Limits on the thermo-elastic coupling of solids

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Abstract

Investigating homogeneous, isotropic, linear elastic, non-viscous solids in a state of equilibrium it has been demonstrated that correlation between the temperature and the pressure can exist only at constant volume. This correlation is irreversible and works in the temperature pressure direction. The limited communication between the temperature and the pressure put constraints on the conversion of the thermal and the mechanical energies. Based on these constraints it is suggested that adiabatic condition should not exist in solid phase in the elastic non-viscous domain.

CONTENTS

1. Introduction

2. Correlation between pressure and temperature

3. The pressure independence of the volume coefficient of expansion

4. The temperature independence of the bulk modulus

5. The irreversibility of the pressure temperature relationship

6. Conclusions

7. Consequences

1. Introduction

The gas laws of Boyle and Charles and Gay-Lussac can be combined into a single equation, which is called the equation of state [EOS]. The EOS for ideal gasses can be written as

nRT = pV       (1)

where n is the number of moles, R is the universal gas constant, T is the temperature, p is the pressure, and V is the volume. The gas constant can be written as the product of the Avogadro number [N_A ] and the Boltzmann constant [k_B ], thus

pV = nN_Ak_BT       (2)

When the gas phase changes to solid then the relationship between the quantities of EOS becomes more complicated and two new thermodynamic parameters are introduced. The function between temperature and volume is characterized by the volume coefficient of expansion [α_V]:

       (3)

The relationship between the pressure and volume is described by the isothermal bulk modulus [K_T] :

       (4)

It is assumed that the solid is homogeneous, isotropic, non-viscous and has linear elasticity. It is also assumed that the stresses are isotropic; therefore, the principal stresses can be identified as the pressure p = σ₁ = σ₂= σ₃. Both the volume coefficient of expansion and the isothermal bulk modulus are pressure and temperature dependent; therefore, it is necessary to know the derivatives of these parameters.

                               (5)

It is generally assumed that beside the introduction of the bulk modulus and the volume coefficient of expansion no additional change is required when a system changes its phase from gas to solid. It is also assumed that the rest of the thermodynamic equations, like work function, state functions, can be used without modification for solid phase. Investigating the physical changes caused by the phase transformation it can be noted that the physical process of the pressure is entirely different in gas and solid phase. In the gas phase the pressure represents the collision density of the atoms/molecules, while in the solid phase the pressure is the result of the elastic resistance of the matter. In the gas phase the relationship between the pressure and the temperature is described by the EOS [Eq. (1)] while in solid phase no direct relationship between these two quantities is known. Besides the lack of direct relationship between the temperature and the pressure it is assumed that a complete thermo-elastic coupling exists in the non-viscous elastic domain of the solid phase. This widely held consensus will be considered in detail.

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2. Correlation between pressure and temperature

Investigating homogeneous, isotropic, non-viscous linear elastic solids in a state of thermodynamic equilibrium it can be seen that both temperature and pressure are the function of the volume:

p = [(V)_T]       and       T = T [(V)_p]       (6)

The functions of the pressure and the temperature to the volume do not necessarily lead to a function between the pressure and the temperature. If a function exists between p and T then one of the conditions which must be satisfied is that the domain of V has to be the same for both the pressure and the temperature. In gas phase this condition is satisfied since the domains are:

+ ∞ > p ≥ 0        + ∞ > V ≥ 0        and        + ∞ > T ≥ 0         + ∞ > V ≥ 0        (7)

In order to investigate the domain of the volume it is necessary to calculate the volume at any pressure and/or temperature. Allowing one of the variables to change while the other one held constant the calculation can be done in two steps (Fig. 1),

       or              (8)

and then

       or              (9)

where V₀ is the initial volume, which is the volume at zero pressure and temperature. These two steps might be combined into one and the volume at a given p, and T can be calculated:

       (10)

The domain of the temperature on the volume at zero pressure is

+ ∞ > T ≥ 0              (11)

The domain of the pressure on the volume at zero pressure is

+ ∞ > p ≥ 0              (12)

In equalities (11) and (12) it was assumed that both the volume coefficient of expansion and the bulk modulus are positive. In this case the temperature will increase the volume while the pressure will decrease it. The domain of these two variables expands into opposite directions and the only common domain left is the initial volume. As long as the volume is constant or the condition

       (13)

satisfied, a function between the pressure and the temperature can exist.

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3. The pressure independence of the volume coefficient of expansion

The expressions of the volume coefficient of expansion and the bulk modulus contain the volume. The volume is related by functions to both the temperature and the pressure. If the volume change caused by the pressure modifies the volume coefficient of expansion or the volume change caused by the temperature modifies the bulk modulus, then there is a correlation between the pressure and the temperature. This possibility will be considered in detail.

Investigating the volume coefficient of expansion at zero pressure can be written as

       (14)

while at pressure p

       (15)

where

       (16)

and

       (17)

Substituting (ΔV_p)_T [Eq. (16)] and ∂(ΔV_p)_T [Eq. (17)] into equation (15), and doing some manipulation on the equation it can be shown that

       (18)

resulting in

       (19)

This equality proves that the pressure does not change the value of the volume coefficient of expansion in equation (3).

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4. The temperature independence of the bulk modulus

The bulk modulus at zero temperature is written as:

       (20)

The differential of the volume at zero temperature is then

       (21)

If the temperature is increased from zero Kelvin to T at constant pressure then the bulk modulus can be written as

       (22)

where (V_T)_p = V _{p, T} and can be calculated by using equations (8)-(10). The differential of the volume at temperature T additionally contains the compressed part of the expanded volume

∂ (V_T)_p = ∂ (V_p)_T=0 + ∂(Δ V_T)_p        (23)

and (Δ V_T)_pcan be calculated as:

       (24)

and the differential of the compressed expansion volume is

       (25)

Substituting the differentials into the expression of the bulk modulus [Eq. (22)] gives

       (26)

By simplifying the equation it can be shown that

K_T = K_T=0       (27)

It also can be shown that the additional volume changes resulting from the temperature change such as the expanded volume and the compressed part of the expanded volume cancel each other out and the value of the bulk modulus remains the same. The bulk modulus at temperature T can be written as

       (28)

Substituting (ΔV_T)_p [Eq. (24)] and (∂ΔV_T)_p [Eq. (25)] into equation (22) gives

       (29)

The same expression for K_T and K_T=0 proves that the volume increase from the expansion is canceled by the compressed volume of the expansion and that the value of the bulk modulus remains the same regardless of the temperature.

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5. The irreversibility of the pressure temperature relationship

The temperature independence of the bulk modulus also means that the pressure volume relationship is independent of the temperature. Changing either the pressure or the volume of a system will not have an effect on the temperature. The temperature pressure relationship, which exists at constant volume, therefore, must be irreversible since the pressure change will not generate temperature change.

The irreversibility can also be demonstrated by calculating the mechanical energy needed to change the pressure. The pressure of a system can be changed by adding to or taking away mechanical energy from the system. The mechanical energy or work between the initial and final state can be approximated as the area of a trapezoid (Fig. 2)

       (30)

Subscript i and f are used for the initial and final conditions respectively. It has been demonstrated that the correlation between the pressure the temperature can exist only if the volume is constant. In order to maintain constant volume the same mechanical energy with opposite sign has to be added to the system (Fig 2) :

(Δw_i-f )_T = -(Δw_f-i )_T        (31)

The quantity of the energy needed to maintain the constant volume is the same as the initial mechanical energy; therefore, the volume restoration energy would use up all of the initial energy. The result is that no energy is left which could be transformed into thermal energy. The energy equivalence between the initial and the volume restoration energies proves that the pressure change will not generate thermal energy and the temperature will remain the same.

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6. Conclusions

It can be concluded that the actual volume comprises three distinct parts. The initial volume, the total elastic and thermal related volume changes (Fig. 1)

       (32)

The correlations between the pressure and volume and between the temperature and the volume are correct only if the following constrains on the volume are satisfied.

p = f (V_{T = cons})       and       T = f (V_{p = const})       (33)

The temperature and pressure conveniently can be chosen to zero.

p = f (V_{T = 0})       and       T = f (V_{p = 0})       (34)

Since

              (35)

the correlations can be written as :

              (36)

The conditions p=0 and T=0 are in parenthesis since and might be calculated when the p=0 and T=0 conditions are not applicable. It can be seen that [Eq. (36)] the pressure and temperature do not correlate to the actual volume but rather only to the elastic and thermal related volume changes. These two different volume changes are independent of each other since either of these volumes can be held constant while the other can still change.

       (37)

Putting a constraint on the volume by not allowing the system to expand freely converts the thermal volume to elastic volume. Using this expansion-compression volume equality the function between the temperature and the pressure can be written as

       (38)

Since this relationship is irreversible an arrow is used to indicate that this equivalence relationship or physical process works only in temperature pressure direction written as

(dp) _V ⇐ (α_V K_T dT ) _V       (39)

The irreversibility is true only for non-viscous elastic solids. In the case of viscous or plastic deformations where frictional forces are present between the atoms/molecules the temperature pressure relationship should be partially or fully reversible.

The exclusive relationship between the pressure and the temperature at constant volume indicates that thermo-elastic coupling in a non-viscous elastic solid phase can exist only at constant volume. This limit on the thermo-elastic coupling prevents the full scale communication between the mechanical and the thermal energies indicating that adiabatic conditions do not exist in the non-viscous elastic solid phase.

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7. Consequences

The limited thermo-elastic coupling of solids has far reaching consequences in every field of science as demonstrated in the following examples from both theoretical and applied fields.

The first law of thermodynamics is based on the convertibility of mechanical and thermal energies,

dU = n c_VdT - pdV       (40)

where U is the internal energy of the system and is the molar volume heat capacity. In the non-viscous elastic solid phase the first law of thermodynamics and the conventional thermodynamic potentials, internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy, which have been derived based on the assumption of free mechanical and thermal energy transfer, have to be modified by taking into account the limitations on thermo-elastic coupling. The work function of non-viscous elastic solids along with the state functions will be discussed in detail in a separate paper.

       (41)

Solid state thermodynamic equations are used to determine the temperature inside the earth and other planets. The temperature distribution as the function of depth (geotherm) is determined by assuming adiabatic conditions for 200 km and deeper regions. Since adiabatic conditions do not exist in the non-viscous elastic solid phase the geotherm must be reconsidered.

Acknowledgement:

I would like to thank Alexandre Laugier for his encouragement and helpful comments on the manuscript.

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