Date post: | 03-Jun-2018 |
Category: |
Documents |
Upload: | carlos-oliveira |
View: | 223 times |
Download: | 0 times |
of 18
8/12/2019 Flory Huggs
1/18
22
2.5 Thermodynamics of Polymer Solutions (1)
Notation: A = solvent; B = solute (polymer)
in case of copolymers or multi-component systems:1 = solvent; 2,3...polymer
Thermodynamic of low molecular weight solution
(revision):Gibbs free energy (Free Enthalpy): G = f(p,T,n)
dG =
G
TdT
G
pdp
G
ndn
p n T n i p T n
i
i i j
+
+
, , , ,
dG = - S dT + V dp + idni;
p = const; T = const:dG = idni
1.+ 2. law of thermodynamics (isothermal condition, dT = 0):dG = dH T dS + idni
partial molar entropy si: si= -(i/T)p,n
partial molar volume vi: vi= (i/p)T,n
Pressure dependence of chemical potential i:
iid
(p) = iid
(po) + RT ln (p/po); iid
(po) = i,o(standard pot.)
ire
(p) = iid
(po) + RT ln (f/fo) ; f = fugacity
Concentration dependence of chemical potential i:
iid
(p,T,xi) = i*(p,T,xi=0) + RT ln xi
ire
(p,T,xi) = i*(p,T,xi=0) + RT ln ai; ai(activity) = xifi
fiactivity coefficient
ire
(p,T,xi) = iid
(p,T,xi) + iexcess
(p,T,xi)
Entropy of mixing: )Sid = -R niln xi= -R nAln xA R nBln xB;
8/12/2019 Flory Huggs
2/18
23
Classification of solutions:
ex
sex
h
Ideal solutions
Athermic solutions
Regular solutions
Irregular solutions
= 0
0
0
0
= 0
0
= 0
0
= 0
= 0
0
0
Entropy of mixing: The Flory-Huggins theory (1)
Deviation of polymer solutions from ideal behavior is mainly due to
low mixing entropy. This is the consequence of the range of difference
in molecular dimensions between polymer and solvent.
Flory (1942) and Huggins (1942)
Calculation of Gm= G(A,B) - {G (A) + G (B)}H = 0Gm= -T Sm
Lattice model
volume of solvent molecule: VA;
each solvent molecule occupies
1 lattice cell
NA= number of solvent molecules
volume of macromolecule: VB
each macromolecule occupies
VB/VA= Llattice cells
NB= number of macromolecules
Number of lattice cells: K = NA + L NBCoordination number: z (two-dimensional: z = 4)
VAVB= L VA=10 VA
8/12/2019 Flory Huggs
3/18
24
Flory-Huggins theory (2)
transfer of the polymer chains from a pure, perfectly ordered stateto a state of disorder mixing process of the flexible chains with solvent molecules
Calculation of the number of possible ways a polymeric chain can be
added to a lattice:
1. Macromolecule
1st Segment K possibilities of arrangement on lattice
2nd
Segment z possibilities of arrangement on lattice
3rd Segment z 1 possibilities of arrangement on lattice
L segments of 1. macromolecule:
1= K z (z 1)L 2
i. Macromolecule
number of vacant cells: K (i - 1)L
probability to find a vacant cell: (K (i-1)L)/K(mean-field theory)
L segments of i. macromolecule:
i = (K (i-1)L z (K (i-1)L)/K {(z-1) (K (i-1)L)/K}L - 2
thermodynamic probability = (NB! 2NB
)-1
i
entropy (Boltzmann): S(NA,NB) = kBln (solvent: only 1 arrangement
Sm= S(NA,NB) - {S(NA) + S(NB)}
Sm= -R (nAln A+ nB ln B)
A= volume fraction solvent = NA/K = nAVA/( nAVA+ nBVB)
B= volume fraction polymer = L NB/K = nBVB/( nAVA+ nBVB)
8/12/2019 Flory Huggs
4/18
25
Flory-Huggins theory (3); chemical potential
A= RT ln aA = RT (ln A+ (1 - VA/VB) B)
B= RT ln aB = RT (ln B+ (1 VB/VA) A)
A= f (M) !
VA/VB= [BMA/AMO] 1/PVA/VB~ 1/P ~1/MB
Enthalpy change of mixingquasichemical process: (A-A) + (B-B)(A-B)
(A-B): solvent-polymer contact
interchange energy per contact: u = AB= AB (AA+ BB)
U = H if no volume change takes place on mixing
H = q AB; q = number of new contacts
calculation of number of contacts can be estimated from the lattice
model assuming that the probability of having a lattice cell occupied
by a solvent molecule is simply the volume fractionA, by a polymerB.
q = ABz K
H = ABz KAB
with: := z AB/kBT (definition of !)
and K = NLnA/A; R = NLkB
H = RT nAB
= Huggins interaction parameter
8/12/2019 Flory Huggs
5/18
26
Gibbs enthalpy based on Flory-Huggins theory:
Gm= RT (nAln A+ nB ln B + nAB)
(often in literature Gm/mol (monomer and solvent));
Gm/(nA+ PnB0) = RT (Aln A+ (B/P)ln B + AB)
A= RT (ln A+ (1 - VA/VB) B+ B2)Meaning of
combinatorial
comb
= entropy according F.-H.residualR= difference to the combinatorial solution, and
excessex
= difference to the ideal low-molecular weight solution
term of chemical potential
A= Acomb+ A
R
= AR/RTB
enthalpic and entropic parts of AR:
AR= TsA
R + hA = H+ S
with H= h/RTB; S= sAR/RB
Determination of H and S: H= -T(/T)p
S = d(T)/dT
S= 0 (combinatorial solution; F.-H. equ. valid)= a/T
experiments: = a + b/TS0
in most cases: S, H > 0; S> H;< 0 means: contacts between A and B are preferred (good solution)
8/12/2019 Flory Huggs
6/18
27
Theta-temperature and Phase separation (1)
phase stability conditions:
temperature (g/T)p < 0pressure (g/p)T < 0
concentration (g/x)p,T > 0binodal curve (local minima):
spinodal curve (reflection point):
Application on Flory-Huggins:
binodal curve
AB p T B
A
BBRT
V
V
=
+
+
,
1
11 2
(1)
spinodal curve( )
2
2 2
1
12
A
B p T B
RT
=
+
,
(2)
critical point: (1); (2) = 0
B cB A
cA
B
A
BV V
V
V
V
V,
/;=
+= + +
1
1
1
2 2
polydispers polymer:
( )( ) B c w z c z w zP PP P P
, / ; / /= + = + +
1
1
1
21 1 1
g
x
g
x
m
p T
m
p T
=
>
, ,
;0 02
2
2
2
3
30 0
g
x
g
x
m
p T
m
p T
=
, ,
;
8/12/2019 Flory Huggs
7/18
28
Theta-temperature and Phase separation (2)
( )
AR
Acomb
A A A
RT RT RT
h
RT
s
R= + = = = +B H s B
2 2
critical point: T = Tc; = c
(*)
cA
B
A
B
A
c
A
c
V
V
V
2 V
h
R T
M )=1
hR T M )
= + + = +
= +
1
2
2
12
2
2
B
s
c
B
s
(
(
s
1
2
(*)
1
2
1
22+ + = +
V
V
V
2 V
h
R T
A
B
A
B
A
c
B
h R T M )A c= B2 (
R T (M
R T
c B
c B
A
B
A
B
V
V
V
V
= + +
+)
2
2 2
1
T
1
T M
1
T Mc c c=
+
+
( ) ( )
V
V
V
V
A
B
A
B2
T M Tc ( ) =
Theta-temperature
8/12/2019 Flory Huggs
8/18
29
Second virial coefficient and
A= Aid+ A
ex
Osmosis: A= -VAreal solution, virial expression:
/cB= RT (1/M + A2cB+ A3cB+ )
A= f(cB)
expanding ln A= ln (1 - B) as far as the second term in a Taylorseries, B = cB/B
A2= (1/2 - )/(B2VA)
Aex
= - RTA2cBVA= - RT(1/2 - )BBVA/(B2VA)
Aex= - RT(1/2 - )B
Second virial coefficient and Thetatemperatur
A2= (1/2 - )/(B2VA);
= H+S= T/T + S= T/T + -
A2= ( 1 - T/T) /(B2VA)
T = T: A2= 0
pseudo-ideal solution
8/12/2019 Flory Huggs
9/18
30
FloryKrigbaum theory
to overcome the limitations of the lattice theory resulting from the
discontinuous nature of a dilute polymer solution
solution is composed of areas
containing polymer which were
separated by the solvent
Polymer areas: Polymer segments
with a Gaussian distribution about the center of mass
chain segments occupy a finite volume from which all other chainsegments are excluded (long range interaction)
see Excluded Volume Theory
Introduction of two parameters
enthalpy parameter
entropy parameter
to describe long range interaction effects:
A= Aid+ A
ex;
Aex
= - RT(1/2 - ) B = hex
-T sex
hex= RT B ; sex = R B
(1/2 - ) = (- )Theta condition A
ex= 0
= hex=T sex;
T:= T (/) ; Aex= - RT( 1 - T/T) B
Deviations from ideal (pseudoideal) behavior vanish
when T = T!
8/12/2019 Flory Huggs
10/18
31
8/12/2019 Flory Huggs
11/18
32
8/12/2019 Flory Huggs
12/18
33
2.5 Thermodynamic of Polymer Solution (2)
Solubility Parameter
The strength of the intermolecular forces between the polymermolecules is equal to the cohesive energy density (CED),
which is the molar energy of vaporization per unit volume.
Since intermolecular interactions of solvent and solute must be
overcome when a solute dissolves, CED values may be used to
predict solubility.
1926, Hildebrand showed a relationship between solubilityand the internal pressure of the solvent;
1931, Scatchard incorporated the CED concept intoHildebrands eq.
= HV2 (nonpolar solvent; )H= heat of vaporization)heat of mixing: Hm= VAB(A- B) = nAVA(A- B)
Acc. solubility parameter concept any nonpolar polymer will
dissolve in a liquid or a mixture of liquids having a solubility
parameter that does not differ by more than 1.8 (cal cm-3)0.5.Small:
F = Fi; Fi= molar attraction constant [in (Jcm)1/2mol-2]
-CH3 438 -CH2- 272 CH- 57
=C= -190 -O- 143 -CH(CH3)- 495
-HC=CH 454 -COO 634 -CO- 563
Like dissolves like is not a quantitative expression!
Problems: polymers with high crystallinity;polar polymers hydrogen-bonded solvents or polymers
additional terms
= = =E
VE F V V
Mcoh
B o
cohB o B o
o
B amorph,, ,
,
; / ;2
8/12/2019 Flory Huggs
13/18
34
The square root of cohesive energy density is called
solubility parameter. It is widely used for correlating
polymer solvent interactions. For the solubility of polymer Pin solvent S ( P- S) has to be small!
8/12/2019 Flory Huggs
14/18
35
Excluded-Volume-Effect
Dilute gas of random flight chains:
it is physically impossible to occupy the same volume elementin space at the same time
the conformations in which any pair of beads
(segments) overlap were avoided
when a pair of beads come close they exert a repulsion
force F on each other
Dilute solution of random flight chains:
The force that acts between a pair of beads becomes no longer
equal to F.interaction bead-solvent > interaction bead-bead => good solvent
interaction bead-solvent < interaction bead-bead => poor solvent
=> solvent-bead (segment) interactions: F
good solvent: F repulsive
bad solvent: F attractiveThe term excluded volume-effect is used to describe any effect
arising from intrachain or interchain segment-segment
interaction.
Excluded-volume of two hard spheres:
( ) = = =4
32R
4
3R Vsphere
3 38 8
excluded volume
Second virial coefficient A2and the
excluded volume:
A2M
R ~ M2 22
32
32=
Nhard sphereL
; : ~
A2~ M-1/2
8/12/2019 Flory Huggs
15/18
36
Excluded Volume Theory
volume of segments interaction between segments (repulsion forces)excluded volume depends on space-filling effects and interaction
forcesshort range, long range interactions
Problems:
Calculation of excluded volume in dependence on molecular
properties;
relation between interaction (A2) and excluded volume.
Excluded volume and lattice theory:
number of possibilities, that the molecule mass center is in the volume
V, excluded volume/molecule , proportionality constant k:
1.molecule: 1= k V2.molecule: 2= k (V - )i. molecule: i= k [V (i 1)]
A= - RT VAcB[1/MB+ ((NL)/(2MB))cB]
Qualitative Discussion:
excluded
volume
r
hard sphere < 0; A2< 0
= 0; A2= 0
> 0; A2> 0
8/12/2019 Flory Huggs
16/18
37
Scaling Law
o= nsls
eq. tell us, how o "scales" with ns
Global (universal) Propertiesproperties of polymer chains, which do not depend on
local properties (independent of the monomer structure, nature
of solvent, etc.)=> very large characteristic lengths
=> small frequenciesIt has been found that in the appropriate variables all
macroscopic polymer properties can be plotted on universal
curves (power laws, characteristic exponents).
The Blob-chain, () of a labeled chain
the labeled chain is made
of n/g blobs each of length (screening length) containing gsegments
a blob is an effective stepalong the contour of the chain
contains g segments
we assume:
- the segments inside the blob
obey the excluded volume
chain statistics, ~ g3/5
- the n/g blobs obey the random walk statistics such that = (n/g) is the distance up to which the native self-avoidance due to theexcluded volume interaction is completely correlated and beyond
which it is totally uncorrelated; since g ~ 5/3 ~ (n/g) ~ n 1/3 ~ n -1/4(see = f (c))
8/12/2019 Flory Huggs
17/18
38
Scaling Laws for polymer solutions(good solvent at nonzero concentrations)
We are in search of a dimensionless variable in order to applythe scaling method.
fundamental concentration to make the polymerconcentration dimensionless:
We introduce a reduced concentration (/*)with: = segment concentration (number of
segments/volume); N chains with nssegments* = overlap concentration~ ( N ns)/( N RF) ~ ns
(1-3)~ ns-4/5; (RF~ ns
3/5)
scaling laws: concentration dependence of (Radius of gyration)
= *: 1/2= RF
solid amorphous polymer, > *: ~ ns ~ RF (/*)
x
= *: ~ RF (/*)x~ RF
> *: ~ RF (/*)x~ ns
since: *~ ns-4/5; RF ~ns
3/5; x = - *)
1/2~ -1/4 ~ cB-1/4
concentration dependence of the screening length= *: = RF> *: ~ ns
0(no molar mass dependence)
~ RF(/*)y
> *: ~ RF(/*)y~ ns
0
since: *~ ns-4/5; RF ~ns
3/5 y = - 3/4
~ -3/4 ~ cB-3/4
*) ~ ns~ xns(6/5+4x/5); 6/5 + 4x/5 = 1 ; x = -1/4
8/12/2019 Flory Huggs
18/18
39
osmotic pressuredilute solution:
R T k T nB s
= =c
M
> *: /cBshow no molar mass dependence
k T nB s~
*
z
> *:
k T
n
B
s0~ ; *~ n
s
-4/5 z = 5/4
~ ~94 ns
0
experiments, poly--methyl-styrene in toluene, different molar mass