Article pubs.acs.org/Langmuir
Contact Mechanics of Nanoparticles Jan-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 detachment of nanoparticles from a substrate. The critical detachment force, f *, is obtained as a function of the nanoparticle radius, Rp, shear modulus, G, surface energy, γp, and work of adhesion, W. The magnitude of the detachment force is shown to increase from πWRp to 2.2πWRp with increasing nanoparticle shear modulus and nanoparticle size. This variation of the detachment force is a manifestation of neck formation upon nanoparticle detachment. Using scaling analysis, we show that the magnitude of the detachment force is controlled by the balance of the nanoparticle elastic energy, neck surface energy, and energy of nanoparticle adhesion to a substrate. It is a function of the dimensionless parameter 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 a deformed nanoparticle. In the case of small values of the parameter δ ≪ 1, the critical detachment force approaches a critical Johnson, 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 *∝ γp3/2 Rp1/2G−1/2. Simulation data are described by a scaling function f *∝ γp3/2Rp1/2G−1/2δ−1.89.
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INTRODUCTION In everyday life, we face numerous manifestations of adhesion and contact phenomena from the stickiness of chewing gum to the sole of a shoe, to the peeling of scotch tape from a piece of paper, to a gecko running on the walls and ceiling.1−7 These phenomena are also important for cell adhesion,8 colloidal stabilization,9,10 nanomolding and nanofabrication,11−13 and drug delivery. These examples illustrate the true interdisciplinary nature of adhesion and contact mechanics problems that are of interest to scientists working in the areas of colloidal physics and chemistry, soft matter physics, biophysics, polymer science and technology, and cell biology. It is well understood that adhesion between macroscopic deformable bodies is a result of the fine interplay between an elastic energy of contact and adhesion in the contact and the vicinity of the contact area.1−7 It was shown by Johnson, Kendall, and Roberts (JKR)14,15 and by Derjaugin, Muller, and Toporov (DMT)16 that because of dispersion (van der Waals) forces two bodies in contact can have a finite contact area even in the case of zero load forces.9 Whereas JKR and DMT theories were originally considered to be competitive, it turns out that they represent two limiting cases of deformation of large compliant solids (JKR) and small rigid solids (DMT), corresponding to large and small values of the so-called Tabor parameter, defined as the ratio of the elastic deformation to the range of action of the dispersion forces.5,17 During the last 40 years, numerous experimental, theoretical, and numerical studies confirmed the assumptions and validity of the JKR and DMT models for macroscopic objects.1,2,5,6 However, it was shown recently that a JKR/DMT-like approach fails to describe the contact and adhesion of nanoscale-size objects correctly.18 For such objects in addition © 2012 American Chemical Society
to the elastic and adhesion energies, one has to account for the changes in the surface energy induced by the shape deformation of nanoparticles upon contact. There are two different asymptotic regimes of nanoparticle deformation. In the first regime, which corresponds to large, hard nanoparticles, the deformation of nanoparticles in contact with a substrate is obtained by balancing the nanoparticle elastic energy with the particle adhesion energy, as is done in the JKR or DMT theories (JKR regime).5 In the second regime, corresponding to small, soft nanoparticles, nanoparticle deformation is determined by the balance of the nanoparticle adhesion and interfacial energies (nanoparticle wetting regime). The crossover between these two regimes is governed by the dimensionless parameter β ∝ γp(GRp)−2/3W−1/3, where G is the particle shear modulus, Rp is the particle radius, γp is the polymer interfacial energy, and W is the particle work of adhesion. This parameter is proportional to the ratio of the nanoparticle surface energy to the elastic energy of the deformed nanoparticle. Here we study the contact mechanics of the nanoscale-size objects in the wide interval of pulling forces and show that pulling forces for nanoparticle detachment depend on material properties such as the nanoparticle elastic modulus, surface energy, and work of adhesion. In particular, we demonstrate that the detachment of nanoparticles occurs through neck formation and that the critical detachment force is 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, 2012 Revised: June 11, 2012 Published: June 19, 2012 10881
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energy of a neck and the elastic energy of the deformed nanoparticle.
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RESULTS AND DISCUSSION Molecular Dynamics Simulations of Nanoparticle Contact Mechanics. We have performed molecular dynamics simulations19 of the detachment of nanoparticles from substrates. In our simulations, we used a coarse-grained representation 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 chains were modeled by Lennard-Jones particles (beads) with diameter σ. The connectivity of monomers in polymer chains and cross-linking bonds between chains were maintained by FENE bonds.20 Nanoparticles were prepared by confining a polymer melt in cavities with radii of R0 = 11.3σ, 16.4σ, and 20.5σ (preparation size).18 The elastic properties of nanoparticles were controlled by changing the cross-linking density. After a nanoparticle was cross-linked, the confining cavity was removed and the nanoparticle was relaxed by performing an MD simulation run. The size of the relaxed nanoparticle Rp varied between 10.5σ and 21σ depending on the cross-linking density. (See ref 18 for actual nanoparticle sizes.) For each cavity size, R0, we performed simulations of nanoparticles with six different cross-linking densities. The adsorbing (adhesive) substrate was modeled by an external 3−9 potential acting along the normal to the surface z direction. The magnitude of the interaction potential εW was equal to 1.5kBT, 2.25kBT, and 3.0kBT, where kB is the Boltzmann constant and T is the absolute temperature. We performed two sets of simulations. In the first set of simulations, the pulling force was modeled by applying a constant force fz to each monomer forming a nanoparticle. In the second set of simulations, we used a weighted histogram analysis method (WHAM)21 to obtain a potential of the mean force between the nanoparticle and substrate as a function of the distance between the nanoparticle center of mass and the substrate 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 radius on the applied force. We used red symbols to show the force obtained from the differentiation of the potential of the mean force. The differentiation was performed by fitting the potential of the mean force by the Bezzier spline and differentiating the obtained fitting curve. The blue symbols in the insets in Figure 1 a,b represent the results of the constant force simulations. Note that by differentiating the potential of mean force we can obtain both the stable and unstable branches of the contact radius function, a( f). 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, f *, corresponding to nanoparticle detachment is determined 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 soft nanoparticles, we see the formation of the neck as the distance between the nanoparticles and substrate increases. In this case, the height of the nanoparticle can be larger than the diameter of the undeformed nanoparticle, 2Rp. For hard nanoparticles, there is no neck formation, and nanoparticle detachment takes place abruptly when the value of the force exceeds a critical value.
Figure 1. Potential of mean force as a function of the displacement of the nanoparticle center of mass along the z axis for soft (G = 0.0233kBT/σ3) and hard (G = 0.591kBT/σ3) nanoparticles with size R0 = 20.536σ, interacting with the substrate with εW = 1.5kBT. Insets show the dependence of the contact radius a on the applied force f. Blue symbols correspond to constant force simulations; red symbols represent the effective force obtained from the differentiation of the potential of mean force.
In Figure 3, we plot the dependence of the reduced nanoparticle radius as a function of the reduced force. The values of the contact radius a0 were obtained from zero force simulations when the nanoparticle deformation is controlled only by the long-range 3−9 potential (Appendix A). The work of adhesion W = εwρpσ3/2d02 was calculated by using the average nanoparticle density ρp and the distance of closest approach d0 = 0.886σ obtained from simulations. It follows from this figure that the critical value of the detachment force decreases with decreasing nanoparticle shear modulus G. For the hard nanoparticles, the value of the detachment force, f *, could be larger than 2πRpW. It is larger than both the critical detachment force for the DMT model, f DMT = 2πRpW, and that for the JKR model, f JKR = 1.5πRpW.5 However, the shape of the curve is topologically similar to those obtained for the JKR model15 ⎛ a ⎞3 ⎛ a ⎞3/2 = −6⎜ ⎟ + 6⎜ ⎟ πR pW ⎝ a0 ⎠ ⎝ a0 ⎠ f
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Figure 2. Deformation of nanoparticle shape for soft (G = 0.0233kBT/σ3) (top row) and hard (G = 0.591kBT/σ3) (bottom row) nanoparticles with size R0 = 20.536σ adsorbed on a substrate with εW = 1.5kBT.
detachment force, f *, shifts toward larger force values with increasing both the nanoparticle shear modulus, G, and the work of adhesion, W, of nanoparticles to substrate. For soft nanoparticles with G = 0.023kBT/σ3, the value of the detachment force f * approaches πRpW. It is important to point out that significant deviations from the JKR model are seen for the large nanoparticle interaction energy with a substrate (εW = 3.0kBT). In this case at zero force, soft nanoparticles are strongly deformed (Figure 4). Such nanoparticles experience large nonlinear deformations as the applied force approaches the detachment force. The shape of these nanoparticles resembles those of pendant droplets.10 The softer the nanoparticle, the thicker the neck it grows during the pulling process (Figure 4).
Figure 3. Dependence of the reduced contact radius, a/a0, on the reduced force, f/πWRp, for nanoparticles with a preparation radius of R0 = 16.4 σ and different values of the shear modulus G = 0.023kBT/σ3 (open squares), 0.21kBT/σ3 (crossed squares), and 0.86kBT/σ3 (filled squares) at different magnitudes of the wall interaction potential, εw. a0 is the contact radius at zero applied force, and W is the work of adhesion. The solid line corresponds to eq 1.
where a0 is the nanoparticle contact radius at zero applied force. Note that in the JKR model particle detachment occurs at finite values of the contact radius whereas in the DMT model the contact radius gradually decreases to zero with increasing magnitude of the force such that at the detachment point the critical value of the force is equal to the adhesion force of the undeformed (rigid) particle, f DMT = 2πWRp. The value of the
Figure 4. Snapshots of nanoparticle shape deformation for nanoparticles of size R0 = 16.4σ adsorbed to a substrate with εW = 3.0kBT. f W = πWRp and f * is the detachment force. 10883
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Model of Nanoparticle Contact Mechanics. Our simulations show that there are two different regimes in nanoparticle detachment from a substrate. Nanoparticles with a large value of the shear modulus G detach from the substrate almost without neck formation. We have called such nanoparticles “hard” nanoparticles. However, nanoparticles with small values of the shear modulus detach with neck formation. We referred to these nanoparticles as “soft” nanoparticles. We will 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 nanoparticle with size Rp can be expressed in term of the change in
absolute value of the adhesion force between undeformed nanoparticles and a substrate. Note that the values of the numerical coefficients depend on the particular nanoparticle shape and deformation that were used for the calculation of the different terms in the nanoparticle free energy. (See Appendix B and ref 18 for details.) Using these dimensionless variables, we can reduce eq 3 to a simple cubic form 0 = − 1 + β (f )y 2 + y 3
(5)
The deformation of the nanoparticles is described by a positive root of this equation 2/3 ⎛ W (1 − (f /f )) ⎞ Δh DMT ⎟⎟ = A⎜⎜ 2R p GR p ⎝ ⎠
⎛ ⎜3 r + ⎝
q3 + r 2 +
3
r−
q3 + r 2 −
β (f ) ⎞ ⎟ 3 ⎠
2
(6)
where ⎛ β (f ) ⎞ 3 ⎛ β (f ) ⎞ 2 1 r= −⎜ ⎟ and q = −⎜ ⎟ ⎝ 3 ⎠ ⎝ 3 ⎠ 2
Figure 5. Schematic representation of a deformed nanoparticle without a neck.
It follows from eq 6 that we should be able to collapse our simulation data for hard nanoparticles by plotting (Δh/ 2Rp)(GRp/Wf)2/3 as a function of γp(GRp)−2/3Wf−1/3, where Wf = W(1 − (f/f DMT)) is a renormalized value of the work of adhesion (Figure 6). Indeed, we see a reasonably good collapse of the simulation data for zero and small values of the pulling forces. However, simulation data begin to deviate from the universal curve with increasing magnitude of the pulling force.
nanoparticle height (Figure 5) (The derivation details are given in Appendix B.) F(Δh) ∝ f Δh − 2πWR pΔh + πγpΔh2 +
28 2 GR p1/2Δh5/2 15
(2)
The first term on the r.h.s of eq 2 represents the contribution due to the applied pulling force. The positive sign of this term reflects the pulling nature of the applied force. The second term describes the contribution of the long-range van der Waals interactions of the deformed nanoparticle with a substrate where W is the work of adhesion. The third term represents the increase in the surface energy of the deformed nanoparticle with the interfacial energy γp in comparison with that of a sphere with radius Rp. Finally, the last term is the elastic energy contribution due to the deformation of a nanoparticle with shear modulus G. The equilibrium deformation of a nanoparticle is obtained by minimizing the nanoparticle free energy (eq 2) with respect to the height deformation, Δh. This results in the following equation describing nanoparticle deformation: 0 = f − 2πWR p + 2πγpΔh +
14 2 GR p1/2Δh3/2 3
(7)
(3)
To obtain an exact solution of eq 3, it is useful to introduce dimensionless variables ⎞2/3 GR p Δh ⎛ ⎜ ⎟ y = 2R pA ⎜⎝ W (1 − f /fDMT ) ⎟⎠
Figure 6. Dependence of the reduced nanoparticle deformation on the value 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/σ3 (right-half-filled symbols), 0.37kBT/σ3(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), wall interaction potential with εw = 1.5kBT (circles), 2.25kBT (squares), and 3.0kBT (diamonds). The green triangles show zero force simulations from ref 18. The solid line is the best fit to eq 6 with A = 0.419 and B = 0.84.
2
(4a)
⎛ ⎞2/3 γp3/2 ⎟ β(f ) = B⎜⎜ 1/2 1/2 ⎟ GR W (1 − f / f ) p ⎝ ⎠ DMT 2/3
(4b) 2/3
where A = (3π/28) = 0.484 and B = 2(3π/28) = 0.968 are numerical coefficients and f DMT = 2πWRp is a critical detachment force for hard nanoparticles that is equal to the 10884
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conservation upon nanoparticle deformation (details are given in Appendix B). In this approximation, the free energy of nanoparticles is equal to
In Figure 7, we show how this departure happens for nanoparticles adsorbed on substrates with different strengths
F(Δh , a) ∝ f Δh − Wπa 2 ⎛ 4 ⎛ a2 ⎞⎞ a ⎜ ⎟⎟⎟ + γpπ ⎜⎜ + − Δ 2 a h ⎜ 2R 2 ⎟ ⎝ p ⎠⎠ ⎝ 4R p ⎡ ⎤ 2 Δha3 1 a5 ⎥ + 4G⎢Δh2a − + ⎢⎣ 3 Rp 5 R p2 ⎥⎦
(8)
The equilibrium values of the contact radius a and height deformation Δh are obtained by solving equations ∂F(Δh, a)∂a = 0 and ∂F(Δh, a)/∂Δh = 0. To analyze these equations, we introduce dimensionless variables 2 ⎞1/3 ⎛ ⎛ 3 ⎞2/3⎜ f JKR ⎟ 3 f JKR R p 3 a ̂ , Δh = ⎜ ⎟ ⎜ 2 ⎟ Δh ̂, f = f JKR f ̂ a = ⎝4⎠ 4 G ⎝ G Rp ⎠ 3
Figure 7. Dependence of the reduced nanoparticle deformation on the value 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 (red squares), and 3.0kBT (blue diamonds). The solid line is the best fit to eq 6 with A = 0.419 and B = 0.84.
(9)
where f JKR = 1.5πWRp. In these new variables, the equations are written as follows [Δh ̂ − a 2̂ ]2 −
of the surface potential, εW. A comment has to be made here concerning the value of the fitting parameter B = 0.84 and its relation to the value of the fitting parameter BG used in ref 18. There was a typographical error in ref 18 in reporting the value of 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 reason for such a deviation is the formation of the neck connecting a nanoparticle to a substrate. This transformation takes place for very small values of the applied forces for nanoparticles adsorbed on substrates with εw = 2.25kBT and 3.0kBT (Figure 4). However, in the case of a weaker substrate potential for which εw = 1.5kBT, the simulation data first follow the universal curve and then start to deviate (Figure 7). The deviation from the universal curve occurs when the contribution from the surface energy of the neck to the total nanoparticle free energy becomes comparable to the change in the surface energy of the deformed spherical cap. Below we will generalize our model to account 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 height deformation Δh (Figure 8). The height of a neck connecting a nanoparticle to a substrate is related to these parameters as hn ≈ (a2/2Rp) − Δh. This is due to the requirement of volume
4 a ̂ + β1a 3̂ + δ(3a 2̂ − 2Δh)̂ = 0 9
⎡ f̂ 1 ⎤ a⎢̂ Δh ̂ − a 2̂ ⎥ − δa ̂ + =0 ⎣ 3 ⎦ 6
(10)
(11)
where two dimensionless parameters β1 =
δ=
γp π 2/3 2 × 32/3 (GR p)2/3 W1/3
γp π 1/3 4/3 1/3 2/3 3 (GR p) W
(12a)
(12b)
determine the relative strengths of the adhesion, interfacial, and elastic energies. There is a simple physical meaning of parameters β1 and δ. Parameter β1 corresponds to the ratio of the change in the surface energy of a spherical cap with respect to the elastic energy of the deformed nanoparticle, and parameter δ is proportional to the ratio of the surface energy of a neck to the nanoparticle elastic energy. For our systems, the ratio 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 function of the contact radius a f ̂ = −4a 3̂ + 2a ̂ 9δ 2 + 4a ̂ − 9δa 2̂ − 9β1a 3̂
(13)
Equation 13 gives both stable, ∂a/∂f < 0, and unstable, ∂a/∂f > 0, branches of the nanoparticle deformation curves. Furthermore, eq 13 is correct only when a macroscopic neck (hn > σ) is beginning to form, which takes place at finite values of the applied force, f. In the interval of the applied forces where there is no macroscopic neck (hn < σ), the deformation of a nanoparticle 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 of parameters β1 (β1 ≪ 1) and δ (δ ≪ 1, JKR regime), the detachment force corresponding to the minimum in eq 13, ∂f/
Figure 8. Schematic representation of a deformed nanoparticle with a neck of radius a. 10885
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∂a = 0, is equal to f * ≈ 1.5πWRp, with the contact radius at the detachment point being equal to â* ≈ 2−2/3. In the opposite limit of large values of parameter δ (δ ≫ 1, necking regime), by differentiating eq 13 with respect to contact radius a one obtains for the contact radius at the detachment point â* ≈ (δ/ 2)1/2 and for the detachment force f *∝ γp3/2 Rp1/2G−1/2. (Note that to obtain this result we kept only the δ2 term under the square root in eq 13.) Thus, we can plot a normalized detachment 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
In Figure 10, we summarized different nanoparticle deformation regimes in the W/γp, W/(GRp) plane. In the
Figure 10. Diagram of states of a nanoparticle at an adhesive surface on logarithmic scales.
undeformed nanoparticle regime, adhesion forces are too weak to cause noticeable nanoparticle deformation. In this regime, the deformation of the nanoparticle height Δh is smaller than the characteristic length scale of the short-range interactions (LJ potential) or the bead diameter σ, Δh < σ. In the case of soft nanoparticles, the nanoparticle height deformation is determined by the balance of the change in the nanoparticle interfacial energy, γpΔh2, and its adhesion energy to a substrate, −WRpΔh (eq 2 in the zero force limit). The deformation of nanoparticle height, Δh ∝ RpW/γp, is larger than the bead diameter σ when W/γp > σ/Rp. We refer to this regime as the wetting 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 (eq 2). This results in Δh ∝ Rp(W/GRp)2/3. Therefore, the crossover to the nanoparticle deformation regime for Δh ∝ σ occurs at W/GRp ∝ (σ/Rp)3/2. It is interesting that this crossover 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 DMT regime in the diagram of states). However, in the case of large values of Tabor’s parameter μ ≫ 1, the deformation of nanoparticles is described in the framework of the JKR theory (the JKR regime in Figure 10). In the JKR regime, the nanoparticle detachment force is equal to f * ≈ 1.5πWRp, and in the DMT and undeformed nanoparticle regimes, it is f * = 2πWRp. In a crossover, the detachment force depends on the value 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 nanoparticle deformation 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 deformation resembles that of a liquid nanodroplet. In the wetting regime, a
Figure 9. Dependence of the normalized detachment force on the value of the parameter γp(GRP)−1/3W−2/3 for nanoparticles with different preparation radii R0 = 11.3 σ (open symbols), 16.4σ (crossed symbols), and 20.5σ (filled symbols) interacting with substrates with εw = 1.5kBT (black circles), 2.25kBT (red squares), and 3.0kBT (blue diamonds) and different values of the shear modulus G. The solid line is given by the equation f(x) = 5.34x −1.89.
reduced force on parameter γp(GRp)−1/3W−2/3 for all of our systems. Simulation data have collapsed into one universal curve, proving that the normalized detachment force is a universal function of parameter δ. Unfortunately, the studied interval of the system parameters is not sufficient to cover pure asymptotic regimes. Our simulation data correspond to a crossover between the JKR and necking regimes. In this interval of parameters simulation, data can be fitted by power law 5.34x −1.89.
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CONCLUSIONS We have shown that nanoparticle detachment from an adhesive substrate under the action of the pulling force occurs through neck formation. The critical detachment force depends on the nanoparticle adhesion energy with a substrate, W, the nanoparticle interfacial energy, γp, and the nanoparticle shear modulus, G. With decreasing nanoparticle shear modulus, G, the critical force required for nanoparticle detachment decreases from f * ≈ 2.2πWRp to πWRp (Figure 3). Our simulations confirmed that the value of the normalized detachment 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 strength of the adhesion energy εW and with decreasing nanoparticle shear modulus, G (Figures 2 and 4). 10886
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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 the growing nanoparticle neck provides a dominant contribution to the nanoparticle interfacial energy and controls nanoparticle detachment (necking regime in Figure 9). Along this line, the value of parameter δ (eq 12a) is on the order of unity. In this range of parameters, the nanoparticle detachment force scales as f *∝ γp3/2Rp1/2G−1/2. Our analysis of different nanoparticle deformation regimes is correct only in the case of small nanoparticle deformations, Δh/Rp < 1. This condition determines boundaries for the linear deformation 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 as the substrate elasticity. These topics will be the subject of future publications.
space. The system was periodic in the x and y directions with dimensions of Lx = Ly = Lz = 4R0, where R0 is the nanoparticle radius (details in ref 18). We have modeled nanoparticles with radii Rp varying between 10.5σ and 20.6σ. The actual nanoparticle sizes as a function of the nanoparticle crosslinking density and confining spherical cavity R0 are listed in the appendix of ref 18. To model nanoparticle pulling, we have performed two sets of simulations. In the first set of simulations, we applied a constant force fz pointing along the positive z direction. Simulations were carried out for a constant number of particles and temperature ensemble. The constant temperature was maintained by coupling the system to a Langevin thermostat. The equation of motion of the ith particle with mass m was
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m
APPENDIX A: SIMULATION DETAILS We have performed coarse-grained molecular dynamics simulations of the pulling of a spherical nanoparticle from a substrate. The nanoparticles were prepared by cross-linking polymer chains with a degree of polymerization of N = 32 confined in a spherical cavity with size R0. By varying the number of cross-links per chain, we have been able to control the nanoparticle elastic modulus. The details of the nanoparticle preparation procedure are described in ref 18. In our simulations, the interactions between monomers forming a nanoparticle were modeled by the truncation-shifted LennardJones (LJ) potential
(A.1)
U (zcm , z*) =
where rij is the distance between the ith and jth beads and σ is the bead (monomer) diameter. The cutoff distance for the polymer−polymer interactions was set to rcut = 2.5σ, and the value of the Lennard-Jones interaction parameter was equal to 1.5kBT. This choice of parameters corresponds to poor solvent conditions for the polymer chains. The connectivity of the beads into polymer chains and the cross-link bonds were maintained by the finite extension nonlinear elastic (FENE) potential20
1 K sp(zcm − z*)2 2
(A.5)
with the value of the spring constant being Ksp = 100kBT/σ2. During these simulation runs, we have varied the location of the tethering points with increment Δz* = 0.25σ. For each location of the tethering point, the system was equilibrated for 103τLJ followed by 4 × 103 τLJ, during which we have calculated the distribution of the center-of-mass location for the WHAM calculation of the potential of mean force. In these simulations, we have applied the Langevin thermostat to control the system temperature in all three directions.
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APPENDIX B: VARIATIONAL APPROACH TO NANOPARTICLE DEFORMATION Let us consider a small deformation of the nanoparticle with radius Rp and approximate the deformed nanoparticle by a hemisphere of radius R1 connected to a substrate by a cylindrical neck of height hn (Figure 11). The nanoparticle is pulled from the surface by a force with magnitude f pointing along the normal to the substrate direction. We will assume that the radius of the contact area between nanoparticles and substrate is equal to a and that the displacement of the nanoparticle surface at the point of the closest approach to the substrate is Δh = 2Rp − h, where h is the height of the deformed nanoparticle. The total free energy of the deformed nanoparticle includes the elastic energy of the nanoparticle deformation, the van der Waals energy contribution due to the interaction between the nanoparticle and substrate, the surface
(A.2)
with a spring constant of kspring = 30kBT/σ2 and a maximum bond length of Rmax = 1.5σ. The repulsive part of the bond potential is given by the truncation-shifted LJ potential with rcut = (2)1/6σ and εLJ = 1.5kBT. In our simulations, we have excluded 1−2 LJ interactions such that monomers connected by a bond interacted only through the bonded potential. The presence of the substrate was modeled by the external potential ⎡ 2 ⎛ σ ⎞ 9 ⎛ σ ⎞3 ⎤ U (z) = εw ⎢ ⎜ ⎟ − ⎜ ⎟ ⎥ ⎝z⎠ ⎦ ⎣ 15 ⎝ z ⎠
(A.4)
where vi⃗ (t) is the ith bead velocity and F⃗i(t) is the net deterministic force acting on the ith bead. F⃗Ri (t) is the stochastic force with zero average value ⟨F⃗Ri (t)⟩ = 0 and δ-functional correlations ⟨F⃗Ri (t)F⃗Ri (t′)⟩ = 4ξkBTδ(t − t′). Note that the stochastic force had only nonzero x and y components. Friction coefficient ξ was set equal to m/(7τLJ), where τLJ is the standard LJ time τLJ = σ(m/εLJ)1/2. The velocity Verlet algorithm with a time step of Δt = 0.01τLJ was used to integrate the equations of motion (eq A.4). All simulations were performed using LAMMPS.22 The duration of each simulation run was up to 5 × 104τLJ. In the second set of simulations, we used the weighted histogram analyzes method21 to calculate the potential of mean force between a nanoparticle and substrate. In these simulations, a z coordinate of the center of mass, zcm, of a nanoparticle was tethered to z* by a harmonic spring
⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎛ ⎞12 ⎛ ⎞6⎤ ⎪ σ σ σ σ ⎪ 4εLJ⎢⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ − ⎜ ⎟ + ⎜ ⎟ ⎥⎥ r ≤ rcut rij ⎠ rij ⎠ rcut ⎠ rcut ⎠ ULJ(rij) = ⎨ ⎝ ⎝ ⎝ ⎝ ⎣ ⎦ ⎪ ⎪ 0 r > rcut ⎩
⎛ 1 r2 ⎞ ⎟ UFENE(r ) = − kspringR max 2 ln⎜1 − 2 R max 2 ⎠ ⎝
dvi⃗(t ) R = Fi ⃗(t ) − ξvi⃗(t ) + fz nz⃗ + Fi⃗ (t ) dt
(A.3)
where εw was equal to 1.5kBT, 2.25kBT, and 3.0kBT. The longrange attractive part of the potential z−3 represents the effect of van der Waals interactions generated by the substrate half10887
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a 2 = (2R1hR − hR 2) and hR = 2R p − Δh − hn
(B.6)
Equations B.5 and B.6 determine the radius of the hemispherical cap R1 as a function of the contact radius a, neck height hn, nanoparticle height deformation Δh, and initial size Rp. Solving eqs B.5 and B.6 together, we can derive a quadratic equation for a as a function of the nanoparticle shape. ⎛4 ⎛ hR ⎞ h 3⎞ ⎜ + h n ⎟a 2 − ⎜ R p 3 − R ⎟ = 0 ⎝2 ⎠ 6 ⎠ ⎝3
The physical solution of this equation is
Figure 11. Schematic representation of nanoparticle deformation.
a=
energy contribution due to the change in the surface area of a nanoparticle as it comes in contact with a substrate, and the work done by displacing the center of mass of the nanoparticle by 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. The stress exerted by a deformed nanoparticle on the surface of a semi-infinite half space is given by15 −1/2 1/2 ⎛ ⎛ ρ2 ⎞ ρ2 ⎞ σ(ρ) = σ0̃ ⎜1 − 2 ⎟ + σ1̃ ⎜1 − 2 ⎟ a ⎠ a ⎠ ⎝ ⎝
R p2 − ρ2 ) ≈ Δh −
2R p(Δh + hn)
(B.2) (1) U vdw (Δh , a , hn) ≈ −εw ρp σ 3π
ρ2 2R p
(B.8)
(B.9)
∫0
hR
r 2(z) dz (z + h)̃ 3
= −εw ρp σ 3π
∫0
hR
(hR − z)(2R1 − hR + z) dz (z + h)̃ 3 (B.10)
(B.3)
where r(z) is the radius of the spherical cap a distance z from the substrate and d0 is the distance of closest approach between the nanoparticle and substrate. After integration, we arrive at
Comparing eqs B.2 and B.3, one obtains the following expressions for parameters σ̃0 = K(Δh/a − a/Rp)/π and σ̃1 =K(2a/Rp)/π. Taking this into account, the elastic energy of the deformed nanoparticle is estimated as follows: 1 d2x uz(x) σ(x) Uel(Δh , a) = 2 π 2a3 ⎡ 2 2 2 2⎤ = σ1̃ ⎥ ⎢⎣σ0̃ + σ0̃ σ1̃ + K 3 15 ⎦ ⎡ ⎤ 2 Δha3 1 a5 ⎥ ≈ K ⎢Δh2a − + ⎢⎣ 3 Rp 5 R p2 ⎥⎦ (B.4)
(1) (Δh , a , hn) ≈ −εw ρp σ 3π U vdw
⎡ (2R h − h 2 + h h ̃ + 2h 2̃ )h ⎛ h ⎞⎤ 1 R R R R ×⎢ − log⎜1 + R ⎟⎥ 2 ⎝ ⎢⎣ h ̃ ⎠⎦ 2h ̃ (hR + h)̃
∫
(B.11)
This expression reduces to the expression for the interaction potential of the rigid sphere (R1 = Rp, and hR = 2Rp) located a distance h̃ from a substrate. The contribution from the cylindrical neck of height hn located a distance d0 from the substrate is calculated in a similar way
The surface energy contribution includes the contribution from the spherical cap with radius R1 and a cylindrical neck. To evaluate different contributions to the nanoparticle surface energy, we will assume that the deformation of the adsorbed nanoparticle occurs at a constant volume such that h 3 4 3 R p = R1hR 2 − R + hna 2 3 3
≈
where the surface energy also includes a contribution from the nanoparticle contact area with a substrate with radius a (the first term in parentheses). An evaluation of the contribution from the van der Waals interactions of the deformed nanoparticle of average density ρp with a substrate requires the integration of the long-range van der Walls potential of magnitude εw over the particle height. We will separate this contribution from contributions from the spherical cap and cylindrical neck. The contribution from the spherical cap located a distance h̃ = hn + d0 from the substrate is given by
where K = 2G/(1 − υ) is the nanoparticle rigidity. The parameters for the displacement field σ̃0 and σ̃1 can be found from the displacement of the surface of the nanoparticle. The displacement of the surface of the nanoparticle for small deformations is given by (Figure 11) uz(ρ) = Δh − (R p −
3(hR + 2hn)
Us(Δh , a , hn) = γpπ (a 2 + 2ahn + 2R1hR )
(B.1)
σ̃ ⎛ ρ 2 ⎞⎞ πa ⎛ ⎜⎜σ0̃ + 1 ⎜1 − 2 ⎟⎟⎟ K⎝ 2⎝ 2a ⎠⎠
8R p3 − hR 3
The surface energy of the deformed nanoparticle with interfacial energy γp is equal to
The displacement field of the nanoparticle that generates this stress is uz(ρ) =
(B.7)
(2) U vdw (Δh , a , hn) ≈ −εw ρp σ 3π
=− (B.5)
∫0
hn
a2 dz (z + d0)3
εw ρp σ 3 ⎛ d0 2 ⎞ 2 ⎜ ⎟π a 1 − 2 2d0 2 ⎝ h̃ ⎠
(B.12)
For the systems with only van der Waals interactions, we can relate the surface and interfacial energies to the system density
where we have introduced parameters 10888
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It is important to point out that the neck formation, hn > d0, is related to the value of the Tabor parameter, μ. Our simulations show that d0 ∝ σ; therefore, we can use a bead diameter σ, which describes the characteristic length scale of the LennardJones interaction potential, instead of d0 for the neck formation regime. For soft nanoparticles, the typical nanoparticle contact radius a according to the JKR theory14 is on the order of (WRp2/G)1/3. Thus, for the neck height to be larger than the bead diameter σ, the following must be true:
and strength of the van der Waals interactions. In this case, the surface energy of the polymer−substrate interface γps is equal to γps = γp + γs −
εw ρp σ 3 2d0 2
(B.13)
where γp is the polymer surface energy and γs is the substrate surface energy. Using eq B.13, we can relate the parameters of the van der Waals interactions to the work of adhesion9 W = γp + γs − γps =
⎛ W 2R ⎞1/3 hn p ≈ ⎜⎜ 2 3 ⎟⎟ ≫ 1 μ≡ σ σ G ⎝ ⎠
εw ρp σ 3 2d0 2
(B.14)
Taking this into account, we can rewrite the total van der Waals interaction energy as follows:
Tabor’s parameter defines a crossover between the JKR and DMT regimes.5,17 Large values of the Tabor parameter correspond to the JKR regime14 whereas for small values of μ≪1 the particle deformation is described by the DMT theory.16 In our variational approach, Tabor’s parameter μ defines regimes for nanoparticle detachment with and without neck formation. Now we are in a position to write a scaling expression for the nanoparticle free energy. For hard nanoparticles, the contact radius a and deformation Δh are not independent and are related by a2 ≈ 2RpΔh (eq B.8). In this approximation, eq B.16 reduces to
⎡ (a 2 + h h ̃ + 2h 2̃ )h R R Uvdw(Δh , a , hn) ≈ −Wπd0 2⎢ 2 ̃ ̃ ⎢⎣ h (hR + h) ⎡ ⎛ d 2⎤ h ⎞⎤ − 2 log⎜1 + R ⎟⎥ − Wπa 2⎢1 − 02 ⎥ ⎝ h ̃ ⎠⎦ ⎣ h̃ ⎦ (B.15)
The equilibrium nanoparticle shape is obtained from a minimization of the total nanoparticle free energy ⎡ ⎤ 2 Δha3 1 a5 ⎥ + F(Δh , a , hn) ≈ K ⎢Δh2a − ⎢⎣ 3 Rp 5 R p2 ⎥⎦
F(Δh) ∝ f Δh − 2πWR pΔh + πγpΔh2 +
+ Uvdw(Δh , a , hn) + f Δh + γpπ (2a 2 + hR 2 + 2ahn)
F(Δh , a) ∝ f Δh − Wπa 2 ⎛ 4 ⎛ a2 ⎞⎞ a ⎜ ⎟⎟⎟ + γpπ ⎜⎜ + − Δ 2 a h ⎜ 2 ⎟ ⎝ 2R p ⎠⎠ ⎝ 4R p ⎡ ⎤ 2 Δha3 1 a5 ⎥ + 4G⎢Δh2a − + ⎢⎣ 3 Rp 5 R p2 ⎥⎦
■ ■
ACKNOWLEDGMENTS This work was supported by the National Science Foundation under grant DMR-1004576 and by the American Chemical Society Petroleum Research Fund under the grant PRF#49866ND.
Note that we can neglect the term describing the interaction of a soft nanoparticle with a substrate in comparison with the leading interaction term Wπa2 as long as parameter Rpd02/hna2 is smaller than unity. Using eq B.8, we can estimate the neck height as hn ∝ a2/Rp. Thus, for the parameter to be small it requires that hna 2
a4
d0 2 hn 2
AUTHOR INFORMATION
The authors declare no competing financial interest.
(B.17)
≈
(B.21)
Notes
2πWRpd02/hn
≈
(B.20)
See the main text for an analysis of different scaling regimes.
Uvdw(Δh , a , hn) ≈ −Wπa 2 − Wπd0 2 ⎧ 2R p ⎛ 2R p ⎞ ⎪ + 2 log⎜ ⎟ , hn < d 0 ⎪ d0 ⎝ d0 ⎠ ⎨ ⎛ 2R p ⎞ ⎪ 2R p + 2 log⎜ ⎟ , hn > d 0 ⎪ ⎝ hn ⎠ ⎩ hn
R p 2d0 2
28 2 GR p1/2Δh5/2 15
where we omitted terms independent of Δh. In the case of soft nanoparticles with a neck, hn > σ, the free energy is
(B.16)
with respect to nanoparticle deformation Δh and radius of contact a, taking into account the relation among the neck height, nanoparticle deformation, and contact radius (eq B.8). A scaling analysis of eq B.16 can be performed in two limiting cases of deformation of hard and soft nanoparticles. In the case of hard nanoparticles, nanoparticle detachment occurs without neck formation, hn < d0, while during the detachment of soft nanoparticles a neck is formed, hn > d0. By expanding the van der Waals interaction energy in these two limiting cases, one has
R pd0 2
(B.19)
■
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