Density Of High Angle Grain Boundaries Research Articles

Crystallographic study on microstructure evolution and mechanical properties of coarse grained heat affected zone of a 500 MPa grade wind power steel

The effect of different welding heat inputs on the microstructure and mechanical properties of the simulated coarse grained heat affected zone (CGHAZ) of high-strength wind power steel with yield strength of 500 MPa has been investigated using scanning electron microscopy (SEM) and electron backscatter diffraction (EBSD). Charpy impact tests have demonstrated that there exists an optimum heat input of ∼20 kJ/cm that allows optimum impact toughness to be obtained for the CGHAZ. It was shown that this is related to the refined bainitic structure and the highest density of high-angle grain boundaries (HAGBs) with misorientation angle of more than 45°. In crystallographic visualization studies, it was shown that the weakest variant selection occurs for the bainite transformation in the optimal heat input, leading to the highest density of HAGBs with each Closed-packet group containing two or three Bain groups and showing a staggered arrangement structure. The contribution that can effectively deflect and prevent crack propagation during impact experiments has to come from the block boundary. However, it was also found that the center segregation induced by C and Mn reduces the low-temperature impact toughness of the core sample before and after simulated welding, and affects the fluctuations of impact toughness and fatigue performance of simulated CGHAZ. Mn segregation can have a genetic effect on the welding heat affected zone, inducing a lower temperature martensitic transformation, which in turn leads to a decrease in low-temperature toughness and fatigue crack arrest performance.

Effect of Pre-Weld Heat Treatment on the Microstructure and Properties of Coarse-Grained Heat-Affected Zone of a Wind Power Steel after Simulated Welding

The effect of pre-weld heat treatment on the microstructure and low-temperature impact toughness of the coarse-grained heat-affected zone (CGHAZ) after simulated welding was systematically investigated through the utilization of scanning electron microscopy (SEM) and electron back-scattering diffraction (EBSD). The Charpy impact test validated the presence of an optimal pre-weld heat treatment condition, resulting in the highest impact toughness observed in the CGHAZ. Three temperatures for pre-weld heat treatment (690, 720 and 750 °C) were used to obtain three different matrices (Steel 1, Steel 2, Steel 3) for simulated welding. The optimal pre-weld heat treatment is 720 °C for 15 min followed by water quench. Microstructure characterization showed that there is an evident microstructure comprising bainite (B) in Steel 1 and Steel 2 after pre-weld heat treatment, while the addition of martensite (M) with the pre-weld heat treatment temperature exceeds Ac1 by almost 60 °C (Steel 3). These differences in microstructures obtained from pre-weld heat treatment influence the refinement of high-temperature austenite during subsequent simulated welding reheating processes, resulting in distinct microstructural characteristics in the CGHAZ. After the optimal pre-weld heat treatment, Steel 2 subjected to single-pass welding thermal simulation demonstrates a refined microstructure characterized by a high density of high-angle grain boundaries (HAGBs) within the CGHAZ, particularly evident in block boundaries. These boundaries effectively prevent the propagation of brittle cracks, thereby enhancing the impact toughness.

Effect of Boron on Microstructures and Low-Temperature Impact Toughness of Medium-Carbon CrMo Alloy Steels with Different Quenching Temperatures

The effect of boron (B) on the microstructures and low-temperature impact toughness of medium-carbon CrMo steel quenched at 870~1050 °C and tempered at 600 °C was studied via Charpy impact testing and microstructure characterizations. The results showed that with an increasing B content from 0 to 50 ppm, the low-temperature impact toughness deteriorated significantly at quenching temperatures (Tq) lower than 950 °C but increased at a higher Tq of 1050 °C. Undissolved M2B particles remained and coarsened during the holding process due to the low Tq, decreasing the critical stress required for crack initiation and deteriorating the impact toughness accordingly. However, this detrimental effect of B could be mitigated by a higher Tq, and the favorable influences on the impact toughness improvement could be attributed to (1) the finer M2B particles formed during quenching effectively pinning the austenite grain boundaries (GBs), leading to a finer block size and a high density of high-angle grain boundaries, which reduced the critical stress for crack initiation; and (2) the fact that the coarsening of M23C6 on the GBs during tempering was slightly suppressed by the segregated B, eventually increasing the energy required for crack propagation. However, the degree of the favorable effect due to B was still lower than the negative effect of a high Tq.

Postweld Heat Treatment on Toughness of Electric- Resistance Welded X70 Steel

The effect of postweld heat treatment (PWHT) on Charpy V-notch impact toughness of a highfrequency electric-resistance welded (HF-ERW) grade X70 pipeline steel is investigated. PWHT thermal cycles are simulated using the Gleeble on the as-welded specimens. Microstructure, along with crystallographic texture and microhardness, is characterized in specimens that are impact tested at –5°C (23°F), –30°C (–22°F), and –45°C (–49°F). The impact toughness values show a wide scatter band and decrease with the increasing peak temperature of the PWHT. For higher peak temperatures, the microstructure of the heat-affected zone (HAZ) gradually changed from equiaxed ferrite to bainitic ferrite. Furthermore, the prior austenite grain size (PAGS) increases with the increasing peak temperature. The density of high-angle grain boundaries decreases, and the fraction of cleavage planes {100} parallel to the impact fracture plane increases for higher peak temperatures of the PWHT.

Unraveling the Mechanism of Toughness Fluctuation in Ultra-High-Strength Casing from the Perspective of Crystallography

The microstructure and impact toughness in an ultra-high-strength casing were investigated, attempting to reveal the reason for toughness fluctuations along the casing in terms of crystallography. The morphological structures at the head of the casing are lath bainite, while those at the end are granular bainite and lath bainite. The head exhibits a higher density of high-angle grain boundaries dominated by block boundaries and a higher impact toughness than the end, showing an inhomogeneous microstructure and causing toughness fluctuations in the casing. The higher density of block boundaries in the head resulted from weaker variant selection because of its higher cooling rate than the end. Therefore, the underlying reason for the toughness fluctuations lies in the varying densities of the block boundaries along the casing triggered by uneven cooling.

Crystallography of transformation products with different cooling rates in low-carbon alloy steel and its effect on low-temperature toughness uniformity of heavy plates

The crystallography of transformation products with different cooling rates has been investigated in a typical low-carbon alloy steel, and its effect on low-temperature impact toughness uniformity in the thickness direction of heavy steel plates has been elucidated. The results show that both the full lath martenite (LM) microstructure transformed at the cooling rate of 45 K s−1 and granular bainite (GB) + lath bainite (LB) + LM microstructure transformed at the cooling rates of 3∼6 K s−1 have higher density of high-angle grain boundaries (HAGBs). The density of HAGBs in the full GB microstructure transformed at the cooling rate of 0.5 K s−1 is the lowest, and that in the LB + LM microstructure transformed at the cooling rates of 15∼22 K s−1 is the middle. In the GB + LB + LM microstructure, the division of austenite by multi-stage transformation reduces the LB and LM transformation temperatures, contributing to a weakening of variant selection and a tendency to close-packed plane grouping. Therefore, the GB + LB + LM microstructure achieves the refinement degree similar to the full LM microstructure, but slightly higher than that of the LB + LM microstructure. For the 60 mm thick heavy steel plate with ultra-fast cooling, the cooling rate decreases from 41 K s−1 at the subsurface to 14 K s−1 at the 1/4-thickness and then to 4 K s−1 at the 1/2-thickness. The full LM microstructure, LB + LM microstructure, and GB + LB + LM microstructure are obtained in sequence, resulting in a more uniform density of HAGBs. Consequently, the relatively uniform impact toughness at 143 K in the thickness direction is achieved with a small difference of ∼30 J.

Variations in microstructure and mechanical properties along thickness direction in a heavy high strength low alloy steel plate

A thermal-mechanical control process (TMCP) followed by tempering at 600 °C was applied to produce 420 MPa grade high strength low alloy (HSLA) steel heavy plates. The microstructure and mechanical properties through the thickness were analyzed using various techniques including optical microscopy (OM), scanning electron microscopy (SEM), transmission electron microscopy (TEM), electron back-scattered diffraction (EBSD), tensile and impact tests. Microscopy analyses showed that a fine-grained structure with an average grain size of 6–10 μm could be obtained by TMCP. After subsequent tempering, the microstructure near the surface of the plate was dominated by bainite, while the microstructure of the core was mainly characterized by aggregated carbides and ferrite. In addition, the density of high-angle grain boundaries (HAGB) on the surface was significantly higher than that in the core. Tensile tests showed a decrease in both yield and tensile strengths from 530.1 and 626.9 MPa on the surface to 384.3 and 496.7 MPa in the core, respectively. The contribution of different strengthening mechanisms to the variations in strengths along the thickness was clarified. In addition, the deformation of ferrite and the initiation of micro-voids during tensile tests were also analyzed.

Local measurement of geometrically necessary dislocation densities and their strengthening effect in ultra-high deformed pearlite

The strength of pearlitic wires can be increased by cold-drawing to a world record level for bulk ductile materials of 7 GPa. The underlying strengthening mechanisms are not fully understood, as the application of usual characterization is challenging because of the small grain sizes and the high degree of deformation. Here we demonstrate that the microstructure of the wires can be directly probed by nano-beam diffraction (NBD) orientation mapping in a transmission electron microscope even after a drawing strain of 6.52. We observe a highly fragmented microstructure with a high density of low-angle grain boundaries (LAGBs) within the ferrite lamella. That makes it difficult to define grain sizes in the ordinary way. We thus calculate an equivalent grain size based on the density of high-angle grain boundaries (HAGBs) per measurement area and an average density of geometrically necessary dislocations (GNDs) calculated from all local misorientation gradients below 15° misorientation. Total strengths calculated from a summed Hall-Petch and Taylor effect of the latter values as well as carbon solid solution hardening are in good agreement with the strengths as measured by tensile tests. Our results show that the GNDs are similarly important as HAGBs in terms of their contributions to the total strength. On this basis, the experimental evidence of the strengthening mechanism with emphasis on GNDs, particularly in ultra-high deformed materials is highlighted. The present results also validate the application of NBD to assessing mechanical properties of other ultra-high deformed materials where mechanical tests often are not feasible.

Hot deformation mechanisms of dual phase high entropy alloys

The microstructure of high entropy alloys can finally be designed via thermomechanical treatments to tune the mechanical properties. This work investigates the modification of the microstructure after treatments at 1100 °C for three hypo-eutectic high entropy alloys. Two phases were indexed according to the BCC and FCC crystal structures using electron backscattered diffraction. Their microstructure is investigated for three hot deformation tests: at a constant strain rate of 0.001s−1, at a strain rate jumps from 0.001s−1 to 1s−1 and from 1s−1 to 0.001s−1. The BCC size and fraction strongly influence the deformation of the FCC matrix. Due to its typical semi-interconnected hypo-eutectic structure, the BCC phase carries the load at the beginning of the deformation. Progressively, the FCC phase deforms to accommodate the plastic strain due to the bending and fragmentation of the BCC phase. Fine particles of the BCC phase are formed within the FCC matrix at high temperatures, and they pin the high-angle grain boundaries formed by continuous dynamic recrystallisation. The fragmentation of the BCC phase occurs faster for thinner eutectic BCC particles, and it is a consequence of I) the formation of boundaries during plastic deformation via dynamic recovery followed by continuous dynamic recrystallisation; II) the movement of phase boundaries consuming the formed boundaries within the BCC phase, fragmenting them. A fine substructure with a high density of high-angle grain boundaries is formed at 1100 °C for the alloy with initial fine BCC eutectic particles.

Microstructure characteristics and mechanical properties of deposited metals with different types of bainite

In this work, three types of bainite microstructure have been gained in the deposited metals by adjusting the Ni content (1 wt %, 2.5 wt %, and 4 wt %). The microstructure feature of different types of bainite has been characterized by optical microscope, scanning electron microscope, transmission electron microscope, and electron backscattering diffraction. Microstructure changed from granular bainite with different morphology to lath bainite with the increase of Ni content. The number of elongated Martensite-Austenite constituents in granular bainite increased with the increasing Ni content from 1 wt % to 2.5 wt % due to the increase in hardenability. The complex orientation in the deposited metal with 4 wt % Ni content attributed to the numerous autocatalytic nucleation occurrence during the bainite transformation. After crystallographic characterization, the bainite variant selection tendency decreased with the bainite type change from granular bainite to lath bainite with autocatalytic nucleation, and the density of high-angle grain boundaries increases obviously. The deposited metal with numerous autocatalytic nucleation has an outstanding match of high strength and excellent low-temperature toughness due to the effective grain-refinement strengthening. The large-sized elongated Martensite-Austenite constituents in the deposited metals with 1 wt % and 2.5 wt % Ni contents deteriorate low-temperature toughness in two aspects, promoting crack nucleation and reducing the crack propagation resistance. Interlaced bainite is the most optimized among different bainite types. This work guides the microstructure design in high-strength deposited metals.

Role of Carbon Content on Microstructure Evolution and Impact Toughness in Coarse-Grained Heat-Affected Zone of High-Strength Steel

The effect of carbon content in the base metals of high-strength steel on the microstructure and impact toughness of simulated welding focusing on a coarse-grained heat-affected zone (CGHAZ) at different heat inputs was systematically investigated by using scanning electron microscopy (SEM) and electron back-scattering diffraction (EBSD). The Charpy impact test confirmed that there was an optimal heat input, which caused the CGHAZ to obtain the highest impact toughness. The optimal heat input is ~20 kJ/cm and remains unchanged with an increase in carbon content from 0.04 to 0.12 wt.%. However, the impact toughness of the CGHAZ decreases with the increase in carbon content at each heat input. Microstructure characterization showed that a CGHAZ with 0.04 wt.% carbon gradually changed from lath bainite (LB) to granular bainite (GB) with an increase in heat input, while it changed from lath martensite (LM) to LB and then to GB for a CGHAZ with 0.12 wt.% carbon. Although the density of high-angle grain boundaries (HAGBs) obtained at 20 kJ/cm in the high-carbon sample is higher than that of the low-carbon sample, its impact toughness is lower, which is related to the parallel structure of the lath bundles and the morphology the austenite penetration.

Carbides Dissolution in 5Cr15MoV Martensitic Stainless Steel and New Insights into Its Effect on Microstructure and Hardness.

The dissolution behavior of carbides in martensitic stainless steel and its effect on microstructure and hardness were investigated by using X-ray diffractometer (XRD) and field emission scanning electron microscopy (FE-SEM) with energy dispersive spectrometer (EDS) and electron backscattering diffraction (EBSD). The results indicated that the microstructure after austenitizing heat treatment and oil quenched consisted of martensite, M23C6 carbides and retained austenite. The temperature and particle size had great influence on the dissolution of carbides. The EBSD results showed that the twin-related variant pair V1/V2 governed the phase transformation. Meanwhile, the density of high-angle grain boundaries (HAGBs) increased with the increase of austenitizing temperature from 950 to 1150 °C. The hardness test results indicated that the hardness first increased and then decreased with the increase of the austenitizing temperature, and the peak appeared at 1050 °C with a Rockwell hardness value of 59.8 HRC. A model was established to quantitatively explain the contribution of different microstructures to hardness. The contribution to hardness came mainly from martensite. The retained austenite had a negative effect on hardness when the volume fraction was more than 10%. In contrast, carbides contributed less to hardness due to their small content.

Bainite nucleation mechanism and mechanical properties in low carbon bainite deposited metals with different nickel additions

Exploring the transformation mechanism of bainite is necessary to achieve the matching of high-strength and accepted damage tolerance for low carbon bainite deposited metals. In this work, the effect of Ni content on the bainite transformation mechanism and mechanical properties has been investigated systematically with Ni contents change from 4 wt. % to 9 wt. %. Scanning electron microscope, transmission electron microscope, and electron backscatter diffraction were used for the characterization of the microstructure. Results indicated that the nucleation method of bainite ferrite (BF) laths changed from autocatalytic nucleation to classical nucleation when the Ni content exceeded 5.5%. The deposited metals with BF laths nucleated by autocatalysis had excellent combinations of strength and toughness while the toughness severely deteriorated as the classical nucleation replaced the autocatalytic nucleation. The presence of autocatalytic nucleation resulted in the refinement of microstructure, and the density of high-angle grain boundaries had been improved effectively. In particular, apparent variant selection tendency in microstructure occurred accompanied by the disappearance of autocatalytic nucleation. Moreover, the strong variant selection tendency resulted in numerous identical variants growing next to each other and coarsening the microstructure. In particular, the key factor for the generating of autocatalytic nucleation was that the lower interfacial energy for the new BF laths nucleated through autocatalytic nucleation, and resulted in a lower nucleation activation energy. Therefore, autocatalytic nucleation tends to occur in the deposited metals with low bainite transformation speed due to enough time and space for new BF laths nucleates at existing BF lath.

Identification of the critical microstructural parameters on the corrosion behaviour of commercially pure aluminium alloy

The influence of microstructure on the corrosion behaviour of commercially pure Al alloy in a chloride-containing sulphate solution was systematically investigated. An isothermal annealing at 300 °C led to an increase in the density of high-angle grain boundaries (HAGBs) and decrease in number density of intermetallic particles (IMPs). The grain size did not affect the pitting potential, but the corrosion current decreased with increase in the density of HAGBs. IMPs acted as pit initiation sites and influenced the pitting potential. After pit initiation, pit propagation occurred along HAGBs; pit width and depth were strongly dependent on the density of HAGBs.

Influence of cooling conditions on the microstructure and impact strength of welded joints made by automatic submerged multi-arc welding

The aim of the work was to study the effect of cooling rate on the formation of the structure and impact strength of a welded joint of 0,07C-1,6Mn-0,2Cr-Nb pipe steel. The cooling rate w8/5 was varied from 0.9 to 12.4 °/s in the temperature range of 800–500 °/s by changing the heat input of welding, the groove width, and the temperature of 22 mm thick plates welded on a laboratory welding stand. The conducted studies confirm the previously obtained results on the weldability of pipe steels, that a decrease in the heat input of welding and, as a result, an increase in the cooling rate of the w8/5 welded joint is accompanied by an increase in impact toughness. For the studied steel, a significant increase in the impact toughness of KCV-40 (more than 2.5 times) occurs in the range of cooling rates w8/5 from 0.9 to 7.3 °/s and is typical not only for tests along the fusion line, but also in the center of the seam weld. It is shown that with an increase in the cooling rate of the welded joint in the structure of the weld metal, the maximum length and width decrease, and, at the same time, the proportion of grain boundary ferrite crystallites. At high cooling rates, the matrix of the weld structure consists entirely of acicular ferrite, which is favorable in terms of increasing the density of high-angle grain boundaries, toughness, and cold resistance. In the heat-affected zone, with an increase in the cooling rate, the maximum and average grain sizes decrease; grain-boundary and Widmanstatt ferrite is eliminated; a completely bainite structure is formed and the proportion of lath bainite increases, which also leads to an increase in the density of high-angle boundaries and contributes to an increase in impact strength. With an increase in the cooling rate, the proportion of the MA-constituents in the structure of the seam and the HAZ are increases. It is assumed that an increase in the cooling rate, simultaneously with a favorable effect on the structure of the weld and the weld pool, reduces the time for the processes occurring in the weld pool, which can be an unfavorable factor in terms of the toughness of the weld.

Titania coating formation on hydrostatically extruded pure titanium by micro-arc oxidation method

• Ti grade 4 after 3-pass HE and RS presents strong microstructure anisotropy. • MAO coating is uniform on side walls and top surface of extruded rod. • Na 2 HPO 4 electrolyte produced coating of mixed amorphous and crystalline nature. • Large HAGB density allowed to form continuous amorphous layer close to substrate. • This layer eliminates effect of anisotropic substrate on coating microstructure. In this work, the microstructure of titania coating fabricated on the surface of hydrostatically extruded titanium grade 4 with the use of the micro-arc oxidation method was studied. The surface topography and microstructure investigations performed with atomic force microscopy and scanning and transmission electron microscopy revealed that, by using an Na 2 HPO 4 electrolyte, a well-adherent porous coating is produced on the top surface and side walls of the extruded rod. The distribution of chemical elements was analyzed by using energy dispersive X-ray spectroscopy. The chemical elements dissolved in the electrolyte (Na, P and O) incorporated into the coating. Sodium locates preferentially in the outer part of the coating, while phosphorus and oxygen are distributed throughout the whole coating. The most relevant finding shows that a grain refinement caused by a hydrostatic extrusion provoked an increase in density of high-angle grain boundaries (HAGB), which in turn secured the formation of a continuous amorphous layer close to the substrate. The presence of this layer compensates for the effect of anisotropic substrate, producing a comparable and homogenous microstructure with a large number of micropores.

The Relationship Between Microstructure and Hardness in the Heat-Affected Zone of Line Pipe Steels

Welding is an essential fabrication step during the construction of pipelines using high-strength low-alloy steels. The base metal experiences a rapid thermal excursion in the heat-affected zone, resulting in modification of the microstructure and mechanical properties. In particular, the coarse-grain heat-affected zone closest to the weld pool is of significance for the integrity of pipelines. In the present work, the microstructure and hardness were quantified from laboratory-simulated samples, obtained at different cooling rates for 12 steels, where their chemistry was systematically varied. Electron backscattered diffraction was used to quantify the bainitic microstructures based on the density of high-angle grain boundaries (HAGBs). It was observed that an increase of the cooling rate results in (1) a reduction of the transformation start temperature, (2) a larger density of HAGBs, and (3) an increase in hardness. A Hall–Petch-type relationship is proposed to link the hardness with the HAGB density.

The Effects of Prior Austenite Grain Refinement on Strength and Toughness of High-Strength Low-Alloy Steel

The effects of prior austenite grain (PAG) refinement on the mechanical properties of bainitic/martensitic steels not only come from itself, but also have more complex effects by affecting the substructure formed by coherent transformation. In this study, the samples of a low-alloy steel were water quenched from different austenitizing temperatures and the bainitic/martensitic microstructures with different PAG sizes were obtained. Electron back-scattered diffraction was used to characterize the microstructure and different types of boundaries were identified and quantitatively analyzed. The tensile tests and series temperature Charpy impact tests of different heat treatment were also carried out and comprehensively analyzed with microstructure characterization works. The results show that the uniform refinement of prior austenite grain can increases the density of packet boundary and block boundary, which leads to microstructure refinement with higher density of high-angle grain boundaries with misorientation >45°. The contribution of this microstructure refinement to toughness is significant, but its contributions to strength and elongation are relatively limited. Compared to uniform refined PAG, if the PAGs are mixed crystal, the density of block boundary will be reduced, which leads to a lower density of the high-angle boundary with misorientation >45° and the positive effects of microstructure refinement on toughness improvement are weakened. The observation of fracture surface of impact specimens indicates that refining the PAG can delay the tendency of brittle fracture with the decrease in test temperature, and even in the case of brittle fracture, the cleavage facet of the fracture surface is relatively smaller. This result also verifies that PAG refinement can effectively improve toughness by inhibiting cleavage fracture.

Effect of matrix carbon content and lath martensite microstructures on the tempered precipitates and impact toughness of a medium-carbon low-alloy steel

Different austenitizing temperatures were used to obtain medium-carbon low-alloy (MCLA) steels with different initial states of tempering, including lath martensite microstructures and matrix carbon contents. The effects of these factors on the tempered carbides and impact toughness of the MCLA steels were investigated via optical microscopy (OM), scanning electron microscopy (SEM), electron backscatter diffraction (EBSD), X-ray diffraction (XRD) and transmission electron microscopy (TEM). The results showed that the matrix carbon content and lath martensite substructure sizes could be effectively controlled by the austenitizing temperature. The morphologies of the tempered carbides precipitated at the boundaries were controlled by both the matrix carbon content and the lath martensite substructure sizes. Higher matrix carbon content and boundary densities increased the nucleation rate and promoted the formation of spherical carbides. The spheroidization of carbides was conducive to improving the impact toughness. However, the decrease in the density of high-angle grain boundaries and the appearance of long strip-shaped carbides drastically reduced the impact toughness.

Correlation of creep fracture lifetime with microstructure evolution and cavity behaviors in G115 martensitic heat-resistant steel

In order to deepen the understanding towards the creep rupture lifetime of G115 martensitic heat-resistant steel, a series of creep tests using 130–200MPa loading stresses at 923K were carried out to investigate the as-crept microstructural characteristics and cavity behaviors. The significant microstructure deformation and a dramatic increase in the areal density of high-angle grain boundaries occur when the stress loading is over 160MPa. Apart from the strain concentration at elongated grain boundaries and refined grains, the elevated stress also leads to the appearance of a strong α-fiber texture with {001}<110>//RD. The nucleation and growth of cavities can be effectively accelerated by the stress elevation, and ductile fracture is determined as the dominate mechanism within the test stresses. However, a transition towards brittle intergranular fracture is observed due to the rapid coarsening of Laves phase. Furthermore, a good prediction for creep lifetimes of G115 steel has been obtained based on the constrained diffusional cavity growth model.