Hg Workshop Talk Arndt

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 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



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)



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



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/

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

segment number m

dispersion energy

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

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



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)



- 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

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 AiBi

Proton-

Donator

Proton-

Acceptor



association energy

association volume AiBi


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

= +

PC-SAFT association contribution

Association i

i

ii

iassoc AA 1ln

2 2A

a Xx X

kT

= +



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


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

=



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


Electrostatic interactions

reference system:

hard-sphere fluid

perturbation: Debye-Hckel charge forces

22

12

elec

i i ieli

a ex z

kT kT

pi

=

+



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


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



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



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)



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 by iteration chemical potential

fugacity coefficient Entropy S

internal energy U

T

Ap

V

=

AS

T

=

ijT,V,nii n

A

=



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 i = i

vapour phase

liquid phase



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



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]

[

k

g

/

m

]

*

*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: 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|



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

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



[ ]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



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



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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