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Origin and evolution of cup-shaped structures on leached nuclear waste containment glass surfaces
J O U R N g L OF
ELSEVIER
Journal of Non-Crystalline Solids 171 (1994) 290-298
Origin and evolution of cup-shaped structures on leached nuclear waste containment glass surfaces C. Dubois a,,, F. Villa a, A. Chambaudet a, E. Vernaz b a Laboratoire de Microanalyses Nucl~aires, La Bouloie, Route de Gray, 25030 Besan~on cddex, France b Rhone Valley Research Center, DPR/SCD, BP 171, 30207 Bagnols-sur-Cdze c~dex, France
Received 16 June 1993;revised manuscript received 14 January 1994
Abstract A three-dimensional surface microanalysis system equipped with a sensitive topographical probe was used to quantify the evolution of cup-shaped structures formed by aqueous leaching of nuclear waste containment glass. A model of the dissolution phenomenon provides satisfactory correlation between calculated and measured cup radius and depth. Dissolution cups form from cracks on the initially cut glass surface. Large cracks control the phenomenon by forming the largest cups, which gradually absorb smaller ones. The evolution of the size and shape of the dissolution cups was described by a model that assumes a constant dissolution rate on the surface, diminishing with crack depth. The best fit with the experimental data was obtained with a dissolution rate one hundred times lower at the bottom of the crack than at the surface. Moreover, it is predictable that all the cups will gradually disappear as they grow larger and flatter over a leaching period of some 2 years, for the glass composition and experimental leaching procedures used in this work.
I. Introduction The long-term behaviour of nuclear waste containment glass is predicted from models based on glass durability [1,2]. A large body of published experimental research exists concerning aqueous leaching of natural or synthetic silicate glasses, containing fission products or not, involving chemical analysis of the leachate and investigation of the solid alteration products that form on the glass surface [3]. The glass surface finish
* Corresponding author. Tel: +33 81 66 65 02. Telefax: +33 81 66 65 22.
before and after leaching has also been investigated [4]. The glass reaction front sometimes exhibits small concavities known as 'dissolution cups' (Fig. 1) with a surface diameter ranging from a few micrometers to a few tens of micrometers. An investigation of three borosilicate glasses and a basalt glass [5] demonstrated that the cups form only over superficial glass flaws that existed prior to leaching. Such flaws may occur when glass specimens are cut (cracks and abrasions), by mechanical marking of the glass with a microhardness tester or by latent damage induced by fission fragments, etc. This paper discusses an experimental assessment of the evolution of dissolution cups on the
0022-3093/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0022-3093(94)00232-C
C. Dubois et aL /Journal of Non-Crystalline Solids 171 (1994) 290-298
291
Table 1 Specimen surface roughness measurements Specimen
Leaching duration (30-day months)
Ra (~m)
Standard deviation (~m)
Rt (~m)
Standard deviation (~m)
R7A R7B R7C R7D R7E R7F R7G R7H
0 0.93 2 3 4 5 6 12.4
1.60 3.09 3.49 4.62 4.27 3.89 3.33 1.34
0.26 0.40 0.21 0.50 0.48 0.43 0.57 0.13
18.10 22.35 26.15 34.85 29.79 23.64 25.11 9.73
4.50 3.40 2.15 5.09 3.44 2.55 5.84 1.76
as-cut surface of French R7T7 reference glass specimens during dynamic leach testing at 100°C, and the development of a model describing the phenomenon. R7T7 glass is a non-radioactive analog of the containment glass used for nuclear waste generated by reprocessing of 'light water' reactor fuel.
2. Experimental procedure Test coupons were cut from R7T7 glass rods with 25 X 25 mm 2 cross-sections using a dia-
mond-impregnated saw rotating at 500 rpm. The coupons were leached in a conventional Soxhlet device consisting of a boiler containing 300 ml of double-distilled water. Each coupon was placed in a 25 ml sample boat into which condensed steam flowed continuously at a rate of about 200 mlh -1, Six specimens were tested for periods ranging from 28 to 372 days (Table 1). The specimen surfaces were characterized before leaching (R7A) and after different leaching periods following removal of the alteration film. Each surface was examined by scanning electron microscopy (SEM) using a Jeol 35 CF system, as
Fig. 1. SEM view of cups on the glass surface.
292
C. Dubois et al. /Journal of Non-Crystalline Solids 171 (1994) 290-298
= , , 1 1
as cut
0.9 month
R7C 2 months
R7D 3 months
i,~~,~~
6 monms Fig. 2. Surface images (Williamson perspectives).
C. Dubois et al. / Journal of Non-Crystalline Solids 171 (1994) 290-298
I
3 Months
293
12.4 Months
Fig. 3. Surface isocurves.
well as by 3D surface scanning and 2D profile measurements using a system developed in the laboratory [6,7]. The glass surface was analyzed with a 2.5 Ixm radius diamond-tip probe capable of obtaining a
three-dimensional surface analysis covering 256 points on the x-axis and 256 points on the y-axis. The pitch between points may range from one to several tens of micrometers, and different pitches may be applied on the x- and y-axes. The z-axis
Fig. 4. SEM view of cups on the section normal the surface.
294
C D u b o i s et a L / J o u r n a l
o f N o n - C r y s t a l l i n e Solids 171 ( 1 9 9 4 ) 2 9 0 - 2 9 8
resolution was about 0.01 txm for the smoothest surfaces. A computer data base was compiled from the (x,y,z) coordinates of 65536 surface points for use by various programs to obtain perspectives of the surface from different points of view. The system may also be used as a surface roughness gauge by recording a linear profile of up to 4000 points along one axis. A computer program then computes the roughness parameters after determining the mean line. The roughness values calculated in this test were the mean values of 35 profiles. The following parameters may be defined: R~ is the average roughness (the arithmetic mean profile deviation from the mean line) R a = (1)/(L)fo L I z(x)l dx, where L is the length over which profile z(x) is evaluated; R t is the total profile roughness deviation (i.e., maximum peakto-root dimension).
Note that the angles selected for the perspective result in an anamorphic coefficient. Since the perspective angles are identical for all six images, the anamorphosis is the same and does not prevent visual comparison. The images clearly show the formation of dissolution cups from defects on the as-cut surface, and trace their increasing size and decreasing frequency during leaching. This suggests that the largest cups are formed from the deepest cracks, and that they progressively 'swallow up' the smaller adjacent cups. The surface isocurves (constant height locus lines) for two specimens (R7D and R7H) in Fig. 3 confirm this hypothesis, and show that the cracks from which the cups originated tend to evolve into hemispherical craters, as indicated in Fig. 4. The crest lines (limits between cups) in Fig. 5 were plotted after extracting the corresponding points from the data base with a program implementing the algorithm proposed by Peucker and Douglas [10]. On both surfaces shown (R7G and R7H), the crest lines (or, more accurately, their projection on the x-y plane) form a polygonal distribution. The same evolution of the size and number of the cups as the leaching progressed was again observed.
3. Results
3.1. 3D surface analysis The x- and y-axis pitch was 2 txm for all the surfaces; thus, the scanned area was 512 × 512 ixm2.
3.2. 2D profile analysis
The Williamson perspective images [8,9] are shown in Fig. 2 for six characteristic surfaces.
Fig. 6 shows the variations in R t and 5R, (expanded scale) versus the leaching time. The
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295
C. Dubois et aL / Journal of Non-Crystalline Solids 171 (1994) 290-298
5Ra ta Ignored Values of 5Ra
6(]
•
o Ignored Values of RI E 4(I
-~ 3(I
(I
S
ID
15 21) Leaching Time (Mol/th~,)
25
Fig. 6. R t and 5 R a variation versus leaching time. increasing values for both parameters during the first two months is attributable to the fact that the initial surface cracks and abrasions had a high depth-to-diameter ratio, and the feeler was unable to reach the bottom of the cavities. The diameter increased in time, and by the third month was large enough with a sufficient bottom radius to allow accurate recording of the z ( x ) coordinates (Fig. 2); this explanation was confirmed by the SEM observations. The curves were therefore plotted by exponential and polynomial regression from the R t and 5R~ p a r a m e t e r values beginning with the third month of leaching: R, = 50.44
× 10 -5"72e-2x
R = 0.988,
5R~ = 30.5 - 2.55x - 0.05x 2 R = 1.00. The y-intercept values of R t and R~ are probably close to the actual values. Both curves are consistent with gradual surface flattening as the cup depth continues to decrease. It may therefore be estimated that, after 2 years of leaching, the cups will no longer be identifiable and the glass surface will be practically flat (at the scale of the recording system z-axis sensitivity) for the experimental leaching procedures used during this work.
(a) The mathematical analysis and computer simulation are performed in a plane normal to each crack. (b) Glass corrosion begins in relatively closed cracks. (c) Corrosion progresses at a rate corresponding to the rate at which the pristine glass recedes in a direction normal to each point on the surface (this hypotheses is used and confirmed in nuclear track models for various solid detectors [11,12]). (d) The corrosion rate, V, decreases as a linear function of the depth inside the crack. Assuming a rate V0 at the surface and an initial corrosion rate percentage, k, at the bottom of the crack, then the rate V at a depth z - z 0 in the analysis frame of reference is expressed as .
V =
.
[ Z() --
.
.
Zrnin
+ 1 V0 = ~rVa,
1 100
where or is the corrosion rate coefficient at depth Z -- Z 0 .
Glass dissolution in the crack is simulated on the computer by a series of calculation steps. At this stage, the corrosion rates are expressed in pixels per calculation step (ppcs) to provide a graphical display of the recession of the pristine glass reaction front during leaching. The calculation step becomes the time unit. Tests showed that the maximum rate V0 = 0.1 ppcs simulates continuous dissolution on the screen image. During each iteration, the new straight walls of the crack are computed by linear interpolation from the end points. Absolute Frame
Computil
ZO
i=O
v~,
"
,' ,'
2
Frame
0
v~, i=1
i=2
4. Discussion Zmi
~Vo
ZO-Zmi n
4.1. Two-dimensional model of cup formation
The model is based on the following hypotheses (Fig. 7).
t
n=l
Fig. 7. Graphic simulation of the first two calculation steps.
C. Dubois et al. /Journal of Non-Crystalline Solids 171 (1994) 290-298
296
On the basis of the 3D and 2D cup analysis results and the SEM cross-sectional views (Fig. 4), the bottom of the crack was assumed to be circular in the plane normal to the crack (Fig. 7). The coordinates of a point are (x~, z~), where i designates the ith step and the simulated glass profile after i steps, and n is a number indicating the filiation of points on the simulated profiles after each calculation step. For the first step, the development of the circular crack bottom is obtained by calculating a glass regression, o'~V0, for three filiations n = 1,2,3 as indicated in Fig. 7. Dissolution occurs from a single point (x °, z°). The first cup is then computed from the three points (x~, ZzL),(x~, z l) and (x31, z~). After the ith step ( i > 1), the glass has regressed by triVo from the points at the same depth (x~-', z~-1) and (x~-', z~ 1) and by ~i'V0 from point (x1-1, zi-1). Since the latter point is lower, crCVo is less than triVo: as a result, the cup bottom circle radius increases with the number of calculation steps after the second step. After the ith step, the coordinates of the center of the circle are (x~, zi), where X ci ~ X l 0 i
Zc i
+
i
i
k=lO0%
i
The circle radius is
Ric=[(xi2--xi)23t-(Z~--Z~)2]'/2" A computer program implementing this calculation mode allows the cup formation to be simulated in two dimensions. Fig. 8 shows two simulated cups in formation from two vertical cracks (k = 50% in Fig. 8(a) and 0.01% in Fig. 8(b)). The glass regression front is shown after each series of 200 calculation steps. The small cups disappear first. Simulations with oblique cracks produced similar results using a suitable algorithm based on the same hypotheses.
4.2. Correlation between the cup formation model and three-dimensional measurements After ensuring that the cup radius could be simulated, the mean cup radii were measured on specimen surfaces leached for 3 months or more (after 3 months, the cups may be considered as regular concavities). The profiles passing through the cup bottoms were plotted by extracting the x,z or y,z coordinates of the relevant points from the set of 65 536 points whose coordinates were recorded during the surface analysis. The simulated cup profiles were found to be virtually circular. The circle containing the points of each profile was then computed and its radius was determined. Thirty cups were analyzed for each
k=O,Ol%
Fig. 8. S i m u l a t e d d i s s o l u t i o n c u p s in f o r m a t i o n .
297
C. Dubois et al. ~Journal of Non-Crystalline Solids 171 (1994) 290-298
Table 2 Calculated dissolution cup parameters Specimen Leaching Mean duration radius (months) (Ixm)
Standard deviation (Ixm)
R7D R7E R7F R7G R7H
9.3 14.0 10.5 7.8 63.0
3 4 5 6 12.4
42.3 51.0 60.8 80.5 454.2
surface; for each cup, the radii were calculated for two circles in the orthogonal planes x , z and y , z . The mean radii and standard deviations are indicated in Table 2. The mean cup radius did not vary in a linear manner with the leaching time. This validates the hypothesis of a corrosion rate gradient as a function of the cup depth. Extrapolating the R t c u r v e to the y-intercept (Fig. 6) suggests that the largest initial cracks were approximately 50 ~ m deep. As indicated above, the large cracks control the cup formation phenomenon by creating large cups which progressively envelop the smaller ones. We therefore simulated the corrosion of a cup 50 Ixm deep to check for a correlation with the experimental values. Fig. 9 shows the simulated evolution of the radius of a cup 50 ~m deep (280 pixels on the
screen) assuming a constant corrosion rate of 0.1 ppcs. Three values were simulated for the k coefficient: 100, 50 and 0.01%. A correlation was observed with the mean experimental cup radii when k = 0.01%. Simulations using time-dependent variable corrosion rates did not correlate, regardless of the k value. Measurements have shown that during the first few days the mean dissolution rate of as-cut R7T7 glass is 2.5 times higher than after 28 days of leaching, at which point it becomes virtually constant [4]. The phenomenon therefore does not appear to be a determining factor in long-term glass dissolution; moreover, the cup radii were measured experimentally after 3 months or more of leaching. Based on the correlation observed in Fig. 9, 4500 calculation steps correspond to an actual leaching time of 12.4 months. As noted above, each calculation step corresponds to the simulated dissolution of 0.1 pixel of glass, which is equivalent to a dissolved glass thickness of 0.1 × 50/280 = 0.018 Ixm. The dissolved glass thickness after 12.4 months of simulated leaching is therefore 4500 x 0.018 = 81 Ixm, which corresponds to a simulated daily mean leached glass thickness of 0.22 ixm for the experimental procedures used during this work. These results are consistent with experimental observations [13]. Fig. 10 shows the simulated variation of R t = z 0 -Zmi n for the same k values and with V0
50
k = 100% 500'
g
4O
400'
3O 300'
k = t00%
E
2O 200-
c a.
(.)
=
,
,
I0
100
0 v
•
i 2
A : Measured
•
i 4
.
r 6
.
i
Leaching , i
8
10
Time •
(Months) , , 12
Values
Fig. 9. Simulated cup bottom radius variation versus leaching time (initial crack depth 50 ~m).
Leaching Time (Months) (I
2 A : Measured
4
6
8
I0
12
VaLues
Fig. 10. Simulated R t variation versus leaching time (initial crack depth 50 ~m).
298
C. Dubois et al. /Journal of Non-Crystalline Solids 171 (1994) 290-298
constant in time. Once again, the satisfactory correlation with k = 0 . 0 1 % confirmed our hypotheses. R t and the cup radii were also simulated using an algorithm allowing for a linear drop in the corrosion rate with the leaching time. As before, the measured radii and R t values did not correlate with the calculated values.
5. Conclusion This investigation confirmed the role of initial cracks in the formation of dissolution cups on the surface of R7T7 glass. It also revealed the dominant effect of the deepest cracks in the evolution of the cups. During this evolution, the large cups grew more quickly than the smaller ones, which were progressively eliminated. The diminishing values of the R a and R t parameters during the leaching period corroborate this evolution. Extrapolating the roughness parameters suggests that the cups will completely 'disappear' after two years of leaching, under the conditions of these experiments. A cup development model based on these observations was qualified by correlation with topographical probe measurements of the increasing cup radius and surface flattening due to the decreasing total profile roughness, R t. This correlation demonstrates that the dissolution rate at the top of the cracks - and subsequently at the top of the cups - may be considered constant at 0.22 txm per day over a 12.4 month period for the
glass composition and experimental leaching procedures used in this investigation.
References [1] B. Grambow, in: Scientific Basis for Nuclear Waste Management VIII, ed. by C. Jantzen, J. Stone and E. Ewing, Mater. Res. Soc. Symp. Proc. 44 (1985) 15. [2] E. Vernaz and J.L. Dussossoy, Appl. Geochem. Special issue No. 1 (1992) 13. [3] J.L. Nogu~s, PhD thesis, Universit6 de Montpellier (1982). [4] J.L. Dussossoy, C. Dubois, E. Vernaz and A. Chambaudet, in: Scientific Basis for Nuclear Waste Management XV, ed. C. Sombret, Mater. Res. Soc. Symp. Proc. 257 (1992) 109. [5] A. Chambaudet, C. Dubois, J.C. Petit and J.C. Dran, in: Proc. Conf. SFEN-RECOD '87, Paris (1987). [6] M. Assoul, C. Brugger, A. Chambaudet, C. Dubois, M. Fromm, N. Jacquet-Francillon, J. Mignot, E. Painchaud and E. Vernaz, poster, CEA/IPSN exhibit, Besan~on, 1990. Ext. Abstracts, Restricted Diffusion. [7] M. Assoul, PhD thesis, Universit6 de Franche-Comt~ (1991). [8] J.B.P. Williamson, in: Proc. NASA Symp. on Interdisciplinary Approaches to Friction and Wear, San Antonio, TX, Nov. 28-30, 1967 (NASA, Washington, DC, 1968). [9] M. Chuard, A.C. Roudot and J. Mignot, Wear 96 (1984) 31. [10] T.K. Peucker and D.H. Douglas, Comp. Graphics Image Proc. 4 (1975) 375. [11] M. Fromm, PhD thesis, Universit6 de Franche-Comt~ (1990). [12] M. Fromm, F. Membrey, A. Chambaudet and R. Saouli° Nucl. Tracks Rad. Meas. 19 (1991) 163. [13] F. Delage and J.L. Dussossoy, in: Scientific Basis for Nuclear Waste Management XIV, ed. C. Sombret, Mater, Res. Soc. Symp. Proc. 212 (1991) 41.
ELSEVIER
Journal of Non-Crystalline Solids 171 (1994) 290-298
Origin and evolution of cup-shaped structures on leached nuclear waste containment glass surfaces C. Dubois a,,, F. Villa a, A. Chambaudet a, E. Vernaz b a Laboratoire de Microanalyses Nucl~aires, La Bouloie, Route de Gray, 25030 Besan~on cddex, France b Rhone Valley Research Center, DPR/SCD, BP 171, 30207 Bagnols-sur-Cdze c~dex, France
Received 16 June 1993;revised manuscript received 14 January 1994
Abstract A three-dimensional surface microanalysis system equipped with a sensitive topographical probe was used to quantify the evolution of cup-shaped structures formed by aqueous leaching of nuclear waste containment glass. A model of the dissolution phenomenon provides satisfactory correlation between calculated and measured cup radius and depth. Dissolution cups form from cracks on the initially cut glass surface. Large cracks control the phenomenon by forming the largest cups, which gradually absorb smaller ones. The evolution of the size and shape of the dissolution cups was described by a model that assumes a constant dissolution rate on the surface, diminishing with crack depth. The best fit with the experimental data was obtained with a dissolution rate one hundred times lower at the bottom of the crack than at the surface. Moreover, it is predictable that all the cups will gradually disappear as they grow larger and flatter over a leaching period of some 2 years, for the glass composition and experimental leaching procedures used in this work.
I. Introduction The long-term behaviour of nuclear waste containment glass is predicted from models based on glass durability [1,2]. A large body of published experimental research exists concerning aqueous leaching of natural or synthetic silicate glasses, containing fission products or not, involving chemical analysis of the leachate and investigation of the solid alteration products that form on the glass surface [3]. The glass surface finish
* Corresponding author. Tel: +33 81 66 65 02. Telefax: +33 81 66 65 22.
before and after leaching has also been investigated [4]. The glass reaction front sometimes exhibits small concavities known as 'dissolution cups' (Fig. 1) with a surface diameter ranging from a few micrometers to a few tens of micrometers. An investigation of three borosilicate glasses and a basalt glass [5] demonstrated that the cups form only over superficial glass flaws that existed prior to leaching. Such flaws may occur when glass specimens are cut (cracks and abrasions), by mechanical marking of the glass with a microhardness tester or by latent damage induced by fission fragments, etc. This paper discusses an experimental assessment of the evolution of dissolution cups on the
0022-3093/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0022-3093(94)00232-C
C. Dubois et aL /Journal of Non-Crystalline Solids 171 (1994) 290-298
291
Table 1 Specimen surface roughness measurements Specimen
Leaching duration (30-day months)
Ra (~m)
Standard deviation (~m)
Rt (~m)
Standard deviation (~m)
R7A R7B R7C R7D R7E R7F R7G R7H
0 0.93 2 3 4 5 6 12.4
1.60 3.09 3.49 4.62 4.27 3.89 3.33 1.34
0.26 0.40 0.21 0.50 0.48 0.43 0.57 0.13
18.10 22.35 26.15 34.85 29.79 23.64 25.11 9.73
4.50 3.40 2.15 5.09 3.44 2.55 5.84 1.76
as-cut surface of French R7T7 reference glass specimens during dynamic leach testing at 100°C, and the development of a model describing the phenomenon. R7T7 glass is a non-radioactive analog of the containment glass used for nuclear waste generated by reprocessing of 'light water' reactor fuel.
2. Experimental procedure Test coupons were cut from R7T7 glass rods with 25 X 25 mm 2 cross-sections using a dia-
mond-impregnated saw rotating at 500 rpm. The coupons were leached in a conventional Soxhlet device consisting of a boiler containing 300 ml of double-distilled water. Each coupon was placed in a 25 ml sample boat into which condensed steam flowed continuously at a rate of about 200 mlh -1, Six specimens were tested for periods ranging from 28 to 372 days (Table 1). The specimen surfaces were characterized before leaching (R7A) and after different leaching periods following removal of the alteration film. Each surface was examined by scanning electron microscopy (SEM) using a Jeol 35 CF system, as
Fig. 1. SEM view of cups on the glass surface.
292
C. Dubois et al. /Journal of Non-Crystalline Solids 171 (1994) 290-298
= , , 1 1
as cut
0.9 month
R7C 2 months
R7D 3 months
i,~~,~~
6 monms Fig. 2. Surface images (Williamson perspectives).
C. Dubois et al. / Journal of Non-Crystalline Solids 171 (1994) 290-298
I
3 Months
293
12.4 Months
Fig. 3. Surface isocurves.
well as by 3D surface scanning and 2D profile measurements using a system developed in the laboratory [6,7]. The glass surface was analyzed with a 2.5 Ixm radius diamond-tip probe capable of obtaining a
three-dimensional surface analysis covering 256 points on the x-axis and 256 points on the y-axis. The pitch between points may range from one to several tens of micrometers, and different pitches may be applied on the x- and y-axes. The z-axis
Fig. 4. SEM view of cups on the section normal the surface.
294
C D u b o i s et a L / J o u r n a l
o f N o n - C r y s t a l l i n e Solids 171 ( 1 9 9 4 ) 2 9 0 - 2 9 8
resolution was about 0.01 txm for the smoothest surfaces. A computer data base was compiled from the (x,y,z) coordinates of 65536 surface points for use by various programs to obtain perspectives of the surface from different points of view. The system may also be used as a surface roughness gauge by recording a linear profile of up to 4000 points along one axis. A computer program then computes the roughness parameters after determining the mean line. The roughness values calculated in this test were the mean values of 35 profiles. The following parameters may be defined: R~ is the average roughness (the arithmetic mean profile deviation from the mean line) R a = (1)/(L)fo L I z(x)l dx, where L is the length over which profile z(x) is evaluated; R t is the total profile roughness deviation (i.e., maximum peakto-root dimension).
Note that the angles selected for the perspective result in an anamorphic coefficient. Since the perspective angles are identical for all six images, the anamorphosis is the same and does not prevent visual comparison. The images clearly show the formation of dissolution cups from defects on the as-cut surface, and trace their increasing size and decreasing frequency during leaching. This suggests that the largest cups are formed from the deepest cracks, and that they progressively 'swallow up' the smaller adjacent cups. The surface isocurves (constant height locus lines) for two specimens (R7D and R7H) in Fig. 3 confirm this hypothesis, and show that the cracks from which the cups originated tend to evolve into hemispherical craters, as indicated in Fig. 4. The crest lines (limits between cups) in Fig. 5 were plotted after extracting the corresponding points from the data base with a program implementing the algorithm proposed by Peucker and Douglas [10]. On both surfaces shown (R7G and R7H), the crest lines (or, more accurately, their projection on the x-y plane) form a polygonal distribution. The same evolution of the size and number of the cups as the leaching progressed was again observed.
3. Results
3.1. 3D surface analysis The x- and y-axis pitch was 2 txm for all the surfaces; thus, the scanned area was 512 × 512 ixm2.
3.2. 2D profile analysis
The Williamson perspective images [8,9] are shown in Fig. 2 for six characteristic surfaces.
Fig. 6 shows the variations in R t and 5R, (expanded scale) versus the leaching time. The
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Fig. 6. R t and 5 R a variation versus leaching time. increasing values for both parameters during the first two months is attributable to the fact that the initial surface cracks and abrasions had a high depth-to-diameter ratio, and the feeler was unable to reach the bottom of the cavities. The diameter increased in time, and by the third month was large enough with a sufficient bottom radius to allow accurate recording of the z ( x ) coordinates (Fig. 2); this explanation was confirmed by the SEM observations. The curves were therefore plotted by exponential and polynomial regression from the R t and 5R~ p a r a m e t e r values beginning with the third month of leaching: R, = 50.44
× 10 -5"72e-2x
R = 0.988,
5R~ = 30.5 - 2.55x - 0.05x 2 R = 1.00. The y-intercept values of R t and R~ are probably close to the actual values. Both curves are consistent with gradual surface flattening as the cup depth continues to decrease. It may therefore be estimated that, after 2 years of leaching, the cups will no longer be identifiable and the glass surface will be practically flat (at the scale of the recording system z-axis sensitivity) for the experimental leaching procedures used during this work.
(a) The mathematical analysis and computer simulation are performed in a plane normal to each crack. (b) Glass corrosion begins in relatively closed cracks. (c) Corrosion progresses at a rate corresponding to the rate at which the pristine glass recedes in a direction normal to each point on the surface (this hypotheses is used and confirmed in nuclear track models for various solid detectors [11,12]). (d) The corrosion rate, V, decreases as a linear function of the depth inside the crack. Assuming a rate V0 at the surface and an initial corrosion rate percentage, k, at the bottom of the crack, then the rate V at a depth z - z 0 in the analysis frame of reference is expressed as .
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where or is the corrosion rate coefficient at depth Z -- Z 0 .
Glass dissolution in the crack is simulated on the computer by a series of calculation steps. At this stage, the corrosion rates are expressed in pixels per calculation step (ppcs) to provide a graphical display of the recession of the pristine glass reaction front during leaching. The calculation step becomes the time unit. Tests showed that the maximum rate V0 = 0.1 ppcs simulates continuous dissolution on the screen image. During each iteration, the new straight walls of the crack are computed by linear interpolation from the end points. Absolute Frame
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4.1. Two-dimensional model of cup formation
The model is based on the following hypotheses (Fig. 7).
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Fig. 7. Graphic simulation of the first two calculation steps.
C. Dubois et al. /Journal of Non-Crystalline Solids 171 (1994) 290-298
296
On the basis of the 3D and 2D cup analysis results and the SEM cross-sectional views (Fig. 4), the bottom of the crack was assumed to be circular in the plane normal to the crack (Fig. 7). The coordinates of a point are (x~, z~), where i designates the ith step and the simulated glass profile after i steps, and n is a number indicating the filiation of points on the simulated profiles after each calculation step. For the first step, the development of the circular crack bottom is obtained by calculating a glass regression, o'~V0, for three filiations n = 1,2,3 as indicated in Fig. 7. Dissolution occurs from a single point (x °, z°). The first cup is then computed from the three points (x~, ZzL),(x~, z l) and (x31, z~). After the ith step ( i > 1), the glass has regressed by triVo from the points at the same depth (x~-', z~-1) and (x~-', z~ 1) and by ~i'V0 from point (x1-1, zi-1). Since the latter point is lower, crCVo is less than triVo: as a result, the cup bottom circle radius increases with the number of calculation steps after the second step. After the ith step, the coordinates of the center of the circle are (x~, zi), where X ci ~ X l 0 i
Zc i
+
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i
k=lO0%
i
The circle radius is
Ric=[(xi2--xi)23t-(Z~--Z~)2]'/2" A computer program implementing this calculation mode allows the cup formation to be simulated in two dimensions. Fig. 8 shows two simulated cups in formation from two vertical cracks (k = 50% in Fig. 8(a) and 0.01% in Fig. 8(b)). The glass regression front is shown after each series of 200 calculation steps. The small cups disappear first. Simulations with oblique cracks produced similar results using a suitable algorithm based on the same hypotheses.
4.2. Correlation between the cup formation model and three-dimensional measurements After ensuring that the cup radius could be simulated, the mean cup radii were measured on specimen surfaces leached for 3 months or more (after 3 months, the cups may be considered as regular concavities). The profiles passing through the cup bottoms were plotted by extracting the x,z or y,z coordinates of the relevant points from the set of 65 536 points whose coordinates were recorded during the surface analysis. The simulated cup profiles were found to be virtually circular. The circle containing the points of each profile was then computed and its radius was determined. Thirty cups were analyzed for each
k=O,Ol%
Fig. 8. S i m u l a t e d d i s s o l u t i o n c u p s in f o r m a t i o n .
297
C. Dubois et al. ~Journal of Non-Crystalline Solids 171 (1994) 290-298
Table 2 Calculated dissolution cup parameters Specimen Leaching Mean duration radius (months) (Ixm)
Standard deviation (Ixm)
R7D R7E R7F R7G R7H
9.3 14.0 10.5 7.8 63.0
3 4 5 6 12.4
42.3 51.0 60.8 80.5 454.2
surface; for each cup, the radii were calculated for two circles in the orthogonal planes x , z and y , z . The mean radii and standard deviations are indicated in Table 2. The mean cup radius did not vary in a linear manner with the leaching time. This validates the hypothesis of a corrosion rate gradient as a function of the cup depth. Extrapolating the R t c u r v e to the y-intercept (Fig. 6) suggests that the largest initial cracks were approximately 50 ~ m deep. As indicated above, the large cracks control the cup formation phenomenon by creating large cups which progressively envelop the smaller ones. We therefore simulated the corrosion of a cup 50 Ixm deep to check for a correlation with the experimental values. Fig. 9 shows the simulated evolution of the radius of a cup 50 ~m deep (280 pixels on the
screen) assuming a constant corrosion rate of 0.1 ppcs. Three values were simulated for the k coefficient: 100, 50 and 0.01%. A correlation was observed with the mean experimental cup radii when k = 0.01%. Simulations using time-dependent variable corrosion rates did not correlate, regardless of the k value. Measurements have shown that during the first few days the mean dissolution rate of as-cut R7T7 glass is 2.5 times higher than after 28 days of leaching, at which point it becomes virtually constant [4]. The phenomenon therefore does not appear to be a determining factor in long-term glass dissolution; moreover, the cup radii were measured experimentally after 3 months or more of leaching. Based on the correlation observed in Fig. 9, 4500 calculation steps correspond to an actual leaching time of 12.4 months. As noted above, each calculation step corresponds to the simulated dissolution of 0.1 pixel of glass, which is equivalent to a dissolved glass thickness of 0.1 × 50/280 = 0.018 Ixm. The dissolved glass thickness after 12.4 months of simulated leaching is therefore 4500 x 0.018 = 81 Ixm, which corresponds to a simulated daily mean leached glass thickness of 0.22 ixm for the experimental procedures used during this work. These results are consistent with experimental observations [13]. Fig. 10 shows the simulated variation of R t = z 0 -Zmi n for the same k values and with V0
50
k = 100% 500'
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400'
3O 300'
k = t00%
E
2O 200-
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•
i 2
A : Measured
•
i 4
.
r 6
.
i
Leaching , i
8
10
Time •
(Months) , , 12
Values
Fig. 9. Simulated cup bottom radius variation versus leaching time (initial crack depth 50 ~m).
Leaching Time (Months) (I
2 A : Measured
4
6
8
I0
12
VaLues
Fig. 10. Simulated R t variation versus leaching time (initial crack depth 50 ~m).
298
C. Dubois et al. /Journal of Non-Crystalline Solids 171 (1994) 290-298
constant in time. Once again, the satisfactory correlation with k = 0 . 0 1 % confirmed our hypotheses. R t and the cup radii were also simulated using an algorithm allowing for a linear drop in the corrosion rate with the leaching time. As before, the measured radii and R t values did not correlate with the calculated values.
5. Conclusion This investigation confirmed the role of initial cracks in the formation of dissolution cups on the surface of R7T7 glass. It also revealed the dominant effect of the deepest cracks in the evolution of the cups. During this evolution, the large cups grew more quickly than the smaller ones, which were progressively eliminated. The diminishing values of the R a and R t parameters during the leaching period corroborate this evolution. Extrapolating the roughness parameters suggests that the cups will completely 'disappear' after two years of leaching, under the conditions of these experiments. A cup development model based on these observations was qualified by correlation with topographical probe measurements of the increasing cup radius and surface flattening due to the decreasing total profile roughness, R t. This correlation demonstrates that the dissolution rate at the top of the cracks - and subsequently at the top of the cups - may be considered constant at 0.22 txm per day over a 12.4 month period for the
glass composition and experimental leaching procedures used in this investigation.
References [1] B. Grambow, in: Scientific Basis for Nuclear Waste Management VIII, ed. by C. Jantzen, J. Stone and E. Ewing, Mater. Res. Soc. Symp. Proc. 44 (1985) 15. [2] E. Vernaz and J.L. Dussossoy, Appl. Geochem. Special issue No. 1 (1992) 13. [3] J.L. Nogu~s, PhD thesis, Universit6 de Montpellier (1982). [4] J.L. Dussossoy, C. Dubois, E. Vernaz and A. Chambaudet, in: Scientific Basis for Nuclear Waste Management XV, ed. C. Sombret, Mater. Res. Soc. Symp. Proc. 257 (1992) 109. [5] A. Chambaudet, C. Dubois, J.C. Petit and J.C. Dran, in: Proc. Conf. SFEN-RECOD '87, Paris (1987). [6] M. Assoul, C. Brugger, A. Chambaudet, C. Dubois, M. Fromm, N. Jacquet-Francillon, J. Mignot, E. Painchaud and E. Vernaz, poster, CEA/IPSN exhibit, Besan~on, 1990. Ext. Abstracts, Restricted Diffusion. [7] M. Assoul, PhD thesis, Universit6 de Franche-Comt~ (1991). [8] J.B.P. Williamson, in: Proc. NASA Symp. on Interdisciplinary Approaches to Friction and Wear, San Antonio, TX, Nov. 28-30, 1967 (NASA, Washington, DC, 1968). [9] M. Chuard, A.C. Roudot and J. Mignot, Wear 96 (1984) 31. [10] T.K. Peucker and D.H. Douglas, Comp. Graphics Image Proc. 4 (1975) 375. [11] M. Fromm, PhD thesis, Universit6 de Franche-Comt~ (1990). [12] M. Fromm, F. Membrey, A. Chambaudet and R. Saouli° Nucl. Tracks Rad. Meas. 19 (1991) 163. [13] F. Delage and J.L. Dussossoy, in: Scientific Basis for Nuclear Waste Management XIV, ed. C. Sombret, Mater, Res. Soc. Symp. Proc. 212 (1991) 41.