RESPONSE OF RADIATION-INDUCED EVAPORATION TO THERMAL AND HYDRODYNAMIC PERTURBATIONS

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RESPONSE OF RADIATION-INDUCED

EVAPORATION TO THERMAL AND

HYDRODYNAMIC PERTURBATIONS

A. Korotchenko, A.A. Samokhin

To cite this version:

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JOURNAL DE PHYSIQUE ColZoque C9, suppl6ment au n o l l , Tome 41, novembre 1980, page C9-259

-RESPONSE OF RADIATION-INDUCED EVAPORATION _TO

THERMAL

AND. HYDRODYNAMIC PE.RT.URB.ATJQ!S

A.I. Korotchenko and A.A. Samokhin.

Academy of Sciences, P. I. Lebedev Physical Institute, 140scow, U. S. S. R.

.

RQsum6.- On analyse la rdponse lindaire du procQd6 stationnaire de l'dvaporation pour des liquides qui absorbent l'irradiation intense. On examine des perturbations thermiques et hydrodynamiques. En cas de perturbations thermiques qui sont conditionndes par des modulations harmoniques de l'intensi- t6 absorbante, la rdponse de la pression qui a lieu au cours du proc6dh de 1'Evaporation d6couvre-la conduite de rdsonance par rapport 2 la frdquence de la modulation. Les perturbations hydrodynamiques msnent 2 la modulation spatiale de la pression du recul qui joue un rale essentiel dans le problsme de la stahilit6 de la surface dvaporante. Une Qquation dispersive pour des ondes de surface (ripplond du liquide Qvaporant d'une manisre intense est resue.

Abstract.- A linear response of the steady-state evaporation process is investigated for liquids absorbing intense radiation. Thermal and hydrodynamic perturbations are considered. In the case of the thermal p'erturbations due to intensity modulation the vaporization pressure response shows a resonant behaviour with respect to the modulation frequency. The hydrodynamic, (ripplon-like) perturbation re- sults in differential pressure recoil which plays an essential role.in the stability problem of evaporating liquid surface. Dispersion relation is obtained for ripplons in rapidly evaporating liquids.

One dimensional heat conduction eq- If the radiation intensity

L+I~pc'a)

uation is widely used for description of is modulated near the constant valueI,~I radiation-induced phase transitions in then the linearized conduction problem absorbing condensed matter (see, e.g.

LI]

for the disturbed part of the temperatu- and references therein). However, appli-

reTW)

can be written as follows

cability limits of the approach are not

'9t - v a

-j@T

-

W k f i

+=&~erp(-.st+iW

well defined. This approach also involves

at

a2

W'qt

f

some essential hypothesis about nonequi-

_Cj(fl

\

_=&vY$

_T(@,t

_1.

₀

{

41 a2

0

librium phase transitions which require

further investigations both theoretical where,Tgc3) is unperturbed temperat'ure and experimental. For this reason it is distribution:

useful to study such properties of the

T ~ T ~ [ @ - A ) ~

~ ~ p ( - d o ~ ~ ,

pup

model which can directly manifest them-

A-

4(4+~/i16r)/

( q - b ) ,

( 2 )

selves in experiments.

1,

=

~ N ( L +

~TQs)

.

Vaporization pressure behaviour is

In (I)

the reference frame is used which one of the sensitive characteristic of moves together with the vaporization

h

rqdiation-induced vaporization process. front, its velocity l$+*Tf being sup-

In this paper an expression for linear posed to depend only on surface tempera- pressure response to harmonic modulation ture

TS

.

Heat capacity

C

,

thermal of the radiktion intensity is derived. conductivity * a

XpC

end dif fusivity

This expression shows some specific fea-

x

are approximated by the constants tures pertinent to the conduction appro- the latent heat of vaporiaation

L

de- ach and can be verified experimentally. pends on the surface temperature, prime

r )

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~9-260 JOURNAL DE PHYSIQUE

means d i f f e r e n t i a t i n g with respect t o

Tot

CTW

o.~s(L+cT&)

I

&=do)

&

a %/A ) The absorption c o e f f i c i e n t s do and

4

E o -

9

/ d b ( 5 )

f o r t h e main and modulated p a r t s of the

r a d i a t i o n may be d i f f e r e n t from each i n C G S u n i t s . other.

For the steady -st a t e surf ace t empe- r a t u r e o s c i l l a t i o n s

Ts

from (1) i t f o l - lows

r,

=

,

IU~~.~),'~[V(L+

.%dl'

,

+

=(9.-d3h[~ry/c+qr(9.0-9.)n,~qO-ab)n~:

n,=

~ U ( L +

~T~]Yc(~(s

+

dv-%

d2)

8 (3

h=

**ATesr*d/(i~+&v--**

v-&)).

~4

=

Cc-A)

,

~ t ~ [ 4 + +

ixp]

0/1

5 -

4ul$/zr2

and f o r t h e surface pressure v a r i a t i o n _Pig.₁

$0

_{behaviour a t d i f f e-} r e n t values of

&=

10(1), 5(2), 1(3),

one obtains

The f a c t o r

$ 8 4 d i f f e r s from unity only

s l i g h t l y . Its value can be determined a l s o experimentally in t h e a d i a b a t i c r e - gime where

\6

<<

{ m d

f

=

4 .

The case of s t r o n g absorbing substa- nces d-=23(o*

9

was considered i n some d e t a i l i n [I]

.

It was shown t h e r e t h a t modulus

T

of t h e complex function behaves resonantly with r e s p e c t t o t h e

0.2(4), O(5) f o r rl(bu1k l i n e s ) and 0.97(dotted l i n e s

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A s

&

i n c r e a s e s maximum

S;(

diminishes f i r s t l y and then begins t o increase. Por l a r g e values 6 )

&,,

diverges a s

a t

8.4

and remains f i n i t e i f

-

_g)#4

The divergence of

f

i s due t o the ohe dimensional i n s t a b i l i t y of t h e temperatu- r e d i s t r i b u t i o n a t

L1+ C

&

0

and

l r ) 4 .

In p a r t i c u l a r , i f

L'+c

>o,

~4

d

and

ce,y>

6 ,

then from ( 3 ) i t follows

( 6 )

TJ=

~ r r ~ ~ W / g v ( ~ + c )

modulation frequency. This expression shows e x p l i c i t l y t h e thre-

The resonant behaviour of t h e modu- ahold meaning of the condition

L'+c=

0 .

-

l u s

f

,

remains a l s o i n t h e bulk absorp- A thermal expansion of i r r a d i a t e d t i o n case a s it c l e a r from Pig.1 where l i q u i d c o n t r i b u t e s a s well t o t h e t o t a l

2p/$=

Y ~ ~ ~ Q ~ ~ z . s ( ~ - T ~ / ~ & + ~ ' ]

_a

Pressure r e c o i l . This c o n t r i b u t i o n

pe

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P,=.,&T~L

where ,is thermal expan- s i o n c o e f f i c i e n t and

h

denotes an ave- rage t h i c k n e s s of t h e expanded l a y e r . For t h e s u r f a c e a b s o r p t i o n case

3-

11/,

s o t h a t

Qe

grows approxima- t e l y a s t.4

Y'

while t h e s u r f a c e prss- s u r e

P

diminishes l i k e f,tjh i n t h e h i g h frequency l i m i t , These v a l u e s of

P

and

p,

become comparable w i t h each o t h e r a t frequency W P P & ~ which can be a s

8

-4

h i g h a s

( 0

S

,

i f , e-g.,

pa40

,

~3

140 ,

4 x.Q(

, p 4 0 ,

i n CGS u n i t s .

The corresponding frequency can be considerably lower f o r t h e bulk a b s o r p t i o n

v & / ~

provided t h e a b s o r p t i o n

a

c o e f f i c i e n t i s s u f f i c i e n t l y s m a l l &C

W / j L

In t h i s c a s e , however, t h e main f e a t u r e s of v a p o r i z a t i o n process depend s t r o n g l y on t h e behaviour of superheated metastable

The a v a i l a b l e experimentaf d a t a on r a d i a t ion-induced p r e s s u r e behaviour [2-61 a r e t o o s c a r c e t o give a d e f i n i t e p i c t u r e of t h e phase t r a n s i t i o n in superheated l i q u i d w i t h a f r e e s u r f a c e . F u r t h e r in- v e s t i g a t i o n s a r e needed, e.g,, t o d i s t i n - . g u i s h between s u r f a c e and bulk e v a p o r a t i o n processes. The p r e s s u r e response t o t h e i n t e n s i t y modulation can be probably used

f o r t h i s purpose,

It i s w e l l known a l s o t h a t a p l a n a r phase f r o n t becomes u n s t a b l e f o r t h e phase t r a n s i t 5 ons which i n v o l v e m e t a s t a b l e s t a t e s ( s e e , e.g.,

[7]

1.

Surface temperature response t o t h e time- and space-modulated i n t e n s i t y

feFP(i~t-i~)

can be considered i n t h e

same way a s described above. An expres- s i o n f o r .f(bl,~) i s obtained s t r a i g h t - forwardly from eq. ( 3 ) a f t e r replacement

r3

by W -

i%C

and m u l t i p l i c a t i o n of h.4 and b z by t h e f a c t o r

4 +

i m / % ~ ~

.

An e x i s t a n c e c o n d i t i o n f o r n o n t r i v i a l s o l u t i o n T S ( ~ , ~ ) + b a t

I=O

g i v e s a d i s p e r s i o n r e l a t i o n f o r thermal p e r t u r b a t i o n s of t h e p l a n a r v a p o r i z a t i o n f r o n t . For l i q u i d s , however, t h e hydrodynamic mot i o n of t h e s u r f ace a d j a c e n t l a y e r should be t a k e n i n t o account [I]

.

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C9-262 JOURNAL DE PHYSIQUE

where

U~CK)

is the unperturbed frequency of the surf ace waves.

Equation (9) can be analysed in gene- ral only numerically, though in some limi- ting cases stable and unstable analytic solution are also obtainable.

~ f , e.e. *q~+l<$and q L o C then from (81, (9) one has

This perturbation is stable because d > O

.

In the case of small absorptivity

A4

(C,

one finds instead of (11)

This corresponds to the unstable case provided

L'+c

>

0 .

An absolute value of the parameter & is much greater than unity because

pb>

f

.

For inten- sive evaporation process the factor

P/?$

is proportional to the liquid-

vapor density ratio f/fy

.

This factor shows clearly the impor- tant role of hydrodynamic movement in the stability problem of evaporating liquid surface. There

is

no such factor in the expression for thermal perturbations which are obtained without proper account of hydrodynamics.

It should be mentioned in conclusion that in the linearized approach the con-

sidered instabilities do not contribute to the surface pressure averaged over sufficiently large irradiation spot.

1. A.P. Gus'kov, A.I. Korotchenko,

A.A. Samokhin,Preprint No 203 P.N. Le- bedev Physical institute 1979

2.

N.I.

Popov, A.I. Korotchenko, A.A. Sa- mokhin, A.B. Uspensky,Preprint No 103

P.N. Lebedev Physical Institute 1979

Phys. Lett's

73A

No5, 6 1979

3. B.M. Zhirgakov, N.I. Popov, A.A. _- _. Samok- hin,Zh

~k

Teor. Fis.

4

19 8

4. M.W. S I ~ P ~ ~ Z , PrK. Kneub6z14jASA 24

1052 1978

5. A.I. ~orotchenko, A.A. Samokhin, A.V. Sidorin,Lebedev Institute Reports No3 35 - - . 1979

6. P. Giovanneschi,

D.

Dufresne, 3.P. Ca- ressa, Ph. Bournot,Appl. Phys. Lett's

36 882 1980