Dispersion, Phase Separation, and Self-Assembly of Polymer-Grafted

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

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Dispersion, Phase Separation, and Self-Assembly of Polymer-Grafted Nanorod Composites Vaishnavi Gollanapalli, Anirudh Manthri, Uma K. Sankar, and Mukta Tripathy* Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076 Maharashtra, India

ABSTRACT: A systematic understanding of the miscibility of grafted nanorods in polymer melt is required in order to synthesize materials and make devices with controllable properties. While there have been a number of studies on the effect of graft length and graft density on the miscibility of grafted nanorods, the effect of graft arrangement and nanorod geometry remains to be explored. We use integral equation theory to study the dispersion, macrophase separation, and self-assembly of sparsely grafted nanorods in polymer melt. This phase behavior is studied as a function of nanorod diameter, aspect ratio, and density as well as the length and arrangement of the polymer grafts. The phase behavior of these systems is a result of a competition between matrix-induced depletion attraction between nanorods and the steric stabilization provided by grafts. Because of steric shielding of the grafts, nanorod miscibility usually increases with graft length, and trans-grafted rods are more soluble than cis-grafted rods. Depletion attraction is stronger between larger nanorods. Therefore, shorter and thinner nanorods with longer grafts are found to microphase separate, while longer and thicker nanorods with shorter grafts tend to macrophase separate from the matrix polymer. While miscibility of bare nanorods is a monotonically decreasing function of aspect ratio, the miscibility of grafted nanorods can also be a nonmonotonic or a monotonically increasing function of aspect ratio. Grafted nanorods become less soluble in the matrix polymer as their diameter increases. Thus, the effect of rod geometry on the phase behavior of these composites is subtle and complex. of semiconducting CdSe28 or ZnO29 nanorods and conjugated polymers have applications in photovoltaic devices. Nanorods can also be used to seed the self-assembly of block copolymers30 and induce a kinetic stabilization of incompatible heteropolymers.31 At low grafting densities (σ), the matrix polymer and nanoparticles phase separate because of van der Waals attraction as well as a matrix-induced depletion attraction between the particles. This phenomenon known is as “allophobic dewetting”.22,32 With increasing grafting density, these effects are mitigated. The short-range van der Waals attractions are screened by the grafted layer, allowing chemically similar grafts (of length N) and matrix polymer (of length P) to mix. Self-consistent field (SCF) theories have estimated the thickness of this layer of penetration between grafted and matrix polymer for flat33,34 as well as curved

I. INTRODUCTION Composites of polymer and asymmetric nanoparticles have interesting mechanical1,2 and structural properties.3,4 Solubilizing nanorods and nanotubes in a polymer melt is currently a major challenge of materials science. Improvements in strength,5 conductivity,6−8 and rheological9−11 and optical properties12−16 can be achieved when such quasi-1-dimensional objects are dispersed in a polymer matrix. However, a polymermediated entropic depletion attraction can induce phase separation even for noninteracting nanorods.17−19 Toward this end, grafting polymer of identical chemistry as the matrix polymer onto nanospheres is emerging as a method for obtaining dispersed as well as structured composites.20−24 Similar methods have been applied to gold nanorods,25,26 which have two different surface plasmon resonance (SPR) frequencies corresponding to their diameter and length, respectively. Furthermore, the orientation and spacing of the nanorods can cause a shift in the SPR peaks. This makes composites of polymer and Au nanorods likely candidates for use in biosensing27 and optoelectronic devices.13,14 Composites © XXXX American Chemical Society

Received: August 14, 2017 Revised: October 11, 2017

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DOI: 10.1021/acs.macromol.7b01754 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. (a) Schematic of matrix polymer and trans-grafted nanorods. (b) Schematic of matrix polymer and cis-grafted nanorods. (c) The peak values of nanoparticle−nanoparticle partial static structure factors of a bare rod (red pluses) and a rod with graft length N = 8 (green crosses) as a function of polymer−particle interaction strength. In both cases the rods are of AR = 4, D/d = 2, and ϕn = 0.1. For the bare rod, the peak is located at k* = 0, and for the grafted rod, the peak is located at k* ≠ 0. (d) The nanoparticle−nanoparticle structure factor of nanospheres with D/d = 4 and ϕn = 0.1 with different graft lengths. The horizontal black dashed line represents the Hansen−Verlet criterion.

surfaces.35 These as well as scaling36 theories have found that at high enough graft densities the grafts stretch to become a brush, and beyond a critical grafting density the matrix polymer is expelled from the brush. This expulsion of the matrix polymer from the brush is called “autophobic dewetting” and has been discussed in the context of systems with flat surfaces,37 polymer−nanosphere composites,22,23 and polymer nanorod composites.26,38 The origin of this behavior is entropic and occurs at high grafting density (σN1/2> 1), particularly when the grafted polymer is significantly shorter than the matrix polymer. Scaling as well as SCF theories based on flat surfaces have predicted an autophobic dewetting boundary given by σN1/2 ∼ (N/P)γ, where γ is a positive scaling exponent which depends on the theoretical approach taken.32,36,39,40 A more recent SCF theory approach35 shows that the high curvature of nanoparticles gives more space for the grafted polymer chains to splay out. This allows the matrix polymer to wet the grafted polymer without significant entropic loss and shifts the demixing boundary to lower N/P. In this paper, we report results for grafted nanorods in the low grafting density regime (σ < 0.1/nm2 and σN1/2 < 1), where autophobic dewetting is not expected. Nearly a decade ago, Frischknecht38 used fluids density functional theory to determine the free energy between two infinitely long, moderately grafted cylinders in a polymer melt and found an attractive minimum in the free energy when the N/P < 1. Moreover, this minimum was found to become deeper as the grafting density increases. Dissipative particle dynamics (DPD) simulations41 of sparsely to moderately grafted nanorods reveal a phase diagram which is divided into three regions: dispersed, partially aggregated, and aggregated. Both DPD41,42 and molecular dynamics43 simulations show an increased miscibility of nanorods as the graft length and/or graft density increase. Experiments25 with polystyrene-grafted gold nanorods in polystyrene melt show that the nanorods become less miscible as their aspect ratio increases, and N/P decreases. Complementary25 SCF theory calculations also show that the inter-rod free energy minimum deepens as the rod lengthens. It is observed through experiments25 as well as simulations41 that nanorods are found to preferentially aggregate in a side-by-side (rather than end-to-end) manner. However, a recent SCF theory study44 shows that by taking advantage of the tendency of nanoparticles to aggregate at low N/P and grafting the nanorods with short polymers on its ends

and long polymers on its sides, a preferential end-to-end arrangement of nanorods can be obtained. The calculations also reveal that grafted nanorods with larger diameters have a higher tendency to aggregate. Recently, there have been efforts to study grafted nanoparticles in a chemically dissimilar matrix polymer using theory and molecular dynamics simulations.43,45 In our previous work, we have used Polymer Reference Interaction Site Model (PRISM) theory to determine the phase behavior of bare nanorods in a polymer melt.18 In this article, we look at how grafting polymer (of the same chemistry as the matrix polymer) onto the nanoparticles changes its phase behavior. We explore low grafting densities and two different graft arrangements.

II. MODEL AND THEORY We model the grafted and matrix polymers as freely jointed chains of N and P = 100 segments, respectively. Each segment has a diameter d, and the distance between consecutive segments is l = 1.33d. The nanorods are treated as a collection of linearly and tangentially attached spherical sites of diameter D. The thickness of the nanorod is varied by this site diameter, and three different nanorod diameters are studied (D/d = 2, 4, and 10). This article presents a study of nanorods with aspect ratios 1 (nanosphere), 2, 4, 10, 25, 50, and 100. A graft is attached to each of these spherical sites, and two different kinds of grafted nanorods are the subject of this study. The trans arrangement of grafts is such that the grafts on two consecutive nanorod sites are attached on diametrically opposite sides, as is illustrated by Figure 1a. In the cis arrangement, all the grafted polymers are attached on the same side of the nanorods (Figure 1b). While we are not aware of the fabrication of such structures, there has been significant recent progress in the synthesis of DNA46,47 and polypeptide48 helix based bottlebrush structures. Given the regularity of a helix turn, and the existence of fine control over nucleotide and amino acid sequence, we think that it is possible to synthesize rodlike molecules with graft arrangements of nearly arbitrary specificity. At high grafting densities, the grafted polymer stretch into what is known as the “brush configuration”, while at low grafting densities, the grafted polymer retain their natural (solvent dependent) statistics and are in what is called the “mushroom configuration”. Sparse grafts immersed in a polymer melt maintain ideal random walk statistics.36,49 We B


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correlation function between the monomers of the matrix polymer is given by

model the grafts as freely jointed chains and work in the low grafting density regime such that distance between two adjacent grafts is greater than or equal to the radius of gyration of the graft (Rg,graft = ( N/6 )l). In the trans arrangement, the distance between adjacent grafts is 2D, while in the cis arrangement it is D. Keeping these restrictions in mind, Table 1 lists the graft lengths (N) studied for the nanorods of different thicknesses and different graft arrangements.

ωpp(k) = [1 − f 2 − 2P−1f + 2P−1f P + 1 ]/(1 − f )2

In eq 2, f = sin(kl)/kl. The Fourier space intramolecular correlation function between the sites of a nanorod is as follows:17,18,61 ωnn(k) =

Table 1. A List of All Combinations of Rod Thickness (D/d) and Graft Arrangements (Cis or Trans) and the Corresponding Graft Lengths (N) Studied D/d D/d D/d D/d D/d D/d

= = = = = =

2, cis 2, trans 4, cis 4, trans 10, cis 10, trans

N N N N N N

= = = = = =

0, 0, 0, 0, 0, 0,

8, 8, 8, 8, 8, 8,

1 ⎡⎢ Nn + 2 Nn ⎢⎣

Nn − 1

∑

(Nn − λ)

λ= 1

sin λkD ⎤⎥ λkD ⎥⎦

(3)

Here, Nn is the number of spherical sites on the nanorod. The nanorod−graft intramolecular correlation function is given by ωgn(k).

and 12 12, 25, and 50 25, and 50 25, 50, 100, 150, and 200 50, 100, 150, and 200 50, 100, 150, 200, and 500

ωgn(k) =

⎡ ⎛ sin k(D + d)/2 ⎞ 1 − fN ⎢Nn⎜ ⎟ Nn(N + 1)(1 − f ) ⎢⎣ ⎝ k(D + d)/2 ⎠

Nn − 1

+2

∑ m=1

The total volume fraction of the system is ηt = 0.4. The fractions of this volume that are occupied by nanoparticles, grafts, and matrix polymer are given by ϕn, ϕg, and ϕp, respectively. The phase behavior of the grafted nanorod composites are studied at low (ϕn = 0.01) and moderate (ϕn = 0.1) rod densities. Therefore, the fractional space occupied by the nanorods becomes ηn = 0.004 and ηn = 0.04, respectively. At rod density ϕn = 0.1, we examine the phase behavior of grafted nanorods only up to an aspect ratio (AR) of 50, so that all calculations are performed far from the Onsager liquid-crystal transition density. We use Polymer Reference Interaction Site Model (PRISM)51−55 integral equation theory to determine the structural correlations within the grafted-nanorod composite. The Reference Interaction Site Model (RISM) theory56 generalizes the Ornstein−Zernike equation for spherical particles to systems of aspherical particles described as a collection of spherical sites. PRISM theory57−59 simplifies the RISM matrix equation by treating all chemically similar sites within the same molecule (or the same species) as equivalent. In this study, this means treating all rod sites, all grafted polymer sites, and all matrix polymer sites as equivalent to each other. This reduces the problem to a 3 × 3 matrix equation, the three components being the nanorod (n) sites, the grafted polymer (g) sites, and the matrix polymer (p) sites. The PRISM equation, in Fourier space, is given as follows: ̲ k) ̲ k) = Ω̲(k)C( ̲ k)H( ̲ k)Ω̲(k) + Ω̲(k)C( H(

(2)

(Nn − m)

sin kxm ⎤ ⎥ kxm ⎥⎦

(4)

In eq 4, xm = (mD)2 + [(D + d)/2]2 . Finally, the graft− graft intramolecular distribution function is given by ωgg (k) = +

⎡ 2f N + 1 2f 1 ⎢1 − f 2 + − 2 N N (1 − f ) ⎢⎣

2 (1 − 2f N + f 2N ) NnN

Nn − 1

∑ m=1

(Nn − m)

sin kym ⎤ ⎥ kym ⎥⎦

(5)

where y m = mD for the cis arrangement, y m = (mD)2 + (D + d)2 for the trans arrangement when m is odd, and ym = mD for the trans arrangement when m is even. Derivations for eqs 4 and 5 are given in the Appendix. In an initial attempt to solve the PRISM equations for randomly arranged grafts, we have found them difficult to converge for long rods. Equation 1 is solved self-consistently along with molecular closures using the KINSOL algorithm.62,63 The Percus−Yevick (PY) closure is used for all correlations except the nanoparticle−nanoparticle correlations. It relates the direct correlation function to the pair distribution function through the interaction potential, Uij(r).

Cij(r ) = (1 − e βUij(r))(1 + hij(r ))

(6)

The hypernetted-chain (HNC) closure is used for nanoparticle−nanoparticle correlations.

(1)

The first term represents the direct correlation between two sites through all the sites within the respective species. The second term, which includes many-body effects, represents the indirect correlation through other species within the system. The components of matrix H(k) are Hij(k) = ρiρjhij(k). Here, ρi and ρj are the number densities of sites of type i and j and hij(k) is the Fourier transform of pair distribution function, which is the nonrandom part of the radial distribution function, hij(r) = gij(r) − 1. The elements of matrix Ω(k) are Ωij(k) = ρi ρj ωij(k), where ωij(k) are the Fourier transforms of the

Cnn(r ) = hnn(r ) − βUnn(r ) − ln(1 + hnn(r ))

(7)

The closures to integral equations represent somewhat uncontrolled approximations, and their validity is largely determined by comparison to simulations. If the commonly used Percus−Yevick approximation is employed for all correlations, then the solution for nanoparticle−nanoparticle radial distribution functions may have negative values just outside the hard core. Using the HNC closure overcomes such an unphysical result. The use of the PY closure for polymer− polymer and polymer−nanoparticle correlations and the HNC closure for nanoparticle−nanoparticle correlations is well established.17,64−66 Hooper et al.64 have shown that this combination of closures compares well against simulations of

intramolecular correlation functions. The components of C(k) are Fourier transforms of direct correlation functions. Within the PRISM equivalent site approximation and the freely jointed chain model for polymers,58,60 the intramolecular C


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Macromolecules polymer−nanosphere composites. Two of us18 have previously demonstrated the same closure scheme to compare qualitatively well against dissipative particle dynamics simulations19 of polymer nanorod composites. As in the previous study of polymer-bare nanorod composites,18 the polymer and nanoparticles interact through a short-range hard core and a longer range exponentially decaying attraction. Upn(r ) = ∞ ,

rising peaks (as can be seen in Figure 1c), we estimate the location of the phase boundary as the limit of convergence of the integral equations. If the convergence limit is approached before the “microphase peak” satisfies the Hansen−Verlet criterion, as is the case for N = 100 and N = 200 in Figure 1d, we do not classify the system as microphase separated and do not indicate it in the phase diagrams. A microphase length scale is determined by the relation L* = 2π/k*. At meltlike densities, polymer chains dimensions are characterized by the freely jointed chain model. However, with the introduction of nanoparticles, chains close to the nanoparticles may expand. The ensemble averaged intramolecular distribution function of polymer close to nanoparticles can be determined through course-grained simulations.50 It is a function of the polymer length, nanoparticle size, and the polymer−nanoparticle interaction. Using such corrections (obtained from simulations) to the intramolecular distribution functions employed in the PRISM equation (eq 1) is called the self-consistent PRISM approach. In this study we have approximated the polymer dimensions to be those determined by the freely jointed chain model, rather than determining the intramolecular distribution through simulations. This approximation restricts the chain dimensions according to the freely jointed chain model and does not allow it to expand or shrink when a polymer is close to a nanoparticle. Since the intramolecular distribution of polymer chains is thus constrained, we expect the intermolecular distribution functions (the radial distribution function) to be qualitatively, rather than quantitavely, correct. We may also expect that this will have some effect on the exact location of the phase boundaries. We, however, do not expect qualitative changes in the trends of the phase boundaries.

r ≤ (D + d)/2

Upn(r ) = −εpne−(r − (D + d)/2)/ αpn ,

r > (D + d)/2

(8)

Since the grafted and matrix polymers are of the same chemistry, eq 8 accounts for the interactions between the nanoparticles and both grafted and matrix polymers. In this equation, εpn = εgn is the strength of the attractive interactions between the polymer (grafted and ungrafted) and nanoparticles, and αpn = αgn = 0.133(D + d)/2 is the range of the interactions. The range of interactions is chosen to be the same as that used in the previous study of mixtures of polymer and ungrafted nanrorods.18 A negative value of εpn represents a repulsive interaction between the polymer and nanoparticles. Interactions between all other sites are of a purely hard core nature. The matrix of pair correlation functions which are obtained by self-consistently solving eqs 1, 6, and 7 can be used to determine the partial static structure factors through the following matrix equation:

̲ k) S̲ ′(k) = Ω̲(k) + H(

(9)

The density-multiplied static structure factors make up the components of S′(k) through the relationship Sij′ (k) = ρi ρj Sij(k). The nanoparticle−nanoparticle static structure factor is used to determine the dominant density fluctuations. The location of the maximum of the structure factor is given by k*. A microphase separation is characterized by a significant and rapidly rising peak at a small, but nonzero wavevector (k* ≠ 0). We call the system microphase separated only if this “microphase peak” also satisfies the Hansen−Verlet67 freezing criterion (Snn(k≠0) ≥ 2.85). Macrophase separation is indicated by a sharply rising peak at k* = 0. While in meanfield theories a spinodal phase separation is signified by a divergence in the static structure factor, PRISM theory preserves fluctuations effects, and therefore such divergences are not expected. Instead, the solution of the PRISM and closure equations yields a rapidly rising peak in the structure factor with decreasing εpn. Figure 1c illustrates an example of such rapidly rising peak values of the nanoparticle−nanoparticle structure factors as a function of the polymer−nanoparticle interaction strength for a system of nanorods (with AR = 4 and D/d = 2). The largest peak appears at k* = 0 for bare nanorods, indicating macrophase separation. It appears at k* ≠ 0 for tethered nanorods with graft length N = 8, indicating microphase separation. For even lower values of εpn, the equations do not reach a convergent solution. As illustrated by Figure 1c, the peak values of the nanoparticle−nanoparticle static structure factor rise rapidly as the convergence limit is approached. Some PRISM studies extrapolate the inverse peak value of the static structure factor to zero in order to locate the phase boundary.68,55 In other studies,17,18,69 the peak of the structure factor rises so quickly that an extrapolated estimation of the phase boundary is no different from the convergence limit (in terms of εpn). Since our calculations reveal such rapidly

III. STRUCTURAL CORRELATIONS To understand the phase behavior of the grafted nanorods, we first discuss their radial distribution functions. Figure 2 plots the

Figure 2. (a) Nanoparticle−nanoparticle radial distribution function for rods of AR = 10 and D/d = 4 at ϕn = 0.01, with grafts of length N = 25, with polymer nanoparticle interaction strengths of εpn/kBT = 1 (solid red curve), εpn/kBT = 0 (dashed green curve), and εpn/kBT = −0.033 (dotted blue curve). The inset presents the graft−graft radial distribution function under the same conditions. (b) Nanoparticle− graft radial distribution function of the same three systems as in (a). D


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Macromolecules nanoparticle−nanoparticle, graft−graft, and nanoparticle−graft radial distribution functions for a system of grafted nanorods with AR = 10 in a polymer melt. Under the equivalent-site approximation, Figure 2a represents the radial distribution functions between and average nanoparticle site on one rod and the average nanoparticle site on a different rod. It is not the distribution function between the centers of mass of the rods. Similarly, the inset of Figure 2a plots the radial distribution functions between the average monomer on a graft of a rod and the average monomer on the graft of a different rod, and Figure 2b represents the radial distribution functions between the average nanoparticle site on one rod and the average monomer in the graft of a different rod. Nanoparticle−nanoparticle radial distribution functions (Figure 2a) indicate a high degree of contact clustering even when the polymer and nanorod are noninteracting (εpn = 0). The clustering increases for a slightly repulsive interaction between the polymer and nanoparticle (εpn/kBT = −0.033) and can be dramatically reduced for modestly attractive interactions (εpn/kBT = 1). The contact value of the radial distribution functions between graft monomers (see inset of Figure 2a) is low due to two reasons: first, the self-shielding effect of graft polymers and, second, the steric hindrance posed by the attached nanorod. However, at longer length scales, a broad peak is seen for εpn = 0 and εpn/ kBT = −0.033. This broad peak is indicative of clustering between grafts as the nanorods are driven together due to depletion attraction. Figure 2b shows that the contact value of the nanoparticle− graft radial distribution function decreases as the polymer− nanoparticle interaction changes from attractive to repulsive. This is because the graft and matrix polymer are chemically identical (εpn = εgn), and as the polymer−nanoparticle interaction changes from attractive to repulsive, so does the interaction between the graft and the nanoparticle. However, as the nanorods cluster due to entropic depletion, the grafted polymer are also brought close to the nanoparticles. This behavior is reflected by the broad and shallow peak between r/d = 5 and r/d = 11 (at εpn/kBT = −0.033), in spite of the contact repulsion. Taken together, the radial distribution functions illustrate clustering between nanorods and the attached grafts at low (and negative) attraction strengths.

Figure 3. Spinodal phase boundaries of D/d = 2 rods as a function of the polymer−nanoparticle attraction strength, aspect ratio, and graft length. The fluid phase lies above the curve while the microphase separated (hollow symbols) or macrophase separated (filled symbols) region lies below the curves. (a) trans-grafted rods at ϕn = 0.01. (b) trans-grafted rods at ϕn = 0.1. (c) cis-grafted rods at ϕn = 0.01. (d) cisgrafted rods at ϕn = 0.1.


IV. DEPLETION PHASE BOUNDARIES A. Bare Nanorods. The depletion phase boundaries are determined by estimating a divergence in the structure factor (as illustrated by Figure 1c). These phase boundaries are presented in Figures 3, 4, and 5 at two nanorod densities (ϕn = 0.01 and 0.1) for both the cis and trans configurations of grafts. The red curves indicate the phase boundaries of bare (ungrafted) rods in polymer melt. They show that the location of the depletion phase boundary is an increasing function of rod aspect ratio19 and becomes nearly constant at high aspect ratios.18 While the phase boundary rises as a function of nanosphere diameter, it is a weakly nonmonotonic function of nanorod diameter.18 The depletion boundary of nanospheres with D/d = 2 lies below εpn = 0 (Figure 3), meaning that for such small nanospheres a repulsion is required between them and the matrix polymer to induce a phase separation. The phase diagrams also show that bare nanospheres are less miscible in the polymer matrix as their density increases (from ϕn = 0.01 to ϕn = 0.1). However, bare nanorods become slightly more miscible as the rod density increases; i.e., their depletion phase boundary falls. This results in a flattening of

the phase boundary at higher density. The decrease in the miscibility of bare nanospheres with increasing density becomes less prominent with increasing D/d. However, the increase in bare nanorod miscibility becomes more prominent with increasing rod thickness. B. Grafted Nanorods. Figures 3, 4, and 5 show that even with short oligomeric grafts (N = 8, N/P = 0.08), the phase boundary falls compared to that of bare nanoparticle composites. The spinodal boundary in thin rod (D/d = 2) composites is pushed below εpn = 0, even for short graft lengths of N = 8. This means that a repulsion would be required between polymer and nanoparticle to induce a phase separation. For rods of intermediate thickness (D/d = 4) and graft lengths of N = 50 and longer, no attraction is required to form a dispersed composite. For thick rods (D/d = 10), the E


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transition from a fluid to a microphase separated state as the polymer−nanoparticle attractions decrease while the filled symbols indicate a fluid to macrophase separation transition. The figures demonstrate that there is a greater tendency to form a microphase separated state for short, thin rods with longer grafts, at higher rod densities. Microphase separation may be the result of two competing phenomena: the depletion attraction between nanorods that causes the nanorods to aggregate and steric shielding provided by the grafts which prevents aggregation. Entropic depletion is weaker for smaller nanoparticles,70−72 and longer grafts provide more steric stabilization. Hence we observe microphase separation for short and thin rods with longer grafts and macrophase separation for longer and thicker rods with shorter grafts. As can be seen in Figures 3b, 4b, and 5b, the crossover from microphase to macrophase separation occurs at shorter graft lengths for thinner and shorter rods. Since cis-grafted nanorods have less steric stabilization, their phase boundaries are higher (i.e., less miscible) than those of trans-grafted rods. Recent dissipative particle dynamics41 and molecular dynamics43 simulations for thin nanorods with aspect ratios of 8.8 and 11, respectively, report an increase in miscibility with increasing graft lengths. Figure 3b presents the phase diagram for trans-grafted thin nanorods at comparable densities. It shows that the miscibility of rods with AR = 10 increases with increasing graft length and qualitatively confirms the results of these simulations. In experiments of polystyrene-grafted gold nanorods in polystyrene melt performed by Wang et al.25 for rod aspect ratios of 2.5−6.3 and D/d ∼ 8.3, they find that the nanorods become less soluble with increasing aspect ratio and decreasing N/P. This general trend is also predicted by the calculations presented in Figure 5 for D/d = 10. As discussed earlier, the crossover from microphase to macrophase separation appears at shorter graft lengths for thinner rods (and smaller spheres). To examine this behavior, Figure 6 presents the spinodal boundaries as a function of the ratio of the nanoparticle diameter to the grafted polymer radius


depletion boundaries lie above εpn = 0, at all the graft lengths explored in this study. Thus, even though the location of the bare nanorod phase boundary is a weakly nonmonotonic function of nanorod thickness, the miscibility of grafted nanorods decreases greatly with increasing rod diameter.18 The fall in the miscibility of grafted nanorods with increasing rod thickness has also been reported by self-consistent field theory calculations.44 The low miscibility of the D/d = 10 nanorods is likely due to a combination of large nanoparticle size and low grafting density (σ < 0.004/nm2). At short aspect ratios (particularly at D/d = 2 and 4), the phase boundary becomes a nonmonotonic function of the graft length. This behavior is discussed in more detail later in the article. The trans-grafted rods show better dispersion than cis-grafted rods, for all rod thicknesses, aspect ratios, densities, and graft lengths. At longer graft lengths, the spinodal boundary is no longer a monotonically increasing function of rod aspect ratio (particularly for thin nanorods). It can also be a nonmonotonically varying function (as for the trans graft arrangement with N = 50 and D/d = 4 at ϕn = 0.01 in Figure 4a) or a monotonically decreasing function of aspect ratio (as for the trans graft arrangement with N = 25 and D/d = 2 at ϕn = 0.1 in Figure 3b). As is the case for bare nanoparticles, grafted nanospheres become less miscible with increasing density, while grafted nanorods become more miscible with increasing density. The only exception we found to this is for short rods (AR = 2) of intermediate thickness (D/d = 4), with short grafts (N = 8). Broadly speaking, as εpn is lowered, a large degree of nanoparticle clustering (as seen in Figure 2a) can signal a transition from a homogeneous fluid to two different kinds of aggregated states. The composite may macrophase separate between nanoparticle-rich and matrix polymer-rich regions. This is what is found for the bare nanorods in a polymer melt. Alternatively, it is also possible that the nanoparticles locally cluster and self-assemble into many smaller nanoparticle-rich domains within the polymer melt. The latter can be considered as a microphase separated state, which is signaled by a large peak at a short (nonzero) wavevector of the static structure factor. The open symbols in Figures 3, 4, and 5 signal the

Figure 6. Spinodal phase boundaries of grafted nanoparticles as a function of the polymer−nanoparticle attraction strength and the ratio of the nanoparticle diameter to the radius of gyration of the graft (D/ Rg,graft) at D/d = 2 (red circles), D/d = 4 (green squares), and D/d = 10 (blue triangles). The fluid phase lies above the curve while the microphase separated (hollow symbols) or macrophase separated (filled symbols) region lies below the curves. (a) For grafted (monotailed) nanospheres at ϕn = 0.1. (b) For trans-grafted nanorods of AR = 4 at ϕn = 0.1. F


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Macromolecules of gyration (D/Rg,graft) at ϕn = 0.1. While a short wavevector peak appears for larger nanospheres with longer grafts (at D/d = 10 and D/Rg,graft < 1.4), we have not characterized it as a microphase peak, since the solution to the integral equations fail to converge before the Hansen−Verlet criterion is satisfied. However, this short wavevector peak is the largest peak in the nanoparticle static structure facto, and rises quickly with decreasing εpn. Plotted in this manner, it can be seen that the crossover from microphase separation to macrophase separation occurs within a narrow range of D/Rg,graft for both nanospheres (∼1.4−1.8) and nanorods of AR = 4 (∼1.1−1.5). It therefore seems as though short nanorods prefer to microphase separate when their diameter is smaller than the radius of gyration of the graft and macrophase separate when their diameters are significantly larger than Rg,graft. The crossover shifts to longer graft lengths as the aspect ratio increases (see also Figures 3b and 4b). As mentioned earlier, in some cases the spinodal boundary of nanospheres and short nanorods is a nonmonotonic function of the graft length. Figure 6a shows that the phase boundary of nanospheres with D/d = 2 and 4 is a nonmonotonic function of the graft length, and the minimum of the phase boundary roughly coincides with the crossover from microphase separation to macrophase separation. It is clear that the nonmonotonic character of the phase boundary diminishes with increasing nanoparticle size. It is likely due to this reason that the behavior has not been reported in the experimental literature surveyed by us. Most experiments observe an increase in miscibility with increasing graft length for attractive nanospheres with multiple grafts.20,23 A possible explanation is that multiple grafts provide better steric shielding against the clustering of attractive nanoparticles and hence against microphase separation. A suppression of the microphase peak with increasing number of grafts has also been observed in previous PRISM studies of grafted nanospheres with sticky interactions.66,73 The nonmonotonic dependence of the phase boundary on graft length (particularly for small nanospheres) may be a result of two competing phenomena: (a) increase in miscibility with increase in N due to steric shielding from depletion attraction and (b) an increase in the fraction of the shapeamphiphilic55,74−76 species with increasing N (particularly for small nanospheres). As the length of the graft increases, the fraction of grafted-nanoparticle species in the system increases as ϕn + ϕg = ϕn(1 + N(D/d)−3). For instance, when ϕn = 0.1, D/d = 2, and N = 50, the fraction of grafted-nanoparticle species is ϕn + ϕg = 0.725. A recent study has shown that shapeamphiphilic species show an increased tendency for microphase separation with decreasing D/Rg,graft even when the system is completely noninteracting.55 Thus, small nanoparticles with long grafts may be reproducing the phase behavior of such shape-amphiphiles and showing an increasing tendency to microphase separate. For short grafted nanorods (AR = 4), Figure 6b shows that the phase boundary is slightly nonmonotonic at D/d = 2, but not for thicker rods. Figure 7 illustrates the dominant microphase length scale (L* = 2π/k*) and shows that L* is a nonlinearly increasing function of the rod length. L* also tends to be longer for the cis arrangement than the trans arrangement. This indicates that although both cis and trans-grafted rods show a crossover from microphase to macrophase behavior at roughly similar rod lengths and graft lengths, the microphase morphologies may be quite different. Figure 7c (D/d = 4) shows that the microphase

Figure 7. Microphase length scale as a function of rod aspect ratio. The filled symbols represent ϕn = 0.1, and the hollow symbols represent ϕn = 0.01. (a) trans-grafted nanorods with D/d = 2. (b) cisgrafted nanorods with D/d = 2. (c) trans-grafted nanorods with D/d = 4. (d) cis-grafted nanorods with D/d = 2.

length scale is a decreasing function of graft length, while Figure 7a (D/d = 2) shows that L* first decreases and then increases slightly. Both Figures 7a and 7c show that L* at ϕn = 0.01 is much larger than at higher nanoparticle density of ϕn = 0.1, indicating larger nanoparticle-rich domains at lower densities.

V. CONCLUSIONS We have used PRISM integral equation theory to study the phase behavior of sparsely grafted nanorods in polymer melt. The arrangement (cis or trans) and length of the grafts as well as the aspect ratio, thickness, and density of the nanorods are varied, while the matrix polymer length is kept constant. Two competing phenomena dictate the phase behavior of these systems: (a) a polymer matrix induced entropic depletion attraction between the nanorods and (b) steric stabilization provided by grafted polymer. An additional smaller effect may be that of shape-anisotropy-induced self-assembly of polymergrafted monotailed nanospheres, at higher volume fraction of the grafted-nanoparticle species. These three physical phenomena act in concert, resulting in a rich phase diagram of dispersed, macrophase separated, and microphase separated systems. At low polymer−nanoparticle attractions, the grafted nanorods either microphase separate or macrophase separate from the matrix. (Sometimes, this phase boundary occurs at negative attraction strengths, i.e., when the polymer and nanoperticle share a repulsive interaction beyond the hard core.) Table 2 summarizes the effect of geometrical features on whether the systems micophase separate or macrophase separate. Shorter rods with smaller diameters and longer grafts have lower matrix-induced depletion attraction and better steric shielding. They, therefore display microphase separation. Table 2. A Summary of the Influence of the Geometric Features of Grafted Nanorods on the Preference between Microphase and Macrophase Separation

G

microphase separation

macrophase separation

shorter rods thinner rods longer grafts

longer rods thicker rods shorter grafts DOI: 10.1021/acs.macromol.7b01754 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

Here, NnN is the total number of grafted monomer sites and ωNfjc(k) is the intramolecular distribution function of a freely jointed chain of N monomers as given in eq 2. The graft−graft intramolecular distribution function is a sum of intra-graft and inter-graft terms and can be written as follows:

Longer and thicker rods with shorter grafts experience a higher depletion attraction and less steric stabilization and therefore tend to macrophase separate. Because of higher steric shielding, trans-grafted nanorods are universally more miscible that cisgrafted nanorods. The miscibility of the grafted nanorod usually increases with graft length. However, small nanospheres (or short and thin rods) with longer grafts and higher density defy this trend. In these composites, the tendency to microphase separate increases (and hence the tendency to form stable dispersion decreases) as the graft length increases. We speculate that this might be due to larger fractional volume being occupied by a shape-anisotropic species, which have an increasing tendency to self-assemble into alternating polymer-rich and nanoparticlerich domains as the graft length increases. The results presented in this article, in a qualitative sense, compare well against SCF theory,44 simulations,41,43 and experiments25 performed in different regions of the multidimensional phase diagram. At low grafting densities, while some studies report microphase separated structures,21 others report macrophase separation between the matrix polymer and the grafted nanoparticles.22 In this article, we report that indeed both microphase separation and macrophase separation are possible in grafted nanosphere composites. In qualitative agreement with experiments, meanfield theory, and Monte Carlo simulations,20,77 we find that macrophase separation occurs for shorter graft lengths and larger particles, while microphase separation is predicted for smaller particles and longer grafts. While this work predicts the microphase separation of nanorods within a polymer melt under the conditions discussed above, experiments and simulations are required to determine the morphologies of the self-assembled states. Recently, there have been active and systematic attempts to explore the effect of grafting density and the relative lengths of matrix and graft polymers on the dispersion of nanorods.26,41−43,78 In this article, we show that the combined influence of length and arrangement of grafts as well as the geometry of the nanorods results in a rather rich and complex phase diagram.

N ⎡ 1 ⎢ NnNωfjcN + 2 ∑ NnN ⎢⎣ α=1

ωgg (k) = +2

2N

NnN

∑

∑

NnN − N

ωij(k) = ωi , i + 1(k)ωi + 1, i + 2(k)...ωj − 1, j(k)

NnN

∑ α=N+1

=

2N

3N

∑

∑

N

ωαβ (k) =

α = N + 1 β = 2N + 1

∑

N

⎛

⎞α − 1 sin(kD)

∑ ∑ ⎜ sin(kl) ⎟ α=1 β=1

⎝

kl

⎠

kD

2 ⎛ sin(kl) ⎞ β − 1 ⎛ 1 − f N ⎞ sin(kD) ⎟ ⎟ ×⎜ =⎜ ⎝ kl ⎠ ⎝ 1 − f ⎠ kD

(A4)

In eq A4, f = sin(kl)/(kl). Adding such inter-graft terms, we have the following relationship: 2N N 1 ⎡⎢ 2(1 + f − 2f × 2 NnN ⎢⎣ (1 − f ) Nn − 1

∑

(Nn − m)

m=1

(A1)

ωgg (k) = NnN

NnN

∑ ωαβ + ∑ ∑

ωαβ +

α=1 β=N+1

N

∑ ∑ ωαβ+ α=N+1 β=1

+

⎤ ωαβ ⎥ ⎥⎦ β=N+1 NnN

∑

⎛ 2N ⎜ ∑ ω + αβ ⎜ α=N+1 ⎝ β=N+1 NnN

+

(A3)

where ωi,i+1(k) = sin(kLi,i+1)/(kLi,i+1) and Li,i+1 is the scalar distance between the ith and (i + 1)th monomer. The intergraft term between two neighboring grafts in the cis arrangement can, therefore, be written as follows:

∑ ∑ ωαβ(k)

⎡ N 1 ⎢ NωfjcN (k) + 2 ∑ NnN ⎢⎣ α=1

if i < j

sin(mkD) ⎤ ⎥ mkD ⎥⎦

(A5)

Adding the intra-graft and inter-graft terms, and generalizing for the cis and trans arrangements, the following equation holds:

α=1 β=1

β=1

(A2)

In eq A2, the first term represents a sum of intragraft terms, while the remaining represent terms are inter-graft terms. In the freely jointed chain model, the distance between neighboring monomers is of fixed length, but each bond vector is independent of all other bond vectors. Therefore

NnN NnN

N

⎤ ωαβ ⎥ ⎥⎦ β = N (Nn − 1) + 1

∑

α = N (Nn − 2) + 1

APPENDIX Within the site-equivalent approximation of PRISM theory, the graft−graft intramolecular distribution function is written as follows:

N

ωαβ + ... NnN

∑

+2

■

⎡ N 1 ⎢ = ∑ NnN ⎢⎣ α = 1

ωαβ

β=N+1

α = N + 1 β = 2N + 1

ωinter‐graft(k) =

1 ωgg (k) = NnN

NnN

∑

⎡ 2f 2f N + 1 1 ⎢1 − f 2 − + N N (1 − f )2 ⎢⎣

2(1 − 2f N + f 2N ) NnN

Nn − 1

∑

(Nn − m)

m=1

sin kym ⎤ ⎥ kym ⎥⎦

(A6)

In eq A6, ym = mD for the cis arrangement, y m = (mD)2 + (D + d)2 for the trans arrangement when m is odd, and ym = mD for the trans arrangement when m is even. Similarly, the graft-nanorod intramolecular distribution function, within the equivalent-site approximation, is written as follows:

NnN

∑

ωαβ

β=N+1

⎞⎤ ωαβ ⎟⎟⎥ ⎥ β = 2N + 1 ⎠⎦ NnN

∑

ωgn(k) = H

1 NnN + Nn

Nn NnN

∑ ∑ ωαβ(k) α=1 β=1

(A7) DOI: 10.1021/acs.macromol.7b01754 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules =

⎡ Nn ⎛ N 1 ⎢ ∑ ⎜ ∑ ω (k ) + αβ Nn(N + 1) ⎢⎣ α = 1 ⎜⎝ β = 1

=

Nn ⎡ sin[k(D + d)/2] α − 1 1 ⎢Nn ∑ f Nn(N + 1) ⎢⎣ α = 1 k(D + d)/2 Nn − 1

+2

t

∑ ∑ t=1 m=1

=

sin(kxm) kxm

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⎤

∑ f α−1 ⎥ α=1

⎥⎦

1 − f N ⎡ sin[k(D + d)/2] 1 ⎢Nn Nn(N + 1) 1 − f ⎢⎣ k(D + d)/2 Nn − 1

+2

∑

(Nn − m)

m=1

In eq A7, xm =

■

N

⎞⎤ ( k ) ω ∑ αβ ⎟⎟⎥⎥ ⎠⎦ β=N+1 NnN

sin(kxm) ⎤ ⎥ kxm ⎥⎦

(mD)2 + [(D + d)/2]2 .

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (M.T.). ORCID

Mukta Tripathy: 0000-0001-6954-9876 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the Department of Science and Technology, Government of India, through project SB/S3/CE/ 072/2013 and IIT Bombay Seed Grant through project 12IRCCSG039.

■

REFERENCES

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