Random fcc structure

The statistical approach to the microscopic crystal structure offers a straightforward approach to substitutional disorder as in the random fcc structure. The equilibrium densities n^(s) at freezing were again determined following [RH90]. The evaluation of eq. (7.11) differs from the previous NaCl case in two respects: Firstly both species occupy the same fcc lattice such that b^(s) = 0 in eq. (7.13). Secondly, as a consequence, the occupancies ÷(s) < 1 are shared among the two species. In the defect-free case still holds ÷₍₁₎ +÷₍₂₎ = 1, with phase separation ÷₍₁₎ ”= ÷₍₂₎ allowed (compared to the equimolar reference liquid). The terms ln÷_(s) in eq. (7.6) signify a mixing entropy contribution. The results for the freezing transition are shown in figure 7.4. The minimization of ∆œ_HR is performed over Á₍₁₎,Á₍₂₎ and also both solid densities n^(s)_s . The plot in [RH90] is for the large sphere densities, and figure 7.4(a) shows excellent agreement. The additional parameter‰^crit_s for the solid stoichiometry Na‰Cl1 (used synonymous with “Small‰Large₁-NaCl structure”) is shown on the right-hand axis. Î = 1corresponds to the one-component HS crystal and yields an equimolar crystal phase ‰^crit_s (Î = 1) = 1 as one would expect from mixing entropy. For Î <1 the large species quickly prevails in the crystal. It seems plausible to think of a lattice site occupied by the small species as a “partial vacancy”:

The space available in each unit cell is bounded from below by the large species and is not filled to the same extent by the small. This makes it energetically favorable for the large species to enter the crystal while mixing entropy still favors equimolarity.

Not the entire range in Î was recomputed: In figure 7.4(b)Á(1) increases more and more strongly for Î .0.33. To treat larger values of Á₍₁₎, a third approximation for

the integral (7.3) would be needed which remains to be implemented. Interestingly, also in figure 7.2(b) a jump in Á₍₁₎ begins near Î = 0.33. A potential physical mechanism behind this drop in the localization of the small particles could be hopping across tetrahedral gaps. At close packing, the tetrahedral gap has a diameter of 0.225‡. With the lattice constants at Î = 0.3, the tetrahedral gap has diameters of approximately0.244‡ and 0.233‡ in the NaCl respectively the random fcc crystal.

This is about 4/ 5of the small sphere diameter. In [Fil+11] hopping processes of the small species in an Na‰Cl1 structure through tetrahedral gaps were analyzed by molecular dynamics (MD) simulations. It was found that, while keeping a constant diameter ratio Î = 0.3, an increase in the occupancy ÷(s) of the small sphere sublattice facilitated the hopping processes. While a direct study of this effect requires the consideration of irreversible dynamics the so-called Na‰Cl1 “interstitial solid” offers a system where finally point defects are expected to play a major role.

The following paragraph7.1.4 generalizes the DFT approach from paragraph 7.1.2to a partly occupied Na sublattice which will be compared to the evaluation of Monte Carlo (MC) simulation data in section7.2.

0 125

» L

∆ ≈ X W À K

≈ ÊËÒ kBT/(m‡2 )È

Polarization within À LOTOz

TOxy

LATAz

TAxy

(a) weakly bidisperse,Î = 0.9

» L

∆ ≈ X W À K

≈ 0

125

ÊËÒ kBT/(m‡2)È K/ 7

≈

(b) strongly bidisperse,Î = 0.3

Figure 7.5.: Dispersion relations for the random fcc structure at(a)Î = 0.9 and (b)Î= 0.3.

The wave vectors are again from the path shown in figure7.3(b). The polarizations of the corresponding eigenvectors along the À segment are given in the key of(a) for both plots. The inset in (b)magnifies the crossover region between acoustic and optical modes. For the full set of equilibrium density parameters cf. table7.1.

The dispersion relations for the random fcc crystal are shown in figure 7.5at (a) Î = 0.9 and (b) Î = 0.3. Unlike in the NaCl evaluation, small and large particles

Table 7.1.: Equilibrium structure parameters for the dispersion relations shown in this chapter

Structure

Parameter Î m⁽¹⁾ ‰_¸ n^crit,(2)_¸ ‰s n^crit,(2)s Á₍₁₎ Á₍₂₎ [m] [‡^≠3] [‡^≠3] [10^≠2‡] [10^≠2‡]

Na1Cl1 fig. 7.3(a) 0.445 1 1 1.00692 1 1.13285 3.9943 4.3804 rand. fcc 7.5(a) 0.9 γ 1 0.56783 0.56852 0.78768 7.8255 3.0350 rand. fcc7.5(b) 0.3 γ 1 0.97073 0.14276 1.11065 39.371 2.1111 Na‰Cl1 – DFT 0.3 1 3.6 0.91533 0.66880 1.24844 7.4643 2.4853 and sim. fig. 7.9 0.3 1 — — 0.66699 1.35000 not determined

were not given identical masses but identical mass densities, i.e. m^(1)Om⁽²⁾ = γ. This better represents an experimental setup where both species consist of the same material. While for the NaCl dispersion relations figure 7.3(a) all of the 6 six eigenmodes showed significant variations in their frequency throughout the BZ, the frequency scale is clearly split in figure 7.5. For easier reference, the 6 curves will be associated with the polarization of the corresponding eigenvector along the À path which are the same in figures 7.5(a) and (b). Only close to the ≈ point do the 3 acoustic curves show the expected linear slope and then quickly saturate to a more or less constant plateau value. A way to understand this feature seems to be via the weaker localization of the small spheres: The dispersion relations can be seen as the result of probing the linear response of a sample to a plane-wave deformation of various wave length and direction. The weaker the localization to the lattice sites, the more a species will appear as a liquid on short length scales. This seems clearly connected to the observation that there is roughly a factor of 5 between Á₍₁₎ ¥0.078 in figure 7.5(a) and Á₍₁₎ ¥0.394 in figure7.5(b). Further note that two of the three acoustic modes decay more strongly from figure 7.5(a) to (b). These were identified as transversal modes (along À,∆and ≈) which will be absent in a homogeneous liquid. The remaining longitudinal acoustic mode is the one affected by the ideal-gas contribution id in . It is obtained from settingc^(ab)©0 in eq. (7.14).

Diagonalization of »⁽¹¹⁾_id ^OÍ0 yields a linear dispersion relation Êid along À. With the values provided in table 7.1 follows Êid(q=K)¥ 37.46[kBT /(m‡²)]^1/2 which is slightly above the LA frequency in figure 7.5(b). This seems counterintuitive as one would expect interaction to increase the eigenfrequencies compared to the ideal gas limit. A role in that might play the coupling to the heavier large spheres:

Figure 7.6 shows the eigenvector components of the modes polarized in thexy plane for both species. While the large species asymptotically decouples from the TAxy

mode (cf. figure 7.6(a)) and the TAz mode (not shown) shows qualitatively the same behavior (also for q up to the K point), a sign change of the eigenvector

≈ K 3

-1 ^Sˆjxy,transversal 0 ^Sˆjxy,transversal 1 small optical

large optical small acoustic large acoustic

(a) transversal in xy plane

-1 ^Sˆjxy,longitudinal 0 ^Sˆjxy,longitudinal 1≈ K

3

≠₁₀₀¹ 0 ₁₀₀¹≈ K small optical 3

large optical small acoustic large acoustic

(b) longitudinal

Figure 7.6.: Components of the normalized eigenvectors ^Sˆj of ^S corresponding to 4 of the 6 modes shown in theÀ segment of figure7.5(b). “small/large” refers to the species considered. The color code and labelling of modes are adopted from figure 7.5(a).

The inset in (b) highlights the sign change of large acoustic and small optical component for the longitudinal mode.

component of the large species occurs for the LAxy mode (inset of figure 7.6(a)).

The limit limÁ₍₁₎ æ Œ of a complete small sphere delocalization remains to be studied. It corresponds to the “fast sphere” structure (5) introduced in [SH87]. More generally, figure7.6 shows the crossover of acoustic and optical modes visible in the inset of figure 7.5(b) in terms of the eigenvectors of ^S . Each mode is associated with two graphs for the normalized amplitude of the small and the large species.

Towards the ≈ point, i.e. q =0, the optical mode shows antiparallel movement of both species. Reassuringly, the ratio of amplitude between small and large species is found to be determined by

ˆj

S (1) xy,optical

O^Sˆj_xy,optical⁽²⁾ (q=0) =

ˆı

ıÙn^crit,(2)s m⁽²⁾ n^crit,(1)s m⁽¹⁾ .

With eq. (5.24), this matches the physical requirement of a resting center of mass for the optical modes at q=0,

Í⁽²⁾₀ ˆj⁽¹⁾+Í⁽¹⁾₀ ˆj⁽²⁾ =0 (as a statistical average).

For the acoustic modes atq=0, both species move parallel and contribute to the total momentum also according to Í⁽¹⁾₀ ^OÍ⁽²⁾₀ . After the crossover, the dominant contributions of small and large species to optical and acoustic mode have exchanged their roles. The originally optical mode with largely moving small spheres is now mainly defined by a movement of the large species and vice versa. This coincides with the observation that the original optical modes in figure 7.5 resemble in shape the acoustic modes in a one-component fcc crystal (cf. e.g. [NL11, figure 5]). In that sense, the small-q optical modes take the role of the acoustic modes in most of the Brillouin zone, especially for weakly localized small spheres.

0.9 Figure 7.7.: Coexistence parameters for the Na‰Cl1 structure plotted versus the liquid

stoi-chiometry ‰_¸ at constant diameter ratio Î = 0.3. (a) and (b) show the same as figures 7.2 and 7.4 where however Î was varied at constant ‰¸. (c) shows the stoichiometry of the solid phase ‰_s. It can here alternatively be read as the Na-sublattice occupancy ÷₍₁₎. The freezing parameters at‰_s= 2/3 will be used for a comparison to simulation in section 7.2.