Contact Mechanics of Nanoparticles: Pulling Rigid Nanoparticles from

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Contact Mechanics of Nanoparticles: Pulling Rigid Nanoparticles from Soft, Polymeric Surfaces Zhen Cao and Andrey V. Dobrynin* Department of Polymer Science, University of Akron, Akron, Ohio 44325-3909, United States ABSTRACT: Detachment of rigid nanoparticles from soft, gel-like polymeric surfaces is studied by using a combination of the molecular dynamics simulations and theoretical calculations. Simulations show that detachment of nanoparticles from soft surfaces proceeds through a neck formation. Analysis of the simulation results demonstrates that the magnitude of the detachment force f* depends on the nanoparticle radius Rp, shear modulus of substrate Gs, surface tension of substrate γs, and work of adhesion W. It is controlled by the balance of the elastic energy, surface energy of the neck, and nanoparticle adhesion energy to a substrate and depends on the dimensionless parameter δ ∝ γs(GsRp)−1/3W−2/3. In the case of small values of the parameter δ ≪ 1, the critical detachment force approaches a critical detachment force calculated by Johnson, Kendall, and Roberts for adhesive contact, f * = 1.5πWRp. However, in the opposite limit, corresponding to soft substrates, for which δ ≫ 1, the critical detachment force 1/2 −1/2 1/2 −1/2 −1.89 f * ∝ γ3/2 . All simulation data can be described by a scaling function f * ∝ γ3/2 δ . s Rp Gs s R p Gs this technique was used to study cell elasticity,12−14 nanoparticle-cell interactions,15−17 and interactions between nanoparticles and elastic substrates.18−23 The analysis of experimental data to extract system’s elastic and surface properties is usually done in the framework of JKR or DMT − like models.15 However, it was demonstrated that application of the macroscopic models to describe micro- and nanocontact problems is questionable.24−26 Experimental, simulation, and theoretical studies26−32 of the interactions between nanoparticles and substrates have pointed out that the JKR model fails to describe the results. It was proven that in order to correctly describe contact phenomena at micro- and nanoscales, one has to take into consideration variation of the surface free energy outside the contact area upon deformation. For example, in the case of a soft nanoparticle in contact with a rigid substrate, one has to account for change in the surface free energy of nanoparticle due to its shape deformation as it comes in contact with a substrate.24 The classical JKR model of the adhesive contact fails when the elastocapillary number, γp/Gpa (where γp is the surface free energy of nanoparticle with shear modulus Gp and having radius, a, of contact), becomes larger than unity, γp/Gpa > 1. Note that the elastocapillary number, γp/Gpa, is proportional to the ratio of the change in the surface free energy of nanoparticles to its elastic energy of deformation upon contact. For JKR contact at zero external force, the contact radius a is a function of the work of adhesion W and scales with the system parameters as a ∝ (WR2p/Gp)1/3. This results in elastocapillary number for a nanoparticle with size Rp in contact with a substrate to be γp/(GpRp)2/3W1/3.

1. INTRODUCTION Understanding contact phenomena between deformable bodies is an important area of research with tremendous technological implications.1−6 The first solution of the contact problem was obtained by Hertz in 1882,7 who showed that the deformation of the two elastic spheres in contact scales with the magnitude of the applied force as f 2/3. Weaker than linear dependence of the deformation on the load force is a manifestation of the fact that, with increasing deformation, the area of contact increases as well, and therefore the restoring force becomes a nonlinear function of produced indentation. In the 1970s, the Hertz’s solution of the contact problem was extended by Johnson, Kendall and Roberts (JKR),8 and Derjaugin, Muller, and Toporov (DMT)9 to account for dispersion (van der Waals) forces. These forces are acting between two bodies in contact, even in the case of zero loading force, f, resulting in a finite contact area. These two models of the adhesive contact describe two limiting cases of deformation of large compliant particles (JKR) and small rigid particles (DMT) in a contact with an elastic half space. The crossover between these two solutions of the contact problem is governed by the so-called Tabor’s parameter10 defined as a ratio of the elastic deformation to the characteristic length scale of dispersion forces. The closed form solution of the adhesive contact problem covering both the JKR and DMT regimes was developed by Maugis.11 He has applied a Dugdale potential by approximating the Lennard-Jones potential as a square well. This model describes a crossover between JKR and DMT regimes as a function of the Tabor parameter. In recent years, there has been a rejuvenation of interest to contact phenomena due to a widespread application of atomic force microscopy (AFM) to study mechanical and surface properties of soft matter and biological systems. In particular, © 2015 American Chemical Society

Received: August 28, 2015 Revised: October 28, 2015 Published: October 28, 2015 12520

DOI: 10.1021/acs.langmuir.5b03222 Langmuir 2015, 31, 12520−12529

Article

Langmuir Although the limitations of the JKR model to describe contact at micro- and nanoscales have been demonstrated, it is still widely used in analysis of the AFM data of soft and biological materials. To demonstrate the failure of the JKR model to describe detachment of a rigid nanoparticle from an elastic substrate, the most commonly used geometry in AFM set up to study soft and biological systems, in this paper we use a combination of the molecular dynamics simulations and theoretical calculations to elucidate factors determining interactions between rigid nanoparticles with soft gel-like polymeric substrates. The rest of the paper is organized as follows. In the next section, we present simulation results. In section 3, we develop a theoretical model of contact of rigid particles with soft substrates and compare model predictions with simulation data.

2. SIMULATION RESULTS To study detachment of rigid nanoparticles from gel-like (soft) polymeric substrates, we have performed coarse-grained molecular dynamics simulations33 of rigid nanoparticles in contact with soft surfaces (see Figure 1). Rigid nanoparticles

Figure 2. Snapshots of substrate deformation during nanoparticle detachment.

Figure 1. Snapshots of a rigid nanoparticle and an elastic substrate (a) before indentation, (b) in equilibrium contact, and (c) during the detachment process.

with radius Rp = 17.9σ were modeled by spherically shaped assembly of Lennard-Jones beads of diameter σ arranged in a hexagonal closed-packed (HCP) lattice and connected by the springs. The gel-like polymeric substrates were prepared by cross-linking bead−spring chains34 with the number of monomers N = 32. The elastic modulus of the substrate was varied by changing the number of cross-links between chains. The strength of pairwise interactions between beads forming substrates and nanoparticles was controlled by changing the value of the Lennard-Jones interaction parametr, εLJ. Simulation details are described in the Simulation Methods section below. Figure 2 shows snapshots of the typical nanoparticle/ substrate configurations during nanoparticle pulling from a substrate. For soft surfaces, we see a well-developed neck as the nanoparticle detaches from the substrate. This should not be surprising since for soft substrates the elastic energy penalty for deformation of the polymeric strands in the neck decreases with decreasing the substrate shear modulus, Gs, and the main contribution to the free energy of the system comes from the surface free energy of the neck. Also the neck height increases with increasing the affinity between nanoparticle and substrate (increasing value of the Lennard-Jones interaction parameter, εsp). Favorable nanoparticle/substrate interactions offset the elastic energy penalties associated with formation of the elongated neck. For more rigid substrates, our simulations show the formation of short necks connecting the nanoparticle with the substrate. To quantify the nanoparticle detachment process in Figure 3, we show results of the potential of the mean force calculations for interactions of nanoparticles with soft, Gs = 0.072kBT/σ3,

Figure 3. Potential of mean force as a function of displacement of nanoparticle center of mass along the z-axis for soft (Gs = 0.072kBT/ σ3) and hard (Gs = 0.833kBT/σ3) substrates with value of interaction parameter for nanoparticle/substrate pairs εsp = 1.20 kBT. Insets show dependence of the contact radius, a, on the magnitude of the pulling force, f, obtained from differentiation of the potential of mean force.

and hard, Gs = 0.833kBT/σ3, substrates. In both cases, we see that the neck thickness at nanoparticle contact gradually 12521


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Figure 4. Dependence of the normalized contact radius, a/Rp, on normalized nanoparticle separation distance, Δh/Rp, for nanoparticles interacting with substrates with shear modulus Gs = 0.024 kBT/σ3 (black squares), Gs = 0.072 kBT/σ3 (red circles), Gs = 0.162 kBT/σ3 (blue triangles), Gs = 0.252 kBT/σ3 (yellow inverted triangles), Gs = 0.498 kBT/σ3 (pink pentagon), Gs = 0.833 kBT/σ3 (cyan diamonds), and two different values of the nanoparticle/substrate interaction parameter εsp = 1.2kBT (a) and εsp = 0.75kBT (b). Inset: Schematic representation of nanoparticle in contact with substrate and definition of length scales.

Figure 5. Dependence of the normalized contact radius a/a0 (a0 is contact radius at zero force) on the normalized pulling force f/f JKR ( f JKR = 1.5πWRp, where W is the work of adhesion) for interaction of nanoparticles with substrates with shear modulus Gs = 0.024 kBT/σ3 (black squares), Gs = 0.072 kBT/σ3 (red circles), Gs = 0.162 kBT/σ3 (blue triangles), Gs = 0.252 kBT/σ3 (yellow inverted triangles), Gs = 0.498 kBT/σ3 (pink pentagon), Gs = 0.833 kBT/σ3 (cyan diamonds) and two different values of the nanoparticle/substrate interaction parameter εsp = 1.2kBT (a) and εsp = 0.75kBT (b). The solid lines correspond to predictions of the JKR model.

strength of the nanoparticle/substrate interaction parameter, εsp. Displacement of nanoparticle results in gradual decrease of the contact radius followed by a jump at critical Δh* when contact between nanoparticle and substrate disappears. The abrupt transition seen in Figure 4 is a signature of a first-order transition. Usually nanoparticle detachment is analyzed in applied force−contact radius coordinates. In Figure 5 we plot the dependence of the reduced nanoparticle contact radius, a/a0, as a function of the reduced applied force. The value of contact radius a0 corresponds to the minimum of the potential of the mean force (zero external force). For normalization of the applied force, we choose the critical detachment force calculated in the framework of the JKR model, f JKR = 1.5πWRp,35 where W is the work of adhesion. According to the JKR model, the normalized force scales with the contact radius as

decreases with pulling nanoparticle from substrate. Insets in Figure 3 show dependence of the contact radius, a, on the magnitude of the pulling force, f. The magnitude of the pulling force was obtained by differentiating potential of the mean force with respect to location of the nanoparticle center of mass. The differentiation of the potential of mean force was performed by fitting the potential by a quartic function and differentiating the obtained fitting curve. Note that by differentiating potential of the mean force, we have been able to obtain evolution of the contact radius, a, on both stable and unstable branches. Along the stable branch, the derivative of the contact radius with respect to force is negative, ∂a/∂f < 0, but on the unstable branch, it is positive, ∂a/∂f > 0. The critical force, corresponding to nanoparticle detachment, is determined by the divergence of the derivative, ∂a/∂f = ∞. Figure 4 shows dependence of the normalized contact radius a/Rp on normalized nanoparticle indentation or separation Δh/ Rp with respect to undeformed substrate surface. For nanoparticles interacting with soft substrates, the radius of indentations produced by nanoparticle in substrates is approaching the size of nanoparticle. Increasing substrate rigidity makes indentations in the substrate shallow at the same

f f JKR 12522

⎛ a ⎞3 ⎛ a ⎞3/2 = −4⎜ ⎟ + 4⎜ ⎟ ⎝ a0 ⎠ ⎝ a0 ⎠

(1) DOI: 10.1021/acs.langmuir.5b03222 Langmuir 2015, 31, 12520−12529

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related by the following expression: Δ2 + a2 = 2RpΔ. Expression eq 2 for surface free energy change due to nanoparticle contact with substrate can be rewritten in terms of a, Δ, and Δh:

Analysis of the data shown in Figure 5 indicates that the critical detachment force does not follow the JKR model predictions and is a strong function of the substrate modulus and strength of the nanoparticle/substrate interactions. These figures also point out that it is not sufficient to consider adhesive contact between nanoparticle and substrate in the physical contact zone, as it is done in the JKR model, to correctly predict critical detachment force and its dependence on the materials parameters. It is important to point out that similar trends were observed in simulations of detachment of soft nanoparticles from rigid substrates.26

ΔFsurf (a , Δ, Δh) = 2πγsa(Δ + Δh) + πγsΔ2 − 2πWR pΔ (4)

where we introduced work of adhesion W = γs + γp − γsp. Note that, in the limit of zero neck height, hn = 0, Δ = −Δh, eq 4 reproduces the expression of surface free energy change due to nanoparticle indentation (see ref 29). Elastic Energy. In addition to surface free energy contribution, there is also elastic energy contribution due to substrate deformation. To evaluate elastic energy of the deformed substrate, we will use a variational approach and approximate stress generated in the contact between particle and substrate by a function of the following form:35

3. MODEL OF A NANOPARTICLE IN CONTACT WITH A SOFT SURFACE Analysis of the snapshots of nanoparticle and substrate during the nanoparticle detachment process (see Figures 2 and 3) shows that nanoparticle detachment proceeds through the formation of a neck connecting the nanoparticle with a substrate. In our model of the nanoparticle/substrate interactions, this feature of nanoparticle/substrate contact will be taken into account explicitly. Consider a rigid nanoparticle with radius Rp having surface tension (free energy) γp in contact with an elastic substrate of initial area A and surface free energy γs. The center of mass of the nanoparticle is located at distance H from the substrate surface as shown in Figure 6. The particle

1/2 −1/2 ⎛ ⎛ ρ2 ⎞ ρ2 ⎞ σ(ρ) = σ0̃ ⎜1 − 2 ⎟ + σ1̃ ⎜1 − 2 ⎟ a ⎠ a ⎠ ⎝ ⎝

where ρ is a distance from vertical z axis (see Figure 6). This stress distribution corresponds to a parabolic displacement field of the substrate surface in the contact with nanoparticle uz(ρ) =

uz(ρ) = H −

is connected to a substrate by a neck of catenoid-like shape with height hn. The catenoid has a contact radius a1 with a particle and contact radius a2 with a substrate. Surface Free Energy. The change of the surface free energy of the system due to nanoparticle contact with a substrate is equal to

(6)

R p2 − ρ2 ≈ Δh +

ρ2 2R p

(7)

Comparing eqs 6 and 7, we obtain σ0̃ =

ΔFsurf (a1 , a 2 , hn) = γs(Sneck + A − πa 22) + γsp2πR pΔ

K ⎛⎜ Δh a ⎞⎟ + π ⎜⎝ a R p ⎟⎠

and

σ1̃ = −

2Ka πR p

(8)

Taking this into account, the elastic energy of the deformed substrate can be estimated as follows

(2)

where γsp is the surface tension of the substrate/nanoparticle interface, Δ = hn − Δh is indentation produced by nanoparticle in the neck, and Sneck is the surface area of a neck connecting nanoparticle to substrate. The surface area of a catenoid is equal to ⎡h ⎛ 2h ⎞⎤ 1 Sneck(a1 , hn) = πa12⎢ n + sinh⎜ n ⎟⎥ ⎢⎣ a1 2 ⎝ a1 ⎠⎥⎦

⎛ ρ 2 ⎞⎞ πa ⎛ ⎜⎜σ0̃ + σ1̃ ⎜1 − 2 ⎟⎟⎟ K⎝ 2a ⎠⎠ ⎝

where K = 2Gs/(1 − v) is the substrate rigidity, Gs is the substrate shear modulus, and v is the Poisson ratio. Note that for deformation of polymeric gels with Poisson ratio v = 0.5, the substrate rigidity K = 4Gs. The parameters for the displacement field σ̃0 and σ̃1 can be expressed in terms of neck height Δh and contact radius a. The displacement of the substrate surface along the z-direction normal to the substrate surface in the limit of small deformations is given by

Figure 6. Schematic representation of a rigid particle in contact with an elastic substrate.

+ γp(4πR p2 − 2πR pΔ) − Aγs − 4πR p2γp

(5)

Uel =

1 2

∫ d2xuz(x)σ(x) =

π 2a3 ⎡ 2 2 2 2⎤ σ1̃ ⎥ ⎢⎣σ0̃ + σ0̃ σ1̃ + K 3 15 ⎦

⎡ ⎤ 2 Δha3 1 a5 ⎥ = K ⎢Δh2a + + 2 ⎢⎣ 3 Rp 5 R p ⎥⎦ (9)

(3)

Total Free Energy. The change of the total free energy of the system due to the contact of a nanoparticle and an elastic substrate includes both elastic (eq 9) and surface (eq 4) free energy contributions. Combining together eqs 4 and 9, we arrive at the following expression for small substrate deformations:

Note that for catenoid surface a2 = a1 cosh(hn/a1). At small nanoparticle/substrate separations such that hn ≪ a1, we can set a = a1 ≈ a2. In this approximation, the surface area of catenoidal neck can be approximated by that of a cylinder, Sneck ≈ 2πahn. Note that a and Δ are not independent and are 12523


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Langmuir ΔFtotal(a , Δ, Δh) ≈ 2πγsa(Δ + Δh) + πγsΔ2 − 2πWR pΔ ⎡ ⎤ 2 Δha3 1 a5 ⎥ + 4GS⎢Δh2a + + ⎢⎣ 3 Rp 5 R p2 ⎥⎦

(10)

For small indentations, we can approximate a2 ≈ 2RpΔ. In this approximation, eq 10 transforms to ⎛ a2 ⎞ a4 ΔFtotal(a , Δh) ≈ 2πγsa⎜⎜ + Δh⎟⎟ + πγs 2 − πWa 2 4R p ⎝ 2R p ⎠ ⎡ ⎤ 2 Δha3 1 a5 ⎥ + 4Gs⎢Δh2a + + ⎢⎣ 3 Rp 5 R p2 ⎥⎦ (11)

This equation is similar to the one derived for detachment of soft nanoparticle from rigid substrates (see ref 26). Following the approach developed in this paper, it is useful to introduce dimensionless variables:

Figure 7. Dependence of the dimensionless free energy ΔF̂total on the normalized radius of contact area, â, calculated for δ = β = 0.1 and different values of the nanoparticle−substrate separation, Δĥ: Δĥ = 0.01 (black line), Δĥ = 0.1 (red line), Δĥ = 0.2 (blue line), and Δĥ = 0.4 (green line).

⎛ 9f 2 ⎞1/3 ⎛ 3f JKR R p ⎞1/3 JKR ⎟ ⎜ ̂ ⎟⎟ a ̂, Δh = a = ⎜⎜ ⎜ 16G 2R ⎟ Δh , 4 G ⎝ s ⎠ s p⎠ ⎝ γs γs , β=B , δ=D 1/3 2/3 2/3 1/3 (GsR p) W (GR p) W ΔFtotal =

1/3 2/3 37/3π 2/3 f JKR R p W ̂ ΔFtotal 4 Gs 2/3

simulations of dynamics of soft nanoparticle detachment from rigid surfaces.32 The value of the contact radius â as a function of the nanoparticle−substrate separation Δĥ minimizing free energy eq 13 is obtained by solving equation ∂ΔF̂total(â, Δĥ)/∂â = 0. 4 δ(3a 2̂ + 2Δh)̂ + βa 3̂ − a ̂ + [Δh ̂ + a 2̂ ]2 = 0 (14) 9 Solution of this equation is easy to write for Δĥ as a function of â

(12)

where f JKR = 1.5πWRp is the critical detachment force in the JKR model. D = (π/81)1/3 and B = 0.5 × (π/3)2/3 are numerical coefficients. In these new variables, eq 12 can be rewritten as ̂ (a ̂, Δh)̂ = δ(a 3̂ + 2Δhâ )̂ + 1 βa 4̂ − 2 a 2̂ ΔFtotal 4 9 ⎡ 2̂ 2 ̂ 3 1 5⎤ + ⎢Δh a ̂ + Δha ̂ + a ̂ ⎥ ⎣ 3 5 ⎦

Δh ̂ = −δ +

δ 2 − δa 2̂ − βa 3̂ +

4 a ̂ − a 2̂ 9

(15)

This solution is plotted in Figure 8a, where solid lines show the interval of contact radius where this solution corresponds to a minimum of the free energy at the interval â ≥ 0. Dash-dotted lines show the location of the first-order transition, and dotted lines represent an interval of nanoparticle substrate separations where solution with finite contact radius, â > 0, is metastable. This is in agreement with the abrupt nanoparticle detachment transition observed in Figures 4a,b. In Figure 8b, we compare analytical solution (see eq 15) and simulation data. To account for small gel shape deformations outside contact area and improve agreement with simulation data, we introduce a numerical correction factor A renormalizing values of parameters β → Aβ and δ → Aδ in eq 15. The results of fitting procedure considering A as an adjustable parameter are shown in Figure 8b by lines. The value of the fitting parameter A is equal to 0.85. The largest deviation between simulation data and eq 15 is observed close to nanoparticle detachment point. This could be explained by a nonlinear network deformation in the neck region. Analysis of the eq 15 shows that there are two asymptotic scaling regimes for the dependence of the contact radius â. In the limit of small values of the parameters δ ≪ 1 and β ≪ 1, the equilibrium contact radius is obtained by balancing the work of adhesion and elastic energy term resulting in 2 Δh ̂ ≈ a1/2 ̂ − a 2̂ (16) 3

(13)

In our simulations, parameter β was varied between 0.14 and 1.5, and parameter δ assumed values between 0.17 and 0.7. Figure 7 shows evolution of the free energy eq 13 for different nanoparticle separations from the substrate, Δĥ. For small separations, Δĥ, this function has a minimum at finite â. However, as the separation between nanoparticle and substrate increases, this minimum becomes shallow. If the distance between nanoparticle and substrate increases further, the function eq 13 has a minimum at â = 0 at the boundary of the interval â ≥ 0, while the solution with finite â corresponding to nanoparticle in contact with substrate becomes metastable. Finally, the solution with finite â becomes unstable and nanoparticle detaches from the substrate. Thus, the form of the free energy function eq 13 indicates that detachment of nanoparticle from the substrate is a first-order transition such that two states with â = 0 and finite â could coexist. However, the actual detachment of nanoparticle from the substrate seen in experiments or simulations will depend on the height of the barrier which separates a state corresponding to nanoparticle in contact with substrate and that corresponding to the detached nanoparticle. For energy barriers larger than kBT, the nanoparticle will be trapped in a metastable state, and detachment will occur when the state with finite contact radius, â, becomes unstable. This is exactly what was observed in 12524


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Figure 8. (a) Dependence of normalized radius of contact area, â, on normalized nanoparticle separation distance, Δĥ, for different values of the parameter δ and β. δ = β = 0.0 (black line), δ = β = 0.01 (red line), δ = β = 0.1 (blue line), δ = β = 0.2 (pink line), δ = β = 0.35 (yellow line), δ = β = 0.5 (cyan line), and δ = β = 1.0 (purple line). Solid lines correspond to globally stable states with â ≥ 0, dot lines correspond to metastable states, and dash-dotted lines show location of a first order detachment transition. (b) Comparison between simulation data and analytical calculations of â and Δĥ for systems with δ = 0.212, β = 0.274 (yellow inverted triangles), δ = 0.184, β = 0.190 (pink pentagon), δ = 0.167, β = 0.145 (cyan diamonds). Lines correspond to eq 15, where we substituted β → Aβ and δ → Aδ with A = 0.85 (see text for details).

Figure 9. Landscape of function Γ(â, Δĥ, f)̂ at different values of the applied external force, f,̂ for nanoparticles in contact with substrates having values of the dimensionless parameters δ = 0.239 and β = 0.358 (left column), and δ = β = 0 (JKR regime) (right column).

This corresponds to JKR solution of the contact problem (JKR regime). In the opposite limit of large values of the parameter δ ≫ 1, the equilibrium contact radius is determined by balancing the work of adhesion and surface energy of the neck

Δh ̂ ≈

β 2 â 3 − a 2̂ − a 3̂ δ 9δ 2

(17)

This is the so-called Necking regime. Taking the derivative of Δh with respect to contact radius a, we can find the location of the critical point where solution 12525


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force corresponding to the minimum of the eq 22, ∂f/∂a = 0, is equal to f* ≈ 1.5πWRp. In this interval of parameters, the contact radius at the detachment point is equal to â* ≈ 2−2/3. In the opposite limit of the large values of the parameters δ ≫ 1 and β ≫1 (Necking regime), taking the derivative of eq 22 with respect to contact radius a, we obtain for the contact radius at the detachment point â* ≈ (δ/2)1/2 and for the 1/2 −1/2 detachment force f * ∝ γ3/2 . Note that these results are s Rp Gs similar to the ones derived in ref 26 for pulling of the soft nanoparticle from the rigid substrate if one substitutes shear modulus Gs and surface tension γs of substrates by those of soft nanoparticles. Figure 10 verifies the scaling relation for critical detachment force f * as a function of the parameter δ. In this figure, we have

with finite contact radius becomes unstable. In two limits (see eqs 16 and 17), we obtain the following scaling relations for contact radius a* a*JKR ∝ W 1/3R p2/3/Gs1/3

and

* ∝ WR p/γ aneck s

(18)

and Δh* at nanoparticle detachment Δh*JKR ∝ W 2/3R p1/3/Gs2/3

and

* ∝ W 2R p/γ 2 Δhneck s (19)

Therefore, crossover between these two asymptotic regimes takes place for nanoparticles with size R̃ *p ∝ γ3s /W2Gs. External Force Dependence. Fixing the position of the nanoparticle at a certain distance Δh from the substrate surface is equivalent to applying a pulling force f acting on the nanoparticle and balancing the elastic and capillary forces generated in a system upon nanoparticle displacement. Therefore, we can use a Legendre transformation of the total free energy of the system (eq 12) to find dependence of the nanoparticle substrate separation, Δh, and contact radius, a, as a function of the applied external force, f. The external force corresponds to the rate of change of the system free energy with variation of nanoparticle substrate separation Δh, ∂ΔFtotal(a, Δh)/∂Δh = f. Note that the force, f, is pointing along the normal to the substrate direction away from its surface. Performing Legendre transformation, we can define a function ̂ ̂ ̂ (a ̂, Δh)̂ − f Δh Γ(a ̂, Δh ̂, f ̂ ) = ΔFtotal 3

(20)

This function defines equilibrium â and Δĥ as a function of the external force, f.̂ Figure 9 shows the evolution of the function Γ(â, Δĥ, f)̂ at different values of the applied external force. As the magnitude of the external force increases, the minimum of the function Γ(â, Δĥ, f),̂ which corresponds to a nonzero value of the contact radius, becomes shallow. Finally, at a large applied force, the function Γ(â, Δĥ, f)̂ has a minimum at a = 0, which corresponds to detachment of nanoparticle from the substrate. The location of the minimum of the function Γ(â, Δĥ, f)̂ at different values of the applied external force is obtained by ̂ solving the system of equations ∂Γ(â, Δĥ, f)/∂â = 0 and ∂Γ(â, ̂ ̂ = 0. Taking derivatives of the function Γ(â, Δĥ, f)̂ Δh,̂ f)/∂Δh with respect to Δĥ and â, we arrive at ⎡ 1 ⎤ 1 a⎢̂ Δh ̂ + a 2̂ ⎥ + δa ̂ − f ̂ = 0 ⎦ ⎣ 3 6 4 δ(3a 2̂ + 2Δh)̂ + βa 3̂ − a ̂ + [Δh ̂ + a 2̂ ]2 = 0 9

Figure 10. Dependence of the normalized detachment force on the value of parameter γ(GRp)−1/3W−2/3 for nanoparticles interacting with substrates with εsp = 1.2kBT (filled symbols) and 0.75kBT (open symbols), and different values of the shear modulus G of the substrate. Notations are the same as in Figure 4. Green stars represent data for critical detachment force obtained for pulling soft nanoparticles from rigid substrates (see ref 26). The solid line is given by the equation f(x) = 5.34x−1.89.

combined the data points calculated for detachment of rigid nanoparticles from soft substrates and soft nanoparticles from rigid substrate. As one can see, all data sets have collapsed into one universal curve indicating similarity in detachment of nanoparticles in both these systems. The solid line, f(x) = 5.34x−1.89, in this figure corresponds to the best fit to the simulation data.

(21a)

4. CONCLUSIONS We have performed coarse-grained molecular dynamics simulations of the detachment of rigid nanoparticles from soft substrates. These simulations mimic the AFM set up used for measurements of the mechanical and surface properties of soft materials. Analysis of simulations data clearly shows break down of the classical JKR theory describing the deformation of substrates upon nanoparticle detachment. In particular, our simulations highlight that detachment of nanoparticles from soft substrates proceeds through neck formation. We have modified the JKR theory to account for formation of the neck taking into account contribution of the surface free energy of the deformed substrate into the total free energy of the system. The equation describing deformation of the substrate upon nanoparticle detachment (see eqs 11 and 13) includes

(21b)

Solving together eqs 21a and 21b, we obtain an expression for the applied force f as a function of the contact radius a: f ̂ = −4a 3̂ + 2a ̂ 9δ 2 + 4a ̂ − 9δa 2̂ − 9βa 3̂

(22)

It is important to point out that eq 22 gives both stable, ∂a/∂f < 0, and unstable, ∂a/∂f > 0, branches of the nanoparticle deformation curves (see Figure 5). Simple scaling relations for dependence of the critical detachment force on the system parameters can be obtained in asymptotic regimes corresponding to small and large values of the parameters δ and β. In the limit of small values of the parameters β ≪ 1 and δ ≪ 1 (JKR regime), the detachment 12526


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where rij is the distance between the ith and jth beads, and σ is the bead diameter. The cutoff distance for the bead−bead interactions was set to rcut = 2.5 σ, and the value of the Lennard-Jones interaction parameter εLJ was set to 1.5kBT between the beads of the same type, and εsp was set to 1.2 and 0.75kBT for interactions between beads belonging to substrates and nanoparticles, where kB is the Boltzmann constant and T is the absolute temperature. The connectivity of the beads into polymer chains, the cross-link bonds, and bonds between beads forming nanoparticles were modeled by the finite extension nonlinear elastic (FENE) potential34

contribution from surface free energy changes associated with deformation of the substrate surface. The magnitude of these free energy contributions in comparison with those considered in the JKR model is governed by two dimensionless parameters, β and δ. The parameter β is proportional to the ratio of the change in the surface energy of substrate outside the neck to the elastic energy of the deformed substrate in the contact zone, and the parameter δ corresponds to the ratio of the surface energy of the neck and the substrate elastic energy. In the limit when both parameters approach zero, our expression for the dependence of the contact radius, a, on the magnitude of the applied force, f, reduces to the classical JKR solution (see eq 22). Note that this model also describes detachment of soft nanoparticles from rigid substrates by substituting the substrate shear modulus and surface tension by those for soft nanoparticles pulled from rigid substrates (see Figure 10). Recently Hui et al.28 have studied detachment of rigid nanoparticles from soft substrates and proposed an alternative expression for the detachment force. Our expression for the detachment force (eq 22) is different from the one derived in ref 28. The main difference comes from explicit consideration of the surface free energy contribution of the neck into the total free energy of the system. This contribution becomes important in the limit of longer necks connecting nanoparticle and substrate. The results presented here could have significant implications for analysis of the AFM data obtained for the soft biological and polymeric substrates using rigid tips attached nano- and microparticles as AFM probes. For soft materials which have shear modulus Gs on the order of 0.1 MPa, surface tension γs ∼ 20 mN/m, and work of adhesion W ∼ 40 mN/m, the probe size Rp should be larger than 2 μm for detachment curves to be described by the JKR expression. (In our estimate, we set the parameter δ = 0.1.) We hope our work will inspire a detailed calibration of the AFM technique to establish the range of applicability of the contact mechanics based on the JKR and DMT models and pinpoint the range of parameters where surface free energy controls the probe/substrate interactions.

⎛ 1 r2 ⎞ 2 UFENE(r ) = − kspringR max ln⎜1 − 2 ⎟ 2 R max ⎠ ⎝

with the spring constant kspring = 30kBT/σ2 and the maximum bond length Rmax = 1.5σ. The repulsive part of the bond potential was modeled by the LJ-potential with rcut = 21/6σ and εLJ = 1.5kBT. The gel-like substrates of thickness Hgel were placed on rigid half-space, which was represented by the external potential ⎡ 2 ⎛ σ ⎞ 9 ⎛ σ ⎞3 ⎤ U (z) = εw ⎢ ⎜ ⎟ − ⎜ ⎟ ⎥ ⎝z⎠ ⎦ ⎣ 15 ⎝ z ⎠

m

dvi⃗(t ) R = Fi ⃗(t ) − ξvi⃗(t ) + Fi⃗ (t ) dt

(26)

where m is the bead mass set to unity for all beads in a system, vi⃗ (t) is the bead velocity, and F⃗i(t) is the net deterministic force acting on the ith bead. The stochastic force F⃗Ri (t) has a zero average value and δ-functional correlations ⟨F⃗Ri (t)F⃗Ri (t′)⟩ = 6kBTξδ(t − t′). The friction coefficient ξ is set to ξ = m/τLJ, where τLJ is the standard LJ-time τLJ = σ(m/εLJ)1/2. The velocity−Verlet algorithm with a time step Δt = 0.01τLJ was used for integration of the equations of motion. In our simulation, we used periodic boundary conditions in x and y directions, where Lx = Ly = 82.1σ. Dimension in the z-direction is 110σ. All simulations were performed using LAMMPS.36 Nanoparticles. The spherically shaped nanoparticles were made by arranging beads with diameter 1.0σ into HCP lattice and connecting 12 closest neighbors by the FENE springs. Nanoparticles were relaxed by performing MD simulation runs lasting 100 τLJ. After equilibration, the radius of the nanoparticle was equal to 17.9σ, and the bond length between beads forming the nanoparticle was equal to 0.97σ. Gel Substrates. To create elastic substrates, polymer chains were placed inside a slab and cross-linked by the FENE bonds. After cross-linking, the system was equilibrated for 104τLJ in the external potential given by eq 25. The modulus of the substrates, Gs, and their equilibrium thicknesses, Hgel, are summarized in Table 1. The shear modulus of substrates was obtained from 3-D simulations of gels with the same beads’ and cross-links’ densities (see ref 24 for details). Nanoparticle Indentation and Pulling. Simulations of nanoparticles interacting with soft substrates began with placing a nanoparticle at distance 2.0σ from the substrate. A harmonic

⎛ σ ⎞6 ⎛ σ ⎞12 ⎜⎜ ⎟⎟ − ⎜ ⎟ rij ≤ rcut ⎝ rcut ⎠ ⎝ rij ⎠

0

(25)

where εw is equal to 1.0kBT. Simulations were carried out in a constant number of particles and temperature ensemble. The constant temperature was maintained by coupling the system to a Langevin thermostat33 implemented in LAMMPS.36 In this case, the equation of motion of the ith bead is

5. SIMULATION METHODS We have performed coarse-grained molecular dynamics simulations of rigid nanoparticles in contact with soft gel-like substrates. Nanoparticles were modeled by spherically shaped assembly of Lennard-Jones beads of diameter σ arranged in a hexagonal closed-packed (HCP) lattice structure and connected to their nearest neighbors by the elastic springs. The substrates were made by cross-linking bead−spring chains each consisting of N = 32 monomers. In our simulations, we have varied the number of cross-links per chain to cover a wide range of the substrate shear modulus. The interactions between all beads in a system were modeled by the truncated-shifted Lennard-Jones (LJ) potential33 ⎧ ⎡⎛ ⎞12 ⎪ ⎢ σ ⎪ 4εLJ⎢⎜⎜ r ⎟⎟ − ⎪ ⎣⎝ ij ⎠ ⎪ ULJ(rij) = ⎨ ⎛ σ ⎞6 ⎤ ⎪ +⎜ ⎟⎥ ⎝ rcut ⎠ ⎥⎦ ⎪ ⎪ ⎪ ⎩

(24)

rij > rcut (23) 12527


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Langmuir ξ

Table 1. System Parameters: Shear Modulus, Density, Surface Tension, and Work of Adhesion 3

Gs [kBT/σ ] ρσ3 Hgel [σ] γs [kBT/σ2] W (εsp = 1.2kBT) [kBT/σ2] W (εsp = 0.75kBT) [kBT/σ2]

γs =

0.024 0.962 44.3 1.84 2.65

0.072 0.972 22.1 1.88 2.68

0.162 0.983 21.9 1.97 2.73

0.252 0.993 21.8 2.20 2.75

0.498 1.004 21.5 2.21 2.77

0.833 1.014 21.2 2.40 2.78

1.30

1.31

1.31

1.32

1.34

1.35

∫−ξ (PN(z) − PT(z)) dz

(28)

where 2ξ is the thickness of the interface that was determined from the monomer density profile as an interval within which the monomer density changes from zero to its bulk value. The results of surface tension calculations are presented in Table 1.

■

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

■

spring with a spring constant Ksp = 400 was applied to nanoparticle center of mass for 100τLJ to bring a nanoparticle into contact with a substrate. After that the spring was removed, and the system was equilibrated for 5 × 104τLJ. In order to calculate interaction potential between a nanoparticle and a substrate, we used the weighted histogram analyzes method (WHAM).37 In these simulations, the zcoordinate of the center of mass, zcm, of the nanoparticle was tethered at location z* by a harmonic potential 1 U (zcm , z*) = K sp(zcm − z*)2 (27) 2 2 where Ksp = 400 kBT/σ is the spring constant. The initial tethering point z0* was set to the location of the center of mass of nanoparticle in equilibrium contact with the substrate. The location of the tethering point was moved with increment Δz* = 0.1σ until the nanoparticle was completely detached from the substrate. To prevent a gel substrate from moving with nanoparticle, we have tethered the z-coordinate of the center of mass of the substrate at its initial location by a harmonic potential with spring constant equal to 3 × 104 kBT/σ2. For each location of the tethering point, the system was equilibrated for 103τ followed by a production run lasting 103τ, during which the average contact radius and distribution of the center of mass of the nanoparticle were collected for WHAM calculations of the potential of mean force. Work of Adhesion. The work of adhesion between a nanoparticle and substrate was calculated from the potential of the mean force between a crystal slab and soft (gel-like) substrate by using WHAM.37 In these simulations, two slabs having the structure of a crystal and gel with dimensions 10σ × 10σ × 10σ were pushed toward each other. In these simulations we set the z-component of the velocity of the crystal slab to zero, thus fixing te location of its center of mass. The center of mass of the gel, zcm, was tethered at point z* by a harmonic potential (see eq 27). The value of the spring constant Ksp was set to 200kBT/σ2 and 500kBT/σ2. The location of the tethering point was moved with increment Δz* = 0.1σ. For each location of the tethering point, we performed simulation runs lasting 5 × 103τLJ, during which we calculated the distribution of the center of mass of the gel, and thus calculated the potential of the mean force between two films via WHAM. The potential of the mean force was used to calculate the work of adhesion, W = ΔF/A, as a function of the interaction parameter and the crosslinking density of the gel-like substrates (see Table 1). Surface Tension of Substrate. The surface free energy (surface tension) of gel-like substrate was evaluated by integrating the difference of the normal PN(z) and tangential PT(z) to the interface components of the pressure tensor.24 Note that in our simulations, the z direction was normal to the interface.

ACKNOWLEDGMENTS This work was supported by the National Science Foundation under the Grant DMR-1409710.

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REFERENCES

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Contact Mechanics of Nanoparticles: Pulling Rigid Nanoparticles from

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