Article pubs.acs.org/Macromolecules
Nanoparticles as Adhesives for Soft Polymeric Materials Zhen Cao and Andrey V. Dobrynin* Department of Polymer Science, University of Akron, Akron, Ohio 44325, United States S Supporting Information *
ABSTRACT: The ability of nanoparticles to remain adsorbed at an interface between two soft materials makes it possible to use them as efficient adhesives. Using a combination of the molecular dynamics simulations and theoretical calculations, we establish conditions for different regimes of interfacial confinement of nanoparticles between two polymeric gels. Depending on the relative strength of the capillary and elastic forces acting on a nanoparticle in contact with substrates, a nanoparticle could be in the bridging, the Pickering, or the submerged state. The work of adhesion for a nanoparticle reinforced interface was derived analytically and obtained in simulations from the potential of mean force for separation of two gels. Simulations show that the work required for separation of two gels with nanoparticles confined at interface could be up to 10 times larger than the work of adhesion between two neat gels without the nanoparticle reinforcement. These results provide a valuable insight in understanding the mechanism of gluing soft materials, including gels and biological tissues, by nano- and microsize particles.
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our model recovers a classical Pickering solution25 of particle at interface between two liquids. However, in the opposite limit of small elastocapillary numbers, our analysis show that the elastic response of the deformed substrates determines the adhesive strength of nanoparticles’ reinforced interface. These model predictions are confirmed by molecular dynamics simulations of reinforcement of the interface between two soft gel-like surfaces by nanoparticles.
INTRODUCTION How to glue two surfaces together? The answer to this question has significant ramifications for science and technology.1−7 Polymeric adhesives offer many advantages in binding unlike materials, in distributing mechanical stress and in preventing the fracture of the adhesive joints. However, despite of a long history of the development of adhesives, gluing together two gel-like surfaces is still challenging. The current techniques involve utilization of copolymers for interface reinforcement,8 modification of gel surfaces9 with associating or covalent groups capable of reacting in response to external stimuli such as pH,10 light,11 or heat.12 Typically, these techniques require a complex design of specific adhesives on molecular level. Recently, it has been shown that nanoparticles could serve as simple and versatile adhesives capable of gluing together various soft materials such as elastomers, gels, and biological tissues.13−17 This approach takes advantage of nanoparticles intrinsic ability to adhere to an interface between two soft materials creating adhesive contact of high strength and enhancing the contact area. However, the mechanism of reinforcement of the interface by nanoparticles is far from being understood. Here we will demonstrate that unique adhesive properties of nanoparticles are manifestations of the distinct features of the contact phenomena at nanoscale.18−24 Through a combination of the theoretical calculations and molecular dynamics simulations, we show that gluing two soft gel-like surfaces by nanoparticles is due to a subtle interplay between capillary and elastic forces acting in the contact area. We establish thermodynamic conditions for different adhesion regimes when nanoparticles could reinforce an interface between soft gel-like materials. In the limit of large elastocapillary numbers20 (describing relative strength of the capillary and elastic forces) © XXXX American Chemical Society
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RESULTS AND DISCUSSION Model of Nanoparticle in Contact with Soft Gel-Like Surfaces. Nanoparticle in contact with two elastic surfaces can: (i) form a bridge connecting two substrates together (Figure 1a) which we will call below the “bridging state”; (ii) be partitioned between two soft substrates in contact, “Pickering state” (Figure 1b); or (iii) stay inside one of the gel phases when there is no affinity between nanoparticle and the interface, “submerged state” (Figure 1c). In all these cases, the equilibrium system configuration is a result of a fine interplay between elastic energy of the indentations produced by nanoparticle in substrates and surface free energy change due to changes in the contact surface area of nanoparticle and substrates. Bridging State. We will first consider a nanoparticle with radius Rp bridging together two elastic gel-like surfaces as shown in Figure 1a. The indentations produced by the nanoparticle in two gels are Δh1 and Δh2 for the top and bottom substrates, respectively. In the bridging state, Received: March 1, 2016 Revised: April 8, 2016
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DOI: 10.1021/acs.macromol.6b00440 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules
Figure 1. Schematic representation of a rigid nanoparticle in contact with gel surfaces: (a) nanoparticle in “bridging state”, (b) nanoparticle in “Pickering state”, or (c) nanoparticle in “submerged state”. A lighter color of the gel corresponds to a softer substrate with a lower value of the shear modulus.
2Rp − Δh1 − Δh2 > 0, this inequality implies that there is a gap between two surfaces. The free energy of the system with surface area per nanoparticle A and nanoparticle bridging two gels together is equal to
2/3 ⎛ W ⎞2/3 ⎛ 4π ⎞2/3⎛ Wi ⎞ Δhi ⎟ ∝ ⎜⎜ i ⎟⎟ , for i = 1, 2 = ⎜ ⎟ ⎜⎜ ⎝ 5C ⎠ ⎝ GiR p ⎟⎠ Rp ⎝ GiR p ⎠
(5)
This solution corresponds to an adhesive (JKR-like)26 contact between substrate and nanoparticle. Another asymptotic solution of the eq 4 can be derived by balancing the work of adhesion and substrate free energy terms (the first and the second terms in the rhs of the eq 4). This gives
FB(Δh1 , Δh2) = F1(Δh1) + F2(Δh2) + 2πR p(2R p − Δh1 − Δh2)γp
(1)
where γp is the surface energy of nanoparticle. The free energy of nanoparticle in contact with a substrate is a sum of contributions from surface free energy of nanoparticle− substrate contact and elastic energy of deformation of the substrate18,19
Δhi W = i , for i = 1, 2 Rp γi
It is important to point out that for rigid nanoparticles this expression is exact even in the case of large substrate indentations. We call this type of contact between substrate and nanoparticle−wetting contact.20 Equating eqs 5 and 6 one can show that crossover between adhesion and wetting contact regimes takes place for the values of the dimensionless parameter, βi ∝ γi(GiRp)−2/3Wi−1/3 ≈ 1.18,20 (Note that this parameter describes a relative strength of capillary and elastic forces.18) In particular, this crossover occurs for indentation depths Δhi* ≈ Wi1/2γi1/2/Gi.20 For interval of system parameters such that Δhi < Δh*i (or βi < 1) we have an adhesive (JKR-like) contact between substrate and nanoparticle while for the range of parameters when the opposite inequality holds,Δhi > Δhi*, the depth of indentation is controlled by the capillary forces. The exact solution of the eq 4 is obtained by using the approach developed in ref 18
Fi(Δhi) = γi(A − πai 2) + 2πR pΔhiγi p + CGiR p1/2Δhi 5/2 , for i = 1, 2
(2)
where γi is the surface tension of the substrate i with shear modulus Gi, and ai is the contact radius of indentation produced by nanoparticle in substrate i, which for rigid nanoparticles is equal to ai2 = 2RpΔhi − Δhi2. In eq 2 parameter C is a numerical constant which value depends on the model used to describe stress distribution in a substrate.18,19 Taking eq 2 into account we can write the following expression describing total free energy of the system due to formation of the nanoparticle bridge: FB(Δh1 , Δh2) = 4πR p2γp + (γ1 + γ2)A − 2πR p(W1Δh1 + W2Δh2) + π (γ1Δh12 + γ2Δh2 2) + CR p1/2(G1Δh15/2 + G2Δh2 5/2)
⎛ W ⎞2/3 Δhi = AG⎜⎜ i ⎟⎟ 2R p ⎝ GiR p ⎠
(3)
⎛ ⎜ 3 ri + ⎝
where Wi = γi + γp − γip is the work of adhesion between nanoparticle and substrate i. At equilibrium indentations produced by nanoparticles in substrates are obtained by minimizing eq 3 with respect to indentations Δh1 and Δh2. This results in the following system of equations 0 = −2πR pWi + 2πγiΔhi + for i = 1, 2
(6)
qi 3 + ri 2 +
3
ri −
qi 3 + ri 2 −
βi ⎞ ⎟ 3⎠
2
(7)
where r i = 1 / 2 − (β i /3) 3 , q i = −(β i /3) 2 , β i = BGγi(GiRp)−2/3(Wi)−1/3, numerical coefficient AG = (√2π/ 5C)2/3, and BG = 2AG. For the value of parameter AG = 0.416 obtained from fitting simulation and experimental data in ref 19, a numerical constant C is equal to 3.31. Figure 2 summarizes different nanoparticle−substrate interaction regimes. The region in diagram of nanoparticle states belonging to the envelop of parameters for which Δh1 + Δh2 < 2Rp corresponds to the bridging state where nanoparticle connects two substrates. Depending on the values of the parameter βi ≈ γi(GiRp)−2/3Wi−1/3 the contact between nanoparticle and substrate could be (i) adhesive when for both contacts Δhi < Δh*i , (ii) adhesive-wetting contact for Δh1
5 CGiR p1/2Δhi 3/2 , 2 (4)
There are two simple asymptotic solutions of the eq 4. The equilibrium indentation produced by nanoparticle could be obtained by balancing the elastic energy of the substrate deformation (the last term on the rhs of eq 4) and work of adhesion term (the first term on the rhs of eq 4) describing change in the surface free energy due nanoparticle−substrate contact. B
DOI: 10.1021/acs.macromol.6b00440 Macromolecules XXXX, XXX, XXX−XXX
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Figure 2. Diagram of states of a nanoparticle in contact with two soft gel-like surfaces. Insets show typical nanoparticle−substrate configurations in different nanoparticle−substrate contact regimes.
< Δh*1 and Δh2 > Δh*2 , or Δh1 > Δh*1 and Δh2 < Δh*2 , and (iii) wetting contact when for both indentations Δhi > Δhi*. Pickering State. In the Pickering state, the gap between two substrates disappears such that Δh1 + Δh2 = 2Rp and nanoparticle is partitioned between two substrates (see Figure 1b). Analyzing diagram of states shown in Figure 2 we can conclude that in the Pickering state, we have to include one of the contacts as wetting contact, while the second one can be either adhesive or wetting contact. Here we will assume that Δh1 < Rp, this warrants applicability of small substrate deformation approximation used in evaluation of the elastic energy of the substrate indentation. In this approximation the system free energy due to nanoparticle partitioning between two gels is
surfaces for different nanoparticle−substrate contact regimes described in Figure 2. Note that in describing crossover between bridging and Pickering states to ensure continuity of the free energy expressions eq 3 and eqs 8−10 one has to take into account contribution from long-range van der Waals interactions acting across the gap between two surfaces. In the limit of small gap when contact radii of both indentations are close a1 ≈ a2 we could substitute in eq 3 γ1 + γ2 → γ1 + γ2 − W12d02/(δh + d0)2, where d0 is a distance of the closest approach between two surfaces, δh = 2Rp − Δh1 − Δh2 In describing crossover it is more convenient to rewrite free energy eq 3 in terms of smallest indentation and δh.
FP(Δh1) = 4πR p2γ2p + γ12A − 2πR pWp21Δh1 + πγ12Δh12 + CG1R p1/2Δh15/2
(8)
where we introduced the work of adhesion Wp21 = γ2p + γ12 − γ1p. As before, we can identify two nanoparticle contact regimes with a substrate. There is an adhesive contact of nanoparticle with gel 1 for range of parameters such that Δh1 < Δh*p1 ≈ Wp211/2γ121/2/G1 and wetting contact when the opposite inequality holds Δh1 > Δhp1 * (see Figure 2). We can rewrite expression eq 8 for small indentation produced by a nanoparticle in a second gel (gel 2), Δh2 < Rp, as follows
Figure 3. Initial and final states of separation of two substrates from bridging (left) and Pickering (right) states.
To quantify the effect of reinforcement of interface between two gels by nanoparticles, we calculate the work required to separate two gels with nanoparticles binding them together as shown in Figure 3. We begin our calculations of the work required for two substrate separation by first considering separation from bridging state. Bridging State. The work required to separate two gels connected by nanoparticles is equal to change of the system free energy upon pulling apart two substrates. Note that our calculations give a work of separation per nanoparticle. In the case when after two substrate separation a nanoparticle ends up as being attached to the gel i, the free energy difference between final and initial states can be written as follows
FP(Δh2) = 4πR p2γ1p + γ12A − 2πR pWp12Δh2 + πγ12Δh2 2 + CG2R p1/2Δh2 5/2
(9)
where parameter Wp12 = γ1p + γ12 − γ2p. The crossover between wetting and adhesion regimes occurs at Δh2 ≈ Δhp2 * ≈ Wp121/2γ121/2/G2. In the limit of a classical Pickering contact, when nanoparticle is trapped at interface between two soft gels, elastic energy contribution to the system free energy can be neglected. In this approximation eq 8 reduces to FP(Δh1) = 4πR p2γ2p + γ12A − 2πR pWp21Δh1 + πγ12Δh12
ΔFBi = FB(Δhi , 0) − FB(Δh1 , Δh2), for i = 1, 2
(10)
(11)
where for calculation of the system free in the initial and final state we use eq 3 describing a nanoparticle bridging two surfaces together. Note that in the case when contact of the nanoparticle with gel i remains unaltered, its contribution to the free energy difference cancels out resulting in the following expression
This situation is similar to the case of the nanoparticle at interface between two liquids. We call this type of contact wetting−wetting Pickering contact. Minimizing eq 10 with respect to indentation Δh1, and solving resultant equation, we obtain Δh1 = RpWp21/γ12.Table 1 summarizes the equilibrium depth of indentations produced by a nanoparticle in gel C
DOI: 10.1021/acs.macromol.6b00440 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules Table 1. Equilibrium contact and work of separation for interface reinforced by nanoparticles gel 1 state
contact
bridging
adhesive wetting adhesive wetting
Pickering
submerged
gel 2 Δh1
Δh2
contact
⎛ W ⎞ 0.83R p⎜ G R1 ⎟ ⎝ 1 p⎠ RpW1/γ1
2/3
adhesive
⎛ W ⎞ 0.83R p⎜ G R1 ⎟ ⎝ 1 p⎠ RpW1/γ1
2/3
wetting wetting adhesive
⎛ W ⎞ 0.83R p⎜ G R2 ⎟ ⎝ 2 p⎠ RpW2/γ2 RpW2/γ2
work of separation 2/3
ϕNPWi (Wi/GiRp)2/3 ϕNPWi2/γi ϕNPW22/γ2
⎛ W ⎞2/3 0.83R p⎜ G R2 ⎟ ⎝ 2 p⎠ 2Rp − Δh1
ϕNPW12/γ1
⎛ ⎛ Wp21 ⎞2/3 ⎛ W ⎞2/3⎞ − 4ϕNP(γ2p − γp) + W12 + ϕNP⎜⎜Wp21⎜ G R ⎟ − W1⎜ G R1 ⎟ ⎟⎟ ⎝ 1 p⎠ ⎝ 1 p⎠ ⎠ ⎝
adhesive
⎛ Wp21 ⎞2/3 0.83R p⎜ G R ⎟ ⎝ 1 p⎠
wetting
wetting
2Rp − Δh2
adhesive
⎛ Wp12 ⎞2/3 0.83R p⎜ G R ⎟ ⎝ 2 p⎠
⎛ ⎛ Wp12 ⎞2/3 ⎛ W ⎞2/3⎞ − 4ϕNP(γ1p − γp) + W12 + ϕNP⎜⎜Wp12⎜ G R ⎟ − W2⎜ G R2 ⎟ ⎟⎟ ⎝ 2 p⎠ ⎝ 2 p⎠ ⎠ ⎝
wetting
RpWp21/γ12
wetting
2Rp − Δh1
⎛ Wp(3 ‐ i)i2 − 4ϕNP(γ(3 − i)p − γp) + W12 + ϕNP⎜ γ − ⎝ 12
submerged no contact
>2Rp
