Universality of the Entanglement Plateau Modulus of Comb and

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Universality of the Entanglement Plateau Modulus of Comb and Bottlebrush Polymer Melts Heyi Liang,† Benjamin J. Morgan,‡ Guojun Xie,§ Michael R. Martinez,§ Ekaterina B. Zhulina,∥ Krzysztof Matyjaszewski,§ Sergei S. Sheiko,*,‡,∥ and Andrey V. Dobrynin*,† †

Department of Polymer Science, University of Akron, Akron, Ohio 44325, United States Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290, United States § Department of Chemistry, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, United States ∥ Institute of Macromolecular Compounds, Russian Academy of Sciences, St. Petersburg 199004, Russia Macromolecules Downloaded from pubs.acs.org by COLUMBIA UNIV on 12/03/18. For personal use only.

‡

S Supporting Information *

ABSTRACT: A combination of scaling analysis and rheological experiments was used to study correlations between the entanglement plateau modulus and grafting density of graft polymers in a melt. Using the crowding parameter Φ, which describes overlap of side chains belonging to neighboring macromolecules, we identified two classes of graft polymerscombs and bottlebrushesthat demonstrate distinct conformational and rheological behaviors. In comb systems, both the backbones and sparsely grafted side chains are coiled that allow side chains of neighboring macromolecules to overlap (Φ < 1). In bottlebrush systems, steric repulsion between densely grafted side chains causes chain extension and inhibits side chain interpenetration (Φ ≥ 1). The ratio Ge,gr/Ge,lin ≅ φ3(1 + (Φ/ 0.7)3) of the plateau modulus of a graft polymer melt, Ge,gr, to that of a melt of linear chains, Ge,lin, is a universal function of the crowding parameter Φ ≅ φ−1nsc−1/2 and graft polymer composition φ = ng/(ng + nsc), where nsc and ng are the degrees of polymerization of side chains and a spacer separating two consecutive side chains along the polymer backbone, respectively. This dependence of the plateau modulus is verified for poly(n-butyl acrylate) combs and other graft polymer systems reported in the literature. In a special case of graft polymers with long entangled side chains, the Ge,gr/Ge,lin ratio is proportional to φ2.

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Architectural disentanglement of graft polymers was first discovered by Fetters et al. in melts of poly(α-olefins) by demonstrating distinct scaling relations for the decrease of the entanglement plateau modulus with linear mass density.3,4 They obtained two empirical relationships that allowed for estimation of the entanglement plateau modulus GN0 of polyolefins from the average molecular weight per backbone bond (mb) as

INTRODUCTION Brushlike macromolecules with a long backbone and multiple side chains constitute a unique class of polymers that combine the properties of molecules and filaments. Loosely grafted combs overlap and entangle similar to regular linear chains, whereas densely grafted bottlebrushes behave as semiflexible filaments with negligible overlap and markedly lower entanglement density.1 Unlike linear polymers, which are defined by a single parameterthe degree of polymerization (DP) of a linear chaingraft polymers have three architectural parameters that control their physical properties: DPs of the backbone (nbb), side chains (nsc), and spacer between side chains (ng) (Figure 1a). Independent variation of these parameters allows for substantial control over the entanglement plateau modulus and zero shear viscosity of polymer melts1−7 as well as Young’s modulus and elongation at break of polymer networks.1,8−20 This ability to control chain entanglements through macromolecular architecture without changing the chemical composition paved the way for the design of supersoft hyperelastic solvent-free materials for low voltage actuators,17 vibration damping,21 soft robotics,22,23 and materials mimicking mechanical properties of biological tissues.18,19 © XXXX American Chemical Society

GN0 = 24.82mb−3.49

for mb = 14−28

(loose grafts) (1)

GN0 = 41.84mb−1.58

for mb = 35−56

(dense grafts) (2)

These equations have been used in the literature to describe various types of graft polymers with little explanation of the physics behind them. Herein we expand upon previously published theory on bottlebrush and comb polymer melts1,12,24 to validate eqs 1 and 2. We also demonstrate universality of our Received: August 15, 2018 Revised: October 25, 2018

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DOI: 10.1021/acs.macromol.8b01761 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. (a) Schematics of a graft macromolecule defined by the degrees of polymerization of the backbone nbb, side chains nsc, and spacer between neighboring side chains ng. (b) A graft macromolecule is considered as a chain of blobs with size Rsc that host monomeric units of the surrounding macromolecules (bleached lines). (c) Diagram of states of graft polymers in a melt with monomer projection length l, Kuhn length of the backbone and side chains b, and monomer excluded volume v (logarithmic scales). SBB: stretched backbone subregime; SSC: stretched side chain subregime; RSC: rodlike side chain subregime. Insets illustrate typical macromolecular conformations in different regimes. The accessible per regimes of the diagram of states are determined by a chemistry of the backbone which restricts a maximum number of the side chains 1/nmax g + 1, which is plotted backbone monomer. Because of this constraint, the upper boundary of accessible regimes is given by φ−1 ≤ φmax−1 = nsc/nmax g = 1. for nmax g

model by testing it against different types of graft polymers synthesized here and reported previously.1,3,4,7,25,26

Φ≡

■

ng ng + nsc

(3)

which corresponds to the volume fraction of backbone monomers in a graft macromolecule. Note that φ−1 is equivalent to the “swelling ratio” of a polymer gel as α≡

ng + nsc Vbb + Vsc ≅ = φ −1 Vbb ng

(5)

In this definition, polymer melts of overlapped and segregated macromolecules correspond to Φ < 1 and Φ = 1, respectively. As illustrated in Figure 1b,24 a graft macromolecule can be viewed as a chain of blobs with a size of the side chains Rsc, which is on the order of the square root of the mean-square end-to-end distance. The pervaded volume occupied by side chains of a given macromolecule can be calculated as Vp ≅ Rsc3nbb/g, where g is the number of backbone monomers per blob. For the low grafting density, both side chains and backbone maintain their ideal conformations, which has two implications for estimating dimensions of graft macromolecules. First, the side chains have ideal chain dimensions with Rsc ≅ (blnsc)1/2, where l is the monomer projection length and b is the Kuhn length assumed to be the same for both side chains and backbone. Second, the number of backbone monomers per blob equals to that of the side chain, i.e., g ≅ nsc. As such, the pervaded volume can be estimated as Vp ≅ (bl)3/2nbbnsc1/2. The volume occupied by monomers from the same macromolecule is Vm ≅ vnbbφ−1 (where v is the monomer excluded volume). By substituting these relations into eq 5, we obtain the following expression for the crowding parameter

CONFORMATIONAL REGIMES OF GRAFT POLYMERS Conformations of graft macromolecules in a melt are determined by (i) the effective flexibility of the backbone and (ii) degree of side chain interpenetration of neighboring macromolecules. Both properties are controlled by nsc and ng (Figure 1a).24 Here we consider a case when both backbone and side chains are made of chemically identical monomers. If the backbone and side chains are composed of chemically different monomers, the corresponding difference in the chain stiffness and the possibility of microphase separation should be considered. Previously, we have developed a diagram of states in terms of nsc and ng.1 However, a more accurate description is based on the compositional parameter24 φ=

Vm Vp

Φ≅ (4)

v φ−1nsc−1/2 (bl)3/2

(6)

In our classification of graft polymers, the comb regime is characterized by overlap of side chains from neighboring macromolecules, which means that the fraction of their own (i.e., host) monomers is smaller than unity, i.e., Φ ≤ 1.24 With increasing grafting density, the fraction of the host monomers increases and reaches the limit of Φ = 1, where the pervaded volume of a side chain contains only monomers from the same macromolecule. Upon further increase of the grafting density, the system enters the so-called bottlebrush regime characterized by Φ > 1. The unphysical value of the crowding parameter less than one macromolecule in its pervaded volume (Φ > 1)

wherein we view side chains as effective solvent for their backbones with volumes Vsc and Vbb, respectively. The gel analogy is useful when comparing the rheological properties of graft polymer melts and linear polymer gels as discussed below. To describe interpenetration of side chains in graft polymers, we have introduced the crowding parameter Φ, which is defined as a volume fraction of monomers of a graft polymer, Vm, in the pervaded volume, Vp, of its side chains along the polymer backbone (Figure 1b): B


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Macromolecules corresponds to a hypothetical system where backbones and side chains maintain their ideal conformations even in densely grafted macromolecules. However, in real systems, an increase of grafting density leads to extension of both backbone and side chains to maintain constant density of monomers in a melt. It is important to point out that eq 6 corresponds to graft polymers with flexible side chains (nsc > b/l). For short and/or rigid side chains (nsc < b/l and Rsc ≅ nscl), the crowding parameter is estimated as Φ ≅ vl−3φ−1nsc−2. Figure 1c outlines the corresponding diagram of states in terms of φ−1 and nsc.24 In the comb regime, neighboring graft polymers interpenetrate whereby both side chains and backbones maintain their unperturbed ideal chain conformations. The backbones remain flexible with a Kuhn length equal to that of the linear chains (bK ≅ b). However, in the bottlebrush regime, the excluded volume interactions between densely grafted side chains cause (i) extension of backbone and side chains, (ii) stiffening of a graft macromolecule (bK > b), and (iii) withdrawal of its side chains from neighboring macromolecules. Free energy calculations suggest that backbone extension is followed by extension of side chains with increasing grafting density. Therefore, depending on the grafting density and side chain length (or the value of φ parameter), the bottlebrush regime has three subregimes. Bottlebrushes with extended backbones belong to the stretched backbone (SBB) subregime, whereas macromolecules with stretched side chains constitute the stretched side chain (SSC) subregime. Bottlebrushes with fully stretched side chains define the rodlike side chain (RSC) subregime. It is important to point out that chemical structure of graft polymers sets the maximum number of side chains which is possible to graft to a backbone monomer 1/nmax g . This imposes + 1 on accessible an upper boundary φ−1 ≤ φmax−1 = nsc/nmax g regimes in the diagram of states in Figure 1c. The line φ−1 = φmax−1 runs above and parallel to the crossover line to SSC regime and intersects crossover line to RSC subregime at a 3 max 2 point with coordinates (v/l3nmax g ,v/l (ng ) ). This chemical constraint prohibits significant section of the SSC and RSC bottlebrush subregimes, which makes the Comb and SBB bottlebrush regimes as the most common ones. For conventional graft polymers with an aliphatic backbone, a carbon atom allows grafting of two side chains, which corresponds to = 0.5.27 It is possible to “barbwire” bottlebrushes with nmax g synthesize brushlike structures with a higher grafting density by incorporating dendritic monomers to the backbone. However, this will increase persistence length of the bare backbone, which may exceed bK and thus outweigh the effect of the side chains. The outlined conformational regimes and subregimes are characterized by different effective Kuhn length, bK, of the brush backbone.24 Similar to the crowding parameter Φ, the Kuhn length can be expressed in terms of nsc and φ as summarized in Table 1.

Table 1. Conformational Regimes Shown in Figure 1c and the Corresponding Effective Kuhn Lengths of Graft Polymers regime boundariesa

regime comb, Φ < 1

φ

bottlebrush, Φ>1

SBB SSC RSC

−1

3/2 −1

≤ (bl) v nsc , for nscl > b φ−1 ≤ l3v−1nsc2, for nscl < b (bl)3/2v−1nsc1/2 ≤ φ−1 ≤ bl2v−1nsc bl2v−1nsc ≤ φ−1 ≤ l3v−1nsc2 l3v−1nsc2 ≤ φ−1, for nscl > b l3v−1nsc2 ≤ φ−1, for nscl < b 1/2

Kuhn length, bKb b vl−3/2b−1/2φ−1nsc−1/2 v1/2l−1/2φ−1/2 lnsc b

l − monomer projection length, b − Kuhn length of a linear polymer chain, and v − monomer excluded volume defined as v = M0/(cNA), where c is mass density of a polymer melt. bFor the SBB and SSC subregimes, the effective Kuhn length corresponds to the brush diameter (bK ≅ Rsc) which was calculated in ref 24. In the RSC regime, the effective Kuhn length equals to Rsc or the backbone Kuhn length b depending on which is larger. For the SBB and SSC subregimes, Rsc depends on extension of the backbone and side chains.24 For longer side chains, the Kuhn length may exceed the brush diameter. However, this regime is not reachable in conventional systems. Condition nscl < b corresponds to graft polymers with rodlike side chains while nscl > b represents graft polymers with flexible side chains. a

melts with unentangled side chains, and graft polymer melts with entangled side chains (see Figure 2). In polymeric systems, the entanglement plateau modulus is of the order of the thermal energy kBT (kB is the Boltzmann constant and T is the absolute temperature) for each entanglement strand per unit volume Ge ≅ ρe kBT ≅

ρkBT ne

(7)

where ρe and ρ are respectively number densities of entanglement strands and monomers in a melt and ne is the average number of monomeric units in an entanglement strand. For melts of linear polymer chains (system I), ne corresponds to ne,lin, the DP of the polymer strand between entanglements, which gives the entanglement plateau modulus as Ge,lin ≅

ρkBT ne,lin

(8)

The relationship between ne,lin and chain’s molecular parameters follows from the Kavassalis−Noolandi conjecture stating that there is a fixed number of entanglement strands, Pe ≅ 20, within a confinement volume a3 occupied by a strand with DP = ne,lin.1,29,30 For a melt of linear chains, the size and excluded volume of an entanglement strand are a ≅ blne,lin and Ve = vne,lin. Therefore, the number of overlapping entanglement strands inside the volume, a3, is estimated as

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ENTANGLEMENT PLATEAU MODULUS OF GRAFT POLYMERS (SCALING ANALYSIS) Entanglements of polymer chains result in a plateau of storage modulus as a function of frequency.28 The plateau modulus depends on the average size (volume) of entanglement strands, which in turn is determined by the size and grafting density of side chains. Herein we consider three distinct systems of entangled polymer melts: linear chain melts, graft polymer

Pe,lin ≅

(bl)3/2 a3 ne,lin ≅ Ve v

(9)

which determines ne,lin and Ge,lin as a function of the molecular parameters l, b, and v as ne,lin ≅ Pe,lin 2 C

v2 (bl)3

(10a) DOI: 10.1021/acs.macromol.8b01761 Macromolecules XXXX, XXX, XXX−XXX

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Figure 2. Three distinct polymer melt systems. The equations above the cartoons correspond to the number of monomeric units in the corresponding entanglement strands (eqs 10a, 14, and 16). The backbone fraction φ and the effective Kuhn length of the backbone bK are defined by eq 3 and in Table 1, respectively.

Ge,lin ≅

i φb y Ge,gr ≅ Ge,linjjj K zzz k b {

ρkBT (bl)3 Pe,lin 2 v 2

(10b)

Both properties exhibit minor variations with chemical composition (l, b, and v), which explains why the majority of conventional linear polymers have the entanglement plateau modulus of the order of 105 Pa. In contrast, modification of polymer architecture results in significantly greater shifts in the entanglement properties as discussed below. In melts of graft polymers (system II in Figure 2), grafting side chains to a polymer backbone results in dilution of the backbone monomers by a factor φ (eq 3). Therefore, the entanglement plateau modulus (eq 7) of melt of graft polymers with entangled backbones decreases with increasing “swelling ratio” α ≡ φ−1 (eq 4) as Ge,gr ≅

(14)

The specifics of chain architecture enter through the graft polymer composition φ and dependence of the Kuhn length bK on nsc and φ (Table 1). It is important to point out that the φbK/b factor in eq 14 includes two distinctly opposite effects of the side chains on the entanglement plateau modulus: (i) dilution of the backbones by side chains (φ < 1) promotes plateau modulus decrease as Ge,gr ∼ φ3 and (ii) effective stiffening of the backbones due to steric repulsion between their side chains (bK/b > 1) causes the modulus to increase such as Ge,gr ∼ (bK/b)3. In the comb regime (bK ≅ b), the dilution effect prevails resulting in overall modulus decrease, whereas in the bottlebrush regime (bK > b), stiffening of the backbone weakens the decrease in modulus. To develop a universal picture for all graft polymer melts, we separate the dilution and stiffening effect and rewrite eq 14 as

ρkBT ne,bbφ−1

3

(11)

where n e,bb is DP of the backbone strand between entanglements. Note that for a melt of linear chains (nsc = 0 and φ = 1) eq 11 is equivalent to eq 8, i.e., Ge,gr = Ge,lin and ne,bb = ne,lin. In addition to dilution of the backbones, side chains lead to an increase of the effective Kuhn length bK of graft macromolecules (Table 1). This in turn changes the tube diameter a ≅ bK lne,bb and the excluded volume of a brush section with ne,bb backbone monomers between entanglements to Ve = vne,bbφ−1. In this case the number of the entangled chains within volume a3 is1,29,30

Ge,gr φ3Ge,lin

ib y ≅ jjj K zzz kb {

3

(15)

which presents the normalized entanglement modulus as a function of the normalized Kuhn length bK/b. In polymer combs (Φ < 1, bK ≅ b), the DP of the backbone of an entanglement strand eq 13 increases with dilution as ne,bb ≅ φ−2ne,lin

(16)

3/2

Pe,gr ≅

(b K l )

ne,bb

vφ −1

and the normalized entanglement modulus approaches unity: ij Ge,gr yz jj zz ≅1 jj 3 z j φ Ge,lin zz k {comb

(12)

This expression determines ne,bb as a function of both the molecular (l, b, v) and architectural (nsc, φ) parameters. From eqs 10a and 12, the entanglement DP of the backbone of graft polymers can be expressed in terms of that of linear chains as ne,bb

ij Pe,gr yz zz ≅ jjjj zz P e,lin k {

2

ij b yz −2 i y jj zz φ n ≅ jjj b zzz φ−2n e,lin e,lin jj b zz jj b zz k K{ k K{ 3

(Φ < 1) (17)

This equation demonstrates the dominating role of the dilution factor Ge,gr ∼ φ3 in the comb regime. In the stretched backbone (SBB) subregime (Φ > 1), the steric repulsion of densely grafted side chains results in backbone stretching and an increase of the Kuhn length. According to Table 1 and eq 6, the normalized Kuhn length of graft polymers in the SBB subregime increases linearly with the crowding parameter Φ as bK/b ≅ Φ. Therefore, the normalized entanglement modulus can be expressed in terms of crowding parameter Φ as

3

(13)

Note that the final form of the eq 13 is only correct if the number of entanglement strands is a universal number, Pe,lin ≈ Pe,gr. The modulus of the melts of graft polymers with entangled backbones is obtained by substituting eq 13 into eq 11 and taking into account eq 8 for Ge,lin D


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ij Ge,gr yz jj zz ≅ Φ3 jj 3 z j φ Ge,lin zz k {SBB

qualitatively similar to the plateau modulus change in semidilute polymer solutions Ge,lin ∼ φa, where scaling exponent a depends on the solvent quality.30 However, in contrast to polymer solutions, disentanglement of macromolecules in melts of graft polymers is achieved without using any solvent.

(Φ > 1) (18)

This shows that the dilution effect (Ge,gr ∼ φ ) is counterbalanced by the stiffening effect (Ge,gr ∼ Φ3), which weakens the decrease in modulus caused by the dilution of backbones by side chains. In the subsequent SSC subregime, the stiffening factor (bK/b) can be calculated from the corresponding bK equations in Table 1, which results in a different expression for the normalized modulus. However, this subregime takes place at high grafting densities (ng < 2), which, in practical terms, corresponds to a solitary bottlebrush system with ng = 1, i.e., one data point in the diagram of states (Figure 1c). The RSC subregime also has marginal significance as it corresponds to even higher grafting densities with two and more side chains per backbone monomer. To summarize, eqs 17 and 18 provide a universal representation of the entangled plateau modulus of conventional graft polymers with unentangled side chains. For graft polymers with long side chains in the comb regime (system III, Φ < 1, bK ≅ b), both backbones and side chains are entangled (Figure 2c). Because there is no distinction between backbone and side chain strands, the system effectively behaves as a melt of linear chains with an entanglement plateau modulus defined by eq 8. However, the DP of the entanglement strand is given by eq 16 as ne,comb ≅ φ−2ne,lin because the packing condition (eq 12) counts the total number of the other chain sections (including side chains and backbones) within a pervaded volume of an entanglement strand. By substituting eq 16 as ne,lin in eq 8, we obtain the following expression for the entanglement plateau modulus of combs with entangled side chains: 3

Ge , entcomb ≅ Ge , linφ 2

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COMPARISON WITH EXPERIMENTS To verify the scaling predictions, we have synthesized a series of graft poly(n-butyl acrylates) (PBA) with similar nsc ≅ 14 and systematically varied ng from 3 to 68 through copolymerization of nonfunctional n-butyl (meth)acrylates with trimethylsilyl protected (meth)acrylates by atom transfer radical polymerization (ATRP).31,32 The synthetic scheme is shown in Figure 3. To avoid microphase separation between the backbone and side chains in polymers with low grafting density (ng ≥ 6), all graft-polymer sections have the same chemical composition, i.e., n-butyl acrylate. However, the densely grafted polymers (ng < 6) have slightly different chemical compositions of their backbones and side chains: poly(n-butyl methacrylate) and poly(n-butyl acrylate), respectively. This unfortunate mismatch is the result of a compromise between rheological requirements and synthetic constraints. Because of steep increase of the entanglement length with grafting density in the bottlebrush regime (eq 12), rheological observation of the entanglement plateau requires macromolecules with ne,bb > 3000. Such long bottlebrushes are more feasible using methacrylate monomers. Despite this mismatch, we did not observe any indication for microphase separation for two reasons: (i) PBA and PBMA have virtually identical solubility parameters33 of 17.9 MPa1/2 and (ii) volume fraction of the methacrylate backbone in the ng = 3 and ng = 5 polymers is relatively low: ∼5% and 8%, respectively. As shown below, mechanical properties of the graft polymer with PBMA spacers in the backbone are consistent with that of their PBA counterparts. Table 3 outlines molecular parameters of all graft polymers with PBA side chains studied in this paper and previously reported PBA bottlebrushes with 1.6 ≤ ng ≤ 2.0 and different DPs of the side chains.1 Side chain cleavage of ng = 3 provided an initiation efficiency of 98%. It is anticipated that for larger values of ng the initiation efficiency remains equally high due to low steric hindrance from neighboring chains. The corresponding diagram of states shows that our samples cover both comb and bottlebrush regimes (Figure 4a). Note that the SSC and SBB bottlebrush subregimes are relatively narrow in terms of φ−1. For a side chain DP nsc = 25, the SSC and SBB subregimes are confined between ng = 1 and ng = 4. Comprehensive coverage of the bottlebrush regime requires high grafting density of ng ∼ 1 and variation of the side chain length within a range of nsc ∼ 10−100 (red circles in Figure 4a). To demonstrate universality of our data analysis, we combined them with the corresponding literature data for polysterene (PS)7 and polyolefin3,4,25,26 graft polymers. Figure 4b shows the diagram of states, which identifies the studied polymers with respect to the graft polymer regimes. Unlike Figure 4a, the diagram is plotted in terms of normalized φ and nsc, which account for the difference between graft polymers with rodlike (nscl < b) and flexible (nscl > b) side chains. In these new variables, graft polymers of different chemical compositions with flexible side chain can be described by one diagram of states, whereas the crossover between the comb and RSC bottlebrush regimes for the systems with rodlike side

(19)

Note that we can also express the normalized plateau modulus of the entangled combs in terms of the crowding parameter such as Ge,entcomb 3

φ Ge,lin

≅ φ −1 ≅

nsc Φ

(Φ < 1) (20)

Below we will test applicability of eqs 17, 18, and 19 to rheological data of melts of graft polymers. Table 2 combines expressions of the plateau modulus of graft polymer melts in all regimes of the diagram of states in Figure 1c. As follows from this table, depending on the regime, the variation of the plateau modulus in melts of graft polymers can follow different scaling laws ranging from Ge,gr ∼ φ1.5 to Ge,gr ∼ φ3. This behavior is Table 2. Ratio of the Entanglement Plateau Shear Modulus in a Melt of Graft Polymers to That of Linear Chains in Different Conformation Regimes regimea comb, Φ < 1 bottlebrush, Φ > 1

Ge,gr/Ge,linb

equation

φ unentangled side chains φ2 entangled side chains (φΦ)3 ≅ v3(bl)−9/2nsc−3/2 (φv/b2l)3/2 (φlnsc/b)3, for nscl > b φ3, for nscl < b

17 19 18 14 and Table 1 14 and Table 1

3

SBB SSC RSC

a

Regime boundaries are given in Table 1. bGe,lin is defined by eq 10b. In the SBB subregime, Ge,gr depends only on nsc while remains invariant with respect to ng. E


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Figure 3. Synthesis of combs with poly(n-butyl acrylate) side chains and poly(n-butyl acrylate) spacers between the consecutive side chains. The degrees of polymerization of the side chains and backbone are nsc and nbb = nb1 + nb2, respectively. The molar fractions of n-butyl acrylate and HEATMS in the backbone determine the grafting density ng−1, where ng = nb1/nb2 + 1.

Table 3. Parameters of n-Butyl Acrylate Graft Polymers nga

nsc

b

nbb

c

d

Mn [kg/mol]

Đ

d

groups. For all PBA samples, the three polyolefin bottlebrushes (s-PPEN, s-PHEX, and s-POCT) prepared by Fetters et al.3 and all polyolefin samples (PH, PO, PN, PD, PDD, PTD, and POD) prepared by Lopez-Barron et al.,26 their side chains are shorter than the corresponding entanglement strands of linear PBA34 (ne,lin = 219) and PE (ne,lin = 36).4 Therefore, these systems generally follow a scaling function Ge,gr ∼ φα with the exponent varying between −3/2 and −3, which corresponds to the bottlebrush and comb regimes (Figure 6a) and agrees with the empirical relations by Fetters et al.3 (eqs 2 and 1). However, the side chains in the PS7 and PE combs4 (PEC) are longer than the corresponding entanglement DPs of linear chains ne,lin = 140 for PS7 and ne,lin = 36 for PE.4 These systems belong to system III: comb polymers with entangled side chains (Figure 2c). In agreement with the eq 19, the data points for PS and PEC samples have collapsed into one universal curve with a slope of −2 in logarithmic scales (Figure 6b). However, because of the proximity of crossovers between the different graft polymer regimes, it is difficult to distinguish the corresponding scaling behaviors when plotting Ge,gr as a function of nsc/ng + 1 (Figure 6c). To address this uncertainty, we replotted the data as a function of the crowding parameter Φ (Figure 6d). The dashed line at Φ* = 0.7 corresponds to the crossover from the comb to bottlebrush regimes as discussed elsewhere.24 The PBA combs and bottlebrushes with shorter side chains follow the respective scaling laws given by eqs 17 and 18. The broadening of the scaling dependence of the shear modulus on the crowding parameter (Ge,gr ∼ (φΦ)3) to SSC subregime (see Figure 4a) is due to the fact that for the densely grafted side chains with ng ∼ 1 the crowding parameter Φ ≅ φ−1nsc−1/2 ≅ φ−1/2, and the effective Kuhn length of the bottlebrush bK has the same scaling dependence on the crowding parameter Φ in both SBB and SSC subregimes (see Table 1). Note that a similar trend was observed in computer simulations.24 It is important to point out that there is a deviation of the experimental data points from the universal curve close to the crossover value of the crowding parameter. The possible origin of this deviation will be discussed below. In contrast, the normalized entanglement plateau modulus of the PS and PE combs with entangled longer side chains deviates from unity eq 17 and exhibits the scaling exponent of 1. This behavior is consistent with eq 20 for constant DP of the side chains (Table 5). Note that the three polyolefin comb (sPPEN, s-PHEX, and s-POCT) samples by Fetters et al.3 belong to the RSC subregime, and samples from Lopez-Barron et al.26 cover both RSC and SBB subregimes of the bottlebrush regime (Figure 4b). In the RSC subregime an entanglement plateau modulus approaches Ge,gr ∼ φ3 scaling behavior while

e

Ge [kPa]

series 1: variation of grafting density at constant side chain length 68 14 2100 300 1.22 75.5 27 13 2600 476 1.38 44.4 16 15 2700 474 1.31 32.0 11 13 3300 520 1.26 31.2 6 15 3000 420 1.57 9.8 5* 13 5000 972 1.14 8.1 3*f 14 5700 1210 1.24 2.7 series 2: variation of side chain length at constant grafting density1 2.0 17 2040 2900 1.50 2.42 1.6 23 2040 4700 1.60 1.31 1.6 34 2040 5800 1.60 0.94 1.9 130 2040 17900 1.50 0.18 a

Calculated based on the monomer to initiator ratio and conversion determined by 1H NMR in copolymerization of HEA-TMS (or *HEMA-TMS) and n-BA (or *n-BMA). bCalculated based on ratio of peaks by 1H NMR spectrum of backbones. cCalculated based on monomer to initiator ratio and conversion determined by 1H NMR polymerization of n-butyl acrylate. dDetermined by SEC using linear polystyrene standards (series 1) (see Figure SI1) and by the AFM-LB method (series 2) (see ref 1). eCalculated as the storage modulus at the minimum of tan(δ). fInitiation efficiency calculated by side chain cleavage was 98%, ng,app = 3.2, nsc,app = 14.7. Quantitative initiation was assumed for all other samples.

chains depend on the system parameters (b, l, and v). This is shown in Figure 4b as three curved lines in the interval nscl/b < 1. We will use this classification of the different systems for interpretation of the rheological data below. Rheological master curves measured over a broad range of frequency were used to observe the entanglement plateaus of the synthesized graft polymers. For consistency, graft polymers with PBMA and PBA spacers between PBA side chains are presented separately in Figures 5a and 5b, respectively. At low angular frequencies, all samples show terminal relaxation, which is preceded by a distinct entanglement plateau in the G′(ω) curves between 10−2 and 103 rad/s. The grafted side chains are well below the entanglement molar mass for pBA, which indicates that the entanglement plateau corresponds to the entanglement of neighboring graft polymers. The plateau modulus was measured as the storage modulus at the minimum of tan(δ) as a function of the G′(ω) in this range (Table 3). Figure 6a−c plots the normalized entanglement modulus in terms of the graft polymer composition φ−1 = 1 + nsc/ng (eq 3) for different graft polymers including those reported by other F


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Figure 4. (a) Diagram of states of PBA combs (blue ●) and bottlebrushes (red ●) (logarithmic scales). According to Figure 1c, the solid black line corresponds to a crossover between comb and bottlebrush regimes and is calculated by setting parameters v(bl)−3/2φ−1nsc−1/2 = 0.7 and vl−3φ−1nsc−2 = 0.7 for flexible, nsc > b/l, and rodlike, nsc < b/l, side chains, respectively. The coefficient 0.7 has been determined in computer simulations (see ref 24). In this calculation we used l = 0.25 nm, b = 1.7 nm, and v = 0.197 nm3. The location of the intersection point is obtained by substituting nsc = b/l = 6.8 into expression for φ−1 at the crossover between comb and bottlebrush regimes: φ−1 ≅ 0.7b2l/v ≅ 2.6. The boundary between comb and bottlebrush regime for nsc < b/l = 6.8 (rodlike side chains) is given by a crossover expression φ−1 = 1 + 0.43l3v−1nsc2, which is obtained by taking into account scaling relation for the crowding parameter at a crossover Φ ≅ vl−3φ−1nsc2 ≈ 1 ⇒ φ−1 ≈ l3v−1nsc2 and requiring function φ−1 to pass through the points with coordinates (6.8, 2.6) and (0, 1) corresponding to a linear chain limit. The upper boundary for the accessible regimes is given by φ−1 + 1 (solid red line). (b) Diagram of states of PBA combs (blue ●), PBA bottlebrushes (red ●), PE bottlebrushes (red ▲), PE combs (blue ▲) with entangled side chains, PS combs (blue ■), and PS bottlebrushes (red ■), where φ−1 and nsc coordinates were respectively normalized by b2l/v and b/l at the crossover between rigid and flexible side chains. Data for this figure are summarized in Tables 4 and 5. The solid black line corresponds to a crossover between comb and bottlebrush regimes for flexible (long) side chains. The solid, dashed, and dotted black lines correspond to a crossover between comb and bottlebrush regimes with short (rodlike) side chains for PE, PS, and PBA systems, respectively. Dashed red lines mark crossovers between different bottlebrush subregimes (Figure 1c). Boundaries of the corresponding “forbidden regions” are not shown.

Figure 5. Dynamic master curves of the storage modulus G′ (black lines) and loss modulus G″ (red lines) were measured for graft polymers with (a) poly(butyl methacrylate) and (b) poly(butyl acrylate) spacers between side PBA chains. The curves correspond to a temperature of 25 °C, where αT is frequency shift factor for a reference temperature of 70 °C (Supporting Information). For distinction, the curves are vertically shifted by a factor between 10 and 104 as indicated.

in the SBB subregime it is expected to follow Ge,gr ∼ (φΦ)3 (Table 2). This is clearly seen in Figure 6d where the combined polyolefin data set approaches a universal scaling dependence Ge,gr ∼ (φΦ)3 with increasing the DP of the side chains. At last, we compare the behavior of graft homopolymers with graft copolymers with dissimilar backbone and side chains.36,37 For graft block copolymers, the Kuhn length, monomer projection length, and excluded volume of the backbone and side chains are different, which shifts the crowding parameter at the comb−bottlebrush crossover. The effect of chemical heterogeneity can be explicitly considered by our model through graft polymer composition and crowding parameter as described in the Supporting Information and demonstrated for recently studied poly(norbornene)-graf tpoly(lactide) (PNB-g-PLA) systems.37 As shown in Figures 7a,

one can collapse both homopolymer and copolymer data by normalizing the crowding parameter by its crossover value. An interesting feature of this plot is that both data sets demonstrate deviation from the universal curve in the crossover region. In this interval of parameters, side chains belonging to the same graft macromolecule begin to overlap and to push out the side chains belonging to neighboring macromolecules. Effectively, we deal with a transition from polymer chains to mesoscopic (brushlike) filaments. In addition, the interplay between the side chains and backbone may change the backbone conformation and result in illdefined variation of the backbone persistence length.37 While such explanation is plausible, since the normalized ratio Ge,gr/ Ge,linφ3 strongly depends on bK (eq 15), the relatively low grafting density of the side chains in this range (φ−1 ≈ 3) makes this scenario unlikely (Figures 7b). Also, computer G


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Figure 6. Normalized entanglement modulus as a function of the compositional parameter φ−1 = nsc/ng + 1 for different graft-polymer systems: (a) PBA combs (blue ●), PE bottlebrushes (red ▲), and PBA bottlebrushes (red ●). (b) PS combs (blue ■) and PE combs (blue ▲) with entangled side chains. (c) Combined plot of the normalized entanglement modulus as a function of graft polymer composition for melts of combs and bottlebrushes from (a) and (b). (d) Normalized entanglement modulus as a function of crowding parameter. The crossover equation y = 1 + (x/ 0.7)3 between comb and bottlebrush regimes is shown by the dashed line.

function of the graft polymer composition φ = ng/(ng + nsc). In the case of unentangled side chains, the modulus follows Ge,gr ∼ φ3 (see eq 17) and Ge,gr ∼ nsc−3/2 ∼ φ3/2 (for longer side chains nsc > ng with φ ≈ ng/nsc) in the comb and bottlebrush regimes, respectively (Figure 6a). Note that in the bottlebrush regime the expression for the modulus is derived by taking into account explicit expression for the effective Kuhn length bK ∼ nsc3/2 ∼ φ−3/2 (see Table 1) and substituting this expression for bK into eq 15. The plateau modulus of combs with long entangled side chains is given by Ge,gr ∼ φ2 (Figure 6b). In addition to the generic increase of modulus with φ, our model enabled a more distinct identification of the graft polymers regimes in terms of a crowding parameter Φ, which describes degree of the interpenetration of the side chains belonging to neighboring macromolecules. By plotting the normalized plateau modulus Ge,gr/φ3Ge,lin as a function of Φ, we separated the conventional graft polymer regimes (combs and bottlebrushes) from combs with entangled side chains and bottlebrushes with rigid or short side chains (Figure 6d). We have shown that our model can be extended to graft copolymers with chemically different backbone and side chains. Figures 6d and 7a reveal systematic deviation of the experimental data points from universal scaling curve in the crossover between the comb and bottlebrush regimes. A possible explanation of this behavior could be variation of the number of entanglement strands Pe within confining volume upon the polymer−filament crossover. Detailed analysis of this situation will be considered in a separate publication.

Table 4. Structural Parameters of Graft Polymers Used in Calculations of the Crowding Parametera polymer b

PBA PSc PEc

l [nm]

b [nm]

c [g/mL]

M0 [g/mol]

v [nm3]

v/(bl)3/2

0.25 0.25 0.25

1.7 1.8 1.4

1.08 0.97 0.78

128 104 28

0.197 0.178 0.060

0.72 0.60 0.29

a

l is monomer length and b is Kuhn length of the linear polymer strand. The monomer values of v were calculated from the molecular weight of monomer M0, mass densities c, and Avogadro’s number NA: v = M0/(cNA). PBA: poly(butyl acrylate); PS: polystyrene; PE: polyethylene. bData from ref 35. cData from ref 30.

simulations of graft polymers demonstrate monotonic increase of the effective Kuhn length with the crowding parameter.24 This suggests a different reason for this peculiar behavior dependence of the number of entanglement strands Pe within confining volume on the grafting density in the crossover between comb and bottlebrush regime. The physical reason for such change could be variation in the monomer density in the vicinity of the grafting points. At the moment, we do not have irrefutable proof to support this explanation.

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CONCLUSIONS We developed a scaling model for the analysis of the entanglement plateau modulus of graft polymer melts as a function of their molecular architecture. The model was tested for graft polymers of different chemical compositions (PBA, PS, and PE) with different grafting densities and DP of the side chains. It is important to emphasize that backbones and side chains of the selected polymers have similar chemical compositions. Depending on the grafting density (n−1 g ) and DP of the side chains (nsc), the studied polymers exhibited distinct scaling laws for the entanglement plateau modulus as a

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METHODS

Materials. (2-Trimethylsiloxy)ethyl methacrylate (HEMA-TMS, 96%, Aldrich), n-butyl acrylate (BA, ≥99%, Aldrich), and n-butyl methacrylate (BMA, 99%, Aldrich) were purified by passing the monomer through a column filled with basic alumina to remove H


Article

Macromolecules Table 5. Structural Parameters and Rheological Properties of Melts of Graft Polymers sample ID

nsc

linear PBA BA-17 BA-23 BA-34 BA-130 ng-68 ng-27 ng-16 ng-11 ng-6 ng-5 ng-3

17 23 34 130 14.4 12.9 14.9 13 15 12.6 14.4

linear PE PEC(101)-g-(7)30 PEC(97)-g-(23)26 PEC(100)-g-(5)2 PEC(100)-g-(5)12 PEC(108)-g-(6)12 s-PPEN s-PHEX s-POCT PH PO PN PD PDD PTD POD

232 839 161 189 207 1.5 2 3 2 3 3.5 4 5 6 8

linear PS PS290-3-44 PS290-10-44 PS290-14-44 PS290-30-44 PS290-60-44 PS290-120-44 PS290-190-44

433 423 433 413 423 423 423

ng

2.0 1.6 1.6 1.9 68.3 26.9 15.9 11 6 5 3.1

119 128 1071 274 299 1 1 1 1 1 1 1 1 1 1

697 253 186 90 46 23 15

Ge [kPa] PBA 130 2.42 1.31 0.94 0.18 75.5 44.4 32.0 31.2 9.8 8.1 2.7 PEa 2000 240 25 1310 768 894 200 130 76 138 72.5 61.9 51.1 35.9 28.2 20.1 PSb 200 90 55 20 7.2 2.4 1.5 0.25

φ

Φ

Ge,gr/(φ3Ge,lin)

1.00 0.10 0.07 0.05 0.014 0.83 0.68 0.52 0.46 0.29 0.29 0.18

1.67 2.27 2.70 4.35 0.23 0.29 0.36 0.43 0.64 0.70 1.06

16.85 35.99 78.24 470.52 1.03 1.11 1.79 2.49 3.23 2.68 3.77

1.00 0.34 0.13 0.87 0.59 0.59 0.40 0.33 0.25 0.33 0.25 0.22 0.20 0.17 0.14 0.11

0.06 0.08 0.03 0.04 0.03 4.24 2.86 1.70 2.86 1.70 1.40 1.19 0.92 0.82 0.92

3.08 5.39 1.00 1.85 2.17 1.56 1.76 2.43 1.86 2.32 2.82 3.19 3.88 4.84 7.33

1.00 0.62 0.37 0.30 0.18 0.10 0.05 0.03

0.05 0.08 0.09 0.16 0.29 0.56 0.84

1.92 5.25 3.69 6.28 12.72 54.69 31.12

a Data from refs 3, 4, 25, and 26. The values of the crowding parameter for polyolefin systems s-PPEN, s-PHEX, s-POCT, PH, PO, PN, PD, and PDD are calculated by assuming rodlike side chains since their DP is below the DP of the Kuhn monomer, Φ ≅ vl−3φ−1nsc−2. bData from ref 7.

inhibitors. Ethylene bis(2-bromoisobutyrate) (2f-BiB) and (2trimethylsiloxy)ethyl acrylate (HEA-TMS) were synthesized based on the literature.38,39 Ethyl α-bromoisobutyrate (EBiB, 98%, Acros), copper(I) bromide (CuBr, 99.999%, Aldrich), copper(I) chloride (CuCl, ≥99.995%, Aldrich), copper(II) bromide (CuBr2, 98%, Acros), copper(II) chloride (CuCl2, ≥99.995%, Aldrich), potassium fluoride (KF, 99%, Aldrich), tetrabutylammonium fluoride (TBAF, 1.0 M in THF, Aldrich), 2-bromoisobutyryl bromide (98%, Aldrich), 2,6-di-tert-butylphenol (DTBP, 99%), triethylamine (TEA, ≥99%, Aldrich), 4,4′-dinonyl-2,2′-bipyridyne (dNbpy, 97%, Aldrich), tris[2dimethylamino)ethyl]amine (Me6TREN, 99+%, Alfa Aesar), and solvents were used as received without further purification. Synthesis of Polymethacrylate Backbones (ng = 3 and ng = 5). The procedure below describes synthesis of the ng = 3 backbone. The stoichiometric ratios used to prepare the ng = 5 backbone are shown in Table 6. HEMA-TMS (5.00 mL, 17.0 mmol), n-butyl methacrylate (7.30 mL, 45.9 mmol), CuCl2 (0.58 mg, 4.3 μmol, via stock solution), dNbpy (0.169 g, 0.413 mmol), ethylene bis(2bromoisobutyrate) (2f-BiB, 1.6 mg, 4.3 μmol), and anisole (8.2 mL, 40% v/v) were added to a Schlenk flask equipped with a magnetic stir

bar. The reaction mixture was purged for 20 min and then further degassed by three freeze−pump−thaw cycles. During the final cycle, the flask was filled with nitrogen and CuCl (20.0 mg, 0.202 mmol) was quickly added to the frozen mixture. The flask was sealed, evacuated, and backfilled with nitrogen five times before it was allowed to thaw to room temperature. The reaction was allowed to stir in an oil bath set to 70 °C until its target conversion of ∼30% was reached, as determined by 1H NMR spectroscopy. Dichloromethane (DCM) was added to the reaction mixture, and the resulting polymer solution was purified by passing through a neutral alumina column into a tared round-bottom flask. DCM was evaporated from the vessel using rotary evaporation. The apparent molecular weight of the obtained poly P(HEMA-TMS-stat-BMA) determined by THF SEC using linear PS standards. Synthesis of Polyacrylate Backbones (ng = 6 through ng = 68). The procedure described below is for the synthesis of the ng = 6 backbone. The stoichiometric ratios used to prepare all other acrylic backbones are shown in Table 7. The acrylic macroinitiator was prepared by adding HEA-TMS (3.28 g, 17.0 mmol), n-butyl acrylate (10.0 mL, 69.8 mmol), CuBr2 (0.78 mg, 3.48 μmol), and anisole I


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Macromolecules

Figure 7. (a) Normalized entanglement modulus as a function of the normalized crowding parameter. PBA combs (blue ●), PBA bottlebrushes (red ●), PNB-g-PLA combs (blue ▼), and PNB-g-PLA bottlebrushes (red ▼). Φ* = 0.4 for PNB-g-PLA and Φ* = 0.7 for PBA. The crossover equation y = 1 + x3 between comb and bottlebrush regimes is shown by the dashed line. (b) Normalized Kuhn length obtained from plateau shear modulus ratio (eq 15) as a function of graft polymer composition. Here we use Kuhn length of the backbone bb as a normalization factor due to difference between Kuhn length of the backbone and side chains for PNB-g-PLA. In the SSC regime, the Kuhn length follows the bK ∼ φ−1/2 (Table 1). At lower grafting density, it approaches the zero slope due to the independence of grafting density of the effective Kuhn length (bK ≅ bb). The scaling behavior in the intermediate (crossover) region is complex due to the transition from the molecular to filament-like behavior.

Table 6. Synthesis of P(BiBEMA-stat-BMA) Backbonesa parameters sample ID

[BMA]:[HEMA-TMS]:[2f-BiB]:[CuCl]:[CuCl2]:[dNbpy]

conv (%)

nbbb

ng-3-bb ng-5-bb

5330:2670:0.5:23.5:0.5:48 6400: 1600:0.5:23.5:0.5:48

72 62

5660 4940

ngc

Mn,appd [kg/mol]

Đe

3.09 5.03

579 528

1.26 1.20

a T = 70 °C, anisole 40% v/v, ng-x-bb is the backbone for synthesizing ng-x graft polymers. bCalculated based on the monomer-to-initiator ratio and conversion determined by 1H NMR in copolymerization of HEMA-TMS and BMA. cCalculated based on ratio of peak ratios in the 1H NMR spectrum of backbones. dMolar mass is calculated based on molecular parameters. eDetermined by SEC using linear PS standards.

Table 7. Synthesis of P(BiBEA-stat-BA)a Backbones parameters sample ID

[BA]:[HEA-TMS]:[EBiB]:[CuBr2]:[Me6TREN]

VBA (mL)

LCu wire (cm)

conv (%)

nbbb

ngc

Mn,appd [kg/mol]

Đe

ng-6-bb ng-11-bb ng-16-bb ng-27-bb ng-68-bb

8000:2000:1:0.4:2 8750:1250:1:0.4:2 9170:830:1:0.40:2 7600:400:1:0.4:2 3900:100:1:0.4:2

10.0 20.0 20.0 15.0 22.4

0.7 1.3 1.3 1.0 1.0

27 33 27 32 53

2983 3270 2687 2570 2100

6.1 11.0 15.9 26.9 63.8

165 296 302 387 207

1.50 1.21 1.43 1.42 1.16

a

T = room temperature, anisole 50% v/v, diameter of Cu wire = 1 mm. ng-x-bb is the backbone for synthesizing ng-x graft polymers. bCalculated based on the monomer-to-initiator ratio and conversion determined by 1H NMR in copolymerization of HEMA-TMS and BMA. cCalculated based on ratio of peak ratios in the 1H NMR spectrum of backbones. dMolar mass is calculated based on molecular parameters. eDetermined by SEC using linear PS standards. (10.0 mL, 50% v/v) to a Schlenk flask equipped with a magnetic stir bar. The mixture was purged for 20 min then followed by addition of EBiB (1.3 μL, 8.7 μmol) and Me6TREN (4.7 μL, 0.0176 mmol). The reaction mixture was then degassed by three freeze−pump−thaw cycles. During the final cycle, the flask was filled with nitrogen and Cu wire (0.67 cm length, d = 1 mm) was quickly added to the frozen mixture. The flask was sealed, evacuated, and backfilled with nitrogen five times before it was allowed to thaw to room temperature. After ∼30% conversion, by 1H NMR spectroscopy, the polymerization was stopped by exposing the reaction mixture to air. DCM was added to the reaction mixture and passed through a neutral alumina column. DCM was evaporated using a rotary evaporation. The apparent molecular weights of the obtained poly(HEA-TMS-stat-BA) were determined by THF SEC using linear PS standards.

Preparation of Macroinitiators. This procedure describes deprotection of the ng = 3 backbone. Synthesis of all other backbones was scaled to the relative amount of protected TMS functional groups along their respective polymer backbones. KF (1.30 g, 22.0 mmol) and 2,6-di-tert-butylphenol (0.378 g, 1.84 mmol) were placed into a 100 mL round-bottom flask containing the protected the ng = 3 backbone (assuming 18.4 mmol of TMS groups). The flask was flushed with nitrogen, and dry THF (40 mL) and a 1.0 M solution of tetrabutylammonium fluoride in THF (0.18 mL) were added. 2Bromoisobutyryl bromide (2.73 mL, 22.0 mmol) was added dropwise, and then the reaction mixture was stirred overnight at room temperature to yield poly(BiBEMA-stat-BMA) macroinitiator. The functionalized polymer was precipitated in cold methanol and dried J


Macromolecules

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overnight. 1H NMR was used to quantify the ratio of poly(BiBE(M)A) groups to poly(B(M)A) on all macroinitiators. Synthesis of Graft Polymers. The below procedure outlines the synthesis of the ng = 6 graft polymers. Stoichiometric conditions were scaled to bromoisobutyrate groups along a polymer backbone and are outlined in Table 8 for all other P[(BiBE(M)A-graft-PBA)-stat-

[BA]:[BiBE(M)A]:[CuBr]: [CuBr2]:[ligand]

ng-3 ng-5 ng-6

280:1:0.33:0.017:0.7a 280:1:0.33:0.017:0.7a 50:1:0.08:0.04:0.14b

ng-11

40:1:0.062:0.038:0.100b

ng-16 ng-27 ng-68

183:1:0.21:0.017:0.46a 200:1:0.233:0.017:0.5a 140:1:0.332:0.018:0.7a

solvent 50% v/v anisole 50% v/v anisole 16% v/v DMF, 64% v/v anisole 16% v/v DMF, 64% v/v anisole 30% v/v anisole 30% v/v anisole 78% v/v anisole

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b01761. SEC spectra of BA graft polymers; WLF equation parameters; plateau modulus determination methodology; calculation of parameters for (PNB-g-PLA) systems (PDF)

Table 8. Synthesis of P[(BiBE(M)A-graf t-PBA)-statB(M)A] Graft Polymers sample ID

Article

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conv (%) 5.1 4.6 30.0

AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (A.V.D.). *E-mail [email protected] (S.S.S.).

32.5

ORCID

Heyi Liang: 0000-0002-8308-3547 Krzysztof Matyjaszewski: 0000-0003-1960-3402 Sergei S. Sheiko: 0000-0003-3672-1611 Andrey V. Dobrynin: 0000-0002-6484-7409

8.1 6.5 10.3

T = 70 °C, ligand = dNbpy. bT = room temperature, ligand = Me6TREN.

a

Author Contributions

H.L., B.J.M., and G.X. equally contributed to the paper. Notes

The authors declare no competing financial interest.

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B(M)A] brushes. P(BiBEA-stat-BA) macroinitiator (0.3286 g of polymer, assuming BiBEA contributes 31.5 wt % of overall polymer), BA (2.24 g, 15.6 mmol), anisole (7.2 mL, 64% v/v), DMF (1.79 mL, 16% v/v), and CuBr2 (0.0034 g, 0.015 mmol) were added to a Schlenk flask with a magnetic stir bar. The mixture was purged for 20 min followed by the addition of Me6TREN (0.010 mL, 0.039 mmol). The reaction mixture was degassed by three freeze−pump−thaw cycles. During the final cycle, the flask was filled with nitrogen, and CuBr (0.0034 g, 0.024 mmol) was quickly added to the frozen mixture. The flask was sealed, evacuated, and backfilled with nitrogen five times before it was allowed to thaw to room temperature. The reaction was stirred at room temperature before reaching the side chain DP of ∼14, at which point the polymerization was stopped by exposing the reaction mixture to air and diluting with THF. The resulting polymer solution was purified by passing through a neutral alumina column then precipitated into cold methanol and dried under vacuum at room temperature for 24 h. The apparent molecular weight of the obtained polymers was determined by THF SEC using linear PS standards (see Figure SI1). Cleavage of Side Chains. 50 mg of polymer brush was dissolved in THF (2 mL) and 1-butanol (99+%) (12 mL). Concentrated sulfuric acid (5 drops) was added, and the solution was heated at 100 °C for 5 days. Afterward, solvents were removed and the residue was dissolved in THF (∼3 mL) and passed through basic alumina. The molecular weight and dispersity of the cleaved linear polymer (cleaved side chains) were determined by THF SEC using linear PS standards (see Figure SI2). Initiation efficiency of ng = 3 was calculated to be 98% (initiation efficiency = Mn,BA*([BA]0/[BiBEM]0)*(monomer conversion)/Mn,GPC,SC). Rheological Characterization. The effect of ng on the graft polymers plateau modulus, was measured using an ARES-G2 rheometer from TA Instruments fitted with 8 mm parallel plates. The frequency dependence of the dynamic moduli was measured at a frequency window between 0.01 and 100 rad s−1 over a range of temperatures and strains. The time−temperature superposition principle was used to construct master curves of the modulus versus frequency using the Williams−Landel−Ferry equation with a reference temperature of 70 °C. The plateau modulus was reported as the storage modulus at the minimum of tangent delta.

ACKNOWLEDGMENTS The authors gratefully acknowledge funding from the National Science Foundation (DMR 1407645, DMR 1436201, DMR 1436219, and DMR 1624569) and from the Ministry of Education of the Russian Federation within State Contract N 14.W03.31.0022.

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REFERENCES

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