Article pubs.acs.org/JPCB
Effect of the Titanium Nanoparticle on the Quantum Chemical Characterization of the Liquid Sodium Nanofluid Ai Suzuki,*,† Patrick Bonnaud,† Mark C. Williams,†,‡ Parasuraman Selvam,†,§ Nobutoshi Aoki,† Masayuki Miyano,† Akira Miyamoto,† Jun-ichi Saito,∥ and Kuniaki Ara∥ †
New Industry Creation Hatchery Center (NICHe), Tohoku University, 6-6-10, Aramaki, Aoba ku, Sendai 980-8579, Japan AECOM Technology Corporation, Morgantown, West Virginia 26501, United States § National Centre for Catalysis Research (NCCR) and Department of Chemistry, Indian Institute of Technology-Madras, Chennai 600 036, India ∥ Japan Atomic Energy Agency (JAEA), 4002 Narita, Oarai-machi, Ibaraki 311-1393, Japan ‡
ABSTRACT: Suspension state of a titanium nanoparticle in the liquid sodium was quantum chemically characterized by comparing physical characteristics, viz., electronic state, viscosity, and surface tension, with those of liquid sodium. The exterior titanium atoms on the topmost facet of the nanoparticle were found to constitute a stable Na−Ti layer, and the Brownian motion of a titanium nanoparticle could be seen in tandem with the surrounding sodium atoms. An electrochemical gradient due to the differences in electronegativity of both titanium and sodium causes electron flow from liquid sodium atoms to a titanium nanoparticle, Ti + Na → Tiδ− + Naδ+, making the exothermic reaction possible. In other words, the titanium nanoparticle takes a role as electron-reservoir by withdrawing free electrons from sodium atoms and makes liquid sodium electropositive. The remaining electrons in the liquid sodium still make Na−Na bonds and become more stabilized. With increasing size of the titanium nanoparticle, the deeper electrostatic potential, the steeper electric field, and the larger Debye atmosphere are created in the electric double layer shell. Owing to electropositive sodium-to-sodium electrostatic repulsion between the external shells, naked titanium nanoparticles cannot approach each other, thus preventing the agglomeration. and is relatively immiscible.7 However, when the liquid sodium embrace titanium nanoparticles, the formation enthalpy, i.e., ΔH f Ti/ S − Na , was experimentally found to be around −19.8 kJ mol−1Na1− for the probable exothermic reaction: Na + Ti → Na−Ti. This formation enthalpy was derived from the difference of the heat of reaction with water (−177 kJ mol−1) for the liquid sodium and −157 kJ mol−1 for Ti/S -Na nanofluid.2 This means that there exists an electrostatic interaction between a titanium nanoparticle and the liquid sodium in the Ti/S -Na nanofluid. It is also noteworthy here that the agglomeration of titanium nanoparticles does not occur, and the nanoparticles were found to be well-dispersed and stable with the particle size remaining the same in the liquid sodium. So far, the atomic interaction between a titanium nanoparticle and the sodium atoms at the surface of the nanoparticle has been investigated by the solid cluster model in the density functional theory (DFT) calculation.1 However, in order to begin a fundamental investigation, much more realistic models are essential. Therefore, in this work, the solid (Ti
1. INTRODUCTION Liquid sodium (S -Na) is used as a coolant in the sodium-cooled fast reactor, which is under development as a practically applicable next-generation nuclear power system. Although liquid sodium has superior heat removal characteristics, it exhibits a high chemical reactivity with liquid water or even water vapor in the atmosphere. Thanks to the recent advancement of nanotechnology, wherein the dispersion of titanium nanoparticles in a liquid sodium medium (Ti/S -Na nanofluid) can be achieved, such an effort has attracted significant interest as a candidate for a less explosive safer coolant. The Ti/S -Na nanofluid has experimentally been demonstrated not only to enhance the liquid stability itself and suppress the rate of sodium-water exploding reaction but also to reduce agglomeration.1−4 However, the mechanism of balancing the suppressive explosion while avoiding agglomeration is unknown. Furthermore, the mechanism of the solid sodium clusters explosive reaction with water has only recently been elucidated.5,6 When the solid sodium meets water molecules, sodium is enclosed by water molecules immediately,5 and Coulomb explosion between sodium cations was found to occur in the early stage.6 On the other hand, the Na− Ti binary system is known not to have an intermediate phase © 2016 American Chemical Society
Received: November 24, 2015 Revised: February 23, 2016 Published: March 23, 2016 3527
DOI: 10.1021/acs.jpcb.5b11461 J. Phys. Chem. B 2016, 120, 3527−3539
Article
The Journal of Physical Chemistry B
step, and the end of the simulation. The electronic state of the system was calculated self-consistently by solving the Schrödinger equation (HC = εSC; where H, C, ε, and S refer to the Hamiltonian matrix, eigenvector, eigenvalues, and overlap integral matrix, respectively) with the diagonalization condition (CTSC = I ; I refers to identify matrix). In the TBQC scheme, the total energy of the system, ET, is expressed by
nanoparticle)-liquid (Na atoms) interfacial chemical structure was considered in the liquid sodium medium with a suspended titanium cluster and calculated by ultra-accelerated quantum chemical molecular dynamics (UAQCMD). Indeed, UAQCMD method is computationally low-load and satisfies required accuracy. Such an approach has successfully been applied so far in order to explore the industrially important fields such as catalytic reaction, atomic surface mobility and tribological behavior.8−10 In this study, we, therefore, employed the UAQCMD simulation to quantify the self-diffusion of sodium atoms and the diffusivity of a titanium nanoparticle in the Ti/S -Na nanofluid under the realistic conditions, viz., with an initial operation temperature at 473 K and a steady state of sodium-cooled fast reactor at 773 K. The UAQCMD simulations, thus, provide a detailed insight about the quantum chemical characteristics of the Ti/S -Na nanofluid, which helps to understand the experimental observations such as diffusion coefficient, viscosity, Na−Ti formation enthalpy, and surface tension.
occ
ET =
k
j>1
rij
+
⎛ aij − rij ⎞ ⎟⎟ ⎝ bij ⎠
∑ ∑ bijexp⎜⎜ i
j>1
where the first, second, and third terms on the right-hand side of the eq 2 refer to the molecular orbital (MO) energy, Coulombic energy, and exchange-repulsion energy, respectively. To avoid loading heavy computational time and keeping the required accuracy, in the part of quantum chemical calculation, first-principles parametrization was done for the valence-state ionization potential, Hrr and the exponents of Slater-type atomic orbital functions, ζr. Hrr and ζr must be optimized to set the Hamiltonian H and overlap integral S matrices for each atomic orbital. 2.2. Numerical Solution of the Poisson’s Equation. The electrostatic potential, φ(i, j, k) can be expressed with the charge distribution, ρ(i, j, k), and the relative permittivity of the medium, ε in the three-dimensional Cartesian grid, which is given by Poisson’s equation. ∂ 2ϕ(i , j , k) ∂x
2
+
∂ 2ϕ(i , j , k) ∂y
2
+
∂ 2ϕ(i , j , k) ∂z
2
=−
ρ (i , j , k ) ε (3)
The first term in the left-hand side of the eq 3, i.e., ∂ ϕ/∂x2 can be rewritten as 2
⎡ Z Z e2 ⎛ ai + aj − rij ⎞⎤ i j ⎟⎟⎥ + f0 (bi + bj)exp⎜⎜ U = ∑∑⎢ ⎢ ⎝ bi + bj ⎠⎥⎦ i j > 1 ⎣ rij
∂ 2ϕ(i , j , k)
∑ ∑ Dij[exp{−2βij(rij − r0)} − 2exp{−βij(rij − r0)}] i
i
ZiZje 2
(2)
2. COMPUTATIONAL METHODS 2.1. Ultra Accelerated Quantum Chemical Molecular Dynamics. The thermodynamics of the nanofluid and the chemical reactions with electron transfer can be calculated by the UAQCMD simulations. The program is based on a tightbinding quantum chemical (TBQC) calculation program and a molecular dynamics method, which permits both chemical reactions and fluid dynamics of a complex system such as nanofluid to be studied. The following potential functions were employed to consider the ionic, covalent, and van der Waals interactions among the various atoms.
+
∑ nkεk + ∑ ∑
∂x
2
⎡ ∂ϕ ⎤ ∂ϕ (i , j , k)⎥ /∂x = ⎢ (i + 1, j , k) − ⎣ ∂x ⎦ ∂x
⎡ ϕ(i + 1, j , k) − ϕ(i , j , k) ϕ(i , j , k) − ϕ(i − 1, j , k) ⎤ ⎥ /δx =⎢ − δx δx ⎣ ⎦
j>1
(1)
The first term in eq 1 corresponds to the Coulomb potential and the second to the short-range exchange repulsion potential, where f 0 is a constant for unit adjustment, a is the size, and b is the stiffness, which gives a good account of the repulsive interactions arising from the overlap of electronic clouds. Zi and Zj are the charges on the atoms, e is the elementary charge, and Rij is the internuclear distance. The third term in eq 1 corresponds to the Morse-type potential function, which represents covalent interactions, where Dij is the bond energy, βij is the form factor, and r0 is the bond length at the minimum energy. A classical MD simulation11 was performed using the described potentials. This system can solve the equation of motion for large sets of atoms. In this simulator, a Verlet algorithm12 was employed to integrate the equation of motion. A temperature scaling method was also implemented in the system, which is similar to the Woodcock algorithm.13 The temperature in the simulation was maintained by scaling the atomic velocities. On the other hand, the UAQCMD methodology consists of two parts: the first is the quantum chemical calculation, and the second involves a MD simulation. To reflect the Morse-type potential function and atomic charges during the simulation process, the TBQC calculations are performed at the beginning, every specified intermediate
=
ϕ(i + 1, j , k) + ϕ(i‐1, j , k)‐2ϕ(i , j , k) δx2
(4)
Hence, the discretization of eq 4 can be totally formulated as ϕ(i + 1, j , k) + ϕ(i − 1, j , k) − 2ϕ(i , j , k) δx2 + + =−
ϕ(i , j + 1, k) + ϕ(i , j − 1, k) − 2ϕ(i , j , k) δy2 ϕ(i , j , k + 1) + ϕ(i , j , k − 1) − 2ϕ(i , j , k) δz2 ρ(i , j , k) ε
(5)
φ(i + 1, j , k) + φ(i − 1, j , k) δx2 +
+
δy2
ϕ(i , j , k + 1) + ϕ(i , j , k − 1) δz2
⎛ ⎞ 1 1 1 = 2ϕ(i , j , k)⎜⎜ 2 + 2 + 2 ⎟⎟ δy δz ⎠ ⎝ δx 3528
ϕ(i , j + 1, k) + ϕ(i , j − 1, k)
+
ρ(i , j , k) ε
(6) DOI: 10.1021/acs.jpcb.5b11461 J. Phys. Chem. B 2016, 120, 3527−3539
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The Journal of Physical Chemistry B
STO Figure 1. Radial distributions of electron density for: (a) Na 3s orbital; ΦSTO Na3s(ζ = 0.96), and Na 3p orbital; ΦNa3p(ζ = 0.78). (b) Ti 3d orbital; STO STO STO (ζ = 1.7) + 0.5Φ (ζ = 3.4), Ti 4s orbital; Φ (ζ = 1.35), and Ti 4p orbital; Φ (ζ = 1.2). Dotted and solid lines denote the results of 0.5ΦSTO Ti3d 1 Ti3d 2 Ti4s Ti4p DFT and TBQC calculations, respectively.
An iterative method to obtain the electrostatic potential, φ(i, j, k) numerically starts with an initial guess for solving eq 6, and it successively continues until the potential remains unchanged. To confirm the accuracy of numerical solution, the analytical solutions for the electric potential and electric field distribution are also studied. A static electric field, E, can be derived from the obtained scalar electrostatic potential, φ(i, j, k) which obeys Poisson’s law, E = −∇φ(i, j, k) by taking unbiased central difference as Ex(i , j , k) = −
ϕ(i + 1, j , k) − ϕ(i − 1, j , k) 2δx
(7)
Ey(i , j , k) = −
ϕ(i , j + 1, k) − ϕ(i , j − 1, k) 2δy
(8)
Ez(i , j , k) = −
ϕ(i , j , k + 1) − ϕ(i , j , k − 1) 2δz
(9)
Figure 2. Charge dependency of orthogonal Hamiltonian term, Hrr for orbitals in Na and Ti atoms. The solid lines correspond to the Hrr of Na 3s and Na 3p for TBQC method dependent on atomic charge, q as Hrr (Na3s) = −4.85 − q and Hrr (Na3p) = −0.45 − q. The dotted lines correspond to the Hrr of Ti 4s, 4p and 3d for TBQC method dependent on atomic charge, q as Hrr (Ti4s) = −5.5 − 5.5q, Hrr (Ti4p) = −2 − 4q, and Hrr(Ti3d) = −4 − 5.5q.
condition were considered. The cell parameters of the hcpTi48 model is a = 10.22, b = 8.85, and c = 9.36 Å with interaxial angles α = β = γ = 90°, whereas for the bcc-Na16 bulk model, a = b = c = 8.58 Å with interaxial angles of α = β = γ = 90°. Table 1 lists the binding energy and electronic compositions for bcc-
A scalar static electric field can be expressed as in eq 10. Escalar =
Ex 2 + Ey 2 + Ez 2
(10)
Table 1. Energetic and Electronic Profiles of Atomic Orbitals of Sodium and Titanium Metals
3. RESULTS AND DISCUSSION 3.1. First-Principles Parameterization. The suitability of quantum chemical parameters, Hrr and ζr for TBQC calculation was confirmed by comparing with several DFT results of ADF program and DMol3 code.14 According to the previously established first-principles parametrization procedure,15 Figure 1a,b depict the respective radial distributions of electron density for the sodium (Na 3s and Na 3p) and titanium (Ti 3d, Ti 4s, and Ti 4p) atomic orbitals with neutral charge calculated by ADF program and TBQC programs. Applied single-ζ or the double-ζ basis sets of Slater type orbital of TBQC program are also noted in the caption. It is clear from this figure that we can observe a good agreement between the results obtained by both DFT and TBQC methods. At this juncture, it is important to note that we have ignored the inner electron density because it does not influence the interactions with surrounding atoms significantly. Figure 2 shows the valence-state ionization potential, Hrr for both Na and Ti atoms calculated by the Dmol3 program and that for the TBQC method. The charge dependency of Hrr for TBQC calculation shows a good agreement with that of DFT results. For further validation of energetic and electronic profiles, hexagonal close-packed (hcp)-Ti48 and body-centered cube (bcc)-Na16 crystal models under the periodic boundary
bcc - Na16 method DFT TBQC experiment
hcp - Ti48
binding energy
electronic composition
binding energy
electronic composition
(kcal/mol)
s, p (%)
(kcal/mol)
s, p, d (%)
− 408 − 411 − 411
87, 13 97, 3
− 5171 − 5319 − 5426
21, 8, 71 23, 6, 71
Na16 and hcp-Ti48 structures obtained by both TBQC and DFT methods. It can be noticed from this table that the binding energies and electronic compositions calculated by UAQCMD agree with those obtained by DFT, and the values are also fairly close to the experimental cohesive energies, as the standard molar enthalpy of formation at gas condition for sodium and titanium are 107.5 and 473.0 kJ/mol, respectively.16 Accordingly, UAQCMD method with first-principles parametrization guarantees the reasonable accuracy with DFT level to reproduce energy and electronic profiles. 3.2. Model of the Liquid Sodium and Liquid Sodium with a Titanium Nanoparticle. The model of liquid sodium medium for each temperature applied in this study accurately satisfies the appropriate density dependent on the temperature. 3529
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The Journal of Physical Chemistry B
Figure 3. (a) Model of a liquid sodium, S -Na410 at 773 K with a density of 0.90 g/cm3. (b) The sphere radius, r (m), of a diffusing sodium atom in liquid sodium.18 (c) The cubic lattice, v1/3 (m), occupied by a single sodium atom including the free volume in liquid sodium.19,20 (d) The model of liquid sodium containing a spherical Ti114 nanoparticle with a diameter of 1.5 nm (Ti114/S -Na410) and a density of 0.88 g/cm3 at 773 K, having 78 surface titanium atoms on the exterior. (e) The model of liquid sodium containing a spherical Ti57 nanoparticle with a diameter of 1.0 nm (Ti57/ S -Na410) having 44 topmost titanium atoms on the exterior facet with a density of 0.87 g/cm3 at 773 K. The cubic cell lattice parameters for (d) and (e) are a = b = c = 3.0 nm with α = β = γ = 90° and a = b = c = 2.84 nm with α = β = γ = 90°, respectively.
Table 2. Structural Details and Various Properties of Liquid Sodium at Different Temperaturesa
a
Cell dimension, atomistic sphere radius (Figure 3b), atomistic cubic lattice (Figure 3c), liquid density, and molar volume are noted. Simulation results of temperature dependence of vaporization enthalpies, MSD, and trajectories of sodium atoms are also noted.
The density, d (g/cm3), and molar volume, VM (m3/mol), of the liquid sodium under the atmospheric pressure are expressed as a function of temperature, T (K) in eq 11 and eq 12, respectively17 d = 0.001 × [dm − 2.35 × (T − Tm)]
(11)
VM = Vm[1 + 24.8 × (T − Tm)]
(12)
cubic cell model to express atomic self-diffusion proposed by Eyring19 and Houghton20 (see eq 16 for details). Figure 3d shows the case for liquid sodium containing a spherical titanium nanoparticle of 1.5 nm diameter (Ti114/S -Na410). In order to investigate the influence of the nanoparticle size at steady-state condition of sodium cooled fast reactor, 773 K, a smaller nanoparticle is also modeled, as shown in Figure 3e, with a spherical Ti57 nanoparticle of 1.0 nm diameter in the liquid sodium (Ti57/S -Na410), which satisfies the appropriate density at 773 K. Both these models are under a three-dimensional periodic boundary condition. At this juncture, it is to be noted here that the adjusted numbers of sodium atoms in the cell were the same as the model for liquid sodium for respective temperatures when a spherical nanoparticle is contained. Experimentally measured density of the Ti/S -Na nanofluid, d Ti/ S − Na (kg/m3) is empirically expressed by a linear function of the absolute temperature, d Ti/ S − Na = −0.311 T + 1120. These data are compiled in Table 3 to show the appropriateness of the model.
where, Tm is the melting temperature of sodium (371 K), and dm and Vm are the corresponding density (0.927 g/cm3) and molar volume (24.8 × 10−6 m3/mol), respectively. Liquid sodium medium contains 470, 450, 430, and 410 atoms at 473, 573, 673, and 773 K, respectively, and they are initially arranged in a cubic cell where the volume is fitted to the appropriate liquid density for each temperature. Figure 3a is a liquid sodium structure (S -Na410) which has the appropriate density at 773 K. The densities and molar volumes of the structures of liquid sodium are compared with that of the calculated ones by eq 11 and eq 12, respectively, and are listed in Table 2. Figure 3b shows a spherical atomic self-diffusion model as suggested by Sutherland19 (see eq 15 for details), whereas Figure 3c shows a 3530
DOI: 10.1021/acs.jpcb.5b11461 J. Phys. Chem. B 2016, 120, 3527−3539
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The Journal of Physical Chemistry B 3.3. Self-Diffusion of Liquid Sodium. For the analysis of the diffusion of sodium atoms and titanium nanoparticle in the liquid sodium medium, the UA-QCMD simulations were performed in the canonical (NVT) ensemble, where the constant numbers of containing atoms (N), volume (V), and temperature (T) are fixed to satisfy the density for the temperature. We stored atomic trajectories generated from the dynamics simulations over a sufficiently long period, 100 000 time steps with 0.1 fs/step and calculated the time-evolution of the mean square displacements (MSD) as a function of diffusing positions of liquid sodium atom, i, as Δri(t )2 = (x i(t ) − x i(0))2 + (yi (t ) − yi (0))2 + (z i(t ) − z i(0))2
(13) Figure 4. Time evolution of atomic diffusion with linear slopes for pure liquid sodium and a titanium nanoparticle in the liquid sodium. The slopes of plots correspond to the diffusion coefficients of liquid sodium and titanium nanoparticle in liquid sodium at various temperatures. Both S -Na410 (solid line at 773 K, d = 0.83 g/cm3 and cell: x = y = z = 2.66 nm) and S -Na686 (open circles at 773 K, d = 0.83 g/cm3 and cell: x = y = z = 3.07 nm) have identical slopes. Similarly, both S -Na470 (solid line at 473 K, d = 0.90 g/cm3 and cell: x = y = z = 2.71 nm) and S -Na686 (open circles at 473 K, d = 0.90 g/cm3 and cell: x = y = z = 3.15 nm) systems have also identical slopes.
The MSD and estimated diffusion coefficients, D, are given by the well-known relation due to Einstein as D=
1 d lim ⟨Δri(t )2 ⟩ 6 t →∞ dt
(14)
2
where is the average MSD of diffusing objects (sodium atoms or titanium nanoparticle) at time t. The MSD values of sodium atoms with their trajectories are listed in Table 2. The MSD of a titanium nanoparticle is listed in Table 3. Table 3. Structure Details of the Ti/S -Na Nanofluida temperature (K) model cell (nm)/volume (nm3) density, d Ti/ S − Na (g/cm3) experiment model MSD (Å2) titanium
473
573
673
773
Ti114/ Na470 3.0/27
Ti114/ Na450 3.0/27
Ti114/ Na430 3.0/27
Ti114/ Na410 3.0/27
0.97 0.97
0.95 0.94
0.91 0.91
0.88 0.88
6.3
12.3
17.7
20.6
following previously established relations between diffusion coefficient, D, and viscosity, η. When the spherical radius of a diffusing particle, r = (3Vm/ 4πNA)1/3 (NA: Avogadro number) is equal to that of the liquid sodium (Figure 3b), a spherical model for self-diffusion known as Sutherland−Einstein formula18 is expressed as D=
kBT 4πηr
(15)
where, kB is the Boltzmann coefficient, and T is absolute temperature. Another self-diffusion model is the Eyring− Houghton cubic cell model.19,20
a
Setting conditions of cell volume, cell lattice, and density, d Ti/ S − Na . Obtained temperature dependence of atomic MSD values for a titanium nanoparticle is also noted.
D=
RTdv 2/3 6ηM w
(16) 1/3
3
where the volume, v = (Mw/dNA) (m ) (Mw: molecular weight, 22.9 g/mol for sodium and d: the density of the liquid sodium) occupied by a sodium atom including the free volume20 is illustrated in Figure 3c. The numerical values of r in eq 15 and v in eq 16 are listed in Table 2. On the other hand, the underlying assumption of the Stokes−Einstein formula25 (eq 17) is not valid for hydrodynamic self-diffusion.
Figure 4 presents the MSD values of the liquid sodium and titanium nanoparticle in liquid sodium. It can be seen from this figure that the slope of MSD values for diffusing objects versus time, t yields the diffusion coefficient, D. The slopes of solid and dotted lines correspond to the diffusion coefficient of liquid sodium, and a titanium nanoparticle, respectively. The slopes of open circles for 473 and 773 K correspond to the larger cell systems containing 686 sodium atoms those have the same liquid densities as the systems containing 410 sodium atoms at 773 K and 470 sodium atoms at 473 K. The linear slopes of the MSD for S -Na686 systems at the 473 and 773 K are identical to those of S -Na470 at 473 K, and S -Na410 for 773 K. Accordingly, the atomic diffusion coefficient is not affected by the cell size. Figure 5 shows the relationship between diffusion coefficient and temperature through Arrhenius equation for the sodium atoms in the liquid state. The diffusion coefficient obtained by the MSD of the UAQCMD method positioned in the vicinity of experimental measurements21−23 as well as the power law, D = Dm (T/Tm)n where Tm is melting temperature, 371 K, Dm is corresponding diffusion coefficient, 4.35 × 10−8 m2/s, and n is 2.093 for liquid sodium.24 Additionally, diffusion coefficients were obtained by inputting measured viscosity data in the
D=
kBT 6πηr
(17)
where r is the radius of a suspending titanium particle larger than the atomic radius of the liquid sodium. However, there was the circumstance when the Stokes− Einstein formula could give a satisfactory description of the experimental viscosity of liquid mercury.26 Hence, we investigate the applicability of Stokes−Einstein formula to the case of liquid sodium. As a result of the temperature dependence of self-diffusion coefficient of liquid sodium in Figure 5, the Sutherland−Einstein formula and Eyring− Houghton cubic cell model are close to the diffusion coefficient obtained by the MSD of the UAQCMD method. In case of liquid sodium, the Sutherland−Einstein formula and Eyring− 3531
DOI: 10.1021/acs.jpcb.5b11461 J. Phys. Chem. B 2016, 120, 3527−3539
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Figure 5. Temperature dependence of self-diffusion coefficient of liquid sodium. Previously established theoretical formula to obtain diffusion coefficients by inputting the experimentally measured viscosity are Sutherland−Einstein formula (circles, eq 15), Eyring−Houghtpon cubic cell model (squares, eq 16), and Stokes−Einstein formula (triangles, eq 17). Diffusion coefficients obtained from the MSD values from the UAQCMD simulation are denoted as filled diamonds. The experimental data by Meyer and Nachtrieb21 from 380 to 550 K is expressed as thin dotted line, whereas the experimental measurements by Ozelton and Swalin22 are shown by a thick broken line. On the other hand, the experimental diffusion coefficients presented by Balucani et al.23 were shown by plus signs at 378, 403, 602, 803, and 900 K. A solid line is the empirical power law for the self-diffusion coefficient of the liquid sodium.24
Figure 6. Temperature dependence of simulated viscosity for (a) the pure liquid sodium and (b) the Ti/S -Na. The experimental measurements4 for both (a) and (b) were noted as gray filled circles. Andrade’s empirical formula (eq 18) is noted as a broken line. Theoretical estimation were obtained by inputting simulated diffusion coefficients in the Sutherland−Einstein formula (circles, eq 15), the Eyring−Houghton cubic cell model (squares, eq 16), and the Stokes−Einstein formula (diamonds, eq 17).
sodium at the equilibrium state of the Brownian motion, a viscosity of the medium can be obtained via the Stokes− Einstein D (diffusion coefficient) and η (viscosity) relationship. Diffusion coefficients of the titanium nanoparticle were obtained from the slopes of MSD, and inputted in the Stokes−Einstein formula (eq 17), and viscosity of the Ti/S -Na nanofluid can be obtained. The simulated viscosity is in line with the experimentally measured one. The viscosity of Ti/S -Na nanofluid is theoretically demonstrated to be well retained relative to pure liquid sodium, and it becomses one of the practical advantages. 3.5. Interaction between a Titanium Nanoparticle and Liquid Sodium. The interaction between a titanium and sodium atoms was quantitatively investigated at the interface of liquid sodium and titanium nanoparticle. Figure 7 depicts the Na and Ti trajectories, and the diffusion of a titanium nanoparticle in liquid sodium at different temperatures. Translational diffusion of a titanium nanoparticle could be seen in Figure 7a,b for 473 and 773 K, respectively. The titanium nanoparticle moves in tandem with proximate sodium atoms. Diffusivity of a titanium nanoparticle increased as the temperature rises. The binding energy and the bond population of Na−Ti bonds in the Ti/S -Na nanofluid are shown in Figure 7c. The surface Ti atoms interacts with surrounding sodium atoms to form exclusively Na−Ti bonds. It can also be seen
Houghton cubic cell model show superior qualitative reproduction of hydrodynamic self-diffusion to the Stokes− Einstein formula. Diffusion coefficients obtained from the UAQCMD simulations are justified by qualitative agreements between the previously established theories as well as experimental measurements including empirical power law. 3.4. Comparison of Viscosities. Figure 6a shows the viscosity of liquid sodium derived from the Sutherland− Einstein equation (η = kBT/(4πDr), eq 15) and the Eyring− Houghton cubic cell model (η = RTdv2/3/(6DMw), eq 16) by inputting calculated self-diffusion coefficient, D. The viscosity obtained from the Stokes−Einstein equation, (cf. eq 17) was also included for comparison. We have also used Andrade’s empirical formula of viscosity (Ns/m2)27 with arbitrary constants determined for the liquid sodium28 which is expressed as η = 1.183 × 10‐3d1/3exp(716.5d /T )
(18)
where d is the liquid density of sodium listed in Table 2. As Andrade’s viscosity formula is independent from diffusivity, we plot it as an empirical proof line in Figure 6a. Figure 6b shows the viscosity of the Ti/S -Na nanofluid. If a nanoparticle is suspended in a liquid, it is subject to be in constant erratic motion due to the resultant force from many atomistic collisions. For spherical nanoparticles moving in the liquid 3532
DOI: 10.1021/acs.jpcb.5b11461 J. Phys. Chem. B 2016, 120, 3527−3539
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The Journal of Physical Chemistry B
Figure 7. Na and Ti trajectories are drawn by thin and thick lines, respectively. Diffusion of a titanium nanoparticle in liquid sodium with atomic trajectories at different temperature: (a) 473 K and (b) 773 K. (c) Na−Ti binding energy and its bond population. (d) Na−Na binding energy before and after Na−Ti interaction in the liquid sodium with a Ti114 nanoparticle at 773 K. The binding energy of Na−Na bond is previously estimated as −12.0 kcal/mol,1 which is close to TBQC results.
from this figure that the Na−Ti binding energies are larger than those of Na−Na bonds. The binding energy is increasing in the following order: Na−Na < Na−Ti < Ti−Ti, and that the Ti−Ti binding energy in the nanoparticle is −27.3 kcal/mol on the average, which is close to the experimental dissociation energy (−25.0 kcal/mol).16 Figure 7d compares the Na−Na binding energy in the Ti/ S -Na nanofluid before, i.e., at the initial states of the simulation (open circles), and the after (filled circles) of the Na−Ti reaction. The Na−Na bond distance of immediate neighbors among liquid sodium is experimentally confirmed around 3.8 Å,29 which is close to the simulation result. Further, the Na−Na binding energy, after the reaction, is found to be stabilized uniformly for all investigated temperatures about −18 kJ/mol−1 Na −1 , or, the formation enthalpy of Na−Ti bonds, ΔH f (Ti/ S − Na). In fact, by taking into account the allowance for experimental margin of error,2 this value is in reasonable agreement with experimental value, −19.8 kJ/mol−1 Na−1. The free electrons in Na−Na bonds around the Na−Ti bonds were localized to compete for deficient electrons, and the Na−Na binding energies were stabilized through the formation of Na− Ti bonds, and it results in a corresponding gain to surface tension. Accordingly, the addition of titanium nanoparticles into the liquid sodium is confirmed to be effective in stabilizing the Na−Na bonds. 3.6. Partial Density of States. Figure 8a,b show the partial density of states (DOS) for bcc-sodium at 298 K and liquid sodium at 773 K calculated by the UAQCMD method, respectively. The energy level of the highest occupied molecular orbital (HOMO) for bcc-Na and liquid sodium consists of mainly 3s and partially 3p orbitals. Increasing temperature brought the disorder or symmetry-breaking to the Na−Na
Figure 8. Partial DOS for Na3s and Na3p orbitals in (a) bcc-solid state sodium at 298 K, (b) liquid state sodium at 773 K, (c) Ti4s, Ti4p, and Ti3d orbitals in the Ti nanoparticle. The energy level is relative to the HOMO for Ti nanoparticle.
bonds, and it results in broadening of the p band below the Fermi level.30 Figure 8c shows the DOS for a Ti114 nanoparticle, and the HOMO for the titanium nanoparticle is deeper than that of liquid sodium. Therefore, the sodium atoms can donate electrons to the unoccupied molecular orbitals in the nanoparticle due to the energy gap between their frontier orbitals. This energy gap would be an origin of the electron transfer from sodium atom to titanium atom, which would result in making chemical reaction possible between these 3533
DOI: 10.1021/acs.jpcb.5b11461 J. Phys. Chem. B 2016, 120, 3527−3539
Article
The Journal of Physical Chemistry B
Figure 9. Temperature dependence of the surface tension for the following: (a) the liquid sodium and (b) the liquid sodium with the titanium nanoparticle. The filled gray diamonds are experimentally measured results of surface tension.2 The broken line is Grosse’s empirical law for liquid sodium.40 The open circles and plus signs are theoretical and simulated values obtained from eq 20 for pure liquid sodium and eq 22 for liquid sodium with a titanium nanoparticle.
Figure 10. Atomic charge distribution in upper figures (Q means the charge degree) and projection transformation of electrostatic potential φ (V) and the electric field, E(r) (V/nm) for (a) pure liquid sodium (b) a titanium nanoparticle of 1.0 nm diameter in the liquid sodium, and (c) a titanium nanoparticle of 1.5 nm diameter in the liquid sodium at 773 K. The numerical solutions by the three-dimensional Poisson’s solver are noted as diamonds for electric field and circles for electrostatic potential, respectively. Analytical solutions for the electrostatic potential (solid line, eq 23) and the static electric field (dotted line, eq 24) are also sketched.
liquid sodium.34 The values of theoretical and simulated vaporization enthalpy, Δgl Hvap are listed in Table 2. Theoretical one is determined by eq 19, and simulated one corresponds to the divided values of total binding energies obtained from the UAQCMD calculation by the numbers of sodium atoms contained. Stephan35 first links surface tension to the heat of evaporation, and it is well corrected by Overbury et al.36 as
elements. As d−s bonding was assumed, if the bond is created between titanium and sodium,31 mainly 3d−3s orbital superposition is natural when sodium atoms are attracted to the surface of titanium nanoparticle. Accordingly, the direction of major electron flow, i.e., the total chemical reaction can be expressed as Ti + Na → Tiδ− + Naδ+, and it is truly due to the difference of electronegativities of the constituent elements. 3.7. Comparison of Surface Tension. The vaporization enthalpy can be expressed as eq 19, when the boiling point is set as a reference temperature to estimate the vaporization enthalpy of liquid metals32 ⎛ T ‐T ⎞0.38 Δgl H vap = ΔHb⎜ c ⎟ ⎝ Tc‐Tb ⎠
−2/3 σ = f Δgl H vapN A−1/3V M
(20)
where, f is the configuration factor of atoms in the liquid, i.e., 1.12 for body-centered cubic structure.37 The eq 20 clearly shows the general tendency that the liquid metal shows lower surface energy than their heat of vaporization. The surface tension can be determined via the vaporization enthalpy, Δgl Hvap with molar volume, VM. In conjunction with the fact that the enthalpy and entropy of vaporization, the surface tension and surface energy all vanish at the critical temperature.
(19) 33
where, Tc is the critical temperature, 2573 K, and Tb is the boiling point, 1154.55 K and ΔlgHb is the liquid−vapor transition enthalpy at boiling temperature, 89.3 kJ/mol of the 3534
DOI: 10.1021/acs.jpcb.5b11461 J. Phys. Chem. B 2016, 120, 3527−3539
Article
The Journal of Physical Chemistry B In this context, Eötvös38 formula is quite natural to express the temperature-dependent surface tension with defining the reference to the critical temperature, as well as expressing the tendency of surface tension, which drops linearly with increasing temperature. However, the coefficient of Eötvös formula is overestimated for liquid metal, and Guggenheim39 modified it in applicable formula for liquid metal; subsequent empirical correction was done by Grosse40 to simplify the formula. The molar surface tension of the liquid sodium can be expressed by Grosse’s empirical law as σ = σc(1 − T /Tc)
The electrostatic potential shows how much the titanium nanoparticle can withdraw electrons from sodium atoms and store them. The electric field shows the electrochemical gradient in the electric double layer, and how large the Debye shielding atmosphere can be spread just around the liquid sodium−solid titanium interface. Sodium charge in liquid sodium is electroneutral as shown in Figure 10a. However, liquid sodium enclosing a electronegative titanium nanoparticle became electropositive in the Ti/S -Na nanofluid. The Debye atmosphere becomes larger as increasing the size of the titanium nanoparticle. According to the charge distribution as shown in Figure 10b and 10c, Q is almost uniformly distributed in the sphere inside of radius r0 for the titanium nanoparticle. Hence, the analytical solutions of the electrostatic potential, φ(r) at position r inside and outside the nanoparticle are given by eq 23, making the surface area a spherical Gaussian surface, 4πr2.
(21)
where, Tc is the critical temperature, 2573 K, and σc is the corresponding surface tension of the sodium, 0.22 N/m.40 The agreement between theory and experiment is fairly good, as shown in Figure 9a. −2/3 σ = f (Δgl H vap + ΔH f (Na(Ti)))N A−1/3V M
⎧ 2⎞ ⎛ ⎪ Q 1 ⎜3 − r ⎟ : r < r0 ⎪ 4πεε0 2r0 ⎝ r02 ⎠ ⎪ ⎪ Q : r = r0 ϕ(r ) = ⎨ ⎪ 4πεε0r0 ⎪ ⎛ r − r0 ⎞ ⎪ Q ⎪ 4πεε r exp⎜⎝ − λ ⎟⎠ : r0 < r ⎩ 0 D
(22)
Figure 9b shows the simulated surface tension of the liquid sodium containing a titanium nanoparticle with experimental measurements.2 Figure 9b also includes theoretically estimated surface tension by applying eq 22. Since addition of Na−Ti cohesive forces among liquid sodium is responsible for the corresponding increase of surface tension, surface tension is theoretically obtained by adding measured Na−Ti formation enthalpy, ΔHf (Ti/S -Na), − 19.8 kJ/mol−1Na1− in Stephan’s law as eq 22. Simulated surface tension is obtained by adding calculated Na−Ti formation enthalpy which corresponds to the stabilization enthalpy of Na−Na binding energy per sodium atom through the Na−Ti interaction. Namely, Na−Na binding energy for all investigated temperatures becomes stabilized uniformly, ca. − 18 kJ/mol−1Na1− and the value is inputted in the ΔHf (Ti/S -Na) in eq 22. Then, simulated surface tension noted as plus signs in the Figure 9b was obtained. Compared with Figure 9a, surface tension is increased when the liquid sodium contains a titanium nanoparticle. Increase of surface tension means the addition of stabilization energy in Na−Na bonds. 3.8. Electrostatic Potential and Electric Field around Liquid−Solid Interface. Figure 10 illustrates the atomic charge distributions (upper part), the electrostatic potential φ (V), and the electric field E (r) (V/nm) (lower part) for pure liquid sodium (S -Na410, Figure 10a), a titanium nanoparticle of 1.0 nm diameter in liquid sodium (Ti57/S -Na410, Figure 10b), and a titanium nanoparticle (1.5 nm diameter) in liquid sodium (Ti114/S -Na410, Figure 10c). The transferring amount of electrons per one titanium atom is about −0.05e and almost unchanged between the different-sized nanoparticles. In fact, this expected value fairly agrees with the value, −0.048e by previous theoretical approach.2 The numerical Poisson solutions were drawn in lower parts with the graphics of projection transformations. Analytical solution lines presented by eq 23 for electrostatic potentials are noted as solid lines, and those for the electric field presented by eq 24 are also noted as dotted lines. The electric field became maximum at immediate periphery of the titanium nanoparticle. The deeper electrostatic potential and the steeper electric field are found to be created as the size of the nanoparticle is increased. The agreement between analytical solution lines and the numerical Poisson solution lines are qualitatively reasonable for both the electrostatic potentials and the electric field.
(23)
The analytical solution of the electric field at position r, E (r) inside and outside the nanoparticle are given by ⎧ ⎛ ⎞ ⎪− Q ⎜ r ⎟ : r < r0 ⎪ 4πεε0 ⎝ r03 ⎠ ⎪ Q ⎪ : r = r0 ⎪− 2 E(r ) = ⎨ 4πεε0r0 ⎪ ⎛ r − r0 ⎞ 1 ⎛ r − r0 ⎞⎤ ⎪ Q ⎡1 ⎢ 2 exp⎜− ⎟ + exp⎜− ⎟⎥ ⎪− r λD ⎠ λD ⎠⎥⎦ ⎝ ⎝ ⎪ 4πεε0 ⎢⎣ r ⎪ :r
