Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013 (30–35)

Theorization and Modeling Ⅲ(Review)

Diffusion of Mass in Liquid Metals and Alloys Recent Experimental Developments and New Perspectives
Andreas MEYER and Florian KARGLAbstract
Despite its tremendous importance for the understanding of underlying mechanisms and for the input in modeling and simulation ofprocesses alike, accurate experimental diffusion data in liquid metals and alloys are rare. Common techniques exhibit several drawbacks that inmost cases prevent an accurate measurement of diffusion coefficients - convective contributions during diffusion annealing are the mostprominent ones. Recently, we advanced the field of liquid diffusion experiments through the use of quasielastic neutron scattering (QNS) on
levitated metallic droplets for accurate measurements of self-diffusion coefficients in high-temperature metallic liquids. For the accurate
measurement of interdiffusion coefficients we combine long-capillary experiments with an in-situ monitoring of the entire interdiffusion
process by the use of X-ray and neutron radiography. These experiments are accompanied by diffusion experiments in space in order to benefit
from the purely diffusive transport under microgravity conditions. Recent experimental results are discussed in the context of the relation ofself- and interdiffusion (Darken’s equation) and of the relation of self-diffusion and viscosity (Stokes-Einstein relation).
Keyword(s): liquid metals, self-diffusion, interdiffusion, long-capillary, shear-cell, X-ray radiography, quasielastic neutron scattering.

1. Introduction
The study of diffusion processes in melts is vital for anunderstanding of liquid dynamics, nucleation, vitrification, andcrystal growth. Diffusion coefficients are an essential input tothe modeling of microstructure evolution and serve as a soundbenchmark to molecular dynamics (MD) simulation results. In
general, the self-diffusion coefficients of the individualcomponents of a multicomponent liquid are related to the meansquare displacement of the tagged atoms, whereas theinterdiffusion coefficients are related to collective transport ofmass driven by gradients in the chemical potential.
A common method to measure diffusion coefficients in liquidalloys is the long-capillary (LC) technique and its variations.
There, a diffusion couple of different composition, in the case ofinterdiffusion, or containing a different amount of isotopes, inthe case of self-diffusion, is annealed in the liquid state andsubsequently quenched to ambient temperature. The diffusion
profiles are analyzed post mortem. This technique exhibits
several drawbacks that in most cases prevent an accurate
measurement of diffusion coefficients.
Convective flow during diffusion annealing is recognized to
be a severe problem in capillary experiments. Already small
gradients in density, inherently present because of concentration
gradients in interdiffusion experiments or caused by small
gradients in temperature, result in additional contributions to thetransport of mass between the diffusion couple due to flow. As

has been shown by comparison of LC diffusion experiments on
ground with experiments under microgravity conditions 1),
where buoyancy driven convective flow is suppressed, or withquasielastic neutron scattering (QNS) 2), where the presence offlow does not alter the measurement, resulting LC values onground are systematically larger by several 10% to 100%
compared to the real value (Fig. 1).
Relying on a post mortem analysis not only how convective
flow alters the diffusion profile is unknown, but also thesolidification of the sample itself poses a problem. In general, inalloys a more or less coarse grained microstructure is forming
during crystallization that depends on the alloy composition,quench rates, and resulting temperature gradients. This alters theresulting concentration profiles on mm length scales and inmany cases renders a post mortem analysis impossible. In
addition, one has to retransform the coordinate of thedetermined concentration profile from the crystalline state atambient temperature to the liquid state, which requires theknowledge of thermal expansions or absolute values of densities.
If known at all, their experimental errors significantly reduce theprecision of the diffusion coefficient. In long capillary diffusion
experiments, the uncertainty in the knowledge of the absolutevalue of the annealing time poses an additional source of error.
Therefore, accurate experimental diffusion data in liquid metals
are rare and as a consequence a thorough understanding ofatomic diffusion and its fundamental relations to otherproperties is still missing.

Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 51170 Köln, Germany(E-mail: andreas.meyer@dlr.de, florian.kargl@dlr.de)− 30 −

Diffusion of Mass in Liquid Metals and Alloys - Recent Experimental Developments and New Perspectives

Self-diffusion coefficients in liquid copper fromquasielastic neutron scattering (QNS) and from longcapillary (LC) tracer experiments 3). Figure
reproduced from Ref. 2).

Fig. 2 Quasielastic signal of liquid titanium as measured
with neutron time-of-flight spectroscopy on an electromagnetically levitated sample (reproduced from Ref. 9)).
From the width of the quasielastic line, 𝛤q, the selfdiffusion coefficient can be obtained via D Ti = 2ℏ𝛤q /q2
for small q.

For the last six years we have invented / further developed an
entire suite of new techniques for the investigation of diffusion
in liquid metals and alloys at the DLR Institute of Materials
Physics in Space. These techniques allow circumventing above
mentioned restriction and as a consequence addressing thefollowing open issues :dynamic range of QNS or appear as a flat background (central
Rayleigh line) in the measured signal. In the case of asignificant incoherent scattering contribution, e.g. from a liquidcontaining an adequate amount of Ni, Ti, or Cu, the QNS signal
is also in most alloys dominated by the incoherent contributions.
From the resulting incoherent intermediate scattering function
the self-diffusion coefficient can be obtained on an absolute
scale (Fig. 2). The combination of QNS with electromagnetic
and electrostatic levitation of the sample enables the in-situ
investigation of chemically reactive as well as of deeply
undercooled metallic liquids 6),7),8),9).
Cu self-diffusion coefficients in liquid Cu are shown in Fig. 1
Our QNS results are compared to the LC data set by Henderson
and Yang 3) used as the text book reference for simulations inthis system. In their LC experiment a relatively large scatter canbe observed in the data and values are significantly above the
QNS values. Their data set also exhibits an activation energythat is about 25% larger than for the QNS results. This indicatesthat the LC data are hampered by convective flow contributions
during the diffusion annealing. Diffusion coefficients reported inclassical and ab-initio MD simulations on liquid Cu exhibit ascatter of about +-50% around our experimental values 2). Thisshows that, albeit MD simulation is a tool to reveal microscopic
mechanisms, it is in general not applicable for the determinationof absolute values of materials properties. On the other hand,accurate values of liquid diffusion coefficients from experiments
can serve as a benchmark for the improvement of model
potentials as has been recently shown for liquid Ti 10).
Figure 3 displays self-diffusion coefficients in bulk-glass
forming Pd40Ni10Cu30P20 above its liquidus temperature. The
temperature dependence of the Ni diffusion coefficient asmeasured with QNS exhibits a deviation from the usual
Arrhenius-type (Fig. 1) and follows the prediction of mode

Fig. 1

Relation between self- and interdiffusion.



Relation between self- and impurity diffusion.

Elemental self-diffusion coefficients of an alloy.



Dependence of diffusion on temperature.



Dependence of diffusion on alloy composition.



Relation of diffusion to viscous flow.

Relation of diffusion to crystal growth.

In this review recent results of our work are highlighted andnew perspectives for diffusion experiments under microgravity
conditions are discussed.

2.

New Experimental Techniques

2.1 Quasielastic Neutron Scattering
Recently, the field of liquid diffusion experiments advanced
through the use of quasielastic neutron scattering (QNS) in
combination with container-less processing for accurate
measurements of self-diffusion coefficients in metallic liquids 4).
QNS probes the dynamics of a liquid on atomic length scales
and on a picosecond time scale, which is short enough to be
undisturbed by the presence of convective flow (Fig. 1). It has
been recently shown for the case of liquid Ge 5), that coherent
contributions are separated in energy in the intermediatewavenumber (q) range of QNS from incoherent contributions.
Coherent contributions are either outside (Brillouin lines) theInt. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013− 31 −

Andreas MEYER and Florian KARGL

Fig. 3 Self-diffusion coefficients in liquid Pd-Ni-Cu-P as
measured with QNS (Ni) 11) and in a shear-cellfurnace aboard FOTON M2 (Ni and Pd) 13). Within
experimental error bars the diffusion coefficients
exhibit no size or mass dependence.

coupling theory 11). Self-diffusion coefficients in this alloy havealso been measured under microgravity conditions in a shearcell by applying isotopic substitution 12). Within error bars the
Ni self-diffusion coefficient is equal to the QNS one. The
comparably large and heavy Pd atoms exhibit a similar diffusion
coefficient. Apparently, at least in this alloy, the transportmechanism is rather collective 13).

2.2 X-ray and Neutron Radiography
For the accurate measurement of interdiffusion we are
currently establishing another technique: We combine LC
experiments with an in-situ monitoring of the entire
interdiffusion process by the use of X-ray and neutron
radiography 14),15). After penetration of the X-ray or the neutron
beam, respectively, through sample, holder and furnace,transmitted beam images are recorded by a flat panel detector.
The intensity profile along the long axis of the diffusion
couple is transformed to a concentration profile via
normalization to the intensity values of two reservoirs
containing two standard reference compositions placed justabove and below the sample. The detector is read out and resetin intervals of a few seconds thus resulting in a series ofdiffusion profiles. Hence, the interdiffusion process can bemonitored as a function of time, from melting of the sample toits solidification. The concentration profiles obtained on the AlCu systems displayed in Fig. 4 do not show obvious influencesof convective flow. Apparently, the small capillary diameter in
combination with the diffusion couples used here, that provide a
sufficiently large difference in density, results in a stable density
layering that in turn suppresses convective flow. A more detailed
discussion on perspectives and especially on limitations of thisnew technique can be found in Ref. 16).
A first successful neutron experiment has just been performedat the ANTARES radiography beamline of the FRM-II 15). Using

Int. J. Microgravity Sci. Appl. Vol. 30 No.1 2013

Fig. 4 In-situ monitoring of interdiffusion in Al-Cu. Every 5 s
an entire diffusion profile is measured (top). Analysis
with the corresponding error function results in the
diffusion length L as a function of time. Theinterdiffusion coefficient DAlCu can be obtained from theslope of the square of L (bottom). Figure reproduced
from Ref. 14).

neutrons, the radiographic contrast is in most cases
complementary to that of X-rays and can even be changed by
the use of proper isotopes. The in-situ monitoring paves the way
for accurate measurements of transport coefficients, where one
can rule out or eventually even correct for, convective
contributions to the resulting concentration profiles, and where
the experiment is not altered by the microstructure evolution
during solidification. X-ray radiography of diffusion processes
in liquid alloys is currently extended to the use on synchrotron
radiation facilities, where the radiographic contrast of different
components in an alloy depends on the selected wavelength ofthe monochromatic beam. This will allow the measurement of
the different interdiffusion coefficients in a ternary alloy.
Figure 5 shows self- and interdiffusion coefficients in
Al80Cu20. The Cu self-diffusion in this alloy was measured with
QNS - the interdiffusion with X-ray radiography. Theinterdiffusion coefficient is about a factor of 3 larger than the
self-diffusion coefficient. A simple Ansatz to relate self- andinterdiffusion goes back to Darken 17). The kinetic contributions
to the interdiffusion coefficient DAB are expressed by the selfdiffusion coefficients DA and DB of the two components of thebinary system, however, neglecting dynamic cross correlations 18):
DAB = (CB DA + CA DB) 𝛷(1)

CA and CB represent the alloy composition in at%. The
thermodynamic factor 𝛷 is the 2nd derivative of the Gibbs freeenergy with respect to concentration and represents the− 32 −

Diffusion of Mass in Liquid Metals and Alloys - Recent Experimental Developments and New Perspectives

Fig. 5

Self- and interdiffusion coefficients in Al80Cu20. The
prediction of interdiffusion coefficients via Darken’s
equation (dashed line) fails to describe theexperimental data under the assumption of DAl⋍DCu.
Figure reproduced from Ref. 14).

contribution from a thermodynamic driving force. In Al80Cu20
the resulting 𝛷 = 1.6 is not large enough to account for thedifference between self- and interdiffusion. Dynamic crosscorrelations further enhance interdiffusion with respect to theself-diffusion of the atoms by about a factor of 2. This showsthat interdiffusion coefficients have to be measured and cannot
be obtained from the knowledge of self-diffusion coefficientsand the thermodynamic factor alone 14).
Another common relation that is often taken for granted in orderto calculate required self-diffusion coefficients of atoms or
molecules in a liquid from the viscosity, or vice versa, is the
Stokes-Einstein (SE) relation, 19):𝐷 𝜂 = (𝑘𝐵 𝑇)/(6 𝜋 𝑅𝐻 ),(2)

where η is the viscosity, D the self-diffusion coefficient of the
diffusing particles, RH their hydrodynamic radius, T theabsolute temperature, and kB the Boltzmann constant.
The Stokes-Einstein relation was derived in order to study the
diffusive motion of a mesoscopic sphere in a viscous medium 20).
However, in liquid metals and alloys the diffusing objects are of
atomic size. Using the technique of electrostatic levitation, we
measured Ni self-diffusion and viscosity of liquid Zr64Ni36 insitu with high precision and accuracy 21), over a broad
temperature range spanning more than 800 K in which thevalues of mass transport are changing by 1.5 orders ofmagnitude. The inverse of the viscosity, measured via theoscillating drop technique, and the self-diffusion coefficient,obtained from quasielastic neutron scattering experiments,
exhibit the same temperature dependence. It was found that Dη= const. for the entire temperature range. This is contradicting
the Stokes-Einstein relation, where Dη ∝ T (Fig. 6).

Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013

Fig. 6 Ni self-diffusion coefficient times viscosity in Zr64Ni36
versus temperature (symbols). The solid line is a guideto the eye. The dashed and dotted lines correspond to
the Stokes-Einstein relation, Eq. (2), with different
choices of the hydrodynamic radius (RH ⋍RNi for c = 1).
Figure reproduced from Ref. 21).

3. Perspectives for Experiments in SpaceThe use of these in-situ techniques plays an essential role foradvancements in the field of liquid diffusion: X-ray and neutron
radiography for an in-situ monitoring of interdiffusion processes
in long-capillary experiments, and quasielastic neutron
scattering for accurate measurements of self-diffusion
coefficients. However, both techniques are limited to selective
systems, in which (only) one component exhibits a sufficientincoherent neutron scattering cross section, and in which stabledensity layering can be realized in long-capillary experiments,
respectively.
Therefore, we initiated a series of post-mortem and in-situ
diffusion experiments under microgravity conditions in order toreduce systematic errors due to gravity-induced flow- andsedimentation effects. Experiments under microgravity
conditions should serve as a first benchmark for our earth-boundLC experiments and, in addition, should enable measurementsof self- and interdiffusion coefficients on systems in which
diffusion couples only exhibit an insufficient difference indensity, and consequently, convective effects are expected toalter the measurement.
In this context we initiated the development of a compact Xray apparatus (Fig. 6) 22). This apparatus was designed as asecond generation insert for the Materials Science Lab aboard
the International Space Station (ISS) and currently adapted to
our DLR sounding rocket MAPHEUS. This development
includes a new high-temperature shear-cell (Fig. 7) 23). It allows
in-situ measurements via X-ray radiography due to the use of Xray transparent materials as graphite and boron-nitrite and its
flat design. The shear-cell permits the samples to homogenize
and equilibrate at the desired temperature before the diffusion
process is initiated. Possible disturbing flow effects during− 33 −

Andreas MEYER and Florian KARGL

Fig. 6 Compact and fully MSL compatible X-ray radiography
insert for in-situ monitoring of interdiffusion processes
under microgravity conditions – on the MAPHEUS
sounding rocket and aboard the ISS.

melting are hereby avoided. A compact and sufficiently rugged
design also enables experiments on a sounding rocket which
inhibits buoyancy-driven fluid flow. A total of six samples witha diameter of 1.0 to 1.5 mm can be processed simultaneously.
For fast impurity and interdiffusion processes, where
diffusion coefficients approach a value of 10-8m2s-1, a few
minutes diffusion annealing suffice to obtain analyzablediffusion profiles. We built and space-qualified the ATLAS-M
module (Atomic Transport in Liquid Alloys andSemiconductors) 24). With ATLAS-M (Fig. 8) we succeeded todesign a furnace set-up that combines fast heating, excellent
isothermal conditions, and forced gas cooling at temperatures ashigh as 1100°C. On MAPHEUS ATLAS-M provides about 3
min annealing time in microgravity for measurements ofimpurity diffusion and interdiffusion in Ge and Al based alloys.
Building on the successful development of the shear-cell
technique by Griesche and coworkers 12), we are currently
extending the accessible temperature range from 900°C to
1600°C (Fig. 9) 25). Within the shear cell concept the long
diffusion capillary is divided into segments, each adding a data

Fig. 7 Ultra-small shear cell (40 mm x 40 mm x 19 mm) forinterdiffusion experiments using X-ray radiography.
Liquid Al-Ni capillary samples of different Ni
concentration shortly after initiating the diffusion
process by shearing the upper half to the right.

Int. J. Microgravity Sci. Appl. Vol. 30 No.1 2013

Fig. 8MAPHEUS short-time diffusion module ATLAS-M
with 8 diffusion furnaces and the MAPHEUS
experiment control unit (top). The module’s outerradius is 356 mm.

point in a concentration profile. The two parts of the diffusion
couple are molten and homogenized each in its part of the shear
cell. This avoids sedimentation of small crystals during melting(as observed e.g. in Al-Ni with X-ray radiography). For the
diffusion annealing the two parts of the diffusion couple are
sheared together and subsequently the liquid diffusion profile isfrozen in by segmenting the long capillary in the liquid. In sucha way the diffusion profile is not altered by the microstructureformation upon crystallization and errors due to a length
correction are not introduced. Applied under microgravity
conditions, convection effects are suppressed and accurate self-,
impurity, and interdiffusion coefficients can be obtained.
For the experimental investigation of diffusion of mass inliquid metals and semiconductors we already use or we intend touse in the future the following advanced and complementary
techniques:

QNS in combination with dedicated EML and ESL.



X-ray and neutron radiography for the monitoring ofinterdiffusion processes.

Fig. 9 Prototype of a 1600°C shear cell comprising of 30, 3
mm thick, shear segments (displayed up front) andbores for up to 6 long-capillary samples.

− 34 −

Diffusion of Mass in Liquid Metals and Alloys - Recent Experimental Developments and New Perspectives

Short-time
experimentsoninterdiffusion on MAPHEUS.



X-ray radiography on sounding rockets and in MSL
aboard the ISS.



Shear-cell experiments in new high-temperaturefacilities on ground and in space.

impurity-

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4. Conclusions
Quasielastic neutron scattering on levitated metallic droplets
and long-capillary experiments in combination with X-ray orneutron radiography are new experimental tools that allow foran accurate and precise in-situ measurement of self- andinterdiffusion coefficients. Experiments under microgravity
conditions – shear-cell and in-situ with X-ray radiography –serve as a benchmark for earth-bound long-capillaryexperiments and, in addition, enable measurements on systems
in which diffusion couples not exhibit a sufficient difference indensity, and as a consequence, gravity driven convective
contributions render the diffusion measurement impossible. First
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general. In the upcoming years research on diffusion of mass inliquid metals and alloys will further expand and new
fundamental discoveries will be made that will also be crucial
for the envisioned materials design from the melt. In thiscontext, experiments in space will pose a central part of thisdevelopment.

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Acknowledgement
We thank our co-workers for a fruitful collaboration, theFRM-II for providing beam time for quasielastic neutron
scattering and neutron radiography campaigns, and the German
Ministry for Economy and Technology (BMWi) for funding
through the Economic Stimulus Package I.

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(Received 29 Oct. 2012; Accepted 6 Jan. 2013)

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