The root of the clogging problem in microfluidics begins with events that take place at the molecular scale, while keeping in mind that much larger scales of tens-to-hundreds of microns ultimately result in device failures. If the thermodynamics predict reactive particle formation, then nucleation theory is the appropriate starting point to understand how to control particles in microfluidics [15,34,35,36,70]. The solubility product of each compound should be estimated for a given reaction solvent and the corresponding supersaturation ratio, the driving force for nucleation, approximated. Precipitation can be anticipated when supersaturation exists, where the primary rate of nucleation is dependent on the induction time; the time from when supersaturated conditions have been achieved to the appearance of the first detectable nucleus. It is common for nucleation kinetic predictions to vary by orders of magnitude due to heterogeneous nucleation and other molecular level variations. The presence of impurities, the reactor surface characteristics, axial and radial concentration and temperature gradients, and molecular interactions between compounds can all influence the prediction of the nucleation kinetics. If a reasonable estimate for the induction time has been experimentally validated, then a quantity useful in evaluating the potential for reactor fouling is the ratio of the induction time, t_ind, relative to the time necessary for a molecule to flow from the entrance to the exit of a reactor (i.e., the residence time τ), α, which can be expressed as [62]:

Such a dimensionless quantity is, as a worst case, useful for reducing the risk of particle accumulation on continuous reactor surfaces. In continuous operation, the large surface-to-volume ratios of microreactors are exposed to infinite residence time. Evaluation of Equation (1) can offer insight, from a heterogeneous nucleation perspective, on whether or not production goals can be met before any material accumulation begins. Classical nucleation theory and extensive models derived from it have been developed for inorganic and organic crystallization [71,72,73,74].

A reasonable starting point once bulk particles grow to a critical size (or in the absence of particle growth) is the consideration of inertial forces, which influence the motions of particles in microscale laminar flow. Stokes number, the ratio of the viscous to the inertial forces acting on a flowing particle, written as [75]

can be estimated to approximate the potential for particle-to-wall interactions. When St < 1, particles do not spend enough time in the vicinity of a microchannel wall to undergo inertial impaction. When St > 1, particle trajectories along streamlines are modified and collisions with microreactor walls possible. When the inertial forces are favorable for impaction, or the onset of crystal growth on reactor surfaces takes place, estimation of the constriction rate, r_c, gives the ratio of the constriction rate to the convective transport rate through the reactor and in the axial direction, written as

Equation (3) is useful to approximate the accumulation time scale relative to the residence time, which similar to Equation (1) can help to identify a process window in which the accumulation influences production. Comprehensive understanding of each materials class, crystalline or amorphous and organic or inorganic, is needed to most appropriately engineer continuous reactors that undergo steady-state operation while manufacturing microfluidic products.

Colloidal attraction and repulsion theory describes the significance of sub-micron particle interaction energies and their role on aggregate formation and/or wall deposition. Consider two colloidal particles flowing along laminar streamlines in a microfluidic device (i.e., Reynolds number (Re) < 1000) and in the axial direction (i.e., the z-direction) according to Figure 1. Neglecting body forces such as gravity, an assumption that generally holds for sub-micron particles, approximates the settling time relative to the axial particle motion through the microfluidic device to be large. Thus, particle-to-particle, particle-to-wall, and hydrodynamic interactions represent the dominant forces acting on each particle, as illustrated in Figure 1.

The total particle-to-particle colloidal force, F_col^p-p, is made up of three components with the first being van der Waals attraction for particles of identical surface chemistry, V_A(h), [76,77]

The particle-to-particle colloidal force is also a function of the electrostatic interaction energy, V_R(h), [75] and the Born interaction energy, V_B(h) [78]. Each of the three interaction energies are influenced by the interparticle distance, h, and the particle diameters, a. When particle-to-wall attraction or repulsion exists, quite possible in complex synthesis routes that often involve ionic species, the particle-to-wall colloidal force should be considered, F_col^p-w, expressed in terms of the electrostatic free energy (G_R) [79] and the free energy of adhesion (G_A) [80], by

where κ is the Debye parameter, ε the permittivity, k Boltzmann constant, e the electronic charge, T the temperature, and x₀ the minimum equilibrium distance between the particle and the wall. Here, note that the particle-to-wall separation distance, x, influences the magnitude of the force, and thus only particles flowing near the reactor wall are expected to encounter such an energy field. Although our example is a simplified scenario, estimation of the magnitudes of the particle-to-particle and the particle-to-wall interaction forces yields important information for the design of continuous reactor operating conditions. As an example, the ratio of the interparticle and the particle-to-wall colloidal attraction magnitudes,

offers mechanistic understanding of the potential solids handling challenges. When λ > 1, aggregation may take place, leading to microscopic blockages. When λ < 1, there exists the potential for the accumulation of particles on surfaces. In a system of particles, Equation (6) can be expressed as the sum of the particle-to-particle and particle-to-wall attractive interactions. A force balance in equilibrium and without gravitational forces can be rearranged as

One observes that η< −1 when the net hydrodynamic force acting on a particle exceeds the colloidal forces. For η > −1 values, low Reynolds number flow for instance, colloidal particle forces can generate agglomerates or accumulate on microreactor surfaces. The relationship of Equation (7), however, yields a deeper understanding when a comparison of the important time scales is made. When α< 1, defining the work time for each force reveals the expression,

The hydrodynamic work time relative to the colloidal attraction work time, ω, only considering particle attraction, describes the probability of either aggregation or wall accumulation in terms of the overall residence time. For example, when λ ~ 1 and t_x >> t_h (or the distance x >> h), then ω_w = (|F^h|Lτ)/(|F_col^p-w|xt_x) and ω_w values < 1 imply that particles are transported axially throughout a microreactor before accumulation has had enough time to occur. Similarly, when λ~1 and t_h >> t_x (or the distance h >> x), then ω_p = (|F^h|Lτ)/(|F_col^p-p|ht_h) and ω_p values < 1 imply that particles are transported axially throughout a microreactor before aggregation has had enough time to occur. Such a relationship is critical in understanding when particle aggregates that remain in the bulk flow exit a microreactor in time to eliminate any pressure losses. Additional theoretical models relating particle aggregation physics can be applied to predict the growth rates for a population of suspended particles.

Three fundamentally different mechanisms, (1) bridging, (2) constriction, or (3) random detachment, can result in pressure losses during the micro-to-meso-scale laminar flow of particulates. A combination of any of the three mechanisms can propagate when chemical reactions occur. Each of the three mechanisms may also arise in the absence of particle formation or growth.

Particle motions tracking lamellae are possible when the net hydrodynamic forces are greater than the colloidal and inertial forces (i.e., η < −1, ω < 1, and St < 1). A reduction in the cross-sectional flow path under such conditions can lead to the simultaneous arrival of particles at the constriction point, and in turn generate bridging events; a statistical or velocity controlled phenomenon depending on the operating regime [76,81]. Bridging can also occur in the absence of reduced cross-sections. As shown in Figure 2a, bridging constrains fluid flow (not considering attraction or repulsion) for channel width-to-particle size aspect ratios, D/a < 10. The bridging, also shown in the microreactor image of Figure 2b for NaCl formation, is imminent when St > 1, η > −1, and ω > 1. Under such conditions particles traveling near the reactor wall deposit, forming dendrites that eventually bridge.

Fluid flow may also be constrained when microreactor cross-sections diminish with time, [82] illustrated in Figure 2(c,d). The constriction occurs when particles deposit on surfaces or when nucleation followed by growth takes place. In either scenario, wall build-ups present a challenge to continuous production in microfluidic systems because process shutdowns are needed to remediate accumulated material. The random detachment of the accumulated material, another potential challenge, is virtually unpredictable yet can generate macroscopic blockages that severely limit fluid flow.

Passive techniques are those that do not impose any external forces beyond the steady-state energy required to commence and sustain fluid flow. Hydrodynamic flow focusing, an example of passive particle manipulation, has been demonstrated as an effective technique to order micron-sized particles in microfluidics. The streaming of particle suspensions (e.g., polystyrene microspheres) is made possible via flow through spiral geometries that establish Dean flow [83]. The existence of hydrodynamic force vectors tangential to the flow cross-section enables centerline particle equilibrium (i.e., the ratio of the inertial lift to the drag force magnitudes is optimized), and thus the longitudinal ordering and lateral flow focusing of random particles is possible [83]. The Dean number, expressed as

estimates the existence of cross-sectional velocity components where R is the radius of microchannel curvature. In straight microchannels, De = 0, indicating the absence of a drag force due to the transverse Dean flows [83]. The magnitude of the transverse Dean flow increases, as seen in Equation (9), with a decreasing radius of curvature, and thus the microchannel geometry can be microfabricated to spatially position particles of different momentums [47]. A challenge with passive flow focusing, however, is its ability to manage particle concentrations encountered in fine chemical production when particle-to-particle and particle-to-wall attraction exists (i.e., η > −1, and ω > 1). Furthermore, particle momentum can lead to migration across streamlines, creating particle-to-particle and particle-to-wall collisions.

The isolation of particles that tend to agglomerate or stick to surfaces has been accomplished by the encapsulation of reactions within dispersed droplets, as highlighted in Figure 3(a,b). Protein crystals (Figure 3a), [67] or other insoluble compounds such as polymers (Figure 3b), [38,39] are limited in their interaction with microchannel surfaces in the presence of an immiscible barrier between the phases. Interestingly, immiscible gas-liquid interfaces [84,85,86] have been demonstrated as possible separation techniques; micron-sized particles accumulate at the lower surface-free-energy interfaces (see Figure 3c) [87,88,89]. In a system of excess particles, shown in Figure 3d, armored bubbles form when liquid-liquid immiscible interfaces are occupied entirely by particulates [90,91]. Establishing the equilibrium illustrated in Figure 3d is possible when the colloidal forces are stabilized, which depends on the ionic strength of the liquid media; an important design consideration and a challenge to control in complex heterogeneous reaction pathways exposed to convective forces.

Active particle techniques involve external forces beyond the steady-state energy required to commence and sustain fluid flow. Pulse flow could be considered an active particle manipulation technique and has been exploited to engineer dry powder injections within microfluidic devices [31]. For D/a aspect ratios that do not bridge, inert gas has been used to carry dry pharmaceutical powders within microscale devices [31]. The working principle allows the formation of a fluidized particle bed in microfluidics when particle-to-particle attractive interactions do not exist [31].

The use of light and electron gradients also represents active particle manipulation techniques. For example, optical particle displacement has been accomplished by generating bubbles with sufficiently intense laser light [92]. Single and tandem bubbles generated by infrared and green lasers displaced particles such as polystyrene microspheres [92]. As shown in Figure 4a, particles were displaced tens-of-microns and within microseconds [92]. Optical force switching, another active technique utilizing light, has been engineered for the microfluidic separation of mammalian cells [93]. The diminishing light intensity with increasing cross-sectional dimensions presents a challenge, in general, for scaling up optical particle control techniques, especially when considering fine chemical manufacturing. Dielectrophoretic particle separations and cell isolations are also possible when an electric field is imposed upon a system of charged particles traveling through a microfluidic device. The governing principles that establish a force balance for the steady-state separation of particles have previously been described [41,94]. As seen in the schematic of Figure 4b, the fabrication of devices with deflection electrodes enables particle sorting where the threshold velocity for the displacement of different sized latex microspheres scales with the voltage amplitude (Figure 4c) [94].

Magnetism, another active technique, also enables the separation of particulates in microfluidics [2,24,30,32,33,95,96]. When particles flow by a magnetic field (see Figure 5a), the magnetically induced velocity vector, u_mag, is described by the magnetic force, F_mag, exerted on the particle relative to the viscous drag force, F_vis, by [30]

where Δχ is the magnetic susceptibility difference between the particle and the liquid medium, V_p the particle volume, B the externally applied magnetic flux density, ∇B the flux density gradient, μ the liquid viscosity, and K the permeability of a vacuum [30]. One can infer from Equation (10) that magnetically driven particle motions occur when the magnitude of the magnetic force exceeds the viscous force. Equation (10) additionally illustrates the critical dependence of magnetically driven particle separations on the particle radius, a, and the magnetic characteristics of the particle (i.e., Δχ by χ_p). There exists potential for magnetic separations in fine chemical and some inorganic crystal manufacturing. Aggregates (i.e., agglomerates) behave as larger particle sizes in a magnetic field, and thus the continuous, laminar separation of the aggregates is possible [30]. In practice, a distribution of particle sizes (superparamagnetic particles, e.g., Dynabeads), shown in Figure 5b, were deflected each corresponding to the parallel outlets of Figure 5b [30]. Scaling up magnetic particle separations for the continuous flow manufacturing of many microfluidic processes remains a challenge as the viscous force magnitude increases with Re and the delivery of adequate magnetic fields, a capital cost consideration, requires design innovations.

It is also readily known that acoustic waves displace particles traveling along lamellae [18,20,21,22,23,25,26,27,97,98,99,100,101,27,97]. High frequency (e.g., MHz) acoustic standing waves force particles to pressure nodes in laminar flow, which has been demonstrated for the separation of red blood cells from lipids [23,25,26,102]. A particle traveling through an acoustic standing wave experiences a primary force, F_a¹, described by

where p₀ is the acoustic pressure amplitude, V_c the particle volume, λ_a the acoustic wavelength, k the wavenumber, and x the distance from a pressure node. The primary force is considered significant when a phase difference parameter, φ, expressed as

is non-zero [23,25,26,102]. The terms ρ_i and β_i are the density and compressibility of the solvent (s) and the particle (p), respectively.When particle diameters are less than half the wavelength, the acoustic streaming of suspended particles is possible on-chip, shown in Figure 6a, by integrating ultrasonic transducers with microfluidic devices. Secondary forces, also significant when a concentrated system of particles scatters acoustic standing waves, have been described in a more detailed review [22].

Acoustic waveforms in the range of kHz, for example ultrasonic cleaning equipment, have been demonstrated as effective in carrying out microchemical transformations that generate solids [15,35,64]. On-chip ultrasound has been engineered to manage the NaCl formation during palladium-catalyzed bond forming reactions [13]. As an example, the by-product MnO₂ formed during the Nef oxidation leads to flow-induced clogging, which can be prevented using ultrasound (shown in Figure 6b) [103]. Overall, the integration of acoustics with microreactors has tremendous potential that fosters a need for scale-up innovations.

Advancements continue to be made towards the innovation of new techniques for solids handling in microreactors. Continuous-stirred tank reactors and mechanically agitated flow cells [104] have been demonstrated as effective methods to carry out particle forming reactions. Nevertheless, the design of solids handling reactors only represents part of the challenge; an entire chemical process is made up of many different unit operations each where solids in continuous flow could be problematic.