RELATING SURFACE TEXTURE AND ADHESION WITH AREA-SCALE FRACTAL ANALYSIS

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1 RELATING SURFACE TEXTURE AND ADHESION WITH AREA-SCALE FRACTAL ANALYSIS Christopher A. Brown, Professor of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, MA Abstract Quantitative understanding of the relation between substrate roughness and adhesion leads to texture characterization parameters that can be used in product and process design and in quality control. A straightforward physical model, based on surface available for bonding, is used to relate the results of area-scale fractal analysis with adhesive strength. Area-scale fractal analysis is based on virtual tiring to determine the area as a function of scale on' a rough surface. A series of regression analyses over a range of scales can be used to determine experimentally the scale of the bonding. Results support the discrete bonding model and show that correlations with adhesive strength can have regression coefficients as high as Introduction The objective of this work is to better understand the influence of surface roughness on adhesion. While it is generally accepted that certain kinds of roughness can enhance certain kinds of adhesion, the appropriate quantitative characterization of the roughness and its quantitative correlation with adhesive strength have been lacking. Area-scale fractal analysis by the patchwork method can be used as a method of quantifying substrate roughness to help elucidate the influence of surface roughness on adhesion on rough substrates. This paper examines this approach theoretically and experimentally. The discrete bonding model (DBM) is used for modeling rough-substrate - adhesive interaction. The current supposition is that the DBM should be applicable to a wide variety of adhesive systems. Experimental support for the DBM in metallic, thermal-spray coating system is presented. While the thermal-spray system may seem far removed from pressure sensitive adhesives, there are powerful novel experimental methods reported here with good potential to identify critical scales in any adhesive system. These critical scales are crucial for the prediction of adhesive strength as a function of substrate roughness in the DBM. 1.1 The discrete bonding model (DBM) The DBM attempts to predict the adhesive strength on a rough surface by summing the contribution of discrete, individual bonds. The DBM assumes that each bond requires the same force to break it, and each takes up the same amount of space, or discrete area, on the surface. The macroscopic adhesive strength is due, thereby, to summing the contribution of a large number of individual bonds (Brown 1994). The view of the substrate surface in the DBM is based on fractal properties that are common to many engineering surface textures (Brown et al. 1998). The key element of the fractal properties for the DBM is that practically the surface area available for bonding is not unique - it depends on the scale, or discrete area, of the bond. Smaller bonds see more of the details in the topographic features than the larger bonds. For fractal surfaces with dimension greater than 2, which indicates a perfectly smooth " ~..,ii I 51

2 of the adhesive, as well as the method of its application. In some cases the characteristic scale may be derived from first principles applied to the adhesive system. It can also be determined or verified experimentally, as shown below. Knowing the characteristic scale is necessary for applying the DBM. In some adhesive systems it may be possible to suggest a characteristic scale from some characteristic size of the adherents. In thermal spray, the complexity of the interaction of a liquid droplet and a solid substrate is complex, and an experimental approach seems necessary to determine the characteristic scale. 1.2 Area-scale fractal analysis and the determination of relative areas Specialized software ( is used to determine the relative areas as a function of scale from measurements of a rough surface. The software uses the patchwork method, a virtual tiling algorithm (Brown 1993, Brown et al. 1993), applying triangular tiles, or patches, with the same area, but not necessarily the same shape, to virtually tile a measured surface (Fig. 1). The area of the tile represents the scale. The virtual tilings are repeated so that a range of scales is represented' The apparent or calculated area at a particular scale is equal to the number of tiles used in the tiling multiplied by the area of the tile. The relative area is determined by dividing the calculated area by the nominal area of the surface that has been covered in the tiling. i... ~~:~:~.~!~.~!!!!~!~!!!!~!i!~.!!~!~!ill!~i~i!~i!ii!~!~!~i~i~i~iiii~ii~iii~!ii~iii~iiiiii~iiiiii~iiiii~!~!~iiiiiiiii~i~iii~ii~i~i~i~iiiiiiii~iiiiiii~iiii!~i~i~;ii~~i;~iiiiiiiii~iii!i~!~i~i~ii~ ~ii~i~i~i~i~i~iii~i~ii~i~iiii~iiiiiiiiiiiiii~iiii~i~!~ii~ii~i~i~i~i~i~!~!;~;~:~..~.~... [ ~t ~" ,-... [... ~... :.. :~,..~.~:~..~.~.~.~:::,:;;::2 ~2~[:~:i:~::::a::::,,::::::..~.????:::::.~:,:.,:.v.-~.::::~:~.'2~. y:.u.;::;:::: ~.:::::::::::::::::::::::::::::::::: ::::::::::... ~=... :... [... "....:-::v::~..:,.;v..~ [... ~:::~:~:~:~~2~?~2~?~i~Li;~?~i??~i~i2~2~?~i~2~?~iL~~~;~~22~2~~2~~::~:~:~:~:::~:::,:~...! /... [ :.;:;:~.:~.~.:7:;;!:!:':::?i.:!2::::-.::::... :::::::::::::::::::::::::::::::::::::::.....~...:~.::.~=~... ::.=:::.:::7:~:::-... %,,,,.,Tj.~ [...,~/i... ::7:::?:::::::~:-...,~.:~...;:.::::... ]... ~.~!ii~.:`~.~.`..;.~..~.~..:...:...`..~.~`.~..~...~.~.~!~~~... x:... ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::...,~".~,..~,.,::i:i:i::i:~:.?:: : -.-~:, ~, ~. ~ :,.. :~ ~. -:-,..,...1 :~:::..v ::v:.~,,~ "..... ~...;~.~,,... ~... ~'~; ~.. "... ~,~,...~.:,-.~,,, -... '...%.:~....v.....-x :... : f... ::::... ~.~ ~......;~...-.,::.:,~,.:~...~,~... "". :~..~::-~-~..~+.~-,~.,..~:.~.~."~"-.... :::::::.-:-~-?~.~... ~:~-~~:..,,~::::.':~.:.:~:.-.:~:~.~-~:~.~:~.../ -~-~::x:::...:--~;-~.?:~!~:~:... ~.... r...l.j ~:.~i~...,~7-::.::... ~... ::::::::,.,,~... ~~::~,~-~~ J ,,,~./ " ~;~:-~'~ ":~]"1 ~... : ' ~ " ~"~ " " "~["... ~ "... ~ " "... [[';;:~-JJ[[~ " -- "... ~ ~',"... ':..-~."~"~... ".::"... ~"~\"~:... : ~... ~ ~ -:~2~.~;~;~ "... :::..:.-~-~:'~~.":,~.~..~:~:,.'~,,.Z.... ::.~.:;':::...,~:,~.....!.":!:!:!::::~:=:~ ~.:..- :,,,.~,.... :.,.,,.~ ~.... ~.~ ~...~.-~......' ~i.!~:.,~.~...-.:~,. ""...~.,....!~ : '":':: ::.~ ~. "....~,..~.... ~...?=..~....= <~ 2.5mm / Fig 1. Six virtual tilings illustrated on a measured machined surface measured by a scanning laser microscope. The top tiling uses 32 triangular tiles with areas of 0.174mm 2 to calculate a relative area of at that scale. The bottom one uses 5,236 triangles with areas of mm 2 to calculate a relative area of :-~ k 1.3 Area-scale plots and the calculation of the fractal dimension The dependency of the calculated area on the scale of measurement or observation is a fractal property. The fractal dimension, which could be used to characterize the complexity of the measured surface over 52 [.,-[

3 surface, if the scale, or area of the bond is reduced by one half the number of available bond sites will more than double. In a conventional calculus approach the macroscopic adhesive strength St, in units of force per area, might be represented as the integral of the strength over the total surface area: St = 1/Ap fgdss [ 11, where s~ is the differential adhesive strength, and Ap is the nominal area over which the bonding is taking place (i.e., the projection of the rough substrate surface onto a nominally horizontal plane). The use of a differential element of area, however, ignores the known fractal properties of engineering surfaces. Over some range of scales, specifically those where the surface textures have fractal properties, the surface area is not differentiable. In the DBM the bonding phenomenon is not considered to be infinitely divisible, as in a classic calculus approach. The DBM supposes that the bonds are discrete, i.e., individual and separable. It supposes that each bond has a finite strength, Ss, and size, As. The size of the bond, As, is the characteristic scale. Then overall adhesive strength is a function of the number of bonds, N. Assuming that each bond has the same strength, then: St-- N Ss/Ap + C [2]. ~... In this expression C is a kind of constant of integration, with units of force per area. The physical interpretation of C is discussed below with the experimental results. The number of bonds is a function of the available area Ats, at the scale of the individual bond, As" N = m Ats/As [3 ]..i The total area of the surface determined at the scale of the bond, Ats, includes the area due to the topographic features at and above the scale of the individual bond, As. The total number of potential bonding sites Ats/As is likely to be large since As is expected to be small. Ats is evaluated, or measured, at the scale of As, since rough surface areas vary with the scale of measurement. Some of the potential bonding might not be available, therefore the equation includes the factor m, which would be between 0 and 1, accounting for the fraction of available bond sites on the rough surface that is occupied. It could be that m is a function of the surface geometry, making it a function of position, m(x,y). Substituting Eq. [3] into Eq. [2] and rearranging gives: St = m (Ss/As) (Ats/Ap) + C [4]. The discrete bond is characterized by term (Ss/A~), which has units of force per area. The term (Ats]Ap) is called the relative area, and it characterizes the surface texture, or roughness at the scale As. The relative area is the ratio of the total area at the scale of the bond to the nominal area. Since Ats varies with scale, the relative area also varies with the scale of evaluation on a rough surface. The relative area, (Ats]Ap), is always equal to or greater than one. At some sufficiently large scale most surfaces of technical interest appear smooth, and at this scale and larger the relative areas will be one. The relative areas as a function of scale for a rough surface can be determined by area-scale fractal analysis using the virtual tiling or patchwork method (Brown et al. 1993). The appropriate areal scale for the Virtual tiles used for determining the relative area, so that Eq. [3] is valid, is As, the characteristic scale for the discrete bonds in that adhesive system. It should be expected that this scale would be a function of the composition and microstructure of the substrate and 53 i, <. r- v i... ~.

4 .. r some particular range of scales, can be determined from the slope of a log-log plot of the relative area versus scale, i.e., area-scale plot (Fig.2): Das = 2-2(slope) [5],.i--.. where (Das) is the fractal dimension determined from area-scale analysis. The fractal dimension itself, however, is not used in this method for determining the characteristic scale of adhesion, nor is it used Specifically in the DBM for determining the adhesive strength. According to the DBM, the relative area at the characteristic scale for that adhesive system is one of the necessary parameters for characterizing the surface for understanding the roughness component of adhesive strength. Relative area t.5 X X X 0 t.4 x t.3 x x t.2 ; x,,, x x t.t!i slope " ~ '~ t.o t, 000 t O, 000 i too, 000 t, 000, 000 to, 000, 000 i Scale of observation (nm 2) Fig. 2. Area-scale plot illustrated using a measurement by a scanning tunneling microscope of a diamond coating on a silicon substrate (courtesy of UNCCCC). The slope of the region between the two vertical dashed lines is about so the fractal dimension would be about Note that the relative areas indicate that at scales of about 1000nm 2 there is 50% more area available for bonding than at scales of 10,000,000nm 2 (=10~tm2). Nonetheless the essential element for the DBM is the relative area at the characteristic scale. 2. Methods.:.. i L A verification of the DBM has been done with thermal spray coating on four metallic substrates. The substrates were prepared by grit blasting with a number of different rough surface textures. These textures were measured and the area-scale relations are determined over an appropriate range of scales. The surfaces were then coated by thermal spray and the adhesive strengths are determined,i 54 :...

5 experimentally according to ISO standard (1999). The scale where the relative area best correlates with the adhesive strength is determined from a series of plots of relative area versus adhesive strength. 2.1 Substrate surface preparation A variety of textures were prepared on specimens made from four different steel alloys. First they were polished to a mirror finish, then grit blasted varying impact angles, stand-off distances, pressures and number of passes (Seigmann and Brown 1999). 2.2 Surface texture measurement A scanning stylus instrument (UBM, Sunnyvale, CA) was used to measure the surfaces. The stylus radius was 51am, the sampling interval 21am and the scanned region was 500x5001am. Area-scale analyses were performed on the measurements from each substrate prior to coating by vacuum plasma spraying. The relative areas were determined for scales between 2 ~tm 2 and 125,000 l.tm 2. Preliminary area-scale plots were used to optimize the measurement parameters. The size of the scanned region was selected so that large scales, those above which the relative areas greater than one, would just be included in the analyses of the measured surfaces, avoiding measuring larger regions than necessary. The sampling interval was selected as small as reasonable with a 5 ~tm stylus radius, although it is not certain to be small enough to include the characteristic scale for these adhesive systems. 2.3 Coating application A 200 gm thick NiCr coating was applied at a chamber pressure of 100 mbar, a standoff distance of 275 mm and 44kW of electrical power. The primary gas was argon at'46 1/min and the secondary gas was hydrogen at 6.5 1/min. The powder grain size was ~m. Two identical adhesion test specimens were made for each substrate material and grit blast combination. Preliminary tests indicated that the variation in adhesion strength for any given condition would be small, typically about +5 MPa, thereby encouraging the use of a small number of specimens in each sample. 2.4 Determination of the characteristic scales The experimental determination of the characteristic scale in an adhesive system is a crucial stem. Dr. Seigmann at the Swiss Federal Testing and Material Laboratories (EMPA) in Thun developed the method. The method examines adhesive strengths as a function of relative areas at a particular scale, and then looks for patterns over a range of scales. By S iegmann's method first one substrate material and one scale are considered at a time. The adhesive strengths of the coatings on the several different surface textures are plotted versus the relative areas for those textures, at that scale. Then the least square regression coefficients, R z, are determined for the adhesive strengths versus relatives areas at that scale. And, then the process is repeated at other scales so R z values are determined over a significant range of scales. Finally the regression coefficients are plotted versus scale to see if there is a tendency for a maximum. From the DBM it is postulated that there should be a maximum on this final plot of R 2 versus scale. If the DBM is correct, then relative areas at the large scales should tend to underestimate adhesive strength, and the relative areas at the fine scales should overestimate adhesive strength, and between :..".i ~.(i :, - 55

6 I these there should be a maximum. The scale with the highest regression coefficient, i.e., where R 2 is a maximum, should be the characteristic scale for this adhesive system. 3. Results The adhesive strengths versus relative areas at four different scales on one substrate are shown in Fig. 3 The linear regressions on plots like these are used to generate the plots of R 2 versus scale plots for each substrate material (Fig. 4). In Fig. 1 for the plot generated at the scale of 25 ~tm 2, the equation for the best-fit linear regression line is: St = 1422(Relative area)@25rtm, [4], where (Relative area)o25rtm, = and the value of m (Ss/As) is 1422 MPa. scale = pm = scale = pm = ,m < oooo i R = I Adhesive Strength (MPa) o0oo t [] R2= 0.38 [] [] [] [] Adhesive Strength (MPa) scale = 3791 pm = scale = 25 pm = (D ~, ml.004 ~rl.002 J R 2 = 0.64 & 1.12 m 1.10 (D < 1.08 (D > N 1.o6 rr ~ R =0.88 QQ 1.ooo i Adhesive Strength (MPa) Adhesive Strength (MPa) +~ Fig. 3. Relative areas at four different scales (indicated on the plots) versus adhesive strength, for the 100Cr6 substrate. 56

7 The regression coefficients generally increase as the scale decreases from the largest scales, l.tm 2 to about 100 gm 2, albeit with somewhat erratic behavior. The plot of R 2 versus scale for adhesion on the St 37 substrate shows what could be interpreted as a broad maximum from about 200 tam 2 to about 20 ILtm 2, as R 2 exceeds 0.70, although the maximum R 2 exceeds 0.80 at a scale of gm 2. The regression coefficients for relative areas versus adhesion on the XCrNil 8/10 substrate follow St37 X C rn il n [ i i i i Illl[ I I I I 1 III [ I I,,, I II Scale (pm =) Scale (pm =) 42CrMo4 100Cr n I I I I I IIII I I I I I l III n ~, F, ', ' ', '"' I Scale (pm =) t S c a le (p rn =) Fig. 4. Regression coefficients, R 2, for relative area as a function of adhesive strengths on substrates with different textures, versus scale for the four substrate materials. a similar pattern, although the relative maximum for R 2 is higher, and at a finer scale. The values of R 2 versus scale for adhesion on the 42CrMo4 and 100Cr6 substrates still seem to be increasing at the finest scale, 2 ILtm 2. If there is some kind of maximum, as might be expected, it could be in the region of erratic behavior above 200 ILtm 2, or it could be below 2 ILtm 2. 57

8 i 4. Discussion This method of plotting the regression coefficient as a function of scale appears to indicate a characteristic scale, where the regression coefficient is the greatest. The characteristic scale depends on the substrate material. The stylus radius limits the resolution of the measuring instrument starting at about 100~tm 2, which may explain why there is not a clear maximum on all the R 2 - scale plots in Fig. 2. Practically it appears that a sampling interval of about 10~tm is all that is necessary to achieve good regression coefficients, and establish a strong functional correlation between adhesive strength and substrate roughness as characterized by the relative area at the characteristic scale. Assuming that the characteristic scale for the coating on the 100Cr6 substrate is about 25~m 2, then an examination of Eq. 4 can be used to interpret the coefficient for (Relative area)~zs~m, and the constant. The coefficient should represent the strength of a characteristic bond. If the value of the constant were zero then 1422MPa would seem unreasonably high. However from the plot in Fig. 3 it can be seen that there is no adhesive strength at a relative area of about Therefore some roughness is needed to make this coating stick at all. The constant, MPa, represents the adhesive, or rather repulsive, strength at a relative area of zero. Since the minimum relative area is 1, the zero intercept is of no practical interest. Still at a relative area of 1, theresulting adhesive strength is still negative. One interpretation of this negative strength is that surface texture plays an integral role in the adhesive mechanism, since without it there is no adhesion. 5. Conclusions A functional correlation is established between the relative area at characteristic scales with adhesive strength, with R 2 values as high as Discrete bonding is a reasonable basis for a model for understanding the role of substrate roughness on adhesive strength. Relative areas determined by area-scale fractal analysis using the patchwork method of virtual tiling can be used to determine the characteristic scales of adhesion, and to predict the texture component of adhesive strength. : References 1. C.A.Brown, "Surface Roughness Characterization," OberflachenWerkstoffe 34/12 (1993) C.A.Brown, "Scale-Area Analysis and Roughness: A Method for Understanding the Topographic Component of Adhesive Strength," The Adhesion Society, Proceedings of the Seventeenth Annual Meeting and the Symposium on Particle Adhesion, Orlando Florida February 1993, K.M.Liechti, Ed. The Adhesion Society, Library of Congress (1994) C.A.Brown, P.D. Charles, W.A. Johnsen, S. Chesters, "Fractal Analysis of Topographic Data by the Patchwork Method," Wear 161 (1993) C.A.Brown, W.A.Johnsen, K.M.Hult, "Scale-sensitivity, Fractal Analysis and Simulations," International Journal of Machine Tools and Manufacturing 38, 5-6 (1998) S.Siegmann, C.A.Brown, "Surface texture correlations with tensile adhesive strength of thermally sprayed coatings using area-scale fractal analysis," United Thermal Spray Conference- Proceedings, E.Lugschneider, P.A.Kammer, eds., DVS Verlag, Dusseldorf (1999) i.,,, ~.:7,. LI, 58