Many common elastomeric products, including nitrile gloves, are manufactured by coagulant dipping. This process involves the destabilization and gelation of a latex dispersion by an ionic coagulant. Despite widespread application, the physical chemistry governing coagulant dipping is poorly understood. It is unclear which properties of an electrolyte determine its efficacy as a coagulant and which phenomena control the growth of the gel. Here, a novel experimental protocol is developed to directly observe coagulant gelation by light microscopy. Gel growth is imaged and quantified for a variety of coagulants and compared to macroscopic dipping experiments mimicking the industrial process. When the coagulant is abundant, gels grow with a *t*^{1/2} time dependence, suggesting that this phenomenon is diffusion-dominated. When there is a finite amount of coagulant, gels grow to a limiting thickness. Both these situations are modeled as one-dimensional diffusion problems, reproducing the qualitative features of the experiments including which electrolytes cause rapid growth of thick gels. We propose that the gel thickness is limited by the amount of coagulant available, and the growth is, therefore, unbounded when the coagulant is abundant. The rate of the gel growth is controlled by a combination of a diffusion coefficient and the ratio of the critical coagulation concentration to the amount of coagulant present, which in many situations is set by the coagulant solubility. Other phenomena, including diffusiophoresis, may make a more minor contribution to the rate of gel growth.

## I. INTRODUCTION

Coagulant dipping is a process used for the industrial-scale manufacture of elastomeric products including the disposable gloves used as personal protective equipment in laboratories, medicine, and food preparation.^{1} With the advent of the Covid-19 pandemic in 2020, the annual global demand for disposable protective gloves reached 300 × 10^{9} pieces.^{2} Despite coagulant dipping being widely employed for low cost manufacturing, industrial procedures have been developed in a primarily phenomenological manner through trial and error, rather than from a detailed understanding of the underlying physical chemistry.^{3}

In this article, we describe a new, microscopy-based, experimental procedure for quantifying latex gel growth and aim to understand the physicochemical principles governing coagulant gelation. These data are supplemented with and compared to macroscopic dipping experiments that model the industrial process. Simple, diffusion-based models are developed to explain the experimental data in terms of a handful of physical properties of the coagulants and latex. In addition, the potential role of diffusiophoresis, the directed motion of colloidal objects as a consequence of solution concentration gradients,^{4,5} is assessed.

### A. The coagulant dipping process

Coagulant dipping is illustrated in Fig. 1. A mold, known as a former, is dipped into an aqueous coagulant solution, then withdrawn and dried, coating it with a thin layer of the coagulant [Fig. 1(a)]. The former is typically made of porcelain and the most commonly used coagulant is calcium nitrate [Ca(NO_{3})_{2}].^{1,3} The coagulant-coated former is dipped into the latex formulation and remains immersed for a duration known as the dwell time [Fig. 1(b)]. The latex formulation is an aqueous latex dispersion, typically between 20% and 50% solids by volume, and often contains small amounts of pigment, vulcanization chemicals, antioxidants, and stabilizers. The latex may be either synthetic or natural and can include rubber, acrylates, urethanes, and nitriles.^{1} Coagulant enters the solution near the interface with the former, and the high coagulant concentration in this region destabilizes the latex particles, causing them to aggregate into a wet gel film coating the former^{3} [Figs. 1(d) and 1(e)]. After the dwell time, the gel-coated former is withdrawn [Fig. 1(c)] and dried.

Latex stability is provided by either direct electrostatic repulsion between particles or a surfactant additive, such as sodium dodecyl benzene sulfonate (SDBS). Coagulant ions in solution screen the electrostatic repulsion, allowing latex particles to approach sufficiently closely that their mutual van der Waals attraction causes them to aggregate, as described by the DLVO theory of colloidal stability.^{6}

The minimum electrolyte concentration above which a latex dispersion is destabilized is known as the critical coagulation concentration,^{7} *c**. The value of *c** depends on the properties of both the latex and the electrolyte and can range from millimolar concentrations for multivalent electrolytes up to near molar concentrations for monovalent salts, roughly following the Schulze–Hardy rule,^{8,9} which says that *c** ∝ *Z*^{−6} for a symmetric *Z*:*Z* electrolyte. In coagulant dipping, it is expected that the wet gel forms in the region of space close to the former in which *c** is exceeded.

### B. Previous research on coagulant gelation

Measurements of film thickness and material properties following coagulant dipping in natural rubber latices have investigated numerous parameters: the choice of coagulant; dwell time; immersion and withdrawal speeds; latex solid content; amount of coagulant; and latex dispersion viscosity.^{10–14} Fundamentally, coagulant dipping results in a film with improved tensile and elastic properties compared to dipping without coagulant.^{10} Latex films are invariably thicker when the amount of coagulant on the former is increased^{12–14} and calcium salts (nitrate and chloride) produce the thickest films^{12,13} with better elastic, tensile, and puncture-resistant properties.^{11,15}

Over short timescales of up to 5 min, the coagulant dipped natural rubber latex film thickness increases with the dwell time.^{11–14} This is initially very rapid and then slows at longer dwell times. The former’s immersion speed has little effect on film thickness, but thicker films are obtained when the former is withdrawn from the latex dispersion more quickly.^{13,14} Furthermore, thicker films are formed by dipping into a more viscous latex formulation,^{13,14} suggesting that understanding may be advanced by treating coagulant dipping as a Landau–Levich-type problem, considering the thickness of the film formed on a plate withdrawn from a fluid bath.^{16–18}

A theoretical treatment was given by Stewart in 1973 and tested with experiments using polychloroprene latices in contact with a Ca(NO_{3})_{2} solution.^{19} Stewart considered hindered coagulant diffusion through the growing gel and developed a description based on latex film rheology to relate particle coalescence to the cross-sectional area available for diffusion through the film. Films of softer latices, which are able to coalesce quickly, grow more slowly than those of hard latices, which resist coalescence. The coalesced soft latex forms an impermeable barrier to diffusion, which arrests film growth. By contrast, hard latex particles coalesce more slowly and form a porous gel through which diffusive transport is maintained. Consequently, films of these latices grow for many hours and can attain thicknesses of over a centimeter. Here, latex “softness” was assessed from the relaxation modulus of a dry film penetrated by a weighted steel ball.

Most recently, Groves and Routh presented a thorough study of coagulant dipping into nitrile latex and natural rubber.^{3} Their coagulant was calcium nitrate, and dwell times of up to 20 min were considered. Wet gel thickness was estimated from its mass. Consistent with earlier research, wet gel thickness initially increases rapidly with dwell time. For dwell times longer than 1 min, this growth slows down and the gel approaches a limiting thickness at ∼5 min. Groves and Routh model the coagulant dipping process as a diffusion problem and consider a number of corrections to explain the limiting wet gel thickness. The phenomena considered are: hindered diffusion through a porous gel; the kinetics of coagulation and the consumption of the finite amount of coagulant on the former. The authors conclude that the most significant effect in limiting gel growth is the kinetics of coagulation.

Researchers have focused on calcium salts,^{3,12,13,19} as these are found to produce the thickest films. For manufacturing purposes, the consequences of varying dwell time, dipping speed, amount of coagulant, and formulation viscosity have all been cataloged. However, with the exception of a few scientific studies,^{3,19} the coagulant gelation literature remains phenomenological, with little understanding of the physical chemistry governing the process.

### C. The potential role of diffusiophoresis in coagulant gelation

A solution concentration gradient can induce directed motion of colloidal particles due to diffusiophoresis.^{20,21} The theory of electrolyte diffusiophoresis is described in the Appendix. Importantly, diffusiophoresis may be directed either up or down an electrolyte gradient, depending on the particle surface charge and the relative diffusivities of the anion and cation.

When a coagulant-coated former is immersed in a latex dispersion, coagulant diffuses away from the former and establishes a concentration gradient. It is, therefore, reasonable to assume that latex particles respond to this gradient diffusiophoretically. Indeed, diffusiophoresis was originally described by Derjaguin in the context of latex deposition and film formation.^{22} Thus, under conditions for which diffusiophoresis is directed up-gradient, diffusiophoretic transport of latex may enhance gel growth during coagulant dipping. Conversely, if diffusiophoresis is directed down-gradient, gel growth may be inhibited.

### D. Article overview

One shortcoming of dipping experiments such as described in Sec. I B is the sparsity of data they produce. Each point on a graph of film thickness vs dwell time represents an experiment or multiple repeated experiments. Therefore, generating data over dwell times up to 20 min for a range of coagulants, coagulant concentrations, and latex formulations is a time-consuming task. In addition, due to their macroscopic nature, coagulant dipping experiments consume large volumes of latex. The gel thickness vs dwell time datasets presented by Groves and Routh^{3} each contain ∼10 measurements acquired over 20 min, while others report fewer than five measurements in 5 min or less.^{11–14}

Such data reveal trends but are difficult to compare or fit to theoretical models. For this reason, we developed a new technique to directly observe coagulant gelation *in situ* with a fluorescence microscope. A single experiment provides a complete gel thickness vs time dataset at a rate limited only by image acquisition. Measuring gel thickness *in situ* has the added benefit that the former is not withdrawn from the latex before characterization, and therefore, any potential changes induced by the withdrawal process are absent. These data facilitate a more meaningful comparison to theory. In addition, these experiments are based on direct imaging, rather than an indirect, *post hoc* thickness estimate derived from accumulated latex mass. Imaging colloidal gels is a common technique to assess their structure and rheology during gelation and aging,^{23–27} but to the best of our knowledge, ours are the first experiments to directly image unidirectional gel growth. These novel microscopic experiments are supplemented with macroscopic dipping experiments replicating previous research and industrial practice.

The fastest gel growth is invariably reported to occur when Ca(NO_{3})_{2} is the coagulant.^{12,13,15} Ca(NO_{3})_{2} is also the coagulant of choice for manufacturing by coagulant dipping. However, why should Ca(NO_{3})_{2} be the best coagulant? Many trivalent salts, such as aluminum nitrate [Al(NO_{3})_{3}], have a lower *c** than Ca(NO_{3})_{2}. Therefore, Al(NO_{3})_{3} destabilizes latex at a lower concentration than Ca(NO_{3})_{2} and might be expected to form a thicker gel. We explore this by measuring coagulation gelation due to Al(NO_{3})_{3}, Ca(NO_{3})_{2}, and sodium nitrate (NaNO_{3}), which have valence 3^{+}, 2^{+}, and 1^{+}, respectively. Furthermore, to investigate the role of coagulant solubility, we choose calcium sulfate (CaSO_{4}) and calcium carbonate (CaCO_{3}) and compare their performance to Ca(NO_{3})_{2}.

Other coagulants are chosen based on their diffusiophoretic properties. The Appendix explains how the direction of diffusiophoresis depends on the parameter $\beta =D+\u2212D\u2212D++D\u2212$, encoding the relative diffusivities of the cation and anion. Monovalent salts with *β* < 0 include sodium chloride (NaCl), sodium nitrate (NaNO_{3}), and tetrabutylammonium bromide (TBAB). Monovalent salts with *β* > 0 include potassium acetate (KOAc) and sodium tetraphenylborate (NaTPB). Finally, potassium chloride (KCl) is chosen as it has *β* ≈ 0.

The remainder of this article is structured as follows: The materials and experimental procedures are provided in Sec. II. Experimental results are presented in Sec. III and theoretical models are described in Sec. IV. Implications and limitations are discussed in Sec. V and conclusions are drawn in Sec. VI.

## II. MATERIALS AND METHODS

### A. Materials

An aqueous, colloidal dispersion (45% solids) of carboxylated butadiene-acrylonitrile copolymer latex (Synthomer™ 6338, henceforth referred to as latex) was provided by Synthomer UK, Ltd. and used in all experiments, either as stock or diluted with deionized (DI) water as detailed. Characterization of the latex particles by dynamic light scattering and scanning electron microscopy is described in the supplementary material. The latex particle diameter is 130 nm and its *ζ* potential is measured to be −40 mV at neutral pH. The stock latex dispersion pH reported by the manufacturer is 8.2.

Sodium chloride (99.9% pure), sodium nitrate (99.0% pure), calcium nitrate tetrahydrate (99.0% pure), aluminum nitrate nonahydrate (98.0% pure), potassium acetate (99.0% pure), tetrabutylammonium bromide (98.0% pure), and sodium tetraphenylborate (99.5% pure) were purchased from Sigma-Aldrich and used as received. Potassium chloride (99% pure) and calcium carbonate (99.8% pure) were purchased from Fisher Scientific and used as received. Calcium sulfate dihydrate (gypsum, 99% pure) was purchased from The Malt Miller and used as received. Coagulant solutions or suspensions were prepared in either deionized water [NaCl, NaNO_{3}, Ca(NO_{3})_{2}, Al(NO_{3})_{3}, KCl, KOAc, CaSO_{4}, CaCO_{3}] or isopropyl alcohol (TBAB, NaTPB). Isopropyl alcohol (99.8% pure) was purchased from Sigma-Aldrich and used as received.

Fluorescently labeled polystyrene particles of diameter 1 *μ*m (Invitrogen FluoSpheres™, carboxylate-modified, yellow-green fluorescent) were purchased from ThermoFisher as a 2% solids suspension and mixed into the latex as required at a concentration of 1 drop $(\u223c30\mu L)$ per 5 ml.

### B. Coagulant properties

In this work, a single latex is employed, and therefore, *c** is treated as though it were a property of the electrolyte coagulant alone. *c** is estimated visually as described in the supplementary material. These measurements are given in Table I. In the case of CaCO_{3}, no coagulation is observed. Despite not restricting ourselves to symmetric electrolytes, these measurements of *c** approximately follow the Schulze–Hardy rule in cation valence,^{8,9} $c*\u221dZ+\u22126$.

Coagulant . | c* (mol m^{−3})
. | c_{sat} (mol m^{−3})
. |
---|---|---|

NaCl | 600 ± 100 | 6 160 |

NaNO_{3} | 600 ± 100 | 10 730 |

KCl | 700 ± 100 | 4 556 |

KOAc | 600 ± 100 | 27 369 |

TBAB | 100 ± 50 | 1 861 |

NaTPB | 350 ± 100 | 1 373 |

Ca(NO_{3})_{2} | 7 ± 1 | 5 460 |

Al(NO_{3})_{3} | 2 ± 0.5 | 1 794 |

CaSO_{4} | 6 ± 0.5 | 15.1 |

CaCO_{3} | 0.13 |

Coagulant . | c* (mol m^{−3})
. | c_{sat} (mol m^{−3})
. |
---|---|---|

NaCl | 600 ± 100 | 6 160 |

NaNO_{3} | 600 ± 100 | 10 730 |

KCl | 700 ± 100 | 4 556 |

KOAc | 600 ± 100 | 27 369 |

TBAB | 100 ± 50 | 1 861 |

NaTPB | 350 ± 100 | 1 373 |

Ca(NO_{3})_{2} | 7 ± 1 | 5 460 |

Al(NO_{3})_{3} | 2 ± 0.5 | 1 794 |

CaSO_{4} | 6 ± 0.5 | 15.1 |

CaCO_{3} | 0.13 |

The solubility limit or saturation concentration of a coagulant in water, *c*_{sat}, may affect its performance in coagulant gelation. Table I provides *c*_{sat} for the chosen coagulants at room temperature.

During coagulant dipping, the coagulant must diffuse into the latex dispersion from a concentrated source on the former. A salt in solution dissociates into its cation and anion, which have individual diffusion coefficients at infinite dilution, *D*_{+} and *D*_{−}, reported in the literature.^{21,28} A more detailed discussion of diffusion is provided in Sec. V A, but for the purposes of this article, cation and anion diffusivities at infinite dilution are employed. These are reported in Table II. In addition, for the symmetric electrolytes, the diffusiophoretic parameter *β* is reported.

Coagulant . | D_{−} (×10^{−10} m^{2} s^{−1})
. | D_{+} (×10^{−10} m^{2} s^{−1})
. | β
. |
---|---|---|---|

NaCl | 20.3 | 13.3 | −0.21 |

NaNO_{3} | 19.0 | 13.3 | −0.18 |

KCl | 20.3 | 19.6 | −0.02 |

KOAc | 10.9 | 19.6 | 0.28 |

TBAB | 20.8 | 5.2 | −0.6 |

NaTPB | 5.6 | 13.3 | 0.41 |

Ca(NO_{3})_{2} | 19.0 | 7.9 | |

Al(NO_{3})_{3} | 19.0 | 5.4 | |

CaSO_{4} | 10.7 | 7.9 | −0.15 |

CaCO_{3} | 9.2 | 7.9 | −0.08 |

Coagulant . | D_{−} (×10^{−10} m^{2} s^{−1})
. | D_{+} (×10^{−10} m^{2} s^{−1})
. | β
. |
---|---|---|---|

NaCl | 20.3 | 13.3 | −0.21 |

NaNO_{3} | 19.0 | 13.3 | −0.18 |

KCl | 20.3 | 19.6 | −0.02 |

KOAc | 10.9 | 19.6 | 0.28 |

TBAB | 20.8 | 5.2 | −0.6 |

NaTPB | 5.6 | 13.3 | 0.41 |

Ca(NO_{3})_{2} | 19.0 | 7.9 | |

Al(NO_{3})_{3} | 19.0 | 5.4 | |

CaSO_{4} | 10.7 | 7.9 | −0.15 |

CaCO_{3} | 9.2 | 7.9 | −0.08 |

Tables I and II elucidate our coagulant choices. *c** for the multivalent electrolytes, Ca(NO_{3})_{2}, CaSO_{4}, and Al(NO_{3})_{3}, is two orders of magnitude smaller than for monovalent electrolytes. This is likely part of the reason that Ca(NO_{3})_{2} (a divalent electrolyte) is the coagulant of choice for manufacturing by coagulant dipping. A natural question to arise is whether the further reduction in *c** afforded by using a trivalent electrolyte such as Al(NO_{3})_{3} results in faster gelation and the formation of thicker films. Furthermore, the solubility of CaSO_{4} and CaCO_{3} are two and five orders of magnitude smaller than that of Ca(NO_{3})_{2}, respectively. The comparison of the three calcium coagulants reveals the role of solubility in coagulant gelation.

To investigate the potential role of diffusiophoresis in coagulant gelation, monovalent electrolytes with a range of diffusivities are chosen. For sodium chloride and sodium nitrate, the cation diffuses more slowly than the anion, giving *β* ≈ −0.2. The diffusiophoretic motion of a negatively charged colloid is expected to proceed up the concentration gradient, as previously observed.^{32} Conversely, for KOAc, the anion diffuses more slowly than the cation (*β* > 0). A concentration gradient of KOAc is predicted to drive diffusiophoresis of a negatively charged colloid down gradient. The diffusivities of the K^{+} and Cl^{−} ions are approximately the same, meaning that KCl has *β* ≈ 0, reducing Eq. (A1) to just the chemiphoretic term.

TBAB and NaTPB are chosen as extreme cases. Tetrabutylammonium is a large, slowly diffusing cation, giving a large negative *β* when paired with the fast diffusing Br^{−} ion. Conversely, tetraphenylborate is a large, slowly diffusing anion, giving a large positive *β* when paired with the Na^{+} ion. Therefore, any diffusiophoretic contributions to coagulant gelation may be most clearly evident with this pair of coagulants.

### C. Macroscopic dipping

Macroscopic coagulant dipping experiments are performed to mimic the industrial process. This is a gravimetric approach in which the wet gel thickness is estimated from the change in mass of a coagulant-coated former following its immersion in and withdrawal from a latex dispersion.

The coagulant is deposited on a glass substrate as described in the supplementary material and the coagulant-coated former is attached to the clamp of a TA.XTplus Texture Analyzer (Stable Microsystems), which measures the downward force. The former is immersed in the latex dispersion, diluted to 20 wt. % solid content, following the protocol described in the supplementary material. A typical force–displacement curve is shown in Fig. 2(a), and the process is illustrated in [(b)–(d)].

During immersion, the coagulant induces gelation of the latex on the former [Fig. 2(c)]. Following withdrawal, the measured force is greater than before immersion [Fig. 2(a)]. This excess force is due to the wet gel mass and is used to estimate gel thickness under the assumption that the latex density is the same as that of water.^{3}

Dipping experiments are performed with Ca(NO_{3})_{2}, Al(NO_{3})_{3}, KCl, NaNO_{3}, TBAB, and NaTPB. Control experiments are performed without a coagulant to measure the mass of latex picked up by the glass substrate alone. With each coagulant, four areal coagulant densities, *N*, are targeted: 0.01, 0.1, 0.25, and 0.45 mol/m^{2}. For each experiment, the actual coagulant density is measured from the mass of the coagulant-coated former. For each coagulant at each areal density, experiments are performed with dwell times of 1, 5, 10, 15, and 20 min. Each dipping experiment is performed twice, with the exception of KCl, which is not repeated.

### D. Microscopic observation

A new technique is developed to directly visualize coagulant gelation of latex *in situ*. Since the latex particles are small (diameter $\u223c130nm$) and dispersed at a relatively high volume fraction, they cannot be observed via light microscopy with single particle resolution. Therefore, the latex dispersion is doped with a low concentration of fluorescently labeled polystyrene colloids of diameter 1 *μ*m, which are clearly visible in epifluorescence microscopy. These tracer particles have *ζ* potential ∼−50 mV^{33} and are, therefore, expected to undergo diffusiophoresis in the same direction as the latex particles.

Thin cells incorporating a coagulant source at one side are constructed following the protocol in the supplementary material. Side view schematics of the experimental geometry are shown in Figs. 3(a) and 3(b) and the coagulant source is visible on the left hand side of the micrograph in (c). The thickness of the coagulant layer is exaggerated for clarity in Fig. 3(a). In reality, it is unknown and likely varies between experiments but is only a few *μ*m thick.

The empty sample chamber is placed on a light microscope (Olympus BX3M) and positioned with the edge of the coagulant source oriented approximately vertically at the left of the field of view, as shown in Fig. 3(c). Image acquisition takes place under blue light epifluorescence illumination at 20× magnification at either 0.5 or 1 fps depending on experimental duration.

Figures 3(a) and 3(b) illustrate the experiment. Immediately following loading, the suspension of latex (small magenta) and tracer (large green) particles contact the coagulant source. Coagulant diffuses from the source, forming a concentration gradient. As a consequence, latex coagulates near the source, and a gel grows outwards. The existence of the gel is inferred from tracer dynamics. Within the gel, tracers are fixed in place, while outside the gel they are mobile. This is evident in the trajectories shown in Fig. 3(c) and the supplementary material—movies.

Directional gel growth is quantified using tracer trajectories extracted from the micrograph series using standard algorithms^{34} implemented in the R programming language. Each image sequence is rotated and cropped such that the coagulant source is vertically oriented at *x* = 0. The time *t* = 0 is defined as the first frame of the video in which the latex dispersion fills the field of view.

The images are divided into vertical strips of width 5 *μ*m, illustrated by the vertical dashed lines in Fig. 3(c). Within each strip, the average tracer displacement, Δ*r*, in an interval Δ*t* = 10 s is computed as a function of time. Figure 3(d) shows example displacement profiles as a function of distance from the coagulant source, *x*, at three different times in a single experiment. All three profiles show small mean displacements at small *x*, close to the coagulant source. This is indicative of the gel fixing tracers in place. Outside the gel, the average displacement grows, indicating mobile tracers in the ungelled latex dispersion. The transition between the two regions is not always sharp, indicating some spatial heterogeneity in gel growth and perhaps that gelation is not an instantaneous process, but that tracers become increasingly confined as the latex gel forms and strengthens around them.

The tracer displacement profiles are reduced to a single curve of gel thickness, *L*, as a function of time for each experiment. A moving average of each displacement profile is computed over three points [solid lines in Fig. 3(d)] to smooth the data. A displacement threshold is chosen at Δ*r* = 0.75 *μ*m [the black horizontal dashed line in Fig. 3(d)], and the largest value of *x* at which the moving average displacement profile intersects this threshold from below is taken as the estimate of gel thickness corresponding to a given displacement profile. In this way, the gel thickness as a function of time, *L*(*t*), is estimated for each experiment. The full *L*(*t*) corresponding to the displacement profiles shown in Fig. 3(d) is shown in Fig. 3(e).

## III. RESULTS

### A. Macroscopic dipping

Gel thicknesses measured in macroscopic dipping experiments are summarized in Fig. 4, where (a) shows *L* as a function of dwell time, *t*, at coagulant density *N* = 0.25 ± 0.05 mol/m^{2} and (b) shows *L*(*N*) for *t* = 1200 s. The thickest gels are formed using Ca(NO_{3})_{2}, followed by Al(NO_{3})_{3}, then the monovalent coagulants. Ca(NO_{3})_{2} forms thicker gels more quickly than Al(NO_{3})_{3}, despite the lower *c** of Al(NO_{3})_{3}. Therefore, *c** cannot be the only parameter controlling coagulant gelation. This will be explored in Sec. IV.

For Ca(NO_{3})_{2} and Al(NO_{3})_{3}, *L*(*t*) increases over dwell times between 1 and 20 min. However, monovalent coagulants show little change in *L* after 1 min. Comparison with the measurements made without the coagulant (black) shows that a gel *is* formed with TBAB, NaNO_{3}, and KCl. Existing data show that rapid increases in *L* occur at very short dwell times and a limiting thickness is approached within a few minutes.^{3,12,14} This is consistent with our measurements using monovalent coagulants, which show that *L* is independent of *t* for *t* > 60 s.

The data for NaTPB are indistinguishable from those measured without coagulant, suggesting that NaTPB does not form a gel for *t* up to 20 min. NaTPB is expected to induce latex diffusiophoresis away from the former, which may explain this inhibition of gel formation. Furthermore, of the monovalent coagulants, the thickest gels are formed with TBAB, which is expected to cause the strongest up-gradient diffusiophoresis.

### B. Microscopic observation

Gel thicknesses as a function of time, measured via microscopic observation of tracer particle motion, are shown in Fig. 5. Data are colored according to the coagulant, as indicated in the legends, and are averaged over at least 3 and up to 11 repeated experiments. Panel (a) shows the series of nitrate coagulants: NaNO_{3}, Ca(NO_{3})_{2}, and Al(NO_{3})_{3}, for which cation valences are 1^{+} for NaNO_{3}, 2^{+} for Ca(NO_{3})_{2}, and 3^{+} for Al(NO_{3})_{3}. In this series, *c** decreases from 600 mol m^{−3} for NaNO_{3} to 2 mol m^{−3} for Al(NO_{3})_{3}. Panel (b) shows the series of calcium coagulants: Ca(NO_{3})_{2}, CaSO_{4}, and CaCO_{3}, for which *c*_{sat} decreases from 5460 mol m^{−3} for Ca(NO_{3})_{2} to 0.13 mol m^{−3} for CaCO_{3}. Panel (c) shows the remaining monovalent coagulants with a range of anticipated diffusiophoretic properties. The maximum measurable gel thickness is limited by the microscope field of view and is ∼600 *μ*m.

Consistent with dipping experiments, the thickest and most quickly growing gels are obtained using Ca(NO_{3})_{2}. Figure 5(a) shows that NaNO_{3} and Al(NO_{3})_{3} induce slower gel growth than Ca(NO_{3})_{2}, but at approximately the same speed as one another. This is despite Al(NO_{3})_{3} having *c** three hundred times smaller than NaNO_{3}, reaffirming that *c** alone cannot be controlling coagulant gelation.

Figure 5(b) shows that coagulant solubility also affects gel growth. CaSO_{4} is significantly less soluble than Ca(NO_{3})_{2} and causes significantly slower gel growth. CaCO_{3} is only marginally soluble and causes no observable gelation. This is likely because CaCO_{3} is sufficiently insoluble that coagulant concentration never exceeds *c**.

Figure 5(c) shows that data for KCl and NaCl are approximately superposed and cause gel growth at approximately the same rate as NaNO_{3} and Al(NO_{3})_{3}. TBAB is slightly slower, and KOAc and NaTPB are slower again. Unlike CaSO_{4}, the slowest coagulants in Fig. 5(c) are both soluble to molar concentrations, so it is unlikely to be solubility that limits their gel growth. Both NaTPB and KOAc have *β* > 0 and are predicted to induce latex diffusiophoresis down the concentration gradient, away from the coagulant source. This may inhibit gel growth with these coagulants.

A qualitative comparison between Figs. 5 and 4 reveal some important differences between the experimental procedures. Macroscopic experiments show no change in *L* with dwell time for monovalent coagulants and increase toward a plateau for multivalent coagulants. This contrasts with microscopic experiments, in which none of the coagulants reach a thickness plateau, despite the longer experimental duration. When dipped, monovalent coagulants reach their maximum thickness at *t* ∼ 1 min, while in the microscopic experiments, the same coagulants cause gel growth for tens of minutes.

The key difference between these experiments is the coagulant source. In dipping experiments, the coagulant source is only a thin layer. By contrast, the microscopic experiments have a large reservoir of available coagulant. It is proposed that the gel thickness plateau in the dipping experiments is a consequence of the finite amount of coagulant quickly diffusing away from the former and the coagulant concentration everywhere dropping below *c**, depleting the source. In the microscopic experiments, the coagulant source is not depleted within the experimental duration and so the gels continue to grow for up to 1 h. This is explored in Sec. IV.

When the coagulant is not depleted, it is anticipated that *L*(*t*) is determined by coagulant diffusion. Diffusive transport typically scales with *t*^{1/2}, and therefore, the data in Fig. 5 are fitted with the form *L*(*t*) = *A*_{exp}*t*^{1/2}, where *A*_{exp} is a constant. These fits are shown as solid or dashed black lines in Fig. 5. For most of the coagulants, these fits do an acceptable job describing gel growth. However, for instance, the data for CaSO_{4} in Fig. 5(b) are more closely described by the *t*^{1/3} form (the red dashed line). Diffusion in this system is discussed in Sec. V A, but since the *t*^{1/2} fits are acceptable for most of the data, we suggest that coagulant diffusion controls gel growth when the coagulant is abundant.

## IV. MODELING

Two simple models are developed to describe the two experimental procedures. Both models are predicated on simple solutions to the one-dimensional diffusion equation for coagulant concentration, *c*, treating the electrolyte coagulant as though it were a single species whose diffusive transport is governed by a single diffusion coefficient, *D*. The key assumption is that latex instantaneously forms a gel wherever *c* ≥ *c**. In the microscopic experiments, the concentration gradient is initially two-dimensional, extending vertically in *z* as well as laterally in *x*. The time at which the root mean squared displacement of a molecule with diffusion coefficient *D* ∼ 10^{−9} m^{2} s^{−1} is equal to the chamber depth is ∼16 s, and we measure gel thickness over lengthscales larger than the chamber depth. Therefore, beyond a few tens of seconds, it is reasonable to consider the coagulant concentration to be independent of *z* and the problem to be one dimensional.

### A. Macroscopic dipping: Finite coagulant model

Groves and Routh^{3} modeled coagulant dipping as a diffusion problem by assuming fixed *c* at the surface of the former, *x* = 0. If gelation occurs everywhere that *c* ≥ *c**, then the gel *L* grows with *t*^{1/2}. This assumes an infinite source of the coagulant, which the authors recognize is unrealistic and does not match their experimental observations. In their work, coagulant depletion is treated by calculating the time for complete dissolution of coagulant, and assuming that *c* evolves according to the infinite coagulant model on timescales shorter than this dissolution time. On longer timescales, they alter the *x* = 0 boundary condition from fixed concentration to zero flux. This gives a numerical description of dipping with a finite coagulant, which is found to not match their experimental observations. Consequently, Groves and Routh develop an alternative approach employing an infinite coagulant source and treating latex aggregation as being limited by a reaction between coagulant and a stabilizing surfactant. The resulting reaction-diffusion equations are solved numerically. Introducing a coagulant “sink” in this way can account for the measured gel thickness plateau. However, this approach predicts a limiting gel thickness in both finite and abundant coagulant scenarios, contrary to our experimental observations.

Since the hypothesis here is that the limiting gel thickness is primarily a consequence of coagulant depletion, we take a different approach. Our initial condition is an infinitesimal coagulant film of areal density *N* mol/m^{2} at *x* = 0 and *c* = 0 everywhere else. Space is restricted to positive *x* and the total amount of coagulant is conserved. Under these conditions, the one-dimensional diffusion equation for coagulant concentration is solved by

This expression is a good approximation as long as $t>N2/\pi Dcsat2$ but produces unphysical concentrations, *c* > *c*_{sat}, at shorter times. However, for the coagulants and *N* investigated, $N2/\pi Dcsat2\u226445s$, and therefore, this timescale is shorter than the shortest experimental dwell time. Consequently, we expect this form to be a good approximation for the experimental conditions. The key advantage of Eq. (1) is that it is analytic and produces a tractable model.

Concentration profiles of the form in Eq. (1) are illustrated in Fig. 6(a) at three times. Curves are labeled *t*_{1}, *t*_{2}, and *t*_{3}, where *t*_{3} > *t*_{2} > *t*_{1} > 0. As time proceeds, the concentration at *x* = 0 drops, modeling the depletion of coagulant at the former. Irreversible gelation occurs instantaneously wherever *c* > *c**. Figure 6(a) indicates *c** with a horizontal line. At *t*, the gel thickness, *L*(*t*) is the location at which *c*(*x*) = *c**. As the coagulant diffuses from the former, this location initially moves outwards. However, since a finite amount of coagulant is diffusing into infinite space, at sufficiently long times, the concentration everywhere drops below *c**, as is the case with the green *t*_{3} line in Fig. 6(a).

Setting Eq. (1) equal to *c** and rearranging gives

This function initially increases to a maximum *L*_{max} at *t* = *t*_{max} and then decreases toward 0. For long times, *L*(*t*) has no real solution, representing the situation where *c* < *c** everywhere. However, since gelation is irreversible, only the increasing part of this function is important, and once *L* = *L*_{max}, it remains at this plateau for all *t* > *t*_{max}. The inset in Fig. 6(a) shows the finite coagulant model prediction for *L*(*t*).

The limiting gel thickness predicted by this model is

and is attained at *t*_{max}, given by

*L*_{max} and *t*_{max} depend only on the characteristics of the coagulant–latex system. Therefore, given infinite time, thicker gels will be formed when more coagulant is used (large *N*) and when *c** is small. However, coagulant dipping in experiment or manufacturing does not occur at infinite *t*. The dwell time required to reach *L*_{max} increases with $(N/c*)2$, more quickly than *L*_{max}, and also depends on the diffusion coefficient governing coagulant transport.

Consequently, the model reveals that comparing experiments using different coagulants at fixed *t* is not the most insightful way to assess their performance. At fixed *t* < *t*_{max}, two coagulants with different *c** and *D* will have progressed through their gelation processes by different amounts—one will be closer to *L*_{max} and *t*_{max} than the other. Therefore, the experimental data should be compared in nondimensional form, rescaling *L*(*t*) by *L*_{max} and *t* by *t*_{max}. The model then predicts a master curve given by

This curve is shown as the black solid line in Fig. 6(b).

*L*_{max} and *t*_{max} are calculated for each experiment using the measurements of *c** in Table I, the values of *D*_{+} in Table II, and measurements of *N* derived from the mass of the coagulant-coated formers. Using the cation diffusion coefficient at infinite dilution likely overestimates the rate of diffusive transport and will be discussed in Sec. V A. The points in Fig. 6(b) show the nondimensionalized experimental data. These are compared to the master curve predicted by the model (black line). NaTPB is omitted as no gel was formed with this coagulant. Also omitted are data for the lowest coagulant coverage, *N* ∼ 0.01 mol/m^{2}, as the mass of such a small amount of coagulant could not be accurately measured.

The experimental data follow the shape of the theoretical master curve, increasing for *t*/*t*_{max} < 1 and rolling over to a plateau for *t*/*t*_{max} > 1. The comparison between the theory and experiment does not require any fitting and comes directly from characteristics of the coagulant–latex system. This agreement suggests that our simple model captures the dominant physics of coagulant dipping and that depletion of a finite coagulant source is the most important phenomenon governing the growth of the gel to a limiting thickness. The additional processes considered by Groves and Routh^{3} likely have a smaller effect on gel growth.

However, the experimental data do not fall perfectly onto the theoretical master curve. Nondimensionalization is based on two independent quantities: *N*/*c** and *D*. *N* and *c** are directly measured and are considered reliable. We, therefore, propose that the offset between the model and the data is a consequence of not knowing the relevant diffusion coefficient. In Fig. 6, the cation diffusion coefficient at infinite dilution, *D*_{+}, is used. However, this is unlikely to be correct for diffusion at high coagulant concentration through a growing porous gel. This is discussed further in Sec. V A.

### B. Microscopic observation: Semi-infinite coagulant source model

Since the microscopic experiments do not exhibit a plateau in gel thickness, a simpler modeling approach is taken. The initial condition for *c*(*x*) is a step function,

This defines a semi-infinite coagulant source at initial concentration *c*_{0}. *x* is defined between −*∞* and *∞*, and *x* = 0 is the edge of the source. The one-dimensional diffusion equation for *c* is solved by

where erf is the error function. This is identical to the infinite coagulant solution of Groves and Routh,^{3} with *c*(0, *t*) = *c*_{0}/2 and is illustrated in Fig. 7(a) for three different times, *t*_{3} > *t*_{2} > *t*_{1} > 0.

Again assuming instantaneous gelation wherever *c* > *c**, this model predicts

This form is shown in the inset in Fig. 7(a) and represents a *t*^{1/2} scaling, *L*(*t*) = *A*_{mod}*t*^{1/2}. The constant $Amod=4Derf\u221211\u22122c*c0$ is a combination of a diffusion coefficient and the ratio *c**/*c*_{0}.

*c** is measured and given in Table I. The simplest estimate for the relevant diffusion coefficients is the cation diffusion coefficients at infinite dilution, given in Table II. In the experiments, the coagulant source is confined to a very thin region above the source coverslip. The exact depth of this region is unknown but is assumed to be a few micrometers. Therefore, a reasonable estimate of *c*_{0} is the saturation concentration, *c*_{sat}, as given in Table I.

Combining these values according to Eq. (8) for each coagulant gives the model predictions shown in Figs. 7(b)–7(d). Data are grouped in the same way as Fig. 5 and lines are colored according to the legends. The experimental data mostly follow a *t*^{1/2} growth, as predicted by the model. The comparison between experimental data and model prediction is provided by the bar charts in Figs. 7(e) and 7(f), which show the prefactor $Amod=4Derf\u221211\u22122c*c0$ predicted by the model, (e), and the experimental fit parameter *A*_{exp}, defined by *L*(*t*) = *A*_{exp}*t*^{1/2}, (f).

Aspects of the data are accurately captured by the model. Ignoring the anomalous prediction for KOAc (discussed below), the fastest growth is predicted for Ca(NO_{3})_{2}. Despite its lower *c**, slower growth is predicted with Al(NO_{3})_{3}, which is consistent with the experiment. With the exception of KOAc, the sequence of growth rates is accurately predicted. No gel growth is predicted for CaCO_{3} as its *c*_{sat} means *c* never exceeds *c**. This matches the experiments. Slow growth is predicted for CaSO_{4} and NaTPB, also matching the experiments. TBAB is predicted to cause the next fastest growth, followed by KCl and NaCl, which are predicted to cause gelation at approximately the same rate. These predictions also match the experimental measurements. Finally, Al(NO_{3})_{3} and NaNO_{3} are predicted to drive growth at an intermediate rate between NaCl and Ca(NO_{3})_{2}. This does not quite match the experimental observations, which show Al(NO_{3})_{3} and NaNO_{3} causing growth at approximately the same rate as NaCl and KCl. Thus, there are a few minor inconsistencies between the model and experiments, but the trend is generally captured.

Although the model is qualitatively relatively successful, it does not quantitatively match the experimental data. The model consistently over-predicts gel thickness by 4–5 times, suggesting that it does not account for some aspects of coagulant gelation. However, given the simplicity of the model, we consider less than an order of magnitude disparity to be a good agreement. As noted previously, the most uncertain physical quantity is the diffusion coefficient, and using *D*_{+} is again likely an overestimate.

Figure 5 shows that KOAc is a poor coagulant in the experiment, resulting in slow gel growth. This is at odds with the model, which predicts the fastest gel growth for KOAc. The origin of this difference is likely the assumption that *c*_{0} = *c*_{sat}. Table I shows that KOAc is significantly more soluble than any of the other coagulants—its *c*_{sat} is nearly five times that of NaCl. Therefore, the amount of KOAc in the experiment is likely insufficient to create a saturated solution and *c*_{0} < *c*_{sat} in the KOAc experiments. However, since the exact depth of the coagulant source region of the experimental sample cell is unknown, a more accurate estimate of *c*_{0} for KOAc cannot be made.

The fact that such a simple model predicts the *t*^{1/2} scaling measured in the experiment and also largely captures the qualitative differences between so many different coagulants indicates that this model contains the key physics of coagulant gelation. When gelation is driven by a semi-infinite source of coagulant, gel growth is dominated by coagulant diffusion and follows a *t*^{1/2} scaling. The differences between coagulants can largely be described by differences in their diffusion coefficient, *c**, and saturation concentration.

## V. DISCUSSION

Both sets of experiments show that Ca(NO_{3})_{2} forms the thickest gels most quickly. This is aligned with industrial practice. This article began by naïvely speculating that Al(NO_{3})_{3} might perform better than Ca(NO_{3})_{2} by virtue of its lower *c**, but experiments have shown this is not the case.

When dipping with finite coagulant, depletion of coagulant leads to a maximum gel thickness, *L*_{max}, attained at dwell time *t*_{max}. *L*_{max} and *t*_{max} depend on the amount of coagulant, *c**, and the relevant diffusion coefficient [Eqs. (3) and (4)]. Ca(NO_{3})_{2} outperforming Al(NO_{3})_{3} is a question of timescales. Using *c** and *D*_{+} given in Tables I and II and choosing *N* = 0.1 mol/m^{2}, the model predicts that Ca(NO_{3})_{2} will reach *L*_{max} = 6.9 mm in *t*_{max} ≈ 8.4 h, while Al(NO_{3})_{3} will grow to *L*_{max} = 2.4 cm by *t*_{max} ≈ 6.3 days. Thus, Al(NO_{3})_{3} may outperform Ca(NO_{3})_{2} on very long timescales, but such long timescales are irrelevant to manufacturing by coagulant dipping.

When the coagulant is abundant, as in the microscopic experiments, Ca(NO_{3})_{2} outperforms Al(NO_{3})_{3} by virtue of Eq. (8), which shows that the rate of gel growth does not simply increase as *c** decreases. Instead growth depends on $4Derf\u221211\u22122c*c0$. Therefore, it is the combination of the diffusivity, *c**, and the coagulant source concentration that determines the growth rate. The inverse error function diverges at 1, and therefore, the growth rate is maximized when 2*c**/*c*_{0} is minimized. When the coagulant source is a saturated solution, the fastest growth is predicted for coagulants that have both a low *c** and high solubility. Behavior is controlled by the ratio *c**/*c*_{sat}, rather than by *c** alone. Tables I and II show that although Al(NO_{3})_{3} has a lower *c** than Ca(NO_{3})_{2}, it is also less soluble and the Al^{3+} ion diffuses more slowly than the Ca^{2+} ion. The combination of all parameters results in the prediction that the gel grows most rapidly when using Ca(NO_{3})_{2} as the coagulant.

The experimental data shown in Fig. 5(b) and the model predictions in Fig. 7(c) reinforce the importance of solubility in coagulant gelation. Ca(NO_{3})_{2} and CaSO_{4} have the same *D*_{+} and very similar *c**, but CaSO_{4} causes gel growth at a much slower rate than Ca(NO_{3})_{2}, by virtue of its greatly reduced solubility. Coagulant solubility is not a part of the finite coagulant model, and, as mentioned previously, unphysically high coagulant concentrations are produced for *t* < *N*^{2}/*π Dc*_{sat}. As a consequence, the finite coagulant model increasingly over-predicts the rate of gel growth the less soluble a coagulant is.

Related to solubility is the dynamic process of dissolution, which is not considered in either model. In the experiment, some salts (NaCl, NaNO_{3}, KCl, CaSO_{4}, CaCO_{3}, and NaTPB) remain crystalline and dry after being deposited onto glass, while the others [Ca(NO_{3})_{2}, Al(NO_{3})_{3}, TBAB, and KOAc] absorb moisture from the atmosphere and are, therefore, already in concentrated solution at the start of the experiment. Dissolution of the dry coagulants is not instantaneous. A complete theory of coagulant dipping should consider the time it takes to dissolve an initially solid coagulant source. This modification would reduce the predicted gel growth rate for dry coagulants.

### A. Diffusion coefficient

Since both models are based on the diffusion equation, the most important parameter to discuss is the relevant diffusion coefficient. This is also the most difficult parameter to estimate. In Sec. IV, the cation diffusivity at infinite dilution, *D*_{+}, has been employed, as the cation screens the negative surface charge of the latex particles, and *D*_{+} is easily obtained from the literature.^{28} However, there are multiple problems with this assumption.

First, coagulant gelation does not occur at infinite dilution and diffusivity depends on solution concentration. This may be fairly moderate, such as a reported decrease in the diffusivity of Na^{+} and Li^{+} ions of around 10% up to concentrations of 1M.^{35} Or this may be more severe, such as a reported threefold decrease in Al^{3+} diffusivity in a 2M solution.^{36} Incorporating a concentration dependent diffusivity into the models presented here makes them analytically intractable, as the constant and uniform *D* must be replaced by a diffusivity that varies in space and time. In addition, the concentration dependence of diffusivity for each of the coagulants is not readily available in the literature.

A further shortcoming is considering the diffusive transport of only the cation when the coagulant consists of a cation–anion pair. Although both ions have their respective infinite dilution diffusion coefficients, at finite concentration, they interact with one another through electrostatics.^{37} Due to this interaction, the faster diffusing ion is slowed down due to its interaction with the slower diffusing ion, and vice versa. The result is the ions diffusing together at a rate determined by an effective ambipolar diffusivity.

Perhaps, the most significant reason to be skeptical of using *D*_{+} is the existence of the gel. A gel is a porous material with a tortuous microstructure, which hinders diffusion. As the gel grows during the experiment, accounting for this effect would also give a diffusion coefficient that varies in time and space. This effect was a key observation in the research of Stewart.^{19}

Diffusion in a porous medium is often treated by reducing the diffusion coefficient by a factor between 2 and 6.^{3,38} This factor is related to the microstructure of the gel, which is unknown in our experiments. However, Fig. 8 shows the effect of reducing *D*_{+} by a factor of 3 in the finite coagulant model. This plausible modification of the diffusion coefficient improves the agreement between the model (solid line) and experiments (points), suggesting that our use of *D*_{+} is indeed the weakest assumption of our modeling. However, since there is no independent justification of this reduction, this is presented only as a plausible explanation of the quantitative discrepancy between our models and experiments. Groves and Routh^{3} find that reducing the diffusion coefficient in their infinite coagulant source model to account for tortuosity cannot reconcile their experimental data and their model. However, Fig. 8 shows that combining a simple tortuosity correction with our analytical treatment of a finite coagulant source results in a significantly improved agreement between the experiment and theory.

Figure 5 shows that the *t*^{1/2} scaling is not a good fit for CaSO_{4}. If the rate of diffusive coagulant transport varies in space and time due to the aforementioned reasons, the gel growth is not expected to follow the simple *t*^{1/2} scaling derived assuming a constant and uniform diffusion coefficient. In particular, the rate of gel growth may decrease with time, as diffusion is increasingly hindered by the growing gel. This is qualitatively consistent with the discrepancy between experimental data and *t*^{1/2} fits observed for, e.g., CaSO_{4}. It is, therefore, anticipated that both models should become increasingly inaccurate for longer times.

### B. Diffusiophoresis?

Finally, we come to the question of whether diffusiophoresis plays a role in coagulant gelation. Our results are inconclusive in this regard. The models in Sec. IV do not consider diffusiophoresis and are successful in reproducing many of the experimental observations. Since microscopic experiments visualize tracers and not the latex particles, it is impossible to directly observe latex particle diffusiophoresis.

The slowest gel growth is observed with NaTPB and KOAc, which both have *β* > 0 and are anticipated to drive latex away from the coagulant source by diffusiophoresis. This could be considered evidence for diffusiophoresis, but in the case of NaTPB, the model also predicts slow growth without appealing to diffusiophoresis. On the other hand, the model predicts rapid gel growth with KOAc, contradicting the experimental measurements. It was previously suggested that this was a consequence of the high solubility of KOAc, but it is also possible that a repulsive diffusiophoretic motion also contributes to this disparity between modeling and experiment. Further experiments with a modified protocol will be required to definitively determine the role of diffusiophoresis in coagulant dipping.

## VI. CONCLUSION

Two sets of experiments investigating the gelation of latex due to a source of electrolyte coagulant are reported. Macroscopic dipping experiments, mimicking industrial coagulant dipping, are compared to novel microscopic experiments directly imaging gel growth. In dipping experiments, gels grow to a limiting thickness. For monovalent coagulants, this occurs within 1 min but takes longer for multivalent coagulants. In microscopic experiments, gel growth mostly follows a *t*^{1/2} scaling, indicative of growth dominated by coagulant diffusion. This difference is attributed to having a finite amount of coagulant in dipping experiments and an abundant source of coagulant in microscopy experiments. Both sets of experiments show that, of the coagulants considered, Ca(NO_{3})_{2} results in the fastest growing and thickest gel films.

Two simple models based on coagulant diffusion are developed to describe the experiments. These models successfully reproduce many of the features of the experimental data, including the thickness plateau in dipping experiments and the superiority of Ca(NO_{3})_{2} as a coagulant. This suggests that coagulant gelation of latex is primarily governed by coagulant diffusion, *D*, and the ratio of the critical coagulation concentration, *c**, to the amount of coagulant available, *c*_{0}. The fastest gel growth occurs when the ratio *c**/*c*_{0} is minimized, and when the coagulant is abundant, *c*_{0} = *c*_{sat}, the coagulant saturation concentration. Experiments and modeling suggest that the plateau in gel thickness with an increasing dwell time that is regularly observed^{3,12–14} is primarily a consequence of coagulant depletion at the former.

More accurate determination of the relevant physicochemical properties of the latex–coagulant system is anticipated to further improve the quantitative predictive power of the models. In particular, the relevant diffusion coefficient cannot be the cation diffusion coefficient at an infinite dilution that has been employed throughout this article but must account for diffusion through a growing porous medium, ambipolar diffusion of both anion and cation, and diffusion at high concentration.

Phenomena including diffusiophoresis, coagulant dissolution, and the kinetics of latex aggregation are also expected to have an effect on the rate of gel growth. Although this research was initiated to investigate the effect of diffusiophoresis on coagulant gelation, the results are inconclusive in that regard, and the experimental data can largely be explained without appealing to diffusiophoresis. The beauty of the models we have developed is that they are simple and analytical. Their accuracy in predicting the qualitative behavior of the experiments suggests that coagulant diffusion is the dominant effect in coagulant gelation, and the depletion of a finite amount of coagulant is the limiting effect in coagulant dipping.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the following: dynamic light scattering and electron microscopy characterization of latex particles; determination of critical coagulation concentrations; additional details of macroscopic dipping experiments; details of sample cell construction for microscopy; control experiments without coagulant; and sample movies showing data from microscopic experiments with and without coagulant.

## ACKNOWLEDGMENTS

The authors are grateful for insightful conversations with Dr. Robert Groves and Professor Alex Routh (University of Cambridge). The authors acknowledge the contribution of Yuxiu (Phil) Chen, who made *ζ* potential measurements, and Alexia Beale for critical reading of the manuscript. Acknowledgment is made to the donors of the American Chemical Society Petroleum Research Fund for partial support of this research. The authors thank Synthomer for the donation of the latex dispersions. Funding for I.W. was provided by the EPSRC through a New Horizons grant (Grant No. EP/V048473/1).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

J.L.K., R.P.S., and I.W. conceived of the research project. S.N. performed macroscopic dipping experiments and made critical coagulation concentration measurements. I.W. developed and performed microscopic imaging experiments and theoretical modeling and wrote the first draft of the manuscript. All authors analyzed and discussed data and contributed to writing the manuscript.

## DATA AVAILABILITY

The data that support the findings of this study are openly available in Figshare at http://doi.org/10.6084/m9.figshare.19665069.

### APPENDIX: ELECTROLYTE DIFFUSIOPHORESIS

Coagulants used for dipping are electrolytes, and the diffusiophoretic velocity of a charged, spherical particle in a linear concentration gradient of symmetric electrolyte is given by^{5,32,39}

where *c* is the electrolyte (coagulant) concentration; *ζ* is the particle zeta potential, related to its surface charge density; *ɛ* is the absolute permittivity of the solvent; *η* is the solvent viscosity; *k*_{B} is Boltzmann’s constant; *T* is the absolute temperature; *Z* is the electrolyte valence; *e* is the elementary electronic charge; and *β* is a parameter that encodes the difference in diffusivity between the cation and anion, defined by

where *D*_{+} and *D*_{−} are the diffusion coefficients of the cation and anion, respectively. *β* may be positive or negative depending on whether the cation or anion diffuses more quickly, and a larger difference in ion diffusivities results in a larger absolute value of *β*.

Examining Eq. (A1) reveals that there are two contributions to *U*_{DP} for an electrolyte gradient. The first term is known as the electrophoretic term and is proportional to *β*. This is a consequence of the spontaneous electric field (diffusion potential) generated to maintain electroneutrality when the cation and anion diffuse at different rates. Since *β* can be positive or negative depending on which ion diffuses more quickly, the electrophoretic contribution to *U*_{DP} may be directed either up or down the electrolyte gradient.

The second term is known as the chemiphoretic term and arises due to the osmotic pressure gradient along the particle surface. This term is always positive and directed up the electrolyte gradient. Thus, the direction of the net diffusiophoretic velocity is set by the ion diffusivity difference parameter, *β*, and the relative magnitudes of the two contributions.

It is important to bear in mind that Eq. (A1) is strictly valid only for dilute solutions in which the Poisson–Nernst–Planck equations accurately describe the electrolyte concentration, and for an infinitesimally thin electrical double layer (compared to the particle radius).^{40} Furthermore, Eq. (A1) describes only symmetric electrolytes, for which the cation and anion have the same valence, *Z*. This is not the case for calcium nitrate, the industrial coagulant of choice. More sophisticated treatments of diffusiophoresis for concentrated^{37,41} and multivalent electrolytes^{42} exist but are less insightful for the purposes of this article. Therefore, while Eq. (A1) informs our thinking about diffusiophoresis, we do not necessarily expect it to exactly describe any diffusiophoretic effects observed during coagulant dipping.

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