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Critical radius for interface separation of a compressively stressed film from a rough surface
PII:
Acta mater. Vol. 47, No. 6, pp. 1749±1756, 1999 # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(99)00078-6 1359-6454/99 $20.00 + 0.00
CRITICAL RADIUS FOR INTERFACE SEPARATION OF A COMPRESSIVELY STRESSED FILM FROM A ROUGH SURFACE D. R. CLARKE1{ and W. POMPE2 Materials Department, College of Engineering, University of California, Santa Barbara, CA 931065050, U.S.A. and 2Technical University of Dresden, Dresden, Germany
1
(Received 19 November 1998; accepted 12 February 1999) AbstractÐSurfaces are often roughened so that coatings under residual compression will remain intact and adherent. By representing surface roughness as a sinusoidal surface and comparing the elastic strain and surface energies of a compressively stressed ®lm, it is shown that there exists a critical radius of curvature below which interface separation is energetically favored. As with many other residual stress problems in thin ®lms, separation depends on a dimensionless group, in this case (R2 Eÿ=h3 b2 ) where R is the local radius of curvature, G the fracture resistance of the interface and E, h and sb are the ®lm modulus, ®lm thickness and residual stress, respectively. # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
The mechanics of failure of thin, highly stressed ®lms on ¯at surfaces has been well developed in the last decade and quantitative relations have been established between ®lm stress, ®lm thickness and interface fracture resistance for a variety of failure modes [1±4]. However, there are many practical situations in which a stressed ®lm is formed on a rough rather than on a ¯at surface. Indeed, it is common practice in the manufacture and application of a wide variety of coatings to roughen the surface prior to coating in order to increase the ®lm adhesion, particularly for ®lms under compression. The reason for the increased resistance to ®lm decohesion is not fully understood but is believed to be associated with the fact that the roughness causes a crack propagating along the interface to propagate under mixed mode conditions thereby increasing the energy dissipated during ®lm decohesion. Roughness, however, also causes local stress concentrations to arise when the ®lm is under compression, for instance as a result of thermal expansion mismatch with the underlying substrate. Such stress concentrations can result in local tensile forces [5] across the interface between the ®lm and the substrate, as described below, and lead to local separations that may, under prescribed conditions, grow to cause decohesion. In this work we introduce the notion that there exists a critical local curvature for local separation of a compressively stressed ®lm from a surface. The critical curvature is related to the ®lm thickness and ®lm stress and is {To whom all correspondence should be addressed.
estimated in the following using a fracture mechanics approach based on the energies associated with extension of an interface crack (separation). The analysis described in this work is motivated by our recent observations of localized separation of aluminum oxide ®lms, formed by high-temperature oxidation of FeCrAl, from the crests of a wrinkled surface [6]. An example is shown in the optical micrograph of Fig. 1(a) in which the local separation is visible as the white patches on account of their enhanced back-scattering. In this particular alloy, an originally ¯at, smooth surface wrinkles during oxidation with a wavelength and amplitude that depends on oxidation time and temperature [6, 7]. Four pertinent observations made were that (a) the separations only occurred after a certain oxidation time, (b) they spontaneously ``popped'' in on cooling and then arrested, (c) the size of the separations was larger at longer oxidation times, and (d) there was little stress relaxation associated with the separation. The last observation was made by piezospectroscopy using an optical probe method and is illustrated by the series of measurements shown in Fig. 2 recorded from the larger regions shown in Fig. 1. The measurements were made using a probe with a diameter of 03 mm and were taken from the oxide that had separated at the crests of the wrinkles as well as from the crests and valleys where no separation had occurred. Substantial variation in the frequency shift from one crest to another is evident but nevertheless the frequency shift from the large regions of separated oxide is about 10% lower. This corresponds to a reduction in elastic strain energy of about 15% in the oxide [8].
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CLARKE and POMPE: INTERFACE SEPARATION OF A FILM
Fig. 1. (a) Optical micrograph of local separation of the alumina scale formed on an FeCrAl alloy at the crests of the surface wrinkles after 100 h oxidation at 10008C. (b) Scanning electron micrograph of the same sample as in (a) illustrating the three-dimensionality of the surface wrinkling. Atomic force microscopy indicates that the average wavelength of the wrinkling is 10 mm.
It is emphasized that these local separations between the oxide and the underlying metal are quite distinct from the buckling mode of failure that occurs in compressively stressed ®lms. Buckling requires a ®nite region of debonding between the ®lm and substrate, a region much larger than the separation shown in Fig. 1, and is associated with uplift of the ®lm from the surface [1]. The separ-
ations shown in Fig. 1 and described in detail in Ref. [6] appear to be associated with elastic debonding of the alloy away from the oxide ®lm on cooling with little or no detectable vertical displacement of the ®lm away from the surface. (Buckling is also seen but is occasional and does not occur over the entire surface as the wrinkling induced ®lm separation does. It is considered to be associated with
CLARKE and POMPE: INTERFACE SEPARATION OF A FILM
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Fig. 2. A series of measurements, using luminescence piezospectroscopy, taken at the crests and valleys of a wrinkled scale, such as shown in Fig. 1. The results are presented in terms of the shift in frequency, from the unstressed state, which is proportional to the local elastic strain energy in the oxide. Despite substantial variations from place to place, there is a small decrease in frequency shift from the oxide that has separated at the crests of the wrinkles. The data are from a scale formed on an FeCrAl alloy after 100 h oxidation using a 3 mm optical probe.
local impurities on the surface and so is an artifact when it occurs at early oxidation times.) From quantitative measurements of the surface wrinkling, made by atomic force microscopy [6, 7], the wrinkling in Fig. 1(b) is characterized by an average wavelength of 10 mm and average roughness amplitude of 0.2 mm. 2. APPROXIMATE ANALYSIS
We idealize surface roughness by a sinsuoidal variation in height and consider that the ®lm forms, under compression, on this surface. On a ¯at surface, the compressive stress would be biaxial and have a value of sb. However, on the sinusoidal surface, the height variation causes a geometrical transformation so that at the crests the interface is under tension and at the valleys the interface is under compression [5, 8]. This state of stress is shown schematically, in one dimension, in Fig. 3. Finite element computations [8] indicate that, to a very good approximation, the maximum values of the tensile (and compressive) stresses on an axisymmetric hemispherical surface are given by a soap®lm (Laplace equation) analogy, namely the maximum stress normal to the interface, sn, is sn
2sb h R
1
where R is the local radius of curvature. (This approximation breaks down when the ®lm thickness is
similar to the roughness amplitude [8].) In moving along the interface from the crests to the valleys, the normal stress decreases in magnitude, changes sign, and then increases again until the valley is reached. This normal stress distribution would suggest that if an interface crack were to form, it would preferentially nucleate at the crests but could not propagate far until entering a region in which the stress on the interface is compressive and arresting. In the following we calculate the crack driving force acting on a crack nucleated at a hemispherical-shaped crest and compare it with the crack resistance associated with the formation of the free surfaces. The calculation is an approximate one since we are assuming that the interface stresses dominate the energy release rate during extension of the separation crack and so are not including any relaxation of the in-plane residual stress. Prior to any local separation of the ®lm from the underlying substrate, elastic strain energy is stored in both the ®lm and the substrate material in the hemispherical cap region. Although the stress, and hence the strain energy density, varies throughout the ®lm, we make two assumptions in calculating the elastic strain energy available for driving a separation. The ®rst is that the elastic strain energy in the vicinity of a single crest is proportional to the product of the square of maximum normal stress and the volume of strained material in the cap region. The second is that the region in the vicinity of the crest can be represented as a hemispherical
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CLARKE and POMPE: INTERFACE SEPARATION OF A FILM
Fig. 3. State of residual stress across the interface as a result of the ®lm being under compression. The local tensile stress across the interface at the crests of the sinusoidal wrinkle motivates local separation (lower ®gure).
cap or shell with radius R and thickness h. Using these assumptions, the elastic strain energy that motivates separation can then be written as Estrain BR2 h
s2n E
2
where B is a proportionality constant which takes into account both the diering elastic constants of the ®lm and metal as well as a weak additional dependence on the h/R ratio. (In the limit of hÿ ÿ40 ÿ the elastic energy goes to zero and in the limit of very large R, a ¯at surface, the elastic energy, per unit area, is proportional to the ®lm thickness, h.) When a separation crack forms at the crest, the material in the hemispherical cap relaxes principally in the z-direction [Fig. 3(b)]. Only a portion of the total elastic strain energy is relieved when the separation occurs, as shown in Fig. 2, largely because the stress in the plane of the ®lm is essentially unrelieved. (Even at a much later stage, when buckling occurs, the in-plane stress is not substantially relieved.) To establish whether the separation crack can form and extend, we consider the energetics of crack extension in which the elastic strain energy associated with the normal component of the interface stress in the cap region is converted to surface energy during separation. This requires a functional
form for the elastic strain energy in the ®lm and underlying metal in terms of the crack length. As far as the authors are aware this has not been computed or analyzed previously. In the absence of such a function, we assume that it is a continuous function and conclude from dimensional analysis that it is a function of the two characteristic length ratios, f c=h,c=R. The elastic strain energy will then vary with crack length as s2 c c Estrain BR2 h n f , : 3 h R E In establishing a form for the function, we are guided by the behavior in two asymptotic limits, one at small length scales where the dominant dependence should be given by the length scale c/h and the other at a length scale of c/R. The ®rst limit corresponds to a very small crack, smaller than the ®lm thickness. In this limit, the crack is, in essence in bulk material and the strain energy released will scale as c3. The other limit corresponds to when the crack size is large compared to the ®lm thickness. For a ¯at interface, the energy release scales as c2. Furthermore, due to the variation of the residual stress along the curved interface, it must have a similar form to the normal stress variation across the ®lm/metal interface varying con-
CLARKE and POMPE: INTERFACE SEPARATION OF A FILM
1753
tinuously from a maximum positive value at the crest through zero and to a maximum negative value at the valley. [The negative portions of the function will not be used in the calculation. The cosine function is obtained from the angular variation in normal stress along the interface. This can be constructed, in two dimensions, by representing the ®lm in the crest and valley by two superimposed shells (Fig. 4) each having an inner radius R. One consists of a circular core of metal with an outer shell of ®lm under stress and the other a circular hole lined with a shell of ®lm thickness h inside a continuum.] A continuous function that embodies both length scales and also satis®es the anticipated behavior in both limits is c f c,h,R 1 ÿ 1 ÿ eÿc=h sin2 : 4 R
sociated with incremental extension of crack area, A, exceeds the fracture resistance, R, namely
The strain energy relieved by the crack extension is assumed spent in forming new surfaces over the hemispherical cap: c Esurface 4pR2 1 ÿ cos G 5 R
For large crack sizes, the inequality approaches
where G is the interface fracture energy in J/m2. (For small crack size, this scales with c2 as is expected for a penny-shaped crack.) Combining the two energies and using equation (1) to relate the normal interface stress to the biaxial stress, the total energy of the system as a function of crack length can be written as s2 c ET 4Bh3 b 1 ÿ 1 ÿ eÿc=h sin2 R E c 4pR2 1 ÿ cos G: R Thermodynamically, fracture can be considered to occur when the strain energy release rate, G, as-
GrR i:e:
@ Estrain Esurface R0: @A
6
Evaluating this inequality, using the relationship dA 2pc dc and re-arranging pR2 EG c c ÿc=h ÿc=h R e sin : 7 R 1 ÿ e cos R 2h R 2Bh3 s2b Although the value of the proportionality constant, B, is not known it is expected to be of the order of unity. In the limit of the crack length being small compared to both the ®lm thickness and the radius of curvature, the Grith condition is recovered: 4B cs2n B h2 cs2b rG: p E p R2 E 2B h3 s2b c cos rG: p R2 E R
8
9
3. DISCUSSION
Although a ®nite element computation [9] is required for determining the precise shape of the normalized energy curves, the approximate treatment leading to equation (7) presented above provides several key insights into the conditions under which interface separation is favored. These can be seen most clearly in the graphical representation of the energy function for dierent ®lm thicknesses normalized by the local radius of curvature (Fig. 5). According to the inequality of equation (7), separation will occur when the normalized energy, represented by a horizontal line in Fig. 5(b), is lower
Fig. 4. Overlapping shells used to construct, by supposition, the normal stress variation along the sinusoidal interface.
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CLARKE and POMPE: INTERFACE SEPARATION OF A FILM
Fig. 5. (a) Normalized strain energy release rate associated with possible interface separation for dierent ®lm thicknesses normalized by the radius of curvature. (b) For illustrating the existence of a critical ¯aw size a horizontal line corresponding to a particular value of the interfacial fracture resistance is superimposed. Once formed the ¯aw extends until arresting when the strain energy release rate again equals the interface fracture resistance. The critical ¯aw size, c^0 , depends on the ®lm thickness, ®lm stress and substrate curvature.
CLARKE and POMPE: INTERFACE SEPARATION OF A FILM
than the function on the right-hand side of the inequality, indicating that the determining group for interface separation is the dimensionless parameter g R2 EG=h3 s2b . Interface separation is considered to occur when this parameter is equal to a critical value, gcrit. The ®rst insight is that there, correspondingly, exists, for a given ®lm thickness, residual stress and interface fracture resistance, a critical radius, R, of the surface curvature above which interface separation will not occur energetically. This suggests that separation is not energetically favored when the surface roughness is too low. Secondly, the critical radius depends on both the ®lm thickness and the residual stress in the ®lm. Thus, for a given residual stress, there is a critical ®lm thickness above which separation is favored but below which it will not occur. This is consistent with our observations that separation was only seen after several hours of oxidation. Supposing that the oxide thickness varies parabolically with oxidation time, t, as h kp t1=2 then the critical oxidation time for interface separation follows as 2=3 1 R2 EG tcrit 2 2 10 kp sb gcrit where kp is the parabolic rate constant. An additional insight that can be gained from the graphical construction is that for the separation to form there is a nucleation condition. A pre-existing
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¯aw of size, c0/R, must exist. If such a ¯aw exists, then it will propagate unstably until the crack driving force falls to the value of the crack resistance force at which point it arrests and stops [Fig. 5(b)]. This is a general characteristic of cracks nucleated and then propagating in a decreasing stress ®eld, such as cracks formed by point contact and fatigue. When the residual stress in the ®lm is due to thermal expansion mismatch, then the inequality of equation (7) can be speci®cally written in terms of the temperature dierence, DT, on cooling and reexpressed by 2B h3 c DaDT 2 E cos rG: p R2 R
11
The separation behavior on cooling can be followed using the diagram in Fig. 6 for a ®lm of a given thickness with a ¯aw of size, c0, at the interface. As cooling proceeds, the development of thermal mismatch stress is represented by a downward movement of the horizontal line corresponding to the fracture criterion. Once the horizontal line intersects the energy curve at the ¯aw size, c0, the separation abruptly extends and then arrests; the ``pop-in'' behavior referred to above. This corresponds to a critical cooling temperature, DTc, expressed as 1=2 1 R2 G DTc : 12 Da h3 Egcrit
Fig. 6. Interface separation nucleation and growth as a result of thermal expansion mismatch on cooling. Once the interface ¯aw grows and arrests at the critical cooling, it will then continue to extend in a stable manner as the temperature is further decreased.
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CLARKE and POMPE: INTERFACE SEPARATION OF A FILM
As cooling continues, the separation grows stably, as represented in the ®gure as the horizontal line continues to move downwards, until room temperature is reached. What distinguishes the interface separation described here from the cracking of thin ®lms under stress on a ¯at surface is the explicit dependence of both the unstable crack extension criterion and the nucleation condition on the local radius of curvature. The latter is of particular interest since it indicates that the smaller the radius the smaller is the critical nucleus size. It is tempting to suppose that as this size is made smaller, the probability of ®nding such a critically sized ¯aw decreases. This would result then in there being a range in surface roughness over which interface separation will occur; too large a radius and the fracture criterion is not met and too small a radius and the nucleation condition is not satis®ed due to a lack of appropriately sized pre-existing ¯aws. The interface separation discussed in this work is distinct from a buckling instability of a ®lm under compression although the growth of the interface separation may serve as a precursor event. Buckling requires that there already exist a region, of some critical size, of the interface that is not attached. Whilst, the buckling condition for a compressively stressed ®lm attached to a curved surface has not, to the authors' knowledge, been established, the solutions for a ¯at interface suggest that the critical sized region on a curved surface will also be many times that of the ®lm thickness. For a rough interface that is nevertheless macroscopically ¯at, local separations of the type discussed in this work are expected to occur when the local separation condition is satis®ed. This is likely to occur at several places on the interface but the separated regions cannot link-up since they each run into a local region of compressive stress. These regions of attachment would serve as non-linear tractions across the interface and prevent the buckling instability. Two additional factors, however, may give rise to local buckling. One is that rough interfaces must have some statistical distribution in local curvatures and wavelengths. This is expected to modify the local stress at an individual crest allow-
ing some adjacent separations to grow together. (Evidence for this is seen as the roughness increases with oxidation time [6].) As a result of both of these eects, separations may link-up by a percolation process and, at some as yet unde®ned threshold, create the critical size for spontaneous buckling. This process would, necessarily, be statistical in nature and cause buckling to be observed at random over the surface. 4. CONCLUSIONS
An approximate analysis of the energetics of fracture is presented to show that separation of a compressively stressed ®lm from a curved surface is favored if the local curvature exceeds a critical value. The critical value depends on the residual stress in the ®lm and its thickness as well as the fracture resistance of the interface. The principal features of the analysis are consistent with recent observations of interface separation of an aluminum oxide ®lm, formed by high temperature oxidation, from a wrinkled alloy surface. AcknowledgementsÐIt is a pleasure to acknowledge the continued support for our work from the Oce of Naval Research under grant number NOOO14-97-1-0190 and the generosity of Vladimir Tolpygo in allowing us to use his photomicrographs. REFERENCES 1. Hutchinson, J. W. and Suo, Z., Adv. appl. Mech., 1992, 29, 63. 2. Gille, G., in Current Topics in Materials Science, Vol. 12, ed. E. Kaldis, 1985, pp. 420±472. 3. Hutchinson, J. W., Thouless, M. D. and Liniger, E. G., Acta metall. mater., 1992, 40, 295. 4. Thouless, M. D., Hutchinson, J. W. and Liniger, E. G., Acta metall. mater., 1992, 40, 2639. 5. Evans, A. G., Crumbley, G. B. and Demaray, R. E., Oxid. Metals, 1983, 20, 193. 6. Tolpygo, V. K. and Clarke, D. R., Acta mater., 1998, 46, 5167. 7. Suo, Z., J. Mech. Phys. Solids, 1995, 43, 829. 8. Gong, X.-Y. and Clarke, D. R., Oxid. Metals, 1998, 50, 355. 9. He, M.-Y., Clarke, D. R. and Evans, A. G., in preparation.
Acta mater. Vol. 47, No. 6, pp. 1749±1756, 1999 # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(99)00078-6 1359-6454/99 $20.00 + 0.00
CRITICAL RADIUS FOR INTERFACE SEPARATION OF A COMPRESSIVELY STRESSED FILM FROM A ROUGH SURFACE D. R. CLARKE1{ and W. POMPE2 Materials Department, College of Engineering, University of California, Santa Barbara, CA 931065050, U.S.A. and 2Technical University of Dresden, Dresden, Germany
1
(Received 19 November 1998; accepted 12 February 1999) AbstractÐSurfaces are often roughened so that coatings under residual compression will remain intact and adherent. By representing surface roughness as a sinusoidal surface and comparing the elastic strain and surface energies of a compressively stressed ®lm, it is shown that there exists a critical radius of curvature below which interface separation is energetically favored. As with many other residual stress problems in thin ®lms, separation depends on a dimensionless group, in this case (R2 Eÿ=h3 b2 ) where R is the local radius of curvature, G the fracture resistance of the interface and E, h and sb are the ®lm modulus, ®lm thickness and residual stress, respectively. # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
The mechanics of failure of thin, highly stressed ®lms on ¯at surfaces has been well developed in the last decade and quantitative relations have been established between ®lm stress, ®lm thickness and interface fracture resistance for a variety of failure modes [1±4]. However, there are many practical situations in which a stressed ®lm is formed on a rough rather than on a ¯at surface. Indeed, it is common practice in the manufacture and application of a wide variety of coatings to roughen the surface prior to coating in order to increase the ®lm adhesion, particularly for ®lms under compression. The reason for the increased resistance to ®lm decohesion is not fully understood but is believed to be associated with the fact that the roughness causes a crack propagating along the interface to propagate under mixed mode conditions thereby increasing the energy dissipated during ®lm decohesion. Roughness, however, also causes local stress concentrations to arise when the ®lm is under compression, for instance as a result of thermal expansion mismatch with the underlying substrate. Such stress concentrations can result in local tensile forces [5] across the interface between the ®lm and the substrate, as described below, and lead to local separations that may, under prescribed conditions, grow to cause decohesion. In this work we introduce the notion that there exists a critical local curvature for local separation of a compressively stressed ®lm from a surface. The critical curvature is related to the ®lm thickness and ®lm stress and is {To whom all correspondence should be addressed.
estimated in the following using a fracture mechanics approach based on the energies associated with extension of an interface crack (separation). The analysis described in this work is motivated by our recent observations of localized separation of aluminum oxide ®lms, formed by high-temperature oxidation of FeCrAl, from the crests of a wrinkled surface [6]. An example is shown in the optical micrograph of Fig. 1(a) in which the local separation is visible as the white patches on account of their enhanced back-scattering. In this particular alloy, an originally ¯at, smooth surface wrinkles during oxidation with a wavelength and amplitude that depends on oxidation time and temperature [6, 7]. Four pertinent observations made were that (a) the separations only occurred after a certain oxidation time, (b) they spontaneously ``popped'' in on cooling and then arrested, (c) the size of the separations was larger at longer oxidation times, and (d) there was little stress relaxation associated with the separation. The last observation was made by piezospectroscopy using an optical probe method and is illustrated by the series of measurements shown in Fig. 2 recorded from the larger regions shown in Fig. 1. The measurements were made using a probe with a diameter of 03 mm and were taken from the oxide that had separated at the crests of the wrinkles as well as from the crests and valleys where no separation had occurred. Substantial variation in the frequency shift from one crest to another is evident but nevertheless the frequency shift from the large regions of separated oxide is about 10% lower. This corresponds to a reduction in elastic strain energy of about 15% in the oxide [8].
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CLARKE and POMPE: INTERFACE SEPARATION OF A FILM
Fig. 1. (a) Optical micrograph of local separation of the alumina scale formed on an FeCrAl alloy at the crests of the surface wrinkles after 100 h oxidation at 10008C. (b) Scanning electron micrograph of the same sample as in (a) illustrating the three-dimensionality of the surface wrinkling. Atomic force microscopy indicates that the average wavelength of the wrinkling is 10 mm.
It is emphasized that these local separations between the oxide and the underlying metal are quite distinct from the buckling mode of failure that occurs in compressively stressed ®lms. Buckling requires a ®nite region of debonding between the ®lm and substrate, a region much larger than the separation shown in Fig. 1, and is associated with uplift of the ®lm from the surface [1]. The separ-
ations shown in Fig. 1 and described in detail in Ref. [6] appear to be associated with elastic debonding of the alloy away from the oxide ®lm on cooling with little or no detectable vertical displacement of the ®lm away from the surface. (Buckling is also seen but is occasional and does not occur over the entire surface as the wrinkling induced ®lm separation does. It is considered to be associated with
CLARKE and POMPE: INTERFACE SEPARATION OF A FILM
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Fig. 2. A series of measurements, using luminescence piezospectroscopy, taken at the crests and valleys of a wrinkled scale, such as shown in Fig. 1. The results are presented in terms of the shift in frequency, from the unstressed state, which is proportional to the local elastic strain energy in the oxide. Despite substantial variations from place to place, there is a small decrease in frequency shift from the oxide that has separated at the crests of the wrinkles. The data are from a scale formed on an FeCrAl alloy after 100 h oxidation using a 3 mm optical probe.
local impurities on the surface and so is an artifact when it occurs at early oxidation times.) From quantitative measurements of the surface wrinkling, made by atomic force microscopy [6, 7], the wrinkling in Fig. 1(b) is characterized by an average wavelength of 10 mm and average roughness amplitude of 0.2 mm. 2. APPROXIMATE ANALYSIS
We idealize surface roughness by a sinsuoidal variation in height and consider that the ®lm forms, under compression, on this surface. On a ¯at surface, the compressive stress would be biaxial and have a value of sb. However, on the sinusoidal surface, the height variation causes a geometrical transformation so that at the crests the interface is under tension and at the valleys the interface is under compression [5, 8]. This state of stress is shown schematically, in one dimension, in Fig. 3. Finite element computations [8] indicate that, to a very good approximation, the maximum values of the tensile (and compressive) stresses on an axisymmetric hemispherical surface are given by a soap®lm (Laplace equation) analogy, namely the maximum stress normal to the interface, sn, is sn
2sb h R
1
where R is the local radius of curvature. (This approximation breaks down when the ®lm thickness is
similar to the roughness amplitude [8].) In moving along the interface from the crests to the valleys, the normal stress decreases in magnitude, changes sign, and then increases again until the valley is reached. This normal stress distribution would suggest that if an interface crack were to form, it would preferentially nucleate at the crests but could not propagate far until entering a region in which the stress on the interface is compressive and arresting. In the following we calculate the crack driving force acting on a crack nucleated at a hemispherical-shaped crest and compare it with the crack resistance associated with the formation of the free surfaces. The calculation is an approximate one since we are assuming that the interface stresses dominate the energy release rate during extension of the separation crack and so are not including any relaxation of the in-plane residual stress. Prior to any local separation of the ®lm from the underlying substrate, elastic strain energy is stored in both the ®lm and the substrate material in the hemispherical cap region. Although the stress, and hence the strain energy density, varies throughout the ®lm, we make two assumptions in calculating the elastic strain energy available for driving a separation. The ®rst is that the elastic strain energy in the vicinity of a single crest is proportional to the product of the square of maximum normal stress and the volume of strained material in the cap region. The second is that the region in the vicinity of the crest can be represented as a hemispherical
1752
CLARKE and POMPE: INTERFACE SEPARATION OF A FILM
Fig. 3. State of residual stress across the interface as a result of the ®lm being under compression. The local tensile stress across the interface at the crests of the sinusoidal wrinkle motivates local separation (lower ®gure).
cap or shell with radius R and thickness h. Using these assumptions, the elastic strain energy that motivates separation can then be written as Estrain BR2 h
s2n E
2
where B is a proportionality constant which takes into account both the diering elastic constants of the ®lm and metal as well as a weak additional dependence on the h/R ratio. (In the limit of hÿ ÿ40 ÿ the elastic energy goes to zero and in the limit of very large R, a ¯at surface, the elastic energy, per unit area, is proportional to the ®lm thickness, h.) When a separation crack forms at the crest, the material in the hemispherical cap relaxes principally in the z-direction [Fig. 3(b)]. Only a portion of the total elastic strain energy is relieved when the separation occurs, as shown in Fig. 2, largely because the stress in the plane of the ®lm is essentially unrelieved. (Even at a much later stage, when buckling occurs, the in-plane stress is not substantially relieved.) To establish whether the separation crack can form and extend, we consider the energetics of crack extension in which the elastic strain energy associated with the normal component of the interface stress in the cap region is converted to surface energy during separation. This requires a functional
form for the elastic strain energy in the ®lm and underlying metal in terms of the crack length. As far as the authors are aware this has not been computed or analyzed previously. In the absence of such a function, we assume that it is a continuous function and conclude from dimensional analysis that it is a function of the two characteristic length ratios, f c=h,c=R. The elastic strain energy will then vary with crack length as s2 c c Estrain BR2 h n f , : 3 h R E In establishing a form for the function, we are guided by the behavior in two asymptotic limits, one at small length scales where the dominant dependence should be given by the length scale c/h and the other at a length scale of c/R. The ®rst limit corresponds to a very small crack, smaller than the ®lm thickness. In this limit, the crack is, in essence in bulk material and the strain energy released will scale as c3. The other limit corresponds to when the crack size is large compared to the ®lm thickness. For a ¯at interface, the energy release scales as c2. Furthermore, due to the variation of the residual stress along the curved interface, it must have a similar form to the normal stress variation across the ®lm/metal interface varying con-
CLARKE and POMPE: INTERFACE SEPARATION OF A FILM
1753
tinuously from a maximum positive value at the crest through zero and to a maximum negative value at the valley. [The negative portions of the function will not be used in the calculation. The cosine function is obtained from the angular variation in normal stress along the interface. This can be constructed, in two dimensions, by representing the ®lm in the crest and valley by two superimposed shells (Fig. 4) each having an inner radius R. One consists of a circular core of metal with an outer shell of ®lm under stress and the other a circular hole lined with a shell of ®lm thickness h inside a continuum.] A continuous function that embodies both length scales and also satis®es the anticipated behavior in both limits is c f c,h,R 1 ÿ 1 ÿ eÿc=h sin2 : 4 R
sociated with incremental extension of crack area, A, exceeds the fracture resistance, R, namely
The strain energy relieved by the crack extension is assumed spent in forming new surfaces over the hemispherical cap: c Esurface 4pR2 1 ÿ cos G 5 R
For large crack sizes, the inequality approaches
where G is the interface fracture energy in J/m2. (For small crack size, this scales with c2 as is expected for a penny-shaped crack.) Combining the two energies and using equation (1) to relate the normal interface stress to the biaxial stress, the total energy of the system as a function of crack length can be written as s2 c ET 4Bh3 b 1 ÿ 1 ÿ eÿc=h sin2 R E c 4pR2 1 ÿ cos G: R Thermodynamically, fracture can be considered to occur when the strain energy release rate, G, as-
GrR i:e:
@ Estrain Esurface R0: @A
6
Evaluating this inequality, using the relationship dA 2pc dc and re-arranging pR2 EG c c ÿc=h ÿc=h R e sin : 7 R 1 ÿ e cos R 2h R 2Bh3 s2b Although the value of the proportionality constant, B, is not known it is expected to be of the order of unity. In the limit of the crack length being small compared to both the ®lm thickness and the radius of curvature, the Grith condition is recovered: 4B cs2n B h2 cs2b rG: p E p R2 E 2B h3 s2b c cos rG: p R2 E R
8
9
3. DISCUSSION
Although a ®nite element computation [9] is required for determining the precise shape of the normalized energy curves, the approximate treatment leading to equation (7) presented above provides several key insights into the conditions under which interface separation is favored. These can be seen most clearly in the graphical representation of the energy function for dierent ®lm thicknesses normalized by the local radius of curvature (Fig. 5). According to the inequality of equation (7), separation will occur when the normalized energy, represented by a horizontal line in Fig. 5(b), is lower
Fig. 4. Overlapping shells used to construct, by supposition, the normal stress variation along the sinusoidal interface.
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CLARKE and POMPE: INTERFACE SEPARATION OF A FILM
Fig. 5. (a) Normalized strain energy release rate associated with possible interface separation for dierent ®lm thicknesses normalized by the radius of curvature. (b) For illustrating the existence of a critical ¯aw size a horizontal line corresponding to a particular value of the interfacial fracture resistance is superimposed. Once formed the ¯aw extends until arresting when the strain energy release rate again equals the interface fracture resistance. The critical ¯aw size, c^0 , depends on the ®lm thickness, ®lm stress and substrate curvature.
CLARKE and POMPE: INTERFACE SEPARATION OF A FILM
than the function on the right-hand side of the inequality, indicating that the determining group for interface separation is the dimensionless parameter g R2 EG=h3 s2b . Interface separation is considered to occur when this parameter is equal to a critical value, gcrit. The ®rst insight is that there, correspondingly, exists, for a given ®lm thickness, residual stress and interface fracture resistance, a critical radius, R, of the surface curvature above which interface separation will not occur energetically. This suggests that separation is not energetically favored when the surface roughness is too low. Secondly, the critical radius depends on both the ®lm thickness and the residual stress in the ®lm. Thus, for a given residual stress, there is a critical ®lm thickness above which separation is favored but below which it will not occur. This is consistent with our observations that separation was only seen after several hours of oxidation. Supposing that the oxide thickness varies parabolically with oxidation time, t, as h kp t1=2 then the critical oxidation time for interface separation follows as 2=3 1 R2 EG tcrit 2 2 10 kp sb gcrit where kp is the parabolic rate constant. An additional insight that can be gained from the graphical construction is that for the separation to form there is a nucleation condition. A pre-existing
1755
¯aw of size, c0/R, must exist. If such a ¯aw exists, then it will propagate unstably until the crack driving force falls to the value of the crack resistance force at which point it arrests and stops [Fig. 5(b)]. This is a general characteristic of cracks nucleated and then propagating in a decreasing stress ®eld, such as cracks formed by point contact and fatigue. When the residual stress in the ®lm is due to thermal expansion mismatch, then the inequality of equation (7) can be speci®cally written in terms of the temperature dierence, DT, on cooling and reexpressed by 2B h3 c DaDT 2 E cos rG: p R2 R
11
The separation behavior on cooling can be followed using the diagram in Fig. 6 for a ®lm of a given thickness with a ¯aw of size, c0, at the interface. As cooling proceeds, the development of thermal mismatch stress is represented by a downward movement of the horizontal line corresponding to the fracture criterion. Once the horizontal line intersects the energy curve at the ¯aw size, c0, the separation abruptly extends and then arrests; the ``pop-in'' behavior referred to above. This corresponds to a critical cooling temperature, DTc, expressed as 1=2 1 R2 G DTc : 12 Da h3 Egcrit
Fig. 6. Interface separation nucleation and growth as a result of thermal expansion mismatch on cooling. Once the interface ¯aw grows and arrests at the critical cooling, it will then continue to extend in a stable manner as the temperature is further decreased.
1756
CLARKE and POMPE: INTERFACE SEPARATION OF A FILM
As cooling continues, the separation grows stably, as represented in the ®gure as the horizontal line continues to move downwards, until room temperature is reached. What distinguishes the interface separation described here from the cracking of thin ®lms under stress on a ¯at surface is the explicit dependence of both the unstable crack extension criterion and the nucleation condition on the local radius of curvature. The latter is of particular interest since it indicates that the smaller the radius the smaller is the critical nucleus size. It is tempting to suppose that as this size is made smaller, the probability of ®nding such a critically sized ¯aw decreases. This would result then in there being a range in surface roughness over which interface separation will occur; too large a radius and the fracture criterion is not met and too small a radius and the nucleation condition is not satis®ed due to a lack of appropriately sized pre-existing ¯aws. The interface separation discussed in this work is distinct from a buckling instability of a ®lm under compression although the growth of the interface separation may serve as a precursor event. Buckling requires that there already exist a region, of some critical size, of the interface that is not attached. Whilst, the buckling condition for a compressively stressed ®lm attached to a curved surface has not, to the authors' knowledge, been established, the solutions for a ¯at interface suggest that the critical sized region on a curved surface will also be many times that of the ®lm thickness. For a rough interface that is nevertheless macroscopically ¯at, local separations of the type discussed in this work are expected to occur when the local separation condition is satis®ed. This is likely to occur at several places on the interface but the separated regions cannot link-up since they each run into a local region of compressive stress. These regions of attachment would serve as non-linear tractions across the interface and prevent the buckling instability. Two additional factors, however, may give rise to local buckling. One is that rough interfaces must have some statistical distribution in local curvatures and wavelengths. This is expected to modify the local stress at an individual crest allow-
ing some adjacent separations to grow together. (Evidence for this is seen as the roughness increases with oxidation time [6].) As a result of both of these eects, separations may link-up by a percolation process and, at some as yet unde®ned threshold, create the critical size for spontaneous buckling. This process would, necessarily, be statistical in nature and cause buckling to be observed at random over the surface. 4. CONCLUSIONS
An approximate analysis of the energetics of fracture is presented to show that separation of a compressively stressed ®lm from a curved surface is favored if the local curvature exceeds a critical value. The critical value depends on the residual stress in the ®lm and its thickness as well as the fracture resistance of the interface. The principal features of the analysis are consistent with recent observations of interface separation of an aluminum oxide ®lm, formed by high temperature oxidation, from a wrinkled alloy surface. AcknowledgementsÐIt is a pleasure to acknowledge the continued support for our work from the Oce of Naval Research under grant number NOOO14-97-1-0190 and the generosity of Vladimir Tolpygo in allowing us to use his photomicrographs. REFERENCES 1. Hutchinson, J. W. and Suo, Z., Adv. appl. Mech., 1992, 29, 63. 2. Gille, G., in Current Topics in Materials Science, Vol. 12, ed. E. Kaldis, 1985, pp. 420±472. 3. Hutchinson, J. W., Thouless, M. D. and Liniger, E. G., Acta metall. mater., 1992, 40, 295. 4. Thouless, M. D., Hutchinson, J. W. and Liniger, E. G., Acta metall. mater., 1992, 40, 2639. 5. Evans, A. G., Crumbley, G. B. and Demaray, R. E., Oxid. Metals, 1983, 20, 193. 6. Tolpygo, V. K. and Clarke, D. R., Acta mater., 1998, 46, 5167. 7. Suo, Z., J. Mech. Phys. Solids, 1995, 43, 829. 8. Gong, X.-Y. and Clarke, D. R., Oxid. Metals, 1998, 50, 355. 9. He, M.-Y., Clarke, D. R. and Evans, A. G., in preparation.