where is the offset, the final value, and the time constant. Typical values of these parameters are

reported in Table 2. The slope of the graph in Figure 2(b) is negative, indicating the evolution

of toward a stable fixed point [22] . The rate of increase of, i.e. the absolute value of the slope of

the graph, is. The amplitude of, i.e. the magnitude of the interval

along the horizontal scale of the graph in Figure 2(b), is 21.5 mW. In the 50 hours after start-

ing the illumination of the power-meter sensor, the power, shown Figure 2(c), reaches a plateau. However, in multiple data sets, we always observe that undergoes small sinusoidal instabilities shown in Figure 2(d) in which the vertical scale has been expanded. The instability, due to small periodic fluctuations in the closed sample compartment, can be fitted with:

where and are the offset value and half of the separation between and, respectively. The critical time is the point in time in which reaches, while is the amount of time necessary to move from to (practically, 1/4 of the period of the sinusoidal function). The typical values of, , , and are reported in Table 2, labelled in Figure 2(d), and were obtained by placing the zero of the time-scale at the start of the illumination. The instability in is small, as

inferred from. The graph, shown in Figure 2(e), highlights a periodic behavior

with frequency! We observe the sinusoidal instability of, which modulates the amplitudes of the electric and magnetic fields of the IR radiation, to persist beyond the 50 h time interval in Figure 2(d).

We observed that the power of the IR radiation rises exponentially obeying Equation (5) at the start of the illumination, and exhibits a sinusoidal instability in the 50 hours thereafter. For the entire time span, we hypothesize that the IR radiation transfers energy, through electric contribution, to the surface density

of the electrons on the illuminated Cu plate of the PG device, and contributes to producing through the and fields (electric contribution) and (entropic contribution).To prove that a link exists between and, possibly also between and, we sketch the behavior of and relate it to the observed, , and.

To sketch, we hypothesize that, while hitting the surface of the Cu plate, the IR radiation modulates

the electric field through the sinusoidal instability of the IR power. In

turn, and its modulation act on the electrons of the Cu plate with force, where e is the electron’s charge. As in the photoelectric effect [3] , displaces the electrons away from the location in which the IR radiation impinges on the Cu plate, locally decreasing their surface density such that. However, unlike in the photoelectric effect, does not kick the electrons out of the Cu plate. In this process, varies in time t as well as in space, i.e. the 2-dimensional (2D) surface of the Cu plate. To allow us versatility in choosing reference system, orientation and phase, we represent the 2D space variable as the complex variable, where i is the imaginary unit. This choice resembles that adopted to describe light polariza-

tion through Jones matrices [23] -[25] . Thus,. All possible rotations of the refer-

ence system, phases, and positions in the 2D plane can be obtained by selecting magnitude and sign of, , , and.

With this choice of, upon starting the illumination, we picture to exponentially decrease accord-

ing to, where and are the initial and final surface electron den-

sities, the time constant, a vector with units of inverse length, and an arbitrary phase. We note that the exponential behavior is modulated by the oscillatory function.

In the subsequent 30 hours, from Equation (6) we expect to undergo a slow variation in time such

that, where the sine function has the frequency derived in Sec-

tion 3(a). With the choice of discussed above, and utilizing the laws of trigonometric functions for complex variables, we obtain:

Here, and are the instability’s propagation velocities along the x and y directions; and are the lengths of the Cu plate along x and y; finally, and are the critical times of the surface electron density’s instability along x and y. Considering and, the equilibrium electron density and its devia-

tion from equilibrium, respectively, we obtain. The 2D space variable

, therefore unveils a hyperbolic instability in modulated by a sine function.

While requires a spatiotemporal set of variables, the functions, , , and are only time-dependent. To decouple from the effects of on, , , and, and allow time t to be the only effective variable, we integrate over the surface area a of

the Cu plate as. Because of the capacitor-type structure of the PG device, with overall capacitance

C, we expect. This integration causes the loss of correlation between the phase of

and. In the 30 hours after starting the illumination, since where, from Equation (7) we expect the function to determine the behavior in time of. We also envision the capacitor-type structure of the PG device to affect and, while leaving constant.

Summarizing, we expect to obey an exponential behavior in the first minutes after starting the illumination, and to exhibit a hyperbolic-secant-type of instability in the 30 hours thereafter, with no phase relationship with, no periodic behavior, and with constant. We are currently exploring this hypothesis further numerically, which will be the focus of future work.

In this expression, and are the offset and amplitude of the departure of from the offset, respectively. The positive or negative sign of corresponds to a downward or upward concavity, respectively, of the instability. The critical time is the instant in which the maximum value is achieved. Finally, the term H indicates the half width at half maximum (HWHM), or minimum (depending upon the sign of) of the instability. The magnitude of corresponds to the long term equilibrium voltage

reported in Table 3. The typical values of, , , and H are reported in Table 4, labelled in Figure 6(a), and were obtained by placing the zero of the time-scale at the start of the illumination. We highlight that, since in Figure 6(b) does not peak at, it does not correlate with the behavior in time of

. The instability in is evidenced in the graph in Figure 6(c), where stable

and unstable fixed points [20] alternate in a complex fashion without periodicity.

Since the time-dependence is enclosed in a hyperbolic secant function, we name the instability in in Equation (11) and Figure 6(a) as hyperbolic instability. We establish the lack of correlation between, on one hand, and and on the other, as the criterion to identify such instability. Since the hyperbolic instability in is absent in, as expected from Section 3(b), we relate the instability to the sole electric contribution.

Here we highlight the correlations existing between the power and the voltage difference to further support the choice of the fitting function in Equation (11) for based on the hypothesis highlighted in Section 3(b).

We found that Equation (11), used to fit the hyperbolic instability in in the 30 hours after starting the

illumination of the PG device with IR radiation, can be a solution of the equation:

This work was supported by the U.S. Office of Naval Research (awards # N000141410378 N000141512158), JMU 4-VA Consortium (2013), Thomas F. Jeffress and Kate Miller Jeffress Memorial Trust (grant # J-1053), the Madison Trust―Fostering Innovation and Strategic Philanthropy-Innovation Grant 2015, the JMU Program of Grants for Faculty Assistance 2014, the JMU Center for Materials Science, and the JMU Department of Physics and Astronomy. The authors thank Dr. A. V. Zenkevich (Moscow Institute of Physics and Technology), Prof. G. Casati (University of Insubria, Italy), and Prof. D. J. Lawrence (JMU) for fruitful discussions.

Aidan L.Gordon,YosypSchwab,Brian N.Lang,Graham P.Gearhart,Tara R.Jobin,Justin M.Kaczmar,Zachary J.Marinelli,Harkirat S.Mann,Brian C.Utter,GiovannaScarel, (2015) Decoupling the Electrical and Entropic Contributions to Energy Transfer from Infrared Radiation to a Power Generator. World Journal of Condensed Matter Physics,05,301-318. doi: 10.4236/wjcmp.2015.54031

To highlight the effects of the entropic contribution and decouple it from the electric contribution, we collected the voltage difference and the temperature difference from the PG device activated by conductive heat transfer from a 100 W resistor in contact with the surface of the device. A temperature of ≈24˚C was generated by the PG device activated with a B&K Precision 1665 power supply providing 0.02 A and 3.2 V to the resistor. The achieved temperature is of the same order of magnitude as that detected when IR radiation hits the PG device, as can be seen in Figure 5(c) and Figure 5(d). We used a PG device 07111-9L31-04B by Custom Thermoelectric Inc. finished with an AlO plate (not with a Cu plate) and, in selected cases, gaffer tape. We placed the sample holders on both sides of the PG device in the toward architecture to create a capacitor structure with electrodes on both faces and thus avoid the “decay” in shown in Figure 3(d). The experiments investigating the effects of conductive heat transfer were performed in an insulated sample compartment described in Ref. [30] . The PG device was horizontally fixed on the sample holders in all the measurements. When the IR radiation was used to compare the results obtained with conductive heat transfer, the PG device was also positioned horizontally and at an angle of incidence with respect to the IR radiation. The basic experimental parameters are summarized in Table 1.

In the 400 seconds immediately following the start the activation of the PG device by either conductive heat transfer or IR radiation, we measured the rate of increase of () and (), and the ratio between the jumps in voltage and temperature difference in the away and toward architectures. We report the average values of, , and R, alongside their uncertainties, in Figures A1(a)-(c), respectively. Figure A1(a) indicates that the average values obtained by activating the PG device with IR radiation are always larger than those obtained by activating the PG device using conductive heat transfer. Figure A1(b) suggests that the average values have a common value, independently of the source of activation of the PG device. Moreover, the average and values are similar when the PG device is activated by conductive heat transfer. In addition, the values coincide with those derived from Figure 4(b) and Figure 4(d). This finding supports the identification of the contribution to energy transfer with slower rate of increase with the entropic contribution. Figure A1(c) shows that, within the errors, the average

value of the ratio R is larger when the PG device is activated by IR radiation through the electric contribution.

In the 400 seconds immediately following the start the activation of the PG device by either conductive heat transfer or IR radiation, we also measured the jump in voltage obtained performing measurements on the PG device described above and finished with colored gaffer tape. We performed the measurements for the away and toward architectures. Figure A2(a) displays the magnitude of obtained by activating the PG device with conductive heat transfer from a 100 W resistor. The observed trends are neither affected by the color nor by the presence of the tape. On the other hand, Figure A2(b) shows that the magnitude of obtained by activating the PG device with IR radiation, is slightly sensitive to the color of the tape, and exhibits a noticeable drop when the tape is absent. Interestingly, the black-colored tape on the illuminated face of the PG device doubles the magnitude of compared to the case without tape. Values of above 0.5 mV, repre- sented by the horizontal line in Figure A2(b), can be achieved only with tape present on the illuminated face of the PG device. Thus, we conclude that the tape profoundly affects the capacitor-type behavior of the PG device.

In the time span of about 50 hours following the start of the activation of the PG device with conductive heat transfer from a 100 W resistor, the voltage difference, temperature difference, and temperatures and of the activated and non-activated faces, respectively, are shown in Figures A3(a)-(d). We collected the particular set of data chosen using the PG device described above, finished with pink-colored gaffer tape, and set-up in the away architecture. Nevertheless, Figure A3 is representative of all the data collected in the away and toward architectures, with tape of all the available colors, and without tape. In all these cases, we observed, , , and to be featureless and follow the same trends. The trend of is symmetric to that of, , and. This behavior differs from that observed for the PG device activated by the IR radiation and captured in Figure 5. In this case, and the temperatures and of the illuminated and non-illuminated faces are strongly correlated, while is flat and featureless.

The results in Figures A1-A3, together with those of Figures 3-5, further support that the electric and the entropic contribution to energy transfer from low power IR radiation to the PG device are decoupled.

Figure 6 shows the hyperbolic instability revealed in the voltage difference in the time span of about 30 hours following the start of the illumination of the PG device with IR radiation in the away architecture. We identify such hyperbolic instability when no correlation is found between and the temperatures and of the illuminated and non-illuminated faces of the PG device, respectively, and when is flat. To demonstrate that hyperbolic instabilities are common in the interaction between IR radiation and a PG device, we collected a set of measurements with the PG device described in Appendix-1. With the AlO plate or

the layer of tape facing the IR radiation, the surface density refers to the charges in the AlO plate or in the tape. The results, summarized in Figure A4, report the average values of the critical time () and amplitudes () of the revealed hyperbolic instabilities. We found a correlation neither between the values of in Figure A4(a) and the signs of in Figure A4(b), nor between these quantities and the away or toward architectures. However, by observing Figure A4(a), we found that is typically located around, which is about twice the magnitude of at for the sinusoidal instability in of the IR source reported in Table 2. In addition, we noted lower values for in Figure A4(a) located around, i.e. in the neighborhood of. These observations highlight the correlations existing between the instabilities in and as already discussed in Section 3(e). Furthermore, in Figure A4(b) we noted randomness in the sign of, which we attribute to the loss of correlation between the phase of and, and then between and. We predicted the loss of phase relationship in Section 3(b).

We summarize the findings in Figure A4 as follows: the hyperbolic instability in is a common

phenomenon, which is linked with the sinusoidal instability in the power of the IR radiation.