Numerical Simulation of ph-sensitive Hydrogel Response in Different Conditions

Transcription

1 Numercal Smulaton of ph-senstve Hydrogel Response n Dfferent Condtons Bran O. Asmba, Andrew K. Shgol, Kamlesh J. Suthar *, Muraldhar K. Ghantasala, and Derrck. C. Mancn * Department of Mechancal and Aeronautcal Engneerng Western Mchgan Unversty, 1903, West Mchgan Ave., Kalamazoo, MI * Advanced Photon Source, Argonne Natonal Laboratory, 9700, S. Cass Ave, Argonne, IL Abstract: The understandng of ph-senstve hydrogel swellng response n dfferent buffer envronmental condton s essental for ts use n dfferent practcal applcatons. Ths necesstates ts smulaton n steady state and transent condtons. Ths paper manly deals wth the detals of the numercal smulaton performed by developng coupled formulaton of chemoelectro-mechancal behavor of the hydrogel n response to changng ph of the surroundng soluton. Smulatons were performed to determne the response of hydrogel wth varyng ph of the surroundng soluton n a wde range of ph (2-12). The nvestgaton of the responsveness of the hydrogels s focused manly on the study of effect of varaton of pka and Young s Modulus of the gel. The methodology used for ths fnte element based smulaton s presented. The swellng characterstcs of the hydrogel obtaned under steady state condtons n these nvestgatons are compared wth prevous smulatons usng other models/methods. Ths analyss s carred out usng COMSOL and the effects of fxed charge densty, buffer soluton ph and Young s modulus on the swellng were studed n dfferent smulatons. These smulaton results are compared wth avalable expermental evdence to show the accuracy of the model. Keywords: Hydrogel, multphyscs models, COMSOL, swellng, fxed charge densty. 1. Introducton Hydrogels are polymers that respond to envronmental factors such as temperature, ph, electrc potental and lght by ether swellng or deswellng. Ther ablty to absorb water s attrbuted to presence of hydrophlc functonal groups whch are attached to the polymerc network, whle the crosslnkng prevent complete mxng of the hydrogel from dssolvng n the solvent by producng an elastc restorng force that counters the expanson of the network. The study of swellng and deswellng mechansms of the hydrogels has been carredout by few research groups world over, by developng dfferent mathematcal models and usng fnte element based software [1-6]. However, the software n most cases was custom developed for ths applcaton, whch s not avalable to all researchers. Ths has prompted us to start workng on developng the approprate models for usng wdely avalable COMSOL software [6-7]. In ths paper, we modeled hydrogel swellng and dewellng n response to surroundng ph changes usng chemo-electro-mechancal couplng of three partal non-lnear dfferental equatons representng onc dffuson nto the gel, potental changes due to the redstrbuton of charges wthn and outsde the gel and contnuty equaton respectvely. These smulatons were done usng COMSOL Multphyscs and employng movng mesh n a 2-dmensonal feld. The man focus of these smulatons s to understand the effect of varaton of young s modulus of the gel along wth the pka of the buffer. 2. Governng Equatons Smulaton of Hydrogel swellng/ deswellng characterstcs requre consderng the mathematcal representaton of dfferent nteracton mechansms (between the gel and a soluton) and the resultant changes. Three mportant changes normally consdered n these

2 smulatons are chemcal, electrcal and mechancal n nature. The Nernst Planck, Posson and mechancal deformaton equatons are commonly used to determne these changes. Nernst Planck Equaton The Nernst-Planck equaton defnes the relaton between the concentratons of the varous moble speces n the buffer soluton. Applyng contnuty equaton, the change n concentraton flux wth respect to space s equated wth the rate of change of concentraton whch s gven by: c t dv( j ) 0 If we modfy ths equaton to nclude flux due to dffuson of ons whch s manly due to two factors: dffuson due to concentraton gradent and mgraton flux because of electrc potental and hence t can be modfed as: c zf dv( D ( grad( c ) c grad( ))) 0 (1) t RT where D, c, z, F, R, T and ψ are the dffuson co-effcent of the th on, the concentraton of the th on, valence of the th on, the faraday constant, the unversal gas constant, temperature and the electrc potental respectvely. In ths equaton, the frst term represents the dffusve flux due to the concentraton gradent and the second term whch s coupled wth the Posson s equaton s the mgraton flux whch s due to the electrc potental gradent. Snce we are consderng steady state, the Nernst-Planck equaton s wrtten as: D RT Where μ s the moblty of the th on. Posson s Equaton The Posson s equaton s used to understand the spatal dstrbuton of the electrc potental (ψ) and satsfes the electroneutralty condton. It s gven by the equaton: 2 0 (2) Where ρ s the charge densty n the hydrogel, s the delectrc constant of the vacuum and the relatve delectrc constant of the solvent. n F z c z c 1 (3) Where Z f and C f are the valence and fxed charge concentraton n the hydrogel. The fxed charge concentraton wthn the hydrogel s calculated usng the relaton, C f Where, C K, C H and H are the onzable charge s mo 0 s Cmo ( K) H( K C ) f f concentraton, dssocaton constant, hydrogen on concentraton and hydraton, respectvely. The hydraton state of the hydrogel s the rato of the volume of the flud to the volume of the sold n the gel [8]. In ths equaton c f s represented as a functon of the surroundng ph, where both H and C H are used n defnng changng onc condtons wthn the hydrogel, whch s due to the moble ons dffusng nto the hydrogel. Due to the change n the fxed charge concentraton wth changes n ph, the c f and ntal condtons are updated n every teraton. Mechancal Feld Equaton H The gel expanson s represented by a secondorder partal-dfferental equaton of moton n s

3 tme. The nertal term s neglected for steady state models. The mechancal feld equaton s thus represented as: ( C EP I) 0 (4) where [C], E, and I are the materal elastcty matrx, Green stran tensor, and dentty matrx respectvely. At the nterface of the hydrogel mxture and the soluton bath, the osmotc pressure s gven as a force term, whch s represented by: (5) Where n s number of ons, c s the concentraton of th on n the hydrogel, and c s at the th on concentraton outsde the hydrogel at the ntal condton. In ths smulaton, movng mesh was used due to the large hydrogel deformatons whch are due to the hydrogel expanson, whch s up to 300% ts ntal sze. It s also used because the plan stran was used to calculate mechancal deformatons n the hydrogel are based only on small strans. 3. Smulaton Osmotc n P RT c c o Osmotc 1 In ths smulaton, a hydrogel radus of 300µm was used wth the buffer soluton mmersng the gel sample completely. Assumng a symmetrc system, only a quarter of the sample was used nstead of a complete crcular hydrogel. Its chemo-electro-mechancal behavor was smulated n response to the ph of the buffer soluton surroundng t usng COMSOL (ver. 3.5a) wth followng modules: o In ths smulaton two frames were used, namely: the fxed frame and the movng mesh frame. The chemcal dffuson and electrostatc physcs are consdered n the movng mesh to evaluate swellng at dfferent condtons, whle the mechancal equlbrum physcs s consdered n the fxed frame wth large deformaton to calculate the large expanson wth a change n ph. Boundary condtons The followng are the boundary condtons used n the smulaton. Subdoman 1: Materal: Hydrogel Physcs Chemcal Dffuson Electrostatcs Swellng Subdoman 2: Materal: Buffer Physcs Chemcal Dffuson Movng boundares Equaton Nernst-Planck Posson s equaton Mechancal feld Wth Movng mesh Equaton Nernst Planck Mechancal feld equaton, movng mesh frame Boundary 1 Type: Insulaton/symmetrc c k x = 0 : NP equaton 1. Nernst-Planck wthout electro-neutralty (Chemcal Engneerng Module) 2. Conductve Meda DC(AC/DC Module) for Posson s Equaton 3. Plane Stran (Structural Mechancs Module) for Mechancal Feld Equaton 4. Movng Mesh (ALE) = 0 x : Posson s equaton The movng mesh frame equlbrum equatons are: u = 0, v = free; on vertcal symmetrc sde

4 u = free, v = 0; on horzontal symmetrc sde Boundary 2 Type: Interface between the hydrogel and buffer C k = contnuous : NP equaton Input: (1) chemcal and physcal parameters; (2) Boundary condtons = 0 N(+ j ) = : Posson s equaton P_osmotc : Posson s equaton Calculate fxed charge densty c X = x + u, Y = y + v : Movng mesh frame Solve for the Posson- Nernst Planck equatons Boundary 3 Update C k and ψ Type: Buffer far-feld c k = c k u = 0, v = 0 : NP equaton : Posson s equaton The algorthm used for ph smulaton s shown n the fgure 1.The fxed charge densty together wth the Posson s dependent varable (ψ) were used to couple the NP and Posson s equatons. The fxed charge densty s updated after each teraton. The mechancal feld equaton uses the osmotc pressure n the calculaton of the dsplacement, whch s found from the movng mesh n the x and y drecton. Due to the contnuous expanson n response to ph changes, hydraton s also updated after every teraton n movng mesh. No Convergence? Solve for Hydraton from the Mechancal Equlbrum Equaton Update u Convergence? Output Yes Yes Fgure 1: Flow chart of the algorthm used to solve the hydrogel response to ph varaton n steady state. No

5 2 Hydrogel 2 Buffer Soluton Fgure 2: Meshed model of the gel sample. The meshed model of the gel and surroundng buffer soluton s depcted n fgure. 2. The moble ons, Na +, Cl - and H + are consdered n the smulatonconsstng of 10 dependent varables.in ths smulaton, three dependent varables from Nernst-Planck equaton (C Na,C Cl,C H ),the electrc potental (ψ) from the Posson s equaton, the dsplacements (u, v) from the mechancal feld equaton, and the x and y coordnates (X & Y) andtwo weak constrant varables from the movng mesh module. The whole doman conssted of 1810 mesh elements wth degrees of freedom. In the smulaton, electroneutralty condton was satsfed by Posson s equaton wth the hydrogel taken as an sotropc materal. The ph was then vared from 2-12 wth a step sze of wth the error convergence crteron was fxed at 1x10-4. The modulus of elastcty s 0.29 Mpa for ph<5.5 and 0.23 for ph>7.5, wth a lnear varaton profle assumed between these two ph values, except for the case where effect of the modulus of elastcty s nvestgated. A Posson s rato of 0.43 was assumed for the entre range of ph. Statc equlbrum s used as the fnal convergence crteron usng the statonary Drect-PARADISO lnear system solver. Parametrc effects were studed usng plan stran confguraton. 4. Results and Dscusson These studes manly focused on the varaton of concentraton and pka of the buffer soluton, fxed charge densty and young s modulus of the gel on gel expanson. Fgure 3: The effect of pka on hydrogel swellng wth varyng ph. Fgure 3 shows the effect of varaton of the dsassocaton constant (pka) on swellng of the gel. From fgure 3, t can be seen that as the dssocaton constant ncreases, the gel swells rapdly at a gven ph wthn the range 3.5 to 7.5.

6 However, beyond ths ph range (>7.5), the gel becomes saturated resultng n no further swellng. It s well known that the phase transton of the gels occurs close to ts pka, whch s dentcal to the pka of onzable group [9]. The swellng process ceases, when the gel reaches an equlbrum state wthn a gven buffer envronment. charge densty s ncreased, the hydrogel swellng also enhanced. Ths can be explaned by the fact that as t ncreases the avalablty of fxed charge stes ncrease proportonally. Ths possbly attracts more moble ons to assocate wth fxed charge stes n the gel polymer network and hence the ncrease n hydrogel swellng. Fgure 4: Effect of fxed charge densty on hydrogel swellng wth change n ph. Fgure.4 shows the effect of fxed charge densty on the hydrogel swellng at varous ph values. For a gven gel volume, the fxed charge densty s constant. As the ph ncreases, dffuson of moble ons from the buffer soluton to the hydrogel s promoted resultng n hydrogel expanson untl the ph reaches around 8, where all the fxed charge stes mght have reached an equlbrum state wth buffer soluton ons. Hence, there s no further change n gel swellng beyond ths value of ph at any fxed charge densty. However, as the fxed Fgure 5: Gel expanson wth varyng Young s modulus as a functon of ph. Hydrogels wth hgh Young s modulus have exhbted relatvely lower expanson. Ths can be attrbuted to the fact that hydrogels wth hgh Young s modulus has low stran and hence decreased swellng. Young s modulus s used n the calculaton of the elastc force that s due to the osmotc pressure change between the hydrogel and the soluton, thus the degree of swellng of the hydrogel s hghly dependent on the Young s modulus as evdent from fgure.5. It may also be observed from fgure. 5, the degree of swellng s lower

7 at hgher ph values. A smlar varaton was also observed by Hua L et al [4] earler. Ths clearly shows the valdty of our model as well as smulaton. In order to ensure the valdty of our smulatons, as reported earler we have also compared our smulaton results wth exstng expermental values and found an excellent agreement between the two. It should be noted that the smulaton was done at 300 µm dameter HEMA hydrogel to match wth the expermental gel dmensons, wth the buffer concentraton fxed at 300 mm [8, 10]. 5. Conclusons In ths paper, we demonstrated the use of COMSOL for the smulaton of hydrogel swellng behavor wth respect to the varatons n the ph of surroundng buffer soluton, dssocaton constant (pka) and young s modulus(e) of the gel. Our results were compared wth some of the prevous smulatons performed usng other models n the lterature. Further, to ensure the valdaton of our model as well as methodology, smulaton results were compared wth exstng expermental results. These smulatons showed that the varaton of dsassocaton constant as well as Young s Modulus has sgnfcant effect on the gel swellng behavor. 6. References 1.Grmshaw, P.E., Electrcal control of solute transport across polyelectrolyte membranes Grmshaw, P.E., et al., Knetcs of electrcally and chemcally nduced swellng n polyelectrolyte gels. The Journal of Chemcal Physcs, 93(6), 1990, p De, S.K., et al., Equlbrum swellng and knetcs of ph-responsve hydrogels: Models, experments, and smulatons. Journal of Mcroelectromechancal Systems, 11(5), 2002, p L, H., et al., Modelng and Smulaton of the Swellng Behavor of ph- Stmulus-Responsve Hydrogels. Bomacromolecules, 6(1), 2005, p Wallmersperger, T., B. Kropln, and R.W. Gulch, Coupled chemo-electromechancal formulaton for onc polymer gels--numercal and expermental nvestgatons. Mechancs of Materals, 36(5-6), 2004, p Suthar, K.J., M. K. Ghantasala, and D. C. Mancn. Steady State Smulaton of the chemo-electro-mechancal Behavor of Hydrogels, 30(3), 2010, Suthar, K.J., M.K. Ghantasala, and D.C. Mancn. Smulaton of hydrogel mcro-actuaton. In Mcroelectroncs: Desgn, Technology, and Packagng III, 2007, 67980P P-9 ; Kamlesh J. Suthar, Derrck C. Mancn, Muraldhar K. Ghantasala, Smulaton of the effect of dfferent parameters on the swellng characterstcs of a ph-senstve hydrogel Publshed n SPIE Proceedngs Volume 7644: Behavor and Mechancs of Multfunctonal Materals and Compostes, March

8 8. De, S.K. and N.R. Aluru, A chemoelectro-mechancal mathematcal model for smulaton of ph senstve hydrogels. Mechancs of Materals, 36(5-6), 2004, p Andreas Rchter et al., Revew of Hydrogel based ph Sensors and Mcrosensors, Sensors, 8, 2008, Luo, R., H. L, and K. Y. Lam, Modelng and smulaton of the chemoelectro-mechancal behavor of phelectrc-senstve hydrogel, Analytcal and Boanalytcal Chemstry, 389 (3), 2007,