Synthesis of Poly(N-vinylcaprolactam)-Based Microgels by Precipitation Polymerization: Pseudo-Bulk Model for Particle Growth and Size Distribution.

Particle size distribution and in particular the mean particle size are key properties of microgels, which are determined by synthesis conditions. To describe particle growth and particle size distribution over the progress of synthesis of poly(N-vinylcaprolactam)-based microgels, a pseudo-bulk model for precipitation copolymerization with cross-linking is formulated. The model is fitted and compared to experimental data from reaction calorimetry and dynamic light scattering, showing good agreement with polymerization progress, final particle size, and narrow particle size distribution. Predictions of particle growth and reaction progress for different experimental setups are compared to the corresponding experimental data, demonstrating the predictive capability and limitations of the model. The comparison to reaction calorimetry measurements shows the strength in the prediction of the overall polymerization progress. The results for the prediction of the particle radii reveal significant deviations and highlight the demand for further investigation, including additional data.

Introduction

Poly(N-vinylcaprolactam) (PVCL)-based nano- and microgels are thermoresponsive, biocompatible, and easily functionalized.[1] In consequence, PVCL-based microgels are under investigation for versatile applications, ranging from microgel-coated membranes[2] to nanocoatings for biointerfaces.[3] However, each application demands a specific microgel size, typically from a few nanometers to several micrometers, with a narrow particle size distribution (PSD).[4] In conclusion, particle size distribution and particularly the mean particle size are important properties of microgels that need to be controlled by the synthesis operation.

Batch precipitation polymerization is a simple and common method for the synthesis of PVCL-based microgels and allows narrow particle size distributions.[5] All reactants are initially dissolved in a solvent, typically water. The thermal initiator, 2,2′-azobis(2-methylpropionamidine) dihydrochloride (AMPA), initiates the free radical cross-linking copolymerization reaction of VCL and the cross-linker, N,N′-methylenebisacrylamide (BIS). Dissolved oligomers collapse due to a lower solubility of oligomers with higher chain lengths and form precursor particles. At this stage, the homogeneous system with dissolved oligomers transfers into a heterogeneous system with a continuous aqueous phase and disperse polymer particles. The precursor particles continue to grow until the final microgels are obtained. With internal cross-linking by the incorporated cross-linker, stable microgels are obtained, which can undergo reversible swelling and collapse. Growth of the polymer particles thereby is a combination of several simultaneous mechanisms illustrated in Figure : polymerization by absorption of the monomer and cross-linker and chain propagation reactions in the microgels, entry and desorption of oligomers, as well as coagulation among the growing particles.

Figure 1

Illustration of different particle growth mechanisms contributing to microgel growth in a precipitation polymerization: nucleation (rate Rnuc), monomer absorption (partition coefficient DM), radical absorption and desorption (rates Re and Rdes), and particle coagulation (rate Rcoag).

The different growth mechanisms determine the characteristics of the particle size distribution. However, the contributions of the individual growth mechanisms are not fully understood yet. A better understanding of growth mechanisms will enable the targeted synthesis of predefined microgel sizes by a tailored process operation. For this purpose, process modeling and simulation is a valuable tool to complement experimental investigations.

Several experimental studies address the effects of different operation conditions of the PVCL-based microgel synthesis on the obtained microgel sizes. Imaz and Forcada (2008) provide a fundamental study of the impact of concentrations of monomer, cross-linker, initiator, and surfactant as well as temperature in emulsion polymerization. They report that increasing cross-linker and initiator concentrations lead to increasing microgel diameters, whereas an increase of the surfactant concentration results in smaller diameters.[6] In a consecutive study, the effects of different cross-linker types, BIS and the biocompatible cross-linker poly(ethylene glycol) diacrylate, and their respective concentrations on the particle size are compared.[7] Schneider et al. (2014) characterize the particle size in relation to the cross-linker concentration for precipitation polymerization, with particular focus on the radial heterogeneous cross-link distribution within the particles and the resulting swelling behavior.[8] These studies emphasize the multitude of influencing factors of the final microgel sizes.

So far, only a few models have been proposed for microgel synthesis. The focus of these contributions is rather the internal copolymer composition of microgels such as cross-linking density, whereas little attention has been paid to particle size (distribution). Hoare and McLean (2006) propose a kinetic model for the prediction of the internal microgel structure of poly(N-isopropylacrylamide) (PNIPAM)-based microgels. The study correlates the solution polymerization reaction rates of NIPAM with several comonomers with the local polymer composition[9] and employs this approach to further adjust the microgel internal copolymer composition.[10] With the assumptions of a monodisperse particle size distribution and the lacking onset of gelation, solution polymerization kinetics can be applied to describe a surfactant-free emulsion polymerization of PNIPAM-based microgels. Similarly, Acciaro et al. (2011) employ the solution polymerization rate constants of PNIPAM-based microgels to obtain information on the internal distribution as microgel growth is assumed to occur by polymerization on the surface of the particles.[11] Based on reaction rates of the solution copolymerization, a feeding strategy is determined to synthesize homogeneously cross-linked microgels. Likewise, assuming the solution polymerization mechanism, Virtanen et al. (2015) provide empirical equations to predict the particle number density and, hence, particle size of PNIPAM-based microgels depending on the monomer and initiator concentration.[12]

In contrast to these solution-polymerization-based models, a two-phase-based model for the precipitation polymerization of PVCL-based microgels was presented in our previous work.[13] The model distinguishes between an aqueous liquid and a polymer-rich gel phase, whereas the mass transfer among the phases is described by precipitation at critical chain length. Comparing the impact of the polymerization reactions in gel to the liquid phase, the gel phase is identified as the primary reaction locus. The growth of the gel phase related to the cross-linking reaction provides a prediction of the internal cross-linking distribution. In an enhancement, the model is adapted to PVCL–PNIPAM-based microgels for a coarse prediction of the internal comonomer distribution.[14]

All of the previous models for microgel synthesis share in their prediction of particle growth the assumption of a uniform particle size. The narrow particle size distribution is mechanistically explained by a short particle formation phase, after which the formed microgels are considered to grow evenly. However, this assumption lacks a mechanistic description of the microgel formation and, in particular, prevents the description of the period of precipitation and coagulation, the number of formed microgels, and their size and particle size distribution. This information can be captured by population balance equation (PBE) models.

PBE models are widely applied for emulsion polymerization.[15−18] They can be subdivided into 0-1 models and pseudo-bulk models based on the number of radicals present in a polymer particle. In 0-1 models, only polymer particles with either no or a single radical exist as the radical undergoes immediate termination in the presence of a second radical.[19−21] Hence, they are generally recommended for disperse systems with small particle sizes, unless the radical propagation is fast.[17,18]

In contrast, pseudo-bulk models have no limitation regarding the number of radicals. Instead, an average number of radicals for particles of the same radius is assumed. The assumptions of pseudo-bulk models are often controversially discussed due to lack of accuracy for small particles,[17] which has a large impact on the prediction of particle formation, and alternative modifications have been proposed to overcome this issue.[22] However, the pseudo-bulk model is the most general formulation and, hence, more suited to describe the entire conversion range.[18] Araujo et al. (2001) presented a pseudo-bulk model for copolymerization with coagulation. Therein, colloidal stability of the polymer particles is expressed in terms of stable particles, which are unable to undergo coagulation with other stable particles. Immanuel et al. (2002) provide a comprehensive first-principle pseudo-bulk model for the seeded emulsion polymerization of vinyl acetate with butyl acrylate. In addition to micellar nucleation, homogeneous nucleation due to precipitation is considered while coagulation is neglected.[23] In a subsequent contribution, they extend the pseudo-bulk model by coagulation under the impact of a sterical stabilizer.[24] Kiparissides et al. (2002) apply a pseudo-bulk model to describe the impact of oxygen on the particle size distribution of vinyl chloride-based latex particles.[25] Brunier et al. (2017) apply the pseudo-bulk model for emulsion polymerization.[26] The focus of their contribution is on ongoing diffusion limitation within the gel particles, beginning with the gel effect and leading to full stagnation of the polymerization by the emerging glass effect. The drawback of the lacking accuracy is avoided by neglecting the first interval of emulsion polymerization and coagulation, focusing on preexisting and stable particles. In summary, pseudo-bulk models are probably the most applied models for emulsion polymerization systems.

In this work, a pseudo-bulk polymerization model is employed for the synthesis of PVCL-based microgels. Several considerations argue in favor of the pseudo-bulk model for this purpose. First, Wu (1994) and Varga et al. (2001) concluded for NIPAM-based microgels that more than one radical can be present in a growing microgel.[27,28] Further, when the termination is not the rate-determining step in the overall polymerization progress, pseudo-bulk models are favorable over zero-one models.[17] The previously determined termination rate constants in the gel phase were low, as is usually the case with diffusion-controlled termination, and hence, polymerization is rate determining.[13,14] Finally, more explicit models such as 0-1-2, 0-1-2-3, or hybrid 0-1-pseudo-bulk models require population balances for more particle species, distinguished by their respective number of radicals.[22,29,30] This comes with high computational cost, especially when parameter estimations are performed. Hence, the pseudo-bulk model comes in handy for this first-time investigation.

From the well-established emulsion polymerization models, mechanisms such as the homogeneous nucleation by precipitation of oligomers can be adapted to describe precipitation polymerization. The following chapter presents the first pseudo-bulk model to describe precipitation copolymerization with internal cross-linking for the synthesis of PVCL-based microgels. The model uses well-established descriptions for the growth mechanisms’ radical entry, radical desorption, and coagulation. A parameter estimation is performed to determine the contributions of radical entry, desorption, and coagulation from experimental data. Therein, the central question is whether the model can equally be fitted to the experimental data from reaction calorimetry while simultaneously achieving the experimental particle size with a narrow particle size distribution. With the estimated parameter values, simulations for different initial initiator and cross-linker concentrations as well as reaction temperatures are performed and compared to experimental data to evaluate the qualification of the pseudo-bulk model for this purpose.

Modeling Particle Size Distribution for Precipitation Polymerization

Central to the pseudo-bulk model is the population balance equation (PBE)which describes the particle number density F(r, t) in moles as a function of the particle radius r and time t (cf. refs (15, 16)). For simplicity, time dependence of variables is not stated explicitly in the following. The time derivative of F(r, t) is a function of the velocity of radial particle growth v(r), nucleation with rate Rnuc(r), and coagulation among particles with rate Rcoag(r). Particle nucleation therein occurs exclusively at the nucleation radius rnuc, which is expressed by the Kronecker delta function δ. Integration of F(r, t) over the particle radius provides the absolute number of particles Nparwhere NA is the Avogadro constant and V is the volume of the reactor content.

The free radical copolymerization reactions assumed to take place in both phases are listed in Table . Therein, the basic assumptions concerning the polymerization reactions are based on our previous work.[13] The species initiator I, initiator radicals I•, monomers M, radicals R, pendant double bonds PDB, and dead polymer P refer to the respective phase (liquid phase l and gel particles g). The indices i, j correspond to the species of the monomer or terminal end of the radical (i, j = 1 for VCL and i, j = 2 for BIS), and n and m denote the length of the radical or polymer.

Table 1

Reactions of Free Radical Copolymerization Occurring in Both Phases (Liquid Phase l and Gel Particles g)a

reaction
initiator decompositionb
initiation
chain propagation
chain transfer to monomer
terminationc
cross-linking

The indices i, j correspond to the respective species of monomer or terminal end (i, j = 1 for VCL and i, j = 2 for BIS).

For the initiation reaction of initiator radicals and monomers applies the initiator efficiency fI.

Termination modeled by the disproportionation mechanism and kt = (ktkt)0.5.

Based on the reaction mechanisms above, the terms of the pseudo-bulk model are formulated in the following. First, the formulation of particle nucleation, represented by Rnuc in eq , is described as well as the consecutive monomer partitioning between the liquid phase and the recently formed particles. Then, particle growth due to ongoing polymerization, represented by v(r), and, in this context, the average number of particles with radical entry and desorption are described. Further, the formulation for the coagulation rate Rcoag in eq is established, before a population balance equation for the PDB of the cross-linker is introduced to account for the cross-linking reactions in the particles. Finally, the calculations of the reaction enthalpy transfer rate and properties of particle size distribution for the connection of the model and experimental data are described.

Particle Nucleation and Monomer Partitioning

The nucleation rate Rnuc in eq describes the formation of precursor particles. These particles are formed when the dissolved oligomers precipitate due to lower solubility at high chain length. Corresponding to the homogeneous nucleation in emulsion polymerization, this is expressed by the propagation beyond a discrete critical chain length with η repeating units (cf. refs (23, 31)). Hence, the nucleation rate is obtained bywith kpl as the propagation rate constant of monomers of type j with radicals with terminal end i, cRl as the concentration of radicals with length η, pl as the fraction of radicals with terminal end of species i, cMl as concentrations of monomers of species j, and Vl as the volume of the liquid phase. The derivation of pl and cRl based on the quasi-homopolymerization approach[32] and the quasi-steady state assumption is described in the Supporting Information SI I.

The nucleation radius rnuc is calculated assuming spherical precursor particles. Each precursor particle consists of one oligomer with η repeating units, each with the molar mass Mw,unit (cf. Supporting Information SI I, eq S13), and with a residue water mass fraction wWg.The density ρg of the microgel particle is derived from the polymer density ρP and the water density ρW under the assumption of incompressible volumes of water and polymer.The total volume of the reactor content V consists of the volumes of the two phases V = Vl + Vg. The volume of the gel phase Vg is calculated by the integral of the volumes of the individual particles Vp(r)where Vp(r) is calculated by Vp(r) = 4/3πr3.

For smaller molecules involved in the reactions in Table , such as initiator I, monomer M1 and cross-linker M2, transport limitations are neglected and the concentrations in the liquid phase and gel particles are assumed to be determined by phase equilibrium. A similar assumption is proposed by Arosio et al. (2011).[33] This is represented by the partition coefficient D.Hence, the balances for the overall amount of substance n and the concentrations c in the respective phases can be formulated aswhere kd is the initiator decomposition rate and kfm denotes the chain transfer to monomer rate constants of radicals with terminal end i and monomer of species j. cλl represents the concentration of radicals in the liquid phase (cf. Supporting Information SI I), n̅(r) represents the average number of radicals in a particle of radius r, and p is the fraction of radicals of species j in the respective phase. The temperature dependence of the initiation, chain propagation, and chain transfer to monomer reaction rate constants is expressed in terms of the Arrhenius equationwhere A is the frequency factor, EA is the activation energy, R is the universal gas constant, and T is the reaction temperature.

Particle Growth

The particles continue to grow by chain propagation of the radicals inside a particle. Hence, the radial growth rate of particles with radius r results from the addition of polymer volume and waterwhere pg and cMg denote the species of terminal end of the radical and the monomer concentration in the gel phase, respectively. The average number of radicals per particle n̅(r) in eq is derived from the Smith–Ewart theory,[34,35] for which steady state is assumedThe rates of formed and entered radicals per particle, RIp(r) and Rep(r), equal the rates of exited and terminated radicals, Rdesp(r) and Rtp(r), with the last two being functions of n̅(r). The superscript p represents that these rates refer to individual particles of size r. As a consequence of initiator partitioning, eq also includes the radical formation by initiator decomposition in the particlewith initiator efficiency fIg. The termination rate in a particle with the pseudo-bulk assumption and for copolymerization (cf. refs (23, 34))includes the termination rate constants ktg, which is distinguished into the chemical termination contribution (=ktl) and a diffusion-controlled contribution (ktdiff) by ktg–1 = ktl–1 + ktdiff–1.[36] The diffusion-controlled contribution is assumed to be independent of conversion as a high polymer content ensues immediately from precipitation in the particles.[13]

Radical Entry and Desorption

The entry of oligomers from the liquid phase into a particle is described by the entry rate Rep(r). For Rep(r), we employ the diffusion-controlled radical capture model by Smith and Ewart[34]which assumes that diffusion of oligomers in the liquid phase determines radical entry. The diffusion coefficient of monomers in the liquid phase, DW, is adjusted by the average oligomer length navgl (cf. Supporting Information SI I, eq S3) in the liquid phase to describe the diffusion coefficient of oligomers. fe is an efficiency factor to compensate for the overprediction of the radical capture model.[35]

For the radical desorption rate Rdesp(r) from a single particle, a simple form of the radical desorption model proposed by Harada et al. (1971) is applied.[37]Therein, only monomeric radicals formed by chain transfer to monomer can desorb from the particles. Desorption, described by the equilibrium radical desorption coefficient k0 for the monomeric radicals of species i, competes with chain propagation, which is expressed in terms of probability. k0 depends on the diffusion coefficients DW and DP in water and polymer, respectively, and the partition coefficients DM according to the relation established by Harada et al. (1971)[37]

Particle Coagulation

The particle coagulation rate Rcoag(r) combines formation and depletion of particles with radius r. Depletion describes the loss of particles due to their aggregation with other particles. Formation accounts for the gain of particles due to the aggregation of smaller particles. The coagulation rate is employed as described by Immanuel et al. (2003).[38]The radii of the aggregating particles are therein related by the combined volume, which equals the volume of the particle with radius r: r′3 + r″3 = r3. The size-dependent coagulation kernel β(r, r̃) is calculated employing a semiempirical approach. The simplification distinguishes between unstable and stable particles with respective radii below and above a stable particle radius rstable.[30,39] Unstable particles can coagulate with unstable particles as well as stable particles, whereas stable particles can aggregate with unstable particles but not with other stable particles. This is described by an adaptation of the Fuchs modification of the Smucholski equation as used by Vale and McKenna (2009)[30]r* denotes the smaller particle radius of the particles involved, r or r′. kB, μl, and W represent the Boltzmann constant, the viscosity of the liquid phase, and the Fuchs stability ratio, respectively. The distinction between unstable and stable particles ensures that particle aggregation will come to a hold, which is essential for a monodisperse particle size distribution.

Pendant Double-Bond Density

PDBs are tied to a specific polymer particle by reaction. Hence, the PDBs are balanced depending on the particle radius r, comparable to the particle density F(r, t). The distribution of the pseudo-species PDB, FPDB(r, t), is formulated in terms of its moles in particles of radius r. Hence, it can be interpreted as the product of F(r, t), V, and the average number of PDB per particle with radius r.Like the particle number density, FPDB(r, t) depends on precipitation, growth, and coagulation. In addition, FPDB(r, t) is also affected by the formation of new PDB in a particle by reactions of the cross-linker, cross-linking itself with the cross-linking efficiency fPDBg, as well as radical absorption and desorption. For the gain of PDB by precipitation and absorption, the number of entering PDB is calculated from the length of the entering polymer chains, η and navgl, respectively, and the fraction of the incorporated cross-linker, which is approximated by the fraction of radicals with the terminal end of the type cross-linker p2l. For loss of PDB by desorption, the desorption rate for radicals of the type cross-linker is employed and, since only radicals with one repeating unit can desorb, it is not required to consider the chain length here.

With the cross-linking reaction, the terminal end of the reacting oligomer changes to a radical end of the species cross-linker. Hence, the Mayo–Lewis equation to calculate the fraction pg of radicals with species i is extended by the pseudo-species PDBwith the average PDB concentration cPDBg

Particle Characterization

The particle size distribution is characterized by the average microgel radius rh and the standard deviation of the particle size distribution σ. The polydispersity index PDI, as defined for dynamic light scattering (DLS) measurements, is a measure of the width of the particle size distribution.[40]

Reaction Enthalpy Transfer Rate

The reaction enthalpy transfer rate ∑R represents the enthalpy released when double bonds react in a propagation reaction. Considering the propagation reactions in both the liquid phase and the gel particles, and propagation reactions with pendant double bonds, ∑R is calculated according towhere ΔHR represents the enthalpy of the specific propagation reaction of a radical with terminal end j with monomer i. With the enthalpy transfer rate, the model can be linked with experimental data from reaction calorimetry.[13]

Experimental Data

The recipes for the microgel syntheses are listed in Table . All reaction components except the initiator are initially dissolved in the solvent, and the reaction mixture is heated under stirring to the reaction temperature. The reactor is purged for 30 min with nitrogen. When steady state is obtained, the initiator is added and the synthesis runs for 1 h. After synthesis, the reactor content is cooled to room temperature and the microgels are dialyzed. The specifications for monomer and initiator concentrations as well as temperature for the reference experiment are defined according to the common and well-tested recipe for PVCL-based microgels by precipitation polymerization (e.g., refs (8, 41, 42)), whereas the selected cross-linker concentration corresponds to the middle cross-linker concentration investigated in our previous work.[13] The recipe variations include changes of the initial initiator or cross-linker concentrations or temperature, modified one at a time and both lower and higher than the reference value.

Table 2

Recipes for the Reference Microgel Synthesis and the Investigated Experiment Variations of Different Temperatures, Initial Initiator, or Cross-Linker Concentrationsa,b

label	nM1 (t = 0 s) (mol)	nM2 (t = 0 s) (mol)	nI (t = 0 s) (mol)	T (K)
reference	0.0318	7.78 × 10–4	3.69 × 10–4	343
333 K	0.0318	7.78 × 10–4	3.69 × 10–4	333
353 K				353
0.6 mol % I	0.0318	7.78 × 10–4	1.84 × 10–4	343
2.4 mol % I			7.37 × 10–4
1.2 mol % BIS	0.0318	3.89 × 10–4	3.69 × 10–4	343
5.0 mol % BIS		1.56 × 10–3

Information is given as initial conditions for eqs and 9. The labels refer to the initiator or cross-linker to monomer ratios in the respective experiment variations.

All experiments are performed in 0.3 L water as the solvent with 0.0443 g of CTAB as the stabilizer.

From the measurements of the Mettler Toledo reaction calorimeter RTcal in the isothermal control mode, the reaction enthalpy transfer rate ∑R is calculated with the energy balance for the isothermal batch reactor (cf. ref (13)). The experimental data of reaction calorimetry for the reference experiment and the variation of the initial cross-linker concentrations is used as published previously.[14]

The final particle size in the collapsed state is determined by dynamic light scattering at 323 K. It has been reported previously that dialysis does not affect the measured particle radius in the collapsed state;[6] hence, particle characterization is performed after dialysis. For the reference experiment, the particle radius is determined by the mean of 5 experiment repetitions with a standard deviation (σ = 3.8 nm). The particle radii for the experiment variations are determined from single experiments.

Simulation and Parameter Estimation

The model is implemented in gProms Model Builder, version 5.0.1.[43] The discretization of the partial differential equations (PDEs) in eqs and 21 is performed by the gProms intrinsic first-order backward finite differences method. To reduce the error of numerical diffusion of the selected solution method, a fine discretization of 250 intervals is chosen. For dynamic integration, the gProms intrinsic DASOLV solver is selected, which is based on a variable time step and the variable-order backward differentiation formula. The employed parameter values are listed in Table and provided in the Nomenclature. Note that the temperature dependence for diffusion coefficients, partition coefficients, critical chain length, and water density and viscosity is also provided in the Nomenclature. Parameter estimation of the adjustable parameters is conducted with the maximum likelihood method with the same solution parameters as selected for dynamic integration.

Table 3

Parameter Values for the Arrhenius Equation (eq ) Used in This Work to Calculate Reaction Rate Constantsa

	liquid phase		gel phase
rate constant	A	EA	A	EA	ΔHR[13]
kd(45)	9.19 × 1014	1.24 × 105	9.19 × 1014	1.24 × 105
kp11	6.49 × 104	30.81 × 103	1.71 × 104	29.42 × 103	–83.2
kp12	2.23 × 105	18.39 × 103	1.03 × 105	18.35 × 103	–87.4
kp21	3.65 × 104	19.91 × 103	1.48 × 104	20.01 × 103	–74.8
kp22	9.03 × 104	26.63 × 103	3.51 × 104	28.31 × 103	–77.8
kfm11	1.83 × 108	65.00 × 103	1.83 × 108	65.00 × 103
kfm12	0	1	0	1
kfm21	1.57 × 109	76.80 × 103	1.57 × 109	76.80 × 103
kfm22	8.82 × 106	128.0 × 103	8.82 × 106	128.00 × 103

Parameter values for chain propagation and transfer to monomer are calculated as described by Kröger et al. (2017).[44] Dependence of the reaction on the polymer content of the surrounding phase is calculated only for chain propagation (wWg = 0.5).

Six adjustable parameters (W, rstable, fe, DP, kt11diff, and kt22diff) are determined by parameter estimation. The termination rate constants ktdiff have been previously estimated with our two-phase model.[14] Unpublished simulation studies with the previous model have shown that temperature-independent termination rate constants provide sufficiently accurate predictions in the considered temperature range. Hence, a temperature dependence of the termination rate constants ktl and ktdiff is neglected. Nonetheless, considering additional mass transfer among the phases (radical entry and desorption) and distinct particle volumes instead of a continuous gel phase implies a change of the average radical concentration and consequently demands an adjustment of the termination rate constant ktdiff. The previously determined values for ktdiff are used as initial guesses for parameter estimation.

The parameter values are estimated based on the reference experiment (cf. Table ). Experimental data for the parameter estimation includes the reaction enthalpy transfer rate ∑R and the final particle radius rh as well as the condition for a monodisperse PSD (PDI = 0.01). For parameter estimation, the interval of 700 s after initiation (t = 0 s) is used because reaction calorimetry shows only within this time frame significant dynamic behavior. The number of ∑R measurements, which are available in 2 s intervals, outweighs the end-point measurements of rh and PDI. Since the gProms intrinsic parameter estimation does not allow one to weigh the experimental data, the measurements of rh and PDI are duplicated over the interval of t = 600–700 s. Within this interval, calorimetry measurements are in steady state. The introduction of artificial measurement points compensates the high number of data points for ∑R compared with few data points for rh and PDI and will ensure that the measurements of the particle radius are respected.

Results and Discussion

In the following, the simulation of the fitted model is compared to the experimental data. First, the results of the parameter estimation are evaluated by a comparison of simulation with the fitted model and experimental data. This comparison includes the reaction enthalpy transfer rate (eq ), the average particle radius (eq ), and the polydispersity (eq ). Afterward, the fitted model is employed to predict particle growth for the variation of reaction conditions, and the predictions are compared to the corresponding experimental data.

Simulation of the Reference Experiment

Figure shows the comparison of the simulation and experimental data in terms of ∑R over the polymerization time t. The experimental data is used as published previously. t = 0 s represents the time of initiation and the beginning of the simulation. Immediately after initiation, ∑R increases rapidly to a first maximum after approximately t = 25 s, caused by the fast cross-propagation reaction of VCL and BIS.[13] Then, the enthalpy transfer rate decreases due to the consumption of the cross-linker, before it increases again due to the polymerization of the remaining VCL. The simulation with the fitted model agrees with the experimental data within the accuracy of the standard deviation. In conclusion, the pseudo-bulk model is equally suited to fit reaction enthalpy measurements as the two-phase model presented previously.

Figure 2

Comparison of experimental data (cf. Janssen et al. (2018)[14]) and simulation with the fitted model for a polymerization of the reference experiment (cf. Table ). The error bars denote the standard deviations of measurements of three experimental repetitions. For parameter estimation, a constant mean variance model is applied (σ = 0.32 W).

Besides the reaction calorimetry, the simulation satisfies the experimental values for the mean particle radius. Figure shows the predicted growth of the average particle radius over the polymerization time. The average particle radius increases rapidly immediately after initiation, and after approx. t = 400 s, the microgels have obtained their final particle size. At the end of the simulation, the predicted average particle radius shows a very good agreement with the particle radius from DLS, and the deviation is below the experimental error. For further insight, the prediction of the mean particle radius is validated with data from in situ DLS, which is provided in the Supporting Information SI II.

Figure 3

Predicted mean particle radius over polymerization time and measured final radius (DLS) for the reference experiment (cf. Table ). The deviation is below the standard deviation of the measurements. Vertical lines illustrate the predicted ends of (a) nucleation and (b) coagulation.

Finally, the simulation needs to satisfy the condition of a monodisperse PSD. Figure shows the PSD in terms of particle number density F(r, t) over the particle radius at different stages of the synthesis. At early stages (t = 10–50 s), coagulation has a significant effect on particle number density, as the area under curves, proportional to the particle number, decreases rapidly. With further progress (t = 100–400 s), coagulation comes to a hold, and radical and monomer absorption merely shift the mean of F(r, t) to higher radii, whereas the particle number, represented by the area under the curves, remains unaffected. At late stages (t = 400, 700 s), particle growth comes to an end as the particle number densities cannot be distinguished. Throughout the synthesis, the predicted particle density shows that a monodisperse PSD is obtained despite the inhomogeneous consumption of cross-linker and monomer in Figure , since small particles are absorbed instead of leading to a secondary nucleation. Virtanen et al. (2019) recently concluded a comparable formation mechanism based on in situ small-angle neutron scattering measurements of the synthesis of PNIPAM-based microgels.[46]

Figure 4

Prediction of particle size distribution for the reference experiment (cf. Table ) at different stages of the process. For comparison of the final particle size distribution, intensity measurements from DLS for an individual sample (323 K) are included. For parameter estimation, a PDI = 0.01 for the measured rh was used.

For comparison of the final particle size distribution, the intensity measurement from DLS for an individual sample is provided. The intensity measurement reveals a narrow distribution around the mean particle radius. The comparison shows that the simulated F(r, t) provides a good prediction of the monodisperse particle size distribution of the final microgels.

The estimated parameter values, listed in Table , show the general trend of the impacts of the individual growth mechanisms. Three central aspects should be highlighted here. (1) The radical entry efficiency is high compared with values proposed in the literature. It is probably exaggerated due to the inaccuracy of pseudo-bulk models to describe nucleation. Nonetheless, the reported values are determined for emulsion polymerization and might not apply for precipitation polymerization. (2) The diffusion-controlled contributions of the termination rate constants estimated with the pseudo-bulk model provide the same trends as those previously estimated with the two-phase model, but cannot be transferred directly under the given assumptions. The larger value of kt22diff affects only the beginning of the process before the cross-linker is fully consumed. Hence, the diffusion-controlled contribution effectively decreases in value over time (and conversion), which could represent an increasing diffusion limitation from gelation in the particles. (3) The results for the stable particle radius and the Fuchs stability ratio suggest that coagulation has a significant impact on particle growth and is not limited to precursor particles.

Table 4

Estimated Parameter Valuesa

parameter	symbol	unit	parameter value
radical entry efficiency	fe		0.534
diffusion coefficient	DP	m3 s–1	10–16.65
diffusion limited contribution of termination rate constants	kt11diff	m3 (mol s)−1	10–0.90
	kt22diff	m3 (mol s)−1	101.64
stable particle radius	rstable	nm	26.27
Fuchs stability ratio	W		104.44

Additional remarks regarding their interpretation or context of literature values are provided in the Supporting Information SI III.

In this context, it should also be noted that despite the effort to limit the number of adjustable parameters for the first-time investigation of the reaction system, the estimated parameter values reveal in parts significant correlations. This concerns the values of the parameters W, rstable, and fe, but also the combination of parameter values for DP and kt,11diff. These high correlations indicate that parameter values are not identifiable based on the available experimental data. For instance, both coagulation parameters rstable and W have a significant effect on the particle size distribution (cf. Supporting Information SI IV for a small simulation study). Higher rstable leads to larger particle radii since larger particles coagulate, broadening the particle size distributions in terms of higher PDI. Decreasing W increases the coagulation kernel and therefore accelerates particle aggregation, leading to larger particle radii and more narrow PDI. Hence, both an increased rstable and a decreased W result in larger particle radii. The difference in the final predicted PDI can be small, especially in comparison to the experimental error. However, more distinct differences in the PDI can be observed at earlier polymerization times. Hence, when particle size distribution measurements at an early stage of polymerization are provided, correlations among the coagulation parameters can be reduced, and the parameter estimation results can be improved. Also, further model simplification can reduce or eliminate correlations, for example, neglecting radical desorption due to its low impact, which eliminates the parameter DP. Hence, the estimated parameter values need to be treated carefully in terms of model predictions. Nonetheless, the predictive capability of the model will be investigated without further model adjustments in the following.

Prediction of Experiment Variations

To evaluate the predictive capabilities of the fitted model, variations of the microgel synthesis are simulated and compared to experimental data. The comparison is performed in terms of reaction enthalpy transfer rates and final particle size for variations of reaction temperature, initial initiator, and cross-linker concentration.

Variation of Reaction Temperature

The first simulation addresses the impact of reaction temperature on the overall polymerization time as well as the final particle size.

The comparison of predicted and measured ∑R for 333 and 353 K is depicted in Figure . The simulation for both temperatures predicts that an increase of the reaction temperature facilitates the overall polymerization. For 353 K, the maximal ∑R is higher, and it approaches zero after approx. 300 s, indicating the end of synthesis, whereas the simulation for 333 K predicts a duration of almost 1000 s for the synthesis.

Figure 5

Comparison of experimental and predicted reaction enthalpy transfer rates for different reaction temperatures. Experimental data has not been published previously. For improved readability, measurement data is shown only for 4 s intervals.

The experimental data for both temperatures confirm the predictions. Comparing the accuracy of the predictions, the simulation for 333 K matches the experimental data better, whereas the prediction for 353 K exceeds the experimental data around its maximum value. Also, the simulation does not predict the secondary increase in measured ∑R for T = 353 K (t = 200–300 s), which probably results from a delayed heat transfer due to the polymer film on the reactor surface.[47] Nonetheless, in terms of ∑R, the pseudo-bulk model provides a good prediction of the temperature dependence of the process.

Figure shows the corresponding predicted mean particle radii over the progress of polymerization. The trend reveals that with increasing T, microgels initially grow faster but obtain a lower final particle size. At the early stage, more particles are formed by nucleation due to the higher initiator decomposition rate and lower η. Faster chain propagation, radical entry, and coagulation lead to a high number of particles obtaining the stable particle radius. Beyond the stable particle radius, coagulation is limited and the high number of particles competes for absorption of smaller particles and monomers. In contrast to the predicted particle radii, the DLS measurements of microgels synthesized at different temperatures show no significant trend, and deviations are within the experimental error of the reference experiment. There are several explanations for the differences among prediction and experimental data. First, the temperature could affect the coagulation of the particles, which would not be covered by the empirical modeling approach employed in this work. Further, microgels synthesized at different temperatures might have different densities, whereas in this contribution, a constant density is assumed. Also, the measurement error needs to be taken into account, as DLS measurements are obtained from singular synthesis and a temperature dependence has been observed previously in the literature for PVCL-based[6] and more thoroughly for PNIPAM-based microgels.[48,49] In conclusion, the predictions indicate a significant dependence of particle size on temperature. However, a confirmation or disproval of the goodness of the prediction would require further experimental investigation.

Figure 6

Comparison of predicted radii and measured final radii (DLS) for different reaction temperatures (cf. Table ). Markers illustrate the corresponding measured particle radii at the end of synthesis (cyan triangle up open: 333 K; open circle: ref (343 K); red cross: 353 K).

Variation of Initial Initiator Concentration

As illustrated in Figure , the prediction of ∑R increases and shifts to earlier polymerization times with increasing initiator concentrations. The initiator concentration affects the overall polymerization in a similar manner to reaction temperature, as the initiator decomposes faster with increasing temperature. However, unlike the temperature, the initiator concentration does not affect the propagation rate constant and, in consequence, the maximal ∑R for 2.4 mol % I is lower than for 353 K. Again, the predicted ∑R is in good agreement with the experimental data without further parameter adjustment, especially for 0.6 mol % I.

Figure 7

Comparison of experimental and predicted reaction enthalpy transfer rates for different initial initiator concentrations. Experimental data has not been published previously. For improved readability, measurement data is shown only for 4 s intervals.

The predicted particle growth for different initial initiator concentrations is depicted in Figure . Based on the estimated parameter values, the largest final particle radii are obtained for the lowest initial initiator concentration, following the same argumentation as for temperature dependence that fewer particles are formed in the beginning of the process. However, DLS measurements show the opposite behavior. As previously observed for PVCL- and PNIPAM-based microgels, an increase of the initial initiator concentration leads to an increase of the final microgel size.[6,48,49] Hence, based entirely on the reaction rates, the final particle size dependence on the initiator concentration cannot be predicted. An adjustment of the coagulation parameters, such as a decrease of W and/or an increase of rstable, appears to be the apparent approach. However, this bears to this point of too many unknown parameters and factors for a mechanistic modeling approach (e.g., DLVO theory) and too few experimental data points for an empirical description.

Figure 8

Comparison of predicted particle growth and measured final radii (DLS) for different initial initiator concentrations (cf. Table ). Markers illustrate the corresponding measured particle radii at the end of synthesis (cyan triangle up open: 0.6 mol % I; open circle: ref (1.2 mol % I); red cross: 2.4 mol % I).

Variation of Initial Cross-Linker Concentration

The predictions in Figure show a significant impact of the initial cross-linker concentration on the ∑R profile. The fast cross-propagation of VCL and BIS, which is discussed above in the context of Figure , amplifies on increasing the initial cross-linker concentration. This results in a rapid increase of the first peak in the predicted ∑R profile for 5 mol % BIS, whereas for 1.2 mol % BIS, it almost disappears. The comparison to ∑R from measurements shows that the predictive qualities of the model for overall polymerization progress in terms of the reaction enthalpy transfer rate are good, even for changing monomer/cross-linker ratios.

Figure 9

Comparison of experimental (Janssen et al. (2018)[14]) and predicted reaction enthalpy transfer rates for different initial cross-linker concentrations. For improved readability, measurement data is shown only for 4 s intervals.

Although reaction rates represented by ∑R differ significantly, the predicted particle growth for different cross-linker concentrations in Figure differs only insignificantly. The differences in the final predicted particle radii are within the order of magnitude of standard deviation of the reference. In contrast to the variations of temperature and initial initiator concentration, the progress of the particle growth differs insignificantly in the beginning, leading to the prediction of similarly sized final microgels. For 1.2 mol % BIS, the largest final particle size is predicted, and for 5 mol % BIS, microgels are the smallest. This prediction is the opposite trend to the microgel sizes determined by DLS measurements. The measurements show that the differences among the microgel radii are clearly larger than the standard deviation, and further, larger microgels are obtained for higher initial cross-linker concentrations. Again, this leads to the conclusion that reaction rates alone cannot be employed to explain the dependence of microgel size on initial cross-linker concentration. An extension of the coagulation behavior, such as an increasing rstable or preferably a decrease of W as a function of increasing cross-linker concentration in the particles, is recommended. However, in this context, it should also be noted that microgels with a higher cross-linker concentration reveal higher densities due to higher internal cross-linking.[8,41] Higher microgel densities would facilitate the trend that smaller particles are obtained for higher cross-linker concentrations. On the other hand, the higher densities, resulting in higher polymer fractions in the particles, will also impact the reaction rate constants in the gel phase[44] and are accompanied by a higher diffusion limitation of the termination reaction. In consequence, for a predictive model that captures the impact of a cross-linker on the final particle size, quantitative reliable experimental data of the polymer fraction in the collapsed state is required.

Figure 10

Comparison of predicted particle growth and measured final radii (DLS) for different cross-linker concentrations (cf. Table ). Markers illustrate the measured particle radii at the end of synthesis (cyan triangle up open: 1.2 mol % BIS; open circle: ref (2.5 mol % BIS); red cross: 5.0 mol % BIS).

Conclusions

The synthesis of cross-linked PVCL-based microgels by precipitation polymerization was described by a pseudo-bulk model. The purpose of this first-time investigation was to obtain a mechanistic model that cohesively describes the most important quantities of microgel synthesis: the overall polymerization progress as well as the final particle size and a narrow particle size distribution. The model comprises common formulations for all growth mechanisms in microgel synthesis. Precipitation of precursor particles is described by homogeneous nucleation. The particles continue to grow by absorption of monomers and radicals and successive polymerization within particles as well as coagulation of the growing particles.

Based on experimental data from reaction calorimetry and DLS measurements for the final microgel size, the six adjustable parameters of the model are estimated. The comparison of simulation with the fitted model and experimental data shows that the pseudo-bulk model can in principle be fitted to describe the characteristics of the specific synthesis process. The simulation of the reaction enthalpy transfer rate coincides with the experimental data within the order of magnitude of the experimental error while the requirements of the final particle size are met. The estimated parameter values suggest the impact of individual growth mechanisms. High radical entry efficiency and low effective termination rate constants in the gel phase express that the particles are the preferred reaction locus. The stable particle, which is significantly larger than the nucleation radius, suggests that larger particles do not absorb precursor particles exclusively and coagulation has a significant impact on particle growth and particle size distribution.

These contributions result in the following microgel formation process. After initiation, particles are formed in a short nucleation phase only. Particles coagulate until stable, while also growing by polymerization with absorbed monomers and oligomers. Growth continues by absorption of monomers and oligomers and ongoing polymerization in the particles until the monomer is fully consumed. The short nucleation phase and the following coagulation facilitate the narrow particle size distribution.

Testing the predictive capabilities of the fitted model for variations of the experimental recipes demonstrates its potential and current limitations. For variations of temperature, initial initiator, and cross-linker concentration, good predictions of the reaction enthalpy transfer rates were obtained. The prediction of the final microgel radii on the other hand shows the shortcomings and hence requirement of further research in both modeling and experimental investigation. While predictions show the largest impact on final microgel size by variation of temperature, measurements show no significant impact. In contrast, for the cross-linker concentrations, experiments show significant differences in particle radii, whereas simulations predict similar microgel sizes. The deviations of predictions and measurements might indicate the need for model adjustments. However, an advanced and validated modeling will require more quantitative information on the particles’ properties, such as density or water content as well as accurate measurements of particle size throughout the synthesis process. Nonetheless, the good results for the simulation of the standard synthesis and predictions of reaction enthalpy show that the pseudo-bulk model bears the opportunity to describe the precipitation polymerization of PVCL-based microgels.