Elastocapillarity: Adhesion and Wetting in Soft Polymeric Systems

Sep 9, 2014 - We have developed a generalized model of particle-substrate interactions describing both adhesion and wetting behavior. Using a ...
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Elastocapillarity: Adhesion and Wetting in Soft Polymeric Systems Zhen Cao,† Mark J. Stevens,‡ and Andrey V. Dobrynin*,† †

Polymer Program, Institute of Materials Science and Department of Physics, University of Connecticut, Storrs, Connecticut 06269, United States ‡ Center for Integrated Nanotechnologies, Sandia National Laboratories, Albuquerque, New Mexico 87185-1315, United States S Supporting Information *

ABSTRACT: We have developed a generalized model of particle-substrate interactions describing both adhesion and wetting behavior. Using a combination of the molecular dynamics simulations and scaling analysis we have shown that the crossover between adhesion and wetting-like behavior for a particle with size Rp and shear modulus Gp interacting with a substrate of shear modulus Gs is determined by the dimensionless parameter β ∝ γ*(G* Rp)−2/3W−1/3, where G* = GpGs/(Gp + Gs) is the effective shear modulus, W is the work of adhesion between particle and substrate, and γ* = Wgr + γp(1− 2gr) + γspgr2 is the effective surface tension of the particle/substrate system with γp and γs being surface tensions of particle and substrate, γsp − surface tension of the particle-substrate interface, and gr = Gp/(Gp + Gs). This parameter β is proportional to the ratio of elastocapillary length γ*/G* and contact radius a, β ∝ γ*/G*a. In the limit of small values of the parameter β < 1, when the contact radius a is larger than the elastocapillary length γ*/G*, our model reproduces Johnson, Kendall, and Roberts results for particle adhesion on elastic substrates (adhesion regime). However, in the opposite limit, β > 1 (a < γ*/G*), the capillary forces play a dominant role and determine particle-substrate interactions (wetting regime). Model predictions are in a very good agreement with simulation and experimental results.

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opposite limit γp/Gpa > 1 one observes a wetting-like behavior (wetting regime) where equilibrium shape of deformed particle is obtained by balancing variation of the surface free energy and adhesion energy. Similar behavior was observed for interaction between rigid particles and soft substrates.23−25 However, in this case the elastocapillary length, γs/Gs, is determined by the surface tension γs and shear modulus Gs of the substrate. These studies pin pointed a unique feature of contact phenomena at micro- and nanoscales and showed that they are governed by a fine interplay between capillary and elastic forces. However, until now only two extreme cases of interactions between particles and substrates were considered with only one of the “participants” in this process capable of undergoing elastic deformations. In this paper, we use a combination of the molecular dynamics simulations and scaling analysis to develop a generalized model of interactions between particles and substrates of arbitrary rigidity. This new model takes into account the change in the substrate and particle surface energy and elastic energy upon deformation.

INTRODUCTION We see manifestation of elastic and capillary forces in our everyday life as wet hair assemble into bundles. Similar aggregation is observed in arrays of macro- and nanorods in wet conditions.1−4 It is now recognized that the stability of polymeric micropillars and microchannels in microcontact printing stamps is determined by a fine interplay between elastic and capillary forces.5−7 These forces also control friction and adhesion between soft elastic solids.8 The dramatic role of capillary and elastic forces is believed to be behind collapse of the lung airways.9,10 These examples illustrate importance of understanding of the specific role of the capillary and elastic forces in different areas of science (colloidal science, tribology, and biophysics), technology (micro- and nanofabrication), and medicine (see for review11−16). Recent studies of the adhesion of soft nano- and microparticles with rigid substrates17−20 have shown that the classical Johnson−Kendall−Roberts (JKR) approach to contact mechanics21,22 breaks down when the contact radius a becomes smaller than the elastocapillary length defined as a ratio of the particle surface tension γp to its shear modulus Gp, γp/Gp. Contact of such soft particles with rigid substrates is controlled by capillary forces and can be described by Young’s law11 for equilibrium of liquid droplet on solid substrates. It was demonstrated that there are two different interaction regimes between a soft particle and a rigid substrate. For small values of the ratio γp/Gpa < 1 the equilibrium contact radius is determined by balancing elastic and adhesion energies as it is done in the classical JKR model21,22 (adhesion regime). In the © 2014 American Chemical Society

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RESULTS AND DISCUSSION We used molecular dynamics simulations26 to study interactions of elastic nanoparticles with soft (gel-like) substrates (see Figure 1). In our simulations nanoparticles and substrates Received: July 7, 2014 Revised: August 25, 2014 Published: September 9, 2014 6515

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simulation details are discussed in Supporting Information. All simulations were performed using LAMMPS.28 Figure 2 shows evolution of the nanoparticle shape as a function of the interaction parameter εLJ and the shear modulus of nanoparticle Gp and substrate Gs for spherical nanoparticles with size Rp = 25.7 σ. The values of the substrate and nanoparticle shear moduli were obtained from 3D simulations of the polymeric networks with the same degree of cross-linking and interaction parameters (see Supporting Information for detail). For soft nanoparticles and substrates having the same values of the shear modulus an adsorbed nanoparticle adopts a lens-like symmetric shape with both contact angles θ1 and θ2 being equal to each other (see picture in the top left corner in Figure 2). Increasing substrate shear modulus (top row in Figure 2) results in decrease of the indentation depth produced by nanoparticle in the substrate. This correlates with increase of the contact angle θ1 formed by the top part of nanoparticle with the substrate. However, the angle θ2 monotonically decreases toward zero. In the case when the shear modulus of nanoparticle is larger than the shear modulus of substrate the indentation produced by nanoparticle in the substrate is larger than the nanoparticle deformation (left column in Figure 2). For such nanoparticles the value of the contact angle θ2 is larger than 90° while the value of the contact angle θ1 is smaller than 90°. However, when both nanoparticle and substrate are sufficiently rigid (see bottom right corner) the value of the contact angle θ1 becomes larger than 90° and nanoparticle is almost completely expelled from the substrate.

Figure 1. Snapshot of the simulation box showing deformation of nanoparticle and elastic substrate. One quarter of substrate is removed to illustrate substrate and nanoparticle deformation.

were modeled by randomly cross-linked bead−spring chains. Their elastic properties were controlled by changing crosslinking density between chains. The interactions between beads with diameter σ forming polymer chains were described by the truncated-shifted Lennard-Jones (LJ) potential with the value of the interaction parameter εLJ and the cutoff radius rcut. The bonds connecting beads into polymer chains and cross-linking bonds holding chains together were represented as a sum of the FENE potential and purely repulsive LJ-potential.27 The elastic substrate with thickness H was placed on the top of a solid substrate which was modeled by an external 3−9 potential with the value of the interaction parameter εw. The interaction parameters for the potentials, their functional form and

Figure 2. Evolution of the average shape of nanoparticle with size Rp = 25.7 σ and shear modulus Gp and of the contact angles θ1 and θ2 produced by a contact with elastic substrate of shear modulus, Gs.G̃ i = Giσ3/kBT is a reduced shear modulus and εLJ is the value of the LJ-interaction parameter between beads forming nanoparticle and substrate. kB is the Boltzmann constant and T is the absolute temperature. Numbers on the figures show corresponding values of the parameter γ*(G*Rp)−2/3W−1/3 describing relative importance of surface and elastic energies in determining nanoparticle shape (see discussion below). 6516

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Taking into account expression for the work of adhesion, W = γs + γp − γsp, after some algebra we can rearrange eqs 1 as follows

In order to quantify dependence of nanoparticle deformation and values of the contact angles on the system parameters we will first analyze our simulation results by using the Neumann’s triangle equation11 for wetting of liquid surface and JKR theory of the adhesive contact.22 Neumann’s triangle represents a balance of surface tensions at the contact line of the liquid droplet on liquid substrate (see inset in Figure 3a). Equating

1=

⎡ ⎤ sin θ1 (1 − cos θ2)⎥ ⎢(1 − cos θ1) + 2γp − W ⎣ sin θ2 ⎦ γp

(2.a)

In Figure 3a, we test applicability of eq 2.a to our data by plotting the data in a plane X=

Y=

projections of the surface tensions of nanoparticle γp, substrate γs and particle-substrate γsp parallel and normal to the substrate directions we obtain and

2γp − W

(1 − cos θ1)

γp sin θ1 (2γp − W ) sin θ2

and

(1 − cos θ2) (2.b)

Note that the values of the work of adhesion W and surface tension of nanoparticle γp used for this plot were calculated from separate simulations (see Supporting Information for detail). For this plot we used macroscopic values of the contact angles that were obtained by fitting averaged droplet shape by two spherical caps and using obtained values of the caps’ heights and contact radius to calculate contact angles. It follows from this figure that only data corresponding to very soft nanoparticles and substrates approach Neumann’s condition. The majority of our data cannot be described by the Neumann’s relations and require consideration of the elastic forces generated by deformation of the substrate and nanoparticle in the force balance at the contact line. It also follows from this figure that the data set corresponding to more rigid nanoparticles with shear modulus Gp = 0.864 kBT/σ3 shown by blue triangles is closer to the Neumann’s line than the data set with Gp = 0.214 kBT/σ3 (green circles). As one can see from Figures 2 our rigid nanoparticles (bottom row) do not show significant shape deformation thus their contact angles with the substrates are determined by balancing substrate and nanoparticle surface energies. However, for nanoparticles with Gp = 0.214 kBT/σ3 (green circles) there is substantial nanoparticle shape deformation (see Figure 2). In this case, nanoparticle elasticity begins to control the values of the contact angles. This could be a reason for such peculiar nonmonotonic behavior of data sets. In the framework of the JKR theory of the adhesive contact between an elastic substrate and a particle with size Rp, the equilibrium contact radius a is obtained by equating the elastic energy of deformation of substrate and nanoparticle, Uel ∝ G* a5/R2p, where G* = GsGp /(Gs + Gp), and absolute value of the change in the adhesion energy due to a contact between

Figure 3. Comparison of simulation results with Neumann’s triangle condition (solid line) (a) and JKR model (dashed line) (b), for nanoparticles with different sizes Rp, different values of the shear modulus Gp = 0.864 kBT/σ3 (blue triangles), Gp = 0.214 kBT/σ3 (green circles) and Gp = 0.023 kBT/σ3 (red squares) adsorbed on substrates with shear modulus varying between 0.024 and 0.833 kBT/σ3. For definition of X and Y in part a, see eq 2.b. Inset in part a shows Neumann’s triangle.

γs = γp cos θ1 + γsp cos θ2

γp

γp sin θ1 = γsp sin θ2 (1)

Figure 4. Schematic representation of nanoparticle deformation and the substrate indentation produced by a nanoparticle with radius Rp used for derivation of the system surface free energy. 6517

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nanoparticle and substrate with area πa2, πWa2.22 The equilibrium contact radius a scales with the effective modulus G* as follows a≈

⎛ W ⎞1/3 2/3 ⎜ ⎟ R p ⇒ aG* ∝ (G*R p)2/3 ⎝ G* ⎠

δs ≈

(3)

Gs + Gp

R p1/2Δh5/2

(6)

(7)

where we introduced new parameters G* = GsGp /(Gs + Gp), and γ* = Wgr + γp(1−2gr) + γspgr2, gr = Gp /(Gp + Gs). Note that we can transform expression for γ* in a more symmetric form by substituting expression for the work of adhesion, W = γs + γp − γsp. This results in γ* = γs gr + γp(1 − gr) − γspgr(1 − gr). It is important to point out that the particular form of the crossover function γ* and effective shear modulus G* is a result of stress balance approximation used in derivation of eq 6. In these variables eq 7 looks similar to those derived for adhesion of soft nanoparticles on rigid substrates17 and rigid nanoparticles on soft substrates24 and can be reduced to those by taking the appropriate limits: gr = 0 for soft nanoparticles on rigid substrate (Gp ≪ Gs)17 and gr = 1 for rigid nanoparticles on soft substrates (Gp ≫ Gs).24 The advantage of using eq 7 is that it describes deformation of substrates and nanoparticles of arbitrary rigidity. The equilibrium nanoparticle height deformation Δh is obtained by minimizing the change in the system free energy eq 7 with respect to Δh

(4)

In deriving eq 4 we took into account volume conservation of nanoparticle during nanoparticle deformation, and geometrical relations between radii of curvature of spherical caps, contact radius a, Δh and δs (see Supporting Information for derivation detail). In addition to the surface energy contribution there is also elastic energy contribution due to deformation of elastic nanoparticle and substrate. In the limit of sufficiently thick substrates for which a/H < 1 we can consider the gel layer as an elastic half space and approximate the elastic energy contribution as follows22 GsGp

Δh

ΔF(Δh) = −2πR pW Δh + πγ *Δh2 + CGG*R p1/2Δh5/2

ΔFsurf (δs , Δh) = −2πR pW Δh + πWδsΔh + πγspδs 2

Uelast ≈ CG

Gp + Gs

Substituting this equation into the expression for the change of the system surface free energy (eq 5), we can write the total free energy change of the nanoparticle substrate system only as a function of the deformation of nanoparticle height above the substrate Δh:

In Figure 3b, we test how well eq 3 describes our data. For nanoparticles and substrates for which G*Rp > 1, we see a good collapse of the simulation data. However, as a value of the parameter G*Rp becomes smaller than unity our simulation data deviate from the JKR line (dashed line in Figure 3b). This points out that it is not sufficient to take into account only elastic energy and work of adhesion in the contact area to describe deformation of nanoparticle in contact with elastic substrate. Therefore, in order to describe interaction between nanoparticle and substrate we have to consider adhesion, elastic and surface forces simultaneously. We will approximate the shape of the deformed nanoparticle by two spherical caps as shown in Figure 4. The change of the system surface free energy due to deformation of nanoparticle height above the substrate Δh = 2Rp − h and substrate indentation δs produced by nanoparticle contact with substrate in the limit of small deformations, Δh/2Rp ≪ 1 and δs/2Rp ≪ 1, is equal to

+ πγpΔh2 − 2πγpδsΔh

Gp

0 ≈ −2πWR p + 2πγ *Δh +

5 CGG*R p1/2Δh3/2 2

(8)

Analysis of eq 8 shows that there are two asymptotic solutions. In the case when nanoparticle and substrate deformations are controlled by the adhesion and elastic energy terms (the first and the last terms in the rhs of eq 8), we obtain the classical adhesion scaling dependence of the nanoparticle height deformation on the system parameters: Δh ∝ R p(W /G*R p)2/3

(5)

(9)

We called this regime the adhesion regime. Following the analysis presented in ref 24, we can simplify eq 9 by using relationship between the contact radius a and nanoparticle height deformation Δh. For small deformations,a ≈ (2ΔhRp)1/2, we can express Δh as a ratio of W/G* to the contact radius a by substituting Rp1/2 Δh3/2 ≈ Δha in eq 9

where CG is a numerical constant, which value depends on the model used to describe an elastic contact.22 Here we will consider CG as an adjustable parameter to fit our simulation data. A comment has to be made here that eq 5 only describes elastic energy of the incompressible (rubber-like) materials with Poisson’s ratios νs = νp =0.5. In general case one has to use rigidities of substrate and nanoparticle, Ki = 2Gi/(1 − vi), instead of shear moduli in eq 5 and derivations below. In the contact area between nanoparticle and substrate, the elastic stress generated in nanoparticle and substrate should be balanced providing the additional constraint for Δh and δs. The stress at the interface between nanoparticle and substrate can be estimated using the following simple scaling arguments. For indentation depth δs, the typical value of the restoring stress acting in the contact area from the substrate is on the order of Gsδs/a. Nanoparticle deformation Δh − δs produces restoring stress Gp(Δh − δs)/a acting from the nanoparticle in the contact area. At equilibrium these two stresses are balanced resulting in

Δh ∝ R pW /G*a

(10)

In the other asymptotic regime, where the equilibrium nanoparticle deformation is obtained by balancing adhesion and surface free energy terms (the first and the second terms in the rhs of eq 8), this results in the following expression: Δh ∝ R pW /γ *

(11)

We called this regime the wetting regime. The crossover between these two regimes occurs when the contact radius a becomes on the order of the elastocapillary length γ*/G*. We can write down a general crossover expression for nanoparticle height deformation Δh as a function of the contact radius a as follows 6518

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⎛ W ⎞2/3 Δh ⎟⎟ = AG⎜⎜ 2R p ⎝ G*R p ⎠

(12)

⎛ r+ ⎝

where Ca is a numerical constant. In Figure 5, we plot the reduced nanoparticle deformation γ*Δh/WRp as a function of the ratio of the elastocapillary

⎜3

q3 + r 2 +

3

r−

q3 + r 2 −

β⎞ ⎟ 3⎠

2

(14)

where r=

and

⎛ β ⎞2 q = −⎜ ⎟ ⎝3⎠

(15)

The actual values of the parameters AG and BG are model dependent, and we will consider them as adjustable parameters to fit our simulation data. Analysis of the solution eq 14 shows that we recover eq 9 in the limit of small values of the parameter β ∝ γ*(G*Rp)−2/3W−1/3 < 1. This corresponds to the adhesion regime. However, in the interval of large values of this parameter, β > 1, which is usually the case for soft substrates and nanoparticles, the nanoparticle deformation given by scaling relation eq 11 (wetting regime). Figure 6 shows dependence of the reduced particle deformation (G*Rp/W)2/3Δh/2Rp on the value of the

Figure 5. Dependence of the reduced nanoparticle deformation or substrate indentation, γ*Δh/WRp, on the ratio of the elastocapillary length to contact radius, γ*/G*a. The solid line corresponds to eq 12 with numerical constant Ca = 0.816. Simulation results for adhesion of rigid nanoparticles on soft substrates are shown by open green triangles and for adhesion of soft nanoparticles on solid substrates are shown by open pink triangles. Experimental data for adhesion of glass particles on silicon substrates with different values of the shear modulus G are shown as 1 kPa (black stars), 28.3 kPa (pink pentagons), 83.3 kPa (brown hexagons), and 167 kPa (inverted cyan triangles). For our data set we used the same notations as in Figures 3

length and contact radius, γ*/G*a. To confirm that our expression can describe deformation of nanoparticles and substrates with a wide range of elastic properties we have combined in Figure 5 our simulations results with simulation results for adhesion of soft nanoparticles on rigid substrates,17 rigid nanoparticles on soft substrates24 and experimental data sets for adhesion of glass spheres on silicon substrates.23 As expected from eq 12 for small values of the parameter γ*/G*a the nanoparticle deformation shows a linear dependence on the ratio of the elastocapillary length to the contact radius (see eq 10). However, for large values of the parameter γ*/G*a > 1, we see a deviation from the linear dependence. In this interval of parameters the surface free energy term begins to dominate interactions between a nanoparticle and substrate. Following the approach developed in ref 17, we can solve eq 8 for nanoparticle height deformation, Δh. It is convenient to normalize nanoparticle height deformation Δh by its value in the adhesion regime (see eq 9) and introduce dimensionless variables y2 = (Δh/2RpAG)(G*Rp/W)2/3 and β = BG (γ*3/2/ G*RpW1/2)2/3 (where AG = (√2π/5CG)2/3 and BG = 2(√2π/ 5CG)2/3 are numerical coefficients). Dimensionless parameter β has a simple physical meaning: it is proportional to the ratio of the surface free energy of nanoparticle, γ*Rp2(W/G* Rp)4/3, to the free energy of nanoparticle deformation, WRp2(W/G* Rp)2/3, in the adhesion regime, β ∝ γ*(G*Rp)−2/3W−1/3. In these new variables eq 8 can be reduced to a simple cubic equation 0 = − 1 + βy 2 + y 3

⎛ β ⎞3 1 −⎜ ⎟ ⎝3⎠ 2

Figure 6. Dependence of the reduced nanoparticle deformation or substrate indentation, (G*Rp/W)2/3 Δh/2Rp, on the value of the parameter γ*(G*Rp)−2/3W−1/3 for substrates and nanoparticles with different cross-linking densities, different strengths of the particlesubstrate interactions and different nanoparticle sizes. Solid line corresponds to eq 14 with values of the fitting parameters AG = 0.416 and BG = 0.798. Notations are the same as in Figure 5

parameter γ*(G*Rp)−2/3W−1/3 for all data sets shown in Figure 5. To make a connection between a shape deformation of nanoparticle and value of the parameter γ*(G*Rp)−2/3W−1/3 describing crossover between wetting and adhesion regimes, in Figure 2 we have given values of this parameter for each nanoparticle/substrate system. The solid line in Figure 6 represents a best fit to eq 14 with the values of the fitting parameters AG = 0.416 and BG = 0.798. All data sets are in a very good agreement with the theoretical expression eq 14. A comment has to be made here concerning the value of the fitting parameter BG = 0.798 and value of the fitting parameter BG used in ref 17 for fitting simulation data for interaction of nanoparticles with rigid substrates. There is a typo in ref17 in reporting the value of parameter BG. It should be BG/3 = 0.28. Thus, two values of the parameters BG are close. For comparison with experiments it is useful to obtain a master curve representing nanoparticle deformation as a function of nanoparticle size. In order to collapse our simulation data into a master curve in this representation we normalize the nanoparticle height deformation Δh and

(13)

The equilibrium deformation Δh is described by a positive root of this equation 6519

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nanoparticle size Rp by their crossover values between adhesion and wetting regimes. The crossover value for nanoparticle size, R̃ p, is obtained by equating eqs 9 and 11. This results in the following values of the normalization factors for Rp and Δh as functions of the system parameters: R̃ p = γ *3/2 G*−1W −1/2

and

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ASSOCIATED CONTENT

S Supporting Information *

Description of the simulations details, calculation of the work of adhesion and gel surface tension, and derivation of the system free energy.This material is available free of charge via the Internet at http://pubs.acs.org.

Δh ̃ = γ *1/2 W 1/2G*−1

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(16)

Figure 7 represents all data sets shown in Figures 5 and 6 in these new reduced variables Δh/Δh̃ and Rp/R̃ p. In the wetting

AUTHOR INFORMATION

Corresponding Author

*(A.V.D.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation under the Grant DMR-1004576. Computer simulations were performed at the U.S. Department of Energy, Center for Integrated Nanotechnologies, at Los Alamos National Laboratory (Contract No. DE-AC52-06NA25396) and Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract No DE-AC04-94AL85000.

Figure 7. Dependence of the reduced nanoparticle deformation or substrate indentation, Δh/Δh̃, on reduced particle size, Rp/R̃ p, for substrates and nanoparticles with different cross-linking densities, different strengths of the particle-substrate interactions and different nanoparticle sizes. Solid line corresponds to eq 14 with values of the fitting parameters AG = 0.416 and BG = 0.798 which is plotted in these new variables. Notations are the same as in Figure 5

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regime, our data follow Δh/Δh̃ ∝ Rp/R̃ p while in the adhesion regime Δh/Δh̃ ∝ (Rp/R̃ p)1/3as expected from eqs 9 and 11.

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REFERENCES

(1) Lau, K. K.; Bico, J.; Teo, K. B. K.; M, C.; Amaratunga, G. A. J.; Milne, W. I.; McKinley, G. H.; Gleason, K. K. Nano Lett. 2003, 3, 1701−1705. (2) Nguyen, C. V.; Delzeit, L.; Cassell, A. M.; Li, J.; Han, J.; Meyyappan, M. Nano Lett. 2002, 2, 1079−1081. (3) Chakrapani, N.; Wei, B.; Carrillo, A.; Ajayan, P. M.; Kane, R. S. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 4009−4012. (4) Paulose, J.; Nelson, D. R.; Aizenberg, J. Soft Matter 2010, 6, 2421−2434. (5) Tanaka, T.; Morigami, M.; Atoda, N. Jpn. J. Appl. Phys. 1993, 32, 6059−6054. (6) Hui, C. Y.; Jagota, A.; Lin, Y. Y.; Kremer, E. Langmuir 2002, 18, 1394−1407. (7) Sharp, K. G.; Blackman, G. S.; Glassmaker, N. J.; Jagota, A.; Hui, C. Y. Langmuir 2004, 20, 6430−6438. (8) Persson, B. N. J. Sliding Friction. Physical Principles and Applications; Springer: Berlin, 2000. (9) Heil, M.; White, J. P. J. Fluid Mech. 2002, 462, 79−109. (10) Hazel, A. L.; Heil, M. Proc. R. Soc. London, A 2005, 461, 1847− 1868. (11) de Gennes, P.-G.; Brochard-Wyart, F.; Quere, D. Capillarity and Wetting Phenomena; Springer: New York, 2002. (12) Shull, K. R. Mater. Sci. Eng. R-Rep. 2002, 36, 1−45. (13) Shull, K. R. J. Polym. Sci., Part B: Polym. Phys. 2006, 44, 3436− 3439. (14) Szlufarska, I.; Chandross, M.; Carpick, R. W. J. Phys. D: Appl. Phys. 2008, 41, 39. (15) Trepat, X.; Lenormand, G.; Fredberg, J. J. Soft Matter 2008, 4, 1750−1759. (16) Roman, B.; Bico, J. J. Phys.: Condens. Matter 2010, 22, 16. (17) Carrillo, J. M. Y.; Raphael, E.; Dobrynin, A. V. Langmuir 2010, 26, 12973−12979. (18) Carrillo, J. M. Y.; Dobrynin, A. V. Langmuir 2012, 28, 10881− 10890. (19) Carrillo, J. M. Y.; Dobrynin, A. V. J. Chem. Phys. 2012, 137, 214902−1−10. (20) Salez, T.; Benzaquen, M.; Raphael, E. Soft Matter 2013, 9, 10699−10704. (21) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London A 1971, 324, 301−313.

CONCLUSIONS

We developed a generalized approach describing interactions between elastic particles and substrates. In the framework of our approach the interaction of a particle with a substrate is a result of a fine interplay between elastic, adhesion and surface forces. There are two different particle-substrate interaction regimes. The crossover between these two regimes is governed by the ratio of the elastocapillary length γ*/G* and contact radius a. In the limit when the elastocapillary length γ*/G* is smaller than the contact radius a, the particle and substrate deformations are determined by balancing the system elastic energy and work of adhesion between nanoparticle and substrate (adhesion regime). However, when the elastocapillary length γ*/G* becomes larger than the contact radius a, the adhesion and surface energy terms provide dominant contribution into the system free energy (wetting regime). In this range of system parameters the behavior of the system is similar to that of a liquid droplet at liquid/air interface. It is important to point out that our model of particle-substrate interactions reproduces results obtained for soft particles interactions with rigid substrates17 and rigid particle interactions with soft substrates24 in two asymptotic regimes Gs ≫ Gpand Gs ≪ Gp respectively. The predictions of our model of particle-substrate interactions are confirmed by molecular dynamics simulations of interactions of elastic nanoparticles with elastic substrates and are in a very good agreement with experimental results on adhesion of glass spheres with silicon substrates.23 6520

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(22) Johnson, K. L. Contact Mechanics, 9th ed.; Cambridge University Press: 2003. (23) Style, R. W.; Hyland, C.; Boltyanskiy, R.; Wettlaufer, J. S.; Dufresne, E. R. Nat. Commun. 2013, 4, 2728. (24) Cao, Z.; Stevens, M. J.; Dobrynin, A. V. Macromolecules 2014, 47, 3203−3209. (25) Xu, X.; Jagota, A.; Hui, C.-Y. Soft Matter 2014, 10, 4625−32. (26) Frenkel, D.; Smit, B. Understanding Molecular Simulations; Academic Press: New York, 2002. (27) Kremer, K.; Grest, G. S. J. Chem. Phys. 1990, 92, 5057−5086. (28) Plimpton, S. J. Comput. Phys. 1995, 117, 1−19.

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