Computational simulation of biomolecules transport with multi-physics near microchannel surface for development of biomolecules-detection devices

Abstract

Background:

The quantitative evaluation of the biomolecules transport with multi-physics in nano/micro scale is demanded in order to optimize the design of microfluidics device for the biomolecules detection with high detection sensitivity and rapid diagnosis.

Objective:

Methods:

The biomolecules transport with fluid drag force, electric double layer (EDL) force, and van der Waals force was modeled by Newtonian Equation of motion. The model validity was verified in the influence of ion strength and flow velocity on biomolecules distribution near the surface compared with experimental results of previous studies. The influence of acting forces on its distribution near the surface was investigated by the simulation.

Results:

The trend of its distribution to ion strength and flow velocity was agreement with the experimental result by the combination of all acting forces. Furthermore, EDL force dominantly influenced its distribution near its surface compared with fluid drag force except for the case of high velocity and low ion strength.

Conclusions:

The knowledges from the simulation might be useful for the design of biomolecules-detection devices and the simulation can be expected to be applied on its development as the design tool for high detection sensitivity and rapid diagnosis in the future.

Keywords

Biomolecules transport multiphysics microfluidics electric double layer numerical modeling computational simulation

1. Introduction

Micro Total Analysis Systems (μ-TAS) and Lab-on-a-chip (LOC) has been of great interest to researchers in the fields of biology, chemistry, physics, and engineering for several decades [1,2]. In particular, biosensors for the detection of various biomolecules (e.g., viruses and proteins) using microfluidics device are demanded to achieve high detection sensitivity and rapid diagnosis [3]. This is due to the short diffusion distance of biomolecules and the high surface-to-volume ratio by miniaturizing the detection part for biomolecules, as it is called a reaction field (for instance, microchannel surface in microfluidic device), from sizes of the conventional diagnosis device [4,5]. However, specialized phenomena in nano/micro-scale appear dominantly with the miniaturization of a reaction field, such as the convection of buffer solution, biomolecules diffusion and the surface interaction between a reaction field and a biomolecule which is composed of electric double layer (EDL) repulsion and van der Waals attraction [6–9]. These phenomena influence the transport of target biomolecules to the surface of a reaction field in the process of biomolecules detection [10,11]. Conventionally, although various biosensors using microfluidics device has been developed by trial and error, the optimized design for the highly performable device has not been achieved since it is difficult to evaluate these phenomena quantitatively [6–9]. Therefore, the analytic method of multiple phenomena related with the biomolecules transport in a miniaturized reaction field of a microfluidics device is necessary for the development of biomolecules-detection devices with highly sensitivity and rapid diagnosis.

Many researchers have investigated the biomolecules or sphere nanoparticles transport in a microfluidics device for bio-application by the computational simulation analysis which has become increasingly important tool for the study of microfluidics device [12]. Waghmare and Mitra numerically modeled the biomolecules transport in a rectangular microchannel for LOC [13]. Although its transport with only fluid flow and the microscopic distribution was focused, the nanoscopic distribution, near a microchannel surface, was not verified. Jomeh and Hoorfar investigated the biomolecules distribution in microchannel with the biomolecules transport for the biomolecule-capturing device using the convection-diffusion equation with incompressible Navier–Stokes equation. However, the nano-scale phenomenon, as the surface interaction between a microchannel and a biomolecule, was not considered in this numerical model [10]. The nanoscopic distribution of transported biomolecules is the important knowledge for the design of biomolecules-detection devices and the understanding of multi-physics such as the convection of buffer solution and the surface interaction between a microchannel and a biomolecule. In traditional studies the biomolecules transport in micro scale or with single-physics was modeled and, however, the verification of multiple phenomena related with biomolecules transport in nano-scale was not enough. Therefore, the modeling and verification of the biomolecules transport with multi-physics in nano/micro scale is demanded for the detection of various biomolecules using microfluidics device.

This paper aimed to investigate the effectivity of the computational simulation using the numerical model of the biomolecules transport with multi-physics near a microchannel surface on the development of biomolecules-detection devices. In the numerical model, Newtonian Equation of motion was employed as the modeling method of sphere nanoparticles, as biomolecules, transport which can include the various forces acting on the biomolecules transport. This method can evaluate each force quantitatively compared with modeling method in traditional studies. Furthermore, the nano/micro-scale simulation of biomolecules transport can be performed by the combination between this equation and various governing equations of fluid flow, electric field and so on such as Navier–Stokes equation and Poisson–Boltzmann equation [14,15]. In this paper, only Navier–Stokes equation was used for the reduction of the computational cost. The validity of this numerical model was verified using previous studies which investigated the nanoparticle distribution near the channel surface by particle image velocimetry (PIV) process with evanescent light using a fabricated microchannel [16,17]. PIV process with evanescent light is an effective method for the characterization of nanoparticles transport near microfluidics device [18]. The influence of acting forces such as fluid drag and the surface interaction force between a microchannel and a biomolecule on the biomolecules distribution near the microchannel surface was investigated by the computational simulation using the numerical model.

2. Numerical modeling and simulation analysis

2.1. Governing equation for buffer solution flow in a microchannel

The buffer solution was assumed to flow through a microchannel at steady state. Incompressible Navier–Stokes equation was used along with the continuity equation to find the fluid velocity profile throughout the domain. $\begin{matrix} (1) & \begin{matrix} ρ u \cdot \nabla u = - \nabla p + μ \nabla^{2} u \\ \nabla \cdot u = 0 . \end{matrix} \end{matrix}$

In this equation, u is the flow velocity vector which is applied to the drag force for the particle trajectory analysis, p is the pressure, ρ is the density, and μ is the viscosity. Although the viscosity of the fluid including nanoparticles or biomolecules is influenced on by its concentration, the viscosity was assumed to be constant due to the low concentration [19].

2.2. Modeling of the biomolecules transport near a microchannel surface

Biomolecules near the microchannel surface were assumed to be transported by acting forces including fluid drag force, Brownian force, the interaction force between a wall and a biomolecule, or between biomolecules including EDL force and van der Waals force. In this paper, the influence of fluid flow and the surface interaction between a microchannel and a biomolecule was focused on this paper. Furthermore, the low concentration of biomolecules near the microchannel surface was assumed for the development of the diagnosis device which can detect biomolecules in the low-concentration sample. Therefore, Brownian force and the interaction force between biomolecules ware neglected in the numerical model of the biomolecules transport. The governing equation with this assumption is denoted by Newtonian Equation of motion. $\begin{matrix} (2) & m_{p} \frac{d v_{p}}{d t} = F_{D} + F_{EDL} + F_{vdW} . \end{matrix}$

In this equation, $m_{p}$ is the mass of a biomolecules and $v_{p}$ is the biomolecule velocity vector.

$F_{D}$ is the fluid drag force induced by the buffer flow which acts on the sphere-shaped biomolecule. $\begin{matrix} (3) & F_{D} = 6 π μ r_{p} (u - v_{p}) \end{matrix}$ in which μ is the viscosity of the buffer solution, $r_{p}$ is the biomolecule radius, and u is the velocity of the buffer solution calculated in the fluid analysis in the third model.

$F_{EDL}$ is EDL force between the microchannel surface and a biomolecule [20]. $\begin{matrix} (4) & F_{EDL} = 2 π r_{p} ε_{0} ε_{r} κ exp (- κ x_{sp}) (\frac{2 V_{s} V_{p} - (V_{s}^{2} + V_{p}^{2}) exp (- κ x_{sp})}{1 - exp (- 2 κ x_{sp})}) n_{v} \end{matrix}$ where $ε_{o}$ is the permittivity of vacuum, $ε_{r}$ is the relative permittivity of buffer solution assumed to be the same with water, and $x_{sp}$ is the distance between a biomolecule and the microchannel surface. $V_{s}$ and $V_{p}$ are the surface voltages of a microchannel and a biomolecule, relatively. The surface voltage is assumed to be the same with zeta potential. $n_{v}$ is the normal vector from the microchannel surface. κ is the inverse of Debye length given by: $\begin{matrix} (5) & κ = \sqrt{\frac{2 N_{A} e^{2} I_{s}}{ε_{0} ε_{r} k_{B} T}} \end{matrix}$ in which $N_{A}$ is Avogadro constant, e is elementary charge, and $I_{s}$ is the ion strength of the buffer solution.

$F_{vdW}$ is van der Waals force between the microchannel surface and a biomolecule [20]. $\begin{matrix} (6) & F_{vdW} = - \frac{A}{6} (\frac{r_{p}}{x_{sp}^{2}} + \frac{r_{p}}{{(x_{sp} + 2 r_{p})}^{2}} + \frac{1}{x_{sp} + 2 r_{p}}) n_{v} \end{matrix}$

In this equation, A is Hamaker constant of the combination with microchannel, buffer solution and biomolecules.

2.3. Geometry model of a microchannel and boundary conditions for validity of the numerical model

To verify the validity in the numerical model of the biomolecules transport near the microchannel surface, the simulation of the fluid flow and biomolecules transport in a microchannel was performed by finite element method (FEM) and particle trajectory analysis [14,15]. Furthermore, the biomolecules distribution near the microchannel was verified in this simulation compared with experimental results in previous studies [16,17] which investigated the influence of ion strength and flow velocity in the buffer solution, phosphate buffer solution (PBS, $pH = 6.5$ ), on the nanoparticles distribution near a microchannel surface by PIV process with evanescent light. Figure 1 shows the two-dimensional geometry model of a microchannel and the flow chart for the simulation of fluid flow and biomolecules transport in a microchannel. In the experiment in previous studies of PIV process in a microchannel with evanescence light, the inlet-or-outlet width, its depth and the channel length of the used microchannel is 1 mm, 20 μm, and 20 mm, respectively, and the measured range of the channel depth was from 0 nm (the bottom surface of microchannel) to 500 nm. Therefore, the fluid analysis of the microchannel model entails two-steps process of micro-scale and nano-scale for the reduction of the computational cost. The first step is the calculation of the flow distribution in the half model of whole microchannel. The second step is the calculation of the flow distribution in a part model of the microchannel with the depth range from 0 to 500 nm. In this step, the velocity at the depth of 500 nm calculated in the first step was used as the boundary condition of the move wall. The simulation of the biomolecules transport was carried out using the result of fluid flow simulation as boundary conditions in the part model of a microchannel. In these simulations, COMSOL Multiphysics^® (COMSOL, Inc., U.S.) was employed as the simulation software based on FEM.

Fig. 1.

(a) The schematic image of two-dimensional geometry model for a microchannel. (b) The flow chart for the simulation of fluid flow and biomolecules in a microchannel.

In the fluid flow simulation, the density and the viscosity ware assumed to be the same with water as shown in Table 1. In the first step of the fluid flow simulation, the no-slip condition was assumed to be at the bottom surface of a microchannel. The inlet velocity was the same with the experimental condition in previous studies [16,17]. The outlet was assumed to be at zero pressure. In the second step, the top surface was assumed to be the move wall with the velocity at the position of 500 nm from the bottom surface in whole microchannel. The inlet and outlet in the part model of a microchannel were assumed to be open boundary.

Table 1

Conditions of the fluid flow simulation

Density of the buffer solution ρ [kg/m³]	998
Viscosity of the buffer solution μ [Pa·s]	0.001
Inlet velocity in the microchannel model [mm/s]	0.0, 0.2, 0.3

As boundary conditions of the biomolecules transport simulation, the initial position of biomolecules was assumed to be distributed uniformly in a microchannel. When transported biomolecules arrived at all surface of the geometry model (the top surface, the channel-bottom surface, the inlet, and the outlet), particles was assumed to hold these positions uniformly. Table 2 shows the conditions of the biomolecules transport simulation. Experimental conditions in previous studies [16,17] were applied on simulation conditions of the biomolecules transport.

Table 2

Conditions of the biomolecules transport simulation

Particle mass [kg]	$5.50 \times 10^{- 19}$
Particle radius [nm]	50
Relative permittivity of buffer solution	78.3
Absolute temperature [K]	295.15
Surface voltage of microchannel [mV]	$- 44.4$
Surface voltage of particle [mV]	$- 26.1$
Ion strength of buffer solution [mM]	0.38, 0.76
Hamaker constant [J]	$2.13 \times 10^{- 14}$
Time step of the simulation [s]	$1.0 \times 10^{- 4}$
Total simulation time [s]	0.1

2.4. Verification of dominant forces acting for the biomolecules distribution in a microchannel

To verify dominant forces acting to the biomolecules transport for the biomolecules distribution near the channel surface, the simulation of the fluid flow and the biomolecules transport in a microchannel with the wide range of flow velocity and ion strength was carried out. Table 3 shows conditions of the particle trajectory analysis for verification of dominant force. The flow chart of this simulation was the same with the simulation condition for the validity verification. Moreover, other conditions with the exception of inlet velocity and ion strength of buffer solution were the same with the simulation condition for the validity verification. The range of the inlet velocity is from 0.1 mm/s to 10 mm/s while the range of ion strength is from 0.1 mM to 10 mM.

Table 3
Conditions of the particle trajectory analysis for verification of dominant force

Density of the buffer solution ρ [kg/m³] 998

Viscosity of the buffer solution μ [Pa·s] 0.001

Inlet velocity in the microchannel model [mm/s] 0.1, 1.0, 10

Particle mass [kg] $5.50 \times 10^{- 19}$

Particle radius [nm] 50

Relative permittivity of buffer solution 78.3

Absolute temperature [K] 295.15

Surface voltage of microchannel [mV] $- 44.4$

Surface voltage of particle [mV] $- 26.1$

Ion strength of buffer solution [mM] 0.1, 1.0, 10

Hamaker constant [J] $2.13 \times 10^{- 14}$

Time step of the simulation [s] $1.0 \times 10^{- 4}$

Total simulation time [s] 0.1

3. Results and discussions

3.1. Validity in the numerical model of the biomolecules transport near the microchannel surface

To verify the validity in the numerical model of the transport of nanoparticles as biomolecules with multi-physics near a microchannel surface, the comparison between the case of the single force which is only drag force, and the case of multiples force including drag force, EDL force, and van der Waals force was firstly performed by the computational simulation according to the experimental conditions in previous study [16]. Figure 2(a) and (b) present the distribution of the particle number in the depth direction of a microchannel on no fluid flow without and with fluid drag force, respectively. Figure 2(c) shows the experimental result of the previous study on the same condition with the computational simulation. Generally, many biomolecules or nanoparticles are distributed at a certain position from a solid surface in contact with electrolyte liquid such as the case of a microchannel as shown in Fig. 2(c) due to the particles restraint at the position of the equilibrium between EDL repulsion force and van der Waals attraction force [20]. Furthermore, this restraint becomes weaker with lower ion strength and distributed particles separate from a microchannel as shown in Fig. 2(c). The trend of the peak position of distributed particles to ion strength in simulation results was agreement with this experimental result. On the other hand, the shape of the distribution curve in simulation results was different with the curve shape in the experimental result. In Fig. 2(a), most particles in both ion strengths were distributed at 500 nm which is the limited position of the geometry model due to the biomolecules transport by strong electrostatic repulsion by EDL between a microchannel surface and particles without drag force. In Fig. 2(b), particles were locally distributed at the position of the equilibrium between EDL repulsion force and van der Waals attraction force since the surface interaction between particles was not included in this numerical model. However, the trend of the maximum particle number to ion strength was agreement with the experimental results. Therefore, these results indicated that the combination and the balance of acting forces on the particles transport are necessary for the simulation of this particles transport and distribution near the microchannel surface compared with single force.

Fig. 2.

The distribution of the particle number for channel depth in no fluid flow (a) without drag force and (b) with drag force by the particle transport simulation. The insert image in (a) is the extended graph. (c) The distribution of particle concentration for the channel depth in no fluid flow on the experimental result of the previous study [16].

Fig. 3.

(a) The distribution of the particle number for channel depth with fluid flow in low ion strength of 0.38 mM by the particle transport. (b) The distribution of particle concentration for the channel depth on the experimental result of the previous study [17].

Fig. 4.

(a) The distribution of the particle number for channel depth with fluid flow in high ion strength of 0.76 mM by the particle transport. (b) The distribution of particle concentration for the channel depth on the experimental result of the previous study [17].

Then, the influence of ion strength and fluid velocity on the particles distribution near a microchannel surface was investigated by the computational simulation according to the experimental conditions in the previous study [17]. The distribution of the particle number in channel flow and low ion strength (0.38 mM) with drag force, EDL force and van der Waals force by the computational simulation is shown in Fig. 3(a). In the experimental result of low ion strength as shown in Fig. 3(b), maximum particle concentration decreased and the peak position of this concentration became far from the microchannel surface with increase in the flow velocity. However, the result of computational simulation shows that the peak position of the maximum particle number was not changed by flow velocity. Although drag force, EDL force, and van der Waals force were applied on this simulation as the acting force on the particles transport, other forces affect the actual particles transport such as lift force, hydration force, hydrophobic interaction force, and the interaction between particles [21]. Especially, lift force might be necessary to be considered on the particles transport near the microchannel surface with the fluid flow in low ion strength because lift force in this case affect flowed particles in the vertical direction to the microchannel surface which brings in the separation of particles from the microchannel surface. However, in bio-application, the biomolecule size is less than 100 nm and lift force does not affect the material under 100 nm generally [14,15]. Therefore, lift force can be neglected for the simulation application to the development of biomolecules-detection devices. Figure 4(a) presents the distribution of the particle number in channel flow and high ion strength (0.76 mM) with drag force, EDL force and van der Waals force by the computational simulation. In high ion strength with fluid flow, the simulation result was agreement with the experimental result as shown in Fig. 4(b) in point of the trend for both of the maximum particle number and the peak position of this number. Hence, this result indicates other forces than drag force, EDL force and van der Waals force between a particle and the channel surface can be neglected for the particles transport near the channel surface in high ion strength. Therefore, this result shows that this numerical model is agreement with in the case of high ion strength. Furthermore, these results concluded that the selection of acting forces on the particles transport by the value range of phenomenon parameters such as flow velocity and ion strength and the consideration of other forces such as lift force, hydration force, hydrophobic interaction force, and the interaction between particles is necessary for the modeling and the simulation of the particles transport near the microchannel surface.

Fig. 5.

The maximum particle number and the peak position at the maximum particle number in a microchannel to (a), (b) flow velocity, and (c), (d) ion strength, respectively.

3.2. Verification of dominant forces for the distribution of particle concentration in a microchannel

To verify dominant forces acting to the particle transport for the distribution of the particle concentration near the channel surface, the computational simulation using the numerical model of the particle transport with the wide range of flow velocity and ion strength was carried out. Figure 5(a) and (b) show the maximum particle number and the position of this number in a microchannel to flow velocity, respectively. In the range of ion strength from 1 mM to 10 mM, maximum particle concentration was decreased by the highest flow velocity and, however, the point of this particle number was constant although flow velocity was increased. Furthermore, whole maximum particle number was relatively low and this number decreased with increase in flow velocity in the low ion strength of 0.1 mM. Moreover, whole position of maximum particle number in 0.1 mM was relatively far from the channel surface and the position of this number in the highest velocity became close to the channel surface. On the other hand, the maximum particle number and the position of this concentration in a microchannel to flow velocity are presented in Fig. 5(c) and (d), respectively. The maximum particle numbers of all velocity increased and the position of this concentration in all velocity became close to the channel surface with increase in ion strength. These results indicate that EDL force dominantly influences the particle number near the channel surface compared with fluid drag force except for the case of high velocity and low ion strength. The local maximum value of the surface interaction force between a microchannel and a particle such as EDL force and van der Waals force to the channel depth decreases and the position of this value becomes far from the channel surface with decrease in ion strength which is known as DLVO theory [20], while fluid drag force increase with increase of the channel depth and flow velocity in the basis of Hagen–Poiseuille law [22–24]. In the case of high velocity and low ion strength, particle restraint by the surface interaction force near the channel surface and fluid drug force become weak and high, respectively. Therefore, fluid drag force became higher and influenced the particle concentration dominantly compared with the surface interaction force. Moreover, this result shows that the dominant force acting to particle transport for the particle concentration near the microchannel surface varies with the range of physical parameters such as fluid velocity and ion strength. Furthermore, the knowledges from the simulation might be useful for the design of biomolecules-detection devices.

4. Conclusions

This paper aimed to investigate the effectivity of the computational simulation using the numerical model of the biomolecules transport with multi-physics near a microchannel surface on the development of biomolecules-detection devices. In the numerical model, Newtonian Equation of motion was employed as the modeling method of sphere nanoparticles, as biomolecules, transport. The validity of this numerical model was verified using experimental results of previous studies. The influence of dominant forces on the biomolecules distribution near the microchannel surface was investigated by a computational simulation using the numerical model. In the validity verification of the numerical model, the trend of maximum particles number and the position of the maximum particle numbers was agreement with the experimental result by the combination of fluid drag force and the surface interaction force such as EDL force and van der Waals force except for the influence of lift force on the particles transport. Therefore, the numerical model of the particle transport near the channel flow is agreement with the case of high ion strength and the consideration of multi-physics is important for the simulation of the particles transport near the channel surface on electrolyte liquid. In the verification of dominant forces on the particles transport, EDL force dominantly influenced the particle distribution near the channel surface compared with fluid drag force except for the case of high velocity and low ion strength in which fluid drag force became higher and influenced the particle concentration dominantly compared with the surface interaction force. Furthermore, the computational simulation using the numerical model of the particle transport can be expected to be applied on the development of bioanalysis device as a design tool for high detection sensitivity and rapid diagnosis.

Footnotes

Acknowledgement

This work was supported by the Tokyo Metropolitan Government High Technology Research Fund.

Conflict of interest

The authors have no conflict of interest to report.

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