ARTICLE pubs.acs.org/Langmuir
Shape and Eccentricity Effects in Adhesive Contacts of Rodlike Particles Narayan Sundaram* and Srinivasan Chandrasekar Center for Materials Processing and Tribology, Purdue University, West Lafayette, Indiana 47907-2023, United States
bS Supporting Information ABSTRACT: The effects of shape and eccentricity on adhesion and detachment behavior of long, rodlike particles in contact with a half-space are analyzed using contact mechanics. The particles are considered to have cross sections that are squarish, oblate, or prolate rather than circular. Such cross sections are represented very generally by using superellipses. The contact mechanics model allows deduction of closed-form expressions for the contact pressure, load-contact size relation, detachment load, and detachment contact size. It is found that even relatively small deviations in shape from a cylinder have a significant influence on the detachment load. Eccentricity also affects the adhesive behavior, but to a lesser extent, with oblate shapes requiring larger separation loads than prolate shapes. The load-contact size solution reduces to that for a right-circular, cylindrical rod when the appropriate limit is taken. The detachment behavior of right-circular cylinders is also found to be mimicked by an entire family of rod shapes with different cross sections.
’ INTRODUCTION The effect of adhesive forces on contact and detachment behavior of spheres was first addressed by the JKR theory.1 This seminal work subsequently led to an explosion of theoretical and experimental development, driven in recent times by applications in biology and nanotechnology. The recent review by Barthel2 provides a comprehensive look at various developments in JKR theory. Notably, the theory was successfully extended to the contact of a right-circular cylinder and a half-space by Barquins3 and Chaudhury et al.4 A wide range of factors influences the adhesion and detachment behavior of contacting bodies. Consideration of some of these factors requires different models of adhesion. A well-known example of this is the DMT model,5 which accounts for adhesive forces outside the contact zone. The domain of applicability of each of these theories has been explored and commented on extensively in the past.6 However, there are other factors such as friction,7 nonelastic material behavior,8,9 and geometric factors, which may be accounted for within the JKR framework. The latter includes the study of periodic contacts,1012 contact flattening,13 and finite-size corrections.14 While studies of periodic contacts are typically concerned with the influence of roughness on adhesion studies of other, less microscopic shape deviations are important because the ideal geometries (sphere, cylinder, flat punch) assumed for the purposes of adhesive calculations are rarely found in applications. This is especially true in biology, which has lead researchers to explore alternative shapes like the torus and suction-cup.15 The key role of geometry in biological adhesive contacts has been highlighted in a recent book by Kendall et al.16 In a similar vein, the detachment behavior of long, rodlike particles whose cross sections are not perfectly circular is likely to r 2011 American Chemical Society
deviate from that of a right-circular cylinder.17 If such a particle is acted on by forces distributed uniformly over its interior (such as gravity) in addition to adhesive forces, and there are no significant longitudinal forces, its contact with a half-space may be represented well using a plane-strain model. However, the shape change renders the standard plane-strain adhesive solution of Chaudhury et al.4 inapplicable and makes it necessary to find different solutions. Of course, there are innumerable ways of describing shape deviations from the ideal right-circular cylindrical shape, not all useful. In the preent work, the focus is on modeling adhesion and detachment behavior of eccentric, flattened, or squarish symmetric cross sections that are likely to be of use in applications. This is achieved by considering indenters with superelliptical cross sections. The advantage of such an approach is 2-fold. First, it incorporates sufficient parametric flexibility to model a wide family of indenter shapes, as opposed to one shape. Second, both the contact pressure and the equilibrium load-contact size relations for such contacts may be obtained in closed form. This makes it easy to quantify the influence of shape deviations on adhesive behavior. Conversely, it will be seen that shape has a considerable influence on detachment behavior. This indicates possibilities for design of adhesive contacts. For instance, it has been shown that design with multiple contacts enhances the effect of adhesion in patterned surfaces.15 Shape control offers another important means of supplementing such efforts. Shape-influenced adhesion behavior is also likely to be useful for applications in sensing.7 Received: March 22, 2011 Revised: August 29, 2011 Published: September 01, 2011 12405
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Figure 1. (a) Gallery of superelliptic indenter rod cross sections with different F and e. (b) Cross-sectional view of contact geometry and coordinate system.
Other geometric aspects such as relative curvature also influence adhesive behavior, but are not considered here. The contact is assumed to be frictionless in the present work.
’ CONTACT OF RODS WITH SUPERELLIPTIC CROSS SECTIONS The superellipse (see Figure 1a) is a generalized shape, described by the following implicit equation when its center is at (0,Rb) F x R yF b ð1Þ þ ¼1 Ra Rb The eccentricity of the indenter is controlled by the parameter e = Rb/Ra. The parameter F controls the shape. As F is increased beyond 2, the cross-sectional shapes become increasingly “squarish”. Decreasing F below 2 leads to more sharply curved figures, and a perfect “diamond” shape results at F = 1. The derivative of the profile in eq 1 is given by 0 F 11=F 1 F 1 x x dy Rb @ ¼ signðxÞ 1 A ð2Þ Ra Ra dx Ra Now if x , Ra, then one has (1 |x/Ra|F) ≈ 1. This assumption is exactly the same as that used in modeling the contact of a cylinder and a half-space, and is valid under similar conditions, that is, when the contact size is much smaller than the radius (in this instance, semimajor axis Ra) of the indenter. With this assumption, one has dy Rb ¼ F jxjF 1 signðxÞ dx Ra
subject to the additional condition Z
a
pðxÞ dx ¼ P
ð5Þ
Here a is the half-size of the contact, P is the normal force per unit length (line load) transmitted across the contact, and A is the composite elastic material parameter defined as A = (2/π)((1 ν12)/E1 + (1 ν22)/E2). In contact mechanics, compressive p(x) is considered positive. In the JKR adhesive problem, one is interested in unbounded solutions to the governing eq 4. In the present case, as shown in the Appendix, the integral equation may be inverted and the pressure p(x) described using special functions of mathematical physics, that is, the general solution to eq 4 which is unbounded at the ends and satisfies eq 5 is P pðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi π a2 x2
Fþ1 " ! pffiffiffiffiffiffiffiffiffiffiffiffi 2 Rb F 5 x2 2 F1 F 2 x2 F , 2; ; 1 a 1 þ pffiffiffi a 2 1 Fþ2 3a 2 2 a2 π πA Ra F Γ 2 ! a F 3 x2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 F1 , 1; ; 1 2 ð6Þ 2 2 a 2 a2 x2 Γ
where 2F1 is the Gauss hypergeometric function, defined as follows20
ð3Þ
This indicates that the superellipse may be treated as an eccentric power-law indenter for the purposes of contact modeling. Note that when F = 2,Rb = Ra = R, one has the standard locally parabolic profile y = x2/2R for the cylinder (with circular cross section). Now, if a rigid or elastic rod with superelliptical cross-section is in frictionless contact with an elastic half-space (Figure 1b), the contact pressure p(x) obeys the following singular integral equation18,19 Z a dy Rb pðsÞ ¼ F jxjF 1 signðxÞ ¼ A ds a < x < a ð4Þ dx Ra a x s
a
a, b; c; zÞ 2 F 1 ð~
¼
∞ Γð~ ΓðcÞ a þ nÞ Γðb þ nÞ zn Γð~aÞ ΓðbÞ n ¼ 0 Γðc þ nÞ n!
∑
jzj < 1 ∨ jzj ¼ 1 ∧ c ~a b e 1
ð7Þ
’ ADHESIVE LOAD CONTACT-SIZE RELATION According to the Maugis approach,6 the edge of a JKR adhesive contact may be considered as the tip of a mode-I crack which propagates in response to changes in load, and its stress intensity factor calculated. The crack advances when the contact 12406
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retreats and vice versa.3 Using the definition of the mode-I stress intensity factor pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi KI ¼ lim σ yy ð2πða xÞ xfa
¼ lim pðxÞ xfa
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2πða xÞ
ð8Þ
calculating the derivatives dP/da and d2P/da2. Setting dP/da = 0, the critical contact half-size at separation, ac, is given by 2 3 F þ 2 2=ð2F1Þ Γ 6 7 6pffiffiffiffiffiffiffiffiffi RaF 7 2 7 ð13Þ ac ¼ 6 πAw 4 F þ 1 5 FRb Γ 2
Only such terms in the pressure p(x) which are unbounded near the edge of contact ( a need be considered. Substituting p(x) from eq 6 into eq 8, taking limits as x f a and simplifying, F þ 1 Γ P 1 Rb aF 2 p ffiffiffi KI ¼ pffiffiffipffiffiffi þ ð9Þ F þ 2 πA Ra F a π a Γ 2
It is easily verified that a = ac is always a minimum of the P a curve when F > 1. The corresponding detachment load, Pc, is 0 11=ð12FÞ F þ 1 F=ð12FÞ BeFΓ C Ra ð1 2FÞ πwA C B 2 C B Pc ¼ pffiffiffi @ F þ 2 A Ra πFA Γ 2 ð14Þ
In this simplification, the general property that limxfa 2F1(~a,b; c,1 x2/a2) = 2F1(~a,b;c,0) = 1 has been used. Now, according to the JKR theory, of all possible contact sizes, the equilibrium size is the one that satisfies the Griffith criterion G = w, where G is the elastic energy release rate and w is the Dupre energy of adhesion. By using the definition of the elastic energy release rate, this condition may be rewritten as rffiffiffiffiffiffi π 4w ð10Þ G ¼ AKI 2 ¼ w w KI ¼ 4 πA
To visualize equilibrium curves, it is helpful to define normalized contact half size a* and load P* in terms of multiples of the critical contact size and detachment load for a cylinder (i.e., F = 2, e = 1): !1=3 2 1=3 w R a a ¼ a= wARa 2 ð15Þ P ¼ P= 8A
Using the value of KI obtained from eq 9 in eq 10 and simplifying, the adhesive load-contact size relation is obtained in closed form as follows F þ 1 rffiffiffiffiffiffiffiffi Γ 1 Rb 4aw 2 F a ð11Þ P ¼ pffiffiffiffiffiffi F F þ 2 A πA Ra Γ 2 √ Setting F = 2, Ra = Rb, and using Γ(1/2) = π, one recovers the Chaudhury/Barquins solution for the cylinder rffiffiffiffiffiffiffiffi a2 4aw ð12Þ P¼ A 2AR The restriction on the validity of the solution is that F > 1. Profiles with F e 1 have a sharp kink at the origin, and the pressure will, on that account, not be bounded there so that a solution to the integral equation with an additional singularity at x = 0 must be sought.
’ RESULTS AND DISCUSSION Equation 11 describes the contact and detachment behavior of the system completely; it allows calculation of detachment loads as well as the effects of varying the aspect ratio e = Rb/Ra and shape index F. Qualitatively, the superelliptic cross section rod exhibits classical JKR snap-in behavior. This is because, at zero applied load (P = 0), eq 11 always has a nontrivial (a > 0) contact half-size solution in the regime of interest F > 1. On applying a separating force, these indenters also exhibit unstable separation at some nonzero contact half-size ac; that is, the load-contact size curve always has a tensile force minimum. This may be seen by
The normalized load-contact size relation is thus F þ 1 ðF2Þ=3 Γ pffiffiffiffiffi 2 wA 2 P ¼ pffiffiffi eaF 4 a F þ 2 Ra π Γ 2
ð16Þ
When F 6¼ 2, the P* a* relation depends on the dimensionless parameter wA/Ra; a representative value of 108 was used for the results in this paper. Normalized equilibrium curves using eq 16 for different values of the shape parameter F and three different aspect ratios e are shown in Figure 2. The a* P* plots for the adhesive cylinder on half-space solution and nonadhesive cylinder on half-space Hertz solution are also shown for reference. It is evident that shape and eccentricity have a significant quantitative influence on adhesive contact behavior. Rods with squarish shaped cross sections (F > 2) require much larger detachment loads and separate at significantly larger contact sizes than the cylinder. The opposite effect occurs for more sharply curved shapes, for example, F = 1.9. It is notable that the magnitudes of these deviations for what are relatively small changes in F are quite large, and likely to dominate other factors, such as differences in Young’s modulus or Poisson’s ratio. Comparing curves with identical values of F in Figure 2a and b shows that prolate shapes (e > 1) have a smaller contact size a* than oblate shapes (e < 1) for the same load P*. In particular, for F = 2, one observes that a*HERTZ < a*e=1.2 < a*e=1 < a*e=0.8. This means that the effect of adhesion in increasing contact size is relatively diminished for prolate cross-section rods and relatively enhanced for oblate cross-section rods, when compared to noneccentric (e = 1) cross section rods. An interesting consequence of shape change is the crossover observed between the Hertz and adhesive a* P* in Figure 2b and c. This implies that, at large enough P*, an indenter with a slightly reduced F = 1.9 forms contacts of smaller sizes than the adhesionless cylinder, despite the presence of adhesion in the former case. It may be 12407
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Figure 2. Equilibrium curves showing normalized contact half-width a* as a function of the normalized transmitted normal line force P* at various values of the index F. The plot at the top (left) is for e = 0.8, and the plot at the top (right) is at e = 1.2, while the lowermost plot is at e = 1. The Hertz cylinder a* P* curve with no adhesion and the Barquins adhesive a* P* curve (with F = 2.0, e = 1.0 and labeled “CYL”) are also shown.
Figure 3. Normalized detachment/rupture load P*c (a) and corresponding critical contact half-width a*c (b) as a function of indenter cross-sectional aspect ratio e and shape index F. For the right-circular cylindrical particle (e = 1.0, F = 2.0), P*c = 3 and a*c = 1.
shown that this crossover with the Hertz curve occurs only for indenters with F < 2. The plots in Figure 3 show the considerable effect of varying e and F on the normalized detachment load P*c and detachment contact half-size a*c. For instance, a rod with oblate, squarish section (F = 3, e = 0.6) separates at a load P*c = 12, which is nearly 4 times that of the cylinder (P*c = 3 at F = 2, e = 1). It is also clear that the shape parameter has a greater influence on P*c than the aspect ratio. This is because, at a fixed value of F, the detachment load and contact half-size lie within a fairly narrow range when e is varied, but there is considerable variation of these values when F is changed and e held fixed.
Figure 3a also shows that the normalized detachment load P*c = 3 is not unique to the cylinder, but shared by a family of indenter shapes with a range of e and F values. This somewhat surprising result shows that a family of shapes that mimic the detachment behavior of the cylinder exist. It is possible that such mimics may be used as substitutes in applications when other geometric or design constraints (that may not be met by the cylinder) exist. The existence of adhesion mimics may be explained by the fact that while the shape parameter F and the aspect ratio e are independent, a change in F may be nullified (to an extent) by a suitable change in e. The shapes themselves may be quite different; for instance, the highly oblate ellipse with 12408
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Figure 4. Scatter plots of normalized detachment loads P*c and critical contact half-width a*c for a variety of indenter shapes at indicated values of F, with e varying from 0.6 to 1.4. The shapes themselves are also plotted. In plot (a), the parameter wA/Ra = 108; plot (b) uses a higher value wA/Ra = 106. Note that, for the right-circular cylindrical particle, P*c = 3 and a*c = 1 regardless of wA/Ra.
e = 0.6, F = 2 has the same detachment load as the highly prolate, rounded shape with e = 1.4, F = 2.14. The effects of shape on detachment behavior are visualized in the scatter plot Figure 4a, which shows a*c and P*c for 1.6 < F < 2.8 and 0.6 < e < 1.4. The corresponding shapes are also plotted, as a visual aid. The parameter wA/Ra has been mentioned before; it is a normalized measure of the strength of the surface forces (via w). The scatter plot in Figure 4b shows the effect of using a larger value of normalized adhesion parameter, wA/Ra = 106. Qualitatively, all the results are similar to wA/Ra = 108, except that the influence of shape changes on P*c, a*c is relatively reduced. This is inferred from the somewhat tighter clustering of points in the scatter plot about a*c = 1, P*c = 3. Nonetheless, the absolute effects of shape deviation on a*c, P*c are still considerable. Interestingly, despite varying two independent parameters F and e, P*c and a*c seem to lie within a fairly narrow band (almost a curve) in both scatter plots. This observation is seen to hold even when wider ranges of F and e are used, and the parametric range sampled more finely. This finding hints at the existence of a universal, powerlaw type P*c/a*c detachment relation for indenters of any shape; the underlying physical reasons merit further exploration. In keeping with the objective of exploring deviations from the cylinder shape, the range of the parameters e and F has been kept fairly narrow in the present work. It is, however, perfectly feasible to consider smaller e and larger F values as long as the contact size a , Ra. For instance, in the limit as F f 1, ac f π2wA/(4e2) and Pc f πw/(Ae). In the other extreme, as F f ∞, Pc f 2 (wR/A)1/2; however, ac f Ra violating the half-space assumption a , Ra and the limit is not useful. However, for any chosen upper-bound on a/Ra (say 0.05), it is nonetheless possible to determine Pc at a large (but finite) F. Another interesting special case is the indenter with elliptical cross-section (F = 2, e 6¼ 1), in which case the normalized critical contact size a*c = e2/3 and P*c = 3e1/3. This also indicates that the detachment load of elliptical indenters has a simple, reciprocal cube-root dependence on e. Again, oblate (e < 1) elliptical particles have a larger detachment load (in magnitude) than prolate (e > 1) elliptical particles. For highly prolate shapes with very large e, the shape introduces an additional source of instability of equilibrium (since the indenter may “tip over” when P is small) and other considerations are needed.
’ CONCLUSIONS The effects of shape and eccentricity on adhesion and detachment behavior of rod-like particles in contact with a half-space were analyzed using contact mechanics. The particles were be of superelliptic rather than circular cross-section. The contact pressure, load-contact size relation, detachment load, and detachment contact size were derived in closed-form using special functions of mathematical physics. Even small deviations in shape from a cylinder have a significant influence on the detachment load and critical contact size, with squarish cross section indenters having much larger detachment loads than circular ones. Eccentricity affects the adhesive contact behavior to a lesser extent than shape, with oblate shapes requiring larger detachment loads than prolate shapes. The load-contact size solution reduces to that for a rightcircular, cylindrical rod in the appropriate limit. The detachment behavior of right-circular cylinders is mimicked by an entire family of rodlike shapes. Lastly, scatter plots of P*c and a*c for a wide family of shapes at different values of the adhesion parameter wA/Ra indicates the existence of a universal power-law relation between detachment load and detachment contact size. ’ APPENDIX Since the pressure p(x) is an even function, the integral equation eq 4 may be rewritten as Z
a 0
2xpðsÞ 1 Rb F 1 ds ¼ x x2 s2 A RaF
0
