Johannes Gutenberg-University Mainz
Institute for Physics
Chapter 4: POLYMER SOLUTIONS AND MIXTURES
4.1
•
•
•
Excluded volume interaction & Flory theory
idea of excluded volume interactions
θ−temperature
Flory approach for the free energy and scaling behavior of RE with excluded volume
interactions
Up to here (see script Chap. 3) we looked at the properties of a single chain with increasing
concentration polymer will interpenetrate each other. We can distinguish the following
concentration regions:
Range of concentrations
single
polymer
dilute solution
c << c*
overlap concentration
c ≈ c*
semi-dilute solution
c > c* (actual concentration low)
melt
Overlap concentration c∗
pervaded volume V~R3
overlap of single polymer if c∗ =
⇒ c∗ ~ N 1−3ν ~ N −0.8 if ν =
Nvmon Nvmon Nvmon
≈
= ν 3
V
R3
(N l)
(with ν = Flory exponent)
3
for good solvent conditions.
5
For high degrees of polymerization, the overlap concentration can be rather low!
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone:
+49
(0)(0)
6131/
392-3323
phone:
+49
6131/
392-3323
Fax:
+49
(0)(0)
6131/
392-5441
Fax:
+49
6131/
392-5441
email:
[email protected]
email:
[email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Now we look at the thermodynamic properties of polymers in solutions at c ≥ c∗ and in polymer
mixtures (dense systems).
i.e. phase separations & osmotic pressure
Flory-Huggins theory
A mixture is miscible if the free enthalpy of mixing ΔGmix = ΔH mix − T ΔS mix < 0 where ΔH mix
and ΔSmix are the enthalpy and entropy of mixing, respectively.
First we look at a mixture of low molecular weigth substances
• distribution on lattice without change in volume
• both components have lattice volume so in this theory; number fraction=volume fraction
Entropy of mixing
n1 = number of lattice sites for component 1
n2 = number of lattice sites for component 2
n=n1+n2 = total number of lattice sites
Calculation of entropy change
ΔSmix = k B ln Ω
Ω=
where Ω =
number of possible distributions of component 1 & 2 on
lattice sites
n!
the denominator takes permutations of component 1&2 among each
n1 !n2 !
other into account
with ln x ! ≈ x ln x − x
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
⇒
⎡
n
n
⎛n ⎞
⎛ n ⎞⎤
ΔSmix = − k B ⎢ n1 ln ⎜ 1 ⎟ + n2 ln ⎜ 2 ⎟ ⎥ with number fractions c1 = 1 and c2 = 2
n
n
⎝n⎠
⎝ n ⎠⎦
⎣
since component 1&2 have same lattice volume: number fraction=volume fraction
ΔSmix = −nk B ⎡⎣ c1 ln ( c1 ) + c2 ln ( c2 ) ⎤⎦
⇒
⇒
if n=NA
n
ΔSmix
= − R ⎡⎣c1 ln ( c1 ) + c2 ln ( c2 ) ⎤⎦
per mole!
with gas-constant R=kBNA
Calculation of enthalpy change
Every lattice site is in the average surrounded
by
c1 z neighbors of component 1
by
c2 z neighbors of component 2
(z = coordination number)
Looking at averages implies that concentration fluctuations are small. The Flory-Huggins theory
is a mean-field theory.
Enthalpy of mixing
z=6
⇒
interaction enthalpy of component 1:
component 2:
1
nc1 (c1 z w11 + c2 z w12 )
2
1
H 2 = nc2 (c1 z w12 + c2 z w22 )
2
H1 =
with wij=pair-wise interaction energies
The factor 0.5 arises from the fact, that all pairs are counted twice.
The expressions for the pure components are:
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
H i0 =
1
nci z wii , i = 1, 2
2
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1
nzc1c2 (2w12 − w11 − w22 ) with c1 + c2 = 1
2
1
with Δw12 = w12 − ( w11 + w22 )
2
ΔH mix = ( H1 + H 2 ) − ( H10 + H 20 ) =
= nzc1c2 Δw12
⇒
M
ΔH mix
= nzc1c2 Δw12
Δw12 < 0
Δw12 > 0
Δw12 = 0
per mole!
exothermal mixing process
endothermal mixing process
athermal mixing process
introduction of the interaction parameter:
⇒
M
ΔH mix
= RT χ12c1c2
⇒
free enthalpy of mixing:
χ12 = Δw12
1
1
∝
RT
T
M
ΔGmix
= RT [ c1 ln c1 + c2 ln c2 + χ12 c1c2 ]
per mole
Polymer solutions
The calculation of the enthalpy of mixing is the same for mixtures of low molecular weight and
polymeric substances. It was based on pairwise interaction of segments/molecules. Connectivity
of the segment plays no role!
Calculation of entropy change
Entropy of mixing
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
n1=
n2=
N=
n= n1+n2N =
number of solvent molecules
number of polymers
lattice sites per polymer = degree of polymerization
total number of lattice sites
Assume that i polymers are already distributed on the lattice
⇒
⇒
⇒
⇒
number of free sites is:
n-N×i
for the first segment of the (i+1) polymer there are (n-N×i) possibilities to find a site
for the second segment there are z × (number of free sites)
for the third to xth segment there are (z-1) × (number of free sites)
n − N ×i
n
n>>N otherwise there would be a significant reduction with each segment added
number of free sites is replaced by the average number:
•
Entropy of mixing
X
X
1
X
X
2
1
3
4
X
X
not considered:
X
X
2
X
..
1
X
X
⇒
4
excluded volume interaction
3
2
1
X
number of possibilities to distribute the (i+1)th polymer on the lattice:
N −2
(n − Ni )
⎡ (n − Ni ) ⎤ if z>>1
⎛ z −1 ⎞
⋅ ( z − 1) N − 2 ⎢
≈ (n − Ni ) N ⎜
ν i +1 = (1
n − Ni ) ⋅ z ⋅
⎟
⎥
424
3
n 3 1444
n ⎦3
⎣24444
⎝ n ⎠
1424
1st
4
2nd
⇒
N −1
all other (N-2) segments
number of possibilities to distribute all polymers on n2 lattice sites:
ν 1 ⋅ν 2 ⋅ ... ⋅ν n2
P2 =
denominator takes into account indistinguishability of polymers!
n2 !
for n>>N
⇒
⎛ z −1 ⎞
P2 ≈ ⎜
⎟
⎝ n ⎠
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
n2 ( n −1)
n!
(n − n2 N )!n2 !
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desorientation of polymers
entropy of the solution:
⇒
S=kB ln P2
solution of polymers
with n=n1+n2N and ci =
ni
n1 + n2 N
⎧
⎛ z − 1 ⎞⎫
S = k B ⎨−n1 ln c1 − n2 ln c2 + n2 ( N − 1) ln ⎜
⎟⎬
⎝ e ⎠⎭
⎩
The final stage has been reached by
(1)
(2)
disorientation of the polymer (creation of an amorphous state)
dissolution of the disoriented polymer
substraction of the entropy change due to step (1)
⎛ z −1 ⎞
s (n1 = 0) = k B n2 ln N + k B n2 ( z − 1) ln ⎜
⎟
⎝ e ⎠
change solely due to step (2):
ΔS mix = S − S (n1 = 0) = − k B [ c1 ln c1 + c2 ln c2 ]
entropy of a polymer melt
⇒
⇒
⇒
c
⎡
⎤
M
ΔSmix
= − R ⎢c1 ln c1 + 2 ln c2 ⎥
N
⎣
⎦
⎡c
⎤
c
M
ΔS mix
= − R ⎢ 1 ln c1 + 2 ln c2 ⎥
N2
⎣ N1
⎦
polymer in solution
polymer mixture
⎧c
⎫
c
M
ΔGmix
= RT ⎨ 1 ln c1 + 2 ln c2 + χ12 c1c2 ⎬
N2
⎩ N1
⎭
in general Δw12 ∝ χ12 > 0 (endothermal mixing)
The entropy of mixing is for high molecular weights small.
⇒
Polymers are poorly miscible!
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Osmotic pressure & membrane osmometry
polymer solution
pure solvent
ns′ , p′,T
ns, np, p,T
semipermeable membrane: exchange of solvent until the chemical potential of the solvent is
equal on both sides`.
chem. potential pure solvent = μ s0 ( p ', T ) = μ s ( p, T ) =
Δμ s = μ s ( p, T ) − μ s0 ( p, T ) =
∂ΔGmix
∂ns
chem. potential solvent in polymer
solution
enthalpy for dilution(*)
p, T , np
n
c
⎧
⎫
n
remember: ΔGmix = nk BT ⎨cs ln cs + p ln c p + χ cs c p ⎬ with cs = s ; c p = p ; n = ns + n p N
n
n
N
⎩
⎭
∂ΔGmix
1
⎧
⎫
⇒
Δμ s =
= nk BT ⎨ln cs + (1 − )c p + χ c 2p ⎬ = μ s ( p, T ) − μ s0 ( p, T )
∂ns
N
⎩
⎭
1
⎧
⎫
⇒
(**)
Δμ sM = RT ⎨ln cs + (1 − )c p + χ c 2p ⎬ per mole
N
⎩
⎭
The osmotic pressure is the pressure difference p − p ' = Π
In a linear approximation (in equilibrium):
μ s ( p, T ) = μ s0 ( p ', T ) = μ s0 ( p, T ) − ( p − p ')
∂μ s0
= μ s0 ( p, T ) − Π vs
∂p T , n
1
424
3s
= vs
= molar volume
with (*) ⇒
Δμ s = −Π vs
with (**) ⇒
1
⎧
⎫
−Πvs = RT ⎨ln cs + (1 − )c p + χ c 2p ⎬
N
⎩
⎭
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
1
c p << 1: ln cs = ln(1 − c p ) ≈ − c p − c 2p + ...
2
⎧ cp ⎛ 1
⎫
⎞
−Πvs = RT ⎨ + ⎜ − χ ⎟ c 2p + ...⎬
⎠
⎩N ⎝2
⎭
c
van’t Hoff Law
Πvs = RT p
N
for low polymer concentrations
⇒
with
and the other terms are derivations from ideal behavior, they disappear for χ =
remember:
(1)
(2)
1
2
χ ∝ T −1
osmotic pressure is the pressure difference between polymer solution
and pure solution
⇒
1.
2.
3.
1
(high temperatures):
2
osmotic pressure is higher compared to ideal gas condition; solvent is incorporated into the
polymer and results in extension of the polymers; polymers repell each other.
good solvent conditions
for χ <
1
(low temperatures):
2
osmotic pressure is reduced compared to ideal gas condition; solvent is supressed from
polymer and results in strinkage of the polymers; polymers attract each other.
bad solvent conditions
for χ >
1
:
2
osmotic pressure corresponds to ideal gas condition.
θ-conditions
for χ =
Radius of PS in cyclohexane as a function of the solvent quality
bad solvent
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
good solvent
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Dependance of osmotic pressure on polymer concentration and solvent quality
Πv s
c p RT
A2
χ 12 < 12
( A2 > 0 )
χ 12 = 12
( A2 = 0 )
χ 12 > 12
( A2 < 0 )
good solvent
θ solvent
1
N
bad solvent
c
p
Question: How can we determine N and χ ?
Answer: plot
Π vs
versus c p !
c p RT
⇒
axis intercept
⇒
slope
1
N
1
= −χ
2
Π vs
⎡ 1 ⎛1
⎤
⎞
= ⎢ + ⎜ − χ ⎟ c p + ...⎥
c p RT ⎣ N ⎝ 2
⎠
⎦
=
= 2nd virial coefficient A2
Question: How can we measure Π ?
Answer: Membrane osmometry
The membrane is impermeable for the polymer. Solvent permeates through the membrane until
the chemical potential of the solvent is the same in both chambers. The excess column is a
measure of the hydrostatic pressure which is in equilibrium equal to the osmotic pressure.
⎡1 1
⎤
⎛1
⎞
c p + ⎜ − χ ⎟ c 2p + ...⎥
⎝2
⎠
⎣ N vs
⎦
ρ s g Δh = Π = RT ⎢
with
ρ s = solvent density; g = acceleration of gravity; Δh = excess column height;
vs = molar volume solvent
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Δh
μ S ( p '+ Π, T ) = μ S0 ( p ', T )
μS
μ S0
pure solvent
polymer solution
semipermeable membrane
Question: What kind of molecular weight average is measured by membrane osmometry?
[see “Physical chemistry of polymer solutions”, Kamide & Dobast]
1
⎧
⎫
−Πvs = RT ⎨ln cs + (1 − )c p + χ c 2p ⎬
N
⎩
⎭
for very (!) dilute solutions: Πvs = − RT ln cs = − RT ln(1 − ∑ i c p ,i )
with
∑c
i
∑n
n +∑ n
=
p ,i
i
s
i
Π=−
with (*) ⇒
p ,i
≈
∑n
p ,i
i
p ,i
ns
RT ⎛ ∑ i n p ,i
ln ⎜ 1 −
vs ⎜⎝
ns
⎞ (× )
∑ n p ,i
⎟⎟ ≈ RT i
vs ns
⎠
(× )
total concentration of solute v = vs ns + ∑ i n p ,i ≈ vs ns
c=
∑n
i
p ,i
Mi
v
⎛ ∑ n p ,i M i ⎞ ⎛ ∑ i n p ,i
=⎜ i
⎟
⎜ ∑ n p ,i ⎟ ⎜⎜ v
i
⎝14
4244
3⎠ ⎝
=M n
with (×) ⇒
⇒
⇒
Π = RT
∑n
i
p ,i
⎛ ∑ n p ,i
<< ns : ln ⎜1 − i
⎜
ns
⎝
⎞
∑ n p ,i
⎟⎟ = M n i
vs ns
⎠
notice: concentration in
weight
volume
⎞
∑ n p ,i
⎟⎟ ≈ − i
ns
⎠
c
Mn
Answer: Membrane osmometry determines the number averaged molecular weight! (in
very dilute solutions)
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Phase diagrams for polymer solutions and –mixtures
Phase diagrams (the basics)
Φα
Φ
Φβ
Assume a mixture with composition Φ separates into two phases with composition Φαand Φβ
Φ = α Φα + β Φ β with α + β = 1
with volume fractions α and β, respectively.
The original phase is stable, if
αβ
ΔGmix ≤ α ΔGmix (Φα ) + (1 − α ) ΔGmix (Φ β ) = ΔGmix
Gibbs stability criterion
free enthalpy of mixing as a function of volume fraction of one component
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
more complex situation:
convex and concave curve regions
stable, metastable and unstable regions
ΔGmix
at fixed temperature
stable
metastable
metastable
unstable
stable
spinodal
binodal
(1)
The binodal defines the boundary between the one-and two-phase region and can be
determined by tangent construction
⎛ ∂ΔGmix ⎞
⎜
⎟ =0
⎝ ∂Φ ⎠T , p
(2)
(minima)
The spinodal defines the boundary between the metastable and unstable region and can be
determined from the points of inflection
⎛ ∂ 2 ΔGmix ⎞
⎜
⎟ =0
2
⎝ ∂Φ ⎠T , p
unstable:
small fluctuations in concentration lead to spontaneous demixing
metastable:
system is stable in respect to small fluctuations; only if a sufficient large nucleus
with equilibrium composition is formed it growth in size. The nucleus has to be
large enough because of the surface tension between the two phases.
keyword: separation by nucleation
(3)
There is a temperature Tc and a composition Φ c where spinodal and binodal merge called
critical point
⎛ ∂ 3 ΔGmix ⎞
(extremum of the spinodal curve)
⎜
⎟ =0
3
⎝ ∂Φ ⎠T , p
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Construction of a phase diagram
T5
Å free energy
Tc
T3
T2
T1
Å
good solvent
temperature
Tc
bad solvent
T1
concentration
(1)
(2)
Determination of the free enthalpy as a function of Φ at various temperatures
Determination of stable and unstable regions and projection into the (T , Φ) space
⎧Φ
⎫
Φ
M
Flory ΔGmix
= RT ⎨ 1 ln Φ1 + 2 ln Φ 2 + χ12 Φ1Φ 2 ⎬ < 0
N2
⎩ N1
⎭
1
χ12 ∝
T
⇒
as condition for miscibility
In Flory-theory always lower miscibility gap and upper critical point
(miscibility at high temperatures)
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Polymer solution
N1=N ; N2=1 ; Φ1 = Φ ; Φ 2 = (1 − Φ )
(1)
(2)
(3)
M
1
1
∂ ΔGmix
= ln Φ + − ln(1 − Φ ) − 1 + (1 − 2Φ ) χ = 0
∂Φ RT
N
N
M
2
Δ
G
∂
1
1
mix
=
+
− 2χ = 0
spinodal:
2
∂Φ RT
N Φ (1 − Φ )
M
1
1
∂ 3 ΔGmix
=−
+
=0
critical point:
3
2
∂Φ RT
N Φ c (1 − Φ c ) 2
binodal:
⇒ Φc =
1
−
1
≈N 2
N +1
for large N>>1
insertion of Φ c into (2):
M
∂ 2 ΔGmix
1
1
+
− 2χc = 0
(Φ c ) = 0 ⇒
2
∂Φ RT
N Φ c (1 − Φ c )
1
−
1 1
1⎛ 1
1
1
⎞ 1
⇒ χc = (
+ 1) 2 = ⎜ + 2
+ 1⎟ = + N 2 + N −1 ∝ Tc −1
2 N
2⎝ N
2
N
⎠ 2
⇒ Asymmetric miscibility gap which shifts to very small polymer contents for high molecular
weight polymers.
⇒ Tc increases with molecular weight of polymer
Binodal experimentally determined from precipitation temperature
Mw=
•
No good description of
experimental data!
•
The dependence of χ on Φ
has to be taken into account;
also bad description of T
dependence!
binodal (Flory)
binodal
(experimental)
spinodal (Flory)
volume fraction polymer
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Polymer mixtures
Φ1 = Φ ; Φ 2 = (1 − Φ )
⎧Φ
⎫
Φ
M
ΔGmix
= RT ⎨ 1 ln Φ1 + 2 ln Φ 2 + χ12 Φ1Φ 2 ⎬ < 0 as condition for miscibility
N2
⎩ N1
⎭
M
1
1 ln(1 − Φ ) − 1
∂ ΔGmix
=
−
+ (1 − 2Φ ) χ = 0
ln Φ +
(1) binodal:
∂Φ RT
N1
N1
N2
(2)
spinodal:
M
1
1
∂ 2 ΔGmix
=
+
− 2χ = 0
2
∂Φ RT
N1Φ N 2 (1 − Φ )
(3)
critical point:
M
∂ 3 ΔGmix
1
1
=−
+
=0
3
2
N1Φ c N 2 (1 − Φ c ) 2
∂Φ RT
1
N1 N 2 + 1
)
2
1
1 N1 + 1 N 2 ∝ Tc−1
2
Tc increases with increasing N1 or N2
N1 << N2 or N2 << N1 ⇒ Φ c shifts towards very low concentrations of the longer
component.
with (2) ⇒ χ c =
⇒
⇒
(
⇒ Φc =
Special case: Polymer with the same degree of polymerization: N1 = N2 = N
⇒ Φ c = 1 2 and χ c = 2 N ∝ Tc−1
⇒
(χc N ) = 2
(χc N ) < 2
(χc N ) > 2
critical point
one-phase region
two-phase region
⇒ Phase diagram is symmetrical at Φ c for polymer mixtures with identical degree of
Polymerization
Mixture of symmetric polymers
χN=2.
2.
(χN)c=2
χ
temperatur
ΔGmix
two-phase
1.
one-phase
Φ
Φ
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
In summary
•
Connectivity of polymers ⇒ degree of possible disorder is limited ⇒ entropy of mixing is
reduced by prefactor 1 ⇒ monomer-monomer interaction determines mixing behavior.
N
•
As higher the molecular weight as higher the critical temperature; weak repulsive
interaction between the monomers leads to demixing.
•
Miscibility of polymers as significantly reduced compared to low molecular weight
substances.
•
Attractive interaction, such as dipole-dipole interaction / hydrogen bonds / donor acceptor
interaction, is necessary.
Flory-Huggins theory yields bad description of experimental data
The simple lattice model does not describe the behavior of dilute polymer solutions very well
(1)
Approach is based on mean-field-theory, i.e. lattice site occupation is uniform and every
molecule/ monomer sees the same averaged surrounding. It was assumed that the
monomer/ molecule distribution on the lattice is purely statistically.
⇒ only true if Δw12 = 0
⇒ fluctuations in particular relevant in dilute solutions are neglected
(2)
Volume changes due to mixing having an impact on the mobility/ flexibility (⇒ entropy)
are neglected.
⇒ χ is concentration dependent!
(3)
Interaction is assumed to be between nearest neighbors only. Any specific interaction
(longer ranged, inducing order or specific interaction involving only some specific groups)
between component A and B like: dipole-dipole interaction, hydrogen bonding, donoracceptor interaction, solvent orientation in polar solvent close to the chain.
Ansatz:
χ parameter comprises on enthalpic χ H and entropic χ S contribution
χ = χS + χH = A +
B
k BT
B
d ( χT )
ΔS
⎛ dχ ⎞
and χ S =
= A=−
with χ H = −T ⎜
⎟=
dT
kB
⎝ dT ⎠ k BT
from experiments ⇒ major contribution from non-combinatorial change in entropy: χ S
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
UCST = upper critical solution temperature
χ decreases with increasing temperature
e.g. polybutadiene (88% vinyl, protonated; 78% vinyl, deuterated)
mixing leads to increase in volume;
volume for local movements of the chains is larger ⇒increase in mobility/flexibility
⇒ increase in entropy.
⇒ effect more dominant with increasing temperature
⇒ χ S is negative and decreases with T , i.e. increases with 1/T
LCST = lower critical solution temperature
χ increases with increasing temperature
e.g. polyisobutylene (protonated); head-to-head polypropylene (deuterated)
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Mechanism:
(1)
Mixing leads to reduction in volume; volume for local movements of the chains is reduced
⇒ reduction in mobility/ flexibility ⇒ reduction in entropy.
Effect becomes more dominant with increasing temperature ⇒
χ s is positive and increases with temperature/ decreases with 1/T
(2)
Competition between attractive interaction of “special” groups of both components and
repulsive interaction between remaining groups (e.g. copolymers). Attractive interaction:
dipole-dipole interaction; hydrogen bonds; donor acceptor interaction
Increasing temperature weakens bonds and repulsive interaction dominates
⇒ both LCST and UCST
Phase diagram with lower and upper miscibility gap
Flory-Huggins parameter as a function of temperature
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
4.4
Flory-Krigbaum Theory
On the foundation of the excluded volume interactions an enthalpy parameter H and an entropy
parameter Ψ for dilution are defined describing deviations from ideal behaviour.
M
∂ΔGmix
⎡
1⎞
⎤
⎛
ΔG =
= Δμ SM = RT ⎢ln cS + ⎜1 − ⎟ c p + χ c 2p ⎥
Flory-Huggins:
∂nS
⎝ N⎠
⎣
⎦
1 2
with cp << 1 ⇒ ln cS = ln(1 − c p ) ≈ −c p − c p + ... as for osmotic pressure
2
cp ⎛ 1
⎞
⇒ ΔG d = Δμ SM = − RT [ + ⎜ − χ ⎟ c 2p + ...]
2 24
N ⎝14
⎠3
d
deviations from
ideal behavior
Flory-Krigbaum:
dilution enthalpy:
ΔH nid = RTHc 2p
dilution entropy:
ΔS nid = RΨc 2p
( c 2p due to pair contacts)
⇒ free enthalpy of dilution: ΔGnid = RT ( H − Ψ ) c 2p
⎛1
⎞
⇒ comparison Flory-Huggins and Flory-Krigbaum theory: −( H − Ψ ) = ⎜ − χ ⎟
⎝2
⎠
1
θ conditions: χ = ⇔ H = Ψ
; i.e. ΔH nid = T ΔS nid
2
H
define:
θ =T
(only defined if H and Ψ have equal sign)
Ψ
non-ideal behavior disappears for T = θ !
1
⎛ θ⎞
− χ = Ψ ⎜ 1 − ⎟ = 2nd virial coefficient A2 !
2
⎝ T⎠
⎡ cp
⎤
⎛ θ⎞
⇒ ΔG d = Δμ SM = − RT ⎢ + Ψ ⎜1 − ⎟ c 2p + ...⎥
⎝ T⎠
⎣ N
⎦
replace
Determination of the θ temperature in polymer solutions
(A)
From the determination of the critical phase separation temperature Tc as a function of the
degree of polymerization
1
⎛ θ
− χc = Ψ ⎜1 −
2
⎝ T
1 −1
⎞
−1 2
⇒
⎟=N − N
2
⎠
1
1 ⎛ −1 2 1 −1 ⎞ 1
=
⎜N + N ⎟+
Tc Ψθ ⎝
2
⎠ θ
1
⇒ θ can be determined from axis intersection N → ∞ and N −1 2 + N −1 → 0
2
⇒ Ψ from slope
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
(B)
⎛ θ⎞
From A2 measurement (osmotic pressure) as a function of the temperature A2 = Ψ ⎜ 1 − ⎟
⎝ T⎠
intersection with T axis (A2=0) ⇒ T = θ
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
4.5
Phase separation mechanisms
Separation process
time resolution possible due to high viscosity of polymer or by quenching to temperatures below
glas transition temperature
Methods for the determination of the cloud point
Depending on method of investigation different length scales can be resolved.
Phase separation mechanisms
cooling from one-phase region into:
(1) binodal (metastable region)
segregation by nucleation
(2) spinodal (unstable region)
spinodal segregation
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Cooling from one-phase region into two-phase region
Resulting structure depends strongly whether cooling down ends in metastable or unstable
region. In unstable region any small concentration fluctuation can growth. In metastable region
the concentration fluctuation has to be large enough in respect to concentration difference
ΔΦ and size R
(1)
(2)
spinodal decomposition results in interpenetrating, continuous structure with specific
length scale.
binodal decomposition by nucleation, spherical precipitates growthing in time.
Spontaneous local concentration fluctuation
δΦ in volume d 3 r and −δΦ in adjacent volume d 3r
⇒ change of free enthalpy in terms of free enthalpy g per unit volume
1
2
δ G = [ g (Φ 0 + δΦ ) + g (Φ 0 − δΦ )]d 3 r − g (Φ 0 )d 3r
series expansion of g up to second order in Φ yields
δG =
1 ∂2 g
1 1 ∂ 2 ΔGmix
2 3
Φ
Φ
=
d
r
δ
(
)
(Φ 0 )δΦ 2 d 3 r
0
2 ∂Φ 2
2 v ∂Φ 2
∂ 2 ΔGmix
a small fluctuation in concentration leads to
∂Φ 2
an increase or decrease of the free enthalpy (Gibbs stability criterion)
⇒ depending on the sign of the curvature
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
(1)
⎛ ∂ 2 ΔGmix
⎜
2
⎝ ∂Φ
(2)
⎛ ∂ ΔGmix
⎜
2
⎝ ∂Φ
⎞
⎟ > 0 (convex) δ G > 0
⎠
phase is stable against small pertubation;
amplitude of small pertubation decays
2
⎞
⎟ < 0 (concave) δ G < 0
⎠
phase is unstable against small pertubation;
amplitude of small pertubation growth
Separation by nucleation
•
•
•
a small concentration fluctuation decays.
concentration difference of the nucleus must correspond to that of the equilibrium phases.
nucleus must have critical size.
2
π2
r3
σ with Δg = g (Φ 0 ) − g (Φ 2 ) (*)
free enthalpy for nucleation: ΔG (r ) = − 4π r 3Δg + 41
3
14243 Term II
Term I
Term I: volume term corresponds to reduction in enthalpy by generation of the new phases.
Term II: surface term corresponds to increase in enthalpy due to surface generation with
σ being
the free enthalpy per unit surface.
segregation by nucleation
⇒
overcome energy barrier before nucleus can growth
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
⎛ ΔGb ⎞
nucleation rate increases with Arrhenius law: ν nucleus ∝ exp ⎜ −
⎟
⎝ k BT ⎠
16π σ 3
which is the maximum of (*) ΔG (r )
with the barrier height ΔGb =
3 ( Δg ) 2
⇒
⇒
energy barrier ΔGb increases with decreasing distance from the binodal (approaching
⇒ ΔGb → ∞
from spinodal); at the binodal Δg = 0
⇒
mixture needs to be supercooled (or overheated) to achieve relevant nucleation rates.
In summary
•
nucleus growth diffusion controlled; downhill-diffusion towards decreasing concentrations
(chemical potential gradient supports feeding nucleus).
•
decreasing temperature ⇒ diffusion slower (nucleus growth slower); nucleation rate
increases (for lower miscibility gap) ⇒ number of nuclei increases
•
droplet size, number and distance depend on time and temperature
•
at later stages coalescence, coarsening and ripening until there are two large phases with
composition Φ1 and Φ 2
Spinodal Decomposition
•
•
∂ 2 ΔGmix
→ 0 ⇒ spontaneous demixing; no activation energy
at spinodal
∂Φ 2
no sharp transition from phase separation by nucleation to spontaneous demixing since
∂ 2 ΔGmix
vanishes continuously
∂Φ 2
Kinetics of spinodal decomposition in polymer mixtures
•
local chemical potential difference caused by concentration fluctuations
∂ΔGmix
Δμ = μ s − μ s0 =
− 2κ ∇ 2Φ
1
424
3
∂Φ
123
Term II
Term I
Term I: homogenous system
Term II: gradient energy term associated with departure from uniformity
κ = gradient energy coefficient (empirical constant)
• difference in the chemical potentials causes interdiffusion flux. Linear response between flux
and gradient in chemical potential:
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
⎡ ∂ 2 ΔGmix
⎤
− j = Ω∇(Δμ ) = Ω ⎢
∇Φ − 2κ∇ 3Φ ⎥
2
⎣ ∂Φ
⎦
with Ω =Onsager-type phenomenological coefficient
•
substitution into equation of continuity which equates the net flux over a surface with the
loss
⎡ ∂ 2 ΔGmix 2
⎤
∂Φ
= −∇ ⋅ j = Ω ⎢
∇ Φ − 2κ∇ 4 Φ ⎥
2
∂t
⎣ ∂Φ
⎦
or gain of material within the surface
Cahn-Hillard relation
•
comparison to conventional diffusion equation
∂Φ
= D∇ 2 Φ
∂t
∂ 2 ΔGmix
⇒D≈Ω
<0
∂Φ 2
(D=diffusion coefficient)
in unstable region
diffusion against concentration gradient at spinodal (T → TSp ) D → 0 “critical slowing down”
general solution for Cahn-Hilliard relation:
Φ − Φ 0 = ∑ exp(Γ( β ) ⋅ t ) ⋅ [ A( β ) cos( β x) + B( β ) sin( β x)]
β
with λ =
stable region:
unstable region:
2π
β
⎡ ∂ 2 ΔGmix
⎤
2
and Γ( β ) = −Ωβ ⎢
2
κβ
−
2
{⎥
>0 ⎥
⎣⎢ ∂Φ
⎦
2
∂ 2 ΔGmix
> 0 ⇒ Γ( β ) < 0 ⇒ instabilities rapidly decay
∂Φ 2
∂ 2 ΔGmix
< 0 ⇒ Γ( β ) > 0 ⇒ some wavelengths notably the long
∂Φ 2
wavelengths (small β ) will grow
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Spinodal segregation
(1)
(2)
(3)
early stages: amplitudes of fluctuations growth for preferred wavelength
λ = (length scale) −1 ; modes are independent.
intermediate stages: amplitudes growth, modes interact with each other; wavelength
growth and structure coarsens; deviations from cosine wave form.
late stages: fluctuation reaches the upper and lower limit Φ1 and Φ 2 of equilibrium
coexistence curve; wave form becomes rectangular and domain structure appears. Domain
boundaries have finite thickness but extension is small compared to domain size.
Late stages of spinodal segregation
Finally: system tends to minimize its interfacial free energy by minimizing the amount of
interface area. Sphere contains the most volume and the least surface area. ⇒
(1)
(2)
interwoven structure coarsens by an interfacial-energy driven viscous flow mechanism
tendency to spherical precipitates
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
4.6
Summary
Miscibility of mixtures if
ΔGmix = ΔH mix − T ΔSmix < 0
Flory-Huggins Theory (mean field theory)
considers a random distribution of polymer
monomers & solvent molecules on a lattice
calculation of
ΔH mix
in terms of pair interaction energies; every molecule or
monomer sees the same average surrounding.
calculation of
ΔS mix
number possible distributions of both component on the lattice
under preservations of the connectivity of the polymer(s).
further assumptions:
•
distribution on lattice without volume change; both components have same lattice
volume: volume fractions = number fractions.
•
number of lattice sites much larger than the degree of polymerization.
•
large coordination number (nearest neighbors on lattice).
Flory-Huggins Theory
⎡c
⎤
c
M
ΔGmix
= RT ⎢ 1 ln c1 + 2 ln c1 + χ12 c1c2 ⎥
N2
⎣ N1
⎦
z Δw12
1
χ12 =
∝ T −1 with Δw12 = w12 − ( w11 + w22 )
RT
2
for polymers with high degree of polymerization the entropy gain due to mixing very is small
(reduced by factor 1/N)
for most substances endothermal mixing χ12 > 0
⇒ small enthalpic contribution leads to demixing!
in Flory-Huggins theory χ12 ∝ T −1
⇒ mixing at high temperatures & demixing at low temperatures
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Osmotic pressure
is the pressure produced by a solution in a space that is enclosed by a semi-permeable
membrane. Π = pressure polymer solution – pressure pure solvent
1
⎡c
⎤
ΠVs = RT ⎢ P + ( − χ )cP2 + ...⎥
2
⎣N
⎦
1
χ<
good solvent - high temperatures - polymer swells
2
1
χ>
bad solvent - low temperatures - polymer shrinks
2
1
θ solvent
χ=
2
Membrane osmometry (measures excess column height)
hydrostatic pressure = osmotic pressure
dilute solutions:
ΠVs ⎡ 1
1
⎤
= ⎢ + ( − χ )cP + ...⎥
RTcP ⎣ N
2
⎦
in very dilute solution: the number averaged molecular weight is measured
Phase diagrams
Gibbs stability criterion: ΔG mix ≤ α ΔG mix (Φ α ) + (1 − α ) ΔG mix (Φ β ) = ΔG αβ
mix
ΔGmix
concave
unstable
stable
metastable
metastable
convex
convex
at fixed temperature
stable
spinodal
binodal
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Construction of phase diagrams
binodal (metastable region)
T
>
⎛ ∂ΔG mix ⎞
⎜
⎟ =0
⎝ ∂Φ ⎠T,p
>
Å free enthalpy
T
T
>
T
>
spinodal (unstable region)
T
⎛ ∂ 2 ΔGmix ⎞
⎜
⎟ =0
2
∂Φ
⎝
⎠T , p
Å
good
solvent
χ
temperature
T
critical point
bad
solvent
⎛ ∂ 3ΔGmix ⎞
⎜
⎟ =0
3
⎝ ∂Φ ⎠T , p
T1
binodal and spinodal
merge; determined by (Tc,Φc)
concentration Φ
Flory-Huggins Theory : χ ∝
1
always lower miscibility gap & upper critical point
T
Polymer solution
Mw=
1
≈ N −1/ 2
N +1
1
1
χ C = + N −1/ 2 + N −1 ∝ TC−1
2
2
ΦC =
binodal
(experimental)
binodal (Flory)
volume fraction polymer
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
miscibility gap is highly
asymmetric,
shifts to very small polymer
concentrations for high N
Tc increases with N
spinodal
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Polymer mixtures
ΦC =
temperature
χN
χC =
two-phase region
one-phase region
1
N1
N2
+1
2
1 −1/ 2
N1 + N 2−1/ 2 ) ∝ TC−1
(
2
•
for N1<<N2 or N2<<N1: ΦC shifts
towards very concentrations
of the shorter component.
•
TC increases with N1 or N2
•
miscibility gap is symmetric at
critical point for N1=N2
Φ
ΦC = 1
2
χ C = 2 N ∝ TC−1
Limitation of the flory theory
Mean field approach but distribution of the monomers + molecules on the lattice is not purely
statistically due to non-equal interaction between identical and non-identical components: χ
becomes concentration dependent. Concentration fluctuations are also neglected.
Main lack of the Flory Huggins theory originates from an additional non-combinatorial
entropical contribution to the enthalpy of mixing resulting from volume changes due to mixing.
Ansatz: : χ=χS + χH =A(T)+B/(kBT)
(1)
(2)
mixing leads to increase in volume - gain of entropy due to higher chain mobility - χS is
negative and decreases with temperature ⇒ UCST- Lower miscibility gap
mixing leads to decrease in volume - loss of entropy due to higher chain mobility - χS is
positive and increases with temperature ⇒ LCST- Upper miscibility gap
Combinations of enthalpic contribution and (2) to χ can lead to both UCST + LCST
also in combination with:
(3)
competition of attractive bonds of certain groups on the polymer and repulsive interaction
between remaining groups. Weakening of the attractive bonds with increasing temperature
⇒ repulsive interaction dominates.
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Phase diagram with lower and upper miscibility gap
Flory-Huggins enthalpy of mixing:
⎡c
⎤
c
M
ΔGmix
= RT ⎢ 1 ln c1 + 2 ln c1 + χ12 c1c2 ⎥ < 0
N2
⎣ N1
⎦
(LCST)
two-phase
region
one-phase region
(UCST)
Flory Krigbaum theory
introduces an enthaply κ and an entropy ψ parameter for dilution describing deviations from
ideal behavior due to excluded volume interaction,
introduction of the θ temperature: θ=T κ/ψ
∂ΔG M
⎛ θ⎞
mix
= ΔμSM = −RT ( Ψ − κ ) c2P = −Ψ ⎜1 − ⎟ c P2
∂n S
⎝ T⎠
2nd virial coefficient A2 ~ (1/2-χ)= ψ(1-θ/T)
•
•
θ temperature can be determined from A2 measurements
θ temperature and entropy parameter ψ can be determined from molecular weight
dependency on Tc
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Phase separation mechanisms
cooling from one-phase region into:
(1) binodal (metastable region)
segregation by nucleation
results in spherical precipitates
growthing with time.
(2) spinodal (unstable region)
spinodal segregation
results in interpenetrating, continuous
structure with specific length scale.
Separation by nucleation
•
a small concentration fluctuation decays.
•
concentration difference of the nucleus must correspond to that of the equilibrium phases.
and nucleus must have critical size.
•
nucleus growth diffusion controlled; downhill-diffusion towards decreasing concentrations
(chemical potential gradient supports feeding nucleus).
•
decreasing temperature ⇒ diffusion slower (nucleus growth slower); nucleation rate
increases (for lower miscibility gap) ⇒number of nuclei increases
•
droplet size, number and distance depend on time and temperature
•
at later stages coalescence, coarsening and ripening until there are two large phases with
composition Φ1 and Φ 2
Spinodal segregation
•
a small concentration fluctuation growth.
•
diffusion against concentration gradient at spinodal (T → TSp ) D → 0 “critical slowing
down”
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Stages
(1)
early stages: amplitudes of fluctuations growth for preferred wavelength
λ = (length scale) −1 ; modes are independent.
(2)
intermediate stages: amplitudes growth, modes interact with each other; wavelength
growth and structure coarsens; deviations from cosine wave form.
(3)
late stages: fluctuation reaches the upper and lower limit Φ1 and Φ 2 of equilibrium
coexistence curve; wave form becomes rectangular and domain structure appears. Domain
boundaries have finite thickness but extension is small compared to domain size.
(4)
Finally: system tends to minimize its interfacial free energy by minimizing the amount of
interface area. Sphere contains the most volume and the least surface area. ⇒
• interwoven structure coarsens by an interfacial-energy driven viscous flow mechanism
• tendency to spherical precipitates
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/
Literature
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15.
U.W. Gedde, “Polymer Physics”, Chapman & Hall, London, 1995.
G. Strobl, “The Physics of Polymers”, Springer-Verlag, Berlin, 1996.
M.D. Lechner, K. Gehrke, E.H. Nordmeier, “ Makromolekulare Chemie”, Birkhäuser
Verlag, Basel, 1993.
J.M.G. Cowie, “Polymers: Chemistry & Physics of Modern Materials”, Blackie Academic
& Professional, London, 1991.
K. Kamide, T. Dobashi, “Physical Chemistry of Polymer Solutions”, Elsevier, Amsterdam,
2000.
O. Olabisi, L.M. Robeson, M.T. Shaw, “Polymer-Polymer Miscibility”, Academic Press,
San Diego, 1997.
M. Rubinstein, R.H. Colby, “Polymer Physics“, Oxford University Press, New York, 2003.
U, Eisele, “Introduction to Polymer Physics”, Springer Verlag, Berlin. 1990.
M. Doi, “Introduction to Polymer Physics”, Chapman & Hill, London, 1995.
A. Baumgärtner, “ Polymer Mixtures”, Lecture Manuscripts of the 33th IFF Winter School
on “ Soft Matter – Complex Materials on Mesoscopic Scales”, 2002.
T. Springer, “ Entmischungskinetik bei Polymeren”, Lecture Manuscripts of the 28th IFF
Winter School on “ Dynamik und Strukturbildung in kondensierter Materie”, 1997.
K. Kehr, “ Mittlere Feld-Theorie der Polymerlösungen, Schmelzen und Mischungen;
Random Phase Approximation”, Lecture Manuscripts of the 22th IFF Winter School on “
Physik der Polymere”, 1991.
H. Müller-Krumbhaar, “ Entmischungskinetik eines Polymer-Gemisches”, Lecture
Manuscripts of the 22th IFF Winter School on “ Physik der Polymere”, 1991.
T. Springer, “ Dichte Polymersysteme und Lösungen”, Lecture Manuscripts of the 25th
IFF Winter School on “ Komplexe Systeme zwischen Atom und Festkörper”, 1994.
H. Müller-Krumbhaar, “ Phasenübergange: Ordnung und Fluktuationen ”, Lecture
Manuscripts of the 25th IFF Winter School on “ Komplexe Systeme zwischen Atom und
Festkörper”, 1994.
PD Dr. Silke Rathgeber
Johannes Gutenberg-University Mainz
Institute for Physics – KOMET 331
Staudingerweg 7
55099 Mainz
Germany
phone: +49 (0) 6131/ 392-3323
Fax: +49 (0) 6131/ 392-5441
email: [email protected]
webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/