Research ArticlePHYSICAL SCIENCES

Molecular structure of bottlebrush polymers in melts

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Science Advances  11 Nov 2016:
Vol. 2, no. 11, e1601478
DOI: 10.1126/sciadv.1601478
  • Fig. 1 Molecular architecture and conformation of a bottlebrush polymer.

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

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

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

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

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

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

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

  • Fig. 5 Geometry of a bottlebrush polymer.

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

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

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

  • Fig. 7 Size of bottlebrushes in a melt.

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

  • Fig. 8 Extension of a bottlebrush in a melt.

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

  • Fig. 9 Interpenetration of bottlebrushes in a melt.

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

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

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

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

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

Supplementary Materials

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

    The size of a bottlebrush side chains in a melt

    Persistence length of a bottlebrush in a melt

    The size of a bottlebrush in a melt

    Bottlebrush melt preparation

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

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

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

    table S4. Parameters Formula and Formula describing the sizes of backbone sections.

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

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

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

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

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

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

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

  • Supplementary Materials

    This PDF file includes:

    • The size of a bottlebrush side chains in a melt
    • Persistence length of a bottlebrush in a melt
    • The size of a bottlebrush in a melt
    • Bottlebrush melt preparation
    • table S1. Summary of the adjustable parameters sc Csc and Ñsc describing the mean square size of side chains.
    • table S2. Summary of system parameters for simulations of bottlebrush melts and linear chain melts.
    • table S3. Summary of the adjustable parameters A, sf, sp, and ς for the bond angle correlation function.
    • table S4. Parameters Cbb and describing the sizes of backbone sections.
    • fig. S1. The size of side chains of a bottlebrush in a melt.
    • fig. S2. The bond angle correlation functions g(s).
    • fig. S3. Persistent segments of bottlebrushes in a melt.
    • fig. S4. The size of backbones for bottlebrushes in a melt.
    • fig. S5. Distribution of sizes of bottlebrushes in a melt.
    • fig. S6. A scheme demonstrating sample preparation of a bottlebrush melt.
    • fig. S7. Snapshots displaying conformations of bottlebrush molecules in a melt state.

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