Diffusional Instability Research Articles

Gradient-induced instability in tumour spheroids unveils the impact of microenvironmental nutrient changes

Tumours often display invasive behaviours that induce fingering, branching and fragmentation processes. The phenomenon, known as diffusional instability, is driven by differential cell proliferation, migration, and death due to the presence of metabolite and catabolite concentration gradients. An understanding of the intricate dynamics of this spatially heterogeneous process plays a key role in the investigation of tumour growth and invasion. In this study, we developed an in vitro tumour invasion assay to investigate cell invasiveness in tumour spheroids under a chemotactic stimulus. Our method, employing tumour spheroids seeded in a 3D collagen gel within a microfluidic chemotaxis chamber, focuses on the role of diffusive gradients. Using Time-Lapse Microscopy, the dynamic evolution of tumour spheroids was monitored in real-time, providing a comprehensive view of the morphological changes and cell migration patterns under different chemotactic conditions. Specifically, we explored the impact of fetal bovine serum (FBS) gradients on the behaviour of CT26 mouse colon carcinoma cells and compared the effects of varying FBS concentrations to two isotropic control conditions. Furthermore, a finite element in silico model was developed to quantify the diffusive flow of nutrients in the chemotaxis chamber and obtain a detailed understanding of tumour dynamics. Our findings reveal that the presence of a chemotactic gradient significantly influences tumour invasiveness, with higher concentrations of nutrients associated with increased cancer growth and cell migration.

Morphological Instability of Lithium Electrodeposition Induced by Elastic Stress-Driven Diffusion

Lithium metal negative electrodes for liquid cells have high theoretical capacity, but suffer from unstable metal interfaces during charging. Here a model for morphology evolution during plating is presented that focuses on the interface instability during plating at low to moderate current density. It is proposed that the instability is caused by stress due to excess Li atoms incorporated in the metal during electrodeposition. The resulting stress gradients drive diffusion along the lithium surface that amplify surface roughness. The instabilty is modeled by extending the Asaro-Tiller-Grinfel’d diffusional instability for stressed surfaces of elastic solids to interfaces between elastic layers, namely that between lithium metal and the solid-electrolyte interphase (SEI). The assumption of elastic deformation is valid for lithium deposits less than about 1 μm thick. Linear stability analysis revealed that the model predicts the correct scaling between instability length scale and stress: for elastic stress of order 1 MPa as found experimentally, the spacing between interface protrusions is a few μm, similar to typical distances between whiskers on deposit surfaces. The model suggests that the instability can be suppressed by use of elastically stiff current collector substrates.

Lewis number effects on lean premixed combustion characteristics of multi-component fuel blends

Variation in natural gas composition, alongside the potential for H2 enrichment, creates the potential for significant changes to premixed flame behaviour. To strengthen fundamental understanding of lean multi-component alternative fuel blends, an outwardly propagating spherical flame was employed to measure the flame speeds and Markstein lengths of C1C4 hydrocarbons, alongside precisely mixed blends of CH4/C2H6, CH4/C3H8 and CH4/H2. Theoretical relationships between Markstein length and Lewis Number are explored alongside effective Lewis number formulations. Under lean conditions, equal volumetric additions of H2 and C3H8 (30% vol.) to CH4 resulted in similar augmentation of burning velocity, however, opposite susceptibility to preferential diffusional instability was noted. At a fixed equivalence ratio of 0.65, limited changes in composition provide a marked change in the premixed flame response with the addition of C2H6 and C3H8 to CH4. For lean CH4/H2 mixtures, a diffusional based Lewis Number formulation yielded a favourable correlation, whilst a heat-release model resulted in better agreement for lean CH4/C3H8 blends. Modelling work suggests that measured enhancement of lean CH4 flames upon H2 or C3H8 is strongly correlated to changes in volumetric heat release rates and production of H radicals. Furthermore, a systematic analysis of the flame speed enhancement effects (thermal, kinetic, diffusive) of H2 and C3H8 addition to methane was undertaken. Augmented flame propagation of CH4/H2 and CH4/C3H8 was demonstrated to be principally an Arrhenius effect, predominantly through reduction of associated activation energy. Finally, plausible short-term variations in composition with hydrogen-enriched multi-component natural gas flames were investigated experimentally and numerically. At the leanest conditions, small variations in CH4:C3H8 content at a fixed H2 fraction resulted in discernible changes in stretch related behaviour, a reflection of the thermo-diffusive behaviour of each fuel's response.

Investigation on the effects of steam/CO2/N2 on the flame suppression and flame stability of the methane-oxygen mixtures at elevated thermodynamic conditions

This work aimed to study the dilution effect of steam, carbon dioxide, and nitrogen on the unstretched laminar burning velocity and burned gas Markstein length of premixed (100-Z) % (CH4 + (2O2)/ϕ) + Z%(H2O/N2/CO2) mixtures at 300–453 K, 1–4 bar, equivalence ratio (ϕ)- 0.6–1.4, and Z was varied from 40 to 60% for steam and kept constant at 50% for CO2/N2. Expanding spherical flames method was used. Nonlinear stretch extrapolation schemes were used to extract the burning velocity and Markstein length. All conditions were simulated using the 1D planar flame model in CHEMKIN with GRIMech3.0 and FFCM-1 reaction mechanisms. Present mixture formulation led to the comparison of the absolute dilution effect of each diluent at an identical mixture and thermodynamic conditions. Addition of 50% diluents significantly suppressed the flame propagation. It was the highest for CO2 (87% as compared to no dilution) due to the simultaneous reduction in flame temperature and its kinetic effect, and it was followed by steam (69%) and N2 (61%), and their suppression effect was primarily due to the reduction in the flame temperature. All the studied mixtures were stable to preferential diffusional instability as their: (a) Markstein length was positive and (b) the effective Lewis number of the mixtures were well higher than the critical Lewis number. Predicted LBV with GRIMech3.0 compared well with the experiments than FFCM-1. As the LBV increased beyond 150 cm/s, predicted LBV of both GRIMech3.0 and FFCM-1 showed significant deviation with the measurements.

A comparative study of the growth dynamics and tribological properties of nanocrystalline diamond films deposited on the (110) single crystal diamond and Si(100) substrates

Nanocrystalline diamond (NCD) films were grown on the High Pressure High Temperature (HPHT) (110) single crystal (SC) diamond substrates by Microwave Plasma Enhanced Chemical Vapor Deposition (MPCVD) in methane/hydrogen/nitrogen plasma. The thickness of the films was varied between 2.2 and 22.5 μm. The cauliflower-like surface morphology was observed by means of Atomic Force Microscopy (AFM) and Scanning Electron Microscopy (SEM). The scaling behavior of NCD films growth was investigated. The relatively high value of the roughness exponent αs = 1.5–1.6 was found indicating anomalous scaling. Therefore, shadowing and diffusional instabilities can affect the film growth. The tribological properties of the NCD films deposited on the SC(110) diamond were compared with the NCD films prepared on the Si(100) substrates. Both types of specimens were tested under similar Hertzian contact pressure. The lower wear volume losses were observed on the NCD/SC(110) specimens. Therefore, the influence of substrate and substrate/film interface properties on the tribological behavior of the NCD films grown on Si(100) can be expected to cause NCD films deflection.

Dynamics of surface evolution in semiconductor thin films grown from a chemical bath.

Dynamics of surface evolution in CdS thin films grown by chemical bath deposition technique has been studied from time sequence of atomic force micrographs. Detailed scaling analysis of surface fluctuation in real and Fourier space yielded characteristic exponents αloc = 0.78 ± 0.07, α = 2.20 ± 0.08, αs = 1.49 ± 0.22, β = 0.86 ± 0.05 and βloc = 0.43 ± 0.10, which are very different from those predicted by the local growth models and are not related to any known universality classes. The observed anomalous scaling pattern, characterized by power law scaling dependence of interface width on deposition time differently at local and global scale, with rapid roughening of the growth front has been discussed to arise as a consequence of a nonlocal effect in the form of diffusional instability.

Highly-stable silver nanobelts joined via diffusion-free attachment

Silver nanobelts are demonstrated here to undergo inter-particle joining at relatively low temperatures of less than 180 °C. For surface-coated networks of nanobelts this joining reduced the network sheet resistance by 95%. The joining mechanism appears to be non-diffusional oriented attachment, caused by the thermal reactivation of the halted oriented attachment mechanism that occurred originally at room temperature during the rapid nanobelt synthesis. This self-assembly mechanism was explored by in situ electrical and calorimetric experiments, and supported by electron microscopy. Unlike pentagonal silver nanowires, silver nanobelts do not rely on diffusional instability to achieve workably low joining temperatures. The oriented attachment displayed by nanobelts represents a new approach to achieving valuable reductions in network resistance, disentangled from the instability and diffusion-driven failure by nanoparticle degradation displayed by competing silver nanoparticles.

Diffusion-controlled growth and degree of disequilibrium of garnet porphyroblasts: is diffusion-controlled growth of porphyroblasts common?

Rate-limiting processes and the degree of disequilibrium during metamorphic mineral growth are key controls on the rate of dehydration and hydration in the Earth’s crust. This paper examines diffusion-controlled growth and the degree of disequilibrium of garnet porphyroblasts in the Tsukuba metamorphic rocks of central Japan. The analyzed porphyroblasts have irregular and branching morphologies with clear diffusional haloes, indicating that they grew in a diffusion-controlled regime. Mathematical analysis shows that the dominant wavelength of the interface of a garnet porphyroblast is dependent on the extent of supersaturation (Δζ), which is an index for the degree of disequilibrium. Using the calculated upper and lower limits of the dominant wavelength, the value of Δζ is estimated to be 0.05 × 10−1–0.16, which corresponds to a Gibbs free energy (ΔG r ) overstep of 0.9–27 kJ per mole of garnet (12 oxygen atoms) and a temperature overstep (ΔT) of 1.7–50 °C. Using the average value of the dominant wavelength, the following results are obtained: Δζ = 0.15 × 10−1, ΔG r = 2.7 kJ per mole of garnet, and ΔT = 5 °C. These values bring into question the importance of diffusion-controlled growth of garnet porphyroblasts, as highly irregular and branching garnet porphyroblasts are rare in most metamorphic belts. After significant overstepping for the nucleation of garnet, the garnet porphyroblasts grow at a high degree of disequilibrium. However, a high degree of disequilibrium under diffusion-controlled growth would be characterized by diffusional instability. The results indicate that garnet porphyroblasts that lack an irregular and branching morphology may grow at a high degree of disequilibrium under interface-controlled growth, provided they are set in a medium where the diffusion and supply of constituent elements are sufficient, such as a sufficient volume of metamorphic fluid.

Flame-Front Instabilities of Outwardly Expanding Isooctane/n-Butanol Blend–Air Flames at Elevated Pressures

Experiments were conducted in a constant volume bomb filled with isooctane/n-butanol blend–air mixtures over a wide range of n-butanol blending ratios, equivalence ratios, and initial pressures. High-speed schlieren photography was used to measure the critical radius for the onset of flame-front instability. Results show that, for a given n-butanol blending ratio, cellular instabilities appear earlier and the critical Peclet number, which represents the onset of instability, is decreased with the increase of the initial pressure and equivalence ratio. Under fuel-rich conditions, the critical Peclet number increases with the increase of the n-butanol blending ratio. Theoretical calculation of the critical Peclet number was also conducted to interpret the effects of n-butanol on the flame-front instabilities of the blends. Results show that, for rich mixture conditions, with the increase of the n-butanol blending ratio, the overall activation energy of the mixtures is decreased and, thus, the thermal–diffusional instabilities are suppressed. While under fuel-lean conditions, the thermal–diffusional instabilities are favored. As a consequence, with an increasing n-butanol blending ratio, the critical Peclet number is expected to be decreased under fuel-lean conditions and increased under fuel-rich conditions.

Interaction between buoyancy and diffusion-driven instabilities of propagating autocatalytic reaction fronts. I. Linear stability analysis

The interaction between buoyancy-driven and diffusion-driven instabilities that can develop along a propagating reaction front is discussed for a system based on an autocatalytic reaction. Twelve different cases are possible depending on whether the front is ascending or descending in the gravity field, whether the reactant is heavier or lighter than the products, and whether the reactant diffuses faster, slower, or at the same rate as the product. A linear stability analysis (LSA) is undertaken, in which dispersion curves (plots of the growth rate sigma against wave number k) are derived for representative cases as well as an asymptotic analysis for small wave numbers. The results from the LSA indicate that, when the initial reactant is denser than the reaction products, upward propagating fronts remain unstable with the diffusion-driven instability enhancing this instability. Buoyantly stable downward propagating fronts become unstable when the system is also diffusionally unstable. When the initial reactant is lighter than the reaction products, any diffusionally unstable upward propagating front is stabilized by small buoyancy effects. A diffusional instability enhances the buoyant instability of a downward propagating front with there being a very strong interaction between these effects in this case.

Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching

We develop a thermodynamically consistent mixture model for avascular solid tumor growth which takes into account the effects of cell-to-cell adhesion, and taxis inducing chemical and molecular species. The mixture model is well-posed and the governing equations are of Cahn-Hilliard type. When there are only two phases, our asymptotic analysis shows that earlier single-phase models may be recovered as limiting cases of a two-phase model. To solve the governing equations, we develop a numerical algorithm based on an adaptive Cartesian block-structured mesh refinement scheme. A centered-difference approximation is used for the space discretization so that the scheme is second order accurate in space. An implicit discretization in time is used which results in nonlinear equations at implicit time levels. We further employ a gradient stable discretization scheme so that the nonlinear equations are solvable for very large time steps. To solve those equations we use a nonlinear multilevel/multigrid method which is of an optimal order O(N) where N is the number of grid points. Spherically symmetric and fully two dimensional nonlinear numerical simulations are performed. We investigate tumor evolution in nutrient-rich and nutrient-poor tissues. A number of important results have been uncovered. For example, we demonstrate that the tumor may suffer from taxis-driven fingering instabilities which are most dramatic when cell proliferation is low, as predicted by linear stability theory. This is also observed in experiments. This work shows that taxis may play a role in tumor invasion and that when nutrient plays the role of a chemoattractant, the diffusional instability is exacerbated by nutrient gradients. Accordingly, we believe this model is capable of describing complex invasive patterns observed in experiments.

Effects of a constant electric field on the diffusional instability of cubic autocatalytic reaction fronts

An electric field applied in the direction of propagation of a chemical reaction-diffusion front can affect the stability of this front with regard to diffusive instabilities. The influence of an applied constant electric field is investigated by a linear stability analysis and by nonlinear simulations of a simple chemical system based on the cubic autocatalytic reaction A-+2B--->3B-. The diffusional stability of the front is seen to depend on the intensity E and sign of the applied field, and D, the ratio diffusion coefficients of the reactant species. Depending on E, the front can become more or less diffusively unstable for a given value of D. Above a critical value of E, which depends on D, electrophoretic separation of the two fronts is observed.

On the Stability of Miniemulsions in the Presence of RAFT Agents

The colloidal stability of miniemulsions in the presence of RAFT or other control agents for controlled free radical polymerization is examined. A derivation, based on Lifshitz-Slyozov-Wagner (LSW) theory, is proposed here to evaluate the effect of a RAFT agent on the diffusional stability of the miniemulsions before the onset of polymerization. Results indicate that, depending on the hydrophobicity of the control agent, its presence may augment or detract from the effectiveness of the costabilizer in preventing diffusional instability due to Ostwald ripening.

Dispersion curves in the diffusional instability of autocatalytic reaction fronts

A (linear) stability analysis of planar reaction fronts to transverse perturbations is considered for systems based on cubic autocatalysis and a model for the chlorite-tetrathionate reaction. Dispersion curves (plots of the growth rate sigma against a transverse wave-number k) are obtained. In both cases it is seen that there is a nonzero value D0 of D (the ratio of the diffusion coefficients of autocatalyst and substrate) at which sigma(max), the maximum value of sigma for a given value of D, achieves its largest value, with sigma(max) being less for other values of D and becoming small as D decreases to zero. The existence of the optimum value D0 for initiating a diffusional instability is confirmed, in the cubic autocatalysis case, by an asymptotic analysis for small wave numbers.

Turing instability in pioneer/climax species interactions

Systems of pioneer and climax species are used to model interactions of species whose reproductive capacity is sensitive to population density in their shared ecosystem. Intraspecies interaction coefficients can be adjusted so that spatially homogeneous solutions are stable to small perturbations. In a reaction-diffusion pioneer/climax model we will determine the critical value of the diffusion rate of the climax species, below which the equilibrium solution is unstable to non-homogeneous perturbations. For diffusion rates smaller than this critical value, an equilibrium solution remains stable to spatially homogeneous perturbations but is unstable to non-homogeneous perturbations. A Turing (diffusional) bifurcation leads to the formation of spatial patterns in species’ densities. Forcing, interpreted as stocking or harvesting of the species, can reverse the bifurcation and establish equilibrium solutions which are stable to small perturbations. The implicit function theorem is used to determine whether stocking or harvesting of one of the species in the model is the appropriate remedy for diffusional instability. The use of stocking or harvesting by a natural resource manager thus influences the long-term dynamics and spatial distribution of species in a pioneer/climax ecosystem.

Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method

We present a multi-scale computer simulator of cancer progression at the tumoral level, from avascular stage growth, through the transition from avascular to vascular growth (neo-vascularization), and into the later stages of growth and invasion of normal tissue. We use continuum scale reaction-diffusion equations for the growth component of the model, and a combined continuum-discrete model for the angiogenesis component. We use the level set method for describing complex topological changes observed during growth such as tumor splitting and reconnection, and capture of healthy tissue inside the tumor. We use an adaptive, unstructured finite element mesh that allows for finely resolving important regions of the computational domain such as the necrotic rim, the tumor interface and around the capillary sprouts. We present full nonlinear, two-dimensional simulations, showing the potential of our virtual cancer simulator. We use microphysical parameters characterizing malignant glioma cells, obtained from recent in vitro experiments from our lab and from clinical data, and provide insight into the mechanisms leading to infiltration of the brain by the cancer cells. The results indicate that diffusional instability of tumor mass growth and the complex interplay with the developing neo-vasculature may be powerful mechanisms for tissue invasion.

Morphological Stability during Electrodeposition: I. Steady States and Stability Analysis

We develop a formalism for predicting morphology evolution during electrodeposition as a function of the deposition parameters, composition of the electrolyte, and the species being deposited. Our model explicitly couples the electrostatic fields and the metal cations and spectator ions of arbitrary concentrations. We first perform a mixed asymptotics analysis to predict the self-consistent, uniform, steady-state electrostatic, and concentration fields. Morphology evolution is analyzed within the framework of perturbation theory, where we linearize around the uniform, steady-state fields. We find that the surface is unstable at long length scales due to a diffusional instability, in agreement with previous results. Furthermore, we find that both increasing the deposition rate and the spectator ion concentration within the electrolyte at fixed deposition rate increases surface roughness, also in agreement with common experience. We provide an approximate analytical formula for the perturbation growth rates as a function of the spectator ion concentration. The formalism developed here provides a rigorous, self-consistent foundation upon which the effects of additives on surface morphology are analyzed in a companion paper. © 2003 The Electrochemical Society. All rights reserved.

Single bubble sonoluminescence: Unstable diffusion and the kinetics of chemical reactions

Sonoluminescence of a single bubble in water driven by ultrasound is accompanied by long-term stability of the radial bubble wall oscillation. However, in a certain parameter range instabilities of the phenomenon are observed. They are characterized by growing of the ambient (and maximum) radius of the bubble and a sudden microbubble split-off with spatial dislocation (recoil). Experimental results using images of shock waves shed into the surrounding water at bubble collapse are shown to visualize the effect. Numerical calculations show that the observed nonlinear dynamical effects can be interpreted by the influence of diffusional, chemical, and spatial instabilities of the bubble.

Kinetically Driven Growth Instability in Stressed Solids

We report a new stress-induced kinetically driven morphological instability for driven systems. The effect of stress on the interfacial mobility couples to stress variations along a perturbed planar growth front. Comparison of theory and experiment for solid phase epitaxy at a corrugated Si(001) interface, with no free parameters, indicates that the new mechanism is required to account for the observed growth of the corrugation amplitude. This mechanism operates in conjunction with known diffusional and elastic strain energy-driven instabilities in determining morphological evolution. {copyright} {ital 1998} {ital The American Physical Society }

Lauroyl Peroxide as a Cosurfactant in Miniemulsion Polymerization

Miniemulsion polymerizations have been carried out using only an oil-soluble initiator (lauroyl peroxide) as the cosurfactant. Diffusional instability was reduced to the point where nucleation in the monomer droplets and polymerization could be carried out before significant diffusional degradation of the droplets took place. The polymerization behavior was that expected in miniemulsion polymerization with more conventional cosurfactants.