2.1. Numerical Modeling of the Perforated Microcantilever
Figure 1 presents a 3D mold of the perforated microcantilever covered with a layer of CMs to improve the performance of the reported microcantilever sensors. The 3D structure of the perforated microcantilever, shown in
Figure 1a, is composed of two layers. The bottom layer is the perforated microcantilever, and its size is
L (length) *
b (width) *
tf (thickness). The top layer, with thickness
tf, has CMs cultured on the surface of the microcantilever. The oval-shaped CMs are treated as cubes for a better understanding of the contractile force of the CMs, as shown in
Figure 1c, whose length (
Ls), width (
bs), and thickness (
ts) are reported as 100 µm, 30 µm, and 10 µm, respectively [
19,
22]. Further, the contractile force in the other directions (except longitudinal direction) has little effect due to the vertical deflection of the microcantilever [
23]. It is reported that the microgroove structures in the longitudinal direction can produce an accumulated action of the cultured CMs. However, when the CMs are arranged in the longitudinal direction, it can enhance the microcantilever’s macroscopic bending motion [
24]. The CMs connect tightly to improve the microcantilever’s bending deflection, as shown in
Figure 1d, with a magnitude of contraction of 2~5 nN/µm
2 [
25].
Figure 1e shows a perforated microcantilever section with a circular aperture, where
d is the aperture size, and
q1 and
q2 are the spacing of the hole based on the direction of the length and width, respectively.
The CM layer consists of a thin film attached to the surface of the plain microcantilever. The contractile force of the CMs is equivalent to the residual stress of the thin film. Based on this model, Stoney’s equation is used to calculate the free-end deflection of the plain microcantilever, as shown in Equation (1) [
26,
27].
where
and
present the curvature radius of the substrate before and after bending;
is Young’s modulus of the substrate;
is Poisson’s ratio of the substrate;
is the substrate thickness;
is the film thickness; and
is the film stress, considered as the contractile force of the cardiomyocytes. Equation (1) presents the relationship between the substrate deflection and film stress. Berry conducted a more detailed mechanical analysis of the deformation of the coated cantilever and believed that it is more reasonable to replace the biaxial modulus
Es/(1
−vs) with the plate modulus
Es/(1
−vs2) of the substrates in the Stoney equation [
28]. Thus, the modified Stoney equation can be expressed by the following equation:
Based on geometric theory, the relationship between the deflection of a plain cantilever and the curvature radius of the substrate can be achieved using Equation (3)
where
L is the length of the cantilever; and
δ is the vertical deflection of the cantilever. After substituting Equation (3) into Equation (2), the vertical displacement at the free end of a plain cantilever can be calculated by the following equation:
Figure 1 presents a schematic of the perforated microcantilever to improve the bending effect. The reason is that the perforation can reduce the rigidity of the microcantilever. The substrate material of the plain microcantilever is considered a porous material that is dense. Several equations for the relationship between Young’s modulus and porosity have been reported [
29,
30], in which the most frequently used equation is as follows:
where
E presents the equivalent Young’s modulus of the porous materials;
Es is Young’s modulus of the dense materials;
p is the volume fraction of porosity (porosity factor); and a and b are constants and can be determined by experimental data fitting. By combining Equations (4) and (5), the following equation can calculate the vertical displacement at the free end of the perforated cantilever:
In the present study, the experimental data can be obtained by using the FEM solution, whereas the relevant results are shown in
Figure S1 (Supplementary Materials). The porosity factor,
p, is decided by the aperture size and the hole spacing (
q1 and
q2). Constants a and b were calculated to be 0.49 and 5.86, using the MATLAB fitting tool.
2.2. Plain Microcantilever
In order to confirm the efficiency of the proposed numerical model, the bending effect of the microcantilever without imperfection was first investigated by the modifier Stoney equation, as shown in
Figure 2. Compared with the simulation results obtained by COMSOL Multiphysics software (COMSOL Multiphysics 5.4, COMSOL Inc, Stockholm, Sweden), the numerical results calculated by Equation (4) show that the modifier Stoney equation has a very high accuracy. The maximum deflection of the plain microcantilever under different conditions was studied, such as the size of the plain microcantilever, Young’s modulus of the substrate material, and the contractile force of the cardiomyocytes, studied based on numerical and simulation methods.
Figure 2a shows the vertical displacement plot (obtained by COMSOL Multiphysics software) when the plain microcantilever lengths are 1500, 3000, 6000, and 9000 μm, respectively; the aspect ratio is 3:1, the thickness is 100 µm, and the contractile force is 2 nN/μm
2. Here, Young’s modulus and Poisson’s ratio of PDMS (Polydimethylsiloxane) was found to be 750 kPa and 0.49, respectively [
31]. Young’s modulus and Poisson’s ratio of the CMs are 188 kPa and 0.49, respectively [
32,
33].
Figure 2b compares the maximum displacements generated by the plain microcantilever with different sizes (the lengths are 1500, 3000, 6000, and 9000 μm; the aspect ratio is 3:1; and thickness is 100 um) when the contractile force is changed from 2 to 5 nN/μm
2. The simulation results show that the maximum displacement of the plain microcantilever with a 1500 μm * 500 μm * 100 μm/3000 μm * 1000 μm * 100 μm/6000 μm * 2000 μm * 100 μm/9000 μm * 3000 μm * 100 μm size is 14 μm/54 μm/222 μm/500 μm, respectively, after a 2 nN/μm
2 contraction force. When the contraction force is 5 nN/μm
2, the maximum displacement of the plain cantilever reaches 34 μm, 135 μm, 574 μm, and 1249 μm, respectively. Compared with the simulation, the displacement generated by the numerical method is slightly lower; however, the two methods are consistent. A linear increase in the maximum bending displacement with an increase in contractile force is observed in our proposed numerical model. An error of both the numerical and simulation results may come from the thickness of the substrate, which is not much greater than the thickness of the film.
In order to further validate the accuracy of the present numerical method, the maximum displacements of the plain microcantilever with different substrates were characterized, as shown in
Figure 3a. Here, Young’s modulus of the substrate material depends on the ratio of the PDMS base and curing agent. When the ratio is 10:1, Young’s modulus is 750 kPa, and the contractile force is 2 nN/μm
2. Four sizes of a plain microcantilever are mentioned above. The simulation and calculated maximum displacement of a plain microcantilever is nonlinearly decreased when Young’s modulus is over 750 kPa. However, the rate of decrease in the maximum displacement slows down significantly. The simulation and numerical results are closer when Young’s modulus of the substrate material is lower, agreeing with the Stoney equation when Young’s modulus has a low value by assuming that Young’s modulus of the film and substrate are equal.
Figure 3b shows the effect of substrate thickness on the maximum displacement when the microcantilever length is 9000 μm and the contractile force is 2 nN/μm
2. The simulation and numerical results are in good agreement and show that the substrate thickness greatly influences the maximum displacement. When the thickness of the substrate is increased from 70 μm to 130 μm, with intervals of 10 μm, the maximum displacement is decreased from 1030 μm to 303 μm in the simulation analysis; however, for the numerical analysis, it is from 1005 μm to 291 μm.
2.3. Perforated Microcantilever
The structure of a perforated microcantilever was covered with a layer of CMs to improve the bending effect. After designing a perforated microcantilever, the numerical model is proposed to investigate the bending effect. While the same size was proposed for a plain microcantilever, four different sizes were chosen for the perforated microcantilevers:
L is 1500, 3000, 6000, and 9000 μm; aspect ratio is 3:1;
d is 60 μm;
q1 is 300 μm;
q2 is 100 μm;
tf is 100 μm; and d is 60 μm. The holes are regularly arranged on the microcantilever, and the number of holes in the four cases increases in proportion to ensure the same porosity.
Figure 4 presents the numerical and simulation results of the maximum displacements at the free end of a perforated microcantilever under different conditions. Further,
Figure 4a depicts a vertical displacement plot of the perforated microcantilever when the contractile force is 2 nN/μm
2 (
L is 1500, 3000, 6000, and 9000 μm, respectively; the aspect ratio is 3:1; d is 60 μm;
q1 is 300 μm;
q2 is 100 μm; and
tf is 100 um). Since the aperture of the perforation is smaller than the size of the self-organized CMs, it is assumed that the perforation would not affect the arrangement and contractile force of the CMs. The perforated microcantilever has a lower stiffness due to holes in the body. The maximum displacement of the perforated microcantilever is observed at 299 μm, increasing by about 35% compared to a plain microcantilever with the same size, as shown in
Figure 3a and
Figure 4a.
Figure 4b shows the effect of the contractile force on the maximum displacements of the perforated microcantilever with different sizes (
L is 1500, 3000, 6000, and 9000 μm; the aspect ratio is 3:1;
d is 60 μm;
q1 is 300 μm;
q2 is 100 μm; and
tf is 100 um). When the contractile force of the CMs is increased from 2 to 5 nN/μm
2, the simulation results show that the ranges of maximum bending displacement of the cantilever are 16~41 μm, 75~187 μm, 299~747 μm, and 671~1678 μm in the above four cases, respectively. Similarly, for the perforated microcantilever, the numerical results agreed well with simulation results using the FEM model in the four cases; however, the maximum bending displacement of the perforated microcantilevers is about 30% larger than the plain microcantilevers with lower stiffness. These results show the application potential of the perforated cantilever due to a higher bending displacement. Similarly, with the increase in contraction force, the maximum bending displacement increases linearly. For better visualization, 3D representation in
Figure S2 regarding vertical displacement vs Contractile force.
Young’s modulus on the bending displacement of the perforated cantilever was also examined, wherein Young’s modulus increased from 500 to 1000 kPa with intervals of 100 kPa, as shown in
Figure 5a. Similarly, the numerical results agreed well with the simulation results, and both show a nonlinear, inverse relationship between Young’s modulus and bending displacement. However, when Young’s modulus exceeds 750 kPa, the bending displacement reduction rate decreases; the same tendency is also observed in a plain microcantilever. With an increase in Young’s modulus, the gap between the simulation and numerical results increases, as the larger the Young’s modulus of the substrate, the greater the error in Stoney’s calculation. Simultaneously, perforation can also further increase the bending displacement. Compared with the plain microcantilever, as shown in
Figure 3a and
Figure 5a, the maximum bending displacement in the simulation results shows an increase of 30%. In the numerical results, it offers an increase of 22% in all size cases. Therefore, the perforated microcantilever can obtain a lower bending displacement at a higher Young’s modulus. Furthermore, the effect of the perforated microcantilever thickness on the bending displacement was studied, as shown in
Figure 5b, where the microcantilever length is 9000 μm and the CMs’ contractile force is 2 nN/um
2. The numerical results agreed well with the simulation solution with an increase in the thickness of the substrate from 70 to 130 μm, with intervals of 10 μm. The maximum bending displacement decreased from 1292 to 403 μm in the simulation; however, in the numerical results, it is from 1224 to 355 μm. As we can see, the numerical results agreed well with the simulation results, and the thickness of the substrate has a significant effect on the bending displacement of the microcantilever. The perforated microcantilever has a larger bending displacement than the plain microcantilever; the maximum displacement increased by 21~33% compared with the plain microcantilever. Therefore, the perforated microcantilever can achieve a greater bending displacement on a thinner substrate. Due to the small Young’s modulus and the self-weight, the end of the microcantilever has a bending effect. Thus, the minimum thickness should be selected to ensure the stability of the structure in the experiment. In general, the error between the numerical and simulation results increases with an increase in microcantilever size, as shown in
Figure 2 and
Figure 4. For the plain microcantilever, the error between the numerical and simulation is a little higher than the other cases when the microcantilever length is 6000 µm, compared with the perforated microcantilever. The reason can be explained by the equivalent Young’s modulus and the porosity factor. As seen from
Figure 4, the holes are regularly arranged on the microcantilever and the number of holes in the four cases increases in proportion to ensure the same porosity in theory. However, the porosity in the four cases is slightly different due to the constraint of the microcantilever size. When the microcantilever length is changed from 1500 µm to 9000 µm, the porosity is 0.094, 0.076, 0.068, and 0.065, respectively. For better visualization, 3D representation in
Figures S3 and S4 regarding vertical displacement vs Young’s modulus and substrate thickness respectively.
In order to further study the influence of perforation on the bending displacement of the perforated microcantilever, the effect of aperture and hole spacing on displacement was characterized, as shown in
Figure 6. Here, the length of the microcantilever is 3000 μm, the width is 1000 μm, and the thickness is 100 μm, whereas the contraction force is 2~5 nN/μm
2. The proposed numerical model and FEM model were used to characterize the effect of different apertures and spacing on the bending displacement of a perforated microcantilever. The aperture, d, increased from 50 to 80 μm, with intervals of 10 μm, whereas the volume fractions of porosity were found to be 0.053, 0.076, 0.104, and 0.136, respectively (
q1: 300 μm;
q2: 100 μm), as shown in
Figure 6a. When the contractile force of the CMs is 5 nN/μm
2 and the aperture is 80 μm, the bending displacements were calculated to be 151.1 μm and 146 μm from the numerical and simulation solution, respectively.
Figure 6b shows the effect of hole spacing on the bending displacement of a perforated microcantilever with an aperture
d of 50 μm and
q2 of 100 μm. The spacing
q1 increased from 100 to 400 μm, with intervals of 100 μm, and the volume fractions of porosity were 0.147, 0.076, 0.053, and 0.041, respectively. When the contractile force of the CMs is 5 nN/μm
2 and the spacing
q1 is 100 μm, the bending displacements are 151.1 μm and 146 μm based on the numerical and simulation results, respectively. The ratio of increase in aperture and decrease in spacing can directly increase the porosity factor of a perforated microcantilever. It is assumed that porosity is an important factor affecting the stiffness of a perforated microcantilever. When the porosity factor increases to 0.147, Young’s modulus increases; however, the maximum error between the numerical and simulation results becomes 3.4% (
d: 50 μm;
q1: 100 μm;
q2: 100 μm; contractile force: 5 nN/μm
2), as presented in
Figure 6. Therefore, the numerical results agreed well with the simulation results, confirming that the numerical model can work well.
2.4. Transient Analysis for Microcantilever Sensor
Figure 7 presents a transient analysis of a plain microcantilever and a perforated microcantilever to validate the feasibility of a numerical model for the microcantilever sensors. Here, the length of the microcantilever is considered to be 3000 μm, the width is 1000 μm, and the thickness is 100 μm. For the perforated microcantilever,
d is 80 μm,
q1 is 300 μm, and
q2 is 100 μm. In order to simulate the contraction and expansion of the CMs, a time-varying contraction force based on a sine function was applied, as shown in the equation
, in which the period is 1 s and amplitude is 5 nN/μm
2.
Figure 7 presents the displacement–time function for a period of 1 s. The center point of the free end of the microcantilever is selected to obtain the displacement–time function diagram. With a period of 1 s, along with the change in contraction force, the displacement of the microcantilever reaches its maximum value at 0.25 s and 0.75 s, whereas the maximum displacement is almost identical in both the plain and perforated microcantilever, as observed in the static analysis. The displacement–time function of the microcantilever agreed well with the sine function in surface contraction, explaining that the displacement of the microcantilever directly reflected the transient change in surface contraction in the CMs. Besides, the perforated microcantilever has a higher displacement; the same results can be obtained from the transient analysis of the perforated microcantilever, as shown in
Figure 7. Based on a static and transient analysis, our proposed model can detect the transient contractile force of CMs.
After the numerical analysis and simulation by COMSOL Multiphysics, the neonatal mouse ventricular cardiomyocytes (NMVCs) will be cultured on this type of perforated microcantilever to further validate the efficiency of the proposed numerical model in the next step. The microcantilever will be designed with a length (L), width (b), and thickness (ts) of 9000 μm, 3000 μm, and 100 μm, respectively, accompanied with holes whose diameter (d), length (q1), and width (q2) spacing are 60 μm, 300 μm, and 100 μm, respectively, uniformly distributed on the microcantilever. The PDMS prepared with the ratio of 10:1 PDMS base to curing agent will be chosen as the material to fabricate the microcantilever. To measure the displacement of the microcantilever, the end of the microcantilever is deposited onto an Au layer with a size of about 1000 μm in length, 1000 μm in width, and 100 μm in thickness. Therefore, the signal of the microcantilever bending can be detected with a laser amplifier. The accuracy of the proposed model in describing the bending effect can be further obtained by comparing it with the experimental values. We believe that the proposed numerical model of the perforated microcantilever is a precise model to forecast the bending effect and then to calculate the contraction force of the cardiomyocytes.