A Model for polymorphic melting in binary solid solutions. Huaming Li a,b, Mo Li a, * GA 30332

Transcription

1 A Model for polyorphic elting in binary solid solutions Huaing Li a,b, Mo Li a, * a School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 333 b School of Physics, Georgia Institute of Technology, Atlanta, GA 333 We propose a phenoenological theory to describe polyorphic elting in binary solid solutions with teperature and coposition as two external variables. In this fraework, we treat elting as a topological-order-to-disorder transition with the priary order paraeter chosen to represent the loss of the long-range-order and the elastic strain induced by different alloy coponents as the secondary order paraeter. Under polyorphic constraint, the odel gives the elting line and other therodynaic properties at elting that are in very good agreeent with available experiental results in dilute solutions; extrapolation is ade into high concentration regie. Possible icroscopic echanis is discussed for polyorphic elting in binary systes as a firstorder phase transition. PACS Nuber: 64.7.D-, 64.7.dj, 8.6.Lf, 64.7.P- * The corresponding author: o.li@se.gatech.edu 1

2 1. Introduction Melting of crystalline solids is a topological order-to-disorder transition in which a solid phase with long-range translational syetry becoes a liquid with topological disorder. Melting is recognized as a first-order phase transition that involves discontinuous change in latent heat and volue at elting point 1. Although great progress has been ade in the past century, fundaental understanding of kinetics and icroscopic pictures of elting is still not fully coprehended. Fro early experients on pure eleents, hoogeneous elting odels have been proposed. Born described elting as a echanical instability of a solid phase, which iplies that elting is a continuous transition without nucleation and growth of the elt. Melting as he defined 3 is a crystal instability with the loss of shear rigidity. Lindeann 4, on the other hand, relates the elting teperature with the critical root-ean-square displaceent of theral vibrations of atos. Both odels are one-phase theory without any consideration of liquid phase and do not take into account of the discontinuous character of first-order phase transition. Experientally, elastic odulii of solids decrease with increasing teperature but do not go to zero at elting point, while the root-ean-square displaceent does show increase in agnitude as elting is approached. To rationalize elting, ore detailed echaniss, ost of which invoke various defects, are introduced. Tallon 5 odified Born s criterion by stating that the shear odulus of a cubic crystal goes to zero, or a continuous elting would occur, when the volue of the crystal equals to that of the liquid phase at a higher teperature than the therodynaic elting point, causing the echanical instability or catastrophe 6. One way to induce the instability, which requires extra volue expansion 7, is by introducing point defects, such

3 as vacancies 8 and interstitials 9 ; and soeties line defects are also introduced 1. For theral elting, these scenarios ay see far-fetched as it is ipossible to achieve such high defect densities as required by the odels, which is evidenced fro experiental easureents and coputer siulations of vacancies and interstitials in elting processes 11. In practice, elting is found to be a heterogeneous process that nucleates at exterior surfaces or internal defects 1, 13. In alloys systes, coposition is another therodynaic variable beside pressure and teperature that greatly affects elting 14. In fact, solute atos are another type of point defect to the host crystalline phase since different eleents have different physical and cheical properties. Mixing different alloy eleents leads to certain incopatibility which in turn results in various disorders in a crystal. As known, atoic size isatch and cheical disorder induced by the solute atos play an iportant role during the topological disordering processes (see Fig. 1). It is generally found that the elting point in elting of alloy systes is depressed as the concentration of solutes increases, along with the increasing static disorder, internal strain and softening of the elastic constants. Static atoic displaceents obtained fro x-ray Debye-Waller factor of a defective alloy crystal is a generic easure of the static disorder caused by solute atos 15, which is frequently accopanied by increasing anharonicity in lattice vibrations that can give rise to additional theral disorder. These effects undoubtedly contribute to crystalline phase instability and thus elting. In these cases, both static and dynaic coponents of the total ean-square atoic displaceent are a viable easure of crystalline disorder, which was indeed eployed in a generalized Lindeann elting criterion 16. In binary alloys, however, both cheical concentration redistribution and 3

4 topological disordering occur at the sae tie when theral elting occurs, resulting in a two-phase co-existing region that corresponds to a range of elting teperatures in alloys, which is contrast to just one elting point in pure systes. With appropriate heating rate, however, polyorphous elting could occur where a crystalline solid of a specific coposition becoe a liquid without coposition change or partition 17, 18, which were observed by x-ray diffraction and other ethods. This case is very appealing to us for understanding elting as one can focus on the topological disordering exclusively. Two unique features eerge in polyorphic elting in alloy systes that are not present in pure systes. First, under polyorphic constraint on a binary alloy syste that liits the long-range diffusion, the defects once created are localized for the lack of obility on the tie scale as copared with the observation tie 19. The second effect is the eergence of the elastic energy created around these defects. Figure 1 illustrates how strain is created around a point defect, a solute ato. The influence of elastic strain field on the phase stability is significant and should not be overlooked. These two constraints, lack of long-range atoic obility and eergence of internal elastic field, unique to polyorphic phase transition ust be taken into consideration in elting processes in alloy systes. The crucial part of the polyorphic elting process is the creation of static, or iobile defects. The presence of the static defects, i.e. solutes, in the ordered structure that introduces strain field is anifested in the softening of elastic constants. The increasing static disorder easured by the root-ean-displaceent, which is accopanied by lattice softening and increasing anharonicity in lattice vibrations, will raise the free energy of the parent crystal state. Lattice softening and anharonicity will 4

5 give rise to additional dynaic disorder. The cobined effects will lead to elting. The chief purpose of this work is to establish the link between these causes and theral elting in alloy systes. In this paper, we propose a siple odel using Landau theory to explain polyorphic elting in alloy systes. The priary order paraeter is the ean atoic displaceent as defined by Lindeann to represent he structure disorder, along with a secondary order paraeter chosen as the strain induced by the solutes. The identification of these two quantities as the order paraeters allows us to construct a Landau free energy that will be used to describe the transition phenoenologically. The internal strain couples to the ean atoic ean square displaceent to iic the interaction of the topological disorder caused by the presence of the strain field of the solute atos. This coupling is the key to forulate the phenoenological odel as entioned above: When the defect density increases, it will not only cause displaceents of the surrounding atos fro their lattice positions, but also generate internal strain field that leads to weakening of the elastic rigidity of the background. Both of the will contribute to the rise of the free energy. The interaction between the two order paraeters becoes ore intensified as ore defects are created. Eventually the syste reaches a point where the free energy of the syste reaches that of a liquid phase, resulting in polyorphic elting. This paper is organized as follows. In Section, we present a siple analytical expression of Landau free energy for polyorphic elting. The fors of Landau free energy are discussed and various therodynaic properties are derived for pure eleents and binary alloys. In Section 3, using the experiental easured data in Si-As solutions, we obtain the coefficients in the theory, which allow us to calculate the elting line, or 5

6 the phase diagra in T - x plane and the corresponding therodynaic quantities along the elting line. We copare our proposed theory of polyorphic elting with the experiental results. Good agreeent is found between the theoretical results and those fro the published experiental easureent in dilute solute region. In Section 4, we discuss the theoretical results. The validity of prediction for high coposition value is also discussed. The concluding reark is drawn in Section 5.. Landau Free Energy Melting of a crystal to a liquid phase is characterized by any property changes including volue, heat capacity and theral conductivity. But the ost revealing feature is the disappearance of the Bragg diffraction patterns in scattering experients at elting, which signals the eergence of a topologically disordered phase. The underlying reason for this structural change is the rando displaceent of atos fro their equilibriu lattice positions under the theral agitation. Based on this, Lindeann and later Gilvary 4 proposed a therodynaic criterion for elting: elting occurs when the ean square displaceent (MSD) of the vibrating atos reaches soe critical value, or η = u a η, where a is the equilibriu nearest neighbor atoic distance and / c η c is the Lindeann constant, which is about 1% for ost etals. In general, u is the MSD consisting of two parts, u = u + u, where dyn sta u dyn is the dynaic MSD directly related to theral vibration during heating and u sta is the static coponent fro the static disorder induced by the rando displaceent fields associated with the configurations of the defects (Fig. 1). For elting in pure syste, u sta is negligible and u dyn is large, especially at high teperature. For alloy systes, both are iportant. 6

7 Therefore, we could use the MSD as an order paraeter in a Landau theory (or rather a disorder paraeter in our case). u = u + u would increase during heating and dyn sta reach a critical value at elting; it is therefore a viable representation for the state of the syste fro an ordered phase to a disordered one. This is especially appealing in polyorphic elting as u sta, as a priary coponent of the order paraeter, is wellbehaved in both the parent and new phases. After elting, however, u dyn becoes divergent in the liquid phase due to the contribution fro diffusion of the atos for large tie scale. It is sufficient to use this order paraeter if our focus is only on elting. For a crystalline phase subject to theral stiuli as in polyorphic elting, we can therefore write the free energy density as a b 4 c 6 Δ F ( η, T ) = F( η, T ) F ( T ) = η η + η. (1a) 4 6 Unlike the usual definition of an order paraeter, in this work, η is for the ordered state and 1 η for disordered state. a, b, c are coefficients with a a ( T T) a >, and T is a reference teperature with b >, and =, c >. F ( T) = F( η =, T) is the free energy of the defect-free saple. The choice of this expression by including only even ters in F (, T ) η is fro the consideration that only u is physically eaningful, where stands for theral and configuration average. As we show later, this choice also allows us to include the bilinear coupling between the displaceent and the strain field. In crystal state with a sall aount of disorder ( η ), F( η, T) Δ is sall; it becoes larger as η oves to a larger value close to the transition. As known 1, 7

8 Eq. (1a) describes a non-syetry driven first-order phase transition, such as polyorphic elting. As we entioned above, for polyorphic elting in alloys, additional ters are needed to describe the contributions fro the strain energy and the coupling between the strain and MSD. For siplicity, we shall choose to use a binary solution odel. For the polyorphic elting with the absence of long-range diffusion, the solute atos with atoic sizes different fro that of the solvent creates local, atoic size-induced strain, ε ' (Fig. 1). In this work, we shall use only the shear or deviatoric strain, that is, ε = ε ' 1/3 tr( ε '), which is based on the consideration that shear softening is the predoinant ode in elting. The Landau theory therefore has two order paraeters, the priary one, η, and the secondary one, ε. The free energy density is expressed as ( ) ( ) ( ) Δ F( η, ε, T) = F( η, ε, T) F ( η =, ε =, T) =Δ F η, T + f ε + f c η, ε. (1b) ( η, T) Δ F is given in Eq. (1a). The strain energy contribution is given by 1 f ( ε ) = με, (1c) where μ is the bare shear elastic constant of the solvent. c (, ) f η ε = γηε (1d) is the bilinear coupling ter between the position disorder represented by η and shear strain by ε with a coefficient γ..1 Theral elting of pure eleents Equation (1b) is the expression for the free energy for a polyorphic topological order-to disorder transition. In a one-coponent syste, Eqs. (1c) and (1d) do not show 8

9 up. As shown below, theral elting in a pure syste can be described well as a firstorder transition with Eq. (1a). When approaching elting, the co-existing solid and liquid phases requires that Δ F = η F and the therodynaic stability condition, >. Fro these conditions, η along with the equilibriu coexistence, F(, T) F ( T) η = at elting, we obtain the elting point T 3b = T and the relation aong the coefficients 16ac 3b a = at elting 16c b b 4ac point. The order paraeter changes as η + = c 1/ toward the elting point and at 3b T = T there is a finite jup of the MSD, η( T ) =± 4c 1/. The finite change of the heat capacity is ΔS ΔF a Δ c= T T= T= T T= T= T T b. The susceptibility associated η 16c 1 with the order paraeter χ = li T ζ = =, where ζ is the conjugate field to ζ 3b a 3 8 a 1 b η. In addition, the energy barrier at elting is Δ F = = 7 b 96 c, which is deterined by the constants b and c. By fitting experiental values fro MSD, Δ c, and latent heat Δ S syste., etc, we could obtain the coefficients (a, b, and c) for a specific. Theral elting of binary solid solutions under polyorphic constraint Equations (1b)-(1d) describe a polyorphic elting where the defect-induced strain field begins to play a role in the transition. Since there is no external applied stress, under 9

10 Δ F static equilibriu condition, the internal stress should be zero, or = γη + μ ε =, ε γ fro which we obtain ε = η. Substituting this relation into Eqs. (1), we obtain the μ effective free energy where a~ b c F eff Δ = η η η, () ~ γ a = a. (3) μ Eq. () is the free energy that we shall use to describe theral elting in binary syste. To take into account the alloying effect, we consider the Landau free energy in Eq. () with the coefficients ( ) ( ) a = a T, x, b= b x, c= const, (4) where T and x stand for teperature and the alloy coposition (or ipurity concentration). Following the protocols in Landau theory, we shall consider the choice for the functional for for the coefficients. For theral elting of a single coponent syste, we take a = a ( T T), ( x) b =constant, c = constant, which led to the results in the previous section. For alloy syste under polyorphic constraint, i.e. the concentration reains the sae before and after the transition, we can rewrite, fro Eq. (3), ~ γ a ( x, T ) = a μ = a ( T γ ( x) T ) μ ~ = a ( T T ), (5) 1

11 where ~ T γ ( x) = T. As done routinely in Landau theory, we shall choose the a μ functional for for γ (x) and b (x) with the consideration of keeping the lower order dependence of the ipurity concentration x, which allows easy treatent of the proble. Different functional fors could be taken if needed. For the coupling coefficient γ (x), we have 3 γ = γ1x+ γx + γ3 x +..., for x xc (6) γ =, for x > xc γ with appropriate choice of coefficients γ n ( n = 1,, 3, ). γ (x) is chosen where ( x) in this way based on our understanding that the coupling of the MSD to the strain becoes stronger as ore solute is added to the syste. x c is the critical solute or ipurity concentration where phase instability ay occur at a given teperature. The coupling is expected to disappear when a liquid phase fors, which is particularly true for liquid that are known to have fast relaxation of the internal stress. For b (x), we choose b = b + b x+ b x + b x + (7) to ake sure that b (x) is positive always with the appropriate choice of the coefficients b ( n =, 1,, 3, ). The trend in b (x) vs. x is fro the experiental observation that n anharonicity induced by the solvent atos increases with solute addition. We need to ention that such a choice for b (x) ay not hold in high solute concentration where interactions aong solutes and local atoic configuration change ay occur. We shall discuss this point further in Section 4. 11

12 .3. Results The free energy defined in Eq. () with the coefficients specified above leads iediately to several interesting, although not entirely unexpected results. One is the depression of the elting point in the alloy systes with increasing solute or ipurity concentration. Since the syste is kept under polyorphic constraint, the cheical ΔF equilibriu eff = x ay not be necessarily satisfied. However, the polyorphic elting transition described fro Eq. () in the alloy syste is still deterined by the ΔF conditions, ΔF eff = and eff = η. As shown in the pure syste, the relation a~ ( x, T ) T 3b( x) = T = still holds for alloy systes, fro which we can obtain the elting 16c line vs. solute concentration ( x) ~ ~ 3b( x) γ ( x) 3b T ( x) = T = T 16ac aμ 16ac. (8) As seen in Eq. (8), the elting point is depressed by the presence of the internal strains represented by increasing γ (x) and the anharonic effects represented by b (x). The rate of elting point depression with respect to x is described by dt ( x) γ( x) dγ( x) 3 b( x) db( x) =. (9) dx a μ dx 8a c dx As γ (x) is an increasing function of x, the larger the aount of solute or ipurity is added, the lower the elting point becoes. The prediction agrees with the results in ost binary systes 3, except when certain interetallic phases for. 1

13 As shown in section.1, in alloys the MSD at elting in the alloy syste is η ( T 3b( x) ) = 4c 1/, the entropy change Δ s = 3 a b( x) 4c, the heat capacity change a 1 b( x) Δ c =, and the activation energy barrier Δ F = b( x) 96 c 3. The effective elastic constant in the disordered syste can be obtained fro 1 ΔF ΔF ΔF ΔF μ = μ = γ χ ε ε η η η ε, (1) where the order paraeter susceptibility is ΔF χ = η 1. Since μ is the elastic shear odulus in the pure syste at x =, addition of solute to the host lattice causes softening of the elastic odulus. Fro (1), we can see that the softening is caused by two factors, the solute-induced strain field via γ and teperature, through the relation, 1 1. a ( T T ) At a given teperature, the softening becoes ore significant when ore solute is added and the syste becoes ore susceptive o the structural change or disordering. As we know that 1 F Δ χ = η (>) and γ (x) is finite at the elting point, the shear odulus at the elting point still reains finite, which is different fro Born s prediction. 3. Coparison with experients Due to the strict kinetic polyorphic constraint to aintain the syste in a cheically partitionless process, usually the heating rate (or cooling rate) needs to be high, which led to difficulties to easure the transition teperature and elastic constant, 13

14 and deterine the phase diagras with conventional ethods. Progress has been ade however to deterine the polyorphous phase diagras. Therefore available experiental data can be used for coparisons with our theory 4, 5, 6, 7, 8. Several easureents for Si-As alloy by pulse laser 9 or line source electron bea 4 heating technique have been reported. And droplet eulsion technique is also used for Sn-Bi alloy 18. Since direct teperature easureent with several independent ethods gives consistent results in Si-As syste by Kittz et al 4, we will use the reported data to fit the coefficients in our odel. To siplify the fitting process, we noralize the easured polyorphic elting points with the elting teperature of crystalline silicon at 1685K. We also rescale the variables and coefficients, so c = 1, T 1 and μ 1. And the rest of the variables and coefficients becoe as the following: = ( c / 6) = 1/ 6 1/ F F η, ε γ ε, γ, 1/ 1/ 1/ 3 6 1/ μ F c / 1/ μ η 6 ( ) 6 at a, and / 3 1/ 3 F ( c / 6) b b ( F is the reference free energy). F ( c / ) / 3 1/ 3 6 With the chosen fors for γ (x) and bx ( )(Eqs. (6) and (7)), we have for γ, b >, b > 1 and γ = for n > 1, b = for n >, n n 1 > γ ( x) = γ1x, for c x x (1) γ ( x) =, for x > xc ; b ( x) = b + b x. (11) 1 With these relations, the elting point is expressed as ~ ~ T ( x) = T γ 1 μ 3b1 8c + x a (1) 14

15 with ~ 3b = T as the elting point of crystalline silicon. Fro this relation and 16a c T the experiental data for pure Si (Table 1), we have a =. 1 and b =. 15 (and c = 1 as shown above). Through the fitting on the elting line with available experiental data on Si-As (inset in Figure ), we have b =. 1 5 and γ = Fro the obtained coefficients, we now have γ ( x) and ( x) b explicitly. Fro the results, we can see that the elting point is depressed in a linear fashion that is proportional to the coposition. Since the experiental data are only available in dilute solute concentration, the linear fitting appears in good agreeent with the experient. We can extrapolate the fitted elting line to higher solute concentration where few experiental data are available (Figure ). However, we should ention that with ore experiental data available in high solute regie in the future, we could chose to have different choices for γ ( x) and ( x) function of the concentration there. b such that T ( x) ay not be necessarily linear Although there were no direct easureent of the change of elastic constants in polyorphic elting of Si-As, fro the coefficients obtained above and experiental data of pure Si, we can predict the change that ight happen in the alloy. As the variation of elastic constant C 44 of pure silicon is a (linear) decreasing function of increasing teperature 6 between 15K and 1K fro experients (Table 1), using the relation obtained fro the experient data in Table 1, we can obtain the shear odulus of pure silicon at different teperatures μ ( T ). So the values of the shear odulus of Si can be used to estiate the shear odulus of the alloy syste at different concentration x at the sae teperatures through the relation =. For exaple, μ( x, T) μ ( T) χγ( x) 15

16 C = GPa at T = 1556 K and C = GPa at T = 147 K for Si. These teperature values are corresponding to the elting points at x =. 45 and x =. 9 for Si-As alloy as seen fro Figure (a). Specifically, at a given teperature, the predicted elastic constant for the alloy is γ 1x μ = 1 C44( Si)( T ) where the values for a( T ) 1 γ, at ( ) and ( )( ) C Si T are known. Fro the above relation, we can predict the elastic constants of the alloy syste at different As concentrations. Elastic softening is seen in Figure 3 as a linear function of the solute concentration. For instance at the above two teperatures, the shear odulus at elting points is C = GPa at x =. 45 ( T = 1556 K) and C = GPa at x =.9 ( T = 147 K). 44 The latent heat of fusion is defined as L f = T ~ Δs with L ( Si ) =5.719 KJ Mol -1 f for pure crystalline silicon 4 and the calculated entropy change is 3.1 JK -1 Mol -1 at the elting point T ( Si ) = 1685 K. For alloy systes going through polyorphic elting, the predicted entropy change is Δ s( x) ( x = ) 1/ Δ b s = = + x JK -1 Mol -1 Δs b which is a function of the solute concentration x. Siilarly, the predicted latent heat is L f f f ( ) ( = ) ( ) 1/ L x T x b = = + L x b (1 x) T J Mol -1. The heat capacity change at elting for pure crystalline silicon is Δ c( Si) = 1.89 J Mol -1 K -1 that is obtained by using the forula for the crystalline silicon and data for liquid Si (Table 1). The heat capacity 16

17 of the alloy syste is a decreasing function of the coposition as b b 1 1/ Δ c= 1.89(1 + x) J Mol -1 K -1. Both free energy barrier F ( b b x) 3/ 1 Δ = + 1 and the ean square displaceent 96c 1/ 3 η( T ) = ( b + b1x) 4c 1/4 is a onotonic increasing function of coposition. Both quantities show explicit dependence on the choice of the coefficient b ( x). We should ention in passing that the choice of a onotonic increasing function for b ( x) is based on the experiental results in dilute solute liit. This assuption, i.e., linear dependence of b ( ) x on x, ay break down at high solute concentrations, which will render a ore coplex for for b( x) 3. A suary of soe specific predicted properties at elting in specific As concentration and teperature is listed in Table. 4. Discussions The concept of polyorphic elting is originally proposed as a stability liit of undercooled liquids by partitionless solidification or hoogeneous, diffusionless crystals at elting. The polyorphic elting/freezing diagra is expected to be used in deterining the easy glass foration region 31. Exact deterination of polyorphic elting teperatures needs both sufficient reliable therodynaic data and accurate odeling and calculation of free energies of both the stable and etastable phases involved. Systeatic easureents have been ade in such systes as Si-As and Sn-Bi, although there are large differences aong the reported results with various ethods. 17

18 Several therodynaic odels have been constructed that provide reasonable fits for the dilute solutions, including the regular solution odel, but with significant differences in the etastable region 3. On atoic scale, advent has been ade in understanding the etastable systes where cheical partition is liited, in particular, the role of elastic degree of freedo played in phase transitions in solids 33. This aspect is often cast in ters of atoic size difference and the size-isatch induced elastic energy 34. We included this crucial part into a Landau free energy by assuing that the cheically hoogeneous or partitionless syste undergoing polyorphic elting is affected by the strain and elastic energy. The solute-induced disorder couples with the atoic displaceent, eventually causing elting. The result is the elting line (Eq. (8)). In fact, we can further illustrate this connection via Lindeann criterion of elting 16 : As T d kθ = 9 u cri, where u = u + u is the total critical MSD. The effective cri dyn sta Debye teperature θ d for the disordered crystal where the static MSD is doinant is θ u sta d = θ 1 ucri and the average shear odulus is B 4πV a / 3 1 k μ = ρθ d, const h 3 where V a is the atoic volue and ρ is the density. Fro these relations, we have ~ T T θ d = θ μ = = μ 1 u u sta cri, where the ratio u u sta cri defines the contribution fro static disorder. Fro the Landau theory, we have T / T (Eq. (8)) that allow us to obtain the ratio u u sta cri T = 1 T. Fro this relation, we can see that the solute-induced-msd increases with the increasing solute concentration as T / T decreases with increasing x. 18

19 Another link between polyorphic elting in alloy systes and the soluteinduced disordering is through the softening of the shear elastic odulus, μ = μ γ χ. As we entioned in the Introduction, the presence of static disorder anifested as local strain add an additional aount of disordering to the already-existing theral disorder, ainly the dynaic MSD: the interaction between the MSD and the strain field further push the crystal toward liquid state. The nearly universal appearance of the depression of the elting line in alloy systes is the reflection of this effect. The siple Laudau theory for polyorphis elting in binary solid solutions was tested using the experiental input fro a Si-As syste. The calculated results show good agreeent in dilute solute regie. Extrapolation into high concentration regie gives various therodynaic properties that can be copared to the few reported data 4,9 (Table ). For exaple, the prediction at the two calculated elting teperatures corresponding to the concentration at x =. 45 and x =. 9 are very close to the easured data 4. The obtained values of latent heat of fusion show slight dependence on the coposition in the dilute region, while the easured data are independent of the coposition 4,9. The values fro our prediction decrease strongly with the concentration of solute atos in high coposition regie. The predicted elting teperature at high solute concentration, say x =. 145 is 17K which is lower than the experiental result reported by Lee et al 18 who argued that As n -vacancy coplexes ight cause the higher elting point with the slower heating rate. The disagreeent ay also ste fro other causes. But we believe that if enough data fro ore systeatic easureent are available, we should be able to capture these subtle variations. One way we can do is to 19

20 re-select the functional for for the two coefficients, γ ( x) and b ( x). In general, they should not be the siple function of the alloy coposition as we selected in this work. 5. Conclusion In this work, we report a siple, two-order-paraeter Landau theory to describe polyorphic elting in binary solid solutions. The two order paraeters, the ean square atoic displaceent η and shear strain ε, represents the two basic processes underlying the polyorphic elting. η easures the topological order-to-disorder transition as Lindeann proposed originally and ε represents the strain caused by liiting the partition or diffusion of the cheical coposition in the alloys. To deonstrate the idea, we took the siplest for for the Landau free energy where only one coponent of the strain tensor is taken and the coupling of the two order paraeters is bilinear. Fro the odel, we calculated the polyorphic elting line as a function of coposition in Si-As alloy using the easured and estiated therodynaic data. The calculated elting point and other therodynaic properties such as latent heat of fusion are consistent in trend with the reported data in dilute solutions. Extrapolation into high concentration region is also ade and found to be reasonable. We also show the elastic softening caused by alloying. The result is consistent with the nature of the first order transition, that is, there is only a finite drop of elastic odulus at elting. This is different fro Born s original thought and the theories developed subsequently where shear odulus is deeed to vanish at elting. The validity of the prediction at high alloy concentration needs further verification fro experiental results. Acknowledgent

21 The financial support to this work provided partially by NSF NSF (Grant No. NSF- 973) is gratefully acknowledged. M.L. likes to acknowledge the partial support by National Science Foundation under Grant No. NSF PHY and stiulating discussion with Takeshi Egai. 1

22 Table 1. Therodynaic and elastic properties of silicon. Si(crystal) Melting teperature 1685 T (K) 4 Si(liquid) Latent heat of fusion L f 5719 (J Mol -1 ) 4 Specific heat (J Mol -1 K -1 ) 4, T 3 7 T T 7. Elastic constant C 44 (GPa) T Molar volue (c 3 /Mol) Lattice paraeter (f) Table. Predicted therodynaic properties for Si-As alloy. Si-4.5 at. %As Si-9. at. %As Si-14.5at. %As Melting teperature (K) Latent heat of fusion (J Mol -1 ) Specific heat (J Mol -1 K -1 ) Elastic constant C 44 (GPa) Generalized Lindeann s Criterion (MSRD) (f)

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26 Figure captions Figure 1. Scheatic illustration of solute induced local configuration change, or disorder that causes strain field in the saple. (a) The solvent atos arranged in a crystal of square lattice. (b) Introduction of a larger solute ato at the center of the figure of the crystalline lattice causes local distortions. The circle with solid line is the new position of the solvent atos after displaceent by the solute and the ones with dashed line are the solvent atos on the perfect crystal lattice as in (a). Figure. The polyorphic elting diagra for Si-As solid solution. Solid line is the polyorphic elting line deterined by this odel and noralized by the elting teperature of crystalline silicon.,,, + indicate the easured data By Kittz et al 4 The inset is the easured elting points within the dilute liit and our predicted elting line within this regie. The relevant paraeters used are a =. 1, b =. 15, b =.5 1, and γ = Figure 3. The shear elastic constants of Si-As solid solution as a function of the solute (As) concentration. The solid line is the elastic shear odulus at T = 147 K and dot line is at T = 1556 K. The end of each line corresponds to elting in the alloy where the shear elastic constants are still finite: x =. 45 ( T = 1556 K) and x =. 9 ( T = 147 K). 6

27 Figure 1 7

28 Figure 8

29 Figure 3 9