Nabarro-Herring and Coble Creep

The class of creep mechanisms of interest here are those that are mediated by stress-biased diffusion. If we are to consider the vacancy flux in a given grain within a material that is subjected to an applied stress, it is argued that the vacancy formation energy differs in different parts of the grain, and hence that there should be a gradient in the vacancy concentration leading to an associated flux. This 

11 Points, Lines and Walls Defect Interactions and Material Response 

Our ambition in the paragraphs to follow is to illustrate an approximate heuristic model which shows how the presence of an applied shear stress can result in mass transport mediated deformation. We begin with the notion that the crystal of interest is subjected to a shear stress. From a two-dimensional perspective, for example, we then make use of the fact that the shear stress 

In principle, we are thus faced with solving a boundary value problem for a region like that shown in fig. 11.6 with the aim being to evaluate the mass transport that arises by virtue of the spatially varying chemical potential, and hence, mass currents, that arise in response to the nonuniform stress. 

In practice, since we are only interested in an estimate of the effect, we resort to an approximate analysis in which the relevant chemical potentials and mass flux, and attendant strain rate are all evaluated heuristically. The argument begins with reference to fig. 11.6 with the claim that the vacancy formation energy for the faces subjected to tensile stresses differs from that on the faces subjected to compressive stresses. Again, a rigorous analysis of this effect would require a detailed calculation either of the elastic state of the crystal or an appeal to atomistic considerations. We circumvent such an analysis by asserting that the vacancy concentrations are given by 

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In general, one should not suppose that the properties of bulk materials will apply to materials at the nanoscale level. With respect to the mechanical properties of small-scale solids, it is known that the elastic behaviour, due to bond stretching and twisting, does not vary significantly in nanoparticles compared with that in the bulk. Other properties are more sensitive. For example, the rate of diffusion creep (Nabarro-Herring and Coble creep) is dependent on grain size. Hence, creep will be enhanced in compacts of nanoparticles and in thin films. 

Nabarro-Herring and Coble creep can take place in parallel so that actual creep rates will involve both components and both diffusion coefficients. In ceramics we also have a situation in which both anions and cations are diffusing adding further complications to the creep rate equations. If there is a large difference in the diffusion rates then the creep rate is controlled by the slower diffusing species along the faster diffusing path. 

FIG. 39 A schematic diagram of the cross section of a polycrystalline Al thin film under compression, (a) The arrows indicate the flow of atoms to relieve the stress on the basis of the Nabarro-Herring or Coble creep models in the case where there is no surface oxide, (b) If a surface oxide exists, stress relief is only possible if it can be broken locally and penetrated. (Courtesy of UCLA.)... 

The densification by plastic deformation and power-law creep is, in principle, independent of particle (grain) size. In the case of diffusion (both lattice and grain boundary), on the other hand, densification depends on not only the effective pressure but also the grain size. The densification by diffusion under an external pressure is similar to diffusional creep Nabarro-Herring creep due to lattice diffusion, and Coble creep due to grain boundary diffusion. The dependency of densification on grain size is the same as that of diffusional creep. 

Mass diffusion between grain boundaries in a polycrystal can be driven by an applied shear stress. The result of the mass transfer is a high-temperature permanent (plastic) deformation called diffusional creep. If the mass flux between grain boundaries occurs via the crystalline matrix (as in Section 16.1.3), the process is called Nabarro-Herring creep. If the mass flux is along the grain boundaries themselves via triple and quadjunctions (as in Sections 16.1.1 and 16.1.2), the process is called Coble creep. 

Figure 16.5 Deformation mechanism map for Ag polycrystal a = applied stress, p = shear modulus, grain size = 32 pm, and strain rate = IGF8 s 1. The diffusional creep field is divided into two subfields the Coble creep field and the Nabarro-Herring creep field. From Ashby [20].

Both Coble creep and Herring-Nabarro creep describe ... 

Derive an expression for the critical grain size below which Coble creep dominates and above which Nabarro Herring creep dominates (at constant temperature). 

In an ultra-high vacuum, no surface hillocks were found to grow on aluminum (Al) surfaces under compression (Ref 24). Hillocks grow on Al surfaces only when the Al surface is oxidized, and Al surface oxide is known to be protective. Without surface oxide, the free surface of Al is a good source and sink of vacancies, so a compressive stress can be relieved uniformly on the entire surface of the Al, based on the Nabarro-Herring model of lattice creep or the Coble... 

The total creep rate is the sum of the Nabarro-Herring creep and the Coble creep. In case solids contain more than one species, as in the case of ceramics, one should use a complex diffusion coefficient. This is given approximately by Equation 15.38 [1]. 

The Herring-Nabarro creep appears in ceramics at high temperatures when the grain size is large enough. At lower temperatures and for smaller grain sizes, the creep is of the Coble type. 

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