The dissolution of the vacancy gas and grain boundary diffusion in
  crystalline solids
The dissolution of the vacancy gas and grain boundary diffusion
in crystalline solids
Fedor V.Prigara
Institute of Microelectronics and Informatics, Russian Academy of Sciences,
21 Universitetskaya, Yaroslavl 150007, Russia∗
(Dated: September 12, 2021)
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.
PACS numbers: 61.72.Bb, 66.30.Dn, 68.35.-p
http://arxiv.org/abs/0704.0972v4
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]
dE = TdS − pdV, (1)
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]
n = (P0/T ) exp (−Ev/T ) = (n0T0/T ) exp (−Ev/T ) , (2)
where Ev is the energy of the vacancy formation, P0 = n0T0 is a constant, T0 can be put
equal to the melting temperature of the solid at ambient pressure, and the constant n0 has
an order of magnitude of the number density of atoms in the solid. Here the Boltzmann
constant kB 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 Ev depends linearly on the pressure P (in the region
of high pressures) as given by the formula
Ev = E0 − αP/n0, (3)
where α is a dimensionless constant, α ≈ 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
n = (n0T0/T ) exp (− (E0 − αP/n0) /T ) , (4)
and, finally, the vacancies can condense, forming their own sub-lattice. Such is the explana-
tion 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 nc of vacancies is achieved. In view of the equation (2), it means that the ratio
of the energy of the vacancy formation Ev to the melting temperature Tm of the solid is
approximately constant,
Ev/Tm ≈ α. (5)
The value of the constant α 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]:
E0 ≈ 18Tm, (6)
so that α ≈ 18.
Substituting the expression (3) in the relation (5), we obtain
(E0 − αP/n0) /Tm ≈ α. (7)
The last equation gives the melting curve of the crystalline solid in the region of high
pressures in the form
T + P/n0 ≈ E0/α ≈ T0, (8)
where T0 is the melting temperature of the solid at ambient pressure.
The constant n0 can be determined from the relation between the tensile strength σs and
the melting temperature Tm of a solid [1]
n0 ∼= σs/Tm. (9)
The numerical value of this constant is n0 ≈ 1.1× 10
22cm−3 [1].
Replacing in the relation (4) the pressure P by the absolute value of the stress or tension
σ = 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:
〈n〉 ∼= (n0T0/T ) exp (− (E0 − ασ/n0) /T ) . (10)
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 σb of the mechanical stress in the close vicinity of a grain boundary is
given by the formula
σb ∼= γb/r0, (11)
where γb is the energy of the grain boundary and r0 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
Eb = E0 − αγb/ (n0r0) . (12)
For the small values of misorientation angle θ 6 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 γ of the solid, γb ∼= γ.
Due to the Einstein relation between the mobility of an atom, µ = v/F , where v is the
velocity of the atom and F is the force acting on the atom, and the diffusion coefficient D
µ = v/F = D/T, (13)
the speed of grain boundary motion v is proportional to the diffusion coefficient D⊥ 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⊥ of
self-diffusion across the grain boundary. The last activation energy is equal to the activation
energy Eb of grain boundary self-diffusion in the case of high-angle grain boundaries, and
is approximately equal to the activation energy E0 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 θc of the misorientation angle (θc = 10− 15
◦, 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
θc = 13.6
◦ [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
γ ∼= βn0E0a0, (14)
where a0 = n
∼= 0.45nm has an order of magnitude of the lattice spacing a, and β is a
dimensionless constant which has an order of unity. For hard metals such as Al, Zr, Nb, Fe,
Pt, β ∼= 0.8. In the case of mild metals, β is normally smaller, e.g. for Rb and Sr, β ∼= 1/4.
Substituting the estimation (14) for the energy of the grain boundary γb ∼= γ in the
equation (12), we find
Eb ≈ E0 (1− βαa0/r0) . (15)
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]
Eb ≈ 9Tm ≈ E0/2. (16)
From equations (15) and (16), we find the estimation of the radius of the atomic relaxation
region,
r0 ≈ 2βαa0 ∼= αa0, (17)
since β has an order of unity. The radius of the atomic relaxation region has an order of
r0 ∼= 18n
≈ 8nm. This value is comparable with the diameters of tracks produced by
high energy ions in metals [15-17]. The grain boundary diffusion width δ [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 σ decreases linearly with the distance x from the
plane of the grain boundary,
σ = σ0 (1− kx) , (18)
where σ0 is the stress at the boundary of the atomic relaxation region with the width r0 in
the grain boundary (this value is smaller than σb ∼= γ/r0 ∼= (1/2)n0Tm and has an order of
magnitude σ0 ∼= (1/2)n0T ), then the equation (10) gives the distribution of vacancies in the
vicinity of the grain boundary in the form
n ∼= (n0T0/T ) exp (− (E0 − ασ0 (1− kx) /n0) /T ) = nbexp (−ασ0kx/ (n0T )) , (19)
where nb 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:
c ∼= cbexp (−x/l) , (20)
where cb is the concentration of the diffusing species at the boundary of the relaxation region,
and the scale l is given by the formula
l = n0T/ (ασ0k) . (21)
Here k has an order of magnitude of 1/d, d being the size of the grain, so that l ∼= d/α. 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.
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[12] K.J.Draheim and G.Gottstein, in APS Annual March Meeting, 17- 21 March 1997,
Abstract D41.87.
http://arxiv.org/abs/cond-mat/0701148
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[16] F.V.Prigara, E-print archives, cond-mat/0406222.
[17] M.Toulemonde, C.Trautmann, E.Balanzat, K.Hjort, and A.Weidinger, Nucl. In-
strum. Meth. B 217, 7 (2004).
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∗ Electronic address: fvprigara@rambler.ru
http://arxiv.org/abs/cond-mat/0406222
mailto:fvprigara@rambler.ru
	References
