major concepts - school of chemistry and biochemistryrh143/courses/wordup/15c/lec/26.pdf · 6755...

6755 Lecture #26 —Rigoberto Hernandez

Entangled Polymers 1

Major Concepts •  Light Scattering

– Structure Factor (S(q)) – Dynamic Structure Factor (S(q,t))

•  Dilute or Rarefied limits – Point scatterers – Anisotropic (or macromolecular) structures – Polymers (intramolecular segments)

•  Concentrated Polymers – Scattering between segments of different

polymers

Light Scattering •  Experimental Set-Up

–  Incident Plane Wave, kin – Scattering Region, Rj(t) – Scattered Wave, kout

•  Measure: – Total Scattered Wave

•  Light Scattering – Correlation in Scattered Wave

•  Dynamic Light Scattering





Structure Factor

€

g(r) = 1N δ r − Rm − Rn( )( )

m≠n

N

∑n=1

N

∑

€

g(q) = dr ∫ eiq⋅rg(r)

≠ 1N eiq⋅ Rm −Rn( )

n,m∑ ≡ S(q)

S(q) = 1+ ρg(q)

•  Pair correlation function

€

c(r) = δ r − Ran( )n=1

N

∑a=1

np

∑

where Ran is the location of a segment

€

ρ(r) = δ r − Rn( )n=1

N

∑

€

cq = dr ∫ eiq⋅rc(r) = eiq⋅Rann=1

N

∑a=1

np

∑

€

g(q) =1cV

cq (t)c−q (0)

=1npN

eiq⋅ Ran−Rbm( )

n,m=1

N

∑a,b=1

np

∑ where c ≡npNV

Notation mismatch



Pair Correlation Function, g(q) •  Particle-particle

correlation function: •  (n-m)th segment pair

correlation function: •  Pair correlation

function: •  FT of Chandler’s g(r) is

related to the structure function S(q) plus the self-term

€

gnm (r) = δ r − Rm − Rn( )( )

gn (r) = δ r − Rm − Rn( )( )m≠n

N

∑

g(r) = 1N δ r − Rm − Rn( )( )

m≠n

N

∑n=1

N

∑

€

g(q) = dr ∫ eiq ⋅rg(r)

= 1N eiq⋅ Rm−Rn( )

n≠m∑



Recall: S(q) for Gaussian chains •  Evaluating Doi’s g(q)—viz. S(q)

Resummation/Renormalization to ensure that g(q)→0 as q→∞

A better approximation to observed result:



Dynamic Structure Factor

€

g(r, t) = 1N δ r − Rm (t) − Rn (0)( )( )

m≠n

N

∑n=1

N

∑

€

g(q, t) = dr ∫ eiq⋅rg(r, t)

≠ 1N eiq⋅ Rm ( t)−Rn (0)( )

n,m∑ ≡ S(q, t)

•  Pair correlation function

€

c(r, t) = δ r − Ran (t)( )n=1

N

∑a=1

np

∑

where Ran (t) is the location of a segment

€

ρ(r) = δ r − Rn( )n=1

N

∑

€

cq (t) = dr ∫ eiq ⋅rc(r, t) = eiq⋅Ran (t )

n=1

N

∑a=1

np

∑

€

g(q,t) =1cV

cq (t)c−q (0)

=1npN

eiq ⋅ Ran (t )−Rbm (0)( )

n,m=1

N

∑a,b=1

np

∑

Notation mismatch



g(q,t)—Dilute Polymers @ SA

€

g(q, t) = 1N eiq⋅ Rm ( t)−Rn (0)( )

n,m∑

≈ N eiq ⋅ RG (t)−RG (0)( ) at small angles (SA) – viz. qRG << 1

•  Doi 4.4

€

g(q, t) = N dreiq⋅r∫ 4πDGt( )−3 / 2 exp − r2

4DGt&

' (

)

* + ≈ Ne−DGq

2t

C.o.M. distribution is Gaussian with variance, 2DGt

€

⇒Γq = −d lng(q, t )

dt%

& ' (

) * = DGq

2



Initial Decay in g(q,t) —Dilute Polymers —

€

Let Γq = −d lng(q, t )

dt$

% & '

( ) t= 0

€

g(q,t) =1N

cq (t)c−q (0) =1N

eiq⋅ Rn ( t )−Rm (0)( )

n,m=1

N

∑

€

ddt g(q,t)

t= 0=

1N

kBT∂cq∂Rn

⋅Hnm ⋅∂c-q∂Rmn,m

∑

= - 1N

kBT q ⋅Hnm ⋅ qeiq⋅ Rn−Rm( )

n,m∑

≈ - kBTN

q ⋅Hnm ⋅ q at small angles—viz. qRG <<1n,m∑

= - kBTq2

6πηsN1

Rn − Rmn,m∑

€

g(q,t) t= 0 = N

⇒Γq =kBTq

2

6πηsN2

1Rn − Rmn,m

∑

⇒ DG =kBT

6πηsN2

1Rn − Rmn,m

∑

•  See Doi for long-time g(q,t) behavior



g(q,t)—Concentrated Polymers •  Assume each segment is in a mean field of all other

segments (from varying polymers!)

€

cq (t) = cq (t)e−Dcq

2t

g(q,t) = g(q,0)e−Dcq2t

€

g(q,t) =1npN

eiq ⋅ Ran (t )−Rbm (0)( )

n,m=1

N

∑a,b=1

np

∑

€

c(r, t) = δ r − Ran (t)( )n=1

N

∑a=1

np

∑ : the segment concentration€

⇒ w(r) = νkBTc(r)

€

V (r) = −1ζ∇w(r) = −

νζ

kBT∇c(r,t)

∂∂tδc = −∇ ⋅Vc —the equation of continuity

∂∂tδc = Dc∇

2δc where δc(r,t) ≡ c(r,t) − c

Dc ≡νc ζ

kBT

* + ,

- ,

.

/

, , , ,

0

, , , ,

€

ν : the excluded volume parameter

Dc is the co-operative diffusion constant

•  Doi 5.1



g(q,t)—Concentrated Polymers •  The behavior of the

diffusion constant, D,… –  Co-operative Diffusion

dominates (at intermediate high c’s):

–  Dilute limit

–  Self-Diffusion Dominates(in reptation limit)

€

D ≈ Dc ≡νc ζ

kBT ∝ c1

Dc =kBT6πηsξ

∝ c34

DG ≈ Dreptation ∝ c −1.75

Hervet, Leger, Rondelez, Phys. Rev. Let. 42, 1681 (1979)

€

Dc ∝c1


RGT in 2D & Coarse-Graining 12

Major Concepts

• Renormalization Group Theory (RG) in 2D – Ising Model as example…

• Coarse-Graining • Examples


Polymer - Ideal Chains 13

Major Concepts – Doi 1.1

•  Ideal Chains –  Ideal polymers don’t interact with anything

else… –  Ideal chains do not even see themselves?!

•  Random Walk Models – Phantom Walks – Walks with short-ranged repulsion – Self-Avoiding Walks (SAWs)


Polymer - Gaussian Chains 14

Major Concepts – Doi 1.2 •  Gaussian (Phantom) Chains

– Motivated by phantom random walks – Give rise to the Bead-Spring Model

•  Statistics: – Radius of Gyration, Rg

– Pair Correlation Function


Polymer - NonIdeal Chains 15

Major Concepts •  Beyond Ideal Polymers in the Dilute Limit

–  I.e., Single polymer chain in solvent – But include:

•  interaction with between monomers •  interaction with solvent

•  Excluded Volume Effects – Scaling Law for Radius of Gyration with N

•  Solvent Effects (MFT-1.3 or RG-1.4) – Good vs. Bad Solvents – Theta Solvents


Polymer Solutions - Flory-Huggins 16

Major Concepts •  Polymer Solutions

– Before: 1 chain (very dilute) – Now: many chains

•  Do they interact or not? •  If they are long enough, even small concentration is

enough to have them overlap •  “Concentrated” solutions are entangled and

eventually lead to polymer melts – Flory-Huggins Theory

•  Overlapping (and entangled) chains •  Mean-Field Theory including chain-chain interactions •  Chemical potential, osmotic pressure, phase behavior


Polymer Solutions: Structure Functions

17

Major Concepts •  Polymer Solutions

–  Before: Flory-Huggins Theory •  Overlapping (and entangled) chains (all poly-“A”-mers) •  Mean-Field Theory including chain-chain interactions •  Chemical potential, osmotic pressure, phase behavior

–  Concentration fluctuations •  Model: Solution of “A”-mers and “B”-mers, no solvent •  Structure Function:

–  Correlation functions & linear response –  Random phase approximation (RPA) –  High-Concentration Limit agrees with MFT/Flory-Huggins

•  Scaling theory –  Π: Osmotic pressure –  ξ: Correlation length –  Rg: polymer size


Blends, Copolymers & Gels 18

Major Concepts •  Polymer Blends

– Flory-Huggins Theory tells us that they usually don’t want to mix!

•  Block copolymers – Use chemistry to force monomers to mix – Morphology tends toward concentrated domains


Monte Carlo Methods 19

Major Concepts

•  Naïve Monte Carlo –  E.g., calculating π"

•  Metropolis Monte Carlo –  Detailed Balance

•  Improved Sampling Techniques –  Importance Sampling –  Umbrella Sampling — Torrie and Valleau (1977) –  The Wolff Algorithm — PRL 62, 361 (1989) –  Parallel Tempering & Replica Exchange — Swendsen & Wang (1986) –  Wang-Landau — (PRL 2001)

•  Monte Carlo of Polymers


Blends, Copolymers & Gels 20

Major Concepts •  Polymer Gels

– Rubber elasticity & its free energy – Kuhn’s Theory

•  (stress-strain relationship) •  Elasticity arises because of polymer’s thermal motion •  Under deformation, connections sites move affinely

From Wikipedia… “A gel (from the lat. gelu—freezing, cold, ice or gelatus—frozen, immobile) is a solid, jelly-like material that can have properties ranging from soft and weak to hard and tough. Gels are defined as a substantially dilute cross-linked system, which exhibits no flow when in the steady-state.[1] By weight, gels are mostly liquid, yet they behave like solids due to a three-dimensional cross-linked network within the liquid. It is the cross-links within the fluid that give a gel its structure (hardness) and contribute to stickiness (tack). In this way gels are a dispersion of molecules of a liquid within a solid in which the solid is the continuous phase and the liquid is the discontinuous phase.”

Cross-links may be formed through physical bonds or chemical bonds.


Stress Optical Law 21

Major Concepts •  Stress Optical Law •  Chain Interactions:

– Excluded volume interactions – Nematic interactions – Entanglement interactions

•  Gel Swelling – From neat (“dry”) gel to solvent rich – Need to account for polymer-solvent

interactions


Linear Response 22

Major Concepts •  Linear Response Theory

– Time-dependent observables – Response to a perturbation

•  Onsager’s Regression Hypothesis – Relaxation of a perturbation – Regression of fluctuations

•  Fluctuation-Dissipation Theorem – Proof of FDT – & relation to Onsager’s Regression Hypothesis – Response Functions


Brownian & Langevin Dynamics 23

Major Concepts •  Langevin Equation

–  Model for a tagged subsystem in a “solvent” –  Harmonic bath with temperature, T –  Friction & Correlated forces (FDR)

•  Markovian/Ohmic vs. Memory

•  Chemical Kinetics –  Master equation & Detailed Balance –  Relaxation rate & Inverse Phenomenological Rate

•  Brownian Motion –  Langevin Equation –  Fokker-Planck Equation

⇒  Boltzmann Distribution at Equilibrium –  Diffusion Constant


Polymer Dynamics—Rouse 24

Major Concepts •  Polymer Dynamics

–  Rouse Theory •  Bead-Spring Model (discrete in n) in a Brownian bath

–  Langevin Equation (in the Smoluchowski Limit)

•  Rouse Model (continuous in n) •  Segment Dynamics:

–  Rotational Relaxation Time: τR –  Short time: sub-diffusive (not ballistic!) –  Long time: diffusive


Polymer Dynamics—Zimm 25

Major Concepts •  Polymer Dynamics

–  Introducing hydrodynamics (many-body interactions) •  The mobility matrix •  Navier-Stokes equation •  Zimm Theory

–  Preaveraging approximation in the mobility matrix –  Corrected τ’s and exponents



Major Concepts •  Light Scattering

– Structure Factor (S(q)) – Dynamic Structure Factor (S(q,t))

•  Dilute or Rarefied limits – Point scatterers – Anisotropic (or macromolecular) structures – Polymers (intramolecular segments)

•  Concentrated Polymers – Scattering between segments of different

polymers


Reptation 27

Major Concepts •  Dynamics of Concentrated Polymers

– Theory #1: Cooperative motion between segments as the diffusion-determining process •  segments move in mean field of other segments •  includes hydrodynamics

– Theory #2: Motion along “tubes” •  tubes defined by entanglements!

– Which is correct? •  Tube Model (Edwards & de Gennes)

–  “reptation” is motion through tubes

major concepts - school of chemistry and biochemistryrh143/courses/wordup/15c/lec/26.pdf · 6755...

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