modeling solute diff gels 0

7/24/2019 Modeling Solute Diff Gels 0

1/14

Solute Diffusion w ithin Hy drogels. Mecha nisms an d Models

Brian Amsden

Facul t y of P har macy and P harm aceut i cal S ci ences, Uni versi t y of A l bert a, E dm ont on, A B , Canada

Received M ay 13, 1998; Revised M anu scri pt R eceived Augu st 5, 1998

ABS TRACT: Solute diffusion in h ydrogels is important in ma ny biotechnology fields. Solute behaviorin hydrogels ha s been explained in terms of reduction in h ydrogel free volume, enha nced hydr odynamicdrag on the solute, increased path length due to obstruction, and a combination of hydrodynamic dragand obstruction effects. In t his ar t icle the va rious math emat ical m odels derived to explain a nd predictsolute diffusion in hydrogels are reviewed and tested aga inst litera ture da ta . These models ca n be dividedinto t hose applicable t o hydrogels composed of flexible polymer chain s (i.e., homogeneous hydrogels) andthose composed of rigid polymer cha ins (i.e., heterogeneous hydrogels). For homogeneous hydrogels itwas determined that a scaling hydrodynamic model provided the best explanation for solute diffusion,while for heterogeneous hydrogels obstruction m odels w ere more consistent w ith t he experimenta l da ta .Both the scaling hydrodynamic model and the most appropriate obstruction model contain undefinedparameters which must be clarified in order for these models to gain widespread acceptance.

Introduction

The diffusion of solutes in hy drogels ha s a pplica tion

i n a w i de v a r ie t y o f f i el ds . H y d r og el s a r e u s ed i nseparation processes such as chromatography, 1 for th eencapsulation of cells for both biomedical and fermenta-tion purposes,2 in biosensors,3,4 an d a s biomat erials fort he del i v ery of b i oac t i v e agent s t o t he b ody 4-6 a ndprosthetic a pplica tions.3,4 The one feature of hydrogelstha t a ll these applications capita lize upon is the a bilityof the hydrogel to restrict the diffusive movement of asolute. It is therefore important t o have an understa nd-ing of the para meters governing solute diffusion w ithinhydrogel s as w el l as t he m eans b y w hi c h t hey a f fec tdi ffusion. For this reason, a number of mat hematicalexpressions ha ve been developed in a n effort to modelsolute diffusion in hydr ogels. It is t he purpose of thisart i c le t o exam i ne t he m ost prev al ent m odels and

c om pare t hem t o dat a t aken from t he l i t erat ure, t oassess their predict ive abi li t ies. Such a n examinat ionw as undert aken a num b er of years ago b y Muhr andB l a n s h a r d ;7 however, a number of models ha ve sincebeen proposed.

Before introducing the different models, it would bebeneficial to review the properties of hydrogels. Hy-drogels a re cross-linked h ydrophilic polymers swolleni n w a t er or a n aqueous env ironm ent . Their t hree-dimensional structure is often described as a mesh, withthe spaces between the polymer chains filled with water.The cross-links ca n be formed by covalent or ionic bonds ,v an der Waa l s forces, hydrogen b onds, or physi calentanglements.3,4 The cross-link sites form microre-

gions of local polymer precipita tion from solution. Theoveral l structure of hydrogels therefore ranges fromhomogeneous, where the polymer chains have a highdegree of mobility, to heterogeneous, in wh ich t here isa great deal of inter-polymer interaction and the poly-m er c hai ns are v i r t ual l y i m m ob i l e a t t he m ol ec ul arlevel.7 Examples of homogeneous hydrogels includepoly(ethylene oxide), poly(acrylamide), and poly(vinyla lcohol). Exa mples of het erogeneous hyd rogels includecalcium alginat e, agar ose, and -carrageenan.3,7

Solute transport within hydrogels occurs primari lywithin the water-filled regions in the space delineated

by the polymer chains. Any fa ctor w hich reduces th esize of these spaces will have an effect on the movementof the solute. Such fa ctors include the size of the solute

in relat ion t o the size of the openings betw een polymerchains, polymer chain mobil i ty, and the existence ofc harged groups on t he pol ym er w hi c h m ay b i nd t hesolute molecule. P olymer chain mobility is a n import an tfactor governing solute movement with in the hyd rogel.For homogeneous gels, the mesh openings are neitherconstant in size nor location, whereas for heterogeneoushydrogels the openings between chains can be consid-ered consta nt in size and location.7 I n general , t hediffusivity of a solute t hrough a physically cross-linkedhydrogel decreases as cross-linking density increases,as the size of the solute increases, and as the volumefraction of w at er within the gel decreases.4,8,9

As will be shown in the following sections, the polymer

chains have been proposed to retard solute movementb y reduci ng t he av erage free v ol um e per m ol ecul eava ilable to the solute, by increasing the hydr odynam icdrag experienced by the solute, and by acting as physicalobstructions t hereby increasing the path length of thesolute. As well , models ha ve been proposed w hichcombine the effects of increased h ydrodyna mic drag an dthe existence of physical obstructions. For succinctness,this review will be restricted to hydrophilic, generallyspherical solutes an d polymer-solute intera ction effectsare neglected.

Mechanisms and Models

Free Volume Theory. These models are based on

t h e t h eor y p ut f or w a r d b y C oh en a n d Tu r n bu ll t oexplain t he process of solute diffusion in a pure liquid. 10

In t his th eory, t he solute diffuses by jumping into voidsformed in the solvent s pace by the redistr ibution of thefree volume within the l iquid. I t is assumed tha t thefree volume can be redistributed without any energychange. The voids ar e pictured a s being formed by ageneral w ithdra wa l of the surrounding liquid moleculesdue to ran dom thermal motion. These holes a re thenfilled in by the reverse process.11

Solute diffusion is dependent on t he jumping dista nce,the t hermal veloci ty of the solute, a nd the probabili ty

8382 M acromolecul es1998, 31 , 8382-8395

10.1021/ma 980765f CCC: $15.00 1998 America n Chemica l SocietyP ubl ish ed on Web 10/20/1998


2/14

tha t t here is a h ole free volume adja cent t o the molecule.At a given t emperatur e, the ra te of di ffusion is deter-m i ned b y t he prob ab il it y of a v oi d b eing form ed ofsufficient volume t o a ccommoda te the solute molecule.The diffusion coefficient of the solute in the liquid atinfinite dilution, D0, is then expressed a s

in which Vis the a verage therma l velocity, is the jumplength roughly equivalent to the solute dia meter, v* isthe critical local hole free volume required for a solutemolecule to jump int o a new void, is a numerical factorused to correct for overlap of free volume available tomore than one molecule (0.5 e e 1), and vf i s t heaverage hole free volume per molecule in the liquid.

Yasuda et a l .12 were the first t o apply th is theory t osolute di ffusion in gels. In t heir derivation, they as-sumed tha t only t ra ce amounts of solute were presentso tha t t he free volume per molecule within t he gel wa sgiven by t he summa tion of the free volumes per mol-ecule of the w at er, vf,w, an d polymer, vf,p , within t he gel,w ri t t en as

wherein is the volume fra ction of polymer in th e gel.In other words, the available free volume for diffusionin the gel arises not only from the ra ndom redistribut ionof the water molecules but also by the random redis-tribution of sections of the polymer molecules, calledpolymer jumping units. The contr ibution of the polymerchains to the avai lable free volume per molecule isconsidered to be sma ll, and so t he free volume ava ilableto the solute is expressed as

Wi t hi n t he gel , i n addi t i on t o t he requi rem ent offinding a sufficient free volume within t he l iquid, thesolute molecule also needs to find an opening betweent he sol ut e c hai ns l arge enough t o a l l ow i t s passage.Thus, t he di ffusivi ty of a solute in the gel is given bythe product of th e probabilities of finding a proper freevolume and a proper opening wit hin the cha ins. U singthis concept a nd combining eqs 1-3, yields th e followingexpression for solute diffusion in a gel, Dg:12

I n e q 4 Po is the probabil i ty of f inding an opening

betw een the polymer chains, a* is the effective cross-sect i onal area of t he sol ut e m olecule, a nd B i s a nundefined consta nt of proportional ity. The t erm Porepresents the sieving action of the polymer chains onthe solute molecule. Yasuda et al . 12 chose the solutecross-sectional ar ea as the d etermining par ameter forPoand examined a number of sieving factor probabilityfunctions but did not a pply t he functions to experimen-t a l d a t a .

Variat ions of eq 4 ha ve been derived by others. Forexample, Peppas and Reinhart 13 suggested the form

in which k1 a nd k2 are undefined structural constantsfor a given polymer-solvent system, rs is the ra dius ofthe solute, Mh c is the number average molecular weightbetw een polymer cross-links, Mh n is the number averagemolecular weight of the uncross-linked polymer, andMh c

* is a critical molecular weight between cross-linksrequired to al low solute passa ge. In th eir derivation,P eppas and Reinhart considered t he v ol um e of t hesolute t o be the critica l geometrical pa ra meter decidingwhether a solute will pass through the polymer chains.I n a sub sequent paper, Rei nhart and P eppas14 foundtha t a more accurat e expression of their model is givena s

Despite this finding, this research group recently usedeq 5 t o explain solute diffusion w ithin poly(vinyl a lcohol)/poly(a crylic a cid) interpenetra ting netw orks,15 suggest-ing an inconsistency with this model.

Another version of the free volume theory expressionwas given by Lustig and Peppas, 16 who introduced t he

concept of the scaling correlation length between cross-links, . By a rguing tha t a solute will pass through thepolymer chains only if its effective radius is sma ller th a n, t hey assum ed t ha t t he s ievi ng fac t or expressionshould be (1 - rs/). Applying this a rgument t o the free-volume expression ga ve

w here Y ) rs2/vf, w. Y is therefore the rat io of thecrit ical volume required for a successful tra nslat ionalmovement of the solute molecule and the average freevolume per molecule of the l iquid. The correlat ionlength is related to polymer volume fraction, the func-t i onal i t y of t h i s dependence v aryi ng w i t h pol ym ervolume fraction and polymer-solvent intera ction.17 F orexample, in the si tuation where the polymer and thesolvent ha ve a high a ffinity for each other, t he polymerchains are just touching and no entanglements occur(i .e. , high water fract ion in a homogeneous gel), -0.75. Lustig and P eppas also sta te that for correlat ionpurposes a good approximat ion for Y is unity. However,again there is some confusion as to the proper form ofthe sieving factor. The same a uth ors also give (1 -(rs/)2) as the a ppropriate form of the sieving factor.18,19

Another version of the free volume theory has beengiven by Hennink et al.20 In their model, the reduction

in solute diffusivity is given by

in w hich accounts for the sieving effect . However,t hese aut hors assum e t ha t t he s i ev ing fac t or i s i nde-pendent of polymer volume fra ction. This a ssumptioncould only be considered approximat ely valid a t very lowpol ymer v ol ume frac t ions . As t he concent rat i on ofpolymer cha ins per given volume increas es, the degreeof obstruction or sieving a solute experiences wil lnecessa rily increase. Therefore, this model is unrea l-istic.

Dg

D0) k1(Mh c - Mh c

*

Mh n - Mh c*)

2

exp(-k2rs2( 1 - )) (6)

Dg

D0) (1 - rs )exp(-Y(

1 - )) (7)

ln (Dg

D0) ) ln () - k2rs2( 1 - ) (8)

D0 Vex p(- v*vf) (1)

vf ) (1 - )vf, w+ vf, p (2)

vf (1 - )vf, w (3)

Dg

D0) P0exp(- B a*vf, w (

1 - )) (4)

Dg

D0) k1(Mh c - Mh c

*

Mn - Mh c*)ex p(-k2rs2( 1 - )) (5)

M acromolecul es, Vol. 31, N o. 23, 1998 Solute Diffusion wit hin Hy drogels 8383


3/14

Hydrodynamic Theory. Hydrodynamic descrip-tions of solute tra nsport through gels are ba sed on t heStokes-Einst ein equa tion for solute diffusivity. In theStokes-Einstein derivation, the solute molecule is as-sumed to be a har d sphere which is large compared tothe solvent in w hich it moves.21 The solut e is consideredto move at a constant velocity in a continuum composedof the solvent and is resisted by frict ional dra g. Thediffusivity at infinite dilution, Do, is expressed as21

in wh ich kB is B oltzma nns consta nt , Tis temperature,a nd f is the frictional drag coefficient.

Almost al l hydrodynamic, and obstruction, modelsnecessa rily a ssume a n idealized pictur e of the polymerstructur e. Within the hydrogel, the polymer cha ins areconsidered to be centers of hydrodynamic resistance,fixed in place relative to t he moving solute by enta ngle-ments an d physical cross-l inks. The polymer chainsenhance the frict ional drag on the solute by slowingdown the fluid near the polymer chain. Hydrodyna micmodels of solute diffusion through h ydrogels ar e there-

fore concerned w ith describing f.The frict iona l coefficient ha s been calculated by

summing t he frictiona l contribution of each chain to them ov em ent of t he sol ut e m ol ecul e. I n t hi s m a nner,Altenberger et al .22 derived t he expression

in which

a nd R2 is a constant determined by the fluctuation ofthe force of intera ction betw een the polymer cha ins a ndthe solute molecule. In eq 11 rf i s t he radi us of t hepolymer fiber.

Using t he same model as a basis, Cukier23 describedthe d ecrea se in diffusivity for heterogeneous hy drogels,which have very rigid chains, as

wherein L cis the length of the polymer chain, M fis them ol ecular w ei ght of t he pol ym er chai n , and NA isAvogad ros number. For homogeneous hyd rogels, Cuk-

ier23

proposed an equation based on scaling concepts,

i n w hi ch kC i s a n u n de fi ne d con s t a n t f or a g iv enpolymer-solvent system.

P hillips et al .24 calculated the frictional coefficient byusing Brinkmans equation for f low through a porousmedium and assuming no slip at the solute surface andconsta nt fluid velocity fa r from the solute surfa ce. Themedium is considered to be composed of st ra ight, r igidfibers, oriented in a ra ndom thr ee-dimensiona l fash ion.They thereby obtained the expression

in which kis th e hydra ulic permeability of th e medium.The hydraulic permeabil i ty is est imated using a cor-relation derived by J ackson and J ames, 25

Although derived di fferently, Kosar and Phil l ips 26have recently demonstrated that eqs 10, 13, and 14 arem at hem at i cal l y equi val ent . They t herefore can b einterpreted in the same manner.

Obstruction Theory. Models based on obstructiontheory assume that the presence of impenetrable poly-mer chains causes an increase in the path length fordiffusive tra nsport. The polymer cha ins act as a sieve,a llowing pa ssa ge of a solute molecule only if i t can pa ssbetween th e polymer chains. P erhaps due to i ts con-ceptua l appeal, a number of obstru ction t heory modelsha ve been developed.

Mac ki e and Meares 27 w e r e a m o n g t h e e a r l i e s t t odevelop an obstruction expression for solute diffusion

in a heterogeneous medium. They assumed a lat t icemodel for th e wa ter-polymer hydrogel system, with thepolymer blocking a fra ction, , of the sites. The solutem o l e c u l e w a s a s s u m e d t o b e t h e s a m e s i z e a s t h epolymer segments and solute transport to occur onlywith in the free sites. The expression they derived wa s

Although this expression has been used to analyzetransport through hydrogels,28 i t is of l imited uti l i tybecau se it does not ta ke into considerat ion a ny proper-ties of the hydrogel or of the solute.

The sieving behavior of the polymer chains has alsobeen incorporat ed into free volume theory. Yasud a eta l. 12 used a di fferential hole distribution function todescribe the sieving a ction of the polymer chains. Thereduction in solute diffusivity du e to this sieving a ctionwas given by

in which p(rs) i s t he prob ab il it y of a sol ut e passi ngthrough a given hole in the mesh, f(a) is t he hole areadistribution function, a is t he a rea of t he openingsbetween polymer chains, a nd a* is the effective cross-sectiona l area of th e solute molecule. These researchersshowed the qualitative effects of several distributions,including a G au ssian distr ibution, but did not a pply thedistributions to experimenta l data . Lustig and P eppas 16

assumed th at the probability of a solute passing througha given hole in the mesh w as

This expression is based on the argument that solutesof equal cross-sectional ar ea could h ave di fferent hy-drodynamical ly equivalent radi i .

A more phenomenological approach was taken byOgston et al .29 They assumed that solute diffusion inthe hydrogel occurs by a succession of directional ly

Do )kB T

f (9)

De

Do) 1 - R1

1/2- R2. .. (10)

R1 rsrf (11)

De

Do) exp[-(

3L cNA

Mfl n(L c/2rf))rs1/2] (12)

Dg

Do) exp(-kcrs

0.75) (13)

De

Do) [1 + (rs

2

k)

1/2

+13

rs2

k]-1

(14)

k) 0.31rf2-1.17 (15)

Dg

Do) (1 - 1 + )

2(16)

p(rs) ) a*

f(a) d a (17)

p(rs) ) 1 -rs

(18)

8384 Amsden M acromolecul es, Vol. 31, N o. 23, 1998


4/14

ran dom unit steps and t hat the unit step does not ta keplace i f the solute encounters a polymer chain. Thecross-linked polymer is assumed to exist as a randomnetw ork of stra ight, long fibers of negligible width , andthe solute is considered to be a ha rd sphere. The units t ep i s t a k e n t o b e t h e r oot -m ea n -s q u a r e a v er a g ediam eter of spherica l spaces residing between th e fibernet w ork. From such a n a nal ysis , t hey expressed t heratio of the di ffusion coefficient in the gel to that ininfini te di lution in w at er a s

Although conceptually appealing, eq 19 provides only aqua l ita t ive a greement to experimental observations.7,9

J ohansson et al . 30 developed an obstruction modelbased on the idea that the gel can be viewed as beingcomposed of a nu mber of cylindrical cells. Ea ch cylin-drical cell consists of an infinite polymer rod centeredin a cylinder of solvent of a given ra dius. The avera gediffusivity of the solute with in th is cell ca n be found bysolving Ficks f irst la w. The global di ffusivi ty of thesolute (i .e. , throughout the gel) is then calculated by

summing up the number of cells having a given radiust i m es t he a v erage di f fusiv it y w i t hi n t ha t cell . Thedistribution of the cel l radi i was calculated using anexpression for t he distr ibution of spherical spa ces w ithina ra ndom network of stra ight fibers.31 Their express ionfor the reduction in solute diffusivity is

w here

a nd E1 is t he exponential integra l .The m odel produced a sat i s fact ory agreem ent t o

simulation and experimental results for solute diffusionin both polymer solutions and polymer gels. However,the theory predicted di ffusivi t ies greater than thosefound from simulation for large solutes a t high polymervolume fra ctions (>0.01).32 The Brownian motion simu-lation data were, however, well-described by the follow-ing argument, obtained through a r egression to the da ta :

Moreover, as a result of the a ssumption of an infini tepolymer chain, the analytical expression (eq 20) doesnot w ork well for th e condit ion where the persistencelength of the polymer, which is the average projectionof the end-to-end distance of an infini te chain in thedirection of the first cha in segment, is less tha n 10 timesgreater tha n the solute ra dius. For this si tuat ion, it isnecessary to calculate the diffusivity numerically.

The ra ndom net work of overlapping fibers pictur e ofhydrogel structure has also been invoked as a physicalmodel in the study of tra nsport properties within ma nytypes of porous media, w hether the tra nsport be ther-ma l, electrical, or diffusiona l in na ture. 33 Ea ch of theseprocesses is ma themat ical ly a na logous, a nd so results

ob t ai ned i n one f i el d a re a ppl icab l e i n a not her . Re-cently, models d eveloped t o describe tra nsport proper-ties in ra ndom ar ra ys of overlapping fibers have beenrevi ew ed a nd com pared t o t he resul t s o f a Brow ni anmotion simulation.33 The results indicate t ha t, of thesem odel s , t hat o f Tsai and St r i eder 34 a d a p t e d f o r t h ediffusive process a s

provided the best a greement to Brown ian motion simu-lat ion da ta for cylinder volume fra ctions less tha n 0.40.In eq 23, R is as described in eq 21. In t he derivat ionof this equa tion, the volume fraction of the fiber netw orkexcl uded t o t he sol ut e , , w a s g iv en b y O gs t on sexpression,31

Recently, a new obstruction model was developed. 35

In this model, solute movement through t he hydr ogelis considered t o be a stochast ic process, w ith successfulmovement through the gel determined by the solute

molecule finding a succession of openings within thepolymer chains large enough to accommodate i ts hy-drodynamic radius. Again, Ogstons expression w asused to describe the distribution of openings betweenstraight, randomly oriented polymer fibers. 31 The m odelis expressed as

in which rj is the a verage ra dius of the openings betweenthe polymer cha ins. The avera ge radius of the openingsbetween the polymer chains was taken to be one-halfof the average end-to-end distance between the polymer

chains, . With employment of scaling concepts, t hisdistance is given by 36

w here ks i s a const ant for a g i ven pol ym er-solventsystem, dependent on t he flexibi li ty of the polymerch a i n . D u e t o t h e a s s u mp t ion of s t r a ig h t p ol ym erchains, this model is applicable to diffusion in hetero-geneous hy dogels, a l though t he use of scal ing la ws todescribe the average distance between polymer chainsmay make it applicable to homogeneous hydrogels.

Combined Obstruction and Hydrodynamic Ef-fects. Recently, Bra dy 37 proposed th a t t he obstruction

an d hydr odynamic influences on solute tra nsport withina gel fiber ma trix ar e multiplica tively rela ted. J ohnsonet al .38 used t his concept a nd combined t he obstructionexpression of J ohansson et al .32 (eq 22) with eq 14,resulting in

w hereR is given by eq 21.Eq ua tion 27 wa s shown to be a better predictor of the

effect of a n d r s on solute di ffusivi ty within agarosegels tha n either the Ogst on et a l. expression (eq 19) or

Dg

Do) exp[-

(rs + rf)

rf] (19)

Dg

Do) e-R + R2eRE1(2R) (20)

R )

(

rs + rf

rf

)

2

(21)

Dg

Do) exp[-0.84R1.09] (22)

Dg

Do

)

(1 +

2

3

R

)

-1(23)

) 1 - exp(-R) (24)

Dg

Do) exp(- 4(

rs + rf

rj + rf)

2

) (25)

) ks-1/2 (26)

Dg

Do)

exp[-0.84R1.09]

[1 + (rs2

k)

1/2

+13

rs2

k] (27)



5/14

the expression of Ph illips et al . (eq 14). However, th em odel consis t ent ly underest i m at ed t he decrease i nrelatively small solute diffusivity and provided only aqua l ita t ive approximation to the diffusivi ty data at thehighest polymer volume fraction tested.

A more rigorous combined hy drody na mic/obstru ctionsimulation model which does not depend on an effectivem edi um cal cul at i on w a s developed b y C l ague a ndPhillips.39 In t his model the hydrodyna mic intera ctionswere calculat ed by representing t he solute a s a collectionof point singulari t ies a nd accounting for the fibers byusing a num eri cal v ersion of s l ender-b ody t heory .Clague a nd P hil lips39 performed simulat ions a t va rioussolute radius to f iber radius rat ios and fi t a stretchedexponential function to th ese data of the form

in wh ich aa nd are fitted para meters and a re functionsof rf/rs. This simulation model was tested aga inst theeffect of polymer volume fraction on BSA diffusivi tywithin a gar ose gels a nd w as shown to provide a closeragreement to the data tha n tha t given by the model of

J ohnson et al.38 However, the model was only testedaga inst one other solute, myoglobin, and t hus its a bilityt o a ccount for t he effect of sol ut e radi us i s a s yetunresolved. Un fortuna tely, the simulat ion na ture of themodel does not easily lend itself to compa rison.

To use their model it is necessary to ha ve a m eans ofdetermining aa nd for va rious rf/rs values. However,the exponential nature of the dat a mea ns tha t there aremultiple possible solutions to th is exponentia l fit . Thepara meters returned by the fitting algorithm depend ont he i nit i a l guesses suppli ed. Revi ew i ng t heir f it t edpara meters indicated tha t the values of a returned wereall approximately equal to. Therefore, their data wa srefit by fixing a) . The results a re displayed in Figure

1. The values of ob t ai ned w ere t hen pl ot t ed as afunction of rf/rs (Figure 2). A regression an alysis ofthese da ta yields t he following relat ionship

wh ich provides very good a greement (R2 ) 0.976). Thusthe hy drodyna mic contribution to the reduction in solutediffusivity can be wri t ten a s

Clague and Phil l ips combined their hydrodynamicterm with the obstruction expression of Tsai and Stre-ider (eq 23), and so their model can be written as

Due to the assumptions made in the development of thismodel, it can only strictly apply to solute diffusion inheterogeneous hydrogels composed of very stiff poly-mers.

Models Summary. The suitabi l i ty of each of them odels descri bed depends t o a l arge ext ent on t henat ure of the polymer ma king up the hydrogel . Diffu-

sion in homogeneous hydrogels could not expected to

be aptly described by a model which assumes tha t t hepolymer chains are rigid, straight, and motionless, andneither w ould the converse be expected t o be tr ue. Thehydrogel class for which each of the models discusseda bove is suita ble is listed in Ta ble 1, along w ith t he formof the model equa tion w hich will be used for compa risont o l it erat ure dat a .

Methods

Data for solute diffusion in hydrogels were taken froma number of sources.9,20,28,35,40-47 These sources arelisted in Table 2, along with the polymer and solute(s)used. In order for the literat ure source to be considereduseful the following information had to be provided: thevolume fraction of polymer in the hydrogel (given orreadi l y c al c ul at ed b ased on l i t erat ure dat a) and t hediffusivity of the solute in a queous medium used in thestudy . The method used for determining diffusivity inthe gel a nd t he classi ficat ion of th e hydrogel as ei therhomo- or heterogeneous a re a lso listed in Table 2. It isbeyond the scope of this paper to compare methods ofdetermining di ffusivity. The values obta ined in thel iterat ure were a ccepted a s a ccura te. The interestedreader can see Westrin et al . 48 for a comparison of theestima ted a ccura cys of these methods. A final criteriafor use of the data wa s tha t more than three data pointsbe provided in the study.

As all the models require an estimation of the solutera dius, the Stokes-Einstein hydrodynamic rad ius of the

Dg

Do) exp(-a) (28)

) 0.174 ln(59.6rf

rs) (29)

Dg

Do) exp(-0.174ln(59.6rf/rs)) (30)

Dg

Do) (1 + 23R)

-1exp(-0.174ln(59.6rf/rs)) (31)

Figure 1. C lag u e a nd P hil lips s imu lat ion r e su lt s of hy d r o-dyna mic effects on spherical solute diffusion through a ra ndomne t wor k of c y lind r ic al f ibe r s .39 (A) E ffect of f iber volumefr ac t ion, f ib e r r a d iu s, and solu t e r a d iu s. I n t he le g end , )rf/rs. The lines represent the regression results of eq 28 w itha) . (B) Relationship between fit ted values of a nd rf/rs.The line represents the regression result given by eq 29.



6/14

solute wa s used, a s calculat ed from21

in wh ich kB is B oltzma nns consta nt , Tis temperature,

a nd is the viscosity of water at T. The radi i obtainedfor the solutes examined a re listed in Ta ble 3, along w iththe l i terat ure source used t o obta in Do. The exceptionto this definition of effective solute ra dius wa s the ra diusof gyrat ion, rg, used for th e poly(ethy lene glycol) (P EG )sol ut es . The ra di us of gyra t i on w a s used for t hese

Figure 2. Applicat ion of the Cukier hyd rodyna mic-scaling model23 to literat ure dat a sh owing th e effect of polymer volume fractionon solute diffusivity w ithin various homogeneous hydrogels: (A) polyacryla mide gels; (B) poly(vinyl alcohol) gels;42 (C) poly-(ethylene oxide) (PEO) and poly(hydroxyethyl methacrylate) gels (HEMA); (D) dextran (Dex) and hydroxypropyl methylcellulose(HPMC) gels. The lines represent regression results.

Table 1. Summary of Diffusion Models and the Hydrogels for Which They are Suited

model expression ref hydrogel cla ss

free volume theory Dg

D0) (1 - k1rs

0.75) exp(-k2rs2( 1 - )) Lustig and Peppas 16 homogeneous

hydrodynamic Dg

Do) exp(-kCrs

0.75)Cukier 23 homogeneous

hydrodynamic Dg

Do) [1 + (rs

2

k)

1/2

+13

rs2

k]-1 Phillips et al.24 heterogeneous

obstruction Dg

Do) exp[- (rs + rf)rf ]

Ogston et al. 31 heterogeneous

obstruction Dg

Do) exp(-0.84R1.09)

J ohansson et al.30 heterogeneous

obstruction Dg

Do) (1 + 23R)

-1 Tsai an d St reider34 heterogeneous

obstruction Dg

Do) exp[-

( r

s

+ rf

ks1/2 + rf)

2

] Amsden 35 heterogeneous

combined Dg

Do)

exp[-0.84R1.09]

[1 + (rs2

k)

1/2

+13

rs2

k]J ohnson et al. 38 heterogeneous

combined Dg

Do) (1 +

23R)

-1exp(-0.174ln(59.6rf/rs))

Clague an d P hil lips39 heterogeneous

rs ) kB T

6Do(32)



7/14

solutes because i t w as reported t o be a better est imateof the effective radius of the solute tha n t he hydrody-namic radius.30

The l i terature models were applied to the data andanalyzed via a nonlinear regression procedure (Leven-berg-Marq uar dt n onlinear regression a lgori thm usingKa leidaG ra ph softwa re). In some cases, the di ffusivi tydat a w ere pl ot t ed as l n(Dg/Do) versus either polymervolume fraction, , or solute radius, rs. Th e n a t u r a llogarithm of the diffusivity ratio was used to provide amore robust regression procedure. The success of therespective model wa s determined by a n a na lysis of botht he v al ue of t he f i t t ed param et er( s) ret urned, t hei rc onf i denc e i nt erv al s , and t he sum of squares of t heresiduals (2) obtained by the fit . In th e majority of the

cases, estimates of experimental error were not given,and so confidence intervals were determined using thesum of squa res of the r esiduals divided by t he degreesof freedom a s a n est imat e of the varian ce.

Results and Discussion

The results of the a pplica tion of the var ious descrip-tive models to th e l iterat ure da ta wil l be discussed asfollows. The models found to be best a t describing theentirety of the dat a w ill be presented first, with t he lesssuccessful models discussed a fter.

Homogeneous Hydrogels. Of the models derivedto predict solute diffusion in homogeneous hydrogels,the hydrodynamic-scaling model of Cukier 23 providedthe best f i t to the l i terature da ta . This model (eq 13)wa s applied to the Dg/Doversus polymer volume fra ctiondat a shown in Figure 2a-d. The regression results ar elisted in Ta ble 4. The free volume model expressionderived by Lustig and Peppas (eq 7) was applied to theD g/D o versus polymer volume fraction data shown in

Table 2. Literature Sources Used

polym er solut e(s)a met hod ref

Homogeneouspoly(a cr yla m ide) sucrose, urea 0.046-0.238 d iffus ion in to sla b Wh it e a nd D or ion 40

poly(a cryla m ide) RNa se, B S A 0.021-0.138 FRAP b Tong and Anderson 41

poly (vin yl a lcoh ol) TP L ,R-LA, LYS, vit B 12 0.043-0. 189 p er m ea t ion a cr os smembrane

M a t s uya m a et a l .42

poly(et hylene oxide) ca ffeine 0.092-0.317 relea se from sla b G ra ha m et a l.43

poly (e th y len e ox id e) L YS , v it B 12, C H Y, OVA 0.062, 0.084 d iffus ion in to sla b Mer rill et a l.44

poly(hydroxyethyl

methacryl ide)

P P A 0.19-0.60 relea se from sla b K ou et a l.45

hydroxypropylmethylcellulose

AM 0.000-0.131 P FS G E-NMRc Ga o a nd Fa gernes s28

dext ra n L YS , B S A, I gG 0.048-0. 227 r elea s e f rom cy lin der H en nin k et a l.20

Heterogeneousca lcium a lgina t e dext ra ns, C H Y, B SA 0.003 la ser light sca t t er ing S ellen 58

ca lcium a lgina te glucose, R-LA, B SA,I g G , F B G

0.012 relea se from spheres Ta na ka et a l.46

calcium a lginate -L G , OVA, P E P , B S A 0.005-0.050 relea se from sla b Am sden 35

a ga rose B S A, MYO, C 12E 8a ndC 12E 10 micelles

0.010-0. 050 h olog ra ph icinterferometry

Kong et al .47

-ca ra geen an P E G 326, 678, 1118,1822, 2834, 3978

0.005, 0.010 sla b-sect ion in g J oh a nsson et a l.9

a Solute abbreviat ions: AM, adina zolam mesylate; B SA, bovine serum albumin; CH Y, chymotrypsinogen; FB G, fibrinogen; IgG ,immunoglobulin G; R-LA, R-lactalbumin; -LG, -lactoglobulin; L YS, lysozyme; MYO, myoglobin; OVA, ovalbumin; P EG , poly(ethyleneglycol); P EP , pepsin; P PA, P henylpropanolamine; RNase, r ibonuclease; vi t B 12, v i t a m i n B 12. b FRAP: fluorescence recovery after

photobleaching. c

PF SG E-NMR: pulsed field gradient spin-

echo nuclear magnetic resonance.

Table 3. Diffusivity and Radius of the Solutes Examined

soluteDo(106

cm 2/s)temp(C)

radius() ref

urea 18.1 37 1.9 Wh it e a n d D or ion 40

glucose 6.4 23 3.6 Ta n a ka et a l.46

t h eoph yllin e 6.54 25 3.8 Ma tsuy a ma et a l.42

sucrose 6.97 37 4.8 Wh it e a n d D or ion 40

ca ffein e 6.3 37 5.3 G ra ha m et a l.43

a d in a zola m m es yla t e 4.30 23 5.4 G a o a n d F a ger nes s28

ph en ylpr opa n ola m in e 5.51 37 6.0 K ou et a l.45

vi tamin B 12 3.79 37 8.7 C olt on et a l.61

P E G 326 4.9 25 7.5 J oh a n sson et a l.9

P E G 678 3.5 25 10.5P E G 1118 2.8 25 13.1P E G 1822 2.2 25 16.7

P E G 2834 1.8 25 20.4P E G 3978 1.5 25 24.5

r ibon uclea se 0.131 20 16.3 Ty n a nd G usek62

my oglobin 0.113 20 18.9ly sozyme 0.112 20 19.1R-la ct a lbumin 0.106 20 20.2ch ym ot ry psin ogen 0.095 20 22.5pepsin 0.090 20 23.8ova lbumin 0.073 20 29.3b ov in e s er um a l bu min 0. 060 20 36. 3-la ct oglobulin A 0.042 20 51.0im mu nog lobulin G 0.040 20 56.3fibr in ogen 0.020 20 107.0C 12E 8micelle 0.08 21 27.4 K on g et a l.47

C 12E 10micelle 0.069 21 31.7 K on g et a l.47

Table 4. Regression R esults of Application ofHydrodynamic Model of Cukier23 to Solute Diffusion in

Homogeneous Hydrogels (Eq 13)a

p ol y m er s ol u t e rs, kc (S E 2 R2 d a t a r e f

P AAM u rea 1.9 1.12 0.11 0.255 0.874 Wh it e a n dDorion 40

s u cr os e 4 .7 5 1. 06 0 .0 9 0 .1 16 0 .9 29P A AM R N a se 1 6. 3 0 .5 5 0 .0 25 0 .0 70 0 .9 45 Ton g a n d

Anderson 41

B S A 36. 3 0. 45 0. 020 0. 116 0. 929D ext r a n L YS 19. 1 0. 57 0. 020 0. 26 0. 921 H en n in k

et al .20

B S A 36.3 0.58 0.034 2.52 0.903I gG 56.3 0.66 0.027 3.17 0.921

P VA vit B 12 8 .7 0 .6 19 0 . 05 0 0 .5 00 0 . 87 4 M a t s u y a m aet al .42

L YS 1 9. 1 0 .3 98 0. 01 3 0 .0 07 0. 960P E O ca f fe in e 5 .2 5 0 .8 83 0. 02 0 0 .0 30 0. 974 G r a h a m

et al .43

P H E MA P P A 6.0 1.10 0.06 2.36 0.904 Kou et a l.45

H P MC AM 5.4 1.16 0.009 0.006 0.998 G a o a n dFagerness28

a S E ) standard error (95%confidence interval). 2 ) sum ofsquares of the residuals. R) correlation coefficient.



8/14

Figure 3a-d. This expression wa s chosen over eq 6

because it conta ins para meters for wh ich da ta is readilyobta inable from other l iterat ure sources. The resultsof the nonlinear fitting procedure to the data are givenin Table 5.

What is immediat ely noticeable about t he va lues ofthe fi t ted parameters obtained using the free volumem od el of L u st i g a n d P e pp a s ,16 i s t h a t i n t h e v a s tmajori ty of the cases neither k1 nor k2 is sta t ist ical lysignifica nt. In other words, the confidence interva l foreach para meter is greater tha n the value of the para m-et er , i ndi cat i ng t hat z ero i s a s l ikel y a v al ue for t hepara meter as t he returned value. This result impliest hat t he s t ruc t ural para m et ers are hi ghl y c orrelat ed;that is, there is some functional dependence of one ofthe para meters on the other. As neither of the struc-tura l par ameters ha s been defined in any w ay, the factt h a t t h ey a r e cor r el a t ed d em on s t r a t es t h a t f ur t h err ef in em en t t o t h is m od el i s r eq u ir ed i f i t i s t o b econsidered a viable explanation for solute transport inhomogeneous hydr ogels.

B y removal of the sieving term from eq 7, the fi t tedva lues for k2become st a tist ically releva nt (Ta ble 6). Thefi t t ing equation used was

The l ines in Figure 3a-d are t he resul t ant c urv esfrom fi t t ing eq 33 to the data . An examination of the

2 a nd Rva lues in Ta ble 6 suggests t ha t eq 33 providesa r ea s on a b ly g ood f it t o t h e d a t a . H ow e ve r, t h e 2

values obta ined using the hy drodyna mic-scaling modelwere, in most cases, lower than those obtained usingthe free volume models indicating that the hydrody-na mic expression is a better model. Also, the kcvaluesobtained using the hydrodynamic-scaling model are allsta tistically significant . Furthermore, sincekcis defined

as a constant for a given polymer-

solvent condition, th efi t ted values for t his para meter should be sta t ist ical lyequal for s i t uat i ons w here m ore t han one solut e w asinvestigated in a part icular hydrogel . An examinationof Ta ble 6 shows t ha t, w ith t he exception of the r esultsfrom the data of Matsuyama et al . ,42 this is indeed thecase. The fi t ted values for kc a re ei ther consistent orsta t ist ical ly equal . Thus, it a ppears a t this point tha tthe hydr odyna mic-scaling model of Cukier23 is a bettermodel to use for homogeneous hydrogels than the freevolume models.

Further support for this conclusion comes from ama thema tical ana lysis of the two model equa tions. Thefree volume model a s expressed by eq 33 is an a na lyticfunction wherea s the hydr odyna mic model (eq 13) is not.

Analytic functions represent a small class of possiblefunct i ons and so plac e a rest r i ct i on on t he t ype offunct i on required t o represent t he phenom ena. Ex-panding eq 33 for results in

For < 0.10, eq 34 reduces to just t he first tw o terms,a nd so a n ana lytic function is unnecessary. This resulti ndicat es t hat f ree v ol ume t heory cannot representdiffusion in h ydrogels of low volume fraction. Thehydr odyna mic-scaling model does not pose this t ype ofrestriction and thus can represent diffusion in hydrogelsof a wider polymer volume fraction range.

The finding that hydrodynamic phenomena are im-port a nt comes in contr a diction to the findings of a r ecentpaper 19 which compar ed t he predict ions of th e hy dro-dynam i c m odel of Anderson and Quinn 49 a n d t h epredict ions of free volume theory to the di ffusion ofsol ut es i n polyac rylam i de gels . On t he b a sis of t h i scompar ison, it wa s concluded tha t hyd rodyna mic modelsdo not describe solute di ffusion well . However, t hemodel of Anderson and Quinn is not a valid model fordiffusion in hydrogels, a nd wa s not considered in thisreview, because it assumes that the hydrogel is a porousmembrane where the pores are long, cyl indrical , andconnect both surfa ces of the membra ne. This depiction

Table 5. Regression Results of the Application of the Lustig and Peppas16 Free Volume Model to Solute Diffusion inHomogeneous Hydrogels (eq 7)a

polymer solut e rs, k1 (S E k2 (S E 2 R2 d a t a r ef

P AAM urea 1.9 0.070 0.491 0.728 0.337 0.0080 0.970 Whit e a nd D orion 40

sucrose 4.75 0.066 0.050 0.263 0.014 0.0005 0.998P AAM RNa se 16.3 0.143 0.605 0.043 0.097 0.0976 0.924 Tong a nd Anderson 41

B S A 36.3 0.071 1.486 0.019 0.115 0.3550 0.783D ext ra n LYS 19.1 0.082 1.02 0.031 0.120 0.945 0.726 H ennink et a l.20

B S A 36.3 0.061 0.087 0.017 0.004 0.403 0.984I gG 56.3 0.026 0.073 0.016 0.024 0.986 0.978

P VA vit B 12 8.7 0.176 0.977 0.026 0.246 0.0126 0.925 Ma tsuya ma et a l.42

LYS 19.1 0.158 0.005 0.022 0.002 0.0130 0.996P E O ca ffeine 5.25 0.260 1.36 0.129 0.53 0.0081 0.904 G ra ha m et a l.43

P H E MA P P A 6.0 0.175 1.10 0.059 0.298 3.07 0.878 K ou et a l.45

H P MC AM 5.4 0.391 0.862 0.193 0.419 0.076 0.970 G a o a nd F a gerness28

a S E ) standard error (95%confidence interval). 2 ) sum of squares of the residuals. R2 ) correlation coefficient.

Table 6. Regression Results of the Application of theFree Volume Model without a Preexponential SievingTerm to Solute Diffusion in Homogeneous Hydrogels

(eq 33)a

polymer solute rs () k2 (S E 2 R2 d a t a r e f

P A AM u r ea 1. 9 0 .7 74 0. 03 9 0 .0 080 0 .9 70 Wh it e a n dDorion 40

sucrose 4.75 0.281 0.002 0.0010 0.998P A AM R Na s e 1 6. 6 0 .0 60 0. 00 3 0 .1 21 0 .9 04 Ton g a n d

Anderson41

B S A 36. 3 0. 023 0. 002 0. 399 0. 755d ex tr a n L YS 19. 4 0. 038 0. 003 1. 08 0. 677 H en n in k

et al .20

B S A 3 6. 3 0 .0 21 0. 00 05 0 .4 33 0 .9 84I g G 5 6. 5 0 .0 16 0. 00 04 1 .0 24 0 .9 78

P VA vit B 12 8 .7 0 .0 61 0 .0 05 0 .0 51 0 .7 19 M a t s u y a m aet al .42

L YS 19. 4 0. 044 0. 003 0. 255 0. 937P E O ca f f ei ne 5. 25 0 .1 79 0. 01 1 0 .2 08 0 .8 19 G r a h a m

et al .43

P H E M A P P A 6. 0 0. 081 0. 006 4. 87 0. 803 K ou e t a l .45

H P M C AM 5. 4 0. 338 0. 0125 0. 122 0. 952 G a o a n dFagerness 28

a S E ) standard error (95%confidence interval). 2 ) sum ofsquares of the residuals. R) correlation coefficient.

ln (Dg

Do) ) -k2rs2( 1 - ) (33)

Dg

Do) 1 - k2rs

2 -k2rs

2

2 2 -

k2rs2

6 3 ... (34)



9/14

of hydrogel morphology is unr eal ist ic, an d so their

conclusion tha t hydrodynamic effects are not a viableexpla na tion for the observed decrea se in solute diffusioni s unw arrant ed.

Thus far, the hydrodynamic model has been appliedon ly t o d a t a f or w h i ch t h e r a n g e o f s o lu t e s i ze i srelat ively nar row. The model wa s therefore a pplied toliterature data in which the solute size ranges from 3.8to 36.3 (Figure 4). The r egression results ar e dis-played in Ta ble 7.

From t hi s ra t her l i m i t ed dat a , i t appears t hat t hehydrodyna mic model provides a n adequa te fit . Furt hersupport for th e model comes from the result t ha t t he kcpara meters obta ined for t he Merri ll et a l . data 44 a t t w odifferent polymer volume fra ctions a re sta t ist ical lyidentical. The k

c parameter reflects polymer-solvent

interaction and thus would be expected t o be consta ntfor a given polymer-solvent system. However, the datain Figure 4, part icularly t hat of Merri l l et a l . , suggestsome convex curvature that is not accounted for by thehydr odyna mic model. Hence, the dependence of ln(Dg/Do) on solute size seems to be at least a weak powerfunction an d not simply the l inear dependence sug-gested by t he hydr odynamic model. This dependencehas also been noted by other researchers.9,50 To verifythis observation a study needs to be done to f ind thediffusivities of solutes of a w ide size range using a well-characterized homogeneous hydrogel.

A few a dditional comments on th e free volume modelare w ar ran ted at this point , part icular ly because of i ts

populari t y i n t he f ield of cont roll ed drug deli very .Exa mination of the da ta in Table 6 indicat es tha t k2 i ss t a t i s t i cal l y relevant i n a l l c ases. What needs t o b eascertained is whether the values for k2 m ake sensephysical ly. Compar ing the fi t ted values obta ined forcases wher e the polymer a nd solvent used w ere identicaland t he sol v ent v ari ed show s t hat , i n eac h c ase, t hevalue of k2 decreased as the solute size increased.

Figure 3. Application of free volume theory models to literat ure da ta showing t he effect of polymer volume fraction on solutediffusivity w ithin va rious homogeneous hydrogels: (A) polyacryla mide gels; (B ) poly(vinyl a lcohol) gels;42 (C) poly(ethylene oxide)(P EO) a nd poly(hydroxyethyl m ethacryla te) gels (HEMA); (D) dextran (Dex) and hydroxypropyl methy lcellulose (HP MC) gels.The lines represent regression results.

Figure 4. Regression results of the implementation of theCukier hy drodynam ic-scaling m odel23 (eq 13) to literature dat ashowing t he e ffec t s of solu t e r ad iu s on d if fu sivit y wit hinhomogeneous hydr ogels. The lines represent the regressionresults.



10/14

The structural parameter k2, as defined, is equal to/vf,w. The avera ge free volume of bulk wa ter, vf, w, isconstant for a given temperature, and so the effect ofsolute size ca nnot be expla ined by changes in its va lue.It would be unreasona ble to expect tha t the overlap infree volume parameter, , would be a function of solutesize and so it t oo can be discounted. What remains isthe jump length, , of the solute molecule moving fromone free volume to a nother. One would expect tha t th elarger the molecule, the greater the length required inorder for a successful t ra nslat i onal jum p t o occur.Therefore, the result th a t k2a ctually decreases a s solutesize increases is in direct contr a diction to th e underlyingtheory.

Another fundamental weakness of the free volumeapproach is t he use of eq 3 to describe the contr ibutionsof b ot h t h e w a t e r a n d t h e p ol ym er t o t h e t ot a l f r eevolume of the system. In order for this equation to bevalid, th e specific volume a nd molecular weight s of thepolymer jumping units a nd w at er must be equal .11 Atpolymer concentra tions near unity, t hese conditions ma ybe approximated, depending on the polymer, but theycannot hol d ov er a w i de ra nge of pol ym er v ol um efractions. The models derived t o dat e based on free-volume theory ar e thus str ictly va lido n l yfor polymer-

diluent systems with large polymer volume fractions.7Despite t his l imita t ion, th e theory ha s been a pplied tohydrogels of high water content .11,13-16,20,42,51-53 Thebasis for t his w idepsread us e of the free volume modelsi s t he general ly l inear rela t i onshi p w hi ch t ypical l yr es u lt s f r om a p lot of l n(Dg/Do) v ersus /(1 - ).However, the fact that such a functional dependence ispredicted by the theory does not necessarily mean thetheory is a ccurat e. Other models, such a s the obstruc-t i on m odels and hydrodynam i c m odels deriv ed forheterogeneous hydr ogels, also predict a similar func-t i onal dependenc e and yet are deri v ed usi ng a c om -pletely inappropriate basis.

Finally, the Cohen and Turnbull free volume modelof solute tra nsport in simple l iquids10 i s know n t o b ean improper model of the mechanism involved.54,55 Thefree volume model also ca nnot a dequa tely expla in solutediffusivi ty in si tuations where the solute molecule islarg er tha n th e solvent molecule, a situa tion commonlyencountered in a pplicat ions of hydr ogels.56,57 Since thefree volume model fai ls to explain aqueous solutiondi ffusi vi t ies, t here i s no reason t o expect i t t o b eextendable to hydrogels.

Heterogeneous Hydrogels. There ar e a relat ivelylarge n umber of mecha nist ic models which ha ve beenderived to describe solute diffusion in heterogeneoushydr ogels (Ta ble 1). It would therefore appear to be alengthy task to determine which of these models is mostapplicable. However, the t ask can be reduced by ex-

am i ni ng t he ef fec t o f sol ut e s i z e on di f fusi v i t y i n aheterogeneous hydrogel system first . Figure 5 displayssuch a dependence for various solutes in calcium algi-nat e. An examina tion of the experimental ly observedrelat ionship between solute ra dius a nd the di ffusivi ty

rat i o Dg/Do reveals t ha t t he di ffusivi ty ra t io displays aGa ussian dependence on solute ra dius, such a s w ouldb e g i v e n b y a n e x p (-rs2) funct i on. Thi s resul t i m -mediat ely rules out th e a pplicabili ty of t he Ogston eta l. obstr uction expression (eq 19) a nd t he hydr odyna micmodels represented by eq 14. The rema ining modelswere applied to the data shown in Figure 5 and otherl it erat ure dat a for sol ut e di ffusi on i n b ot h cal ci umal ginat e a nd -carageenan. The calcium alginat e datareflects tw o different ty pes of algina te, a h igh guluronica cid r esidue content 35 and a low guluronic acid residuecontent.46,58 The high guluronic acid residue a lgina tehas very st i ff polymer chains, w hile t he low guluronicacid residue alginat e ha s more flexible chains.59

The remaining models, with the exception of theAmsden obstruction model , were fi t to the l i teraturedat a b y considering t he pol ym er radi us , r f, t o b e a na d ju s t a bl e p a r a m e t er . Th e v a lu es r et u r n ed b y t h eregression procedure were then compared to availablelitera ture dat a. For calcium alginat e, which gels throughan electrostat ic at t ract ion for tw o guluronic acid resi-dues on opposi ng pol ymer chai ns for t he di val entcalcium ion, the polymer chain radius was est imatedt o b e 4.2 . Thi s v al ue i s a num b er av erage of t hera dius of the unreacted residues w hich ma ke up about2/3of the hy drogel cha ins (3.6 ) a nd t ha t of the ra diusof two electrostatically bonded guluronic acid residues(5.4 ).60 To account for the strongly bonded watermolecules w hich hydra te t he polymer, to this a verage

Table 7. Regression R esults for Application of Obstruction Models to Dg/Doversus Solute Radius Li terature Data forHeterogeneous Hydrogels

model

J ohansson et al.30 Tsai and Streider 34 Amsden a,35

polymer (ref) rf fit () (S E 2 R2 rf fit () (S E 2 R2 ks fit () (S E 2 R2

a l gi na t e35 3.46 0.05 0.001 0.992 2.10 0.26 0.029 0.801 5.63 0.24 0.005 0.964a l gi na t e46 7.31 0.46 0.013 0.974 5.13 0.50 0.019 0.962 13.02 1.05 0.020 0.960a l gi na t e58 7.28 0.40 0.036 0.931 5.68 0.55 0.065 0.878 13.62 0.87 0.043 0.918carageenan 9 ( ) 0.005) 2.66 0.08 0.001 0.964 2.38 0.04 0.001 0.992 6.56 0.25 0.001 0.951

carageenan9

()

0.010) 3.72 0.13 0.004 0.912 3.08 0.07 0.001 0.974 7.31 0.10 0.001 0.987a Regression done using rf ) 8.0 . SE ) sta nda rd error (95%confidence interva l).2 ) sum of squares of the residuals. R) correlation

coefficient.

Figure 5. Lit eratur e data showing the effect of solute radiuson diffusivity w ithin heterogeneous (calcium algina te) hydro-gels.



11/14

c h a i n r a d i u s c a n b e a d d e d t h e d i a m e t e r o f a w a t e rmolecule (3.8 ). This r esults in a n effective polymerra dius of 8.0 . For -cara geena n, which in the gel stat ehas some of i ts chains in an aggregated double hel ix(ra dius 5.1 ), the a verage polymer rad ius used wa s 4.2. This value represents the situa tion where half of thepol ymer c hai ns a re i n t he a ggregat ed s t a t e .30 Again,the hydr a ted polymer chain ra dius would be 8.0 . Theregression results a re illustra ted in Figures 6 and 7 andlisted in Tables 7 and 8.

Exa minat ion of the figures shows tha t t he obstructionmodels provided the best fits to th e da ta (Figure 6). All

the obstruction models a re consistent with the observedGa ussian trend in the dat a. This result indicates thatthe obstruction models have potential for wide rangea pplicat ion. The polymer ra dius values obtained (Ta ble7) were rea sonable an d consistent for both the J oha ns-son et a l . expression 30 a n d t h e Ts a i a n d S t r eid ermodel,34 but were unr ealistically small for the Tsa i andSt rei der m odel a ppl ied t o t he hi gh guluroni c a ci dal ginat e and t he c arageenan hydrogel s. Thi s resul t ,coupled with the fact that i t produced the greatest 2

values, implies that this model is the weakest of theobstruction models. The J ohan sson et a l. expression

Figure 6. Implementation of obstruction models to literature data showing the relationship between solute radius and solutediffusivity w ithin heterogeneous hy drogels: (A) model of Tsai an d S treider34 (eq 23) versus solute diffusion in calcium algina tegels; (B) model of Tsai and Streider 34 (eq 23) versus solut e diffusion in -cara geenan ; (C) model of Amsden 35 (eq 25) versus solutediffusion in calcium alginate gels; (D) model of Amsden 35 (eq 25) versus solute diffu sion in -carageenan; (E) model of J ohansson

e t a l .30 (eq 22) versus solute diffusion in calcium algina te gels; (F) model of J oha nsson et al. 30 (eq 22) versus solute diffusion in-carageenan.



12/14

produced a smaller polymer radius estimate for the highguluronic acid content alginate hydrogel than for thelow gu luronic a cid residue hyd rogels. This is a reflec-tion of the assumed polymer chain stiffness.

The results from th e regression of the J ohnson et a l.38

and t he Clague and P hil lips39 combined hydrodynamicand ob st ruct i on m odels t o t he l i t erat ure dat a can b eseen in Figure 7 a nd a re listed in Ta ble 8. The 2 valuesfor each model are very close, with the J ohnson et al.expression providing sl ightly lower values. Exa mina-tion of Table 8 shows that the obtained fi t ted valuesfor rf for di ffusion in the same polymer (i .e. , for theTana ka et a l .46 and Sellen 58 al gi nat es and t he -cara-geenan of J ohansson et al .9) were consistent . The rfv al ues ob t ai ned from t he regression t o t he cal ci uma l g in a t e d a t a of Ta n a k a e t a l .46 and Sellen 58 appearsomewhat high but can be rationalized by the possibility

of the electrosta t ic interaction of more tha n 2 a lgina techai ns . Al so, neit her m odel m a t c hes t he ob serv edGa ussian tr end in the data . This lack of fit for the lowerrs values indicat es tha t t hese models a re perhaps bestsuited for si tuat ions where the solute size approachesthe a verage opening size in t he hydr ogel . Therefore,these models appear to be applicable only to the diffu-sion of lar ge molecules (rs greater than approximately20 ) in very st iff hydr ogels of rela tively high polymerfraction.

On the basis of these results, the models of J ohanssonet a l . and Amsden w ere then applied to li terature da tadescribing the effect of polymer volume fra ction onsolute diffusivity for two hydrogels, calcium alginate anda ga rose (Figure 8). The regression results ar e listed inTa ble 9. B oth the models provide good agreement w itht he dat a , as i ndi c at ed b y t he l ow 2 values returned.

Figure 7. Implementation of combined hydrodynamic and obstruction models to literature data of the relationship betweensolute ra dius a nd solute diffusivity w ithin h eterogeneous hydr ogels: (A) model of J ohnson et al. 38 (eq 27) versus solute diffus ionin calcium a lginate gels; (B) model of J ohnson et a l.38 (eq 27) versus solute diffusion in -cara geena n gels (data from J ohanssonet al.9); (C) model of Cla gue an d P hillips39 (eq 31) versus solute diffusion in ca lcium algina te gels; (D) model of Clague a nd P hillips39

(eq 31) versus solute diffusion in -carageenan gels (data from J ohansson et al. 9).

Table 8. Regression Results for Application of C ombined H ydrodynamic and Obstruction Models to Dg/Doversus SoluteRadius Literature Data for Heterogeneous Hydrogels

modela

J ohnson et a l.38 Clague an d P hil lips39

polymer (ref) rf fit () (S E 2 R2 rf fit () (S E 2 R2

a l gi na t e35 5.44 0.50 0.018 0.872 4.09 0.58 0.032 0.780a l gi na t e46 13.95 1.81 0.033 0.937 11.15 2.15 0.055 0.893a l gi na t e58 14.87 2.35 0.121 0.774 13.72 2.20 0.120 0.774carageenan 9 ( ) 0.005) 8.00 0.66 0.002 0.912 7.21 0.69 0.002 0.891carageenan 9 ( ) 0.010) 9.56 0.39 0.002 0.956 8.73 0.60 0.004 0.904

a S E ) standard error (95%confidence interval). 2 ) sum of squares of the residuals. R) correlation coefficient.



13/14

The J ohansson et al . expression aga in produces rfest i m at es w hi c h are c onsi s t ent for t he agarose dat a .However, the est imated va lues of about 6 are muchl ow er t han t he a v erage v al ue giv en i n t he l it erat ure.Agarose gels by forming physically cross-linked fiberbundles of R-helica l chains. As reported by J ohnson eta l. 38 the distribution of the sizes of these bundles isbimodal , w ith 87% ha ving a ra dius of 15 and 13%hav i ng a radi us of 45 . Thus, t he a v erage pol ym erchain ra dius for a ga rose is ta ken to be 19 . This valuewa s used in applying the Amsden model. The ksvalueswhich were obtained for a garose were very consistent ,as predicted by the model.

On the basis of these compar isons, it appears t ha t t heobstruction model of Amsden provides the most consis-tent explanation for solute diffusion in heterogeneoushydrogels. However, the model contains a para meter

which is as yet undefined and must be for the model tobecome completely verifiable.

Conclusions

Models for solute diffusion in hydrogels can be dividedinto t hose a pplica ble to h ydrogels composed of flexiblepolymer cha ins (i.e., homogeneous hyd rogels) a nd t hosecomposed of rigid polymer chains (i .e. heterogeneoushydrogels). Of those used to describe solute diffusion inhomogeneous hyd rogels, the model most consistent w itht h e d a t a a n d w i t h i t s p h y s i c a l p a r a m e t e r s w a s t h ehydrodynamic scal ing model of Cukier.23 This modelhowever, suffers from conta ining a n undefined polymer-solvent int era ction para meter. The free volume models,which are general ly invoked, are typical ly applied tosi t uat i ons for w hi c h t he assum pt i ons m ade i n t hei rderivat ion a re not va l id a nd a re not physical ly consis-tent . Of those models derived to expla in solute diffusionin heterogeneous hydrogels, the obstruction models werebest at providing agreement to the experimental datata ken from the literatur e. The combined hydrodynamica nd obstruction effect models a ppea red t o be limited inapplicabil i ty to si tuations of large solute di ffusion inrelat ively high polymer fraction hydrogels. In terms ofa ccounting for va ria tion due to polymer cha in flexibility,

the best obstruction model wa s tha t of Amsden. Likethe Cukier model , which is also ba sed on scal ing con-cepts, this model conta ins a n a s yet undefined polymer-solvent intera ction para meter. For these models to beconsidered truly predictive, a means of calculating thesepara meters must be derived.

Acknowledgment. I thank Drs. Amir Shojaei andMart i n Gua y of t he U ni versi t y of A lb ert a for t heirconstructive comments. This work was supported by ag r a n t f r om t h e N a t i on a l S c ie nce a n d E n g in ee r in gResear ch Council of Ca nad a.

References and Notes

(1) Moussaoui, M.; Benylas, M.; Wahl, P. J . Ch r o ma t o gr . 1991,558, 71.(2) J en, A. C.; Wake, M. C.; Mikos, A. G. Biotech. Bioeng.1996,

50, 357.(3) Rat ner, B. D . Biomedical Applicat ions of Synt hetic Polymers.

I n Compr ehensive Polymer Science; A gga r w a l , S . L . , E d . ;P ergam on P ress: Toronto, 1989, Vol. 7, p 201.

(4) P eppas, N. A., Ed. H ydrogels in M edicine and Pharm acy; CRCPress: Boca Rat on, FL, 1987.

(5) K im, S. W.; B ae, Y. H .; Okano, T. Pharm. Res. 1992, 9 (3),283.

(6) Lee, P. I. S ynth etic hydrogels for drug delivery. In Contr olledRelease Systems: Fabr icat ion Techn ology; Hsieh, D., Ed.; CRCPress: Boca Rat on, FL, 1988.

(7) Muhr, A. H.; Bla nshard, J . M. V. Polymer 1982, 23(J uly),1012.

(8) Yasuda , H.; Lama ze, C. E.; Ikenberry, L. D. M akromol. Chem.1968, 118, 19.

Table 9. Regression Results for Application of Obstruction Models to Dg/Doversus Polymer Volume Fraction LiteratureData for Heterogeneous Hydrogels

modela

J ohansson et al.30 Amsden b,35

p ol ym er (r ef ) s ol ut e (r a d iu s , ) rf fit () (S E 2 R2 ks fit () (S E 2 R2

a l gi na t e35 B S A (36.3) 3.48 0.05 0.018 0.951 5.73 0.11 0.018 0.940agarose47 MYO (18.9) 6.02 0.23 0.005 0.960 11.63 1.39 0.030 0.776agarose47 B S A (36.3) 7.9 0.50 0.033 0.753 12.45 0.31 0.003 0.978agarose47 C 12E 8micelle (27.4) 6.26 0.50 0.023 0.284 10.19 0.36 0.003 0.939

agarose47

C 12E 10 micelle (31.7) 6.24 0.31 0.010 0.876 9.72 0.64 0.009 0.908a S E ) standard error (95%confidence interval). 2 ) sum of squa res of the residuals. R) correlation coefficient. b Regression done

using rf )8.0 .

Figure8. Applicat ion of obstruction models to data displayingthe dependence of solute diffusivity on polymer volume fractionin het erogeneous hy drogels: (A) model of Amsden 35 (eq 25);(B ) model of J ohan sson et al. 30 (eq 22).



14/14

(9) J ohan sson, L. ; S kant ze, U .; Lofroth, J .-E. M acrom olecules1991, 24, 6019.

(10) Cohen, M. H.; Turnbull, D. J. Chem. Phys. 1959, 31, 1164.(11) Vrenta s, J . S. ; Duda , J . L. J . P ol y m. S c i .1977, 15, 403.(12) Yasuda , H.; P eterlin, A.; Colton, C. K.; Smith, K . A.; Merrill,

E. W. M a k r o mo l . Ch em. 1969, 12 6, 177.(13) P eppas, N. A.; Reinha rt, C . T. J . M emb r . S c i .1983, 1 5, 275.(14) Reinha rt, C . T.; Peppa s, N. A. J . M emb r . S c i .1984, 1 8, 227.(15) Peppas, N. A.; Wright, S. L. M acromolecul es1996, 29(27),

8798.(16) Lustig, S . R.; Peppas, N. A.J . A p p l . P o l y m. S ci .1988, 3 6(4),

735.

(17) Scha efer, D. W. Polymer 1984, 25, 387.(18) Peppas, N. A.; Lustig, S. R. A n n . N. Y . A c a d . S ci .1985, 446,

26.(19) am Ende, M. T. A.; Peppas, N. A. J. Controlled Release1997,

48 (1), 47.(20) Henn ink, W. E.; Talsma , H.; B orchert, J . C. H.; DeSm edt, S.

C.; Demeester, J . J . Contr olled Release1996, 39 (1), 47.( 2 1 ) B i r d , R. B . ; S t ew a r t , W . E . ; L i ght fo o t , E . N . T r a n s p o r t

Phenomena; J ohn Wiley an d S ons: Toronto, 1960.(22) Altenberger, A. R.; Tirrell, M.; Da hler, J . S. J. Chem. Phys.

1986, 84(9), 5122.(23) Cukier, R. I. M acromolecul es1984, 17, 252.(24) P hillips, R. J .; Deen, W. M.; Bra dy, J . F. A I C h E J .1989, 3 5,

5 (11), 1761.(25) J ackson, G . W.; J ames, D. F. Ca n . J . Ch em. E n g .1986, 64,

362.(26) Kosar, T. F.; P hillips, R. J ., A I C h E J .1995, 41, 1 (3), 701.(27) Mackie, J . S.; Meares, P. Pr oc. R. Soc. L ondon 1955, A232,

498.(28) Ga o, P. ; Fa gerness, P. E. Phar m. Res. 1995, 1 2(7), 955-964.(29) Ogston, A. G.; P reston, B . N.; Wells, J . D. Proc. R. Soc. London

Ser. A 1973, 33 3, 297-316.(30) J oha nsson, L.; E lvingston, C.; L ofroth, J .-E. M acromolecul es

1991, 24, 6024.(31) Ogston, A. G. Trans. Faraday Soc. 1958, 54, 1754.(32) J ohansson, L.; Lofroth, J .-E. J . Ch e m. P h y s . 1993, 98 (9),

7471.(33) Tomada kis, M. M.; S otirchos, S. V. J. Chem. Phys.1993, 98

(1), 616.(34) Tsai, D. S.; Strieder, W. Chem. Eng. Commun.1985, 4 0, 207.(35) Amsden, B. Polym. Gels Networks, in press.(3 6) S ko ur i, R. ; S chos s el er , F . ; M unch, J . P . ; C a nd a u, S . J .

M acromolecul es1995, 28, 197.

(37) Brady, J . A I Ch E A n n . M eet .1994, 320.(38) J ohns on, E. M.; B erk, D. A.; J ain, R. K .; Deen, W. M. Biophys.

J . 1996, 70(2), 1017.(39) Clague, D. S.; Phillips, R. J . P h y s . F l u i d s 1996, 8(7), 1720.(40) White, M. L.; Dorion, G. H. J . P ol y m. S c i .1961, 55, 731.(41) Tong, J .; Anderson, J . L. Biophys. J.1996, 70(3), 1505.(42) Matsuyama, H.; Teramoto, M.; Urano, H. J . M e m b r . S c i .

1997, 126, 151.(43) Gra ham, N. B . ; Zulfiqar , M.; MacDonald, B . B . ; McNeill , M.

E . J . Contr olled Release1988, 5(3), 243.(44) Merrill, E. W.; Dennison, K . A.; Sung, C . B i o ma t er i a l s 1993,

14(15), 1117.(45) Kou, J . H.; Amidon, G. L .; Lee, P. I . Phar m. R es.1988, 5(9),

592.(46) Tana ka, H.; Ma tsumura, M.; Veliky, I . A. Biotech. Bioeng.

1984, 26, 53.(47) Kong, D. D.; Kosar , T. F.; Dunga n, S. R.; P hillips, R. J .A I C h E

J . 1997, 43 (1), 25.(48) Westr in, B . A.; Axelsson, A.; Zacchi, G . J. Controlled Release

1984, 30(3), 189.(49) Anderson, J . L.; Quinn, J . A. Biophys. J.1974, 15, 130.(50) Phillies, G. D. J . J. Phys. Chem. 1989, 93, 3 (13), 5029.(51) Wisniewski, S.; Kim, S. W. J . P o l y m . S ci . , P ar t A : P ol y m .

Chem.1988, 26(4), 1179.(52) Sun g, K.; Topp, E. M. J . Contr olled Release1995, 37(1-2),

95.(53) Schlick, S.; Pila r, J .; Kweon, S.-C.; Vacik, J .; Ga o, Z.; Labsky ,

J . Ma cromolecul es1995, 28(17), 5780.(54) Hildebrand, J . H.; Scott, R. L. Free volume theory. Regular

S o l u t i o n s ; P rentice Ha ll Inc.: En glewood Cliffs, NJ , 1962; p60.(55) Limoge, Y. S cr . M et a l l . M a t er . 1992, 26, 809.(56) Wesselingh, J . A.; Bollen, A. M. Chem. Eng. Res. Des., Part

A : T r a n s . I n s t . Ch em. E n g . 1997, 75, 590.(57) Vogel, H.; Weiss, A. Ber Bunsen-Ges. Phys. Chem.1981, 85,

1022.(58) Sellen, D. B. N AT O Ad v. Sci. I nst. Ser., Ser. A. 1983, 5 9, 209.(59) Whittington, S. G. Biopolymers1971, 10, 1481.(60) Nilsson, S. Biopolymers1992, 32, 1311.(61) Colton, C. K .; Smith, K. A.; Merril l , E. W.; Fa rrel l, P . C. J .

Bi omed. M ater. Res.1971, 5, 459.(62) Tyn, M. T.; Gusek, T. W. Biotech. Bioeng.1990, 35, 327.

MA980765F


modeling solute diff gels 0

Documents