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An analytical solution for non-steady-state diffusion through thin films
ELSEVIER
Thin Solid Films 272 ( 1996) 124-l 3 I
An analytical solution for non-steady-state diffusion through thin films A. M6zin a, J. Lepage a, P.B. Abel b aLuboratoire de Science et G&e des Surfaces, INPL-Universitede Nancy I-CNRS (IRA 1402. Ecole de.7 Mines, Part de Saurupl, 54042 Nancy cedex, France h National Aeronautics and Space Admbtistration. Lewis Research Center. 21000 Brookpark Rood3 Cleveland, OH 44135, USA
Received 12 April 1995; accepted 25 July I995
Abstract We present an analytical solution for non-steady-state diffusion in coated materials which assumes a constant gradient in the coating. The application domain and degree of validity are determined as a function of two appropriate parameters. Possible surface and interface effects are taken into account, as far as is consistent with boundary conditions of the first kind. The solution applies generally to thin films in which the diffusion equation prevails, whether to describe diffusion of a species or heat transfer by conduction. For example, this solution provides an easy tool for studying diffusion processes in multilayer materials in situ which is we11adapted to the case of protective coatings. Keywords: Diffusion; Coatings;
Hydrogen;
Molybdenum
1. Introduction An analytical
solution
of the diffusion
equation
is available
of situations with various boundary conditions, geometries and material properties [ I ,2]. Nevertheless, a situation of practical interest in the domain of coated materials seems not to have been considered. As an illustration of this general problem, consider a planar sheet coated on both surfaces by a thin deposit as a barrier layer. With the aim of testing the efficiency of the barrier layer, a given concentration of diffusant is imposed on the external surfaces of the specimen. The quantity of diffusant absorbed during a given time by the substrate can then constitute a quantitative measure of the coating “tightness”, as far as suitable modeling allows the relevant quantities to be determined. Assuming a constant gradient in the deposit permits an analytical solution, which, depending on experimental conditions, may be an acceptable approximation to the exact solution of the general problem. This formulation is general in character, even though only developed for thin films. The solution can be applied to diverse situations where diffusion phenomena prevail (species diffusing in solids or liquids, heat conduction, etc.). For convenience, the presentation can be restricted without loss of generality to the take up of a species by bulk material in the presence of barrier layers. In the following presentation, surface and interface effects are first discussed. They can be included analytically, as far as is consistent with boundary conditions of the first kind for a great number
0040-6090/96/$15.00 0 1996 Elsevier Science S.A. All rights reserved SSDIOO40-6090(95)06969-O
(fixed concentration). After deriving the approximated analytical solution, the domain of applicability and degree of validity of the approximation are considered for limiting cases, then in general.
2. Surface and interface effects on permeation In permeation problems, boundary conditions of the physical problem often limit the modeling validity. This explains in many cases the wide variation in solubilities and diffusion coefficients reported in the literature. Two types of boundary conditions can be distinguished. The first type is that of a free surface in contact with diffusing matter (liquid or gas under some pressure), and involves a pseudo-equilibrium between a gaseous or liquid phase and a dissolved phase, through surface phenomena such as adsorption or dissociation. The second type relates to solid-solid interfaces and involves a pseudo-equilibrium between a phase dissolved in each of two different materials, taking into account interface phenomena which may be present. It is convenient to refer to the case of dynamic steady state, which is the most common in permeation studies [ 3-5 1, e.g. where a constant pressure difference is established across a film. Mass flow passing through a homogeneous membrane of thickness b and unit surface area is then:
A. M&in et al. /Thin Solid Films 272 (19961 124-131
where D denotes the diffusion coefficient of the gas in the material, and ci and cZ are the volume concentrations within the first atomic layers of the two membrane faces. Relation ( 1) arises solely from the hypothesis of Fickian diffusion. In the case of a diatomic gas, Sieverts law gives:
c,=s&
(2)
where S is the Sieverts constant, equivalent to the gas solubility in terms of pressure units, and where pj is the gas (diatomic) pressure imposed on the face j. In so far as Sieverts law is valid, gas flux obeys the relation:
~=~t&xd
(3)
The permeation property of the material is then well characterized by the permeation constant P: P=DS
(4)
where D and S can each be considered an intrinsic volume property. The strict applicability of Sieverts law implies a local thermodynamic equilibrium at the interface between the gaseous and the dissolved phases. In the same way, under the assumption of thermodynamic equilibrium at the interface of two materials, the ratio of the concentrations on either side of an interface (material fmaterial s) equals the ratio of the solubilities [ 61:
c,r si -=Cl,
(5)
SS
To illustrate this, consider at the interface between the two materials a geometrical discontinuity such as a pit, inside of which is imposed a constant pressure p of a diatomic gas (Fig. 1) Applying Sieverts law at the two inside free surfaces and taking into account that, due to thermodynamic equilibrium, the concentration gradients are zero in the materials, the interface concentrations obey Eq. (5).
The permeation constant P of a two-layered membrane, for example substrate s and deposit f of thicknesses u and b respectively, is then simply given [ 61 by: a+b -=_+_
a
b
P
P,
P,
Sf g(p)
\,
cs= ssg(P)’
(7)
where S’ is an “effective solubility”, as opposed to S of Eq. (2), which is an intrinsic volume property. Here, we account for the surface effect by using a phenomenological parameter (T, which is concentration and flux independent like S’: c= a%(p)
Fig. 1. Thought experiment showing that the ratio of concentration, c,Ic,, at the interface of two materials f and s at thermodynamic equilibrium equals the ratio of solubilities. .7,/S,, with p the equilibrium gas pressure in an adjoining void.
(6)
where P, and Pf are the permeation constant of the materials s and f respectively. When the variable S is regarded as an intrinsic volume property, Eqs. (3)-( 6) are strictly applicable only if flux is in no way limited by a surface or interface phenomenon. It is well established that surface phenomena may affect absorption and permeation processes. The synthesis paper of Kompaniets and Kurdyumov [7] gives many examples where large discrepancies from Eq. (3) have been observed experimentally for mass-how dependence either on the inverse of film thickness or on the square root of pressure. Increasing the hydrogen permeability of a material is sometimes accomplished merely by coating the material with a very thin layer of a more hydrogen-reactive material (Pd, Ni) [S]. The permeability increase observed, when this is done, is inconsistent with Eq. (6) and indicates the presence of surface effects. In general, the surface effects may result from an intrinsic property of the material (e.g. reactivity to the ambient gas) and/or experimental conditions (e.g. surface roughness, cleanliness, state of strain and stress) [ 81. Though surface phenomena have not always been well characterized, they have been widely studied [4,7,9]. Discrepancies between experiment and Eqs. ( 3 ) -( 6) have typically been attributed to surface effects [ 71. Interfaces, on the other hand, have been almost ignored from both experimental and theoretical points of view. In the cast of a reactive metal deposit which reduces the surface barrier of the bare substrate to hydrogen, the deposit-substrate interface does not seem to hinder permeation. This perhaps results from the energy involved at an interface (solute state-solute state) being lower than the surface energy (gas phase-solute state). Taking into account possible surface and interface effects, and making the assumption that they lead to a pseudo-equilibrium consistent with boundary conditions of the first kind (fixed concentration), the surface effect is sometimes described [ 61 with a modified Eq. ( 2) : c=S’g(pf
Cf =
12s
(8)
with S remaining a property of the volume. Similarly, we describe the interface effect by using a concentrationand flux-independent parameter y, so that Eq. (5) becomes:
c,r_ s,,,-s xYsr_=k c s
(9)
I26
A. M&in et al. /Thin Solid Films 272 (1996) 124-131
Sfg(Po)i
Cof =
3. Analysis
,
c 00
I
h4
f
\
PO \
\
\
r,
\
.
‘!
cof = Of Sf g(p,)
Pl
I
I
\
\
3. I. Preliminary
Cls = s, g(p1) .
\
cts = bs Ss g(p1)
I
Fig. 2. Concentration of diffusant in a two-layer system with constant external pressures pO and p,
Introducing the (T and y parameters has the advantage of distinguishing between the surface and interface effects. It also seems more reasonable than attributing false properties to the volume. Subsequently, we will have to compare our results (nonsteady state) with the case of the dynamic steady state (DSS) (Fig. 2) in a two-layer system. The volume concentrations in the first atomic layers of the free surface of the coating, f, and of the substrate, s, are constant and equal to respectively. The flux car= afS,g(pc) and cls = a&(p,), (DSS) through the bimaterial can then be written in the form:
considerations
For clarity, the analysis can be restricted without loss of generality to the case of the experiment that consists of measuring the amount of a diffusing substance absorbed by a covered specimen as a function of the charging time. The deposit covers all faces of the specimen, which is thin ribbon. The specimen initially contains none of the diffusing substance and the temperature is held constant during absorption, The amount of diffusing substance absorbed depends on the tightness of the deposit, which in this case is the quantity to be determined. If the specimen is thin enough, the substrate can be considered a planar sheet of thickness 2a with infinite dimensions in the plane (directions y and z in Fig. 3). The substrate is coated on both surfaces with a deposit of thickness b. S, and D,, Sf and Df denote the solubility and the diffusion coefficient of the substrate and the coating, respectively. For our conditions in the two media we need to solve the one-dimensional diffusion equation:
wx,t> -=
a*c(x,t)
at
a2
(15)
with the initial condition:
(10) where
aDf
tL=;Ek
The effect of the coating is revealed by comparing the flux (p, +f through the coated substrate with the flux & which under ,cls) would pass the same conditions (cos = a&&,) through the bare substrate:
(P
P
s+f -=-_ A
cof
1+ CLkc,,
I-
(c,Jco,)
1-
(kco,lco,)
The decrease in permeation then is quantitatively described with the help of the two parameters p and kcoslcof Using relation (8) demonstrates that the parameter:
=o
where -a-b
(16)
The possible surface and interface effects on the absorption of diffusant are assumed to be concentration and flux independent (Section 2). A local equilibrium at the surface and interface is assumed, which leads to boundary conditions of the first kind (Dirichlet). Given the experimental conditions, the diffusant volume concentration within the very first atomic layers of the two external faces of the sample is assumed to be time-independent and equal to: c(x=a+b,t) c(x=
( 12)
%~%s
kc,, 0s C0f = OYsf
c(x,t=O)
=cof
-a-b,t)
(17a)
=cof
(17b)
The boundary condition at the interface is determined by assuming that the ratio of the concentrations on each side of the interface is concentration and flux independent (see Section 2)) and obeys relation ( 18) where k is time independent.
(13)
f D
depends on the two possible surface and interface effects. On the other hand, the parameter:
SS
Cof
A
Sf
Y
OS
a
Cof
Of X
(14) does not depend on any surface effect.
.
-s-b
-a
0
a
a+b
-
Fig. 3. Schematic diagram of the coated specimen. Specimen is much larger in y and z dimensions than in 1.
127
A. M&in et al. /Thin Solid Films 272 (1996) 124-131
CA+a,t) c,( ka,t)
Sf
(18)
=S,ysf=k
Continuity
of diffusant flux at the interface is given by:
The roots’ of Eq. (26) are tabulated in standard books [ 1,2] for certain values of p. Note that knowledge of these roots is usually not needed for a comparison with experiment, because, as will be shown later, the experiment gives the quantity cr:, from which p is deduced. If cri is a solution of Eq. (26)) then:
(19) I
From the symmetry of the problem we need only consider the positive x domain. Substituting for condition ( 17b) :
I
$(XJ) -yg-I,=,=0
This relation allows the coefficients giving:
Dimensional analysis shows that the problem is governed by four independent parameters, which may be chosen:
c,(u) k_= car
cos ((Y;u)cos (cu,u) d,=;(l+~)S,,
Xcos(&)exp(
(21) For the general solution, this problem would normally be solved numerically. The aim of this study is to find an approximate analytical solution, and to determine the domain where this solution is an acceptable approximation to the exact solution of the general problem.
(27)
0
-$$i)
H, to be determined,
(28)
The relative amount of solute absorbed by the total substrate is given by:
Q(t) -= Q=
I-
?A;
I==,
exp
(29)
where: 3.2. Analytic solution assuming a constant gradient in the film (CGF) This approximation is based on the hypothesis of a constant gradient in the film (CGF) . Under the CGF hypothesis, the flux in the coating (a < x < a + b) is given by:
(22) Taking into account relations ( 18) and (22)) conservation of flux at the coating-substrate interface (Eq. ( 19)) yields:
(23) where the only position- and time-dependent quantity is G(W). Although many problems of the same type have already been treated [ 1,2], the problem defined by the relations (15)-( 18), (20) and (23) has not yet, to our knowledge, been solved. Assuming separation of variables and using symmetry (Eq. (20) ) , we look for a solution of the form:
(-sff+
(24)
Taking into account relation (23) leads to: a,=0
with Ho = c,,/k
(25) for i>O
Lyjtan(cr;) =2$=k s
(26)
(30) The deposit and substrate are free of diffusant initially (t = 0). In the real case, since the diffusant has to pass through the deposit before diffusing into the substrate, a certain time is necessary before the flux into the substrate reaches its maximum value. Consequently, if the effect of the coating only is taken into account, the resulting absorbed amount increases very slowly during small times, and then evolves into a regular, increasing value which corresponds to the first term (leading term) in the series that gives the absorbed amount. The associated delay time, typical for non-steady state, is referred to in general as lag time. It is conventionally defined by the intercept of the leading term with the time axis, and would be positive in this case where the effect of ’ For approximately /L < 0.5, the roots of a tan a = CLcan be approximated by a,=(~--_z/3+4~3/45)“2,~d a,=(i-l)~+&[(i-I)T]+{$ [(i-l)n])*fori>l. For I*> 1, no unique power series seems possible, because the roots then are close to k/2 for small i, and to (i - 1) T for large i. Practically, provided the time is not too small, only the first few terms in the series (Eqs. (24) and (29) ) are usually needed. Only for IL> 6, the approximation a,=(i~-?r/2)(l-1/~)(l+1I~L?+(i?r-~/2)~(l-l/~)l(3~3)) may then be used, which gives within 1% the first two roots for + > 6, the first three roots for F> 9, etc. In any case, the most convenient method probably consists of using an iterative numerical calculation, e.g. the Raphson-Newton method (in this case, the initial value for LI, can be chosen a,a = rr( i - I ) + T+/ (2 + 2~), for any value of p and i)
128
A. MPzin et al. /Thin Solid Films 272 (1996) 124-131
the coating only is taken into account (see Section 3.3, ZGS solution). On the other hand, due to boundary conditions, the large difference in concentration between the surface and interior at small times causes the flux to be initially very important. Next, the flux is rapidly reduced as diffusion smoothes the concentration gradient. The corresponding absorbed amount increases quickly from t = 0, before evolving into a regular, less rapidly increasing value (leading term). In this case, the lag time is negative (see Section 3.3, HM solution). In general, the lag time is positive or negative, according to the relative importance of coating and boundary condition effects. Considering the CGF approximation, keeping only the first term of relation (29) gives:
Q,(t) QW
-=
1
-A,
exp
(31)
The lag time, defined by Q, (T) = 0, is then: r=$AlnA, The CGF approximation consists of neglecting the lag time in the deposit. The net lag time T then is negative (A, < 1) . Note that the CGF problem is governed by the two parametersP,=D,la*and (P,IP2)P,=p. 3.3. Analytic solution with zero gradient in the substrate (ZGS) and for a homogeneous medium (HIV} Ignoring the CGF solution, an exact analytic solution exists both in the case when the concentration gradient is zero in the substrate and in the case of a homogeneous medium. Knowledge of these solutions will permit us to test the approximation degree of the CGS solution. Consider first the case when diffusion in the sample is limited only by the coating. Flux passing through the deposit spreads uniformly and immediately in the substrate (ac,/ ax = O), which is the ZGS hypothesis. According to relation ( 19), this condition corresponds to infinite D,, i.e. p = 0. The conservation of flux at the interface can be expressed:
and pi is the ith root of: Pitan P, = rl
(36)
The relative amount of diffusant absorbed by the substrate is written:
Q,(t) =1-CB,ex ffi
QP
#=I
where: (37) Using a procedure similar to that for Eq. (32)) the lag time is: b2 1 r,,=-lln D,P,
B,
(38)
Since only the effect of the coating is taken into account, the lag time rq is always positive. Note that the present problem is governed by the two parameters P, = D,l b2 and P, = 77,which are not independent of the two CGF parameters ( p, P2), since: w,
=
(39)
PP2
Determining the relevant quantities in the case of homogeneous diffusion follows from the classical solution for a homogeneous medium [ 1,2]. A homogeneous medium corresponds to k = 1 and DflD, = 1, which leads to p= I/ 77.The problem is governed by two parameters which can be chosen as D/a2 and bla. The amount of diffusion absorbed by the substrate is written:
Xsin(i$$-f)exp[
-(sfg]
(40)
and note that, particularly for small times, expressions other than Eq. (40) [ 121 may be more suitable. The corresponding lag time is given by: rh=[~(l+j~ln[(l+~)$sin(i-&-$]
(41)
(33) This problem, defined by Eqs. ( 15)-( 18), (20) and (33), has been solved [ 10,111. The concentration in the deposit (a
(34) where: q= (bla)k
(35)
The lag time is negative only for b/a < 0.3 approximately, because, for convenience (see below), only the amount absorbed by the substrate is considered. The total absorbed amount simply corresponds to bla=O. In that case, the lag time is negative and depends on a2/D only. It should be noted here that in each case the amount of diffusant absorbed by the substrate alone is being considered, while the amount of absorbed diffusant measured experimentally evolves from the entire specimen. This approximation simplifies the analysis considerably. It can be verified that other than for lag times for low values of p and q, the approximation is quite acceptable. This is because, in the main of
A. M&in ef al. /Thin SnIid Films 272 (1996) 124-131
interest, the amount of diffusant absorbed by the deposit is small (q < 0.05) and its evolution with time is not very different for the various models. For a given application, this quantity can be easily taken into account.
129
R, = cos p,
(42)
which tends toward 1 - 77/2 for small values of 7. Then the condition: T= (bla)k<0.05
3.4. Degree of validity of the constant gradient in the film (CGF) approximation The domain of validity of the approximation can be described with the help of the two parameters CLand q. The value p= 0 and q=O correspond to the two extreme cases, respectively, of a strictly zero gradient in the substrate and of strictly constant gradient in the deposit. The condition 0 that does not fix the value F= (cllb)(D,ID,)(S,IS,)y,,_= of 7” (b/a) (S,/S,)y,,-, results from D, infinite, in accordance with the basic relation ( 19). The case 77= 0 has to be considered the theoretical limit when (b/a) (&/S,) yS,-tends towards 0. The quantity 77 can be written 77= (b/a)*( D,/ Dr) yS,p, and then, for a given p, can be fixed to an arbitrarily small value by choosing (bla)*( D,lD,-) yS,-small enough. The parameters p and v appear in the experiment analysis by Zuchner (cited in Ref. [ 71) concerning permeation through a trilayer of the same geometry as is considered in this work. The quality of the approximation does not depend on the value of the parameters D,-lb’ and DSla2, since these parameters always appear in the form (Drlb2)t and (D,/a*)t. 3.4.1. Degree of validity of the CGF approximation in the case of zero gradient in the substrate (ZGS) It is useful to examine first the case when p= 0, keeping in mind that in this case the exact solution of the general problem is the ZGS solution. With the flux of absorbant decreasing continuously from the free surface of the deposit to the interface, we choose as a measure of the gradient uniformity the ratio of the fluxes at the coating surface and interface, i.e. R(t) = #‘lC’fa’C/@~T”e. Fig. 4 illustrates the evolution of the ratio R(t) compared with the ratio Q,(t) / Q,, For ~=0.05. The non-constant concentration gradient corresponds to the existence of a lag time as well as to an inequality of the absorbant flux at the surface and interface of the coating, even for very large times since it is non-steady state. For a large times, the flux ratio tends toward: 1
Fig. 4 Evolution with time of the ratio of the exit and entry fluxes in the deposit (dashed line), compared with the corresponding amount of diffusant absorbed by the substrate (full line), in the case of a zero gradient in the substrate.
(43)
ensures that the fluxes are equal to within less than 2.5% (0.5% for 77< 0.01) after long times. Moreover, the ratio of fluxes does reach a value close to the extreme value R, before a significant amount of solute enters the substrate. To demonstrate this, it is convenient to define the lag time in a way different from that suggested by Eq. (32) or Eq. (38). Let us define a pseudo-lag time T, as the time required for the ratio of fluxes to reach a certain fraction X of R,. If 77is lower than 0.1 with X greater than about 0.5, keeping only the first two terms in each of the series for the surface and interface fluxes respectively, gives an approximation to better than 0.3% of the fluxes and their ratio R(t). Keeping only the first two terms in each series then permits the lag time T to be approximated by: (44) The dependence of Ton q is neglected in Eq. (44), since for X greater than 0.5, the approximation is better than I .7% and 0.9% for 77< 0.1 and 0.05 respectively. If 77 is lower than 0.05, the surface flux into the deposit then equals the interface flux to better than 1% (X= 0.99) before the amount of solute absorbed by the substrate reaches 2.2% of its maximum value (0.44% with r) lower than 0.01). This means the lag time condition is implicitly included in condition (43). Finally, in the ZGS case the conditions necessary to assume a constant gradient in the deposit are reduced to inequality (43) alone, which determines the degree of validity of the approximation. As a consequence, for r_~= 0 there is no question of determining a lag time using the CGF solution, since the lag time is assumed zero in the CGF hypothesis. When an experimental lag time can be measured for small values of p, the ZGS solution is certainly a better approximation than the CGF. This point will be explored more quantitatively later. 3.4.2. Degree of validity of the CGF approximation in the general case (CLand q > 01 We now have to examine the degree of validity of the CGF solution in a realistic case, in which p and 77are not equal to zero. In order to obtain a homogeneous measurement of the degree of validity of the approximation over the entire domain, we determine initially the degree of validity using a mathematical distance A between the two functions Q,,(t) and QaP( t), where Q,,(t) and Q,,(t) are the exact and approximate (CGF) solution respectively: A=
”
“’
(45)
A. M&in et (II./Thin Solid Films 272 (1996) I24-13I
130
0
10
60
- 100% by definition, since the CGF lag time T is zero for /.L= 0 while the real lag time is typically non-zero for p = 0 and 77> 0. Keeping in mind that the CGF approximation neglects any diffusion lag time in the deposit, the approximation is not appropriate when the substrate diffusion lag time is less than or of the same order of magnitude (in absolute value) as the deposit lag time, which occurs for low values of p (when 17is small). The domain of validity for determining the lag time using the CGF solution can be estimated by asking that the substrate lag time be N times larger (in absolute value) than the deposit lag time. N is then given by: P
Fig. 5. Error in the constant gradient in the film approximation (CGF), asa function of w = (alb)S,D,/SSD,) ysr and v= (b/a) (S,/S,) ysP The exact solution is respectively, the constant gradient in the film (CGF) solution on the line q=O, zero gradient in the substrate (ZGS) solution on the line p = 0, and homogeneous medium (HM) solution on the line /.L= 1/T.
t, and f2 being defined by Q,,(t) /Qexm= 0.1 and 0.9 respectively. The discrepancy A between the exact and approximate solutions can be calculated exactly on the line p = 0 (ZGS) and p = 1/r], as shown in Fig. 5 (solid thick lines) (note also that A = 0 for 77= 0). For p = 0 and 17< 0.05, the quantity A increases fairly linearly with 77.Over the entire domain of p and 11,the quantity A can be seen to be a continuous and monotonic function of p and 7. From its value for p = 0 and p = 1/q, it is quantitatively estimated (dashed lines) on the domain of interest by assuming, for every value of P, the same sort of variation of A with q as for /.L= 0. The discrepancy A increases with q and decreases with J.L For all ?I < rll, A(p,q) is less than A(0, r],). With regard to experiments, the two parameters D,/a’ and p should usually be determinable by mathematically fitting the CGF solution, Q( rj) (Eqs. (29) and (30) ), to the experimental points QeXP(rj). For small values of /..Lhowever, in practice only the parameter (D,la) CY:E ( 1lab) (D&IS,) ‘ysf can be determined, as will be seen later. More commonly two parameters are determined from the slope of In ( 1 - Q_,(t) /Qm for long times, i.e. (D,la2> a:, combined with the value of the lag time r= (u2/D,) ( l/ CY:)In A,. In this case, the theoretical error relative to (D,/ a’)a: is due to the use of the CGF solution instead of the exact solution. This theoretica error, A’ = 1 - (approximate value/exact value), exhibits the same qualitative behaviour of A (Fig. 5). On the line /.L= 0, the error increases fairly linearly with 7~ (A’ = - 7713- 71~145). For a given 7, the error falls very abruptly from its value at p = 0 (equal to = 1.67% for ~=0.05 for instance) as p increases ( - 0.0176% at 7)= 0.05 and TV= 1/ 77= 20). As an approximationfor~=1/~>6,A’=-(~/2)2(2/3$)(1-11/~). The error relative to the lag time behaves qualitatively the same (for instance - 1.62% at v= 0.05 and I_L= 1/r) = 20, and - 0.24% at 7 = 0.02 and p = I/ 71 = 50) _ An important exception is that for p = 0 and 7 > 0, the error is equal to
N_pP:ln4 7)
(46)
a:InB,
As an example, for r) = 0.01,0.05 or 0.1, N is greater than 10 when p exceeds about 0.94,2.4 or 3.6, respectively. In practice, we can consider the approximate condition r~/ /_L’<~x 10-4N. In practice, this point is not of great importance because, for small values of p and v, the real lag time is so smal1 that it is generally not measurable due to experimental uncertainty. Note also that for very small values of p and 17,a very simple modeling approach can be used. This consists of assuming that the gradient is constant in the deposit and vanishes in the substrate. The corresponding analytic solution can easily be worked out directly: y=
1
_exp($_yq
Alternatively, the analytic the limit of Eq. (29) when of Eq. (37) when r] tends equivalent, since according qDflb2 = pD,la’=
(47) solution Eq. (47) is obtained as p tends toward 0, or as the limit toward 0. These two limits are to Eq. (39) :
( 1 lab) (S,D,y,,IS,)
(48)
Disregarding the lag times which are not considered, the corresponding error can be estimated immediately since a: is replaced by /L, while the true value is given by CL= ff, tan (Y,.
4. Conclusion The CGF solution permits the relative diffusant distribution in the specimen to be predicted as a function of time, provided certain parameters are known, i.e. the coating and substrate thickness (b and 2~)) bulk diffusion coefficients Df and D, and ratio of solubilities multiplied by the interface effect (k= (S,/S,) r,r). Added knowledge of the external imposed surface concentration cof gives the absolute value of the concentration. Alternatively, determination of the two parameters D,la2 and p = (a/b) ( SfDfysf/SsD,) is possible from experiment by using the CGF solution under certain conditions. For small
A. Nzin et al. /Thin Solid Films 272 {19%> 124-131
values of p (N < 10 approximately), only the parameter ( 11 ab) (S&y,,IS,) can be determined. In every case, the quantity 2ac,,lk (or more exactly (2ac,Jk) ( 1 + 71)) is also determined by taking into account the amount of diffusant absorbed by the deposit at equilibrium. By comparing with a complementary experiment using the uncoated substrate, the quantity (ar/(~,~,r) ( 1 + 77) appears. It can be easily verified that these are the parameters involved in the case of steadystate permeation (Section 2). Moreover, the quantity D,la’ can be determined if p is not too small. Note that the quantities D, and S,, in general, cannot be determined independently, whereas the product D&ysf can (assuming that S, is known). This product, which according to definition could be multiplied by the surface effect (+r, can be considered the “effective” in-situ permeation constant of the deposit. The constant gradient in the film (CGF) approximation provides an easy tool for studying in situ the diffusion processes m multilayer materials. In particular, the CGF solution is well adapted to the case of protective coatings, for which the quantity TI= (b/a) (S,/S,) -ys,is generally small. The CGF approximation is best for small values of q and large values of /.L= (a/b) (SfD,/S,D,) -ysnwhich can be set in experimental work by choosing alb large enough. The CGF approximation is restricted in that the solubility and diffusion coefficient cannot be independently determined. The most essential restriction concerns the type of
131
boundary condition considered (fixed concentration), which may presume the nature of the surface and interface effects. It would be of interest to attempt a similar analysis with boundary conditions of the second and third kind. In a companion paper, the CGF approximation is used to determine the permeation properties of a molybdenum coating deposited as a hydrogen barrier. References [ 11 H.S. Carslaw and J.C. Jaeger, Conduction oj Heat in Solid.7,2nd edn., Oxford University Press, New York, 1986. [2] J. Crank, The Marhematics ofDiffusion, 2nd edn.. Oxford University Press, New York, 1975. [3] R. Frauenfelder, J. Chem. Phys., 48 ( 1968) 3955. [41 W.R. Wampler, J. App. Phys.. 65 ( 1989) 4040. [5] T.P. Pemg, M.J. Johnson and C.J. Altstetter, Mefull. Truns. A., 19A (1988) 1187. [6] M. Kano, N. Kagawa, S. Koike, K. Furuyaand T. Suzuki, Acfa Metall., 36 (1988) 1553. [7] T.N. Kompaniets and A.A. Kurdyumov. Prqr. Surf Ser.. 17 (1984) 15. [8] J. Chevallier 219.
and M. Aucouturier,
Ann. Rev. Mater. Sci.. 18 (1988)
[9] P.M. Richards, S.M. Myers, W.R. Wampler and Follstaedt, J. Appl. Phys.. 65 (1988) 180. [IO] D.K. Paul and A.T. Dibenedetto, J. Polym. Sci. C. 10 ( 1965) 17. [ 111 H.S. Carslaw and J.C. Jaeger, Conduction of Hear in Solids. 2nd edn., Oxford University Press, New York, 1986, pp. 128-129. [ 121 J. Crank, The Mathematics cfDi#usiusion,2nd edn., Oxford University Press, New York, 1975, pp. 48,58.
Thin Solid Films 272 ( 1996) 124-l 3 I
An analytical solution for non-steady-state diffusion through thin films A. M6zin a, J. Lepage a, P.B. Abel b aLuboratoire de Science et G&e des Surfaces, INPL-Universitede Nancy I-CNRS (IRA 1402. Ecole de.7 Mines, Part de Saurupl, 54042 Nancy cedex, France h National Aeronautics and Space Admbtistration. Lewis Research Center. 21000 Brookpark Rood3 Cleveland, OH 44135, USA
Received 12 April 1995; accepted 25 July I995
Abstract We present an analytical solution for non-steady-state diffusion in coated materials which assumes a constant gradient in the coating. The application domain and degree of validity are determined as a function of two appropriate parameters. Possible surface and interface effects are taken into account, as far as is consistent with boundary conditions of the first kind. The solution applies generally to thin films in which the diffusion equation prevails, whether to describe diffusion of a species or heat transfer by conduction. For example, this solution provides an easy tool for studying diffusion processes in multilayer materials in situ which is we11adapted to the case of protective coatings. Keywords: Diffusion; Coatings;
Hydrogen;
Molybdenum
1. Introduction An analytical
solution
of the diffusion
equation
is available
of situations with various boundary conditions, geometries and material properties [ I ,2]. Nevertheless, a situation of practical interest in the domain of coated materials seems not to have been considered. As an illustration of this general problem, consider a planar sheet coated on both surfaces by a thin deposit as a barrier layer. With the aim of testing the efficiency of the barrier layer, a given concentration of diffusant is imposed on the external surfaces of the specimen. The quantity of diffusant absorbed during a given time by the substrate can then constitute a quantitative measure of the coating “tightness”, as far as suitable modeling allows the relevant quantities to be determined. Assuming a constant gradient in the deposit permits an analytical solution, which, depending on experimental conditions, may be an acceptable approximation to the exact solution of the general problem. This formulation is general in character, even though only developed for thin films. The solution can be applied to diverse situations where diffusion phenomena prevail (species diffusing in solids or liquids, heat conduction, etc.). For convenience, the presentation can be restricted without loss of generality to the take up of a species by bulk material in the presence of barrier layers. In the following presentation, surface and interface effects are first discussed. They can be included analytically, as far as is consistent with boundary conditions of the first kind for a great number
0040-6090/96/$15.00 0 1996 Elsevier Science S.A. All rights reserved SSDIOO40-6090(95)06969-O
(fixed concentration). After deriving the approximated analytical solution, the domain of applicability and degree of validity of the approximation are considered for limiting cases, then in general.
2. Surface and interface effects on permeation In permeation problems, boundary conditions of the physical problem often limit the modeling validity. This explains in many cases the wide variation in solubilities and diffusion coefficients reported in the literature. Two types of boundary conditions can be distinguished. The first type is that of a free surface in contact with diffusing matter (liquid or gas under some pressure), and involves a pseudo-equilibrium between a gaseous or liquid phase and a dissolved phase, through surface phenomena such as adsorption or dissociation. The second type relates to solid-solid interfaces and involves a pseudo-equilibrium between a phase dissolved in each of two different materials, taking into account interface phenomena which may be present. It is convenient to refer to the case of dynamic steady state, which is the most common in permeation studies [ 3-5 1, e.g. where a constant pressure difference is established across a film. Mass flow passing through a homogeneous membrane of thickness b and unit surface area is then:
A. M&in et al. /Thin Solid Films 272 (19961 124-131
where D denotes the diffusion coefficient of the gas in the material, and ci and cZ are the volume concentrations within the first atomic layers of the two membrane faces. Relation ( 1) arises solely from the hypothesis of Fickian diffusion. In the case of a diatomic gas, Sieverts law gives:
c,=s&
(2)
where S is the Sieverts constant, equivalent to the gas solubility in terms of pressure units, and where pj is the gas (diatomic) pressure imposed on the face j. In so far as Sieverts law is valid, gas flux obeys the relation:
~=~t&xd
(3)
The permeation property of the material is then well characterized by the permeation constant P: P=DS
(4)
where D and S can each be considered an intrinsic volume property. The strict applicability of Sieverts law implies a local thermodynamic equilibrium at the interface between the gaseous and the dissolved phases. In the same way, under the assumption of thermodynamic equilibrium at the interface of two materials, the ratio of the concentrations on either side of an interface (material fmaterial s) equals the ratio of the solubilities [ 61:
c,r si -=Cl,
(5)
SS
To illustrate this, consider at the interface between the two materials a geometrical discontinuity such as a pit, inside of which is imposed a constant pressure p of a diatomic gas (Fig. 1) Applying Sieverts law at the two inside free surfaces and taking into account that, due to thermodynamic equilibrium, the concentration gradients are zero in the materials, the interface concentrations obey Eq. (5).
The permeation constant P of a two-layered membrane, for example substrate s and deposit f of thicknesses u and b respectively, is then simply given [ 61 by: a+b -=_+_
a
b
P
P,
P,
Sf g(p)
\,
cs= ssg(P)’
(7)
where S’ is an “effective solubility”, as opposed to S of Eq. (2), which is an intrinsic volume property. Here, we account for the surface effect by using a phenomenological parameter (T, which is concentration and flux independent like S’: c= a%(p)
Fig. 1. Thought experiment showing that the ratio of concentration, c,Ic,, at the interface of two materials f and s at thermodynamic equilibrium equals the ratio of solubilities. .7,/S,, with p the equilibrium gas pressure in an adjoining void.
(6)
where P, and Pf are the permeation constant of the materials s and f respectively. When the variable S is regarded as an intrinsic volume property, Eqs. (3)-( 6) are strictly applicable only if flux is in no way limited by a surface or interface phenomenon. It is well established that surface phenomena may affect absorption and permeation processes. The synthesis paper of Kompaniets and Kurdyumov [7] gives many examples where large discrepancies from Eq. (3) have been observed experimentally for mass-how dependence either on the inverse of film thickness or on the square root of pressure. Increasing the hydrogen permeability of a material is sometimes accomplished merely by coating the material with a very thin layer of a more hydrogen-reactive material (Pd, Ni) [S]. The permeability increase observed, when this is done, is inconsistent with Eq. (6) and indicates the presence of surface effects. In general, the surface effects may result from an intrinsic property of the material (e.g. reactivity to the ambient gas) and/or experimental conditions (e.g. surface roughness, cleanliness, state of strain and stress) [ 81. Though surface phenomena have not always been well characterized, they have been widely studied [4,7,9]. Discrepancies between experiment and Eqs. ( 3 ) -( 6) have typically been attributed to surface effects [ 71. Interfaces, on the other hand, have been almost ignored from both experimental and theoretical points of view. In the cast of a reactive metal deposit which reduces the surface barrier of the bare substrate to hydrogen, the deposit-substrate interface does not seem to hinder permeation. This perhaps results from the energy involved at an interface (solute state-solute state) being lower than the surface energy (gas phase-solute state). Taking into account possible surface and interface effects, and making the assumption that they lead to a pseudo-equilibrium consistent with boundary conditions of the first kind (fixed concentration), the surface effect is sometimes described [ 61 with a modified Eq. ( 2) : c=S’g(pf
Cf =
12s
(8)
with S remaining a property of the volume. Similarly, we describe the interface effect by using a concentrationand flux-independent parameter y, so that Eq. (5) becomes:
c,r_ s,,,-s xYsr_=k c s
(9)
I26
A. M&in et al. /Thin Solid Films 272 (1996) 124-131
Sfg(Po)i
Cof =
3. Analysis
,
c 00
I
h4
f
\
PO \
\
\
r,
\
.
‘!
cof = Of Sf g(p,)
Pl
I
I
\
\
3. I. Preliminary
Cls = s, g(p1) .
\
cts = bs Ss g(p1)
I
Fig. 2. Concentration of diffusant in a two-layer system with constant external pressures pO and p,
Introducing the (T and y parameters has the advantage of distinguishing between the surface and interface effects. It also seems more reasonable than attributing false properties to the volume. Subsequently, we will have to compare our results (nonsteady state) with the case of the dynamic steady state (DSS) (Fig. 2) in a two-layer system. The volume concentrations in the first atomic layers of the free surface of the coating, f, and of the substrate, s, are constant and equal to respectively. The flux car= afS,g(pc) and cls = a&(p,), (DSS) through the bimaterial can then be written in the form:
considerations
For clarity, the analysis can be restricted without loss of generality to the case of the experiment that consists of measuring the amount of a diffusing substance absorbed by a covered specimen as a function of the charging time. The deposit covers all faces of the specimen, which is thin ribbon. The specimen initially contains none of the diffusing substance and the temperature is held constant during absorption, The amount of diffusing substance absorbed depends on the tightness of the deposit, which in this case is the quantity to be determined. If the specimen is thin enough, the substrate can be considered a planar sheet of thickness 2a with infinite dimensions in the plane (directions y and z in Fig. 3). The substrate is coated on both surfaces with a deposit of thickness b. S, and D,, Sf and Df denote the solubility and the diffusion coefficient of the substrate and the coating, respectively. For our conditions in the two media we need to solve the one-dimensional diffusion equation:
wx,t> -=
a*c(x,t)
at
a2
(15)
with the initial condition:
(10) where
aDf
tL=;Ek
The effect of the coating is revealed by comparing the flux (p, +f through the coated substrate with the flux & which under ,cls) would pass the same conditions (cos = a&&,) through the bare substrate:
(P
P
s+f -=-_ A
cof
1+ CLkc,,
I-
(c,Jco,)
1-
(kco,lco,)
The decrease in permeation then is quantitatively described with the help of the two parameters p and kcoslcof Using relation (8) demonstrates that the parameter:
=o
where -a-b
(16)
The possible surface and interface effects on the absorption of diffusant are assumed to be concentration and flux independent (Section 2). A local equilibrium at the surface and interface is assumed, which leads to boundary conditions of the first kind (Dirichlet). Given the experimental conditions, the diffusant volume concentration within the very first atomic layers of the two external faces of the sample is assumed to be time-independent and equal to: c(x=a+b,t) c(x=
( 12)
%~%s
kc,, 0s C0f = OYsf
c(x,t=O)
=cof
-a-b,t)
(17a)
=cof
(17b)
The boundary condition at the interface is determined by assuming that the ratio of the concentrations on each side of the interface is concentration and flux independent (see Section 2)) and obeys relation ( 18) where k is time independent.
(13)
f D
depends on the two possible surface and interface effects. On the other hand, the parameter:
SS
Cof
A
Sf
Y
OS
a
Cof
Of X
(14) does not depend on any surface effect.
.
-s-b
-a
0
a
a+b
-
Fig. 3. Schematic diagram of the coated specimen. Specimen is much larger in y and z dimensions than in 1.
127
A. M&in et al. /Thin Solid Films 272 (1996) 124-131
CA+a,t) c,( ka,t)
Sf
(18)
=S,ysf=k
Continuity
of diffusant flux at the interface is given by:
The roots’ of Eq. (26) are tabulated in standard books [ 1,2] for certain values of p. Note that knowledge of these roots is usually not needed for a comparison with experiment, because, as will be shown later, the experiment gives the quantity cr:, from which p is deduced. If cri is a solution of Eq. (26)) then:
(19) I
From the symmetry of the problem we need only consider the positive x domain. Substituting for condition ( 17b) :
I
$(XJ) -yg-I,=,=0
This relation allows the coefficients giving:
Dimensional analysis shows that the problem is governed by four independent parameters, which may be chosen:
c,(u) k_= car
cos ((Y;u)cos (cu,u) d,=;(l+~)S,,
Xcos(&)exp(
(21) For the general solution, this problem would normally be solved numerically. The aim of this study is to find an approximate analytical solution, and to determine the domain where this solution is an acceptable approximation to the exact solution of the general problem.
(27)
0
-$$i)
H, to be determined,
(28)
The relative amount of solute absorbed by the total substrate is given by:
Q(t) -= Q=
I-
?A;
I==,
exp
(29)
where: 3.2. Analytic solution assuming a constant gradient in the film (CGF) This approximation is based on the hypothesis of a constant gradient in the film (CGF) . Under the CGF hypothesis, the flux in the coating (a < x < a + b) is given by:
(22) Taking into account relations ( 18) and (22)) conservation of flux at the coating-substrate interface (Eq. ( 19)) yields:
(23) where the only position- and time-dependent quantity is G(W). Although many problems of the same type have already been treated [ 1,2], the problem defined by the relations (15)-( 18), (20) and (23) has not yet, to our knowledge, been solved. Assuming separation of variables and using symmetry (Eq. (20) ) , we look for a solution of the form:
(-sff+
(24)
Taking into account relation (23) leads to: a,=0
with Ho = c,,/k
(25) for i>O
Lyjtan(cr;) =2$=k s
(26)
(30) The deposit and substrate are free of diffusant initially (t = 0). In the real case, since the diffusant has to pass through the deposit before diffusing into the substrate, a certain time is necessary before the flux into the substrate reaches its maximum value. Consequently, if the effect of the coating only is taken into account, the resulting absorbed amount increases very slowly during small times, and then evolves into a regular, increasing value which corresponds to the first term (leading term) in the series that gives the absorbed amount. The associated delay time, typical for non-steady state, is referred to in general as lag time. It is conventionally defined by the intercept of the leading term with the time axis, and would be positive in this case where the effect of ’ For approximately /L < 0.5, the roots of a tan a = CLcan be approximated by a,=(~--_z/3+4~3/45)“2,~d a,=(i-l)~+&[(i-I)T]+{$ [(i-l)n])*fori>l. For I*> 1, no unique power series seems possible, because the roots then are close to k/2 for small i, and to (i - 1) T for large i. Practically, provided the time is not too small, only the first few terms in the series (Eqs. (24) and (29) ) are usually needed. Only for IL> 6, the approximation a,=(i~-?r/2)(l-1/~)(l+1I~L?+(i?r-~/2)~(l-l/~)l(3~3)) may then be used, which gives within 1% the first two roots for + > 6, the first three roots for F> 9, etc. In any case, the most convenient method probably consists of using an iterative numerical calculation, e.g. the Raphson-Newton method (in this case, the initial value for LI, can be chosen a,a = rr( i - I ) + T+/ (2 + 2~), for any value of p and i)
128
A. MPzin et al. /Thin Solid Films 272 (1996) 124-131
the coating only is taken into account (see Section 3.3, ZGS solution). On the other hand, due to boundary conditions, the large difference in concentration between the surface and interior at small times causes the flux to be initially very important. Next, the flux is rapidly reduced as diffusion smoothes the concentration gradient. The corresponding absorbed amount increases quickly from t = 0, before evolving into a regular, less rapidly increasing value (leading term). In this case, the lag time is negative (see Section 3.3, HM solution). In general, the lag time is positive or negative, according to the relative importance of coating and boundary condition effects. Considering the CGF approximation, keeping only the first term of relation (29) gives:
Q,(t) QW
-=
1
-A,
exp
(31)
The lag time, defined by Q, (T) = 0, is then: r=$AlnA, The CGF approximation consists of neglecting the lag time in the deposit. The net lag time T then is negative (A, < 1) . Note that the CGF problem is governed by the two parametersP,=D,la*and (P,IP2)P,=p. 3.3. Analytic solution with zero gradient in the substrate (ZGS) and for a homogeneous medium (HIV} Ignoring the CGF solution, an exact analytic solution exists both in the case when the concentration gradient is zero in the substrate and in the case of a homogeneous medium. Knowledge of these solutions will permit us to test the approximation degree of the CGS solution. Consider first the case when diffusion in the sample is limited only by the coating. Flux passing through the deposit spreads uniformly and immediately in the substrate (ac,/ ax = O), which is the ZGS hypothesis. According to relation ( 19), this condition corresponds to infinite D,, i.e. p = 0. The conservation of flux at the interface can be expressed:
and pi is the ith root of: Pitan P, = rl
(36)
The relative amount of diffusant absorbed by the substrate is written:
Q,(t) =1-CB,ex ffi
QP
#=I
where: (37) Using a procedure similar to that for Eq. (32)) the lag time is: b2 1 r,,=-lln D,P,
B,
(38)
Since only the effect of the coating is taken into account, the lag time rq is always positive. Note that the present problem is governed by the two parameters P, = D,l b2 and P, = 77,which are not independent of the two CGF parameters ( p, P2), since: w,
=
(39)
PP2
Determining the relevant quantities in the case of homogeneous diffusion follows from the classical solution for a homogeneous medium [ 1,2]. A homogeneous medium corresponds to k = 1 and DflD, = 1, which leads to p= I/ 77.The problem is governed by two parameters which can be chosen as D/a2 and bla. The amount of diffusion absorbed by the substrate is written:
Xsin(i$$-f)exp[
-(sfg]
(40)
and note that, particularly for small times, expressions other than Eq. (40) [ 121 may be more suitable. The corresponding lag time is given by: rh=[~(l+j~ln[(l+~)$sin(i-&-$]
(41)
(33) This problem, defined by Eqs. ( 15)-( 18), (20) and (33), has been solved [ 10,111. The concentration in the deposit (a
(34) where: q= (bla)k
(35)
The lag time is negative only for b/a < 0.3 approximately, because, for convenience (see below), only the amount absorbed by the substrate is considered. The total absorbed amount simply corresponds to bla=O. In that case, the lag time is negative and depends on a2/D only. It should be noted here that in each case the amount of diffusant absorbed by the substrate alone is being considered, while the amount of absorbed diffusant measured experimentally evolves from the entire specimen. This approximation simplifies the analysis considerably. It can be verified that other than for lag times for low values of p and q, the approximation is quite acceptable. This is because, in the main of
A. M&in ef al. /Thin SnIid Films 272 (1996) 124-131
interest, the amount of diffusant absorbed by the deposit is small (q < 0.05) and its evolution with time is not very different for the various models. For a given application, this quantity can be easily taken into account.
129
R, = cos p,
(42)
which tends toward 1 - 77/2 for small values of 7. Then the condition: T= (bla)k<0.05
3.4. Degree of validity of the constant gradient in the film (CGF) approximation The domain of validity of the approximation can be described with the help of the two parameters CLand q. The value p= 0 and q=O correspond to the two extreme cases, respectively, of a strictly zero gradient in the substrate and of strictly constant gradient in the deposit. The condition 0 that does not fix the value F= (cllb)(D,ID,)(S,IS,)y,,_= of 7” (b/a) (S,/S,)y,,-, results from D, infinite, in accordance with the basic relation ( 19). The case 77= 0 has to be considered the theoretical limit when (b/a) (&/S,) yS,-tends towards 0. The quantity 77 can be written 77= (b/a)*( D,/ Dr) yS,p, and then, for a given p, can be fixed to an arbitrarily small value by choosing (bla)*( D,lD,-) yS,-small enough. The parameters p and v appear in the experiment analysis by Zuchner (cited in Ref. [ 71) concerning permeation through a trilayer of the same geometry as is considered in this work. The quality of the approximation does not depend on the value of the parameters D,-lb’ and DSla2, since these parameters always appear in the form (Drlb2)t and (D,/a*)t. 3.4.1. Degree of validity of the CGF approximation in the case of zero gradient in the substrate (ZGS) It is useful to examine first the case when p= 0, keeping in mind that in this case the exact solution of the general problem is the ZGS solution. With the flux of absorbant decreasing continuously from the free surface of the deposit to the interface, we choose as a measure of the gradient uniformity the ratio of the fluxes at the coating surface and interface, i.e. R(t) = #‘lC’fa’C/@~T”e. Fig. 4 illustrates the evolution of the ratio R(t) compared with the ratio Q,(t) / Q,, For ~=0.05. The non-constant concentration gradient corresponds to the existence of a lag time as well as to an inequality of the absorbant flux at the surface and interface of the coating, even for very large times since it is non-steady state. For a large times, the flux ratio tends toward: 1
Fig. 4 Evolution with time of the ratio of the exit and entry fluxes in the deposit (dashed line), compared with the corresponding amount of diffusant absorbed by the substrate (full line), in the case of a zero gradient in the substrate.
(43)
ensures that the fluxes are equal to within less than 2.5% (0.5% for 77< 0.01) after long times. Moreover, the ratio of fluxes does reach a value close to the extreme value R, before a significant amount of solute enters the substrate. To demonstrate this, it is convenient to define the lag time in a way different from that suggested by Eq. (32) or Eq. (38). Let us define a pseudo-lag time T, as the time required for the ratio of fluxes to reach a certain fraction X of R,. If 77is lower than 0.1 with X greater than about 0.5, keeping only the first two terms in each of the series for the surface and interface fluxes respectively, gives an approximation to better than 0.3% of the fluxes and their ratio R(t). Keeping only the first two terms in each series then permits the lag time T to be approximated by: (44) The dependence of Ton q is neglected in Eq. (44), since for X greater than 0.5, the approximation is better than I .7% and 0.9% for 77< 0.1 and 0.05 respectively. If 77 is lower than 0.05, the surface flux into the deposit then equals the interface flux to better than 1% (X= 0.99) before the amount of solute absorbed by the substrate reaches 2.2% of its maximum value (0.44% with r) lower than 0.01). This means the lag time condition is implicitly included in condition (43). Finally, in the ZGS case the conditions necessary to assume a constant gradient in the deposit are reduced to inequality (43) alone, which determines the degree of validity of the approximation. As a consequence, for r_~= 0 there is no question of determining a lag time using the CGF solution, since the lag time is assumed zero in the CGF hypothesis. When an experimental lag time can be measured for small values of p, the ZGS solution is certainly a better approximation than the CGF. This point will be explored more quantitatively later. 3.4.2. Degree of validity of the CGF approximation in the general case (CLand q > 01 We now have to examine the degree of validity of the CGF solution in a realistic case, in which p and 77are not equal to zero. In order to obtain a homogeneous measurement of the degree of validity of the approximation over the entire domain, we determine initially the degree of validity using a mathematical distance A between the two functions Q,,(t) and QaP( t), where Q,,(t) and Q,,(t) are the exact and approximate (CGF) solution respectively: A=
”
“’
(45)
A. M&in et (II./Thin Solid Films 272 (1996) I24-13I
130
0
10
60
- 100% by definition, since the CGF lag time T is zero for /.L= 0 while the real lag time is typically non-zero for p = 0 and 77> 0. Keeping in mind that the CGF approximation neglects any diffusion lag time in the deposit, the approximation is not appropriate when the substrate diffusion lag time is less than or of the same order of magnitude (in absolute value) as the deposit lag time, which occurs for low values of p (when 17is small). The domain of validity for determining the lag time using the CGF solution can be estimated by asking that the substrate lag time be N times larger (in absolute value) than the deposit lag time. N is then given by: P
Fig. 5. Error in the constant gradient in the film approximation (CGF), asa function of w = (alb)S,D,/SSD,) ysr and v= (b/a) (S,/S,) ysP The exact solution is respectively, the constant gradient in the film (CGF) solution on the line q=O, zero gradient in the substrate (ZGS) solution on the line p = 0, and homogeneous medium (HM) solution on the line /.L= 1/T.
t, and f2 being defined by Q,,(t) /Qexm= 0.1 and 0.9 respectively. The discrepancy A between the exact and approximate solutions can be calculated exactly on the line p = 0 (ZGS) and p = 1/r], as shown in Fig. 5 (solid thick lines) (note also that A = 0 for 77= 0). For p = 0 and 17< 0.05, the quantity A increases fairly linearly with 77.Over the entire domain of p and 11,the quantity A can be seen to be a continuous and monotonic function of p and 7. From its value for p = 0 and p = 1/q, it is quantitatively estimated (dashed lines) on the domain of interest by assuming, for every value of P, the same sort of variation of A with q as for /.L= 0. The discrepancy A increases with q and decreases with J.L For all ?I < rll, A(p,q) is less than A(0, r],). With regard to experiments, the two parameters D,/a’ and p should usually be determinable by mathematically fitting the CGF solution, Q( rj) (Eqs. (29) and (30) ), to the experimental points QeXP(rj). For small values of /..Lhowever, in practice only the parameter (D,la) CY:E ( 1lab) (D&IS,) ‘ysf can be determined, as will be seen later. More commonly two parameters are determined from the slope of In ( 1 - Q_,(t) /Qm for long times, i.e. (D,la2> a:, combined with the value of the lag time r= (u2/D,) ( l/ CY:)In A,. In this case, the theoretical error relative to (D,/ a’)a: is due to the use of the CGF solution instead of the exact solution. This theoretica error, A’ = 1 - (approximate value/exact value), exhibits the same qualitative behaviour of A (Fig. 5). On the line /.L= 0, the error increases fairly linearly with 7~ (A’ = - 7713- 71~145). For a given 7, the error falls very abruptly from its value at p = 0 (equal to = 1.67% for ~=0.05 for instance) as p increases ( - 0.0176% at 7)= 0.05 and TV= 1/ 77= 20). As an approximationfor~=1/~>6,A’=-(~/2)2(2/3$)(1-11/~). The error relative to the lag time behaves qualitatively the same (for instance - 1.62% at v= 0.05 and I_L= 1/r) = 20, and - 0.24% at 7 = 0.02 and p = I/ 71 = 50) _ An important exception is that for p = 0 and 7 > 0, the error is equal to
N_pP:ln4 7)
(46)
a:InB,
As an example, for r) = 0.01,0.05 or 0.1, N is greater than 10 when p exceeds about 0.94,2.4 or 3.6, respectively. In practice, we can consider the approximate condition r~/ /_L’<~x 10-4N. In practice, this point is not of great importance because, for small values of p and v, the real lag time is so smal1 that it is generally not measurable due to experimental uncertainty. Note also that for very small values of p and 17,a very simple modeling approach can be used. This consists of assuming that the gradient is constant in the deposit and vanishes in the substrate. The corresponding analytic solution can easily be worked out directly: y=
1
_exp($_yq
Alternatively, the analytic the limit of Eq. (29) when of Eq. (37) when r] tends equivalent, since according qDflb2 = pD,la’=
(47) solution Eq. (47) is obtained as p tends toward 0, or as the limit toward 0. These two limits are to Eq. (39) :
( 1 lab) (S,D,y,,IS,)
(48)
Disregarding the lag times which are not considered, the corresponding error can be estimated immediately since a: is replaced by /L, while the true value is given by CL= ff, tan (Y,.
4. Conclusion The CGF solution permits the relative diffusant distribution in the specimen to be predicted as a function of time, provided certain parameters are known, i.e. the coating and substrate thickness (b and 2~)) bulk diffusion coefficients Df and D, and ratio of solubilities multiplied by the interface effect (k= (S,/S,) r,r). Added knowledge of the external imposed surface concentration cof gives the absolute value of the concentration. Alternatively, determination of the two parameters D,la2 and p = (a/b) ( SfDfysf/SsD,) is possible from experiment by using the CGF solution under certain conditions. For small
A. Nzin et al. /Thin Solid Films 272 {19%> 124-131
values of p (N < 10 approximately), only the parameter ( 11 ab) (S&y,,IS,) can be determined. In every case, the quantity 2ac,,lk (or more exactly (2ac,Jk) ( 1 + 71)) is also determined by taking into account the amount of diffusant absorbed by the deposit at equilibrium. By comparing with a complementary experiment using the uncoated substrate, the quantity (ar/(~,~,r) ( 1 + 77) appears. It can be easily verified that these are the parameters involved in the case of steadystate permeation (Section 2). Moreover, the quantity D,la’ can be determined if p is not too small. Note that the quantities D, and S,, in general, cannot be determined independently, whereas the product D&ysf can (assuming that S, is known). This product, which according to definition could be multiplied by the surface effect (+r, can be considered the “effective” in-situ permeation constant of the deposit. The constant gradient in the film (CGF) approximation provides an easy tool for studying in situ the diffusion processes m multilayer materials. In particular, the CGF solution is well adapted to the case of protective coatings, for which the quantity TI= (b/a) (S,/S,) -ys,is generally small. The CGF approximation is best for small values of q and large values of /.L= (a/b) (SfD,/S,D,) -ysnwhich can be set in experimental work by choosing alb large enough. The CGF approximation is restricted in that the solubility and diffusion coefficient cannot be independently determined. The most essential restriction concerns the type of
131
boundary condition considered (fixed concentration), which may presume the nature of the surface and interface effects. It would be of interest to attempt a similar analysis with boundary conditions of the second and third kind. In a companion paper, the CGF approximation is used to determine the permeation properties of a molybdenum coating deposited as a hydrogen barrier. References [ 11 H.S. Carslaw and J.C. Jaeger, Conduction oj Heat in Solid.7,2nd edn., Oxford University Press, New York, 1986. [2] J. Crank, The Marhematics ofDiffusion, 2nd edn.. Oxford University Press, New York, 1975. [3] R. Frauenfelder, J. Chem. Phys., 48 ( 1968) 3955. [41 W.R. Wampler, J. App. Phys.. 65 ( 1989) 4040. [5] T.P. Pemg, M.J. Johnson and C.J. Altstetter, Mefull. Truns. A., 19A (1988) 1187. [6] M. Kano, N. Kagawa, S. Koike, K. Furuyaand T. Suzuki, Acfa Metall., 36 (1988) 1553. [7] T.N. Kompaniets and A.A. Kurdyumov. Prqr. Surf Ser.. 17 (1984) 15. [8] J. Chevallier 219.
and M. Aucouturier,
Ann. Rev. Mater. Sci.. 18 (1988)
[9] P.M. Richards, S.M. Myers, W.R. Wampler and Follstaedt, J. Appl. Phys.. 65 (1988) 180. [IO] D.K. Paul and A.T. Dibenedetto, J. Polym. Sci. C. 10 ( 1965) 17. [ 111 H.S. Carslaw and J.C. Jaeger, Conduction of Hear in Solids. 2nd edn., Oxford University Press, New York, 1986, pp. 128-129. [ 121 J. Crank, The Mathematics cfDi#usiusion,2nd edn., Oxford University Press, New York, 1975, pp. 48,58.