A Statistical Associating Fluid Theory for Electrolyte ... · Thermodynamics & Properties Lab. 1 A...

Thermodynamics & Properties Lab. 1

A Statistical Associating Fluid Theoryfor Electrolyte Solutions

based on SAFT-VRE ( Gil-Villegas et al. Molecular Physics, 2001, 99, 531-546 )

Kim, Yong Soo

Chemical and Biological Engineering, Korea University


Introduction

Application of Electrolyte Equation of State (EoS) high pressure phase equilibria, ionic surfactant systems, gas

hydrate systems containing salts, and so on.

Approach of SAFT-VR to electrolyte systems SAFT-VR

• A second-order high-temperature perturbation expansion of a variable-ranged potential (Square well potential).

• Dispersion term : The mean-field level of van der Waals is used. Contribution due to the ion-ion interaction

• Primitive models : Debye-Hückel theory, Mean Spherical approximation (MSA)


Primitive Models for Electrolyte Solutions

The solvent contribution is described by a constant dielectric constants.

Interaction potential : (1) Potential

• The repulsive interactions with a hard-sphere potential.

• (2)

Electrostatic potential : (3)• The first term : the Coulomb potential between two ions.• The second term : Oppositely charged co-ions surrounding each one

of the interacting ions i and j, as well as the reaction-field potential.

)()()( rrru ijijij ψφ +=

)(rijφ

2)( where, if ,0 if ,

);( jiHSij r

rr σσσ

σσ

σφ +=

><∞

=

)(/)( rDrqqr CSijjiij ψψ +=


Primitive Models(PM) for Electrolyte Solutions

Expressions for An average electrostatic potential is considered as follows

(4) Poisson-boltzmann equation

(5)

Potential of mean force ( PMF ), • It is possible to obtain wij(r) by solving the Ornstein-Zernike (OZ)

equation for the interionic pair potential.• At low density, the PMF reduces to the intermolecular pair potential.

(6)

)(rijψ)(rΨ

)()( rqr jij Ψ=ψ

−−=Ψ∇j

ijjjj kTrwqD

r ]/)(exp[4)(2 ρπ

)(rwij

)()( rurw ijij =


Debye-Hückel Theory (I)

A low density fluid of ions of the same diameter(7)

Boundary condition Continuity of and at vanishes at infinity

Solution of average electronic potential

(8)

)()(4

]/)(exp[4)(

2

2

Ψ+=

−−=Ψ∇

jijjj

jijjjj

rrqDkT

kTruqD

r

κφρπ

ρπ

=j

jqDkT

222 4 where, ρπκ

)(rΨ )(rD Ψ∇ σ=r)(rΨ

+−−=Ψ

κσσκ

1)](exp[)( r

Drqr i


Debye-Hückel Theory (II)

Radial distribution function(9)

Internal energy(10)

Free energy

(11)

]exp[11

)(exp[exp)(

]//)(exp[)(

rDrkT

qqrDrkT

qqrg

kTqkTrrg

jijiHSij

jijij

κκσ

σκ

φ

−−=

+−−−=

Ψ−−=

=i j

ijijji

ION

drrgrurNkT

U )()(2 2ρρρπ

πκ

12

3

−=NkTAION


Debye-Hückel Theory (Summary)

The low density of ions has been assumed.

The size of the ions has been neglected.

The structure of the fluid is governed by the electrostatic interactions.

This approach is applicable up to 0.001 molal.

)()( rurw ijij =


Augmented Debye-Hückel Theory (I)

Electrostatic corrections Consideration for the size of the ions

(12)

(13)

Taking into account the quadratic term in (14)

(15)

+−−−=

κσσκ

1)(exp[1)( r

DrkTqq

rg jiij

+−+−=

2)1ln(

41 22

3σκκσκσ

πσNkTAION

)()(2

1)(11)( 222 rq

kTrq

kTrg jjij Ψ+Ψ−=

)(rgij

+

++−=)1(

1)1ln(24 3

2

κσκσ

πσκ

NkTAION


Augmented Debye-Hückel Theory (II)

Non-electrostatic corrections For higher densities of ions, the non-ionic packing information

must be considered.(16)

The radial distribution function and the free energy can be calculated in the context of perturbation theory as follows.

(17)

(18)

);();()( σεσφφ rrr HS Φ−=

221

1

)(11

)()()(

akT

akT

aNkTA

rgkT

rgrg

HSM

HSij

Mij

++=

++= ε


Mean Spherical Approximation (MSA)

Ornstein-Zernike (OZ) equation Linking the total correlation function to the direct

correlation function according to

(19)

MSA The direct correlation function is given by the intermolecular pair

potential outside the repulsive core of the molecule. The radial distribution function is zero inside the core.

1)()( −= rgrh)(rc

+= )()()()( 321331212 rhrcdrrcrh ρ


MSA-Restricted Primitive Model (MSA-RPM)

Assumption All the ions in the solution are of the same size. The pair potential is given by equations (1) to (3) and .

MSA expressions with RPM

0=CSijψ

(21) for )(

(20) for 1)(

σ

σ

>−=

<−=

rDkTr

qqrc

rrh

jiij

ij


MSA-Restricted Primitive Model (MSA-RPM)

The radial distribution function (Blum et al., 1970)

Where,

The electrostatic residual free energy

(23) )21(

(22) ),1()()(

2

2/12

2

xxxxxτ

DkTrqq

gg jiHSijij

+−+=

−−= τσσ

κσ=x

(24) 12

)21(22633

2/32

πρσxxx

NkTAION +−++−=


MSA-Primitive Model (MSA-PM)

Anions and cations are not restricted to be of the same size. The structural and thermodynamic properties are written as

functions of a characteristic inverse length, Γ.

==

=

=

−=ΔΓ+Δ

+=Ω

Γ+Ω=

Γ+Δ−=

=Γ

n

iiii

n

i ii

ii

n

i ii

iiiin

ii

niiii

n

iii

qPPqQ

QDkT

1

3

1

3

1

2

2/1

1

2

61 ,

121

11 ,

1)2/(

(25)

σρπσ

σπσ

σρσ

σπ

ρπ


MSA-Primitive Model (MSA-PM)

The radial distribution function (Blum et al., 1975)

The residual free energy

Where

(26) )1(2

31

)( 23

2

3 DkTQQ

g jijjiiijijij −

−+

−=

ςσσς

ςσ

σσ

(27) 321

1 3

1

22

ρππ

σρ

ρΓ+

Ω

Δ+

Γ+Γ−=

=

n

in

ii

iiION

PqDkTNkT

A

. and 2/)( 1 ==+= ni

liiiljiij σρςσσσ


SAFT-VRE Approach

SAFT-VRE

Ideal free energy of the mixture

where, Λi is the thermal de Broglie wavelength of species i.

(28) NkTA

NkTA

NkTA

NkTA

NkTA

NkTA IONASSOCCHAINMIDEAL

++++=

(29) 1ln1

3 −Λ= =

n

iiii

IDEAL

xNkT

A ρ


SAFT-VRE Approach

The monomer-monomer interaction

is the hard-sphere free energy of the mixture are the first two terms of the perturbation expansion

associated with the attractive energy.

(30) )(

11221 a

kTa

kTaa

NkTA HSM

M

++==

HSa

21 and aa

(31) ];[1 1

31 1

11 = == =

−==n

i

n

j

effij

HSij

VDWijji

n

i

n

j

ijji gxxaxxa ςσαρ

(32) 3/)1(2 33 −= ijijijVDWij λσπεα

(33) 2 1 1

1

1 122

= == = ∂∂==

n

i

n

j

ij

ijHS

ji

n

i

n

j

ijji

aKxxaxxaρ

ερ


SAFT-VRE Approach

Association term Water-water ( hydrogen bonding ), ion-water ( solvation ), ion-ion

(ion-pairing) interactions.

The effect of the association interactions depends on the range of densities, temperatures, and the nature of the solvent.

Solvation : the formation of hydration shells in aqueous solutions

Ion-pairing is not important in aqueous solutions of strong electrolytes at ambient temperature.

(34) 22

ln1

,,

1

+

−= ==

iS

a

iiaia

n

ii

ASSOC sXXx

NkTA


Model parameter study for a model solution of NaCl

Effect of the dielectric constant Determining the nature of the

solvent The higher the dielectric con-

stant the better the ionic solvent

Using DH and MSA-RPM theories

D=1

D=∞


Solvent-ion attractive interactions : dispersive type Interactions through the

repulsive and Coulombic potentials only.

The vapor pressure is under-predicted as the solvent-ion interaction is strong.

For greater λ, a less deep well and higher vapor pressure.

α1j /α11=0.31For λ1j =1.2

α1j /α11=1

For α1j /α11=1

λ1j =1.2

λ1j =1.8

3/)1(2 33 −= ijijijVDWij λσπεα


Solvent-ion attractive interactions : associative type To describe the solvent-ion

interactions (solvation shell)

Adjustable parameter• Water-ion interations :

• The number of attractive site :][/1 KkHB

jε

iS

For 6 sites

For 12 sites

For 6 sites

For 12 sites

1366K

2448K1224K

1366K

1366K

1224K

1366K2448K


Ion pairing ion-ion interactions

• Strong electrolytes are fully dissociated.

• The ion pairing occur around the critical region.

Three models• Coulombic interaction• Anion-cation attractive square-

well interaction• Association via short-ranged

attractive sites

For supplementing solvent-ion attractive square-well interactions+(a)

Coulombic interaction only(1)Ion-ion association+(1)

Attractive ion-ion square-well interaction+(1)+(2)


Conclusion

Several levels of approximation have been considered for primitive model theories.

The solvent-ion interaction is modeled via an attractive potential of variable-range.

Associative sites of ion-water interactions have been considered.

SAFT-Electrolyte EoS is applicable to phase equilibria containing strong electrolyte.

A Statistical Associating Fluid Theory for Electrolyte ... · Thermodynamics & Properties Lab. 1 A...

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