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Biochemical and
Chemical Engineering
Markus C. Arndt, Gabriele Sadowski
PC-SAFT: Theory and Application
Laboratory of Thermodynamics
Workshop on Hydrogel Modelling
24 November 2011, Stuttgart
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Outline
Idea of PC-SAFT
Its contributions:
hard sphere
hard chain
dispersion
association
dipoles
electrostatic energyConclusion
2Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
electrostatic energy
elastic forces
exemplary modelling results
Conclusion
-
Basic idea of PC-SAFT
PC-SAFT: Perturbed-Chain Statistical Associating Fluid Theory
Developed by Joachim Gro and Gabriele Sadowski
Molecular model from statistical mechanics
Molecules are built of spherical segments
- may compose chains
- with repulsive and attractive interactions
polymer
solvent
3Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
solvent
schematic Flory-Huggins
Lattice Chain Model for a
polymer solution Theory for chain molecules
structure of chain fluid?
intermolecular radial distribution function
interaction between chain fluid?
interaction potentialGross, Sadowski, Fluid Phase Equilib. (168) 2000Gross, Sadowski, Ind. Eng. Chem. Res. (40) 2001
-
Radial distribution function g(r)
gives the relative probability of finding
other molecules surrounding a center
molecule in the distance of r
product of ( g(r)) gives the local densityin the distance of r from the center of
a molecule
0
1
2
0 1 2 3 4 5 6
r/
g(r)
4Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
is a function of density, molecule size and shape,
as well as of interaction potential
calculated from molecular simulations or
integral calculations (statistical mechanics) r
dr
g(r)
-
Interaction potentials
Hard-sphere fluid:
hard-sphere repulsion
no attraction
Square-well fluid:
hard spheres
with attraction well
Modified square-well fluid:
-0.01
0.99
0 1 2 3
r/
u(r)
-101234
0 1 2 3
u/
34
5Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
Modified square-well fluid:
soft spheres
with attraction well
Lennard-Jones fluid:
soft spheres
with soft attraction well
0 1 2 3
r/
-101234
0 1 2 3
r/
u/
-101234
0 1 2 3r/
u/
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Perturbation theory
Problems for real systems:
Analytical function of radial distribution is not available
Interaction potentials for real systems are unknown
Solution: perturbation theory
reference system: hard-sphere fluid
1st perturbation: chain formation
hard-chain fluid
6Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
hard-chain fluid
reference system: hard-chain fluid
2nd perturbation: attraction (dispersive interaction)
hard-chain fluid with attractive interactions
Contributions to Helmholtz energy: a residual = a hard sphere + a chain formation + a dispersion
a hard chain
-
PC-SAFT parameters
Pure-component parameters for molecules (non-polar, non-associating, uncharged):
segment diameter
segment number m
dispersion energy
Mixtures: One-fluid theory
7Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
Mixtures: One-fluid theory
mean segment number
Berthelot-Lorenz combining rules between components i and j:
i i=i
m x m
( )12
ij i j = +
( )1ij i j ijk = kij = binary parameter
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PC-SAFT hard chain contribution
Hard-chain term
with
ii
hc hshs
i i ii1 ln ( )i
a am x (m ) g d
kT kT=
i i
i
m x m=
hs 3 3
1 2 2 20 32 2
0 3 3 3 3
1 3ln 1
1 1
a ( )
kT ( ) ( )
= + +
221 3 2d d d d
mean segment number
hard-sphere term
radial distribution
8Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
22
i j i jhs 2 2ij ij 2 3
3 i j 3 i j 3
1 3 2
1 1 1
d d d d g (d )
( ) d d ( ) d d ( )
= + +
+ +
{ }nn i i 0,1,2,36
i
i
x m d n= =
ii i 1 0.12 exp 3d
kT
=
radial distribution
function
temperature-dependent
segment diameter
-
=
for square-well potential
PC-SAFT dispersion
Perturbation theory of Barker and Henderson
( )mI ,31
...+=
pi1
232 rdrgkT
mmxxkTa chainhard
ijij
i jjiji
disp
r
s
ls
u(r)
9Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
- power function in 3 (reduced density)- simple dependence of coefficients ai upon m (segment number)
- fitted to simulation data of square-well fluids
( )mI ,31
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PC-SAFT equation of state
Contributions to Helmholtz energy
a residual = a hard sphere + a chain formation + a dispersion
10Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
a residual = a hard sphere + a chain formation + a dispersion
a hard chain
but what about other molecular interactions?
-
Further contributions to PC-SAFT
Association hydrogen bonding
association between two (hard-sphere) molecules
with association sites (proton donator and acceptor)
reference system: hard-sphere fluid
perturbation: square-well attraction
Additional pure-component parameters for associating molecules:
association energy AiBi
Proton-
Donator
Proton-
Acceptor
11Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
association energy
association volume AiBi
Mixtures: One-fluid theory
Wohlbach-Sandler combining rules:
no additional binary parameter
( )12
i j j ji iA B A BA B
= + ( )3
12
i j j ji iii jjA B A BA B
ii jj
= +
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PC-SAFT association contribution
Association i
i
ii
iassoc AA 1ln
2 2A
a Xx X
kT
= +
12Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
with 1
1 j i ji
j
B A BA
j
j B
X x X
= +
( ) 3 exp 1i ji j i j A BA B A Bhsij ij ijg dkT
=
fraction of molecules
which are not bonded
association strength
-
Further contributions to PC-SAFT
Dipole and Quadrupole
reference system: two-center Lennard-Jones fluid
perturbation: dipole (Stockmayer potential) or quadrupole
( )DD
ijjijiij
jjii
i j
jjiiji JnnkT
xxa,2
22,,3
33
22=
pi
333224 pi
+
23
2
1 aaa
apolar
=
13Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
applicable for polar components (ketones, esters, ethers, aldehydes, etc.)
no additional pure-component parameters required
direct use of the dipole moment from experiments or quantum mechanics
( )DD
ijkkjikjijkikij
kkjjii
i j
kkjjiikji
kJnnn
kTxxxa
,3222
,,,
333
3
22
3 34 =
pi
n
n
ijijnijn
DDij kT
baJ 34
0,,,2
=
+=
=
=
4
03,,3
n
nijkn
DDijk cJ
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Further contributions to PC-SAFT
Electrostatic interactions
reference system:
hard-sphere fluid
perturbation: Debye-Hckel charge forces
22
12
elec
i i ieli
a ex z
kT kT
pi
=
+
14Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
with
no additional pure-component parameters required
ion-specific, not salt-specific parameters
direct use of the molecules electric charge
2
3
3 3 1ln(1 ) 2(1 ) (1 )
( ) 2 2i i i i
i
= + + + + +
22 2N
i ie li
ez x
kT
=
12 ikT kTpi
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Further contributions to PC-SAFT
Elastic energy of a network
reference system:
hard-chain fluid
perturbation: elastic force due to deformation
=
0
13
2
max
32
0ln11
232
VV
VV
VV
xkT
a
n
np
elast
15Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
with
one new parameter to be adjusted: network functionality n use of experimental setup
3
3
1
6
Pi i i
ip
n NV x m
x
pi=
3
3 2
max
1 109.52 sin
8 2 180c p pV x N m d
pi =
o
o
0max0
2 VVVkT n
-
Adjustable parameter is inevitable
Assumption: tetrahedrally oriented monodisperse chains
Reality: Networks are not homogenously built
Imperfections and network errors may
cause greater stiffness
give greater mesh size
be elastically ineffective
Experimental procedure?
varying chain length
16Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
Accounted for with adjustable parameter n
entanglement
unoccupied binding
sites of cross-linkerdangling endschain loops
-
Contributions and parameters of PC-SAFT
=aresidual ahard-chain adispersion+
+
mseg
hb hb
n+
(+kij)
17Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
2 up to 6 (+1) adjustable pure component parameters to obtain the
Helmholtz energy in arbitrary mixtures
Parameter fitted to experimental data of pure components or a mixture
such as liquid density, vapor pressure, activity coefficient, solubility, ..
=aresidual ahard-chain adispersion+
(+ aassociation) (+ aelastic)(+ aelectrostatic)(+ adipole)
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Why Helmholtz energy A?
PC-SAFT Helmholtz Energy A may be used for
pressure p and compressibility factor Z
density by iteration chemical potential
fugacity coefficient Entropy S
internal energy U
T
Ap
V
=
AS
T
=
ijT,V,nii n
A
=
18Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
internal energy U
enthalpy H
Gibbs energy G
complete thermodynamic description of a system
U A TS= +VT
=
H U pV= +G H TS=
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Modelling approach of phase equilibria
Modelling the thermodynamic equilibrium i = i
vapour phase
liquid phase
19Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
Condition: isofugacity criterion fi = fi
Using the concept xii p = xii p with the fugacity coefficient from PC-SAFT:
-
n-heptane ethanol
temperature dependence of VLE
365
370
375
kij=0,038
T
[K]
Vapour Liquid Equilibria (VLE)
1,0
1,5
kij=0,036
p
[bar]
338,15 K
Acetone n-heptane
pressure dependence of VLE
V
20Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
0,0 0,2 0,4 0,6 0,8 1,0
345
350
355
360
x/yHeptan
[-]
1,0132 bar
0,0 0,2 0,4 0,6 0,8 1,0
0,5
x/yAceton
[-]
313,15 K
Albers, unpublished 2011
V
VL
L
-
Liquid Liquid Equilibria (LLE)
Miscibility gap water ethylacetate narrows with increasing temperature
and with the solubiliser methanol (at 20 C)
L
21Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
Exp: Sorensen; Liquid-liquid Equilibrium Data Collection 1980
L
LL
-
Solid Liquid Equilibria (solubility)
Amino acids in water (binary and ternary systems)
glycine L-alanine
L-valine
25 C 30 C
SL
22Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
L-leucine
Held et al., Ind. Eng. Chem. Res (50) 2011
L-valine
L
-
Solid Liquid Equilibria (solubility)
Ternary sugar solubility in water
SL
35 C
25 C
23Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
Held, unpublished
Exp: Ferreira et al., Ind. Eng. Chem. Res (42) 2003
L
-
PNIPAAm in water
Binary mixture of PNIPAAm-water (without cross-linker)
LLE and density
LL
T [K]
[
k
g
/
m
]
*
*n
NH O
24Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
Exp: Wohlfarth C, CRC Handbook of Liquid-Liquid Equilibrium Data of Polymer Solutions 2008
L
wPNIPAAm [-]
5 C
20 C
25 C
wPNIPAAm [-]
-
PNIPAAm in water
Binary mixture of PNIPAAm-solvent (cross-linked to hydrogel)
Considering elastic force: xii p = xiipxii p = xii(p-pelast)
T [K] y = 0.005y = 0.005y = 0.005y = 0.005
y = 0.010y = 0.010y = 0.010y = 0.010
elastic pressure
p = pelast
p = p|
25Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
Exp: Poschlad, Diss., Berlin, 2011
Exp: Zhi, Chem. Eng. Sci. (65) 2010
m/m0 [-]
y = 0.015y = 0.015y = 0.015y = 0.015
-
Hydrogels: other mixtures
Poly(acrylic acid) PAA water
Loose chains: no LLE
Cross-linked gel: high swelling
without transition
T [K]
1.5
PNIPAAm water/2-propanol (20 C)
Pronounced co-nonsolvency
26Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
Exp: Shin et al., Eur Polym J, 34 (2), 1998
m/m0 [-]
Exp: Miki et al., Mater. Res. Soc. of Japan, 32 (889), 2007
1
0.5
0
-
Ternary swelling: PNIPAAm in water/ethanol
Ternary System: Swelling depends on
Temperature
Concentration of
Water/EtOH
27Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
PNIPAAm in water
PNIPAAm in ethanol
Ternary mixture (25 C)
-
Conclusion (I)
Advantages of PC-SAFT compared to other EOS and activity-coefficient models
physically-based model
accounts for size and shape of molecules
suitable also for complex and large molecules
equations of state account for the density (pressure) dependence
28Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
reliable for extrapolation
to other conditions (T, p, concentration)
to multi-component systems (binary ternary, )
results confirmed the wide applicability of PC-SAFT
all thermodynamic properties can be derived from Helmholtz energy function
-
Conclusion (II)
Hydrogel networks can be modelled with PC-SAFT
by considering a new elastic contribution:
aelast in the Helmholtz energy
Pelast in the isofugacity equation
Gel swelling and the gel concentrations depend on
chain length
29Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
chain length
temperature
concentration of solutes/cosolvents
Further research with focus on more complex
systems, polyelectrolytes and diffusion
m/m0 [-]
T [K]
-
PC-SAFT: Theory and Application
Thank you for your attention!
30Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
Questions?
-
Deduction and explanation of PC-SAFT
Gross, J.; Sadowski, G. Application of perturbation theory to a hard-chain
reference fluid: An equation of state for squarewell chains. Fluid Phase
Equilib. 2000, 168, 183.
Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a
Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40, 1244.
31Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
-
PC-SAFT - dispersion
Dispersion term
with
dispj 3
1 i j i j ij
2j 3
1 2 i j i j ij
2 ,
,
i
i j
i
i j
a I ( m) x x m m
kT kT
m C I ( m) x x m mkT
=
1hc
hc1
12 2 3 4
1
8 2 20 27 12 2 1 (1 )
ZC Z
m m
= + +
+ = + +
defined variable
32Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
[ ]248 2 20 27 12 2
1 (1 )(1 ) (1 )(2 )m m
+ = + +
( ) 6 i1 i0
, ( )i
I m a m =
= ( )6
i2 i
0, ( )
iI m b m
=
= power functions
i 0i 1i 2i1 1 2( ) m m ma m a a a
m m m
= + +
i 0i 1i 2i1 1 2( ) m m mb m b b b
m m m
= + +
defined coefficients
-
PC-SAFT polar contibutions
Pad approximation
Dipolar term
polar 2
3 21a
aa a
=
( )DD
ijjijiij
jjii
i j
jjiiji Jnnkt
xxa,2
22,,3
33
22=
pi
( )DD
ijkkjikjikkjjiikkjjii
kji Jnnnktxxxa
,3222
,,,
333
3
22
3 34 =
pi
33Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
Quadrupolar term
( ) ijkkjikjijkikiji j kjik Jnnnktxxxa ,3,,,33 3 =
( )
pi ijjiji
ij
jjii
i j
jjiiji Jnnkt
xxa,2
22,,7
55
2
2
2 43
=
( )
pi
ijkkjikjijkikij
kkjjii
i j
kkjjiikji
kJnnn
ktxxxa
,3222
,,,333
555
3
22
3 169 =
-
PC-SAFT equations
Dipolar term
Quadrupolar term
with
n
n
ijijnijn
DDij kT
baJ
=
+=
4
0,,,2
=
=
4
0,,3
n
n
ijknDD
ijk cJ
n
n
ijijnijnij kT
baJ
=
+=
4
0,,,2
=
=
4
0,,3
n
n
ijknijk cJ
mmm 211
34Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
with
n
ij
ij
ij
ijn
ij
ijnijn a
m
m
m
ma
m
maa 210,
211 +
+=
n
ij
ij
ij
ijn
ij
ijnijn b
m
m
m
mb
m
mbb 210,
211 +
+=
n
ijk
ijk
ijk
ijkn
ijk
ijknijkn c
m
m
m
mc
m
mcc 210,
211 +
+=
( )21jiij mmm =
( )31kjiijk mmmm =
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