Microfluidic Directed Synthesis of Alginate. Nanogels with Tunable Pore Size for Efficient

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1 Supporting inormat Microluidic Directed Synthei o Alginate Nanogel with Tunable Pore Sie or Eicient Protein Delivery Salime Baban-Shotorbani, Eran Dahtimoghadam, Akbar Karkhaneh, Mohammad Mahdi Haani-Sadrabadi,, *, Karl I. Jacob,, * enter o Excellence in Biomaterial, Department o Biomedical Engineering and Department o Polymer Engineering, Amirkabir Univerity o Technology, Tehran, Iran. Parker H. Petit Intitute or Bioengineering and Biocience, G.W. Woodru School o Mechanical Engineering, Georgia Intitute o Technology, Atlanta 3033, GA, USA. School o Material Science and Engineering, Georgia Intitute o Technology, Atlanta , GA, USA. Laboratoire de Microyteme (LMIS4), Intitute o Microengineering and Intitute o Bioengineering, École Polytechnique Fédérale de Lauanne (EPFL), H-05 Lauanne, Switerland.

2 A microluidic approach or ynthei o alginate nanogel with adjutable pore ie in order to achieve adjutable releae proile o protein i preented. Here we derive equat to calculate average molecular weight between crolink, M c, or ph-enitive alginate nanogel in order to get an inight into the eect o parameter involved in microluidic directed ynthei o particle on their releae proile in imulated gatric and intetinal luid. Equilibrium welling theory: The theory to explain the equilibrium welling o polymeric network wa irt propoed by Flory and Rehner [, ]. According to their theory, the ree energy change ( G) involved in the welling o a polymeric network in a welling agent conit o two part, the ree energy o mixing ( G) mix and the elatic ree energy ( G) el, ( G)= ( G) el ( G) mix () Equilibrium welling theory or polyelectrolyte: In the cae o ic network uch a alginate, the condit i more complicated and the term( G), ic Free energy, hould be added to the Equat [3], ( ) ( ) ( ) G= G G G () el mix Taking the derivative o each term in thi equat with repect to the number o molecule o olvent lead to the Equat 3, ( ) ( ) ( ) µ µ = µ µ µ (3) 0 el mix 0 where µ i the chemical potential o the olvent inide the gel and µ i the chemical potential o the pure olvent. At the equilibrium welling tate, µ mut be equal to the chemical potential

3 o the olvent in the external olutµ. By inerting thi condit in Equat 3 and µ = µ µ can readily be hown that 0 ubtituting ( ) ( µ ) ( µ ) ( µ ) ( µ ) = (4) el mix The elatic contribut: ( µ ) el can be obtained by dierent equat depending on the elatic nature o the network. Equat 5 and Equat 6 are ued or thi purpoe, which were propoed by Peppa or polymer crolinked in the preence o olvent and are valid or Gauian and non-gauian chain length ditribut, repectively [3], [4]. 3 Gauian V M υ, υ c, µ = RT el υ, r M M 0 υ, r υ cυ, r ( ) (5) ( ) 3 3 non Gauian V,,, M υ υ υ c 3 RT, r el, M M 0 υ, r υ cυ, r n υ, r n µ = υ υ 3 (6) where R i the univeral ga contant, T i the abolute temperature, υ, i the polymer volume ract in wollen gel, υ,r i the polymer volume ract in relaxed gel (ormed right ater crolinking and beore welling), and n i the crolink denity calculated rom Equat 7, wherein Mr i the molecular weight o the monomer. M c n= (7) M r The mixing contribut: ( µ ) mix i given by the Flory- Huggin polymer olut theory a ollow [5], 3

4 ( µ ) = RT ( υ ) υ χυ ln mix,,, (8) χ i Flory polymer- welling agent interact parameter that can be expreed a [6] χ = υ, (9) By ubtituting Equat 9 in Equat 8, 3 ( ) RT ln( ) µ = υ, υ mix, υ, υ, (0) The ic contribut: ( µ ) ( µ ) i calculated by relathip that propoed by Flory [5]. In thi repect, when an ic network i placed in an electrolyte olut, the concentrat o mobile inide the gel i i more than the concentrat o them in the external olut i, becaue o the attracting power o the ixed charge on the polymer chain; Thereore, the omotic preure o the olut inide the gel π i greater than the omotic preure o urrounding olutπ. The dierence between omotic preure can be obtained by ( ) i i ( ) ( ) π = RT = RT () ( ) 0 µ µ onidering the thermodynamic relathip π =, the right-hand ide o Equat 4 can be written a V ( µ ) ( µ ) = V π = RTV ( ) ( ) () 4

5 Now, we need to calculate and which are the concentrat o mobile cat and an inide the gel, and and which are the correponding concentrat in the external olut. The exchange o and olvent between the gel and it urrounding i imilar to Donnan membrane equilibrium [5], ince polymer act a it own membrane and prevent the ixed charge group on the chain rom diuing into external olut. Suppoe that polymer contain ubtituent which diociate to mobile cat external olut contain a trong electrolyte M cat M and ixed an remain on the chain and the A υ υ, which completely diociate into υ M and υ an A. Without acriice o generality, the cat o the external electrolyte and the cat o the polyelectrolyte gel are conidered to be ame. Initially, there i no A an inide the gel, o ome o them will diue into the gel. Thee an attract more M rom outer olut into the gel. By neglecting the toichiometric ene, the migrat o an i conidered to be in the ame equivalence with cat. At the equilibrium tate, the concentrat o electrolyte inide and outide o the gel are named and, repectively; hence, the concentrat o mobile negative and poitive outide and inide the gel will be obtained a = υ (3) = υ (4) = υ (5) = υ (6) di The concentrat o mobile an and cat outide the gel a well a the concentrat o mobile an inide the gel are due to the preence o the diociated o the electrolyte, but 5

6 the concentrat o mobile cat inide the gel include the concentrat o diociated φi o electrolyte υ, and concentrat o diociated o the polymer di =, where i the molar concentrat o monomer that can be given by Equat 7, di φi φiυ, = = (7) υm r The term = i i the total concentrat o ixed charge on the poly and φ i the ract o eective ixed charge, which doe not participate in -pairing phenomenon. According to counter condenat (or -pairing) theory o Manning [7], ome o diociated counter condene onto the polyelectrolyte until the charge denity along the polyelectrolyte chain i reduced below a critical value. I the concentrat o diociate which remain bound to the chain i b and the concentrat o the ree charge group i, φ i equal to φ = = b (8) By ue o Equat 3 to 6, the Equat can be rewritten a ollow whereυ = υ υ, ( µ ) ( µ ) = RTV di υ( ) (9) It can readily be hown that Equat 4 to 6, 0 and 9 lead to Equat 0 and, which can be ued or calculat o Mc or Gauian and non-gauian polymer chain, repectively. 3 3 V M υ, υ c, ln( υ, ) 0.06υ, 0.44υ, υ, υ, r V di υ( ) = M M 0 υ, r υ cυ, r (0) 6

7 ( ) 3 3 3,,, 3 V M υ υ c M υ r M r 3,,,,, r, M M0 υ, r υ, r M c υ, r M cυ c ln υ 0.06υ 0.44υ υ υ υ = ( ) V di υ Derivat o an equat or approximat o the quantity o : All term o the derived equat or calculat o M, are meaurable except. Dierent method were propoed or approximat o the quantity o c (). A common approach i uing the thermodynamic ormulat, which i preented in Equat. According to thi ormulat, the activity o the electrolyte i the ame inide and outide the gel. υ υ υ υ a a = a a () In thi equat, a and a are the activity o cat and an inide the gel and a and a are their activity in the external olut. Subtitut o a coeicient, lead to = γ where γ i the activity γ γ = (3) υ υ υ υ υ υ υ υ γ γ The ormulat or calculating the quantity o were developed by Flory by auming an ideal olut environment in which the activitie are equal to concentrat ( γ = ) [5]. Recently, han et al. propoed ome relathip or predicting the concentrat o mobile inide the gel in which the condit i not ideal ( γ ). According to them [8, 9], when a polyan i placed in an environment containing a trong electrolyte, the activity coeicient o 7

8 mobile an inide the polyan γ can be aumed to be equal to the activity coeicient o external an γ, becaue thi do not participate in -pairing phenomenon. Additally, or trong electrolyte the activity coeicient o external i equal to the mean ic activity coeicient, γ = γ = γ (4) ± γ γ = γ (5) ± The activity coeicient o mobile cat inide the gel γ can be obtained by ubtract o activity coeicient o bound cat γ x rom activity coeicient o the external cat, γ = γ γ x (6) Now, we need to calculate the quantity oγ x. Suppoe that one cat with charge o inide the gel can pair with ied group on the chain (we conider that the ixed an o the polymer ha charge o -). The accuracy o thi aumpt depend on the ditance between ixed charge, the diameter o the cat and interact parameter between and olvent. I diameter o the cat i much more than the ditance between the ixed charge and interact between cat and ixed charge are tronger than interact between and olvent, thi aumpt can be conidered to be true. According to our aumpt, the concentrat o cat which bind to the chain i x b =. Since the activity coeicient o concentrat o i equal toγ, the activity coeicient o concentrat o x can be written a γ = = (7) x b γ x γ 8

9 We ee rom Equat 6 and Equat 7 that b γ = γ γ (8) Our aumpt lead alo to the binding contant K bc = (9) b From Equat 8, 9 and 5 we obtain γ = γ± Kbc (30) Thi equat how good agreement with empirical relathip [0]. By ubtituting Equat 3 to 6, 4, 5 and 30 in Equat 3, we would arrive at the ollowing Equat or calculating, υ υ φi υ Kbc υ υ = υ (3) Similarly, or polyelectrolyte chain with charge o we can write υ φi υ υ υ υ υ = K bc υ (3) alculat o * and or alginate nanogel in SIF ater paing SGF: In order to predict the average molecular weight o alginate chain between crolink in imulated intetinal luid (SIF) ater paing the imulated gatric luid (SGF), V i the volume o particle in SIF and V r 9

10 which reer to the volume o particle beore welling, can be conidered to be the volume o them in SGF. Simulated intetinal luid i a olut o monobaic potaium phophate ( KH PO 4 ) and odium hydroxide (NaOH) with the ph o 6.8. KH PO 4 i a weak electrolyte and the concentrat o K and H PO 4 in the olut are very low and we can aume that NaOH i the only eective electrolyte in the welling o particle. The concentrat o NaOH in SIF i mol/cm 3, o we can write odium and carboxylate, * 5 mol =.4 0. The aociat contant or 3 cm K bc, i 90 M - [9], and the ract o omotically active charge, φ, i 0.4 [] and the molecular weight o odium alginate repeating unit ( 5H 7O4OONa ) i 98 g υ, mol. In thi way, Equat 3 can be written a ( ) = υ. Uing two latter, 40 equat lead to the ollowing graph, which can be ued to predict the concentrat o by knowing the volume ract o polymer in the wollen gel. and 0

11 REFERENES. Flory, p.j. and J.R. Rehner, Statitical Mechanic o ro-linked Polymer Network I. Rubberlike Elaticity. journal o chemical phyic, 943. (): p Flory, P.J. and J.R. Rehner, Statitical Mechanic o ro-linked Polymer Network II. Swelling. journal o chemical phyic, 943. (): p Brannon-Peppa, L. and N.A. Peppa, Equilibrium welling behavior o ph-enitive hydrogel. Journal o chemical engineering cience, (3): p Brannon-Peppa, L. and N.A. Peppa, Equilibrium welling behavior o dilute ic hydrogel in electrolytic olut. Journal o controlled releae, 99. 6: p Flory, P.J., Principle o polymer chemitry. 953, New York: Ithaca: ornell Univerity. 6. Vaheghani-Farahani, E., et al., Swelling o ic gel in electrolyte olut. Indutrial and Engineering hemical Reearch, (4): p Manning, G.S., Limiting Law and ounter ondenat in Polyelectrolyte Solut I. olligative Propertie. journal o chemical phyic, (3): p han, A.W., R.A. Whitney, and J.R. Rehner, Kinetic controlled ynthei o phreponive network alginate. Biomacromolecule, : p han, A.W. and R.J. Neueld, Modeling the controllable ph-reponive welling and pore ie o networked alginate baed biomaterial. Biomaterial, : p

12 0. Mae, S., et al., Model or counter-membrane-ixed pairing and Donnan equilibrium in charged membrane. Journal o phyical chemitry, : p Moe, S.T., et al., Swelling o ovalently rolinked Alginate Gel: Inluence o Ionic Solute and Nonpolar Solvent. Macromolecule, : p