“Cooling by Heating”—Demonstrating the Significance of the Longitudinal Specific Heat

Heating a solid sphere at its surface induces mechanical stresses inside the sphere. If a finite amount of heat is supplied, the stresses gradually disappear as temperature becomes homogeneous throughout the sphere. We show that before this happens, there is a temporary lowering of pressure and density in the interior of the sphere, inducing a transient lowering of the temperature here. For ordinary solids this effect is small because $c_p\cong c_V$. For fluent liquids the effect is negligible because their dynamic shear modulus vanishes. For a liquid at its glass transition, however, the effect is generally considerably larger than in solids. This paper presents analytical solutions of the relevant coupled thermoviscoelastic equations. In general, there is a difference between the isobaric specific heat $c_p$ measured at constant isotropic pressure and the longitudinal specific heat $c_l$ pertaining to mechanical boundary conditions that confine the associated expansion to be longitudinal. In the exact treatment of heat propagation, the heat-diffusion constant contains $c_l$ rather than $c_p$. We show that the key parameter controlling the magnitude of the “cooling-by-heating“ effect is the relative difference between these two specific heats. For a typical glass-forming liquid, when the temperature at the surface is increased by 1 K, a lowering of the temperature at the sphere center of the order of 5 mK is expected if the experiment is performed at the glass transition. The cooling-by-heating effect is confirmed by measurements on a glucose sphere at the glass transition.

Popular Summary

When a bulk piece of material is heated at its surface, does its center become hotter or cooler? It turns out that the answer to this seemingly banal question is not obvious at all at a fundamental level. In this paper, we report, first through a theoretical analysis and prediction and then with experimental confirmation, an extraordinary phenomenon of cooling at the center of a supercooled liquid drop heated at its free surface.

Most liquids, when supercooled, eventually end up in the glassy state. Just before the glassy state sets in, such a liquid shows remarkable features. It becomes viscoelastic, or either solid- or liquidlike, depending on whether the time scale of your observation is below or above a threshold time. And, the threshold time increases rather dramatically as the temperature of the liquid is lowered. What is the implication of these properties for the heat-transport process in the supercooled liquid? The ordinary heat-diffusion theory, which treats the heat transport and the mechanical deformation (including expansion and shear deformation) as independent processes, is fundamentally inadequate for describing surface-heating-induced physical processes in the liquid. Based on a theory that couples heat transport to mechanical deformation governed by viscoelasticity, however, we have predicted a measurable, transient center-cooling effect induced by surface heating for a supercooled glucose ball. With a highly sensitive experimental apparatus, we have measured a temperature drop of 7 mK at the center of a glucose ball of 19 mm in diameter in response to a temperature elevation of 5 K at the free surface, confirming both qualitatively and quantitatively the theoretical prediction.

The physics underlying this intriguing phenomenon is not obvious. Heating a piece of material at its surface leads to heat-induced expansion in the material. But, whether that expansion proceeds outward or inward depends on the material’s mechanical properties and how the surface is mechanically constrained. If the expansion proceeds outward, as in the case of the glucose ball in our experiment, a negative pressure change results from that outward expansion and propagates to the interior of the material, causing the temperature there to go down. But, if the material is an ordinary liquid, the pressure change is so short lived that this center-cooling effect is nonexistent for all practical purposes. In a stark contrast, in the case of a supercooled viscoelastic glucose ball, two of its properties combine to make the pressure change measurable: its solidlike characteristic on short, but experimentally accessible time scales, and even more important, a large difference in how the liquid absorbs heat under the condition of either constant pressure or constant volume.

Our work adds a new fascinating page to the fundamental physics of classical materials.

Figure 1

Sketch of overlapping relaxations (absolute values) of $4G/(3M_T)$ (green line) and the thermomechanical coupling $a=T{\alpha }_p^2{K}_T/c_p$ (red line). The longitudinal coupling constant $a_l=(c_l-c_p)/c_p$ (black line) is the product of those two frequency-dependent functions. Near the crossing of the two curves, the difference between $c_l$ and $c_p$ is generally largest. This determines the time scale of experiments on glass-forming liquids, where the cooling-by-heating effect is particularly large.

Figure 2

The real part of the ratio between the complex amplitudes of temperature at the center and the heat supplied at the surface scaled with the isobaric heat capacity. At high frequencies the limit becomes the negative value $-a_l/(1-a_l)=1-c_p/c_l$.

Figure 3

The temperature of the sphere as a function of time for different radii normalized to the final temperature $\Delta T_{eq}=\delta Q_0/(V_0{c}_p)$. After the addition of heat at the surface, the temperature drops instantaneously throughout the sphere, showing adiabatic cooling. The time scale is given by the characteristic heat-diffusion time, $\tau =R^2{c}_l/\lambda$. The longitudinal coupling constant is chosen to be $a_l=0.091$.

Figure 4

The $rr$ component of the stress tensor, $1-F_2$, as a function of time for a number of radii scaled with the initial stress, ${\sigma }_0=\frac{4}{3}\frac{G}{M_T}\frac{{\beta }_V}{V_0{c}_l}\delta Q_0$. After the addition of heat at the surface, ${\sigma }_{rr}$ immediately increases throughout the sphere. The stress is released as heat diffuses toward the center from the surface.

Figure 5

The $\theta \theta$ component of the stress tensor, $1-\frac{3}{2}{F}_2+\frac{1}{2}{F}_1$, as a function of time scaled with the initial stress ${\sigma }_0$. There is an initial adiabatic positive step up in ${\sigma }_{\theta \theta }$ throughout the sphere. The regions that have been reached by the incoming diffusive heat at a certain point in time experience a negative $\theta \theta$-stress component since the inner unheated regions pull the outer heated regions inwards.

Figure 6

The temperature at the center of the sphere as a function of time. After the temperature at the surface is raised, the temperature at the center of the sphere initially decreases. This only happens when $c_l\ne c_p$, i.e., when the longitudinal coupling $a_l$ is not negligible. The temperature step at the boundary is $\Delta T=1\mathrm{K}$, and the time scale is given by the characteristic diffusion time, $\tau =R^2{c}_l/\lambda$.

Figure 7

Electrical equivalent diagram of the interaction of a volume element with its surroundings through two gates. The thermal gate where entropy displacement $\delta S$ or temperature $\delta T$ can be controlled, and the mechanical gate where volume displacement $\delta V$ or pressure $\delta p$ can be controlled. $\delta S$ and $\delta V$ are generalized charge displacements and $\delta T$ and $-\delta p$ are generalized voltages. The relaxational elements $J_A$ and $J_B$ are simple single-relaxation-time elements.

Figure 8

The change in temperature $\delta T$ at the center of a sphere, after a temperature step of $\Delta T=1K$ has been applied at the surface. A model of the thermoviscoelastic relaxation of glucose with realistic limiting thermodynamic parameters has been invoked. Time is scaled by the characteristic diffusion time ${\tau }_0$, which is 800 s for a glucose ball of radius 9.5 mm. The minimum of approximately $-5\mathrm{mK}$ occurs for the 309 K curve just above $T_g$ at $0.017{\tau }_0$, corresponding to 14 s. Notice that although the cooling-by-heating phenomenon is present in the glassy solid state, it becomes more pronounced at the glass transition. The squares marking the minima at each temperature are plotted against temperature in Fig. 10.

Figure 9

A simulation similar to the one shown in Fig. 8 except that ${\alpha }_{p,\infty }$ has been set to 0, forcing cooling by heating to be absent in the glassy solid phase. It is seen that the cooling-by-heating effect still appears as a dynamic phenomenon at the glass transition.

Figure 10

The minima from Fig. 8 as a function of temperature. At high temperatures, the phenomenon of cooling by heating is absent. Going down in temperature, the liquid becomes more viscous and a dip in temperature emerges. Close to the glass transition, the cooling-by-heating phenomenon gets most pronounced. As temperature is decreased further and the liquid enters the glassy state, the effect is still present, but is reduced in strength.

Figure 11

Sketch of the experimental setup. The liquid is molded into a sphere, in which a small NTC-thermistor bead is placed at the center, connected to wires that lead to a multimeter that performs resistance measurements. The sphere is inserted into the cylindrical chamber of a cryostat. The photo shows one of the samples; at the time of the photo shoot (1 month after molding) the sample is no longer transparent because it crystallized.

Figure 12

The temperature at the center of a sphere of glucose, with diameter 19 mm. At time $t=0\mathrm{s}$ a step in temperature from 298 to 303 K is applied with the cryostat (see the setup in Fig. 11). The temperature drops initially by 7.3 mK as result of cooling by heating. The inset shows the minimum temperature $\delta T_{\mathrm{MIN}}$ reached in three measurements, on three different samples, at the same temperature.