^{1,2,3,4,a)}, Hsiao-Ping Hsu

^{1,b)}, Wolfgang Paul

^{5,c)}and Kurt Binder

^{1,d)}

### Abstract

By molecular dynamics simulation of a coarse-grained bead-spring-type model for a cylindrical molecular brush with a backbone chain of *N* _{ b } effective monomers to which with grafting density σ side chains with *N* effective monomers are tethered, several characteristic length scales are studied for variable solvent quality. Side chain lengths are in the range 5 ⩽ *N* ⩽ 40, backbone chain lengths are in the range 50 ⩽ *N* _{ b } ⩽ 200, and we perform a comparison to results for the bondfluctuation model on the simple cubic lattice (for which much longer chains are accessible, *N* _{ b } ⩽ 1027, and which corresponds to an athermal, very good, solvent). We obtain linear dimensions of the side chains and the backbone chain and discuss their *N*-dependence in terms of power laws and the associated effective exponents. We show that even at the theta point the side chains are considerably stretched, their linear dimension depending on the solvent quality only weakly. Effective persistence lengths are extracted both from the orientational correlations and from the backbone end-to-end distance; it is shown that different measures of the persistence length (which would all agree for Gaussian chains) are not mutually consistent with each other and depend distinctly both on *N* _{ b } and the solvent quality. A brief discussion of pertinent experiments is given.

P.E.T. thanks for financial support by the Austrian Science Foundation within the SFB ViCoM (Grant No. F41), and also the Max Planck Fellowship during the time he was in Mainz. H.-P.H. received funding from the Deutsche Forschungsgemeinschaft (DFG), Grant No. SFB 625/A3. We are grateful for extensive grants of computer time at the JUROPA under Project No. HMZ03 and SOFTCOMP computers at the Jülich Supercomputing Centre (JSC).

I. INTRODUCTION

II. MODEL AND SIMULATION METHOD

III. SIDE-CHAIN AND BACKBONE LINEAR DIMENSIONS AND ATTEMPTS TO EXTRACT “THE” PERSISTENCE LENGTH OF BOTTLE-BRUSH POLYMERS

IV. CONCLUSIONS

### Key Topics

- Solvents
- 53.0
- Polymers
- 52.0
- Chemical bonds
- 10.0
- Mean field theory
- 5.0
- Macromolecules
- 4.0

## Figures

Plot of ϕ(*t*) (see Eq. (6)) versus the time *t* in units of τ_{LJ}. Several cases are shown: σ = 0.5, *N* _{ b } = 100, *N* = 10, *T* = 3.0 (a); σ = 1.0, *N* _{ b } = 100, *N* = 20, *T* = 3.0 (b); and σ = 1.0, *N* _{ b } = 200, *N* = 40, *T* = 3.0 (c). The fluctuations of ϕ(*t*) change sign initially at *t* ≈ 1500τ_{LJ} (a), *t* ≈ 10000τ_{LJ}, and *t* ≈ 20000τ_{LJ} (c) as shown in the inserts.

Plot of ϕ(*t*) (see Eq. (6)) versus the time *t* in units of τ_{LJ}. Several cases are shown: σ = 0.5, *N* _{ b } = 100, *N* = 10, *T* = 3.0 (a); σ = 1.0, *N* _{ b } = 100, *N* = 20, *T* = 3.0 (b); and σ = 1.0, *N* _{ b } = 200, *N* = 40, *T* = 3.0 (c). The fluctuations of ϕ(*t*) change sign initially at *t* ≈ 1500τ_{LJ} (a), *t* ≈ 10000τ_{LJ}, and *t* ≈ 20000τ_{LJ} (c) as shown in the inserts.

Selected snapshot pictures of equilibrated configurations of bottle-brush polymers. Backbone monomers (when visible) are displayed in magenta (darker grey) color, side chain monomers in green (lighter grey). Cases shown refer to σ = 0.5, *N* _{ b } = 100, *N* = 10, *T* = 3.0 (a) and *T* = 4.0 (b), as well as σ = 1.0, *N* _{ b } = 100, *N* = 20, *T* = 3.0 (c) and *T* = 4.0 (d).

Selected snapshot pictures of equilibrated configurations of bottle-brush polymers. Backbone monomers (when visible) are displayed in magenta (darker grey) color, side chain monomers in green (lighter grey). Cases shown refer to σ = 0.5, *N* _{ b } = 100, *N* = 10, *T* = 3.0 (a) and *T* = 4.0 (b), as well as σ = 1.0, *N* _{ b } = 100, *N* = 20, *T* = 3.0 (c) and *T* = 4.0 (d).

Log-log plot of mean square radius of gyration of the side chains normalized by their chain length and by the square bond length between successive monomers, , versus side chain length *N*, for σ = 0.5 (a) and σ = 1.0 (b). Data for the bead-spring model at two temperatures (*T* = 3.0 and *T* = 4.0), as indicated in the figure are included, as well as three backbone chain lengths (*N* _{ b } = 50, 100, and 200, respectively). For comparison, also data for the athermal bond fluctuation model for comparable backbone chain length are included. Straight lines indicate effective exponents ν_{eff} ≈ 0.60 (*T* = 3.0) or ν_{eff} ≈ 0.63 (*T* = 4.0) in case (a), and ν_{eff} ≈ 0.60 (*T* = 3.0) or ν_{eff} ≈ 0.64 (*T* = 4.0) in case (b). For the bond fluctuation model under very good solvent conditions the slightly larger effective exponent (ν_{eff} ≈ 0.65 for σ = 0.5, and ν_{eff} ≈ 0.66 for σ = 1.0) than for the off-lattice model with *T* = 4.0 results.

Log-log plot of mean square radius of gyration of the side chains normalized by their chain length and by the square bond length between successive monomers, , versus side chain length *N*, for σ = 0.5 (a) and σ = 1.0 (b). Data for the bead-spring model at two temperatures (*T* = 3.0 and *T* = 4.0), as indicated in the figure are included, as well as three backbone chain lengths (*N* _{ b } = 50, 100, and 200, respectively). For comparison, also data for the athermal bond fluctuation model for comparable backbone chain length are included. Straight lines indicate effective exponents ν_{eff} ≈ 0.60 (*T* = 3.0) or ν_{eff} ≈ 0.63 (*T* = 4.0) in case (a), and ν_{eff} ≈ 0.60 (*T* = 3.0) or ν_{eff} ≈ 0.64 (*T* = 4.0) in case (b). For the bond fluctuation model under very good solvent conditions the slightly larger effective exponent (ν_{eff} ≈ 0.65 for σ = 0.5, and ν_{eff} ≈ 0.66 for σ = 1.0) than for the off-lattice model with *T* = 4.0 results.

Temperature dependence of the normalized mean square radius of gyration for four chain lengths (*N* = 5, 10, 20, and 40), and three backbone lengths (*N* _{ b } = 50, 100, and 200), respectively. All data are for the bead-spring model and σ = 1. Data for bottle-brush polymers with flexible backbone are connected by curves to guide the reader's eyes. Data are only shown for *T* = 3.0 and 4.0 for the rigid backbone case.

Temperature dependence of the normalized mean square radius of gyration for four chain lengths (*N* = 5, 10, 20, and 40), and three backbone lengths (*N* _{ b } = 50, 100, and 200), respectively. All data are for the bead-spring model and σ = 1. Data for bottle-brush polymers with flexible backbone are connected by curves to guide the reader's eyes. Data are only shown for *T* = 3.0 and 4.0 for the rigid backbone case.

Log-log plot of versus side chain length *N*, for the bond fluctuation model with σ = 0.5 (a) and σ = 1.0 (b) including different choices of *N* _{ b } as indicated.

Log-log plot of versus side chain length *N*, for the bond fluctuation model with σ = 0.5 (a) and σ = 1.0 (b) including different choices of *N* _{ b } as indicated.

Plot of the effective exponent ζ, extracted from the slope of the straight lines in Fig. 5, versus the backbone chain length *N* _{ b } for σ = 0.5 (a) and σ = 1.0. All data are for the athermal bond fluctuation model.

Plot of the effective exponent ζ, extracted from the slope of the straight lines in Fig. 5, versus the backbone chain length *N* _{ b } for σ = 0.5 (a) and σ = 1.0. All data are for the athermal bond fluctuation model.

Log-log plot of versus *N*. All data are for the bead-spring model with σ = 1.0 at both *T* = 4.0 (a) and *T* = 3.0 (b). Three backbone lengths are shown as indicated.

Log-log plot of versus *N*. All data are for the bead-spring model with σ = 1.0 at both *T* = 4.0 (a) and *T* = 3.0 (b). Three backbone lengths are shown as indicated.

Log-log plot of versus backbone chain length *N* _{ b }, for σ = 0.5 (a) (b) and σ = 1.0 (c) (d), including both *T* = 3.0 (a) (c) and *T* = 4.0 (b) (d), for the bead-spring model. Several side chain lengths *N* are included as indicated. Straight lines indicate a fit with effective exponents ν_{eff}.

Log-log plot of versus backbone chain length *N* _{ b }, for σ = 0.5 (a) (b) and σ = 1.0 (c) (d), including both *T* = 3.0 (a) (c) and *T* = 4.0 (b) (d), for the bead-spring model. Several side chain lengths *N* are included as indicated. Straight lines indicate a fit with effective exponents ν_{eff}.

Log-log plot of for the bond fluctuation model versus backbone chain length *N* _{ b }, for 67 ⩽ *N* _{ b } ⩽ 259, and several side chain lengths *N*, for σ = 0.5 (a) and σ = 1.0 (b). Part (c) shows the plot of rescaled radius of gyration of the bottle-brush polymers versus *N* _{ b } for σ = 1.0. The persistence length *l* _{ p, Rg } is determined by the values of plateau. Part (d) shows a crossover scaling plot, collapsing for *N* = 6, 12, 18, and 24, σ = 1.0, and all data for 67 ⩽ *N* _{ b } ⩽ 1027 on a master curve, that describes the crossover from rods to swollen coils . For this purpose, *N* _{ b } is rescaled with the blob diameter *s* _{blob}, which has been determined to be *s* _{blob} = 6, 10, 12, and 14 for *N* = 6, 12, 18, and 24, respectively.^{6,11}

Log-log plot of for the bond fluctuation model versus backbone chain length *N* _{ b }, for 67 ⩽ *N* _{ b } ⩽ 259, and several side chain lengths *N*, for σ = 0.5 (a) and σ = 1.0 (b). Part (c) shows the plot of rescaled radius of gyration of the bottle-brush polymers versus *N* _{ b } for σ = 1.0. The persistence length *l* _{ p, Rg } is determined by the values of plateau. Part (d) shows a crossover scaling plot, collapsing for *N* = 6, 12, 18, and 24, σ = 1.0, and all data for 67 ⩽ *N* _{ b } ⩽ 1027 on a master curve, that describes the crossover from rods to swollen coils . For this purpose, *N* _{ b } is rescaled with the blob diameter *s* _{blob}, which has been determined to be *s* _{blob} = 6, 10, 12, and 14 for *N* = 6, 12, 18, and 24, respectively.^{6,11}

Semi-log plot of 〈cos θ(*s*)〉 vs. *s* for the bead-spring model with σ = 0.5. Data are chosen for *N* _{ b } = 50 (a) (b) and *N* _{ b } = 200 (c) (d) at *T* = 3.0 (a) (c) and *T* = 4.0 (b) (d), respectively. Several choices of *N* are shown as indicated. Straight lines illustrate fits to a exp (− *bs*), with constants *a*, *b* quoted in the figure.

Semi-log plot of 〈cos θ(*s*)〉 vs. *s* for the bead-spring model with σ = 0.5. Data are chosen for *N* _{ b } = 50 (a) (b) and *N* _{ b } = 200 (c) (d) at *T* = 3.0 (a) (c) and *T* = 4.0 (b) (d), respectively. Several choices of *N* are shown as indicated. Straight lines illustrate fits to a exp (− *bs*), with constants *a*, *b* quoted in the figure.

Semi-log plot of 〈cos θ(*s*)〉 vs. *s* for the bead-spring model with σ = 1.0 and *N* _{ b } = 100 at the temperatures *T* = 3.0 (a) and *T* = 4.0 (b), respectively. Several choices of *N* are shown as indicated. Straight lines illustrate fits to a exp (− *bs*), with constants *a*, *b* as quoted in the figure.

Semi-log plot of 〈cos θ(*s*)〉 vs. *s* for the bead-spring model with σ = 1.0 and *N* _{ b } = 100 at the temperatures *T* = 3.0 (a) and *T* = 4.0 (b), respectively. Several choices of *N* are shown as indicated. Straight lines illustrate fits to a exp (− *bs*), with constants *a*, *b* as quoted in the figure.

Log-log plot of the effective persistence length as a function of side chain length *N* for *T* = 3.0 (a) (c) and *T* = 4.0 (b) (d) for several choices of *N* _{ b }. Results for σ = 1.0 and σ = 0.5 are shown as indicated and the exponent estimates ζ extracted from the slope are listed. Values of are also listed in Table I.

Log-log plot of the effective persistence length as a function of side chain length *N* for *T* = 3.0 (a) (c) and *T* = 4.0 (b) (d) for several choices of *N* _{ b }. Results for σ = 1.0 and σ = 0.5 are shown as indicated and the exponent estimates ζ extracted from the slope are listed. Values of are also listed in Table I.

Plot of the function ν_{eff}(*L*/*l* _{ p }) versus *L*/*l* _{ p } as predicted from the Kratky-Porod model, Eqs. (16) and (17) (full curves). The numbers for *L*/*l* _{ p } extracted for σ = 0.5 (a) and σ = 1.0 (b) are quoted in the figure for various *N* at the abscissa. The average of the exponents ν_{eff} from Figs. 8(a) and 8(c) and from (not shown) are quoted on the ordinate. Log-log plot of vs. *L*/*l* _{ p }(*N*) for σ = 0.5 (c) and σ = 1.0 (d). Several choices of *N* are shown as indicated. Data plotted by the same symbol correspond to *N* _{ b } = 50, 100, and 200 from left to right. The prediction for the Kratky-Porod model (Eq. (16)) is also shown for comparison. Approximate data collapse are obtained by introducing a factor of *a* _{ r } (*a* _{ r } = 9.2 ± 1.4. and 4.9 ± 0.2 for σ = 0.5 and 1.0, respectively), cf. text. Values of *l* _{ p }(*N*) are also listed in Table I.

Plot of the function ν_{eff}(*L*/*l* _{ p }) versus *L*/*l* _{ p } as predicted from the Kratky-Porod model, Eqs. (16) and (17) (full curves). The numbers for *L*/*l* _{ p } extracted for σ = 0.5 (a) and σ = 1.0 (b) are quoted in the figure for various *N* at the abscissa. The average of the exponents ν_{eff} from Figs. 8(a) and 8(c) and from (not shown) are quoted on the ordinate. Log-log plot of vs. *L*/*l* _{ p }(*N*) for σ = 0.5 (c) and σ = 1.0 (d). Several choices of *N* are shown as indicated. Data plotted by the same symbol correspond to *N* _{ b } = 50, 100, and 200 from left to right. The prediction for the Kratky-Porod model (Eq. (16)) is also shown for comparison. Approximate data collapse are obtained by introducing a factor of *a* _{ r } (*a* _{ r } = 9.2 ± 1.4. and 4.9 ± 0.2 for σ = 0.5 and 1.0, respectively), cf. text. Values of *l* _{ p }(*N*) are also listed in Table I.

## Tables

Estimates of the effective persistence length shown in Fig. 12 are listed for the BS model with the grafting densities σ = 0.5 and 1.0 at temperatures *T* = 3.0 and 4.0. Various values of the backbone length *N* _{ b } and the side chain length *N* are chosen here. All length quoted in this table are given in units of ℓ_{ b } (with ℓ_{ b } ≈ 0.97σ_{LJ}). Note that depends not only on *N* and σ, but also on *N* _{ b } and *T*, and hence does not seem as a characteristic of intrinsic chain stiffness. Estimates of *l* _{ p }(*N*) shown in Fig. 13 are also listed for comparison.

Estimates of the effective persistence length shown in Fig. 12 are listed for the BS model with the grafting densities σ = 0.5 and 1.0 at temperatures *T* = 3.0 and 4.0. Various values of the backbone length *N* _{ b } and the side chain length *N* are chosen here. All length quoted in this table are given in units of ℓ_{ b } (with ℓ_{ b } ≈ 0.97σ_{LJ}). Note that depends not only on *N* and σ, but also on *N* _{ b } and *T*, and hence does not seem as a characteristic of intrinsic chain stiffness. Estimates of *l* _{ p }(*N*) shown in Fig. 13 are also listed for comparison.

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