Poroelasticity of a covalently crosslinked alginate hydrogel under compression

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1 JOURNAL OF APPLIED PHYSICS 8, 54 Pooelsticity of covlently cosslinked lginte hydogel unde compession Sengqing Ci, Yuhng Hu, Xunhe Zho, nd Zhigng Suo, School of Engineeing nd Applied Sciences, Kvli Institute fo Nnobio Science nd Technology, Hvd Univesity, Cmbidge, Msschusetts 8, USA Deptment of Mechnicl Engineeing nd Mteils Science, Duke Univesity, Duhm, Noth Colin 778, USA Received 5 Septembe ; ccepted 8 Octobe ; published online 7 Decembe This ppe studies the pooelstic behvio of n lginte hydogel by combintion of theoy nd expeiment. The gel covlently cosslinked, submeged in wte, nd fully swollen is suddenly compessed between two pllel pltes. The gp between the pltes is held constnt subsequently, nd the foce on the plte elxes while wte in the gel migtes. This expeiment is nlyzed by using the theoy of line pooelsticity. A compison of the elxtion cuve ecoded in the expeiment nd tht deived fom the theoy detemines the elstic constnts nd the pemebility of the gel. The mteil constnts so detemined gee well with those detemined by using ecently developed indenttion method. Futhemoe, duing elxtion, the concenttion of wte in the gel is inhomogeneous, esulting in tensile hoop stesses ne the edge of the gel, nd possibly cusing the gel to fctue. Ameicn Institute of Physics. doi:.6/.5746 I. INTRODUCTION Electonic mil: suo@ses.hvd.edu. A flexible, covlent netwok of polymes cn imbibe lge quntity of solvent, esulting in gel. Gels constitute mny tissues of nimls nd plnts, nd e used in divese pplictions, including dug delivey,, micofluidics,,4 tissue engineeing, 5,6 oilfield mngement, 7,8 nd food pocessing. 9, The mechnicl behvio of gels nd gellike tissues e.g., ctilge 4,5 is time-dependent. The netwok enbles lge nd evesible defomtion, while the solvent in the gel migtes. The concuent defomtion of the netwok nd migtion of the solvent is known s pooelsticity. We hve ecently epoted expeiments on n lginte hydogel pessed by flt plte 6 nd by n indente. 7 In ech expeiment, disk of n lginte hydogel is covlently cosslinked, submeged in wte o queous solution, nd fully swollen. The gel is pessed by suddenly pessing the plte Fig. o the indente Fig. b. The displcement is kept constnt subsequently Fig. c, while the foce on the plte o the indente is ecoded s function of time Fig. d. The foce instntly ises nd then elxes, s wte in the gel migtes nd the gel ppoches new stte of equilibium. This elxtion cuve is used to deduce mteil constnts of the gel the she modulus nd Poisson s tio of the gel, s well s the pemebility of the solvent though the netwok. The min object of this ppe is to scetin tht the two methods compession nd indenttion yield the sme mteil constnts fo the sme gel. To minimize the vibility of the gel used in the two expeiments, hee we conduct both expeiments by using the lginte hydogel peped in the sme btch. The mteil constnts of the gel e detemined by comping the elxtion cuves obtined fom the expeiments to those deived fom the theoy of pooelsticity. Ou pevious ppe 7 hs epoted the theoeticl elxtion cuve fo indenttion, nd this ppe will deive the theoeticl elxtion cuve fo compession. Futhemoe, we will descibe the theoeticl pediction of tnsient fields in the () (b) solvent solvent (c) t gel z F Impemeble, fictionless gel F F indente F F() F( ) b migtion of solvent FIG.. Colo online A disk of gel is submeged in solvent, nd is compessed by fictionless, impemeble, igid pltes. b A disk of gel is submeged in solvent, nd conicl indente is pessed into the gel. c In both expeiments, the displcement is suddenly pescibed nd subsequently held fixed. d The foce is ecoded s function of time. (d) t -8979//8/54/8/$. 8, 54- Ameicn Institute of Physics Autho complimenty copy. Redistibution subject to AIP license o copyight, see

2 54- Ci et l. J. Appl. Phys. 8, 54 compessed gel. In pticul, the tnsient hoop stess is tensile ne the edge of the gel, nd my cuse the gel to fctue. II. GOVERNING EQUATIONS OF POROELASTICITY This section wites Biot s theoy of pooelsticity 8 in fom suitble fo the nlysis of the compession test. The pesenttion will be bief; detils concening ppliction of the theoy to polyme gels my be found elsewhee e.g., Refs.,, 9, nd. Figue illusttes disk of gel, dius nd thickness b, long with the cylindicl coodintes,,z. The disk is pessed veticlly, nd the gel is slippey between the two pltes, so tht the disk is tken to defom unde the condition of genelized plne stin. The xil stin is homogeneous in the gel but cn vy with time. Let z t be the xil stin of the gel s function of time. The defomtion of the disk is tken to be xisymmetic, so tht the dil displcement u is independent of z nd but is function of time nd dil position. Wite the field of the dil displcement s u,t. The hoop stin nd the dil stin e = u/, = u/. All the she stins vnish. The pltes e impemeble to the solvent, nd the solvent in the gel migtes in the dil diection. Let J,t be the flux of the solvent i.e., the numbe of solvent molecules cossing unit e in efeence stte pe unit time. Let C,t be the field of the concenttion i.e., the numbe of solvent molecules pe unit volume of the gel in the efeence stte. The numbe of solvent molecules is conseved: C t + J =. The gel is in mechnicl equilibium t ll time. The dil stess,t nd the hoop stess,t stisfy + =. 4 The xil stess z,t gives ise to the compessive foce: Ft z d. 5 = We dopt the sign convention tht the compessive foce F is positive. All components of the she stesses vnish. The gel, howeve, is not in diffusive equilibium. The chemicl potentil of the solvent in the gel is timedependent field,t. The gdient of the chemicl potentil / dives the flux of the solvent. The two quntities e tken to be linely elted, witten in the fom J = k, 6 whee is the viscosity of the solvent nd the volume pe solvent molecule. Both nd e tken to be the vlues fo the pue liquid solvent e.g., fo wte =. Nsm nd =. 9 m. Consequently, 6 defines phenomenologicl quntity, k, which is known s the pemebility nd hs the dimension of length squed. At ny time, ech diffeentil element of the gel is in stte of themodynmic equilibium. A efeence stte is ssigned when the gel is stess-fee nd the solvent in the gel is in equilibium with the pue liquid solvent. In the efeence stte, the stins of the gel e set to be zeo, the chemicl potentil of the solvent in the gel is set to be zeo, nd the concenttion of the solvent in the gel is denoted by C. When the gel is subject to stte of stess, the gel is in nothe stte of equilibium, in which the gel defoms nd the solvent in the gel my no longe be in equilibium with the pue liquid solvent. This stte of equilibium of the gel is chcteized by the stesses,, z, the stins,, z, the concenttion C, nd the chemicl potentil of the solvent. These themodynmic vibles e connected though the equtions of stte, s descibed below. Becuse the stess in gel is typiclly smll, the polymes nd the solvent molecules e commonly ssumed to be incompessible. Consequently, the incese in the volume of the gel is entiely due to the volume of the bsobed solvent: + + z = C C. The gel is ssumed to be isotopic, nd the stesses e ssumed to be line in stins. Unde these ssumptions, the equtions of stte tke the fom 7 =G +, 8 =G + z =G z z + + z + + z,, 7 9 whee G is the she modulus nd Poisson s tio. When the gel is constined by igid nd pemeble wlls in ll diections, such tht ll stins vnish, n incese in the chemicl potentil of the solvent gives ise to hydosttic pessue, /. The bove equtions specify the theoy of pooelsticity. A combintion of these equtions gives the govening equtions fo the fields C,t, u,t, nd,t: u + zt = C C, u = G, C t = D C, with the diffusivity given by Autho complimenty copy. Redistibution subject to AIP license o copyight, see

3 54- Ci et l. J. Appl. Phys. 8, 54 D = Gk. 4 Eqution tkes the fmili fom of the diffusion eqution. In pooelsticity, howeve, this diffusion eqution cnnot be solved by itself, becuse the boundy conditions typiclly involve the chemicl potentil nd the displcement. Nonetheless, indictes tht ove time t distubnce diffuses ove length. Thoughout the expeiment, the gel is submeged in pue liquid solvent, whose chemicl potentil is set to be zeo. Befoe being compessed, the gel is in equilibium with the extenl solvent stte tken to be the efeence stte of the gel. At time t=, compessive stin of mgnitude is suddenly pescibed by pessing the igid pltes, nd this stin is held constnt in subsequent time. Tht is, z t=, fo t. We dopt the sign convention tht fo compession. The boundy conditions on the edge of the disk e obtined by ssuming tht the gel is loclly in equilibium with the extenl solvent t ll time. Thus, the chemicl potentil of the solvent in the gel, on the edge, equls tht of the extenl solvent t ll time:,t =. 5 Futhemoe, the dil stess on the edge of the gel vnishes t ll time:,t =. 6 Inseting 5 nd 6 into 8, we obtin boundy condition in tems of the displcement: u,t + u,t =. III. SHORT-TIME AND LONG-TIME LIMITS 7 The compession cuses potion of the solvent in the gel to migte out, so tht the field in the gel evolves with time. We fist conside the shot-time limit, t =, instntneously fte the gel is compessed with the stin. The gel undegoes homogeneous defomtion. Instntneously fte the gel is compessed, the solvent in the gel hs no time to migte so tht C,=C, nd the volume of the gel does not chnge, + + z =. The xil stin is z =, nd the dil nd hoop stins e, =, =. The dil displcement is u, =. 8 9 Instntneously fte compession, the dil nd the hoop stesses e zeo,,=,=. The solvent in the gel is out of equilibium with the extenl solvent: the chemicl potentil of the solvent in the gel is homogeneous but is not zeo. Setting,= nd + + z = in Eq. 8, weobtin tht, = G. Fom we obtin the xil stess z, = G. Recll tht the edge of the gel is ssumed to be in locl equilibium with the extenl solvent t ll time, so tht,= instntneously fte compession. This boundy vlue is unequl to the vlue in the inteio of the gel,,=g. Such discontinuity is common in initil/ boundy-vlue poblems subject to suddenly pescibed initil conditions. We now exmine the consequence of this discontinuity in the chemicl potentil. Geometic comptibility equies tht,=/, while mechnicl equilibium equies tht,=. Inseting these conditions, long with,=, into 8, we obtin the instntneous dil stin, = the hoop stess,, = G, nd the xil stess z, = G. 4 The dil stin on the edge diffes fom tht in the inteio of the gel,,=/. Similly, the hoop nd xil stesses lso diffe fom thei countepts in the inteio of the gel. Also note tht the instntneous hoop stess on the edge of the gel is tensile. We next conside the long-time limit, t. Afte being compessed fo long time, the gel eches new stte of equilibium: the chemicl potentil of the solvent eveywhee in the gel equls tht in the extenl solvent,, =. The dil nd hoop stesses vnish,,=, =. Fom Eqs. 8 nd 9, we obtin the dil nd the hoop stins:, =, =. The dil displcement is u, =. Eqution gives the xil stess z, = +G A compison of 8 nd 5 shows tht, s the solvent migtes out the gel, the tnsvese expnsion educes fom the instntneous vlue,=,=/, nd ppoches the vlue of new stte of equilibium,, =,=. Thus, Poisson s tio chcteizes the chemomechnicl intection of the gel. Poisson s tio is esticted in the intevl / by the equiement tht the fee-enegy density is positive definite. When gel is subject to compession nd eches new stte of equilib- Autho complimenty copy. Redistibution subject to AIP license o copyight, see

4 54-4 Ci et l. J. Appl. Phys. 8, 54 u.5... u,..6.8 /. FIG.. Colo online The distibution of the dil displcement t sevel times. μ Ωε G =.8.6. =.6 =. = = = / FIG.. Colo online The distibution of the chemicl potentil of the solvent in the disk t sevel times. ium with the extenl solvent, no solvent in the gel migtes out if /, o potion of the solvent in the gel migtes out if /. IV. TRANSIENT FIELDS The ptil diffeentil Eqs., long with the boundy conditions 5 nd 6 nd the initil conditions 9 nd, cn be solved by the method of seption of vibles. Conside displcement fields of, u,t = + fexp t. 8 The fist tem is the long-time limit, nd the second tem epesents the tnsient devition fom the new stte of equilibium. Inseting 8 into nd, we obtin tht Dd d df + f =,. 9 d The solution to this odiny diffeentil eqution is the Bessel functions. Let J m be the Bessel function of ode m. The displcement field tht solves, 5, 6, 9, nd is u,t = + n= B n J n exp n. The eigenvlues n e detemined by 7, nmely, n J n + J n =. The coefficients B n e detemined by the initil condition 9, giving B n = n J n n J n. Figue plots the displcement field t sevel times. Instntneously fte the gel is compessed, t / =, the disk expnds in the dil diection, nd the displcement is line in the dius, u,=/. As time poceeds, solvent gdully migtes out the gel, nd the disk shinks. When /, the gel ttins the new stte of equilibium, u, =. Figue indictes tht the gel nely ttins the new stte of equilibium when / =. Inseting the displcement field into, nd using the boundy condition,t=, we obtin the field of chemicl potentil:,t G = n= B n nj n J n exp n. Figue plots the chemicl potentil field t sevel times. Immeditely fte compession, the chemicl potentil of the solvent inside the gel is homogenous,,=g. This chemicl potentil exceeds the chemicl potentil of solvent outside the gel, =, nd dives the solvent to migte out. The chemicl potentil of the solvent in the gel t the edge of the disk is tken to equl tht in the extenl solvent t ll time,,t=. As time poceeds, the chemicl potentil of the solvent in the gel gdully deceses. In the long-time limit, the compessed gel equilibtes with the extenl solvent, nd the chemicl potentil of the solvent in the gel ppoches zeo. Inseting the displcement field nd the chemicl potentil field into the equtions of stte 8, we obtin the stesses: G = B n= n J n + n J n + J n exp n, G = B n n= J n + exp n, nj n z G = + + B n n J n n= + J n exp n Autho complimenty copy. Redistibution subject to AIP license o copyight, see

5 54-5 Ci et l. J. Appl. Phys. 8, 54.5 () (,)..5. gel disk / (b).6 FIG. 5. Colo online The compessive foce elxes s function of time. The elxtion cuve vies with Poisson s tio somewht (c)..6 Figue 4 plots the distibution of the hoop stess t sevel times. Instntneously fte compession, the solvent in the gel hs no time to migte, so tht no hoop stess develops in the inteio of the disk. Howeve, tensile hoop stess develops instntneously t the edge of the disk, s discussed befoe. Afte the gel is compessed fo some time, solvent leves the gel gdully, so tht the concenttion becomes inhomogeneous: the concenttion of the solvent ne the edge is lowe thn tht ound the cente. As time poceeds, the tensile stess educes mgnitude but speds ove lge egion. Towd the cente of the disk, the hoop stess is compessive. In the long-time limit, the hoop stess eveywhee in the gel vnishes. Figue 4b plots the distibution of the dil stess t sevel times. The dil stess t the edge vnishes t ll time, s dictted by the boundy condition. Afte the gel is compessed by the pltes, the solvent migtes out, initilly fom the egion ne the edge of the disk. Consequently, the dil stess ound the cente of the disk is compessive. / /.5.5 z (,) FIG. 4. Colo online The evolution of the hoop stess, b the dil stess, nd c the xil stess. The mgnitude of the compessive dil stess initilly ises nd then flls. In the long-time limit, the dil stess eveywhee vnishes. Figue 4c plots the distibution of the xil stess t sevel times. As discussed befoe, instntneously fte the gel is pessed, the xil stess is z,= G / t the edge of the disk, nd is z,= G in the inteio of the disk. These two levels of the xil stess e unequl, so long s.5. The mgnitude of the xil stess t the edge of the disk inceses s time pogesses. The mgnitude of the xil stess t the cente of the disk initilly ises nd then flls. Afte some time, the xil stess homogenizes in the disk, nd ppoches the long-time limit z,= +G. V. USING RELAXATION CURVES TO DETERMINE PROPERTIES OF GELS Integting the xil stess ove the e of the disk, we obtin the xil foce s function of time: Ft G = + B n 4J n n= + nj n exp n. The shot-time limit is F =G. The long-time limit is F =+G Figue 5 plots the elxtion cuve 7 in the fom Ft F = f, F F, 4 The tio on the left-hnd side mesues how f the gel is wy fom the stte of equilibium. The tio depends on Poison s tio wekly, s indicted in Fig. 5. Covlently cosslinked lginte hydogels e peped following the potocol peviously descibed. The gel is sub- Autho complimenty copy. Redistibution subject to AIP license o copyight, see

6 54-6 Ci et l. J. Appl. Phys. 8, 54 FIG. 6. Colo online Photos of fctued lginte hydogel cused by compession. meged in distilled wte fo 4 h until it is fully swollen. Fom piece of the gel, we punch out thee disks of dii, 4, nd 5 mm. These disks e then pessed with stinless steel plte. The plte fist ppoches the sufce of the disk with speed of m/s until the mesued foce stts to incese. At this point, the gp between the top nd bottom pltes is viewed s the thickness of the disk, which is 7.7, 7.8, 7.65 mm, of mm dius, 4 mm dius, nd 5 mm dius smple, espectively. Ech disk is subject to % veticl compessive stin. The totl ising time is bout s, which is negligible comped to the elxtion time bout to8h. While the veticl stin is held t the fixed vlue, the foce on the plte is ecoded s function of time by using the AR heomete fom TA Instuments. The esolution of the foce is.5 N, nd dt e tken t the te of 6 points pe second. The covlently cosslinked lginte hydogels e quite bittle, nd sometimes fctue duing the expeiment Fig. 6. The fctue mechnics of gels is inteesting in its own ight but will not be pusued in this ppe. The dt epoted below e tken fom expeiments in which no fctue is obseved. Figue 7 plots the elxtion cuves mesued expeimentlly fom the thee disks. In ech cse, the foce ises shply s the plte is pessed. Subsequently, the plte is held t the fixed position, while the foce elxes nd ppoches new stte of equilibium. The mgnitude of the foce, s well s the elxtion time, is lge when the dius of the disk is lge. Once the foce is divided by the e of the disk, nd the time is divided by, the elxtion cuves mesued fom the disks of the thee dii collpse into single cuve Fig. 7b. This behvio is consistent with the pediction of the theoy of pooelsticity. The nominl stess the foce divided by the e of the disk elxes s the solvent migtes out fom the edge of the gel. The elxtion time is popotionl to the dius of the disk squed. By comping the elxtion cuve mesued expeimentlly with tht deived fom the theoy of pooelsticity, we cn detemine the she modulus, Poisson s tio, nd the diffusivity. In the shot-time limit, compison of the expeimentl dt F/ =.5 kp nd the theoeticl fomul F/ =G gives the she modulus G =4. kp. In the long-time limit, compison of the expeimentl dt F/F=.8 nd the theoeticl fomul F/F=+/ gives Poisson s tio =.. The elxtion cuve clculted fom the theoy of pooelsticity ovelps with the elxtion cuves expeimentlly mesued fom the thee disks when the diffusivity is fit to D=6. 9 m /s Fig. 7b. FIG. 7. Colo online A disk of n lginte hydogel is compessed between pllel pltes, while the foce on the plnes is ecoded s function of time. Relxtion cuves obtined by using disks of n lginte hydogel of thee dii. b Ech of the thee elxtions cuves is plotted gin, with the foce divided by the e of the disk, nd the time divided by the dius squed. Also plotted is the elxtion cuve obtined fom the theoy of pooelsticity. In ecent ppe, 7 we hve used conicl indente to chcteize the lginte hydogel. As illustted in Fig., the gel is submeged in wte nd is fully swollen. The conicl indente, of hlf included ngle, is suddenly pessed into the gel nd is subsequently held t fixed depth h. The foce on the indente is mesued s function of time. This test hs been nlyzed within the theoy of pooelsticity, 7 nd the elevnt esults e summized hee. The dius of contct is given by = h tn. 4 In the shot-time limit, solvent in the gel hs no time to migte, the gel behves like n incompessible elstic solid, nd the foce on the indente is given by F =4Gh. 4 In the long-time limit, potion of the solvent in the gel hs migted out, the gel hs ttined new stte of equilibium with the extenl solvent, nd the foce on the indente is given by F =Gh/. 4 Fo the gel to evolve fom the shot-time limit towd the long-time limit, the solvent in the gel unde the indente must migte. The elevnt length in this diffusion-type poblem is Autho complimenty copy. Redistibution subject to AIP license o copyight, see

7 54-7 Ci et l. J. Appl. Phys. 8, 54 FIG. 8. Colo online A conicl indente is pessed into disk of n lginte hydogel to cetin depth, while the foce on the indente is ecoded s function of time. Relxtion cuves obtined by keeping the indente t thee depths. b The elxtion cuves e plotted gin by using nomlized vibles. Also included is the elxtion cuve obtined fom the theoy of pooelsticity. the dius of contct,, nd the nomlized time tkes the fom =/. The function Ft obeys Ft F F F = g. 44 The dimensionless tio on the left-hnd side of 44 is mesue of how f the gel is wy fom the new stte of equilibium. The function g is detemined by solving the pooelstic boundy-vlue poblem. Ou pevious wok indictes tht g is function of the single vible, given by g = 9 exp exp To minimize the vibility of the mteil, we mke the lginte hydogels fo both tests compession nd indenttion in the sme btch. The solutions e poued into plstic mold of cm dius nd cm thickness. Afte geltion, the gel is submeged in distilled wte fo 48 h until it is fully swollen. We then pess n luminum indente of hlf included ngle =7 into the gel to fixed depth. The foce on the indente is ecoded s function of time by using custom-built lod fme with foce esolution of. N nd displcement esolution of m. The indentes e pogmmed to ppoch the sufce of the smple t the speed of m/s, until the slope of the ecoded focedisplcement cuve stt to be positive. The time used to pess the indente into the lginte gels s is much shote thn the elxtion time to6h, so tht the effect of the initil loding stge is minimized. Figue 8 shows the mesued elxtion cuves ecoded t the thee depths of indenttion. In ech cse, the foce ises shply, nd then elxes s the gel ppoches new stte of equilibium with the extenl solvent. The mgnitude of the foce, s well s the elxtion time, is lge when the depth of indenttion is lge. A compison of the expeimentl vlue F/h= kp nd the nlyticl fomul F/h=4G gives G=.5 kp. A compison of the expeimentl vlue F/F=.56 nd the nlyticl fomul F/F= gives Poisson s tio =.. Figue 8b plots the elxtion cuves mesued with the thee depths indenttion the in dimensionless fom. The thee cuves collpse into single cuve. Futhemoe, these cuves ovelp with the elxtion cuve clculted with the theoy of pooelsticity, g in 45, when the diffusivity is fit to the vlue D=6.6 9 m /s. Comping the mteil popeties mesued by compession nd indenttion, we note 5.% diffeence in the she modulus, 4.6% diffeence in Poisson s tion, nd 6% in the diffusivity. This excellent geement lends suppot to both tests. The two tests hve thei own dvntges nd disdvntges. The compession test equies the smple to be fbicted with pefectly pllel top nd bottom sufces, which my be difficult in pctice. This concen is bsent fo the indenttion test becuse the stting point of the mesuement is edily detected fo conicl o spheicl indentes. The indenttion test, howeve, equies the thickness of the smple to be moe thn ten times lge thn the depth of indenttion. This equiement might be difficult to stisfy in pctice. VI. CONCLUDING REMARKS The compession test is nlyzed within the theoy of pooelsticity. By comping the elxtion cuve deived fom the theoy to tht mesued in the expeiment, we obtin the she modulus, Poisson s tio nd the pemittivity of covlently cosslinked lginte hydogel. The mteil constnts so detemined gee well with those obtined fom ecently developed indenttion method. The geement lends suppot to both methods. Futhemoe, ou clcultion shows tht, s the compessed gel elxes, the concenttion of the solvent in the gel is inhomogeneous, esulting in tensile hoop stesses ne the edge of the gel. While fctue is indeed often obseved in ou expeiments, the mechnics of fctue wits clifiction. ACKNOWLEDGMENTS This wok is suppoted by the NSF Gnt No. CMMI- 86 nd by the MRSEC t Hvd Univesity. R. Duncn, Nt. Rev. Dug Discovey, 47. N. A. Pepps, J. Z. Hilt, A. Khdemhosseini, nd R. Lnge, Adv. Mte. 8, A. Richte, G. Pschew, S. Kltt, J. Lienig, K. Andt, nd H. P. Adle, Sensos 8, P. Clvet, Adv. Mte., K. Y. Lee nd D. J. Mooney, Chem. Rev., S. Vghese nd J. H. Elisseeff, Adv. Polym. Sci., M. Kleveln, R. H. vn Hoot, nd I. Jones, SPE/IADC Dilling Confeence, Amstedm, Nethelnds, 5 Febuy, 5 Society of Petoleum Enginees, Richdson, TX, 5. Autho complimenty copy. Redistibution subject to AIP license o copyight, see

8 54-8 Ci et l. J. Appl. Phys. 8, 54 8 S. Q. Ci, Y. Lou, P. Gnguly, A. Robisson, nd Z. G. Suo, J. Appl. Phys. 7, L. Bngou, C. Dubet, nd E. Foegeding, Food Hydocolloids, J. Zhng, C. Dubet, nd E. Foegeding, J. Food Sci. 7, e45 5. K. Uym, Y. Tok, K. Nkmu, nd T. Tkigw, Polyme 49, C. Y. Hui, Y. Y. Lin, F. C. Chung, K. R. Shull, nd W. C. Ling, J. Polym. Sci., Pt B: Polym. Phys. 44, M. L. Oyen, J. Mte. Res., V. Moel nd T. Quinn, J. Othop. Res., B. Kuz, M. Jin, P. Ptwi, D. Cheng, M. Lk, nd A. Godinsky, J. Othop. Res. 9, 4. 6 X. H. Zho, N. D. Huebsch, D. J. Mooney, nd Z. G. Suo, J. Appl. Phys. 7, Y. Hu, X. Zho, J. Vlssk, nd Z. Suo, Appl. Phys. Lett. 96, M. A. Biot, J. Appl. Phys., M. Doi, J. Phys. Soc. Jpn. 78, 5 9. J. Yoon, S. Q. Ci, Z. G. Suo, nd R. C. Hywd, Pooelstic swelling kinetics of thin hydogel lyes: Compison of theoy nd expeiment, Soft Mtte to be published. K. H. Bouhdi, D. S. Husmn, nd D. J. Mooney, Polyme 4, Autho complimenty copy. Redistibution subject to AIP license o copyight, see