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Edge instabilities in thin plates with spatial variations in thickness
Scripta METALLURGICA et MATERIALIA
Vol. 28, pp. 269-274, 1993 Printed in the U.S.A.
Pergamon Press Ltd. All rights reserved
EDGE INSTABILITIES IN THIN PLATES WITH SPATIAL VARIATIONS IN THICKNESS
R. S. Kedia, T. M. Lillo, Q. Horn, M. R. Plichta and S. A. Hackney Dept. of Metallurgical and Materials Engineering Michigan Technological University Houghton, Michigan 49931 (Received
October
26, 1992)
The morphological stability of two dimensional structures has far-reaching consequences in a variety of materials problems. For example, the time dependent behavior of lamellar composite materials and thin coatings at elevated temperature can be directly related to the morphological integrity of the two dimensional structure. "time dependent behavior can be expected when the two dimensional morphology is not the equilibrium morphology and a thermodynamic driving force exists for the thin layer to undergo a morphological change to a structure with a smaller surface area to volume ratio. In many cases, the change in morphology occurs by a diffusional instability at the edges of two dimensional structures. The edge instabilities of two dimensional structures have been studied in relation to microcrystalline thin film stability [1], thin plate stability [2], intergranular film stability [3], and crack healing in brittle materials [4]. Several studies [1,3,4] have reported that the formation of cylindrical rods perpendicular to the original edge is observed to occur by the growth of a periodic instability in the profile of the edge. It is important to understand this mechanism of shape change as it can control the rate at which the morphological integrity of a two dimensional structure is destroyed. In general, it has been assumed that a uniform edge profile is unstable with respect to perturbations in edge thickness and that the resulting development of the periodic edge instability is associated with the reduction in surface area. The experimental results to date [1,3,4] appear to show that the rod spacing is not a strong function of time when the two dimensional structure has an initial condition of uniform thickness. A steady state analysis assuming time independent rod spacing has been based on this observation [ 1]. However, there have not been any experimental studies focussed on the dynamic stability of this periodic phenomena with respect to spatial variations in the initial thickness or the manner in which the periodicity can be expected to vary with changes in thickness. The purpose of this investigation is to examine the role that film thickness plays in the development of edge instabilities by studying the dynamics of the instability at the edges of thin solid foils exhibiting a wedge shaped cross section using in situ transmission electron microscopy. Such a study allows the examination of the behavior of the fingering instability as the average edge position of the foil moves toward the thicker regions of the foil. Large grain Cu foil (25 micron grain size) was electropolished to perforation using 1 to 1 orthophosphoric acid/ water solution. Electropolishing occurred at 293"K in a Tenupol 3 with an applied voltage of 15 V and a current of 850 mA. The surfaces of the electropolished Cu foils were sputter cleaned for one minute at 5 kV accelerating voltage and a gun tilt of 25 °. The specimen preparation produced a thinned disc with a central perforation. The thickness increased in the radial direction from the edge of the perforation. Although the change in thickness with radial distance varied slightly from point to point in the electron transparent foil, the areas which were studied showed a thickness change/radial distance ratio on the order of 0.04. The thin film specimens of Cu were heated in the JEOL 4000 using a double tilt heating stage to a thermocouple reading of 888°K. Thickness measurements in the Cu were done using the convergent beam electron diffraction technique detailed in reference [6] with the TEM operating at 300 kV. The required two beam conditions were established using the closest low index two beam condition to the tilt which provided the largest projected instability spacing over the two axes of tilt. As suggested in reference [6], a check on the operative extinction distance was performed by recording three separate thickness measurements at widely differing thicknesses using, as near as possible, the same diffracting conditions. 269 0956-716X/93 $6.00 + .00 Copyright (c) 1993 Pergamon Press Ltd.
270
EDGE
INSTABILITIES
Vol.
28, No.
Experimental Results and Discussion The formation of the fingering instability is observed to occur when there is a periodic spatial variation in the velocity of the receding edge, leading to a periodic variation in edge position. The regions of the edge profile which extend past the average edge position because of locally reduced edge recession velocity will be referred to as edge protrusions. Edge intrusions will occur in the edge profile where the edge recession velocity is relatively large compared with the average edge velocity. The edge protrusions which develop because of the periodic variation in edge recession velocity are observed to undergo shape coarsening into cylindrical rods which then undergo spheroidization, as discussed elsewhere [5]. The spacing of the rods is observed to increase as the edge recedes into the thicker areas of the electropolished foils. An example of this type of behavior is shown in Figure 1. Figure la shows the edge instability in the initial stages of development but where a characteristic wavelength is clearly evident, while Figure lb is a micrograph taken from the same region after 30 minutes at temperature. A comparison between Figures la and b reveals a dramatic adjustment in rod spacing occurs as the edge recedes into the thicker areas of the wedge shaped foil. It is apparent from quantitative experimental measurement (Figure 3) and qualitative observation that the rod spacing increases as the receding edge encounters the thicker cross sections on the foil. One prevalent mechanism which is observed to allow an increase in spacing of the instability wavelength is what may be termed a "Y" fault. Examples of this dynamic spacing change mechanism are shown in Figure 2. The "Y" fault develops when one of the intrusions of the edge lags slightly behind the neighboring intrusions. Based on the anomalous absorption contrast, it appears that the diffusion from the tips of the neighboring intrusions causes a buildup of mass at the tip of the lagging intrusion, further reducing the velocity (Figure 2a). The neighboring intrusions grow across the front of the lagging intrusion, resulting in the increase in the effective wavelength. One result of this process is that two of the instability rods are forced to coalesce as the lagging intrusion is forced to stop and the two neighboring intrusions grow across the front of the lagging intrusion. When this event occurs, a characteristic "Y" shape is observed in the rod morphology, as detailed in Figures 2b and c. It is interesting to note the similarity between this phenomenon and spacing adjustments which occur in lamellar eutectic and eutectoid growth processes, producing the lamellar fault [7]. One view of the experimentally observed fingering instability is that it is driven by a reduction in surface area. This argument is certainly supported by the observation that the rod spacing adjusts to a larger value as the edge recedes into the thicker regions of the foil. A test of this hypothesis can be carried out by measuring how the spacing of the rods varies with thickness of the foil area into which the edge is receding. If the spacing of the rods is such that the surface area has been increased during the shape evolution, then an additional driving force must also be involved in the development of the instability. The thickness measurements determined in these experiments were made in regions where it was observed that the edge velocity was nonzero. The convergent beam electron diffraction technique was applied in the manner suggested in reference [6], where it was found to be necessary to obtain three separate thickness measurements with the same dynamical diffracting conditions in order to determine the best value for the acting extinction distance. One of the thickness measurements was taken just beyond (-8 x 10-7 m) the blunted edge at the base of a rod where the thickness had not been affected by mass diffusion, while the other two required measurements were carried out in thicker sections of the same grain. The experimentally determined thickness at the point adjacent to the blunted edge is plotted against the experimentally measured finger spacing in Figure 3. Also in Figure 3 is a line which approximates the thermodynamic limit for the minimum rod spacing as a function of thickness based on the hypothesis that reduction in surface area is the dominant driving force for the development of the instability. This thermodynamic limit is determined by considering the spacing of uniform cylindrical rods of length w created from a unit length of an infinite strip of thin foil of thickness t and width w, with surface area and volume being constant. Conservation of volume is given by l (tw) = n ~ RZ w (1) where R is the rod radius, ! is a unit length, and n is the number of rods. Conservation of surface area is given by I (2t +2w) = 2n rc (Rw +R 2)
(2)
From these two equations it is apparent that the thermodynamic limit on the minimum rod radius is t and the thermodynamic limit on the minimum rod spacing is ~ × t. The experimental results presented in Figure 3 show that the experimentally observed spacings exceed the thermodynamic minimum by approximately a factor of two. This result is in excellent agreement with surface area reduction hypothesis as it is expected that the experimentally observed spacing should be somewhat higher than the thermodynamic minimum because the edge velocities are nonzero. This indicates that some of the driving force associated with reduction of surface area is being used to drive the diffusional process related to the nonzero edge velocity. The experimental observation that the rod spacing is
3
Vol.
28, No. 3
EDGE INSTABILITIES
271
continually readjusted to approach a factor of two above the thermodynamic minimum suggests that a special dynamic stability is associated with this particular wavelength. This special stability would appear to be imposed through the formation of the "Y" fault which allows rapid adjustment of the edge morphology to the optimum value of the rod spacing. This contention is supported by an in situ heating experiment on a single crystal film of silver formed by vapor deposition and supported by a silver grid. The thickness of this film is relatively uniform and does not exhibit the wedge shape in cross section. The spacing/thickness ratio at the morphologically unstable edge is still observed to be between 1.5 and 2 times the thermodynamic limit. Although this is the only published study to attempt to correlate the instability wavelength to the plate thickness, a semi-quantitative comparison is possible between the magnitude of the spacing/thickness ratio observed here and an estimate of that observed for the microcrystalline films on bulk substrates studied in [1]. Such a comparison between the edge behavior of a free standing, single crystal foil and a microcrystalline film on a bulk substrate is of interest because it may reveal the effects of small grain size and contact of the film with the bulk substrate on the overall behavior of the edge instability. By examining figure 2a in reference [1], it is apparent that the rod spacing is on the order of 2 × 10-6 m for a reported film thickness 3 x 10-8 m. This is about an order of magnitude difference between the two studies. Two major differences between the study in [1] and this study which might account for this significant disparity in experimental result are the contact of the film with the bulk substrate and the microcrystalline nature of the films studied in [I]. The authors of [1] model the rnicrocrystalline film as a continuum and imply a contact angle between the continuum film and the substrate of 90°. The effect of the contact angle on the thermodynamics of the edge instability may be analyzed by considering the minimum for the rod spacing when the microcrystalline film is approximated as a continuum with a contact angle of 90°. The constraints to be applied are conservation of volume and conservation of surface/interface energy. The conservation of volume is
l (tw) -
nTtR2w 2
(3)
while the conservation of energy is
Ix (wl + 2tl) + ~ (wl) = Ix ( n ~ R w + n~R 2) + ~ ( (2nRw) + ( l w - 2Rwn) )
(4)
where Ix and 13are the surface energy of the film and the substrate/film interracial energy, respectively. From these two equations, it can be determined that the thermodynamic minimum for the rod spacing is given by - = 2~t. n
At most, a 90° contact angle will double the minimum possible spacing for rod formation in a continuum film relative to that in a free standing foil of the same thickness. The consideration of this contact angle would appear to account for only part of the discrepancy between the behavior of the single crystal foil edges and the microcrystalline film edges. The continuum approximation for microcrystalline films does appear to give some insight into the overall behavior of the microcrystalline film edge, however, a precise treatment of microcrystalline film behavior will likely necessitate a detailed consideration of factors associated with small grain size, such as local surface curvature associated with individual grains. Conclusion The techniques of in situ transmission electron microscopy have been applied to a fundamental phenomenon of morphological stability of the edges of two dimensional structures. A model system involving free standing, electron transparent foils heated in a TEM hot stage has been used to study the details of the morphological changes associated with the edge instability. The hypothesis that the driving force for the development of the edge instability is associated with a reduction in surface area has been examined by studying the relationship between the thickness of the foil and the wavelength of the edge instability. It has been found that the rod spacing is approximately a factor of two greater than the thermodynamic minimum predicted by conservation of surface area. The edge instability was studied in wedge shaped foils where the receding edge encountered a progressively increasing cross sectional thickness. It has been found that the rod spacing associated with the edge instability will continuously adjust to a larger value as the edge of the foil recedes to the thicker sections of the foil. The time dependent wavelength of the instability observed in this study is quite different from the steady state behavior observed in previous studies. This difference may be attributed to the differences in geometry of the two dimensional structure. Previous studies have considered two dimensional structures of uniform cross section where the driving force for surface area reduction was evidently independent of time. The wedge shaped cross section of the free standing plates studied in the experiments presented here will decrease the driving force for spheroidization, and thus rod formation, as the edge recedes into thicker areas of the plate. The mechanism by which this dynamic spacing
272
EDGE
INSTABILITIES
Vol.
28, No.
change occurred was directly observed using ih situ TEM. As the edge intrusions move into progressively thicker regions of the foils, the array of intrusions became unstable with respect to variations in velocity of adjacent intrusions. An inlrusion which lagged behind the neighboring intrusions was forced to zero velocity because of an increase in blunting at the tip of the lagging intrusion associated with the mass flow from the tips of the two neighboring intrusions. This dynamic behavior of the edge as increasing cross sectional thickness was encountered allowed continuous adjustn~ent in the spacing of the rods along the edge so that the periodicity was always approximately a factor of 2 above the thermodynamic minimum. Such observations have not been detailed in previous works. A comparison between the morphology of single crystal, free standing edges studied in this work and that observed for microcrystalline films on bulk substrates in other studies showed significant disparity in the experimental results on rod spacing/thickness ratio. Part of this discrepancy could be accounted for by consideration of the contact angle between the film and the bulk substrate. Acknowledgments
1. 2. 3. 4. 5. 6. 7.
This work was supported by the National Center for Electron Microscopy and by DOE (DE-FG02-87ER45315). References E. Jiran and C.V. Thompson, J. of Electronic Materials 19 (1990) 1153. 1". H. Courmey and J. C. Malzahn Kampe, Acta Met. 37 (1989) 1747. F. E Lange and D. R. Clarke, J. American Ceramic Soc. 65 (1982) 502. J. Rodel and A. M. Glaeser, J. American Ceramic Soc. 73 (1990) 592. S.A. Hackney, Scripta Met. 25 (1991) 799. R.C. Ecob, Scripta Met. 20 (1986) 1001. K. Jackson and J. Hunt, Metall. Trans. 2A (1972) 345. Figure 1 (a) Periodic array of edge intrusions in the thin area of the wedge-shaped foil near the original edge. Co)Tbe same region as that shown in l(a) after the edge has receded into the thicker area of the wedgeshaped foil.
3
Vol.
28, No. 3
EDGE INSTABILITIES
273
Figure 2 (a) The initiation of the "Y" fault. Examination of the absorption contrast implies that the curvature gradient in front of the lagging intrusion is reduced by mass flow from the neighboring intrusion tips. This further reduces the velocity of the lagging inu-asion, allowing the neighboring intrusions to grow across the front of the lagging intrusion. Shape coarsening in this region results in the increase in spacing of the intrusions and the formation of a characteristic "Y" configuration of the rods due to the termination of the lagging intrusion, as shown in figure 2(b). (c) A double "Y" fault associated with an instability wavelength change by the termination of adjacent intrusions
bd
274
EDGE INSTABILITIES
3.50 ~O"
I
I
Vol.
l
I
3.00 2.50
•
~I~Emperical data [
.-
l"y~*x
I
tO 2.00 ~
1.50
~©
1.00
.9 ~
o.5o
_
0.00 0.00
°
0.10
0.20
_
0.30
-6 0.40
0.50
Sample thickness (m*lO) Figure 3
Experimental data on the spacing of the rods as a function of local foil thickness. The thermodynamic minimum predicted by conservation of surface area is also shown for comparison.
28, No. 3
Vol. 28, pp. 269-274, 1993 Printed in the U.S.A.
Pergamon Press Ltd. All rights reserved
EDGE INSTABILITIES IN THIN PLATES WITH SPATIAL VARIATIONS IN THICKNESS
R. S. Kedia, T. M. Lillo, Q. Horn, M. R. Plichta and S. A. Hackney Dept. of Metallurgical and Materials Engineering Michigan Technological University Houghton, Michigan 49931 (Received
October
26, 1992)
The morphological stability of two dimensional structures has far-reaching consequences in a variety of materials problems. For example, the time dependent behavior of lamellar composite materials and thin coatings at elevated temperature can be directly related to the morphological integrity of the two dimensional structure. "time dependent behavior can be expected when the two dimensional morphology is not the equilibrium morphology and a thermodynamic driving force exists for the thin layer to undergo a morphological change to a structure with a smaller surface area to volume ratio. In many cases, the change in morphology occurs by a diffusional instability at the edges of two dimensional structures. The edge instabilities of two dimensional structures have been studied in relation to microcrystalline thin film stability [1], thin plate stability [2], intergranular film stability [3], and crack healing in brittle materials [4]. Several studies [1,3,4] have reported that the formation of cylindrical rods perpendicular to the original edge is observed to occur by the growth of a periodic instability in the profile of the edge. It is important to understand this mechanism of shape change as it can control the rate at which the morphological integrity of a two dimensional structure is destroyed. In general, it has been assumed that a uniform edge profile is unstable with respect to perturbations in edge thickness and that the resulting development of the periodic edge instability is associated with the reduction in surface area. The experimental results to date [1,3,4] appear to show that the rod spacing is not a strong function of time when the two dimensional structure has an initial condition of uniform thickness. A steady state analysis assuming time independent rod spacing has been based on this observation [ 1]. However, there have not been any experimental studies focussed on the dynamic stability of this periodic phenomena with respect to spatial variations in the initial thickness or the manner in which the periodicity can be expected to vary with changes in thickness. The purpose of this investigation is to examine the role that film thickness plays in the development of edge instabilities by studying the dynamics of the instability at the edges of thin solid foils exhibiting a wedge shaped cross section using in situ transmission electron microscopy. Such a study allows the examination of the behavior of the fingering instability as the average edge position of the foil moves toward the thicker regions of the foil. Large grain Cu foil (25 micron grain size) was electropolished to perforation using 1 to 1 orthophosphoric acid/ water solution. Electropolishing occurred at 293"K in a Tenupol 3 with an applied voltage of 15 V and a current of 850 mA. The surfaces of the electropolished Cu foils were sputter cleaned for one minute at 5 kV accelerating voltage and a gun tilt of 25 °. The specimen preparation produced a thinned disc with a central perforation. The thickness increased in the radial direction from the edge of the perforation. Although the change in thickness with radial distance varied slightly from point to point in the electron transparent foil, the areas which were studied showed a thickness change/radial distance ratio on the order of 0.04. The thin film specimens of Cu were heated in the JEOL 4000 using a double tilt heating stage to a thermocouple reading of 888°K. Thickness measurements in the Cu were done using the convergent beam electron diffraction technique detailed in reference [6] with the TEM operating at 300 kV. The required two beam conditions were established using the closest low index two beam condition to the tilt which provided the largest projected instability spacing over the two axes of tilt. As suggested in reference [6], a check on the operative extinction distance was performed by recording three separate thickness measurements at widely differing thicknesses using, as near as possible, the same diffracting conditions. 269 0956-716X/93 $6.00 + .00 Copyright (c) 1993 Pergamon Press Ltd.
270
EDGE
INSTABILITIES
Vol.
28, No.
Experimental Results and Discussion The formation of the fingering instability is observed to occur when there is a periodic spatial variation in the velocity of the receding edge, leading to a periodic variation in edge position. The regions of the edge profile which extend past the average edge position because of locally reduced edge recession velocity will be referred to as edge protrusions. Edge intrusions will occur in the edge profile where the edge recession velocity is relatively large compared with the average edge velocity. The edge protrusions which develop because of the periodic variation in edge recession velocity are observed to undergo shape coarsening into cylindrical rods which then undergo spheroidization, as discussed elsewhere [5]. The spacing of the rods is observed to increase as the edge recedes into the thicker areas of the electropolished foils. An example of this type of behavior is shown in Figure 1. Figure la shows the edge instability in the initial stages of development but where a characteristic wavelength is clearly evident, while Figure lb is a micrograph taken from the same region after 30 minutes at temperature. A comparison between Figures la and b reveals a dramatic adjustment in rod spacing occurs as the edge recedes into the thicker areas of the wedge shaped foil. It is apparent from quantitative experimental measurement (Figure 3) and qualitative observation that the rod spacing increases as the receding edge encounters the thicker cross sections on the foil. One prevalent mechanism which is observed to allow an increase in spacing of the instability wavelength is what may be termed a "Y" fault. Examples of this dynamic spacing change mechanism are shown in Figure 2. The "Y" fault develops when one of the intrusions of the edge lags slightly behind the neighboring intrusions. Based on the anomalous absorption contrast, it appears that the diffusion from the tips of the neighboring intrusions causes a buildup of mass at the tip of the lagging intrusion, further reducing the velocity (Figure 2a). The neighboring intrusions grow across the front of the lagging intrusion, resulting in the increase in the effective wavelength. One result of this process is that two of the instability rods are forced to coalesce as the lagging intrusion is forced to stop and the two neighboring intrusions grow across the front of the lagging intrusion. When this event occurs, a characteristic "Y" shape is observed in the rod morphology, as detailed in Figures 2b and c. It is interesting to note the similarity between this phenomenon and spacing adjustments which occur in lamellar eutectic and eutectoid growth processes, producing the lamellar fault [7]. One view of the experimentally observed fingering instability is that it is driven by a reduction in surface area. This argument is certainly supported by the observation that the rod spacing adjusts to a larger value as the edge recedes into the thicker regions of the foil. A test of this hypothesis can be carried out by measuring how the spacing of the rods varies with thickness of the foil area into which the edge is receding. If the spacing of the rods is such that the surface area has been increased during the shape evolution, then an additional driving force must also be involved in the development of the instability. The thickness measurements determined in these experiments were made in regions where it was observed that the edge velocity was nonzero. The convergent beam electron diffraction technique was applied in the manner suggested in reference [6], where it was found to be necessary to obtain three separate thickness measurements with the same dynamical diffracting conditions in order to determine the best value for the acting extinction distance. One of the thickness measurements was taken just beyond (-8 x 10-7 m) the blunted edge at the base of a rod where the thickness had not been affected by mass diffusion, while the other two required measurements were carried out in thicker sections of the same grain. The experimentally determined thickness at the point adjacent to the blunted edge is plotted against the experimentally measured finger spacing in Figure 3. Also in Figure 3 is a line which approximates the thermodynamic limit for the minimum rod spacing as a function of thickness based on the hypothesis that reduction in surface area is the dominant driving force for the development of the instability. This thermodynamic limit is determined by considering the spacing of uniform cylindrical rods of length w created from a unit length of an infinite strip of thin foil of thickness t and width w, with surface area and volume being constant. Conservation of volume is given by l (tw) = n ~ RZ w (1) where R is the rod radius, ! is a unit length, and n is the number of rods. Conservation of surface area is given by I (2t +2w) = 2n rc (Rw +R 2)
(2)
From these two equations it is apparent that the thermodynamic limit on the minimum rod radius is t and the thermodynamic limit on the minimum rod spacing is ~ × t. The experimental results presented in Figure 3 show that the experimentally observed spacings exceed the thermodynamic minimum by approximately a factor of two. This result is in excellent agreement with surface area reduction hypothesis as it is expected that the experimentally observed spacing should be somewhat higher than the thermodynamic minimum because the edge velocities are nonzero. This indicates that some of the driving force associated with reduction of surface area is being used to drive the diffusional process related to the nonzero edge velocity. The experimental observation that the rod spacing is
3
Vol.
28, No. 3
EDGE INSTABILITIES
271
continually readjusted to approach a factor of two above the thermodynamic minimum suggests that a special dynamic stability is associated with this particular wavelength. This special stability would appear to be imposed through the formation of the "Y" fault which allows rapid adjustment of the edge morphology to the optimum value of the rod spacing. This contention is supported by an in situ heating experiment on a single crystal film of silver formed by vapor deposition and supported by a silver grid. The thickness of this film is relatively uniform and does not exhibit the wedge shape in cross section. The spacing/thickness ratio at the morphologically unstable edge is still observed to be between 1.5 and 2 times the thermodynamic limit. Although this is the only published study to attempt to correlate the instability wavelength to the plate thickness, a semi-quantitative comparison is possible between the magnitude of the spacing/thickness ratio observed here and an estimate of that observed for the microcrystalline films on bulk substrates studied in [1]. Such a comparison between the edge behavior of a free standing, single crystal foil and a microcrystalline film on a bulk substrate is of interest because it may reveal the effects of small grain size and contact of the film with the bulk substrate on the overall behavior of the edge instability. By examining figure 2a in reference [1], it is apparent that the rod spacing is on the order of 2 × 10-6 m for a reported film thickness 3 x 10-8 m. This is about an order of magnitude difference between the two studies. Two major differences between the study in [1] and this study which might account for this significant disparity in experimental result are the contact of the film with the bulk substrate and the microcrystalline nature of the films studied in [I]. The authors of [1] model the rnicrocrystalline film as a continuum and imply a contact angle between the continuum film and the substrate of 90°. The effect of the contact angle on the thermodynamics of the edge instability may be analyzed by considering the minimum for the rod spacing when the microcrystalline film is approximated as a continuum with a contact angle of 90°. The constraints to be applied are conservation of volume and conservation of surface/interface energy. The conservation of volume is
l (tw) -
nTtR2w 2
(3)
while the conservation of energy is
Ix (wl + 2tl) + ~ (wl) = Ix ( n ~ R w + n~R 2) + ~ ( (2nRw) + ( l w - 2Rwn) )
(4)
where Ix and 13are the surface energy of the film and the substrate/film interracial energy, respectively. From these two equations, it can be determined that the thermodynamic minimum for the rod spacing is given by - = 2~t. n
At most, a 90° contact angle will double the minimum possible spacing for rod formation in a continuum film relative to that in a free standing foil of the same thickness. The consideration of this contact angle would appear to account for only part of the discrepancy between the behavior of the single crystal foil edges and the microcrystalline film edges. The continuum approximation for microcrystalline films does appear to give some insight into the overall behavior of the microcrystalline film edge, however, a precise treatment of microcrystalline film behavior will likely necessitate a detailed consideration of factors associated with small grain size, such as local surface curvature associated with individual grains. Conclusion The techniques of in situ transmission electron microscopy have been applied to a fundamental phenomenon of morphological stability of the edges of two dimensional structures. A model system involving free standing, electron transparent foils heated in a TEM hot stage has been used to study the details of the morphological changes associated with the edge instability. The hypothesis that the driving force for the development of the edge instability is associated with a reduction in surface area has been examined by studying the relationship between the thickness of the foil and the wavelength of the edge instability. It has been found that the rod spacing is approximately a factor of two greater than the thermodynamic minimum predicted by conservation of surface area. The edge instability was studied in wedge shaped foils where the receding edge encountered a progressively increasing cross sectional thickness. It has been found that the rod spacing associated with the edge instability will continuously adjust to a larger value as the edge of the foil recedes to the thicker sections of the foil. The time dependent wavelength of the instability observed in this study is quite different from the steady state behavior observed in previous studies. This difference may be attributed to the differences in geometry of the two dimensional structure. Previous studies have considered two dimensional structures of uniform cross section where the driving force for surface area reduction was evidently independent of time. The wedge shaped cross section of the free standing plates studied in the experiments presented here will decrease the driving force for spheroidization, and thus rod formation, as the edge recedes into thicker areas of the plate. The mechanism by which this dynamic spacing
272
EDGE
INSTABILITIES
Vol.
28, No.
change occurred was directly observed using ih situ TEM. As the edge intrusions move into progressively thicker regions of the foils, the array of intrusions became unstable with respect to variations in velocity of adjacent intrusions. An inlrusion which lagged behind the neighboring intrusions was forced to zero velocity because of an increase in blunting at the tip of the lagging intrusion associated with the mass flow from the tips of the two neighboring intrusions. This dynamic behavior of the edge as increasing cross sectional thickness was encountered allowed continuous adjustn~ent in the spacing of the rods along the edge so that the periodicity was always approximately a factor of 2 above the thermodynamic minimum. Such observations have not been detailed in previous works. A comparison between the morphology of single crystal, free standing edges studied in this work and that observed for microcrystalline films on bulk substrates in other studies showed significant disparity in the experimental results on rod spacing/thickness ratio. Part of this discrepancy could be accounted for by consideration of the contact angle between the film and the bulk substrate. Acknowledgments
1. 2. 3. 4. 5. 6. 7.
This work was supported by the National Center for Electron Microscopy and by DOE (DE-FG02-87ER45315). References E. Jiran and C.V. Thompson, J. of Electronic Materials 19 (1990) 1153. 1". H. Courmey and J. C. Malzahn Kampe, Acta Met. 37 (1989) 1747. F. E Lange and D. R. Clarke, J. American Ceramic Soc. 65 (1982) 502. J. Rodel and A. M. Glaeser, J. American Ceramic Soc. 73 (1990) 592. S.A. Hackney, Scripta Met. 25 (1991) 799. R.C. Ecob, Scripta Met. 20 (1986) 1001. K. Jackson and J. Hunt, Metall. Trans. 2A (1972) 345. Figure 1 (a) Periodic array of edge intrusions in the thin area of the wedge-shaped foil near the original edge. Co)Tbe same region as that shown in l(a) after the edge has receded into the thicker area of the wedgeshaped foil.
3
Vol.
28, No. 3
EDGE INSTABILITIES
273
Figure 2 (a) The initiation of the "Y" fault. Examination of the absorption contrast implies that the curvature gradient in front of the lagging intrusion is reduced by mass flow from the neighboring intrusion tips. This further reduces the velocity of the lagging inu-asion, allowing the neighboring intrusions to grow across the front of the lagging intrusion. Shape coarsening in this region results in the increase in spacing of the intrusions and the formation of a characteristic "Y" configuration of the rods due to the termination of the lagging intrusion, as shown in figure 2(b). (c) A double "Y" fault associated with an instability wavelength change by the termination of adjacent intrusions
bd
274
EDGE INSTABILITIES
3.50 ~O"
I
I
Vol.
l
I
3.00 2.50
•
~I~Emperical data [
.-
l"y~*x
I
tO 2.00 ~
1.50
~©
1.00
.9 ~
o.5o
_
0.00 0.00
°
0.10
0.20
_
0.30
-6 0.40
0.50
Sample thickness (m*lO) Figure 3
Experimental data on the spacing of the rods as a function of local foil thickness. The thermodynamic minimum predicted by conservation of surface area is also shown for comparison.
28, No. 3