Polymers Physics Michael Rubinstein
University of North Carolina at Chapel Hill
1. “Real” Chains
2. Thermodynamics of Mixtures
3. Polymer Solutions
Outline
Summary of Ideal Chains Ideal chains: no interactions between monomers separated by many bonds
Mean square end-to-end distance of ideal linear polymer 22 NbR
Mean square radius of gyration of ideal linear polymer
6
22 Nb
Rg
2
22/3
232
3exp
2
3,
Nb
R
NbRNP d
Probability distribution function
Free energy of an ideal chain 2
2
2
3
Nb
RkTF
Entropic Hooke’s Law RNb
kTf
2
3
Pair correlation function 2
13)(
rbrg
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
d
R
Nb*
If chains are ideal 2/1bNR 2/1* dN small for d>2
Number of contacts between pairs of monomers that are far along
the chain, but close in space 2/2* dNN small for d>4
For d<4 there are many contacts between monomers in an ideal chain.
Interactions between monomers change conformations of real chains.
Life of a Polymer is a Balance of entropic and energetic parts of free energy
Fent Feng
Entropic part “wants”
chains to have ideal-like
conformations
Energetic part typically
“wants” something else
(e.g. fewer monomer-
monomer contacts
in a good solvent).
Chain has to find a compromise between these two desires
and optimize its shape and size.
R
Flory Theory Number density of monomers in a chain is N/R3
Probability of another monomer being within excluded
volume v of a given monomer is vN/R3
2
2
3
2
Nb
R
R
NvkTF
Excluded volume interaction energy per monomer kTvN/R3
Excluded volume interaction energy per chain kTvN2/R3
Entropic part of the free energy is of order kTR2/(Nb2)
Flory approximation of the total free energy of a real chain
Free energy is minimum at 5/35/25/1 NbvR
Universal relation NR ~ - Flory scaling exponent
More accurate estimate 588.0
R
M w ( g / m o l e )
1 0 4 1 0 5 1 0 6 1 0 7 1 0 8
Rg (
nm
)
1
1 0
1 0 0
1 0 0 0
1 / 2 0 . 5 9
Radius of gyration of polystyrene chains in a q-solvent (cyclohexane
at 34.5oC) and in a good solvent (benzene at 25oC).
Fetters et al J. Phys. Chem. Ref. Data 23, 619 (1994)
Polymer Under Tension
Rf
x
Unperturbed size 2/1
0 bNR 5/3bNRF
Tension blobs (Pincus blobs) of size x
contain g monomers
2/1bgx5/3bgx
Chains are almost unperturbed on length scales up to x
On larger length scales they are stretched arrays of Pincus blobs
xxx
20
2 RNb
g
NR f
3/2
3/5
3/2
3/5
xxx F
f
RNb
g
NR
Size of Pincus blobs
fR
R20x
2/3
2/5
f
F
R
Rx
Ideal chain Real chain
Polymer Under Tension
Size of Pincus blobs
fR
R20x
2/3
2/5
f
F
R
Rx
Free energy cost for stretching a chain is on order kT per blob 2
0
R
RkT
RkT
g
NkTF
ff
x
2/5
F
ff
R
RkT
RkT
g
NkTF
x
Ideal chain Real chain
Tension force is on order kT divided by blob size x
0020
R
R
R
kTR
R
kTkTf
ff
x
2/32/3
2/5
F
f
Ff
FR
R
R
kTR
R
kTkTf
x
1 3/2
kT/b f
R0 RF Rmax
Rf
linear elasticity non-linear elasticity
even at low forces
0 For b = 1nm at room T
kT/b = 4pN Log – log plot
Polymer Under Tension Ideal chain Real chain
fRRkTf 2
0 2/32/5
fF RRkTf 3/2
RF
1
kT/b
f
R0 Rmax
Rf linear elasticity non-linear elasticity
Challenge Problem 1: Pulling a Ring Consider a pair of forces applied to monomers 1 and 1+N/2
i. Show that the modulus of an ideal ring
is twice the modulus of a linear N/2-mer
ii. How does modulus of a ring in a good solvent Gr
compare to twice the modulus of linear N/2-mer GN/2?
f
f
2f
2f
2f
2f
a. Gr = 2GN/2 similar to ideal case
b. Gr < 2GN/2 because the entropy of sections of a ring is lower
c. Gr > 2GN/2 because ring sections reinforce each other
Biaxial Compression
D
R||
Ideal chain Real chain
D
R||
On length scales smaller than compression blob of size D chain is almost
unperturbed 2
b
Dg
3/5
b
Dg
Occupied part of the tube
2/12/1
|| bNg
NDR
Nb
D
b
g
NDR
3/2
||
Free energy of confinement 2
D
bkTN
g
NkTFconf
3/5
D
bkTN
g
NkTFconf
20
D
RkTFconf
3/5
D
RkTF F
conf
Challenge Problem 2:
Biaxial Confinement of a Semiflexible Chain
Consider a semiflexible polymer – e.g. double-
stranded 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).
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 d<D<RF).
D
R||
RF
Uniaxial Compression
D
Free energy of confinement in a slit is the
same as in the cylindrical pore
Longitudinal size of an ideal chain in a slit is the same as for an
unperturbed ideal chain. 2/1|| bNR
2-dimensional Flory theory for real chain confined in a slit
2
2||
2||
22
)/(
)/(
DgN
R
R
gNDkTF
D2 is the excluded area
of a confinement blob
with g monomers
Longitudinal size of a real chain in a slit
4/14/3
4/3
||
D
bbN
g
NDR
Fractal dimension of real chains in 2-d is D = 4/3
Scaling Model of Real Chains Thermal blob - length scale at which excluded
volume interactions are of order kT kTg
vkTT
T 3
2
x
Chain is ideal on length scales smaller than thermal blob 2/1
TT bgx
Number of monomers in a thermal blob 26 vbgT
Size of a thermal blob v
bT
4
x
xT R
xT
Rgl
5/35/1
3
5/3
Nb
vb
g
NR
TT
x 3/1
3/1
23/1
Nv
b
g
NR
T
T
x
good solvent v>0
poor solvent v<0
Flory Theory of a Polymer in a Poor Solvent
2
2
3
2
Nb
R
R
NvkTF In poor solvent v < 0 and 0R
Cost of Confinement
2NbR Compression blob of size R
with g monomers
2
b
Rg
Confinement free energy 2
2
R
NbkT
g
NkTFconf
2
2
2
2
3
2
R
Nb
Nb
R
R
NvkTF For v < 0 0R
Three Body Repulsion
6
3
3
2
2
2
2
2
R
Nw
R
Nv
R
Nb
Nb
RkTF
Size of a globule 3/1
v
wNRgl
gT
xT
N 1
b
R
3/5
3/5
good
1/3
poor 1/2 q-solvent
1/3 non-solvent
athermal
5/3bNRathermal
3/1bNR solventnon
2/1bNR q
5/3
TTgood
g
NR x
3/1
TTpoor
g
NR x
xT
xT
Real Chains in Different Solvents
Temperature Dependence of Chain Size
Mayer f-function
1exp
kT
rUrf
-1 for r<b where U(r)>>kT { kT
rU for r>b where |U(r)|<kT
Excluded volume
32
0
2
0
2 14
44 bT
drrrUkT
drrdrrrfvb
b
q
Interaction parameter z is related to number of thermal blobs per chain
2/12/1
3N
T
TN
b
v
g
Nz
T
q
Chain contraction in a poor solvent
3/1
2/1
z
bN
R
Chain swelling in a good solvent
12
2/1
zbN
R
2
6
v
bgT
N1/2
(1 - q/T)
-40 -30 -20 -10 0 10 20 30
Rg
2/R
q
2
0.0
0.5
1.0
1.5
2.0
2.5
N1/2
(1 - q/T)
-5 0 5 10 15 20
Rg
2/R
q
2
0.0
0.5
1.0
1.5
2.0
2.5
Monte-Carlo simulations
Graessley et.al., Macromolecules
32, 3510, 1999 & I. Withers
Polystyrene in decalin
Berry, J. Chem. Phys.
44, 4550, 1966
Universal Temperature Dependence
of Chain Size
Summary for Real Chains
2/1bNR q
Size of Linear Chains
nearly ideal in q-solvents
588.018.0
3
12
3N
b
vbN
b
vbRgood
swollen in good solvents
588.0bNbNRathermal in athermal solvents
3/123/1NbvRpoor
collapsed into a globule in poor solvents
In good solvents Stretching a real chain
43.2)1/(1
FF R
RkT
R
RkTF
non-Hookean elasticity
Confinement of a real chain
7.1/1
D
RkT
D
RkTF FF
Excluded volume
T
Tbv
q3
1. Real Chains
2. Thermodynamics of Mixtures
3. Polymer Solutions
Outline
Assume, for simplicity, no volume change on mixing
Consider a mixture with total number n of monomers (sites)
and total volume v0n where v0 – volume of a lattice site
Assume that a monomer of each species occupies the same volume v0
Binary Mixtures Homogeneous – if components are intermixed on a molecular scale
Heterogeneous – if there are distinct phases
Volume fractions
BA
AA
VV
V+ =
VA VB VA+VB
1B
A n = n monomers of type A and B nB = (1- n monomers of type B
Entropy of Mixing
+ =
- volume fraction of A
nv0 1nv0 nv0 v0 – volume of a lattice site
Number of translational states of a molecule
in a mixture is the number of sites n n
v
VV BAAB
0
Entropy change upon mixing of a molecule A
Number of states of molecule A in a pure A phase nv
VAA
0
ln1
lnlnlnln kkkkkSA
ABAABA
nA/NA and nB/NB – number of A and B molecules
Total entropy change upon mixing
1lnln
B
B
A
AB
B
BA
A
A
N
n
N
nkS
N
nS
N
nS
S
0 1
Infinite
slopes
1 - volume fraction of B
z – coordination number
(number of neighbors)
uij – interaction of i with j
Average energy per monomer of pure components
before mixing BBAAi uuz
u 12
Mean Field Theory Average energy per A monomer
ABAAA uuz
u 12
B monomer with probability 1 – A ?
? ? ?
? ?
A monomer with probability
Average energy per B monomer BBABB uuz
u 12
Average energy per monomer in homogeneous mixture
BABBAA
f uun
ununu 1
Energy of mixing ifm uuu
U
BBu
z
2
AAuz
2
1 0
um
um BBABAA uuu
z 22 1122
Energy of a Homogeneous Mixture
Energy of Mixing
kTnuu
uznnuU BBAAABmm
1
21
0 1
Um
Energy change on mixing per site
Probability of AB contact 1
)1( kTum
per site
)1(1ln
1ln
BA
m
NNkT
n
F
Free Energy of Mixing
mmm STUF
2
BBAAAB
uuu
kT
z
Flory interaction parameter
Regular solution: NA = NB = 1 Polymer solution: NA = N >>1, NB = 1
Polymer blend: NA >>1, NB >> 1
At lower T for >0 repulsive
interactions are important and
there is a composition range
with thermodynamically stable
phase separated state.
Flory-Huggins Free Energy of Mixing
11ln
1ln
BA
mix
NNkT
n
F
At high T entropy of mixing dominates,
Fmix is convex and homogeneous
mixture is stable at all compositions.
'''
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0 0.2 0.4 0.6 0.8 1
250o K
” ’
F
mix/(
kn)
300o K
350o K
NA = 200 NB = 100 T
Ko5
This composition range, called
miscibility gap, is determined by
the common tangent line.
Phase Diagrams
11ln
1ln
BA
mix
NNkT
n
F
For cr mixture is stable at all
compositions. 2
11
2
1
>
BAcr
NNFor
there is a miscibility gap for '''
0
1
2
3
4
0 0.2 0.4 0.6 0.8 1
N
Single Phase
Two Phases
Ncr
cr
For a symmetric blend NA=NB=N
cr=2/N cr=1/2
For polymer solutions NA = N, NB = 1
BA
B
crNN
N
Critical composition
NNcr
2
11
2
1
Ncr
1
-0.1
0
’ ” sp1 sp2
N
Fm
ix
kT
n N=2.7
NA = NB
Phase Diagram of Polymer Solutions
M=53.3kg/mole
Polyisoprene in dioxane
Takano et al., Polym J. 17, 1123, 1985
Single Phase
Two Phases
spinodal
binodal
Polymer solutions phase separate upon decreasing
solvent quality below q-temperature
Upper critical solution temperature B>0
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
T
BA
metastable
(nucleation
and growth)
unstable
(spinodal
decomposition)
membrane
h
solvent solution
10
6 P
/cR
T (m
ole
s/g
)
c (g/ml)
0.00 0.01 0.02
0
5
10
15
20
25
30
M n = 70800 g/mole
M n = 200000 g/mole
M n = 506000 g/mole
Intermolecular Interactions
Osmotic pressure
P ...2
2cAM
cRT
n
Osmometer
Poly(a-methylstyrene)
in toluene at 25 oC
Noda et al, Macromol.
16, 668, 1981
A2 – second virial coefficient
Mixtures at Low Compositions
11ln
1ln
BA
mix
NNkT
n
F
...
62
1
2
1ln
32
BBBA
mix
NNNNkT
n
F
Expand ln(1-) in powers of composition
Osmotic pressure
P ...
322
1 32
33
2
BBA
mix
n
mix
NNNb
kTn
F
bV
F
A
Virial expansion in powers
of number density cn = /b3
P ...
2
32nn
A
n wccv
N
ckT
Excluded volume 32
1b
Nv
B
3-body interaction
BN
bw
3
6
In polymer solutions NB = 1 AvNb
MA
T
T
b
v3
202
3
221
q
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.
Excluded volume BB N
bb
Nv
332
1
is very small for NB >> 1
Thermal blob 2
2
6
BT Nv
bg BTT bNgb x
Chains smaller than thermal blob NA < NB2 are nearly ideal.
In monodisperse NA = NB and weakly polydisperse
melts chains are almost ideal.
In strongly asymmetric blends NA > NB2
long chains are swollen 10/1
2
2/1
5/3
2
5/3
B
AA
B
AB
T
ATA
N
NbN
N
NbN
g
NR x
RA
NA
bNA
NB2
1/2
3/5
b 1
Flory Theorem
M. Lang
RA
2/(
b2N
A)
NA/NB2
NB
NB
NB
NB
NB
NB
NB
Long NA-mer in a 3-d Melt of NB-mers
Why doesn’t Flory Theorem work?
5/1
22
2
B
A
A
A
N
N
Nb
R
Challenge Problem 3:
Challenge Problem 4:
v0 – volume of a lattice site = volume of a small B monomer
mv0 – volume of an A monomer m times larger than B monomer
Mixing of Polymers with Asymmetric Monomers
NA – monomers
per A chain
NB – monomers
per B chain
Derive the free energy of mixing
Fmix of A and B polymers.
Calculate the size RA of dilute A-
chains 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 )
1ln
1ln
BA
mix
vvkT
V
ST
Energetic part per unit volume
10v
kTV
Umix
Many low molecular weight liquids are miscible
Some polymer – solvent pairs are miscible
Very few polymer blends are miscible
vA – volume of A chain
vB – volume of B chain
v0 – volume of a lattice site
T
BA Flory interaction parameter
Chains are almost ideal in polymer melts as long as they are
shorter than square of the average degree of polymeriztion NA<N2
1. Real Chains
2. Thermodynamics of Mixtures
3. Polymer Solutions
Outline
Which of These Chains are Ideal?
Quiz #1
C. Chain in a sediment of a polymer solution
in a poor solvent
A. Polymer in a solution of its monomers
B. Polymer dissolved in a melt of identical chains
D. None of the above sediment
Polymer Solutions
T
2-phase
Tc
` ``
1/2 c 0
poor solvent
Theta solvent
021 33
bT
Tbv
q
Chains are nearly ideal
at all concentrations NbR
Overlap concentration
NR
Nb 13
3* q
good solvent
q dilute q *
* * *1
semidilute
qsolvent
Chain is ideal if it is smaller than
thermal blob
v
bT
4
x
Boundaries of dilute q-regime
N
bb
T
Tbv
33321
q
NT
11qTemperatures at which chains begin
to either swell or collapse
v
Poor Solvent
Critical composition
NNcr
2
11
2
1
Ncr
1
Critical interaction
parameter
M = 43.6 kg/mole
Solution phase separates below
the binodal into a dilute supernatant
of isolated globules at ` and
concentrated sediment at ``.
` ``
Sediment concentration `` is determined by the balance of second and
third virial term (similar to the concentration inside globules).
3
3
312''
glR
Nb
b
v
3/1
3/12
3/1
3/1
12 v
NbbNRgl
Polystyrene in cyclohexane ( ) and
polyisobutylene in diisobutyl ketone ( )
Shultz and Flory, J. Am. Chem. Soc.
74, 4760, 1952
M = 1,270 kg/mole
Poor Solvent Dilute Supernatant of Globules
Globules behave as liquid droplets with size
3/1
3/12
v
NbRgl
Surface tension is of order kT per thermal blob
2
2
2
8212
x
b
kTv
b
kTkT
T
v
bT
4
x
Total surface energy of a globule 3/2
4
3/4
2
22 N
b
vkTkTRR
T
glgl
x
Concentration of a dilute supernatant
3/2
4
3/4
3
2
expexp''' Nb
v
b
v
kT
Rgl
is balanced by its translational entropy kTln’
is different from the
mean field prediction.
Rgl
xT
Good Solvent
* dilute
good
(swollen)
semidilute good
conce
ntr
ated
T q
2-phase
dilute q semidilute q c
Tc
` `` dilute
poor
(globules)
v
0
Swollen chain size
in dilute solutions
5/35/1
3N
b
vbR
Overlap concentration
5/4
5/33
3
3*
N
v
b
R
Nb
Semidilute solutions *1
x
Correlation length x
Distance from a monomer to nearest
monomers on neighboring chains.
Correlation Length
x
R
For r < x monomers are surrounded by
solvent and monomers from the same chain.
The properties of this section of the chain
of size x are the same as in dilute solutions.
g – number of monomers inside a correlation
volume, called correlation blob.
For r > x sections of neighboring chains overlap and screen each other.
On length scales r > x polymers are ideal chains - melt
with N/g effective segments of size x.
Correlation blobs are at overlap
5/35/1
3g
b
vb
x
3
3
x
gb 4/3
4/13
x
v
bb
2/1
8/1
3
2/1
Nb
vb
g
NR
x
Semidilute Solutions
R
N g
x
xT
gT 1
b 1/2
1/2
3/5 x
R
On length scales less than xT chain is ideal because excluded volume
interactions are weaker than kT.
r
n
r ~ n1/2
On length scales larger than xT but smaller than x excluded volume
interactions are strong enough to swell the chain. r ~ n3/5
On length scales larger than x excluded volume interactions are
screened by surrounding chains. r ~ n1/2
xT
Scaling Theory of Semidilute Solutions
At overlap concentration 5/4
5/33*
N
v
b
chain has its dilute solution size 5/3
5/1
3N
b
vbRF
In semidilute solutions R is a power of that matches dilute size at *
xxxx
F Nb
vbRR
5/)43(5/)31(
3*
Chains in semidilute solutions are random walks
2
1
5
43
xx = -1/8
2/1
8/1
3
8/1
*N
b
vbRR F
Similarly correlation length yyyy
F Nb
vbR
x 5/)43(
5/)31(
3*
is independent of N in
semidilute solution 3 + 4y = 0 y = -3/4 4/3
4/134/3
*
x
v
bbRF
Concentrated Solutions Correlation length x decreases with concentration, while thermal blob
size xT is independent of concentration.
At concentration ** the two length are equal and
intermediate swollen regime disappears. Txx
Tv
b
v
bb xx
44/3
4/13
3
**
b
v
This concentration is analogous to ’’ in poor solvent at
which two- and three-body interactions are balanced.
In concentrated solutions chains are ideal at all length scales.
On length scales less than x chains are ideal because excluded volume
interactions are weaker than kT.
On length scales larger than x chains are ideal because excluded
volume interactions are screened by surrounding chains.
Polymer Solutions
T q
2-phase
*
dilute q semidilute q c
Tc
` `` dilute
poor
(globules)
dilute
good
(swollen)
semidilute good
conce
ntr
ated
v
0
**
R
-1/8 RF
R0
** *
-3/4
-1
xT
1
b
8/1
**0
RR
for * < < **
In athermal solvent 3bv bT x
1**
4/3x
b
8/1
2/1
bNR
Osmotic Pressure
In dilute solutions < * - van’t Hoff Law Nb
kT 3
P
Osmotic pressure P in semidilute solutions > * is a stronger
function of concentration
P
*3
f
Nb
kT
*
f { 1 for < *
(/*)z for > *
15/45/3
3
1
3*3
P z
zz
z
Nb
v
b
kT
Nb
kT
Osmotic pressure in semidilute solutions
is independent of chain length (4z/5-1=0). z=5/4
3
4/94/3
33 x
kT
b
v
b
kT
P
Neighboring blobs repel each other with energy of order kT.
3.1
**
3
1
P
f
kT
Nb
Poly(a-methylstyrene) in toluene at 25 oC
Noda et al, Macromol. 16, 668, 1981
Concentration Dependence of Osmotic Pressure
Semidilute Theta Solutions
T q
2-phase
*
dilute q semidilute q c
Tc
` `` dilute
poor
(globules)
dilute
good
(swollen)
semidilute good
conce
ntr
ated
v
0
** Chains are almost ideal 2/1bNR
x 2
bg
x
Correlation blobs are space-filling
xx
x
x
bbbgb
3
23
3
3 /
NbR 0xNR
Nb 130
3* at
2/1
*
2/1 xx
x
NbbN
x
Correlation length in semidilute solutions
is independent of chain length (1+x)/2 = 0 x=-1
Scaling assumption for > *
x
b
x
b
What is the meaning of
correlation length in semidilute
theta solutions?
Quiz # 2
How different are semidilute theta
solutions from ideal solutions of
ideal chains?
Osmotic Pressure in Semidilute Theta Solutions
Mean-field prediction
P ...3
63
b
w
Nb
kT
First term (van’t Hoff law) is important in dilute solutions
Three-body term is larger than linear
in semidilute solutions > 1/N1/2 3
3
b
kTP
Scaling Theory
P
*3
h
Nb
kT
*
h { 1 for < *
(/*)y for > *
12/1
3*3
P yy
y
Nb
kT
Nb
kT
Osmotic pressure in semidilute qsolutions
is independent of chain length (y/2-1=0). y=2 3
3
3 x
kT
b
kTP
0.001 0.01 0.1 1
P (
Pa
)
102
103
104
105
3
1
2.31
Osmotic Pressure
Polyisobutylene in benzene
at q24.5 oC (filled circles),
in benzene at 50 oC (filled
squares), in cyclohexane at
30 oC (open circles) and at
8 oC (open squares)
Flory and Daoust J. Polym.
Sci. 25, 429, 1957
Correlation length in semidilute q-solvents is of the order
of the distance between 3-body contacts.
Number density of n-body contacts ~ n/b3
Distance between n-body contacts in 3-dimensional space 3/nn br
x
b
3
33
x b
kTkTP
Summary of Polymer Solutions
T q
2-phase
*
dilute q semidilute q c
Tc
` `` dilute
poor
(globules)
dilute
good
(swollen)
semidilute good
conce
ntr
ated
v
0
**
1/2
In poor solvent part of the
diagram binodal separates
2-phase from 2 single phase
regions:
Dilute globules at low
concentrations < ’
Concentrated solutions
with overlapping ideal
chains at > ”
Near qtemperature there are dilute and semidilute q-regimes
with ideal chains.
Dilute good solvent regime with swollen chains at < * and v > 0.
Semidilute good solvent regime at * < < ** with chains swollen
at intermediate length scales shorter than correlation length x.
Osmotic pressure in semidilute solutions is kT per correlation volume x3.
Challenge Problem 5: Confinement of Polymers in Solutions & Melt
Calculate the pressure between two solid plates fully immersed into
a semidilute polymer solution or melt as a function of separation D
between plates for separations smaller than chain size R.
Assume no attraction between plates and polymer.
Separately consider cases with correlation length x<D<R and b<D<x
semidilute polymer
solution or melt
D
x
Single Chain Adsorption xads
Volume fraction in a chain section of size xads containing g monomers
Ideal chain Real chain
adsads
bgb
xx
3
3
adsorption blob 2/1bgads x5/3bgads x
adsorption blob
3/4
3
3
adsads
bgb
xx
Number of monomers per adsorption blob in contact with the surface
bb
bads
xx
2
3
3/22
3
bb
bads
xx
Energy gain per monomer in contact with the surface is -ekT
Energy gain per adsorption blob
kTb
kT ads x
e kTb
kT ads
3/2
xe
Single Chain Adsorption xads Ideal chain Real chain
Size xads and number of monomers g in an adsorption blob
1badsx
e 1
3/2
badsx
e
ex
bads
2/3ex
bads
Free energy of an adsorbed chain
2ekTNg
NkTF
2/5ekTNg
NkTF
22
e
x
bg ads 2/5
3/5
e
x
bg ads
Flory Theory of Adsorption
D
b Fraction of monomers in direct contact
with the surface is b/D
Number of monomers in direct contact
with the surface is Nb/D
Energy gain per monomer contact with
the surface is -kTe
Energy gain from surface interactions per chain D
bkTNF eint
Total free energy is a sum of interaction and confinement parts
D
bNkT
D
bkTNF e
2
D
bNkT
D
bkTNF e
3/5
intFFF conf Ideal chain Real chain
Optimal thickness
e
bD
2/3e
bD
xads
x(z) z
z de Gennes’ self-similar carpet
Multi-Chain Adsorption
First layer is dense packing of adsorption blobs
Polymer concentration decays from the high value in the first layer.
The correlation length x(z) corresponding to concentration (z) at
distance z from the surface is of the order of this distance.
xads
Single-chain adsorption
2/3ex
bads
3/4
b
zz
z
z
bz
4/3x Coverage is controlled by
the first layer of blobs.
22
2/13/12
3/4
3
3ads
ads
ads
R g
b
bbdz
b
zbdz
b
z
adsx
e
x
x
2/5 eadsg
x
H
s
z
Alexander – de Gennes Brush
Grafting density s number of chains per
unit area.
Distance between sections of chains
x s1/2
Number of monomers per blob
g ~ x1/ ~ s-1/(2)
Thickness of the brush H ~ x N/g ~ Ns(1-)/(2)
Energy per chain Echain ~ kT N/g ~ kTNs1/(2)
Energy per unit volume P3xx
sss kTkT
Hg
NkT
HE
V
Echain
chain
102
103
10-5
10-4
10-3
10-2
10-1
Grafting density s (nm-2)
Bru
sh h
eight
H (
nm
) Polymer Brush Height depends on Grafting Density
If the grafting density is high, chains
repel each other and stretch away from
the surface, forming a polymer brush.
Mushroom Regime
unperturbed size
H ≃ R0
H ~ N s1/3