Microscopic origin of immiscibility and segregation in liquid binary alloys
DDedicated to the birth centenary of the Father of Nation,Bangabandhu Sheikh Mujibur Rahman
MICROSCOPIC ORIGIN OF IMMISCIBILITY ANDSEGREGATION IN LIQUID BINARY ALLOYS
G. M. BHUIYAN
Department of Theoretical Physics, University of Dhaka, Bangladesh ∗ Correspondence E-mail: [email protected]
ABSTRACT
Microscopic description in the study of immiscibility and segregating properties of liquidmetallic binary alloys has gained a renewed scientific and technological interests during thelast eight years for the physicists, metallurgists and chemists. Especially, in understandingthe basic mechanisms, from the point of interionic interaction, and how and why segregationin some metallic alloys takes place at and under certain thermodynamic state specified bytemperature and pressure. An overview of the theoretical and experimental works done bydifferent authors or groups in the area of segregation combining electronic theory of met-als, statistical mechanics and the perturbative approach is presented in this review. Mainattention in this review is focused on the static effects such as the effects of energy ofmixing, enthalpy of mixing, entropy of mixing and understanding the critical behaviour ofsegregation of alloys from the microscopic theoretical approach. Investigation of segregatingproperties from the dynamic effects such as from the effects of shear viscosity and diffusioncoefficient is just becoming available. However, we have restricted this review only on staticeffects and their variation of impacts on different alloys.
PACS:
Keywords:
Segregation, Thermodynamics of mixing, Electronic theory ofmetals, Critical temperature and critical concentration, Perturbative approach. a r X i v : . [ c ond - m a t . o t h e r] D ec G M BHUIYAN
Some advancement in understanding the segregating properties, miscibilitygap, demixing tendency etc. of some metallic binary alloys, has been made,so far, from the empirical models [1-3], and phenomenological theories [4,5] inconjunction with arbitrary concepts of association and dissociation[6-8]. Ex-perimental data for some liquid segregating alloys [9-20] plays the pivotal roleto arouse the interest in theoretical study, in particular to understand the crit-ical behaviours. This knowledge is required to find the possible application ofsegregating materials to innovate technology and to industries for car engines,electrical contacts and switches, separation of impurities from the iron melts,ceramic industries, cosmetic and the food industry.Known signatures of the existence of immiscibility, segregation, miscibilitygap, and critical properties of segregating alloys are deviation from Roult’s law,concave downward of the free energy of mixing profile [21-25], concave upwardof entropy [24,25] and entalpy of mixing [25], large density fluctuation displayedby concentration-concentration structure factors [26], large difference in partialcoordination numbers [27] derived by using partial pair correlation functions g ii ( r ) and g , sudden sharp bending of the atomic transport properties as afunction of concentration [25], positivity of short range order parameter [28-32], exhibiting some sort of scaling laws [25] etc. But, understanding of theactual mechanisms involved behind these signatures is a great challenge tophysicists, metallurgists and the material engineers.Segregating properties of liquid binary alloys may be studied microscopi-cally from the static [21-24]and dynamic effects [25]. The static effects may beobserved from the thermodynamic properties of mixing, coordination numberderived from structural properties etc. The dynamic effects can be seen fromthe atomic transport properties such as coefficient of viscosity and diffusioncoefficient, and the electronic transport properties such as electrical resistivity[26]. In this review, a microscopic theoretical approach that involves the elec-tronic theory of metals [23,33-35], perturbation theory [36,38], the hard spherereference system [39,40] and the statistical mechanics. Electron ion interac-tion is described by a local pseudopotential, the interionic pair interaction isderived from the energy band structure which is finally employed to evaluatestatic structure of liquid metals and their alloys. Specifically, the form factorsof the pseudo potential is used to find effective pair potentials and the volumedependent contribution to the free energy. The knowledge of pair potentials isessential to have pair correlation functions, the energy of the reference systemICROSCOPIC ORIGIN OF IMMISCIBILITY · · · Different theories relevant to the present review article are briefly presentedbelow.
A macrostate of a condensed system may be described by four independentvariables. These are pressure, p , volume, V , temperature, T , and entropy, S .Here, p and V form a pair representing the mechanical degrees of freedom, G M BHUIYANand T and S form another pair representing the thermal degrees of freedom.Any two of the four variables may be chosen in six different ways. Of them fourpair of variables are (p,T), (p,s), (V,T) and (V,S), each of which contains onevariable from the mechanical and another from the thermal degrees of freedom.Thermodynamic functions constructed by these pairs are Gibbs free energy,G(p,T), enthalpy, H(p,S), Helmholtz free energy, F(V,T), and the internalenergy, E(V,S).The free energy in thermodynamics is the amount of energy of the systemfree to work. Internal energy of a system is the sum of kinetic energy, potentialenergy, rotational energy and the vibrational energy etc. Of course, in themagnetic systems the magnetic energy [44,45] and for a finite sized samplethe surface energy correction to be counted in the above functions [46]. For amonoatomic systems (also in random binary alloys) there are no rotational andvibrational energy contribution in general. However, the above thermodynamicfunctions are not independent to each other. They are rather interconnected.The Helmholtz free energy (for the bulk) is (dropping the variables for brevity) F = E − T S . (1)The Gibbs free energy G = F + pV = E − T S + pV . (2)The enthalpy H = E + pV . (3)And the change in the internal energy dE = T dS − pdV . (4)From the theoretical point of view, we can further analyze the above relations.for example, for zero pressure, i.e. at p = 0, G = F (5) H = E . (6)Again, most of the experimental data for thermodynamical quantities avail-able in the literature are at standard temperature and pressure. In oneICROSCOPIC ORIGIN OF IMMISCIBILITY · · · pV appears to be very smallwhen compared with other terms of the above thermodynamic functions. So,in one or two atmospheric pressure or less one can write H ≈ E ; G ≈ F . (7)We note here that other physical quantities such as heat capacity, compress-ibility etc. can be derived from equations (1) to (3) [47-49].For binary alloys the free energy of mixing is∆ F = F alloy − (cid:88) a C a F a (8)where F alloy is the free energy of the alloy, C a is the concentration of the a -th component and, F a is the free energy of the a -th element in the samethermodynamic state. Similarly the enthalpy of mixing may be expressed as∆ H = H alloy − (cid:88) a C a H a , (9)and the entropy of mixing as∆ S = S alloy − (cid:88) a C a S a . (10)It is worth noting that, the thermodynamics is a phenomenological sub-ject because all relations in thermodynamics are obtained just looking at theexperimental results. The only way to have microscopic description of thethermodynamic quantities is through the statistical mechanics [48,49]. Let us consider N ions each of valence Z are there in a volume V in a liquidmetallic system. So, the total number of conduction electrons in this system G M BHUIYANis N Z . The Hamiltonian of the sample may be written as H = H e + H ee + H ei + H i + H ii = NZ (cid:88) i =1 p i M + e NZ (cid:88) i (cid:54) = j | (cid:126)R i − (cid:126)R j | + (cid:88) i,l v ( | (cid:126)R i − (cid:126)r l )+ N (cid:88) l =1 P l m + 12 N (cid:88) l (cid:54) = l (cid:48) w ( | (cid:126)r l − (cid:126)r l (cid:48) | ) (11)where H e , H ee , H ei , H i , and H ii denote contributions from kinetic energyof electrons, electron-electron interactions, electron-ion interactions, kineticenergy of ions and ion-ion interactions, respectively. In equation (11) { R i } and { r l } are electronic and ionic coordination; { p i } and { P l } electronic andionic momenta, and, M and m are corresponding masses. v and w denoteelectron-ion and ion-ion potential energies, respectively.In the canonical ensemble theory the normalized equilibrium probabilitydensity f ( N )0 for a system of homonulear atoms is given by f ( N )0 ( (cid:126)r , · · · , (cid:126)r N , (cid:126)p , · · · , (cid:126)p N ) = exp[ − βH ( (cid:126)r , · · · , (cid:126)r N , (cid:126)p , · · · , (cid:126)p N )] N ! h N Q N ( V, T ) (12)where h denotes Planck’s constant, and Q N ( V, T ) the total partition function, Q N ( V, T ) = Tr e − βH = 1 N ! h N (cid:90) d(cid:126)r · · · d(cid:126)r N (cid:90) d(cid:126)p · · · d(cid:126)p N Tr e e − βH (13)where Tr e refers complete set of electronic states corresponding to a particularionic configuration. The motion of ions is very slow relative to the conductionelectrons, so, ions can be treated classically unlike electrons that must behandled quantum mechanically. As classical particle do not obey uncertaintyprinciple one can integrate over position and momentum independently. Theresult thus obtained is Q N ( V, T ) = 1 N ! (cid:20) πm (cid:126) β (cid:21) N Z N ( V, T ) (14)where the configurational partition function Z N ( V, T ) = (cid:90) · · · (cid:90) d(cid:126)r · · · d(cid:126)r N exp( − βH ii ) (cid:26)(cid:90) · · · (cid:90) d (cid:126)R · · · d (cid:126)R N d (cid:126)P · · · d (cid:126)P N e − β ( H e + H ee + H ei ) (cid:111) (15)ICROSCOPIC ORIGIN OF IMMISCIBILITY · · · U N = H ii + F (cid:48) (16)where F (cid:48) is the Helmholtz free energy of the conduction electrons in the ex-ternal potential H ie . F (cid:48) can be calculated by some approximation schemes.Therefore Z N = (cid:90) d(cid:126)r d(cid:126)r · · · d(cid:126)r N e − βU N , (17)and the L-body probability density n ( L ) N = (cid:82) · · · (cid:82) d(cid:126)r L +1 d(cid:126)r L +2 · · · d(cid:126)r N e − βU N Z N (18)This is related to the L -particle distribution function defined as g ( L ) ( (cid:126)r (cid:126)r · · · (cid:126)r N ) ≡ n ( L ) N n L = N ! n L ( n − L )! (cid:82) · · · (cid:82) d(cid:126)r L +1 · · · d(cid:126)r N e − βU N Z N (19)Now, the two body reduced distribution function stands as g (2) ( (cid:126)r , (cid:126)r ) = N ( N − n (cid:82) · · · (cid:82) d(cid:126)r · · · d(cid:126)r N e − βU N Z N (20)For an isotropic liquid g (2) ( (cid:126)r , (cid:126)r ) = g ( | (cid:126)r − (cid:126)r | ) = g ( r ) , which is also known as pair correlation function and is the central idea in mostliquid state theories.Now if it is assumed that the effective interionic potential is pairwise addi-tive in the following way U N = N E ( V ) + 12 (cid:88) i,j v ( r ij ) , (21)where E ( V ) is the volume dependent (but structure independent) part of en-ergy that includes the free energy of electrons, then all thermodynamic G M BHUIYANfunctions can be expressed in terms of g ( r ) and the pairpotential of interaction.We note here that equation (20) cannot be solved analytically even if Eqn. (21)is used. For this mathematical limitation different approximation methods andcomputer simulation methods are devised to solve for g ( r ). But for the hardsphere (HS) potential v hs ( r ) = (cid:40) ∞ for r < σ r > σ (22) g ( r ) can be evaluated analytically [33,40], here σ denotes the hard spherediameter (HSD). The pair correlation function for a liquid binary alloys [50]may be expressed as g ij ( r ) = 1 + 1(2 π ) ρ (cid:112) C i C j (cid:90) ( S ij ( q ) − δ ij ) e i(cid:126)q · (cid:126)r d r (23)where S ij ( q ) is the static structure factors and q the momentum transfer. The thermodynamic perturbation theory proposed by Weeks-Chandler-Andersen(WCA)[51] splits the interionic potential as core and tails terms v ( r ) = v core ( r ) + v tail ( r ) (24)The core term is related to the HS potential, v hs through the Mayer’s clusterexpansion in the following way. f µ ( r ) = f hs ( r ) + µ ∆ f ( r ) for 0 ≤ µ ≤ , (25)where µ is the coupling parameter, and∆ f ( r ) = f core ( r ) − f hs = (cid:2) e − βv core − e − βv hs (cid:3) (26)The Helmholtz free energy can be expanded now as F core = F hs + E ( V ) − kT ρ σ ξ + O ( ξ ) (27)where ξ = 1 σ (cid:90) ∞ B hs ( r ) d(cid:126)r (28)ICROSCOPIC ORIGIN OF IMMISCIBILITY · · · B ( r ) = y σ ( r ) (cid:8) e − βv core ( r ) − e − βv hs ( r ) (cid:9) . (29)From equation (27) it is clear that, when ξ = 0, F core = F hs + E ( V ). In theWCA theory hard sphere diameter σ is determined following this conditionthat Fourier transform of B ( r ) that is B ( q ) vanishes at r = σ . But in WCAtheory r B ( r ) shows a saw tooth shaped function. If this function is linearzedto have a triangular form one can find an equation [52] βv ( σ ) = ln (cid:18) − βv (cid:48) ( σ ) + X + 2 − βv (cid:48) ( σ ) + X + 2 (cid:19) , (30)here, prime indicates the first derivative of the potential energy at r = σ , and X = σ/σ w g (cid:34)(cid:88) k =0 ξ k +1 ( η w ) n ! (cid:18) σσ w − (cid:19) n − Aσ w σ (1 + µσ ) (cid:35) (31)All symbols are defined in reference [52]. Solution of the transcendental equa-tion (30) yields the effective HSD. The pair correlation function is now evalu-ated using this effective HSD.0 G M BHUIYANAndersen et al.[53] proposed a simplest version of the perturbative schemeknown as exponential approximation, g hs ( r ) = g e − v ( r ) /kT where v ( r ) is the real short range part of the potential; in the present case itis v core . We note that this optimized form gives more realistic description ofthe pair correlation function.Now using the perturbation theory one can calculate the free energy of asystem per ion as F = F unp + 2 πρ (cid:90) v pert g hs d r (32)where F unp = E ( V ) + F hs = F vol + F gas + F hs and v pert = v tail .F vol = 132 π (cid:90) ∞ q (cid:26) (cid:15) ( q ) − (cid:27) | v ( q ) | dq − ZE F Y (33)where Z = C Z + C Z ; Y = χ elec /χ F , subscripts elec and F denoteisothermal compressibilty of the interacting and free electrons, respectively.The values of Y are obtained from [54].The electron gas contribution to the free energy per valence in Rydbergunit is F gas = 2 . r s − . r s + 0 .
31 ln r s − .
115 (34)where r s = (cid:18) πρZ (cid:19) /a ; ρ = ρ ρ C ρ + C ρ .F hs = (cid:88) i (cid:2) − ln (cid:0) Λ i v (cid:1) + ln C i (cid:3) − (cid:18) − y + y + y (cid:19) +(3 y − y ) / (1 − η ) + 32 (cid:16) − y − y − y (cid:17) / (1 − η ) +( y −
1) ln(1 − η ) (35)ICROSCOPIC ORIGIN OF IMMISCIBILITY · · · i = (cid:26) π (cid:126) m C m C kT (cid:27) ,η = (cid:88) i η i ; η i = C i πρ i σ ii ,F tail = D (cid:88) i,j C i C j M ij ,D = 2 πρ , M ij = (cid:90) ∞ σ v ij g ij ( r ) r dr . Now, the energy of mixing∆ F = ∆ F vol + ∆ F gas + ∆ F hs + ∆ F tail (36)∆ F y to be calculated by using equation (8). Now if the experimental densitiesof the alloy at different concentrations are available, and if the difference be-tween calculated density and the experimental ones exists and significant anexcess volume correction to be added with the thermodynamics of mixing as[34] ∆ F = ∆ F vol + ∆ F gas + ∆ F hs + ∆ F tail + ∆ F evc . (37) Enthalpy of alloy :
Enthalpy of the alloy per ion H = E + pV = 32 kT + E ( V ) + ρ (cid:88) i =1 (cid:90) g ij ( r ) v ij ( r ) d r + pV (38) Entropy of alloy :
Within the above perturbation scheme the entropy of alloy (devided by
N k )2 G M BHUIYANreads [38] S = S ref + S tail , (39) S ref = S id + S gas + S η + S σ ,S id = − [ C ln C + C ln C ] ,S gas = 52 ln (cid:34) ρ (cid:18) m C m C kT π (cid:126) (cid:19) (cid:35) ,S η = ln(1 − η ) + 32 [1 − (1 − η ) − ] ,S σ = (cid:20) πC C ρ ( σ − σ )(1 − η ) − (cid:21) ×{
12 ( σ + σ − π ρ [ C σ + C σ ] } , and S tail = 1 k (cid:34)(cid:18) ∂F tail ∂T (cid:19) V,ρ,σ ii + (cid:88) i =1 (cid:18) ∂F tail ∂σ ii (cid:19) V,T (cid:18) ∂σ ii ∂T (cid:19) V,ρ (cid:35) . (40)The temperature dependent HSD as proposed by Protopapas et al. [55] is σ ( T ) = 1 . σ m (cid:40) − . (cid:18) TT m (cid:19) (cid:41) . (41)Entropy of mixing therefore stands∆ S = ∆ S ref + ∆ S tail (42)An alternative way may also be used to evaluate entropy of mixing∆ S = ∆ H − ∆ FT . (43)
The effective electron-ion interaction between a conduction electron and anion may be written as (in atomic unit)[56] w ( r ) = (cid:26) (cid:80) m =1 B m exp( − r/ma ) if r < R c − Z/r if r > R c , (44)ICROSCOPIC ORIGIN OF IMMISCIBILITY · · · Z , R c and a are the effective s -electron occupancy number, core radiusand, the softness parameter, respectively. B m is the coefficient of expansionwhich is independent of r but depends explicitly on parameters Z , R c and a . The pseudopotential theory leads to an expression for effective interionicpotential of an alloy through the energy band structure [33,36], v ij ( r ) = Z i Z j r (cid:20) − π (cid:90) dq F ( N ) ij sin qrq (cid:21) . (45)where the wave number characteristics F ( N ) ij = (cid:34) q πρ (cid:112) Z i Z j (cid:35) w i ( q ) w j ( q ) (cid:20) − (cid:15) ( q ) (cid:21) [1 − G ( q )] − (46) For a condensed state one of the most basic ingredients from which any micro-scopic description begins is the subatomic interaction or interionic interactionderived from the former one. This interaction dictates if the alloy would be anordered or a segregating type. In ordered alloy, the unlike atoms are preferredas nearest neighbours to like atoms, whereas in segregating alloys like atomsare preferred as nearest neighbours to unlike atoms. But direct identification oflike and unlike atoms in the sample is very difficult to achieve experimentally.Indirect ways through some probes assigned with interionic interactions, forexample, structural data, thermodynamics of mixing (viz. energy of mixing,enthalpy of mixing, entropy of mixing), atomic transport properties (viz. coef-ficient of shear viscosity, diffusion coefficient) and electronic transport proper-ties (viz. resistivity) provide good alternative ways. Some of the microscopicparameters used in identifying segregating alloys are(i) downward concavity or positivity of the free energy of mixing vs concen-tration profile at any or some concentrations,(ii) upward concavity or negativity of the entalpy of mixing profile at any orsome concentrations,(iii) upward concavity or negativity of the entropy of mixing profile at any orsome concentrations,(iv) order potential v ord = v ij ( r ) − v ii + v jj > α > w > S cc (0) diverges near the critical temperature and, the sharp increasehappens around the critical concentration.Some alloys such as Li-Na, Al-Bi, Al-Sn, Fe-Cu, Cu-Co, Al-Pb, Bi-Zn, Cd-Ga,Ga-Pb, Ga-Hg, Pb-Zn, Pb-Si, and Cu-Pb are well known systems for whichsome segregating properties are measured. So, it is worth pursuing to under-stand the microscopic origin of segregation from the theoretical point of viewand compare them with the experimental ones. The Gibbs free energy of mixing ( of a sample of N moles) for binary alloys is∆ G = G alloy − (cid:88) i =1 C i G i . (47)In terms of the partial Gibbs energies ∆ G i , one can write∆ G = C i ∆ G i + C j ∆ G j , (48)with ∆ G i = R T ln a i ( i = 1 , a i denotes the thermodynamic activity of the i -th component.The stability of a binary mixture is mostly determined by ∆ G . Figure2( i ) shows a schematic diagram for ∆ G denoted by G M as a function of con-centration C . Here curve a describes a miscible stable state whereas curve bdescribes an immiscible unstable state in the concentration range ∆ C . Thepoints P and Q in Figure 2( i ) give compositions of two segregatedICROSCOPIC ORIGIN OF IMMISCIBILITY · · · P and Q the partial Gibbs energies of the components areequal, ∆ G i ( C ) = ∆ G i ( C ) ( i = A, B ) . The point of inflexion in the curve b for T < T c represents the spinodal line.The critical concentration and critical temperature follow from the followingconditions (cid:18) ∂ ∆ G∂C (cid:19) C = x c = 0 ; (cid:18) ∂ ∆ G∂C (cid:19) C = x c = 0at T = T c . Following the Bhatia-Thronton structure factors [57], which is well( i ) ( ii )Figure 2: ( i ) A schematic diagram of Gibbs free energy of mixing as a functionof concentration and ( ii ) S CC (0) for different temperatures (after Singh andSommer [26]).known for the concentration-concentration fluctuation in the long wave lengthlimit, one can show S CC (0) = RT (cid:18) ∂ G m ∂C (cid:19) − T,p . As C −→ x c , and T −→ T C , S CC (0) −→ ∞ . This property of S CC (0) −→ ∞ signals the phase separation in a binary mix-ture. Figure 2( ii ) shows this behaviour. Other empirical models used6 G M BHUIYANTable 1: Critical concentration and critical for demixing liquid alloys.SystemsA m -B n m n x c,A wkT c A-B 1 1 0.5 2.0A -B -B -B -B ∂ ln a i ∂C i = 0 ; ∂ ln a i ∂C i = 0 , for different clusters suggested by the self association model.Although this empirical theory presents a good prescription to study thecritical properties of segregating alloys, its reliability in predicting critical prop-erties of real binary alloys is yet to be seen.ICROSCOPIC ORIGIN OF IMMISCIBILITY · · · For any microscopic description of a condensed matter the most fundamentalingredient necessary is the knowledge of interionic potential. Direct derivationand application of the N -body potentials to the study of the physical propertiesof condensed matter is a too much difficult job to handle theoretically. In orderto avoid this difficult situation one goes for the effective pairpotentials. Theterm effective indicates that, these potentials take into account the many bodyeffects in an average way following indirect routes. Figure 3 shows the profile −0.006−0.004−0.002 0 0.002 0.004 0.006 2 2.5 3 3.5 4 V i j ( r )( a u ) r(Å) Cu x Al X=0.5 V V V −0.006−0.004−0.002 0 0.002 0.004 2 2.5 3 3.5 4 4.5 5 5.5 V ij (r)( au ) r(Å) Zn x Bi x=0.5(b) V Bi−Bi V Zn−Bi V Zn−Zn
Figure 3: Partial pair potentials for Al x In − x , Cu x Al − x and Bi x Zn − x liquidbinary alloys For x = 0 . x In − x .It is seen that partial pair potential v AlAl ( r ) has the sallowest potential welland v InIn the deepest well. That of v AlIn lies in between. It is also seen thatthe position of the well minima for v AlIn and v InIn shift to large r relative to v AlAl . Similar feature is also observed for transition metal segregation alloys (for example Fe x Cu − x , Cu x Co − x ). In case of Zn x Bi − x the amount of shiftamong different partial pair potentials is significantly small. This shifting islargely associated with the difference in the values of the core radii betweenindividual components of the alloy. We note here that in random alloys v generally lies between v and v . But in the case of compound formingalloys v goes down the well of the v or v whichever has lower value. Wenote here that in the effective pair potential calculations Ichimaru-Utshumidielectric function [59] has been used by Bhuiyan and his group because thisfunction satisfies both compressibility8 G M BHUIYANsum rule and the short range correlation function. The BS pseudopotentialmodel has proven to be successful in the studies of liquid structure [60-63], ther-modynamic properties [34,35,64,65], atomic transport [66-70] and electronictransport properties [71,72] of liquid metals and there alloys. g a e ( r ) r(Å) Al Bi x X=0.1 g g g g a e ( r ) r(Å) Al Bi x X=0.5 g g g g a e ( r ) r(Å) Al Bi x X=0.9 g g g Figure 4: Partial pair potentials for Bi x Zn − x liquid binary alloys for x =0.1, x =0.5, x =0.9 respectively ( after Kasem et al. [24]).The partial pair correlation function, g ( r ), is related to the partial interionicpair potential through the statistical mechanics[73] (see equation (7)). Partialpair correlation functions for three different concentrations are presented infigure 4. In the alloys, rich in component 1, g exhibits the largest peak,while the trends become opposite in alloys rich in component 2; that is g shows the largest main peak. But in both cases peak value of g remains inthe middle of g ( r ) and g ( r ). The physical significance of g ( r ) is that, it givesa measure of the probability of finding the number of nearest neighbours at adistance of the peak from the ion located at the origin. Thus the area under theprincipal oscillation provides the coordination number, a characteristic featureof the condensed matter. Advantage of it is that, g ( r ) can also be derived fromthe X -ray or neutron diffraction data through the Fourier transformation, anddirectly from the computer simulation experiment. In the theoretical study ofliquid metals it plays the central role in describing thermodynamic properties. The free energy of mixing and its effects on the critical properties of segregationare described for different alloys below.ICROSCOPIC ORIGIN OF IMMISCIBILITY · · · Li x Na − x liquid binary alloys: The first attempt to estimate the energy of mixing theoretically for Li − x Na x liquid binary alloys from a microscopic approach was made by Tamaki [74].He also attempted to relate effective pairpotential between ions with the im-miscibility of the segregating alloys (see Figure 5(a)). He was Stroud whomade an attempt systematically for the first time to understand the segregat-ing properties such as critical concentration x c and critical temperature T c ofLi − x Na x liquid binary alloys using a microscopic theoretical approach [23]. Heemployed there the electronic theory of metal based on the empty core model[75], statistical mechanics and the Gibbs-Bogoliubov variational scheme [76]in order to calculate the free energy of mixing. Figure 5(b) shows a schematic(a) (b)Figure 5: Energy of mixing as a function of x for Li x Na − x liquid binaryalloys (after (a) Tamaki [72], (b) Stroud [20])diagram, presented by Stroud in [23], for the ∆ F as a function of concentra-tion for different temperatures. For T > T c the energy of mixing profile areconcave upward for all concentrations, which manifests complete miscibility(i.e. alloy is stable against segregation) at any concentration. But for T < T c the profile becomes concave downward at some concentrations which indicatessegregation of the alloy. The temperature at which spinodal points P and Q coincides is called the critical temperature T c , and the concentration at whichit happens is called the critical concentration x c . Here, in the calculation theHubbard type dielectric function [77] is used. The critical concentration forLi x Na − x segregating alloy was found to be x c =0.7, but the predicted criticaltemperature was overestimated by one third0 G M BHUIYANTable 2: Potential parameters and densities used for elements that formeddifferent alloys under study are listed.Elements ρ (˚ A − ) R c ( au ) a ( au ) Z Al 0.0517 1.91 0.30 3.0In 0.0342 1.32 0.29 3.0Bi 0.0289 1.49 0.36 (0.35) 3 (5)Fe 0.0756 1.425 0.33 1.5Co 0.0787 1.325 0.27 1.5Cu 0.0760 1.510 0.44 1.5[23].(b) Al x In − x liquid binary alloys: This Al x In − x alloy is formed by the elemental metals Al and In. Theseelements belong to the less simple polyvalent metals. Al based alloys are knownto be good candidates for a new advanced anti-friction materials. The inputvalues such as potential parameters R c , a and Z along with number density, ρ , for Al and In, and also for some other elements are shown in Table 2.Faruk and Bhuiyan[21] studied the segregating properties of Al x In − x liquidbinary alloys by using the electronic theory of metals (first principle calcula-tions) along with the statistical mechanics and perturbative approach. Ini-tially, they justified the appropriateness of the potential parameters by cal-culating static structure factors of the elemental liquids at a thermodynamicstate at which experimental data are available [78]. Figure 6(a) shows thebreakdown details of energy of mixing, ∆ F , at T = 1173 K. It is noticed thatthe HS contribution to the energy of mixing is negative and values are thelowest among all other contributions across the whole range of concentrations.The tail part contribution is also negative across the concentration range andvalues are the second lowest among all others. Contribution of the electrongas, ∆ F eg , is positive for the full concentration range and valuesICROSCOPIC ORIGIN OF IMMISCIBILITY · · · x for Al x In − x liquid binary alloys(a) breakdown details at T=1173 K, (b) Temperature dependence.(after Farukand Bhuiyan [21])are very close to zero. The volume dependent (i.e. structure independent)part of the energy of mixing, ∆ F vol , due to electron ion interaction is positiveand large across the full range of concentration. The combined effect of allcontributions, ∆ F , agree well with the corresponding experimental data [79].This signifies the accuracy of the approach for the study of energy of mixingat different temperatures.The temperature dependent energy of mixing for Al x In − x liquid binary al-loys for different concentrations are illustrated in figure 6(b). As temperatureis decreased from 1173 K, ∆ F increases gradually and at 1155 K becomes par-tially positive and partially negative. Further lowering of temperature increasethe miscibility gap and at 1140 K the concentration gap span the whole rangeof concentration. A careful observation finds the first downward concavityor positivity of ∆ F at 1160 K, and the concentration at which it happens is x = 0 .
5. So, the predicted critical temperature and concentration for Al x In − x segregating alloys are T c = 1160 K and x c = 0 .
5, respectively. The experi-mental work by Campbell et al.[80], and Campbell and Wagemann [9] reporta critical temperature of 1220 K. whereas Predel [1] reports 1100 K for x In − x liquid binary alloys. Differential thermal analysis by Sommer et al. [81] re-ports T c = 1112 K. The average of these scattered experimental data is 1144K which is close to the theoretical prediction of Faruk and Bhuiyan[21]. Theexperimental critical concentration [10,1] is x c = 0.5 which is exactly the sameas that of theoretical prediction2 G M BHUIYAN[21]. But the experimental data reported in [9] is 0.34 which largely deviatesfrom 0.5.(c) Bi x Al − x liquid binary alloys: (a) −0.2−0.15−0.1−0.05 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 E n er gy o f m i x i n g , ∆ F ( e V ) Al x Bi Al Bi x T=1187 K ∆ F tail ∆ F ∆ F ∆ F vol ∆ F eg ∆ F hs (b) −0.06−0.04−0.02 0 0.02 0.04 0 0.2 0.4 0.6 0.8 1 E n er gy o f m i x i n g , ∆ F ( e V ) Al x Bi Al Bi x ∆ F ∆ F ∆ F ∆ F ∆ F ∆ F Figure 7: Energy of mixing as a function of x for Bi x Al − x liquid binaryalloys (a) breakdown details at T=1187 K, (b) Temperature dependence (afterAbbas et al. [25]).Bi x Al − x liquid binary alloy is formed by two elements Al and Bi whichbelong to group IIIB and VB in the periodic table, respectively. The meltingpoints of Al and Bi are 933 and 544 K, respectively; the corresponding densitiesare 2.375 and 9.78 gm cm − . Al is a trivalent and Bi is a pentavalent metal.The atomic radii of Al and Bi are 1.82 and 1.63 ˚A, respectively. The largemismatch in their physical properties makes this alloy interesting to studytheoretically.Figure 7(a) illustrates the breakdown details of the energy of mixing at T =1187 K at which some experimental data [79] for ∆ F are available in theliterature. The HS contribution to the energy of mixing is negative for thewhole concentration range as is found for Al x In − x liquid binary alloys. Butunlike Al x In − x , the tail part contribution, ∆ F tail , of Bi x Al − x alloys is positivefor all concentrations with a maximum near equiatomic concentration. Thevolume dependent part, ∆ F vol , in this case, is positive but the magnitudesare much lower than that of ∆ F tail . The electron gas contribution, ∆ F eg , isnearly zero as for Al x In − x . The combined effect of all contributions to thefree energy, however, agrees well with the experimental results at T =ICROSCOPIC ORIGIN OF IMMISCIBILITY · · · x Al − x liquid binary alloys fordifferent temperatures. It appears that the alloy exhibits a complete miscibil-ity at 1350 K, and immiscibility for all concentrations at 1050 K. But at T =1290 K, ∆ F shows a partial positivity with concavity downward near x = 0.15.Further decrease of temperature gradually enhances the concentration gap. Asthe concavity downward (or positivity) of ∆ F manifests onset of segregation,one can conclude that the predicted critical concentration is x c = 0 .
15, andcritical temperature T c = 1290 K, while the corresponding experimental valuesare x c = 0 .
19 [82] and T c = 1310 [82,83].(d) Zn x Bi − x liquid binary alloys: (a) −0.15−0.1−0.05 0 0.05 0.1 0.15 0 0.2 0.4 0.6 0.8 1 ∆ F ( e V ) Zn x Bi
T=873K Zn x Bi ∆ F Tail ∆ F Exp ∆ F ∆ F vol ∆ F eg ∆ F HS (b) −0.06−0.04−0.02 0 0.02 0.04 0 0.2 0.4 0.6 0.8 1 ∆ F ( e V ) Zn x Bi Zn x Bi ∆ F(573K) ∆ F(673K) ∆ F(873K) ∆ F(773K) ∆ F(550K)
Figure 8: Energy of mixing as a function of x for Bi x Zn − x liquid binaryalloys (a) breakdown details at T=873 K, (b) Temperature dependence (afterKasem et al. [24].Figure 8(a) shows the free energy of mixing for liquid Zn x Bi − x alloys at 873K [24]. ∆ F hs is negative for Zn x Bi − x liquid binary alloys for the whole rangeof concentration, this trend is similar to that of the previous alloys, and havingthe smallest values relative to the other components for each concentration.In this case ∆ F is asymmetric in nature where the minimum value is foundto be around x = 0.6 which is located in the Bi rich alloys. ∆ F tail and ∆ F vol contributions are positive for the full range of concentrations, but ∆ F tail showsthe larger values than that of ∆ F vol . ∆ F eg contribution is nearly zero as othersegregating alloys under consideration of4 G M BHUIYAN[21]this review article. The total energy of mixing, however, matches well withcorresponding experimental data [79].Temperature dependence of ∆ F are illustrated in figure 8(b) It is noticedthat at T =773 K and higher temperatures ∆ F is negative for all concentra-tions. This nature indicates that the alloy is completely miscible in the regimeof the above thermodynamic states. But at a lower temperature T = 673 K,∆ F becomes positive i.e. concave downward for some concentrations and neg-ative for others. When temperature is lowered further miscibility gap increasesgradually as previous systems and cover the whole concentration range at 550K. From figure it appears that the critical concentration is x c = 0 . T c = 773 K. The experimental value for x c is 0.83 [16,17],the critical concentration found theoretically by Stroud [36] and Karlhauber etal [84] (from quasi lattice theory) was x c =0.75 and 0.87, respectively. Experi-mental critical temperatures are 856 K [16] and 878 K [17], and a theoreticalstudy shows 438 K [36].(e) Fe x Cu − x liquid binary alloys: (a) x Cu T=1823K
CuFe F HS F tail F F exp F eg F vol F ( e V ) X (b) x Cu F F F F CuFe F ( e V ) X Figure 9: Energy of mixing as a function of x for Fe x Cu − x liquid binary alloys(a) breakdown details at T=1823 K, (b) Temperature dependence (After Faruket al. [22]).Figure 9(a) shows that the HS contribution to the free energy of mixingfor Fe x Cu − x is negative for all concentrations at T = 1823 K [22]. Here thetail part contribution to the energy of mixing is negative for all concentrationsunlike other segregating alloys. The volume dependent term ∆ F vol isICROSCOPIC ORIGIN OF IMMISCIBILITY · · · F eg is almost zero as for all others discussed above. The total energy of mixingobtained summing all four contributions is negative for all concentrations andthe agreement with available measured data [79] is fairly good. At T = 1823K the alloys remain miscible across the full concentration range. As temper-ature is lowered to 100 K, ∆ F becomes partially positive around equiatomicconcentration where the concavity is downward, and the other part of energyof mixing remains negative with upward concavity. The critical temperaturethus found was T c = 1750 K and the critical concentration found was x c =0.5. The experimental values reported by different authors for x c are 0.56 [13],0.538 [14] and 0.538 [15], and the corresponding experimental data for T c are1696, 1704 K and 1694 K, respectively.(f) Co x Cu − x liquid binary alloys: Cu x Co F F F F CoCu F ( e V ) X Figure 10: Temperature dependence of energy of mixing as a function of x for Cu x Co − x liquid binary alloys (after Faruk et al. [22]).In this case the behaviour of various contributions to ∆ F is found to besimilar to that of Fe x Cu − x [22]. But, in the immiscible state ∆ F shows (Figure10) an asymmetric feature with a value of critical concentration x c = 0.58 andcritical temperature T c = 1650 K. The corresponding experimental values are x c = 0.53 and T c = 1547 K [12].Bhuiyan and coworkers carefully investigated why ∆ F varies with6 G M BHUIYANtemperature. They have found that ∆ F hs and ∆ F tail are sensitive to T and aremostly responsible for the variation. While ∆ F vol and ∆ F eg are not sensitiveto T at all. The sensitivity arises, in this case, due to the alteration of σ andconsequently g hs ( r ), with the change of T . −0.1−0.05 0 0.05 0.1 0.15 0 0.2 0.4 0.6 0.8 1 E n t h a l p y o f m i x i n g , ∆ Q ( e V ) Al x Bi Al Bi x ∆ Q ∆ Q ∆ Q exp ∆ Q ∆ Q ∆ Q Figure 11: Temperature dependence of enthalpy of mixing as a function of x for Bi x Al − x liquid binary alloys (after Fysol et al. [25]). (a) Bi x Al − x liquid binary alloys: The enthalpy of mixing, ∆ H , are used as a probe to study the criticalproperties of Bi x Al − x liquid binary alloys. Figure 11 demonstrates that cal-culated values of enthalpy of mixing agree in an excellent way with availableexperimental data for miscible alloys at 1187 K [79]. However, figure also showthat the trends of ∆ H as a function of concentration is just opposite like amirror reflection to that of free energy of mixing discussed above. That is at1350 K ∆ H is positive, and at 1050 K it is negative for the full concentra-tion range, while at the same thermodynamic states ∆ F shows negative andpositive values, respectively. Figure also shows that, at about 1290 K, ∆ H exhibits negative (i.e. concave upward) at low values of x and positive (i.e.concave downward) for the rest. That means segregation of the alloy beginsat 1290 K which is exactly same as found from ∆ F [25]. But in theICROSCOPIC ORIGIN OF IMMISCIBILITY · · · It is interesting to see how another static magnitude the entropy of mixingdescribes the critical properties of segregation for different alloys.(a) Zn x Bi − x liquid binary alloys (a) −1−0.5 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 ∆ S / N k B Zn x Bi
T=873K Zn x Bi ∆ S Tail ∆ S Exp ∆ S ∆ S σ ∆ S gas ∆ S HS (b) −1.5−1−0.5 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 ∆ S / N k B Zn x Bi Zn x Bi
873 K773 K 673 K573 K550 K
Figure 12: Entropy of mixing as a function of x for Bi x Zn − x liquid binaryalloys; (a) breakdown details, (b) temperature dependence (after Kasem et al.[24]).Figure 12(a) shows the breakdown details of different contributions to the to-tal entropy of mixing calculated by Kasem et al. [24]. It is seen from figurethat at T = 873 K, ∆ S hs is negative up to x ≤ . S gas is negative in the concentration interval 0 . < x < .
8, and positivebeyond it. Contribution of HSD mismatch term, ∆ S σ , is almost zero acrossthe whole range of concentration. The tail part contribution, ∆ S tail , is foundto be positive for the full concentration range. However, the combined effectof these contributions that is the total entropy of mixing is positive for all con-centrations and, the agreement between theory and experiment is very goodup to x = 0 .
7, and fairly good for x > . x Bi − x liquid binary alloys [24]. We note here that negativity of ∆ S (i.e.upward concavity) is an indication of segregation. Figure also shows that thecritical temperature and critical concentration are T c = 773 K and x c = 0 . Bi x Al − x liquid binary alloys −2−1.5−1−0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 E n t r o p y o f m i x i n g , ∆ S / N k B Al x Bi Al Bi x ∆ S ∆ S ∆ S exp ∆ S ∆ S ∆ S Figure 13: Temperature dependence of entropy of mixing as a function of x for Bi x Al − x liquid binary alloys ( after Fysol et al. [25]).Figure 13 shows entropy of mixing for Bi x Al − x liquid binary alloys calculatedby Fysol et al. [25]. At T = 1350 K, ∆ S is positive for all concentrations andat T = 1050 K it is negative for the whole range of concentration. For temper-atures in between the entropy of mixing is partly positive and partly negative.Fysol et al. [25] theoretically found the values x c = 0.1 and T c = 1290 K forcritical concentration and critical temperature, respectively. Here the value of T c is the same as that found from ∆ F [25] but x c is somewhat lower in thiscase. Looking at figures of free energy of mixing for Al x In − x , Fe x Cu − x , Cu x Co − x ,Zn x Bi − x , and Bi x Al − x liquid binary alloys one can easily find that the HSICROSCOPIC ORIGIN OF IMMISCIBILITY · · · F hs is always negative for all concentrations and temperatures.This means that, HS liquid alone cannot describe segregation for binary al-loys. This finding agrees with that of Libowitz and Rowlinson [40]. However,∆ F vol becomes positive for the whole range of concentration and dominatesother contributions in the case of Fe x Cu − x , Cu x Co − x , and Al x In − x liquidbinary alloys; this feature directly favours the segregation for these alloys. TheTable 3: Critical temperature and critical concentrations for different segre-gating alloys. x c T c (K)Systems (Theo.) (Expt.) (Theo.) (Expt.) Others(Theo.)AlIn 0.5 0.5, 0.34 1160 1155, 1150, 1145 -FeCu 0.5 0.56, 0.538 1750 1696,1704,1694 -CuCo 0.58 0.53 1650 15473 -ZnBi 0.9 0.83 773 856, 878 438BiAl 0.15 0.19 1290 1310 -contribution of the tail part of the pair potential, ∆ F tail , becomes positive forthe full concentration range for Zn x Bi − x , and Bi x Al − x liquid binary alloysand negative for others. The electron gas contribution ∆ F eg is nearly zero forall systems and for any thermodynamic state characterized by temperature.Energy of mixing for hard sphere liquid and the tail part contribution are verysensitive to temperature unlike ∆ F vol and ∆ F eg . In the case of free energy∆ F hs and ∆ F tail increases with increasing temperature, as a result total en-ergy of mixing becomes concave downward which manifests immiscibility ofthe alloy. The values of the critical temperatures and critical concentrationsfor different alloys are illustrated in Table 3.Understanding of the segregating behaviour of liquid binary alloys from themicroscopic theory for transport properties has just began. Some interestingfeatures exhibited by the coefficient of viscosity and diffusion coefficient as afunction of concentration appears to be spectacular [25]. One of the featuresis the sharp bending in the η vs x (or D vs x ) profile around the0 G M BHUIYANcritical concentration. Competition between the thermal excitation of ions andthe variation of density with temperature is another one. In this case, for T
08, near the critical temperature. These novel features showed by somesegregating liquid binary alloys demands further research to understand thedynamic effects in segregating alloys.ICROSCOPIC ORIGIN OF IMMISCIBILITY · · · The softness parameter a i used in the calculation are determined by fittingexperimental S ( q ) at small q (see Fig.12). S ( q ) q(Å) Al Bi Figure 14: Determination of a i from the best fit of S(q); line theory, closeddots experiment. For AlIn (left)(after Faruk and Bhuiyan [21] and for BiZn(right) (after Kasem et al. [24]).
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