MODELING OF MATERIALS DISINTEGRATION DURING REACTIVE DIFFUSION OF ENVIRONMENTAL COMPONENTS
A. Katsman and L. Levin
Department of Materials Engineering, Technion, Haifa32000, Israel
Abstract
Disintegration of a solid material under the action of environmental components is analyzed. Such components are adsorbed by the surface of the material, diffuse into the solid matrix and form precipitates of new phases at grain boundaries and dislocations. Internal stresses may appear due to different specific volumes of the parent phase and precipitates and to diffusion caused by selective reaction of the active element with the alloy components. Different modes of stress relaxation (diffusional creep, void formation, plastic deformation and fracture) are realized depending on temperature, the mechanical properties of the material and the diffusivities. A model of the disintegration process is put forward and applied to description of such phenomena as dusting of steels, and of nickel- and cobalt alloys, and pesting of nickel aluminides.
1. Introduction
Degradation of materials under the action of active elements from a gaseous environment is a well-known problem in metal technology. The catastrophic disintegration of alloys, namely pesting and dusting have been investigated in detail over the last years. The first process is disintegration of intermetallics during oxidation. The second one is disintegration of metals during carburization, with the solid material transformed in both cases into small particles covered with reaction products (oxides in the case of pesting, graphite in the case of dusting). The disintegration mechanisms proposed by different authors [1-7] have not been fully elucidated.
In the present work, a model is proposed in a form providing a common quantitative description for the different disintegration processes. Generation and relaxation of stresses, onset of diffusion fluxes of the alloy components, generation of vacancies, formation of voids and channels along grain boundaries - are considered as the basic factors in the disintegration process.
2. The model
Let us consider the following model of disintegration of a solid material under the action of an active element from the environment. The latter is adsorbed by the outer surface and diffuses from there into the matrix (Fig.1). The main diffusion path is along grain boundaries (GB’s) and dislocations. We shell consider only GB diffusion. There are two possibilities for formation of a new phase:
One)formation on the outer surface;
Two)intergranular phase formation, which requires a certain supersaturation level of the GB matrix by the solute atoms.
Fig.1. A model of a solid material disintegration under the action of an active element from the environment. New phase and void formation along GBs leads to failure of the contact layer.
Formation of a new phase on the outer surface due to preferential reaction of the environmental component with one component of the alloy may lead to concentration gradients (due to consumption of the less-noble element), and consequently, to diffusion fluxes of the alloy components along grain boundaries. Such fluxes may in turn lead to formation of vacancies due to different diffusion coefficients of the alloy components (Kirkendall effect), and to concentration and diffusion stresses [4]. Formation of a new phase at GB’s may also cause internal stresses due to different specific volumes of the new and old phases. These stresses influence the diffusion process. They may relax by diffusion creep, by dislocation slip if they exceed the yield stress, Y, or by crack formation if the plastic strain reaches the ultimate value.
With a view to quantitative description of these processes, the following equations were used. The concentration of the active element, Ca, at the GB can be represented through the diffusion equation:
where Da is the diffusion coefficient of the active element, is the sink density, which may be a function of time, Cs is the solubility limit.
Let us assume that a new phase is formed in the course of a reaction between the active element, a, and one alloy component, A, while the second component, B, does not participate in the reaction. In this case the concentration profile of A-atoms, CA, along GB’s follows the diffusion equation:
where R is the number of A-atoms corresponding to one atom of the active element in the new phase. The change in the new phase volume fraction is
where q=ph/A, ph and A being the average atomic volume in the new phase and that of the A-atoms in the GB matrix, respectively.
The diffusion equation for vacancies, comprising their sources and sinks, can be written as follows:
where DV and DB are the GB diffusion coefficients of the vacancies and B-atoms respectively, =A-V is the atomic volumes difference of the A-atoms and vacancies, 1-1=rv - the average spacing of the vacancy sinks, CV0 - the equilibrium vacancy concentration, p the coefficient of vacancy generation during new phase formation.
The volume fraction of voids can be determined by integration of the last term in eq.(4), representing the sinks of vacancies:
The coefficient k1 indicates the fraction of the vacancy sinks due to voids, k1<1.
Precipitation of the new phase at GB’s and the climb of dislocations lead to internal stress formation:
where H1=(1-k1)Vvoid/V and H2=(/) are the hydrostatic strains caused by the dislocation climb and by the volume change during formation of the new phase, respectively, K is the elastic modulus.
The evolution of internal stresses can be described by the equation:
where relax is the strain caused by the various possible relaxation processes: diffusion creep, dislocation slip, crack formation, etc. The second term on the right-hand side describes the diffusion stress generation due to dislocation climb under the action of a vacancy flux.
If the stress is below the yield level, the only relaxation mechanism is diffusion creep. In that case:
where is the effective viscosity. It was shown [7] that in a certain temperature interval, depending on the ratio Da/DA and supersaturation of the active element at the outer surface, the stress can reach the yield level, Y, and the relaxation will proceed by plastic deformation through dislocation slip. In that case:
The ultimate plastic strain, lim, can be reached in a thin near-surface layer, in which cracking occurs.
2.1. Disintegration Mechanisms.
Let us consider the following disintegration mechanisms of a solid material:
(a)Grain boundary disintegration: formation of channels along GB’s as a result of vacancy condensation (formation of voids at GB’s and their subsequent merging).
(b)Cracking of the near-surface layer subjected to ultimate plastic deformation.
Since fracture of the near-surface layer permits a contact of the active environmental element with the next layer of the material, both processes may be repetitive.
In the first case the depth of the GB channel, L, was assumed to be correlated with deposition of the void-enriched zone. In addition, the channel surface was assumed to be impermeable to vacancies. The total vacancy flux to the surface equals zero:
The vacancy concentration is not at equilibrium at the material/new-phase interface. The expected outcome is formation of voids immediately beneath the interface. The volume fraction of these voids, s=Vvoid/V, was calculated from eq.(5). When s reaches a certain value close to unity, the interface of the GB channel is advanced to this region.
In the second case, fracture of the near-surface layer occurs when the stress reaches the ultimate value. The stresses formed due to formation of the new phase and to the climb of dislocations at grain boundaries can be calculated from eqs.(1)-(4),(7)-(8). When the stress reaches the yield stress, the plastic deformation follows eq.(9). The layer fracture time, f, was calculated as the time when the ultimate plastic strain, lim, is reached:
where is the volume fraction of the new phase averaged over the layer thickness d=1/.
The disintegration rate under the action of internal stresses arising due to formation of the new phase precipitates at GB’s and dislocations, was calculated in the temperature interval where the yield stress is achieved. The effective disintegration rate was estimated as
where d=-1/2, being the initial dislocation density.
3. Results and Discussion
The set of equations (1)-(5), (7)-(9), was solved numerically by the explicit finite-difference method, with initial and boundary conditions as follows:
where xs is the position of the material/new-phase interface, and l is half the sample thickness.
The dimensionless parameters =x/l and =tDA/l2 were used in the calculations. In the computer simulation of the GB channel growth we used the following procedure: When the volume fraction of voids, s, at =s reaches the value sc (0.1, 0.5, or 1), the material/new-phase interface is advanced one step, d. The time dependences of the interface displacement were calculated for different parameters. Straight lines (Fig.2) average the stepwise dependences obtained in the calculations The most important parameters governing the displacement rate are the ratios DB/DA, Da/DA and the concentration of the active element at the surface.
Fig.2. Channel depth as function of time. The GB channel grows due to void formation beneath the interface. DB/DA=0.1 (1), DB/DA=1 (2), DB/DA=3 (3), Da=0, l=100.
Propagation of channels along GB’s seems to be possible during selective oxidation of intermetallics, when a substantial number of vacancies is generated due to the oxidation reaction itself (for example, due to the release of structural vacancies). Evidence for void and channel formation at grain boundaries of NiAl during oxidation was found [4,8].
Fracture of the near surface layer under the action of internal stresses formed during diffusion and new phase formation is possible in a certain temperature interval, depending on the parameters of the system (Fig.3).
Fig.3. Temperature dependence of maximum stress reached during diffusion and new phase formation. Y=150MPa, ex calculated for different tex = 104s (1) and 105s (2).
At very high temperatures the formation rate of the new phase is high, but stress relaxation by diffusion creep is also very fast. The stress reaches the steady-state value which is below the yield level. In this case the metal matrix is neither plastically deformed nor embrittled. As the temperature decreases, the maximum steady-state stress increases, reaching the yield level at temperature Tpl, which is the upper limit for plastic relaxation. At low temperatures the stress value is mainly controlled by formation of the new phase due to the slow relaxation rate. The steady-state stress is very high and is not achieved during the experiment; the stress level actually reached can be approximated by the expression:
where Cs is the solute supersaturation level at the surface. This value increases with temperature and may reach the yield level at temperature Td (depending on tex), which is the lower limit for plastic relaxation. The temperature interval Td<T<Tpl may be considered as that of plastic deformation of the near-surface layer, terminating in its fracture at the ultimate value). For typical values q=0.1, Da=10-8cm2/s, 2=1/d2=1010…1011cm-2, lim=0.1 and Cs=10-6…10-5, the period f=102…104s and the effective disintegration rate u=(10-9…310-8)cm/s were obtained. These values are of the same order as the experimental rates of dusting observed during carburization of steels and Ni-based alloys [6]. This mechanical fracture mechanism of disintegration is sensitive to the initial density of the grain boundaries and dislocations. According to eqs.(3),(9),(11),(12), the disintegration rate, u, is proportional to (1/d)1/2, which means that preliminary cold deformation of the metal should accelerate the dusting process. This was actually observed during carburization of cold-worked steels [5]. In many cases, however, this effect is suppressed by formation of protective chromia scales, since chromium diffusion to the surface is also controlled by fast diffusion paths.
4. Conclusion
Computer simulation of disintegration of a solid material during diffusion and interaction of an environmental active element with the material components was carried out. Two different mechanisms of disintegration were considered: (a) formation of voids and channels along grain boundaries as a result of vacancy formation and condensation during diffusion and formation of the new phase; (b) mechanical fracture of the near-surface layer embrittled due to plastic relaxation of the internal stresses. Both mechanisms have a stepwise manner, as fracture of the near-surface layer permits contact of the active environmental element with the next layer of the material. The first mechanism is realized in the case of pesting of certain alloys and intermetallics, such as nickel aluminides. The second mechanism may be responsible for dusting of steels and Ni- and Co- based alloys.
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