Michael Rubinstein [PDF]

Polymers Physics Michael Rubinstein University of North Carolina at Chapel Hill

Outline 1. “Real” Chains 2. Thermodynamics of Mixtures 3. Polymer Solutions

Summary of Ideal Chains Ideal chains: no interactions between monomers separated by many bonds Mean square end-to-end distance of ideal linear polymer

R 2  Nb 2

2 Nb Mean square radius of gyration of ideal linear polymer Rg2  6   3 3 / 2  3R 2   exp  Probability distribution function P3d N , R    2 2  2Nb   2 Nb  2 3 R Free energy of an ideal chain F  kT 2 Nb 2  3kT  R Entropic Hooke’s Law f  2 Nb 3 1 g ( r )  Pair correlation function  rb2

Real Chains Include interactions between all monomers Short-range (in space) interactions

Probability of a monomer to be in contact with another monomer for d-dimensional chain d N *  b d R If chains are ideal R  bN 1/ 2

 *  N 1d / 2

small for d>2

Number of contacts between pairs of monomers that are far along the chain, but close in space N *  N 2d / 2 small for d>4 For d 2GN/2 because ring sections reinforce each other

 f  f 2  f 2

Biaxial Compression Ideal chain R||

Real chain D

R||

D

On length scales smaller than compression blob of size D chain is almost 5/3 2 unperturbed D D   g   g   b b Occupied part of the tube 1/ 2 2/3 N b   N R||  D     Nb R||  D   bN 1/ 2  g   D g Free energy of confinement 5/3 2 N b   N b  Fconf  kT  kTN   Fconf  kT  kTN   g  D g  D

R  Fconf  kT  0  D

2

Fconf

RF    kT    D

5/3

Challenge Problem 2: Biaxial Confinement of a Semiflexible Chain Consider a semiflexible polymer – e.g. doublestranded DNA with Kuhn length b=100nm and contour length L=16mm. Assume that excluded volume diameter of double helix is d=3nm (larger than its actual diameter 2nm due to electrostatic repulsions). R|| D

RF Calculate the size RF of this lambda phage DNA in dilute solution and the length R|| occupied by this DNA in a cylindrical channel of diameter D (for db where |U(r)|>1, NB = 1

Polymer blend: NA >>1, NB >> 1

Flory-Huggins Free Energy of Mixing    Fmix 1  kT  ln   ln 1      1    n NB  NA 

At lower T for >0 repulsive interactions are important and there is a composition range  '    '' with thermodynamically stable phase separated state. This composition range, called miscibility gap, is determined by the common tangent line.

Fmix/(kn)

At high T entropy of mixing dominates, Fmix is convex and homogeneous 0 mixture is stable at all compositions. -0.1

250o K

-0.2 -0.3

300o K

-0.4 -0.5 -0.6 -0.7

350o K 0 ’ 0.2

0.4

0.6

0.8

”  5o K NA = 200 NB = 100   T

1

Phase Diagrams    Fmix 1  kT  ln   ln 1      1    n NB  NA 

Critical composition

cr 

NB N A  NB

For a symmetric blend NA=NB=N cr=2/N cr=1/2

For polymer solutions NA = N, NB = 1 1 1 1 1   cr  cr    N 2 N 2N

0

N=2.7

NFmix kTn

there is a miscibility gap for  '     ' '

NA = NB

’ sp1

-0.1

sp2

4 3 Ncr2

Two Phases

1 0

Single Phase

N

For   cr mixture is stable at all compositions. 2   1 1 1    >    For cr  2 NA NB 

0

0.2 0.4 0.6 0.8

cr

”

1



Phase Diagram of Polymer Solutions Polymer solutions phase separate upon decreasing solvent quality below q-temperature

B   A T

Upper critical solution temperature B>0 binodal

Solution phase separates below the binodal in poor solvent regime into a dilute supernatant of isolated globules at ` and concentrated sediment at ``. dilute supernatant concentrated sediment

` ``

Single Phase

`

spinodal

”

unstable (spinodal decomposition)

metastable Two Phases (nucleation and growth) M=53.3kg/mole Polyisoprene in dioxane Takano et al., Polym J. 17, 1123, 1985

Intermolecular Interactions Osmotic pressure  c  2 P  RT   A2c  ...   Mn  A2 – second virial coefficient

h

membrane solvent

Osmometer

Poly(a-methylstyrene) in toluene at 25 oC Noda et al, Macromol. 16, 668, 1981

30

106 P/cRT (moles/g)

solution

25 20

Mn= 70800 g/mole

15 10

Mn= 200000 g/mole

5

0 0.00

Mn= 506000 g/mole 0.01

c (g/ml)

0.02

Mixtures at Low Compositions

   Fmix 1  kT  ln   ln 1      1    n NB  NA  Expand ln(1-) in powers of composition       2  1  3 Fmix 1  kT  ln        2    ...   n NB  2  NB   6N B  NA  Osmotic pressure  Fmix  P

Fmix V

    2 3   n  kT    1  3  3    2     ... b  b  N A  NB  2 3N B  2

nA

 cn v 2  Virial expansion in powers 3 P  kT  c  wc  ...   n n of number density cn = /b3  NA 2 

 1  3 b6  2  b Excluded volume v   3-body interaction w  3N B  NB  v T  q 2 A2 M 02  1  2   3 In polymer solutions NB = 1 3 T b b N Av

Polymer Melts Consider a blend with a small concentration of NA chains in a melt of chemically identical NB chains. No energetic contribution to mixing  = 0.  1  3 b3 is very small for NB >> 1 Excluded volume v    2  b  NB  NB  Flory Theorem 6 b 2 xT  b gT  bN B g   N Thermal blob T B 2 v Chains smaller than thermal blob NA < NB2 are nearly ideal. In monodisperse NA = NB and weakly polydisperse melts chains are almost ideal. RA 3/5 2 In strongly asymmetric blends NA > NB bNA long chains are swollen 1/2 3/ 5 1 / 10 3/ 5  NA   NA   NA  1 / 2 b R A  xT    bN B  2   bN A  2  2 N 1 N N N  gT  B A  B  B

Challenge Problem 3:

RA2/(b2NA)

Long NA-mer in a 3-d Melt of NB-mers NB NB NB NB NB NB NB

1/ 5

 NA  R   2  2 b N A  NB  2 A

NA/NB2

M. Lang

Why doesn’t Flory Theorem work?

Challenge Problem 4: Mixing of Polymers with Asymmetric Monomers NA – monomers per A chain

NB – monomers per B chain

v0 – volume of a lattice site = volume of a small B monomer mv0 – volume of an A monomer m times larger than B monomer Derive the free energy of mixing Fmix of A and B polymers.

Calculate the size RA of dilute Achains in a 2-d of B-chains for m>>1 in the case of =0.

Summary of Thermodynamics of Mixtures Free energy of mixing consists of entropic and energetic parts. Entropic part per unit volume (translational entropy of mixing Smix )

  TSmix 1   kT  ln   ln 1    V vB vA  Energetic part per unit volume U mix   kT  1    V v0

vA – volume of A chain vB – volume of B chain v0 – volume of a lattice site

B Flory interaction parameter   A  T Many low molecular weight liquids are miscible Some polymer – solvent pairs are miscible Very few polymer blends are miscible

Chains are almost ideal in polymer melts as long as they are shorter than square of the average degree of polymeriztion NA