Research ArticlePHYSICAL SCIENCES

Molecular structure of bottlebrush polymers in melts

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Science Advances  11 Nov 2016:
Vol. 2, no. 11, e1601478
DOI: 10.1126/sciadv.1601478

Abstract

Bottlebrushes are fascinating macromolecules that display an intriguing combination of molecular and particulate features having vital implications in both living and synthetic systems, such as cartilage and ultrasoft elastomers. However, the progress in practical applications is impeded by the lack of knowledge about the hierarchic organization of both individual bottlebrushes and their assemblies. We delineate fundamental correlations between molecular architecture, mesoscopic conformation, and macroscopic properties of polymer melts. Numerical simulations corroborate theoretical predictions for the effect of grafting density and side-chain length on the dimensions and rigidity of bottlebrushes, which effectively behave as a melt of flexible filaments. These findings provide quantitative guidelines for the design of novel materials that allow architectural tuning of their properties in a broad range without changing chemical composition.

Keywords
  • polymer physics
  • molecular dynamics simulations
  • molecular bottlebrushes
  • polymer melts

INTRODUCTION

Significant progress in polymerization techniques allows synthesis of hyperbranched molecules with precisely controlled architectures (19). Dense branching results in distinct shape of individual molecules and reduces overlap of neighboring molecules in dense systems (concentrated solutions and melts). These unique features inspire the design of new materials with physical properties that are different from properties of conventional linear polymers. Branched macromolecules were explored as molecular pressure sensors (10), pH-sensitive probes (11), supersoft elastomers (12, 13), and drug delivery agents (1416). They have also been used as components for the construction of mesoscopic systems (6) and controlling conformations of polymer chains (17).

One of the most distinct examples of highly branched macromolecules are molecular bottlebrushes composed of many polymer side chains densely grafted to a linear chain (backbone) (Fig. 1). The high grafting density results in strong steric repulsion between the side chains, causing extension of the backbone (1821) and, in some cases, even scission of its covalent bonds (21, 22). Because of this steric repulsion, bottlebrushes adapt a wormlike conformation controlled by side-chain length and grafting density (23). In bulk melts, this conformation promotes reduction of entanglement density of the wormlike molecules (24), resulting in unusual rheological properties (25, 26) with an ultralow plateau modulus of 102 to 103 Pa (13, 25, 27), which is much lower than the 105 to 106 Pa typically observed in melts of linear polymers. Note that these fundamental changes in physical properties are achieved only through architectural control without changing the chemical composition. Varying length and grafting density of side chains allows for systematic control of conformation of individual molecules as well as overlap and entanglements with neighboring molecules in dense systems.

Fig. 1 Molecular architecture and conformation of a bottlebrush polymer.

(A) Architecture of a bottlebrush molecule consisting of a backbone with Nbb monomers (red beads) and z side chains (blue beads) per backbone monomer. Each side chain is made of Nsc monomers. The total number of monomers of bottlebrush macromolecule is N = Nbb(1 + zNsc). All beads in the simulation are considered to be identical and interact via bonded and nonbonded potential (see Materials and Methods for details). Here, Nbb = 20, Nsc = 4, and z = 2. (B) The bottlebrush molecule in a melt state can be represented as a chain of effective persistence segments of length Embedded Image and thickness Rsc. R denotes end-to-end distance of bottlebrush backbone. Here, Nbb = 150, Nsc = 10, and z = 2.

Given their unique physical properties, molecular bottlebrushes have been an active field for many theoretical (18, 2832), experimental (13, 27, 3340), and numerical investigations (20, 22, 34, 36, 4151). Most of these studies focused on basic structural properties of bottlebrushes in solutions and in the adsorbed state. Particular attention was paid to the bending rigidity of bottlebrush macromolecules, which is characterized by the persistence length Embedded Image and remains a matter of debate in the scientific literature. The major difficulty is the interplay between many length scales in the bottlebrush structure and their impact on Embedded Image. Several theoretical approaches have been proposed to address this problem using scaling analysis (28, 29, 31, 52) and the self-consistent field method (49). For bottlebrushes in dilute solutions, under good solvent conditions, the persistence length was predicted to scale as Embedded Image, with α as low as Embedded Image (28) or as high as 1.11 (49) and Embedded Image (29). The exponent α for bottlebrushes in a θ solvent was predicted to be Embedded Image (28) or 1.01 (49). Significantly less attention (53) has been paid to solvent-free systems. Here, we address the problem of architecture-induced increase of bottlebrush persistence length as the key feature underlying physical properties of bottlebrush melts and elastomers.

In this work, we present the results of systematic coarse-grained molecular dynamics simulations and scaling analysis of the equilibrium structure of bottlebrush polymer melts for a range of degrees of polymerization of the backbone Nbb, side chains Nsc, and backbone spacer between the neighboring side chains. The latter is inversely proportional to the grafting density z, which is the number of side chains per backbone monomer. We show that the persistence length Embedded Image for z = 1 and z = 2 bottlebrushes is on the order of the size of side chains Embedded Image and scales as Embedded Image. This finding suggests that the entanglement plateau modulus of bottlebrush melts decreases as (27) Embedded Image, where Vperv is the volume of the effective bottlebrush Kuhn segment proportional to the pervaded volume of a side chain Embedded Image. The pervaded volume Vperv of a side chain is the volume of a sphere that encompasses this side chain. Our results also indicate that the backbones of bottlebrushes for z = 1 and z = 2 in a melt state obey Gaussian statistics with their size R (root mean square radius of gyration and end-to-end distance) scaling as Embedded Image for NbbNsc. Furthermore, our molecular modeling provided vital insights into the internal organization of bottlebrush melts, including limited interpenetration of side chains of neighboring molecules, radial distribution function of backbone monomers, and the form factor of individual bottlebrushes inside melt. We conclude that bottlebrush melts behave as melts of thick and flexible filaments, with a persistence length proportional to the size of the side chains.

RESULTS

Scaling theory of combs and bottlebrush molecules

Conformations of combs and bottlebrushes depend on the degree of polymerization of the side chains Nsc and their grafting density z. Although most of the paper concentrates on bottlebrushes with z ≥ 1, in the present section, we consider a broader set of parameters, including loosely grafted bottlebrushes (LBs) and loosely grafted combs (LCs) with z < 1. Depending on grafting density, we identify four conformational regimes of comb and bottlebrush melts (27), depicted in Fig. 2. At lower grafting density, we distinguish two comblike regimes characterized by Gaussian conformations of both backbone and side chains: (i) LCs with long backbone spacers between side chains z < 1/Nsc and with strongly interpenetrating neighboring molecules and (ii) densely grafted combs (DCs) for 1/Nsc < z < z* with weak interpenetration between molecules, where z* is defined in Eq. 1 below. There are also two regimes at higher grafting density: (iii) LBs with extended backbones and Gaussian side chains for intermediate grafting density z* < z < z** and (iv) densely grafted bottlebrushes (DBs) with extended backbones and side chains for high grafting density of side chains z > z**, where z** is defined in Eq. 3 below. The boundary between the comb and bottlebrush regimes can be found from the space-filling condition of zNsc side chains with physical volume vNsc, each within their pervaded volume (blNsc)3/2, resulting in reduced interpenetration of side chains from neighboring moleculesEmbedded Image(1)where b is the Kuhn length, l is the monomer length, and v is the monomer volume. The present paper focuses on the melts of DBs, whereas below we briefly review conformations of other types of molecules.

Fig. 2 Diagram of states of combs and bottlebrush molecules.

Molecular conformations are determined by the degree of polymerization Nsc of side chains (blue circles) and the number z of side chains per backbone monomer (red circles). Four conformational regimes are distinguished: LC like polymer with z < 1/Nsc, DC with 1/Nsc < z < z*, LB with z* < z < z**, and DB with z > z** (see Eqs. 1 and 3 for the definitions of z* and z**). The solid lines indicate crossovers between regimes [green, LC-DC boundary at z ≈ 1/Nsc; blue, DC-LB crossover line at Embedded Image; red, LB-DB boundary at z** (see Eq. 3)].

The low grafting density regime with z < z* (combs) includes two subregimes: LC and DC. LCs (LC part of Fig. 2), with spacers between side chains longer than the side chains (z < 1/Nsc) and a high volume fraction of backbones (> 50%), fully interpenetrate each other in melts. DCs (DC part of Fig. 2), with spacers shorter than the side chains 1/Nsc < z < z*, allow only partial interpenetration of the side chains because there is not enough space to accommodate side chains of neighboring molecules near the backbone of a host molecule. Both the side chains and backbones in melts of combs (LC and DC regimes) are in almost unperturbed Gaussian conformations.

Macromolecules with z > z* correspond to the so-called bottlebrush regime, which onsets because of a lack of space for side chains emanating from the unperturbed Gaussian backbone. Interpenetration of these side chains without their significant deformation is only possible upon extension of the backbone. We can estimate z* (Eq. 1) by considering a side chain with an unperturbed Gaussian size Embedded Image and with pervaded volume Embedded Image. This pervaded volume can only fit Embedded Image side chains, each with a physical volume VscvNsc. A section of the backbone of size Embedded Image passing through this pervaded volume contains Nsc monomers if it is in its unperturbed Gaussian conformation (assuming the same conformational statistics of backbone and side chains). Therefore, if grafting density is too high (z > z*), the Nscz side chains grafted to the undeformed section of the backbone with combined physical volume Embedded Image can no longer fit in the pervaded volume Vperv, forcing the backbone to extend.

The backbone extension on the length scale Embedded Image assures a fixed number of grafting points along the backbone section of this size Embedded Image equal to the number of overlapping side chains Embedded Image. On the small length scales, up to the size of the tension blob (54), the backbone remains unperturbed. The size of the tension blob ξ ≈ (blg)1/2 consisting of g monomers is estimated from the condition that gz side chains emanated from this section of the backbone densely fill its pervaded volume ξ3 [there are ξ3/(vg) ≈ gz such overlapping chain sections]. Therefore, the tension blob size is ξ ≈ (lb)2/(vz). There is no crowding issue on length scales r smaller than the tension blob (r < ξ), and bottlebrush backbones maintain the unperturbed Gaussian conformations with bare Kuhn length b. On the intermediate length scales (Embedded Image), a backbone can be visualized as an extended array of tension blobs with a constant average distance between grafting points v/(bl). On larger length scales (Embedded Image), backbones of bottlebrushes in a melt are represented as random walks of these extended arrays of tension blobs. The LB (LB part of Fig. 2) is described as a thick filament with contour length LNbbvz/(bl), thickness equal to the end-to-end distance of its side chains Embedded Image, and persistence length on the same order of magnitude (see detailed derivation in the subsection “Persistence length of a bottlebrush in a melt”). Thus, bottlebrush macromolecules are considered as chains of Embedded Image effective monomers of size Embedded Image. The mean square end-to-end distance of the backbone of an LBEmbedded Image(2)increases with increasing degree of polymerization Nsc and grafting density z of side chains. Considering bottlebrush as a dense “sausage-like” random walk, we can estimate its mean square size from its physical volume VchainvzNbbNsc as Embedded Image.

Side chains begin to extend at the crossover between LB and DB regimes (red line in Fig. 2 at zz**). The crossover value of the grafting density is given byEmbedded Image(3)

This crossover occurs either if the backbone spacer between neighboring grafting points begins to extend (for v > b2l) or if the backbone approached the fully extended state (for v < b2l). In the former case at zz** ≈ (bl)3/v2, the scale associated with the tension blob of the backbone ξ ≈ (lb)2/(vz) becomes comparable to the unperturbed spacer size (bl/z)1/2. At high grafting density (z** < z < l3/2/v1/2), the balance of side chain and backbone spacer stretching leads to the equilibrium size of extended side chains Embedded Image, with the corresponding average spacer length (v/z)1/3 and the contour length of the bottlebrush Lbbv1/3z2/3Nbb. The mean square size of the bottlebrush in this regime isEmbedded Image(4)

The backbone is almost fully stretched in the case of lower monomer volume v < b2l if z > z** ≈ (l2b)/v or, for higher grafting density, z > l3/2/v1/2 in the case of higher monomer volume v > b2l. In this case, the dense packing of side chains forces them to extend to the mean square sizeEmbedded Image(5)

The filament-like bottlebrush with both thickness and persistence length on the order of Embedded Image and bottlebrush contour length on the order of the contour of the backbone lNbb has mean square sizeEmbedded Image(6)

The dependence of backbone and side-chain size of combs and bottlebrushes on z is summarized in Fig. 3.

Fig. 3 Size of combs and bottlebrushes in different regimes.

With increasing grafting density, the dimensions of both backbone 〈R21/2 (red solid line) and side chain Embedded Image (blue dashed line) undergo characteristic variations in the comb (LC and DC) and bottlebrush (LB and DB) regimes. This figure corresponds to the case of lower monomer volume v < b2l. Abbreviations are the same as in Fig. 2. In addition, Embedded Image, Embedded Image, R3 ≡ (blNbb)1/2, Embedded Image, and R5 ≡ (blNsc)1/2.

The size of side chains of DBs with almost fully stretched backbones

The size of side chains increases with degree of polymerization Nsc (see fig. S1 and table S1). Their size also increases with grafting density z along the backbone. This effect is illustrated in Fig. 4A, which exhibits the variation of the ratio of the mean square distance Embedded Image of side-chain monomer s from the grafting point and the corresponding Gaussian size sσ2 as a function of the bond index s for different grafting densities z. Different colors and symbols correspond to bottlebrushes with different values of Nbb, NSC, and z, as shown in Fig. 4B and table S2. To understand the bond index s dependence of the mean square distance 〈Rsc(s)〉, we consider the average of the square of the size Rsc(s) = 〈Rsc(s)〉 + δRsc(s) of these side-chain segments containing s monomersEmbedded Image(7)

Fig. 4 Size of side chains of bottlebrushes in a melt.

(A) Dependence of the rescaled values of the mean square distance of a side-chain monomer s from the grafting point Embedded Image for side chains with Nsc = 10 and Nsc = 16 monomers as a function of the bond index s counting from the grafting point for molecules with different number z of grafted side chains per backbone monomer. The mean square fluctuations of the size of an s-segment Embedded Image are assumed to be equal to their value for linear 16-mer in a melt (z = 0, crosses). The dashed line is the fit to these z = 0 points by Embedded Image with two adjustable parameters Embedded Image and Embedded Image. Curves for z ≥ 1 show theoretical predictions of Eq. 10 with fitting parameter 〈Rsc〉. (Inset) Dependence of 〈Rsc2/(Nscσ2) on parameter z for 〈Rsc〉 obtained from the separate fit to Eq. 10 for each curve. Dashed line represents the theoretical prediction of Eq. 11 with scaling parameter Csc = 0.17. (B) Convention of symbols used in all figures to denote a particular bottlebrush melt. Color and shape of symbols denote the values of Nbb and Nsc, respectively. Crosses represent the data for linear chains (z = 0), solid symbols correspond to bottlebrushes with z = 1, open symbols are for bottlebrushes with z = 2, and plus symbols denote the data for bottlebrushes with z = 4 (see table S2 for more details). (C) Dependence of the rescaled values 〈Rsc(s)〉/(sz1/2σ) of the corresponding mean distance Embedded Image on the bond index s. Dashed lines are the theoretical predictions (see text for details).

We assume that the nontrivial s dependence of Embedded Image observed in Fig. 4A is due to chain extension 〈Rsc(s)〉, whereas the fluctuations Embedded Image of the size of these s-segments can be described by the mean square size of chain sections containing s monomers of a free linear 16-mer (z = 0, red crosses).

For monomers near the free ends of side chains, the mean distance 〈Rsc(s)〉 can be expanded in the Taylor series of the variable 1 − s/Nsc:Embedded Image(8)where 〈Rsc〉 = 〈Rsc(Nsc)〉 is the average size of a side chain. The first coefficient is a1 = 0 due to the boundary condition dRsc/ds = 0 at the free end s = Nsc. The condition 〈Rsc(s)〉 ≪ 〈Rsc〉 for small sNsc leads to the constraint for the sum of all coefficients ∑nan = −1. Note that the asymptotic expressions for the higher-order coefficients an in Eq. 8 can be found by expanding the dependence 〈Rsc(s)〉 ∝ (s/Nsc)1/2 = [1 − (1 − s/Nsc)]1/2 (see Eq. 14 below) in the power series of (1 − s/Nsc). Comparing this expansion with expansion in Eq. 8 term by term, we estimate a3 ≈ −1/16 and coefficients an decay with n as n−3/2. The small values of these coefficients justify omission of the higher-order terms in the expansion in Eq. 8. Thus, we take all an>2 = 0 and a2 = −1 and obtainEmbedded Image(9)

The parameter γ ~ 0.3 to 0.5 defines the lower boundary of the interval of validity of the above approximation. The mean square size of linear chain segments containing s monomers Embedded Image obtained from molecular dynamics simulations is presented by the lowest set of points denoted by × symbols in Fig. 4A. This dependence can be approximated by Embedded Image, where Embedded Image and Embedded Image, as shown by the dashed line (see Eq. 22 below for the similar approximation for bottlebrush backbones). In Fig. 4A, we compare our prediction from Eqs. 7 and 9Embedded Image(10)with the data obtained from simulations of bottlebrushes with grafting density z = 1, 2, and 4 using single fitting parameter 〈Rsc〉 and the value of Embedded Image and Embedded Image from the fit to linear chain data (z = 0). This simple estimate (Eq. 10) demonstrates excellent agreement with the simulation data.

The average side-chain size 〈Rsc〉 can be estimated from the monomer dense packing condition. The transverse slice of a bottlebrush can be approximated by a disc of volume dRsc2 and thickness d ≈ σ of the backbone bond projection onto the contour of the molecule. Assuming that there is no (or limited) overlap between the side chains of neighboring bottlebrushes, the disc volume is occupied by z side chains of volume vNsc each, where v ≈ σ3 is the volume of one monomer. Therefore, the square of the average size of side chains can be estimated asEmbedded Image(11)where Csc is the numerical coefficient accounting for the scaling form of this expression. The inset in Fig. 4A shows good agreement with Eq. 11, with the value of the fitting parameter Csc = 0.17.

Combining Eqs. 9 and 11, we can writeEmbedded Image(12)

In Fig. 4C, we test this prediction by plotting the s dependence of the ratio of the average distance Embedded Image and sσz1/2 using the simulation data presented in Fig. 4A. For larger values of s for γNscsNsc, this rescaled function is z-independent and exhibits linear dependence on s, Embedded Image, with Nsc-dependent negative slope predicted by Eq. 12. The red and black dashed lines in Fig. 4C (for s > 6) have slopes of −0.012 and −0.0064 for Nsc = 10 and Nsc = 16, respectively, which are consistent with the predicted negative slopes Embedded Image and −0.0063 from Eq. 12.

The mean square fluctuations of the size of chain segments containing large number s ≫ 1 of monomers are Gaussian. Therefore, the normalized mean square size of side-chain segments (Eq. 10) can be approximated for large s byEmbedded Image(13)

This equation predicts a maximum at smax = 2Nsc/3. This prediction is in good agreement with the simulations (see Fig. 4A). Note that the position of the maximum (for both points and lines) has a slightly higher value of s than 2Nsc/3 because of the residual s dependence of the ratio Embedded Image for short side chains. The physical explanation of this peak is that not all of the chains extend all the way to 〈Rsc〉. There is a wide distribution of the positions of side-chain ends around their average value 〈Rsc〉. Because fewer side chains extend to larger radial distances from the backbone, they provide an additional contribution to the ratio Embedded Image in Eq. 13 for γNsc < s < smax and a relatively smaller contribution for larger values of s > smax. As a result of side chains that do not extend to large radial distances from the backbone, the crowding of remaining side chains at these large radial distances decreases. This decrease in crowding weakens the stretching of the remote side-chain sections, resulting in a relatively smaller average extension of chain sections with s > 2Nsc/3. The stretching decreases with s and vanishes at the free side-chain ends in the overlapping zone of neighboring bottlebrushes.

Conformations of side-chain segments with s ≲ γNsc near the grafting point are determined by the monomer packing condition due to the limited penetration of monomers with index s′ > s into this zone near the backbone, similar to packing restrictions for the entire side chain (see Eq. 11)Embedded Image(14)

Therefore, the ratio 〈Rsc(s)〉/(sσz1/2) should be independent of Nsc for small s ≲ γNsc, as observed in Fig. 4C. However, note that the s dependence of 〈Rsc(s)〉 for s ≲ 6 differs from our prediction (Eq. 14) because of strong crowding of side-chains near the backbone and the non-Gaussian behavior of these short chain segments. The s dependence of Embedded Image significantly deviates from a constant for s ≲ 6 (see red crosses in Fig. 4A), as described by the crossover expression Embedded Image.

Persistence length of a bottlebrush in a melt

The rigidity of bottlebrush is only due to the mutual repulsion of the crowded side chains. The excluded volume interactions in a melt state are highly screened. In the unrealistic case of complete screening of steric interactions, the resulting persistence length of a bottlebrush is on the order of its monomer size ≈ σ. To estimate the persistence length of “real” bottlebrushes, we have to account for partially screened excluded volume interactions between side chains. The physical volume of spz side chains grafted to a persistent bottlebrush section is vspzNsc, whereas the radius of this section is Embedded Image. Therefore, the length of this cylindrical-like section is Embedded Image and its pervaded volume is Embedded Image. The pervaded volume of a persistent bottlebrush section is the volume of a sphere that encompasses this cylindrical-like persistent segment. The excluded volume interactions between polymer sections in the melt are reduced by the degree of polymerization Pw = spzNsc of these sections (54) see Fig. 5. Thus, the free energy of the excluded volume interactions between these persistent bottlebrush sections within their pervaded volume isEmbedded Image(15)

Fig. 5 Geometry of a bottlebrush polymer.

A bottlebrush is composed of z side chains with Nsc monomers each grafted to every backbone monomer (z = 2 in this figure). RscRsc(Nsc) and Rsc(s) denote instantaneous values of size of side chains (bottlebrush thickness) and distance of a side-chain monomer s from the grafting point, respectively. The number of monomers per persistence segment is sp, and persistence length is Embedded Image. PwspzNsc is the total degree of polymerization of cylindrical-like section composed of sp backbone monomers and spzNsc side-chain monomers. d is average projection of a backbone bond onto the direction of the backbone contour.

The persistent segment sp is determined by the condition that the excluded volume interaction energy Esc is on the order of thermal energy kBT, resulting inEmbedded Image(16)where Eq. 11 Embedded Image was used. In this case, the size of the persistence segment Embedded Image isEmbedded Image(17)

The conformations of bottlebrush backbones at small length scales are similar to those of flexible polyelectrolytes that are almost undeformed on scales up to electrostatic blob size but extended into a linear array of electrostatic blobs on larger length scales with persistence length determined by the screening length (55). By analogy with the polyelectrolytes, bottlebrushes are flexible on small length scales and have large persistence length, induced by side-chain repulsion, on intermediate length scales.

Our simulations confirm the scaling prediction that the persistence length of bottlebrush backbones in a melt state is comparable to the size of side chains. To determine the length of persistence segments sp, we calculated the decay rate of the correlations between bond orientations Embedded Image (see simulation data in Fig. 6A and fig. S2). The cosine of the angle is cos θ = (ri/|ri|) ⋅ (ri+s/|ri+s|), where ri = Ri+1Ri is the bond vector i between monomers i and i + 1, and ri+s = Ri+s+1Ri+s is the bond vector i + s between monomers i + s and i + s + 1. The data obtained from simulations (symbols) have been fitted to the function (dashed lines) of the following formEmbedded Image(18)where A, sf, sp, and ζ are fitting parameters (see table S3) and denote the following: (1 − A) and A are the magnitudes of short- and long-range correlations, respectively, between bond vectors of the backbone; sf is the characteristic number of beads in the backbone, which undergo “local” bond vector correlations; sp is the number of beads per persistence segment; and ζ is the magnitude of long-range interactions induced by the connectivity of backbone bonds (see the section on persistence length in the Supplementary Materials for details). The power-law decay ∝ s−3/2 at s ≫ 1 was reported for polymers melts (56) and for θ solutions (57). The origin of these interactions is explained either by effective compression of polymer coils due to the correlation hole effect (56) or by the shift of the monomeric Mayer f-function due to the finite interaction range and chain connectivity (57). The function tanh [(s/3)3] present in Eq. 18 describes a cutoff at minimal loop size s = 3.

Fig. 6 Persistence segment of a bottlebrush in a melt.

(A) Decay of backbone bond orientational correlations g(s) as a function of the number of monomers s between two bonds for bottlebrushes with various degrees of polymerization Nsc of side chains and number z of side chains grafted per backbone monomer (see Fig. 4B and the corresponding caption for the definition of symbols). Dashed lines represent best fits to the expression for g(s) given by Eq. 18. (B) Persistence segments obtained from the decay of bond orientational correlations plotted as a function of the side-chain polymerization degree Nsc for various backbones Nbb and grafting densities z of side chains, as indicated. The dashed lines represent the best power-law fit for data sets with Nbb = 100: Embedded Image for z = 1 and Embedded Image for z = 2.

In Fig. 6B, the number of monomers along the bottlebrush backbone in the persistent segment sp is presented as a function of the degree of polymerization of side chains Nsc for grafting densities z = 1 and 2 and various backbone degrees of polymerization Nbb. These data demonstrate the effect of backbone straightening with the increasing Nsc and z. The exponents of the observed power lawsEmbedded Image(19)Embedded Image(20)agree with the exponent Embedded Image predicted from the scaling arguments, as presented in the beginning of this subsection (see Eq. 16), and indicate that the number of monomers sp in the persistence segments of the bottlebrush backbones with z = 1 or 2 side chain per backbone monomer and Nbb ≳ 50 is proportional to the brush radius (side-chain size Embedded Image) (see fig. S3).

Size of a bottlebrush in a melt

The size of a bottlebrush in a melt state can be estimated from a simple physical picture of nonoverlapping flexible filaments with Embedded Image. Consider a bottlebrush backbone that is much longer than persistence segment Nbbsp (see Fig. 1B). The bottlebrush conformation can be represented as a chain of effective monomers consisting of sp backbone monomers and spz side chains of size Embedded Image. The mean square end-to-end distance of the backbone can be estimated as the number of these effective monomers Nbb/sp times the square of their size, which is proportional to the mean square size of side chains Embedded Image, resulting in Embedded Image. Because the number of monomers in a persistence segment sp is proportional to the size of a side chain Embedded Image (see Eqs. 17, 19, and 20), we predict that the mean square size of the backbone is also proportional to the size of a side chain Embedded Image. Our simulation results corroborate this prediction (cf. Eq. 6), as discussed below.

The mean square end-to-end distance of the backbone 〈R2〉 and the mean square radius of gyration Embedded Image of the whole bottlebrush are plotted as functions of the degree of polymerization of side chains Nsc and backbone Nbb in Fig. 7 and fig. S4 (for the definition of symbols, see Fig. 4B and the corresponding caption as well as table S2). The mean square size increases with Nbb and Nsc and obeys the power lawEmbedded Image(21)and Embedded Image for z = 1 and Embedded Image for z = 2. Similar to linear chains in melts, the conformations of bottlebrushes with long backbones are well described by the ideal chain statistics, that is, Embedded Image, as verified by the distributions of the end-to-end distances R and the radii of gyration Rg (cf. fig. S5). From Fig. 7A, one can observe that the mean square end-to-end distance of molecules with backbones Nbb = 10 and 20 becomes insensitive to further increase in Nsc. This is attributed to the crossover from the crew-cut bottlebrushes to the starlike configurations.

Fig. 7 Size of bottlebrushes in a melt.

Mean square end-to-end distance 〈R2〉 (A) and mean square radius of gyration Embedded Image (including side chains) (B) of bottlebrushes in a melt normalized by the ideal mean square size of backbones Nbbσ2 as functions of the degree of polymerization of side chains Nsc. (C) Mean square radius of gyration Embedded Image of bottlebrushes normalized by σ times the ideal root mean square size of side chains Embedded Image as a function of the degree of polymerization of backbones Nbb. See Fig. 4B and the corresponding caption for the definition of symbols. In (A) and (B), the number of side chains grafted per backbone monomer is z = 1 and 2. (C) displays data for z = 2. Dashed lines represent fitted scaling laws: (A) Embedded Image for z = 1 and Nbb = 100 and Embedded Image for z = 2 and Nbb = 100, (B) Embedded Image for z = 1 and Nbb = 100 and Embedded Image for z = 2 and Nbb = 100, and (C) Embedded Image for z = 2 and Nsc = 10. The error bars for all data points are smaller than the size of symbols.

The mean square internal distances 〈R2(s)〉 between backbone monomers are plotted in Fig. 8 as a function of the number of bonds s in a backbone section for molecules with various side-chain degrees of polymerization Nsc and different number z of side chains grafted per backbone monomer (the definition of symbols is displayed in Fig. 4B and described in the corresponding caption). The simulation data for all z values were fitted to the crossover expressionEmbedded Image(22)with fitting parameters Embedded Image and Embedded Image plotted in Fig. 8B and reported in table S4. The overall good agreement between the crossover expression and simulation data is observed by the overlap between points (simulation data) and lines (Eq. 22) in Fig. 8A. The backbone stretching (characterized by both parameters Embedded Image and Embedded Image) increases significantly with the grafting density of side chains z and with the degree of polymerization of side chains Nsc because of steric repulsion between densely grafted side chains. The Flory characteristic ratio Embedded Image and parameter Embedded Image increase proportionally to the power of the degree of polymerization of side chains Embedded Image and Embedded Image for z = 1 and Embedded Image and Embedded Image for z = 2 (see Fig. 8B and table S4). The scaling exponents are close to Embedded Image, indicating that s dependence of distances between backbone bonds separated by s monomers for Embedded Image is consistent with the backbone size scaling (Eq. 21). We observe that Embedded Image is 60% and 25% larger than corresponding Embedded Image values (see Fig. 8B and table S4) for z = 1 and 2, respectively, pointing out the wiggling of the backbone 〈R2(s)〉1/2s1/2 σ on length scales smaller than the tension blob. The bottlebrushes undergo a conformational transformation from a random coil to a rod as the grafting density z increases from 1 to 4, which is evidenced by the increase of both the Flory characteristic ratio Embedded Image and the crossover value Embedded Image by more than an order of magnitude. Because the backbone degree of polymerization Embedded Image, the simulated bottlebrush with side-chain grafting density z = 4 is effectively rodlike. Note that it is hardly possible to determine persistence segment sp from correlations of bond orientations (see Eq. 18) for simulated bottlebrushes with short backbones Nbb ≲ 50 and z = 4 because the corresponding g(s) functions do not decay sufficiently.

Fig. 8 Extension of a bottlebrush in a melt.

(A) Dependence of the mean square internal distances 〈R2(s)〉 between bottlebrush backbone monomers normalized by their ideal mean square size sσ2 on the number of monomers s in the backbone sections plotted for various side-chain polymerization degrees Nsc and grafting densities z (see Fig. 4B and its caption for the definition of symbols). Solid lines represent best fits to the crossover expression Embedded Image, with fitting parameters Embedded Image and Embedded Image listed in table S4. (B) Fitting parameters Embedded Image and Embedded Image plotted as a function of side-chain polymerization degree Nsc for z = 1 (full symbols) and z = 2 (open symbols). Dashed lines represent fitted scaling laws: Embedded Image and Embedded Image for z = 1, whereas Embedded Image and Embedded Image for z = 2.

Interpenetration of neighboring bottlebrushes in a melt

As discussed below, reduced interpenetration of side chains from neighboring macromolecules is a distinct feature that distinguishes segregated filaments (bottlebrushes) from overlapped molecules (linear chains and combs). This interpenetration is crucial for the understanding of the friction between these molecules and for the explanation of the complex rheological response of bottlebrush-based materials observed in recent experiments (12, 26, 27, 58, 59). We have analyzed the number of intramolecular Embedded Image and intermolecular gs contacts between side-chain monomers. Contacts were defined between pairs of monomers within distance rshell = 1.5σ from each other. This separation corresponds to the position of the minimum in the interbead correlation function g(r), as discussed below. We denote by 〈hs〉 the average number of intramolecular nonbonded contacts between sth monomer of a side chain and monomers within the same host molecule, whereas 〈gs〉 stands for the average number of intermolecular contacts between the ith monomer of a side chain of a given molecule with monomers belonging to all other molecules. Both quantities were calculated as functions of monomer index s along a side chain and normalized by the average number of nonbonded neighbors per sth monomer Zs = 〈hs〉 + 〈gs〉. The average value of Zs is independent of the monomer index s, Zs ≈ 〈Z〉 ≈ 4.7, except for terminal monomers Z1 ≈ 5.0 and Embedded Image. The results of this analysis are presented in Fig. 9A (for the definition of symbols, see Fig. 4B and the corresponding caption as well as table S2). The average fraction of intermolecular contacts for the first side-chain monomers (s = 1) is low and decreases with increasing grafting density z: 〈g1〉/〈Z1〉 ≈ 0.2 for z = 1, 〈g1〉/〈Z1〉 ≈ 0.1 for z = 2, and 〈g1〉/〈Z1〉 < 0.01 for z = 4. The limited interpenetration is ascribed to dense crowding of the side-chain monomers belonging to the same bottlebrush in the vicinity of its backbone, which hinders penetration of guest monomers to the central region of the host bottlebrush. The probability of encountering guest monomers 〈gs〉/〈Zs〉 increases with s and reaches the maximum value for terminal monomers of the side chains. This maximum value was found to be Embedded Image independent of grafting density z and degree of polymerization of side chains Nsc. The terminal monomers are in the interpenetration zone between two neighboring bottlebrushes. This zone contains 50:50 composition of monomers from both molecules. The low interpenetration of bottlebrushes in a melt state is demonstrated by the “territorial map” (6062) of the simulation box snapshot in Fig. 9B.

Fig. 9 Interpenetration of bottlebrushes in a melt.

(A) Average number of contacts 〈gs〉 between bottlebrush side-chain monomer s and monomers of other molecules. The value of 〈gs〉 is normalized by the average number of nonbonded neighbors 〈Zs〉 and plotted as a function of monomer index i (counting from the backbone) normalized by the degree of polymerization of side chains Nsc. Data for Nsc = 10 and Nsc = 16, with grafting density z = 1, 2, and 4 of side chains per backbone monomer (see Fig. 4B and the corresponding caption for the definition of symbols). The inset displays the average number of contacts 〈gs〉 between sth monomer of a linear chain and monomers of surrounding linear chains in a melt normalized by the average number of nonbonded neighbors. (B) The “map of territories” for an equilibrated melt of bottlebrushes with Nbb = 100, Nsc = 10, and z = 2 demonstrates reduced overlap between neighboring molecules.

For a reference system, we have calculated the number of contacts between monomers of neighboring linear chains in a melt (cf. the inset of Fig. 9A). The interchain contacts for linear polymers are almost uniformly distributed along the backbone except for the chain ends. The estimated fraction of guest monomers for linear chains was found to be 〈gs〉/〈Zs〉 ≈ 0.65 (independent of the degree of polymerization) and is higher than the maximum fraction of 0.5 (encountered by end monomers of bottlebrush side chains). The increase in the fraction of intermolecular contacts of up to ≈ 0.78 is observed at the ends of linear chains. The intrachain nonbonded contacts in linear chains are due to the formation of self-loops. For inner monomers of a linear chain, one can have self-loops at both sides, whereas for end monomers, self-loops can only be formed from one side. This explains the higher number of interchain contacts for end monomers of linear chains in a melt. The main conclusion of this analysis is that the overlap of bottlebrush molecules in the melt is qualitatively different on a monomer level from that of linear chains. Below, we show that melts of bottlebrushes become qualitatively similar to those of linear chains if we describe bottlebrushes as thick flexible filaments, which are chains of “effective monomers” of size Embedded Image.

Local structure

The internal structure of melts composed of linear chains or bottlebrushes is illustrated in Fig. 10 by the pair correlation functions g(r) of backbone monomers. The intrabackbone pair correlation function gintra(r) has several peaks, as shown in Fig. 10A, for both linear chains and bottlebrush melts with different side-chain grafting densities z. There are two peaks in melts of bottlebrushes with short side chains: one at r ≈ 0.96 σ that corresponds to bonded (nearest) neighbors and the peak at r ≈ 1.87 σ that corresponds to second-nearest neighbors. As the degree of polymerization of side chains Nsc increases, one observes that, for the grafting density of z = 2 side chains per backbone monomer (solid lines), new peaks appear at r ≈ 2.6 σ and 3.4 σ, which are due to stretching of the backbone. As the number of side chains attached to a backbone monomer increases to z = 4 (dashed line), the backbone stiffens and more peaks appear, indicating stronger intramolecular correlations. In this case, a series of clearly distinguishable subsequent and equidistant maxima is observed: 1.92σ, 2.79σ, 3.63σ, 4.48σ, 5.32σ, … The existence of local structure for bottlebrush melts is due to long-range interactions between backbone monomers induced by side chains. The strength of the intramolecular interaction increases with grafting density z and with length of side chains. The longer the side chains are, the stronger the interaction is between backbone monomers and the longer the range of these interactions is. The interaction range along the backbone is comparable to the size of the persistence segment, which scales with the size of side chains as Embedded Image.

Fig. 10 Radial distribution function of bottlebrushes in a melt.

Pair correlation functions between (A) intrabackbone monomers gintra(r) and (B) interbackbone monomers ginter(r) normalized by melt density ρ and density of backbone monomers in a melt ρbb, respectively. Correlation functions were plotted for various degrees of polymerization of side chains Nsc. As indicated in the legend, various colors are used to distinguish between the lines with different values of Nsc. Solid gray lines denote results for linear melts, that is, with grafting density z = 0 (Nsc = 0), whereas other solid lines correspond to z = 2. Dashed lines represent results for z = 4 and Nsc = 10.

The information about bottlebrush packing in the melt is contained in the interbackbone correlation function ginter(r) presented in Fig. 10B. In the case of linear chain melts, ginter(r) reveals the structure at very small length scales. ginter(r) is zero for distances r below first peak because of Lennard-Jones (LJ) core repulsion between monomers. The sharp peak at r ≈ 1.1 σ corresponds to the first “solvation shell,” which is the optimal distance between pairs of nearest-neighbor monomers. The second peak at r ≈ 1.9 σ displays the influence from the shell of the second-nearest neighbors. In contrast to melts of linear chains, the large-scale structure is observed in bottlebrush melts. Bottlebrush backbones are shielded by densely grafted side chains and do not approach each other. This fact is clearly demonstrated in the case of bottlebrushes with the long side chains, that is, Nsc = 10 (solid red line for z = 2 and dashed red line for z = 4). At distances comparable to bottlebrush thickness, equal to the average side-chain size Embedded Image for z = 2 and at Embedded Image for z = 4, the corresponding values of ginter(r) are low ginter(r)/ρbb ≈ 0.1 because the “coat” of side chains around a given backbone prevents neighboring backbones from approaching it. Note that the highest probability to find monomers of neighboring backbones does not exactly match the brush diameter Embedded Image but is rather at a shorter distance because of partial interpenetration of side chains. The first maxima are observed at rmax ≈ 7.25 σ for z = 2 and ≈ 8.35 σ for z = 4, respectively. The smaller value of Embedded Image for z = 2 (≈0.87) with respect to z = 4 (≈0.91) implies a stronger overlap of grafted side chains in bottlebrush melts with lower z (see Fig. 9). The inset of Fig. 10B shows the correlation functions ginter(r), with the abscissa rescaled by the position of the first maximum rmax. The positions of the peaks for bottlebrush systems correlate very well with the peaks observed for linear chain, indicating a similar “liquid-like” origin of melt structure (63, 64) but with stronger correlations at correspondingly larger distances. This result justifies representing bottlebrushes in melts by chains of “effective” monomers of size Embedded Image, which are thick flexible filamentous objects.

The form factor of a bottlebrush backbone in a melt

The form factor S(q) of the backbones of bottlebrushes with different grafting densities z of side chains is presented as the Holtzer plot in Fig. 11. The form factor of linear chains (z = 0) in a melt state (solid black line) is well represented by the Debye function (black dashed line) and scales as S(q) ∝ q−2 for large wave vectors q. The backbone form factors of bottlebrushes exhibit different behavior because of the backbone extension induced by side chains. For bottlebrushes with z = 2 (solid red line), the backbone S(q) at intermediate values of wave vector q < 0. 5 σ−1 is similar to the form factor of a semiflexible chain. The simulation data for z = 2 (solid red line) were fitted to the theoretical prediction of the structure factor of a semiflexible chain (blue dashed line) (65). Note that the form factor of a semiflexible chain (blue dashed line) interpolates between the Debye function (red dashed line) at large length scales (small q values) and rigid rod form factor (dotted black line) at small length scales (high q values). The overall very good agreement between simulation data and theoretical prediction (65) is observed. The only difference is the presence of a hump located between q ≈ 0.65 σ−1 and 1.96 σ−1. The nonmonotonicity of Holtzer plot qS(q) for high q values is attributed to the local flexibility of bottlebrushes at small length scales. Flexibility of bottlebrushes at length scales below tension blob size ξ is also confirmed by the existence of the fast initial decay of backbone bond correlations (see Fig. 6A and Eq. 18). The case of bottlebrushes with z = 4 (solid green line) cannot be fitted by the theoretical structure factor of a semiflexible chain. Larger crowding of side-chain monomers around the backbone gives rise to its increased extension for bottlebrushes with z = 4. The persistence length in this case is much larger than bottlebrush thickness, and structure factor is well approximated by the rigid rod–like scattering (dotted green line).

Fig. 11 The form factor of a bottlebrush in a melt.

The Holtzer representation of the backbone form factor of bottlebrush melts with different grafting densities of side chains per backbone monomer z = 2 (red solid line) and z = 4 (green solid line). The black solid line represents the form factor of a linear chain (z = 0). The blue dashed line depicts the theoretical prediction for the form factor of a semiflexible chain (65). The dotted lines (black and green) represent form factors of a rigid rod, whereas black and red dashed lines are Debye form factors of a flexible chain. The inset shows simulation data in the standard, S(q) versus q, representation. The scaling laws of the ideal chain ∝ q−2 for z = 0 and the rigid rod ∝ q−1 for z = 4 are denoted.

DISCUSSION

The conformations of bottlebrush macromolecules in melts depend on the side-chain grafting density z and the degree of polymerization of side chains Nsc. We demonstrated that the increase of z and Nsc leads to the extension of the bottlebrush backbone at intermediate length scales and enhancement of long-range interactions between backbone monomers, causing stronger intra- and intermolecular correlations similar to semidilute polyelectrolyte solutions. These interactions also result in significant reduction of the overlap between side chains of neighboring molecules. For moderate grafting densities with z ≤ 2, the conformation of bottlebrush is similar to a filament composed of “effective monomers” of size comparable with bottlebrush thickness (size of side chains). The bottlebrush size follows ideal (Gaussian) chain statistics for z ≤ 2, and the mean square size of a bottlebrush is proportional to the product of the backbone degree of polymerization Nbb and the persistence length of the molecule Embedded Image, that is, Embedded Image. For high grafting densities z > 3, bottlebrushes with short backbones Nbb ≲ 50 adopt rodlike conformations.

MATERIALS AND METHODS

Simulations of bottlebrush melts were performed using three-dimensional coarse-grained bead-spring model (63). An individual bottlebrush molecule is composed of Nbb backbone monomers (beads) connected by bonds and z side chains of Nsc monomers grafted to every backbone monomer (see Fig. 1A). Thus, the total number of beads in a bottlebrush is N = Nbb + zNbbNsc, where z is the grafting density. The case of Nsc = 0 (or z = 0) corresponds to a linear chain.

The nonbonded interactions between monomers separated by distance r were modeled by the truncated and shifted LJ potentialEmbedded Image(23)where the interaction strength ϵ is measured in units of thermal energy kBT, σ is the monomer diameter, and rc is the cutoff. In the NVT ensemble, we have used Embedded Image and rc = 21/6σ. This choice of LJ potential results in purely repulsive interactions between monomers. The bonded interactions in a molecule were described by the Kremer-Grest potential (63), VKG(r) = VFENE(r) + VLJ(r), with the “finitely extensible nonlinear elastic” (FENE) potentialEmbedded Image(24)where the bond stiffness Embedded Image and the maximum bond length r0 = 1.5 σ (63). All simulations were performed in a cubic box with periodic boundary conditions imposed in all spatial dimensions. In the NVT ensemble, the simulations were carried out at the overall monomer density ρ = 0.85 σ−3 corresponding to the intermolecular pressure 〈P〉 ≈ 4.75 ϵ/σ3. In a separate set of simulations, we have also investigated melts of bottlebrushes with attractive LJ potential with the interaction strength Embedded Image and the cutoff rc = 2.5 σ using NPT ensemble with P = 0, ensuring that the average density 〈ρ〉 ≈ 0.85 σ−3 is the same as in the NVT runs. The static properties obtained from both NVT and NPT simulations, for example, the average bottlebrush size, demonstrate good agreement with each other within the error bars after rescaling by the corresponding average bond length l (bonds in NPT simulations are 1% shorter than those in NVT simulations).

The molecular dynamics simulations were performed by solving the Langevin equation of motion for the position ri = [xi, yi, zi] of each bead (66)Embedded Image(25)which describes the motion of a set of interacting monomers. Forces Embedded Image and Embedded Image in Eq. 25 above are obtained from the LJ (Eq. 23) and FENE (Eq. 24) interaction potentials between the ith monomer and surrounding monomers. The third and fourth terms on the right-hand side of Eq. 25 are a slowly evolving viscous force Embedded Image and a rapidly fluctuating stochastic force Embedded Image, respectively. This random force Embedded Image is related to the friction coefficient ζ by the fluctuation-dissipation theorem Embedded Image. The friction coefficient used in simulations was ζ = 0.5 mτ−1, where m is the monomer mass and Embedded Image is the LJ time. The velocity Verlet scheme (67) was used for numerical integration of equations of motion in Eq. 25. The integration step was taken to be Δτ = 0.01τ. A Langevin thermostat was used to keep the temperature constant. All simulations were carried out using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) (68). Simulation snapshots were rendered using Visual Molecular Dynamics (VMD) (69). Initially, molecules were grown using a self-avoiding random walk technique and placed randomly in the simulation cell. The initial density of all systems was small (≈ 0.03 σ−3). Overlapping monomers in the initial configuration were pushed off using soft potential with slowly ramped interaction strength. To obtain the desired melt density (ρ = 0.85 σ−3), the simulation box was gradually decreased in size (see fig. S6) at constant velocity 10−3σ/τ. Equivalently, a short (≈ 104τ) NPT simulation was performed at pressure Embedded Image. We have verified that the results do not depend on the sample preparation method. Once the target density was reached, simulations were continued for up to at least three relaxation times of the corresponding system. During the equilibration stage, the molecules diffused, on average, at least the root mean square end-to-end distance of their backbones.

Simulations of bottlebrush and linear chain melts were carried out for the following number of backbone monomers Nbb = 10, 16, 20, 50, 100, and 150. The number of side-chain monomers Nsc was varied between 0 and 32 for bottlebrushes with z = 1 and between 0 and 16 for bottlebrushes with z = 2 side chains attached to each backbone monomer. We assign unique symbols to denote data for each particular system. The convention of symbols used throughout the article is displayed in Fig. 4B. The complete list of symbols can be found in table S2. In addition, for molecules with Nbb = 50, the number of side chains per backbone monomer was varied (z = 0, 1, 2 and 4). To avoid the finite size effects, the number of molecules M in a simulation box was changed; thus, the box size a was at least ≈ 2.5 times larger than the root mean square end-to-end distance 〈R21/2 of bottlebrush backbones. Table S2 summarizes all parameters used in our computer simulations.

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/11/e1601478/DC1

The size of a bottlebrush side chains in a melt

Persistence length of a bottlebrush in a melt

The size of a bottlebrush in a melt

Bottlebrush melt preparation

table S1. Summary of the adjustable parameters Embedded Image and Ñsc describing the mean square size of side chains.

table S2. Summary of system parameters for simulations of bottlebrush melts and linear chain melts.

table S3. Summary of the adjustable parameters A, sf, sp, and ζ for the bond angle correlation function.

table S4. Parameters Embedded Image and Embedded Image describing the sizes of backbone sections.

fig. S1. The size of side chains of a bottlebrush in a melt.

fig. S2. The bond angle correlation functions g(s).

fig. S3. Persistent segments of bottlebrushes in a melt.

fig. S4. The size of backbones for bottlebrushes in a melt.

fig. S5. Distribution of sizes of bottlebrushes in a melt.

fig. S6. A scheme demonstrating sample preparation of a bottlebrush melt.

fig. S7. Snapshots displaying conformations of bottlebrush molecules in a melt state.

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.

REFERENCES AND NOTES

Acknowledgments: J.P. would like to thank J. Brock, A. Erbas, A. Galuschko, M. Koch, T. Kreer, P. Lorchat, M. Mateyisi, O. Sariyer, and J.-U. Sommer for many stimulating discussions. Funding: J.P. acknowledged the support from the Polish Ministry of Science and Higher Education (Iuventus Plus: IP2012 005072 and IP2015 059074) and the computational time at PL-Grid. M.R. would like to acknowledge financial support from the NSF (under grants DMR-1122483, DMR-1309892, DMR-1436201, and DMR-1121107), the NIH (under grants P01-HL108808 and 1UH2HL123645), and the Cystic Fibrosis Foundation. S.S. acknowledged the support by NSF under grant DMR-1407645. Author contributions: All authors designed and conceived the research. All authors wrote the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data and materials related to this paper may be requested from the authors and are available online.
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