Conversion of plastic work to heat in crystal Cu: A microscopic view by molecular simulations

Conversion of plastic work to heat has been a controversial and challenging topic since the discovery of metals generating a large amount of heat during plastic failure by Tresca.¹ To investigate this complicated issue, many sub-problems came into being. One of the most mysterious is the micro-mechanism of heat generation.

When atomic level cannot be reached before the prevalence of computer simulation, sequence of heat generation and plastic instability is primary. Besides strain or strain rate effects^2,3 and micro-defects,^4,5 many researchers proposed that plastic failure could also be the consequence of thermal softening.^6–8 Experimentally, one popular way of probing the sequence of heat and plastic instability is measuring the work-heat conversion, universally known as Taylor–Quinney coefficient (TQC), pioneered by Taylor and co-workers and estimated to be around 0.9 at that time.^9,10 This coefficient is now generally written as $β int$⁠.¹¹ 

On one hand, during the past three decades, the values of TQC measured significantly scatter from 0.2 to 0.9 rather than the constant value of Refs. 12–22. TQC is also shown to be dependent on strain rate^{15,18,21,23,24} and ambient temperature.^25–27 Zubelewicz²⁷ supposed that higher temperatures and lower strain rates influence the TQC by softening the path-rerouting constraints but the physical mechanism behind it remains to be discussed.

On the other hand, the microsecond lag of temperature maximum after a plastic failure (e.g., the adiabatic shear band) has been frequently observed in experiments.^{13,17,28–31} Only a few reported the near simultaneity of temperature maximum and plastic failure.^14,32 Recently, Guo et al.²² clarified that temperature rise should be the consequence of formation and propagation of plastic events by a persuasive experiment. So far, these are phenomenological descriptions and lacking a specific microscopic physical mechanism of heat generation.

Apart from heat, the remaining plastic work is the stored energy of cold work (SECW).³³ The ratio of SECW should be $1− β int$⁠.^18,34 Although SECW is stored in various types of crystallographic defects,³⁵ most of mathematical models^25,36–38 calculate the SECW based on the dislocation theory.³⁹ Nieto-Fuentes et al.³⁴ brought to light that the analytical SECW based on dislocation density is far less than experimental estimations under a high strain rate,⁴⁰ suggesting no unique relationship between the final microstructure and the final SECW.³⁴ Lieou and Bronkhorst⁴¹ introduces a dynamic model with finite element analysis (FEA) linking the SECW with disorders of atomic configurations. It is well known that new free surfaces during fracture also contain energy,⁴² but seldom reports considered it when SECW is calculated. As a result, where does SECW go is a micro-scale problem related to the micro-mechanism of work-heat conversion.

Over the past few decades, FEA has become a commonly adopted method for simulating work-heat conversion.^43–47 In contrast, during the 2000s, molecular dynamic (MD) simulations were utilized to explore energy variations in microscopic scale,^48,49 but did not cover the thermal dissipation related to TQC. Although MD faces limitations in capturing mesoscopic complexity due to scale constraints,²⁷ recent years have seen them increasingly address problems involving work-heat conversion. This is due to the distinct advantages of MD for precise energy calculations and visualization of plastic events. Recent applications of MD have demonstrated that dislocation gliding is the primary mechanism underlying TQC.^50,51 Additionally, Xiong et al.⁵² reveal that the thermal dissipation of moving dislocation depends significantly on bulk pressure.

In this work, the micro-mechanism of conversion of plastic work to heat is discussed based on results from MD of a two-dimensional (2D) dislocation and three types of three-dimensional (3D) crystallographic defects in crystal Cu, providing nano-scale explanations for the time lag of temperature rising in experiments and demonstrating micro-mechanism related to heat generation and plastic instability.

Two atomic models A and B shown in Fig. 1 are constructed to investigate microscopic physical mechanism of heat generation under plastic deformation. The simplified 2D model A is used to study heat generation involved in propagation of a single dislocation, which reveals some essentials for understanding related phenomena in complicated cases, while model B is a more realistic 3D system.

The 2D model in Fig. 1(a) contains 2400 atoms with a width (⁠ $L x$⁠) of 26.7 nm and a height (⁠ $L y$⁠) of 5.1  nm. Lennard-Jones (LJ) potential with $ε=0.4912$ eV and $σ=0.23276$ nm is adopted for 2D configuration.⁵³ Fixed boundary condition is imposed on three layers of atoms at the left and right sides, while atoms at the top and bottom surfaces are free to move. The compression loading is applied to the 2D system in an adiabatic way. Specifically, the system is equilibrated at an initial temperature of 50 K within an NPT ensemble, then compressed with a strain rate of $10 8$ s $− 1$ by controlling displacement of fixed boundary at the right side. During loading, free atoms are integrated with an NVE ensemble (adiabatic) in which no temperature control is applied.

Configuration of model B is a 3D single crystal with dimensions of $10.4×30.7×1.8$ nm, containing 48 395 Cu atoms. Lattice orientations along $x$-, $y$-, and $z$-directions are $[110]$⁠, $[ 1 ¯10]$⁠, and $[001]$⁠, respectively. The notch in Fig. 1(b) is used to control the origin of plastic deformation. The potential field of Cu atoms in model B is described by the embedded atom method (EAM) of Mishin et al.,⁵⁴ which has been successfully used in many simulations in predicting mechanical behaviors of 3D Cu atomic systems.^55–57 Similar to model A, fixed and free boundary conditions are applied in $x$- and $y$-directions, while periodic boundary condition (PBC) is applied in the $z$-direction without pressure control. Tensile loading is applied to model B in the same adiabatic way in model A. Only one combination of initial temperature (300 K) and strain rate (⁠ $10 8$ s $− 1$⁠) is considered for model B to clarify the relationship between TQC and types of crystallographic defects. All MD simulations are performed with LAMMPS,⁵⁸ and visualization of atoms is rendered in OVITO.⁵⁹ 

In fact, only atomic kinetic and potential energy, accordingly denoted by $e k$ and $e p$⁠, are obtained during MD simulations. Then, one can derive atomic SECW (⁠ $e s$⁠) as $Δ( e k+ e p)−2Δ e k$ after plastic instability. $E k$ and $E s$ are the sum of $e k$ and atomic SECW, respectively. $e p$ and $e k$ at a specific strain state are obtained by performing time average with a small-time window $t w$ (or an equivalent small-strain window $γ w$⁠) to take account of the randomly thermal vibrations of atoms. $t w$ should be not only large enough to cover vibrational period of an atom (⁠ $10 − 12$– $10 − 13$ s), but also small enough compared with the duration of defect evolution, which is controlled by strain. A strain window of $γ w≤0.1%$ is sufficient to capture the smooth evolution of defects, corresponding to a time window of $γ w/ γ ˙$⁠. Meanwhile, $γ w/ γ ˙$ is larger than 10 $− 13$ s for $γ ˙< 10 − 10 s − 1$ and $γ w=0.1%$⁠. As a result, $t w= γ w/ γ ˙$ is used in this work. In order to investigate the distribution of energies in each system, average is made with a spatial interval of $0.37×0.37$ nm $2$⁠, which only includes the nearest neighbors of one atom, eliminating noise without causing defects to become larger or distorted.

The relationship between 2D dislocation propagation and heat generation is depicted in Fig. 2. The 2D model A in Fig. 2(a) is compressed under a strain rate of $10 8$ s $− 1$ (noting the numerical correspondence between a strain of 0.01% and 1 ps in Fig. 2), showing a linear elastic line followed by a sudden stress drop due to dislocation emission, accompanied by a drastic temperature increase in Fig. 2(a). Temperature is computed by Eq. (2).

Before computations of TQC during the first stress drop, SECW and heat generation should be elucidated. First, SECW is the residual energy stored in newly generated distorted crystal lattices by plastic work. For single dislocation propagation in Figs. 2(c)–2(e), SECW is originated from the potential energy of dislocation atoms. SECW is nearly invariant during the propagation because the dislocation is not annihilated.

Second, the heat generation $Q$ is attributed to activities of defects. One portion of $Q$ stems from the propagation of defects. Two atoms are selected to investigate the micro-process of plastic heat generation from 622.00 to 623.00 ps during propagation of the first single dislocation in Figs. 2(b)–2(e). Atom A is in the propagation line of the first dislocation. Atom B is the nearest neighbor of Atom A. One solid line with dots in Fig. 2(b) is the $Δ e p$ of atom A and the other solid line with a marker “x” is the $Δ e k$ of atom B. As the core of dislocation passes by, atom A is returning to its normal hexagonal lattice, transferring part of atomic potential energy to its adjacent atom B as kinetic energy. Then, the atomic kinetic energy of atom B decreases due to heat diffusion. Consequently, heat generation density is equal to the SECW density of dislocation, and microscopic temperature rise appears at the tail of the dislocation in Fig. 2(d) during propagation, which is recorded in Video S1 attached in the supplementary material. The evolution of potential wells and energy curves of atoms A and B are also recorded in Video S2 in the supplementary material, vividly showing the process of energy transfer and heat generation.

The other portion of $Q$ comes from merging of dislocation into free surfaces. When arriving at the free surface in Fig. 2(f), the dislocation will be merged and annihilated. Besides the heat generated at the tail of dislocation, the potential energy stored in dislocation, i.e., the SECW of dislocation, will also be converted to heat at the head (see also in Video S1 in the supplementary material). With only a new free surface atom left after merging, the heat generation density during the merging and annihilation is twice that of propagation.

Finally, results from the 2D model reveal that heat and TQC are connected to three micro-factors: the propagation distance, the type of defects (different SECW density), and the final state of remaining defects (final SECW). TQC of the first stress drop in the 2D model can be determined by macroscopic SECW and $Q$⁠. Lack of residual defects results in total conversion of plastic work to heat, that is, final TQC is 1.0.

In the context of our simulations, the occurrence of TQC = 1.0 indicates that dislocations do not persist in the crystal but instead are annihilated or merged into free surfaces at last. This final annihilation results in the complete conversion of the potential energy associated with dislocations (SECW) into kinetic energy, manifesting as heat. The physical implication of this assumption is that the crystal undergoes a highly dynamic process where dislocation interactions lead to their final annihilation, thereby maximizing heat generation. The results provide valuable insights into the microscopic mechanisms of work-heat conversion but should be interpreted within the context of these idealized conditions.

It is important to consider that in real-world scenarios, the complete and simultaneous annihilation of all dislocations is unlikely. In most materials, dislocations can be pinned or tangled, leading to residual SECW. Therefore, a TQC value of 1.0 is more representative of an idealized case rather than a typical experimental observation.

The 2D model also reveals clues on the problem of time delay of heat generation related to plastic events.^{13,17,28–31} Heat comes along with the propagation of dislocations. However, the propagation speed and direction of generated heat are quite different from that of dislocation. If the change of microstructure and temperature are detected at the surface of the system, the time lag of heat generation will be observed because of the faster propagation of dislocation. The value of delayed time could fluctuate considering the position of dislocations or shear bands.³⁰ 

Heat generation and plastic deformation of 3D model B are quite distinct, as shown in Fig. 3. First, a plastic flow behavior prevents brittle fracture. Stress gradually drops to zero in Fig. 3(a). There are three strain ranges. After a particular strain (3.25%), the stress curve deviates from a linear tendency but without temperature rising, which is labeled range I ending up to the strain of 3.80%. Several dislocations are nucleated in range I but without propagation. Range II from 3.80% to 7.22% covers the first stress drop and initial dislocation propagation, accompanied by immediate system temperature rising, which is similar to model A. Then follows crack propagation and plastic flow in range III from 7.22% to 19.32%.

Second, SECW is significant and TQC is around 0.7 rather than 1.0. SECW rises and elastic energy deviates from external work in range I without increasing temperature, because dislocations have been nucleated but not propagated. SECW continues rising in range II for a large number of crystal defects being generated. Total external work and heat generation increase by about 600 and 500 eV during range III in Fig. 3(b), respectively. In contrast, SECW rises by around 100 eV. Consequently, TQC increases from 0.6 to 0.75 in range III. Elastic energy tends to zero because of low tensile stress before fracture.

Microscopically, 3D plastic heat is featured by kinds of defects. For a face-centered cubic (FCC) system entering plastic deformation, several kinds of dislocations can be observed: 1/2 $⟨110⟩$ perfect, 1/6 $⟨112⟩$ Shockley partial, 1/6 $⟨110⟩$ stair-rod, 1/3 $⟨110⟩$ Hirth, 1/3 $⟨111⟩$ Frank, etc. Two main kinds of dislocation are explained in the following to represent two typical circumstances of heat generation.

Stair-rod dislocation here is emitted from the tip of the notch and finally merges into a free surface, resembling behaviors of dislocation in the 2D system. The contoured map of $e k$ in the left side of Fig. 4(a) shows an obvious local hotspot at the tail of stair-rod dislocation. A dashed box, whose atomic energy is scattered in the right side of Fig. 4(a), covers the hotspot and part of stair-rod dislocation. The peak of $e k$ at the local hotspot is around 0.02 eV, nearly equal to that of $e p$ at the stair-rod region, considering the difference between $e p$ of stair-rod dislocation and normal FCC atoms, i.e., the SECW of stair-rod. Therefore, the lattice passed by stair-rod is restored to FCC with its SECW of stair-rod, transferring to kinetic energy. Similarly, Fig. 4(b) shows the process of a stair-rod merging into a free surface and the distribution of $e k$ in the corresponding dashed box. As indicated in the variation of local kinetic energy, heat generation due to surface interaction is about 0.04 eV, which is twice that of stair-rod propagation in Fig. 4(a). Similar to heat generation of dislocation annihilation in the 2D system, besides the heat generated at the tail of dislocation, SECW of dislocation will also be converted to heat. With few new atoms on free surface being disordered after merging, the resulting rise of potential energy is too small compared to the generated heat during propagation and annihilation of dislocations. However, it does not mean generation of free surfaces is trivial. In addition to merging and annihilation of dislocations, a large number of lattices will be distorted to generate new free surfaces due to crack propagation, preserving a large amount of SECW.

Structural evolution of Shockley dislocations at 495 ps (strain of 4.95%) is presented in Fig. 4(c). As shown in the right, $e p$ of atoms passed by the Shockley dislocation does not decrease. In fact, they become stacking fault rather than returning to normal FCC. Meanwhile, there is no local hotspot in the contoured map of $e k$⁠, indicating that no obvious heat generation is involved in the propagation of the Shockley dislocation, and $e p$ of stacking faults is similar to that of Shockley dislocation. Therefore, the plastic work during propagation of Shockley dislocation is stored as SECW of stacking faults. There is no lattice restoration and no potential energy drop so that no heat generation.

This suggests that the type of defects plays a crucial role in determining the TQC. Defects that facilitate lattice restoration and subsequent SECW conversion to kinetic energy contribute more to heat generation. Although SECW is stored in various types of crystallographic defects, experiments brought to light that the analytical SECW based on dislocation density with an ad hoc correction factor is approximated as experimental calculations.⁴⁰ In this work, stair-rod dislocations are observed to generate significant heat upon propagation and merging into free surfaces, whereas Shockley partial dislocations primarily stored energy in the form of stacking faults without notable heat generation.

MD simulations in this work are conducted at a constant strain rate of $10 8$ s $− 1$⁠. This high strain rate is typical for simulations due to computational constraints. At this strain rate, we observed that TQC values are around 0.7 in the 3D model, which aligns with the understanding that higher strain rates result in higher TQC values.^15,18,21,23 This work focuses on elucidating the physical processes of the conversion of plastic work to heat. Based on the results of the strain rates that can currently be simulated, there is no fundamental change in the physical process of the conversion of plastic work to heat. Therefore, our results could provide valuable insights into the microscopic mechanisms of work-heat conversion in Cu crystal.

The local hotspots observed in simulations are regions where intense heat generation occurs due to the rapid conversion of SECW into kinetic energy during plastic deformation. The specific contribution of these hotspots can be quantified by comparing the local heat generation density with the total heat dissipated during the entire deformation process. Our simulations indicate that the localized heat generation at hotspots plays a crucial role in the overall work-heat conversion mechanism in Cu crystals. To quantify the contribution of local hotspots, we analyzed the energy distribution and heat generation in both the 2D and 3D models.

In the 2D model, the heat generated at the tail of the dislocation during propagation, as well as at the free surface during dislocation annihilation, is attributed to the localized conversion of SECW into kinetic energy. The heat generation density at the tail of the dislocation was found to be approximately equal to the SECW density of the dislocation. When the dislocation merges into the free surface, the heat generation density is about twice that of the propagation, due to the complete conversion of SECW into heat simultaneously at both head and tail.

In the 3D model, similar behavior is observed with different types of dislocations. For instance, the stair-rod dislocation generates a local hotspot at the tail during propagation and a higher heat generation density during merging into the free surface. In contrast, the Shockley partial dislocation, which results in stacking faults, does not generate significant heat due to the lack of lattice restoration.

Results from both 2D and 3D models reveal that heat generation comes from the lattice restoration after plastic deformation, indicating that temperature rise is the consequence of plastic instability rather than the trigger of plastic instability. This fact was observed in experiments by Guo et al.²² Here, we explained it microscopically and physically. Plastic heat is affected by the type, propagation path, and density of defects. For example, the Shockley dislocation will lead to SECW in the form of stacking faults without distinct heat generation. Heat generation of any defects’ propagation is dependent on their SECW and propagation path.

While our current study focuses on crystal Cu, it is essential to consider the implications of our findings for polycrystal Cu, which are more commonly encountered in macroscopic experiments. Polycrystal Cu consists of numerous grain boundaries (GBs) that significantly influence micro-mechanical behaviors. GBs act as barriers to dislocation motion, leading to higher dislocation densities and complex interactions compared to single crystals. The presence of GBs can increase the SECW due to the accumulation of dislocations at these boundaries. This can reduce the overall TQC as a greater portion of plastic work is stored rather than converted into heat.¹⁸ 

It should be noted that while the fundamental mechanisms of plastic work conversion to heat explored in this work could potentially be applicable to various crystalline materials, our specific results are derived from molecular simulations of crystal Cu. Therefore, the universality of these mechanisms across different materials remains to be verified through further research. The material-specific parameters used in our models primarily reflect the behavior of Cu, and caution should be taken when generalizing these findings to other materials. However, the key physical processes and the magnitudes of the associated time scales involved in plastic deformation are likely similar across different crystalline materials, providing a basis for the broader applicability of our results.

In summary, 2D and 3D molecular simulations are carried out to study the microscopic mechanism of plastic work converting to SECW and heat in crystal Cu. Part of defects’ potential energy beyond normal lattice is stored energy of cold work (SECW). Besides crystallographic defects, the free surfaces generated during crack propagation also contain a lot of SECW. Plastic heat is generated during propagation and evolution of defects. More generally, heat generation comes along with lattice restoration, transferring atomic SECW of defects to atomic kinetic energy, so that specific crystallographic defects generate corresponding amounts of heat. If defects are annihilated during propagation, its heat generation density is around twice the atomic SECW of defects. The simulation results are indispensable complements of experimental research, unveiling how plastic heat is affected by the type, propagation path, and density of defects, and heat generation is the consequence of plastic instability, so that the time lag of heat generation will be observed. The simulation results are also expected to further explain the macroscopic effects of initial temperatures and strain rates on the final Taylor–Quinney coefficient (TQC), plastic work, and SECW.

The additional documents in the supplementary material include two videos. Video S1 is the microscopic temperature rising appearing at the tail of the dislocation during propagation, while the evolution of potential wells and energy curves of atoms A and B are recorded in Video S2.

Support from the National Natural Science Foundation of China (NNSFC) (Grant No. 12302082), the NSFC Basic Science Center Program for “Multiscale Problems in Nonlinear Mechanics” (Grant No. 11988102), the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant no. XDB0620103), the Fundamental Research Fund (Project No. 240617035), the Start-up Project Funding (Project No. 231817007), and the Scientific and Technological Research Project (Project No. 242217018) of Henan Academy of Sciences is gratefully acknowledged.

The authors have no conflicts to disclose.

Ethics approval is not required.

Rong-Hao Shi: Data curation (equal); Formal analysis (lead); Funding acquisition (equal); Investigation (lead); Writing – original draft (lead); Writing – review & editing (equal). Pan Xiao: Conceptualization (equal); Funding acquisition (equal); Methodology (lead); Supervision (lead). Rong Yang: Conceptualization (equal); Data curation (equal); Writing – review & editing (equal). Jun Wang: Conceptualization (equal); Data curation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.