Phonon Modes and Elastic Properties of Zr-Based Bulk Metallic Glasses: A Pseudopotential Approach Phonon Modes and Elastic Properties of Zr-Based Bulk Metallic Glasses: A Pseudopotential Approach

The collective dynamics for longitudinal and transverse phonon modes and elastic prop- erties are studied for bulk metallic glasses (BMGs) using Hubbard-Beeby approach along with our well establishes model potential. The important ingredients in the present study are the pair-potential and local field correlation function. The local field correlation functions due to Hartree (H), Taylor (T), Ichimaru and Utsumi (IU), Farid et al. (F), Sarkar et al. (S) and Hubbard and Sham (HS) are employed to investigate the influence of the screening effects on the vibrational dynamics of Zr-Ti-Cu-Ni-Be, Zr-Cu-Ni-Al-Ta, Zr-Ti-Cu-Ni- Al and Zr-Al-Ni-Cu. The result for the elastic constants like bulk modulus B T , rigidity modulus G, Poisson ’ s ratio ξ , Young ’ s modulus Y, Debye temperature Ɵ D , the propaga-tion velocity of elastic waves and dispersion curves are found to be in good agreement with experimental and other available data. The present results are consistent and confirm the applicability of model potential and self-consistent phonon theory for such studies.


Introduction
Bulk metallic glass-forming liquids are alloys with typically three to five metallic components that have large atomic size mismatch and a composition close to a deep eutectic [1]. Metallic glasses have regained considerable interest due to the fact that new glass forming composition have been found that have a critical cooling rate of less than 100 K s À1 and can be made glassy with dimensions of 1 cm or more. The development of such alloys with a very high resistance to crystallization of the under cooled melt has opened new opportunities for the primary study

Theory
There are three main theoretical approaches used to compute the phonon frequency of binary alloys: one is Hubbard and Beeby (HB) [12], second is Takeno and Goda (TG) [18], and last is Bhatia and Singh (BS) [11]. The HB approach is the random phase approximation according to this theory, a liquid random from a crystalline solid in two principal ways. Initially, the atoms in the metallic glasses do not form a regular array, i.e. they disordered. So the HB theoretical models have been employed to generate the phonon dispersion curve and their related properties of bulk metallic glass alloys in the present computation.
The effective ion-ion interaction is given as where F(q) is the characteristic energy wave number. The first and second terms in the above expression are due to the coulomb interaction between ion and indirect interaction through the conduction electrons, respectively; q is the q-space wave vector, e is the charge of an electron.
In the present study, we have considered the effective atom approach to compute the phonon dispersion curve (PDC). For the present study, we have used the Jani et al. [19][20][21] model potential in q-space is given as, The characteristic feature of this model potential is the single parametric nature. r eff c is the potential parameter. This determination of parameter is independent of any fitting procedure with the observed quantities. The energy wave number characteristics in expression (1) are given by [21][22][23][24][25] Here W eff B q ð Þ is the effective bare ion potential, ε eff H q ð Þis the Hartree dielectric response function and f eff q ð Þ is the local field correction function (LFCF) due to the Hartree (H) [13], Taylor [17] are used here to include the effect of screening on the collective modes of bulk metallic glasses Zr-Ti-Cu-Ni-Be BMG for five different concentration, Zr-Cu-Ni-Al-Ta BMG, Zr-Ti-Cu-Ni-Al BMG and Zr-Al--Ni-Cu BMG for three different concentration. Long-wavelength limits of the phonon modes are then used to investigate the elastic properties, viz. longitudinal sound velocity, transverse sound velocity, Debye temperature, isothermal bulk modulus, modulus of rigidity, Poisson's ratio and Young's modulus. Five different types of LFCF are employed here for the study of the effect of exchange and correlation on the aforesaid properties. Pair potential or effective interaction is realized through interatomic potential, ion-ion potential and electron-electron potential developed between two similar particles like atoms, ion and electrons. The paircorrelation function g(r) is equally important as the pair potential. It contains useful information about the inter particle radial correlation and structure which in turn decides the electrical thermodynamically and amorphous materials.
The effective potential and pair correlation function g r ð Þ are the used to calculate the longitudinal and transverse phonon frequencies. The product of the static pair-correlation function g r ð Þ and the second derivative of the interatomic potential V } eff r ð Þ are peaked at σ, which is the hard sphere diameter. The longitudinal phonon frequency ω l (q) and transverse phonon frequency ω t (q) are given by the expression due to Hubbard and Beeby (HB). and where and Here, r is the number density, M is the atomic mass, g(r) is the pair correlation function, Ω 0 is the atomic volume, F(q) and S(q) be the energy wave number characteristic and the structure factor of the element, respectively.
In the long-wavelength limit, the phonon dispersion curve shows an elastic behavior. Hence, the longitudinal ν l and transverse ν t sound velocities are also calculated by [22][23][24][25] ω L α q and ω T α q, ∴ω L ¼ v l q and ω T ¼ v l q Various elastic properties are determined by the longitudinal and transverse phonon frequencies.
The bulk modulus B, Poisson's ratio 'ξ', modulus of rigidity G, Young's modulus Y and the Debye temperature θ D are calculated using the expression below [22][23][24][25] r is the isotropic density of the solid, and where 'h' is Plank constant and k B is the Boltzmann constant.

Results and discussion
The input parameters and constants used in the present computation are shown in Table 1. Our well-established model potential is used along with five different local field correction functions to computed form factors and thereby effective pair potentials. The phenomenological approach of Hubbard and Beeby [8,12,[21][22][23][24][25] is used further to compute phonon frequencies.
The longitudinal and transverse phonon frequencies show all broad features of collective excitations of all BMGs. It is seen from the results of phonon frequencies that the nature of peak positions are not much affected by different screening functions, but both the longitudinal and transverse frequencies show deviation for H, T and S functions with respect to IU and F screening function for present model potential. At large momentum, phonons from longitudinal branch are found responsible for momentum transfer. Phonons of transverse branch undergo large thermal modulation due to the anharmonicity of lattice vibrations in this branch. The first minimum in the longitudinal branch corresponds to umklapped scattering process. No experimental data of structure factor for these BMGs are available. In the long wavelength limit, the frequencies are elastic and allow us to computed elastic constants.            properties. So we do not offer any concrete remark at this stage, but it is sure that this data is very useful for the further investigation.
3.2. Zr 52.25 -Cu 28.5 -Ni 4.75 -Al 9.5 -Ta 5 BMG Here, Our well established model potential is used along with five different types of local field correction functions due to H, T, IU, F and S for to generate the pair potential for Zr 52.25 Cu 28.5 Ni 4.75 Al 9.5 Ta 5 BMG [4] system. Figure 5 shows the calculated pair potential for Zr 52.25 Cu 28.5 Ni 4.75 Al 9.5 Ta 5 BMG [4]. It is observed that the depth of the pair potential obtained using model potential is highly affected. This depth affects the height and peak of the pair correlation functions. This pair potential is helping to compute phonon frequencies of longitudinal and transverse branch and it is shown in Figure 6(b). No experimental data and other available data are found for comparison for sound velocities. So, we do not put any concert comment on sound velocity at this point.
The absence of experimental data and other information on elastic properties like bulk modulus, Poisson ratio and Debye temperature, so that, present results are compared with calculated values and other available results [5]. From the  Young modulus and Shear modulus using T, IU and S local field correction functions are in good agreement with the results mention in Ref. [5].

Zr 57 -Ti 5 -Cu 20 -Ni 8 -Al 10 BMG
Here, it has been reported for the first time to generate the pair potential for the Zr 57 Ti 5 Cu 20 Ni 8 Al 10 BMG system. The computed pair potential is shown in Figure 7(a) using present model potentials. In this case, the computed pair potential is affected by type of screening used. The pair potential computed using model potential shows first positive minimum. The depth of this minimum is affected by type of screening used and almost at the r value where pair potential shows a positive minimum. The computed pair potential is greatly affected by model potential.
Using a pair potential it has been projected the longitudinal and transverse phonon frequencies for Zr 57 Ti 5 Cu 20 Ni 8 Al 10 BMG are shown in Figure 7(b). From the Figure 7(b) it is understood from the results of phonon frequencies that the nature of peak positions are not much exaggerated by different screening functions, but both the longitudinal and transverse frequencies show small deviation for H, T and S functions with respect to IU and F screening function in Figure 7(b).
On the other hand, the transverse modes undergo larger thermal modulation due to the anharmonicity of the vibrations in the BMGs. In the long wavelength limit, the dispersion curves are linear and confirming characteristics of elastic waves. The PDC for transverse phonons attain maxima at a higher q value than the longitudinal phonon curve. At present calculated elastic properties for Zr 57 Ti 5 Cu 20 Ni 8 Al 10 BMG are listed in Table 8. From Table 8, one can see that by using the T, IU and F screening functions, the results are very close to one another as compared to the H screening function. Modulus of rigidity 'G', Young modulus 'Y' and Debye temperature is showing the better agreement with experimental values [5,6] computed using the T, IU and F screening while obtains due to H and S show the underestimate values than the experimental and other available data. We are sure that this data is very useful for the further investigation.  Figures 8(a), 9(a) and 10(a) it is observed that the study reveals the general trends of the pair potential in all cases, suggesting that the position of the first minimum depth in the pair potential in the present study is obtained due to F screening function.
Using this pair potential it has been computed the longitudinal and transverse phonon frequency for Zr 61.88 Al 10  From the Tables 9-11 it is observed that sound velocity computed using present model potential along with T, IU and F functions are found to be very close to one another and calculated sound velocities using F screening function shows a good agreement with available data [4][5][6]. Computed bulk modulus using model potential along with all local field correction functions is underestimated as compared to the experimental [4] and other available data [5,6]. Presently computed percentile deviation for modulus of rigidity with respect to available data.

Conclusion
The dispersion of longitudinal phonon shows oscillatory behavior for the large q values while lack of thereof in the transverse phonons. The transverse phonon frequencies increase with wave number and get saturated at the first peak of ω T ! q curves with small variations. The ω ! q curve for the transverse phonons achieves maxima at a higher q value than the longitudinal phonon curve. The peak heights of the longitudinal as well as transverse phonon frequencies of these BMGs are nearly the same. Thus, the dispersion curves of these BMGs are found to be similar. It is apparent from the ω L ! q curves of the glassy materials that they are screening sensitive in the low-momentum region. The difference in ω ! q relation begins right from the starting value of q and it's becomes maximum at the first peak of the ω L ! q curve, again, it tends to decrease and both ω L ! q relations seem to converge at the first minima of the ω L ! q curve. The position of the first peak is independent of the screening functions. However, the height of the peak strongly depends on the type of screening employed in the present calculations. The phonon dispersion curve for the Zr-based bulk metallic glasses computed using the IU and F function give higher numerical values than other local field correction functions. Using H-function give the lowest values for the Zr-base bulk metallic glasses. Agarwal has done the longitudinal and transverse modes for the Zr-Ti-Cu-Ni-Be for three different concentrations using the BS-Method. When compared with our present approach, it is found that model potential gives underestimated results.
Presently calculated elastic properties for BMGs are listed in Tables 2-11. It is observed that the computed elastic properties using model potential are in excellent agreement with experimental and other available theoretical data. Among the five different screening functions T, IU and F functions show good agreement for present model potential. While due to H screening function than the other LFCF and computed values using S lying between the F and H screening function. For Zr-Ti-Cu-Ni-Be, Zr-Cu-Ni-Al-Ta, Zr-Ti-Cu-Ni-Al and Zr-Al-Ni-Al BMGs at different concentrations computed using model potential, it is observed that ν l and ν t , Young modulus, modulus rigidity, Debye temperature using the T, IU, F and S local field correction functions show the very good agreement with experimental and other available data. The Zr-based BMGs are observed that the present results obtained due to T, IU and F screening functions are in good agreement with available with other data. Present study clearly reveals that proper description of local field correction function is required for the study of phonon modes of bulk metallic glasses.
Overall, we stated that the phonon dispersion curve generated from the HB approach reproduces satisfactorily the general characteristic of dispersion curves. The well recognized Model potential along with IU, Farid et al.
[F] and Sarkar-Sen et al.
[S] local field correction functions generates consistent results. Hence, our Model-1 potential is suitable for the studying the phonon dynamics of bulk metallic glasses.