Isotropic-Nematic Behaviour of Semiflexible Polymers in the Bulk and under Confinement

Transcription

1 Isotropic-Nematic Behaviour of Semiflexible Polymers in the Bulk and under Confinement S. Egorov 1, A. Milchev 2, P. Virnau 2 and K. Binder 2 1 University of Virginia 2 University of Mainz

2 Outline Microscopic Model Theoretical Methods: Molecular Dynamics and Density Functional Theory Isotropic-Nematic Behaviour in the Bulk Isotropic-Nematic Behaviour between Flat Walls Conclusions

3 Microscopic Model Molecular Dynamics Simulations Coarse-grained bead-spring model of polymers, augmented by a bond-bending potential. Finite extensible nonlinear elastic potential: V FENE (r) =.5kr 2 ln[(1 (r/r ) 2 )], r < r. r = 1.5σ and k = 3k B T/σ 2 Bending potential V bend (θ ijk ) = ǫ b [1 cos(θ ijk )] acts between subsequent bonds of the chain. Persistence length: l p /d ǫ b T. Density Functional Theory Tangent hard-sphere model of polymers, augmented by a bond-bending potential.

4 Theoretical Methods: MD vs DFT Molecular Dynamics Simulations Exact results for a given model High computational cost; Thermodynamic observables are difficult to compute Density Functional Theory Unavoidable approximations Low computational cost; Thermodynamic observables are easy to compute

5 DFT Implementation F(ρ mol ( r)) Nk B T = F id(ρ mol ( r)) Nk B T + F exc(ρ mol ( r)) Nk B T F id (ρ mol ( r)) Nk B T = ln(ρ mol ) 1+ dωf(ω) ln[4πf(ω)] F exc (ρ mol ( r)) Nk B T = a resc 2 dω dω f(ω)f(ω )V exc (ω,ω ) resc = ρ 4 3η mol 4(1 η) 2, DFT CS a PL a VL resc = Fiso exc (ρ mol( r)) Nk B T 2 V iso exc, DFT GFD

6 Phase Diagram: MD vs DFT 1. N=32, =32 P ρ Isotropic (DFT-CS) Nematic (DFT-CS) I-N Coexistence (DFT-CS) MD Isotropic (DFT-GFD) Nematic (DFT-GFD) I-N Coexistence (DFT-GFD).5

7 Order Parameter: MD vs DFT 1.8 MD ( =8) DFT ( =8) Chen ( =8) MD ( =16) MD ( =32) MD ( =64) MD ( =1).6 S.4 N= ρ S = dωf(ω)( 3 2 cos2 θ 1 2 )

8 RMS End-to-End Distance 1.98 (a) I-N R e (ρ)/l I-N I-N N=16 N=32 N= Monomer concentration ρ N = 32, ǫ b T = 1.9 (b) R e (ρ)/l N=16 N=32 N=64 I-N I-N.5 I-N Monomer concentration ρ N = 32, ǫ b T = 8

9 MD Snapshot: Isotropic Phase ǫ b T = 1, N = 32, ρ =.12

10 MD Snapshot: Nematic Phase ǫ b T = 1, N = 32, ρ =.6

11 Isotropic-Nematic Phase Behavior Inverse Persistence Length 1/l p ρ iso, ρ nem (Chen, N=8) ρ iso, ρ nem (MC, N=8) ρ tr (MD, N=8) ρ iso, ρ nem (DFT, N=8) ρ iso, ρ nem (MC, N=16) ρ tr (MD, N=16) N=32 N=64 (a) Monomer density ρ

12 Isotropic-Nematic Phase Behavior Previous Theoretical Work Khokhlov, Semenov, Odijk, Chen J. Z. Y. Chen, Macromolecules, 26, 3419 (1993). ρ I,N l p /d = f(l/l p ) constant (L/l p ) ρ I,N l p /d = f(l/l p ) l p /L (L/l p ) Scaling function f depends on one dimensionless ratio. Present Theory SAE, A. Milchev, K. Binder Phys. Rev. Lett., 116, (216). ρ I,N l p /d = f (L/l p,l p /d) Three lengths (L,l p,d) Two dimensionless ratios L/l p, l p /d. Scaling function f depends on two dimensionless ratios.

13 Isotropic-Nematic Phase Behavior ρ iso πl p /(4d) N=2l p MD Chen Odijk (b) KS SPT DFT-CS DFT-GFD DuPre-Yang l p

14 Isotropic-Nematic Phase Behavior 2 (a) ρ tr (MD, 2.) ρ iso, ρ nem (DFT, 2.) ρ tr πn/4, ρ iso πn/4, ρ nem πn/ l p

15 Theory vs Experiment ρ π l p / (4d) ρ i, ρ n (Chen) ρ ave (PHIC,.34) ρ ave (PYP,.923) ρ tr (MD,.125) ρ tr (MD,.625) ρ tr (MD,.3125) L / l p Poly(n-hexylisocyanate) in toluene: d/l p =.34. Poly(yne)-platinum in trichloroethylene: d/l p =.923. T. Sato and A. Teramoto, Adv. Polym. Sci., 126, 85 (1996).

16 DFT: Planar Confinement F(ρ mol ( r,ω)) k B T = F id(ρ mol ( r,ω)) k B T + F exc(ρ mol ( r,ω)) k B T F id (ρ mol ( r,ω)) k B T = d r dωρ mol ( r,ω)(ln[4πρ mol ( r,ω)] 1) Fexc orient (ρ mol ( r,ω)) k B T = d r dω d r dω a PL resc (ρ iso( r)) (ρ mol ( r,ω) ρ mol /(4π)) (V exc ( r, r,ω,ω ) V iso exc ) (ρ mol ( r,ω ) ρ mol /(4π)) V exc ( r, r,ω,ω ) δ( r r )V exc (ω,ω )

17 Density Profiles: MD vs DFT.2 (a) N=16, ρ=.1.15 ρ(z).1.5 T=1, ρ middle =.115 T=5, ρ middle =.111 T=1, ρ middle =.1143 T=3, ρ middle = z.2 (b) N=32, ρ=.1.15 ρ(z).1.5 MD ( =1), ρ middle =.1116 MD ( =5), ρ middle =.1119 MD ( =1), ρ middle =.1138 MD ( =3), ρ middle = z

18 Surface Tension vs Stiffness.6 (a) ρ b =.1, N=32.4 γ.2 MD DFT T

19 Surface Tension vs Chain Length.25.2 T=16, ρ b =.625 γ MD DFT.4 T=1, ρ b = γ.2.1. DFT (orient) DFT (iso) N

20 Order Parameter: MD (a) -.1 S(z) N=1 N=15 N=2 N=25 N=3 N= z R e (z)/(n-1), R e (z)/(n-1) (b) N1, N1, N15, N15, N2, N2, N25, N25, N3, N3, N4, N4, z

21 Order Parameter: DFT ρ(z)/ρ b (a) ρ b =.1, N=32 T= T=2 T=5 T=1 T=2 T=3 -.1 S(z) z

22 End and Middle Monomers N ρ end (z)/ρ b / (b) ρ b =.1, N=32 T= T=2 T=5 T=1 T=2 T=3. 1. Nρ mid (z)/ρ b / z

23 End Monomers: Stiffness φ e (z) ρ b =.1, N=32 T= T=2 T=5 T=1 T=2 T= z φ e (z) = N 2 ρ end (z) ρ(z)

24 End Monomers: Chain Length φ e (z) ρ b =.625, T=16 N=6 N=1 N=16 N=2 N=26 N= z φ e (z) = N 2 ρ end (z) ρ(z)

25 End and Middle Monomers: MD 16 Nρ end (z)/[2ρ(z)] ρ=.5 ρ=.16 ρ=.2 ρ=.25 ρ=.29 Nρ end (z) / [2ρ(z)] z 1 2 z T=8 T=2 T=15 T=1 T=5 T=1 1.2 (b) Nρ mid (z) / 2ρ(z) z(ρ mid max ) 1 ~ ( T).37 ~ ( T) R e.1 1 l p / L z z(ρ mid max )

26 Surface Tension: Stiffness.2 (b) ρ b =.625 γ.1 N=8 N=2 N=32 N=4 N= ρ b =.2 γ T

27 End Monomers: Stiffness 1 (a) ρ b = φ e () φ e () 5 ρ b =.2 2 N=8 N=2 N=32 N=4 N= T

28 Surface Tension: Chain Length.2 (b) ρ=.625 γ ρ=.2 T= T=1 T=2 γ.6.4 T=4 T=8 T=16 T= N

29 End Monomers: Chain Length 1 (a) ρ=.625 φ e () 5 φ e () ρ=.2 T= T=1 T=2 T=4 T=8 T=16 T= N

30 MD Snapshot: Isotropic Phase ǫ b T = 1, N = 32, = 4, ρ =.1

31 MD Snapshot: Nematic Phase ǫ b T = 1, N = 32, = 4, ρ =.3

32 Order Parameter: MD.8 Order parameter S(z) ρ=.3 ρ=.27 ρ=.25 ρ=.2 ρ= z N = 32, ǫ b T = 32, = 4. Order parameter S =4 =5 =64 =1 PBC Monomer density ρ

33 Order Parameter: MD 1-2P 2 (cosθ z ) z / R g ( ) R g 2 (z)/rg 2 ( ), Rg 2 (z)/rg 2 ( ) ,1,1,64,64,5,5,4, z / R g ( ) N = 32, ǫ b T = 32, ρ =.27.

34 Order Parameter: DFT Bulk (nematic) Bulk (transition) =12 =1 =8 =6 =5 =4 =3 N=32, =32 S µ

35 Order Parameter: DFT 1.5 N=32, =32 1/(dS/dµ) 1..5 =12 =1 =8 =6 =55 =53 =5 =4 = µ

36 Disjoining pressure: Stiffness F( ) T (a) = =1 =2 =3 =5 =8 =16 =24 = Nρ mid (h/2)/(2ρ b ) 1..5 (b) ρ b =.65, N=

37 Disjoining Pressure: Bulk Density.1 F( ) T (c) ρ b =.15 ρ b =.2 ρ b = Nρ mid (h/2)/(2ρ b ) ρ b =.3 =32, N=32 (d)

38 Conclusions Isotropic-Nematic phase behaviour of semiflexible polymers in the bulk was studied via MD and DFT; it was found that the co-existing densities at the isotropic-nematic transition depend on two dimensionless ratios: L/l p and l p /d. DFT for semiflexible polymers under planar confinement between two hard walls was developed and compared with MD; its accuracy for structural (density profiles) and thermodynamic (surface tension) observables was confirmed. Capillary nematization of semiflexible polymers under planar confinement was studied via MD and DFT.

39 Acknowledgment Alexander von Humboldt Foundation