Simple analysis of scattering data with the Ornstein-Zernike equation

Predicting Colloidal Interaction Parameters from Small-Angle X-ray Scattering Curves Using Artificial Neural Networks and Markov Chain Monte Carlo Sampling

Small-angle X-ray scattering (SAXS) is a characterization technique that allows for the study of colloidal interactions by fitting the structure factor of the SAXS profile with a selected model and closure relation. However, the applicability of this approach is constrained by the limited number of existing models that can be fitted analytically, as well as the narrow operating range for which the models are valid. In this work, we demonstrate a proof of concept for using an artificial neural network (ANN) trained on SAXS curves obtained from Monte Carlo (MC) simulations to predict values of the effective macroion valency (Z eff) and the Debye length (κ-1) for a given SAXS profile. This ANN, which was trained on 200,000 simulated SAXS curves, was able to predict values of Z eff and κ-1 for a test set containing 25,000 simulated SAXS curves, where most predicted values had errors smaller than 20%. Subsequently, an ANN was used as a surrogate model in a Markov chain Monte Carlo sampling algorithm to obtain maximum a posteriori estimates of Z eff and κ-1, as well as the associated confidence intervals and correlations between Z eff and κ-1 for an experimentally obtained SAXS profile.

The Role of Energy Scales for the Structure of Ionic Liquids at Electrified Interfaces: A Theory-Based Approach

Ionic liquids offer unique bulk and interfacial characteristics as battery electrolytes. Our continuum approach naturally describes the electrolyte on a macroscale. An integral formulation for the molecular repulsion, which can be quantitatively determined by both experimental and theoretical methods, models the electrolyte on the nanoscale. In this article, we perform a systematic series expansion of this integral formulation, derive a description of chemical potentials in terms of higher-order concentration gradients, and rationalize the appearance of fourth-order derivative operators in modified Poisson equations, as recently proposed in this context. In this way, we formulate a rigorous multiscale methodology from atomistic quantum chemistry calculations to phenomenological continuum models. We apply our generalized framework to ionic liquids near electrified interfaces and perform analytical asymptotic analysis. Three energy scales describing electrostatic forces between ions, molecular repulsion, and thermal motion determine the shape and width of the long-ranging charged double layer. We classify the charge screening mechanisms dependent on the system parameters as dielectricity, ion size, interaction strength, and temperature. We find that the charge density of electrochemical double layers in ionic liquids either decays exponentially, for negligible molecular repulsion, or oscillates continuously. Charge ordering across several ion diameters occurs if the repulsion between molecules is comparable with thermal energy and Coulomb interactions. Eventually, phase separation of the bulk electrolyte into ionic layers emerges once the molecular repulsion becomes dominant. Our framework predicts the exact phase boundaries among these three phases as a function of temperature, dielectricity, and ion size.

Models for randomly distributed nanoscopic domains on spherical vesicles

The existence of lipid domains in the plasma membrane of biological systems has proven controversial, primarily due to their nanoscopic size-a length scale difficult to interrogate with most commonly used experimental techniques. Scattering techniques have recently proven capable of studying nanoscopic lipid domains populating spherical vesicles. However, the development of analytical methods able of predicting and analyzing domain pair correlations from such experiments has not kept pace. Here, we developed models for the random distribution of monodisperse, circular nanoscopic domains averaged on the surface of a spherical vesicle. Specifically, the models take into account (i) intradomain correlations corresponding to form factors and interdomain correlations corresponding to pair distribution functions, and (ii) the analytical computation of interdomain correlations for cases of two and three domains on a spherical vesicle. In the case of more than three domains, these correlations are treated either by Monte Carlo simulations or by spherical analogs of the Ornstein-Zernike and Percus-Yevick (PY) equations. Importantly, the spherical analog of the PY equation works best in the case of nanoscopic size domains, a length scale that is mostly inaccessible by experimental approaches such as, for example, fluorescent techniques and optical microscopies. The analytical form factors and structure factors of nanoscopic domains populating a spherical vesicle provide a new and important framework for the quantitative analysis of experimental data from commonly studied phase-separated vesicles used in a wide range of biophysical studies.