On The Electrostatic Interaction Between Two Nonuniformly Charged Colloidal Particles

The study of electrostatic interaction between like-charged colloidal particlesin an electrolyte becomes a fast growing research area since Grier et al. foundan anomalous electrostatic attraction between these particles in a series experi-ments. Though many theoretical investigations were made in the past decadesand different mechanisms were suggested to explain the experimental observa-tions, a universally accepted explanation is still far reaching. To seek the reasonof this anomalous attraction, in this thesis, we consider the nonuniform distri-bution of surface charges on the colloidal particles and investigate the effect ofthis nonuniformity on the electrostatic interaction between like-charged colloidalparticles. The thesis is mainly composed of three parts to investigate the electro-static interaction between two nonuniformly charged colloidal particles: the elec-trostatic interaction between like-charged spherical colloidal particles in a bulkelectrolyte; the electrostatic interaction between like-charged spherical colloidalparticles confined in a long charged cylinder wall; the electrostatic interactionbetween like-charged ellipsoidal colloidal particles.In the first chapter, we introduces in detail the background and evolvementof our research. To show the influence of anisotropy of surface charge distributionof the particles on the pairwise interaction more clearly, in the second chapter,we consider a simple system of two nonuniformly charged spherical particles im-mersed in a bulk electrolyte. In this chapter, we expand the potential in theelectrolyte in terms of spherical harmonic functions with respect of one chosencoordinates by a method of a general rotation-translation transformation and weobtained the potential of the system easily by using the orthogonality conditionof the spherical harmonic functions. Due to the anisotropy of the distributionof the surface charges on the particles, we find the electrostatic interaction be- tween two like-charged colloid which are in some fixed configuration is sensitiveto the relative orientations of two particles in our calculations. After a thermalaverage with respect to all orientations at fixed sphere-sphere distance, the ef-fective electrostatic interaction between two like-charged colloidal particles stillmake a remarkable change with the change of the nonuniformity of the particles.The effective repulsive interaction after thermal average is weaker than the casewhen charges are uniformly distributed on the spheres, and with the increase ofthe nonuniformity, an attractive interaction between the two spheres emerges,which suggests that the anisotropy of the distribution of surface charges on theparticles is one of important reasons to induce the anomalous electrostatic attrac-tion between two like-charged colloidal particles. The effective interaction of twoparticles is the result of the competition of the effective electrostatic repulsionbetween two effective charges and the effective electrostatic attraction betweentwo effective dipoles. Additionally, we also investigate the relation between theeffective electrostatic interaction of like-charged particles and the consistency ofthe net charges in the electrolyte. And we find, due to the screen effect by thecounterion in the electrolyte, the amplitude of the repulsive (attractive) inter-action was weaken (enhanced) with the densification of the net charges in theelectrolyte, which is consistent with the confined system investigated in our thirdchapter. At the last part of this chapter, we extend our method mentioned in thischapter to a system with a more general distribution of surface charges on thecolloidal particles. And we find the effective dipolar moment is the direct reasonto affect the electrostatic interaction of two like-charged colloidal particles, whilethe explicit form of the distribution of the surface charges is inessential to thispairwise interactions as long as the effective dipolar moments of the particles arefixed.Since in most of experiments about the electrostatic interaction of colloidalparticles are made in a confined system, in the third chapter, we investigatethe electrostatic interaction of colloidal particles in a confined system. In thissystem, two nonuniformly charged spherical colloids confined in a long chargedcylinder wall. Two models of the boundary conditions on the cylinder wall areconsidered. One is the fixed potential model and the other is the fixed surfacecharge density model. The confinement of the charged cylinder wall enhances the effect of the anisotropy of the distribution of the surface charges on theparticles to the electrostatic interaction of two like-charged colloidal particles.And it makes a quantitative even qualitative change for the spheres'interactioncomparing with the unconfined system. A long-ranged attraction emerges in theconfined system though they are repulsive in the unconfined system under thesame environment. That means the confinement of the charged cylinder wall isalso one of important reason to induce the anomalous attraction between like-charged colloidal particles. Additionally, because of the existence of the longcharged cylinder wall, the potential of the system becomes complicated and itcan't be expanded in terms of the spherical harmonic functions in one coordinates.As a result, the method mentioned in the unconfined system will not be availablefor this confined system. In this chapter, we use so called"multipole expansionmethod"to obtain the potential of the system.In the fourth chapter, we investigate the electrostatic interaction of two like-charged ellipsoidal colloidal particles confined in a long charged cylinder wall. Inthis system, we also consider two models of boundary conditions on the cylinderwall. One is the fixed potential model and the other is the fixed surface chargedensity model. With a fixed total charges on the ellipsoidal colloidal particles,fixed shortest"particle-wall"distance and fixed consistency of the net chargesin the electrolyte, the electrostatic repulsive interaction of the like-charged ellip-soidal colloid decreases at first time and then increases with the increasing of theeccentricity of the ellipsoid for both the fixed potential model and fixed chargedensity model. This relation of the electrostatic interaction of like-charged parti-cles and the eccentricity of the ellipsoid is opposite to the relation of the effectivedipolar moment of the particles and the eccentricity of the ellipsoid. This phe-nomenon suggests further that the effective dipolar moment due to the anisotropyof the distribution of the surface charges on the particles is one of the importantreasons to induce the anomalous attraction of the like-charged colloidal particles.For the symmetry-broken of the ellipsoid, the method of the multipole expansionisn't feasible for the ellipsoid system yet. In this chapter, we introduced a"leastsquares multipole expandsion method"to obtain the potential of the system.As a conclusion, the effective dipolar moment due to the anisotropy of thedistribution of the surface charges on the colloidal particles is one of the important reasons to induce the anomalous attraction of the like-charged colloids. And theconfinement to the colloid particles also promotes the emergence of the anomalousattraction of the like-charged colloidal particles. The last chapter is the conclusionand expectation of this thesis.