Fysol Ibna Abbas^1*, Shantanu Chakraborty^2,3, Mousume Samad^4,5
1Department of Electrical & Electronics Engineering, Faculty of Science & Technology, City University, Dhaka, Bangladesh.
²Department of Applied Mathematics and Physics, Valdosta State University, Valdosta, Georgia, USA.
³National High Magnetic Field Laboratory, Tallahassee, Florida, USA.
⁴Department of Electrical and Electronic Systems Engineering, Faculty of Engineering, Saitama University, Saitama, Japan.
⁵Department of Information and Communication Engineering, Bangladesh Army University of Engineering and Technology (BAUET), Qadirabad, Bangladesh.
DOI: 10.4236/msce.2025.138001   PDF    HTML   XML   73 Downloads   1,005 Views   Citations

Abstract

The physicochemical property (PHP) and transport coefficients (TC) of liquid aluminum (Al) have been studied based on how they change with temperature (T), using the microscopic first-order perturbation hard sphere (HS) theory of liquid metals. PHP involves surface tension (S_T) and isothermal compressibility (χ_T) of the surface properties. On the other hand, TC properties such as shear viscosity (η) and diffusion (D) are investigated for the same footing. The effective hard-sphere diameter (σ), along with packing fraction (Ω), and the effective pair potential (V_ij(r)) are the basic ingredients of the first-order perturbation microscopic theory in the present inquisition. To facilitate accurate computational analysis, these constituents are assessed utilizing a local pseudopotential and the linearized Weeks-Chandler-Andersen thermodynamic perturbation theory (LWCA). The calculated results, when juxtaposed with the existing experimental data and estimated theoretical values, indicate that the LWCA predicts a slight temperature-dependent divergence in the study of Al. The root cause of this deviation in the uncertainty level of the results for Al is examined, and possible reasons are explained.

Keywords

Surface Tension (S_T), Isothermal Compressibility (_T), Shear Viscosity (η), Diffusion (D), Bretonnet-Silbert Pseudopotential, LWCA Theory

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1. Introduction

Because of numerous uses of PHP (S_T, χ_T) and TC (η, D) in both industry and academia, S_T [1]-[6], χ_T [7]-[14], η [12], and D [13] have drawn the attention of scientists, technologists, and metallurgists, respectively. S_T markedly affects numerous processes, including the nucleation of gas bubbles, gas absorption, the creation of non-metallic inclusions, and metal reactions [1]-[6]. Legionary physical features associated with χ_T into material processing procedures, such as soldering, brazing, sintering, dyeing, and wetting, necessitate an understanding of surface physics for enhanced insight. Moreover, η and D can also have a significant effect in the process of binary solidification and in the regulation of the rate at which biological and new functionality of material growth procedures occur [12] [13]. This suggests that structural and thermodynamic factors might significantly influence surface tension and, consequently, the atomic transport properties of liquid metals. For instance, shear viscosity and diffusion coefficients are two examples of features that can be affected by these processes. Consequently, examining the potential interconnection among the aforementioned traits presents a compelling challenge. Here, a theory that best explains the liquid structure and thermodynamics of the systems being studied is required.

The purpose of investigations of PHP and TC for different thermodynamic states T = 950, 1050, and 1150 K, respectively, is fourfold. First, even though Al is one of the most often utilized metals, little is known about the inherent usefulness of its S_T. Keene [1] investigation demonstrated that S_T of Al at temperatures close to its melting point (943K) is between 850 - 1100 mN/m. This S_T has been measured by different researchers using different methods. The source of this significant dispersion, markedly above the usual error of S_T surface tension measurements (2% - 3%), is unequivocally the heightened sensitivity of the surface characteristics of molten aluminum to oxygen. In the literature review [1]-[6], two distinct groupings of S_T values can be recognized. S_T values ranging from 850 to 900 mN/m were found from meticulous tests conducted by multiple teams under diverse atmospheric conditions either the sessile drop technique or the maximum bubble pressure (MBP) approach. The temperature coefficient of S_T (dS_T/dT) in these studies was found to range from −0.10 to −0.20 mN m⁻¹ K⁻¹ [1]. Comparable values have recently been acquired via the contactless oscillating droplet tech0nique [2], and those were between 1030 and 1100 mN/m. Goumiri and Joud [3] were the first to quantify a high S_T value (1050 mN/m) with a sophisticated sessile drop experiment conducted within an Auger spectroscopy chamber following in situ cleaning of the aluminum surface. These authors assert that experimental results S_T about 850 mN/m pertain to aluminum surfaces coated with a monolayer of adsorbed oxygen. Subsequently, a significant value of approximately 1090 mN/m was observed in MBP tests [4] [5]. Recently, Sarou-Kanian et al. [6] determined $S_T^{mp}$ = 1024 (±48) mN/m by extrapolating to the melting point of Al using a contactless approach at temperatures ranging from 1500˚C to 1900˚C in various neutral or reducing atmospheres. These analyses all indicated a conclusive point that the limitation of experimental facts. Second, only a small number of theoretical studies have been undertaken to examine S_T, χ_T, η, and D sequentially, utilizing the same ingredients from empirical or semi-empirical models [7]-[14] for liquid Al, with only a handful achieving success. Furthermore, to our knowledge, liquid Al has not been investigated for its thermodynamic states at temperatures of 950, 1050, and 1150 K, utilizing the same parameters from the initial conditions to the end results in accordance with any microscopic theory for S_T, χ_T, η, and D, respectively. Third, the dynamic features of interest for theoretical investigation have already been evaluated by many experimentalists [1]-[6]. Fourth, experimental results for static structure factors for elemental liquid Al are not available in the literature at the thermodynamic state in question [4]. Therefore, the structural requirements are crucial for making accurate assumptions about liquid Al at those temperatures.

At the same time, the literature review says that many studies have looked into the statistical mechanical theory and the S_T of certain simple liquid metals, and these reviews have pointed out significant shortcomings. First, a specific study utilizes the approximate value [8]. This value induces divergence if the potential possesses an extensive Friedel-type oscillatory tail. Avoiding this divergence is crucial for achieving the appropriate ST. It is necessary to truncate the integrand at a specific interionic distance to prevent the divergence of the Friedel-type oscillatory tail. Second, the measurement and calculation of S_T using transport coefficients provide significant challenges from both experimental and theoretical viewpoints [9]-[14]. Mayer’s empirical method [7] was utilized to assess the temperature variation and S_T of liquid Al. A first-order approximation of Percus-Yevick [15]-[21] was then employed for the estimation of χ_T. Sutherland’s [22], latterly development technique was applied for the estimation of the η, and D. σ and Ω were the key ingredients of this estimation. The electronic theory of metals employed in this work is based on a local pseudopotential proposed by Bretonnet and Silbert (BS) [21]. The band structure energy calculated from the pseudopotential theory provides the interionic interaction, which in turn is used to calculate static structure factors as well as other physical properties investigated here. It is worth noting that the BS pseudopotential has three parameters (core radius, R_c, softness parameters, a, and valency, Z) to be fixed to perform effective calculations. Here, values of R_c a, and Z are taken from other published work [18]. Once this is done, the rest of the calculation is completely parameter free. Moreover, the BS pseudopotential model has demonstrated efficacy in characterizing the structural [16], thermodynamics [17]-[19], and transport features [20] [21] of less simple liquid metals and their alloys. Another important ingredient of the present microscopic approach is the partial correlation function, g_ij(r), which describes the liquid structure of the constituent ions in the liquid state. In order to determine this function, we have employed the thermodynamic perturbation theory [23] as simplified further by Meyer et al. [7]. The latter theory is known in literature as the Linearized Weeks-Chandler-Andersen (LWCA) theory [24]-[26].

This paper follows a structured format. Section 2 provides a concise discussion that describes briefly relevant theories that are used in the calculations. The results of the calculation are presented and discussed in Section 3. We conclude this article with some remarks in Section 4.

2. Theory

2.1. Effective Partial Pair Potential

The local pseudopotential for the i-th component of a metallic alloy may be modeled as a superposition of two terms, [21] one inside and another outside the core,

$Y_i\left(r\right)=\{\begin{array}{ll}{\displaystyle \sum }_{m=1}^2{B}_m^i\exp \left(-\frac{r}{m{a}_i}\right)\hfill & \text{if}r<R_{ci}\hfill \\ -\frac{Z_i}{r}\hfill & \text{if}r>R_{ci}\hfill \end{array}$ (1)

where a, R_ci and Z denote the softness parameter, core radius and the effecttive s-electron occupancy number, respectively. The term outside the core is just the bare Coulomb interaction (in atomic units) between a conduction electron and an ion. The contribution inside the core is contributed by the first two terms of the Dirichlet series arising from the inverse scattering approach. For details, see Ref [18]. The coefficients of expansion in the core depends on the parameters a_i, R_ci, and Z_i. Finally, the partial interionic interaction between i-th and j-th ions is

$V_{ij}\left(r\right)=\frac{Z_i{Z}_j}{r}\left[1-\frac{2}{\pi }{\displaystyle \int \text{d}q}{F}_{ij}^N\frac{\sin \left(qr\right)}{q}\right]$ (2)

where the normalized energy wave number characteristics W_i(q), in Equation (3), denotes the unscreened

$F_{ij}^N={\left[\frac{q^2}{\pi a\rho \sqrt{\left(Z_i{Z}_j\right)}}\right]}^2{W}_i\left(q\right)W_j\left(q\right)\left[1-\frac{1}{\epsilon \left(q\right)}\right]\left[\frac{1}{1-G\left(q\right)}\right]$ (3)

form factor of the i-th com ponent obtained from the Fourier transform of W_i(r) (see Equation (9)), ε(q) and G(q) are dielectric function and the local field factor in momentum space, respectively, with q as the amount of momentum transferred. These functions are taken from Ichimaru and Utsumi [27] because their theory satisfies both the compressibility sum rule and the short-range correlation condition.

2.2. LWCA Theory

The LWCA technique [24]-[26] was utilized in this work to calculate HSD Because it is an easily understood and theoretically accessible theory. It is well known that the first principles perturbation theory requires such a reference system which can closely resemble the concerned real system [28]. There are many experimental as well as theoretical evidence that the HS theory of liquid can describe the structure of simple and transition metals [29] [30] and their binary alloys [15]-[19]. Being prompted by the above history of success we employ HS reference system for liquid Al within the LWCA thermodynamic perturbation theory proposed by Meyer et al. [7] was derived from the WCA theory [25]. The Blip function in the theory stands as,

$B\left(r\right)=Y_{\sigma }\left(r\right)\left[\exp \left[-\beta v\left(r\right)\right]-\text{exp}\left[\beta v_{\sigma }\left(r\right)\right]\right].$ (4)

Here $v\left(r\right)$ , and $v_{\sigma }\left(r\right)$ Here, $v\left(r\right)$ and $v\left(r\right)$ denote soft and hard sphere, HS potentials, respectively. $\beta$ is the inverse temperature divided by the Boltzmann constant and $Y_{\sigma }\left(r\right)$ is the hard sphere cavity function and it is continuous at r = σ; σ being the effective HSD. In the linearized version of the WCA, σ is obtained from the solution of the transcendental equation [31]. Our previous work [18] mentions the solution to this equation.

2.3. Pair Distribution Function

In order to have numerical values for partial correlation function, we first calculating the Ashcroft-Langreth (AL) partial static structure factors, [32] S_ij(q), and then take a Fourier trans- form of it,

$g_{ij}\left(r\right)=1-+\frac{1}{{\left(2\pi \right)}^3\rho \sqrt{x\left(1-x\right)}}{\displaystyle {\int }_0^{\infty }\left(S_{ij}\left(q\right)-{\delta }_{ij}\right){\text{e}}^{iq\cdot r}{\text{d}}^{3}q}$ , (5)

where ρ is the ionic density of alloys. We note that calculation of S_ij(q) requires the knowledge of the effective hard sphere diameters, σ_ij, which is obtained by using the linearized WCA thermodynamic perturbation theory [33].

2.4. Surface Tension (S_T), Isothermal Compressibility (χ_T), Shear Viscosity (η) and Diffusion (D)

Liquid metals have a higher density than other common liquids. As a result, clarifying liquid metals requires sophisticated concepts. The idea of repulsive intermolecular forces determines the structure of liquid metals has given rise to various ideas [34]-[37]. Molecular configuration governs intermolecular interactions. A hard-sphere potential is typically, entirely repulsive. Therefore, when modeling the interactions in real liquid systems using a rigid sphere potential, one should only reproduce the repulsive part of the potential [34]. As an alternative to the integral equation approach, a rational function approximation can be used to express the S_T equation obtained from the first-order approximation of the Percus-Yevick solution as follows:

$\gamma =\frac{\delta \eta k_{B}T\left(2+\Omega \right)}{4{\left(1-\Omega \right)}^2}$ (6)

This study employs Equations (6) to compute the S_T of the relevant system. In order to estimate the isothermal compressibility (χ_T), the first-order approximation of the Carnahan and Stirling [38] solution has been used, taking into account the rational function approximation. This approximation can also be adjusted as an alternative to the integral equation method for hard sphere fluid. The mathematical equation is presented below:

${\chi }_T=\frac{{\left(1-\Omega \right)}^4}{\rho k_{B}T\left\{2\Omega \left(4-\Omega \right)+{\left(1-\Omega \right)}^4\right\}}$ (7)

where ρ is the ionic number density, k_B is the Boltzmann constant and T is the temperature. Furthermore, equations (8) and (9) have been utilized to get the transport coefficients η and D. Equation (8) has been examined within the context of the assumptions stated by Born and Green [14]. Conversely, equation (9) was also formulated by Sutherland [22] et al. from the original Stokes-Einstein equation utilizing hydrodynamic theory, as detailed in the following two equations:

$\eta =\frac{16m^{\frac{1}{2}}}{15{\left(k_{B}T\right)}^{\frac{1}{2}}}{S}_T,$ (8)

and,

$D=\frac{15k_{B}T}{32\pi \sigma S_{T}m}\frac{{\left(k_{B}T\right)}^{1/2}}{m^{1/2}}.$ (9)

where, m is the atomic mass of the liquid atoms.

3. Results and Discussion

In this section, we have presented the results of calculations for the S_T, χ_T, η, and D of the simple liquid Al metal at temperatures of 950 K, 1050 K, and 1150 K, respectively. The reason for choosing this liquid system due to its generally accepted lack of sd hybridization effects [37]. Besides, Al is a heavy polyvalent metal, and are sometimes difficult to handle theoretically, in the framework of empirical or semi empirical models. So, we have employed the microscopic self-consistent theory within the conjunction of pseudopotential as BS model potential to describe the interionic interaction of the Al simple metal for revealing the liquid state features with temperature variation effect. Three parameters of the BS model are the core radius, R_c, the softness parameter, a, and the valence Z. The value of R_c is generally fixed by fitting physical properties of the concerned systems [39]. We have taken the values of R_c for Al from Ref [18] [37]. The value for this is 1.91 au. Furthermore, at a core radius of R_c=1.91 au, e pair potential absence a local minimum; instead, it exhibits a principal minimum succeeded by Friedel oscillations [20]. This investigation was undertaken in order to avoid the local minimum

Table 1. Input parameters and calculated results.

Temperature, T (K)	Ionic number Density, ρ (Å-3)	Hard sphere diameter, σ (Å)	Packing fraction, Ω	Potential minimum (eV)	Valency, Zs	Core radius, Rc (a.u.)	Softness parameter, a (a.u.)
950	5.314	2.80545	0.61437	−0.0011	3	1.91	0.49
1050	5.251	2.80245	0.60514	−0.00127	3	1.91	0.49
1150	5.189	2.80954	0.60255	−0.0014	3	1.91	0.49

effect of this R_c, which generates the effective potential pattern near −0.0011 eV. The softness parameter ‘a’ is determined by the best fitting of the LWCA structure factor to the experimental ones. The values of ‘a’ thus found for Al 0.49 au, respectively which is taken from our previous work [18] [37]. Regarding the third parameter called effective s-electron occupancy number, Z_s, we took the chemical valence 3 for Al. We note here that, the dielectric function plays an important role in determining the effective potential profile [20]. In this work we have used the dielectric function proposed by Icimaru and Utsumi [27], because of their theory satisfies the compressibility sum rule and the short-range correlation conditions for the wide range of metallic density as well avoid the local minimum.

Figure 1. Pair potential profile for liquid Al at (a) 950 K, (b) 1050 K, and (c) 1150 K, respectively.

For calculating S_T and χ_T, we have used the relation in equations (6), and (7), sequential order. On the other hand, equation (8), and (9) employed for the estimation of η, and D. The major ingredients of the study of all the equations required are the packing fraction, Ω, and the temperature dependent effective HS diameter, σ(T). We have ascertained them utilizing BS pseudopotential adjacent to LWCA perturbation theory.

Table 1 enumerates the input values that were subsequently obtained. We have applied to the LWCA theories of liquid structures to achieve them. The σ(T) from LWCA is smaller than the one from an earlier study [18] [37] on Al-based alloys at a low temperature (973 K), because the σ(T) in LWCA shows how far the origin is from the point where g(r) starts to be greater than zero. The graphical solution of the transcendental equation of equations used to estimate S_T is utilized to estimate σ(T) for LWCA theory. Ω is then calculated from the relation, Ω = πσ³n/6 [32]. Using these estimated values, the effective interionic interaction, V(r), for the systems under consideration are calculated and shown in Figure 1(a-c). According to the potential profile shown in Figure 1, the potential well’s depth reaches its maximum at T = 1150 K. However, at 950 K, we have seen that the distance to the initial minimum reaches its lowest values. The depth of the well is determined by a subtle combined effect of the ionic density and the value of the valence of the metal. Besides, both repulsive and attractive forces must be exquisitely balanced to determine the potential well’s depth. As a result, the potential well’s depth varies from one metal to another and also showed a strong correlation

Figure 2. Partial pair correlation function for liquid Al at (a) 950 K, (b) 1050 K, and (c) 1150 K, respectively.

with varying parameters. A key factor in defining repulsive and attractive contributions to the interionic effective pair potential is the softness parameter, a. Moreover, in the present study consideration has been taken to avoid the local minima effect in the calculation to the principal minimum succeeded by Friedel oscillations. Perhaps this may be the reason for the potential well depth in the order of 10⁻⁴ (−0.0011eV). It is worth noting here that, the BS-model uses the empty core potential outside the core.

As can be seen in Figure 2(a)-(c), temperature has an effect on pair correlation function. In this work, the pair correlation function, denoted by g(r), of liquid aluminum is demonstrated at various temperatures. Besides we can see from Figure 2(a)-(b), the height of the major peak reduces slightly as the temperature rises, while the peak position stays virtually the same. This observation has been noted previously [19] [20]. We found here that general agreement among g(r) for temperature fluctuations were better than the results obtained by previous researchers Bhuiyan et al. [19].

Figure 3(a)-(d) showed the χ_T, η, and D respectively. After comparing the present results of this study to experimental results, similar deviations have also been observed for them to the liquid polyvalent Al. It is believed that the same phenomena have also reflected for these microstructure properties. Besides, the uncertainty results for χ_T is 60%, on the other hand η, and D possesses 20% and 30 % respectively. Comparing among the investigated results to the sensitivity, it is found that the isothermal compressibility (χ_T) has the highest sensitive tendency which correlates electron density fluctuation has the significant impact on these properties. In addition, the calculated results for η, and D for the temperature range except for χ_T which has compared to the melting point experimental results demonstrated the similar tendency of decreasing (η), increasing (D) and increasing (χ_T) with the following temperature range for their corresponding experimental results [36]-[42]. However, in the present study, we have used microscopic theory involving perturbation scheme and the statistical mechanics. Here, three parameters R_c, a and Z_s are chosen only to calculate effective potentials. Once it is done, the rest of the calculations are completely parameter free. Moreover, calculations from the theoretical point of view are also self-consistent and, as a result the accuracy of calculations and the predictability of the theory is much more reliable than the empirical and semi-empirical methods. For each property measurement that was looked at, there is a difference between the experimental results and the estimated values because the same components and footing were used to get the results. Moreover, the deviation for χ_T was estimated as 10⁻² order in range (Figure 3(d)). Furthermore, the uncertainty to sensitivity rearranges, then the studied properties can be found in the order, χ_T > D > η > S_T.

Three possible manifestations of this divergence from the assessed results may arise. First, perhaps the underlying reason is that the BS potential has both an attraction and a repulsion component, LWCA theory only considers the repulsive portion of the potential profile derived from the BS pseudopotential. Overcoming

Figure 3. (a) Temperature dependence of Surface tension, S_T is plotted as a function of temperature for liquid Al. Square black color present the experimental results [1], and red circles present the calculated results for this study. Other calculated results are plotted in (b) isothermal compressibility (χ_T) [36], (c) shear viscosity (η) [41], and (d) diffusion coefficient (D) [42] for liquid Al. η and D compare with their melting temperature data.

the attractive part of the calculation could be the cause of this slight variation in the surface tension results. Second, the local field modifications suggested by Vashishta and Singwi (VS) [39] in the BS pseudopotential are crucial for obtaining an accurate assessment of the potential profile as illustrated in Figure 1(a)-(c). It (Figure 1(a)-(c)) reveals that the potential depth increases and the first minimum shifts marginally to the right, which correlates with the significance of local field correction in the pseudopotential analysis. So, the field correction is essential for authentic estimation. Third, there is a possibility bridization effect of liquid Al may occur, as demonstrated in Al cluster formation [43] [44]. In the literature [43] [44], it is mentioned that Al has hybridization effect as exhibited in Al Cluster. In addition, this study has been conducted the microstructural properties without considering the hybridization effect. Perhaps, if the hybridization effect may consider then there is a possibility to reduce the discrepancy between the present study and the experimental results.

The calculated values for physicochemical (S_T, χ_T,) and transport coefficients (η, D) are presented in this study. These properties are found to be deviated for the polyvalent liquid Al metallic element. A possible correlation between the hybridization effect and liquid Al was primarily confirmed [43]-[45]. Moreover, a strong connection shows the necessity of local field correlation in the pseudopotential application, which is significant for accurate estimation.

4. Conclusion

The findings of the S_T, χ_T, η, and D calculation for liquid aluminum are presented in this article. The temperature changes of the hard sphere diameter were assessed using the LWCA theory within the BS pseudopotential. Mayer’s method has been utilized to calculate S_T for the liquid Al system across multiple temperature ranges. Carnahan and Stirling first order approximation was utilized for the mechanical profile evaluation as well χ_T. Sutherland et al. modification of the original Stokes-Einstein equation was employed for the estimation of η, and D. The result analysis suggested that the structure involving the full potential profile is significant for accurate numerical calculations. Moreover, a possible hybridization effect may also exist for the formation of the liquid Al structure. Further study is also needed to reveal the microstructural phenomenon of liquid Al.

Acknowledgements

The author expresses his sincere appreciation to Prof. Dr. Abdur Razzaque Khan (Faculty of Social Science, University of Dhaka, Bangladesh), M.M. Rahman, BCS EDUCATION CADER (34^th) and material science researcher for giving fruitful suggestions, and the study environment to finish this project.

Author Contributions

FIA: Conceptualization, Investigation, Writing-original draft, Plotting figures, Methodology, Software, review & editing, Supervision; S.C: Writing-Original, review & editing.

Data Availability

The raw/processed data required to reproduce these findings can be shared upon depending on request.

Conflicts of Interest

The authors of the study assert that there are no conflicts of interest.

References

[1]	Keene, B.J. (1993) Review of Data for the Surface Tension of Pure Metals. International Materials Reviews, 38, 157-192.[CrossRef]
[2]	Egry, I., Schneider, S., Seyhan, I. and Volkmann, T. (2001) Surface Tension Measurements of High Temperature Metallic Melts. Trans JWRI, 30, 95.
[3]	Goumiri, L. and Joud, J.C. (1982) Auger Electron Spectroscopy Study of Aluminium-Tin Liquid System. Acta Metallurgica, 30, 1397-1405.[CrossRef]
[4]	Pamies, A., Garcia Cordovilla, C. and Louis, E. (1984) The Measurement of Surface Tension of Liquid Aluminium by Means of the Maximum Bubble Pressure Method: The Effect of Surface Oxidation. Scripta Metallurgica, 18, 869-872.[CrossRef]
[5]	Garcia-Cordovilla, C., Louis, E. and Pamies, A. (1986) The Surface Tension of Liquid Pure Aluminium and Aluminium-Magnesium Alloy. Journal of Materials Science, 21, 2787-2792.[CrossRef]
[6]	Sarou-Kanian, V., Millot, F. and Rifflet, J.C. (2003) Surface Tension and Density of Oxygen-Free Liquid Aluminum at High Temperature. International Journal of Thermophysics, 24, 277-286.[CrossRef]
[7]	Mayer, S.W. (1963) A Molecular Parameter Relationship Between Surface Tension and Liquid Compressibility. The Journal of Physical Chemistry, 67, 2160-2164.[CrossRef]
[8]	Evans, R. (1974) A Pseudo-Atom Theory for the Surface Tension of Liquid Metals. Journal of Physics C: Solid State Physics, 7, 2808-2830.[CrossRef]
[9]	Sonvane, Y.A., Thakor, P.B. and Jani, A.R. (2012) Atomic Transport and Surface Properties of Some Simple Liquid Metal Using One Component Plasma System. Journal of Theoretical and Applied Physics, 6, 43.[CrossRef]
[10]	Fowler, R.H. (1937) A Tentative Statistical Theory of Macleod’s Equation for Surface Tension, and the Parachor. Proceedings of the Royal Society A, 159, 229.
[11]	Korkmaz, S.D. and Korkmaz, Ş. (2010) Investigation of Surface Properties of Liquid Transition Metals: Surface Tension and Surface Entropy. Applied Surface Science, 257, 261-265.[CrossRef]
[12]	Egry, I. (1993) On the Relation between Surface Tension and Viscosity for Liquid Metals. Scripta Metallurgica et Materialia, 28, 1273-1276.[CrossRef]
[13]	Yokoyama, I. (1998) A Relationship between Excess Entropy and Diffusion Coefficient for Liquid Metals near the Melting Point. Physica B: Condensed Matter, 254, 172-177.[CrossRef]
[14]	Born, M. and Green, H.S. (1947) A General Kinetic Theory of Liquids III. Dynamical properties. Proceedings of the Royal Society A, 190, 455.[CrossRef] [PubMed]
[15]	Abbas, F.I., Bhuiyan, G.M. and Kasem, R. (2020) Critical Properties of Segregation for Al1−xBix Liquid Binary Alloys. Journal of the Physical Society of Japan, 89, Article 114004.[CrossRef]
[16]	Abbas, F.I. and Sugiyama, M. (2025) Zno-thin Film Growth Processes: Correlation between the Structural Properties of Hydrochloric Acid (HCl) and Water (H2O) Solution Effect Using Innovative Electrostatic Spray Deposition (ESD) Technology. Latvian Journal of Physics and Technical Sciences, 62, 68-76.[CrossRef]
[17]	Abbas, F.I. and Bhuiyan, G.M. (2023) Atomic Transport Properties and Liquid-Liquid Phase Separation of ZnxBi1−x Liquid Monotectic Alloys. Journal of Physics: Condensed Matter, 35, Article 324001.[CrossRef] [PubMed]
[18]	Abbas, F.I. and Bhuiyan, G.M. (2022) A Study of Thermodynamics of Mixing for Al1−xZnx Liquid Binary Alloy. Physica B: Condensed Matter, 647, Article 414365.[CrossRef]
[19]	Abbas, F.I., Bhuiyan, G.M. and Ahmed, A.Z.Z. (2016) Energy of Mixing and Entropy of Mixing for Cux Al1-x Liquid Binary Alloys. Materials Science. arXiv:1607.05827.
[20]	Bhuiyan, G.M. and Abbas, F.I. (2019) Local Minimum in Pair Potentials of Polyvalent Metals: A Limitation of Pseudopotential Theory. International Journal of Modern Physics B, 33, Article 1950049.[CrossRef]
[21]	Bretonnet, J.L. and Silbert, M. (1992) Interionic Interactions in Transition Metals. Application to Vanadium. Physics and Chemistry of Liquids, 24, 169-176.[CrossRef]
[22]	Sutherland, W. (1905) A Dynamical Theory of Diffusion for Non-Electrolytes and the Molecular Mass of Albumin. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 9, 781-785.[CrossRef]
[23]	Waseda, Y. (1984) The Structure of Non-Crystalline Materials. McGraw-Hill.
[24]	Chandler, D. and Week, J.D. (1970) Equilibrium Structure of Simple Liquids. Physical Review Letters, 25, 149.
[25]	Weeks, J.D., Chandler, D. and Andersen, H.C. (1971) Perturbation Theory of the Thermodynamic Properties of Simple Liquids. The Journal of Chemical Physics, 55, 5422-5423.[CrossRef]
[26]	Andersen, H.C. and Chandler, D. (1972) Optimized Cluster Expansions for Classical Fluids. I. General Theory and Variational Formulation of the Mean Spherical Model and Hard Sphere Percus-Yevick Equations. The Journal of Chemical Physics, 57, 1918-1929.[CrossRef]
[27]	Ichimaru, S. and Utsumi, K. (1981) Analytic Expression for the Dielectric Screening Function of Strongly Coupled Electron Liquids at Metallic and Lower Densities. Physical Review B, 24, 7385-7388.[CrossRef]
[28]	Bransden, B.H. and Joachen, C. J. (1997) Introduction to Quantum Mechanics. Longman Group UK Ltd., 355.
[29]	Demmel, F. and Jakse, N. (2025) Diffusion in Liquid Metals Is Directed by Competing Collective Modes. Physical Review B, 111, L081104.[CrossRef]
[30]	Finster, K.L., Krebs, E.J., May, C.J., Kreitzberg, P.A. and Roundy, D. (2022) Soft Fundamental Measure Theory Functional for the Weeks-Chandler-Andersen Repulsive Potential. Physical Review E, 106, Article 064134.[CrossRef] [PubMed]
[31]	Rosenfeld, Y. (1986) Comments on the Variational Modified-Hypernetted-Chain Theory for Simple Fluids. Journal of Statistical Physics, 42, 437-457.[CrossRef]
[32]	Ashcroft, N.W. and Langreth, D.C. (1967) Structure of Binary Liquid Mixtures. I. Physical Review, 156, 685-692.[CrossRef]
[33]	Phuong, L.D., Pasturel, A. and Manh, D.N. (1993) Effect of S-D Hybridization on Interatomic Pair Potentials of the 3d Liquid Transition Metals. Journal of Physics: Condensed Matter, 5, 1901-1918.[CrossRef]
[34]	Protopapas, P., Andersen, H.C. and Parlee, N.A.D. (1973) Theory of Transport in Liquid Metals. I. Calculation of Self-Diffusion Coefficients. The Journal of Chemical Physics, 59, 15-25.[CrossRef]
[35]	Shimoji, M. (1977) Liquid Metals: An Introduction to Physics and Chemistry Metals in the Liquid State. Academic Press, 6.
[36]	Molina, J.M., Voytovych, R., Louis, E. and Eustathopoulos, N. (2007) The Surface Tension of Liquid Aluminium in High Vacuum: The Role of Surface Condition. International Journal of Adhesion and Adhesives, 27, 394-401.[CrossRef]
[37]	Abbas, F.I., Bhuiyan, G.M. and Riad Kasem, M. (2018) A Study of Thermodynamics of Mixing for Al1−xSnx Liquid Binary Alloy. Journal of Non-Crystalline Solids, 481, 391-396.[CrossRef]
[38]	Carnahan, N.F. and Starling, K.E. (1969) Equation of State for Nonattracting Rigid Spheres. The Journal of Chemical Physics, 51, 635-636.[CrossRef]
[39]	Hausleitner, C., Kahl, G. and Hafner, J. (1991) Liquid Structure of Transition Metals: Investigations Using Molecular Dynamics and Perturbation-and Integral-Equation Techniques. Journal of Physics: Condensed Matter, 3, 1589-1602.[CrossRef]
[40]	Vashishta, P. and Singwi, K.S. (1972) Electron Correlations at Metallic Densities. V. Physical Review B, 6, 875-887.[CrossRef]
[41]	Assael, M.J., Kakosimos, K., Banish, R.M., Brillo, J., Egry, I., Brooks, R., et al. (2006) Reference Data for the Density and Viscosity of Liquid Aluminum and Liquid Iron. Journal of Physical and Chemical Reference Data, 35, 285-300.[CrossRef]
[42]	Demmel, F., Szubrin, D., Pilgrim, W.-C. and Morkel, C. (2011) Diffusion in Liquid Aluminium Probed by Quasielastic Neutron Scattering. Physical Review B, 84, Article 014307.[CrossRef]
[43]	Li, X., Wu, H., Wang, X. and Wang, L. (1998) s-p Hybridization and Electron Shell Structures in Aluminum Clusters: A Photoelectron Spectroscopy Study. Physical Review Letters, 81, 1909-1912.[CrossRef]
[44]	Ganteför, G. and Eberhardt, W. (1994) Shell Structure and s-p Hybridization in Small Aluminum Clusters. Chemical Physics Letters, 217, 600-604.[CrossRef]
[45]	Abbas, F.I. and Chakraborty, S. (2025) A Systematic Investigation of Temperature Variation Surface Tension of Liquid Al. American Journal of Science Education Research, 4, 1-6.