor their parallel alignment. Parallel spheroids can be arranged into a variety of densely packed configurations. All of these configurations turn out to have the same lattice energy. We discuss the implications of this degeneracy with respect to the stability of photonic crystals consisting of spheroids.
1. INTRODUCTION The stability and structure of suspensions containing large colloidal particles in a solvent are determined by the effective interaction between them. At high densities, excluded volume effects can orient anisometric colloidal particles. Several decades ago, Onsager1 found that long, rodlike particles align to a nematic phase if their density exceeds a critical value. Moreover, a nematic phase was observed in suspensions of platelike particles at moderate and high densities.2,3 However, interactions other than the excluded volume interactions should have an additional effect on the structure of suspensions. For example, van der Waals, electrostatic, capillary, and depletion forces have an influence on the formation of ordered phases and gel structures.46 It is also well known that electrostatic interactions govern the arrangement of platelike particles in clay suspensions. Although the overall surface charge of clay particles is negative, the charge at the edges of the particles may be positive. Thus, positively and negatively charged regions of particle surfaces have a pronounced effect on the flocculation properties of clay suspensions.4 In these suspensions, face-to-edge configurations of charged platelike particles that are perpendicular to each other often dominate. The addition of nonadsorbing polymers to a suspension may facilitate the aggregation of the colloidal particles. A depletion force appears if the gap between two adjacent colloidal particles is comparable to or smaller than the diameter of a polymer coil.7 In this case, the nonadsorbing polymer cannot penetrate this gap, and an osmotic pressure appears between the regions inside and outside the gap. This pressure is accompanied by an attractive force between the surfaces of the colloidal particles. Particle attractions may also result from capillary forces because of small liquid bridges.8 Liquid bridges can be formed by capillary condensation in narrow gaps between adjacent colloidal particles when the suspension contains liquid components that do not mix perfectly. Pressure differences between capillary bridges and the r 2011 American Chemical Society
suspension agent may be very large. In this case, very strong attractive forces between colloidal particles appear. Moreover, in dry systems, water vapor can condense from moist air in the narrow slits between adjacent colloidal particles. In this case, the resulting capillary forces between particles are even stronger than capillary forces in imperfect mixtures. There are known useful empirical potentials for describing the interaction between spheroids.9 In this article, we consider only interaction potentials based on the sum of microscopic pair potentials (e.g., van der Waals interaction) or those derived from macroscopic theories, such as the DebyeH€uckel theory and the theory of capillarity. In many cases, the interaction range of colloidal and macroscopic particles is substantially shorter than the particle dimensions.10 For depletion forces, the interaction range is equal to the diameter of polymer coils of the depletion agent. For electrostatic double layer forces, the interaction range is comparable to the Debye length, and for capillary forces, the range is related to the radius of curvature of the meniscus. In these cases, the interaction range is typically a few nanometers. Hence, a remarkable force between two large colloidal particles with diameters exceeding a few dozen nanometers appears only if both particle surfaces are nearly touching each other. A simplified theory for effective pair potentials utilizes the generally accepted assumption that almost the total interaction energy of large convex particles originates from a relatively small region with a short distance between particle interfaces. In this case, a rather general mathematical approach called Derjaguin’s approximation10 works very well for short distances between particles. For longer distances, the interaction energy is almost zero. Recent Monte Carlo simulations have demonstrated that Received: April 29, 2011 Revised: July 13, 2011 Published: July 22, 2011 10429
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Langmuir depletion forces between cylindrical particles are well described by using Derjaguin’s approximation, even if the diameters of the cylinders are only a few times larger than the interaction range.11 Most applications of Derjaguin’s method are focused on the interaction between spherical or cylindrical particles. However, as shown by White,12 this method should be applicable to any convex particle geometry. In recent years, the self-assembly of colloidal particles into a regular lattice has gained much attention in science and technology.1316 In various optical applications, the formation of photonic crystals in suspensions is of particular interest. Photonic crystals often consist of densely packed spheres that are a few hundred nanometers in diameter. The optical properties of photonic crystals can be modified when the spheres are replaced by spheroids.15 The self-assembly of spheroids into a regular lattice is more challenging because the positional order is always associated with the orientational order of particle axes. Nevertheless, there are successful attempts to form regular lattices of prolate and oblate spheroids.1517 In most cases, it can be assumed that the range of pair interactions between the spheroids is considerably shorter than the length of their principal axes. Hence, the pair potential can be derived by using Derjaguin’s approximation.10,6 By summing the pair potentials, it is possible to evaluate the lattice energy of a photonic crystal. There exist a variety of configurations formed by densely packed spheroids. The evaluation of the lattice energy is useful in the estimation of the stability of different configurations. Recent simulations revealed the existence of arrangements of nonparallel ellipsoids with packing densities higher than the densities in Bravais lattices.18 Here, we consider only Bravais lattices, which have been observed in photonic crystals. In this article, we slightly modify the approach of White12 in evaluating the pair potential of anisometric particles. Using differential geometry, the pair potential is formulated in terms of the mean and Gaussian curvatures of the particle surfaces. This mathematical formulation allows us to obtain simple expressions for elucidating the interaction energies for spheroids dependent on their distances and mutual orientations. The results are applicable to screened electrostatic, depletion, capillary, and van der Waals forces. Attractive forces between prolate or oblate spheroids favor a parallel alignment of their axes of revolution. However, in the case of electrostatic repulsion forces, a perpendicular alignment is preferred. These forces are found to be strong enough to produce preferred configurations of adjacent particles at low densities, where excluded volume interactions have little effect on particle orientations. In dense systems, parallel spheroids can be arranged into colloidal crystals.15 By summing the pair potentials for colloidal crystals, the stability of different lattices and particle configurations can be estimated. Photonic crystals consisting of densely packed spheroids are found to have a variety of configurations with equal energies. This degeneracy could be a source of disorder in such colloidal crystals.
2. INTERACTION OF CONVEX COLLOIDAL PARTICLES If the range of interaction between colloidal particles is much smaller than the particle size, then Derjaguin’s approximation for evaluating the pair potential is applicable.6 A smooth surface can be covered with a net of curvature lines (Figure 1).19 For directions parallel to these lines, the radius of curvature is minimal or maximal. Two curvature lines always meet at each regular point on a surface. These lines are perpendicular to each
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Figure 1. Spheroid with lines of curvature (parallels p and meridians m).
Figure 2. Adjacent prolate spheroids with tangent planes for the points of closest aproach Ta and Tb. At points Ta and Tb, meridians are parallel to axes y and ^y and parallels are parallel to axes x and ^x. If coordinate axes ^x and ^y of the tangent plane for point Tb are projected onto the xy plane, then axes x and ^x enclose angle ϕ.
other. Let a and b be the surfaces of two adjacent convex colloidal particles with diameters that are much larger than their interaction range. Using special coordinate systems associated with the curvature lines, there is a simple way to evaluate the distance between two surfaces. Let d be the shortest distance between surfaces a and b, which is the connecting line between surface points Ta and Tb in Figure 2. The tangent planes to the surfaces at points Ta and Tb are parallel to each other. Furthermore, we introduce a Cartesian coordinate system with the origin located at point Ta so that d coincides with part of the z axis whereas axes x and y define the tangent plane. The x axis may be chosen in such a way that its direction is parallel to a curvature line passing point Ta. Hence, the y axis is parallel to the second curvature line at point a. In a similar way, a Cartesian ^x^y^z coordinate system with its origin located at point Tb is defined. The directions of axes ^x and ^y are chosen to be parallel to the curvature lines at point Tb in a manner similar to the directions chosen for axes x and y. Because the tangent planes to surfaces a and b at points Ta and Tb are parallel, the ^z axis is parallel to the z axis. However, the ^x^y coordinate system is generally turned away by an angle ϕ from the xy coordinate system. Following White,12 the distance between surfaces a and b for a region around Ta and Tb can be evaluated by using the expansion Δz ¼ d þ 10430
x2 y2 ^x2 ^y2 þ þ þ þ Oð3Þ 2ra1 2ra2 2rb1 2rb2
ð1Þ
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where rai (i = 1, 2) denotes the principal radii of curvature at point Ta and rbi denotes the principal radii of curvature at point T b. Symbol O(3) indicates that higher-order terms, such as x3 or x2y, are neglected. Such a truncation of the Taylor series is justified if the interaction range is much smaller than the size of the considered colloidal particles. Taking into account that axes x and ^x enclose angle ϕ, the coordinate transformation ^x ¼ x cos ϕ þ y sin ϕ
leads to x2 a ðk þ kb1 cos2 ϕ þ kb2 sin2 ϕÞ 2 1
y2 a ðk þ kb1 sin2 ϕ þ kb2 cos2 ϕÞ 2 2 þ xyðkb1 kb2 Þ sin ϕ cos ϕ þ Oð3Þ
and the transformation z = d + F2 yields F = 2πW(d)/(λ1λ2)1/2, where Z ∞ WðdÞ ¼ PðzÞ dz ð9Þ is the potential attributed to the force per unit area between two parallel planar surfaces with an intervening distance d. Finally, the potential for the interaction between surfaces a and b is obtained from Z ∞ 2π U ¼ pffiffiffiffiffiffiffiffiffi WðzÞ dz ð10Þ λ1 λ2 d
^y ¼ x sin ϕ þ y cos ϕ
þ
ð2Þ
b 1 where kai = r1 ai and ki = rbi (i = 1, 2) are the principal curvatures. In the spirit of Derjaguin’s approximation,6 the force between adjacent surfaces a and b is evaluated by an integration Z ∞Z ∞ PðΔzðx, yÞÞ dx dy ð3Þ F ¼
By inserting solutions λ1 and λ2 of eq 5 into eq 10 and using the notation Z ∞ Q ðdÞ ¼ WðzÞ dz ð11Þ d
we arrive at 2πQ ðdÞ U ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ka þ Kb þ 2Ha Hb 2Sab cos 2ϕ
∞ ∞
where P(Δz) is the force per unit area between two planar and parallel surfaces at a distance of Δz. F is directed parallel to the z axis. To evaluate the integral in eq 3, it is useful to rotate the xy coordinate system about the z axis in such a way that the quadratic form (eq 2) is transformed into a simpler expression. T T Equation 2 can be written as Δz(x,y) = d + 1/2m B Bm B, where m B = (x, y) is the transpose of column vector m B and B is a symmetric matrix. After a certain rotation of the xy coordinate system, the quadratic form Δz(x, y) is transformed into ZðX, Y Þ ¼ d þ
1 ðλ1 X 2 þ λ2 Y 2 Þ 2
ð4Þ
where λ1 and λ2 are the eigenvalues of matrix B. When the unit matrix is denoted by E, the eigenvalues result from the quadratic equation det(B λE) = 0. This equation can be written as λ2 2ðHa þ Hb Þλ þ Ka þ Kb þ 2Ha Hb qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðHa 2 Ka ÞðHb 2 Kb Þ cos 2ϕ ¼ 0
ð5Þ
where Ha = (ka1 + ka2)/2, Hb = (kb1 + kb2)/2, Ka = ka1ka2, and Kb = kb1kb2 are the mean and Gaussian curvatures for the particles at points Ta and Tb. Because for a rotation of the xy coordinate system around the z axis the Jacobian is equal to 1 (i.e., dx dy = dX dY), the integral for the force (eq 3) is simply replaced by Z F ¼
∞Z ∞ ∞ ∞
PðZðX, Y ÞÞ dX dY
ð6Þ
~ = X (λ1/2)1/2 and Furthermore, applying the transformation X 1/2 ~ ~ ~ ~ ~ Y = Y (λ2/2) , we obtain Z(X ,Y ) = d + X 2 + Y~ 2 and 2 F ¼ pffiffiffiffiffiffiffiffiffi λ 1 λ2
Z
∞Z ∞
∞ ∞
~ ðX ~ , Y~ ÞÞ dX ~ dY~ PðZ
ð8Þ
d
and
Δzðx, yÞ ¼ d þ
~ 2 + Y~ 2 leads to The replacement F2 = X Z ∞ 2 F ¼ pffiffiffiffiffiffiffiffiffi Pðd þ F2 Þ2πF dF λ1 λ2 0
ð7Þ
where Sab ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðHa 2 Ka ÞðHb 2 Kb Þ
ð12Þ
ð13Þ
It should be mentioned that the interaction between surfaces a and b also involves an angular momentum Ld = ∂U/∂ϕ, which may be evaluated by Ld ¼
4πSab Q ðdÞ sin 2ϕ ðKa þ Kb þ 2Ha Hb 2Sab cos 2ϕÞ3=2
ð14Þ
Equations 12 and 14 are not valid for parallel cylindrical particles because in this case one of the two roots of eq 5 is equal to zero. In a manner similar to the previous method, it can be shown that the force between two parallel cylinders with length L is Z ∞ L PðzÞ dz pffiffiffiffiffiffiffiffiffiffi ð15Þ FðdÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ha þ Hb d zd and the angular momentum Ld(d, ϕ) is equal to zero. Again, P(z) is defined as the force per unit area between two planar parallel surfaces separated by distance z. The interaction potential per unit length can be evaluated by a further integration: Z ∞ UðdÞ FðzÞ ¼ dz ð16Þ L L d Generally, the pair potential in eq 12 can be obtained for all forces with interaction ranges much shorter than the particle size. If the interaction energy per unit area W(d) is known for parallel surfaces, then function Q(d) can be evaluated by eq 11. Let us consider depletion, capillary, van der Waals, and electrostatic forces for aqueous suspensions containing ellipsoidal particles. Depletion forces arise if nonadsorbing polymers are added to the suspension. In many cases, if the chain length of polymers is large enough, they form spherical coils in aqueous 10431
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Table 1. Function Q(d) for Some Interaction Types Q(0)/kT or Q(d) depletion force
Q(d = 5 nm)/kT Q(0)/kT = 6.4
1/2FkT(σ d)2
103 nm1
θ(σ d) capillary force
1
/2γR (2R d) 1
2
Q(0)/kT = 29.0 nm1
θ(2R d) electrostatic
2εΦ2 exp(d/l)
complete DLVO-force
Q(5 nm)/kT = 7.7 102 nm1
repulsion 2εΦ2 exp(d/l) H/12πd
Q(5 nm)/kT = 6.4 102 nm1
solutions. Usually, for the purpose of evaluating depletion forces, a polymer coil is considered to be a hard sphere with diameter σ. In the region where the distance between surfaces a and b is smaller than sphere diameter σ, no spherical polymer coil occurs. Hence, an osmotic pressure P is maintained between the regions inside and outside the gap. For low polymer coil densities, the osmotic pressure is obtained from the relation P = FkT,5 where F is the particle number density of the polymer coils, k is the Boltzmann constant, and T is the temperature. The osmotic pressure is zero inside a gap if its width is smaller than the diameter σ of a polymer coil. Otherwise, if the gap width is larger than σ, then coils can move into the gap and the regions inside and outside the gap have the same osmotic pressure. Hence, the potential W(d) for the depletion force per area for two parallel surfaces is5 WðdÞ ¼ FkTðσ dÞ θðσ dÞ
ð17Þ
where Heaviside function θ(σ d) has a value of 1 if σ d g 0 and zero if σ d < 0. Equation 17 may be improved by replacing FkT = P by a proper osmotic pressure equation for the depletion agent.11 However, at large densities of the depletion agent, the simple theory based on eq 17 fails because oscillating density structures appear near the surfaces of colloidal particles.20,21 Capillary forces can appear if an aqueous solution of the suspension contains a small amount of a second fluid, which does not mix perfectly with water. Such fluid additions can condense in narrow gaps between particle surfaces. This effect, called capillary condensation, leads to bridges between adjacent colloidal particles when the gap between them is sufficiently small. These bridges may be accompanied by rather strong attractive forces between colloids. For the sake of simplicity, we assume that the contact angle at the three-phase line, where the solid and both liquid phases meet, is equal to zero or is extremely small. In this case, the force per unit area between two parallel plates is P = γ/R,8 where γ and R denote the surface tension and the radius of curvature of the meniscus for the fluidfluid interface. The potential for the attractive force can be written as WðdÞ ¼
γð2R dÞ θð2R dÞ R
vapor is violated, Derjaguin’s approximation is applicable only to the contact pair potential U(d = 0) but fails for d > 0.22 Suspensions are often stabilized by the repulsion between charged particle surfaces. For two weakly charged parallel surfaces, the interaction energy for the repulsion is 2εΦ2 d exp ð19Þ WðdÞ ¼ l l where ε is the dielectric constant, Φ is the electrostatic potential of the surfaces, l is the Debye length, and d is the distance between the charged surfaces.6 Apart from the electrostatic repulsion, the van der Waals attraction also contributes to the interaction. By adding this contribution, the complete force considered by the DLVO theory6 can be expressed as 2εΦ2 d H exp ð20Þ WðdÞ ¼ l 12πd2 l where H denotes Hamaker’s constant. Functions Q(d), which characterize the interaction strength, can be evaluated from potentials W(d) by using eq 11. The results for different interaction types and typical numerical values are summarized in Table 1. For spherical particles with radii R, eq 12 leads to U = πQ(d)R. Energies are expressed in terms of kT to allow us a comparison with the mean energy of Brownian motion. When |U| > kT, the pair potential should have an influence on the configuration of particles. Let us consider the depletion interaction, which is the weakest force in Table 1. If we increase R from 50 to 1000 nm, then interaction energy |U| varies between kT and 20kT. The interaction energies resulting from capillarity are much larger, exceeding the depletion interaction energies by 3 orders of magnitude. The numerical estimations summarized in Table 1 have been made by using the following values for physical parameters: polymer coil diameter σ = 3 nm, meniscus curvature radius R = 3 nm, surface tension of a fluidfluid interface γ = 20 103 N/m, electrostatic potential of charged surfaces Φ = 25 mV, Debye length l = 5 nm, dielectric constant of water ε = 78ε0, Hamaker constant H = 1020 Nm, particle number density of polymer coils F = 1.41 1024 m3, volume density of polymer coils Ψ = Fπσ3/6 = 0.02, and temperature T = 300 K.
3. INTERACTIONS OF UNIAXIAL ELLIPSOIDS 3.1. Pair Potential. The evaluation of the pair potential for two convex colloidal particles requires the evaluation of the mean and Gaussian curvatures for the points where the particle surfaces have the shortest distance. Let us consider rotational ellipsoids (spheroids) with two equal principal axes 2A and a principal axis of different length 2B suspended in an aqueous solution. The surfaces of these particles can be described by the parameter representation23
xB ðu, υÞ ¼ A cos u sin υ B e 1 þ A sin u sin υ B e2 þ B cos υ B e3
ð18Þ
It should be mentioned that we consider only the case of liquid bridges that have a fixed radius of curvature R in accordance with Kelvin’s equation.10 In this case, the liquid bridge is in thermodynamic equilibrium with the vapor of the environment and Derjaguin’s approximation is applicable to capillary forces. Otherwise, if the phase equilibrium between the bridge and its
ð21Þ
where u = u1 (0 e u < 2π) and v = u2 (0 e υ e π) are surface coordinates (eq A1) and B e i (i = 1, 2, 3) denote the unit vectors parallel to the axes of a Cartesian coordinate system. Curves B x (u, x (u = c2,υ) υ = c1) with constant c1 are called parallels, and curves B with constant c2 are meridians of the surface.19 Parallels and meridians are the curvature lines of the ellipsoid (Figure 1). In 10432
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Figure 3. Geometrical parameters for describing the pair potential of spheroids. For ϕ = 0, the axes of both spheroids lie in the x2 = x20 = 0 plane and an angle η (0 e η < 2π) is enclosed by the axes of revolution. These axes lie along coordinate axes x3 and x30 . Parameters υ^a and υ^b (0 e υ^a, υ^b < 2π) define the points of closest approach of the ellipses (cross sections of parallel spheroids a and b) in the plane x2 = x20 = 0.
the special case of a sphere (A = B), surface coordinates u and v can be considered to be azimuthal and polar angles of a spherical coordinate system. If A , B, then the ellipsoid looks like a needle, whereas for A . B it has a disklike shape. By applying the parametrization equation (eq 21) to two adjacent ellipsoids a and b, the mean and Gaussian curvatures (Appendix A) at the points of shortest distance Ta and Tb (Figure 3) can be written as Hβ ¼
BðA2 þ A2 cos2 υβ þ B2 sin2 υβ Þ 2AðA2 cos2 υβ þ B2 sin2 υβ Þ3=2
ð22Þ
and Kβ ¼
B2 ðA2 cos2 υβ þ B2 sin2 υβ Þ2
ð23Þ
with β = a, b. The evaluation of pair potentials requires the determination of parameter vb of point Tb, which is dependent on parameter va of point Ta and the angles that define the mutual orientation of two ellipsoids a and b. Let us first assume that the lines of curvature of both spheroids are parallel to each other at the points of closest approach Ta and Tb (i.e., the condition ϕ = 0 is satisfied). Then, the axes of revolution (2B) of both spheroids are lying in the same plane, say, in the plane x2 = 0 as shown in Figure 3. Let η (0 e η e 2π) be the angle enclosed by these axes, which lie along coordinate axes x3 and x30 . Spheroid a resides with its center in the origin of the x1x2x3 coordinate system. It is convenient to introduce a parameter^va (0 e ^va < 2π) instead of surface coordinate νa for localizing the point of closest approach Ta of spheroid a in plane x2 = 0 (Figure 3). The relation between υa and υ^a is ( for 0 e υ^a < π υ^a ð24Þ υa ¼ 2π υ^a for π e υ^a < 2π Using parameter υ^a, we can express points Ta for spheroid a as e 1 þ B cos υ^a B e3 Ta ðυ^a Þ ¼ A sin υ^a B
ð25Þ
x10 x20 x30
coordinate system (Figure 3), the Analogously, in the point of closest approach for ellipsoid b is written in terms of a parameter υ^b (0 e ^vb < 2π) as e 01 þ B cos υ^b B e 03 Tb 0 ðυ^b Þ ¼ A sin υ^b B
Figure 4. Scaled pair potential for repulsive electrostatic forces between two oblate spheroids with A = 600 nm and B = 300 nm (Table 1). Q(5 nm)/kT = 7.7 102 nm1; diagram a: ϕ = 0, 0 e υ^a = υ^a < π, 0 e η < π, diagram b: η = 0, 0 e υ^a = υ^a < π, 0 e ϕ < π.
Using the x1x2x3 coordinate system, we represent this point as Tb ðυ^b , ηÞ ¼ B r þ ðA cos η sin υ^b þ B sin η cos υ^b Þ B e1 þ ðA sin η sin υ^b þ B cos η cos υ^b Þ B e3 ð27Þ whereBr localizes the center of gravity of spheroid b. At the points of shortest distance Ta and Tb, the unit normals of surfaces a and b (eq A3) satisfy the condition Bb N Ba ¼ N
ð28Þ
This condition leads to a relation between parameters ^va and ^vb, namely, tan υ^b ¼ S
ð29Þ
where the notation S¼
ð26Þ 10433
AðA sin η cos υ^a B cos η sin υ^a Þ BðA cos η cos υ^a þ B sin η sin υ^a Þ
ð30Þ
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is used. Because the axes of revolution (2B) of both ellipsoids are chosen to lie in the same plane (x2 = 0 in Figure 3), angle ϕ in eqs 12 and 14 is equal to zero. However, if we turn ellipsoid b around the axis ! d ¼ jTa Tb j as depicted in Figure 3, ! then ϕ becomes nonzero. The rotation of ellipsoid b around axis |Ta Tb | does not change parameters υ^a and υ^b. Obviously, the pair potential for spheroids is a function of parameter ^va, angles η and ϕ, and distance d. To evaluate the pair potential, the mean curvature (Hb) and the Gaussian curvature (Kb) of particle b as functions of parameter ^va are needed. By combining eqs 22, 23 and 29, we arrive at rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B½2A2 þ ðA2 þ B2 ÞS2 1 þ S2 ð31Þ Hb ¼ 2A½A2 þ B2 S2 A 2 þ B 2 S2 and B2 ð1 þ S2 Þ2 Kb ¼ ðA2 þ B2 S2 Þ2
ð32Þ
Figure 5. Spheroid in the vicinity of a planar surface. The interaction potential depends on polar angle η and distance d.
where S is a function of η and parameter^va is given by eq 30. Hence, the pair potential for any configuration of two spheroids can be evaluated by using eqs 12, 22, and 23 for β = a (eqs 31 and 32). Figure 4a,b illustrates the pair potential for an attraction force between two oblate spheroids as a function of surface parameter va for the point of closest approach Ta and angles η and ϕ. The diagrams refer to the parameter interval 0 e υ^a < π, where the condition υ^a = υa is satisfied (eq 24). For π e ^va < 2π, the corresponding plots are only slightly different. 3.2. Interaction between Spheroids and Planar Surfaces. Using eq 12, the interaction of a spheroid with a plane surface can also be evaluated. Let us assume that the axis of revolution and the surface normal enclose polar angle η (0 e η e π/2) as shown in Figure 5. It should be noted that the potential for special cases η = 0 and π/2 has already been evaluated by Bhattacharjee et al.24 Taking into account that the curvatures of the planes are zero (Ha = 0 and Ka = 0), eq 12 yields 2πQ ðdÞ U ¼ pffiffiffiffiffi Kb
ð33Þ
where Kb is the Gaussian curvature of the spheroid at the point of closest approach, T Bb. At this point, the surface normal of the spheroid is antiparallel to the normal of the planar interface. In the coordinate system, where the principal axes are parallel to the coordinate axes, the surface normal of the spheroid N Bb is obtained from eq 21 by N Bb = (∂x B/∂u ∂x B/∂ν) for u t ub and v t vb, where λ is a normalization factor. It is useful to introduce a coordinate system fixed to the planar surface with the x3 axis parallel to the surface normal. A tilt of the spheroid can be produced by turning the spheroid by an angle η around the x2 axis. After this rotation, the components of the surface normal N Bb = (Nb1, Nb2, Nb2) of the spheroid are Nb1 ¼ λðB cos ub sin υb cos η þ A cos υb sin ηÞA sin υb Nb2 ¼ λAB sin ub sin2 υb Nb3 ¼ λðB cos ub sin υb sin η A cos υb cos ηÞA sin υb in the x1x2x3 coordinate system associated with the planar surface (Figure 5). Now, the point of closest approach, where the
Figure 6. Potential energy of prolate spheroids adhered to a planar surface (B = 600 nm, Q(0)/kT = 29 nm1, (—) B/A = 2, (---) B/A = 4, (———) B/A = 10).
distance between the spheroid interface and the plane is minimal, can be obtained from the conditions that both the first and second components of N Bb are equal to zero (Nb1 = 0 and Nb2 = 0). The relevant solution of these equations is ub = 0 and vb = π arctan(AB1 tan η). By inserting this solution into eq 33, we arrive at U ¼
A2
2πQ ðdÞA2 B sin2 η þ B2 cos2 η
ð34Þ
Assuming capillary forces (Table 1), Figure 6 illustrates potential eq 34 for prolate spheroids with several aspect ratios B/A. 3.3. Lattice Energies for Densely Packed Spheroids. If anisometric colloidal particles are used in photonic crystals, then the parallel alignment of the particle axes is required in addition to the positional order of the lattice points. Sophisticated procedures are necessary to produce 2D or 3D lattices of well-ordered colloidal particles.16 Recently, Ding et al.15 demonstrated that a large variety of 2D close-packed structures of spheroids can result 10434
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from a self-assembly processes. Figure 7 shows three arrangements of spheroids that have the same surface density as a triangular lattice of close-packed spheres. Let us compare the lattice energies of different configurations. It would be advantageous for the formation of a well-ordered photonic crystal if the final state were a configuration with the lowest lattice energy. The lattice energy can be evaluated by summing pair potentials. A symmetry consideration reveals that for parallel spheroids (η = 0 and ϕ = 0) the mean and Gaussian curvatures of both particles coincide (Hb = Ha and Kb = Ka) at the points of closest approach (Ta and Tb). Thus, the pair potential (eq 12) for two parallel spheroids can be simplified to U = πQ(d)/(Ka)1/2. Using eq 23, we obtain U ¼
πQ ð0Þ 2 ðA cos2 υa þ B2 sin2 υa Þ B
ð35Þ
for d = 0. By placing spheroid a in the origin of the a x1x2x3 Cartesian coordinate system and aligning its axis of revolution (B axis) along the x3 axis, the relation x3 = B cos va (0 e υa e π) is satisfied for points at which spheroid a touches its nearestneighbor spheroids. Hence, eq 35 can be replaced by ! ðA2 B2 Þx23 πQ ð0Þ 2 B þ ð36Þ U ¼ B B2 To evaluate the lattice energy, we need the values of x3 for all contact points. Instead of spheroids, let us first consider a monolayer of close-packed spheres arranged in the plane x2 = 0. Each sphere with radius A touches its six nearest neighbors. Using polar coordinates (r, υ^a) for the lattice plane x2 = 0, we give the six contact points localized at the equator of sphere a by r = A and υ^a(i) = R + iπ/3 + 2mπ (i = 0, 1, 2,.., 5), where υ^a(i) denotes the angle between the position vector of contact point i and the x3 axis of the Cartesian system, angle R can be chosen arbitrarily, and integer m is chosen in such a way that the condition 0 e υ^a(i) < 2π holds. The values of the x3 coordinate for the six contact points of sphere a are obtained from x3(i) = A cos υ^a(i). Densely packed spheroids can be produced by stretching the x3 axis of a monolayer of spheres (Figure 7) (e.g., by applying the coordinate transformation x3 w (B/A)x3). This procedure does not change the values of parameter υ^a(i), although they can no longer be identified by polar angles. Thus, we obtain new coordinates x3(i) = B cos (R + (iπ/3)) for the six contact points i = 0, 1, 2,.., 5. By varying R, any possible densely packed monolayer structure consisting of parallel spheroids with half axes A and B can be generated. Taking into account that a plane lattice with N particles and coordination number Z = 6 has ZN/2 contact points, we obtain the lattice energy by summing the pair potentials (eq 36): ! 2 2Z1 πNQ ð0Þ A B ZB2 þ x23 ðiÞ ð37Þ E¼ 2B B2 i¼0
∑
The evaluation of the sum (eq 37) leads to a rather simple result E¼
3πNQ ð0ÞðA2 þ B2 Þ 2B
Figure 7. There are a variety of possibilities in forming 2D closepacked lattices of parallel spheroids. The lattices differ in the shape of the unit cell.15
In a similar way, the lattice energy E for a 3D lattice of spheroids can be evaluated. We start with a hexagonal closepacked lattice of spheres. Each sphere with radius A touches 12 nearest neighbors. Let P(i) = (x1(i), x2(i), x3(i)) be the contact points (i = 0, 1, 2,.., 11) of sphere a placed in the center of a Cartesian coordinate system. A possible choice of the 12 contact points is (A cos(jπ/3), A sin(jπ/3), 0) with j = 0, 1, 2,.., 5, (cos(π/6 + kπ/3) sin t, sin(π/6 + kπ/3) sin t, cos t) and (cos(π/ 6 + kπ/3) sin(π t), sin(π/6 + kπ/3) sin(π t), cos(π t)) with k = 0, 1, 2 and t = arccos((6)1/2/3). In the second step, the lattice of spheres, and thus the set of contact points, is rotated arbitrarily. Using rotation matrices 0 1 1 0 0 B C B C 0 cos χ sin χ C C¼B @ A 0 sin χ cos χ 0
ð38Þ
cos ψ
B B sin ψ M ¼B @ 0
which turns out to be independent of R. Hence, all close-packed monolayers of parallel spheroids with half axes A and B have the same lattice energy E. 10435
sin ψ cos ψ 0
0
1
C C 0C A 1
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0
B B L¼B @
ARTICLE
cos R
0
0
1
sin R
0
sin R
1
C C 0 C A cos R
any rotation can be described by P(i)T w (LCM)P(i)T, where P(i)T is the transpose of P(i). Such a rotation does not change the position of sphere a in the center of the coordinate system, but it produces a new set of contact points located at the surface of this sphere. After the rotation, we stretch the whole lattice system of spheres along the x3 axis in a similar way as for the 2D case (x3 w (B/A)x3). Finally, when new coordinates x3(i) (i = 0, 1, 2,.., 11) for the contact points of sphere a are used, eq 37 yields the energy of the close-packed hexagonal lattice (Z = 12). By simplifying the result with computer algebra, we obtain E¼
2πNQ ð0ÞðA2 þ 2B2 Þ B
ð39Þ
Surprisingly, the lattice energy (eq 39) is independent of angles χ, ψ, and R, which determine the configuration of the closepacked lattice of ellipsoids. Moreover, if we start with a face-centered lattice of spheres instead of the hexagonal close-packed lattice, we again arrive at the same simple result (eq 39). Hence, all close-packed arrangements of parallel spheroids have the same lattice energy.
4. DISCUSSION Equation 34 describes the potential for the interaction between spheroids and a planar surface. In the case of prolate spheroids, attractive forces favor an orientation with the long axis parallel to a planar interface, whereas repulsive forces favor a perpendicular orientation. In the case of attractive capillary forces (Q(0) = 29 nm1, Table 1), Figure 6 depicts the energy of the interaction for prolate spheroids as a function of tilt angle η. The orientation of the long axis (2B) parallel to the surface (η = π/2) is strongly preferred if the aspect ratio B/A becomes large. More geometrical variables are needed to describe the configurations of two spheroids. When oblate spheroids with a moderate aspect ratio B/A = 0.5 are assumed, Figure 4 depicts the interaction potential associated with repulsive electrostatic forces for the distance d = 5 nm and a Debye length of l = 5 nm (Table 1). Let us restrict our analysis to the parameter interval (0 e υ^a < π and υa = υ^a) for the point of closest approach Ta because the other case (π e υ^a < 2π and υa = 2π υ^a) leads to essentially the same results. The electrostatic surface potential is chosen to be small enough (Φ = 25 mV) for the application of the DebyeH€uckel approximation. At low and moderate densities F < (4πA3/3)1, when excluded volume effects are negligibly small, face-to-edge orientations (η = π/2, 3π/2) of the ellipsoids are more favorable than face-to-face orientations (η = 0,π). Furthermore, repulsion forces also favor configurations with va = π/2 (edge-to-edge position), whereas the pair potential is almost independent of angle ϕ (Figure 4b). General conclusions remain valid if the repulsive electrostatic interaction is supplemented by the van der Waals attraction. As long as the suspension does not flocculate, the electrostatic repulsion predominates. In Table 1, the last row (Q(5 nm)/kT = 6.4 nm1) refers to the combined interaction potential used in the framework of the DLVO theory. The addition of the van der Waals interaction leads to only a slight reduction in the effective repulsive potential.
Attractive capillary forces may be important in photonic crystals if water bridges occur within the cracks between adjacent colloidal particles. Liquid bridges should appear during the drying process of self-assembled photonic crystals that are formed in suspensions. Moreover, when the humidity is sufficiently high, water bridges may spontaneously appear because of capillary condensation. In many cases, attractive forces resulting from liquid bridges are larger than other forces (Table 1). If the spheroids are densely packed and parallel to each other, as observed in experimental systems, then a peculiar degeneracy is found. The lattice energy obtained from summing the pair potentials does not depend on the shape of the unit cell of the lattice. This degeneracy may be a source of disorder in self-assembled photonic crystals consisting of spheroids. In comparison with arrangements of low symmetry (third lattice structure shown in Figure 7), high-symmetry configurations are not preferred. In conclusion, a suspension containing prolate or oblate colloidal particles is subjected to rather strong anisotropic interactions forces (e.g. electrostatic, depletion, or capillary forces) that favor either a parallel or a perpendicular particle alignment. At low and moderate densities, the anisotropy of these interactions should influence the structure of particular sols and gels. Densely packed colloidal crystals of parallel spheroids have a variety of configurations with the same lattice energy.
’ APPENDIX A The equation of a surface may be written in vector form as e 1 þ x2 ðu1 , u2 Þ B e2 xB ¼ x1 ðu1 , u2 Þ B 1 2 þ x3 ðu , u Þ B e3
ðA1Þ
where B e i (i = 1, 2, 3) represent the unit vectors associated with a Cartesian coordinate system and xi(u1, u2) represents differentiable functions of two parameters u1 and u2.19 These parameters are coordinates that can be used to localize points on the surface. 1 2 x 2 = ∂x are tangent vectors of Derivatives B x 1 = ∂x B/∂u and B B/∂u 1 2 the surface at a point point (u , u ). A metric tensor with x1 O B x 2, g12 = components evaluated by scalar products g11 = B x1 O B x 2 and g22 = B x2 O B x 2 can be used to determine g21 = B distances between points on the surface. The expression ds2 = dx B 23 O dx B, which is explicitly written as ds2 ¼ g11 ðdu1 Þ2 þ 2g12 du1 du2 þ g22 ðdu2 Þ2
ðA2Þ
is called the first fundamental form in the theory of surfaces. Its square root ds is equal to the distance between adjacent points (u1 + du1, u2 + du2) and (u1, u2). There is also a second fundamental form that provides a further characterization of the local surface properties. The unit normal N B of the surface is obtained from x xB2 N B ¼ B1 j xB1 xB2 j
ðA3Þ
where the symbol indicates the cross product. By using the derivatives N B1 = ∂N/∂u1 and N B1 = ∂N/∂u2 and defining the matrix elements bik = x Bk (i, k = 1, 2), the second fundaBiON mental form is expressed as d xB o d N B ¼ b11 ðdu1 Þ2 þ 2b12 du1 du2 þ b22 ðdu2 Þ2 10436
ðA4Þ
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The coefficients of both fundamental forms (eqs A2 and A4) can be used to evaluate the mean curvature H = (r11 + r21)/2 and the Gaussian curvature K = r11r21, where r1 and r2 are the curvature radii for a surface point along the directions of principal curvature. Let g and b be the determinants of matrices gik and bik and let the matrix gik = (gik)1 denote the inverse matrix of metric tensor gik. It can be shown that H and K may be obtained from23 K ¼
b g
ðA5Þ
1 ik g bik 2
ðA6Þ
and H ¼
where we sum over i and k (i, k = 1, 2) in accord with Einstein’s summation convention. Equations A5 and A6 for the curvature have the advantage that they are generally valid for any surface defined by eq A1. Using the parametrization equation (eq 21) with u1 = u and u2 = v, we arrive at eqs 22 and 23 for the curvature of spheroid surfaces.
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