General Potential for Anisotropic ColloidâSurface Interactions

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General Potential for Anisotropic Colloid-Surface Interactions Isaac Torres-Díaz, and Michael A. Bevan Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b00051 • Publication Date (Web): 07 Apr 2017 Downloaded from http://pubs.acs.org on April 16, 2017

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General Potential for Anisotropic Colloid-Surface Interactions Isaac Torres-Díaz and Michael A. Bevan∗ Chemical & Biomolecular Engineering, Johns Hopkins University, Baltimore, MD 21218

Abstract A general closed-form, analytical potential is developed for the interaction of planar surfaces with super-ellipsoidal particles (which includes shapes such as spheres, ellipsoids, cylinders, polygons, super-spheres, etc.). The Derjaguin approximation is used with DLVO halfspace interactions (e.g., electrostatics and van der Waals) to yield potentials for arbitrary particlewall separation and orientation. The resulting potential is a function of the minimum distance between surfaces and the particle’s local Gaussian curvature at the minimum distance position. The validity of the solution is reported in terms of the local Gaussian curvature Γ and characteristic interaction range (e.g., Debye length, κ-1, for electrostatics) based on the limits of the Derjaguin approximation. This solution is limited for super-ellipsoids with convex shapes and orientations where the condition κ/Γ1/2>2 is satisfied. The potentials reported in this work should be useful for modelling a wide range of natural and synthetic non-spherical and anisotropic colloidal particles in environmental, biological, and advanced material applications.

Introduction There has been an increase in the variety of colloidal particle shapes that can be synthesized and a commensurate demand in the increase to understand how such objects interact with each other and substrates in applications. Well-established methods for controlling particle shape include the growth of inorganic nanoparticles1 and mechanical deformation of polymer colloids.2 Novel approaches continue to be developed for producing particles with different combinations of shape and surface functionality including rods,3 cubes,4-5 polyhedra,6-9 superspheres,10-12 etc. These exotic and diverse anisotropic particles have been summarized in review articles and classified according to their aspect ratio, patchiness, and branching.13-15 Anisotropic colloidal particles are of great interest as constituents in advanced materials and for use in a variety of technologies including colloidal assembly,16-20 photonic materials,21-22 and drug delivery.23-25 However, accurate models of physically realistic interactions between particles with each other and surfaces are still lacking. While several models have been developed for anisotropic particle interaction potentials, these have been limited to uniaxial ellipsoids,26-27 cylinders,28 hard wall objects,29 and approximate molecular potentials.30-31 In general, closed-form analytical potentials are not available for DLVO interactions (e.g., electrostatic, van der Waals) between different shaped particles and surfaces, but exist only for spherical particles.32-33 A general approach to model interactions between spherical particles and surfaces is to employ the Derjaguin approximation in combination with rigorous parallel flat plate results (e.g., see solutions for electrostatics34 and van der Waals35). This approach is commonly used to model DLVO sphere-sphere and sphere-plate interactions.36 This methodology has been generalized to

∗ To whom correspondence should be addressed: [email protected]

Torres-Díaz & Bevan

ACS Paragon Plus Environment

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describe the interaction energy between two convex anisotropic colloidal particles, taking into account the principal radii of curvature of the surfaces at the points where the distance between surfaces is a minimum.37 This general approach has also been used to derive an analytical potential for the specific case of axisymmetric ellipsoidal particles.26, 38 No analytical expressions have been reported in the literature for the interaction between arbitrarily shaped and oriented anisotropic particles with planar wall surfaces. Here, we present a closed-form expression for the interaction potential for arbitrarily oriented convex anisotropic particles and a planar wall using the Derjaguin approximation, which can be written in a general closed form as37, 39

U ( h,θ ,φ ,ψ ) = 2π Γ (θ ,φ ,ψ )

−0.5

Q ( h) ,

(1)

where h is the distance of closest approach between the particle and wall, Γ(θ, φ, ψ) is the particle’s local Gaussian curvature at the point where the particle wall surfaces are closest, and Q(h) is the integral over the half-space interaction energy per area. The particle geometry is defined by the super-ellipsoid equation (see Fig. 1) given by40

 x′ / r r + y′ / r r  x y  

n/r

n

+ z ′ / rz = 1,

(2)

where rx, ry, and rz are the principal semi-axes of the particle, and n and r are parameters that define the particle shape, which are greater than zero. When any of the values of n or r 1 to generate only convex geometries for consideration with the Derjaguin approximation.

shape. Eq. (7) is valid for particles with convex surfaces (i.e. n>1 and r>1), although similar expressions can be derived for concave shapes (n, r < 1). The Gaussian curvature (Γ) of a surface at point (x’, y’, z’) is the product of the maximum, γ1, and minimum, γ2, values of normal curvature (i.e., principal curvatures47). The Gaussian curvature at any point on a super-ellipsoidal particle surface (using the parametric form of the surface S (Eq. (15)) is given by

( n −1) ( r − 1) rx2ry2rz2 cos ( v ) cos ( u ) 4

Γ=

4+2r r

sin ( v )

4+2r n

sin ( u )

4+2r r

4 4 4 4  2 2  4 4 n cos u r sin v sin u r + r 2 cos v sin v n λ r r cos v ( ) ( ) ( ) ( ) z ( ) ( ) Γ   x y



4

4

2

, (26)



λΓ =  rx2 cos ( u ) r sin ( u ) + ry2 cos ( u ) sin ( u ) r  ,   4

4

which can be evaluated at the point where the distance between surfaces is minimum (Eq. (23)) to give Eq. (9).



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Results & Discussion Super-Ellipsoid Sub-Classes To predict particle-wall interactions for the types of particles illustrated in Fig. 1, Fig. 3 depicts different particle geometries defined by a super–ellipsoid with equal semi-axis lengths (i.e., a super–sphere) vs. the shape parameters n and r in Eqs. (2) and (15). It is shown that the super–sphere (rx = ry = rz) encloses different shapes such as the sphere (n=r=2), octahedron (n=r=1), and cube (n=r ➝ ∞). In the present work, the values of n and r are defined to be larger than 1 to obtain a convex particle surface, which is a required condition to apply the Derjaguin approximation. It is in general possible using the approach outlined here to generate many additional particle shapes with locally convex surfaces. As described by Eq.(4) , the Derjaguin approximation depends on both the minimum distance between surfaces (Eq. (7)) and the local Gaussian curvature at the position where the distance is a minimum (Eq. (9)). These expressions reduce for particular shapes as shown in Table 1, such as when n=r, for ellipsoidal particles when n=r=2, and in the limit when r=n→∞. Ultimately, these expressions reduce to the case for sphere-wall interactions by letting 2 rx=ry=rz=a, n=r=2, and A132 + A23 + A332 = 1 . Eqs. (12) and (14) reduce to the interaction energy between a spherical colloid and wall, where the minimum distance between surfaces is h=z-a and Minimum distance between surfaces (h = z – S

z,min

Gaussian curvature (Γ)

)

n≠r

r−1 n  r n  r r n−1   z −  rx A13 r−1 + ry A23 r−1  + rz A33 n−1      

n=r

n n n   z −  rx A13 n−1 + ry A23 n−1 + rz A33 n−1   

n,r=2

z−

n,r

∞

( rx A13 )2 + ( ry A23 ) + ( rz A33 ) 2

(

z − rx A13 + ry A23 + rz A33

2

n−1 n

n−1 n

( n − 1) ( r − 1) S n+2 n−1 rx2 ry2 rz2

z,min

rx A13

r−2 r−1

ry A23

r−2 r−1

rz A33

n−2 n−1

×

2( n−r )

r r   r( n−1) r−1 + r A r−1 r A y 23  x 13 

( n − 1)2 S n+2 n−1 rx2 ry2 rz2

z,min

rx A13

n−2 n−1

(

ry A23

n−2 n−1

rz A33

n−2 n−1

)

2 2 1  ( rx A13 )2 + ry A23 + ( rz A33 )  rr r 2 2 2 x y z

)

0

Table 1. Columns indicate: shape parameters (n, r) from Eqs. (2) and (15), schematics of each particle class, minimum distance (h) between a particle with arbitrary orientation and a wall, and local Gaussian curvature (Γ). The particle center is located at an elevation z above the wall. Super-ellipsoids are defined by semi-axes rx, ry, rz. A13, A33, A33 are components of the transformation matrix in Eq. (18).



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the Gaussian curvature is Γ=a-2, which yields the familiar result for sphere-wall interactions,

U ( z ) = aZ e−κ ( z−a ) −

aAH . 6 ( z − a)

(27)

Table 1 summarizes expressions for minimum distance between surfaces and local Gaussian curvature for particular geometries. The analytical expression for minimum distance has no restriction on particle orientation for super-ellipsoidal particles with the convex shapes (n>1, r>1) depicted in Fig. 3. They are also valid for surfaces with small (n, r2) of n and r, which correspond to limiting cases with sharp edges (i.e., large values of Gaussian curvature). The expression for Gaussian curvature (n≠r) has a singularity when one of the components of the transformation matrix (A13, A23, A33) is zero. This singularity results when the sharp corner of the particle (n or r < 2) is oriented towards the wall. However, as we discuss in the following section, the Derjaguin approximation is not applicable for all particles geometries and orientations. Ellipsoid-Wall Interactions The minimum distance between the particle and the wall, h, is a linear function of the particle position z, and Sz,min (Eq.(8)). It is clear that due to particle symmetry around the z–axis (lab frame), h is not a function of the azimuthal angle θ. This is consistent with Eq. (18), where

Fig. 4. (a) Schematic of ellipsoidal particles with different orientations and color coded according to points marked on plots in parts (b) and (c). Particle semi-axes are rz/a=1, ry /a=2, rx/a=3, with the minimum axis equal to a = 1µm. (b) Minimum distance between a particle (located at a fixed position z/a=3) and a planar surface, and (c) Gaussian curvature for different particle orientations. (d)-(e) DLVO interaction energy (kBT) between an ellipsoidal particle and a planar wall as a function of minimum separation distance between the surfaces and the azimuthal angle ϕ, with (c) ψ=0, and (d) ψ=π/2. Curve colors correspond to the depicted particle orientations in (a).



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the transformation matrix terms A13, A23, and A33 are also independent of θ. Because there are an infinite particle shapes that can be described by the super-ellipsoid equation, we present results for specific cases to demonstrate the basic features of the resulting interaction potentials. As shown schematically in Fig. 4(a), first, we consider an ellipsoidal particle (n=r=2) with three different semi-axes lengths rx/a=1, ry/a=2, rz/a=3, and with the minimum axis equal to a=1µm. Fig. 4(b) shows the minimum separation distance between the particle and a wall for a range of different orientations. Fig. 4(c) shows the dimensionless curvature (a2Γ) for the same separations and particle orientations depicted in Fig. 4(b). Fig. 4(d), (e) show the DLVO interaction energy in units of kBT as a function of the minimum distance between surfaces (h/a), azimuthal angle (ϕ), and the rotational angle around the z’ axis (ψ). Constants in these theoretical profiles are ε m = 78ε 0 , zv = 1, Ψ p = Ψ w = −75mV , κ −1 = 33nm , and AH = 3k BT (in Eqs. (12)(14) ). The results in Fig. 4(d), (e) indicate the minimum interaction energy between an ellipsoidal particle and a planar wall is when the minimum axis of the particle is perpendicular to the wall (ϕ=ψ=π/2), which is consistent with expectations. Super-Sphere-Wall Interactions To better visualize the spatial functionality of the interaction energy for different particle shapes, in Fig. 5 we report energy landscapes for two representative super-spheroidal particle shapes with (n=r=1.6 and n=r=3.2) and the same semi–axes length (rx/a=ry/a=rz/a=1). In Fig. 5(a), the particles are depicted at different orientations (a.1) n=r=1.6 and (a.2) n=r=3.2 for fixed elevations of z/a=2. The physical constants used to generate the results in Fig. 5 are the same as those used to generate the results in Fig. 4. The colored surface on the particle represents the minimum distance (h) between the particle and the planar wall. The projection of the surface color map is depicted in Fig. 5(b). Since the super-spheroidal particles are orthotropic, h is symmetric over the three axes of the particle, and orientations between 0 ≤ ϕ ≤ π/2 and 0 ≤ ψ ≤ π/2 describe the behavior of the particle. However, the analysis for these particle geometries does not include orientations with ϕ, ψ=0 and ϕ, ψ=π/2 to avoid the zero curvature in n=r=3.2 and singularity in n=r=1.6. Every particle shape with n and r values larger than two has regions of zero curvature at these orientations. Practically, the interaction energy at these orientations can be quantified by the limiting value of the potential at surrounding orientations with non-zero curvature. Fig. 5(c) shows the dimensionless curvature (a2Γ) for the same particle orientations; the surface has a cut-off value of 10 kBT to aid visualization. The electrostatic interaction energy as a function of the angles (ϕ, ψ) and the minimum separation between surfaces (h) is depicted in Fig. 5(d). A section of the interaction potential is cut out to aid visualization of the energy landscape. The representative orientations illustrated in Fig. 5(a) are denoted by points in Fig. 5(b)-(d). Animated renderings of Fig. 5 (see Supporting Information, SI), show in detail via sequential snapshots how particle positions and orientations correspond to coordinates within the multidimensional energy landscapes (Movie S1 shows left-hand-side of Fig. 5 for n=r=1.6 and Movie S2 shows right-hand-side of Fig. 5 for n=r=3.2). For both particle shapes in Fig. 5, results show that for a fixed particle surface-wall separation, the minimum energy occurs when the particle is oriented such that regions of high local Gaussian curvature are closest to the wall. For example, on super-spheres with increasingly sharp edges and corners approaching regular polyhedra (i.e., cubes, octahedra), lower interaction



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Fig. 5. (a) Schematics of super-spherical particles at different orientations with respect to wall. Semi-axes are rz=ry=rx=a=1µm, and shape parameters are equal to (a.1) n=r=1.6 and (a.2) n=r=3.2. (b) Minimum distance between surfaces for particle located at z=2a as a function of ϕ and ψ. (c) Gaussian curvature of particles at the point where the distance between surfaces is a minimum vs. ϕ and ψ. (d) Electrostatic interactions between a particle with arbitrary orientation and wall vs. minimum separation distance between the surfaces and angles ϕ and ψ. The particular orientations represented in (a) are denoted by points. The Supporting Information contains animated renderings of the left and right panels to visualize height, Gaussian curvature, and position on the energy landscape for each orientation.

energies are observed when corners and edges (i.e., higher curvature) have the distance of closest approach to the wall rather than the flatter faces (i.e., lower curvature). This is consistent with Eq. (1), which shows that the interaction energy is inversely proportional to the square root of the Gaussian curvature. The dependence of the Derjaguin approximation on Gaussian curvature provides two important limiting cases. When the curvature is identically zero, a singularity is generated, and the interaction energy should be calculated by the equivalent expression for parallel plates (see Eq. (4)). Since the Derjaguin approximation takes into account a continuously differentiable surface, surfaces with sharp corners or edges are not suitable to calculate the interaction energy. In the following section, we analyze and report estimates of maximum particle curvature that still satisfy the assumptions underlying the Derjaguin approximation.



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Fig. 6. (a) Cartoon representing the Derjaguin approximation for spherical particles with different particle size. γ1 represents the curvature, and γ1-1 is the radius of curvature. For convex anisotropic particles, and considering a reference particle with γ1-1=a and thin electrostatic double layers (κa >> 1), in (b) it is shown that Derjaguin approximation is valid for particle orientations where the equivalent radius of curvature γ1-1>a, but not for particles with γ1-1>a. Here, we aim to quantify when the Derjaguin approximation is valid for an anisotropic particle with local Gaussian curvature, Γ, and an electrostatic double layer interaction characterized by the Debye length, κ –1. For the same super-spheres shown in Fig. 3 (vs. shape parameters n and r), Fig. 7 shows results for a dimensionless characteristic length 1/(a2Γ)1/2 indicated by coloring local surface points with a linear spectrum scale (shown by vertical scale on the right hand plot in Fig. 7(b)). The spherical particle with 1/(a2Γ)1/2 =1 is taken as reference, which is set to a cyan color. For a spherical colloid interacting with a wall, the Derjaguin approximation is accurate for a relative Debye length and particle radius when κa>2.33, 42 The sphere analysis can be extended to anisotropic particles based on local Gaussian curvature to yield κa/(a2Γ)1/2=κ/Γ1/2>2 as the criterion for the validity of the Derjaguin approximation. Using this criterion, Fig. 7(b) shows 1/(a2Γ)1/2 vs. κa when the Derjaguin



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approximation is valid. As such, it is possible using the color scheme in Fig. 7(b) to determine which colors on particle surfaces in Fig. 7(a) are valid to consider using the potential in Eqs. (3)(7) for a given Debye length (κ), minimum anisotropic particle axis (a), and shape parameters (n, r). For example, for essentially all κa in Fig. 7(b), only the violet and darkest blue regions have local Gaussian curvatures that are too high to obtain accurate electrostatic interaction potentials based on the Derjaguin approximation. In summary, the interaction potential for convex shape particles requires, as expressed in Eq. (1), that the local curvature of the particle be non-zero. For particle orientations where the local curvature is zero, the piece-wise potential (Eq. (4)) can be used at all orientations except when the surface is violet. In this case, a different potential could be substituted to approximate or more accurately model particle surface regions with high local Gaussian curvature. Furthermore, in regions with high Gaussian curvature (e.g., edges, corners), the Derjaguin approximation can be used at orientations where violet regions are not the closest point to the wall (see Fig. 7). The potential reported in the work should also be useful in problems where other constraints force low-curvature regions to be preferentially oriented near a wall surface; for example, an external field like gravity could confine many of the particles reported in this paper to have their low-curvature faces parallel to an underlying flat substrate surface. Finally, for clarification, the expressions for distance of closest approach and Gaussian curvature in Eqs. (7), (8), and (9), summarized in Table 1, are always correct; it is only the constraint on the Derjaguin approximation that affects the limits of validity of the final potential.

Conclusions A closed-form interaction potential was developed for super-ellipsoidal particles interacting with planar surfaces using the Derjaguin approximation. Analytical expressions for local Gaussian curvature and minimum distance between an arbitrarily oriented super–ellipsoidal particle and a planar wall are reported. The analytical DLVO potential is applicable using different orientation parameters for the transformation matrix, such as Euler angles or quaternion parameters. We also conclude that the Derjaguin approximation is limited to particle orientations where the condition κ/Γ1/2>2 is satisfied, where Γ is local Gaussian curvature and κ –1 is the Debye length. Ultimately, the approach developed here can be applied to any type of interaction for which the Derjaguin approximation in combination with flat plate results is considered to be valid (as counter-examples, such an approach is invalid if local curvature fundamentally influences the physical mechanism leading to the underlying interaction, which can be the case for capillary, steric, and depletion interactions48).

Acknowledgments We acknowledge financial support by the National Science Foundation (CBET-1434993).

Supporting Information Supporting information is available containing animated renderings to visualize supersphere orientation dependent height at distance of closest approach, local surface Gaussian curvature, and corresponding energy landscape coordinates. This material is available free of charge at http://pubs.acs.org.



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General Potential for Anisotropic Colloid-Surface Interactions Isaac Torres-Díaz and Michael A. Bevan Chemical & Biomolecular Engineering, Johns Hopkins University, Baltimore, MD 21218



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General Potential for Anisotropic ColloidâSurface Interactions

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