International Journal of Thermal Sciences 161 (2021) 106762

1290-0729/© 2020 The Authors. Published by Elsevier Masson SAS. This is an open access article under the CC BY license(http://creativecommons.org/licenses/by/4.0/).

Comparison of molecular heat transfer mechanisms between water and ammonia in the liquid states

Hiroki Matsubara *, Gota Kikugawa, Taku Ohara Institute of Fluid Science, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan

A R T I C L E I N F O

Keywords: Associated liquids Water ammonia Thermal conductivity Molecular dynamics simulation Hydrogen bond

A B S T R A C T

Water and ammonia are both associated liquids with high thermal conductivity, but their degrees of molecular association are characterized differently by strong and weak hydrogen bonds, respectively. Here, we employed non-equilibrium molecular dynamics simulation to clarify and compare the molecular mechanisms of high thermal conductivity of water and ammonia. The molecular-scale heat transfer was analyzed in relation to molecular configuration using the atomistic heat path analysis [J. Heat Mass Transf. 108, 749 (2017)], where a single van der Waals (vdW) interaction and a single Coulomb interaction were considered as a heat path and their contributions to heat transfer were quantified. Both water and ammonia showed that the primary factor of high thermal conductivity is a large amount of heat transfer via Coulomb interaction, which was enabled by a high density of heat paths. These heat paths for outstanding Coulomb heat transfer included not only hydrogen bonds, but also more distant Coulomb interactions. On the other hand, an important role of hydrogen bond was to form a specific coordination structure of molecules that realizes the high heat path density. Such coordination structures differed between water and ammonia because of different strengths of hydrogen bond, which in turn led to different molecular pictures of heat transfer. Thus, water and ammonia achieve high thermal conductivity using somewhat different mechanisms at the molecular scale.

1. Introduction

Heat transfer fluids of high thermal conductivity are in demand for many purposes such as thermal management of engines [1], cooling of electronic devices [2], and improving the thermal response of phase change materials [3]. The properties of heat transfer fluid can be tailored by blending different liquids or adding filler particles, but one is often forced to find appropriate liquids or fillers by trial and error. To-wards more theoretical design of heat transfer fluids on the basis of molecular-scale principles, it is essential to understand how macroscopic heat conduction is affected by microscopic details such as molecular shapes, functional groups, and inter- and intramolecular interactions. In this context, the present study focuses on water and ammonia, which have particularly high thermal conductivity among pure liquids and are in widespread use as traditional heat transfer fluids. Water and ammonia are both associated liquids, but they are differently characterized by strong and weak hydrogen bonds, respectively. Our interest here is to clarify the molecular-scale mechanisms by which these liquids exhibit high thermal conductivity, in relation to the effect of the different

strengths of hydrogen bonds. Molecular dynamics (MD) simulation is suitable to this aim. There

are a number of MD studies that investigated thermal conductivity of water [4–11] and a few such studies for ammonia [12,13]. In many of these studies, the main concern was to evaluate or compare the per-formance of different molecular models, with respect to the reproduc-ibility of experimental thermal conductivity. A few of them, on the other hand, shed light on the microscopic mechanism of heat conduction. Ohara [4] investigated the molecular-scale heat transfer in water by decomposing macroscopic heat conduction into the contributions from the intermolecular energy exchange associated with translational and rotational motions of molecules. In the study of Sirk et al. [10], the thermal energy transfer in frequency domain was discussed to pursue the cause of discrepancy in the calculated values of water thermal conductivity among different authors. In addition to these insights, a clear picture about the relationship between molecular structure (shape and functional group) and heat conduction is necessary to pave the way to the molecular design of heat transfer fluids. In order to obtain such information from MD simulation, we have developed the atomistic heat

* Corresponding author. E-mail address: matsubara@microheat.ifs.tohoku.ac.jp (H. Matsubara).

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International Journal of Thermal Sciences

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https://doi.org/10.1016/j.ijthermalsci.2020.106762 Received 31 August 2020; Received in revised form 11 November 2020; Accepted 23 November 2020

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path (AHP) analysis [14,15], which considers a single atom–atom interaction as a heat path and expresses macroscopic heat conduction as an accumulation of the microscopic heat transfer due to these heat paths. This method has been applied to alkanes [14] and alcohols [15, 16] to understand the effects of alkyl chain lengths and the addition of hydroxyl groups on the heat transfer mechanism and thermal conductivity.

In the present study, we performed non-equilibrium MD (NEMD) simulation for water and ammonia and analyzed the molecular heat transfer in these liquids using the AHP analysis. SPC/E [17] and the model by Eckl et al. [18] were chosen as molecular models for water and ammonia, respectively. In order to compare water and ammonia in an equivalent liquid state, the analysis was carried out for the saturated liquid state at the same reduced temperature T/Tc = 0.7, where Tc is the critical temperature of each liquid. This particular condition is preferred for comparison with the previous results of alkanes [14] and alcohols [15,16] performed under the same condition. However, many of exist-ing MD studies on water thermal conductivity were carried out near the standard state (298 K and 1 atm). Therefore, to make a better connection with these studies, we also performed the same analysis for water at 298 K and 1 atm. In this simulation at the standard state, the TIP5P-Ew [19] model was examined in addition to SPC/E, and the effects of different molecular models and different definitions of hydrogen bonds were discussed.

2. Method

2.1. NEMD simulation for the saturated liquids at 0.7Tc

As molecular models, we chose SPC/E [17] for water and the model developed by Eckl et al. [18] for ammonia. These models well reproduce the vapor–liquid coexistence curve, and they can describe the experi-mental thermal conductivities at the saturated liquid states relatively well [7,13]. These features are preferable for our analysis at saturated liquid states. Both are rigid models, where O atoms or N atoms interact via van der Waals (vdW) and Coulomb interactions whereas H atom interacts with Coulomb interaction only. It has been pointed out that the choice of rigid or flexible molecular models affects the vapor–liquid equilibrium properties [20]. Here, we chose rigid molecular models because it has been pointed out that thermal conductivity is significantly overestimated when the intramolecular vibration of a hydrogen atom was treated by the classical mechanics [21]. The vdW interaction was expressed by Lennard-Jones (LJ) potential with a cutoff radius of 12 Å, and Coulomb interaction was evaluated using the smooth particle mesh Ewald (SPME) method [22]. The equation of motion was solved with a 2.0 fs timestep, and RATTLE algorithm [23] was employed to constrain the bond lengths and angles.

In advance of the NEMD simulations, we calculated the vapor–liquid coexistence curve (saturated density ρ versus temperature T) to deter-mine the temperature and density conditions corresponding to the saturated liquid density at 0.7Tc for SPC/E and the Eckl model. The coexistence curve was modeled by the scaling law ρliq − ρvap = A(Tc −

T)0.32 and the law of rectilinear diameters (ρliq + ρvap)/2 = ρc + B(Tc −

T) [24], where critical temperature Tc, critical density ρc, A, and B are the fitting parameters. The vapor and liquid densities, ρvap and ρliq, respectively, were obtained for four different values of T by conducting equilibrium MD simulations of vapor–liquid coexistence system at these temperatures. The MD system for water contained 3388 molecules in the volume of 52 × 52 × 120 Å3, and that for ammonia contained 2268 molecules in the volume of 45 × 45 × 140 Å3. For both cases, a 2 ns constant temperature run by velocity scaling was followed by a 4 ns NVE run. The first 2 ns of the 4 ns NVE run was used for the relaxation to bring the system to a true equilibrium state near the target temperature. The liquid and vapor densities were obtained from the last 2 ns of the NVE run.

The fitting parameters were determined by fitting the equation of

coexistence curve to these MD results. In the simulations of vapor–liquid coexistence, the cutoff radius of LJ interaction was increased to 20 Å. This long cutoff length is necessary to accurately describe a vapor–liquid interface [25]. The coexistence curves thus obtained for water and ammonia are in good agreement with experimental ones, and the resultant critical constants are listed in Table 1. The remaining fitting parameters were determined as A = 153.83 kg/m3⋅K− 0.32 and B =0.54068 kg/m3⋅K− 1 for water, and A = 135.35 kg/m3⋅K− 0.32 and B =0.66552 kg/m3⋅K− 1 for ammonia. These critical constants also show a reasonable agreement with the existing simulation results, which are Tc = 630.4 K and ρc = 308 kg/m3 for SPC/E water [26], and Tc = 402.21 K and ρc = 228.3 kg/m3 for the Eckl model of ammonia [18].

The systems of our NEMD simulations were constructed based on the saturated liquid density at T = 0.7Tc derived from the coexistence curve. The specific values of the number of molecules, temperature, and den-sity for the water and ammonia systems are listed in Table 2. The pressure values at these states were calculated from the 4 ns NVE simulation described below as 6.7 MPa for water and 15 MPa for ammonia, which are sufficiently low not to influence the liquid prop-erties. An orthorhombic simulation box of the dimensions 45 × 45 ×300 Å3 was used with the three-dimensional periodic boundary condi-tions. The system was first brought to an equilibrium state at 0.7Tc by a 2 ns constant temperature run with velocity scaling and a subsequent 4 ns NVE run. A constant heat flux Jex = 2 × 109 W/m2 was then imposed in the z direction using the scheme of Jund and Jullien [28], where the hot slab, cold slab, and the control volumes were arranged as illustrated in Fig. 1. After a 10 ns run to establish non-equilibrium steady state, a production run was conducted for 20 ns. The value of temperature gradient in the z direction, ∇zT, was derived from linear fit to a local temperature profile inside a control volume computed by dividing the control volume into slabs of 1.0 Å thickness along the z direction. The local temperature Ts of a slab s was computed using the atom-base for-mula [29] as

Ts =1

gnskB

∑ns

i∈slab smiv2

i (1)

where mi and vi are the mass and velocity vector of atom i, respectively, ns is the number of atoms in the slab, kB is the Boltzmann constant, and g is the approximate degrees of freedom of motion per atom. In the case of a rigid body molecule composed of na atoms, g is equal to 6/na, i.e., g = 2 for water and g = 1.5 for ammonia. Thermal conductivity λ was then determined by the Fourier law

Jex = − λ∇zT (2)

We calculated λ separately for the two control volumes and averaged them. The above procedure of NEMD simulation was repeated ten times starting from different molecular configurations; i.e., the total length of production run was 200 ns. The statistical uncertainty of a physical quantity A was estimated as σ(A)/

10

√, where A is the average of A in a

single run, and σ(A) is the standard deviation of A over the ten runs.

Table 1 Critical temperature Tc and critical density ρc obtained from the vapor–liquid MD simulations in comparison with the experimental values.

Tc [K] ρc [kg/m3]

Water Our MD 662.6 315.0 expt. [27] 647.1 322 Relative error 2.4% − 2.2%

Ammonia Our MD 393.0 237.6 expt. [27] 405.4 225.0 Relative error − 3% 6%

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2.2. NEMD simulation for water at 298 K and 1 atm

While we chose the condition of 0.7Tc saturated liquids to make comparisons with different kinds of liquid at the same liquid state under the law of corresponding states, many of other MD studies have reported thermal conductivity of water near room temperature [4–6,8–10]. Therefore, as for water, we also performed NEMD simulations at the standard condition (298K and 1 atm) to make a direct comparison with existing studies. It is well known that near the standard condition, SPC/E and most other water models overestimate the experimental thermal conductivity by 30–50% [8,10]. Some authors reported, however, that the five-site models, TIP5P [30] and TIP5P-Ew [19], can reasonably reproduce it [8,11]. Considering such situation, we performed NEMD simulations at the standard condition employing TIP5P-Ew [19] in addition to that employing SPC/E model in order to examine the effect of different molecular models on the molecular mechanism of heat conduction. For these simulations, the system size was set to 35.7 ×35.7 × 238 Å3 and the length of production run was 60 ns. The specific simulation settings are listed in Table 3. The number of molecules was determined based on the density at 298 K and 1 atm for each water model, which was computed by an equilibrium MD simulation in the NpT ensemble [31]. For TIP5P model, the force on the mass-less charge

sites (M-sites) was remapped onto the real atomic sites using the method described in Ref. [32], which was necessary for using RATTLE algo-rithm. Other details were basically the same as those for the 0.7Tc saturated liquids described in Section 2.1. We note that in the compu-tation of local temperature in Eq. (1), M-sites in TIP5P-Ew are not included in a number of atoms, na. Thus, the degrees of freedom per atom is g = 2 for TIP5P-Ew as well as for SPC/E.

2.3. Analysis of molecular mechanisms of heat transfer

Our analysis is based on the microscopic representation of heat flux vector. For a system of rigid-body molecule, the z component of heat flux vector, Jz, averaged over the control volume VCV can be expressed as [33].

Jz =1

VCV

∑

I∈CVeIvI,z +

∑

XJX (3)

For convenience, we refer to the first and second terms in the right- hand side as the transport and interaction terms, respectively. The transport term represents the thermal energy transfer accompanied by the transport of molecule I having the mechanical energy eI, where eI is the sum of potential and kinetic energies and vI,z is the translational velocity of molecule I in the z direction. The interaction term explains the thermal energy transfer between different atoms via the intermo-lecular interactions, and JX is called the partial heat flux, which accounts for the contribution from a specific interaction type X. In the present case, X is vdW or Coulomb interaction and JX can be represented as [11].

JX =1

2VCV

∑

I

∑

J∕=I

∑

i∈I

∑

j∈J

(fX

ij ⋅ vi

)z*

IJ (4)

where i ∈ I means the summation runs through atom i in molecule I, fXij is

the force vector on atom i from atom j through the interaction of type X, vi the velocity vector of atom i, z*

IJ the portion of zIJ = zI − zJ contained in VCV, and zI the z coordinate of molecule I. The expression of Eq. (3) together with Eq. (4) is equal to the traditional definition of heat flux for rigid-body molecules [4,34], but some molecular variables, including angular velocity and torque, are rewritten using atomic variables for a better matching with constrained dynamics.

The partial heat fluxes due to Coulomb and vdW interactions were subdivided into the contribution from hydrogen bonding molecular pairs and that from other pairs. The geometric definition [35] was used to define hydrogen bond. For water, an intermolecular O⋯H pair was assumed to form a hydrogen bond if the O⋯H and O⋯O distances and the O⋯O–H angle were all less than threshold values. The distance threshold was taken as the 1st minimum position of the radial distri-bution function (RDF) of the relevant atom pair, while the angle threshold was set to 30◦ [35]. From the calculated RDFs (see Fig. 2), we set the distance thresholds for O⋯H and O⋯O as 2.4 and 3.8 Å, respectively. The same definition was used for ammonia except that O atom is replaced by N atom. The distance threshold for N⋯N pair was set to 5.1 Å. For N⋯H pair, it is known [36,37] that the signature of hydrogen bond appears as a shoulder rather than a clear peak as a consequence of weak hydrogen bond (see also Fig. 2(b)). Therefore, as was adopted in the MD study of Tassaing et al. [37], we approximately set the distance threshold of N⋯H pair to be 2.7 Å, which was deter-mined by Boese et al. using first-principles MD simulations [36].

In a liquid state, the interaction term in Eq. (3) is much more dominant than the transport term. To understand how the interaction term is related to the molecular-scale structures, we used the AHP analysis [14,15], which is described in the following. One can apply the Fourier law to the partial heat flux as JX = − λX∇zT to define the partial thermal conductivity λX, which represents the contribution from a spe-cific interaction type X to the total thermal conductivity [38]. Then, λX is expressed as the product of the number density of heat paths, ρX

path, and

Table 2 Number of molecules Nmol, mass density ρ, and thermal conductivity λ for our NEMD simulations on the saturated liquids of water and ammonia at tempera-ture T = 0.7Tc in comparison with the experimental values. The figure in pa-rentheses shows uncertainty of the last digit.

water ammonia

NEMD expt. [27] NEMD expt. [27]

Nmol 17074 – 13476 – T = 0.7Tc [K] 463.8 453.0 275.1 283.8 ρ [kg/m3] 840.8 887.2 627.5 623.8 λ [W/(m⋅K)] 0.745(1) 0.673 0.595(1) 0.527

Fig. 1. Illustration of our NEMD simulation system. For the systems of 0.7Tc saturated liquids of water and ammonia, L = 15.0 Å, and for the water system at 298 K, L = 11.9 Å.

Table 3 Number of molecules Nmol, mass density ρ, and thermal conductivity λ for our NEMD simulations of water at 298 K and 1 atm with the SPC/E and TIP5P models in comparison with the experimental values. The figure in parentheses shows uncertainty of the last digit.

SPC/E TIP5P expt. [27]

Nmol 10068 10139 – ρ [kg/m3] 992.6 1000.0 1000.0 λ [W/(m⋅K)] 0.897(4) 0.848(4) 0.607

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the efficiency of a single heat path, ΛX, as

λX = ρXpathΛX (5)

The path efficiency ΛX quantifies the contribution from a single pair interaction of type X to macroscopic thermal conductivity. With Eq. (5) and λ = ΣX λX, we can understand macroscopic thermal conductivity λ as the accumulation of thermal energy transfer through an interatomic interaction. A single interaction between atoms or group of atoms is considered as a heat path only if the interaction transports a non- negligible amount of thermal energy. The specific definitions for the heat paths associated with vdW and Coulomb interactions, which we shortly call the vdW and Coulomb paths, respectively, were carefully determined in the previous studies of alkanes [14] and alcohols [15]. The same definitions were used in the present study. For the case of water, a vdW path was assigned to an intermolecular O⋯O pair within the first neighbor shell radius r1st because the direct heat transfer due to vdW interaction does not go beyond the first neighbor shell [39]. The corresponding path density was obtained by counting such heat paths as

ρvdWpath =

nO(nO − 1)2V2

CV

∫ r1st

04πr2gOO(r)dr

ρOncor,1st

2(6)

where gOO(r) is the RDF, nO the average number of atoms in VCV, ρOthe number density of atoms, and ncor,1st the coordination number in the first neighbor shell, all of which are with respect to O atom. The heat transfer via Coulomb interaction between charged atoms reaches too far to be dealt with practically. Therefore, the Coulomb path must be defined for a charge-neutral group of atoms and a longer cutoff radius than that of vdW path is required [15]. Here, a Coulomb path was assigned to a molecular pair within the second neighbor shell radius, r2nd. The Coulomb path density ρCl

path was computed from the same equation as Eq. (6) using the cutoff distance r2nd, the number of water molecules nW, and the water–water RDF gWW(r). The vdW and Coulomb paths for ammonia were similarly defined except that O atom is replaced with N atom and water molecule with ammonia molecule.

From ρXpath and λX, we obtained the path efficiency per a single path,

ΛX, by Eq. (5). The path efficiency thus defined by Eqs. (5) and (6) represents an average heat transfer capability of a single interaction

within a certain cutoff radius. It is also possible to evaluate the heat transfer efficiency per single interaction as a function of intermolecular distance r as

ΛX(r)= − JX(r)/[ρX(r)∇zT] (7)

where JX(r) is the contribution to the partial heat flux JX in Eq. (4) from the molecular pair separated by r, and ρX(r) is the number density of such molecular pairs. In the actual calculation, the r-dependent vari-ables were computed by binning r with a thickness Δr = 0.05 Å. On the basis of Eq. (4), it is straightforward to decompose JX(r) into the con-tributions from A⋯B atom pair, JX,AB(r), where r of JX,AB(r) is intermo-lecular distance rather than interatomic one. Correspondingly, ΛX(r) can be written as the sum of atomic pair contributions. For example, in the case of water, a pair Coulomb interaction between water molecules is composed of one O⋯O, four O⋯H, and four H⋯H interactions. Then, the distance-dependent efficiency for Coulomb interaction (Cl) can be written as ΛCl(r) = ΛCl,OO(r) + 4ΛCl,OH(r) + 4ΛCl,HH(r), where ΛCl,AB(r) =–JCl,AB(r)/[ρCl(r)∇zT].

3. Results and discussion

3.1. NEMD results for the 0.7Tc saturated liquids

For the saturated liquid states at 0.7Tc, thermal conductivity was calculated as λ = 0.745 ± 0.001 W/(m⋅K) for SPC/E water, and 0.595 ±0.001 W/(m⋅K) for ammonia, as shown in Table 2. These simulated values overestimate the experimental ones [27] by 11% and 13% for water and ammonia, respectively. For ammonia, Guevara-Carrion et al. [13] computed thermal conductivity using NEMD simulation with the Eckl model for a wide range of temperature and density. Our result is consistent with their result that λ = 0.60 ± 0.02 W/(m⋅K) at T = 273 K and ρ = 641 kg/m3.

The RDFs for intermolecular atom pairs are shown in Fig. 2. The RDF for molecular pair with respect to the center of mass coordinates of molecule was almost the same as that of O⋯O or N⋯N although it is not shown explicitly. Corresponding to the RDFs, typical neighbor shell structures formed in water and ammonia are shown in Fig. 3. As can be seen from these figures, the neighbor shell structures of the two liquids are rather different. In the case of ammonia, the integration of inter-molecular RDF showed that the coordination numbers of molecules in the first and second neighbor shells were 12 and 40 molecules, respec-tively. The coordination number of 12 is commonly seen for liquids of non-polar, near-spherical molecules [40], suggesting that the interac-tion is isotropic. The first N⋯N peak position r = 3.46 Å in Fig. 2(b) is

Fig. 2. Radial distribution functions of intermolecular atom pairs in the 0.7Tc saturated liquids of (a) water and (b) ammonia. The locations of the 1st and 2nd neighbor shell radii that were used to define hydrogen bonds and atomistic heat paths are also indicated.

Fig. 3. Typical coordination structures and the corresponding atomistic heat paths in the 0.7Tc saturated liquids of water and ammonia. The first nearest neighbors (yellow) use the vdW and Coulomb paths to exchange thermal energy with the blue molecule at the center whereas the second nearest neighbors (red) use the Coulomb paths only.

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close to the minimum position of the vdW potential between nitrogen atoms, rmin,NN = 3.79 Å. This is in contrast to the case of water that the location of the first O⋯O peak, r = 2.79 Å, is much less than rmin,OO =

3.55 Å. These results suggest that vdW interaction plays an important role in forming the coordination shells of ammonia.

Compared with ammonia, the neighbor shells of water were smaller in size, and the coordination numbers were half, i.e., 6.2 and 20 mole-cules in the first and second neighbor shells, respectively. Experiments [41] suggested that at ambient temperature, water forms tetrahedral-like coordination due to strong hydrogen bonds and on average 4.7 molecules are coordinated in the first neighbor shell on the basis of the integration over the first peak in O⋯O RDF. Here, we observed a slightly larger coordination number than 4.7 because the temperature is higher and thereby the neighbor shell is slightly expanded. The analysis of hydrogen bond showed that a water molecule has 2.6 hydrogen bonding neighbors out of 6.2 molecules in the first neighbor shell whereas an ammonia molecule has 2.0 hydrogen bonding neighbors out of 12 molecules. In terms of energy, the average inter-action energy of a hydrogen bonding molecular pair was − 4.2 kcal/mol for water and − 1.3 kcal/mol for ammonia. Thus, as expected, the hydrogen bond in water is stronger than that in ammonia both in number and energy.

The analysis in this subsection revealed the difference in neighbor shell structures between the two liquids as follows. Water molecules are arranged so as to strengthen hydrogen bonds or Coulomb interaction as possible, which results in higher liquid density instead of shrinking the neighbor shells and reducing the number of molecules therein. In contrast, ammonia molecules try to minimize vdW potential energy, by which the coordination number per molecule is maximized.

3.2. Heat flux decomposition

The result of heat flux decomposition is shown in Fig. 4. In both liquids, the partial heat flux associated with molecular transport was minor and heat conduction was mostly due to intermolecular in-teractions. The Coulomb heat flux (the HB + non-HB contributions) accounts for 80% and 56% of the total heat flux in water and ammonia, respectively. On one hand, this prominent heat transfer associated with Coulomb interaction is a striking feature of these liquids, as compared with the case of alcoholic associated liquids (ethanol and ethylene gly-col), where the Coulomb heat flux was less than half of the total at the 0.7Tc saturated liquid state [16]. On the other hand, in ammonia, the vdW heat flux also has a relatively large contribution (38%), which distinguishes the heat transfer mechanism in ammonia from that in water. The percentages of heat flux via hydrogen bond, JHB, as the sum of the Coulomb and vdW parts, were similar for both liquids (36% for

water and 33% for ammonia), and again the main difference is that JHB of water is mostly due to the Coulomb part whereas the vdW part is superior in the case of ammonia. It should be noted that the non-HB contribution accounts for a rather large proportion even for the Coulomb part, indicating that heat transfer does not occur solely through hydrogen bonds even in the case of water.

The properties of atomistic heat paths are listed in Table 4. A noticeable feature of both liquids is the high density of Coulomb paths. In our previous study [16], ethylene glycol, a common heat transfer fluid with high thermal conductivity, was also characterized by a high Coulomb path density of 0.198 Å− 3, which was one order of magnitude higher than those in monohydric alcohols. The Coulomb path densities for water and ammonia are more than 1.8 times this value. This high density of Coulomb paths is one reason for the large percentage of Coulomb heat flux. The path density of ammonia was 1.5 times higher than that of water for both the Coulomb and vdW paths, which mainly originates from the difference in molecular coordination structure. That is, the coordination numbers in both the first and second neighbor shells of a water molecule were half those of an ammonia molecule, as dis-cussed in Section 3.1, whereas water has a ~1.3 times higher number density of molecules than ammonia; hence the 1.5 times difference in the path density according to Eq. (6).

The path efficiencies of water and ammonia derived in the present study do not deviate from the typical values seen for other liquids. The previous studies for alcohol and alkanes [14–16] showed that the effi-ciency of Coulomb path ranges 1–5 × 10− 30 Wm2/K, and the efficiency of vdW path is several orders of magnitude less than that of Coulomb path. The low efficiency of vdW path is the result of interference be-tween the dynamics of multiple atoms in the same molecule [14]. For example, thermal energy transfer via vdW interaction between atoms i and j moves these atoms so as to suppress the further energy transfer between i and j (otherwise the energy transfer diverges). If another atom k is covalently bonded to j, atom j pulls atom k in the similar direction to make the vdW energy transfer between i and k more difficult. Thus, the i–j energy transfer interferes the i–k energy transfer. For a molecule that is regarded as a single vdW site, such interference does not occur and the efficiency of vdW path is kept high. Actually, in the case of methane, the vdW path efficiency was as high as 1.3 × 10− 30 Wm2/K [14], which is comparable to the typical values of Coulomb path efficiency. Likewise, water and ammonia, which are both a molecule of single vdW site, have a high vdW path efficiency of the order of 10− 30 Wm2/K as shown in Table 4.

Since the total thermal conductivity of water and ammonia are not very different (see Table 2), the difference in a partial thermal con-ductivity between the two liquids are similar to that in the corre-sponding partial heat flux in Fig. 4. The main difference in heat transfer mechanism between water and ammonia exists in the ratio of Coulomb to vdW partial conductivity, which can be expressed as λCl/λvdW =

ρClpath/ρvdW

path ⋅ΛCl/ΛvdW. Since Table 4 shows that the ratio ρClpath/ρvdW

path is almost the same for the two liquids, the difference in the ratio λCl/λvdW is attributed to the difference in the path efficiency ratio, ΛCl/ΛvdW. For water, ΛCl was higher than ΛvdW and vice versa for ammonia. To

Fig. 4. Breakdown of heat flux in the 0.7Tc saturated liquids of water and ammonia into those due to Coulomb interaction, vdW interaction, and molec-ular transport (Trans). The Coulomb and vdW components are subdivided into the contributions from the molecular pairs with (HB) and without (non-HB) hydrogen bonds.

Table 4 Properties of the vdW (X = vdW) and Coulomb (X = Cl) paths for the saturated liquids of water and ammonia at 0.7Tc, where ρX

pathis the path density in Å− 3, ΛX

the path efficiency in 10− 30 Wm2/K, and λX the partial thermal conductivity in W/(m⋅K). Standard error of mean for the last digit is shown in parentheses.

water ammonia

ρvdWpath 0.0865(1) 0.1298(1)

ρClpath 0.372(1) 0.5700(5)

ΛvdW 0.98(3) 1.714(7) ΛCl 1.60(1) 0.582(2) λvdW 0.084(2) 0.2225(8) λCl 0.595(5) 0.332(1)

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examine the origin of this difference, the distance-dependent efficiency ΛX(r) for intermolecular interaction in Eq. (7) is shown in Fig. 5. The figure also includes the partial contributions from O⋯O and N⋯N atom pairs, but the curves for other atomic pairs are not shown to avoid overcomplicating the figure. We note that ΛvdW(r) is explained only by the O⋯O or N⋯N contributions since H atom has no vdW interaction in the present case. If the intermolecular heat transfer at a specific inter-molecular distance r occurs in the opposite direction to the total heat flux direction, ΛX(r) is negative. It is known [39] that the direction of pair heat transfer at r is correlated with the corresponding RDF g(r). In Fig. 5, g(r) is included in the right axis in terms of the potential of mean force (PMF), u(r) = − kBT ln g(r).

Comparing the curves for water in Fig. 5(a), one can see that ΛX(r) of Coulomb interaction is higher than that of vdW interaction, except some points in the short r region where the PMF is highly repulsive. In this short r region, ΛCl(r) is lower than its partial contribution ΛCl,OO(r), as a result of cancelation between the O⋯O, O⋯H, and H⋯H contributions. The two curves, however, are still close with each other, and as a whole, ΛCl(r) is higher than ΛvdW(r). In the case of ammonia in Fig. 5(b), ΛCl(r) is rather lower than ΛCl,NN(r), indicating that there is a large cancelation between the Coulomb contributions from N⋯N, N⋯H, and H⋯H pairs. As a result, the superiority of ΛCl(r) over ΛvdW(r) is not clear unlike in the case of water. The degree of cancelation in the Coulomb contributions from atom pairs is clearly correlated with molecular structure. That is, the isotropic distribution of H atoms in ammonia molecule effectively screens the Coulomb interaction between N atoms whereas this screening effect is limited in water molecule. These results in Fig. 5 explain the magnitude relationship between the Coulomb and vdW path efficiencies in Table 4.

3.3. Effect of different water models

The thermal conductivity of water calculated from the NEMD simulation at 298 K and 1 atm is shown in Table 3. The thermal con-ductivity of SPC/E water was obtained as λ = 0.897 ± 0.004 W/(m⋅K). Sirk et al. [10] pointed out that the thermal conductivity of water can differ sensitively depending on the methodology of thermal conductivity calculation even if the same molecular model is used. SPC/E usually gives λ = 0.81–0.95 W/(m⋅K) if NEMD simulation is used and the Coulomb force calculation is evaluated by Ewald sum [10,42] or reac-tion field [29,43] rather than simple cutoff method. The value of λ ob-tained here falls in this range.

The thermal conductivity of TIP5P-Ew water was calculated as λ =0.848 ± 0.004 W/(m⋅K). Only a few authors have examined the thermal conductivity of TIP5P-Ew or TIP5P. We note that TIP5P-Ew is an opti-mized version of TIP5P for Ewald sum and the difference between the two models is minor. Kumar [6] found λ ~ 1.5 W/(m⋅K) for TIP5P using NEMD simulation. In contrast, the NEMD simulations of Mao and Zhang [8,9] obtained 0.68 W/(m⋅K) and 0.62 W/(m⋅K) for TIP5P and TIP5P-Ew, respectively, and these values are close to the thermal con-ductivity of real water 0.61 W/(m⋅K) [27]. Using equilibrium MD simulation, English and Tse [11] reported a similar result 0.668 W/(m⋅K) for TIP5P. Our result for TIP5P-Ew appears to be somewhat different from either of the reported values and is not significantly different from the values for SPC/E and other water models. The discrepancy in these results among different authors may be due to the difference in detailed methodologies of NEMD simulations. At least from the aspect of system size, our result is more reasonable because our system size of ~104 molecules is much larger than those in the past studies. Actually, with this system size, we found the similar thermal conductivity values for TIP5P-Ew and SPC/E in spite of the fact that the difference between the two models is relatively large in terms of the number of atomic sites and whether or not the model has massless sites. Therefore, the use of a sufficiently large simulation system might resolve some discrepancies in the calculated thermal conductivity of water re-ported by different authors and by different water models. However, further study is necessary to find the exact reason for the discrepancy.

Fig. 5. (Left axis) Distance-dependent efficiency of heat transfer per single Coulomb (Cl) or vdW interaction for water and ammonia and their partial contributions from O⋯O and N⋯N atom pairs, where r is intermolecular dis-tance. The filled and open circles indicate that ΛX(r) is positive and negative, respectively. (Right axis) Potential of mean force for molecular pair, u(r).

Fig. 6. Breakdown of heat flux in water at 298 K for SPC/E or TIP5P molecular models using the geometric (G) or energetic (E) definitions of hydrogen bond. The meanings of decomposed components of heat flux are the same as those in Fig. 4. The number of hydrogen bonds per molecule, nHB, for each case is also shown.

H. Matsubara et al.

International Journal of Thermal Sciences 161 (2021) 106762

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The breakdown of heat flux is compared between SPC/E and TIP5P- Ew in Fig. 6. The decomposition was made in the same manner to that in Fig. 4, but here the hydrogen bond portions based on the energetic definition (denoted by E in the figure) were also examined in addition to the geometric definition (G) using the same simulation trajectory to see the effect of different hydrogen bond definitions. In the energetic defi-nitions, a hydrogen bond was defined for a molecular pair if the pair potential energy was less than an energy threshold. As is usually done [44], the energy threshold Eth was determined from the first minimum in the pair–energy distribution function of water dimer as Eth = − 3.06 kcal/mol for SPC/E and Eth = − 2.66 kcal/mol for TIP5P-Ew.

As shown in Fig. 6, the four cases gave similar breakdowns. Comparing with the results for the 0.7Tc saturated liquid, the breakdown at 298 K has a lower percentage of the transport contribution and higher hydrogen bond contributions. It can be understood that this change is caused by reduced mobility of molecules and increased number of hydrogen bonds owing to the lower temperature. SPC/E and TIP5P-Ew showed almost the same percentages for the Coulomb and vdW contri-butions, which accounted for 74–75% and 21–22%, respectively. This result supports that the two models gave similar values of thermal conductivity with consistent molecular-scale mechanisms. The number of hydrogen bonds, nHB, calculated for the four cases were nHB =

3.24–3.54, and these values are similar to nHB = 3.34–3.78 that were previously reported for SPC/E using various definitions of hydrogen bonds [44]. Although the HB parts of the Coulomb and vdW heat fluxes slightly changed by different definitions of hydrogen bond, the change was still small. From the above results, we conclude that the effect of specific molecular model and hydrogen bond definition are relatively minor.

4. Conclusions

On the basis of NEMD simulation and the atomistic heat path anal-ysis, we explored and compared the detailed molecular mechanisms of high thermal conductivity in water and ammonia. We conclude that the main reason for the high thermal conductivity of these liquids is a large amount of heat conduction due to Coulomb interaction, which is enabled by high density of Coulomb paths. The secondary reason is the high efficiency of vdW path and this characteristic is particularly important for ammonia. These features were not seen in other polar liquids [15,16] because in general the density of Coulomb paths are much lower than those of other paths and also because the efficiency of vdW path is lowered by interference effect [15]. We found that different strengths of hydrogen bond in water and ammonia result in different ways of realizing the high density of Coulomb paths, which in turn leads to different heat transfer mechanisms at the molecular scale. The two liquids thus exemplify different routes to the molecular design of highly thermally conductive liquids.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.

Acknowledgments

This work was partly supported by JST-CREST Grant Number JPMJCR17I2 Japan and JSPS KAKENHI Grant Number JP20K04300. Numerical simulations were performed on the supercomputer system “AFI-NITY” at the Advanced Fluid Information Research Center, Insti-tute of Fluid Science, Tohoku University.

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