Von Smoluchowski Research Articles

A chemical kinetics theory for interpreting the non-monotonic temperature dependence of enzymatic reactions

Abstract. One notable observation of enzymatic chemical reactions is that, for a given abundance of enzymes and substrates, temperature increases cause reaction rates to first increase consistent with the Arrhenius relationship, then plateau, and finally fall off quickly to zero at high temperatures. While many mathematical functions have been used to describe this pattern, we here propose a chemical kinetics theory which successfully replicates this observation and provides insights into the processes responsible for these dynamics. The chemical kinetics theory combines the law of mass action, von Smoluchowski's diffusion-limited chemical reaction theory, and Eyring's transition state theory. This new theory reveals that the thermally reversible enzyme denaturation ensured by the ceaseless thermal motion of molecules and ions in an enzyme solution explains the plateau and subsequent decrease in chemical reaction rates with increasing temperature. The temperature-dependent affinity parameter (K) that relates enzymes and substrates through their binding also affects the shape of the emergent temperature response. We demonstrate that with an increase in substrate availability, K shifts the optimal temperature, where reaction rates plateau, towards higher values. Further, we show that the chemical kinetics theory accurately represents 12 sets of published enzyme assay data and includes the popular mechanistic model by Ratkowsky et al. (2005) as a special case. Given its good performance and solid theoretical underpinning, we believe this new theory will facilitate the construction of more mechanistic-based environmental biogeochemical models.

Diffusion-Controlled Reactions: An Overview.

We review the milestones in the century-long development of the theory of diffusion-controlled reactions. Starting from the seminal work by von Smoluchowski, who recognized the importance of diffusion in chemical reactions, we discuss perfect and imperfect surface reactions, their microscopic origins, and the underlying mathematical framework. Single-molecule reaction schemes, anomalous bulk diffusions, reversible binding/unbinding kinetics, and many other extensions are presented. An alternative encounter-based approach to diffusion-controlled reactions is introduced, with emphasis on its advantages and potential applications. Some open problems and future perspectives are outlined.

Purification of aquaculture effluent using Picralima nitida seeds

Aquaculture effluent treatment is essential to eliminate the undesirable characteristics of water to ensure cleaner production and environmental sustainability. In an effort to develop green coagulant without compromising cost, this research investigated the feasibility of aquaculture effluent (AQEF) pollutant removal using Picralima nitida seeds extract (PNSC) and its bio-coagulation/adsorption kinetic characteristics with the substrate in water. The coagulative decrease was observed in terms of TD (turbidity), TSS (total suspended solids), COD (chemical oxygen demand), BOD (biochemical oxygen demand), and COLR (color) from AQEF. The active coagulant was extracted from the seeds and analyzed for its spectral and morphological characteristics through FTIR and SEM. The influence of PNSC dosage (0.10–0.50 g L−1), pH (2–10), settling time (0–60 min), and temperature (303–323 K) on the removal of contaminants were surveyed. The process kinetics of coagulation–flocculation were also explored. Maximal TD reduction of 90.35%, COD (82.11%), BOD (82.38%); TSS (88.84%), and COLR (65.77%) at 0.2 g PNSC L−1, pH 4, and 303 K was achieved. Analysis of variance (ANOVA) tests proved that pH, temperature, and settling time had a significant effect on pollutant removal. Results fitted Von Smoluchowski’s perikinetics theory at the optimum conditions, which gave R2 > 0.900. At perikinetics circumstances, the Kb (reaction rate) and {t}_{ffrac{1}{2}} (half-life) correspond to 0.0635 Lg−1 min−1 and 1.9 min. More so, sorption results fitted the Lagergren over the Ho model. Additionally, the net cost of using PNSC to handle 1 L of AQEF (including electricity, material, and labor costs) was evaluated to be €4.81. Overall, the PNSC appears reliable and useful in pretreating AQEF for improved biodegradability and superior effluent quality.

Marian von Smoluchowski's Oeuvre on Diffusion and its Impact from Natural Sciences to Social Sciences

Marian von Smoluchowski's Oeuvre on Diffusion and its Impact from Natural Sciences to Social Sciences

Coagulation-Flocculation of Aquaculture Wastewater Using Green Coagulant from Garcinia kola Seeds: Parametric Studies, Kinetic Modelling and Cost Analysis

To achieve sustainability, it is necessary to use proper treatment methods to reduce the pollutant loads of receiving waters. This study investigates the coagulative reduction of turbidity, COD, BOD and colour from aquaculture wastewater (AW) using a novel Garcinia kola seeds coagulant (GKC). This coagulant was obtained from extraction of Garcinia kola seeds and analysed for its spectral and morphological characteristics through FTIR and SEM. The kinetics of coagulation-flocculation were also investigated in terms of total dissolved and suspended solids (TDSP). The seeds had 11.27% protein and 68.33% carbohydrate, showing usability in adsorption/charges neutralisation as a coagulant to reduce particles. Maximal turbidity reduction = 81.93%, COD = 75.03%, BOD = 72.84% and colour = 56.69% at 0.3 g GKC/L, pH 2, 60 min and 303 K were achieved. Von Smoluchowski’s second-order peri-kinetics theory was used to fit the results, giving R2 > 0.9. At a coagulation order (α) of 2, the reaction rate (KC) and half-life (τS1/2) were 0.0003 L/g·min and 25.3 min at the optimal conditions. The sorption data better fit the Lagergren compared to the Ho adsorption model. Furthermore, the net cost of using GKC to handle 1 L of AW (including electricity and material costs) was calculated to be 1.57 EUR, and the costs of 0.3 g/L GKC preparation and energy were 0.27 and 1.30 EUR, respectively. In summary, these seeds can be used to pre-treat AW.

Numerical investigation of rarefied vortex loop formation due to shock wave diffraction with the use of rorticity

When compressed gas is ejected from a nozzle into a low-pressure environment, the shock wave diffracts around the nozzle lip and a vortex loop will form. The phenomenon has been widely investigated in the continuum flow regime, but how the shock diffraction and vortex behave under rarefied flow conditions has not received as much attention. It is necessary to understand this transient flow in rarefied environments to improve thrust vector control and avoid potential contamination and erosion of spacecraft surfaces. This work provides numerical results of the vortex loop formation caused by shock wave diffraction around a 90° corner using the direct simulation Monte Carlo method and the compressible Navier–Stokes equations with the appropriate Maxwell velocity slip and the von Smoluchowski temperature jump boundary conditions. The Mach number and rarefaction effects on the formation and evolution of the vortex loop are discussed. A study of the transient structures of vortex loops has been performed using the rorticity concept. A relationship of mutual transformation between the rorticity and shear vectors has been discovered, demonstrating that the application of this concept is useful to understand vortex flow phenomena.

Electrocoagulation-flocculation of aquaculture effluent using hybrid iron and aluminium electrodes: A comparative study

This work investigated the treatment of fish pond wastewater (FPW) using electrocoagulation – flocculation (ECF) through aluminium (Al) and iron (Fe) electro-coagulants. The feasibility of using this method has been determined in terms of turbidity (TUR), biochemical demand oxygen (BOD), chemical demand oxygen (COD), and reduction of colour (COL). The study analysed the role of pH, current dosage, electrolysis time, settling time, temperature, and concentration of NaCl on the ECF process. Also, the influence of using the hybrid of Al and Fe in different configurations (Al-Al, Fe-Fe, Fe-Al, and Al-Fe) was taken into account for the ECF process. The perikinetics kinetics of the process flocculation was studied. The coefficient of regression (R2) was used to confirm the theoretical applicability of perikinetics flocculation. The forecast of the time evolution of aggregating particles or species was included. The optimum pH was 2. Al-Al electrode had a higher TUR removal (87.87%). Fe-Fe electrodes were the best for reducing COL (73.56%). Al-Al was proven to be the best electrode configuration for BOD removal (76.90%) and COD removal (86.4 %). The ECF kinetic data partially followed Von Smoluchowski's equation for the Brownian perikinetics flocculation theory. The mechanisms that accounted for the distribution of particles were sweep flocculation and charge neutralisation. The ECF was very effective for reducing FPW pollutants.

Development of a comprehensive understanding of aggregation-settling movement of CeO2 nanoparticles in natural waters

Parameters such as the settling rate, aggregation rate, and collision frequency in predictive models used to describe the fate of nanoparticles (NPs) are very important for the risk assessment of NPs in the environment. In this study, CeO2 NPs were chosen as the model particles to investigate such parameters through aggregation-settling experiments under environmentally relevant conditions. The results indicate that natural colloids (Ncs) have no effect on the settling of NPs in seawaters, whereas they stabilize the NPs at a low initial particle concentration and promote the heteroaggregation of NPs at a high initial particle concentration in lake waters. In all cases, a suspended sediment absorbs the NPs and Ncs as mixed aggregates, resulting in a rapid settling. Furthermore, the calculation results of the model indicate that the shear force increases the collision frequency of the NPs by 4–5 orders of magnitude higher than that in quiescent waters. However, the break-up effect by the shear force is more obvious, namely, the shear force hinders the aggregation of NPs in natural waters, instead of promoting aggregation. Remarkably, a negative value of the dis-heteroaggregation rate based on the combined von Smoluchowski–Stokes equation can reflect the hindering effect on the aggregation process. The results of this study will provide scientific and accurate guidance for the parameter selection in the existing prediction model and contribute to a prediction of the fate and transport of NPs in the environment.

Self-Assembly of Fibrinogen in Aqueous, Thrombin-Free Solutions of Variable Ionic Strengths.

Fibrinogen not only forms fibrin networks if assisted by thrombin but also exhibits self-assembly in dilute aqueous solutions in the absence of thrombin. It could be shown that self-assembly can be triggered in a controlled way by diluting the ionic strength set to a value of 0.14 M NaCl in the starting solutions. The present work unravels the mechanism of this self-assembly process by means of a combination of time-resolved multiangle static and dynamic light scattering and atomic force microscopy. Analysis was carried out as a function of the ionic strength adjusted by the drop in ionic strength and at variable salt compositions at a given final ionic strength. Composition was varied by changing the ratio of NaCl and phosphate buffer. The self-assembly induced by the drop of the ionic strength depends on the final value. The lower the final ionic strength gets, the faster is the self-assembly process. The variation of the salt composition at a given ionic strength has only a marginal effect, which depends on the ionic strength. The self-assembly obeys a step-growth process, where any intermediate cluster can coalesce with any other cluster. Interpretation of the data with a kinetic model based on the approach of von Smoluchowski follows a diffusion-limited cluster aggregation at ionic strength values lower than 30 mM. At an ionic strength of 30 mM, the model has to take into account a size dependence of the rate constant, and at 60 mM a transition is observed to a reaction-limited cluster aggregation.

Statistical thermodynamics of casein aggregation: Effects of salts and water

Salts, when added to milk, profoundly influence casein aggregation. Even though this well-known phenomenon has been widely exploited, there are still many unanswered questions. How do salts affect casein aggregation? Does water contribute significantly to the aggregation change? The key to answering these questions comes from statistical thermodynamics, i.e. the principles of physics that can link macroscopic data to the collective behaviour of molecules. We present two theoretical approaches. A rigorous approach which demands far more measurements than reported hitherto; and an approximate, pragmatic approach. It bases on stoichiometric models (isodesmic model for aggregation equilibria and von Smoluchowski model for kinetics) that can yield information on protein-water and protein-salt interactions from ‘real-life’ experimental measurements on model systems available in a variety of formats. Using experimental data from the literature, casein aggregation, in the absence of κ-casein, has been shown to be modulated by protein-salt interaction, while the contribution from water structure changes has been shown to be negligible.

Effect of Hydrodynamic Interactions on Reaction Rates in Membranes

The Brownian motion of two particles in three dimensions serves as a model for predicting the diffusion-limited reaction rate, as first discussed by von Smoluchowski. Deutch and Felderhof extended the calculation to account for hydrodynamic interactions between the particles and the target, which results in a reduction of the rate coefficient by about half. Many chemical reactions take place in quasi-two-dimensional systems, such as on the membrane or surface of a cell. We perform a Smoluchowski-like calculation in a quasi-two-dimensional geometry, i.e., a membrane surrounded by fluid, and account for hydrodynamic interactions between the particles. We show that rate coefficients are reduced relative to the case of no interactions. The reduction is more pronounced than the three-dimensional case due to the long-range nature of two-dimensional flows.

Stability of co-existing ZnO and TiO2 nanomaterials in natural water: Aggregation and sedimentation mechanisms

The use of diverse engineered nanomaterials (ENMs) potentially leads to the coexistence of multiple ENMs in the natural environment. The fate such as colloidal stability, transport, and transformation of individual ENMs are dedicated to the coexistence of other types of ENMs in the environment. Here, we for the first time investigated the sedimentation behaviors of two most widely used ENMs (i.e. ZnO and TiO2 nanomaterials, nZnO and nTiO2) copresented in the natural water of China. Sedimentation rates (Vs), homo-aggregation (khom, crit) and hetero-aggregation (khet, crit) rate of nZnO and nTiO2 were calculated based on Von Smoluchowski–Stokes equation and the sedimentation mechanisms were systematically analyzed. The results showed that the coexistence of like negative charge nZnO and nTiO2 effectively enhanced the stability of either ENM by competing hetero-aggregation with natural colloids (NCs) and reducing to form homo-aggregates by the hindrance effect on particle collision. In the natural water, homo-aggregation, hetero-aggregation between ENMs and NCs, as well as the hetero-aggregation between nZnO and nTiO2 were the main aggregation and sedimentation mechanisms. The coexistence of nZnO and nTiO2 made Vs of nZnO decreased by 30.7–49.1% and Vs of nTiO2 decreased by a factor of 42.4%. Value of khet.crit between nZnO and NCs was 0.084–0.132 L mg−1 day−1, was 0.038 L mg−1 day−1 between nTiO2 and NCs, and was 0.011–0.014 L mg−1 day−1 between nZnO and nTiO2.

Ecology: Science or philately? An interdisciplinary analysis of sustainability by exploring if it is possible to get more and more information by reducing collateral environmental damages

We herein explore the connections between the current condition of ecology concerning to sustainable development and the statement of Rutherford regarding the importance of physics to understand sustainability and biological conservation. The recent emergence of organic biophysics of ecosystems (OBEC) may constitute a feasible alternative to fill the gap between conventional ecological thinking and physics, especially thermodynamics. However, our comprehension of sustainability and biological conservation is influenced by the interactions between information and entropy, because we tend to exclude parts of the biosphere as well as their relationships among them. We explore the use of a holistic analysis of sustainability and biological conservation using physics, and also establish a parallelism between Maxwell's demons and human beings. Lastly, the ecological meaning of the hypothetical feasibility of Maxwell's demon at the anthroposphere scale is analyzed starting from the objections of von Smoluchowski, Szilard and Bennet.

Markov Chains as Models in Statistical Mechanics

The Bernoulli [Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae 14 (1769) 3–25]/Laplace [Théorie Analytique des Probabilités (1812) V. Courcier] urn model and the Ehrenfest and Ehrenfest [Physikalische Zeitschrift 8 (1907) 311–314] urn model for mixing are instances of simple Markov chain models called random walks. Both can be used to suggest a probabilistic resolution to the coexistence of irreversibility and recurrence in Boltzmann’s H-Theorem. Marian von Smoluchowski [In Sitzungsberichte der Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse (1914) 2381–2405 Hölder] also modelled by a simple Markov chain, with analogous properties, have fluctuations over time in the number of particles contained in a small element of volume in a solution.This paper explores the themes of entropy, recurrence and reversibility within the framework of such Markov chains. A branching process with immigration, in this respect like Smoluchowski’s model, is introduced to accentuate common features of the spectral theory of all models. This is related to their reversibility, a key issue.

Smoluchowski Reaction Kinetics for Reactions of Any Order

In 1917, Marian von Smoluchowski presented a simple mathematical description of diffusion-controlled reactions on the scale of individual molecules. His model postulated that a reaction would occur when two reactants were sufficiently close and, more specifically, presented a succinct relationship between the relative proximity of two reactants at the moment of reaction and the macroscopic reaction rate. Over the past century, the Smoluchowski reaction theory has been applied widely in the physical, chemical, environmental, and, more recently, biological sciences. Despite the widespread utility of the Smoluchowski theory, it only describes the rates of second order reactions and is inadequate for describing higher order reactions for which there is no equivalent method for theoretical investigation. In this paper, we derive a generalized Smoluchowski framework in which we define proximity in the context when more than two reactants are involved. We derive the relationship between the macroscopic reaction rate and the critical proximity at which a reaction occurs for higher order reactions. Using this theoretical framework and through numerical experiments, we explore various peculiar properties of multimolecular diffusion-controlled reactions which, due to there being no other numerical method of this nature, have not been previous reported.

Adsorption and flocculation of bentonite by chitosan with varying degree of deacetylation and molecular weight

Chitosans with different degrees of deacetylation (DD) and molecular weights (Mw) were tested for the flocculation of bentonite suspensions prepared with demineralized water (DW) and tap water (TW), respectively. Flocculation kinetics model of particles collisions combining zeta potential and turbidity measurements was employed to investigate the effects of the DD and Mw of the chitosans on the flocculation properties. The results indicated that the chitosan (Mw 232kDa) dosages required for maximum flocculation are 20mg/L in DW and 5mg/L in TW, respectively, regardless of DD from 54.6% to 95.2% and pH of bentonite suspension. Chitooligomers (Mw 1.5kDa, DD 95.2%) failed to reach the required residual turbidity (i.e., 10 NTU) in DW under all investigated conditions, whereas good results were obtained using 5–20mg/L in TW. The polymer-induced flocculation processed obeyed Von Smoluchowski's bimolecular rate equation. The flocculation performance in TW was very different from that in DW due to the presence of salts in TW. The results were consistent with the destabilization of bentonite by the combined mechanisms of charge neutralization and bridging.

Modelling of strongly coupled particle growth and aggregation

The mathematical modelling of the dynamics of particle suspension is based on the population balance equation (PBE). PBE is an integro-differential equation for the population density that is a function of time t, space coordinates and internal parameters. Usually, the particle is characterized by a unique parameter, e.g. the matter volume v. PBE consists of several terms: for instance, the growth rate and the aggregation rate. So, the growth rate is a function of v and t. In classical modelling, the growth and the aggregation are independently considered, i.e. they are not coupled. However, current applications occur where the growth and the aggregation are coupled, i.e. the change of the particle volume with time is depending on its initial value v0, that in turn is related to an aggregation event. As a consequence, the dynamics of the suspension does not obey the classical Von Smoluchowski equation. This paper revisits this problem by proposing a new modelling by using a bivariate PBE (with two internal variables: v and v0) and by solving the PBE by means of a numerical method and Monte Carlo simulations. This is applied to a physicochemical system with a simple growth law and a constant aggregation kernel.

Mean-field theory for coarsening faceted surfaces

A mean-field theory is developed for the scale-invariant length distributions observed during the coarsening of generic one-dimensional faceted surfaces. This theory closely follows the Lifshitz-Slyozov-Wagner theory of Ostwald ripening in two-phase systems, but the mechanism of coarsening in faceted surfaces requires the addition of convolution terms recalling work on particle coalescence, and induces an unexpected coupling between the convolution and the rate of facet loss. As a generic framework, the theory concisely illustrates how the universal processes of facet removal and neighbor merger are moderated by the system-specific mean-field velocity describing average rates of length change. For a simple, example facet dynamics associated with the directional solidification of a binary alloy, agreement between the predicted scaling state and that observed after direct numerical simulation of 40,000,000 facets is reasonable given the limiting assumption of noncorrelation between neighbors; relaxing this assumption is a clear path forward toward improved quantitative agreement with data in the future.

Temperature Effect on the Aggregation Kinetics of CeO2 Nanoparticles in Monovalent and Divalent Electrolytes

Aggregation of engineered nanoparticles (NPs) plays a crucial role in their environmental transport, fate,bioavailability and biological effects. This study investigated the temperature effect on the aggregation kinetics of CeO2 NPs in KCl and CaCl2 solutions using time-resolved dynamic light scattering. The results show that in KCl and CaCl2, the aggregation rate became faster as the temperature increased. The critical coagulation concentration (CCC) of CeO2 NPs went down from approximately 100 to 10 mM in KCl and from approximately 10 to 2 mM in CaCl2 solutions when the temperature increased from 4 to 37°C. The observations were analyzed in the framework of extended Derjaguin-Landau-Verwey-Overbeek (EDLVO) theory in order to find out the mechanisms underlying the temperature effect. Moreover, a theoretical model developed on the basis of EDLVO theory and von Smoluchowski’s population balance equation was used to predict the aggregation kinetics of CeO2 NPs under different temperature. The model predictions agreed well with experimental data, suggesting that the model could be employed to predict the size change of NPs in solution. Overall, this work provides insights into NP aggregation using experimental and modeling approaches, and allows people to better understand and theoretically predict the environmental behavior and risk of NPs.

Aggregation kinetics of CeO2 nanoparticles in KCl and CaCl2 solutions: measurements and modeling

To characterize the environmental transport and health risks of CeO2 nanoparticles (NPs), it is important to understand their aggregation behavior. This study investigates the aggregation kinetics of CeO2 NPs in KCl and CaCl2 solutions using time-resolved dynamic light scattering (TR-DLS). The initial hydrodynamic radius of CeO2 NPs measured by DLS was approximately 95 nm. Attachment efficiencies were derived both from aggregation data and predictions based on the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory. The deviations of the DLVO predictions were corrected by employing the extended DLVO (EDLVO) theory. The critical coagulation concentration (CCC) of CeO2 NPs at pH = 5.6 is approximately 34 mM for KCl and 9.5 mM for CaCl2. Furthermore, based on the EDLVO theory and the von Smoluchowski’s population balance equation, a model accounting for diffusion-limited aggregation (DLA) kinetics was established. For the reaction-limited aggregation (RLA) kinetics, a model that takes fractal geometry into account was established. The models fitted the experimental data well and proved to be useful for predicting the aggregation kinetics of CeO2 NPs.