2.2. Hamaker’s Microscopic Approach

Hamaker [33] determined the van der Waals interaction of two macroscopic bodies by carrying out an integration of all the intermolecular interactions. First, van der Waals interactions between two particles were considered as pairwise inter-particle interactions and tended to be generalized on interactions between large bodies. These large body “macroscopic body” interactions were developed by Hamaker by summation of pairwise interactions over all constituent atoms. The Hamaker approach is widely used because of its simplicity and applicability. General dispersion pair interaction potentials between two interacting molecules separated by a distance r can be assumed to be of the power-law form:

where C_d = 3α²I/4(4πε₀)² is the Lifshitz-van der Waals potential coefficient (α = polarizability; I = certain characteristic potentials of the atom) and n depends on the specific potential model used for inter-molecular potential, (for r > molecule diameter, d, n = 6) [34].

According to Hamaker’s microscopic approach, the interaction between a molecule and a nearby planar surface of a solid is assumed to be purely attractive. Initially, the volume of the ring containing interacting molecules in the solid is calculated as shown in Figure 1. The radius of the ring, R_ring, and the distance of the ring to the side of the solid, x, may be extended to infinity during the integration process to cover all the molecules in the solid slab [35]. Geometrically, the total number of molecules in a circular ring of volume V = 2πR_ringdR_ringdx is n = ρ2πR_ringdR_ringdx, where ρ is the number density of the molecule (molecule/m³). By assuming the sum of the interactions of all molecules present in the solid body with this single molecule, and approximating this summation process with an integral, the net interaction for a molecule of the same material at a distance, D, away from the surface of the solid slab becomes:

Interaction between a molecule and a solid slab. D = distance between the molecule and the surface of the solid plane; R_ring = radius of the ring; x is the distance of the ring to the side of the solid; and r = distance between the molecule and the ring of molecules in the planar surface.

By applying Pythagoras’s theorem, r² = (D + x)² + R²_ringr2=(D+x)2+Rring2r2=(D+x)2+Rring2, and since 2R_ringdR_ring = dR²_ring, .

and by inserting this into Equation (2), we obtain:

From Equation (5), it is shown that the interaction potential energy of a molecule and a macroscopic surface decreases proportionally to D⁻³, instead of D⁻⁶ for molecule-molecule interaction. This integration was simplified by several assumptions used by Hamaker: The multibody interactions are discounted and the interactions are only pair-wise; the intervening medium is a vacuum; the molecule and the solid body are not distorted by the attractive forces; the interactions due to the Coulomb forces and permanent dipoles are neglected; all dispersion force attractions are due to a single dominant frequency; the interactions of molecular electron clouds are instantaneous; and the solid body is assumed to have uniform density right to the interface.

By applying a similar procedure for a sphere particle to flat plate, the interaction potential energy between a large spherical particle with radius, R_sph, and the ring in the plate surface (Figure 2) can be calculated. The volume of the circular slice in the sphere is:

Interaction between a large sphere and a solid flat surface.

R_slice is related to R_sph by using chord theorem of plane geometry by applying Pythagoras’s theorem to all triangles in Figure 3. From the figure, [(AC)² = (AB)² + (BC)²] giving (AC)² = [(AD)² + (BD)²] + [(BD)² + (CD)²]. After simplification and rearrangement, this yields [(2R)² = (2R − x)² + 2h² + x²] giving [h² = x(2R − x)].

Right-angled triangle used in the proof of the chord theorem of plane geometry.

Thus, corresponding to Figure 3:

Then the total volume of the thin circular slice in the sphere becomes:

Therefore, the total number of molecules in this slice is πρ(2Rsph−x)xdx. Since all the molecules in this sphere are at a distance of (D + x) from the flat surface, the total sphere to surface dispersion interaction potential can be found by multiplying the number of molecules within the spherical particle and the dispersion interaction potential of a single molecule to a flat surface (Equation (8)) and becomes:

Therefore, by derivation of this equation (Equation (9)), the van der Waals, vdW interaction potential of a large spherical particle can be calculated. This equation can then be applied into the calculation of the vdW interaction potential for the capsule-shaped particle to flat plate interaction. The capsule model is defined by the combination of a spherical and a cylindrical-shaped particle.

Once the spherical particle to flat surface van der Waals, vdW interaction potential has been derived, the vdW interaction potential for a cylindrical particle to flat plate interaction can be derived in order to justify and satisfy the suggested capsule model. The vdW interaction between a cylindrical particle and a flat plate can be both horizontal and vertical. By considering these geometrical orientations, it can be shown that the effective area of interaction for sphere to flat surface is the ring of molecule’s element inside the interacting bodies.

As referred to Figure 4, the van der Waals interaction potential between a horizontal cylinder and a flat surface is the sum of all molecule elements in a cylinder of radius R_c and length L. For an infinitesimal differential volume element of cross-sectional area, rdϕdr,and vertical height dz, the element volume is rsdϕdrsdz and the total number of molecules in the element is ρrsdϕdrsdz where ρ is the number density of molecules in the cylinder. Thus, this enables us to calculate the overall interaction potential of molecules within the cylindrical particle. Geometrically, the distance between a differential volume element (the blue cube in Figure 4) and any molecule in ring in the flat surface is equal to:

Interaction between a horizontal cylindrical particle and a solid flat surface.

First, the interaction potential between the cylindrical particle (summation of overall molecules in the cylinder) to a molecule in ring in the flat surface will be:

Second, to calculate the total van der Waals interaction potential between a horizontal cylinder and the ring in the flat surface as shown in Figure 4, the interaction of a cylinder and the circular element ring of cross-sectional area πR_ring² and thickness dx in the surface is considered. The volume of the ring is πR_ring²dx and the number of molecules in this volume element is ρ_surfπR_ring²dx, where ρ_surf is the number density of molecules in the flat surface. Since all the molecules in the ring are at the same separation distance (D + x) away from the cylinder surface, the overall van der Waals interaction potential between this element ring and the cylinder is v(D + x)ρ_surf πR_ring²dx, where v(D) is given in Equation (11). Thus, the total interaction potential between the horizontal cylindrical particle and the flat surface is equal to:

The total dispersion interaction for a vertical cylinder can be found in the same way as that for the spherical particle interaction, using the ring of elements from a slice of the cylindrical particle’s cross-sectional area (Figure 5). Except for dx, where x is changed from 0 to the cylinder length, L. From the volume of the cylinder, V = πR_c²L; the total dispersion interaction potential for a vertical cylinder can be given as:

Interaction between a vertical cylindrical particle and a solid flat surface.

The derived van der Waals interaction potential equations for the geometrical shapes described earlier are unretarded (without retardation effect) interaction equations for sphere, cylinder, and rod-to-solid flat surfaces and are derived based on an implicit assumption that the speed of light is infinite. “Retardation effect” is the correction factor that must be made because of the reduction in the van der Waals interaction potential due to the finite speed of light [26,36]. Owing to the electromagnetic nature of the interaction, the actual van der Waals potential between two interacting bodies is reduced at large separation distances because the time it takes for the electric field to propagate from one body to another and back is such that the fluctuating electric moments become slightly out of phase. Such an effect may become very pronounced for macroscopic bodies at separation distances larger than about 5 nm [24]. For a large particle with diameter of 1 µm or greater, omission of the retardation has proved to lead to a serious overestimate of the van der Waals interaction[19]. In principle, the Hamaker approach can be modified to account for this retardation by multiplying the unretarded vdW interaction potential by a correction factor:

Where the correction factor, f(p), depends on the reduced distance, p= 2πrλ, and λ is the “London characteristic wavelength” of the interaction. λ is assumed to be about 100 nm for most materials and the retardation only becomes significant when the separation distance between particles is of the same order as the characteristic wavelength [19] as shown in Figure 6. Since the rod particles are so much smaller than the flat plate, the existing correction factor for the sphere-plate interaction potential may be most suitable to account for the retardation effect on the rod-flat plate interaction system. The correction factor given by [37] as reported by [19] is as follows:

where, λ is a retardation parameter and “s” is a constant (s = 11.116). Multiplying the unretarded van der Waals interaction potentials by f (D, λ) yields the approximate retarded van der Waals interaction potential. When using a bacterium as the macroscopic body, there is only a small difference (up to 10–30J) between retarded and non-retarded vdW forces, and therefore will be ignored in this study.

The van der Waals forces with non-retarded and retarded regimes [23].