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(d) Nolan, C. M.; Serpe, M. J.; Lyon, L. A. Biomacromolecules 2004, 5, 1940−1946. [Crossref] ...... Madeline J. SimpsonBrandon CorbettAna ArezinaTod...
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J. Phys. Chem. B 2006, 110, 20327-20336

20327

Kinetic Prediction of Functional Group Distributions in Thermosensitive Microgels Todd Hoare* and Daniel McLean Department of Chemical Engineering, McMaster UniVersity, 1280 Main Street West, Hamilton, Ontario L8S 4L7, Canada ReceiVed: July 10, 2006; In Final Form: August 4, 2006

A kinetic model accounting for the copolymerization of up to four comonomers is applied to predict both chain and radial functional group distributions in carboxylic-acid-functionalized poly(N-isopropylacrylamide) (NIPAM)-based microgels. The model can accurately predict the experimentally observed radial distributions of functional monomers in microgels prepared using a variety of different carboxylic-acid-functionalized monomers with significantly different hydrophobicities, copolymerization kinetics, and reactivities, without requiring the use of adjustable parameters. Multimodal distributions can both be predicted and experimentally generated by copolymerizing two -COOH-containing monomers with widely different reactivities. Chain distributions and monomer block formation can also be probed using the kinetic model, allowing for qualitative predictions of the potentiometric titration behavior of the microgels. The kinetic model reported herein therefore provides the first available analytical method for semiquantitatively predicting and controlling functional group distributions in bulk-polymerized microgel systems.

Introduction Poly(N-isopropylacrylamide) (PNIPAM)-based microgels functionalized with reactive functional groups have great technological potential. Coupling the thermal phase transition response of PNIPAM with the chemical reactivity and pH-responsiveness of functional groups such as amines and carboxylic acids has permitted the design of a range of environmentally responsive devices such as sensors,1 rheology modifiers,2 and molecular delivery vehicles.3 The successful implementation of microgels in all these applications is contingent on the designer’s ability to control and predict the swelling responses of the microgel upon the application of a desired stimulus, responses which are strong functions of the distribution of the reactive functional groups within the three-dimensional microgel matrix. Indeed, our previous work showed that both the radial and the chain distributions of functional groups have significant impacts on the swelling4 and electrophoretic5 behaviors of functionalized microgels. Up to this point, the ability to predict or control functional group distributions in gel networks has largely been limited to cases in which a priori knowledge of the distribution was known (i.e., the work of Kokufuta et al. using a poly(acrylic acid)based macromonomer to fix a desired chain distribution in a macrogel6) or a multistep polymerization approach was used to enforce a specific outcome (i.e., Jones and Lyon’s work preparing core-shell microgels to fix a desired radial distribution7). Given the comparatively cumbersome nature of these synthetic procedures compared to the classical bulk polymerization method used for microgel preparation,8 the development of a method for predicting and thus designing functional group distributions in bulk-polymerized PNIPAM microgels would be greatly beneficial. In our previous work,4 we observed a correlation between the radial functional group distribution in functionalized mi* Author to whom correspondence should be addressed. Phone: (905) 525-9140 ext. 27342. Fax: (905) 528-5114. E-mail: [email protected].

crogels and the copolymerization kinetics of the functional monomer: The slower the functional monomer reacts relative to NIPAM, the more localized the functional groups are on the microgel surface. Such a relationship has also been suggested to control the radial distribution of the N,N-methylene(bis) acrylamide (MBA) cross-linker typically used to prepare PNIPAM-based microgels. Kinetic studies have suggested that MBA reacts faster than NIPAM under microgel polymerization conditions;9 by extension, MBA has been proposed to be localized in the core of the microgel as opposed to the surface despite its significantly higher hydrophilicity compared to NIPAM.10 This apparent correlation between polymerization kinetics and radial functional group distributions is physically realistic given the mixed precipitation-emulsion polymerization mechanism used for microgel preparation.11 Since all monomers used in the gel preparation are water-soluble, polymerization occurs primarily in the solution phase. This is consistent with the absence of an observed gel effect on the conversion rate of NIPAM in microgel synthesis compared to linear polymer synthesis.9 Given the dilute conditions used for microgel preparation (∼1 wt %), low-concentration solution polymerization kinetics should therefore be largely applicable to predicting instantaneous chain compositions. Furthermore, as the NIPAM-rich chains grow, the elevated reaction temperature (70 °C) induces a thermal phase transition, causing the growing polymer to precipitate on the nucleated gel particles. Thus, a link may exist between the time of polymer chain formation in solution and its radial position within the growing microgel particles. Microgels therefore seem potentially well-suited to a kinetic approach to compositional distribution prediction, at least on a macroscopic level. Toward exploring this link between copolymerization kinetics and radial and chain monomer distributions, we have reported the development of an analytical four-component terminal copolymerization kinetics model to simulate the conversion profiles and chain distributions generated via the copolymeri-

10.1021/jp0643451 CCC: $33.50 © 2006 American Chemical Society Published on Web 09/22/2006

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Hoare and McLean

zation of NIPAM with a variety of functional comonomers.12 In this work, the observed link between copolymerization kinetics and radial functional group distributions in microgels is applied together with this kinetic model to predict functional group distributions in microgels. Such modeling work is also a key first step in the development of strategies to theoretically predict microgel microstructures under a range of different environmental conditions. The kinetic models allow for the estimation of the gel composition in the “fully collapsed”, dry state while gel swelling thermodynamic models could be used to extrapolate this local compositional information to predict microgel pore sizes and swelling responses to environmental stimuli. It must be emphasized that this work is not intended to fully capture the precise local chain morphologies in microgels13 or kinetically describe the precise nucleation/aggregation/ emulsion mechanism by which microgels are formed. Rather, this work aims to understand the macroscopic distributions of monomer residues in functional microgels and develop methods for predicting these distributions. Experimental Methods Kinetic Modeling Framework. Three- and four-component copolymerizations are modeled using a terminal copolymerization methodology to track the conversion of each monomer as a function of time.14 This model assumes that the penultimate monomer residue on a growing chain has only a minimal impact on the reactivity of the chain end radical and that the chains are sufficiently long that termination and initiation do not significantly influence the composition of the polymer, both reasonable assumptions for these acrylamide copolymer systems.15 For a given monomer component i, radical mass balances can be written using eq 1

d(VRi) dt

)

∑j kji Rj Ni - ∑j kij Ri Nj

where i,j ) 1,2,3,4 (1)

where Ni is the concentration of monomer type i (NIPAM - i ) 1; MBA - i ) 2; functional monomer #1 - i ) 3; functional monomer #2 - i ) 4), V is the reaction volume, Ri is the concentration of radicals of type i, and kij is the rate constant of monomer j propagating with radical i. The steady-state hypothesis [d(VRi)/dt ) 0] can then be applied to generate a system of four equations in four unknowns. For the threecomponent case, N4 ) 0 and the radical fractions φi can be expressed according to eqs 2-4

φ1 )

φ2 )

( (

k21k31 f12 + k21k32 f1 f2 + k23k31 f1 f3 k21k31 f12 + k21k32 f1 f2 + k23k31 f3 f1 + k12k31 f2 f1 + k12k32 f22 + k13k32 f3 f2 + k12k23 f2 f3 + k13k21 f3 f1 + k13k23 f32 k12k31 f2 f1 + k12k32 f22 + k13k32 f3 f2 k21k31 f12 + k21k32 f1 f2 + k23k31 f3 f1 + k12k31 f2 f1 + k12k32 f22 + k13k32 f3 f2 + k12k23 f2 f3 + k13k21 f3 f1 + k13k23 f32

) )

(2)

(3)

φ3 )

(

k12k23 f2 f3 + k13k21 f3 f1 + k13k23 f32 k21k31 f12 + k21k32 f1 f2 + k23k31 f3 f1 + k12k31 f2 f1 + k12k32 f22 + k13k32 f3 f2 + k12k23 f2 f3 + k13k21 f3 f1 + k13k23 f32

)

(4)

The parallel, explicit 64-term solution to the four-component model is derived in our previous work and will not be reproduced here for simplicity.12 Monomer consumption and polymer fraction expressions can then be written for each monomer component i using the pseudo-kinetic rate constant approach,16 as shown in eq 5

dNi dPi )) κi Ni Rtotal i ) 1,2,3,4 dt dt

(5)

Here, the pseudo-kinetic rate parameter κi ) ∑j kjiφj (i,j ) 1,2,3,4) and Rtotal is the total concentration of radicals in the system (equal to ∑j Rj, j ) 1,2,3,4). Conversion versus time profiles acquired based on the concurrent integration of the system of monomer and polymer concentration equations for each component i (MathCad 13, Mathsoft) can be converted to radial functional group distributions by assuming that the particles are formed by sequential aggregation of newly polymerized polymer chains onto the growing polymer particles. Polymer chains formed first in solution are assumed to be localized in the core of the particle with polymer chains formed later in the reaction sequentially incorporated at larger radial distances from the particle core. The monomer concentrations calculated as a function of conversion are averaged at intervals of 5% NIPAM conversion, effectively dividing the microgel into 20 concentric shells of defined compositions. The dry volumes of each shell can be estimated according to the predicted monomeric composition of each shell, allowing for the calculation of the average functional group density (moles of monomer/unit volume) as a function of radial distance. The average distance between two charged monomers can be estimated based on the average monomer compositions within each of the shells. The mole fractions of each monomer within a shell are used to estimate the average number of NIPAM or MBA monomers incorporated between functional monomer residues using the expression

(

dCC ) bo

)

n1 + n 2 + n 3 n4

(6)

where ni is the number of moles of each monomer component incorporated into the volume considered and bo is the effective length of a monomer unit, equal to the x-axis projection of two carbon-carbon bond lengths (taken as 2.55 Å as per Katchalsky and Michaeli’s average estimate for vinylic polymers17). Alternately, the average volumetric spacing between charged monomers can be calculated based on this shell-specific radial functional group density. The equivalent radius required to pack the model-estimated number of charged sites into a close-packed configuration within the calculated shell volume can be estimated using the equation

3 1/3 d ) 2rsphere ) 2 ΦVc 4π

(

)

(7)

where d is the average charge-charge separation distance, Φ is the packing fraction of equivalent spheres (0.74 in their closest-packed configuration), and Vc is the volume per unit

Kinetic Microgel Modeling

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TABLE 1: Initial Concentrations of Monomers, Initiator, and Surfactant Used for Both the Experimental Synthesis of the Copolymer Microgels and the Model Predictionsa functional monomer microgel

NIPAM (mol/L)

MBA (mol/L)

#1 (mol/L)

#2 (mol/L)

SDS (mol/L)

APS (mol/L)

particle size (nm) (pH 10, 1 mM KCl)

AA-NIPAM MAA-NIPAM FA-NIPAM MA-NIPAM VAA-NIPAM AM-NIPAM MAA-FA-NIPAM AA-VAA-NIPAM

0.077 0.077 0.077 0.077 0.077 0.077 0.077 0.077

0.0041 0.0041 0.0041 0.0041 0.0041 0.0041 0.0041 0.0041

0.0049 0.0049 0.0043 0.022 0.022 0.0049 0.0025 0.0025

0 0 0 0 0 0 0.0022 0.011

0.0011 0.0011 0.0011 0.0011 0.0011 0.0011 0.0011 0.0011

0.0027 0.0027 0.0027 0.0027 0.0027 0.0027 0.0027 0.0027

428 ( 7 379 ( 3 442 ( 4 448 ( 9 396 ( 7 295 ( 10 399 ( 5 418 ( 7

a

The particle sizes of the resulting microgels are also included for comparison.

charge in the microgel, equal to the total volume of the shell divided by the total number of charged monomers incorporated within the shell. Chain functional group distributions can also be approximated by estimating the average length of monomer blocks according to the probabilistic relationship of eq 818

1 1 Vavg4 ) ) 1 - p44 p41 + p42 + p43

(8)

where Vavg4 is the average number of consecutive monomer 4 units incorporated into the polymer and pab is the probability of monomer b reacting with a polymer radical capped with monomer a, defined by

pab )

kabNaNb

∑b kabNaNb

b ) 1,2,3,4

(9)

The kinetic-model-calculated average monomer concentrations in each shell can be used in conjunction with this model to estimate p44. Average block lengths for both functional monomer units and NIPAM units can be calculated using this strategy. Microgel Preparation. Microgels were prepared by aqueous free radical polymerization according to the recipes given in Table 1. N-Isopropylacrylamide (NIPAM), the N,N-methylene(bis)acrylamide (MBA) cross-linker, one or two of the functional monomers acrylic acid (AA), methacrylic acid (MAA), acrylamide (AM), vinylacetic acid (VAA), fumaric acid (FA), and maleic acid (MA), and a small quantity of sodium dodecyl sulfate (SDS) surfactant were dissolved in 150 mL of Millipore water and heated to 70 °C under N2 purge. The functional monomer loading was adjusted using a trial-and-error method until a total of 6.5 mol % functional monomer is present in the bulk of each functionalized microgel, as measured via independent potentiometric and conductometric titrations (ManTech Burivar II automatic titrator).19 The ammonium persulfate (APS) initiator is then injected in a 10 mL Millipore water solution and the polymerization is carried out overnight. Suspensions are purified via successive ultracentrifugation to remove residual monomer and surfactant prior to analysis. Model Parameters. Table 2 lists the experimental copolymerization ratios20-24 and homopolymerization rate constants25-30 used in the model calculations in this work. Functional monomer copolymerization data is reported in aqueous solvent under acidic conditions, consistent with the acidic reaction conditions used experimentally to prepare the microgels. All other twocomponent reactivity ratios required for modeling are estimated using the Price-Alfrey scheme.31 Homopolymerization rate constants are normalized to the 70 °C experimental reaction

TABLE 2: Literature Copolymerization Ratios r12 ) k11/k12 and r21 ) k22/k21 for Pairs of Monomers Used in Microgel Synthesis and Homopolymerization Rate Constants k22 for Each of the Monomers Used for Microgel Preparation monomer 1 AM AM AM AM AM AM AM AM

monomer 2

r12

AA (pH4) 0.57 MAA (pH4) 0.2 VAA(pH4) 16.7 FA (pH4) 7.0 MA (pH4) 12.8 NIPAM 0.95 MBA 0.57 AM

r21 0.32 2.8 0.002 0.09 0.01 1.04 3.4

k22 ref (L/(mol s)) 20 20 21 22 22 23 24

182000 122000 180 100 0 93100 120000 131000

ref 26 27 21 28, 29 30 25 9 23

temperature using the Arrhenius equation. Of particular note, FA, MA, and VAA all react very slowly compared to the acrylamide-based NIPAM and cross-linker monomers. This is consistent with the experimental observation that significant excesses of these monomers must be added to the bulk copolymerization mixture to achieve a functional monomer loading of 6.5 mol %.19 While AA, MAA, and AM react essentially quantitatively with NIPAM and MBA, molar excesses of 1.7 × FA, 4.4 × VAA, and 8.8 × MA are required to achieve the target functional group loadings with the slowerreacting monomers. Microgel Characterization. Transmission electron microscopy (TEM) images were obtained using a JEOL 1200EX TEMSCAN microscope operating at 80 kV. Microgels were suspended in a 1 mM uranyl acetate solution for 2 h; uranyl acetate binds to anionic sites in the microgel to selectively stain the -COOH groups. A single drop of a stained 0.05 wt % microgel suspension was dropped on a Formvar-coated copper TEM grid and dried overnight. The resulting stained TEM images were subjected to image analysis using Matlab. For any given microgel image, the microgel-solution interface is defined manually, allowing for computation of the particle centroid. The program then extrapolates 8 rays (separated by 45°) out from the centroid, subdivides each ray into 25 units, and records the average pixel intensity (on the 0-255 black-white-gray scale) within each of these units. This analysis is repeated for 20 different particles within each TEM image; reported results represent the average pixel density of all rays measured for a given microgel. Error bars represent the standard error of the pixel intensities. Particle sizes were measured via dynamic light scattering using a Lexel argon ion laser operating at 100 mW and a BI-9000 autocorrelator (Brookhaven Instruments Corporation). Electrophoretic mobilities were recorded in pH 10, 1 mM KCl solutions using a ZetaPlus instrument (Brookhaven Instruments Corporation) operating in PALS mode. Error bars represent the standard error of 10 replicates.

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Figure 1. Radial density profiles of functional monomer and crosslinker in a microgel containing acrylic acid (AA): (a) kinetic model predictions of cross-linker (dashed line) and carboxylic monomer (solid line) distributions; (b) experimental carboxylic monomer distribution determined via image analysis of the dry-state TEM image in which functional groups are selectively stained with uranyl acetate (c).

Results and Discussion Radial Functional Group Distributions. The radial distribution of functional groups in microgels prepared using one functional monomer can be predicted using the three-component kinetic model to track the relative conversions of NIPAM (N1), MBA (N2), and the functional monomer (N3). This model is the most directlyapplicablesimulationforunderstandingtheradialfunctional group distributions since monomer consumption is directly related to the polymer composition within any conversion range. Figures 1-6 show the radial density predictions for each of the monoacid-functionalized microgels prepared for this work. The model results are given both graphically (panel a in each of the figures) and diagrammatically via the radial density plots (the inset images in panel a). The radial density plots represent the model results by varying the pixel density of the image; the maximum calculated shell density of -COOH groups in any the microgels (3000 mol/m3) is represented by 100% black shading while zero -COOH density is represented by 100% white shading. Corresponding experimental results are also provided in Figures 1-6, both in terms of the raw TEM images (panel c in each figure) and graphs reporting the gray scale pixel intensity across the diameters of the imaged particles, detected via image analysis (panel b in each Figure). Pixel intensity is plotted on a

Hoare and McLean

Figure 2. Radial density profiles of functional monomer and crosslinker in a microgel containing maleic acid (MA): (a) kinetic model predictions of cross-linker (dashed line) and carboxylic monomer (solid line) distributions; (b) experimental carboxylic monomer distribution determined via image analysis of the dry-state TEM image in which functional groups are selectively stained with uranyl acetate (c).

dimensionless scale since the different background lighting in each of the TEM images prevents relevant comparisons between the absolute gray scales of the different microgel images; however, the y-axis in each graph spans an 80-pixel range such that the relatiVe changes in pixel intensity in the different microgel results in Figures 1-6 can be quantitatively compared. Functional Group Distributions. The kinetic model performs extremely well in terms of predicting the experimentally identified radial functional group distributions. The modelpredicted radial density plots consistently give qualitative matches with the relative gray scale shadings observed in the TEM images. Furthermore, the graphical radial functional group profile defined by TEM image analysis consistently mirrors the kinetic prediction. Indeed, semiquantitative predictions appear to be possible given the correlation observed between the relative changes in pixel density as a function of radial distance and -COOH density predictions from the kinetic model. It must be emphasized that no fitting parameters are used to generate these model results; the predictions are generated based solely on the experimentally measured copolymerization ratios and homopolymerization constants and the defined initial monomer concentrations used to prepare the microgels.

Kinetic Microgel Modeling

Figure 3. Radial density profiles of functional monomer and crosslinker in a microgel containing vinylacetic acid (VAA): (a) kinetic model predictions of cross-linker (dashed line) and carboxylic monomer (solid line) distributions; (b) experimental carboxylic monomer distribution determined via image analysis of the dry-state TEM image in which functional groups are selectively stained with uranyl acetate (c).

Two small differences consistently observed between the experimental image analysis and theoretical modeling results require special comment. First, image analysis generally yields a somewhat broader, more continuous -COOH profile than does the kinetics model. This difference is likely attributable to small errors in the experimental copolymerization ratios applied in the model, particularly for the order-of-magnitude slowerreacting FA, MA, and VAA monomers that give the largest deviations between experiment and theory in this context. The baseline uranyl acetate uptake by the anionic sulfate initiator end groups will also increase the pixel density in regions of the TEM image containing little functional monomer. Second, in particles containing highly surface-localized carboxylic acid groups (VAA-NIPAM, FA-NIPAM, and MA-NIPAM), the kinetic model predicts a continually increasing -COOH density up to the particle-solution interface while image analysis predicts a slight plateau for the -COOH density in the outer shells. This discrepancy is an artifact related to the difficulty inherent in defining the “real” particle interface when performing the image analysis on stained TEM images. This is particularly problematic with microgels given that the lightly cross-linked, “hairy” chains near the microgel surface dry on the TEM grid

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Figure 4. Radial density profiles of functional monomer and crosslinker in a microgel containing fumaric acid (FA): (a) kinetic model predictions of cross-linker (dashed line) and carboxylic monomer (solid line) distributions; (b) experimental carboxylic monomer distribution determined via image analysis of the dry-state TEM image in which functional groups are selectively stained with uranyl acetate (c).

to form irregular interfaces which are difficult to define. Consequently, some of the rays extending from the centroids of the microgel images extend slightly beyond the actual periphery of the particle, skewing the accuracy of the image analysis results in the near-surface region. This effect is demonstrated experimentally by the significantly higher error bars observed at r/ro > 0.9. However, neither of these effects significantly alters the macroscopic predictive power of the model for estimating radial functional group distributions in microgels. Cross-Linker Distributions. In general, the cross-linker MBA reacts faster than NIPAM (consistent with the experimental data of Wu et al.9) and is thus predominantly localized in the core of each of the functionalized microgels. This is consistent with experimental small-angle neutron scattering and light scattering observations reported for microgels prepared using a similar synthetic approach.10,32,33 However, in copolymerizations that require the addition of large molar excesses of functional monomer to achieve the desired functional group loading, MBA is consumed progressively slower relative to NIPAM as the reaction proceeds and the monomer mixture becomes enriched with excess functional monomer.12 The result of this kinetic effect is evident in the near-surface region of the

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Figure 5. Radial density profiles of functional monomer and crosslinker in a microgel containing hydrolyzed acrylamide (H-AM): (a) kinetic model predictions of cross-linker (dashed line) and carboxylic monomer (solid line) distributions; (b) experimental carboxylic monomer distribution determined via image analysis of the dry-state TEM image in which functional groups are selectively stained with uranyl acetate (c).

predicted radial density profiles of the microgels shown in Figure 7. In each of the copolymerizations in which functional monomer is present in a significant excess (FA, VAA, and MA), the cross-linker content increases in the outermost shell. The magnitude of this increase corresponds directly to how much excess functional monomer is present and, by extension, to what degree the MBA conversion is slowed relative to that of NIPAM. FA-NIPAM (monomer excess of ∼70%) shows a slight increase in cross-linking at the surface, while VAA-NIPAM (∼340% excess) and MA-NIPAM (∼780% excess) show much larger increases. These apparently different cross-linker densities at the microgel surface are expected to have significant impacts on the swelling responses of the microgels. This is particularly true in the FA-, MA-, and VAA-functionalized systems since the additional cross-linker is present in the same localized nearsurface region as the majority of the functional monomers that predominantly drive swelling in polyelectrolyte hydrogels. Indeed, preliminary light scattering data indicates that FANIPAM (which contains a lower surface cross-link density) swells 40% more upon ionization than MA-NIPAM (which

Hoare and McLean

Figure 6. Radial density profiles of functional monomer and crosslinker in a microgel containing methacrylic acid (MAA): (a) kinetic model predictions of cross-linker (dashed line) and carboxylic monomer (solid line) distributions; (b) experimental carboxylic monomer distribution determined via image analysis of the dry-state TEM image in which functional groups are selectively stained with uranyl acetate (c).

Figure 7. Near-surface cross-linker concentration profiles for each of the functionalized microgels

contains a higher surface cross-link density) despite the apparently similar radial and chain functional group distributions in these two diacid-functionalized microgels. This comparison is

Kinetic Microgel Modeling

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TABLE 3: Electrophoretic Mobilities of Carboxylic-Acid-Functionalized Microgels at pH 10 in 0.001 M KCl Suspensions microgel

electrophoretic mobility (× 10-8 m2/(V s))

H-AM-NIPAM MAA-NIPAM AA-NIPAM VAA-NIPAM FA-NIPAM MA-NIPAM

-0.72 ( 0.03 -0.99 ( 0.04 -1.47 ( 0.04 -1.87 ( 0.03 -2.17 ( 0.05 -2.50 ( 0.03

facilitated by the statistically identical particle sizes observed for the two swollen microgels, as shown in Table 1. The kinetic model predictions (relating to the radial distribution of charges in the “dry” state) are also consistent with the general trends observed in the swollen state. Electrophoretic mobility data for each of the functionalized microgels in the fully ionized state are shown in Table 3. In general, microgels predicted to be primarily surface-functionalized (VAA-NIPAM, FA-NIPAM, and MA-NIPAM) have significantly higher electrophoretic mobilities than microgels predicted to be primarily core-functionalized (MAA-NIPAM and AA-NIPAM). Thus, the kinetic predictions are relevant for qualitatively predicting functional group distributions in both the swollen and the collapsed states. The data in Table 3 also provide insight regarding the comparative near-surface cross-linker densities in the different microgels. For soft particles such as microgels, the electrophoretic mobility depends on both the density of charges at the surface of the microgel and the electrophoretic softness, which quantifies how freely draining the polymer chains are in the near-surface electrophoretic region of the microgels.5 The softness parameter is related to both the water content and the cross-linking density near the microgel surface. The higher the cross-linker and/or functional monomer concentration at the surface, the less free-draining the gel will be to electrolyte solutions, reducing the softness and increasing the absolute electrophoretic mobility. The statistically identical particle sizes of the MA-NIPAM and FA-NIPAM microgels (Table 1) facilitate direct comparisons between these microgels in terms of the relative softness of their electrophoretic shells. While MANIPAM and FA-NIPAM both contain similar, highly surfacelocalized charge distributions, the local increase in the crosslinking density at the MA-NIPAM surface suggested by the kinetics model would make the MA-NIPAM surface less soft and account for its higher observed absolute mobility. Therefore, even accounting for the minor differences observed between theory and experiment, these data suggest that kinetics can be used to predict the macroscopic radial functional group distributions in microgels with a reasonable degree of qualitative and even semiquantitative accuracy. Bimodal Radial Distributions. Given the direct correspondence between kinetics and functional group distributions in microgels prepared with a single carboxylic-acid-containing monomer, more exotic gel morphologies could be created simply by manipulating the kinetics of the functional monomer. To test this hypothesis, we prepared microgels containing two different -COOH-containing monomers of widely different kinetic reactivities and used the four-component kinetic model to predict the expected functional group distributions. Figures 8 and 9 show that bimodal -COOH distributions can be both predicted and achieved experimentally using a two-monomer strategy. The experimental TEM images and image analysis results closely correspond to the kinetic-model-predicted radial functional

Figure 8. Radial density profiles of functional monomer and crosslinker in a microgel containing equal amounts of acrylic acid (AA) and vinylacetic acid (VAA): (a) kinetic model predictions of crosslinker (dashed line) and carboxylic monomer (solid line) distributions; (b) experimental carboxylic monomer distribution determined via image analysis of relative pixel density from the dry-state TEM image in which functional groups are selectively stained with uranyl acetate (c).

monomer densities. When one very fast-reacting (MAA) and one very slow-reacting (FA) monomer are used (Figure 8), the bimodal distribution is very well-defined; as the reactivity gap between the monomers is narrowed (i.e., AA and VAA, Figure 9), the bimodality becomes less pronounced. This result suggests that the kinetic model may be used in conjunction with experimental kinetic data to design microgels with targeted functional group distributions and, by extension, specific swelling and surface properties by mixing defined quantities of two (or more) different monomers with different kinetic reactivities but the same chemical functionality. This approach may be particularly useful in uptake and release applications given that well-defined bimodal pore size distributions may be designed. Chain Distributions. The kinetic model results can also be applied to estimate the chain distribution of different monomers within microgel subchains based on the average block length calculation in eq 8. This information is of practical importance since the clustering of -COOH-containing monomers influences both the effective pKa values of functional groups and the availability of reactive sites for conjugation reactions. The key chain distribution parameter of interest is Vavg4, the average length of functional monomer blocks. Figure 10 shows

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Figure 10. Correlation between the overall change in pKa over the full ionization range (from potentiometric titration) and the average functional monomer block length Vavg4 predicted by the kinetics model. The inset graph is an expansion of the monoacid region of the main figure.

Figure 9. Radial density profiles of functional monomer and crosslinker in a microgel containing equal amounts of methacrylic acid (MAA) and fumaric acid (FA): (a) kinetic model predictions of crosslinker (dashed line) and carboxylic monomer (solid line) distributions; (b) experimental carboxylic monomer distribution determined via image analysis of relative pixel density from the dry-state TEM image in which functional groups are selectively stained with uranyl acetate (c).

the correlation between the average functional monomer block length prediction from the kinetic model and the overall change in the effective pKa observed in the potentiometric titration of each of the functionalized microgels. The average block length was estimated by weight-averaging the Vavg4 values calculated for each of the individual shells with the percentage of the total number of functional monomers present within these shells. It must be emphasized that Vavg4 is only an average value; calculation of the full distribution yields nonzero probabilities for the formation of two, three, and four monomer blocks (with successively decreasing probabilities) within each of the functionalized microgels. A systematic correlation is observed between the average value of Vavg4 and the overall change in the apparent pKa over the full ionization range for each of the functional microgels: As Vavg4 increases, the overall ∆pKa also increases. This trend is physically reasonable since the clustering of functional groups in blocks will enhance the impact of the polyelectrolyte effect on microgel ionization. As the number of clustered functional monomers increases, both the magnitude of the pKa shift observed for a single functional group in the block and the

proportion of functional groups impacted by the polyelectrolyte effect increase. The shape of the correlation in Figure 10 can also be understood based on the polyelectrolyte effect. The clustering of functional groups in two-unit blocks as opposed to isolated entities has a much higher relative impact on the ∆pKa value than the subsequent formation of blocks containing three or more functional monomer units. By extension, the polyelectrolyte effect has a proportionately lower impact on inhibiting functional group ionization after at least one ionized functional group is present within the Bjerrum length (or, more specifically in this case, the fixed intracharge spacing within a functional monomer block) and the local electric field around the still-protonated monomer residue is perturbed. As a result, the slope of the ∆pKa versus Vavg4 correlation decreases systematically as Vavg4 increases. Thus, not only the radial functional group distribution but also the general potentiometric titration behavior of a functionalized microgel can qualitatively be predicted according to the kinetic model. The average charge separation distance is also important for understanding the ionization response of functionalized microgels. The closer the proximity of the charges in space, the more local charge-charge repulsion can occur upon ionization to drive changes in microgel swelling. The average charge separation distance can be predicted using two strategies: based on the average number of carbons between functional monomer sites (average chain separation, eq 6 and Figure 11a) or based on the average volumetric density of charges smeared over a given shell (average radial separation, eq 7 and Figure 11b). The dotted lines on both plots at 0.7 nm represent the Bjerrum length, an estimate of the distance over which two charges significantly interact. The general trends in both plots are related to the relative rates of reaction between the functional monomers and NIPAM. As the reactivity difference between the functional monomer and NIPAM increases, the difference between the maximum and minimum functional monomer separation distances also increases. For the average chain separation calculation (Figure 11a), the calculated chain functional monomer separation distances are typically less than the Bjerrum length, particularly within the shells containing the most functional monomer (i.e.,

Kinetic Microgel Modeling

J. Phys. Chem. B, Vol. 110, No. 41, 2006 20335 will be analyzed in a future publication in which gel swelling models are applied to the kinetic compositional results described in this paper to predict swelling in heterogeneous microgels. Conclusions (1) Monomer distributions in multicomponent microgel particles can be predicted by dividing the microgel into a finite number of shells containing the same amount of the primary monomer (NIPAM) and using a terminal copolymerization model to predict the amount of each monomer present in each shell. (2) Radial functional group distributions in carboxylic-acidfunctionalized microgels can be predicted via copolymerization kinetics modeling. TEM images in which -COOH groups are selectively stained can be reproduced by radial density diagrams constructed based on kinetic model predictions. This is the first analytical tool reported to enable semiquantitative prediction of functional group distributions in bulk-polymerized microgels. (3) Multimodal functional group distributions can both be achieved experimentally and predicted theoretically using the kinetics model when two different carboxylic-acid-containing monomers with different copolymerization kinetics are used to prepare a functionalized microgel. (4) Both the chain distributions of functional monomer (i.e., the average block length of functionalized microgels) and the average volumetric spacing between functional groups can be predicted and verified experimentally via ionization-driven swelling, electrophoretic mobility, and ∆pKa potentiometric titration methods.

Figure 11. Average separation distance of adjacent carboxylic acid functional groups in functionalized microgels: (a) calculated assuming that all charges present in each shell are uniformly distributed over the full volume of the shell; (b) calculated according to average separation distance between two adjacent carboxylic-acid-containing monomers along the length of a single polymer chain.

lower-numbered shells in MAA-NIPAM and AA-NIPAM and higher-numbered shells in FA-NIPAM, MA-NIPAM, and VAANIPAM). Thus, the charges are close enough in space such that direct charge-charge repulsion can drive intrachain repulsion between different subchains within the gel network to induce hydrogel swelling. The volumetric charge smearing approach (Figure 11b) gives the same general trend, although the average charge separation distances are significantly larger. In this case, the only shell in which the volumetric -COOH separation distance is less than the Bjerrum length is the outermost shell of VAA-NIPAM; correspondingly, VAA-NIPAM swells the most of any of the tested microgels upon ionization.4 This analysis, however, gives only a preliminary assessment of the effect of local functional group densities on gel swelling. Increasing the volumetric charge density in microgels has several competing effects: increasing the contribution of direct chargecharge repulsion (driving swelling), increasing the stiffness of the polymer subchains (limiting swelling), and reducing the osmotic coefficient and thus the effective degree of ionization (reducing the Donnan contribution to gel swelling). The osmotic coefficient is a particularly significant factor for interpreting swelling results for the diacid monomer-functionalized microgels that contain two -COOH groups covalently bound in close proximity on the same polymer network subchain. These effects

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