Universal diagram for the kinetics of particle deposition in microchannels

Why do particles, suspended in a fluid and traveling in a channel, deposit onto walls? The question has far-reaching implications in different domains (filtering, syringeability, fouling, etc.). Close to a channel wall, particles are subject to a variety of effects, which control their trajectories: hydrodynamic forces, diffusion, van der Waals adhesion forces, and electrostatic forces. The existing theories and phenomenologies, due to their inherent limitations, and the numerical and experimental studies, due to their scarcity, did not allow thus far to establish a general description of the deposition process. By coupling microfluidic experiments, theory, and numerics, we succeed in establishing a general description of the phenomenon. We discover that the problem is particularly rich. We show the existence of three regimes: van der Waals, Debye, and diffusive, each including various subregimes. Within each main regime, particle deposition is dominated respectively by (attractive) van der Waals forces, (repulsive) electrostatic forces, or diffusion. We establish the scaling laws governing the collector efficiency, $S$, in each regime. The ensemble of the regimes and their transitions can be displayed in diagrams. We focus on the case $\frac{A}{kT}\sim 1$ (the case of most practical interest), in which the diagram involves two dimensionless numbers, $P$ (incorporating the Debye layer characteristics) and ${\xi }_L$ (a function of the flow speed, diffusion constant, and the geometry). In this case, the three main regimes organize around a cantilever beam, in which the vertical support is defined by the condition $P=4e^{-2}$, and the horizontal beam is located at ${\xi }_L=\frac{A}{kT}$, where $A, k$, and $T$ are the Hamaker constant, the Boltzmann constant, and the absolute temperature, respectively. The present work allows understanding of empirical observations thus far left unexplained and provides a paradigm enabling engineering of devices in a way that reduces or enhances particle deposition.

Figure 1

Scheme showing a concentration of particles, $C(x,z)$, of radius $r$ inside a rectangular channel of height $h$. Particles and channel walls in aqueous solutions exhibit an electrostatic double layer of length, ${\lambda }_D$, with $\zeta$ potentials ${\zeta }_w$ (wall) and ${\zeta }_p$ (particle). Fluid flow follows a parabolic profile with near-wall velocity, $U_r$, calculated from a Poiseuille profile of average velocity, $U_P$, and shear effects. The van der Waals adhesion forces act when the particle is within $z$ distance from the wall.

Figure 2

Phase diagram of the four subregimes: (i) diffusive 1, where $S\sim {\xi }_L^{1/3}$; (ii) diffusive 2, where $S\sim {\xi }_L$; (iii) van der Waals 1, where $S={\left(\frac{2A}{3kT}{\xi }_L\right)}^{1/4}$; and (iv) van der Waals 2, where $S={\left(\frac{A}{2.1kT}{\xi }_L\right)}^{1/2}$. The frontier between diffusive 1 and diffusive 2 is ${\xi }_L=1$. The frontier between diffusive 1 and van der Waals 1 is ${\xi }_L\sim {\left(\frac{A}{kT}\right)}^3$. The frontier between diffusive 2 and van der Waals 2 is ${\xi }_L\sim \frac{A}{kT}$. The frontier between van der Waals 1 and van der Waals 2 is ${\xi }_L\sim {\left(\frac{A}{kT}\right)}^{-1}$.

Figure 3

Sketch of $F\left(\eta \right)$ [Eq. (27)] as a function on $\eta$ based on numerical simulation results. The condition $P>P_C$ (dash-dotted red line), where $P_C=4\exp (-2)$, demonstrates a negative force resulting in attractive interaction between two surfaces. The condition $P<P_C$ (solid blue line) demonstrates a positive force within a range of $\eta$. Finally, the condition $P=P_C$ (dashed black line) represents a critical case between two regimes.

Figure 4

The universal diagram of particle deposition represented as ${\xi }_L$ [Eq. (14)] in the $y$ axis as a function of dimensionless number $P$ [Eq. (28)] in the $x$ axis. The transition from diffusive 1 to van der Waals 2 is marked by ${\xi }_L\sim 1$ and that from Debye to diffusive 1 or van der Waals 2 corresponds to $P=4e^{-2}$.

Figure 5

Scheme of the PDMS-based microfluidic chip consisting of parallel rectangular channels, of dimension $hwL$. The channels are flanked by inlet and outlet reservoirs of height $h_{\mathrm{res}}$. The channels are 150 $\mu \mathrm{m}$ apart to avoid cake formation so each channel acts independently from one another.

Figure 6

Experimental results of different deposition regimes. Error bars reflect $95\%$ confidence levels. In all cases, black broken lines represent respective theoretical expressions while solid grey lines represent numerical simulation results performed under the same conditions as the experiments. (a1) van der Waals 2 (vdW) regime showing deposited microparticles at ${\xi }_L=0.0078, 1M$ NaCl $d=5 \mu \mathrm{m}, h=20 \mu \mathrm{m}, w=100 \mu \mathrm{m}, L=400$ $\mu \mathrm{m}, U=1$ mm/s, $\phi =0.3\%$, and $A=8\times {10}^{-21}$ J [14]. Images have been treated to show immobile particles as black spots (a2) $N_A\left(t\right)$ for PS-hydrophobic PDMS, where data follow $S_{vdW}$ [Eq. (16)]. (a3) Histogram distribution (normalized density in semilogarithmic format), showing deposited particles in both channel floor and ceiling, where data follow $y\sim x^{-1/2}$ [Eq. (17)]. (a4) Plot of $S\left(t\right)$ [Eq. (31)] for vdW, where $S\approx 0.09$ ($S<1$), compared with $S_{vdW}$ [black broken line, Eq. (16)] and $S_{\mathrm{diff}}$ [red broken line, Eq. (23)]. (b1) Diffusive 1 (diff) regime showing deposited nanoparticles at ${\xi }_L\sim 35.06, 1M$ NaCl, $d=0.5 \mu \mathrm{m}, h=10 \mu \mathrm{m}, w=100$ $\mu \mathrm{m}, L=400$ $\mu \mathrm{m}, U=1$ mm/s, $\phi =0.003\%, A=8\times {10}^{-21}$. (b2) $N_A\left(t\right)$ for carboxylated PS-hydrophobic PDMS, where data follow $S_{\mathrm{diff}}$ [Eq. (23)]. (b3) Histogram distribution (in semilogarithmic format), where data only show particles deposited on the floor (see Supplemental Material SI-2 [39]). The data follow $y\sim x^{-2/3}$ [Eq. (20)]. (b4) $S\left(t\right)$ [Eq. (31)] for diffusive, where $S\approx 2.7$ ($S>1$), compared with $S_{\mathrm{diff}}$ [red broken line, Eq. (23)] and $S_{vdW}$ [black broken line, Eq. (16)]. (c1) Debye regime showing deposited microparticles on both the floor and the ceiling at ${\xi }_L=0.0078, d=5 \mu \mathrm{m}, h=20 \mu \mathrm{m}, w=100 \mu \mathrm{m}, L=400$ $\mu \mathrm{m}, U=1$ mm/s, $\phi =0.3\%$ at $0.007M$ NaCl. (c2) $N_A\left(t\right)$ for PS-hydrophobic PDMS at 0.007M NaCl. (c3) Histogram distribution at ${\xi }_L=0.0078$ at the end of the experiment. Results show sporadic profiles with few deposited particles. (c4) $S\left(t\right)$ [Eq. (31)] for Debye regime where the theoretical expectation is $S=0$ [black broken line, Eq. (10)].

Figure 7

(a) Universal diagram of particle deposition with both experimental and numerical results presented as ${\xi }_L$ [Eq. (14)], as a function of $P$ [Eq. (28)], as defined by the two-dimensional (2D) theoretical diagram in Fig. 4. In two dimensions, the vertical line is defined by $P=4e^{-2}$, and the horizontal line in terms of ${\xi }_L=1$. In all cases, solid symbols represent experimental points while open symbols are numerics. (b) The universal diagram presented in practical or operational terms in three dimensions with the deposition ratio $S/S_{vdW}$ as a function of $C_{\mathrm{salt}}$ (salt concentration in $M$) and ${\xi }_L$. The parameter $C_{\mathrm{salt}}$ can be extracted from the parameter $P$ [Eq. (28)]. Note that the vdW regime is associated with ${\xi }_L<1$ while the diffusive regime is associated with ${\xi }_L>1$. (c, top) Slice of the plane along $S/S_{vdW}$ as function of $C_{\mathrm{salt}}$, showing the transitions from Debye to van der Waals (blue) and Debye to diffusive (red). Grey lines are theoretical predictions and the grey dashed line is drawn to guide the eye. (c, bottom) Slice of the plane along $S/S_{vdW}$ as function of ${\xi }_L$ showing $S/S_{vdW}\approx 1$ in the vdW regime while showing $S/S_{vdW}\approx {\xi }_L^{-1/6}$ in the diffusive regime. Legend: Circles and inverted triangles denote the Debye regime, where numerics and experiments are performed with these parameters: $h=20 \mu \mathrm{m}, w=100 \mu \mathrm{m}, L=400 \mu \mathrm{m}, \phi =0.3\%$, while varying $U, r$, and relative low $C_{\mathrm{salt}}$ (${10}^{-4}M$–${10}^{-1}M$ NaCl). Inverted red triangle denote the diffusive regime, where numerics and experiments are performed with these parameters: $h=10 \mu \mathrm{m}, w=100 \mu \mathrm{m}, L=400 \mu \mathrm{m}, \phi =0.003\%$, while varying $U, r$, and at relatively high $C_{\mathrm{salt}}$ (0.$1M$–$1M$ NaCl). Blue circles denote the van der Waals regime, where numerics and experiments are performed with these parameters: $h=20 \mu \mathrm{m}, w=100 \mu \mathrm{m}, L=400 \mu \mathrm{m}, \phi =0.3\%, A=8\times {10}^{-21}$ J, while varying $U, r$, and relative high $C_{\mathrm{salt}}$ (0.$1M$–$1M$ NaCl).

Figure 8

Numerical checks of the theoretical predictions for the four subregimes discussed in preceding sections: (a) diffusive 1, (b) diffusive 2, (c) van der Waals 1, and (d) van der Waals 2. Solid lines in all four graphs correspond to theoretical predictions, while inset sketches show the position of the subregime with respect to the diagram shown in Fig. 2. Legend: (a) $\begin{array}{c}▴\end{array}, h=20 \mu \mathrm{m}, \phi =0.3\%, L=400 \mu \mathrm{m}, A=8\times {10}^{-21}$ J, $U=1$ mm/s, $0.1\le r\le 0.5$ $\mu \mathrm{m}$; $\begin{array}{c}◊\end{array}, h=20 \mu \mathrm{m}, \phi =0.3\%, A=8\times {10}^{-21}$ J, $U=1$ mm/s, $r=0.25$ $\mu \mathrm{m}, 40\mu \mathrm{m}\le L\le 10\mathrm{cm}$; $\begin{array}{c}\square \end{array}, \phi =0.3\%, L=400$ $\mu \mathrm{m}, A=8\times {10}^{-21}$ J, $U=1$ mm/s, $r=0.25 \mu \mathrm{m}, 5\mu \mathrm{m}\le h\le 1\mathrm{mm}$. (b) $\begin{array}{c}★\end{array}, h=20 \mu \mathrm{m}, \phi =0.3\%, A=1\times {10}^{-24}$ J, $r=2.5 \mu \mathrm{m}, 0.001\le U\le 100$ mm/s. (c) $\begin{array}{c}◂\end{array}, h=20 \mu \mathrm{m}, \phi =0.3\%, L=400$ $\mu \mathrm{m}, 1\times {10}^{-21} \le A\le 8\times {10}^{-17}$ J, $U=1$ mm/s, $r=0.25 \mu \mathrm{m}, {\xi }_L=6.8$. (d) $\begin{array}{c}•\end{array}, h=20 \mu \mathrm{m}, \phi =0.3\%, L=400 \mu \mathrm{m}, 2\times {10}^{-22} \le A\le 5\times {10}^{-20}$ J, $U=1$ mm/s, $r=2.5 \mu \mathrm{m}, {\xi }_L=0.0078$.

Figure 9

Numerical checks of the theoretical predictions for the four subregimes showing ${\xi }_L\left(\frac{A}{kT}\right)$. Solid lines correspond to theoretical expressions of the frontiers (see Fig. 2 caption), while symbols are different numerical simulation results. Common parameters for these simulations include $h=20 \mu \mathrm{m}, \phi =0.3\%$, and $L=400$ $\mu \mathrm{m}$. Certain other parameters vary with respect to the associated subregime. Legend: Diffusive 1, $\begin{array}{c}▴\end{array}$ (left column), $A=4\times {10}^{-23}$ J, $r=0.25 \mu \mathrm{m}, 0.1\le U\le 10$ mm/s; (right column) $A=8\times {10}^{-21}$ J, $U=1$ mm/s, $0.1\le r\le 0.5 \mu \mathrm{m}$. Diffusive 2, $\begin{array}{c}★\end{array}, h=20 \mu \mathrm{m}, \phi =0.3\%, L=400 \mu \mathrm{m}, A=1\times {10}^{-24}$ J, $r=2.5 \mu \mathrm{m}, 0.001\le U\le 100$ mm/s. van der Waals 1, $\begin{array}{c}◂\end{array}$ (topmost row), $2.5\times {10}^{-19} \le A\le 5\times {10}^{-19}$ J, $U=2$ mm/s, $r=0.125 \mu \mathrm{m}, {\xi }_L=544$; (second row from top) $5\times {10}^{-20} \le A\le 2.5\times {10}^{-17}$ J, $U=1$ mm/s, $r=0.25 \mu \mathrm{m}, {\xi }_L=68$; (middle row) $8\times {10}^{-21} \le A\le 1\times {10}^{-19}$ J, $U=10$ mm/s, $r=0.25 \mu \mathrm{m}, {\xi }_L=6.8$; (second row/point from bottom) $A=5\times {10}^{-20}$ J, $U=100$ mm/s, $r=0.25 \mu \mathrm{m}, {\xi }_L=0.68$; (bottommost row/point) $A=5\times {10}^{-19}$ J, $U=0.1$ mm/s, $r=2.5$ $\mu \mathrm{m}, {\xi }_L=0.078$; van der Waals 2, $\begin{array}{c}•\end{array}$ (top row/point), $A=8\times {10}^{-21}$ J, $U=0.1$ mm/s, $r=2.5 \mu \mathrm{m}, {\xi }_L=0.078$; (middle row) $2\times {10}^{-22} \le A\le 5\times {10}^{-20}$ J, $U=1$ mm/s, $r=2.5 \mu \mathrm{m}, {\xi }_L=0.0078$; (bottom row) $7\times {10}^{-24} \le A\le 5\times {10}^{-20}$ J, $U=10$ mm/s, $r=2.5$ $\mu \mathrm{m}, {\xi }_L=0.00078$.