Input parameters and constants.

## Abstract

The collective dynamics for longitudinal and transverse phonon modes and elastic properties 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 BT, rigidity modulus G, Poisson’s ratio ξ, Young’s modulus Y, Debye temperature ƟD, the propagation 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.

### Keywords

- bulk metallic glass
- pseudopotential
- local field correction function
- phonon modes
- elastic properties

## 1. 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^{−1} 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 of both the liquid state and the glass transition. The ability of bulk specimens has enabled measurements of various physical, particularly mechanical, properties that were previously impossible. Furthermore, these alloys are progressively being used for engineering applications. This new class of material is normally referred to as “bulk metallic glasses.” [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

Bulk metallic glass an excellent glass forming ability and high thermal stability have attracted much attention in the recent years. It has also occupied an important position due to their unique physical properties and technological importance. Zr-based BMGs are the coming up candidates to replace the usual materials used in the industry as a result of their superior mechanical properties, for example strength, hardness, toughness and elasticity [2, 3, 4, 5, 6, 7, 8, 9, 10]. Recently formation of Zr-Ti-Cu-Ni-Be, Zr-Ti-Cu-Ni-Be, Zr-Cu-Ni-Al-Ta, Zr-Ti-Cu-Ni-Al and Zr-Al-Ni-Cu bulk metallic glass (BMGs) alloys distinguished itself from other metallic alloys by extraordinary glass forming ability (GFA) [4, 5, 6, 7, 8, 9]. These alloys have opened the possibility to study the nature of glass transition in BMGs, offered the potential of BMGs as new generation engineering materials and controllable properties is one of the central issues in the condensed matter physics and material sciences [2, 3, 4, 5, 6, 7, 8, 9, 10]. Metallic glass alloys are normally regarded as elastically isotopic, and they behave as an elastic continuum at low temperatures, with well-defined dispersion relation so that ultrasonic waves propagated through it. Perker and Johnson [9] are made an important process of the design of the Zr–Ti–Cu–Ni–Be (Zr_{41}Ti_{14}Cu_{12.5}Ni_{10}Be_{22.5,} Zr_{45.4}Ti_{9.6}Cu_{10.15}Ni_{8.6}Be_{26.25,} and Zr_{46.75}Ti_{8.25}Cu_{7.5}Ni_{10}Be_{27.5}) five different concentration, Zr-Cu-Ni-Al glass-forming alloy family. This family has distinct glass transition, very high stability of supercooled liquid state and exhibits high thermal stability against crystallization. Dissolution of minute amount of metalloid elements into the Zr-base system can enlarge the thermal stability and hardness of the BMG. Zr-based BMGs are used in manufacturing of industrial production [4, 5, 6, 7, 8, 9]. Recently, Agarwal has reported the phonon dispersion curves of Zr-Ti-Cu-Ni-Be BMGs using Bhatia and Singh (BS) approach [11]. Wang et al. [4, 5, 6] have represented elastic properties using ultrasonic method. In the present study, we have investigated theoretically, phonon dispersion curve and elastic properties of Zr-based [4, 5, 6, 7, 8, 9, 10] bulk metallic glasses with the help of pseudopotential formalism and along with Hubbard and Beeby [12] approach and employing pseudopotential formalism along with the five different local field correction functions due to Hartree [13], Taylor [14], Ichimaru and Utsumi [15], Farid et al. [16], Sarkar et al. [17].

## 2. 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*(

*) 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,*q

The characteristic feature of this model potential is the single parametric nature.

Here

The effective potential and pair correlation function _{l} (q) and transverse phonon frequency ω_{t}(q) are given by the expression due to Hubbard and Beeby (HB).

and

where

and

Here, ρ 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]

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]

and

where ‘h’ is Plank constant and k_{B} is the Boltzmann constant.

## 3. Results and discussion

The input parameters and constants used in the present computation are shown in Table 1. Pair potential (Figures 1(a), 2(a), 3(a), 4(a), 5(a), 6(a), 7(a), 8(a), 9(a), 10(a)) and the phonon frequencies of longitudinal and transverse branch (Figures 1(b), 2(b), 3(b), 4(b), 5(b), 6(b), 7(b), 8(b), 9(b), 10(b)) of various bulk metallic glasses are shown in Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 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.

### 3.1. Zr-Ti-Cu-Ni-Be BMG (Zr_{41}Ti_{14}Cu_{12.5}Ni_{10}Be_{22.5,} Zr_{45.4}Ti_{9.6}Cu_{10.15}Ni_{8.6}Be_{26.25,} Zr_{46.75}Ti_{8.25}Cu_{7.5}Ni_{10}Be_{27.5,} Zr_{44}Ti_{11}Cu_{10}Ni_{10}Be_{25,} Zr_{38.5}Ti_{16.5}Ni_{9.75}Cu_{15.25}Be_{20})

Equation to compute phonon frequency and longitudinal and transverse sound velocity are taken from Thakore et al. [21, 23]. Pair potential and the phonon modes computed for Zr_{41}Ti_{14}Cu_{12.5}Ni_{10}Be_{22.5} BMG, Zr_{45.4}Ti_{9.6}Cu_{10.15}Ni_{8.6}Be_{26.25} BMG, Zr_{46.75}Ti_{8.25}Cu_{7.5}Ni_{10}Be_{27.5} BMG, Zr_{44}Ti_{11}Cu_{10}Ni_{10}Be_{25} BMG, Zr_{38.5}Ti_{16.5}Ni_{9.75}Cu_{15.25}Be_{20} BMG are shown in Figures 1, 2, 3, 4, 5 respectively. It is observed that the study reveals the general trends of the pair potential for BMGs considered here, suggesting that the position of the first minima is affected by the type of screening functions. The maximum depth in the pair potential in the present study is obtained due to S screening function for Zr_{41}Ti_{14}Cu_{12}Ni_{10}Be_{22.5}, Zr_{45.4}Ti_{9.6}Cu_{10.15}Ni_{8.6}Be_{26.25} and Zr_{46.75}Ti_{8.25}Cu_{7.5}Ni_{10}Be_{27.5} BMGs. From Figures 2, 3, 4 it is seen from the results of phonon frequencies that the general nature and peak positions are not affected very much by different screening functions, but magnitude of both the longitudinal and transverse frequencies show deviation with respect to screening function. From the Figures 2, 3, 4, 5, 6 that the oscillations are prominent in the longitudinal mode as compared to transverse mode, which indicates that the collective excitations at larger wave vector transfer due to the dispersion of longitudinal excitation. The maximum and minimum percentile deviation for the elastic properties of bulk modulus, modulus of rigidity, Young modulus and Debye temperature with respect to the experimental data from Tables 2, 3, 4. For bulk modulus, the computed percentile maximum deviation with respect to experimental data is found in H functions, the values are 43.82, 42.42 and 43.48% and minimum deviation is found in F functions, the values are 26.69, 24.92 and 26.25% for Zr_{41}Ti_{14}Cu_{12}Ni_{10}Be_{22.5}, Zr_{45.4}Ti_{9.6}Cu_{10.15}Ni_{8.6}Be_{26.25} and Zr_{46.75}Ti_{8.25}Cu_{7.5}Ni_{10}Be_{27.5} BMGs respectively. The computed percentile deviation for modulus of rigidity with respect to experimental data, the maximum deviation is found to be 35.54, 35.12 and 42.55% of F functions and minimum deviation is found in H function, the values are 3.47, 3.47 and 9.26% for Zr_{41}Ti_{14}Cu_{12}Ni_{10}Be_{22.5}, Zr_{45.4}Ti_{9.6}Cu_{10.15}Ni_{8.6}Be_{26.25} and Zr_{46.75}Ti_{8.25}Cu_{7.5}Ni_{10}Be_{27.5} BMGs respectively. For Young modulus, the computed percentile maximum deviation with respect to experimental data is found in F functions, the values are 24.72, 24.90 and 24.74% and minimum deviation is found in H functions, the values are 4.49, 4.22 and 7.47%, for Zr_{41}Ti_{14} Cu_{12}Ni_{10}Be_{22.5}, Zr_{45.4}Ti_{9.6}Cu_{10.15}Ni_{8.6}Be_{26.25} and Zr_{46.75}Ti_{8.25}Cu_{7.5}Ni_{10}Be_{27.5} BMGs respectively. Similarly for Debye temperature, the computed percentile maximum deviation with respect to experimental data is found in F function, the values are 16.39, 16.90 and 16.33% and minimum deviation is found in H function, the values are 1.86, 2.36 and 2.37%, for Zr_{41}Ti_{14} Cu_{12}Ni_{10}Be_{22.5}, Zr_{45.4}Ti_{9.6}Cu_{10.15}Ni_{8.6}Be_{26.25} and Zr_{46.75}Ti_{8.25}Cu_{7.5}Ni_{10}Be_{27.5} BMGs respectively.

BMGs | ^{3} | ||||
---|---|---|---|---|---|

Zr_{38.5}Ti_{16.5}Cu_{15.25}Ni_{9.75}Be_{20} | 3.100 | 115.447 | 1.0565 | 60.237 | 2.0716 |

Zr_{41}Ti_{14}Cu_{12.5}Ni_{10}Be_{22.5} | 3.100 | 119.893 | 1.0699 | 59.945 | 2.0978 |

Zr_{44}Ti_{11}Cu_{10}Ni_{10}Be_{25} | 3.100 | 119.809 | 1.0696 | 59.882 | 2.0974 |

Zr_{45.4}Ti_{9.6}Cu_{10.15}Ni_{8.6}Be_{26.25} | 3.100 | 123.501 | 1.0805 | 59.875 | 2.1187 |

Zr_{46.75}Ti_{8.25}Cu_{7.5}Ni_{10}Be_{27.5} | 3.100 | 124.510 | 1.0834 | 59.710 | 2.1245 |

Zr_{52.25}Cu_{28.5}Ni_{4.75}Al_{9.5}Ta_{5} | 2.957 | 128.860 | 1.1133 | 80.173 | 2.1830 |

Zr_{57}Ti_{5}Cu_{20}Ni_{8}Al_{10} | 3.040 | 127.676 | 1.0385 | 74.493 | 2.1560 |

Zr_{61.88}Al_{10}Ni_{10.12}Cu_{18} | 3.056 | 127.039 | 1.0960 | 76.143 | 2.1459 |

Zr_{64.13}Al_{10}Ni_{10.12}Cu_{15.75} | 3.124 | 130.140 | 1.0968 | 76.874 | 2.1505 |

Zr_{65}Al_{10}Ni_{10}Cu_{15} | 3.150 | 135.985 | 1.1099 | 77.395 | 2.1762 |

Properties | H [13] | T [14] | IU [15] | F [16] | S [17] | Exp. [4, 5, 6] | Other [4, 5, 6] |
---|---|---|---|---|---|---|---|

υ_{L}×10^{5} (cm s^{−2}) | 4.553 | 5.177 | 5.176 | 5.203 | 4.776 | — | 5.174 |

υ_{T}×10^{5} (cm s^{−2}) | 2.629 | 2.989 | 2.988 | 3.004 | 2.757 | — | 2.472 |

B_{T}×10^{12} (dyne/cm^{2}) | 0.645 | 0.834 | 0.834 | 0.842 | 0.701 | 1.148 1.141 | 1.150 |

G×10^{12} (dyne/cm^{2}) | 0.387 | 0.501 | 0.500 | 0.505 | 0.426 | 0.374 | 0.471 |

ξ | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.35 | 0.352 |

Y×10^{12} (dyne/cm^{2}) | 0.966 | 1.251 | 1.251 | 1.263 | 1.065 | 1.013 1.012 | 1.055 |

θ_{D} (K) | 333.0 | 378.7 | 378.7 | 380.6 | 349.4 | — | 327 |

Properties | H [13] | T [14] | IU [15] | F [16] | S [17] | Exp. | Other [4, 5, 6] |
---|---|---|---|---|---|---|---|

υ_{L}×10^{5} (cm s^{−2}) | 4.621 | 5.253 | 5.249 | 5.277 | 4.850 | — | 5.171 |

υ_{T}×10^{5} (cm s^{−2}) | 2.668 | 3.033 | 3.031 | 3.047 | 2.800 | — | 2.485 |

B_{T}×10^{12} (dyne/cm^{2}) | 0.644 | 0.833 | 0.832 | 0.840 | 0.709 | — | 1.119 |

G×10^{12} (dyne/cm^{2}) | 0.386 | 0.499 | 0.499 | 0.504 | 0.426 | — | 0.373 |

ξ | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | — | 0.35 |

Y×10^{12} (dyne/cm^{2}) | 0.966 | 1.249 | 1.247 | 1.260 | 1.065 | — | 1.009 |

θ_{D} (K) | 334.7 | 380.5 | 380.2 | 382.2 | 351.3 | — | 327 |

Properties | H [13] | T [14] | IU [15] | F [16] | S [17] | Exp. [5, 6] | Other [4, 5, 6] |
---|---|---|---|---|---|---|---|

υ_{L}×10^{5} (cm s^{−2}) | 4.634 | 5.266 | 5.265 | 5.294 | 4.866 | — | 5.182 |

υ_{T}×10^{5} (cm s^{−2}) | 2.675 | 3.040 | 3.0403 | 3.056 | 2.810 | — | 2.487 |

B_{T}×10^{12} (dyne/cm^{2}) | 0.641 | 0.828 | 0.828 | 0.836 | 0.707 | 1.134 | 1.137 1.119 |

G×10^{12} (dyne/cm^{2}) | 0.385 | 0.497 | 0.496 | 0.502 | 0.424 | 0.352 | 0.476 0.372 |

ξ | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.35 | 0.352 |

Y×10^{12} (dyne/cm^{2}) | 0.961 | 1.242 | 1.241 | 1.255 | 1.060 | 0.957 | 1.039 1.005 |

θ_{D} (K) | 334.7 | 380.4 | 380.4 | 380.4 | 351.5 | — | 327 |

Presently calculated elastic properties for Zr_{44}Ti_{11}Cu_{10}Ni_{10}Be_{25} and Zr_{38.5}Ti_{16.5}Cu_{15.25}Ni_{9.75}Be_{25} BMG are listed in Tables 4 and 5 respectively. From this table, one can see that the results are obtained due to T, IU and F screening functions are very close to one another as compared to the H screening function. Due to lack of experimental data and other information of elastic 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.

Properties | H [13] | T [14] | IU [15] | F [16] | S [17] | Exp. | Other |
---|---|---|---|---|---|---|---|

υ_{L}×10^{5} (cm s^{−2}) | 4.544 | 5.171 | 5.180 | 5.207 | 4.765 | — | — |

υ_{T}×10^{5} (cm s^{−2}) | 2.624 | 2.986 | 2.991 | 3.006 | 2.751 | — | — |

B_{T}×10^{12} (dyne/cm^{2}) | 0.643 | 0.832 | 0.835 | 0.843 | 0.706 | — | — |

G×10^{12} (dyne/cm^{2}) | 0.386 | 0.499 | 0.509 | 0.506 | 0.424 | — | — |

ξ | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | — | — |

Y×10^{12} (dyne/cm^{2}) | 0.964 | 1.248 | 1.252 | 1.265 | 1.059 | — | — |

θ_{D} (K) | 332.5 | 378.3 | 379.1 | 380.9 | 348.7 | — | — |

Properties | H [13] | T [14] | IU [15] | F [16] | S [17] | Exp. | Other |
---|---|---|---|---|---|---|---|

υ_{L}×10^{5} (cm s^{−2}) | 4.347 | 4.942 | 4.954 | 4.954 | 4.561 | — | — |

υ_{T}×10^{5} (cm s^{−2}) | 2.510 | 2.853 | 2.860 | 2.880 | 2.633 | — | — |

B_{T}×10^{12} (dyne/cm^{2}) | 0.614 | 0.793 | 0.797 | 0.805 | 0.675 | — | — |

G×10^{12} (dyne/cm^{2}) | 0.368 | 0.476 | 0.478 | 0.483 | 0.405 | — | — |

ξ | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | — | — |

Y×10^{12} (dyne/cm^{2}) | 0.920 | 1.189 | 1.196 | 1.208 | 1.013 | — | — |

θ_{D} (K) | 322.0 | 366.1 | 367.0 | 368.9 | 337.9 | — | — |

### 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 Table 7, it is observed that F function is in good agreement with the calculated values and other available results [5]. On the other hand, 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].

Properties | H [13] | T [14] | IU [15] | F [16] | S [17] | Exp. | Other [5] |
---|---|---|---|---|---|---|---|

υ_{L}×10^{5} (cm s^{−2}) | 3.5505 | 4.0124 | 4.0041 | 4.4025 | 3.7471 | — | — |

υ_{T}×10^{5} (cm s^{−2}) | 2.0499 | 2.3166 | 2.3118 | 2.324 | 2.1634 | — | — |

B_{T}×10^{12} (dyne/cm^{2}) | 0.4879 | 0.6230 | 0.6205 | 0.6271 | 0.5434 | — | — |

G×10^{12} (dyne/cm^{2}) | 0.2927 | 0.3738 | 0.3723 | 0.3762 | 0.3260 | — | — |

ξ | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | — | — |

Y×10^{12} (dyne/cm^{2}) | 0.7317 | 0.9346 | 0.9307 | 0.9406 | 0.8151 | — | 0.90 |

θ_{D} (K) | 253.57 | 286.56 | 285.97 | 287.48 | 267.61 | — | — |

### 3.3. 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.

Properties | H [13] | T [14] | IU [15] | F [16] | S [17] | Exp. [4, 5, 6] | Other [4, 5, 6] |
---|---|---|---|---|---|---|---|

υ_{L}×10^{5} (cm s^{−2}) | 3.2622 | 3.6125 | 3.6212 | 3.6375 | 3.3707 | 4.623 | — |

υ_{T}×10^{5} (cm s^{−2}) | 1.8834 | 2.0857 | 2.0907 | 2.1001 | 1.9461 | 2.149 | — |

B_{T}×10^{12} (dyne/cm^{2}) | 0.3867 | 0.4738 | 0.47608 | 0.4804 | 0.4125 | 0.992 | — |

G×10^{12} (dyne/cm^{2}) | 0.2318 | 0.2843 | 0.2857 | 0.2882 | 0.2475 | 0.301 | 0.301 |

ξ | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | — | — |

Y×10^{12} (dyne/cm^{2}) | 0.5795 | 0.7107 | 0.7141 | 0.7206 | 0.6188 | 0.82 | 0.82 |

θ_{D} (K) | 233.70 | 258.79 | 259.92 | 260.58 | 241.47 | 270.3 | 270.1 |

### 3.4. Zr-Al-Ni-Cu BMG (Zr_{61.88}Al_{10}Ni_{10.12}Cu_{18}, Zr_{64.13}Al_{10}Ni_{10.12}Cu_{15.15}, Zr_{65}Al_{10}Ni_{10}Cu_{15})

Our well recognized model potential is used along with five different local field correction function for the generate pair potential for Zr_{61.88}Al_{10}Ni_{10.12}Cu_{18}, Zr_{64.13}Al_{10}Ni_{10.12}Cu_{15.75} and Zr_{65}Al_{10}Ni_{10}Cu_{15} BMG system. From 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}Ni_{10.12}Cu_{18}, Zr_{64.13}Al_{10}Ni_{10.12}Cu_{15.75} and Zr_{65}Al_{10}Ni_{10}Cu_{15} BMG are shown in Figures 8(b), 9(b) and 10(b) respectively. Phonon mode graphs show that the nature of peak position is not much affected by different type of screening functions. The longitudinal and transverse frequencies show deviation for H, T and S functions with respect to F screening for Zr-Al-Ni-Cu BMG systems. It is obvious from the figures that the oscillations are protruding in the longitudinal mode as compared to transverse mode, which indicates that collective excitations at larger wave vector transfer due to the dispersion of longitudinal excitation. The influence of the results of phonon frequencies is clearly observed on the elastic constants, as we have calculated these elastic constants from the long wavelength limit of phonon frequencies.

From the Tables 9, 10, 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. The maximum deviation is found in F screening function, the values are 25.5, 26.9 and 6.2% for Zr_{61.88}Al_{10}Ni_{10.12}Cu_{18}, Zr_{64.13}Al_{10}Ni_{10.12}Cu_{15.75} and Zr_{65}Al_{10}Ni_{10}Cu_{15} respectively, and minimum deviation is found for in H function, the values are found 2.57 and 4.0% for Zr_{61.88}Al_{10}Ni_{10.12}Cu_{18} and Zr_{64.13}Al_{10}Ni_{10.12}Cu_{15.75} respectively.

Properties | H [13] | T [14] | IU [15] | F [16] | S [17] | Exp. [6] | Other [6] |
---|---|---|---|---|---|---|---|

υ_{L}×10^{5} (cm s^{−2}) | 3.6701 | 4.1883 | 4.1731 | 4.1990 | 3.8565 | 4.693 | 4.704 |

υ_{T}×10^{5} (cm s^{−2}) | 2.1189 | 2.4181 | 2.4043 | 2.4243 | 2.2265 | 2.046 | 2.092 |

B_{T}×10^{12} (dyne/cm^{2}) | 0.4975 | 0.6479 | 0.6432 | 0.6513 | 0.5493 | 1.077 | 1.083 |

G×10^{12} (dyne/cm^{2}) | 0.2985 | 0.3888 | 0.3899 | 0.3908 | 0.3296 | 0.293 | 0.291 |

ξ | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.375 | 0.37 |

Y×10^{12} (dyne/cm^{2}) | 0.7463 | 0.9719 | 0.9649 | 0.9769 | 0.8240 | 0.805 | 0.801 |

θ_{D} (K) | 263.355 | 300.544 | 299.451 | 301.31 | 276.73 | 0.263 | 262.9 |

Properties | H [13] | T [14] | IU [15] | F [16] | S [17] | Exp. | Other [6] |
---|---|---|---|---|---|---|---|

υ_{L}×10^{5} (cm s^{−2}) | 3.6694 | 3.4202 | 4.1873 | 4.2100 | 3.8666 | — | 4.679 |

υ_{T}×10^{5} (cm s^{−2}) | 2.1186 | 2.4262 | 2.4175 | 2.4306 | 2.2324 | — | 2.079 |

B_{T}×10^{12} (dyne/cm^{2}) | 0.4940 | 0.6479 | 0.6432 | 0.6502 | 0.5481 | — | 1.066 |

G×10^{12} (dyne/cm^{2}) | 0.2964 | 0.3887 | 0.3859 | 0.3901 | 0.5481 | — | 0.285 |

ξ | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | — | 0.37 |

Y×10^{12} (dyne/cm^{2}) | 0.7410 | 0.9718 | 0.9649 | 0.9754 | 0.8227 | — | 0.785 |

θ_{D} (K) | 261.2 | 299.1 | 298.06 | 299.68 | 275.24 | — | 259.2 |

Properties | H [13] | T [14] | IU [15] | F [16] | S [17] | Exp. [4, 5, 6] | Other [4, 5, 6] |
---|---|---|---|---|---|---|---|

υ_{L}×10^{5} (cm s^{−2}) | 3.3964 | 3.9328 | 3.9199 | 3.9440 | 3.6025 | — | 5.050 |

υ_{T}×10^{5} (cm s^{−2}) | 1.9609 | 2.2706 | 2.2632 | 2.2771 | 2.0799 | — | 2.393 |

B_{T}×10^{12} (dyne/cm^{2}) | 0.4086 | 0.5478 | 0.5443 | 0.5510 | 0.4597 | 1.067 | 1.066 1.034 1.120 |

G×10^{12} (dyne/cm^{2}) | 0.2452 | 0.3267 | 0.3265 | 0.3306 | 0.2758 | 0.31 | 0.303 0.359 |

ξ | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | — | 0.35 |

Y×10^{12} (dyne/cm^{2}) | 0.6129 | 0.8218 | 0.8164 | 0.8265 | 0.6895 | 0.791 | 0.83 |

θ_{D} (K) | 238.2 | 275.9 | 274.9 | 276.6 | 252.7 | 267 | 267 292.9 |

Presently calculated Young’s modulus and Debye temperature using local model potential along with the S screening function is found in good agreement with available results for Zr_{61.88}Al_{10}Ni_{10.12}Cu_{18}, and Zr_{64.13}Al_{10}Ni_{10.12}Cu_{15.75} and IU function is found in good agreement with experimental as well as other available results [6]. On the other hand, the calculated values of Young’s modulus and Debye temperature using model potential along with T, IU and F screening functions are found very close to each other and good agreement with the other theoretical data for Zr-Al-Ni-Cu BMG system.

## 4. 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, 3, 4, 5, 6, 7, 8, 9, 10, 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.