arXiv:cond-mat/0501173v3 [cond-mat.stat-mech] 26 Aug 2005
Atomic-scale surface demixing in a eutectic liquid BiSn alloy
Oleg G. Shpyrko,1, ∗Alexei Yu. Grigoriev,1Reinhard Streitel,1Diego Pontoni,1
Peter S. Pershan,1Moshe Deutsch,2Ben Ocko,3Mati Meron,4and Binhua Lin4
1Department of Physics and DEAS, Harvard University, Cambridge MA 02138
2Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
3Department of Physics, Brookhaven National Laboratory, Upton NY 11973
4CARS, University of Chicago, Chicago, IL 60637
(Dated: February 2, 2008)
Resonant x-ray reflectivity of the surface of the liquid phase of the Bi43Sn57 eutectic alloy reveals
atomic-scale demixing extending over three near-surface atomic layers. Due to the absence of un-
derlying atomic lattice which typically defines adsorption in crystalline alloys, studies of adsorption
in liquid alloys provide unique insight on interatomic interactions at the surface. The observed com-
position modulation could be accounted for quantitatively by the Defay-Prigogine and Strohl-King
multilayer extensions of the single-layer Gibbs model, revealing a near-surface domination of the
attractive Bi-Sn interaction over the entropy.
PACS numbers: 61.20.–p, 61.10.–i, 68.03.–g
The widely-accepted Gibbs adsorption rule  pre-
dicts the surface segregation of the lower surface en-
ergy component of a binary mixture. Liquid metals are
ideal systems for studying Gibbs adsorption due to the
nearly spherical shape of interacting particles, relative
simplicity of the short-range interactions and the avail-
ability of bulk thermodynamic data for many binary al-
loys. While certain aspects of Gibbs theory can be tested
through macroscopic measurements of surface tension or
adsorption isotherms, very few direct measurements of
the atomic-scale composition profiles of the liquid-vapor
interface were reported [2, 3, 4]. In addition to funda-
mental questions related to surface thermodynamics of
binary liquids, BiSn-based alloys have been widely stud-
ied as substitutes for Pb-based low-melting solders .
Thus, understanding their wetting, spreading, alloying,
reactivity and other surface-related properties is of great
practical importance. Moreover, interfacial phenomena
dominate the properties of the increasingly important
class of nanoscale materials, as demonstrated recently in
studies of the liquid-solid phase stability of nanometer-
sized BiSn particles .
Synchrotron-based x-ray reflectivity (XR) can mea-
sure the surface-normal density profile of a liquid with
˚ Angstr¨ om-scale resolution. Over the last decade XR re-
vealed the long-predicted surface-induced atomic layer-
ing at the liquid-vapor interface for a number of elemen-
tal liquid metals [7, 8, 9, 10, 11]. Resonant XR near
an absorption edge resolved the density profile of each
component in GaIn , HgAu  and BiIn  liquid bi-
nary alloys. The enhancement of the concentration of the
low-surface-tension component was invariably found to
be confined to the topmost surface monolayer, with sub-
sequent layers having the composition of the bulk, in ac-
cord with the simplest, and widely used, interpretation of
the Gibbs rule. By contrast, we find here an atomic-scale
phase separation extending over at least three atomic
layers. This is unexpected, considering the near-perfect-
solution nature of the Bi43Sn57alloy in the bulk [12, 13].
A liquid Bi43Sn57sample (99.99% purity, Alfa Aesar)
was prepared under UHV conditions (P< 10−9Torr).
Atomically clean liquid surfaces were obtained by me-
chanical scraping and Ar+ion sputtering, as described
previously [10, 14, 15]. Measurements were done using
the liquid surface diffractometer at the ChemMatCARS
beamline, Advanced Photon Source, Argonne National
Laboratory at a sample temperature of T = 142◦C,
4◦C above the Bi43Sn57 alloy’s eutectic temperature,
The reflected intensity fraction, R(qz), of an x-ray
beam impinging on a structured liquid surface at a graz-
ing angle α, is given by the Born approximation as:
R(qz) = RF(qz) · |Φ(qz)|2· CW(qz)(1)
where qz = (4π/λ)sinα, λ is the x-ray wavelength,
RF(qz) is the Fresnel XR of an ideally abrupt and flat
surface, CW(qz) is due to thermal surface capillary waves
[9, 10], and the surface’s structure factor is :
Here z is the surface-normal axis, ρ∞ and ρ(z) are the
bulk and surface electron densities, respectively, and ?..?
denotes surface-parallel averaging. As RF is a univer-
sal function depending only on the known critical angle
for total external reflection, and CW(qz) is known accu-
rately from capillary wave theory, the intrinsic density
profile, ?ρ(z)?, is obtained by computer fitting the mea-
sured R(qz) by a physically motivated model described
The (forward) atomic scattering factor of a Z-electron
atom varies with energy as : Z′= Z+f′(E)−if′′(E),
where f′(E) and f′′(E) = µ(E)λ/(4π), are the real and
FIG. 1: Dispersion corrections (a) f′(E) and (b) f′′(E) of
Bi near the L3 absorption edge at EL3 = 13.418 keV. The
right scale of (b) is the electron density contrast ∆Z′/Z′
imaginary parts of the dispersion correction, and µ(E) is
the photoelectric absorption coefficient.
The effect of f′′on the analysis can be neglected and
the changes in f′are significant only near an absorp-
tion edge. Fig. 1(a) shows f′′(E) near the Bi L3 edge
as obtained from an absorption measurement in a Bi
foil. Fig. 1(b) is the corresponding f′(E), calculated
from f′′(E) using the Kramers-Kronigrelation . Both
agree well with theory [18, 19]. The composition depen-
dence of ?ρ(z)? was obtained by fitting the measured XR
by the distorted crystal (DC) model for a layered liquid
surface [7, 8]:
1 + δnZ′
The progressive increase in the Gaussian width parame-
describes the layering amplitude’s decay below the sur-
face . The layer spacing d is kept constant in this
model due to similarity in size between Bi and Sn atoms.
The bulk’s average effective electron number per atom
ference in the Bi fraction between the n-th layer, x′
and the bulk, x. The corresponding atomic densities,
cn and c, are determined from the atomic volumes vBi
and vSn: cnx′
∞, varies strongly near the edge due to
the variation of Z′
Bi: from 0.43 at E=12.00 keV to 0.27
at E=13.418 keV (right axis in Fig. 1). This is the basis
for the resonant XR method which allows to separate out
the density profiles of the two species [4, 15].
Fig. 2 shows Fresnel-normalized XRs R(qz)/RF(qz)
0+ (n − 1)σ2with increasing layer number n
Bi+ (1 − x)Z′
Sn, and δn = x′
n− x is dif-
nvBi+ cn(1 − x′
n)vSn = 1. The contrast,
FIG. 2: XR measured at the indicated energies, with fits by
the three-layer model (lines). The dashed line is the XR of a
uniform-composition surface. Inset: The E=12.00 keV mea-
sured R/RF with fits by the three models discussed in the
text (lines). Error bars are smaller than the symbols’ size.
measured near the Bi L3 edge at the four energies marked
by triangles in Fig. 1(b). The dashed line is calculated
from the DC model for a layered interface assuming a
uniform composition (δn= 0). The strong enhancement
of the measured R(qz)/RF(qz) over this line, evidenced
by the peak at qz ≃ 1.0˚ A−1, and the strong energy-
dependence of the low-qz reflectivity, clearly indicate a
significant surface segregation of Bi, and its variation
Three fits of the data by the DC model, Eq. 3, were
carried out, assuming that only one (δ1?= 0, δn≥2= 0),
two (δ1,2?= 0, δn≥3= 0), or three (δ1,2,3?= 0, δn≥4= 0)
surface layers deviate from the bulk composition. All fits
employed d = 2.90˚ A, σ0= 0.30˚ A and σ = 0.57˚ A, de-
rived from the energy-independent position, shape and
intensity of the layering peak at qz= 2.0˚ A−1. The mea-
sured R(qz)/RF(qz) of all four energies were fitted simul-
taneously, using the experimentally determined f′(E).
Fig. 2 exhibits an excellent agreement of the three-layer
model (solid lines) with the measurements, but a very
poor agreement for the one- and two-layer models (inset).
Table I lists the best-fit values of x′
and the corresponding 95% non-linear confidence inter-
vals Y (x′
n ) determined from a six-parameter
support plane analysis . The most striking result is
the non-monotonic deviation δnof Bi from the 43% bulk
value, showing an enhanced composition of 96% and 53%
in the first and third layers, and depletion down to 25%
in the second layer (see Fig. 3). Beyond the third layer
entropy effects dominate the Gibbs adsorption and the
layer and bulk concentrations can not be distinguished.
While demixing has not been previously reported in liq-
uid alloys, similar decaying oscillatory composition pro-
n) and Y (δFit
FIG. 3: Electron density profiles as derived from the fits to
the reflectivities shown in Fig. 2. Inset: the bulk-normalized
differences in electron density of Bi and Sn, (ρBi− ρSn)/ρ∞.
files were discovered in several crystalline alloys such as
Cu3Au . However, the properties, formation mech-
anism, and strong temperature dependence of the com-
position modulations in Cu3Au alloys were found to be
intimately related to, and largely dominated by, the long-
range fcc order in the bulk crystal, and the severe packing
strains resulting from the big mismatch in the atomic di-
ameter of the two components. As none of these exist
in our liquid alloy, surface-induced segregation and the
Gibbs rule can be studied in a pure short-range-order in-
teraction context, free from the complicating influence,
or even dominance, of other effects. We now compare
our experimental observations with theory.
Guggenheim’s  application of Gibbs theory  to
regular solutions assumes the surface segregation to be
restricted to a single surface monolayer.
nearest neighbors for each bulk atom in a layered lat-
tice model, lp are within, and mp are in the adjacent,
layers. For a close-packed lattice, for example, p = 12,
l = 0.5 and m = 0.25. The surface tension of the regular
solution, γAB, follows from those of the pure components,
γAand γB, as :
?1 − x′
1 − x
aA[l(1 − x′)2− (l + m)(1 − x)2].
aB[lx′2− (l + m)x2] (4)
Here, x and (1−x) are the bulk concentrations of atoms
A (Bi) and B (Sn), while x′?= x and (1−x′) are the corre-
sponding surface concentrations, aAand aBare the two
atomic areas, and ω = 2ωAB−ωAA−ωBBis the interac-
tion parameter, defined by the A-B, A-A and B-B atomic
interaction energies. Extrapolated down to T = 142◦C,
γBi= 398 mN/m and γSn= 567 mN/m, while aBiand
aSn are calculated from the atomic radii rBi = 1.70˚ A
and rSn= 1.62˚ A assuming hexagonal close packing .
TABLE I: Density model parameters x′
and confidence intervals Y (x′
three-layer model fits compared to theoretical δDP
derived from the Defay-Prigogine and Strohl-King models.
n ) obtained from the
n) and Y (δFit
Treating Bi43Sn57as a perfect solution (ω/kT = 0), the
Gibbs theorem, Eq. 4, yields γAB = 444 mN/m and
x′= 0.904, below the experimental value x′
I. However, assuming a regular solution behavior with
ω/kT = 1 yields γAB = 432 mN/m, and x′= 0.941,
which agrees very well with the experimentally derived
1(Bi) in Table I. Both γABagree well with experiment
and theory . Note that γABand x′are only weakly de-
pendent on ω/kT due primarily to the logarithmic func-
tional behavior and large surface tension difference of the
two components, aSnγSn− aBiγBi≈ 2 kT. This intro-
duces a large uncertainty of ω/kT calculated from mea-
surements of surface tension or surface monolayer compo-
sition. Resonant XR measurements of sub-surface layer
composition therefore present a unique opportunity to
probe the nature of atomic interactions at the surface.
In spite of the good agreement above, confining the
surface excess to a single monolayer is correct for perfect
solutions only, but not for our case of a regular solution,
as Defay and Prigogine  point out. They provide a
correction for regular solutions, where the surface excess
extends over two layers, the γAB values above do not
change significantly and the layers’ δnare related by:
1(Bi) in Table
1 + δ2/x
1 − δ2/(1 − x)−2ω
Expanding Eq. 5 to first order in δ2:
(δ1− 2δ2) = 0. (5)
2ωmx(1 − x)δ1
kT − 2ωlx(1 − x). (6)
For nearly perfect solutions (ω/kT ≪ 1) Eq. 5 yields
a negligible δ2: 0 < δ2 ≪ δ1.
ever, δ2 and δ1 are of opposite signs and |δ2| may be-
come comparable to |δ1|. This prediction is qualitatively
consistent with the demixing observed here. For exam-
ple, when ω/kT ≫ 1, Eq. 6 can be simplified further:
δ2= −(m/l)δ1. For Bi43Sn57, m/l ≈ 0.5 and the Gibbs-
1= 0.90 (or δ1 = 0.47) yields δ2 = −0.23,
δ3= 0.12 and δ4= −0.06 . These values, shown as
in Table I, agree well with δFit
three-layer model fits. The smallest value of the inter-
action parameter ω/kT for which satisfactory agreement
with the Defay-Prigogine model could be obtained (by
treating m as an adjustable parameter) is ω/kT = 2.3.
For ω/kT ? 1, how-
obtained from the
4 Download full-text
Strohl and King  suggest a multilayer, multicompo-
nent model, where no expansion is used, and x′
tained iteratively, until convergence to a self-consistent
composition profile is reached. A good agreement of this
theory with our BiSn measurements is obtained when
ω/kT = 1.0 − 1.7.
Table I. As observed, the Strohl-King model provides
composition profiles very similar to those of the Defay-
Prigogine model, albeit with slightly different δnvalues,
thus supporting our overall conclusions.
Theoretically, ω and the enthalpy of mixing, ∆Hm,
are related by ω = ∆Hm/[x(1 − x)]. In practice, how-
ever, bulk thermodynamic quantities were often found to
yield inaccurate values for surface quantities. For exam-
ple, organic  and metallic  mixtures exhibit sig-
nificant disagreements between ω values derived empiri-
cally from surface tension measurements and from bulk
calorimetry. For BiSn, reported values of ∆Hm range
from endothermic values of 80 to 140 J/mol  to an
exothermic value of -180 J/mol . These values lead
to |ω/kT| < 0.2, i.e. an almost perfect solution, and an
insignificant |δ2| < 0.01. On the other hand, the value
of ω/kT ≈ 10 that we previously found necessary to ac-
count for the observed 35% Bi concentration enhance-
ment at the surface monolayer at the BiIn eutectic is of
the same order of magnitude as the value we find nec-
essary to account for the present observation of surface
segregation in BiSn, ω/kT ≈ 1.0 − 2.3. Unfortunately
we do not have an explanation for the origin of the dis-
crepancy in the values of ω/kT and this suggests an ur-
gent need for both further theoretical studies of surface
demixing as well as experimental investigations of simi-
lar effects in other binary alloys. In particular, the BiSn
system appears to be the only liquid alloy for which clear
evidence for multilayer surface demixing has been found.
The case for new studies is strongly reinforced by the
existence of a growing class of surface-induced ordering
phenomena that have been observed in metallic liquids.
In addition to the surface demixing reported here, these
include layering [7, 8, 9, 10, 11], relaxation , segrega-
tion [2, 3, 4, 30], wetting transitions [14, 31], and surface
freezing . Finally, there is a basic unresolved question
of whether the surfaces of liquid metals are fundamen-
tally different from those of non-metallic liquids .
This work has been supported by the U.S. DOE grants
and the U.S.-Israel Binational Science Foundation,
Jerusalem. We gratefully acknowledge useful discussions
with E. Sloutskin at Bar-Ilan as well as assistance from T.
Graber, D. Schultz and J. Gebhardt at ChemMatCARS
Sector 15, principally supported by the NSF/DOE grant
No. CHE0087817. The Advanced Photon Source is sup-
ported by the U.S. DOE contract No. W-31-109-Eng-38.
values are listed in
∗Electronic address: email@example.com; present ad-
dress:Center for Nanoscale Materials, Argonne National
Laboratory, Argonne, IL, 60439
 J. W. Gibbs et al., The collected works of J. Willard Gibbs
(Longmans, New York, 1928).
 M. J. Regan et al., Phys. Rev. B 55, 15874 (1997).
 E. DiMasi et al., MRS Symp. Ser. 590, 183 (2000).
 E. DiMasi et al., Phys. Rev. Lett. 86, 1538 (2001).
 K. N. Tu et al., J. Appl. Phys. 93, 1335 (2003); J. Glazer,
Int. Mat. Rev. 40, 65 (1995).
 J. G. Lee and H. Mori, Phys. Rev. B 70, 144105 (2004).
 O. M. Magnussen et al., Phys. Rev. Lett. 74, 4444 (1995).
 M. J. Regan et al., Phys. Rev. Lett. 75, 2498 (1995).
 H. Tostmann et al., Phys. Rev. B 59, 783 (1999).
 O. G. Shpyrko et al., Phys. Rev. B 67, 115405 (2003).
 O. G. Shpyrko et al., Phys. Rev. B 70, 224206 (2004).
 N. A. Asryan and A. Mikula, Inorg. Mat. 40, 386 (2004);
R. Hultgren et al., Selected Values of the Thermodynamic
Properties of Binary Alloys (Metals Park, OH, 1973);
R. L. Sharkey and M. J. Pool, Metall. Trans. 3, 1773
(1972); H. Seltz et al., J. Am. Chem. Soc. 64, 1392
 S. A. Cho and J. L. Ochoa, Metall. Mater. Trans. B 28,
 P. Huber et al., Phys. Rev. Lett. 89, 035502 (2002).
 O. G. Shpyrko, Ph.D. thesis, Harvard University, 2003
 J. Als-Nielsen and D. McMorrow, Elements of Modern
X-ray Physics (Wiley, New York, 2001).
 G. Evans and R. F. Pettifer, J. Appl. Cryst. bf 34, 82
(2001); J. J. Hoyt et. al, J. Appl. Cryst. 17, 344 (1984).
 E. A. Merritt,Anomalous
 M. Newville, computer code IFEFFIT, (1997-2004).
 P. R. Bevington and D. K. Robinson, Data Reduction and
Error Analysis for the Physical Sciences (McGraw-Hill,
New York, 1992) p. 212.
 H. Reichert et al., Phys. Rev. Lett. 78, 3475 (1997); ibid
74, 2006 (1995) ; ibid 90, 185504 (2003).
 E. A. Guggenheim, Trans. Faraday Soc. 41, 150 (1945).
 CRC Handbook of Chemistry and Physics,
D. R. Lide (CRC, Boca Raton, 1996).
 S. W. Yoon et al., Script. Mat. 40, 297 (1999).
 R. Defay and I. Prigogine, Trans. Far. Soc. 46, 199
 Eq. 6 yields δ3 from δ2 and δ4 from δ3.
 J. K. Strohl and T. S. King, J. Catal. 118, 53 (1989).
 G. L. Gaines, Trans. Far. Soc. 65, 2320 (1969).
 T. P. Hoar and D. A. Melford, Trans. Far. Soc. 53, 315
 E. B. Flom et al., Science 260, 332 (1993).
 H. Shim et al., Surf. Sci. 476, L273 (2001); A. H. Ayyad
and W. Freyland, ibid. 506, 1 (2002).
 A. Issanin et al., J. Chem. Phys. 121, 12005 (2004);
B. Yang et al., Phys. Rev. B 62, 13111 (2000); O. G. Sh-
pyrko et al., in preparation (2005).
 E. Chac´ on and P. Tarazona, Phys. Rev. Lett. 91, 166103
(2003); D. X. Li and S. A. Rice, J. Phys. Chem. B 108,
19640 (2004); O. G. Shpyrko et al., Phys. Rev. B 69,