Diffusion-controlled Growth Model Research Articles

Theoretical analysis of protein effects on calcium phosphate precipitation in simulated body fluid

The effects of protein on the nucleation and growth rate of calcium phosphate precipitation, including hydroxyapatite (HA) and octacalcium phosphate (OCP) precipitation, in simulated body fluid were analyzed based on the classical nucleation and diffusion-controlled growth theory. To obtain quantitative results for analysis, a model system of bovine serum albumin and lysozyme added to revised simulated body fluid was studied. The parameters, such as the concentration of protein binding to calcium and phosphorous ions and the viscosity of the revised simulated body fluid with protein additions, were experimentally measured to assist quantitative analysis. The analysis focused on the effects of protein on calcium phosphate nucleation including changes in the ion concentration, interfacial energies and pH values of the simulated body fluid system. The protein effects on calcium phosphate growth were also analyzed based on a diffusion-controlled growth model. The analysis results indicate that proteins present in simulated body fluid act to increase the free energy of nucleation and reduce the nucleation rates by reducing the effective level of supersaturation in simulated body fluid. The proteins also affect calcium phosphate nucleation by changing the interfacial energies in simulated body fluid. However, the analysis indicates that the effects of protein on the interfacial energies could be to either promote or retard the nucleation. The protein effects on the nucleation induced by altering the pH value of the simulated body fluid can be neglected. The analysis also indicates that protein addition in revised simulated body fluid does not change the basic thermodynamic nature of HA and OCP formation; that is, HA precipitation is more thermodynamically favourable than OCP while OCP precipitation is more kinetically favourable than HA. Proteins presenting in simulated body fluid always exhibit an inhibitory effect on the growth rate of calcium phosphate precipitates because proteins increase the diffusion resistance of calcium and phosphate ions by increasing the revised simulated body fluid viscosity.

Morphological transition in crystallization of Si from undercooled melt

Using CO2 laser equipped electro-magnetic levitator, we carried out the crystallization of Si at undercoolings from 0 K to 200 K. From the point of the interface morphologies, the relationship between growth velocities and undercoolings was classified into two regions, I and II, respectively. In region I where the undercooling is approximately less than 100 K, needle-like thin plate crystals whose interface consists of faceted plane were observed. In region II, the morphology of growing crystals changed to massive dendrites. Although the interface morphologies look quite different between region I and II, the growth velocities are expressed by two dimensional (2D) nucleation-controlled growth model, and at undercoolings larger than 150 K, the growth velocities asymptotically close to the analysis of the mono-parametric linear kinetics growth model. In this stage, the kinetic coefficient of 0.1 m/sK is equivalent to that derived by the diffusion-controlled growth model. This result means that with increase of undercooling, the rate-determining factor changes from 2D nucleation on the faceted interface to random incorporation of atoms on the rough interface.

Hydrogen Transport and Rationalization of Porosity Formation during Welding of Titanium Alloys

The transport of hydrogen during fusion welding of the titanium alloy Ti-6Al4V is analyzed. A coupled thermodynamic/kinetic treatment is proposed for the mass transport within and around the weld pool. The modeling indicates that hydrogen accumulates in the weld pool as a consequence of the thermodynamic driving forces that arise; a region of hydrogen depletion exists in cooler, surrounding regions in the heat-affected zone and beyond. Coupling with a hydrogen diffusion-controlled bubble growth model is used to simulate bubble growth in the melt and, thus, to make predictions of the hydrogen concentration barrier needed for pore formation. The effects of surface tension of liquid metal and the radius of preexisting microbubble size on the barrier are discussed. The work provides insights into the mechanism of porosity formation in titanium alloys.

Siの過冷凝固におけるTwin-related ⇔ Twin-free遷移

Using CO2 laser equipped electro-magnetic levitator, we carried out the crystallization of Si at undercoolings from 0 K to 200 K. From the point of the interface morphologies, the relationship between growth velocities and undercoolings was classified into two regions, Iand II, respectively. In regionIwhere the undercooling is approximately less than 100 K, thin plate crystals whose interface consists of faceted plane were observed. In region II, the morphology of growing crystals changed to massive dendrites. Although the interface morphologies in region II look quite different from that in regionI, the growth velocities are expressed by two dimensional (2D) nucleation-controlled growth model, and at undercoolings larger than 150 K, the growth velocities asymptotically close to the analysis of the mono-parametric linear kinetics growth model. In this stage, the kinetic coefficient of 0.1 m/sK is equivalent with that derived by the diffusion-controlled growth model. This result means that with increase of undercooling, the rate-determining factor changes from 2D nucleation on the faceted interface to random incorporation of atoms on the rough interface.

Modeling of soft impingement effect during solid-state partitioning phase transformations in binary alloys

The soft impingement effect at the later stage of partitioning phase transformations has been modeled both for the diffusion-controlled growth model and for the mixed-mode model. Instead of the linear and exponential approximations for the concentration gradient in front of the interface used in the past, a general polynomial method of dealing with the diffusion field is proposed. The linear and exponential diffusion field approximations are two specific cases of the polynomial diffusion field approximation. The effect of soft impingement on the overall partitioning phase transformation is only related to the degree of the super-saturation in case of the diffusion-controlled growth model, while it is determined by both the growth mode and the degree of super-saturation in case of the mixed-mode model.

Early stages of cobalt electrodeposition on FTO and n-type Si substrates in sulfate medium

The mechanism of cobalt electrocrystallization on fluorine-doped tin oxide (FTO)-coated conducting glass substrate has been studied and compared with results obtained on n-type Si electrodes. The initial stages of metal deposition from a solution of 0.25 M CoSO 4, 1 M Na 2SO 4 and 0.5 M H 3BO 3 (pH 3.5) were investigated using cyclic voltammetry, and chronoamperometry measurements. Important kinetics parameters were estimated from the analysis of current transients on the basis of the Scharifker–Hills model for electrochemical nucleation and diffusion-controlled growth. From the analysis of the experimental current transients, it was found that the process of cobalt deposition involves a three-dimensional (3D) electrochemical nucleation and diffusion-controlled growth. A strong dependence of the number density of active sites N 0 with applied potential was observed on both FTO and n-type Si electrodes.

Electrocrystallization of Au nanoparticles on glassy carbon from HClO 4 solution containing [AuCl 4] −

The mechanism and kinetics of electrocrystallization of Au nanoparticles on glassy carbon (GC) were investigated in the system GC/1 mM KAuCl 4 + 0.1 M HClO 4. Experimental results show that the gold electrodeposition follows the so-called Volmer–Weber growth mechanism involving formation and growth of 3D Au nanoparticles on an unmodified GC substrate. The analysis of current transients shows that at relatively positive electrode potentials ( E ≥ 0.84 V) the deposition kinetics corresponds to the theoretical model for progressive nucleation and diffusion-controlled 3D growth of Au nanoparticles. The potential dependence of the nucleation rate extracted from the current transients is in agreement with the atomistic theory of nucleation. At sufficiently negative electrode potentials ( E ≤ 0.64 V) the nucleation frequency becomes very high and the nucleation occurs instantaneously. Based on this behaviour is applied a potentiostatic double-pulse routine, which allows controlled electrodeposition of Au nanoparticles with a relatively narrow size distribution.

Crystallization kinetics of CdSe nanocrystals synthesized via the TOP–TOPO–HDA route

An effective trioctylphosphine–trioctylphosphine oxide–hexadecylamine (TOP–TOPO–HDA) route was used to investigate the crystallization kinetics of CdSe nanocrystals. A new diffusion-controlled growth model was founded to simulate the CdSe nanocrystals (NCs) growth process. It was found that the population density of the CdSe NCs in solution remained constant during the early stage of the growth before ripening. About 28% of the Cd was consumed during nucleation. The concentration of Cd decreased after nucleation and the Cd monomer was almost exhausted after 300 s growth. The diffusion coefficient was found to be ∼150 nm 2/s. The apparent critical nucleus was estimated to be 1.64 nm by extrapolation of the modeling curve.

Crystallization Experiments on Amorphous Silicates with Chondritic Composition: Quantitative Formulation of the Crystallization

In order to make clear crystallization process of silicates in circumstellar environments of oxygen-rich young stars, we have performed laboratory experiments on crystallization of a silicate material by use of a synthetic sample with the chondritic composition for the first time. The aim of this work is to analyze the crystallization process quantitatively using the amorphous material with the chondritic composition. The starting amorphous material was synthesized by the sol-gel method. The sample was heated at 660°-1200°C for 0.5-12 hr to investigate the temperature and time dependence of the crystallization. The run products were analyzed using infrared absorption spectroscopy, X-ray diffraction, scanning electron microscopy and transmission electron microscopy. Olivine [(Mg, Fe)2SiO4] was mainly crystallized from the starting amorphous material. We performed infrared spectral fittings of the heated samples using individual spectra of olivine and amorphous silicate, and estimated the degree of crystallization quantitatively. The time-dependent crystallization process could be formulated using the Johnson-Mehl-Avrami equation with the power of about 1.2, which is consistent with theoretical crystallization model of three-dimensional diffusion-controlled growth from a state that a number of nuclei is constant. The constant number of nuclei corresponds to the starting material, which contains crystallites of magnetite (Fe3O4) and ferrihydrite (5Fe2O3 9H2O) as nucleation sites of olivine crystals. From the quantitative analyses, we suggest that crystallization processes in circumstellar regions should depend on properties of the interstellar amorphous silicates such as existence of crystallites and/or FeO content.

Effect of H2SeO3 on the early stages of nucleation and growth during Cu electrocrystallization in acidic medium on glassy carbon electrode

The early stages of Cu electrodeposition onto a GC electrode were investigated in 0.5 M H2SO4 + 0.01 M CuSO4 solution without or with H2SeO3 when a molar concentration ratio [Cu(II)]/[Se(IV)] was 1.104 to 2.102. The H2SeO3 solution in 0.5 M H2SO4 was also used. The electrochemical techniques such as cyclic voltammetry and chronoamperometry and structural investigation using ex situ AFM were applied to study the nucleation and growth of Cu onto a GC electrode. Chronoamperometric results were shown to follow an instantaneous 3D nucleation and diffusion-controlled growth model by Scharifker and Hills. The values of number of Cu nuclei N and average nuclei radius r av were calculated. It was shown that, in the presence of H2SeO3 in amounts of 0.001 to 0.005 mM, N increases and r av decreases. At higher concentrations of the additive, the changes of these parameters with the deposition potential E dep were shown to be somewhat more complex. The dependences of N and r av on the concentration of H2SeO3 in different regions of Cu overpotentials were also revealed.

A bioreaction–diffusion model for growth of marine sponge explants in bioreactors

Marine sponges are sources of high-value bioactives. Engineering aspects of in vitro culture of sponges from cuttings (explants) are poorly understood. This work develops a diffusion-controlled growth model for sponge explants. The model assumes that the explant growth is controlled by diffusive transport of at least some nutrients from the surrounding medium into the explant that generally has a poorly developed aquiferous system for internal irrigation during early stages of growth. Growth is assumed to obey Monod-type kinetics. The model is shown to satisfactorily explain the measured growth behavior of the marine sponge Crambe crambe in two different growth media. In addition, the model is generally consistent with published data for growth of explants of the sponges Disidea avara and Hemimycale columella. The model predicted that nutrient concentration profiles for nutrients, such as dissolved oxygen within the explant, are consistent with data published by independent researchers. In view of the proposed model's ability to explain available data for growth of several species of sponge explants, diffusive transport does play a controlling role in explant growth at least until a fully developed aquiferous system has become established. According to the model and experimental observations, the instantaneous growth rate depends on the size of the explant and all those factors that influence the diffusion of critical nutrients within the explant. Growth follows a hyperbolic profile that is consistent with the Monod kinetics.

Spheroidal particle stability in semisolid processing

A model for diffusion-controlled spherical particle growth is presented and solved numerically, showing how, on cooling at a sufficient rate from a given fraction solid, growth velocity first increases and then decreases rapidly when solute fields of adjacent particles overlap. An approximate analytical solution for the spherical particle growth velocity is then developed and shown to be valid until the solute fields begin to overlap. A particle stability model is next presented, building on the preceding analytic solution. This model permits prediction of the maximum cooling rate at which a semisolid slurry or reheated semisolid billet can be cooled while still retaining the spherical growth morphology. The model shows that particle stability is favored by high particle density, high fraction solid, and low cooling rate. The predictions of the stability model are found to be in good quantitative agreement with experimental data collected for Al-4.5 wt pct Cu alloy. Engineering applications of the results obtained are discussed.

Nucleation and growth of Cu onto polycrystalline Pt electrode from acidic CuSO4 solution in the presence of H2SeO3

The early stages of Cu electrodeposition onto Pt(poly) have been investigated in 0.5 M H2SO4 + 0.01 M CuSO4 solution without or with H2SeO3 when a molar concentration ratio [Cu(II)]/[Se(IV)] ≥ 2×102 using electrochemical and ex situ AFM techniques. The overpotential deposition of Cu has been performed onto a Pt surface precovered independently with Cu in amount close to an equivalent monolayer. Chronoamperometric results have been shown to follow an instantaneous 3D nucleation and diffusion-controlled growth model. The values of diffusion coefficient D for Cu2+, number of nuclei N and average nuclei radius r av have been calculated. In the local regions of the surface, the separate large agglomerates composed of the different diameter clusters have been revealed in both cases, but, in the presence of the H2SeO3, they attained a distinct chain-like configuration. Some morphological characteristics have been reported.

Chemical composition of a binary dilute solid solution during growth from its melt: effect of kink kinetics

Effective distribution coefficients for an interface of a binary dilute solid solution (or alloy) growing from its melt are formulated for the steady state using a kink kinetics model. Solutions to the set of equations show that the effective distribution coefficients of energetically different sites in the crystal and the melt will depart from their respective equilibrium values as the degree of dis-equilibrium increases. These results indicate that the compositional change across the interface is gradual or diffuse under both equilibrium and non-equilibrium conditions. The direct exchange of atoms among energetically different sites in the melt adjacent to the interface strongly influences the effective distribution coefficients. The approximate equations of the effective distribution coefficients for an interface are also derived. A diffusion controlled growth model is generalized by incorporating the kink kinetics.

Electrochemical nucleation of lead and copper on indium-tin oxide electrodes

The nucleation process of Pb and Cu onto indium-tin oxide electrodes was studied by single potentiostatic steps using 0.1×10−4 mol L−1 Cu(NO3)2 and 2×10−2 mol L−1 Pb(NO3)2 solutions in aqueous 1.0 mol L−1 NaNO3. The current, I max, and the time, t max, corresponding to the maximum, were evaluated by 3D nucleation with a diffusion-controlled growth model for instantaneous and progressive nucleation. The non-dimensional plots of (I/I max)2 vs. t/t max showed that Cu and Pb nucleation closely followed the response predicted for instantaneous nucleation. Other parameters, such as I vs. t 1/2 and I 2 max t max, as well as nuclei micrographs, were analyzed, corroborating the results obtained.

A model of the growth of intermediate phase islands in multilayers

Recent experiments on reactive diffusion in multilayer films have shown that the first phase formation includes two steps: (i) nucleation and lateral growth of islands, and (ii) classical diffusion controlled growth. We present a model of diffusion-controlled growth of the phase 1 island on the interface AB. In this model a capillary effect and corresponding drift term related to the gradient of the curvature of the interface as the well as the chemical driving force are taken into account. This model predicts an asymptotic maximal thickness of growing island, the shape of island, and gives Avrami exponents between 1 and 2, which correspond to experimental data.

Deposition of Cu onto polycrystalline Pt electrode from acidic CuSO4 solution in the presence of H2SeO3

SUMMARYThe first stages of electrodeposition of Cu onto polycrystalline Pt in an acidic 0.1 M CuSO4 solution containing H2eO3 were studied combining electrochemical and surface analytical techniques. The molar ratio of the concentrations of Cu2+’ and H2SeO3 was from 1 × 105 to 1 × 103. The deposition kinetics was investigated by cyclic voltammetry and chronoamperometry; the elemental composition of deposits was studied by means of XPS, and the surface morphology by SEM. The instantaneous 3-D nucleation with diffusion-controlled growth model in the absence and in the presence of H2eO3 was elucidated. The XPS investigation revealed that the crystallites are comprised mainly of metallic copper and no more than 10 at.% elemental selenium. The number of nuclei per unit area calculated from the chronoamperometric data was of the order of 1 × 105 to 1 × 106 cm2 and that estimated from the SEM images was about two orders of magnitude larger.

Study on the lateral growth of silicon films from metal solutions with temperature gradient

The growth of Si film from Cu–Si solution with temperature gradient was conducted experimentally and examined by a simplified diffusion-controlled growth model calculation. It was found that the relation between the grown film thickness distribution along the temperature gradient direction and the cooling rate could be explained by the model calculation, and values for binary diffusion coefficient in the Cu–Si solution was estimated. The experimental results showed that a uniform film growth at the cooling rate as high as 0.1°C/s was possible.

Hydrogen porosity in directionally solidified aluminium–copper alloys: a mathematical model

A combined continuum and stochastic model of diffusion-controlled growth was developed to simulate the formation of porosity during the solidification of aluminium alloys. The whole population of pores was tracked, rather than just the average values. A finite difference solution of the diffusion equations was used, combined with a stochastic model of nucleation. The growth of each individual pore was simulated, assuming the shape to be spherical until it impinged on the developing dendrites, at which point the growth was modelled as a hemispherically capped segmented cone, with the growth radius limited by the liquid space between the dendrites. A previously published model by one of the authors was used to predict the dendritic spacing as a function of the thermal conditions. The model was compared with in situ observations of the formation of porosity during the solidification of aluminium–copper alloys, where the size, distribution and morphological evolution of pores were measured as a function of temperature/time. The predicted development of the porosity, including the distribution in size and morphology, compared well with that observed experimentally. The qualitative agreement between the model predictions and experimental results supports the hypothesis that the effect of hydrogen and its diffusion must be incorporated into any accurate model of pore formation in aluminium alloys.

Activation energies of intermetallic growth of Sn-Ag eutectic solder on copper substrates

Intermetallic phases formed along a Sn-Ag eutectic solder/Cu interface during solid-state aging have been characterized and the activation energies of Cu3Sn and Cu6Sn5 growth have been calculated. Diffusion couples consisting of Cu/ 96.5Sn-3.5Ag/Cu were aged at 110 to 208°C for 0 to 32 days. After aging, the Cu/ solder interfaces were examined using scanning electron microscopy and energy dispersive x-ray spectroscopy. The growth rate constants for each intermetallic layer were calculated assuming a simple parabolic diffusion-controlled growth model. The activation energy for Cu3Sn growth is 0.73 eV/atom and the activa-tion energy for Cu6Sn5 growth is 1.11 eV/atom.