# Optimizing the synthesis of monodisperse colloidal spheres using holographic particle characterization

Christine Middleton    Mark D. Hannel    Andrew D. Hollingsworth    David J. Pine    David G. Grier Department of Physics and Center for Soft Matter Research, New York University, New York, NY 10003
###### Abstract

Holographic particle characterization measures the sizes and compositions of individual colloidal particles dispersed in fluid media and rapidly amasses statistics on the distributions of these properties, even for complex heterogeneous dispersions. This information is useful for analyzing and optimizing protocols for synthesizing colloidal particles. We illustrate how holographic characterization can guide process design through a case study on a particularly versatile model system composed of an aqueous dispersion of micrometer-scale spheres synthesized from the organosilane monomer 3-(trimethoxysilyl)propyl methacrylate (TPM).

## I Introduction

Since the serendipitous discovery that emulsion polymerization can produce colloidal dispersions with very small polydispersity in size [1, 2], techniques for synthesizing monodisperse colloidal particles have progressed through trial-and-error experimentation inspired by principles of chemistry and physics and validated by batteries of particle-characterization measurements [3, 4, 5, 6, 7, 8]. How process choices influence the properties of synthetic colloids typically can be gauged only after synthesis is complete; assessing the outcome often requires multiple orthogonal measurement techniques. Scanning electron microscopy, for example, is used to measure solid particles’ size distribution and surface texture [9]. Mercury porosimetry and gas adsorption gauge their porosity [10] and surface area [11]. Refractometry and light scattering probe their optical properties [12]. Using such techniques to amass a representative set of characterization results takes time and requires expertise. Preparing samples for measurement, moreover, often involves transferring particles out of their native medium, and so can change their properties. Correlations among particles’ properties are especially difficult to measure, particularly in heterogeneous dispersions.

Holographic particle characterization can streamline the design and optimization of colloidal synthesis processes by providing particle-resolved assays of samples’ size distributions and compositions rapidly and with minimal sample preparation. Holographic characterization works equally well for fluid droplets and solid particles [13, 14]. It naturally accommodates heterogeneous samples [15, 16], and reveals correlations between size and composition [13, 17]. Extensions to the technique provide particle-resolved morphology measurements from the same underlying data. All of this information can be obtained with a single measurement in about ten minutes, which expedites systematic assays and can be fast enough to provide feedback for process control.

To illustrate the use of holographic particle characterization to guide process design, the present work explores the roles of emulsion stoichiometry, initiator choice, and agitation conditions in the synthesis of monodisperse spheres of 3-(trimethoxysilyl)propyl methacrylate (TPM) [18], a model system with increasingly widespread applications in soft-matter research [19, 20, 21]. We apply holographic characterization to identify factors that influence size selection, polydispersity and composition and validate the results with conventional particle characterization techniques. This study therefore builds on the work of van der Wel, et al. [18], which introduced the synthesis of TPM colloids and deployed conventional techniques to systematically characterize the emulsion polymerization protocol. Our holographic measurements confirm that a straightforward tabletop synthesis yields exceptionally monodisperse spheres, and reproduces the observation [18] that smaller particles form at higher pH. Holographic analysis also confirms that increasing monomer concentration tends to form larger spheres, but reveals that this enhancement decreases as pH increases. Holographic particle characterization also reveals that more vigorous mixing during emulsification yields larger particles without necessarily increasing polydispersity. This surprising observation runs counter to the usual trend in emulsion polymerization processes. Also surprising is the observation that the choice of free-radical initiator has no discernable effect on the final particles’ properties, at least not in the range of particle sizes we considered. These observations help to make our central point that holographic characterization provides a rich source of information for choosing among synthesis options in real time without requiring special sample preparation.

## II Materials and methods

### II.1 Materials

The TPM monomer (3-(trimethoxysilyl)propyl methacrylate, $98\,\mathrm{\char 37}$) used in particle synthesis was purchased from Sigma Aldrich. Ammonium hydroxide solution ($29.3\,\mathrm{\char 37}$ NH${}_{3}$ in water) was added to adjust the pH of the synthesis environment. Two water-insoluble initiators, 2,2’-azobis(2-methylpropionitrile) (AIBN) and 1,1’-azobis(cyclohexanecarbonitrile) (ACHN) were purchased from Sigma Aldrich. Two water-soluble initiators, ammonium persulfate (APS) and potassium persulfate (KPS), also were purchased from Sigma Aldrich. All reagents were used as delivered.

### II.2 Emulsion polymerization

The protocols analyzed in this work focus on the influence of stir rate, pH, emulsion stoichiometry, and choice of radical initiation on the size, polydispersity, and refractive index of synthesized particles. While this list is not exhaustive, it illustrates choices made in optimizing synthetic protocols and the role that holographic characterization can play in making successful choices.

The synthesis begins with the formation of an emulsion of monodisperse TPM droplets. Monomeric TPM is insoluble in water but hydrolyzes into soluble monomers in the basic environment of an aqueous ammonium hydroxide solution ($\text{pH}>9$): where Hydrolyzed monomers then condense into dimers, and higher-order branched silsesquioxanes. These oligomers are functionalized with propyl methacrylate groups and so are insoluble in water. They therefore coalesce homogeneously into monodisperse droplets, which continue to grow in a well-mixed environment until the hydrolyzed monomer is depleted.

At this point, the droplets still are fluid, but are largely stable against coarsening because of surface charging [18]. They can be transformed into solid spheres by warming the emulsion to $80\,\mathrm{\SIUnitSymbolCelsius}$ and adding a heat-activated free radical initiator to polymerize the methacrylate moieties of the condensed oligomers.

All samples are prepared in identical $12\,\mathrm{mL}$ glass vials to produce $5\,\mathrm{mL}$ of colloidal dispersion. The vials are sealed to ensure consistent evaporation of ammonia from run to run. Identical stir bars are used for all syntheses to ensure consistent flow properties. Each synthesis is repeated at least once, and characterization data compared to ensure consistent results.

### II.3 Holographic particle characterization

We perform holographic characterization measurements with a Spheryx xSight, a commercial holographic particle characterization system. The xSight draws a small sample of colloidal suspension through the observation volume of an in-line holographic microscope where it is illuminated by a laser operating at a vacuum wavelength of $532\,\mathrm{nm}$. Light scattered by a colloidal particle interferes with the rest of the beam in the focal plane of the microscope’s objective lens. The objective lens relays this interference pattern to a tube lens that focuses it onto the sensor of a digital video camera. The intensity of the recorded interference pattern is a hologram of the particle that can be analyzed [15] to obtain information about the particle’s position and composition. The instrument’s analytical software fits each hologram to predictions of the Lorenz-Mie theory of light scattering [22, 23, 24] to measure the associated particle’s diameter and refractive index, as well as its three-dimensional position relative to the center of the focal plane [15]. Each particle is recorded and analyzed multiple times during its transit through the observation volume. These time-resolved observations are linked into trajectories using a maximum likelihood algorithm [25], both to map the particle’s transit through the sample volume and also to combine multiple independent measurements of that particle’s diameter and refractive index for improved accuracy and precision [26]. Typical measurements on micrometer-scale spheres yield a particle’s diameter with a precision of $5\,\mathrm{nm}$ and its refractive index to within five parts per thousand [27, 28]. Holographic analysis also yields the particle’s in-plane position to within a nanometer and its axial position to within $5\,\mathrm{nm}$ [26, 27]. These measurements’ accuracy and precision have been validated with measurements on standard particles [15, 28], by comparison with orthogonal techniques [28], and through analysis of well-understood physical processes [27].

Originally demonstrated with dispersions of model colloidal spheres [15], holographic particle characterization has been applied successfully to porous particles [13], dimpled spheres [29], and fractal aggregates [17]. Time-resolved holographic characterization has been used to monitor the growth of colloidal polydimethylsiloxane (PDMS) spheres [30], the response of colloidal sensors to changing environmental conditions [31] and molecular binding to the surface of functionalized beads [26]. Practical applications include monitoring protein aggregation in biopharmaceuticals [28, 32], nanoparticle agglomeration in semiconductor polishing slurries [33] and oil droplet concentration in wastewater [14].

xSight measurements are limited to particle concentrations below $10^{6}\,\mathrm{mL}^{-1}$ to minimize interference of overlapping single-particle holograms. TPM synthesis, however, produces samples with number densities of $10^{10}\,\mathrm{mL}^{-1}$. We therefore dilute each sample by a factor of $10^{4}$ with deionized water before analysis. This should not affect the particles’ properties because oligomerized TPM droplets and polymerized spheres are hydrophobic [18].

We analyze each sample by pipetting $100\,\mathrm{\SIUnitSymbolMicro L}$ of the diluted dispersion into an xCell microfluidic sample cell and loading the xCell into the xSight. The xSight draws $3\,\mathrm{\SIUnitSymbolMicro L}$ of the fluid through the xCell’s $50\,\mathrm{\SIUnitSymbolMicro m}$ deep observation volume. A ten-minute measurement provides characterization data for a few thousand colloidal spheres per sample. A typical example is presented in Fig. 1. Each point in this plot represents the measured diameter, $d_{p}$, and refractive index, $n_{p}$, of a single particle, averaged over multiple measurements. Points are colored by the relative density of observations. This coloring reveal a peak in the distribution at $d_{p}=1.5\,\mathrm{\SIUnitSymbolMicro m}$ and $n_{p}=1.495$. The cross superimposed on the plot indicates the median particle diameter and median refractive index, together with the median absolute deviation in those properties. The spread in measured properties is much larger than the measurement uncertainty, and so represents the polydispersity in the sample’s actual properties.

The measured trajectories of colloidal particles can be used to monitor the success of a measurement by mapping the Poiseuille flow profile within the xCell microfluidic channel [26, 34]. Each point in Fig. 2(a) represents the speed, $v_{p}(z_{p})$, for a single particle’s transit at its mean axial position, $z_{p}$. The width of the observed distribution of transit speeds is dominated by variations in the pump speed over the $10\,\mathrm{min}$ course of the measurement. These variations do not affect the precision or accuracy of characterization measurements because the flow speed is always low enough to avoid artifacts due to motion blurring [26, 35]. The dashed curve in Fig. 2(a) is a fit to a the expected parabolic flow profile. Extrapolating this fit to $v(z)=0$ yields estimates for the axial positions of the channel’s walls, assuming no-slip boundary conditions. The measured $51\,\mathrm{\SIUnitSymbolMicro m}$ range is consistent with the xCell’s nominal channel depth. Because the particle’s axial position is measured relative to the microscope’s focal plane, the measured range of $z_{p}$ can be used to confirm that the xCell is seated properly within the xSight during measurement.

Most observations fall neatly onto the parabolic profile. Some, however, deviate markedly, presumably because of tracking errors that result in particles being misidentified in a sequence of holograms. Tracking errors are more common in holograms containing multiple particles, particularly when faster-moving particles near the midplane of the channel overtake slower-moving particles near the walls. Characterization data obtained from such faulty tracks are likely to combine results from different particles rather than reflecting the properties of just one particle. We therefore reject any features that deviate from the parabolic profile by more than one median absolute deviation, as depicted in Fig. 2(b). In this case, $72$ out of $6807$ trajectories were cut. The remaining results are used to characterize the sample, as is the case in Fig. 1.

### II.4 Orthogonal validation methods

Holographic particle characterization is a comparatively new technique and is not yet widely adopted. We therefore validate results from holographic particle characterization with orthogonal measurement techniques. Specifically, we use scanning electron microscopy to provide baseline estimates for the particles’ diameters and Abbe refractometry to measure their refractive indexes.

#### II.4.1 Scanning electron microscopy

Each sample of polymerized spheres was imaged with a field emission scanning electron microscope (MERLIN SEM, Carl Zeiss). Typical images are presented in Fig. 3. SEM images provide a visual check of the spheres’ surface texture and can be used to estimate their mean diameter and polydispersity. While a well calibrated scanning electron microscope provides $1$ to $10\,\mathrm{nm}$ spatial resolution, sample preparation, particularly exposure to vacuum and sputter coating, can shrink or even swell the sample [9, 36]. TPM emulsion droplets are fluid and so are not amenable to SEM analysis.

The diameter of an individual sphere is obtained by analyzing its image with ImageJ [37], specifically by drawing a tight-fitting oval around its image and computing the average of the oval’s major and minor axes. The distribution of sphere diameters is estimated by analyzing $50$ spheres’ images for each sample.

#### II.4.2 Abbe refractometry

We estimate the refractive index of the TPM particles by suspending them in water and measuring the mean refractive index of the suspensions, $n(\phi)$, as a function of the particles’ volume fraction, $\phi$. Linearly extrapolating to $\phi=1$ yields an estimate for the spheres’ refractive index, $n_{p}$, in their native medium at room temperature $21.5\,\mathrm{\SIUnitSymbolCelsius}$ [38]. Figure 4 shows an application of this technique to two similarly prepared samples of TPM spheres.

The initial number density of particles in each sample is obtained with the xSight by counting the particles in $3\,\mathrm{\SIUnitSymbolMicro L}$ of fluid. This approach yields particle concentrations with an accuracy of $10\,\mathrm{\char 37}$ in the range from $10^{3}\,\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{l}% \mathrm{e}\mathrm{s}\mathrm{/}\mathrm{m}\mathrm{L}$ to $10^{7}\,\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{l}% \mathrm{e}\mathrm{s}\mathrm{/}\mathrm{m}\mathrm{L}$ [28]. Each sample is diluted to $5$ different volume fractions to provide a total of $12$ different suspensions, including the two stock samples. The refractive index of each suspension was measured with an Abbe refractometer at room temperature $21.5\,\mathrm{\SIUnitSymbolCelsius}$. Linear extrapolation yields $n_{p}=1.506\pm 0.007$ with $95\,\mathrm{\char 37}$ confidence. This range includes uncertainty in the samples volume fractions as well as any differences between the two samples. This result demonstrates both the reproducibility of the synthesis and the reproducibility of this measurement technique.

This method is complementary to the approach used by van der Wel, et al., who index matched their particles to a solution of pyridine ($n=1.509$) and $2$-ethylhexyl $4$-methoxycinnamate ($n=1.545$) and identified the refractive index of the particles with that of the best index matching solution [18]. They report the refractive index to be between $1.512$ and $1.513$, presumably at a wavelength of $589\,\mathrm{nm}$.

### II.5 Optimization strategies

The immediate goal of this study is to design a synthesis protocol that reproducibly yields TPM spheres with a selected size and the smallest degree of polydispersity in size. Factors governing size selection include ammonia concentration, emulsion stoichiometry, and mixing conditions. These factors determine the nucleation rate of oligomer droplets, the rate and duration of growth, and the homogeneity of these processes, respectively.

The choice of free-radical initiator does not affect the size distribution of the oligomer droplets, but may influence the course of polymerization, and thus the density and size of the final spheres. Oil-soluble initiators might be expected to polymerize the spheres uniformly thereby shrinking the spheres as the material becomes more dense. Water-soluble initiators, by contrast, might polymerize the spheres from the surface inward, perhaps inhibiting shrinkage and producing a larger and less dense product.

To guide protocol design, we first monitor the effect of mixing conditions on size selection during droplet formation. Using holographic characterization data to monitor polymerization speeds the optimization process by identifying when processing has run to completion. We then use the optimal mixing conditions for a binary search through parameter space of ammonia concentration and TPM stoichiometry. Finally, we assess the influence of initiator choice on particle properties.

## III Results and Discussion

Each holographic particle characterization measurement requires roughly $15\,\mathrm{min}$ including sample preparation and yields a comprehensive view of the joint distribution of the sizes and refractive indexes of the particle in the sample. Orthogonal techniques yield information about the particles’ size distribution and refractive index separately. They cannot identify correlations between size and composition. We therefore average holographic characterization data over the complementary characteristic when comparing with orthogonal techniques.

### III.1 Effect of stirring rate on particle size

Stirring the sample while the oligomers condense into droplets promotes homogeneous nucleation by uniformly dispersing the hydrolyzed monomer and droplet nuclei, thereby ensuring that all droplets grow under comparable conditions. Beyond simply mixing the sample, however, stirring creates shear forces [39] that can alter droplets’ size distribution [40, 41, 42]. The overall effect of stir rate on the resulting particle size is best assessed experimentally.

Figure 5 summarizes how stirring rate influences the size of TPM spheres for a given set of synthesis conditions. The four emulsions represented in this plot were produced in identical cylindrical vials with identical stir bars. Each vial was charged with $15.0\pm 0.3\,\mathrm{\SIUnitSymbolMicro L}$ of $29\,\mathrm{\char 37}$ ammonia ($29.3\,\mathrm{\char 37}$ NH${}_{3}$(aq) by HCl titration) using an Eppendorf Research plus pipette, followed by $200\,\mathrm{\SIUnitSymbolMicro L}$ of TPM monomer and was immediately brought up to $5\,\mathrm{mL}$ with DI water. These conditions yield $\text{pH}=11.0$ and $[\text{TPM}]=0.16\,\mathrm{M}$. The four samples were stirred for $2\,\mathrm{h}$ using magnetic stir plates set at $500$, $700$, $900$, and $1100\,\mathrm{min}^{-1}$. The resulting droplets were polymerized by adding an excess ($1\,\mathrm{m}\mathrm{g}$) of AIBN as a radical initiator and heating the sample to $80\,\mathrm{\SIUnitSymbolCelsius}$ for $2\,\mathrm{h}$. These samples then were analyzed by both HPC and SEM.

The mixing conditions are set by the geometry of the vial, the shape of the stir rod, and the stir rate. The $12\,\mathrm{mL}$ cylindrical vials have an inner diameter of $16\,\mathrm{mm}$ and a height of $60\,\mathrm{mm}$. Each $5\,\mathrm{mL}$ sample filled approximately one third of the cylinder’s volume. Each pill-shaped magnetic stir rod has a long axis length of $12\,\mathrm{mm}$ and short axis length of $7\,\mathrm{mm}$. The nature of the flow in the vial can be gauged with the dimensionless impeller Reynolds number,

 $N_{Re}=\frac{\Omega d^{2}\rho}{\mu},$ (1)

where $\Omega$ is the rotation speed in rotations per second, $d$ is the impeller diameter, and $\rho$ and $\mu$ refer to the density and dynamic viscosity of water, respectively. The transition from laminar flow to turbulent flow occurs for $N_{Re}>1000$. In our system, the impeller Reynolds number ranges from $3200$ at a stir rate of $500\,\mathrm{min}^{-1}$ to $7000$ at $1100\,\mathrm{min}^{-1}$, and so is turbulent [39]. The meniscus of the driven vortex ranges in depth from $5\,\mathrm{mm}$ at $500\,\mathrm{min}^{-1}$ to $20\,\mathrm{mm}$ at $1100\,\mathrm{min}^{-1}$, and does not reach the stir bar until $2000\,\mathrm{min}^{-1}$. These considerations suggest that samples should be well mixed over the entire range of stirring rates, the principal difference among them being the intensity of the shear forces encountered by the dispersed droplets.

The results in Fig. 5 reveal that the smallest mean particle size is obtained at the lowest stirring speeds. This surprising observation runs counter to the usual trend [40, 41] in emulsion polymerization [43] and suspension polymerization [44]. Increasing stirring speed also increases the particles’ polydispersity, particularly increasing the proportion of undersized particles. It is possible that increasing shear rate favors collision-induced droplet coalescence in this system [42, 45, 46]. The samples’ median diameters, $\left$, are reported in Table 1 together with their standard deviations, $\sigma_{d_{\text{p}}}$.

Size distributions obtained holographically are consistent with results of SEM analysis on the same samples, including the images in Fig. 3. These results also are plotted in Fig. 5 and reported in Table 1. For all samples, the median particle diameter obtained holographically agrees with the median SEM result to within $40\,\mathrm{nm}$, which is smaller than the sample standard deviation. This agreement supports our use of HPC for rapid sample characterization.

Although unpolymerized droplets are not amenable to SEM analysis, they can be by analyzed holographically. Figure 6 presents the median size and refractive index of TPM emulsion droplets prepared at the four stirring rates, both before and after polymerization. Overall the influence of polymerization on median droplet diameter is consistent with the $2\,\mathrm{\char 37}$ reduction in size reported by van der Wel, et al. based on dynamic light scattering [18].

Based on the outcome of this survey, we synthesize the remainder of the particles in this study at a stirring rate of $700\,\mathrm{min}^{-1}$, which reproducibly yields particles with $3\,\mathrm{\char 37}$ relative polydispersity in diameter.

### III.2 Effect of stirring rate on refractive index

Figure 6 summarizes holographic characterization results for the influence of stirring rate on the properties of TPM emulsions and their associated polymerized spheres. Each dot represents the median size and refractive index for one sample. Error bars represent the median absolute deviations of those properties.

Whereas stirring has a substantial influence on droplet size, its effect on refractive index is slight. This is reasonable because the refractive index reflects the composition of the condensed phase, which should not vary substantially with local flow conditions. Polymerized spheres have an average refractive index of $n_{p}=1.501\pm 0.009$, which is consistent with the value obtained by Abbe refractometry.

Unpolymerized emulsion droplets have a systematically lower refractive index than polymerized spheres Polymerization changes the chemical makeup of the droplet and generally increases the density. Both effects might reasonably account for the observed change in refractive index.

Oligomerized TPM droplets and polymerized spheres both have higher refractive indexes than bulk TPM oil, which is indicated by a dashed line in Fig. 6. Hydrolysis and oligomerization account for a refractive index shift of roughly $0.05$ and polymerization accounts for the remaining increase of $0.02$.

### III.3 Monitoring polymerization

The data in Fig. 7 illustrate how changes in droplet size and refractive index can be used to monitor the progress of polymerization. This is useful both for ensuring that polymerization has run to completion before analysis, and also for minimizing the time required for combinatorial studies of processing choices. This sample consists of an emulsion of TPM droplets undergoing polymerization with AIBN as the initiator. The suspension is continuously stirred at $1100\,\mathrm{min}^{-1}$ and is thermally coupled to a heat bath set at $80\,\mathrm{\SIUnitSymbolCelsius}$. Starting when the initiator is added, we draw $1\,\mathrm{\SIUnitSymbolMicro L}$ of the suspension after $5$, $10$, $15$, $20$, $40$, $60\,\mathrm{min}$. Each aliquot is added immediately to $10\,\mathrm{mL}$ of room temperature deionized water to arrest polymerization through thermal quenching. The diluted sample then is transferred to the xSight for analysis.

The median diameter of the TPM droplets plotted in Fig. 7(a) decreases by $2\,\mathrm{\char 37}$ on average during polymerization. Although this is within the measured polydispersity of each sample, it is consistent with the $7\,\mathrm{\char 37}$ reduction in volume reported in Ref. [18]. The change in size is correlated with a simultaneous increase in the refractive index from $1.485$ to $1.495$ that is plotted in Fig. 7(b). Dashed curves in Fig. 7 are exponentials with a $10\,\mathrm{min}$ characteristic time that are superimposed on the data as a guide to the eye. Changes in dispersion properties have largely run their course in $20\,\mathrm{min}$.

All four free-radical initiators have half-lives at $80\,\mathrm{\SIUnitSymbolCelsius}$ that are substantially longer than the time required to polymerize the droplets. The shortest-lived initiator is APS, with a half-life of $1.8\,\mathrm{h}$ [47]. KPS has a half-life of $2\,\mathrm{h}$ [48], AIBN has a half-life of $6\,\mathrm{h}$ and ACHN has a half-life of $30\,\mathrm{h}$. The results in Fig. 7 therefore suggest that heating for more than half an hour will not substantially alter these particles’ properties. Because holographic characterization can be performed in a matter of minutes, this information can be used to minimize heating time and thus energy cost in emulsion polymerization. We use this information to ensure that all syntheses have run to completion in assessing the role of emulsion stoichiometry and initiator selection on particle properties.

### III.4 Effect of emulsion stoichiometry and initiator solubility

The amount of hydrolyzed TPM oil in solution influences both the number of droplets that nucleate and the size to which they grow. The outcome also depends on the pH of the solution, which influences the rate of oligomerization. To identify conditions that minimize polydispersity, we prepare four batches of droplets corresponding to binary choices in the concentrations of TPM oil and ammonia. In each case, TPM oil is injected into an ammonia solution with stirring. Two samples are prepared with $81\,\mathrm{mmol}\,\mathrm{L}^{-1}$ TPM and another two with $122\,\mathrm{mmol}\,\mathrm{L}^{-1}$. In each case, one of the samples is prepared with $31.8\,\mathrm{mmol}\,\mathrm{L}^{-1}$ of ammonia, and the other with $61.5\,\mathrm{mmol}\,\mathrm{L}^{-1}$, which correspond to pH $10.87$ and pH $11.02$, respectively.

How the droplets polymerize may depend on the solubility of the initiator. Water-soluble initiators, for example, are believed to polymerize particles from the surface inward, leading to inhomogeneous solidification [19]. Such particles might differ appreciably from those polymerized with oil-soluble initiators that presumably would be more homogeneously solidified. To investigate this effect, each of the four emulsions is polymerized with four different initiators: two oil-soluble initiators, AIBN and ACHN, and two water-soluble initiators, potassium persulfate and ammonium persulfate. This yields a total of $16$ samples.

Both types of initiators are at least slightly soluble in both phases. The oil-soluble initiators therefore reach the droplets through the aqueous phase. The water-soluble initiations, moreover, can permeate the spheres.

Figure 8 presents characterization data from all $16$ data sets plotted to highlight different binary choices. Each point represents the median properties of one of the samples and is amassed from several thousand individual holographic characterization measurements. Figure 8(a) emphasizes the influence of TPM concentration on the polymerized particle diameter for fixed concentration of ammonia and choice of initiator. Increasing TPM content increases particle diameter at low concentration of ammonia, but has less influence at higher concentration of ammonia. Increasing the concentration of ammonia therefore improves reproducibility by reducing sensitivity to variations in TPM concentration.

Figure 8(b) shows the complementary projection of the data that emphasizes the influence of ammonia concentration. All eight assays convey the same message: increasing ammonia concentration decreases the average particle diameter, presumably because increasing pH increases surface charge and so promotes stability of nucleated particles [18, 42]. This, in turn, favors higher particle number density and smaller particle size for a given amount of TPM monomer. Increasing the number and stability of nuclei has the additional effect of improving predictability of the particle radius by reducing sensitivity to the amount of TPM.

Choice of initiator has no appreciable influence on the polymerized particles’ average diameter. Size selection is predominantly determined by nucleation and growth of oligomerized TPM droplets. The concentration of ammonia increases the pH and therefore increases the rate at which TPM hydrolizes and oligomerizes. Faster nucleation presumably yields more droplets and reduces the average particle size. At low concentrations of ammonia, oligomerization proceeds more slowly and yields fewer of the larger droplets.

The choice of initiator also has no substantial influence on the particles’ refractive indexes, as shown in Fig. 9. This is surprising because larger TPM droplets are known to form dimples when polymerized with the water-soluble initiators [19]. Dimpling is believed to result from surface hardening followed by ejection of low-molecular-weight oligomers [19]. Smaller TPM spheres polymerized with these initiators do not form dimples [18] and so might be expected to have nonuniform densities with possible signatures in the measured refractive index. In fact, no significant differences are observed.

## IV Summary and Conclusions

We have demonstrated some of the ways in which holographic particle characterization can be used to guide the development of synthesis protocols for colloidal particles. Our specific study of monodisperse TPM spheres examines the influence of particular protocol choices on the size and refractive index of the resulting polymerized spheres. Specifically, we investigate the roles of stirring rate, incubation time, stoichiometry, and choice of free radical initiator.

Increasing stirring rate is found to increase particle size at the cost of increasing polydispersity. Increasing pH through the concentration of ammonia decreases particle size and enhances reproducibility.

Unlike orthogonal techniques, holographic characterization requires minimal sample preparation and provides results in minutes, and so can be used for feedback at all stages in the synthesis. Periodically sampling the reaction vessel, for example, is useful for determining when droplet growth has run to completion.

These results demonstrate that holographic characterization can be a valuable addition to the arsenal of techniques used to design and control colloidal synthesis protocols.

## Acknowledgements

This work was supported primarily by the MRSEC program of the National Science Foundation through Award Number DMR-1420073. Additional support was provided by the SBIR program of the National Science Foundation through Award Number IPP-1519057, and in part by NASA under Award Number NNX13AR67G. The Spheryx xSight holographic characterization instrument used in this study was acquired by the NYU MRSEC as a shared instrument. The Zeiss scanning electron microscope used in this study was acquired under NSF Award Number DMR-0923251 and is maintained as a shared facility by the NYU MRSEC.

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