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A Mechanical Model of Brownian Motion for One Massive Particle Including Slow Light Particles
We provide a connection between Brownian motion and a classical mechanical system. Precisely, we consider a system of one massive particle interacting with an ideal gas, evolved according to non-random mechanical principles, via interaction potentials, without any assumption requiring that the initial velocities of the environmental particles should be restricted to be “fast enough”. We prove the convergence of the (position, velocity)-process of the massive particle under a certain scaling limit, such that the mass of the environmental particles converges to 0 while the density and the velocities of them go to infinity, and give the precise expression of the limiting process, a diffusion process.
Understanding of Void Formation in Cu/Sn-Sn/Cu System During Transient Liquid Phase Bonding Process Through Diffusion Modeling
Transient Liquid Phase (TPL) bounding of Sn foil sandwiched between two Cu foils involves, in the temperature range above the melting point of Sn (232 °C) and below 350 °C, the formation and the growth of two intermetallic compounds (IMCs) Cu{sub 6}Sn{sub 5} and Cu{sub 3}Sn and mostly unintended micro-pores. The present study aims to analyze the mechanism of void development during the soldering process through an experimental and modeling approach of diffusion-controlled IMC transformation. This modeling couples the diffusion process and the interface motion with the volume shrinkage induced by the difference of partial molar volumes of atoms between each phase. We also consider two types of inter-diffusion transports: (i) inter-diffusion based on the exchange of Cu and Sn atoms and (ii) inter-diffusion of Sn atoms with vacancies allowing Kirkendall void formation. The simulations of IMC growth performed correspond to a sequence of planar phase layers, where the distinctive scallop morphology of the Cu{sub 6}Sn{sub 5} layer is described through an analytical function allowing to quantify the grain boundary diffusion pathway. We take into account of the volume diffusion mechanism for Cu{sub 3}Sn intermetallic. For Cu{sub 6}Sn{sub 5} intermetallic two mechanisms are considered, volume diffusion and grain boundary diffusion, limited by grain growth. The simulations of IMC growth kinetics, for different transport scenarios, are compared to the experimental evolving morphologies to determine the most likely mechanism of micro-void formation.
Blackout Diffusion: Generative Diffusion Models in Discrete-State Spaces
Typical generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used to generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, including many scientific applications. Here, we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process using exact (as opposed to variational) analysis. We relate the theory to the existing continuous-state Gaussian diffusion as well as other approaches to discrete diffusion, and identify the corresponding reverse-time stochastic process and score function in the continuous-time setting, and the reverse-time mapping in the discrete-time setting. As an example of this framework, we introduce “Blackout Diffusion”, which learns to produce samples from an empty image instead of from noise. Numerical experiments on the CIFAR-10, Binarized MNIST, and CelebA datasets confirm the feasibility of our approach. Generalizing from specific (Gaussian) forward processes to discrete-state processes without a variational approximation sheds light on how to interpret diffusion models, which we discuss.
Boundary Conditions for Diffusion-Mediated Processes within Linear Nanopores: Exact Treatment of Coupling to an Equilibrated External Fluid
In this paper, we consider a variety of diffusion-mediated processes occurring within linear nanopores, but which involve coupling to an equilibrated external fluid through adsorption and desorption. By determining adsorption and desorption rates through a set of tailored simulations, and by exploiting a spatial Markov property of the models, we develop a formulation for performing efficient pore-only simulations of these processes. Coupling to the external fluid is described exactly through appropriate nontrivial boundary conditions at the pore openings. This formalism is applied to analyze the following: (i) tracer counter permeation (TCP) where different labeled particles adsorb into opposite ends of the pore and establish a nonequilibrium steady state; (ii) tracer exchange (TE) with exchange of differently labeled particles within and outside the pore; (iii) catalytic conversion reactions where a reactant in the external fluid adsorbs into the pore and converts to a product which may desorb. The TCP analysis also generates a position-dependent generalized tracer diffusion coefficient, the form of which controls behavior in the TE and catalytic conversion processes. We focus on the regime of single-file diffusion within the pore which produces the strongest correlations and largest deviations from mean-field type behavior. Finally, behavior is quantified precisely via kinetic Monte Carlo simulations but is also captured with appropriate analytic treatments.
Minimum stabilizing energy release for mixing processes
Diffusive operations, which mix the populations of different elements of phase space, can irreversibly transform a given initial state into any of a spectrum of different states from which no further energy can be extracted through diffusive operations. We call these ground states. Here, the lower bound of accessible ground-state energies represents the maximal possible release of energy. This lower bound, sometimes called the diffusively accessible free energy, is of interest in theories of instabilities and wave-particle interactions. On the other hand, the upper bound of accessible ground-state energies has escaped identification as a problem of interest. Yet, as demonstrated here, in the case of a continuous system, it is precisely this upper bound that corresponds to the paradigmatic “quasilinear plateau” ground state of the bump-on-tail distribution. Although for general discrete systems the complexity of calculating the upper bound grows rapidly with the number of states, using techniques adapted from treatments of the lower bound, the upper bound can in fact be computed directly for the three-state discrete system.
Product selectivity and mass transport in levulinic acid transfer hydrogenation by monolithic MIL-100, MIL-88B and ZIF-8@Pd MOFs
The diffusion processes between adsorbent and adsorbate naturally play a significant role in the efficiency and selectivity of the heterogenous catalytic process. This paper considers the importance of diffusion processes in the transfer hydrogenation reaction of levulinic acid to γ-valerolactone by MIL-88B, MIL-100, and ZIF8@Pd monolithic catalysts. Over a period of five catalytic cycles, it was shown that the Fe-based catalysts can achieve similar conversions to the ZIF-8 supported Pd, with the only current limitation being the lower aqueous stability of these MOFs. Diffusion constants were calculated using the ZLC method, with micropore diffusion limitation found for ZIF-8 and MIL-100 monolithic frameworks at 2.7 x 10 -8 and 4.6 x 10 -8 cm 2 s -1 respectively. This diffusion limitation was also confirmed by IR spectroscopy with an increasing concentration of C—H bands on the MOF substrate post-reaction. Mass transfer coefficients, also calculated by ZLC, revealed increased mass transport for the hydrophobic ZIF-8 framework, which perhaps aids in the γ-valerolactone selectivity over side products that are produced in the absence of catalytic material, as seen for MIL-88B and MIL-100 after multiple uses.
Theoretical Analysis of Experimental Data of Sodium Diffusion in Oxidized Molybdenum Thin Films
In this work, the diffusion process of sodium (Na) in molybdenum (Mo) thin films while it was deposited on soda lime glass (SLG) was studied. A small amount of oxygen was present in the chamber while the direct-current (DC) magnetron sputtering was used for the deposition. The substrate temperatures were varied to observe its effect. Such molybdenum films, with or without oxidations, are often used in thin film solar cells, either as back contact or as hole transport layers. Secondary ion mass spectrometry (SIMS) was used to quantify the concentration of the species. A grain diffusion mechanistic model incorporating the effect of grain and grain boundary geometrical shape and size was developed. The model was used to provide an in-depth theoretical analysis of the sodium diffusion in molybdenum thin films that lead to the measured SIMS data. It was observed that not only diffusion coefficients should be considered when analyzing diffusion processes in thin films but also the ratio of grain boundary size to grain size. Both depend on substrate temperature and directly affect the amount of diffused species in the film. The data were analyzed under the light of the film growth speed versus diffusion front speed, the effect of oxygen content, and the effect of substrate temperature on the overall diffusion process. The temperature inversely affects the ratio of grain boundary size and grain size and directly affects the diffusion coefficient, which leads to a preferable temperature at which the highest amount of alkali can be found in the film.
Effect of indium concentration on electrochemical properties of electrode-electrolyte interface of CuIn{sub 1-x}Ga{sub x}Se{sub 2} prepared by electrodeposition
Highlights: • CIGS thin film was electrodeposited on a molybdenum (Mo) electrode from a non-pH buffered chloride-based electrolyte. • The electrodeposited time and the concentration of In{sup 3+} control the growth of CIGS thin film prepared in this electrodeposition system. • The kinetic of electrodeposition system was gradually dominated by the diffusion process with the electrodeposition time. - Abstract: In this study, the electrochemical impedance spectroscopy (EIS) technique was used to investigate the electrochemical behavior of the interface between the Mo electrode and the Cu-In-Ga-Se electrolyte. In this electrodeposition system, the main parameters include (1) various deposition times, and (2) the effect of In{sup 3+} concentration on the diffusion processes, electrical double layer, charge transfer resistance, and Warburg impedance. Experimental results revealed that the electrochemical kinetic behavior of CIGS thin film is strongly influenced by (1) the electrical double layer existing between the substrate, and (2) the electrolyte with different concentrations of In{sup 3+}. With an increase in the electrodeposition time, the kinetic behavior of this electrodeposition system was gradually dominated by the diffusion process rather than charge transfer process.
Diffusion behavior of Ag in TiO2 nanofilms
Highlights: • We prepared Ag/TiO{sub 2} bilayer nanofilms and heat treatment at 400 °C lead to the formation of Ag nanoparticles, which evenly distributed inside TiO{sub 2} nanofilms. • Ag is not oxidized during the heat treatment. • The morphology and size of Ag nanoparticles after heat treatment is temperature related, higher temperature lead to larger Ag nanoparticles on the surface of the film. • The diffusion process is of Ag much easier in amorphous than in anatase structure. - Abstract: Using magnetron sputtering, Ag/TiO{sub 2} bilayer nanofilms were deposited on glass substrates. Heat-treating the Ag/TiO{sub 2} nanofilms at 400 °C led to the formation of Ag nanoparticles, which dispersed inside the TiO{sub 2} films as well as the free surface of the TiO{sub 2} films. After heat treatment the sample at 500 °C, larger Ag nanoparticles could be observed on the surface, which means the particle size after heat treatment is temperature related. Anatase TiO{sub 2} phase was formed after the Ag/TiO{sub 2} films were heat-treated at 400 °C. SEM and TEM were used to investigate the morphology and structure of the films and it was found that Ag was not oxidized during heat treatment. The diffusion process of Ag in TiO{sub 2} film is also related to the order of Ag layer and TiO{sub 2} layer on glass substrate. Moreover, the diffusion process is much easier in amorphous than that in anatase structure.
Aging power spectrum of membrane protein transport and other subordinated random walks
Single-particle tracking offers detailed information about the motion of molecules in complex environments such as those encountered in live cells, but the interpretation of experimental data is challenging. One of the most powerful tools in the characterization of random processes is the power spectral density. However, because anomalous diffusion processes in complex systems are usually not stationary, the traditional Wiener-Khinchin theorem for the analysis of power spectral densities is invalid. Here, we employ a recently developed tool named aging Wiener-Khinchin theorem to derive the power spectral density of fractional Brownian motion coexisting with a scale-free continuous time random walk, the two most typical anomalous diffusion processes. Using this analysis, we characterize the motion of voltage-gated sodium channels on the surface of hippocampal neurons. Our results show aging where the power spectral density can either increase or decrease with observation time depending on the specific parameters of both underlying processes.
Diffusion in intact secondary cell wall models of plants at different equilibrium moisture content
Secondary plant cell walls are composed of carbohydrate and lignin polymers, and collectively represent a significant renewable resource. Leveraging these resources depends in part on a mechanistic understanding for diffusive processes within plant cell walls. Common wood protection treatments and biomass conversion processes to create biorefinery feedstocks feature ion or solvent diffusion within the cell wall. X-ray fluorescence microscopy experiments have determined that ionic diffusion rates are dependent on cell wall hydration as well as the ionic species through non-linear relationships. In this work, we use classical molecular dynamics simulations to map the diffusion behavior of different plant cell wall components (cellulose, hemicellulose, lignin), ions (Na + , K + , Cu2 + , Cl - ) and water within a model for an intact plant cell wall at various hydration states (3-30 wt% water). From these simulations, we analyze the contacts between different plant cell wall components with each other and their interaction with the ions. Generally, diffusion increases with increasing hydration, with lignin and hemicellulose components increasing diffusion by an order of magnitude over the tested hydration range. Ion diffusion depends on charge. Positively charged cations preferentially interact with hemicellulose components, which include negatively charged carboxylates. As a result, positive ions diffuse more slowly than negatively charged ions. Measured diffusion coefficients are largely observed to best fit piecewise linear trends, with an inflection point between 10 and 15% hydration. These observations shed light onto the molecular mechanisms for diffusive processes within secondary plant cell walls at atomic resolution.
Toward a Universal Model of Hyporheic Exchange and Nutrient Cycling in Streams
In this paper we demonstrate that several ubiquitous hyporheic exchange mechanisms can be represented simply as a one-dimensional diffusion process, where the diffusivity decays exponentially with depth into the streambed. Based on a meta-analysis of 106 previously published laboratory measurements of hyporheic exchange (capturing a range of bed morphologies, hydraulic conditions, streambed properties, and experimental approaches) we find that the reference diffusivity and mixing length-scale are functions of the permeability Reynolds Number and Schmidt Number. These dimensionless numbers, in turn, can be estimated for a particular stream from the median grain size of the streambed and the stream's depth, slope, and temperature. Application of these results to a seminal study of nitrate removal in 72 headwater streams across the United States, reveals: (a) streams draining urban and agricultural landscapes have a diminished capacity for in-stream and in-bed mixing along with smaller subsurface storage zones compared to streams draining reference landscapes; (b) under steady-state conditions nitrate uptake in the streambed is primarily biologically controlled; and (c) median reaction timescales for nitrate removal in the hyporheic zone are 0.5 and 20 hr for uptake by assimilation and denitrification, respectively. While further research is needed, the simplicity and extensibility of the framework described here should facilitate cross-disciplinary discussions and inform reach-scale studies of pollutant fate and transport and their scale-up to watersheds and beyond.
Electrochemical force microscopy
A system and method for electrochemical force microscopy are provided. The system and method are based on a multidimensional detection scheme that is sensitive to forces experienced by a biased electrode in a solution. The multidimensional approach allows separation of fast processes, such as double layer charging, and charge relaxation, and slow processes, such as diffusion and faradaic reactions, as well as capturing the bias dependence of the response. The time-resolved and bias measurements can also allow probing both linear (small bias range) and non-linear (large bias range) electrochemical regimes and potentially the de-convolution of charge dynamics and diffusion processes from steric effects and electrochemical reactivity.
Atomic Diffusion, Segregation, and Grain Boundary Migration in Nickel-Based Alloys from Molecular Dynamics Simulations
Grain boundary diffusion and metal mobility in alloys control material performance in many applications and yet remain poorly understood at a mechanistic level. With advances in accessible time and length scales for computational molecular simulations, and recent force field developments, we now possess tools to help unravel those mechanisms. Using large-scale molecular dynamics simulations, here we examined vacancy-mediated diffusion processes in Ni-5Cr alloy with low and high-energy grain boundaries. We show that atomic diffusion inside the grain boundary plane is about four times higher than bulk diffusion, at any temperature, and exhibits a typical Arrhenius behavior with a very small energy barrier (0~.8 eV for Cr and 0.7 eV for Ni within 1300-1600 K). Additionally, the fastest diffusing species inverts; Cr diffusion was faster than Ni in the bulk but slower in the grain boundaries. This is attributed to the creation of high cohesive energy clusters of Cr at the grain boundary. Grain boundary migration was also observed to be temperature dependent and appears to be two times higher in the 5% Cr alloy than in pure Ni, highlighting the important role of the alloying element on grain boundary motion.
Tritium Diffusion Pathways in y-LiAlO2 Pellets Used in TPBAR: A First-Principles Density Functional Theory Investigation
With superior thermophysical and thermochemical properties, gamma-LiAlO2 has high compatibility with other blanket materials and is used in the form of an annular pellet in tritium-producing burnable absorber rods (TPBARs) to produce tritium by thermal neutron irradiation of Li-6. In radiation damaged gamma-LiAlO2, different types of vacancies, defects of its constituent elements, and other trapping sites hinder the diffusion process of tritium. In this study, the first-principles density functional theory approach is used to study the diffusion mechanisms of tritium defect and its species, such as interstitial and substitutional tritium defects, oxygen-tritium vacancy defects, and interaction of tritium with oxygen vacancies in defective and nondefective gamma-LiAlO2. The obtained results provide an understanding of how such defects hamper the diffusivity and solubility of tritium. By calculating several different diffusion pathways, our results show that the smallest activation energy barrier is 0.63 eV for substitutional tritium diffusion with a diffusion coefficient of 3.25 X 10(-12) m(2)/s. The smallest oxygen-tritium diffusion barrier is found to be 2.17 eV, which is around 3.5 times higher than the tritium diffusion barrier alone.
Exploring the energy landscape of RBMs: reciprocal space insights into bosons, hierarchical learning and symmetry breaking
Deep generative models have become ubiquitous due to their ability to learn and sample from complex distributions. Despite the proliferation of various frameworks, the relationships among these models remain largely unexplored, a gap that hinders the development of a unified theory of AI learning. In this work, we address two central challenges: clarifying the connections between different deep generative models and deepening our understanding of their learning mechanisms. We focus on Restricted Boltzmann Machines (RBMs), a class of generative models known for their universal approximation capabilities for discrete distributions. By introducing a reciprocal space formulation for RBMs, we reveal a connection between these models, diffusion processes, and systems of coupled bosons. Our analysis shows that at initialization, the RBM operates at a saddle point, where the local curvature is determined by the singular values of the weight matrix, whose distribution follows the Marc̆enko-Pastur law and exhibits rotational symmetry. During training, this rotational symmetry is broken due to hierarchical learning, where different degrees of freedom progressively capture features at multiple levels of abstraction. This leads to a symmetry breaking in the energy landscape, reminiscent of Landau’s theory. This symmetry breaking in the energy landscape is characterized by the singular values and the weight matrix eigenvector matrix. We derive the corresponding free energy in a mean-field approximation. We show that in the limit of infinite size RBM, the reciprocal variables are Gaussian distributed. Our findings indicate that in this regime, there will be some modes for which the diffusion process will not converge to the Boltzmann distribution. To illustrate our results, we trained replicas of RBMs with different hidden layer sizes using the MNIST dataset. Our findings not only bridge the gap between disparate generative frameworks but also shed light on the fundamental processes underpinning learning in deep generative models.
Chemical imaging and diffusion of hydrogen and lithium in lithium aluminate
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