Combined dry and wet adhesion between a particle and an elastic substrate

Journal of Colloid and Interface Science 483 (2016) 321–333

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Journal of Colloid and Interface Science journal homepage: www.elsevier.com/locate/jcis

Regular Article

Combined dry and wet adhesion between a particle and an elastic substrate Jin Qian a,b,⇑, Ji Lin a, Mingxing Shi c,⇑ a

Department of Engineering Mechanics, Soft Matter Research Center, Zhejiang University, Hangzhou, Zhejiang 310027, China Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou, Zhejiang 310027, China c Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China b

g r a p h i c a l a b s t r a c t

a r t i c l e

i n f o

Article history: Received 4 March 2016 Revised 15 August 2016 Accepted 19 August 2016 Available online 21 August 2016 Keywords: Combined dry and wet adhesion Dry contact Liquid bridge Capillary interaction Adhesive strength Scaling effect Hysteresis

a b s t r a c t We theoretically model the combined dry and wet adhesion between a rigid sphere and an elastic substrate, where the dry contact area is surrounded by a liquid meniscus. The influence of the liquid on the interfacial adhesion is twofold: inducing the Laplace pressure around the dry contact area and altering the adhesion energy between solid surfaces. The behavior of such combined dry and wet adhesion shows a smooth transition between the JKR and DMT models for hydrophilic solids, governed by the prescribed liquid volume or environmental humidity. The JKR-DMT transition vanishes when the solids become hydrophobic. An inverse scaling law of adhesive strength indicates that size reduction helps to enhance the adhesive strength until a theoretical limit is reached. This study also demonstrates the jumping-on and jumping-off hysteresis between the combined dry-wet adhesion and pure liquid bridge in a complete separation and approach cycle. Ó 2016 Published by Elsevier Inc.

1. Introduction It is widely known that capillary bridges may form at the interface between two solid surfaces when they are in contact or proximity, as a consequence of either intentionally added liquid or ⇑ Corresponding authors at: Department of Engineering Mechanics, Soft Matter Research Center, Zhejiang University, Hangzhou, Zhejiang 310027, China (J. Qian). E-mail addresses: [email protected] (J. Qian), [email protected] (M. Shi). http://dx.doi.org/10.1016/j.jcis.2016.08.049 0021-9797/Ó 2016 Published by Elsevier Inc.

condensation of water from a humid atmosphere. Such liquidmediated capillary interaction may contribute attraction between solid surfaces in addition to the van der Waals interactions, and influence the resultant interfacial energy in various physical or technological systems, particularly at small scales [1,2]. For example, the capillary force formed between micro- or nano-machined structures can be vital to cause the structures to collapse [3]. The effects of water condensation on probe-substrate interaction in atomic force microscope (AFM) or nano-indentation technique

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are always an outstanding issue that requires cautious interpretation of the measurements [4,5]. Wet adhesive interaction may also be responsible for the aggregation of granular materials exposed to humid environments [6], in which the presence of liquid smoothens the irregularities and thus increases the effective contact area between solids surfaces [7]. The crucial importance of capillary interaction is also exemplified by many biological attachment systems. Capillary-based wet adhesion has been shown to play a dominant role in contributing to the attachment systems of beetles [8], blowflies [9], ants [10,11] and tree frogs [12], in contrast to other animals that use elaborate hierarchical hairs in their feet to achieve superior dry adhesion without the presence of liquid, purely relying on van der Waals forces [13,14]. The presence of secretory fluids on foot pads has been reported for various insects [8–11], and special mechanism for the release of secretion to individual pads has been confirmed by recent experiments with flies [15]. This liquid secretion has been shown to play a critically important role for successful attachment: animals catastrophically lost their ability to adhere on surfaces after the treatment of removing secretion from their feet [16]. The quantitative modeling of how the presence of liquid would affect the overall adhesion between deformable solid surfaces should provide inspiration and guidance for the development of biomimetic adhesive devices [17–20]. Wet adhesion is based on the capillary forces through a liquid bridge, with or without dry contact area between solid surfaces. For the case of liquid meniscus surrounding a dry contact area (Fig. 1a), wet adhesion may introduce additional Laplace pressure in the annulus, causing deformation in the compliant solids and modifying the adhesion energy between the solids. For the other case of pure wet bridge without dry contact area (Fig. 4), the region of Laplace pressure induced by wet adhesion becomes a circle acting on the solid surfaces. In both cases, additional capillary force is induced by the relative pressure change within the liquid, which may have a significant influence on the resultant adhesion between the solids. A simple equation has been widely used to calculate the capillary-based adhesive force between a sphere and a flat substrate: Pad ¼ 2pRcðcos h1 þ cos h2 Þ [21], where R is the radius of the sphere, c is the surface tension of the liquid, and h1 ; h2 are the contact angles of liquid on individual surfaces. This formula simplifies the problem in a number of aspects: neither material deformation nor solid–solid interaction is considered, and it implies that the resultant adhesive force is independent of liquid volume and humidity. More rigorous analysis and modeling of the combined dry and wet adhesion are crucial for advancing conceptual insights and the development of capillary-based adhesive devices. Early theories by Fogden and White [22], as well as Maugis and

Gauthier-Manuel [23] studied the elastic contact between a sphere and a deformable substrate in the presence of liquid meniscus, showing that an effective Dugdale-type cohesive zone [24] owing to the uniform capillary attraction within the liquid meniscus leads to a transition between Johnson-Kendall-Robert (JKR) model [25] and Derjaguin-Muller-Toporov (DMT) model [26]. However in their studies, the capillary attraction of meniscus was approximated by a phenomenological cohesive law, and the role of liquid volume in affecting the cohesive zone was not explicitly considered. Fan and Gao [27] studied the adhesion problem that couples sphere-plane contact and liquid-bridging forces, where Hertzian distribution of interfacial pressure was assumed in the area of dry contact. Chen and Yu [28] considered the contact problem between a spherical punch and a piezoelectric base in the presence of capillary annulus. These analyses ignored the adhesion energy of solid-solid interface immersed in liquid. Moreover, wet adhesion mediated by pure liquid bridges between rigid surfaces has been studied with emphasis on the size effect [29] and shape effect [30–32], without considering the possibility of forming dry contact area. Wexler et al. [33] also studied the liquid bridge that is trapped in the thin gap between two soft planar surfaces, without considering solid-solid contact. As evident from the numerous studies above, the issues of combined dry-wet contact, effects of solid deformation, solid-solid interaction surrounded by liquid medium, and how liquid volume at various thermodynamic conditions may influence the combined adhesion were often treated separately, and a coherent theoretical modeling is still lacking. Built upon previous progress, the present work aims to address the aforementioned issues by considering a classical JKR set-up where a sphere is in combined dry and wet adhesion with an elastic substrate. The scaling law of adhesive strength, effects of liquid volume on force-displacement relation, jumping transitions between the combined dry-wet adhesion and pure liquid bridge, and other adhesive performance of the problem will be discussed in a coherent framework. Besides the wet adhesive attachment from natural or artificial devices where the liquid is intentionally added, the problem under investigation should be relevant to asperity-surface interactions (e.g., those in AFM and nanoindentation measurements) as well, where liquid bridges inevitably emerge from vapor condensation at the solid gaps in humid environments. 2. Model We consider a classical JKR-type problem where a rigid sphere of radius R is in adhesive and frictionless contact with the planar surface of a semi-infinite elastic body under a tensile force P ad (Fig. 1a). The half-space material has Young’s modulus E and

Fig. 1. (a) Schematic of the adhesion problem with combined dry contact area and liquid meniscus between a rigid particle and an elastic substrate. (b) The analytical model of combined dry and wet adhesion, as a superposition of external Mode I crack subjected to load F and uniform Laplace pressure DP, respectively.

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Poisson’s ratio m, and a reduced modulus can be defined through E ¼ E=ð1  m2 Þ. The circumstance here is complicated due to the existence of a small amount of liquid around the contact, which fills the gap between solid profiles from contact edge a up to radius c, referring to Fig. 1b. A constant pressure difference DP relative to the atmosphere is therefore induced within the meniscus due to the curved liquid profile. For the present case with combined dry and wet adhesion, we may consider it as the superposition of two configurations, caused by the solid-solid interaction (denoted by the subscripts ‘I’) and the liquid pressure (denoted by the subscripts ‘m’), respectively (Fig. 1b). The adhesion problem mediated purely by liquid bridge without solid-solid contact will be treated separately in Section 2.5.

contact should be compensated by the reduction in interfacial energy at equilibrium, and the settled edge of dry contact should be determined by

2.1. Receding and advancing of dry contact edge

Noticing that the newly created surfaces are immersed in liquid, DcSL should be expressed as ðc1L þ c2L  c12 Þ, where the subscripts 1, 2 and L denote solid 1 (sphere), solid 2 (substrate) and the liquid, respectively. According to the Dupré equation [38], the interfacial energy between the sphere and substrate plane is

Regarding the solid-solid interaction, we assume that a force F is required on the sphere to achieve system equilibrium, and we follow the convention of contact mechanics that negative F represents tensile forces, as indicated in Fig. 1b. The change in solidsolid contact area between the sphere and the substrate can be equivalently treated as receding or advancing of an exterior circular crack in Mode I. Following the approach of fracture mechanics, the stress intensity factor at the edge of dry contact area induced by F and solid-solid contact is [34]

KI ¼

4E a3 =ð3RÞ  F pffiffiffiffiffiffi ; 2a pa

ð1Þ

where a is the radius of dry contact area. The term 4E a3 =ð3RÞ can be recognized as the repulsive force due to the sphere-plane mismatch in contact shape from Hertz model [35]. Because of K I , the opening gap between the sphere and the deformed substrate surface is [36]

uI ¼

pffiffiffiffiffiffi   pa 1 1 a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2  1 þ ðq2  2Þ cos1 ; cos þ q pE q pR q

2K I

ð2Þ

where q ¼ r=a P 1 is the radial coordinate r beyond the dry contact area, normalized by the contact radius a of solid surfaces. The liquid forms a meniscus surrounding the solid-solid contact. A constant pressure change relative to atmosphere pressure, DP in magnitude, is induced up to radius c and tends to bring the solid surfaces together for hydrophilic solids, or separate them apart for hydrophobic ones (Fig. 1b). This uniformly distributed Laplace pressure leads to another stress intensity factor K m , namely,

  DP  a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ; K m ¼  pffiffiffiffiffiffi m2  1 þ m2 cos1 m pa

ð3Þ

where m ¼ c=a is the normalized outer radius of the ring-like meniscus, which is always greater than unity. The displacement of the substrate plane due to the pressure difference DP is therefore [36]

pffiffiffiffiffiffi 2K m pa 1 cos1 pE q pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  4DP  a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m2  1  q2  1  cos1  pE q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! Z minðq;mÞ q2  t 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dt : m2 1 t 2 m2  t 2

um ¼

ð4Þ

The system equilibrium requires a balance between the elastic energy and solid-solid interfacial energy. According to Griffith relation [37], the change in elastic energy per unit area of solid-solid

ðK I þ K m Þ2 =2E ¼ DcSL ;

ð5Þ

where DcSL is the adhesion energy, i.e. the energy required to create a unit area of new surfaces in liquid from the solid-solid contact. Substituting the expressions of K I and K m into Eq. (5) leads to

4E a3 =ð3RÞ  F DP  a pffiffiffiffiffiffi  pffiffiffiffiffiffi  A1 ¼ 2a pa pa

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2E DcSL ;

ð6Þ

where A1 is a function of m (i.e., c=a) given in Appendix A. 2.2. Solid-solid adhesion energy in the presence of liquid

c12 ¼ c1V þ c2V  DcSV ; c12 ¼ c1L þ c2L  DcSL ;

ð7Þ ð8Þ

when the interface is surrounded by vapor and liquid, respectively. Here the subscripts S, V, L denote the solid, vapor and liquid, respectively. In addition to the energetic condition associated with the release of elastic energy and creation of new wet surfaces as the dry contact area changes, we have extra relations according to the equilibrium at the contact edge of triple-phase: the liquid profile should meet both solid surfaces at certain contact angles, denoted as h1 and h2 for the sphere and the substrate plane, respectively. Moreover, the contact angles on individual solid surfaces should follow the Young’s equation (Fig. 1b), namely,

c1V ¼ c1L þ c cos h1 ; c2V ¼ c2L þ c cos h2 ;

ð9Þ ð10Þ

where c is the surface tension of liquid. Combining Eqs. (7)–(10) leads to

DcSL ¼ DcSV  cðcos h1 þ cos h2 Þ;

ð11Þ

which quantitatively relates the solid-solid adhesion energy immersed in liquid (DcSL ) to the counterpart commonly measured in vapor (DcSV ), in terms of liquid surface tension (c) and solidliquid contact angles (h1 and h2 ). Essentially, the adhesion energy associated with the dry contact area is modified by introducing liquid around it. For hydrophilic situation where h1 ; h2 < 90 , the formation of a liquid meniscus reduces the adhesion energy between the solid surfaces, as a compromise of additional Laplace pressure within the meniscus. 2.3. Laplace pressure within the liquid bridge We may approximate the liquid profile as a segment of a circle if gravitational effects are negligible. This treatment is not unreasonable since the opening gap is generally much smaller than any lateral sizes for small contact radii and small liquid volume, as considered in the present problem. Fig. 2 shows the geometry of the liquid profile meeting the opposing solid surfaces at h1 and h2 , in which M and N are assumed to be the intersecting points of tri-phase with horizontal coordinates c and d, respectively. a1 and a2 are the local slope angles of the sphere and deformed substrate plane, respectively. The two dashed lines passing M and N are parallel to the horizontal plane, with h representing the gap distance between them. The geometric relation in Fig. 2 leads to

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  h1  h2 a1 þ a2 cd : tan þ ¼ h 2 2

ð12Þ

The difference of pressure within liquid bridge relative to the atmospheric pressure is determined by the Young-Laplace equation [39], which can be related to the curved liquid profile as

0

1

  1 1 B2cosððh1 þ h2 þ a1  a2 Þ=2Þ sinðh1 þ a1 ÞC qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ¼ c@ DP ¼ c  A; r1 r2 d 2 2 ðc  dÞ þ h ð13Þ where r1 is the meridional radius of the meniscus, r2 is the slant distance from the meniscus surface to the axis of symmetry, and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 h ¼ uI ðmÞ þ um ðmÞ  R2  d  R2  c2 , which can be explicitly written in terms of F and DP as

Here A2 to A5 are all coefficients in terms of m, which are summarized in Appendix A. The local slope angles a1 and a2 in Eqs. (12) and (13) are simply

 dwðrÞ ; dr r¼c

pffiffiffiffiffiffi   pa 1 1 a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 sin  q2  1 þ ð2  q2 Þ sin1 ;  pE q pR q

2K I

wm ¼ um ;

ð16Þ ð17Þ

where um has been explicitly given in Eq. (4). Substituting wI and wm into the expression of a2 in Eq. (15), we obtain its explicit form as



a2 ¼ tan1 

 4E a3 =ð3RÞ  F a 2DP 4DP  A6   A7 þ  2   A8 þ   A9 ; pE a pR pE pE ð18Þ

where A6 to A9 are functions of m (referring to Appendix A).

The liquid fills the opening gap between the two solids from a to c, as demonstrated in Fig. 3. The volume of this ring-like liquid bridge can be integrated as

ð14Þ

d R

wI ¼

2.4. Liquid volume: conserved or varying with environmental humidity

  3  4E a =ð3RÞ  F a2 2 DP  a 4 DP  a h¼  A4   A5  A2 þ  A3     pE a pR pE pE qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  R2  d  R2  c 2 :

a1 ¼ sin1 ; a2 ¼ tan1

when it points downwards in the present problem, which can be calculated by parts as [40]

ð15Þ

where wðrÞ ¼ wI ðrÞ þ wm ðrÞ is the superposed deformation of the substrate surface, with wI ðrÞ and wm ðrÞ representing the surface deformation caused by solid-solid interaction (K I ) and liquid pressure (K m ), respectively. wðrÞ is assumed to take positive values

Fig. 2. Geometry of the liquid profile between two hydrophilic surfaces.

Z V ¼ 2pa2

1

m

q  ðuI ðqÞ þ um ðqÞÞdq;

ð19Þ

where uI ðqÞ þ um ðqÞ is the total opening gap between the solid surfaces, induced by K I and K m together. Substituting the expressions in Eqs. (2) and (4) into Eq. (19) leads to

  3 4E a =ð3RÞ  F a2 a2 2DP  a  A13  A10 þ A þ A  pE  a pR 11 pR 12 pE  4 DP  a 4 DP  a ð20Þ  A14 þ  A15 ;  pE pE

V ¼ 2pa2 

where A10 to A15 are all functions of m, as shown in Appendix A. We caution the simplification of neglecting the access liquid volume associated with the concave shape of the meniscus, so we have conducted numerical calculation to exactly evaluate this neglected part of liquid volume and compared it to that from Eq. (20). It turns out that the difference associated with this simplification is negligible. This will be discussed later in Section 4.3. For the limiting case where liquid volume is conserved, V should be kept constant and the outer radii of wet adhesion region, c and d, recede or advance as the applied force Pad changes.

Fig. 3. The amount of liquid volume that fills the gap between the opposing solids.

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325

For the opposite limit that the system rapidly achieves thermodynamic equilibrium with environmental humidity, the ambient vapor spontaneously condenses into the liquid meniscus and the resultant liquid volume V should be determined by the Kelvin equation [38,41]:

DP ¼ 

RG T P ln ; Vm P0

ð21Þ

where RG is the gas constant, T is the absolute temperature in degree Kelvin, V m is the molar volume of the liquid, and P=P 0 is the relative humidity defined as the ratio of the environmental vapor pressure P to the saturation vapor pressure P 0 . P=P 0 6 1 indicates that the liquid meniscus is subjected to negative pressure compared to the atmospheric pressure, with DP being the magnitude. 2.5. Pure wet adhesion without dry contact The analyses in preceding sections are based on the scenario that there exists a nonzero area of dry contact surrounded by ring-like liquid bridge. When the radius of dry contact vanishes, the adhesive interaction can still exist through a pure liquid bridge between the two solid surfaces (Fig. 4). In this case, a constant negative pressure DP induced by the liquid meniscus acts on the substrate surface in a circular area with radius c, and the normal displacement of the substrate surface under this evenly distributed load is [42]

wðrÞ ¼

4DPc Eðr=cÞ pE

ð22Þ

within the loaded area (0 6 r 6 c), and

wðrÞ ¼

   4DPr c2 Eðc=rÞ  1  2  Kðc=rÞ  pE r

ð23Þ

out of the loaded region (r > c), where KðÞ and EðÞ are the complete elliptic integrals of the first and the second kind, respectively. The gap between the sphere surface and the deformed substrate plane is therefore

uðrÞ ¼ d þ R 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2  r 2  wðrÞ;

ð24Þ

where d is the height of sphere’s lower apex relative to the undeformed substrate plane, as indicated in Fig. 4, the term pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R  R2  r 2 represents the spherical profile, and wðrÞ is the deformation of substrate surface as described in Eqs. (22) and (23). Now the liquid fills the gap between the two surfaces from zero up to c, and the liquid volume should be calculated as

Z V¼ 0

c

2pr  uðrÞdr:

ð25Þ

Inserting the explicit expression of uðrÞ into Eq. (25), we obtain

Z V ¼ pdc2 þ 2p

c

r  ðR  0

Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8DPc c R2  r 2 Þ dr   r  Eðr=cÞdr: E 0 ð26Þ

We also realize that the geometric relation in Eq. (12) and the Young-Laplace equation in Eq. (13) perfectly apply for the present case of pure wet adhesion, with minor adjustment on h:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 h ¼ d þ R  R2  d  wðcÞ: Here wðcÞ can be calculated by Eq. (22).

ð27Þ

Fig. 4. Schematic of pure wet adhesion where dry contact area vanishes. The formation of pure liquid bridge causes solid deformation by exerting uniformly distributed Laplace pressure DP on a circular region of substrate surface (the deformation is not drawn to scale).

2.6. Adhesive force and stress under pulling The total adhesive force Pad is contributed by three components, which are induced by the solid-solid interaction (for combined dry-wet adhesion only), the pressure difference DP and the axial component of liquid surface tension acting along the liquid periphery on the sphere, respectively. The force balance leads to 2

Pad ¼ lF þ pðd  la2 Þ  DP þ 2pd  c sinðh1 þ a1 Þ;

ð28Þ

in which l ¼ 1 if dry contact area is nonzero and l ¼ 0 for the case of pure wet adhesion. The nominal adhesive stress is therefore

rad ¼ Pad =ðpR2 Þ. Essentially, Pad is the externally applied force that is in balance with the overall adhesion contributed by the solid-solid interaction as well as the capillary interaction. As Pad is changed, the spheresubstrate interface adapts and achieves new equilibrium by adjusting the radius of dry contact and the configuration of liquid bridge. We can use d, the height of the sphere’s lower apex relative to the undeformed substrate plane, as the measure of ’displacement’ as the sphere is quasi-statically pulled away from the substrate. Of course d could be negative if the sphere penetrates into the substrate plane under force. At the sphere-substrate interface, the total opening gap between solid surfaces (i.e., uðrÞ) plus the displacement of substrate surface (i.e., wðrÞ), should recover the height of a point on the sphere surface (i.e., d þ f ðrÞ), namely,

uðrÞ þ wðrÞ ¼ d þ f ðrÞ:

ð29Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here f ðrÞ ¼ R  R2  r 2 represents the profile of the sphere. When the dry contact is present, uðrÞ and wðrÞ are in explicit expressions, and d can be determined as

326

d¼

J. Qian et al. / Journal of Colloid and Interface Science 483 (2016) 321–333

a2 F ;  3R 2aE

ð30Þ

depending on the unknown quantities a and F that need to be solved. For the case of pure wet adhesion, d itself is an unknown quantity to be solved from the governing equations. 3. List of governing equations and normalization scheme In the presence of dry contact, we proceed by normalizing the unknown quantities in the governing equations (Eqs. (6), (12), (13) and (20)) using

^¼a a

4E 3pDcSV R2

!1=3 ; b F¼

F

pDcSV R

c d ; m¼ ; n¼ ; a a

ð31Þ

which are chosen in consistency with the normalization scheme of b ¼ DP  R=Dc is unknown for the case of JKR and DMT models. D P SV b ¼ V=R3 should be a prescribed conserved liquid volume, in which V parameter. When the liquid volume changes with the environmenb can be directly determined by the Kelvin equation tal humidity, D P

b should be an unknown quantity to be calculated in Eq. (21), and V by Eq. (20). In either case of conserved or varying liquid volume, m > 1 can be selected as an independent variable, and specifying ^, b the value of m point by point will lead to the solution of a F , n, b (or V b ) from the four governing equations with prescribed sysDP tem parameters. The dimensionless expressions of the governing equations are listed in Appendix B. For the case of pure wet adhesion, because the dry contact radius a vanishes, all the unknown quantities in the governing equations (Eqs. (12), (13) and (26)) are normalized according to: b ¼ DP  R=Dc (for conserved liq^ d ¼ d=R, m ¼ c=R, n ¼ d=R, and D P

where R and E have been described above (sphere radius and reduced modulus), W a is the work of adhesion which corresponds to DcSL (solid-solid adhesion energy in liquid) of our study, and e refers to a length scale of interatomic spacing. Later, Muller et al. [45] confirmed that the JKR to DMT transition was indeed observed as the Tabor number decreased from large to small value. In our study, the circumstances are complicated by the presence of liquid bridges and its role in modifying the surface energy DcSL . According to the analysis in Section 2.2, the work of adhesion in the presence of liquid is given by DcSL ¼ DcSV  cðcos h1 þ cos h2 Þ, implying that the surface energy (DcSL ) can be varied from large to small, as done by Muller et al. [45] for decreasing Tabor number, by reducing solid-liquid contact angles (h1 ; h2 ). In Fig. 5b, we have calculated b ad ^ and adhesive force P the relation between dry contact radius a for different contact angles. Evidently, the maximum adhesive force is lower than the JKR prediction for hydrophobic cases b ad ¼ 1:5) and approaches DMT (h1 ; h2 > 90 ), passes JKR result ( P b ad ¼ 2:0) when the contact angles h1 ; h2 change from result ( P 120 to 10 , consistent with the governing role of the Tabor number. On the other limit of fast thermodynamic equilibrium, the relative humidity P=P 0 is chosen as a prescribed system parameter and the liquid volume becomes an unknown variable. For different ^ as a function of environmental humidity, the dry contact radius a adhesive force is shown in Fig. 6a, which approaches the JKR solution at low relative humidity (e.g., P=P 0 ¼ 0:05) and becomes close

SV

b ¼ V=R3 for liquid volume varying with environuid volume, or V mental humidity), and specifying m will lead to the solution of the rest three quantities. These governing equations in dimensionless form are also listed in Appendix B. 4. Results and discussion 4.1. JKR-DMT transition as a result of changing liquid volume, contact angles or humidity For the case of combined dry and wet adhesion, we first explore ^ and adhethe quantitative relation between dry contact radius a b sive force P ad for conserved liquid volume. The parameters adopted

in the calculation are the following: E ¼ 1 GPa, R ¼ 1 lm, h1 ¼ h2 ¼ 30 , c=DcSV ¼ 0:5 and c ¼ 0:072 N=m. These selected values are used throughout our calculations unless stated otherwise. b varies from 106 to 103 , Interestingly, as the liquid volume V

b ad exhibit a smooth transition from the ^ versus  P the curves of a JKR to the DMT results (Fig. 5a). Here the JKR and DMT solutions for dry adhesion are plotted for comparison in the same normalization scheme. The JKR result [25] reduces to a simple expression of ^ and v (v ¼ ð3pDcSV =ð4E RÞÞ1=3 ), i.e., adhesive force in terms of a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b JKR ¼ 6ða ^=vÞ3  ða ^=vÞ3 , and the DMT result [26] is simply P ad b DMT ¼ 2  ða ^=vÞ3 . In our calculation, the maximum adhesive force P ad of each curve also increases with more liquid volume, between 1.5 and 2 which are the JKR and DMT limits, respectively. Similar JKRDMT transition has been predicted for sphere-substrate dry adhesion in the presence of material gradation [43]. Tabor [44] was able to unify the JKR and DMT models by iden 1=3 tifying a governing dimensionless parameter: RW 2a =E2 e3 ,

Fig. 5. (a) Liquid volume dependence and (b) contact angle dependence of the b ad for the combined dry ^ and adhesive force P relation between dry contact radius a and wet adhesion, indicating a smooth transition between JKR and DMT results. The adopted parameters are: E ¼ 1 GPa, R ¼ 1 lm, c=DcSV ¼ 0:5 and c ¼ 0:072 N=m; the contact angles are fixed at h1 ¼ h2 ¼ 30 in (a) and the liquid volume is fixed at V=R3 ¼ 106 in (b).

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to the DMT result at high humidity (e.g., P=P0 ¼ 0:95). Higher relative humidity generally leads to larger adhesive force, and this trend is similar to the dependence of adhesive force on conversed b in Fig. 5. This can be understood by the fact that liquid volume V increased amount of liquid will be condensed to the spheresubstrate gap in a more moist environment, as evident by the calculated liquid volume that varies with different relative humidity in Fig. 6b. It should be noted that Xu et al. [46] also developed a quantitative description of the JKR–DMT transition in terms of relative humidity, by including the Kelvin meniscus radius as one governing parameter.

4.2. Scaling law of dry-wet adhesive strength Now we consider that the sphere is quasi-statically pulled away from the substrate against the adhesive force provided by the liquid bridge and possible solid-solid interaction. The forceseparation curves in Fig. 7a record the non-dimensional adhesive b ad ) as a function of the sphereforce acting on the sphere ( P b under the condition of consubstrate separation (^ d) for different V served liquid volume. The starting points of the curves represent the equilibrium condition at which the total adhesive force is zero. As ^ d increases, a transition from combined dry-wet adhesion to pure wet adhesion occurs until the liquid bridge breaks up. Fig. 7b shows several snapshots of equilibrium profiles of the liquid bridge and the substrate surface at different sphere-substrate separations

b ¼ 104 ). It is seen that the liquid profile meets the (when V deformed substrate and the sphere at prescribed contact angles. As the separation is reduced, the liquid spreads outward due to the condition of conserved volume. Above the equilibrium point, the sphere-substrate interaction is always attractive (Fig. 7a). The adhesive force increases at the beginning until a maximum value is reached, and then decreases with continuously increasing separation. There exists a critical separation beyond which a stable liquid bridge no longer exists due to configurational instability, corresponding to the ultimate failure. Throughout the pulling process, there generally exists a maximum adhesive stress, which corresponds to the adhesive strength:  ad

r ¼

F ad

pR2

¼

c R



b F

a

þ

ðn2  1Þ

a

! b þ 2na ^v sinðh1 þ a1 Þ ^2 v2 D P a

: ð32Þ

Here a ¼ c=DcSV . Eq. (32) indicates an inverse scaling law of the combined dry and wet adhesion, implying that contact splitting into smaller R can result in significant strength enhancement and is therefore an important principle in designing artificial adhesive devices. The principle of contact splitting has been pointed out by several other studies on fibrillar structures in biological attachment systems [47,48]. It should be noted that the strength enhancement by size reduction is ultimately limited by a saturation level of stress within the liquid bridge, which is set by the ability of liquid to withstand negative pressure [49]. Possible existence of small-scale bubbles or cavities leads to a much lower level of saturation stress. Qian and Gao [29] demonstrated that decreasing the size of fibrillar spatula down to micrometer scale would actually hinder wet adhesion in biological attachment systems. 4.3. Instabilities and hysteresis in a separation and approach cycle As the sphere is separated from or brought toward the substrate surface, the force-separation curve shows two solution branches for the accessible range of separation, as demonstrated in Fig. 7a. One can expect two instabilities during the separation and approach process: one occurs at the instability of the solid-solid contact, and the other takes place at the stability limit of pure liquid meniscus, corresponding to points B and D in Fig. 7, respectively. As the transition between the combined dry-wet adhesion to pure wet adhesion occurs, the turning points B and D indicate the jumping-off and jumping-on of dry contact area when the sphere-substrate separation is increased or decreased infinitesimally (Fig. 7b). These jumping-off and jumping-on generally follow different paths, and this hysteretic behavior in force-separation curves becomes more pronounced as the liquid volume is increased, as shown in Fig. 7a. The substrate modulus E plays an important role in governing the hysteretic loop of such force-separation curves. Comparing the separation and approach process at two different moduli (E ¼ 1 GPa and 10 MPa), we find that the hysteresis is much larger when the substrate becomes softer (Fig. 8a), which can be understood by the fact that the softer substrate can accommodate the external load by producing larger deformation. For a given separation ^ d ¼ 0, dry contact adhesion exists for both cases, but the dry contact area of the softer substrate is much larger than that of ^ is increased the stiffer one (Fig. 8b). As the normalized separation d

b ad for ^ versus adhesive force P Fig. 6. (a) JKR-DMT transition of dry contact radius a the scenario that achieves rapid thermodynamic equilibrium with various environmental humidity P=P 0 . (b) The change of liquid volume at the indicated ^. The adopted parameters are the same as those in humidity levels as a function of a Fig. 5a.

to 0.03, the stiffer case of E ¼ 1 GPa already loses the dry contact and enters the region of pure wet adhesion, while the combined dry and wet adhesion is still present for the softer case of E ¼ 10 MPa (Fig. 8c). The sudden transition between the combined dry-wet adhesion and pure wet adhesion is also exhibited by tracking the outer radii

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Fig. 7. (a) The force-separation curves of the quasi-static pulling process with different conversed liquid volume, showing two points of instability corresponding to the transition between the combined dry-wet adhesion and pure liquid bridge. (b) Snapshots of the equilibrium profiles of solid surfaces as well as liquid bridge at four transition points (A, B, C, D), as indicated in (a).

of liquid bridge as well as the negative pressure within the meniscus. Fig. 9a shows the two outer radii (c and d) of the liquid profile meeting the two solid surfaces during the complete separation and approach cycle, and there exists a sudden change when the combined adhesion jumps to pure liquid bridge due to the instability, and vice versa. The pressure difference DP within the liquid meniscus shows similar behavior at the transition points (Fig. 9b). We also examine the calculation of liquid volume in Eq. (20) by neglecting the access liquid volume associated with the complex shape of the meniscus, so we have conducted detailed calculation to exactly evaluate this neglected part of liquid volume. It turns out that, the relative error of the liquid volume induced by this treatment is below 9% for the entire force-separation curve (Fig. 9c). We also examine the force-separation curve when the liquid volume is changing by achieving thermodynamic equilibrium with the environmental humidity (Fig. 10a), and it is found that the transition from the combined adhesion to pure wet adhesion becomes smooth, owing to the elevated liquid volume from the environment before and after the transition (referring to the curves of liquid volume in Fig. 10a). This feature differs from the case of conversed liquid vol-

ume. Fig. 10b shows the snapshots of system configuration with or without solid-solid contact at different ^ d, where the change of liquid volume is remarkable around the transition points. Wet adhesion can have a much longer range of effective interaction compared to dry adhesion based on van der Waals interaction that is of the order of atomic scale. For example, the effective interaction range can be hundreds of nanometer for the forceseparation curves in Figs. 7a, 8a and 10a, which allows the wet adhesive systems achieve considerable amount of adhesion energy against external pulling. This behavior of liquid bridges is quite analogous to the concept of ’long bonds, not strong bonds’ proposed for elastic fibers adhering to a substrate [50].

4.4. Modifications by considering real contact area, solid surface tension, hydrophobic and more compliant substrates In considering the equilibrium between the elastic energy and solid-solid interfacial energy at dry contact edge (Eq. (6)), the interfacial energy is calculated based on the projected flat area instead

J. Qian et al. / Journal of Colloid and Interface Science 483 (2016) 321–333

Fig. 8. (a) The force versus separation curves for different values of substrate modulus (E ¼ 10 MPa and E ¼ 1 GPa). The liquid volume is conserved at b ¼ 104 . (b and c) Snapshots of the deformed substrate surface when the substrate V stiffness adopts different values, at two separations of (b) ^ d ¼ 0 and (c) ^ d ¼ 0:03.

of real contact area. For small scale contacts to soft materials, the difference between the two areas could be considerable, as analyzed by Lu and coworkers [51,52]. The real contact surface between the sphere and substrate is a spherical cap, whose actual  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi area is 2pRH, H ¼ R  R2  a2 being the height of dry contact edge relative to the sphere’s lower apex. Therefore, the product 2pRHDcSL represents the reduced interfacial energy associated with real contact area. Furthermore, the solid surface tension

329

Fig. 9. The variation of (a) the outer radii of liquid profile and (b) Laplace pressure during a quasi-static pulling process. (c) The relative error of using Eq. (20) in calculating the liquid volume.

may also result in a significant part of energy penalty for small scale contacts to soft materials, which is ignored in the preceding analysis. According to the previous analysis [53,54], this energy term is equal to the solid surface tension c2L multiplied by the additional surface area pH2 , where c2L is the work needed to create additional surface area in liquid by stretching (the subscripts 2 and L denoting the substrate and liquid, respectively). Therefore, the combined effects of real contact area and solid surface tension can be described by the energy term 2pRHDcSL þ pH2 c2L ,

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Fig. 10. (a) The force-separation curve, together with the change of liquid volume, in a quasi-static pulling process when the system achieves rapid thermodynamic equilibrium with the environmental humidity (P=P 0 ¼ 0:85). (b) The equilibrium profiles of solid surfaces and liquid bridge at indicated separation.

consistent with the result by Style et al. [53], and considering this combined energy term will modify the equilibrium condition that determines the settled edge of dry contact (i.e., Eq. (6)), namely,

4E a3 =ð3RÞ  F DP  a pffiffiffiffiffiffi  pffiffiffiffiffiffi  A1 ¼ pa 2a pa

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2E DcSL ð1 þ q1 þ q2 Þ;

Fig. 11. The modified force-separation curves by considering (a) real contact area, (b) solid surface tension and (c) the combined effects of real contact area and solid surface tension. The adopted parameters are: E ¼ 10 MPa, V=R3 ¼ 104 , R ¼ 1 lm, h1 ¼ h2 ¼ 30 , c=DcSV ¼ 0:5, c2L =DcSV ¼ 0:5 and c ¼ 0:072 N=m.

ð33Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where q1 ¼ 1= 1  a2 =R2  1 and q2 ¼ q1  c2L =DcSL account for the effects of real contact area and solid surface tension, respectively. It is noticed that these two factors (q1 ; q2 ) carry different signs that cancel the effects of each other, and the modified relation in Eq. (33) reduces to Eq. (6) when q1 ¼ q2 ¼ 0. The modified forceseparation curves by considering real contact area, solid surface

tension and the combined effects of the two are shown in Fig. 11a–c, respectively. It is also interesting to examine the combined dry and wet adhesion between hydrophobic sphere and substrate. For hydrophobic cases, the contact angles larger than 90 result in the convex shape of the meniscus, as demonstrated in Fig. 12a. The geometric relation and Laplace equation in Eqs. (12) and (13) remain valid for hydrophobic cases. We take the parameters

J. Qian et al. / Journal of Colloid and Interface Science 483 (2016) 321–333

331

b ad when the ^ and adhesive force P Fig. 13. The relation between dry contact radius a substrate modulus is reduced to E ¼ 1 MPa. The adopted parameters are the same as those in Fig. 5a except that E ¼ 1 MPa.

5. Conclusion

Fig. 12. (a) Geometry of the liquid profile between two hydrophobic surfaces. (b) ^ and Liquid volume dependence of the relation between dry contact radius a b ad when the solid surfaces are hydrophobic. The adopted paramadhesive force P eters are: h1 ¼ h2 ¼ 100 , E ¼ 10 MPa, R ¼ 1 lm, DcSV ¼ 0:055 N=m and c ¼ 0:072 N=m.

h1 ¼ h2 ¼ 100 , DcSV ¼ 0:055 N=m and c ¼ 0:072 N=m (for water) to achieve the experimentally measured value 0.08 N/m for solid-solid adhesion energy underwater [55], referring to DcSL ¼ DcSV  cðcos h1 þ cos h2 Þ. Fig. 12b shows that the transition from JKR to DMT vanishes when the solid surfaces become hydrophobic. Therefore, the transition from JKR to DMT governed by liquid volume is specific to hydrophilic condition. We have also performed more calculations for substrate modulus down to E ¼ 1 MPa. The curves of adhesive force versus dry contact radius are shown in Fig. 13 for different liquid volume. We can still find that the maximum adhesive force varies between b ad ¼ 2, respectively), b ad ¼ 1:5 and P the JKR and DMT limits ( P influenced by the liquid volume. Of course, the errors caused by approximating the spherical profile by a parabola within dry contact region are expected to arise for substrate modulus as low as 1 MPa, and the exact spherical geometry may be required for the analysis [56,57]. The proper modeling in exploring the extremity of small particles in contact to soft materials, in the presence of liquid, certainly warrants a more sophisticated study in future.

The present study develops a coherent modeling framework to describe the combined wet and dry adhesion between a spherical surface and an elastic substrate in the presence of a liquid bridge. The effects of liquid on the interfacial adhesion are twofold: it induces Laplace pressure that tends to contribute to the sphere-substrate interaction within the wet region, and it alters the adhesion energy between solid surfaces within the dry contact area. Our results show a smooth transition between the JKR and DMT models for hydrophilic solids, governed by the prescribed liquid volume or the environmental humidity for two distinct thermodynamic conditions. Such JKR-DMT transition vanishes when the solids become hydrophobic. The adhesive strength of such combined dry and wet adhesion is investigated in a quasi-static separation process, where an inverse scaling law of adhesive strength is obtained and enables strength optimization by size reduction. There generally exists a hysteresis during the cycle of separation and approach process, which is more pronounced when the liquid volume is larger and/or the substrate is softer. The present study is based on a continuum-level description of capillary force, solid deformation and interfacial interaction, and it is known that some of these assumptions may fail when the characteristic system size is on the order of several nanometers [58,59]. A recent study showed that continuum-level predictions on capillary adhesion match well with molecular dynamic simulations for liquid bridges down to 5–10 nm [60]. Adhesion down to nanometer scale might also lead to emerging statistical features caused by thermal fluctuations [61]. This study also neglects the rate dependent factors such as the viscosity from both liquid and solid materials. These important aspects certainly warrant more detailed investigations in future.

Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 11321202), the Zhejiang Provincial Natural Science Foundation of China (No. LR16A020001), and the Open Fund of Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province.

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Appendix A The coefficients in the modeling part, which are functions of m ¼ c=a, are listed as follows.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m2  1 þ m2 cos1 ; m 1 ¼ cos1 ; m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ m2  1 þ ðm2  2Þ cos1 ; m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 1  cos1 ; ¼ m2  1 þ m2 cos1 m m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ m  1  m2  1 cos1 ; m 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; m m2  1 m 1 1 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ðm2  2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2m  sin ; 2 2 m m 1 m m 1 1 m 1 ¼ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos1 ; m m m2  1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 10 Z minðq;mÞ q2  t2 1 2 @ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dtA ¼m m  ; m 1 t2 m2  t 2

A1 ¼ A2 A3 A4 A5 A6 A7 A8

A9

q¼m

Z

m

A10 ¼

q cos

1

Z

1 m

A15 A16 A17

dq;

qðq2  2Þ cos1

1

q

0

Appendix B In the presence of dry contact area, the non-dimensional set of governing equations is the following.

pffiffiffiffiffiffi v b ^2 ^3=2 ; ^3  b a  a 2A1 ¼ 6b a F  DP p   h1 h2 a1 þ a2 va^ ¼ ðmnÞ; þ tan ^ 2 2 h 2

v b ^ 2cosððh1 þh2 þ a1  a2 Þ=2Þ sinðh1 þ a1 Þ D P  a ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ; 2 n a ^ va ^Þ ðmnÞ2 þðh= b 3V 3 3 v2 b ^2 ^ ðA11 þA12 Þþ D P ^3  Fb ÞA10 þ a ¼ða  a ð2A13 4A14 þ4A15 Þ; 4 ^ 4 p 8v a where

a1 ¼ sin1 ðva^nÞ; 

a2 ¼ tan1 

Here v ¼ ð3pDcSV =ð4E RÞÞ , a ¼ c=DcSV , h1 , h2 are the prescribed parameters of the physical system. From Eq. (11), we have b ¼ DcSL =DcSV ¼ 1  aðcos h1 þ cos h2 Þ. For the case of conserved liqb ¼ V=R3 is a prescribed parameter in solving the equauid volume, V 1=3

tions. Instead, when the liquid volume changes with environmental b is determined by the Kelvin equation in Eq. (21) and humidity, D P

b should be solved from Eq. (20). V In the region of pure wet adhesion without dry contact, the non-dimensional governing equations are:

  h1  h2 a1 þ a2 mn ; þ ¼ ^ 2 2 h sinðh1 þ a1 Þ 1 þ h2 þ a1  a2 Þ=2Þ b ¼ 2cosððh qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  DP ; n 2 ^2 ðm  nÞ þ h tan

b ¼ p  ^d  m2 þ 2p  A16  8D P b  k  A17 ; V where

a1 ¼ sin1 n;

 dwðrÞ ; dr r¼c b pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ¼ h ¼ ^d þ ð1  1  n2 Þ  4D P  m  k: h R p

a2 ¼ tan1

dq; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Z m 1 1  ¼ m2  1 þ m2 cos1 q cos1 dq; m q 1  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z m pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ m2  1  q q2  1  cos1 dq; q 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z mZ q 2 2 q t ¼ m2  q 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi dtdq; 1 1 t m2  t Z m pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ q  ð1  1  q2 Þdq; 0 Z m ¼ m q  Eðq=mÞdq: 1

A14

q

q q2  1dq;

A12 ¼ A13

1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi

m

A11 ¼ Z

1

  2 3 3 v2 b ^ 2 ^ ¼ h ¼ 4v ^3  b ^  A3 þ D P h F Þ  A2 þ a  a ð2A4  4A5 Þ ða ^ R 3pa 4 p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ^nÞ  1  ðva ^mÞ :  1  ðva

  4v 3 3 v2 b ^ 2 ^3  b ^  A7 þ D P ða F Þ  A6  a  a ð2A8 þ 4A9 Þ ; 2 ^ 4 p 3pa

We have three equations and four unknowns in total. Specifying any one of the four unknowns, m for example, will determine the rest ones. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

[14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

H.J. Butt, M. Kappl, Adv. Colloid Interface Sci. 146 (2009) 48. Z.L. Peng, C. Wang, S.H. Chen, Colloid Surf. B-Biointerfaces 122 (2014) 662. Y.P. Zhao, L.S. Wang, T.X. Yu, J. Adhes. Sci. Technol. 17 (2003) 519. Q.Y. Li, T.E. Tullis, D. Goldsby, R.W. Carpick, Nature 480 (2011) 233. S.H. Chen, A.K. Soh, Int. J. Solids Struct. 45 (2008) 3122. S. Herminghaus, Adv. Phys. 54 (2005) 221. V.N. Samoilov, I.M. Sivebaek, B.N.J. Persson, J. Chem. Phys. 121 (2004) 9639. T. Eisner, D.J. Aneshansley, Proc. Natl. Acad. Sci. U.S.A. 97 (2000) 6568. G. Walker, A.B. Yule, J. Ratcliffe, J. Zool. 205 (1985) 297. W. Federle, W. Baumgartner, B. Holldobler, J. Exp. Biol. 207 (2004) 67. D. Labonte, W. Federle, Philos. Trans. R. Soc. B-Biol. Sci. 370 (2015) 20140027. T.A. Ba-Omar, J.R. Downie, W.J.P. Barnes, J. Zool. 250 (2000) 267. K. Autumn, M. Sitti, Y.C.A. Liang, A.M. Peattie, W.R. Hansen, S. Sponberg, T.W. Kenny, R. Fearing, J.N. Israelachvili, R.J. Full, Proc. Natl. Acad. Sci. U.S.A. 99 (2002) 12252. H.J. Gao, H.M. Yao, Proc. Natl. Acad. Sci. U.S.A. 101 (2004) 7851. S.N. Gorb, Proc. R. Soc. B-Biol. Sci. 265 (1998) 747. A.F.G. Dixon, P.C. Croghan, R.P. Gowing, J. Exp. Biol. 152 (1990) 243. L.J. Xue, A. Kovalev, A. Eichler-Volf, M. Steinhart, S.N. Gorb, Nat. Commun. 6 (2015) 6621. M.J. Vogel, P.H. Steen, Proc. Natl. Acad. Sci. U.S.A. 107 (2010) 3377. H. Chen, T. Tang, A. Amirfazli, Langmuir 31 (2015) 11470. M. Mastrangeli, Adv. Mater. 27 (2015) 4254. F.M. Orr, L.E. Scriven, A.P. Rivas, J. Fluid Mech. 67 (1975) 723. A. Fogden, L.R. White, J. Colloid Interface Sci. 138 (1990) 414. D. Maugis, B. Gauthier-Manuel, J. Adhes. Sci. Technol. 8 (1994) 1311. D.S. Dugdale, J. Mech. Phys. Solids 8 (1960) 100. K.L. Johnson, K. Kendall, A.D. Roberts, Proc. R. Soc. London Ser. A-Math. Phys. Sci. 324 (1971) 301. B.V. Derjaguin, V.M. Muller, Y.P. Toporov, J. Colloid Interface Sci. 53 (1975) 314. H. Fan, Y.X. Gao, J. Appl. Phys. 90 (2001) 5904. Z.R. Chen, S.W. Yu, J. Appl. Phys. 94 (2003) 6899. J. Qian, H.J. Gao, Acta Biomater. 2 (2006) 51. E. Hsiao, M.J. Marino, S.H. Kim, J. Colloid Interface Sci. 352 (2010) 549. L.F. Wang, S. Regnier, Model. Simul. Mater. Sci. Eng. 23 (2015) 12. Y.W. Su, B.H. Ji, Y.G. Huang, K.H. Hwang, J. Mater. Sci. 42 (2007) 8885. J.S. Wexler, T.M. Heard, H.A. Stone, Phys. Rev. Lett. 112 (2014) 066102.

J. Qian et al. / Journal of Colloid and Interface Science 483 (2016) 321–333 [34] D. Maugis, Contact, Adhesion and Rupture of Elastic Solids, Springer, New York, 2000. [35] H. Hertz, J. Reine Angew. Math. 92 (1882) 156. [36] H. Tada, P.C. Paris, G.R. Irwin, The Stress Analysis of Cracks Handbook, American Society of Mechanical Engineers, New York, 2000. [37] A.A. Griffith, Philos. Trans. R. Soc. London, A 221 (1921) 163. [38] J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, 2011. [39] A.W. Adamson, Physical Chemistry of Surfaces, John Wiley & Sons Inc., New York, 1976. [40] D. Maugis, J. Colloid Interface Sci. 150 (1992) 243. [41] J.Z. Wang, J. Qian, H.J. Gao, Langmuir 25 (2009) 11727. [42] K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1985. [43] F. Jin, X. Guo, H.J. Gao, J. Mech. Phys. Solids 61 (2013) 2473. [44] D. Tabor, J. Colloid Interface Sci. 58 (1977) 2. [45] V.M. Muller, Y.S. Yushchenko, B.V. Derjaguin, J. Colloid Interface Sci. 77 (1980) 91.

[46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61]

333

D.W. Xu, K.M. Liechti, K. Ravi-Chandar, J. Colloid Interface Sci. 315 (2007) 772. E. Arzt, S. Gorb, R. Spolenak, Proc. Natl. Acad. Sci. U.S.A. 100 (2003) 10603. H.J. Gao, X. Wang, H.M. Yao, S. Gorb, E. Arzt, Mech. Mater. 37 (2005) 275. W.T. Pockman, J.S. Sperry, J.W. Oleary, Nature 378 (1995) 715. B.N.J. Persson, Journal of Chemical Physics 118 (2003) 7614. P. Lu, Y.L. Foo, L. Shen, D.W.C. Cheong, S.J. O’Shea, J. Mater. Res. 26 (2011) 1406. P. Lu, Tribol. Lett. 58 (2015) 6. R.W. Style, C. Hyland, R. Boltyanskiy, J.S. Wettlaufer, E.R. Dufresne, Nat. Commun. 4 (2013) 2728. Z. Cao, M.J. Stevens, A.V. Dobrynin, Macromolecules 47 (2014) 3203. M.K. Chaudhury, G.M. Whitesides, Langmuir 7 (1991) 1013. D.S. Rimai, L.P. Demejo, W.B. Vreeland, R.C. Bowen, Langmuir 10 (1994) 4361. D. Maugis, Langmuir 11 (1995) 679. P.G. de Gennes, Rev. Mod. Phys. 57 (1985) 827. R. Digilov, Langmuir 16 (2000) 1424. S.F. Cheng, M.O. Robbins, Phys. Rev. E 89 (2014) 062402. J. Lin, Y. Lin, J. Qian, Langmuir 30 (2014) 6089.