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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
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 Groß 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 (ρ • g(r)) gives the local density
in the distance of r from the center of
a molecule
�
0
1
2
0 1 2 3 4 5 6
r/σ
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)ρρρρ
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/σ
u(r)
-101234
0 1 2 3
u/ε
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/σ
-101234
0 1 2 3
r/σ
u/ε
-101234
0 1 2 3r/σ
u/ε
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 σ
• segment number m
• dispersion energy ε
� 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
( )1
2ij i jσ σ σ= +
( )1ij i j ijkε ε ε= ⋅ − kij = binary parameter
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 ζ ζ ζ ζζ ( ζ )
kT ζ ( ζ ) ζ ( ζ ) ζ
= + + − ⋅ − − −
221 3 2d d d dζ ζ
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 dζ ζg (d )
( ζ ) d d ( ζ ) d d ( ζ )
= + + − + − + −
{ }n
n i i 0,1,2,36
i
i
πζ ρ x m d n= ⋅ =∑
ii i 1 0.12 exp 3d
kT
ε = σ − ⋅ − ⋅
radial distribution
function
temperature-dependent
segment diameter
=
for square-well potential
PC-SAFT – dispersion
� Perturbation theory of Barker and Henderson
( )mI ,31 ζ
...+⋅⋅⋅⋅−= ∫∑∑λ
σε
πρ1
232 rdrgkT
mmxxkTa chainhard
ijij
i jjiji
disp
r
s
l·s
u(r)
σ
λσ
9Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
- power function in ζ3 (reduced density)
- simple dependence of coefficients ai upon m (segment number)
- fitted to simulation data of square-well fluids
( )mI ,31 ζ
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 εAiBi
Proton-
Donator
Proton-
Acceptor
11Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
• association energy ε• association volume κAiBi
� Mixtures: One-fluid theory
• Wohlbach-Sandler combining rules:
no additional binary parameter
( )1
2
i j j ji iA B A BA B= +ε ε ε
( )
3
12
i j j ji iii jjA B A BA B
ii jj
σ σκ κ κ
σ σ
⋅ = ⋅ +
PC-SAFT association contribution
� Association i
i
ii
iassoc AA 1
ln 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
−
= + ⋅ ⋅∆
∑ ∑ρ
( ) 3 exp 1i j
i j i j
A BA B A Bhs
ij ij ijg dkT
εκ σ
∆ = ⋅ ⋅ ⋅ −
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 Jnn
kTxxa ,2
22,,3
33
22∗∗∑∑−= µµ
σσσεε
πρ µµ
333224 σσσεεερπ
δ+δ−
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
k
JnnnkT
xxxa ,3222
,,,
333
3
22
3 34 ∗∗∗∑∑∑−= µµµ
σσσσσσεεερπ
µµµ
n
n
ijijnijn
DDij kT
baJ 3
4
0,,,2 ζ
ε∑
=
+= ∑
=
=4
03,,3
n
nijkn
DDijk cJ ζ
Further contributions to PC-SAFT
� Electrostatic interactions
• reference system:
hard-sphere fluid
• perturbation: Debye-Hückel charge forces
22
12
elec
i i ieli
a ex z
kT kT
κ χπ ε
= − ∑
+
–
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
χ κ σ κ σ κ σκ σ
= + + ⋅ − + ⋅ + + ⋅ ⋅
22 2N
i ie li
ez x
kT
ρκε
= ∑
12 ikT kTπ ε ∑
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
0
ln11232
VV
VV
VV
xkT
a
n
np
elast
ϕϕ
15Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
with
• one new parameter to be adjusted: network functionality ϕn
• use of experimental setup
3
3
1
6
Pi i i
ip
n NV x m
x
π σζ
= ∑
3
3 2
max
1 109.52 sin
8 2 180c p pV x N m d
π = ⋅
o
o
0max02 VVVkT nϕ
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 ϕn
entanglement
unoccupied binding
sites of cross-linkerdangling endschain loops
Contributions and parameters of PC-SAFT
=aresidual ahard-chain adispersion+
+–
σmseg
εεhb κhb
ϕn
δ+δ−
(+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)
Why Helmholtz energy A?
� PC-SAFT � Helmholtz Energy A may be used for…
• pressure p and compressibility factor Z
• density ρ by iteration
• chemical potential µ
• fugacity coefficient ϕ• Entropy S
• internal energy U
T
Ap
V
∂ = − ∂
AS
T
∂ = − ∂
ijT,V,nii n
Aµ
≠
∂∂=
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= −
Modelling approach of phase equilibria
� Modelling the thermodynamic equilibrium µi‘ = µi‘‘
vapour phase
liquid phase
19Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
• Condition: isofugacity criterion fi‘ = fi‘‘
• Using the ϕ−ϕ concept xi‘ϕi‘ p‘ = xi‘‘ϕi‘‘ p‘‘
• with the fugacity coefficient ϕ 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]
ρ [k
g/m
³]
**
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: xi‘ϕi‘ p‘ = xi‘‘ϕi‘‘p‘‘
xi‘ϕi‘ p‘ = xi‘‘ϕi‘‘(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
2
j 31 2 i j i j ij
2 ,
,
i
i j
i
i j
aπρ I (η m) x x m m
kT kT
πρ m C I (η m) x x m mkT
ε = − ⋅ ⋅ σ
ε − ⋅ ⋅ ⋅ ⋅ σ
∑∑
∑∑1hc
hc1
12 2 3 4
1
8 2 20 27 12 2 1 (1 )
ZC Z
m m
−
−
∂= + + ρ ∂ρ
η − η η − η + η − η = + + −
defined variable
32Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
[ ]24
8 2 20 27 12 2 1 (1 )
(1 ) (1 )(2 )m m
η − η η − η + η − η = + + − − η − η − η
( )6
i1 i
0
, ( )i
I m a mη η=
= ⋅∑ ( )6
i2 i
0
, ( )i
I m b mη η=
= ⋅∑ power functions
i 0i 1i 2i
1 1 2( )
m m ma m a a a
m m m
− − −= + +
i 0i 1i 2i
1 1 2( )
m m mb m b b b
m m m
− − −= + +
defined coefficients
PC-SAFT – polar contibutions
� Padé approximation
� Dipolar term
polar 2
3 21
aa
a a=
−
( )DD
ijjijiij
jjii
i j
jjiiji Jnn
ktxxa ,2
22,,3
33
22∗∗∑∑−= µµ
σσσεε
πρ µµ
( )DD
ijkkjikjikkjjiikkjjii
kji Jnnnkt
xxxa ,3222
,,,
333
3
22
3 3
4 ∗∗∗∑∑∑−= µµµσσσσσσεεερπ
µµµ
33Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
� Quadrupolar term
( ) ijkkjikjijkikiji j
kjik
Jnnnkt
xxxa ,3,,,33 3 ∑∑∑−= µµµσσσ µµµ
( )θθ
θθ θθσ
σσεερπ ijjiji
ij
jjii
i j
jjiiji Jnn
ktxxa ,2
22,,7
55
2
2
2 4
3 ∗∗∑∑
−=
( )θθ
θθθ θθθσσσσσσεεερπ
ijkkjikjijkikij
kkjjii
i j
kkjjiikji
k
Jnnnkt
xxxa ,3222
,,,333
555
3
22
3 16
9 ∗∗∗∑∑∑−=
PC-SAFT equations
� Dipolar term
� Quadrupolar term
with
n
n
ijijnijn
DDij kT
baJ ηε
∑=
+=
4
0,,,2 ∑
=
=4
0,,3
n
nijkn
DDijk cJ η
n
n
ijijnijnij kT
baJ ηεθθ ∑
=
+=
4
0,,,2 ∑
=
=4
0,,3
n
nijknijk cJ ηθθ
mmm 211 −−−
34Laboratory of Thermodynamics
Prof. Dr. G. Sadowski
with
nij
ij
ij
ijn
ij
ijnijn a
m
m
m
ma
m
maa 210,
211 −−+
−+=
nij
ij
ij
ijn
ij
ijnijn b
m
m
m
mb
m
mbb 210,
211 −−+
−+=
nijk
ijk
ijk
ijkn
ijk
ijknijkn c
m
m
m
mc
m
mcc 210,
211 −−+
−+=
( )2
1
jiij mmm =
( )3
1
kjiijk mmmm =