Interaction energy between spherical colloids

Use schematic diagrams to describe the influence of electrolyte concentration, type of electrolyte, magnitude of surface electrostatic potential and strength of the Hamaker constant on the interaction energy between two colloidal-sized spherical particles in aqueous solution. What theory did you use to obtain your description Briefly describe the main features of this theory. 

In DL V O theory the total interaction energy between two spherical colloids is given as a function of distance, under some simplifying assumptions, by Equation 10.4. 

Spherical colloidal particles of diameter 10 m are dispersed in 10 mol aqueous NaCl solution at 25 °C. The Hamaker constants of the particles and the dispersion medium are 1.6 x 10 J and 0.4 x 10 J, respectively, and the zeta potential is -40 mV. Under certain assumptions, the total potential interaction energy between the particles versus the interparticle distance H is given by the plot shown in Figure 11.1. 

The total DLVO interaction energy (Vs) between two spherical colloids (each of radius a and separated by distance H) is given by the following approximate equation ... 

That is, the potential energy of attraction is identical in the two cases. This is an important result as far as the extension of molecular interactions to macroscopic spherical bodies is concerned. What it says is that two molecules, say, 0.3 nm in diameter and 1.0 nm apart, interact with exactly the same energy as two spheres of the same material that are 30.0 nm in diameter and 100 nm apart. Furthermore, an inspection of Equation (49) reveals that this is a direct consequence of the inverse sixth-power dependence of the energy on the separation. Therefore the conclusion applies equally to all three contributions to the van der Waals attraction. Precisely the same forces that are responsible for the association of individual gas molecules to form a condensed phase operate —over a suitably enlarged range —between colloidal particles and are responsible for their coagulation. 

When the radius of the spherical colloidal particles a is much larger than the shortest distance between the surfaces of two colloidal particles, the free energy between two identical spherical particles, due to double-layer, steric, and depletion interactions, can be calculated using the Derjaguin approximation 2... 

Consider next two interacting spherical colloidal particles 1 and 2 of radii ai and 02 having surface potentials ij/oi and ij/oi, respectively, at separation R = H+ai+a2 between their centers in a general electrolyte solution. According to the method of Brenner and Parsegian [6], the asymptotic expression for the interaction energy V (R) is given by... 

Equations (2)-(4) show that the total potential energy of interaction between two colloidal spherical particles depends on the surface potential of the particles, the effective Hamaker constant, and the ionic strength of the suspending medium. It is known that the addition of an indifferent electrolyte can cause a colloid to undergo aggregation. Furthermore, for a particular salt, a fairly sharply defined concentration, called critical aggregation concentration (CAC), is needed to induce aggregation. 

This entry is organized in the following paragraphs First, the advanced determination of van der Waals interaction between spherical particles is described. Second, the relevant approximate expressions and direct numerical solutions for the double-layer interaction between spherical surfaces are reviewed. Third, the experimental data obtained for AFM tips having nano-sized radii of curvature and the DLVO forces predicted by the Derjaguin approximation and improved predictions are compared. Finally, a summary of the review and recommended equations for determining the DLVO interaction force and energy between colloid and nano-sized particles is included. 

Glendinning, A.B. Russel, W.B. The electrostatic repulsion between charged spheres from exact solutions to the linearized Poisson-Boltzmann equation. J. Colloid Interface Sci. 1983, 93, 95-111 Carnie, S.L. Chan, D.Y.C. Interaction free energy between identical spherical colloidal... 

Figure 24 Total free energy of interaction between solid colloidal panicles inmersed in solution, obtained as a sum of three contributions electrostatic (EL), Lifshitz-van der Waals (LW). and acid-base (AB), following the extended DLVO model, (a) Spherical hydrophilic panicles of radius 2(X) run in 10 M solution of ttidlfferent I. I clcclruiyle and neutral pH potential 22 mV Hamaker constant A 10" J and AC(H ) = 5,. 4 mJ/m (b) Identical hydrophobic particles but in this case AG(ffu) = -.10 mJ/m ...

As model samples for the verification of the PBFFF as a separation technique, colloidal samples of hematite and titanium dioxide with submicron monodisperse spherical particles were used. In the first example, the fractionation of titanium dioxide [Ti02 with the nominal diameter obtained by a transmission electron microscope (TEM) of 0.298 jLm] and hematite-I [a-Fe203(I) with nominal diameter obtained by TEM of 0.148 xm] spherical particles was succeeded by the PBSdFFF technique. Fig. la shows the fractionation of the Ti02 and a-Fe203(I) particles by the normal SdFFF mode of operation, and Fig. lb shows the fractionation of the same particles by the potential barrier mode of SdFFF. The latter is based on the difference of the total potential energy of interaction between the colloidal particles and the channel wall due to the variation of the ionic strength of the suspending medium. 

The gap between two colliding particles (bubbles, droplets, solid particles, surfactant micelles) in a colloidal dispersion can be treated as a film of uneven thickness. Then, it is possible to utilize the theory of thin films to calculate the energy of interaction between two colloidal particles. Deijaguin [276] has derived an approximate formula which expresses the energy of interaction between two spherical particles of radii and i 2 through integral of the excess surface free energy per unit area, f h), of a plane-parallel film of thickness h [see Eq. (161)] ... 

A quantitative treatment of the effects of electrolytes on colloid stability has been independently developed by Deryagen and Landau and by Verwey and Over-beek (DLVO), who considered the additive of the interaction forces, mainly electrostatic repulsive and van der Waals attractive forces as the particles approach each other. Repulsive forces between particles arise from the overlapping of the diffuse layer in the electrical double layer of two approaching particles. No simple analytical expression can be given for these repulsive interaction forces. Under certain assumptions, the surface potential is small and remains constant the thickness of the double layer is large and the overlap of the electrical double layer is small. The repulsive energy (VR) between two spherical particles of equal size can be calculated by ... 

Partial adhesion on the SdFFF channel wall was achieved by using monodisperse spherical particles of polymethyl methacrylate (PMMA) with nominal diameter 0.358 p.m." " The extent of the PMMA particles adhesion and detachment on and from the channel wall depends on the concentration of the indifferent electrolyte Ba(N03)2 added to the suspending medium to influence the total potential energy of interaction between the PMMA particles and the channel wall. When the concentration of the electrolyte exceeds a given value, which is called critical electrolyte concentration (CEC), total adhesion of the colloidal particles occurs at the beginning of the SdFFF channel wall. 

The strength of the van der Waals interactions is determined by the size and the shape of the colloidal particles and by the chemical composition of the system, which is described by the Hamaker constant A. Between two similar particles, the van der Waals forces are always attractive and /I is a positive constant. For spherical particles of radius R at separation d, the van der Waals energy is given as... 

The experiments in the first series were conducted with methylated surfaces immersed in aqueous solutions of alkylbenzenesulfonates. The free energy of interaction between methylated glass and quartz spherical surfaces having radii of about 1 mm was studied as described in Section 1.2. Parallel to these studies, the colloid stability of 10 mn particles of methylated Aerosil (i.e., hydrophobized nanoparticles of quartz) suspensions was monitored via turbidity measurements. A characteristic sharp increase in turbidity was observed at the coagulation threshold. This behavior was reversible an increase in the surfactant concentration resulted in a decrease in turbidity, while dilution of the solutions caused the turbidity to increase. 

Modern synthetic methods allow preparation of highly monodisperse spherical particles that at least approach closely the behavior of hard-spheres, in that interactions other than volume exclusion have only small influences on the thermodynamic properties of the system. These particles provide simple model systems for comparison with theories of colloidal dynamics. Because the hard-sphere potential energy is 0 or 00, the thermodynamic and static structural properties of a hard-sphere system are determined by the volume fraction of the spheres but are not affected by the temperature. Solutions of hard spheres are not simple hard-sphere systems. At very small separations, the molecular granularity of the solvent modifies the direct and hydrodynamic interactions between suspended particles. 

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