2.4 Nonlocal Phase-Lag Heat Conduction in a Finite Strip
2.4.1 Molecular Dynamics to Determine Correlating
Determining the value of correlating nonlocal length is one of the most intricate challenges for application of recently developed nonlocal non-Fourier heat con- duction models to nanoscale materials. Since the correlating nonlocal length is an intrinsic property of material, it is required to be determined for each material. The experimental testing is a cumbersome task for measuring the thermal nonlocal length. While the experimental testing is inevitable for accurate determination of nonlocal length, molecular dynamics (MD) and atomistic simulation are feasible methods for determining nonlocal length by comparing the characteristics of thermal wave in nonlocal non-Fourier heat conduction and MD thermal results.
Herein, we introduce the MD approach for measuring thermal nonlocal length of 2.4 Nonlocal Phase-Lag Heat Conduction in a Finite Strip 37
copper. In specific, we present the results of MD simulation for a relatively-long nano-slab subjected to thermal excitation on its left side, an example which resembles thermal wave propagation in one-dimensional heat transport.
The MD simulation is conducted by LAMMPS software [9] for a copper single-crystalline nano-slab of 361.5 nm length, 7.23 nm width, and 7.23 nm height as shown schematically in Fig.2.10. The steps required to be taken for the MD simulation are [7]:
(1) Creating the copper nano-slab by face-centred-cubic lattices.
(2) Initializing the atoms with random velocities.
(3) Equilibrating the nano-slab at room temperature 300 K for 20 picosecond (ps) under Noose-Hoover thermostat (NVT) ensembles [10]. Wefix the tem- perature of both hot and cool zones of the equilibrated nano-slab by rescaling their atoms at each time step.
(4) Increasing the temperature of the fixed hot zone of the nano-slab to 1000 K temperature, a condition that replicates the thermal boundary condition of one-dimensional continuum NL FTPL heat conduction.
We apply MD simulation for time steps of 1 femtosecond (fs) and for a total time period of 10 ps before thermal wavefront reaches the right end of nano-slab. Heat transport in solids is carried out by electrons and phonons. The atomistic interac- tions between atoms and electrons are introduced into the MD simulation through the embedded atom method (EAM) potential defined as [11]:
E¼X
i
Fi X
i6¼j
qi rij
! þ1
2 X
ij;i6¼j
/ij rij ð2:28Þ
whererijrepresents the distance between atomsiandj,qiis the contribution of the electron charge density,Fiis the summation of individual embedding function of atomi, and/ij represents a pairwise potential function between atoms. To deter- mine temperature distribution along the slab length, we divide the slab to finite numbers of segments (here 100 segments) and obtain the average temperature of each segment as:
Fig. 2.10 A nano-slab considered for MD simulation of thermal wave propagation [7]
Tseg¼ P
imm2i
2NsegkB ð2:29Þ
In this equation,m,mi,Nseg, andkB are, respectively, atomic mass, velocity of atomi, number of atoms in each segment, and Boltzmann constant andTsegpresent the average value at each segment. As seen in Eq. (2.29), average thermodynamic temperature is related to the mean square velocity of atoms.
Figure2.11shows temperature profile in the copper nano-slab at different time.
Similar to the temperature profile observed in a nano argonfilm [12], temperature evolves in the nano-slab in the form of thermal wave. Thermal wave is observed to travel from the hot surface on the left side of the nano-slab towards the cold surface on the right with estimated thermal wave speed ofCMD¼2325103m/s. If we correlate the thermal wave speed estimated by MD simulation with the speed of sharp thermal wavefront in NL C-V model, the correlating length kq defined in Eq. (2.14) can be estimated as:
kq¼sq CNL C-V C2C-V
CNL C-V
sq CMDCC2-V
CMD
¼5:797:11 nm ð2:30Þ
whereCC-VandCNL C-Vrepresent the thermal wave speed predicted by the C-V model and NL C-V models, respectively. While MD results in Fig.2.11show the propa- gation of temperature disturbance in the form of thermal wave, slight temperature rise in atoms (locations) ahead of thermal wavefront is observed in temperature distri- bution. This observation is compatible with the characteristics of thermal wave pre- dicted by the NL FTPL heat conduction and those reported in Ref. [8] for phase-lag heat conduction in homogenous and heterogeneous porous materials.
Fig. 2.11 Temperature distribution in a nano-slab at different time obtained by MD simulation [7]
2.4 Nonlocal Phase-Lag Heat Conduction in a Finite Strip 39
2.4.1.1 Nonlocal Heat Conduction in Functionally Graded Materials Most of biological materials with extreme mechanical and thermochemical prop- erties, e.g. Moso culm bamboo [13], dento-enamel-junction of natural teeth [14], and the Humboldt squid beak [15], reveal a multi-scale hierarchical and function- ally graded (FG) microstructure. Examples of the extreme properties of functionally graded materials (FGMs) are resistant to contact damage, cracking, deformation, thermal stresses, and heatflow due to the gradation of microstructural morphology, porosity, and chemical/material ingredients in FGMs [16–18]. FGMs enable the engineering of advanced materials with tuned multiphysics properties to satisfy mechanical, hygrothermal, electrical, and biological requirements for structural design in a wide range of applications as thermal barriers, bone tissues and implants, thermoelectric generators, and energy harvesters. Advances in powder metallurgy [19], laser cutting [20], and additive manufacturing/3D printing [21]
have also facilitated fabrication and the arbitrary variation of material composition and micro-architecture of FGMs.
Due to the importance of FGMs, we present here the temperature evolution and thermal wave propagation in an FGM nano-slab. The material properties of FGMs can be arbitrarily tailored within FGMs through the variation of volume fraction of constituent solid components (Two-phase solid FGMs) or relative density of porous materials (Single-phase porous FGMs). To be able to obtain closed-form solutions for transient temperature in the Laplace domain, we adopt here an exponential function for variation of thermal conductivity ð Þ, material constant of the TPLk theory ð Þ, and specik fic heat per volume ð Þqc through the length of FGM nano-slab:
k¼k0enGx;k ¼k0enGx;qcp
¼C0enGx ð2:31Þ where nG represents the FGM exponential index for the variation of material properties [22]. We assume that other thermosphysical properties are constant throughout the nano-slab. The wave-like NL FTPL heat conduction equation for the exponentially graded medium in the absence of heat source can be obtained by using the NL FTPL heat conduction Eq. (2.21) along with the energy Eq. (1.6) and non-dimensional parameters (2.16):
1þn0L
ð Þ@2h f;ð bÞ
@b2 þL@3h f;ð bÞ
@b2@f þ@3h f;ð bÞ
@b3 þ1
2
@4h f;ð bÞ
@b4
¼IaF1CK@2h f;ð bÞ
@f2 þn0IaF1@2h f;ð bÞ
@b@f þIaF1ð1þCKZ Þ@3h f;ð bÞ
@f2@b þIaF1Z@4h f;ð bÞ
@f2@b2
ð2:32Þ
wheren0¼nG ffiffiffiffiffiffiffi asq
p . The closed-form solution of Eq. (2.32) in the Laplace domain is the same as the one provided in Eq. (2.24) with the characteristics roots modified as follows:
r1;2¼
Ls2n0SaFþ2
ð Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ls2n0SaFþ2
ð Þ
þ4saFþ3ðCKþð1þCKZ ÞsþZs2Þ 1þn0Lþsþs22
vu uu ut
2saFþ1ðCKþð1þCKZ ÞsþZs2Þ ð2:33Þ Temperature can then be retrieved in the time domain by using a numerical Laplace inversion technique.
Figure2.12 shows the effect of FGM exponential index nG [presented in Eq. (2.31)] on the characteristics of NL FTPL thermal wave at the non-dimensional timeb¼1. As seen in thisfigure, the NL FTPL thermal wave speed is constant and independent of non-homogeneity indexnGfor exponential type of FGM materials;
a phenomenon caused by the absence of non-homogeneity index parametern0¼ nG ffiffiffiffiffiffiffi
asq
p in terms of the highest order of temperature derivatives in NL FTPL
Fig. 2.12 Effect of material non-homogeneity index of FGM nano-slab on non-dimensional temperature distribution at non-dimensional timeðb¼1Þusing NL FTPL heat conduction model
L¼1;CK¼0:3;aF¼0:95;Z¼10;Z ¼10
ð Þ[7]
2.4 Nonlocal Phase-Lag Heat Conduction in a Finite Strip 41
differential equation of heat conduction [Eq. (2.32)]. It is important mentioning that while the exponential material gradation does not alter the thermal wave speed based on the NL FTPL model for this specific FGM medium, thermal wave speed can vary within an FGM medium with the FGM non-homogeneity index for FGM materials with an arbitrary variation of material properties [8,23,24]. Figure2.12 shows that material gradation can effectively tailor temperature within the thermal affected zone of FGM medium. In addition, increasing the value of FGM expo- nential indexnG from−0.5 to 0.5 can remarkably reduce temperature within the thermal affected zone of the FGM medium. Interestingly, decreasing the FGM exponential index nG can also magnify temperature at the thermal wavefront causing that the temperature within the FGM medium exceeds the temperature at the boundaries; for example, the maximum temperature within an FGM medium with nG¼ 0:5 is about 30% higher than the maximum temperature occurred within a homogenous medium nG¼0. Consequently, material gradation can potentially improve the performance of advanced materials used in extreme envi- ronmental conditions if the material gradation index is optimized.