Contactless and spatially structured cooling by directing thermal radiation

In recent years, radiative cooling has become a topic of considerable interest for applications in the context of thermal building management and energy saving. The idea to direct thermal radiation in a controlled way to achieve contactless sample cooling for laboratory applications, however, is scarcely explored. Here, we present an approach to obtain spatially structured radiative cooling. By using an elliptical mirror, we are able to enhance the view factor of radiative heat transfer between a room temperature substrate and a cold temperature landscape by a factor of 92. A temperature pattern and confined thermal gradients with a slope of ~ 0.2 °C/mm are created. The experimental applicability of this spatially structured cooling approach is demonstrated by contactless supercooling of hexadecane in a home-built microfluidic sample. This novel concept for structured cooling yields numerous applications in science and engineering as it provides a means of controlled temperature manipulation with minimal physical disturbance.

1. Transmission spectrum of the germanium window. Figure S1: Transmission of the germanium window placed in front of the cryostat. The data is kindly provided by Thorlabs GmbH.

Mathematical analysis of the change of the view factor
With the values of the mirror dimensions as provided by the manufacturer, the following calculations have been performed to estimate the improvement of the view factor (Fv) by the addition of an elliptical mirror. The calculated values for each situation (with and without the mirror) are summarized in Table S1. From the focus point F1 on the sample, the entire surface area Atotal of a half-sphere with radius 10.16 cm (focal length between F1 and F2) is 648.59 cm 2 . Figure S2: Depiction of the view factor without a) and with b) the elliptical mirror. The sample emits thermal radiation equally in all directions. We are considering the view factor FV for the focal point F1 on the sample. The light red describes the section of the view factor where radiation leaving the warm sample at F1 does not strike the cold bull's eye on the cryostat window. a) Without the elliptical mirror, only the radiation emitted within the opening angle α is reaching the cold sample. The spherical cap of the half sphere corresponding to the opening angle is shown on the right, illustrating the surface areas of warm (Awarm) and cold (Acold) that determine the view factor from point F1. b) Introducing an elliptical mirror improves the view factor significantly, since a larger proportion of the radiation emitted at a solid angle is reflected to the cold bull's eye. In this configuration, only the thermal radiation emitted with the angle β-α does not contribute to the view factor. View factor without the elliptical mirror: When placing the sample at the focal point of the elliptical mirror, the only "cold region" that the sample is exposed to is the surface area of the germanium window of the cryostat. In this case, there is no structured cooling taking place as the sample cools down homogeneously. Considering the dimensions of the window, only 0.78% of thermal radiation emitted at the focus F1 strikes the cold substrate (Table S1).
View factor with the elliptical mirror: With an elliptical mirror, the proportions of the warm and cold radiation that determine the view factor change significantly. By considering the focal length and mirror dimensions, only a small fraction of thermal radiation that is emitted at an angle β-α is not directed by the mirror. This results in a large increase of Fv to 72.14%.

Value
Without  Table S1: Calculated values for the surface areas of the cold and warm sections of the view factor emphasize significance of the elliptical mirror.

Analysis of temperature decay curves.
For the measurement of the hexadecane in the microfluidic chamber, the temperature decay was fitted bi-exponentially. The fit equation is given by: ( ) = 1 * − 1 + 2 * − 2 +  The bi-exponential fit describes both, the exponential Newton's law of cooling due to convection and conduction, as well as the radiative heat transfer part. The weights a1 and a2 for both fits are of similar values and allowing us to compare the fits. Looking at the cooling rates κ1 and κ2, it is clearly evident that the fast rate κ1 is approximately 5 times faster for position 1 than it is at position 2. The slower rate κ2 and the added constant c is almost identical for both curves. High R 2 values for both curves validate the fit.
As shown in Fig. S3, the data is fitted until the crystallization of hexadecane takes place, which is indicated by the vertical temperature jump. The temperature will further decrease in both positions 1 and 2. By considering a mono-exponential fit for the later time points, the offset values to which the exponents converge to have been extracted. For position 1 this value amounts to 11.08 °C, whereas a value of 11.94 °C is obtained for position 2. Position 1 is slightly colder, as it is located at the center of the focus of the elliptical mirror. At this position, the influence of the radiative cooling is strongest. Figure S3: Bi-exponential fits applied to the temperature curves measured at position (Pos) 1 (a) and 2 (b). Only the blue (Pos 1) or green (Pos 2) points of temperature measurements are fitted (fit shown in red). The greyedout data points are not considered for the fit. The colored-in areas show the region considered for the calculation of the half-life times.

Determination of the half-life time:
The half-lives were calculated by considering the time it takes for the sample after spontaneous crystallization to cool down back to half the value of the temperature before crystallization.