Modified Diffusion Model Research Articles

Cooling rate effects on microstructure and diffusion behaviour in Ti65 alloy: Insights from a modified diffusion model

In the continuous cooling process, the growth of the equiaxed α-phase grains in near-α high-temperature titanium alloys is controlled by the diffusion of alloying elements. Establishing a specific connection between the cooling rate and the diffusion behaviour of alloying elements aids in the precise prediction of the evolution of equiaxed α-phase grain size. This study meticulously controlled the cooling rate during the two-phase region annealing treatment of the Ti65 alloy. Using EPMA technology, the diffusion behaviour of solute elements during cooling was accurately characterized. The study found that slowing the cooling rate promotes the coarsening of the lamellar secondary α-phase grains and the growth of the primary equiaxed α-phase grains. At higher annealing temperatures, the growth of equiaxed α-phase grains can occur at faster cooling rates, while coarse lamellar secondary α-phase grains can be retained at slower cooling rates. The diffusion behaviour of solute elements Al, Ta, Mo, and W between the α-phase and transformed β-phase matrix is significantly influenced by the cooling rate, thus they are considered as the controlling elements for the growth of the equiaxed α-phase grains. Based on the diffusion behaviours of these controlling elements, their single-element diffusion models were categorized and integrated for predicting the grain size of the equiaxed α-phase. The predictions from the revised diffusion model show an excellent agreement with the actual results, with an error margin of about 5%.

Decay characteristics and application predictions of far-field high-pressure water: Jet velocity and volume fraction

In the field of deep mining engineering, high-pressure water jets progress toward larger diameters and higher velocities to enhance their impact performance. Understanding the decay characteristics as well as their implications remains challenging. In this article, the jet spread coefficients (k and c) within the empirical model of jet diffusion are determined by series of lab experiments, while simulations using the Eulerian method are conducted by implementing the modified diffusion model as a subroutine. The effects of spread coefficient, jet velocity, and nozzle diameter on the degree of jet decay are studied. The results show that the jet spread coefficient increases logarithmically with increasing jet velocity. With the increase in nozzle diameter, the growth of spread coefficient k of the jet gradually decreases. Increasing the nozzle diameter and spread coefficient k can effectively reduce the decay degree of jet axial velocity and water volume fraction. Notably, although the increase in jet velocity does not impact the decay of axial velocity, it exacerbates the decay of water volume fraction. Similarly, the increase in spread coefficient c has no effect on the reduction of water volume fraction, but it intensifies the decay of jet axial velocity. The combined effects of increased jet velocity and jet spread coefficient weaken the degree of jet decay. The research presents a comprehensive and innovative study of jet diffusion and attenuation phenomena. These insights not only expand the application of jet diffusion models but also provide theoretical support for understanding and optimizing the application of jets.

Vertical matrix advection dominates transient anomalous diffusion within fracture-matrix systems using a modified diffusion model

Pollutant transport in fractured media has been well analyzed using process-based diffusion models and concise, time-nonlocal models like the tempered fractional derivative (TFD) model. While traditional diffusion models are limited to specific transport processes, TFD model parameters lack hydrogeological meaning. In addressing these knowledge gaps, a modified diffusion model is proposed to augment the traditional diffusion model and offer hydrogeological interpretations for TFD parameters. This modified diffusion model incorporates transverse advection in low-permeability matrices, allowing for the quantification of transient anomalous diffusion in a single fracture-matrix system. This model offers a semi-analytical solution, systematically validated for its applicability. Memory function analysis establishes a quantitative connection between the modified diffusion model and the TFD model, uncovering a pivotal discovery in this study. Parameter sensitivity analysis further shows that the matrix characteristic rate defined as a function of (Vm)2/(4Dm), where Vm is the transverse matrix velocity and Dm is the transverse matrix diffusion coefficient, controls the temporal evolution of anomalous sub-diffusion in the fracture. Applications are performed to check the feasibility of the modified diffusion model. The developed modified diffusion model sheds light on quantifying complex transport in fracture-matrix systems and enhances the predictability of nonlocal transport models.

Super-resolution reconstruction of ultrasound image using a modified diffusion model

Objective. This study aims to perform super-resolution (SR) reconstruction of ultrasound images using a modified diffusion model, designated as the diffusion model for ultrasound image super-resolution (DMUISR). SR involves converting low-resolution images to high-resolution ones, and the proposed model is designed to enhance the suitability of diffusion models for this task in the context of ultrasound imaging. Approach. DMUISR incorporates a multi-layer self-attention (MLSA) mechanism and a wavelet-transform based low-resolution image (WTLR) encoder to enhance its suitability for ultrasound image SR tasks. The model takes interpolated and magnified images as input and outputs high-quality, detailed SR images. The study utilized 1,334 ultrasound images from the public fetal head-circumference dataset (HC18) for evaluation. Main results. Experiments were conducted at 2× , 4× , and 8× magnification factors. DMUISR outperformed mainstream ultrasound SR methods (Bicubic, VDSR, DECUSR, DRCN, REDNet, SRGAN) across all scales, providing high-quality images with clear structures and rich detailed textures in both hard and soft tissue regions. DMUISR successfully accomplished multiscale SR reconstruction while suppressing over-smoothing and mode collapse problems. Quantitative results showed that DMUISR achieved the best performance in terms of learned perceptual image patch similarity, with a significant decrease of over 50% at all three magnification factors (2× , 4× , and 8× ), as well as improvements in peak signal-to-noise ratio and structural similarity index measure. Ablation experiments validated the effectiveness of the MLSA mechanism and WTLR encoder in improving DMUISR’s SR performance. Furthermore, by reducing the number of diffusion steps, the computational time of DMUISR was shortened to nearly one-tenth of its original while maintaining image quality without significant degradation. Significance. This study demonstrates that the modified diffusion model, DMUISR, provides superior performance for SR reconstruction of ultrasound images and has potential in improving imaging quality in the medical ultrasound field.

Integrated Experimental and Mathematical Exploration of Modular Tissue Cultures for Developmental Engineering.

Developmental engineering (DE) involves culturing various cells on modular scaffolds (MSs), yielding modular tissues (MTs) assembled into three-dimensional (3D) tissues, mimicking developmental biology. This study employs an integrated approach, merging experimental and mathematical methods to investigate the biological processes in MT cultivation and assembly. Human dermal fibroblasts (HDFs) were cultured on tissue culture plastics, poly(lactic acid) (PLA) discs with regular open structures, or spherical poly(methyl methacrylate) (PMMA) MSs, respectively. Notably, HDFs exhibited flattened spindle shapes when adhered to solid surfaces, and complex 3D structures when migrating into the structured voids of PLA discs or interstitial spaces between aggregated PMMA MSs, showcasing coordinated colonization of porous scaffolds. Empirical investigations led to power law models simulating density-dependent cell growth on solid surfaces or voids. Concurrently, a modified diffusion model was applied to simulate oxygen diffusion within tissues cultured on solid surfaces or porous structures. These mathematical models were subsequently combined to explore the influences of initial cell seeding density, culture duration, and oxygen diffusion on MT cultivation and assembly. The findings underscored the intricate interplay of factors influencing MT design for tissue assembly. The integrated approach provides insights into mechanistic aspects, informing bioprocess design for manufacturing MTs and 3D tissues in DE.

Demand forecasting of technological products considering re-purchasing and dynamic pricing: a modified diffusion model

Demand forecasting of technological products considering re-purchasing and dynamic pricing: a modified diffusion model

Neutrally- and stably-stratified boundary layers adjustments to a step change in surface roughness

In this work, we study the development of the internal boundary layer (IBL) induced by a surface roughness discontinuity, where the downstream surface has a roughness length greater than that upstream. The work is carried out in the EnFlo meteorological wind tunnel, at the University of Surrey, in both thermally neutral and stable cases with varying degrees of stability. For the neutrally-stratified boundary layer, the IBL development in the log-law region shows good agreement with the diffusion model proposed by Panofsky and Dutton (Atmospheric turbulence, Wiley, New York, 1984) provided that a modified origin condition is introduced and its growth rate is dictated by a constant diffusion term. However, the model over-predicts the growth of the IBL in the outer layer, where the IBL depth grows slowly with fetch following a power function with exponent n being 0.61 (whereas the original model prescribes napprox 0.8). For the stably-stratified boundary layers, n is found to further reduce as the bulk Richardson number, textrm{Ri}_textrm{b}, increases. The analysis of the top region of the IBL shows that the slow growth rate is due to a combination of the decay of the diffusion term and a significantly negative mean wall-normal velocity, which transports fluid elements towards the wall. Considering these two effects, a modified diffusion model is proposed which well captures the growth of the IBL for both neutrally and stably-stratified boundary layers.Graphical abstract

The role of chloride binding mechanism in the interpretation of chloride profiles in concrete containing limestone powder

The effect of limestone powder addition on the chloride binding capacity of concrete was evaluated. The results indicate that the chloride binding capacity becomes weaker as the limestone content increases due to decreased physical adsorption and chemical binding. XRD patterns indicate that adding limestone restrains the formation of monosulfate (AFm), thereby undermining the chemical binding. Thermogravimetric analyses (TGA) validate that despite limited participation of calcite in the hydration reaction, it is significant enough to change the stoichiometry of the cementitious systems and restrain the formation of Friedel's salt. The ratio between the apparent and effective chloride diffusion coefficients increases with decreasing chloride binding. A modified diffusion model was proposed based on Fick’s second law coupled with Freundlich adsorption isotherm on water-accessible pore surfaces to predict the chloride profile. The model exhibits a consistent trend that an enhanced chloride binding, higher tortuosity and lower porosity contribute to a lower chloride content.

Characteristics of Modified Diffusion Analysis of Squeezed Hydro-magnetic Nanofluid Flow

This communication based on nanofluid theory is exhibited to study the squeezing mechanism. Hydro-magnetic phenomenon is incorporated in the analysis of nanofluid flow. The flow of heat and mass transport is interpreted by the constitutive relationship of modified diffusion model which overcomes the deficiencies in Fourier’s and Fick’s classical theories. The nano-fluid theory also incorporates the features of two important slip mechanisms namely Brownian diffusion and thermophoresis. Resulting heat and mass transport problems are figure out to develop the series solutions. Interval of convergence is established for computed series solutions. Behavior of emerging parameters in flow distributions are graphed and discussed. The major founds in this attempt is that higher Brownian diffusion strengthens the both heat and mass fluxes, while larger thermophoresis parameter enhances the heat flux but it weakens the mass flux. Moreover, both heat and mass flow field decays for enlarge estimation of thermal and solutal time relaxation parameters, respectively. Velocity field shows dual behavior for dominant magnetic parameter.

A time-dependent chloride diffusion model for predicting initial corrosion time of reinforced concrete with slag addition

The effect of granulated blast furnace slag (GBFS) addition on the threshold chloride concentration (TCC) for rebar corrosion was investigated. A modified diffusion model, coupled with a time-dependent effective diffusion coefficient and surface chloride concentration, was proposed to predict the chloride profile. The corrosion initiation time was estimated based on the model predictions and the measured TCC. The results indicate that adding GBFS decreases the TCC by lowering the pH value of the pore solution. The evolution of corrosion potential and current density is found to obey a 3-parameter Weibull distribution. MIP tests show that adding GBFS contributes to refinement of pore structure by decreasing the fraction of large capillary pores. The time-dependent model exhibits good predictive strength and helps understand how GBFS addition delays the corrosion initiation by retarding the chloride diffusion, though a lower TCC is obtained.

Treatment of wastewater from a dairy plant by adsorption using synthesized copper oxide nanoparticles: kinetics and isotherms modeling optimization

Dairy plants produce 1 to 4 L of wastewater per 1 L of processed milk. The wastewater contains high values of chemical oxygen demand (COD) and biochemical oxygen demand (BOD) concentrations, in addition to high levels of dissolved solids. In this study, synthesized copper oxide nanoparticles (CuONPs) coupled with Sophora Japonica fruit, were used as an adsorbent, for the first time, to treat the effluent of dairy plants in a batch adsorption process. The analysis techniques, Fourier-transform infrared spectroscopy (FTIR), X-ray diffraction (XRD) and scanning electron microscopy (SEM) were utilized to characterize the adsorbent. The COD removal, using (CuONPs)-based adsorbent, was investigated by varying contact time, masses of the adsorbent, initial COD value and temperatures. The optimum conditions for highest removal percentage were contact time of 120 min, a temperature of 25 °C, pH value of 7.5, and 1 g of adsorbent. The initial COD values used were in the range of 100-700 ppm. The COD percent removal was in the range of 77 to 95%. Freundlich isotherm exhibited the best fitting for the results (R2 = 0.998) with a favorable spontaneous exothermic adsorption process. Based on the calculated normalized deviation value, the modified diffusion model, intra-diffusion, and pseudo-second-order kinetics all showed very good fitting for the adsorption data as indicated by the kinetics study.

Tuning the Δn and scattering in Bayfol® HX based holograms

Self-developing photopolymers are crucial materials for making high efficiency volume holograms. A key feature is the tuning of the refractive index modulation (Δn), but at the same time, it is crucial to minimize the light scattering which reduces the efficiency and increase the unwanted light. Here, we study Bayfol® HX materials in terms of scattering formation and control of the Δn by means the realization of Volume Phase Holographic Gratings (VPHGs), which are intended for the application in spectroscopic instrumentation. We highlight the dependency of the Δn and scattering with the laser light intensity, showing that the increase of the intensity reduces the amount of scattering. In addition, the fine tuning of the Δn with the incoherent light pre- and co-exposure is studied and the full dynamic range can be addressed controlling the scattering. A modified diffusion model is also used to describe and predict the experimental results.

The effect of hydrogen bonding on diffusion and permeability in UV-cured Polyacrylate-based networks for controlled release

Polyacrylates are important polymers widely used in pharmaceutical industry such as drug coatings due to their low cost, processability and ease of functionalisation. Chemical functionalities (e.g. H-bonding) can be easily included to modulate the transport of low molecular weight drug-like entities through the network. Understanding how such microscopic structural modifications determine macroscopic diffusion is critical for designing next generation responsive polymers. In this study pulsed field gradient (PFG) 1H NMR measurements of the self-diffusion of a dye molecule (Eosin Y) in a series of polyacrylate networks with differing H-bonding strength were undertaken; it was found that the diffusion of Eosin Y is significantly reduced in networks with H-bonding. Detailed analyses by 1H NMR relaxometry and double quantum (DQ) NMR show that H-bonding can also reduce polymer chain mobility. Furthermore, DSC thermoporometry showed a significant increase in the average network mesh size potentially due to the pre-organization of H-bonding containing monomer during network curing. By introducing the H-bonding disrupter, LiClO4, it was found that the diffusivity of solute becomes positively correlated to the average mesh size across the series of networks. Hence, a modified diffusion model based on hydrodynamic theory is proposed to separate the direct (solute-network) H-bonding contribution to solute diffusion from the indirect contribution arising from monomer pre-ordering induced mesh size reduction. Finally, it is shown that the same direct and indirect contributions to microscopic diffusivity, arising from the H-bond strength of the co-monomers, also contribute significantly to the macroscopic membrane permeability which is similarly subject to H-bond disruption.

Modeling constitutive relationship of sulfate-attacked concrete

Sulfate attack is one of the most common factors responsible for the performance deterioration of concrete exposed to aggressive environment. The transport of sulfate in concrete is a gradual process causing a gradient sulfate attack in concrete. Nominal stress-strain relationship which is directly based on the whole cross-sectional area is not applicable in practical engineering because of its dependence on geometrical dimensions. In this paper, a method is developed to obtain a real constitutive model for sulfate-attacked concrete. To investigate the transport behavior of sulfate ion in concrete, 16 cubic specimens were immersed in 10% sodium sulfate solution for eight immersion durations. Based on the development of sulfate ion content distribution, a modified diffusion model was developed to simulate the transport of sulfate ion in concrete. To clarify the effects of sulfate attack on mechanical properties of concrete, 24 prism specimens were immersed in 10% sodium sulfate solution, and another 24 specimens were immersed in pure water as control group to compensate the influence of further hydration. Based on the sulfate ion content distribution and nominal stress-strain curve for concrete in different immersion durations, real stress-strain curves of concrete with different sulfate ion contents were obtained by an innovative method. After analyzing the effects of sulfate ion content on the mechanical properties of concrete, a real constitutive model was proposed, which can well predict the structural performance of sulfate-attacked concrete members. With the increase of sulfate immersion duration, the peak stress and elastic modulus increase first and then decrease, while the peak strain decreases first and then increase.

A novel comprehensive evaluation method for the corrosion initiation life of RC hollow piles in chloride environments

A modified diffusion model of chloride ions in reinforced concrete (RC) hollow piles is proposed by taking into account the ageing of the pile concrete, loading level, environmental temperature and relative humidity. The corrosion initiation life of RC hollow piles is predicted by deterministic and probabilistic methods. In addition, by combining deterministic and probabilistic methods, a novel comprehensive method is proposed for evaluating the corrosion initiation life of RC hollow piles in chloride environments. A sensitivity analysis on the main influencing parameters is conducted for the prediction of the corrosion initiation life by the comprehensive method. Finally, the comparative analysis of three different prediction methods is also performed. The results show that the corrosion initiation life calculated by the comprehensive method decreases with increasing environmental temperature, relative humidity, water-cement ratio and initial surface chloride concentration, while it increases with increasing concrete cover depth and chloride threshold level. The comprehensive method is more reasonable than the deterministic and probabilistic methods for predicting the corrosion initiation life of RC hollow piles in chloride environments.

A modified diffusion model for characterization of real I–V curve and charge collection efficiency of CdZnTe detectors

In the present paper, a modified diffusion model describing I–V curve of CZT detectors is proposed and used to characterize the electric field distribution. The detectors are considered to be fully depleted at high bias and incompletely depleted at low bias. For incompletely depleted condition, the inner electric field of CZT detector was separated into depletion region and neutral region. The neutral region was usually neglected in the traditional model. Two fitting parameters VD and α are included in the proposed model, which can be fitted from I–V curve. The model is also extended to describe the electric field distribution. Furthermore, we figured out the calculated charge collection efficiency equations, which are consistent with experimental results. Experimental data from planar detectors indicate that the model could describe I–V curve precisely throughout a broad bias range. Serious electric field distortion in some planar detectors is attributed to the Schottky barrier. The relationship between charge collection efficiency (CCE) and bias voltage in these devices is not consistent with the Hecht equation. The difference may arise from the non-uniform electric field. The stronger effect of the Schottky barrier on electron collection efficiency is found in pixel detectors according to the non-uniform weighting potential, because for pixel detectors the induced charge is mainly contributed by the near anode region, which has low electric field under high cathode barrier.

Microstructures evolution of nano TiB2/7050Al composite during homogenization

The microstructures evolution of nano TiB2/7050Al composite during homogenization was studied by in-situ scanning electronic microscopy, transmission electron microscope, energy dispersive spectrometer, electron backscattered diffraction and differential scanning calorimeter. The results showed that many secondary phases (SPs) and TiB2 particles coexisted at grain boundaries (GBs). These SPs mainly consisted of S(Al2CuMg) phase and T(Mg(Zn, Cu, Al)2) phase. During homogenization, a transformation from T to S phase occurred, and the homogenization was hindered. Furthermore, the strip-type arrangement of TiB2 at Type-I GBs should accelerate homogenization. Comparably, the shell-type arrangement of TiB2 at Type-II GBs can decelerate homogenization, since the aggregated SPs were hard to be dissolved herein. To clarify the effects of the phase transformation and particles arrangement on the homogenization, a modified diffusion model was proposed. Besides, the Vickers hardness increased and the electronic conductivity decreased with the elevated homogenization temperature and prolonged holding duration. Therefore, a multiple-stage homogenization process should be reasonable for nano TiB2/7050 Al composite.

Grain boundary-constrained reverse austenite transformation in nanostructured Fe alloy: Model and application

Reverse austenite transformation (RAT) is critical for designing advanced high-strength steels (AHSS), which, however, has not been sufficiently studied in nanostructured (NS) steels or Fe alloys, and hence not fully understood yet. Herein, the RAT (e.g. ferrite to austenite) kinetics in the NS Fe alloy upon continuous heating was experimentally and theoretically investigated, where, the ultrafine austenite characterized by a sluggish growth velocity and a high thermal stability, and additionally, an appreciable solute partitioning detected using atom probe microscopy, indicate the diffusion-controlled mechanism of RAT. The double-edged role of grain boundaries (GBs) in the NS alloy is elucidated, i.e. enhancing the diffusivity due to the type-A kinetics, and simultaneously, facilitating the formation of constrained diffusion field mainly due to the segmented effect of GB nucleation. On this basis, a modified diffusion model incorporating the effect of GBs is implemented to understand the GB-constrained austenite growth and the associated partitioning behavior, and further complemented with Cahn model, an austenite growth model is applied to predict the overall kinetics of RAT in the NS Fe alloy. It then follows that a strategy by combination of diffusion-controlled growth model and microstructure model could serve as a framework to predict the kinetics of RAT in the NS alloys. Regarding the RAT kinetics in the NS alloys, the present work uncovers the ‘GB-constrained’ mechanism, which is expected to offer the potential application for nanostructure manipulation in the development of AHSS.

An efficient detection algorithm based on anisotropic diffusion for low-contrast defect

In this paper, we propose an efficient algorithm based on the anisotropic diffusion model to detect defect in a low-contrast surface image, especially aimed at anti-reflective (AR) glass. The proposed algorithm has two important procedures: (1) a modified anisotropic diffusion model and (2) morphological directivity filter. The modified diffusion model based on an adaptive edge threshold, the standard score (Z-score), is proposed to quickly and efficiently enhance the low-contrast defects. It acts as the adaptive enhancement process. The pixels with both the low gradient and the high Z-score or the high gradient and the low Z-score will generate a high diffusion coefficient. It enhances the gray levels of suspected defective edges and also preserves the original gray levels of an internal area of suspected defects, as well as generates the slight smoothing process for the noisy background. A simple and efficient method can easily segment the defects, followed by the effective morphological directivity filter, which removes noise from the thresholding image. The proposed algorithm is evaluated by a set of 23 low-contrast surface images of AR glass in this study. The experimental results show that the proposed algorithm is superior to the four competitive approaches. The defect detection results demonstrate that the proposed algorithm can segment the complete defects. In addition, the computational advantage compared to the Perona and Malik model, C-T model 1, C-T model 2, and histogram statistics model are about 4.48, 2.26, 19.53, and 26.63 times, respectively. It can be concluded that the proposed algorithm not only provides reliable inspection results but also improves the inspection efficiency over 2 times.

Chloride transport in cracked RC beams under dry–wet cycles

The coupled effect of transverse cracking and cyclic drying–wetting conditions on chloride penetration in concrete is explored in this paper. The transport mechanisms of chloride in cracked concrete subjected to drying–wetting cycles are described and simplified. An experimental investigation was conducted for sustained cracked concrete beams and the results are reported here. It is found that the depth interval of the convection region in cracked concrete varies from approximately 10 to 20 mm, which is slightly greater than that found in sound concrete. Also, it increases with the increase in crack width and decreases with the increase in fly ash content. At the cracked sections, the speed of chloride penetration is increased by the presence of cracks. It is shown that the modified diffusion model is still practicable for describing the chloride distribution in transversely cracked concrete when the crack width is within a range close to 0·30 mm. Finally, a regressive relationship between equivalent chloride diffusion coefficient, DCleq, and crack width, wcr, is proposed with a quadratic function of crack effect.