Download Validation of Molecular Dynamics simulations of evaporation and

Validation of Molecular Dynamics simulations of evaporation and
condensation of more complex molecules
E.A.T. van den Akkera , A.J.H. Frijnsa , A.A. van Steenhovena , P.A.J. Hilbersb
a Department of Mechanical Engineering, Eindhoven University of Technology, the Netherlands
b Department of Biomedical Engineering, Eindhoven University of Technology, the Netherlands
Summary
New cooling methods are needed to keep up with the heat production in processors. Evaporation in micro
channels is a promising method. To analyze this method, Molecular Dynamics (MD) simulations are used. The
validity of MD simulations in phase transitions is checked thermodynamically. The thermodynamic properties
of MD with bonds agree well with experimental results from literature.
Keywords: Molecular Dynamics, evaporation, micro channels, thermodynamics
1.
Introduction
Since the invention of the integrated circuit, the power consumption increases with a factor of 10 every 6
years [1]. It is widely believed that this trend will continue [2]. If no energy is removed, the processor will
heat up; this causes malfunction or even breakdown. Processors can be cooled by forced convection by gases;
the heat removal here is in the range 25 − 250 kW/m2 [3]. Current processors approach the limits of air fan
cooling, and new cooling methods are needed. One method is micro channel cooling [4], which can remove a
heat flux of 500 kW/m2 [5]. The heat removal is increased by flow boiling, because the heat of evaporation can
be used: experiments have shown a critical heat flux of 3183 kW/m2 , at the cost of less stability [6].
To be able to use flow boiling in micro channels for practical applications, the operational stability needs to be
improved. To do this, a fundamental understanding of the flow boiling phenomena is needed [7].
On the small scales of evaporation inside micro channels, the continuum approximation fails. In this regime,
Molecular Dynamics simulations are useful. The thermodynamical properties of evaporation and condensation
in Molecular Dynamics have already been tested for mono-atomic Argon molecules and show good resemblance [8]. In real applications with micro channel cooling, the coolant consists of complex molecules with
many atoms. The intramolecular interactions, such as bonds between molecules and angle bends, should be
modeled correctly. To test these interactions, simple molecules are used. In this paper, the bond between two
molecules is tested for a nitrogen molecule (N2 ), a molecule with only two atoms and one bond.
1.1
Bond testing
From a classic point of view, a diatomic molecule can translate, vibrate, and rotate. However, the vibrational
contribution to the energy is absent at room temperatures, due to quantum-mechanical effects[9]. In general,
the vibrational energy Evib of a bond between two atoms is quantized. A characteristic temperature Tr can be
found, below which the vibrational energy is limited to the ground state:
h̄
Tr =
kB
s
k
,
m
(1)
where h̄ is Planck’s reduced constant, k is the spring constant of the bond between the two atoms and m is the
mass of the atoms. Because the spring constant k is in the order of 100 N/m[10] and the mass of atoms m
Corresponding author is E.A.T. van den Akker.
E-mail: [email protected], Telephone: +31 (0)40 247 2768
is in the order 10−26 kg, this characteristic temperature is in the order of 1000 K. In micro channel cooling
applications, the working temperature is lower, so the vibrational contributions can be neglected, and the two
atoms are rigidly connected. This rigid connection is integrated in the Molecular Dynamics simulation with the
Rattle method [11].
2.
Molecular dynamics simulations for nitrogen
To simulate the phase transition for N2 from liquid to gas, the Molecular Dynamics code Pumma[12] is
used. For intermolecular interactions the Lennard-Jones potential is used. The mass of one nitrogen atom
is 2.326 · 10−26 kg, and the molecular diameter σ = 0.31 nm. The bond length between the two nitrogen
atoms is 0.110 nm, and non-bonded interaction is with the Lennard-Jones potential with well depth ε/kB =
35.4 K[10]. The Lennard-Jones potential is cut off at 4.5σ, a value that gives good results for intermolecular interactions in phase transitions[8]. The size of the simulation box is 71 × 18 × 18 atomic diameters (Figure 1). In
all directions, periodic boundary conditions were used, so an infinite plane of evaporation/condensation is simulated. The simulation is set up such that the liquid is centered on the boundary by applying an extra force φ0
on both sides, given by
(
0.4x
if x ≤ σ;
σ2
φ0 (x) =
(2)
0
if x > σ,
and an mirrored force on the right side. This creates a potential well around x = 0, and will act as a condensation place.
m
5n
5.85 nm
5.8
22.0 nm
Figure 1: Schematic view of the simulation box with periodic boundary conditions in all directions. On the right
of x = 0 nm there is a small force to the left, on the left of x = 22.0 nm there is a small force to the right.
Initially, 4000 N2 -molecules are put into the simulation box in a fcc-lattice on the liquid side. To reach equilibrium between liquid and gas, the temperature of the simulation should be between the triple point (63.1 K)
and the critical point (126.2 K)[13] The simulation is then started with a Berendsen thermostat[14] close to the
condensation places on the left and the right to set the temperature of the particles in the simulation box at the
desired temperature (T = 71.4 K) and the simulation runs for millions of time steps.
3.
Results for nitrogen molecules
3.1
Temperature
The molecular dynamics simulations can be interpreted thermodynamically with the use of statistical mechanics.
Because the temperature at the boundaries was set to be 71.4 K and the system is in thermodynamic equilibrium,
it was expected that the temperature in the liquid and gas are both 71.4 K. The numerical results showed a mean
liquid temperature of 70.7 K and a mean gas temperature of 76.0 K. The uncertainty in the gas temperature is
relatively high, due to the low density, so although this value is too high, it is still acceptable.
3.2
Density
1000
900
100
Enthalpy h [kJ/kg]
Density n [kg/m3 )
800
700
600
500
400
300
50
0
−50
200
100
0
−100
0
5
10
Location x [nm]
15
20
0
5
10
15
Location x [nm]
20
Figure 2: Density (left) and enthalpy (right) profiles in the equilibrium situation of nitrogen vapor and nitrogen
gas at 71.4 K.
In the liquid phase, the density is 869 kg/m3 , which is close to the literature-value[16] of 833.56 kg/m3 .
In the vapour phase, the simulation gives a density of 16.5 kg/m3 , whereas the literature gives a value of
2.207 kg/m3 . The density in the gas phase is therefore overestimated in MD simulations. The density profile is
shown in Figure 2.
3.3
Pressure
In thermodynamic equilibrium between the liquid phase and the gas phase, the pressures in both phases should
be equal. The pressure in the liquid part is fluctuating more, and the gas pressure is more stable; this is due
to the effect that the adiabatic bulk modulus in the liquid is higher[15]. The mean pressure in the simulation
is 771 kPa, more than to the experimental result for 71.4 K found in literature[16], 45.48 kPa; this is due to the
higher density in the gas phase. The simulations results show a drop in pressure at the phase transition , which
is also found in other literature[17].
3.4
Enthalpy
Also shown in Figure 2 is the enthalpy profile. The (constant) bond contribution to the enthalpy can not be calculated from the MD simulations, so this is a relative enthalpy. In literature, the enthalpy goes from −134.6 kJ/kg
in the liquid to 72.10 kJ/kg in the gas phase, so the enthalpy jump is 206.7 kJ/kg[16]. The MD simulation
show an enthalpy in the liquid of −79 kJ/kg, and in the gas, of 113 kJ/kg, so the jump is 192 kJ/kg.
4.
Conclusions
The behavior of atomic bonds in Molecular Dynamics simulations was tested. This was done by analyzing
thermodynamic quantities in gas-vapor equilibrium of nitrogen, where the N –N bond in nitrogen was modeled
by the Rattle method. Although there is no full agreement between the Molecular Dynamics simulations and
experimental results, especially not in the gas phase, the agreement is good enough to use the Rattle method for
molecular dynamics simulations with phase transitions .
Acknowledgements
This work was financially supported by MicroNed grant 4-A-6.
References
[1] R. R. Schmidt, and B. D. Notohardjono, “High-end server low-temperature cooling” IBM Journal of Research and Development, vol. 46, p. 739-751, 2002.
[2] Semiconductor Industry Association, “The International Technology Roadmap for Semiconductors,” 2007
edition.
[3] F. P. Incropera, and D. P. DeWitt, “Fundamentals of heat and mass transfer,” Wiley, 2001.
[4] D. B. Tuckerman, and R. F. W. Pease, “High-Performance Heat Sinking for VLSI,” IEEE Electron Device
Letters, vol. 2, p. 126-129, 1981.
[5] W. K. Kuan, and S. G. Kandlikar, “Experimental Study and Model on Critical Heat Flux of Refrigerant-123
and Water in Microchannels,” Journal of Heat Transfer, vol. 130, p. 034503, 2008.
[6] J. Lee, and I. Mudawar, “Fluid flow and heat transfer characteristics of low temperature two-phase microchannel heat sinks – Part 1: Experimental methods and flow visualization results,” International Journal of
Heat and Mass Transfer, vol. 51, p. 4315-4326, 2008.
[7] S. G. Kandlikar, “Hight Flux Heat Removal with Microchannels - A Roadmap of Challenges and Opportunities,” Heat Transfer Engineering, vol. 26, p. 5-14, 2005.
[8] E. A. T. van den Akker, and A. J. H. Frijns, and A. A. van Steenhoven, and P. A. J. Hilbers, “Thermodynamics analysis of molecular dynamics simulation of evaporation and condensation,” 5th European ThermalSciences Conference, Eindhoven, 2008.
[9] M. Kaviany, Principles of Heat Transfer. John Wily and Sons, Chicester, 2002.
[10] A. K. Rappe, C. J. Casewit, K. S. Colwell, W. A. Goddard, and W. M. Skiff, “UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations,” Journal of the American
Chemical Society, vol. 114, p. 1002410035, 1992.
[11] H. C. Andersen, “Rattle: A ”Velocity” Version of the Shake Algorithm for Molecular Dynamics Calculations,” Journal of Computational Physics, vol. 52, pp. 24–34, 1983.
[12] A. J. Markvoort, and P. A. J. Hilbers, and S. V. Nedea, “Molecular dynamics study of the influence of
wall-gas interactions on heat flow in nanochannels,” Physical Review E, vol. 71, pp. 066702, 2005.
[13] P. Atkins and J. de Paula, Physical Chemistry. Oxford, 7th ed., 2002.
[14] H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola, and J. R. Haak, “Molecular dynamics
with coupling to an external bath,” Journal of Chemical Physics, vol. 81, pp. 3684–3690, 1984.
[15] J. Copley and S. Lovesey, “The dynamic properties of monatomic liquids,” Reports on Progress in Physics,
vol. 38, pp. 461–563, 1975.
[16] VDI Gesellschaft Verfahrenstechnik und Chemieingenieurswesen, VDI-Wärmeatlas. Springer, 9 ed., 2002.
[17] S. Marayuma, Molecular dynamics methods in microscale heat transfer, ch. 2.13.7. Begell House, 2004.