Stokes-Einstein relation and excess entropy scaling law in liquid Copper
We report an ab initio study of structural and dynamic properties of liquid copper as a function of temperature. In particular, we have evaluated the temperature dependence of the self-diffusion coefficient from the velocity autocorrelation function as well the temperature dependence of the viscosit...
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Jakse, N. Pasturel, A. 2019-06-16T15:20:30Z 2019-06-16T15:20:30Z 2015 Stokes-Einstein relation and excess entropy scaling law in liquid Copper / N. Jakse, A. Pasturel // Condensed Matter Physics. — 2015. — Т. 18, № 4. — С. 43603: 1–11 . — Бібліогр.: 52 назв. — англ. 1607-324X DOI:10.5488/CMP.18.43603 arXiv:1512.07795 PACS: 61.25.Mv, 61.20.Ja, 66.10.-x https://nasplib.isofts.kiev.ua/handle/123456789/155274 We report an ab initio study of structural and dynamic properties of liquid copper as a function of temperature. In particular, we have evaluated the temperature dependence of the self-diffusion coefficient from the velocity autocorrelation function as well the temperature dependence of the viscosity from the transverse current correlation function. We show that LDA based results are in close agreement with experimental data for both the self-diffusion coefficient and the viscosity over the temperature range investigated. Our findings are then used to test empirical approaches like the Stokes-Einstein relation and the excess entropy scaling law widely used in the literature. We show that the Stokes-Einstein relation is valid for the liquid phase and that the excess entropy scaling law proposed by Dzugutov is legitimate only if a self-consistent method for determining the packing fraction of the hard sphere reference liquid is used within the Carnahan-Starling approach to express the excess entropy. У цiй статтi представлено ab initio дослiдження структурних i динамiчних властивостей рiдкої мiдi як функцiї температури. Зокрема, проведено розрахунок температурної залежностi коефiцiєнта самодифузiї з автокореляцiйної функцiї швидкостей, а також температурної залежностi швидкостi з поперечної кореляцiйної функцiї струму. Показано, що результати, базованi на наближеннi локальної густини, добре узгоджуються з експериментальними даними як для коефiцiєнта дифузiї, так i для в’язкостi в межах дослiджуваного дiапазона температур. Цi результати використанi для перевiрки емпiричних пiдходiв типу спiввiдношення Стокса-Ейнштейна та скейлiнгового закону для надлишкової ентропiї, якi широко використовуються в лiтературi. Показано, що спiввiдношення Стокса-Ейнштейна задовiльняється для рiдкої фази i скейлiнговий закон для надлишкової ентропiї, запропонований Дзугутовим, є легiтимним лише за умови використання самоузгодженого методу для визначення фракцiї упакування твердосферної рiдини, коли пiдхiд Карнагана-Старлiнга використано для отримання виразу надлишкової ентропiї. We acknowledge the CINES and IDRIS under Project Number INP2227/72914 as well as PHYNUM CIMENT for computational resources. This work was performed within the framework of the Centre of Excellence of Multifunctional Architectured Materials “CEMAM” number ANR-10-LABX-44-01 funded by the “Investments for the Future” Program. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Stokes-Einstein relation and excess entropy scaling law in liquid Copper Спiввiдношення Стокса-Ейнштейна i скейлiнговий закон для надлишкової ентропiї у рiдкiй мiдi Article published earlier |
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| title |
Stokes-Einstein relation and excess entropy scaling law in liquid Copper |
| spellingShingle |
Stokes-Einstein relation and excess entropy scaling law in liquid Copper Jakse, N. Pasturel, A. |
| title_short |
Stokes-Einstein relation and excess entropy scaling law in liquid Copper |
| title_full |
Stokes-Einstein relation and excess entropy scaling law in liquid Copper |
| title_fullStr |
Stokes-Einstein relation and excess entropy scaling law in liquid Copper |
| title_full_unstemmed |
Stokes-Einstein relation and excess entropy scaling law in liquid Copper |
| title_sort |
stokes-einstein relation and excess entropy scaling law in liquid copper |
| author |
Jakse, N. Pasturel, A. |
| author_facet |
Jakse, N. Pasturel, A. |
| publishDate |
2015 |
| language |
English |
| container_title |
Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Спiввiдношення Стокса-Ейнштейна i скейлiнговий закон для надлишкової ентропiї у рiдкiй мiдi |
| description |
We report an ab initio study of structural and dynamic properties of liquid copper as a function of temperature. In particular, we have evaluated the temperature dependence of the self-diffusion coefficient from the velocity autocorrelation function as well the temperature dependence of the viscosity from the transverse current correlation function. We show that LDA based results are in close agreement with experimental data for both the self-diffusion coefficient and the viscosity over the temperature range investigated. Our findings are then used to test empirical approaches like the Stokes-Einstein relation and the excess entropy scaling law widely used in the literature. We show that the Stokes-Einstein relation is valid for the liquid phase and that the excess entropy scaling law proposed by Dzugutov is legitimate only if a self-consistent method for determining the packing fraction of the hard sphere reference liquid is used within the Carnahan-Starling approach to express the excess entropy.
У цiй статтi представлено ab initio дослiдження структурних i динамiчних властивостей рiдкої мiдi як функцiї температури. Зокрема, проведено розрахунок температурної залежностi коефiцiєнта самодифузiї з
автокореляцiйної функцiї швидкостей, а також температурної залежностi швидкостi з поперечної кореляцiйної функцiї струму. Показано, що результати, базованi на наближеннi локальної густини, добре
узгоджуються з експериментальними даними як для коефiцiєнта дифузiї, так i для в’язкостi в межах дослiджуваного дiапазона температур. Цi результати використанi для перевiрки емпiричних пiдходiв типу
спiввiдношення Стокса-Ейнштейна та скейлiнгового закону для надлишкової ентропiї, якi широко використовуються в лiтературi. Показано, що спiввiдношення Стокса-Ейнштейна задовiльняється для рiдкої
фази i скейлiнговий закон для надлишкової ентропiї, запропонований Дзугутовим, є легiтимним лише за
умови використання самоузгодженого методу для визначення фракцiї упакування твердосферної рiдини,
коли пiдхiд Карнагана-Старлiнга використано для отримання виразу надлишкової ентропiї.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/155274 |
| citation_txt |
Stokes-Einstein relation and excess entropy scaling law in liquid Copper / N. Jakse, A. Pasturel // Condensed Matter Physics. — 2015. — Т. 18, № 4. — С. 43603: 1–11 . — Бібліогр.: 52 назв. — англ. |
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2025-11-24T15:49:15Z |
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2025-11-24T15:49:15Z |
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| fulltext |
Condensed Matter Physics, 2015, Vol. 18, No 4, 43603: 1–11
DOI: 10.5488/CMP.18.43603
http://www.icmp.lviv.ua/journal
Stokes-Einstein relation and excess entropy scaling
law in liquid Copper
N. Jakse, A. Pasturel
Sciences et Ingénierie des Matériaux et Procédés, UMR CNRS 5266, Grenoble Université Alpes,
BP 75, 38402 Saint-Martin d’Hères Cedex, France
Received September 8, 2015, in final form September 29, 2015
We report an ab initio study of structural and dynamic properties of liquid copper as a function of temperature.
In particular, we have evaluated the temperature dependence of the self-diffusion coefficient from the veloc-
ity autocorrelation function as well the temperature dependence of the viscosity from the transverse current
correlation function. We show that LDA based results are in close agreement with experimental data for both
the self-diffusion coefficient and the viscosity over the temperature range investigated. Our findings are then
used to test empirical approaches like the Stokes-Einstein relation and the excess entropy scaling law widely
used in the literature. We show that the Stokes-Einstein relation is valid for the liquid phase and that the excess
entropy scaling law proposed by Dzugutov is legitimate only if a self-consistent method for determining the
packing fraction of the hard sphere reference liquid is used within the Carnahan-Starling approach to express
the excess entropy.
Key words: liquid copper, Stokes-Einstein relation, universal scaling laws, ab initio molecular dynamics
PACS: 61.25.Mv, 61.20.Ja, 66.10.-x
1. Introduction
The study of the relationship between structural and dynamic properties of metallic liquids is a long-
standing open problem in condensed matter physics. For instance, transport properties such as the shear
viscosity (η) and the diffusion coefficients (D) of liquid metals are key parameters in the study of the crys-
tal nucleation and growth in metallic melts [1, 2].The temperature dependences of the viscosity and the
relaxation time also play a very important role in studying the liquid-glass transition in a glass-forming
system. Therefore, finding such a link should be also of practical interest as it relates a quantity like the
self-diffusion coefficient that can be difficult to measure accurately to an experimentally more accessible
dynamic, thermodynamic or structural quantity.
One alternative is to approximate diffusion coefficients from viscosities through the Stokes-Einstein
(SE) relation [3], DSE = kBT /2πRη, where kB is Boltzmann’s constant and R is an effective diameter of
particles. The interest of this relation lies on the fact that it is much more difficult to measure the diffu-
sion coefficients of liquids than their viscosities since the influences of convective flow on the diffusion
profile during annealing can be important using the long-capillary (LC) technique and its variation. At the
same time, viscosity data are often available from standard data collections with a weak scatter between
measurements made using different techniques. For instance, it is only ±6% for copper [4]. First derived
for the motion of a macroscopic particle in a homogeneous viscous medium, its use for atomic diffusion
in liquid alloys requires defining R at the atomistic scale. It is well accepted in the common literature [3]
to use the first position in the pair-correlation function obtained either experimentally or by means of
atomistic simulations.
Recently, the accurate experimental determination of self-diffusion coefficients of species like Ni, Ti,
or Cu [5–7] from the use of quasielastic neutron scattering (QNS) has made it possible to extensively
discuss the applicability of the Stokes-Einstein relation. Themain result is that some experiments support
© N. Jakse, A. Pasturel, 2015 43603-1
http://dx.doi.org/10.5488/CMP.18.43603
http://www.icmp.lviv.ua/journal
N. Jakse, A. Pasturel
its validity even for complex alloys [8] while others evidence its failure [9, 10], indicating the importance
of the chemical composition of the melt. Moreover, it seems that the temperature also plays an important
role in its use [10]. These experiments raise many questions and among them the choice of the effective
diameter of particles as well as in a more general aspect, the relation of transport properties with the
melt structure.
Another alternative is to relate the dynamic properties to the structure and thermodynamics of melts
using the universal scaling relationship like that proposed by Dzugutov [11]. Within this scheme, the pair
excess entropy of a liquid, S2, is related to a dimensionless form of the diffusion coefficient,D∗
. The quan-
tity S2 can be easily computed from pair correlation functions. However, if this approach is legitimate for
model fluids like Lennard-Jones or Hard-Sphere (HS) fluids, a coupled Monte Carlo/molecular dynamics
study has shown that this two-body correlation approximation does not hold for liquid metals and alloys
modelled by EAM potentials [12].
More generally, molecular-dynamics (MD) simulations represent a powerful means for getting a mi-
croscopic picture of structural and dynamic properties in the liquid state and for studying their relation-
ships but such studies are fruitful at the condition that atomic interactions are accurately modelled.
For liquid Cu, structural and dynamic properties have been extensively studied using the semi-empi-
rical embedded-atom model (EAM) formalism [13–18], as well as by the orbital free ab initio method
[19]. However, if structural properties are in very good agreement with experimental data whatever the
EAM potential, diffusion coefficients reported in classical MD simulations show an important scatter, de-
pending on the specific implementation of EAM potentials. Another difficulty holds with the fact that the
calculated diffusion coefficients are mainly referred to experimental LC data [20], with the consequence
that they exhibit a scatter of about ±50% around the most recent QNS values [7]. Only simulated results
provided by Alemany et al. [14] disagree with LC data and these authors suggest that the LC values may
be erroneous. Let us mention that very recent MD simulations [18] refer to experimental QNS values and
give a better agreement with them.
Within the density functional theory (DFT) framework, ab initioMD (AIMD) simulations performed at
1500 K either by Pasquarello et al. [21] or by Kresse and Hafner [22] give calculated self-diffusion coeffi-
cients that differ by a factor of 2. Whereas Pasquarello et al. reported a value of (2.8±0.2)×10−9
m
2
s
−1
,
Kresse and Hafner gave a value of (5.6± 0.4)× 10−9
m
2
s
−1
. At the same temperature, the QNS value is
(4.33±0.06)×10−9
m
2
s
−1
[7].
All these results show that it is necessary to revisit the determination of the structural and dynamic
properties of liquid Cu using ab initiomolecular dynamics simulations.
In a first step, we show that the determination of the self-diffusion coefficient as well as the viscosity
depend on the exchange-correlation functional used in DFT calculations, namely the local-density ap-
proximation (LDA) and the generalized gradient approximation (GGA). Our study shows that GGA based
calculations of the diffusion coefficient underestimates both LDA based values and experimental data
while we obtain the opposite trend for the viscosity. We evidence that these results can be explained by a
more important cage effect using GGA as compared to LDA, due to the increase of the icosahedral motifs.
In a second step, our results are used to discuss the Stokes-Einstein relation and its validity as a func-
tion of temperature. Finally, we test the universal scaling law relating the diffusion coefficient and the
excess entropy of a liquid as proposed by Dzugutov [11]. We show that the scaling law is obeyed provided
a self-consistent method for determining the packing fraction of the hard sphere reference fluid is used
within the Carnahan-Starling approach to express the excess entropy. In order to strengthen the present
discussion, we shall present our results together with those recently published for liquid Al [23].
2. Simulation details
The AIMD simulations have been performed by using the Vienna ab initio simulation package [24]
using the projected augmented-wave method to describe the electron-ion interaction [25]. A plane-wave
basis set with an energy cutoff of 273 eV is used. The exchange-correlation energy is described using
either the local-density approximation [26, 27] or the generalized gradient approximation in the Perdew-
Burke-Ernzerhof form [28]. Only the Γ-point sampling is considered to represent the supercell Brillouin
zone.
43603-2
Stokes-Einstein relation and excess entropy scaling law in liquid Copper
Equations of motion are solved using Verlet’s algorithm in the velocity form with a time step of 1 fs.
Atomic motions are carried out in the NVT ensemble by means of a Nose thermostat to control tempera-
ture. For each temperature, we use a cubic cell with periodic boundary conditions containing 256 atoms.
The liquid sample is first equilibrated at 2000 K (well above the experimental melting temperature of
1358 K) and then cooled to 1800, 1600, and 1398 K, successively with a rate of 1013
K/s. At each temper-
ature, we adjust the volume V of the supercell to reproduce the experimental densities [4] and after an
equilibration of 10 ps, the run is continued further during 80 ps for production of the structural and
dynamic quantities.
A number of 2000 configurations are collected to calculate the static structure factor, the pair-correla-
tion function, as well as the diffusion coefficient and the viscosity. Among these configurations, we select
ten independent ones, to compute their inherent structures [29] for the purpose of studying the local
ordering. This is done numerically by carrying out a conjugate gradient energy-minimization in order to
suppress the kinetic energy.
3. Results and discussion
3.1. Self-diffusion coefficient
In a first step, we consider the single-atom dynamics through the velocity auto-correlation function,
and its time integration that gives access to the self-diffusion coefficient, D [4]. Let us mention that this
method requires shorter simulation times than that based on the mean-square displacement to have
statistically meaningful results [4, 30].
We report the temperature dependence of D using the LDA and GGA approximations in figure 1.
We can see that LDA calculations are close to the experimental QNS values [7] while GGA calculations
underestimate LDA calculations by about 20% above the melting point. We also compute an activation
energy assuming an Arrhenius-type behaviour for the diffusion process in liquid copper. As shown in
figure 1, the data are well fitted in the whole range of temperatures and the derived activation energy is
ED = 370±5 and 475±5meV respectively for the LDA and GGA approximations. Let us mention that the
LDA value is in close agreement with the experimental value, namely 337±5meV [7].
Figure 1. (Color online) Evolution of the self-diffusion coefficient of liquid Cu as a function of inverse
temperature. The solid circles are AIMD results with LDA and the open ones are those with GGA. The
dashed lines are their respective Arrhenius fit. The full squares are experimental data of reference [7]
along with their Arrhenius fit (dash-dotted lines). AIMD results of references [21] (full circle) and [22]
(triangle up) are also plotted for comparison.
43603-3
N. Jakse, A. Pasturel
3.2. Viscosity
To compute the viscosity, we use a direct method based on the transverse current correlation function
(see reference [31] for a detailed description of this technique), which has the advantage of yielding a
generalized q-dependent shear viscosity from which the hydrodynamic limit can be evaluated [32, 33].
In figure 2 we compare LDA and GGA results with the assessed values [3]. As for the temperature
dependence of the self-diffusion coefficient, we observe that the temperature dependence of the viscos-
ity obtained by LDA is close to the temperature dependence of the assessed viscosity while GGA values
overestimate LDA and the assessed values. More particularly, LDA values are consistent with experimen-
tal data of Kehr et al. [34] and Brillo et al. [35], both experimental sets being used in the assessment.
The derived activation energy assuming an Arrhenius-type behaviour is Eη = 370± 5 and 475± 5 meV,
respectively, for the LDA and GGA approximations.
Figure 2. (Color online) Viscosity of liquid Cu as a function of temperature. The full circles and full squares
correspond to our AIMD results, respectively with LDA and GGA, as obtained with the transverse current-
correlation functions. The open circles and open squares correspond to viscosities inferred from the
self-diffusion coefficients, respectively with LDA and GGA, determined from our AIMD results using the
Stokes-Einstein relation. The triangles, diamonds, and dash line correspond to the experimental values
of references [34] and [35], and the assessment of reference [3], respectively.
Both computed self-diffusion coefficients and viscosities are then used to test the validity of the Stokes-
Einstein relation. In figure 2, we report viscosities computed via the Stokes-Einstein relation using R
given by the position of the first peak in the computed pair-correlation function. At T = 1600 K, the LDA
value afforded by the Stokes-Einstein relation is 2.78 mPa·s, close to the direct one, 2.71 mPa·s and the
assessed value of 2.81 mPa·s. We obtain a similar agreement at higher temperature, i.e., T = 1800 K,
while at lower temperature, namely T = 1400 K, we observe a small discrepancy, the value afforded by
the Stokes-Einstein relation being 10% smaller than the direct one. GGA calculations follow the same
trend. For instance at T = 1600 K, the value obtained from the SE relation is 3.45 mPa·s which can be
compared to that obtained by the direct method, namely 3.55 mPa·s. Note that the present study does
not support the conclusions of the EAM based study [18] claiming that the SE relation is not obeyed in
liquid Cu.
3.3. Discussion
In order to understand the significant discrepancies between GGA and LDA calculations of the self-
diffusion coefficient and the viscosity, we inspect structural properties of liquid copper using both func-
tionals.
In figure 3, we report the pair-correlation function g (r ) for all the three temperatures in the liquid
range using GGA and LDA. As already obtained for liquid aluminum [23], we do not see any appreciable
43603-4
Stokes-Einstein relation and excess entropy scaling law in liquid Copper
Figure 3. (Color online) Evolution of the pair-correlation function with temperature. Open circles and
triangles correspond to experimental values of references [36] (1393 K) and [37] (1600 K), respectively.
The curves for 1600 K, and 1398 K are shifted upwards by an amount of 1 and 2, respectively.
difference between the two sets of calculations. Let us mention that our g (r ) at T = 1600 K compares well
with g (r ) interpolated from X-ray diffraction experiments [36], while our g (r ) at T = 1398 K compare
also well with the results from neutron diffraction experiments at T = 1393 K [37]. The calculated g (r )
from previous ab initio [21, 22] or EAMmodel [13–18] are also consistent with our curves.
Coordination numbers are obtained from the integration of the computed pair-correlation functions
g (r ) up to its first minimum. We find an increase of coordination number from Nc = 11.8± 0.1 at T =
1800 K to Nc = 12.4± 0.1 at T = 1398 K while GGA calculations give Nc = 12.0± 0.1 at T = 1800 K and
Nc = 12.5±0.1 at 1398 K. We obtain that LDA calculations give coordination numbers smaller than those
obtained using GGA ones, but the differences remain small and cannot explain the differences obtained
in the dynamic properties.
A deeper insight into the presence and the nature of the local atomic environments can be obtained
from the common-neighbor analysis introduced in reference [38] and described in some detail in refer-
ence [39]. In figure 4, we display the most abundant pairs, averaged over the ten inherent configurations
Figure 4. (Color online) Evolution of abundances of the most important pairs for liquid Cu with tempera-
ture. The solid symbols are the results of LDA and the open ones are those of GGA. Error bars are typically
of the order of 0.01.
43603-5
N. Jakse, A. Pasturel
for each temperature. For both LDA and GGA calculations, 142x (sum of 1422 and 1421), 1431 and 15xx
(sum of 1551 and 1541) pairs are found to be the main ones. We can also put forward two important
remarks. 15xx pairs are the most abundant ones and they are the only pairs that display a linear increase
as the temperature decreases. Let us mention that our results are in agreement with the previous ab
initio based analysis [40, 41]. The 15xx bonded pairs characterize the icosahedral order [42–44] while
the 142x bonded pairs are related to close packed structures like the face-centered cubic and hexago-
nal close-packed structures [39]. The 1431 pairs can be considered as distorted icosahedra or distorted
close-packed structures [45]. However, they cannot be responsible for the differences between dynamic
properties using either GGA or LDA since they are similar in both approximations. The same comment
holds for the 142x pairs. On the contrary, a significant increase of the number of 15xx pairs using the
GGA approximation is a strong indication that the icosahedral short range order is more pronounced
when using the GGA approximation.
To study the influence of ISRO on the dynamic properties, we inspect the velocity auto-correlation
function from which the self-diffusion coefficient is obtained. In figure 5 we report this function com-
puted at T = 1398 K using the LDA and GGA approximations. The function decays quickly first and gets
to a minimum value after nearly 0.1 ps, which is known as the cage effect due to the temporary trapping
of atoms by their neighbours. We can note that this first minimum becomes deeper using GGA, which
reveals an increase of the cage effect. This minimum is also less pronounced when the temperature in-
creases, as shown in the inset for LDA (a similar behaviour is seen for the GGA). This behaviour can
be related to the increase of the icosahedral motifs using GGA as compared to LDA for the same tem-
perature or when the temperature decreases as seen in figure 5. Indeed, as discussed in reference [41],
the backscattering regime, that is predominant for liquid Cu, is more pronounced in the phase which
presents the highest ISRO and consequently the diffusivity is smaller.
Figure 5. (Color online) Velocity auto-correlation function obtained for liquid Cu at 1398 Kwith LDA (solid
lines) and GGA (dashed lines). Inset: Comparison in the vicinity of the main minimum with the LDA for
1398 K, 1600 K and 1800 K.
We now discuss the universal scaling relationship proposed by Dzugutov [11], that relates a dimen-
sionless form of the diffusion coefficient,D∗
to the excess entropy of the liquid phase, SE. The relationship
can be written as:
D∗ = 0.049exp(SE), (3.1)
where D∗
is reduced with respect to the diffusion coefficient D using uncorrelated binary collisions de-
scribed by the Enskog theory as:
D∗ = DΓ−1σ−2. (3.2)
In equation (3.2), Γ is the Enskog collision frequency which can be calculated for the temperature T and
43603-6
Stokes-Einstein relation and excess entropy scaling law in liquid Copper
Figure 6. (Color online) The scaled diffusion coefficient of Cu and Al (a) vs the S2 approximation using ab
initio calculations, (b) vs the excess entropy calculated by the Carnahan Starling (CS) equation with the
simple choice for σ and g (σ) (see text) as well as (c) with self-consistent values for σ and g (σ).
the number density ρ within the framework of the hard-sphere fluid as:
Γ= 4σ2g (σ)ρ
(
πkBT
m
)1/2
, (3.3)
wherem and σ are the atomic mass and the hard-sphere diameter. Parameter g (σ) is the value of the HS
pair-correlation function g (r ) at the contact distance. In a first step, we study the scaling law assuming
Dzugutov’s idea that the excess entropy can be represented by the two-body correlation approximation
denoted by:
S2 =−2πρ
∞∫
0
[
g (r ) ln g (r )− g (r )+1
]
r 2dr. (3.4)
Note that σ and g (σ) are approximated by the position and height of the first peak in the ab initio com-
puted pair-correlation function. Results for liquid Cu and previous ones for Al [23] are displayed in fig-
ure 6 (a) and show that the S2 approximation is also not sufficiently accurate when utilizing either LDA
or GGA calculations. It is worth mentioning that the truncation errors in equation (3.4), due to the lim-
ited range of g (r ) calculated by AIMD, do not exceed 2%, and, therefore, do not affect the results for
Dzugutov’s law.
To go beyond the two-body correlation approximation, we use a more reliable determination of SE
but still keeping the framework of the HS reference fluid. It is based on the Carnahan-Starling equation
which is known to give a quasi-exact equation of states for HS fluids in a range of values of packing
fraction characteristic of melts [46]. In this case, the excess entropy can be written as:
SE = ξ(3ξ−4)
(1−ξ)2 − 2(2−ξ)
(1−ξ)3 T
(
∂ξ
∂T
)
V
, (3.5)
43603-7
N. Jakse, A. Pasturel
where the packing density is given by ξ=πρσ3/6. All entropy calculations can be done using the packing
fraction ξ and its evolution with temperature. To determine ξ, we use the experimental densities [4] and
σ values obtained previously. Note that the temperature dependence of ξ is mainly related to that of the
density, as σ does not vary in the investigated temperature range. From figure 6 (a), one can see that the
agreement with the original fit of Dzugutov is better since the scatter in data becomes smaller.
To explain the remaining discrepancy, we turn back to the determination of σ and g (σ). It is well
known from perturbation theories [4] that the effective HS diameter adjusted to represent the structure
and thermodynamics of a real system is different from the simple choice for σ and g (σ)made above. In
particular, this method leads to a lack of thermodynamic consistency between the HS reference fluid and
the real one. To evidence this failure, we compare in table 1 the isothermal compressibility calculated
by the Carnahan Starling expression, i.e., ρkBTχT = (1− ξ)4/[2ξ(4− ξ)+ (1− ξ)4] (values in parenthe-
sis) with that obtained from the extrapolation of S(q) at q = 0, S(q) being the structure factor obtained
from ab initio calculations. Table 1 indicates for both Al and Cu that the thermodynamic consistency, i.e.
ρkBTχT = S(0), via calculations of the isothermal compressibility is not respected. We present only LDA
calculations for comparison as we obtain the same trend for GGA based results.
Table 1. Values of the packing fraction ξ and isothermal compressibility χT as a function of temperature
T for Cu and Al (reference [23]) obtained from the self-consistent procedure using the integral equation
method (LDA calculations). The values in parenthesis correspond to the simple choice forσ and g (σ). The
last column corresponds to ab initio values (see text). The typical error bars on the AIMD results are 0.005.
T (K) ξ=πρσ3/6 (1−ξ)4
2ξ(4−ξ)+(1−ξ)4 ρkBTχT = S(0)
Cu
1398 0.461 (0.565) 0.027 (0.0090) 0.030
1600 0.434 (0.545) 0.034 (0.0112) 0.035
1800 0.416 (0.534) 0.039 (0.0125) 0.041
Al
875 0.466 (0.563) 0.024 (0.0093) 0.029
1000 0.446 (0.553) 0.029 (0.0103) 0.031
1125 0.431 (0.547) 0.033 (0.0109) 0.034
1250 0.415 (0.538) 0.038 (0.0119) 0.040
Therefore, to enforce the thermodynamic consistency between the HS fluid and the liquid described
from AIMD, we impose that the pair-correlation entropy of the hard sphere model SHS2 , is equal to the
pair-correlation entropy provided by LDA or GGA calculations, respectively SLDA2 and SGGA2 . To obtain
SHS2 , we compute the pair-correlation function of the hard sphere model, gHS(r ), using accurate integral
equation method [47]. This method is based on the exact Ornstein-Zernike convolution equation
h(r ) = c(r )+ρ
∫
h(r ′)c(|r− r′|)dr′ = c(r )+γ(r ), (3.6)
where h(r ), c(r ) and γ(r ) are respectively the total, direct and indirect correlation functions between two
atoms, separated by a distance r in a liquid composed of N atoms with number density ρ [4]. In order to
determine the pair-correlation function g (r ) = h(r )+1, equation (3.6) should be solved together with a
closure relation whose formal expression is as follows:
h(r ) = exp
[−βuHS(r )+γ(r )+B(r )
]−1, (3.7)
In equation (3.7), B(r ) is the so-called bridge function composed of an infinite series of elementary dia-
grams [4], and uHS(r ) is the hard-sphere potential. For B(r )we use the efficient approximate formulation
for the hard-sphere model proposed by Rodgers and Young [48]:
B(r ) =−γ(r )+ ln
[
1+ Aγ(r )−1
A
]
, (3.8)
which ensures an accurate description of thermodynamic properties of the HS fluid through the opti-
mization of parameter A [49]. The set of equations (3.6), (3.7), and (3.8) are solved numerically using the
43603-8
Stokes-Einstein relation and excess entropy scaling law in liquid Copper
algorithm of Labik et al. [50], which combines the Newton-Raphson method and the traditional iterative
technique. Following the authors of reference [51, 52], we use the tangent linear differentiation tech-
nique that is an essential ingredient for improving the accuracy of the integral equation method and for
computing thermodynamic functions of the HS fluid that involve the derivative of g (r ).
From table 1, we can note that our approach leads to similar values of the isothermal compressibility
calculated using either the Carnahan Starling expression, i.e. ρkBTχT = (1−ξ)4/[2ξ(4−ξ)+(1−ξ)4], or the
extrapolation of S(q) at q = 0, i.e. ρkBTχT = S(0), S(q) being obtained from LDA. This close correspon-
dence is a strong indication of the thermodynamic consistency of our approach to determine the excess
entropy.
Taking advantage of this simple analytical expression of the excess entropy, we plot in figure 6 (c)
the relationship between the excess entropy and the dimensionless form of the diffusion coefficient, D∗
for both Cu and Al data. It is found that the scaling law proposed by Dzugutov is legitimate with the
excess entropy derived from the Carnahan-Starling approach. Let us mention that this approach does not
require any adjustable parameter.
4. Conclusion
In summary, we have computed the temperature dependence of the self-diffusion coefficient as well
the viscosity of liquid copper by ab initiomolecular dynamics using two different exchange and correla-
tion potentials, LDA and GGA. The comparison of the calculated self-diffusion coefficients with the most
recent QNS experimental data and the comparison of the calculated viscosities with the assessed values
favor the LDA approximation to compute the dynamic properties of liquid copper. We show that the ori-
gin of the discrepancy using GGA is due to an enhancement of the icosahedral short range order (ISRO)
that directly impact the dynamic properties of liquid copper. Our results allow us to discuss the applica-
bility of the SE relation to the description of the relation between diffusivity and viscosity and to the firm
establishment of its validity for liquid copper over the temperature range investigated. Finally, we show
that the proposed scaling law of Dzugutov which relates the scaled diffusivity to the excess entropy can
be successfully applied if a self-consistent method for determining the pair-correlation function of the HS
fluid is used as well as the Carnahan Straling approach for expressing the excess energy is introduced.
Note that the excess entropy based on the Carnahan Starling equation is given by an analytical expres-
sion and does not contain any adjustable parameter. How this formulation can be applied to liquids such
as silicon, which presents loosely packed structures, is an important open question to be addressed in a
future work.
Acknowledgements
We acknowledge the CINES and IDRIS under Project Number INP2227/72914 as well as PHYNUM CI-
MENT for computational resources. This work was performed within the framework of the Centre of
Excellence of Multifunctional Architectured Materials “CEMAM” number ANR-10-LABX-44-01 funded by
the “Investments for the Future” Program.
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Stokes-Einstein relation and excess entropy scaling law in liquid Copper
Спiввiдношення Стокса-Ейнштейна i скейлiнговий закон
для надлишкової ентропiї у рiдкiй мiдi
Н. Жакс, А. Пастурель
Наука i технiка матерiалiв i процесiв, UMR CNRS 5266, Альпiйський унiверситет, Гренобль, Францiя
У цiй статтi представлено ab initio дослiдження структурних i динамiчних властивостей рiдкої мiдi як фун-
кцiї температури. Зокрема, проведено розрахунок температурної залежностi коефiцiєнта самодифузiї з
автокореляцiйної функцiї швидкостей, а також температурної залежностi швидкостi з поперечної коре-
ляцiйної функцiї струму. Показано, що результати, базованi на наближеннi локальної густини, добре
узгоджуються з експериментальними даними як для коефiцiєнта дифузiї, так i для в’язкостi в межах до-
слiджуваного дiапазона температур. Цi результати використанi для перевiрки емпiричних пiдходiв типу
спiввiдношення Стокса-Ейнштейна та скейлiнгового закону для надлишкової ентропiї, якi широко вико-
ристовуються в лiтературi. Показано, що спiввiдношення Стокса-Ейнштейна задовiльняється для рiдкої
фази i скейлiнговий закон для надлишкової ентропiї, запропонований Дзугутовим, є легiтимним лише за
умови використання самоузгодженого методу для визначення фракцiї упакування твердосферної рiдини,
коли пiдхiд Карнагана-Старлiнга використано для отримання виразу надлишкової ентропiї.
Ключовi слова: рiдка мiдь, спiввiдношення Стокса-Ейнштейна, унiверсальнi скейлiнговi закони, ab initio
молекулярна динамiка
43603-11
Introduction
Simulation details
Results and discussion
Self-diffusion coefficient
Viscosity
Discussion
Conclusion
|