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Effect of surfaces on glass waste form leaching
Journal of Non-Crystalline Solids 49 (1982) 379-388 North-Holland Publishing Company
379
EFFECTS OF SURFACES ON GLASS WASTE F O R M LEACHING C. PESCATORE and A.J. MACHIELS Nuclear Engineering Program, University of Illinois, Urbana, IL 61801, USA
A promising method for providing backup protection against the dispersion of radioactive waste by water is to incorporate the waste in a glass matrix. Although diffusion and network dissolution are generally considered to be rate-controlling, rates of attack of glass by water and aqueous solutions are best rationalized when the composition and microstructure of the glass surface are taken into account. In order to account for those effects, a leaching model emphasizing the processes occurring at the solid waste form-leachant interface is presented. Although the model leads rapidly to situations which are no longer easily amenable to analytical solutions, valuable physical insight can still be gained by determining the asymptotic behaviors of leach rates in the short- and long-term.
1. Introduction
Geologic disposal of nuclear wastes presently constitutes the most practical approach to isolate the radioactive wastes from the biosphere. Although envisioned burial depths are of the order of 1000 m, it is generally accepted that, at some time after burial, portions of the waste will be exposed to groundwater. Therefore it has been proposed to incorporate the waste in glass in order to provide additional protection against the dispersion of radioactivity by water. Measurements of rates of attack of glass by water and aqueous solutions and identification of the controlling leaching mechanisms have been the object of many experimental and theoretical programs [1-3]. Although diffusion and network dissolution are generally considered to be rate-controlling, leaching behaviors are best rationalized when composition and microstructure of the glass surface are taken into account. Since the latter depends upon many factors including test samples thermal history and treatment procedures prior to experimentation [2,3], it is not surprising that past and on-going investigations have uncovered a wide range of behaviors. In order to account better for all effects, a leaching model emphasizing the role of interfacial processes is presented in section 2. When interface kinetics are neglected, the model reduces to the classical diffusion equation (section 3). In section4, an analytical solution to a model discussed previously [4] is presented. In section 5, a method to determine how leach rates vary in the short and long terms, without requiring the full analytical expressions, is presented. 0022-3093/82/0000-0000/$02.75 © 1982 North-Holland
C. Pescatore, ,4.J. Machiels / Effects of surfaces on glass waste
380
2. Reference model
Leaching of a specified species, "A ", is assumed to proceed by the following steps: kl
A(surface)
~-
A(solution),
(1)
A(surface),
(2)
A(surface layer).
(3)
kd H
A (surface layer)
=
A(bulk)
=
D
Step (1) represents the surface processes leading to the exchanges of A between the waste glass and the leachant. The atoms of A present on the waste form outer surface that is in contact with the leachant (surface species) are released into the leachant at a rate characterized by a phenomenological rate constant k l, while the atoms of A present in the leachant (solution species) can re-deposit on the glass surface at a rate characterized by a mass transfer coefficient k d. Step (2) assumes the existence of a chemical eqilibrium between the atoms of A exposed to the leachant (surface species) and those present in a layer of solid adjacent to the glass-leachant interface. The equilibrium constant is represented by a solubility coefficient, H. Bulk diffusion of A in the waste glass is represented by step (3) in terms of a diffusion coefficient, D. The leach rate, L ( t ) , is proportional to J(t), the net flux of A per unit area at the specimen-leachant interface, given by: J( t ) = k , n ( t ) - kdCso.( t ),
(4)
where t designates the time; n ( t ) represents the surface concentration of speciesA expressed in atoms/unit surface area of waste glass while Csoj(t ) represents the concentration of A in the leachant expressed in atoms/unit volume of leachant. To obtain J(t), the following system of equations has to be solved:
at
az o
dn(t) dt - - k i n ( t )
az
'
(OC) + koCso,( t ) + D -~z s'
(5) (6)
Cs(/) = H n ( t ) ,
(7)
dC~o](t) dt - f l J ( t ) - dpCso](t),
(8)
C(z,O)
(9)
and
C~ol(0):specified.
Eq. (5) is the diffusion equation; for simplicity a one-dimensional formulation
C. Pescatore,A.J. Machiels / Effects of surfaces on glass waste
381
has been retained; C(z, t) represents the concentration of A in the waste glass at some arbitrary space coordinate z and time t. Eq. (6) constitutes one boundary condition to the diffusion equation and is obtained by performing a mass balance on the surface species; D(OC/Oz)s represents the bulk diffusive flux at the waste glass outer surface. It is to be noted that the second boundary condition results from the choice of a geometry for the glass specimen. Eq. (7) is the mathematical representation of the condition of equilibrium represented by step(2); C~(t) denotes the concentration of A in the layer of solid adjacent to the glass-leachant interface. Eq. (8) is obtained by performing a mass balance on the solution species; J ( t ) is given by eq. (4) while: •= S A / V ,
(10a)
and
r/v,
(10b)
where SA is the surface area of the glass specimen; V is the volume of leachant contacting the specimen and F represents the leachant volumetric flow rate. fl and ff are important leach parameters. Initial conditions for C and C~o1 also have to be specified explicitly [eq. (9)].
3. Diffusion control
When interface kinetics are fast, leach rates are controlled by the bulk diffusion process. Assuming a uniform initial distribution of A in the glass C(z, 0), and constant Cs, leach rates are governed by the familiar square root-of-time dependence regardless of the dependence of the diffusion coefficient on the concentration C, i.e. L ( t) ~: t - t / 2
(11)
when the sample approximates a semi-infinite geometry, which is generally the case [5]. This result was pointed out long ago by Doremus [6] and can be simply obtained by using a Lie group approach [7] or by using the Boltzmann transformation directly [6,8]. Non-uniform initial solute distributions a n d / o r time-dependent Cs do not generally result in a simple square root-of-time dependence.
4. Surface and diffusion controls: an example of analytical solution
The influence of the surface step can be illustrated simply by considering the leaching behavior under dynamic leaching and uniform initial distribution
C. Pescatore,A.J. Machiels/ Effects of surfaces on glass waste
382 conditions, or:
C~o,(t)~ 0
(12a)
C(z,O)=Co"
(12b)
Furthermore let us assume that the specimen can be approximated by a semi-infinite geometry with the z-coordinate representing the distance from the solid surface-leachant interface to any point in the solid; the second boundary condition to eq. (5) now becomes:
(13)
c ( + o ,t) = Co.
Let us furthermore assume that D, H and k I are constant. Under these conditions, applying the Laplace transformation to eqs. (4), (5), (6) and (13), taking eqs. (12) into account, yields:
.f = nD1/ZQ, Q
Co =7
(14a) */Co
(14b)
p[pl/2 + c p + * / ] '
where J and Q represent the image functions of J and Cs, and p is the parameter of the transformation. Moreover,
c-- I / H D 1/2,
(15a)
71= k l / H D 1/2.
(15b) m
In order to invert Cs, it is expedient to re-write eq. (14b) as:
=Co P
.Co P~ - P 0
1
~2p'/2(p-po ) ,/
1
c(P-Po)
c2pl(p-p,)
1
c2p'/2(p-p,)
1
+c(p-p,)
,/
c2po(p-po)
2p
Po
P 1
'
(16) where P0 and P n are the two zeroes of the polynomial:
p ( p ) __ ~2p2
_
_
(1 -- 2~T/)p + 7/2,
(17)
namely, (1 --
Po ----
2c 2 (1 -
Pl -
2c~/) + (1 -- 4c*/) '/2
2c7/) - (1 - 4c,/) '/2 2c 2
(18a)
(18b)
C. Pescatore. A.J. Maehiels / Effects of surfaces on glass waste
For Po 5~:Pl :
C~(t) _
ePo Z
r/Co
'(Po-Pl)
383
,)
(1 e r f ( p o t ) , / 2 _ ~__~____ ¢p10/2 cPo
e ?'' (1_~__ erf(plt)t/2 - - - -71 c ( p o - p l ) ~ Cpll/2 Epj
1).
(19a)
For Po = P l :
cs(t) nCo
- (l --
2tpo ) e ?°' f i r ~0
epo t
+ 2 t ePo t - - - ¢
__
71
I/2 e - e : d r 2t3/2
(19b)
,//-1/2¢ 2
Moreover, kI
J(t) = k , n ( t ) = ~ C~(t).
(20)
Physical insight can be gained by determining how leach rates vary in the short and long term. Operating on eq. (19a) a n d / o r eq. (19b) the short-term behavior of J(t) is found to be: J( t) ~ k , n o ( 1 -
4
kit + 3--~klnDl/2t3/2
- ... )
(21a)
and
J(O) = k,n o.
(21b)
The long-term behavior reduces to the classical result:
J(t) ~ Co(D/Irt) '/2
for t large enough.
(21c)
Therefore chemical exchange at the surface is the rate determining process in the early stages of leaching, there being
J(t) -~ kin o e -k''
for t small enough.
(21d)
5. Complex models: the use of asymptotic solutions
As the model becomes increasingly complex, e.g. by taking into account the effect of flow rate, a fully analytical inversion of the expression for ar becomes impractical. For the case where all parameters are constant we obtain:
J ( p ) = k l n o f ( p ) -- k2nog( p ),
(22a)
where:
f(P)
=
P+ ~
p(P+*I') '
(22b)
C Pescatore, A.J. Maehiels / Effects of surfaces on glass waste
384 ~--
g(P)
( p ...~_q~)2
,
(22C)
p ( p + r / ' ) [ p ( p + r/') + k , ( p + qa) + y p , / 2 ( p + ~/,)]
where
rt'= q~+ k,j/3
(22d)
y = HD I/2.
(22e)
and Eqs. (22b) and (22c) correctly predict that the two cases kd,8 = 0 and ,# = oa are indeed equivalent, and eq. (22a) reduces to eq. (14a). The full analytical expression for g(t) using the inversion theorem is practically useless, even if it can be formally derived by using the theory of residues [9]. In fact it turns out that the function g ( p ) only possesses two simple poles at p = 0 and p = - rt' in the complex plane cut along the negative real axis, where - ~ ' < arg p ~ or. This can be seen by applying Hurwitz's theorem [9] to the term in square brackets in eq. (22c), and noticing that R e { p ~/2 ) > 0 for -~r < arg p ~< ~r. Nevertheless helpful information can be obtained by resorting to the theory of asymptotic expansions from the point of view of functional transforms [10,11], i.e. we determine the asymptotic behavior of the function J(t) as t --, 0 or as t ~ oo, by a knowledge of the asymptotic behavior of J ( p ) as p --, ~ , or as p tends to some special point, p*, respectively. This technique is commonly employed in the theory of heat transfer [12,13] and of electric circuits, for instance.
5.1. The behavior of the function g(t) for small values of time Following ref. [10], one can deduce that the asymptotic expansion of g ( p ) as p ~ ~ reads as: N
g(p)=
~ A,p-~"+o(p-~),
0
.... asp---, ~
(23)
n=l
The term o ( p -"N) in eq. (23) implies that the difference between the function g ( p ) and the indicated partial sum is of smaller order of magnitude than the last included element as p --, oc. The first four values of p, and A, are: u1=2
A)=I,
~'2=~
A 2 = - - H D '/2,
P3 = 3
A 3 = H2D - 2kdfl -- k),
g4 = 7
An=HD'/2(2kd,8+k,-H2D).
(24)
Accordingly, for small values of time, if we assume that g(t) can be written
385
C. Pescatore, A.J. Machiels / Effects of surfaces on glass waste
down term by term from the above result, we find that:
4HDt/Zt3/2 + I ( H Z D - k
g(t)=t
,
2kdfl)t 2
3qrl/2
8 I4D,/Z(H 15~rl/2
D_2k _2kdfl)ts/2+o(tS/2) '
t ---~O.
(25)
5.2. The behavior of the function g(t) for large values of time One can deduce that g ( p ) possesses the asymptotic expansion
g(p)
~
=
A, --+o(p-~N),
v , > v z > .... a s p s 0 .
(26a)
n ~ l pV,,
Indeed, when ~ -- 0 corresponding to static leaching conditions:
v 1 = --½
A 1= 1
HD 1/2 k d
,
/"2 ~- -- l
A2 ~-. __ (kdfl q'- kl)/[H2D(kdfl)3],
(268)
and when q~4= 0: b'lzl
A I --- ( ~ / [ k l ( k d f l ~- ~ ) ] ,
v2= +½
A2 = _ H D , / Z / k ( .
(26c)
The function g ( p ) is many-valued in the complex plane at p = 0, which is the singular point with the larger real part. Therefore [ 11 ] the original function g(t) possesses the asymptotic representation: U A, g(t) = 2 -F-(t +" "v-,')+ o ( t " ~ - ' ) ,
ast~oo,
(27)
n=l
where 1 - - - - 0 ,
for v , = 0 , - 1 , - 2 ....
5.3. The behavior of J(t) for large and small values of time It is formally
J( t ) = k,nof( t ) - k2nog( t ),
any t,
(28a)
where:
di, karl e-~kd~+~.), f ( t ) -- k ofl +~----~ -~ k dfl + dp and g(t) is given by eqs. (25) or (27a).
(288)
C. Pescatore, A.J. Machiels / Effects of surfaces on glass waste
386
Summarizing we can write that k d fl ~ e -(k~t~+~)' ) - k 2 n o g ( t ) , J ( t ) = k , n o ( d fl + d? ~-knfl+--------
(29)
where the different asymptotic expressions for g(t) are given in table 1; in those expressions, o(t ~) represents the order of magnitude of the first excluded term in the expansion. Using eq. (29) along with the expressions of g(t) shown in table 1, we obtain:
J(t) ~Co(D/~rt) '/2, J(t) ~ Co(D/~rt) t/z
dpvLO,t--, o0, k, 2H2D(kdfl)zt,
(30a)
ep = O, t --. o0,
(30b)
i.e. bulk diffusion eventually becomes rate controlling when 4' v~ O. Similarly, short-term leaching behaviors are described by: ( 4 J( t ) ~ k,n o l - ( k, + k dfl )t + 3 - - ~ k,HD' /2t3 /2
-½[k,nED-
k 2 - 2kdflk , - kdfl(kdfl+ ~b)]t 2 + . . . } as t---,0. (31)
Eqs. (30a) and (31) reduce to eqs. (21c) and (21a) respectively, thereby providing a direct validation of the asymptotic analysis that we have performed. The relative importance of the various physical processes that appear in the model is reflected by their position in the short-term asymptotic series. As t "approaches zero, the rates at which t"-containing terms approach zero depend upon ~,. A larger value for p results in a faster approach to zero. By inspection of eq. (31), it can be seen that the process described by kj appears instantly followed by the processes described by k a fl (re-deposition), D (diffusion), and Table 1 Asymptotic expressions for g(t) t~0
t --
t~ 4
HDI/2t 3/2
3rrl/2
+ l ( H 2 D - k I --2kdfl)t 2 ,#~0
8
15~1/2
HDW2(H2D_2kl _2kdfl)ts/2
4,
H(DI~/2
+ o(t 3/2)
+ o( t ~) 4, = 0
same as above
t-3/2 2(~rD)l/2(kafl) 2 +0(t - 5 / 2 )
C. Pescatore, A.J. Machiels / Effects of surfaces on glass waste
387
finally ~ (leachant renewal frequency). Once again surface processes dominate in the early stages of leaching, and we extract the limiting behavior J( t ) ~- k i n o e -lk'+kJB)t
for t small enough.
(32)
This relatively fast release of surface species is reported in the literature [14,15], and is commonly referred to as the initial wash-off. By reasoning in an analogous manner, asymptotic expansions can also be developed for the other functions of interest, e.g. Cs(t) and Csol(t).
6. Conclusion
The proposed leach model is formulated to emphasize the surface processes occurring at the waste form-leachant interface. Although network dissolution can be readily added to our model as long as a linear formulation is retained, we have chosen not to include it at this time in order to facilitate the presentation of our considerations. Short- and long-term leaching trends can be obtained by using asymptotic expansions. At present the usefulness of this technique is limited by the requirement that the parameters used to describe the various physico-chemical processes involved in leaching remain constant, leading to a linear model. In fact, it is hardly possible in the nonlinear case to find an integral representation for J(t), e.g. the original of a Laplace transform, on which we can operate with an asymptotic technique. A computer would be the only viable alternative. However, unlike an asymptotic expansion, this would not provide a fresh insight into the factors which influence leaching, whereas the physical relevance of each process with regard to leaching is reflected by the position occupied by the various physical parameters in the asymptotic expressions which have been provided for the leach rate. This work has been supported by the United States Department of Energy through the Office of Nuclear Waste Isolation.
References [1] J.E. Mendel, PNL-2764, Pacific Northwest Laboratory (1978). [2] D.E. Clark, C.G. Pantano and L.L. Hench, Corrosion of glass (Books for Industry and the Glass Industry, New York, 1979). [3] G.H. Frischat, Ionic diffusion in oxide glasses (Trans. Tech. Publications, 1975). [4] A.J. Machiels and C. Pescatore, in: Scientific basis for nuclear waste management, Vol. 3 (Plenum, New York, 1981) p. 371. [5] C. Pescatore and A.J. Machiels, J. Nucl. Tech. 56 (1982) 297. [6] R.H. Doremus, in: Modern aspects of the vitreous state, ed., J.D. Mackenzie, Vol. 2 (Butterworths, London, 1962). [7] G.W. Bluman and J.D. Cole, Similarity methods for differential equations, Applied Mathematical Sciences, Vol. 13 (Springer-Verlag, Berlin, 1974).
388
C. Pescatore, A.J. Machiels / Effects of surfaces on glass waste
[8] J. Crank, The mathematics of diffusion, 2nd Ed. (Clarendon, Oxford, 1975). [9] M. Lavrentiev and B. Chabat, M6thodes de la th~orie des fonctions d'une variable complexe (MIR editions, Moscow). [10] G. Doetsch, Istituto Nazionale Applicazioni del Calcolo, Publ. no. 420, Rome 0954). Ill] G. Doetsch, Introduction to the theory and application of the Laplace transformation (Springer-Verlag, Berlin, 1974). [12] H.S. Carslaw and J.C. Jaeger, Operational methods in applied mathematics, 2nd Ed. (Oxford University Press, Oxford, 1948). [13] H,S. Carslaw and J.C. Jaeger, Conduction of heat in solids (Clarendon Press, Oxford, 1959). [14] A.J. Machiels, Workshop on Alternate nuclear waste forms and interactions in geologic media, Gatlinburg (TN), May 13-15, 1980 (DOE-CONF-8005107, April 1981). [15] H.W. Godbee et al., Nucl. Chem. Waste Management 1 (1980) 29.
379
EFFECTS OF SURFACES ON GLASS WASTE F O R M LEACHING C. PESCATORE and A.J. MACHIELS Nuclear Engineering Program, University of Illinois, Urbana, IL 61801, USA
A promising method for providing backup protection against the dispersion of radioactive waste by water is to incorporate the waste in a glass matrix. Although diffusion and network dissolution are generally considered to be rate-controlling, rates of attack of glass by water and aqueous solutions are best rationalized when the composition and microstructure of the glass surface are taken into account. In order to account for those effects, a leaching model emphasizing the processes occurring at the solid waste form-leachant interface is presented. Although the model leads rapidly to situations which are no longer easily amenable to analytical solutions, valuable physical insight can still be gained by determining the asymptotic behaviors of leach rates in the short- and long-term.
1. Introduction
Geologic disposal of nuclear wastes presently constitutes the most practical approach to isolate the radioactive wastes from the biosphere. Although envisioned burial depths are of the order of 1000 m, it is generally accepted that, at some time after burial, portions of the waste will be exposed to groundwater. Therefore it has been proposed to incorporate the waste in glass in order to provide additional protection against the dispersion of radioactivity by water. Measurements of rates of attack of glass by water and aqueous solutions and identification of the controlling leaching mechanisms have been the object of many experimental and theoretical programs [1-3]. Although diffusion and network dissolution are generally considered to be rate-controlling, leaching behaviors are best rationalized when composition and microstructure of the glass surface are taken into account. Since the latter depends upon many factors including test samples thermal history and treatment procedures prior to experimentation [2,3], it is not surprising that past and on-going investigations have uncovered a wide range of behaviors. In order to account better for all effects, a leaching model emphasizing the role of interfacial processes is presented in section 2. When interface kinetics are neglected, the model reduces to the classical diffusion equation (section 3). In section4, an analytical solution to a model discussed previously [4] is presented. In section 5, a method to determine how leach rates vary in the short and long terms, without requiring the full analytical expressions, is presented. 0022-3093/82/0000-0000/$02.75 © 1982 North-Holland
C. Pescatore, ,4.J. Machiels / Effects of surfaces on glass waste
380
2. Reference model
Leaching of a specified species, "A ", is assumed to proceed by the following steps: kl
A(surface)
~-
A(solution),
(1)
A(surface),
(2)
A(surface layer).
(3)
kd H
A (surface layer)
=
A(bulk)
=
D
Step (1) represents the surface processes leading to the exchanges of A between the waste glass and the leachant. The atoms of A present on the waste form outer surface that is in contact with the leachant (surface species) are released into the leachant at a rate characterized by a phenomenological rate constant k l, while the atoms of A present in the leachant (solution species) can re-deposit on the glass surface at a rate characterized by a mass transfer coefficient k d. Step (2) assumes the existence of a chemical eqilibrium between the atoms of A exposed to the leachant (surface species) and those present in a layer of solid adjacent to the glass-leachant interface. The equilibrium constant is represented by a solubility coefficient, H. Bulk diffusion of A in the waste glass is represented by step (3) in terms of a diffusion coefficient, D. The leach rate, L ( t ) , is proportional to J(t), the net flux of A per unit area at the specimen-leachant interface, given by: J( t ) = k , n ( t ) - kdCso.( t ),
(4)
where t designates the time; n ( t ) represents the surface concentration of speciesA expressed in atoms/unit surface area of waste glass while Csoj(t ) represents the concentration of A in the leachant expressed in atoms/unit volume of leachant. To obtain J(t), the following system of equations has to be solved:
at
az o
dn(t) dt - - k i n ( t )
az
'
(OC) + koCso,( t ) + D -~z s'
(5) (6)
Cs(/) = H n ( t ) ,
(7)
dC~o](t) dt - f l J ( t ) - dpCso](t),
(8)
C(z,O)
(9)
and
C~ol(0):specified.
Eq. (5) is the diffusion equation; for simplicity a one-dimensional formulation
C. Pescatore,A.J. Machiels / Effects of surfaces on glass waste
381
has been retained; C(z, t) represents the concentration of A in the waste glass at some arbitrary space coordinate z and time t. Eq. (6) constitutes one boundary condition to the diffusion equation and is obtained by performing a mass balance on the surface species; D(OC/Oz)s represents the bulk diffusive flux at the waste glass outer surface. It is to be noted that the second boundary condition results from the choice of a geometry for the glass specimen. Eq. (7) is the mathematical representation of the condition of equilibrium represented by step(2); C~(t) denotes the concentration of A in the layer of solid adjacent to the glass-leachant interface. Eq. (8) is obtained by performing a mass balance on the solution species; J ( t ) is given by eq. (4) while: •= S A / V ,
(10a)
and
r/v,
(10b)
where SA is the surface area of the glass specimen; V is the volume of leachant contacting the specimen and F represents the leachant volumetric flow rate. fl and ff are important leach parameters. Initial conditions for C and C~o1 also have to be specified explicitly [eq. (9)].
3. Diffusion control
When interface kinetics are fast, leach rates are controlled by the bulk diffusion process. Assuming a uniform initial distribution of A in the glass C(z, 0), and constant Cs, leach rates are governed by the familiar square root-of-time dependence regardless of the dependence of the diffusion coefficient on the concentration C, i.e. L ( t) ~: t - t / 2
(11)
when the sample approximates a semi-infinite geometry, which is generally the case [5]. This result was pointed out long ago by Doremus [6] and can be simply obtained by using a Lie group approach [7] or by using the Boltzmann transformation directly [6,8]. Non-uniform initial solute distributions a n d / o r time-dependent Cs do not generally result in a simple square root-of-time dependence.
4. Surface and diffusion controls: an example of analytical solution
The influence of the surface step can be illustrated simply by considering the leaching behavior under dynamic leaching and uniform initial distribution
C. Pescatore,A.J. Machiels/ Effects of surfaces on glass waste
382 conditions, or:
C~o,(t)~ 0
(12a)
C(z,O)=Co"
(12b)
Furthermore let us assume that the specimen can be approximated by a semi-infinite geometry with the z-coordinate representing the distance from the solid surface-leachant interface to any point in the solid; the second boundary condition to eq. (5) now becomes:
(13)
c ( + o ,t) = Co.
Let us furthermore assume that D, H and k I are constant. Under these conditions, applying the Laplace transformation to eqs. (4), (5), (6) and (13), taking eqs. (12) into account, yields:
.f = nD1/ZQ, Q
Co =7
(14a) */Co
(14b)
p[pl/2 + c p + * / ] '
where J and Q represent the image functions of J and Cs, and p is the parameter of the transformation. Moreover,
c-- I / H D 1/2,
(15a)
71= k l / H D 1/2.
(15b) m
In order to invert Cs, it is expedient to re-write eq. (14b) as:
=Co P
.Co P~ - P 0
1
~2p'/2(p-po ) ,/
1
c(P-Po)
c2pl(p-p,)
1
c2p'/2(p-p,)
1
+c(p-p,)
,/
c2po(p-po)
2p
Po
P 1
'
(16) where P0 and P n are the two zeroes of the polynomial:
p ( p ) __ ~2p2
_
_
(1 -- 2~T/)p + 7/2,
(17)
namely, (1 --
Po ----
2c 2 (1 -
Pl -
2c~/) + (1 -- 4c*/) '/2
2c7/) - (1 - 4c,/) '/2 2c 2
(18a)
(18b)
C. Pescatore. A.J. Maehiels / Effects of surfaces on glass waste
For Po 5~:Pl :
C~(t) _
ePo Z
r/Co
'(Po-Pl)
383
,)
(1 e r f ( p o t ) , / 2 _ ~__~____ ¢p10/2 cPo
e ?'' (1_~__ erf(plt)t/2 - - - -71 c ( p o - p l ) ~ Cpll/2 Epj
1).
(19a)
For Po = P l :
cs(t) nCo
- (l --
2tpo ) e ?°' f i r ~0
epo t
+ 2 t ePo t - - - ¢
__
71
I/2 e - e : d r 2t3/2
(19b)
,//-1/2¢ 2
Moreover, kI
J(t) = k , n ( t ) = ~ C~(t).
(20)
Physical insight can be gained by determining how leach rates vary in the short and long term. Operating on eq. (19a) a n d / o r eq. (19b) the short-term behavior of J(t) is found to be: J( t) ~ k , n o ( 1 -
4
kit + 3--~klnDl/2t3/2
- ... )
(21a)
and
J(O) = k,n o.
(21b)
The long-term behavior reduces to the classical result:
J(t) ~ Co(D/Irt) '/2
for t large enough.
(21c)
Therefore chemical exchange at the surface is the rate determining process in the early stages of leaching, there being
J(t) -~ kin o e -k''
for t small enough.
(21d)
5. Complex models: the use of asymptotic solutions
As the model becomes increasingly complex, e.g. by taking into account the effect of flow rate, a fully analytical inversion of the expression for ar becomes impractical. For the case where all parameters are constant we obtain:
J ( p ) = k l n o f ( p ) -- k2nog( p ),
(22a)
where:
f(P)
=
P+ ~
p(P+*I') '
(22b)
C Pescatore, A.J. Maehiels / Effects of surfaces on glass waste
384 ~--
g(P)
( p ...~_q~)2
,
(22C)
p ( p + r / ' ) [ p ( p + r/') + k , ( p + qa) + y p , / 2 ( p + ~/,)]
where
rt'= q~+ k,j/3
(22d)
y = HD I/2.
(22e)
and Eqs. (22b) and (22c) correctly predict that the two cases kd,8 = 0 and ,# = oa are indeed equivalent, and eq. (22a) reduces to eq. (14a). The full analytical expression for g(t) using the inversion theorem is practically useless, even if it can be formally derived by using the theory of residues [9]. In fact it turns out that the function g ( p ) only possesses two simple poles at p = 0 and p = - rt' in the complex plane cut along the negative real axis, where - ~ ' < arg p ~ or. This can be seen by applying Hurwitz's theorem [9] to the term in square brackets in eq. (22c), and noticing that R e { p ~/2 ) > 0 for -~r < arg p ~< ~r. Nevertheless helpful information can be obtained by resorting to the theory of asymptotic expansions from the point of view of functional transforms [10,11], i.e. we determine the asymptotic behavior of the function J(t) as t --, 0 or as t ~ oo, by a knowledge of the asymptotic behavior of J ( p ) as p --, ~ , or as p tends to some special point, p*, respectively. This technique is commonly employed in the theory of heat transfer [12,13] and of electric circuits, for instance.
5.1. The behavior of the function g(t) for small values of time Following ref. [10], one can deduce that the asymptotic expansion of g ( p ) as p ~ ~ reads as: N
g(p)=
~ A,p-~"+o(p-~),
0
.... asp---, ~
(23)
n=l
The term o ( p -"N) in eq. (23) implies that the difference between the function g ( p ) and the indicated partial sum is of smaller order of magnitude than the last included element as p --, oc. The first four values of p, and A, are: u1=2
A)=I,
~'2=~
A 2 = - - H D '/2,
P3 = 3
A 3 = H2D - 2kdfl -- k),
g4 = 7
An=HD'/2(2kd,8+k,-H2D).
(24)
Accordingly, for small values of time, if we assume that g(t) can be written
385
C. Pescatore, A.J. Machiels / Effects of surfaces on glass waste
down term by term from the above result, we find that:
4HDt/Zt3/2 + I ( H Z D - k
g(t)=t
,
2kdfl)t 2
3qrl/2
8 I4D,/Z(H 15~rl/2
D_2k _2kdfl)ts/2+o(tS/2) '
t ---~O.
(25)
5.2. The behavior of the function g(t) for large values of time One can deduce that g ( p ) possesses the asymptotic expansion
g(p)
~
=
A, --+o(p-~N),
v , > v z > .... a s p s 0 .
(26a)
n ~ l pV,,
Indeed, when ~ -- 0 corresponding to static leaching conditions:
v 1 = --½
A 1= 1
HD 1/2 k d
,
/"2 ~- -- l
A2 ~-. __ (kdfl q'- kl)/[H2D(kdfl)3],
(268)
and when q~4= 0: b'lzl
A I --- ( ~ / [ k l ( k d f l ~- ~ ) ] ,
v2= +½
A2 = _ H D , / Z / k ( .
(26c)
The function g ( p ) is many-valued in the complex plane at p = 0, which is the singular point with the larger real part. Therefore [ 11 ] the original function g(t) possesses the asymptotic representation: U A, g(t) = 2 -F-(t +" "v-,')+ o ( t " ~ - ' ) ,
ast~oo,
(27)
n=l
where 1 - - - - 0 ,
for v , = 0 , - 1 , - 2 ....
5.3. The behavior of J(t) for large and small values of time It is formally
J( t ) = k,nof( t ) - k2nog( t ),
any t,
(28a)
where:
di, karl e-~kd~+~.), f ( t ) -- k ofl +~----~ -~ k dfl + dp and g(t) is given by eqs. (25) or (27a).
(288)
C. Pescatore, A.J. Machiels / Effects of surfaces on glass waste
386
Summarizing we can write that k d fl ~ e -(k~t~+~)' ) - k 2 n o g ( t ) , J ( t ) = k , n o ( d fl + d? ~-knfl+--------
(29)
where the different asymptotic expressions for g(t) are given in table 1; in those expressions, o(t ~) represents the order of magnitude of the first excluded term in the expansion. Using eq. (29) along with the expressions of g(t) shown in table 1, we obtain:
J(t) ~Co(D/~rt) '/2, J(t) ~ Co(D/~rt) t/z
dpvLO,t--, o0, k, 2H2D(kdfl)zt,
(30a)
ep = O, t --. o0,
(30b)
i.e. bulk diffusion eventually becomes rate controlling when 4' v~ O. Similarly, short-term leaching behaviors are described by: ( 4 J( t ) ~ k,n o l - ( k, + k dfl )t + 3 - - ~ k,HD' /2t3 /2
-½[k,nED-
k 2 - 2kdflk , - kdfl(kdfl+ ~b)]t 2 + . . . } as t---,0. (31)
Eqs. (30a) and (31) reduce to eqs. (21c) and (21a) respectively, thereby providing a direct validation of the asymptotic analysis that we have performed. The relative importance of the various physical processes that appear in the model is reflected by their position in the short-term asymptotic series. As t "approaches zero, the rates at which t"-containing terms approach zero depend upon ~,. A larger value for p results in a faster approach to zero. By inspection of eq. (31), it can be seen that the process described by kj appears instantly followed by the processes described by k a fl (re-deposition), D (diffusion), and Table 1 Asymptotic expressions for g(t) t~0
t --
t~ 4
HDI/2t 3/2
3rrl/2
+ l ( H 2 D - k I --2kdfl)t 2 ,#~0
8
15~1/2
HDW2(H2D_2kl _2kdfl)ts/2
4,
H(DI~/2
+ o(t 3/2)
+ o( t ~) 4, = 0
same as above
t-3/2 2(~rD)l/2(kafl) 2 +0(t - 5 / 2 )
C. Pescatore, A.J. Machiels / Effects of surfaces on glass waste
387
finally ~ (leachant renewal frequency). Once again surface processes dominate in the early stages of leaching, and we extract the limiting behavior J( t ) ~- k i n o e -lk'+kJB)t
for t small enough.
(32)
This relatively fast release of surface species is reported in the literature [14,15], and is commonly referred to as the initial wash-off. By reasoning in an analogous manner, asymptotic expansions can also be developed for the other functions of interest, e.g. Cs(t) and Csol(t).
6. Conclusion
The proposed leach model is formulated to emphasize the surface processes occurring at the waste form-leachant interface. Although network dissolution can be readily added to our model as long as a linear formulation is retained, we have chosen not to include it at this time in order to facilitate the presentation of our considerations. Short- and long-term leaching trends can be obtained by using asymptotic expansions. At present the usefulness of this technique is limited by the requirement that the parameters used to describe the various physico-chemical processes involved in leaching remain constant, leading to a linear model. In fact, it is hardly possible in the nonlinear case to find an integral representation for J(t), e.g. the original of a Laplace transform, on which we can operate with an asymptotic technique. A computer would be the only viable alternative. However, unlike an asymptotic expansion, this would not provide a fresh insight into the factors which influence leaching, whereas the physical relevance of each process with regard to leaching is reflected by the position occupied by the various physical parameters in the asymptotic expressions which have been provided for the leach rate. This work has been supported by the United States Department of Energy through the Office of Nuclear Waste Isolation.
References [1] J.E. Mendel, PNL-2764, Pacific Northwest Laboratory (1978). [2] D.E. Clark, C.G. Pantano and L.L. Hench, Corrosion of glass (Books for Industry and the Glass Industry, New York, 1979). [3] G.H. Frischat, Ionic diffusion in oxide glasses (Trans. Tech. Publications, 1975). [4] A.J. Machiels and C. Pescatore, in: Scientific basis for nuclear waste management, Vol. 3 (Plenum, New York, 1981) p. 371. [5] C. Pescatore and A.J. Machiels, J. Nucl. Tech. 56 (1982) 297. [6] R.H. Doremus, in: Modern aspects of the vitreous state, ed., J.D. Mackenzie, Vol. 2 (Butterworths, London, 1962). [7] G.W. Bluman and J.D. Cole, Similarity methods for differential equations, Applied Mathematical Sciences, Vol. 13 (Springer-Verlag, Berlin, 1974).
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C. Pescatore, A.J. Machiels / Effects of surfaces on glass waste
[8] J. Crank, The mathematics of diffusion, 2nd Ed. (Clarendon, Oxford, 1975). [9] M. Lavrentiev and B. Chabat, M6thodes de la th~orie des fonctions d'une variable complexe (MIR editions, Moscow). [10] G. Doetsch, Istituto Nazionale Applicazioni del Calcolo, Publ. no. 420, Rome 0954). Ill] G. Doetsch, Introduction to the theory and application of the Laplace transformation (Springer-Verlag, Berlin, 1974). [12] H.S. Carslaw and J.C. Jaeger, Operational methods in applied mathematics, 2nd Ed. (Oxford University Press, Oxford, 1948). [13] H,S. Carslaw and J.C. Jaeger, Conduction of heat in solids (Clarendon Press, Oxford, 1959). [14] A.J. Machiels, Workshop on Alternate nuclear waste forms and interactions in geologic media, Gatlinburg (TN), May 13-15, 1980 (DOE-CONF-8005107, April 1981). [15] H.W. Godbee et al., Nucl. Chem. Waste Management 1 (1980) 29.