Brannon-Peppas theory of swelling in ionic hydrogels

Transcription

1 BEH.46/3.96J Mlecular Principles f Bimaterials Spring 3 Brannn-Peppas thery f swelling in inic hydrgels Original thery fr elastic netwrks develped by Flry and Mehrer 1-3, refined fr treatment f inic hydrgels by Brannn-Peppas and Peppas 4,5 Other theretical treatments 6 Derivatin f inic hydrgel swelling Mdel structure f the system: Mdel f system: Inrganic anin, e.g. Cl - Inrganic catin, e.g. Na water System is cmpsed f permanently crss-linked plymer chains, water, and salt We will derive the thermdynamic behavir f the inic hydrgel using the mdel we previusly develped fr neutral hydrgels swelling in gd slvent Mdel parameters: a activity f catins in gel a activity f catins in slutin a - activity f anins in gel a - activity f anins in slutin c cncentratin f catins in gel (mles/vlume) c cncentratin f catins in slutin (mles/vlume) c - cncentratin f anins in slutin (mles/vlume) c - cncentratin f anins in slutin (mles/vlume) cncentratin f electrlyte c cncentratin f inizable repeat units in gel (mles/vlume) µ 1 chemical ptential f water in slutin µ 1 chemical ptential f water in the hydrgel µ 1 chemical ptential f pure water in standard state M Mlecular weight f plymer chains befre crss-linking M c Mlecular weight f crss-linked subchains n 1 number f water mlecules in swllen gel χ plymer-slvent interactin parameter k B Bltzman cnstant T abslute temperature (Kelvin) v m, 1 mlar vlume f slvent (water, vlume/mle) v m, mlar vlume f plymer (vlume/mle) v sp, 1 specific vlume f slvent (water, vlume/mass) v sp, specific vlume f plymer (vlume/mass) V ttal vlume f plymer V s ttal vlume f swllen hydrgel V r ttal vlume f relaxed hydrgel ν number f subchains in netwrk ν e number f effective subchains in netwrk ν stichimetric cefficient fr eletrlyte catin stichimetric cefficient fr eletrlyte anin φ 1,s vlume fractin f water in swllen gel φ,s vlume fractin f plymer in swllen gel φ,r vlume fractin f plymer in relaxed gel x 1 mle fractin f water in swllen gel x 1 mle fractin f water in slutin Asterisks dente parameters in slutin Free energy has 3 cmpnents: free energy f mixing, elastic free energy, and inic free energy Eqn 1 G ttal = G mix G el G in Lecture 9 plyelectrlyte hydrgels 1 f 6

2 BEH.46/3.96J Mlecular Principles f Bimaterials Spring 3 Eqn Eqn 3 Eqn 4 Eqn 5 Eqn 6 Eqn 7 At equilibrium, the chemical ptential f water inside and utside the gel are equal: µ 1 = µ 1 µ 1 - µ 1 = µ 1 µ 1 Slutin cntains ins s µ 1 is nt equal t µ 1 ( µ 1 ) TOTAL = ( µ 1 ) TOTAL ( µ 1 ) in = ( µ 1 ) mix ( µ 1 ) el ( µ 1 ) in The equatin we ll try t slve is a rearrangement f this: ( µ 1 ) in - ( µ 1 ) in = ( µ 1 ) mix ( µ 1 ) el Cntributins t the free energy: Free energy f mixing: G mix = H mix T S mix We previusly derived the cntributin frm mixing using the Flry-Rehner lattice mdel: Eqn 8 G mix = k B T[n 1 ln (1-φ,s ) χn 1 φ,s ] ( 1 ) = mix ( G mix ) Eqn 9 µ = k B T[ln(1 φ,s ) φ,s χφ,s ] = RT[ln(1 φ,s ) φ,s χφ,s ] n 1 T,P Secnd expressin puts us n a mlar basis instead f per mlecule Elastic free energy: Eqn 1 G el = (3/)k B Tν e (α 1 ln α) Eqn 11 ( ) µ = ( G el ) = ( G el ) α v m,1 φ,s 1/3 1 φ,s 1 el n 1 α M T,P T,P n 1 = RTν 1 M c V T,P r φ rs φ rs v m,1 = RT v sp, M c M 1 c φ,s 1/3 1 φ,s M φ,r φ rs φ rs Last equality uses: ν = V /v sp, M c (n handut) V r = V /φ, r (n handut) Thus ν/v r = φ, r /v sp, M c Inic free energy: Term driving dilutin f ins diffusing int gel t maintain charge neutrality Chemical ptential change in slutin: Eqn 1 ( µ ) 1 1 all _ slutes 1 in = µ µ = RT ln a 1 RT ln x 1 = RT ln(1 x ) apprximatin in third equality is used fr dilute slutins Lecture 9 plyelectrlyte hydrgels f 6

3 BEH.46/3.96J Mlecular Principles f Bimaterials Spring 3 Eqn 13 ( µ 1 ) all _ ins all _ ins all _ ins all _ ins RT in x = RT n = v RT m,1 n v RT m,1 c n v m,1 n The first apprximatin hlds if Σx is small Furth equality hlds because we assume in the liquid lattice mdel that the mlar vlume f all species is the same, thus v m, 1 n = V, the ttal vlume f the system Chemical ptential change in gel: Eqn 14 all ins ( µ 1 ) in = µ 1 µ 1 = RT ln a 1 v m,1 RT c all ins ( ) Eqn 15 µ ( 1 ) = v RT in m,1 (c c ) 1 in µ The electrlyte disslved in water prvides mbile catins and anins in the slutin and in the gel: E.g. NaCl: Na νcl - ν (s) ν Na (aq) ν - Cl - (aq) ν = ν - = 1 stichimetric cefficients z z Eqn 16 C ν A ν C z A z e.g. CaCl : ν = 1, ν - =, z =, z - = 1 Eqn 17 ν = νˆ fr a 1:1 electrlyte ˆ Eqn 18 ν = = ν fr a 1:1 electrlyte ˆ Eqn 19 c c = (ν ) = ν ttal cncentratin f ins Eqn We will derive equatins fr an aninic netwrk Assuming activities ~ cncentratins Inside gel: c = ν Eqn 1 c - = ν - ic /z - [ RCOO ][ H ] Eqn K a = [RCOOH] Eqn 3 i = c is the mles f inizable repeat grups n gel chains per vlume First term cmes frm electrlyte anins in gel, secnd term frm inized grups n the plymer chains The degree f inizatin i can be related t the ph f the envirnment and the pka f the netwrk grups: [RCOO ] K a H [RCOO ] [RCOOH] [ ] K a K a 1 pk a = = = = = [RCOOH] [RCOO ] [RCOO ] K 1 a [ ] K a 1 ph K a 1 ph 1 pk a 1 [RCOOH] [ H ] H Lecture 9 plyelectrlyte hydrgels 3 f 6

4 BEH.46/3.96J Mlecular Principles f Bimaterials Spring 3 Outside gel: Eqn 4 c = ν Eqn 5 c - = ν - Our relatinship fr the inic chemical ptentials is nw: ( ) ( ) all ins ) Eqn 6 µ µ = v 1 in m,1rt (c c 1 )=v m,1 RT(c c c c in Using Eqn, Eqn 1, Eqn 4, and Eqn 5, Eqn 6 becmes: Eqn 7 ( µ ) ( ) µ = v 1 in m,1rt ν ic ˆc ν s = v m,1 RT νˆ ic 1 ˆc ν s in = v m,1 RT ic ν(c ˆ s ) Hw can we relate and? We can make simplificatins fr a 1:1 catin:anin electrlyte: The chemical ptentials f the mbile ins must als be equilibrated inside/utside the gel: Eqn 8 µ = µ Eqn 9 µ - = µ - Add Eqn 9 t Eqn 8: Eqn 3 µ µ - = µ µ - ν Eqn 31 RT ln a ν RT ln a ν = RT ln a ν RT ln a Therefre we can write: ν ν Eqn 3 a a = a a Assuming dilute slutins where the activities are apprximately equal t the cncentratins: Eqn 33 Eqn 34 c c = c c ν ν ν c s = ic ν Eqn 35 ν cs s c = ic Lecture 9 plyelectrlyte hydrgels 4 f 6

5 BEH.46/3.96J Mlecular Principles f Bimaterials Spring 3 Eqn 36 c s ν c s c 1 ic = 1 sic c = 1 sic ic = c s ν ˆz z ν ˆ Derivatin f this equatin in appendix Nw Eqn 7 becmes: Eqn 37 ( µ ) ( ) i µ = v c 1 1 in m,1rt z ν in ˆ But definitin f initrength I is: Eqn 38 all _ ins I = 1 z i c i = z ν i ˆ fr a 1:1 electrlyte Therefre: Where z i is the charge n in i i φ ( ),s Eqn 39 µ ( ) = v i µ c 1 1 in m,1rt = v m,1 RT in 4I 4Ivsp, M φ,s (Using relatin c = vsp, M =mles inizable grups/vlume) Eqn 39 can be re-cast in terms f the slutin ph: ( ) 1 in ( µ ) = v RT m,1 K a φ,s K a φ Eqn 4 µ,s 1 in 4I 1 ph K a v sp, M = v RT m,1 1 ph K a 4Iv sp, M Returning t the equilibrium criterin: Eqn 41 1 pk a φ,s v m,1 M 1 c φ 1/3,s 1 φ,s v m,1 1 ph 1 pk a 4Iv sp, M = ln(1 φ,s ) φ,s χφ,s φ,r vsp, M c M φ,r φ,r Brannn-Peppas paper analyzes Plyacrylates/plymethacrylates: In water ph 7. with I =.35 χ =.8 pk a = 6. v sp, =.8 cm 3 /g M = 75, g/mle M c = 1, g/mle M = 9 g/mle φ,r =.5 Lecture 9 plyelectrlyte hydrgels 5 f 6

6 BEH.46/3.96J Mlecular Principles f Bimaterials Spring 3 References 1. James, H. M. & Guth, E. Simple presentatin f netwrk thery f rubber, with a discussin f ther theries. J. Plym. Sci. 4, (1949).. Flry, P. J. & Rehner Jr., J. Statistical mechanics f crss-linked plymer netwrks. I. Rubberlike elasticity. J. Chem. Phys. 11, 51-5 (1943). 3. Flry, P. J. & Rehner Jr., J. Statistical mechanics f crss-linked plymer netwrks. II. Swelling. J. Chem. Phys. 11, (1943). 4. Brannnpeppas, L. & Peppas, N. A. Equilibrium Swelling Behavir f Ph-Sensitive Hydrgels. Chemical Engineering Science 46, (1991). 5. Peppas, N. A. & Merrill, E. W. Plyvinyl-Alchl) Hydrgels - Reinfrcement f Radiatin-Crsslinked Netwrks by Crystallizatin. Jurnal f Plymer Science Part a-plymer Chemistry 14, (1976). 6. Ozyurek, C., Caykara, T., Kantglu, O. & Guven, O. Characterizatin f netwrk structure f ply(n-vinyl - pyrrlidne/acrylic acid) plyelectrlyte hydrgels by swelling measurements. Jurnal f Plymer Science Part B- Plymer Physics 38, (). Lecture 9 plyelectrlyte hydrgels 6 f 6