Bulk-mediated Surface Diffusion Research Articles

Bulk-mediated surface transport in the presence of bias.

Surface transport, when the particle is allowed to leave the surface, travel in the bulk for some time, and then return to the surface, is referred to as bulk-mediated surface transport. Recently, we proposed a formalism that significantly simplifies analysis of bulk-mediated surface diffusion [A. M. Berezhkovskii, L. Dagdug, and S. M. Bezrukov, J. Chem. Phys. 143, 084103 (2015)]. Here this formalism is extended to bulk-mediated surface transport in the presence of bias, i.e., when the particle has arbitrary drift velocities on the surface and in the bulk. A key advantage of our approach is that the transport problem reduces to that of a two-state problem of the particle transitions between the surface and the bulk. The latter can be solved with relative ease. The formalism is used to find the Laplace transforms of the first two moments of the particle displacement over the surface in time t at arbitrary values of the particle drift velocities and diffusivities on the surface and in the bulk. This allows us to analyze in detail the time dependence of the effective drift velocity of the particle on the surface, which can be highly nontrivial.

Nuclear-magnetic-resonance relaxation due to the translational diffusion of fluid confined to quasi-two-dimensional pores.

Nuclear-magnetic-resonance (NMR) relaxation experimentation is an effective technique for nondestructively probing the dynamics of proton-bearing fluids in porous media. The frequency-dependent relaxation rate T_{1}^{-1} can yield a wealth of information on the fluid dynamics within the pore provided data can be fit to a suitable spin diffusion model. A spin diffusion model yields the dipolar correlation function G(t) describing the relative translational motion of pairs of ^{1}H spins which then can be Fourier transformed to yield T_{1}^{-1}. G(t) for spins confined to a quasi-two-dimensional (Q2D) pore of thickness h is determined using theoretical and Monte Carlo techniques. G(t) shows a transition from three- to two-dimensional motion with the transition time proportional to h^{2}. T_{1}^{-1} is found to be independent of frequency over the range 0.01-100 MHz provided h≳5 nm and increases with decreasing frequency and decreasing h for pores of thickness h<3 nm. T_{1}^{-1} increases linearly with the bulk water diffusion correlation time τ_{b} allowing a simple and direct estimate of the bulk water diffusion coefficient from the high-frequency limit of T_{1}^{-1} dispersion measurements in systems where the influence of paramagnetic impurities is negligible. Monte Carlo simulations of hydrated Q2D pores are executed for a range of surface-to-bulk desorption rates for a thin pore. G(t) is found to decorrelate when spins move from the surface to the bulk, display three-dimensional properties at intermediate times, and finally show a bulk-mediated surface diffusion (Lévy) mechanism at longer times. The results may be used to interpret NMR relaxation rates in hydrated porous systems in which the paramagnetic impurity density is negligible.

A new approach to the problem of bulk-mediated surface diffusion.

This paper is devoted to bulk-mediated surface diffusion of a particle which can diffuse both on a flat surface and in the bulk layer above the surface. It is assumed that the particle is on the surface initially (at t = 0) and at time t, while in between it may escape from the surface and come back any number of times. We propose a new approach to the problem, which reduces its solution to that of a two-state problem of the particle transitions between the surface and the bulk layer, focusing on the cumulative residence times spent by the particle in the two states. These times are random variables, the sum of which is equal to the total observation time t. The advantage of the proposed approach is that it allows for a simple exact analytical solution for the double Laplace transform of the conditional probability density of the cumulative residence time spent on the surface by the particle observed for time t. This solution is used to find the Laplace transform of the particle mean square displacement and to analyze the peculiarities of its time behavior over the entire range of time. We also establish a relation between the double Laplace transform of the conditional probability density and the Fourier-Laplace transform of the particle propagator over the surface. The proposed approach treats the cases of both finite and infinite bulk layer thicknesses (where bulk-mediated surface diffusion is normal and anomalous at asymptotically long times, respectively) on equal footing.

Tuning the Flight Length of Molecules Diffusing on a Hydrophobic Surface.

Transport at solid-liquid interfaces is critical to self-assembly, biosensing, and heterogeneous catalysis, but surface diffusion remains difficult to characterize and rationally manipulate, due to the inherent heterogeneity of adsorption on solid surfaces. Using single-molecule tracking, we characterized the diffusion of a fluorescent long-chain surfactant on a hydrophobic surface, which involved periods of confinement alternating with bulk-mediated "flights". The concentration of methanol in solution was varied to tune the strength of the hydrophobic surface-molecule interaction. The frequency of confinement had a nonmonotonic dependence on methanol concentration that reflected the relative influence of anomalously strong adsorption sites. By carefully accounting for the effect of this surface heterogeneity, we demonstrated that flight lengths increased monotonically as the hydrophobic attraction decreased, in agreement with theoretical predictions for bulk-mediated surface diffusion. The theory provided an accurate description of surface diffusion, despite the system being heterogeneous, and can be leveraged to optimize molecular search and assembly processes.

Anomalous Surface Diffusion of Protons on Lipid Membranes

The cellular energy machinery depends on the presence and properties of protons at or in the vicinity of lipid membranes. To asses the energetics and mobility of a proton near a membrane, we simulated an excess proton near a solvated DMPC bilayer at 323 K, using a recently developed method to include the Grotthuss proton shuttling mechanism in classical molecular dynamics simulations. We obtained a proton surface affinity of −13.0 ± 0.5 kJ mol−1. The proton interacted strongly with both lipid headgroup and linker carbonyl oxygens. Furthermore, the surface diffusion of the proton was anomalous, with a subdiffusive regime over the first few nanoseconds, followed by a superdiffusive regime. The time- and distance dependence of the proton surface diffusion coefficient within these regimes may also resolve discrepancies between previously reported diffusion coefficients. Our simulations show that the proton anomalous surface diffusion originates from restricted diffusion in two different surface-bound states, interrupted by the occasional bulk-mediated long-range surface diffusion. Although only a DMPC membrane was considered in this work, we speculate that the restrictive character of the on-surface diffusion is highly sensitive to the specific membrane conditions, which can alter the relative contributions of the surface and bulk pathways to the overall diffusion process. Finally, we discuss the implications of our findings for the energy machinery.

Enhanced transport through desorption-mediated diffusion

We present a master equation approach to the study of the bulk-mediated surface diffusion mechanism in a three-dimensional confined domain. The proposed scheme allowed us to evaluate analytically a number of magnitudes that were used to characterize the efficiency of the bulk-mediated surface transport mechanism, for instance, the mean escape time from the domain, and the mean number of distinct visited sites on the confined domain boundary.

Bulk-mediated surface diffusion along a cylinder: Propagators and crossovers

We consider the effective motion along a cylinder of a particle that freely diffuses in the bulk and intermittently binds to the cylinder. From an exact approach we derive the different regimes of the effective motion along the cylinder characterized by physical rates for binding/unbinding and the bulk diffusivity. We obtain a transient regime of superdiffusion and, interestingly, a saturation regime characteristic for the cylindrical geometry. This saturation in a finite system is not terminal but eventually turns over to normal diffusion along the cylinder. The first passage behavior of particles to the cylinder surface is derived. Consequences for actual systems are discussed.

Bulk-mediated surface diffusion: return probability in an infinite system

We analyse the dynamics of adsorbed molecules within the bulk-mediated surface diffusionframework. We consider that the particle’s desorption mechanism is characterized by anon-Markovian process, while the particle’s motion in the bulk is governed by Markoviandynamics, and also include the effect of a Markovian absorption probability on the surface.We study this system for the diffusion of particles in a semi-infinite lattice, analysing thereturn probability to the reference absorptive plane as well as the mean return time to sucha surface. Comparisons with numerical simulations show an excellent agreement.

Bulk-mediated surface diffusion: non-Markovian desorption and biased behaviour in aninfinite system

We analyse the dynamics of adsorbed molecules within the bulk-mediated surface diffusionframework. We consider that the particle’s desorption mechanism is characterized by anon-Markovian process, while the particle’s adsorption and its motion in the bulk aregoverned by Markovian dynamics, and include the effect of an external field in the form ofa bias in the normal motion to the surface. We study this system for the diffusion ofparticles in a semi-infinite lattice, analysing the conditional probability to find the systemon the reference absorptive plane as well as the surface dispersion as functions of time.The agreement between numerical and analytical asymptotic results is discussed.

Effect of hydrodynamic flow on low-field spin-lattice relaxation in liquids in the nanoscopic vicinity of solid surfaces: Theory and Monte Carlo simulations of model pore spaces

It is shown that slow hydrodynamic flow with velocities of a few millimeters per second reduces the spin-lattice relaxation rate of fluids confined to pores of a diamagnetic, polar, solid material. The effect is predicted by an analytical theory and Monte Carlo simulations of model pore spaces. Adsorbate molecules diffusing in the vicinity of pore surfaces can perform adsorption, desorption, and readsorption cycles, effectively leading to displacements along the surface (also termed "bulk mediated surface diffusion" or BMSD). Since the surface determines the orientation of the adsorbed molecule relative to the external magnetic field, desorption at one site and readsorption at another site of a nonplanar surface will cause molecular reorientation. This is the basis of the "reorientation mediated by translational displacements" (RMTD) relaxation mechanism. If hydrodynamic flow is superimposed on diffusion, the RMTD process will be accelerated in a sort of rotational analog to translational hydrodynamic (or Taylor-Aris) dispersion. This reveals itself by a prolongation of spin-lattice relaxation times at low frequencies. The flow-relaxation effect takes place in the vicinity of the pore surfaces on the order of nanometers. The conclusions are (i) the BMSD and RMTD relaxation mechanism of fluids in porous materials is corroborated, (ii) hydrodynamic dispersion affects molecular displacements at surfaces, and (iii) interfacial slip in the sense of a molecular hopping, i.e., a desorption-readsorption process takes place.

Pulsed Field Gradient NMR Study of Anomalous Diffusion in a Lecithin-Based Microemulsion

Self-diffusion measurements in microemulsion systems composed of a naturally occurring soybean lecithin mixture, an aqueous phase, either water or a 1% aqueous PDADMAC solution, and isooctane were accomplished by pulsed field gradient (PFG) 1H NMR spectroscopy at oil dilution lines of low and intermediate water/lecithin ratios. The concentration-dependent diffusion data reveal water-in-oil (W/O) reverse micellar aggregates with dimensions on the nanometer scale being slightly smaller at low water content. With increasing micellar volume fractions, both hydrodynamic as well as direct interactions between particles significantly slow aggregate diffusion. The surfactant mean square displacements (msd's) in dilute and concentrated polymer-free systems studied as a function of diffusion time (20-1000 ms) are characterized by a crossover from Gaussian diffusion, due to slow aggregate motion, to anomalously enhanced diffusion, due to fast surface-bulk surfactant exchange at intermediate times revealing weak, barrier-controlled adsorption behavior. Upon addition of the polycation PDADMAC, the diffusion characteristics change to exclusively superdiffusive behavior with surfactant msd scaling with time as t(3/2) over the entire time range studied. This is caused by surfactant molecules performing Levy walks along the surface of reverse micelles mediated by the dilute bulk. The bulk-mediated surface diffusion is a consequence of the diffusion-controlled micelle-bulk exchange dynamics induced by interactions of PDADMAC with surfactant headgroups.

Flow-Enhanced Molecular Reorientations and Interfacial Slip Probed by Field-Cycling NMR Relaxometry in Microscopic Pores

It is shown that hydrodynamic flow has an effect on spin-lattice relaxation in water filled into a porous monolithic silica material. This is a rotational analogue of translational hydrodynamic (or Taylor-Aris) dispersion arising from incoherent Brownian motion in combination with coherent flow. The effect is demonstrated with the aid of field-cycling NMR relaxometry and confirmed by theoretical considerations. The results directly verify bulk mediated surface diffusion and reveal interfacial slip at fluid-solid interfaces.

Bulk mediated surface diffusion: non Markovian desorption with�finite first moment

Here we address a fundamental issue in surface physics: the dynamics of adsorbed molecules. We study this problem when the particle’s desorption is characterized by a non Markovian process, while the particle’s adsorption and its motion in the bulk are governed by a Markovian dynamics. We study the diffusion of particles in a semi-infinite cubic lattice, and focus on the effective diffusion process at the interface z = 1. We calculate analytically the conditional probability to find the particle on the z=1 plane as well as the surface dispersion as functions of time. The comparison of these results with Monte Carlo simulations show an excellent agreement.

Bulk-mediated surface diffusion: non-Markovian desorption dynamics

Here we analyse the dynamics of adsorbed molecules within the bulk-mediated surface diffusion framework, when the particle's desorption mechanism is characterized by a non-Markovian process, while the particle's adsorption as well as its motion in the bulk is governed by Markovian dynamics. We study the diffusion of particles in both semi-infinite and finite cubic lattices, analysing the conditional probability to find the system on the reference absorptive plane as well as the surface dispersion as functions of time. The results are compared with known Markovian cases showing the differences that can be exploited to distinguish between Markovian and non-Markovian desorption mechanisms in experimental situations.

Bulk mediated surface diffusion: the infinite system case

An analytical soluble model based on a Continuous Time Random Walk (CTRW) scheme for the adsorption-desorption processes at interfaces, called bulk-mediated surface diffusion, is presented. The time evolution of the effective probability distribution width on the surface is calculated and analyzed within an anomalous diffusion framework. The asymptotic behavior for large times shows a sub-diffusive regime for the effective surface diffusion but, depending on the observed range of time, other regimes may be obtained. Montecarlo simulations show excellent agreement with analytical results. As an important byproduct of the indicated approach, we present the evaluation of the time for the first visit to the surface.

Bulk mediated surface diffusion: finite bulk case

Within the framework of a Master Equation scheme, we address the dynamics of adsorbed molecules (a fundamental issue in surface physics) and study the diffusion of particles in a finite cubic lattice whose boundaries are at the z = 1 and the z = L planes where L = 2; 3; 4; ..., while the x and y directions are unbounded. As we are interested in the effective diffusion process at the interface z = 1, we calculate analytically the conditional probability for finding the particle on the z = 1 plane as well as the surface dispersion as a function of time and compare these results with Monte Carlo simulations finding an excellent agreement. These results show that: there exists an optimal number of layers that maximizes 〈r 2(t)〉 on the interface; for a small number the layers the long-time effective diffusivity on the interface is normal, crossing over abruptly towards a subdiffusive behavior as the number of layers increases.

Propagator representation of anomalous diffusion: the orientational structure factor formalism in NMR.

The radial Fourier transform for the isotropic space with a fractal dimension is discussed. The moments of diffusive displacements with non-Gaussian propagators arising as solutions of fractional diffusion equations are calculated. The Fourier propagator is applied to NMR correlation and spectral density functions in context with the orientational structure factor formalism. It is shown that the low-frequency molecular fluctuations of liquids in porous media with strong or forced adsorption at surfaces are due to reorientations mediated by translational displacements caused by surface diffusion of the adsorbate molecules. In terms of this formalism, field-cycling NMR experiments provide information on the static and dynamic fractal dimensions related to surface diffusion. The experimental results for liquids in porous silica glass can be explained by a surface fractal dimension df=2.5, where the mean squared displacement scales as <r(2)(t)> proportional, variantt(2/dw) with dw=1 (ballistic transport), if the surface population can exchange with the bulklike phase in the pores, and with dw=2, if the bulklike phase is frozen. The former dynamics is interpreted in terms of bulk-mediated surface diffusion.

Field-cycling NMR relaxometry of water in synthetic saponites: Lévy walks on finite planar surfaces

Aqueous suspensions of synthetic saponite clays have been studied using field-cycling NMR relaxometry. The saponite platelets form planar surfaces on which water molecules are adsorbed. As the strong-adsorption limit applies, “bulk mediated surface diffusion” as originally suggested by Bychuk and O’Shaughnessy is expected to occur. That is, within the so-called retention time, adsorbate molecules tend to perform Lévy walks along the surfaces. A corresponding formalism (the “pancake model”) is derived and compared with the experimental data. The low-frequency spin-lattice relaxation dispersion can perfectly be described on this basis. The platelet diameter can be estimated from the peculiar adsorbate dynamics on the surfaces in accordance with literature data.

Field-cycling nuclear magnetic resonance relaxometry of molecular dynamics at biological interfaces in eye lenses: The Lévy walk mechanism

We have studied the water dynamics in whole rabbit lens and fragments using H1-field-cycling nuclear magnetic resonance relaxometry. We have measured the proton spin–lattice relaxation time T1 as a function of the Larmor frequency. The data can be interpreted well using the reorientation mediated by translational displacement model in combination with the mechanism of bulk-mediated surface diffusion, where individual water molecules perform Lévy walks with a Cauchy diffusion propagator. This gives evidence of anomalous water surface diffusion on proteins in the rabbit lens. We assume that the lens protein surface topology can be modeled by a polyhedral structure of randomly oriented faces with individual correlation lengths s0. For the whole lens we obtained the most frequent s0 value of 2.5 nm whereas for fragments from the rabbit cortex and the nucleus we obtained values of 3 and 0.3 nm, respectively. The correlation length values obtained for the lens can be attributed to the short-range order of the lens proteins necessary for maintaining lens transparency.

Proton and deuteron field-cycling NMR relaxometry of liquids confined in porous glasses

Polar and non-polar liquids in porous glasses have been studied by proton and deuteron field-cycling NMR relaxometry. The mean pore diameters were 4 nm, 30 nm and 208 nm. The frequency dependence of the spin-lattice relaxation time T 1 is strongly influenced by the polarity of the adsorbate. Non-polar liquids show a flat T 1 dispersion compared with polar species. We designate the two cases as “weak” and “strong” adsorption respectively. Molecules of liquids at surfaces are known to adopt a preferential orientation, whereas they retain a high diffusivity. Therefore, reorientations mediated by translational displacements (RMTD) must contribute to the correlation function which decays on a timescale of up to eight orders of magnitude greater than in the bulk. The high diffusivity at the surface is made plausible by a mechanism called bulk-mediated surface diffusion recently proposed by Bychuk and O'Shaughnessy [O.V. Bychuk and B. O'Shaughnessy, J. Chem. Phys., 101 (1994) 772]. These authors found that the displacements effective on the surface can be described as the result of Lévy walks. They therefore obey a Cauchy distribution. This work employs the Cauchy distribution for a numerical derivation of a surface correlation function, g( r c ) , which correctly reproduces the typical correlation lengths of the substrate and thus renders the notion of the liquid molecules performing Lévy walks reasonable. The difference in T 1 dispersion behaviours of polar and non-polar adsorbates disappears when the free liquid is frozen while the approximately two monolayers thick surface film remains liquid. The T 1 dispersions are then equally steep irrespective of the polarity. This indicates that a non-polar liquid confined to a thin, topologically two-dimensional layer on a polar surface undergoes the same relaxation mechanism as a strongly adsorbed polar liquid which is dominated by RMTD processes.