Roughness of glass surfaces formed by sub-critical crack growth

Journal of Non-Crystalline Solids 353 (2007) 1582–1591 www.elsevier.com/locate/jnoncrysol

Roughness of glass surfaces formed by sub-critical crack growth Sheldon M. Wiederhorn a,*, Jose M. Lo´pez-Cepero Jean-Pierre Guin b, Theo Fett c a

a,1

, Jay Wallace a,

National Institute of Standards and Technology, Gaithersburg, MD 20899-8500, United States b CNRS-University of Rennes I, Rennes, France c Forschungszentrum Karlsruhe and the University of Karlsruhe, Karlsruhe, Germany Received 13 October 2006; received in revised form 18 January 2007

Abstract This paper presents a study on the roughness of glass fracture surfaces formed as a consequence of sub-critical crack growth. Doublecantilever-beam specimens were used in these studies to form fracture surfaces with areas under well-defined crack velocities and stress intensity factors. Roughness depends on crack velocity: the slower the velocity, the rougher the surface. Ranging from approximately 1 · 1010 m/s to approximately 10 m/s, the velocities were typical of those responsible for the formation of fracture mirrors in glass. Roughness measurements were made using atomic force microscopy on two glass compositions: silica glass and soda lime silica glass. For silica glass, the RMS roughness, Rq, decreased from about 0.5 nm at a velocity of 1 · 1010 m/s to about 0.35 nm at a velocity of 10 m/s. For soda lime silica glass, the roughness decreased from about 2 nm to about 0.7 nm in a highly non-linear fashion over the same velocity range. We attributed the roughness and the change in roughness to microscopic stresses associated with nanometer scale compositional and structural variations within the glass microstructure. A theory developed to explain these results is in agreement with the data collected in the current paper. The RMS roughness of glass also depends on the area used to measure the roughness. As noted in other studies, fracture surfaces in glass exhibit a self-affine behavior. Over the velocities studied, the roughness exponent, f, was approximately 0.3 for silica glass and varied from 0.18 to 0.28 for soda lime silica glass. The area used for these measurements ranged from (0.5 lm)2 to (5.0 lm)2. These values of the roughness exponent are consistent with values obtained when the scale of the measurement tool exceeds a critical size, as reported earlier in the literature.  2007 Elsevier B.V. All rights reserved. Keywords: Mechanical properties; Crack growth; Fracture; Microscopy; Atomic force and scanning tunneling microscopy

1. Introduction Examination of fracture surfaces in glass by atomic force microscopy (AFM) suggests that these surfaces are never flat in a Euclidian sense, despite their appearance under the optical or scanning electron microscope. Gupta et al. [1] have shown that the root-mean-square (RMS) roughness of fracture surfaces ranged from 0.34 nm to *

Corresponding author. Tel.: +1 301 975 5772. E-mail address: [email protected] (S.M. Wiederhorn). 1 Now at the University of Seville, Department of Condensed Matter Physics, Seville, Spain. 0022-3093/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2007.01.029

0.83 nm depending on the glass composition and the method of making the glass. The variation in the fracture path through the glass was attributed to the structure of the glass, specifically to inhomogenieties in the structure. In an AFM study on silica glass, Poggemann et al. [2] obtained results similar to those obtained by Gupta et al. The spacing of the high points on the surface of their glass correlated with molecular dimensions within the glass structure: e.g. the Si–O and the O–O distances. Hence, the authors attributed the distances to typical features of the silica network. Other AFM studies concentrated on the self-affine nature of fracture surfaces in glass. Various techniques were

S.M. Wiederhorn et al. / Journal of Non-Crystalline Solids 353 (2007) 1582–1591

used to extract the roughness exponent, f, from fracture surfaces formed under a variety of conditions [3,4]. The roughness exponent depended on the length scale of the measurement. For length scales less than a critical value, n, the exponent is about 0.8 and is purportedly the same for all fracture surfaces. For length scales greater than n, f takes on the value 0.4. The critical length scale, n, depends on the crack velocity, ranging from about 20 nm to about 80 nm for crack velocities ranging from about 1010 m/s to 105 m/s [4]. Theoretical justifications for these values have been given [4]. Although AFM studies have quantified the roughness of glass fracture surfaces, only one systematic study has investigated the effect of crack velocity on the roughness of glass for fracture surfaces formed by sub-critical crack growth [4]. Usually, specimens are simply broken and roughness measurements made on the flat area surrounding the fracture origin. This flat area is known as the ‘mirror’ region and is bounded by an area of fine scale roughness (the ‘mist’ region) that can just be detected optically. Beyond the mist region is the ‘hackle’ region where macroscopic roughness can be detected [5,6]. It is generally thought that the RMS roughness of glass is relatively constant in the mirror region, but increases as the glass fracture approaches and then passes through the mist into the hackle region [6]. Within the mirror region, near the crack origin, surfaces are optically smooth and featureless [6]. At approximately one-third the distance from the origin to the boundary between the mirror and mist regions, rm, small scale roughness increases [7], but is still too small to be seen with an optical microscope so that the surface still looks smooth. At rm the RMS roughness further increases so that features are now large enough to be seen optically. A similar scenario was reported for a brittle, glassy, isotropic epoxy resin [6]. In this paper, we examine topographic features in fracture surfaces for cracks moving at crack velocities ranging from about 1 · 1010 m/s to approximately 10 m/s. All studies are carried out in water on double-cantilever-beam specimens (DCB); by using the DCB specimen, crack velocities can be closely correlated with topographic features detected by atomic force microscopy. Results show that the RMS roughness decreases with increasing crack velocity within the range of velocities studied. Furthermore, the RMS roughness over the range of velocities studied is greater for soda lime silica glass than for silica glass. 2. Experimental procedure All crack velocity measurements on soda lime silica glass microscope-slides were made under deadweight loading on double-cantilever-beam specimens, (75 · 25 · 1) mm in size using a diamond scribed scratch to guide the crack [8]. The composition of the soda lime silica glass by mass fraction (%), determined by X-ray fluorescence analysis, is as follows [9]: Na2O – 14; MgO – 3.7; Al2O3 – 1.8; SiO2 – 74;

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K2O – 0.36; CaO – 5.9. The silica glass specimens were Corning C79802 slides, (75 · 25 · 1.5) mm, with midline notches approximately 0.5 mm deep to guide the direction of crack growth; the crack width was about 1 mm [9]. Crack velocities ranged from approximately 1 · 1010 m/s to 10 m/s. All experiments were carried out in water at room temperature (22 C). Crack velocities, v, were measured using either a traveling telescope (accuracy ±10 lm), or a digital camcorder (accuracy ±10 lm). The camera was capable of taking individual pictures at fixed intervals. In the course of the study, crack growth curves were obtained over a velocity range from 1 · 1010 m/s to 1 · 102 m/s. Given an applied load, the experimentally determined crack growth curves, log v versus KI, where KI is the applied stress intensity factor, were used to calculate crack velocity as a function of position along the surface on the specimen. Initially, a load was applied to the specimens such that the crack velocity was in the 1 · 1011–1 · 109 m/s range. The crack was then permitted to grow to produce a large identifiable region that could be later studied by atomic force microscopy. Fracture surfaces were produced at higher velocities by increasing the load on the crack so that the crack velocity was in the range 1 · 109–1 · 108 m/s. The crack was then allowed to grow under a constant load until the specimen failed. Since the load was constant, the value of KI could be calculated as a function of crack length alone. Then, using the experimental v  KI curve, the crack velocity, v, could be calculated at every point along the crack trajectory. With this information, the fracture surface roughness could be determined as a function of crack velocity, or stress intensity factor at every point on the fracture surface of the slide. The highest velocities on the slide were not measured, but estimated as 10 m/s.3 This value is consistent with the fact that the fracture surface was optically smooth to the very end of the specimen. After the specimens were fractured in two, the crack velocity was determined as a function of distance along the slide, and optical micrographs were taken to identify the areas associated with the velocities. Height images were then made as a function of crack velocity for square areas having edge lengths of 0.5 lm, 1 lm, 2 lm and 5 lm. Regardless of AFM scan-area size, the number of scan lines was the same, 512, as the number of points in each scan line, also 512. The scan area was square; therefore, in the discussion below, the length of the edges of the square scan-areas, L0, were used to represent the roughness data. The RMS roughness, Rq, of each area was determined by using the software package included with the AFM (Digital 3100, Veeco Metrology Group, Santa Barbara, CA,). Prior to measuring the roughness all images were flattened to 0 2

The use of commercial names is only for purposes of identification and does not imply endorsement by the National Institute of Standards and Technology. 3 The estimate was made from the intersection of the crack growth curve in water with that in dry nitrogen, using Ref. [8].

S.M. Wiederhorn et al. / Journal of Non-Crystalline Solids 353 (2007) 1582–1591

3. Results 3.1. Appearance of fracture surfaces Earlier investigations of microscopically ‘smooth’ glass fracture surfaces by atomic force microscopy revealed that they are ‘rough’ on a nanometer scale [1,2]. Studies carried out in our laboratory support these earlier findings. Fig. 1 shows a fracture surface approximately 1 lm square that has a height variation of about 2 nm over the entire image. As indicated both by the three dimensional view of the surface and by the lightening of the high points of image, the surface is ‘rough’ on the nanometer scale. To display this ‘roughness’, the magnification perpendicular to the fracture surface, the z direction, has been set to approximately 1000 times the lateral dimension of the image. By quantifying this roughness we can show that the roughness of glass depends on the velocity of the crack, the chemical composition of the glass and the area used to measure the roughness. The two parameters used to characterize the roughness are the RMS roughness, Rq, and the roughness exponent, f, which tells how Rq varies with the size of the measurement area [10]. 3.2. RMS roughness The RMS roughness, Rq, is shown in Fig. 2 for soda lime silica glass and silica glass as a function of measure-

Fig. 1. Fracture surface in silica glass formed during crack propagation at a velocity of 8 · 1011 m/s, in water. The fracture surface of the glass is irregular varying as much as 1 nm from the fracture surface mean plane. The x-axis is in the direction of crack growth and is also the scan direction; the y-axis is in the fracture plane but perpendicular to the growth direction; z is perpendicular to the apparent fracture plane. This is the AFM convention, but is contrary to the fracture mechanics convention, for which z and y are interchanged.

0.8 Silica Glass

RMS Roughness, nm

order and plane fit to 2nd order. The roughness exponent, f, was determined from the slope of a log–log plot (base e) of Rq versus L0. All measurements were made using the contact mode with tips having a nominal radius of 20 nm (DNP, Veeco Metrology Inc, Camarillo, CA).

0.6 2 µm

0.4

1 µm 0.5 µm

0.2

0.0 10 -1010 -910 -810 -710 -610 -510 -410 -310 -210 -1 10 0 10 1 10 2 Crack Velocity, m/s

2.5 Soda-Lime-Silicate Glass

RMS Roughness, nm

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2.0

5 µm 2.5 µm

1.5

1 µm

1.0

0.5 µm

0.5 Silica Glass

0.0 10-10

10-8

10-6

10-4

10-2

100

102

Crack Velocity, m/s Fig. 2. RMS roughness of two glasses tested in water; silica glass and soda lime silica glass. The number on each curve indicates the length, L0 in lm, of the side of the square area used to measure the roughness. The silica glass data is repeated in the lower figure for a direct comparison of the roughness data. Where the error bars (the standard deviation of the data) were larger than the symbol they were placed on the figure, where smaller, they were left out.

ment area and crack velocity. The RMS roughness is defined as the standard deviation of the height of the surface from the average height of a plane passing through the surface [11]. The size, L0, of an edge of the square area used for the measurement is indicated in these figures by the number on each curve. The silica curves have been repeated in the lower figure so that the two sets of data can be compared easily. Both glasses increase their roughness as the crack velocity decreases. The change in roughness with crack velocity is much greater for the soda lime silica glass than for the silica glass. Over a crack velocity range of about 11 orders of magnitude, the roughness of the silica glass is linear on this semi-log plot and increases in value by about 30% as the velocity decreases. By contrast, the roughness of the soda lime silica glass is non-linear. At high velocities, the roughness of the soda lime silica glass is higher than that of the silica glass, and increases by a factor of at least three as the crack velocity is decreased.

S.M. Wiederhorn et al. / Journal of Non-Crystalline Solids 353 (2007) 1582–1591

3.3. The roughness exponent, f For any given velocity, Rq increases as L0 increases. This increase in roughness can be expressed as a power function of the length, L0, [10]: Rq ¼ a  L10 ;

ð1Þ

where f is the roughness exponent. For our data, f ranges between approximately 0.2 and 0.3 for the range of L0 shown in Fig. 3. Roughness exponents reported in the literature for silica glass have been shown to depend on the scale of the measurement [4]. For L0 less than the critical value, n (20– 80 nm), the roughness exponent, f, reported in Ref. [4] has a value of about 0.8, which is larger than the values determined in the present study, Table 1. For L0 greater than the critical value, n, the value of f is reported to range from approximately 0.4 to 0.5 [3,4]. In Appendix A we provide a further discussion of the role of scale of measurement on roughness exponent.

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the RMS roughness of the soda lime silica glass was greater than that of the silica glass and had a larger dependence on crack velocity. The magnitude of the roughness is a consequence of compositional and/or structural variations within glasses [1], which give rise to local stresses that cause the tip of a moving crack to deviate from the projected crack plane. Two sources of stress can be envisioned both due to small localized variations in the structure and composition of the glass: stresses due to thermal expansion mismatch between different parts of the glass, and stresses due to small local differences in the elastic moduli of the glass from one location to the other. In silica glass, the composition is nominally the same throughout the structure. However, density fluctuations due to the stochastic arrangement of the silica tetrahedra within the glass structure can also deflect the crack from its projected plane. Soda lime silica glass shows a greater variation in roughness because of its more complex microstructure and chemical composition, which leads to a greater variation in the local structure.

4. Discussion of results This paper presents data in which the roughness of fracture surfaces of silica and soda lime silica glass were characterized by atomic force microscopy as a function of crack velocity. The behavior of the two glasses was very different;

RMS Roughness, Rq nm

3

0.18 Soda Lime Silica

0.25

1

0.1

0.28 Silica

0.29

0.30

0.3

1

10

Length,Lo, µm Fig. 3. RMS Roughness versus L0 as a function of crack velocity: soda lime silica glass – d 10 m/s; + 3 · 104 m/s; . 1 · 1010 m/s; silica glass – h 10 m/s; s 1 · 1010 m/s. The number next to each line is the roughness exponent, f.

Table 1 Estimation of roughness exponent, f, using a logarithmic plot of RMS roughness versus L0 Glass

Velocity, m/s

Range of L0, lm

Roughness exponent, f

Soda–lime–silica

1 · 1010 3 · 104 10 1 · 1010 16

0.5–5

0.18 ± 0.02 0.25 ± 0.03 0.28 ± 0.04 0.29 ± 0.02 0.30 ± 0.01

Silica

0.5–2

The uncertainty of f is the standard error of the roughness exponent.

4.1. Theory of roughness4 In deriving a theory to explain the roughness of glass, we assume that the crack passes through a glass matrix that has an average composition and structure. Above and below the projected crack plane, inhomogenieties within the glass are envisioned as having slightly different compositions and/or structures, Fig. 4, resulting in slightly different values for thermal expansion coefficients and elastic moduli.5 These inhomogenieties are modeled as spherical inclusions. As the glass is cooled from the processing temperature to room temperature, residual stresses develop as a consequence of thermal expansion coefficient differences between the inclusions and the matrix. These thermal stresses are present throughout the material whether a crack is there or not. If the elastic constants of the inclusion differ from the matrix, stresses will develop locally upon approach of the crack tip as a consequence of mechanical loading of the inclusions by the stress field of the crack. These stresses, too, can cause the crack to deviate from its mean plane. 4.2. Residual stresses A difference in thermal expansion coefficient between that of a spherical inclusion, ai, and that of a matrix, am, gives rise to both radial, r0 ; ð2Þ rrr ¼  3 ðr=RÞ 4

A more complete discussion of this derivation is planned. The toughness or moisture resistance of the glass within the inclusions might also vary spatially and cause the crack to deviate from the mean fracture plane. This possibility is not treated in this paper. 5

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" rhh ¼ r1 1 

high density

low density

and tangential stresses,

2ðr=RÞ

3

3

þ

ðr=RÞ

5

ð5Þ

;

4.4. Stress intensity factors

Fig. 4. Distribution of density and/or composition fluctuations in the glass, modeled by spherical inclusions.

r0

ðr=RÞ

#

9C 2

where C1 and C2 are complex algebraic functions of the elastic constants. A similar relation holds for the stresses ruu along the equator. Note that the stress given by Eq. (5) depends on the remote tensile stress, r1, and only exists in the presence of the remote stress. In the current theory, the remote stress is provided by the stresses about the crack tip.

matrix of average density

rhh ¼

C1

ð3Þ

;

Let us now consider a crack of length, a, which is loaded by a mode-I distribution of applied forces and stresses. We are interested in the region ahead of the tip and in the mode-II stress intensity factor contribution caused by the shear stresses in the prospective crack extension plane szx(x, y). For shear stresses, szx(x, y), it holds that Z szx ðx; yÞdS ! 0 ð6Þ ðSÞ

in the matrix outside of the inclusion (see e.g. [12,13]); r is the distance from the center of the spherical inclusion, and R is the radius of the inclusion. In these relations the coefficient r0 is given by r0 ¼

ðai  am ÞðT 2  T 1 Þ ; 1þmm i þ 12m 2Em Ei

ð4Þ

where T2 is the room temperature and T1 is a high temperature at which the material is free of residual stresses. The parameter, a, is the thermal expansion coefficient and m is Poisson’s ratio; the subscript ‘i’ denotes inclusion and ‘m’ denotes matrix. The parameter, E, is Young’s modulus.

for a sufficiently large crack area increment, S = D‘ · t (Fig. 5a). The distribution of shear tractions is in general a three dimensional problem as stresses fluctuate also in the depth direction. In the following derivation we reduce the problem to a simpler two dimensional analysis, neglecting any changes in depth direction, y. If szx(x) is the shear stress in the projected crack plane of the uncracked region D‘, Fig. 5, the weight function method [15] allows us to compute the influence of the shear stress fluctuations according to Z K II ¼ hszx dx ð7Þ ðSÞ

4.3. Interaction of inclusions with remote stresses Stresses at a spherical inclusion under a remote tensile stress are given by the stress relations derived by Goodier [14]. Here the elastic constants of the inclusion differ from that of the matrix. Under a remote tensile stress r1 the tangential stresses rhh at the equator of the sphere within the matrix is given by

z

z

t

y

with h being the weight function. The total mode-II stress intensity factor is the superposition of the contribution due to the thermal mismatch and the stress redistribution by the inclusions. The thermal part is, of course, independent of the external mechanical load. The stress redistribution part is proportional to the stresses ahead of the crack, i.e. to the crack tip stress intensity factor KI,tip, since r1 / KI,tip. Consequently, KII is given by

τzx x

x

a

a Δ

Fig. 5. Computation of mode-II stress intensity factor: (a) 3-d and (b) 2-d model.

Δ

S.M. Wiederhorn et al. / Journal of Non-Crystalline Solids 353 (2007) 1582–1591

K II ¼ A þ BK I;tip ;

1587

ð8Þ

where A and B are constants. The deviation angle between the crack and the straight plane, H, is given as [16] H ¼ 2

K II ; K I;tip

which then yields   A H ¼ 2 þB : K I;tip

ð9Þ

ð10Þ

The angle of local crack plane deviation is interpreted as a measure of roughness. Since for small angles H = dz/dx, the amplitude z is given by a similar relation z¼

A0 þ B0 K I;tip

ð11Þ

as will be shown in Appendix B. From Eq. (10) or (11) it has to be expected that a plot of roughness versus 1/KI,tip should result in a straight line. Roughness is determined by direct measurement; KI,tip is determined from a v  KI plot. The v  KI curves for silica and soda lime glass were fitted by the following power law equation: v ¼ CK nI;tip

ð12Þ

with n = 43.7, C = 4.4 · 106 for silica and n = 14.3, C = 0.5 for soda lime silica glass (v in m/s for K in MPa m0.5), see Fig. 6. For the silica glass, KI,tip is calculated directly from the v  KI curve in Fig. 6, because the data can be fitted by a straight line. By contrast, the data from the soda lime silica glass appears to approach a static fatigue limit as the crack velocity decreases. Several authors have suggested that this decrease in velocity is a consequence of ion exchange between the crack tip environment and the freshly formed fracture surfaces near the crack tip [18–20]. Sodium ions leaving the glass for the solution at the crack tip are replaced by hydronium ions, which results in a compressive stress of the glass near the crack tip, and, therefore, a reduction of the stress intensity factor [9]. This reduction in crack tip stress intensity factor is taken into account by extending the straight line portion of the v  KI curve for soda lime silica glass into the slow crack growth region. The crack tip stress intensity factor is then evaluated from the straight line extension of the v  KI curve, at the same crack velocity as measured experimentally on the growing crack. Following this procedure, the crack tip stress intensity factor is determined, and Rq versus 1/KI,tip is plotted in Fig. 7. The data on this plot is in good agreement with the form predicted by Eq. (11), provided Rq can be assumed to be proportional to z.

Fig. 6. Crack growth in glass as a function of the applied stress intensity factor; taken from Ref. [17]. The glass for the soda lime silica experiment is the same as that used in this paper. The same grade of silica glass was also used here, although the lot was 40 years newer.

2.5

Rq nm 2

5

soda-lime glass

1.5

2.5 1 0.5

1

Silica

0.5

0 0

1

2

3

4

5

1/KI,tip Fig. 7. Roughness parameter Rq as a function of the reciprocal crack tip stress intensity factor.

4.5. Harmonic analysis of the stresses near the crack tip In Appendix B we carry out a harmonic analysis of the stresses near the crack tip. The distribution of stresses on the projected fracture plane is expressed in terms of a series

of sines and cosines, which can be used to calculate KII. From KII, a crack surface profile is calculated and shown to have the same fluctuation lengths as the inhomogeneities

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in the glass. This is an important finding; by evaluating the fluctuation patterns on the fracture surfaces, we can also determine the fluctuation of the inhomogeneities within the glass. The roughness exponent, f, can be calculated, in principle, from the wavelength distribution of fluctuation pattern. By contrast, the RMS roughness of the glass comes directly from the strength of the interaction between the crack and the microstructure of the glass, via Eq. (11). This conclusion can be tested through a variety of experiments that include modification and characterization of the glass structure by small angle X-ray and neutron scattering, and by comparing these results with those obtained by surface roughness studies.

[11,21]. In both, the following equation is used to characterize surface geometry6:   ðA:1Þ GðxÞ ¼ 2 x2  CðxÞ ;

5. Summary

where h(x 0 ) is the height of the surface at point x 0 while h(x 0 + x) is the height of the surface a distance x away from point x 0 and L is the section length. The distance, x, is sometimes called the lag. When x = 0, C(0) = x. The pair correlation function, G(x), is given by the following equation: Z 1 L 2 GðxÞ ¼ lim ½hðx0 þ xÞ  hðx0 Þ dx0 : ðA:3Þ L!1 L 0

This paper presents a study on fracture surfaces in glasses using fracture mechanics specimens. We show that the roughness of glass depends on the crack velocity and the composition of the glass. In the velocity range 1010 m/s to about 10 m/s, the RMS roughness decreases with increasing velocity. RMS roughness values of soda lime silica glass are much higher than for silica glass and have a much stronger dependence on crack velocity. At any given velocity, the roughness was also found to depend on the area used to measure the roughness. The RMS roughness of glass scaled as a power law function of the area. The roughness exponent of silica glass was approximately 0.3, whereas that of the soda lime silica glass appeared to increase slightly from 0.18 to 0.28 as the velocity increased from 1010 m/s to 10 m/s. These values are lower than the value of 0.8 reported in the literature for silica glass. An explanation of this difference is given in terms of the sensitivity of roughness exponent to the scale of the measurement tool (Appendix A). The roughness behavior was modeled by assuming that the glasses contained inclusions of the order of 10 nm in dimensions. These inclusions were responsible for residual thermal stresses, and for local modification of the elastic constants, both of which interacted with the crack front causing it to deviate from the mean fracture plane. A Fourier analysis of stresses along the projected mean fracture plane (Appendix B) suggests that the fracture surface of the glass has to be quantitatively related to the structure of the glass, at the nanometer scale. Based on this conclusion, we suggest that systematic studies of the roughness of glass fracture surfaces as a function of chemical composition and annealing treatment may be used to develop a better understanding of the structure of the glass at the nanometer level. Appendix A. Effect of measurement scale on roughness measurement The effect of measurement scale on surface roughness has been discussed in both the literature on fractal geometry [3,4,10] and in the literature on surface roughness

where x is the RMS roughness of the surface, i.e. Rq. C(x) is the height autocorrelation function and G(x) is the pair correlation function. In Refs. [4,10] these are defined along the profile of a vertical section through the surface. The mean height of the surface is defined as zero. The height autocorrelation function is given by the following equation: Z 1 L CðxÞ ¼ lim ½hðx0 þ xÞhðx0 Þdx0 ; ðA:2Þ L!1 L 0

For small dimensions of x, the fractal dimension, D 0 , of the surface can be evaluated [10] from the following equation: 0

GðxÞ / x42D ¼ x2f

ðA:4Þ

since f = 2  D 0 [10]. The roughness exponent can be obtained from the slope of a logarithmic plot of G(x) versus x: f¼

1 d log GðxÞ : 2 d logðxÞ

ðA:5Þ

This is the equation used by Bonamy et al. [4] to obtain the roughness exponent of silica glass. Another technique is also suggested in Ref. [10]. This technique also uses a perpendicular section through the fracture surface, but instead of determining the pair correlation coefficient, the RMS roughness is determined for the section profile. Milman et al. [10] show that a logarithmic plot of the RMS roughness versus the length of the section, L0, should give a curve with a slope equal to the roughness exponent. For small lengths, the roughness exponent should be equal to the fractal dimension, D 0 , i.e., f = 2  D 0 . For this technique, Eq. (A.6) was used to calculate f: f¼

d logðxÞ ; d logðL0 Þ

ðA:6Þ

where x is the RMS roughness, Rq. This technique is similar, but not identical to the one we used in our paper, in which the roughness was obtained from an entire surface instead of a single section. When the roughness is deter6

The terminology and equations from Ref. [10] are used here. The equations above are stated without proof and are used to relate our data to that in Refs. [3,4].

S.M. Wiederhorn et al. / Journal of Non-Crystalline Solids 353 (2007) 1582–1591

Loge of Pair Correlation Function (nm)

0 2 µm

-1

5 µm

-2 -3 -4 -5 -6

Pair Correlation

-7 0

1

2

3

4

5

6

7

9

8

Loge of Lag Length, x, (nm) Fig. 1A. Estimation of the roughness exponent by the pair correlation function technique, Eq. (A.5). Estimates were made for line lengths 2 lm and 5 lm long. Results of the estimation of the roughness exponent, f, are in Table 1.A.

Loge of RMS Roughness (nm)

0 2 µm 5 µm

-1

-2

-3 Line Roughness

0 Loge of RMS Roughness (nm)

mined for a square area, L0 is the length of an edge of the square. In the present paper, we used Eq. (1) to obtain the roughness exponent of silica glass; Bonamy et al. [4] used Eq. (A.5). The results were not the same, hence one might question whether the two techniques are equivalent. To settle this question, we carried out a number of computer studies on AFM data from a fracture surface. The same grade of silica glass was used as in the present study. First we applied Eq. (A.5) to see if we could duplicate the results obtained by Bonamy et al. Then we measured the roughness of line profiles as a function of profile length, L0, in a glass surface taken from the identical area as used for Eq. (A.5). Finally, we measured the roughness the same way as was done in this paper, using areas of different edge lengths, L0. Here again, the area of analysis was the same as that used for Eq. (A.5). The results of this study are presented in Figs. 1A, 2A, 3A.

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2 µm 5 µm

-1

-2

-3 Area Roughness -4 0

1

2

3

4

5

6

7

8

9

10

Loge of Lo (µm) Fig. 3A. Estimation of the roughness exponent, f, by the area roughness technique, Eq. (1). Estimates were made for square areas 2 lm and 5 lm on a side. Results of the estimation are in Table 1.A.

In Fig 1A, a logarithmic plot (base e) of the pair correlation function versus the lag length, x, of the section indicates a curve that starts out in a linear fashion but then decreases its slope as the lag length, x, increases in size. The calculation was carried out for two section lengths, L = 2 lm and L = 5 lm, as defined in Eq. (A.2). With a minor exception at the long length end of the data, the two curves overlap over most of their trajectory. By fitting a straight line to the linear short length end of the curve by the method of least squares, we obtained a slope of 0.686 (0.003) and 0.714 (0.002) for the 2 lm and 5 lm curves, respectively. The numbers in brackets are the standard errors of the fit. The lower two points of the 2 lm curve were not used in the fit as these points lie outside of the resolution of our AFM.7 Data used for the straight line portion of the fit ranged from loge(x) = 2.4 to loge(x) = 4.0, which approximately corresponds to 12 nm < L < 55 nm. The slopes obtained are close to the values, 0.75 and 0.61 obtained by Bonamy et al. [4]. The slopes and standard errors of all of the figures are summarized below in Table 1.A. In Fig. 2A, we have a logarithmic plot of the RMS roughness as a function of the length of the section, L0, on which the roughness was measured. The figure is similar to that in Fig. 1A. The values of roughness overlap over most of the graph. Below a loge(x) of 3, the data gently decreases in slope. As the resolution of the AFM is about 0.05 nm, the data lying below log(0.05) = 3.00 can safely be ignored in fitting a line to the lower portion of the curves in Fig. 2A. A straight line fitted to the linear portion of the two curves gives slopes of 0.751 (0.006) for the 2 lm curve

-4 0

1

2

3 5 6 4 Loge of Length, Lo, (nm)

7

8

9

Fig. 2A. Estimation of the roughness exponent by the line roughness technique, Eq. (A.5). Estimates were made for line lengths 2 lm and 5 lm long. Results of the estimation of f are in Table 1.A.

7 For the AFM used, the variance of the noise of a measurement is 0.0025 (nm)2. Since two measurements are made, the variance of the pair correlation should be twice that of a single measurement, 0.005. Hence, loge(0.005) = 5.3, which is larger than the lowest point in Fig. 1A and very close to the second lowest point. Therefore, these two points were dropped from the linear least squares fit for the line.

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Table 1.A Determination of roughness exponent (standard deviation)

2 lm images 5 lm images

Pair correlation function

Line roughness

Area roughness

0.686 (0.003)

0.751 (0.006)

0.92 (0.013)

0.714 (0.002)

0.781 (0.008)

0.95 (0.018)

and 0.781 (0.008) for the 5 lm curve. The linear line portion of the curve in Fig. 2A ran from about loge(L0) = 3.4 to loge(L0) = 4.35, corresponding to a L0 of 30–80 nm. The values of the roughness exponent lie close to those obtained by the pair correlation technique, and are similarly close to the values published by Bonamy et al. [4]. Fig. 3A is a roughness plot similar to Fig. 2A, but for the entire area examined. Because the surface images were all obtained by line scans, we thought that the same functionality would apply for the entire figure as for an individual line. Apparently, this is not the case. Although the overall shape of the curve is the same as those in Figs. 1A and 2A, the slope of the curves for small roughness values is steeper than that for the other techniques of obtaining the roughness exponent. The slopes obtained for Fig. 3A were 0.92 (0.013) for L0 = 2 lm and 0.95 (0.018) for L0 = 5 lm. Here the slopes were measured over loge(L0) values of 3.94–5.24, corresponding to about 52–188 nm. With these figures in hand, it is relatively easy to rationalize the striking difference between the exponents we obtained, Fig. 3, and those reported by Bonamy et al. [4]. Simply put, the roughness exponents obtained by Bonamy et al. were calculated by using very small values of the characteristic length, with an upper limit of about 100 nm. In our graphs, this means that all values with loge (L0) > loge(100)  4.5 would be discarded, retaining only the high slope, linear region of the graph. This explains the good agreement between the exponents found for the linear region of our graphs and the Bonamy et al. results. In our experiments, exponents in Fig. 3 were calculated by using much larger values for the characteristic length, from several hundred nanometers to a few microns. This means that the relevant region in the graphs is around an x-axis value of about loge (1000)  7. At this value of loge(L0) we are no longer in the linear region. Hence, the slopes are much smaller than those obtained by Bonamy et al. Thus, the reason for the original discrepancy on the exponent values stems from the fact that they were calculated for different characteristic scales: one in the tens of nm range, the other in the 1000 nm range. It should be pointed out that, even if our measurements had been made at the same characteristic lengths as the ones in the paper by Bonamy et al., a second discrepancy would still exist, which arises from the fact that our original exponents were calculated by using area RMS measurements, whereas the ones in the Bonamy et al. paper were calculated along line profiles. As we have already shown, the exponents calculated using areas, Fig. 3A, are some-

what higher than the ones calculated along line profiles, Figs. 1A and 2A. In our case, however, the importance of this second effect is in all likelihood small compared to the effect of the prominent difference in the characteristic scale. It should be noted that this difference can be attributed exclusively to the choice of method, area versus line profile, since calculations made using line profiles, Figs. 1A and 2A, closely match the exponents reported by Bonamy et al.

Appendix B. Harmonic analysis of the stresses near the crack tip In order to proceed further in the analysis of our roughness data, more detailed information on the nature of the stress field interacting with the moving crack is needed. The distribution of inclusions and information on their thermal expansion coefficients and their elastic constants are desirable to estimate the magnitude of the stress fields in the projected fracture plane. Alternatively, an estimate of the residual strains in the material could be used to estimate the stress distribution. Short of these kinds of data, however, we can still provide some information on the nature of the crack surface given an arbitrary stress field in front of the moving crack. In order to evaluate Eq. (7), the unknown shear stress distribution in the prospective crack plane of the uncracked specimen, Fig. 1B, is expanded in a Fourier series. Since the constant term in this expansion must disappear due to the equilibrium condition, it holds that for periodic boundary conditions szx ¼

1  X x x ak cos 2kp þ bk sin 2kp : W W k¼1

ðB:1Þ

The total mode-II stress intensity factor results from the shear stresses in the crack plane as well as on the deviation of the crack path from the plane crack as derived by Cotterell and Rice [16] dz K II T  z0 ðaÞ ¼ 2 þ2 da KI KI

Z

a

hðx; aÞz0 ðxÞdx: 0

Fig. 1B.

ðB:2Þ

S.M. Wiederhorn et al. / Journal of Non-Crystalline Solids 353 (2007) 1582–1591

The so-called ‘T-stress’ term, T, is the first higher-order term of the crack tip stress field. In order to show the principal influence of the stress intensity factor on the roughness, we neglect the higher-order term, T. Substituting Eq. (B.1) into Eq. (7) and using the following crack tip weight function [15], sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 h¼ : ðB:3Þ pða  xÞ Eq. (B.4) is obtained: rffiffiffiffiffiffiffi" rffiffiffiffiffi! X 2W ka a a K II ¼ C 2 ak sin 2kp þ bk cos 2kp kp W W W ðkÞ # rffiffiffiffiffi! ka a a S 2 ak cos 2kp  bk sin 2kp ; ðB:4Þ W W W

1591

Since the two Fourier coefficients ak and bk in (B.1) are linearly dependent on KI,tip ak ; bk / ak þ bk K I;tip ) z /

1 þ const: K I;tip

ðB:9Þ

It should be mentioned that the same type of relation must also result by considering the T-stress because the correction term is also proportional to 1/K as the dominating first term in Eq. (11). The values of the constants A, B in Eq. (10) will then slightly depend on T/KI,tip, indicating that for the crack surface profile, an equation of the type 12 must hold. Thus, the fracture surface of the glass should be quantitatively related to the structure of the glass, at the nanometer scale. References

where C and S are the Fresnel integrals. The distances between the inhomogeneities are much smaller than a and W. Consequently, only high numbers of k will contribute to the sum of (B.4). The Fresnel integrals under this circumstance simplify to 1 1 CðuÞ ! ; SðuÞ ! for u ! 1 2 2 and Eq. (B.4) reduces to rffiffiffiffiffiffiffiffih  1 X W a a ak sin 2kp  cos 2kp K II ¼ 2kp W W k¼1  a a i þ bk sin 2kp þ cos 2kp : W W

ðB:5Þ

ðB:6Þ

For small angles, H, Eq. (9) can be written as dz K II ¼ 2 da K I;tip pffiffiffiffiffiffiffiffiffiffiffiffi 1 rffiffiffih  2W =p X 1 a a ¼ ak sin 2kp  cos 2kp W W K I;tip k¼1 k  a a i þ bk sin 2kp þ cos 2kp ðB:7Þ W W and in integrated form as pffiffiffi  3=2 1 X sin kp a 2 W W z¼ 3=2 K I;tip p k k¼1 h a ai  ðak  bk Þ cos kp  ðak þ bk Þ sin kp ; W W

ðB:8Þ

i.e., a crack surface profile with the same lengths of fluctuations as the fluctuations of the inhomogenieties.

[1] P.K. Gupta, D. Inniss, C.R. Kurkjian, Q. Zhong, J. Non-Cryst. Solids 262 (1–3) (2000) 200. [2] J.-F. Poggemann, A. Goß, G. Heide, E. Ra¨dlein, G.H. Frischat, J. Non-Cryst. Solids 281 (2001) 221. [3] E. Bouchaud, J. Phys.: Condens. Matter 9 (1997) 4319. [4] D. Bonamy, L. Ponson, S. Prades, E. Bouchaud, C. Guillot, Phys. Rev. Lett. 97 (2006) 135504. [5] J.W. Johnson, D.G. Holloway, Phil. Mag. 14 (1966) 734. [6] D. Hull, J. Mater. Sci. 31 (1996) 1829. [7] D.M. Kulawansa, L.C. Jensen, S.C. Langford, J.T. Dickinson, J. Mater. Res. 9 (2) (1994) 476. [8] S.M. Wiederhorn, J. Am. Ceram. Soc. 50 (8) (1967) 407. [9] J.-P. Guin, S.M. Wiederhorn, T. Fett, J. Am. Ceram. Soc. 88 (3) (2005) 652. [10] V.Y. Milman, N.A. Stelmashenko, R. Blumenfeld, Prog. Mater. Sci. 38 (1994) 435. [11] T.R. Thomas (Ed.), Rough Surfaces, Longman, London, 1982, p. 92, Chapter 5. [12] D. Weyl, Ber. Deut. Keram. Ges. 36 (1959) 319. [13] J. Selsing, J. Am. Ceram. Soc. 44 (1959) 419. [14] J.N. Goodier, J. Appl. Mech. 1 (1933) 39–44. [15] T. Fett, D. Munz, Stress Intensity Factors and Weight Functions, Computational Mechanics Publications, Ashurst Lodge, Ashurst, Southampton, UK, 1997. [16] B. Cotterell, J.R. Rice, Int. J. Fract. 16 (2) (1980) 155. [17] S.M. Wiederhorn, L.H. Bolz, J. Am. Ceram. Soc. 53 (10) (1970) 543. [18] T.A. Michalske, B.C. Bunker, in: K. Salama, K. Ravi-Chandrar, D.M.R. Tablin, P. Ramao Rao (Eds.), Advances in Fracture Research, Vol. 6, Pergamon, New York, 1989, pp. 3689–3701. [19] T.A. Michalske, B.C. Bunker, K.D. Keefer, J. Non-Cryst. Solids 120 (1990) 126. [20] T. Fett, J.-P. Guin, S.M. Wiederhorn, Eng. Frac. Mech. 72 (18) (2005) 2774. [21] M.R. Stoudt, J.B. Hubbard, Acta Mater. 53 (2005) 4293.