Biochemical and Chemical Engineering PC-SAFT: Theory and ...icp/mediawiki/...PC-SAFT:...

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



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)



� 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



� 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



� 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




• 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



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



- 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



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



• association energy ε• association volume κAiBi


• 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

= − +

∑ ∑



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


� 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

−=



• 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


� 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

= − ∑

+

–



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 ∑


� 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



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



� 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)



� 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

≠

∂∂=



• 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



• 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



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



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



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



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



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‘‘|



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



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



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



• 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



• 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!



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.



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



[ ]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



� 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 −−−



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 =

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