Structure of liquid metals determined by scattering techniques

Materials Science and Engineering, A178 (1994) 9-14

9

Structure of liquid metals determined by scattering techniques J. T e i x e i r a

Laboratoire L~on Brillouin (CNRS-CEA), CE Saclay, 91191 G!f-sur-Yvette Cedex (France)

Abstract Liquids structure is characterized by a short range order, which can be determined by X-ray and neutron scattering. The important function measured by these techniques is the structure factor. It is the Fourier transform of the pair correlation function, which describes the instantaneous spatial positions of the atoms. Partial pair correlation functions describe the local arrangements of the atoms constituting the alloy. They can be determined, in particular by neutron scattering. Metastable states are usually related to nucleation processes. Their study is also possible by small angle scattering (SAS). The kinetics of phase separation in binary systems can be followed in real time by SAS. Some fundamental notions of scattering techniques applied to the study of liquid metals structure and to the kinetics of phase separation during nucleation processes are presented.

1. Introduction

2. Scattering theory

Liquid and glassy structures are characterized by isotropic short range or local order which can be described by a pair correlation function g(r), a function of the distance r from an arbitrary chosen origin, occupied by an atom of the material. More exactly, g(r) plays the role of a probability function and is related to the probability of finding an atom within a distance r of another reference atom assumed to be at the origin ( r = 0). One can envisage the physical meaning of g(r) by supposing that an observer sits on an atom and does an extremely short-time snapshot of the surrounding atoms, measuring the distances between him and these atoms. Clearly, snapshots taken from different origins or at different times give slightly different results, even for a stationary system, g(r) is an average of all the snapshots taken at a given time from all the possible atoms taken as origin (ensemble average). This can also be seen as taking the average over infinite time of all the snapshots taken from one atom. If the system is ergodic, the two averages coincide. This is the case for common liquids. In the case of glasses, some degrees of freedom are frozen and the system does not cover all the space of phases, thus the two averages may be different. Of course, the same is true for systems out of equilibrium. Scattering experiments determine almost exactly the Fourier transform of g(r), called the structure factor S(Q). Consequently, they are powerful tools for the study of liquids structures. In this paper, we focus on the particular case of metallic liquids; in contrast with molecular liquids, we have only to consider atoms interacting through spherical potentials.

In a scattering experiment, a plane wave associated either with electromagnetic radiation (X-rays) or with the movement of a neutron interacts with either the electronic cloud (X-rays) or the nuclei (neutrons) of the atoms of the sample (we do not consider the magnetic scattering of neutrons here). Due to electrostatic or nuclear forces, the momentum and energy of the incident wave may change. The interference of the different scattered waves corresponds to the coherence in the spatial distribution of the atoms. Two extreme cases are the perfect crystal and the perfect gas. In the first case, the well defined periodicity of different layers of atoms implies that constructive interference takes place only in well defined orientations. Then, the scattered intensity vanishes in all directions except a few, for which the intensity is extremely large (Bragg diffraction). In the second case, the absence of coherence and the short range of the nuclear interaction give a scattered intensity which is constant (independent of the scattering angle). This is due to the absence of correlations between the positions of the atoms of the perfect gas, and because the scattered wave is spherical. Liquids are between these two extreme situations, and one observes an angular distribution of the scattered intensity with more or less pronounced broad peaks. A careful mathematical treatment of the scattering theory can be found in many textbooks (e.g. ref. 1 ). For the moment, it is sufficient to note that the total scattered intensity S(Q) is related to the spatial Fourier transform of the pair correlation function g(r) by S ( Q ) = 1 +/3 f [ g ( r ) - 1]exp(iQ'r) dr

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(1)

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10

J.

Teixeira / Structureof liquidmetals

where Q = k~- kr represents the momentum exchange, i.e. the difference between the momentum ki of the incident wave and that, k r of the scattered wave and p is the number density (number of atoms per unit volume of the sample). For elastic scattering, i.e. without energy exchange, [kil = [kf] = k and Q =]Q[ = 4~r sin( 0/2)/2, where 2 = 2~r/k is the wavelength of the radiation and 0 is the scattering angle (note that many authors prefer to call 2 0 the scattering angle, following an old practice of diffraction studies). Because the sample is isotropic, the integral can be written as a function of r

S(Q) =1 + 4zrp f r2[g(r)-

1] sin(Qr)/(Qr)dr

(2)

0

From this expression, it is clear that for a perfect gas S(Q) is identically equal to unity. In practice, in a scattering experiment one measures an intensity I(Q), from which, and within some hypothesis and corrections, it is possible to extract S(Q). g(r) is then obtained by the inversion of eqn. ( 1):

pig(r)-

1] = oo

f Q2[S(Q)-I)] sin(Qr)/(Qr)dQ

(3)

0

Very often, it is interesting to use the function defined by

G(r)

G(r) = 4Jrrp[g(r)- 1]

IsAs(Q) =(N/Vo)bZS(Q)

/SAs(Q) has the dimension (length)-1, and is currently expressed in cm- 1. Another important parameter of a scattering experiment is the wavelength ~, of the incident radiation. Obviously, the spatial resolution is related to it. More precisely, and taking account of the properties of the Fourier transform, one averages the spatial distribution of atoms over volumes of the order of (2~r/Q) 3, where Q is the momentum exchange defined above. Because the interatomic distances are typically of the order of 0.1 nm, wavelengths of the same order of magnitude are appropriate to determine the structure of condensed matter at the atomic level. It is important to consider also the large wavelength limit or, equivalently, the small Q transfer, commonly called SAS. In this limit, the spatial resolution is poor and one typically averages over several nm. If the sample is homogeneous, the only differences between elementary scattering volumes are due to thermal fluctuations, related to the isothermal compressibility. An easy derivation gives S(0) = 1 + p J- [g(r) - 1] dr

Q[S(Q)-1]

(6)

oo

co

= (2/Jr)J-

where the sum extends over all the atoms and N is the total number of atoms. I(Q) has the dimensions of an area and goes to b 2 at large Q, when S(Q) approaches the perfect gas limit (S(Q)-- 1 ). In SAS experiments, it is usual to normalize by dividing by the volume V0 of the sample, writing

sin(0r)dQ

(4)

0

=pkBTzx

(7)

0

where kB is the Boltzmann constant, T is the absolute temperature and ZT is the isothermal compressibility. This quantity is normally small, except close to the critical point, when ZT diverges (critical opalescence or forward scattering). The preceding expressions can be easily generalized to multicomponent liquids. In particular, eqn. (5) must be written as

The coupling between radiation and matter depends on the nature of the interactions. As we have seen above, X-rays, sensitive to the electronic cloud, interact over a relative large volume around the atomic nuclei. The scattering amplitude increases linearly with the total charge. As a consquence, light atoms are less "visible" than heavy atoms. Absorption increases with increasing atomic number, and for most metals one must work with very thin samples. In contrast, the scattering amplitudes for neutron scattering are independent of the charge and, except for some rare cases, are of the same order of magnitude for most of the nuclei. Note that the neutron scattering amplitudes depend on the isotope and can be negative. The quantity really measured is the cross-section per atom

Now, the scattering amplitudes b i are not the same for all the atoms. In the particular case where there is a mixture of two kinds of atoms (or of isotopes in the case of neutron scattering), the sum in eqn. (8) splits into three terms corresponding to the relative arrangements of the two kinds of atoms. The generalized structure factor is defined by

I(Q)=(1/N2)(~b2exp(iQ'(r,-r,)) I

S~b(Q)= 1 + p f [ga~(r)- 1] exp(iQ.r)dr

= b2S(Q)= b 2+b2[S(Q) -

1]

(5)

I(Q)=(l/N2)(i,~/bibjexp[iQ'(r,-ri)] )

(8)

(9)

where the indexes a and fl refer to the different species. For a binary mixture of atoms A and B, there

J. Teixeira / Structureof liquid metals are three partial pair correlation functions: gaA(r), gBB(r) and gAB(r)= gBA(r),with obvious meanings. Consequently, the normalized structure factor is given by

S(Q)-I=(,b 2) I[cA2bAE(SAA(Q)--I)+CB2bB2(SBB(Q)--I) + 2 CACBbAbB(SAB(Q ) - 1 )]

( 10)

where the formalism of Faber and Ziman [2] is followed. In this expression, ci represents the atomic concentrations of species i, with correspondent coherent scattering lengths bi, and {b 2) = CAbs: + ~'~bB~-

(1 1 )

The other possible formalism is from Bhatia and Thornton [3] S(Q) ={b2) - ~[(b)2SNN(Q) + 2Ab{b)SNc(Q) +(A b )2Scc ( Q)]

(12)

where A b = bA- b Band

(b)= CAbA+ CBbB

(13)

The three structure factors are given as a function of the partial pair correlation functions by

SyN(Q)=cAZSAA(Q)+cB2SBB(Q)+2CACBSA~(Q)

(14)

SNC(Q) = CACB{CA(SAA(Q) -- SA,(Q)) - c,[S,B (Q) - SAB( Q)]}

(15)

Scc (Q) = cAc,{ 1 + CACB[SAA(Q) + SAB(Q) -- 2SAB( Q)]}

(16) The physical meaning of these structure factors is as follows. SNN(Q) represents the global structure factor of the alloy. If A b = 0 , and taking into account that CA+cB = 1, S(Q)=SNN(Q). At large Q, SyN(Q)~I. Scc(Q ) is related to concentration fluctuations. For a "zero-alloy", i.e. an alloy for which (b) = 0, only the concentration fluctuations are observed. Indeed, in this case, the scattered amplitude is due to the difference (or "contrast") A b between the two scattering lengths. If there is segregation, Scc(Q ) will show a large SAS. At large Q, Scc(Q ) oscillates around CACB. Its Fourier transform yield Pcc(r)

Q2[Scc(Q)/(cAcB)-l]sin(Qr)/(Qr)dQ

Pcc(r) =(2/~) f

(17) The integral of 4~rZPcc(r), called the "radial concentration-correlation function" [14], gives the Warren-Cowley short range order parameter ap

4~rr2Pcc(r)dr

ap=(Zp) ~ rp

~"

(18)

11

where rp is the mean distance from an atom to its coordination shell of order p, which contains Zp atoms within the thickness 2 e. The value of the parameter ap is related to the tendency for segregation. When ap is positive, there is self-coordination; if a < 0, there is hetero-coordination within the p-shell, a = 0 corresponds to the statistical distribution imposed by the alloy composition. In particular, the parameter a~ is directly related to the chemical short range order in substitutional alloys. Bhatia and Thornton [3] have shown that Scc(O) is proportional to the composition fluctuation {Ac2), and related to the curvature of the Gibbs energy G, plotted vs. composition, c = CA, by Scc(0) = NkB T(O2G/Oc:) 'T,P N

(1 9)

Figure 1 shows an example of application of the two formalisms to the study of amorphous Nis~B~,) alloys [6], where variable amounts of several isotopes of nickel have been used, in order to separate the different partials of the structure factor. In particular, because the coherent scattering length of 62Ni is negative, it is possible to prepare an alloy with {bNi)= 0, i.e. a sample for which only the boron atoms contribute to the scattered intensity. After several corrections, it is possible to derive the partials SNiNi(Q), SBB(Q ) and SNiB(Q) , within the Faber-Ziman formalism, and the partials SNN(Q), Scc(Q ) and SNc(Q), within the Bathia-Thornton formalism. The oscillations of SNc(Q) are a result of the different sizes of the atoms of nickel and boron, i.e. of the fact that NiB is not a substitutional alloy. The oscillations of Scc (Q) are related to the chemical ordering present in the glass. The width AQ of the first peaks of SNN(Q ) and Scc(Q ) are, in principle, related to the correlation lengths of density or number fluctuations and to concentration fluctuations respectively by ~ii=2~/AQii. In this particular example, because ~NN> ~CC, one may deduce that topological short range order is larger than chemical short range order [5]. In segregating systems a strong SAS is present, due to important concentration fluctuations. This is the case, for example, for the liquid mixture Na39Li,~ (Fig. 2) at a temperature close to the consolute point [6, 7]. Small angle scattering (SAS) is particularly adapted to the study of aggregation and phase separation processes, as it averages the sample structure over volumes typically of the order of several nm. Moreover, because the kinetics of phase separation in metallic binary systems may be slow, real time experiments are often possible. The theory of SAS is mainly used in the context of chemical physics, to study either the size and shape of macromolecules or their aggregation. In many cases, an ensemble of quasi-monodisperse centrosymmetric

12

J. Teixeira / Structure of liquid metals 4

i

I

i

3 a

590 K

2

2 1 0 7

0

6 5 4 3 2 I 0 -I

i I

i 2

i 3

i Z,

i 5

Q (Aq) Fig. 2. Scc(Q )/(cLicN. ) for liquid Li~lNa~9 (from ref. 6).

0

I

I

i

I

I

I

2

/,

6

8

10

12

I

I

the volumes occupied by these atoms. For example, if the system A-B separates into two phases, one with concentration cA= q and the other with concentration CA= CII, the scattering length density of phase I is given by

Q (.&-~)

(a/ i

i

I

I

p [ = Navdi[cibA +(1 - Cl)bB]/[CIMA +(1 -- q)MB]

(22)

2

and similarly for phase II. In this expression, M i are the atomic masses and Nay is the Avogadro number. Another extreme situation occurs when concentration fluctuations are present in the system: a common case for a binary system, as different species have different mutual energies of interaction. In the one phase region of the phase diagram, concentration fluctuations cannot be at the origin of a growing embryo and of phase separation and they are generally well described by a structure factor of the Ornstein-Zernike form

1 0

"0"5/~-

i

I

0

2

4

(b)

I

I

6 8 Q (j~-t)

r

I

10

12

I(Q)°cK2kBT/(1

Fig. 1. Partial (a) Faber-Ziman and (b) Bathia-Thornton structure factors of amorphous NisiBl9 (from ref. 5). particles interacting in a continuous surrounding medium allows the separation of the total scattering intensity into two Q-dependent terms [8] I ( Q ) = f~KZV2p(Q)S(Q)

(20)

where ~bis the volume fraction of the sample occupied by the particles, P(Q) is a form factor depending on the size and shape of the particles, and S(Q) is the Fourier transform of the pair correlation function of the centres of mass of the particles. K is the important factor that allows the observation of the clusters. It is called "contrast", and is the difference between the two scattering length densities, pi and pn, present in the sample K =pI

_plI

= (~-~.b i i / v i ) _

(~,biil/Vli)

(21)

where the summations are over all the atoms of species A and B present in phases I and II, and vt and vii are

+

Q 2~cc2)

(23)

where ~cc is the correlation length of the concentration fluctuations. Note that this structure factor superimposes to the density correlation fluctuations but, except close to a thermodynamic critical point, it dominates the scattered intensity. During the process of phase separation, i.e. when the system has been quenched into the metastable or unstable regions, clusters of one type of atoms coalesce forming larger clusters as time elapses. Depending on the chemical composition, one can imagine the system either as an ensemble of polydisperse clusters embedded in a continuous phase of a different composition, or like two interpenetrated continuous media. The simple decomposition of eqn. (20) cannot apply, essentially because there is an extremely large polydispersity and because concentration fluctuations may be superimposed. The scattered intensity is consequently not simply related to the structure of the sample. In almost all cases, the scattered intensity is characterized by a broad and asymmetric peak. The position and the width of this peak change with composition,

J. T e i x e i r a

/

Structure of liquid metals

temperature, time and quenching rate, but some scaling laws have been derived and are rather well observed. The inverse of the position QM of the peak position in Q space is related to some characteristic length scale of the system. Sometimes, QM is supposed to represent the average distance between clusters. This interpretation can, at best, give an order of magnitude of such a distance. However, some features of S(Q) can be interpreted directly. When the interface between the two phases is clear-cut, S(Q) decreases with Q-4 and the constant of proportionality is a measure of the total area of the interface. Consequently, when a region with a Q-4 dependence can be identified at large Q, one has a precise way to determine the total area of the total interface between the two phases (Porod's law)

I(Q)=2:rK2(S/V)Q

4

S(Q)=const.

Q~ t)s

=I(Q,)/(2~rK=Q, 4)

(26)

v~, may be interpreted as the area of the rough interface as measured with a mesh of size 2Jr/Q]. Clearly, the larger is Q~, the smaller is the mesh and the larger is the effective area of the interface. All the preceding considerations apply, of course, to porous samples, for which the contrast is equal to the scattering length density of the solid material.

• 25

o 50

300

* 100

P'~]a

0 200

i

100

o

,~200

//;, I! i/ ,// ~.<

LJ

1

0

(a)

2

Q l

[

3

4

(10-2~,-~) I

0,8

I

/ ~ i ~

0.6

I

o 12.5 t (min)° 25 o 50

• 100 o 200

c~ 0.4

0,2

0 / (b)

; 0.2

1 0.6

I 1.0

I 1.4

I 18

Q/QM

Fig. 3. Time evolution of the structure factor of the glass system B20~-PbO-A12Q and its scaling (from ref. 10). have seen above, on the total volume of the clusters and on the contrast, QM(t) is the time-dependent position of the peak and d is the dimension of the system, f(x) is a time-independent scaling function. For three-dimensional systems, it is useful to plot QMBI(Q; t) vs. Q/QM in order to obtain the universal function f(x). In order to indicate the agreement that may be found, we show in Fig. 3 the results obtained for the glass system BRO~-PbO-AI20 ~[10]. Furukawa [11] proposed the following expression for the universal function f(x)

f(x) = [(d+ 3)x2]/(d+ 1 + 2x J+3]

3. Kinetics of phase separation During the phase separation process, the scattered intensity I(Q; t) changes with time. (Note that I(Q; t) represents the scattered intensity at time t, and is distinct from the dynamic structure factor S( Q, t).) The time dependence follows in general scaling laws. The purpose of many experiments is to test such laws or to make comparisons with computer simulations. Very good reviews exist on the subject (e.g. ref. 9). We indicate here only some general rules. The most important scaling law can be written

I( Q; t)= Af[ Q/QM(t)]/ QM(t)a

t (min) o125 /~,~

(25)

where D~ is the fractal dimension of the interface (a number between 2 and 3). The meaning of the constant can be understood if one identifies eqns. (24) and (25) for a particular value Q1 of Q. Then

(S)

400

(24)

where (S/V) represents the total area (per unit volume of the sample) of the interface between the two phases. This expression can be generalized to rough interfaces and written as

13

(27)

where A is the integrated intensity, depending, as we

(28)

Note that for small x, f(x) = x: and for large Q, in a three-dimensional system, f(x)=x 4, in agreement with Porod's law. The maximum o f f ( x ) is at x = 1 and f ( 1 ) = 1. This equation fits well the experimental data for off-critical concentrations. For critical concentrations, Furukawa [11] proposed a different expression off(x)

f(x) =[(d+ 1)x2]/ D + x 2 + 2d]

(29)

Actually, in this regime, the kinetics of the aggregation processes is better described by percolation theories [12]. The time dependence of S(Q; t) has been studied by Langer [13] and is given by the differential equation

J. Teixeira

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/

Structure of fiquid metals

0 Ot S( Q; t) = - 2MQZ[ B Q 2 + A( t)]S( Q; t) + 2 M k B TQ 2

(30) where M represents the mobility, B is a constant related to the gradient energy and A(t) is a timedependent function. The stationary solution (assuming A(t) = A0) gives the Ornstein-Zernike equation S ( Q ) =kBT/(A o +BQ 2)

(31)

Concerning QM(t), a power law is generally observed QM(t) =

(32)

t -m

Assuming that QM(t)- 1 represents the typical length scale of the system at time t, the preceding eqn. (32) expresses the fact that the coalescence process, determined by the cluster diffusion, depends essentially on this characteristic length. At the beginning of the phase separation, the efficient mechanisms for cluster formation and growth follow the law [14] QM(t) = t -1/6. Closer to the phase separation, the cluster growth is dominated by atomic evaporation from the small clusters, which decreases their size, and atomic condensation which increases the size of the larger clusters. This process, called "LifshitzSlyozov-Wagner ripening", yields the scaling law [15] QM(t) "~ t- 1/6. As an example of the application of the scaling laws, Glas et al [16] showed that, for polycrystalline samples Aua0Pt60 quenched within the miscibility gap, the

I

I

I

I

I

100 Z.O

I

-

Cl

31D

-

E 20

0

I

I

I

I

I

20

~0

60

80

100

eIPt) [at %]

Fig. 4. Exponents 1/m of the time dependence QM oct -m (from ref. 16).

scattering pattern is isotropic and eqn. (27) applies with d = 3. Instead, single crystals show a strong anisotropy, with peaks located on [002] cubic directions. Then, the same equation applies, but with d = 1, describing well the one-dimensional lattice modulation. These authors have also shown that the power law (eqn. (32)) applies rather well, but m takes the value 1/3 only near the boundaries of the miscibility gap. In contrast, near the symmetric concentration (60% Pt), 1/m is very large (Fig. 4). In this paper, we have tried to prove that X-rays and neutron scattering are powerful methods for structure characterization of liquid metals and alloys. The high flux sources available at present allow the time evolution of scattering patterns during nucleation processes to be followed. The uses of isotopes in neutron scattering allows the separation of different partials and enables precise comparison with both theory and computer simulations.

Acknowledgments The author is grateful to Dr. P. Chieux and Dr. R. Glas for their valuable help during the preparation of the manuscript.

References 1 G.L. Squires, Introduction to the Theory of Thermal Neutron Scattering, Cambridge University Press, London, 1978. 2 T.E. FaberandJ. M. Ziman, Phil. Mag., 11 (1965) 153. 3 A. B. Bhatia and D. E. Thornton, Phys. Rev., B2 (1970) 3004. 4 H. Ruppersberg and H. Egger, J. Chem. Phys., 63 (1975) 4095. 5 P. Lamparter, W. Sperl, S. Steeb and J. Bl6try, Z. Naturforsch., 37A (1982) 1223. 6 H. Ruppersberg and W. Knoll, Z. Naturforsch., 32A (1977) 1374. 7 P. Chieux and H. Ruppersberg, J. Phys. Coll., C8 (41)(1980) 145. 8 J. Teixeira, in S.-H. Chen, J. S. Huang and E Tartaglia (eds.), Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution, Kluwer, Dordrecht, 1992, p. 635. 9 S. Komura, in S. Komura (ed.), Phase Transitions, 12 (1988) 3. 10 A. Craievich and J. M. Sanchez, Phys. Rev. Lett., 47 (1981) 1308. 11 H. Furukawa, Physica, 123A (1984) 497. 12 E Family, in H. E. Stanley and N. Ostrowsky (eds.), On Growth and Form, Martinus Nijhof, Dordrecht, 1986, p. 231. 13 J.S. Langer, Ann. Phys., 65 (1971 ) 53. 14 K. Binder and D. Stauffer, Phys. Rev. Lett., 33 (1974) 1006. 15 I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids, 19 (1961)35. 16 R. Glas, O. Blaschko and L. Rosta, Phys. Rev., BIO (1992) 5972.