Chapter 5 Ph D (Thesis) # 5.1

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1 Chapter 5 Ph D (Thesis) # Structure and properties of liquid metals Introduction As said earlier, developing an understanding of the liquid state (l state) of matter is a formidable challenge, especially for simple metals (mono valant) and non simple (poly valant) liquids. These metals have been extensively studied in the crystalline phase, but the liquid properties for these have not received similar attention. Although they are technologically important, the high melting points, and the high reactivity of liquids make experimental examinations much more difficult. Theoretical studies of liquids have also been somewhat limited owing to the complexity of the problem, i.e, large numbers of atoms in motion with little symmetry [5.1 5.]. One can envision two limiting states of matter, an ideal gas and a perfect crystal. In the ideal gas limit, the atoms interact rarely and randomly. As a first approximation, the atoms can be treated as point particles undergoing elastic collisions. In the perfect crystal limit, atoms are associated with a lattice site. They can be described by a fixed periodic array of interacting point particles. Liquids reside between these two limits. The atoms may strongly interact and do so without long range correlations. They may exhibit local order as for a solid state, but such ordering is temporal and transient fluctuations occur as in a gaseous state. This temporal nature of the l state can be described by statistical measures [5.3]. As described earlier, the microstructure of the liquid can be characterized by a radial distribution or pair correlation function, which gives an average interatomic distance. The aim of physics of liquid is to understand why particular phases are stable in particular ranges of temperature, density, atoms, ions with structural and dynamical properties of fluids phases to size and shape of the molecules, and the nature of force between them. A century of efforts since the pioneering work of Van der Waals has led to the fairly complete basic understanding of the static and dynamical physiochemical properties of liquids. The advances in stastical mechanics and (the fundamental formulations of Gibbs and Boltzmann, integral equations and perturbation theories, computer simulations) in knowledge of intermolecular force and in experimental techniques have all contributed to this. Existence of liquid seemed little mysterious fifty years ago, but today one can make fairly precise prediction of the solid liquid, gas phase

2 Chapter 5 Ph D (Thesis) # 5. diagram and of the microscopic and macroscopic, static and dynamical properties of liquid [5.4]. Liquid metals are differing from the other classes of liquids considered so far primarily through the presence of conduction electrons. Although the theory of liquid metals much common with that of the other ionic liquids, but the problem is complicated by the fact that the electronic components requires a quantum mechanical treatment. Many elements show metallic behaviour in the liquid state, but their electronic band structures can differ widely. We shall restrict ourselves to the so called simple liquids metals. This class of simple metals comprises those in which the electronic valance states are well separated in energy from the tightly bound core states: their properties are reasonably well describes by the nearly free electron model. Liquid metals are different from the others by long range Coulomb forces, short range order, softer over lap effects with electrical screening effects. Given the nature of the liquid state, it is imperative to construct accurate and representative ensembles [ ]. The ensemble must be sufficiently large to capture the essential features of the l state, but it should not be so large as to preclude an accurate numerical simulation. The ensemble should also contain accurate interatomic interactions at a user specified temperature and density. Replicating the interatomic interactions is the most difficult aspect in the process of simulating an accurate ensemble. Ideally, one has to treat the interactions in a fully quantum mechanical fashion. In this way, interatomic interactions involving metallic and covalent bonding, hybridization changes, or charge transfers must be handled accurately and correctly. Only recently it has been possible to apply quantum methods to complex systems such as liquids. Early theoretical studies attempted to use empirical, many body interatomic potentials, or semi empirical approximations [ ]. Typically, three approximations are made for quantum based simulations: (1) The Born Oppenheimer approximation. This approximation treats the nuclei as classical particles and allows one to focus only on the electronic degrees of freedom [5.7]. () Density functional theory [5.8] with different base functions. This theory allows one to map the fully many electron theory onto a one particle problem. (3) The pseudopotential approximation [5.1, 511].

3 Chapter 5 Ph D (Thesis) # 5.3 The pseudopotential approximation reflects the physical content of the Periodic Table. Namely, the chemical nature of matter is determined solely by the valence electron configuration. These approximations are also captured within the pseudopotential density functional method (PDFM) [5.1]. The PDFM allows one to treat relatively large ensembles of atoms in the liquid state. It has opened up a new frontier of so called ab initio simulations, i.e., simulations without any a priori assumptions about the nature of the interatomic forces. Several fundamental issues can be addressed with ab initio simulations. For example, structural properties of liquids cannot be directly extracted from experimental studies. Only distribution functions or correlation functions can be measured. One cannot obtain snap shots of the atomic structure of the melt from experiment [5.13]. In contrast, theoretical simulations provide a direct window on the microscopic structure of the l state at any instant of time. Simulations allow one to assess the microstructure, the dynamical and electronic properties, and the nature of the bonds in the liquid state. Moreover, by including an explicit description of the electronic states, it is possible to explore questions about the electronic properties of the liquid such as the transport and dynamical properties of the liquid. Such issues are difficult, if not impossible, to address via classical simulations as these methods do not include a description of the electronic degrees of freedom. [ ] One of the important property while characterizing a liquid metal is its static structure factor. The static structure factor S(q) is a measure of particle correlation in the reciprocal space, and from the result one can answer the question that what structural feature changes as material passes through a transition and hence accurate knowledge of this quantity is vital for studying numerous properties of liquid metals [5.14]. Given the discussion, in the present study calculation of structural and transport properties of liquid Mg, Zn, Al, In, and Tl metals. Since all these metals are hybridizing in a more or less similar fashion and having more than one valance electrons in the outer cell, it is important to focus the attention in the computation of atomic transport properties of above listed metals of group II to VI. Most of the earlier attempts to compute structure factor S(q) of liquid metals has the system of neutral hard sphere as their reference system. These models are quite

4 Chapter 5 Ph D (Thesis) # 5.4 useful for studying inert gas liquids, mono and poly valent metals. This model [5.15] is very effortless and more generally used in early period of time Percus Yevick (PY) Hard sphere model One of the earliest attempts to compute structure factor S(q) of the liquid metals is the system of neutral hard sphere as a reference system [5.15]. Among these the most straightforward and extensively used model and is obtained from the exact solution of the Percus Yevick equation for Hard Sphere (PY) [5.16] diameter(σ ), which is obtained, to get best fit of S(q) with the experimental data also known as Ashcroft Lekner model [5.15]. The PY integral equation to yield the structure factor using the expression as ( σ ) S q = 1 ( 1 nc( qσ )) (5.1) Where σ is Hard sphere diameter and the direct correlation function c qσ in momentum space is yield as follows. 1 3 sin( sqσ ) cq ( σ ) = 4πσ dss ( α + β s+ γ s ) (5.) sq 0 The parameter α, β and γ are the functions of packing density parameter η, the function of total fluid volume occupied by the sphere defined as, π 3 η = 6 n σ (5.3) ( 1+ η ) α = 4 ( 1 η) (5.4) η 6η ( 1+ ) β = 4 ( 1 η) and 1 η( 1+ η) γ = 4 ( 1 η) (5.5)

5 Chapter 5 Ph D (Thesis) # 5.5 On integrating the equation (5.) over the volume of the sphere, the expression for the structure factor turns out to be 4η S( q) = 1+ 4 (1 ) η y 6 3 (1 + η ) y (sin y y cos y) η 6η 1 + y η + (1 + η ) { ysin y ( y )cos y } 3 {(4 y 4y)sin y ( y 1y + 4) cos y + 4} 1 (5.6) In which y = qσ. In the PYHS reference system no effect is accounted for the softness of potential which plays significant role in determining the structures of liquid metals. Since the hard sphere diameter (σ) is left undetermined in the perturbation theory, gives us the freedom to use it for fitting the computational results with the experimental results Charged hard sphere (CHS) model Another method to predict the structure factor is the charged hard sphere model. This model has some attractive features as in this liquid metals are treated as a system of charged hard sphere (CHS) in back ground of interacting electrons. Such a system of CHS in uniform background was first studied by Palmer and Weeks [5.17] in a mean spherical approximation inside the core. With the effect of this model as a reference, the effect of responding electrons is taken in to account in a linear response approximation. The resulting expression for the structure factor S(q) involves the CHS diameter as the only free parameter. If the charge on the hard sphere is taken as zero, then the expression turns out to be familiar Ashcroft Lekner hard sphere [5.15] model. In the framework of mean spherical approximation inside the core and outside the core, a perturbation in the form of Coulomb interaction is assumed to act by Palmer and Weeks [5.17] the direct correlation function is given by [ ].

6 Chapter 5 Ph D (Thesis) # r r r r 0 ( ) = < σ, C r A B C D E r σ σ σ σ γ = r > σ. r σ ( ) (5.7) Ze In which γ = β, gives the Coulomb interaction potential in which σ is charged ε σ 0 hard sphere diameter, Ze the ionic charge, and β is the reciprocal of the product of Boltzmann constant k β and absolute temperature T, ε is the static dielectric constant of the medium. Since the electron background is uniform, it is taken as unity. Hoverever the iron cores have finite dimensions so their polarization should be slightly greater than unity. But for sake of simplicity in most of the application it is taken as unity [5.]. The coefficients A, B, C, D and E appearing in the equation (5.7) are expressed in terms of the packing fraction η as ( 1+ η) Q ( 1+ η) QK ( 5 + η ) A = + 4 ( 1 η) 41 ( η) 1η 60η (5.8) B 6 = ηm, K 6 C =, D η ( )( A K U) = + and E = ηk (5.9) ( 1+ η ) 1 ( η) K With Q = ( 1 η) ( 1+ η) (5.10) η ( 1+ ) Q M = and 4η ( 1 η) η 1+ η ( 1 η)q U = 1η 1ηK (5.11) The K is given by K = (4ηγ) 1/ is the inverse screening length due to Debye Huckel [ ]. The static structure factor is related to the Fourier transform of the direct correlation function in the following way. S 0 ( q) 1 = 1 ρ C q 0 ( ) The Fourier transform of the above expression leads to the (5.1)

7 Chapter 5 Ph D (Thesis) # 5.7 ρ C 0 4η 6 q [ 3 ( q ) = A q ( sin q q cos q ) + B q { q sin q ( q ) cos q } + C q + D {( 3q 6 ) sin q ( q 6 )} 4 {( 4 q 4 ) q sin q ( q 1 q + 4 ) cos q + 4} (5.13) E + q 4 6 ( q 0 q + 10 ) q sin q 6 4 ( q 30 q q 70 ) cos q 70 γ q 4 cos q ]. 1 Here, q is expressed in units of σ. For a given temperature and density the only parameter involved in the above is the diameter σ of the CHS, which makes application of this approximation simple. Taking into consideration the weak coupling between valance electrons and ions through the well known pseudopotential formalism the optimized expression for the structure factor is given by ( ) S q S ( q) 0 = + ρβv qs 0 q [ 1 ( ) ( )] (5.14) In equation (5.14) V(q) is the attractive screening correlation to the direct ionion potential of the form ( ) V q W ( q) 1 = 1 φ( q) ε( q) (5.15) The W(q) is the electron ion pseudopotential and ( ) 4π e φ q = is the Fourier q transforms of bare Coulomb interaction between two electrons. The ε(q) is the wave vector dependent function of the interacting electrons. This function cares about dielectric screening as explained earlier.

8 Chapter 5 Ph D (Thesis) # 5.8 The Charged Hard Sphere (CHS) model is extremely useful in the evaluation of structure factor of metals in liquid state. Inspite of simplicity, the model yields quite satisfactory results and has been utilized to study the structures of liquid metals by various authors [ ]. It was also pointed out by Kumaravadivel and Evans [5.9] and, McLaughlin and Young [5.30] that CHS model provides consistently satisfactory results, in close agreement with results obtained from neutral hard sphere model and offers an alternative reference system for the structural studies of liquid metals One component plasma (OCP) model One component plasma (OCP) is straightforward and effective model to understand the dense fluid consists of very soft potential such as the Coulomb liquids (simple and some non simple). In contrast to the PYHS and CHS reference systems the OCP scheme can be viewed as a model liquid metal in which the ions are regarded as point like charges, while electron gas is assumed to form uniform and rigid background, hence this ensures the overall neutrality of the fluid. [ ] The above features make OCP an idealized system of the ions immersed in a uniform sea of electrons so that the whole system is characterized by a single parameter ( ) Ze called Plasma Parameter Γ, which is defined as Γ = where a k T radius of sphere whose volume equals to the volume per electrons. B 1 3 a 3 = 4 π ρ is a The plasma parameter Γ measures the ratio between the average kinetic energy per particle. The choice of the parameter is an important aspect as it governs the structural and thermodynamical properties including stability of reference fluid. This allows us to use it directly for the real liquid metals which includes the Coulomb repulsion between the bare ions but omits the direct ion ion interaction arising from the electronic screening. Historically, Bretonnet and Derouchie [5.33] have developed an analytical expression for the direct correlation C(r), which is useful to determine the structure of bare OCP reference system. The structure factor S(q) using OCP can be derived by

9 Chapter 5 Ph D (Thesis) # 5.9 S0 ( q) S( q) = * 1 + ρβu q S q (5.16) ( ) ( ) In above expression 0 ρ = 1 is the number density, Ω 1 β = where, k B and k β T T are the Boltzmann constant and temperature of the system respectively. S 0 ( q) is the static structure factor of the bare OCP reference system. The simple analytical expression of S 0 ( q) is obtained as [ ], S 0 ( q) 1 = 1 ρ C q 0 ( ) (5.17) with, ρ ( q) = cos( qaα ) + cos( qaα ) ( qaα ) 3 Γ 3 sin 1 C (5.18) q a α qaα1 The α 1 and α in the expression (5.18) are dimensionless parameters and are given by Bretonnet and Derouchie [5.33] depends on the valancy of system. u * ( q) expression (5.16) is the Fourier transform of the perturbative potential u * () r can be considered as the difference between the true interionic potential u() r a Γ OCP reference potential. Thus the perturbation can be written in q space as β r in, which and the * 4π Z e 4π a u ( q) = 1 F N ( q) q Γ. β q (5.19) Here, F N ( q) is the normalized energy wave number characteristic of the form q 1 1 FN( q) = 1 WB q 4π Ze ε ( q) 1 f( q) ( ), (5.0) function. W B is the bare ion pseudopotential and ( q ) Here ( q) ε the modified dielectric

10 Chapter 5 Ph D (Thesis) # 5.10 Presently, three reference systems i.e. PYHS, OCP and CHS are employed to investigate the structural properties of some simple liquid metals viz; Mg, Zn, Al, In, Tl and Pb. Among these systems, the PYHS method is independent of model potential while other two methods have the necessary feature of taking account of pseudopotential contribution and are frequently used in predicting the structural information and thermodynamic properties. It has been also shown that the OCP yields a lower variational upper bound to the free energy than does the PYHS model and CHS yields a higher variational upper bound to the free energy than either the PYHS or the OCP fluid [ ]. To incorporate temperature dependency, the following relation between temperature and packing fraction is used [5.14]. ( ) = exp( ) η T A BT (5.1) Where, T is the absolute temperature and the parameters A and B are listed in Waseda [5.14]. It is obvious that as temperature changes the number density changes. As the number density varies with the temperature the Fermi wave vector k F also varies and ultimately the potential parameter r (which is the function of k ) is also affected by c F temperature. dependent. Thus strength of the interatomic potential is also made temperature The input parameters used to investigate the temperature dependency of the structural properties of liquid metals are tabulated in Table 5.1. For the structural studies of liquid Mg, Zn, Al, In, and Tl metals the modified pseudopotential [ ] is employed along with local field correction function f ( q) due to Ichimaru Utsumi [5.37] is used for exchange and correlation effect in the screening. The IU screening function is chosen because it satisfies all the important sum rules for a response function, in particular, the compressibility sum rule which is most important for obtaining the repulsive diameter of the ion correctly [5.37]. We have used the effective pair potential based on well established model potential [ ] which has proved

11 Chapter 5 Ph D (Thesis) # 5.11 its ability previously by providing excellent results of structure factor and its dependent properties [5.43]. The expression for the radial distribution function is given by [ ] 1 g r = 1+ q S q 1 qr dq ( ) ( ) { } sin( ) π ρr (5.) Results of structure factor and pair correlation function for liquid metals Results of structure factor for liquid Mg, Zn, Al, In, and Tl metals are shown in Figures 5.1 to 5.5 with the experimental structure factor by using all the three approaches. Tables 5. and 5.3 represents the first peak position and second peak position and related magnitudes analysis for structure factor S(q) along with experimental values [5.14]. The corresponding calculated g ( r ) is shown in Figure 5.6 to Tables 5.4 and 5.5 represent the first peak position and second peak position and corresponding magnitudes for pair distribution function g ( r ) along with experimental values [5.14]. The first peak position represents the inter atomic distance r 1 of the first nearest neighbor atoms, which corresponds to the first maxima of g( r ) curve.

12 Chapter 5 Ph D (Thesis) # 5.1 Table 5.1: Input Parameters and Constants used for S(q) and g(r) calculation Metal Z ( K ) q 0 T k F k F (Å 1 ) r c (Å) Ω (Å 3 ) η Γ Mg Zn Al 3 In 3 Tl

13 Chapter 5 Ph D (Thesis) # 5.13 Figure 5.1: Temperature dependent structure factor S(q) of liquid Mg

14 Chapter 5 Ph D (Thesis) # 5.14 Figure 5.: Temperature dependent structure factor S(q) of liquid Zn

15 Chapter 5 Ph D (Thesis) # 5.15 Figure 5.3: Temperature dependent structure factor S(q) of liquid Al

16 Chapter 5 Ph D (Thesis) # 5.16 Figure 5.4: Temperature dependent structure factor S(q) of liquid In

17 Chapter 5 Ph D (Thesis) # 5.17 Figure 5.5: Temperature dependent structure factor S(q) of liquid Tl

18 Chapter 5 Ph D (Thesis) # 5.18 Table 5.: Peak position analysis of S(q) using CHS method for liquid elements Metal T ( K ) Mg Zn Al In Tl First Peak position and Related magnitude in S ( q) Peak position q 1 in (Å 1 ) PYHS OCP CHS Expt [5.14] Related magnitude PYHS OCP CHS Expt. [5.14]

19 Chapter 5 Ph D (Thesis) # 5.19 Metal T ( K ) Mg Zn Al In Tl Table 5.3: Second Peak position and related magnitude in Sq ( ) Second Peak position and Related magnitude in S ( q) Peak position q in (Å 1 ) PYHS OCP CHS Expt. [5.14] Related magnitude PYHS OCP CHS Expt. [5.14]

20 Chapter 5 Ph D (Thesis) # 5.0 Figure 5.6: Temperature dependent pair distribution function g ( r ) of liquid Mg

21 Chapter 5 Ph D (Thesis) # 5.1 Figure 5.7: Temperature dependent pair distribution function g ( r ) of liquid Zn

22 Chapter 5 Ph D (Thesis) # 5. Figure 5.8: Temperature dependent pair distribution function g ( r ) of liquid Al

23 Chapter 5 Ph D (Thesis) # 5.3 Figure 5.9: Temperature dependent pair distribution function g ( r ) of liquid In

24 Chapter 5 Ph D (Thesis) # 5.4 Figure 5.10: Temperature dependent pair distribution function g ( r ) of liquid Tl

25 Chapter 5 Ph D (Thesis) # 5.5 Table5.4: First Peak position and related magnitude in ( ) g r Metal T ( K ) First Peak position and Related magnitude in ( ) g r Peak position r 1 in (Å 1 ) PYHS OCP CHS Expt. [5.14] Related magnitude PYHS OCP CHS Expt. [5.14] Mg Zn Al In Tl

26 Chapter 5 Ph D (Thesis) # 5.6 Table 5.5: Second Peak position and related magnitude in ( ) g r Metal T ( K ) Mg Zn Al In Tl Second Peak position and Related magnitude in ( ) g r Peak position r in (Å 1 ) PYHS OCP CHS Expt. [5.14] Related magnitude PYHS OCP CHS Expt. [5.14] During the study of the temperature dependent structural properties, some remarkable outcomes have been obtained. The main effect of the temperature variation is the change in the amplitude of the oscillations in the structure factor and pair distribution function. Namely, a decrease around the peak and an increase elsewhere are observed. It is also observed that at higher temperatures the first main peak tends to become flatter and broader with increasing temperature, but the position of peak is usually unchanged.

27 Chapter 5 Ph D (Thesis) # 5.7 The pair correlation function gives the idea of order parameter, which is function of radial distance r, which vanishes when the desire form is absent and rise up from the zero as soon as it is present. The vanishing of g ( r ) at small values of r is due to the fact that as other atom approaches the central one closely, strong repulsive forces arise which push other atoms away. This repulsive force therefore prevents the other atoms from overlapping the central atom. The later fact is evident from the g(r) Figure as g(r)= 0 at small values r. the peak is due to, at very short distances, atoms attract each other hence the force therefore tends to pull other towards the center, resulting in a particularly large density at a certain specific distances. The damping oscillation in the curve arises from the interaction between the forces of central atom and near neighbours still farther away. For the large values of r, the concentration approaches the constant value due to short range order of liquid, the distribution of atoms are completely random and independent of the position of the central atom, i.e. independent of r, hence g(r) 1 as r is a situation corresponding to the absence of correlation between atoms. The major effect of the temperature on the g(r) is a gradual filling of the gap between the first and the second peak, together with a small broadening of the first peak towards smaller distances From the comparison, it is seen that the CHS method along with the present form of model potential is able to explain more accurately the structural behaviour than PYHS and OCP methods. The discrepancy in the outcome of OCP with experimental observations is found higher than those with CHS and PYHS. Our study leads to conclude that CHS reference system is found better than PYHS and OCP in explaining the structure of liquid metals. The results also confirm the use of our single parametric model potential in explaining the structure of liquid metals Resistivity and shear Viscosity of liquid metals (1) Resistivity The study of electrical transport properties of liquid metal alloys remain one of the favourite quantities either experimentally or theoretically [ ]. One of the significant results of understanding of liquid metals has been the success of calculations

28 Chapter 5 Ph D (Thesis) # 5.8 of the electrical resistivity in explaining experimental data. In the Ziman s [ ] neutral pseudo atom theory of liquid metals, the electrical resistivity is written (in Born approximation) in terms of the product of a structure factor term and the Fourier component of the screened ionic potential or pseudopotential. In the preliminary applications of the theory, before more reliable pseudopotentials or structure data were available, Ziman [ ] suggested that the resistivity might be divided into two contributions: plasma and structure. Although the separation was somewhat arbitrary, the idea was that in the small wave vector (k) region the known structure factor at k = 0 could be used to approximate the small wave vector or 'plasma region'. At the other extreme of wave vector k >k F, the value of the potential at this wave vector is used to approximate the large wave vector region, combined with the measured structure factor, a(k). Thus, the large wave vector contribution depends in detail on the structure of a(k) near k k F and the pseudopotential in this region this was designated the ' structure' resistivity. With the advent of more accurate values for the pseudopotential, the tendency has been either to combine the potential calculations with structure data to extrapolate to the low wave vector region in which accurate data do not exist, or to make use of model structure calculations. In the polyvalent metals the structure part of the resistivity dominates the plasma contribution and there is little dependence of the calculated resistivity on the extrapolated low wave vector behaviour. For the monovalent, alkali metals, the first peak in the structure factor does not appear in the resistivity calculation and the low wave vector region or plasma contribution plays an important role. In the early years, Paskin et al. [5.46] had given a relation between the structure factor and the interatomic potential which is valid at small wave vector. We shall here make use of this relationship over the entire wave vector region to obtain a plasma like contribution to the resistivity of the liquid metals. In the past, despite the rich accumulation of experimental studies, the atomistic approach to the problem of the liquid metals had been very slow in progress; untill Ziman [ ] proposed the theory of electrical resistivity of liquid metals. Basically, there are three approaches for the theoretical investigations of transport properties of liquid metals. One is based on the nearly free electron picture [ ], second one is the finite mean free path approach [ ] and the third is based on tight binding approximation [ ]. The tight binding approach usually involves

29 Chapter 5 Ph D (Thesis) # 5.9 either the average T matrix approximation or the coherent potential approximation. A self consisting approach corresponding to the finite mean free path is taking account of finite uncertainty in the electron momentum. In this chapter, we intend to report the electrical resistivity of liquid metals and their binary alloys based on the presently formulated local pseudopotential. Hence both the second and third approaches are beyond the confines of present objectives of the thesis as well as pseudopotential theory. It is to perceive here that experimentally the electrical resistivity of liquid metals has been measured either by the rotating magnetic field method or by the fourprobe method for temperatures up to 1500 K [5.56]. The Faber Ziman [ ] approach of investigating electrical resistivity of liquid metals assumes the model of a gas of conduction electrons, which interact with and are scattered by irregularly placed metal ions. As an external electric field drives the electron through the disordered medium, the scattering determines the electrical resistance which can be calculated using perturbation theory: the transition rate from an initial state k to the final state N FE (E F ) is given by k + q on the Fermi level with the density of state π 1 P V N E h ( θ ) = k+ q k ( ) FE F. (5.3) Where θ is the angle between k and k+q, the factor 1/ arises from the fact that electron spin does not change on scattering. Now the conductivity in the relaxation time approximation is given by 1 3 ( E ) σ = e vf τ NFE F. (5.4) Here e is electronic charge, the relaxation time τ. The relaxation time τ is given by v F velocity of the electrons at the Fermi level and 1 = ( θ) ( θ) Ω τ 1 cos P d, (5.5)

30 Chapter 5 Ph D (Thesis) # 5.30 Here θ is scattering angle, Ω is solid angle and P ( θ) is probability for scattering through the angle θ. Now assuming the free electron distribution, an expression for the electrical resistivity of liquid metal in terms of the average of the product of the structure factor and pseudopotential matrix element can be written as 3π m 3 ρ = S( q) W( q) q dqθ( kf q) 4e h nk. (5.6) 3 6 F 0 Here n is the electron density related to Fermi wave number and θ(k F q) is the unit step function that cuts of the q integration at k F corresponding to a perfectly sharp Fermi surface. The S(q) is the structure factor and W(q) is the screened ion pseudopotential form factor. We have used the modified model potential [ ] which has proved its ability previously by providing excellent results of structure factor and its dependent properties [5.43]. Using the discussed approach we have calculated the resistivity of liquid Mg, Zn, Al, In, and Tl metals after structural studies. The input parameters and constants are same as described in Table 5.1. To study the influence of the screening on the resistivity of liquid metals five screening functions are employed. The structure factor S(q) generated from the CHS approached is incorporated. The computed result of electrical resistivity ρl using all the screening functions Hartree (H) [5.57], Taylor (T), [5.58] Ichimaru Utsumi (IU), [5.59] Sarkar Sen (SS) [5.60] and Farid (F) et al [5.61] are tabulated in Table 5.6. It is immediately evident from Table 5.6 that calculated values of electrical resistivity ρl using CHS model matches with different screening functions matches qualitatively well whenever experimental findings [5.14] are available. The temperature dependence of resistivity enters through the change in screened potential and the structure factor with temperature. It is also concluded here that the trends in the computed electrical resistivity of all the metals Mg, Zn, Al, In, and Tl shows metallic behaviour.

31 Chapter 5 Ph D (Thesis) # 5.31 Table 5.6: Electrical resistivity (μω cm) of liquid metals Electrical Resistivity (micro ohm cm) Mg Zn Al In Tl Z T (K) H [5.57] T [5.58] IU [5.59] F [5.60] SS [5.61] Expt. [5.14] () Shear Viscosity One of the most exciting properties of a liquid is its shear viscosity. The study of viscosity of liquid metals is important for various metallurgical, industrial and biophysical applications. In the recent years considerable efforts have been done to study the shear viscosity of liquid [ ]. In the course of theoretical investigation Molecular Dynamics (MD) also provides substantial understanding of microscopically physical mechanism underlying the dynamics of liquids [5.64, 5.70]. Besides this, the distribution function as developed by Rice and Allnatt [5.71] is very convenient for calculation of shear Viscosity due to its simple form.

32 Chapter 5 Ph D (Thesis) # 5.3 The essential ingredients in calculation of the shear viscosity are pair potential and pair correlation function. Proper incorporation of model potential in describing these two properties are important. The shear viscosity of Hard sphere liquid is be written as [5.14] n = nh ns n k (5.7) The first term on right hand side of above expression is the contribution of the hard part of the potential expressed by σ 8σ 96 ( ) 1 ( ) σ σ n = KT + + h ( ) g σ 5 D g KT y g σ 5 Ω π (5.8) Where σ is the hard sphere diameter and D y,ω, ξ s are constants defined in [5.14]. The contribution of soft part of the potential in shear Viscosity is given by 4π Mn 4 v 4 v ns = + ( ) 30ξ r g r dr s 0 r r r (5.9) The kinetic contribution may be written as 8σ g( σ ) 5ξ s nk = 5KT 1+ 8σ g( σ ) 5 Ω + 4nMg( σ ) (5.30) g(σ) in the above expression (5.30) is the explicit value of pair correlation function at hard sphere diameter, the final expression of shear viscosity is η = C (σ) (5.31) s n 0 The C s (σ) in above expression is the scaling correction term which takes into account the effect of multiple scattering. The necessary effective inter ionic pair potential in above expression (5.9) is calculated using the expression (.59) given earlier as

33 Chapter 5 Ph D (Thesis) # 5.33 Z V ( r) = 1 FN ( q) Sin( qr) dq r π (5.3) 0 as Where F N ( q), the normalized energy wave number characteristics is expressed F N B ( qw ( q)) 1 1 ( q) = 1 4π Zn ε( q) 1 G( q) (5.33) Here, W B (q ) is the Fourier transform of the corresponding bare ion form factor in the momentum space, while Z, q and r c are respectively valency, magnitude of the momentum transfer vector and the parameter of the potential. In equation (5.33), ε(q) denotes the dielectric function and G(q) takes care for the local field correction functions. For the present study the local field correction due to Ichimaru Utsumi (IU) [5.59] is used for the proper incorporation of the exchange and correlation among the conduction electrons in dielectric screening. The effective inter ionic pair potential V(r) for metals has been computed using the same procedure as described in chapter. And pair correlation functions using charged hard sphere (CHS) reference systems are utilized here for the same purpose. Recently, Shear Viscosity of liquid metals is also be reported by using different formalism by Zahid et al [5.68]. They employed the self consistent Variational Modified Hypernetted Chain (VMHNC) integral equation based on Ornstein Zernike equation. The presently generated results for the shear viscosity are narrated in Table 5.7 with available experimental [5.14] results at temperatures near and above melting. It is evident from the present study of the pair potential and pair correlation function derived from the CHS model that the computed values of Shear Viscosity match qualitatively well with the experimental and other theoretical findings. It was concluded [5.69] earlier that attractive part and oscillation on the higher q values are not so significant in pair potential [5.55]. It is also observed that with increase in temperature the calculated shear viscosity decreases.

34 Chapter 5 Ph D (Thesis) # 5.34 Table 5.7: Shear Viscosity of Liquid metals Element T (K) Shear Viscosity η ( in cp) Present [5.7] Expt. [5.14] Others [5.69] Mg Zn Al In Tl The estimated results are free from fitting procedures and confirm the applicability of established model potential as well as CHS method for predicting the proper structural behaviour and shear viscosity of liquid elements. As in the calculation of structure factor CHS approximations are employed, it shift the position of the principal peak of S(q) towards smaller values of q as compared with Hard Sphere approach bringing the calculated results closer to the experimental data for the elements [5.7]. As major contribution of shear viscosity of elements as from the expression (5.7) comes from the hard part, pair correlation is equally important as pair potential in this kind of study [5.7].

35 Chapter 5 Ph D (Thesis) # Structure and properties of liquid binary alloys Introduction The outstanding properties of binary liquid alloys with strong non ideal mixing behavior have been of longstanding interest in the past decade and are useful in many diversified fields. The study of alloys is of immense importance for not only physicists but also chemists and engineers. This study is important for metallurgical chemistry and geology which involve the study of core of earth and astrophysics which involves the study of interior of planets (the core of the earth and interior of the planet are made up in part of a liquid mixture). Alloys with miscibility gap in the liquid state are especially interesting for advanced bearing materials. The structure factor of alloys provides information about the detail internal structure of binary sample, i.e. internal arrangement of atoms inside the solid. There are two types of structure factors, static and dynamic. The static structure factors are only wave vector dependent, i.e. S(q) and are useful when static or bulk properties are to be calculated, such as electrical resistivity etc. whereas dynamic structure factors are wave vector as well as frequency ω dependent, i.e. S(q, ω) and is important when the determination of time dependent properties (i.e. vary in time and reciprocal space) will be required. As per the scope of thesis, we discussed only static structure factor. If we consider particular case of binary mixture where two types of atoms are present say A and B. Then the partial structure factor S ij (q) in momentum space q associated with A type, B type and AB type combinations are respectively S 11 (q), S (q) and S 1 (q). So in the case of binary mixture we have S AA, S BB, S AB shows probability of finding A atom from A, B atom from B and A atom from B respectively. Hence, our main objective is to calculate this partial structure factors S ij (q) for binary alloys and total structure factor of one component liquid metal as well as for less simple binary liquid alloys. From the literature survey it is established that there are very few models or approaches through which these accurate partial structures and hence total structures can be yield. Among this the Ashcroft Langreth (AL) partial structure factor is the most popular in past [5.73].

36 Chapter 5 Ph D (Thesis) # Ashcroft Langreth (AL) Structure factor The structure factors for binary alloys proposed by Ashcroft and Langreth [3.58] using the hard sphere momentum space solutions of the Percus Yevick (PY) equation for the binary mixtures [5.73] are following three partial structure factors S ( y ) = ( 1 nc ( y) ) ( )( 1 ) S ( y ) = and nc ( y) nc ( y) n n C ( y) ( 1 nc 1 11( y) ) ( )( 1 ) 1 1 S ( y ) = nc ( y) nc ( y) n n C ( y) 1/ 1/ ( n1 n C1( y )) ( )( 1 ) nc ( y) nc ( y) n n C ( y) (5.34) (5.35) (5.36) ij where ni s represent the respective number densities of ith component and C is the direct correlation functions in momentum space q with y = qσ which are given by a1 [ sin( αy) ( αy)cos( αy) ] η1 β1 nc 1 11( α y) = 4 ( )sin( ) (( ) )cos( ) 3 + ( ) ( ) αy αy αy αy αy αy 3 4 γ (( 4 αy ) 4( αy))sin( αy) (( αy) ( αy) 1( α y) + 4)cos( α y) + 4 a [ sin( y) ( y) cos( y) ] η β nc ( α y) = 4 ( )sin( ) ( )cos( ) 3 + ( ) ( ) y y y y y y 3 γ ( 4y 4y)sin( y) ( α y ) ( y 1y + 4)cos( y) + 4 and (5.37) (5.38)

37 Chapter 5 Ph D (Thesis) # 5.37 n n C ( y ) 1 / 1 / / 1 / 1 / 1 / 3 3 ηx ( 1 x) sin( y λ) y λcos( y ) λ x ( 1 x) α = 31 ( α) a1 4η x + ( 1 x) α ( y λ ) x + ( 1 x) α sin( y ) γ β ( α y )cos( α y ) + (( α y ) )sin( α y )] + ( ( α y ) )cos( α y ) ( α y ) ( α y ) λ γ1 + (( α y ) 6( α y ))sin( α y ) + 6] + ( 4( α ) 4( α ))cos( α ) + (( α ) 1( α ) ( α ) y y y y y y { cos( y ) + 4)sin( α y ) + β ( α y )sin( α y ) + (( α y ) )cos( α y ) + λ ]} 1 ] 1 4 ( α y ) ( α y ) 3 γ 3 ( 3( α y ) 6)sin( α y ) + (( α y ) 6( α y ))cos( α y )] + 1 ( 4( α y ) 4( α y ))sin( ( α y ) α y ) a sin( α y ) ( α y )cos( α y ) + (( α y ) 4 1( α y ) + 4)cos( α y ) + 4]} + 1 cos( y λ ) ( α y ) ( α y ) γ 1 α 1 cos( α y ) cos( α y ) ( α y )sin( α y ) 1 1 α sin( α y ) + + sin( y λ ) + α ( α y ) ( α y ) α ( α y ) (5.39) Here, α, β and γ are defined as given by the expression (5.6) For the mixture of hard spheres whose diameters are in ratio ά =σ1/σ 1 by choice with σ being the diameter of larger hard sphere. The total packing fraction of mixture is η = η1 + η, where ηi = (π/6)niσi3 with i= 1, and ( 1 x ) α ( 1 x) 3 η= 1 η 3 + α x and x η = x + ( 1 x) α 3 η (5.40) (5.41)

38 Chapter 5 Ph D (Thesis) # 5.38 Here, x is the concentration of the second metallic component of A 1 X B X binary alloys. Using the same approach initially applied to the liquid metals, the approach restructured to investigate the resistivity of A 1 x B X liquid binary alloys [3.8, 3.16, 3.6]. The electrical resistivity of alloys can be written as ρ L alloy 3π = ζ 4h Ω Ο m 3 6 ekf k F 0 ( ) 3 q q dq (5.4) The Ω 0 is atomic volume of alloy, V ij (q) denotes the screened model potentials of binary alloys, k F is the Fermi wave number, x is the concentration of the second metallic component of A 1 X B X binary alloys, S ij (q) are partial structure factors with ζ( q ) in expression (5.4) is given by ( q) ( 1 x) S ( q) V ( q) x( 1 x) S ( q) V ( q) xs ( q) V ( q) ζ = + + (5.43) Here V 1 (q) and V (q) denote the electron ion screened pseudopotential of metals A and B. Sij is the partial structure factors as defined in expression ( ). The x denotes the concentration of the second species B in A 1 x B x liquid binary. The calculated partial structure factor using above described method is shown in Figure 3.11 for Al Mg alloy with equal concentration. In the present study the electrical resistivity of the liquid Al Mg binary alloys has been investigated. The concentration dependence of the resistivity has examined by varying x = 0 to x = 1 in the step of 0.1. The influence of local field correction functions is also evaluated by incorporating various f(q)s in the screened form factors. Previously, it was found that the electrical resistivity of alloys obeys two rules: (i) Nordheim s rule [3.11] and (ii) Linde s rule [3.1]. According to the Nordheim s rule, the resistivity of liquid alloys depends directly on the product of the atomic concentrations in the percentage of the guest and host elements [3.11]. While according to the Linde s rule the resistivity derivative with respect to atomic concentration is directly proportional to the difference of the variables of the guest and host elements [3.1].

39 Chapter 5 Ph D (Thesis) # 5.39 Figure 5.11: Partial structure factor of binary Al Mg liquid of equal composition

40 Chapter 5 Ph D (Thesis) # 5.40 Figure 5.1: Temperature dependent total structure factor of binary Al Mg liquid at equal composition

41 Chapter 5 Ph D (Thesis) # 5.41 Table 5.8: Input parameters for Aluminum (Al) and Magnesium (Mg) Aluminum T ( o C) R C (au) Ω (au 3 ) K F (au) η R S (au) Magnesium T ( o C) R C (au) Ω (au 3 ) K F (au) η R S (au) Figure 5.13: Concentration dependence of resistivity of Al Mg binary alloy

42 Chapter 5 Ph D (Thesis) # 5.4 Figure 5.14: Temperature dependent resistivity of Al Mg alloy The total structure factor is also shown here in Figure 3.1 for equal concentration of Al Mg alloys at different temperatures. The input parameters used in calculation is tabulated in Table 5.6. The Figures 5.13 and 5.14 comprises the outcome of ρ for Al 1 x Mg x alloys with the available experimental data. Following points are inferred from the critical examinations of these Figures. The outcome of ρ for binary alloys due to T, IU, and S screening are lying between those obtained by employing H and F screening functions. The various f(q)s influence the ρ of liquid binary vary drastically. Such influence is predicted maximum up to 300% for a particular concentration in Al Mg system. The maximum resistivity is obtained for Al 1 x Mg x at x = 0.58 using F local field function. For Al 1 x Mg x the comparison with experimental available data [5.74] is good. The results of resistivity of binary alloys in comparison with the experimental data Al 1 x Mg x confirm the trend of the present computations. As concentration x of the element B increases, initially the ρ increases and after a critical value of x, the further increase in x decreases the values of resistivity ρ of the system Al 1 x Mg x. The overall picture of the present computations thus confirms the applicability of our model potential for studying the electrical resistivity of metallic complexes of aluminum magnesium alloys.