Time-dependent Diffusion Research Articles

Homogeneous turbophoresis of heavy inertial particles in turbulent flow

Heavy particles suspended in turbulent flow possess inertia and are ejected from violent vortical structures by centrifugal forces. Once piled up along particle paths, this small-scale mechanism leads to an effective large-scale drift. This phenomenon, known as ‘turbophoresis’, causes particles to leave highly turbulent regions and migrate towards calmer regions, explaining why particles transported by non-homogeneous flows tend to concentrate near the minima of turbulent kinetic energy. It is demonstrated here that turbophoretic effects are just as crucial in statistically homogeneous flows. Although the average turbulent activity is uniform, instantaneous spatial fluctuations are responsible for inertial-range inhomogeneities in the particle distribution. Direct numerical simulations are used to probe particle accelerations, specifically how they correlate to local turbulent activity, yielding an effective coarse-grained dynamics that accounts for particle detachment from the fluid and ejection from excited regions through a space- and time-dependent non-Fickian diffusion. This leads to cast fluctuations in particle distributions in terms of a scale-dependent Péclet number ${\textit {Pe}}_\ell$ , which measures the importance of turbulent advection compared with inertial turbophoresis at a given scale $\ell$ . Multifractal statistics of energy dissipation indicate that $ {\textit {Pe}}_\ell \sim \ell ^\delta /\tau _{p}$ with $\delta \approx 0.84$ . Numerical simulations support this behaviour and emphasise the relevance of the turbophoretic Péclet number in characterising how particle distributions, including their radial distribution function, depends on $\ell$ . This approach also explains the presence of voids with inertial-range sizes, and the fact that their volumes have a non-trivial distribution with a power-law tail $p(\mathcal {V}) \propto \mathcal {V}^{-\alpha }$ , with an exponent $\alpha$ that tends to 2 as ${\textit {Pe}}_\ell \to 0$ .

Revealing membrane integrity and cell size from diffusion kurtosis time dependence.

The nonmonotonic dependence of diffusion kurtosis on diffusion time has been observed in biological tissues, yet its relation to membrane integrity and cellular geometry remains to be clarified. Here we establish and explain the characteristic asymmetric shape of the kurtosis peak. We also derive the relation between the peak time , when kurtosis reaches its maximum, and tissue parameters. The peak shape and its position qualitatively follow from the adiabatic extension of the Kärger model onto the case of intra-cellular diffusivity time-dependence. This intuition is corroborated by the effective medium theory-based calculation, as well as by Monte Carlo simulations of diffusion and exchange in randomly and densely packed spheres for various values of permeability, cell fractions and sizes, and intrinsic diffusivity. We establish that is proportional to the geometric mean of two characteristic time scales: extra-cellular correlation time (determined by cell size) and intra-cellular residence time (determined by membrane permeability). When exchange is barrier-limited, the peak shape approaches a universal scaling form determined by the ratio . Numerical simulations and theory provide an interpretation of a specific feature of kurtosis time-dependence, offering a potential biomarker for in vivo evaluation of pathology by disentangling the functional (permeability) and structural (cell size) integrity in tissues. This is relevant as the time-dependent diffusion cumulants are sensitive to pathological changes in membrane integrity and cellular structure in diseases, such as ischemic stroke, tumors, and Alzheimer's disease.

Stimulated echo acquisition mode (STEAM) diffusion tensor imaging with different diffusion encoding times in the supraspinatus muscle: Test-retest reliability and comparison to spin echo diffusion tensor imaging.

Diffusion tensor imaging (DTI) provides insight into the skeletal muscle microstructure and can be acquired using a stimulated echo acquisition mode (STEAM)-based approach to quantify time-dependent tissue diffusion. This study examined diffusion metrics and signal-to-noise ratio (SNR) in the supraspinatus muscle obtained with a STEAM-DTI sequence with different diffusion encoding times (Δ) and compared them to measures from a spin echo (SE) sequence. Ten healthy subjects (mean age 31.5 ± 4.7years; five females) underwent 3-Tesla STEAM and SE-DTI of the shoulder in three sessions. STEAM was acquired with Δ of 100/200/400/600 ms. The diffusion encoding time in SE scans was 19ms (b = 500s/mm2). Region of interest-based measurement of fractional anisotropy (FA), mean diffusivity (MD), and SNR was performed. Intraclass correlation coefficients (ICCs) were computed to assess test-retest reliability. ANOVA with post-hoc pairwise tests was used to compare measures between different Δ of STEAM as well as STEAM and SE, respectively. FA was significantly higher (FASTEAM: 0.38-0.46 vs. FASE: 0.26) and MD significantly lower (MDSTEAM: 1.20-1.33 vs. MDSE: 1.62×10-3mm2/s) in STEAM compared to SE (p < 0.001, respectively). SNR was significantly higher for SE (72.3 ± 8.7) than for STEAM (p < 0.001). ICCs were excellent for FA in STEAM (≥0.911) and SE (0.960). For MD, ICCs were good for STEAM100ms-600ms (≥0.759) and SE (0.752). STEAM and SE exhibited excellent reliability for FA and good reliability for MD in the supraspinatus muscle. SNR was significantly higher in SE compared to STEAM.

Fast and Accurate Numerical Integration of the Langevin Equation with Multiplicative Gaussian White Noise.

A univariate stochastic system driven by multiplicative Gaussian white noise is considered. The standard method for simulating its Langevin equation of motion involves incrementing the system's state variable by a biased Gaussian random number at each time step. It is shown that the efficiency of such simulations can be significantly enhanced by incorporating the skewness of the distribution of the updated state variable. A new algorithm based on this principle is introduced, and its superior performance is demonstrated using a model of free diffusion of a Brownian particle with a friction coefficient that decreases exponentially with the kinetic energy. The proposed simulation technique proves to be accurate over time steps that are an order of magnitude longer than those required by standard algorithms. The model used to test the new numerical technique is known to exhibit a transition from normal diffusion to superdiffusion as the environmental temperature rises above a certain critical value. A simple empirical formula for the time-dependent diffusion coefficient, which covers both diffusion regimes, is introduced, and its accuracy is confirmed through comparison with the simulation results.

Assessing tumor microstructure with time-dependent diffusion imaging: Considerations and feasibility on clinical MRI and MRI-Linac.

Quantitative imaging biomarkers (QIBs) can characterize tumor heterogeneity and provide information for biological guidance in radiotherapy (RT). Time-dependent diffusion MRI (TDD-MRI) derived parameters are promising QIBs, as they describe tissue microstructure with more specificity than traditional diffusion-weighted MRI (DW-MRI). Specifically, TDD-MRI can provide information about both restricted diffusion and diffusional exchange, which are the two time-dependent effects affecting diffusion in tissue, and relevant in tumors. However, exhaustive modeling of both effects can require long acquisitions and complex model fitting. Furthermore, several introduced TDD-MRI measurements can require high gradient strengths and/or complex gradient waveforms that are possibly not available in RT settings. In this study, we investigated the feasibility of a simple analysis framework for the detection of restricted diffusion and diffusional exchange effects in the TDD-MRI signal. To promote the clinical applicability, we use standard gradient waveforms on a conventional 1.5 T MRI system with moderate gradient strength (Gmax=45 mT/m), and on a hybrid 1.5 T MRI-Linac system with low gradient strength (Gmax=15 mT/m). Restricted diffusion and diffusional exchange were simulated in geometries mimicking tumor microstructure to investigate the DW-MRI signal behavior and to determine optimal experimental parameters. TDD-MRI was implemented using pulsed field gradient spin echo with the optimized parameters on a conventional MRI system and a MRI-Linac. Experiments in green asparagus and 10 patients with brain lesions were performed to evaluate the time-dependent diffusion (TDD) contrast in the source DW-images. Simulations demonstrated how the TDD contrast was able to differentiate only dominating diffusional exchange in smaller cells from dominating restricted diffusion in larger cells. The maximal TDD contrast in simulations with typical cancer cell sizes and in asparagus measurements exceeded 5% on the conventional MRI but remained below 5% on the MRI-Linac. In particular, the simulated TDD contrast in typical cancer cell sizes (r=5-10µm) remained below or around 2% with the MRI-Linac gradient strength. In patients measured with the conventional MRI, we found sub-regions reflecting either dominating restricted diffusion or dominating diffusional exchange in and around brain lesions compared to the noisy appearing white matter. On the conventional MRI system, the TDD contrast maps showed consistent tumor sub-regions indicating different dominating TDD effects, potentially providing information on the spatial tumor heterogeneity. On the MRI-Linac, the available TDD contrast measured in asparagus showed the same trends as with the conventional MRI but remained close to typical measurement noise levels when simulated in common cancer cell sizes. On conventional MRI systems with moderate gradient strengths, the TDD contrast could potentially be used as a tool to identify which time-dependent effects to include when choosing a biophysical model for more specific tumor characterization.

Charge transport in organic semiconductors from the mapping approach to surface hopping.

We describe how to simulate charge diffusion in organic semiconductors using a recently introduced mixed quantum-classical method, the mapping approach to surface hopping. In contrast to standard fewest-switches surface hopping, this method propagates the classical degrees of freedom deterministically on the most populated adiabatic electronic state. This correctly preserves the equilibrium distribution of a quantum charge coupled to classical phonons, allowing one to time-average along trajectories to improve the statistical convergence of the calculation. We illustrate the method with an application to a standard model for the charge transport in the direction of maximum mobility in crystalline rubrene. Because of its consistency with the equilibrium distribution, the present method gives a time-dependent diffusion coefficient that plateaus correctly to a long-time limiting value. The resulting mobility is somewhat higher than that of the relaxation time approximation, which uses a phenomenological relaxation parameter to obtain a non-zero diffusion coefficient from a calculation with static phonon disorder. However, it is very similar to the mobility obtained from Ehrenfest dynamics, at least in the parameter regimes we have investigated here. This is somewhat surprising because Ehrenfest dynamics overheats the electronic subsystem and is, therefore, inconsistent with the equilibrium distribution.

Time-Dependent Diffusion MRI Helps Predict Molecular Subtypes and Treatment Response to Neoadjuvant Chemotherapy in Breast Cancer.

Background Time-dependent diffusion MRI has the potential to help characterize tumor cell properties; however, to the knowledge of the authors, its usefulness for breast cancer diagnosis and prognostic evaluation is unknown. Purpose To investigate the clinical value of time-dependent diffusion MRI-based microstructural mapping for noninvasive prediction of molecular subtypes and pathologic complete response (pCR) in participants with breast cancer. Materials and Methods Participants with invasive breast cancer who underwent pretreatment with time-dependent diffusion MRI between February 2021 and May 2023 were prospectively enrolled. Four microstructural parameters were estimated using the IMPULSED method (a form of time-dependent diffusion MRI), along with three apparent diffusion coefficient (ADC) measurements and a relative ADC diffusion-weighted imaging parameter. Multivariable logistic regression analysis was used to identify parameters associated with each molecular subtype and pCR. A predictive model based on associated parameters was constructed, and its performance was assessed using the area under the receiver operating characteristic curve (AUC) and compared by using the DeLong test. The time-dependent diffusion MRI parameters were validated based on correlation with pathologic measurements. Results The analysis included 408 participants with breast cancer (mean age, 51.9 years ± 9.1 [SD]). Of these, 221 participants were administered neoadjuvant chemotherapy and 54 (24.4%) achieved pCR. The time-dependent diffusion MRI parameters showed reasonable performance in helping to identify luminal A (AUC, 0.70), luminal B (AUC, 0.78), and triple-negative breast cancer (AUC, 0.72) subtypes and high performance for human epidermal growth factor receptor 2 (HER2)-enriched breast cancer (AUC, 0.85), outperforming ADC measurements (all P < .05). Progesterone receptor status (odds ratio [OR], 0.08; P = .02), HER2 status (OR, 3.36; P = .009), and the cellularity index (OR, 0.01; P = .02) were independently associated with the odds of achieving pCR. The combined model showed high performance for predicting pCR (AUC, 0.88), outperforming ADC measurements and the clinical-pathologic model (AUC, 0.73 and 0.79, respectively; P < .001). The time-dependent diffusion MRI-estimated parameters correlated well with the pathologic measurements (n = 100; r = 0.67-0.81; P < .001). Conclusion Time-dependent diffusion MRI-based microstructural mapping was an effective method for helping to predict molecular subtypes and pCR to neoadjuvant chemotherapy in participants with breast cancer. © RSNA, 2024 Supplemental material is available for this article. See also the editorial by Partridge and Xu in this issue.

Detecting Cellular Microstructural Changes of Liver Fibrosis with Time-Dependent Diffusion MRI.

Background The potential of time-dependent diffusion MRI in imaging the progression from liver fibrosis to cirrhosis has not been established. Purpose To assess the effectiveness of time-dependent diffusion MRI in mapping the microstructure and characterizing cellular attributes during the progression of liver fibrosis to cirrhosis and to investigate its potential in grading liver fibrosis. Materials and Methods This prospective study, performed between December 2022 and October 2023, used 60 rats to establish a liver fibrosis model by means of diethylnitrosamine administration, with five additional rats serving as control animals. Time-dependent diffusion MRI was performed with equivalent diffusion time of 5.4, 10.7, and 69.3 msec on a 3.0-T scanner. Time-dependent diffusion MRI-based microstructural parameters, including cell diameter, intracellular volume fraction (ICVF), cellularity, and extracellular diffusivity, were estimated with use of the imaging microstructural parameters using limited spectrally edited diffusion, or IMPULSED, model. The fitted microstructural parameters were validated with histopathologic measurements. Results All 60 rats developed liver fibrosis, with a noticeable decrease in cell diameter and an increase in ICVF and cellularity observed as liver fibrosis progressed. The diameter measured at pathologic examination ranged from 11.4 μm to 35.4 μm, aligning with the range of 12.4-33.4 μm observed in time-dependent diffusion MRI, which indicated a strong correlation (r = 0.84; P < .001). The quantified ICVF at pathologic examination ranged from 0.28 to 0.89 and varied from 0.23 to 0.85 at time-dependent diffusion MRI, showing a high correlation (r = 0.62; P < .001). The cellularity observed at pathologic examination increased from 0.74 to 5.85, while the cellularity measured at time-dependent diffusion MRI ranged from 0.77 to 3.70, showing a correlation (r = 0.44; P < .001). Conclusion This study revealed the changes in quantitative microstructural mapping across the spectrum from liver fibrosis to cirrhosis. Cell diameter, ICVF, and cellularity are reliable markers for liver fibrosis, with diameter and ICVF presenting good discrimination ability. © RSNA, 2024 Supplemental material is available for this article. See also the editorial by Matos and Metens in this issue.

A POD based reduced-order local RBF collocation approach for time-dependent nonlocal diffusion problems

A fast algorithm based on reduced-order model (ROM) is proposed for unsteady nonlocal diffusion models. It combines proper orthogonal decomposition (POD) approach and collocation method with local radial basis functions (RBFs), which makes it possible for using ROM to solve nonlocal models. Several numerical experiments showed that this approach significantly reduce the computational cost of nonlocal models while keep the similar convergent behavior compared with the RBF collocation methods.

The modelling error in multi-dimensional time-dependent solute transport models

Starting from full-dimensional models of solute transport, we derive and analyze multi-dimensional models of time-dependent convection, diffusion, and exchange in and around pulsating vascular and perivascular networks. These models are widely applicable for modelling transport in vascularized tissue, brain perivascular spaces, vascular plants and similar environments. We show the existence and uniqueness of solutions to both the full- and the multi-dimensional equations under suitable assumptions on the domain velocity. Moreover, we quantify the associated modelling errors by establishing a-priori estimates in evolving Bochner spaces. In particular, we show that the modelling error decreases with the characteristic vessel diameter and thus vanishes for infinitely slender vessels. Numerical tests in idealized geometries corroborate and extend upon our theoretical findings.

Time-dependent diffusion magnetic resonance imaging: measurement, modeling, and applications

Time-dependent diffusion magnetic resonance imaging: measurement, modeling, and applications

Methodology of integrated measurements with radon diffusion chamber

Calibration of Solid-State Nuclear Track Detectors - SSNTD in the diffusion chamber in most cases assumes a constant radon concentration outside the diffusion chamber and a steady state. Although this has been proven to be a valid approximation in most cases, the fact is that the radon concentration outside the chamber is not constant. In this way, the detectors are calibrated to measure the mean radon concentration. For this reason, there is doubt if the time-dependent concentration in dwellings results in the same number of tracks on the detector as the time-averaged radon concentration for the same exposure period. To obtain an answer, time-dependent and steady-state diffusion equations for radon and its progeny were solved. Then the number of tracks on the detector was calculated for these two cases and compared. For that goal, software for solving the time-dependent diffusion equation for time-varying external radon concentration was developed. Results show that concentrations measured using a diffusion chamber are the real concentrations at the measuring site. However, it was also found that the validity of results is not a universal property, as shown in the current paper. In some cases, results can be wrong. For some extreme conditions when diffusion coefficients are disrupted calibration using stationary conditions model will lead to incorrect results of real measurements in experiments where ambiental concentration is time-varying.

The Diffusion Exchange Ratio (DEXR): A minimal sampling of diffusion exchange spectroscopy to probe exchange, restriction, and time-dependence.

Water exchange is increasingly recognized as an important biological process that can affect the study of biological tissue using diffusion MR. Methods to measure exchange, however, remain immature as opposed to those used to characterize restriction, with no consensus on the optimal pulse sequence(s) or signal model(s). In general, the trend has been towards data-intensive fitting of highly parameterized models. We take the opposite approach and show that a judicious sub-sample of diffusion exchange spectroscopy (DEXSY) data can be used to robustly quantify exchange, as well as restriction, in a data-efficient manner. This sampling produces a ratio of two points per mixing time: (i) one point with equal diffusion weighting in both encoding periods, which gives maximal exchange contrast, and (ii) one point with the same total diffusion weighting in just the first encoding period, for normalization. We call this quotient the Diffusion EXchange Ratio (DEXR). Furthermore, we show that it can be used to probe time-dependent diffusion by estimating the velocity autocorrelation function (VACF) over intermediate to long times (~ 2-500 ms). We provide a comprehensive theoretical framework for the design of DEXR experiments in the case of static or constant gradients. Data from Monte Carlo simulations and experiments acquired in fixed and viable ex vivo neonatal mouse spinal cord using a permanent magnet system are presented to test and validate this approach. In viable spinal cord, we report the following apparent parameters from just 6 data points: , , , and , which correspond to the exchange time, restricted or non-Gaussian signal fraction, an effective spherical radius, and permeability, respectively. For the VACF, we report a long-time, power-law scaling with , which is approximately consistent with disordered domains in 3-D. Overall, the DEXR method is shown to be highly efficient, capable of providing valuable quantitative diffusion metrics using minimal MR data.

New numerical models of atomic diffusion in the atmospheres of cool Ap stars, including ambipolar diffusion of hydrogen

Ambipolar diffusion of hydrogen gives an additional upward thrust to metals that diffuse in the atmosphere of Ap stars. Its quantitative effect on the build-up of abundance stratification due to atomic diffusion that produces the observed abundance anomalies in Ap stars has not been evaluated so far. The purpose of this work is to quantify this effect throughout the stratification process of metals inside the atmosphere. We used our code caratmotion to compute the time-dependent atomic diffusion of four metals (Mg, Ca, Si, and Fe) in the atmosphere of a main-sequence star with an effective temperature of $8\,500$\,K, which is a typical temperature of Ap stars. The results, including ambipolar diffusion of H, are compared to results obtained without this process. Our main result is that ambipolar diffusion must be included in any calculation of atomic diffusion in Ap star atmospheres, at least for stars with $T_ eff 10\,000$\,K. We show that this concerns all metals, even those that are well supported by the radiation field, such as Fe. The crucial role of the stellar mass-loss rate is confirmed; it remains a determining parameter that is constrained, but still free in our calculations. We also present 3D calculations of Ca distributions in magnetic atmospheres. Questioning the interest of systematic searches for stationary solutions (which can often only be reached after a long evolutionary process), we note that remarkable behaviour can occur during the transient phases of the stratification build-up.

In vivo mapping of cellular resolution neuropathology in brain ischemia with diffusion MRI.

Noninvasive mapping of cellular pathology can provide critical diagnostic and prognostic information. Recent advances in diffusion magnetic resonance imaging enabled in vivo examination of tissue microstructures well beyond the imaging resolution. Here, we proposed to use diffusion time-dependent diffusion kurtosis imaging (tDKI) to simultaneously assess cellular morphology and transmembrane permeability in hypoxic-ischemic (HI) brain injury. Through numerical simulations and organoid imaging, we demonstrated the feasibility of capturing effective size and permeability changes using tDKI. In vivo MRI of HI-injured mouse brains detected a shift of the tDKI peak to longer diffusion times, suggesting swelling of the cellular processes. Furthermore, we observed a faster decrease of the tDKI tail, reflecting increased transmembrane permeability associated with up-regulated water exchange or necrosis. Such information, unavailable from a single diffusion time, can predict salvageable tissues. Preliminary applications of tDKI in patients with ischemic stroke suggested increased transmembrane permeability in stroke regions, illustrating tDKI's potential for detecting pathological changes in the clinics.

Time-dependent diffusion MRI-based microstructural mapping for differentiating high-grade serous ovarian cancer from serous borderline ovarian tumor

PurposeTo investigate the value of microstructural characteristics derived from time-dependent diffusion MRI in distinguishing high-grade serous ovarian cancer (HGSOC) from serous borderline ovarian tumor (SBOT) and the associations of immunohistochemical markers with microstructural features. MethodsTotally 34 HGSOC and 12 SBOT cases who received preoperative pelvic MRI were retrospectively included in this study. Two radiologists delineated the tumors to obtain the regions of interest (ROIs). Time-dependent diffusion MRI signals were fitted by the IMPULSED (imaging microstructural parameters using limited spectrally edited diffusion) model, to extract microstructural parameters, including fraction of the intracellular component (fin), cell diameter (d), cellularity and extracellular diffusivity (Dex). Apparent diffusion coefficient (ADC) values were obtained from standard diffusion-weighted imaging (DWI). The parameters of HGSOCs and SBOTs were compared, and the diagnostic performance was evaluated. The associations of microstructural indexes with immunopathological parameters were assessed, including Ki-67, P53, Pax-8, ER and PR. ResultsIn this study, fin, cellularity, Dex and ADC had good diagnostic performance levels in differentiating HGSOC from SBOT, with AUCs of 0.936, 0.909, 0.902 and 0.914, respectively. There were no significant differences in diagnostic performance among these parameters. Spearman analysis revealed in the HGSOC group, cellularity had a significant positive correlation with P53 expression (P = 0.028, r = 0.389) and Dex had a significant positive correlation with Pax-8 expression (P = 0.018, r = 0.415). ICC showed excellent agreement for all parameters. ConclusionTime-dependent diffusion MRI had value in evaluating the microstructures of HGSOC and SBOT and could discriminate between these tumors.

Anomalous diffusion of scaled Brownian tracers.

A model for anomalous transport of tracer particles diffusing in complex media in two dimensions is proposed. The model takes into account the characteristics of persistent motion that an active bath transfers to the tracer; thus, the model proposed here extends active Brownian motion, for which the stochastic dynamics of the orientation of the propelling force is described by scaled Brownian motion (sBm), identified by time-dependent diffusivity of the form D_{β}∝t^{β-1},β>0. If β≠1, sBm is highly nonstationary and suitable to describe such nonequilibrium dynamics induced by complex media. In this paper, we provide analytical calculations and computer simulations to show that genuine anomalous diffusion emerges in the long-time regime, with a time scaling of the mean-squared displacement t^{2-β}, while ballistic transport t^{2}, characteristic of persistent motion, is found in the short-time regime. We also analyze the time dependence of the kurtosis, and the intermediate scattering function of the position distribution, as well as the propulsion autocorrelation function, which defines the effective persistence time.

Microstructural and transport characteristics of triply periodic bicontinuous materials

Three-dimensional (3D) bicontinuous two-phase materials are increasingly gaining interest because of their unique multifunctional characteristics and advancements in techniques to fabricate them. Because of their complex topological and structural properties, it still has been nontrivial to develop explicit microstructure-dependent formulas to predict accurately their physical properties. A primary goal of the present paper is to ascertain various microstructural and transport characteristics of five different models of triply periodic bicontinuous porous materials at a porosity ϕ1=1/2: those in which the two-phase interfaces are the Schwarz P, Schwarz D and Schoen G minimal surfaces as well as two different pore-channel structures. We ascertain their spectral densities, pore-size distribution functions, local volume-fraction variances, and hyperuniformity order metrics and then use this information to estimate certain effective steady-state as well as time-dependent transport properties via closed-form microstructure–property formulas. Specifically, the recently introduced time-dependent diffusion spreadability is determined exactly from the spectral density. Moreover, we accurately estimate the fluid permeability of such porous materials from a closed-form formula that depends on the second moment of the pore-size function and the formation factor, a measure of the tortuosity of the pore space, which is exactly obtained for the three minimal-surface structures. We also rigorously bound the permeability from above using the spectral density. For the five models with identical cubic unit cells, we find that the permeability, inverse of the specific surface, hyperuniformity order metric, pore-size second moment and long-time spreadability behavior are all positively correlated and rank order the structures in exactly the same way. We also conjecture what structures maximize the fluid permeability for arbitrary porosities and show that this conjecture must be true in the extreme porosity limits by identifying the corresponding optimal structures.

Time-dependent gas dynamic diffusion process in shale matrix: model development and numerical analysis

Gas diffusion is a pivotal process during shale gas recovery, which is determined by diffusion coefficient to a large extent. In previous studies, the gas diffusion coefficient is generally assumed as a constant. However, increasing experiments prove that the diffusion coefficient of shale gas is strongly time-dependent. Therefore, to perfect the theory of shale gas diffusion, this paper proposes a time-dependent diffusion model for shale gas, which incorporates time-dependent gas diffusion coefficient, composing of the bulk diffusion coefficient for free gas in organic and inorganic pores, as well as the surface diffusion coefficient for adsorbed gas in organic pores. To validate the accuracy of the new theory, we calibrate the theoretical results against experimental data, and the results show that they have strong correlation, and the time-dependent diffusion model is superior to classical model. Finally, the numerical analysis of gas dynamic diffusion process in shale matrix is conducted. The results show that at the end of diffusion, a large amounts of shale gas remain trapped in the matrix core due to the attenuation of gas diffusion coefficient. In addition, neglecting the time-dependent nature of gas diffusion in shale matrix leads to a significant overestimation of gas production.

Rescaled mode-coupling scheme for the quantitative description of experimentally observed colloid dynamics.

We describe experimentally observed collective dynamics in colloidal suspensions of model hard-sphere particles using a modified mode coupling theory (MCT). This rescaled MCT is capable of describing quantitatively the wave-vector and time-dependent diffusion in these systems. Intermediate scattering functions of liquidlike structured dispersions are determined by means of static and dynamic light-scattering experiments. The structure and short-time dynamics of the systems can be described quantitatively employing a multicomponent Percus-Yevick ansatz for the partial structure factors and an effective, one-component description of hydrodynamic interactions based on the semianalytical δγ expansion. Combined with a recently proposed empirical modification of MCT in which memory functions are calculated using effective structure factors at rescaled number densities, the scheme is able to model the collective dynamics over the entire accessible time and wave-vector range and predicts the volume-fraction-dependence of long-time self-diffusion coefficients and the zero-shear viscosity quantitatively. This highlights the potential of MCT as a practical tool for the quantitative analysis and prediction of experimental observations.