A comparison between MC and PB results showing the ion densities around a Janus dipole for N ± = 300, λB = d, and a = 10λB. (a) The contour plot shows the net charge density : the MC result (left) and the nonlinear PB solution (right). (b) The same data are represented using a Fourier-Legendre mode expansion of the charge density ρ±, ℓ(r), for ℓ = 0, 1, 2, and 3. Note that the negative modes in (b) can in this case be mapped onto the positive modes by multiplication with (− 1)ℓ.
A comparison of the data obtained by PB theory and by MC simulations of homogeneously charged spheres according to the difference function f 0 of Eq. (18), which quantifies the deviation in the distribution of charge in the ionic double layer. We show f 0 as a function of κμ, the ratio of the Gouy-Chapman and the Debye length, and Ξ, the strong-coupling parameter, for several of the systems we studied. The field-theoretical prediction of Refs. 57,58 for homogeneously charged flat surfaces partitions parameter space into three regimes, as is indicated by the continuous and the dashed line. The Debye-Hückel (DH), Poisson-Boltzmann (PB), and Strong-Coupling (SC) approximations should be used to obtain acceptable results in the respective domains. The value of the PB-SC divide is denoted by Ξ* ≈ 10. (b)–(d) Three samples of the ion profiles that are obtained by PB theory (full curves) and MC simulations (dashed curves), showing cation and anion densities. Here f 0 = 0.03, 0.39, and 0.99. The discontinuity in the first derivative of the MC ρ+, 0 profile in (d) is caused by the positive ions condensing on the surface of the colloid in combination with the binning procedure we applied to average the ion densities. A similar discontinuity is present in the ρ−, 0 results, although this is not visible on the scale of this plot.
The deviation f ℓ in the double layer, determined using the MC simulations and PB theory, for a Janus dipole as a function of the modified charge density and the modified strong-coupling parameter ΞΣ. The subgraphs show the results for the first two odd FL modes (ℓ = 1, 3), corresponding to the dipole and octupole terms, respectively.
(a) A sketch of two interacting particles with radius a, both having charge on only one hemisphere. To eliminate the dipole moment we relocate the centre of the charge distribution at a distance b (along the particle's rotational symmetry axis) from the geometrical (hard core) centre of the particle. This choice results in a distance R ij between the charge distributions. Graphs (b) and (c) show the suggested value of b and the Yukawa weight factor C(κa), respectively, as a function of the colloid radius a in terms of the Debye screening length κ−1, for a wide range in εc/ε, the ratio between the relative dielectric constant of the particle and that of the surrounding medium. In (c), we indicate C = 1, which is the weight in case of the regular DLVO equation, using a dashed line.
A contour plot of the net charge density around an antisymmetric Janus particle for the parameters Z = 10, N ± = 425, λB = d, and a = 10λB, showing the profile that follows from a mode expansion up to ℓ = 6 on the left, on the right only the dipole mode is plotted.
A sketch of two interacting charge distributions (grey) and , inside the enclosing volumes V i and V j , respectively. These volumes are rod-like in this particular example to resemble rods with one charged “head” and have boundaries that are indicated by dashed instead of solid lines since we do not consider hard cores whilst calculating the ion densities connected to these charge distributions. The centres of the charge distributions can be chosen arbitrarily and are indicated by and .
A sketch of two interacting charge distributions q i and q j with hard-core volumes V i and V j , respectively, separated by a distance . We also show the vectors and , which have their origins at and , respectively.
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