Distinct Diffusion Coefficients Research Articles

Generalities on a delayed spatiotemporal host-pathogen infection model with distinct dispersal rates

We propose a general model to investigate the effect of the distinct dispersal coefficient {for the} infected and susceptible hosts on the pathogen dynamics. The mathematical challenge lies in the fact that the investigated model is partially degenerate and the solution map is not compact. The spatial heterogeneity of the model parameters and the distinct diffusion coefficients induce infection in the low-risk regions. In fact, as infection dispersal increases, the reproduction of the pathogen particles decreases. The dynamics of the investigated model is governed by the value of the basic reproduction number $R_0$. {If $R_0\leq1$, then the} pathogen particles extinct, and for $R_0>1$ the pathogen particles persist, and we guarantees of the existence of at least one positive steady state. The asymptotic profile of the positive steady state is shown in the case when one or both diffusion coefficients for the host tends to zero or infinity.

Effect of Relative Mass on Ion Velocity Cross-Correlations in Ionic Liquids and Molten Salts: Different Perspectives in Different Reference Frames.

In electrolytes, the self- and interdiffusion coefficients, transport numbers, and electrical conductivity can be used to determine velocity cross-correlation coefficients (VCC) that are also accessible through molecular dynamics simulations. In an ionic liquid or molten salt, there are only three, corresponding to correlations between the velocities of distinct ion pairs (cation-anion, cation-cation, and anion-anion) averaged over both the ensemble and time, calculable from experimental ion self-diffusion coefficients and the electrolyte conductivity. Most usually, the mass-fixed frame of reference (with velocities relative to that of the center of mass of the system) is used to discuss the VCC and the distinct diffusion coefficients (DDC) derived from them. Recent work in the literature has suggested a dependence of the DDC on the ratio of the anion to cation mass. Here, we demonstrate, using our own and literature transport property data for a large number of ionic liquids and molten salts, that the trends observed depend on the particular choice of velocity reference frame, mass-, number-, or volume-fixed. The perception of ion-ion interactions may be distorted in the mass- and volume fixed frames when the co-ions have very different masses or volumes, particularly for systems containing light lithium ions.

Ionogels based on protic ionic liquid - lithium salt mixtures

The effect of the addition of lithium nitrate on the liquid range and electrical conductivity of mixtures of a protic ionic liquid, ethylammonium nitrate (EAN) with LiNO3 is reported in this work. Moreover, changes in these properties upon confinement in silica-based matrices are also analysed and related to variations on the microstructure of the mixtures. Our results show that gelation of the mixtures induces amorphous behaviour in all the samples, but no significant changes in the thermal stability are detected. However, important differences on the electrical conductivity of the liquid and gel states of the mixtures were observed. While this magnitude decreases with the addition of salt in the liquid samples for all the temperatures it increases at low concentrations of LiNO3 up to a maximum in the gelled samples with the lowest concentration of added salt (0.1 mol of salt by kg of EAN). High resolution magic angle spinning (HRMAS) NMR shows that the addition of lithium nitrate increases the mobility (both rotational and translational motion) of the cation EA+ within the ionogel and the diffusion of the cation provides evidence of the presence of two different populations of the same molecules experiencing two distinct apparent diffusion coefficients (bi-exponential decay of magnetization in diffusion NMR experiments), while it is mono-exponential in the pure ionic liquid. The Li+ cation is shown to display a higher mobility (both rotational and translational motion) in the ionogel than in the neat ionic liquid. The frustration of the formation of solvation complexes due to the confinement is suspected to be behind this behaviour.

UNDERSTANDING THE ROLE OF DIFFUSION IN THE SEPARATION OF RARE EARTH ELEMENTS IN WATER-n-HEXANE SYSTEMS: A MOLECULAR DYNAMICS SIMULATION STUDY

Liquid-liquid extraction is one of the methods for separating rare earth elements (REE) in the presence of an extractant. The separation of REE ions, complex with an extractant, involves interfacial migrations that are influenced by the diffusion of the respective ions. Therefore, we performed molecular dynamics simulations on REE ions (La3+, Sm3+, Eu3+, Gd3+, and Tb3+) in a water-n-hexane system to determine if each ion exhibited a distinct diffusion coefficient. By employing molecular dynamics simulations, we calculated diffusion coefficients for these ions based on the Einstein relation. The diffusion coefficients for La3+, Sm3+, Eu3+, Gd3+, and Tb3+ were found to be 4.21×10–6, and 3.96×10–6, 4.57×10–6, 4.17×10–6, and 5.19×10–6 cm2/s, respectively. However, statistical analysis via the Kruskal-Wallis test revealed no significant variance in the diffusion coefficients (p-value greater than 0.05), indicating that diffusion is not a rate-limiting factor in REE separation. The findings suggest that effective mixing during extraction can eliminate the role of diffusion as a differentiating factor in REE separation. Overall, this study offers critical insights into optimizing REE extraction processes

Temperature and Pressure Dependence of the Transport Properties of the Ionic Liquid Triethyloctylphosphonium Bis(trifluoromethanesulfonyl)amide, [P222,8][Tf2N

Viscosities of triethyloctylphosphonium bis(trifluoromethanesulfonyl)amide, [P222,8][Tf2N], are reported as a function of temperature [(273 to 363) K] and pressure (maximum 302 MPa) with a falling-body viscometer together with electrical conductivities (κ) measured by impedance spectroscopy at (283 to 348) K, to 250 MPa maximum pressure. pVT data were determined with a vibrating tube densimeter from (298 to 353) K to 50 MPa. Ion self-diffusion coefficients (DSi) were measured by steady-gradient spin-echo NMR [(313 to 365) K], and densities [(273 to 363) K] were determined with a vibrating tube densimeter, both at atmospheric pressure. The results were correlated with Walden and Stokes–Einstein–Sutherland relations. Velocity cross-correlation coefficients (VCC), distinct diffusion coefficients (DDC) and Laity resistance coefficients (LRC) were calculated from DSi and κ at 0.1 MPa. The DDC and LRC for [P222,8][Tf2N] and the pentyl-substituted analogue, [P222,5][Tf2N], showed differences for cation–cation and cation–anion velocity anti correlations, presumably due to the different cation structures. The high-pressure viscosities were used to predict the pressure dependence of the glass-transition temperature for [P222,5][Tf2N] and [P222,8][Tf2N]. Density scaling was applied to the high-pressure viscosities and conductivities of [P222,8][Tf2N] for comparison with [P222,5][Tf2N]. The scaling parameters are consistent with the theoretical treatment of Knudsen et al. for ionic liquids.

Hydrogen Diffusion in Clay Slit: Implications for the Geological Storage

We determined the self-diffusion coefficients of hydrogen in clay (montmorillonite) nanopores using molecular dynamics under subsurface conditions. We explored the effects of temperature, pressure, pore size, moisture content, and salinity. Our results show that the self-diffusion coefficient of hydrogen is on the order of magnitude of 10–8 m2/s. The diffusivity of confined hydrogen increases moderately with temperature and slit aperture but declines with pressure. The estimated density profile suggests that only one dense layer of hydrogen molecules is adsorbed near the slit surface. The distinct diffusion coefficients in the parallel and perpendicular directions to the basal surfaces confirm the confinement effect of the substrates. As the volume ratio of hydrogen increases, the existing pattern of hydrogen changes from a droplet to a layer sandwiched by the aqueous solution. The water bridge will act as a piston for the hydrogen droplet and impede hydrogen diffusion. However, when the hydrogen and brine form a stratified structure, the self-diffusion coefficient of hydrogen sandwiched by two brine films is similar to that of confined pure gas at the same pressure and temperature conditions. If the brine salinity reaches some extent, part of brine and hydrogen molecules will mix as a new phase, which slightly inhibits the hydrogen diffusion. This work provides a better insight into hydrogen diffusion through the clay nanopores, which is critical for reliably assessing the risk of hydrogen leakage through caprocks.

Investigation of Molecular Diffusion at Block Copolymer Thin Films Using Maximum Entropy Method-Based Fluorescence Correlation Spectroscopy and Single Molecule Tracking.

Fluorescence correlation spectroscopy (FCS) has been widely used to investigate molecular diffusion behavior in various samples. The use of the maximum entropy method (MEM) for FCS data analysis provides a unique means to determine multiple distinct diffusion coefficients without a priori assumption of their number. Comparison of the MEM-based FCS method (MEM-FCS) with another method will reveal its utility and advantage as an analytical tool to investigate diffusion dynamics. Herein, we measured diffusion of fluorescent probes doped into nanostructured thin films using MEM-FCS, and validated the results with single molecule tracking (SMT) data. The efficacy of the MEM code employed was first demonstrated by analyzing simulated FCS data for systems incorporating one and two diffusion modes with broadly distributed diffusion coefficients. The MEM analysis accurately afforded the number of distinct diffusion modes and their mean diffusion coefficients. These results contrasted with those obtained by fitting the simulated data to conventional two-component and anomalous diffusion models, which yielded inaccurate estimates of the diffusion coefficients. Subsequently, the MEM analysis was applied to FCS data acquired from hydrophilic dye molecules incorporated into microphase-separated polystyrene-block-poly(ethylene oxide) (PS-b-PEO) thin films characterized under a water-saturated N2atmosphere. The MEM analysis revealed distinct fast and slow diffusion components attributable to molecules diffusing on the film surface and inside the film, respectively. SMT studies of the same materials yielded trajectories for mobile molecules that appear to follow the curved PEO microdomains. Diffusion coefficients obtained from the SMT data were consistent with those obtained for the slow diffusion component detected by MEM-FCS. These results highlight the utility of MEM-FCS and SMT for gaining complementary information on molecular diffusion processes in heterogeneous material systems.

Output Feedback Control of Coupled Linear Parabolic ODE–PDE–ODE Systems

This article deals with the backstepping design of observer-based compensators for parabolic ODE–PDE–ODE systems. The latter consist of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol {n}$</tex-math></inline-formula> coupled parabolic PDEs with distinct diffusion coefficients and spatially-varying coefficients, that are bidirectionally coupled to ODEs at both boundaries. The actuation and sensing appears through these ODEs resulting in a challenging control problem. For this setup a systematic backstepping approach is proposed, in order to determine a state feedback controller and an observer. In particular, the state feedback loop and the observer error dynamics are mapped into stable ODE–PDE–ODE cascades by making use of a sequence of transformations. With this, the design can be traced back to the solution of kernel equations already found in the literature as well as initial and boundary value problems, that can be solved numerically. Exponential stability of the closed-loop system is verified, wherein the decay rate can be directly specified in the design. The results of the article are illustrated by the output feedback control of an unstable ODE–PDE–ODE system with two coupled parabolic PDEs.

Mathematical Model for Estimating Parameters of Swelling Drug Delivery Devices in a Two-Phase Release.

Various swelling drug delivery devices are promising materials for control drug delivery because of their ability to swell and release entrapped therapeutics, in response to physiological stimuli. Previously, many mathematical models have been developed to predict the mechanism of drug release from a swelling device. However, some of these models do not consider the changes in diffusion behaviour as the device swells. Therefore, we used a two-phase approach to simplify the mathematical model considering the effect of swelling on the diffusion coefficient. We began by defining a moving boundary problem to consider the swelling process. Landau transformation was used for mitigating the moving boundary problem. The transformed problem was analytically solved using the separation of variables method. Further, the analytical solution was extended to include the drug release in two phases where each phase has distinct diffusion coefficient and continuity condition was applied. The newly developed model was validated by the experimental data of bacterial cellulose hydrogels using the LSQCURVEFIT function in MATLAB. The numerical test showed that the new model exhibited notable improvement in curve fitting, and it was observed that the initial effective diffusion coefficient of the swelling device was lower than the later effective diffusion coefficient.

Multivariable Extremum Seeking for PDE Dynamic Systems

In this article, we address the design and analysis of multivariable Newton-based extremum seeking (ES) for static locally quadratic maps subject to actuation dynamics governed by diffusion partial differential equations (PDEs). Multi-input systems with distinct diffusion coefficients in each individual input channel are dealt with. The phase compensation of the dither signals is handled as a trajectory generation problem and the inclusion of a multivariable diffusion feedback controller with a perturbation-based (averaging-based) estimate of the Hessian's inverse allow to obtain local exponential convergence results to a small neighborhood of the optimal point. The stability analysis is carried out using backstepping transformation and averaging in infinite dimensions, capturing the infinite-dimensional state due the diffusion PDEs. In addition, the generalization of the results for different classes of parabolic (reaction-advection-diffusion) PDEs, wave equations and/or first-order hyperbolic (transport-dominated) PDEs is also discussed. As for ordinary differential equations case, the proposed Newton approach removes the dependence of the algorithm's convergence rate on the unknown Hessian of the nonlinear map to be optimized, being user-assignable unlike the gradient algorithm. A numerical example illustrates the performance of our multivariable ES for compensating PDE-based systems.

Backstepping Control of Coupled Linear Parabolic PDEs With Space and Time Dependent Coefficients

In this article, the backstepping design of stabilizing state feedback controllers for coupled linear parabolic PDEs with spatially varying distinct diffusion coefficients as well as space and time dependent reaction is presented. The selected target system is a cascade of exponentially stable, time-invariant systems, with time dependent couplings, that is uniformly exponentially stable with a prescribed rate of convergence. To determine the state feedback controller, the kernel equations are derived, which results in a set of coupled PDEs for a time dependent and spatially varying kernel. For this, the method of successive approximations is extended from the time-invariant case to the present problem. The applicability of the method is demonstrated by the stabilization of two coupled unstable parabolic PDEs with space and time dependent coefficients.

Output regulation for coupled linear parabolic PIDEs

This paper presents a backstepping-based solution of the output regulation problem for coupled parabolic partial integro-differential equations (PIDEs) with spatially-varying coefficients and distinct diffusion coefficients. The considered setup assumes in-domain as well as boundary disturbances, while the output to be controlled can be defined distributed in-domain, pointwise in-domain or at the boundaries and need not be measured. By assuming a finite-dimensional signal model, which may also be non-diagonalizable, a systematic solution of the output regulation problem is presented by making use of observer-based feedforward control. Existence conditions for the corresponding regulator are formulated in terms of the plant transfer behaviour. For the resulting closed-loop system, exponential stability with a prescribed decay rate is verified. The regulator design for two unstable coupled parabolic PIDEs demonstrates the results of the paper.

Backstepping Control of Coupled Linear Parabolic PIDEs With Spatially Varying Coefficients

This paper considers the backstepping design of state feedback controllers for coupled linear parabolic partial integro-differential equations (PIDEs) of Volterra-type with distinct diffusion coefficients, spatially varying parameters and mixed boundary conditions. The corresponding target system is a cascade of parabolic PDEs with local couplings allowing a direct specification of the closed-loop stability margin. The determination of the state feedback controller leads to kernel equations, which are a system of coupled linear second-order hyperbolic PIDEs with spatially varying coefficients and rather unusual boundary conditions. By extending the method of successive approximations for the scalar case to the considered system class, the well-posedness of these kernel equations is verified by providing a constructive solution procedure. This results in a systematic method for the backstepping control of coupled parabolic PIDEs as well as PDEs. The applicability of the new backstepping design method is confirmed by the stabilization of two coupled parabolic PIDEs with Dirichlet/Robin unactuated boundaries and a coupled Neumann actuation.

Nondestructive investigation on the nanocomposite ordering upon holography using Mueller matrix ellipsometry

Ordered materials, which hold organized structures of heterogeneous matter and thus always present superior performance than their non-ordered counterparts, have been constantly pursued. Nevertheless, the direct, precise and nondestructive observation of the ordering process, which is especially critical for evaluating the quality of consecutive manufacturing, remains a formidable challenge. Herein, we introduce Mueller matrix ellipsometry (MME) as a nondestructive method to quantitatively investigate the nanocomposite ordering process upon holography. This nondestructive investigation directly offers the exact width of, refractive index and nanoparticle fraction in each bright (constructive) and dark (destructive) interference area, which is impossible to be implemented using other existing techniques. Interestingly, the width of dark regions in the formed holographic gratings is observed to decrease while the width of bright regions increases with an augmentation of holographic recording time, distinct from previous width-equal assumption. Meanwhile, an apparent diffusion coefficient of 2 × 10−15 m2 s−1 for nanoparticles is determined on the basis of time dependent grating parameter variation, which is 3 orders of magnitude lower than the initial value theoretically predicted by the Stokes-Einstein diffusion equation. The distinct diffusion coefficient is attributed to the rapid increase of viscosity driven by polymerization during holography. No depolarization is observed in these holographic polymer nanocomposites, indicating uniform dispersion of nanoparticles in the polymer matrices. The proposed protocol herein is envisioned to pave the way for precisely and nondestructively understanding the formation of ordered structure in electronics, photonics, photovoltaics, biomaterials and other disciplines.

Temperature and Density Dependence of the Transport Properties of the Ionic Liquid Triethylpentylphosphonium Bis(trifluoromethanesulfonyl)amide, [P222,5][Tf2N

Viscosities (η), conductivities (κ, Λ), and ion self-diffusion coefficients (DSi) for triethylpentylphosphonium bis(trifluoromethanesulfonyl)amide, ([P222,5][Tf2N]), have been measured as a function of temperature and pressure in the ranges ([273–363] K, 243 MPa max), ([273–353] K, 251 MPa max), and (298–363 K, 225 MPa max), respectively. pVT data are also reported from (298 to 353) K to 50 MPa. The ratio of the ion self-diffusion coefficients is constant, independent of temperature and pressure. The results are discussed using velocity correlation, distinct diffusion, and resistance coefficients, and the Stokes–Einstein–Sutherland (SES: relating DSi and η) and Nernst–Einstein equations (NE: relating Λ and DSi). As is usual for ionic liquids the SES and NE plots for high-pressure isotherms and the atmospheric pressure isobar overlap quantitatively, that is the self-diffusion and distinct diffusion coefficients, and the molar conductivity, are functions of the viscosity. This can be used for the interpolat...

Demixing of active particles in the presence of external fields.

Self-propelled active particles are inherently out of equilibrium as they collect energy from their surroundings and transform it into directed motion. A recent theoretical study suggests that binary mixtures of active particles with distinct effective diffusion coefficients exhibit dynamical demixing when their diffusion coefficients differ by more than one order of magnitude. Here, we show that this difference may be reduced drastically in the presence of external fields even when the response to the field is the same for both species. We investigate this demixing as a function of the ratio of the diffusion coefficients and discuss the implications of the results for active systems.

Inter-diffusion and its correlation with dynamical cross correlation in liquid Ce80Ni20

We reported the inter-diffusion coefficients in liquid Ce $$_{80}$$ Ni $$_{20}$$ measured by the sliding cell technique. Combined with the self-diffusion data of Ni measured by quasi-elastic neutron scattering in the literature, it was found that the relationship between inter-diffusion and self-diffusion in liquid Ce $$_{80}$$ Ni $$_{20}$$ was strongly deviated from the standard Darken equation with an abnormally small dynamical cross correlation factor S (the so called Manning factor) in a range of 0.6–0.8, less than unity in standard systems. Through the calculated distinct diffusion coefficient and its deviation from the standard one, it was discovered that the small S value was directly originated from enhanced distinct diffusion between Ce and Ni atoms and reduced distinct diffusion between Ni and Ni atoms. Because the inter-atomic interaction was not considered in the standard liquids, the present small S factor and intrinsic distinct diffusion coefficients were believed to be resulted from the chemical interaction between Ce and Ni in the liquid. The results provide new evidence of the dynamic cross correlation in liquid diffusion, and thus shed light on the understanding of the correlation between dynamics and structure in liquid alloys.

Investigating RNAP Search Dynamics in Live E. Coli Cells using Single Molecule and Statistical Methods

Transcription is an essential component of gene regulation and one of the most highly regulated processes in the cell. Because bacteria lack a nuclear envelope, genes transcribed by RNA polymerase (RNAP), the key enzyme of transcription, are essentially immediately translated into protein, leaving the kinetics of transcription significantly influencing the resulting protein expression timing and levels. However, while the kinetics of transcription is greatly studied in vitro, the in vivo dynamics of RNAP, including the typical search time and transcription time, are difficult to obtain. Our lab takes advantage of the sensitivity and fast time scale of single-molecule tracking to determine the kinetics of RNAP binding and transcription in live E. coli cells. We used a strain of E. coli in which the β’ subunit (rpoC) of RNAP is tagged with a photoactivatable fluorescent protein, PAmCherry, replacing the endogenous chromosomal copy. We discovered there are three states of RNAP with distinct diffusion coefficients. Using genetic manipulations and drug treatments, we were able to assign each diffusion state to a particular function of RNAP, i.e. RNAP that is bound to the DNA, RNAP that is undergoing rapid association and dissociation from the nucleoid, and freely diffusing RNAP. Further, using a Markov chain Monte Carlo algorithm, we were able to form a hidden Markov model of our data to parse out the dynamics of RNAP switching states, which enabled us to parse out a complete dynamical model of RNAP in living cells. We obtainined a wide variety of important parameters, including the non-specific dwell time, the typical promoter search time, and the typical transcription time for an average gene. These parameters will be useful for future gene regulations studies of all kinds, as well as to help understand general protein-nucleic acid interactions and dynamics within the crowded cytoplasm of a live cell.

Revised and Extended Values for Self-Diffusion Coefficients of 1-Alkyl-3-methylimidazolium Tetrafluoroborates and Hexafluorophosphates: Relations between the Transport Properties.

Earlier measurements of the self-diffusion coefficients of 1-alkyl-3-methylimidazolium (or [RMIM], R = alkyl) tetrafluoroborates and hexafluorophosphates have been revised and extended to 90 °C. The main changes are to DS+ and DS- for [HMIM][PF6] ([C6C1Im][PF6]) and DS- for [OMIM][BF4] ([C8C1Im][BF4]). New atmospheric pressure self-diffusion, density, and conductivity data are provided for [HMIM][BF4] ([C6C1Im][BF4]). Velocity cross-correlation, distinct diffusion, and Laity resistance coefficients have been calculated. There is no evidence for ion association. A new relation between the Nernst-Einstein deviation parameter (Δ) and resistance coefficients (rij) is derived; Δ tends toward 0.5 when the like-ion rii are much smaller than the unlike-ion rij, i.e., when the counterion interactions dominate. [OMIM]+ ion salts approach this limit. Stokes-Einstein-Sutherland and Walden plots overlap almost quantitatively for [BF4]-, [PF6]-, and Cl- salts with a common [RMIM]+ cation. That is, in thermodynamic states that have the same viscosity, the salt molar conductivities and hence ionic electrical mobilities of, say, [BMIM][BF4] and [BMIM][PF6] are almost equal, as are the corresponding Brownian or diffusive mobilities, (DSi/RT), for the cation, and also for these three small anions.

Induced polarization and pore radius — A discussion

Permeability estimation from induced polarization (IP) measurements is based on a fundamental premise that the characteristic relaxation time [Formula: see text] is related to the effective hydraulic radius [Formula: see text] controlling fluid flow. The approach requires a reliable estimate of the diffusion coefficient of the ions in the electrical double layer. Others have assumed a value for the diffusion coefficient, or postulated different values for clay versus clay-free rocks. We have examined the link between a widely used single estimate of [Formula: see text] and [Formula: see text] for an extensive database of sandstone samples, in which mercury porosimetry data confirm that [Formula: see text] is reliably determined from a modification of the Hagen-Poiseuille equation assuming that the electrical tortuosity is equal to the hydraulic tortuosity. Our database does not support the existence of one or two distinct representative diffusion coefficients but instead demonstrates strong evidence for six orders of magnitude of variation in an apparent diffusion coefficient that is well-correlated with [Formula: see text] and the specific surface area per unit pore volume [Formula: see text]. Two scenarios can explain our findings: (1) the length scale defined by [Formula: see text] is not equal to [Formula: see text] and is likely much longer due to the control of pore-surface roughness or (2) the range of diffusion coefficients is large and likely determined by the relative proportions of the different minerals (e.g., silica and clays) making up the rock. In either case, the estimation of [Formula: see text] (and hence permeability) is inherently uncertain from a single characteristic IP relaxation time as considered in this study.