New Model for Heat Transfer in Nanostructures

Copyright © 2012 American Scientific Publishers All rights reserved Printed in the United States of America

Journal of Computational and Theoretical Nanoscience Vol. 9, l-12, 2012

New Model for Heat Transfer in Nanostructures R. Stana1, I. Casian Botez2· *, V. P. Paun 3, and M. Agop 4• 5 1

Faculty of Physics, "Al. I. Cuza" University, Blvd. Carol I no.11, lasi 700506, Romania 2 Department of Electronics, Telecommunication and Information Technology, "Gh. Asachi" Technical University, Blvd. Copou no. 22, lasi, 700029, Romania 3 Faculty of Applied Sciences, Politehnica University of Bucharest, Department of Physics, Bucharest, 060042, Romania 4 Laboratoire de Physique des Lasers, Atomes et Molecules (UMR 8523), Universite des Sciences et Technologies de Lille, Villeneuve d'Ascq, 59655, France ;:;, 5 Physics Department, Technical "Gh. Asachi" University, Blvd. Mangeron no.56, lasi, 700029, Romania A new model on the heat transfer processes in nanostructures considering that heat flow paths take place on fractal curves is established. In the dissipative approximation of this process, for Peano type heat flow paths and synchronous movements at differentiable and non-differentiable scales, the heat transfer mechanism is of diffusive type. In the dispersive approximation of the same process, both at differentiable and non-differentiable scales, the thermal transfer mechanism is given through the cnoidal oscillation modes of the temperature fie ld (ballistic thermal transport). In such conjecture, two flow regimes result: one by means of waves and wave packets and the other by means of solitons and soliton packets. Practically, we discuss about an unique mechanism of thermal transfer in which the usual one can be seen as approximations of it.

Keywords: Fractals, Nanostructures, Nano-Fluides, Transport Phenomena.

1. INTRODUCTION If at a macroscopic scale the heat transfer mechanism implies either diffusion type conduction or phononic type conduction, 1· 2 at a microscopic scale the situation is completely different. This happens because the macroscopic familiar concepts cannot be applied at a microscopic scale, e.g., the concept of a distribution function of both coordinates and momentum used in the Boltzmann equation. 3 Moreover, fundamental concepts such a temperature cannot be defined in the conventional sense, i.e., as a measure of thermodynamic equilibrium. 4 Thus anomalies might occur: the thermal anomaly of the nanofluids, 5- 7 etc. According to our opinion, anomalies become normalities if their specific measures depend on scales: heat conduction in nanostructures differs significantly from that in macrostructures because the characteristic length scales associated with heat carriers, i.e., the mean free path and the wavelength, are comparable to the characteristic length of nanostructures. 4 Therefore, we expect to replace the usual mechanisms (ballistic thermal transport, etc.) by something more fundamental: a unique mechanism in which the physical measurea should depend not only on spatial coordinates and time, but olso on scales. This new way will be possible through the Scale Relativity •Author to whom correspondence should be addressed. J Comput. Theoc Nanosci. 2012, Vol. 9, No. 1

(SR) theory.ll- 11 Some applications of the SR theory at the nanoscale was given in Refs. [12, 13]. In the present paper, ~ a new model of the heat transfer on nanostructures, considering that the heat flow paths take place on continuous but non-differentiable curves, i.e., an fractals, is established. The structure of the paper is the following: mathematical and physical implications of the fractality in the heat transfer processes are presented in paragraphs 2 and 3; dissipative and dispersive approximations of the heat transfer processes are analysed in paragraphs 4 and 5, respectively.

2. CONSEQUNCES OF NON-DIFFERENTIABILITY IN THE HEAT TRANSFER PROCESSES Let us suppose that the heat flow take place on continuous but non-differentiable curves (fractal curves). The nondifferentiability implies the followings: 8- 11 (i) A continuous and a non-differentiable curve (or almost nowhere differentiable) is explicitly scale dependent, and its length tends to infinity, when the scale interval tends to zero. In other words, a continuous and non-differentiable space is fractal, in the general meaning given by Mandelbrot to this concept; 14 (ii) There is an infinity of fractals curves (geodesics) relating any couple of its points (or starting from any point), and this is valid for all scales;

1546-1955/2012/9/00 l/012

doi: 10.l 166/jctn.2012.1996

New Model for Heat Transfer in Nanostructures

Stana et al.

(i ii ) The breaking of local differential time reflection invariance. The time-derivative of the temperature field T can be written two-fold: dT . T(t + dt) - T(t) -=hm-----dt dH O dt (la, b) dT . T(t) - T(t - dt) = hm --'-'------'----'dt d1--+ 0 dt Both definitions are equivalem in the differentiable case. In the non-differentiable situation these definitions fail, since the limits are no longer defined. In the framework of fractal theory, the physics is related to the behavior of the function during the "zoom" operation on the time resolution 01, here identified with the differential element dt ("substitution principle"), which is considered as an independent variable. The standard temperature field T(t) is therefore replaced by a fractal temperature field T(t , dt), explicitly dependent on the time resolution interval, whose derivative is undefined only at the unobservable limit dt ---* 0. As a consequence, this lead us to define the two de1ivatives of the fractal temperature field as explicit functions of the two variables t and dt,

d+T = lim T(t+dt, dt)-T(t, dt) dt d H O+ dt dJ dt

.

- - = hm

d1--+0_

(3a, b) where d±x(t) is the "classical part" and d±t;(t, dt) is the "fractal part." (v) the differential of the "fractal part" of d±X satisfies the relation (the fractal equation)

d±g; = A ~ (dt) 1 f Dp

(4a, b)

where >..~ are some constant coefficients, and DF is a constant fractal dimension. We note that for the fractal dimension we can use any definition (Kolmogorov, Hausdorff,8• 14 etc.); (vi) the local differential time reflection invariance is recovered by combining the twO, derivatives, d+ / dt and d _/ dt, in the complex operator: =

V= V+ +V _ 2

(7a, b)

U= V+ -V 2 The real part, V , of the complex speed, V, represents the standard classical speed, which is differentiable and independent of resolution, while the imaginary part, U, is a new quantity arising from fractality, which is nondifferentiable and resolution-dependent; (vii) the average values of the quantities must be considered in the sense of a generalized statistical fluid like description. Particularly, the average of d±X is (8a, b) with (9a, b) (viii) in such an interpretation, the "particles" are indentified with the geodesics themselves. As a consequence, any measurement is interpreted as a sorting out (or selection) of the geodesics by the me'asuring devices.

(2a, b)

T(t , dt)-T(t-dt, dt) dt

The sign, +, corresponds to the forward process and, - , to the backward process; (iv) the differential of the fractal coordinates, d±X(t , dt), can be decomposed as follows:

;

with

~ ( d+:id_ ) - ~ ( d+~ d_)

(5)

3. COVARIANT TOTAL DERIVATIVE IN THE HEAT TRANSFER PROCESSES Let us now assume that the curves describing the heat flow (continuous but non-differentiable) is immersed in a •)\ 3-dimensional space, and that X of components X ;(i = "\ U) is the position vector of a point on the curve. Let us also consider the fractal temperature fluid T (X , t ), and expand its total differential up to the third order:

ar J a2 T . . d±T = -dt+ '\IT -d±X + --. - d±X'd±X 1 a1 2 ax· ax1 +~

3

aT d X id Xi d xk ± ± 6 axiax1axk ±

( lOa, b)

where only the first three terms were used in the Nottale's theory (i.e., second order terms in the equation of motion). The relations ( !Oa, b) are valid in any point of the space manifold and also for the points X on the fractal curve which we have selected in rel ations ( 1Oa, b). From here, the forward and backward average values of this relation take the form:

Applying this operator to the "position vector" yields a complex speed

V = dx

=

dt

~(d+X +d_X)- ~(d+X-d_X) 2

dt

2

V + + V - - i V + - V - = V - iU

2 2

2

(Ila, b)

dt

(6)

We make the followin g stipulation: the mean value of 'th_e function f and its derivatives coi ncide with themselves, J. Comput. Theor. Nanosci. 9, 1-12, 2012

New Model for Heat Transfer in Nanostructures

Stana et al.

and the differentials d ±X; and dt are independent, therefore the average of their products coincide with the product of averages. Thus, the Eqs. (I la, b) become: 2

oT I o T ( . •) d±T= - dt+\7T· (d±X) + - -.- . d±X'd±X 1 ar 2 ax•ax1

Under these circumstances, let us calculate (a T / ot ). Taking into account Eqs. (18a, b), (5) and (6) we obtain:

aT = ~[d+ T ar 2 dt

1 oT

1 ()3T ( . 1. k) + -6 ax;ax1axk d ± X'd ± X d ± X

(l 2a, b)

.

+

(

. .)) d±t'd±e

;

03 T

2 -

=

-aT +

1

o3 T

d

dt

(v++v_ Y+ -v-) 2

+ + +

a,

+

b)

-1

2

(dr )c 21op). \ f T + - - --

1

4

)-i(,\;+ ,.\+}- ,.\;- ,\l- )] ~ ax•ax1

- - -

. 0 T.

ax•ax1axk

,.\ ~,.\ ~ + ,.\ ~ ,.\ ~ )- i( ,.\ ~,.\ ~ - ,.\ ~,.\ ~) Ja:~;XJ

(dt) (3/Dp)-1 [ ( ,.\; ,.\} ,\k +..\; ,.\} ,\k ) 12 + + + - - (1 9a, b)

This relation also allows us to define the fractal operator:

a

a

A

'

(dt )- 1 = -+ V ·\7T+---ot 4

If we divide by dt and neglect the terms which contain differential factors (for· details on the method see Refs . [12,13]), the Eqs. (16a, b) are reduced to: d T oT 1 o2T . . _±_ = + V . \TT+ --.-.,.\' ,\1 (dt)(2 /Dp)- l dt at ± 2 ax•ax1 ± ±

+ -6 ax;ax1axk ,.\;± ,.\}± ,\k± (dt)( 3/Dp)-\

~(dt) (2/Dp)- 1 ~

- -2

(dt) (3/Dp)- \ [ ( ,\;,\} ,\k+,\; ,\j ,\k ) 12 + + + - - -

x [( (16

i i)T 2 at

3

.

(3/Dp ·) - 1 dt

a2T .. ax•ax1

-

a3 T i aT + __ ax;a x 1axk 2 at

- i ( ,.\'· ,.\]· ,.\ k. - ,.\'· ,\l· ,.\ k. ) ]

+ 6 oX;oXloXk d ±x'd±xld±x k k

.

. \TT+,.\; ,\l

ot

+

±

·

i

x [ ( ,.\; ,\1 + ,.\; ,.\} + + - -

---,.\;A} (dt)( 2/Dp) - ldt

·

i

+..\; ,\l ,\k _!_ ( dt) C3/l'Jpl- 1 .o3T - - - 12 ax•ax1a x k

2

o3T

F

- -4

+ ~V

oT I o2 T . . I d±T = -dt+d±x·\7T+- - . - .d ±x'd±x 1+ot 2 ax•ax1 2

I

o T 1 iJT ax ;oXl ()X k + 2at

(J/ D ) - I

. .1 c21v ) . \TT+,.\' ,\1 - (dt) F -

.

Then, Eqs. (13a, b) may be written under the form:

+ -6 iJX;oXliJ Xk ,.\'± A]± A± (dt)

3

1

+ + + 12

(15a, b)

±

k

_ ,.\; ,\1 ,\k -(dt )(3/Dp)-1

Then, let us consider the mean (d±g;d±gid±gk). If i =f j =f k, this average is zero due the independence of d±g; on d ±g1 and d ±e. Now, using Eqs. (4a, b), we can write:

o3 T

J

i i ~T - -V ·\TT-,.\; ,\1 -(dt) C2/ Dp )-l _ _. - . 2 + + +2 ax•ax1

(14a, b)

I

o2 T

. .I

,\l ,\k _!_(dt) (3/Dp) -I ()3T - - -12 aj ioXl oXk

Even the average value of the fractal coordinate, d±g\ is null (see (9a, b)), for the higher order of the fractal coordinate average the situation can be different. First, let us focus on the mean (d±td±gJ). If i =f j, this average is zero due the independence of d ±g; and d±gi. So, using ( 4a, b ), we can write:

oT ax;ax1

ddt_T)]

+ ,.\;

+ 6 ax;ax1axk

x (d ±x;d±x 1d ±xk+(d±td±g1d±e)) ( 13a, b)

X

I

+ -21 V 1

-i ( d+ T _ dt

+ A+A+A+ u(dt)

oT I o2 T d ±T = - dt+'lT·d±x+ - - ot 2 ax;ax1 .

dt

= - - +-V ·\TT+,.\' ,\1 -(dt) (2/Dp) - I __ . -. 2 ai 2 + + +4 ax•a x 1

or more, using Eqs. (3a, b) with the property (9a, b ),

x ( d ±x'd±x1

+ d_ T

x [ ( ..\; ..\1

+ +

+ ..\;- ..\1- )

- i(..\; ..\1 - ..\; ..\1 + + - -

)J .a2-. ax•ax1

a 1 a2 . . -+V · \!+---.- .,.\' ,\1 (dt) (2/Dp)-1 at ± 2 ax•ax1 ± ± + -1

a3

6 ax; ax 1axk

,.\; ,\1. ,\k (dt)(3/Dp )-l ±

±

(18a, b)

±

J. Comput. Theor. Nanosci. 9, 1-12, 2012

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New Model for Heat Transfer in Nanostructures

Stana