R. L. Coble

A Model for Boundary Diffusion Controlled Creep in Polycrystalline Materials

The creep rate (ė) predicted by the boundary diffusion (Db) model is ė≃150σDbWΩ/(GS)3kT, where σ is the stress, W is the boundary width, (GS) is the average grain size, and Ω is vacancy volume. The stress dependence is the same as the lattice diffusion model, given by C. Herring, while the grain size dependence and the numerical constant are greater for boundary diffusion. Discussion of the mechanism of creep in polycrystalline alumina is based on the differences between the lattice and boundary diffusion models.

Sintering Crystalline Solids. I. Intermediate and Final State Diffusion Models

Photomicrographs of pore and grain boundary structures in sintered powder compacts are presented to provide the basis for qualitative description of the important phases of the course of densification. From this guide, appropriate grain shapes and pore shapes and locations are selected for the formulation of diffusion sintering models. The principle models presented are for bulk diffusion transport with the grain boundaries as vacancy sinks when the pore phase is continuous and coincident with three grain edges, and also when the pore phase is discontinuous and located at four-grain corners. These models predict that the rate of density change is constant when the diffusion coefficient and grain size are constant. The need for simultaneous isothermal densification and grain growth data is indicated. The explicit change in densification rate with discontinuous grain growth is predicted in terms of pore spacing and grain size.

Sintering Crystalline Solids. II. Experimental Test of Diffusion Models in Powder Compacts

During sintering in alumina powder compacts, the density has been found to increase linearly with the logarithm of time, and the grain size increases with the one-third power of time. Incorporation of the time dependence of grain size increase into late-stage bulk diffusion sintering models (from Part I )[R. L. Coble, J. Appl. Phys., 32, 787 (1961)] leads to corrected models by which a semilogarithmic behavior is predicted. The presence of density gradients in normally fabricated pellets makes impossible the deduction of whether theoretical density will be achieved from the early stages of the course of densification. Diffusion coefficients calculated from the intermediate and later stages of sintering bear order-of-magnitude agreement with those calculated from the initial-stage sintering measurements in alumina. All diffusion coefficients from sintering data are higher than Kingery’s measured diffusion coefficients for oxygen. It is hypothesized that the sintering process must then be controlled by bulk diffusion of aluminum ions while the oxygen transport takes place along the grain boundaries. In controlling the sinterability of alumina to theoretical density, it appears that magnesia does not ‘inhibit’ discontinuous grain growth, but instead increases the sintering rate such that discontinuous growth nuclei do not have time to form.

Diffusion Models for Hot Pressing with Surface Energy and Pressure Effects as Driving Forces

Models for initial‐, intermediate‐, and final‐stage densification under pressure have been developed, which explicitly include both the surface energy and applied pressure as driving forces. For the initial stage, the time dependences and size effects given by the integrated equations are identical to those reported earlier for surface energy (alone) as the driving force. The only modification is that the surface energy (γ) is expanded into (γ+PaR/π), where Pa is the applied pressure and R is the particle radius. For the intermediate stage of the process, the Nabarro‐Herring and Coble creep models may be adapted to give approximate (∼4×) densification rates for lattice and boundary diffusion models, respectively. In these cases the complex driving force is written as: (Pa/D+γk), where D is the relative density, and k is the pore surface curvature. At the final stage of the process those models are invalid; an alternate model is developed based on diffusive transport between concentric spherical shells whi...

Microstructure development during final/intermediate stage sintering—I. Pore/grain boundary separation

Abstract The motion of pores attached to two grain interfaces has been analyzed. It is shown that pores exhibit a maximum steady state velocity that varies with the dihedral angle and that the onset of non-steady state pore motion results in grain boundary convergence and the separation of the pore from the grain boundary. The peak steady pore velocity has been compared with grain boundary velocities for several grain configurations, in order to identify a critical condition for the onset of separation. This comparison indicates that the pore size must be maintained below a critical value to ensure attachment.

The role of grain size distributions in diffusional creep

Abstract The relationship between the strain rate and the stress in diffusional creep has usually been derived with the assumption that all the grains have the same size. In a normal polycrystaltine solid, however, grains of widely different sizes are usually present. In order to investigate to what extent the grain boundary diffusion creep rate (for a given stress and metallographic grain size) depends on the grain size distribution and grain arrangement, the strain rates of simple regular arrays of square grains (2 dimensions) with bimodal size distributions were calculated. Depending on the particular grain arrangement employed, variations of the strain rate by up to a factor of 4.4 were found. Also, internal stresses as high as 3.5 times the applied shear stress occur for particular distributions. The grain shape influences the diffusional strain rate in two ways. Changing the shape of approximately equiaxed grain results in a variation of the strain rate by up to a factor of ~13.5. Changing the aspect ratio of the grains while preserving their geometric shape (i.e., the grain boundaries are not allowed to rotate during elongation) reduces the strain rate only moderately, for example by a factor of 1.27 for a strain of 50%. The influence of different grain size distributions and grain arrangements is intermediate between these two effects.

Dynamic dislocation behavior in iron‐doped magnesium oxide crystals

Edge‐ and screw‐dislocation velocities in iron‐doped magnesium‐oxide single crystals containing 150 ppm Fe+3 have been measured as a function of stress and temperature in order to elucidate the rate‐controlling drag mechanism for dislocation mobility. Edge dislocations have been found to move faster than the screw dislocations over the stress and temperature regimes investigated. From the analysis of the edge‐ and screw‐dislocation velocity data in terms of the activation parameters (activation volume, activation enthalpy, total activation enthalpy, and the stress exponent of dislocation velocity), it is suggested that the edge‐ and screw‐dislocation mobilities in MgO single crystals containing 150‐ppm Fe+3 dopants are controlled by the interaction of dislocations with the nonsymmetric distortions due to (FeMg.−VMg″) defects. The strain Δe owing to such defects has been calculated and is related to the observed hardening of ion‐doped (Fe+3) MgO single crystals.

Mobility of Edge Dislocations in Single‐Crystal Calcium Fluoride

The mobility ν of edge dislocations in calcium fluoride has been measured over an applied stress τ range of 0.1–1.0 kg/mm2 and from temperatures between 25° and 100°C. The data are fitted to the relationship v=6×104exp[−(0.54 eV/kT)−(0.89 kg/mm2/τ)]cm/sec. The activation energy observed for dislocation mobility agrees with that for migration of fluorine ion vacancies in CaF2. Oxygen impurity is observed to harden the crystals. Interactions between oxygen ion and fluorine ion‐vacancy dipoles and the dislocation strain fields are considered responsible for some of the velocity‐stress‐temperature behavior.

Overview no. 84: An analysis of diffusion and diffusional creep in stoichiometric and hyperstoichiometric uranium dioxide

Abstract Diffusion and diffusional creep mechanisms are analyzed for stoichiometric and hyperstoichiometric UO 2+ x . A variety of criteria are used in the evaluation based on a critical review of literature data: 1. (1) agreement between measured tracer diffusivities and those calculated from creep rates or reported for sintering; 2. (2) self-consistent values of the activation energies for various diffusion controlled processes; 3. (3) self-consistent and theoretically reasonable stoichiometry dependencies for self-diffusion, creep, and sintering; 4. (4) a satisfactory grain size correlation in the diffusional creep regime. It is conclusively shown that Coble creep is the dominant mechanism for any practical grain size where viscous creep processes are diffusion limited. This conclusion applies at stoichiometric and all oxygen-rich compositions. At low stress interface controlled diffusional creep is an important mechanism. A compilation of the “best” set of uranium grain boundary and lattice diffusivities is presented for the entire range of UO 2+ x stoichiometries.

On the removal of pores from castings by sintering

The causes, sizes, and distribution of porosity in castings have been reviewed and quantitatively evaluated for several important modes of alloy solidification. In general, gas exsolution is found to be the most probable cause of porosity in castings which solidify in either a cellular or dendritic fashion. On the other hand, solidification alone may cause porosity creation if the interdendritic liquid metal cannot feed the solidification shrinkage. This effect may be enhanced by gas exsolution.Removal of porosity by “sintering” after solidification requires that the grain size be of the order of, or smaller than, the pore spacing, and that the pores be small (>1 µ) for removal within reasonable times (tens of hours). When gas exsolution is the cause of pore creation, the gas must be diffused out of the sample to permit pore shrinkage. Small ingot sizes (>10 cm) and rapidly diffusible gases (H2) are required for pore elimination within reasonable times (tens of hours).The application of low pressure (>20 atm) during sintering increases the rate, or the size (to >10 mµ) of the pores which can be eliminated within >20 hr.