Accurate Estimation of Polymer Coverage of Hairy Nanoparticles PDF

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Soft Matter View Article Online PAPER View Journal | View Issue Accurate estimation of the polymer coverage of hairy nanoparticles Published on 04 September 2018. Downloaded by Brock University on 10/23/2024 6:07:30 PM. Cite this: Soft Matter, 2018, 14, 7906 Makoto Asai, Dan Zhao and Sanat K. Kumar * Understanding and predicting the mechanisms underpinning the self-assembly of polymer-grafted nanoparticles (PGNPs) are important for controlling the engineering applications of these novel materials. The self-assembly of these materials is driven by their surfactancy, i.e., by the fact that the (inorganic) nanoparticles energetically dislike the (organic) polymer tethers. In previous work we developed a model in which a grafted polymer chain was treated as a rigid equivalent sphere (ES) which was impenetrable to the NPs, but completely penetrable to other ESs. This description, along with a geometric analogy with patchy particles, allowed us to facilely explain the self-assembly of PGNPs. However, since we model an ES as being completely penetrable to other ESs but impenetrable to the NPs the physical correspondence between a ‘‘real’’ grafted polymer and an ES is not clear. The application of the ES model to experiments and to computer simulations has therefore seen limited success, and only qualitative agreement has been obtained. Received 27th June 2018, In this paper, we develop a more realistic description, termed the modified ES (mES) model, based on the Accepted 4th September 2018 work of Daoud and Cotton on curved polymer brushes, which takes the impenetrability of the individual DOI: 10.1039/c8sm01311j chain monomers into account. While this approach increases the complexity of our formalism, we find that the resulting mES model quantitatively captures computer simulation results on the structure of the PGNPs rsc.li/soft-matter-journal and also quantitatively explains their self-assembly over a broad range of conditions. Introduction ‘‘bare’’ NP. This is reasonable in the low grafting density regime studied in the current paper, where the self-assembly of PGNPs A central challenge in improving the properties of polymer most likely occurs. However, there was no excluded volume nanocomposites is to control the spatial dispersion of the interactions between two (or more) ESs, either on a single NP or nanoparticles (NPs).1–4 One particularly facile approach is to across multiple NPs. This last assumption is based on the fact graft the NPs with polymer chains.5–8 It has been found that that, while excluded volume interactions apply strictly at the these grafted NPs behave akin to surfactants due to the dislike level of two monomers, the centers of mass of two chains can between the typically hydrophilic cores and the hydrophobic overlap with only a small free energy cost.25 The fraction (g*) of corona. This surfactant-like nature causes these tethered NPs to the NP surface that is excluded to a second bare NP due to an ES assemble into a large range of superstructures especially in the low (or a single grafted chain) can then be derived: grafting density, s, limit (typically for s t 0.1 chains per nm2, a see ref. 9), i.e., where the cores are not completely shielded from g ¼ (1) 2ð1 þ aÞ other cores by the corona.9–23 To understand this behavior, we have previously developed Here, a  R/Rn. When a - N, g* - 1/2, which means that one the equivalent sphere (ES) model.24 A spherical NP of radius Rn ES can cover half of the surface of a NP in this limit. For f grafts randomly grafted with f chains was considered, where each the fraction (S*) of the NP surface that is excluded to a second chain was comprised of N catenated monomers. As a signifi- bare NP can then be described by analogy to random sequential cant simplification we modeled each grafted polymer as a rigid adsorption: ES of radius R. We further assumed that the ES served to dS exclude a ‘‘patch’’ on the grafting NP’s surface where the core ¼ 1  S ð0  S   1Þ (2) df g of another bare NP cannot contact. It is noted that the second NP has grafted polymers on its surface as well but for simplicity That is, we write that S* can only increase if the newly we only calculated the excluded area on the NP to another grafted ES shields unexcluded parts of the surface. This yields S* = 1  efg*, which we have found to be in good agreement Department of Chemical Engineering, Columbia University, New York, 10027, USA. with our simulation results on NPs literally grafted with ESs. E-mail: [email protected] Thus, we have a means of describing the excluded surface area 7906 | Soft Matter, 2018, 14, 7906--7915 This journal is © The Royal Society of Chemistry 2018  View Article Online Paper Soft Matter In ref. 24 we postulated that R = bRg where Rg is the radius of gyration of the grafted polymer chain in a good solvent and b is an empirical fitting parameter. Fig. 2(a) uses the naive ansatz that b = 1 and we see that the ES model only provides qualitative agreement with experimental data and simulation results from the literature.24 For example, it is clear that regions where strings are formed in the Monte Carlo simulations Fig. 1 Different particle morphologies with increasing polymer surface are predicted to form clumps etc. Clearly, there is room for coverage, following Asai et al. [ref. 24] (Aggregate) three-dimensional and improvement in terms of this model prediction. (2D-Sheet) two-dimensional aggregates (coordination number Z 4); Instead, we have empirically found that we need to use b E 0.46 Published on 04 September 2018. Downloaded by Brock University on 10/23/2024 6:07:30 PM. (String) one dimensional linear aggregate (coordination number = 2–3); (Clump) small aggregates including dimers, trimers and tetramers (coordina- to get better agreement with experiments and simulations.24,26 tion number = 1–3); (Dispersed) isolated particles with full surface coverage However, it is unclear why we have to use an R that is less than Rg (coordination number = 0). The formula for S* for each morphology is and whether the factor b = 0.46 is universal. When this information presented in ref. 24. is absent, the conventional ES model is limited in terms of structure prediction. Additionally, the model has a major simpli- fication in that the excluded volume interactions between two afforded by the grafted polymers on the surface of the NP with (or more) grafted chains are ignored since the ES are assumed to variations in f and a. be fully penetrable to each other but completely impenetrable to We then drew a geometric analogy between NPs grafted with the core of another NP. We conjecture that dispensing with these ES and patchy particles. Regions on the NP surface from which assumptions, by modeling the grafts more realistically, should a second NP is excluded (due to the presence of the ES) are allow for a more reliable representation for these systems.27 So, defined as repulsive patches while the remaining NP surface is in this paper, we introduce this improvement to the conventional attractive to a second NP. In this representation, thus, the ES model. In particular, we use the Daoud–Cotton model for polymer chains (or the ESs) are abstracted away and only polymer brushes to more accurately model the polymer chains in manifest themselves as an effective (angle and distance depen- this situation. By validating against computer simulations, we dent) inter NP potential. With this mapping we can predict the show that this model provides an improved description of the self-assembled structures formed by a particular NP with structure of these NPs and hence their self-assembly behavior. knowledge of the S* and the geometric structure of the PGNPs. Thus, we propose that this modified ES (mES) model can be used Our idea, which has been previously discussed in ref. 24, is to reliably understand the self-assembly of this class of materials. sketched in Fig. 1. When the polymer surface coverage S* is large, we get well dispersed NPs. As the ES coverage decreases we first see the formation of small clumps comprised of Results and discussion 2–4 NPs. Further decreases in S* yield linear strings of NPs and finally two and three-dimensional aggregates. Since this mES model argument is purely geometrical, we can provide precise values The basic strategy for constructing the mES model is to more of S* where these ‘‘structural’’ transitions occur. We do not accurately account for the structural properties of real polymer have the ability to decide if these are thermodynamic transi- brush chains than in the ES model, i.e., to account for the tions or not. excluded volume interactions at the level of two monomers. Fig. 2 Phase diagram of self-assembled structures of PGNPs: (a) based on ES model and (b) based on mES model. f is the number of chains grafted to a NP, and a is defined in the text. This journal is © The Royal Society of Chemistry 2018 Soft Matter, 2018, 14, 7906--7915 | 7907  View Article Online Soft Matter Paper According to the Daoud–Cotton picture of a star polymer,28 which can be extended to describe polymer statistics on curved surfaces,29 the grafting process only weakly changes the effec- tive chain size, R, i.e., by a factor f 1/5, where f is the number of grafts. Since R should be a function of N (the degree of polymerization for the grafted chain) and f, we assume that an appropriate size of the brush chain follows R(N, f ) = R(N)  R( f ), that is we assume that the N and f dependences are separable. Before considering the f dependency, first, we investigate the Published on 04 September 2018. Downloaded by Brock University on 10/23/2024 6:07:30 PM. statistics of a single grafted polymer ( f = 1). Let us consider one free (ungrafted) polymer chain with radius of gyration Rg and degree of polymerization N. The monomer density at a distance r from the polymer’s center of mass is30 Fig. 3 Examples of C. Rn = 7.0. Rg = 3.5, 7.8 and 15.0 for N = 30, 100 ND D3 and 300. rðrÞ ¼ r (3) 4pRD g Here D is the fractal dimension of the chain, and we use the While the discussion above focuses on the overlap between a Ð 2p Ð p ÐR polymer chain grafted onto a NP1 and the core of a NP2, the mass balance condition 0 dj 0 sin ydy 0 g r2 rðrÞdr ¼ N. If the more important quantity is what fraction of the NP1 surface is chain were fully collapsed then D = 3 and eqn (3) suggests that inaccessible to NP2 due to the presence of the grafted chain. To the monomer density profile is constant, as expected. For a make this calculation we place NP2 in contact with the surface Gaussian chain D = 2 and so the monomer density decreases of NP1 and calculate the number C of monomer units of the with increasing r. We assume that a single grafted chain on the polymer overlapping with NP2 (see Fig. 4(a)). The distance NP (NP1) surface has the same density distribution about between the center of mass of the polymer chain and NP2 is L. the center of mass as the ungrafted analog and calculate the When C Z 1 then this point corresponds to an overlap between number C(L,Rn,Rg) of monomer units of the polymer over- NP2 and the graft – this point is thus excluded to NP2. We now lapping with a second bare NP (NP2) of radius Rn which is place the NP2 on several points on the surface: the ratio of the placed at a distance L from the center of mass of the polymer number of points with C Z 1 to the total number sampled should chain (see Appendix Fig. 9). That is, we take a NP1 with a be the exclusion area g* for a polymer chain for NP2. Therefore, grafted chain and ask as to how much grafted chain-NP2 we define g* as follows: overlap this system has when a NP2 is brought to a distance L from the grafted polymer chain’s center of mass. To calculate ð 2p ðp 1   sin ydy this quantity, the following volume integration should be g ¼ dj C (5) 4p 0 0 performed on the region where the NP2 and the polymer grafted on the NP1 overlap. ( ð 1; C  1     C (6) C L; Rn ; Rg ¼ rðrÞdV (4) 0; C o 1 While the calculation of this integration is straightforward, it is Then, we can determine R using eqn (1). The geometric concept is mathematically tedious and deferred to Appendix A. However, shown in Fig. 4(b). We evaluated eqn (5) numerically since it is illustrative numerical examples of the behavior of this function difficult to analytically calculate it (Fig. 5(a)–(c)). As a reference, we are shown in Fig. 3. We assumed 2Rg = N1/D and D = 5/3, which also show g* calculated by the ES model. In this case, we used describes the radius of gyration of coarse-grained Kremer–Grest eqn (1) with b = 1 (R = Rg). This figure critically illustrates the chains in good solvent.31 Some general comments are in order. qualitative errors in our previous ES model. In the ES model, g* In general, the overlap function C(L,Rn,Rg) increases with monotonically increases with increasing N and eventually reaches decreasing L, till it reaches a plateau value at small L. Note the theoretical maximum value (= 1/2). This result simply follows that C(Rn + Rg,Rn,Rg) = 0 in the situation where the distance from the fact that there is strict impenetrability between an ES and between the NP2 and the center of mass of the grafted polymer the core of a NP2. On the other hand, in the mES model, g* starts chain is L = Rn + Rg or for any larger separations since there is to decrease when N becomes sufficiently large. This is because, no overlap. In addition, when Rg r Rn, the chain is completely when Rn { Rg, the NP2 cannot overlap with the whole polymer inside NP2 when L = 0 and thus C(0,Rn,Rg) = N. On the other chain. To illustrate this point, we consider the case when NP1, NP2 hand, when Rg 4 Rn, the maximum value inC is less than N, and the polymer are collinear, but with NP1 and NP2 being in because the chain is not completely inside the NP2 even when contact. Under these conditions, L = Rg  Rn c 0. Therefore, the their centers of mass coincide: thus, the NP2 can only overlap center of NP2 will only experience the very low-density perimeter of with a part of the polymer chain. the polymer coil. This means that, in the long chain limit, the NP2 7908 | Soft Matter, 2018, 14, 7906--7915 This journal is © The Royal Society of Chemistry 2018  View Article Online Paper Soft Matter Published on 04 September 2018. Downloaded by Brock University on 10/23/2024 6:07:30 PM. Fig. 4 Determination of the effective sphere diameter in the mES models. (a) Searching around a grafted polymer to determine the distribution of C. In the mES model we define the area where C Z 1 as the excluded area, and the ratio of the total excluded area to the NP surface area is g*. (b) Determining the ES corresponding to g*. Grey area indicates the excluded area, 4pRn2g* where NP2 cannot contact NP1 in. The dashed line represents the corresponding ES whose size is determined using eqn (1): R = 2g*Rn/(1  2g*). Y is the contact angle between NP1, NP2 and ES, and it follows from cos Y = Rn(Rn + Rg)1. Fig. 5 Comparison of mES model with ES model. Comparison of g*: (a) Rn = 3.0. (b) Rn = 7.0. (c) Rn = 14.0. (d) Comparison of R derived from g* and eqn (1). and the polymer chain interact minimally leading to small g* limit of large N, R approaches zero, which corresponds to g* values. approaching zero. From the above discussion, it is clear that Moreover, we estimated R corresponding to these g* using the initial assumption built into the ES model that R is simply eqn (1) (Fig. 5(d)). In the conventional ES model, since R proportional to Rg, R = bRg, is incorrect. was assumed to be related linearly to Rg, R depends on only To prove the validity of the mES model, we performed N (R p Rg p N1/D). However, in the mES model we find that R coarse-grained Molecular Dynamics simulations using the depends, not only on N, but also on Rn. Furthermore, in the Kremer–Grest model.31 We directly measured the excluded area This journal is © The Royal Society of Chemistry 2018 Soft Matter, 2018, 14, 7906--7915 | 7909  View Article Online Soft Matter Paper provided by one grafted chain, g*, to the core of a NP2 in the measured in simulations, especially as N becomes larger. On the range of 5 r N r 104 for up to 107–108 MD time steps, long other hand, we found that g* calculated in the mES model was in enough to achieve equilibrium in all cases. We set Rn = 7.0. good agreement with those measured in simulations over the Simulation details are described in the Methods Section. First, whole range of N, including the non-monotonic dependence of g* we calculate the N-dependence of the Rg of one grafted polymer on N observed in simulations. Furthermore, we converted g* to an chain. We obtained Rg = 0.33N 0.70 (N r 50) and Rg = 0.49N 0.60 effective R, the size of an ES, using eqn (1) as shown in Fig. 6(c). (N 4 50). Next, we directly measured g* in the simulations by The size of an ES thus does not monotonically depend on the Rg of tessellating the surface of the NP1 using 4112 points placed at the chain. We thus have a full understanding of the surface the vertices of a spherical crystal following the symmetry of a coverage afforded by a single grafted chain. (20, 20) icosadeltahedron. We fixed the center of NP1 and Next, we account for the effect of multiple grafted chains by Published on 04 September 2018. Downloaded by Brock University on 10/23/2024 6:07:30 PM. performed MD simulations of a single tethered polymer. For each assuming the validity of the Daoud–Cotton ansatz.28 Note that MD snapshot, we assign pi = 1 if there are polymer beads which the separation of the N and f dependence inherent in our overlap with a NP2 located on the ith point (i = 1,2,...,4112) of the approach is only reasonable when the grafting density is surface of NP1, otherwise pi = 0. By taking the time-average h piit of relatively low, i.e., when the chains are not significantly dis- P 4112  torted. We thus estimate the f-dependence of R( f ) as follows. pi, we calculate the excluded area ratio as g ¼ h pi it 4112. i R = R(N,Rn)f 1/5 (7) Fig. 6(a) shows an example of the surface distribution of h piit. We see that there is a spherical cap-shaped excluded area To confirm the validity of eqn (7), we calculated the f-dependence formed by a grafted chain on the surface of NP1, indicating that of g* of a polymer chain in the range 1 r f r 100 and N = 5, 100, the geometric concept expressed by eqn (1), commonly used in 200 using MD simulations. The images in Fig. 7(a) show examples the ES and mES models, is reasonable. However, because a of the surface distribution of h piit. We found that as f becomes polymer is treated as a rigid sphere in the original ES model, larger, the distribution of hpiit becomes narrower. Note that this is h piit should be 1 within the spherical cap-shaped excluded area. a plot of the probability density associated with one representative In fact, h piit varies gradually as shown in Fig. 6(a). Fig. 6(b) shows chain out of the f that are grafted to the NP1 surface: as f increases the N-dependence of g*. In the ES model, we calculated g* using the distribution of a single chain narrows in space as may be eqn (1) assuming b = 1 (R = Rg). On the other hand, in the mES expected. We converted these g* to R using eqn (1). Fig. 7(a) and (b) model, we calculated g* using eqn (5) without any assumptions. show the f-dependence of g* and R as determined from the As a result, the ES model overestimates g* compared to those simulations, respectively. We confirm the scaling law: R p f 1/5 Fig. 6 Excluded area of single grafted chain on NP1 surface to a second NP (NP2). (a) An example of the surface distribution of hpiit. We set the following parameters: Rn = 7.0, N = 200, f = 1, Rg = 11.3. We measured g* = 0.23. Color indicates value of hpiit. (b) N-Dependence of g*. (c) N-Dependence of R derived using the g* and eqn (1). Error bars show standard error. 7910 | Soft Matter, 2018, 14, 7906--7915 This journal is © The Royal Society of Chemistry 2018  View Article Online Paper Soft Matter Published on 04 September 2018. Downloaded by Brock University on 10/23/2024 6:07:30 PM. Fig. 7 Effect of adding grafted polymer chains on the NP1 surface on g* and R. (a) f-Dependence of g*. Note that g* represents the fraction of the NP1 surface that is excluded to NP2 due to presence of a single grafted chain. Pictures shows the surface distribution of hpiit for N = 200 and Rn = 7.0. The color scheme is the same as in Fig. 6(a). (b) f-Dependence of R. The dashed lines represent eqn (7). in the large N (Z200)/relatively large f (420) limit. On the other fraction of the NP1 surface that is inaccessible to the NP2 due hand, the scaling law does not work for smaller N and f. According to the presence of the grafted chains. In addition, we compared to Daoud–Cotton theory, polymers are not influenced by the S* with those predicted by the ES and mES models. In the ES curvature of the NP surface when the end-to-end distance of the and mES models, we first calculated R using the hypothetical chains Re { Rn. Indeed, we find that Re E 4.2Rn for N = 200, Re E relationship, R = bRg and eqn (7), respectively. Note that since 2.8Rn for N = 100, and Re E 1.8Rn for N = 50, respectively, suggesting eqn (7) does not work in the region of small f and small N as that we are approaching the limits of this theory for small N. discussed in Fig. 7, we used the fitted functions obtained from We directly measured the total excluded area ratio S* in Fig. 7(b) as follows: for N = 50, R = 3.45f 0.046 ( f o 30). For the simulations with f grafted chains – this corresponds to the N = 100, R = 4.39f 0.128 ( f o 20). By substituting R into eqn (1), Fig. 8 f-Dependence of S*. Rn = 7.0. (a) N = 50. (b) N = 100. (c) N = 200. This journal is © The Royal Society of Chemistry 2018 Soft Matter, 2018, 14, 7906--7915 | 7911  View Article Online Soft Matter Paper Table 1 Experimental conditions and parameters. R, g* and S* are calculated by the mES model System Rna (nm) Na fa Morphologya Rg (nm) R (nm) g* S* a (I) PS-g-silica NPs ref. 10 7.0 489.68 6.16 String/clump 4.52 2.78 0.14 0.58 0.40 7.0 1517.04 6.16 String/clump 7.95 4.80 0.20 0.71 0.69 7.0 1538.46 30.80 Dispersed 8.01 3.51 0.17 0.99 0.50 7.0 1017.76 30.80 Dispersed 6.51 2.87 0.15 0.99 0.41 7.0 432.07 61.60 Dispersed 4.24 1.62 0.09 1.00 0.23 7.0 1480.77 61.60 Dispersed 7.85 3.02 0.15 1.00 0.43 (II) PS-P2VP-g-silica NPs ref. 32 7.0 998.56 12.32 Clump 6.45 3.45 0.17 0.87 0.49 7.0 1507.44 30.79 Dispersed 7.93 3.49 0.17 0.99 0.50 Published on 04 September 2018. Downloaded by Brock University on 10/23/2024 6:07:30 PM. 7.0 1488.24 67.73 Dispersed 7.87 2.97 0.15 1.00 0.42 (III) PS-b-P2VP-a-silica NPs ref. 33 7.0 1425.83 0.62 Aggregate 7.71 6.75 0.25 0.11 0.96 7.0 1425.83 1.23 Sheet/connected 7.71 6.75 0.24 0.27 0.92 7.0 1425.83 6.16 String/clump 7.71 6.75 0.20 0.86 0.67 7.0 1056.17 6.16 String/clump 6.63 5.70 0.18 0.81 0.57 a These parameters were obtained from references. we calculated g*. Finally, using eqn (2), we calculated S* as matrix (PS-g-silica NPs),10,32 (II) mixed bimodal polystyrene- shown in Fig. 8(a)–(c). The mES model shows good agreement poly(2-vinylpyridine) brush coated silica NPs in a polystyrene with simulation values of S*, and the difference between the ES matrix (PS-P2VP-g-silica NPs),32 (III) polystyrene-b-poly(2- and mES models becomes larger with increasing N. This is due vinylpyridine) block copolymer physically absorbed silica NPs to the fact that the difference of R between the ES and mES in a polystyrene matrix (PS-b-P2VP-a-silica NPs),33 (IV) coarse- models is small until N E 50 (see Fig. 6(c)). grained bead-spring polymer-grafted onto NPs studied by Monte Carlo simulation (CG simulation).10 In the case of Self-assembly experiments using polystyrene, we estimated Rg as Rg E a(N/6)1/2, Finally, we examine whether the mES model can quantitatively which is the unperturbed radius of gyration in the melt and a is the explain experimental results and simulation findings for the segment length of a polystyrene chain, which was estimated to be self-assembly of this class of PGNPs. To this end, we use the 5 Å.34 In case (III), we assumed that since P2VP adsorbs completely information on experimental conditions (N, f, Rn) and calculate on the surface of NP, Rg of BCP is calculated by only taking the PS the effective R for each condition using the mES model and eqn (1). block into account. We show all necessary parameters in Tables 1 Then, using the geometric analogy with patchy particles,24 and 2. With given Rn, N and f we estimate R by eqn (7), and then discussed above in the context of Fig. 1, we predicted the self- calculate a as a  R/Rn. Fig. 2b shows each sample plotted on the assembled structures formed and compared them with the a–f plane of the theoretical phase diagram of the self-assembly of morphologies reported. We refer to the following different PGNPs. We find that self-assembled structures found in simula- systems: (I) polystyrene-grafted silica NPs in a polystyrene tions of coarse grained models and also three different classes of Table 2 Simulation conditions and parameters. R, g* and S* are calculated by the mES model System Rna (s) Na fa Morphologya Rg(s) R(s) g* S* a (IV) MC simulations ref. 10 3.75 0 0 Aggregates 0 0 0 0 0 3.75 3 4 Sheet 0.71 0.34 0.04 0.15 0.09 3.75 6 4 Sheet 1.16 0.63 0.07 0.25 0.17 3.75 10 4 Sheet 1.65 0.92 0.10 0.33 0.25 3.75 12 4 Sheet 1.88 1.03 0.11 0.35 0.27 3.75 4 6 Sheet 0.87 0.45 0.05 0.27 0.12 3.75 5 6 Sheet 1.02 0.53 0.06 0.31 0.14 3.75 6 6 Sheet 1.16 0.63 0.07 0.35 0.17 3.75 8 3 Sheet 0.71 0.34 0.04 0.28 0.09 3.75 8 4 Sheet 0.87 0.45 0.04 0.35 0.12 3.75 2 12 Sheet 0.54 0.23 0.03 0.29 0.06 3.75 3 12 Sheet 0.71 0.34 0.04 0.39 0.09 3.75 6 8 String 1.41 0.78 0.09 0.40 0.21 3.75 8 6 String 1.16 0.63 0.07 0.44 0.17 3.75 6 10 String 1.65 0.92 0.10 0.45 0.25 3.75 8 8 String 1.41 0.78 0.09 0.50 0.21 3.75 12 4 String 0.87 0.45 0.05 0.47 0.12 3.75 14 6 Clump 2.09 1.17 0.12 0.51 0.31 3.75 10 8 Clump 1.65 0.92 0.10 0.55 0.25 3.75 6 12 Clump 1.16 0.63 0.07 0.58 0.17 a These parameters were obtained from references. 7912 | Soft Matter, 2018, 14, 7906--7915 This journal is © The Royal Society of Chemistry 2018  View Article Online Paper Soft Matter Published on 04 September 2018. Downloaded by Brock University on 10/23/2024 6:07:30 PM. Fig. 9 Schematic of geometric relation between the grafted polymer coil and the NP2. The dashed and solid lines represent a grafted polymer coil (radius of Rg) and a NP2 (radius of Rn). NP1 which the polymer coil is grafted on is not shown. experiments are in good agreement with the mES – based theore- Methods tical predictions over a broad range of a and f values. We therefore Simulation model believe that the mES model allows us to capture the self-assembly behavior of these PGNPs without the use of any adjustable Grafted polymers are represented using the coarse-grained parameters. bead-spring model of Kremer and Grest.31 Each chain contains N beads of mass m = 1. All beads interact via the Lennard-Jones (LJ) potential. 8  12  6  Conclusions < 4e s  s > ; r  rc U p ðrÞ ¼ r r (9) We propose a new calculation method for the surface coverage > : afforded by polymer chains grafted on to spherical NP surfaces. 0; r 4 rc This calculation method can accurately predict the area of the NP surface that is excluded to another NP by the presence of the where r is the distance between two beads, e is the Lennard- grafted chain. A simple extension of this model by adopting Jones unit of energy, and s is the bead diameter. We set rc = 21/6s. ideas from the Daoud–Cotton approach allows us to model Beads along the chain are connected by an additional unbreakable NPs with multiple grafts. All of these results are in quantita- finitely extensible nonlinear elastic (FENE) potential UFENE(r) = tive agreement with coarse grained simulations. Further, we 1/2klmax2 ln[1  (r/lmax)2], with lmax = 1.5s and k = 30e/s2. draw an analogy of these grafted particles to patchy NPs, and We use the expanded LJ potential for pair interactions between from there predict the self-assembled structures that are colloid–colloid and colloid–polymer beads as follows; formed. These results, which therefore have no adjustable 8   > s 12  s 6 parameters, are in excellent agreement with appropriately < 4e  ; r  rc þ D U ðrÞ ¼ rD rD (10) curated previous experiments and simulations. We therefore > : propose that the mES model can apparently be used to 0; r 4 rc þ D quantitatively understand the structure and the anisotropic self-assembly of this class of polymer grafted nanoparticles. here, we choose D = 4s and D = 2s for colloid–colloid and Importantly, although in the current paper we only focus colloid–polymer bead interactions, respectively. One end bead on the case of good solvent conditions, our model can be of the grafted polymer is fixed on the surface of the colloid naturally extended to other solvent qualities (e.g. theta solvent (grafting point). f grafting points are randomly located on the or poor solvent) by reformulating the Daoud–Cotton model surface. NNP(= 43) colloids have different patterns of grafting and the associated scaling laws. points arrangements. This journal is © The Royal Society of Chemistry 2018 Soft Matter, 2018, 14, 7906--7915 | 7913  View Article Online Soft Matter Paper Molecular dynamics simulation right side. So cos ym1 = (L2 + Rg2  Rn2)/2RgL and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi All simulations are carried out using the LAMMPS parallel MD rmax ¼ L cos y  ðL cos yÞ2 ðL2  Rn2 Þ. As a result, we package. NVT MD simulations are performed in an orthogonal obtain: cubic simulation box. Temperature T is set to 1.0e/kB and is  2  maintained by a Langevin thermostat with a damping constant N L þ Rg2  Rn2 C ¼ 1 G = 0.01s1(m/e)1/2. kB is Boltzmann’s constant. The NPs’ 2 2Rg L positions are fixed and only the dynamics of grafted polymers h N   is enumerated. The simulations are run for 106–108 time steps þ RD1 g ðD þ 1Þ Rn2  L2 þ ðD  1ÞRg2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 4RDg Lð D 2  1Þ of length dt ¼ 0:005 ms2 =e to equilibrate the system and then i Published on 04 September 2018. Downloaded by Brock University on 10/23/2024 6:07:30 PM. another 107–108 time steps for each observation. 2ðRn  LÞD ðDRn þ LÞ (A4) Conflicts of interest Rn r L r Rn + Rg: the polymer coil and the NP2 are partially The authors declare no competing financial interests. overlapping and the center of the polymer coil (Op) is outside the NP2 and vice versa (Fig. 9(c)). In this case, eqn (4) becomes: Appendix   ND ð 2p ð ym ð Rg C L; Rn ; Rg ¼ dj sin ydy rD1 dr (A5) We will explain the derivation of C, which depends on Rg, Rn and 4pRD g 0 0 rmin L. Fig. 9(a)–(e) show 5 different geometric situations to consider. Here a polymer chain grafted on the surface of a NP1 is drawn as a Here cos ym = (L2 + Rg2  Rn2)/2RgL and rmin ¼ L cos y qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi spherical sphere with radius of Rg (hereinafter called ‘‘a polymer ðL cos yÞ2 ðL2  Rn2 Þ. As a result, we can derive: coil’’) and it has fractal structure inside the sphere. The integration   2  range is the overlapping range of two spheres (the polymer coil N L þ Rg2  Rn2 D2 and the NP2), and the coordinates within the integration range are C ¼ 1 2 2Rg LðD2  1Þ denoted by (r,y,j). ym, rmax and rmin are the maximum angle and  2  #   the maximum and the minimum length in the integration range L  Rn D DRn þ L L  Rg2  Rn2 D þ  of y and r, respectively. Rg ðD2  1ÞL 2Rg LðD2  1Þ (I) Rg r R n (A6) In this case, we have to consider the following three situations (II) Rn r Rg r 2Rn for the range of L: 0 r L r Rg  Rn: the NP2 is completely inside the polymer coil 0 r L r Rn  Rg: the polymer coil is completely inside the and the center of the polymer coil (Op) is inside the NP2. As an NP2. As an example, Fig. 9(a) shows a case where L = Rn  Rg. example, Fig. 9(d) shows a case where L = Rn  Rg. In this case, In this case, eqn (4) becomes: eqn (4) can be described as: ð ðp ð Rg   ND 2p ð 2p ðp ð rmax C L; Rn ; Rg ¼ dj sin ydy rD1 dr (A1)   ND 4pRD g 0 0 0 C L; Rn ; Rg ¼ dj sin ydy rD1 dr (A7) 4pRD g 0 0 0 As a result, we derive: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C=N (A2) Here, rmax ¼ L cos y þ ðL cos yÞ2 ðL2  Rn2 Þ. As a result, we derived: Rn  Rg r L r Rn: the polymer coil and the NP2 are partially N n o overlapping and the center of the polymer coil (Op) is inside the C¼ ðRn þ LÞD ðDRn  LÞ  ðRn  LÞD ðDRn þ LÞ 2RD 2 NP2 (On) and vice versa. As an example, Fig. 9(b) shows a case g LðD  1Þ where L = Rn. In this case, eqn (4) becomes: (A8) ð 2p ð ym1 ð Rg   ND C L; Rn ; Rg ¼ dj sin ydy rD1 dr Rg  Rn r L r Rn: the polymer coil and the NP2 are partially 4pRD g 0 0 0 overlapping and the center of the polymer coil (Op) is inside the (A3) ð 2p ð p ð rmax NP2 (On) and vice versa. This case is geometrically same with a þ dj sin ydy rD1 dr case of Fig. 9(b), but just different of relative size of Rg and Rn. 0 ym1 0 The obtained result is equal to eqn (A4). Here, the integration range should be divided into two, corres- Rn r L r Rn + Rg: the polymer coil and the NP2 are partially ponding to the first and second term of the right side, respec- overlapping and the center of the polymer coil (Op) is outside tively. The first and second integration range are represented by the NP2 and vice versa (Fig. 9(c)). This case is geometrically filled and dotted areas, respectively in Fig. 9(b). Here, ym1 is the same with a case of Fig. 9(c), but just relative size of Rg and Rn. maximum angle of integration range of y in the first term of The obtained result is equal to eqn (A6). 7914 | Soft Matter, 2018, 14, 7906--7915 This journal is © The Royal Society of Chemistry 2018  View Article Online Paper Soft Matter (III) 2Rn r Rg 7 C.-K. Wu, K. L. Hultman, S. O’Brien and J. T. Koberstein, 0 r L r Rn: the NP2 is completely inside the polymer coil and J. Am. Chem. Soc., 2008, 130, 3516–3520. the center of the polymer coil (Op) is inside the NP2. This case 8 S. E. Harton and S. K. Kumar, J. Polym. Sci., Part B: Polym. is geometrically same with a case of Fig. 9(d), just different Phys., 2008, 46, 351–358. of relative size of Rg and Rn. The obtained result is equal to 9 S. K. Kumar, N. Jouault, B. Benicewicz and T. Neely, Macro- eqn (A8). molecules, 2013, 46, 3199–3214. Rn r L r Rg  Rn: the NP2 is completely inside the polymer 10 P. Akcora, H. Liu, S. K. Kumar, J. Moll, Y. Li, B. C. coil and the center of the polymer coil (Op) is outside the NP2. Benicewicz, L. S. Schadler, D. Acehan, A. Z. Panagiotopoulos, As an example, Fig. 9(e) shows a case where L = Rg  Rn. In this V. Pryamitsyn, V. Ganesan, J. Ilavsky, P. Thiyagarajan, R. H. Colby and J. F. Douglas, Nat. Mater., 2009, 8, 354–359. Published on 04 September 2018. Downloaded by Brock University on 10/23/2024 6:07:30 PM. case, eqn (4) becomes: 11 M. Belkin, A. Snezhko, I. Aranson and W.-K. Kwok, Phys. ð ð ym ð rmax   ND 2p Rev. Lett., 2007, 99, 158301. C L; Rn ; Rg ¼ dj sin ydy rD1 dr (A9) 4pRDg 0 0 rmin 12 M. Seul and D. Andelman, Science, 1995, 267, 476. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 Z. Tang, Z. Zhang, Y. Wang, S. C. Glotzer and N. A. Kotov, Here sin ym = Rn/L, rmax ¼ L cos y þ ðL cos yÞ2 ðL2  Rn2 Þ, Science, 2006, 314, 274–278. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 14 K. Van Workum and J. F. Douglas, Phys. Rev. E: Stat., and rmin ¼ L cos y  ðL cos yÞ2 ðL2  Rn2 Þ. As a result, we Nonlinear, Soft Matter Phys., 2006, 73, 031502. can derive: 15 S. N. Fejer and D. J. Wales, Phys. Rev. Lett., 2007, 99, 086106. 16 E. Rabani, D. R. Reichman, P. L. Geissler and L. E. Brus, N C¼ 2 Nature, 2003, 426, 271–274. 2RDg LðD  1Þ 17 S. Gupta, Q. Zhang, T. Emrick, A. C. Balazs and T. P. Russell, n o