Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Guilherme Volpe
Bossa
^{a},
Joseph
Roth
^{a},
Klemen
Bohinc
^{b} and
Sylvio
May
*^{a}
^{a}Department of Physics, North Dakota State University, Fargo, ND 58108-6050, USA. E-mail: sylvio.may@ndsu.edu
^{b}Faculty of Health Sciences, University of Ljubljana, Zdravstvena 5, SI-1000 Ljubljana, Slovenia

Received
9th February 2016
, Accepted 24th March 2016

First published on 25th March 2016

Charged spherical nanoparticles trapped at the interface between water and air or water and oil exhibit repulsive electrostatic forces that contain a long-ranged dipolar and a short-ranged exponentially decaying component. The former are induced by the unscreened electrostatic field through the non-polar low-permittivity medium, and the latter result from the overlap of the diffuse ion clouds that form in the aqueous phase close to the nanoparticles. The magnitude of the long-ranged dipolar interaction is largely determined by the residual charges that remain attached to the air- (or oil-) exposed region of the nanoparticle. In the present work we address the question to what extent the charges on the water-immersed part of the nanoparticle provide an additional contribution to the dipolar interaction. To this end, we model the electrostatic properties of a spherical particle – a nanoparticle or a colloid – that partitions equatorially to the air–water interface, thereby employing nonlinear Poisson–Boltzmann theory in the aqueous solution and accounting for the propagation of the electric field through the interior of the particle. We demonstrate that the apparent charge density on the air-exposed region of the particle, which determines the dipole potential, is influenced by the electrostatic properties in the aqueous solution. We also show that this electrostatic coupling through the particle can be reproduced qualitatively by a simple analytic planar capacitor model. Our results help to rationalize the experimentally observed weak but non-vanishing salt dependence of the forces that stabilize ordered two-dimensional arrays of interface-trapped nanoparticles or colloids.

Classical works by Stillinger^{6} and Pieranski^{7} highlight the long-ranged, dipole-like nature of the electrostatic interactions between charged particles at dielectric interfaces. Clearly, within a bulk aqueous solution the interaction is screened by mobile ions (salt, or H^{+} and OH^{−} ions in the absence of added salt) and thus decays exponentially whereas in a uniform dielectric medium without mobile ions a bare 1/r-Coulomb potential as function of the distance r emerges. When trapped at an air–water (or oil–water) interface, image charges and the presence of mobile ions in the aqueous medium are expected to render the long-range part of the interaction dipole-like, 1/r^{3}, and inversely proportional to the salt concentration.^{8}

Charged colloids or nanoparticles often carry dissociable groups (phosphate or carboxyl moieties^{9–12}) that allow the surface charge density to adjust. When immersed in water the particle's surface charge density tends to be much larger as compared to being exposed to air or oil.^{13} Indeed, water has a large dielectric constant and contains mobile ions that effectively screen electrostatic interactions and thus reduce the energy needed to establish a highly charged surface. In contrast, the high cost of forming electrostatic fields in media of low permittivity and the absence of mobile ions tend to oppose the accumulation of charge at charge-regulated surfaces. This results in charge densities that are high and low in the water-exposed and air- (or oil-) exposed regions of the particle, respectively. However, the latter and not the former mediate long-ranged particle–particle interactions. In line with this, Aveyard et al.^{13} have reported that the ordered pattern formed by polystyrene latex particles covered by sulfate groups was insensitive to the electrolyte when they were placed at an oil–water interface. A similar observation was presented by Law et al.^{14} for silica particles trapped at an octane–water interface. In all these cases the authors concluded that the electrostatic repulsion emerged exclusively due to residual charges at the oil-exposed surface regions of the particles.

Based on the initially observed^{13} electrolyte insensitivity and the perceived importance of the charges exposed to the medium of low dielectric constant, Danov et al.^{15–17} have modeled the electrostatic force acting on charged colloids at the oil–water interface thereby imposing a scenario of no penetration of the electric field into the aqueous medium. Computational studies with a similar scope have been presented by Zhao et al.^{18} and Majee et al.^{19} Both consider spherical particles that partition equatorially into an oil–water interface and compute electrostatic fields inside the aqueous and oil phases for asymmetric charge distributions on the particles (that is, different uniform surface charge densities on the oil- and water-exposed particle regions). While the two models accurately account for the particle shape and charge distribution, neither of them allows the electrostatic field to penetrate into the particle interior and thus to couple the electrostatic properties in the aqueous phase with those in the apolar medium.

Recent experiments have shown, however, that the repulsion between charged colloids at an oil–water interface is weakly dependent on the electrolyte concentration,^{20–23} putting into question the sole responsibility of the oil-exposed charges for the long-range dipole interactions. As pointed out by Frydel et al.,^{24} there is a possibility of the electric field produced by the charges on the water-facing side of a particle to propagate into the oil phase by passing through the particle interior instead of spreading exclusively into the aqueous medium. This idea has been pursued using a renormalization approach,^{12,24,25} where the charges on the water-facing side of a colloid give rise to an electrostatic potential in the air (or oil) phase that far away from the colloid can be matched with the potential produced by a dipole with an apparent dipole moment. The apparent dipole moment was determined from numerical solutions of the Poisson–Boltzmann and Laplace equations for spherical particles.^{24} Yet, what was not accounted for is the possibility that, first, not only the water-facing side of the colloid but also its oil-facing side is charged and, second, the dielectric constant inside the colloid can be different from that in the air.

In the present work we analyze the electrostatic properties of a spherical particle (i.e., a colloid or a nanoparticle) that partitions equatorially into the interface between water and a medium of low dielectric constant (we focus on air but the model applies similarly to an oil phase). We allow the particle to have different uniform charge densities on its water- and air-exposed regions. In contrast to previous studies we explicitly include into our model the dielectric properties inside the particle. That is, we allow the electric field to propagate into the particle interior and thus to either enhance or diminish the electric field in the air. Hence, our model is designed to predict the salt dependence of the long-ranged dipolar particle–particle interactions. Our calculations are carried out on the level of mean-field electrostatics. To this end, we solve Laplace equations in the air and inside the spherical particle, and the nonlinear Poisson–Boltzmann equation in the water phase. Two approximations are adopted. First, we assume the radius of the particle is much larger than the Debye screening length of the aqueous solution and, second, the surface potential at the air–water interface is fixed and constant. We express our results in terms of an apparent surface charge density at the air-exposed region of the particle. This apparent charge density lumps the bare charge density and the contribution of the electric field that propagates through the particle into an apparent value, thus expressing the degree of coupling between the air-exposed and water-exposed particle regions. Equivalently, the apparent charge density produces the same electric field in the air while assuming the particle interior is impenetrable to the electric field than the bare charge density does for the same, yet field-penetrable, particle. Note that this concept exactly preserves the image charges needed to produce the electric field in the air.^{26} Our calculations demonstrate that even with a small dielecric constant inside the particle, the charges at its water-exposed region can make a significant contribution to the long-ranged dipolar interactions between interfacially trapped particles. We analyze this behavior in terms of several parameters: the dielectric constant inside the particle, the potential difference across the air–water interface, and the salt content in the aqueous solution. We also show that a simple approximation – that of a planar capacitor with appropriate boundary conditions – yields an explicit expression for the apparent charge density at the air-exposed region of the particle, which is in qualitative agreement with our detailed numerical calculations for the spherical particle geometry.

The present work is based on mean-field electrostatics, expressed in terms of the commonly used dimensionless electrostatic potential Ψ = eΦ/k_{B}T, where Φ is the electrostatic potential, k_{B} the Boltzmann constant, T the absolute temperature, and e the elementary charge. Note that Ψ = 1 corresponds to an electrostatic potential of Φ = 25 mV at room temperature. We use indices “a”, “w”, and “n” to label the three regions: air, water, and the inside of the nanoparticle. Hence Ψ_{a}, Ψ_{w}, and Ψ_{n}, denote the dimensionless potential in the air, water, and particle interior, respectively. With this, the electrostatic free energy, in units of k_{B}T, of the charged particle at the air–water interface can be expressed as

(1) |

We note that the electrostatic potential is believed to change when passing from air into the bulk of an aqueous solution. The potential difference likely reflects both the adsorption of ions (OH^{−}versus H^{+}) and the dipole potential from interface-induced water ordering. The magnitude and sign have been a matter of debate,^{27,28} but a change from a more negative potential in the air to a more positive potential in bulk water finds wide experimental^{29,30} and some computational^{31} support. In the present work we use the potential in the air, far away from the air–water interface and from the particle, as reference that we define as zero. Hence, in the bulk of the aqueous phase, the potential adopts a non-vanishing constant value that we denote by Ψ^{(b)}_{w}. The final term in eqn (1) introduces Ψ^{(b)}_{w} as a fixed (and yet unspecified) external potential in the aqueous medium. Indeed, minimization of F_{el} with respect to the local ion concentrations yields the Boltzmann distributions

(2) |

l_{D}^{2}∇^{2}Ψ_{w} = sinh(Ψ_{w} − Ψ^{(b)}_{w}), | (3) |

∇^{2}Ψ_{a} = 0, ∇^{2}Ψ_{n} = 0, | (4) |

(5) |

(6) |

(7) |

(8) |

Due to the presence of salt and the large dielectric constant of water it is a reasonable approximation to treat the aqueous solution as a perfect conductor, implying the condition Ψ_{a} = 0 at the air–water interface.

We also wish to calculate the electrostatic free energy. To this end, we insert the distributions for n_{±} from eqn (2) into eqn (1), and then re-express F_{el} exclusively in terms of the particle's surface potentials using the Poisson–Boltzmann equation (eqn (3)) and the Laplace equations (eqn (4)) as well as the boundary conditions according to eqn (5) and (8),

(9) |

The electrostatic problem is now fully defined; we need to solve the two Laplace equations in eqn (4), each in a medium with uniform but different dielectric constant, subject to the boundary conditions in eqn (5) and (8) (the latter one being nonlinear), and Ψ_{a} = 0 at both the air–water interface and at very large distance away from the particle. Once the potential at the surface of the particle is known, we may calculate the corresponding electrostatic free energy using eqn (9). In order to find explicit solutions for the potential we need to specify the shape of the particle. We will focus on a spherical particle of radius R that partitions equatorially to the air–water interface. However, prior to considering the spherical geometry explicitly, we investigate a planar capacitor-like geometry that serves us as an approximation for the spherical geometry and allows to compute simple analytical solutions for the potential.

(10) |

(11) |

(12) |

Upon inserting the potentials Ψ_{a}(x) and Ψ_{n}(x) we obtain two algebraic equations for the two surface potentials

(13) |

(14) |

(15) |

We are interested in the apparent surface charge density σ^{app}_{a} of the particle as observed from the air. We define σ^{app}_{a} as the surface charge density at the air-exposed region of the particle that preserves the electric field in the air while imposing ε_{n} = 0. Similar to eqn (11), we define a dimensionless apparent charge density ^{app}_{a} = 4πl_{B}l_{D}σ^{app}_{a}/(eε_{w}), which we can compute according to eqn (12) through ^{app}_{a} = _{a} + H^{n}RΨ_{n}′(−R), or equivalently using eqn (13), ^{app}_{a} = H^{a}Ψ^{(a)}_{0}. Hence, the surface potential at the air-exposed region of the nanoparticle determines the apparent surface charge density (in units of the elementary charge e)

(16) |

(17) |

We note that the surface potentials in eqn (15) become an exact solution of eqn (13) if the scaled surface charge density _{w} ≪ 1 is sufficiently small. This case corresponds to the linear Debye–Hückel limit of Poisson–Boltzmann theory, leading to

(18) |

(19) |

(20) |

(21) |

(22) |

with

(23) |

Fig. 3 The same system as illustrated in Fig. 1 but for a spherical particle of radius R that partitions equatorially to the air–water interface; θ is the polar angle measured with respect to the normal direction as indicated. |

In order to determine the coefficients C_{l}, which contain all the information needed to specify the electrostatic potential everywhere, we proceed as for the planar capacitor approximation (see the preceding Section 2.1) by expanding the surface potential at the water-exposed region of the particle in terms of H^{n} up to first order. The two boundary conditions in eqn (5) and (8) then read

(24) |

(25) |

To derive eqn (25) we have also used orthogonality , where δ_{mn} denotes the Kronecker delta, and symmetry of the Legendre polynomials.

Solutions of eqn (25) can be found numerically for a finite set of coefficients C_{l} with l = 0, 1, 2,…, l_{max}. The choice of l_{max} will determine the accuracy of the electrostatic potential. We will determine l_{max} such that the free energy F_{el} = F_{el}(l_{max}), calculated on the basis of the surface potential , converges to F_{el}(l_{max} → ∞) up to a certain numerical accuracy. Recall that F_{el} is fully determined by the surface potential; see eqn (9). For spherical particle geometry, eqn (9) reads

(26) |

(27) |

(28) |

Clearly, in the hypothetical limit of ε_{n} = 0 the inside of the particle becomes impenetrable to the electric field; this renders the electrostatic properties of the air-exposed and water-exposed regions of the particle independent from each other. Increasing ε_{n} allows the electric field to enter the particle and thus decreases the free energy. The free energy also decreases with l_{max} because each C_{l} adds a degree of freedom to the system. Most importantly, F_{el}(l_{max}) converges to a fixed constant (within the thickness of the printed symbol) for a value of l_{max} smaller than about 70. Consequently, we have carried out all our calculations for l_{max} = 70. That is, we have solved eqn (25) (in the nonlinear regime) and eqn (27) (in the linear regime) for l = 0…70, yielding the dimensionless surface potential and thus, using eqn (21), the dimensionless potentials (with r ≤ R and −1 ≤ s ≤ 1) inside the spherical particle and (with r ≥ R and 0 ≤ s ≤ 1) in the air.

The main objective of the present work is to predict the apparent charge of the spherical particle on its air-exposed region. In addition to the bare charge, this renormalized charge contains a contribution from the electric field that penetrates through the particle's interior and determines the salt dependence of the long-ranged dipolar interactions among interface-trapped particles. For the planar capacitor approximation we have already defined in eqn (16) the apparent surface charge density σ^{app}_{a}. In a similar manner we define the average apparent surface charge density

(29) |

(30) |

Fig. 5 shows two contour plots of the dimensionless electrostatic potential, calculated for particle radius R = 50 nm, Debye length l_{D} = 5 nm, surface charge density at the water-exposed region of the particle σ_{w} = 3.2 μC cm^{−2} = 0.2 e nm^{−2}, dielectric constant inside the particle ε_{n} = 2, and Ψ^{(b)}_{w} = 0. The two diagrams are computed for surface charge densities at the air-exposed particle region σ_{a} = 0 (left) and σ_{a} = 3.2 nC cm^{−2} = 0.0002 e nm^{−2} (right). At the water-exposed region, both particles possess an almost identical constant potential of Ψ_{0}(s) = 4.36, which is slightly smaller than the prediction from the Poisson–Boltzmann model for a planar isolated surface Ψ_{0}(s) = Ψ^{(b)}_{w} + 2arsinh(_{w}/2) = 4.37 (the close proximity is expected and, in fact, was our motivation for the expansion of the surface potential with respect to H^{n}, employed in the derivations of eqn (24) and (25)). At the air-exposed surface, however, Ψ_{0}(s) adopts a minimum at s = 1 when σ_{a} = 0 (left), whereas it adopts a maximum at s = 1 when σ_{a} = 3.2 nC cm^{−2} (right). These differences result from the interplay between the charges attached to the water-exposed and air-exposed faces of the particle and the vanishing potential at the air–water interface. This interplay is also reflected in the apparent surface charge densities at the air-exposed region, for which we obtain according to eqn (29)σ^{app}_{a} = 3.6 nC cm^{−2} (left) and σ^{app}_{a} = 5.9 nC cm^{−2} (right). We can thus state that, according to eqn (30), the ≈3200 charges attached to the water-exposed region of the particle cause an increase in the number of apparent charges on the air-exposed particle region from zero to 3.6 for the left diagram in Fig. 5 and from 3.2 to 5.9 for the right diagram in Fig. 5. Of course, an isolated consideration of the two arbitrarily selected systems in Fig. 5 does not yield a systematic understanding of the relation between σ_{a} and σ^{app}_{a}. In the following we provide a more comprehensive analysis.

In Fig. 6 we show the results of a detailed analysis of σ^{app}_{a} as function of ε_{n} for eleven different choices of σ_{a} in each diagram. All results in Fig. 6 refer to a Debye screening length l_{D} = 5 nm (that is, a 4 mM salt concentration in the aqueous medium). Each diagram corresponds to a specific combination of σ_{w} and Ψ^{(b)}_{w}, with σ_{w} = 0 in the left column and σ_{w} = 3.2 μC cm^{−2} in the right column, as well as Ψ^{(b)}_{w} = −2 in the upper row of diagrams, Ψ^{(b)}_{w} = 0 in the middle row, and Ψ^{(b)}_{w} = +2 in the bottom row. All solid lines refer to calculations based on the nonlinear Poisson–Boltzmann model; see eqn (25). The dashed lines, visible only in the right column of diagrams are computed for the linearized Debye–Hückel model; see eqn (27). On the left column of diagrams, the dashed lines coincide with the solid lines and are thus not visible individually. In the limit ε_{n} = 0 there is no interaction between the air- and water-exposed regions of the particle, implying σ^{app}_{a} = σ_{a}. Hence, the value of σ_{a} for which each curve in Fig. 6 is derived corresponds to the value of σ^{app}_{a} at ε_{n} = 0. Note also that the two specific systems represented in Fig. 5 are marked in Fig. 6 by the symbol ●.

Fig. 6 Apparent charge density σ^{app}_{a} at the air-exposed surface of a spherical particle as function of the particle's dielectric constant ε_{n}. Solid and dashed lines correspond, respectively, to results in the nonlinear Poisson–Boltzmann and the linear Debye–Hückel regimes. Different curves in each diagram refer to different σ_{a} = σ^{app}_{a}(ε_{n} = 0). The two columns of diagrams are computed for σ_{w} = 0 (left) and σ_{w} = 3.2 μC cm^{−2} (right); the three rows refer to Ψ^{(b)}_{w} = −2 (top), Ψ^{(b)}_{w} = 0 (middle), and Ψ^{(b)}_{w} = 2 (bottom). All results are derived for R = 50 nm and l_{D} = 5 nm. The two bullets in the middle-right diagram refer to the contour plots displayed in Fig. 5. |

Let us now discuss the findings in Fig. 6. Consider first the middle diagram on the left column, derived for σ_{w} = 0 and Ψ^{(b)}_{w} = 0. For σ_{a} = 0 the particle is completely uncharged, the potential is zero everywhere, and thus σ^{app}_{a} = 0 for any choice of ε_{n}. For σ_{a} > 0 the apparent value σ^{app}_{a} decreases with growing ε_{n} because a part of the electric field propagates through the inside of the particle and interacts with negative charges in the aqueous solution that are polarized at the water-exposed region of the particle. This is more favorable than passing exclusively through the air and interacting with negative charges in the aqueous solution that are polarized at the air–water interface. We note that the ratio σ^{app}_{a}/σ_{a} reaches 50% roughly at ε_{n} ≈ 4. We also note that the potential inside the aqueous phase, which is only caused by the few charges at the air-exposed region of the particle, is small so that it practically makes no difference to use the linear Debye–Hückel model or the nonlinear Poisson–Boltzmann approach.

Next, we consider the middle diagram on the right column, derived for σ_{w} = 3.2 μC cm^{−2} and Ψ^{(b)}_{w} = 0. For σ_{a} = 0 all charges carried by the particle (about 3200) are attached to the water-exposed region. These charges are very effectively screened by the mobile salt ions in the aqueous solution, which are present with a bulk concentration of 4 mM. However, as ε_{n} grows, a small (but increasing) part of the electric field produced by σ_{w} is able to propagate through the particle interior into the air and thus appears as an apparent charge density σ^{app}_{a}. For example, at ε_{n} = 2, we find σ^{app}_{a} = 3.6 nC cm^{−2}, corresponding to an apparent number of 3.6 elementary charges attached to the air-exposed particle region. This, in fact is the example already presented in Fig. 5 (left diagram) and marked by the lower of the two bullets in the middle-right diagram of Fig. 6. Although few in number, these apparent charges are unscreened and thus highly effective in influencing the long-ranged interactions between interface-trapped particles. As σ_{a} grows, the increase in σ^{app}_{a}(ε_{n}) becomes weaker and eventually reverses into a decreasing function. Indeed, with growing σ_{a} the particle-propagating part of the electric field produced by the air-exposed charges becomes stronger and eventually reverses the direction of the total electric field in the particle interior. The reversal occurs roughly at σ_{a} = 12 nC cm^{−2}. At this particular combination of charge densities – about 3200 charges at the water-exposed region and 12 charges at the air-exposed region of the particle – the dielectric constant ε_{n} becomes practically irrelevant and thus does not affect the interactions between interface-trapped particles. It is one of the central conclusions of the present work that the ability of the electric field to propagate into the particle interior can enhance or diminish the interaction strength of particles at the air–water (and similar for oil–water) interface. That is, already a few air-exposed charges will reverse the direction of the electric field inside the particle and thus qualitatively change the influence of the particle's dielectric constant on the long-ranged particle–particle interactions.

As pointed out in the Introduction, sign and magnitude of the change in electrostatic potential upon crossing from air into bulk water have received significant attention in recent years.^{27–31} The implications of this potential difference on the electrostatic properties of interface-trapped particles, however, have not been analyzed previously. We have therefore incorporated the presence of an arbitrary bulk potential Ψ^{(b)}_{w} into our theoretical approach (recall that Ψ^{(b)}_{w} denotes the difference of the dimensionless electrostatic potential in bulk water and in air, both far away from the air–water interface). Note that we have not introduced an additional change in potential when passing from the interior of the particle into the aqueous medium. In fact, there is no need to introduce such an additional change in potential if we interpret Ψ^{(b)}_{w} as the difference in the change of the (dimensionless) electrostatic potential at the bare air–water interface and particle–water interface. We do not know the sign and magnitude of Ψ^{(b)}_{w} but we can analyze its general impact on σ^{app}_{a}. This is shown in the upper and lower rows of Fig. 6 for Ψ^{(b)}_{w} = −2 and Ψ^{(b)}_{w} = 2, respectively. Our motivation to use the specific magnitude |Ψ^{(b)}_{w}| = 2 for the displayed examples goes back to a suggestion of Gehring and Fischer.^{30} Yet, we emphasize that the actual value and sign of Ψ^{(b)}_{w} remain a matter of debate. A negative value of Ψ^{(b)}_{w} mimics the presence of additional negative charges at the water-exposed region of the particle, implying more negative slope of the function σ^{app}_{a}(ε_{n}). This is most clearly seen for the case σ_{w} = σ_{a} = 0, where the increase of ε_{n} from 0 to 2 changes σ^{app}_{a} from 0 to about −1.8 nC cm^{−2}; see the upper-left diagram of Fig. 6. Hence, even a completely uncharged particle carries a small apparent negative charge on its air-exposed face. All curves (solid lines) in the two top and two bottom diagrams of Fig. 6 can be rationalized by translating a negative or positive bulk potential Ψ^{(b)}_{w} into, respectively, an additional negative or positive charge at the water-exposed particle region. We add two comments. First, changing the magnitude of |Ψ^{(b)}_{w}| from 0 to 2 (which corresponds to a change of 50 mV) typically causes Q^{app}_{a} to adjust by 2–5 elementary charges for a fixed ε_{n} in the region 2 < ε_{n} < 5. Second, the relation σ^{app}_{a}(ε_{n}) can pass through a local maximum (which, however, is not very pronounced). This implies that, perhaps somewhat unexpectedly, the apparent charge Q^{app}_{a} may be observed to first increase and then decrease as function of increasing ε_{n}.

We have carried out calculations of σ^{app}_{a} on the basis of the nonlinear Poisson–Boltzmann model (solid lines in Fig. 6) and the linearized Debye–Hückel approximation (dashed lines in Fig. 6). For σ_{w} = 0 (left column of diagrams in Fig. 6) both models yield virtually identical results, but for σ_{w} = 3.2 μC cm^{−2} (right column of diagrams in Fig. 6) this is no longer the case. Indeed, the surface potential at the water-exposed region of the particle is only slightly smaller than 4.37, implying that the linearization of the Poisson–Boltzmann equation is a poor approximation and, in fact, overestimates the magnitude of the surface potential.^{33} Hence, in the linearized model, we expect the more positive surface potential at the water-exposed particle region to cause a larger σ^{app}_{a} than the nonlinear model predicts, and this is indeed what we observe in Fig. 6. Despite this overestimation, however, the qualitative nature of the results for σ^{app}_{a} is preserved in the linear Debye–Hückel approximation; this includes the reversal of the slope of the function σ^{app}_{a}(ε_{n}) for sufficiently large σ_{a} as can be observed directly in the top-right diagram of Fig. 6 (at about σ_{a} = 16 nC cm^{−2}).

Numerical results like those in Fig. 6 are computed for a specific set of parameters, of which some are kept constant and others varied across a small set of discrete values. Analytic expressions offer the advantage of allowing a systematic analysis and hence a clearer understanding of the relationships between parameters. In Section 2.1 we have proposed a planar capacitor approximation and derived a simple expression for σ^{app}_{a}; see eqn (16) (as well as eqn (17) for large σ_{w} and eqn (19) for small σ_{w}). Recall that the planar capacitor approximation is based on representing the interface-trapped spherical particle by the geometry of a planar capacitor; see Fig. 2. In Fig. 7 we present predictions for σ^{app}_{a} as function of ε_{n} according to the planar capacitor approximation for exactly the same set of parameters as in Fig. 6. Here too, solid lines refer to nonlinear Poisson–Boltzmann theory (calculated using eqn (16)), whereas the dashed lines correspond to the linear Debye–Hückel limit (calculated using eqn (19)). A comparison of Fig. 6 and 7 reveals good qualitative agreement. This includes (i) the slope-reversion of σ^{app}_{a} (that is, σ^{app}_{a} being a decreasing function for sufficiently large σ_{a} and an increasing function for sufficiently small σ_{a}), (ii) the down-shift of the point where the slope-reversion occurs for negative Ψ^{(b)}_{w} and its up-shift for positive Ψ^{(b)}_{w}, (iii) the excellent agreement between the nonlinear and linear models for σ_{w} = 0, and (iv) the overestimation of σ^{app}_{a} for large σ_{w} when comparing the linear and nonlinear models. There are also notable differences between Fig. 6 and 7. First, the dependence of σ^{app}_{a} on ε_{n} tends to be stronger in the planar capacitor approximation as compared to the spherical geometry. For example, for σ_{w} = 0, Ψ^{(b)}_{w} = 0, σ_{a} = 16 nC cm^{−2}, and ε_{n} = 5 our calculations predict σ^{app}_{a} = 7 nC cm^{−2} for spherical geometry and σ^{app}_{a} = 3 nC cm^{−2} for the planar capacitor approximation. A second difference is the lack of any local maxima of the function σ^{app}_{a}(ε_{n}). Instead, at one specific value for σ_{a} (the slope-reversion point) the function σ^{app}_{a}(ε_{n}) becomes independent of ε_{n}; the corresponding locations are marked by pairs of open circles in Fig. 7 (the two pairs of open circles on the diagrams refer to the nonlinear and linear models). From eqn (16) we find the condition σ^{app}_{a} = σ_{a} to be fulfilled for

(31) |

Fig. 7 Apparent surface charge density σ^{app}_{a} at the air-exposed surface as function of ε_{n} according to the planar capacitor approximation, calculated according to eqn (16) (solid lines) on the level of nonlinear Poisson–Boltzmann theory and according to eqn (19) (dashed lines) in the linear Debye–Hückel limit. All results are computed for exactly the same set of parameters as in Fig. 6. Specifically, different curves in each diagram refer to different σ_{a} = σ^{app}_{a}(ε_{n} = 0). The two columns of diagrams are computed for σ_{w} = 0 (left) and σ_{w} = 3.2 μC cm^{−2} (right); the three rows refer to Ψ^{(b)}_{w} = −2 (top), Ψ^{(b)}_{w} = 0 (middle), and Ψ^{(b)}_{w} = 2 (bottom). All results are derived for R = 50 nm and l_{D} = 5 nm. |

This marks the point where for ε_{n} = 0 the potential produced by σ_{a} at the air-exposed surface is equal to the potential produced by σ_{w} at the water-exposed surface. The electrostatic properties of the air-exposed and water-exposed regions are then decoupled and thus do not depend on ε_{n}. A similar rationale applies to the slope reversion of spherical particles observed in Fig. 6.

As discussed in the Introduction, experimental investigations of how the salt concentration in the aqueous medium affects the observed long-ranged repulsive forces between interface-trapped colloidal particles have not led to conclusive results. A number of studies suggest the interaction is insensitive to the salt concentration,^{13,35,36} while others report a weak dependence.^{20–23,37} Note that the force between two interface-trapped particles is proportional to the square of the apparent surface charge density σ^{app}_{a}, which depends on the salt concentration. In Fig. 8 we display the dependence of σ^{app}_{a} on the Debye screening length l_{D} for spherical particle geometry (left diagrams) and for the planar capacitor approximation (right diagrams). The two sets of curves in each diagram (σ_{a} = 0 for dashed lines in upper diagrams, σ_{a} = 3.2 nC cm^{−2} for solid lines in upper diagrams, σ_{a} = 16 nC cm^{−2} for dashed lines in lower diagrams, and σ_{a} = 32 nC cm^{−2} for solid lines in lower diagrams) refer to ε_{n} = 0 (symbol ○), ε_{n} = 1 (◁), ε_{n} = 2 (●), ε_{n} = 5 (▷). We have placed the symbols ○, ◁, ●, ▷ at the position l_{D} = 5 nm, which corresponds to the results in Fig. 6 and 7. Note that for spherical geometry we only consider Debye lengths up to l_{D} = 10 nm to ensure l_{D} ≪ R.

Fig. 8 Apparent surface charge density σ^{app}_{a} at the air-exposed particle region as function of the Debye screening length l_{D} for fixed σ_{w} = 3.2 μC cm^{−2}, R = 50 nm, and Ψ^{(b)}_{w} = 0. Dashed and solid lines in the upper two diagrams refer to σ_{a} = 0 and σ_{a} = 3.2 nC cm^{−2}, respectively. Dashed and solid lines in the lower two diagrams refer to σ_{a} = 16 nC cm^{−2} and σ_{a} = 32 nC cm^{−2}, respectively. Left and right diagrams correspond, respectively, to calculations for the spherical geometry (see Section 2.2) and the planar capacitor approximation (see Section 2.1). The four different curves for each set are derived for ε_{n} = 0 (symbol ○), ε_{n} = 1 (◁), ε_{n} = 2 (●), ε_{n} = 5 (▷). We have placed the symbols at position l_{D} = 5 nm, for which all calculations in Fig. 6 and 7 were carried out. |

All curves in Fig. 8 indicate nondecreasing behavior of σ^{app}_{a} as function of l_{D}. That is, adding salt is never predicted to increase the apparent particle charge (yet, it could do so in the hypothetical case that the bare charge densities σ_{a} and σ_{w} were of different sign). Let us discuss decreasing the salt concentration from 100 mM (l_{D} = 1 nm) to 1 mM (l_{D} = 10 nm) for a particle of dielectric constant ε_{n} = 5. For σ_{a} = 0 this induces an increase of σ^{app}_{a} from 2.0 nC cm^{−2} to 7.3 nC cm^{−2} and thus a 13.5-fold increase in the force between two particles (the force increase calculated within the planar capacitor approximation is 13.1). For large σ_{a} the absolute increase in σ^{app}_{a} is similar but the relative increase in the force is much lower. For example, σ_{a} = 32 nC cm^{−2} leads to an increase in σ^{app}_{a} from 14.1 nC cm^{−2} to 19.5 nC cm^{−2}, implying a 1.9-fold increase of the force (and a 1.6-fold increase predicted by the planar capacitor approximation). Because the planar capacitor model makes reasonable predictions, we may insert the parameters used in our specific example into eqn (17) (namely σ_{w} = 3.2 μC cm^{−2}, R = 50 nm, l_{B} = 56 nm, ε_{a} = 1, ε_{w} = 80, ε_{n} = 5, and Ψ^{(0)}_{w} = 0), yielding

σ^{app}_{a} = c_{1}σ_{a} + c_{2}ln(c_{3}l_{D}), | (32) |

To be specific, we attempt to model the salt concentration dependence of the force F ∼ (σ^{app}_{a})^{2}/r^{4} between charge-stabilized polystyrene particles (R = 1.5 μm, ε_{n} = 2.5, σ_{w} = 9.1 μC cm^{−2}) at a decane–water interface (ε_{a} = 2.0, ε_{w} = 80) as measured by Park et al.^{23} For this system we obtain c_{1} = 0.44, c_{2} = 2.03 × 10^{−6} nC cm^{−2}, and c_{3} = 5.0/nm. Decreasing the salt concentration from 1 mM (implying l^{(1)}_{D} = 10 nm) to 0.01 mM (implying l^{(2)}_{D} = 100 nm) at a particle-to-particle separation of r = 9 μm was reported to increase the force from about F_{1} = 0.2 pN to about F_{2} = 0.6 pN. Using the dependence of the force F on σ^{app}_{a} together with eqn (32) yields

(33) |

We finally point out that in Fig. 6 we had discussed the possibility of adjusting σ_{a} to render σ^{app}_{a} virtually independent of ε_{n}. Fig. 8 reveals that this may also be accomplished by adjusting the salt concentration. For example for σ_{a} = 3.2 nC cm^{−2} (see the upper left diagram in Fig. 8) the solid lines all intersect in a region close to l_{D} = 1 nm, implying σ^{app}_{a} does not depend on ε_{n}. Note that the planar capacitor approximation also predicts such a point, yet fails to correctly predict the corresponding salt concentration.

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