A Micromachined Picocalorimeter Sensor for Liquid Samples with Application to Chemical Reactions and Biochemistry

Abstract Calorimetry has long been used to probe the physical state of a system by measuring the heat exchanged with the environment as a result of chemical reactions or phase transitions. Application of calorimetry to microscale biological samples, however, is hampered by insufficient sensitivity and the difficulty of handling liquid samples at this scale. Here, a micromachined calorimeter sensor that is capable of resolving picowatt levels of power is described. The sensor consists of low‐noise thermopiles on a thin silicon nitride membrane that allow direct differential temperature measurements between a sample and four coplanar references, which significantly reduces thermal drift. The partial pressure of water in the ambient around the sample is maintained at saturation level using a small hydrogel‐lined enclosure. The materials used in the sensor and its geometry are optimized to minimize the noise equivalent power generated by the sensor in response to the temperature field that develops around a typical sample. The experimental response of the sensor is characterized as a function of thermopile dimensions and sample volume, and its capability is demonstrated by measuring the heat dissipated during an enzymatically catalyzed biochemical reaction in a microliter‐sized liquid droplet. The sensor offers particular promise for quantitative measurements on biological systems.

we minimize the noise equivalent power (NEP) of the thermopile, which can be written as the ratio of the Johnson noise [2] ( 4 , where is Boltzmann's constant, T is temperature, R is the electrical resistance of the thermopile, and B is bandwidth) to the thermopile responsivity (Σ ∆ ∆ ∆ , where P is the power dissipated by the sample, ∆ is the difference between the sample and a given reference, and ∆ is the difference between the Seebeck coefficients of the materials in the thermopile): (S1) To evaluate the NEP, the relationship between the power P dissipated by the sample and the resulting ∆ needs to be estimated using a thermal model of the sensor. The details of the model are given in the Analytical thermal model section. Here, we provide a brief synopsis. The model has two components: derivation of 1) the temperature difference between the sample and the ambient, and 2) the temperature distribution along the thermopile. For typical sensor dimensions, the heat loss from the sample occurs predominantly by conduction through air; only a very small fraction of heat is lost via conduction through the thermopiles or the membrane because of their small cross-sectional area. Thus, the temperature difference between the sample and ambient is set solely by the power dissipated by the sample and the rate of heat loss to the ambient air. Assuming a temperature distribution that is approximately spherically symmetric around the sample area of the sensor, the temperature difference between sample and ambient is given by Here is the thermal conductivity of air and a is the radius of the sample. This temperature difference creates a temperature profile inside the thermopile. If the thermopile is sufficiently long, the reference side of the thermopile is at ambient temperature and the thermopile generates the strongest possible signal, but it also has greater electrical resistance and more Johnson noise.
If the thermopile is short, the signal is weaker because of the smaller temperature difference, but it has lower electrical resistance and thus less noise. The temperature distribution along the length of the thermopile may be estimated by modeling the thermopile as a cooling fin that loses heat both by conduction through the fin and through air. Taking into account the temperature distribution along the length of the thermopile, the noise equivalent power can then be approximated as where l is the length of the thermopile, k is the effective thermal conductivity of the thermopile, ℎ is the thickness of the thermopile, and is the distance from the sensor membrane to the wall of the chamber that encloses the sensing portion of the sensor. Writing the electrical resistance of the thermopile in terms of the dimensions of the sensor, we finally find the following expression for the noise equivalent power of the sensor,

Analytical thermal model
Assuming heat loss occurs by conduction through air only, the steady-state temperature distribution around a spherically symmetric sample that is generating heat at a rate P is given by , where TA is the ambient temperature, r the distance to the center of the sample, and g the thermal conductivity of air. The temperature difference between the sample and its surroundings is then where a the radius of the sample. To determine the temperature distribution in the thermopile, the thermopile is modeled as a heat fin, a distance Lo from a wall at temperature TA. In the steady state, the temperature in the thermopile is then approximately where and ℎ are the effective thermal conductivity and thickness of the thermopile.
Finally, the temperature drop across a thermopile of length l is approximately The material properties used to calculate Figure 1c are listed in Table S1. Table S1. Material properties for calculating NEP as a function of thermopile length [3][4][5] Material Seebeck coefficient relative to platinum (μV/K)

FEM simulation
Finite element simulations were performed using the commercial software package COMSOL Multiphysics®. The parameters used in the model are listed in Table S1. To reduce the size of the model, the individual legs of the thermopiles were not resolved, but were modeled using effective material properties taking into account the geometry of the thermopiles. The effective thermal conductivity of the thermopiles was taken as 5.81W/m K in the direction of the temperature gradient, and 3.95 W/m K in the perpendicular direction, for an effective thermopile thickness of 1.25 µm.
The material properties used in the FEM model are listed in

Fabrication of the picocalorimetric sensors
The sensors were fabricated using standard microfabrication techniques. Figure S1 illustrates the fabrication sequence. Figure S2 shows the residual stress in the nichrome and constantan coatings as a function of Ar working gas pressure. While the majority of the processes used to fabricate the sensors are fairly standard microfabrication steps, integration of the thermopile on the Si3N4 membrane presents a particular challenge. Constantan and nichrome films with a thickness of 500 nm sputter deposited under typical conditions of Ar working gas pressure (3-5 mTorr) on a Si3N4-coated Si substrate show extensive fracture and delamination. Moreover, as soon as the sensor membrane is made freestanding, even intact constantan and nichrome films fracture because of the greatly increased energy release rate associated with a more compliant substrate [10] . Successful deposition of these films requires careful management of the Ar pressure to limit the residual stresses in the coatings ( Figure S2).  Constantan is deposited without cracks at a working gas pressure of 3 mTorr, pressures of 4 mTorr or greater result in extensive cracking during deposition. Nichrome was deposited without cracking at argon pressures below 1.5 mTorr.  Table 2  are connected in series, the resolution improves by a factor of two.

Effective radius and time constant
The data in Figure 5a can be used to determine an effective radius aeff that can be used to evaluate the effect of sample volume on the responsivity of the sensor. Assuming the response of the thermopiles is given by where and are 25 µV/K and -35 µV/K, the Seebeck coefficient of Nichrome and Constantan, respectively. We define the effective sample radius, aeff, such that the temperature difference, ∆ , between sample and reference is given by where α is a numerical factor that brings Equation (1) in quantitative agreement with the FEM simulations for a blank sensor and that has a value of approximately 0.35. Combining Equations (S10) and (S11) yields the following expression for aeff: (S12) Equation (S12) allows direct calculation of aeff as a function of sample volume Vs from the data in Figure 5a. The effective radius is then well represented by 791 µm 109 µ µ for the sensor used to collect the data in Figure 5a (Sensor 22). Once the effective radius is known for a given sample volume, a combination of Equations (S10) and (S11) yields the responsivity of the sensor for that sample volume.
The effective radius also elucidates how the response time of the sensor changes with sample volume and characteristics. On the assumption that heat loss happens mainly by conduction through air and that the thermal mass of the sensor is negligible compared to that of the sample, a simple lumped thermal model results in the following expression, , for the time constant  of a sensor loaded with a sample of volume Vs, density , and heat capacity . The linear relationship between the time constant and the sample volume divided by the effective sample radius is indeed borne out in Figure S4, with a slope that is in good agreement with Equation (S13) [11,12] .  Figure S5 shows the evolution of the sensor signal as a function of time. Figure S6 shows the phosphate calibration curve. Figure S5. Effect of evaporation of aqueous solutions. Calorimetric signal obtained as a result of differential evaporation between a sample and a reference droplet. After evaporation of the droplet in the reference area, the signal changes sign as the temperature of the sample area decreases below that of the reference area. Data were obtained using a sampling frequency of 30PLC with auto-zero and power line synchronization enabled, and the built-in analog filter turned off. A 10-reading moving average digital filter was used to smooth the data. phosphate were fit to a quadratic function to generate calibration curves for each day.