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Formation of the universal distribution function in the dimension space for new-phase particles in the diffusive decomposition of the supersaturated solid solution
I Phyr Chem. Solids. 1978. Vol. 39. pp 367-374
Pergamon Press.
Printed in Great Britain
FORMATION OF THE UNIVERSAL DISTRIBUTION FUNCTION IN THE DIMENSION SPACE FOR NEW-PHASE PARTICLES IN THE DIFFUSIVE DECOMPOSITION OF THE SUPERSATURATED SOLID SOLUTION v. v. SLEZOV Physico-Technical Institute, Kharkov, U.S.S.R. (Received21 March 1971; accepted 19 August 1977) Abstract-The methodof time-asymptoticsolution of nonlinearequationsdescribingthe decompositionof supersaturatedsolid solutions is developed.The distributioncurve is plottedfor any dimensionat the given momentof time. The results are generalizedfor the case of volume sources of the solute, for the multicomponentmultiphase solid solutions and for the differentmass-transfermechanisms.
1.
INTRODUCTMlN
The theory of diffusive decomposition in the later stage
(coalescence) in zeroth approximation was developed by Lifshitz and Slezov[l]. In this approximation they qualitatively analyzed the equations describing the diffusive decomposition and showed that at sufficiently large times the contribution to the material from the new-phase particles whose size exceeds the “blocking” point is negligible. They also showed that the supersaturated solution must have an exact time behavior. The growth rate of particles in relative dimensions and its derivative vanish at the blocking point u,,. This approximation uniquely leads to a universal solution, which isindependent of the initial conditions, for the size distribution function of particles for u I u* In the region u 2 u,, the distribution function in zeroth approximation is zero. However, the dynamics of plotting the universal distribution function and its relation to the initial distribution function cannot be determined in this approximation. It is also not possible to determine the distribution function in the region of the blocking point u,, and beyond it, nor a more precise asymptotic behavior of the supersaturated solution with time and the time TVat which the asymptotic time description of the diffusive decomposition can be used with sufficient accuracy. Since all these values are closely related to the initial distribution function, in order to calculate them a systematic theory must be formulated for the solution of the equations describing the diffusive decomposition of the supersaturated solution with allowance for the initial conditions. This paper is devoted to the formulation of this theory.
unit volume: f(R, t) dR = dN is the number of particles with dimensions R 5 R’ I R t dR per unit volume. Like in Ref. [l], we use more convenient relative dimensions of the particles U = RIR,tir and the times R4t) = T = In V&,(t)/&,(O)1 = 3 In (A(O))/(A(t)). u/A(l), where a = 2(4kT)c,, u is the interface surface tension, o is the volume per atom of the dissolved component, k is the Boltzmann constant, and T is the absolute temperature. The relation f(R, 0 dR = cp(s 7) du
Let us examine a one-component supersaturated solu-
is between the distribution functions in the absolute and relative variables. These relative dimensions of particles must be introduced because an ordered dissolution of these particles always takes place in these variables, beginning with a certain time, and the distribution function at the given time is determined by the asymptotic behavior of the initial distribution function. We note that if the particles have too much excess material, they will be slightly dissolved after ‘some finite time, causing the level of supersaturation of the solution to initially increase for a certain time, dddr < 0, and then to decrease monotonically, ddddr>O. We investigate the diffusive decay at d7/dt>O. This situation always occurs at low supersaturation, irrespective of the initial conditions, which determine only the time when dT/dt begins to increase monotonically with time. Moreover, as demonstrated by Liishift and Slezov[ll, at A4 1 the equilibrium concentration and the steady diffusive flow will have sufficient time to establish at the boundary of the particles. Under these conditions the growth rate of the particle resulting from diffusive mass transfer across the bulk of the material is [ I]
tion characterized by supersaturation A(r) = E - c,, where E = E(t) is the average concentration of the solution at the given time and c, is the equilibrium concentration at the plane boundary, and by the distribution function of the new-phase particles at the given time in the dimension space R, j(R, t), which is normalized to
where D is the diffusion coefficient of the dissolved component.
2. CANONICAL FOPMOF ‘IliE FUNDAMENTAL EQUATIONS
v. v.
368
SLUOV
Using the variables u and r the growth rate can be expressed as follows: Taking into account y(z) and the fact that when 7 = y. du/dr at’ the point u0 = 3/2 has a second-order zero, we can rewrite eqn (4) as follows:
du7(7HU-1)-U’
dr
3u2
$=-g(u, T _ RL(0). aD’
y = 27 ’ 7’
= -&
this ‘form of the 7(r) function is required for ordered motion from right to left in relative dimensions, which corresponds to du/dr
cpl.4= fduh ul.-0 = u.
(2)
Since we are analyzing a later stage of the diffusive decay when Ati I, the fluctuating production of the new-phase particles with R > a/A may be neglected and the right-hand side of the equation of continuity can be set to zero. This corresponds to the “hydrodynamic” approximation whose general solution is
(P(u.I) = fdu(u, ?)I$
(3)
D = u(u, r) is the integral of the characteristic d+7(u;f!-u’
(u + 3)(u - ua)2 + Y(u
(4)
The dissolved component can be written as’follows:
I-
fduht’b. 7) du e,o.r,
_g=_(u+3Ku-3*
excess quantity of the material.
I
.
(6)
du -= dr
I
u
3u’
-
;(u - uCJ*++*(r)] [ _g=_(u+3Ku-32 3u’
u*cucu,
(7)
U>U,>U*
Here the region near the point u,,, where the E*(T) function has an important role, has been isolated. The boundaries of the transformation region u, and u2 satisfy the condition:
l2(7)(U2
1,
-
uJ2
Q
I.
We determine the flu) function for u 7 u0 and u < u0 whose difference at the two u points is the time required to go from the larger value of u to the smaller one
(8)
(5)
where A(0) is the initial saturation and Q0 is the total
- l)r2(7)
When r+m l*(r)+O. This behavior of E*(T) This behavior of c*(r) is attributable to the fact that as r increases as a result of ordered motion from right to left, the more remote regions of the initial distribution function, which is a decreasing function of its argument fdu) 4 I/U”; n > 4, u +m, from the distribution function at a given time to the left of the blockina point, which is the main contribution to the matenat balance. Since duldr ~0, the number of particles to the right of the blocking point decreases with increasing time; hence their leakage rate across the transformation region near the blocking point will decrease so that all the excess material can settle on them. This means that the distribution function will condense in a self-consistent manner. Since the continuity equation has a general solution, the equation for the material balance (5). together with eqn (4), is a nonlinear equation for determining the E*(T) function, the characteristic o(u,r), and the distribution function. Taking into account the fact that C*(T) + 0 as T + Q), we can write the equation for the characteristic as follows:
e*(r)@, - I&J-**
UIV_O = 0.
= K e’
c*(T))
In Ref. [I] the function determined
$=
O”$=s(~)-#O) was I only for u c u, The exact solution of the
The dimension space for new-phase particlesin the diffusive decomposition of the supersatur;
Thus we obtain the values of o(u, T) for u s u2, which coincide with zeroth approximation determined in Ref.
continuity equation in these three regions is (p(a T) = f&a.
(9)
TN:.
U(U,T) for u t Y, can be determined by solving the characteristic equation which in this region applies to all times: 11au,
S(u) = 06(u)+ 7.
(10)
(11) tic) = &u,)+ TAU.7. ur’
(P(u. 7) = fdufu,.
(12)
u2cu~u,
7,
u3, UA
Y4Uz
(14)
T#= T,(u. T, u,) is determined by solving the equation for the characteristic in the region u1 s Y s u, du -z-w dT
:(
g-&Y
-$2(T)
(IS)
ul.-Tj= Ul
r2 = T*(u, T, uJ is determined by solving the equation for the characteristic in the region Y su2, which can be written to include all the times: 1, = 7 t $(u)-
ill. p(u,r)=fdv)~~~=jd~)--
I
do
2
#(UT).
(16)
The e’(T) function and the distribution function can be determined when practically all the excess material is in the particles with dimensions u < ul. The conservation of the material at this time is given by
This expression omits the material in the solution and in the particles with u > uz, which can be taken into account in the following approximation, and gives exponentially small correctionsfor T. Substituting in eqn (13) the solution in (II) for (p at u c ul. we obtain
I
"'e*jdo(r,(rJ]).~~~u3du
0
Using tl in (16) we obtain
To satisfy eqn (19) we must have
-(+*+WN =ce+WL
(21) g(u)
.
The relation of the motion integral at u < u2 u = ~(73 to the asymptotic behavior of the initial distribution function, which can be written allowing for the fact that ~(72) is the initial relative size of the particle which at the time 7 has the value u, increases without restriction as T + 00
Determining a(Q) from (22) and using (l4), we determine 7, = T,(TJ for us uT. The motion integral 7, representsthe arrival time of the particle at the point 14, in the dimensionspace,if it was at the point u at the time 7. By setting u = u2 and r2 = 7. we can determine the relation for the particle’s arrival time at the boundary of the region uI, if it was at the boundary u2 at the time T W(T)) = T,(T)+ flu,).
= I.
(18)
(23)
It can be used for times satisfying the condition U(T)Z u,. This relation, together with qn (IS) for the region u2s u s II,, yields the equation for the r2(r) function. It can be written in canonical form after the substitution -$u
- uO)+u and &T-T du _=_a* dr’
4 r’-t(-+,(.,=
K
I
a72Au)
(13)
T,(U& T2IU.T, uzl, u,),)~~~
$tu) = flu,)+ TdU2.du.
369
d solid solution
- 647’) Ul.
ul+_, = u2.
(24)
Equation (22). in which q = T, (23), and (24) comprisethe canonical equations determining the C’(T) function, the characteristic and the distriiution function in the region u 2 u2. If a Riccati-type solution of eqn (24) can be found for any ~~(1) function, then the boundary conditions will yield the functional equation for the t’(r) function. Although this cannot be accomplished analytically. the asymptotic behavior of the initial physical distribution functions. which decrease according to size, leads to such dependencesof T, = T,(T) that the whole problem can be reduced to solving ordinary equations with sufficient accuracy. The time AT = 7 - T,(T) needed to traverse the region U~SYSU, increases with T, which monotonkauy decreases 8(r) with increasing T. This means that tW = t42 + To), where the parameter 7, is determined bytbeinahl(z=O)v~ueof6~c(r~. Equrtion (24) has 8 group of functional transformations that do not change it. In fact, after making the
370
SLEZOV
following substitutions:
du ;i;=-u2-E*(T+T0) Uld(r+m,=
111,
(2%
uI,+*, = h.
After transformation of 5 = U(T+ TJ
l/2, we obtain
-
di = - 52- s= d In (7 + 7”) we obtain
s* = e2(7 t To)*- a
~‘(7 + ro) + 8’ = ~‘(7 + T# - t+ 1’ = S’(ln (7 t 7J)’ - t .
=
ill
1 2
u,(7,+70)--+a,
I”(r,+r”,=l~~ro)-l”o Equation (24) remains the same. Note that since the equations always include the relations [d In (T + To)]/[ln (T + TJ], etc. their modulus should be used in the region of negative logarithms. Note that for infinitely extended initial distribution functions the right-hand side of eqn (24), after such transformations, cannot vanish, and the leakage across the blocking point will cease, while will disturb the equilibrium of the material. This means that et, 6’ and /* in this case are bounded below. After determining the functional transformation for 7, = T,(T), which would make the right-hand side of eqn (24) independent or nearly independent of its argument during a certain time, we can easily solve eqn (24).
I =14*(T+To)---+-m.
&I Gz$
2
To satisfy these boundary conditions S must be a constant. Thus, r+ro _ arctan C,lS = - In ---In-j T(T)+ 70
f arctan [,/I? -i
(31)
After substituting f,,(u) = Bu-” and B = AR;:(O) eqn (22). we obtain ~(7)s I
in
and r+m,
and as &+-00, J,++~
we
have
-*=lnYjd-ln(n~l)/3 2(T)=
3. SOLUTION OF THE EQUATIONS WtTtt ASYMPTOTIC BEHAVtOR OF TttR INITIAL DtSTRIBUTtON F’UNCTION OVER TRE DtMENStONS &,(R) = AR-”
(30)
4(T;
To)2U
(3.2)
+
(33)
4e
The following substitutions must be made if we return to the previous variables: T+To'-(TtTo),Tj+To+~(T,+To),u+(u-u)~
$3
d/3
2 3
I = -(u - u())(TtT~)--.
OV/3 1
2
Thus, Substitution of eqn (25) in eqn (23) yields: $(e “(“-“(r+‘“D)) = T,(T)+ $(u,).
2(T
3
1 4(TtT,)'[lt
+ To)=
4SZ].
(34)
(26)
If E*(T) is known, we can determine 7, = T,(&T) within the region u2 s u 5 u, and hence the distribution funcSince T,(T)zO, eqn (26) can be used when T? tion. To do this, 5 should be substituted for & in eqn (31) (n - 1) In u, - In D.Using the asymptotic form of 4(u) = and In I/d + In [T + TJ[T,([, T) t ~~1to the right, because 3Inuforo%l,wedetermineas~+m the boundary [, is reached at the time 7, = T,(& T), if the 3 point is reached at the time T. It seems that 7,(L 7)lc++CC = T. Thus we obtain /3 =dln D-t,b(u,). d=&cl: T,(T) = dT + /3, (27) Since 7Oand /3 depend on the time origin, it should be chosen so as to satisfy the relation /3=-(1-d)Tw
Thus we obtain becomes
In
7 f To
= 8-l
[
7,(5.7) + 70
t-arctan
l/S
I
(35)
Thus,
(28)
T,(T)+ TO= d(T + TO) and eqn (23)
p
=
i
I -
jf
arctan f
(36)
The dimension space for new-phase particles in the diffusive decomposition of the supersaturated
solid solution
371
obtained can be used. By definition, the parameter 7c is a function of R,,(O) = ~/(A(O));therefore, as noted above, it depends on the time origin. After substituting the constants u, @ and c, r. becomes (37) Using the asymptotic form of Jt(v) = 3 In v, a % 1 from (I 1) and (12) Ju/ar, = u/3, and the explicit expression for TVand D in the constant factor, we determine for relatively large values of 7: cp(u,7) = ffJ(r(~,(5,7)))g(a(r& =fBexp
[
a71 7))) *aid
-$r,(&r)+Ijl(~,)
I
.$
= c e--i 2 exp {-(d-’ - I)(7 + z~)). q=k
*+$arctanf [
1
&CUGt&,.
(38)
3(n-3) r0=n_41n
1 ii@-Al”(
cP(n A- l)Q0 > (43)
-&(U2)f~W,).
Al~ou~ the initial equations determine E*(T)when T & I, but since c*(~+ TJ is a monotonic function of time and the time origin is chosen arbitrarily, these expressions also determine e*(r) when T 2 0 if the initial conditions are such that c*(z-J
Using the asymptotic form of e(u) = 3 In v, tr % 1, we obtain from (9) and (IO) for ti 2: II,: 9(U,
?f = Bv
-*~=~Bexp
-$(7+4((u) .A. I 1 l&au,.
(39)
Thus, eqns (21), (38) and (39) determine the dimensional distribution function in the whole range of variation of II at sufficiently large times T B I. At T = 0 and u -P 01,we have cp(u,0) --, BU-” and u --, m. The solution obtained in the region u2 I u 5 u, smoothly matches the solutions for u S u2 and u 2 u,, and the values of the matching points drop out asymptotically from the result. In fact,
The small terms have been dropped because asymptotically when practically all the excess material is in the particles
Wzf
1
1
444)
1
~-(u2_11o)q-y~-y--ul--o~o
1
<*
and A(0)4 Q,,. Asymptotic analysis can be used when this expression 1 becomes a constant value. Any time beginning with this a7( &I zd-. (40) time can be assumed the initial time (7 = 0) and all the au IL=tZ4-m J?od duIt=C,-=& I TIdimensional values can be related to Rcrit= a/A(O). In Substituting these values in eqn (38) and using the explicit this case the boundary conditions assume the canonical form of TO,we obtain at the boundaries, form (29). Investigation of the distribution function beyond the blocking point will make it possible to determine the value of n for the initial distribution function and to &, T) = ;B e-WXT+W). _& 9(u, 7) = c e-‘I. compare it with eqn (44). F?(U)’ AU) U-+UI U-+U2 Note that if the time origin is not chosen in a special (41) manner (44). i.e. if a solution is sought after arbitrarily As shown in Ref. [I], the average size of the particles choosing the time, requiring only that e’(r) 4 1, then the R(t) coincides with R,& for the mass -transfer solution obtained for u > u2 will coincide with eqn (38) mechanism being examined. Using a more precise value with accuracy to terms of the order of @dr-rO and of Y(T), we determine A(l) and Rcrit(t): 7+m. Determination of the amount of excess material in the region of the blocking point and beyond it yields ‘(‘) = R,:(t) Q’=“KuQ3c=Q0e-“‘@+~Q,. (45) 4n’ I+ Rl,,(r)=R~~i,(O)+~~f. t t In? Since this analysis applies to the case in which the region ( > of the blocking point becomes small asymptotically, . In (4/9)aaf+ , + 7 -2 -’ which corresponds to tfrfDo)* I. we have Q’ Q Qo. Thus, 0 (42) ( [ R,:i,(O) I >I asymptotically the distribution function is determined from eqns (21), (38). (39) and (42) according to the Let us determine the time when the asymptotic functions dimensions of the initial dist~bution function.
1-$(
3x7
v. v.
SLEZOV
4. SOLUTION OF THE EQUATIONS WITH EXPONENTIAL ASYMPTOTIC BEHAVlOR OF THE INITIAL SUE FUNCTlON
cpk 7) =fo(u(u, 7)). g
In the general form f@(R) = A e-“R-Ro)“cl’“’. i/R”; m > 0. In relative
variables
-.AA”(O) on ,
A
B=R,:,o= Inserting
it in
Jr(u)= 7
f,(v) = B e~p”“u~“o”“( I/t’“),
where P=?=-
&i,(O)
+
(52)
uau,.
Ii(u),
The distribution function in the region u > uz can be constructed by using eqns (46-52). For this time interval
a hA(O)’
Jr1 = arl al au a[ au
eqn (22) we obtain for 7 b 1
is determined by eqn (37) after substituting
u(~~)=vg+$((~*+loD)-~ln(T*+lnD) I/m
1 _+7+70
-~ln(T2+InD+pv,) ) D_
BP”_’ MC
d
(46)
The logarithmic terms are important only when u(r) is present in the exponent; in the other cases they can be neglected when r+m. Inserting eqn (46) in eqn (23) and assuming, as before, that u = uz and 7 = r2, we obtain the condition at the boundary
of the transformation
T,(T)
region
= 4(U(T))
-
at the point u = u,
ILCU,).
(47)
Under these conditions u, and uz cannot be reduced at the boundaries of the transformation region to canonical form for which the characteristic equation can be solved exactly in this region. The characteristic can be determined with suficient accuracy in this region for different time intervals, if eqn (24) is transformed in such a manner that its right-hand side would be independent of time within certain time intervals. First, we can use eqn (31) after inserting in it T,(T) from eqn (47). Thus we obtain from eqn (30), after 7 + To
having substituted in it $+---
T,(T)
T,(T)+
T,,
Note that the distribution function in the region uz 5 u 5 ur, which always matches the distribution function in the region u 2 u,, is independent of the accuracy of the characteristic. This is attributable to the exact fulfillment of the boundary condition T,(T, [)-+T when [-+ +a(~-, u,), and the characteristic to the left of u, goes over smoothly to the characteristic to the right of u,, rr(r, I) + 4(u,)+ T + 4(u) when u -+ II,. The requirement of smooth matching at the point u2 sets the condition for the time interval for which our solution is applicable. Inserting ~(7, I) in eqn (51) when u + uz and l-)-m, we obtain
p(u.
Thus
T) = c em’--!-.
g(u)
our
r,(r) + 70 mp(r+ln D)(m-‘)‘mg(u)----7 + 70 (53) u -+ u2, .C$+--m.
solution can be used for the time interval when
f(7) = mp(r + In D)‘m-‘)‘mg(~(T))- T,(Tj + TO)
+ To’
7
t
To
(54)
S= ?r(ln*)-’
3 e2(7 + 70)= ____ 4(7 + r0)2. (I t 4SZ) r,(&r)+%=(*y
(48)
.(7-eT0)
(49)
, >
00)
where C=f
(
I-?arctanj
7T
4L(1)(7,0)= 7,(4-.7) + $.(u,). Hence
where u(r) is determined by (46). The accuracy of the solution obtained is wholly determined by the accuracy of the characteristic rather than by the initial distribution function. In fact, if the approximate solution of (31) with T,(T) taken from (47) is inserted in the equation for the characteristic, then it will be accurate to the term that is small in this time interval: d mln(7
-I = 1-f’(7)“0. 0
(55)
This condition coincides exactly with the requirement (54). Let us determine this time interval. First, we note thatwhenr*landu%l g(v)zu
3’
g(u) av ?!_=I ( a7> 7+7,,
dlnu 3dIn(7+~,J=?
1
3 $(0)=3lno==-InA m and 4(u,)=Oand
u,;=2.
P
The dimension
space for new-phase
particles
in the diffusive
Thus the solution obtained is suitable for the time -Jln-$+Y=
decomposition
r2(7t7J=--
of the supersaturated
3
1
1t4/* 3 1 In (7 t To))2I = 4 ’ 7.
l+
4 (T + To)[
1.
373
solid solution
(59)
In 7% I.
We now determine the parameter T,,.It seems that eqn (47) can be used when T,(T)2 0 or when ~(7) 2 u,. The condition ~~(7,~~)= 0 or u(T,,& = u, determines the time interval measured from the time origin chosen when the distribution function at u 2 u, forms the function in the interval u2 5 u 5 u,. We use the following equation to determine TV:
Inserting T,([, T) from (58) in eqns (51) and (52), we determine the distribution function. It can be assumed that at such large times T,(T)= (3/m)In(~/~~), and r0 can be neglected. The joining of the distribution function in the region u,, as noted above, is accomplished smoothly. In the region u = uZ, 7 +--co substitution of (58) in (51) yields
(m-l)/mg(u)% = e-~_.Lln
cp(u,T) = c em’mpr
c
ln
T
g(u)
I/WI
V/3 1 4 = 2 (7,in + 70)
4rr* [In (T,~"+ 7J7J2
’ +
1
'('@
T&AJ = (puo - u,)” - In D >>1.
Thus, to determine 70 the derivative [(d/d?)(l/A’)]-’ should be determined experimentally when A(t) = A(0)e-’ min’*d. If the initial distribution function has exponential asymptotic form, then the nonlinear eqn (47) cannot determine the time 7,(O) when the particles with dimensions u2 had dimensions u,. This is because eqns (48-52), which are suitable for 7~ T,,,~“,are greatly deformed with time, and they cannot be extended into the region 7< Tmin. We perform one more functional transformation with eqn (30) for m-’ In (T/P”‘) 9 I and In 7 = 1 using a new argument (time scale) In In (T t TV) and the function 7 = J In (T t T,,)- l/2. Thus we obtain =_&/2 + TJ
I’= (2(T+To)~(T+T$-i)ln(T+Td)*-i.
3
3P
1 In In 7 7 ’
dq %$ 3 a71= _a In 7, --=___ au
At sufficiently late stage
d InIn (T
v==u7In7;g(u)=-V=+
"
a7 agaU
m g(u)
(60)
It follows that the solution obtained is suitable for the time interval in which In In T = I. Equation (57) is satisfied with accuracy to the small term:
- d -7r +-!-zc. dIn7 0 1
(61)
In In 7
Thus, in the time interval In In T = 1 the accuracy of the distribution for u > u2 is determined solely by the accuracy of the characteristic. At u > u,cp(u, T)exp [-m exp (T t e(u)] the distribution function is very small and contributes practically noting to the material balance. Note that if the initial size distribution function of the particles has an end point, i.e. it vanishes at R 2 R”, then the time it is transformed into a universal function and ~~(7) = 0 and be easily determined by analyzing the change in supersaturation of the solution with time,
(57)
(62) When I’= const the solution (5) coincides with (3l), where In (7
Under these initial conditions and at this time
+ 7”)
In [T,(T)
1 I A
cp(u,7) =
~~1
+
e-,r+44”))~
_
I
g(u) 0
u b ug
should be substituted for l/d and ln(7 +TJ l=?r In [ In [T,(T) +
(63)
1 _'
~~1
with T,(T) taken from (47). Thus we obtain
5. GENERAL
In (T,(T)
In (7,G 7) + ro) =
In [
CT
+
+ To To)
OSU~U”
I
c = i I - z arctan T ) C r
.
In (7
+ To)
(58)
CONCLUSIONS
The solution obtained above, in which the mass-transfer mechanism consisting of the diffusion across the material was examined, can be easily applied to other mass-transfer mechanisms[2-41. The principal canonical equations remain the same. Only the Q(u) function, which is determined by the mass-transfer mechanism, is
374
v.
v.
different. An iteration method can be developed for obtaining corrections for these solutions. To determine the first correction for the characteristic, we must take into account the amount of material in the solution A - er’3 and in the tail of the distribution function when u 2 u2, which is calculated in zeroth approximation. This procedure reduces to substituting in eqn (17) the factor e-T+e-T[l + O(e-T”)l. The integrand with allowance for the correction can be written as follows: fO(uOt So) = fO(vO)+ [df(uo)/duo]Sv. Retaining the next order small terms, we can easily show that the corrections for the characteristic and for the e*(r) function are O(em”3), i.e. they are exponentially small. Generally, the hydrodynamic approximation need not be refined because the local fluctuations in concentration of the particles acquire greater importance with time. This is attributable primarily to direct diffusion interaction (“collisions”) of the new-phase particles at distances smaller than their size. As shown in Refs. [5] and [6), this leads to the formation of an additional tail of the distribution function when u > uO, which is governed primarily by the distribution function when u < uO, and by the tendency of e2(r) = Ay/rO to approach a constant value of l*-- (In QJ’. This means that the universal distribution function is formed primarily in the “hydrodynamic” mode at u I ua and the particle “collisions” at u < u0 will in time become the main contribution to the tail of the distribution function at u > uO. A large contribution from the collisions will begin when 8, which is determined in the hydrodynamic approximation, is (In Q,,)*. Note that the maximum of the distribution function does not coincide with the point of the lowest velocity but is shifted to the left. This is attributable to the fact that in any approximation the distribution function at u > u0 is a decreasing function. Therefore, after having leaked across the region of the blocking point uo. it will have the maximum value only at some distance from uO, which is also independent of the initial conditions when T +m.
SLEZOV
The results obtained above can be easily extended to the case in which the sources of the dissolved component are present, because the growth rate of the new-phase particles, as demonstrated in Refs. [7-lo], ‘is independent of the existence of the sources, and in the equation for the material balance the factor e’+e”‘. where the exponent n I I depends on the power of the component’s source. The values of n, which depend on the mass-transfer mechanism, are bounded above, for which A(t)+0 when 7-m and the universal distribution function has time to form[9]. These results can be extended to the multicomponent, multiphase systems[l I-131, because the continuity equations for these systems in the later phase break down exponential accuracy into a set of independent equations for each phase [ I I]. Only the algebraic equations, which determine the regions of coexistence of the phases and the phase distribution of the components, are “coupled.” In this system the corrections for the concentration of any component are determined by the initial distribution functions of the phases containing the given component.
REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. IO. I I. 12. 13.
Lifshitz 1. M. and Slezov V. V., I.E.T.P. 35, 2 (1958). Slezov V. V., Fiz. Tverd. Tela 9, 4 (1%7). Slezov V. V. and Levin D. M., Fir. Tuerd. Tela 12, 6 (1970). Slezov V. V. and Saralidze Z. K., Fiz. Tverd. Tela 7.6 (1%5). Lifshitz I. M. and Slezov V. V., Fiz. Tverd. Tela 1. 9 (1959). Lifshitz I. M. and Slezov V. V., J. Phys. Chem. Solids 19,35 (l%l). Kosevich A. M., Saralidze Z. K. and Slezov V. V.. J.E.T.P. 52.64, 1079 (1967). Slezov V. V., Fiz. Tverd. Tela 9, 4 (1%7). Slezov V. V. and Shikin V. B., Fir. Tuerd. Tela 6, 1 (1964). Slezov V. V. and Shikin V. B., Fiz. Tverd. Tela 2, 3 (1965). Slezov V. V. and Sagalovich V. V.. Fiz. Tverd. Tela 17. 1497 (1975). Slezov V. V. and Sagalovich V. V.. Fiz. Tverd. Tela 17, 9 (1975). Slezov V. V.. Fiz. Tverd. Tela 17. 9 (1975).
Pergamon Press.
Printed in Great Britain
FORMATION OF THE UNIVERSAL DISTRIBUTION FUNCTION IN THE DIMENSION SPACE FOR NEW-PHASE PARTICLES IN THE DIFFUSIVE DECOMPOSITION OF THE SUPERSATURATED SOLID SOLUTION v. v. SLEZOV Physico-Technical Institute, Kharkov, U.S.S.R. (Received21 March 1971; accepted 19 August 1977) Abstract-The methodof time-asymptoticsolution of nonlinearequationsdescribingthe decompositionof supersaturatedsolid solutions is developed.The distributioncurve is plottedfor any dimensionat the given momentof time. The results are generalizedfor the case of volume sources of the solute, for the multicomponentmultiphase solid solutions and for the differentmass-transfermechanisms.
1.
INTRODUCTMlN
The theory of diffusive decomposition in the later stage
(coalescence) in zeroth approximation was developed by Lifshitz and Slezov[l]. In this approximation they qualitatively analyzed the equations describing the diffusive decomposition and showed that at sufficiently large times the contribution to the material from the new-phase particles whose size exceeds the “blocking” point is negligible. They also showed that the supersaturated solution must have an exact time behavior. The growth rate of particles in relative dimensions and its derivative vanish at the blocking point u,,. This approximation uniquely leads to a universal solution, which isindependent of the initial conditions, for the size distribution function of particles for u I u* In the region u 2 u,, the distribution function in zeroth approximation is zero. However, the dynamics of plotting the universal distribution function and its relation to the initial distribution function cannot be determined in this approximation. It is also not possible to determine the distribution function in the region of the blocking point u,, and beyond it, nor a more precise asymptotic behavior of the supersaturated solution with time and the time TVat which the asymptotic time description of the diffusive decomposition can be used with sufficient accuracy. Since all these values are closely related to the initial distribution function, in order to calculate them a systematic theory must be formulated for the solution of the equations describing the diffusive decomposition of the supersaturated solution with allowance for the initial conditions. This paper is devoted to the formulation of this theory.
unit volume: f(R, t) dR = dN is the number of particles with dimensions R 5 R’ I R t dR per unit volume. Like in Ref. [l], we use more convenient relative dimensions of the particles U = RIR,tir and the times R4t) = T = In V&,(t)/&,(O)1 = 3 In (A(O))/(A(t)). u/A(l), where a = 2(4kT)c,, u is the interface surface tension, o is the volume per atom of the dissolved component, k is the Boltzmann constant, and T is the absolute temperature. The relation f(R, 0 dR = cp(s 7) du
Let us examine a one-component supersaturated solu-
is between the distribution functions in the absolute and relative variables. These relative dimensions of particles must be introduced because an ordered dissolution of these particles always takes place in these variables, beginning with a certain time, and the distribution function at the given time is determined by the asymptotic behavior of the initial distribution function. We note that if the particles have too much excess material, they will be slightly dissolved after ‘some finite time, causing the level of supersaturation of the solution to initially increase for a certain time, dddr < 0, and then to decrease monotonically, ddddr>O. We investigate the diffusive decay at d7/dt>O. This situation always occurs at low supersaturation, irrespective of the initial conditions, which determine only the time when dT/dt begins to increase monotonically with time. Moreover, as demonstrated by Liishift and Slezov[ll, at A4 1 the equilibrium concentration and the steady diffusive flow will have sufficient time to establish at the boundary of the particles. Under these conditions the growth rate of the particle resulting from diffusive mass transfer across the bulk of the material is [ I]
tion characterized by supersaturation A(r) = E - c,, where E = E(t) is the average concentration of the solution at the given time and c, is the equilibrium concentration at the plane boundary, and by the distribution function of the new-phase particles at the given time in the dimension space R, j(R, t), which is normalized to
where D is the diffusion coefficient of the dissolved component.
2. CANONICAL FOPMOF ‘IliE FUNDAMENTAL EQUATIONS
v. v.
368
SLUOV
Using the variables u and r the growth rate can be expressed as follows: Taking into account y(z) and the fact that when 7 = y. du/dr at’ the point u0 = 3/2 has a second-order zero, we can rewrite eqn (4) as follows:
du7(7HU-1)-U’
dr
3u2
$=-g(u, T _ RL(0). aD’
y = 27 ’ 7’
= -&
this ‘form of the 7(r) function is required for ordered motion from right to left in relative dimensions, which corresponds to du/dr
cpl.4= fduh ul.-0 = u.
(2)
Since we are analyzing a later stage of the diffusive decay when Ati I, the fluctuating production of the new-phase particles with R > a/A may be neglected and the right-hand side of the equation of continuity can be set to zero. This corresponds to the “hydrodynamic” approximation whose general solution is
(P(u.I) = fdu(u, ?)I$
(3)
D = u(u, r) is the integral of the characteristic d+7(u;f!-u’
(u + 3)(u - ua)2 + Y(u
(4)
The dissolved component can be written as’follows:
I-
fduht’b. 7) du e,o.r,
_g=_(u+3Ku-3*
excess quantity of the material.
I
.
(6)
du -= dr
I
u
3u’
-
;(u - uCJ*++*(r)] [ _g=_(u+3Ku-32 3u’
u*cucu,
(7)
U>U,>U*
Here the region near the point u,,, where the E*(T) function has an important role, has been isolated. The boundaries of the transformation region u, and u2 satisfy the condition:
l2(7)(U2
1,
-
uJ2
Q
I.
We determine the flu) function for u 7 u0 and u < u0 whose difference at the two u points is the time required to go from the larger value of u to the smaller one
(8)
(5)
where A(0) is the initial saturation and Q0 is the total
- l)r2(7)
When r+m l*(r)+O. This behavior of E*(T) This behavior of c*(r) is attributable to the fact that as r increases as a result of ordered motion from right to left, the more remote regions of the initial distribution function, which is a decreasing function of its argument fdu) 4 I/U”; n > 4, u +m, from the distribution function at a given time to the left of the blockina point, which is the main contribution to the matenat balance. Since duldr ~0, the number of particles to the right of the blocking point decreases with increasing time; hence their leakage rate across the transformation region near the blocking point will decrease so that all the excess material can settle on them. This means that the distribution function will condense in a self-consistent manner. Since the continuity equation has a general solution, the equation for the material balance (5). together with eqn (4), is a nonlinear equation for determining the E*(T) function, the characteristic o(u,r), and the distribution function. Taking into account the fact that C*(T) + 0 as T + Q), we can write the equation for the characteristic as follows:
e*(r)@, - I&J-**
UIV_O = 0.
= K e’
c*(T))
In Ref. [I] the function determined
$=
O”$=s(~)-#O) was I only for u c u, The exact solution of the
The dimension space for new-phase particlesin the diffusive decomposition of the supersatur;
Thus we obtain the values of o(u, T) for u s u2, which coincide with zeroth approximation determined in Ref.
continuity equation in these three regions is (p(a T) = f&a.
(9)
TN:.
U(U,T) for u t Y, can be determined by solving the characteristic equation which in this region applies to all times: 11au,
S(u) = 06(u)+ 7.
(10)
(11) tic) = &u,)+ TAU.7. ur’
(P(u. 7) = fdufu,.
(12)
u2cu~u,
7,
u3, UA
Y4Uz
(14)
T#= T,(u. T, u,) is determined by solving the equation for the characteristic in the region u1 s Y s u, du -z-w dT
:(
g-&Y
-$2(T)
(IS)
ul.-Tj= Ul
r2 = T*(u, T, uJ is determined by solving the equation for the characteristic in the region Y su2, which can be written to include all the times: 1, = 7 t $(u)-
ill. p(u,r)=fdv)~~~=jd~)--
I
do
2
#(UT).
(16)
The e’(T) function and the distribution function can be determined when practically all the excess material is in the particles with dimensions u < ul. The conservation of the material at this time is given by
This expression omits the material in the solution and in the particles with u > uz, which can be taken into account in the following approximation, and gives exponentially small correctionsfor T. Substituting in eqn (13) the solution in (II) for (p at u c ul. we obtain
I
"'e*jdo(r,(rJ]).~~~u3du
0
Using tl in (16) we obtain
To satisfy eqn (19) we must have
-(+*+WN =ce+WL
(21) g(u)
.
The relation of the motion integral at u < u2 u = ~(73 to the asymptotic behavior of the initial distribution function, which can be written allowing for the fact that ~(72) is the initial relative size of the particle which at the time 7 has the value u, increases without restriction as T + 00
Determining a(Q) from (22) and using (l4), we determine 7, = T,(TJ for us uT. The motion integral 7, representsthe arrival time of the particle at the point 14, in the dimensionspace,if it was at the point u at the time 7. By setting u = u2 and r2 = 7. we can determine the relation for the particle’s arrival time at the boundary of the region uI, if it was at the boundary u2 at the time T W(T)) = T,(T)+ flu,).
= I.
(18)
(23)
It can be used for times satisfying the condition U(T)Z u,. This relation, together with qn (IS) for the region u2s u s II,, yields the equation for the r2(r) function. It can be written in canonical form after the substitution -$u
- uO)+u and &T-T du _=_a* dr’
4 r’-t(-+,(.,=
K
I
a72Au)
(13)
T,(U& T2IU.T, uzl, u,),)~~~
$tu) = flu,)+ TdU2.du.
369
d solid solution
- 647’) Ul.
ul+_, = u2.
(24)
Equation (22). in which q = T, (23), and (24) comprisethe canonical equations determining the C’(T) function, the characteristic and the distriiution function in the region u 2 u2. If a Riccati-type solution of eqn (24) can be found for any ~~(1) function, then the boundary conditions will yield the functional equation for the t’(r) function. Although this cannot be accomplished analytically. the asymptotic behavior of the initial physical distribution functions. which decrease according to size, leads to such dependencesof T, = T,(T) that the whole problem can be reduced to solving ordinary equations with sufficient accuracy. The time AT = 7 - T,(T) needed to traverse the region U~SYSU, increases with T, which monotonkauy decreases 8(r) with increasing T. This means that tW = t42 + To), where the parameter 7, is determined bytbeinahl(z=O)v~ueof6~c(r~. Equrtion (24) has 8 group of functional transformations that do not change it. In fact, after making the
370
SLEZOV
following substitutions:
du ;i;=-u2-E*(T+T0) Uld(r+m,=
111,
(2%
uI,+*, = h.
After transformation of 5 = U(T+ TJ
l/2, we obtain
-
di = - 52- s= d In (7 + 7”) we obtain
s* = e2(7 t To)*- a
~‘(7 + ro) + 8’ = ~‘(7 + T# - t+ 1’ = S’(ln (7 t 7J)’ - t .
=
ill
1 2
u,(7,+70)--+a,
I”(r,+r”,=l~~ro)-l”o Equation (24) remains the same. Note that since the equations always include the relations [d In (T + To)]/[ln (T + TJ], etc. their modulus should be used in the region of negative logarithms. Note that for infinitely extended initial distribution functions the right-hand side of eqn (24), after such transformations, cannot vanish, and the leakage across the blocking point will cease, while will disturb the equilibrium of the material. This means that et, 6’ and /* in this case are bounded below. After determining the functional transformation for 7, = T,(T), which would make the right-hand side of eqn (24) independent or nearly independent of its argument during a certain time, we can easily solve eqn (24).
I =14*(T+To)---+-m.
&I Gz$
2
To satisfy these boundary conditions S must be a constant. Thus, r+ro _ arctan C,lS = - In ---In-j T(T)+ 70
f arctan [,/I? -i
(31)
After substituting f,,(u) = Bu-” and B = AR;:(O) eqn (22). we obtain ~(7)s I
in
and r+m,
and as &+-00, J,++~
we
have
-*=lnYjd-ln(n~l)/3 2(T)=
3. SOLUTION OF THE EQUATIONS WtTtt ASYMPTOTIC BEHAVtOR OF TttR INITIAL DtSTRIBUTtON F’UNCTION OVER TRE DtMENStONS &,(R) = AR-”
(30)
4(T;
To)2U
(3.2)
+
(33)
4e
The following substitutions must be made if we return to the previous variables: T+To'-(TtTo),Tj+To+~(T,+To),u+(u-u)~
$3
d/3
2 3
I = -(u - u())(TtT~)--.
OV/3 1
2
Thus, Substitution of eqn (25) in eqn (23) yields: $(e “(“-“(r+‘“D)) = T,(T)+ $(u,).
2(T
3
1 4(TtT,)'[lt
+ To)=
4SZ].
(34)
(26)
If E*(T) is known, we can determine 7, = T,(&T) within the region u2 s u 5 u, and hence the distribution funcSince T,(T)zO, eqn (26) can be used when T? tion. To do this, 5 should be substituted for & in eqn (31) (n - 1) In u, - In D.Using the asymptotic form of 4(u) = and In I/d + In [T + TJ[T,([, T) t ~~1to the right, because 3Inuforo%l,wedetermineas~+m the boundary [, is reached at the time 7, = T,(& T), if the 3 point is reached at the time T. It seems that 7,(L 7)lc++CC = T. Thus we obtain /3 =dln D-t,b(u,). d=&cl: T,(T) = dT + /3, (27) Since 7Oand /3 depend on the time origin, it should be chosen so as to satisfy the relation /3=-(1-d)Tw
Thus we obtain becomes
In
7 f To
= 8-l
[
7,(5.7) + 70
t-arctan
l/S
I
(35)
Thus,
(28)
T,(T)+ TO= d(T + TO) and eqn (23)
p
=
i
I -
jf
arctan f
(36)
The dimension space for new-phase particles in the diffusive decomposition of the supersaturated
solid solution
371
obtained can be used. By definition, the parameter 7c is a function of R,,(O) = ~/(A(O));therefore, as noted above, it depends on the time origin. After substituting the constants u, @ and c, r. becomes (37) Using the asymptotic form of Jt(v) = 3 In v, a % 1 from (I 1) and (12) Ju/ar, = u/3, and the explicit expression for TVand D in the constant factor, we determine for relatively large values of 7: cp(u,7) = ffJ(r(~,(5,7)))g(a(r& =fBexp
[
a71 7))) *aid
-$r,(&r)+Ijl(~,)
I
.$
= c e--i 2 exp {-(d-’ - I)(7 + z~)). q=k
*+$arctanf [
1
&CUGt&,.
(38)
3(n-3) r0=n_41n
1 ii@-Al”(
cP(n A- l)Q0 > (43)
-&(U2)f~W,).
Al~ou~ the initial equations determine E*(T)when T & I, but since c*(~+ TJ is a monotonic function of time and the time origin is chosen arbitrarily, these expressions also determine e*(r) when T 2 0 if the initial conditions are such that c*(z-J
Using the asymptotic form of e(u) = 3 In v, tr % 1, we obtain from (9) and (IO) for ti 2: II,: 9(U,
?f = Bv
-*~=~Bexp
-$(7+4((u) .A. I 1 l&au,.
(39)
Thus, eqns (21), (38) and (39) determine the dimensional distribution function in the whole range of variation of II at sufficiently large times T B I. At T = 0 and u -P 01,we have cp(u,0) --, BU-” and u --, m. The solution obtained in the region u2 I u 5 u, smoothly matches the solutions for u S u2 and u 2 u,, and the values of the matching points drop out asymptotically from the result. In fact,
The small terms have been dropped because asymptotically when practically all the excess material is in the particles
Wzf
1
1
444)
1
~-(u2_11o)q-y~-y--ul--o~o
1
<*
and A(0)4 Q,,. Asymptotic analysis can be used when this expression 1 becomes a constant value. Any time beginning with this a7( &I zd-. (40) time can be assumed the initial time (7 = 0) and all the au IL=tZ4-m J?od duIt=C,-=& I TIdimensional values can be related to Rcrit= a/A(O). In Substituting these values in eqn (38) and using the explicit this case the boundary conditions assume the canonical form of TO,we obtain at the boundaries, form (29). Investigation of the distribution function beyond the blocking point will make it possible to determine the value of n for the initial distribution function and to &, T) = ;B e-WXT+W). _& 9(u, 7) = c e-‘I. compare it with eqn (44). F?(U)’ AU) U-+UI U-+U2 Note that if the time origin is not chosen in a special (41) manner (44). i.e. if a solution is sought after arbitrarily As shown in Ref. [I], the average size of the particles choosing the time, requiring only that e’(r) 4 1, then the R(t) coincides with R,& for the mass -transfer solution obtained for u > u2 will coincide with eqn (38) mechanism being examined. Using a more precise value with accuracy to terms of the order of @dr-rO and of Y(T), we determine A(l) and Rcrit(t): 7+m. Determination of the amount of excess material in the region of the blocking point and beyond it yields ‘(‘) = R,:(t) Q’=“KuQ3c=Q0e-“‘@+~Q,. (45) 4n’ I+ Rl,,(r)=R~~i,(O)+~~f. t t In? Since this analysis applies to the case in which the region ( > of the blocking point becomes small asymptotically, . In (4/9)aaf+ , + 7 -2 -’ which corresponds to tfrfDo)* I. we have Q’ Q Qo. Thus, 0 (42) ( [ R,:i,(O) I >I asymptotically the distribution function is determined from eqns (21), (38). (39) and (42) according to the Let us determine the time when the asymptotic functions dimensions of the initial dist~bution function.
1-$(
3x7
v. v.
SLEZOV
4. SOLUTION OF THE EQUATIONS WITH EXPONENTIAL ASYMPTOTIC BEHAVlOR OF THE INITIAL SUE FUNCTlON
cpk 7) =fo(u(u, 7)). g
In the general form f@(R) = A e-“R-Ro)“cl’“’. i/R”; m > 0. In relative
variables
-.AA”(O) on ,
A
B=R,:,o= Inserting
it in
Jr(u)= 7
f,(v) = B e~p”“u~“o”“( I/t’“),
where P=?=-
&i,(O)
+
(52)
uau,.
Ii(u),
The distribution function in the region u > uz can be constructed by using eqns (46-52). For this time interval
a hA(O)’
Jr1 = arl al au a[ au
eqn (22) we obtain for 7 b 1
is determined by eqn (37) after substituting
u(~~)=vg+$((~*+loD)-~ln(T*+lnD) I/m
1 _+7+70
-~ln(T2+InD+pv,) ) D_
BP”_’ MC
d
(46)
The logarithmic terms are important only when u(r) is present in the exponent; in the other cases they can be neglected when r+m. Inserting eqn (46) in eqn (23) and assuming, as before, that u = uz and 7 = r2, we obtain the condition at the boundary
of the transformation
T,(T)
region
= 4(U(T))
-
at the point u = u,
ILCU,).
(47)
Under these conditions u, and uz cannot be reduced at the boundaries of the transformation region to canonical form for which the characteristic equation can be solved exactly in this region. The characteristic can be determined with suficient accuracy in this region for different time intervals, if eqn (24) is transformed in such a manner that its right-hand side would be independent of time within certain time intervals. First, we can use eqn (31) after inserting in it T,(T) from eqn (47). Thus we obtain from eqn (30), after 7 + To
having substituted in it $+---
T,(T)
T,(T)+
T,,
Note that the distribution function in the region uz 5 u 5 ur, which always matches the distribution function in the region u 2 u,, is independent of the accuracy of the characteristic. This is attributable to the exact fulfillment of the boundary condition T,(T, [)-+T when [-+ +a(~-, u,), and the characteristic to the left of u, goes over smoothly to the characteristic to the right of u,, rr(r, I) + 4(u,)+ T + 4(u) when u -+ II,. The requirement of smooth matching at the point u2 sets the condition for the time interval for which our solution is applicable. Inserting ~(7, I) in eqn (51) when u + uz and l-)-m, we obtain
p(u.
Thus
T) = c em’--!-.
g(u)
our
r,(r) + 70 mp(r+ln D)(m-‘)‘mg(u)----7 + 70 (53) u -+ u2, .C$+--m.
solution can be used for the time interval when
f(7) = mp(r + In D)‘m-‘)‘mg(~(T))- T,(Tj + TO)
+ To’
7
t
To
(54)
S= ?r(ln*)-’
3 e2(7 + 70)= ____ 4(7 + r0)2. (I t 4SZ) r,(&r)+%=(*y
(48)
.(7-eT0)
(49)
, >
00)
where C=f
(
I-?arctanj
7T
4L(1)(7,0)= 7,(4-.7) + $.(u,). Hence
where u(r) is determined by (46). The accuracy of the solution obtained is wholly determined by the accuracy of the characteristic rather than by the initial distribution function. In fact, if the approximate solution of (31) with T,(T) taken from (47) is inserted in the equation for the characteristic, then it will be accurate to the term that is small in this time interval: d mln(7
-I = 1-f’(7)“0. 0
(55)
This condition coincides exactly with the requirement (54). Let us determine this time interval. First, we note thatwhenr*landu%l g(v)zu
3’
g(u) av ?!_=I ( a7> 7+7,,
dlnu 3dIn(7+~,J=?
1
3 $(0)=3lno==-InA m and 4(u,)=Oand
u,;=2.
P
The dimension
space for new-phase
particles
in the diffusive
Thus the solution obtained is suitable for the time -Jln-$+Y=
decomposition
r2(7t7J=--
of the supersaturated
3
1
1t4/* 3 1 In (7 t To))2I = 4 ’ 7.
l+
4 (T + To)[
1.
373
solid solution
(59)
In 7% I.
We now determine the parameter T,,.It seems that eqn (47) can be used when T,(T)2 0 or when ~(7) 2 u,. The condition ~~(7,~~)= 0 or u(T,,& = u, determines the time interval measured from the time origin chosen when the distribution function at u 2 u, forms the function in the interval u2 5 u 5 u,. We use the following equation to determine TV:
Inserting T,([, T) from (58) in eqns (51) and (52), we determine the distribution function. It can be assumed that at such large times T,(T)= (3/m)In(~/~~), and r0 can be neglected. The joining of the distribution function in the region u,, as noted above, is accomplished smoothly. In the region u = uZ, 7 +--co substitution of (58) in (51) yields
(m-l)/mg(u)% = e-~_.Lln
cp(u,T) = c em’mpr
c
ln
T
g(u)
I/WI
V/3 1 4 = 2 (7,in + 70)
4rr* [In (T,~"+ 7J7J2
’ +
1
'('@
T&AJ = (puo - u,)” - In D >>1.
Thus, to determine 70 the derivative [(d/d?)(l/A’)]-’ should be determined experimentally when A(t) = A(0)e-’ min’*d. If the initial distribution function has exponential asymptotic form, then the nonlinear eqn (47) cannot determine the time 7,(O) when the particles with dimensions u2 had dimensions u,. This is because eqns (48-52), which are suitable for 7~ T,,,~“,are greatly deformed with time, and they cannot be extended into the region 7< Tmin. We perform one more functional transformation with eqn (30) for m-’ In (T/P”‘) 9 I and In 7 = 1 using a new argument (time scale) In In (T t TV) and the function 7 = J In (T t T,,)- l/2. Thus we obtain =_&/2 + TJ
I’= (2(T+To)~(T+T$-i)ln(T+Td)*-i.
3
3P
1 In In 7 7 ’
dq %$ 3 a71= _a In 7, --=___ au
At sufficiently late stage
d InIn (T
v==u7In7;g(u)=-V=+
"
a7 agaU
m g(u)
(60)
It follows that the solution obtained is suitable for the time interval in which In In T = I. Equation (57) is satisfied with accuracy to the small term:
- d -7r +-!-zc. dIn7 0 1
(61)
In In 7
Thus, in the time interval In In T = 1 the accuracy of the distribution for u > u2 is determined solely by the accuracy of the characteristic. At u > u,cp(u, T)exp [-m exp (T t e(u)] the distribution function is very small and contributes practically noting to the material balance. Note that if the initial size distribution function of the particles has an end point, i.e. it vanishes at R 2 R”, then the time it is transformed into a universal function and ~~(7) = 0 and be easily determined by analyzing the change in supersaturation of the solution with time,
(57)
(62) When I’= const the solution (5) coincides with (3l), where In (7
Under these initial conditions and at this time
+ 7”)
In [T,(T)
1 I A
cp(u,7) =
~~1
+
e-,r+44”))~
_
I
g(u) 0
u b ug
should be substituted for l/d and ln(7 +TJ l=?r In [ In [T,(T) +
(63)
1 _'
~~1
with T,(T) taken from (47). Thus we obtain
5. GENERAL
In (T,(T)
In (7,G 7) + ro) =
In [
CT
+
+ To To)
OSU~U”
I
c = i I - z arctan T ) C r
.
In (7
+ To)
(58)
CONCLUSIONS
The solution obtained above, in which the mass-transfer mechanism consisting of the diffusion across the material was examined, can be easily applied to other mass-transfer mechanisms[2-41. The principal canonical equations remain the same. Only the Q(u) function, which is determined by the mass-transfer mechanism, is
374
v.
v.
different. An iteration method can be developed for obtaining corrections for these solutions. To determine the first correction for the characteristic, we must take into account the amount of material in the solution A - er’3 and in the tail of the distribution function when u 2 u2, which is calculated in zeroth approximation. This procedure reduces to substituting in eqn (17) the factor e-T+e-T[l + O(e-T”)l. The integrand with allowance for the correction can be written as follows: fO(uOt So) = fO(vO)+ [df(uo)/duo]Sv. Retaining the next order small terms, we can easily show that the corrections for the characteristic and for the e*(r) function are O(em”3), i.e. they are exponentially small. Generally, the hydrodynamic approximation need not be refined because the local fluctuations in concentration of the particles acquire greater importance with time. This is attributable primarily to direct diffusion interaction (“collisions”) of the new-phase particles at distances smaller than their size. As shown in Refs. [5] and [6), this leads to the formation of an additional tail of the distribution function when u > uO, which is governed primarily by the distribution function when u < uO, and by the tendency of e2(r) = Ay/rO to approach a constant value of l*-- (In QJ’. This means that the universal distribution function is formed primarily in the “hydrodynamic” mode at u I ua and the particle “collisions” at u < u0 will in time become the main contribution to the tail of the distribution function at u > uO. A large contribution from the collisions will begin when 8, which is determined in the hydrodynamic approximation, is (In Q,,)*. Note that the maximum of the distribution function does not coincide with the point of the lowest velocity but is shifted to the left. This is attributable to the fact that in any approximation the distribution function at u > u0 is a decreasing function. Therefore, after having leaked across the region of the blocking point uo. it will have the maximum value only at some distance from uO, which is also independent of the initial conditions when T +m.
SLEZOV
The results obtained above can be easily extended to the case in which the sources of the dissolved component are present, because the growth rate of the new-phase particles, as demonstrated in Refs. [7-lo], ‘is independent of the existence of the sources, and in the equation for the material balance the factor e’+e”‘. where the exponent n I I depends on the power of the component’s source. The values of n, which depend on the mass-transfer mechanism, are bounded above, for which A(t)+0 when 7-m and the universal distribution function has time to form[9]. These results can be extended to the multicomponent, multiphase systems[l I-131, because the continuity equations for these systems in the later phase break down exponential accuracy into a set of independent equations for each phase [ I I]. Only the algebraic equations, which determine the regions of coexistence of the phases and the phase distribution of the components, are “coupled.” In this system the corrections for the concentration of any component are determined by the initial distribution functions of the phases containing the given component.
REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. IO. I I. 12. 13.
Lifshitz 1. M. and Slezov V. V., I.E.T.P. 35, 2 (1958). Slezov V. V., Fiz. Tverd. Tela 9, 4 (1%7). Slezov V. V. and Levin D. M., Fir. Tuerd. Tela 12, 6 (1970). Slezov V. V. and Saralidze Z. K., Fiz. Tverd. Tela 7.6 (1%5). Lifshitz I. M. and Slezov V. V., Fiz. Tverd. Tela 1. 9 (1959). Lifshitz I. M. and Slezov V. V., J. Phys. Chem. Solids 19,35 (l%l). Kosevich A. M., Saralidze Z. K. and Slezov V. V.. J.E.T.P. 52.64, 1079 (1967). Slezov V. V., Fiz. Tverd. Tela 9, 4 (1%7). Slezov V. V. and Shikin V. B., Fir. Tuerd. Tela 6, 1 (1964). Slezov V. V. and Shikin V. B., Fiz. Tverd. Tela 2, 3 (1965). Slezov V. V. and Sagalovich V. V.. Fiz. Tverd. Tela 17. 1497 (1975). Slezov V. V. and Sagalovich V. V.. Fiz. Tverd. Tela 17, 9 (1975). Slezov V. V.. Fiz. Tverd. Tela 17. 9 (1975).