Limitations of Bulk Diamond Sensors for Single-Cell Thermometry

Abstract

The present paper reports on a Finite Element Method (FEM) analysis of the experimental situation corresponding to the measurement of the temperature variation in a single cell plated on bulk diamond by means of optical techniques. Starting from previous experimental results, we have determined-in a uniform power density approximation and under steady-state conditions-the total heat power that has to be dissipated by a single cell plated on a glassy substrate in order to induce the typical maximum temperature increase ΔTglass=1 K. While keeping all of the other parameters constant, the glassy substrate has been replaced by a diamond plate. The FEM analysis shows that, in this case, the maximum temperature increase is expected at the diamond/cell interface and is as small as ΔTdiam=4.6×10-4 K. We have also calculated the typical decay time in the transient scenario, which resulted in τ≈ 250 μs. By comparing these results with the state-of-the-art sensitivity values, we prove that the potential advantages of a longer coherence time, better spectral properties, and the use of special field alignments do not justify the use of diamond substrates in their bulk form.

Keywords: bio-sensing; diamond temperature sensors; finite element analysis.

Figure 1

(a) Sketch of the model used for the cell–water–borosilicate experimental setup: (i) the cell, colored in orange, modeled as a hemisphere of 40 $\mathsf{\mu }$m of diameter; (ii) the water environment, colored in blue, above the cell, modeled as a cylinder with a base of 400 $\mathsf{\mu }$m and a height of 200 $\mathsf{\mu }$m; and (iii) the borosilicate sample, colored in dark blue below the cell, modeled as a cylinder with a base of 400 $\mathsf{\mu }$m and a height of 150 $\mathsf{\mu }$m. The total dissipated heat power integrated over the whole cell is fixed at $P_{diss}=1.04\times {10}^{-4}W$ and corresponds to a heat power density Q that is constant throughout the cell. The temperature of the boundaries is fixed at 310.15 K (37.00 ${}^{\circ }$C). (b) Map of the temperature increase in the z-r plane. (c) Temperature increase profile along the positive z-axis, i.e., inside the cell and then in the water, and along negative z (d). In (c,d), the radial position is always $r=0$ μm. All of the temperatures are represented in terms of their increase $\Delta T$ with respect to the temperature of the boundaries (37.00 ${}^{\circ }$C).

Figure 2

Results of the stationary scenario. (a) Sketch of the model using cell–water–diamond: (i) the cell, colored in orange, modeled as a hemisphere of 40 $\mathsf{\mu }$m of diameter; (ii) the water environment, colored in blue, above the cell, modeled as a cylinder of a base of 400 $\mathsf{\mu }$m and a height of 200 $\mathsf{\mu }$m; and (iii) the bulk diamond sample, colored in light blue below the cell, modeled as a cylinder with a base of 400 $\mathsf{\mu }$m and a height of 300 $\mathsf{\mu }$m. The origin of the frame of reference is located at the center of the cell hemisphere. The total dissipated heat power integrated over the whole cell is fixed at $P_{diss}=1.04\times {10}^{-4}W$ and corresponds to a heat power density Q that is constant throughout the cell. The temperature of the boundaries is fixed at 310.15 K (37.00 ${}^{\circ }$C). (b) Map of the temperature increase in the z-r plane. (c) Temperature increase profile along the positive z-axis, i.e., inside the cell and then in the water, and along negative z (d), i.e., inside the diamond. (c,d) have different scales. In (c,d), the radial position is always $r=0$ μm. (e) Spatial decay along r of the temperature increase for z = −0.01 $\mathsf{\mu }$m, z = −0.1 $\mathsf{\mu }$m, and z = −1 $\mathsf{\mu }$m, i.e., at 3 different depths inside diamond. All of the panels show the temperature increase with respect that of the boundaries (37.00 ${}^{\circ }$C).

Figure 3

Results of the transient scenario for cell–water–diamond. This scenario starts, at $t=0$, with the temperature increase calculated in the previous stationary scenario. The temperature increases inside the diamond are plotted as a function of time for the points (r = 0 $\mathsf{\mu }$m, z = −0.01 $\mathsf{\mu }$m), (r = 0 $\mathsf{\mu }$m, z = −0.1 $\mathsf{\mu }$m), and (r = 0 $\mathsf{\mu }$m, z = −1 $\mathsf{\mu }$m) for the first 300 $\mathsf{\mu }$s (a) and for the first 10 $\mathsf{\mu }$s (b).