Structure Analysis of Bottle-Brush Polymers: Simulation and Experiment

Transcription

1 Jacob Klein Science 323, 47, 2009 Manfred Schmidt (Mainz) Structure Analysis of Bottle-Brush Polymers: Simulation and Experiment Hsiao-Ping Hsu Institut für Physik, Johannes Gutenberg-Universität Mainz, Germany with Wolfgang Paul (Mainz, Halle) Silke Rathgeber, Kurt Binder (Mainz) Bottle-brush polymers p.1

2 Model and Algorithm Bond fluctuation model: self-avoiding walks on a simple cubic lattice with bond constraints 108 bond vectors r b are from: [2,0,0], [2,1,0], [2,1,1], [2,2,1], [3,0,0], [3,1,0] 2 r b 10 linear polymer chain bottle-brush polymer Bottle-brush polymers p.2

3 Geometry of bottle-brush polymers: σ = 1 σ = 1/2 N tot = N b + n c N: total # of monomers N b = [(n c 1)/σ + 1] + 2: monomers in a backbone N: monomers in a side chain n c : # of side chains, σ: grafting density Algorithm: L26 move, pivot algorithm Bottle-brush polymers p.3

4 Autocorrelation functions c(a, t) c(a, t) = < A(t 0)A(t 0 + t) > < A(t 0 ) >< A(t + t 0 ) > < A(t 0 ) 2 > < A(t 0 ) > 2. A = R 2 g,c : square radius of gyration of side chain monomers A = R 2 g,b : square radius of gyration of backbone monomers Bottle-brush polymers p.4

5 Snapshots: N b = 131, σ = 1 N = 6 N = 24 N = 12 N = 48 backbone chain length N b, side chain length N grafting density σ n c /N b (n c : number of side chains) solvent quality: temperature, PH,... (good solvent) Bottle-brush polymers p.5

6 Structural characterization Coarse-grained description of a bottle-brush polymer as a flexible spheocylinder: L cc : length of the axes of the cylinder L cc R e,bb R e,bb : end-to-end vector of the backbone chain ls R cs lp R cs : cross-sectional radius l p : persistence length R e : end-to-end vector of the side chain lb R e l s : bond vector connecting monomers in the side chains l b : bond vector connecting monomers in the backbone chain Bottle-brush polymers p.6

7 Structure factor S w (q) The structure factor for the bottle-brush polymers S w (q) = 1 N tot N tot < c( r i )c( r j ) > sin(q r i r j ) N tot q r i r j i=1 j=1 c( r i ) = 1 (0) if r i is occupied (unoccupied) N = 48 N = 24 N = 12 N = 6 Rathgeber et. al., J. Chem. 122, (2005) S w (q) 100 N tot (1-q 2 <R 2 g >/3) q -1/ N b = 259 q q Bottle-brush polymers p.7

8 Structure factor S w (q) The structure factor for the bottle-brush polymers S w (q) = 1 N tot N tot < c( r i )c( r j ) > sin(q r i r j ) N tot q r i r j i=1 j=1 c( r i ) = 1 (0) if r i is occupied (unoccupied) S w (q) N tot (1-q 2 <R 2 g >/3) N b = q N = 48 N = 24 N = 12 N = 6 q -1 q -1/0.588 small q (Guinier regime): S w (q) N tot [1 q 2 < R 2 g,bb > /3] intermediate q-value: S w (q) q 1/ν, ν = large q: almost rigid segments Bottle-brush polymers p.7

9 Experimental results: (Rathgeber et. al., J. Chem. 122, (2005)) (σ 1, good solvent) Structure formula of the bottle brush polymer (n butyl acrylate) (hydroxyethylmetacrylate) Techniques: static light scattering, small angle neutron scattering, x-ray scattering Bottle-brush polymers p.8

10 Experimental results: (Rathgeber et. al., J. Chem. 122, (2005)) (σ 1, good solvent) Exp Backbone chain length exp b Side chain length exp ( N N ) Simulation ( N N ) b Techniques: static light scattering, small angle neutron scattering, x-ray scattering Bottle-brush polymers p.8

11 Exp. and sim.: S(q) S exp (q 0) = 1 q qr, R exp g,bb 30.5 nm (Guinier formula) 1 S exp (q) exp 1-(q R g,bb ) 2 /3 B2: b400s q R g R g = R 2 g 1/2 Bottle-brush polymers p.9

12 Exp. and sim.: S(q) 1 S exp (q), S(q) 0.1 B2: b400s b259s48 b195s48 b131s b99s48 b67s q R g Exp. (b400s62): N b = 400, N = 62, R exp g,bb = 30.5 nm Sim. (b259s48): N b = 259, N = 48, R g,bb = lattice spacing 1 nm 3.79 lattice spacing Bottle-brush polymers p.9

13 Exp. and sim.: S(q) 1 S exp (q), S(q) 0.1 B4: b188s b259s24 b195s24 b131s b99s24 b67s q R g Exp. (b188s58): N b = 188, N = 58, R exp g,bb = 21 nm Sim. (b99s24): N b = 99, N = 24, R g,bb = lattice spacing 1 nm lattice spacing Bottle-brush polymers p.9

14 Radial density profile ρ cs (r) Assumptions in the analysis of experimental data: Gaussian profile: ρ cs (r) exp( r 2 / R 2 g,cs ) R 2 g,cs = 2π 0 rdrρ cs (r)r 2, 2π 0 rdrρ cs (r) = 1 Decoupling approximations for the scattering data: S w (q) S b (q)s cs (q) Scattering function of the backbone S b (q): S b (q) = [1 χ(q)]s SAW chain (q) + χ(q)s rod (q) Cross sectional scattering S cs (q): S cs (q) = [2π 0 drrρ cs (r)j 0 (qr)] 2 J 0 (r): the zeroth order Bessel function Bottle-brush polymers p.10

15 Simulations: ρ(r) ^ x j l c ^ y j C.M. ^ z j backbone: undulating line On a coarse-grained scale l c cylindrical-like object (straight rigid backbone of l c + 1 monomers) direction ẑ = 1 l c lc i êi going through C.M. length: R c e = r n+l c r n Monomers located within this cylinder segment are counted Bottle-brush polymers p.11

16 ρ(r) = 1 ns n s j=1 ρ j (r), n s = N b l c ρ(r) l c = 130 l c = 64 l c = 48 l c = 28 l c = 18 l c = 12 l c = 3 (N b = 131, N = 48) For each cylindrical-like object j: ρ j (r) = N(r) r N(r) = N N r, N(r): # of monomers in the interval [r, r + dr] N r : # of lattice sites (x j, y j ) satisfying the constraint r 2 = x 2 j + y2 j l c l p (persistence length) 1e r Bottle-brush polymers p.12

17 Persistence length of backbone l (k) p : the projection of the end-to-end vector, R e,b, on the segment vector r k : l (k) p = rk r k R e,b R e,b k 1 r k k l s : the length over which correlations in the direction of the tangent are lost. < cos θ(s) > = e s/l s 1 N b 1 s N b 1 s i=1 u i u i+s (worm likechain), u i = r i r i Bottle-brush polymers p.13

18 Persistence length l p : Scaling law of mean square end-to-end distance of the backbone: R 2 e,b (N) = 2l pl b N 2ν b as N b 40 N = 24 > / ( l b N b 2 ν ) 2 < R e,b N = 18 N = 12 N = 6 N = 0 N: side chain length ν (3D SAW) N b Local intrinsic stiffness of the backbone of bottle-brush polymer Bottle-brush polymers p.14

19 Exp. and Sim.: S(q), ρ cs (r), R g,cs experiment simulation Gaussian b259s48 S(q) 0.01 ρ cs (r) e q R g r [nm] Experiment (N b = 400, N = 62): R g,cs = 6.30 nm Simulation (N = 259, N = 48): R g,cs = 5.04 nm Bottle-brush polymers p.15

20 References: "How to Define Variation of Physical Properties Normal to an Undulating One-Dimensional Object, phys. rev. lett. 103, (2009). "Characteristic Length Scales and Radial Monomer Density Profiles of Molecular Bottle-Brushes: Simulation and Experiment, preprint (2009). "Standard Definitions of Persistence Length Do Not Describe the Local Intrinsic Stiffness of Real Polymer Chains, preprint (2009). Deutsche Forschungsgemeinschaft (DPG), grant No. SFB 625/A3 European Science Foundation (ESF) under the STIPOMAT program Bottle-brush polymers p.16