Chapter 3 Polymer Solutions

0

3.1 Interactions in Polymer System

3.7 Thermodynamics of Gels

3.4 Thermodynamics and Conformations of Polymers in Dilute Solution

3.6 Conformations of Polymer in Semi-dilute Solution

3.3 Thermodynamics of Polymer Solutions: Flory-Huggins Theory

3.8 Polyelectrolytes Solution3.9 Hydrodynamics of Polymer Solutions

3.5 Scaling Law( ) of Polymers

3.2 Criteria of Polymer Solubility

Chapt. 3 Polymer SolutionsThe solution process (linear Polymer)

This process is usually slower compared with small molecules, andstrongly dependent on the chemical structures and condensed states ofthe samples.

1

Swelled Sample Polymer Solution

Crosslinked polymers: only can be swelled

2

Crystalline polymers

3

Crystalline PE: dissolve at the temperature approached toits melting temperature.

Crystalline Nylon 6,6: dissolved at room temperature by using thesolvent with strong hydrogen bonds.

3.1 Interactions in Polymer System

4

van der Waals interactions1. electrostatic interaction

2. Induction (polarization) interaction

3. Dispersion interaction

Keesom force

Debye force

London force

23 between permanent charges

between a permanent multipoleon one molecule with aninduced multipole on another=

+

= 3: Dipole moment : Polarizability

between any pair ofmolecules, including non-polar atoms, arising from theinteractions of instantaneousmultipoles.

32 +

I: Ionization energy

Long-range Coulomb interactionssee 3.8

Hydrogen bond interactions

5

21 2

04 B

Q Q eU rk Tr

The Lennard-Jones or Hard Core PotentialThe L-J (6-12) Potential is often used as anapproximate model for a total (repulsion plusattraction) van der Waals force as a function ofdistance.

6== 0 >=

Mayer f-Function and Excluded Volume

7

The Probability P(r) of finding two monomers atdistance r:

Boltzmann factor

Mayer f-function: difference of Boltzmannfactor for two monomers at r and at

= exp -1

Exclude Volume v:= = 1 exp

exp

3.2 Criteria ( ) of Polymer Solubility

Solubility occurs only whenthe Gmix is negative.

0mixS

mix mix mixG H T S

Gibbs free energy of mixing

Entropy of mixing for ideal solution

1 1 2 2ln ln 0imixS k N X N X

8

???mixH

1. Hildebrand enthalpy of mixing ( )

9

21/2 1

2

2

/

1

2

1mix

Ev

H Ev

( E/v) cohesive energy density

( )

=( E/v)1/2 : solubility parameter( )

1-1 + 2-2 1-2 + 1-2

1

21

212

21

2mixE EE E

v vv vH

1: solvent; 2: polymer

P12

2

2

2

1/2 1

1 21

/

1

2

mEv

Ev

V

22

1 2 1mV

2 22

m

n VV1

m

m

VV

P12 total pairs of [1-2]

2 2n V 1 11

m

nVV

21/2 1

1

/

2

2

2

1 Ev

Ev 1 1nV 2

m

m

VV

2. Huggin’s Enthalpy of mixing

Mixing process: 1 [2 2]2

1 [2

[ 21 1] 1 ]

1 212 211212

Different pairs in solution:solvent-solvent molecule: [1-1], 11solute-solute segment: [2-2], 22solvent-solute: [1-2], 12

12 12mixingH P 1 2 12( 2)mixingH Z N

12( 2)ZkT

1 2mixingH kT N

Flory-Huggins parameter:(interaction parameter)

12 ( 2) 2P Z x

volume fraction of solvent ~ Possibilityof the cell occupied by solvent.

cells surroundinga polymer

number of polymers

P12 total pairs of [1-2]

111

1 2

s

m

V NNN xN V

222

1 2

s

m

xN VxNN xN V

and

N1N2

10

1 1 2( 2)Z N2N 1 2RT n

1 2= m

s

V kTV

21 2 1 2 /mV N

21 2sVRT

12 1 2( 2)Z xN

s psV V

with xsegments

11

211 11 1

1

s

H r drV

111 21

s

r r drV

11 111 1 2

s s

r dr r r drV V

222 22 2

m

s

VHV 22 2 11 m

s

VV 22 2 22 1 2

m

s

VV

12 12 1 2m

s

VHV

12 11 2212mixingH H H H

11 1 11 1 2m

s

VV

12 11 22 1 21 .2

m

s

V constV

1 2 .m

s

V constV

12

211 11 1

12 s

H r drV

111 21

2 s

r r drV

11 111 1 22 2s s

r dr r r drV V

222 22 2 2

m

s

VHV 22 2 11

2m

s

VV 22 2 22 1 22

m

s

VV

12 12 1 2m

s

VHV

12 11 22mixingH H H H

11 1 11 1 22m

s

VV

12 11 22 1 212

m

s

VV 1 2

m

s

V kTV

0 011 22H H

00 111 11 112 2

s

s

V NHV

00 222 22 222 2

p

s

V xNHV

0 0s p mV V V

11

s

m

V NV

22

s

m

xN VV

3.3 Thermodynamics of Polymer Solutions

13, > 0

+

0confS

2

2

3( , )2conf g B

g

S N k CN lhh

?

3.3 Thermodynamics of Polymer Solutions

(1) Entropy of mixing for ideal solution

1 1 2 2ln lnimixS k N X N X

(2) Entropy of mixing for polymer solutionsThe lattice model assumes that the

volume is unchanged during mixing.

Each repeating unit of the polymer(segment) occupies one position in thelattice and so does each solvent molecule.

The mixing entropy is stronglyinfluenced by the chain connectivity ofthe polymer component.

14

Flory-Huggins theory (Lattice Model ( ))

N1 + N2 x

: N=N1+xN2

j , N-xj , j+1Wj+1 ???

1. j+1 1N-xj

2. j+1 2Z

3. j+1 3(Z-1) ….

(Z-1)x. x

15

(N-xj-1)/N

(N-xj-2)/N

(N-xj-x+1)/N

11

1 2

NN xN

22

1 2

xNN xN

2211

210

2 2 2

! !1 1 1 ! ...! ! ! 2 ! !

N xN

jj

N x N xN xZ NWN N N N x N x N xN

1st 2st N2st chain

Entropy of mixing from FH theory1

1 2 3 11 1 1jN xj N xj N xj N xj xW N xj Z Z Z Z

N N N N

xth segment1st 4th3rd2nd

Z Z – 1

2 ( 1)

2 2

1 1 !! !

N xZ NN N N xN

2 2 21ln ( 1) ln ln ! ln ! ln( )!solution

ZS k k N x N N N xNN

Entropy of solution:

16

1

1

!1!

x

j

N xjZWN N xj x

Entropy of mixing from FH theory

eZxN

xNNNN

xNNNNkS solution

1ln)1(lnln 221

22

21

11

Using Stirling’s approximation (lnx! xlnx –x), we have:

0solventS

( )mixing solution solvent polymerS S S S

Entropy of the pure solventand pure polymer: 2

1ln ( 1) lnpolymerZS kN x x

e(N1 = 0) and

Therefore,

1 11 11

1 2 1 2

s s

m m

N V NnVN nN xN n xn V V

2 22 22

1 2 1 2

s s

m m

xN V Nxn VxN xnN xN n xn V V

where

17

1 21 2

1 2 1 2

ln lnN xNk N NN xN N xN

1 1 2 2ln lnmixingS k N N 1 1 2 2ln lnR n n

21 1 2ln lnm

s

VkV x

0confS

Free Energy of FH Theory1 2 1 2 1 2= m

mixings

VH kT N RT n kTVHuggins Enthalpy:

Gibbs Free Energy

1 1 2 2 1 1 2ln lnRT n n x n

1 1 2 2 1 1 2ln lnmixingG kT N N x N

1 21 2 1 2

1 2

ln lnm

s

VkTV x x

1 1 1 11 1 1 11

1 1 2 2 1 1 2 2

s s

m m

x N V Nx nVx N x nx N x N x n x n V V

2 2 2 22 2 2 22

1 1 2 2 1 1 2 2

s s

m m

x N V Nx n Vx N x nx N x N x n x n V V

For Polymer Solutions x1=1

mixing mixing mixingG H T S

s mm

m

V GFV kT

1 21 2 1 2

1 2

ln lnx x

Chemical potentials ( ):

22221 2

11lnln

2 31 2 2 2

1 12

RT wx

2

21 1 2 2

1 , ,

1ln 1m

T P n

GRT

n x

1

22 2 1 1

2 , ,

ln 1m

T P n

GRT x x

n

(for solvent)

(for polymer)

In the case of << 1,

< 1/2, good solvent= 1/2, theta solvent> 1/2, poor solvent

=1/2, ”???

19

s mm

m

V GFV kT

21 1 2 1 2ln ln

x

= =

= =

Osmotic pressure ( ):

Polymersolution pure solvent

P P

20m p s sV n x n V

, , ,p s m m m p s sG n n P T G PV G P n x n V

2 , ,, , , , ,

ps p s T P n

s

P T G n n P Tn

02 2

2

, , ms s m s

FP T RT F PV

0

, , p

ms s

s T P n

GPV

n

22, , p

m mm

s T P n

G FRT Fn

2 , , 0, ,s sP T P T

22

mm

s

RT F FV

s mm

m

V GFV kT

2 , , 0, ,s sP T P T

Osmotic pressure ( ):

21

22

mm

s

RT F FV

2>>1/x, 22>> 2 /x

2<<1/x(0.5- ) =1/22

1

s

RTV x

22 ???

second Virial coefficent2

12

s

AV

22 2

1 12s

RTV x

21 1 2 1 2ln lns m

mm

V GFV kT x

22

,s

c MxV 2

2cRT cM

A2

2 21

12

cAV

Polymer Shapes in Dilute Solutions

concentration

/RTc

A2 < 0, > 1/2;poor solvent

A2 > 0, < 1/2;good solvent

A2 = 0, = 1/2;condition

Coil-globuletransition

Expanded, unperturbed, andcollapsed chains

The coil-globule transition in a solution ofpolystyrene in cyclohexane. The radius ofgyration Rg and the hydrodynamic radiusRh of the polymer show a dramatic changeas temperature passes through thetemperature. (Sun, S.T.; etc. J. Chem. Phys.1980, 73, 5971.)

~1/kT !!!

22 2

1

1 12

RTV x

22

3.4 Chain Conformations in Dilute Solutions(1) Flory-Krigbaum’s Theory

N2 U ”

20 1 11 1 ... 1/ / / NU V U V U V2

2 2 21 /2 /2

1 1N N N

V VU U

UV

lnF T S kT ln 1 iU iUV V

1

1

222

FV V

c NRT U cM M

A2 23

3~ ~ ??gU R T

2 1

0

1N

i

iUV

or

22

2 ln2

NkT N VVU

3.4 Chain Conformations in Dilute Solutions(2) Flory

3/ 2 22

0 2 2

3 3, exp 42 2

hW h x hxl xl

ideal chain

real chain insolutions

0, , expE h

W h x W h x hkT

24

25

h3 vc=l3(1) (h)

m

n

hvc/h3

(1-vc/h3)

1

3 3 30

1 1 1 2 ... 1x

c c c

i

v v vh ih h h

1 1

3 30 0

exp ln 1 exp ln 1x x

c c

i i

v vi ih h

ln 1 iu iuV V

21

3 30

exp exp2

xc c

i

v v xih h

2 3cxv

h

W(h,x)(2) E(h)

2 22

2 33, exp 1 22 2

ch x vW h x hxl h

=1/2,26

2 3cxv

h

1/2, ???

-

, = , exp

1 2m

s

H VkT V

E hkT

3

2 21c

hv

3 32

2 2c c

h hv v

3 2 2

6c

c

h x vxv h

2

3cx vconst

h

3mV h s cV v

1. Polymer chain in good solvents – method I

,0

W h xh

2 2

2 3

, 3exp 1 22 2

cW h x h x vh xl h

*2 2

2 *3

3 31 1 2 02 4

cvh xxl h

W0(h,x) h0*=(2xl2/3)1/2

51/ 2

3* *

* * 30 0

9 6 1 216

cvh h xh h l

, =1/2

1/51/51/2* *

0 3 3

1 21 2 cc vx vh h x l

l l

27

2 22

2 33, exp 1 22 2

ch x vW h x hxl h

1 2c cu v v

1 2

22

2 43 32 1 2

2ch x vh h

xl h

1/5vx

1. Polymer chain in good solvents - method II

Two body interactions are important in good solvents!

~ lnBG k T 2

2

3

2

, exp 1 2 ...2

32

cx vxl

hh

hx

2

32

2

~ 1 22

3 ...2

cB

x vG k Th

hxl

0Gh

<1/2

two body interaction:excluded volumerepulsion

solvent-segmentinteraction

/ BG k T

second Virial coefficent

2 3cxv

h

conformationentropy

+

28

1/53/5 1 2h x

32 +

12 +

Temperature & Solventsh2~

21 212( 2) 1

2sVZ

kT RT Solution

29

2121 2

2 2 2 sZ VTk R

Temperature

T T

Excluded Volume

30

1 2 1 11/ 2

TT

31 2c eu v l 1/* 5 1/51 2h x x

2121 2

2 2 2 sZ VTk R

Temperature

1 , 02

u 1 , 02

u 1 , 02

u

When does the freely jointed chain works

“Coarse-grained” ( ) picture:

R

(2) -

Ne le

chain head

chain end

(1)<h2>0~N

2 6/5 2 /5~h N2 1

0~h N

2 2

0~ e eh N l

31

Crossover from Gaussian to Swollen

32

3 31 35 5

1/2 10 5T T T

T T

x xR g l g x lg g

1/2T Tg l

1

33/5

/5ux ll

1 2c cu v v

2

3Tugl

1 4

3Tu lll u

23B TT

uE k Tg B

RNumber of Segments in a Chain: x

Number of Segments in a Blob: gT

Numbers of Blob in a Chain: x/gT

Each polymer chaincan be divided intomany blobs. Polymerswithin blobs are ideal.

Thermal Blob

2. Polymer chain in poor solvents-Method I: blob model

33

1/2T Tg l

22

3Tugl

11

3Tu l ll

Blobs:

13 3globR l N3 3

glob TT

NRg

In a Globule: Blobsare densely packed

23B T BT

uE k Tg k T

3 31 2u l l

Globule:

Interaction Energyin a blob:

Excluded Volume:

1/3 1/3globR N l

2int B B

T

NF k T k TNg

Interaction Energyin a globule:

Blob size:

2. Polymer chain in poor solvents-Method II: chemical potential

2 31 2 2 2

1 12

RT wx

21~ 32

"

T

2’ ”

1 1 "'2

011 01 '

' 0

23

1 2 22" " "1 1 0,"

2RT w x

x

In Polymer rich Phase:Concentration in Coil: 3 2 "c

inxvh

1/31/31/3 1/3 1/3~ 1/ 2 ~ch v x x l

34

w=1/3

2. Polymer chain in poor solvents-Method III: three body interactions

2

2

3

2

, exp 1 2 ...2

32

cx vWl

hh

hx

x

13 ~ ch x v

h<<h02

32

2

~ 1 22

3 ???2

cB

x vG k Th

hxl

=xvc/h3

three body repulsionthird Virial coefficent G=0

two body interaction:excluded volume repulsionsolvent-segment interaction

second Virial coefficent+

Three body interactionsbecome important

>1/2

2 332 3

2 3 32 3

1 12 2/ ~ vc

Bc c

xvG k T h hxv v

vw wh h

35

Flory Formula in Dilute Solutions

36

2 3

3 6BN NG k T u wR R

R h1 2cu v

2 2 3

2 3 6BR N NG k T u wNl R R N: Number of

Segments in a chain

l: Length of theKuhn Segment

In poor solvents

In good solvents2 2

2 3BR NG k T uNl R

Polymer Shapes in Dilute Solutions

37

athermal solvents =0

Good solvents

=1/2

theta( )-solvents

positive negativezero

~N3/5

12

>1/2

poor solvents

~N1/2 ~ N1/3|

collapsed coil

<h>

Coil-globule transitionswelling chain unperturbed chain

12

Excluded Volume

T

From dilute to concentrated

38

T

Semi-dilute

Anomalous phenomenon in Semi-Dilute PolymerSolution

39

MRTcwcw

1lim0

independenton M

/c~c5/4

Osmotic pressure measured for samples ofpoly( -methylstyrene) dissolved in toluene(25 C). Molecular weight vary betweenM = 7 104 (uppermost curve) and M =7.47 106 (lowest curve). (Noda,I.; et al.Macromolecules 1981, 14, 668.)

/c~c1

/c~c5/4

22 2

1

1 12

RTV x

2<<1 22>> 2 /x

/c~c0

21

MxV2

2

c1

2cxVM

12 2

1 2 1 2

1 1 12 2

VRT c cRTc V M M V

???

40

<Rg>~Nv

Rg N

Rg

2 2 6/5SAW ~ ~R N N

3/2 2

2 2

3 3, exp2 2

NNl Nl

hh2 1RW ~R N

???

-

4141

3.5 Scaling Law( )- (1)

42

/N N g

( , )gh R F l N , ( , )/l gF lFN N g

?/( , ) . .aN NF cl ons gl g Nlt const

(1a)

1/ 2h lN

a=1/2

l l g

(1b)

43

/N N g

?, ./ /. alg lgh F const constN g N g N l

h lN

v-a=0, a=v

Energy of each blobThermal fluctuation energy kBT

Blob Model

~ g l

, /,h l N gF NF

~ B BG k Tn kg

T N

Numbers of blobs: n=N/g

l lg

(2)

44

2 21 / 2g

Ng NpR

k kk

,gF kR N ,F kl NgR l

l

(2)

45

exp( ) d ig k k rr r

( ) ( ) /g g gk k1( , )/ ( , )k N glg kF F Nlg

( ) ( , ) ( , ) ( )1 gkl klNN N Ng F F F kRk

( )) . ( )( .a a agg const coN NkR ns N kt klk

kkRg>>1 g(k N

22 2 2 22 / 1g gg N R k Rk k k

1 /k

( ) ( , )kg NlFkg N

v=1/2 idealv=3/5 real

1 0a1 /a

( , ),k l N NF k r

Form Factor of a Real Chain

46

1/( ) kp kkRg>>1

=3/5

(3) Blob Model - a. Athermal Chain Stretching

47

~ TgT

xNhg

~ TB

NG k Tg

1/1

1/1~x

TTNg

h

5 20. ~ BT T

hk TN

f1/1

~ BTT

hk Tfh NG

v=1/2 ideal v=3/5 athermal

1/1

~ BT T

hk TN

Energy of one blob thermal blobThermal fluctuation energy kBT

23T

Bk TN

hf

~ TB

NG k Tg

Number of large blobs: n=NT/gNumber of Thermal Blobs in a large blob: g

Total Free Energy

5/2

3/2

3/2~atherma

TTl B

hkN

f T

v=3/5 thermal solution Energy of large blob thermal blob T

3/2

3/2 5/2/~ B

T T

hk TN g

1/2T Tg l

Thermal Blobs

/T TN N gNumber of Thermal blobs

48

3/2

3/2 5/20.6 /~ B

T T

hk TN

fg

2/3~h f

1 4

3Tu lll u

** /B Tkf T

* /Bf k T l

3/2

3/2 5/2/~ B

T T

hk Tg

fN

2~/B

T T

hk TN

fg

to

** /B Tkf T

** / TNh g

T l

(b)

49

1/1

~ BhG k T

N l1/1

2, ~ exp ~ expR

G hh NkT h

2h h N l

2, ~R

hh Nh

2h h0.28

0.28 2.50.278 exp 1.206R x x x2/x h h

:

=3

2exp

32

=3

2 exp 1.5

(2) Blob Model – c. Biaxial Compression

50

D g l

2/31/

// 1/~ ~ ~real

N Nl lR D D Nlg D D

1/ 21/ 2 21/ 2

// 2~ ~ideal

N NlR D D N lg D

2 20

conf B B Bideal

N l RG k T k TN k Tg D D

1/1/ 5/3r

conf B B B Breal

N l N l RG k T k TN k T k Tg D D D

v=1/2 idealv=3/5 athermal

1/20R N l

3/5rR N l

TD gor

d. Uniaxial compression ??

51

3/4R N l

Answers

2

, 2/conf blob BRG k T

N g D

22

int, 2

/blob B

N gG k TD

R

, int, 0conf blob blobG GR

Entropy of anideal blob chain

,

32d

drealR N l12

idealR N l

d: dimensions( )

2 2

2 20

conf B BR RG k T k TNl R

Entropy of anideal chain

2

int 3BNG k TuR

3 1 2u l

Interactionbetween blobs

I

52

1. 2 1RW ~R N

3/2 2

2 21

33 exp2 2

gNi

ii l l

hh

23 /21

2 21

33 exp2 2

g gN Ni i

nb bR R

2

2 0

3exp2

gN nconst dnb n

R

2 2l b

Path Integral( )Edward’s Minimum Model

interaction energyih H Mean-field Free Energy

,=

6,, =

32

exp3

2

53

Gaussian Chain Model

Entropy of Chain Conformations

Mechanic Properties

Scattering Theory (R> /20)

h, (h, N)

h

h

2

2

3( , )2conf g B

g

S N k CN lhh

20 eR l L

2 2 2~R N l

Scaling Concept & Blob Model

-

2 20

R NlIn solution:

2 2

11 / 2g

pR

kk

23 Bk TNlhf

~ BNG k Tg

, =3

2exp

32

II

54

3/5SAW ~ ~R N N

(N,l,k)

blob model

2 Unique Features of Polymers

55

(1) Large Spatial Extent

(2) Connectivitya. Tacticity –

b. Polymer Topology

c. Flexible vs. Rigid

d. Multiple Confirmations (Entropy)

(3) Multiple Interactions (Enthalpy)

(4) Entanglement

Thermodynamics:

Dynamics:

(5) Responsive Moleculesa. Large-scale Relaxation Time Spectrum

b &c . Temp, Rate and Time Dependent Behavior

Spatial Extent & Connectivity

56

(Monomer) (Segment) (Blob) (Chain)

( )

Entropy

57

2

2conf BS kNlR 2

2/conf BS kN g

RA Chain A Blob Chain

Multiple Confirmations (Entropy)

Flory-Huggins Entropy of Mixing for Multi Chains

1 1 2 2ln lnmixing BS k N N 1 21 2

1 2

ln lnmB

s

VkV x x

Enthalpy

58

Multiple Interactions (Enthalpy)

23

2,int 3

/B

N gH k T

R

2

,int2 3BNH k TuR

2

3T

BT

gE k T u

In a 3d Chain In a 3d Blob Chain

within a Blob

22

2,int 2

/B

N gH k T

R

2

2,int 2BNH k TuR

2d chain

1 2c cu s s

3cv l1 2c cu v v

2cs l

Flory-Huggins Enthalpy of Mixing for Multi Chains 1 2m

mixing Bs

VH k TV

2d Blob Chain

2 1Bk T xN

Two-body interaction

Three-body 3d 3 33,int 2/ BH k T w R

3

6

NwR

3.6 Semi-dilute Solutions of Polymers

MRTcwcw

1lim0

independenton M

/c~c5/4

Osmotic pressure measured for samples ofpoly( -methylstyrene) dissolved in toluene(25 C). Molecular weight vary betweenM = 7 104 (uppermost curve) and M =7.47 106 (lowest curve). (Noda,I.; et al.Macromolecules 1981, 14, 668.)

/c~c1

/c~c5/4

22 2

1

1 12

RTV x

2<<1 22>> 2 /x

/c~c0

59

21

MxV2

2

c1

2cxVM

12 2

1 2 1 2

1 1 12 2

VRT c cRTc V M M V

(1) overlap concentration c*

Apparent correlation length( ) Rg> app> T >l

Semi-dilute regime: c > c*Overlap concentration: c = c*

<Rg>

3*

3

g

NlR

For good solvent, v = 3/5

c

60

1 2 1 11/ 2

TT

1/5 1/51 2g lR N N l

3 3/5 3 1 3/51~ N

N N

dilute c: < c*

c

4/5 3/5~ N3 1/21N

1/2 1/5

0

gRN

R

3 3/2 3/5 1/21

N N

* *2c

(2) Osmotic pressure of semi-dilute solution

61

21

MxV

2

122

12

c cRTM V 1 cV Nv

3

1

x

gi

u i U R

23g

c cRTM M

NR

1 2cu v

*3g

McR

2

'**

1 ...c c cRT B BM cc

21

221

2xcR cM

TM

V

2

1

x

i

c cRT N uM M

i

2

1 2

x

i

xi

*c cRTfM c

3

1 gcRNRM

T cM

RTc M

Van’t Hoff relation of osmotic pressure

22

212 c

c cRTM

Nv xM

22

2c cRT NM M

x u

osmotic pressurefrom Flory-HugginsTheory

*

mc cRTM c

f c

Scaling Law of semi-dilute solution

*Bc ck TfN c

(3 1) *m m vB

c k Tc N c cN

01)13( vm

1 9/4mc c

In semi-dilute regime, isindependent on N:

For good solvent,v = 3/5, therefore m = 5/4.

Osmotic pressure:

* 1 3~c N5/4

1

*Bc k T c cN

*

m

Bc ck TN c

62

< <

(3) Apparent correlation length app:

1+3 /5(3 1) *mv m mN c N l c c

*app gcR fc

*vgR N l c c

0)13( vmv

3/4 1/4app c l

app is independent on N:

For good solvent, v = 3/5,therefore m = -3/4.

app

1/5*app

mv cN l

c

1 3 3/5* vc N

63

(4) Polymer Shapes in Semi-dilute Solutions

app~c-3/4 --1/4l

g , N/g, app

3/5 1/5app

5/3 1/3 5/4 3/4app

~

~ ~

g l

g c

2 2 6/4 2app 5/4semi-dilute

2 1/4 1/4 2

semi-dilute

~ ~

~

g

g

N NR c lg c

R Nc l

blob model:N/g

- g ( )

2 2/g appR N g

2 6/5 2 2/5~app g l

64

3 3~ / appc glapp~c-3/4 --1/4l

3/5 1/5app ~ g l

Gaussian Chain

2 2

dilute~gR N 2 1/4

semi-dilute~gR Nc 2

concentrated~gR N

The presence of monomers from the other chains begins to “screen”( ) the intramolecular excluded volume interactions.

app gR app gR app ~ gu R

: ,

::

65

/

0

gRR

(5) Regions of the Polymer-Solvent Phase Diagram

6666

1/ 2~gR NI’

II 1/ 2 1/8 1/8~gR N c

III 1/ 2~gR N

V 1/31/ 3~gR N

3/5 1/5~gR NI

IV:1/3~ ( / )gR N 1/ 2~gR N

c* : 4/5 3/ 5*c N

BM c

EF 1/ 2N

AB 1/ 2N

I II

IIII’

IV

Other Interesting Topics:

67

Grafted Layerfrom Single Chain

l

Adsorption:

to Multichain

(6) Concentrated Solutions

68

1. Polymer-Plasticizer

2. Spinning Solution

3. Gel

Fiber Spinning

69Wet Spinning Dry Spinning

(6) Concentrated Polymer SolutionsConcentrated regime c**

70

1/2T Tg l

app

35

TT

NRg

35

1/2T

T

Ng lg

1 310 5

Tg N l

1

33/5

/5uN ll

1 2c cu v v

3cv l2

3Tugl

1

3Tu ll

1l

Dilute

Semi-dilute

Concentrated

3/4 1/4**app c l

3/4 1/4app c l

app gR

T

3** ucl

3

3T

T

g l2

3B TT

uE k Tg B

*c c

*c c

**c c

(7) Polymer Melts

71

Polymer A + Polymer B21 2

2A A

cA B

c ckT vN N

Polymer A + Solvent B2

1 22

A A

c A

kTv N

31 2 1 2cu v l

1 2cB

u vN

Excluded Volume

03

c

B B

v luN N

1/2T Tlg 2

3B TT

uE k Tg

62

2T Blg Nu

1/2T T Blg lN

2

3 3/2T

B BT

u gk T k Tl g

cA= A/vc2

1 22

A Ac

A

c ckT vN

A

Polymer Chain in Melts

72

AA T

T

NRg

35

Overlap parameter P3 3/2 3

3 3AR N lP N

Nl Nl3D

2D2 2

2 2 1R NlPNl Nl

Flory

B ( )

2A

BB

NlNN

1/21/2

2A

AB

NlNN

T BlN2T Bg N

3.7 Flory-Huggins free energy of a gel

02 3

1VV

V0

<h2>1/2

swelling3V0

<h2>1/2

mixing elasticF F F

1 1 2 2 1 2ln lnmixingF RT n n n

2 2 2 2 2 220

2 2 220

32

32

elastic elasticF T S

NkT x y zh

NkT x y zh

ln ~ ln ,S k k NR

2 0 2 /32

3 12elastic

c

VF RTM

2 2 2 20

13

x y z h

0 20 0

/ /

c

MN N VN V N V NM M

21 3 32

NkT

73

Basic Equation of Gel Swelling

2

2

1 11

0elasticgel

m

nnFFF

n

5/32 02

2

elastic

c

F V RTM

22 2 2

1

1ln 1 1mF RTn x

gel gel

22 2 2

1ln 12 x

1/32 12

21 2

12

/ 0c

VM

RT

22

12

RT

0

20 12 022

1 1 00 1 1

s ss

VV nV V VV

n n VV nV

Q=V/V0=1/ 2= 3

5/3

2

12

c

s

M QV

(1)

(2)

1/322

2

2

1

elastic s

c

F V RTMn

74

gel

10

3/5cQ M

Volume Phase Transition of Gels

Theory, =1- ~ 1-T /T

Experiment

75

LCST

UCST

/A T B

0A

0A

Flory-HugginsParameter

T

T

Problem: semi-dilute solution???

76

2 1/4 1/4 22fluctR N l

2 20R Nl

9/4 ?c

3.8 Solutions of Polyelectrolytes

+C

+

+

+

+

+

+ +

++

+

+

++

+

+

+

+

+

+

+

+ +

+

+

+++

++

+

+

++

++

+

++

+

C+

77

Charge Reversal & Layer by Layer Assembly

78

Oppositely charged plane

Polyion of charge Z

Li-ion Battery

79

Wrapping a chain on/in a sphere

80

Appendix: Introduction to the theory ofelectrolytes

81

The force between two spherically symmetric charges in vacuum

rQ1 Q2

21 2

304

Q Q efr

rr

21 2

0

/4B

Q Q eU r k Tr

The interaction energy (in unit of kBT) between two sphericallysymmetric charges in vacuum

Long-range Coulomb interaction

/ =Compared to van der Waals Interaction

2

3/ 1 22c

Bv NE k T

RSummation of two-body Interactionswithin a Chain or a Blob

Linearized Poisson-Boltzmann Equation- Debye-Hückel Theory

82

When the electric potential is low ( <25mV), we can makethe expansion and only keep the linear term. Thus the Debye-Hückel equation (linearlized Poisson-Boltzmann equation) isobtained. The Debye-Hückel treatment gives a simple (mean-field) description to the many-body interactions between ions.

32 22 20 0

2

8 1 8 1...3!B B B D

e n e nek T k T k T

r rr r r

2 2Dr r 2

0 0

18 8

BD

B

k Te n l nwhere

In the limit of a strong electrolyte, the surface potential s is smallenough so a linearization of the P–B equation can be justified.

2

BB

elk T

lB: Bjerrumn0 is the number density (per unit volume) of the ions.

Debye-Hückel potential

83

The effect of chargescreening is dramaticallydifferent from thepresence of a polarizableenvironment. As hasbeen shown by Debyeand Hückel 80 years ago,screening modifies theelectrostatic interactionsuch that it falls offexponentially withdistance.

D=3Å (1M NaCl)

D=1 m (Water)

exp /B Dl rr

r

Charge screening

84

Salt ions of opposite charge are drawn to charged objects andform loosely bound counter-ion clouds and thus effectively reducetheir charges. This process is called screening.

BoundedCounterions

Negativelycharged object

Polyelectrolyte in solution

85Semi-dilute concentrated

Theoretical Model of A Single Polyelectrolyte Chain

86

confe e electr e

B B B

F R F R F Rk T k T k T

2conf

2e e

B

F R Rk T Nl

2B

1/2lnelectr e e

e

F R l fN RkT R N l

lB Bjerrum 2B / Bl e k T /B Bu l l

0e

e

F RR

2Belectr e

e

F R l fNkT R

efN

or

or

l: Kuhn segment

Guassian Chain

(1) A Single Polyelectrolyte Chain in good solvents

87

1/32/31/3 2/3 2lne B BR Nlu f eN u f

Good solvent, <1/2:1/3 2/3

e BR Nlu f or

1/2eR N

(2) A Single Polyelectrolyte in poor solvents

88

Poor solvent, >1/2:

N is small

N is large

Rayleigh instability of a charged droplet – 1882

89

the break-up of a charged drop of liquid into droplets due toelectrostatic repulsion

22/Rayleigh B drop

drop

QF k T RR

3/2crit dropQ R

2 2/surf B glob TF k TR

2 /electro B B globF k Tl fN R 1/2

critB

fu N

/ 0Rayleigh dropF R

1T l

1/3 1/3globR l N

Blob Bead (globule) Beads+ Strings = necklace

Computer Simulations

90

Topics not Discussed: Hydration andAssociating in Polymer Solutions

91

Hydration

92

and Mixed SolventsPNIPAM (N - ) in Water

Topics not Discussed:Associating Polymer Solutions

93

3.9 Hydrodynamics Properties of Polymer Solutions

94

F

v

v + dv

A

Tv v

For Newtonian fluidsv

ddyv

3D2D

' ' 'd H Fv r r r r r

n nm mm

v r H F

Correlation function: ornmH 'H r r

1D v

/AFA

Diffusion of Suspensions in Solution

95

6 RF v v

/6

BB

k TD k TR

v

F

Stokes-Einstein relation

Stokes formula

0 6b B

Dh

k TD k MR

13

aMHK Mab

0 1 ...DD D k chydrodynamics radius: Rh

2

2

cJ Dr

c cJ Dt r

Fick’s law

flux

6 R

Effective viscosity of suspensions

0

0 0cc

For the solution of impenetrable spheres of radius R, Einstein derivedthe Effective viscosity of suspensions

0 1 2.5 : volume fraction occupied bythe suspensions in the solution.

If each sphere consists of n particles (monomer units) ofmass m, and their density is c , we have

3 34 4/3 3

AN cN R V RM

0: viscosity of pure solvent

32.5 4 / 32.5 2.5 hA A

VRN Nc M M

NA : Avogadro Number

Vh hydrodynamics volume96

nm=M

Instrinsic viscosity ( )

[ ] dependence of MW: Flory-Foxequation

3/ 23/ 22 21/ 2

h hM

M M

3/ 220 1/ 2

0

hM

M

Mark-Houwink Relation aKM

For flexible chain

For stiff chain

a=0.5~0.8

a=0.8~1.2

For flexible chain in good solvent

2 2~h M 2~ M

2 6/5~h M 0.8~ M

For stiff chain

For solution0.5~ M2 1

0 ~h M

23 10 2.84 10 mol

0 is calculated by Rouse-Zimm Theory and confirmedby experiments

1/ 22 2 0.50/ ~h h N

3/ 220 1/ 2 3

hM

M

=1/2

97

Rouse-Zimm Model

98

3 30.425 6A

gN N a RM M

230(RZ)

230 exp

0.425 2.56 10

2.2 ~ 2.87 10AN

0

0.1966 6

B B BG

H

k T k T k TDR Nb

1 3 0.664678 2H gR Nb R

Chapter 4 Multi-component Polymer Systems

4.1 Thermodynamics of Polymer Mixtures4.2 Properties of Polymer Interface4.3 Thermodynamics of Block Copolymers

99

Mixing?

100

Polymer A Polymer B Polymer Blends

4.1 Thermodynamics of Polymer MixturesWhy are two kinds of polymers not compatible?

Entropy of Mixing

,1 2

1 21 2

1 2 1 2

ln lnN N

s N Nk N NN N N N

S

N1Polymer/N2Polymer

N1Solvent/N2Solvent

xN1Solvent/xN2Solvent

,1 2

1 21 2

1 2 1 2

ln lnxN xN

p xN xNS k N NxN xN xN xN

,1 2

1 21 2

1 2 1 2

ln lnxN xN

s xN xNk xN xNxN xN xN x

SN

,, , 21 1 122 N N xN NxN xxN

s spS SS101

1 1 2 2 1 1 2ln lnmixingG kT N N x N

The importance of multi-component, multi-phase materials

102

prop

ertie

s

Polymer A Polymer BcontentsHIPS, ABS Polyolefins

prop

ertie

s

Polymer A Polymer Bcontents

homogenous

heterogeneous

Polymer blends

crystallinity

Typical Phase Diagram of One Component

103

Typical Phase Diagram of Mixture I:Liquid-Liquid

104UCST LCST LCST + UCST

Typical Phase Diagram of Mixture II:Liquid-Liquid & Liquid-Vapour

105There is no Vapour Phase in Polymer System

Typical Phase Diagram of Mixture III:Liquid-Solid & Solid-Solid

106

& Liquid-Liquid

There may exit Solid Phase in Polymer System

Relation between phase diagram and morphology

107

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

quench

TWO PHASE REGION

ONE PHASE REGIONbinodal curvespinodal curve

T

Phase Diagrams of Polymer Blends

108

UCST LCST

12( 2)Z ATk T

0A 0???A

Phase diagram of aqueous solutions ofPEO

109

UCST/LCST

Endothermic symmetricalpolymer mixture

Exothermal symmetricalpolymer mixture

Lower critical solutiontemperature (LCST)Upper critical solution

temperature (UCST)

AT

0A 0A

1 2G T S

TA B

,T ,T

110

> 0

111

or

Universal Phase Diagram

Why UCST or LCST ?ln lnA B

mix A B A BA B

G RTVN N

is the key issue.

0T

disp

0T

f.v

0T

.s.i

Dispersion force:~1/T monotonic decreasing,

disp 0 as T .

Free volume effect:Monotonic increasing with T,small, but positive at low T.

eff di s..sp i.f.v

Effective interaction parameter eff:Dispersion forcesFree volume effectsSpecific interactions

Specific interaction:Always < 0, decreasingmagnitude with increasing T.

A>0 UCST

A<0 LCSTA BT

112

The Phase Behavior of Polymer MixturesHow to judge it‘s homogeneous state or inhomogeneous state ?

What is the mechanism of phase separation?

G

2V1 g1+(1-V1) g2- G<0?

V1 g1+(1-V1) g2- G>0?

G~ A

G~ A

The shape of the mixing free energy curvewith one local minimum

114

P P Q Q

P Q

QP

Q P

PQ

Q P

mix P m P Q m QG kT g g

mg

Q Pm P m Q

Q P Q P

kT g g

Phase Separation will not occur if free energy curve has one local minimum

'G R G R

Free energy curve with two local minima

115A

mg

mg

B

mg m B m A

B A

g g

Phase Separation can take place if free energy curve has two local minima

Phase Diagram and Phase Equilibrium

212, 2 2, 2 1 2, 2

2

222, 2 2, 2 2 2

212, 1 2, 1 1 2, 1

2

222, , 21 2, 1 2 2, 1

1 1

ln 1 1

ln 1 1 1

ln 1 1

ln 1 1 1

p p p

p

p p

p p p p p

px xx

x x

x xx

xxxx

1, 2

2,

1

2

1,

2, 1p

p

p

p

116

Phase 1 Phase 2

1, 1 2

1, 2 2

,

2

1

,

1

1p p

p p

1, 1

2, 1

p

p

1, 2

2, 2

p

p

mixN GVkT

I. Phase diagram of symmetric mixtures

117

1ln ln 1 1mix N NG RTV 0mixG

ln 1ln 1 1 1 2bN N NRTV

N

1 ln 1 2 01 bRTV

N

1 ln2 1 ln 1bN

bb

A BT

1 ln2 1 1

bAT

BN

II. Phase diagram of asymmetric mixtures

1 1 2 2

1 1 2 2

d G dn dnG n nPhase 1 Phase 2

2111 2

22

1

2 V121 V2

22

+

T1 1 2 2

1 2,s s

m m

n x V n x VV V 1 21

1

1 1

1 11 1 2 2 2

1 1 11 2 2 2 1

m m

s s

m

s

V VG x xV V

V x x xV

G Y=A+B 2, 2*

G

22*

*

B 1

1 1 *2 2 1

2

m

s

VGB x xV

11 1

m

s

VA xVG*=A+B 2

*

A

21

1ln ln 1 1mix NG RTV

N

Finding the phase equilibrium conditions

119

G

22* 2**

G1 * 1 * 1 *

1 1 2 2 1 1 2m m

s s

V VY x x xV V

1 *2 0 1 1

1 *2 1 2 2

0,

1,

m

s

m

s

VY xV

VY xV

1 *1 1

m

s

VxV

1 *2 2

m

s

VxV

1 **1 1

m

s

VxV

2111 2

22

1

* **1 1* **2 2

1 **2 2

m

s

VxV

*1

*1

*

*2

*2

*

Phase * Phase **

Search the phase equilibrium point

*1

*1

*

*2

*2

*

120

121

Relations between Free Energy Chemical Potentials and Phase Diagram

0.0 0.2 0.4 0.6 0.8 1.0

1/RT

,2/

RT

2

0.0 0.2 0.4 0.6 0.8 1.0

2G

/kBT

*1

*1

* *2

*2

*

Phase * Phase **

1 22

2

G0

(1) Phase equilibrium curve – binodal

T ( ) ~ [ 2*(T), 2**(T)] binodal curve

122

(2) Metastable/unstable limits - spinodal

G

22

22

0G Spinodal curve2

22 1 2 2 2

1 1 2 01

Gx x

Phase 1 Phase 2

2 V121 V2

22+ unstable

metastable

(3) Critical pointCritical point

21/ 21

2, c1/ 2 1/ 2 1/ 2 1/ 21 2 1 2

1 1 1,2c

xx x x x

3

32 2 1 2 2 2

1 1 2 01

Gx x

Spinodal binodal :

For symmetricdi-blocks f=0.5 c 10.5N

For symmetricblends x1=x2

c 2N x=N

For polymersolutions c

12

x=N

Critical points dependence of N

2 2c , 0,x

c , cx T21/ 2

12, c1/ 2 1/ 2 1/ 2 1/ 2

1 2 1 2

1 1 1,2c

xx x x x

For blends

2

2, c1/ 2 1/ 22 2

1 1 1, 11 2c x x

For solutions x1=1

125

T

2

T

x2

Poor solvent>0.5

solvent=0.5

c12

126

Phase Separation Dynamics

(1) , , spinodal decomposition

(2) spinodal , , nucleationand growth

127

T

quenching( )

T

Phase diagram and phase separationmechanisms

(1) , ,spinodal decomposition

Spinodal decomposition

Nucleation and growth

(2) spinodal ,

, nucleation and growth

128

Phase Separation Mechanisms1. Nucleation and growth ( ) mechanism

* **

In metastable region,separation canproceed only byovercoming thebarrier with a largefluctuation incomposition.

Nucleation Growth

3 24( ) 43

G r r g r 0( ) ( '')g g gNucleation barrier: with

r: radius of the nuclear; : excess free energy per unit surface area.

droplet-dispersedphase

129

Phase Separation Mechanisms2. Spinodal decomposition ( ) mechanism

In unstable region, separation can occurspontaneously and continuously withoutany thermodynamic barrier.

’ **Early stageNo sharp interface

Late stageWell-established interfaces

Coarsening process*

Co/bi-continuousphase 130

Examples of Phase Separation Dynamics

131Nucleation & Growth Spinodal Decomposition

Interfaces between weakly immiscible polymers

132

, ,T p n

GA

int int intA A B BG A n n

int

A A B BG

A

A A B Bi i i i idG S dT V dp dn dn

A A B Bi i iG n n

A-rich B-rich

1 2

1 2 intA A A

1 2 intB B B

for each i phase

for interfacial region

intA

AnA

intB

BnA

interfacial tension is defined as theincrease in Gibbs free energy ofthe whole system per unit increasein interfacial area

and the unfavorable energy of interaction between the twodifferent species, which favors a narrow interface.

133

The width of the interface is determined by a balance of chainentropy, favoring a wider interface,

Suppose a loop of the Apolymer with Nloop unitsprotrudes into the B side ofthe interface

ABloop loop BG N k T Bk T AB 1loopN

Interfacial width:

S

1/2

,

22

6loop

g loop

bNR

AB

26

b

b: Kuhn segment

AB AB /A BV k T S / 2V S Segment density: 3AN

Nb

ABAB 2 6

Bk Tb

(Helfand, E.; Tagami, Y. JPS, PL, 1971, 9, 741)

A-rich B-rich

4.3 Theromodynamics of Block Copolymers

diblock triblock random multiblock

four arm starblock graft copolymer

134

Thermoplastics ( ) PU, SBS

Applications:

Compabilitzier ( ) of Polymer Blends

Nanotechnology, Biomaterials…

Self-assembly of Diblock Copolymers

Melts

Solids

Solutions

Microphase (mesophase, nanophase) separation( ) is driven by chemicalincompatibilities between the different blocksthat make up block copolymer molecules.

Micellization ( ) occurs when block copolymerchains associate into, often spherical, micelles ( ) indilute solution in a selective solvent ( ). Inconcentrated solution, micelles can order into gels ( ).

Crystallization of the crystalline block from meltoften leads to a distinct (usually lamellar ( ))structure, with a different periodicity from themelt . 135

Microphase Separation of DiblockCopolymers (BCPs)

Phase diagram

f is the volume fraction ofone component. f controlswhich ordered structuresare accessed beneath theorder-disorder transition.

N expresses the enthalpic-entropic balance. It is used toparameterize block copolymerphase behavior, along with thecomposition of the copolymer.

HomogeneousState

order

disorder

order

order

StructurallyOrderState

StructurallyOrderState

136

Microphase Separation of TriblockCopolymers

137

Thermodynamics of Microphase Separation

Minimize interfacial areaand Maximize chainconformational entropy(MIN-MAX Principle)

F: free energy per chainN: number of segments (=NA + NB)b: Kuhn length vb b3, bA bBL: domain periodicity

: interfacial area per chainAB: interfacial energy per areaAB: segment-segment interaction parameter

, AB L/2Lamellar structure

2 1 ( )2AB AB AA BB

ZkT AB

A BT

or

2 6AB

ABkTb

138

entropic spring term

2

2

3 ( / 2)2LAM AB

LF kTNb

Thermodynamics of Microphase Separation

Free energy of lamellae:

enthalpic termUsing 3

2LNb V

we have3 2

22 2

3 ( / 2)6 ( / 2) 2AB

LAMkT Nb LF kT Lb L Nb L

1/3 1/31.2 14AB ABkT N kTN

3/ 2( ) (4.8) ~ 10.5cN

0L

FLAM2 2 0opt

opt

LL

Thus, the optimum period of the lamellae and the lamellar free energy are:3/13/12.1 ABLAM kTNF2/3 1/63

2opt ABL bN 2)( optopt

optLAM LL

LFand

Assume mdisorder AB A B AB A B

s

VF kT N kTV

At the order-disorder transition:For a 50/50 volume fraction, A B =1/4, so:

Critical point

See Appendix

Symmetric blendsBCPs

disor rLA eM de dFF

139c 2N

Appendix3/2 2

2 2

3/2 2

2 2

2

2

2 2 22

2 2

3 3( ) ln ln , , ln exp2 2

3 3 ln ln exp2 2

3 .2

/ 2 / 2 / 23 3 3( / 2) 12 2 2

el

el

hS h k k h N h N dh kNb Nb

hk kNb Nb

hk constNb

L Nb L LF h L kT kT kT

Nb Nb Nb2

140

141

:

142

(segment)

(Thermal Blob)

(Single Chain)

2

3Tugl

1

3Tu ll

23T BT

uE kTg k T

20 /el h L

eh N l

( )

143

(1)

(2)

(3)

(5)

(4)

144

G H T S

2

2

32conf BS k

xlh0H

2

3cx vH

h

2

2

32conf BS k

xlh

23

3

3

3/ ~ c

c

vxvwh

xvH hv h

0confS

2

3ln2

cx vP hh

,

0Gh

0Gor

ln 0P h

2

2

32conf BS k

xlh

+blob model

+blob model

145

2

2

32 BS k

Nlh

2B

1/2ln eelectr e B

e

l fN RH R k TR N b

2B

electr e Be

l fNH R k T

Ror

( )

( )

1 21 2

1 2

ln lnmM B

s

VS kV x x1 2

mM B

s

VH k TV

Flory-Huggins Theory

1 2m

M Bs

VH k TV

1 21 2

2

0 20

ln ln1

32

mB

s

B

VS kV x

N khh N0:

0confS

145

146

1 21 2

1 2

ln lnmM B

s

VS kV x x1 2

mM B

s

VH k TV

int erface ABH2

2

3 ( / 2)2conf

LS kTNa

0confS

0FL optL

mAB A B AB A B

s

VH kT N kTV

disoeder orderF F

0confS

10.5c N

,1 ,2 1,2i i i2

2 0MG

Phase equilibrium(binodal)Phase stability(spinodal)

M M MG H T S

2cN 1 / 2c

Critical Point:

Critical Point3

3 0MG

Symmetric Blends: Solutions:

Phase Diagram