# Transport properties in liquids from first principles: the case of liquid
water and liquid Argon

Pier Luigi Silvestrelli  Dipartimento di Fisica e Astronomia “G. Galilei”,
Università di Padova, via Marzolo 8, I-35131 Padova, Italy

###### Abstract

Shear and bulk viscosity of liquid water and Argon are evaluated from first
principles in the Density Functional Theory (DFT) framework, by performing
Molecular Dynamics simulations in the NVE ensemble and using the Kubo-
Greenwood equilibrium approach. Standard DFT functional is corrected in such a
way to allow for a reasonable description of van der Waals (vdW) effects. For
liquid Argon the thermal conductivity has been also calculated. Concerning
liquid water, to our knowledge this is the first estimate of the bulk
viscosity and of the shear-viscosity/bulk-viscosity ratio from first
principles. By analyzing our results we can conclude that our first-principles
simulations, performed at a nominal average temperature of 366 K to guarantee
that the systems is liquid-like, actually describe the basic dynamical
properties of liquid water at about 330 K. In comparison with liquid water,
the normal, monatomic liquid Ar is characterized by a much smaller bulk-
viscosity/shear-viscosity ratio (close to unity) and this feature is well
reproduced by our first-principles approach which predicts a value of the
ratio in better agreement with experimental reference data than that obtained
using the empirical Lennard-Jones potential. The computed thermal conductivity
of liquid Argon is also in good agreement with the experimental value.

##  I Introduction

Transport properties are among the most important and useful features of
condensed-matter systems, particularly for characterizing the dynamical
behavior of liquids, since they play an important role in many technical and
natural processes. Therefore their estimate represents one of the most
relevant goal of Molecular Dynamics (MD) simulation techniques which become
particularly useful in cases where experimental data are not available or
difficult to obtain. Different theoretical approaches can be adopted with a
varying degree of accuracy (see, for instance, refs. Allen, ; Helfand, ;
Alder, ; Gosling, ; Ciccotti, ; Ciccotti78, ; Ciccotti79, ; Schoen, ;
Hoheisel, ; Erpenbeck, ; Balasubramanian, ; Alfe, ; Guo, ; Viscardy, ; Jones,
; Lishchuk, ; Kim, ; Kirova, ; Shi, ; Chen, ; Rabani, ; Kusudo, ; Torres, ;
Vogelsang, ; Fan, ; Kang, ; French, ; Fernandez, ; Marcolongo, ; Ercole, ;
Marcolongo21, ; Tisi, ; Malosso, ; Ercole17, ; Ercole22, , and further
references therein).

Basically, in MD simulation transport properties can be evaluated either
through a genuine nonequilibrium approach by applying an explicit external
perturbation (such as a shear flow or a temperature gradient), which is
clearly direct and intuitive but is affected by non-trivial technical issues
(in particular the need to generate nonequilibrium steady states in typical
systems characterized by finite-size supercells with periodic boundary
conditions and to extrapolate to the limit of zero driving force).
Alternatively, the transport coefficients can be more easily estimated from
equilibrium MD simulations by using the Green-Kubo relationsGreen ; Kubo ;
McQuarrie  of statistical mechanics (dissipation-fluctuation theorem) which
allow the calculation of transport coefficients by integration of suitable
autocorrelation functions. This latter approach is simpler because standard
equilibrium MD simulations can be easily carried out and estimated transport
coefficients exhibit a weaker system-size dependence.Kang  An equivalentKim
equilibrium method exploits the Einstein–Helfand expressionsHelfand  to get
transport coefficients directly from the particle displacements and
velocities;Kirova  for instance, the shear viscosity can be computed in terms
of the mean-square $x$ displacement of the center of $y$ momentum, while the
thermal conductivity is proportional to the mean square $x$ displacement of
the center of energy.

The shear viscosity describes the resistance of a fluid to shear forces and is
a measure of the shear stress induced by an applied velocity gradient,Allen
while the bulk viscosity refers to the resistance to dilatation of an
infinitesimal volume element at constant shape and measures the resistance of
a fluid to compression. It is closely connected with absorption and dispersion
of ultrasonic waves in a fluid, so it can provide valuable information about
intermolecular forces. Moreover, the role of the bulk viscosity is acquiring
more and more importance, for instance in the area of surface and interface-
related phenomena and for the interpretation of acoustic sensor data.Hafner
In spite of its relevance, bulk viscosity has received less experimental and
theoretical attention, partly due to the greater difficulties in obtaining
accurate measurements and estimates. In principle it should be evaluated in
the microcanonical (NVE) ensemble where there is no need to evaluate an
additional term which would be required if, for instance, the canonical NVT
ensemble were used.Guo ; Hafner  Moreover, bulk viscosity is subject to much
larger statistical error caused by the fact that it must be calculated by the
regression of fluctuations about a nonzero mean.Alder  While the shear
viscosity is associated with changes in water Hydrogen-bond network
connectivity and is mostly related to translational molecular motion, the bulk
viscosity is associated with local density fluctuations and reflects the
relaxation of both rotational and vibrational modes.Yahya ; Dukhin  The
thermal conductivity describes instead the capability of a substance to allow
molecular transport of energy driven by temperature gradients.

In general dynamical properties such as the transport coefficients are much
more dependent on the simulation size and timescale than structural
properties.Torres  One must also point out that shear and bulk viscosities,
and thermal conductivity are even more difficult to be evaluated accurately
than, for instance, the diffusion coefficient (a single-particle property)
since they are collective transport properties involving all the
particles.Viscardy  In fact, for estimating the diffusion coefficient one can
perform a statistical average over the particles in addition to the average
over time because every particle diffuses individually but any stress or
energy fluctuation is an event involving the system as a whole. As a
consequence, in order to obtain the same statistical accuracy, collective
properties need much longer runs than single particle properties by a factor
proportional to the size of the system.Alfe

We here estimate from first principles simulations, in the framework of the
Density Functional Theory (DFT), the shear and bulk viscosity of liquid water
and Argon. For liquid Argon the thermal conductivity is also calculated. By
analyzing our results we can conclude that our first-principles simulations,
performed at a nominal average temperature of 366 K to guarantee that the
systems is liquid-like, actually describe the basic dynamical properties of
liquid water at about 330 K. Our approach is also able to reproduce well the
bulk-viscosity/shear-viscosity ratio of liquid Ar which is much smaller than
that of liquid water.

##  II Method

We have performed first principles MD simulations of liquid water using the
CPMD package,CPMD  at constant volume, considering the experimental density of
water at room temperature. The computations were performed at the
$\Gamma$-point only of the Brillouin zone, using norm-conserving
pseudopotentialsTroullier  and a basis set of plane waves to expand the
wavefunctions with an energy cutoff of 250 Ry; we have explicitly tested that
this energy cutoff, much higher than that used in standard DFT simulations of
liquid water, is required to have a good convergence also for the stress
tensor components.

We have adopted the gradient-corrected BLYPBLYP  density functional augmented
by van der Waals (vdW) corrections, hereafter referred to as
DFT-D2(BLYP).Grimme06  This choice is motivated both by the fact that BLYP has
been shownSprik96 ; PRL ; JCP ; Boero1 ; Boero2  to give an acceptable
description of Hydrogen bonding in water, and because it represents a good
reference DFT functional to add vdW corrections.PSIL1 ; PSIL2 ; PSIL3 ;
Kannemann  A good description of Hydrogen bonding is essential here since, in
liquid water, the shear viscosity mostly originates from covalent interactions
in the Hydrogen-bond dynamics of water molecules.Shi  Moreover, vdW
corrections to BLYP are important because it was shown that BLYP significantly
underestimates (by 25%) the equilibrium density of liquid water; the
experimental density can be recovered by adding the vdW corrections proposed
by Grimme,Grimme06  which have the further effect of making the oxygen-oxygen
radial distribution function in better agreement with experiment.Schmidt ;
Wang  Our system consists of 64 water molecules contained in a supercell with
simple-cubic symmetry and periodic boundary conditions. Hydrogen nuclei have
been treated as classical particles with the mass of the deuterium isotope
which allows us to use larger time steps. The effective mass determining the
time scale of the fictitious dynamics of the electrons was 700 a.u. and the
equations of motion were integrated with a time step of 3 a.u. (=0.073 fs).

Our simulation consisted of an initial equilibration phase, lasting about 0.15
ps, in which the ionic temperature was simply controlled by velocity
rescaling, followed by a much longer (about 22 ps) canonical (NVT) MD
simulation (using suitable thermostats for a Nosé-Hoover dynamics), followed
by a final 22 ps microcanonical (NVE) production MD run. A common drawback of
most standard DFT functionals applied to liquid water at room temperature is
their tendency to ”freeze” the system which therefore exhibits an ice-like
behavior. By applying vdW corrections the problem is reduced but it still
present. In particular, since the melting temperature of water estimated by
DFT-D2(BLYP) is 360 KYoo  (while it is 411 K with BLYP), following a common
strategy,Marcolongo ; Tisi ; Malosso ; Grossman  we performed NVT simulations
with an average ionic temperature of 380 K to be sure that the system is
indeed liquid-like. This use of artificially increased temperature also serves
to mimic Nuclear Quantum Effects in simulations of liquid water.Torres  The
average ionic temperature of the subsequent NVE MD simulation was 366 K.
Several data (atomic coordinates, velocities, stress-tensor components,…)
relevant for characterizing structural and dynamical properties of the system
were recorded every 20 steps in the production stage.

As far as liquid Ar is concerned, before starting MD simulations, we have
performed extensive preliminary calculations to choose optimal parameters and
a suitable DFT functional. Clearly in this case even an empirical Lennard-
Jones potential reference could probably give reasonable results but here we
are interested in studying transport properties using DFT functionals in a
first-principle framework, which has the advantage of explicitly accounting
for the electronic structure of matter. Application to the face-centered cubic
(fcc) Ar crystal (considering a fcc supercell with 32 Ar atoms) and comparison
with experimental reference values for the equilibrium Ar-Ar distance and the
cohesive energy, suggests that, among many tested, vdW-corrected DFT
functionals, DFT-D2(PBE)Grimme06 ; PBE  is the most adequate to describe
extended systems made by Ar atoms, hence we mainly use it for the MD
simulations of liquid Ar. In this case we have checked that a suitable energy
cutoff to get a good convergence for the stress tensor components is 110 Ry.

The liquid Ar sample was prepared starting from an initial (unfavorable)
simple cubic lattice configuration with 64 Ar atoms and considering the
experimental Ar density (1.4 g/cm3) at melting point (84 K). Then the systems
was heated by gradually increasing the ionic temperature (by velocity
rescaling) to 500 K (in a time of 1.3 ps) to be sure that the system was truly
melted; then the temperature was gradually decreased (in 1.0 ps) to 150 K,
which is a temperature sufficiently higher than the experimental melting point
that it can be assumed that the system is indeed in a liquid phase; this has
been explicitly checked looking at the translational order parameter.Allen

Then a 60 ps canonical (NVT) MD simulation (with a ionic temperature of 150 K)
was performed, followed by a 60 ps microcanonical (NVE) MD production runs
with an average ionic temperature of 129 K. In this case the electronic
effective mass was 700 a.u. and the equations of motion were integrated with a
time step of 5 a.u. (=0.121 fs). Data (atomic coordinates, velocities, stress-
tensor components,…) relevant for structural and dynamical properties of the
system were recorded every 10 steps in the production stage.

As mentioned above, different approaches exist for the calculation of shear,
$\eta_{S}$, and bulk, $\eta_{B}$, viscosity from MD simulations.Allen ; Guo ;
Balasubramanian ; Erpenbeck ; Schoen ; Hoheisel  The most used technique is
based on the evaluation of the autocorrelation functions of stress-tensor
components; in particular,Allen

$\eta_{S}={V\over{k_{B}T}}\int_{0}^{\infty}dt\langle
P_{\alpha\beta}(0)P_{\alpha\beta}(t)\rangle\;,$ (1)

$\eta_{B}={V\over{9k_{B}T}}\int_{0}^{\infty}dt\langle\delta
P_{\alpha\alpha}(0)\delta
P_{\beta\beta}(t)\rangle={{V}\over{k_{B}T}}\int_{0}^{\infty}dt\langle\delta
P(0)\delta P(t)\rangle\;,$ (2)

where, in practice the upper limit of integration ($\infty$) is replaced by a
reasonably-long simulation time, $t_{max}$, $\langle...\rangle$ denotes
average over different time origins, $V$ is the system volume, $T$ the ionic
temperature, $k_{B}$ the Boltzmann constant, $P_{\alpha\beta}$ quantities
denote the components of the stress tensor, the instantaneous pressure is
given by $P(t)=1/3\sum_{\alpha}P_{\alpha\alpha}$ (that is the average of the
diagonal elements of the stress tensor), and the fluctuations are defined as:

$\delta P_{\alpha\alpha}(t)=P_{\alpha\alpha}(t)-\langle
P_{\alpha\alpha}\rangle=P_{\alpha\alpha}(t)-P\;,\delta P(t)=P(t)-\langle
P\rangle=P(t)-P\;,$ (3)

where $P$ is the system pressure obtained as the ensemble average of $P(t)$.
In isotropic fluids (with rotational invariance) there are only 5 independent
(and equivalent) components of the traceless stress tensor: $P_{xy}$,
$P_{yz}$, $P_{zx}$, $(P_{xx}-P_{yy})/2$, and $(P_{yy}-P_{zz})/2$, so that it
is convenient to compute the shear viscosity $\eta_{S}$ by averaging over
these 5 components to get better statistics.

Instead, the bulk viscosity $\eta_{B}$ has only one component, moreover the
diagonal stresses must be evaluated carefully since a non-vanishing
equilibrium average must be subtracted. In oder to get more accurate
evaluations of transport properties and also reliable estimates of the
associated statistical errors, we adopt the block-average technique,Frenkel
which consists of dividing the whole simulation into a sequence of several
shorter intervals (“blocks”), each with an equal number of samples; then block
averages are calculated which allow to estimate means and variances.Jones  In
the case of the bulk-viscosity calculation, to reduce the error, it is
convenient to take for the system pressure the average value of the pressures
over all blocks.Guo  Clearly the choice of the block size must be made with
care; in fact, samples become uncorrelated as the block size increases so for
small block sizes, the error is underestimated while for large block sizes the
error estimate is inaccurate due to insufficient sampling (see detailed
discussion below).

Typically transport coefficients are estimated from classical MD simulations
based on empirical interatomic potentials. The practical feasibility of
calculating transport coefficients in liquids using instead first principles
MD simulations, was demonstrated by D. Alfé and M. J. Gillan,Alfe  who used
the Green-Kubo relations to compute the shear viscosity of liquid iron and
aluminum, with a statistical error of about 5%. However, the simulations of D.
Alfé and M. J. GillanAlfe  were performed in the NVT ensemble, while our
simulations have been carried out using the NVE ensemble, since the NVE
simulations also allow the evaluation of the bulk viscosity without any
correction term (see above). Subsequently, the problem of atomic heat
transport and the development of a microscopic theory for the calculation of
the thermal conductivity within the Green–Kubo formalism, by a first-
principles DFT approach, has been successfully addressed by Marcolongo et
al.Marcolongo ; Ercole ; Marcolongo21  Efficient, first-principles evaluations
of thermal conductivity and shear viscosity have been also recently
proposed.Tisi ; Malosso ; Ercole17 ; Ercole22

A simpler alternative method exists (valid for temperatures that are not too
lowHerrero ) to obtain an approximate estimate of the shear viscosity, by
exploiting its connection with the self-diffusion coefficient $D$ via the
Stokes-Einstein relation:Gosling ; Alfe

$\eta_{S}={{k_{B}T}\over{2\pi aD}}\;,$ (4)

where $a$ is an effective atomic diameter. Such relation is exact for the
Brownian motion of a macroscopic particle of diameter $a$ in a liquid of shear
viscosity $\eta_{S}$, but it is only approximate when applied to atoms;
however if $a$ is chosen to be the radius $r_{1}$ of the first peak in the
radial distribution function, the relation usually predicts $\eta_{S}$ to
within 40%.Alfe  Here we take for $r_{1}$ the position of the first peak in
the O-O and Ar-Ar radial distribution function for liquid water and liquid Ar,
respectively, while the diffusion coefficient $D$ can be computedAllen  from
the mean square displacement of the oxygen atoms (for liquid water) or Ar
atoms (for liquid Ar). The validity of the Stokes–Einstein relation has been
recently discussed in detail by Herrero et al.Herrero  who also explored the
connection between structural properties and transport coefficients.

For liquid Argon the thermal conductivity has been also calculated, using the
formula:Allen

$\lambda_{T}={V\over{k_{B}T^{2}}}\int_{0}^{\infty}dt\langle
j_{\alpha}^{E}(0)j_{\alpha}^{E}(t)\rangle\;,$ (5)

where $j_{\alpha}^{E}$ is the $\alpha$ component of the energy current defined
as the time derivative of

$\delta E_{\alpha}={1\over V}\sum_{i}r_{i\alpha}(E_{i}-\langle
E_{i}\rangle)\;,$ (6)

and $E_{i}$ is the energy of the $i-th$ Ar atom (located at coordinates
$r_{ix}$, $r_{iy}$, $r_{iz}$), which can be evaluated as

$E_{i}=p_{i}^{2}/2m_{i}+1/2\sum_{j\neq i}v(r_{ij})\;,$ (7)

by assuming a pairwise interatomic potential. In order to obtain a pair
potential for evaluating the thermal conductivity of liquid Ar using
configurational data from our first-principles DFT simulations, we have
adopted a strategy similar to that proposed in ref. Kang, : we assume for the
pair potential a Lennard-Jones analytical form:

$v(r)=a(b^{2}/r^{12}-b/r^{6})\;,$ (8)

where $a$ and $b$ are parameters optimized by fitting the potential-energy
curve of the Ar dimer (at different interatomic distances) obtained by using
our DFT approach. Admittedly, such an approach is rather approximated and
could represent only a rough estimate of the thermal conductivity. A more
fundamental and accurate method has been recently proposed.Marcolongo ; Ercole
; Marcolongo21

##  III Results and Discussion

In Fig. 1 and 2 we plot the behavior of the temperature and pressure as a
function of time in the NVE simulation for liquid water and Ar, respectively.
As can be seen, these quantities turn out to be stable and exhibit only
moderate oscillations around the average values, which are, for liquid water,
0.132 GPa and 366 K for the pressure and the temperature, respectively, while
for liquid Ar the values are 0.173 GPa and 129 K.

Figure 1: Temperature and pressure of liquid water plotted as a function of
the simulation time.

Figure 2: Temperature and pressure of liquid Ar plotted as a function of the
simulation time.

Figure 3: Auto-correlation functions (ACFs) used for the evaluation of the
shear and bulk viscosities of liquid water (see text) plotted as a function of
the simulation time.

Figure 4: Auto-correlation functions (ACFs) used for the evaluation of the
shear and bulk viscosities of liquid Ar (see text) plotted as a function of
the simulation time.

In Fig. 3 and 4 we instead plot the auto-correlation functions (ACFs),
corresponding to the integrands (considering the average over the components
for the shear viscosity) of eqs. (1) and (2). Differently from what observed
in monatomic systems (such as liquid Ar) or in classical MD simulations where
waters are modeled by rigid molecules, in first-principles simulations of
liquid water, high-frequency intermolecular vibrations lead to corresponding
high-frequency oscillations in the pressure and in related ACFs. In order to
better appreciate the global decay behavior of ACFs, in the case of liquid
water, high-frequency components have been cut by Fourier-transforming the
ACFs. A quantitative estimate of the ACFs relaxation times can be obtained
assuming a global exponential decay ($\simeq e^{-t/\tau}$) of the integrands
and computing:

$\tau_{S}=\int_{0}^{\infty}dt{{\langle
P_{\alpha\beta}(0)P_{\alpha\beta}(t)\rangle}\over{\langle
P_{\alpha\beta}(0)P_{\alpha\beta}(0)\rangle}}\;,$ (9)

and

$\tau_{B}=\int_{0}^{\infty}dt{{\langle\delta P_{\alpha\alpha}(0)\delta
P_{\beta\beta}(t)\rangle}\over{\langle\delta P_{\alpha\alpha}(0)\delta
P_{\beta\beta}(0)\rangle}}$ (10)

For liquid water we find $\tau_{S}\simeq 6$ fs and $\tau_{B}\simeq 4$ fs,
while for liquid Ar $\tau_{S}\simeq 340$ fs and $\tau_{B}\simeq 410$ fs.

Figure 5: Shear and bulk viscosity of liquid water plotted as a function of
the upper limit of the integrals of the ACFs.

Figure 6: Shear and bulk viscosity of liquid Ar plotted as a function of the
upper limit of the integrals of the ACFs.

Figure 7: Thermal conductivity of liquid Ar plotted as a function of the upper
limit of the integral of the ACF.

The shear and bulk viscosity, computed using eqs. (1) and (2), are plotted as
a function of the upper limit of the integrals in Fig. 5 and 6, while the
thermal conductivity of liquid Ar is reported in Fig. 7. From these curves an
approximate estimate of the shear and bulk viscosity can be obtained
considering the values of the quantities corresponding to the position of the
first pronounced maximum-plateau; in fact this indicates that the running
integral starts becoming nearly independent of time implying that the
corresponding ACF has decayed to zero and is fluctuating along the horizontal
time axis. Clearly, considering longer times only introduces additional noise
to the signal and the beginning of a plateau represents the desired value of
the viscosity with the smallest uncertainty. As can be seen, the maximum-
plateau is reached at about $t=0.8$ ps for both the shear and bulk viscosity
of liquid water, while the corresponding values for liquid Ar are 3.0, 5.0 ps,
and 0.5 ps for the shear viscosity, the bulk viscosity, and the thermal
conductivity, respectively. As expected, these times are much larger than the
corresponding relaxation times $\tau_{S}$ and $\tau_{B}$ estimated above.

As already discussed, a more accurate evaluation, with also a reliable
estimate of the associated statistical error, can be obtained by adopting a
block-average technique. In this case a proper choice of the block size is
crucial: with many, small-size blocks, the statistical error is small but the
blocks are probably correlated and the viscosity is typically underestimated
(not yet converged); on the contrary, with just a few, large-size blocks,
these are probably uncorrelated and the viscosity is converged but the
statistical error is large.

In Figs. 8, 9, 10, and 11 we plot the values of the shear and bulk viscosity
of liquid water and Ar evaluated by using different numbers of blocks (keeping
constant the total number of configurations) with the relative statistical
errors. The dashed horizontal lines indicate the corresponding values inferred
by considering the maximas-plateaus of the curves in Figs. 5 and 6. As can be
seen, in the case of liquid water, the maxima of the shear and bulk
viscosities are obtained considering 16 blocks, each equivalent to a
simulation time of about 1.4 ps. Interestingly, taking statistical
uncertainties into account, these maxima are compatible with the rough
estimates obtained before and, for the shear viscosity, also with the values
obtained using the Stokes-Einstein formula (Eq. (4)). As already described
above, in the Stokes-Einstein estimate the shear viscosity is obtained in
terms of the diffusion coefficient $D$ and the radius of the first peak in the
radial distribution function (see Eq. (4), for liquid water we have considered
the first peak in the oxygen-oxygen radial distribution function, see below).
Actually our reported Stokes-Einstein estimated values are corrected by
finite-size effects: in fact $D$ can be extrapolated to infinite size of the
simulation box (see, for instance, ref. Herrero, ) by just considering the
shear-viscosity value:

$D_{\infty}=D+2.837{{k_{B}T}\over{6\pi\eta_{S}L}}\;,$ (11)

where $L$ is the size of the cubic simulation box. Therefore, by
simultaneously taking into account Eqs. (4) and (11), one can get a “self-
consistent”, finite-size corrected Stokes-Einstein estimate for $\eta_{S}$ :

$\eta_{S}^{*}={{k_{B}T}\over{2\pi aD}}-2.837{{k_{B}T}\over{6\pi LD}}\;.$ (12)

Figure 8: Shear viscosity of liquid water evaluated by using different numbers
of blocks (the smaller is the block number the larger is the number of
configurations of each block) with the relative statistical errors. The dashed
horizontal line indicates the position of the first-pronounced maximum-plateau
of the corresponding curve of Fig. 5. The asterisk denotes the value obtained
by the Stokes-Einstein formula (Eq.12), while the triangles indicate
experimental estimates at different temperatures.

Figure 9: Bulk viscosity of liquid water evaluated by using different numbers
of blocks (the smaller is the block number the larger is the number of
configurations of each block) with the relative statistical errors. The dashed
horizontal line indicates the position of the first-pronounced maximum-plateau
of the corresponding curve of Fig. 5.

Quantitative data are collected in Table I where they are also compared with
some theoretical and experimental reference values.

Table 1: Shear and bulk viscosity of liquid water and Ar, in $10^{-4}$ Pa s,
compared with theoretical and experimental reference data. Statistical errors
are in parenthesis. $\eta_{S}^{*}$ indicates the shear viscosity estimate
obtained by the Stokes-Einstein relation (see text).

system | $\eta_{S}$ | $\eta_{S}^{*}$ | $\eta_{B}$ | $\eta_{B}/\eta_{S}$ | $3/4\eta_{B}/\eta_{S}+1$
---|---|---|---|---|---
water (366 K) | 4.8(0.7) | 5.7 | 11.3(2.9) | 2.4(0.8) | 2.8(0.6)
water DFT SCANa (300K) | 23 | — | — | — | —
water DFT SCANa (330K) | 6 | — | — | — | —
water DFT SCANa (360K) | 5 | — | — | — | —
water DFT OPTB88-vdWa (300K) | 30 | — | — | — | —
water DFT OPTB88-vdWa (330K) | 15 | — | — | — | —
water DFT OPTB88-vdWa (360K) | 8 | — | — | — | —
water force fielda (300K) | 8 | — | — | — | —
water force fielda (330K) | 5 | — | — | — | —
water force fielda (360K) | 3.5 | — | — | — | —
water force fieldb (303K) | 6.5(0.4) | — | 15.5(1.6) | 2.4(0.3) | 2.8(0.2)
water expt.c (298 K) | 8.90 | — | — | — | —
water expt.b,d,e (303 K) | 7.97 | — | 21.5 | 2.7 | 3.0
water expt.f (323 K) | 5.47 | — | 14.8 | 2.7 | 3.0
water expt.c (333 K) | 4.67 | — | — | — | —
Ar (129 K) | 3.7(1.6) | 2.0 | 4.0(2.2) | 1.1(0.8) | 1.8(0.6)
Ar expt.g (90 K) | 2.33 | — | 1.82 | 0.8 | 1.6
Ar expt.h (90 K) | 2.57 | — | — | — |

11footnotetext: ref.Herrero, .22footnotetext: ref.Guo, .

33footnotetext: ref.Harris, .44footnotetext: ref.Litovitz, .

55footnotetext: ref.Sengers, .66footnotetext: ref.Holmes, .

77footnotetext: ref.Cowan, .88footnotetext: ref.NIST, .

As far as the shear viscosity is concerned, for liquid water our estimated
value, obtained from the NVE simulation at an average temperature of 366 K,
agrees with the experimental reference data at a lower temperature of about
330 K. This is in line with the performances of other DFT functionals; for
instance (see Table I), in recent simulationsHerrero  of liquid water based on
the SCAN functional,SCAN  the shear viscosity estimate is close to that
obtained from a force-field approach (that, for this quantity, well reproduces
the experimental behavior) only between 330 and 360 K, while it is severely
overestimated at 300 K. A similar behavior has been found for the SCAN
functional by Malosso et al.,Malosso  who computed the shear viscosity of
liquid water from first-principles and deep-neural-network simulations. With
the OPTB88-vdW functionalOPTB88-vdW  reasonable agreement with experimental
data at room temperature is only foundHerrero  at 360 K, while with the PBE
functionalPBE  this is only true at 454 K.Malosso

Figure 10: Shear viscosity of liquid Ar evaluated by using different numbers
of blocks (the smaller is the block number the larger is the number of
configurations of each block) with the relative statistical errors. The dashed
horizontal line indicates the position of the first-pronounced maximum-plateau
of the corresponding curve of Fig. 6. The asterisk denotes the value obtained
by the Stokes-Einstein formula (Eq.4), while the triangle indicates the
experimental estimate at 90 K.

Figure 11: Bulk viscosity of liquid Ar evaluated by using different numbers of
blocks (the smaller is the block number the larger is the number of
configurations of each block) with the relative statistical errors. The dashed
horizontal line indicates the position of the first-pronounced maximum-plateau
of the corresponding curve of Fig. 6.

Figure 12: Thermal conductivity of liquid Ar evaluated by using different
numbers of blocks (the smaller is the block number the larger is the number of
configurations of each block) with the relative statistical errors. The dashed
horizontal line indicates the position of the maximum-plateau of the
corresponding curve in Fig. 7.

For liquid Ar the behavior is qualitatively similar (see Figs. 10, 11, and 12
for the shear viscosity, the bulk viscosity, and the thermal conductivity,
respectively). In this case both the maxima of the shear and bulk viscosities
are obtained considering 5 blocks, each equivalent to a simulation time of
about 12.0 ps. Even in this case, taking statistical uncertainties into
account, these maxima are compatible with the plateau positions and, for the
shear viscosity, also with the estimate from the Stokes-Einstein formula. The
maximum of the thermal conductivity is instead reached with 25 blocks, each
equivalent to a simulation time of about 2.4 ps, and its value ($0.11\pm 0.02$
W/m K) is again compatible with that estimated considering the maximum-plateau
position and in good agreement with the literature reference value (0.12 W/m
K) at 90 KYounglove  and that obtained by classical MD simulations based on
the Lennard-Jones potential (0.119 W/m K).Hoheisel87  Accurate first-
principles calculations of the thermal conductivity of liquid Ar have been
carried out, at higher temperatures (250 and 400 K), by Marcolongo et
al.Marcolongo , while the thermal conductivity of liquid water (adopting
different DFT functionals) has been evaluated by Tisi et al.Tisi  also using a
deep neural network trained on DFT data.

An interesting physical quantity is represented by the ratio between bulk and
shear viscosity, which can be related to the ratio of observed to classical
absorption coefficients in ultrasonic absorption experiments.Guo  In fact,
under the condition that the heat conductivity contribution to the ultrasonic
absorption may be neglected,

${{\alpha}\over{\alpha_{class}}}=3/4{{\eta_{B}}\over{\eta_{S}}}+1\;,$ (13)

and water belongs to the group of the so-called ”associated liquids”,
characterized by a ratio from 1 to 3, where structural relaxation is dominant.

Classical MD simulations based on the SPC/E semiempirical potential predictGuo
a ${\eta_{B}}\over{\eta_{S}}$ ratio of 2.4, leading to a
${\alpha}\over{\alpha_{class}}$ ratio of 2.79, in reasonable agreement with
the experimental value of 3.0.Litovitz  Instead normal liquids, such as
monatomic liquids (for instance liquid Ar) are characterized by a ratio no
greater than 1.2. Although in general the ratio varies with temperature and
pressure, in liquid water it is found to remain constant within 20% in the
temperature range 0-90 C (273-363 K).Davis  By taking statistical errors into
account, our estimated value of the ${\alpha}\over{\alpha_{class}}$ ratio
($2.8\pm 0.6$) is compatible with the available experimental data at ambient
temperature (3.0). This is a remarkable result, considering that most of the
reported classical MD simulationsGuo  predict a bulk viscosity lower than the
the experimental one, leading to an underestimated value of the
${\alpha}\over{\alpha_{class}}$ ratio.

One should also point out that a proper comparison with experimental data
requires a careful analysis taking into account the pronounced temperature
dependence of shear and bulk viscosity. In fact, according to a common
empirical model,Reynolds ; Holmes  the viscosity strongly decreases with
increasing temperature following an exponential decay. By fitting experimental
dataHolmes  with an exponential function and taking statistical errors into
account, our estimated values of the shear and bulk viscosity of liquid water
are compatible with experimental data in the temperature range of 323-344 K.
One should also consider that also the bulk-viscosity/shear-viscosity ratio
for liquid water tends to decrease slightly with temperature,Holmes
suggesting an even better agreement between our estimated value and the
experimental data.Holmes  We remind that our simulations have been carried out
at temperatures higher than ambient temperature to guarantee that the systems
is liquid-like. By considering that our estimate (after finite-size
correction) for the diffusion coefficient, $D=5.02\times 10^{-5}$ cm2/s,
corresponds to the experimental value measured at about 336 K,diffcoeff  we
can conclude that, our DFT simulations based on the DFT-D2(BLYP) functional
and performed at a nominal average temperature of 366 K, actually describe the
basic dynamical properties of liquid water at about 330 K. One should also
mention that bulk-viscosity measurements are indirect and affected by
considerable errors.Guo ; Fernandez ; Holmes ; Dukhin ; L-D ; Xu  In summary,
we can conclude that our adopted BLYP-D2 functional is able to describe
reasonably well the density fluctuations of liquid water; the discrepancy with
respect to experimental data at ambient conditions can be to a large extend
explained in terms of the pronounced temperature dependence of both shear and
bulk viscosity and the need to perform first-principles MD simulations at
temperatures higher than ambient temperature, in line with the conclusions of
other investigations.Malosso ; Herrero

As far as liquid Ar is concerned, our shear and bulk viscosities, computed by
first-principles at a nominal average simulation temperature of 129 K, turn
out to be somehow overestimated with respect to the reference experimental
values at 90 K, although they are compatible with them if statistical errors
are taken into account. Moreover our bulk-viscosity/shear-viscosity ratio
(close to unity) agrees well with the reference estimate, while interestingly
this is not the case if a standard Lennard-Jones empirical potential is
adopted using classical MD simulations that predict instead a very low
valueGuo ; Hoheisel87  of the ratio (0.17-0.35 at high densities), thus
showing that this popular potential cannot properly reproduce all the
dynamical properties of liquid Ar and underlining once again the superiority
of first-principles approaches.

We conclude our study by reporting some basic structural properties of our
investigated systems. In particular, in Fig. 13, for liquid water we plot our
computed O-O pair correlation function, $g_{OO}(r)$, compared with that
obtained experimental from X-ray diffraction measurements at ambient
conditions.Skinner ; Skinnerbis ; Daru  The main features of the $g_{OO}(r)$
curves are reported in Table II. As can be seen, there is a good agreement
between the two curves; the fact the oscillations of our computed curve are
slightly reduced with respect to the experimental one can again be related to
the higher effective temperature of our simulation.

Figure 13: O-O pair correlation function, $g_{OO}(r)$, compared with that
obtained experimentally from X-ray diffraction measurements at ambient
conditions.Skinner ; Skinnerbis ; Daru

Figure 14: Ar-Ar pair correlation function, $g(r)$, compared with that
obtained experimentally from neutron-scattering measurements at 85K.Yarnell

Table 2: Main features of the O-O pair correlation function, $g_{OO}(r)$, of
liquid water and of the Ar-Ar pair correlation function, $g(r)$ of liquid Ar
compared with experimental reference data, obtained from X-ray diffraction
measurements at ambient conditions for liquid water and neutron-scattering
measurements for liquid Ar. $r_{max}$ and $r_{min}$ indicate the position of
the first maximum (the main peak) and the first minimum of $g_{OO}(r)$ and
$g(r)$, respectively, and $g_{max}$ and $g_{min}$ the corresponding values of
the $g_{OO}(r)$ and $g(r)$ functions.

system |  $r_{max}$(Å) | $g_{max}$ |  $r_{min}$(Å) | $g_{min}$
---|---|---|---|---
water (366 K) | 2.79 | 2.42 | 3.66 | 0.88
water expt.a (293 K) | 2.80(1) | 2.55(5) | 3.41(4) | 0.85(2)
Ar (129 K) | 3.70 | 2.80 | 5.29 | 0.64
Ar expt.b (85 K) | 3.68 | 3.05 | 5.18 | 0.56

11footnotetext: ref.Skinner, ; Skinnerbis, ; Daru, .22footnotetext:
ref.Yarnell, .

In Fig. 14, for liquid Ar our computed Ar-Ar pair correlation function,
$g(r)$, is compared with that obtained experimentally from neutron-scattering
measurements at 85 K,Yarnell  while again the main features of the $g(r)$
curves are reported in Table II. Even in this case there is a reasonable
agreement between the simulation and experimental curve, by considering that
simulations for liquid Ar have been performed at significantly higher
temperature (129 K) than experiments (85 K) (note that the experimental
melting and boiling points of Ar are at 84 and 87 K, respectively). After
applying the same finite-size correction adopted above for liquid water, our
estimated diffusion coefficient for liquid Ar, $D=3.82\times 10^{-5}$ cm2/s,
evaluated at a nominal simulation temperature of 129 K is significantly higher
than the reference value ($1.6\times 10^{-5}$ cm2/s) reported at 84 K.Hansen
Again this discrepancy can be explained in terms of the higher temperature of
the liquid Ar simulation.

##  IV Conclusions

Shear and bulk viscosity of liquid water and Argon have been evaluated,
together with other structural and dynamical properties, from first principles
by adopting a vdW-corrected DFT approach, by performing Molecular Dynamics
simulations in the NVE ensemble and using the Kubo-Greenwood equilibrium
approach. For liquid Argon the thermal conductivity has been also calculated.
Concerning liquid water, to our knowledge this is the first estimate of the
bulk viscosity and of the shear-viscosity/bulk-viscosity ratio from first
principles. By analyzing our results and comparing then with reference
experimental data, we can conclude that our first-principles simulations,
performed at a nominal average temperature of 366 K to guarantee that the
systems is liquid-like, actually describe well the basic dynamical properties
of liquid water at about 330 K. In comparison with liquid water, the normal,
monatomic liquid Ar is characterized by a much smaller bulk-viscosity/shear-
viscosity ratio (close to unity) and this feature is well reproduced by our
first-principles approach which predicts a value of the ratio in better
agreement with experimental reference data than that obtained using the
empirical Lennard-Jones potential. The computed thermal conductivity of liquid
Argon is also in good agreement with the experimental value.

##  V Acknowledgements

We acknowledge S. Baroni for useful disucssions and funding from Fondazione
Cariparo, Progetti di Eccellenza 2017, relative to the project: ”Engineering
van der Waals Interactions: Innovative paradigm for the control of Nanoscale
Phenomena”.

##  VI Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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Heidelberg, Germany 53Instituto de Radioastronomía y Astrofísica (IRyA),
Universidad Nacional Autónoma de México (UNAM), Antigua Carretera a Pátzcuaro,
8701, Ex-Hda. San José de la Huerta, Morelia, Michoacán, 58089, México 54INAF-
Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, I-50125 Firenze,
Italy 55National Radio Astronomy Observatory, Socorro, NM, 87801, USA
56University of Milan Bicocca, Piazza della Scienza 3, Milan, Italy
57Astrophysics Department, Instituto Nacional de Astrofísica, Óptica y
Electrónica, Luis E. Erro 1, Tonantzintla, Puebla, C.P. 72840, México
58National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA
22903, USA 59School of Physics and Astronomy, University of Leicester,
Leicester, LE1 7RH, UK 60Center for Solar-Terrestrial Research, New Jersey
Institute of Technology, Newark, NJ, 07102, USA 61Astronomical Institute,
Czech Academy of Sciences, 25165, Ondřejov, Czech Republic 62Department of
Astrophysics, University of Vienna, Türkenschanzstrasse 17, A-1180 Vienna,
Austria 63Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences,
Rabiańska 8, 87-100, Toruń, Poland 64Division of Geological and Planetary
Sciences, California Institute of Technology, Pasadena, California, 91125, USA
65School of Physics & Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK
66National Taiwan Normal University, Taipei City, 116, Taiwan 67Kavli
Institute for Particle Astrophysics & Cosmology (KIPAC), Stanford University,
Stanford, CA 94305, USA 68Aix Marseille Univ, CNRS, CNES, LAM, Marseille,
France 69NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 70The
Catholic University of America, Washington, DC 20064, USA 71LESIA,
Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université
Paris Cité, 5 place Jules Janssen, 92195 Meudon, France 72Central Astronomical
Observatory at Pulkovo of Russian Academy of Sciences, St. Petersburg, 196140,
Russia 73Ioffe Institute, Polytekhnicheskaya, 26, St. Petersburg, 194021,
Russia 74INAF Osservatorio Astronomico di Cagliari, Via della Scienza 5, 09047
Selargius (CA), Italy 75Instituto de Estudios Astrofísicos, Facultad de
Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejército Libertador
441, Santiago, Chile 76Kavli Institute for Astronomy and Astrophysics, Peking
University, Beijing 100871, China 77School of Physics and Astronomy,
University of Leeds, Leeds LS2 9JT, UK 78Institut de Ciències de l’Espai (ICE,
CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193, Bellaterra (Barcelona),
Spain 79Institut d’Estudis Espacials de Catalunya (IEEC), 08860, Castelldefels
(Barcelona), Spain 80National Astronomical Observatory of Japan, Mitaka,
Tokyo, 181-8588, Japan 81Graduate University of Advanced Studies, SOKENDAI,
Mitaka, Tokyo 181-8588, Japan 82IAPS-INAF, Rome, I-00133, Italy 83Dipartimento
di Fisica e Astronomia ’Augusto Righi’, Università degli Studi di Bologna, Via
Gobetti 93/2, I-40129 Bologna, Italy 84Air Force Research Laboratory, Space
Vehicles Directorate, Kirtland AFB, NM 87123, USA

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