PC-SAFT: Theory and Application

Size: px

Start display at page:

Download "PC-SAFT: Theory and Application"

Maurice McLaughlin
6 years ago
Views:

1 Biochemicl nd Chemicl Engineering PC-SAFT: Theory nd Appliction Mrkus C. Arndt, Gbriele Sdowski Workshop on Hydrogel Modelling 4 November 011, Stuttgrt

2 Outline Ide of PC-SAFT Conclusion Its contributions: hrd sphere hrd chin dispersion ssocition dipoles electrosttic energy elstic forces exemplry modelling results

3 Bsic ide of PC-SAFT PC-SAFT: Perturbed-Chin Sttisticl Associting Fluid Theory Developed by Jochim Groß nd Gbriele Sdowski Moleculr model from sttisticl mechnics Molecules re built of sphericl segments - my compose chins - with repulsive nd ttrctive interctions polymer solvent Theory for chin molecules structure of chin fluid? intermoleculr rdil distribution function interction between chin fluid? interction potentil schemtic Flory-Huggins Lttice Chin Model for polymer solution Gross, Sdowski, Fluid Phse Equilib. (168) 000 Gross, Sdowski, Ind. Eng. Chem. Res. (40) 001

4 Rdil distribution function g(r) gives the reltive probbility of finding other molecules surrounding center molecule in the distnce of r g(r) 1 product of (ρ g(r)) gives the locl density in the distnce of r from the center of molecule is function of density, molecule size nd shpe, s well s of interction potentil r/ clculted from moleculr simultions or integrl clcultions (sttisticl mechnics) ρ g(r) dr r 4

5 Interction potentils Hrd-sphere fluid: hrd-sphere repulsion no ttrction Squre-well fluid: hrd spheres with ttrction well Modified squre-well fluid: soft spheres with ttrction well Lennrd-Jones fluid: soft spheres with soft ttrction well 0.99 u(r) u/ε u/ε u/ε r/ 0 1 r/ 0 1 r/ 0 1 r/ 5

6 Perturbtion theory Problems for rel systems: Anlyticl function of rdil distribution is not vilble Interction potentils for rel systems re unknown Solution: perturbtion theory reference system: hrd-sphere fluid 1 st perturbtion: chin formtion hrd-chin fluid reference system: hrd-chin fluid nd perturbtion: ttrction (dispersive interction) hrd-chin fluid with ttrctive interctions Contributions to Helmholtz energy: residul = hrd sphere + chin formtion + dispersion hrd chin 6

7 PC-SAFT prmeters Pure-component prmeters for molecules (non-polr, non-ssociting, unchrged): segment dimeter segment number m dispersion energy ε Mixtures: One-fluid theory men segment number m = i xm i i Berthelot-Lorenz combining rules between components i nd j: 1 = i + j ( ) ( 1 k ) ε = ε ε k = binry prmeter i j 7

8 PC-SAFT hrd chin contribution Hrd-chin term with m = i xm i i hc kt hs hs = m x(m i i 1) ln gii ( dii) kt men segment number i hs 1 ζ1ζ ζ ζ = + + ζ 0 ln ( 1 ζ ) kt ζ 0 ( 1 ζ ) ζ ( 1 ζ ) ζ hs 1 dd i j ζ dd i j ζ = + + ( 1 ζ ) di + d j ( 1 ζ ) di + d j ( 1 ζ ) g (d ) hrd-sphere term rdil distribution function π ζ ρ xmd n { } n n = i i i = 0,1,, 6 i εi d i = i exp kt temperture-dependent segment dimeter 8

9 PC-SAFT dispersion Perturbtion theory of Brker nd Henderson kt disp λ ε hrd chin = πρ xi x jmim j g r dr kt i j u(r) I 1( ζ,m) = for squre-well potentil s λ l s r ( ) I,m 1 ζ -power function in ζ (reduced density) -simple dependence of coefficients i upon m (segment number) - fitted to simultion dt of squre-well fluids 9

10 PC-SAFT eqution of stte Contributions to Helmholtz energy residul = hrd sphere + chin formtion + dispersion hrd chin but wht bout other moleculr interctions? 10

11 Further contributions to PC-SAFT Assocition hydrogen bonding ssocition between two (hrd-sphere) molecules with ssocition sites (proton dontor nd cceptor) reference system: hrd-sphere fluid perturbtion: squre-well ttrction Proton- Dontor Proton- Acceptor Additionl pure-component prmeters for ssociting molecules: ssocition energy ε AiBi ssocition volume κ AiBi Mixtures: One-fluid theory Wohlbch-Sndler combining rules: AB AjB i i j ( ) AB 1 i j ε = ε + ε AB A B i j AB i i j j κ κ κ = 1 ii jj ( ii + jj ) no dditionl binry prmeter 11

12 PC-SAFT ssocition contribution Assocition ssoc A i A X 1 i = xi ln X + kt i A i with j j B Ai j i j X = 1 + ρ x X j B AB 1 frction of molecules which re not bonded AB i j AB i j AB hs i j ε = g ( d ) κ exp 1 kt ssocition strength 1

13 Further contributions to PC-SAFT Dipole nd Qudrupole polr = 1 reference system: two-center Lennrd-Jones fluid perturbtion: dipole (Stockmyer potentil) or qudrupole δ+ δ ε ε ii jj ii jj DD = πρ xi x j n, in, j i j J µ µ µ µ, i j ( kt ) 4 ε ε ε = µ 4 π ρ ii jj kk ii jj kk xi x j xk n in jn µ, µ, µ, kµ i µ j k i j k ( kt ) ik jk ε 4 DD = J, k = c n= 0 4 DD n J, n, + bn, ζ n= 0 kt pplicble for polr components (ketones, esters, ethers, ldehydes, etc.) no dditionl pure-component prmeters required direct use of the dipole moment from experiments or quntum mechnics n, k ζ n J DD, k 1

Further contributions to PC-SAFT Electrosttic interctions reference system: hrd-sphere fluid perturbtion: Debye-Hückel chrge forces + kt elec κe = el xz i i χi 1 π ε kt i with κ ρ e N = el zi

14 Further contributions to PC-SAFT Electrosttic interctions reference system: hrd-sphere fluid perturbtion: Debye-Hückel chrge forces + kt elec κe = el xz i i χi 1 π ε kt i with κ ρ e N = el zi xi ε kt i 1 χ = + ln(1 + κ ) (1 + κ ) + (1 + κ ) i i i i ( κ i) no dditionl pure-component prmeters required ion-specific, not slt-specific prmeters direct use of the molecules electric chrge 14

15 Further contributions to PC-SAFT Elstic energy of network reference system: hrd-chin fluid perturbtion: elstic force due to deformtion kt elst = x p ϕn ϕ ϕ n V V 1 1 V V 1 ln V V 0 mx 0 with V npn π 1 = xi i m x 6 ζ p i i o π Vmx = xcn sin m d o p p one new prmeter to be djusted: network functionlity ϕ n use of experimentl setup 15

16 Adjustble prmeter is inevitble Assumption: tetrhedrlly oriented monodisperse chins Relity: Networks re not homogenously built Imperfections nd network errors my cuse greter stiffness give greter mesh size be elsticlly ineffective Experimentl procedure? vrying chin length Accounted for with djustble prmeter ϕ n entnglement chin loops dngling ends unoccupied binding sites of cross-linker 16

17 Contributions nd prmeters of PC-SAFT m seg ε (+k ) ε hb κ hb δ+ δ + ϕ n residul = hrd-chin + dispersion (+ ssocition ) (+ dipole ) (+ electrosttic ) (+ elstic ) up to 6 (+1) djustble pure component prmeters to obtin the Helmholtz energy in rbitrry mixtures Prmeter fitted to experimentl dt of pure components or mixture such s liquid density, vpor pressure, ctivity coefficient, solubility,.. 17

18 Why Helmholtz energy A? PC-SAFT Helmholtz Energy Amy be used for pressure p nd compressibility fctor Z density ρ by itertion chemicl potentil µ fugcity coefficient ϕ Entropy S internl energy U enthlpy H Gibbs energy G A p = V A µ i = n i A S = T U = A + TS H = U + pv G = H TS T T,V,n j V i complete thermodynmic description of system 18

19 Modelling pproch of phse equilibri vpour phse liquid phse Modelling the thermodynmic equilibrium µ i = µ i Condition: isofugcity criterion f i = f i Using the ϕ ϕconcept x i ϕ i p = x i ϕ i p with the fugcity coefficient ϕ from PC-SAFT: 19

20 Vpour Liquid Equilibri (VLE) n-heptne ethnol temperture dependence of VLE Acetone n-heptne pressure dependence of VLE T [K] k =0,08 V 1,01 br VL p 1,5 [br] 1,0 0,5 k =0,06 8,15 K 1,15 K 45 0,0 0, 0,4 0,6 0,8 1,0 x/y Heptn [-] L 0,0 0, 0,4 0,6 0,8 1,0 x/y Aceton [-] Albers, unpublished 011 0

21 Liquid Liquid Equilibri (LLE) Miscibility gp wter ethylcette nrrows with incresing temperture nd with the solubiliser methnol (t 0 C) L L LL Exp: Sorensen; Liquid-liquid Equilibrium Dt Collection

22 Solid Liquid Equilibri (solubility) Amino cids in wter (binry nd ternry systems) 5 C 0 C glycine L-lnine SL L-leucine L L-vline Held et l., Ind. Eng. Chem. Res (50) 011

23 Solid Liquid Equilibri (solubility) Ternry sugr solubility in wter 5 C 5 C SL L Held, unpublished Exp: Ferreir et l., Ind. Eng. Chem. Res (4) 00

24 PNIPAAm in wter Binry mixture of PNIPAAm-wter (without cross-linker) LLE nd density * HN n O * T [K] LL ρ [kg/m³] L 5 C 0 C 5 C w PNIPAAm [-] w PNIPAAm [-] Exp: Wohlfrth C, CRC Hndbook of Liquid-Liquid Equilibrium Dt of Polymer Solutions 008 4

25 PNIPAAm in wter Binry mixture of PNIPAAm-solvent (cross-linked to hydrogel) Consideringelsticforce: x i ϕ i p = x i ϕ i p x i ϕ i p = x i ϕ i (p -p elst ) p = p elstic pressure T [K] y = y = y = p = p elst Exp: Poschld, Diss., Berlin, 011 Exp: Zhi, Chem. Eng. Sci.(65) 010 m/m 0 [-] 5

26 Hydrogels: other mixtures Poly(crylic cid) PAA wter Loose chins: no LLE Cross-linked gel: high swelling without trnsition PNIPAAm wter/-propnol (0 C) Pronounced co-nonsolvency T [K] m/m 0 [-] Exp: Shin et l., Eur Polym J, 4 (), 1998 Exp: Miki et l., Mter. Res. Soc. of Jpn, (889), 007 6

27 Ternry swelling: PNIPAAm in wter/ethnol Ternry System: Swelling depends on Temperture Concentrtion of Wter/EtOH PNIPAAm in wter PNIPAAm in ethnol Ternry mixture (5 C) 7

28 Conclusion (I) Advntges of PC-SAFT compred to other EOS nd ctivity-coefficient models physiclly-bsed model ccounts for size nd shpe of molecules suitble lso for complex nd lrge molecules equtions of stte ccount for the density (pressure) dependence relible for extrpoltion to other conditions (T, p, concentrtion) to multi-component systems (binry ternry, ) results confirmed the wide pplicbility of PC-SAFT ll thermodynmic properties cn be derived from Helmholtz energy function 8

29 Conclusion (II) Hydrogel networks cn be modelled with PC-SAFT by considering new elstic contribution: elst in the Helmholtz energy P elst in the isofugcity eqution Gel swelling nd the gel concentrtions depend on chin length temperture concentrtion of solutes/cosolvents T [K] Further reserch with focus on more complex systems, polyelectrolytes nd diffusion m/m 0 [-] 9

30 PC-SAFT: Theory nd Appliction Thnk you for your ttention! Questions? 0

31 Deduction nd explntion of PC-SAFT Gross, J.; Sdowski, G. Appliction of perturbtion theory to hrd-chin reference fluid: An eqution of stte for squrewell chins. Fluid Phse Equilib. 000, 168, 18. Gross, J.; Sdowski, G. Perturbed-Chin SAFT: An Eqution of Stte Bsed on Perturbtion Theory for Chin Molecules. Ind. Eng. Chem. Res. 001, 40,

32 PC-SAFT - dispersion Dispersion term with C hc 1 = 1+ Z + ρ kt disp Z ρ hc ε = πρ I (η, m) x x m m 1 i j i j i j kt εi j πρ m C1 I (η, m) xi xj mi mj i j kt 1 defined vrible 8 η η 0 η 7 η + 1 η η = 1 + m + (1 m) (1 η) 4 [(1 η)( η) ] i 1 ( η, ) = i( ) η ( η ) i i= 0 i= 0 I m m I, m = b( m) m 1 m 1 m ( m) = + + m m m i 0i 1i i m 1 m 1 m b ( m) = b + b + b m m m i 0i 1i i i η power functions defined coefficients

33 PC-SAFT polr contibutions Pdé pproximtion Dipolr term = 1 polr ε ε ii jj ii jj DD = πρ xix j n, in, j i j J µ µ µ µ, i j ( kt) ε ε ε = µ 4π ρ ii jj kk ii jj kk x i x j x k n in jn k i µ, µ, µ, µ µ j k i j k ( kt ) ik jk J DD, k Qudrupolr term ε 5 5 ii jj ii jj θθ π = ρ xi x j nθ, inθ, jθi θ j J 7, 4 i j ( kt) ε = ε ε ε π ρ ii jj kk ii jj kk xi x j xk nθ inθ jnθ kθi θ j θ,,, k 16 i j k ( kt) ik jk J θθ, k

34 PC-SAFT equtions Dipolr term J ε 4 4 DD n DD = n + bn η n= kt,,, J, k = 0 n= 0 c n, k n η Qudrupolr term J ε 4 4 θθ n = n + bn η n= kt θθ,,, J, k = c n, k 0 n= 0 n η with m 1 m 1 m 1n + m m m n, = 0n + n m = 1 ( mim j ) m 1 m 1 m bn, = b0n + b1 n + bn m m m m 1 m 1 m k k k cn, k = c0n + c1n + cn mk mk mk m = k 1 ( mim jmk ) 4

7/19/2011. Models of Solution Chemistry- III Acids and Bases

7/19/2011. Models of Solution Chemistry- III Acids and Bases Models of Solution Chemistry- III Acids nd Bses Ionic Atmosphere Model : Revisiting Ionic Strength Ionic strength - mesure of totl concentrtion of ions in the solution Chpter 8 1 2 i μ ( ) 2 c i z c concentrtion