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Nanoparticles Organization Controls Their Potency as Universal Glues for Gels Nicola Molinari, and Stefano Angioletti-Uberti Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b00586 • Publication Date (Web): 01 May 2018 Downloaded from http://pubs.acs.org on May 1, 2018

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Nano Letters

Nanoparticles Organization Controls Their Potency as Universal Glues for Gels Nicola Molinari†,‡ and Stefano Angioletti-Uberti∗,¶,§ †Department of Physics, Imperial College London, SW72AZ London, UK ‡Thomas Young Centre, Imperial College London, SW72AZ London, UK ¶Department of Materials, Imperial College London, SW72AZ London, UK §Beijing Advanced Innovation Centre for Soft Matter Science and Engineering, Beijing University of Chemical Technology, 100099 Beijing, PR China E-mail: [email protected]

Abstract Nanoparticles have been recently shown to act as universal glues for both synthetic and biological gels, providing a tunable, cheap and general solution to the centuries-old problem of sticking soft materials together. The design of new adhesive solutions based on this platform, however, requires understanding how nanoparticles’ design parameters concur to determine the final adhesion strength. Here, we use coarse-grained modelling and Molecular Dynamics simulation to investigate such link. Our main result is to show that at experimentally relevant concentrations adhesion is strongly influenced by the way nanoparticles organise at the interface, resulting in non-monotonous reinforcements behaviour. Our findings represent an important step towards rationalising this new class of nanoparticles-based adhesives.

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Keywords Nanoparticles-Based Glues, Polymer Adhesives, Soft Matter, Coarse-Grained Modelling, Molecular Dynamics

Developing the perfect adhesive to stick two materials together represents an important technological challenge in materials science. Due to their stress dissipation properties, their ability to adapt to different surface morphologies and their extremely low cost, polymers are often the primary choice for various materials. Given their exceptional properties as adhesives, it is peculiar instead that gluing polymer gels together has proven rather difficult 1 . In practice, this issue is typically tackled on an ad-hoc basis via the design of smart but complex molecular glues 2,3 . Recently, Rose et al. 4 have shown that an elegant and general solution to this problem is to exploit the ability of nanoparticles to form multiple and reversible bonds with polymers. In particular, these authors showed that painting the surface of different types of both synthetic and natural polymer-based gels allows them to strongly stick at room temperature without the need of any further treatment. Despite the apparent simplicity of this solution, there are still various parameter that must be carefully tuned in order to provide good adhesion. For example, the chemistry of the nanoparticles must be compatible with that of the gels 1 , or in other words there should be a net attraction between the gel monomers and nanoparticles. Another important parameter that have been speculated to play a role is nanoparticles’ size 1,4 . In this case, optimal adhesion is thought to occur, based on scaling principles for polymers 5,6 , when the nanoparticle’s diameter is comparable to the mesh of the gel, i.e., the average distance between cross-linkers. That not all choices of parameters lead to adhesion is well exemplified by the fact that, as shown experimentally 7 , a non-monotonic behaviour of interface reinforcement with respect to nanoparticle concentration can be observed. In fact,

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in certain cases the use of nanoparticles can lower the adhesion energy compared to that of the bare gel. This latter aspect should certainly not come as a surprise, considering that powders like talc, where a poly-disperse distribution of particles of multiple size are present, are commonly used as lubricant to prevent otherwise self-adhesive polymeric surfaces from sticking 8 . Overall, experimental evidence points to the fact that nanoparticles-mediated adhesion only happens within a certain range of parameters in terms of nanoparticle size, chemistry and their concentration at the interface 7 . However, identification of these sweet spots is everything but trivial, especially considering the lack of a detailed microscopic picture for the adhesion process. In fact, with the notable exception of the work of Cao and Dobrynin 9 , which addressed the case of a single nanoparticle between two gels, previous modelling efforts focused on the role of nanoparticle-polymer interactions in bulk systems rather than at interfaces, e.g., to understand reinforcement in composites 10–13 . To shine some light on this issue, here we performed Molecular Dynamics (MD) simulations using a coarse-grained model and explored the tearing mechanism of an interface between two permanently cross-linked gels which adhere due to nanoparticles adsorbed on their surface. In this way, we systematically obtained stress-strain curves as a function of polymernanoparticle interaction, nanoparticle size as well as nanoparticles planar concentration, from which trends in the adhesion strengthening can be analysed. Furthermore, the use of MD allowed us to establish a link between this macroscopic behaviour and the microscopic details of the system. In this regard, our major point is to drive the attention towards the way in which nanoparticle-polymer and polymer-polymer interactions, together with geometrical constraints, drive nanoparticles organisation at the interface. As it turns out, this organisation is crucial to rationalise trends in the strength of adhesion. In fact, we show here that not only an optimal choice for the parameters exists that maximises mechanical strengthening, but that outside a certain range weakening can also occur, thereby providing a robust explanation for recent experimental observations 7 .

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We consider a system comprising two permanently cross-linked polymer gels glued together by the addition of a variable number of spherical particles at their interface. In these particular simulations we concentrate on the case of a polymer in bad solvent conditions, i.e., the network is in a collapsed state 14 . Throughout the paper, we work with Lennard-Jones reduced units, in which the mass m, and the parameters of the inter-particle interaction σ and are set to unity. Using this choice all other quantities, e.g., pressure or time, are made dimensionless by scaling them with the proper combination of σ, and m, see 15 for reference. The number of nanoparticles inserted at the interface NP , is controlled by imposing the ratio between the area obtained by projecting all the particles onto the interface and the area of the interface Aint = Lx Ly , Lx(y) being the equilibrium values for the simulation cell along the x(y) directions (i.e., those perpendicular to the strain axis z), at zero external pressure. 2 /Aint , where RNP is the radius of the nanoparticle. In this We call this ratio AP = NP πRNP

work AP ranges from 0.0 (corresponding to no particles) to 1.2. Note that with this definition, and considering that nanoparticles are not confined to move exactly on a plane, AP can have values larger than that expected at the maximal packing fraction for spheres on a plane. Stress-strain curves are obtained following the recipe in 16 . Briefly, the structures are first equilibrated 17,18 and then strained at constant rate along the z axis in the NTLz σxx σyy ensemble, in which the total number of particles N , temperature T , and normal stresses (σxx and σyy ) perpendicular to the straining direction are kept at zero 16 . During the straining procedure Lz , the length of the simulation box along the straining axis, is increased at a uniform strain rate (details in the supplementary information). The stress along the straining direction, σzz , is computed from the pressure along the z axis, σzz = −Pzz . Throughout this work, pressure and temperature are fixed at P = 0.0 and T = 1.0, respectively, and the stresses σxx and σyy are held at zero. The strain rate is kept constant at 3.27 % every 103 timesteps as in previous studies with similar models 10,19,20 . All the molecular dynamics simulations are performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) package 21 . Nanoparticles (NPs) have a radius RNP = 1.0, 2.0 and 4.0 σ

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depending on the system under investigation and interact purely via repulsive, excluded volume interactions, implemented as a truncated and double-shifted Lennard-Jones potential (see details in the supplementary information). Polymers are represented using a model based on the bead-spring coarse-grained description developed by Kremer and Grest 22,23 , modified to include nanoparticles 10 . Polymer chains comprises 30 beads each with a radius of r = 0.5 σ. Such model proved successful in studying polymers and polymer nanocomposites under various stress conditions 10,24 , and it is therefore adopted here. Intra-chain and cross-link bonds are represented with the widely-used finite extensible nonlinear elastic (FENE) potential with R0 = 1.5 σ and k = 30/σ 2 to avoid chain-crossing and high frequency modes 22 . Similarly to the NP-NP interaction, the non-bonded interactions between polymer beads and between polymer beads and NPs are also implemented via a truncated and double-shifted Lennard-Jones potential, albeit with different parameters to represent the different size of NPs compared to the polymer beads. Notably, unlike those between NPs, interactions involving at least one polymer bead possess an attractive region. Throughout this work, we keep the strength of attraction between polymer beads constant by setting BB = = 1.0, while varying instead the strength of attraction between polymer beads and nanoparticles by changing BN . It should be noted that in this coarse-grained description this latter parameter measures the effective interaction strength between the polymer and the nanoparticle, and thus include possible effects due to the nanoparticle’s surface functionalisation. In fact, use of different stabilizing ligands on the nanoparticle surface could be an effective way to tune BN in experiments. Details on the structure generation as well as the exact form and parameters for all interaction potentials used can be found in the electronic supplementary information.

Figure 1 shows a representative evolution of the structures during the simulated mechanical test. From a) to d), the system is held at increasingly higher strains. In a) and b) the system is still able to withstands moderate strains and is held together by the presence of

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(b) (c) (d) Figure 1: Evolution of a structure during the straining procedure. The red spheres represents the nanoparticles while the polymer chains are made partially transparent for clarity. From top to bottom: (a,b) the cross-linked, reinforced with nanoparticles, and equilibrated structure is ready to undergo the straining procedure. The NPs act as glue between the two cross-linked polymer gels improving the overall mechanical properties. For larger strains, (c), the system starts to fail; voids and cavities weaken the mechanical response of the system. Finally, (d), complete mechanical failure is reached and two interfaces coated with nanoparticles are obtained. the nanoparticles at the interface. At larger strains, c), voids and cavities appear and the structure is weakened up to the point of complete failure, d), where two surfaces coated with nanoparticles are obtained. Figure 2 presents the stress-strain response at low planar concentration of particles, AP = 0.4, as a function of (a,b) nanoparticle radius, and (c) strength of the interaction between the NPs and the polymer melt. This specific value of AP = 0.4 was chosen to illustrate the behaviour in a regime where nanoparticle-nanoparticle interactions can be neglected, as will be clarified later. It should be noticed that in both (a) and (b) BN > BB = 1.0 so that in both cases we are dealing with nanoparticles whose chemistry is tuned to induce polymer adsorption on their surface. However, in (a) BN = 2.0 whereas in (b) BN = 5.0. For (a) and (b), two different behaviours are observed. In (a), nanoparticles are detrimental to the stress-strain response of the system, the more so the larger their size. Whereas for RNP = 1.0 σ (hence twice the radius of a polymer bead) the system shows almost the same mechanical properties as in the unfilled case, nanoparticles with larger radii weaken the interface adhesion, both in term of the maximum strain as well 6


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Figure 2: Mechanical response for different (a,b) nanoparticle radius and (c) nanoparticle interaction with the polymer melt. The black curve shows the stress-strain for the unfilled case. For both (a,b) AP = 0.4 while the polymer-NP interaction is set at 2.0 and 5.0 for (a) and (b), respectively. In (a) a higher effectiveness of small particles is observed while (b) shows a non-monotonic behaviour as a function of NP radius. In (c) NP radius is set at 2.0 σ and AP = 0.4. Note that a weakening of the mechanical response is observed even for a polymer-nanoparticle interaction exceeding the polymer-polymer parameter. The shade around each curve represents the standard deviation amongst eight independently generated structures. The small difference in the stress-strain response of the unfilled structures is due to the way the systems are equilibrated before the straining, details in the SI. We have carefully checked that in all cases such small differences do not affect any of the trends reported. as the maximum stress at rupture. On the other hand, when the BN is “high enough” as in (b), a non-monotonic reinforcement as a function of particle size is observed. RNP = 1.0 σ results in small improvement with respect to the unfilled case and for RNP = 2.0 σ we have the best mechanical response, while for RNP = 4.0σ we observe a moderately weakened curve 7


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compared to RNP = 2.0 σ, where not only the maximum stress but also the strain at the breakage point are reduced. Figure 2(c) helps to further clarify the difference in the results observed between (a) and (b). Purely based on the difference between polymer-nanoparticle and polymer-polymer attraction strength, which we remind in our coarse-grained description is measured by BN and BB , respectively, stronger adhesion between the two gel surfaces as obtained in the green curve for BN = 5.0 would be expected for any BN > BB = 1.0, albeit with different magnitudes. However, for a polymer-NP interaction twice as strong as the polymer-polymer one the system shows a weaker response to the straining action compared to the unfilled case. The results from Figure 2 show that the polymer-nanoparticle interaction alone is not a good predictor of interfacial mechanical reinforcement, which instead requires some form of coupling with the nanoparticle size. In practice, one needs to consider the fact that although a strongly attractive nanoparticle-polymer interaction can reinforce the gel-gel interface, forming this contact also requires the polymer to accommodate the presence of the nanoparticle. More precisely, the very strong repulsion between polymer bead and nanoparticle at distances shorter than RNP means a decrease in the available volume for the polymer. The consequent reduction in the allowed configurations for the polymer, compared to the case where such nanoparticles were not present, leads to a standard steric repulsion, e.g., see 25 . In macroscopic terms this can simply be understood as the work required to compress the polymer to make space for the nanoparticle. In contrast to the attractive contribution due to polymer adsorption at the surface of the nanoparticles, which stabilises the gel-gel interface, this polymer compression implies a positive mechanical work that will weaken the adhesion (free-)energy of the interface Fadhesion . This latter quantity, defined for our system as the minimum work required to separate the two polymer gels to infinity, is a well-defined measure for the mechanical strength of an interface 26 . In this regard, it is worth highlighting that despite the strain rate used is several orders of magnitude higher than experimental rates, satisfying the quasi-static condition necessary to discuss the property of the system in terms of equilibrium free-energies only requires the strain rate to be small compared to

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the relaxation rate of the system. For our system and strain rate, this condition is met and several studies using this strain rate can be found in the literature 10,19,20 . In fact, the observations at low AP are well-described by adapting a model to calculate adhesion free-energies recently presented by Cao and Dobryinin 9 . In practice, one can measure the interface reingel NP+gel , the difference in the quasi-static work required − Fadhesion forcement by Wreinforce = Fadhesion

to separate two polymer gels connected by nanoparticles compared to the case for two bare gels. By doing this, we obtain the following expression: "

Wreinforce

# 2/3 γBB + γNN − γBN = AP (γBB + γNN − γBN ) − 2γBB G RNP # " 2/3 BN − BB − 2βBB , ∝ AP α (BN − BB ) G RNP

(1) (2)

where α and β are two proportionality coefficients of order 1, and we assume the nanoparticles to be in the bridging regime 9 . G is the gel shear modulus and in going from Equation 1 to Equation 2 we use a simple broken-bond argument to connect the microscopic interaction parameters, i , to the interfacial free-energy, γi , of the polymer (i ≡ BB) the nanoparticle (i ≡ NN) and their interface (i ≡ BN) 27 . Although a strict proportionality between and γ is a strong assumption of broken-bond models, requiring no surface relaxation compared to the bulk, one can safely expect these two quantities to at least show the same trend. In other words, we expect one to be a strictly monotonic function of the other. To aid the interpretation of Equation 2, Figure 3 shows a “reinforcement phase-diagram” for G = α = β = 1. Considering that reinforcement compared to the bare interface is only observed if Wreinforce > 0 and the opposite otherwise, Equation 2 is in good qualitative agreement with the trends we observe: a lower critical value for the polymer-nanoparticle interaction parameter BN below which the interface is weakened rather then reinforced, a monotonically increasing reinforcement as BN increases, and the fact that larger particles require stronger interaction energies to show the same reinforcement as smaller ones. In principle, Equation 2 cannot explain the non-monotonic behaviour as a function of size ob9


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-1

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Figure 3: “Reinforcement phase diagram” in the limit of low nanoparticles surface concentration (AP 1), as predicted by Equation 1. The colour codes for the work of reinforcement per projected area of the nanoparticles normalised by the bead-bead interaction parameter Wreinforce /(AP BB ) as a function of both the (normalised) polymer-nanoparticle interaction strength BN /BB and the nanoparticle radius. The model shows the presence of a size-dependent critical value for the interaction energy below which adding nanoparticles actually decreases the interfacial strength, as observed in the simulations. The values of the constants are set to G = α = β = 1. served in Figure 2(b). However, it should be noticed that the weak reinforcement observed for smaller nanoparticles is due to the fact that these, due to a size that allows them to easily diffuse through the polymer network, do not stay at the interface, but rather move towards the bulk of the gel (see the electronic supplementary information). In practice, this means that for very small nanoparticles reinforcement will be a time-dependent pro2 cess, but for t RNP /DNP (DNP being the diffusion coefficient of the particle in the gel)

the interface will become the same as a clean/bare interface between two gels, for which then Wreinforcement = 0. These results are compatible with experimental studies of similar 10


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nanoparticles-elastomer systems, where it was indeed shown that depending on the chemical compatibility between nanoparticles and polymer, or in other words the interaction strength measured by BN , nanoparticles can be used to both prevent or strengthen adhesion 28,29 .

When the number of nanoparticles inserted at the gel-gel interface is small, i.e., much smaller than the maximum planar packing fraction achievable, their mutual interactions can be neglected. In this concentration regime, a single layer of nanoparticle is formed and inserting more of them will monotonically strengthen or reduce the interfacial adhesion between the two gels depending on whether or not BN is lower or higher compared to a radius-dependent critical value, as described by Equation 2. However, a more complex picture arises when the planar packing fraction of nanoparticles is high enough that they feel each other presence. More precisely, the increasing importance of excluded volume effects between nanoparticles at higher AP means that they do not all dispose into a single monolayer bridging the two gels (see also the supporting information). In fact, in the high AP regime nanoparticles organization at the interface plays a crucial role, producing peculiar effects. In this regard, it is worth noting that AP is linked to the planar packing fraction of spheres in two-dimension, η, only if all the NPs initially inserted at the interface between the two polymer gels were confined there, which is (approximately) correct only in the low AP regime where AP ηmax , ηmax being the maximal planar packing. Regardless of this, we use AP to discern between different scenarios because this quantity would be the control parameter in an experiment, where one can only control the concentration of the NPs solution and, hence, the number of NPs initially deposited on the gel surface. Given this introduction, let us now describe what happens in different AP regimes. In Figure 4, we present the stressstrain curves for different AP values and nanoparticle radii, RNP = 2.0 σ and RNP = 4.0 σ for (a) and (b), respectively. In both cases, we consider BN = 5.0, for which in the low AP regime the presence of nanoparticles monotonically increases gel-gel adhesion as a function of AP .

In Figure 4(a) we observe no statistically significant difference in the mechanical

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Figure 4: Mechanical response at strong polymer-nanoparticle interaction, BN = 5.0, for: (a) RNP = 2.0 σ and (b) RNP = 4.0 σ. The top (a.1/2 and b.1/2 ) molecular dynamics snapshots present the distribution of NPs between the two polymer melts - not shown for clarity. For each snapshot two nanoparticles and their coordinating chains are highlighted. The shade around each stress-strain curve represents the standard deviation amongst eight independently generated structures. response upon increasing the number of inserted nanoparticles. The initial insertion of NPs, corresponding to a AP = 0.4, leads to a stress at failure almost twice stronger than the one of the unfilled system. However, additional insertion of nanoparticles does not result in further reinforcement and a plateau is reached. The only observed difference for different AP is a small increase in the toughness of the plastic hardening regime. To clarify the origin of this plateau, (a.1 ) and (a.2 ) present molecular dynamics snapshots representative of the local environment of the NPs in the structure; the dashed line is placed at the interface between the two polymer networks. In both snapshots we highlight two nanoparticles and their coordinating chains, whereas the rest of the polymer beads are ignored for the sake of clarity. 12


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For both AP = 0.4, (a.1 ), and AP = 1.2, (a.2 ), on average the chains coordinating NPs at the interface belong to both networks, i.e., there is no significant coordination predominance from a network from either side of the interface. However, at large values of AP not all nanoparticles can sit at the interface and they are pushed away from it. Since these latter particles do not contribute to linking the two gels, the mechanical reinforcement obtained plateaus to the observed value. It is important to highlight the fact that for this particle size passive diffusion through the polymer gel is very slow, unlike in the case of small particles (RNP = 1.0 σ). Hence once the osmotic pressure due to their elevated concentration at the interface decreases as some of them move towards the bulk of the gel, there is no more driving force for their diffusion away from the interface and they remain localized there. In other words, the observed plateau arises from self-regulation of their planar density at the interface to a critical value (see also the supporting information). Incidentally, this value seems to be AP = 0.4 for our system, although in practice it will be a function of the interaction parameters , as well as particle and polymer network mesh size, (i.e., cross-link density). Figure 4(b) presents the same study as Figure 4(a) but for particles of larger radius, which further complicates the observed picture. In fact, the reinforcement plateau reported for smaller nanoparticle radius is not observed. Instead, increasing the number of inserted NPs above a certain value leads to a weakening of the mechanical response. In other words, an optimal reinforcement condition emerged. To understand the origin of this non-monotonicity we observe that nanoparticles with RNP = 4.0 σ are hindered in their ability to reach a “good coordination” condition whereby particles are evenly coordinated from polymer chains belonging to both gels. Moreover, nanoparticles are also increasingly trapped at the interface because of their size, larger then the polymer mesh. This results in the observed formation of contiguous layers of nanoparticles for AP values that exceed optimal planar packing, impeding an homogeneous chain coordination by both sides of the interface and therefore weakening the stress-strain response (see also the supporting information). This observation does not hold for AP = 0.4 instead, where a single layer of nanoparticles

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is always present, leading to “good coordination” and, hence, more effective mechanical reinforcement compared to higher AP values. (b.1 ) and (b.2 ), corresponding to AP = 0.4 and 1.2, respectively, present molecular dynamics snapshots of the NPs with RNP = 4.0 σ in the structure. (b.1 ) shows a coordination environment similar in nature to the one presented in (a.1 ) and (a.2 ). On the other hand, (b.2 ) highlights the asymmetric origin of the coordinating chains for two nanoparticles. In order to support our qualitative discussion of the different coordinating environments of nanoparticles and their connection to the mechanical response with a more quantitative description, we finally report here two quantities. The first is the average number of polymer beads coordinating a nanoparticle, hCi. A polymer bead is marked as “coordinating” if its centre is within a distance of RNP + σ from the centre of a nanoparticle, and the average is taken over all nanoparticles for each simulation. Furthermore, we also define and compute a new parameter, which we dub the “bridging number” B: hCi |hC1 − C2 i| + 1 hCi = , |hC∆ i| + 1

B=

(3) (4)

where C1 and C2 are the number of beads coordinating the nanoparticle and belonging to either gel one or two, respectively (see Figure 5 for reference). In the definition of Equation 4, the absolute value of C∆ is taken as we are only interested in how symmetrical the coordination is. In other words, ±C∆ indicate the same degree of asymmetric coordination, and, therefore, are not distinguished. The +1 addition to the denominator of Equation 4 ensures B is well-defined in the case C∆ = 0, corresponding to perfectly symmetric coordination. We compute B for eight independent simulations starting with different initial configurations and then average its value to obtain an ensemble average, hBi. In the limit where the nanoparticle coordination is perfectly symmetrical, C∆ = 0, B and C would have the same value. At the opposite limit, i.e., where the coordination is entirely provided by one gel only,

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|C∆ | = C, and, consequently, B ≈ 1. In our system, for low strain values and AP = 0.4, the nanoparticles should be symmetrically coordinated by both gels, what we called a good coordination condition. As a result hBi is expected to have a value considerably greater than one and, as the straining in the structure increases, hBi → 1 since one of the two coordinating gels is progressively detached from the nanoparticles, for all nanoparticles. On the other hand, at AP = 0.8, 1.2 the coordination of the nanoparticles should be dominated by either one of the polymer gels. Consequently, hBi is expected to have a smaller value, close to unity even at the beginning of the straining procedure. The aforementioned behaviour is exactly what we observe in our simulations, as shown in Figure 5. Panel (a) and (b) of this figure report hCi and hBi, respectively, as a function of strain for different values of AP . As an example, we discuss the case RNP = 4.0 σ and BN = 5.0, corresponding to the curves in panel (b) of Fig. 4. In (a) we first notice that the nanoparticles possess the same average Strain [%] 0 240

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Figure 5: (a) Average coordination hCi, and (b) bridging number hBi as a function of the straining for AP = 0.4, 0.8, and 1.2, RNP = 4.0 σ and BN = 5.0 BB . The shading indicates the standard deviation among eight structures. Note, in (a) and (b) RNP is the same, and the bigger representation of the nanoparticle in the sketches of panel (b) compared to panel (a) is purely to exacerbate and hence better clarify the differences between the two cases. bead coordination at all AP , and that they share the same strain dependence for strains 15


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lower than approximately 70 %, corresponding to the onset of mechanical failure observed in Figure 4. After a strain value of 70 %, the decrease in average coordination experienced for AP = 0.8 and 1.2 almost stops, while it keeps decreasing for AP = 0.4. These observations provide an indirect confirmation of the “good coordination” condition experienced by AP = 0.4 as opposed to the coordination condition for AP = 0.8, 1.2: for AP = 0.4 there is the need to break a larger number of links with the polymer beads to tear the two gels apart, since both are participating to the coordination. On the other hand, at larger AP values the coordination is already predominantly coming from one of the polymer networks, and fewer beads detach from the nanoparticles. Panel (b), provides a more direct proof of the “good coordination” concept. As expected from earlier considerations, at low strain values, hBiAP =0.4 > hBiAP =0.8,1.2 , signalling a higher symmetry in coordination for AP = 0.4 than AP = 0.8, 1.2, i.e., a more equal contribution to coordination between the two gels. As the strain increases hBi decreases and, as failure is reached, hBiAP =0.4 ≈ hBiAP =0.8,1.2 ≈ 1, confirming that once the two gels are separated the nanoparticles are coordinated exclusively by either one of the two. A non-monotonous behaviour in the interfacial adhesion strength as a function of the number of inserted nanoparticles has been very recently observed experimentally by Kim et al. 7 . In fact, in order to explain their results these authors speculated that a layering mechanism might occur in their system. The detailed microscopic picture arising from our coarse-grained simulations clearly confirms their hypothesis. At the same time, it shines light on the dependency of this effect on nanoparticles’ size. It has been previously speculated by Rose et.al. 4 that the optimal size of the nanoparticles is close to the polymer network mesh size, or in other words the average distance between cross-linkers, on the ground that in this case several different network strands adsorb on the same particle, allowing it to act as an effective connector between surfaces 6 . Our simulations as well as the analytical model in Equation 2 show that different regimes should be considered, depending on nanoparticles concentration. The latter determines their organisation at the interface and thus modifies this simple pic-

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ture. Overall, the understanding gained from a microscopic view of the problem allowed us to help reconcile various different trends observed in this peculiar system, thereby providing a more complete understanding which will aid nanoparticles design for different applications.

In conclusion, using coarse-grained molecular dynamics simulations we studied how nanoparticles properties such as size and strength of the interaction couple together with those of the polymer, controlling their organisation at the interface and thereby affecting their potency as adhesive. Our investigation resulted in the following two important findings: i) The existence, at nanoparticles concentration far away from maximal planar packing, of a size-dependent threshold in the NP-polymer interaction required to observe any interface reinforcement compared to the bare gel. This and other results are qualitatively captured by a simple analytical model, Equation 2, derived by adapting an earlier model of Cao and Dobrynin 9 to our system. ii) We highlighted the presence and explained the origin of plateauing and non-monotonicity in the mechanical response at high nanoparticle concentrations, and the different pictures arising from different NPs size. In practice, nanoparticles self-organisation in mono- or bi-layers has a direct implication on how the structure reacts under uni-axial strain. This organisation depends on the concentration-dependent effective interactions between nanoparticles, as well as on geometrical constraints affecting their diffusion. For this reason, and using simple generalising arguments from soft matter physics 30 , it is tempting to speculate that peculiar and exploitable effects could be observed by playing with the shape rather than the chemistry of the nanoparticles. In particular, shapes that allow for a denser surface packing fraction at high density compared to spheres, e.g., nanosheets and nanotubes, could provide stronger adhesion compared to spherical ones. Future work along this line is ongoing and will be presented in a future publication. Thanks to the coarse-grained approach we used, our results are based on the general physics of crosslinked polymer gels rather than their specific chemistry. In this regard, although our result should be considered qualitative rather than quantitative, we expect the trends observed and

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their underlying microscopic origins to be rather general and observable in various type of cross-linked polymer gels, at least as long as certain conditions are met. For example, it is worth remarking that these findings are strictly valid for a polymer network in bad solvent conditions, or in other words for gels in a collapsed state (e.g., see 31 ). In good solvent conditions, the gel will be swollen to different degrees and this, as pointed out by Rose et al. 4 , can have an impact on the adhesive power of the nanoparticles. Similarly, if shear stress is applied a different response might arise, although based on our results we would still expect their organisation at the interface to play an important, and qualitatively similar, role. An exact quantification of nanoparticles adhesivity for these different scenarios is currently not present, and would certainly constitute an interesting avenue for future research. In this regard, the computational approach we presented in this manuscript can easily be adapted to treat these different cases. Overall, we believe the methodology and results presented here provide solid and general foundations for future design of nanoparticles-based polymer adhesives.

Supporting Information Available Supporting information: details on structure generation and equilibration, straining of the structure, force-field functional form and parameters used, and polymer as well as nanoparticle density profiles. This material is available free of charge via the Internet at http://pubs.acs.org/.

Acknowledgement We particularly thank Bortolo M. Mognetti and Fernando Bresme for a critical reading of our manuscript. We also both acknowledge support from the Thomas Young Centre under grant TYC-101; NM also thanks Arash A. Mostofi for the opportunity to work on this project; S.A-U acknowledges funding from the Beijing Advanced Innovation Centre for 18


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Soft Matter Science and Engineering (BAIC-SMSE), PR China. Finally, we thank Imperial College High Performance Computing Service, doi: 10.14469/hpc/2232 and the UK’s HEC Materials Chemistry Consortium, which is funded by EPSRC (EP/L000202).

References (1) Appel, E. A.; Scherman, O. A. Gluing gels: A nanoparticle solution. Nature materials 2014, 13, 231–232. (2) Ahn, Y.; Jang, Y.; Selvapalam, N.; Yun, G.; Kim, K. Supramolecular velcro for reversible underwater adhesion. Angewandte Chemie 2013, 125, 3222–3226. (3) Holten-Andersen, N.; Harrington, M. J.; Birkedal, H.; Lee, B. P.; Messersmith, P. B.; Lee, K. Y. C.; Waite, J. H. pH-induced metal-ligand cross-links inspired by mussel yield self-healing polymer networks with near-covalent elastic moduli. Proceedings of the National Academy of Sciences 2011, 108, 2651–2655. (4) Rose, S.; Prevoteau, A.; Elziere, P.; Hourdet, D.; Marcellan, A.; Leibler, L. Nanoparticle solutions as adhesives for gels and biological tissues. Nature 2014, 505, 382. (5) De Gennes, P.-G. Scaling concepts in polymer physics; Cornell university press, 1979. (6) Netz, R. R.; Andelman, D. Neutral and charged polymers at interfaces. Physics reports 2003, 380, 1–95. (7) Kim, J.-H.; Kim, H.; Choi, Y.; Lee, D. S.; Kim, J.; Yi, G.-R. Colloidal Mesoporous Silica Nanoparticles as Strong Adhesives for Hydrogels and Biological Tissues. ACS applied materials & interfaces 2017, 9, 31469–31477. (8) Uyama, Y. Lubricating polymer surfaces; CRC Press, 1998. (9) Cao, Z.; Dobrynin, A. V. Nanoparticles as Adhesives for Soft Polymeric Materials. Macromolecules 2016, 49, 3586–3592. 19


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(19) Gao, J.; Weiner, J. Simulated polymer melt stress relaxation. II. Search for entanglements. J. Chem. Phys. 1995, 103, 1621. (20) Liu, J.; Shen, J.; Gao, Y.; Zhou, H.; Wu, Y.; Zhang, L. Detailed simulation of the role of functionalized polymer chains on the structural, dynamic and mechanical properties of polymer nanocomposites. Soft Matter 2014, 10, 8971–8984. (21) Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. 1995. (22) Kremer, K.; Grest, G. Dynamics of entangled linear polymer melts: A moleculardynamics simulation. The Journal of Chemical Physics 1990, 92, 5057–5086. (23) Grest, G. S.; Kremer, K. Molecular dynamics simulation for polymers in the presence of a heat bath. Physical Review A 1986, 33, 3628. (24) Everaers, R.; Kremer, K. Elastic properties of polymer networks. Molecular modeling annual. 1996; pp 293–299. (25) Varilly, P.; Angioletti-Uberti, S.; Mognetti, B.; Frenkel, D. “A general theory of DNAmediated and other valence-limited colloidal interactions”. The Journal of Chemical Physics 2012, 137, 094108–094122. (26) Jones, R. A. L. Polymers at surfaces and interfaces; Cambridge University Press, 1999. (27) Geoghegan, M.; Hadziioannou, G. Polymer electronics; OUP Oxford, 2013; Vol. 22. (28) Gent, A.; Hamed, G.; Hung, W.-J. Adhesion of elastomer layers to an interposed layer of filler particles. The Journal of Adhesion 2003, 79, 905–913. (29) Nah, C.; Jose, J.; Ahn, J.-H.; Lee, Y.-S.; Gent, A. Adhesion of carbon black to elastomers. Polymer Testing 2012, 31, 248–253.

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(30) Angioletti-Uberti, S. Theory, simulations and the design of functionalized nanoparticles for biomedical applications: A Soft Matter Perspective. npj Computational Materials 2017, 3, 48. (31) Kim, W. K.; Moncho-Jord´a, A.; Roa, R.; Kanduc, M.; Dzubiella, J. Cosolute partitioning in polymer networks: Effects of flexibility and volume transitions. Macromolecules 2017, 50, 6227–6237.

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Table of Content Graphics Original Strain cutting plane

Strain

(a.1)

(a.2)

(b.1)

PA = 0.8 PA = 1.2

4

(a)

2 1 0

(b.

PA = PA =

4 3 Stress

3

No Nps PA = 0.4

RNP = 2.0 RNP = 4.0

No Nps PA = 0.4

Stress

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2 1 0

0

50

100 Strain [%]

23

150


0

50 1 Strain [%]

Nanoparticle Organization Controls Their Potency ... - ACS Publications

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