Measuring mechanical properties of coatings - Materials Technology

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been proposed in the literature, and develops them further to be useful tools in the .... hardness, fracture toughness of coating and interface, and residual stress. ... scratch testing, and finally, discuss the application of scratch testing to coatings.
Materials Science and Engineering R 36 (2002) 47±103

Measuring mechanical properties of coatings: a methodology applied to nano-particle-filled sol±gel coatings on glass J. Malzbendera,*, J.M.J. den Toonderb, A.R. Balkenendeb, G. de Witha a

Laboratory of Solid State and Materials Chemistry, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands b Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands Accepted 26 July 2001

Abstract The main aim of this paper is to demonstrate the practical use of nano-indentation and scratch testing in determining mechanical properties of thin coatings. We place our emphasis on how information obtained using both techniques can be combined to give a more complete representation of the properties of a coating±substrate system. Part I of the paper gives an overview of approaches to determine mechanical properties of thin coatings that have been proposed in the literature, and develops them further to be useful tools in the analysis of coatings. This results in methods for measuring the mechanical properties of thin coatings. We particularly emphasise the determination of the elastic modulus, hardness, coating and interfacial fracture toughness and residual stress using indentation and scratch testing. Part II of the paper illustrates the application of these methods to relatively soft coatings of methyltrimethoxysilane (MTMS) filled with colloidal silica or alumina particles on glass. The coatings were prepared using a sol±gel process. We report results of the dependence of the mechanical properties on the filler particle content, illustrating that microstructural changes can also be tracked using these techniques. The effects of the nature and volume fraction of the filler particles are discussed. # 2002 Published by Elsevier Science B.V. Keywords: Nano-indentation; Scratch testing; Mechanical properties

1. General introduction Thin coatings are increasingly being applied in today's advanced products. Examples of applications are: optical filters on low-E window glazing and displays, protective topcoats and stacks of recording films in optical storage disks, functional layers in semiconductor chips, thin layers in multilayer capacitors, and many more [1,2]. The thickness of such coatings can range from nanometers to several hundred micrometers. The mechanical properties of the coatings play a crucial role in the reliability of the products to which they are applied. If a coating fails, or experiences substantial deformation, its function may be deteriorated. This is particularly obvious for those coatings that are actually applied to offer mechanical protection, such as topcoats for optical disks. Relevant mechanical quantities are: hardness, elastic modulus, residual stress, coating fracture toughness and interfacial fracture toughness (or adhesion energy) [3±5]. For thick coatings, it is sometimes possible to detach the coating from the substrate and to measure properties such as the *

Corresponding author. Present address: Forschungszentrum JuÈlich, IWV-2, 52425 JuÈlich, Germany. Tel.: ‡49-2461-616964; fax: ‡49-2461-613699. E-mail address: [email protected] (J. Malzbender). 0927-796X/02/$ ± see front matter # 2002 Published by Elsevier Science B.V. PII: S 0 9 2 7 - 7 9 6 X ( 0 1 ) 0 0 0 4 0 - 7

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elastic modulus and the fracture toughness by carrying out mechanical tests with this ``freestanding'' coating [5,6]. However, it is usually impossible to detach the coating. Then, an obvious way to measure the mechanical properties of the coating is to deform it on a very small scale. A convenient way to accomplish this is by indentation testing on a nanometer scale, commonly referred to as nano-indentation [5,7,8], combined with scratch testing [9]. Hence, these techniques have become the most widely used techniques for measuring the mechanical properties of thin coatings, although a complete and sound methodology to determine the full set of relevant mechanical properties is still lacking. The main aim of this paper is to demonstrate the practical use of nano-indentation and scratch testing in determining mechanical properties of thin coatings. Part I of the paper gives an overview of approaches for measuring mechanical properties of thin coatings that have been proposed in the literature. It is not our intention to cover fundamental aspects of indentation-theory, as these have been treated, e.g. by Fischer-Cripps [8] and in the various papers to which we refer in the text. Nor will we deal with nano-indentation instrumentation, which is covered, e.g. by the review by Bhushan [10]. Instead, we will concentrate on those methods, which are of use in our daily practice of coating characterisation. We present various existing and new methods to measure the fracture toughness and the adhesion energy of coatings with the use of indentation and scratch testing. All the methods are based on the occurrence of cracking during indentation or scratch testing, either in the coating itself or along the interface between the coating and the substrate. That means that the methods can only be used for materials responding in a brittle manner to the applied indentation or scratching loads. We pay particular attention to the use of the load±displacement curve and the dissipated energy to determine the fracture energy. Part II illustrates the application of the methods for a particular system, namely, nano-particle-filled methyltrimethoxysilane (MTMS)-based coatings on glass that were prepared using a sol±gel process and in which the particle filler content was varied. The second aim of the paper is to show the quantitative influence of the filler content on the mechanical properties of the sol±gel coatings, as obtained from the measured results. 2. Part I: A methodology to determine mechanical properties of coatings 2.1. Introduction Part I of this paper reviews methods to measure mechanical properties of coatings, focusing on the use of indentation and scratch testing. We will discuss methods to measure the elastic modulus, hardness, fracture toughness of coating and interface, and residual stress. The remaining part of this chapter is divided into four sections. We start with an explanation of the general principle of indentation, followed by the application of indentation to coatings. We explain the principle of scratch testing, and finally, discuss the application of scratch testing to coatings. 2.2. Principle of indentation testing 2.2.1. General principle Main contributions into the understanding of the contact of elastic solids have been made by Hertz [11], Boussinesq [12] and Sneddon [13]. The principle of recording indentation testing is straightforward. As illustrated in Fig. 1, the material surface is indented with a tip (the indenter) loaded with a force P, resulting in a penetration depth h of the indenter into the material. During indentation, the force P and the penetration depth h are recorded as a function of time, and in this

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Fig. 1. Schematic representation of a section through an indentation using a conical indenter.

way the load±displacement relationship is obtained (Fig. 2). A recording indentation experiment commonly consists of a single loading±unloading cycle. As the specimen is loaded to a maximum force Pmax, the indentation depth increases to a maximum hmax. If plastic deformation occurs, a different curve is followed upon unloading and the final depth is not zero but some finite value hf, due to the plastic deformation of the material during indentation. Several types of indenter are used. Mostly pyramidal indenters are used, but spheres of various diameter and conical indenters are also used. In indentation testing of coatings, sharp indenters are preferred since they allow the properties of very thin coatings to be measured (Section 2.3). It is, therefore, common practice to use a Berkovich indenter [14], made of diamond, which is a threesided pyramidal shape. Berkovich indenters can have a tip radius in the order of 50±100 nm [15]. The four-sided pyramidal Vickers indenter is less popular since, practice, it is more difficult to fabricate with such a small tip radius. For a spherical indentation during which no plastic deformation or cracking occurs, the indentation load±displacement curve can be described analytically using Hertz's solution [11]. This solution is only valid if the ratio of contact radius a to the indenter radius R is small enough. Johnson

Fig. 2. Schematic representation of a typical indentation load±displacement curve.

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[16] suggests that this ratio should be less than 0.3. The Hertz solution reads: P ˆ 43 R1=2 Er h3=2

(1)

In this equation, Er is the reduced elastic modulus, which is given by [7]: 1 1 ni 2 1 n2 ˆ ‡ Er Ei E

(2)

where Ei and ni are the elastic modulus and Poisson's ratio of the indenter material, and E and n are the elastic modulus and Poisson's ratio of the indented material. It is easy to see that Eq. (1) allows the elastic modulus of the indented material to be determined from the load±displacement curve for elastic spherical indentations. In the case of indentation with an infinitely sharp, ideal, Berkovich or Vickers indenter, a relationship between the load P and the displacement h during the loading part can also be derived, yielding [17,18]: Er h2 P ˆ h p p ppi2 …1= 24:5† Er =H ‡ e p=4 H=Er

(3)

In this equation, e is a constant that depends on the geometry of the indenter. The parameter e ranges from 1 for a flat punch to 0.75 for a paraboloid, and 0.72 for a conical indenter. The value 0.75 is commonly used for Berkovich indenters [7]. Furthermore, H is the hardness of the indented material, which will be defined more precisely in Section 2.2.3. Eq. (3) allows either Er or H to be estimated from the loading curve when the other is known. Since indenters are not infinitely sharp, in practice, and might possess an angle that changes with the indentation depth, Eq. (3) must be modified to account for this, as is shown in references [17,18]. Note, that the square dependence in Eq. (3) is the consequence of the assumption of the indenter being sharp, and is true even for a purely elastic contact where yielding does not occur. Although Eqs. (1) and (3) state a dependency of the load on the displacement, it should be noted that, in practice, the displacement is usually the dependent variable. 2.2.2. Elastic modulus The standard way of estimating the elastic modulus from the indentation load±displacement curve is not by using Eq. (1) or (3), but rather by analysis of the unloading curve (see Fig. 2). The basic idea is that even for materials, which exhibit plastic deformation during loading, the initial unloading is elastic. Thus, the initial slope of the unloading curve is directly related to the elastic modulus. Based on Sneddon's work [13,19], the following formula has been derived for the reduced elastic modulus, Er, to the unloading slope [7]: p p Smax p (4) Er ˆ 2 A In this equation, Smax is the slope of the unloading curve at the point of maximum load (i.e. at the start of unloading, see Fig. 2), and A is the projected area of contact between the indenter and the material at that point. The slope of the unloading curve Smax can be determined directly from the unloading curve. The projected contact area is generally expressed in terms of the contact depth, hc, defined in Fig. 1.

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If the geometry of the indenter is known precisely, the relationship between A and hc is known and the problem is reduced to determining hc from the measurement. The following expression has been derived for hc [7]: hc ˆ hmax

e

Pmax Smax

(5)

This equation expresses hc in measurable quantities. Its graphical interpretation is sketched in Fig. 2. It is obvious that hc < hmax always. The contact area is generally a function of the contact depth [7]: A ˆ A…hc †

(6)

For a perfectly sharp Berkovich or Vickers indenter, A…hc † ˆ 24:5h2c . Using the above equation, and knowledge of the material's Poisson ratio n, the elastic modulus E can be determined. Similar relationships can be derived for different indenter shapes, i.e. cones and spheres. This theory has some limitations. In the first place, when a material exhibits substantial pile-up or sinking-in, the estimated contact depth, and hence, the projected contact area, is either underestimated or overestimated. This happens, e.g. for metals that show strain hardening. The effect obviously leads to an error in the elastic modulus [5,8]. Secondly, the equations are not applicable for materials that creep substantially. We can avoid this problem to some extent by performing several loading±unloading cycles on the same position, instead of one single load±unload cycle, and by using the final unloading curve in the analysis [7]. Finally, it should be noted that the use of the reduced elastic modulus for sharp indenters is currently under discussion [20]. When applying the above theory, some important practical issues have to be taken into consideration. Firstly, the measurements will be influenced by the finite stiffness of the frame of the apparatus [7]. Thus, the frame compliance, Cf, must be determined, and the measurements should be corrected for this effect. Secondly, no indenter is perfectly sharp, in practice [7]. Even very small deviations from the ideal indenter geometry will give deviations from the ideal relationship for Berkovich indenters, particularly at small indentation depths. The area function (see Eq. (6)) relating A to hc, therefore, has to be determined for each individual indenter, by carrying out measurements on a reference material [7]. Fused silica is commonly used as a reference material since its elastic modulus is well known and it is independent of the indentation depth. area function is P The 1=n commonly expressed as a polynomial function of hc, i.e. A ˆ 24:5h2c ‡ n Cn hc . 2.2.3. Hardness The hardness is traditionally measured by performing an indentation at a certain indentation load, removing the load and optically examining the surface to determine the area of the plastic residual imprint. The hardness is then defined as the ratio of the maximum indentation load and the measured area. A Vickers indenter is usually used, however, spheres and other geometries are also used. In modern nano-indentation, the definition of hardness is somewhat different. In this case, the area used in the definition of hardness is actually the projected contact area at maximum load, which is not necessarily equal to the area of the final residual imprint. It is, therefore, determined by [7]: Hˆ

Pmax A

(7)

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where A is estimated using the indenter area function. The advantage of this nano-indentation definition is that the surface does not need to be imaged. Instead, A as defined by the area function is directly determined from the load±displacement curve. It should be emphasised that hardness is not a fundamental property of materials. Hardness is related to material properties, in particular to the yield strength and the elastic modulus, but this relationship depends on the indenter geometry. We should, therefore, be careful when comparing hardness values from different sources. For metals, the following semi-empirical relationship is used to relate the hardness to the yield stress Y [21]: H ˆ 2:8Y

(8)

This relationship is valid for a representative yield stress at a representative strain value of approximately 8±10%. However, the equation is also frequently used to approximate the yield strength of inorganic materials. However, the actual relationship between hardness and the yield of materials is more complicated. Some relationships for different indenter geometries have been derived in [22,23]. 2.2.4. Fracture toughness and residual stress For sufficiently large indentation loads, brittle materials exhibit cracking during indentation. Several crack patterns may occur [24]. Radial/median cracks and lateral cracks are common features for sharp pyramidal indenters. Radial/median cracks emerge from the edges of the indenter, as sketched in Fig. 3, and can be observed as traces in the indented surface. Lateral cracks initiate under the indenter tip at the edge of the plastically deformed zone, and grow underneath and parallel to the surface, which can lead to chipping of the material [24]. During spherical indention, the formation of cone cracks has been observed (see Fig. 4). These grow into the material at a certain cone angle, and can be seen as ring-shaped cracks on the indented surface. The initiation and subsequent growth of the cracks is determined by the elastic and plastic properties of the indented material (i.e. the size of the plastic-zone), and also by the fracture toughness KIc of the material and the residual stress sr present in the indented surface. Thus, the crack features may be used to estimate fracture toughness and residual stress. Lawn and Evans [25] analysed initiation of median-cracking for monolithic materials under a Vickers indenter in an elastic±plastic indentation field. On the basis of the expanding cavity model, they derived the following formula for the critical load for the crack initiation, P [25]: P ˆ 21:7  103

KIc4 H3

Fig. 3. Schematic image of the formation of radial cracking under a Berkovich indenter.

(9)

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Fig. 4. Schematic representation of a section through a Hertzian cone crack showing critical geometrical parameters.

The pre-factor of 21:7  103 was theoretically derived by considering a Vickers indenter geometry [25]. Since the projected contact area of a Vickers and Berkovich indenter is identical at equal hc [7], this relationship can be used for both indenter types. By measuring the critical load for radial crack initiation and using our knowledge of the hardness, the fracture toughness is determined by Eq. (9). Because of the statistical nature of defect distribution and fracture events, the use of Eq. (9) requires taking a number of measurements per load and over a range of loads. The critical load is then defined in a statistical sense [26]. In our experiments, the critical loads were determined from the change in slope reflected in the load±displacement curve (see Section 3.3.1). Another, more common, method to estimate the fracture toughness and the residual stress is to consider the length of the radial cracks due to indentation with a Berkovich or a Vickers indenter. Various expressions have been suggested to relate the fracture toughness to the load, crack length and materials properties [27±29]. The most commonly used relationship for the formation of radial cracks is [30]: KIc ˆ wr

P c3=2

‡ Zsr c1=2

(10)

where c is the crack length and Z the crack shape factor. The parameter wr, which is related to the size of the plastic-zone, is given for a Berkovich or Vickers type indenter by [31]:  1=2 E wr ˆ w H

(11)

with w ˆ 0:016 for both a Berkovich and Vickers type indenter. The results are only reliable when the radial cracks are significantly larger than the lateral size of the plastically deformed zone. Furthermore, if the crack size is comparable to the scale of the microstructure of the indented material (e.g. grain size), the method is not applicable due to the interaction between the crack and the microstructure. The parameter Z in Eq. (10) depends on the shape of the radial cracks. The appropriate expression for Z is approximately given by [30]: p Z ˆ 1:12 p

d=c …3p=8† ‡ …p=8†…d=c†2

(12)

It is usually impossible to determine the depth of the radial cracks d from an experiment, although it may be done by post-fracturing the sample. It is commonly assumed that the crack has a semi-circular

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shape, i.e. the depth of the crack d is equal to the crack length c. In that case, Z takes the value 1.26. In the case of cracking in coatings, on the other hand, there is another way to estimate d, which will be discussed in Section 3.3.4. Returning to Eq. (10), it is clear that both the fracture toughness and the residual stress can be determined by carrying out indentations at various loads and plotting wr P=c3=2 against Zc1=2 . A relationship similar to Eq. (10) also holds for cone cracks (Fig. 4), where the crack length has to be substituted by an imaginary cone depth equal to c ‡ R0 =cos a [32], where c is the depth of the cone crack, R0 is the radius of the surface cone and a the cone angle (see Fig. 4). However, the parameter wc, which has to be substituted for wr, has not been calibrated for cone cracks and a general derivation of the crack shape parameter Z is not available due to the complicated crack shape (see Section 2.4.4 for more details). Finally, based on the energy-balance concept proposed by Griffith [33], we know that the measured fracture toughness can be transformed to the fracture energy G, which yields: Gˆ

KIc2 E

(13)

2.3. Indentation of coatings 2.3.1. General behaviour A general overview on the indentation technique has been given in Section 2.2. In this section, we will look more specifically at the application of indentation to thin coatings. The indentation is carried out into a coating of thickness t, which is adherent to a substrate. The resulting load± displacement curve can be analysed as outlined in the earlier sections. In the case of a thin coating, however, the response will be a combination of both coating and substrate behaviour. This means that the parameters obtained are actually a combination of the properties of the coating and the substrate, i.e. the coating properties will dominate for shallow indentation depths, and the substrate will dominate for deeper indentations. As a rule-of-thumb, the indentation depth should be less than one tenth of the coating thickness in order to measure the properties of the coating without significant interference of the substrate. This is just a rough guideline, since blunt indenters, such as spheres, have a larger contact area at that particular indentation depth, so that the substrate has more influence on the measured properties than for sharper indenters [3]. The rule-of-thumb is certainly not true for hard, stiff coatings on soft, compliant substrates: for these systems the substrate effect is already present at much smaller relative indentation depths [3]. Other effects that can influence the results are cracking and delamination of the coating, occurring in response to the indentation stress [3,4,18]. On the other hand, these effects can be used to determine the fracture toughness of the coating and interface, and residual stress in the coating, as will be shown in the following sections. An example of possible fracture events is shown in Fig. 5. This figure depicts load± displacement curves as well as optical micrographs of the surface of a sol±gel coating on a glass substrate, indented using a Berkovich indenter. At low loads radial-cracking occurs in the coating. As the load is increased, delamination of the coating from the glass substrates occurs, as is clearly visible as white areas in the micrograph. The initiation of radial-cracking and delamination is visible in the load±displacement curve as slight discontinuities (see Section 3.3.1). Then, at a certain critical load, chipping occurs, i.e. pieces of coating are removed. This is clearly reflected as a discontinuity in the load±displacement curve. An AFM image of a radial crack is shown in Fig. 6, where deformations due to crack closure after unloading can also be seen.

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Fig. 5. Images observed using optical microscopy and load±displacement curves for a 2 mm thick sol±gel coating after loading to 40 mN (a); 100 mN (b) and 300 mN (c) [3].

The critical loads at which these fracture events occur depend on the material properties of the coating and the substrate, on the coating thickness and on the indenter geometry. For some coating±substrate combinations, fracture may not occur at all. In the following sections, we will also discuss how the cracking events influence the effective, i.e. measured, elastic modulus and hardness. The load±displacement curve itself can also be used to quantify the mechanical behaviour of the coating±substrate system. As mentioned in Section 2.2.1, the load±displacement behaviour for an ideal Berkovich or Vickers type indenter can be described using the following relationship: P ˆ Kh2

(14)

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Fig. 6. AFM image of indentation at a load of 200 mN into a 4.5 mm thick sol±gel coating [4]. Deformations inside the radial crack are due to closure of the crack after unloading.

where the parameter K depends on the elastic modulus and hardness, see Eq. (3). In the case of depth-independent material properties and an ideal indenter, K is a constant. For indentations in coatings, K is not constant because of the transition from a coating-dominated to a substratedominated response with increasing depth. The observed change in K allows this transition to be quantified [34,35]. In fact, in the transition range the parameter K is a function of the indentation depth, thus, the parameter K incorporates the dependency on the indenter shape and the tip radius as well as the dependency on the systems properties, and it can be stated that the square dependency is generally valid also for coated systems for small or large indentation depths where either the coating or the substrate dominates the behaviour. Fig. 5 shows that the load±displacement behaviour is also sensitive to cracking. In fact, the ratio P/h2 was used by Hainsworth et al. [36] and the derivative @P=@h2 was used by McGurk and Page [34] to detect cracking and yielding events during indentation. These events show up as peaks in the @P=@h2 versus h curve, as will be illustrated using a particular example in Section 3.3.1. Eq. (14) becomes more complicated in the case of a non-ideal indenter geometry [17,18]. However, the effect on the derivative @P=@h2 is merely a monotonous change as a function of indentation depth. This effect can be discarded when only peaks in this derivative are used to analyse fracture or yielding events [18] (see Section 3.3.1). 2.3.2. Elastic modulus The elastic modulus can be deduced from indentation experiments as discussed in Section 2.2.2. As mentioned earlier, this elastic modulus is a combination of the moduli of the coating and substrate, with the substrate having an increasing influence for larger loads, since the elastically deformed volume underneath the indenter will then increasingly penetrate into the substrate. The onset of the substrate influence obviously depends on the elastic moduli, hardness, coating thickness, applied load and indentation depth, as well as the indenter shape. The spatial extent of the

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deformation field (and, hence, of the stress field) scales roughly with the size of the contact area between the indenter and the surface. This means that we can estimate the material properties of thin coatings most accurately by using very small indentation depths (relative to the coating thickness) and a sharp indenter. In the literature, various models have been proposed to describe the combined influence of the coating and substrate on the effectively measured elastic modulus as a function of the indentation depth. One model that is widely used is the empirical relationship proposed by Doerner and Nix [37]: 1 FDN 1 FDN ˆ ‡ E Ec Es

(15)

in which the subscripts c and s indicate coating and substrate properties, respectively. The asterisk denotes an effective, composite property. The substrate properties can be determined separately by carrying out experiments on the uncoated substrate. The weight function FDN is defined as [37]:   bt FDN ˆ 1 exp (16) h where t is the coating thickness and b is a constant. Fig. 7 shows an example of a fit to indentation measurements with various indentation loads, on a relatively compliant sol±gel coating on a glass substrate. It should be noted that the constant b in Eq. (16) appears to depend on the coating thickness [4,38]. In addition, the Doerner and Nix fit may become inaccurate at low indentation loads when the elastic response is dominated by the coating [3,4]. If the relationship is scaled with respect to the contact radius instead of indentation depth, a more universal description is possible [3,38]. Relationships similar to the Doerner and Nix [37] have been proposed; an overview is given by Mencik and Swain [39]. It is always preferable to determine the elastic modulus of the coating from the measured plateau value at low loads and not from an overall fit. As mentioned earlier, cracking affects the measured effective elastic modulus. The effective stiffness of the coating±substrate system decreases due to cracking, so that the measured modulus is

Fig. 7. Effective elastic modulus E vs. hc/t measured using a Berkovich indenter for a sol±gel coating with a thickness of 0.5 mm (triangles), 2 mm (squares) and 4 mm (stars), respectively [3]. The line is a fit to the Doerner and Nix model [30] to the data of the 4 mm thick coating.

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lowered [3]. Cracking often results in discontinuities in the measured elastic modulus versus depth curve, so that Eq. (15) cannot be applied. 2.3.3. Hardness As in the case of the elastic modulus, the combined responses of the coating and the substrate are actually measured when determining the hardness of thin coatings using indentation. This is caused by the fact that plastically deformed volume under the indenter extends into the substrate when the load (and, hence, the displacement) reaches a certain level. In general, the influence of the substrate on the measured hardness will be less than its influence on the measured elastic modulus at equal indentation depth, since the plastic deformation field has a smaller spatial extent than the elastic deformation field. Several relationships have been proposed for modelling the combined effect of substrate and coating on the measured composite hardness. For example, Bhattacharya and Nix [40] proposed the following relationship:   n  h  (17) H ˆ Hs ‡ …Hc Hs † exp an t For hard coatings on soft substrates, n ˆ 1, whereas for soft coatings on hard substrates n ˆ 2. The constant an can be expressed in terms of the elastic moduli and yield strength of the coating and the substrate, but, in practice, it is used as a fitting parameter. Eq. (17) is a purely empirical relationship, which was found to correlate well with finite element simulations [40]. The equation does not take the influence of fracture into account. Another expression was developed by Ahn and Kwon [41] on the basis of the plastic-zone volume of the coating (Vc) and the substrate (Vs) using elastic±plastic indentation-theory. Their final formula takes the form [41]: H ˆ

Vc Vs Hc ‡ Hs V V

(18)

where V is the total plastically deformed volume. Explicit expressions for these ratios, in terms of coating and substrate properties, are given by Ahn and Kwon [41]. Similar models have been proposed by others, e.g. see [42]. The drawback of this approach is that the explicit expression for the ratios is quite complicated, and the fitted coating hardness is sensitive to the quality of input data. A direct experimental estimate of the respective deformed volumes is quite laborious. Nevertheless, the approach has been successful in separating out the coating hardness when plasticity dominates, i.e. when there is no fracture [41]. Finally, Korsunsky et al. [42] proposed the following for the composite hardness of a hard coating on a soft substrate: H  ˆ Hs ‡

Hc

Hs

1 ‡ k…h=t†2

(19)

This relationship, which was derived considering work of indentation, should according to Korsunsky et al. apply equally well to coatings with and without cracking. The physical meaning of the parameter k depends on whether the deformation is plasticity-dominated or fracture-dominated, see [42]. In practice, it is used as a fitting parameter. Although originally derived for a hard coating on a soft substrate, Eq. (19) was also successfully applied to soft coatings on hard substrates [43].

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Fig. 8. Comparison of fits of Eq. (17) (line), Eq. (18) (small dashing) and Eq. (19) (large dashing) to the effective hardness H as measured using a Berkovich indenter for a 0.5 mm thick sol±gel coating [3].

In fact, it has been observed previously that only the data for very thin sol±gel coatings (thickness 0.5 mm) could be described using the above equations [3], as shown in Fig. 8, where the three models given by Eqs. (17)±(19) are fitted to the measurements. For thicker coatings (2 and 4 mm), in which radial-cracking occurred, the data could not be fitted by the equations, although the model by Korsunsky et al. [42] should explicitly take fracture of the coating into consideration. This shows that we should be careful in applying the models to coatings in which cracking occurs. 2.3.4. Fracture toughness One method of determining the fracture toughness of the coating, KIc, and the residual stress in the coating, sr, is to use the method based on radial-cracking outlined in Section 2.2.4. This model was developed for monolithic materials, and a suitable extension to coatings on substrates is not available. The model can strictly only be applied to coatings if the stress intensity is not influenced by the substrate, i.e. if the radial cracks are confined to the surface of the coating. The model may also be used for deeper cracks if we assume, as a first approximation, that the main influence of the substrate is to change the shape of the radial cracks. In that case, we can no longer assume that the radial cracks are semi-circular, and the parameter Z in Eq. (10) will not be 1.26. If there is a way to determine the ratio between the crack depth and the crack length, d/c, for a particular coating, then the proper parameter Z can be determined from Eq. (12), and the model can be applied (see also Section 3.3.4). We stress that this procedure is only approximate but it is the best one can do presently. On the other hand, knowing the correct value of Z is only important when determining the residual stress: from Eq. (10) we can see that the fracture toughness can be determined without knowing the parameter Z and the correct crack shape is, therefore, not needed. Eq. (9) was derived using the hardness as defined as load divided by area. If now the area of contact is modified by the fact that the compliance is influenced by the underlying substrate, the hardness as defined by load divided by area is changed and, thus, the effective hardness has to be incorporated into Eq. (9). This interpretation is tested and proven later in Section 3.3. Thus, the fracture toughness can also be estimated using Eq. (9), provided that the hardness and the load at which radial-cracking initiates are known. However, since the derivation of this relationship is then based on the effective hardness, i.e. H, when considering coatings, H has to be substituted for H in

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the equation (see Section 3.3.4). Another method to determine the fracture toughness is based on the critical thickness and will be outlined later (see Section 2.3.5). An alternative approach to assess the coating fracture toughness is based on the analysis of the energy dissipated during indentation. In general, the definition of work, which also applies to indentation, is Z W ˆ P dh (20) During indentation, this work is transformed into elastic strain energy Uel, energy dissipated due to fracture Ufr, energy dissipated due to plastic deformation (yield) Upl and thermal energy (heat) D, giving W ˆ Uel ‡ Ufr ‡ Upl ‡ D

(21)

The elastic strain energy is reversible while the other contributions are irreversible. The amount of energy dissipated during a complete loading±unloading cycle equals Wirr ˆ Ufr ‡ Upl ‡ D, and is equivalent to the area between the loading and unloading curves. Plastic deformation usually initiates before cracking. In fact, for sharp indenters, fracture initiates at the edge of the plastic-zone [16]. The energy dissipated in plastic deformation increases with the plastically deformed volume, and therefore, after the onset of yield with increasing load. Fig. 5 shows how different fracture features may occur during indentation: in our experiments, they had a specific sequence as follows: first radial-cracking, then delamination, and finally chipping. Energy is dissipated in each of these fracture events. By considering the dissipated energy Wirr for indentation curves with increasing maximum depth, it is possible to separate these components. We consider dissipation into heat D to be negligible. Consider the example in Fig. 9, which shows the total dissipated energy Wirr during indentation as a function of maximum indentation load for a particular sol±gel coating on glass [44]. Thus, each

Fig. 9. Irreversibly dissipated energy Wirr as a function of the applied indentation load into a 4 mm thick sol±gel coating showing the changes introduced due to radial cracking, delamination and chipping [37].

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point in the figure was calculated from the area between the loading and the unloading curve. For the smallest loads, energy is dissipated in plastic deformation only. At a load of approximately P ˆ 15 mN, radial-cracking already starts, and acts as an additional mechanism for energy dissipation. Delamination adds another contribution to the total dissipated energy, here at approximately 40 mN. Finally, chipping occurs somewhere between 100 and 200 mN. It is clear from Fig. 9 that the chipping results in a sudden jump of the dissipated energy. Radial-cracking and delamination result in rather gradual changes of Wirr, since both of these cracking types grow gradually during the increase of the load. The onset of delamination changes the slope of the Wirr±P curve. An estimate of the fracture energy of the coating can be obtained by considering the energy dissipated into chipping, Ufrc , since chipping is associated with crack growth through the coating. Ufrc is obtained by determining the difference between Wirr before and after chipping from the jump observed in Fig. 9. The fracture energy, G, is then obtained by dividing Ufrc by the area of the crack in the coating created by chipping. This leads to Gˆ

Ufrc Npt0 Cd

(22)

where Cd is the diameter of the delamination crack that initiated the chipping and N is the number of chipped areas (usually three in experiments with a Berkovich indenter and one for scratch tests). The quantity t0 is the effective coating thickness, which accounts for crack propagation through the coating not perpendicular to the coating but at an angle d. Thus, t0 equals the coating thickness divided by sin d, where d is the average angle of the chipping edge. From Eq. (22), the fracture toughness of the coating can then be determined using Eq. (13), with the coating modulus Ec substituted by E. The energy dissipated in the propagation of the delamination crack can be related to the interfacial fracture energy of the coating±substrate interface. This will be discussed in Section 2.3.6. When the experiments are carried out with a controlled load, sudden fracture events will result in a change in displacement at a relatively constant load, as illustrated in Fig. 10a for chipping of a sol±gel coating on glass. The measurement was obtained using a scratch tester. A lower, and upper bound for the energy dissipated during this event, respectively can also be obtained by assuming that the material either behaves in a fully elastic or fully plastic manner before and after fracture. This is equivalent to assuming that either Upl or Uel equals zero in Eq. (21). If we also assume that, as a firstorder approximation, P  h2 during loading and unloading, this leads to the following limits for the energy dissipated during the fracture: 2 3 Pcp …hcp

h0 †  Ufr  Pcp …hcp

h0 †

(23)

where h0 and hcp are the indentation depths before and after the fracture event, respectively. The lower limit follows from the assumption of fully elastic deformation (see Fig. 10b) and the upper limit from assuming fully irreversible behaviour (Fig. 10c). Thus, in the case of chipping, Eq. (23) can be used to obtain limits for the fracture energy of the coating. When the displacement is controlled during the experiment, on the other hand, sudden fracture results in a change in the load at a constant displacement (see Fig. 10d). In that case, no work is done by the indenter during fracture, i.e. W ˆ 0 in Eq. (21). The energy dissipated in the fracture must then be supplied by a decrease in elastic strain energy Uel. Assuming fully elastic behaviour during fracture, i.e. U pl ˆ 0, Eq. (21) leads to the following approximation for the upper bound of the energy dissipated in the fracture: Ufr  13 hcp …Pcp

P0 †

(24)

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Fig. 10. (a) Curve of the normal load vs. the normal displacement obtained during a scratch test (f ˆ 0:46, thickness 11 mm). The horizontal displacement in the curve is an effect of chipping of the coating; (b) definition of parameters for load-controlled systems. The shaded area corresponds to the energy on the basis of elastic deformation of the system; (c) the shaded area corresponds to the energy on the basis of plastic deformation of the system; (d) load±displacement curve under normal indentation defining the parameters for displacement-controlled systems (f ˆ 0:52, thickness 7 mm). Each individual load drop corresponds to one chipping event.

where Pcp and P0 are the loads before and after the fracture event, respectively and hcp is the associated depth. Again, we have assumed that P  h2 . In the case of chipping, Eq. (24) can subsequently be used to obtain an upper limit for the fracture energy of the coating. A somewhat different approach was followed by Li and co-workers [45,46]. Their method is also based on the determination of the energy dissipation during chipping in a load-controlled measurement. Their procedure is shown schematically in Fig. 11. The loading curve is extrapolated in a tangential direction from the starting point of the discontinuity (A) (which can be associated with the start of chipping), to the indentation depth corresponding to the end of the discontinuity (C). The difference between the extrapolated and the measured load displacement curve, i.e. the area ABC, is taken as a measure of the dissipated energy U. The fracture toughness is then calculated using " KIc ˆ

! #1=2 U 2 t0 nf † 2pCd

Ec N…1

(25)

where the parameters are defined as in Eq. (22). Eq. (25) has been used by various authors to determine coating fracture toughness [4,45,46]. We should note, however, that U is in principle not the dissipated energy. For example, if we consider

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Fig. 11. Schematic load±displacement curve showing the relationship between the energy estimated using the model by Li and co-workers [45,46] (area ABC) and the energy really dissipated in the purely elastic case during the loading cycle (area OAB).

purely elastic deformation, the dissipated energy equals OAB (Fig. 11), which, in general, is not equal to ABC. However, Eq. (25) often seems to give reasonable values for KIc, since the area ACB will actually be directly related to the area OAB. Based on geometrical analysis, the relationship between the areas for the elastic case, as illustrated in Fig. 11, is ACB/OAB ˆ …b2 =2a2 † 1. Thus, U is directly related to the lower bound of Eq. (23) (Fig. 10b). 2.3.5. Residual stress The residual stress in the coating can be estimated using the length of the radial cracks, as discussed in Section 2.2.4. The remarks made in Section 2.3.4 on the possible influence of the radial crack geometry also apply here. However, in contrast to the case of the fracture toughness, the crack shape parameter Z is critical for the determination of the residual stress using this method. Another way to assess the stress in thin coatings is to utilise the critical thickness of coatings. Although this method is not based on indentation or scratch testing, it is incorporated in the current section due to its ease of use and its relationship to the indentation data, allowing a comparison and calibration of the crack shape factor Z (Section 2.2.4). Coatings deposited on a substrate can exhibit a certain critical thickness. This is due to tensile intrinsic stresses, originating from the reactions during formation of the coating, and tensile thermal stresses, caused by differences in thermal expansion coefficients. In other words, if the thickness of the coating exceeds this value, the coating will fracture during deposition or thermal treatment, depending on whether the stresses are intrinsic or thermal. An example of a spontaneously cracked coating is shown in Fig. 12. The cracked coating shown in Fig. 12 is just one example of a pattern that can occur due to a thickness in excess of the critical thickness. Other failure modes that may occur are depicted in Fig. 13. Failure may occur within the coating, along the interface between the coating and substrate, or within the substrate. Which of these failure modes actually occurs depends on the relative fracture toughness of the coating, interface and substrate. The value of the critical thickness tc of a particular coating±substrate system can be determined using the following relationship [6,47,48]: tc ˆ

1 Ecf Gcf l s2r

(26)

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Fig. 12. Optical micrograph of the crack pattern that occur for a coating that exceeds the critical thickness as observed for a 10 mm thick sol±gel coating cured at 250 8C [44].

where l is a configuration dependent, non-dimensional cracking number and Gcf and Ecf are the relevant fracture energy and elastic modulus, respectively of the coating, interface or substrate. Eq. (26) gives a lower bound for cracking. The equation actually expresses that failure cannot occur as long as the elastic strain energy stored in the coating per unit area, given by ts2r =2Ec, does not exceed the fracture energy G multiplied by a proportionality factor of 1/2l. The value of l depends on the failure mode and on the elastic moduli of coating and substrate. Some indicative values are listed in Fig. 13. These values are approximations to more rigorous solutions for each case represented in the figure, but they may be used as ``rule-of-thumb'' values [6]. We should keep in mind that the incidence of cracking and delamination for tc < t is probabilistic in its nature and depends on the critical flaw size, e.g. on manufacturing defects. The following energies can be associated with Eq. (26), based on Fig. 13. For surface cracking and channelling, i.e. simultaneous perpendicular and in-plane extension, the relevant fracture energy in Eq. (26) is the coating fracture toughness, provided that there is no delamination at the interface; for delamination, it is the interface fracture energy, and for spalling, it is the substrate fracture energy. The substrate damage mode involves both the coating and the substrate fracture energies. Beuth [49] analysed the channelling case more rigorously, and obtained the following expression for the critical thickness: tc ˆ

2KIc2 g…Da ; Db †ps2r

(27)

The parameter g(Da, Db) takes the elastic mismatch between the coating and the substrate into consideration. It depends on the Dundurs parameters Da and Db, given by [50]: Da ˆ

E c Es E c ‡ Es

(28)

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Fig. 13. Sketches of possible crack patterns in a coated material and the associated cracking number l [6].

and Db ˆ

1 mc …1 2ns † ms …1 2 mc …1 ns † ‡ ns …1

2nc † nc †

(29)

with Ec  Ec …1 nc 2 †, Es  Es …1 ns 2 †, and where m is the shear modulus. The parameter g(Da, Db) is tabulated by Beuth [49] for a range of values of the Dundurs parameters. This theory is based on an elastic analysis, so if substantial plastic deformation occurs, the equations are not valid. Nevertheless, if the elastic moduli of the coating and substrate and the fracture toughness of the coating (or the fracture energy) are known, the residual stress can be estimated from the critical thickness. It is obviously also possible to estimate the fracture toughness of the coating from the critical thickness, provided that the elastic moduli of the coating and the substrate and the residual stress are known [51]. For transparent coatings, an estimate of the residual stress can also be obtained by using

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the dimensions of the cracked and delaminated areas, and by considering interference fringes as observed using optical microscopy [51]. 2.3.6. Adhesion Adhesion is an important property of thin coatings, but is also difficult to assess. This is reflected in the huge number of different tests to characterise adhesion: over 250 have been counted [52]. None of these techniques is perfect. In this section, we will discuss methods to establish adhesion based on indentation techniques. We define adhesion using a fracture mechanics approach: the adhesion energy or interfacial fracture energy of a coating±substrate system is the energy needed to propagate a crack along the interface between the coating and the substrate by a unit area, denoted by the fracture energy Gi [33]. In practical systems, it is not always clear whether coating±substrate failure is truly interfacial or is actually cohesive or partially cohesive in its nature (i.e. failure within the coating or the substrate) [6,53]. What often seems to be a pure interfacial failure from optical observation, turns out, after a chemical analysis, to be cohesive cracking: the crack has not propagated along the interface, but within the coating or the substrate, parallel to the interface and quite close to it. In other words, there is actually an ``interphase''. We will also categorise growth of cracks in this interphase as adhesion failure. The growth of a crack along an interface can occur under various loading modes, in particular the opening mode I and the shearing mode II [53]. We can associate interfacial stress intensity factors KI,i and KII,i with these modes. The relationship with the energy release rate that is available for interfacial cracking, Gi is: Gi ˆ

1 2 …K 2 ‡ KII;i † Eint I;i

(30)

where Eint, which can be interpreted as an interfacial elastic modulus, is defined by Hutchinson and Suo [6] as:   1 1 1 1 ˆ ‡ (31) Eint 2 Ec Es The phase angle, C, also called the mode mixity parameter, is often used as a measure of the relative amount of shearing mode II to opening mode I loading at the interface. It is defined by [6]:   KII;i C ˆ tan KI;i 1

(32)

Hence, for pure shear loading we have C ˆ p/2, and for pure mode I loading we have C ˆ 0. The condition for a crack to advance along the interface is Gi …C†  Gi [6]. The adhesion strength is the stress that the interface can withstand before fracturing. This strength is not only determined by the adhesion energy Gi, but also by the nature and the size of flaws that are present at the interface. Hence, although the adhesion energy may be high for a certain coating±substrate combination, for example, the adhesion strength can nevertheless be low if large flaws are present at the interface, due to processing. If interfacial cracking (delamination) occurs during indentation, it is possible to extract information about the interfacial fracture energy or interfacial fracture toughness from the indentation results. Let us consider the failure mode of the system in Fig. 5 as a first example. At low

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loads, radial-cracking is observed and, on increasing the load, delamination occurs. The radialcracking and the delamination are related in the sense that delamination most probably occurs due to radial cracks that reach the coating±substrate interface, and subsequently get deflected along the interface, thus, causing the delamination [4]. The delaminated areas, which have the form of a sector of a disk limited by the radial cracks, grow upon increasing the indentation load. At some point, the delaminated part of the coating will buckle in response to the in-plane stress caused by the indentation. This changes the mode mixity of the stress at the interface in such a way that the crack has a tendency to kink out of the interface and to grow through the coating towards the surface. At that point, chipping occurs [54]. Thouless [54] derived a model to determine the interfacial fracture energy for this failure mode. The basic assumptions are: (1) the growth of the delamination area is caused by the in-plane load on the delaminated sector due to indentation, and (2) the coating chips at the moment of buckling of the sector due to the same in-plane load. Hence, at chipping, the load driving the delamination equals the critical load for buckling. Thouless's derivation results in an expression for the interfacial fracture energy. This expression contains the geometry of the chipped area and the elastic modulus of the coating. Modifying Thouless's expression to account for the geometry of the chip as in Fig. 5, which is drawn schematically in Fig. 14, results in:     Ec t5 a=L ‡ bp=2 2 t…1 n†s2r 3:36…1 n†t3 sr a=L ‡ bp=2 ‡ ‡ Gi ˆ 1:42 4 L Ec L2 a=L ‡ bp a=L ‡ bp

(33)

in which L, a, and b define the geometry of the chipped piece in radians (see Fig. 14). The interfacial fracture toughness KC,i can be defined by invoking the interfacial modulus Eint, given by Eq. (31) as follows [6]: p (34) KC;i ˆ Gi Eint Comparison with Eq. (30) shows that KC,i is in fact a combination of the mode I and II fracture toughness. In principle, the mode mixity can be estimated from the angle of the chipping crack [53], so that KIc,i and KIIc,i can be separated. The interfacial fracture energy can also be estimated on the basis of the energy dissipated during indentation. This method has already been discussed with respect to the fracture toughness of the coatings in Section 2.3.4. The propagation of the delamination crack can be related to the interfacial fracture energy of the coating±substrate interface, Gi. The energy dissipated into delamination cracking can be estimated as follows. As mentioned in Section 2.3.4, the initiation of delamination results in a slight change in the slope of the Wirr±P curve (see Fig. 9). The curve before this point is extrapolated to larger loads and the energy dissipated into delamination Ufrd , at a certain delamination

Fig. 14. Sketch of a chipped area introducing the quantities used in Eq. (33).

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Fig. 15. Schematic cross-section for the type of delamination of a coating on which Eq. (36) is based.

diameter Cd, is approximated by the difference between the measured Wirr and the extrapolated value. The interfacial fracture energy is then given by: Gi ˆ

Ufrd pCd2

(35)

Another type of failure mode has been analysed in models proposed by Ritter and co-workers [55,56] on the basis of a derivation by Matthewson [57]. They assume that the delamination crack is axisymmetric, and no radial-cracking occurs, as illustrated in Fig. 15. The volume under the indenter is plastically deformed and the associated pressure is assumed to act on the partially delaminated coating. This pressure results in an in-plane stress in the coating that drives the delamination. Ritter and co-workers did not account for residual stresses in the coating. Their final relationship reads [55,56]: Gi ˆ

0:627H 2 t…1 Ec

n2f †

1 …1 ‡ nf ‡ …1

nf †HCd2 =P†2

(36)

where Cd is the diameter of the delamination crack, which has to be determined by (optical) microscopy. The fracture energy can be translated into the interfacial fracture toughness using Eq. (34). The failure mode assumed by Ritter and co-workers is sometimes observed in specific coating±substrate systems under a spherical indenter, but also under a Vickers indenter [55,56]. However, delamination due to a pyramidal indenter is usually accompanied by radial-cracking, so that Eq. (36) is not applicable. In conclusion, various models are available to determine the interfacial fracture energy from indentation-related delamination of coatings. However, the models have been designed for specific modes of failure, and we should take care when applying them to other situations. 2.3.7. Subcritical crack growth It is well known that cracks may grow slowly with time under subcritical conditions, i.e. when the energy available for cracking (the energy release rate G), is smaller than the fracture energy G. This is due to chemical effects at the crack tips in combination with a residual stress [58±60]. After a certain time, cracks will reach a critical length, whereupon catastrophic failure occurs in the form of channelling of cracks [49]. This channelling can be accompanied by delamination [51,61]. Subcritical crack growth (SCG) is an important effect with respect to the lifetime of products. SCG in thin coatings can be studied by observing the growth rate of radial cracks introduced in the coating by indentation. Thus, the length of the radial cracks has to be measured immediately after the indentation and at different times after the removal of the load [61].

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Fig. 16. Images of radial cracks in the surface of a sol±gel coating directly after the indentation, after 162 days and after 517 days. The coating thickness was approximately 9 mm [54]. The indents are marked by arrows.

Fig. 16 shows typical indentation marks and radial cracks for a particular sol±gel coating on glass, aged for various times t, after the removal of the load in ambient atmosphere. Here, we can see that the crack length indeed increases with aging time. The figure also shows that cracks prefer to grow away from each other due to the stress relaxation in the vicinity of an existing crack. Since no additional load has been applied the driving stress for crack growth, in this case is merely the residual stress in the coating [61]. The crack growth rate v is related to the crack length c and the time t(i) after indentation by vˆ

@c @t…i†

(37)

Under steady state conditions, v can be determined as vˆ

Dc Dt…i†

(38)

From the theory of rate processes and neglecting bond healing, the crack growth rate v can be described by the velocity for a reaction-controlled mechanism, which is given by [62±64]: 

 a0 …G G† v ˆ v0 exp kT

(39)

where v0 is a function of the bonding energy and temperature T, k is Boltzmann's constant and a0 is the activation volume. For G  G and a0 G=kT @ 1, Eq. (39) can be reduced to the widely used relationship [64]:  v

KI KIc

n

(40)

where n ˆ 2a0 G=kT which is generally supposed to be a constant for a certain materials system. For cracks that are significantly larger than the residual stress field around an indentation it follows from Eq. (10) that: KI ˆ Zsr

p c

(41)

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Since the fracture toughness is related to the critical thickness via Eq. (27) it follows from Eq. (40): !n p Z c p 0:5g…Da ; Db †ptc

v

(42)

p The further simplification to v  … c=tc †n is not generally valid. It has to be considered that the ratio d/c changes with progressive crack growth. Therefore, the parameter Z changes. In fact, Z ˆ Z…d=c†, where d is a function of d0, the initial depth of the crack after the indentation, which is in itself a function of load p P  and mismatch Ec/Es, i.e. d0 ˆ d0 …P; Ec =Es †. If c @ d and d ˆ t the relationship becomes v  … t=tc †n , as implicitly given in [49]. Furthermore, the parameter n is proportional to the fracture energy, thus, complicating the real dependency. 2.4. Principle of scratch testing 2.4.1. General principle In a scratch test, a well-defined tip is drawn over the surface of the coating while applying a particular normal-load P [65]. This normal-load can be constant during the entire scratch, however, it may also be increased during scratching from a low initial value to a maximum value at the end of the scratch. Various types of scratch tip are in use: spherical tips of various radii, conical tips of various top angles and pyramidal indenters. Lateral force transducers can be used to measure the lateral force F acting on the scratch tip, thus, allowing us to determine the friction coefficient using [66]: mˆ

F ˆ mA ‡ mP P

(43)

where mA and mP are the adhesion and ploughing friction, respectively. A reasonable estimate for mA is the measured friction coefficient for P ! 0, since ploughing is absent in that case and, thus, mP ˆ 0. For the case of purely elastic deformation of a monolithic material scratched with a spherical tip, explicit equations for the induced stress field have been derived by Hamilton [67]. Hamilton and Goodman [68] showed that, in the case of elastic sliding indentation, the lateral stress field is significantly influenced by the frictional force. In the case of normal frictionless spherical indentation, the Hertz solution is applicable (see Section 2.2.1). This solution shows a maximum tensile stress in the surface at the perimeter of the contact that may lead to cracking in the shape of a cone extending from the surface. In fact, Hertz [11] observed the formation of such cone cracks in glass specimens subjected to a normal-loaded spherical asperity. If an additional tangential loading component is applied, a partial cone crack can initiate due to the maximum tensile stress at the trailing edge of the spherical indenter [69]. The formation of these partial cone cracks will repeat with a regular interval (see Fig. 17) that depends on the load, the critical flaw size and the fracture toughness of the material [70,71]. The tensile stress at the trailing edge is intensified due to the sliding contact, whereas it is decreased at the front edge of the scratch tip [67]. This is illustrated in Fig. 18. According to Hamilton [67], the peak value is given by: sT ˆ

  3P 1 2n 4 ‡ n ‡ pm 2pa2 3 8

(44)

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Fig. 17. Partial cone cracks and associated delamination formed in a coating due to scratch testing with a 150 mm sphere (f ˆ 0:67, thickness 11 mm); (b) enlargement of the cracks formed at higher loads. The scratch direction is from left to right.

The occurrence of this maximum can obviously have a significant effect on cracking induced during scratching. Some examples of this are given in Section 2.5.1. Mechanical parameters such as elastic modulus, hardness and fracture toughness can be estimated from scratch test results. However, this is currently possible only when considerable simplifications are allowed: the elastic modulus is determined under the assumption of a purely elastic response, whereas for an estimate of the hardness, it is assumed that the behaviour is perfectly plastic. The results are obviously only approximations and indentation is preferred for a better quantitative determination of the parameters. We will nevertheless review the scratch testing methods in the following sections, to be able to compare the results with the indentation results. Another reason for applying scratch testing is that it offers an alternative for the determination of interfacial fracture energy of thin coatings (Section 2.5.4).

71

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Fig. 18. Ratio of tensile stress s to normal stress s0 vs. the position x relative to the contact radius a for elastic sliding with a spherical indenter [60,61].

2.4.2. Elastic modulus One result from Hamilton's analysis [67] is that the contact area, as well as the relation between the normal-load and the normal displacement, is hardly influenced by the additional stresses due to friction, for the purely elastic case (see Johnson [16]). Hence, Hertz's classical solution [11] for the deformation of a monolithic material under a sliding spherical tip can still be applied, and the elastic modulus can be determined from the loading curve using Eq. (1). However, the contact area will be increased by the adhesion between the sphere and the substrate at very low loads [72,73]. Moreover, plastic deformation will almost always occur at high loads, which also leads to an increased contact area [74]. If the calculation of the contact radius is based on the indentation depth, the use of the formula for elastic deformation will lead to an over-estimation of the elastic modulus. The result should, therefore, be considered to be an approximation. 2.4.3. Hardness Two parameters were suggested to quantify the hardness during scratch testing, namely, the scratch hardness HS and the ploughing hardness HP. Both have been shown to lead to values close to the indentation hardness of materials [75]. The scratch hardness, which is a measure of the resistance of the material to normal penetration, is defined as [66,75]: HS ˆ

P ALB

(45)

where ALB is the projected load bearing area. In the definition of ALB it is assumed that the material behaves perfectly plastic, thus, ALB ˆ 18 pw2

(46)

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where w is the local width of the scratch and it is assumed that the load is carried only by the front half of the spherical tip. The ploughing hardness, on the other hand, is defined as [66,75]: HP ˆ

mP P AP

(47)

in which AP is the projected contact area seen in the direction normal to that of the relative motion of the indenter. HP is a measure of the resistance of the material against ploughing. For perfectly plastic deformation: s  2  w  w w 2 1 2 R (48) AP ˆ R sin 2R 2 4 The geometrical relationship between the track width w and the penetration depth h is [76]: p (49) w ˆ 2 2Rh h2 In practice, the material's response will not be perfectly plastic, but will be affected by elastic effects as well as by ploughing. Hence, Eqs. (46) and (48) are just rough approximations of the actual contact areas. The influence of elastic effects is to decrease the contact areas, whereas ploughing will have the opposite effect. 2.4.4. Fracture toughness and residual stress Indentation under normal-loading with a sphere can lead to the formation of cone cracks (Fig. 4). Lawn et al. [77] suggested and confirmed by experimental observation that the addition of a frictional force on the stress field can be represented by a rotation of the cone crack axis. This gives rise to the observed shape of a partial cone crack during scratch testing (see Fig. 19). The dimensions of the normal cone crack are related to the fracture toughness (Section 2.2.4), although the application of the relation is hampered by the difficulty of determining the crack depth c. For partial cone cracks, a similar relationship can be derived following Lawn et al. [77]. Friction causes the cone axis to be rotated over an angle b relative to the free surface (see Fig. 19). The angle b is related to the friction coefficient as b ˆ arctan m. If b is larger than the cone angle a, this will lead to partial cone cracks which become visible as traces at the surface. The relationship between the cone crack depth c and the in-plane dimension cs is c ˆ cs sin b=sin a ˆ cs m…1 ‡ m2 † 1=2 =sin a.

Fig. 19. Schematic section through a cone crack for sliding contact of a sphere. The effect of the frictional force is to rotate effectively the load axis over an angle b [70].

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Thus, an approximate value of the cone crack depth c can be obtained from a measurement of cs of the partial cone crack. Using the approach of Lawn et al. [77], the following equation is then obtained for partial cone cracking during scratching: KIc ˆ wc

p 1 ‡ m2

 a 1=2  1=2 ‡ Zs …c † 1 ‡ r c cos a …1 ‡ …a=c cos a††3=2 …c †3=2 P

1

(50)

where it has been assumed that the cone crack radius R0 can be approximated by the contact radius a as given by the Hertz solution: a3 ˆ

3 PR 4 Er

(51)

In addition to the need to determine the parameters wc and Z, it remains difficult to determine the cone crack angle a. For monolithic materials angles between 20 and 308 have been reported [78]. Nevertheless, this shows that a plot similar to the one given in Section 2.3.4 for the formation of a radial crack may be used to estimate the fracture toughness and the residual stress from the formation of partial cone cracks. 2.5. Scratch testing of coatings 2.5.1. General behaviour In the scratch testing of thin coatings, the usual procedure is to move the scratch tip across the coated surface under an increasing load until, at a certain load referred to as the critical load Pc, a well-defined failure event occurs [65]. In conventional scratch testing, the critical load is often determined either by visual examination of the scratch track or by acoustic emission [65,79]. In modern systems, measurement of the load±displacement characteristics provides another simple means to determine the critical load, since the failure or detachment of the coating usually results in an abrupt change in the load±displacement characteristics [79]. This is particularly convenient, since imaging of the scratch is not necessary. An example is shown in Fig. 20. This figure depicts results of a scratch carried out on a particular sol±gel coating on glass. The normal-load was increased linearly during scratching. The top picture clearly shows the point where detachment of the coating was initiated during the scratch. The lower picture shows the normal displacement measured during scratching. The displacement clearly shows a discontinuity that corresponds to the onset of detachment, as can be seen by comparing the top and bottom pictures in Fig. 20. If the failure event represents coating detachment, as in the previous example, then the critical load can be used as a qualitative measure of the coating±substrate adhesion, as we will discuss in Section 2.5.4. However, a range of possible failure modes can occur. Bull [9] has made a classification of possible failure modes induced by scratching. The failure mode and the value of the critical load depend on the various parameters, namely, the properties of the substrate, the coating, and the coating±substrate interface, the coating thickness, the shape of the scratch tip, the loading rate, and the friction between the scratch tip and the coating surface [9]. The influence of friction on the critical load was discussed by Blees et al. [65]. Basically, we can again distinguish between two types of failure, namely, cohesive failure of the coating or substrate, and interfacial failure of the coating±substrate interface. Cohesive coating failure is often in the form of partial cone cracking, which sometimes penetrates the thickness of the coating, induced at the perimeter at the trailing edge of the scratch tip [69] (Section 2.4.1).

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Fig. 20. Optical micrograph of the scratch track (top) and the scratch depth vs. horizontal displacement curve (bottom) for a particle-filled sol±gel coating on glass. The length of the scratch is 1 mm while the normal load was increased linearly from 0 to 1 N during scratching. A spherical scratch tip with radius 150 mm was used.

An example of this kind of failure is shown in Fig. 17. Another type of cohesive coating failure is the so-called conformal cracking, where cracks form in front of the moving indenter, i.e. at the leading edge, rather than at the trailing edge [9]. This latter type of cohesive failure is often observed in relatively hard coatings on much softer substrates, in which case the coating is bent into the substrate. A frequently observed failure mode is chipping of the coating in front of the scratch tip, which is obviously related to interfacial failure. It is important to note, however, that the very moment of

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Fig. 21. Typical example of chipping that occurred during a scratch test of a 2 mm thick sol±gel coating in front of a 150 mm radius sphere [54].

chipping may not be directly correlated with interfacial properties (unless a ring crack in the coating occurs before the event), since the event of chipping itself is due to the kinking of a delamination crack back into and through the coating. The driving force for both the growth of a delamination crack and chipping is most commonly the in-plane compressive stresses generated in the coating in front of the scratch tip. An illustration of a chipped coating due to scratching is shown in Fig. 21. We can conclude that the failure that is actually observed depends on many aspects. A combination of failure modes usually occurs. Depending on the nature of the failure, coating or interface properties can be determined. 2.5.2. Elastic modulus and hardness The elastic modulus can be estimated from scratch testing as discussed in Section 2.4.2. When applied to coatings, the determined elastic modulus will be influenced by both the coating and the substrate properties, with the substrate having an increasing effect for larger scratch depths. The definitions of scratch hardness and ploughing hardness (Section 2.4.3) can be applied to scratch tests on coatings. The determined quantities are once again affected by both coating and substrate properties and are, therefore, effective values. 2.5.3. Fracture toughness At least two methods can be used to determine the fracture toughness from the load± displacement curve obtained using scratch testing with normal-load control. One is the model proposed by Li and co-workers [45,46], Eq. (25), and the other is Eq. (24). Both methods are based on determination of the energy dissipation due to crack growth, and have already been discussed for indentations (Section 2.3.4). The model proposed by Bhushan leads to a value that is proportional to the fracture toughness, whereas Eq. (24) gives an upper bound. Both methods rely on the occurrence of a clear chipping event during scratching. We should also note that in case of scratch testing, the work done by the scratch tip (W in Eq. (20)) now consists of two contributions, one due the normalload and another due to the tangential load: Z Z W ˆ P dh ‡ F dx (52)

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where x is the tangential coordinate. In practice, the tangential displacement (and not the tangential load) is usually controlled. Hence, a fracture event will not result in a significant change in the tangential load±displacement curve, and the second contribution in Eq. (52) is, therefore, usually negligible with respect to the first. 2.5.4. Adhesion Various models have been developed to determine the fracture energy of the interface from scratch testing. This technique is mostly used for qualitative purposes [9,80±82]. A review of existing models is given by Blees et al. [65]. The applicability of these models to scratch test results depends on the failure mode actually observed. To assess the actual applicability of the models, we must carefully establish whether the failure is interfacial or cohesive. In addition to the aspects discussed on the determination of the adhesion energy using indentation (Section 2.3), the location of the failure also has to be considered, i.e. compressive failure in front of the stylus or tensile failure behind the stylus. Most models in literature attempt to use chipping of the coating to estimate the adhesion energy. However, unless chipping occurs due to interfacial debonding after cohesive failure of the coating, these models will only give a rough estimate of the adhesion energy or a combination of the fracture energy of the coating and interface. In all existing models, a critical normal-load Pc is coupled to the interfacial fracture energy Gi. Three of the models proposed are outlined in the following sections. Burnett and Rickerby derived [83]: Gi ˆ

32 tP2c p2 Ec w4c

(53)

where wc represents the width of the scratch track at the critical load. In this model, the elastic± plastic indentation stress is considered to be dominant, which, according to Burnett and Rickerby [83], is only the case for a relatively low friction coefficient and large coating thickness. In the model by Bull et al. [82], it is assumed that coating detachment occurs when the in-plane compressive stresses in the coating in front of the indenter induce a critical stress normal to the coating±substrate interface due to Poisson's effect. The interfacial fracture energy from their model is given by [82]:   1 t n f mc Pc 2 Gi ˆ 2 Ec Ac

(54)

in which the cross-sectional area of the track at the critical load, Ac is defined by Eq. (48) at wc . Eq. (52) is derived for thin hard coatings on soft substrates, whereas the total indentation depth must be at least twice the coating thickness for the model to be applicable [82]. Attar and Johannesson [80] modified the model of Bull et al. [82] by assuming that the tangential force responsible for coating removal acts on the cross-section of the coating only, which results in [80]:   1 1 nf mc Pc 2 Gi ˆ 2 tEc wc

(55)

These three models basically assume that failure occurs due to chipping in front of the indenter, and that the elastic strain energy stored in the coating above the chipped area, right before chipping, is released by interfacial fracture at the critical load. However, as mentioned above, fracture usually

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also occurs in the coating at the moment of chipping, which leads to additional energy dissipation. The obtained fracture energies are, therefore, at least a combination of both the interface and coating fracture energy. The models are still useful in qualitative, rather than quantitative, assessment of film adhesion. Finally, the model derived by Thouless [54] (already discussed in Section 2.3.6) can also be used to determine the interfacial fracture energy from sliding indentation for the specific case of a coating that chipped due to buckling after the formation of radial (in-plane through-the-thickness) cracking. 3. Part II: Application to nano-particle filled sol±gel coatings on glass 3.1. Introduction In Part I of this paper we discussed methods to determine mechanical properties of thin coatings using indentation and scratch testing. In this Part II, we will apply these methods to coatings consisting of an organically modified silicate matrix filled with particles that were applied on glass substrates using a sol±gel process. Sol±gel-based silicate coatings are widely used to modify the surface properties of glass. However, the preparation of thick sol±gel coatings is hampered by tensile residual stresses that build up due to the thermal treatment during processing of the coating. These stresses can lead to cracking and delamination when the coating thickness exceeds a critical value [51,84]. This critical thickness can be increased by modifying the coating composition. One method is to bond organic groups to the silicon atom, which limits the extent of cross-linking, thus, reducing the residual stress and enabling the preparation of thicker coatings. Removal by thermal decomposition takes place at temperatures in the range of 200±600 8C [85], depending on the nature of the organic group [86]. The methylmodified silicate used here decomposes in air at 450 8C [87,88]. Another method to increase the critical thickness is to introduce colloidal particles into the coating. This not only leads to the reduction of residual stress, but can also increase the strength of the final coating [89]: the incorporation of particles may increase the energy dissipation and, hence, the strength of a material by crack deflection [90], micro-cracking [91], or differences in residual stress between matrix and particle [92]. In the coatings used in the present study, both methyl-groups and colloidal particles were incorporated to modify the structure of the coatings. We used silica particles with average diameters varying between 12 and 100 nm, and alumina particles with a diameter of about 200 nm. The volume fraction of the particles varied from 0 to 0.69. We determined the mechanical properties of the coatings using most of the methods outlined in Part I of this paper. We also discuss the influence of the nature and volume fraction of the filler particles on the mechanical properties of the coating. 3.2. Experimental The experiments were carried out using float-glass that was coated with a hybrid organic± inorganic coating. A hydrolysis mixture containing MTMS, tetraethylorthosilicate (TEOS), acidic acid and water was prepared. The total amount of MTMS and TEOS was 0.15 mol (0.147 mol MTMS and 0.03 mol TEOS for most experiments) to which 3:3  10 3 mol of acidic acid and 0.33 mol of water was added during vigorous stirring. Shortly after the addition of the water, the temperature rose up to 50 8C, and after about 5 min, the temperature slowly decreased. The solution was hydrolysed for 10 min.

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LUDOX TM-50 (previously Dupont, now Grace Davison, Columbia) was used in most experiments as the source of silica. This is a stable dispersion of small silica particles (50 wt.% silica, spherical particles of 20 nm diameter according to the manufacturer, 35 nm according to dynamic light scattering experiments). Ludox HS-40 (12 nm particles), Nyacol Grade 830 (12 nm, Eka Chemicals, Maastricht) and Nyacol Grade 9950 (100 nm) were also used to study the effect of particle size. Since the results for the different sets of silica fillers were observed to agree with each other, they are not mentioned separately. All as-received silica dispersions have a pH of 9±10. The silica dispersions were acidified by rapidly adding 10 wt.% of acidic acid to the silica dispersion to avoid immediate precipitation of the MTMS hydrolysis mixture due to the addition to a basic dispersion. The final pH of the silica dispersion was about 4. Depending on the silica to MTMS ratio, additional water was added to the silica dispersion to achieve a water to MTMS molar ratio of at least 6 in the final coating solution. The alumina dispersion was prepared as follows. Starting from 30 vol.% Al2O3 in water, the pH was adjusted to 1.5 by adding nitric acid, which equilibrated in time to a pH of 4. The alumina dispersion was prepared by ultrasonic treatment for 5 days using a Branson LTH 610-6 sonifier. The alumina used was TaiMei TM-DAR. The average particle size after dispersion was 200 nm according to dynamic light scattering experiments. The coating solution was obtained by adding the hydrolysis mixture to the dispersion during vigorous stirring. The amount of dispersion was adapted to obtain the desired particle volume fraction in the final coating. Shortly after mixing, the coating solution was centrifuged for 5 min to remove agglomerates and air bubbles. The coating was then deposited within 10 min. Coatings with a silica to MTMS weight ratio of 0 to 2.2 were produced; the corresponding particle volume fraction (assuming comparable densities for trimethylsiloxane and colloidal silica) is estimated to range from 0 to 0.69. The particle volume fraction for the alumina coatings ranged from 0 to 0.62. The coatings were applied to square B270 glass substrates (Schott, Jena, composition in wt.%: 70% SiO2, 17% Na2 O ‡ K2 O, 9% CaO, 1% BaO, 2% ZnO, 0.5% TiO2, 0.5% Sb2O3) of 100 cm2, which were approximately 2 mm thick. The substrates were cleaned by immersion for at least 24 h in a neutral soap solution (5% Extran). Just before deposition, the glass plates were rinsed with demineralised water, ethanol and isopropanol, respectively. The substrates were then dried in a furnace at 70 8C. In one set of experiments, the glass substrates were also pre-treated with aminosilane. Before depositing the solution by spin coating, the glass substrates were pre-wetted with ethanol for 15 s at 1000 rpm. The coating solution was then dispensed during 10 s at 100 rpm. The substrate was subsequently rotated for 120 s at 100±1600 rpm to obtain coatings of varying thickness. After deposition the coatings were dried on a hot plate at 90 8C for 1 min. The coatings were then cured at 160 8C for 30 min followed by curing at 350 8C for 1 h. The final coating thickness was between 1.5 and 18 mm. At least three coatings of different thickness were prepared at each particular volume fraction. The roughness of the layers (Ra) is 0.1 mm for the 1.5 mm coating and about 0.3 mm for coatings exceeding 5 mm. Roughness variations have a periodicity of about 100 mm. The measurements of the elastic, plastic and fracture related properties were not influenced by the roughness due to the low periodicity of the roughness variations. The coated glass pieces were cut into smaller samples for further experiments. Indentation experiments were carried out at room temperature and ambient atmosphere using an instrument designed and constructed at Eindhoven University of Technology. The maximum indentation loads ranged from 2 to 1000 mN. A Berkovich-type diamond indenter was used in most of the experiments. The instrument only allowed experiments under displacement control to be performed. A normal displacement rate of 10 nm/s was used in the experiments.

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The frame compliance Cf and the area function of the indenter tip were calibrated using the procedure suggested by Oliver and Pharr [7], applied to indentations on B270 glass. For this material, the elastic modulus was determined independently as E ˆ 75 GPa using the pulse-echo method. The compliance of the system was measured to be C f ˆ 300  10 nm/N. The area function of the Berkovich indenter was determined to be: A ˆ ah2c ‡ bhc , with a ˆ 24:5  0:5 and b ˆ 5:71  0:09 mm [4]. The scratch experiments were performed on a scratch tester that was designed and constructed at Philips Research Laboratories in Eindhoven. Spherical shaped sapphire indenters with a radius of 20 and 150 mm were used in the experiments. The compliance of the system was 710 nm/N. In the experiments reported here, the normal load was increased at a constant rate of 10 mN/s from 0 to 1000 mN, while the indenter was moved tangentially at a constant displacement rate of 10 mm/s from the relative position x ˆ 0±1000 mm. The tangential position, the scratch depth and the corresponding forces were recorded. The following procedure was used to determine the initial position of the sample surface relative to the indenter. The indenter was moved towards the surface in position-controlled mode with a constant normal displacement rate (0.5 mm/s), until a load of 2 mN was reached. This normal load was maintained for 2 s before continuing with the scratch test, in order to let the indenter settle on the sample surface. The corresponding position is considered to be the initial position of the sample surface. It is evident that this introduces a small error in the initial scratch depth. 3.3. Results and discussion 3.3.1. General observations Coatings of various thickness were deposited with a silica volume fraction ranging from f ˆ 0 to 0.69, and an alumina volume fraction ranging from 0 to 0.62. The properties of the silica-filled coatings were determined by most of the methods discussed in Part I. For the alumina-filled coatings, only the elastic modulus, hardness and fracture toughness were determined, to illustrate the influence of the filler properties. The critical thickness of the coatings increased with the filler fraction, as shown in Table 1. The alumina coatings were non-transparent, due to the relatively large particle size. An AFM image showing the silica particles embedded in the matrix is shown in Fig. 22. The silica-filled coatings were transparent for most of the filler fractions, but became milky at filler contents exceeding 0.67. Scanning electron microscopy revealed areas of lower density at f ˆ 0:67, shown in Fig. 23. A further increase to f ˆ 0:69 resulted in a white scattering coating with pores of about 0.5 mm in diameter (Fig. 24). A random close packing of spheres has a relative packing density of approximately 0.64, suggesting that the observed features are related to an insufficient amount of MTMS in the coating solution to fill the gaps between the particles. Various fracture phenomena were observed during indentation of the coatings using the Berkovich indenter. Radial cracking occurred at small loads, followed by delamination at higher loads and finally chipping (Fig. 5). Radial cracking, delamination and chipping resulted in discontinuities in Table 1 Critical thickness fsilica tc (mm) falumina tc (mm)

0 7 0 7

0.09 8 0.2 11

0.2 8±11 0.28 >11

0.36 11 0.32 >11

0.46 >11 0.42 >11

0.53 >11 0.43 >11

0.58 >11 0.5 >11

0.63 16 0.52 >11

0.64 >11 0.63 >11

0.67 >11 ± ±

0.69 >11 ± ±

J. Malzbender et al. / Materials Science and Engineering R 36 (2002) 47±103

Fig. 22. An AFM image showing the silica particles embedded in the matrix (f ˆ 0:52) [81].

the load±displacement curves, as illustrated in Fig. 25. The critical indentation loads for the various cracking phenomena were dependent on the coating thickness. Although delamination occurred at a lower load for the thinner coatings, the length of the radial cracks as a function of the applied load was constant, as long as the radius of the delaminated area was smaller than the length of the radial cracks. Chipping of the coating occurred at an indentation depth that corresponded to approximately the coating thickness. That the cracking occurred during loading was furthermore clarified for some measurements using in situ observation using a Vickers type indenter. The ratio P/h2 and the derivative @P=@h2 can be used to analyse indentation data in more detail (see Section 2.3.1). This is illustrated in Fig. 25 [93]. Although kinks can be seen in the ratio P/h2, they show more clearly as minima in the @P/@h2 curve at values of h2 of approximately, 1.4 mm2 (P  30 mN), 2.5 mm2 (P  50 mN) and 3.8 mm2 (P  75 mN) (Fig. 25c). Only the feature at 50 mN was clearly evident from the original data. The features, marked with arrows in the figure, are associated with the onset of radial cracking, delamination and, probably, substrate yielding. Cracking and delamination were established from microscopic analysis of indentations made at various loads

Fig. 23. SEM image showing local void formation in a silica-filled coating of f ˆ 0:67.

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Fig. 24. SEM image showing multiple void formation in a silica-filled coating of f ˆ 0:69.

Fig. 25. (a) Load±displacement (P±h) curve for a 3.5 mm thick coating (f ˆ 0:52); (b) P/h2 vs. h2 and (c) @P/@h2 vs. h2. The minima in the @P/@h2 curve are indicated by arrows [86].

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between 0 and 1000 mN [3,4,18]. The third minimum is probably due to substrate yield. This is supported by the following rough estimate. The substrate will deform plastically if the stress exceeds the yield strength of the material, which can be approximated as H/2.8 [21]. This leads to a value of approximately 2 GPa for the glass substrate. In fact, in our measurements we observed that the indentation pressure was approximately 1 GPa at the contact depth that corresponds to the supposed substrate yield. The measurement was influenced by the fracture of the coating, however, which leads to a displacement of the indenter at constant load and thereby to an underestimation of the indentation pressure [3]. As in the case of indentation, different fracture features were observed during scratch testing. Partial cone cracks formed in the surface at low loads (Fig. 17). The in-plane length and the radius of these cracks changes with increasing load. At higher loads, the cone cracks were accompanied by delamination of the coatings, which was visible as bright spots for the silica-filled coatings (Fig. 17). The ratio of in-plane length to coating thickness at which the coatings delaminated was approximately constant. At still higher loads, the coating buckled and chipped (Fig. 21). Close observation of cone cracks suggests that their average radius decreases as the load is increased. In fact, at higher loads, the radius of curvature of the cone cracks (Fig. 17a and b) appears to be smaller near their centre. Recent in situ observations of scratch-testing using an optical microscope, clearly showed that the cone cracks do not form at the trailing edge at higher loads, but rather initiate under the indenter. Since the maximum tensile stress occurs at the trailing edge of a sliding sphere, we might expect cone cracks to initiate at this edge [67,68]. However, with an increasing load and friction coefficient, the stress under the trailing half of the indenter becomes also tensile (Fig. 18). When the corresponding stress intensity factor reaches the critical value, a cone crack initiates. As the in-plane movement of the indenter continues, the crack extends, where the direction of the crack tip extension always reflects the current contact radius of the indenter at this position. The outer radius of the partial cone crack should, therefore, correspond to the contact radius at the applied load, as might be estimated using Eq. (1), whereas the inner radius gives information on the applied stress intensity during crack initiation and, thus, on the critical flaw size. The exact position of crack initiation under the indenter is required to estimate the critical flaw size. This can be obtained via in situ observation. 3.3.2. Elastic modulus The procedure outlined in Section 2.3.2 was used to determine the elastic modulus from the indentation measurements at different indentation loads. In the analysis, a Poisson's ratio of 0.225 was assumed for the coatings. Typical results are shown in Fig. 26. Results are plotted as a function of the ratio of contact depth to coating thickness (hc/t). The data shown in Fig. 26 corresponds to unfilled coatings of 2 and 5 mm thickness, respectively. All of the coatings showed qualitatively similar results. Starting at a relatively small value at very low loads, the elastic modulus increases with increasing indentation depth due to the presence of the substrate. The increase becomes discontinuous at the moment of chipping. Fig. 26b is a magnification of the curve in Fig. 26a, for small indentation depths. The measured elastic modulus clearly reaches a plateau for hc/t < 0:1 for the 5 mm coating. This plateau value of E, therefore, corresponds to the coating elastic modulus Ec. Various models can be used to estimate the elastic modulus of a coating from the change of the apparent elastic modulus with the indentation depth (see Fig. 7). However, the most accurate value of the elastic modulus of a coating is obtained from a measurement not affected by the underlying substrate, typically implying that hc/t should be 0:5. Various models have been suggested to estimate the effect of filler particles on the composite elastic modulus [94]. The simplest ones are the Voigt [95] and the Reuss [96] model, based on assuming either equal strains or equal stresses in the particle and matrix, respectively. They read: E ˆ Em …1

f† ‡ fEf

(56)

and Eˆ

 1

f Em

f ‡ Ef



1

(57)

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Fig. 27. Elastic modulus E as a function of f for the alumina filled coating. The full line corresponds to a fit to the data of Eq. (56), the large dashing to Eq. (57), the smallest dashing to Eq. (58), the dashing of intermediate size to Eq. (59).

Here, Em is the elastic modulus of the matrix, and Ef that of the filler particles. In fact, these equations constitute an upper bound (Voigt, Eq. (56)) and a lower bound (Reuss, Eq. (57)) for the composite modulus. Ishai and Cohen [97,98] have a different model. These authors derived relationships by analysis of a cubic inclusion within a cubic matrix, with a uniform strain applied at its boundary. The model reads: " # f (58) E ˆ Em 1 ‡ m=…m 1† f1=3 where m ˆ Ef =Em . A similar model has been derived by Paul [99], who analysed a cubic inclusion within a cubic matrix, at whose boundary a uniform stress is applied. " # 1 ‡ …m 1†f2=3 (59) E ˆ Em 1 ‡ …m 1†…f2=3 f†

Fig. 28. Elastic modulus E as a function of f for the silica filled coating. The lines correspond to the fits defined at Fig. 27.

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Various investigations on glass particle-filled polymeric materials reported that the data could be described well by either of these models [94,100±103]. In our measurements, we assume that Em equals the value we measured for the unfilled MTMS coating, i.e. Em ˆ 5 GPa. For the alumina and silica filler particles, an Ef ˆ 400 GPa [104] and 72 GPa [105], respectively may be taken as an estimate. It has been reported that LUDOX gels cured at 1000 8C have an elastic modulus of 72 GPa [106]. In Figs. 27 and 28, we have plotted Eqs. (56)±(59), substituting the values given in the previous paragraph. The results for the alumina particles appear to be well described by Ishai and Cohen's model and by Paul's model for values of f up to 0.35, where a relationship to the particle percolation limit of the particles might be suggested. After this point, the measured modulus increases faster with f than predicted by the models. At the largest f, the measured modulus is approximately constant. This suggests that the maximum packing density is reached, implying that pores would be introduced upon a further increase of the volume fraction: the amount of binder material is then no longer sufficient to completely fill the voids in-between the particles. The data for the silica particles shown in Fig. 28 falls below the Reuss limit, which indicates that the value substituted for Ef is too large. There is evidence that LUDOX particles possess a microstructure showing interwoven filaments (width 0.1±1 nm), and that the particle size reduces due to thermal treatment [107], thus, indicating a porous structure. This results in a lower value for Ef. A fit of the Reuss model to the data leads to Ef ˆ 40 GPa, which corresponds to a porosity of between 19 [108] and 23% [109] of the silica particles. A fit of the Ishai and Cohen model leads to Ef ˆ 15 GPa, corresponding to a porosity of between 34 [108] and 53% [109]. This latter value of the porosity is unreasonably high, suggesting that the low value of the elastic modulus is a result of weak adhesion between the particles and the matrix. It is interesting to compare the elastic modulus of the coatings obtained from indentations with the value obtained from scratching. The effective elastic modulus obtained using Eq. (1) for silicafilled coatings of varying thickness is shown in Fig. 29 as a function of h. The results obtained using the 150 mm sapphire sphere show a relatively constant elastic modulus. Similar values are initially observed with the 20 mm sapphire sphere. However, a sudden decrease is observed at larger h values. This is again related to chipping of the coatings, which did not occur under the 150 mm sapphire sphere. The constant level agrees well with the values obtained in indentation experiments. We should also note the following. Firstly, the relatively high values at low indentation depths can be attributed to the non-ideal indenter tip radius, or to an increase of the contact area due to adhesion. Secondly, we did not observe an increase of the effective elastic modulus towards the value expected for the substrate. This is probably related to the formation of partial cone cracks in the surface of the coating. 3.3.3. Hardness The hardness was determined from the indentation measurements using Eq. (7). Typical results for the hardness as a function of hc/t are shown in Fig. 30. These results were again obtained for unfilled coatings of 2 and 5 mm thickness. Like the modulus, the hardness starts at relatively small values at a small indentation depth. Fig. 30b is a magnification of Fig. 30a, at low indentation depths. For hc =t  0:1, a relatively constant hardness value is measured. This means that the measured hardness H is equal to the coating hardness Hc in this range. Chipping results in a decrease in the apparent hardness, as is clear from Fig. 30a. For intermediate values of hc/t, the hardness increases with depth. However, the increase is lower than expected, due to the formation of radial cracks and delamination. In a similar way as for the elastic modulus, this is mainly due to the indenter displacement associated with the fracture event, but since the hardness is inversely proportional to the depth squared, the effect will be more pronounced. All coatings exhibited a similar behaviour.

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Fig. 29. Effective elastic modulus E as a function of the scratch depth obtained using Eq. (1) for coatings of different thickness (f ˆ 0:52) measured using a 20 mm sphere (a) and a 150 mm sphere (b).

For the hardness, the preferred value is once again the measured value at low indentation depths and not that of one of the models described in Section 2.3.3, since this requires no fitting parameters and the fitting will be sensitive to coating failure (see Fig. 8). In the following, the hardness value for hc/t  0:1 is, therefore, taken as the coating hardness. In Figs. 31 and 32, the coating hardness is plotted as a function of f for the alumina and silicafilled coatings. For the silica-filled coatings, the hardness shows an increase with f up to a volume fraction of about 0.58, whereas for the alumina-filled coating, the hardness increases up to 0.52. The effect on the hardness is more clearly evident for both the alumina and the silica-containing coatings, due to the stronger dependency of the hardness on the displacement. This decrease is related to the structure of the coatings, which was reported in Section 3.3.1. Scanning electron microscopy revealed the occurrence of porosity at a silica volume fraction of 0.67 (Fig. 23). A further increase of the volume fraction to 0.69 even resulted in a white scattering coating with pores of about 0.5 mm (Fig. 24). The decreasing hardness is, thus, directly related to the occurrence of porosity inside the coating. Porosity is indeed expected at these high filler contents. For example, the packing density of random close packing is 0.64 and for simple cubic packing it is 0.52. These values are in the range of filler contents for which we observed the start of the decrease in hardness. As in the case of the elastic modulus, various models exist to describe the hardness of particlefilled matrices [110±112]. However, most of these models lack a sound physical foundation and may be used merely as mathematical fitting models, since a simple combination of the hardness of the filler and the matrix is often used, without considering both the filler and the matrix yield. A model that does take yielding into account is given by Jancar et al. [113], who studied the yield strength of particulate reinforced thermoplastic composites. If we assume that the hardness is proportional to

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Fig. 30. Effective hardness H vs. hc/t for an unfilled coating of 2 mm (squares) and 5 mm (triangles) thickness, respectively; (b) enlargement for shallow indentation depths.

the yield strength (Eq. (8)), we may apply this model to our hardness measurements. Jancar et al. [113] considered both the case of no-adhesion and perfect adhesion between the filler and the matrix. Based on a modification of the Nicolais and Narkis model [114], they proposed the following for the no-adhesion case: H ˆ S…f†Hm …1

1:21f2=3 †

(60)

Fig. 31. Hardness H as a function of f for the alumina filled coatings. The line corresponds to a fit of Eq. (61).

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Fig. 32. Hardness H as a function of f for the silica filled coatings. The line corresponds to a fit of Eq. (61).

The `strength reduction factor' S(f) was determined by finite element analysis, and varies between 1 and 0.2, respectively for low and high volume fraction. The model Jancar et al. [113] proposed for perfect adhesion leads to H ˆ Ff2 H max ‡ Hm …1

Ff2 †

(61)

in which F is a pre-factor that depends on the average yielded volume of the matrix per particle, and is used in practice as a fitting parameter. Using finite element simulations, Jancar et al. [113] determined that below a critical interparticle separation (i.e. above a certain f), a maximum yield strength/hardness Hmax was achieved, for which they obtained H max ˆ 1:33H m for their specific matrix-filler combination. A fit of Eq. (61) to the results for the alumina filled coatings up to f ˆ 0:52 is shown in Fig. 31. In this fit, the maximum measured hardness of 1.6 GPa was used for Hmax, and F ˆ 3:5 was determined from the fit. A fit of Eq. (61) to the results for the silica-filled coatings up to f ˆ 0:58 is shown in Fig. 32. Here, the maximum measured hardness of 1.2 GPa was used for Hmax, and F ˆ 2:5 was determined. The values for F are similar to the value of 3.2 obtained by Jancar et al. [113]. As stated above, the change of the elastic modulus with the filler fraction suggests that there is no perfect adhesion between the matrix and the silica particles. The fact that Eq. (61) agrees well with the experimental data of the hardness suggests that the model is also valid for limited adhesion. The increase of the hardness with filler fraction for both the alumina and the silica filler particles is due to the fact that the elastic moduli and yield strength of both types of particle are larger than those of the matrix. Since alumina has a higher modulus and yield strength than silica (alumina, hardness ˆ 20 GPa [104], silica, hardness ˆ 6 GPa [105]), the increase in hardness is larger. When using scratching to determine the hardness, we have to distinguish between the scratching hardness HS and the ploughing hardness HP (see Section 2.4.3). In order to determine the ploughing hardness it is necessary to separate the adhesion and ploughing friction coefficient. At very low loads the friction coefficient showed a maximum, then it decreased. This maximum is related to the adhesion of the indenter to the surface, however, the current set-up of the scratch tester did not permit to analyse this peak in more detail. The friction coefficient showed a nearly constant level at loads between approximately 20 and 100 mN, which can be seen as the adhesive friction, which superimposes to the ploughing friction coefficient at higher loads. At higher loads the friction coefficient started to increase. The exact load/depth where this increase starts depends on the coating thickness which has been interpreted in our previous publication [115]. The coefficient of adhesive

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Fig. 33. Effective scratch hardness HS (a) and ploughing hardness HP (b) as a function of the depth for coatings of different thickness (f ˆ 0:52).

friction was determined as the average value for small loads and the ploughing friction coefficient was determined from the difference with the total friction coefficient. For silica-filled coatings of different thickness the effective values of HS and HP are shown in Fig. 33 as a function of h. All curves are similar. After an initial increase, a maximum is reached and, in a similar way as for the effective elastic moduli determined from scratch experiments, the effective hardness decreases abruptly due to chipping of the coating. A subsequent increase due to increasing penetration of the glass substrate is observed. However, the values are severely underestimated, as they are determined on the basis of the surface of the coating being the reference plane. At small scratch depths, the effective scratch hardness has a value comparable with the value expected for the indentation hardness. The effective ploughing hardness, on the other hand, is essentially zero at low loads. At higher loads, the values become comparable. This agrees with data for monolithic materials [75], where an approximately one-to-one relationship between scratch, ploughing and indentation hardness was reported. A comparison between hardness values obtained from scratch and indentation measurements would not be very useful. The hardness values obtained from scratch testing approach asymptotically zero for small loads, since in that regime elastic deformation under the sphere dominates. At higher loads the hardness increases but to a lesser extend since cone cracks appear behind the stylus. For scratch testing these hardness values are, thus, a measure of the stress. For a Berkovich indenter, however, even for the smallest loads plastic deformation dominated. For higher loads only the values expected for the coatings can be determined since for the Berkovich indenter radial cracks appeared under the indenter, leading to a constant measured hardness. 3.3.4. Fracture toughness and fracture energy If the modulus and hardness of the coating are known, the fracture toughness and the residual stress can be determined from the length of the radial cracks that form during an indentation experiment (see Section 2.2.3). However, it was not possible to determine KIc and sr using the data of the thin coatings, since the data range in that case is too limited due to the small load of chipping. Thus, only the thickest coatings of each composition have been used to determine the fracture toughness from the length of the radial cracks. The crack shape will be influenced by the presence of the substrate, and this should be taken into consideration in the crack shape factor Z (see Section 2.2.3). We estimated the value of Z from the onset of delamination. In our experiments, delamination can be considered to take place once the

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Fig. 34. wrP/c3/2 as a function of Zc1/2 for an 11 mm thick silica filled coating, f ˆ 0:63. The line represents a linear fit.

depth of the crack reaches the interface. By measuring the length of the radial cracks at this point, we obtained a value of d/c ˆ 0:21  0:02 for the ratio of crack depth to crack length. Thus, using Eq. (12), a value for the parameter Z ˆ 0:34  0:04 is obtained. A typical plot of (wrP/c3/2) versus (Zc1/2) is shown in Fig. 34 for an 11 mm thick silica-filled coating. The intercept with the ordinate axis for a linear fit to the data is KIc, and the slope is equal to sr, where a negative slope implies a tensile stress. The obtained fracture toughness values KIc are shown in Figs. 35 and 36 for the alumina and silica-filled coatings, respectively. The residual stress for the alumina and silica-filled coatings is plotted in Figs. 37 and 38, respectively. These latter results will be discussed in more detail in Section 3.3.5. For the silica-filled coating with f ˆ 0:18, the fracture toughness and residual stress could not be determined from the radial cracks, since it was not possible to prepare sufficiently thick coatings. This was due to the low critical thickness (as a result of the relatively low fracture toughness), the low residual stress, which led to comparatively short radial cracks, and the low interfacial fracture toughness, leading to large delaminated areas at comparatively low loads. Below, we will show that the fracture toughness of these coatings can be estimated using models based on energy dissipation. The effect of the particle filler fraction f on the fracture toughness will be discussed in Section 3.3.4.1. The values for the fracture toughness as determined from the radial cracks can be corroborated using Eq. (9), which gives a relationship between the critical load for radial cracking and the fracture toughness. The coating without filler particles showed a critical load for radial cracking of

Fig. 35. Fracture toughness KIc vs. f for the alumina filled coatings.

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Fig. 36. Fracture toughness KIc vs. f for the silica filled coatings.

Fig. 37. Residual stress sres vs. f for the alumina filled coatings.

Fig. 38. Residual stress sres vs. f for the silica filled coatings as determined from radial cracking, Eq. (10), and critical thickness, Eq. (27). The average uncertainty in the residual as determined using the radial cracking is 18%, whereas it is 50% for the data determined using the critical thickness, where the larger uncertainty for this method is a result of the difficulty to determine the critical thickness accurately.

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100  20 mN for a 1.5 mm thick coating, and 300  50 mN for a 4 mm thick coating. In the case of silica-filled coatings, the critical load was approximately 15  5 mN for a coating thickness of 1.5 mm, and increased to 33  5 mN for a coating thickness of 11 mm. These values can be compared with those calculated from the fracture toughness and the hardness of the coatings using Eq. (9). A critical load of 200  60 mN is calculated for the coatings without particles, and for the silicafilled coatings, a load of 30  12 mN. These values agree well with the critical loads observed experimentally for relatively thick coatings. For the thinner coatings, we must take into consideration that the stress field is modified by the underlying substrate even at the relatively low critical load, i.e. a part of the load is carried by the substrate. We determined experimentally, that the composite hardness for a 1.5 mm thick filled coating already increased by 30% at the critical load of 15 mN. When the composite hardness is used in Eq. (9) instead of the coating hardness, the agreement is also good for thin coatings. It seems reasonable to use the composite hardness, since the original derivation of the equation is based on the hardness as defined by Eq. (7). The thick coatings were not influenced by this effect, since fracture occurred at such a low indentation depth that the measured hardness was not affected by the substrate. The use of the composite hardness is supported by the fact that the measured critical loads and the values found from the determined fracture toughness agree well with each other. Determination of the fracture toughness should also be possible from the size of the partial cone cracks that are formed during scratching using Eq. (50). Again, the crack shape factor Z is determined from the onset of delamination (see Fig. 17), leading to a ratio of d/c ˆ 0:32  0:05. Thus, a value of Z ˆ 0:52  0:09 is obtained using Eq. (12), which is approximately 1.5 times larger than the value obtained for normal indentation. However, it should be noted that Eq. (12) is not derived for cone cracks, and thus, can only be used as an approximation. This implies that only a rough estimate is expected for the residual stress. The parameter c can be determined from the experimental relation d/c ˆ 0:32  0:05 at the onset of delamination. The parameter cs is approximately c/2 (see Fig. 17) and c  d, thus, c/ cs  1. Since b can be determined from the friction coefficient, i.e. b ˆ arctan m ˆ 25  38, this allows to estimate a as approximately 418, which is slightly larger than the values between 20 and 308 reported for monolithic p materials [78]. A plot of wc P 1 ‡ m2 =……1 ‡ …a=c cos a††3=2 …c †3=2 as a function of Z…c †1=2 …1 ‡ …a=c cos a††1=2 is shown in Fig. 39 for an 11 mm thick silica-filled coating (f ˆ 0:58), where it can be seen that the

Fig. 39. wc P…1 ‡ m2 †1=2 =…c3=2 …1 ‡ …a=c cos a††3=2 † vs. Zc1=2 …1 ‡ …a=c cos a††1=2 for scratch testing experiments for an 11 mm thick silica filled coating (f ˆ 0:63). The line represents a linear fit.

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Fig. 40. Fracture toughness for the silica filled coatings vs. f as determined using different methods. The data marked `A' correspond to the lower limit obtained using Eq. (23) from the scratch testing normal-load±displacement; `B' represents the change in Wirr from P±h curves before and after chipping; `C' is determined from the Berkovich indentation using Eq. (24), `D' is obtained from radial cracking. The average uncertainty in the fracture toughness determined using the radial cracks is 18%. For the energy methods labelled A, B and C the uncertainties are 15, 10 and 12%, respectively.

data is well described by this relationship. On the basis of the results obtained using the radial cracking under the Berkovich indenter, it was possible to determine that a factor of wc ˆ 0:011  0:002 leads to values of the fracture toughness and residual stress that agree with the results obtained using the radial cracking within 30%. Considering the approximate nature of the assumptions made in the calculation, this shows that a rough estimate of the fracture toughness and the residual stress can be obtained from scratch experiments. A calibration of the parameter wc is still necessary for monolithic materials, however, and it has to be proven that this factor is also valid for coated systems. The fracture toughness can also be estimated with the various methods based on the energy dissipation during loading and unloading (Section 2.3.4). The results of the methods are compared in Fig. 40. The data marked `A' was determined from the scratch testing normal-load±displacement, using the lower limit of Eq. (23), which should be adequate for the load-control that was used during this test. Curve `B' represents the results obtained from the method of calculating the change in Wirr from the indentation load±displacement curves before and after chipping (Eq. (22)). The data marked `C' is determined from the Berkovich indentation load±displacement curve using Eq. (24), the appropriate equation for displacement control used during these experiments, which gives an upper bound for the fracture energy. The results obtained from the radial cracking (already presented in Fig. 36) are marked `D' in Fig. 40. The results show a similar tendency for all four methods, however, the minimum in KIc is reached at a lower filler fraction (f ˆ 0:1) for the energy methods than for the radial cracking method (f ˆ 0:35). Although curve `A' should represent a lower limit for KIc, and curve `C' should represent an upper limit, the values are quite similar for the three energy methods, but are larger than the radial cracking results. We should note, that the model of Li and Bhushan (Eq. (25)), which is not depicted in the figure, gives a curve similar to the energy methods, but with values that are a factor of 1.5 larger. A possible cause of the energy methods overestimating KIc is the occurrence of additional energy dissipation, which cannot be separated from the fracture energy. Possible mechanisms for this are energy dissipation into plastic deformation of the coating due to buckling, substrate elasto± plastic deformation that occurs during the chipping, a contribution of interfacial fracture, and, particularly for scratch testing, friction, however, the agreement between the indentation and the

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scratch testing results suggests that friction is of minor importance. In fact, it was suggested in Section 2.3.6 that the buckling of the coating is the origin of the chipping. The buckling will lead to a significant bending of the coating and, thus, also a significant displacement of the indenter. It is, therefore, likely that the displacement, which is observed during the loading, is related to the buckling of the coating and not the entire energy is released in the chipping. Another source of error may be introduced by the assumption that the load is proportional to the displacement squared for both loading and unloading. Although the approximation is correct during loading for bulk materials, the relationship is just an approximation for the unloading curve and for coated materials. In any case, the methods appear to give a good indication of the fracture toughness, and they certainly are capable of tracking changes in fracture toughness. The methods should be further refined for an even better quantitative result. Some preliminary experiments we have carried out with an alternative method, i.e. four-point bend adhesion testing, give values within 30% of the data presented here, but more research is needed to verify this. 3.3.4.1. Influence of filler particles. The fracture toughness, as measured from the radial cracks, shows a remarkable behaviour as a function of filler content, as depicted in Figs. 35 and 36: for the alumina-filled coatings it is approximately constant, whereas for the silica-filled coatings there is a minimum at about f ˆ 0:35. The fracture energy G of the coating can be estimated using Eq. (13), using the measured fracture toughness in combination with the measured elastic modulus (Figs. 27 and 28). The result is shown in Fig. 41 for the alumina and in Fig. 42 for the silica-filled coatings. The fracture energy decreases with f for the alumina-filled coatings, whereas for the silica-filled coatings, it exhibits a minimum at approximately f ˆ 0:30. This means that the increase of the fracture toughness with f for the larger volume fractions is due to both the increase in the elastic modulus and fracture energy. The effect of filler particles on fracture toughness and fracture energy has been analysed in various reports [94,116]. In the literature, both a decrease [117±122] and an increase [123±125] of the fracture energy of composites with increasing filler content have been reported. Various models have been proposed to describe the effect (see [97,114,116,126] for some examples). The formulation of such a model is complicated by the fact that the influence of the particles is a combination of different effects and the available models are, therefore, quite specific. Thus, we will restrict ourselves to a qualitative explanation of the observed trends.

Fig. 41. Fracture energy G vs. f for the alumina filled coatings. The line represents a guide to the eye.

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Fig. 42. Fracture energy G vs. f for the silica filled coatings. The line represents a guide to the eye.

The following possible effects will decrease the fracture energy G. Firstly, the spatial limitation caused by the presence of the particles can result in a decrease of the plastic zone size at the crack tip, which is known to lead to a lower fracture energy [127]. Secondly, if the adhesion between the particles and the matrix is weak, preferential cracking through the particle-matrix interface can lead to a decrease in the composite fracture energy. Thirdly, if the particles themselves are significantly weaker than the matrix, cracking through the particles will result in a lower fracture energy. The particles in our coatings are relatively strong, so that fracture through the particles is quite improbable. This eliminates the third effect as an explanation for the observed decrease in G. Since both the silica and the alumina particles are relatively hard, a decrease of the plastic zone size may well occur. Cracking along the particle-matrix interface can also contribute to the observed decrease in G, particularly for the silica particles for which, as we have already suggested in Section 3.3.2, a relatively weak particle-matrix adhesion is present. For the alumina particles, occurrence of the latter effect is less certain. There are also effects that can increase the fracture energy. One such effect is the deflection of cracks around the particles, either through the matrix itself or along the particle-matrix interface. The fracture surface, hence, becomes rougher on a microscopic scale, which is known to result in an increase in the effective fracture energy. Another cause for enlargement of G is enhanced (elastic or plastic) deformation of the particles. A third effect that can increase G is friction between the particles and the matrix, which absorbs energy. Finally, the interaction between the crack front and the particles, known as pinning and bowing of cracks, is a further possible effect that can increase the fracture energy (see [128]). It is most probable that it is the first effect, i.e. crack deflection that occurs in our coatings. Large deformation of the particles will not occur, since they have relatively large elastic modulus and hardness. Friction plays only a role in systems with elongated filler particles, i.e. fibres. Crack pinning and bowing could also have an effect, however, this will only give a substantial contribution when the particles are well bonded to the matrix material [124]. This is not the case for the silica particles, for which the increase in G is observed. To summarise, the following explanation of the observed trends appears to be the most probable. The decrease of G with f for both systems appears to be due to a combination of two effects. The first effect is a decrease of the plastic zone around the crack tip. This effect becomes stronger with decreasing inter-particle distance, i.e. with increasing f. The second effect is an increasing contribution of particle-matrix interfacial cracking, at least for the silica-filled coatings,

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which initially decreases G, since the bonding between the silica particles and the matrix is relatively weak. In the case of the silica particles, the fracture energy starts to increase again when fracture around the particles increases the total fracture surface, i.e. the fracture surface becomes progressively rougher. The adhesion between the alumina particles and the matrix appears to be stronger, and therefore, less particle-matrix interfacial cracking occurs. Hence, the increase in fracture energy due to an increase in the fracture surface area that would accompany interfacial fracture does not occur for the alumina particles. 3.3.5. Residual stress The residual stress as obtained from the radial crack length is shown in Figs. 37 and 38. For the silica-containing coatings, we also determined the residual stress from the observed critical thickness using Eq. (27). The results based on the critical thickness are a factor of about 1.5 larger, however, the data sets show the same qualitative behaviour. It should be noted that the residual stress determined using the radial cracking is sensitive to errors in the crack shape factor Z. The residual stress determined from the critical thickness is proportional to the fracture toughness, thus, an overestimated residual stress might be an indication that the fracture toughness determined from the radial cracking is overestimated. Furthermore, a recent analysis of cracked coatings using boundary element methods led to 30% lower values for the proportionality factor between stress intensity and critical thickness [129], thus, raising doubts on the accuracy of the residual stress obtained using Eq. (27). An estimate based on this more recent analysis leads to values of the residual stress that agree, to within 10%, with those obtained using the radial cracking. Residual stress develops in the coating during processing due to intrinsic factors and due to differences in thermal expansion between the coating and the substrate during thermal processing. The latter contribution can be estimated as [130]: sr ˆ …ac

as †

Ec DT 1 nc

(62)

in which ac and as are the thermal expansion coefficient of the coating and substrate, respectively and DT is the temperature difference during processing, which is 330 K in our case. The quantities ac and Ec are determined by a combination of the filler and the matrix properties, and change with the filler volume fraction. At f ˆ 0, ac equals the thermal expansion coefficient of the unfilled MTMS/TEOS coating, which approximately equals 15  10 6 K 1. This value was determined from measuring the change in stress as a function of temperature in an unfilled MTMS/TEOS coating on a silicon wafer (determined from the measurement of the variation of the radius of curvature of the wafer). The stress changed linearly and reproducibly with temperature in the range of the measurement (50±3508C). Combining this with the layer thickness of the coating (ellipsometry) and the elastic modulus (indentation) gives the expansion coefficient. Since, in addition, as ˆ 10  10 6 K 1 [105], Ec ˆ 5 GPa, and nc ˆ 0:2, a residual stress of 10 MPa results from Eq. (62). Comparing this to the measured value of about 80 MPa, it is clear that a substantial part of the residual stress has an intrinsic origin. Addition of either silica or alumina particles to the MTMS/TEOS matrix causes a decrease in ac, which lowers the difference (ac as ). This will result in a smaller residual stress due to thermal mismatch, and therefore, lead to a larger critical thickness, provided that this effect is not compensated by the increase in elastic modulus with the particle volume fraction. Indeed, Fig. 37 shows a decrease in sr with the alumina filler volume fraction. The effect of the silica particles, as depicted in Fig. 38, is an initial decrease in residual stress followed by an increase

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above approximately f ˆ 0:35. The reason for this increase is as yet unclear, particularly since the contribution of intrinsic factors to the total residual stress was not quantified. Various models consider the effect of filler particles on the thermal expansion coefficient [130±132], but since the stresses are largely intrinsic their use here would not be adequate. 3.3.6. Interfacial fracture toughness Most existing models cannot be used to determine the interfacial fracture toughness (Sections 2.3.6 and 2.5.4). The failure mode must be considered before allowing the application of a particular model. For the sol±gel coatings, chipping occurred at a critical load during the indentation and scratch experiments in response to the induced in-plane compressive stress. The chipped areas of the coating had the form of disk sectors, as shown in Fig. 5 for indentation and in Fig. 21 for scratching experiments. In addition, chipping is preceded by radial crack formation and delamination, and the area of the chip is limited by the radial cracks. Thus, Eqs. (33) and (34), which have been derived to determine the interfacial fracture toughness under such circumstances, should be applicable. The apparent elastic modulus of the interface Eint, as defined by Eq. (31), is also used. The interfacial fracture toughness of the silica-filled coatings determined in this way is shown in Fig. 43 for both indentation and scratch experiments, which agree well with each other. For the sake of comparison, the results obtained using the model by Attar and Johannesson [80], Eq. (55), are also shown in Fig. 43. The values according to Eq. (55) are significantly larger than the fracture toughness of the coating. Possible reasons are that the radial cracks are not considered, and thus, the stress is overestimated, and that the model by Attar and Johannesson [80] estimates a combination of coatings and interfacial fracture toughness (Section 2.5.4). The interfacial fracture toughness for the coating on clean glass is, on average, 20% smaller than the coating fracture toughness, and shows the same dependency on the filler fraction, which suggests that the mechanisms are related to the same bonding mechanism. We repeated the measurements at a limited number of filler fractions for samples for which the glass substrate was pre-treated with aminosilane. The interfacial fracture toughness was estimated using Eq. (33). The results, depicted in Fig. 44, show a clear influence. The interfacial fracture toughness is significantly decreased for the pre-treated substrates. However, at the same time, the coating fracture toughness

Fig. 43. Interfacial fracture toughness Kint vs. f for the silica filled coatings as determined using Eq. (33) and the values determined using Eq. (55). The average uncertainty in the values determined using Eq. (33) on the basis of the radial cracks is 27%, whereas it is 25% for the scratch testing results. For the data determined using Eq. (55) the average uncertainty is 20%, however, the value determined using this method is only proportional to the fracture toughness (see text).

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Fig. 44. Interfacial fracture toughness vs. f obtained using Eq. (33) for coatings on clean (marked as triangles) and aminosilane pre-treated (stars) glass substrates. The lines are a guide to the eye.

turned out to be unaffected. This supports the assumption that the methods to determine the coating fracture toughness and the interfacial fracture toughness are independent. 3.3.7. Aging In order to determine the influence of the environment on the properties of the coatings, silicafilled coatings possessing a value of f ˆ 0:48 were aged for 30 and 90 days under ambient conditions. The properties of the coatings were then determined again as outlined in the earlier sections. From these measurements we can conclude that aging reduced the hardness as well as the elastic modulus of the coating by approximately 10% after aging for 30 days. Aging for 90 days reduced the elastic modulus by approximately 17%, whereas the reduction of the hardness remained at 10%. The fracture toughness of the coating and interface remained unchanged within the limits of uncertainty. This indicates that the indentation methods are capable of following changing properties with time. 3.3.8. Subcritical crack growth During storage of the coatings after indentation, under ambient conditions, the radial cracks grow due to so-called SCG (see Fig. 16). This was analysed for the silica-filled coatings. A general observation in relationship to the crack growth is that the size of the delaminated area does not increase with aging time. This suggests that the energy release rate at the interface was too low to result in interfacial crack growth. Optical images rather suggest that delaminated areas are partly closed during the aging process. This is not as surprising as it sounds in the first instance, since the direction of the bending due to the indentation stress field opposes the direction of the bending due to the residual stress. The cracked coating appears to be fully relaxed and the residual stress is not sufficient to result in a bending of the large cracked areas. The length of the radial cracks was measured immediately after the indentation and after 140 and 230 days at ambient conditions. The rate of crack propagation was estimated using Eq. (38). The dependence of the rate of crack propagation on the particle volume fraction for 7 and 11 mm thick coatings is shown in Fig. 45. It can be seen that the rate of crack propagation decreases with increasing filler content and decreasing thickness. It was shown Section 2.3.7 that the rate of crack propagation is related to the ratio of crack depth to crack length and to the critical thickness, however, the relationship is too complex to be analysed in more detail.

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Fig. 45. Crack growth rate vs. f for 7 and 11 mm thick silica filled coatings.

4. Conclusions Nano-indentation and scratch testing are powerful techniques to determine mechanical properties of thin coatings. The attractiveness of these techniques can be found in the simplicity of the measurement principle in combination with the wealth of information that can be extracted from the measurements. For nano-indentation, in particular, there are many methods to analyse the experimental results, which provide quantitative information about a range of mechanical properties. The elastic modulus and hardness of the coatings can be determined quantitatively by analysis methods that are now widely accepted as standard. For brittle coatings, the fracture toughness and fracture energy can be determined from nano-indentation results using various approaches. Relatively new in this respect are methods based on the calculation of the energy dissipated during the indentation process, as introduced and applied in this article. The latter methods are still open to further refinement, as indicated by our results, but the values obtained provide a good indication, and they can certainly be used to track changes in fracture toughness or fracture energy. The fracture toughness of the interface between the coating and the substrate can also be determined from nano-indentation on brittle materials in a way comparable to the determination of the fracture toughness of the coating itself. More work is needed to refine the models further. On the other hand, the agreement is within 50% and this is quite reasonable considering that the radial-cracking method itself is thought to be not much more accurate than that, even for bulk materials. Our measurement results show a good similarity between the indentation and the scratch results, which gives some confidence in the method. Also, a surface pretreatment of the substrate before applying the coating (which influences the adhesion) is clearly reflected in the results. However, it is still uncertain whether the obtained values are also quantitatively correct, and more research is needed to investigate this aspect further. Finally, nano-indentation results can be used to estimate the residual stress in coatings, where again different methods lead to results that differ by 50%, and to characterise their SCG behaviour. Analysis methods for scratch testing are still less well developed, due to the larger complexity of the test caused by the additional frictional force. The technique is, therefore, less reliable than nano-indentation for quantitative determination of parameters. Scratch testing is nevertheless very useful for comparative, qualitative studies and is, therefore, often employed in the analysis of the consistence of the properties of coatings during industrial production processes. The technique also

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offers an alternative to determine the fracture toughness of the coating±substrate interface, which is particularly convenient for coating±substrate systems for which the measurement of this parameter is impossible by nano-indentation. The practical use of nano-indentation and scratch testing was demonstrated for particle-filled sol±gel coatings. The mechanical properties of these coatings depend on the nature and volume fraction of the filler particles, i.e. on the microstructure. The influence of these microstructural changes can be tracked using nano-indentation and scratch testing. We were, therefore, able to map out the dependence of mechanical properties on the filler content for the coatings. An attempt to describe the dependence with the use of models taken from the literature shows both the merits and the limitations of these models. Acknowledgements The authors would like to thank the Philips Research Laboratories, Eindhoven, for the financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

H. Bach, D. Krause, Thin Films on Glass, Springer, Berlin, 1997. G.L. Smay, Glass Technol. 26 (1985) 46. J. Malzbender, G. de With, J.M.J. den Toonder, Thin Solid Films 372 (2000) 134. J. Malzbender, G. de With, J.M.J. den Toonder, Thin Solid Films 366 (2000) 139. G.M. Pharr, W.C. Oliver, MRS Bull. 7 (1992) 28. J.W. Hutchinson, Z. Suo, Adv. Appl. Mech. 29 (1992) 63. W.C. Oliver, G.M. Pharr, J. Mater. Res. 7 (1992) 1564. A.C. Fischer-Cripps, Vacuum 58 (2000) 569. S.J. Bull, Surf. Coat. Technol. 50 (1991) 25. B. Bhushan, Handbook of Micro/Nano Tribology, 2nd Edition, CRC Press, Boca Raton, 1999, p. 433. H. Hertz, Miscellaneous Papers, Mac Millan, London, 1896. J. Boussinesq, Application des Potentiels a l'Etude de l'Equilibre et du Movement des Solids Elastiques, GauthierVillars, Paris, 1885. I.N. Sneddon, Fourier Transforms, McGraw-Hill, New York, 1951. E.S. Berkovich, Ind. Diamond Rev. 11 (1951) 129. Product Information: Nano Instruments, 1001 Larson Drive, Oak Ridge, TN 37830, USA. K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1985. J. Malzbender, J.M.J. den Toonder, G. de With, J. Mater. Res. 5 (2000) 1209. J. Malzbender, G. de With, Surf. Coat. Technol. 127 (2000) 266. I.N. Sneddon, Proc. Cambridge Philos. Soc. 44 (1948) 492. M.M. Chaudhri, J. Mater. Res. 16 (2001) 336. D. Tabor, Hardness of Metals, Oxford Academic Publishers, Oxford, 1951. W. Yu, J.B. Blanchard, J. Mater. Res. 11 (1996) 2358. Y.-T. Cheng, C.M. Cheng, J. Appl. Phys. 84 (1998) 1284. R.F. Cook, G.M. Pharr, J. Am. Cerm. Soc. 73 (1990) 787. B.R. Lawn, A.G. Evans, J. Mater. Sci. 12 (1977) 2195. I.J. McColm, Ceramic Hardness, Plenum Press, New York, 1990. D.K. Shetty, A.R. Rosenfield, W. Duckworth, J. Am. Cer. Soc. 68 (1985) C65. Z. Li, A. Ghosh, A.S. Kobayashi, R.C. Bradt, J. Am. Cer. Soc. 72 (1989) 904. C.B. Ponton, R.D. Rawlings, Mater. Sci. Techol. 5 (1989) 865. D. Broek, Elementary Engineering Fracture Mechanics, Kluwer Academic Publishers, Dordrecht, 1997. D.B. Marshall, B.R. Lawn, J. Am. Ceram. Soc. 60 (1977) 86. B.R. Lawn, J. Am. Ceram. Soc. 81 (1998) 1977. A.A. Griffith, Philos. Trans. Roy. Soc. London A221 (1921) 163. M.R. McGurk, T.F. Page, J. Mater. Res. 14 (1999) 2283.

101

102

J. Malzbender et al. / Materials Science and Engineering R 36 (2002) 47±103

[35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91]

K. Tunvisut, N.P. O'Dowd, E.P. Busso, Int. J. Solids Struct. 38 (2001) 335. S.V. Hainsworth, H.W. Chandler, T.F. Page, J. Mater. Res. 11 (1995) 1987. M.F. Doerner, W.D. Nix, J. Mater. Res. 1 (1986) 6501. R.B. King, Int. J. Solids Struct. 23 (1987) 1657. J. Mencik, M.V. Swain, J. Mater. Res. 10 (1995) 1491. A.K. Bhattacharya, W.D. Nix, Int. J. Solids Struct. 24 (1988) 1287. J.-H. Ahn, D. Kwon, J. Appl. Phys. 82 (1997) 3266. A.M. Korsunsky, M.R. McGurk, S.J. Bull, T.F. Page, Surf. Coat. Technol. 99 (1998) 171. J.R. Tuck, A.M. Korsunsky, D.G. Bhat, S.J. Bull, Surf. Coat. Technol. 139 (2000) 63. J. Malzbender, G. de With, Surf. Coat. Technol. 135 (2000) 68. X. Li, B. Bhushan, Thin Solid Films 315 (1998) 214. X. Li, D. Diao, B. Bhushan, Acta Mater. 11 (1997) 4453. M.S. Hu, A.G. Evans, Acta Metall. 37 (1989) 917. M.S. Hu, A.G. Evans, M.D. Thouless, Acta Metall. 36 (1988) 1301. J.L. Beuth, Int. J. Solids Struct. 29 (1992) 1657. J. Dundurs, J. Appl. Mech. 36 (1969) 650. J. Malzbender, G. de With, Thin Solid Films 359 (2000) 210. K.L. Mittal, Adhesion Measurement of Films and Coatings, VSP, Utrecht 1995. J. Mencik, Mechanics of Components with Treated or Coated Surfaces, Kluwer Academic Publishers, Dordrecht, 1996. M.D. Thouless, Eng. Fract. Mech. 61 (1998) 75. J.E. Ritter, T.J. Lardner, L.G. Rosenfeld, M.R. Lin, J. Appl. Phys. 66 (1989) 3626. J.E. Ritter, L.G. Rosenfeld, J. Adhesion Sci. Technol. 4 (1990) 551. M.J. Matthewson, Appl. Phys. Lett. 49 (1986) 1426. T.A. Michalske, in: R.C. Bradt, R.E. Tressler (Eds.), Fractography of Glass, Plenum Press, New York, 1994, p. 115. R.F. Cook, E.G. Liniger, J. Am. Ceram. Soc. 76 (1993) 1096. T. Sung Oh, R.M. Cannon, R.O. Ritchie, J. Am. Ceram. Soc. 70 (1987) C±352. J. Malzbender, G. de With, J. Non-Cryst. Solids 275 (2000) 135. B. Lawn, Fracture of Brittle Solids, Cabridge University Press, New York, 1993. T. Sung Oh, R.M. Cannon, R.O. Ritchie, J. Am. Ceram. Soc. 70 (1987) C±352. M.A.H. Donners, L.J.M.G. Dortmans, G. de With, J. Mater. Res. 15 (2000) 1377. M.H. Blees, G.B. Winkelman, A.R. Balkenende, J.M.J. den Toonder, Thin Solid Fims 359 (2000) 1. S. Jacobsson, M. Olsson, P. Hedenqvist, O. Vingsbo, in: P.J. Blau (Ed.), American Society of Metals (ASM) Handbook, Am. Tech. Publ. 18 (1992) 430. G.M. Hamilton, Proc. Inst. Mech. Eng. 197C (1983) 53. G.M. Hamilton, L.E. Goodman, J. Appl. Mech. 88 (1966) 371. B.R. Lawn, Proc. R. Soc. 299 (1967) 307. M. Keer, C.H. Kuo, Int. J. Solids Struct. 29 (1992) 1819. A.F. Bower, N.A. Fleck, J. Mech. Phys. Solids 42 (1994) 1375. K.L. Johnson, K. Kendall, A.D. Roberts, Proc. R. Soc., Lond. A 324 (1971) 301. J. Malzbender, G. de With, Surf. Coat. Technol. 124 (2000) 66. J. Malzbender, G. de With, Wear 236 (2000) 355. J.A. Williams, Tribol. Int. 29 (1996) 675. I.N. Bronstein, K.A. Semendjajew, Taschenbuch der Mathematik, Harri Deutsch, Frankfurt, 1987. B.R. Lawn, S.M. Wiederhorn, D.E. Roberts, J. Mater. Sci. 19 (1984) 2561. C. Kocer, R.E. Collins, J. Am. Ceram. Soc. 81 (1998) 1736. T.W. Wu, J. Mater. Res. 6 (1991) 407. F. Attar, T. Johannesson, Surf. Coat. Technol. 78 (1996) 87. M.J. Laugier, J. Mater. Sci. 21 (1986) 2269. S.J. Bull, D.S. Rickerby, A. Matthews, A. Leyland, A.R. Pace, J. Valli, Surf. Coat. Technol. 36 (1988) 503. P.J. Burnett, D.S. Rickerby, Thin Solid Films 154 (1987) 403. A. Atkinson, R.M. Guppy, J. Mater. Sci. 26 (1991) 3869. N. Tohge, K. Tadanaga, H. Sakatani, T. Minami, J. Mater. Sci. Lett. 15 (1996) 1517. M.R. Bohmer, A.R. Balkenende, T.N.M. Bernards, M.P.J. Peeters, M.J. van Bommel, E.P. Boonekamp, M.A. Verheijen, L.H.M. Krings, Z.A.E.P. Vroon, in: H.S. Nalwa (Ed.), Handbook of Advanced Electronic and Photonic Materials and Devices, Vol. 5, Academic Press, San Diego, 2001, p. 220. H. Okazaki, T. Kitagawa, S. Shibata, T. Kimura, J. Non-Cryst. Solids 116 (1990) 87. J. Malzbender, G. de With, J. Mater. Sci. 35 (2000) 4809. R.D. Shoup, Colloid Interf. Sci. 3 (1976) 63. W.J. Clegg, K. Kendall, N.M. Alford, J.D. Birchall, T.W. Button, Nature 347 (1990) 455. W.J. Clegg, Science 286 (1999) 1097.

J. Malzbender et al. / Materials Science and Engineering R 36 (2002) 47±103

[92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132]

M.P. Rao, A.J. Sanchez-Herencia, G.E. Beltz, R.M. McMeeking, F.F. Lange, Science 286 (1999) 102. J. Malzbender, G. de With, Surf. Coat. Technol. 137 (2001) 72. S. Ahmed, F.R. Jones, J. Mater. Sci. 25 (1990) 4933. W. Vogt, Lehrbuch der Kristallphysik, Teubner, Leipzig, 1910. A. Reuss, Z. Angew. Math. Mech. 9 (1929) 49. O. Ishai, L.J. Cohen, Int. J. Mech. Sci. 9 (1967) 539. O. Ishai, L.J. Cohen, J. Comp. Mater. 2 (1968) 302. B. Paul, Trans. A.I.M.E. 218 (1960) 36. L. Nicolais, E. Droli, R.F. Landel, Polymer 14 (1973) 21. L. Lekatou, S.E. Faidi, S.B. Lyon, R.C. Newmann, J. Mater. Res. 11 (1996) 1293. J. Spanoudakis, R.J. Young, J. Mater. Sci. 19 (1984) 473. J.Z. Liang, R.K.Y. Li, Polym. Comp. 19 (1998) 698. E. DoÈrre, H. HuÈbner, Alumina, Springer, Berlin, 1984. H. Scholze, Glas: Natur, Struktur und Eigenschaften, Springer, Berlin, 1977. D. Ashin, R.A. Haber, J. Wachtman, J. Am. Ceram. Soc. 73 (1990) 3376. M.W. Shafer, V. Castano, W. Krakow, R.A. Figat, G.C. Ruben, Mater. Res. Soc. Symp. Proc. 73 (1986) 331. L.J. Broutman, R.H. Kock, Modern Composite Materials, Addison-Wesley, Reading, Mass., USA, 1967. F. Gatto, Alluminio (Milan, Italy) 19 (1950) 19. A. Flores, J. Aurrekoetxea, R. Gensler, H.H. Kausch, F.J. Balta Celleja, Colloid Polym. Sci. 276 (1998) 276. R.W. Rice, J. Mater. Sci. 14 (1979) 2768. P. Etienne, J. Phalippou, P. Sempere, J. Mater. Sci. 33 (1998) 3999. J. Jancar, A. Dianselmo, A.T. Dibenedetto, Polym. Eng. Sci. 32 (1992) 1394. L. Nicolais, M. Narkis, Pol. Eng. Sci. 11 (1971) 194. J. Malzbender, G. de With, Wear 236 (1999) 355. B. Pukansky, F.H.J. Maurer, Polymer 36 (1995) 1617. O. Ishai, L.J. Cohen, J. Comp. Mater. 2 (1968) 302. L. Nicolais, L. Nicdemo, Pol. Eng. Sci. 13 (1973) 469. F.P. Kehoe, G.A. Chadwick, Mater. Sci. Eng. A135 (1991) 209. K.W.-Y. Wong, R.W. Truss, Comp. Sci Technol. 52 (1994) 361. M.L. Shiao, S.V. Nair, P.D. Garrett, R.E. Pollard, J. Mater. Sci. 29 (1994) 1739. S. Hashemi, K.J. Din, P. Low, Pol. Eng. Sci. 36 (1996) 1807. K. Kendall, Br. Polym. J. 10 (1978) 35. A.J. Kinloch, D.L. Maxwell, R.J. Young, J. Mater. Sci. 20 (1985) 4169. B. Pukanszky, F.H.J. Maurer, Polymer 36 (1995) 1617. C.B. Bucknall, Adv. Polym. Sci. 27 (1978) 121. G.R. Irwin, Appl. Mater. Res. 3 (1964) 65. F.F. Lange, J. Am. Ceram. Soc. 54 (1971) 614. Y.-L. Cheng, C.-F. Pon, Int. J. Solids Struct. 38 (2001) 75. P.S. Turner, J. Res. NBS 37 (1946) 239. E.H. Kerner, Proc. Phys. Soc. 69B (1956) 808. A.A. Fahmy, A.N. Ragai, J. Appl. Phys. 41 (1970) 5108.

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