Additive Group Contribution Methods for
Predicting Properties of Polymer Systems
Grozdana Bogdanić
INA-Industrija nafte, d.d., Technology Development and Project Management
Department, Zagreb, Croatia
1. VLE
1.1. Group contribution methods for predicting the properties of
polymer–solvent mixtures
 Activity coefficient models
 Equations of state
2. LLE
2.1. Group contribution methods for predicting the properties of
polymer–solvent mixtures
 Activity coefficient models
 Equations of state
2. 2. Group contribution methods for predicting the properties of
polymer–polymer mixtures (polymer blends)
3. Conclusions
Group Contribution Methods for Predicting
Properties of Polymer – Solvent Mixtures (VLE)
wi 
mi
 mj
j

x iM i
 x jM j
j
ai = xi  i = w i  i
The UNIFAC-FV model
T. Oishi, J.M. Prausnitz, 1978.
ln  i = ln  i
comb
combinatorial
+ ln  i
resid
residual

 ~ 1/3 - 1 
vi

FV

 - Ci 
ln i
= 3 C i ln 

1/3
 ~v M
- 1 
 
+ ln  i
FV
free-volume

~v
i - 1   1 - 1

~v
~v 1/3

M
i




 1



The Entropic-FV model
H.S. Elbro, Aa. Fredenslund, P. Rasmussen, 1990.
G.M. Kontogeorgis, Aa. Fredenslund, D.P. Tassios, 1993.
ln  i = ln  i
ln  i
entr
= ln
i
FV
+1-
xi
i
entr
+ ln  i
attr
FV
xi
i
attr
 ln  i
The free-volume definition:
v f, i
= v i - v *i
*
i
v = v w, i
attr
( UNIFAC )
The GC-Flory EOS
F. Chen, Aa. Fredenslund, P. Rasmussen, 1990.
G. Bogdanić, Aa. Fredenslund, 1994.
attr
n RT  ~v 1/3 + C 
E
P =
 1/3
 +
V  ~v - 1 
V
comb
+ ln  i
combinatorial
FV
ln  i = ln  i
FV
+ ln  i
attr
attractive
N. Muro-Suñé, R. Gani, G. Bell, I. Shirley, 2005.
ln  i
FV
ln  i
attr
ln  i
comb
= ln
i
xi

= 3(1 + C i ) ln 

+ 1 -
i
xi
~v 1/3 - 1 
i
 - C i ln
~v 1/3 - 1

 1
= 1/2 z q i 
[  ii ( ~v ) -  ii ( ~v i )] + 1 - ln
 RT
-

j




exp
(
/RT)
k
 ki


 j exp (-   ji /RT)

k
j
j
~v
i
~v
exp (    ji / RT )
The GC-lattice-fluid EOS
M.S. High, R.P. Danner, 1989; 1990.
~
2
~
P
z  ~v + q/r - 1 
 v 

=
ln
+
ln
~



~
~
~
T
2 
v
T
 v -1

~v
 2  i, p  
 ~v ( ~v i - 1) 
z qi
i


ln  i = ln  i - ln w i + ln ~ + q i ln  ~
+
q
+
ln  ii
~
i


~
~v
v
(
v
1)
T
2
i


 Ti

B.C. Lee, R.P. Danner, 1996.
ln  i = ln  i - ln w i
~v
 2  i,p    
z qi
 ~v ( ~v i - 1) 
i


+
ln  ii
+ ln ~ + q i ln  ~
- ~
 + qi 
~

~
2
v
T
vi 
Ti
 ( v - 1)

G. Bogdanić, Aa. Fredenslund, 1995.
UNIFAC-FV
Entropic-FV
GC-Flory
GC-LF (1990)
Prediction of infinite dilution activity coefficients versus experimental values
for polymer solutions (more than 120 systems)
B.C. Lee, R.P. Danner, 1997.
Prediction of infinite dilution activity coefficients versus experimental
values for systems containing nonpolar solvents (215-246 systems)
B.C. Lee, R.P. Danner, 1997.
Predictions of infinite dilution activity coefficients versus experimental
values for systems containing weakly polar solvents (cca 60 systems)
B.C. Lee, R.P. Danner, 1997.
Predictions of infinite dilution activity coefficients versus experimental
values for systems containing strongly polar solvents (cca 30 systems)
G. Bogdanić, Aa. Fredenslund, 1995.
T = 383 K
T = 373 K
T = 322 K
Activity of 2-methyl heptane
in PVC
(Mn = 30000; Mn = 105000)
Activity of ethyl benzene
in PBD (Mn = 250000)
Activity of MEK in PS
(Mn = 103000)
LLE
 Polymer solutions
 Polymer blends
  2G

2
 1


 0
 T ,P
  2G

2
 2

  3G
  
3

 2

  0

  ln  1

 2
  2 ln  1

  

  2

  0

The segmental interaction
UNIQUAC-FV model(s)
G. Bogdanić, J. Vidal, 2000.
G.D. Pappa, E.C. Voutsas, D.P. Tassios, 2001.
ln  i = ln  i
ln  i
entr
= ln
entr
i
+ ln  i
FV
xi
+1-
i
resid
FV
xi
ncomp

Xk 
xik
(i)
i
ncomp nseg
 
j
x j m
( j)
m


 nseg
ln k  Q k  1  ln    m  mk

 m






nseg

m
 m  km
nseg

n
a n m  a n m , 1  a n m , 2  T  T0 
ln 
resid
i


k
(i)
k
ln 
k
 ln 
(i)
k

n
 nm






J. Vidal, G. Bogdanić, 1998.
Correlation (  ) of LLE PEG/water system by the UNIQUAC–FV model
G. Bogdanić, J. Vidal, 2000.
Mv=65000 g/mol,  correlation
 Mv=135000 g/mol,   prediction
 Mw=44500 g/mol, - - - - prediction

 poly(S0.54-co-BMA0.46), Mw=40000 g/mol,  correlation
 poly(S0.80-co-BMA0.20), Mw=250000 g/mol, - - - - prediction
Correlation and prediction of LLE for
PBD/1-octane by the UNIQUAC-FV model
Correlation and prediction of LLE for
poly(S-co-BMA)/MEK by the UNIQUAC-FV
model
The GC-Flory EOS
G. Bogdanić, Aa. Fredenslund, 1994.
G. Bogdanić, 2002.
LLE parameters
εnn , Δεnm
G. Bogdanić, 2002.
T /K
T /K
4 00
42 0
3 75
3 50
M v =98000
41 0
M v = 191000
3 25
M v = 380000
3 00
40 0

M n =60400, M w = 82600
2 75
M n = 97700, M w =135900
 M w = 180000
39 0
0.00
0.02
0.04
0.06
0.08
0.10
M a s s fra c tio n o f p o ly m e r
Coexistence curves for HDPE/n-hexane systems
as correlated by the GC-Flory EOS (  )
2 50
0.00
0.05
0.10
0.15
0 .2 0
0.25
M a s s fra c tio n o f p o ly m e r
Coexistence curves for PIB/n-hexane
systems as correlated
by the GC-Flory EOS (  )
The mean-field theory
R.P. Kambour, J.T. Bendler, R.C. Bopp, 1983.
G. ten Brinke, F.E. Karasz, W.J. MacKnight, 1983.
 GM
R T
=
1
N1
ln  1 +
2
N2
combinatorial
ln  2 +  blend  1  2
residual
(A1-xBx)N1/(C1-yDy)N2:
 blend = ( 1 - x ) ( 1 - y )  AC + ( 1 - x ) y  AD + x ( 1 - y )  BC + x y  BD
- x ( 1 - x )  AB - y ( 1 - y )  CD
G. Bogdanić, R. Vuković, et. al., 1997.
Miscibility of poly(S-co-oClS)/SPPO
Miscibility of poly(S-co-pClS)/SPPO
() one phase; () two phases; (  ) predicted miscibility/immiscibility boundary by the mean-field
model
G. Bogdanić, 2006.
Miscibility behavior PPO/poly(oFS-co-pClS)
system
( ------ ) correlated by the UNIQUAC-FV model
Miscibility of SPPO/poly(oBrS-co-pBrS)
system
(  ) correlated by the UNIQUAC-FV model
 Why so many different models have been
developed for polymer systems?
 The choice of a suitable model depends on:
 the actual problem and on the type of mixture
 type of phase equilibrium (VLE, LLE, SLE)
 conditions (temperature, pressure, concentration)
 type of calculation (accuracy, speed, yes/no
answer, or complete design)
Many databases and reliable GC-methods are
available for estimating:
 pure polymer properties
 phase equilibrium of polymer solutions
VLE:
 GC - models based on UNIFAC + FV
 GC - EOS
LLE
 simple FV expression + local composition
energetic term (UNIQUAC)