COUPLED EQUATIONS OF MODULATED PHOTOTHERMAL EFFECTS, HYPOTHESES AND SOLUTIONS

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COUPLED EQUATIONS OF MODULATED

PHOTOTHERMAL EFFECTS, HYPOTHESES AND SOLUTIONS

F. Lepoutre

To cite this version:

F. Lepoutre. COUPLED EQUATIONS OF MODULATED PHOTOTHERMAL EFFECTS, HY- POTHESES AND SOLUTIONS. Journal de Physique Colloques, 1983, 44 (C6), pp.C6-3-C6-8.

�10.1051/jphyscol:1983601�. �jpa-00223160�

COUPLED EQUATIONS OF MODULATED PHOTOTHERMAL EFFECTS, HYPOTHESES AND SOLUTIONS

F. Lepoutre

Laboratoire d'Gptique Physique, ESPCI, 10, rue VauqueZin, 75231 Paris Cede= 05, France

C. N.R.S., GR

EcoZe Centraze, Chatenay-~a~abry, France

Resume - Les equations coupl6es de la photothermique sont resolues en regime sinusoidal. Les hypotheses habituelles et les solutions correspon- dantes sont discutees pour fournir des points de vue physiques simples.

Abstract - The coupled photothermal equations are solved for a sinusoidal time dependence. The usual hypotheses and the corresponding solutions are discussed in order to provide simple physical insights.

It is interesting to recall that the equations of the photothermal effects can be obtained in any material from five basic laws /I/:three laws of conservation for mass, momentum and energy and two thermodynamical laws, the equation of state and the variation of entropy. In these five equations some assumptions are usually done

- the effect of viscosity is neglected,

- the energy transfers are only due to conduction,

- the sample under study can be considered as a fluid.

It has been shown /I/ /2/ that the first two conditions are generally well ful- filled experimentally ; the last one is not, when dealing with a solid medium / 3 / . Nevertheless, to begin this study we retain the three above hypotheses.

Considering the previous laws for each medium i of the system under study, one obtains two coupled differential equations /4/

The usual notations have been used

is the heat capacities ratio,

CL =

(aP/3T)V, po is the density, B the bulk modulrls, k; the heat conductivity,

K;

the heat diffusivity. Following the arrows on eq. 1 and 2, we see that the source term h (heat power deposited per unit volume in the absorbing medium produces a temperature variation (eq. 1). This temperature variation acts in eq. (2) as a source term and produces a pressure variation. Finally, the pressure variation modifies the temperature distribution in eq. (1).

In order to understand the physical meaning of the coupling terms, let us assume that the medium (index i) is not conductive. Then the thermal diffusivity ~i equals zero, so that the term d~/dt-{(~-l)/~@ }(dP/dT) in eq. (1) must equal zero too. Using this relation between (dT/dt) and (dP/dt) in eq. (2), one obtains

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983601

JOURNAL DE

where Ci is the adiabatic sound velocity.

Equation

is exactly the classical sound propagation equation. Thus the coupling term in eq. (2) represents the modification of the sound propagation due to heat conduction in the medium.

The second coupling term is proportional to (y-l)/ya which can be written also T a /poCp where aT is the thermal expansion coefficient. If the medium is condensed,

0

T

aT is very weak (or y is close to 1) and the coupling term in eq. (1) can be neglected

in such a medium, the temperature distribution is then given by a

simple heat diffusion equation

Thus, the coupling term of eq. (2) represents the generation of heat by alternative compressions and expansions of the medium.

The two differential equations are of course three dimensional equations which depend upon r,

z and

(see fig. 1). Though the 3-D solutions can be found without further hypotheses, the physical insights are easier to obtain, when the problem can be solved with a unique spatial variable, i.e., when the temperature and pressure distributionsare only functions of z and

To be valid, this hypothesis requires at least that

be independent of r, i.e., that the surface of the absorbing medium (s) receives a uniform light beam. Even in this case, the I-D hypothesis is not straightforward

we shall discuss this problem later.

Assuming both P and T to be proportional to exp j(kz

wt), the two differen- tial coupled equations (1) and (2) become a linear system

its solution is the sum of the general homogeneous solution (h

and a particular solution depen- ding upon the function h(z). The homogeneous solution (i.e., the solution for the medium (i) different from ( s ) , and especially for (g))is

The cwo first terms of each equation are called acoustic terms (ac) since they are characterized by the acoustic wave number

where Ai is the sound wavelength.

The two last terms of

and P. are called thermal terms (th) since they are .haracterized by a wavenumber depending only upon the thermal properties in the medium i

(l+j)/ui

(I+j) ( ~ 1 2 ~ ~ ) 112 where pi is the thermal diffusion length.

Since this wavenumber is complex, the thermal terms are damped. It is also impor-

tant to note that the pressure and the temperature of each mode are proportional

dth

j way K;/c~ (1 0) Table I gives some orders of magnitude of the quantities characterizing the two modes, Four remarks about these numerical values have to be noted

- the sample wavelength Ai is large, - the thermal diffusion length f ~ i is small,

- the ratio dac is very large, so that even a small acoustic temperature rac can be associated with a large acoustic pressure Pa,.

- the ratio dth is very weak, especially in a liquid, so that even a large thermal temperature rth is unable to create an appreciable thermal pressure Pth.

All these orders of magnitude do not depend upon the particular case under study.

They remain also valid when 3-D or viscosity effects are taken into account.

On the contrary, the constants A,

C, D of eq.

are determined by the boundary conditions (continuity of temperature, heat flow, displacement) at the various interfaces b-s, s-g, g-w, so that it is then necessary to choose a parti- cular example to carry out this work. We have chosen the simplest, but also the most. oftenly encountered one which is characterized by the following conditions in the fluid (g) (length (R

e

and a motionless interface L-g.Eq. (11) is easily satisfied while the second condition ne9ds a sample (s) having a small thermal expansion coefficient /5y (when eq.(l and 2) are valid). Within these two hypotheses, one can calculate A, B and D as functions of C and one obtains

The first term of -rg which decreases with

is the thermal term rth and the second one, independent of z, is the acoustical term

Since pg << Rg,

(z=O)is almost equal to C, i.e., C is the sample surface temperature o0.

This result is general

in all cases, r can always be calculated as a g

function of e0and this explains why the photothermal methods are measurements of surface temperatures.

Though pg is small with respect to Lg, it is not obvious that

is negligible with respect to rth

actually, table I (forRg

1 cm)shows that it occurs only when g is a Liquid (y 1). This result is straightforward when one calculates the average values of r and

th ( < T ~ ~ > and <rth>) over the volume of g

ac

< T ~ ~ > / < T ~ ~ >

y-I (14) Eq. (14) clearly shows that <T > and < T ~ ~ > have the same order of magnitude in

ac

a gas. An experimental verification of this theoretical temperature distribution

is given in poster 1.3.

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Eq. (13) shows that Pg equals Pa, which is, of course, proportional to rac. This expression is identical to that deduced from the thermal piston model of

Rosencwaig and Gersho 161.

The sample surface temperature oo ^(or

is easy to obtain since it is sufficient to solve a simple heat diffusion equation (eq.

We do not develop this solution which can be found in many papers (see for instance ref. / 4 / ) and we come back now to eq. (1) and (2) written for (g) (without the source term h) to examine the three dimensional (3-D) effects

let us integrate all the terms of these equa- tions over the volume of the gas 171. The terms thus averaged have interesting physical meanings

for instance, the first term of eq. (1) multiplied by

leads to the net heat flow Gg in the medium (g). The final result of this integration can be written in the following form

where AVg is the gas volume variation. The first term of eq. (15) is due to heat diffusion in (g) and is called the thermal piston 151, 161. The second one is proportional to the sample (s) volume variation AVs when the walls of the cell are rigid. It is called the mechanical piston /5/ or the acoustic piston because it is due to the acoustic mode in (s). Eq. (15) is known as the composite piston of McDonald and Wetsel 171.

We firstly discuss the second term, i.e., the mechanical piston. When the boundary conditions of the fastening of (s) are such that no stresses appear in (s), the equation of state of (s) is identical to that of a fluid

AV, is then simply the free thermal expansion of (s). We have seen that, in a solid, this contribution is negligible because y is close to 1. In such cases the mechanical piston is negli- gible. However, there exist situations where the mechanical piston can become important

- when (s) is a liquid, at low modulation frequencies /5/,

- when (s) is a powder 181. The interstitial gas which has a large thermal expan- sion coefficient

1 1 ~ ~ ) can produce at the interface (s-g) an important mechanical effect,

- when the boundary conditions for the fastening of a solid sample are such that stresses occur 191, /lo/. In such cases, eq.

and (2) become invalid for (s) since (s) can be no longer considered as a fluid. Many papers in this confe- rence have discussed such thermoelastic effects /Ill.

Let us discuss now the first term of eq. (15), i.e., the thermal piston. If one assumes that no heat is transfered from (g) to the outside of the cell, then

Gg is equal to the heat Q(s-rg) which diffuses from

to

Now (b(s-g) depends only upon the quantity of energy deposited per second in (s) if, once again, no heat loss from (s) to the outside of the cell occurs. @(s+g) is then independent of the radial distribution of the light beam. The pressure Pg can be deduced from a I-D calculation using the average value of the beam profile over r. This physical insight is well confirmed by a complete 3-D calculation /12/,/13/

if the cylindrical symmetry is respected in the cell, one finds thatthe temperature distribution in

can be written as a series 1131. When no heat losses occur, only the first element of the series (which is produced by the average value of the light beam over the sample surface) leads to a pressure which is not damped.

This pressure is exactly that calculated using the I-D model. The other terms of the series lead to a pressure which equals exactly zero.

This result is also valid when a part of the heat flow is transfered to the window w 1141. On the contrary, when radial heat losses occur in one of the media, important discrepancies between 3-D and 1-D results appear. Figure 2

(from paper 1.2 of these ~roceedings) shows experimental ~ointsobtained by

Quimby and Yen 1151 a few years ago. Curve @ has been calculated with the I-D

necessary to calculate the 3-D effects with the complete expression of the tempera- ture

+

as it was suggested by the results of the 1-D calculation

th ac previously discussed.

Figure 2 and the papers presented in session 1 prove that, now, the theory of modulated photothermal effects is well established. Nevertheless, the mathematical results are not always straightforward. This paper has recalled the usual hypothe- ses which lead to the simplest results. Some physical insights have tried to point out the effects which can occur when some of these hypotheses are not valid.

Table I - Numerical values for air and benzene at 10

and 100 Hz and a lcm length cell.

Fig. 2 - 3.D effects on the photo- acoustic pressure P

(from HcDonald). g +:experimental points in He (Quimby and Yen 1151)

curve

l.D The- ory

curve @ :3.D theory without acoustic coupling

curve @

3.D Theory with acoustic coupling.

Pig. 1 - Schematic represen- -- tation of the photoacoustic

cell.

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