Diffusion Exponent Research Articles

Construction of a Tungsten Trioxide Modified Smart-Polymers as a NIR-Responsive Platform for Photo-Thermal Therapy on Hepatocellular Cancer

Cancer remains a predominant contributor to both mortality and morbidity on a global scale, with over 10 million new cases diagnosed annually. The aim of this work was to prepare and evaluate erlotinib hydrochloride loaded tungsten trioxide modified with smart polymers for their anticancer potential. The morphology, crystallinity, chemical bonding, and thermal behavior of the nanoadsorbent were characterized using field emission scanning electron microscopy, transmission electron microscopy, X-ray diffraction, Fourier-transform infrared spectroscopy, and thermogravimetric analyses. The optimization of the drug on the synthesized adsorbent was investigated utilizing response surface methodology-central composite design. The maximum sorption efficiency of 90.18% was achieved at the initial drug concentration of 32.82 mg L-1, a pH of 8.17, a temperature of 32.93 °C, and a contact time of 16.08 min. The drug sorption process follows the Langmuir isotherm and it displays pseudo-first-order kinetics. The nanocarrier showed a significant release profile for 6 h with a maximum release percentage of 99.81% in simulated cancer fluid at high temperatures. The nanocarrier showed low drug release without near-infrared laser irradiation and a rapid release rate over 15 min of near-infrared laser irradiation. The drug release properties fit the Korsmeyer-Peppas model, with a diffusion exponent indicative of both diffusion and erosion release mechanisms. In-vitro cytotoxicity of the nanocarrier in normal cells indicated that it had no significant cytotoxicity. The cytotoxicity of the nanocarrier in Hep-G2 hepatocellular carcinoma cell lines was evaluated by MTT assay, and the data showed that the nanocarrier has excellent cytotoxic activity (75.45%).

Synthesis and characterization of novel carboxymethyl tamarind kernel gum - Poly (vinyl alcohol)/guar gum-based hydrogel film loaded with ciprofloxacin for biomedical applications

The current study delineates the synthesis of hydrogel films comprised of carboxymethyl tamarind kernel gum (CMTKG), poly (vinyl alcohol) (PVA), and guar gum (GG) using glutaraldehyde (GTA) as cross-linker. The hydrogel films were evaluated in terms of equilibrium swelling ratio (ESR), moisture content, thickness, wetting analysis, thermal, and mechanical analysis. The tensile strength value lies in the range of 95.80 to 149.07 MPa while elongation at break value lies in the range of 1.51 to 5.20 %. The FTIR spectroscopy confirmed the presence of hydrogen bonding between CMTKG, PVA, and GG components of hydrogel film. FE-SEM micrographs indicated the rough surfaces of hydrogel film. TGA-DTA analysis confirmed that the thermal stability of hydrogel film was found to be increased by incorporating the ciprofloxacin (CFX) drug into the hydrogel matrix. CFX was embedded in the best-swelled hydrogel film and in-vitro drug release behavior was studied at alkaline pH 7.4 phosphate buffer solution. It was found that the maximum drug release was to be 73 % after 24 h at pH 7.4. Moreover, the release data was fitted in various kinetic models such as the First-order, Higuchi, and Korsmeyer-Peppas models. The best-fitted Korsmeyer-Peppas model suggested that the release of the drug follows Fickian diffusion and the value of diffusion exponent (n) was determined to be 0.38. The cytocompatibility of the hydrogel film was analyzed by MTT assay while the antibacterial behavior of the hydrogel film against E. coli and S. aureus showed clear zone of inhibition area. Thus, the overall results indicated that CMTKG/PVA/GG hydrogel film have potential to be used in the biomedical applications.

Scaled Brownian motion with random anomalous diffusion exponent

The scaled Brownian motion (SBM) is regarded as one of the paradigmatic random processes, characterized by anomalous diffusion through the diffusion exponent. It is a Gaussian, self-similar process with independent increments and has found applications across various fields, from turbulence and stochastic hydrology to biophysics. In our paper, inspired by recent single particle tracking biological experiments, we introduce a process called scaled Brownian motion with random exponent (SBMRE), which retains the key features of SBM at the level of individual trajectories, but with anomalous diffusion exponents that vary randomly across trajectories.We discuss the main probabilistic properties of SBMRE, including its probability density function (pdf) and the qth absolute moment. Additionally, we present the expected value of the time-averaged mean squared displacement (TAMSD) and the ergodicity breaking parameter. Furthermore, we analyze the pdf of the first hitting time in a semi-infinite domain, the martingale property of SBMRE, and its stochastic exponential. As special cases, we consider two distributions for the anomalous diffusion exponent, namely the mixture of two point distributions and beta distribution, and explore the asymptotics of the corresponding characteristics. Theoretical results for SBMRE are validated through numerical simulations and compared with the analogous characteristics of SBM.

Change-point detection in anomalous-diffusion trajectories utilising machine-learning-based uncertainty estimates

Abstract When recording the movement of individual animals, cells or molecules one will often observe changes in their diffusive behaviour at certain points in time along their trajectory. In order to capture the different diffusive modes assembled in such heterogeneous trajectories it becomes necessary to segment them by determining these change-points. Such a change-point detection can be challenging for conventional statistical methods, especially when the changes are subtle. We here apply Bayesian Deep Learning to obtain point-wise estimates of not only the anomalous diffusion exponent but also the uncertainties in these predictions from a single anomalous diffusion trajectory generated according to four theoretical models of anomalous diffusion. We show that we are able to achieve an accuracy similar to single-mode (without change-points) predictions as well as a well calibrated uncertainty predictions of this accuracy. Additionally, we find that the predicted uncertainties feature interesting behaviour at the change-points leading us to examine the capabilities of these predictions for change-point detection. While the series of predicted uncertainties on their own are not sufficient to improve change-point detection, they do lead to a performance boost when applied in combination with the predicted anomalous diffusion exponents.

Modeling catalyst effectiveness factor with space-fractional derivative using Haar wavelet collocation method

Abstract Mass transfer limitations may considerably affect the rate of a heterogeneous catalytic process. The catalyst effectiveness factor is a quantitative measure of the impact of the diffusion process inside a catalyst particle. The effectiveness factor is derived from the solution of the steady-state reaction-diffusion problem. Herein, we simulate the steady-state reaction-diffusion equation with space-fractional derivative and linear reaction kinetics. The solution to the problem is obtained numerically using the Haar wavelet collocation method. The effect of the anomalous diffusion exponent on the catalyst effectiveness factor and process parameters, e.g. reactor volume and catalyst mass, is demonstrated. We anticipate that the process efficiency will be notably improved by changing the diffusion regime from standard to superdiffusive.

A pH-Responsive Hydrogel for the Oral Delivery of Ursolic Acid: A Pentacyclic Triterpenoid Phytochemical.

In this study, poly(HEMA-PEGxMEM-IA) hydrogels were prepared by radical copolymerization of poly(ethylene glycol) methyl ether methacrylate (PEGxMEM), 2-hydroxyethyl methacrylate (HEMA), and itaconic acid (IA). The reaction was carried out in ethanolic solution using N,N'-methylenebisacrylamide (MBA) as a crosslinking agent and 1-hydroxycyclohexyl phenyl ketone (HCPK) as a photo-initiator. The poly(HEMA-PEGxMEM-IA) hydrogels (HGx) were evaluated as a delivery system for ursolic acid (UA), a phytochemical extracted from the plant Clinopodium revolutum, "flor de arena". The hydrogels were characterized by Fourier-transform infrared spectroscopy (FTIR-ATR), Raman spectroscopy, X-Ray diffraction (XRD), thermogravimetric analysis (TGA), and scanning electron microscopy (SEM). The swelling behavior was studied in buffer solutions from pH 2 to 10, specifically at pH 2.2 (gastric environment) and 7.4 (intestinal environment). It was found that the hydrogels studied showed sensitivity to pH. At pH 2.2, the degree of swelling for HG5 and HG9 hydrogels was 0.45 and 0.93 (g water/g hydrogel), respectively. At pH 7.4, the degree of swelling for HG5 and HG9 hydrogels was 1.97 and 2.64 (g water/g hydrogel), respectively. The SEM images show the variation in pore size as a function of pH, and the UA crystals in the pores of the hydrogels can also be observed. The in vitro UA release data best fit the Korsmeyer-Peppas kinetic model and the diffusion exponent indicates that the release mechanism is governed by Fickian diffusion.

Preformulation of curcumin and eugenol nanoemulsion in-situ mucoadhesive gel

In situ mucoadhesive gel provides the drug release at a controlled rate for prolong period of time directly to the target site which reduces side effects, thus improving patient compliance. The main advantages of the implants of formation in situ are as follows: they can easily be injected into periodontal pockets, harden to form a solid implant with customized geometry, the time-controlled release of drugs and no need to remove the empty remnants. For the formulation of Curcumin and Eugenol Nanoemulsion in-situ Mucoadhesive gel poloxamer-407 was used as thermo reversible agent and surfactant for nanoemulsion while carbopol-934 was used as mucoadhesive and pH sensitive agent. Curcumin and Eugenol Nanoemulsion in-situ Mucoadhesive gel were prepared and evaluated for zeta potential, globule size, size distribution, viscosity, syringeability, gelation temperature, mucoadhesive strength, in-vitro release study. Batch F3 showed good in vitro drug release and also shown good result for all parameters when compared with all other formulations. Hence, Batch F3 was considered as the optimized formulations. Batch F3 was subjected for four different models viz. Zero order, First order, Higuchi matrix and Peppas model equations and the formulations best fit in to the Peppas model by giving the value of diffusion exponent (n) 1.596 for curcumin and 1.592 for Eugenol respectively, that indicate the formulation had release the drug by case II transport.

The Influence of the Structural Architecture on the Swelling Kinetics and the Network Behavior of Sodium-Alginate-Based Hydrogels Cross-Linked with Ionizing Radiation.

The present work discusses the influence of the structural architecture of sodium alginate-co-acrylic acid-poly(ethylene) oxide hydrogels, crosslinked through electron beam (e-beam) radiation processing. The most important properties of the hydrogels were studied in detail to identify a correlation between the architecture of the hydrogels and their properties. Furthermore, the effect of sodium alginate (NaAlg) concentration, the amounts of the polymer blend, and the size of the samples on hydrogel properties were investigated. The results show that the hydrogels cross-linked (0.5% and 1% NaAlg) with 12.5 kGy exhibit improved physicochemical properties. High gel fraction levels (exceeding 83.5-93.7%) were achieved. Smaller hydrogel diameter (7 mm) contributed to a maximum swelling rate and degree of 20.440%. The hydrogel network was dependent on the hydrogels' diameter and the amount of polymer blend used. The hydrogels best suited the first-order rate constants and exhibited a non-Fickian diffusion character with diffusion exponent values greater than 0.5. This study indicates that the cross-linked hydrogel has good properties, particularly because of its high degree of swelling and extensive stability (more than 180 h) in water. These findings show that hydrogels can be effectively applied to the purification of water contaminated with metals, dyes, or even pharmaceuticals, as well as materials with a gradual release of bioactive chemicals and water retention.

Formulation Design and Characterization of Nilotinib Polymeric Nanoparticles by Nanoprecipitation Technique for the Improved Drug Solubility and Dissolution Rate

Introduction: Nilotinib is a BCS class-IV poorly water-soluble kinase inhibitor drug, that was used for this study to prepare the polymeric nanoparticles by nanoprecipitation technique using Eudragit RL-100 and RS-100 as polymers, Killophore P-188 as a surfactant, and PEG 400 used as a non-volatile, and nontoxic solvent for the improvement of the drug solubility and dissolution rate. Method: The initial process and formulation variables are screened out based on the selected critical quality attributes such as drug release (%), particle size (nm), zeta potential (mV), and polydispersity index. The FT-IR and DSC studies reveal that the drug has no compatibility between the selected drug and the polymers and does not show any additional drug peaks after physical mixing and formulations. The prepared nanoparticles were further characterized to evaluate the particle size (nm), polydispersity index (PDI), zeta potential (mV), entrapment efficiency (%), and in-vitro drug release (%). From the in vitro drug release study, Eudragit RL-100 and Killophore P-188-based formulations showed optimum drug entrapment efficiency with improved drug solubility and dissolution rate in PEG 400 compared to Eudragit RS-100-based formulations. The accelerated stability data for the optimized formulation batch (F6) before and after storage conditions at 40±2 0C and 75±5% RH indicates that the optimized formulation (F6) is more stable for up to 6 months without changes in drug entrapment efficiency and in vitro dissolution rate. Dissolution kinetic data and diffusion exponent values suggested that optimized formulation followed the Higuchi model with a non-Fickian transport mechanism. Result: According to the results, the preparation method proposed in this study is the most suitable for generating polymeric nanoparticles of nilotinib for improved drug solubility and dissolution rate. Conclusion: The nilotinib-based polymeric nano-formulation proved a potential alternative for better drug release with an enhanced solubility rate.

Braess's Paradox Analog in Physical Networks of Optimal Exploration.

In stochastic exploration of geometrically embedded graphs, intuition suggests that providing a shortcut between a pair of nodes reduces the mean first passage time of the entire graph. Counterintuitively, we find a Braess's paradox analog. For regular diffusion, shortcuts can worsen the overall search efficiency of the network, although they bridge topologically distant nodes. We propose an optimization scheme under which each edge adapts its conductivity to minimize the graph's search time. The optimization reveals a relationship between the structure and diffusion exponent and a crossover from dense to sparse graphs as the exponent increases.

Synthesis and characterization of cellulose fibers modified zinc phosphate/hydroxyapatite core-shell as enhanced carrier of cisplatin: Loading, release, and cytotoxicity

The uncontrolled administration of the cisplatin drug (CPTN) resulted in numerous drawbacks. Therefore, effective, affordable, and biocompatible delivery systems were suggested to regulate the loading, release, and therapeutic effect of CPTN. Zinc phosphate/hydroxyapatite hybrid form (ZP/HP) and core-shell nano-rod morphology, as well as its functionalized derivative with cellulose (CF@ZP/HP), were synthesized by the facile dissolution precipitation method followed by mixing with cellulose fibers, respectively. The developed CF@ZP/HP displayed remarkable enhanced CPTN loading properties (418.2 mg/g) as compared to ZP/HP (259.8 mg/g). The CPTN loading behaviors into CF@ZP/HP follow the Langmuir isotherm properties (R2 > 0.98) in addition to the kinetic activities of the pseudo-first-order model (R2 > 0.96). The steric assessment validates the notable increase in the existing loading receptors after the functionalization of ZP/HP with CF from 57.7 mg/g (ZP/HP) to 90.5 mg/g. The functionalization also impacted the capacity of each existing receptor to be able to ensure 5 CPTN molecules. This, in addition to the loading energies (<40 kJ/mol), donates the loading of CPTN by physical multi-molecular processes and in vertical orientation. The CPTN releasing patterns of CF@ZP/HP exhibit slow and controlled properties (95.7 % after 200 h at pH 7.4 and 100 % after 120 h at pH 5.5), but faster than the properties of ZP/HP. The kinetic modeling of the release activities together with the diffusion exponent (>0.45) reflected the release of CPTN according to both erosion and diffusion mechanisms. The loading of CPTN into both ZP/HP and CF@ZP/HP also resulted in a marked enhancement in the anticancer activity of CPTN against human cervical epithelial malignancies (HeLa) (cell viability = 5.6 % (CPTN), 3.2 % (CPTN loaded ZP/HP), and 1.12 % (CPTN loaded CF@ZP/HP)).

Universal superdiffusion of random walks in media with embedded fractal networks of low diffusivity.

Diffusion in composite media with high contrasts between diffusion coefficients in fractal sets of inclusions and in their embedding matrices is modeled by lattice random walks (RWs) with probabilities p<1 of hops from fractal sites and 1 from matrix sites. Superdiffusion is predicted in time intervals that depend on p and with diffusion exponents that depend on the dimensions of matrix (E) and fractal (D_{F}) as ν=1/(2+D_{F}-E). This contrasts with the nonuniversal subdiffusion of RWs confined to fractal media. Simulations with four fractals show the anomaly at several time decades for p≲10^{-3} and the crossover to the asymptotic normal diffusion. These results show that superdiffusion can be observed in isotropic RWs with finite moments of hop length distributions and allow the estimation of the dimension of the inclusion set from the diffusion exponent. However, displacements within single trajectories have normal scaling, which shows transient ergodicity breaking.

Simulations of Stochastic Fluid Dynamics near a Critical Point in the Phase Diagram.

We present simulations of stochastic fluid dynamics in the vicinity of a critical endpoint belonging to the universality class of the Ising model. This study is motivated by the challenge of modeling the dynamics of critical fluctuations near a conjectured critical endpoint in the phase diagram of quantum chromodynamics (QCD). We focus on the interaction of shear modes with a conserved scalar density, which is known as model H. We show that the observed dynamical scaling behavior depends on the correlation length and the shear viscosity of the fluid. As the correlation length is increased or the viscosity is decreased we observe a crossover from the dynamical exponent of critical diffusion, z≃4, to the expected scaling exponent of model H, z≃3. We use our method to investigate the time-dependent correlation function of non-Gaussian moments M^{n}(t) of the order parameter. We find that the relaxation time depends in a nontrivial manner on the power n.

Characterizing localization effects in an ultracold disordered Fermi gas by diffusion analysis

Disorder can fundamentally modify the transport properties of a system. A striking example is Anderson localization, suppressing transport due to destructive interference of propagation paths. In inhomogeneous many-body systems, not all particles are localized for finite-strength disorder, and the system can become partially diffusive. Unraveling the intricate signatures of localization from such observed diffusion is a longstanding problem. Here, we experimentally study a degenerate, spin-polarized Fermi gas in a disorder potential formed by an optical speckle pattern. We record the diffusion through the disordered potential upon release from an external confining potential. We compare different methods to analyze the resulting density distributions, including a new approach to capture particle dynamics by evaluating absorption-image statistics. Using standard observables, such as diffusion exponent and coefficient, localized fraction, or localization length, we find that some show signatures for a transition to localization above a critical disorder strength, while others show a smooth crossover to a modified diffusion regime. In laterally displaced disorder, we spatially resolve different transport regimes simultaneously, which allows us to extract the subdiffusion exponent expected for weak localization. Our work emphasizes that the transition toward localization can be investigated by closely analyzing the system's diffusion, offering ways of revealing localization effects beyond the signature of exponentially decaying density distribution. Published by the American Physical Society 2024

Nanorod Diffusion near the Solid-Liquid Interface with Varied Wall Nonuniformity.

The complex diffusion behaviors of rod-shaped nanoparticles near the solid-liquid interface are closely related to various biological processes and technological applications. Despite recent advancements in understanding the diffusion dynamics of nanoparticles near some specific solid-liquid interfaces, systematical studies to tune the interfacial interaction or fabricating nonuniform wall to see their effects on the nanorod (NR) diffusion are still lacking. This work utilized molecular dynamics simulations to investigate the rotational and translational diffusion dynamics of a single NR near the solid-liquid interface. We constructed a patterned wall featuring adjustable nonuniformity, which was accomplished by modifying the interaction between NR and the wall, noting that the resulting nonuniformity limits both the translational and rotational diffusion of NR, evident from decreases in diffusion coefficients and exponents. By trajectory analysis, we categorized the diffusion modes of NRs near the patterned wall with varied nonuniformities into three types: Fickian diffusion, desorption-mediated flight, and in-plane diffusion. Furthermore, energy analysis based on the adsorption-desorption mechanism has demonstrated that the three diffusion states are driven by interactions between the NR and the wall, which are primarily influenced by rotational diffusion. These results could significantly deepen the understanding of anisotropic nanoparticle interfacial diffusion and would provide new insights into the transport mechanisms of nanoparticles within confined environments.

Continuous data assimilation for the 3D and higher-dimensional Navier–Stokes equations with higher-order fractional diffusion

We study the use of the Azouani-Olson-Titi (AOT) continuous data assimilation algorithm to recover solutions of the 3D Navier–Stokes equations modified to have finer-order fractional diffusion. The fractional diffusion case is of particular interest, as it is known to be globally well-posed for sufficiently large diffusion exponent α. In this work, we prove that the assimilation equations are globally well-posed, and we demonstrate that the solutions produced by the AOT algorithm exhibit exponential convergence in time to the reference solution, given a sufficiently high spatial resolution of observations and a sufficiently large nudging parameter. We note that the results hold in arbitrary spatial dimensions d where d≥2, so long as α≥12+d4. Though the cases d>3 are likely only a mathematical curiosity, we include them as they cause no additional difficulty in the proof.

Quantum diffusion induced by small quantum chaos.

It is demonstrated that quantum systems classically exhibiting strong and homogeneous chaos in a bounded region of the phase space can induce a global quantum diffusion. As an ideal model system, a small quantum chaos with finite Hilbert space dimension N weakly coupled with M additional degrees of freedom which is approximated by linear systems is proposed. By twinning the system the diffusion process in the additional modes can be numerically investigated without taking the unbounded diffusion space into account explicitly. Even though N is not very large, diffusion occurs in the additional modes as the coupling strength increases if M≥3. If N is large enough, a definite quantum transition to diffusion takes place through a critical subdiffusion characterized by an anomalous diffusion exponent.

The viscoelasticity of high concentration monoclonal antibodies using particle tracking microrheology.

The viscoelasticity of monoclonal antibodies (mAbs) is important during their production, formulation, and drug delivery. High concentration mAbs can provide higher efficacy therapeutics (e.g., during immunotherapy) and improved efficiency during their production (economy of scale during processing). Two humanized mAbs were studied (mAb-1 and mAb-2) with differing isoelectric points. Using high speed particle tracking microrheology, we demonstrated that the mAb solutions have significant viscoelasticities above concentrations of 40 mg/ml. Power law viscoelasticity was observed over the range of time scales (-1 s) probed for the high concentration mAb suspensions. The terminal viscosity demonstrated an exponential dependence on mAb concentration (a modified Mooney relationship) as expected for charged stabilized Brownian colloids. Gelation of the mAbs was explored by lowering the pH of the buffer and a power law scaling of the gelation transition was observed, i.e., the exponent of the anomalous diffusion of the probe particles scaled inversely with the gelation time.

Antiepileptic drug-loaded and multifunctional iron oxide@silica@gelatin nanoparticles for acid-triggered drug delivery

The current study developed an innovative design for the production of smart multifunctional core-double shell superparamagnetic nanoparticles (NPs) with a focus on the development of a pH-responsive drug delivery system tailored for the controlled release of Phenytoin, accompanied by real-time monitoring capabilities. In this regard, the ultra-small superparamagnetic iron oxide@silica NPs (IO@Si MNPs) were synthesized and then coated with a layer of gelatin containing Phenytoin as an antiepileptic drug. The precise saturation magnetization value for the resultant NPs was established at 26 emu g-1. The polymeric shell showed a pH-sensitive behavior with the capacity to regulate the release of encapsulated drug under neutral pH conditions, simultaneously, releasing more amount of the drug in a simulated tumorous-epileptic acidic condition. The NPs showed an average size of 41.04 nm, which is in the desired size range facilitating entry through the blood–brain barrier. The values of drug loading and encapsulation efficiency were determined to be 2.01 and 10.05%, respectively. Moreover, kinetic studies revealed a Fickian diffusion process of Phenytoin release, and diffusional exponent values based on the Korsmeyer-Peppas equation were achieved at pH 7.4 and pH 6.3. The synthesized NPs did not show any cytotoxicity. Consequently, this new design offers a faster release of PHT at the site of a tumor in response to a change in pH, which is essential to prevent epileptic attacks.

Concentration-Dependent Diffusion of Monoclonal Antibodies: Underlying Mechanisms of Anomalous Diffusion.

The development, storage, transport, and subcutaneous delivery of highly concentrated monoclonal antibody formulations pose significant challenges due to the high solution viscosity and low diffusion of the antibody molecules in crowded environments. These issues often stem from the self-associating behavior of the antibody molecules, potentially leading to aggregation. In this work, we used a dissipative particle dynamics-based coarse-grained model to investigate the diffusion behavior of IgG1 antibody molecules in aqueous solutions with 15 and 32 mM NaCl and antibody concentrations ranging from 10 to 400 mg/mL. We determined the coarse-grained interaction parameters by matching the calculated structure factor with the computational and experimental data from the literature. Our results indicate Fickian diffusion for antibody concentrations of 10 and 25 mg/mL and anomalous diffusion for concentrations exceeding 50 mg/mL. The anomalous diffusion was observed for ∼0.33 to 0.4 μs, followed by Fickian diffusion for all antibody concentrations. We observed a strong linear correlation between the diffusion behavior of the antibody molecules (diffusion coefficient D and anomalous diffusion exponent α) and the amount of aggregates present in the solution and between the amount of aggregates and the Coulomb interaction energy. The investigation of underlying mechanisms for anomalous diffusion revealed that in crowded environments at high antibody concentrations, the attractive interaction between electrostatically complementary regions of the antibody molecules could further bring the neighboring molecules closer to one another, ultimately resulting in aggregate formation. Further, the Coulomb attraction can continue to draw more molecules together, forming larger aggregates.