Colloids and Surfaces A: Physicochemical and Engineering Aspects 165 (2000) 143 – 156 www.elsevier.nl/locate/colsurfa
DLVO interaction energy between spheroidal particles and a flat surface Subir Bhattacharjee, Jeffrey Y. Chen, Menachem Elimelech * Department of Chemical Engineering, Yale Uni6ersity, P.O. Box 208286, New Ha6en, CT 06520 -8286, USA
Abstract The orientation dependent interaction energy between a spheroidal particle and an infinite planar surface is determined using the surface element integration (SEI) technique. The interaction energy predictions of SEI are shown to be considerably more accurate than the corresponding predictions based on Derjaguin’s approximation (DA). Comparison with the Hamaker approach for evaluating the non-retarded van der Waals interaction energy reveals that SEI predicts the orientation dependent interaction energy for spheroidal particles with remarkable accuracy. It is further shown that both SEI and DA give nearly identical predictions of the electrostatic double layer interaction energy between a spheroidal particle and a flat plate at high electrolyte concentrations. However, at low electrolyte concentrations, considerable deviations are noted between the predictions of SEI and DA, particularly for very small aspect ratios of the particle (aspect ratio=length of minor axis/length of major axis). It is also noted that when the spheroidal particle is oriented with its major axis parallel to the planar surface, DA incorrectly predicts the interaction energy as that of a spherical particle with a radius equal to the semi-major axis of the spheroid. This limitation of DA is avoided in SEI, which accounts for the dependence of the interaction energy on the actual shape (aspect ratio) of the particle at any orientation. Predictions of the DLVO interaction energy based on SEI indicate that, at high electrolyte concentrations, the orientation dependence of the interaction energy is not significant at large separation distances, and assumption of an equivalent spherical particle may be sufficient. However, significant deviation of the interaction energy from that of a spherical particle is observed at small separation distances, particularly at low electrolyte concentrations. At these small separation distances, where the correct orientation dependence of the interaction energy must be considered for proper calculations of particle interaction phenomena with flat surfaces (e.g. particle deposition), SEI provides a facile route to perform such calculations. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Spheroidal particle; Surface element integration; Derjaguin’s approximation; DLVO interaction; Particle deposition
Abbre6iations: DA, Derjaguin’s approximation; DLVO, Derjaguin – Landau – Verwey – Overbeek; EDL, electrostatic double layer; SEI, surface element intergration; VDW, van der Waals. * Corresponding author. E-mail address:
[email protected] (M. Elimelech) 0927-7757/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 9 9 ) 0 0 4 4 8 - 3
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1. Introduction The interaction energy between colloidal particles and planar surfaces forms the basis of our theoretical understanding of a wide range of colloidal phenomena, including particle deposition [1], heterocoagulation [2 – 4], and transport of colloidal particles and macromolecules through porous media [4,5]. Frequently, calculations of the interaction energy are performed using the classical Derjaguin–Landau – Verwey – Overbeek (DLVO) theory [6,7] for simple geometrical shapes of the interacting surfaces, like spheres, cylinders, and flat planar surfaces [8]. Naturally occurring colloidal systems, however, abound with particles of various geometrical shapes. It is quite common to encounter colloidal particles, particularly biocolloids such as proteins, viruses, and bacteria [8 – 10], that can roughly be represented as spheroids or ellipsoids. For such particles, the interaction energy becomes orientation dependent, and gives rise to a torque [11]. This torque orients the particles in an energetically favorable configuration. In a colloidal system comprising spheroidal particles, presence of such orientational ordering can lead to a substantially broader spectra of the dispersion properties that cannot be explained merely on the basis of spherically symmetric interaction potentials. Yet, most colloidal systems are treated by assuming the particles therein to be spherical or by considering them to be ‘equivalent spheres’. Studies on particle deposition and adsorption primarily rely on accurate information regarding the interaction energy between a colloidal particle and an infinite planar surface [1,12]. Typically, the deposition rate of a colloid on a planar surface is strongly affected by the interaction energy between the particle and the plane [12]. In this context, it is of particular interest to know the extent to which orientation dependence of the interaction energy affects the deposition of non-spherical colloidal particles onto flat surfaces [12]. Given the nature of the DLVO interaction potential, which involves multiple maxima and minima at various separation distances, there is a possibility that the particle orientations are considerably modified as it approaches the planar surface. In a
repulsive environment, the particle will expose its smallest surface area near the point of closest approach to the planar surface, while in presence of attraction, the particle will reorient to present the maximum possible surface area to the plane. Such a behavior can drastically affect the extent of surface coverage during particle deposition and adsorption, a phenomenon that is possibly beyond the scope of theoretical models for particle deposition dynamics that assume spherical particles [12]. Evaluation of the DLVO interaction potential for arbitrary geometries is considerably restricted owing to the difficulties associated with the evaluation of its two components, namely, the van der Waals (VDW) and the electrostatic double layer (EDL) interactions. The van der Waals interaction energy is commonly evaluated using Hamaker’s approach [13], in which two volume integrals need to be evaluated. For particle geometries where no form of symmetry could be utilized to reduce the order of integrations, the calculations can become considerably bulky owing to the evaluation of six nested integrals over a complex geometry. Similarly, employing rigorous numerical techniques to solve the Poisson–Boltzmann equation appears to be the only means of evaluating the EDL interaction energy for complex geometries. A common alternative to detailed numerical calculations for evaluating the interaction energy is the Derjaguin approximation (DA) technique [14], which approximately scales the interaction energy per unit area between two infinite flat surfaces to the corresponding interaction energy between two curved surfaces. The primary appeal of the technique lies in the separation of the geometrical effects from the interactions [12], thus leading to facile analytical expressions for the interaction energy [8,11,12]. Application of the technique is, however, limited to surfaces that are separated by small distances and for cases where the range of the interaction energy is substantially smaller than the radii of curvature of the surfaces [8,11]. For non-spherical particles — for instance, spheroidal particles — it is common to observe wide variations of the curvature at different regions of the particle surface. Consequently, the
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error in evaluation of the interaction energy using DA technique will be different for every orientation of the surface. In this paper, the orientation dependent interaction energy between a spheroidal colloidal particle and an infinite flat surface is evaluated using the surface element integration (SEI) technique [15– 17]. Surface element integration may be considered as an extension of the basic Derjaguin’s principle of expressing the interaction energy between surfaces of various geometries in terms of the corresponding plate-plate interaction energy per unit area. The primary difference between SEI and DA stems from the consideration of the exact curvatures of the surface and performing the integration over the exact contour of the surface in SEI [15]. Such a procedure is somewhat analogous to the extended Derjaguin’s technique [18,19], which also performs the integration of the interaction energy per unit area over the exact contours of the interacting surfaces facing each other. In the SEI technique, however, the integration is performed over the exact closed contours of the surface (enclosing the entire particle volume) [15,16]. This leads to a considerable improvement in the prediction of the interaction energy. It was indeed shown that such a procedure, when applied to the case of a particle interacting with an infinite flat-plate, is analogous to the Hamaker integration technique for the van der Waals interaction, and yields the same interaction energy [15]. The van der Waals and electrostatic double layer interaction energies between a spheroidal particle and an infinite flat plate are evaluated for different separation distances and orientations of the particle using SEI. The results are compared with corresponding estimates of the interaction energy based on DA technique. Furthermore, comparisons with the Hamaker integration technique for van der Waals interactions are presented to highlight the accuracy and efficiency with which SEI can evaluate the interaction energy. Finally, the DLVO interaction energy between a spheroidal particle and a planar surface is obtained for different electrolyte concentrations. Based on these results, the accuracy of the ‘equivalent sphere’ assumption is assessed.
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2. Model of an ellipsoidal particle In this section, the general expressions representing the shape and local curvatures of an ellipsoidal particle are presented. Fig. 1 shows the essential geometric parameters considered in this section. For convenience, two coordinate systems are used: a set of body-fixed coordinates to account for the internal geometric properties of the ellipsoidal particle, and a space-fixed coordinate system to account for the orientation of the particle relative to the infinite flat plate. The equation of an ellipsoid with its center at the origin of a Cartesian coordinate system is x2 y2 z2 + + =1 a2 b2 c2
(1)
where a, b, and c are the semi axes of the ellipsoid directed along the x, y, and z-axes, respectively. This equation can be represented in a spherical coordinate system using the following set of transformations between Cartesian and spherical (r, u, f) coordinates x=r sin ucos f;
y= r sin u sin f;
z= r cos u
(2)
yielding, r2=
sin2u cos2 f sin2 usin2 f cos2 u + + 2 a2 b2 c
n
−1
(3) The position vector of any point P on the surface of the ellipsoidal particle is represented by r= xi+ yj +zk = ir sin ucos f+ jr sin u sin f+ kr cos u
(4)
The tangent vectors to the u and f curves at point P are dr/du and dr/df, respectively. Using these derivatives, the unit normal to the surface at point P can be expressed as dr dr × du df n= (5) dr dr × du df We note that the unit normal is defined with respect to the body-fixed coordinate system of the
)
)
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particle. To account for the orientation of the particle relative to the space-fixed coordinate system (Fig. 1), the unit normal should be expressed using this space-fixed system. The orientation of the ellipsoidal particle can be accounted for by separately considering the rotation of the bodyfixed coordinates relative to a space fixed coordinate system (X, Y, Z) where the two sets of coordinates are related by
nˆ = (n1l11 + n2l21 + n3l31)i. + (n1l12 + n2l22 + n3l32)j. + (n1l13 + n2l23 + n3l33)k.
(7)
where i. , j. , and k. represent the unit vectors along the X, Y and Z directions, respectively, and nˆ is the unit normal to the surface at point P represented in the space-fixed coordinate system.
3. Surface element integration and Derjaguin’s approximation
X=l11x +l12y + l13z Y= l21x +l22y + l23z Z= l31x +l32y + l33z
(6)
Here, the terms lij (i, j =1 – 3) represent the direction cosines of the X, Y, and Z-axes with respect to the x, y, and z-axes, respectively. The direction cosines are directly related to the orientation angle 8. Any vector quantity n =n1i+ n2j+ n3k in the body-fixed coordinate system can then be transformed to the space-fixed coordinate system using the expression
3.1. Surface element integration The principles of surface element integration (SEI) have been discussed in detail elsewhere [15– 17]. Here, we briefly delineate the basic governing equation, followed by the specific modifications required to account for the spheroidal (or ellipsoidal) geometry. In SEI, integration of the differential interaction energy of every differential area element over
Fig. 1. Schematic representation of an ellipsoid and an infinite flat plate depicting the geometrical considerations used in the mathematical modeling. The semi-major axis c of the particle is directed along the z-axis of the body fixed coordinate system. The space-fixed and body-fixed coordinates both originate at the point O. The body-fixed coordinates are rotated at an arbitrary angle 8 relative to the space-fixed coordinates. The two required geometrical properties for the surface element integration technique are the local unit normal, n, to the surface of the ellipsoidal particle (at point P), and the local distance h of the point P from the infinite flat plate. The distance of closest approach between the particle surface and the flat plate is denoted by H.
&&
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the exact particle surface leads to the final expression for the total interaction energy between a particle and an infinite flat plate [16]. U(D)=
&
nˆ·k. E(h)dS =
S
&
A
E(h)
nˆ·k. dA nˆ·k.
(8)
Here nˆ is the outer unit normal vector on the surface element dS and k. is a unit vector normal to the flat plate (directed along the positive Z-axis of the space-fixed coordinates). The terms E(h) and U(D) represent the interaction energy per unit area between two planar surfaces located at a distance h and the total interaction energy between the particle and the infinite flat plate when the center of the particle is located at a distance D from the infinite flat plate, respectively. The last expression in Eq. (8) is obtained by projecting the surface of the particle S on a plane parallel to the infinite flat plate. In other words, the area A is a projection of the actual curved surface of the particle on a plane facing the infinite flat plate (the XY plane in Fig. 1). It is, however, important to note that the first integral in Eq. (8) is evaluated over the closed surface S of the particle, which implies that the quantity nˆ·k. can assume both positive and negative values. For a particle of arbitrary shape (in this case, an ellipsoidal particle), the differential area of a surface element around a point P located on the particle surface can be written in spherical coordinates as dS = r 2 sin u du df
(9)
where r 2 is obtained from Eq. (3). Using Eqs. (4) – (7) from Section 2, the unit normal to the surface at point P can be determined, which then leads to the evaluation of the quantity nˆ·k. in Eq. (8). Finally, the distance of the point P from the surface of the infinite flat plate is given by h= D −Z
(10)
in the space fixed coordinate system, where D is the distance of the particle center from the infinite planar surface. Using the above information, we can write the surface element integral Eq. (8) as
U(D)=
0
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p
nˆ·k. E(D− Z)r 2sinu du df
(11)
0
Note that the integration in Eq. (11) is performed over the two spherical angles in the spherical coordinate system with the quantities nˆ·k. and Z determined after taking into consideration the relative orientation of the ellipsoidal particle with the infinite flat plate.
3.2. Derjaguin’s approximation technique In this section, application of the Derjaguin approximation (DA) technique for particles of arbitrary geometrical shapes is briefly described. The DA procedure relates the interaction energy per unit area between two flat plates E and the interaction energy between two curved surfaces U by
&
U(D):
E(h) dA: f([a1],[a2])
A
&
E(h) dh
(12)
H
where H is the distance of closest approach between the two curved surfaces, E(h) is the interaction energy per unit area between two infinite flat plates separated by a distance h, dA is the differential area of the surfaces facing each other, [a1] and [a2] represent the sets of principal radii of curvature of the surfaces 1 and 2, respectively, at the distance of closest approach, and f([a1],[a2]) is a function of the radii of curvature of the surfaces. The term f([a1],[a2]) is given by [8,11] f([a1],[a2])= where l1l2 =
2p
(13)
l1l2
1 1 + a11 a21
+ sin28
1 1 + a12 a22
1 1 − a11 a12
1 1 − a21 a22
(14)
Here, a11 and a12 represent the two principal radii of curvature of particle 1, a21 and a22 represent the corresponding radii for particle 2, and 8 represents the angle between the principal axes of the two curved surfaces.
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The principal curvatures can be obtained from a parametric representation of the surface (15)
r =[x,y,I(x,y)]
where the last term results from rearranging Eq. (1) and writing it as z= I(x,y). The components of the vector r acting along the x and y directions are given by rx = (1,0,p) and ry =(0,1,q)
(16)
where p and q are the partial derivatives of I(x,y) with respect to x and y, respectively. Now, defining r = px, s =py =qx, and t =qy
(17)
the principal curvatures k1 and k2 can be obtained as the roots of the following quadratic equation in k H(1+p 2)k −r
Hpqk − s
Hpqk− s
H(1 +q 2)k − t
=0
(18)
(19)
Finally, the principal radii of curvature (for instance, a11 and a12) can be obtained as 1 1 and a12 =k − a11 =k − 1 2
Although the generalized theoretical developments in the previous sections were derived for ellipsoidal particles, where all three principal semi-axes were different, the calculations from this section onward are performed solely for spheroidal particles. In this section, the SEI technique is first employed to evaluate the van der Waals interaction energy between a spheroidal particle and an infinite flat plate and the resulting orientation dependent interaction energy is compared with the corresponding predictions of Hamaker’s approach. Following this, the predictions of SEI are compared to those obtained using DA, highlighting the limitations of the latter. Finally, the orientation-averaged interaction energy is compared with the interaction energy of an equivalent spherical particle to assess the validity of the equivalent sphere approximation.
4.1. Predictions using SEI: comparison with Hamaker’s approach
where H 2 =1+ p 2 +q 2
4. van der Waals interaction between a spheroidal particle and an infinite flat plate
(20)
Application of DA involves determination of the principal radii of curvature for a given orientation using the appropriate equation for the particle surface in Eqs. (18) – (20). These radii can then be substituted in Eqs. (12) – (14) to obtain the interaction energy between the two curved surfaces. It should be noted that the principal radii of curvature are determined at the point of closest approach between the two interacting surfaces. This poses a major limitation toward implementation of DA for surfaces of arbitrary geometrical shapes, as one should first locate the point of minimum separation between the surfaces. This leaves very little scope for obtaining analytical expressions for the interaction energy except for highly symmetric surfaces and for a few specific orientations.
The non-retarded van der Waals interaction energy per unit area between two infinite flat plates separated by a distance h is given by [1,8] E(h)= −
AH 12ph 2
(21)
where AH is the Hamaker constant of the interacting media. Substituting this in Eq. (11) yields the interaction energy between a spheroidal particle and an infinite planar surface. The integrals were evaluated numerically using Simpson’s quadrature rules. The computer code developed for this purpose returned an error estimate after each integration, which was then used to refine the integration step size until the desired accuracy limit was attained. Most of the calculations were performed using spheroidal particles having a semi-major axis of length 50 nm (c, directed along the z-axis) and by adjusting the semi-minor axis length (for spheroidal particles, a= b) to yield various aspect ratios (As =a/c) ranging from 0.2 to 1. An aspect ratio of 1 represents a spherical particle, and this limiting
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Fig. 2. Variation of the scaled van der Waals interaction energy between a spheroidal particle and an infinite flat plate with the orientation angle for three different separation distances (D) between the particle center and the infinite flat plate. The interaction energy was calculated using a Hamaker constant of 1 × 10 − 20 J, particle semi-major axis of 40 nm, and an aspect ratio of 0.5. The lines represent predictions of surface element integration (SEI) and the symbols represent predictions based on Hamaker’s approach.
case was used to check the accuracy of the numerical calculations. Although considerably more cumbersome, Hamaker’s approach may be used to evaluate the van der Waals interaction energy between a spheroidal particle and an infinite flat plate. This was performed by numerical integration of the van der Waals interaction energy between an infinite flat plate and a point (energy per unit volume) [8] over the volume of an arbitrarily oriented spheroidal particle. The procedure involves evaluation of three nested integrals. To achieve a reasonable degree of accuracy in the numerical integration, a few hundred points are required to evaluate each of these integrals. The calculations become considerably error prone for very small
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aspect ratios of the particle (As B 0.1). For As =1 (sphere), the van der Waals interaction energy can be expressed analytically [1,8], and the analytical result was used to determine the accuracy of the numerical calculations for this limiting case. Fig. 2 compares the magnitude of the scaled van der Waals interaction energy, UVDW/kT, predicted using SEI (lines) and Hamaker’s approach (symbols) for a spheroidal particle with a semimajor axis of 40 nm and an aspect ratio of 0.5. The interaction energy was obtained for various orientations of the spheroid with respect to the plate ranging from 8 = 0 to 90o, after fixing the distance D of the particle center from the flat plate. For this situation, the distance of closest approach between the particle and the plane will be minimum when the orientation angle (8) is zero (this configuration will be referred to as the end-on configuration), while it is maximum when the angle is 90o (side-on configuration). It is interesting to note that the magnitude of the van der Waals interaction corresponding to 8= 0o is larger even when the surface area facing the plate near the distance of closest approach is small. This is due to the fixed position of the particle center. If the particle center is allowed to move, a torque will orient the particle to the side-on configuration under the influence of the attractive interaction, effectively bringing the particle center closer to the planar surface. Furthermore, it is observed that as the particle center is moved away from the planar surface, the influence of orientation on the interaction energy becomes less significant. Fig. 2 also demonstrates the excellent agreement between the predictions of SEI and Hamaker’s approach. The two results are generally within 6% of each other over the ranges of separations and orientations considered in this study. The slight deviations arise due to the errors in the numerical integration in both SEI and Hamaker’s approaches. The deviations become more prominent for smaller aspect ratios. However, in real colloidal and macromolecular systems, aspect ratios smaller than 0.2 are rarely encountered [12]. The SEI result was additionally compared with the analytical result for a sphere (As = 1) flat plate interaction based on Hamaker’s approach [1], and
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identical values for the interaction energy were obtained. Finally, it should be noted that the application of SEI is not restricted to the non-retarded van der Waals interaction energy alone, and the technique can be used for predicting the retarded van der Waals interaction provided an appropriate expression of the latter for planar geometries is available.
4.2. Comparison with Derjaguin’s approximation technique The application of DA for arbitrary orientations of the spheroidal particle requires numerical evaluation of the radii of curvature of the surfaces. Furthermore, as mentioned earlier, the radii of curvature are determined at the point of closest approach, which involves complex numerical procedures for arbitrary geometries. These problems limit the applicability of DA to specific orientations utilizing symmetries of the particle shape. For the case of spheroidal particles, two limiting cases for the interaction can be expressed analytically. Table 1 depicts these two limiting cases, referred to as the end-on and the side-on configurations. Here, we consider a spheroidal particle with semi-major axis c and semi-minor axis a. The two principal radii of curvature of the particle at the distance of closest approach in the end-on configuration are both equal to a 2/c. In the sideon configuration, the two principal radii of curvature are c 2/a and a. On the other hand, both the principal radii of curvature of the flat plate are infinite. Using these information, the parameter l1l2 in Eq. (14) was determined for the two Table 1 An assessment of the interaction energy between a spheroidal particle and an infinite flat plate in two limiting orientations based on the Derjaguin geometrical factor
orientations shown in Table 1. Furthermore, the interaction energy of an equivalent spherical particle (based on equivalent volumes, which results in an equivalent radius rp = c 1/3a 2/3) is compared with the corresponding interaction energy of the spheroidal particle for the two orientations. Based on the values of l1l2 shown in Table 1, it is evident that when c \ a, the interaction energy for the end-on configuration will be much lower than the corresponding energy for the side-on configuration for a given distance of closest approach between the surfaces. Consequently, the side-on configuration will be favored in an attractive environment, while the end-on configuration will be favored in a repulsive environment. Secondly, it is noted that the interaction energy becomes lower and higher than the corresponding interaction energy of an equivalent spherical particle for the end-on and side-on configurations, respectively. This implies that the orientation-averaged interaction energy of the spheroidal particle might be quite close to the interaction energy of the equivalent sphere. This issue will be elaborated upon later. Fig. 3(a) and Fig. 3(b) show the comparison between the predictions of SEI and DA for the end-on and side-on orientations, respectively, corresponding to different aspect ratios of the spheroidal particles. The figures depict the variations of the scaled van der Waals interaction energy with the distance of closest approach between the particle and the infinite flat plate. The DA predictions in the end-on configuration (Fig. 3a) are quite accurate near contact, but deteriorate considerably at larger separation distances. The DA predictions overestimate the van der Waals interaction by about one order of magnitude at large separation distances. Furthermore, the DA predictions become worse for small aspect ratios. In the side-on orientation, application of DA leads to grossly erroneous predictions of the interaction energy. In this orientation, DA approximates the energy as the interaction energy of a sphere with radius equal to the semi-major axis of the spheroid (Fig. 3b). Consequently, the interaction energy predicted by DA becomes independent of the aspect ratio. Predictions based on SEI,
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Fig. 3. Comparison of the predictions of the scaled van der Waals interaction energy between a spheroidal particle and an infinite flat plate in the (a) end-on and (b) side-on orientations obtained using SEI and Derjaguin’s approximation (DA). The variation of the interaction energy is plotted against the minimum surface-to-surface separation distance between the particle and the planar surface for different aspect ratios. An aspect ratio of 1 represents a spherical particle. For the side-on configuration DA predictions are independent of the aspect ratio. The calculations were performed for a particle semi-major axis of 50 nm and a Hamaker constant of 1 × 10 − 20 J.
on the other hand, accurately depict the dependence of the interaction energy on the aspect ratio. This leads to a vast overestimation of the interaction energy by DA in the side-on configuration. Although the error in DA diminishes for larger aspect ratios of the particles, it is still nearly an order of magnitude greater than the corresponding SEI prediction at large separation distances.
4.3. The equi6alent sphere model As shown in the previous subsection (Table 1), DA predicts that the interaction energy of a sphere having the same volume as the spheroidal particle is bounded by the interaction energies of the spheroidal particle in the end-on and side-on configurations. This may suggest that the orienta-
tion-averaged interaction energy of the spheroidal particle might be approximated reasonably well by the equivalent sphere energy. It is, however, quite difficult to estimate the orientation dependent interaction energy using DA for arbitrary values of 8. Consequently, the SEI technique was employed to obtain the orientation-averaged interaction energy. The comparison between the orientation-averaged interaction energy for spheroidal particles of different aspect ratios and the corresponding interaction energies of the equivalent sphere is shown in Fig. 4. In this figure, the solid lines represent the orientation-averaged interaction energy U( (D)=
2 p
&
p/2
0
U(D,8) d8
(22)
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where D is the distance of the particle center from the planar surface and 8 is the orientation angle. The dashed lines represent the corresponding interaction energy of the equivalent sphere with its center located at a distance D from the planar surface. It is evident that the two interaction energies are quite similar at large separation distances when the aspect ratio is ]0.4. At small separation distances, however, the equivalent sphere approach predicts a considerably lower energy than the orientation dependent interaction energy of the spheroidal particle. Therefore, the equivalent sphere approach may lead to a considerable underestimation of the magnitude of the interaction energy when the particle is close to the planar surface.
5. Electrostatic double layer and DLVO interaction energy
5.1. Electrostatic double layer interaction The electrostatic double layer (EDL) interaction energies obtained in this section are based on the expression for the interaction energy per unit area between two infinite flat plates given by Hogg et al. [20]. This constant surface potential expression is an exact analytical solution of the linearized Poisson–Boltzmann equation for the planar geometry, and is valid for small surface potentials (cB 25 mV) and symmetrical electrolytes. The expression for the interaction energy per unit area between two infinite flat plates is ECP(h)=
o0ork 2 (c s + c 2p) 2 × +
Fig. 4. Comparison of the orientation-averaged interaction energy of a spheroidal particle with the corresponding interaction energy of an equivalent spherical particle for different aspect ratios. The interaction energy is plotted as a function of the scaled separation distance D/c. The calculations were performed for a particle semi-major axis (c) of 50 nm and a Hamaker constant of 1 × 10 − 20 J.
1− coth(kh)
2cscp cosech(kh) (c 2s + c 2p)
(23)
Here cs and cp are the surface potentials of the two flat surfaces representing the materials of the particle and the flat plate, respectively, and h represents the separation distance between the two planar surfaces. Application of SEI for determination of the EDL interaction energy between the sphere and the flat plate involves substitution of Eq. (23) in Eq. (11). The calculations were performed for constant surface potentials of 25 mV on both the particle and the planar surface, with the background electrolyte (1:1) concentration ranging from 1 mM to 0.1 M. The EDL interaction was calculated for different aspect ratios of the spheroidal particles with the semi-major axis fixed at 50 nm, as done previously for the VDW interactions. The interaction energy predictions obtained using SEI and DA techniques are compared in Figs. 5 and 6 for the limiting cases of end-on and side-on configurations, respectively. Fig. 5 shows the variation of the EDL interaction energy between a spheroidal particle in the end-on configuration and a planar surface with the separation
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Fig. 5. Variation of the scaled electrostatic double layer interaction energy between an infinite flat plate and a spheroidal particle in the end-on configuration with separation distance. The calculations were performed for three aspect ratios. The solid lines were obtained using SEI while the symbols represent DA results. The surface potentials on both the particle and the plate were assumed to be 25 mV and the electrolyte (1:1) concentration was (a) 0.1 M and (b) 0.001 M. The particle semi-major axis length was 50 nm.
distance (H = D −c) for different aspect ratios of the particles. The predictions of SEI (lines) and DA (symbols) corresponding to a high electrolyte concentration of 0.1 M (Fig. 5a) are in remarkably close agreement for all the aspect ratios. Even for an aspect ratio of 0.2, the deviation between the two results is not significant. The high background electrolyte concentration of 0.1 M results in a small Debye screening length k − 1 thereby rendering the electrostatic double layer interactions very short ranged. Under these conditions, DA is expected to be quite accurate. Thus, it may be concluded that for highly screened EDL interactions, the interaction energy for the end-on configuration can be predicted equally well by SEI and DA. The predictions of SEI and DA for a low electrolyte concentration of 1 mM, however, differ considerably (Fig. 5b). The longer range of the EDL interactions in this case imparts a significant error in the interaction energy predictions
obtained using DA. The deviation between the SEI and DA predictions increases with decreasing the aspect ratio. The corresponding predictions of the interaction energy for the side-on configuration are depicted in Fig. 6. Here, the separation distance H= D− a. Once again, it is observed that the predictions of SEI (lines) and DA (symbols) are in reasonably good agreement at high electrolyte concentrations (corresponding to electrolyte concentration of 0.1 M). For an aspect ratio of 0.5, the SEI and DA predictions are nearly identical for this case. It should be noted, however, that although DA predicts the EDL interaction energy for the side-on configuration reasonably well at high electrolyte concentrations, it is qualitatively incorrect since it does not predict any variation of the interaction energy with the aspect ratio of the particle. This error is clearly manifested at low electrolyte concentrations, when the EDL interac-
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tion becomes considerably long ranged. As shown in Fig. 6, for an electrolyte concentration of 0.001 M, the predictions of SEI and DA differ substantially for small aspect ratios. In this case, while SEI predicts an increase in the interaction energy with the aspect ratio of the particle, the predictions of DA remain independent of the aspect ratio, and represent the interaction energy of a sphere with an infinite planar surface.
5.2. The DLVO interaction energy The DLVO interaction energy is obtained by summing the VDW and the EDL interactions. The predictions of the DLVO potential between a spheroidal particle and an infinite planar surface is shown in Fig. 7 for a particle having an aspect ratio of 0.5 and a semi-major axis of 50 nm. The
Fig. 7. Comparison of the DLVO interaction energy for the end-on and side-on configurations at two different electrolyte (1:1) concentrations (0.001 and 0.1 M) as predicted by SEI. The interaction energy was calculated assuming a Hamaker constant of 1×10 − 20 J, a surface potential of 25 mV for the particle and the flat plate, particle semi-major axis of 50 nm, and an aspect ratio of 0.5.
Fig. 6. Variation of the scaled electrostatic double layer interaction energy between an infinite flat plate and a spheroidal particle in the side-on configuration with separation distance. The solid lines and symbols represent the SEI and DA predictions, respectively. The calculations were performed using the same surface potentials and particle size as in Fig. 5, for two different electrolyte (1:1) concentrations: 0.001 and 0.1 M.
two extreme cases of side-on and end-on configurations are shown. At a high electrolyte concentration (0.1 M), the EDL interaction becomes quite short ranged, thereby yielding a shallow secondary minimum predominantly governed by the magnitude of the attractive VDW interaction. The secondary minimum is more prominent for the side-on configuration. As the particle approaches closer to the planar surface, the energy profile encounters a repulsive barrier, the height of which is larger for the side-on configuration. At a low electrolyte concentration (0.001 M), the repulsive barrier becomes considerably broader. In this case, the interaction energy of the particle remains greater in the side-on configuration over the entire range of separation distance shown. From Fig. 7, it may be concluded that for the set of parameter values used in this study, the
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end-on configuration will be energetically favorable (lower repulsive energy) in presence of repulsive double layer interactions at low ionic strength. It should be noted that the interaction energy at any given separation distance will vary only between the extreme limits of the end-on and side-on configurations. At high electrolyte concentrations, the difference in the interaction energy between the end-on and side-on configurations is quite small, and consideration of orientation dependence may not be necessary except at very small separation distances between the particle and the planar surface.
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clearer picture of the potential energy profiles between a particle and a flat plate, leading to the determination of the energetically favorable pathways for particle diffusion from a bulk dispersion toward a flat surface. Furthermore, accurate interaction energy calculations can also shed light on the preferred orientations in which spheroidal particles deposit on planar surfaces.
Acknowledgements The work reported in this paper was supported by the National Science Foundation under Research Grant BES-9996240.
6. Concluding remarks The surface element integration (SEI) technique can be used to calculate the DLVO interaction energy between a planar surface and an arbitrarily oriented spheroidal particle quite accurately. The technique is considerably simpler than either Hamaker’s approach for evaluating van der Waals interaction or the detailed numerical solutions of the Poisson – Boltzmann equation leading to numerical estimates of the electrostatic double layer interactions. Furthermore, SEI provides very accurate predictions of the interaction energy between a particle and an infinite flat plate for any orientational configuration of the particle. For particles of spheroidal shape, Derjaguin’s approximation (DA) can be applied to obtain analytical expressions for the interaction energy only for specific orientations and symmetries in particle shape. Furthermore, DA may lead to considerably erroneous predictions of the interaction energy, specifically when dealing with the side-on configuration and for long-range interactions. The calculations shown here indicate that the assumption of a spherical particle may lead to reasonably accurate predictions of the interaction energy of a spheroidal particle with aspect ratio as low as 0.4 at large separation distances from a planar surface. However, consideration of the actual spheroidal shape and the ensuing orientation dependent interaction energy is necessary for small separation distances and long-range interactions. Such detailed calculations may lead to a
Appendix A. Nomenclature List of symbols a, b, c semi-axes of an ellipsoidal particle aij principal radii of curvature of surface at distance of closest approach, Eq. (14) A area AH Hamaker constant As aspect ratio of spheroidal particle dS differential area of a surface element D separation distance of infinite flat plate from the particle center E interaction energy per unit area between two infinite planar surfaces h local separation distance between a surface and a flat plate H distance of closest approach between the particle surface and a flat plate i, j, k unit vectors in a Cartesian coordinate system Boltzmann constant (1.38×10−23 k JK−1) k1, k2 principal curvatures of a surface at a given point P n unit outward normal vector to the surface in body fixed coordinates nˆ unit outward normal to the surface in space fixed coordinates
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n1, n2, n3
r r S T x, y, z X, Y, Z U( UDLVO UEDL UVDW
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direction cosines of the unit normal vector n with respect to the body fixed coordinates radial coordinate in a spherical coordinate system position vector at a point on the surface of a particle surface area of a particle temperature (K) body-fixed coordinate axes space-fixed coordinate axes radial average interaction energy of a spheroidal particle DLVO interaction energy electrostatic double layer interaction energy van der Waals interaction energy
Greek symbols or dielectric constant of solvent o0 dielectric permittivity of vacuum (8.854×10−12 C2N−1m−2) 8 orientation angle of the spheroidal particle k inverse Debye screening length f, u angular coordinates in a spherical coordinate system cp surface potential of flat plate cs surface potential of spheroidal particle
.
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