Double layer overlap

Carrique F, Arroyo FJ, Jimenez ML, Delgado Av. Influence of double-layer overlap on the electrophoretic mobility and DC conductivity of a concentrated suspension of spherical particles. J. Phys. Chem. B 2003 107 3199-3206. 

For ii) and iii) loosely structured layers are required, and the chains must protrude into the solution over a distance exceeding the thickness of the electrical double layer so that on approach of the surfaces, the adsorbed layers interfere before the electrical double layers overlap. 

So far, we have used the Maxwell equations of electrostatics to determine the distribution of ions in solution around an isolated, charged, flat surface. This distribution must be the equilibrium one. Hence, when a second snrface, also similarly charged, is brought close, the two surfaces will see each other as soon as their diffuse double-layers overlap. The ion densities aronnd each surface will then be altered from their equilibrinm valne and this will lead to an increase in energy and a repulsive force between the snrfaces. This situation is illustrated schematically in Fignre 6.12 for non-interacting and interacting flat snrfaces. 

Theories of electrical double layers and forces due to double-layer overlap (Chapter 11)... 

Two charged particles approaching each other sense the presence of each other through the overlap of their electrical double layers. This double-layer overlap results in a repulsive force between similarly charged particles. 

This chapter focuses on some of the basic theories of electrical double layers near charged surfaces and develops the expressions for interaction energies when two electrical double layers overlap ( interact ) with each other. 

In addition to all these, it is also important to keep in mind that the results depend also on what types of surface equilibrium conditions exist as the double layers interact. For example, when two charged surfaces approach each other, the overlap of the double layers will also affect the manner in which the charges on the surfaces adjust themselves to the changing local conditions. As the double layers overlap and get compressed, the local ionic equilibrium at the surface may change, and this will clearly have an impact on the potential distribution and on the potential energy of interaction. 

In Chapter 5 we learned that, in water, most surfaces bear an electric charge. If two such surfaces approach each other and the electric double layers overlap, an electrostatic double-layer force arises. This electrostatic double-layer force is important in many natural phenomena and technical applications. It for example stabilizes dispersions.7... 

For small electric double layer overlap, such that exp [—kH] < 1, these expressions both reduce to... 

The role of electrostatic repulsion in the stability of suspensions of particles in non-aqueous media is not yet clear. In order to attempt to apply theories such as the DLVO theory (to be introduced in Section 5.2) one must know the electrical potential at the surface, the Hamaker constant, and the ionic strength to be used for the non-aqueous medium these are difficult to estimate. The ionic strength will be low so the electric double layer will be thick, the electric potential will vary slowly with separation distance, and so will the net electric potential as the double layers overlap. For this reason the repulsion between particles can be expected to be weak. A summary of work on the applicability or lack of applicability of DLVO theory to non-aqueous media has been given by Morrison [268],... 

Several repulsive and attractive forces operate between colloidal species and determine their stability [12,13,15,26,152,194], In the simplest example of colloid stability, dispersed species would be stabilized entirely by the repulsive forces created when two charged surfaces approach each other and their electric double layers overlap. The overlap causes a coulombic repulsive force acting against each surface, which will act in opposition to any attempt to decrease the separation distance (see Figure 5.2). One can express the coulombic repulsive force between plates as a potential energy of repulsion. There is another important repulsive force causing a strong repulsion at very small separation distances where the atomic electron clouds overlap, called Born repulsion. 

In the simplest example of colloid stability, suspension partides would be stabilized entirely by the repulsive forces created when two charged surfaces approach each other and their electric double layers overlap. The repulsive energy VR for spherical particles, or rigid droplets, is given approximately as ... 

When two droplets or particles approach a distance h that is smaller than twice the double-layer thickness, repulsion occurs due to double-layer overlap (the double layers on the two particles cannot develop completely). 

Electroosmotic flow is generally reported to be independent of the size of the packing, and consequently the size of the interstitial voids between the particles, unless this size is so small that the electrical double layers overlap [74]. The ability to independently control both the pore size and level of charged functionalities of the methacrylate ester monolithic capillaries enables the direct investigation of the net effect of transport channel size on flow velocity. Recent results clearly demonstrates a... 

A general scheme, based on a rigorous statistical mechanical formulation, for obtaining the interaction between two colloidal particles in a fluid has been outlined. The implementation of the theory is in its early stages. In the DLVO theory and the theory of HLC, it is assumed that the various contributions can be added together. In the MSA, the hard core and electrostatic terms will be additive. However, it is only at low electrolyte concentration that the effect of dipole orientation and the repulsive contribution of the double layer overlap will be additive. There is no reason to believe (or disbelieve) that the van der Waals term should also be additive. 

Boundary effects on the electrophoretic migration of a particle with ion cloud of arbitrary thickness were also investigated by Zydney [46] for the case of a spherical particle of radius a in a concentric spherical cavity of radius d. Based on Henry s [19] method, a semi-analytic solution has been developed for the particle mobility, which is valid for all double layer thicknesses and all particle/pore sizes. Two integrals in the mobility expression must be evaluated numerically to obtain the particle velocity except for the case of infinite Ka. The first-order correction to the electrophoretic mobility is 0(A3) for thin double layer, whereas it becomes 0(A) for thick double layer. Here the parameter A is the ratio of the particle-to-cavity radii. The boundary effect becomes more significant because the fluid velocity decays as r l when the double layer spans the entire cavity. The stronger A dependence of the first order correction for thick double layer than that obtained by Ennis and Andersion [45] results from the fact that the double layers overlap in... 

Electrostatic. Virtually all colloids in solution acquire a surface charge and hence an electrical double layer. When particles interact in a concentrated region their double layers overlap resulting in a repulsive force which opposes further approach. Any theory of filtration of colloids needs to take into account the multi-particle nature of such interactions. This is best achieved by using a Wigner-Seitz cell approach combined with a numerical solution of the non-linear Poisson-Boltzmann equation, which allows calculation of a configurational force that implicitly includes the multi-body effects of a concentrated dispersion or filter cake. 

The pores within the HPLC particles used in CEC have a distribution in the range of 8-10 nm. This will not in general support EOF, owing to the doublelayer overlap that occurs in these narrow pores, and in CEC using these particles flow is assumed to occur in the interstitial region of the space between particles. However, if the packing material contains fine particles of the order of <2 pm, then these may pack into the interstitial space and so in turn cause double-layer overlap and so prevent EOF. 

Electrical Double-Layer Overlap and Pore Flow... 

For relatively wide channels with negligible electrical double-layer overlap (r/8 > 10), a nearly flat flow profile is expected. It has often been stated that when the channel size and the Debye length are of similar dimensions (r 8), complete electrical double-layer overlap occurs and the EOF is negligible. However, when r 8, a significant EOF can still be created the EOF velocity in the central part of the channel is approximately 20% of that in an infinitely wide channel. Only at conditions where r/8 1 is the EOF fully inhibited by double-layer overlap [25], It should be noted here that the approximations made by using the Rice and Whitehead theory at r/8 < 10 may lead to significant errors in the calculation of the velocity distribution and magnitude of the EOF [17] compared to more sophisticated models. 

Relevant for the discussion of the effects of pore flow in CEC is the total or average EOF through narrow channels. The effect of electrical double-layer overlap on EOF is usually expressed in an electroosmotic flow screening factor P, which is defined as the ratio of the EOF velocity to that obtained without double-layer overlap, as can be found from the Smoluchowski equation ... 

Method (a), the use of the position of the coulombic attraction theory minimum with the Od = 0 value for g, leads to the same mathematical formula for s as that expressing the Donnan equilibrium. However, we cannot say that this constitutes a derivation of the Donnan equilibrium from the coulombic attraction theory because it does not correspond to a physical limit. If Od = 0 really were the case, there would be no reason for the macroions to remain at the minimum position of the interaction potential. Nevertheless, the identity of the two expressions is an interesting result. Because Equation 4.20 is derived in the case in which there is no double layer overlap and Equation 4.1 (the Donnan equilibrium) is likewise derived without reference to the overlap of the double layers, it is precisely in this limit that the calculation should reproduce the Donnan equilibrium. The fact that it does gives us some confidence that our approximations are not too drastic and should lead to physically significant results when applied to overlapping double layers. 

The complete double-layer is electrically neutral, so electrical interactions between particles only occur when the double layers overlap. As a result, an increased concentration of ions reduces the range and effect of electrical stabilization. It is this effect that causes milk to flocculate if you add an acid such as vinegar. 

When two charged colloidal particles approach each other, their electrical double layers overlap so that the concentration of counterions in the region between the particles increases, resulting in electrostatic forces between them (Fig. 8.2). There are two methods for calculating the potential energy of the double-layer interaction between two charged colloidal particles [1,2] In the first method, one directly calculates the interaction force P from the excess osmotic pressure tensor All and... 

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