Heat Transport Equations in Elastic Nanosystems

The classical Fourier theory of heat conduction is widely applied in thermal processes also because of its simplicity and reliability in many thermal processes. It is based on Fourier’s law for the heat flux q,

where θ is the (absolute) temperature, Δ denotes the gradient and κ is the conductivity. Yet there are many contexts where Fourier’s theory is inadequate. Conceptually, it is paradoxical that a theory predicts propagation with infinite speed as is the case for Fourier’s theory. The Maxwell-Cattaneo (MC) equation [1-3].

where τ is a time constant and ∂_t is the partial time derivative avoids the paradox of infinite speed of propagation. Indeed, a wide literature has been developed as to the proper formulation in that eq. (1) is viewed as a constitutive equation and accordingly it has to satisfy the objectivity principle and to be thermodynamically consistent. Among others, refs [4-8] give exhaustive accounts of these topics and their developments in the literature.

Now, though the MC equation remedies the paradox, there are experiments on heat propagation in graphene [9] whereby an equation more involved than (1) is appropriate to fit the results. Indeed a good fit has been obtained with the equation

where α is the thermal diffusivity, τ_q,τ_θ two relaxation times or phase lags, and Δ is the Laplacian operator. Indeed, the fact that the fitting indicates comparable values for τ_qandτ_θ, respectively 1.85 and 1.01 ps, makes eq. (2) worthy of attention. Accordingly, though eq. (2) is again plagued by the paradox of infinite speed of propagation, it is of interest to clarify the origin of (2) and to cast it in a context of continuum physics.

Other experiments [10] show a good fit with the solution of the Guyer-Krumhansl equation,

a linear approximation of the nonlinear Guyer-Krumhansl equation [11, 12]

where c is the specific heat and l a parameter justified by phonon processes. Nanodevices suffer from dissipation of heat at a rate that is not explained by the previous equations. Consistently, the decrease of the thermal conductivity (see, e.g., [10, 13, 14]), which hinders heat exchange, calls for more involved materials models. Furthermore, nanoscale systems with dimensions comparable to the mean-free path of particles (or phonons) non-local effects are required to be inserted in the model. In addition, in microdevices working at high frequencies also relaxation effects occur so that realistic models need to account for the time delay of relaxation processes.

Based on these observations, this paper has a twofold purpose. First to investigate the appropriate continuum framework for equations of the form (2). Secondly, to examine non-local generalizations of (4) and to establish the restrictions placed by the thermodynamic consistency.

The body under consideration occupies a time-dependent region Ω. A point of the body is labelled by the position vector X in a reference configuration R. Hence we let X = χ (X, t ) provide the position x in Ω in terms of the position X∈ R and the time t. The body is deformable and we let, F,F_ik=∂_Xkχ_i, be the deformation gradient. The velocity v is defined by v=∂_tχ while ∇=∂_x is the gradient operator. The tensor L denotes the velocity gradient,L_ij= ∂_xjv_i. We let ρ be the mass density, T the Cauchy stress, ε the internal energy density, and η the entropy density (per unit mass). For any pair of vectors a,b we let a ⋅b denote the inner product; likewise for any pair of tensors A,B we let A⋅B =A_ijB_ij For any function f (Χ,t ) we denote by a superposed dot the material derivative and we have f=∂_tf+(v⋅∇)f. The superscript T means transpose of a tensor, sym the symmetric part, and skw the skew-symmetric part; 1 denotes the unit second-order tensor and ⊗ the dyadic product.

By the conservation of mass, the mass density ρ is subject to the continuity equation

The equation of motion is given the form

where b is the body force and (∇ ⋅T)_i=∂_xj T_ij By the balance of angular momentum it follows T = T^T . The balance of energy leads to the equation

where r is the (external) energy supply, q is the heat flux, D = symL is the stretching tensor. The balance of entropy is written in the form

which means that q/θ + k is the entropy flux, r/θ the entropy supply and γ the rate of entropy production; the assumption γ ≥ 0 specifies the law of increase of entropy. Equation (8) is referred to as Clausius-Duhem (CD) inequality. We state the entropy principle as follows: physically admissible models are required to satisfy the CD inequality (8) subject to the balance equations (5)-(7).

Substitution of (∇ ⋅ q −ρ r )/θ from (7) yields

Using the Helmholtz free energy

The unusual form of (2) calls for a possible physical model. Let the body be rigid and assume

the specific heat c being a positive constant. Hence the balance of energy results in

where no heat supply is allowed. The heat conduction consists of two fluxes, q₁,q₂, with

The flux q₁ represents heat conduction due to light particles (electrons) and then modelled by a Fourier law,

The flux q₂ describes heat conduction of heavy particles (ions) and hence is affected by a relaxation property; we model this property by the MC law

Both κ₁ and κ₂ are constant, symmetric, non-singular tensors. Time differentiation of (11) gives

while (10) leads to

Hence by (10) plus τ_q times (14) we obtain

and then

In isotropic materials we have

and hence (16) becomes

If, as we prove in a while, κ₁ and κ₂ > 0 we can define

and hence eq. (2) follows with α =κ/ρc

We now investigate the thermodynamic consistency of the model (11)-(12). body is undeformable (D = 0) then the CD inequality simplifies to

the assumption k = 0 will be shown to be consistent with the constitutive equations. Owing to eqs. (11)-(12) we let

be the set of variables. Time differentiation of ψ (θ,q₂,∇θ) and substitution in (17) results in

Substitution of ∂_t q₂ from eq. (12) leads to

The linearity and arbitrariness of,∇∂_tθ,∂_tθ imply

Assume q₁→0 as∇θ →0 (which is obviously true for (11).

Hence eq. (19) implies that

The obvious integration of the first condition in (20) yields

and, by the inequalities in (20) we find

Consequently, the model (10)-(12) with two different fluxes is consistent with thermodynamics provided only that the conductivity tensors κ₁,κ₂ are positive definite.

A simpler derivation of (2) is obtained by considering a formal generalization of the MC equation in the form

if τ₀=0 we recover the MC equation. To obtain the differential equation (2) we assume the balance of energy in the form (10) so that

Computing the divergence of (21) and using (22) we find (2) with α = λ/ρc

The possible large values of spatial gradients, of temperature and heat flux, in nanosystems motivate both non-local terms and the time derivative of the heat flux in the modelling of heat conduction. To fix ideas we observe that quite a general equation has been considered in [1] in the form

where q² = q ⋅ q and the coefficients m₁,m₂,...m₇ are allowed to depend on temperature.

This indicates that a thermodynamically consistent scheme embodying in some way eq. (23) needs θ ,q,∇θ ,∇q,∇∇q among the variables.

As a further generalization we let the nanosystem be deformable and hence we let the material properties be affected by the deformation gradient F. Hence we let

be the variables describing the nanosystem. Indeed, the scalars ψ and ε should depend on invariants of F; for definiteness we then assume that ψ depends on F through the Green-Lagrange strain

By the objectivity principle ([6], ch. 4; [7], § 1.9) the constitutive equations are required to be form-invariant under Euclidean transformations. This means that the constitutive equations involve scalars, vectors, and tensors relative to Euclidean transfor-mations. The time derivative is not form invariant; the transform of the time derivative is different from the time derivative of the transform. That is why objective time deriva-tives are used for rate equations in continuum mechanics. The simplest objective time derivative is the corotational one, namely

Equation (24) can then be viewed as a constitutive equation for the rate q in terms of the variables θ ,F,q,∇θ ,∇q,∇∇q, where

m₁,m₂,...m₇ may depend on θ and F, possibly through J = det F. Notice that eq. (24) implies

We then start with the assumption

Compute.ψ , substitute in the CD inequality (9) and divide by θ to obtain

where _qδ ψ is the variational derivative,

The derivative q is not arbitrary in thatq = q+ q W is given by eq. (24). Furthermore.E and D are related by the identity E = FT F.

Hence upon substitution of.q from (24) and multiplication by θ we have

This form of the CD inequality allows us to see the possible arbitrariness and hence to find the corresponding consequences. First we notice that (∇θ ) ,(∇∇q) ,θ can take arbitrary values and meanwhile without affecting the constitutive functions ψ ,η,T,k,γ . Owing to the linearity it follows that

The linearity and arbitrariness of W and D imply that

Hence we are left with

Definite consequences of inequality (28) follow in particular cases. Assume m₁,m₂,m₃ = 0. We notice that

where α ,β are positive constants. Assuming it is so we can use (29) and write eq. (28) in the form

Inequality (30) holds if

The reduced inequality (33) implies that

Equation (32) then simplifies to

whence

As it follows by letting q →0 and q →∞ we find the necessary and sufficient conditions

Consequently we find that

If b < 0 then it follows

If, instead, b = 0 then

Since we have assumed m₁,m₂,m₃ = 0 then the non-local model is thermodynamically consistent if it is settled as follows. The heat flux q is governed by the rate equation

where m₄,m₅,m₆,m₇ are allowed to depend on temperature. The entropy principle ulti-mately requires that

where q = q . For definiteness we assume ψ in the form (34) and then we find that if

are constants then

is the extra-entropy flux and

This paper deals with two model equations for heat propagation in nanosystems. The equation with two phase lags (2) is shown to arise from a physical model with two heat fluxes, one for the Fourier- type for light particles (possibly electrons) and one for heavy particles (possibly ions) which involve a relaxation term. The whole physical scheme is found to be thermodynamically consistent.

The second equation is non-local in character in that involves higher-order spatial derivatives of the heat flux thus generalizing the linear Guyer-Krumhansl equation (3). For generality the body is allowed to be elastically deformable. The thermodynamical consistency is then proved for the non-linear objective rate equation (37) subject to

with q = q . The dependence on b generalizes previous models by allowing for the non-linear terms parameterized by m₆ and m₇ . If 0 b = then ψ is given by (36).

the stress emerge through the formal dependence of the free energy on ∇q. Hence the stress behaviour is not affected by letting the objective derivative be the corotational one and the free energy be independent of ∇q. This indicates the guideline for a simple model accounting for heat conduction with non-local and non-linear properties.

The research leading to this work has been developed under the auspices of INDAM-GNFM.

None.

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