Coarsening Exponent Research Articles

Coarsening kinetics of alloy-doped nanoporous metals

Due to their high surface-to-volume ratios, nanoporous metals are being explored for a range of catalytic and structural applications. However, these materials have thermodynamically unstable morphologies and degrade via coarsening at elevated temperatures. One potential mitigation strategy is to introduce atomic species that inhibit diffusional transport, but there is limited mechanistic understanding. To begin addressing this knowledge gap, the impact of a slow-diffusing dopant on the coarsening behavior of a nanoporous metal is studied using kinetic Monte Carlo simulations. The simulations were analyzed using reaction models and isoconversional analyses to extract constitutive coarsening laws, which confirm a coarsening exponent associated with classical surface diffusion. In addition, a rate equation is derived for the role of alloying dopants. It is found that only a few atomic percent is needed to stymie coarsening over experimentally relevant timescales, which has broad implications for the future design and tailoring of these materials.

Hyperuniformity in phase ordering: the roles of activity, noise, and non-constant mobility

Hyperuniformity emerges generically in the coarsening regime of phase-separating fluids. Numerical studies of active and passive systems have shown that the structure factor S(q) behaves as q ς for q → 0, with hyperuniformity exponent ς = 4. For passive systems, this result was explained in 1991 by a qualitative scaling analysis of Tomita, exploiting isotropy at scales much larger than the coarsening length. Here we reconsider and extend Tomita’s argument to address cases of active phase separation and of non-constant mobility, again finding ς = 4. We further show that dynamical noise of variance D creates a transient ς = 2 regime for q^≪q^∗∼Dt[1−(d+2)ν]/2 , crossing over to ς = 4 at larger q^ . Here, ν is the coarsening exponent for the domain size ℓ , such that ℓ(t)∼tν , and q^∝qℓ is the rescaled wavenumber. In diffusive coarsening ν=1/3 , so the rescaled crossover wavevector q^∗ vanishes at large times when d⩾2 . The slowness of this decay suggests a natural explanation for experiments that observe a long-lived ς = 2 scaling in phase-separating active fluids (where noise is typically large). Conversely, in d = 1, we demonstrate that with noise the ς = 2 regime survives as t→∞ , with q^∗∼D5/6 . (The structure factor is not then determined by the zero-temperature fixed point.) We confirm our analytical predictions by numerical simulations of continuum theories for active and passive phase separation in the deterministic case and of Model B for the stochastic case. We also compare them with related findings for a system near an absorbing-state transition rather than undergoing phase separation. A central role is played throughout by the presence or absence of a conservation law for the centre of mass position R of the order parameter field.

Slow coarsening of tetragonal zirconia nanocrystals in a phase-separated sodium borosilicate glass

A Zr4+-containing sodium borosilicate glass (ZNBS) and a zirconium-free reference glass (NBS) were prepared by melt quenching. Thermal analysis revealed two glass transitions, indicating that both glasses exhibit phase separation into boron- and silica-rich domain structures already after quenching to room temperature. Electron micrographs showed that annealing of NBS and ZNBS glasses initiated the coarsening of these domains and the precipitation of metastable t-ZrO2 in the boron-rich matrix of the ZNBS glass. The coarsening kinetics of the silica-rich domains followed the theoretical predictions (power law exponent = 1/3), while the coarsening of the t-ZrO2 crystallites was much slower. The time evolution of the average size of t-ZrO2 nanocrystallites from small angle X-ray scattering (SAXS) and high-temperature X-ray diffraction (HTXRD) experiments revealed a coarsening exponent of ∼1/6. This slow coarsening of t-ZrO2 is assumed to be confined by the domain structure of the primary phase separation.

Dealloying kinetic and catalytic applications of hierarchical porous copper via hot-dip galvanization and subsequent vapor phase dealloying

Hierarchical porous copper (HPC) was synthesized using a Cu-Zn liquid phase alloyed on a Cu tube followed by vapor pressure dealloying. This study demonstrates that HPC with ligament sizes of 0.61–1.97 µm and residual Zn contents of 29–2 at% can be synthesized with a dealloying time of 0.5–30 min and temperature of 773–973 K. The coarsening exponent of 4.099 suggested that the ligament formation and coarsening are attributable to surface diffusion. At the same time, the activation energy of 0.29 eV further confirmed that the Cu atoms diffused on the Cu surface dominated the coarsening of Cu ligaments in this vapor phase dealloying system. The double-layer capacitance of the HPC tube is 34 times higher than that of the electropolished Cu tube. Further, the current density of the CO2 reduction reaction for HPC is two times that of the electropolished Cu tube. With the interaction of Cu and Zn and the use of HPC as an electrocatalyst, the CO2 reduction products transformed from formic acid to carbon monoxide and ethanol.

On stabilizing an α/α′/α″ microstructure in ferritic superalloys

Ferritic superalloys with an α/α′ (A2/B2) microstructure constitute an auspicious materials system for high-temperature application and could help satisfy the need for more sustainable creep resistance materials. An exciting way to improve their creep resistance further through alloy development and microstructure engineering is introducing a hierarchical network consisting of α, α′, and α″ (L2 1 ). This work investigates the formation and evolution of this microstructure in Fe-Al-Ni-Co(Cr-Mo-Ti-Cu) alloys. It has been identified that the substitution of Fe by Cr leads to the stabilization of γ (fcc, A1) at intermediate temperatures. This amount of γ was found to decrease in the presence of Mo, and the transformation is entirely avoidable by carefully adjusting the α - and γ-stabilizing elements. The observations in quinary and senary alloys led to the development of two promising hierarchical α/α′/α″ alloys with additions of Co and Cu. Both solutes are enriched in the precipitates and are expected to act as strengthening elements. Microstructural analysis of the particle size evolution in the range of 800–950 °C indicates that the precipitates undergo a non-classical ripening process at early stages with coarsening exponent deviating from the t 1/3 -relationship predicted by the Lifshitz-Slyozov-Wagner (LSW) theory for multicomponent alloys. Furthermore, after furnace cooling from 950 °C, a pronounced reprecipitation phenomenon was found to occur not only in the matrix but also inside the primary precipitates. • Two promising bcc-derived α/α′/α″ superalloys (A2/B2/L2 1 ) were developed. • Their spinodal network evolves to isolated α′/α″ particles in an α-matrix. • Adding Cr by Fe in the Fe-Al-Ni-Co system was found to stabilize the fcc γ-phase. • In the α″ containing alloys, γ formation could be avoided successfully. • Various phase transformations allow designing a large variety of microstructures.

Thin film nanostructuring at oblique angles by substrate patterning

It is demonstrated that, besides classical nanocolumnar arrays, the oblique angle geometry induces the growth of singular structures in the nanoscale when using wisely designed patterned substrates. Well-ordered array of crosses, cylindrical nanorods or hole structures arranged in square or hexagonal regular geometries are reported as examples, among others. The fundamental framework connecting substrate topography and film growth at oblique angles is presented, allowing the use of substrate patterning as a feasible thin film nanostructuring technique. A systematic analysis of the growth of TiO2 thin films on 4 different lithographic patterned substrates in 4 different scale lengths is also presented. A first conclusion is the existence of a height-based selective growth in the initial stages of the deposition, by which the film preferentially develops on top of the tallest substrate features. This behavior is maintained until the film reaches a critical thickness, the so-called Oblivion Thickness, above which the film topography becomes gradually independent of the substrate features. A general formula relating the spatial features of the pattern, the coarsening exponent and the Oblivion Thickness has been deduced.

Step barrier effects during early stages of the kinetic roughening of fcc(111) surfaces

The influence of the additional Ehrlich–Schwoebel step barrier and temperature on the early stages of kinetic roughening and mound formation of fcc(111) surfaces is studied by means of kinetic Monte Carlo simulations. Increasing the Ehrlich–Schwoebel barrier, the growth mode develops from nearly layer-by-layer growth to statistical (Poisson) growth mode with the formation of wedding-cake-like shaped mounds. The evolution of the growth morphology is characterized by scaling laws with effective critical exponents. On the one hand, coarsening for growth without step barrier follows exponents neff=0.35 and βeff=0.20 for the characteristic lateral distance (coarsening exponent) and the rms roughness, respectively. On the other hand, coarsening is strongly suppressed for large step barriers (neff=0.05, βeff=0.52, Poisson growth) in agreement with different experimental results, e.g., for the formation of mounds during the homoepitaxy of Ag(111) and Pt(111). The lateral roughness on short distances is governed by the low roughness exponent α=0.58 for all growth conditions independently of the growth mode.

Co-strengthening of dislocations and precipitates in alumina-forming austenitic steel with cold rolling followed by aging

The microstructural evolutions of the alumina-forming austenitic (AFA) steel subjected to different cold-rolled reductions and then aging at 700 °C were investigated by multiple characterization methods, and the mechanical properties were also examined. The results show that the increased dislocations (especially, edge dislocations) after cold rolling can effectively enhance the volume fraction and refine the particle size of precipitates during initial aging and subsequently accelerate the coarsening rate of precipitates in the AFA steel, resulting in a high coarsening exponent. The coarsening rate of B2–NiAl phase is higher than that of Laves phase. Excellent mechanical properties with a combination of yield strength (1152 ± 32 MPa) and elongation (44.8 ± 4.1%) of cold rolled AFA steel after aging for 2 h can be obtained due to co-strengthening contribution derived from dislocations and precipitates. With the duration of aging, the dislocation strengthening constantly decreases, the effect of precipitation strengthening is contrarily larger than that of dislocations.

Scaling laws for step bunching on vicinal surfaces: Role of the dynamical and chemical effects.

We study the evolution of step bunches on vicinal surfaces using a thermodynamically consistent step-flow model. By accounting for the dynamics of adatom diffusion on terraces and attachment-detachment at steps (referred to collectively as the dynamical effect), this model circumvents the quasistatic approximation that prevails in the literature. Furthermore, it generalizes the expression of the step chemical potential by incorporating the necessary coupling between the diffusion fields on adjacent terraces (referred to as the chemical effect). Having previously shown that these dynamical and chemical effects can explain the onset of step bunching without recourse to the inverse Ehrlich-Schwoebel (iES) barrier or other extraneous mechanisms, we are here interested in the evolution of step bunches beyond the linear-stability regime. In particular, the numerical resolution of the step-flow free boundary problem yields a robust power-law coarsening of the surface profile, with the bunch height growing in time as H∼t^{1/2} and the minimal interstep distance as a function of the number of steps in the bunch cell obeying ℓ_{min}∼N^{-2/3}. Although these exponents have previously been reported, the novelty of the present approach is that these scaling laws are obtained in the absence of an iES barrier or adatom electromigration. In order to validate our simulations, we take the continuum limit of the discrete step-flow system via Taylor expansions with respect to the terrace size, leading to a novel nonlinear evolution equation for the surface height. We investigate the existence of self-similar solutions of this equation and confirm the 1/2 coarsening exponent obtained numerically for H. We highlight the influence of the combined dynamical-chemical effect and show that it can be interpreted as an effective iES barrier in the setting of the standard Burton-Cabrera-Frank theory. Finally, we use a Padé approximant to derive an analytical expression for the velocity of steadily moving step bunches and compare it to numerical simulations.

Control of Phase Morphology of Binary Polymer Grafted Nanoparticle Blend Films via Direct Immersion Annealing.

While the phase separation of binary mixtures of chemically different polymer-grafted nanoparticles (PGNPs) is observed to superficially resemble conventional polymer blends, the presence of a "soft" polymer-grafted layer on the inorganic core of these nanoparticles qualitatively alters the phase separation kinetics of these "nanoblends" from the typical pattern of behavior seen in polymer blends and other simple fluids. We investigate this system using a direct immersion annealing method (DIA) that allows for a facile tuning of the PGNPs phase boundary, phase separation kinetics, and the ultimate scale of phase separation after a sufficient "aging" time. In particular, by switching the DIA solvent composition from a selective one (which increases the interaction parameter according to Timmerman's rule) to an overall good solvent for both PGNP components, we can achieve rapid switchability between phase-separated and homogeneous states. Despite a relatively low and non-classical power-law coarsening exponent, the overall phase separation process is completed on a time scale on the order of a few minutes. Moreover, the roughness of the PGNP blend film saturates at a scale that is proportional to the in-plane phase separation pattern scale, as observed in previous blend and block copolymer film studies. The relatively low magnitude of the coarsening exponent n is attributed to a suppression of hydrodynamic interactions between the PGNPs. The DIA method provides a significant opportunity to control the phase separation morphology of PGNP blends by solution processing, and this method is expected to be quite useful in creating advanced materials.

Improvement of γ' coarsening model in high γ' volume fraction Ni-base superalloys containing different Ta/W ratio

The current research was carried out to improve the growth model of γ' precipitates in Ni-base superalloys with the high volume fraction of γ' precipitates and large lattice misfit (δ) of γ/γ'. The alloys were designed based on the Ni-base superalloys of CM247 LC with various Ta/W weight ratios from 0.15 to 2.3. Many studies performed so far about the kinetics of the γ' coarsening have been related to the NiCrAl or NiTiAl systems which incorporate insignificant lattice misfit of γ/γ'. However, in the current work, the coarsening kinetics of the γ' precipitates with the composition of Ni3(Al,Ta) was investigated in the superalloys containing high contents of Ta and high δ. For this purpose, the designed superalloys were exposed to long-term aging at 1000 °C for 200 h after being cast by Vacuum Induction Melting (VIM) followed by direct aging heat treatment. Microstructural observations were performed by an SEM, while lattice parameters of the γ and γ' phases were determined using XRD. The results indicate that cubic coarsening law is not valid for the γ' precipitates in high volume fraction condition, while the coarsening exponent (n) in the superalloys having a high volume fraction of γ' and large δ is a non-integer exponent satisfying 1.3 < n < 1.6 criterion. Meanwhile, under long-term aging at 1000 °C, the minimum coarsening rate (k) occurs for the γ' precipitates with the Ta/W ratio of about 0.5, because a combined effect of the maximum values of δ and high ordering degree in the γ' phase is recorded at this ratio. In fact, the elastic coherency strains in the γ matrix around the γ' precipitates increase with increasing the absolute value of δ and consequently the γ' coarsening rate is hindered. Moreover, the variations in the coarsening rate coefficient (k) in terms of δ revealed that the γ/γ' interface is coherent in the range of 0.1%< δ < 0.3%, while this interface becomes incoherent over 0.3%.

Static coarsening behavior of equiaxed α phase in Ti−8Al−1Mo−1V alloy

To get a deep understanding of the evolution behavior of equiaxed α phase in Ti−8Al−1Mo−1V alloy during annealing, its static coarsening mechanism was studied based on the calculations of coarsening exponent n and the activation energy for α boundary migration. The result of n=6 indicates a special coarsening mechanism of equiaxed α phase. The activation energy for α boundary migration is calculated to be 138 kJ/mol, which is close to the activation energy for grain growth of pure α-Ti. It is revealed that the coarsening of equiaxed α can be mainly attributed to the self-diffusion of Ti atoms across the α/α boundaries. Based on the experimental findings, a static coarsening kinetics model of equiaxed α grains in the α+β field is established. At last, the effects of the coarsening behavior of equiaxed α on tensile properties were studied.

Chromatin Mechanics Dictates Subdiffusion and Coarsening Dynamics of Embedded Condensates

DNA is organized into chromatin, a complex polymeric material which stores information and controls gene expression. An emerging mechanism for biological organization, particularly within the crowded nucleus, is biomolecular phase separation into condensed droplets of protein and nucleic acids. However, the way in which chromatin impacts the dynamics of phase separation and condensate formation is poorly understood. Here, we utilize a powerful optogenetic strategy to examine the interplay of droplet coarsening with the surrounding viscoelastic chromatin network. We demonstrate that droplet growth dynamics are directly inhibited by the chromatin-dense environment, which gives rise to an anomalously slow coarsening exponent, beta∼0.12, contrasting with the classical prediction of beta=1/3. Using scaling arguments and simulations, we show how this arrested growth can arise due to subdiffusion of individual condensates, predicting beta∼alpha/3, where alpha is the diffusive exponent. Tracking the fluctuating motion of condensates within chromatin reveals a subdiffusive exponent, alpha∼0.5, which explains the anomalous coarsening behavior and is also consistent with Rouse-like dynamics arising from the entangled chromatin. Our findings have implications for the biophysical regulation of the size and shape of biomolecular condensates and suggest that condensate emulsions can be used to probe the viscoelastic mechanical environment within living cells.

Chromatin mechanics dictates subdiffusion and coarsening dynamics of embedded condensates

DNA is organized into chromatin, a complex polymeric material that stores information and controls gene expression. An emerging mechanism for biological organization, particularly within the crowded nucleus, is biomolecular phase separation into condensed droplets of protein and nucleic acids. However, the way in which chromatin impacts the dynamics of phase separation and condensate formation is poorly understood. Here we utilize a powerful optogenetic strategy to examine the interplay of droplet coarsening with the surrounding viscoelastic chromatin network. We demonstrate that droplet growth dynamics are directly inhibited by the chromatin-dense environment, which gives rise to an anomalously slow coarsening exponent, β ≈ 0.12, contrasting with the classical prediction of β = 1/3. Using scaling arguments and simulations, we show how this arrested growth can arise due to subdiffusion of individual condensates, predicting β ≈ α/3, where α is the diffusive exponent. Tracking the fluctuating motion of condensates within chromatin reveals a subdiffusive exponent, α ≈ 0.5, which explains the anomalous coarsening behaviour and is also consistent with Rouse-like dynamics arising from the entangled chromatin. Our findings have implications for the biophysical regulation of the size and shape of biomolecular condensates and suggest that condensate emulsions can be used to probe the viscoelastic mechanical environment within living cells. Biomolecules in the cell nucleus form condensates at a rate slower than that predicted by the theory of droplet growth. Experiments on living cells attribute this anomalous coarsening behaviour to subdiffusive dynamics in the crowded nucleus.

Coarsening kinetics of γ′ precipitates in a Re-containing Ni-based single crystal superalloy during long-term aging

The morphological evolution and coarsening kinetics of γ′ precipitates in a Re-containing Ni-based single crystal superalloy were investigated during isothermal aging at 900, 950 and 1000 °C. After heat treatment, well-defined cuboidal γ′ precipitates with low misfit was obtained within the experimental alloy. Then coarsening rate constants and particle size distribution (PSD) of γ′ phases were calculated and specified based on the measured precipitate sizes for varying periods of aging times from 100 to 2000 h. After aging for 2000 h, γ′ precipitates maintained cubical shape at 900 °C, while exhibited sphere at 950 and 1000 °C. Coarsening models based on diffusion-controlled process with a functional relationship of r3 vs. t (classic Lifshitz-Slyozov-Wagner coarsening model) and interface-controlled model with a function of r2 vs. t (trans-interface diffusion-controlled coarsening model) were investigated to fit between the experimental results and theoretical analysis. It was found that Re as the slowest diffusing solute in the alloy constituted the rate-limited step for coarsening based on LSW model, while the process limiting coarsening as governed by an interface diffusion process could possibly be related to the Al diffusion through the γ/γ′ interface. The PSDs and coarsening exponent were discussed by comparing the experimental data with predictions of LSW and TIDC models. Finally, coarsening mechanism could be divided into four regimes: (i) coarsening by diffusion-controlled; (ii) coarsening by diffusion and interface co-controlled; (iii) coarsening by interface-controlled; (iv) coarsening by interface-controlled accompanied with γ′ coalescence.

A novel solid-state dealloying method to prepare ultrafine ligament nanoporous Ti

Nanoporous Titanium (NPT) is an attractive material with high corrosion resistance, good biocompatibility, and large surface area for many new functional applications. The solid dealloying reaction was explored using Ti–Cu and Mg diffusion couples to prepare NPT powders. The dealloying temperature in this novel method was much lower than that in the traditional liquid metal dealloying. In addition, the NPT with finer ligament width below 200 nm was obtained when the temperature was lower than 485 °C. A diffusion growth model was applied to explain the quantitative relationship between the morphology and various dealloying parameters. The coarsening exponent and activation energy were also estimated to reveal the difference between dealloying in Mg melts and Mg solids. The quantity as well as the formation ability of the Mg–Cu liquid phase played a crucial role. The NPT with smaller ligament size and larger surface area exhibits higher electrochemical capacitance performances.

A three-dimensional Monte Carlo model for coarsening kinetics of the bi-continuous system via surface diffusion and its application to nanoporous gold

The coarsening of the two-phase, bi-continuous microstructure via surface diffusion is simulated using a three-dimensional Monte Carlo (MC) model. The coarsening exponent obtained from simulations initially decreases and then saturates at ~4 as the temperature increases. Kinetic analysis on nanoporous gold (np-Au) samples, annealed at 450 and 600 °C, confirms the prediction of the model. The coarsening probability ratio of np-Au between two annealing temperatures, calculated from the reported activation energy for coarsening of np-Au, is comparable to that of the model, signifying that the simulation successfully mimics the kinetic evolution of the thermally coarsened np-Au via surface diffusion.

Bidirectional universal dynamics in a spinor Bose gas close to a nonthermal fixed point

We numerically study the universal scaling dynamics of an isolated one-dimensional ferromagnetic spin-1 Bose gas. Preparing the system in a far-from-equilibrium initial state, simultaneous coarsening and refining is found to enable and characterize the approach to a non-thermal fixed point. A macroscopic length scale which scales in time according to $L_{\Lambda}(t)\sim t^{\, \beta}$, with $\beta\simeq 1/4$, quantifies the coarsening of the size of spin textures. At the same time kink-like defects populating these textures undergo a refining process measured by a shrinking microscopic length scale $L_{\lambda}\sim t^{\, \beta'}$, with $\beta'\simeq-0.17$. The combination of these scaling evolutions enables particle and energy conservation in the isolated system and constitutes a bi-directional transport in momentum space. The value of the coarsening exponent $\beta$ suggests the dynamics to belong to the universality class of diffusive coarsening of the one-dimensional XY-model. However, the universal momentum distribution function exhibiting non-linear transport marks the distinction between diffusive coarsening and the approach of a non-thermal fixed point in the isolated system considered here. This underlines the importance of the universal scaling function in classifying non-thermal fixed points. Present-day experiments with quantum gases are expected to have access to the predicted bi-directional scaling.

Phase field simulation of microstructural evolution and thermomigration-induced phase segregation in Cu/Sn58Bi/Cu interconnects under isothermal aging and temperature gradient

Sn-Bi based alloys are widely used as thermal interfacial materials, solders in electronic packaging, thermal fuse and low temperature resistant performance electronic components. However, the inherent tendency of microstructure coarsening and phase segregation in Sn-Bi alloys under thermal loading can easily lead to the inhomogeneous mechanical and thermal response behavior, which can promote crack initiation and expansion near the interface between two major phases (i.e., Bi-rich and Sn-rich phases), and this has brought about serious reliability concern for the above applications and thus attracted increasing attention in recent years. In this study, a phase field model is developed to simulate the microstructural evolution and phase segregation behavior of the eutectic Sn58Bi solder in a line-type Cu/Sn58Bi/Cu interconnect under the condition of isothermal aging and temperature gradient respectively. Results show that the Bi-rich phase and Sn-rich phase distribute inhomogeneously in the Sn58Bi alloy matrix of the solder interconnect during thermal aging; the large size Bi-rich phase particles grow up at the expense of the small size ones, and the value of coarsening exponent n is 0.17. Under temperature gradient, Bi atoms migrate along the direction of the heat flux, and consequently a Bi-rich phase segregation layer forms on the cold end, while a Sn-rich phase area is left on the hot end. The relationship between the temperature gradient distribution and the microstructural characteristics is well revealed. Moreover, the phase segregation under temperature gradient induces the decrease of thermal conductivity, which in turn deteriorates the heat transfer performance of solder interconnects. In addition, comparing the coarsening behavior of the Bi-rich phase under temperature gradient with that during isothermal aging, it is clear that the temperature gradient leads to faster coarsening of the Bi-rich phase, and there is a linear relationship between the mean equivalent radius of the Bi-rich phase and nondimensional time. Finally, the study of thermomigration kinetics of the Bi-rich phase shows that the thickness of the Bi-rich phase segregation layer increases almost linearly with nondimensional time, which is consistent with the theoretical analysis.

Gaining new insights into nanoporous gold by mining and analysis of published images

One way of expediting materials development is to decrease the need for new experiments by making greater use of published literature. Here, we use data mining and automated image analysis to gather new insights on nanoporous gold (NPG) without conducting additional experiments or simulations. NPG is a three-dimensional porous network that has found applications in catalysis, sensing, and actuation. We assemble and analyze published images from among thousands of publications on NPG. These images allow us to infer a quantitative description of NPG coarsening as a function of time and temperature, including the coarsening exponent and activation energy. They also demonstrate that relative density and ligament size in NPG are not correlated, indicating that these microstructure features are independently tunable. Our investigation leads us to propose improved reporting guidelines that will enhance the utility of future publications in the field of dealloyed materials.