Combs and Bottlebrushes in a Melt - UAkron Blog

Apr 11, 2017 - length and chain size of graft polymers in a melt state are universal functions .... considered as unperturbed ideal chains with the me...
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Combs and Bottlebrushes in a Melt Heyi 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-grained molecular dynamics simulations and scaling analysis to study conformations of bottlebrush and comb-like polymers in a melt. Our analysis shows that a crossover between comb and bottlebrush regimes is controlled by the crowding parameter, Φ, describing overlap between neighboring macromolecules. In comb-like systems characterized by a sparse grafting of side chains (Φ < 1), the side chains and backbones belonging to neighboring macromolecules interpenetrate. However, in bottlebrushes with densely grafted side chains (Φ ≥ 1), the interpenetration between macromolecules is suppressed by steric repulsion between side chains. In this regime, bottlebrush macromolecules can be viewed as filaments with diameter proportional to size of the side chains. For flexible side chains, the crowding parameter is given by Φ ≈ [v/(lb)3/2][(nsc/ng + 1)/nsc1/2], which depends on both the architectural parameters (degree of polymerization of the side chains, nsc, and number of backbone bonds between side chains, ng) and chemical structure of monomers (bond length l, monomer excluded volume v, and Kuhn length, b). Molecular dynamics simulations corroborate this classification of graft polymers and show that the effective macromolecule Kuhn length, bK, and the mean-square end-to-end distance of the backbone, ⟨R2e,bb⟩, are universal functions of the crowding parameter Φ for all studied systems.

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INTRODUCTION Graft polymers consisting of linear polymer backbones with grafted side chains are called either combs or bottlebrushes depending on grafting density of the side chains.1−4 The brushlike architecture allows for efficient control over materials’ properties through independent variation of the side chain length and their grafting density.5−16 In a bottlebrush melt, for example, side chains suppress the entanglement threshold and decrease the melt viscosity, making such polymers easier to process.8−11 The elimination of entanglements also opens a possibility for the design of supersoft and superelastic materials11−14 with modulus as low as 100 Pa and tensile strain at break up to 800% in the solvent-free states.11 The unique combination of the elastic softness and inherent strain hardening of graft polymers was utilized in the design of dielectric elastomers for free-standing electroactuation under low applied fields.15 In parallel, synthesis of graft block copolymers created a new class of thermoplastic materials with well-controlled mechanical and optical properties.17−21 Despite substantial experimental, 5−16,22−30 theoretical,11,16,29,31−34 and computational16,28,29,34−43 efforts to establish accurate correlations between the brush architecture and physical properties, the complete solution of this problem still remains elusive. This is in part due to the large number of structural (chemical and architectural) parameters describing brush-like molecular architecture, which make detailed mapping of structure−property relationships for these materials extremely difficult. Here we use a combination of the scaling analysis and coarse-grained simulations to provide general © 2017 American Chemical Society

frameworks for classification of graft polymers into comb and bottlebrush classes that exhibit distinct conformational and physical properties. Specifically, we demonstrate that the Kuhn length and chain size of graft polymers in a melt state are universal functions of the crowding parameter, describing interpenetration between side chains and backbones belonging to different macromolecules. We also outline a diagram of states in terms of two independently controlled parameters: degree of polymerization of side chains nsc and molar fraction of the backbone monomers φ, which describes partitioning of monomers between 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 graft polymer melts in terms of nsc and φ, (ii) the scaling model predictions are compared with results of the molecular dynamics simulations for effective Kuhn length, backbone, and side chain dimensions of graft polymers, and (iii) we show that our simulations data can be collapsed into universal plots as predicted by the scaling model.

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RESULTS AND DISCUSSION Scaling Analysis. Consider a graft polymer consisting of a linear chain backbone of the degree of polymerization Nbb with grafted side chains of the degree of polymerization nsc Received: February 17, 2017 Revised: March 31, 2017 Published: April 11, 2017 3430

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

v nsc /ng + 1 , l3 nsc 2

for rod‐like side chains nsc < b /l (2)

If Φ < 1, monomers of a test macromolecule occupy only a fraction of its pervaded volume. To maintain a constant monomer number density in a melt (ρ ≈ ν−1), the pervaded volume of a test macromolecule “hosts” monomers from neighboring macromolecules (Figure 2b). As the crowding parameter Φ increases (with increasing nsc and/or grafting density ng−1), the guest monomeric units are pushed out of the pervaded volume. At Φ ≈ 1, the pervaded volume is occupied only by the monomers belonging to a test macromolecule. This distinction between systems possessing different crowding parameters defines our classification of the comb-like (Φ < 1) and bottlebrush (Φ > 1) polymers. Note that the value of the parameter Φ > 1 corresponds to a hypothetical system, where bottlebrush macromolecules maintain ideal conformations of side chains and backbone even at infinitely (unreasonably) large grafting density. In real systems, however, in the range of system parameters with Φ > 1 the backbone and side chains will stretch to maintain the melt density (ρ ≈ ν−1). The crossover between combs and bottlebrushes can be defined by setting Φ ≈ 1 and solving eqs 1 and 2 for a composition parameter ng φ= ng + nsc (3)

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

(Figure 1). The side chains are equally spaced with ng bonds between two neighboring side chains along the polymer backbone. Here we assume that both the backbone and side chain monomers 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

which describes partitioning of monomers between a side chain and backbone spacer between two neighboring side chains (i.e., “dilution” of the backbone). Note that the selection of nsc and φ as variables for the diagram of state is more useful than nsc and ng in the previous representations,11,37 as it provides more distinct deconvolution of the inherent cross-correlations between the molecular dimensions and the architectural parameters of graft polymer as explained below. After some algebra, the crossover condition separating comb-like polymers from bottlebrushes is written as

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

of its own and potentially monomers of the neighboring macromolecules. Herein, our classification of graft polymers as comb-like and bottlebrush macromolecules is based on the extent of mutual interpenetration (overlap) of neighboring molecules. To quantify the degree of interpenetration and establish how it depends on the molecular architecture, we calculate the volume fraction of monomers of a test macromolecule within its own pervaded volume. At low grafting density, both the side chains and backbone display statistics of a random walk, whereby the side chain size is described by Rsc ≈ (blnsc)1/2 (assuming flexible side chains with nsc ≥ b/l). Considering a test macromolecule as a chain of blobs with size Rsc each containing nsc backbone bonds the pervaded volume is estimated as V ≈ NbbRsc3/nsc. Now, we can define a crowding parameter, Φ, as a volume fraction of the monomers of test macromolecule within a pervaded volume

3/2 1/2 ⎧ nsc , for nsc ≥ b/l ⎪(bl) φ−1 ≈ v−1⎨ ⎪ 3 2 for nsc < b/l ⎩ l nsc ,

Figure 3 summarizes different regimes of graft polymer in nsc and φ−1 coordinates. 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 to be equal to 2 ⟨R e,bb ⟩ ≈ Nbblb

(5)

and the effective Kuhn length defined as bK ≡

Nbb(nsc /ng + 1)v V v nsc /ng + 1 Φ≡ m ≈ ≈ , 3 V NbbR sc /nsc (lb)3/2 nsc1/2 for flexible side chains nsc ≥ b /l

(4)

2 ⟨R e,bb ⟩

Nbbl

≈b

(6)

In the interval of parameters, for which Φ > 1 (bottlebrush regime in Figure 3), the backbone stretches to decrease the number of the side chains within the volume Rsc3 and thus keep the constant monomer density ρ ≈ ν−1. From the packing

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The effective Kuhn length of bottlebrushes in the SSC regime is bK ≈ R sc ≈

v , φl

for nsc ≥ b/l

(14)

The side chains become fully stretched when Rsc ≈ lnsc. This happens for φ −1 ≈

l3 2 nsc , v

for nsc ≥ b/l

(15)

Above this line both side chains and backbone are fully stretched on the length scales smaller than the side chain size Rsc. The bottlebrush backbone remains flexible on the length scales larger than the side chain size, resulting in the following expression for the mean-square end-to-end distance 2 ⟨R e,bb ⟩ ≈ R sc 2

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

R sc

3

nR Φ≈1 nsc

≈

(7)

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

Nbb ≈ lbNbbΦ, nR

for nsc ≥ b/l

(8)

The effective Kuhn length of the bottlebrushes in this regime is defined as bK ≡

2 ⟨R e,bb ⟩

≈ Φb ,

lNbb

for nsc ≥ b/l

(9)

In Figure 3, this regime is designated as stretched backbone (SBB) regime to emphasize stretching of the backbone inside a cylindrical envelope of a bottlebrush macromolecule. Eventually, the section of the backbone with nR monomers becomes fully extended when nRl ≈ Rsc. This determines an upper boundary for the SBB regime as bl 2 nsc , v

φ −1 ≈

for nsc ≥ b/l

(10)

Above this line, the side chains begin to stretch to satisfy the constant density condition ρν ≈ 1. Correspondingly, we designate this regime as the stretched side chains (SSC) regime. The packing condition in this regime is given by R sc(nsc /ng + 1)v lR sc 3

≈

v ≈ 1, φlR sc 2

for nsc ≥ b/l

(11)

which can be solved for size of the side chains as R sc ≈

v , φl

for nsc ≥ b/l

(12)

Using eq 12, the mean-square end-to-end distance of the bottlebrush can be written as 2 ⟨R e,bb ⟩

≈ R sc

2 lNbb

R sc

≈ Nbb vl /φ ,

for nsc ≥ b/l

(16)

The effective Kuhn length in this regime is equal to bK ≈ lnsc. We call this regime the rod-like side chain (RSC) regime in the bottlebrush region of the diagram of states in Figure 3. Then effect of the Kuhn length b on the transformation of the diagram of 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 polymers with the rod-like side chains (nsc < b/l).30,43,44 In this case, eq 15 describes the crossover between combs and bottlebrushes, which is located in the bottom-left corner of the diagram in Figure 3. Unlike systems with flexible side chains, the backbone in a comb-like macromolecule in the crossover region (Φ ≈ 1) is already almost fully stretched. Therefore, with increasing grafting density (increasing crowding parameter Φ), the Comb regime is directly followed by the RSC regime, i.e., extension of the side chains. Comparison with Simulations. To verify predictions of the scaling model, we performed molecular dynamics simulations45 of graft polymers in a melt using the LAMMPS simulation package.46 Macromolecule backbones and side chains are modeled as bead−spring chains composed of beads with diameter σ interacting through truncated shifted Lennard-Jones (LJ) potential.47 The connectivity of monomers into graft polymers is maintained by the combination of the FENE and truncated shifted LJ potentials. We performed simulations of macromolecules with FENE potential spring constants equal to 30 kBT/σ2 and 500 kBT/σ2, where kB is the Boltzmann constant and T is the absolute temperature. The set of structural parameters for studied systems is summarized in Table 1. In the case of ng = 0.5, two side chains are grafted to each backbone monomer. For all studied systems, the monomer density is set to ρσ3 = 0.8. The simulation details are described in the Supporting Information. The effective Kuhn length of graft polymers in a melt is obtained from bond−bond correlation function of the backbone bonds. This function describes the decay of the orientational correlations between two unit bond vectors ni⃗ and ni⃗ + s pointing along backbone bonds and separated by s bonds and is defined as

condition (ρν ≈ 1), the number of backbone monomers nR within the volume Rsc3 decreases with increasing Φ as nR (nsc /ng + 1)v

lNbb ≈ l 2nscNbb R sc

G (s ) =

1 nbb − s

nbb − s

∑ i=1

⟨ni⃗ ·ni⃗ + s⟩

(17)

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

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With the given bond−bond correlation function, the meansquare end-to-end distance of a backbone section with s bonds can be written as

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

⟨R e 2(s)⟩ =

nbb

1 −s+1

nbb − s + 1

∑

i+s−1

⟨l 2(

∑

i=1

nj⃗ )2 ⟩

j=i

2

= l ((1 − α)g (λ1 , s) + αg (λ 2 , s))

(19)

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

−s / λ 1 + e−1/ λ −1/ λ 1 − e − 2e 1 − e−1/ λ (1 − e−1/ λ)2

(20)

Therefore, the effective Kuhn length bK of the comb or bottlebrush macromolecules can be calculated from fitting parameters of the bond−bond correlation function as bK =

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

⟨R e 2(s)⟩ sl

= l((1 − α)h(λ1) + αh(λ 2)) s →∞

(21)

where we introduced function h(λ) h(λ) =

1 + e−1/ λ 1 − e−1/ λ

(22)

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

Figure 4. Typical bond−bond correlation functions of the backbone bonds for graft polymers with kspring = 30 kBT/σ2, ng = 4, and the degree 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 a double-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 solid lines show scaling predictions in comb and bottlebrush regimes. Symbol notations are summarized in Table 1.

correlation function obtained in our simulations. Here we follow the approach developed in refs 34 and 48−50 to analyze the bond−bond correlation function. In the framework of this approach the simulation data are fitted by the doubleexponential function of the following form ⎛ |s| ⎞ ⎛ |s| ⎞ G(s) = (1 − α) exp⎜ − ⎟ + α exp⎜ − ⎟ ⎝ λ2 ⎠ ⎝ λ1 ⎠

α, λ1, and λ2 from fitting of the bond−bond correlation function. In accordance with the scaling model predictions (see eqs 6 and 9), we have plotted reduced Kuhn length bK/b as a function of the crowding parameter Φ. The value of the Kuhn length b for this plot was obtained from analysis of the simulation data for the bond−bond correlation function of the linear chains. The value of the bond length is equal to l = 0.985σ for 30 kBT/σ2 and 0.837σ for 500 kBT/σ2. The monomer excluded volume is estimated from the monomer density ν = ρ−1. As expected, all our simulation data have collapsed into one universal plot. In the comb regime the effective Kuhn length saturates at b. With increasing the crowding parameter, Φ, the interaction between side chains results in stiffening of macromolecules which is manifested in increase of the effective Kuhn length bK. In the range of crowding parameter Φ > 1 we recover a scaling dependence for the effective Kuhn length of the bottlebrush, bK ≈ bΦ

(18)

where α, λ1, and λ2 are fitting parameters. The existence of the two different correlation lengths λ1 and λ2 is the evidence of two different mechanisms of chain deformation. At short length scales, the decay of the orientational correlation is due to local chain tension, while at long length scales, it is a result of interactions between neighboring side chains. Note that at even longer length scales the bond−bond correlation function deviates from the double-exponential function and should have a power law decay.38,42,51 For this reason, the bond−bond correlation function was only fitted in the range 0 ≤ |s| ≤ 20. 3433

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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 of bonds in a section, for macromolecules with ng = 4, values of the FENE potential spring constants kspring = 30kBT/σ2 (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 (magenta diamond), and 32 (cyan pentagons). The solid and dashed lines in this figure correspond to eq 19. (b) Dependence of the normalized mean-square end-to-end distance of the section of the graft polymer backbone with n bonds on the number of Kuhn segments in such sections. Symbol notations are summarized in Table 1.

(see eq 8). Note that the universal scaling relation bK ≈ bΦ indicates that all our simulation data in the bottlebrush regime could belong to the stretched backbone (SBB) regime in Figure 3. Figure 6a shows simulation results for the mean-square endto-end distance of the section of the backbone with n bonds, ⟨R2e,bb(n)⟩. The stronger than linear growth of the mean-square average end-to-end size of the section of the backbone with number of bonds n < 10 indicates that these sections of the backbone are stretched. Also for this backbone section, the section size is independent of the degree of polymerization of the side chains and is controlled by the local packing condition of beads. However, for larger backbone sections approaching the degree of polymerization of the full backbone ⟨R2e,bb(n)⟩ follows linear scaling dependence. This is exactly what one would expect for a linear chain with the effective Kuhn length bK. This is confirmed in Figure 6b showing collapse of the data when ⟨R2e,bb(n)⟩ is normalized by square of the effective Kuhn length, bK2. Note that broadening of the crossover and data spreading in the crossover region in Figure 6b is a manifestation of 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 of states shown in Figure 3. In the comb regime corresponding to interval of crowding parameter Φ < 1 the data points saturates indicating that statistics of the graft polymer backbone is that of a linear chain. However, in the interval of the crowding parameter Φ > 1, there is an increase of normalized graft polymer size with crowding parameter, ⟨R2e,bb⟩1/2 ∝ Φ1/2. This behavior is in agreement with prediction of the scaling model (see eq 8). Note that the plots shown in Figures 5 and 7 look similar. This points out on the fact that the statistics of the graft polymer backbone is controlled by the local monomer packing and interactions between side chains. Figure 8a confirms that for almost all our systems the side chains maintain their ideal chain conformations, and these data sets correspond to comb and stretched backbone regimes. Note that for bottlebrush systems with two side chains grafted to each backbone monomer the crowding of the monomers forces

Figure 7. Dependence of the normalized mean-square end-to-end distance, ⟨R2e,bb⟩1/2/⟨R2e,0⟩1/2, of graft polymers as a function of the crowding parameter Φ. Normalization factor ⟨R2e,0⟩ is the mean-square end-to-end distance of the linear chain with the same degree of polymerization as a backbone. Thin solid lines show scaling predictions in comb and bottlebrush regimes. Symbol notations are summarized in Table 1.

stretching of the side chains to maintain the monomer volume fraction as shown in Figure 8b. Thus, these bottlebrushes belong to the stretched side chain regime. Note that the side chains are nonuniformly stretched with stretching first increasing for short sections of the side chains and then begins to decrease as the number of bonds in the side chains increases further. This classification of studied systems is further corroborated by diagram of states shown in Figure 9. The crossover line from comb to bottlebrush regime is calculated by setting the value of the crowding parameter at crossover to Φ = 0.7 (see Figures 5 and 7). Note that it would be difficult to separate different bottlebrush regimes just looking at the dependence of the effective Kuhn length or the meansquare end-to-end distance of the backbone as a function of the crowding parameter Φ. In both SBB and SSC regimes the side chain size has an identical scaling dependence, Rsc ∝ nsc1/2, for large nsc. This explains a good collapse of the data shown in Figures 5 and 7 even though data sets for ng = 0.5 are in SSC regime as follows from Figure 9. 3434

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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 number of Kuhn segments in such sections. Solid lines show simulation results for linear polymer chains with FENE potential spring constant kspring = 30 kBT/σ2 (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 ⟨R2e,0 (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 in Table 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/σ2 (b) in a melt. SBB = stretched backbone regime, SSC = stretched side chain regime, and RSC = rod-like side chain regime. Intersect point of the crossover lines between different graft polymer regimes is set at nsc = 1.0. Symbol notations are summarized in Table 1.

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CONCLUSIONS We demonstrate that classification of graft polymers in a melt could be done according to the crowding parameter describing mutual interpenetration between different macromolecules (see eqs 1 and 2). For values of the crowding parameter Φ < 1, the side chains and backbones of graft polymers interpenetrate and remain in their unperturbed ideal chain conformations. This regime is called the comb regime in the diagram of states shown in Figure 3. With increasing the value of the crowding parameter Φ the degree of interpenetration between macromolecules decreases. At Φ > 1 the monomers belonging to surrounding macromolecules are expelled from a pervaded volume of a test macromolecule (see Figure 2) such that it could be considered as a filament with the effective size equal to Rsc. This regime is referred to as the bottlebrush regime in Figure 3. It is important to point out that in the interval of crowding parameter Φ > 1 in order to maintain a constant melt density backbone or side chains should undergo stretching. The different modes of bottlebrush deformation are manifested in appearance of the different subregimes in the bottlebrush region of the diagram of states as shown in Figure 3.

Our analysis of the different conformation regimes is confirmed by molecular dynamics simulations covering both comb and bottlebrush regimes. The simulation data for the effective Kuhn length obtained from analysis of the bond−bond correlation function are shown to be a universal function of the crowding parameter as illustrated in Figure 5. In the comb regime 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 stiffening resulting in linear increase of the effective Kuhn length with the crowding parameter, bK ∝ Φ, in the bottlebrush regime. Similar universal behavior is observed for mean-square end-to-end distance of the backbone across studied interval of crowding parameter (see Figure 7). The analysis of different conformation regimes presented here could be extended to describe properties of macromolecules with grafted side chains in solutions.22,29,30,42 In this case one should use thermal blobs as new effective monomers. The detailed analysis of this situation will be a subject of future study. 3435

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S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b00364. Examples of the diagram of states of graft polymers and simulation details (PDF)

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AUTHOR INFORMATION

Corresponding Author

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

Heyi Liang: 0000-0002-8308-3547 Zilu Wang: 0000-0002-5957-8064 Sergei S. Sheiko: 0000-0003-3672-1611 Andrey V. Dobrynin: 0000-0002-6484-7409 Notes

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-1409710, DMR1407645, DMR-1624569, and DMR-1436201.

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REFERENCES

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