United States
Environmental Protection
Agency
Office of
Radiation Programs
Washington DC 20460
EPA 520/8-81-001
January 1981
Radiation
&EPA
Leaching of
Radioactive Isotopes
from Waste Solids
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EPA 520/8-81-001
Leaching of Radioactive Isotopes
from Waste Solids
Donald S. Cohen
Office of Radiation Programs
U.S. Environmental Protection Agency
Washington, D.C. 20460
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Abstract
The most commonly used empirical model for the quantity Q of
a radioactive isotope leached from a solidified waste as a function
112
of time t is Q = at + bt, where a and b are empirically
determined constants for a specific leach process involving a
specific isotope. This formula works well for many solid wastes and
can be derived theoretically from a model employing Fickian
diffusion. However, the formula is known to be totally inaccurate
for many other solids and also in particular for glassy solids which
devitrify. Since devitrification and other symptoms of aging are
commonly-occurring processes in the long term storage of radioactive
waste, it is important to have a correct alternative formula for Q
and even more important to understand the physical processes
involved in leaching.
A theoretical model involving a generalized, non-Fickian
mechanism for diffusion is derived in this paper and applied to
determine Q as a function of time t. It is found, on the basis
of this type of diffusion which occurs in devitrified glassy solids
and other solid waste materials, that Q=At '4 + Bt3/4. Here A
and B are constants which can be determined empirically, and they
can also be determined phenomenologically in terms of the
fundamental parameters of the diffusion model. When both formulas
are normalized to a common value, at some instant the new formula
predicts larger initial quantities and faster initial leach rates.
m
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CONTENTS
Abstract iii
1. Introduction 1
2. Generalized Diffusion Model 4
3. The Time Law for Leaching 10
4. Accounting for Radioactive Decay 17
References 18
Appendix A
Generalized Diffusion Equations 21
Appendix B
Boundary Conditions for the Generalized Diffused Equation 29
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-1-
1. Introduction
It is essential for safety, engineering, and economic considerations that
we have reliable estimates of the amount of radioactivity that enters the en-
vironment as a result of the treatment, storage, transport, and disposal of
radioactive waste. We shall consider the effects of leaching on solidified
radioactive waste materials. Leaching of a waste deposit by groundwater
followed by transport in an aqueous solution is a commonly occurring situation.
The leach resistance of solidified waste materials in groundwater is often cited
as the critical factor for the decision to use such methods of disposal for
the long term isolation of high level radioactive waste [l]-[3]. In particular,
because of its assumed high resistance to leaching, glass (and various other
forms of vitreous, ceramic, and plastic materials) are currently popular for
storing high level wastes [3]. The storage periods of interest range from
tens of years to periods of approximately 100,000 years.
The most important information sought is a reliable estimate of the
quantity Q of a radioactive isotope leached from a solidified waste as a
function of time t, and the most commonly used empirical model for Q is
(1.1) Q = at1* + bt.
Here a and b are empirically determined constants for a specific leach
process involving a specific isotope. There are some other competing expres-
sions [1] for Q which attempt to account for on-site factors which would not
occur in standard laboratory tests. The formula (1.1) was developed from
studies on the leaching of commercial glasses by aqueous solutions [2], [4]-[7],
and it seems to be very accurate for many glasses as long as the glass matrix
does not devitrify [2], [4]. When devitrification occurs (as in certain
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phosphate-glass products, for example,[2], [8]), the Teachability is increased
by a factor of approximately one thousand. Apparently, no expression (em-
pirical or theoretical) is given in any of the open literature for the quantity
Q of this type of material which is leached. Godbee and Joy [2] state that their
mechanisms can not account for these situations and recommend further studies
exploring alternative combinations of fundamental mechanisms to account for
and predict the leach behavior of these materials. We present one possible
explanation here together with a time law to replace (1.1).
Normally,in such a critical situation as the storage and disposal of
high level radioactive waste,it would obviously be prudent to recommend as
containers vitreous solids which are known to be highly resistant to devit-
rification under the circumstances assumed to be encountered. Recently
reported experimental results [3], [9], however, show that the radioactive
isotopes embedded in a vitreous solid can interact in subtle ways with the
glass to enhance the "aging" (i.e., devitrification, increased porosity,
phase separation, etc.) of a vitreous container made from glasses previously
thought particularly stable to leaching. Thus, in addition to further ex-
perimental studies on the nature of these phenomena it is desirable to have
a reliable formula to replace (1.1) for describing the amount of radioactive
isotope leached from a solid in circumstances where (1.1) does not apply
(for example, in a devitirfied state, or a state of increased porosity due to
increased chemical activity as a result of the embedded radioactive isotope
[9]). If leaching and leach rates are dramatically increased in these situa-
tions, then several possibilities are suggested: (i) don't use these materials,
(ii) attempt to develop ways to "treat" these materials to resist these
"aging" processes, or (iii) accept the aging and increased leach amounts but
perhaps embed smaller amounts per given amount of container.
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In any event reliable estimates of the amount Q of a radioactive isotope
leached from a solidified waste as a function of time t are needed. These
estimates and formulas are usually derived by empirical means although there
are a few theoretical studies [2], [7]. Laboratory experiments to simulate
actual (or what are thought to be actual) conditions have been performed
(see [2] and the references there) in ways thought to correspond to storage
times equivalent to a given number of years. Empirical relations are deduced
and extrapolated to longer times and other situations assuming that the system
maintains the same physical mechanisms and time dependencies. In addition
to experimental studies,theoretical studies should be made with attempts to
model the system using fundamental mass transport processes. Such studies
have several obvious advantages. For example, the basic physical processes
involved will become better understood, empirical constants can be interpreted
in terms of the basic parameters of the system, and extrapolation of time
laws can be performed with a greater degree of confidence. Godbee and Joy [2]
have carried out an excellent theoretical study which substantiates formula
(1.1) in many pertinent cases, and they carefully point out situations in which
(1.1) does not apply and for which their theory is not applicable. For ex-
ample, they cite phosphate ceramic products (which readily devitrify) and
for which highly elevated leaching amounts are observed as being unexplained
according to their theory. For these and other solids we present a theory
based on the pioneering work of J.W. Cahn [10]-[12] on various glasses and
metal alloys to derive an alternative to (1.1)
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2. Generalized diffusion model
Many phenomena influence leach resistance, and the general problem of
mass transport through waste solids is very complicated. For example, among
the pertinent factors are porosity and fractures in the solid, temperature,
chemical composition and reactivity of the storage matrix and its stability
when subjected to surrounding conditions and irradiation by the contained
radioactive isotopes, solubility of the isotope being leached, characteristics
of the leaching solution, possible transport paths which include the inside
of crystal lattices, along crystal grain boundaries, etc. When mass transport
is known to occur via several mechanisms, most theoretical studies invoke an
effective diffusion coefficient and Fickian diffusion. This is the method
used in the theoretical studies cited in the Introduction.
The use of an effective diffusion coefficient is an entirely acceptable
way to lump many detailed and complicated physical effects into one phenomeno-
logical constant which can be experimentally measured. Even if we allow space,
time, and concentration-dependent diffusion coefficients, we can still experi-
mentally determine the dependence of the diffusion coefficient on these variables
phenomenologically. More fundamental, however, is the assumption of Fickian
diffusion. We now show that a more generalized diffusive mechanism is likely
to be operating, and we analyze the equations resulting from this assumption.
It is known [5], [10], [13], [14] that many glassy polymers and certain
metal alloys exhibit non-Fickian diffusion. Chapter 11 of Crank's book [14]
and the excellent survey and expository paper of Cahn [10] describe many ex-
perimental observations and various attempts to theoretically and mathematically
model these non-Fickian situations. The models of Cahn [10]-[12] seem the most
promising, and indeed, they incorporate most of the other models as special
cases. Before presenting the derivation appropriate for the present problem
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of leaching from waste solids, it may be helpful to briefly recall some of
major periods and attempts to generalize Pick's law.
In various fields of physics, chemistry, and engineering in the last
one hundred years Pick's law of diffusion has been inadequate to describe
diffusive processes. In many cases generalizations and alternatives, based
on fundamental physical processes, have been proposed which successfully de-
scribe and predict the experimental observations. Three main periods and
problems have provided the major impetus for this development, (i) The work
of Gibbs and van der Waals around the turn of the century on coexistence near
the critical point led to the concepts of nucleation, metastability, and spin-
odal decomposition in an attempt to understand the simultaneous existence of
two spatially distributed states of matter (i.e., pattern formation). The
fundamental role of diffusion and its physical nature was carefully studied.
(ii) The work of Cahn and his co-workers in the period 1950-1970 on the struc-
ture of metal alloys (again, pattern formation) led to a successful analytical
formulation incorporating the concepts of a negative diffusion gradient which
has been experimentally observed by x-ray techniques and which is necessary
in the theory of metal alloys to account for molecular clumping (or aggregation).
(iii) The recent work (1970-1980) on striations or patterns (in the phonon or
quasi-particle density) on super-conducting thin films led to a re-examination
and further development of the Gibbs-van der Waals theories this time based on
the pioneering work of Landau. At about the same time as the work on metal
alloys was going on, the Landau-Ginzburg theory of non-equilibrium thermodynamics
was proposed to account for the phase co-existence patterns. Inherent in this
theory is a non-Fickian mechanism of diffusion. The recent work in super-
conductivity has led to a satisfying theory based on the Landau-Ginzburg
description. Excellent papers describing the work on metal alloys and the
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-6-
earlier phase co-existence problems are the major survey paper of Cahn [10]
and his paper [12]. The more recent work in super-conductivity can be found
in [15],[16].
The present problem of leaching from waste solids is different from those
just mentioned. However, it is just another application of the generalized
diffusion model. As Cahn [10] points out, the generalized diffusion model
can and has been verified as an appropriate modification of the classical
Fickian diffusion equation, and he further states that his particular problem
is but one application arid that the model should be used in all kinds of
diffusion problems. In particular, quite detailed verification [10] of the
validity of the generalized model has been carried out for certain metal alloys
and various glasses (silica glasses).
We now derive the generalized diffusion equation with a brief plausibility
arqument. A more detailed derivation is given in Appendix A. Recall that
Pick's law states that the diffusive flux J_ (or equivalently the the mass
current J_) is proportional to the concentration gradient; that is,
(2.1) J^ = -D grad C,
where C is the concentration of the diffusing species, and the proportionality
constant D is known as the diffusivity (or diffusion coefficient). In general,
D can depend on space, time, and even concentration, that is D = D(x_,t,C).
The basic equation of continuity (i.e., conservation of mass) then becomes
(2.2) || = - div J_ = div (D grad C),
which is the familiar heat or diffusion equation.
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Alternatively, following Cahn [10] we can derive (2.2) phenomenologically
from a basic thermodynamic or internal energy argument (see Appendix A) by
defining a chemical potential y, the gradient of which drives the mass current
vL That is,
(2.2) J_ = - grad y
If the potential is a function of only the state variable C, then y = y(C),
and
(2.4) J_ = - grady(C) = - y'(C) grad C.
The equation of continuity then becomes
(2.5) || = - div J. = div (y'(c) grad C) ,
which is identical with (2.2) if we make the identification D = y'(C), a
phenomenological definition of diffusion coefficient.
This derivation (in particular, the assumption that y = y(C)) assumes that
concentration gradients are negligible. If a glassy solid changes from a
vitreous to a crystalline condition (i.e., if devitrification occurs), then
concentration gradients may not be negligible. (Cahn [10] substantiates this
for borosilicate glass.) In this case, the chemical potential y will depend
on both C and gradients of C. In Appendix A we shall show that this dependence
is given by
(2.6) y = f(C) - kV2C,
where f(C) is a certain energy density. Now, the mass current J^ driven by
the gradient of y is given by
2
(2.7) J. = - grady = - gradf'(C) + k gradV C .
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-8-
In this case, the basic continuity equation becomes
(2.8) IT = - divJ = div grady
O L —
= V2(f (C) - kV2C)
= - kV4C + div (f" (C) gradC) .
Finally, if we incorporate chemical dynamics (or reaction terms) or concentra-
tion dependent dissolution rates as in [2] and [4], we obtain
(2.9) |£ = - kV4C + div (f" (C) gradC) + G(C).
O L
In the one-dimensional case, which we study in the next section, equation
(2.9) becomes
(2.10) C._kc+
Equation (2.10) is highly nonlinear, and thus, solving specific problems
could be a formidable task (although much could probably be learned about the
structure of various solutions by sophisticated perturbation and bifurcation
procedures). Certainly, in the early stages of any leaching problem it is
valid to consider the appropriate linearized equation. Depending upon the
time scales of interest and the relative sizes of various parameters and con-
centration amplitudes, this early stage may mean large numbers of years, but,
of course, such an estimate would need to be substantiated either by corrob-
orating experiments, an assessment of the neglected nonlinear terms, or appro-
priate asymptotic methods on the full nonlinear problem.
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Clearly, equation (2.10) implies that any linearized problem to be
considered in the early leaching stages involves an equation of the form
(2.11) |£ = _ al_C + 3 9C + h(c c)
c ^ ^ s
where a, 3, h, and Cs are constants. In the next section we examine an
appropriate problem involving (2.11) to find what leaching time law is implied.
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3. The time law for leaching:
We study the transport model of Godbee and Joy [2] replacing their
mechanism of Fickian diffusion with the generalized diffusive mechanism de-
veloped in Section 2. Thus, we consider the following linearized problem of
diffusion from a semi-infinite solid waste:
4 2
(3.1) |£= - a^+ 6 ^4+ h(C -C),
dt 8x 3X
(3.2) C(o,t) = 0,
(3.3) 92C(o t) = Q>
8X
(3.4) C(x,o) = Cs .
Here C is the concentration of the mobile form of some diffusing species.
The initial concentration Cg of this species is assumed at saturation level
for this species in the waste solid, and the rate at which less mobile forms
are converted to more mobile forms is proportional to the difference (C - C)
between the saturation value C and instantaneous concentration C. Thus,
h is a dissolution rate constant, a and 3 are constants determined from
the phenomenological constants and linearization of ((2.10). The justification
and derivation of the boundary condition (3.3) is given in appendix B.
Let u(x,t) = Cs-C(x,t). Then, u(x,t) satisfies
(3.5) !£=-« ^+e4-h" .
3t 3x4 ^
(3.6) u(o,t) = Cs ,
(3.7) A(o2lL= 0 s
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-ii-
(3.8)
u(x,o) = 0 .
The problem (3.5)-(3.8) can be solved efficiently by employing a Fourier sine
transform in x. Thus, define
00 00
(3.9) u(k,t) = / u(x,t) sin kx dx, u(x,t) = f / u(k,t) sin kx dk.
Then, apply this transform in the usual way to obtain
(3.10) ^r + (ak4 + 3k2 + h)u = (ak3 + 3k)C$ ,
(3.11)
u(k,o) = 0 ,
the solution of which is
(3.12) u(k,t) =
(akj + 3k)Ce
1- e
-(ak4 + Sk2 + h)t
Therefore,
00
r 1 e-(<*k4+3k2+h)t
(3.13) u(x.t) - ^ J (ak+^..4 12
2C
sin kx dk .
ak + 3k + h
It is interesting to note that the integral representation (3.13) has
several non-uniformities. For example,
00 OO
lim / , / lim
X-H» / ' J x-*30
lim / j_ / Mm
o o
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Nevertheless, we can assess the large x behavior by an appropriate asymptotic
analysis as follows: First, write (3.13) as
(3.14)
u(x,t) =
2C_
IT
// 4 L-
ak.4+3k2 +
h - h) sjn kx
dk
2C
4 , „, 2
-(ak4+3k2+h)t
oo
2Cs r
" 4
2CS
TT
,2°s
b | \UCK i PN i ii - M y e ''!/•'
17 J ak4 + 3k2 + h
00
sin kx ,,, 2Csh f sin kx .,,
i QK — 1 A A
K u ^ k(ak4 + 3k4 + h)
£ -(ak4+3k2+h)t
e sin kx ..
k dk
0
00 A 9
h f -(air+eir+h)t
/ e sin kx .,
dk
k(ak4+3k2+h)
Now, in the integrals let kx = s and use the fact that
to obtain that as x -> °°,
2C h
/sin s
5
j
ds = 75-
(3.15) u(x,t) =
sin s
S TT
2C
s4 s2 s
OX X
n i
^r+B^
x x
ds
s
„,
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-13-
— dS
-ht °° 00 -ht »
s
s
0
ds
= 0.
Therefore, u(x,t) ->• 0 as x •> °° for fixed t which is, of course, consistent
with our originally posed boundary value problem (3.5)-(3.8).
To determine the amount of material leached in the early stages we shall
first carry out an asymptotic expansion of the solution (3.13) for small t and
then for small x. Let kx = s in (3.13), so that (3.13) becomes
2C
(3.16) u(x,t)
+ t2 + ht
sinsds
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-14-
f/as3 A +
In3 *
as\
n4
Bs
n
n
t%\ e n2
I;*5 + ht
sins ds,
where n = -T- • Tnus, for fixed n, we have that asymptotically as t •* 0,
(3.17) u(x,t)
2C.
TTX
-cs-
2CC -ht
Ht^4 sinsds --^—
5 I* }\
w.
/
-as
e
2Cs(l-ht)
TTX
?- t4 sins ds
-as
2C
sinsds
ds
The last integral in (3.17) can not be evaluated in closed form, but clearly
it is some function of n. Denote it by 90(Tl). Therefore,
w
u(x,t) ^ C - g_(n) as t -> 0, where n = -r- .
so t4
Clearly, from (3.17) the general functional form of the small time asymptotic
expansion is
(3.18) u(x,t) ^ C - g (n) -
and gQ(n) and
+ ... as t * 0, where n = : »
t
are certain integrals defining functions of fixed n.
Before we discuss (3.18) let us examine (3.13) asymptotically for small x.
Again we start by letting kx = s in the integrand, so that (3.13) becomes
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(3.19)
2C_
u(x.t) - s
irx
f a^
I X
I jj 2 sin s ds
J a\ + BVh
x x
Therefore,
(3.20) u(x,t)
2C
2C
0
I
§
I
Jr\
sins
2C
s -ht
7T
2C
sins ds
00 A
M'
/ e x s1jis.ds
«/r>
cs -
1
J
s l"r(M x r(%) / x \ 1
—Tt~ J- ~ + 3/1 ~ I I
4a4 t* 24a/1* It4/
L \ / J
as x •> 0, fixed t.
The asymptotic estimates (3.18) and (3.20) clearly show that in the
early stages of leaching u(x,t) depends on the simularity variable
n = x/Js. Thus, at any concentration level we have a diffusion front with
t
position proportional to t4 rather than the standard t5 found in problems
involving the standard second order diffusion equation. To assess the time
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dependence of the quantity Q leached from the solidified waste we must
calculate the total amount of diffusing species which has left the medium
at time t. To do this we first calculate the flux J^ across the surface
at x = 0. Now, from (2.4) and (2.6), or as we show in greater detail in
Appendix A,
J3C(o.t) 03C(o,t)
(3.21)
-
Upon using the asymptotic expansion (3.20), we see immediately that
3/ J,
(3.22) J_ ^ (constant)t" '" + (constant)t 4 ,
x=0
so that the total amount Q is given by
t
/}- y
J. dt -\> (constant)t4 + (constant)tA.
o x=0
Therefore, the time law for the early stages of leaching implied by the
generalized diffusion model is of the form
(3.24) Q = At3* + Bt3/4 .
The constants A and B can be determined empirically in any given application,
and also, clearly, by more detailed asymptotics on this or more realistic
geometries they can be determined phenomenologically in terms of the fundamental
parameters of the diffusion model. By normalizing both (1.1) and (3.24) to
a common value at some instant we see immediately that (3.24) predicts both
larger initial quantities and faster initial leach rates than (1.1) thus pro-
viding a possible mechanism to account for the observed larger leached qu?nti"Mes
and leach rates for various devitrified glassy materials and other waste solids.
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4. Accounting for radioactive decay
So far we have not considered radioactive decay. We have assumed that
the isotope being removed from the waste product is a stable one. To account
for radioactive decay we can proceed as follows: If the radioactive isotope
under consideration has a simple one-step decay to a stable form, then a
simple standard exponential factor with the relevant half-life constant can
easily be incorported into our calculations. This is done in [2]. However,
many isotopes (the actinides, for example) have complex decay chains with
daughters first displaying a growth with time followed eventually with decay.
In these cases an effective decay factor between two given times can be de-
duced from experimental data and then applied to the appropriate time law
(1.1) or (3.24). This approach is used in [4]. In any event the appropriate
formulas for Q is of paramount importance.
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References
1. Technical support of standards for high-level radioactive waste management,
Volume C, Migration pathways, U.S. Environmental Protection Agency
Report EPA 520/4-79-007C, Contract No. 68-01-4470, 1977.
2. H.W. Godbee and D.S. Joy, Assessment of the loss of radioactive isotopes
from waste solids to the environment. Part I: Background and theory,
Oak Ridge National Laboratory, ORNL-TM-4333, 1974.
3 J.C. Dran, M. Maurette, and J.C. Petit, Radioactive waste storage materials:
Their a-recoil aging, Science, 209 (1980) 1518-1520.
4. Development and application of a risk assessment method for radioactive
waste management, Volume I: Generic description of AMRAW-A model,
U.S. Environmental Protection Agency Report EPA 520/6-78-005,
Contract No. 68-01-3256, 1978.
5. T. Alfrey, E.F. Gurnee, and W.G. Lloyd, Diffusion in glassy polymers,
J. Polymer Science: Part C, 12^ (1966) 249-261.
6. H.L. Frisch, T.T. Wang, T.K. Kwei, Diffusion in glassy polymers. II,
J. Polymer Science: Part A-2, 1_ (1969) 879-887.
7. T.T. Wang and H.L. Frisch, Diffusion in glassy polymers. Ill, J. Polymer
Science, Part A-2, (1969) 2019-2028.
8. J.E. Mendel and J.L. McElroy, Waste solidification program, Volume 10,
Evaluation of solidified waste products, Battelle Northwest
Laboratories Report BNWL-1666, 1972.
9. E.H. Hirsch, A new irradiation effect and its implications for the disposal
of high-level radioactive waste, Science, 209 (1980) 1520-1522.
10. J.W. Cahn, Spinodal decomposition, The 1967 Institute of Metals Lecture,
Trans. Mettalurgical Soc. of AIME, 242 (1968) 166-180.
11. J.W. Cahn, On spinodal decomposition, Acta Metallurgical, 9_ (1961) 795-801.
12. J.W. Cahn, The later stages of spinodal decomposition and the beginnings
of particle coarsening, Acta Metallurgica, J.4 (1966) 1685-1692.
13. J. Crank, A theoretical investigation of the influence of molecular
relaxation and internal stress on diffusion in polymers, J. Polymer
Science, U (1953) 151-168.
14. J. Crank, The Mathematics of Diffusion, Oxford University Press, Second
Edition, 1975.
15. B.A. Huberman, Striations in chemical reactions, J. Chem. Phys. 6>5_
(1976) 2013-2019.
16. K.F. Berggren and B.A. Huberman, Peierls state far from equilibrium,
Physical Review B, JL^ (1978) 3369-3375.
17. P.M. Morse and H. Feshbach, Methods of Theoretical Physics. McGraw-Hill, 1953
18. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Inter-
Science, 1953.
19. R. Weinstock, Calculus of Variations, McGraw-Hill, 1952.
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APPENDIX A
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-21-
Appendix A. Generalized Diffusion Equations
Rather than derive the standard classical diffusion equation by postulating
Pick's law as done in most presentations, it is possible to derive the equation
and Pick's law itself from a basic energy argument. It is this appeal to the
fundamentals of the physics which allows and even suggests the necessary gen-
eralizations.
Perhaps the best way to present the derivation is first to discuss the
Classical Pick's law in the relevant notation. Thus, suppose C(_x,t) rep-
resents the concentration of some diffusing species. Let f(C) represent
the free energy density (i.e., internal energy per unit volume) of the system,
so that the total energy FtCl in a volume V is
(A.I) FEC] = ( f(C) d^<.
V
r n
The variational derivative -^ (that is, the change in energy or work done
in changing states by an amount <$C) defines a (chemical) potential u(C);
that is,
(A.2) y(C) =|£= f'(C)
Now, a gradient of the potential y will drive a current J_, or equiv-
alently, flux J^ is proportional to the gradient of u. This is Pick's law,
Thus,
(A.3) J. = -D grad y(C),
where D is a proportionality constant. The basic equation of continuity
(i.e., conservation of mass) then becomes
(A.4) | = - div J. = div (D grad y(C)) = div (Df" (C) grad C)
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-22-
Hence,
(A.5) |£ = div (D(C) grad C),
where
(A.6) D(C) = Df"(C)
In the case of the simple heat equation or a situation with constant
diffusion, the internal energy density is the standard quadratic form
f(C) = ^C2. Then, V(C) = C, and (A.5) becomes
(A.7) || = D V2C
with D(C) = D representing the constant diffusion coefficient. Classical
Fickian nonlinear diffusion is represented by (A.5) with a nonlinear diffusivity
D(C). Clearly, the derivation remains the same if we allow D to have spatial
and temporal dependence, if we allow C to be a vector of concentrations, and
if we add sources and sinks (i.e., dynamics or reaction terms) to the continuity
equation to obtain the standard reaction-diffusion system
8C ~
(A-8) 3== div(D(£,)i,t)grad C) + 6(£)
in place of (A.5). Here J3(C) represents the dynamics (or reaction terms)
or concentration dependent dissolution rates. This completes our derivation
of Fickian diffusion.
Most derivations of (A.5) or (A.8) start at equation (A.3); that is,
Fick's law as given in (A.3) is simply postulated. All we have done is to
start with the internal energy (consistent with Fick's law) necessary to
maintain states described by Fick's law. The pertinent feature of this energy
functional (A.I) is that it depends only on the state C of the system through
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the density f(C). This assumes that diffusion distances and concentration
amplitudes are such that concentration gradients are negligible. If a glassy
solid changes from a vitreous to a crystalline condition (i.e., if devitrifi-
cation occurs), then concentration gradients may not be negligible, and a
gradient energy may be necessary to describe the system. That is, phenomen-
ologically, instead of (A.I) a more realistic energy functional is
(A.9) F[C] = / [f(C) + |k(VC)2 + kjV2C + ...Idx .
The form of the energy density is consistent with the requirements that the
energy density must be invariant under reflections (x. -> -x.) and rotations
(x. -> x.). Here f(C) represents the energy density which this volume would
* J
have in a composition in which gradients are negligible, and the other terms
represent the energy density (a "gradient" energy) which is contributed by
the now non-neglibible concentration gradients. This is the crucial step in
assessing the physical role of diffusion. Our derivation most closely follows
the work of Cahn [10]-[12] and Landau-Ginzburg (see [15], [16]).
Now, we simply re-trace the steps in going from (A.I) to (A.8). The
potential y induced by our energy functional (A.9) is given by
(A.10) u = y(C,vC) = |£ = - kV2C + f'(C),
so that the flux ^ is given by
(A. 11) J. = - D grad y(C,VC).
The equation of continuity then becomes
(A. 12) = - div J. = div (D grad y)
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-24-
D V2(- kV2C + f (C))
= - kDV4C + div (D f" (C) grad C) .
The basic Landau-Ginzburg assumption for f(C) is that
(A. 13) f(C) =
Only even powers of C appear because the energy density can not depend on
the sign of C. Thus, (A. 12) becomes
(A. 14) |£ = - DkV4C + DAV2C + DBV2C3 .
ot
Finally, if we incorporate the dynamics (or reaction terms), we obtain
(A. 15) IT = - DkV4C + D A V2C + D B V2C3 + G( C) .
ai,
The equations (A. 14), (A. 15) are the generalizations of equations (A. 5),
(A. 8) respectively. In the one-dimensional case equation (A. 15) becomes
4 2
(A. 16) I?- = - Dk 2-% + D(A+3Bn2) ^ + 6DBn(|^-)2 + G(n) .
3t 8x4 8x^ Sx
In the context of the thermodynamics of phase transition the constant A
in (A. 13) is given by
(A. 17) A =-AQ(T-Tc) ,
where AQ is some constant, T is the temperature, and T is the critical
temperature. Thus, the parameter A can assume positive or negative values.
Therefore, if we designate the coefficient DA of the second derivative in
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-25-
(A.14) as the "diffusion coefficient," then negative diffusion is possible.
The stabilizing mechanism then becomes the fourth order term. In applications
to problems involving metal alloys where the effects of elasticity may be
important Cahn [11] has added elastic strain energy terms to the basic energy
functional (A.9). In these applications for problems with both positive and
negative values of A the theory has been successful both qualitatively and
quantitatively.
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APPENDIX B
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Appendix B. Boundary conditions for the generalized diffusion equation
The boundary conditions appropriate for a given problem can be determined
from the variational principle governing the problem [17]- [19], For the
spatial boundary conditions given by (3.2) and (3.3) it is sufficient to
consider the equilibrium (or steady state) equation, namely
(B.I)
3
( _ c) = 0>
s
This is the Euler equation which comes from the Lagrangian L(C, Cv, Cvv)
X XX
given by
(B.2)
L(C,CX,CXX) =a
That is equation (B.I) is the Euler equation which results from the variational
calculation
(B.3)
:, cx, cxx)dx = o.
For arbitrary variations 6C equation (B.3) becomes
°° . Thus, our admissible class of variations
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must require that 6C = 0 at x = 0 and as x + °°. Therefore, (B.4) implies
that
(B.6) Lf 6C I = 0
Lxx X|o
In the calculus of variations (B.6) is known as a natural boundary condition.
Since 6C is arbitrary within the class of admissible variations, we must
take LC = 0 at x = 0 and as x ->• °° to have a well-posed boundary
xx
value problem. Therefore, C = 0 at x = 0, so that
/\n
2
(B.7) 9 c<*t) = o
is the necessary second boundary condition.
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA 520/8-81-001
2.
4. TITLE AND SUBTITLE
Leaching of Radioactive Isotopes from Wast
7. AUTHOR(S)
Donald S. Cohen
9. PERFORMING ORGANIZATION NAME AND ADDRESS
12. SPONSORING AGENCY NAME AND ADDRESS
Office of Radiation Programs
U.S. Environmental Protection Agency
Washington, D.C.
3. RECIPIENT'S ACCESSION NO.
5. REPORT DATE
January 1 981
e SolldS 6. PERFORMING ORGANIZATION CODE
8. PERFORMING ORGANIZATION REPORT NO.
10. PROGRAM ELEMENT NO.
11. CONTRACT/GRANT NO.
13. TYPE OF REPORT AND PERIOD COVERED
14. SPONSORING AGENCY CODE
AN R- 4 58
15. SUPPLEMENTARY NOTES
16. ABSTRACT
The most commonly used empirical model for the quantity Q of a radioactive
isotope leached from a solidified waste as a function of time t is Q=atl/2+ut,
where a and b are empirically determined constants for a specific leach process
involving a specific isotope. This formula works well for many solid wastes and can
be derived theoretically from a model employing Fickian diffusion. However, the formu
la is known to be totally inaccurate for my other solids and also in particular for
glassy solids which devitrify. Since devitrification and other symptoms of aging are
commonly occurring processes in the long term storage of radioactive waste, it is
important to have a correct alternative formula for Q and even more important to
understand the physical processes involved in leaching.
A theoretical model involving a generalized, non-Fickian mechanism for diffusion
is derived in this paper and applied to determine Q as a function of time t. It is
found, on the basis of this type of diffusion which occurs in devitrified glassy
solids and other solid waste materials that Q=At1/4+Bt3/4. Here A ^and B are con-
stants which can be determined empirically, and they can also be determined phenom-
enologically in terms of the fundamental parameters of the diffusion model. When both
formulas are normalized to a common value, at some instant the new formula predicts
larger initial quantities and faster initial leach rates.
17.
KEY WORDS AND DOCUMENT ANALYSIS
a. DESCRIPTORS
radioactive isotope
leaching
radioactive waste
12. DISTRIBUTION STATEMENT
Unlimited
b. IDENTIFIERS/OPEN ENDED TERMS C. COSATI Reid/Group
19. SEC-JRITY CL^f.S: (Tins report/ 21. NO. OF PAGES
Uprl^s^if i Pd i 31
20. SECUR'TY CLASS /< his aegtj 22. PRICE
Unclassified j
FPA Form ?220-1 ;Rev. 4-~7; PF.EvlGUS EDITION 'S OBSOLETL1
«US. GOVERNMENT PRINTING OFFICE: 1981 341-082/223 1-3
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