1
Thermodynamique of polymers in solution
1- Flory Huggins theory
1.1 Athermal macromolecular solutions
The liquid crystalline network model of Flory and Huggins
Molecule of solvent
Segment of chain
v1 = volume of segment
volume of segment = volume of solvent of molecule (v1)
if v2 is the polymer volume
v2 = x.v1
and x the number of segment per chain
if n1 is the number of molecule of solvent and n2 the number of macromolecules,
it is possible using this model to calculate the ΔS (configuration) of a mixture
polymer –solvent.
If the macromolecular solution is athermal, ΔS(conf.) ≈ ΔSM
ΔSM  k(n1Log1  n 2 Log 2 )
with
1 
n1
n1  x  n 2
and macromolecules number = x .n2
and
2 
(1)
x  n2
n1  x  n 2
2
If N1 and N2 is respectively the mole number of solvent and solute, R = k x A, R =
gas constant.
ΔSM  R(N1Log1  N 2 Log 2 )
In the case of athermal solution
(2)
ΔS EM  0 and H M  0 ,
ΔSEM is the excess enthropy of the mixture
The free enthalpy of the mixture is ΔGM = -TΔSM
this value take also the following equation:
ΔG M  RT(N1Log1  N 2Log 2 )
(3)
1-2. The regular macromolecular solutions
A regular solution is characterized by a small value of the mixture enthalpy
(ΔHM) compared with the thermal agitation energy which is alone responsible of
the molecular repartition in the liquid network. So, the mixture entropy of
regular solution (polymer -solvent) has the same value that attributed to
athermal macromolecular solution. This constitutes the ideal model of the real
polymer solutions.
1-2.1. Heat of mixtures, Interaction parameter χ 1,2
In the Flory –Huggins theory, the calculation of the mixture heat ΔHM is exactly
the same that those evaluated for the mixture enthalpy of the regular simple
solution. The segment macromolecular notion drive to the consideration of the
elementary interactions solvent-solvent, solvent-segment and segment-segment
respectively characterized by the energies ε11,ε12 and ε22.
ΔH M
 N1ε11 N 2 xε 22
N12ε11
N 22 x 2ε 22
xN1N 2ε12 
 z (

)(


)
2
2
2(N

xN
)
2(N

xN
)
N

xN
)
1
2
1
2
1
2


(4)
3
z is the cell adjacent at the segment number in the liquid crystalline network
model of Flory.
This expression is usually writing as fellow
ΔHM = RT χ12N1φ2
(5)
χ12 is a coefficient without dimension, called interaction parameter of the
polymer-solvent couple or parameter of Flory and also giving by the following
relationship:
χ12  (z  2)
Δε12
RT
(6)
The energetic quantity RTχ12 present theoretically the thermal effect
accompanied the transfer of one mole of solvent from the solvent to pure
polymer.
The mixture free enthalpy (ΔGM) of the regular macromolecular solution which
indicates practically the case of a real solution at weak thermal effect is:
ΔGM= RT(N1Logφ1 + N2Logφ2 +χ12N1φ2)
(7)
1-2.2. Notion of good solvent and mediocre solvent
The dissolution phenomenon of solute (2) in solvent (1) is effective when the
free enthalpy of mixture (ΔGM) take a negative value. As the mixture entropy is
always positive, the dissolution is conditioned essentially by the sign and the
heat intensity of the mixture ΔHM.
Therefore, the interaction parameter χ12 is measures the quality of a solvent
towards a determined polymer.
By analogy with the theory of the simple regular solutions, the expression of the
mixture heat ΔHM calculated from the Flory –Huggins theory is affected by a
positive sign (endothermic mixture). This is practically true for the majority of
macromolecular solutions. Some time, we observe some exceptions when the
constituents(solvent and polymer) have a polar structures.
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Example: the dissolution of the poly(methylmethacrylate) in trichloroethylene
give a negative value for ΔHM.
The endothermic macromolecular solutions which represent the major part of the
real cases are, in opposite, characterized by the positive values of the interaction
parameters χ12.
For a endothermic mixture realized at temperature T, the complete
dissolution of the macromolecular solute is the best with the weak interaction
parameters. The solvent with the weak interaction parameters is designed by good
solvent for the polymer considered. In this way, the natural rubber-carbon
tetrachloride, poly(vinyle-tetrahydrofuran chloride) and cellulose nitrate are
respectively characterized by χ12 of 0.28, 0.14 and 0.27. On opposite, the solvent
with the χ12 near 0.5 is designed by the mediocre solvent for the polymer
considered. The polyisobutylene is dissolved easily at ordinary temperature in the
cyclohexane (χ12= 0.44) than in benzene (χ12 = 0.5). The good solvent and the bad
solvent are closely lied to the temperature.
1-3.
Real macromolecular solutions
In general, in the case of a real macromolecular solution which the heat of
mixture ΔHM take an notable value or which the constitutes of the mixture are more
polar the regular macromolecular solution concept is not applicable. In this fact, for
this type of solutions it is impossible to identify the entropy of configuration ΔSconf.
with its entropy of mixture ΔSM. Because the disposition of the macromolecular
segments in solution do not only governed by the statistical considerations, and if
ΔHM is positive, the segments of macromolecules tempt to withdraw into oneself
for favorite the segment-segment interactions.
ΔSM = -R[N1Logφ1 + N2Logφ2 +
ΔHM = -RT2.
dχ12
.N1 2
dT
d(χ12 T)N1 2
dT
(8)
(9)
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ΔGM = ΔHM - TΔSM
Now, It is possible to resume the
macromolecular solutions.
Athermal
macromolecular
solution
Regular
macromolecular
solution
Real macromolecular
solution
(10)
definitions of the different categories of
ΔSEM  0
ΔH M  0
ΔSEM  0
ΔH M  RTχ
ΔSEM 
N
12 1 2
dχ
d
(χ 12 T)N1 2 ΔH M  RT2 12 N1 2
dT
dT
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2- Macromolecular diluted solutions
In general, we admit that the macromolecular chains in solution has practically a
homogeny repartition when the fraction volume of solute are superior to 0.04.
Before this limit, the effective volumes occupied by the chains can be more or less
separated and the Flory-Huggins do not experimentally verified. In particular, the
partial molar entropy of solvent take clearly the weak values than the deducted by
the calculation. For the such as diluted solutions, the partial molar entropy of
solvent can write as fellow.
 22
ΔS1  R(
2

 23
3
 ...)
(11)
The ΔS1/R  2 value go to 0.5 when the macromolecular solution became more
2
and more diluted.
2.1 Notion of the excluded volume
The excluded volume is the space non occupied by the macromolecules in solution
Each macromolecule have a excluded volume ( domain delimited by the repulsion of
two adjacent macromolecules)
molecule of solvent
macromolecular chain
Excluded volume
7
o
If u is the excluded volume by each macromolecule, V1 the molar volume of
solvent, N the Avogadro number and x is the number of segment the ΔSM take the
followed equation:
ΔSM  RN2 (Log 2 
Nu
1 )
2V1o x
(12)
Because the diluted macromolecular solution is practically considered as athermal,
the free enthalpy of this solution can be expressed as fellow:
G M  T.SM  RTN2 ( Log 2 
Nu
1  ...)
2V1o x
(13)
In this present case the variation of the chemical potential is ΔG M  Δμ1 .
The relationship
2
Δμ1  RT(
x

Nu
 2  ...)
o 2 2
2V1 x
(14)
In case of the isomolecular sample of molecular weight value is M2, if c2 is the
concentration of macromolecular solute (g/ml)
2 
(15)
c2V1o x
M2
The Δμ1 expression can be expressed as fellow
Δμ1  c2RTV1o (
1
Nu

c2  ...)
M 2 2M 22
(16)
The details of the osmotic pressure of a macromolecular solution allow to represent
the previous equation as fellow
Δμ1  c2RTV1o (A1  A 2c2  ...)
with
A1 
1
M2
and
(17)
A2 
Nu
2M 22
A1 and A2 are respectively the first and second Virial coefficient
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This equation constitutes the base of the molecular weight determinations, the A1
and the interaction parameters A2, realized by the classical physic-chemistry
macromolecular techniques.
2.2 Flory and Krigbaum Theory
The spherical and stick forms of macromolecules proposed by Zimm and Huggins are
in reality two models very simplified. This proposition constitutes two cases
diametrically opposed of macromolecules in solution which depended to the
solvatation of the solvent.
Flory and Krigbaum have calculated the general expressions of the excluded volume
u and the second virial coefficient (A2) which characterize the macromolecules
gifted by a good flexibility. In the case of a very diluted macromolecular solution ,
each isolated macromolecule can be definite by a radial function of segments
distribution since the mass center of the molecule.
ρ(R) radial function of segments distribution
ρ(R)  x(
2
3 3/2 3R 2 /2R G
)
e
2
2ππ G
(18)
with RG is the gyration radius
This function presents the density of segments occupied by a macromolecule . This
representation is habitually called the “Gaussian ball”. This is applicable only in the
particular case of the θ-conditions.
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2.2.1 Volume excluded by a flexible macromolecular chain
Considering in the solution exist two macromolecules isolated as fellow:
In these conditions, the second Virial coefficient A2 of the previous general
equation which characterizes the flexible macromolecular chain in diluted solution
according the Flory – Krigbaum theory is:
2
Nu
1
v2
A2 

(


)
F(X)
12
2
2M 22
V1o
(19)
and the excluded volume (u) can take the fellowed expression
1
u  2(  12 ) x 2 v1o F(X)
2
(20)
v1o is the volume of the molecule of solvent and F(X) is a complex integral that is
possible to develop in series as:
10
X2
F(X)  1  3/2  3/2  ...
2 2! 3 3!
X
(21)
With
2
1
v2M 2
3
X  2(  χ12 ) o 2 (
.)3/2
2
2
V1 N 4π  R G
(22)
The variation of the chemical potential Δμ1 previously determined can be writing as
fellow:
2
Δμ1 
c2RTV1o [
1
1
v2
 (  χ12 ) o c2  ...]
M2
2
V1
(23)
2.1.2 θ-Solvent and θ-temperature
One of the principal consequences of the Flory –Krigbaum theory is that the
excluded volume (u) and the second virial coefficient (A2) are both the functions of
the interaction parameter χ12.
When χ12 is negative, the case of certain exceptional couples polymer-solvent, the
quantity (1/2- χ12) is always positive and the excluded volume (u) and the second
coefficient of Virial (A2) increase with the absolute value of χ12. In this fact, the
segment-solvent intermolecular interactions are energetically more favored than
the segment-segment intramolecular interactions, this fact induce a strong
deployment of each macromolecular ball.
We know that, in opposite, the almost of the macromolecular solutions are
characterized by a positive interaction parameter (χ12). In so far as this parameter is
inferior at 0.50, the equation (20) shows that the excluded volume (u) and the
second coefficient of virial (A2) are always positives and more weak in the case of a
mediocre solvent (χ12 ≈ 0.40 to 0.50) than the good solvent toward the polymer
considered (χ12 ≈ 0 to 0.30).
When the interaction parameter is exactly 0.5, the relation (20) shows that the
excluded volume is nil. The absolute temperature to which the value of the
interaction parameter is 0.5 is called θ-temperature or Flory temperature. A Such as
solvent necessary a mediocre solvent for this polymer taking in the same conditions
of temperature is designed as θ-solvent. θ-solvent can be a pure liquid or a mixture
of liquid.
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Other definition, the θ-temperature can be defined more rigourosely as the
temperature of the macromolecular solution which the second coefficient of Virial
A2 is nil.
2.3 The coefficient of molecular expansion