DOI: 10.1021/la101977c 12973Langmuir 2010, 26(15), 12973–12979 Published on Web 07/06/2010

pubs.acs.org/Langmuir

© 2010 American Chemical Society

Adhesion of Nanoparticles

Jan-Michael Y. Carrillo,† Elie Raphael,‡ and Andrey V. Dobrynin*,†

†Polymer Program, Institute of Materials Science, and Department of Physics, University of Connecticut,Storrs, Connecticut 06268, and ‡PCT Lab., UMR CNRS Gulliver 7083, ESPCI, 10 rue Vauquelin,

75231 Paris Cedex 05, France

Received May 17, 2010. Revised Manuscript Received June 17, 2010

We have developed a new model of nanoparticle adhesion which explicitly takes into account the change in thenanoparticle surface energy. Using combination of the molecular dynamics simulations and theoretical calculations wehave showed that the deformation of the adsorbed nanoparticles is a function of the dimensionless parameter β �γp(GRp)

-2/3W-1/3, where G is the particle shear modulus, Rp is the initial particle radius, γp is the polymer interfacialenergy, andW is the particle work of adhesion. In the case of small values of the parameter β<0.1, which is usually thecase for strongly cross-linked large nanoparticles, the particle deformation can be described in the framework of theclassical Johnson, Kendall, and Roberts (JKR) theory. However, we observed a significant deviation from the classicalJKR theory in the case of the weakly cross-linked nanoparticles that experience large shape deformations upon particleadhesion. In this case the interfacial energy of the nanoparticle plays an important role controlling nanoparticledeformation. Our model of the nanoparticle adhesion is in a very good agreement with the simulation results andprovides a new universal scaling relationship for nanoparticle deformation as a function of the system parameters.

Adhesion phenomena play an important role in different areasof science and technology including tribology,1-3 colloidal science,4,5

materials science,6-10 biophysics, andbiochemistry.11-13They are ofparamount importance for colloidal stabilization, drug delivery, in-terfacial friction and lubrication, nanofabrication and nanomolding,cell mechanics and adhesion, and contact mechanics. The modernapproach to adhesion between elastic bodies in a contact is based onthe classical work by Johnson, Kendal and Roberts (JKR)14 thatextended the Hertz theory15 of the elastic contact by accounting forthe effect of the adhesion in the contact area.According to thismodela contact radius a of an elastic sphere of radius Rp with the shearmodulusG in a contact with a solid substrate is proportional to a�Rp(W/GRp)

1/3, where W is the work of adhesion (for reviews seerefs 2, 7, and 15). The analysis of this scaling relation indicates thatthe importance of the adhesive forces increases for “soft” highlycompliant materials such as elastomers, living cells and soft tissues.

However, the JKR theory14,15 only takes into account thechange in the particle surface energy in the contact area with thesubstrate and completely ignores contribution from the change in

the surface energy of a particle due to its shape deformation. Thiseffect is small for the large particles for which the work of adhesionW is smaller thanGRp.Unfortunatelywhen the opposite inequalityholds,W>GRp, a particle undergoes large deformations with thecontact radius being on the order of the initial particle size. Forexample, a latex particlewith a shearmodulusG≈ 106 Pa adsorbedona siliconwafer surfacewith theworkof adhesionW≈100mJ/m2

will undergo significant deformation when its size is smaller thanW/G ≈ 100. Thus, for nanoparticles the elastic energy of particledeformation GRp

3 becomes comparable with its surface energyγpRp

2 ≈ WRp2 which has to be taken into account in calculating

the equilibrium particle shape. In this paper we will consideradhesion of a soft nanoparticle on a rigid substrate and willdevelop a general approach to describe nanoparticle deformationwhich in addition to the JKRmodel takes into account the changein the nanoparticle surface energy outside the contact area.

The total free energy of the deformed nanoparticle includesnanoparticle surface energy, van derWaals energy5 of interaction ofthe nanoparticlewith a substrate and the elastic energy contributiondue to particle deformation. The particle deformation can bedescribed by approximating a deformed nanoparticle by a sphericalcap with radiusR1 and height h (see Figure 1). In addition we havealso assumed that deformation of the nanoparticle occurs at aconstant volume. This constraint allowed us to relate the radius ofthe deformed nanoparticle R1 with the nanoparticle height h andinitial nanoparticle sizeRp. In the framework of this approximationone can calculate the nanoparticle free energy as a function ofthe nanoparticle deformation 1-h/2Rp = Δh/2Rp. The details ofcalculations are given in the Appendix A.

In the case of small deformations Δh/Rp , 1 the nanoparticlefree energy has a simple analytical form

FðΔhÞ� - 2πWRpΔhþπγpΔh2 þ 7

ffiffiffi2

p

15KRp

1=2Δh5=2 ð1Þ

The first term in the r.h.s of the eq 1 represents contribution of thelong-range van der Waals interactions of the deformed nanopar-ticle with a substratewhereW is thework of adhesion. The second

*To whom correspondence should be addressed.(1) Crosby, A. J.; Shull, K. R. J. Polym. Sci., Part B: Polym. Phys. 1999, 37(24),

3455–3472.(2) Johnson, K. L. Tribol. Int.l 1998, 31(8), 413–418.(3) Mo, Y. F.; Turner, K. T.; Szlufarska, I. Nature 2009, 457(7233), 1116–1119.(4) de Gennes, P.-G.; Brochard-Wyart, F.; Quere, D. Capillarity and Wetting

Phenomena; Springer: New York, 2002.(5) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: London,

1992.(6) Gerberich, W. W.; Cordill, M. J. Rep. Prog. Phys. 2006, 69(7), 2157–2203.(7) Barthel, E. J. Phys. D: Appl. Phys. 2008, 41, 163001.(8) Luan, B. Q.; Robbins, M. O. Nature 2005, 435(7044), 929–932.(9) Shull, K. R. Mater. Sci. Eng. R-Rep. 2002, 36(1), 1–45.(10) Shull, K. R.; Creton, C. J. Polym. Sci., Part B: Polym. Phys. 2004, 42(22),

4023–4043.(11) Arzt, E.; Gorb, S.; Spolenak, R.Proc. Natl. Acad. Sci. U.S.A. 2003, 100(19),

10603–10606.(12) Lin, D. C.; Horkay, F. Soft Matter 2008, 4(4), 669–682.(13) Liu, K. K. J. Phys. D: Appl. Phys. 2006, 39(11), R189–R199.(14) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London A 1971,

324, 301–313.(15) Johnson, K. L. Contact Mechanics, 9th ed.; Cambridge University Press:

New York, 2003.

12974 DOI: 10.1021/la101977c Langmuir 2010, 26(15), 12973–12979

Article Carrillo et al.

term describes the increase of the surface energy of the sphericalcap (deformed nanoparticle) with the interfacial energy γp incomparison with that of a sphere with radius Rp. Finally, the lastterm is the elastic energy contribution16 due to deformation of ananoparticle with rigidityK=2G/(1- ν), whereG is the particleshear modulus and ν is the Poisson ratio. Note, that for deforma-tion at a constant volume ν = 0.5 which results in K = 4G.

The equilibrium deformation of the adsorbed nanoparticle isobtained byminimizing nanoparticle free energy eq 1 with respectto Δh. This results in the following equation for the nanoparticledeformation

0 ¼ - 2πWRp þ 2πγpΔhþ7ffiffiffi2

p

6KRp

1=2Δh3=2 ð2Þ

Introducing dimensionless variables y2 ¼ Δh2RpA

KRp

W

� �2=3and

β¼ Bγp

3=2

KRpW1=2

� �2=3

(where A=(3π/7)2/3 and B=2(3π/7)2/3 are

numerical coefficients) we can reduce eq 2 to a simple cubicequation

0 ¼ - 1þ βy2 þ y3 ð3ÞThe deformation of the nanoparticle is described by a positive

root of this equation

Δh

2Rp¼ A

W

KRp

!2=3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiq3 þ r2

p3

qþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir-

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiq3 þ r2

p3

q-β

3

� �2

ð4Þwhere

r ¼ 1

2-

β

3

� �3

, and q ¼ -β

3

� �2

ð5Þ

Note, that the values of the numerical coefficients in the eq 4,A=(3π/7)2/3 and B = 2A = 2(3π/7)2/3, depend on the particularnanoparticle shape and deformation that was used for calculationof the different terms in the nanoparticle free energy (see Appen-dix A). Thus, we expect the values of these coefficients to bemodel dependent. However, we do not expect the form of theequation and scaling relations to change. One can also expressreduced parameters in terms of the shear modulus G. In this casethe values of the numerical coefficients are AG=A((1- ν)/2)2/3=(3π(1 - ν)/14)2/3 and BG = 2AG.

The analysis of eq 3 points out that the dimensionless para-meter β determines the relative strength of the interfacial andelastic energy contributions. For small values of the parameter

β≈ γp(KRp)-2/3W-1/3, 1 the equilibrium nanoparticle height is

determined by balancing the first and the last terms in the rhs ofthe equation eq 3. In this case the penalty for the elastic deforma-tion of the nanoparticle stabilizes its height. This corresponds tothe JKR-scaling14 dependence of the nanoparticle deformationΔh � Rp(W/KRp)

2/3. This is usually the case for large stronglycross-linked nanoparticles. In the opposite limit of large values ofthe parameter β (small weakly cross-linked nanoparticles), thedeformation of the nanoparticle height is proportional to the ratioof the interfacial and surface energies Δh � RpW/γp. This corre-sponds to the droplet wetting regime when interfacial and surfaceenergies determine the equilibrium droplet shape at the substrate.4

To test our model we have performed molecular dynamicssimulations17 of the adhesion of spherical polymeric nanoparti-cles on a solid substrate. In our simulations nanoparticles weremade of cross-linked linear chains. We used a coarse-grainedrepresentation of polymer chains. In this representationmonomerswere modeled by the Lennard-Jones particles with diameter σ.The connectivity of monomers into polymer chains wasmaintained by the FENE potential. The same bond potentialwas used to model cross-links between polymer chains forming ananoparticle. We set the value of the Lennard-Jones interactionparameter for the polymer-polymer pair to εLJ= 1.5kBT (wherekB is the Boltzmann constant and T is the absolute temperature)which corresponds to a poor solvent conditions for the polymerbackbone. The elastic properties of the nanoparticles were variedby changing the density of the cross-links between linear chainsforming a nanoparticle. The values of the shearmodulusG for thenanoparticles with different cross-linking densities Fc were ob-tained by performing simulations of deformation of the 3-Dnetworks with the same polymer density and number density ofthe cross-links.18 The interaction of a solid substrate with mono-mers forming a nanoparticle was modeled by the 3-9 potentialwith the value of the interaction parameter εw. We performedsimulations with εw= 1.5kBT, 2.25 kBT and 3.0 kBT. The explicitform of the interaction potentials, bond potential, and simulationdetails are given in the Appendix B.

Figure 2 shows dependence of the nanoparticle shape on thestrength of the long-range van der Waals interactions with thesubstrate and the particle shear modulus, G. For the stronglycross-linked nanoparticles corresponding to the large values ofthe shear modulus G the increase in strength of the substrate-nanoparticle interactions leads only to small particle deformations.For this range of the interaction parameters and nanoparticle shearmodulus the value of the contact angle θ >90�. The elasticity ofthe nanoparticle leads to the effective surface-phobic interactionsof a nanoparticle with a substrate. The value of the contact angledecreaseswith increasing the strengthof the substrate-nanoparticleinteractions and decreasing the value of the shear modulus.Decrease in the nanoparticle shear modulus lowers the elasticenergy penalty for particle deformation allowing large particleshape deformations at the same strength of the substrate-nanoparticle interactions. Weakly cross-linked nanoparticlesshow high affinity to the substrate which is manifested in smallervalues of the contact angle. In some cases the value of the contactangle could be θ < 90�. Note, that the changes in the particlecontact angle are coupled with changes in the nanoparticle heightand particle contact radius with the substrate. Thus, there exists apeculiar interplay of the interfacial and elastic energies in deter-mining the equilibrium shape of a nanoparticle.

Figure 1. Snapshot of the deformed nanoparticle and schematicrepresentation of approximation of its shape by a spherical capwith radiusR1 (solid line). The shape of the undeformed nanopar-ticle with radius Rp is shown by a dashed line.

(16) Lau, A. W. C.; Portigliatti, M.; Raphael, E.; Leger, L. Europhys. Lett.2002, 60(5), 717–723.

(17) Frenkel, D.; Smit, B.UnderstandingMolecular Simulations; Academic Press:New York, 2002.

(18) Carrillo, J. M. Y.; Dobrynin, A. V. Langmuir 2009, 25(22), 13244–13249.

DOI: 10.1021/la101977c 12975Langmuir 2010, 26(15), 12973–12979

Carrillo et al. Article

Figure 3 shows dependence of the reduced particle deformation(GRp/W)2/3Δh/2Rp on the value of the parameter γp(GRp)

-2/3W-1/3.For this plot the values of the nanoparticle surface energy γp wereobtained by performing simulations of the polymeric networkswith the same value of the cross-linking density, LJ-interactionparameter εLJ=1.5kBT, and average density as those for nanopar-ticles (see Appendix B). The value of the parameter d0=0.867σ andaverage monomer density Fp used for calculations of the work ofadhesion W = εwFpσ3d0-2/2 (see eq A.6) were obtained fromsimulations. The solid line in Figure 3 is the best fit to the eq 4.The agreement between simulation results and theory is a verygood. Thus, our simple scaling model correctly accounts for theeffect of particle adhesion, surface energy and energy of the elasticdeformation in determining the equilibrium particle shape. Note,that the numerical coefficients obtained from the fitting procedureare different from the numerical coefficients in the eq 4. This shouldnot be surprising since we used an approximated expression for theelastic energy of the nanoparticle deformation (see Appendix A)which can explain this discrepancy.Using our simulation results wecan set numerical coefficients in the eq 4 to be equal toAG= 0.419and BG = 0.28 and consider this equation as an interpolation for-mula describing a crossover from JKR to nanoparticle wettingregime. This equation can be used to estimate the adhesion energyby studying the adhesionof nanoparticles of different sizes by fittingthe experimental data for nanoparticle deformation to the eq 4and considering dimensionless parameters γp(GRp)

-2/3W-1/3 and(GRp/W)2/3 as adjustable parameters. In addition one can alsoestablish the boundary of the JKR regime14where the elastic energyplays the dominant role in controlling nanoparticle deformation.The dashed-line in Figure 3 corresponds to the JKR-like nanopar-ticle deformation. It follows from this figure that a crossover to theJKR regime can be estimated to occur at γp ≈ 0.1(GRp)

2/3W1/3.Thus, knowing the material properties one can estimate for whatnanoparticle sizes the deformation of a nanoparticle at a substrate

Figure 2. Dependence of the average shape of a nanoparticle and a contact angle on the value of the nanoparticle shear modulus G and thestrength of the nanoparticle substrate interaction potential εw.

Figure 3. Dependence of the reduced nanoparticle deformationon the value of the parameter γp(GRp)

-2/3W-1/3 for nanoparticleswith different cross-linking densities, different strengths of thesubstrate-polymer interaction parameter εw = 1.5kBT (filledsymbols), 2.25kBT (crossed symbols), 3.0kBT (open symbols) anddifferent nanoparticle sizes:R0= 11.3σ (squares), 16.4σ (rhombs),20.5σ (hexagons), 26.7σ (triangles), 31.8σ (inverted triangles), and36.0σ (stars). (R0 is the sizeof the cavity inwhichananoparticlewasprepared (see Appendix B for details).) The solid line is the best fitto the eq 4 by considering numerical coefficients as fitting para-metersAG= 0.419,BG= 0.28. For this fit we setK=4G becausedeformation of a nanoparticle occurs at almost constant volume(the Poisson ratio ν ≈ 0.5). The dashed line corresponds to JKRlimit of nanoparticle deformation.

12976 DOI: 10.1021/la101977c Langmuir 2010, 26(15), 12973–12979

Article Carrillo et al.

will be controlled by the nanoparticle surface energy, Rp<103/2γp

3/2/W1/2G.One can also derive a scaling relation for the nanoparticle

contact angle. For small deformations the nanoparticle contactangle can be approximated as

cosðθÞ � - 1þΔh

Rpð6Þ

It is interesting to point out, that in the case when the nanoparticleshear modulus approaches zero,Gf 0, solution of the eq 2 reducestoΔh/Rp≈W/γp and eq 6 reproduces the Young-Dupre equation,4

W ≈ γp(1 þ cos(θ)). In Figure 4 we used reduced variables to col-lapse our simulation data into one universal plot.While the collapseof the data is good there is a difference by a factor of 2 from the valueof (GRp/W)2/3(1þcos(θ)) expected from eq 6 and from the plot inFigure 3 for the same values of the parameter γp(GRp)

-2/3W-1/3.This tells us that theactual valueof the contact angle is different fromthe macroscopic value which is obtained by fitting the nanoparticleshapebya spherical cap.This indicates that local forces actingon thecontact line control the value of the contact angle. One can identifytwo different scaling regimes inFigure 4. In the range of small valuesof the parameter γp(GRp)

-2/3W-1/3< 1, we see a saturation ofthe value of (GRp/W)2/3(1þ cos(θ)) which corresponds to the cross-over to the JKR regime with Δh/Rp ≈ (W/GRp)

2/3. However, withincreasing the value of the parameter γp(GRp)

-2/3W-1/3 the datapoints approach a line with slope-1. This represents a crossover tothe nanoparticle wetting regime where the value of the contact angleis given by the Young-Dupre equation.4

We presented results of the molecular dynamics simulationsand analytical calculations and showed that adhesion of ananoparticle to a substrate is a result of the fine interplay betweenparticle elastic, interfacial and adhesion energies. Depending onthe value of the dimensionless parameter γp(GRp)

-2/3W-1/3 ananoparticle deformation canbedescribed either bybalancing theparticle elastic energy with a particle adhesion energy (the JKRregime γp(GRp)

-2/3W-1/3 , 1) or by balancing the nanoparticle

adhesion energy and interfacial energy (the nanoparticle wettingregime γp(GRp)

-2/3W-1/3. 1). We expect that a newmechanismfor particle adhesion will play an important role in understandingcontact mechanics of nanoscale objects, stability of soft nano-particle in solutions, and friction and lubrication of nanostruc-tured surfaces.

Acknowledgment. This work was supported by the Donors ofthe American Chemical Society Petroleum Research Fund underthe Grant PRF # 49866-ND7.

Appendix A: Model of Nanoparticle Adhesion

The equilibrium shape of the particle is a result of optimizationof the nanoparticle surface energy, its interaction with the sub-strate, and the elastic energy due to deformation of the particleshape. To evaluate different contributions to the nanoparticle freeenergy we assume that deformation of the adsorbed nanoparticleoccurs at a constant volume and the deformed particle has shape ofa spherical cap with radius R1 and height h (see Figure 1).

R1 ¼ 4Rp3

3h2þ h

3ðA.1Þ

The surface energy of the spherical cap with interfacial energyγp is equal to

UsðhÞ ¼ γpðπa2 þ 2πR1hÞ ¼ πγpð4R1h- h2Þ ðA.2Þ

where the surface energy include contribution from the nanopar-ticle contact areawith the substratewith radius a=(2R1h- h2)1/2

and the surface energy of the side area of the cap with height h.Evaluation of contribution of the van der Waals interactions ofthe deformed nanoparticle of average density Fp with a substraterequires integration of the long-range van der Walls potential5 ofmagnitude εw over the particle height

UvdwðhÞ � - εwFpσ3π

Z h

0

F2ðzÞðzþ d0Þ3

dz

¼ - εwFpσ3π

Z h

0

ðh- zÞð2R1 - hþ zÞðzþ d0Þ3

dz ðA.3Þ

where F(z) is the radius of the spherical cap at the distance z fromthe adsorbing surface, d0 is the distance of the closest approach.Performing integration we arrive at

UvdwðhÞ � - εwFpσ3π½ð2R1 - hÞh2ðh þ 2d0Þ

2d02ðhþ d0Þ2

þ ðh-R1Þh2ðhþ d0Þ2d0

- logh

d0þ 1

� �-ð4hþ 3d0Þd02ðhþ d0Þ2

þ 3

2� ðA.4Þ

For the systems with only van der Waals interactions one canrelate the surface and interfacial energies to the systemdensity andstrength of the van der Waals interactions. In this case surfaceenergy of the polymer-substrate interface γps is equal to

γps ¼ γp þ γs -εwFpσ

2

2d02

ðA.5Þ

where γp is the polymer surface energy and γs is the substratesurface energy. Using eq A.5 we can relate the parameters of the

Figure 4. Dependenceof the reducedvalueof thecosineof thenano-particle contactangleonthevalueof theparameterγp(GRp)

-2/3W-1/3

for nanoparticles with different cross-linking densities, differentstrengths of the substrate-polymer interaction parameter εw anddifferent nanoparticle sizes. Notations are the same as in Figure 3.The dashed lines show two different asymptotic regimes.

DOI: 10.1021/la101977c 12977Langmuir 2010, 26(15), 12973–12979

Carrillo et al. Article

van der Waals interactions with the work of adhesion5

W ¼ γp þγs - γps ¼ εwFpσ2

2d02

ðA.6Þ

The final contribution to the nanoparticle free energy is due toelastic energy of the particle deformation. Unfortunately, evenwiththe simplifying assumptions of a spherical cap deformation, arigorous computation of the displacement field for such deforma-tion is a difficult task. Thus, we will apply a simplified approachdeveloped in ref 16 toobtain the stress and the displacement fields inthe deformed nanoparticle. The stress exerted by a deformednanoparticle on the surface of semi-infinite half space is

σðFÞ ¼ ~σ0ð1-F2=a2Þ- 1=2 þ ~σ1ð1-F2=a2Þ1=2 ðA.7ÞThedisplacement field of the nanoparticlewhich generate this stressis equal to

uzðFÞ ¼ πa

K~σ0 þ

~σ1

21-

F2

2a2

!0@

1A ðA.8Þ

where σ~0 = K(Δh/a - a/Rp)/π, σ~1 = K(2a/Rp)/π, and K = 2G/(1 - ν) is the nanoparticle rigidity. Taking this into account theelastic energy of the nanoparticle is estimated as follows

UelðhÞ ¼ 1

2

Zd2xuzðxÞσðxÞ

¼ K Δh2a-2

3

Δha3

Rpþ 1

5

a5

Rp2

" #ðA.9Þ

The equilibrium nanoparticle height is obtained from mini-mization of the total particle free energy

FðhÞ ¼ UsðhÞþUvdwðhÞþUelðhÞ ðA.10Þwith respect to the nanoparticle height h. The scaling analysis ofthe nanoparticle deformation can be performed in the limit ofsmall particle deformations for whichΔh/Rp, 1. In this limit thenanoparticle free energy reduces to

FðΔhÞ� - 2πWRpΔhþπγpΔh2 þ 7

ffiffiffi2

p

15KRp

1=2Δh5=2 ðA.11Þ

Appendix B: Simulation Details

We have performed molecular dynamics simulations of theadhesion of a spherical nanoparticle to a substrate. The nano-particlewasmadeofNch-chainswith the degree of polymerizationN = 32. The chains were cross-linked at different cross-linkingdensities in order to control the nanoparticle elastic modulus. Inour coarse-grained molecular dynamics simulations the interac-tions betweenmonomers forming a nanoparticleweremodeled bythe truncated-shifted Lennard-Jones (LJ) potential19

ULJðrijÞ ¼ 4εLJσ

rij

!12

-σ

rij

!6

-σ

rcut

� �12

þ σ

rcut

� �624

35 re rcut

0 r > rcut

8><>:

ðB.1Þwhere rij is the distance between ith and jth beads and σ is thebead (monomer) diameter. The cutoff distance for the polymer-polymer interactions was set to rcut=2.5σ and the value of theLennard-Jones interaction parameter was equal to 1.5 kBT. This

choice of parameters corresponds to poor solvent conditions for thepolymer chains.

The connectivity of the beads into polymer chains and thecross-link bonds were maintained by the finite extension non-linear elastic (FENE) potential

UFENEðrÞ ¼ -1

2kspringRmax

2 ln 1-r2

Rmax2

!ðB.2Þ

with the spring constant kspring=30kBT/σ2 and the maximum

bond lengthRmax=1.5σ. The repulsive part of the bond potentialis given by the truncated-shifted LJ potential with rcut=(2)1/6σand εLJ= kBT. In our simulations we have excluded 1-2 LJ-interactions such that monomers connected by a bond onlyinteracted through the bonded potential.

The presence of the substrate was modeled by the externalpotential

UðzÞ ¼ εw2

15

σ

z

� �9

-σ

z

� �3" #

ðB.3Þ

where εw was equal to 1.5 kBT, 2.25 kBT, and 3.0 kBT. The long-range attractive part of the potential z-3 represents the effect ofvan der Waals interactions generated by the substrate half-space.The systemwas periodic inx and ydirectionswith dimensionsLx=Ly = Lz = 4R0 where R0 is the radius of a cavity for nanoparticlepreparation (see discussion below).

Simulations were carried out in a constant number of particlesand temperature ensemble. The constant temperature was main-tained by coupling the system to a Langevin thermostat imple-mented inLAMMPS.19 In this case, the equation ofmotion of theith particle with mass m is

mdvBiðtÞdt

¼ FBiðtÞ- ξνBiðtÞþFBR

i ðtÞ ðB.4Þ

where νBi(t) is the ith bead velocity and FBi(t) is the net deterministicforce acting on ith bead. FBi

R(t) is the stochastic force withzero average value ÆFBi

R(t)æ = 0 and δ-functional correlationsÆFBi

R(t)FBiR(t0)æ=6ξkBTδ(t- t0). The friction coefficient ξwas set to

ξ=m/τLJ, where τLJ is the standardLJ-time τLJ= σ(m/εLJ)1/2. The

velocity-Verlet algorithmwith a time stepΔt=0.01τLJwas used forintegration of the equations of motion (eq B.4). All simulationswere performed using LAMMPS.19

The simulations began with creating cross-linked nanoparti-cles. To create a nanoparticle we first placed Nch noninteractingchains inside a cavity with radius R0 made of bead-like over-lapping particles with diameter σ. The radius of the sphericalcavity, R0, and the system sizes are listed in Table 1.

The interaction between beads forming a cavity and polymerswaspure repulsivewith rcut= (2)1/6σand εLJ=kBT.During the ini-tial simulation run lasting t=104 τLJwe set thediameter of thebeadfor polymer-polymer interactions to zero such that chains couldeasily pass through each other and equilibrate their conforma-tions. This followed by the simulation runwith duration t=104 τLJduring which we increased the bead diameter for polymer-polymer interactions from zero to σ. After the growth of the beadswas completed the system was equilibrated for additional t = 104

τLJ (see Figure 5a).The next step involved cross-linking of polymer chains within a

cavity. During this procedure neighboring monomers were ran-domly cross-linked by the FENE bonds if they are within 1.5 σcutoff distance from each other. One cross-link bond was allowedper eachmonomer, thus creating a randomly cross-linked network.The elastic properties of the network were varied by changingthe number of cross-links per particle. After completion of the(19) Plimpton, S. J. Comput. Phys. 1995, 117(1), 1–19.

12978 DOI: 10.1021/la101977c Langmuir 2010, 26(15), 12973–12979

Article Carrillo et al.

cross-linkingprocess the confining cavitywas removedanda systemwas relaxed by performing NVT molecular dynamics simulationrun lasting 104 τLJ. This completed the nanoparticle preparationprocedure (see Figure 5b). The nanoparticle size Rp was obtainedfrom themean-square average value of the particle radius of gyra-tion,Rp=(5ÆRg

2æ/3)1/2. The equilibriumnanoparticle sizes, averagemonomer and cross-linking densities are summarized in Table 1.

Modeling of the nanoparticle adhesion to the substrate wasinitiated with placement of the nanoparticle center of mass atdistance z=R0 þ σ from the substrate. After that the system wasequilibrated for t=2� 104 τLJ and the final 104 τLJ were used forthe data collection. Figure 5c shows the snapshot of the adsorbednanoparticle at the substrate.

The values of the nanoparticle shearmodulus were obtained byusing interpolation expression for G as a function of the cross-linking density18

Gσ3

kBT¼ aðFcσ3Þ2 þ bFcσ

3 ðB.5Þ

with the numerical coefficients a = 3.98 and b = 0.38.A surface energy of a nanoparticle interface was evaluated by

performing molecular dynamics simulations of the 3-D periodicsystem of a slab of a polymeric network with the same set of theLJ-interaction and bond parameters, monomer and cross-linkingdensity as our nanoparticles. For these simulations the box sizewas 14.537 σ � 14.537 σ � 29.074σ and the total number of

monomers was equal to 3072 (96 chains with the degree ofpolymerization 32). The simulation runs continued for 2� 104

τLJ with the last 104 τLJ used for the data analysis. The surfaceenergy then was obtained by integrating the difference of thenormal PN(z) and tangential PT(z) to the interface componentsof the pressure tensor across the interface. Note, that in our

Table 1. System Sizes and Nanoparticle Parameters

R0 [σ] Nch Rp [σ] Fpσ 3 Fcσ3 G [kBT/σ3] R0 [σ] Nch Rp [σ] Fpσ3 Fcσ3 G [kBT/σ

3]

11.304 161 10.904 0.968 0.043 0.023 26.740 2127 25.687 0.963 0.042 0.02311.304 161 10.847 0.980 0.115 0.096 26.740 2127 25.588 0.973 0.114 0.09611.304 161 10.802 0.990 0.189 0.214 26.740 2127 25.494 0.983 0.188 0.21211.304 161 10.758 1.002 0.265 0.379 26.740 2127 25.397 0.994 0.263 0.37511.304 161 10.714 1.014 0.342 0.597 26.740 2127 25.304 1.005 0.340 0.58911.304 161 10.670 1.023 0.421 0.864 26.740 2127 25.213 1.016 0.418 0.85516.400 491 15.775 0.966 0.043 0.023 31.836 3590 30.579 0.962 0.042 0.02316.400 491 15.710 0.976 0.115 0.096 31.836 3590 30.465 0.972 0.114 0.09616.400 491 15.644 0.987 0.189 0.213 31.836 3590 30.349 0.984 0.188 0.21216.400 491 15.585 0.998 0.264 0.378 31.836 3590 30.237 0.994 0.263 0.37616.400 491 15.522 1.009 0.341 0.593 31.836 3590 30.131 1.004 0.340 0.58816.400 491 15.464 1.019 0.420 0.860 31.836 3590 30.020 1.015 0.418 0.85520.536 964 19.737 0.964 0.042 0.023 35.973 5179 34.549 0.962 0.042 0.02320.536 964 19.659 0.975 0.115 0.096 35.973 5179 34.420 0.972 0.114 0.09620.536 964 19.582 0.986 0.188 0.213 35.973 5179 34.293 0.983 0.188 0.21220.536 964 19.508 0.996 0.264 0.377 35.973 5179 34.171 0.993 0.263 0.37520.536 964 19.437 1.007 0.341 0.591 35.973 5179 34.047 1.004 0.340 0.58820.536 964 19.367 1.016 0.418 0.855 35.973 5179 33.925 1.014 0.418 0.853

Figure 5. Snapshots of the simulation box during (a) equilibration of chains inside a cavity, (b) relaxation of cross-linked nanoparticle, and(c) adhesion of nanoparticle to a substrate.

Figure 6. Dependence of the polymer surface energy on polymerdensity. The line is given by the equation y= 1128.5- 3299.5xþ3204.42 x2 - 1031.1 x3.

DOI: 10.1021/la101977c 12979Langmuir 2010, 26(15), 12973–12979

Carrillo et al. Article

simulations the z direction was normal to the interface.

γ ¼Z ξ

- ξðPNðzÞ-PTðzÞÞ dz ðB.6Þ

where 2ξ is the thickness of the interface that was determinedfrom the monomer density profile as an interval within which the

monomer density changes from zero to the bulk value. Figure 6show dependence of the surface energy as a function of thepolymer density. Note that a cross-linking density is relatedto polymer density (see Table 1). The surface energy increaseswith increasing the number of cross-links. This is due to increaseof the average monomer density with increasing the cross-linkingdensity.