Review article
Heat Transport Equations in Elastic Nanosystems
Angelo Morro*
DIBRISDepartment of informatics, Bioengineering, Robotics and system Engineering University/Organization: university of Genoa, Italy
Angelo Morro, DIBRISDepartment of informatics, Bioengineering, Robotics and system Engineering university/Organization: University of Genoa, Italy
Received Date:February 07, 2024; Published Date:February 20, 2024
Abstract
The paper is devoted to some modelling equations for heat propagation in nanosystems. The purpose is to investigate the thermodynamic consistency so that appropriate constitutive restrictions are established. First a secondorder differential equation is considered, with two phase lags, that fits experimental data on heat propagation in graphene. Next a nonlocal model is examined through a ratetype equation with secondorder spatial derivatives of the heat flux. For generality the body is allowed to be elastically deformable. The thermodynamic consistency leads to some simplifications and relates the whole model to free energy function of temperature, strain, and heat flux.
Keywords:Heat propagation in graphene; Relaxation in nanosystems, Thermal wave equations; Nonlocal equations; Thermodynamic consistency
Introduction
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 MaxwellCattaneo (MC) equation [13].
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 [48] 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_{q}andτ_{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 GuyerKrumhansl equation,
a linear approximation of the nonlinear GuyerKrumhansl 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 meanfree path of particles (or phonons) nonlocal 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 nonlocal generalizations of (4) and to establish the restrictions placed by the thermodynamic consistency.
Balance equations and entropy principle
The body under consideration occupies a timedependent 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}=∂_{Xk}χ_{i}, be the deformation gradient. The velocity v is defined by v=∂_{t}χ while ∇=∂_{x} is the gradient operator. The tensor L denotes the velocity gradient,L_{ij}= ∂_{xj}v_{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_{ij}B_{ij} For any function f (Χ,t ) we denote by a superposed dot the material derivative and we have f=∂_{t}f+(v⋅∇)f. The superscript T means transpose of a tensor, sym the symmetric part, and skw the skewsymmetric part; 1 denotes the unit secondorder 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 ClausiusDuhem (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
Heat transport equation with two phase lags
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, q_{1},q_{2}, with
The flux q_{1} represents heat conduction due to light particles (electrons) and then modelled by a Fourier law,
The flux q_{2} describes heat conduction of heavy particles (ions) and hence is affected by a relaxation property; we model this property by the MC law
Both κ_{1} and κ_{2} are constant, symmetric, nonsingular 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, κ_{1} and κ_{2} > 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,q_{2},∇θ) and substitution in (17) results in
Substitution of ∂_{t} q_{2} from eq. (12) leads to
The linearity and arbitrariness of,∇∂_{t}θ,∂_{t}θ imply
Assume q_{1}→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 κ_{1},κ_{2} are positive definite.
A simpler derivation of (2) is obtained by considering a formal generalization of the MC equation in the form
if τ_{0}=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
Nonlocal models of heat conduction in elastic nanosystems
The possible large values of spatial gradients, of temperature and heat flux, in nanosystems motivate both nonlocal 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 q^{2} = q ⋅ q and the coefficients m_{1},m_{2},...m_{7} 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 GreenLagrange strain
By the objectivity principle ([6], ch. 4; [7], § 1.9) the constitutive equations are required to be forminvariant under Euclidean transformations. This means that the constitutive equations involve scalars, vectors, and tensors relative to Euclidean transformations. 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 derivatives 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
m_{1},m_{2},...m_{7} 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 _{q}δ 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 m_{1},m_{2},m_{3} = 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 m_{1},m_{2},m_{3} = 0 then the nonlocal model is thermodynamically consistent if it is settled as follows. The heat flux q is governed by the rate equation
where m_{4},m_{5},m_{6},m_{7} are allowed to depend on temperature. The entropy principle ultimately 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 extraentropy flux and
Conclusions
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 nonlocal in character in that involves higherorder spatial derivatives of the heat flux thus generalizing the linear GuyerKrumhansl equation (3). For generality the body is allowed to be elastically deformable. The thermodynamical consistency is then proved for the nonlinear objective rate equation (37) subject to
with q = q . The dependence on b generalizes previous models by allowing for the nonlinear terms parameterized by m_{6} and m_{7} . 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 nonlocal and nonlinear properties.
Acknowledgement
The research leading to this work has been developed under the auspices of INDAMGNFM.
Conflict of Interest
None.
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Angelo Morro*. Heat Transport Equations in Elastic Nanosystems. Mod Concept Material Sci. 5(5): 2024. MCMS. MS.ID.000621.

Heat propagation in graphene, Relaxation in nanosystems, Thermal wave equations, Nonlocal equations, Thermodynamic consistency

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