Microscopic dynamic analysis of heat and mass transfer

Nonlinear

Analysis,

Theory,

Methods

& Applicarionr, Vol. 30, No. 5, pp. 2797-2802, 1997 Proc. 2nd World Congress of Nonlinear Amlysts

0 1997 Elsevier Science Ltd Britain. All rights reserved 0362-546x/97 S17.M) + 0.00

F’tinted inGreat PII: SO362-546X(97)00369-6

MICROSCOPIC

DYNAMIC

ANALYSIS

OF HEAT AND MASS TRANSFER

- Thermalenergypropagationnearthe critical point and electronicdynamicsin a mesoscopic systemKOJI ISHII, YUJI KODAMA andTORU MAEKAWA Depertment

Key words andphrmes: Electronic tunnelling,

of Mechanical

Engineering,

Toyo University, Japan

2100, Kujirai,

Kawagoe,

Saitama 350

Molecular dynamics method, Thermal energy transfer. Critical fluid, Quantum Electronic oscillation, Mesoscopic system

dynamics method,

1. INTRODUCTION

Thanksto recent remarkabledevelopmentsandprogressin computerhardwareand software,microscale simulationssuchasthe moleculardynamicsmethod(MD) andMonte Carlo method(MC) arebecomingquite commonin microscaleanalysesandplaying animportantrole in estimatingtransportcoefficientsandanalysing microscopicstructures. Thoseclassicalmethodshave beenwidely acceptedasa third type of approach, comparablewith theoreticalor experimentalapproaches, to understandmicroscalephenomena. When electrons’dynamicsbecomeessential,the time-dependentcomplex quantumequationshave to be solved,which is what we call the quantumdynamics(QD) method.This QD methodisbeingdevelopednow to analysethe particles’inelasticcollisionproblem,the electroniccharacteristicsin a mesoscopicsystem,and molecularstructuresanddynamics. In this paper,we presentmicroscopicdynamic analysisof heat andmasstransferfocusingon (i) energy transferin a supercriticalfluid and (ii) electronictransferin a mesoscopicsystem. 2. ENERGY

TRANSFER

IN A SUPERCRITICAL

FLUID

The specific heat at constantpressureand isothermalcompressibilityshowtheir diversity at the critical pointof fluid. It hadbeenbelievedthat thermalenergycouldnot be transferrednearthe critical point sincethe thermaldiffusivity becomesvery smallandthat thermalenergytransfernearthe critical point observedunder terrestrialconditionswasdue to naturalconvection inducedby a strongtemperaturefluctuation. However, there have been severalreports about a fast heat transfer near the critical point even under microgravity conditionsin which naturalconvectionis suppressed [l-5]. This fast thermalenergytransfernearthe critical point is analysedby the moleculardynamicsmethodin this section. 2.1 Molecular

dynamics

analysis

Wecarriedout anMD simulationof Xe nearthecritical point using10,976molecules.The potentialenergy is expressedby the Lennard-Jones potential.

(2.1) wherethe potentialparameters &and oare 3.20x lO*’ J and3.98x lO-‘Om [6], respectively.The equationsof motionwereintegratedby the leap-frogmethod. The systemwasrelaxedasa canonicalensemblefor the first 100psscalingthe velocity of eachatomandwaschangedto a microcanonicalensemble.Periodicboundary conditionswere employedto overcomesurfaceproblems. 2797

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2.2 Result and discussion

The specific heat at constant volume was calculated by the fluctuation of kinetic energy. The critical exponent obtained by this calculation was 0.08, which agrees with that obtained by experiment [7]. Snapshots of the largest cluster formed in the system are shown in Fig. 1. Large clusters are formed near the critical point, while the cluster size decreases as the system deviates from the critical point. The relation between the fractal dimension of clusters and temperature is shown in Fig.2. As the system approaches the critical point, the fractal dimension decreases and reaches a constant value. According to the three dimensional Ising model, the fractal dimension is 2.43 at the critical point. The agreement is quite good. We calculated the mean kinetic energy of atoms contained in two cubes of 8.6 x 8.6 x 8.6 A3 which are 8.6 or 17.2 A apart. The Fourier power cross-spectra are shown in Fig.3. When the distance between two cubes is 8.6 A, there are two main peaks. The high frequency corresponds to molecular diffusion and the low frequency corresponds to acoustic waves. Those frequencies coincide with those obtained by macroscopic analysis [8]. On the other hand, there is only one peak when the distance is 17.2 8, as the propagation time by acoustic wave becomes the same as that by molecular diffusion. At a macroscopic level, temperature propagation time by acoustic wave is much shorter than molecular diffusion time, which explains why thermal energy propagates even near the critical point.

1.0 ps

1.5 ps

2.0 ps

1.5 ps

2.0 ps

(i) 4.3 K above the critical temperature Large clusters are formed and dissociate.

0.0 ps

0.5 ps

1.0 ps (ii) 144 K above the critical temperature

The cluster size is much smaller compared to case (i).

Fig. 1. Time variation

of cluster formation

( Only large clusters are illustrated

and dissociation

in this picture. )

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1.0 0.8 0.6

2.6

0.4 0.2

n

0.0

1o’O

109

IO”

10’2

10”

Frequency Hz (i) Two cubes are 8.6 A apart

n

0.8 0.6

w

2.3

1o‘5

IO4

Normalized

1o-3 temperature

1o-2

10-t

z 2

0.4

g

0.2

10s

1

E

10” 10'2 Frequency Hz (ii) Two cubes are 17.2 A apart 1o'O

10'3

Fig. 3. Fourier power cross-spectra (The temperature is 4.3 K above the critical temperature.

Fig. 2. Fractal dimension

3. ELECTRONIC

A

0.0

DYNAMICS

IN A MESOSCOPIC

)

SYSTEM

In a mesoscopic region, particles show their quantum characteristics. An electron’s tunnelling is a typical example of mesoscopic physics’ phenomena [9]. Nanodevices which make use of this electronic tunnelling are being studied intensively and considered as new devices for the next generation. We developed a quantum dynamics calculation method and analysed single electron tunnelling and two-electron oscillation in a mesoscopic system. The effect of electric field and size of the system on the electronic tunnelling and oscillation mode in a mesoscopic system is discussed. 3.1 Calculation

method

The Hartree-Fock equation shown below is solved numerically.

where i is the imaginary unit and a and fl representup- anddown-spinelectron, respectively. Lagrange matricescan’andePEareintroducedsothat the orthonormalconditionsaresatisfied. HamiltonianHa and HP areexpressedasbelow.

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(3.2)

where the first term is the kinetic energy operator, the second and fourth the Coulomb repulsion energy and the third the exchange term. We solved eq(3.1) by the finite difference method where the second order central difference formula was employed for both the time and spatial derivatives.

Electric field

Fig.4 Calculation

300 0

loo

200

model

300 0

100

300 0

200

X - coordinate (A) E = 0.3774 (MV/m) Fig.5 Time propagation

of electronic

probability

density

1st excited state 2nd excited state 3rd excited state 4th excited state 0

9th excited state 10th excited state

0

Fig.6

1

2 3 4 5 6 Electric field (MVlm)

Dependence

7

of electron’s traverse time on electric field

100

200

300

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3.2Resultanddiscussion We analysed single electron tunnelling in a system as shown in Fig.4 where a heterostrncture of GaAs/ AlGaAs/GaAs is modelled. The time propagation of electronic probability density when resonant electronic tunnelling occurs is shown in Fig.5. The electric fields in which tunnelling occurs coincided with those predicted by steady state analysis

[lOI.

We defined the electron’s traverse time through the barrier as the time when the probability of a tunnelled electron reaches its maximum. The dependence of the electron’s traverse time on the electric field is shown in Fig.6 As the electronic mode and electric field increase, the electron’s traverse time decreases.

Fig.7 Calculationmodelof two electrons

3

50

50 3 Y 2 0 38 100 ai

B .e 0 1 ‘00 X

50

50

0

0 0.1

0.15

Time(ns) (i) E = 0 (V/m)

0.2

0.1

0.15

0.2

Time(ns) (ii) E = 0.5 (MV/m)

Fig.8 mmevariationof averagepositionsof electronsin thecaseof parallelspins

Oscillations of two electrons are analysed, the calculation system of which is shoti in Fig.7. Dvo electrons are placed separately at the beginning as shown in the figure. The time variation of the average position of each wave function is shown in Fig.& The two electrons keep changing their positions in the case of parallel spins, while each electron oscillates around its average position in the case of anti-parallel spins. The oscillation mode is altered by a dc electric field (Fig.8 (ii)). The power spectra of the two electrons’ oscillations are shown in Fig.9. The main frequency increases with electric field in the case of parallel spins. In the case of anti-parallel spins, on the other hand, the main frequency does not change even if the electric field increases, but a higher frequency component appears however.

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(1) E=O(V/m)

(2) E = 1.0 (MV/m)

q-y-y-j 0

q--J 0.5

1

1.5 Frequency

0 (THz)

10

20

30

(3) E = 2.0 (MVlm) (i) Parallel spin Fig.9

(ii) Anti-parallel

spin

Power spectra of electron’s oscillation

4. CONCLUDING

REMARKS

Thermal energy transfer in a supercritical fluid was analysed by the molecular dynamics method. It was found that temperature propagates as acoustic waves rather than molecular diffusion near the critical point. Electron’s dynamics in a mesoscopic system was analysed in the next section. The electron’s traverse time was calculated and the two electrons’ oscillations were made clear. REFERENCES 1. Klein H., Schmitz, G. and Woermann, D., Temperature Propagation in near-critical Fluids prior to and during Phase Separation, Phys. Rev. A, Vo1.43, No.8, pp.4562-4563 (1991) 2. Guenoun, P, Khalil, B., Beysens, D., Garrabos, Y., Kammoun, E. Neindre, B.L. and Zappoli, B., Thermal Cycle around the critical Point of Carbon Dioxide under Reduced Gravity, Phys. Rev. E. Vo1.47, No.3, pp.1531-1540 (1993) 3. Bonetti, M., Perrot, F., Beysens, D. and Garrabos, Y., Fast Thermalization in Supercritical Fluids, Phys. Rev. E, Vo1.49, No.6, pp.4779-4782 (1994) 4. Beysens, D., New Critical Phenomena Observed under Weightlessness, Mareriuls and Fluids under Low Gravity, Vo1.464. pp.325, Springer (1995) 5. Straub, J., Either, L. and Haupt, A., Dynamic Temperature Propagation in a Fluid near its Critical Point Obseved under Microgravity during the German Spacelah Mission D-2, Phys. Rev. E, Vo1.51, No.6, pp.5556-5563 (1995) 6. Bemardes, N., Theory of Solid Ne, A, Kr, and Xe at 0°K. Phys. Rev. 63, Vol. 112, No.5, pp. 1534-1539 (1958) 7. Swinney, H.L. and Henry. D., Dynamics of Fluids near the Critical Point: Decay Rate of Order-Parameter Fluctuations, Phys. Rev. A, Vo1.8. No.5, pp.2587-2617 (1973) 8. Ishii, K., Masuda, S. and Maekawa, T., Thermofluid Dynamics and Molecular Dynamics Analyses of Thermal Energy Transfer near the Critic&l Point, Intemakmal Cenrre for Hear and Mass Tmnsfer Symposium on Molecular and Microscale Hear Transfer in Mamicds Processing and Other Applicakms (1996). in print 9. Mizuta. H., The Thysics and Applications of Resonant Tunnelling Diodes, Combridge .%&es in Semiconduclor Physics and Microelectronic Engineering, Cambridge University Press (1995) 10. Kodama, Y. and Maekawa, T., Single Electron Tunneling through an Insulator in a Quantum Well, Inrern&onal Cenrre for Hear and Moss Transfer Symposium on Molecular and Microscale Heat Transfer in Materials Processing and Other Applications (1996). in print