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Interaction Between Colloidal Particles Literature Review

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Interaction Between Colloidal Particles Literature Review Interaction between colloidal Technical Report TR-10-26 particles – Literature Review Interaction between colloidal particles Literature Review Longcheng Liu and Ivars Neretnieks Department of Chemical Engineering and Technology Royal Institute of Technology February 2010 Svensk Kärnbränslehantering AB Swedish Nuclear Fuel and Waste Management Co Box 250, SE-101 24 Stockholm Phone +46 8 459 84 00 TR-10-26 CM Gruppen AB, Bromma, 2010 CM Gruppen ISSN 1404-0344 Tänd ett lager: SKB TR-10-26 P, R eller TR. Interaction between colloidal particles Literature Review Longcheng Liu and Ivars Neretnieks Department of Chemical Engineering and Technology Royal Institute of Technology February 2010 This report concerns a study which was conducted for SKB. The conclusions and viewpoints presented in the report are those of the authors. SKB may draw modified conclusions, based on additional literature sources and/or expert opinions. A pdf version of this document can be downloaded from www.skb.se. Abstract This report summarises the commonly accepted theoretical basis describing interaction between colloidal particles in an electrolyte solution. The two main forces involved are the van der Waals attractive force and the electrical repulsive force. The report describes in some depth the origin of these two forces, how they are formulated mathematically as well as how they interact to sometimes result in attraction and sometimes in repulsion between particles. The report also addresses how the mathematical models can be used to quantify the forces and under which conditions the models can be expected to give fair description of the colloidal system and when the models are not useful. This report does not address more recent theories that still are discussed as to their applicability, such as ion-ion correlation effects and the Coulombic attraction theory (CAT). These and other models will be discussed in future reports. TR-10-26 3 Sammanfattning Denna rapport sammanfattar de vanligaste etablerade teorierna och modellerna som beskriver samverkan mellan kolloidala partiklar i elektrolytlösningar. De två viktigaste krafterna är den attraktiva van der Waals-kraften och kraften orsakad av elektrostatiska effekter. Rapporten går igenom och beskriver detaljerat hur dessa krafter uppstår, hur de kan formuleras i matematiska modeller liksom hur de sam- verkar för att ibland attrahera och ibland repellera partiklar. Rapporten beskriver hur de matematiska modellerna kan användas för att kvantifiera krafterna och under vilka omständigheter modellerna kan förväntas ge en rimlig beskrivning av kolloidala system och när modellerna inte är användbara. Denna rapport behandlar inte nyare teorier vilkas tillämpbarhet fortfarande diskuteras såsom jon-jon korrelationseffeter och Coulombisk attraktions teori (CAT). Dessa och andra modeller kommer att behandlas i kommande rapporter. 4 TR-10-26 Contents 1 Introduction 7 2 Attraction between molecules 11 2.1 Attraction in a vacuum 11 2.1.1 Keesom-Van der Waals forces 11 2.1.2 Debye-Van der Waals forces 12 2.1.3 London-Van der Waals forces 13 2.1.4 Properties of Van der Waals forces 15 2.1.5 Superposition of Van der Waals forces 16 2.2 Attraction in a medium 16 2.2.1 The McLachlan formula 16 2.2.2 Dielectric permittivity 17 2.2.3 Polarizability 18 3 Interaction between macrobodies 21 3.1 The Hamaker-De Boer approximation 21 3.1.1 Attraction in a vacuum 21 3.1.2 Attraction in a medium 26 3.1.3 Retarded attraction 27 3.2 The Lifshits theory 28 3.2.1 Non-retarded attraction 29 3.2.2 Retarded attraction 29 4 Electric double layers 31 4.1 General description 31 4.1.1 The surface charge 31 4.1.2 The countercharge 32 4.1.3 The Gibbs energy 32 4.2 The Poisson-Boltzmann model 34 4.2.1 Electrolyte mixture and the Debye-Hückel approximation 36 4.2.2 Single electrolyte and the Gouy-Chapman theory 39 4.3 The Stern model 43 4.3.1 The zeroth-order Stern model 43 4.3.2 The triple layer model 46 4.3.3 Variant form of the triple layer model 48 4.3.4 Specific adsorption of ions 49 5 Overlapping double layers 51 5.1 Homo-interaction 51 5.1.1 Interaction at constant potential 52 5.1.2 Interaction at constant charge 60 5.1.3 Interaction between Gouy-Stern double layers 68 5.2 Hetero-interaction 72 5.2.1 Qualitative analysis 72 5.2.2 Quantitative analysis 73 6 Solvent structure-mediated interactions 77 7 Extended DLVO theory 79 7.1 Potential energy curves 79 7.2 The c.c.c. and the Schulze-Hardy rule 82 8 Discussion and conclusions 87 9 Notation and constants 89 10 References 91 TR-10-26 5 1 Introduction In this report, we are concerned with particle-particle interactions, and the focus is put on the main forces that may operate between colloidal particles dispersed in a liquid. By the term, “pair-interaction”, we mean interaction between two particles, embedded in an infinitely large amount of electrolyte solution acting as the environment. Basically, we consider the components of the Gibbs and/or Helmholtz energy and the disjoining pressure, respectively, quantifying them as far as possible for plate-like particles for a range of conditions. For two parallel plates, at a distance s apart, the disjoining pressure Π(s) is the amount by which the normal component of the pressure tensor exceeds the outer pressure. If both plates are infinitely large, it would be the force between unit area of the one plate and the other infinite large plate. Thermodynamically, depending on the process conditions, the Gibbs energy Ga(s) or the Helmholtz energy Fa(s) is the isothermal reversible work of bring these two surfaces from an infinite distance to distance s apart. From that we find, for parallel flat plates, ∂G (s) Π(s) = − a (1-1) ∂s p,T or ∂F (s) Π(s) = − a (1-2) ∂s V ,T For isolated pair interactions in incompressible systems, these two functions are identical. Therefore, we shall generally consider the Gibbs energy Ga due to the process conditions chosen. However, if we want to consider pair interactions in confined geometries or the interaction between a pair selected from a large collection of particles, the Helmholtz energy is the appropriate choice. Then Ga and Fa may differ significantly. With this knowledge at hand, we shall now briefly review and discuss some types of interactions in some depth. London-Van der Waals or dispersion interaction These forces are ubiquitous; they depend on the nature of the particles and the medium, and on the geometry of the particles. As a first approximation, we can write the Van der Waals contribution to the Gibbs energy of interaction between two particles, a distance s apart, as, Ga,VdW = –A12 (3) f (geometry, s) (1-3) where A12(3) is the Hamaker constant for the interaction between particles of nature 1 and nature 2, respectively, across the medium 3. For homo-interaction (material 1 identical to material 2), with Hamaker constants of the type A11(3), Ga,Vdw < 0 (attractive). For hetero-interaction the Hamaker constant can, in a few situations, be negative. In practice, such situations occur most often when one of the components is a vapour. Electrostatic interaction The origin of these forces is double layer overlap. One of the most striking features of double layers is the very strong influence that indifferent electrolytes exert: they reduce ψd, the potential of the outer Helmholtz plane (i.e. the potential of the diffuse part of the double layer), and compress that layer (i.e. the Debye length κ–1 is reduced). As electro-static interaction is mainly determined by the diffuse parts of the double layers, this synergistic electrolyte effect makes itself strongly felt in the stability of hydrophobic colloids. This is the origin of the Schulze-Hardy rule. TR-10-26 7 The trend is that two isolated particles with the same charge sign repel each other. An exception to this rule takes place when one particle is highly charged, but the other only slightly. In this case, upon approach, even when both charges have the same sign, the higher charged one may induce a reverse charge onto the other, followed by attraction. Unlike Van der Waals attraction, the electrostatic contribution to the Gibbs energy of interaction, Ga,el, is independent of the nature of particles, at a given charge or potential; on the other hand, Ga,VdW is virtually insensitive to electrolytes and, for that matter, insensitive to the presence of a double layer. Expressions for Ga,el vary widely, depending on the geometry of the system, strong or weak overlap, high or low electrolyte concentration, etc, but for weak overlap and low potentials many of them have this shape: d 2 Ga, el = f (DL)(ψ ) exp(–κh) (1-4) where f (DL) contains properties of the two double layers, and solution- and geometrical quantities (such as the dielectric permittivity and particle size), whereas h is the distance between the two outer Helmholtz planes, which is shorter than s in Equation 1-3 by an amount of twice the Stern layer thickness. From Ga,el, the disjoining pressure Πel can be obtained by differentiation with respect to h, but there are also ways to compute Πel directly. Obviously, the exponential factor in Equation 1-4 stems from the exponential potential decay of the isolated diffuse double layer. On the other hand, equations such as the one above often contain ψd rather than the surface potential ψ0; this is so, because it is the overlap of the diffuse parts which is most important. This has a historical background.
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