Combs and Bottlebrushes in a MeltHeyi Liang,† Zhen Cao,† 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

*S Supporting Information

ABSTRACT: We use a combination of the coarse-grainedmolecular dynamics simulations and scaling analysis to studyconformations of bottlebrush and comb-like polymers in a melt.Our analysis shows that a crossover between comb andbottlebrush regimes is controlled by the crowding parameter,Φ, describing overlap between neighboring macromolecules. Incomb-like systems characterized by a sparse grafting of side chains(Φ < 1), the side chains and backbones belonging to neighboringmacromolecules interpenetrate. However, in bottlebrushes withdensely grafted side chains (Φ ≥ 1), the interpenetration betweenmacromolecules is suppressed by steric repulsion between sidechains. In this regime, bottlebrush macromolecules can be viewed as filaments with diameter proportional to size of the side chains. Forflexible side chains, the crowding parameter is given by Φ ≈ [v/(lb)3/2][(nsc/ng + 1)/nsc

1/2], which depends on both the architecturalparameters (degree of polymerization of the side chains, nsc, and number of backbone bonds between side chains, ng) and chemicalstructure of monomers (bond length l, monomer excluded volume v, and Kuhn length, b). Molecular dynamics simulationscorroborate this classification of graft polymers and show that the effective macromolecule Kuhn length, bK, and the mean-squareend-to-end distance of the backbone, ⟨Re,bb

2 ⟩, are universal functions of the crowding parameter Φ for all studied systems.

■ INTRODUCTIONGraft polymers consisting of linear polymer backbones withgrafted side chains are called either combs or bottlebrushesdepending on grafting density of the side chains.1−4 The brush-like architecture allows for efficient control over materials’properties through independent variation of the side chainlength and their grafting density.5−16 In a bottlebrush melt,for example, side chains suppress the entanglement thresholdand decrease the melt viscosity, making such polymers easier toprocess.8−11 The elimination of entanglements also opens apossibility for the design of supersoft and superelasticmaterials11−14 with modulus as low as 100 Pa and tensilestrain at break up to 800% in the solvent-free states.11 Theunique combination of the elastic softness and inherent strainhardening of graft polymers was utilized in the design ofdielectric elastomers for free-standing electroactuation underlow applied fields.15 In parallel, synthesis of graft blockcopolymers created a new class of thermoplastic materialswith well-controlled mechanical and optical properties.17−21

Despite substantial experimental,5−16,22−30 theoreti-cal,11,16,29,31−34 and computational16,28,29,34−43 efforts to estab-lish accurate correlations between the brush architecture andphysical properties, the complete solution of this problem stillremains elusive. This is in part due to the large number ofstructural (chemical and architectural) parameters describingbrush-like molecular architecture, which make detailed mappingof structure−property relationships for these materialsextremely difficult. Here we use a combination of the scalinganalysis and coarse-grained simulations to provide general

frameworks for classification of graft polymers into comb andbottlebrush classes that exhibit distinct conformational andphysical properties. Specifically, we demonstrate that the Kuhnlength and chain size of graft polymers in a melt state areuniversal functions of the crowding parameter, describinginterpenetration between side chains and backbones belongingto different macromolecules. We also outline a diagram of statesin terms of two independently controlled parameters: degree ofpolymerization of side chains nsc and molar fraction of thebackbone monomers φ, which describes partitioning of monomersbetween backbone and side chains.The rest of the paper is organized as follows: (i) we use a

scaling approach to construct a diagram of states of graftpolymer melts in terms of nsc and φ, (ii) the scaling modelpredictions are compared with results of the moleculardynamics simulations for effective Kuhn length, backbone,and side chain dimensions of graft polymers, and (iii) we showthat our simulations data can be collapsed into universal plotsas predicted by the scaling model.

■ RESULTS AND DISCUSSION

Scaling Analysis. Consider a graft polymer consistingof a linear chain backbone of the degree of polymerization Nbbwith grafted side chains of the degree of polymerization nsc

Received: February 17, 2017Revised: March 31, 2017Published: April 11, 2017

Article

pubs.acs.org/Macromolecules

© 2017 American Chemical Society 3430 DOI: 10.1021/acs.macromol.7b00364Macromolecules 2017, 50, 3430−3437

(Figure 1). The side chains are equally spaced with ng bondsbetween two neighboring side chains along the polymer back-bone. Here we assume that both the backbone and side chainmonomers are of the same type with the excluded volume v,bond length, l, and Kuhn length b.As shown in Figure 2a, each macromolecule occupies a

pervaded volume V, which includes Nbb(nsc/ng + 1) monomers

of its own and potentially monomers of the neighboringmacromolecules. Herein, our classification of graft polymers ascomb-like and bottlebrush macromolecules is based on the extentof mutual interpenetration (overlap) of neighboring molecules.To quantify the degree of interpenetration and establish how itdepends on the molecular architecture, we calculate the volumefraction of monomers of a test macromolecule within its ownpervaded volume. At low grafting density, both the side chains andbackbone display statistics of a random walk, whereby the sidechain size is described by Rsc ≈ (blnsc)

1/2 (assuming flexible sidechains with nsc ≥ b/l). Considering a test macromolecule as a chainof blobs with size Rsc each containing nsc backbone bonds thepervaded volume is estimated as V ≈ NbbRsc

3/nsc. Now, we candefine a crowding parameter, Φ, as a volume fraction of themonomers of test macromolecule within a pervaded volume

Φ ≡ ≈+

≈+

≥

VV

N n n v

N R nv

lb

n n

n

n b l

( / 1)

/ ( )

/ 1,

for flexible side chains /

gm bb sc g

bb sc3

sc3/2

sc

sc1/2

sc (1)

Following the same arguments, we can calculate volume frac-tion of monomers belonging to a test macromolecule withrod-like side chains, nsc < b/l. In this case Rsc ≈ lnsc, and eq 1transforms to

Φ ≈+

‐ <vl

n n

nn b l

/ 1, for rod like side chains /

g3

sc

sc2 sc

(2)

If Φ < 1, monomers of a test macromolecule occupy only afraction of its pervaded volume. To maintain a constantmonomer number density in a melt (ρ ≈ ν−1), the pervadedvolume of a test macromolecule “hosts” monomers from neigh-boring macromolecules (Figure 2b). As the crowding parameterΦ increases (with increasing nsc and/or grafting density ng

−1),the guest monomeric units are pushed out of the pervadedvolume. At Φ ≈ 1, the pervaded volume is occupied only bythe monomers belonging to a test macromolecule. This distinc-tion between systems possessing different crowding parametersdefines our classification of the comb-like (Φ < 1) andbottlebrush (Φ > 1) polymers. Note that the value of theparameter Φ > 1 corresponds to a hypothetical system, wherebottlebrush macromolecules maintain ideal conformations ofside chains and backbone even at infinitely (unreasonably)large grafting density. In real systems, however, in the range ofsystem parameters with Φ > 1 the backbone and side chainswill stretch to maintain the melt density (ρ ≈ ν−1). Thecrossover between combs and bottlebrushes can be defined bysetting Φ ≈ 1 and solving eqs 1 and 2 for a compositionparameter

φ =+n

n ng

g sc (3)

which describes partitioning of monomers between a side chainand backbone spacer between two neighboring side chains (i.e.,“dilution” of the backbone). Note that the selection of nsc and φas variables for the diagram of state is more useful than nsc andng in the previous representations,11,37 as it provides moredistinct deconvolution of the inherent cross-correlationsbetween the molecular dimensions and the architecturalparameters of graft polymer as explained below. After somealgebra, the crossover condition separating comb-like polymersfrom bottlebrushes is written as

φ ≈≥

<− −

⎪⎪⎧⎨⎩

vbl n n b l

l n n b l

( ) , for /

, for /1 1

3/2sc

1/2sc

3sc

2sc (4)

Figure 3 summarizes different regimes of graft polymer in nscand φ−1 coordinates. In the comb regime, the backbones can beconsidered as unperturbed ideal chains with the mean squareend-to-end distance of the comb backbone to be equal to

⟨ ⟩ ≈R N lbe,bb2

bb (5)

and the effective Kuhn length defined as

≡⟨ ⟩

≈bR

N lbK

e,bb2

bb (6)

In the interval of parameters, for which Φ > 1 (bottlebrushregime in Figure 3), the backbone stretches to decrease thenumber of the side chains within the volume Rsc

3 and thus keepthe constant monomer density ρ ≈ ν−1. From the packing

Figure 1. A graft polymer in a melt (a), single chain (b), and definitionof structural parameters ng and nsc (c). Backbone monomers arecolored in orange, and side chain monomers are shown in blue.

Figure 2. Schematic representation of graft polymers as chains of blobsof size Rsc (a). Side chains and backbone of the test macromolecule areshown in red, and surrounding macromolecules are colored in gray.(b) Conformations of graft polymers and the overlap between chainswithin the pervaded volume with size equal to that of the side chains,Rsc, in different regimes.

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condition (ρν ≈ 1), the number of backbone monomers nRwithin the volume Rsc

3 decreases with increasing Φ as

+≈ Φ ≈

n n n v

Rnn

( / 1)1

R sc g

sc3

R

sc (7)

On the length scales larger than the side chain size, a bottlebrushcan be considered as a flexible chain of blobs each of size Rsc

⟨ ⟩ ≈ ≈ Φ ≥R RNn

lbN n b l, for /e,bb2

sc2 bb

Rbb sc

(8)

The effective Kuhn length of the bottlebrushes in this regime isdefined as

≡⟨ ⟩

≈ Φ ≥bR

lNb n b l, for /K

e,bb2

bbsc

(9)

In Figure 3, this regime is designated as stretched backbone(SBB) regime to emphasize stretching of the backbone inside acylindrical envelope of a bottlebrush macromolecule. Eventually,the section of the backbone with nR monomers becomes fullyextended when nRl ≈ Rsc. This determines an upper boundary forthe SBB regime as

φ ≈ ≥− blv

n n b l, for /12

sc sc (10)

Above this line, the side chains begin to stretch to satisfy theconstant density condition ρν ≈ 1. Correspondingly, wedesignate this regime as the stretched side chains (SSC) regime.The packing condition in this regime is given by

φ

+≈ ≈ ≥

R n n v

lRv

lRn b l

( / 1)1, for /

sc sc g

sc3

sc2 sc

(11)

which can be solved for size of the side chains as

φ≈ ≥R

vl

n b l, for /sc sc(12)

Using eq 12, the mean-square end-to-end distance of thebottlebrush can be written as

φ⟨ ⟩ ≈ ≈ ≥R RlNR

N vl n b l/ , for /e,bb2

sc2 bb

scbb sc

(13)

The effective Kuhn length of bottlebrushes in the SSC regime is

φ≈ ≈ ≥b R

vl

n b l, for /K sc sc(14)

The side chains become fully stretched when Rsc ≈ lnsc. Thishappens for

φ ≈ ≥− lv

n n b l, for /13

sc2

sc (15)

Above this line both side chains and backbone are fully stretchedon the length scales smaller than the side chain size Rsc. Thebottlebrush backbone remains flexible on the length scales largerthan the side chain size, resulting in the following expression forthe mean-square end-to-end distance

⟨ ⟩ ≈ ≈R RlNR

l n Ne,bb2

sc2 bb

sc

2sc bb

(16)

The effective Kuhn length in this regime is equal to bK ≈ lnsc. Wecall this regime the rod-like side chain (RSC) regime in thebottlebrush region of the diagram of states in Figure 3. Theneffect of the Kuhn length b on the transformation of the diagramof states is discussed in the Supporting Information.Note that our analysis of the properties of graft polymers

in the RSC regime should be applicable to graft polymerswith the rod-like side chains (nsc < b/l).30,43,44 In this case,eq 15 describes the crossover between combs and bottle-brushes, which is located in the bottom-left corner of thediagram in Figure 3. Unlike systems with flexible side chains,the backbone in a comb-like macromolecule in the crossoverregion (Φ ≈ 1) is already almost fully stretched. Therefore,with increasing grafting density (increasing crowding parameterΦ), the Comb regime is directly followed by the RSC regime,i.e., extension of the side chains.

Comparison with Simulations. To verify predictions ofthe scaling model, we performed molecular dynamics simula-tions45 of graft polymers in a melt using the LAMMPS simula-tion package.46 Macromolecule backbones and side chains aremodeled as bead−spring chains composed of beads withdiameter σ interacting through truncated shifted Lennard-Jones(LJ) potential.47 The connectivity of monomers into graftpolymers is maintained by the combination of the FENE andtruncated shifted LJ potentials. We performed simulations ofmacromolecules with FENE potential spring constants equal to30 kBT/σ

2 and 500 kBT/σ2, where kB is the Boltzmann constant

and T is the absolute temperature. The set of structuralparameters for studied systems is summarized in Table 1. In thecase of ng = 0.5, two side chains are grafted to each backbonemonomer. For all studied systems, the monomer density isset to ρσ3 = 0.8. The simulation details are described in theSupporting Information.The effective Kuhn length of graft polymers in a melt is

obtained from bond−bond correlation function of the back-bone bonds. This function describes the decay of the orienta-tional correlations between two unit bond vectors ni and +ni spointing along backbone bonds and separated by s bonds and isdefined as

∑=−

⟨ · ⟩=

−

+G sn s

n n( )1

i

n s

i i sbb 1

bb

(17)

where nbb = Nbb − 1 is the number of bonds in the backboneand the brackets ⟨...⟩ denote averaging over backbone

Figure 3. Diagram of states of graft polymers in a melt. SBB −stretched backbone regime, SSC − stretched side chain regime, andRSC − rod-like side chain regime. Logarithmic scales.

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configurations. To avoid chain end effects, we neglected 20bonds on each backbone end when calculating the bond−bondcorrelation function. Figure 4 illustrates a typical bond−bond

correlation function obtained in our simulations. Here wefollow the approach developed in refs 34 and 48−50 to analyzethe bond−bond correlation function. In the framework of thisapproach the simulation data are fitted by the double-exponential function of the following form

αλ

αλ

= − − | | + − | |⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟G s

s s( ) (1 ) exp exp

1 2 (18)

where α, λ1, and λ2 are fitting parameters. The existence of thetwo different correlation lengths λ1 and λ2 is the evidence oftwo different mechanisms of chain deformation. At short lengthscales, the decay of the orientational correlation is due to localchain tension, while at long length scales, it is a result ofinteractions between neighboring side chains. Note that at evenlonger length scales the bond−bond correlation functiondeviates from the double-exponential function and shouldhave a power law decay.38,42,51 For this reason, the bond−bondcorrelation function was only fitted in the range 0 ≤ |s| ≤ 20.

With the given bond−bond correlation function, the mean-square end-to-end distance of a backbone section with s bondscan be written as

∑ ∑

α λ α λ

⟨ ⟩ =− +

⟨ ⟩

= − +

=

− +

=

+ −

R sn s

l n

l g s g s

( )1

1( )

((1 ) ( , ) ( , ))

i

n s

j i

i s

je2

bb 1

12

12

21 2

bb

(19)

where l is the bond length and function g(λ,s) is defined as

λ = +−

− −−

λ

λλ

λ

λ

−

−−

−

−g s s( , )1 e1 e

2e1 e

(1 e )

s1/

1/1/

/

1/ 2(20)

Therefore, the effective Kuhn length bK of the comb or bottle-brush macromolecules can be calculated from fitting parametersof the bond−bond correlation function as

α λ α λ=⟨ ⟩

= − +→∞

bR s

sll h h

( )((1 ) ( ) ( ))

sK

e2

1 2(21)

where we introduced function h(λ)

λ = +−

λ

λ

−

−h( )1 e1 e

1/

1/ (22)

Figure 5 combines our simulation results for the effectiveKuhn length obtained using eq 21 and values of the parameters

α, λ1, and λ2 from fitting of the bond−bond correlation function.In accordance with the scaling model predictions (see eqs 6 and9), we have plotted reduced Kuhn length bK/b as a function ofthe crowding parameter Φ. The value of the Kuhn length b forthis plot was obtained from analysis of the simulation data for thebond−bond correlation function of the linear chains. The valueof the bond length is equal to l = 0.985σ for 30 kBT/σ

2 and0.837σ for 500 kBT/σ

2. The monomer excluded volume isestimated from the monomer density ν = ρ−1. As expected, allour simulation data have collapsed into one universal plot. In thecomb regime the effective Kuhn length saturates at b. Withincreasing the crowding parameter,Φ, the interaction between sidechains results in stiffening of macromolecules which is manifestedin increase of the effective Kuhn length bK. In the range ofcrowding parameter Φ > 1 we recover a scaling dependencefor the effective Kuhn length of the bottlebrush, bK ≈ bΦ

Table 1. Summary of Studied Systems (Data Sets withNbb = 100 Are from Ref 34)

Figure 4. Typical bond−bond correlation functions of the backbonebonds for graft polymers with kspring = 30 kBT/σ

2, ng = 4, and thedegree of polymerization of the side chains nsc = 2 (red circles),4 (green triangles), 8 (blue inverted triangles), 16 (magenta diamond),and 32 (cyan pentagons). Solid lines show the best fit curves using adouble-exponential function (eq 18). Figure 5. Dependence of the normalized Kuhn length, bK/b, of the

graft polymers on the crowding parameter Φ (see eq 1). Thin solidlines show scaling predictions in comb and bottlebrush regimes.Symbol notations are summarized in Table 1.

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(see eq 8). Note that the universal scaling relation bK ≈ bΦindicates that all our simulation data in the bottlebrushregime could belong to the stretched backbone (SBB) regimein Figure 3.Figure 6a shows simulation results for the mean-square end-

to-end distance of the section of the backbone with n bonds,⟨Re,bb

2 (n)⟩. The stronger than linear growth of the mean-squareaverage end-to-end size of the section of the backbone withnumber of bonds n < 10 indicates that these sections of thebackbone are stretched. Also for this backbone section, thesection size is independent of the degree of polymerization ofthe side chains and is controlled by the local packing conditionof beads. However, for larger backbone sections approachingthe degree of polymerization of the full backbone ⟨Re,bb

2 (n)⟩follows linear scaling dependence. This is exactly what onewould expect for a linear chain with the effective Kuhn lengthbK. This is confirmed in Figure 6b showing collapse of the datawhen ⟨Re,bb

2 (n)⟩ is normalized by square of the effective Kuhnlength, bK

2. Note that broadening of the crossover and dataspreading in the crossover region in Figure 6b is a manifestationof the multiscale nature of the bond−bond correlation function(see eq 18 and Figure 4).Figure 7 shows our data set for normalized size of the graft

polymer backbone as a function of the crowding parameter,Φ, in both comb and bottlebrush regimes of the diagram ofstates shown in Figure 3. In the comb regime corresponding tointerval of crowding parameter Φ < 1 the data points saturatesindicating that statistics of the graft polymer backbone is thatof a linear chain. However, in the interval of the crowdingparameter Φ > 1, there is an increase of normalized graftpolymer size with crowding parameter, ⟨Re,bb

2 ⟩1/2 ∝ Φ1/2. Thisbehavior is in agreement with prediction of the scaling model(see eq 8). Note that the plots shown in Figures 5 and 7 looksimilar. This points out on the fact that the statistics of the graftpolymer backbone is controlled by the local monomer packingand interactions between side chains.Figure 8a confirms that for almost all our systems the side

chains maintain their ideal chain conformations, and these datasets correspond to comb and stretched backbone regimes. Notethat for bottlebrush systems with two side chains grafted toeach backbone monomer the crowding of the monomers forces

stretching of the side chains to maintain the monomer volumefraction as shown in Figure 8b. Thus, these bottlebrushesbelong to the stretched side chain regime. Note that the sidechains are nonuniformly stretched with stretching firstincreasing for short sections of the side chains and then beginsto decrease as the number of bonds in the side chains increasesfurther. This classification of studied systems is furthercorroborated by diagram of states shown in Figure 9. Thecrossover line from comb to bottlebrush regime is calculatedby setting the value of the crowding parameter at crossoverto Φ = 0.7 (see Figures 5 and 7). Note that it would be diffi-cult to separate different bottlebrush regimes just looking atthe dependence of the effective Kuhn length or the mean-square end-to-end distance of the backbone as a function of thecrowding parameter Φ. In both SBB and SSC regimes the sidechain size has an identical scaling dependence, Rsc ∝ nsc

1/2, forlarge nsc. This explains a good collapse of the data shown inFigures 5 and 7 even though data sets for ng = 0.5 are in SSCregime as follows from Figure 9.

Figure 6. (a) Dependence of the mean-square end-to-end distance of the section of the graft polymer backbone with n bonds on the number ofbonds in a section, for macromolecules with ng = 4, values of the FENE potential spring constants kspring = 30kBT/σ

2 (filled symbols) and 500kBT/σ2

(open symbols), and degree of polymerization of the side chains nsc = 2 (red circles), 4 (green triangles), 8 (blue inverted triangles), 16 (magentadiamond), and 32 (cyan pentagons). The solid and dashed lines in this figure correspond to eq 19. (b) Dependence of the normalized mean-squareend-to-end distance of the section of the graft polymer backbone with n bonds on the number of Kuhn segments in such sections. Symbol notationsare summarized in Table 1.

Figure 7. Dependence of the normalized mean-square end-to-enddistance, ⟨Re,bb

2 ⟩1/2/⟨Re,02 ⟩1/2, of graft polymers as a function of the

crowding parameter Φ. Normalization factor ⟨Re,02 ⟩ is the mean-square

end-to-end distance of the linear chain with the same degree ofpolymerization as a backbone. Thin solid lines show scalingpredictions in comb and bottlebrush regimes. Symbol notations aresummarized in Table 1.

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■ CONCLUSIONS

We demonstrate that classification of graft polymers in amelt could be done according to the crowding parameterdescribing mutual interpenetration between different macro-molecules (see eqs 1 and 2). For values of the crowdingparameter Φ < 1, the side chains and backbones of graftpolymers interpenetrate and remain in their unperturbed idealchain conformations. This regime is called the comb regime inthe diagram of states shown in Figure 3. With increasing thevalue of the crowding parameter Φ the degree of inter-penetration between macromolecules decreases. At Φ > 1 themonomers belonging to surrounding macromolecules areexpelled from a pervaded volume of a test macromolecule (seeFigure 2) such that it could be considered as a filament withthe effective size equal to Rsc. This regime is referred to as thebottlebrush regime in Figure 3. It is important to point out thatin the interval of crowding parameter Φ > 1 in order to maintaina constant melt density backbone or side chains should undergostretching. The different modes of bottlebrush deformation aremanifested in appearance of the different subregimes in thebottlebrush region of the diagram of states as shown in Figure 3.

Our analysis of the different conformation regimes isconfirmed by molecular dynamics simulations covering bothcomb and bottlebrush regimes. The simulation data for theeffective Kuhn length obtained from analysis of the bond−bondcorrelation function are shown to be a universal function ofthe crowding parameter as illustrated in Figure 5. In the combregime the effective Kuhn length is that of a linear chain,bK ≈ b. With increasing the value of the crowding parameter Φ,the interaction between side chains induces backbone stiffeningresulting in linear increase of the effective Kuhn length with thecrowding parameter, bK ∝ Φ, in the bottlebrush regime. Similaruniversal behavior is observed for mean-square end-to-enddistance of the backbone across studied interval of crowdingparameter (see Figure 7).The analysis of different conformation regimes presented

here could be extended to describe properties of macro-molecules with grafted side chains in solutions.22,29,30,42 In thiscase one should use thermal blobs as new effective monomers.The detailed analysis of this situation will be a subject of futurestudy.

Figure 8. (a) Dependence of the normalized mean-square end-to-end distance of the section of the side chain with n bonds on the numberof Kuhn segments in such sections. Solid lines show simulation results for linear polymer chains with FENE potential spring constantkspring = 30 kBT/σ

2 (red) and 500 kBT/σ2 (blue) in a melt. (b) Normalized mean-square end-to-end distance of the section of the side chains with

n bonds for graft polymers. Normalization factor ⟨Re,02 (n)⟩ is the mean-square end-to-end distance of the section of the linear chain staring from the

point nsc from the chain end with n bonds in it and its end point locating between nsc and linear chain end. Symbol notations are summarized inTable 1.

Figure 9. Diagram of states of graft polymers with values of the FENE potential spring constants kspring = 30 kBT/σ2 (a) and kspring = 500 kBT/σ

2

(b) in a melt. SBB = stretched backbone regime, SSC = stretched side chain regime, and RSC = rod-like side chain regime. Intersect point of thecrossover lines between different graft polymer regimes is set at nsc = 1.0. Symbol notations are summarized in Table 1.

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■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on the ACSPublications website at DOI: 10.1021/acs.macromol.7b00364.

Examples of the diagram of states of graft polymers andsimulation details (PDF)

■ AUTHOR INFORMATIONCorresponding Author*E-mail adobrynin@uakron.edu (A.V.D.).ORCIDHeyi Liang: 0000-0002-8308-3547Zilu Wang: 0000-0002-5957-8064Sergei S. Sheiko: 0000-0003-3672-1611Andrey V. Dobrynin: 0000-0002-6484-7409NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThe authors are grateful to the National Science Foundation forthe financial support under Grants DMR-1409710, DMR-1407645, DMR-1624569, and DMR-1436201.

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