Friction, Free Axes of Rotation and Entropy

Abstract

Friction forces acting on rotators may promote their alignment and therefore eliminate degrees of freedom in their movement. The alignment of rotators by friction force was shown by experiments performed with different spinners, demonstrating how friction generates negentropy in a system of rotators. A gas of rigid rotators influenced by friction force is considered. The orientational negentropy generated by a friction force was estimated with the Sackur-Tetrode equation. The minimal change in total entropy of a system of rotators, corresponding to their eventual alignment, decreases with temperature. The reported effect may be of primary importance for the phase equilibrium and motion of ubiquitous colloidal and granular systems.

1. Introduction

Interrelations between a friction and entropy are perplexing and ambiguous. Friction forces usually result in irreversibility and the generation of entropy [1,2,3,4]. However, as it was already demonstrated by Nosonovsky et al., friction force may lead to self-organization, patterning and the production of negentropy [5,6,7,8,9]. The pattern formation may involve self-organized criticality and reaction-diffusion systems [6]. The self-organization and patterning inspired by friction forces give rise to rich physical content in multi-scaled, hierarchical systems [7]. In certain cases, the entropy production at a particular scale level may be compensated for by the entropy consumption at another level [7]. Possible mechanisms of the decrease of entropy under dry friction have been discussed in Reference [8]. From the practical point of view, friction-induced self-organization is important due to the self-healing and self-lubrication arising from negentropy generation, which is inherent for these phenomena [9].

Our paper is devoted to the friction-induced orientation of rotating rigid spinners, resulting in the decrease of the entropy of a gas containing such spinners; in other words, in the production of orientational negentropy. The effects related to orientational entropy have already been extensively addressed in the context of phase transitions in polymers and liquid crystals [10,11]. Entropic contributions to common ordering transitions are essential for the design of self-assembling systems with addressable complexity [12]. However, the friction-induced generation of negentropy, occurring under the re-orientation of rotating bodies, is first discussed in this paper. The reported effect may be essential for the phase equilibrium and motion of colloidal and granular systems, widespread in nature and engineering.

2. Friction-Induced Orientation of a Rotator

The friction-induced orientation of a rotating rigid body is demonstrated in the following series of experiments.

2.1. Friction-Induced Orientation of Hollow (Light) and Filled (Heavy) Tops of the Same Shape

The first series of experiments was performed with the hollow top depicted in Figure 1A. The mass of the top was varied from 8.650 to 99.330 g by filling its conical cavity with steel balls, as shown in Figure 1B. The mass and diameter of the balls were m = 0.465 ± 0.0005 g and D = 4.8 ± 0.05 mm. The angular velocities of spinning of the full (i.e., filled with steel balls) and hollow tops were ${\omega }_f=25\pm 0.5 {\mathrm{s}}^{-1}$ and ${\omega }_h=32\pm 0.5 {\mathrm{s}}^{-1}$, respectively. Thermal imaging of the rotation of the top was carried out with the Therm-App TAS19AQ-1000-HZ thermal camera (manufactured by Opgal Optronic Industries, Karmiel, Israel) equipped with a LWIR (long-wave-length infrared) 6.8 mm f/1.4 lens. A typical image, representing the temperature field observed under the rotation of the heavy top, is supplied in Figure 2. A distinct thermal track from the heavy spinning top’s tip neb is present on the surface, whereas in the case of a light top the track is beyond the measurement sensitivity range. The thermal field arising from the spinning of the top is supplied in Figure 3. The maximal temperature jump established for the hollow top with a mass of 8.650 g was 0.3 ± 0.07 °C, whereas the top filled by steel balls (m = 99.330 g) gave rise to a temperature jump close to one centigrade, as shown in Figure 3. These results are well expected, due to the fact that the heavier tops led to larger friction forces, resulting in higher heat dissipation. The qualitative observation should be emphasized: when the hollow light top started its rotation on a rough paper when inclined from vertical to θ = 26° ± 0.5° (θ is one of the three Euler angles, see Reference [13]), it continued the inclined spinning until reaching a complete stop (during 30 s), whereas the heavy top, inclined at θ = 26° ± 0.5°, was re-oriented vertically after ca. 8 s of rotation. It was observed that the orientation time decreases with the surface roughness increase. It took 14 s to align the heavy top from θ = 26° ± 0.5° to the vertical orientation on a polished marble plate, and almost 20 s on a glass slide.

2.2. Friction-Induced Orientation of Tops Filled by Agate and Steel Balls

The second series of experiments was performed with the hollow transparent top depicted in Figure 1A, filled by a number of agate (m = 1.340 g, D = 9.7 ± 0.05 mm) and steel balls. The number of balls was varied from one to four, as depicted in Figure 4 and Figure 5. The angular velocities of the tops at quasi-steady spinning time-spans were $\omega =50 \pm 0.5 {\mathrm{s}}^{-1}$ for agate balls and $\omega =80 \pm 0.5 {\mathrm{s}}^{-1}$ for steel ones. The use of the transparent top allowed direct visualization of the location of the steel and agate balls within the top. It was observed that spinning ball-filled tops always adopted a symmetric configuration of balls, where the axis of symmetry was the free axis rotation of the system comprised of the top and balls. When a top was filled by a single agate ball, it chose the lowest possible position at the free axis of the top, as depicted in Figure 4A,B. A pair and triad of agate balls were located symmetrically relative to the axis of rotation at the equator of the top, as shown in Figure 4C,D. Similar results were obtained with the top filled by steel balls, as illustrated in Figure 5. It should be emphasized that when the top was filled by four steel balls, two symmetrical configurations were possible, as depicted in Figure 5C,D; both of them were experimentally observed. One more experimental observation is noteworthy, namely that when the hollow top was greased from inside by silicone oil (polydimethilsiloxane, Mn = 17,000, supplied by Aldrich), the spatial orientation of the agate and steel balls within the cavity of the spinner remained the same. This means that viscous friction also promotes symmetrization of the rotating system.

2.3. Friction-Induced Orientation and Re-Orientation of Coin-Like Tops

The third series of experiments was performed with a coin (m = 6.250 g; diameter D = 25.9 ± 0.05 mm; thickness b = 1.7 ± 0.05 mm; possessing two holes with a diameter of d = 3 ± 0.1 mm, drilled symmetrically at the distance 2l = 11 ± 0.05 mm one from another, as depicted in Figure 6) rotated on a flat horizontal surface around the vertical axis. In this series of experiments, a more complicated rotation was observed: the coin switched the location of holes between two symmetrical orientations admitted by the vertical axis of rotation, as shown in Figure 6A,B. The mechanism of this switching is beyond the scope of our paper. However, consider that the orientation of the coin, depicted in Figure 6A, corresponds to the maximal moment of inertia I of the coin relative to the vertical axis. This kind of rotation was registered for the angular velocity $\omega \cong 8-15\pm 0.5 {\mathrm{s}}^{-1}$ $\omega$, whereas the spatial orientation of the coin depicted in Figure 6B (that is, the minimal I respective to the same axis) took place for $\omega \cong 17-30\pm 0.5 {\mathrm{s}}^{-1}$. In both of these experiments, the coin chose the free axis of rotation. The difference between the momenta of inertia of the coin, oriented as shown in Figure 6A,B, equals $\Delta I=I_{\max }-I_{\min }=2m_0{l}^2$, where m₀ is the mass of the “disk” filling the hole and l = 5.5 mm is the distance from the symmetry axis to the hole center. The same difference was related to the moment of inertia of the non-drilled coin $\Delta I/I=8(m_0/M)\times {(l/R)}^2=8{(rl/R^2)}^2\cong 0.02$ where r = 1.5 mm and R = 13 mm are the radii of the hole and the coin, respectively.

3. Estimation of the Entropy Generation Due to the Friction-Inspired Orientation of a Rotator

Consider a rigid rotator possessing a number of free axes of rotation. Recall that free axes of rotation are those keeping their spatial orientation in the absence of external forces [13]. When external forces are absent, a stable rotation occurs around the principal axes of a rigid body, corresponding to maximal and minimal momenta of the rotator [13,14]. However, when friction acts on the rotator, the stable motion is observed around the axis corresponding to the maximal moment of inertia, as demonstrated by the experiments and reported in Section 2.1, Section 2.2 and Section 2.3. In solid rotor systems, the effect of friction can be thought of as loss of energy from the rotational mode. However, friction has one more effect: namely, it orients the rotator. It should be stressed that friction force is the only lateral force establishing the symmetrical location of agate and steel balls placed into the transparent spinner, as shown in Figure 4 and Figure 5. This is the friction force that also defines the spatial location of the holes drilled in the rotating coin, depicted in Figure 6. The same dry friction establishes the vertical orientation of a “tippe-top” (also known as the Thompson top), as shown in References [15,16].

The mechanism of orientation of spinners by friction is discussed in detail in the classical course by Levi Civita and Amaldi (see Reference [17]). Friction force dictates the orientation of the axis of rotation for a variety of tops, and thus decreases the number of rotational degrees of freedom in the system [15,16,17]. A rectangular parallelepiped falling through air, depicted in Figure 7, supplies another illustrative example of the orientation of a rotator by air friction forces. Owing to the viscous friction, the body chooses to rotate around the main axis OO₁, corresponding to the maximal moment of inertia [18]. This orientation effect leads to the generation of negentropy in a system of rotators.

Consider a gas of identical rigid rotators influenced by either kind of friction (it seems that such systems were introduced by W. Thomson [19,20]). This gas may be a dust cloud (including the dust plasma [21]), a colloidal suspension built of rigid particles [22], or a capillary cluster of water droplets [23,24]. If the system is thermally isolated, the total change of entropy with time $\Delta S$ should be necessarily positive, following the second law of the thermodynamics:

$$\Delta S=\Delta S_{\mathrm{H}}+\Delta S_{ord}>0$$

(1)

where $\Delta S_{\mathrm{H}}={\displaystyle \int \frac{\delta Q}{T}}>0$ is the term describing the generation of entropy in the system of N identical rotators due the heat dissipation caused by friction (either dry or viscous), addressed experimentally in Section 2.1, and illustrated with Figure 2 and Figure 3; $\Delta S_{ord}$ is the negentropy production due to the orientation of the rotators by the same friction force (again, dry or viscous; see Section 2.2 and Section 2.3 for the experimental study of the orientation of spinners by dry and viscous friction forces). Accurate calculation of the orientational contribution to entropy is an extremely challenging mathematical task [10,25]. Thus, we propose to estimate the upper boundary of negentropy generation due to the eventual elimination of i-degrees of freedom by friction (namely $\Delta S_{\max }^{ord}<0$) according to the Sackur-Tetrode formula, supplying the entropy of an ideal gas [26,27,28,29]:

$$\Delta S_{\max }^{ord}\cong -k_{B}N\ln \left[\frac{V}{N}{\left(\frac{2\pi m{k}_{B}T}{h^2}\right)}^{\raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]$$

(2)

where m is the mass of the rotator, V is the volume of the system, k_B and h are the Boltzmann and Planck constants, respectively, i is the number of degrees of freedom “frozen” by friction, and T is the temperature, introduced as the mean kinetic energy of the rotators (which is a subtle point in the discussed case [30]; however, for the sake of simplicity, we neglect problems related to the “thermalization” of the gas of rotators.

When friction eliminates two degrees of freedom, we obtain:

$$\Delta S_{\max }^{ord}\cong -k_{B}N\ln \left[\frac{V}{N}\left(\frac{2\pi m{k}_{B}T}{h^2}\right)\right]$$

(3)

It should be mentioned that the entropy jump due to the loss of rotational/translational degrees of freedom has been intensively discussed in biophysics, regarding bioinformatics of proteins and other complex molecules [31,32]. The maximal orientational entropy given by Equation (3) corresponds to the minimal possible total entropy change $\Delta S_{\min }$ observed when all rotators are completely aligned, given by:

$$\Delta S_{\min }\cong {\displaystyle \int \frac{\delta Q}{T}}-k_{B}N\ln \left[\frac{V}{N}\left(\frac{2\pi m{k}_{B}T}{h^2}\right)\right]$$

(4)

when dry friction force acts on rotators, it is reasonable to assume $\delta Q\ne \delta Q(T)$; thus, Equation (4) may be simplified as follows:

$$\Delta S_{\min }(T)\cong \frac{\Delta Q}{T}-k_{B}N\ln \left[\frac{V}{N}\left(\frac{2\pi m{k}_{B}T}{h^2}\right)\right]$$

(5)

where $\Delta Q$ is the total heat dissipated within the process of complete alignment (re-orientation) of the rotators. Equation (1) (arising from the second law of thermodynamics) combined with Equation (5) yields the non-trivial inequality:

$$\Delta Q>k_{B}TN\ln \left[\frac{V}{N}\left(\frac{2\pi m{k}_{B}T}{h^2}\right)\right]$$

(6)

The temperature at which $\Delta S_{\min }(T)=0$ is of particular interest, hinting at the reversibility of processes occurring in a thermally isolated system of rotators, at the temperature $T^{\ast }$, supplying the solution to the transcendental equation:

$$\frac{\Delta Q}{T}=k_{B}N\ln \left[\frac{V}{N}\left(\frac{2\pi m{k}_{B}T}{h^2}\right)\right]$$

(7)

It is also instructive to elucidate the temperature dependence of $\Delta S_{\min }$. It is easily seen from Equation (5) that:

$$\frac{\partial (\Delta S_{\min })}{\partial T}=-\frac{\Delta Q}{T^2}-\frac{k_{B}N}{T}<0$$

(8)

It is recognized from Equation (8) that the minimal change in total entropy, corresponding to the eventual complete alignment of rotators, is decreased with the temperature (i.e., the mean kinetic energy) of the rotators. If the friction is viscous, and $\delta Q$ is proportional to the mean value of ${\omega }^2$, and correspondingly it is proportional to the temperature T [33], we derive:

$$\frac{\partial (\Delta S_{\min })}{\partial T}=-\frac{k_{B}N}{T}<0$$

(9)

Again, the minimal jump in total entropy of the system of rotators decreases with the temperature. The thermal capacity of a system of N completely oriented spinning tops will obviously be given by:

$$c_v=\frac{1}{2}N{k}_B$$

(10)

which is governed by the single possible degree of freedom.

4. Conclusions

Friction forces may produce both entropy and negentropy, as discussed in References [5,6,7,8,9]. We address the situation where friction forces promote the orientation of rotating rigid bodies, and thereby eliminate (freeze) a number of their degrees of freedom (the well-known orientation of falling bodies or a tippe-top illustrates an alignment of rotators by dry and viscous frictions [15,16,17,18]). We demonstrated the friction-inspired orientation of tops by a series of experiments performed with a variety of rotators, including hollow tops filled with rigid balls and rotating drilled coins. In all the studied spinners, dry friction promoted the orientation of a spinner around the free axis of rotation. Thus, the effect of friction results not only in loss of energy, giving rise to heating, but also leads to the orientation of rotators.

A gas of rotators (representing dust plasmas, colloidal and granular systems) influenced by friction was treated theoretically. Friction forces, by aligning rotators, “freeze” a number of degrees of freedom of rotators, and in parallel heat the system (the effect of heating was addressed experimentally). The negative change in the orientational entropy of the gas of rotators was calculated with the Sackur-Tetrode formula. The minimal change in total entropy corresponding to the complete alignment of rigid bodies decreases with the temperature, defined as the mean kinetic energy of the rotators. The essential effect of the orientational degrees of freedom on the phase equilibrium and motion of systems built from colloidal particles was reported recently [34,35]. In the broader context, the friction-induced negentropy, owing to the elimination of rotational degrees of freedom, is expected to be significant for the phase equilibrium and deformation of granular materials [36,37]. The discussed generation of negentropy by friction may be of primary importance for so-called artificial molecular rotors [38].