Model-based prediction of the hydrodynamic radius of collapsed microgels and experimental validation
Introduction
Stimuli-responsive microgels are functional colloidal polymer networks that swell reversibly in reaction to outer stimuli like pH, temperature, or electric potential. This property makes them attractive for diverse applications such as drug delivery [1], switchable membranes [2], or phase separation [3]. The various applications require microgels of different radii; for example the release kinetics for drug delivery depends on the microgel size [4]. The microgel size is determined by the synthesis, as microgels are products by process [5], [6], [7], [8].
Pich and Richtering [9] describe the established batch synthesis of microgels. Commonly, microgels are synthesized in batch via radical precipitation polymerization. The monomers, cross-linker, and possibly surfactant are first dissolved in a solvent and heated to the reaction temperature of 330 K to 370 K. Then, an initiator is added to the solution that initiates the radical polymerization, active polymer chains start to grow, and precursor particles form. The coagulation of polymer chains in the solution and the incorporation of cross-linker units into the growing polymer chains enable the cross-linking of multiple chains to a microgel network. The effect of various synthesis conditions on the microgel radius has been explored experimentally. The most studied effects on the microgel radius are:
Surfactant Concentration: McPhee et al. [10] and Wu et al. [11] found that a higher concentration of the surfactant sodium dodecyl sulfate (SDS) reduces the radius of microgels based on N-isopropylacrylamide (NIPAM). The effect of the concentration of SDS on the microgel radius has since been confirmed by other authors [12], [13], [14]. Wolff et al. [15] investigated the effect of the surfactant hexadecyltrimethylammonium bromide (CTAB) on the microgel radius in a batch and a continuous polymerization reactor. They found that microgel radius decreases, when the surfactant is added to the polymerization. Nessen et al. [16] measured the effect of the concentration of CTAB on the radius and the swelling behavior of microgels based on NIPAM. They found that the microgel size decreases with an increase in the surfactant concentration, while the swelling properties remain almost unaffected.
Monomer concentration: Virtanen and Richtering [17] found that microgel radius can be increased by increasing the monomer concentration in the batch, as the number concentration of microgels does not change significantly with respect to the initial monomer concentration. Hence, microgels get larger with an increase in initial monomer concentration.
Cross-linker concentration: Wu et al. [11] investigated the effect of the cross-linker N,N’-methylenebisacrylamide (BIS) on microgels based on NIPAM and found that a higher concentration of cross-linker increases microgels radius. Balaceanu et al. [18] and Schneider et al. [19] measured the radius of microgels based on N-vinylcaprolactam (VCL) with respect to the concentration of cross-linker BIS using dynamic light scattering and small angle neutron scattering (SANS). They found that microgel radius increases with the amount of cross-linker added to the synthesis. Virtanen et al. [20] measured the effect of the cross-linker on the growth of microgels based on NIPAM using in situ SANS. They found that an increased cross-linker concentration decreases the concentration of microgels, possibly because of an increase in aggregation.
Initiator concentration: The effect of the initiator concentration on the microgel radius has been investigated by multiple authors, however there is no general agreement between the different investigations. Imaz and Forcada [21] reported that the microgel radius increases with an increase in the concentration of the initiator. Virtanen and Richtering [17] found that a change in the initiator concentration does not affect the microgel radius significantly. Chiu and Lee [22] reported that microgel radius decreases with a higher concentration of initiator.
Reactor temperature: Wu et al. [11] report that microgel radius decreases with an increase in reactor temperature. Their findings are supported by Virtanen and Richtering [17], who also investigated the effect of the reactor temperature on the microgel radius. However, the effect of the reactor temperature on the microgel radius is more prominent in the non-stirred system [17].
Reactor type / Reynolds number (Re): Virtanen and Richtering [17] investigated microgel growth in unstirred micro reactors and obtained reasonably large particles at low concentrations of monomer and initiator. Kather et al. [23] measured the dependence of the microgel radius on the Reynolds number () in different reactor concepts for surfactant-free polymerization. They found that for high Reynolds numbers microgel radius decreases and that microgels in the 50 nm range can be obtained.
While the effects of the synthesis conditions on the microgel size have been investigated experimentally, a mechanistic model describing the radius of microgels with respect to all of the above synthesis conditions has not yet been proposed. Available models of microgel synthesis focus on the kinetics and conversion of reactants during the microgel synthesis. Hoare et al. [24], [25], [26] investigated the incorporation of functional groups in microgels based on NIPAM and present kinetic parameters for the functional groups. Janssen et al. [27], [28], [29] investigated the kinetics of the polymerization of microgel based on VCL and NIPAM with the cross-linker BIS and modeled the microgel size distribution based on the population balance approach. However, the influence of the surfactant and Reynolds number on the microgel size were not taken into account. We presented a polymerization model for microgels based on VCL and NIPAM and performed a parameter identifiability analysis of the kinetic parameters [30]. We showed that for microgels based on VCL, the cross-linker BIS is expected to accumulate in the center of the microgels, because the cross-linking kinetics are faster than the homo-polymerization kinetics of VCL. However, for microgels based on NIPAM, the parameters describing the cross-linking kinetics were not identifiable based on the considered experimental data. Hence, while quantitative models of the kinetics of microgel polymerization are available, the microgel growth considering all of the above synthesis conditions has not yet been modeled quantitatively.
In the more general context of heterogeneous polymerization, the growth of colloidal systems has been modeled by multiple authors; commonly a model requires descriptions of the particle nucleation, growth, and coagulation. Two mechanisms for particle nucleation are commonly considered in emulsion polymerization: micellar nucleation occurring due to the surfactant concentration exceeding the critical micelle concentration, and homogeneous nucleation occurring due to the polymer chains exceeding the critical chain length [31], [32]. In microgel syntheses, the surfactant concentration is typically below the critical micelle concentration, which is why homogeneous nucleation is expected to be the dominant nucleation mechanism. Particle growth takes into account the propagation reactions within a growing particle, which depends on the average number of radicals per particle [33]. Three different cases are distinguished: , and . For microgel syntheses experimental investigations hint towards being large compared to unity [11], [34]; hence termination reactions within the growing microgels are expected to be slow in comparison to influx of new radicals into the microgel by diffusion or coagulation. Coagulation of particles is commonly assumed to be dependent on Brownian diffusion and inter-particle interactions, caused by van der Waals and electrostatic forces [35]. Based on this assumption, the coagulation rate can be determined with the Stokes–Einstein equation and the Fuchs stability ratio [36]; however, several model parameters are required for this approach, which are often unknown for little explored systems. In applications concerned with polymerization, a cut-off value for the particle radius is commonly determined based on available experimental data above which particles are assumed to be stable [29], [36], [37]. This limits the predictive capabilities of a model when it is applied to an alternate reactive system. Therefore, we introduce a novel semi-empirical approach towards the modeling of the coagulation rate that does not require the estimation of parameter specific to the system under investigation.
We present an extension of our previously presented kinetic model [30] to predict the hydrodynamic radius of microgels in the collapsed state. We account for the influence of the reactor type, cross-linker concentration, varying reactor temperature, surfactant concentration, and monomer concentration on the hydrodynamic microgel radius. First, we present the model of the hydrodynamic microgel radius with the modeling assumptions. We also explain model parameters and introduce values of the kinetic parameters. Second, we provide an overview of the available experimental data for the hydrodynamic radius of microgels synthesized under different experimental conditions. Additionally, we present new measurement data from a batch experiment with low initiator concentration that provides large microgels for model validation. Third, we compare model predictions with experimental data and discuss the obtained results. Last, we present final conclusions and give an outlook to future use of the proposed model.
Section snippets
Mechanistic model and model parameters
In this section, we introduce the model of microgel synthesis in two steps. First, we review the previously presented kinetic model of microgel polymerization that allows to quantitatively predict the conversion of individual reactants [30]. We review the important kinetic parameters of the VCL-BIS and NIPAM-BIS systems and introduce the required assumptions for their estimation based on conversion measurements presented by [11], respectively. Second, we present a new approach to model the
Experimental data
We review the available experimental data for the hydrodynamic radius of microgels for various reactor concepts. We group experiments from different authors by the used reactor concepts, as they affect the only parameter in the model to be estimated.
Results and discussion
We first present the results of the parameter estimation and identifiability analysis of the previously unknown kinetic parameters of the NIPAM-BIS system. Second, we present the determined parameter values for the coagulation parameter for the various considered reactor concepts and discuss the agreement of model predictions and experimental data. We also identify limitations of the proposed model that are revealed by comparison with the available experimental data and make suggestion
Conclusions
We presented a mechanistic model for the prediction of the dependence of the hydrodynamic radius of collapsed microgels on various synthesis conditions. The model accounts for microgel growth by propagation reactions and by coagulation with other polymer aggregates in the reactor. We model the coagulation rate based on the assumption that propagation reactions of active polymer chains are a necessary condition for the coagulation of microgels, as it enables their cross-linking. The model only
Acknowledgments
This work was funded by the German Research Foundation (DFG) within project B4 Microgel synthesis: Kinetics, particle formation and reactor modelling of the Collaborative Research Center SFB 985 Functional Microgels and Microgel Systems. We thank Franca Janssen, Leif Kröger, Michael Kather, Johannes Faust, Preet Joy, Walter Richtering, and Kai Leonhard for valuable discussions.
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