Influence of interstitials on the shear moduli

Some time ago it was shown that for the bulk melting transition a strong correlation exists between the volume dependence of shear elastic coefficient and melting. It was discovered [16,67] that the volume of the melt at the melting point can be predicted by a continuous extrapolation of the shear modulus to zero as a function of volume (at zero external stress). In another words, a bulk melting transition will occur in an infinite crystal when it will be expanded up to a critical volume ( molar volume of the liquid phase) at which the shear moduli (or at least one of them) will vanish. This observation lends considerable credibility of the Born mechanism of melting. According to the Born criterion, mechanical melting at zero external stress, is described by the criterion $C'=0$, either in perfect or imperfect crystal containing point defects. This could be verified directly in MD simulations by calculation of the shear elastic moduli at various concentrations of point defects. For any concentration of defects the number of atoms and unit cells in the sample is kept constant in our simulations, so that the number of externally introduced defects is conserved. Under this constraint, atoms and point defects are free to diffuse, agglomerate, etc.

The simulations at various temperatures and concentrations showed that the shear moduli $C'$ and $C_{44}$ decrease when temperature or/and concentration is increased (See Figs. 4.21 -4.24).

Figure: Variation of $C'$ with temperature.

Figure: Variation of $C_{44}$ with temperature.

Figure: Variation of $C'$ with self-interstitial concentration. The dashed lines to guide the eye.

Figure: Variation of $C_{44}$ with self-interstitial concentration. The dashed lines to guide the eye.

The common trend is that self - interstitials induce a noticeable softening of the shear moduli. This softening is anisotropic, in the sense that $C_{44}$ softens more than $C'$. The absolute change of the shear elastic coefficients per percent of interstitals is larger for $C_{44}$ at the high temperatures, as summarized in Table 4.2.

Table: Change in shear elastic moduli $C'~and~C_{44}$ of vanadium per percent of self - interstitials ($S$) at various temperatures.

Temp(K)	$\Delta C'/\Delta S~~~(GPa/\% inters)$	$\Delta C_{44}/\Delta S~~~(GPa/\% inters)$
2100	$-8.1 ~\pm 2.6$	$-11.3 ~\pm 0.5$
2300	$-6.27 ~\pm 1.8$	$-8.97 ~\pm 1.0$
2400	$-5.25 ~\pm 2.4$	$-10.07~\pm 0.7$

Table: Change in the shear elastic moduli $C',~C_{44}$ of copper per percent of self - interstitials ($S$) , obtained by A. Kanigel et al. [67].

Temp(K)	$\Delta C'/\Delta S~~~(GPa/\% inters)$	$\Delta C_{44}/\Delta S~~~(GPa/\% inters)$
1400	$-3.5~ \pm 2.0$	$-23~\pm 1.7$
1450	$-4.1 ~\pm 2.0$	$-18~\pm 2.5$
1480	$-7.8 ~\pm 3.5$	$-15.5~\pm 1.3$

The same effect of anisotropy in softening of the shear moduli was observed for copper [67] (See Table 4.3). Qualitatively, a similar dependence of the ${C'}$ and $C_{44}$ on point defects concentration is obtained (See Figs. 4.25 and 4.26), yet quantitatively bcc lattice is less softened by the presence of self - interstitials, especially for the shear modulus $C_{44}$.

Figure: Dependence of $C_{44}$ on the self-concentration of interstitials.

Figure: Dependence of $C_{44}$ on the concentration of defects at T=1400K, obtained by A. Kanigel et al. [67].

We compare the change of the shear elastic moduli caused by point defects with the similar change induced by thermal expansion of a perfect sample, i.e. we heated a sample without point defects up to a temperature at which its volume is equal to the volume of a sample with point defects but at a lower temperature and compared the shear moduli. We knew from the simulations in canonical ensemble (NVT), how the shear moduli depend on the concentration of interstitials. Dependence of the specific volume on the concentration of defects was obtained by means of the PR method. If we take into account that there is a one-to-one correspondence in both cases, then we can find how the shear moduli depend on the specific volume (See Fig. 4.27).

Figure: Variation of the elastic modulus $C'$ with specific volume. Solid line is a quadratic extrapolation.

We found that the dependence of the shear moduli on the specific volume is the same whether the volume expansion is induced by the thermal expansion or by point defects. That is to say that effect of interstitials is mainly to expand the bcc lattice of vanadium, and when the critical volume is reached the solid melts. It was obtained in the PR simulation of melting that specific volume per atom at the melting point is $v_{cr}=15.02 ~\pm 0.1 ~\dot A^3$. An identical value (within the stated accuracy) of $v_{cr}=14.95 ~\pm 0.08 ~\dot A^3$ is obtained by extrapolating the dependence of $C'$ on specific volume up to the point at which $C'(T_b)=0$. The values of the critical specific volume are close to the specific volume of liquid vanadium at the thermodynamic melting temperature [74] is $v_{liq}=15.431 ~\dot A^3$

Similar results were obtained for copper (fcc lattice) by A. Kanigel et al.[67,92](See Fig 4.28), and J. Wang et al.[15]. Using the molecular dynamics method, it was shown that the Born's melting criterion is valid for the mechanical melting which occurs when the free energy based heterogeneous melting starting from surface or grain boundaries is kinetically suppressed. It was predicted that the incipient instability $C'=0$, occurring at the observed lattice strain $a/a_0=1.024$, where $a$ lattice parameter at $T=1350K$, and $a_0$ lattice parameter at $T=0K$ . Therefore the specific volume ratio of copper is $(a/a_0)^3=1.074$ which is very close to the value obtained for Va $v(T_m)/v(T_0)=1.075$. This result hints that the ratio $a/a_0$ could be a univesal and independent of lattice structure, but to check that assumption it is necessary to investigate the bulk melting transion of other bcc crystals.

Figure: Dependence of $C'$ on the lattice constant for bulk Cu, showing the equivalent effect of addition of point defects at constant T and heating without point defects, from ref.[67,92].