Jacob Klein Science 323, 47, p. 2

Transcription

1 p. 1

2 Jacob Klein Science 323, 47, 2009 p. 2

3 BOTTLEBRUSH POLYMERS bond fluctuation model simulation SNAPSHOT Backbone chainlength N b = 131 grafting density σ = 1 side-chain length N = 6 p. 3

4 N b = 131 N = 12 σ = 1 p. 4

5 N b = 131 N = 24 σ = 1 p. 5

6 N b = 131 N = 48 σ = 1 Varying the grafting density σ and the chain length N of the side chains the STIFFNESS and hence the PERSISTENCE LENGTH of these WORMLIKE objects can be CONTROLLED! p. 6

7 LENGTH SCALES IN BOTTLEBRUSH POLYMERS L cc R e,bb coarsegrained view: flexible cylindrical worm" contour length L cc R cs lp (backbone) end-to-end distance R e,bb cross-sectional radius R cs ls persistence length l p lb R e atomistic view: backbone chain length N b backbone bond vector l b side-chain length N, bond vector l s side-chain end-to-end distance R e p. 7

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14 < cos Θ(s) > POLYMERS IN DILUTE SOLUTION, GOOD SOLVENT CONDITIONS cos θ(s) = e-05 1e-06 1 N b s N b = 6400 N b = 3200 N b = 1600 N b = 800 N b = 400 N b s i=1 slope = s cos θ i,i+s s β, s s N b β = 2(1 ν) (ν 0.588) L. Schäfer et al. (1999) SAW model, simple cubic lattice PERM algorithm p. 14

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16 FLORY s LOCAL persistence length l p (k) = l b a k R e / a k 2 SELF-AVOIDING WALKS (good solvent conditions) Schäfer & Elsner (2004) 6 l p (k) l b = α [ k(nb k) N b ] 2ν 1 l p (k) N b = 6400 N b = 3200 N b = 1600 N b = 800 N b = k / N b l max p l p = l b 0 l b α(n b /4) 2ν 1, N b ds cos θ(s) does not exist, since cos θ(s) s 2(1 ν) decay too slowly p. 16

17 BOTTLEBRUSH POLYMERS (good solvent conditions) Schäfer-Elsner (2004) formula provides good fit! N b = 131 N = 48 N = 36 l p (k) l b = α [ ] 2ν 1 k(nb k) N b l p (k) / l b N = 24 N = 12 N = k p. 17

18 FLORY s LOCAL persistence length l p (k) for bottlebrush polymers l p (k) / l b N = k / N b N b good fit to l p(k) l b = α [ ] k(nb k) 2ν 1 obtain α! N b p. 18

19 prefactor α of the Schäfer-Elsner formula for BOTTLEBRUSH POLYMERS with side-chain length N = 24 α N b [ ] (Nb k)k 2ν 1 l p (k) = αl }{{} b N b measure of intrinsic stiffness" of semiflexible macromolecules under good solvent conditions p. 19

20 > / ( 2 l b N b 2 ν ) 2 < R e,b BOTTLEBRUSH-POLYMERS: END-TO-END DISTANCE SHOWS SAW-LIKE SCALING! possible definition of persistence length l p : Re,b 2 = 2l p l bnb 2ν σ = 1 N = 24 N = 18 N = 12 N = 6 N = 0 ( for Gaussian chains ν = 1/2 and l p = l p ) N b p. 20

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22 BOTTLEBRUSH-POLYMERS: END-TO-END DISTANCE SHOWS SAW-LIKE SCALING! > / ( 2 l b N b 2 ν ) 2 < R e,b σ = 1/2 N = 24 N = 18 N = 12 N = N b pre-asymptotic behavior: NON-GAUSSIAN, at variance with Netz & Andelman (2003) p. 22

23 SAW model + PERM p. 23

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28 FURTHER EVIDENCE FOR THE LACK OF PREASYMPTOTIC GAUSSIAN BEHAVIOR: cos θ(s) = exp( sl b /l p ) 1 sl b l p <cos Θ(s)> N = 48 N = 36 N = 24 N = 12 N b = if one fits a function const. exp( sl b /l p ) to cos θ(s) for some intermediate range of s, the resulting length l p strongly depends on N b as well! s p. 28

29 STANDARD DEFINITION OF PERSISTENCE LENGTH APPLIED TO A BOTTLEBRUSH POLYMER DOES NOT DESCRIBE THEIR LOCAL INTRINSIC STIFFNESS, SINCE IT SYSTEMATICALLY INCREASES WITH N b 100 l p 10 N = 48 N = 24 N = 12 N = N b p. 29

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32 S exp (q), S(q) B2: b400s62 b259s48 b195s48 b131s48 b99s48 b67s48 1e S exp (q), S(q) B4: b188s58 b259s24 b195s24 b131s24 b99s24 b67s24 1e q R g q R g p. 32

33 S exp (q), S(q) B1: b400s22 b259s12 b195s12 b131s12 b99s12 b67s12 1e S exp (q), S(q) B1: b400s22 b259s24 b195s24 b131s24 b99s24 b67s24 1e q R g q R g p. 33

34 qs(q) / (qs(q)) max q b =0.005 q b =0.01 q b =0.02 q b =0.03 q b =0.05 slope = -1 slope = -(1/ν-1) slope = -1/5 (q) max (qs(q)) max slope = -1/ q / q max l p p. 34

35 q* q* q* q* q* q* p. 35

36 SEARCH FOR HOLTZER PLATEAU IN KRATKY PLOTS qs(q) vs. q (SIMULATION for N b = 259, N = 24) m qs(q)/s cs(q) qs b (q) qs(q)/s cs (q) qs(q) q b259s24 S(q) total scattering from bottlebrush S b (q) scattering from BACKBONE ONLY S cs (q) cross-sectional scattering, computed from radial density profile S m cs (q) cross-sectional scattering expression, fitted to S(q)/S b (q) scattering wave number p. 36

37 SEARCH FOR HOLTZER PLATEAU in KRATKY PLOTS (SIMULATION) b259s48 N b = 259 N = m qs(q)/s cs(q) qs b (q) qs(q)/s cs (q) qs(q) q p. 37

38 ESTIMATION OF THE WAVENUMBER q DESCRIBING THE ONSET OF THE HOLTZER PLATEAU q S b (q) q N b = 131 N b = 195 N b = 259 N b = 387 N b = 515 N b = 771 side-chain length N = q N b = 1027 experiment on scattering from backbone only: Lecommandoux et al. (2002) only one value of N b! p. 38

39 WAVENUMBER q DESCRIBING ONSET OF HOLTZER PLATEAU no significant dependence on N b measure of intrinsic stiffness" but q is somewhat ill defined, because onset is gradual and NOT SHARP! 0.12 q* l p = 1/q l p l p from total end-to-end distance of the backbone N b p. 39

40 POLYMERS IN DILUTE SOLUTION, THETA SOLVENT cos θ(s) s 3/2, s s N b Shirvanyants et al. (2008) <cos Θ(s)> e-05 1e-06 N b = 6400 N b = 3200 N b = 1600 N b = 800 N b = s slope = -1.5 SAW model, simple cubic lattice energy ǫ is won if two monomers are nearest neighbors Θ-point for ǫ = k B T ln q θ, q θ = PERM algorithm p. 40

41 FLORY s LOCAL persistence length l p (k) = l b a k R e / a k 2 for SAW s at the THETA POINT 2 l p (k) R e = N b k=1 N b = 6400 N b = 3200 N b = 1600 N b = 800 N b = k / N b a k l p 1 N b l p (k) = N b k=1 N b k=1 N b i=1 l b cos θ ki N b = l b l p exists since s is finite dss 3/2 Gaussian chains: l p = l p! 0 ds cos θ(s), N b p. 41