2.2. Resistive Heater Structure Design
A resistive heater structure will be placed at the bottom side of the chip using shadow masks and a metal deposition method capable of being used for large-scale production. A meandering heater design is chosen, as this minimizes the input power required to heat up the heater. This is evident from Equation (1), which is the relation between Joule’s law, Ohm’s law, and Pouillet’s law. (1) P=Across−section∗V2ρres,i∗lheater Here, P is the input power, Across−section is the cross-sectional area of the resistor, V is the input potential, ρres,i is the resistivity of resistor material i, and lheater is the length of the resistor. This makes a meandering structure, or any other narrow line structure, a quite often used pattern for heaters or electrodes within micro-electromechanical structures and microfluidics [26,29,54,55].
MDA is being done at temperatures of around 30 °C [50], which is lower than, for example, temperatures required for HDA (64 °C) [9] or LAMP (65 °C) [11] and the required PCR temperatures of Chung et al. (95 °C, 54 °C, and 72 °C) [28]. However, most amplification methods require a DNA denaturation step at 95 °C. Equation (2) is used to make an estimation of the required heating powers for a COC–H2O–COC stack (in the real device, the upper plate is an adhesive PCR foil, but the thermal properties of this foil are unknown). (2) P=ΔTRth=ΔT∗Aheated∗κCOClCOC,1+κH2OlH2O+κCOClCOC,2+h Here, Rth is defined as the sum of all thermal resistances in series: (3) Rth=1h×Aheated+∑isubstancesliκi×Aheated Here, P is the required power, ΔT is the temperature difference, Rth is the thermal resistance, Aheated is the heated area, h is the convective heat transfer coefficient (being 10 W m−2 K−1 for convection to air [56]), κi is the thermal conductivity of substance i, and li is the thickness of substance i. Values for κi can be found in Appendix B. From Equation (3), the product Rth×A can be defined as the sum of 1/h and li/κi. Based on this summation, one can conclude that the convective heat transfer to the air is the most present heat transfer mechanism within the system (begin almost a factor 100 higher than the heat lost in the COC and H2O). This is also evident from solving Equation (2) for every individual temperature differences within the system and also including convective heat transfer directly from the heater into the air. If a heated area of 7.7 mm by 10.1 mm is assumed, which covers both the reaction chamber and the temperature monitor chamber, and a system consisting of 1 mm COC–0.5 mm H2O–0.1 mm COC is assumed, than the heater temperatures and powers in Table 1 are required. These are all in the workable range when a COC of a proper grade is chosen (e.g., TOPAS 6017 has a Tg of 170 °C). The only side note here is that at higher temperatures, the temperature gradient through the system also becomes larger. This can be eliminated by using double-sided heating, like Chung et al. [28].
To determine the optimal heater width and heater spacing in the heated area, a parametric study using COMSOL Multiphysics 5.3a finite element method simulations with the Heat Transfer in Solids (ht) package is done (COMSOL Inc., Burlington, MA, USA). The model is designed such that it consists of two parallel rectangles of COC (in the real device, the upper plate is an adhesive PCR foil) with H2O in between. The meandering heater are assumed to be lines at the bottom side of the layer stack. This reduces the required complexity of the mesh tremendously, as the heater in the real device will be approximately 100 nm in thickness. The heater temperature is set at a constant temperature of 303.15 K. This makes the heater material independent and the model purely focused on the heat transfer inside the COC–H2O–COC stack. All used values and equations are given in Appendix B. The layer stack is meshed with an extremely fine mapped mesh consisting of 280.650 elements with average quality of 0.9966. A parametric sweep from 0.3 mm to 2.0 mm, in steps of 0.1 mm, is done for both the heater width (wheater) and the heater spacing (sheater), giving 324 combinations. The simulations are solved by using the fully coupled, direct Pardiso solver on a custom-build and 40% CPU overclocked simulation computer, containing the equipment listed in Table 2.
To validate whether the metal tracks can withstand the required current, a quick analysis is done for the four extreme cases (i.e., wheater of 0.3 mm and 2.0 mm and sheater of 0.3 mm and 2.0 mm). In the same heated area of 7.7 mm by 10.1 mm a 100 nm (theater) thick heater track consisting of rectangles is assumed. The total amount of large and smaller interconnecting rectangles for all 4 cases is estimated in Table 3. Based on the polynomial approximation equations for the resistivity of Au and Pt (ρres,i, where i is either Au or Pt) which COMSOL MultiPhysics 5.3a uses (Equations (4) and (5)) and Equation (6) an estimation is made for the required input currents and the created current densities (defined as Ii/Across−section, in A m−2) when the heater is operated at 129.4 mW to get a temperature of 95 °C. These estimations are also given in Table 3. (4) ρres,Au(T)=−2.210068×10−9+9.057611×10−11∗Tfor60≤T<400−4.632985×10−14∗T2+6.950205×−17∗T3 (5) ρres,Pt(T)=−1.927892×10−8+5.233699×10−10∗Tfor160≤T<600−4.107885×10−13∗T2+6.694129×−16∗T3−4.447775×10−19∗T4 (6) Ii=PRi=P∗Across−sectionρres,i∗lheater
In which ρres,i is the resistivity, T is the temperature, Ii is the current going through the resistor, Pi is the input power, R is the resistance of the resistor, lheater is the length of the resistor, and Across−section is the cross-sectional area of the resistor defined as width times thickness (wheater×theater). The subscript i denotes the material, being Au or Pt.
All these current densities are below the critical current densities for Au and Pt, which are around 1010 A m−2 [57] and 1011 A m−2 [58], respectively. Therefore, any possible combination of heater width and heater spacing will give a resistive that can withstand its operation.