A coupled experimental and mathematical modeling investigation was undertaken to explore nanoscale fullerene aggregate (nC60) transport and deposition in water-saturated porous media. Column experiments were conducted with four different size fractions of Ottawa sand at two pore-water velocities. A mathematical model that incorporates nonequilibrium attachment kinetics and a maximum retention capacity was used to simulate experimental nC60 effluent breakthrough curves and deposition profiles. Fitted maximum retention capacities (S(max)), which ranged from 0.44 to 13.99 microg/g, are found to be correlated to normalized mass flux. The developed correlation provides a means to estimate S(max) as a function of flow velocity, nanoparticle size, and mean grain size of the porous medium. Collision efficiency factors, estimated from fitted attachment rate coefficients, are relatively constant (approximately 0.14) over the range of conditions considered. These fitted values, however, are more than 1 order of magnitude larger than the theoretical collision efficiency factor computed from Derjaguin-Landau-Verwey-Overbeek (DLVO) theory (0.009). Data analyses suggest that neither physical straining nor attraction to the secondary minimum is responsible for this discrepancy. Patch-wise surface charge heterogeneity on the sand grains is shown to be the likely contributor to the observed deviations from classical DLVO theory. These findings indicate that modifications to clean-bed filtration theory and consideration of surface heterogeneity are necessary to accurately predict nC60 transport behavior in saturated porous media.

Experimental and mathematical modeling studies were performed to investigate the transport and retention of nanoscale fullerene aggregates (nC60) in water-saturated porous media. Aqueous suspensions of nC60 aggregates (95 nm diameter, 1 to 3 mg/L) were introduced into columns packed with either glass beads or Ottawa sand at a Darcy velocity of 2.8 m/d. In the presence of 1.0 mM CaCl2, nC60 effluent breakthrough curves (BTCs) gradually increased to a maximum value and then declined sharply upon reintroduction of nC60-free solution. Retention of nC60 in glass bead columns ranged from 8 to 49% of the introduced mass, while up to 77% of the mass was retained in Ottawa sand columns. When nC60 suspensions were prepared in deionized water alone, effluent nC60 BTCs coincided with those of a nonreactive tracer (Br-), with minimal nC60 retention. Observed differences in nC60 transport and retention behavior in glass beads and Ottawa sand were consistent with independent batch retention data and theoretical calculations of electrostatic interactions between nC60 and the solid surfaces. Effluent concentration and retention profile data were accurately simulated using a numerical model that accounted for nC60 attachment kinetics and a limiting retention capacity.

Single-metal site catalysts have exhibited highly efficient electrocatalytic properties due to their unique coordination environments and adjustable local structures for reactant adsorption and electron transfer. They have been widely studied for many electrochemical reactions, including oxygen reduction reaction (ORR) and oxygen evolution reaction (OER). However, it remains a significant challenge to realize high-efficiency bifunctional catalysis (ORR/OER) with single-metal-type active sites. Herein, we report atomically dispersed Fe–Co dual metal sites (FeCo–NC) derived from Fe and Co co-doped zeolitic imidazolate frameworks (ZIF-8s), aiming to build up multiple active sites for bifunctional ORR/OER catalysts. The atomically dispersed FeCo–NC catalyst shows excellent bifunctional catalytic activity in alkaline media for the ORR (E 1/2 = 0.877 V) and the OER (E j=10 = 1.579 V). Moreover, its outstanding stability during the ORR and the OER is comparable to noble-metal catalysts (Pt/C and RuO2). The atomic dispersion state, coordination structure, and the charge density difference of the dual metal site FeCo–NC were characterized and determined using advanced physical characterization and density functional theory (DFT) calculations. The FeCo–N6 moieties are likely the main active sites simultaneously for the ORR and the OER with improved performance relative to the traditional single Fe and Co site catalysts. We further incorporated the FeCo–NC catalyst into an air electrode for fabricating rechargeable and flexible Zn–air batteries, generating a superior power density (372 mW cm–2) and long-cycle (over 190 h) stability. This work would provide a method to design and synthesize atomically dispersed multi-metal site catalysts for advanced electrocatalysis.

The potential toxicity of nanoscale particles has received considerable attention, but the fate of engineered nanomaterials in the environment has been studied only under a limited set of conditions. In the present study, batch and column experiments were performed to assess the aggregation and transport of nanoscale fullerene (nC60) particles in water-saturated quartz sands as a function of electrolyte concentration and species. As the electrolyte concentration increased from 1 to 100 mM, the change in nC60 particle diameter was minimal in the presence of NaCl but increased by more than sevenfold in the presence of CaCl2. The latter effect was attributed to the agglomeration of individual nC60 particles, consistent with a net attractive force between particles and suppression of the electrical double layer. At low ionic strength (3.05 mM), nC60 particles were readily transported through 40- to 50-mesh quartz sand, appearing in the column effluent after introducing less than 1.5 pore volumes of nC60 suspension, with approximately 30% and less than 10% of the injected mass retained in the presence of CaCl2 or NaCl, respectively. At higher ionic strength (30.05 mM) and in finer Ottawa sand (100-140 mesh), greater than 95% of the introduced nC60 particles were retained in the column regardless of the electrolyte species. Approximately 50% of the deposited nC60 particles were recovered from 100- to 140-mesh Ottawa sand after sequential introduction of deionized water adjusted the pH to 10 and 12. These findings demonstrate that nC60 transport and retention in water-saturated sand is strongly dependent on electrolyte conditions and that release of deposited nC60 requires substantial changes in surface charge, consistent with retention in a primary energy minimum.

Experimental and mathematical modeling studies were performed to examine the effects of stabilizing agents on the transport and retention of fullerene nanoparticles (nC(60)) in water-saturated quartz sand. Three stabilizing systems were considered: naturally occurring compounds known to stabilize nanoparticles (Suwannee river humic acid (SRHA) and fulvic acid (SRFA)), synthetic additives used to enhance nanoparticle stability (Tween 80, a nonionic surfactant), and residual contaminants resulting from the manufacturing process (tetrahydrofuran (THF)). The results of column experiments demonstrated that the presence of THF, at concentrations up to 44.5 mg/L, did not alter nC(60) transport and retention behavior, whereas addition of SRHA (20 mg C/L), SRFA (20 mg C/L), or Tween 80 (1000 mg/L) to the influent nC(60) suspensions dramatically increased the mobility of nC(60), as demonstrated by coincidental nanoparticle and nonreactive tracer effluent breakthrough curves (BTCs) and minimal nC(60) retention. When columns were preflushed with surfactant, nC(60) transport was significantly enhanced compared to that in the absence of a stabilizing agent. The presence of adsorbed Tween 80 resulted in nC(60) BTCs characterized by a declining plateau and retention profiles that exhibited hyperexponential decay. The observed nC(60) transport and retention behavior was accurately captured by a mathematical model that accounted for coupled surfactant adsorption-desorption dynamics, surfactant-nanoparticle interactions, and particle attachment kinetics.

The effect of particle shape on its transport and retention in porous media was evaluated by stretching carboxylate-modified fluorescent polystyrene spheres into rod shapes with aspect ratios of 2:1 and 4:1. Quartz crystal microbalance with dissipation (QCM-D) experiments were conducted to measure the deposition rates of spherical and rod-shaped nanoparticles to the collector (poly-l-lysine coated silica sensor) surface under favorable conditions. The spherical particles displayed a significantly higher deposition rate compared with that of the rod-shaped particles. Theoretical analysis based on Smoluchowski-Levich approximation indicated that the rod-shaped particles largely counterbalance the attractive energies due to higher hydrodynamic forces and torques experienced during their transport and rotation. Under unfavorable conditions, the retention of nanoparticles in a microfluidic flow cell packed with glass beads was studied with the use of laser scanning cytometry (LSC). Significantly more attachment was observed for rod-shaped particles than spherical particles, and the attachment rate of the rod-shaped particles showed an increasing trend with the increase in injection volume. Rod-shaped particles were found to be less sensitive to the surface charge heterogeneity change than spherical particles. Increased attachment rate of rod-shaped particles was attributed to surface heterogeneity and possibly enhanced hydrophobicity during the stretching process.

Commercial production and use of fullerene (C60) nanomaterials will inevitably lead to their release into the environment, where knowledge of C60 fate and transport is limited. In this study, a series of one-dimensional column experiments was conducted to assess the transport and retention of nanoscale fullerene aggregates (nC60) in water-saturated soils. Under the experimental conditions, complete retention of nC60 was observed in columns (2.5 cm inside diameter x 11 cm length) packed with Appling or Webster soil, which contain 0.75 and 3.33% organic carbon by weight, respectively. When the volume of aqueous nC60 suspension (approximately 4.5 mg L(-1)) applied to Appling soil was increased from 5 to 65 pore volumes, the travel distance increased from 3 to 8 cm, and the retention capacity approached a limiting value of 130 microg g(-1), although nC60 was not detected in the column effluent. The addition of 20 mg C L(-1) Suwannee River humic acid to the influent suspension increased the nC60 transport in Appling soil but did not resul in breakthrough. Attempts to simulate the experimental data using clean-bed filtration theory were not satisfactory, yielding retention profiles that failed to match observed data. Subsequent incorporation of a limiting retention capacity expression into the mathematical model resulted in accurate predictions of the measured nC60 retention profiles and transport behavior. The sizable retention capacities observed in this study suggest that transport of nC60 is limited in relatively fine-textured soils containing appreciable amounts of clay minerals and organic matter, with substantial accumulation of nC60 aggregates near the point of release.

In this paper, we study a constrained utility maximization problem following the convex duality approach. After formulating the primal and dual problems, we construct the necessary and sufficient conditions for both the primal and dual problems in terms of FBSDEs plus additional conditions. Such formulation then allows us to explicitly characterize the primal optimal control as a function of the adjoint process coming from the dual FBSDEs in a dynamic fashion and vice versa. Moreover, we also find that the optimal primal wealth process coincides with the adjoint process of the dual problem and vice versa. Finally we solve three constrained utility maximization problems, which contrasts the simplicity of the duality approach we propose and the technical complexity of solving the primal problems directly.