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