# Second Law Violation By Magneto-Caloric Effect Adiabatic Phase Transition of Type I Superconductive Particles

## Abstract

**:**

## INTRODUCTION

^{−4}to 10

^{−5}centimeters.

## DISCUSSION

#### **A. Description of the Adiabatic Phase Transition of Bulk Dimensioned Specimens**

_{n}and C

_{s}are, respectively, the normal and superconductive specific heats, V

_{n}and V

_{s}are, respectively, the instantaneous normal and superconductive phase volumes, T

_{1}, and T

_{2}are, respectively, the starting and ending temperatures of the transition process of the elemental volume, ΔV, and S

_{n}and S

_{s}are, respectively, the normal and superconductive phase entropies at T

_{1}.

_{1}and H

_{2}are, respectively, the critical magnetic fields at the initial temperature, T

_{1}, and the final temperature, T

_{f}, attained by the transition of the entire specimen volume, and where M is the specimen magnetization and is related to H by the expression:

_{S}

_{S}is defined from equation (1) and the parabolic relation (Tuyn Curve) for the thermodynamic critical field:

_{2}= H

_{0}(1 − (T

_{2}/T

_{c})

^{2})

_{2}is the critical field at T

_{2}, H

_{0}is the critical field at absolute zero, T

_{c}is the critical temperature in zero field, and T

_{2}is defined by equation (1).

_{c}. The result, showing the change in proportion of the specimen in the superconductive phase versus reduced temperature of the specimen, is summarized in Figure 1. Comparison with the experimental results of Yaqub [9] shows good agreement.

#### **B. Description of the Adiabatic Phase Transition of Particle Dimensioned Specimens**

_{f}, is found from the expression:

_{1}, and involves the entire volume, rather than a series of temperatures and infinitesimal volumes as occurs in the bulk dimensioned adiabatic magneto-caloric process. Notice that this expression is similar to equation (1) except here the entire volume undergoes simultaneous phase transition. Accordingly, the process is not isentropic, as is the case for a bulk dimensioned specimen, but rather, the end point entropy at T

_{f}is less than the beginning point entropy at T

_{1}, in clear circumvention of the Second Law. This net lowering of entropy is explained by an absence of external energy contributions to the (particle sized) specimen during the transition in the form of magnetodynamic work performed against remaining diamagnetic regions (as would occur in bulk specimens) as the field is increased. More particularly, the difference in entropy of phase for a superconductor is given by the relation:

_{n}− S

_{s}= −(1/4)πH

_{c}dH

_{c}/dT

_{b}is the final temperature achieved for a magneto-caloric process performed on a bulk dimensioned specimen.

#### **C. Factors Affecting the Transition of Particle Dimensioned Specimens**

_{0}) = Ψ(H)/Ψ(H

_{0})), and a special case thereof, Φ

_{c}, where Φ

_{c}= Ψ(H

_{c})/Ψ(H

_{0}). Finally, Douglass predicts Φ

_{c}as a function of specimen cross-section.

_{c}for cross-sections less than 10λ(T), the latent heat evolution will be reduced likewise. Thus, for particles having dimension of √5λ(T) < d, no adiabatic magneto-caloric effect can occur; for particles having dimension of √5λ(T) < d < 10λ(T), a first order phase transition will occur, but the observed adiabatic magneto-caloric effect will be reduced dependent on the affect of Φ

_{c}on the magnitude of the latent heat evolution at T

_{1}; and for particles having dimension of d > 10λ(T), no diminution in the adiabatic magneto-caloric effect will be observed due to size effects, the process being defined by equation (6).

_{c}for reduction in the order parameter at H

_{c}, as follows:

_{1}, is arbitrarily .65T

_{c}.

_{1}is very close to T

_{c}. Douglass [19] has experimentally measured the energy gap at H

_{c}for aluminum, where ξ(T)/λ(T) = 32, as a function of film thickness, and found that for d > 5λ(T), the order parameter remains essentially unchanged at H

_{c}. Thus, Type I superconductors should have Φ

_{c}behaviors less sensitive to cross-section than the Ginzburg-Landau Theory would predict. As such, the range of specimen cross-section over which T

_{f}is below that for the bulk dimensioned specimen is probably more pronounced than the range shown in Figure 2.

_{1}on the specimen temperature, causing T

_{f}to be closer to T

_{1}than Figure 2 would predict. Coupling between particles in an ensemble would have a similar effect on T

_{f}due to adjacent particle heat exchange, which, in effect, would suggest a macroscopic magneto-caloric effect behavior for the ensemble, not unlike that of a bulk dimensioned specimen. This view is based on the fact that at any one time a minimum of ensemble free energy can be attained during an adiabatic magneto-caloric process if some of the particles become normal while others remain superconductive as the external field is slowly increased - essentially the situation in the bulk dimensioned specimen adiabatic magneto-caloric effect phase transition. Particle isolation, therefore, is a prerequisite to observing the effects predicted in Figure 2.

_{f}predicted in Figure 2, since the diamagnetic energy will exceed the reversible Meissner effect energy, (H

_{cb})

^{2}/8π. Also of note is that the value of the critical field will be lower in the event the particle has a non-zero demagnetizing coefficient. This would follow from the above reported experiments by Lutes and Maxwell on tin whiskers. Since the exhibition of hysteresis is a maximum at absolute zero [20], where no latent heat is associated with the phase transition, any potential barriers inhibiting the particle phase variation can be predicted from isothermal transition models, as well as means to modify those expectancies [11,17,20,21,22,23].

## CONCLUSION

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**Figure 1.**The thermodynamic state of a bulk dimensioned tin specimen during an adiabatic phase transition is illustrated using equations developed in part A of the Discussion, where the starting temperature is .65T

_{C}. Shown is the variation in superconductive phase volume as a function of reduced temperature during a bulk size specimen adiabatic magneto-caloric process starting in the superconductive phase and ending in the normal phase. The final temperature achieved is 0.416T

_{C}.

**Figure 2.**The adiabatic magneto-caloric effect phase transition dependency on specimen cross-section is indicated by plotting the variation in the process reduced final temperature, T

_{f}, as a function of specimen thickness in units of ξ(T). For reference, the final temperature for a bulk dimensioned specimen adiabatic magneto-caloric effect is 0.348T

_{c}, and the lowest T

_{f}plotted is 0.174T

_{c}. For all T

_{f}lower than 0.348T

_{c}, the Second Law is violated.

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**MDPI and ACS Style**

Keefe, P. Second Law Violation By Magneto-Caloric Effect Adiabatic Phase Transition of Type I Superconductive Particles. *Entropy* **2004**, *6*, 116-127.
https://doi.org/10.3390/e6010116

**AMA Style**

Keefe P. Second Law Violation By Magneto-Caloric Effect Adiabatic Phase Transition of Type I Superconductive Particles. *Entropy*. 2004; 6(1):116-127.
https://doi.org/10.3390/e6010116

**Chicago/Turabian Style**

Keefe, Peter. 2004. "Second Law Violation By Magneto-Caloric Effect Adiabatic Phase Transition of Type I Superconductive Particles" *Entropy* 6, no. 1: 116-127.
https://doi.org/10.3390/e6010116