Design and parametric study of a tapered polymer-based suspended microfluidic channel for enhanced detection of biofluids and bioparticles

Abstract

Advancements in biotechnology and related fields continue to compel the development of new sensing techniques for more efficient miniature bioelement detections at low concentrations. Microsystems have gained huge interest and produced fascinating results in cell studies, owing to their miniature sensing area, low fabrication costs, label-less detection, and ease of integration with lab-on-a-chip (LoC) applications. In this study, an innovative tapered polymer-based microchannel-embedded microcantilever termed the ‘T-SPMF3’ is presented and computationally analyzed. The working of the device is based on the deflection of the microcantilever tip by flow forces produced within the embedded microchannel. Using the fluid–structure interaction (FSI) module in COMSOL Multiphysics, the sensitivity of the biosensor is investigated relative to a series of parameter tweaks. By increasing the characteristic parameter of the model, the lateral displacement angle $\theta$ from ${0}^{0}$ to ${20}^{0}$, sensitivity increased by ~ 90%. The same magnitude of improvement was shown through each of the test fluids (water, milk, saline solution, acetone, and blood) considered in the study, with the most and the least viscous of them producing the largest and smallest deflections respectively. Furthermore, a particle flow analysis is carried out – using three microparticle parameters that represent red blood cells (RBCs), white blood cells (WBCs), and circulating tumor cells (CTCs) – to gain a better understanding of how the microparticles impact the microchannel walls and as a result, produce different beam deflections. Overall, the study shows how the sensitivity of the proposed model can be tuned to meet the demands of different bio-diagnostic applications. This research could greatly push MEMS capabilities and multifunctionality in cell mechanobiological to new heights.

Appendices

Appendix

A. Theoretical analysis

To further validate the improved sensitivity characteristic of the T-SPM3 design, we theoretically compared its stiffness to that of the SPMF3.

(See Table 7)

Table 7 Comparison of the spring stiffness of the SPM3 and T-SPMF3 platforms

The SPMF3 is modeled as a fixed-free beam with loading force normal to the substrate.

Moment of inertia, $I =\frac{w{t}^{3}}{12}$, where w = width, and t = thickness of the cantilever.

Force constant, $k= \frac{F}{x}$ = $\frac{3EI}{{l}^{3}}$= $\frac{Ew{t}^{3}}{{4l}^{3}}$, where F = loading force, x = deflection, E = young’s modulus of material, I = moment of inertia, l = length of the beam.

= $\frac{(700 kPa) ({2 mm) (0.6 mm)}^{3}}{{4 (6 mm)}^{3}}$ ${= 0. 35 Nm}^{-1}$

The T-SPMF3 on the other hand is modeled as two-fixed free beams connected in parallel, with loading force also normal to the substrate.

The moment of inertia of each arm is $I=\frac{w{t}^{3}}{12}$,and overall force constant is, $k=2\left(\frac{F}{x}\right)$ = $2\left(\frac{3EI}{{l}^{3}}\right)$ = $\left(\frac{Ew{t}^{3}}{{2l}^{3}}\right)$

$= \frac{(700 kPa) ({0.55 mm) (0.6 mm)}^{3}}{{2 (5.5 mm)}^{3}}$ ${= 0. 19 Nm}^{-1}$ [l and w values are consistent with the 3D (T-SPMF3- 20⁰) model dimensions].

The analysis shows that the T-SPMF3 exhibits nearly half (0.542) the stiffness of the SPMF3 for the same material (PDMS, E = 700 kPa), thickness (600 um), and perimeter (16 mm), further justifying the former's greater sensitivity characteristic.

B. Cover layer thickness

The cover layer describes the layer above or below the top and bottom microchannels respectively. A range of thicknesses of the layer, obtained by changing the nozzle height while the microcantilever height is fixed, is analyzed for a visualization of the magnitude of stress developed in them from the bending of the microcantilever structure under the impact of flow through the microchannel.

Fig. 11

Analysis of stress in the microcantilever structure. a Section of focus of the T-SPFM3 featuring a z-x cut plane set halfway across the length b Velocity magnitude (m/s) and von-Misses (N/m2) stress profiles at the cut plane; Stress distribution in the microcantilever with a cover layer thickness, T_c, c 150um, d 100um, e 50um, and f 25um from. Inlet velocity was maintained at 0.03 m/s in all cases

Figure 11 presents a summary of the results of the stress analysis conducted on the T-SPMF3 structure. The study aimed to show the effect of different thicknesses of the cover layer on the sensitivity of the beam as well as the overall structural integrity. All boundaries conditions and material and structural parameters remained the same, save the vertical distance between the two channels (equivalent to the nozzle height), which was varied to leave thicknesses of 150 um, 100 um, 50 um, and 25 um equally above and below the top and bottom channels respectively. Figure 11c–f show the stress distributions around the channels and the corresponding impacts on the cover layers (from T_c = 150 um to 25 um). Maximum stress was 1.63 kPa with a corresponding tip deflection of 11um for T_c = 150 um 1.64 kPa and 15 um for T_c = 100 um; 4.69 kPa and 25 um for T_c = 50 um; and finally, 11.1 kPa and 33 um for T_c = 25 um. Overall, sensitivity increases as T_c decreases. However, as can be seen in the figure, particularly Fig. 11(e, f), overly small thicknesses, can impact the overall integrity of the structure as well as the reliability of the result. We used a 100um thickness in our reference model (Fig. 6) because of the good trade-off it provides between sensitivity and structural integrity (compared to 150um thickness, where stress is less (0.6%) but at a considerable sensitivity cost (36.2%)).

C. Governing equations

• a) Fluid–Structure Interaction (FSI) analysis

The analysis of the biosensors utilized the built-in FSI module available within the COMSOL Multiphysics environment. The flow domain, consisting of the whole process liquids (light-water, milk, saline solution, acetone, and blood) used was considered incompressible and is governed by the Naiver-Stokes equations. The derivations of these equations are well documented in the literature, but for the sake of completeness, we provide them here.

$$\rho \frac{D{u}_{f}}{Dt} = - \nabla p + \mu {\nabla }^{2}{{\varvec{u}}}_{f} + {{\varvec{F}}}_{f}$$

(1)

and

$$\nabla {. {\varvec{u}}}_{f} = 0$$

(2)

In Eq. (1), there is the conservation of momentum and Eq. (2), the conservation of mass. The subscript, f, denotes flow domain.

The solid domain (the microcantilever body) is modeled as a linearly elastic material governed by the following equations:

$${{\varvec{F}}}_{s} = - \nabla . \sigma$$

(3)

$$\epsilon = \frac{1}{2} [\nabla {{\varvec{u}}}_{s} + {(\nabla {{\varvec{u}}}_{s})}^{2} ]$$

(4)

$$\sigma = C : \epsilon$$

(5)

where Eq. (3) is the equation of motion- from Newton’s 2nd law, Eq. (4) is the strain–displacement equation, while Eq. (5), describes the Constitutive equations, based on Hooke’s law. Meanwhile, the subscript s stands for the solid domain.

Altogether, the fluid–structure interfaces are bound by the following conditions:

$${{\varvec{u}}}_{f} = \frac{\partial {{\varvec{u}}}_{s}}{\partial t}$$

(6)

$$\sigma . n =\Gamma . n$$

(7)

where$\Gamma = -p\mathbf{I}+\upmu (\nabla {{\varvec{u}}}_{f } + {(\nabla {{\varvec{u}}}_{f }) }^{T}$).

• b) Fluid-Particle Interaction analysis

The equations governing the study are described by partial differential equations based on the balancing of mass (8), momentum (9) within a small element of volume. Equation (10) on the other hand, describes Newton’s 2nd law applied to each particle. The governing equations for the incompressible single-phase laminar flow (Naiver – stokes equations) as well as the particle motion in the fluid are described as:

$$\nabla . {\varvec{u}} = 0$$

(8)

$$\rho \frac{\partial {\varvec{u}}}{\partial t} + \rho ({\varvec{u}}.\nabla )\boldsymbol{ }{\varvec{u}}\boldsymbol{ }=\boldsymbol{ }\nabla .[ -p\mathbf{I}+\upmu (\nabla {\varvec{u}} + {(\nabla {\varvec{u}}) }^{T})] + \mathbf{F}$$

(9)

$$\frac{d}{dt}({m}_{p}v) = {\mathbf{F}}_{{\varvec{D}}}\boldsymbol{ }+\boldsymbol{ }{\mathbf{F}}_{{\varvec{g}}}\boldsymbol{ }+\boldsymbol{ }{\mathbf{F}}_{{\varvec{e}}{\varvec{x}}{\varvec{t}}}$$

(10)

where ${\varvec{u}}$ is the fluid velocity, $p$ is pressure, $\rho$ is density, $\upmu$ is dynamic viscosity, and F is total volume force; ${m}_{p}$ is particle mass, v is the velocity of particles, ${\mathbf{F}}_{{\varvec{D}}}$ is the drag force (modeled with Stokes formulation which takes into account the turbulent dispersion using u′ that describes turbulent velocity fluctuation), ${\mathbf{F}}_{{\varvec{g}}}$ is the gravitational force, and ${\mathbf{F}}_{{\varvec{e}}{\varvec{x}}{\varvec{t}}}$ is external forces.