Atomistic Simulation of Ultrasonic Welding of Copper

Abstract

Molecular dynamics simulations of ultrasonic welding of two blocks of fcc copper containing asperities under the conditions of a constant clamping pressure and sinusoidal shear displacements were performed. Two different atomistic models of blocks were simulated: Model I with no misorientation between the lattices, and Model II with a special misorientation of 78.46°. Alternating shearing results in a plastic deformation of the interface layers and is accompanied by the emission of partial dislocations. Misorientation between the joined blocks contributes significantly to an interface sliding, interface migration, and pores healing during ultrasonic processing. A significantly larger increase in temperature occurs during shearing in Model II than in Model I. The applied pressure has almost no effect on the interface temperature in both studied models. The temperature increases almost up to maximum values after the first shear cycle, and then practically does not undergo changes in the next four cycles. The temperature at the interface in Model II is significantly higher than that in Model I. The change in the porosity of the interface and its structure are analyzed. The results obtained in the present work contribute to a deeper understanding of the processes occurring at the atomic level during ultrasonic welding of metals.

1. Introduction

Ultrasonic welding (USW) is a relatively new method of solid-state joining of metallic materials [1,2]. This method is widely used in electrical, automotive, medical, aerospace, and other industries to produce joints of foils, sheets, and wires of relatively soft metals such as aluminum, copper, and their alloys. During USW of metals, high-frequency shear vibrations are applied to the sheets or wires to be joined which are clamped together to an anvil under a static clamping force. In this process, the joining of metals occurs due to the friction of contacting surfaces, destruction of the oxide layers on them, deformation of asperities, and the appearance and expansion of bonding areas. Intensive friction, high strain rate shear deformation of asperities in the initial stage and bonded zone in the final stage generate heat and high dislocation activity that results in a significant structure evolution of materials in joints and nearby zones of the sheets.

A number of experimental studies carried out on different metals and alloys have given an insight into the main microstructure changes occurring during USW and their relation to the mechanical properties of joints [3,4,5,6,7,8,9]. Most of the studies show that recrystallization with grain refinement near the weld interface occurs at intermediate welding energies imparted during USW and with grain coarsening at high energies. These changes depend on many welding parameters such as the amplitude of vibrations and welding time, clamping force, and the initial structure of materials. For example, during the welding of nickel sheets with an ultrafine-grained structure, a significant grain growth is observed in the bulks of sheets even at regimes at which no significant changes in the microstructure of coarse-grained nickel occur [10]. During welding of titanium requiring long welding times, the temperature increase can be as high as to induce a reversible α→β→α phase transformation in the central region of the thermomechanically affected zone with the formation of grains, the sizes of which significantly exceed the initial coarse grain size [11].

Experimental methods allow one to study only the final microstructures formed during USW but do not allow for in situ studies due to the very fast character of this process. Finite-element calculations based on the mechanics of solids can be used for investigation of the stress–strain states and temperature evolution during ultrasonic treatment, but due to the peculiarities inherent in this method cannot look at the microstructure evolution. Molecular dynamics simulations are the best-suited methods to obtain an insight into the mechanisms of USW, since they allow revealing the structure evolution on the atomic scale and describe in detail the dislocation activity, temperature changes, diffusion processes, etc.

To date, there have been a few molecular dynamics studies of the process of USW. Long et al. [12] used this method to study the microweld formation in Cu. The authors considered two models of wire bonding: (1) the contacting surfaces were totally flat (Model I), and the contacting surfaces contained semi-spherical protuberances modeling the surface roughness (Model II). In both models, the atomic configuration was divided into three parts: rigid upper and bottom layers, thermostat layers, and Newton layer in which the atoms were free to move. The bottom layer was fixed, the upper one was allowed to move parallel and normal to the contacting surfaces thus simulating the movements of the upper wire during welding. The thermostat layers under these rigid layers had a constant temperature of 300 K to remove heat generated in the Newton layer due to the processes occurring there. In general, such a three-layer structure of atomistic models was adopted in other works, which will be discussed below, when modeling USW. Long et al. considered two modes of the motion of the fixed layer of the upper wire with respect to the bottom one: down motion until a given value of displacement followed by the horizontal reciprocal movement and simultaneous vertical and horizontal movements. Samanta et al. [13] simulated the joining of Cu and Al blocks with flat surfaces and developed a multiscale model to study the formation of a diffusion layer at the joint interface. Mostafavi et al. [14,15] simulated the bonding process during sliding between two Al blocks with flat surfaces. The relative movement of blocks, which also was driven by a displacement of the rigid layer of the upper block, consisted of three stages: simultaneous compression and sliding movements for 96 ps, pure sliding for the next 50 ps, and finally equilibration for 50 ps without displacements. In these works, the temperature evolution at the interface region and mean square displacement of atoms were calculated. Both characteristics were shown to increase with an increase in the sliding velocity. Flattening of micro-asperities during USW was simulated by Ma et al. [16,17].

Most realistic molecular dynamics simulations of the USW process were carried out by Yang et al. [18]. They considered a configuration in which contacting surfaces of the bottom block consisting of Al atoms and the upper one consisting of Cu atoms both had a sinusoidal-shape roughness. Initially, a 1 GPa pressure was applied to the upper rigid layer, which resulted in a complete disappearance of interfacial gaps. Then sinusoidal vibrations $S=A\sin \left(2\pi ft\right)$ with the amplitude of A = 1 nm and frequency of f = 100 GHz were applied for a time interval of t = 5 ns, i.e., the number of sliding cycles was N = 500. A transformation of the fcc structure to disordered non-fcc arrangements and increasing diffusion layer were observed with time and the diffusion coefficient was found to be much larger than in the case of stationary diffusion. It should be pointed out, however, that in the real USW process, pores are not eliminated completely in the initial compression process. Bonded zones appear and evolve with time resulting in healing of the pores during the sliding stage, and at certain parameters of welding (even after the whole process) some pores can remain. Therefore, an even more realistic simulation should consider a model in which the gaps are not closed immediately in the pressing stage. Moreover, most of the earlier studies, except for [18], modeled the sliding stage either combined with a continuing vertical displacement or at a fixed distance between the blocks. Meanwhile, USW is carried out under constant clamping force, i.e., under a given pressure.

In this paper, the process of USW of copper under the conditions of a constant pressure and sinusoidal shear displacements is studied via atomistic simulations. The evolution of the atomic structure of joints with time at different clamping pressures, dislocation processes, and temperature evolution in the interface region during welding are analyzed.

2. Atomistic Models and Simulation Methods

Two atomistic models of copper blocks are adopted for simulations. These models differ from each other by the mutual orientations of crystal lattices of the blocks as schematically illustrated in Figure 1. In Model I, the $\left[\overline{1}10\right]$ directions of both blocks are parallel to the x-axis, so that there is no misorientation between their lattices. In Model II, the lattices of the upper and bottom blocks are rotated to angles +39.23° and −39.23° with respect to the reference plane $\left[\overline{1}10\right]$ from the orientation of Model I, respectively, so that between these lattices there is a special misorientation Σ = 5/78.46° (or 101.54°). In both models, crystallographic axes [112] of the fcc blocks are parallel to the z-axis of the coordinate frame (normal to the plane of the figure). The symmetrical misorientation between the lattices provides the possibility of using the same period of the computational cell along the x-axis without any artificial deformation of the blocks. Periodic boundary conditions are applied along the x and z coordinate directions. Periods of the computational cells along the x-axis are $H_x=62a_0/\sqrt{2}=15.848$ nm for Model I and $H_x=48a_0/\sqrt{5/6}=15.840$ nm for Model II. Along the z-axis, the period of the cell is chosen to be equal to $H_z=(3/2)a_0\left|112\right|=3\sqrt{6}{a}_0/2$= 1.328 nm.

Similar to earlier studies [12,15,18,19], the computational cells consist of three types of regions, i.e., constrained, thermostat, and Newton layers as displayed in Figure 1. The thicknesses of constrained and thermostat layers of both the bottom and upper blocks are equal to 1.252 nm and 1.293 nm in Models I and II, respectively. The Newton layers (containing freely moving atoms) in each block consist of a solid region with a thickness of 12.522 nm in Model I and 12.933 nm in Model II and semi-cylindrical grooves having the same radius of 3.406 nm. The total numbers of atoms in Models I and II are respectively equal to 53,172 and 54,384.

The choice of the [112] axis normal to the direction of pressure (y-axis) and sliding (x-axis) has a significant advantage in analyzing the evolution of the defect structure of models under external loading. Indeed, only one slip system $a_0/2\left[\overline{1}10\right]\left(11\overline{1}\right)$ is activated in this case and the dislocation lines are parallel to the [112] direction. This allows for an easy visualization of the generation, motion, and rearrangement of dislocations under loading. Earlier, quasi-two-dimensional configurations with such an orientation of the [112] axis were successfully used to simulate the effect of ultrasonic vibrations on non-equilibrium grain boundaries [20,21] and nanocrystals containing non-equilibrium grain boundaries [22,23].

Simulation of ultrasonic welding includes two different stages. At the first stage, the two blocks are compressed at a constant external pressure. At the second stage, the upper block is shifted relative to the other along the x-axis according to a periodic law, which mimics the action of alternating displacements associated with ultrasound.

Atomistic simulations are carried out using the molecular dynamics code XMD developed by Rifkin [24]. Atomistic interactions are described by the many-body potential for copper based on the embedded atom method (EAM) by Foiles et al. [25].

The zero-temperature lattice parameter reproduced by the used interatomic potential is $a_0=$ 0.3615 nm. Models I and II constructed with this value of lattice parameter are initially relaxed at zero temperature under the specified value of pressure, which is applied to the atoms of the bottom-constrained layer when keeping the top-constrained layer rigidly fixed. During this process, the atoms of the bottom constrained layer are not allowed to move along the x-axis, i.e., they can move only in the yz-plane. After the relaxation procedure, the coordinates of all atoms and sizes of the computational cell are rescaled by a factor of 1.004 to adjust to the temperature of T = 300 K. Thereafter both models are equilibrated for a time interval of 100 ps at the constant temperature with fixed borders of the computational cell and the same constraints on the atoms of the outer layers. In these and further molecular dynamics runs the time step is set to be equal to 2 fs. After equilibration under pressure, oscillating sliding displacements are assigned to the top-constrained layer, while the atoms of the bottom layer are allowed to move only in the yz-plane to keep the pressure constant. The displacements of the top layer obey the relation $x=x_0\sin \frac{2\pi }{T}t$, where the amplitude is $x_0=$ 8 nm, and the vibration period is T = 500 ps. In order to apply these displacements, the whole period is divided into 1000 time intervals, and in each k-th interval the top layer is displaced to a distance of $\Delta x=x_0\left\{\sin \left[2\pi k/1000\right]-\sin \left[2\pi \left(k-1\right)/1000\right]\right\}$, and then 250 molecular dynamics runs are performed.

The chosen value of the simulated ultrasound frequency, $f=1/T=2$ GHz, is 50 times less than that used in Ref. [18]. This is carried out on the basis that during ultrafast straining, the processes can significantly depend on the strain rate and as a result proceed in a different way, in contrast to the processes during straining at conventional rates. Therefore, keeping the frequency as low as possible is important when simulating USW.

Up to k = 5 periods of oscillating displacements are applied to both simulated models. After each half-period, the molecular dynamics state, atomic coordinates, and velocities are written. From these data, the kinetic energies of atoms and temperature profiles near the welding interface are analyzed.

In equilibrium molecular dynamics simulation, the temperature of a configuration of N atoms is calculated based on the equipartition law, i.e., according to the formula

$$T=\frac{1}{3N{k}_B}{\displaystyle \sum }_{i=1}^N{m}_i{v}_i^2 ,$$

where k_B is the Boltzmann constant and m_i and v_i are the mass and the velocity of the i-th atom, respectively. This formula was used, for instance, in Ref. [15], when modeling interface atomic diffusion between two welded Al crystallites.

During fast deformation processes, a significant amount of the kinetic energy of atoms is contributed by their directed movement under the applied forces, and therefore there is a need to extract this part from the total kinetic energy. For the case of non-equilibrium configurations characterized by strong spatial and slight temporal inhomogeneities, Hayashi et al. [26] developed the concept of local quasi-temperature. The condition of a slight temporal inhomogeneity is fulfilled, when the relative sliding speed of two surfaces is significantly lower than that of the thermal motion of particles. In the case modeled in this work, the maximum speed of the displacement of the top layer can be evaluated as $v_0=\frac{2\pi }{T}{x}_0\approx 100$ m/s, which is more than three times less than the root mean square thermal velocity of atoms at T = 300 K.

The quasi-temperature of an atom i is defined as [26]

$$T\left(t\right)=\frac{2}{3k_B}{K}_j\left(t\right),$$

where the kinetic energy originating from the fluctuations of the velocity is calculated as

$$K_j\left(t\right)=\frac{1}{2}{m}_i[\langle v_i{\left(T\right)}^2\rangle -\langle v_i\left(T\right){\rangle }^2].$$

Here, time averaging is carried out over a time interval, which is sufficiently long as compared to the periods of lattice vibrations. In the present study, averaging over 250 successive time steps is executed. A separate test calculation for the case of a perfect fcc lattice at T = 300 K demonstrates that such averaging provides an accuracy in the quasi-temperature calculation of not worse than 5 K.

Simulations for both models are performed for three values of the clamping pressure applied along the y-axis, namely p = 135, 270, 540 MPa for Model I and p = 67, 135, 270 MPa for Model II. The lower pressure values for Model II are justified by the fact that the change in the crystal structure at its interface begins to occur earlier.

Visualization of atomic configurations and their structural analysis are performed by means of OVITO software [27]. Classification of atoms according to their local crystalline structure, i.e., common-neighbor analysis [28], significantly facilitates identification of partial dislocation activity in the modeled blocks. With the use of this approach, four different classes of atoms were defined. Atoms having fcc environment are colored in green, atoms with hcp local environment are colored in red, rarely occurring bcc atoms are colored in blue, and atoms that have a local environment different from the three crystal structures listed above are displayed in white.

3. Results

3.1. Structural Changes during Simulated Ultrasonic Welding

Figure 2a,b represent the snapshots of the atomic structures of Model I and II after molecular dynamics equilibration at T = 300 K and sinusoidal displacements under the pressures of p = 270 MPa and 67 MPa, respectively, at several time moments of the first cycle. The formation of double rows of red atoms, i.e., the stacking faults located on crystallographic planes $\left(11\overline{1}\right)$, are observed quite often in both models and in all stages of straining. This fact indicates that plastic deformation occurs via the emission of partial dislocations. Let us recall that when gliding, a leading partial dislocation changes locally an fcc environment to hcp one and thus leaves a trace of hcp atoms on the slip plane (two adjacent layers of atoms highlighted in red in Figure 2). If a trailing partial dislocation propagates on microscopically the same slip plane, then it eliminates this layer of hcp atoms resulting in a disappearance of the stacking fault. If a trailing partial propagates along an adjacent (parallel) slip plane, then it eliminates one layer of hcp atoms originating from the leading partial and creates another layer of hcp atoms, which is one atomic layer away from the first one. As a result, two single layers of hcp atoms turn out to be separated by one layer of fcc atoms. Such a defect is called a nanotwin and is often observed in modeling the deformation behavior of fully three-dimensional fcc nanocrystalline metals [29,30,31,32].

The most typical dislocation process is the emission of a partial dislocation with one of the Burgers vectors, ${\overrightarrow{b}}_1=1/6\left[\overline{1}21\right]$ or ${\overrightarrow{b}}_2=1/6\left[\overline{2}1\overline{1}\right]$, which glides over the crystal and is absorbed at the nearest sink. In Model I, the dislocations are mostly emitted from a surface of a pore, glide over the interface region, and are absorbed at the surface of a neighboring pore or, due to periodic boundary conditions, at the nearest periodic image of this pore. This results in the formation of a stacking fault between the two pores, which remains for some time unless it is eliminated by the trailing partial. In Model II, the dislocations are emitted from an interface, which is formed after healing of the pores, and glide across the crystallite along $\left(11\overline{1}\right)$ planes inclined to the interface plane. In the other typical case, the trailing partial is emitted after the leading one before it reaches a sink. These cases are illustrated in Figure 3 for Models I and II. As the rarest event, a homogeneous nucleation of a dipole of partial dislocations occurs in the bulks of blocks as shown in Figure 3c.

More completely, the evolution of the atomic structures of Models I and II during the first cycle of sinusoidal sliding simulating the action of an ultrasonic field is presented in Video Files S1–S6 given in Supplementary Materials.

A visual analysis of the further evolution of Model I under USW (up to the end of the fifth cycle) demonstrates that the structure does not evolve qualitatively and is similar to that shown in Figure 2a at t = 1.0T for all three values of the applied clamping pressure. Up to the pressure of p = 540 MPa, no healing of the gaps between the grooves is observed (Figure 4). It should be noted that at different values of the clamping pressure, there are no systematic changes in the sizes of residual pores. For example, at p = 67 and 270 MPa, two small pores are maintained after five cycles of straining (see Figure 4a,b), while at p = 540 MPa, only one larger pore remains (Figure 4c). It is clearly seen that the majority of partial dislocations either nucleate on the surface of the pores and glide through the entire computational cell in the xz-plane, or then sink in the same pores (due to periodic boundary conditions). This is because dislocation nucleation from a free surface occurs at a much lower applied stress as compared to the homogeneous bulk dislocation nucleation [33].

On contrary, in Model II, the sliding displacements greatly reduce the porosity and close the gaps after one cycle of sliding already at the smallest applied clamping pressure of p = 67 MPa (see Figure 2a). The typical structure transformation occurring during subsequent oscillations is the upwards migration of the interface. Evidently, the oscillating shear strain of the whole bicrystal, which is formed after the elimination of pores, occurs by closely correlated mechanisms of the emission and glide of lattice dislocations and interface sliding. The emission of a dislocation by a grain boundary typically forms a residual grain boundary dislocation with a related step, while the sliding along a grain boundary carried by this dislocation results in the grain boundary migration [34]. As seen in Figure 2b and Figure 5, the dislocation activity is higher in the upper block, i.e., non-symmetric with respect to the interface plane, and this results in a preferential migration of the interface in one direction despite the symmetric straining. During USW, the migration of Cu atoms in both blocks from their equilibrium lattice sites to new ones takes place. Thus, the atoms of the blocks mutually penetrate each other, providing the formation of a bonded zone.

3.2. Temperature Evolution during Simulated Ultrasonic Welding

As one can see from the structural evolution during USW described in the previous section, dissipative processes related to the dislocation activity occur mainly in the interface region, and this results in an increase in the temperature in this region. Due to the thermal conductivity, there is a heat flow from this region towards the thermostat layers which are kept at the temperature of 300 K. One can expect that on expiration of a certain time interval, a stationary temperature field will be established.

Due to the high thermal conductivity of copper and small sizes of the simulated welding blocks, this stationary state is established very quickly. In order to analyze the temperature distribution, the computational cell is divided into 30 layers with an equal thickness parallel to the interface, and the temperatures of these layers are calculated by averaging corresponding atomic quasi-temperatures after each welding cycle. The temperature profiles calculated as described above for Models I and II at different clamping pressures are presented in Figure 6a,b, respectively. The vertical dashed lines indicate the centers of the cells; that is, the places where two simulated blocks come into contact when pressure is applied. Note that due to the presence of constrained atoms at the bottom of the computational cell, the specified vertical line is somewhat shifted from its geometric center. In addition, the kinetic energy of the constrained atomic layers is not taken into account when calculating the quasi-temperature, and therefore the data points in Figure 6a,b start at a certain value, but not from zero.

In Model I, after equilibration under the applied pressure and before the onset of ultrasonic treatment, the quasi-temperature is distributed approximately equally over the thickness of the cell and only slightly deviates from 300 K as clearly demonstrated in Figure 6a. Right after the application of cyclic displacements, the quasi-temperature significantly increases up to circa 400 K at the place of the contact of two blocks and almost linearly decreases towards thermostats. The maximum temperature at the welding region is set immediately after the first cycle and practically does not change during subsequent cycles, regardless of the applied clamping pressure. There is a moderate asymmetry in the distribution of the quasi-temperature over the cell thickness. Namely, its maximum is somewhat shifted upwards from the geometric center of the computational cell (see Figure 2a). This fact is related to a more pronounced formation of the defect structure in the upper block and the presence of larger pores in contrast to the bottom block and, as a consequence, a larger amount of atoms with higher kinetic energy resulting in an increase of the quasi-temperature.

In Model II, the distribution of the quasi-temperature over the thickness of the cell is drastically asymmetric as shown in Figure 6b. The quasi-temperature in the welding region is in the range of 650–700 K, which is significantly higher as compared to Model I. Similar to the previous case, the quasi-temperature maxima remain unchanged after the first and subsequent sliding cycles. The maxima of these distributions exactly correspond to the position of the interface formed as a result of the clamping of two crystalline blocks (compare Figure 5 and Figure 6b). This suggests that the less densely packed atoms contribute the most to the increase in the quasi-temperature.

Figure 7 demonstrates the distribution of quasi-temperature in the entire computational cell after the first and fifth cycles in Models I and II. For a more visual comparison, all the figures use color coding for the same temperature range of 300–700 K. The distribution of quasi-temperature in Model I remains practically unchanged during five cycles of alternating shearing as can be seen from the comparison of Figure 7a,b. The main heating occurs at the interface, but its quasi-temperature does not differ much from that inside the blocks. In Model II, the main concentration of kinetic energy and, accordingly, quasi-temperature is in the interface. In this case, as already mentioned above, the latter heats up much more strongly compared to Model I. With a sinusoidal sliding of the interface, as displayed in Figure 7c,d, the region of maximum quasi-temperature shifts towards the top of the computational cell, which is directly associated with the migration of the interface in the upward direction.

3.3. Porosity Change and Interface Structure during Simulated Ultrasonic Welding

The dependences of the porosity of the configurations on the simulation time for the two studied models at different values of the clamping pressure are presented in Figure 8. The porosity was calculated as the ratio of the current value of free volume to the initial pore volume, i.e., to the volume of the empty space in the initial relaxed structure prior to sliding. For both models, after a slight increase, a sharp reduction in porosity down to 0.2–0.5 occurs in the time interval 0.1–0.3 ns during the first cycle of sinusoidal sliding and then fluctuates around a certain steady value, while continuing to decrease slightly. Such a behavior is primarily associated with the closing of the pores existing in the configurations after the application of clamping pressure. For Model I, the behavior of the relative porosity over time is practically similar for the pressures of 135 and 270 MPa, while for higher values of 540 MPa, the residual porosity is significantly higher. As was already pointed out, this is due to one large pore remaining in the structure at 540 MPa (see also Figure 4c). For Model II, the curves practically coincide for all three values of the applied pressure, as demonstrated in Figure 8. The latter speaks in favor of the fact that the magnitude of the clamping pressure does not have a noticeable effect on the residual porosity in the case of misorientation between the welded crystallites.

After a sharp decrease, the change in porosity occurs cyclically with a period equal to a half of the period of sinusoidal shearing along the interface. This is due to the fact that in the first half of the period there is an increase in porosity at first and its subsequent partial accommodation, and in the second half of the period, when shifting from the initial equilibrium state in the opposite direction, the whole process is repeated again. It is interesting to note that during five simulated shearing periods, the amplitude of porosity fluctuations practically does not change. Figure 8 also confirms that in Model I there is no complete disappearance of pores in contrast to Model II. This is related to a presence of a misorientation between the two grains and the emission of partial dislocations from the interface resulting in a decrease of interface stresses and, as a consequence, of the porosity. The question of how the misorientation angle and the size of the grooves affect the residual porosity and the interface structure remains open and requires a separate study.

As one can see from Figure 2b, no visible pores remain at the interface already by the end of the first cycle in Model II. However, the free volume does not diminish down to zero. This means that it is distributed in the interface region and the latter is expected to acquire a structure different from that of an equilibrium grain boundary. Therefore, an analysis of the interface structure during the process of simulated USW is of interest for this case.

Figure 9 displays the radial distribution functions (RDFs) calculated for a crystalline region of the configuration subjected to the pressure of 67 MPa at 300 K without sliding and for interfaces formed during sliding at the three values of the clamping pressure. The RDF of the crystalline part at other values of the pressure is similar to the one obtained for the pressure of 67 MPa. This function is characterized by high narrow peaks corresponding to a well-ordered crystal structure. The peaks are slightly broadened due to the thermal fluctuations of atoms. The RDFs for the interface were calculated by selecting regions which contain as large as a possible fraction of interface atoms in the structures saved during the fifth cycle of straining. Independently of the value of the applied clamping pressure, the intensities of all peaks of these functions are significantly decreased, the widths are increased, and high-order peaks are completely disappeared. This shows that in the interface region the well-ordered crystal structure is destroyed and an amorphous-like disordered structure is formed during USW.

4. Discussion

In Model II, sinusoidal shear displacements lead to the migration of the interface upwards regardless of the applied clamping pressure as shown in Figure 5a–c. This direction of the interface migration is not accidental. Such a phenomenon was studied in the past decades in numerous publications both experimentally [35,36,37,38] and with the help of computer modeling in bicrystals [39,40,41,42,43] and in three-dimensional nanocrystalline structures [44,45,46]. In particular, the authors of [42,43] for a large number of $\rangle 100\langle$, $\rangle 110\langle$ and $\rangle 111\langle$ symmetrical grain boundaries established that depending on the misorientation angle, the grain boundary migration could occur in opposite directions, i.e., either to the upper edge of the simulation cell, or to the lower one. Moreover, such a change in the direction of grain boundary migration has an abrupt character and occurs at a certain misorientation angle (for instance, $\theta \approx 35°$ for $\langle 100\rangle$ misorientation axis [43]) and for a certain temperature range. All investigated grain boundaries were found to be able to migrate in both directions. The latter depends on the critical stresses, Schmid factors, and symmetry-breaking imperfections in the grain boundary structure.

Thus, since the interface misorientation angle in Model II is always the same, the direction of the interface migration also does not change regardless of the pressure and number of cycles. Special studies also show that the direction of the interface migration does not depend on the initial direction of the sinusoidal shear displacements. One can suppose that this effect might be associated with some asymmetry of the border conditions applied to the system, since the upper constrained layer is moved as a rigid whole, while the atoms of the bottom one can move in vertical planes allowing for the transfer of external pressure. However, a detailed analysis of the atomic structures of the upper and bottom layers shows that they remain unchanged and identical to each other until the end of five cycles. Therefore, more research is needed to fully understand the phenomenon and its relation to the geometry of the interface and other factors.

The initial misorientation between the blocks to be welded determines the presence of interface sliding. In the created interface with a misorientation angle of 78.46°, sliding occurs more easily and therefore more intensively than between the $\left(\overline{1}\overline{1}1\right)$ planes in the perfect crystal lattice as clearly seen in Figure 2a,b. The latter fact is confirmed both experimentally [47,48] and via atomistic modeling [49,50,51]. As previously established, grain boundary sliding is closely related to the grain boundary energy. Namely, high-energy grain boundaries have a greater ability to slide in contrast to low-energy ones. A high-energy grain boundary is more disordered, and, consequently, has a lower energy barrier for movement, which results in an easier sliding and migration [49]. Since the energy of the interface in Model II is a priori higher than the energy of a perfect lattice in Model I, this explains the higher ability of the interface to slide and migrate under alternating shear stress in Model II (see Figure 5). Fast healing of pores between two blocks in Model II can also be rationalized in terms of the higher ability of interfaces to slide and migrate as opposed to the blocks in Model I.

Another important feature of Model II is related to the free volume and atomic structure of the interface. Although no localized pores remain there already after the first sliding cycle, the excess volume does not diminish down to zero, i.e., it is distributed in the interface. This is due to the disordered atomic structure of the grain boundary, which is formed during this highly dynamic process. Whereas tilt grain boundaries in the ordinary equilibrium state have a highly ordered crystal-like structure [34], during the simulated USW, the formation of a disordered amorphous-like atomic structure of the interface is observed.

It seems interesting to compare the temperature distributions in the modeled configurations subjected to USW with the results of previous studies. Mostafavi et al. [14,15] simulated USW of two Al blocks using methods of molecular dynamics. In contrast to the present study, instead of applying constant pressure, the cited authors moved the two Al blocks towards each other at a constant compression rate. The latter means that the pressure gradually increased with time, which violates the conditions of pressure constancy during USW. Only one-directional shear along the interface plane was applied instead of alternating shear stress corresponding to ultrasonic action. When calculating the temperature distribution in the cell, the kinetic energy of atoms related to their directed movement during shearing has not been subtracted. Although such a subtraction of the kinetic energy somewhat reduces the temperature of the layers, it does not qualitatively change the overall temperature distribution. In addition, the authors found that the temperature at the interface increased mainly only during sliding, while an increase in pressure at the interface did not lead to an increase in temperature. The latter fact is in full agreement with the results obtained in the present work.

It is worth noting that the authors of [15] found a dependence of the interface temperature on the rotation angle of the upper block around the z-axis. It turned out that the maximum increase in temperature up to 800 K under the given conditions occurred at a rotation angle of 15°, while the temperature reached only 600 K at a rotation angle of 45°. The temperature at the interface has a rather complex dependence on the orientation angle (see, for example, Figure 8 in Ref. [15]). Such a dependence of the interface temperature can apparently be explained by the peculiarities of its crystal structure. The present study indirectly confirms this fact. Namely, in Model I, where the $\left[\overline{1}10\right]$ directions of both blocks are parallel to the x-axis, the maximum temperature increase at the interface is 400 K, while in Model II, i.e., when the lattices of the two blocks are rotated relative to each other, the temperature increases up to 700 K. Although a more detailed answer to this question requires extensive research.

As was noted in the Introduction, a relatively realistic simulation of the USW process between Al and Cu blocks was undertaken by Yang and coauthors [18]. Molecular dynamics modeling was carried out at a constant external pressure and sinusoidal oscillations along the boundary plane. Initially, it was assumed that the welding blocks have a sinusoidal distribution of asperities. However, in contrast to the present work, the mutual arrangement of the asperities in the two blocks was such that when a compressive stress was applied, they completely overlapped each other without the formation of interface gaps. The authors [18] found that after 500 sliding cycles, the maximum increase in temperature to about 375 K occurred at the welding region. This value is somewhat lower than that shown in Figure 6 when welding two Cu blocks. In addition, it should be taken into account that in Ref. [18], the kinetic energy associated with the translational motion of atoms during shear was not subtracted from the total kinetic energy, and therefore the temperature shown in Figure 9 of Ref. [18] turned out to be slightly higher than the actual values. Besides, it remains not entirely clear why the temperature in the Cu block, when moving from the welding region to the upper edge of the block, does not gradually decrease to the temperature of the thermostat as in the Al block, but remains approximately equal to 320 K.

It is worth noting that diffusion processes in the case considered in this work do not play such an important role in contrast to the case studied by Yang et al. [18]. This is primarily due to the fact that a much lower frequency of cyclic straining (2 GHz vs. 100 GHz) and number of cycles (5 vs. 500) was used here. In addition, it is important that although the temperature rises, it is not high enough for a significant activation of self-diffusion in the welding region.

5. Conclusions

In the present paper, the process of ultrasonic welding of two blocks of fcc copper under the conditions of a constant clamping pressure and sinusoidal shear displacements was studied using atomistic simulations. Two different atomistic models of blocks were simulated: Model I with no misorientation between the lattices of the blocks, and Model II with a special misorientation of Σ = 5/78.46°. The evolution of the atomic structure of joints with time at different clamping pressures, dislocation processes, and temperature evolution in the interface region were investigated. The main conclusions can be formulated as follows.

Alternating shearing causes plastic deformation of the interface layers and is associated with the emission of partial dislocations from the surfaces of the pores. The presence of misorientation between the joined blocks contributes to a more pronounced interface sliding, interface migration, and pores healing. The magnitude of the applied clamping pressure does not have any noticeable effect on the interface quasi-temperature in both models. The latter immediately increases almost up to maximum values after the first shear cycle, and then does not change significantly over the next four cycles. The temperature at the interface in Model II turns out to be almost twice higher than in Model I, which is explained by the presence of a misorientation between the joined blocks.

For both models, after the onset of USW, a significant reduction in porosity up to 50–80% of the initial value during very short time intervals takes place, which is related to a closing of pores existing in the configurations. After that, the porosity fluctuates with a frequency twice the shear frequency along the interface. For Model I, the magnitude of the clamping pressure has a pronounced effect on the behavior of the relative porosity over time, while almost no effect for Model II is found.

The radial distribution functions for the interface suggest that the destruction of its initially well-ordered crystal structure and the formation of an amorphous-like disordered structure occur during five cycles of sinusoidal shear displacements.

During atomistic simulations, a relatively small computational cell was used. Due to the small thickness of the Newtonian layer, the temperature from the contact point of the two blocks is removed faster, which is related to the high thermal conductivity of copper. For a more detailed study of the temperature distribution, it is necessary to use larger computational cells, which is planned to be done in the foreseeable future.

It should be noted that the results obtained in this work via atomistic modeling cannot be directly transferred to experimental observations due to the obvious limitations of the method used. First of all, the perfect copper single crystals of a rather small size with idealized semi-cylindrical grooves of the same radius were adopted. In connection with this, the width of the layer between the welding region and the thermostat is quite small resulting in a fast temperature removal and, accordingly, to slightly different conditions compared to those typical for USW. Due to the fact that the time step must be smaller than the period associated with the fastest vibrational frequency in the modeled system, the typical simulation time achievable in classical molecular dynamics is of the order of several nanoseconds, i.e., many orders of magnitude less than those used experimentally. In this regard, it is necessary to keep in mind that, in view of the abovementioned limitations on the simulation time, the study of diffusion requiring much longer time intervals turns out to be difficult within the molecular dynamics approach. In addition, since it is necessary to achieve high strains in a sufficiently short time, there is a need to use very high shear strain rates comparable to the shock loading. However, despite the obvious shortcomings of the molecular dynamics method, it allows one to take a deeper insight into the processes occurring at the atomic level during USW.