Statistical Pull Off of Nanoparticles Adhering to Compliant Substrates

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Statistical Pull Off of Nanoparticles Adhering to Compliant Substrates Ji Lin,† Yuan Lin,‡ and Jin Qian*,† †

Department of Engineering Mechanics, Soft Matter Research Center, Zhejiang University, Hangzhou, Zhejiang 310027, China Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, China

‡

S Supporting Information *

ABSTRACT: It is widely known in adhesive contact mechanics that a spherical particle will not detach from an elastic half-space unless a critical level of pulling force is reached, as already revealed by JKR- or DMT-type deterministic models. This article focuses on the scenario of particle−substrate adhesion where the size of particles is down to the nanometer scale. A consequence of particle size reduction to this range is that the energy scale confining the state of system equilibrium becomes comparable to the unit of thermal energy, leading to statistical particle detachment even below the critical pull-off force. We describe the process by Kramers’ theory as a thermally activated escape from an energy well and develop a Smoluchowski partial differential equation that governs the spatial−temporal evolution of the adhesion state in probabilistic terms. These results show that the forced or spontaneous separation of nanometer-sized particles from compliant substrates occurs diffusively and statistically rather than ballistically and deterministically as assumed in existing models.

1. INTRODUCTION A number of theoretical models have been developed to understand adhesive contact mechanics since the pioneering work by Hertz1 on nonadhesive contact between two elastic bodies. A particle can adhere to a compliant substrate spontaneously because of the ubiquitous adhesive interactions between two objects, reflecting the competition between elastic energy and surface energy. With respect to the Hertz contact model, Johnson, Kendall, and Roberts (JKR)2 incorporated the effects of surface adhesion with the criterion that the stored elastic energy due to the formation of the contact area should be compensated for by the reduction in surface energy at equilibrium, by which the critical tensile load leading to ballistic particle pull off from a substrate was predicted to be P* = −1.5πwR, with the negative sign indicating tensile force according to the convention of contact mechanics, w representing the Dupré work of adhesion between solid surfaces, and R being the radius of the particle. Derjaguin, Muller, and Toporov (DMT)3 proposed an alternate theory for particle−substrate adhesion, where the critical pull-off force was P* = −2πwR. Maugis4 was able to show the JKR−DMT transition using a Dugdale cohesive description5 of surface interaction, with the two models serving as opposite limiting cases. More extensions of 2D cylindrical or 3D spherical profiles in adhesive contact with compliant substrates have involved material anisotropy,6−9 elasticity gradation,10−12 viscoelasticity,13,14 and large deformation in nonlinear elastomers.15−17 Interestingly, it has also been found that JKR theory is applicable to the adhesion between living cells,18 and the critical force needed to detach a thin-walled vesicle adhering © 2013 American Chemical Society

to a substrate assumes the same form as the JKR prediction, albeit with a slightly different numerical coefficient.19 All of these existing results share the viewpoint that a deterministic level of pulling force, commonly referred to as the pull-off force, is needed for particles to detach from substrates. Below the critical pull-off force, particle detachment never occurs. The present study focuses on the scenario of particle− substrate adhesion when the size of particles is reduced to the nanometer scale. Synthetic nanoparticles with sizes from a few nanometers to hundreds of nanometers have been fabricated for a variety of biomedical applications such as drug delivery20,21 and cellular uptake.22−24 It is therefore imperative to develop a quantitative understanding of how these particles of nanometer dimensions behave in processes of attaching to and detaching from deformable surfaces. These processes often take place in aqueous environments for which the Dupré work of adhesion should be modified to solid−solid interaction mediated by liquid.25 More importantly, because the size of particles falls in the nanometer range, the energy variation in the competition between elastic and surface energy approaches the scale of thermal energy, on the order of several to tens of kBT in the practical range of parameters, where kB is the Boltzmann constant and T is the ambient temperature in Kelvin. The adhesive contact between nanoparticles and compliant substrates should therefore be described as a thermally excited distribution among different configurations, Received: October 27, 2013 Revised: December 10, 2013 Published: December 12, 2013 6089

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(2a(πa)1/2), where K = 4E/3(1 − v2) is an alternate way of representing substrate stiffness. The term a3K/R can be readily recognized as the repulsive force due to solid deformation from the Hertz model.1 The energy release rate, defined as the change in elastic energy per unit contact area, is therefore

with the possibility for the bound state to diffuse across a confining energy barrier, which leads to statistical particle detachment in contrast to the deterministic view of particle pull off in existing models.

2. ENERGY SCALE ASSOCIATED WITH NANOPARTICLE ADHESION Here we consider the classical model of adhesive contact mechanics where a rigid spherical nanoparticle of radius R adhering to the planar surface of a semi-infinite elastic body with Young’s modulus E and Poisson’s ratio ν resulted in a certain contact radius a in the presence of an externally applied force P, as schematically shown in Figure 1a. Negative values in

G JKR =

(( a3K /R ) − P)2 1 − ν2 KI,JKR 2 = 2E 6πa3K

(1)

The JKR solution in eq 1 has implicitly assumed that the spherical shape in contact is well represented by the parabolic approximation at small contact radii. However, in the present study, it is anticipated that small particles adhering to compliant substrates may result in rather large contact radii for which the extensions of JKR theory to the exact spherical geometry are required, as performed for the axisymmetric case26 and the plane strain case,27 respectively. Following Maugis,26 the energy release rate in eq 1 should be modified to G=

2 3K ⎛⎜ a R + a ⎞⎟ d − ln 8πa ⎝ 2 R − a⎠

(2)

for the present setup, where d=

2P R R2 − a 2 R+a + − ln 3aK 2 4a R−a

(3)

is the penetration depth of the particle’s lower apex into the substrate surface at load P. The contact radius at equilibrium, if such a state exists, can be solved via the balance between the elastic energy and surface energy G = w, where G is the elastic energy released when the solid−solid contact recedes by the unit area and w = γ1 + γ2 − γ12 is the Dupré work of adhesion that quantifies the energy investment in reducing the unit area of contact, with γ1 and γ2 being the surface energies of the sphere and substrate and γ12 being the interfacial energy. Furthermore, the total energy of the system is the sum of the elastic part and surface part, namely,

Figure 1. (a) Schematic of a rigid spherical nanoparticle in adhesive contact with an elastic half-space. The sphere has radius R, the substrate is characterized by modulus K, and a contact area with radius a is formed in the presence of an externally applied force P. (b) Total system energy U̅ as a function of the contact radius A at fixed values of external load P̅, calculated from the following parameters: K = 1 MPa, R = 50 nm, and w = 0.001 J/m2. Below a critical pulling force P̅*, the energy landscape generally exhibits a local minimum that corresponds to the equilibrium contact radius, separated from an energy barrier. The larger the magnitude of the tensile load, the lower the energy barrier relative to the well, until the local minimum vanishes when the magnitude of load is increased to above the critical value.

U=

∫ G(a)2πada − πwa2

(4)

The classical JKR result is immediately recovered as UJKR =

a5K Pa 2 P2 − − − πwa 2 + constant 3R 3aK 15R2

(5)

by the power expansion ln[(R + a)/(R − a)] ≈ 2(a/R + a3/ 3R3) taking place in eqs 2 and 3 simultaneously when a/R ≪ 1. We proceed by normalizing the involved variables and parameters in the energy expression of eq 4 according to the following scheme:

load (P < 0) indicate tensile forces that tend to pull the particle away from the surface. The exact profile of the spherical particle can be described by z = R − (R2 − r2)1/2, where z represents the height of a surface point relative to the particle’s lower apex and r is the planar radius of the point. The JKR model approximates the exact spherical profile by a parabola (i.e., z = r2/(2R)) with the understanding that the contact radius a is usually much smaller compared to the particle size R. The change in contact area between the opposing solid surfaces can be equivalently treated as the receding or advancing of an exterior circular crack in mode I.4 Following the approach of fracture mechanics, the stress concentration index corresponding to the JKR solution is simply KI,JKR = ((a3K/R) − P)/

A=

a U P , U̅ = , P̅ = , πwR kBT (πwR2/K )1/3

m=

⎛ KR ⎞1/3 πwR2 ⎜ ⎟ , n= ⎝ πw ⎠ kBT

(6)

One may notice that the ratio of the contact radius to the particle size (i.e., a/R) is simply A/m. The dimensionless energy as a multiple of the thermal energy unit kBT then takes the form 6090

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Article

3m2n 4 2 ⎡ 2P ̅ m ⎞⎟ m + A ⎤ 1 1 ⎛⎜ A × ⎢ + − + ln ⎥ ⎣ 3Am2 A⎠ m − A⎦ 2 4⎝m n × dA − 2 A2 (7) m

∫

Figure 1b shows the total system energy U̅ varying with contact radius A in eq 7 calculated from K = 1 MPa, R = 50 nm, and w = 0.001 J/m2 at two different loads (P̅ = −1 and −1.385, respectively). The value w = 0.001 J/m2 is chosen because the solid−solid interaction occurring in a liquid medium is normally about 1 order of magnitude smaller than that in vacuum or air, according to Leckband and Israelachvili.25 The results of the total system energy from the JKR model (eq 5), which is based on the parabolic approximation, are also normalized (by kBT) and plotted for comparison. The difference between the two cases is rather large, indicating that the parabolic approximation is no longer valid in the present regime of parameters. We should point out that the constant term in eq 5 or the one from the integration in eq 7 is arbitrary and has been chosen in a way that the bottom of energy wells starts at zero, with respect to Figure 1b. Usually, energy landscape U̅ (A) exhibits a well that corresponds to the local minimum solved from ∂U̅ /∂A = 0, where the equilibrium radius of the contact area (termed Amin) is located. The energy well is separated at a certain distance from an energy barrier, corresponding to local maximum Amax by ∂U̅ /∂A = 0. Therefore, an Arrhenius-type “activation energy”, denoted as ΔE̅, can be defined as the barrier height relative to the bottom of the well. However, as the magnitude of pulling force P̅ is increased to above a threshold, say P̅*, the local energy minimum and maximum vanish, as indicated in Figure 1b for the case of P̅* = −1.385. P̅* is commonly known as a pull-off force that leads to ballistic particle detachment from the substrate in the deterministic models2−4,6−19 and can be determined through the critical condition that the two roots of ∂U̅ /∂A = 0 become identical. As the most important parameter in the landscape of total system energy, the activation energy ΔE̅ arises from the competition between the elastic and surface energy, which has no analytical expression and is generally a function of particle size R and substrate modulus K and is strongly influenced by the external pulling force P̅, as numerically determined in Figure 2 for various cases. ΔE̅ monotonically decreases with increasing magnitude of P̅ until the energy barrier vanishes at P̅*. When the K value is fixed (Figure 2a, K = 100 kPa for example), the rate of change in ΔE̅ with increasing magnitude of P̅ is faster for larger values of particle size R, as reflected by different slopes of the curves in Figure 2a. In other words, ΔE̅ is more sensitive to force P̅ for larger particles. In contrast, when R is fixed at a constant value (Figure 2b, R = 5 nm), ΔE̅ is found to be more sensitive to force variation when the substrate is softer (Figure S1 in the Supporting Information), and increasing substrate stiffness results in a larger force regime for the existence of positive ΔE̅. More importantly, the magnitude of activation energy ΔE̅ becomes comparable to the scale of kBT in certain combinations of particle size R, substrate modulus K, and external load P̅, and hence the state of adhesive contact between the nanoparticle and the substrate, represented by the contact radius A (or a in real dimensions), is susceptible to fluctuations of the thermal environment. Obviously, in the

Figure 2. Activation energy ΔE̅ (in units of kBT) defined as the height of the energy barrier relative to the well, influenced by the tensile load P̅ (ranging from −1.5 to 0) for various values of (a) particle size R and (b) substrate modulus K. The substrate modulus K is fixed at 100 kPa in plot a, the particle size R is fixed at 5 nm in plot b, and the adopted value for the work of adhesion is w = 0.001 J/m2.

range where the activation energy is orders of magnitude higher than kBT, the motion of the adhesion state is restricted to the proximity of the well’s bottom and the probability of barrier crossing is vanishingly small, which reduces to the usual deterministic description of the equilibrium contact radius.

3. THEORETICAL BASIS: DIFFUSIVE PARTICLE DETACHMENT Following the classical approach by Kramers28 and a recent analysis developed by Freund29 to describe the breaking of molecular bonds, we consider the detachment of the particle from the substrate at various pulling forces as the flux of the probabilistic adhesion state outward passing over an energy barrier in the coordinated energy landscape. We assume that a large ensemble of nominally identical systems can be represented by the probability density function ρ(a, t) defined as the state probability of having the particle−substrate adhesion at contact radius a at time t. In this description, the detachment of the particle is regarded as the distributed ρ(a, t) beyond the energy well, and the intact fraction of the ensemble remaining in the well is the survival probability of the adhesion, defined as S(t) = ∫ Ramaxρ(a, t) da that varies with time t, with amax being the particular contact radius corresponding to the local maximum of system energy. We proceed by denoting the flux of the adhesion state in coordinate a at time t by j(a, t). For the local conservation of states, the rate of change in ρ(a, t) has to balance the negative of the flux divergence, namely, ∂tρ(a , t ) = −∂aj(a , t ) 6091

(8)

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With the constitutive assumption that the flux of the adhesion state is linearly proportional to the local gradient of the chemical potential, which arises from the spatial distribution of the adhesion state as well as the energy landscape of the system (in reference to the general derivations by Freund29), j(a, t) can be related to the probability density function ρ(a, t) and the landscape of potential energy U(a) (in eq 4) through29 ⎛ ⎞ ρ (a , t ) j(a , t ) = −D⎜∂aρ(a , t ) + ∂aU (a)⎟ kBT ⎝ ⎠

(9)

where D = kBT/(6πηR) is the diffusivity of the nanoparticle in the detachment process according to the Stokes−Einstein relation. R is the particle size, and η is the viscosity of the surrounding medium. Throughout the following calculations, η = 1 mPa·s30 is adopted for nanoparticle−substrate adhesion in aqueous environments unless stated otherwise. Combining eqs 8 and 9 and using the normalization scheme in eq 6, we obtain the Smoluchowski partial differential equation governing the spatiotemporal evolution of the probability density function ∂tρ ̅ (A , t ) =

D ∂A(∂Aρ ̅ (A , t ) + ρ ̅ (A , t ) (πwR2/K )2/3 × ∂AU̅ (A))

(10)

where ρ̅ = (πwR2/K)1/3ρ and U̅ (A) is explicitly given in eq 7. The governing equation in eq 10 is subjected to the conditions ρ̅(0, t) = 0 (absorbing boundary) and j(m, t) = 0 (reflecting boundary), where A = 0 and A = m (recalling that a/R = A/m) are the lower and upper limits of possible contact radius. The initial condition is set as ρ̅(A, 0) = δ(A − Amin), where δ is the Dirac delta function and Amin represents the position of the well’s bottom. The finite element method was adopted to obtain solutions to the boundary/initial value problem described in eq 10. Specifically, discrete values of the probability distribution function ρ̅(A, t) at 1001 equally spaced points within the interval 0 ≤ A ≤ m were calculated and updated at every time step. As pointed out earlier, for a large number of nominally identical particle−substrate systems, the probabilistic particle detachment can be extracted by calculating the fraction of all events still being confined within the energy well (from Amax to m) after a total elapsed time of t denoted as S(t). Notice that S(t) equivalently represents the probability that a single particle−substrate pair remains adhered after time t, and hence the likelihood of forcible separation taking place is simply 1 − S(t). Choosing K = 1 MPa, R = 10 nm, and w = 0.001 J/m2 as a representative example, the energy profiles corresponding to different levels of pulling force P̅ are shown in Figure 3a. Evolutions of the probability distribution function ρ̅(A, t), under these energy landscapes were calculated for sequential time points using the numerical scheme described above, and the results for P̅ = −0.5 are illustrated in Figure 3b. The value of survival probability S at any particular time point t is the area under the corresponding curve of ρ̅(A, t) versus A. Obviously, S(t) has an initial value of unity and asymptotically approaches zero when time t is sufficiently large, and its timevarying behavior reflects the frequency of particle detachment from the substrate (Figure 3c). When the S(t) versus t curves that are obtained are used, the average detachment time (i.e., the reciprocal off rate of the particle) is expected to be

Figure 3. (a) Landscapes of the total system energy at various levels of applied load P̅ when K = 1 MPa, R = 10 nm, and w = 0.001 J/m2. (b) Time evolution of the probable contact radius A when the applied load P̅ is −0.5. (c) Semilogarithmic plots of the survival probability S of nanoparticle−substrate adhesion vs time t, corresponding to the energy profiles in plot a, respectively. The symbols in plot c represent the certain time points appearing in plot b.

Td =

∫0

∞

−S(̇ t ) ·t dt

(11)

which is a kinetic description of the diffusive and statistical process of particle detachment, similar to those in the study of force-induced dissociation of molecular bonds29,31−33 but intrinsically different from the ballistic and deterministic description of particle pull off in previous models.2−4,6−19 We further examine the dependence of detachment time Td on the pulling force P̅ for various values of particle radius R (Figure 4a) and substrate modulus K (Figure 4b). Interestingly, our results suggest that particle detachment can occur even if P̅ is below the critical pull-off force, predicted from Maugis26 and marked by the dashed lines for individual cases (Figure 4). The average time for detachment to take place increases almost 6092

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delivery and self (or forced) separation in cell−nanoparticle interaction. Several important aspects of the problem have been neglected in the present study, which certainly warrant future investigation. For example, we have assumed that the diffusivity of nanoparticles obeys the Stokes−Einstein relation, which is expected to be influenced when the nanoparticles and substrate are in close proximity or in contact, as suggested by Banerjee and Kihm.34 Moreover, whether the adhesive interaction between nanoparticles and the substrate can be described by the Dupré work of adhesion remains to be fully clarified on the nanometer scale. It should also be pointed out that the present study is based on the 1D configuration of the adhesion state, which may be extended to higher dimensions to reflect the possibility of nanoparticle−substrate separation by rolling. The development of more sophisticated models to address these issues is currently underway.

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ASSOCIATED CONTENT

S Supporting Information *

Numerical plots of the landscapes of system energy U̅ (A) for different values of substrate modulus K as the pulling force P̅ varies from −0.1 to −0.2 to demonstrate that the activation energy ΔE̅ is more sensitive to force variation when the substrate is softer. This material is available free of charge via the Internet at http://pubs.acs.org/.

■

Figure 4. Average time of particle detachment, denoted as Td, as a function of the applied load P̅ for various values of particle size R in plot a and substrate modulus K in plot b. The substrate modulus is fixed at 100 kPa in plot a, the particle size is fixed at 5 nm in plot b, and w = 0.001 J/m2 is adopted for the calculations. P̅* values corresponding to the critical pull-off forces from the deterministic Maugis model26 are indicated by dashed lines for comparison.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the Thousand Young Talents Program of China, the National Natural Science Foundation of China (11202184 and 11321202), and the Fundamental Research Funds for Central Universities (2012QNA4023) of China.

exponentially with decreasing magnitude of pulling force, as indicated in the semilogarithmic plots. Furthermore, the difference between the present statistical description and the previous deterministic model26 becomes more and more pronounced as the particle size becomes smaller. When R = 5 nm, the particle spontaneously separates from the substrate in less than 1 s even without any disruptive force, irrespective of substrate modulus (Figure 4b), which is in direct contrast to what one would expect from deterministic considerations.

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4. CONCLUSIONS We examine the statistical failure of the adhesive contact between nanoparticles and compliant substrates by taking into account the exact spherical geometry and contributions from both the applied pulling force and thermal excitations. Because the combined elastic and surface energy associated with nanometer-sized particles becomes comparable to the scale of thermal energy (kBT), their detachment from the substrates must be interpreted in probabilistic rather than deterministic terms. Furthermore, we show that the diffusive process can be described by the Smoluchowski partial differential equation in closed form. The time-varying survival probability of nanoparticle−substrate adhesion naturally leads to an expected detachment time that, despite being influenced by factors such as particle size, substrate compliance, and applied load, can typically be on the order of seconds, highlighting the relevance and implications of the present study in problems such as drug 6093

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