Comb and Bottlebrush Graft Copolymers in a Melt | Macromolecules

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

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Comb and Bottlebrush Graft Copolymers in a Melt Heyi Liang,† Zilu Wang,† 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 27599-3220, United States

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‡

ABSTRACT: We have studied properties of comb-like and bottlebrush-like graft copolymers made of chemically different backbones and side chains in a melt. A diagram of states for this class of copolymers was calculated in terms of architectural parameters (degree of polymerization of the side chains, nsc, and the number of backbone bonds between grafting points of side chains, ng) and structural parameters (backbone and side-chain monomer projection lengths, excluded volumes, and Kuhn lengths). We apply the concept of the crowding parameter, Φ, describing overlap between neighboring macromolecules for classification of graft copolymers into combs and bottlebrushes. In this classification, the sparsely grafted side chains of comb-like macromolecules allow interpenetration of both side chains and backbones belonging to neighboring macromolecules (Φ < 1). However, in the case of bottlebrush-like macromolecules, the densely grafted side chains preclude molecular interpenetration because of steric repulsion, resulting in Φ ≥ 1. Coarse-grained molecular dynamics simulations corroborate this classification of graft copolymers and show that the effective macromolecule Kuhn length, bK, is a function of the crowding parameter, Φ. An increase in the backbone Kuhn length or monomer size results in a shift of the boundaries between different regimes in the diagram of states in comparison with those obtained for graft homopolymers with chemically identical backbones and side chains. Graft copolymers with backbones stiffer than side chains are comb-like in a broader range of parameter space. However, by increasing the excluded volume of the backbone monomers, one expands the parameter space where macromolecules demonstrate bottlebrush-like behavior. of the side chains from the backbone)2,32−34 (see ref 4 for review). Despite flexibility in strategy selection, special care should be taken during the polymerization process to produce combs and bottlebrushes with chemically identical monomers.5,16,18 More commonly, however, the outcomes of the synthesis process are comb and bottlebrush macromolecules with side chains and backbones consisting of chemically different monomers, for example, poly(norbornene)-graf tpoly(lactide) (PNB-g-PLA) copolymers.11,14,17,18,35 The difference in properties of backbones and side chains may significantly affect polymer conformations, particularly, in the case of loosely grafted side chains. For example, side chains grafted to a relatively rigid (stiff) backbone will have a marginal effect on the overall macromolecular conformation and related physical properties until the side-chain-induced stiffness overcomes the intrinsic backbone stiffness. To describe properties of graft copolymers with chemically different backbones and side chains, one has to explicitly account for the differences in their monomers’ excluded volumes, projection lengths, and Kuhn lengths. Herein, we present the results of scaling analysis and molecular dynamics simulations

1. INTRODUCTION Molecular combs and bottlebrushes (graft polymers) are made by grafting side chains to linear chain backbones.1−4 These macromolecules, depending on the grafting density and the degree of polymerization (DP) of the side chains, can behave either as linear backbone chains dispersed in an effective solvent of side chains or as flexible filaments in which steric repulsion between the side chains controls filament diameter and stiffness.5−7 Therefore, side chains play a dual role of effective diluents and stiffeners of backbones. This architecture-empowered comb-to-bottlebrush transformation has vital implications for the physical properties of solutions, melts, and networks of these polymers.5,6,8−13 In particular, variation in the DP of the side chains and their grafting density can significantly decrease the entanglement plateau modulus in melts of combs and bottlebrushes.5,8,14−18 This suppression of entanglements opens a pathway for synthesis of supersoft and hyperelastic networks with mechanical properties similar to those of gels and biological tissues.9,19−23 The softness of networks of brush-like strands together with a myriad of chain ends also plays a crucial role in controlling their adhesive and surface properties.24−27 Combs and bottlebrushes are made by implementing one of three synthesis strategies: “grafting-through” (polymerization of macromonomers),28−30 “grafting-to” (attachment of the side chains to the backbone),31 and “grafting-from” (polymerization © XXXX American Chemical Society

Received: March 26, 2019 Revised: May 1, 2019

A

DOI: 10.1021/acs.macromol.9b00611 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules of static properties of such copolymers and their diagrams of states in a melt.

to its pervaded volume, V ≈ NbbR sc 3/g

2. SCALING MODEL OF GRAFT COPOLYMERS Consider graft copolymers consisting of a linear chain backbone made of monomers of type A having the DP Nbb and side chains made of monomers of type B with DP nsc (Figure 1a). Each macromolecule has side chains that are

(3)

of a chain of Nbb/g backbone blobs with the size equal to that of the side chains, Rsc, and each having g backbone monomers (Figure 1b) Φ≡

gv φ−1 Vm ≈ b 3 V R sc

(4)

In order to write down an explicit expression for the crowding parameter in terms of the molecular parameters of graft copolymers, we consider three different cases of backbone and side-chain conformations on length scales of the blob size Rsc: (i) both backbones and side chains are flexible; (ii) rigid backbones and flexible side chains; and (iii) both backbones and side chains are rigid. 2.1.1. Flexible Backbones and Side Chains (nsc > bb2/bsls). At low grafting densities, both the side chains and backbones display statistics of a random walk, therefore, the size of side chains is equal to R sc ≈ (bslsnsc)1/2

Because of different rigidities of the backbones and side chains, there are g ≠ nsc monomers of the backbone inside of the blob with the size of the side chain Rsc ≈ (bblbg)1/2, which results in the number of backbone monomers in the blob g ≈ nscbsls/bblb. Using the definition of the crowding parameter, Φ, eq 4, and the expression for the side-chain size and number of backbone monomers in a blob, we obtain

Figure 1. (a) Schematic representation of a graft copolymer chain showing definition of architectural parameters ng and nsc. (b) Schematic representation of the graft copolymer chain as a chain of blobs with size Rsc. Side chains and the backbone of the test macromolecule are shown in blue and red, respectively, and surrounding macromolecules are colored in gray.

Φ≈

equally spaced with ng bonds along the backbone between two neighboring side chains. To account for two different types of monomers, we assume that the backbone and side-chain monomers have excluded volumes vb and vs, monomer projection lengths lb and ls, and Kuhn lengths bb and bs, respectively. We only consider graft copolymers with backbones stiffer than the side chains, bb > bs, which are pertinent to the experimentally relevant situations.17,35 It is also important to note that studied graft copolymers do not undergo microphase separation of the side chains and backbones.36,37 This condition is satisfied if the product of the Flory−Huggins parameter χ and net DP of the repeating motif of the graft copolymer nsc + ng is smaller than the corresponding value of the critical point of the spinodal for microphase separation, χ(ng + nsc) < 10.79.38 Partitioning of monomers between side chains and backbones (composition of graft copolymers) is quantified by the volume fraction of backbone monomers, defined as v bng

Φ≈

1/2

(lsbs)

lbb b nsc1/2

for nsc > b b 2 /bsls

(6)

v b φ −1 lsbslb nsc

for bs /ls ≤ nsc ≤ b b 2 /bsls

(7)

2.1.3. Rigid Backbones and Side Chains (nsc ≤ bs/ls). Finally, when the side chains become shorter than their Kuhn length, both the backbones and side chains are rigid on the length scale of the blob size, such that Rsc ≈ lsnsc ≈ lbg. In this case, the crowding parameter is equal to

−1

Vm ≡ v bNbb + vsnscNbb/ng = v bNbbφ−1

φ −1

vb

The boundary for applicability of this expression is given by the cross-over condition to rigid backbones on the length scales of the blob size, Rsc ≈ bb. This takes place for the DP of the side chains nsc ≈ bb2/bsls. 2.1.2. Rigid Backbones and Flexible Side Chains (bs/ls < nsc ≤ bb2/bsls). A decrease in the DP of the side chains nsc below bb2/bsls leads to a smaller blob size Rsc in comparison with the Kuhn length of the backbones, bb. For shorter flexible side chains, the number of the backbone monomers in the blob R sc ≈ l b g is estimated as g ≈ bslsnsc /lb. Using this relationship, the volume fraction of monomers belonging to a test macromolecule with flexible side chains is estimated as

ij v n yz = jjjj1 + s sc zzzz j v bng + vsnsc v bng z (1) k { 2.1. Crowding Parameter. Our classification of the graft copolymers into “combs” and “bottlebrushes” is based on the concept of the crowding parameter, Φ,7 describing mutual interpenetration of the graft copolymers in a melt. This parameter is defined as a ratio of the volume occupied by monomers of a test macromolecule φ=

(5)

Φ≈

v b φ −1 ls 2lb nsc 2

for nsc ≤ bs /ls

(8)

Note that the definition of the crowing parameter introduced above is only correct in the interval of parameters where Rsc ≥ Rng. If the opposite inequality holds, Rsc ≤ Rng, one should use the size of the spacer between side chains with the

(2) B


Article

Macromolecules ⟨R e,bb 2⟩ ≈ Nbblbb b

number of bonds ng for the blob size. In this case, the crowding parameter is defined as Φ≈

ng v bφ−1 R ng 3

(11)

and the effective Kuhn length for R sc ≤ R ng

bK ≡

(9)

However, this correction in the blob size definition is only important in the small portion of the diagram of states as we will explain at the end of the next section. It is also interesting to point out that the value of the crowding parameter in the case of linear chains (φ = 1) is estimated as Φ ≈ Nbb−1/2vb/ (bblb)3/2 and corresponds to the volume fraction of monomers in a polymer coil. 2.2. Diagram of States of Graft Copolymers. Following our analysis of graft homopolymers with chemically identical backbones and side chains,7 we use the crowding parameter to separate different regimes of graft copolymers in the diagram of states in the (nsc, φ−1) plane as shown in Figure 2a. For

⟨R e,bb 2⟩

≈ bb

Nbblb

(12)

In the bottlebrush regime, the value of the crowding parameter Φ > 1 (Figure 2a). Note that these values of Φ correspond to a hypothetical system, where graft copolymers maintain ideal conformations of side chains and backbones even for large grafting densities. In real systems, for the range of parameters where Φ > 1, the backbone should stretch to decrease the number of the side chains within the volume Rsc3 to keep the monomer number density constant. The number of backbone monomers g within the volume Rsc3 is determined from the following packing condition 1≈

gv bφ−1 R sc

3

g lbb b Φ nsc lsbs

≈

(13)

On length scales larger than the side chain size, a bottlebrush macromolecule can be considered as a flexible chain of blobs each of size Rsc ⟨R e,bb 2⟩ ≈ R sc 2

Nbb ≈ lbb bNbbΦ g

for nsc > b b 2 /bsls (14)

The effective Kuhn length of the bottlebrushes in this regime is bK ≡ Figure 2. (a) Diagram of states of graft copolymers with bb > bs in a melt. Black solid lines are boundaries between comb and bottlebrush regimes and red dashed lines show boundaries of different bottlebrush subregimes. SBBstretched backbone regime, SSCstretched side chain regime, and RSCrod-like side-chain regime. The part of the diagram filled by red-white stripes corresponds to bottlebrushes with an effective backbone Kuhn length bK ≈ bb. The upper boundary of the accessible region is given by φ−1 ≤ φmax−1 = nscvs/nmax g vb + 1, = 1. (b) Diagram of which is shown as the red solid line for 1/nmax g states for graft homopolymers with chemically identical backbones and side chains (b = bb = bs, l = lb = ls, and v = vb = vs) adapted from ref 7. Notations are the same as in panel (a). Logarithmic scales.

φ −1

≈ Φb b

lbNbb

for nsc > b b 2 /bsls

(15)

In Figure 2a, this regime is designated as SBB regime emphasizing stretching of the backbone. Eventually, the section of the backbone with g monomers becomes fully extended when glb ≈ Rsc. This determines an upper boundary for the SBB regime in terms of molecular parameters of graft copolymers φ −1 ≈

bslslb nsc vb

for nsc ≥ bs /ls

(16)

Above this line, the side chains begin to stretch to satisfy the packing condition. Therefore, we named this regimeas the SSC regime. By taking into account the monomer packing condition, the size of the side chains in this regime is

comparison, Figure 2b presents the diagram of states for melts of graft homopolymers with identical backbones and side chains.7 A cross-over between combs and bottlebrushes corresponds to Φ ≈ Φ* ≈ 1, where Φ* is the cross-over value of the crowding parameter, and eqs 6, 7, and 8 are solved for φ−1 describing “dilution” of the backbone by the side chains as a function of the DP of the side chains nsc. After some algebra, the cross-over conditions separating comb-like copolymers from bottlebrushes are written as l o (bsls)1/2 b blbnsc1/2 for nsc > b b 2 /bsls o o o o o ≈ v b−1o m lsbslbnsc for bs /ls ≤ nsc ≤ b b 2 /bsls o o o o o 2 2 o o n ls lbnsc for nsc < bs /l

⟨R e,bb 2⟩

R scφ−1v b lbR sc 3

≈

vb φlbR sc 2

≈ 1 ⇒ R sc ≈

vb lbφ

for nsc ≥ bs /ls (17)

Using eq 17, the mean square end-to-end distance of the bottlebrush can be written as ⟨R e,bb 2⟩ ≈ R sc 2

lbNbb vl ≈ Nbb b b φ R sc

for R sc ≥ b b

(18) −1

Note that the condition Rsc ≥ bb is satisfied as long as φ ≥ bb2lb/vb. In this case, the effective Kuhn length of bottlebrushes in the SSC regime is

(10)

In the comb regime, the backbones can be considered as unperturbed ideal chains with the mean square end-to-end distance of the comb backbone

bK ≈ R sc ≈ C

vb φl b

for φ−1 ≥ b b 2lb/v b

(19)


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Macromolecules However, when the side chain size, Rsc, becomes smaller than the backbone Kuhn length bb, which takes place for φ−1 < bb2lb/vb, the effective Kuhn length of the bottlebrush is equal to that of the backbone bK ≈ bb. For significantly stiffer backbones (bb ≫ bs), conformation of bottlebrush graft-copolymers is insensitive to nsc and ng variations similar to macromolecules in the comb regime. Finally, the side chains are fully stretched Rsc ≈ lsnsc when the graft polymer composition is on the order of φ −1 ≈

ls 2lb 2 nsc vb

for nsc ≥ bs /ls

copolymers indicates that they can be reduced to those derived for the graft polymers with identical side chains and backbones by setting bb = bs, lb = ls, and vb = vs (Figure 2b).7 At the end of this section, we would like to comment on using the side chain size, Rsc, to define the blob dimensions for calculations of the crowding parameter and determining crossover between comb and bottlebrush regimes. This definition of the blob size breaks down in the small region of the diagram of states where the volume fraction φ ≈ 1 and DP of the side chains nsc ≈ 1. Note that our boundary (see eq 10) between combs and bottlebrushes in this range of parameters should be viewed as a lower bound for an actual cross-over (see also discussion in ref 18 for possible form of the cross-over expression). However, this substitution in the blob size definition does not change expressions for the chain size and the effective Kuhn length summarized in Table 1.

(20)

Thus, above this line, both side chains and backbone are fully stretched on the length scales smaller than the side chain size Rsc. We call this regime as the RSC regime in the bottlebrush region of the diagram of states in Figure 2a. The bottlebrush backbone remains flexible on the length scales larger than the side chain size as long as Rsc ≥ bb. This results in the following expression for the mean square end-to-end distance ⟨R e,bb 2⟩ ≈ R sc 2

lbNbb ≈ lslbnscNbb R sc

3. COMPARISON WITH SIMULATIONS The scaling model of graft copolymers in a melt was tested in coarse-grained molecular dynamics simulations.7,39 We modeled systems with different bending rigidities of the backbones and side chains and different sizes of the backbone and side chain monomers. In these simulations graft macromolecules were represented by bead-spring chains. All beads in a system interacted through modified truncated-shifted Lennard−Jones (LJ) potentials. In particular, we performed two sets of simulations. In the first set of simulations both backbone and side chain beads had the same diameter σ, while in the second set of simulations the diameter of the backbone beads was increased to db = 1.5σ. The connectivity of the beads into backbones and side chains was maintained by the FENE bonds.40 The explicit form of the interaction potentials is given in the Methods section below. Each graft copolymer consisted of a linear chain backbone with the number of monomers Nbb = 129 and grafted side chains having nsc monomers with ng backbone bonds between nearest grafting points of the side chains (Figure 1a). Graft copolymers were central symmetric ending with two linear chain sections with the number of bonds in each to be the same as between the grafted side chains. The bending rigidity of the backbone was introduced through a bending potential with bending constant K.39 The side chains were modeled as flexible chains for which K = 0. In our simulations, the DP of the side chains, nsc, was varied between 8 and 40, and ng between 0.5 and 16 (Table 2). In the case of ng = 0.5, two side chains were grafted to each backbone monomer. We performed simulations of melts of M graft copolymers. The monomer density in simulations with identical backbone and side chain beads was set to ρ = 0.8σ−3. Simulations of melts of graft copolymers with thicker backbones were performed at constant pressure, which value was set to that of the systems with identical beads as described in the Methods section. 3.1. Bond−Bond Correlation Function. We begin our discussion of the simulation results by elucidating the effect of the graft copolymer structure, intrinsic backbone stiffness and backbone bead size on the effective Kuhn length of graft copolymers. The stiffening of backbones induced by steric repulsion between side chains can be quantified by monitoring the bond−bond correlation function of the backbones, G(s). This function describes the decay of the orientational correlations between two unit backbone bond vectors ni and ni+s separated by s bonds and is defined as follows7

for nsc ≥ b b /ls (21)

The effective Kuhn length in this regime is equal to bK ≈ lsnsc. The renormalization of the Kuhn length due to side-chain interactions continues until the side chain size becomes on the order of the backbone Kuhn length, nsc ≈ bb/ls. For shorter side chains nsc < bb/ls, the effective Kuhn length of a graft copolymer is equal to that of the backbone, bb. Note that while there are no theoretical limitations on selecting ng, the particular chemical structure of graft copolymers dictates the maximum number of side chains which is possible to graft to a backbone monomer 1/nmax g . This defines an upper boundary φ−1 ≤ φmax−1 = nscvs/nmax g vb + 1 on accessible regimes (forbidden region in Figure 2). The results for the effective Kuhn length of the graft copolymers in different regimes of the diagram of the state are shown in Figure 2a and corresponding regime boundaries are summarized in Table 1. Analysis of the expressions for the effective Kuhn length and regime boundaries obtained for graft Table 1. Effective Kuhn Length of Graft Copolymers in Different Regimes and Regime Boundaries regime

−1

comb

bottlebrush

Kuhn length, bK

regime boundaries

SBB SSC

RSC

vbφ ≤ (bsls) bblbn1/2 sc , for nsc > bb2/bsls −1 vbφ ≤ lsbslbnsc, for bs/ls ≤ nsc ≤ bb2/bsls vbφ−1 ≤ ls2lbnsc2, for nsc < bs/ls (bsls)1/2bblbnsc1/2 ≤ vbφ−1 ≤ lsbslbnsc lsbslbnsc ≤ vbφ−1 ≤ ls2lbnsc2, for vbφ−1 ≥ bb2lb lsbslbnsc ≤ vbφ−1 ≤ ls2lbnsc2, for vbφ−1 < bb2lb 2 ls lbnsc2 ≤ vbφ−1, for nscls/bb ≥ 1 ls2lbnsc2 ≤ vbφ−1, for nscls/bb < 1 1/2

bb

vb(lsbs)−1/2lb−1φ−1nsc−1/2 (vb/φlb)1/2 bb lsnsc bb

D


Macromolecules

Article

ij |s| yz ij |s| yz G(s) = (1 − α) expjjj− zzz + α expjjj− zzz j λ z j λ z k 1{ k 2{

Table 2. Summary of Studied Systems and Symbol Notations

(23)

where α, λ1 and λ2 are fitting parameters. This functional form captures two mechanisms of the induced chain rigidity on different length scales with corresponding correlation lengths λ1 and λ2. At short length scales, the rigidity of the backbone is due to local backbone tension, while at long length scales it is a result of interactions between neighboring side chains. Note, that with increasing the backbone intrinsic bending rigidity the correlation function G(s) transforms into a single exponential function (shown by open symbols in Figure 3). This (single exponential) form of the bond−bond correlation function is a characteristic feature of semiflexible linear polymer chains. 3.2. Effective Kuhn Length. Knowing the backbone bond−bond correlation function we can use its explicit form to calculate the mean-square end-to-end distance of a backbone section with s bonds 1 ⟨R e (s)⟩ = Nbb − s + 1 2

Nbb − s + 1

∑ i=1

ij i + s − 1 yz z l jj ∑ njzzz jj j = i zz k { 2j j

2

(24)

where l is the bond length. The effective Kuhn length bK of graft polymers can be expressed in terms of the fitting parameters of the bond−bond correlation as7,41 bK =

G (s ) =

1 Nbb − s

i=1

= l((1 − α)h(λ1) + αh(λ 2)) s →∞

(25)

where function h(λ) is

Nbb − s

∑

⟨R e 2(s)⟩ sl

⟨ni ·ni + s⟩

(22)

h(λ) =

where the brackets ⟨...⟩ indicate the ensemble average of backbone conformations during simulation runs. To minimize the chain end effects, 20 bonds on both backbone ends are neglected when evaluating G(s). Figure 3 shows typical bond−

1 + e−1/ λ 1 − e−1/ λ

(26)

The results of these calculations are summarized in Figure 4, showing dependence of the effective Kuhn length of combs and bottlebrushes on the crowding parameter Φ. To illustrate the universality of this representation we have also included simulation data for graft homopolymers with identical backbones and side chains for which the bending constant K

Figure 3. Bond−bond correlation functions G(s) of graft copolymer backbones with the DP of side chains nsc = 8, different intrinsic backbone stiffness K and side chain grafting density, 1/ng. Solid lines represent the best fits to eq 23 using α, λ1 and λ2 as fitting parameters. Symbol notations are summarized in Table 2.

Figure 4. Dependence of the reduced Kuhn length, bK/bb, of graft copolymers on the crowding parameter, Φ. In calculating crowding parameter we have used eqs 6−9. Symbol notations are summarized in Table 2. Gray circles represent graft homopolymers with K = 0 and identical backbones and side chains.7 Solid black lines show pure scaling regimes of the effective Kuhn length dependence on the crowding parameters for combs bK ≈ bb and bottlebrushes, bK ≈ bbΦ. The vertical dashed line corresponds to cross-over between combs and bottlebrushes, Φ = Φ* ≅ 0.7.

bond correlation functions of graft copolymers with different values of DP of spacer between side chains ng and intrinsic backbone stiffness. To quantify the effect of the backbone stiffening the simulation results are fitted by a sum of two exponential functions6,7,41 E


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Macromolecules

Figure 5. Diagrams of states of graft copolymers with backbone bending constants K = 1.5 (bb = 2.87σ) (a), K = 4.0 (bb = 7.22σ) (b) and K = 1.5 and backbone bead diameter 1.5σ (bb = 4.26σ) (c). To construct these diagrams of states we used for backbone (lb = 0.985σ, vb = 1.25σ3) and for side chains (ls = 0.985, bs = 1.93σ, vs = 1.25σ3) in panels (a,b) and for backbone (lb = 1.48σ, vb = 3.28σ3) and for side chains (ls = 0.985, bs = 1.93σ, vs = 1.25σ3) in panel (c). Black solid lines are boundaries between comb and bottlebrush regimes, and red dashed lines show boundaries of different bottlebrush subregimes. The part of SSC subregime below the horizontal black dashed line and the part of the RSC subregime on the left from the vertical black dashed line correspond to bottlebrushes with the effective Kuhn length being on the order of backbone Kuhn length. Symbol notations are summarized in Table 2. (d) Diagram of states of melts of graft homopolymers with identical side chains and backbones (l = 0.985, b = 1.93σ, and v = 1.25σ3) adapted from ref 7. Colored dotted curves represent combs and bottlebrushes with the same ng calculated by using eq 1.

(Figure 4) and using this expression to rewrite eq 10 for φ−1 as a function of the side chain DP. Figure 5 shows diagrams of states for systems with the backbone bending rigidity K = 1.5 (a) and 4.0 (b) and backbone bending rigidity K = 1.5 and backbone bead diameter 1.5σ (c). For comparison in Figure 5d we show the diagram of states of graft homopolymers with identical backbones and side chains having K = 0 obtained in ref 7. It follows from these figures that by increasing the backbone stiffness the range of parameters where graft copolymers are comb-like broadens. For example, in order to synthesize bottlebrushes with the fixed spacer length ng = 2 (green dotted curves in Figure 5), the DP of the side chains should be larger than nsc ≈ 15 and nsc ≈ 117 respectively for graft copolymers with backbone bending constant K = 1.5 (Figure 5a) and K = 4.0 (Figure 5b). Increase of the intrinsic backbone stiffness opens a “window” in which the effective Kuhn length is controlled by the backbone bending rigidity. In this “window” the steric repulsion between side chains is too weak to generate significant backbone stiffening. This is illustrated by systems of graft copolymers with K = 4.0, nsc = 8, ng = 0.5 and ng = 1.0 shown by open red and orange squares in Figure 5b. Furthermore for these systems, the effective Kuhn length is smaller than that for other systems with the same value of crowding parameter (Figure 4). For bigger backbone beads (Figure 5c) the cross-overs to SBB and SSC subregimes shift to the left corner of the diagram of states shrinking the comb regime in this interval of parameters.

= 0 (gray circles in Figure 4).7 It follows from this figure that for small values of the crowding parameter, Φ ≪ 1, interactions between side chains are too weak to influence change in the backbone conformations, such that the effective Kuhn length is on the order of the bare Kuhn length of the backbone, bK ≈ bb. At intermediate values of the crowding parameter (0.3 < Φ < 1), repulsions between side chains results in local backbone stretching and stiffening manifested in an increase of the effective Kuhn length. For densely grafted side chains such that the value of the crowding parameter becomes larger than unity, Φ > 1, the steric repulsion between side chains dominates backbone bending rigidity. In this regime the effective Kuhn length scales linear with the crowding parameter, bK ≈ bbΦ. This is in agreement with the scaling expression given by eq 15. Note that analysis of the different scaling regimes in the effective Kuhn length dependence allows us to establish the cross-over value of the crowding parameter between comb and bottlebrush regimes. This cross-over corresponds to value of the crowding parameter Φ ≈ Φ* ≅ 0.7.7 It is important to point out, however, that not all points collapse into universal curve. The deviation from universal behavior is observed for systems with bending constant of the backbone K = 4.0. We will illustrate below that these systems belong to the bottlebrush subregime with SSCs and a weak renormalization of the effective Kuhn length (Figure 2a). 3.3. Diagram of States. In this section we construct diagrams of states for the simulated systems by setting the cross-over value of the crowding parameter Φ ≈ Φ* ≅ 0.7 F


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Macromolecules

4. COMPARISON WITH EXPERIMENTS We apply the scaling approach to construct a diagram of states of the PNB-g-PLA system.17 For these graft copolymers, the backbones are stiffer than the side chains. Molecular parameters for the poly(norbornene) backbone (lb = 0.61 nm, bb = 1.7 nm, vb = 0.349 nm3) and for poly(lactide) side chains (ls = 0.37 nm, bs = 1.0 nm, vs = 0.095 nm3) required for construction of the diagram of states are calculated in ref 18. The value of the Kuhn length of the PNB backbone was estimated from the packing parameter and value of the shear modulus of PNB. The cross-over between combs and bottlebrushes was determined to be Φ ≈ Φ* ≅ 0.4.18 This value of the cross-over crowding parameter was obtained by collapsing entanglement shear modulus data for PNB-g-PLA and poly(n-butyl acrylates) into one universal curve in the bottlebrush regime as described in ref 18. Note that at this time, there is no undisputed explanation why one should change the cross-over value of the crowding parameter for PNB-g-PLA systems from 0.7 to 0.4. Figure 6 presents the

1), the side chains play the role of a backbone diluent and graft copolymers behave similar to linear chain backbones. In the bottlebrush regime (Φ > 1), steric repulsions between side chains expel monomers belonging to the surrounding graft copolymers from the pervaded volume occupied by the test macromolecule. In this regime, graft copolymers behave as mesoscopic filaments with an effective Kuhn length proportional to the side chain size Rsc.5−7 In the framework of the scaling approach, we have constructed the diagram of states for this class of graft copolymers and compared it with the corresponding diagram of states of graft homopolymers (Figure 2). Predictions of the scaling model were tested in molecular dynamics simulations of graft copolymers with semiflexible backbones and flexible side chains. As in the case of the graft homopolymers,7 our simulations confirmed dependence of the normalized Kuhn length of graft polymers, bK/bb, on the crowding parameter, Φ. In the comb regime, Φ < 1, the backbone remains almost unperturbed, such that the effective Kuhn length is similar to that of the backbone, bK ≈ bb (Figure 4) In the bottlebrush regime, Φ > 1, the repulsion between side chains stiffens the backbone and the effective Kuhn length increases linearly with the crowding parameter, bK ≈ bbΦ. However, this relationship breaks down for systems with stiff backbones and short side chains as illustrated in Figure 4. The results for the effective Kuhn length were used to evaluate a cross-over value of the crowding parameters between comb and bottlebrush regimes, Φ* ≅ 0.7, and to construct diagrams of states of graft copolymers in a melt (Figure 5). In contrast to graft homopolymers (Figure 5d), copolymers with backbones stiffer than side chains (bb > bs) display a broader range of molecular and architectural parameters where graft copolymers are comb-like (Figure 5a,b), while increasing the size of the backbone beads (vb > vs) widens the bottlebrush regime in the diagram of states (Figure 5c). We have applied the scaling approach to classify graft copolymers synthesized by the ROMP technique (Figure 6). In particular, our analysis shows that the majority of the studied systems belong to the comb regime and the bottlebrush subregime with SBB. The scaling model developed here will eliminate ambiguity in classification of graft copolymer systems with assorted backbones and side chains.

Figure 6. Diagram of states of PNB-g-PLA graft copolymers. Red and blue filled circles represent bottlebrushes and combs respectively. Black solid lines are boundaries between comb and bottlebrush regimes. Dashed red lines separate different bottlebrush subregimes. Colored dotted curves represent graft copolymers with constant ng calculated by using eq 1. The filled area corresponds to the forbidden region due to the chemistry limitation on the maximum number of = 1. To construct this side chains per backbone monomer, 1/nmax g diagram of states we used for poly(norbornene) backbone (lb = 0.61 nm, bb = 1.7 nm, vb = 0.349 nm3) and for poly(lactide) side chains (ls = 0.37 nm, bs = 1.0 nm, vs = 0.095 nm3) (see ref 18).

6. METHODS

diagram of states of PNB-g-PLA graft copolymers. It follows from this figure that for majority of the diagram of states is occupied by combs and bottlebrushes with SBBs. By increasing the side chain DP, the forbidden region boundary approaches a cross-over line between SSC and SBB regimes. Thus, for these graft copolymers, in order to explore the SSC subregime, one should synthesize systems with shorter side chains.

6.1. Simulation Details. Combs and bottlebrushes were represented by bead-spring chains. To model systems with different bead sizes, we used the modified truncated-shifted LJ-potential, which accounts for increased size of the beads through offsetting the interaction range by Δ. In this representation, the interaction potential between two beads separated by a distance r is written as follows39

5. CONCLUSIONS We developed a scaling model of comb-like and bottlebrushlike macromolecules, the backbones and side chains of which are made of chemically different monomers. Using the previously developed concept of the crowding parameter Φ,7 which describes the mutual interpenetration between neighboring macromolecules, we characterized two classes of graft copolymers: combs and bottlebrushes. Specifically, we derived expressions for the crowding parameter as a function of the excluded volumes, projection lengths, and Kuhn lengths of backbone and side-chain monomers. In the comb regime (Φ
rcut + Δ o n (27)

G


Article

Macromolecules beads into chains was described by the finite extensible nonlinear elastic potential (FENE)40

UFENE(r ) = − 0.5kspringR max 2 ln(1 − (r − Δ)2 /R max 2)

Zilu Wang: 0000-0002-5957-8064 Sergei S. Sheiko: 0000-0003-3672-1611 Andrey V. Dobrynin: 0000-0002-6484-7409

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Notes

2

with the values of the spring constant kspring = 30kBT/σ and the maximum bond length Rmax = 1.5σ. The repulsive part of the bond potential was represented by the truncated-shifted LJ potential with εLJ = 1.5kBT and rcut = 21/6σ (see eq 27). For the identical backbone and side-chain beads, the parameter Δ = 0 such that their size was equal to 1.0σ. In simulations with the bulkier backbones, the value of the parameter Δ = 0.5σ for backbone−backbone pairs, Δ = 0.25σ for backbone−sidechain pairs, and Δ = 0 for sidechain−sidechain pairs. With such parameter settings, the diameters of the backbone and sidechain beads are 1.5σ and 1.0σ, respectively. The bending rigidity of the backbone was introduced by imposing the bending potential controlling mutual orientations between consecutive along the polymer backbone unit bond vectors niand ni+1

Uibend , i + 1 = kBTK (1 − (ni · ni + 1))

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to the National Science Foundation for the financial support under Grants DMR-1624569, DMR1407645 and DMR-1436201.

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(1) Sheiko, S. S.; Sumerlin, B. S.; Matyjaszewski, K. Cylindrical Molecular Brushes: Synthesis, Characterization, and Properties. Prog. Polym. Sci. 2008, 33, 759−785. (2) Rzayev, J. Molecular Bottlebrushes: New Opportunities in Nanomaterials Fabrication. ACS Macro Lett. 2012, 1, 1146−1149. (3) Verduzco, R.; Li, X. Y.; Peseka, S. L.; Steinc, G. E. Structure, Function, Self-Assembly, and Applications of Bottlebrush Copolymers. Chem. Soc. Rev. 2015, 44, 2405−2420. (4) Xie, G.; Martinez, M. R.; Olszewski, M.; Sheiko, S. S.; Matyjaszewski, K. Molecular Bottlebrushes as Novel Materials. Biomacromolecules 2019, 20, 27−54. (5) Daniel, W. F. M.; Burdyńska, J.; Vatankhah-Varnoosfaderani, M.; Matyjaszewski, K.; Paturej, J.; Rubinstein, M.; Dobrynin, A. V.; Sheiko, S. S. Solvent-Free, Supersoft and Superelastic Bottlebrush Melts and Networks. Nat. Mater. 2016, 15, 183−189. (6) Cao, Z.; Carrillo, J.-M. Y.; Sheiko, S. S.; Dobrynin, A. V. Computer Simulations of Bottle Brushes: From Melts to Soft Networks. Macromolecules 2015, 48, 5006−5015. (7) Liang, H.; Cao, Z.; Wang, Z.; Sheiko, S. S.; Dobrynin, A. V. Combs and Bottlebrushes in a Melt. Macromolecules 2017, 50, 3430− 3437. (8) Fetters, L. J.; Lohse, D. J.; García-Franco, C. A.; Brant, P.; Richter, D. Prediction of Melt State Poly(α-olefin) Rheological Properties: The Unsuspected Role of the Average Molecular Weight per Backbone Bond. Macromolecules 2002, 35, 10096−10101. (9) Neugebauer, D.; Zhang, Y.; Pakula, T.; Sheiko, S. S.; Matyjaszewski, K. Densely-Grafted and Double-Grafted PEO Brushes via ATRP. A Route to Soft Elastomers. Macromolecules 2003, 36, 6746−6755. (10) Cao, Z.; Daniel, W. F. M.; Vatankhah-Varnosfaderani, M.; Sheiko, S. S.; Dobrynin, A. V. Dynamics of Bottlebrush Networks. Macromolecules 2016, 49, 8009−8017. (11) Abbasi, M.; Faust, L.; Riazi, K.; Wilhelm, M. Linear and Extensional Rheology of Model Branched Polystyrenes: From Loosely Grafted Combs to Bottlebrushes. Macromolecules 2017, 50, 5964− 5977. (12) Dalsin, S. J.; Hillmyer, M. A.; Bates, F. S. Linear Rheology of Polyolefin-Based Bottlebrush Polymers. Macromolecules 2015, 48, 4680−4691. (13) Daniels, D. R.; McLeish, T. C. B.; Crosby, B. J.; Young, R. N.; Fernyhough, C. M. Molecular Rheology of Comb Polymer Melts. 1. Linear Viscoelastic Response. Macromolecules 2001, 34, 7025−7033. (14) Hu, M.; Xia, Y.; McKenna, G. B.; Kornfield, J. A.; Grubbs, R. H. Linear Rheological Response of a Series of Densely Branched Brush Polymers. Macromolecules 2011, 44, 6935−6943. (15) López-Barrón, C. R.; Brant, P.; Eberle, A. P. R.; Crowther, D. J. Linear Rheology and Structure of Molecular Bottlebrushes with Short Side Chains. J. Rheol. 2015, 59, 865−883. (16) López-Barrón, C. R.; Tsou, A. H.; Hagadorn, J. R.; Throckmorton, J. A. Highly Entangled α-Olefin Molecular Bottlebrushes: Melt Structure, Linear Rheology, and Interchain Friction Mechanism. Macromolecules 2018, 51, 6958−6966. (17) Haugan, I. N.; Maher, M. J.; Chang, A. B.; Lin, T.-P.; Grubbs, R. H.; Hillmyer, M. A.; Bates, F. S. Consequences of Grafting Density

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where K is the bending constant which values were equal to 1.5 and 4.0. Simulations of graft copolymers with the backbone and side-chain beads of identical sizes were carried out in the canonical (NVT) ensemble. The constant temperature was maintained by coupling the system to a Langevin thermostat implemented in LAMMPS.42 In this case, the equation of motion of the i-th bead is given by the following equation

m

dvi(t ) = Fi(t ) − ξ vi(t ) + FiR (t ) dt

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where m is the beads mass set to unity for all beads, vi(t) is the bead velocity, and Fi(t) is the net deterministic force acting on the ith bead. The stochastic force FRi (t) acting on the i-th bead has a zero average and δ-function correlation ⟨FRi (t)·FRj (t′)⟩ = 6kBTξδijδ(t − t′). The friction coefficient ξ coupling a system to a Langevin thermostat is set to ξ = 0.1m/τLJ, where τLJ = σ(m/kBT)1/2 is the standard LJ-time. The velocity-Verlet algorithm with a time step Δt = 0.005τLJ was used for integration of the equation of motion. All simulations were performed using LAMMPS42 under 3-D periodic boundary conditions. Simulations were performed in accordance with the following procedure. Macromolecules were randomly placed in a simulation box with monomer number density equal to 0.8σ−3. A simulation run lasting 250τLJ with the nonbonded interactions between beads turned off was performed in order to relax the macromolecules’ initial conformations. The strength of LJ-interaction parameter εLJ between beads was then gradually increased to 1.5kBT. This was followed by a simulation run continued until the mean square end-to-end distance of the backbones reach an equilibrium (saturates as a function of time). The equilibration run was followed by a product run lasting 5 × 105τLJ which was used for data collection. Simulations of graft copolymers with bigger backbone beads were first performed at a constant pressure P = 4.5kBT/σ3 (NPT) ensemble to relax the conformation of macromolecules. The value of the pressure was set to that in the melt of graft copolymers with an identical backbone and side-chain beads. The constant pressure and temperature was maintained by coupling the system to the Nose− Hoover barostat and thermostat39 with the pressure and temperature damping parameter both equal to 10τLJ. The relaxation run was performed until the mean square end-to-end distance of the backbones reached an equilibrium (saturates as a function of time). The production run was then performed in the NVT ensemble as described above for duration 5 × 105τLJ.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Heyi Liang: 0000-0002-8308-3547 H


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Comb and Bottlebrush Graft Copolymers in a Melt | Macromolecules

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