5.2. Theoretical modelling/studies

5.1.4. Effect of external field on particle deposition

120 Microfluidics and Nanofluidics

dering transport or attachment.

5.1.5. Effect of temperature on particle deposition

In the previous literature, transport of micron-size particles is normally simplified as a mass transfer process that is majorly affected by Brownian diffusivity. However, gravity or a constant body force exerted on the particles could have significant influence on the deposition process, even for such tiny colloidal particle size (less than 1 μm). Yiantsios and Karabelas [88] reported that gravity played a key role for the deposition rate of spherical Ballotini glass particles (diameter: 1.8 μm). Gravity could control the particle transport boundary layer in a horizontal narrow channel under laminar flow over a fairly wide range of flow rate. They [89] conducted further experiments on the effects of physicochemical and hydrodynamic conditions by using dilute microsized glass particle suspensions (diameter: 1.5 μm) in a parallel-plate channel. They concluded that gravity was a determining factor for deposition at low wall shear stresses. While the hydrodynamic wall shear stress was increased, particle deposition rates were noticeably decreased because of hydrodynamic lift or drag forces hin-

Stamm et al. [90] experimentally examined the initial stage of cluster growth in a particle-laden flow in a microchannel and investigated the parametric effects of a void fraction, flow shear strain rate and channel height to particle diameter ratio. Thereafter, Gudipaty et al. [91] studied the cluster formation of colloidal particles in a PDMS microchannel and found that the clusters were initiated by the attachment of individual flowing particle onto the bottom surface. However, they have not either addressed the physical mechanism of the initial particle attach-

Unni and Yang [92] experimentally investigated the dynamics of particle deposition in an electroosmotic flow using video microscope, and reported that the increased surface coverage at higher salt concentrations resulted from weakened EDL repulsion with particles being adsorbed onto the channel surface. Hydrodynamic blocking became relatively weaker with lower electric

Most of the researches about deposition of micro-/nanoparticles in microchannel are conducted in room temperature environment, seldom with high bulk temperature or temperature gradient, which is a crucial factor for heat exchangers in reality. Yan et al. [73] investigated the effect of bulk solution temperature on particle deposition in a microchannel under wellcontrolled temperature conditions using a microfluidic temperature control device. To the best knowledge of the authors, this was the first attempt to study the thermal effect on the deposition of colloidal particles in an aqueous dispersion onto a microchannel wall. It is found that the temperature of solution has a considerable effect on the particle deposition in microchannels. The static particle deposition rate (Sherwood number) has been measured over a range of temperatures between 20 and 70C. It is found that the Sherwood number is monotonically increased up to 265%, with the solution temperature within the test range. They developed a deterministic model based on the Derjaguin-Landau-Verwey-Overbeek theory with consideration of temperature dependence, and found that by increasing the solution

field strengths because the surface blocking was majorly caused by electrical interactions.

ment to the surface or observed the adherence process in the experiments.

Spielman and Friedlander [93] theoretically analysed the effect of the electrical double layer on particle deposition based on the equation of convective diffusion in an external force field. They reported that the deposition process of Brownian particles was equivalent to ordinary convective diffusion in the bulk with a first-order surface reaction at the collector. With respect to the net interaction potential, a formula can be derived for the surface reaction coefficient.

Another analytical model was developed by Adamczyk and Van De Ven [94] for particle deposition kinetics onto the surfaces of parallel-plate and cylindrical channels (Figure 12). As governing equation, the mass transport equation was formulated with consideration of electrical double layer force, van der Waals force, and external forces such as gravity. The 'perfect sink' boundary condition was applied to solve the mass transport equation. Different dimensionless parameters (Ad, Pe, Dl, and Gr) were proposed to account for dispersion, convection/ diffusion, electrical double layer, and gravity, respectively.

Studies on particle deposition onto permeable surfaces are practically meaningful for membrane filtration industry. Song and Elimelech [95, 96] theoretically studied this phenomenon in

Figure 12. Schematics of parallel-plate and cylindrical channels [94].

a system of parallel-plate channel. The convection-diffusion equation was solved numerically with consideration of lateral transport. The lateral transport would be induced by inertial lift, permeation drag, and transport that are determined by the collective effect of gravity force and surface forces. Parametric studies were systematically carried out regarding the initial particle deposition rate, including the effects of permeation velocity, cross flow velocity, particle size, and solution ionic strength.

Since the distribution of surface charge is not uniform in practice, Nazemifard et al. [97, 98] performed a trajectory analysis of particles close to a micropatterned charged plate based on the radial impinging jet setup for the influences of surface charge heterogeneity on deposition efficiency and particle trajectory. The surface charge heterogeneity was controlled by concentric bands with varied properties, such as geometric dimension and types of surface charges. In their analysis, the van der Waals, electrostatic double layer force, hydrodynamic force, and gravity, have been taken into consideration. Due to the coupled effects of colloidal and hydrodynamic forces, the deposition efficiencies and particle trajectories were remarkably influenced by surface charge heterogeneity when a particle flowed radially away from the stagnation point in the radial jet impingement setup. This analysis demonstrates how the existing particle transport models could possibly be modified in consideration of chemical heterogeneity and additional surface interactions. Similarly, Chatterjee et al. [99] applied the convection-diffusion-migration equation (Eulerian model) with fully developed Poiseuille flow velocity profile to investigate the transport of particles in patchy heterogeneous cylindrical microchannels (Figure 13). They evaluated the effects of surface chemical heterogeneity on particle transport and deposition, and found that particles tend to travel further along the microchannel in the heterogeneous channels compared to the homogeneously favourable channels.

Using a soft-sphere discrete element method, Marshall [100] studied the caption of particles by wall and particle aggregation in a microchannel. According to their results, the particle lift-off from the wall was caused by adhesion and collision with particle aggregate or a passing particle when a single particle with a large size was attached to the wall. The fluid forces were not the direct reason for the particle lift-off.

Unni and Yang [92] developed an electrokinetic particle transport model in a parallel-plate microchannel (Figure 14) based on the Stochastic Langevin equation. They incorporated random Brownian motion of colloidal particles and the hydrodynamic, electrical, DLVO colloidal interactions into the equation. Based on the developed model, particle trajectories can be stochastically simulated using Brownian dynamics simulation and the surface coverage was calculated under a range of electrical and physicochemical conditions.

In the absence of repulsive energy barrier, Jin et al. proposed concurrent modelling for the effects of interaction forces and hydrodynamics on particle deposition on rough spherical surfaces [101]. The model considered the hydrodynamic retardation functions and flow field profiles. Their works showed that the hydrodynamic effects remarkably affect the particle deposition behaviours which were different from the predictions based on DLVO forces alone. In addition, surface roughness played an important role in particle deposition experiment/ simulation. They conducted another study on deposition of colloidal particles and reported a non-monotonic, non-linear effect of nanoscale roughness on particle deposition without

energy barrier using both the convection-diffusion model and parallel-plate chamber experimental system [102]. Their results showed particle deposition flux could reach the minimum

Figure 13. Schematic of patchy heterogeneous cylindrical microchannels. (a) 3D schematic representation of positively charged particles deposition along the the microchannel. (b) 2D axisymmetric view of the microchannel geometry. (c) The

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value when a critical roughness size was provided.

zoomed view of one pitch length [99].

a system of parallel-plate channel. The convection-diffusion equation was solved numerically with consideration of lateral transport. The lateral transport would be induced by inertial lift, permeation drag, and transport that are determined by the collective effect of gravity force and surface forces. Parametric studies were systematically carried out regarding the initial particle deposition rate, including the effects of permeation velocity, cross flow velocity, particle size,

Since the distribution of surface charge is not uniform in practice, Nazemifard et al. [97, 98] performed a trajectory analysis of particles close to a micropatterned charged plate based on the radial impinging jet setup for the influences of surface charge heterogeneity on deposition efficiency and particle trajectory. The surface charge heterogeneity was controlled by concentric bands with varied properties, such as geometric dimension and types of surface charges. In their analysis, the van der Waals, electrostatic double layer force, hydrodynamic force, and gravity, have been taken into consideration. Due to the coupled effects of colloidal and hydrodynamic forces, the deposition efficiencies and particle trajectories were remarkably influenced by surface charge heterogeneity when a particle flowed radially away from the stagnation point in the radial jet impingement setup. This analysis demonstrates how the existing particle transport models could possibly be modified in consideration of chemical heterogeneity and additional surface interactions. Similarly, Chatterjee et al. [99] applied the convection-diffusion-migration equation (Eulerian model) with fully developed Poiseuille flow velocity profile to investigate the transport of particles in patchy heterogeneous cylindrical microchannels (Figure 13). They evaluated the effects of surface chemical heterogeneity on particle transport and deposition, and found that particles tend to travel further along the microchannel in the heterogeneous channels

Using a soft-sphere discrete element method, Marshall [100] studied the caption of particles by wall and particle aggregation in a microchannel. According to their results, the particle lift-off from the wall was caused by adhesion and collision with particle aggregate or a passing particle when a single particle with a large size was attached to the wall. The fluid forces were

Unni and Yang [92] developed an electrokinetic particle transport model in a parallel-plate microchannel (Figure 14) based on the Stochastic Langevin equation. They incorporated random Brownian motion of colloidal particles and the hydrodynamic, electrical, DLVO colloidal interactions into the equation. Based on the developed model, particle trajectories can be stochastically simulated using Brownian dynamics simulation and the surface coverage was

In the absence of repulsive energy barrier, Jin et al. proposed concurrent modelling for the effects of interaction forces and hydrodynamics on particle deposition on rough spherical surfaces [101]. The model considered the hydrodynamic retardation functions and flow field profiles. Their works showed that the hydrodynamic effects remarkably affect the particle deposition behaviours which were different from the predictions based on DLVO forces alone. In addition, surface roughness played an important role in particle deposition experiment/ simulation. They conducted another study on deposition of colloidal particles and reported a non-monotonic, non-linear effect of nanoscale roughness on particle deposition without

calculated under a range of electrical and physicochemical conditions.

and solution ionic strength.

122 Microfluidics and Nanofluidics

compared to the homogeneously favourable channels.

not the direct reason for the particle lift-off.

Figure 13. Schematic of patchy heterogeneous cylindrical microchannels. (a) 3D schematic representation of positively charged particles deposition along the the microchannel. (b) 2D axisymmetric view of the microchannel geometry. (c) The zoomed view of one pitch length [99].

energy barrier using both the convection-diffusion model and parallel-plate chamber experimental system [102]. Their results showed particle deposition flux could reach the minimum value when a critical roughness size was provided.

curves are varied with the bulk solution temperature and the energy barrier reduces consider-

Figure 15. Dimensionless particle-wall interaction potential curves for three different solution temperatures (T = 293.15, 323.15, and 343.15 K). Curves are obtained with the temperature dependent Hamaker constant, zeta potential and the thickness of electric double layer. Inset shows the energy barrier that polystyrene particles should overcome to achieve

Furthermore, Yan et al. [71] developed a simplified one-dimensional mass transport model (i.e., Eulerian model) for calculating the particle deposition rate in microchannel flows at elevated temperatures. A schematic of microparticles flowing through a microchannel at an elevated temperature (T) is shown in Figure 16. For a dilute spherical particle monodispersion in the absence of chemical reactions, the interactions between particles are neglected. The deposition rate of particles from solution onto the microchannel surface at the steady state

where j is the particle flux vector (the number of particles per unit area per second), the particle flux comprises three components as shown in Eq. (12): Brownian diffusion, fluid convection, and migration under external forces. Besides, the particle flux j can be decomposed into two

concentration, D is the diffusion coefficient tensor, u is the particle velocity, F indicates external forces exerted on the particles, kB is the Boltzmann constant, and T<sup>0</sup> is the reference temperature. By an appropriate scaling with the dimensionless parameters given in Table 1, the mass

DF kBT<sup>0</sup> n 

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<sup>x</sup>x⃑<sup>þ</sup> <sup>j</sup>

¼ 0, (12)

<sup>y</sup>y⃑, <sup>n</sup> is the local particle number

ably with increasing the bulk solution temperature as shown in the inset of Figure 15.

can be described by the general convection-diffusion equation as

deposition onto the PMMA surface at different temperatures for the DI water case [73].

portions along the x-direction and y-direction, j ¼ j

transport equation can be expressed in a dimensionless form as

∇ � j ¼ �∇ � ð Þþ D � ∇n ∇ � ð Þþ un ∇

Figure 14. (a) Schematic of electrokinetic transport of a particle in a parallel-plate channel. (b) Forces acting on a moving particle in the vicinity of an attached particle and the channel wall [92].

Based on the Derjaguin-Landau-Verwey-Overbeek theory, Yan et al. [73] developed a qualitative theoretical model with consideration of the temperature effect for the first time. All the driving forces during the whole particle deposition process are temperature dependent, so the particle deposition is determined by the collective effects of the variations of the forces caused by temperature changes. By plotting interaction potential curves and energy barriers, the effect of bulk solution temperature can be clearly seen in Figure 15 that the interaction potential

Figure 15. Dimensionless particle-wall interaction potential curves for three different solution temperatures (T = 293.15, 323.15, and 343.15 K). Curves are obtained with the temperature dependent Hamaker constant, zeta potential and the thickness of electric double layer. Inset shows the energy barrier that polystyrene particles should overcome to achieve deposition onto the PMMA surface at different temperatures for the DI water case [73].

curves are varied with the bulk solution temperature and the energy barrier reduces considerably with increasing the bulk solution temperature as shown in the inset of Figure 15.

Furthermore, Yan et al. [71] developed a simplified one-dimensional mass transport model (i.e., Eulerian model) for calculating the particle deposition rate in microchannel flows at elevated temperatures. A schematic of microparticles flowing through a microchannel at an elevated temperature (T) is shown in Figure 16. For a dilute spherical particle monodispersion in the absence of chemical reactions, the interactions between particles are neglected. The deposition rate of particles from solution onto the microchannel surface at the steady state can be described by the general convection-diffusion equation as

$$\nabla \cdot \mathbf{j} = -\nabla \cdot (\mathbf{D} \cdot \nabla n) + \nabla \cdot (\mathbf{u}n) + \nabla \left(\frac{D\mathbf{F}}{k\_B T\_0} n\right) = 0,\tag{12}$$

where j is the particle flux vector (the number of particles per unit area per second), the particle flux comprises three components as shown in Eq. (12): Brownian diffusion, fluid convection, and migration under external forces. Besides, the particle flux j can be decomposed into two portions along the x-direction and y-direction, j ¼ j <sup>x</sup>x⃑<sup>þ</sup> <sup>j</sup> <sup>y</sup>y⃑, <sup>n</sup> is the local particle number concentration, D is the diffusion coefficient tensor, u is the particle velocity, F indicates external forces exerted on the particles, kB is the Boltzmann constant, and T<sup>0</sup> is the reference temperature. By an appropriate scaling with the dimensionless parameters given in Table 1, the mass transport equation can be expressed in a dimensionless form as

Based on the Derjaguin-Landau-Verwey-Overbeek theory, Yan et al. [73] developed a qualitative theoretical model with consideration of the temperature effect for the first time. All the driving forces during the whole particle deposition process are temperature dependent, so the particle deposition is determined by the collective effects of the variations of the forces caused by temperature changes. By plotting interaction potential curves and energy barriers, the effect of bulk solution temperature can be clearly seen in Figure 15 that the interaction potential

Figure 14. (a) Schematic of electrokinetic transport of a particle in a parallel-plate channel. (b) Forces acting on a moving

particle in the vicinity of an attached particle and the channel wall [92].

124 Microfluidics and Nanofluidics


Table 1. Dimensionless parameters utilised for the mass transport equation [71].

$$-f\_4(H)\mathbb{R}^2\frac{\partial^2 \overline{n}}{\partial X^2} + \left\{ \text{Pf}\_3(H)(H+1)[2-(H+1)\mathbb{R}] \right\} \frac{\partial \overline{n}}{\partial X} + \frac{\partial}{\partial H} \left( -f\_1(H)\frac{\partial \overline{n}}{\partial H} + f\_1(H)\overline{F}\overline{n} \right) = 0. \tag{13}$$

All the forces acting on particles are along the vertical direction (y-axis in Figure 16), and they are colloidal forces (van der Waals force Fvdw and electric double layer force Fedl) and external forces (gravity FG, hydrodynamic lift force FL, and thermophoretic force FT). The thermophoretic force is neglected because of its low magnitude compared to other forces in the study [73]. The non-DLVO forces are excluded for simplicity. Hence, the total force (F = Fy) can be treated as a scalar in the following sections. The dimensionless total force (F) is calculated as

$$
\overline{F} = \overline{F}\_y = \overline{F}\_{cdl} + \overline{F}\_{vdw} + \overline{F}\_G + \overline{F}\_L. \tag{14}
$$

along the x-direction in downstream was found to be insignificant [96, 103]. Consequently, the

Figure 16. Schematic of microparticle transport in a microchannel. The forces on the particle are van der Waals force (Fvdw), gravity force (FG), electric double layer force (Fedl), thermophoretic force (FT), and hydrodynamic lift force (FL). The radius of the particle is ap, the minimum separation distance between the particle surface and the bottom surface of the microchannel is h, the flow velocity distribution is U(y), and the applied temperature gradient in the microchannel is ∇ T

Eq. (17) refers to the 'Perfect sink' boundary condition. H<sup>0</sup> indicates the minimum dimensionless particle-wall distance (h0/ap). This boundary condition has been widely used in particle deposition studies, and it assumes that all particles are irreversibly adhered to the solid surface when they move into the primary energy minimum (PEM) region. It can be explained by that the attractive van der Waals force in the PEM region becomes much stronger than the repulsive electric double layer force. Thus, the particles would deposit onto the solid surfaces. The second boundary condition, given by Eq. (18), states a natural boundary condition for the particle concentration. The particle concentration gets close to that in the bulk phase when

<sup>∂</sup><sup>H</sup> <sup>þ</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>H</sup> Fn 

¼ 0: (16)

n ¼ 0, at H ¼ H<sup>0</sup> (17)

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n ¼ 1, at H ¼ H∞: (18)

mass transport equation can be further simplified to one-dimensional as

<sup>∂</sup><sup>H</sup> �<sup>f</sup> <sup>1</sup>ð Þ <sup>H</sup> <sup>∂</sup><sup>n</sup>

∂

Eq. (16) can be solved with the boundary conditions as

(the figure is not drawn to scale) [71].

the particle-wall distance becomes an 'infinite' distance.

The potential of each force is computed by integrating the force over the separation distance between a particle and the channel surface (H). The total potential (V) can be obtained by using superposition of individual potentials: the van der Waals potential (Vvdw), the EDL potential (Vedl), and the potentials contributed by gravity and lift force (VG, VL),

$$
\overline{V} = \overline{V}\_{\text{eff}} + \overline{V}\_{\text{vdw}} + \overline{V}\_{\text{G}} + \overline{V}\_{\text{L}} = \left\{ \overline{F}\_{\text{eff}} dH + \int \overline{F}\_{\text{vdw}} dH + \int \overline{F}\_{\text{G}} dH + \int \overline{F}\_{\text{L}} dH \right\} \tag{15}
$$

As the scaled particle ratio (<sup>R</sup> <sup>¼</sup> ap=w) in Eq. (13) is about the order of 10�<sup>3</sup> (ap <sup>≪</sup> <sup>w</sup>) for their experiment system, the particle diffusion term (first term in Eq. (13)) and the particle convection term (second term in Eq. (5)) in x-direction can be neglected for the low Peclet number (<1) in the present study. Besides, the variation of the deposition rate for a dilute particle solution Particle Deposition in Microfluidic Devices at Elevated Temperatures http://dx.doi.org/10.5772/intechopen.78240 127

Figure 16. Schematic of microparticle transport in a microchannel. The forces on the particle are van der Waals force (Fvdw), gravity force (FG), electric double layer force (Fedl), thermophoretic force (FT), and hydrodynamic lift force (FL). The radius of the particle is ap, the minimum separation distance between the particle surface and the bottom surface of the microchannel is h, the flow velocity distribution is U(y), and the applied temperature gradient in the microchannel is ∇ T (the figure is not drawn to scale) [71].

along the x-direction in downstream was found to be insignificant [96, 103]. Consequently, the mass transport equation can be further simplified to one-dimensional as

$$\frac{\partial}{\partial H}\left(-f\_1(H)\frac{\partial \overline{n}}{\partial H} + f\_1(H)\overline{F}\overline{n}\right) = 0.\tag{16}$$

Eq. (16) can be solved with the boundary conditions as

�<sup>f</sup> <sup>4</sup>ð Þ <sup>H</sup> <sup>R</sup><sup>2</sup> <sup>∂</sup><sup>2</sup>

126 Microfluidics and Nanofluidics

n

<sup>∂</sup>X<sup>2</sup> <sup>þ</sup> Pef <sup>3</sup>ð Þ <sup>H</sup> ð Þ <sup>H</sup> <sup>þ</sup> <sup>1</sup> ½ � <sup>2</sup> � ð Þ <sup>H</sup> <sup>þ</sup> <sup>1</sup> <sup>R</sup> � � <sup>∂</sup><sup>n</sup>

Table 1. Dimensionless parameters utilised for the mass transport equation [71].

Dimensionless parameters Expression Scaled particle flux Jx <sup>¼</sup> ap <sup>j</sup>

Scaled particle concentration <sup>n</sup> <sup>¼</sup> <sup>n</sup>

Scaled external force <sup>F</sup> <sup>¼</sup> apF

Scaled interaction energy <sup>V</sup> <sup>¼</sup> <sup>V</sup>

Scaled particle-wall separation distance <sup>H</sup> <sup>¼</sup> <sup>h</sup>

Scaled distance from channel entrance <sup>X</sup> <sup>¼</sup> <sup>x</sup>

Scaled particle ratio <sup>R</sup> <sup>¼</sup> ap

Peclet number Pe <sup>¼</sup> <sup>3</sup>Uavg <sup>a</sup><sup>3</sup>

in the following sections. The dimensionless total force (F) is calculated as

(Vedl), and the potentials contributed by gravity and lift force (VG, VL),

V ¼ Vedl þ Vvdw þ VG þ VL ¼

∂X þ

x n∞, pD<sup>∞</sup>

n∞, <sup>p</sup>

kBT<sup>0</sup>

kBT<sup>0</sup>

w

w

ap <sup>¼</sup> <sup>y</sup>�ap ap

p 2w2D<sup>∞</sup> Hydrodynamic retardation functions [101] <sup>f</sup> <sup>1</sup>ð Þ¼ <sup>H</sup> <sup>1</sup> � <sup>0</sup>:399 exp ð Þ� �0:14869<sup>H</sup> <sup>0</sup>:601 exp �1:2015H<sup>0</sup>:<sup>92667</sup> � �

, Jy <sup>¼</sup> ap <sup>j</sup> y n∞, pD<sup>∞</sup>

All the forces acting on particles are along the vertical direction (y-axis in Figure 16), and they are colloidal forces (van der Waals force Fvdw and electric double layer force Fedl) and external forces (gravity FG, hydrodynamic lift force FL, and thermophoretic force FT). The thermophoretic force is neglected because of its low magnitude compared to other forces in the study [73]. The non-DLVO forces are excluded for simplicity. Hence, the total force (F = Fy) can be treated as a scalar

The potential of each force is computed by integrating the force over the separation distance between a particle and the channel surface (H). The total potential (V) can be obtained by using superposition of individual potentials: the van der Waals potential (Vvdw), the EDL potential

FedldH þ

As the scaled particle ratio (<sup>R</sup> <sup>¼</sup> ap=w) in Eq. (13) is about the order of 10�<sup>3</sup> (ap <sup>≪</sup> <sup>w</sup>) for their experiment system, the particle diffusion term (first term in Eq. (13)) and the particle convection term (second term in Eq. (5)) in x-direction can be neglected for the low Peclet number (<1) in the present study. Besides, the variation of the deposition rate for a dilute particle solution

ð

FvdwdH þ

ð

FGdH þ

ð

FLdH (15)

ð

∂

<sup>∂</sup><sup>H</sup> �<sup>f</sup> <sup>1</sup>ð Þ <sup>H</sup> <sup>∂</sup><sup>n</sup>

<sup>f</sup> <sup>3</sup>ð Þ¼ <sup>H</sup> <sup>1</sup> � <sup>0</sup>:3752 exp ð Þ� �3:906<sup>H</sup> <sup>0</sup>:625 exp �3:105H<sup>0</sup>:<sup>15</sup> � � <sup>f</sup> <sup>4</sup>ð Þ¼ <sup>H</sup> <sup>1</sup> � <sup>1</sup>:23122 exp ð Þþ �0:2734<sup>H</sup> <sup>0</sup>:8189 exp �0:175H<sup>1</sup>:<sup>2643</sup> � �

F ¼ Fy ¼ Fedl þ Fvdw þ FG þ FL: (14)

<sup>∂</sup><sup>H</sup> <sup>þ</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>H</sup> Fn � �

¼ 0: (13)

$$
\overline{n} = 0, \text{at } H = H\_0 \tag{17}
$$

$$
\overline{n} = 1 \text{, at } H = H\_{\approx}. \tag{18}
$$

Eq. (17) refers to the 'Perfect sink' boundary condition. H<sup>0</sup> indicates the minimum dimensionless particle-wall distance (h0/ap). This boundary condition has been widely used in particle deposition studies, and it assumes that all particles are irreversibly adhered to the solid surface when they move into the primary energy minimum (PEM) region. It can be explained by that the attractive van der Waals force in the PEM region becomes much stronger than the repulsive electric double layer force. Thus, the particles would deposit onto the solid surfaces. The second boundary condition, given by Eq. (18), states a natural boundary condition for the particle concentration. The particle concentration gets close to that in the bulk phase when the particle-wall distance becomes an 'infinite' distance.

Having obtained dimensionless concentration distribution (n) along the dimensionless separation distance (H) in Eq. (6), the particle deposition flux to the channel surface can be found as

$$J\_0 = -f\_1(H\_0) \left(\frac{d\overline{n}}{dH}\right)\_{H=H\_0}.\tag{19}$$

along the direction of temperature gradient with a single-particle resolution. Moreover, a simplified mass transport model (Eulerian model) with consideration of thermal effects has been presented to describe the particle deposition phenomena in microchannels at elevated temperatures based on the Derjaguin-Landau-Verwey-Overbeek theory. Both the theoretical modelling and experimental measurements have shown that the thermal effects have pro-

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Future research in this field lies in the development of investigations on coupling effects of thermal field and other external fields, such as optical, acoustic, and magnetic fields [104–106]. Microfluidic devices and systems offer ideal experimental platforms which provide wellcontrolled external fields applied to the particle deposition process. Especially, the dynamic behaviour of particle deposition under complex external fields can be observed directly with a resolution of micrometre even nanometre (e.g., single molecule detection). With such unique information, trans-scale theoretical modelling which can bridge the gap between particle kinetics in microscale and fouling phenomena in macroscale will be highly appreciated. The current investigations at elevated temperatures rely on either ensemble average value of a population of particles (i.e., average particle deposition rate) or bulk property of fouling (i.e., fouling resistance). When the deposition of single nanoparticle at elevated temperatures can be reliably investigated, researchers will be able to understand the dynamic processes of nanoparticle deposition in microchannels which elucidates particulate fouling of nanofluids in heat

This work has been partially supported by the National Key Research & Development Program of China (2016YFB0401502), the National Natural Science Foundation of China (61574065, 51561135014, and U1501244), Science and Technology Planning Project of Guangdong Province (2016B090906004), Special Fund Project of Science and Technology Application in Guangdong (2017B020240002), PCSIRT Project No. IRT\_17R40, the National 111 Project, the MOE International Laboratory for Optical Information Technologies, and the Cultivation Project of National Engineering Technology Center of Optofluidic Materials and Devices (2017B090903008). Zhibin Yan also acknowledges the financial support by the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China and the Research and Cultivation Fund for Young Faculty supported by South China Normal University (SCNU). Xiaoyang Huang and Chun Yang appreciate the financial support from the Ministry of Educa-

found the impact on particle deposition in microchannels.

exchangers and nanomaterial drug delivery in vivo in details.

tion of Singapore under Grant No. RG80/15 and RG97/13, respectively.

Acknowledgements

Nomenclature

ap particle radius (m)

a<sup>1</sup> radius of interacting particle 1 (m)

Here, J<sup>0</sup> is the particle number flux at H ¼ H0. The negative sign on the right hand side of Eq. (9) indicates that the particle number flux is toward the solid surface. Moreover, the dimensionless particle deposition rate onto the channel surface can be quantified by the Sherwood number

$$\mathbf{Sh}\_{\text{num}} = -\frac{j\_0}{\left(D\_{\text{oc}}n\_{\text{oc},p}/a\_p\right)} = f\_1(H\_0) \left(\frac{d\overline{n}}{dH}\right)\_{H=H\_0}.\tag{20}$$

The thermal effects on the particle deposition rate (i.e., Sherwood number) are influenced by the temperature dependences of all the forces (Fvdw, Fedl, FG, and FL) acting on the particles. For van der Waals force, the Hamaker constant is a temperature dependent parameter. For EDL force, EDL thickness, zeta potential, and relative dielectric constants of materials are varied with the temperature. For gravity and hydrodynamic lift forces, the magnitude of forces is changed due to the variations of density and viscosity of liquid. Details of the calculations for the temperature dependences can be found in [71, 73].

For the first time, based on the DLVO theory with considering the temperature-dependent interactions, a simplified one-dimensional mass transport model was developed and it can serve as a semi-quantitative approach for describing particle deposition phenomena in microchannel flows at elevated temperatures.

### 6. Summary and future prospects

Particle deposition and particulate fouling have been ubiquitous phenomena in natural and industry processes. Thermal effects (i.e., temperature and temperature gradient) on particle deposition are important but always 'ignored' in literatures. Most of the published research works about micro-/nanoparticles deposition in a microchannel were conducted in the room temperature environment, seldom with consideration of elevated bulk temperature or temperature gradient, which is a crucial factor for thermal driven fouling phenomena in reality. Especially, the microscale mechanism of particle deposition in microchannel at elevated temperature was still in its infancy. In this chapter, researches on particle deposition and particulate fouling on surfaces have been extensively reviewed both theoretically and experimentally from the published works. This chapter has summarised relevant concepts of particle deposition, key parameters, and experimental techniques (e.g., device design) as well as theoretical methodologies (e.g., modelling). The physics of particle deposition phenomena under different parametric influences has been discussed in detail. The authors have presented a new microfluidic temperature-gradient device that can be used to directly observe particle deposition along the direction of temperature gradient with a single-particle resolution. Moreover, a simplified mass transport model (Eulerian model) with consideration of thermal effects has been presented to describe the particle deposition phenomena in microchannels at elevated temperatures based on the Derjaguin-Landau-Verwey-Overbeek theory. Both the theoretical modelling and experimental measurements have shown that the thermal effects have profound the impact on particle deposition in microchannels.

Future research in this field lies in the development of investigations on coupling effects of thermal field and other external fields, such as optical, acoustic, and magnetic fields [104–106]. Microfluidic devices and systems offer ideal experimental platforms which provide wellcontrolled external fields applied to the particle deposition process. Especially, the dynamic behaviour of particle deposition under complex external fields can be observed directly with a resolution of micrometre even nanometre (e.g., single molecule detection). With such unique information, trans-scale theoretical modelling which can bridge the gap between particle kinetics in microscale and fouling phenomena in macroscale will be highly appreciated. The current investigations at elevated temperatures rely on either ensemble average value of a population of particles (i.e., average particle deposition rate) or bulk property of fouling (i.e., fouling resistance). When the deposition of single nanoparticle at elevated temperatures can be reliably investigated, researchers will be able to understand the dynamic processes of nanoparticle deposition in microchannels which elucidates particulate fouling of nanofluids in heat exchangers and nanomaterial drug delivery in vivo in details.
