The dissolution of the vacancy gas and grain boundary diffusion in crystalline solids

TL;DR

This paper develops a model for grain boundary diffusion in crystalline solids, linking vacancy density, surface tension, and activation energy, providing insights into vacancy and diffusing species distributions near grain boundaries.

Contribution

It introduces a new model connecting vacancy density, surface tension, and diffusion activation energy in crystalline solids.

Findings

Derived activation energy dependent on surface tension

Predicted vacancy and diffusing species distributions near grain boundaries

Established a formula for vacancy density under stress

Abstract

Based on the formula for the number density of vacancies in a solid under the stress or tension, the model of grain boundary diffusion in crystalline solids is developed. We obtain the activation energy of grain boundary diffusion (dependent on the surface tension or the energy of the grain boundary) and also the distributions of vacancies and the diffusing species in the vicinity of the grain boundary.

Equations

$$dE=TdS-pdV,$$

$$n=\left(P_0/T\right)exp\left(-E_v/T\right)=\left(n_0{T}_0/T\right)exp\left(-E_v/T\right),$$

$$E_v=E_0-\alpha P/n_0,$$

$$n=\left(n_0{T}_0/T\right)exp\left(-\left(E_0-\alpha P/n_0\right)/T\right),$$

$$E_v/T_m\approx \alpha .$$

$$E_0\approx 18T_m,$$

$$\left(E_0-\alpha P/n_0\right)/T_m\approx \alpha .$$

$$T+P/n_0\approx E_0/\alpha \approx T_0,$$

$$n_0\cong {\sigma }_s/T_m.$$

$$⟨n⟩\cong \left(n_0{T}_0/T\right)exp\left(-\left(E_0-\alpha \sigma /n_0\right)/T\right).$$

$${\sigma }_b\cong {\gamma }_b/r_0,$$

$$E_b=E_0-\alpha {\gamma }_b/\left(n_0{r}_0\right).$$

$$\mu =v/F=D/T,$$

$$\gamma \cong \beta n_0{E}_0{a}_0,$$

$$E_b\approx E_0\left(1-\beta \alpha a_0/r_0\right).$$

$$E_b\approx 9T_m\approx E_0/2.$$

$$r_0\approx 2\beta \alpha a_0\cong \alpha a_0,$$

$$\sigma ={\sigma }_0\left(1-kx\right),$$

$$n\cong \left(n_0{T}_0/T\right)exp\left(-\left(E_0-\alpha {\sigma }_0\left(1-kx\right)/n_0\right)/T\right)=n_{b}exp\left(-\alpha {\sigma }_{0}kx/\left(n_{0}T\right)\right),$$

$$c\cong c_{b}exp\left(-x/l\right),$$

$$l=n_{0}T/\left(\alpha {\sigma }_{0}k\right).$$

Full text

Recently, it was shown that sufficiently high pressures as well as mechanical stresses applied to a crystalline solid lead to the decrease in the energy of the vacancy formation and create, therefore, an additional amount of vacancies in the solid [1]. The last effect enhances self-diffusion in the crystal which is normally vacancy-mediated, at least in simple metals. Since large mechanical stresses are normally present in grain boundaries, these new results can elucidate the mechanisms of grain boundary diffusion which have remained so far unclear [2].

According to the thermodynamic equation [3]

[TABLE]

where E is the energy, T is the temperature, S is the entropy, p is the pressure, and V is the volume of a solid, the energy of a solid increases with pressure, so the pressure acts as the energy factor similarly to the temperature. Therefore, the number of vacancies in a solid increases both with temperature and with pressure.

The thermodynamic consideration based on the Clausius- Clapeyron equation gives the number density n of vacancies in a solid in the form [1]

[TABLE]

where $E_{v}$ is the energy of the vacancy formation, $P_{0}=n_{0}T_{0}$ is a constant, $T_{0}$ can be put equal to the melting temperature of the solid at ambient pressure, and the constant $n_{0}$ has an order of magnitude of the number density of atoms in the solid. Here the Boltzmann constant $k_{B}$ is included in the definition of the temperature T.

The formula (2) describes the thermal expansion of the solid. It should be taken into account that the dissolution of the vacancy gas in a solid causes the deformation of the crystalline lattice and changes the lattice parameters.

The energy of the vacancy formation $E_{v}$ depends linearly on the pressure P (in the region of high pressures) as given by the formula

[TABLE]

where $\alpha$ is a dimensionless constant, $\alpha\approx 18$ for sufficiently high pressures. On the atomic scale, the pressure dependence of the energy of the vacancy formation in the equation (3) is produced by the strong atomic relaxation in a crystalline solid under high pressure.

With increasing pressure, the number density of vacancies in a solid increases, according to the relation

[TABLE]

and, finally, the vacancies can condense, forming their own sub-lattice. Such is the explanation of the appearance of composite incommensurate structures in metals and some other elemental solids under high pressure [4-7].

Further increase of the number density of vacancies in a solid with increasing pressure leads to the melting of the solid under sufficiently high pressure (and fixed temperature). Such effect has been observed in sodium [6]. In general, such behavior is universal for solids, though the corresponding melting pressure is typically much larger than those for sodium.

We assume that the melting of the crystalline solid occurs when the critical number density $n_{c}$ of vacancies is achieved. In view of the equation (2), it means that the ratio of the energy of the vacancy formation $E_{v}$ to the melting temperature $T_{m}$ of the solid is approximately constant,

[TABLE]

The value of the constant $\alpha$ in the last relation can be determined from the empirical relation between the activation energy of self diffusion (which is approximately equal to the energy of vacancy formation) and the melting temperature of a solid [8]:

[TABLE]

so that $\alpha\approx 18$.

Substituting the expression (3) in the relation (5), we obtain

[TABLE]

The last equation gives the melting curve of the crystalline solid in the region of high pressures in the form

[TABLE]

where $T_{0}$ is the melting temperature of the solid at ambient pressure.

The constant $n_{0}$ can be determined from the relation between the tensile strength $\sigma_{s}$ and the melting temperature $T_{m}$ of a solid [1]

[TABLE]

The numerical value of this constant is $n_{0}\approx 1.1\times 10^{22}cm^{-3}$ [1].

Replacing in the relation (4) the pressure P by the absolute value of the stress or tension $\sigma=F/S$, applied to a solid, where *F *is the applied force and S is the cross-section area of the solid in the plane perpendicular to the direction of the applied force, we can estimate the mean number density of vacancies in the solid under the stress or tension:

[TABLE]

The dissolution of the vacancy gas in a solid under the stress or tension is responsible for the low values of the elastic limit and the tensile strength of solids as compared with theoretical estimations not taking into account this process [9].

As indicated above, large mechanical stresses are normally present in grain boundaries. The absolute value $\sigma_{b}$ of the mechanical stress in the close vicinity of a grain boundary is given by the formula

[TABLE]

where $\gamma_{b}$ is the energy of the grain boundary and $r_{0}$ is the radius of the atomic relaxation region (around a vacancy) which will be estimated below.

According to the relation (10), the energy of the vacancy formation in the close vicinity of the grain boundary is given by the formula

[TABLE]

For the small values of misorientation angle $\theta\leqslant 10-15$ degrees, the energy of the dislocation structure contributes to the energy of the grain boundary [10]. However, for larger misorientation angles, the energy of the grain boundary is approximately constant and is determined by the surface tension $\gamma$ of the solid, $\gamma_{b}\cong\gamma$.

Due to the Einstein relation between the mobility of an atom, $\mu=v/F$, where v is the velocity of the atom and F is the force acting on the atom, and the diffusion coefficient D [8]:

[TABLE]

the speed of grain boundary motion v is proportional to the diffusion coefficient $D_{\bot}$ for self-diffusion in the direction perpendicular to the plane of a grain boundary. Therefore, the activation energy E of grain boundary motion is equal to the activation energy $E_{\bot}$ of self-diffusion across the grain boundary. The last activation energy is equal to the activation energy $E_{b}$ of grain boundary self-diffusion in the case of high-angle grain boundaries, and is approximately equal to the activation energy $E_{0}$ of bulk self-diffusion for low-angle grain boundaries. Thus, there is a step of the activation energy for grain boundary motion at some critical value $\theta_{c}$ of the misorientation angle ($\theta_{c}=10-15^{\circ}$, as indicated above). Such a step of the activation energy for grain boundary motion has been observed experimentally in high-purity aluminium, the critical value of the misorientation angle being in this case $\theta_{c}=13.6^{\circ}$ [11].

The driving force for grain boundary motion is provided by the distribution of mechanical stresses in a crystalline solid [12].

Assuming that the free surface of a crystalline solid is formed by the plane of vacancies, we can estimate the surface tension of the solid as follows

[TABLE]

where $a_{0}=n_{0}^{-1/3}\cong 0.45nm$ has an order of magnitude of the lattice spacing $a$, and $\beta$ is a dimensionless constant which has an order of unity. For hard metals such as Al, Zr, Nb, Fe, Pt, $\beta\cong 0.8$. In the case of mild metals, $\beta$ is normally smaller, e.g. for Rb and Sr, $\beta\cong 1/4$.

Substituting the estimation (14) for the energy of the grain boundary $\gamma_{b}\cong\gamma$ in the equation (12), we find

[TABLE]

Due to the atomic relaxation and thermal motion of atoms, the migration barriers are small [2,13], and the activation energy of self-diffusion is approximately equal to the energy of the vacancy formation. The analysis of experimental data on the activation energy of grain boundary self-diffusion gives an empirical relation [14]

[TABLE]

From equations (15) and (16), we find the estimation of the radius of the atomic relaxation region,

[TABLE]

since $\beta$ has an order of unity. The radius of the atomic relaxation region has an order of $r_{0}\cong 18n_{0}^{-1/3}\approx 8nm$. This value is comparable with the diameters of tracks produced by high energy ions in metals [15-17]. The grain boundary diffusion width $\delta$ [14] is smaller than the radius of the atomic relaxation region due to the non-uniform distribution of vacancies inside the atomic relaxation region in the grain boundary.

If we assume that the mechanical stress $\sigma$ decreases linearly with the distance x from the plane of the grain boundary,

[TABLE]

where $\sigma_{0}$ is the stress at the boundary of the atomic relaxation region with the width $r_{0}$ in the grain boundary (this value is smaller than $\sigma_{b}\cong\gamma/r_{0}\cong\left(1/2\right)n_{0}T_{m}$ and has an order of magnitude $\sigma_{0}\cong\left(1/2\right)n_{0}T$), then the equation (10) gives the distribution of vacancies in the vicinity of the grain boundary in the form

[TABLE]

where $n_{b}$ is the number density of vacancies at the boundary of the atomic relaxation region.

Due to the trapping by vacancies [18], the distribution of the concentration c of the diffusing species in the vicinity of the grain boundary follows the same law:

[TABLE]

where $c_{b}$ is the concentration of the diffusing species at the boundary of the relaxation region, and the scale l is given by the formula

[TABLE]

Here k has an order of magnitude of $1/d$, d being the size of the grain, so that $l\cong d/\alpha$. The penetration profiles described by the equation (20) have been indeed observed experimentally in the case of grain boundary diffusion in metals [8, 18], the measured penetration depth $l$ having an order of a few micrometers [8].

To summerize, we obtained the dependence of the activation energy of grain boundary self-diffusion on the energy of the grain boundary, the estimation of the surface tension of a solid and of the energy of the grain boundary, and the width of the atomic relaxation region in the grain boundary (or the radius of the atomic relaxation region around a vacancy). We obtained further the distributions of vacancies and the diffusing species in the vicinity of the grain boundary. The obtained radius of the atomic relaxation region is consistent with the diameters of tracks produced by high energy ions in metals.

—————————————————————

[1] F.V.Prigara, E-print archives, cond-mat/0701148.

[2] A.Suzuki and Y.Mishin, J. Mater. Sci. 40, 3155 (2005).

[3] S.-K.Ma, Statistical Mechanics (World Scientific, Philadelphia, 1985).

[4] R.J.Nelmes, D.R.Allan, M.I.McMahon, and S.A.Belmonte, Phys. Rev. Lett. 83, 4081 (1999).

[5] M.I.McMahon, S.Rekhi, and R.J.Nelmes, Phys. Rev. Lett. 87, 055501 (2001).

[6] O.Degtyareva, E.Gregoryanz, M.Somayazulu, H.K.Mao, and R.J.Hemley, Phys. Rev. B 71, 214104 (2005).

[7] V.F.Degtyareva, Usp. Fiz. Nauk 176, 383 (2006) [Physics- Uspekhi 49, 369 (2006)].

[8] B.S.Bokstein, S.Z.Bokstein, and A.A.Zhukhovitsky, Thermodynamics and Kinetics of Diffusion in Solids (Metallurgiya Publishers, Moscow, 1974).

[9] G.I.Epifanov, Solid State Physics (Higher School Publishers, Moscow, 1977).

[10] A.A.Smirnov, Kinetic Theory of Metals (Nauka, Moscow, 1966).

[11] M.Winning, G.Gottstein, and L.S.Shvindlerman, Acta Mater. 49, 211 (2001).

[12] K.J.Draheim and G.Gottstein, in APS Annual March Meeting, 17- 21 March 1997, Abstract D41.87.

[13] B.P.Uberuaga, G.Henkelman, H.Jonsson, S.T.Dunham, W.Windl, and R.Stumpf, Phys. Stat. Sol. B 233, 24 (2002).

[14] I.Kaur and W.Gust, Fundamentals of Grain and Interphase Boundary Diffusion (Ziegler Press, Stuttgart, 1989).

[15] F.F.Komarov, Usp. Fiz. Nauk 173, 1287 (2003) [Physics- Uspekhi 46, 1253 (2003)].

[16] F.V.Prigara, E-print archives, cond-mat/0406222.

[17] M.Toulemonde, C.Trautmann, E.Balanzat, K.Hjort, and A.Weidinger, Nucl. Instrum. Meth. B** 217**, 7 (2004).

[18] W.P.Ellis and N.H.Nachtrieb, J. Appl. Phys. 40, 472 (1969).