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Next: Summary and conclusions Up: Results: surface melting Previous: Distance between layers

Layer occupation and energetics

The occupation of the topmost surface layers at various temperatures is calculated. The first surface layer is the most affected by disordering effects, i.e. adatom-vacancy pair formation. Since the layers beneath the first are fully occupied at low temperatures, this leads to the formation of an adlayer on the top of the first layer (See Fig. 5.29). The adatoms leave behind an equivalent number of vacancies.

Figure: Snaphot of Va(001) at 1800K, notice the adatoms (white color) on the top of the surface layer.
\begin{figure}\centerline{\epsfxsize=7.0cm \epsfbox{/home/phsorkin/Diploma/Pict/van_3300.eps } }\end{figure}

The occupation of the surface layers at different temperatures for the various faces of vanadium are shown in Figs. 5.30-5.32.

Figure: Layer occupation of the $Va(001)$ sample as a function of temperature for the adatom layer.
\begin{figure}\centerline{\epsfxsize=7.0 cm \epsfbox{/home/phsorkin/Diploma/Surface/Chap3/num_001.eps } }\end{figure}
The formation of an adatom layer on the least packed surface Va(111) begins at around T = 800 K. The onset of an adlayer involves generation of vacancies in the first surface layer, while at higher temperatures ($T \ge 1600 K $) vacancies in the underlying layers (the second and third layers) begin to appear via promotion of atoms to vacant places of the first surface layer, i.e. so called interlayer vacancy migration mechanism. Atom migration from the deeper layers increases significantly as the temperature approaches the melting point. In contrast to the Va(111) sample, an adlayer formation and generation of adatom-vacancy pairs in the other samples becomes observable at more elevated temperatures (about T=1400 K for the Va(001) and T = 2000K for the Va(011)). Practically, all adatoms come from the first surface layer.

Figure: Layer occupation of the $Va(111)$ as a function of temperature for the adatom layer and the first few crystal layers.
\begin{figure}\centerline{\epsfxsize=8.5 cm \epsfbox{/home/phsorkin/Diploma/Surface/Chap3/num_111.eps } }\end{figure}
Figure: Layer occupation of the Va(011) as a function of temperature for the adatom layer and the first few crystal layers.
\begin{figure}\centerline{\epsfxsize=8.5 cm \epsfbox{/home/phsorkin/Diploma/Surface/Chap3/num_011.eps } }\end{figure}
The same effect of disordering and gradual thickening of the surface region was observed in computer simulations of surface premelting of fcc metals [24,79,80,86,87,89]. The least packed face (110) of fcc metals (in analogy with the (111) bcc face) begins to disorder first, while the close packed (111) face (in analogy with the (011) bcc face) preserves its ordered crystalline structure almost up to the melting point.

Knowledge of the equilibrium averaged number of atoms in the adlayer allows us to estimate the surface defect formation energy $E_s $ according to the formula:

\begin{displaymath}
n=exp(-E_s/k_BT)
\end{displaymath} (6.14)

where $n$ is the adlayer occupation, e.g. the equilibrium averaged number of atoms in the adlayer at a current temperature divided by the number of atoms in the layer at a zero temperature. The adlayer occupation vs. temperature fitted to the Boltzmann factor $n=exp(-E_s/k_BT)$ are shown in Figs. 5.33-5.35 for the Va(111), Va(001) and Va(011), respectively.

The calculated surface defect formation energies of the various low-index faces of vanadium are tabulated in Table 5.3. Data for various faces of copper (fcc lattice) [80] are given for comparison. The surface defect formation energy is largest for the close packed surfaces Va(011) in case of a bcc lattice, and Cu(111) in case of a fcc lattice, and the lowest one for the least packed surfaces, the Va(111) and Cu(011), respectively.

Table: Energy of surface defects (adatom-vacancy pairs) is calculated using the adlayer occupation data of the various surfaces of vanadium as a function of temperature. The data for copper is from ref. [80]
Surface $(111)$ $(001)$ $(011)$
$Va~(bcc)~E_s$ $0.27 \pm 0.04~eV $ $0.84 \pm 0.02~eV $ $2.36 \pm 0.01~eV $
$Cu~(fcc)~E_s$ $1.92~eV $ $0.86~eV $ $0.39~eV $


Figure: Equilibrium adlayer occupation vs. temperature for the $Va(111)$.
\begin{figure}\centerline{\epsfxsize=8.0 cm \epsfbox{/home/phsorkin/Diploma/Surface/Chap3/en_111.eps } }\end{figure}
Figure: Equilibrium adlayer occupation vs. temperature for the $Va(001)$.
\begin{figure}\centerline{\epsfxsize=8.5 cm \epsfbox{/home/phsorkin/Diploma/Surface/Chap3/en_001.eps } }\end{figure}
Figure: Equilibrium adlayer occupation vs. temperature for the $Va(011)$.
\begin{figure}\centerline{\epsfxsize=8.5 cm \epsfbox{/home/phsorkin/Diploma/Surface/Chap3/en_011.eps } }\end{figure}

It is possible, that the adatom-vacancy pairs creation in the first crystal layers at high temperature is not the only one surface defect formation mechanism. Recent calculations carried out for fcc metals [79,89] indicate that it is energetically more favorable for the least packed surface to form a pair consisting of a divacancy and two adatoms, than to form two independent vacancy-adatom pairs, i.e. surface defect formation is a kind of a cooperative phenomenon. Therefore, surface defect formation energy, calculated on the basis of information about the adlayer occupation number, may be related to more complicated mechanism of defect formation. We did not study in detail the adatom-vacancy formation mechanism and used the obtained equilibrium averaged adlayer occupation numbers for a straightforward estimation of the adatom-vacancy surface defect formation energy.

Surface defect formation energy, calculated for the least packed surface, can be related to the thermodynamic melting point $T_m$ according to E. Polturak et al.[92], i.e. $E_{def}\simeq c k_B T_m $, where $c$ is a constant which is determined below and $k_B$ is Boltzmann factor. In this scenario, the nucleation of a liquid phase on the free surface of a solid can be associated with the same mechanism which describes melting in a bulk of a solid (the Born criterion). Bulk melting occurs once the specific volume of a crystalline phase attains a critical value, which is close enough to the specific volume of a corresponding liquid phase. The same conditions, leading to melting in a bulk of a solid, are thought to occur at the surface, but at a lower temperature. In line with this assumption, we observe that the quasiliquid film appears first on the least packed surface Va(111), where surface defect formation is minimal with respect to the other low-index faces. Formation of the surface defects is considered to be a very effective mechanism of a stress release. The structure order parameters, plane radial distribution function, transport properties of the top surface layers in the premelting region exhibit liquid-like properties. We can assume that at high temperatures the shear elastic stress at the surface is insignificant. Therefore the Born criterion (at zero external stress) can be applied to the surface layers, and in this way it is possible to give explanation why do melting of a crystal begins at the free surface.

To obtain a linear relation between the melting temperature and surface defect formation energy, let us consider the evolution of the volume of a surface layers with temperature [92]. At low temperature, the volume of the first crystal layer is given by $V=v_0N$, where $v_0$ is the volume per atom, and $N$ is the number of atoms in the layer. While the temperature increases, the layer expands, and the atoms are displaced of the layer forming adatom-vacancy pairs. The temperature dependence of $V(T)$ can be written as:

\begin{displaymath}
V(T)=v_0(T)N+v_dN_d=v_0(T)N\left(1+\frac{v_dN_d}{v_0(T)N}\right)
\end{displaymath} (6.15)

here $v_d$ is the volume added to the layer by a creation of one defect, and $N_d$ is the number of surface defects, which dependence on temperature is given by the Boltzmann factor:
\begin{displaymath}
\frac{N_d}{N}=exp\left(-\frac{E_s}{k_BT}\right)
\end{displaymath} (6.16)

where $E_s $ is the surface formation energy. According to the Born criterion, at $T=T_m$ the $V/N$ reaches its critical value that of the liquid phase (melt), then volume per atom is $V/N \approx v_{liq}$. Taking a natural logarithm of $V/N$ (eq. 5.15 ) and rearranging this expression, we can restate the Born criterion as a linear relation between $E_s $ and $T_m$.
\begin{displaymath}
E_s=\left\{ ln \left(\frac{v_d}{v_0(T_m)}\right)- ln \left(\frac{v_{liq}}{v_0(T_m)}-1\right) \right\} k_BT_m = ck_BT_m
\end{displaymath} (6.17)

It is assumed, in the lowest order approximation, that $v_d \simeq v_0(T_m)$. Indeed, upon creation of an adatom, one atomic volume is added to the first surface layer. To be more precise, since the adatom is weakly bounded to the surface, its vibration amplitude and therefore effective volume should be somewhat larger than $v_0(T)$. However, due to the reconstruction of the atoms around the created vacancy, it is expected that the volume of the vacancy to be smaller than the average atomic volume $v_0(T)$. Within the logarithmic accuracy of the equation (5.16) it is expected that these correlations cancel each in the first approximation, i.e. $v_d \simeq v_0(T_m)$ and consequently
\begin{displaymath}
E_s=\left\{- ln \left(\frac{v_{liq}}{v_0(T_m)}-1 \right) \right\} k_BT_m = ck_BT_m
\end{displaymath} (6.18)

The right hand side of this equation predicts the value of $E_s $ without any adjustable parameters. To check it validity, we compare value of $E_s $ (for the least packed surface Va(111)) calculated using this equation and $E_s $ obtained directly in the MD simulations of bulk and surface melting. According to the results of the MD study of mechanical melting the volume per atom in the bulk at $T_m$ is $v_0(T_m)=14.7 \pm 0.3~ \dot A^3$, while the volume per atom in the melt is $v_0(T_m)=15.47 \pm 0.3 ~\dot A^3$. The thermodynamic melting point is found to be $T_m=2220 \pm 18~ K $ by using the method proposed by Lutchko et al. [77]. Hence we obtain $E_s=0.52 \pm 0.04~ eV$ by means of the formula (5.30). We can also estimate the surface defect formation energy using experimental data: the melting temperature $T_m=2183~K$, the volume per atom in the melt is $v_0(T_m)=15.431~\dot A^3$, and the volume per atom in the crystalline bulk at $T_m$ can be estimated approximately as
\begin{displaymath}
v_b(T_m) \approx v_0(T_0)\exp(\alpha(T_m-T_0))
\end{displaymath} (6.19)

where $T_0=300~K$, the volume at $v_0(T_0)= 14.03~\dot A^3$ and $\alpha=8 \times 10^{-6} 1/K$ is the thermal expansion coefficient of vanadium measured at $T_0$. Thus we can calculate in approximation $v_b(T_m) \approx14.25~\dot A^3 $ and consequently $E_s \approx 0.468~ eV$. The result of the MD simulations of surface melting the surface defect formation energy $E_s $ of the least packed surface Va(111) is found to be $E_s = 0.27 \pm 0.05~eV$ relying on the analysis of the adlayer occupation as a function of temperature. We may conclude that agreement between the theoretically predicted and calculated values of $E_s $ is encouraging.

Finally we remark, that the Born melting criterion was applied in a study of surface melting of fcc metals [92]. It was found that a linear relation between the surface defect formation $E_s $ energy and the melting temperature $T_m$ is valid for metals with the fcc lattice structure (See Fig. 5.36 and Tab. 5.4) This linear relation agrees quantitatively with the experimental and simulation values of the activation energy of the surface defects, without any additional adjustable parameters. Our results imply that the Born criterion correctly describes both surface and bulk melting, and may well provide the ``missing link'' which will finally tie together these two scenarios for the melting transition.

Figure: The activation energy of surface defects $E_s $ vs. $T_m$ for Pb, Al, Ag, Au, Cu, Ni and Pt. The solid line is a linear regression. From ref. [92]
\begin{figure}\centerline{\epsfxsize=8.0 cm \epsfbox{/home/phsorkin/Diploma/Pict/Chapt2/Bulk/intro/emil.eps } }\end{figure}

Table: The activation energy of surface defects $E_s $ vs. the melting temperatures for fcc metals. The data from ref. [92]
Metal Pb Al Ag Au Cu Ni Pt
$T_m~K$ 600 933 1235 1337 1356 1726 2045
$E_{s}~eV$ $0.18\pm0.04$ $0.26\pm0.02$ $0.37\pm0.03$ $0.32\pm0.03$ $0.41$ $0.49\pm0.04$ $0.49\pm0.05$



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Next: Summary and conclusions Up: Results: surface melting Previous: Distance between layers
2003-01-15