Ganguly, Ashim Kumar

Syam, D

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Indian ]. Phys. 64A (3), 259-262 (1990)Etch-pit formation-a convective diffusion processD SyamKrishnagar G ovt. C o lleg e , Krishnagar, W est Bengal, Ind ia  andAshim  Kumar GangulyDepartm ent of P h y s ic s , C a lcu tta  U n ive rs ity , C a lcu tta -7 0 0  0 0 9 , IndiaEtch-pit formation is one of the numerous physical phenomena which can be understood as convective diffusion processes. The general theory of such processes is given by Levich (Levich 1962 : see also Daccord 1987). Preliminary investiga­tions in this direction were carried out by the present authors a few years ago.We begin by writing down the differential equations for convective-diffusion processes (v is the fluid velocity, c the concentration of the etchant, other symbols are standard) :Psol,| j  +  (v.|7)c =  D p“c a t(Navier-Stokes eqn.) (1)(eqn. for convective diffusion) (2)We choose a coordinate system with origin at the vertex of the pit in the SSNTD. The track w ill be assumed to be perpendicular to the flat surface of the detector, and the (-I-)  ve direction of the Z-axis oriented back along the track. We can now specify the boundary conditions :(i) c--------►  Co(2->00)(ii) J =  Kc, on the reaction front (walls of the pit),whereK =  Ko[1 +  <l],l- l( r ,  t) =  ionization density in the detector, and Kp, <  are two constants.An approximate solution of this convective-diffusion problem can be worked out in the quasi-stationary lim it:^ ~ 0 ,at ' at(3)259Actually, inside the pit, v ^  0, since the track etch rate Vt ~  few /xm/hr. Thus we are essentially left to solve the differential eqn, fr’‘c= 0 ,  i.e.,^  , 1 d p^  P d p2 6 0  0  S a y m  a n d  A s h i n t  K u m a r  G a n g u l y(4)in the cylindrical polar coordinates.Now, on general physical grounds, we expect a narrow pit, conical in shape in a first approximation. Since a cone is characterized by a constant value of the ratio (p/z), the following appears to be a plausible solution of the problem :Eqn. (4) remains unchanged when the variables are transformed as follow s ;p Ap', z -► Az' (5)In view of this, a solution of eqn. (4) may be sought such that a combination of variables p and z remains unchanged by transformations (5 ); such a combination isSetting c — c{rf). we can write eqn. (4) in the form( , + r t ^ + ( 2 , +  l ) l ; = 0This equation can be solved easily (for rj > 0). We havedc Az‘dv p(p® +  Z<“)i'=*where A{ < 0) is to be fixed from the boundary condition (il). Finally, for fixed z,•nC(p, z ) -  c ( t; ) -  ^  +  ^  0.1 ^m]Now  the rate (dR;dt) at which the radius (R) of the pit, for a given h (depth, referred to the point of entry of the particle into the SSNTD), changes with time, is propor­tional to the diffusional flux rate (jai„) of the etchant on to the surface of the pit. AgainDARanddt ^1 +R * D I A (6)Etch-pit fo rm a tio n -a  convective diffusion process 261Since, on physical grounds, z = ^ V td t -h  (where V  ^ is the track etch rate) is much larger than R, eqn. (6) reduces toThusdR 1____ —dt RR t^'® (for fixed h)Actually, this solution does not correspond exactly to a conical etch p i t ; but we should remember that a cone is an approximation for the shape of the pit.If 1 ^  be negligible, then the eqn. for convective-diffusion, near the walls of the pit, can be written asV,* az b p * p dpi(7)Scaling arguments show that we may express c(p, z) as a function of This time the differential eqn. (7) reduces towhereThusk A . -  1 ^/C r Sidc AeFinally,dt z i ' * \  4z*/df k ,  (*» ii) dt RConsequently we again getR ~  t*'® (at fixed h).Since z t (for fixed h), therefore R — z^ '*. If the initial boundary condi­tion Is something like c ~ 0  on C=Co. i-e. c(Co) —  0, then, because R ~  z®'®, c(Co) ^ O a t  all later times. Thus R* ~  {®z represents the etch pit for large times.We, therefore, conclude that in either case the etch pit should look somewhat like a*paraboloid rather than a cone.262 D Sya m  and  A sh im  K u m a r  G a n g u ly  ReferencesDaccord G 1987 Phys. R ev. L e tt. 58 479Levich V G 1962 P hysico -C h em ica l H yd ro d yn a m ics  ed L E Scriven Prentice-Hall) (Englewood Cliffs N J

Etch-Pit Formation-a Convective Diffusion Process

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Etch-Pit Formation-a Convective Diffusion Process

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