Dynamic Forces Between Two Deformable Oil Droplets in Water
Title: Dynamic Forces Between Two Deformable Oil Droplets in Water
Authors: R.R. Dagastine, R. Manica, S.L. Carnier, D.Y.C. Chan, G.W. Stevens, and F. Grieser
Journal: Science, 313 (2006): 210-213
dynamic forces, colloidal suspensions, interfacial deformation, static surface forces
Interactions between static surfaces are well studied, however, dynamic interactions in biological and other suspended soft-matter systems is still not completely understood. Dynamic forces are key to manipulating and controlling soft-matter systems (such as emulsions and complex fluids). Understanding dynamic droplet-droplet interactions is very challenging. This paper presents the dynamic interactions between two deformable oil droplets. The methodology presented in the paper is widely applicable to all soft-matter systems and has important implications for studying collision forces from Brownian motion.
Dynamic interactions between soft matter particles, such as emulsion droplets, is difficult because every interface in deformable. Recently, studies have demonstrated direct force measurements of interactions between droplets with radii of around 40 microns (intermediate size droplets). This study focusses on developing a model for droplet-droplet interactions for intermediate size droplets where deformation, hydrodynamic drainage, and interaction forces are all important. This study also shows how traditional concepts of drainage as a two-stage process are not appropriate for droplet sizes relevant to emulsions.
Two decane droplets with radii of 43 and 90 microns, in a surfactant solution, were immobilized on an AFM cantilever and substrate. The dynamic interaction force between the droplets was measured as a function of piezo drive motion of the substrate (see figure 1)
A) the experiment between two oil droplets - one immobilized on the cantilever and the other immobilized on the substrate of an AFM
B - D) The dynamic interaction force (F) versus piezo drive motion (X) between two oil droplets in aqueous solution. The points refer to the experimental data and the solid lines are the calculated force curves from a model of the dynamic droplet interactions. The open symbols refer to approach velocities and the filled symbols refer to retract velocities.
The velocity range spans the likely velocities of an emulsion droplet of comparable size when undergoing Brownian motion. The measurements show that hydrodynamic interactions between droplets of this size are important even when describing emulsion stability, where equilibrium forces are assumed to dominate.
A quantitative analysis of the data to determine the interfacial separation is based on the Young-Laplace equation and the techniques developed for static force measurements between a rigid particle and droplet.
The dynamic problem contains three disparate length scales:
1) droplet radii (~ 50 microns)
2) the axial length scale of interaction forces (10-100nm)
3) radial length scale of the interaction (1-5 microns)
Two PDE were used in the analysis - the Reynolds drainage equation and the normal stress balance:
h(r,t) is the interdroplet separation, p is the hydrodynamic pressure, <math>\Pi</math> is the equilibrium disjoining pressure, Ro is the unperturbed droplet radii
The disjoining pressure was calculated using the Poisson-Boltzmann equation to describe the electrostatic double-layer repulsion between the negatively charged surfactant-laden interfaces. This required a surface potential for the system to be known.
In a dynamic system, the droplets flatten whenever the normal pressure is on the order of the droplet Laplace pressure - regardless of the contribution to the pressure from either equilibrium surface forces or hydrodynamic drainage. The total radial pressure profile changes as a function of radii - for small radii the radial pressure is positive and for larger radii, it is negative. This is due to the combination of pressure with different length scales from a positive equilibrium surface force at these interfacial separations and the negative hydrodynamic drainage pressure. Unlike the case involving only equilibrium interactions, for the dynamic interaction, droplet coalescence can occur as the droplets more apart.
This paper demonstrated how the interactions for dynamic droplets in the 50-100 micron range are much more complicated than for the interactions of static droplets of the same size - hydrodynamic forces and surface forces are coupled. The relative length scales that the forces act over are very important and will influence which force dominates the interaction.