How To Measure Work of Adhesion and Surface Tension of Soft

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Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

How To Measure Work of Adhesion and Surface Tension of Soft Polymeric Materials Yuan Tian,† Maria Ina,‡ Zhen Cao,† Sergei S. Sheiko,*,‡ and Andrey V. Dobrynin*,† †

Department of Polymer Science, University of Akron, Akron, Ohio 44325, United States Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3220, United States

‡

S Supporting Information *

ABSTRACT: Knowledge of the work of adhesion and surface tension directs the design of new materials for coatings, adhesives, and lubricants. We develop an approach to determine both properties from analysis of equilibrium indentations of rigid particles in contact with soft polymeric materials. In accord with coarse-grained molecular dynamics simulations, the indentation depth is described by the crossover expression combining together the adhesion and wetting models, which takes into account both the elastic energy of the contact and full surface free energy change outside and inside the contact area. The crossover expression is applied to obtain the work of adhesion and substrate surface tension for polystyrene (PS), carboxylmodified polystyrene (PS-COOH), and poly(methyl methacrylate) (PMMA) particles in contact with poly(dimethylsiloxane) (PDMS) networks made of brushlike and linear chains. This analysis results in the work of adhesion W = 48.0 ± 2.9 mN/m for PS/PDMS, W = 268.4 ± 27.0 mN/m for PS-COOH/PDMS, and W = 56.2 ± 2.4 mN/m for PMMA/PDMS and the surface tension of the PDMS substrate to be γs = 23.6 ± 2.1 mN/m.

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INTRODUCTION Understanding interfacial interactions between rigid particles and soft substrates has important implications for surface science and technology.1−5 It explains why geckos could defy gravity and climb the vertical walls6 and is broadly used for design of lubricants,7 coatings,8 paints, and adhesives.9−11 The current advances in this area are based on realization that the contact phenomena are controlled by interplay between capillary, adhesive, and elastic forces.1,12 Johnson, Kendall, and Roberts (JKR)13 and Derjaguin, Muller, and Toporov (DMT)14 elucidated particle indentation as a balance of elastic deformation and the long-range van der Waals forces acting in the contact area. The JKR model describes contact of relatively large particles with soft or rigid substrates, whereas the DMT theory describes the behavior of small particles on rigid substrates.15,16 Over the years, the JKR and DMT models have been routinely used to measure the work of adhesion between compliant surfaces in probe detachment experiments.17−19 However, the obtained values depend on the pulling rate, which is particularly significant for soft substrates with modulus R*), the contact behavior is described by the JKR model (adhesion regime). The behavior of smaller particles with Rp < R* is determined by optimization of the surface free energies (wetting regime). Eventually, the particle will submerge in the substrate when Δh = 2Rp. Figure 2 summarizes different particle−substrate interaction regimes. Crossover Expression: From Wetting to Adhesion. Using results for the indentation depth obtained in the wetting and adhesion regimes in terms of Δh (eqs 6 and 11, respectively), we can write the following crossover expression which accounts for both wetting and adhesion contributions: 0 = −2πWR p + 2πγsΔh +

16 GR p1/2Δh3/2 3

0=

(13)

yh = Δh/Δh*

yh =

0 ≈ − x R + yh + δx R

yh

x R = (δyh

+

2

3

for yΔ ≥ 3πτ /4

(21)

2 ⎛ 3π τ ⎞ 1/2 16 2 3/2 ⎛ 3π τ ⎞ ⎟y ⎟ B= y ⎜1 − τ ⎜1 − + 3 2 ⎜⎝ 28 yΔ ⎟⎠ Δ 9π Δ ⎜⎝ 16 yΔ ⎟⎠

(22)

3/2

For system parameters with yΔ = 3πτ/4, the neck disappears and the indentation depth is equal to Δh = Δ = a2/2Rp. In this case, we have only one equation to solve to obtain Δh

(15)

2

δ yh + 4yh ) /4

0 = γsΔh − WR p +

(16)

1/2

for Δh < 3πγs/4G

In terms of dimensionless variables, eq 23 reduces to 0 ≈ −x R + yh + ϕx R 1/2yh 3/2

for yh < 3πτ /4

(24)

where numerical constant ϕ = 7√2/3π ≈ 1.05. By solving eq 24 for xR as a function of yh, we obtain x R = (ϕyh 3/2 +

ϕ2yh 3 + 4yh )2 /4

for yh < 3πτ /4 (25)

2

− ΔhΔ ) + πγsΔ

Combination of eqs 20, 21, and 25 allows plotting the reduced indentation depth yh as a function of xR in the entire interval of the substrate deformations (Figure 3). Figure 3 reveals that the explicit consideration of the neck contribution into the system free energy breaks down universality of the crossover expression (eq 16). It is important to point out that the largest deviation from the universal line occurs in the region of the moderate substrate deformations for which the expression for the elastic energy derived under assumption of small indentations is invalid. In this case, we can consider curves calculated using eqs 18 and 20 as an upper

Equilibrium indentations Δ and Δh are obtained by minimizing the system free energy eq 17 resulting in the following system of equations: 2 πγsR p1/2Δ−1/2 (3Δ − Δh) + 2π (γsΔ − WR p)

+ 2 2 GR p1/2Δ−1/2 (Δh − 2Δ)2

7 2 GR p1/2Δh3/2 3π

(23)

⎡ ⎤ 4 4 − 2πWR pΔ + 4 2 GR p1/2⎢Δh2Δ1/2 − ΔhΔ3/2 + Δ5/2⎥ ⎣ ⎦ 3 5 (17)

0=

B2 + 4yΔ )2 /4

7

This universal function describes a crossover between wetting and adhesion regime with increasing the particle size.36 From Wetting to Adhesion: Generalized Model. The JKR approach to the contact problem does not take into account the surface free energy of the neck connecting particle with substrate. To resolve the issue, we combine the elastic (eq 8) and surface (eq 7) energy contributions as ΔFtotal(Δ, Δh) ≈ 2

(20b)

where parameter B is defined as

(14)

2 πγsR p1/2(Δ3/2

π 2 yΔ + τ 3 4

x R = (B +

where numerical constant δ = 8/π√3 ≈ 1.47. By solving this equation for xR, we obtain 3/2

(20a)

where we introduced dimensionless parameter τ = (γs/W)1/2. Solving eq 20a for xR, we obtain

In these variables, eq 13 reduces to 1/2

⎛ 3π τ ⎞ 1/2 1/2 ⎟x R y τ ⎜⎜1 − + yΔ − x R Δ 3 2 ⎝ 28 yΔ ⎟⎠ 7

2 16 2 3/2 1/2⎛ 3π τ ⎞ ⎟ + y x R ⎜⎜1 − 9π Δ 16 yΔ ⎟⎠ ⎝

The indentation depth Δh is a solution of eq 13. The numerical coefficient in front of the elastic term is obtained from the condition that the solution of the eq 13 in the adhesion regime should reproduce expression for Δh obtained in the JKR approximation (see eq 11). It is convenient to introduce new dimensionless variables by normalizing the indentation depth Δh and particle size Rp by Δh* and R*, respectively, as x R = R p/ R * ;

(18b)

(18a) C

DOI: 10.1021/acs.macromol.8b00738 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules ⎡ ⎛ Rp R p2 − a2 ⎛ R p + 0 = 8a⎢G⎜⎜Δh − + ln⎜⎜ ⎢ ⎝ 2 4a ⎝ Rp − ⎣

a ⎞⎞ π ⎤ ⎟⎟ − γs⎥ a ⎟⎠⎟⎠ 4 ⎥⎦ (28b)

Introducing dimensionless variable for contact radius za = a /a*

a* =

and

Δh*R * = γsG−1

(29)

Similar to eqs 20, we can rewrite eqs 28 in dimensionless form as 0 = yΔ − x R +

+

Figure 3. Comparison between approximate solution eq 16 (dashed line) and exact solutions for yh as a function of xR for different ratios of γs/W: 0.5 (red line), 1.0 (purple line), and 2.0 (blue line). Inset magnifies the difference between solutions in the interval of 1 < xR < 500.

yh = τ 2

za =

za =

ysΔ,

(31)

the neck height is equal

2 τ 2x R − yh z τ ⎛ τx + za ⎞⎞ 2⎛ ⎜⎜yh − a ln⎜ R ⎟⎟⎟ πτ ⎝ 2 ⎝ τx R − za ⎠⎠ 2τ 2x y − y 2 R h h

τ 2x 2 − za 2 ⎛ τx R + za ⎞⎤ x 4 ⎡ za⎢yh − τ 2 R + τ R ln⎜ ⎟⎥ π ⎢⎣ 2 4za ⎝ τx R − za ⎠⎥⎦

2x R yh − yh 2 /τ 2

(32a) (32b)

Figure 4 shows comparison between exact solution using Maugis approximation for yh as a function of xR for different ratios of γs/W and approximate solution eq 16. As evident from Figure 4, the maximum deviation between our crossover expression eq 16 and the solution based on the Maugis expression for the elastic energy of indentation is also observed in the interval of system parameters where xR ≈ 1. Thus, this discrepancy could not be explained by the approximation of the contact and should be viewed as a result of the nonlinear effects in the substrate deformation when Δh approaches Rp as being pointed out by Greenwood.43 It is important to point out that accounting for nonlinear effects in the substrate deformation will increase a penalty for creating deeper indentations and will move a curve closer to crossover expression eq 16. To demonstrate significance of the nonlinear effects in substrate deformation, we have replotted our previous coarsegrained MD simulations data for spherical particles26,27 and supplemented them with new MD simulations of a rigid indenter with a lens-like shape of height hL and radius of curvature Rp = 36.0σ, 60.0σ, 100.0σ, 200.0σ, 300.0σ, 500.0σ, and 1000.0σ in contact with dense networks having a shear modulus of G = 0.498, and 0.833kBT/σ3 (where σ is coarse-

ΔFtotal(Δ, Δh) ≈ 2πγs(Δ − Δh)a + πγsΔ2 − 2πWR pΔ (27)

By minimizing this equation with respect to Δ and Δh, we obtain 0 = 2π (γsΔ − WR p) + 2πγs

a ⎞⎞ ⎟⎟ a ⎟⎠⎟⎠

τ 2x 2 − z s ⎛ τx + zas ⎞ π xR − τ R s a ln⎜ R τ s⎟ + 2 4za 4 ⎝ τx R − za ⎠

0 = yh − x R +

In the limit a/Rp ≪ 1 this expression reduces to the elastic energy in the JKR approximation (see eq 8). Since the indentation may be large, we have to use the exact expression for relation between contact radius a and indentation Δ, a2 = 2RpΔ − Δ2, to rewrite eq 8 as

⎛ a ⎛ Rp + × ⎜⎜Δh − ln⎜⎜ 2 ⎝ Rp − ⎝

(30c)

In the opposite limit when yΔ = yh < to zero such that

(26)

2R pΔ − Δ2

(30b)

2

∫

+ 4G

2

2x R yΔ − yΔ2 /τ 2

yΔs = τ 2

⎤ a/R p ⎛ 1 + x ⎞⎥ ⎟ dx x 2 ln 2⎜ ⎝ 1 − x ⎠⎥ 0 ⎦

R p(3Δ − Δh) + Δ(Δh − 2Δ)

⎛ τx + za ⎞ π τ x − za xR ln⎜ R −τ R ⎟+ τ 2 4za 4 ⎝ τx R − za ⎠ 2

Note that this system of equations is only valid for nonzero (finite) neck dimensions or for yΔ ≥ ysΔ where

+

M + Uelast (a , Δh)

2 τ 2x R − yΔ z τ ⎛ τx + za ⎞⎞ 2⎛ ⎜⎜yh − a ln⎜ R ⎟⎟⎟ 2 ⎝ τx R − za ⎠⎠ 2τ 2x y − y 2 πτ ⎝ RΔ Δ

(30a)

⎡ ⎛ R p2 − a2 M (Δh , a) = 4G⎢Δh2a − Δh⎜⎜R pa − Uelast ⎢ 2 ⎝ ⎣ a ⎞⎞ R 3 ⎟⎟⎟ + p a ⎠⎟⎠ 4

2τ 2x R yΔ − yΔ2

2

bound for dependence of the indentation depth produced by a particle in a substrate as a function of the particle size. Either considering the exact indentation shape or accounting for nonlinear effects in substrate elastic energy should result in shallower substrate indentations and bring the corresponding curves closer to the crossover expression. In the following section, we use Maugis’ approximation42 for the elastic energy of indentation. Analysis of Indentation in Maugis’ Approximation. Maugis used an exact expression for the indentation shape produced by a particle in an elastic substrate in evaluating the elastic energy contribution.42 The elastic energy of the deformed substrate in this approximation is

⎛ Rp + × ln⎜⎜ ⎝ Rp −

τ 2x R (3yΔ − yh ) + yΔ (yh − 2yΔ )

Rp − Δ 2R pΔ − Δ2

2

(28a) D

DOI: 10.1021/acs.macromol.8b00738 Macromolecules XXXX, XXX, XXX−XXX

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from equilibrium indentation data of particles of different sizes on substrates with different values of the shear modulus, G.

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ANALYSIS OF EXPERIMENTAL DATA In this section, we show how the work of adhesion W and substrate surface tension γs can be obtained by fitting indentation data to eq 13. For data analysis, we apply the least-squares method for set of N data points by minimizing the function J(W , γs) =

1 N

N

∑ (−W + γsεi + δGiR pi εi 3/2)2 i=1

(33)

Δhi/Rip

where εi = and parameter δ = 8/π√3 ≈ 1.47. Minimization of this function with respect to W and γs results in the following system of equations: ∂J(W , γs)

Figure 4. Comparison between approximate solution eq 16 (dashed line), exact solutions using Maugis’s approximation for yh as a function of xR for different ratios of γs/W: 0.5 (red line), 1.0 (purple line), and 2.0 (blue line), and MD simulations of rigid nanoparticles (purple squares) and lens (red stars) on gels. Inset magnifies the difference between solutions and simulations data in the interval of 0.2 < xR < 500.

∂γs +

δ N

∂W

W N

N

N

∑ εi + γs 1 ∑ εi 2 N

i=1

i=1

N

∑ GiR pi εi 5/2

(34)

i=1

∂J(W , γs)

grained bead diameter, kB is the Boltzmann constant, and T is absolute temperature). The simulation methodology is described in the Supporting Information and is based on the approach developed in refs 26 and 27. The new simulations cover an additional 2 orders of magnitude of xR between 10 and 500 to improve agreement with a crossover expression. To ensure that the shape of lens-like indenter does not influence contact properties, we have compared the results of the MD simulations of spherical particles with lens-like indenters in the range of xR values between 10 and 15. In all simulations, we independently measured the substrate shear modulus, surface tensions, and work of adhesion. Figure 4 shows simulation results plotted in terms of the universal variables, which are consistent with the analytical expression with no adjustable parameters (eq 14). Therefore, we can apply this expression to obtain the work of adhesion W and substrate surface tension γs

=0=−

= 0 = −W + γs

1 N

N

∑ εi + i=1

δ N

N

∑ GiR pi εi 3/2 i=1

(35)

Solutions of this system of equations for γs and W are γs = δ

⟨GR pε 3/2⟩⟨ε⟩ − ⟨GR pε 5/2⟩ ⟨ε 2⟩ − ⟨ε⟩2

(36)

W = γs⟨ε⟩ + δ⟨GR pε 3/2⟩

(37)

where we introduced an average value of a parameter N ⟨A⟩ = N −1 ∑i = 1 Ai over the entire data set. We have studied three types of polymer particles: (i) monodisperse polystyrene (PS) (Rp = 0.225, 0.5, 1.0, 1.3, 1.75, 3.3, and 45 μm), monodisperse carboxyl-modified polystyrene (PS-COOH) (Rp = 0.25, 0.5, 1.0, 1.5, 3.0, and 10 μm), and

Figure 5. SEM micrographs of silica, PS, and PS-COOH particles with radius Rp = 0.5 μm residing on soft PDMS substrates with different modulus. E

DOI: 10.1021/acs.macromol.8b00738 Macromolecules XXXX, XXX, XXX−XXX

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Figure 6. Normalized particle height GΔh increases with normalized particle size GRp for (a) PS (blue triangles) and silica (red triangles) particles, (b) PMMA (orange triangles), and (c) PS-COOH (green triangles) particles in contact with PDMS substrates. Panel (a): lines are the best fit to eq 13 with the surface tension of the PDMS substrate γs = 23.6 ± 2.1 mN/m and work of adhesions WPS/PDMS = 48.0 ± 2.9 mN/m (blue line), WSiO2/PDMS = 47.4 ± 3.0 mN/m (red line). Panel (b): line is the best fit to eq 13 with the surface tension of the PDMS substrate γs = 23.6 ± 2.1 mN/ m and work of adhesions WPMMA/PDMS = 56.2 ± 2.4 mN/m. Panel (c): thick black line is the best fit to eq 13 with γapp = 177.0 ± 24.3 mN/m and Wapp = 252.3 ± 29.3 mN/m. Solid green line is the best fit to eq 13 for interval GRp > 200 mN/m with WPS‑COOH/PDMS= 268.4 ± 26.5 mN/m and fixed γs = 23.6 ± 2.1 mN/m. It continues as a dotted line in the engulf ing regime.

Table 1. Work of Adhesion and Surface Tension substrate

G [kPa]

glass

PDMS

glass PS PS-COOH PMMA

PDMS PDMS PDMS PDMS

PS glass

PU PU

1.0 28.3 83.3 166.7 3.3−583.0 3.3−583.0 3.3−583.0 3.3−32 E [MPa] 5.0 5.0 0.045 0.25, 0.7 3.83, 41.7

particle

a

Rp [μm] 3−30

W [mN/m]

ref

W [mN/m]

γs [mN/m]

24 72 80 61

25

54.6 ± 1.0

25.8 ± 1.1b

0.2−1.5 0.225−45 0.25−10 0.58−52.1 171a 170a 44a

1.0−6.25 0.5−60 4−100 0.5−125 11, 20

120a

± ± ± ±

3.0b 2.9c 26.5c 2.4c

36

47.4 48.0 268.4 56.2

37 38 40 41 39

142.7 ± 10.2d 109.3 ± 3.8d

23.6 23.6 23.6 23.6

± ± ± ±

2.1b 2.1 2.1 2.1

Poisson’s ratio ν = 1/3 was used in the JKR equation. Surface tension is calculated from eqs 36 and 37. cW is calculated for γs = 23.6 ± 2.1 mN/m. Work of adhesion is calculated from eq 38 which is derived for Poisson’s ratio ν = 0.5. b

d

polydisperse poly(methyl methacrylate) (PMMA) (Rp = 0.58− 52.1 μm) particles. To measure indentation depth as a function of substrate stiffness, the particles were deposited on substrates made of bottlebrush, comb-like, and linear chain PDMS networks with different shear moduli (G = 3.3, 8.1, 15.9,

32.0, 54.7, and 583.0 kPa). The studies are performed at room temperature below Tg = 100 °C of polystyrene and Tg = 125 °C of PMMA; i.e., particles are rigid. The contact properties of the polymeric particles are compared with the corresponding data obtained for silica particles on similar substrates.36 Figure 5 F

DOI: 10.1021/acs.macromol.8b00738 Macromolecules XXXX, XXX, XXX−XXX

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Figure 7. Dependence of the parameter Ga on (GRp)2/3 for glass (a) and polystyrene (b) particles in contact with polyurethane substrates with shear modulus varying between 0.015 and 13.9 MPa. Solid lines are the best fit to the JKR equation with the work of adhesion W = 109.3 ± 3.8 mN/m (a) and W = 142.7 ± 10.2 mN/m (b) in the interval 102 < (GRp)2/3 < 104 (mN/m)2/3.

This suggests that stabilization of the particle indentation in the wetting regime GRp < 10 is due to nonlinear deformation of network strands, which results in effective renormalization of both work of adhesion, Wapp, and surface tension, γapp. Since both work of adhesion and surface tension for GRp < 100 are controlled by substrate deformation, it is more appropriate to refer to these quantitates as apparent work of adhesion, Wapp, and apparent surface tension, γapp. Note that recovered scaling relation Δh ∝ R p in the wetting regime means that renormalization of the work of adhesion, Wapp, and surface tension, γapp, occurs within a layer which is thinner than the particle size. Because of the bulk contribution, the apparent surface tension, γapp, cannot be regarded as a surface stress in the Shuttleworth’s definition.47 Also, Wapp deduced from the fitting procedure may be different from the value obtained by the JKR-like experiments in the limit of linear substrate deformation. To relate with the JKR approach and account for a correction due to the surface tension effects, we apply eq 37 to a subset of the data with GRp > 100 mN/m (adhesion regime). Using a fixed value of γs = 23.6 ± 2.1 mN/m, we obtain WPS‑COOH/PDMS = 268.4 ± 26.5 mN/m (solid green line in Figure 6c). In the wetting regime (GRp < 10) we obtain a particle−substrate affinity parameter Wapp/γapp = 0.69 ± 0.03, which is close to Wapp/γapp = 0.7 calculated from fitting all of the data. This example shows that our approach can account for renormalization of the work of adhesion and surface tension by considering them in the wetting and adhesion regime as apparent values. We can also apply the approach developed here to reanalyze data by Rimai et al.37−41 obtained from SEM micrographs of glass particles (Rp = 0.5−125 μm) on polyurethane substrates G = 1.5 × 10−2−13.9 MPa and polystyrene particles with Rp = 1.0−6.25 μm on polyurethane substrates with G = 1.67 MPa. To demonstrate breakdown of the JKR model for this system in Figure 7a, we replot data from refs 38−41 in terms of Ga vs (GRp)2/3 which is correct scaling dependence in the adhesion regime. For the softest substrates with G = 1.5 × 10−2 MPa (see Figure 7a), we observe deviation from the JKR scaling due to crossover to the wetting regime. However, there are not enough data points deep in the wetting regime to extract surface tension from the data fit. Therefore, we use the JKR expression for a contact radius

shows snapshots of the particle−substrate configurations for particles with Rp = 0.5 μm residing on the substrates with different stiffness. All of the studied systems exhibit an increase of particle height h with increasing G. However, the magnitude of indentation depends on the particle chemical composition. PS-COOH particles have the strongest interactions with substrate as seen from the lowest height of the particles above the substrate. PS and silica particles display nearly identical indentation properties that are also very similar to the behavior of PMMA particles (see Supporting Information). The SEM images are analyzed to obtain Δh = 2Rp − h from the height of the particle h above the substrate surface as described elsewhere.36 To collapse data for different values of the substrate shear modulus and particle sizes, we plot GΔh as a function of GRp (Figures 6a−c). The plots clearly show two characteristic regimes. In the wetting regime, the indentation depth is proportional to the particle size, Δh ∼ Rp, while in the adhesion regime, Δh depends on the substrate modulus as Δh ∼ Rp1/3G−2/3. Applying eqs 36 and 37 to the data sets in Figure 6a, we obtain the work of adhesion W and surface tension γs of the PDMS substrate, which are summarized in Table 1. For the PS/ PDMS system, the work of adhesion is equal to WPS/PDMS = 48.0 ± 2.9 mN/m for surface tension γs = 23.6 ± 2.1 mN/m, which are consistent with WPS/PDMS = 49 ± 2 mN/m44 and 45 mN/m45 obtained from the JKR experiments and literature data for the surface tension for PDMS networks.17,46 Given the close overlap of the PS and silica data points in the entire range of GRp in Figure 6a, the work of adhesion of the silica/PDMS system36 WSiO2/PDMS = 47.4 ± 3.0 mN/m is close to WPS/PDMS. Furthermore, the work of adhesion of PMMA partciles WPMMA/PDMS = 56.2 ± 2.4 mN/m obtained at fixed γs = 23.6 ± 2.1 mN/m is in excellent agreement with WPMMA/PDMS = 57 ± 1 mN/m reported in ref 44. Analysis of the data for PS-COOH/PDMS systems (see Figure 6c) using eqs 36 and 37 leads to a large value of the surface tension for PDMS substrate γapp = 177.0 ± 24.3 mN/m and work of adhesion Wapp = 252.3 ± 29.3 mN/m. As explained below, these surface tension and work of adhesion are considered as apparent quantities. Using the known surface tension of PDMS γs = 23.6 ± 2.1 mN/m, we conclude that PSCOOH particles should submerge long before a crossover to a wetting regime is reached (Δh ≈ WappRp/γs ≈ 10.7Rp > 2Rp). G

DOI: 10.1021/acs.macromol.8b00738 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Ga =

⎛ 9π ⎞1/3 ⎜ W ⎟ (GR p)2/3 ⎝ 8 ⎠

deformation regimes. In the nonlinear deformation regime, the obtained values reflect renormalization of the materials’ contact properties. Furthermore, this analysis can be extended to particles and substrates of arbitrary rigidity following methodology developed in refs 26 and 27. This method will become a useful addition to the classical probe detachment technique.

(38)

and obtain the work of adhesion for glass/polyurethane systems to be W = 109.3 ± 3.8 mN/m where Poisson’s ratio of the rubbery polyurethane substrate is assumed to be ν = 0.5.48 The errors are estimated using the jackknife method49 by removing data points and recalculating the values of the fitting parameters. As follows from Figure 7b, the entire data set for PS/polyurethane (PU) systems is in the adhesion regime. Therefore, eq 38 can also be applied to obtain the work of adhesion of PS/PU systems W = 142.7 ± 10.2 mN/m. The results are summarized in Table 1.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00738. Simulation details, sample fabrication and characterization, data sets (PDF)

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CONCLUSIONS We have shown that an approach based on the simple crossover expression (see eq 13) can be used to obtain work of adhesion and surface tension of the polymeric substrates by analyzing dependence of the substrate indentation produced by particles of different sizes. Collapse of both the experimental and simulation data in Figure 8 validates the crossover expression

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (A.V.D.). *E-mail [email protected] (S.S.S.). ORCID

Yuan Tian: 0000-0002-7277-1408 Zhen Cao: 0000-0001-5499-3130 Sergei S. Sheiko: 0000-0003-3672-1611 Andrey V. Dobrynin: 0000-0002-6484-7409 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors express their gratitude to Dr. R. Style and Prof. E. Dufresne for providing original data sets. The authors gratefully acknowledge funding from the National Science Foundation (DMR 1407645, DMR 1436201, and DMR 1624569).

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REFERENCES

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Figure 8. Dependence of the reduced substrate indentation depth, Δh/Δh*, on the reduced particle size, Rp/R*. The experimental data by Style et al.25 for silica microspheres on PDMS substrates are shown by brown filled triangle, and data by Ina et al.36 for silica microspheres on PDMS substrates are represented by red filled inverted triangles. Data sets for PS microspheres, PMMA microspheres, and PS-COOH microspheres on PDMS substrates are shown by blue filled inverted triangles, orange filled inverted triangles, and green filled inverted triangles, respectively. The MD simulations data for elastic nanoparticles on elastic substrates are shown by green filled squares, data for nanoparticles on rigid substrates are shown by blue filled squares, and data for rigid nanoparticles and lenses on elastic substrates are shown by purple filled squares and red filled stars, respectively. The dashed line corresponds to eq 16.

and its universality for describing particle−substrate interactions. There are two distinct scaling regimes in the indentation plots. For small particles with sizes Rp < R*, the capillary forces dominate equilibrium contact properties (wetting regime) such that the data follow scaling relation Δh/ Δh* ≈ Rp/R*. However, for large particles with radius Rp > R*, the elastic and adhesion forces acting in the contact area become dominant (adhesion regime), resulting in Δh/Δh* ≈ (Rp/R*)1/3. The developed here approach provides an alternative for obtaining the surface tension and the work of adhesion of soft polymeric materials in both linear and nonlinear substrate H

DOI: 10.1021/acs.macromol.8b00738 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.8b00738 Macromolecules XXXX, XXX, XXX−XXX