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* Correspondence address: Department of Chemical Engineering,

University of Colorado, Boulder, Engineering Centre, ECCH 111,

Campus Box 424, Boulder, CO 80309-0424, USA. Tel.: #1-303-492-

7471; fax: #1-303-492-4341.

E-mail address:[email protected] (C. N. Bowman).

Chemical Engineering Science 56 (2001) 3173}3184

Kinetic modeling of the e!ect of solvent concentration on primarycyclization during polymerization of multifunctional monomers

Jeannine E. Elliott, Jay W. Anseth, Christopher N. Bowman*

Department of Chemical Engineering, University of Colorado, Boulder, Engineering Centre, ECCH 111, Campus Box 424,

Boulder, CO 80309-0424, USA

Dental School, University of Colorado Health Science Center, Denver, CO 80045-0508, USA

Received 19 May 2000; received in revised form 28 November 2000; accepted 5 December 2000

Abstract

Controlling the swelling ratio, di!usion rate, and mechanical properties of a crosslinked polymer is important in hydrogel design forbiomedical applications. Each of these factors depends strongly on the degree of crosslinking. Primary cyclization, where a propagat-

ing radical reacts intramolecularly with a pendant double bond on the same chain, decreases the crosslinking density and increases the

molecular weight between crosslinks. Processing conditions, speci"cally the solvent concentration, strongly a!ect the extent of

primary cyclization. In this work the e!ects of solvent concentration and comonomer composition on primary cyclization are

investigated using a novel kinetic model and experimental measurement of mechanical properties. Two divinyl crosslinking agents

were investigated, diethyleneglycol dimethacrylate (DEGDMA) and polyethyleneglycol 600 dimethacrylate (PEG(600)DMA), and

each was copolymerized with hydroxyethyl methacrylate (HEMA) and octyl methacrylate (OcMA). The model is further used to

predict the gel point conversion and swelling ratio of PAA hydrogels polymerized in the presence of varying amounts of water. Model

results show how increasing the solvent concentration during the polymerization increases the molecular weight between crosslinks

by nearly a factor of three and more than doubles the swelling ratio. Where possible, experimental results provide quantitative

agreement with model predictions. 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Polymer; Gels; Crosslinking; Cyclization; Solvent e!ects; Simulation

1. Introduction

The use of crosslinked polymer hydrogels as bio-

materials is a growing area of biomedical technology:

consequently, research on the network formation process

occurring in the presence of solvents is important. Free

radical copolymerization of multivinyl monomers with

hydrophilic monovinyl monomers leads to the formation

of hydrogels that swell signi"cantly but do not dissolve in

the presence of water. Because of their biocompatibilityand hydrophilic nature, hydrogels have biomedical

applications in contact lenses, wound bandages and

dressings, bioadhesives, cell immobilization, tissue engin-

eering, and drug delivery systems (Wichterle & Lim,

1960; Peppas, 1987; Bae & Kim, 1993; Ende & Peppas,

1996; Jen, Wake, & Mikos, 1996; Wheeler, Woods, Cox,

Cantrell, Watkins, & Edlich, 1996; Kao, Manivannan

& Sawan, 1997). Understanding of the polymer network

formation in the presence of a solvent and the resulting

network structure and properties is essential to develop

hydrogels with controlled swelling and properties for

speci"c biomedical applications.

During polymerization or copolymerization involving

multivinyl monomers, primary cyclization can occurwhen a pendant double bond reacts with the radical on

the same propagating chain that created the pendant.

The degree of primary cyclization strongly e!ects the

network structure created and its resulting properties.

The mechanical integrity of a hydrogel is obtained from

the crosslinks in the network, particularly in solution

where `physicalacrosslinks are hardly present. The mesh

size of a polymer, i.e., the distance between crosslinks,

controls the degree of swelling and di!usion in the

polymer, which are important to many applications,

especially in the development of drug delivery materials.

0009-2509/01/$- see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 09 - 2 5 0 9 ( 0 0 ) 0 0 5 4 7 - 9

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Fig. 1. Polymer networks with low and high degrees of cyclization.

As drug release rates are a function of the degree of

crosslinking of the polymer, larger mesh sizes correlate

with greater di!usion of drug molecules through the

polymer (Peppas & Khare, 1993; Lehr, Bouwstra,

Vanhal, Verhoef, & Junginger, 1992). By reducing

the crosslinking density, primary cyclization changes the

mechanical properties, swelling, and di!usion through

the hydrogel from what would be predicted in an idealhomogeneous polymer. Because the crosslinking density

controls so many important hydrogel properties, under-

standing what a!ects the extent of crosslinking is key in

developing biomaterials for speci"c applications.

This work investigates how the comonomer composi-

tion and the amount of solvent (generally water in hydro-

gel formation) used during polymerization in#uence the

degree of primary cyclization using a numerical modeling

approach and experimental measurements of mechanical

properties. Two sets of experiments were performed

for this study using diethyleneglycol dimethacrylate

(DEGDMA) or polyethyleneglycol 600 dimethacrylate(PEG(600)DMA) as the crosslinking agents. DEGDMA

was chosen because it is a commonly used crosslinking

agent in soft contact lenses. To compare the e!ect of the

crosslinking molecule size PEG(600)DMA was also

evaluated. In the "rst set of experiments hydroxyethyl

methacrylate (HEMA) copolymers were photo-

polymerized using methanol as a solvent. HEMA was

studied because, due to its biocompatibility and

hydrophilic properties, it is used in many biomedical

applications including contact lenses, wound bandages,

and cell immoblization (Wichterle & Lim, 1960; Mon-

theard, Chatzopoulos, & Chappard, 1992; Jen et al.,

1996; Ng & Tighe, 1976). Methanol was selected as thesolvent because the monomers, initiator, and polymer all

were highly miscible with it. In the second set of experi-

ments octyl methacrylate (OcMA) copolymers were poly-

merized in varying amounts of hexanol. Although OcMA

is not a typical biopolymer, it was chosen for the second

set of experiments because it will thermally polymerize to

complete conversion, and thus allowed for more control-

led experiments and better comparisons of molecular

weight between crosslinks. Hexanol was used in these

experiments because the polymer, monomer, and initator

were miscible with it. In conjunction with the experi-

ments, this research utilizes a kinetic model to investigatethe e!ect of solvent concentration during polymerization

on the structure and properties of polymer hydrogels.

The model is compared with the experimental results and

further used to predict properties of polyacrylic acid

(PAA) hydrogels.

The model solves the di!erential kinetic balances on

the reacting species to determine the relative formation of

crosslinks and cycles during the polymerization. When

a multivinyl monomer is incorporated into the polymer

chains, a pendant double bond is formed. This pendant

double bond can react with a radical in the bulk solution

to form a crosslink or react with the radical on its own

propagating chain to form a primary cycle. In the di!er-

ential kinetic balances, the rates of crosslinking and pri-

mary cyclization are controlled by the concentration of

radicals in the bulk solution (bulk radical concentration)

and the e!ective concentration of radicals on the same

propagating chain as the pendant (local radical concen-

tration), respectively. These radical concentrations

change with time as radicals are created through initia-

tion reactions and terminate with each other. Addition-

ally, the local radical concentration varies from the timethe pendant was formed, as the proximity of radicals on

the propagating chain is a function of how long the

pendant has existed and how far the radical which for-

med it has propagated. Using the idea of pendant birth

time developed by Tobita (Tobita & Hamielec, 1989;

Tobita, 1992), each pendant double bond is tracked sep-

arately to incorporate the varying reactivity with birth

time. By inclusion of the two radical concentrations (i.e.

the local and bulk concentration), the model predicts

how the rate of crosslinking and cyclization change with

conversion.

Development of a model that includes primary cycliz-ation and varying pendant reactivity is important for

modeling hydrogels and predicting network structure.

Cyclization causes a much more loosely crosslinked ma-

terial to be formed than would be predicted by the

conversion if only crosslinking was occurring. Fig. 1 gives

a visual representation of how crosslinking and cycliz-

ation a!ect the subsequent swelling of the polymer

network. When a polymer system is more highly crosslin-

ked, the overall structure is more tightly held together,

adding rigidity and enhanced mechanical strength, while

reducing the subsequent swelling as shown in Fig. 1a.

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When a polymer is more cyclized, as shown in Fig. 1b, the

overall network structure is "lled with ring structures,

and the backbone polymer chains are able to swell

further apart. The mesh size or molecular weight between

crosslinks (Mc) is signi"cantly increased by primary cycl-

ization, leading to increased swelling and reduced

modulus as well as numerous changes in other network

properties. Evidence of cyclization is observed indirectly,as it cannot be explicitly measured experimentally. The

presence of cyclization in hydrogels is indicated by the

heterogeneity which is seen with small-angle neutron

scattering in a poly(acrylic acid) and methylene bisac-

rylamide copolymers (Moussaid, Candau, & Joosten,

1994). Further, the delayed gel-point conversion from

what is predicted by the classical Flory}Stockmayer the-

ory is evidence of the existence of signi"cant cyclization

reactions (Walling, 1945; Galina, Dusek, Tuzar, & Stokr,

1980; Dusek & Spevacek, 1980; Dusek, 1982; Boots,

Kloosterboer, & Hei, 1985). Similarly, primary cycliz-

ation has been used to explain the increased swellingratio, beyond what is predicted for an ideal network, that

is measured in polyacrylamide gels (Okay, Balimtas,

& Naghash, 1997).

In studying cyclization it is important to use both

experimental and modeling techniques. Numerical

approaches for studying cyclization abound in part

because of the di$culty determining cyclization rates

experimentally. Experimental techniques to measure

primary cyclization are limited to measuring the amount

of extractable, unreacted monomer or measuring the

mechanical properties, and deducing the degree of pri-

mary cyclization (Kloosterboer, 1988; Anseth, Bowman,

& Brannon-Peppas, 1996). Numerical models previouslydeveloped generally fall into three categories: statistical,

space-based simulations, and kinetic approaches. Statist-

ical models include work by Dusek and Ilavsky and

others (Flory, 1953; Stockmayer, 1943; Gordon, 1962;

Dusek & Ilavsky, 1975; Macosko & Miller, 1976; Miller

& Macosko, 1976; Dusek & Spevacek, 1980; Miller

& Macosko, 1988; Gordon & Malcolm, 1966) where

reaction probabilities control the growth of radical

chains. More recently, combined statistical and kinetic

models have been developed by Dusek and Somvarsky as

well as others to model the network formation of cross-

linked polymers (Dusek & Somvarsky, 1996; Luo, Weng,Huang, & Pan, 1997).

Space-based simulations (Monte Carlo, percolation,

kinetic gelation) which use a lattice structure to simulate

network formation have also been widely used over the

last two decades to model chain polymerizations

(Manneville & Seze, 1981; Boots & Pandey, 1984; Bansil,

Herrmann, & Stau!er, 1984; Kloosterboer, 1988; Simon,

Allen, Bennett, Williams, & Williams, 1989; Bowman

& Peppas, 1992; Anseth & Bowman, 1994; Chiu & Lee,

1995; Schroder & Oppermann, 1997). These simulations

are useful for modeling and describing the structural

evolution of highly crosslinked polymers in which the

e!ects of heterogeneity are prominent.

Additionally, work on kinetic models has been done by

Tobita and Hamielec and others (Tobita & Hamielec,

1988; Tobita & Hamielec, 1989; Okay, 1993; Okay, Kurz,

Lutz, & Funke, 1995; Naghash, Okay, & Yildririm, 1995;

Naghash, Yagci, & Okay, 1997) using a pseudokinetic

approach where the non-ideal spatial e!ects are generallyaveraged into the rate constants. Early models speci"-

cally on gel formation developed by Okay and Naghash

neglected cyclization (Okay, 1993; Naghash et al., 1995).

In later research, the rate constant for primary cycliz-

ation was assumed constant throughout the polymeriz-

ation (Okay et al., 1995).

Landin and Macosko developed a mathematically

simpler kinetic model that included cyclization (Landin

& Macosko, 1988) by using a proportionality factor for

the fraction of pendants that are consumed in primary

cyclization reactions. Cyclization is again assumed to

occur at a constant rate and model parameters for cycliz-ation and pendant reactivity are determined from experi-

mental data such that cyclization rates are approximated

rather than predicted (Dusek, 1998). To predict primary

cyclization rates more accurately, the variation of pen-

dant reactivity should be included. The model developed

for this work assumes pendant reactivity for cyclization is

controlled by the local radical concentration. Its "rst

principles approach also gives it the #exibility to predict

behavior in a large variety of experimental systems.

2. Computational methods

The numerical kinetic model has been described in

detail in a previous paper (Elliott & Bowman, 1999a,b).

In general, the model is unique because it develops and

solves the di!erential kinetic equations accounting for

the di!erence in reactivity of the pendant double bonds

spatially and during the polymerization. Monomeric and

pendant double bonds are tracked separately to capture

the local dynamics and reactivity of the pendant double

bonds. Calculation of the rate of consumption of mono-

meric double bonds is based on the kinetic expression

for a bimolecular collision, using the kinetic parameter

k times the concentrations of monomeric double bondsand radical species in bulk solution [R

]. The con-

centration of bulk radicals [R

] is calculated using the

pseudo-steady-state assumption. Once a multifunctional

monomer is consumed, a pendant double bond is cre-

ated, which can react either by crosslinking or cycliz-

ation. As shown in Fig. 2, both of these two mechanisms

of propagation of pendant double bonds (R

) are con-

sidered: the reaction of pendant double bonds with the

radical on the same propagating chain (local radicals) to

form cycles and the reaction of pendant double bonds

with bulk radicals to form crosslinks. Secondary cycles

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Fig. 2. Mechanism of monomeric- and pendant-double-bond reac-

tions.

Fig. 3. Radius containing the local radical for a particular pendant.

can also be produced, but for this work they are con-

sidered equivalent to crosslinks. The di!erence in reactiv-

ity of the two competing mechanisms is incorporated

into the apparent radical concentrations relevant to thecrosslinking and cyclization reactions.

The local radical concentration is the apparent con-

centration of the radical on the same propagating chain

for a speci"c pendant double bond. Pendant double

bonds that have existed for di!erent lengths of time each

have their own local radical concentration, which is

a function of when the pendant was created (birth time

[t

] and the current time. When the pendant is "rst

created, it is very close to the propagating radical and the

local radical concentration (and therefore the rate of

primary cycle formation) is high. Conversely, after long

times the local radical concentration diminishes dramati-

cally. To calculate the local radical concentration, a vol-ume is de"ned which includes both the pendant vinyl and

the propagating radical on the same kinetic chain that

initially formed the pendant as shown in Fig. 3. This

volume has a radius, which represents the distance be-

tween the pendant double bond and the radical and can

be calculated using statistics. The expression for the local

radical concentration [R(t, t

)], as function of time and

pendant birth time, including termination of local rad-

icals with bulk radicals is

[R(t, t

)]"exp[!k

[R

](t!t

)]

1N

[4/3(r

#C

n(t, t

)l)]

. (1)

Here, k

is the termination kinetic constant, N

is

Avogadro's number, r

is the monomer size, C

is the

characteristic ratio (Flory, 1969), n is the number of

carbon}carbon bonds between the pendant double bond

and radical, andlis the length of a carbon}carbon bond.

The assumptions included in this equation are the follow-

ing: (1) the distance the chain propagates can be cal-

culated using statistics assuming an unperturbed chain;

(2) the molecular size of the crosslinking agent (with the

pendant double bond on one end) and the length the

radical propagates can be combined additively to ap-

proximate the distance between the pendant and the

radical; and (3) local radicals terminate with bulk radicals

by a second-order bimolecular kinetic reaction.

As shown below in Eq. (2), the rate of pendant (Pen)consumption,R

, is the sum of the rate of reaction with

bulk monomer and local radicals. The consumption of

pendants by cyclization is evaluated for pendants of all

birth times and summed:

R

(t)"k

[ Pen(t)][R

]

#k

N

exp[!k[R

](t!t

)][Pen(t, t

)]

[4/3(r#C

n(t, t

)l)]

.

(2)

In Eq. (2) the kinetic constant for propagation that leads

to crosslinking and cyclization are k and k , respec-tively, which are both generally assumed to be equal to

k

, the propagation kinetic constant. Using these kinetic

expressions, the numerical model tracks the formation of

pendant double bonds and their subsequent reaction

with bulk or local radicals. In this manner we are able to

predict the extent of cyclization and crosslinking.

The useful quantity to calculate for comparison with

experimental data is the molecular weight between cross-

links. The molecular weight between crosslinks is de"ned

as the polymer density, , (total weight of polymer/vol)

divided by the moles of crosslinked chains per unit vol-

ume,v, as shown in Eq. (1):

Mc"

v. (3)

To"rst calculate the theoreticalMcfor an ideal crosslin-

ked network with complete conversion and no cycliz-

ation, the maximum number of crosslinked chains must

be known. The relationship between the concentration of

crosslinking agent and the moles of crosslinked chains

per unit volume, v can be generalized as the number of

double bonds (ndb) of the crosslinking agent times the


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concentration of crosslinking monomer, [M

] (Hasa

& Janacek, 1967):

v"ndb[M

]. (4)

For a loosely crosslinked system where the concentration

of crosslinking agent is much less then the concentration

of monovinyl monomer, the density will be approxim-

ately the initial monovinyl concentration times the mo-lecular weight of the monovinyl monomer. When the

weight fraction of the crosslinking agent is not negligible,

as in a more highly crosslinked system, the density of the

polymer network will be the initial double bond concen-

tration, [DB

] times the average molecular weight of

a repeat unit on a double-bond basis, Mr. Thus, the

theoretical Mc for the divinyl/monovinyl copolymeriz-

ation system like DEGDMA/OcMA will be the follow-

ing:

Mc"

Mr[DB

]

2[M] . (5)

Using the simulation results, Mc is determined as

a function of conversion, X, and the degree of crosslink-

ing for the non-ideal cases which include cyclization. The

polymer density in Eq. (3) is calculated based on the

concentration of double bonds that have been incorpor-

ated into the network, [DB

]X. This result assumes that

all monomers with at least one double bond reacted are

part of the network and contribute to the density, a rea-

sonable assumption at high conversions. The concentra-

tion of crosslinks, v, in the network will be a function of

both conversion and the extent of cyclization. Every fully

reacted divinyl molecule like DEGDMA will contributetwo crosslinked chains. Divinyl molecules that cycle or

have an unreacted pendant will form a linear structure

and will not contribute to the number of crosslinked

chains. The total concentration of crosslinked chains

formed for the divinyl copolymerization will then be

twice the number of pendants that react in crosslinking

reactions, [ Pen

], and Mc is calculated as follows

Mc"Mr[DB

]X

2[Pen

] . (6)

Several parameters are needed for the model to specify

the polymer system being simulated when determining

Mc. The molecular size of the crosslinking molecule, r

,

are input as 4.5 and 6.7 As to represent DEGDMA andPEG(600)DMA, respectively. These values are calculated

from the molecular weights of monomers, assuming

a spherical molecule with the double bonds on the radius.

The length of the carbon}carbon bond, l , is 1.54 As. Thecharacteristic ratio, C

, relates the mean-squared end-

to-end distance calculated for a freely jointed chain to the

mean-squared end-to-end distance for the actual unper-

turbed chain. The characteristic ratio is related to how

extended the chains will be in solution and varies with

solvent quality and the degree of solvation of the polymer

chains. The value ofC

is speci"c to the polymer com-

position and solvent used and extremely di$cult to de-

termine experimentally in crosslinked polymers. For

these reasons the value ofC

is "t to experimental data.

In the"rst experiments methanol was used with HEMA

copolymers. A characteristic ratio of 3.2 was used to "tthe model data to the experimental results. In the second

set of experiments with OcMA copolymers and hexanol,

5.9 was used for the characteristic ratio. For the simula-

tions with PAA, the characteristic ratio was set to 4.3.

The kinetic parameters, k

, k, k

, k

, all remain con-

stant throughout the simulation. The rate constant for

cyclization (k

) and crosslinking (k

) are assumed to be

equivalent to the kinetic constant for propagation (k

), as

the varying pendant reactivity is captured in the radical

concentrations. The chemical reactivity of the monovinyl

and divinyl crosslinking agent is assumed to be equiva-

lent. The values of the kinetic parameters for k andkwere taken from experimental data for HEMA for all

simulations (Goodner, Lee, & Bowman, 1997).

3. Experimental methods

The monomers used in the experimental work were

diethyleneglycol dimethacrylate (DEGDMA), polyethy-

leneglycol 600 dimethacrylate (PEG600DMA), hydroxy-

ethylmethacrylate (HEMA) and octyl methacrylate

(OcMA). DEGDMA and OcMA were purchased from

Polysciences (Warington, PA). PEG(600)DMA was

obtained from Sartomer (West Chester, PA) and HEMAwas purchased from Aldrich (Milwaukee, WI). The

monomers were used as received without additional

purifying or inhibiting. The solvent used with the

HEMA copolymers was methanol from Fisher (Fair

Lawn, NJ) and the photoinitiator was ,-dimethoxy--

phenylacetophenone (DMPA) from Ciby-Geigy (Haw-

thorne, NY). Solutions of HEMA containing 2 or 10 mol%

crosslinking agent (DEGDMA or PEG(600)DMA) as

well as 0, 50, or 80 vol% solvent were photopolymerized

with 0.1 wt% (relative to the monomers) DMPA using an

ultraviolet light source which operated at approximately

18 mW/cm

for 30 min. In the second set of experimentswith OcMA the solvent used was hexanol from Aldrich

(Milwaukee, WI). Analogous to the "rst set of

experiments, samples were created with 2 and 10 mol%

crosslinking agent (DEGDMA or PEG(600)DMA)

copolymerized with OcMA. Samples were thermally

polymerized with 1.0 wt% (relative to the monomers)

2,2-azobisisobutyronitrite (AIBN) at 703C for 90 min in

Te#on molds sealed with vacuum grease. Solutions were

made with 0, 20 and 50% solvent by volume.

Time}temperature scans were performed using a dy-

namic mechanical analyzer (DMA). A sinusoidal tensile


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Fig. 4. Model prediction of normalized pendant concentrations for 2%

DEGDMA/98% DEGDMA with 0% solvent, (**); 50% solvent,

(- - - - - -); and 80% solvent, ( } } -) (monomer size"4.5 As , lightintensity"18 mW/cm).

force was applied to the sample while raising the temper-

ature 53C/min to obtain the storage modulus of the

polymer system while in the rubbery region. By knowing

the storage modulus of the polymer, the average molecu-

lar weight between crosslinks,Mc, is calculated using the

following equation:

Mc"3E

. (3)

In the above equation, is the density of the polymer

system, is the temperature in Kelvin where the

modulus was obtained, andE is the storage modulus of

the polymer in the rubbery region. This equation is valid

assuming the material behaves as an ideal rubber, chain

ends can be neglected, i.e. the kinetic chain length is much

greater than the distance between crosslinks, and the

storage modulus is much greater than the loss modulus.

All of the assumptions should be valid for the systems

studied.

To determine the "nal conversion of the HEMAcopolymer samples, infrared (IR) spectra were obtained

between 4000 and 400/cm. For each polymer sample, the

peak area of the carbonyl group in the IR spectra was

compared to the peak area of the C"C (stretch) bond at

1637/cm. The ratio of these two areas was then nor-

malized by the ratio of the peak areas of these two bonds

in the monomer solution. The conversion of the system

can be directly obtained by comparing these ratios since

the C"O bond (1720/cm) is una!ected by the polymeriz-

ation reaction whereas the radical involved in the polym-

erization reaction propagates through the C"C bond.

IR experiments were performed on a portion of eachHEMA polymer sample both before and after the DMA

experiments to measure the amount of additional curing

that occurred during heating. Knowing the double bond

conversion is essential to compare the experimental

values for Mc with the values for Mc predicted by the

numerical model. The average molecular weight between

crosslinks strongly depends on the double bond conver-

sion. IR experiments veri"ed that OcMA samples achieve

nearly 100% conversion as polymerized.

4. Results and discussion

Using the kinetic simulations, DMA, and IR experi-

mental techniques, the e!ects of crosslinking agent

concentration, crosslinking agent size, and solvent

concentration on primary cyclization were investigated.

Numerical modeling of copolymerization in the presence

of varying amounts of solvent was performed. As the

numerical model tracks the consumption and creation of

each species as a function of time, the change in the

pendant reactivity with varying amount of solvent is

easily observed. Model results of the total pendant

double bond concentration as a function of polymeriz-

ation time are shown in Fig. 4 for 2% DEGDMA/98%

HEMA. Results were normalized by the initial cross-

linking agent concentration. The addition of solvent

decreases the amount of pendants that are building up by

increasing the rate at which they react away by cycliz-

ation. These results show how the model captures the

varying pendant reactivity that results from increasing

dilution of monomeric double bonds. With no solvent

present during the polymerization, the normalized

pendant concentration at 200 s is almost eleven times as

high as the normalized pendant concentration in the

polymerization performed in 80% solvent.The increased reactivity is caused by the dilution of

monomeric double bonds with the addition of solvent.

The rate of monomeric double-bond consumption

decreases with solvent as it is directly proportional to the

monomeric double bond and bulk radical concentration.

Similarly, the rate of pendant crosslinking also decreases

with the dilution of the radical concentration in the bulk

solution. Consequently, the propagation rate will

dramatically decrease with solvent addition because

monomer units are added more slowly. The local rate

radicals propagate away from the pendant double bonds

also decreases as the propagation rate is reduced. Thedistance between a radical and pendants on its chain will

increase less rapidly and the apparent local radical con-

centration will drop o! more slowly when solvent is

present. This phenomena is also illustrated in the two

pictures in Fig. 5. In Fig. 5a, where the growing radical is

surrounded by more monomer units when little or no

solvent is present, the radical is able to add repeat units

rapidly and does not have a signi"cant amount of time to

react with the pendant double bond to cyclize. The pen-

dant double bond will then have an increased chance of

crosslinking. In Fig. 5b, when solvent is present, the


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Fig. 5. Solvent concentration e!ect on cyclization.

Fig. 6. Integral fraction of reacting pendants which form primary cycles

at each time for DEGDMA with 0% solvent, (**); 50% solvent,

(- - - - - -); and 80% solvent (} } -); (monomer size"4.5 As , lightintensity"18 mW/cm).

Fig. 7. The e!ect of solvent on Mcfor DEGDMA/HEMAcopolymers.

Simulation data for 2% (), and 10% DEGDMA, (). Experimental

data for 2% (), and 10% DEGDMA () (monomer size"4.5 As, light

intensity"18 mW/cm). Theoretical Mc assuming no cyclization and

100% conversion for 2%, (**) and 10%, (- - - - -) crosslinking agent.

concentration of unreacted monomeric double bonds

will be diluted and the slowly growing radical chain will

have an increased chance of encountering the pendant

double bonds, causing more cyclization. Therefore,

primary cyclization, unlike crosslinking reactions, is

facilitated by increasing with solvent concentration. The

increased pendant reactivity with increasing solvent con-

centration during polymerization seen in Fig. 6 is at-

tributed to greater degrees of primary cyclization.

The simulation is also used to investigate the extent ofprimary cyclization for di!erent solvent amounts. The

numerical model can predict the fraction of pendants

that react by primary cyclization at each time during the

polymerization. Fig. 6 shows the model results for the

integral fraction of reacting pendants forming cycles as

a function of conversion for 2% DEGDMA/98%

HEMA with varying amounts solvent. The highest frac-

tion of pendant cyclization occurs at the beginning of the

reaction because of the limited propagation of radicals

away from newly created pendants. These model results

correlate well with recent experimental and modeling

data of Okay and coworkers (Okay et al., 1995; Naghash

et al., 1997). Their work found high cyclization rates at

low conversions for methyl methacrylate and ethylene

glycol dimethacrylate copolymers (Naghash et al., 1997).

As reaction time goes on, the fraction forming cycles

decreases. At higher solvent concentrations (lower mono-

mer concentrations) the pendant double bonds form

more cycles because the e!ective local radical concentra-

tion drops o!more slowly while the monomeric double

bond concentration is dramatically decreased by the sol-

vent. The fraction of pendant double bonds reacting

away by cyclization drops o!less steeply the more sol-vent that is added, showing higher cyclization rates

throughout the reaction and further con"rming why the

pendant double bond concentration does not increase

with higher solvent amounts in Fig. 4.

Primary cyclization cannot be measured directly to be

compared with the simulation data. As discussed

previously, one method for measuring the degree of

cyclization is to determine the mechanical properties of

the polymer, speci"cally the average molecular weight

between crosslinks (Mc). Experimental DMA and simu-

lation results for the Mc of 2 and 10% DEGDMA

copolymerizations with HEMA are shown in Fig. 7. Thetheoretical Mc, assuming no cyclization and 100%

double bond conversion, is also plotted. The experi-

mental results are higher than the theoretical, showing

that incomplete conversion and cyclization are preven-

ting the polymer from reaching its crosslinking potential.

Cyclization causes the distance between crosslinks to be

greater (higherMc) as pendants are consumed by cycliz-

ation reactions. For systems with both the 2 and 10%

crosslinking agent increasing the solvent amount in-

creases the Mc. As expected, the Mc is lower for the


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Fig. 8. The e!ect of solvent concentration on Mc for

PEG(600)DMA/HEMA copolymers. Simulation data for 2% (), and

10% PEG(600)DMA, (). Experimental data for 2% (), and 10%

PEG(600)DMA () (monomer size"6.7 As, light intensity"

18 mW/cm). Theoretical Mc assuming no cyclization and 100% con-

version for 2%, (**) and 10%, (- - - - -) crosslinking agent.

Table 1

Final conversion measured by IR for HEMA copolymers

Solvent (%) DEGDMA (%) PEG(600)DMA (%)

2 10 2 10

0 0.9 0.88 0.8 0.8

50 0.86 0.87 0.82 0.8280 0.75 0.7 0.65 0.62

Fig. 9. The e!ect of solvent onMc for DEGDMA/OcMA copolymers.

Simulation data for 2% (), and 10% DEGDMA, (). Experimental

data for 2% (), and 10% DEGDMA () (monomer size"4.5 As,

thermally polymerized). Theoretical Mc assuming no cyclization and

100% conversion for 2%, (**) and 10%, (- - - - -) crosslinking agent.

copolymer with 10% crosslinking agent because of the

increased crosslinking potential. Fig. 7 also shows that

model predictions simulated for the same "nal conver-

sion are consistent with the experimental results. Experi-

mental error associated with the measurement of the

rubbery modulus can contribute error of up to 20%.

Approximately 10% error is also associated with deter-

mining the "nal conversion of polymer with IR. Within

the error, the kinetic model is thus accurately predicting

the cyclization and crosslinking reaction in this

copolymer system. Results for PEG(600)DMA/HEMAcopolymers are similar (Fig. 8) to those for the DE-

GDMA crosslinked with HEMA. The experimental

values ofMc were greater than the theoretical for all but

one case. The deviation from the theoretical value again

demonstrates the importance of cyclization as the aver-

age Mc increases with increasing solvent amount. The

presented model data for Mc is at the same conversion

that was measured by the IR for each run. IR results for

DEGDMA and PEG(600)DMA are presented in

Table 1.

Similar trends are seen with the OcMA copoly-

mer crosslinked with DEGDMA in Fig. 9 and

PEG(600)DMA in Fig. 10. As before, the theoretical Mc

at 100% conversion assuming no cyclization is plotted

on the "gure along with the experimental and modeling

results. All Mc results are above the theoretical line,

indicating that cyclization is occurring. As OcMA reach-

ed near 100% conversion at these polymerization condi-

tions, the elevated Mc observed cannot be attributed to

incomplete conversion. Model and experimental results

again match very well within experimental error. Here,

error from di!erences in conversion is minimized.

When the DMA results of DEGDMA andPEG(600)DMA are compared at the same crosslinking

agent concentration, the e!ect of crosslinking molecule

size can be ascertained. Figs. 11 and 12 shows the results

for 2 and 10% crosslinking agent with OcMA. In both

"gures the higher molecular weight crosslinking agent

leads to a more crosslinked polymer. These results are

consistent with other researchers' where increasing cross-

linking agent size decreased swelling (Gonzales, Fan,

& Sevoian, 1996). Primary cyclization decreases with

larger crosslinking molecule size because the end-to-end

distance is larger; the pendant is further from the

propagating radical. As cyclization is most likely to occurwhen the pendant double bond has just been formed,

a change in the initial distance between the pendant and

the radical strongly a!ects cyclization.

In the two sets of experiments performed, model re-

sults have been shown to be consistent with experimental

measurements ofMc in polymer gels. The model is #ex-

ible enough to simulate a variety of monomers and their

properties. Besides predicting the degree of primary cycl-

ization and the Mc, the model can predict other impor-

tant properties of hydrogels that are related to primary

cyclization. For instance, the e!ect of cyclization on the


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Fig. 10. The e!ect of solvent concentration on Mc for

PEG(600)DMA/OcMA copolymers. Simulation data for 2% (), and

10% PEG(600)DMA (). Experimental data for 2% (), and 10%

PEG(600)DMA () (monomer size"6.7 As , thermally polymerized).

TheoreticalMc assuming no cyclization and 100% conversion for 2%,

(**) and 10%, (- - - - -) crosslinking agent.

Fig. 11. The e!ect of monomer size on Mc for a copolymer with 2%

crosslinking agent. Simulation data for 2% DEGDMA (), and 2%

PEG(600)DMA (), copolymerized with OcMA. Experimental data

for 2% DEGDMA (), and 2% PEG(600)DMA ().

Fig. 12. The e!ect of solvent on Mc for a copolymer with 10% cross-

linking agent. Simulation data for 10% DEGDMA (), and 10%

PEG(600)DMA (), copolymerized with OcMA. Experimental data

for 10% DEGDMA (), and 10% PEG(600)DMA ().

Fig. 13. Gel-point conversion prediction as a function of crosslinking

agent concentration and solvent concentration for PAA copolymerized

with DEGDMA. (rate of initiation"110mol/l s).

gel-point conversion and subsequent equilibrium swell-ing can also be predicted. As stated in the introduction,

the presence of cyclization was noted when the gel-point

conversion during polymerization was higher then

predicted by classical Flory}Stockmayer theory. In

Fig. 13 the model prediction for the gel-point conversion

as a function of solvent amount and crosslinking agent

concentration is shown in a three-dimensional plot for

polyacrylic acid (PAA) and DEGDMA with a rate of

initiation equal to 110mol/l s. The gel point in-

creases with solvent concentration due to the increase in

cyclization. The most dramatic increase in gel-point con-

version occurs at about 85% solvent, and at very high

solvent amounts (above 93%) no gelation occurs. The gel

point decreases with increasing crosslinking agent

concentration as expected because of the increased cross-linking.

The experimental and numerical results demonstrate

that monomer size, monomer concentration, and solvent

concentration during polymerization, all e!ect the cross-

linking reaction and the hydrogel properties. This result

has signi"cant implications for biomedical applications

where the mechanical, di!usion, and swelling properties

of the polymer are critical. The relationship betweenMc

and solvent concentration is especially important when

speci"c swelling properties are critical to an application.

Using the model predictions for Mc and M

for the 2%


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Fig. 14. Predicted swelling ratio of 2% DEGDMA/98% PAA

copolymer polymerized in varying amounts of solvent (rate of initia-

tion"110mol/l s).

DEGDMA/98% PAA system, the equilibrium swelling

ratio (volume swollen/volume dry) of the polymers was

predicted by the Flory}Rehner equation (Flory, 1953).

The chi factor was assumed to be 0.25 for a generic

thermodynamically good solvent. Results shown in

Fig. 14 demonstrate that increasing the amount of sol-

vent during polymerization increased the equilibrium

swelling ratio of the resulting polymer, especially at sol-

vent amounts over 60%. Increasing the solvent amount

from 60 to 80% changes the equilibrium swelling ratio

from 16 to 36. This di!erence in swelling ratio is caused

by the decreased crosslinking when the solvent concen-

tration during polymerization is increased.Mc increasedfrom 6000 to 16,000 g/mol when the solvent concentra-

tion present during polymerization was increased from

50 to 80 vol% solvent. The fact that the amount of

solvent present during polymerization changes the equi-

librium swelling has important implications on hydrogel

design.

5. Conclusions

The kinetic model and experimental results presentedprovide insight into the e!ect of solvent concentration,

crosslinking agent size, and crosslinking agent concentra-

tion on the extent of crosslinking and primary cyclization

during the photopolymerization of multifunctional

monomers. Simulation results using the kinetic model

demonstrate how pendant reactivity varies during the

polymerization. Adding solvent to the reaction increases

the probability of cycling, due to the diluted concentra-

tion of monomer, and slowed rate of polymerization

causing the local radical on its own chain to remain

longer in close proximity to pendant double bonds. In

both the modeling and experimental results, cyclization

rates change with monomer size and monomer concen-

tration (solvent concentration). Simulation results also

show that increasing the amount of solvent will increase

the cumulative amount of reacting pendants that are

consumed by cyclization. Further, because the fraction of

pendant double bonds that form cycles is highest initially

after it is formed, the size of the monomer strongly e!ectscyclization rates. With a smaller monomer, a pendant is

closer to the propagating radical that created it, and can

more readily react intramolecularly to form a cycle. The

model is able to predict experimental data quantitatively

for the e!ect of solvent concentration during polymeriz-

ation on the average molecular weight between cross-

links for DEGDMA and PEG(600)DMA copolymerized

with HEMA or OcMA. The agreement of the model

with experimental data validates the model as a

useful tool in studying polymerization of highly cross-

linked monomers, and the model can be used to predict

the gel point and swelling ratio of more typical PAAhydrogels. The increase in cyclization and the average

Mcis enough to change the predicted swelling properties

of the polymer for solvent amounts above 60% of the

volume. The solvent concentration during polymeriz-

ation and crosslinking agent concentration will also

change the gel-point conversion of polymer because of

varying degrees of cyclization. These changes are impor-

tant when designing hydrogels for speci"c biomedical

applications.

In conclusion, accounting for primary cyclization is

key to understanding the e!ects of solvent concentration

and comonomer composition on hydrogel crosslinking

density. Experiments and modeling show that greaterprimary cyclization results with more solvent and smaller

crosslinking molecules. Good agreement between the

model and experiments validate it as a valuable

predictive tool in determining the crosslinking density,

swelling, and gel-point conversion of polymer gels and

hydrogel systems.

Acknowledgements

The authors would like to acknowledge Jason Brown

for help with the DMA experiments, the Camille DreyfusTeacher-Scholar Program; National Institutes of Health

for its support through a research grant (DE10959-01A2);

and the Presidential Faculty Fellow Program at the

National Science Foundation for "nancial support.

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