Non-contact adhesion to self-affine surfaces: A theoretical model

Non-contact adhesion to self-affine surfaces: A theoretical model

Physics Letters A 377 (2013) 2806–2809 Contents lists available at ScienceDirect Physics Letters A Non-contact adhesion...

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Physics Letters A 377 (2013) 2806–2809

Contents lists available at ScienceDirect

Physics Letters A

Non-contact adhesion to self-affine surfaces: A theoretical model Maxim A. Makeev Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI 48109-2136, United States

a r t i c l e

i n f o

Article history: Received 4 August 2013 Accepted 16 August 2013 Available online 23 August 2013 Communicated by V.M. Agranovich

a b s t r a c t Strength of adhesion between materials is known to be strongly influenced by interface irregularities. In this work, I devise a perturbative approach to describe the effect of self-affine roughness on noncontact adhesive interactions. The hierarchy of the obtained analytical solutions is the following. First, analytical formulae are deduced to describe roughness corrections to the van der Waals interaction energies between a hemi-space adherend, bounded by a self-affine surface, and a point-like adherent. Second, the problem of two hemi-spaces, one of which has a planar surface, and the other is bounded by a self-affine surface, is solved analytically. In the latter case, a numerical analysis is performed to delineate the behavior of the roughness corrections as a function of the parameters, characterizing selfaffine fractal surface roughness. The problem of two hemi-spaces, both bounded by self-affine fractal surfaces, is also addressed in this work. The model’s predictions are compared with previously reported theoretical results and available experimental data. © 2013 Elsevier B.V. All rights reserved.

Interfacial adhesion is of significant importance for a wide spectrum of scientific and technological problems [1,2]. Research areas, where an in-depth understanding of the phenomenon is imperative, range from the science of colloids [3] to newly emerged research endeavors, such as the studies of gecko’s locomotion [4] and friction phenomena at nanoscales [5]. Furthermore, adhesion plays an important role in determining properties of thermal interface [6] and van der Waals materials [7], as well as devices, based thereupon. Whilst the problem of adhesion is quite intricate per se, the fact that corresponding surfaces can be asperous adds even more to the intrinsic problem’s complexity. In this context, two main directions of research can be distinguished. The first direction encompasses approaches focusing on mechanical contacts between pairs of asperities [8–13]. These studies have addressed the problem via a modified model of contact mechanics, due to Hertz [14]. The focus of the second direction has been on multiasperity surfaces in contact. The first theoretical model, dealing with the multi-asperity systems, was developed in Ref. [15]. The rough surfaces, considered therein, were assumed to have roughness on a single length-scale. The most elaborate treatment of adhesion between rough surfaces in contact, developed to date, is due to Persson [16]. In Ref. [16], a model of adhesion between rough surfaces was developed, which not only provides the most rigorous representation of the multi-length-scale nature of surface roughness, but also includes the effect of interactions between asperities. Recently, a major step forward was made in understanding the roughness effect on adhesion. In Ref. [17] – a study performed

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in the context of MEMS – the authors have shown that there exist two different regimes in adhesion, depending on the surface roughness scale. It was found that, when spatial scale of the surface roughness is small, the adhesive behavior is largely defined by the van der Waals (vdW) forces, acting across extensive noncontacting areas. When the r.m.s. roughness length-scale matches that of the separation gap, the regime of contacting asperities takes place. In the regime of non-contact adhesion, an adherend – adherent system can be viewed as two semi-infinite dielectric media, separated by a vacuum gap. In the case of planar surfaces, the problem of vdW interaction for two- and three-layer media was solved, using the methods of fluctuational electrodynamics [18], in Refs. [19,20]. It is of essence to emphasize that the results on separation distance dependence of the vdW energy were found to be in a qualitative correspondence with the evergreen model of adhesion, developed by Hamaker in the year of 1937 [21]. The latter model is based upon the assumption of pairwise additivity of the vdW inter-atomic interactions. The issues pertained to this assumption will be addressed in this Letter. It should also be noted that, recently, the qualitative predictions of the Hamaker’s model were tested in Ref. [22] via extensive atomistic simulations. An excellent agreement between the theory and simulation results on the separation distance dependence of vdW interaction energies between rigid-rod polymer adherent and silicon substrate was revealed by the study, down to the separation distances of the order of the average inter-atomic spacings. In the past, the roughness effect on the vdW interactions between hemi-spaces, separated by a vacuum gap, has been addressed via a perturbative approach in Ref. [23], for the case of one randomly-rough surface; while, in Ref. [24], the case, when both surfaces are statistically rough, was

M.A. Makeev / Physics Letters A 377 (2013) 2806–2809

scrutinized. The major advantages of the present framework over the ones of previously reported studies are addressed below. In this work, the Hamaker’s approach is employed to devise an analytical model to compute the roughness corrections to vdW interaction energies between rough extended bodies. The surface topography description, employed herein, is based on the modern theory of surface roughness, evoking the concept of the self-affine fractality of rough-surface profiles [25]. Specifically, rough surfaces are characterized by the saturation value of the surface width and the correlation length. Yet another noteworthy feature of the present approach is that, while it considers the case of small-scale surface roughness – much alike the prevailing majority of previously developed models – it does not impose any restriction on the surface profile correlation length. The implications of this feature for the model’s predictive capabilities are discussed in this Letter. The vdW interaction energy between two extended bodies, separated by a vacuum gap, is given by the following integral form [21]

U vdW = −ρa ρs C 6

dV a Va

dV s |rs − ra |−6 ,



where, |rs − ra | is the distance between volume elements of the substrate, dV s , and the adherent, dV a , V a and V s are volumes occupied by the adherent and the substrate, ρa and ρs are densities of the two materials, and C 6 is the van der Waals constant. The system under consideration consists of a homogeneous semiinfinite medium, bounded by a self-affine surface, and a point-like or an extended adherent, interacting with the substrate via the vdW forces. The self-affine surface profiles are described as follows. The width of a self-affine surface is defined as w (r⊥ , ξ ) = ([h(r⊥ ) − h¯ ]2 )1/2 and obeys the following scaling relation [25,26]

w 2 (r⊥ , ξ ) = w 2s


2α  f






where r⊥ = (x, y ), α is the roughness exponent, h(r⊥ ) is the surface profile, h¯ is the mean height of the surface profile (h¯ = 0, by the choice of reference frame), and · · · denotes both ensemble and space averages. The scaling function f (u ) possesses the following properties: f (u → 0) = 1 and f (u → ∞) = u −2α . Without any loss in generality, the roughness exponent is fixed at α = 0.5 [27]. The vdW interaction energy between a point-like adherent ( p ), located at a distance l p above the self-affine surface baseline, and the hemi-space substrate (h), U hp ( p ), can be obtained from Eq. (1) by taking the integral over V s . The upper bound of integration over z is the rough surface profile, h(x, y ). The resultant integral form is, thus, given by

+∞ U hp (l p ) = −ρs C 6 −∞


dx −∞

h (x, y )

dz x2 + y 2 + ( z − l p )2





(3) It is well known, however, that fractal profiles cannot be described in terms of simple analytical functions. Here, a statistical approach is employed to circumvent the problem. At the basis of the present approach is an averaging over all possible realizations of the surface roughness. This can be achieved by utilizing a surface-height distribution function. In the case of self-affine fractal surfaces, the surface-height distribution function [i.e., the probability that surface height at r⊥ = (x, y ) is equal to h(x, y )] is of the following Gaussian form [25]

1 h2 P (r⊥ , h) =  exp − . 2w 2 (r⊥ , ξ ) 2π w 2 (r⊥ , ξ )



Consequently, the average vdW interaction energy, U hp (l p ) P , can readily be computed by taking averages over all surface profile realizations, each of which is weighted by the probability P (r⊥ , h). By combining Eqs. (3) and (4), the following integral form for the average vdW interaction energy is obtained

U hp (l p ) P = −ρs C 6





× exp



+∞ −∞

2π w 2 (r⊥ , ξ )


h2 2w 2 (r


⊥, ξ )






+ ( z − l p )2 ]3

. (5)

After taking the integral over z in Eq. (5), the resultant integrand is given by

 ( p − h) (x, y ,  p , h) = − 2 arctan  8(x + y 2 )5/2 (x2 + y 2 ) ( p − h) − 4(x2 + y 2 )[(x2 + y 2 ) + ( p − h)2 ]2 3( p − h) − 8(x2 + y 2 )2 [(x2 + y 2 ) + ( p − h)2 ] 3


3π 16(x2

+ y 2 )5/2



In what follows, we employ the approximation of small-scale surface roughness. In other words, we consider the case, when the characteristic length-scales of the self-affine roughness are much smaller than those of the nominal separation gap,  p . Under this assumption, (x, y ,  p , h) can perturbatively be expanded in powers of the small parameter (h/l p ) up to the fourth order, inclusive. The resultant series expansion is represented by the following convenient form

(x, y , h) = Υ f (x, y ,  p ) +


Υkh (x,

 k h y, p )

k =1

+ O (h/ p )5 ,



where Υ f (x, y ,  p ) corresponds to the planar-surface contribution. Note that hereinbefore no assumptions regarding the correlation length, ξ , have been made. This is of essence, as roughness of surface profiles can have characteristic length-scales up to tens of nanometers, in the case of self-affine profiles [25], or even be close to a micrometer, as in the case of periodically undulated surfaces [27]. Further, by employing Eq. (7), the corresponding integrals over h in Eq. (5) were taken analytically. This allows for straightforward calculations of the expansion coefficients in Eq. (7). Note that, as can be deduced from the structure of Eqs. (5) and (7), all odd powers of (h/l p ) cancel out by integrating over h. The complete set of coefficients Υkh (x, y ,  p ) [k = 1, . . . , 4] is given by

⎧ ⎪ Υmh (r⊥ ,  p ) = 0, if m is odd, ⎪ ⎪ ⎪ ⎪ ⎪ 12l3p ⎪ ⎨ Υ h (r ,  ) = , p 2 ⊥ 2 (r⊥ + l2p )4 ⎪ ⎪ ⎪ 2 ⎪ l5p (−6.0r⊥ + 14.0l2p ) ⎪ h ⎪ ⎪ Υ ( r ,  ) = . ⊥ p ⎩ 4 2 (r⊥ + l2p )6

Next, the integral over r⊥ = (x, y ) in Eq. (5) is split into two parts in order to take into account the scaling behavior of the surface width [see Eq. (2)]. Finally, by evaluating the corresponding integrals over r⊥ , we readily arrive at the following expressions for the roughness corrections to the vdW interaction energy


M.A. Makeev / Physics Letters A 377 (2013) 2806–2809

U hp (l p )


 f ¯ 2s + 4 (, ξ ) w ¯ 4s = U hp ( p ) 1 + 2 (, ξ ) w 6  ¯s , +O w


f U hp ( p )

¯ s = w s /l p , and where w = −πρs C 6 /6l3p is the corresponding planar-surface solution. Note that the planar-surface solution, given above, exactly reproduces Eq. (19) of Ref. [21]. The corresponding expansion coefficients are given by

 2 ( p , ξ ) = 3



+3  4 ( p , ξ ) = 9




ξ lp


5(l p /ξ )4 + 8(l p /ξ )2 + 3

[(l p /ξ )2 + 1]3   ξ , arctan lp

 (l p /ξ )6 (1 − (l p /ξ )4 ) . 1+ [(l p /ξ )2 + 1]5


In principle, the set of Eqs. (8)–(9) allows for computing the roughness correction to adhesive interactions between a semi-infinite medium, bounded by a self-affine surface, and an adherent of an arbitrary shape. Let us consider the case of a planar hemispace adherent, separated from a self-affine hemi-space substrate’s baseline by a vacuum gap of varied thickness . For this system, the vdW interaction energy can be obtained following the procedure, described in Ref. [21]. Specifically, the reader is referred to Eq. (20) of this reference. In brief, the procedure involves integration over l p , the lower and upper integration bounds being l and +∞, correspondingly. The resultant average vdW interaction energy, U hh (l) P , between two hemi-spaces is then given by the following expression

 f ¯¯ 2 + Ω4 (, ξ ) w ¯¯ 4 U hh () P = U hh () 1 + Ω2 (, ξ ) w s s 6  ¯¯ +O w s ,


¯¯ s = w s /, and U () is the corresponding planar-surface where w hh solution. Note that, in the planar-surface limit (that is, w s → 0), the vdW energy, obtained via the aforedescribed procedure, was f


found to be U hh () = −(πρs ρa C 6 )/122 = −C H /12π 2 . This is exactly the Hamaker’s result, obtained for the vdW interaction energy between two smooth extended bodies, C H being the Hamaker’s constant [21]. The coefficients in Eq. (10), accounting for the second- and fourth-order roughness corrections, are given by

 2      1  ξ ξ , Ω2 (, ξ ) = 6 arctan + ξ   [(/ξ )2 + 1]  2   (/ξ )4 (1 − 3(/ξ )2 ) 3  Ω4 (, ξ ) = . 3+ 2 ξ [(/ξ )2 + 1]3


In what follows, Eqs. (10)–(11) are used for gaining quantitative insights into the effect of self-affine roughness on non-contact adhesive energies between a pair of macroscopic bodies, separated by a vacuum gap. A numerical analysis, however, requires the knowledge of the Hamaker’s constant for corresponding material combinations. Note that the Hamaker’s constant of the form deduced above might not be suitable for precise quantitative description of the real-world systems, as its derivation is based on the additivity ansatz. Indeed, as according to Ref. [28], the non-additivity correction to the vdW interaction energies can be as large as 10%. However, there exists a rigorous methodology, which allows for computing C H for two- (separated by air) or three-layer structures [29]. The results, obtained thereby, can then be used in Eq. (10) to the end of computing the corresponding roughness corrections. In the numerical analysis, performed herein, the value of 12.8 × 10−20 J (corresponding to the vdW interactions between planar mica and Si3 N4 ) was used. The value was adapted from

Fig. 1. (Color online.) (a) The vdW interaction energy versus  is plotted at ξ = 8 nm and different values of w s , as explained in the figure legend. (b) The same quantity is plotted as a function of , at w s = 1.5 nm, and different values of ξ . In both panels of the figure, lines correspond to the total roughness corrections to the vdW energy [as given by Eq. (10)], while solid symbols account for the quadratic corrections.

Ref. [29] merely for the purpose of illustrating the model’s application. In Fig. 1(a), the vdW energy versus  is plotted at five different values of w s . As can be observed in the figure, surface roughness significantly enhances the non-contact adhesion. Note that Ω2 (, ξ ) and Ω4 (, ξ ) are both monotonous functions of . This observation holds for any set of the roughness parameters, ( w s , ξ ). Furthermore, as the numerical analysis shows, at /ξ values below 1.0, the total roughness correction is largely defined by the quadratic term in the expansion Eq. (10). However, when the ratio exceeds 2.0, the quartic contribution becomes significant, being as large as 10%. Shown in Fig. 1(b) is the vdW interaction energy, plotted as a function of  at five different values of ξ . As expected, an increase in ξ causes a significant decrease in the roughness corrections. Indeed, an increase in ξ effectively leads a flattening of the surface profile, as the surface width reaches its saturation value at larger lateral distances. In the limit ξ → +∞, the surface becomes nominally planar and the vdW energy asymptotically approaches the corresponding planar-surface solution. The uncovered behavior (as described hereinabove) is qualitatively consistent with previously reported experimental results. Indeed, in Ref. [13], it was demonstrated that surface irregularities enhance the contact adhesion at small values of r.m.s. roughness, the increase in adhesive energy being a factor of 2.5. A quantitative comparison with the results of this study is not possible due to an incomplete characterization of the surface topographies and lack of data on the non-contact regime in adhesion. A more elaborate comparison was made possible by a recent experimental study of Ref. [17], which combines surface topography characterization with high accuracy measurements of the non-contact adhesive energies. In Ref. [17], the systems under consideration were comprised of rough poly-silicon microcantilevers, for which systems accurate measurements of the vdW interaction energies were performed, alongside detailed studies of the surface topographies. To accomplish the goal of testing the present model against the experimental data of Ref. [17], first, approximate formulae should be obtained for the case of two rough surfaces. This task can be achieved as follows. First, let us assume that the surface

M.A. Makeev / Physics Letters A 377 (2013) 2806–2809


Casimir–Polder interactions requires a different treatment and will be considered elsewhere. In summary, a model was developed to compute surface roughness corrections to the non-contact adhesive energies. Analytical formulae were deduced for the cases of a point-like and an extended adherents interacting with hemi-spaces, bounded by a selfaffine fractal surface. The roughness corrections were deduced as a function of experimentally measurable parameters characterizing the self-affine roughness profiles. The case of two rough hemispaces is discussed in the work, and the model predictions are compared with available experimental data. The approach will be of use for predicting the effect of surface roughness on adhesive behavior, thereby providing a theoretical guidance for design of a wide variety of technological applications. Fig. 2. (Color online.) The absolute value of the adhesion energy versus  are plotted for different values of w s , ranging from zero (solid line) to 12.0 nm, in increments of 2 nm (dashed lines, marked by Roman numbers). The solid line corresponds to parallel-plate model prediction (see Eq. (1) of Ref. [17]). Filled circles (also marked by capital letters) are the experimental data, adapted from Ref. [17]. The vertical dotted line marks the length-scale corresponding to the transition between mixed and retarded regimes in the vdW interactions.

profiles of the two interacting bodies are not correlated. In general, a correlation between surface profiles might have a significant impact on roughness corrections, as was recently demonstrated by the same author for periodically undulated surfaces [30]. Here, to the end of performing an order-of-magnitude comparison of the model predictions with the experimental data, let us assume that the two roughness profiles are similar, i.e., possess similar average properties, but not correlated. Under this premise, f

the following expression can be deduced [31]: U hh ()/U hh () 2Ω2 (, ξ )( w s /)2 + 4Ω4 (, ξ )( w s /)4 + O (( w s /)6 ). Note that the structure of the dominant term is consistent with the theoretical results, obtained in Refs. [24,30,32]. To assess the predictive capabilities of the model, numerical solutions were obtained using the above formula. The results are presented in Fig. 2 alongside the experimental data of Ref. [17]. The procedure, leading to the results in Fig. 2, is as follows. First, numerical data on the planar-surface solution were exactly reproduced, using Eq. (1) of Ref. [17]. The result is shown in Fig. 2 as a solid line. Second, the above approximate formula and Eq. (10) were used to produce a set of curves for rough surfaces, with w s magnitude increasing in increments of w s = 2 nm. This was done for the same material parameters as the ones used in the planar surface limit (dashed lines). It should be noted that, in the comparative analysis, the correlation length of the surface profiles should be deemed as a fitting parameter of the model, rather than a derived from an experimental study quantity. The value used, however, is typical for the self-affine surfaces. Note that, although the data of Ref. [17] correspond to the regimes, where the vdW forces are either retarded (  50 nm) or a combination of the non-retarded and retarded ones (10    50 nm) [2], there still exists a good quantitative agreement between the theory and experiment. The behavior at larger than 50 nm separation distances, however, naturally remains unaccounted for by the model. At these length-scales, the attractive forces are retarded and described by the Casimir–Polder’s theory [2]. The case of the

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