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In our previous publication (Ref. 1) we have shown that the data for the normalized diffusion coefficient of the polymers, $D_p/D_{p0}$, falls on a master curve when plotted as a function of $h/lambda_d$, where $h$ is the mean interparticle distance and $lambda_d$ is a dynamic length scale. In the present note we show that also the normalized diffusion coefficient of the nanoparticles, $D_N/D_{N0}$, collapses on a master curve when plotted as a function of $h/R_h$, where $R_h$ is the hydrodynamic radius of the nanoparticles.
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In order to characterize the geometrical mesh size $xi$, we simulate a solution of coarse-grained polymers with densities ranging from the dilute to the concentrated regime and for different chain lengths. Conventional ways to estimate $xi$ rely either on scaling assumptions which give $xi$ only up to an unknown multiplicative factor, or on measurements of the monomer density fluctuation correlation length $xi_c$. We determine $xi_c$ from the monomer structure factor and from the radial distribution function, and find that the identification $xi=xi_c$ is not justified outside of the semidilute regime. In order to better characterize $xi$, we compute the pore size distribution (PSD) following two different definitions, one by Torquato et al. (Ref.1) and one by Gubbins et al. (Ref.2). We show that the mean values of the two distributions, $langle r rangle_T$ and $langle r rangle_G$, both display the behavior predicted for $xi$ by scaling theory, and argue that $xi$ can be identified with either one of these quantities. This identification allows to interpret the PSD as the distribution of mesh sizes, a quantity which conventional methods cannot access. Finally, we show that it is possible to map a polymer solution on a system of hard or overlapping spheres, for which Torquatos PSD can be computed analytically and reproduces accurately the PSD of the solution. We give an expression that allows $langle r rangle_T$ to be estimated with great accuracy in the semidilute regime by knowing only the radius of gyration and the density of the polymers.
Using molecular dynamics simulations we study the static and dynamic properties of spherical nanoparticles (NPs) embedded in a disordered and polydisperse polymer network. Purely repulsive (RNP) as well as weakly attractive (ANP) polymer-NP interactions are considered. It is found that for both types of particles the NP dynamics at intermediate and at long times is controlled by the confinement parameter $C=sigma_N/lambda$, where $sigma_N$ is the NP diameter and $lambda$ is the dynamic localization length of the crosslinks. Three dynamical regimes are identified: i) For weak confinement ($C lesssim 1$) the NPs can freely diffuse through the mesh; ii) For strong confinement ($C gtrsim 1$) NPs proceed by means of activated hopping; iii) For extreme confinement ($C gtrsim 3$) the mean squared displacement shows on intermediate time scales a quasi-plateau since the NPs are trapped by the mesh for very long times. Escaping from this local cage is a process that depends strongly on the local environment, thus giving rise to an extremely heterogeneous relaxation dynamics. The simulation data are compared with the two main theories for the diffusion process of NPs in gels. Both theories give a very good description of the $C-$dependence of the NP diffusion constant, but fail to reproduce the heterogeneous dynamics at intermediate time scales.
We investigate the dynamics of nanoparticles in semidilute polymer solutions when the nanoparticles are comparably sized to the polymer coils using explicit- and implicit-solvent simulation methods. The nanoparticle dynamics are subdiffusive on short time scales before transitioning to diffusive motion on long time scales. The long-time diffusivities scale according to theoretical predictions based on full dynamic coupling to the polymer segmental relaxations. In agreement with our recent experiments, however, we observe that the nanoparticle subdiffusive exponents are significantly larger than predicted by the coupling theory over a broad range of polymer concentrations. We attribute this discrepancy in the subdiffusive regime to the presence of an additional coupling mechanism between the nanoparticle dynamics and the polymer center-of-mass motion, which differs from the polymer relaxations that control the long-time diffusion. This coupling is retained even in the absence of many-body hydrodynamic interactions when the long-time dynamics of the colloids and polymers are matched.
The organization of nano-particles inside grafted polymer layers is governed by the interplay of polymer-induced entropic interactions and the action of externally applied fields. Earlier work had shown that strong external forces can drive the formation of colloidal structures in polymer brushes. Here we show that external fields are not essential to obtain such colloidal patterns: we report Monte Carlo and Molecular dynamics simulations that demonstrate that ordered structures can be achieved by compressing a `sandwich of two grafted polymer layers, or by squeezing a coated nanotube, with nano-particles in between. We show that the pattern formation can be efficiently controlled by the applied pressure, while the characteristic length--scale, i.e. the typical width of the patterns, is sensitive to the length of the polymers. Based on the results of the simulations, we derive an approximate equation of state for nano-sandwiches.
The proliferation of optical, electron, and scanning probe microscopies gives rise to large volumes of imaging data of objects as diversified as cells, bacteria, pollen, to nanoparticles and atoms and molecules. In most cases, the experimental data streams contain images having arbitrary rotations and translations within the image. At the same time, for many cases, small amounts of labeled data are available in the form of prior published results, image collections, and catalogs, or even theoretical models. Here we develop an approach that allows generalizing from a small subset of labeled data with a weak orientational disorder to a large unlabeled dataset with a much stronger orientational (and positional) disorder, i.e., it performs a classification of image data given a small number of examples even in the presence of a distribution shift between the labeled and unlabeled parts. This approach is based on the semi-supervised rotationally invariant variational autoencoder (ss-rVAE) model consisting of the encoder-decoder block that learns a rotationally (and translationally) invariant continuous latent representation of data and a classifier that encodes data into a finite number of discrete classes. The classifier part of the trained ss-rVAE inherits the rotational (and translational) invariances and can be deployed independently of the other parts of the model. The performance of the ss-rVAE is illustrated using the synthetic data sets with known factors of variation. We further demonstrate its application for experimental data sets of nanoparticles, creating nanoparticle libraries and disentangling the representations defining the physical factors of variation in the data. The code reproducing the results is available at https://github.com/ziatdinovmax/Semi-Supervised-VAE-nanoparticles.
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