How to Measure Work of Adhesion and Surface Tension of Soft

Home , Adhesion

HOW TO MEASURE WORK OF ADHESION AND SURFACE TENSION OF SOFT

MATERIALS

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Master of Polymer Science

Yuan Tian

May 2018

ABSTRACT

Knowledge of the work of adhesion and surface tension plays an important role in design of new materials for applications such as coatings, adhesives and lubricants. We develop an approach for obtaining work of adhesion and substrate surface tension from analysis of the equilibrium indentation data of rigid particles in contact with elastic surfaces. By comparing predictions of different models of a rigid particle in contact with a soft elastic surface we show that a crossover expression taking into account contributions of the elastic energy of the contact and surface free energy change in the contact area and outside is the best in describing the results of the coarse-grained molecular dynamics simulations. This crossover expression is applied to obtain work of adhesion and surface tension of polystyrene (PS) and poly(methyl methacrylate)

(PMMA) particles on poly(dimethyl siloxane) (PDMS) substrates. This is achieved by studying depth of indentations produced by PS and carboxyl group modified PS particles with radii Rp between 0.2 and 45.0

μm and PMMA particles with sizes between 0.58 and 52.1 μm in substrates made of super-soft, solvent-free

PDMS elastomers with brush-like and linear chain network strands having modulus from 3 to 600 kPa.

Analysis of the experimental data 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 for W=56.2±2.4 mN/m PMMA/PDMS. The surface tension of the PDMS substrate is found to be γs = 23.6±2.1 mN/m.

iii

ACKNOWLEDGEMENTS

First and foremost, I would like to express the greatest appreciation to my advisor, Prof. Andrey

Dobrynin. For the first year, I was attracted and affected by Andrey’s patience, motivation, enthusiasm, and immense knowledge. He can always explain profound theories in simple and clear words. Andrey shows his professional perspective and careful attitude, which has inspired me in every aspect of science. Through a solid and highly personalized training, I have learnt a lot from conducting scientific research: thinking a lot before starting, finding important while solving questions, understanding the basic knowledge deeply, establishing a reasonable and minimum model, setting up simulations in an efficient way, analyzing data thoroughly, and discovering meaningful results. I specially appreciate Andrey for his generosity in supporting me to attend conferences and give presentations. It’s my first time to gain opportunities to present my research and communicate with many great scientists to enrich my understanding of the field. More importantly, his guidance and attitude have a deep influence on my professional and personal life. As Andrey said, “stay serious, stay hungry”, I will always keep in my mind.

I would like to thank my thesis committee Prof. Ali Dhinojwala, for his insightful comments on my

thesis. In addition, I have enjoyed his classes which allowed me to gain a historical view on what I have been

studying. Special thanks go to my collaborators Prof. Sergei Sheiko at the University of North Carolina. I

also would like to thank the department chair Prof. Coleen Pugh for her support and help in my master studies.

These acknowledgements would not be complete without mention to my former and current group

members: Dr. Zhen Cao, Mr. Heyi Liang, and Dr. Zilu Wang for their continuing support and encouragement over the years.

Last, my deepest appreciation is reserved for my family. My parents have worked hard for decades to create better educational opportunities for me and support me to finish my master program. I would not be able to achieve anything without their boundless care and love. My grandfather, one of the most important person in my life, has always encouraged me in my pursuit of learning and offered me his best love and wishes. Unfortunately, I cannot stay with him at the last moments of his life. But I will study hard and keep going with the company of his soul.

DEDICATION

My Grandfather

For being the most important person in my life

Chen

For her encouragement and love

TABLE OF CONTENTS Page

LIST OF TABLES ...... vii

LIST OF FIGURES ...... viii

CHAPTER

I. INTRODUCTION ...... 1

II. THEORETICAL ANALYSIS: PARTICLES IN CONTACT WITH A SURFACE ...... 3

2.1 Introduction ...... 3

2.2 Surface Free Energy ...... 3

2.3 Analysis of Indentation in JKR Approximation ...... 4

2.4 Crossover Expression: From Wetting to Adhesion ...... 6

2.5 From Wetting to Adhesion: Generalized Model ...... 7

2.6 Analysis of Indentation in Maugis’ Approximation ...... 9

III. MOLECULAR DYNAMICS SIMULATION ...... 11

3.1 Simulation Details ...... 11

3.2 Simulation Results ...... 13

IV. ANALYSIS OF EXPERIMENTAL DATA...... 16

4.1 Introduction ...... 16

4.2 Experimental data fitting ...... 17

4.3 Discussion ...... 19

4.4 Validation and deviation of classical model ...... 21

V. CONCLUSIONS ...... 23

REFERENCES ...... 25

vi LIST OF TABLES

Table Page

Table 3.1 Interaction Parameters ...... 12

Table 3.2 Systems’ Dimensions ...... 12

Table 4.1 Work of adhesion and surface tension...... 22

vii

LIST OF FIGURES

Figure Page

2.1 Schematic representation of substrate indentation produced by a particle with radius Rp ...... 3

2.2 Schematic representation of different particle-substrate interaction regimes. Darker color of the substrate corresponds to more rigid substrates...... 6

2.3 Comparison between approximate solution eq 2.4.4 (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), 2.0 (blue line). Inset magnifies the difference between solutions in the interval of 1 < xR < 500 ...... 9

3.1 Simulation snapshots of a rigid lens (RL) and an elastic substrate (a) before contact, and (b) in equilibrium state...... 11

3.2 Comparison between approximate solution eq 2.4.4 (dashed line), exact solutions using Maugis’s

approximation for yh as a function of xR for different ratios of γ s /W: 0.5 (red line), 1.0 (purple line), 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 ...... 15

4.1 SEM micrographs of silica, PS and PS-COOH particles with radius Rp=0.5 µm residing on soft PDMS substrates with different modulus ...... 17

4.2 Dependence of the normalized particle height GΔh on normalized particle size GRp for PS (blue triangles) and silica (red triangles) particles (a), PMMA (orange triangles) (b) and PS-COOH (green triangles) (c) particles in contact with PDMS substrates ...... 19

2/3 4.3 Dependence of the parameter Ga on (GRp) for glass (a) and polystyrene (b) particles in contact with polyurethane substrates with shear modulus varying between 0.015 and 13.9 MPa...... 21

5.1 Dependence of the reduced substrate indentation depth parameter Δh/Δh* on the reduced particle size, Rp/R* ...... 24

viii

CHAPTER I INTRODUCTION

Understanding interactions between surfaces and interfaces has an important implication for science and technology.1-5 It explains why geckos could defy gravity and climb the vertical walls,6 and it 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 manifestations of a fine interplay between capillary, adhesive and elastic forces.1, 12 This was quantified by Johnson, Kendal and Roberts (JKR)13 and by Derjaguin, Muller and Toporov (DMT)14 by accounting for contributions of both the elastic response of the deformed interface and the long-range van der Waals forces acting in the contact area. In particular, the JKR model describes contact of relatively large particles or asperities with soft or rigid substrates, while DMT theory describes

15-16 contact of small particles with rigid substrates.

Over the years the JKR and DMT models are routinely used to measure work of adhesion between compliant surfaces from a probe detachment experiments.17-19 However, the obtained values of work of adhesion depend on the pulling rate which is particular important for soft substrates with modulus <105 Pa.20-

21 For such systems the rate dependent energy dissipation in the contact area is due to the crack propagation at the interface and is a result of excitation of the different substrate relaxation modes during the probe detachment process.22-23 Furthermore, it was shown recently that the classical JKR theory breaks down for soft surfaces when the elastic forces acting in the contact area become comparable with capillary forces acting outside the contact area.24-25 This was confirmed experimentally,25 theoretically and in computer simulations.24, 26-31 This new development in our understanding of the contact phenomena in soft matter creates an opportunity to use a new model of the equilibrium contact between a particle and a substrate to simultaneously measure work of adhesion between particle and substrate and substrate surface tension. We have tested this approach by analyzing data for glass particles in contact with poly(dimethyl siloxane) (PDMS) substrates25 and for silica spheres in contact with brush-like32-37 and linear chain networks.36 Here we apply our approach to analysis of the contact properties of polystyrene (PS) and poly(methyl methacrylate) (PMMA) particles.

1 In Chapter II, we examine the contact of particles with soft gel-like surfaces and compare different models of the particle substrate interactions. We introduce our crossover expression which is the best to describe the equilibrium contact for a broad range of particle sized and substrate elastic and interfacial properties.

In Chapter III, we investigate the contact of rigid particles and lenses on elastic surfaces by the molecular dynamics (MD) simulations. The simulation results show that there is a very good agreement between simulation results and the analytical expression with no adjustable parameters.

In Chapter IV, we apply the crossover expression to obtain work of adhesion and surface tension for a new data set of polystyrene (PS), PS-COOH and PMMA particles in contact with PDMS networks of brush-like and linear strands. We have extended this analysis to the data for glass and PS particles in contact

38-42 with polyurethane (PU) substrates.

2 CHAPTER II

THEORETICAL ANALYSIS: PARTICLE IN CONTACT WITH AN ELASTIC

SURFACE

2.1 Introduction

Particle with radius Rp in contact with a substrate (see Figure 2.1) can: (i) reside at substrate interface

if indentation depth produced by a particle in a substrate, ∆h < 2Rp ; or (ii) be submerged inside a substrate,

∆h ≥ 2Rp . In all these cases the equilibrium system configuration is a result of a fine interplay between elastic energy of the indentation produced by a particle in an elastic substrate and surface free energy change due to changes in the contact surface area of particle and substrate.

Rp a hn a ∆h s ∆

Figure 2.1 Schematic representation of substrate indentation produced by a particle with radius Rp.

2.2 Surface Free Energy

In order to quantify the effect of adhesion, elastic and capillary forces in particle/substrate systems, we will approximate an indentation produced by particle in a substrate by a spherical cup–like indentation shown in Figure 2.1 and use approach developed in refs [24, 26]. Consider a rigid spherical particle in contact with soft substrate with total surface area A (see Figure 2.1). The surface free energy of this particle/substrate configuration can be written as follows

2 F (a, ∆, h ) = (A − π a + S (a, h ))γ + 2π R (2R − ∆)γ + 2π R γ ∆ (2.2.1) surf n s neck n s p p p p ps where γp is the surface energy of the particle, γs is the surface tension of a substrate, and γsp is the surface tension particle/substrate interface. In obtaining eq 2.2.1 we used the expression for surface area of a spherical

3 cup with height h, Asp (h) = 2πRp h . Sneck is the surface area of a catenoid shaped neck connecting particle

to substrate. The surface area of a catenoid of height hn = ∆ − ∆h is equal to

2  h 1  2h  S (a, h ) = πa n + sinh n (2.2.2)   neck n    a 2  a 

Note that for a catenoid its radius as a function of the neck height is given by as = a cosh(hn / a). The change of the surface free energy due to particle adsorption is equal to

∆F (a, ∆, hn ) = Fsurf (a, ∆, hn )− Fsurf (0,0,0) ≈ 2  h 1  2h  2  h  2 πγ a n + sinh n + 1 − cosh n − πa γ − 2πR ∆γ + 2πR ∆γ (2.2.3)

s  a 2     s p p p ps a a      ∆ 2 2 Substituting into eq 2.2.3 relationship between contact radius a and indentation depth h: a = 2R ∆p - ∆ , we arrive at

πγ 2  2 ∆ ∆ = a  2h  2h + πγ + − − (2.2.4) F (a, , h ) s n n ∆ − π ∆γ − π ∆γ + π ∆γ n  1 exp   s 2 Rp s 2 Rp p 2 Rp ps 2  a  a 

Introducing work of adhesion, W = γp+γs −γps, we obtain

2 πγ a  2h  2h 2 ∆ ∆ = s n + − − n + πγ ∆ F (a, , hn )  1 exp  s − 2πRpW∆ (2.2.5) 2  a  a 

It follows from the eq 2.2.5 that the minimum of the free energy change as a function of the hn and ∆ is achieved at hn=0 for which

∆h = ∆ = RpW / γ s (2.2.6)

Note that in the case of short necks connecting particle with substrate, hn << a, we can expand the terms in the square brackets

2 ∆F(a, ∆, h )≈ πγ ∆ − 2πWR ∆ + 2πγ ah (2.2.7) n s p s n

2.3 Analysis of Indentation in JKR Approximation

In the JKR model the equilibrium particle indentation in a substrate is result of optimization of the elastic

12 energy due to substrate deformation

4  3 5  ( ∆ ) = 2 2 ∆ha 1 a ∆ − + Uelast a, h 4G h a (2.3.1) 2   3 Rp 5 R p  where G is shear modulus of the substrate (here we assume that the Poisson ratio of the substrate v=1/2). The

surface energy term, − 2πWRp ∆ (the second term in the rhs of eq 2.2.7), accounts only for changes occurring in the contact area. Note that the JKR approximation for the elastic energy is only correct for small

2 indentations, ∆<

∆F (∆, ∆h) ≈ −2πWR ∆ + 4 1/ 2  2 1/ 2 4 3/ 2 4 5 / 2  (2.3.1)

2GR ∆h ∆ − ∆h ∆ + ∆ total p p  3 5  Minimizing eq 2.3.1 with respect to ∆ and ∆h one obtains

1/ 2 −1/ 2 2 0 = −2πWRp + 2 2GRp ∆ (∆h − 2∆) (2.3.2.a)

1/ 2 1/ 2  2  0 = 8 2R ∆ G ∆h − ∆ (2.3.2.b)

p   3   The solutions of eqs 2.3.2 are

2 ∆h = ∆ and 2 / 3 2 / 3 ∆ ≈ Rp (9π /16 2 ) (W / GR ) (2.3.3) 3 p Comparing eqs 2.2.6 and 2.3.3 we can conclude that crossover between JKR solution for indentation depth

(eq 2.3.3) to that obtained from balancing the surface energy of particle-substrate contact (eq 2.2.6) takes

27 place for

3/ 2 −1 −1/ 2 1/ 2 1/ 2 −1 R p≈ R* = γ s G W and ∆h ≈ ∆h* = γ s W G (2.3.4)

Thus, for particles with sizes Rp>R* we can describe contact between a particle and substrate in the framework of the JKR approach. This is so-called - adhesion regime. For small particles with Rp

wetting regime. Finally, the particle will submerge in the substrate when ∆h = 2Rp . Figure 2.2 summarizes different particle substrate interaction regimes.

Figure 2.2 Schematic representation of different particle-substrate interaction regimes. Darker color of the substrate corresponds to more rigid substrates.

2.4 Crossover Expression: From Wetting to Adhesion

Using results for indentation depth obtained in the wetting and adhesion regimes we can write a crossover expression that accounts for both wetting and adhesion phenomena. In this case the indentation depth ∆h is a solution of the equation

16 1/ 2 3/ 2 0 = −2πWR + 2πγ ∆h + GR ∆h (2.4.1) p s p 3 The numerical coefficient in the elastic term is obtained from the condition that the solution of the eq 2.4.1 in the adhesion regime should reproduce expression for ∆h obtained in the JKR approximation (see eq 2.3.3).

To proceed further it is convenient to normalize the indentation depth ∆h and particle size Rp by ∆h* and R*, and introduce new dimensionless variables

xR = Rp / R * ; yh = ∆h / ∆h * (2.4.2)

In these variables eq 2.4.1 reduces to

0 ≈ -x 1/ 2 3/ 2 (2.4.3) R + y h+ δx R y h where numerical constant δ = 8 /π 3 ≈ 1.47 . Solving this equation for xR we obtain

2 3/2 2 3 x R = (δy h + δ y +h 4 y h ) / 4 (2.4.4) Thus, universal function given by the eq 2.4.4 describes a crossover between wetting and adhesion regime

36 with increasing the particle size.

6 2.5 From Wetting to Adhesion: Generalized Model

Analysis of the contact problem between particle and elastic substrate presented above does not take into account the surface free energy of the neck connecting particle with substrate. In order to account for surface energy of the neck the expression for change in the system free energy upon particle contact with substrate should combine the elastic (eq 2.3.1) and surface (eq 2.2.7) energy contributions ∆F (∆, ∆h) ≈ 2 2πγ R1/ 2 (∆3/ 2 − ∆h∆1/ 2 )+ πγ ∆2 − 2πWR ∆ total s p s p  4 4  + 4 2GR1/ 2 ∆h2 ∆1/ 2 − ∆h∆3/ 2 + ∆5 / 2 (2.5.1)

p  3 5  Equilibrium indentations ∆ and ∆h are obtained by minimizing the system free energy eq 2.5.1

1/ 2 −1/ 2 1/ 2 −1/ 2 2 0 = 2πγ R ∆ (3∆ − ∆h)+ 2π (γ ∆ − WR )+ 2 2GR ∆ (∆h − 2∆) (2.5.2.a) s p s p p  ∆ π  0 = 8 1/ 2 1/ 2  2  (2.5.2.b) 2R ∆ p  G ∆h −  − γs    3  4  Solving eq 2.5.2.b for ∆h as a function of ∆ we can write the following expression for the neck height

∆ π γ s h = ∆ − ∆h = − (2.5.3) n 3 4 G

Analysis of this equation shows that a neck disappears when the dimensionless parameter 3πγs / 4G∆ ≈ 1 . In the case of the classical JKR solution for ∆, ∆ ≈ R (9π /16 2 ) (W / GR ) , the value of this p 2 / 3 2 / 3 p

s p 1/ 3 2 / 3 s p 1/ 3 2 / 3 parameter scales as γ /(GR ) W . Thus, the parameter γ /(GR ) W becomes on the order of

3s 2 unity when the particle size Rp becomes smaller than γ / GW .

We can rewrite eqs 2.5.2 a and b in terms of new dimensionless variables

2 7  3π τ   3π τ  =  − 1/ 2 1/ 2 3/ 2 1/ 2 − + 0 τ 1  x y + y x 16 2 y x 1 −  (2.5.4.a) R ∆ ∆ R ∆ R 3 2  28 y∆  9π  16 y∆  2 π y = y + τ ∆ (2.5.4.b) h 3 4 where we introduced y∆ = ∆ / h * and dimensionless parameterτ = γ s /W . Solving eq 2.5.4.a for xR we obtain

2 = + 2 + ≥ (2.5.5) xR (B B 4 y ∆ ) / 4 , for y∆ 3πτ / 4 and parameter B is defined as

2  7  3π τ  3/ 2  3π τ =  − 1/ 2  −  B τ 1 16 2 y 1 (2.5.6)  y∆ + ∆ 9π  16 y 3 2  28 y∆  ∆ 

In the range of system parameters where y∆ = 3πτ / 4 the neck disappears and indentation depth

2 ∆h = ∆ = a / 2R .p In this case we have only one equation to solve to obtain ∆h

7 2 1/ 2 3/ 2 0 = γ ∆h − WR + GR ∆h , for ∆h < 3πγ / 4G (2.5.7) s p p s 3π In dimensionless variables eq 2.5.7 reduces to

0 ≈ -x 1/ 2 3/ 2 , for y < 3πτ / 4 (2.5.8) R + y h+ ϕx R y h h where numerical constant ϕ = 7 2 / 3π ≈ 1.05 . Solving eq 2.5.8 for xR as a function of yh we have

2 3/2 2 3 x R = (ϕy h + ϕ y +h 4 y h ) / 4 , for yh < 3πτ / 4 (2.5.9) Thus, combination of the eqs 2.5.4, 2.5.5, and 2.5.9 allows us to plot dependence of the reduced indentation depth yh as a function of xR in the entire interval of the substrate deformations.

8 Figure 2.3 Comparison between approximate solution eq 2.4.4 (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), 2.0 (blue line). Inset magnifies the difference between solutions in the interval of 1 < xR < 500.

Figure 2.3 shows comparison between exact solution for yh as a function of xR for different ratios

of γs/W and approximate solution eq 2.4.4. It follows from this figure that the explicit consideration of the neck contribution into the system free energy breaks down universality of the particle-substrate interactions observed for crossover expression (eq. 2.4.4). 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 2.5.2 and 2.5.4 as an upper bound for dependence of the indentation depth produced by a particle in a substrate as a function of the particle size. Consideration either of the exact shape of the indentation produced by a particle in a substrate or accounting for nonlinear effects in substrate elastic energy should result to shallower substrate indentations and bring the corresponding curves closer to the crossover expression. In the following section we use Maugis’ approximation43 for the elastic energy of indentation produced by a particle in a substrate.

2.6 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.43 The elastic energy of the deformed substrate in this approximation is: 2 2    R − a  R + a  3 a / Rp  1 + x  R M 2  p p  p 2 2 U (∆h, a) = 4G∆h a − ∆h R a − ln   + dxx ln   (2.6.1) p

elast  2  R − a  4 ∫ 1 − x     p  0 

In the limit a/Rp<<1 this expression reduces to the elastic energy in the JKR approximation (see eq 2.3.1 above). Since indentation produced by a particle in a substrate could be large we have to use the exact

∆ 2 2 expression for relation between contact radius a and indentation , a = 2R p∆ - ∆ . Taking this into account, we can rewrite eq 2.3.1 as:

2 M ∆F (∆, ∆h) ≈ 2πγ (∆ − ∆h)a + πγ ∆ − 2πWR ∆ + U (a, ∆h) (2.6.2) total s s p elast

Minimizing this equation with respect to ∆ and ∆h we obtain

9 Rp (3∆ − ∆h)+ ∆ (∆h − 2∆) 0 = 2π γ ∆ −WR + 2πγ ( s p ) s 2 2R ∆ − ∆ p2 (R − ∆)   + a   (2.6.3.a) p a Rp +4G  ∆h − ln    2    ∆ − ∆ 2 R − 2R p   p   2 2   −  +    R p Rp a R p a  π 0 = 8aG ∆h − + ln   − γ  (2.6.3.b)

 s 2 4a  R − a  4    p   Introducing dimensionless variable for contact radius

−1 za = a / a * and a* = ∆h * R * = γ s G (2.6.4) we can rewrite eqs 2.6.3 in dimensionless form similar to eqs 2.5.4:

2 τ x R (3y∆ − yh )+ y∆ (yh − 2 y∆ ) 0 = y∆ − xR + + 2 2 2 τ x yR − ∆ y ∆ 2 (2.6.5.a) 2  z τ  τx + z  τ 2 x − y a R a  −   R ∆ y ln πτ h τ − 2 2  2  xR za  2 τ x yR − ∆ y ∆

2 2 2 2 x τ x − z τx + z  π = τ R −τ R a  R a  + τ yh ln (2.6.5.b)

2 4za τxR − za  4

2 2 za = 2x Ry −∆ y /∆τ (2.6.5.c)

s Note that this system of equations is only valid for nonzero neck or for y∆ ≥ y ∆ where

2 2 s 2 s s 2 x R τ R xa − z R τx + z a  π y = τ − τ ln   + τ (2.6.7)

∆ s  s  2 4z a  τxR − z a  4

s In the opposite limit when y∆ = yh < y ∆ the neck height is equal to zero such that

2  z τ  τx + z  2 τ 2 x − y a R a = − +  −   R h + 0 yh xR yh ln πτ 2 2 2 τx − z   R a  2τ xR y h− y h 2 2 2 (2.6.8.a) 4  2 x τ x − z  τx + z  − τ R R a R a z a  yh π + τ ln    2 4za  τxR − za 

2 2 za = 2x Ry −h y /τh (2.6.8.b)

Chapter III MOLECULAR DYNAMICS SIMULATIONS

3.1 Simulation Details

We performed coarse-grained molecular dynamics simulations of interaction of rigid lenses (RL) with elastic substrates (see Figure 3.1). A lens with height hL is sliced from the spherical particle of radius Rp made of beads arranged into hexagonal closed-packed (HCP) lattice. For small lenses with Rp < 500σ, each bead is connected with its closest neighbors to form a rigid body; while for large lenses with Rp = 500σ, 1000σ, to reduce computational cost, the lenses are treated as rigid bodies explicitly during simulations (see LAMMPS manual).44 The elastic (gel-like) substrates are made of crosslinked bead-spring chains with the number of monomers N = 32. The elastic modulus of the substrate is controlled by changing degree of cross- linking between chains.

Figure 3.1 Simulation snapshots of a rigid lens (RL) and an elastic substrate (a) before contact, and (b) in equilibrium state.

In our simulations, the interactions between all beads in a system are modeled by the truncated- 45 shifted Lennard-Jones (LJ) potential 6 12 6  12    σ   σ     σ  ε σ      +    r ≤ r 4 − − LJ   cut U LJ =    lj  rcut   rcut   (3.1.1)  rlj r        0 r > rcut

where rij is the distance between the ith and jth beads and σ is the bead diameter. The values of the cutoff distance rcut and the value of the Lennard-Jones interaction parameter εLJ are summarized in Table 3.1. The connectivity of the beads into polymer chains, the cross-link bonds and bonds belonging to beads forming 46 lenses are modeled by the finite extension nonlinear elastic (FENE) potential 2 1 2 r

U FENE (r) = − kspringRmax ln(1 − 2 ) (3.1.2) 2 Rmax

11 2 with the spring constant kspring = 30kBT/σ and the maximum bond length Rmax = 1.5σ. The repulsive part of 1/6 the bond potential is modeled by the LJ-potential with rcut = 2 σ and ԑLJ= 1.5kBT. The elastic substrate made of cross-linked polymer chains is placed on a solid substrate which is modeled by the external potential 9 3  2  σ   σ   − U (z) = ε w       (3.1.3) 15  z   z  

-3 where ԑw is set to 1.0kBT. The long-range attractive part of the potential z represents the effect of van der Waals interactions generated by the wall half-space.

Table 3.1 Interaction Parameters

Bead Types ԑ [kBT] rcut [σ]

RL-RL 1.5 2.5

RL-Gel 1.5 2.5

Gel-Gel 1.5 2.5

The system is periodic in x and y directions with system sizes, thickness of rigid lenses (hL), thickness of gels

(hgel), total number of atoms (Nbeads), and total number of bonds (Nbonds) listed in Table 3.2.

Table 3.2 Systems’ Dimensions

3 Rp [σ] G [kBT/σ ] hL [σ] Lx = Ly [σ] hgel [σ] Nbeads Nbonds

36.0 29.1 143.9 22.9 516,763 1,124,885 60.0 29.3 143.9 22.9 611,557 1,676,372 100.0 29.4 200.0 26.9 1,327,788 3,401,695 200.0 0.498 29.0 200.0 53.1 2,687,921 6,812,808 300.0 33.8 240.0 54.3 4,347,099 12,113,118 500.0 34.2 360.0 67.5 9,175,029 17,305,264 1000.0 38.9 560.0 77.8 23,348,492 36,576,599 36.0 29.1 143.9 22.4 516,763 1,165,309 60.0 29.3 143.9 22.4 611,557 1,716,796 100.0 29.4 200.0 26.2 1,327,788 3,306,698 200.0 0.833 29.0 200.0 52.8 2,687,921 7,002,802 300.0 33.8 240.0 37.5 3,374,269 11,057,247 500.0 34.2 360.0 67.5 9,175,029 17,893,164 1000.0 33.7 500.0 73.4 19,253,109 36,695,844

12 Simulations are carried out in a constant number of particles and temperature ensemble. The constant temperature is maintained by coupling the system to a Langevin thermostat45 implemented in LAMMPS.44 In this case, the equation of motion of the ith particle is

dvi (t) R m = Fi (t) − ξ vi (t) + Fi (t) (3.1.4) dt

where m is the bead mass set to unity for all particles in a system, vi (t) is the bead velocity, and Fi (t)

R denoted the net deterministic force acting on the ith bead. The stochastic force Fi (t) had a zero-average

R R

value and δ-functional correlations < Fi (t) Fi (t') > = 6kBTξδ (t − t') . The friction coefficient ξ is equal to

1/2 7.0 m/τLJ, where τLJ is the standard LJ-time τLJ = σ(m/εLJ) . The velocity-Verlet algorithm with a time step

Δt = 0.01τLJ was used for integration of the equation of motion. All simulations were performed using 44 LAMMPS. Simulations started with placing a lens at distance 2.0σ above the top of an elastic substrate. To 2 accelerate the contact to reach equilibrium, a strong harmonic spring with ksp = 1000 kBT/σ for small lenses 2 and with ksp = 5000 kBT/σ for large lenses is applied to the center of mass of the lens. The spring is removed 2 3 after 10 τLJ for small lenses with Rp < 500σ and 10 τLJ for large lenses with Rp = 500σ and 1000σ. This step 4 followed by a production run with duration 10 τLJ. The equilibrium indentation depth and radius of contact 3 were obtained through averaging the configurations during the last 3 × 10 τLJ.

26 Gel Substrates: The gels are prepared according to simulation procedure described in previous publication. 3 3 The shear moduli of elastic gels used in current simulations are G=0.498 kBT/σ and 0.833 kBT/σ .

3.2 Simulation Results

Figure 3.2 shows comparison between exact solution using Maugis approximation for yh as a

function of xR for different ratios of γ s /W and approximate solution eq 2.4.4. As evident from Figure 3.2, the maximum deviation between our crossover expression eq 2.4.4 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 ∆h approaches Rp as being pointed out by Greenwood.47 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 2.4.4.

13 To demonstrate that accounting for nonlinear effects in substrate deformations will improve agreement with a crossover expression we have replotted our data for sets of coarse grained molecular dynamics simulations of particles interactions with substrates26-27 and supplemented them by the results of new molecular dynamics simulations of rigid indenter with 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 with

3 shear modulus G = 0.498, and 0.833 kBT/σ . The simulation methodology described in this chapter is based on the approach developed in refs [26, 48-49]. Our new simulations cover additional two orders of magnitude of xR between 10 and 500. 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. Note that in all simulations we independently measured the substrate shear modulus, surface tensions and work of adhesion. The simulation results are plotted in universal variables (see eq 2.4.2) in Figure 3.2. There is a very good agreement between simulation results and the analytical expression with no adjustable parameters. Therefore, we could apply this expression to obtain the

work of adhesion W and substrate surface tension γ s from equilibrium indentation data of particles of different sizes on substrates with different values of the shear modulus, G.

14 Figure 3.2 Comparison between approximate solution eq 2.4.4 (dashed line), exact solutions using Maugis’s approximation for yh as a function of xR for different ratios of γ s /W: 0.5 (red line), 1.0 (purple line), 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.

CHAPTER IV ANALYSIS OF EXPERIMENTAL DATA

4.1 Introduction

In this section we will describe how the work of adhesion W and substrate surface tension γ s can be obtained by fitting indentation data to eq 2.4.1. In our data analysis we apply a least-squares method by minimizing function N ( ) 1 i 3/ 2 2 = ∑( ) J W ,γ s − W + γ sεi + δGi Rpεi (4.1.1) N i=1

i where ε = ∆h / R and parameter δ = 8 /π ≈ 1.47 . Minimization of this function with respect to W i i p 3

and γs results in the following system of equations:

N N 2 δ N ∂J (W ,γ ) W 1 ε + s ∑ i 5 / 2 i= 0 = − ∑ε + γ s ∑ i G R ε i p i ∂γ s N i=1 N i=1 N i=1 (4.1.2) ∂J (W ,γ s ) N N = 0 = −W + γ 1 δ i 3 / 2 ε + s ∑ ∑G R ε ∂ i i p i W N i=1 N i=1 (4.1.3)

Solutions of this system of equations for γs and W are

3 / 2 5 / 2 GR pε ε − GR ε p γ s = δ 2 2 ε − ε (4.1.4) = γ ε + δ ε W s GR 3 / 2 p (4.1.5) N = −1 where we introduced an average value of a parameter A N ∑ Ai over a data set with N data points. i =1

Figure 4.1 SEM micrographs of silica, PS and PS-COOH particles with radius Rp=0.5 µm residing on soft PDMS substrates with different modulus.

4.2 Experimental data fitting

We have applied developed above approach to analysis of the indentation data of monodisperse polystyrene (PS) (Rp = 0.225, 0.5, 1.0, 1.3, 1.75, 3.3, 45 μm) and carboxyl group modified polystyrene (PS-

COOH) (Rp = 0.25, 0.5, 1.0, 1.5, 3.0, 10 μm) and polydisperse poly(methyl methacrylate) (PMMA) (Rp= 0.58

~ 52.1 μm) particles deposited on substrates made of bottlebrush, comb-like and linear chain PDMS networks with different rigidity (G = 3.3, 8.1, 15.9, 32.0, 54.7, and 583.0 kPa). The studies are performed at room temperature below Tg=90 of polystyrene and Tg=105 of PMMA thus particles can be considered as rigid.

These three new data sets℃ are compared with the data ℃for silica particles deposited on similar substrates.36

Figure 4.1 shows snapshots of the particle-substrate configurations for particles with Rp=0.5 µm residing on the substrates with different rigidity. For all sets the height h of a particle above substrate monotonically increases with increasing substrate modulus. Comparison between three different types of particles shows that PS-COOH particles have the strongest interactions with substrate as seen from the lowest height of the particles above substrate. PS and silica particles appear to have similar properties that are also qualitatively

17 similar to behavior of PMMA particles in contact with PDMS substrates. The SEM images are analyzed to obtain ∆h = 2Rp - h from the height of the particle h above the substrate surface. The methodology of this approach is described in detail in ref [36]. The results of this analysis are summarized in Figures 4.2 a-c. To collapse data for different values of the substrate shear modulus and particle sizes we have plotted G∆h as a function of GRp. The data clearly show two characteristic regimes. In the wetting regime, the indentation depth is proportional to the particle size, ∆ ~ , however, in the adhesion regime it varies with particle size

𝑝𝑝 ∆ ~ 1/3 1/3 ℎ 𝑅𝑅 and modulus as G . Also, in agreement with the images shown in Figure 4.1 the strongest

𝑝𝑝 substrate particle interactionsℎ 𝑅𝑅 𝐺𝐺 are observed for PS-COOH systems while the PS and silica particles show similar affinity to a PDMS substrate.

18 Figure 4.2 Dependence of the normalized particle height GΔh on normalized particle size GRp for PS (blue triangles) and silica (red triangles) particles (a), PMMA (orange triangles) (b) and PS-COOH (green triangles) (c) particles in contact with PDMS substrates. Panel (a): Lines are the best fit to eq 2.4.1 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 2.4.1 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 2.4.1 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 2.4.1 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 engulfing regime.

4.3 Discussion

Applying eqs 4.1.4 and 4.1.5 to the data sets shown in Figure 4.2a, we obtain the values of the work of adhesion W and surface tension γ s of the PDMS substrate. From this analysis work of adhesion for

PS/PDMS is equal to WPS/PDMS =48.0±2.9 mN/m for surface tension γ s = 23.6±2.1 mN/m. The obtained

50 value of the work of adhesion is consistent with the work of adhesion values WPS/PDMS =49±2 mN/m and 45 mN/m51 obtained from the JKR experiments. Note that the value of the works of adhesion for PS/PDMS

36 system is also close to that for silica/PDMS systems WSiO2/PDMS= 47.4±3.0 mN/m (see for details ref [ ]).

This should not be surprising since both data sets overlap throughout the entire range of GRp. Furthermore, the value of the surface tension for PDMS networks is close to the literature data.52 Similar analysis of the data for PMMA partciles in contact with PDMS susbtrates (see Figure 4.2b) results in WPMMA/PDMS

52 =56.2±2.4 mN/m which is in agreement with WPMMA/PDMS =57±1 mN/m reported in ref [ ].

Analysis of the data for PS-COOH/PDMS systems (see Figure 4.2c) using eqs 4.1.4 and 4.1.5 leads to large value of the surface tension for PDMS substrate γ app = 177.0±24.3 mN/m and work of adhesion

Wapp= 252.3±29.3 mN/m. Here we call surface tension and work of adhesion as apparent quantities this change in notation will be clarified below. This value of surface tension is much larger than the value of the

surface tension of the PDMS γ s = 23.6±2.1 mN/m obtained from fitting silica data. Using a value of the surface tension for the PDMS substrate γs =23.6±2.1 mN/m we can establish that in the wetting regime

∆h ≈ Wapp Rp / γ s ≈ 10.7Rp > 2Rp and PS-COOH particles should submerge long before a crossover to a wetting regime is reached. This points out that stabilization of the particle height in the wetting regime

GRp<10 is not due to surface tension but rather is due to nonlinear network deformation resulting in effective

19 renormalization of both work of adhesion Wapp and surface tension γ app . 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 γ app . Note that recovered scaling

relation ∆h ∝ Rp in the wetting regime means that renormalization of the work of adhesion Wapp and surface

tension γ app occurs within a layer which thickness is smaller than the particle size and it could be much

thicker than an interface layer. Therefore, the apparent surface tension γ app is not a surface stress in the

53 Shuttleworth’s definition since it could have bulk contributions. Also, the value of Wapp obtained from the fitting procedure may be different from the value evaluated from JKR-like experiments in the limit of linear substrate deformations. To make connections with JKR experiments and account for a correction due to surface tension effects we apply eq 4.1.5 to a subset of the data with GRp>100 mN/m. Using a fixed value of

γ s = 23.6±2.1 mN/m we have WPS-COOH/PDMS=268.4±26.5 mN/m (solid green line in Figure 4.2c).

Furthermore, analysis of the data in the interval GRp<10 corresponding to wetting regime gives a particle-

substrate affinity parameter Wapp/ γ app = 0.69 ± 0.03 which is close to 0.7 estimated from a ratio Wapp/ γ app

using work of adhesion Wapp and surface tension γ app obtained from the all data fitting. This example shows what modifications in application of our approach to the systems with large ratio of the work of adhesion and surface tension should be made to obtain meaningful results.

2/3 Figure 4.3 Dependence of the parameter Ga on (GRp) 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 2 2/3 4 2/3 (b) in the interval 10 < (GRp) < 10 (mN/m) .

4.4 Validation and deviation from JKR model

We can also apply developed here approach to reanalyze data by Rimai et al38-42 obtained from SEM micrographs image analysis of glass particles with sizes varying between 0.5 µm and 125 µm on polyurethane substrates with shear modulus in interval 1.5x10-2 MPa and 13.9 MPa and polystyrene particles with sizes varying between 1.0 µm and 6.25 µm on polyurethane substrates with shear modulus G = 1.67 MPa. To demonstrate breakdown of the JKR model for this system in Figure 4.3a we replot data from refs [39-42] in

2 / 3 terms of Ga vs (GR p ) which is expected to be the right scaling dependence in the adhesion regime. For the softest substrates with shear modulus 1.5x10-2 MPa (see Figure 4.3a) there is a clear deviation from the

JKR scaling which could be explained by the 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

1/ 3  9π  2 / 3 Ga =  W  (GR p ) (4.4.1)  8  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 = 0.5.54 The error estimates are obtained by

Jackknife method55 of removing data points and recalculating the values of the fitting parameters. As follows

21 from Figure 4.3b, the entire data set for PS/polyurethane (PU) systems is in the adhesion regime. Therefore, eq 4.4.1 can also be used to obtain the work of adhesion of PS/PU systems W = 142.7±10.2 mN/m. The results are summarized in Table 4.1.

Table 4.1 Work of adhesion and surface tension

Particle Substrate G [kPa] Rp [μm] W [mN/m] Ref. W [mN/m] γs [mN/m]

1.0 24

28.3 72 25 a Glass PDMS 3 ~30 [ ] 54.6±1.0 25.8±1.1 83.3 80

166.7 61

36 a a Glass PDMS 3.3 ~ 583.0 0.2 ~1.5 [ ] 47.4±3.0 23.6±2.1

b PS PDMS 3.3 ~ 583.0 0.225 ~ 45 48.0±2.9 23.6±2.1

b PS-COOH PDMS 3.3 ~ 583.0 0.25 ~ 10 268.4±26.5 23.6±2.1

PMMA PDMS 3.3 ~ 32 0.58 ~ 52.1 56.2±2.4 b 23.6±2.1

E [MPa]

* 38 c PS PU 5.0 1.0 ~ 6.25 171 [ ] 142.7±10.2

5.0 0.5 ~ 60 170* [39]

0.045 4 ~ 100 44* [41] c Glass PU 109.3±3.8 0.25, 0.7 0.5 ~ 125 [42]

3.83, 41.7 11, 20 120* [40]

* Poisson’s ratio ν = 1/3 was used in the JKR equation.; a surface tension is calculated from eqs 4.1.4 and b c 4.1.5; W is calculated for γs=23.6±2.1 mN/m; work of adhesion is calculated from eq 4.4.1 which is derived for Poisson’s ratio ν = 0.5.

22 CHAPTER V CONCLUSIONS

We have shown that an approach based on the simple crossover expression (see eq 2.4.1) 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. This highlights universality of the contact phenomena of particles on soft elastic surfaces. Figure 5.1 confirms this universality by presenting experimental data together with simulation results plotted in reduced variables. Collapse of both the experimental and simulation data confirms validity of the crossover expression and the universality of particle-substrate interactions. The dashed line corresponds to eq 2.4.4. There are two different scaling regimes in particle substrate interactions. For small particles with sizes Rp R* the elastic and adhesion forces acting in∆ℎ the/∆ℎ contact≈ 𝑅𝑅 area𝑅𝑅

1/3 begin to provide dominant contribution (adhesion regime) resulting in ( / ) . We hope that ∗ ∗ 𝑝𝑝 developed here approach will become a useful experimental technique∆ℎ/∆ℎ for characterizing≈ 𝑅𝑅 𝑅𝑅 materials surface and contact properties. Note that presented here analysis can be extended to particles and substrates of arbitrary rigidity following methodology developed in refs [26, 27].

Figure 5.1 Dependence of the reduced substrate indentation depth parameter Δh/Δh* on the reduced particle size, Rp/R*. The experimental data by Style et al 26 for silica microspheres on PDMS substrates are shown by brown filled triangle, data by Ina et al 37 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, data for rigid nanoparticles and lens on elastic substrates are shown by purple filled squares and red filled stars respectively. The dashed line corresponds to eq 2.4.4.

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