Contact Mechanics of NanoparticlesJan-Michael Y. Carrillo and Andrey V. Dobrynin*

Polymer Program, Institute of Materials Science, and Department of Physics, University of Connecticut, Storrs, Connecticut 06268,United States

ABSTRACT: We perform molecular dynamics simulations on the detach-ment of nanoparticles from a substrate. The critical detachment force, f*, isobtained as a function of the nanoparticle radius, Rp, shear modulus, G, surfaceenergy, γp, and work of adhesion,W. The magnitude of the detachment force isshown to increase from πWRp to 2.2πWRp with increasing nanoparticle shearmodulus and nanoparticle size. This variation of the detachment force is amanifestation of neck formation upon nanoparticle detachment. Using scalinganalysis, we show that the magnitude of the detachment force is controlled bythe balance of the nanoparticle elastic energy, neck surface energy, and energyof nanoparticle adhesion to a substrate. It is a function of the dimensionlessparameter δ ∝ γp(GRp)

−1/3W−2/3, which is proportional to the ratio of the surface energy of a neck and the elastic energy of adeformed nanoparticle. In the case of small values of the parameter δ ≪ 1, the critical detachment force approaches a criticalJohnson, Kendall, and Roberts force, f* ≈ 1.5πWRp, as is usually the case for strongly cross-linked, large nanoparticles. However,in the opposite limit, corresponding to soft small nanoparticles for which δ≫1, the critical detachment force, f*, scales as f*∝γp

3/2 Rp1/2G−1/2. Simulation data are described by a scaling function f*∝ γp

3/2Rp1/2G−1/2δ−1.89.

■ INTRODUCTION

In everyday life, we face numerous manifestations of adhesionand contact phenomena from the stickiness of chewing gum tothe sole of a shoe, to the peeling of scotch tape from a piece ofpaper, to a gecko running on the walls and ceiling.1−7 Thesephenomena are also important for cell adhesion,8 colloidalstabilization,9,10 nanomolding and nanofabrication,11−13 anddrug delivery. These examples illustrate the true interdiscipli-nary nature of adhesion and contact mechanics problems thatare of interest to scientists working in the areas of colloidalphysics and chemistry, soft matter physics, biophysics, polymerscience and technology, and cell biology.It is well understood that adhesion between macroscopic

deformable bodies is a result of the fine interplay between anelastic energy of contact and adhesion in the contact and thevicinity of the contact area.1−7 It was shown by Johnson,Kendall, and Roberts (JKR)14,15 and by Derjaugin, Muller, andToporov (DMT)16 that because of dispersion (van der Waals)forces two bodies in contact can have a finite contact area evenin the case of zero load forces.9 Whereas JKR and DMTtheories were originally considered to be competitive, it turnsout that they represent two limiting cases of deformation oflarge compliant solids (JKR) and small rigid solids (DMT),corresponding to large and small values of the so-called Taborparameter, defined as the ratio of the elastic deformation to therange of action of the dispersion forces.5,17 During the last 40years, numerous experimental, theoretical, and numericalstudies confirmed the assumptions and validity of the JKRand DMT models for macroscopic objects.1,2,5,6

However, it was shown recently that a JKR/DMT-likeapproach fails to describe the contact and adhesion ofnanoscale-size objects correctly.18 For such objects in addition

to the elastic and adhesion energies, one has to account for thechanges in the surface energy induced by the shapedeformation of nanoparticles upon contact. There are twodifferent asymptotic regimes of nanoparticle deformation. Inthe first regime, which corresponds to large, hard nanoparticles,the deformation of nanoparticles in contact with a substrate isobtained by balancing the nanoparticle elastic energy with theparticle adhesion energy, as is done in the JKR or DMTtheories (JKR regime).5 In the second regime, corresponding tosmall, soft nanoparticles, nanoparticle deformation is deter-mined by the balance of the nanoparticle adhesion andinterfacial energies (nanoparticle wetting regime). The cross-over between these two regimes is governed by thedimensionless parameter β ∝ γp(GRp)

−2/3W−1/3, where G isthe particle shear modulus, Rp is the particle radius, γp is thepolymer interfacial energy, and W is the particle work ofadhesion. This parameter is proportional to the ratio of thenanoparticle surface energy to the elastic energy of thedeformed nanoparticle. Here we study the contact mechanicsof the nanoscale-size objects in the wide interval of pullingforces and show that pulling forces for nanoparticle detachmentdepend on material properties such as the nanoparticle elasticmodulus, surface energy, and work of adhesion. In particular,we demonstrate that the detachment of nanoparticles occursthrough neck formation and that the critical detachment forceis a universal function of a new dimensionless parameter δ ∝γp(GRp)

−1/3W−2/3 that is proportional to the ratio of the surface

Received: April 23, 2012Revised: June 11, 2012Published: June 19, 2012

Article

pubs.acs.org/Langmuir

© 2012 American Chemical Society 10881 dx.doi.org/10.1021/la301657c | Langmuir 2012, 28, 10881−10890

energy of a neck and the elastic energy of the deformednanoparticle.

■ RESULTS AND DISCUSSIONMolecular Dynamics Simulations of Nanoparticle

Contact Mechanics. We have performed molecular dynamicssimulations19 of the detachment of nanoparticles fromsubstrates. In our simulations, we used a coarse-grainedrepresentation of nanoparticles by cross-linking linear bead−spring chains with the number of monomers equal to N = 32.In this representation, monomers belonging to polymer chainswere modeled by Lennard-Jones particles (beads) withdiameter σ. The connectivity of monomers in polymer chainsand cross-linking bonds between chains were maintained byFENE bonds.20 Nanoparticles were prepared by confining apolymer melt in cavities with radii of R0 = 11.3σ, 16.4σ, and20.5σ (preparation size).18 The elastic properties of nano-particles were controlled by changing the cross-linking density.After a nanoparticle was cross-linked, the confining cavity wasremoved and the nanoparticle was relaxed by performing anMD simulation run. The size of the relaxed nanoparticle Rpvaried between 10.5σ and 21σ depending on the cross-linkingdensity. (See ref 18 for actual nanoparticle sizes.) For eachcavity size, R0, we performed simulations of nanoparticles withsix different cross-linking densities. The adsorbing (adhesive)substrate was modeled by an external 3−9 potential actingalong the normal to the surface z direction. The magnitude ofthe interaction potential εW was equal to 1.5kBT, 2.25kBT, and3.0kBT, where kB is the Boltzmann constant and T is theabsolute temperature.We performed two sets of simulations. In the first set of

simulations, the pulling force was modeled by applying aconstant force fz to each monomer forming a nanoparticle. Inthe second set of simulations, we used a weighted histogramanalysis method (WHAM)21 to obtain a potential of the meanforce between the nanoparticle and substrate as a function ofthe distance between the nanoparticle center of mass and thesubstrate surface.Figure 1a,b shows the potential of mean force for the

interaction of soft and hard nanoparticles with a substrate.Insets in Figure 1a,b show the dependence of the contact radiuson the applied force. We used red symbols to show the forceobtained from the differentiation of the potential of the meanforce. The differentiation was performed by fitting the potentialof the mean force by the Bezzier spline and differentiating theobtained fitting curve. The blue symbols in the insets in Figure1 a,b represent the results of the constant force simulations.Note that by differentiating the potential of mean force we canobtain both the stable and unstable branches of the contactradius function, a( f). Along the stable branch, the derivative ofthe contact radius with respect to force is negative, ∂a/∂f < 0,but on the unstable branch, it is positive, ∂a/∂f > 0. The criticalforce, f*, corresponding to nanoparticle detachment isdetermined by the divergence of the derivative, ∂a/∂f = ∞.The magnitude of nanoparticle deformation along the force

deformation curve can be evaluated from Figure 2. For softnanoparticles, we see the formation of the neck as the distancebetween the nanoparticles and substrate increases. In this case,the height of the nanoparticle can be larger than the diameter ofthe undeformed nanoparticle, 2Rp. For hard nanoparticles,there is no neck formation, and nanoparticle detachment takesplace abruptly when the value of the force exceeds a criticalvalue.

In Figure 3, we plot the dependence of the reducednanoparticle radius as a function of the reduced force. Thevalues of the contact radius a0 were obtained from zero forcesimulations when the nanoparticle deformation is controlledonly by the long-range 3−9 potential (Appendix A). The workof adhesion W = εwρpσ

3/2d02 was calculated by using the

average nanoparticle density ρp and the distance of closestapproach d0 = 0.886σ obtained from simulations. It followsfrom this figure that the critical value of the detachment forcedecreases with decreasing nanoparticle shear modulus G. Forthe hard nanoparticles, the value of the detachment force, f*,could be larger than 2πRpW. It is larger than both the criticaldetachment force for the DMT model, f DMT = 2πRpW, and thatfor the JKR model, f JKR = 1.5πRpW.5 However, the shape of thecurve is topologically similar to those obtained for the JKRmodel15

π= − +

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

fR W

aa

aa

6 6p 0

3

0

3/2

(1)

Figure 1. Potential of mean force as a function of the displacement ofthe nanoparticle center of mass along the z axis for soft (G =0.0233kBT/σ

3) and hard (G = 0.591kBT/σ3) nanoparticles with size R0

= 20.536σ, interacting with the substrate with εW = 1.5kBT. Insetsshow the dependence of the contact radius a on the applied force f.Blue symbols correspond to constant force simulations; red symbolsrepresent the effective force obtained from the differentiation of thepotential of mean force.

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where a0 is the nanoparticle contact radius at zero applied force.Note that in the JKR model particle detachment occurs at finitevalues of the contact radius whereas in the DMT model thecontact radius gradually decreases to zero with increasingmagnitude of the force such that at the detachment point thecritical value of the force is equal to the adhesion force of theundeformed (rigid) particle, f DMT = 2πWRp. The value of the

detachment force, f*, shifts toward larger force values withincreasing both the nanoparticle shear modulus, G, and thework of adhesion, W, of nanoparticles to substrate. For softnanoparticles with G = 0.023kBT/σ

3, the value of thedetachment force f* approaches πRpW.It is important to point out that significant deviations from

the JKR model are seen for the large nanoparticle interactionenergy with a substrate (εW = 3.0kBT). In this case at zero force,soft nanoparticles are strongly deformed (Figure 4). Suchnanoparticles experience large nonlinear deformations as theapplied force approaches the detachment force. The shape ofthese nanoparticles resembles those of pendant droplets.10 Thesofter the nanoparticle, the thicker the neck it grows during thepulling process (Figure 4).

Figure 2. Deformation of nanoparticle shape for soft (G = 0.0233kBT/σ3) (top row) and hard (G = 0.591kBT/σ

3) (bottom row) nanoparticles withsize R0 = 20.536σ adsorbed on a substrate with εW = 1.5kBT.

Figure 3. Dependence of the reduced contact radius, a/a0, on thereduced force, f/πWRp, for nanoparticles with a preparation radius ofR0 = 16.4 σ and different values of the shear modulus G = 0.023kBT/σ

3

(open squares), 0.21kBT/σ3 (crossed squares), and 0.86kBT/σ

3 (filledsquares) at different magnitudes of the wall interaction potential, εw. a0is the contact radius at zero applied force, and W is the work ofadhesion. The solid line corresponds to eq 1.

Figure 4. Snapshots of nanoparticle shape deformation for nano-particles of size R0 = 16.4σ adsorbed to a substrate with εW = 3.0kBT.fW = πWRp and f* is the detachment force.

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Model of Nanoparticle Contact Mechanics. Oursimulations show that there are two different regimes innanoparticle detachment from a substrate. Nanoparticles with alarge value of the shear modulus G detach from the substratealmost without neck formation. We have called such nano-particles “hard” nanoparticles. However, nanoparticles withsmall values of the shear modulus detach with neck formation.We referred to these nanoparticles as “soft” nanoparticles. Wewill first present a model of detachment for hard nanoparticles.In the case of hard nanoparticle deformations, Δh = 2Rp − h

≪ Rp, the total free energy of the deformed hard nanoparticlewith size Rp can be expressed in term of the change in

nanoparticle height (Figure 5) (The derivation details are givenin Appendix B.)

π πγΔ ∝ Δ − Δ + Δ

+ Δ

F h f h WR h h

GR h

( ) 2

28 215

p p2

p1/2 5/2

(2)

The first term on the r.h.s of eq 2 represents the contributiondue to the applied pulling force. The positive sign of this termreflects the pulling nature of the applied force. The second termdescribes the contribution of the long-range van der Waalsinteractions of the deformed nanoparticle with a substratewhereW is the work of adhesion. The third term represents theincrease in the surface energy of the deformed nanoparticlewith the interfacial energy γp in comparison with that of asphere with radius Rp. Finally, the last term is the elastic energycontribution due to the deformation of a nanoparticle withshear modulus G.The equilibrium deformation of a nanoparticle is obtained by

minimizing the nanoparticle free energy (eq 2) with respect tothe height deformation, Δh. This results in the followingequation describing nanoparticle deformation:

π πγ= − + Δ + Δf WR h GR h0 2 214 2

3p p p1/2 3/2

(3)

To obtain an exact solution of eq 3, it is useful to introducedimensionless variables

= Δ−

⎛⎝⎜⎜

⎞⎠⎟⎟y

hR A

GR

W f f2 (1 / )2

p

p

DMT

2/3

(4a)

βγ

=−

⎛

⎝⎜⎜

⎞

⎠⎟⎟f B

GR W f f( )

(1 / )p

3/2

p1/2

DMT1/2

2/3

(4b)

where A = (3π/28)2/3 = 0.484 and B = 2(3π/28)2/3 = 0.968 arenumerical coefficients and f DMT = 2πWRp is a criticaldetachment force for hard nanoparticles that is equal to the

absolute value of the adhesion force between undeformednanoparticles and a substrate. Note that the values of thenumerical coefficients depend on the particular nanoparticleshape and deformation that were used for the calculation of thedifferent terms in the nanoparticle free energy. (See AppendixB and ref 18 for details.) Using these dimensionless variables,we can reduce eq 3 to a simple cubic form

β= − + +f y y0 1 ( ) 2 3(5)

The deformation of the nanoparticles is described by a positiveroot of this equation

β

Δ =−

+ + + − + −

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

hR

AW f f

GR

r q r r q rf

2

(1 ( / ))

( )3

p

DMT

p

2/3

3 2 3 22

3 3

(6)

where

β β= − = −

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟r

fq

f12

( )3

and( )3

3 2

(7)

It follows from eq 6 that we should be able to collapse oursimulation data for hard nanoparticles by plotting (Δh/2Rp)(GRp/Wf)

2/3 as a function of γp(GRp)−2/3Wf

−1/3, whereWf = W(1 − ( f/f DMT)) is a renormalized value of the work ofadhesion (Figure 6). Indeed, we see a reasonably good collapseof the simulation data for zero and small values of the pullingforces. However, simulation data begin to deviate from theuniversal curve with increasing magnitude of the pulling force.

Figure 5. Schematic representation of a deformed nanoparticlewithout a neck.

Figure 6. Dependence of the reduced nanoparticle deformation on thevalue of parameter γp(GRP)

−2/3Wf−1/3 for nanoparticles with different

values of the shear modulus G = 0.023kBT/σ3 (open symbols),

0.096kBT/σ3 (crossed symbols), 0.21kBT/σ

3 (right-half-filled symbols),0.37kBT/σ

3(top-half-filled symbols), 0.59kBT/σ3 (checkered symbols),

and 0.86kBT/σ3 (filled symbols), preparation radii R0 = 11.3σ (black

symbols), 16.4σ (red symbols), and 20.5σ (blue symbols), wallinteraction potential with εw = 1.5kBT (circles), 2.25kBT (squares), and3.0kBT (diamonds). The green triangles show zero force simulationsfrom ref 18. The solid line is the best fit to eq 6 with A = 0.419 and B =0.84.

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In Figure 7, we show how this departure happens fornanoparticles adsorbed on substrates with different strengths

of the surface potential, εW. A comment has to be made hereconcerning the value of the fitting parameter B = 0.84 and itsrelation to the value of the fitting parameter BG used in ref 18.There was a typographical error in ref 18 in reporting the valueof parameter BG. It should be BG/3 = 0.28.An analysis of the snapshots of the deformed nanoparticles

points out (e.g., insets in Figures 1 and 4) that the main reasonfor such a deviation is the formation of the neck connecting ananoparticle to a substrate. This transformation takes place forvery small values of the applied forces for nanoparticlesadsorbed on substrates with εw = 2.25kBT and 3.0kBT (Figure4). However, in the case of a weaker substrate potential forwhich εw = 1.5kBT, the simulation data first follow the universalcurve and then start to deviate (Figure 7). The deviation fromthe universal curve occurs when the contribution from thesurface energy of the neck to the total nanoparticle free energybecomes comparable to the change in the surface energy of thedeformed spherical cap. Below we will generalize our model toaccount for neck formation.In the case of soft nanoparticles, the shape of a deformed

nanoparticle can be described by two independent parameters:the nanoparticle contact radius a and nanoparticle heightdeformation Δh (Figure 8). The height of a neck connecting ananoparticle to a substrate is related to these parameters as hn ≈(a2/2Rp) − Δh. This is due to the requirement of volume

conservation upon nanoparticle deformation (details are givenin Appendix B). In this approximation, the free energy ofnanoparticles is equal to

π

γ π

Δ ∝ Δ −

+ + − Δ

+ Δ − Δ +

⎛⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟⎞⎠⎟⎟

⎡⎣⎢⎢

⎤⎦⎥⎥

F h a f h W a

aR

aaR

h

G h aha

Ra

R

( , )

42

2

423

15

2

p

4

p2

2

p

23

p

5

p2

(8)

The equilibrium values of the contact radius a and heightdeformation Δh are obtained by solving equations ∂F(Δh, a)∂a= 0 and ∂F(Δh, a)/∂Δh = 0. To analyze these equations, weintroduce dimensionless variables

= Δ = Δ = ⎜ ⎟⎛⎝

⎞⎠

⎛

⎝⎜⎜

⎞

⎠⎟⎟a

f R

Ga h

f

G Rh f f f

34

,34

,3 JKR p 32/3

JKR2

2p

1/3

JKR

(9)

where f JKR = 1.5πWRp. In these new variables, the equations arewritten as follows

β δΔ − − + + − Δ =h a a a a h[ ]49

(3 2 ) 02 21

3 2

(10)

δ Δ − − +=

⎡⎣⎢

⎤⎦⎥a h a a

f13 6

02

(11)

where two dimensionless parameters

β π γ=

× GR W2 3 ( )1

2/3

2/3p

p2/3 1/3

(12a)

δ π γ=

GR W3 ( )

1/3

4/3p

p1/3 2/3

(12b)

determine the relative strengths of the adhesion, interfacial, andelastic energies. There is a simple physical meaning ofparameters β1 and δ. Parameter β1 corresponds to the ratioof the change in the surface energy of a spherical cap withrespect to the elastic energy of the deformed nanoparticle, andparameter δ is proportional to the ratio of the surface energy ofa neck to the nanoparticle elastic energy. For our systems, theratio of β1/δ is between 0.8 and 3. By solving eqs 10 and 11,one can obtain an expression for an applied force f as a functionof the contact radius a

δ δ β = − + + − − f a a a a a4 2 9 4 9 93 2 21

3(13)

Equation 13 gives both stable, ∂a/∂f < 0, and unstable, ∂a/∂f >0, branches of the nanoparticle deformation curves. Further-more, eq 13 is correct only when a macroscopic neck (hn > σ) isbeginning to form, which takes place at finite values of theapplied force, f. In the interval of the applied forces where thereis no macroscopic neck (hn < σ), the deformation of ananoparticle is described by eq 6.An analysis of eq 13 shows that there are two asymptotic

scaling regimes for the dependence of the detachment force, f*,on the system parameters. In the limit of small values ofparameters β1 (β1 ≪ 1) and δ (δ ≪ 1, JKR regime), thedetachment force corresponding to the minimum in eq 13, ∂f/

Figure 7. Dependence of the reduced nanoparticle deformation on thevalue of parameter γp(GRP)

−2/3Wf−1/3 for nanoparticles with shear

modulus G = 0.21kBT/σ3, preparation radius R0 = 20.5σ, and wall

interaction potential with εw = 1.5kBT (black circles), 2.25kBT (redsquares), and 3.0kBT (blue diamonds). The solid line is the best fit toeq 6 with A = 0.419 and B = 0.84.

Figure 8. Schematic representation of a deformed nanoparticle with aneck of radius a.

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∂a = 0, is equal to f* ≈ 1.5πWRp, with the contact radius at thedetachment point being equal to a * ≈ 2−2/3. In the oppositelimit of large values of parameter δ (δ ≫ 1, necking regime), bydifferentiating eq 13 with respect to contact radius a oneobtains for the contact radius at the detachment point a * ≈ (δ/2)1/2 and for the detachment force f*∝ γp

3/2 Rp1/2G−1/2. (Note

that to obtain this result we kept only the δ2 term under thesquare root in eq 13.) Thus, we can plot a normalizeddetachment force f*G1/2γp

−3/2 Rp−1/2 as a function of parameter

δ ∝ γp(GRp)−‑1/3W−2/3. Figure 9 shows the dependence of the

reduced force on parameter γp(GRp)−1/3W−2/3 for all of our

systems. Simulation data have collapsed into one universalcurve, proving that the normalized detachment force is auniversal function of parameter δ. Unfortunately, the studiedinterval of the system parameters is not sufficient to cover pureasymptotic regimes. Our simulation data correspond to acrossover between the JKR and necking regimes. In thisinterval of parameters simulation, data can be fitted by powerlaw 5.34x −1.89.

■ CONCLUSIONSWe have shown that nanoparticle detachment from an adhesivesubstrate under the action of the pulling force occurs throughneck formation. The critical detachment force depends on thenanoparticle adhesion energy with a substrate, W, thenanoparticle interfacial energy, γp, and the nanoparticle shearmodulus, G. With decreasing nanoparticle shear modulus, G,the critical force required for nanoparticle detachmentdecreases from f* ≈ 2.2πWRp to πWRp (Figure 3). Oursimulations confirmed that the value of the normalizeddetachment force f*G1/2γp

−3/2 Rp−1/2 is a universal function

of interaction parameter δ ∝ γp(GRp)−1/3W−2/3 (Figure 9) and

monotonically decreases with increasing value of parameter δ.The neck height was shown to increase with increasing strengthof the adhesion energy εW and with decreasing nanoparticleshear modulus, G (Figures 2 and 4).

In Figure 10, we summarized different nanoparticledeformation regimes in the W/γp, W/(GRp) plane. In the

undeformed nanoparticle regime, adhesion forces are too weakto cause noticeable nanoparticle deformation. In this regime,the deformation of the nanoparticle height Δh is smaller thanthe characteristic length scale of the short-range interactions(LJ potential) or the bead diameter σ, Δh < σ. In the case ofsoft nanoparticles, the nanoparticle height deformation isdetermined by the balance of the change in the nanoparticleinterfacial energy, γpΔh2, and its adhesion energy to a substrate,−WRpΔh (eq 2 in the zero force limit). The deformation ofnanoparticle height, Δh ∝ RpW/γp, is larger than the beaddiameter σ when W/γp > σ/Rp. We refer to this regime as thewetting regime in Figure 10. If the opposite inequality holds,then there is no noticeable nanoparticle deformation.In the case of hard nanoparticles, their deformation is

determined by equating the elastic and adhesion energies (eq2). This results in Δh ∝ Rp(W/GRp)

2/3. Therefore, thecrossover to the nanoparticle deformation regime for Δh ∝ σoccurs at W/GRp ∝ (σ/Rp)

3/2. It is interesting that thiscrossover is controlled by Tabor’s parameter μ ≈ (W2Rp/G2σ3)1/3. Over the range of values of Tabor’s parameter μ ≤ 1,the nanoparticle deformation obeys DMT theory (the DMTregime in the diagram of states). However, in the case of largevalues of Tabor’s parameter μ ≫ 1, the deformation ofnanoparticles is described in the framework of the JKR theory(the JKR regime in Figure 10). In the JKR regime, thenanoparticle detachment force is equal to f* ≈ 1.5πWRp, and inthe DMT and undeformed nanoparticle regimes, it is f* =2πWRp. In a crossover, the detachment force depends on thevalue of Tabor’s parameter (details in refs 5 and 17).The crossover from the JKR to the wetting regime occurs

when the deformation of the elastically stabilized nanoparticle,Δh ∝ Rp(W/GRp)

2/3, is on the order of the nanoparticledeformation stabilized by the interfacial energy, Δh ∝ RpW/γp.This results in the following expression for the crossover line,W/γp ∝ (W/GRp)

2/3. Above this line, nanoparticle deformationresembles that of a liquid nanodroplet. In the wetting regime, a

Figure 9. Dependence of the normalized detachment force on thevalue of the parameter γp(GRP)

−1/3W−2/3 for nanoparticles withdifferent preparation radii R0 = 11.3 σ (open symbols), 16.4σ (crossedsymbols), and 20.5σ (filled symbols) interacting with substrates withεw = 1.5kBT (black circles), 2.25kBT (red squares), and 3.0kBT (bluediamonds) and different values of the shear modulus G. The solid lineis given by the equation f(x) = 5.34x −1.89.

Figure 10. Diagram of states of a nanoparticle at an adhesive surfaceon logarithmic scales.

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dashed line, W/γp ∝ (W/GRp)1/3, is a crossover line above

which, during nanoparticle peeling, the interfacial energy of thegrowing nanoparticle neck provides a dominant contribution tothe nanoparticle interfacial energy and controls nanoparticledetachment (necking regime in Figure 9). Along this line, thevalue of parameter δ (eq 12a) is on the order of unity. In thisrange of parameters, the nanoparticle detachment force scalesas f*∝ γp

3/2Rp1/2G−1/2.

Our analysis of different nanoparticle deformation regimes iscorrect only in the case of small nanoparticle deformations,Δh/Rp < 1. This condition determines boundaries for the lineardeformation regime, W/γp < 1 and W/GRp < 1.The simulation results presented here can be extended to

account for the roughness of the adsorbing substrate as well asthe substrate elasticity. These topics will be the subject of futurepublications.

■ APPENDIX A: SIMULATION DETAILSWe have performed coarse-grained molecular dynamicssimulations of the pulling of a spherical nanoparticle from asubstrate. The nanoparticles were prepared by cross-linkingpolymer chains with a degree of polymerization of N = 32confined in a spherical cavity with size R0. By varying thenumber of cross-links per chain, we have been able to controlthe nanoparticle elastic modulus. The details of the nano-particle preparation procedure are described in ref 18. In oursimulations, the interactions between monomers forming ananoparticle were modeled by the truncation-shifted Lennard-Jones (LJ) potential

ε σ σ σ σ=

− − + ≤

>

⎧⎨⎪⎪

⎩⎪⎪

⎡

⎣⎢⎢⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥⎥U r r r r r

r r

r r

( )4

0

ij ij ijLJLJ

12 6

cut

12

cut

6

cut

cut

(A.1)

where rij is the distance between the ith and jth beads and σ isthe bead (monomer) diameter. The cutoff distance for thepolymer−polymer interactions was set to rcut = 2.5σ, and thevalue of the Lennard-Jones interaction parameter was equal to1.5kBT. This choice of parameters corresponds to poor solventconditions for the polymer chains.The connectivity of the beads into polymer chains and the

cross-link bonds were maintained by the finite extensionnonlinear elastic (FENE) potential20

= − −⎛⎝⎜

⎞⎠⎟U r k R

rR

( )12

ln 1FENE spring max2

2

max2

(A.2)

with a spring constant of kspring = 30kBT/σ2 and a maximum

bond length of Rmax = 1.5σ. The repulsive part of the bondpotential is given by the truncation-shifted LJ potential with rcut= (2)1/6σ and εLJ = 1.5kBT. In our simulations, we haveexcluded 1−2 LJ interactions such that monomers connectedby a bond interacted only through the bonded potential.The presence of the substrate was modeled by the external

potential

ε σ σ= −⎜ ⎟ ⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤⎦⎥U z

z z( )

215w

9 3

(A.3)

where εw was equal to 1.5kBT, 2.25kBT, and 3.0kBT. The long-range attractive part of the potential z−3 represents the effect ofvan der Waals interactions generated by the substrate half-

space. The system was periodic in the x and y directions withdimensions of Lx = Ly = Lz = 4R0, where R0 is the nanoparticleradius (details in ref 18). We have modeled nanoparticles withradii Rp varying between 10.5σ and 20.6σ. The actualnanoparticle sizes as a function of the nanoparticle cross-linking density and confining spherical cavity R0 are listed in theappendix of ref 18.To model nanoparticle pulling, we have performed two sets

of simulations. In the first set of simulations, we applied aconstant force fz pointing along the positive z direction.Simulations were carried out for a constant number of particlesand temperature ensemble. The constant temperature wasmaintained by coupling the system to a Langevin thermostat.The equation of motion of the ith particle with mass m was

ξ = − + + mv t

tF t v t f n F t

d ( )d

( ) ( ) ( )ii i z z i

R

(A.4)

where vi(t) is the ith bead velocity and Fi(t) is the netdeterministic force acting on the ith bead. Fi

R(t) is the stochasticforce with zero average value ⟨Fi

R(t)⟩ = 0 and δ-functionalcorrelations ⟨Fi

R(t)FiR(t′)⟩ = 4ξkBTδ(t − t′). Note that the

stochastic force had only nonzero x and y components. Frictioncoefficient ξ was set equal to m/(7τLJ), where τLJ is the standardLJ time τLJ = σ(m/εLJ)

1/2. The velocity Verlet algorithm with atime step of Δt = 0.01τLJ was used to integrate the equations ofmotion (eq A.4). All simulations were performed usingLAMMPS.22 The duration of each simulation run was up to5 × 104τLJ.In the second set of simulations, we used the weighted

histogram analyzes method21 to calculate the potential of meanforce between a nanoparticle and substrate. In thesesimulations, a z coordinate of the center of mass, zcm, of ananoparticle was tethered to z* by a harmonic spring

* = − *U z z K z z( , )12

( )cm sp cm2

(A.5)

with the value of the spring constant being Ksp = 100kBT/σ2.

During these simulation runs, we have varied the location of thetethering points with increment Δz* = 0.25σ. For each locationof the tethering point, the system was equilibrated for 103τLJfollowed by 4 × 103 τLJ, during which we have calculated thedistribution of the center-of-mass location for the WHAMcalculation of the potential of mean force. In these simulations,we have applied the Langevin thermostat to control the systemtemperature in all three directions.

■ APPENDIX B: VARIATIONAL APPROACH TONANOPARTICLE DEFORMATION

Let us consider a small deformation of the nanoparticle withradius Rp and approximate the deformed nanoparticle by ahemisphere of radius R1 connected to a substrate by acylindrical neck of height hn (Figure 11). The nanoparticle ispulled from the surface by a force with magnitude f pointingalong the normal to the substrate direction. We will assume thatthe radius of the contact area between nanoparticles andsubstrate is equal to a and that the displacement of thenanoparticle surface at the point of the closest approach to thesubstrate is Δh = 2Rp − h, where h is the height of thedeformed nanoparticle. The total free energy of the deformednanoparticle includes the elastic energy of the nanoparticledeformation, the van der Waals energy contribution due to theinteraction between the nanoparticle and substrate, the surface

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energy contribution due to the change in the surface area of ananoparticle as it comes in contact with a substrate, and thework done by displacing the center of mass of the nanoparticleby Δh under the action of pulling force f, Δhf.To estimate the elastic energy of the deformed nanoparticle,

we will apply a variational approach developed in ref 18. Thestress exerted by a deformed nanoparticle on the surface of asemi-infinite half space is given by15

σ ρ σ ρ σ ρ= − + −−⎛

⎝⎜⎞⎠⎟

⎛⎝⎜

⎞⎠⎟a a

( ) 1 10

2

2

1/2

1

2

2

1/2

(B.1)

The displacement field of the nanoparticle that generates thisstress is

ρ π σσ ρ= + −

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟⎞⎠⎟⎟u

aK a

( )2

12z 0

12

2(B.2)

where K = 2G/(1 − υ) is the nanoparticle rigidity. Theparameters for the displacement field σ0 and σ1 can be foundfrom the displacement of the surface of the nanoparticle. Thedisplacement of the surface of the nanoparticle for smalldeformations is given by (Figure 11)

ρ ρ ρ= Δ − − − ≈ Δ −u h R R hR

( ) ( )2z p p

2 22

p (B.3)

Comparing eqs B.2 and B.3, one obtains the followingexpressions for parameters σ0 = K(Δh/a − a/Rp)/π and σ1=K(2a/Rp)/π. Taking this into account, the elastic energy ofthe deformed nanoparticle is estimated as follows:

∫ σ

π σ σ σ σ

Δ =

= + +

≈ Δ − Δ +

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

U h a u

aK

K h aha

Ra

R

x x x( , )12

d ( ) ( )

23

215

23

15

zel2

2 3

02

0 1 12

23

p

5

p2

(B.4)

The surface energy contribution includes the contributionfrom the spherical cap with radius R1 and a cylindrical neck. Toevaluate different contributions to the nanoparticle surfaceenergy, we will assume that the deformation of the adsorbednanoparticle occurs at a constant volume such that

= − +R R hh

h a43 3p

31 R

2 R3

n2

(B.5)

where we have introduced parameters

= − = − Δ −a R h h h R h h(2 ) and 221 R R

2R p n (B.6)

Equations B.5 and B.6 determine the radius of the hemi-spherical cap R1 as a function of the contact radius a, neckheight hn, nanoparticle height deformation Δh, and initial sizeRp. Solving eqs B.5 and B.6 together, we can derive a quadraticequation for a as a function of the nanoparticle shape.

+ − − =⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

hh a R

h2

43 6

0Rn

2p

3 R3

(B.7)

The physical solution of this equation is

=−+

≈ Δ +aR h

h hR h h

8

3( 2 )2 ( )p

3R

3

R np n

(B.8)

The surface energy of the deformed nanoparticle withinterfacial energy γp is equal to

γ πΔ = + +U h a h a ah R h( , , ) ( 2 2 )s n p2

n 1 R (B.9)

where the surface energy also includes a contribution from thenanoparticle contact area with a substrate with radius a (thefirst term in parentheses).An evaluation of the contribution from the van der Waals

interactions of the deformed nanoparticle of average density ρpwith a substrate requires the integration of the long-range vander Walls potential of magnitude εw over the particle height.We will separate this contribution from contributions from thespherical cap and cylindrical neck. The contribution from thespherical cap located a distance h = hn + d0 from the substrate isgiven by

∫

∫

ε ρ σ π

ε ρ σ π

Δ ≈ −+

= −

− − ++

U h a hr z

z hz

h z R h zz h

z

( , , )( )

( )d

( )(2 )( )

d

n

h

h

vdw(1)

w p3

0

2

3

w p3

0

R 1 R3

R

R

(B.10)

where r(z) is the radius of the spherical cap a distance z fromthe substrate and d0 is the distance of closest approach betweenthe nanoparticle and substrate. After integration, we arrive at

ε ρ σ πΔ ≈ −

×− + +

+ − +

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟⎤⎦⎥

U h a h

R h h h h h h

h h h

hh

( , , )

(2 2 )

2 ( )log 1

nvdw(1)

w p3

1 R R2

R2

R2

R

R

(B.11)

This expression reduces to the expression for the interactionpotential of the rigid sphere (R1 = Rp, and hR = 2Rp) located adistance h from a substrate. The contribution from thecylindrical neck of height hn located a distance d0 from thesubstrate is calculated in a similar way

∫ε ρ σ π

ε ρ σπ

Δ ≈ −+

= − −

⎛⎝⎜

⎞⎠⎟

U h a ha

z dz

dd

ha

( , , )( )

d

21

n

h

vdw(2)

w p3

0

2

03

w p3

02

02

22

n

(B.12)

For the systems with only van der Waals interactions, we canrelate the surface and interfacial energies to the system density

Figure 11. Schematic representation of nanoparticle deformation.

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dx.doi.org/10.1021/la301657c | Langmuir 2012, 28, 10881−1089010888

and strength of the van der Waals interactions. In this case, thesurface energy of the polymer−substrate interface γps is equal to

γ γ γε ρ σ

= + −d2ps p s

w p3

02

(B.13)

where γp is the polymer surface energy and γs is the substratesurface energy. Using eq B.13, we can relate the parameters ofthe van der Waals interactions to the work of adhesion9

γ γ γε ρ σ

= + − =Wd2p s ps

w p3

02

(B.14)

Taking this into account, we can rewrite the total van der Waalsinteraction energy as follows:

π

π

Δ ≈ −+ +

+

− + − −

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

U h a h W da h h h h

h h h

hh

W ad

h

( , , )( 2 )

( )

2 log 1 1

nvdw 02

2R

2R

2R

R 2 02

2

(B.15)

The equilibrium nanoparticle shape is obtained from aminimization of the total nanoparticle free energy

γ π

Δ ≈ Δ − Δ +

+ Δ + Δ

+ + +

⎡⎣⎢⎢

⎤⎦⎥⎥F h a h K h a

haR

aR

U h a h f h

a h ah

( , , )23

15

( , , )

(2 2 )

n

n

n

23

p

5

p2

vdw

p2

R2

(B.16)

with respect to nanoparticle deformation Δh and radius ofcontact a, taking into account the relation among the neckheight, nanoparticle deformation, and contact radius (eq B.8).A scaling analysis of eq B.16 can be performed in two limitingcases of deformation of hard and soft nanoparticles. In the caseof hard nanoparticles, nanoparticle detachment occurs withoutneck formation, hn < d0, while during the detachment of softnanoparticles a neck is formed, hn > d0. By expanding the vander Waals interaction energy in these two limiting cases, onehas

π πΔ ≈ − −

+ <

+ >

⎧

⎨⎪⎪

⎩⎪⎪

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

U h a h W a W d

R

d

R

dh d

R

h

R

hh d

( , , )

22 log

2,

22 log

2,

vdw n2

02

p

0

p

0n 0

p

n

p

nn 0

(B.17)

Note that we can neglect the term 2πWRpd02/hn describing

the interaction of a soft nanoparticle with a substrate incomparison with the leading interaction term Wπa2 as long asparameter Rpd0

2/hna2 is smaller than unity. Using eq B.8, we

can estimate the neck height as hn ∝ a2/Rp. Thus, for theparameter to be small it requires that

≈ ≈ ≪R d

h a

R d

adh

1n

p

n

p 02

2

20

2

40

2

2(B.18)

It is important to point out that the neck formation, hn > d0, isrelated to the value of the Tabor parameter, μ. Our simulationsshow that d0 ∝ σ; therefore, we can use a bead diameter σ,which describes the characteristic length scale of the Lennard-Jones interaction potential, instead of d0 for the neck formationregime. For soft nanoparticles, the typical nanoparticle contactradius a according to the JKR theory14 is on the order of(WRp

2/G)1/3. Thus, for the neck height to be larger than thebead diameter σ, the following must be true:

μσ σ

≡ ≈ ≫⎛⎝⎜⎜

⎞⎠⎟⎟

h W R

G1n

2p

2 3

1/3

(B.19)

Tabor’s parameter defines a crossover between the JKR andDMT regimes.5,17 Large values of the Tabor parametercorrespond to the JKR regime14 whereas for small values ofμ≪1 the particle deformation is described by the DMTtheory.16 In our variational approach, Tabor’s parameter μdefines regimes for nanoparticle detachment with and withoutneck formation.Now we are in a position to write a scaling expression for the

nanoparticle free energy. For hard nanoparticles, the contactradius a and deformation Δh are not independent and arerelated by a2 ≈ 2RpΔh (eq B.8). In this approximation, eq B.16reduces to

π πγΔ ∝ Δ − Δ + Δ

+ Δ

F h f h WR h h

GR h

( ) 2

28 215

p p2

p1/2 5/2

(B.20)

where we omitted terms independent of Δh. In the case of softnanoparticles with a neck, hn > σ, the free energy is

π

γ π

Δ ∝ Δ −

+ + − Δ

+ Δ − Δ +

⎛⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟⎞⎠⎟⎟

⎡⎣⎢⎢

⎤⎦⎥⎥

F h a f h W a

aR

aaR

h

G h aha

Ra

R

( , )

42

2

423

15

2

p

4

p2

2

p

23

p

5

p2

(B.21)

See the main text for an analysis of different scaling regimes.

■ AUTHOR INFORMATIONNotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work was supported by the National Science Foundationunder grant DMR-1004576 and by the American ChemicalSociety Petroleum Research Fund under the grantPRF#49866ND.

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