46569-AC6
Developing Inverse Density Functional Theory as an Analytical Tool for Diffusing Colloidal Probe Microscopy
Recent advances in microscopy, such as total internal reflection and confocal scanning laser techniques, now permit the direct positional tracking of large numbers of colloidal particles, both in the bulk and near interfaces. A novel application of this technology, recently proposed by Professor Michael A. Bevan (Department of Chemical & Biomolecular Engineering, Johns Hopkins University) under the name of Diffusing Colloidal Probe Microscopy (DCPM), is the use of colloidal particles as probes of the energetic characteristics of a surface. The key idea of DCPM is that the potential energy between a colloidal particle and a surface may be extracted from the recorded Brownian trajectories of an ensemble of such particles near the surface. DCPM is a much more sensitive surface probe than traditional techniques, such as atomic force microscopy, which ultimately rely on mechanical transduction. The theoretical challenge in DCPM is to obtain the potential energy of a single colloidal particle in the external field created by the surface, from the measured particle trajectories. For very dilute colloidal systems, a simple Boltzmann inversion of the average density profile is sufficient. However, non-dilute systems are often needed to obtain sufficient statistics in reasonable experimental time. Dense systems require a more complex inversion analysis, since particle-particle interactions influence the profile. We are developing an approach to inversion based on density functional theory (DFT), where we predict the single-particle surface potential from the experimentally measured density profile in a non-dilute colloidal ensemble. We are using a closure-based DFT formulation developed by Zhou and Ruckenstein [J. Chem. Phys. 112, 8079-8082 (2000)] that requires two closures: one to solve the Ornstein-Zernike equation for the bulk direct correlation function and another to capture the higher-order terms in the free energy perturbation expansion. Our initial work was purely computational, employing Monte Carlo simulation of a colloidal fluid near an interface to provide the particle density profile. Gravitational effects were not included. We performed a study of four common classes of colloidal potential: hard sphere, screened electrostatic repulsion, van der Waals attraction, and Asakura-Oosawa (AO) depletion attraction (the latter three included a hard core repulsion at short range). We surveyed a number of potentially useful closures and found that the combination of Rogers-Young (RY) and Verlet-modified (VM) was generally the best choice. This closure combination produced acceptable inversion results (maximum deviation from true potential less than 0.1 kT) at low to moderate bulk densities (dimensionless hard core densities below 0.3) across the different colloidal interaction types. This work resulted in a publication in Langmuir. We subsequently carried out a study that employed actual experimental data from the laboratory of our collaborator, Prof. Bevan. Specifically we used confocal scanning laser microscopy images of the sedimentation of 720-nm fluorescent core-shell silica particles interacting via a short-ranged electrostatic repulsion. This study included two additional complexities beyond our previous simulation-based work. First, the raw data from the confocal experiments was light intensity rather than particle density; in collaboration with Prof. Bevan we developed a simulation-based technique for converting from intensity profiles to density profiles. Second, the presence of a gravitational field required the development of an additional loop in the DFT algorithm and significantly increased the particle density near the interface for a given bulk density. The latter effect led to difficulties with numerical convergence or accuracy for highly dense sediments, but we were able to obtain meaningful results for a more dilute sediment, where our inverse DFT procedure yielded acceptable results for the particle-surface potential (less than 0.5 kT maximum deviation from the true potential). As in the simulation-based study, a combination of the RY and VM closures was found to be the most accurate. This work resulted in a publication in the Journal of Chemical Physics. In addition to the studies related to DCPM, we have also worked on the application of closure-based DFT to fluid-solid transitions. We are interested in this area because (1) many colloidal self-assembly problems will involve such transitions, and (2) previous DFT work on fluid-solid transitions has mainly focused on perturbative or fundamental-measures approaches. We see an opportunity to develop new DFT tools that will have a general impact on colloid science, beyond the DCPM analytical technique. We have published an initial study on the freezing of hard spheres (Journal of Chemical Physics), and we are following up with work on other colloidal potentials.
