Electrostatic interaction between dissimilar colloids at fluid interfaces

Arghya Majee, Timo Schmetzer, and Markus Bier

Phys. Rev. E 97, 042611 – Published 24 April 2018

Abstract

The electrostatic interaction between two nonidentical, moderately charged colloids situated in close proximity of each other at a fluid interface is studied. By resorting to a well-justified model system, this problem is analytically solved within the framework of linearized Poisson-Boltzmann density functional theory. The resulting interaction comprises a surface and a line part, both of which, as functions of the interparticle separation, show a rich behavior including monotonic as well as nonmonotonic variations. In almost all cases, these variations cannot be captured correctly by using the superposition approximation. Moreover, expressions for the surface tensions, the line tensions and the fluid-fluid interfacial tension, which are all independent of the interparticle separation, are obtained. Our results are expected to be particularly useful for emulsions stabilized by oppositely charged particles.

Received 15 January 2018

DOI:https://doi.org/10.1103/PhysRevE.97.042611

Physics Subject Headings (PhySH)

Electrostatic interactions

Charged colloids Liquid-liquid interfaces

Density functional theory

Polymers & Soft Matter

Authors & Affiliations

Arghya Majee^*, Timo Schmetzer, and Markus Bier^†

Max-Planck-Institut für Intelligente Systeme, 70569 Stuttgart, Germany and IV. Institut für Theoretische Physik, Universität Stuttgart, 70569 Stuttgart, Germany

^*majee@is.mpg.de
^†bier@is.mpg.de

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Vol. 97, Iss. 4 — April 2018

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Figure 1
(a) Sketch of the system under consideration. A pair of nonidentical spherical colloids trapped at an oil-water interface (indicated by the horizontal green line) and separated from each other by a distance small compared to their radii. We consider neutral-wetting situations corresponding to $90^{\circ}$ liquid-particle contact angles for all particles. (b) Sketch of the simplified system that models the boxed region in panel (a). Due to their short separation, particles are approximated as flat walls positioned at $z = 0$ and $L$ . The region in between is filled with two immiscible fluids forming an interface at $x = 0$ . Permittivity and the inverse Debye length of fluid “1” (“2”) are given by $ɛ_{1}$ ( $ɛ_{2}$ ) and $κ_{1}$ ( $κ_{2}$ ), respectively. The surfaces carry fixed charge densities ( $σ_{1}^{0}, σ_{1}^{L}, σ_{2}^{0}$ , and $σ_{2}^{L}$ ) which, in general, may not be the same for the two walls and can vary over each wall depending on the fluid phase it is in contact with.
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Figure 2
Variation of the surface interaction energy densities $ω_{γ, 1}^{e} (L)$ and $ω_{γ, 1}^{s} (L)$ , both expressed in units of $ɛ_{r, 1} κ_{1} / (2 β ℓ_{B})$ , as functions of the scaled separation $κ_{1} L$ between the walls for (a) $σ_{1}^{0} σ_{1}^{L} > 0$ , (b) $σ_{1}^{0} σ_{1}^{L} = 0$ with $σ_{1}^{0} \neq σ_{1}^{L}$ , (c) $σ_{1}^{0} σ_{1}^{L} < 0$ with $σ_{1}^{0} \neq - σ_{1}^{L}$ , and (d) $σ_{1}^{0} = - σ_{1}^{L}$ . As shown by the plots, $ω_{γ, 1}^{e} (L)$ diverges in the limit of vanishing separation between the walls except for the case considered in panel (d), whereas $ω_{γ, 1}^{s} (L)$ is always finite in this limit. For the special case of $σ_{1}^{0} = - σ_{1}^{L}$ considered in panel (d), both $ω_{γ, 1}^{e} (0)$ and $ω_{γ, 1}^{s} (0)$ are finite and $ω_{γ, 1}^{e} (0) = ω_{γ, 1}^{s} (0)$ . In the opposite limit, i.e., in the limit of infinitely large separations between the walls, both $ω_{γ, 1}^{e}$ and $ω_{γ, 1}^{s}$ , if nonzero, decay exponentially which in each case is confirmed by the semilogarithmic plots in the insets. However, when one of the two charge densities are zero [panel (b)], $ω_{γ, 1}^{s} (L)$ is zero for any separation between the walls, whereas $ω_{γ, 1}^{e} (L)$ is nonzero but decays twice as slow compared to the other cases. The overall decay of the surface interaction energies are monotonic except when the walls are oppositely charged with $σ_{1}^{0} \neq - σ_{1}^{L}$ In this case, the exact surface interaction energy shows a minimum before decaying to zero at large separations.
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Figure 3
Line interaction energy $ω_{τ} (L)$ within both the exact and the superposition calculations expressed in units of $ɛ_{r, 1} / (2 β ℓ_{B})$ as functions of the scaled separation length $κ_{1} L$ for different combinations of the charge densities $σ_{1}^{0}, σ_{1}^{L}, σ_{2}^{0}$ , and $σ_{2}^{L}$ specified in Table 1. As it is clear from the plots, both $ω_{τ}^{e}$ and $ω_{τ}^{s}$ can vary monotonically as well as nonmonotonically depending on the parameters and in most cases the superposition approximation fails to capture the correct behavior properly. In the limit of vanishing separation between the walls, $ω_{τ}^{s}$ stays finite, whereas $ω_{τ}^{e}$ usually diverges unless the two walls are exactly oppositely charged (e). The decay at large separations is always exponential [shown, for example, in the inset of panel (e) but the decay rate for $ω_{τ}^{e}$ and $ω_{τ}^{s}$ differs by a factor of 2 when one of the walls is uncharged as shown by the semilogarithmic plot in the inset of panel (f)]. Moreover, even when the superposition expression predicts the correct decay rate it always underestimates the magnitude by a factor of 2 if none of the four charge densities are zero. This can be seen from the corresponding curves in the inset of panel (e), which, instead of falling on top of each other, are parallel to each other. However, when either one of the four or two diagonally opposite charge densities are zero, the ratio $ω_{τ}^{e} (L \to \infty) / ω_{τ}^{s} (L \to \infty)$ converges to a value which depends on the system parameters. For the parameters used in panels (g) and (h), it is close to unity.
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Figure 4
Variation of the exact line interaction energy density $ω_{τ}^{e} (L)$ [expressed in units of $ɛ_{r, 2} / (2 β ℓ_{B})$ in the left panels and $ɛ_{r, 1} / (2 β ℓ_{B})$ in the right panels] as function of the scaled separation $κ_{2} L$ (left panels) and $κ_{1} L$ (right panels) for varying (a) inverse Debye length $κ_{1}$ in medium “1,” (b) inverse Debye length $κ_{2}$ in medium “2,” (c) relative permittivity $ɛ_{r, 1}$ of medium “1,” and (d) relative permittivity $ɛ_{r, 2}$ of medium “2.” The other parameters used for the plots are as follows: $σ_{1}^{0} = 0.02 {e / nm}^{2}, σ_{1}^{L} = - 0.03 {e / nm}^{2}, σ_{2}^{0} = - 0.0004 {e / nm}^{2}, σ_{2}^{L} = 0.0002 {e / nm}^{2}$ , and $β e Ψ_{D} = 1$ . Unless otherwise stated, $κ_{1} = 0.1 {nm}^{- 1}, κ_{2} = 0.03 {nm}^{- 1}, ɛ_{r, 1} = 80$ , and $ɛ_{r, 2} = 2$ are considered. As one can see, $ω_{τ}^{e} (L)$ always show at least one maximum. However, with decreasing $ɛ_{r, 1}$ , a single minimum can occur while with increasing $ɛ_{r, 2}$ , a minimum and a second maximum can occur.
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Figure 5
Total interaction energy $Δ \tilde{Ω} (L)$ $[= ω_{γ, 1} (L) A_{1} + ω_{γ, 2} (L) A_{2} + ω_{τ} (L) ℓ]$ between the two surfaces in the units of 1/β as a function of the scaled separation $κ_{1} L$ for a system with $σ_{1}^{0} = 0.001 {e / nm}^{2}, σ_{1}^{L} = - 0.1 {e / nm}^{2}, σ_{2}^{0} = 0.0001 {e / nm}^{2}, σ_{2}^{L} = - 0.01 {e / nm}^{2}, κ_{1} = 0.1 {nm}^{- 1}, κ_{2} = 0.03 {nm}^{- 1}, ɛ_{r, 1} = 80, ɛ_{r, 2} = 2, β e Ψ_{D} = 1, ℓ_{B} = 55.7 nm, A_{1} \approx 7600 nm, A_{2} \approx 22 000 nm$ , and $ℓ \approx 240 nm$ . The total effective surface areas $A_{i}$ and the total effective length $ℓ$ of the three-phase contact lines are calculated as explained in the main text for particles of radii $R = 100 nm$ . As one can see, whereas the superposition approximation (black dashed line) predicts a deep minimum at vanishing separation between the surfaces leading to aggregation, the exact result (blue solid line) is mostly positive with a shallow minimum (see the inset) at around $κ_{1} L \approx 13$ and thus, rules out the possibility of aggregation.
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