accepted for publication in Phys. Rev. B
Self–diffusion and Interdiffusion in Al80Ni20 Melts: Simulation and Experiment
J. Horbach1,2, S.K. Das1,3, A. Griesche4, M.-P. Macht4, G. Frohberg5, and A. Meyer2
(1)Institut für Physik, Johannes–Gutenberg–Universität Mainz, Staudinger Weg 7, 55099 Mainz, Germany
(2)Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft– und Raumfahrt, 51170 Köln, Germany
(3)Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA
(4)Hahn–Meitner–Institut Berlin, Dept. Materials (SF3), Glienicker Str. 100, 14109 Berlin, Germany
(5)Institut für Werkstoffwissenschaften und –technologien,
Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
(Dated: November 4, 2018)
A combination of experimental techniques and molecular dynamics (MD) computer simulation
is used to investigate the diffusion dynamics in Al80Ni20 melts. Experimentally, the self–diffusion
coefficient of Ni is measured by the long–capillary (LC) method and by quasielastic neutron scat-
tering. The LC method yields also the interdiffusion coefficient. Whereas the experiments were
done in the normal liquid state, the simulations provided the determination of both self–diffusion
and interdiffusion constants in the undercooled regime as well. The simulation results show good
agreement with the experimental data. In the temperature range 3000K≥ T ≥ 715K, the interdif-
fusion coefficient is larger than the self–diffusion constants. Furthermore the simulation shows that
this difference becomes larger in the undercooled regime. This result can be refered to a relatively
strong temperature dependence of the thermodynamic factor Φ, which describes the thermodynamic
driving force for interdiffusion. The simulations also indicate that the Darken equation is a good
approximation, even in the undercooled regime. This implies that dynamic cross correlations play
a minor role for the temperature range under consideration.
PACS numbers: 64.70.Pf, 61.20.Ja, 66.30.Hs
I. INTRODUCTION
Multicomponent liquids exhibit transport processes
due to concentration fluctuations among the different
components. In the hydrodynamic limit, these processes
are described by interdiffusion coefficients [1, 2, 3]. In the
simplest case of a binary AB mixture, there is one inter-
diffusion coefficient DAB. This quantity plays an impor-
tant role in many phenomena seen in metallic mixtures,
such as solidification processes [4], the slowing down near
the critical point of a liquid–liquid demixing transition [5]
or glassy dynamics [6].
Many attempts have been undertaken for different bi-
nary systems to relate DAB to the self–diffusion con-
stants DA and DB via phenomenological formula (see
e.g. [7, 8, 9, 10, 11, 12]). An example is the Darken equa-
tion [13] that is widely used to estimate the interdiffusion
constant of simple binary fluid mixtures. This equation
expresses DAB as a simple linear combination of the self–
diffusion coefficients, DAB = Φ(cBDA+ cADB) (with cA,
cB the mole fractions of A and B particles, respectively).
Here, the so–called thermodynamic factor Φ contains in-
formation about static concentration fluctuations in the
limit of long wavelength.
The relationship between one–particle transport and
collective transport properties is a fundamental question
in undercooled liquids [6]. In the framework of the mode–
coupling theory of the glass transition, Fuchs and Latz
[14] have studied a binary 50:50 mixture of soft–spheres
with a size ratio of 1.2. Their numerical data indicate
that the Darken equation is a good approximation for the
latter system in the undercooled regime. However, from
experiments or computer simulations, not much is known
about the validity of the Darken equation for undercooled
liquids. This is due to the lack of experimental data for
interdiffusion coefficients in this case. Moreover, most of
the computer simulation studies on the relation between
self–diffusion and interdiffusion have been only devoted
to the normal liquid state. In this case, the Darken equa-
tion often seems to work quite well [1, 10, 12, 15, 16].
In this work, a combination of experiment and molecu-
lar dynamics (MD) simulation is used to study the diffu-
sion dynamics in the metallic liquid Al80Ni20. In the MD
simulation, the interactions between the atoms are mod-
elled by an embedded atom potential proposed by Mishin
et al. [17]. The present work is a continuation of a recent
study [18], where a combination of quasielastic neutron
scattering (QNS) and MD simulation was applied to in-
vestigate chemical short–range order and self–diffusion in
the system Al–Ni at different compositions. In the lat-
ter study, we have shown that the MD simulation yields
good agreement with the QNS data, both for structural
quantities and the Ni self–diffusion constant, DNi. In the
present work, an additional experimental method, the
long–capillary (LC) technique, is used. This method al-
lows to determine simultaneously the self–diffusion con-
stant DNi and the interdiffusion coefficient DAB (see be-
low).
Above the liquidus temperature (i.e. in the normal liq-
uid state), thermodynamic properties as well as structure
and dynamics of Al80Ni20 have been studied by different
approaches (see, e.g., [15, 19, 20, 21, 22, 23]). The Al–
http://arxiv.org/abs/0704.0534v1
Ni system is an ordering system which is manifested in
a negative enthalpy of mixing [24]. Thus, it does not
exhibit a liquid–liquid miscibility gap where one would
expect that the interdiffusion coefficient vanishes when
approaching the critical point, whereas the self–diffusion
constants are not affected by the critical slowing down
(see [25] and references therein). Such a behavior is not
expected for the system Al–Ni.
In the computer simulation, the Al80Ni20 melt can be
undercooled to an arbitrary extent avoiding the occur-
rence of crystallization processes. Therefore, we were
able to study a broad temperature range in our MD sim-
ulations, ranging from the normal liquid state at high
temperature to the undercooled liquid at low tempera-
ture. In the experiments presented below crystallization
occurs due to heterogeneous nucleation. Thus, the ex-
periments were performed above the liquidus tempera-
ture TL ≈ 1280K. The combination of experiment and
simulation presented in this work allows for a test of the
validity of the Darken equation in Al80Ni20. We will see
below that this equation is indeed a good approximation,
even in the undercooled regime.
In the next section, we summarize the basic theory on
self–diffusion and interdiffusion. The details of the exper-
iments and simulation are given in Sec. III and Sec. IV,
respectively. In Sec. V we present the results. Finally,
we give a summary of the results in Sec. VI.
II. SELF–DIFFUSION AND INTERDIFFUSION:
BASIC THEORY
Consider a three–dimensional, binary AB system of
N = NA + NB particles (with NA, NB the number of
A and B particles, respectively). The self–diffusion con-
stant Ds,α (α = A,B) is related to the random–walk mo-
tion of a tagged particle of species α on hydrodynamic
scales. It can be calculated from the velocity autocorre-
lation function [1],
Cα(t) =
(t) · v
(0)〉 , (1)
via a Green–Kubo integral:
Ds,α =
Cα(t)dt . (2)
In Eq. (1), v
j (t) is the velocity of particle j of species
α at time t.
The self–diffusion constant can be also expressed
by long–time limit of the mean–squared displacement
(MSD):
Ds,α = lim
(t)− r
. (3)
Here, r
j (t) is the position of particle j of species α at
time t. Note that Eq. (3) is equivalent to the Green–
Kubo formula (2).
Interdiffusion is related to the collective transport of
mass driven by concentration gradients. The transport
coefficient that describes this process is the interdiffusion
constant DAB which can be also expressed by a Green–
Kubo relation, i.e. by a time integral over an autocorre-
lation function. The relevant variable in this case is the
concentration or interdiffusion current [1] given by
JAB(t) =
i (t)− cA
i (t) +
i (t)
where cA ≡ NA/N = 1 − cB is the total concentration
(mole fraction) of A particles. As a matter of fact, the au-
tocorrelation function of the variable JAB(t) depends on
the reference frame and fluctuations of JAB(t) have to be
adapted to the ensemble under consideration. Whereas
experiments are usually done in the canonical ensemble,
in a molecular dynamics simulation, the natural ensemble
is the microcanonical ensemble with zero total momen-
tum [26]. Thus,
i = −
i (5)
follows, where mA and mB denote the masses of A and
B particles, respectively. Introducing the “centre of mass
velocity of component α (α = A,B)” by
Vα(t) =
i (t) , (6)
we can use expression (5) to simplify the formula for the
interdiffusion current,
JAB(t) = NcBcA
VA(t) . (7)
Thus, we have to consider only the velocities of one
species to compute JAB(t).
Now, the autocorrelation function for the interdiffusion
current is given by
CAB(t) = 〈JAB(t) · JAB(0)〉 (8)
= N2 (cBcA)
〈VA(t) ·VA(0)〉 .
The Green–Kubo formula for DAB reads
DAB =
3NScc(0)
CAB(t) dt (9)
where Scc(0) is the concentration–concentration struc-
ture factor in the limit q → 0. The function Scc(q) is the
static correlation function associated with concentration
fluctuations. It can be expressed by a linear combination
of partial static structure factors Sαβ(q) (α, β = A,B) as
follows [1]:
Scc(q) = c
BSAA(q) + c
ASBB(q)− 2cAcBSAB(q) (10)
Sαβ(q) =
〈exp [iq · (rk − rl)]〉 . (11)
Using elementary fluctuation theory [1], Scc(0) can be
related to the second derivative of the molar Gibbs free
energy g,
∂cA∂cB
, (12)
Scc(q = 0)
. (13)
In Eq. 12, kB is the Boltzmann constant and T the tem-
perature. In the following, we will refer to Φ as the ther-
modynamic factor.
We note that the total structure factor for the number
density, Snn(q), and cross correlation between number
density and concentration, Snc(q), can also be written as
a linear combinations of partial structure factors. These
functions are given by [1]
Snn(q) = SAA(q) + 2SAB(q) + SBB(q) , (14)
Snc(q) = cBSAA(q)− cASBB(q) + (cB − cA)SAB(q) .(15)
The typical behavior of these functions for a liquid mix-
ture will be discussed in the result’s section. The func-
tions Snn(q), Snc(q) and Scc(q) are often called Bhatia–
Thornton structure factors [27]. In principle, these func-
tions can be determined in neutron scattering experi-
ments, either by using isotopic enrichment techniques
(see, e.g., Ref. [21]) or by applying a combination of neu-
tron scattering and X–ray diffraction [28].
With Eqs. (9) and (13), the interdiffusion constant can
be written as
DAB = N
cAcBΦ
)2 ∫ ∞
〈VA(t) ·VA(0)〉 dt .
Alternatively, DAB can be also easily related to the self–
diffusion constants to yield
DAB = Φ(cADB + cBDA
+ cAcB
[ΛAA + ΛBB − 2ΛAB] dt) , (17)
where the functions Λαβ(t) denote distinct velocity cor-
relation functions,
Λαβ(t) =
3Ncαcβ
l 6=k if α=β
(t) · v
Note that the three functions Λαβ(t) can be expressed
by the “centre–of–mass” correlation function CAB(t) and
the velocity autocorrelation functions Cα(t) (the latter,
multiplied by 1/cα, has to be subtracted in the case of
ΛAA(t) and ΛBB(t)) [11]. Thus, the functions Λαβ(t)
do not contain any additional information compared to
CAB(t) and Cα(t) and so we do not consider them sepa-
rately in the following.
If one denotes the distinct part in (17) by
∆d = cAcB
[ΛAA(t) + ΛBB(t)− 2ΛAB(t)] dt (19)
one can rewrite Eq. (17),
DAB = ΦS (cADB + cBDA) , (20)
S = 1 +
cADB + cBDA
The quantity S measures the contribution of cross cor-
relations to DAB. If S = 1 holds, the interdiffusion con-
stant is determined by a linear combination of the self–
diffusion constants. In this case, Eq. (20) leads to the
Darken equation [13]. Note that, in the context of chem-
ical diffusion in crystals, S is called Manning factor [29].
As in the case of self–diffusion, the interdiffusion con-
stant can be also expressed via a mean–squared displace-
ment which involves now the centre–of–mass coordinate
of species A,
RA(t) =
j (t) . (22)
Then, the “Einstein relation” for DAB reads
DAB = lim
NcAcBΦ
[RA(t)−RA(0)]
This formula can be used to determine DAB in a com-
puter simulation, where the system is located in a simu-
lation box with periodic boundary conditions. However,
in this case one has to be careful because the difference
RA(t) − RA(0) has to be calculated in an origin inde-
pendent representation [30]. This can be achieved by
computing this difference via the integral
′)dt′.
III. EXPERIMENTAL METHODS
A. Long–capillary technique
The long–capillary technique (LC) has been used to
measure interdiffusion and Ni self–diffusion in liquid
Al80Ni20. The sample material production is similar to
that of Al87Ni10Ce3, which is described in Ref. [31]. The
-2.0 -1.0 0.0 1.0 2.0
x (10
FIG. 1: Typical concentration profiles of a combined interdif-
fusion and self–diffusion experiment. The squares denote the
Al and Ni concentrations measured by energy-dispersive X–
ray spectrometry (EDS) and the dots denote the 62Ni concen-
tration measured by inductively–coupled plasma mass spec-
trometry (ICP–MS). The lines represent the best fit (least–
square method) of the appropriate solution of Fick’s diffusion
equations to the measured concentrations.
experimental apparatus, the measurement of the concen-
tration profiles and the evaluation of the concentration
profiles, including the determination of Fick’s diffusion
coefficients, are also described elsewhere [32, 33]. Thus,
here the experimental technique is reported only briefly.
In more detail we describe an improved diffusion cou-
ple setup, which has been used in this work. This setup,
with a vertical diffusion capillary of 1.5mm diameter, has
an increased stabilization against natural convection and
minimizes the systematic error of convective mass flow
contributions to the total mass transport.
The improvement of the diffusion couple setup implies
the combination of interdiffusion and self–diffusion mea-
surements in one experiment. An Al80Ni20 slice of 2mm
thickness, containing the enriched stable 62Ni isotope,
is placed between both rods of an interdiffusion couple.
The interdiffusion couple consists of a 15mm long rod of
Al85Ni15, placed above the slice, and a 15mm long rod of
Al75Ni25, placed below the slice. This configuration al-
lows the development of an error function shaped chem-
ical interdiffusion profile simultaneously to the develop-
ment of a Gauss function shaped self–diffusion profile. In
a first approximation the diffusion of the enriched stable
isotope takes place at the mean concentration Al80Ni20
without influence of the changing chemical composition
of the melt in the diffusion zone. The only necessary cor-
rection results from the mass spectrometric measurement
of the self–diffusion profile. Here the measured isotope
incidences i(62Ni) of 62Ni have to be corrected for the
overlaying chemical concentration profile of natural Ni,
cNi, by using the following formula:
c(62Ni) = cNi
i(62Ni)− i(62Ni0)
with i(62Ni0) the natural incidence of 62Ni and c(62Ni)
the concentration of this Ni isotope with respect to all
Ni isotopes. Typical concentration profiles of a diffusion
experiment are given in Fig. 1.
The diffusion couple configuration minimizes the risk
of convection compared to conventional self–diffusion ex-
periments in pure melts because of the solutal stabilized
density profile of the melt column. This stabilizing ef-
fect has been described in Refs. [34, 35]. In a standard
self–diffusion experiment without chemical gradient the
solutal stabilization effect is only due to the enrichment
of a tracer.
As a test for other mass transport processes we mea-
sured the mean–square penetration depth x̄2 of interdif-
fusion as a function of time t. We found a deviation from
the linear behavior x̄2 = 2DABt. This has been iden-
tified as sedimentation of Al3Ni2 during solidification of
the diffusion sample. This additional mass transport was
simply corrected by subtracting this contribution as an
off–set of the measured total mass transport. This pro-
cedure adds a 5-10% error to the uncertainty of the dif-
fusion coefficient. The total error in the long–capillary
measurements of the self– and interdiffusion coefficients
is about 30–40%.
B. Neutron scattering experiments
The second experimental technique used in this work is
quasielastic neutron scattering. In this case, the Al80Ni20
alloy was prepared by arc melting of pure elements un-
der a purified Argon atmosphere. The measurements
were done at the time–of–flight spectrometer IN6 of the
Institut Laue-Langevin. The standard Nb resistor high
temperature vacuum furnace of the ILL exhibits a tem-
perature gradient over the entire sample at 1800K that
was less than five degrees and a temperature stability
within one degree. For the scattering experiment we
used a thin–walled Al2O3 container that provides a hol-
low cylindrical sample geometry of 22mm in diameter
and a sample wall thickness of 1.2mm.
An incident neutron wavelength of λ = 5.1 Å yielded
an energy resolution of δE ≃ 92µeV (FWHM) and an
accessible wave number range at zero energy transfer
of q = 0.4 − 2.0 Å
. Measurements were performed
at 1350K, 1525K, 1670K and 1795K in 2 hour runs
each. A run at room temperature provided the instru-
mental energy resolution function. The scattering law
S(q, ω) was obtained by normalization to a vanadium
standard, accompanied by a correction for self absorp-
tion and container scattering, and interpolation to con-
stant wave numbers q. Further, S(q, ω) was symmetrized
with respect to the energy transfer ~ω by means of the
detailed balance factor.
-1.0 0.0 1.0 2.0
hω (meV)
T=1525K, q=1.0Å
T=300K, q=0.8Å
FIG. 2: Normalized scattering law of liquid Al80Ni20. The
data at 300K represent the instrumental energy resolution
function. The line is a fit with a Lorentzian function that is
convoluted with the instrumental energy resolution function.
Diffusive motion of the atoms leads to a broadening of the
quasielastic signal from which the Ni self–diffusivity can be
obtained on an absolute scale.
Figure 2 displays S(q, ω) at q = 1.0 Å
of liquid
Al80Ni20 at 1525K and the crystalline alloy at 300K at
q = 0.8 Å
. Diffusive motion in the liquid leads to a
broadening of the quasielastic signal. The data were fit-
ted with an Lorentzian function that is convoluted with
the instrumental energy resolution function. From the
full width at half maximum of the quasielastic line Γ a q–
dependent diffusion coefficient D(q) can be computed via
D(q) = Γ/(2~q2). Towards small q incoherent scattering
on the Ni atoms dominates the signal and the diffusion
coefficient D(q) becomes constant yielding an estimate
of Ds,Ni. Thus, the self–diffusion constant Ds,Ni can be
determined on an absolute scale [36, 37].
IV. DETAILS OF THE SIMULATION
For the computer simulations of the binary system
Al80Ni20, we used a potential of the embedded atom type
that was recently derived by Mishin et al. [17]. In a re-
cent publication [18], we have shown that this potential
reproduces very well structural properties and the self–
diffusion constant of Al–Ni melts at various compositions.
The present simulations are performed in a similar way
as the ones in the latter work: Systems of N = 1500
particles (NNi = 300, NAl = 1200) are put in a cubic
simulation box with periodic boundary conditions. First,
standard Monte–Carlo (MC) simulations in the NpT en-
semble [38] were used to fully equilibrate the systems at
zero pressure and to generate independent configurations
for MD simulations in the microcanonical ensemble. In
the latter case, Newton’s equations of motion were in-
tegrated with the velocity Verlet algorithm using a time
step of 1.0 fs at temperatures T ≥ 1500K and 2.5 fs at
lower temperatures.
The masses were set to 26.981539amu and 58.69 amu
for aluminum and nickel, respectively. At each tempera-
ture investigated, we made sure that the duration of the
equilibration runs exceeded the typical relaxation times
of the system. The temperatures considered were 4490K,
2994K, 2260K, 1996K, 1750K, 1496K, 1300K, 1100K,
998K, 940K, 893K, 847K, 810K, 777K, 754K, 735K,
715K, 700K, 680K, and 665K. In order to improve the
statistics of the results we averaged at each temperature
over eight independent runs. At the lowest tempera-
ture, the duration of the microcanonical production runs
were 40 million time steps, thus yielding a total simu-
lation time of about 120ns. The latter production runs
were used to study the tagged particle dynamics. For
the calculation of the interdiffusion constant DAB addi-
tional production runs were performed in the tempera-
ture range 4490K ≥ T ≥ 715K that extended the pro-
duction runs for the tagged particle dynamics by about a
factor of ten. This was necessary in order to obtain a rea-
sonable statistics for DAB. Note that DAB is a collective
quantity that does not exhibit the self–averaging prop-
erties of the self–diffusion constant and thus it is quite
demanding to determine transport coefficients such as
the interdiffusion constant or the shear viscosity from a
MD simulation.
V. RESULTS
In Eq. (20), the interdiffusion constant DAB is ex-
pressed as a linear combination of the self–diffusion con-
stants. The prefactor in this formula is a product of
the thermodynamic factor Φ and the Manning factor S.
Whereas Φ can be computed from structural input, the
Manning factor contains the collective dynamic correla-
tions in the expression for DAB (see Sec. II). In the
following, we compare the simulated diffusion constants
for Al80Ni20 to those from experiments. Moreover, the
simulations are used to disentangle differences between
self–diffusion constants and the interdiffusion constants
with respect to the thermodynamic quantity Φ and the
dynamic quantity S.
First, we discuss static structure factors at different
temperatures, as obtained from the MD simulation. Fig-
ure 3 displays the different partial structure factors at
the temperatures T = 2000K and T = 750K. At both
temperatures, a broad prepeak around the wavenumber
q = 1.8 Å−1 emerges in the NiNi correlations, which
indicates the presence of chemical short–ranged order
(CSRO). This feature is absent in the AlAl correlations.
In a recent work [18], we have found that the prepeak in
SNiNi(q) is present in a broad variety of Al–Ni compo-
sitions, ranging from xNi = 0.1 to xNi = 0.9. However,
0.0 2.0 4.0 6.0 8.0 10.0 12.0
, T=2000Ka)
0.0 2.0 4.0 6.0 8.0 10.0 12.0
, T=750Kb)
FIG. 3: Partial structure factors, as obtained from the MD
simulation, for a) T = 2000K and b) T = 750K. The multipli-
cation by 1/(cαcβ)
1/2 is introduced to increase the amplitude
of SNiNi(q) relative to that of SAlAl(q). Note that the factor
1/(cαcβ)
1/2 leads also to the asymptotic value Sαα(q) = 1 for
q → ∞.
the width of the prepeak decreases significantly with in-
creasing Ni concentration and, in melts with a high Ni
concentration, it appears also in SAlAl(q). The prepeak
in Sαβ(q) describes repeating structural units involving
next–nearest αβ neighbors which are built in inhomoge-
neously into the structure. Of course, for the Al rich sys-
tem Al80Ni20 considered in this work, only next–nearest
Ni–Ni units exhibit the CSRO that is reflected in the
prepeak.
From the partial static structure factors, the Bhatia–
Thornton structure factors can be determined according
to Eqs. (10), (14) and (15). These quantities are shown in
Fig. 4, again at T = 2000K and at T = 750K. Although
these structure factors look very different for q > 2 Å−1,
they are essentially identical in the limit q → 0. As
we have indicated before, the static susceptibility, asso-
ciated with concentration fluctuations, can be extracted
from the structure factor Scc(q) in the limit q → 0. As
we can infer from Fig. 4, at the temperature T = 750K
the value of this susceptibility is very small. The small
0.0 2.0 4.0 6.0 8.0 10.0 12.0
, T=2000Ka)
0.0 2.0 4.0 6.0 8.0 10.0 12.0
, T=750Kb)
FIG. 4: Bhatia–Thornton structure factors, as obtained from
the MD simulation, at a) T = 2000K and b) T = 750K.
value of Scc(q = 0) reveals that concentration fluctua-
tions on large length scales are strongly suppressed. This
is the typical behavior of a dense fluid that exhibits a
strong ordering tendency. In contrast, at a critical point
of a demixing transition a divergence of Scc(q = 0) is
expected.
As we have seen in Sec. II, the ratio DAB/Φ can be
expressed as a linear combination of the self–diffusion
constants, provided S = 1 holds. In order to quantify
the temperature dependence of S, we first define the fol-
lowing mean–squared displacements:
〈r2(t)〉int =
NcAcB ×
×〈[RA(t)−RA(0)]
〉 (25)
〈r2(t)〉self = cA
j (t)− r
j (0)
j (t)− r
j (0)
〉 (26)
Whereas the interdiffusion constant can be calculated
t (ps)
interdiffusion
selfdiffusion
3000K
FIG. 5: Simulation results of mean squared displacements
(MSD) for self–diffusion (dashed lines) and interdiffusion
(solid lines) for the temperatures T = 3000K, 2000K, 1500K,
1000K, 850K, 750K, 715K, and 665K (corresponding to the
curves from left to right. Note that for T = 665K only
〈r2(t)〉self was calculated. For the definitions of the MSD’s
see Eqs. (25) and (26).
via DAB = limt→∞Φ〈r
2(t)〉int/(6t), the equation DAB =
limt→∞Φ〈r
2(t)〉self/(6t) is only correct for S = 1. Figure
5 shows the quantities 〈r2(t)〉int and 〈r
2(t)〉self for the dif-
ferent temperatures. Both MSD’s show a very similar be-
havior. At high temperature, a crossover from a ballistic
regime (∝ t2) at short times to a diffusive regime (∝ t) at
long times can be seen. At low temperature, a plateau–
like region develops at intermediate times, i.e. between
the ballistic and the diffusive regime. With decreasing
temperature, the plateau becomes more pronounced. In
〈r2(t)〉self , the plateau indicates the so–called cage ef-
fect [6]. The tagged particle is trapped by its neighbors
on a time scale that increases with decreasing tempera-
ture. Although the MSD for the interdiffusion, 〈r2(t)〉int,
describes also collective particle transport, the plateau
in this quantity has the same origin: The particles are
“arrested” on intermediate time scales. Moreover, the
differences between 〈r2(t)〉self and 〈r
2(t)〉int are anyway
very small in the whole time and temperature range un-
der consideration. This means that the cross correlations
do not give a large contribution to 〈r2(t)〉int.
From the MSD’s in Fig. 5, the Manning factor S can
be extracted using Eq. (21). In Fig. 6 we see that the
Manning factor varies only slightly over the whole tem-
perature range, located around values between 0.8 and
1.0. Also shown in Fig. 6 is the thermodynamic factor Φ
and the product ΦS. We have extracted Φ from the ex-
trapolation of the structure factors Scc(q) toward q → 0
[see Eq. (13)]. In contrast to the Manning factor S, the
thermodynamic factor Φ increases significantly with de-
750 1250 1750 2250 2750
T (K)
S(T)φ(T)
FIG. 6: Thermodynamic factor φ, “Manning” factor S(T ),
and the product of both as obtained from the simulation.
creasing temperature and thus, also the change in the
product ΦS is dominated by the change in Φ. There-
fore, differences in the qualitative behavior between the
self–diffusion constants and the interdiffusion constant
are dominated by the thermodynamic factor.
An Arrhenius plot of the diffusion constants as ob-
tained from simulation and experiment is shown in Fig. 7.
The self–diffusion constants DNi and DAl from the simu-
lation are very similar over the whole temperature range
4490K≥ T ≥ 665K. In a recent publication [18], we have
found that, in the framework of our simulation model,
this similarity of the self–diffusion constants occurs in Al
rich compositions of the system Al–Ni, say for cAl > 0.7.
Whether this is also true in real systems is an open ques-
tion. However, the neutron scattering results for DNi as
well as the single point obtained from the LC measure-
ment is in very good agreement with the simulation data.
Asta et al. [15] have computed the concentration de-
pendence of the self–diffusion constants at T = 1900 K
using two different embedded atom potentials, namely
the one proposed by Voter and Chen [39] and the one
proposed by Foiles and Daw [40]. For both potentials,
they find very similar values forDNi andDAl in Al80Ni20,
in agreement with our results. However, their results for
the Ni diffusion constant are significantly higher than
the ones found in our quasielastic neutron scattering ex-
periment and our simulation. They report the values
DNi ≈ 1.5 · 10
−8m2/s and DNi ≈ 1.9 · 10
−8m2/s for the
Voter–Chen potential and the Foiles–Daw potential, re-
spectively, whereas we obtainDNi ≈ 10
−8m2/s from sim-
ulation and experiment. Thus, the potential proposed by
Mishin et al. [17], which is used in this work, leads to
a better agreement with the experiment, as far as self–
diffusion in Al80Ni20 is concerned.
We emphasize that the statistical error in both the
neutron scattering data and the simulation data for the
self–diffusion constants is relatively small. In both cases,
the error bars for the corresponding data points in Fig. 7
0.3 0.6 0.9 1.2 1.5
1000/T (K
sim. D
sim. D
sim. D
QNS D
0.5 0.6 0.7 0.8
1000/T (K
sim. D
sim. D
sim. D
QNS D
LC exp. D
FIG. 7: Arrhenius plot of interdiffusion and self–diffusion con-
stants, as obtained from experiment and simulation, as indi-
cated. The experimental results are measured by quasielastic
neutron scattering (QNS) and by the LC technique. The lines
through the data points are guides to the eye. The vertical
dotted line in a) marks the location of the experimental liq-
uidus temperature, TL ≈ 1280K. The vertical dashed line is
at the location of the critical temperature of mode coupling
theory, Tc ≈ 700K, as estimated by the MD simulation [41].
Panel b) is an enlargement of the data of panel a) in a tem-
perature range above TL. The error bars of simulation and
QNS data are of the order of the size of the symbols.
are smaller than the size of the symbols.
Due to the lack of self–averaging, it is much more dif-
ficult to yield accurate results for DAB from the simu-
lation. Therefore, in this case we considered a smaller
temperature range than for the self–diffusion constants
to yield results with reasonable accuracy. As we can in-
fer from Fig. 7, the interdiffusion constant is larger than
the self–diffusion constants over the whole temperature
range. The difference becomes more pronounced with
decreasing temperature. At T = 715K, the diffusion co-
efficient DAB is about a factor of 3 larger than DNi and
DAl. This behavior is of course due to the increase of
the thermodynamic factor Φ at low temperature. Also
included in Fig. 7 are the results of the LC measurements
of DAB and DNi. These results are much less accurate
than those of the quasielastic neutron scattering experi-
ments for the determination of DNi (see the error bars for
the LC data in Fig. 7b). Nevertheless, the LC data show
that DAB > DNi holds, in agreement with the simulation
results.
VI. CONCLUSION
A combination of experiment and molecular dynamics
(MD) simulation has been used to investigate the diffu-
sion dynamics in liquid Al80Ni20. We find good agree-
ment between simulation and experiment. Both in ex-
periment and in simulation, the interdiffusion constant
is higher than the self–diffusion constants. This is valid
in the whole temperature range considered in this work,
i.e. in the normal liquid state as well as in the undercooled
regime. In the latter regime (which is only accessible by
the simulation), the difference between the interdiffusion
constant and the self–diffusion constants increases with
decreasing temperature.
All these observations can be clarified by the detailed
information provided by the MD simulation. Both the
thermodynamic factor Φ and the Manning factor S have
been estimated directly and accurately over a wide tem-
perature range, as well as self–diffusion and interdiffusion
coefficients. The central result of this work is shown in
Fig. 6. Whereas the thermodynamic factor Φ increases
significantly by lowering the temperature, the Manning
factor S shows only a weak temperature dependence.
Moreover, the value of S is close to one which means that
dynamic cross correlations are almost negligible and thus,
even in the undercooled regime, the Darken equation is
a good approximation. The temperature dependence of
Φ is plausible for a dense binary mixture with a strong
ordering tendency. The situation is similar to the case of
the isothermal compressibility which normally decreases
with temperature in a densely packed liquid leading to
very low values in the undercooled regime. In the same
sense, the response to a macroscopic concentration fluc-
tuation described by Scc(q = 0) tends to become smaller
and smaller towards the undercooled regime which corre-
sponds to an increase of Φ with decreasing temperature
(since Φ ∝ 1/Scc(q = 0)).
We note that the data shown for DAB are all above the
critical temperature Tc of mode coupling theory which
is around 700K for our simulation model (see Fig. 7)
[41]. Since it is expected that the transport mechanism
changes below Tc [6], it would be interesting to see how
such a change in the transport mechanism is reflected in
the interdiffusion constants. This issue is the subject of
forthcoming studies.
Acknowledgments
We are grateful to Kurt Binder for stimulating dis-
cussions and a critical reading of the manuscript. We
gratefully acknowledge financial support within the SPP
1120 of the Deutsche Forschungsgemeinschaft (DFG) un-
der grants Bi314/18, Ma1832/3-2 and Me1958/2-3 and
from DFG grant Gr2714/2-1. One of the authors was
supported through the Emmy Noether program of the
DFG, grants Ho2231/2-1/2 (J.H.). Computing time on
the JUMP at the NIC Jülich is gratefully acknowledged.
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D. de Fontaine, and S.M. Foiles, Phys. Rev. B 59, 14271
(1999).
[16] F. Faupel, W. Frank, M.–P. Macht, H. Mehrer, V. Naun-
dorf, K. Rätzke, H.R. Schober, S.K. Sharma, and H. Te-
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[17] Y. Mishin, M. J. Mehl, and D. A. Papaconstantopoulos,
Phys. Rev. B 65, 224114 (2002).
[18] S. K. Das, J. Horbach, M. M. Koza, S. Mavila Chatoth,
and A. Meyer, Appl. Phys. Lett. 86, 011918 (2005).
[19] G.L. Batalin, E.A. Beloborodova, and V.G. Kazimirov,
Thermodynamics and the Constitution of Liquid Al Based
Alloys (Metallurgy, Moscow, 1983).
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Chem. 43, 2756 (1969).
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Rev. B 42, 1598 (1990).
[22] S. Sadeddine, J.F. Wax, B. Grosdidier, J.G. Gasser, C.
Regnaut, and J.M. Dubois, Phys. Chem. Liq. 28, 221
(1994).
[23] M. Asta, V. Ozolins, J.J. Hoyt, and M. van Schilfgaarde,
Phys. Rev. B 64, 020201(R) (2001).
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plavy No. 4, 97 (1990); U.K. Stolz, I. Arpshoven, F.
Sommer, and B. Predel, J. Phase Equilib. 14, 473 (1993);
K.V. Grigorovitch and A.S. Krylov, Thermochim. Acta
314, 255 (1998).
[25] S.K. Das, J. Horbach, and K. Binder, J. Chem. Phys.
119, 1547 (2003); S.K. Das, J. Horbach, K. Binder, M.E.
Fisher, J.V. Sengers, J. Chem. Phys. 125, 024506 (2006);
S.K. Das, M. E. Fisher, J.V. Sengers, J. Horbach, K.
Binder, Phys. Rev. Lett. 97, 025702 (2006).
[26] F.O. Raineri and H.L. Friedman, J. Chem. Phys. 91,
5633 (1989); F.O. Raineri and H.L. Friedman, J. Chem.
Phys. 91, 5642 (1989).
[27] A.B. Bhatia and D.E. Thornton, Phys. Rev. B 52, 3004
(1970).
[28] D. Holland–Moritz, O. Heinen, R. Bellissent, T. Schenk,
and D.M. Herlach, Int. J. Mat. Res. 97, 948 (2006).
[29] J.R. Manning, Phys Rev. 124, 470 (1961).
[30] M. P. Allen, D. Brown, and A. J. Masters, Phys. Rev.
E 49, 2488 (1994); M. P. Allen, Phys. Rev. E 50, 3277
(1994).
[31] A. Griesche, F. Garcia Moreno, M.P. Macht, and G.
Frohberg, Mat. Sc. Forum 508, 567 (2006).
[32] A. Griesche, M.P. Macht, J.P. Garandet, and G. Froh-
berg, J Non–Cryst. Solids 336, 173 (2004).
[33] A. Griesche, M.P. Macht, and G. Frohberg, unpublished.
[34] J.P. Garandet, C. Barrat, and T. Duffar, Int. J. Heat
Mass Transfer 38, 2169 (1995).
[35] C. Barrat and J.P. Garandet, Int. J. Heat Mass Transfer
39, 2177 (1996).
[36] A. Meyer, Phys. Rev. B 66, 134205 (2002).
[37] S. Mavila Chathoth, A. Meyer, M. M. Koza, and F. Yu-
ranji, Appl. Phys. Lett. 85, 4881 (2004).
[38] D. P. Landau and K. Binder, A Guide to Monte Carlo
Simulations in Statistical Physics (Cambridge University
Press, Cambridge, 2000).
[39] A.F. Voter and S.P. Chen, in Characterization of Defects
in Materials, edited by R.W. Siegel et al., MRS Sym-
posia Proceedings No. 82 (Materials Research Society,
Pittsburgh, 1978), p. 175.
[40] S.M. Foiles and M.S. Daw, J. Mat. Res. 2, 5 (1987).
[41] S.K. Das, J. Horbach, and K. Binder, unpublished.
	Introduction
	Self–diffusion and interdiffusion: Basic theory
	Experimental Methods
	Long–capillary technique
	Neutron scattering experiments
	Details of the simulation
	Results
	Conclusion
	Acknowledgments
	References

  A combination of experimental techniques and molecular dynamics (MD) computer
simulation is used to investigate the diffusion dynamics in Al80Ni20 melts.
Experimentally, the self-diffusion coefficient of Ni is measured by the
long-capillary (LC) method and by quasielastic neutron scattering. The LC
method yields also the interdiffusion coefficient. Whereas the experiments were
done in the normal liquid state, the simulations provided the determination of
both self-diffusion and interdiffusion constants in the undercooled regime as
well. The simulation results show good agreement with the experimental data. In
the temperature range 3000 K >= T >= 715 K, the interdiffusion coefficient is
larger than the self-diffusion constants. Furthermore the simulation shows that
this difference becomes larger in the undercooled regime. This result can be
refered to a relatively strong temperature dependence of the thermodynamic
factor \Phi, which describes the thermodynamic driving force for
interdiffusion. The simulations also indicate that the Darken equation is a
good approximation, even in the undercooled regime. This implies that dynamic
cross correlations play a minor role for the temperature range under
consideration.

Introduction to Their Statistical Mechanics
(World Scientific, London, 2005).
[7] R. Vogelsang and C. Hoheisel, Phys. Rev. A 38, 6296
(1988); H.P. van den Berg and C. Hoheisel, Phys. Rev.
A 42, 2090 (1990).
[8] K.W. Kehr, K. Binder, and S.M. Reulein, Phys. Rev. B
39, 4891 (1989).
[9] W. Hess, G. Nägele, and A.Z. Akcasu, J. Polymer Sc. B
28, 2233 (1990).
[10] J. Trullàs and J.A. Padrò, Phys. Rev. E 50, 1162 (1994).
[11] A. Baumketner and Ya. Chushak, J. Phys.: Condens.
Matter 11, 1397 (1999).
[12] J.–F. Wax and N. Jakse, Phys. Rev. B 75, 024204 (2007).
[13] L.S. Darken, Trans. AIME 180, 430 (1949).
[14] M. Fuchs and A. Latz, Physica A 201, 1 (1993).
[15] M. Asta, D. Morgan, J.J. Hoyt, B. Sadigh, J.D. Althoff,
D. de Fontaine, and S.M. Foiles, Phys. Rev. B 59, 14271
(1999).
[16] F. Faupel, W. Frank, M.–P. Macht, H. Mehrer, V. Naun-
dorf, K. Rätzke, H.R. Schober, S.K. Sharma, and H. Te-
ichler, Rev. Mod. Phys. 75, 237 (2003).
[17] Y. Mishin, M. J. Mehl, and D. A. Papaconstantopoulos,
Phys. Rev. B 65, 224114 (2002).
[18] S. K. Das, J. Horbach, M. M. Koza, S. Mavila Chatoth,
and A. Meyer, Appl. Phys. Lett. 86, 011918 (2005).
[19] G.L. Batalin, E.A. Beloborodova, and V.G. Kazimirov,
Thermodynamics and the Constitution of Liquid Al Based
Alloys (Metallurgy, Moscow, 1983).
[20] G.D. Ayushina, E.S. Levin, and P.V. Geld, Russ. J. Phys.
Chem. 43, 2756 (1969).
[21] M. Maret, T. Pomme, A. Pasturel, and P. Chieux, Phys.
Rev. B 42, 1598 (1990).
[22] S. Sadeddine, J.F. Wax, B. Grosdidier, J.G. Gasser, C.
Regnaut, and J.M. Dubois, Phys. Chem. Liq. 28, 221
(1994).
[23] M. Asta, V. Ozolins, J.J. Hoyt, and M. van Schilfgaarde,
Phys. Rev. B 64, 020201(R) (2001).
[24] V.S. Sudovtseva, A.V. Shuvalov, N.O. Sharchina, Ras-
plavy No. 4, 97 (1990); U.K. Stolz, I. Arpshoven, F.
Sommer, and B. Predel, J. Phase Equilib. 14, 473 (1993);
K.V. Grigorovitch and A.S. Krylov, Thermochim. Acta
314, 255 (1998).
[25] S.K. Das, J. Horbach, and K. Binder, J. Chem. Phys.
119, 1547 (2003); S.K. Das, J. Horbach, K. Binder, M.E.
Fisher, J.V. Sengers, J. Chem. Phys. 125, 024506 (2006);
S.K. Das, M. E. Fisher, J.V. Sengers, J. Horbach, K.
Binder, Phys. Rev. Lett. 97, 025702 (2006).
[26] F.O. Raineri and H.L. Friedman, J. Chem. Phys. 91,
5633 (1989); F.O. Raineri and H.L. Friedman, J. Chem.
Phys. 91, 5642 (1989).
[27] A.B. Bhatia and D.E. Thornton, Phys. Rev. B 52, 3004
(1970).
[28] D. Holland–Moritz, O. Heinen, R. Bellissent, T. Schenk,
and D.M. Herlach, Int. J. Mat. Res. 97, 948 (2006).
[29] J.R. Manning, Phys Rev. 124, 470 (1961).
[30] M. P. Allen, D. Brown, and A. J. Masters, Phys. Rev.
E 49, 2488 (1994); M. P. Allen, Phys. Rev. E 50, 3277
(1994).
[31] A. Griesche, F. Garcia Moreno, M.P. Macht, and G.
Frohberg, Mat. Sc. Forum 508, 567 (2006).
[32] A. Griesche, M.P. Macht, J.P. Garandet, and G. Froh-
berg, J Non–Cryst. Solids 336, 173 (2004).
[33] A. Griesche, M.P. Macht, and G. Frohberg, unpublished.
[34] J.P. Garandet, C. Barrat, and T. Duffar, Int. J. Heat
Mass Transfer 38, 2169 (1995).
[35] C. Barrat and J.P. Garandet, Int. J. Heat Mass Transfer
39, 2177 (1996).
[36] A. Meyer, Phys. Rev. B 66, 134205 (2002).
[37] S. Mavila Chathoth, A. Meyer, M. M. Koza, and F. Yu-
ranji, Appl. Phys. Lett. 85, 4881 (2004).
[38] D. P. Landau and K. Binder, A Guide to Monte Carlo
Simulations in Statistical Physics (Cambridge University
Press, Cambridge, 2000).
[39] A.F. Voter and S.P. Chen, in Characterization of Defects
in Materials, edited by R.W. Siegel et al., MRS Sym-
posia Proceedings No. 82 (Materials Research Society,
Pittsburgh, 1978), p. 175.
[40] S.M. Foiles and M.S. Daw, J. Mat. Res. 2, 5 (1987).
[41] S.K. Das, J. Horbach, and K. Binder, unpublished.
	Introduction
	Self–diffusion and interdiffusion: Basic theory
	Experimental Methods
	Long–capillary technique
	Neutron scattering experiments
	Details of the simulation
	Results
	Conclusion
	Acknowledgments
	References
