Determination of the elastic modulus and hardness of sol gel coatings on glass: influence of indenter geometry

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1 Ž. Thin Solid Films Determination of the elastic modulus and hardness of sol gel coatings on glass: influence of indenter geometry J. Malzbender a,, G. de With a, J.M.J. den Toonder b a Laboratory of Solid State and Materials Chemistry, Eindho en Uni ersity of Technology, P.O. Box 513, 5600 MB Eindho en, The Netherlands b Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindho en, The Netherlands Received 13 December 1999; received in revised form 7 May 2000; accepted 14 May 2000 Abstract Indentations have been carried out on methyltrimethoxysilane coated float glass by using a spherical and a Berkovich indenter. The composite hardness as well as the effective elastic modulus were determined as a function of indentation depth for coatings of a thickness 0.5, 2 and 4 m. Using the Berkovich indenter, the coating exhibited radial cracking Žonly for the two thicker coatings., delamination and chipping, whereas no cracking occurred during indentations using a spherical indenter. For the experiments in which the coatings exhibited radial cracking, we measured a constant composite hardness over the whole depth range instead of the expected increase. The elastic modulus, on the other hand, turned out to be insensitive to the radial cracking. All results show the expected increase of the modulus with indentation depth. The spherical indenter showed a much steeper rise of the effective elastic modulus with indentation depth than the Ž sharp. Berkovich indenter. By re-scaling the results with respect to contact area instead of indentation depth, the Berkovich and the sphere elastic moduli are comparable. Only at the point where delamination occurred in the Berkovich indentations, the re-scaled results of the two indenter geometries start to deviate Elsevier Science S.A. All rights reserved. Keywords: Coatings; Elastic properties; Hardness; Stress 1. Introduction Sol gel coatings are widely used to modify the functional behavior of glass components or to protect the underlying substrate from environmental influences such as particle impact and moisture 1 6. It is obvious, that in developing such coatings, their mechanical properties are crucial 7,8. Indentation testing is widely used to assess the me- Corresponding author. Tel.: ; fax: address: j.malzbender@tue.nl Ž J. Malzbender.. chanical properties of coatings, such as the elastic modulus and the hardness 9. A complication in determining the coating properties is that, as opposed to bulk materials, the effectively measured properties are dependent on the indentation depth due to the combined response of coating and substrate. To determine the coating properties, either a very small indentation depth Ž with respect to the coating thickness. must be used, or some model accounting for the substrate influence on the measurement must be applied. Another complicating factor is, that cracking may occur during the indentation which also affects the effectively measured properties. In this paper we present an investigation on the $ - see front matter 2000 Elsevier Science S.A. All rights reserved. Ž. PII: S

2 ( ) J. Malzbender et al. Thin Solid Films influence of both the indentation depth and the cracking on the measured elastic modulus and hardness of a particular sol gel coating on a glass substrate. A comparison is made between results for different indenter shapes. 2. Experimental The experiments were carried out using float glass that was coated with an organic inorganic hybrid coating. The coating fluid contained methyltrimethoxysilane Ž MTMS. and colloidal silica in a ratio of 1:1 with approximately 50% of water. The coating fluid was allowed to hydrolyze for 60 h at room temperature. The rectangular glass substrates of 100 cm 2 were 0.2 cm thick. The substrate material was soda lime glass Ž Glaverbel.. The substrates were cleaned using a soap solution, followed by rinsing in demineralized water and alcohol. The coatings were applied by spinning at speeds between 100 and 1600 rev. min to obtain various coating thicknesses. After spinning, the coatings were dried by heating them on a hot plate at 70 C for 1 min. The coatings were then cured at 250 C for 1 h. The resulting coating layer thicknesses were 0.5, 2 and 4 m. The surface roughness of the coatings was approximately 20 nm 10 and was significantly lower than the indentation depth in all experiments. The indentation experiments were carried out at room temperature and ambient atmosphere using an instrument built at the Eindhoven University of Technology. The apparatus permitted up to 25 indentations to be made in one run at loads ranging from 0 to 1000 mn. A Berkovich-type indenter and a sphere were used. The reduced elastic modulus E and the hardness H were determined from the indentation loaddisplacement curve as proposed by Oliver and Pharr 9. The reduced elastic modulus was determined with 9: ' S E Ž 1. 2 ' A where S is the slope of the unloading curve at maximum load, and A is the projected area of contact between indenter and the specimen at maximum load. The reduced modulus is given as 9 : i Ž 2. E E E i where E and represent the materials elastic modulus and Poisson s ratio and the suffix i refers to the parameters of the indenter. The hardness is defined as 9: P H Ž 3. A where P is the maximum indentation load. The calibration procedure suggested by Oliver and Pharr 9 was used to correct for the load frame compliance of the apparatus and the imperfect shape of the indenter tip. The area function of this indenter was calibrated using B270 glass Ž Schott., whose elastic modulus was determined independently as 75 1 GPa using the pulse-echo method. The compliance of the system was determined to be 300 nm N and the projected area of the indenter, A, for the Berkovich indenter was related to the contact depth, h c, of the indentation by: 2 A ah bh Ž 4. c c with a 24.5 and b 5.71 m. This implies that the Berkovich indenter has a spherical tip with a radius of approximately 0.9 m. For the spherical indenter, we approximated the indenter area function with: Ž c c. 2 A 2 Rh h Ž 5. where we obtained for the radius R 45 m. The calibration was performed for a depth range of m for the Berkovich indenter and m for the spherical indenter, where the maximum indentation depth was restricted by the load limitation of the apparatus. Finally, the contact depth was determined from the indentation load-displacement curves with the formula: P h h Ž 6. c S where h is the total indentation depth at maximum load P, and is a geometrical constant, which is 0.75 for a Berkovich indenter and 0.72 for the spherical indenter Results In this section, we present the results of the indentation experiments using either the Berkovich or the spherical indenter. We consider the general observations, the hardness and the elastic modulus. A discussion of the results will be given in Section 4. The results shown in the following sections represent the averages of 10 independent measurements General obser ations During indentation of the coatings using the

3 136 ( ) J. Malzbender et al. Thin Solid Films Ž.Ž. Ž. Fig. 1. Images observed using optical microscopy and load-displacement curves for 2- m-thick coatings in a, b and c after loading to 40 mn, 100 mn and 300 mn, respectively. Berkovich indenter, various phenomena were observed. This is illustrated in Fig. 1, which shows typical load displacement curves along with the optical micrographs of the indentation marks for a 2- m-thick coating. The phenomena that can be seen in this figure are representative for the 2 and 4 m coatings. Radial cracks were observed followed by delamination and chipping at higher loads. For these coatings a distinct change is observable in the slope of the loaddisplacement curve at a load of approximately mn Ž Fig. 1a., which is related to the onset of radial cracking in the coating. The length of the radial cracks increased with increasing load Ž Fig. 1b., but this caused no further abrupt changes in the load displacement curve. The load at which the radial cracking started was found to be independent of the coating thickness. Delamination at the interface occurred at higher loads, i.e. for 2- m-thick coatings Ž see Fig. 1. at mn Ž Fig. 1b. and for a 4- m-thick coating at mn. The delamination led to buckling and eventually chipping of the coating Ž Fig. 1c. at high loads, i.e. for 2- m-thick coatings at mn and for 4- m-thick coatings at mn, which resulted in more abrupt changes in the load displacement curve Ž Fig. 1c.. It was confirmed by in-situ observation of the indentations, by optical microscopy through the glass substrate during indentation, that the delamination and chipping occurred during loading. The 0.5- m-thick coatings did not show radial cracks, but only delamination and chipping 8. No fracture features were visible in the coatings after the indentations using the sphere up to an inden-

4 tation load of 1000 mn. At first glance, the indentation load displacement curves obtained with the sphere suggested purely elastic behavior. However, as will be shown later, irreversible plastic deformation occurred. ( ) J. Malzbender et al. Thin Solid Films Hardness Eq. Ž. 3 can be used to determine the indentation pressure. For a sphere this parameter does not necessarily correspond to the hardness at low loads, but is related to the elastic deformation of the materials and is usually annotated as Meyers hardness. Due to this fact and the effect of the fracture on the measured hardness the results obtained using Eq. Ž. 3 is an apparent hardness. Moreover, the obtained values are in fact effective values due to the combined response of the coating and the substrate and thus it is termed composite hardness in the following. Fig. 2 shows the results obtained for the composite hardness as a function of contact depth as obtained using the Berkovich indenter the 0.5-, 2- and 4- m-thick coatings. In this figure, the contact depth is scaled with the coating thickness t. The composite hardness of the glass substrate was determined independently as H s GPa using the Berkovich indenter. The composite hardness for the 0.5- m-thick coatings exhibits the behavior that is characteristic for soft coatings on harder substrates. Fig. 2 shows that the hardness increases with indentation depth, and that its value approaches that measured for the glass substrate at larger depths. However, the hardness of the 0.5- m-thick coatings seems to be influenced by the substrate hardness even at the shallowest indentation depths. The thicker coatings Ž 2 and 4 m. showed completely different behavior when indented with the Berkovich indenter. The composite hardness stays at a Fig. 3. Composite hardness vs. hc t measured using a spherical indenter for a coating thickness of 0.5 Ž triangles., 2 Ž squares. and 4 m Ž stars., respectively. The line is a fit of H const. h t. constant level of approximately 1.05 GPa and does not show the expected increase as the substrate is approached. In the figure, radial cracking starts at approximately hc t 0.45 for the 2 m coating, and hc t 0.23 for the 4 m coating. Thus, it can be considered that the value of 1.05 GPa corresponds to the hardness of the coating, since this composite hardness was observed at small indentation depths before cracking occurred. It seems plausible to relate the difference in behavior between the thinnest and the thicker coatings to the formation of radial cracks, which were not observed for the 0.5- m-thick coating as discussed in Section 3.1. During indentation with the sphere, in none of the coatings cracking occurred, so the measured hardness was not influenced by this effect. The results for the coatings are shown in Fig. 3. The measured hardness value increases monotonously with indentation depth, as expected. At the smallest indentation depths, the measured value is very low, compared with the results of the Berkovich indenter Ž see Fig. 2.. It is noteworthy, that the hardness of the glass substrate is not reached yet even at the largest indentation depths. c 3.3. Elastic modulus Fig. 2. Composite hardness vs. hc t measured using a Berkovich indenter for a coating thickness of 0.5 Ž triangles., 2 Ž squares. and 4 m Ž stars., respectively. The line represents a guide for the eye. The elastic modulus was determined from the unloading curves using Eq. Ž. 1. The results for the coatings are shown in Fig. 4 for the Berkovich and Fig. 5 for the spherical indenter, respectively. The curves are fits to the data that will be discussed in Section 4. In both cases the measured elastic modulus increases monotonically with indentation depth, showing the increasing influence of the stiffer glass substrate. The elastic modulus of the substrate was measured in a separate experiment to be ES 75 1 GPa. As can be seen the elastic modulus at a particular depth is sig-

5 138 ( ) J. Malzbender et al. Thin Solid Films Fig. 4. Effective elastic modulus vs. hc t measured using a Berkovich indenter for a coating thickness of 0.5 Ž triangles., 2 Ž squares. and 4 m Ž stars., respectively. The line is a fit to the Doerner and Nix model 20 for the 4- m data. nificantly larger for the spherical indenter than for the Berkovich. Also, above we have shown that cracking occurs under the Berkovich indenter above a load of approximately 25 mn, corresponding to hc t 0.23, which might influence the elastic modulus at larger contact depth. 4. Discussion In this section we will show that the results can only partly be described by existing models and, furthermore, we will discuss the reasons for this discrepancy between general theories and the particular experimental results Composite hardness Berko ich Three different models could be fitted to describe the composite hardness contact depth dependency of Fig. 6. Comparison of fits of Eq. Ž.Ž 7 line., Eq. Ž.Ž 8 small dashing. and Eq. Ž.Ž 9 large dashing. to the composite hardness as measured using a Berkovich indenter for a 0.5- m-thick coating. the 0.5- m-thick coating. In these fits, we used the measured value of 5.8 GPa for hardness of the substrate H s, whereas for the coating hardness Hf we used the value measured near the surface for the 2- and 4- m-thick coatings, namely Hf 1.05 GPa. First, the data could be described by an equation similar to the function suggested by Bhattacharya and Nix 11, i.e.: Ž. Ž h c t. Ž. H H H H e 7 s f s where t is the thickness of the coating and a constant that depends on the elastic modulus and yield strength of the film and substrate. A fit of Eq. Ž. 7 to the experimental results is shown in Fig. 6. The single fitting parameter,, was determined experimentally as Secondly, Ahn and Kwon 12 developed a formula to describe the composite hardness on the basis of the plastic-zone volume of coating and substrate using elastic plastic indentation theory. Their final formula took the form 12 : Vf Vs H H H Ž 8. V f V s Fig. 5. Effective elastic modulus vs. hc t measured using a spherical indenter for a coating thickness of 0.5 Ž triangles., 2 Ž squares. and 4 m Ž stars., respectively. The line is a fit to the Doerner and Nix model 20 for the 4- m-thick coating. where Vs V is the ratio of plastically deformed sub- strate volume to total plastically deformed volume, and V V is the same ratio for the film 12 f. The formulas given by Ahn and Kwon 12 were developed for a hard coating on a soft substrate and were therefore modified to take into consideration the current configuration. A fit of Eq. Ž. 8 to the data of the 0.5- m-thick coating is also shown in Fig. 6. Thirdly, Korsunsky et al. 13 proposed for the composite hardness of a hard coating on a soft substrate: Hf Hs H H Ž 9. s 2 1 kž h t. c

6 ( ) J. Malzbender et al. Thin Solid Films The relation should according to Korsunsky et al. 13 apply equally well to films with and without cracking. The physical meaning of the parameter k depends on whether the deformation is plasticity- or fracturedominated. In practice it is used as a fit parameter. A fit of Eq. Ž. 9 to the data can also be seen in Fig. 6. For the fit a constant k was found. Opposite to Eqs. Ž. 7 and Ž. 9, Eq. Ž. 8 needs no fitting parameters. It can be seen from Fig. 6 that Eq. Ž. 8 gives the best fit to the data. It has to be re-emphasized that we have used a fixed value of Hf 1.05 GPa in the fits, and that using Hf as a fitting parameter will lead to much better fits for all three models. The data for the thicker coatings Ž 2 and 4 m. could not be fitted by the above equations. These coatings exhibited a virtually constant composite hardness value, as shown in Fig. 2. This behavior might be explained by assuming that the radial cracks which occurs in these coatings, opposite to the 0.5 m coating, led to a decrease in composite hardness which counteracted the increasing influence of the substrate. To investigate this assumption more quantitatively, we use an argument that was used by Li and Bradt 14. These authors reason that the occurrence of Žradial median. cracking leads to an additional displacement of the indenter. Indeed, as can be seen in Fig. 1 in the load-displacement curves, the formation of the radial cracks led to a small sudden displacement at constant load. Furthermore, the length and depth of the radial cracks increased as the applied load increased, which presumably led to additional displacements of the indenter with respect to the non-fractured case, although this did in our case not lead to further sudden changes in the indentation curves. As a first-order assumption, it might be suggested that the additional displacement hi is linearly proportional to the radial crack length C. With use of the conventional definition for the formation of radial cracking 15 this leads to: P h C Ž 10. i 2 3 ž r K IC / where KIC is the fracture toughness of the coating, Ž E H for a Berkovich-type indenter 16 r and is the proportionality factor. If there is substantial residual stress present in the specimen, Eq. Ž 10. must be modified. However, a preliminary analysis of this effect showed that, for our coatings, the above equation was not significantly modified by this effect for the considered crack lengths. If the coating delaminates or chips additional terms should be added to Eq. Ž 10.. In order to check this approach, we fitted the data of the 2- and 4- m-thick coatings by combining Eqs. Ž. 7 and Ž 10. and Eqs. Ž 8. and Ž 10., respectively. For K IC, we used a value of 0.09 MPa m 1 2 8, and for H and E we used 1.05 GPa and 2 GPa, respectively Žsee Section The equations indeed allow the fitting of the measured constant composite hardness. However, the model based on Eqs. Ž. 7 and Ž 10. required a fitting parameter of 1 and Eqs. Ž. 8 and Ž 10. a factor of This means that the additional indentation depth due to fracture would be larger than the actual radial crack length and also larger than the indentation depth, which is unrealistic. Therefore, this simple approach can not be used to give a scientifically sound description of the data and the additional effect of a change in materials properties has to be taken into consideration, i.e. the cracking reduces the hardness of the system Yield As stated above the Berkovich pyramid used in these experiments was not perfectly sharp. For shallow indentation depths the indenter can be approximated by a sphere possessing a radius of 0.9 m. In order to determine whether the substrate will first yield or if the coating will show radial cracks we can use an approximation based on the Hertzian solution. The pressure exerted in a monolithic material by a circular contact is given by 17 : ž / r z a Ž 1. 1 arctan a z and 0 0 ž / z Ž a ž / 2 z z 1 1 Ž a where 0 is the contact pressure. The principle shear stress for the Tresca yield criterion is: Ž max r z The substrate will deform plastically if this shear stress exceeds the shear yield strength of the material k which is approximated as k H 6 18, where H is the hardness of the substrate. The coating will lead to a modification of the stress field and a discontinuity at the interface. Thus, the current analysis can only be seen as approximation. Considering that the Berkovich can be approximated as a sphere, Eq. Ž. 5 can be used to compute the area of contact. Radial cracks were at a load of approximately

7 140 ( ) J. Malzbender et al. Thin Solid Films Fig. 7. Stress according to the Tresca criterion as a function of a z ratio. 25 mn and an indentation depth of approximately 1.05 m. The contact pressure is: 3P Ž a Thus, the contact pressure 0 has a value of 6.3 GPa at fracture. Using Eq. Ž 13. the change of the principle shear stress with the ratio a z can be calculated, which is shown in Fig. 7. It can be seen from this figure that this shear stress will exceed the shear strength of the substrate at a depth ratio of a z Considering the three different coating thicknesses of 0.5, 2 and 4 m the figure reveals values of a 0.33, 1.34 and 2.68 m. The coating will fracture at a contact area of 1.34 m. Thus, for the 0.5- m-thick coating the substrate will yield before fracture occurs. For a 2- m-thick coating, fracture and yield occur at approximately the same contact radius, whereas for the 4- m-thick coating, fracture will occur first. This explains the fact that only for the 0.5- m-thick coating no radial cracking was observed. Fig. 8. Ratio of the irreversibly dissipated energy to total energy as a function of contact depth for the 4- m coating. energy dissipation of 8%. The irreversibly dissipated energy increases linearly at a contact depth larger than 0.15 m, corresponding to a load of 10 mn, which suggests an increase of the volume of plastically deformed material with increasing contact depth. Since no cracking was observed after the indentations using a sphere, all the irreversible work can be related to plastic deformation. As can be determined from Figs. 2 and 3 the composite hardness measured using a Berkovich and a sphere agree reasonably well above a hc t ratio of approxi- mately 1.5 for the 0.5- m-thick coating. This might be related to a domination of the substrate properties above this depth. As shown in Fig. 9 the coating deforms also plastically even at the lowest loads. Thus, a Hertzian analysis does not appear to be fully applicable in the current case. It can be expected that the composite hardness as a measure of the indentation pressure tend to zero as the indentation pressure tends to zero for elastic defor Composite hardness sphere No cracking occurred during the indentation using the sphere and the measured composite hardness increased monotonously with increasing indentation depth for all coatings, as shown in Fig. 3. However, the values of the composite hardness are very low at the lowest indentation depths, suggesting that the deformation remains primarily elastic in that regime. This can be verified by determining the ratio of the irreversibly dissipated work to the total work done by the indenter W W. These energies were determined by integration ir of the loading and unloading curves. The ratio of W W vs. contact depth for a 4- m-thick coating is ir shown in Fig. 8. The lowest load used in this experiment was 5 mn, which corresponds already to an Fig. 9. Comparison of the effective elastic modulus vs. contact area of a 4- m-thick coating as obtained using a sphere Ž stars. and a Berkovich indenter Ž triangles.. The line is a fit to the Doerner and Nix model 20.

8 ( ) J. Malzbender et al. Thin Solid Films mation thus an agreement between indentation pressure as obtained using the sphere and the Berkovich should not be obtained at low loads. Pharr et al. 19 determined that for a soft coating on a compliant substrate pile-up can be expected. They analyzed the pile-up of coated samples for a conical indenter. Considering the worst case, which occurs for a ideal plastic material, an increase of the real contact area of 50 70% was found. The maximum pile-up occurs when the indenter reaches the interface. The elastic modulus is inversely proportional to the square root of the contact area and the hardness inversely proportional to the contact area. At the moment the indenter reaches the interface, the ratio of contact depth to thickness of approximately 0.5. At this depth the hardness as determined using a sphere and the Berkovich indenter agree. Thus pile-up appears to be negligible Elastic modulus In all measurements the elastic modulus increased with indentation depth due to the increasing influence of the substrate Ž see Figs. 4 and 5.. It is remarkable, that this is also the case for the 2- and 4- m coatings indented with the Berkovich indenter, for which the composite hardness was presumably strongly influenced by radial cracking Ž see Fig. 2., leading to a virtually constant hardness value over the total depth. It appears that the elastic modulus is influenced differently by the occurrence of cracking. The reduced elastic modulus E can be decomposed into the substrate modulus Es and the film modulus E f. For this we used the following relation proposed by Doerner and Nix 20 : 1 Ž 1. Ž 15. E E E f s The parameter is a weight function defined as: t h c Ž. 1 e 16 where is a constant. The elastic modulus of the substrate Ž E 75 1 GPa for soda lime glass. s was measured in a separate experiment and was used as a fixed parameter in the fitting process. The elastic modulus of the coating could only be determined with reasonable reliability only from the 4 m coating results obtained using the Berkovich indenter. For the thinner films indented using both the Berkovich and the spherical indenter, the substrate influence was too large already at the smallest indentation depths to enable a reliable estimate of E f. Fig. 4 shows the fit of Doerner and Nix s model to the 4 m coating results with the Berkovich indenter. From this fit, the values Ef GPa and 0.05 were found. Using this value of 2 GPa, Eq. Ž 15. was also fitted to the other coating results. All results could be fitted well. Fig. 5 shows the fit to the data for the 4 m coating indented using the sphere. For the Berkovich indenter, a slight dependence of the factor on the coating thickness was determined Ž 0.02 for the 0.5- m-thick coating, for the 2- m-thick coating, and 0.05 or the 4- m-thick coating.. A dependence of the factor on the coating thickness has already been predicted by King 21 using a theoretical analysis. However, it can be questioned whether his analysis applies to our results, since King s analysis is purely elastic, while our measurements contained cracking and plastic deformation, which may also influence the estimated. The spherical indentation data could all be fitted with the same, namely Hence, the value of for the spherical indentations is approximately 10 times smaller than that for the Berkovich. This reflects the much steeper rise of E with hc t for the results obtained using a sphere. The difference in the rate of increase of the elastic modulus with indentation depth between the Berkovich and the spherical indenter can be explained by the fact that indentation with the sphere causes a larger volume of material to be deformed at a particular indentation depth than the Berkovich indenter. This leads, for the sphere, to a larger substrate contribution, and hence to a larger effective elastic modulus at a certain indentation depth. The volume of deformed material can be related to the contact area, and therefore we have plotted in Fig. 9 a comparison of the elastic modulus as a function of contact area for the 4- m-thick coating for the sphere and the Berkovich indenter. Indeed, both data sets show good agreement at low contact areas. At larger contact areas the sphere yields larger moduli. This is probably related to a modification of the system s elastic properties by fracture under the Berkovich indenter at higher loads. The contact area at which the curves start to divert corresponds to a load of 100 mn, which was the threshold of delamination under the Berkovich indenter Ž see Section The formation of radial cracks, which occurred at lower loads Ž mn., did not result in a significant diversion. In fact it was possible to obtain a universal fit to the elastic modulus for both the sphere and the Berkovich indenter up to the load of delamination using Eq. Ž 15. by replacing the contact depth by the contact area in Eq. Ž 16.. This resulted in the modified weight function: t A c Ž. 1 e 17 where m independent of the coating

9 142 ( ) J. Malzbender et al. Thin Solid Films thickness. Furthermore, it was attempted to use a weight function of the form: t 2 A c Ž. 1 e 18 which was successful in fitting the data for to 2- and 4- m-thick coatings resulting in a value of For the 0.5- m-thick coating a value of was necessary to fit the data using this function. The difference in fitting parameters may reflect the influence of the different fracture features in the thinnest and the thicker coatings. Lastly, we tried the suggestion of King 21 to use the square root of the projected contact area instead of the contact depth in Eq. Ž 16., but it turned out to be impossible to obtain a reasonable fit to the data using this approach. In any case, our results show that for a sound comparison of indentation results on coatings using various indenter geometries, the results should be plotted against contact area and not indentation depth. Thus, fits like that of Doerner and Nix s relation 20 have to be modified to incorporate the contact area rather than the indentation depth. Another important result from our measurements is that the effective elastic modulus as measured using the Berkovich indenter is influenced by radial cracking of the coating in a totally different manner than the effective hardness. Particularly, whereas the composite hardness is changed drastically by the radial cracking, the elastic modulus seems to be influenced hardly. The reason for this difference will be the subject of future investigations. 5. Conclusions We carried out indentations on methyltrimethoxysilane coatings on float glass using a spherical and a Berkovich indenter. The composite hardness as well as the effective elastic modulus were determined as a function of indentation depth for coating thicknesses 0.5, 2, and 4 m. With the Berkovich indenter, the coating exhibited radial cracking Žonly for the two thicker coatings., delamination and chipping. During spherical indentation, no cracking occurred. The effect of the radial cracking could clearly be seen in the composite hardness obtained using the Berkovich indenter for the 2- and 4- m coatings: instead of exhibiting the expected increase, the effective hardness was constant over the whole depth range. In contrast, the 0.5- m coating, which showed no radial cracking, did exhibit the expected rise, and the results could be described using various literature models. The elastic modulus, however, turned out not to be sensitive to radial cracking. All results showed the expected increase with indentation depth. The main difference between the results obtained using the spherical and the Berkovich indenter was the much steeper rise of the effective elastic modulus with depth for the former. By re-scaling the results with respect to contact area instead of indentation depth, the moduli were comparable. Only at the point where delamination occurred, the re-scaled results of the two indenter geometries started to deviate. Hence, the most important results of our study are the following. First, for a comparison of indentation results on coatings using various indenter geometries, the results should be plotted against contact area and not indentation depth. Also, fits like that of Doerner and Nix s relation 20 have to be modified to incorporate the contact area rather than the indentation depth. Another important result from our measurements is that the effective elastic modulus is influenced by radial cracking of the coating in a totally different manner than the effective hardness. Particularly, whereas the composite hardness is changed drastically by the radial cracking, the elastic modulus seems to be influenced hardly. The reason for this difference will be the subject of future investigations. Acknowledgements The authors would like to acknowledge Philips Re- Ž. search Eindhoven for the financial support. References 1 A. Paul, Chemistry of Glasses, Chapman and Hall, London, H. Bach, D. Krause, Thin Films on Glass, Springer, Berlin, G.L. Smay, Glass Technol. 26 Ž F.H. Wang, X.M. Chen, B. Ellis, R.J. Hand, A.B. Seddon, J. Mater. Sci. Technol. 13 Ž T.H. Wang, P.F. James, J. Mater. Sci. 26 Ž H. Schmidt, J. Sol-Gel Sci. Technol. 1 Ž A. Atkinson, R.M. Guppy, J. Mater. Sci. 26 Ž J. Malzbender, G. de With, J. Non-Cryst. Solids 265 Ž W.C. Oliver, G.M. Pharr, J. Mater. Res. 7 Ž J. Malzbender, G. de With, J. Mater. Sci. Ž in press.. 11 A.K. Bhattacharya, W.D. Nix, Int. J. Solids Struct. 24 Ž J.-H. Ahn, D. Kwon, J. Appl. Phys. 82 Ž A.M. Korsunsky, M.R. McGurk, S.J. Bull, T.F. Page, Surf. Coat. Technol. 99 Ž H. Li, R.C. Bradt, J. Mater. Sci. 31 Ž D.B. Marshall, B.R. Lawn, J. Am. Ceram. Soc. 60 Ž G.M. Pharr, D.S. Harding, W.C. Oliver, in: M. Nastasi, D.M. Parkin, H. Gleiter Ž Eds.., Mechanical Properties and Deformation Behavior of Materials Having Ultra-Fine Mikrostructures, Proceedings of the NATO Advanced Study Institute, Porto Novo, Portugal, June 28 July 10, 1992, Kluwer, Dordrecht, 1993, p. 449.

10 ( ) J. Malzbender et al. Thin Solid Films S.J. Bull, A.M. Korsunsky, Tribol. Int. 31 Ž D. Tabor. The Hardness of Metals, Clarendon Press, Oxford, 1951, p G.M. Pharr, A. Bolshakov, T.Y. Tsui, J.C. Hay, in: R.C. Cammarata, M. Nastasi, E.P. Busso Ž Eds.., Thin films: stresses and mechanical properties VII, Boston, MA, USA, Materials Research Society Symposium Proceedings, 505,, 1998, p M.F. Doerner, W.D. Nix, J. Mater. Res. 1 Ž R.B. King, Int. J. Solids Struct. 23 Ž