CC BY
51
УДК 539.4
Molecular dynamics investigations of the strengthening of Al-Cu alloys during thermal ageing
W. Verestek, A.-P. Prskalo, M. Hummel, P. Binkele, and S. Schmauder
Institute for Materials Testing, Materials Science and Strength of Materials, University of Stuttgart, Stuttgart, 70569, Germany
Classical molecular dynamics simulations of the interaction of edge dislocations with solid soluted copper atoms and Guinier-Preston zones (I and II) in aluminium are performed using embedded atom method potentials. Hereby, the strengthening mechanism and its modulus are identified for different stages of thermally aged Al-Cu alloys. Critical resolved shear stresses are calculated for different concentrations of solid soluted copper. In case of precipitate strengthening, the Guinier-Preston zone size, its orientation and offset from the dislocation plane are taken as simulation parameters. It is found that in case of solid soluted copper, the critical resolved shear stress is proportional to the copper concentration. In case of the two subsequent aging stages both the dislocation depinning mechanism as well as the depinning stress are highly dependent on the Guinier-Preston zone orientation and to a lesser degree to its size.
Keywords: molecular dynamics, Al-Cu alloys, Guinier-Preston zones, dislocation, critical resolved shear stress, precipitate strengthening, embedded atom method potential
Молекулярно-динамическое исследование упрочнения сплавов Al-Cu в процессе термического старения
W. Verestek, A.-P. Prskalo, M. Hummel, P. Binkele, S. Schmauder
Штутгартский университет, Штутгарт, 70569, Германия
Проведено классическое молекулярно-динамическое моделирование взаимодействия краевых дислокаций с атомами твердого раствора меди и зонами Гинье-Престона (I и II) в алюминии с применением потенциалов погруженного атома. Выявлен механизм упрочнения и определен его модуль для разных стадий термического старения сплавов Al-Cu. Рассчитаны приведенные критические касательные напряжения для различных концентраций твердого раствора меди. Для случая дисперсионного упрочнения в качестве параметров моделирования использованы размер зоны Гинье-Престона, ее ориентация и смещение относительно плоскости дислокации. Обнаружено, что для твердого раствора меди приведенное критическое касательное напряжение пропорционально концентрации меди. На двух последующих стадиях старения механизм движения дислокаций, а также напряжение движения сильно зависят от ориентации зоны Гинье-Престона и в меньшей степени от ее размера.
Ключевые слова: молекулярная динамика, сплавы Al-Cu, зоны Гинье-Престона, дислокация, приведенное критическое касательное напряжение, упрочнение осадка, потенциал погруженного атома
1. Introduction
Aluminium is, next to iron, one of the most widely used metals in engineering. However, in its pure form it has rather poor material properties. By adding various other elements such as e.g. Cu, Mn, Zn, and/or Si, it is possible to enhance its mechanical properties, depending on the type of the alloying material and the thermal ageing state. In this work, the mechanical properties of Al-Cu alloys during thermal ageing are investigated.
The ageing process of an Al-Cu alloy can be classified into five distinct stages. At the first stage, a supersaturated solid solution is present, with substitutionally disolved copper atoms embedded in the aluminium matrix. Individual copper atoms interact with dislocations, hindering their pro© Verestek W., Prskalo A.-P., Hummel M., Binkele P., Schmauder S., 2017
pagation and hereby diminishing plastic deformation under external loading. On the macroscale, this is observed as material strengthening. During continuous thermal ageing, vacancy exchange mechanisms lead to precipitate formation [1]. Extensive theoretical investigations of copper precipitation in aluminium are available [2, 3]. In practical ageing experiments, the precipitate formation occurs in a time frame of several hours. In the binary Al-Cu material system the precipitate formation results in coherent single layer Cu discs, known as the Guinier-Preston (GP) zones [4, 5]. The Guinier-Preston zones generally occupy {100} crystallo-graphic planes (metastable disc shaped particles along {111} planes have also been reported [6]), their diameter reaches up to 10 nm. Theoretical investigations [7] indicate that
the copper concentration within a GP I zone correlates positively with its size. Experimental investigations by tomographic atom probe-field microscopy (TIP-FIM) support this idea, showing that in an Al matrix, GP I zones of different copper concentrations ranging from 40 % up to 100 % coexist [8, 9]. In addition,TIP-FIM shows that the concentration of solid soluted copper is zero in the vicinity of a GP I zone, supporting the model presented in Fig. 1. In the subsequent ageing state of Al-Cu alloys, further Cu discs are formed in the vicinity of the existing ones. This results in the formation of GP II zones. The diamater of GP II zones can reach up to 150 nm. The internal structure of a GP II zone consists of several GP I zones parallel to each other, in between which several Al layers are found [10, 11]. The two final stages of the Al-Cu ageing process are represented by the formation of disc shaped, semicoherent 6'- and spherical, incoherent 6-precipitates, both having a AlCu2-stoichiometry. These will, however, not be investigated here.
The objective of this work is to identify the underlying atomistic processes of the strenghtening of Al-Cu alloys, associated with the precipitate formation as well as to quantify its magnitude. For this purpose, the computational method of molecular dynamics has proved itself as method of choice.
Computational results currently available on the topic of materials strengthening in the Al-Cu system are rare. In the field of solid solution strengthening, extensive results can be found dealing with iron [12, 13] or nickel [14]. However, we are unaware of results dealing with solid solution strengthening in aluminium-based alloys. Similarly, results on precipitation strenghtening are focused mainly on spherical precipitates and/or voids [15, 16]. To our knowledge, computational investigations on precipitate strengthening via GP zones in the Al-Cu material system have been performed by Singh et al. [17-19] solely. While a hierarchical multiscale model of the material strengthening by GP I zones is presented in [19], no investigations have been done on
Fig. 1. Simulation cell. On the left hand side three GP I zones 13 are depicted, having a diameter of 13 nm and being orthogonal to each other. On the right hand side a dislocation dissociated into two Shockley partial dislocations is shown 4 with a stacking fault of 1.9 nm width located between the individual partials and presented 5. The double arrow indicates directions of dislocation propagation (in the negative or positive direction x). The system size is indicated by grey lines and crystallographic orientations are denoted
the contribution of the GP II zones in the same model. It is our intention not only to fill the missing link, but also to perform a comprehensive computational investigation on the material strengthening for the three intial stages of the ageing process of Al-Cu alloys.
2. Computational details
Using classical molecular dynamics (MD) has proven as a suitable method for simulations of mechanical properties of materials at the atomic scale for large systems (number of atoms > 100 000) where molecular dynamics atoms are considered as point particles interacting by Newton's laws of classical potential functions. Classical molecular dynamics simulations are performed using the IMD code [20], which allows massively parallel computations. The code has been developed at the former Institute for Theoretical and Applied Physics (ITAP) at the University of Stuttgart. For the description of interactions within the Al-Cu materials system, the embedded atom method (EAM) potential, developed by Cheng et al. [21] is used.
The EAM potential used [21] is excellent with respect to the representation of the dislocation relevant properties of Al, such as equilibrium lattice constant, elastic constants, vacancy formation energy and stacking fault energies [22]. In addition, various crystal structures and stoichiometry of the Al-Cu phase diagram are represented with an excellent agreement with experimental observations.
The two Shockley partial dislocations are aligned parallel to each other with the stacking fault in between having an equilibrium width D of 1.9 nm. The stacking fault width corresponds well to investigations made by Kuksin et al. [23] (stacking fault width 1.5 nm) using two different EAM potentials: a further development of the potential [24] by Liu and Ercolessi [25] and the EAM potential developed by Mishin et al. [26]. Recently, Singh et al. [17] presented similar results, using another potential for the Al-Cu system [27]. For a detailed discrete Fourier transform analysis of the dislocation relevant properties of Al the reader is referred to [22].
The system under investigation is a rectangular simulation box with edge lengths of Lx = 110 nm, Ly = Lz = 18 nm as presented in Fig. 1 and consists of approximately two million atoms. Cartesian axes x, y, z of the system correspond to {1 10}, {112} and {111} crystallographic axes of the face-centered cubic Al-matrix system. Periodic boundary conditions are applied in directions x and y, while in direction z free boundary conditions are used.
The initial nonrelaxed structure can be imagined as two half-crystals in which two (111) planes in the upper halfsystem are removed. A detailed description of the procedure can be found in [23]. The structure is stabilized using the global convergence relaxation integrator and a time step of 2 fs. An axial-independent minimization of both hydrostatic and shear stresses is performed during the relaxation. Hence, on a global scale, all residual stresses which could
influence subsequent shearing simulations, are eliminated. During the relaxation, the newly formed edge dislocation dissociates into an energetically favored state of two Shockley partial edge dislocations, as expected for the fcc aluminum structure. The Burgers vector b of the dislocation changes according to the equation for a dislocation splitting into two partial dislocations in fcc crystals:
b = a/2[110] ^ a/2[2U] + a/2[121]. (1)
For the dislocation identification, coordination number COORD and common neighbor analysis CNA [28] with a designated cut-off radius of 0.35 nm are used. While COORD enables the identification of individual dislocation partials (coordination smaller than 12), CNA is necessary to detect stacking faults. Both tools proved to be reliable in structural analyses. No difference could be noticed between the reliability of COORD/CNA analysis and the centrosymmetry parameter CSP [29], also used in detecting structural defects.
3. Preliminary investigations
At first, preliminary investigations are performed to determine the shear stress necessary to initiate dislocation propagation (Peierls stress tp) in Al. A constant displacement increment in the direction x on the two uppermost z-layers is used, while holding two bottom z-layers fixed. In this way, shear stress is introduced into the system. The Peierls stress of tp = 3 MPa is calculated. Similar values of tp are reported by other groups [17] using a different potential by Apostol et al. [27]. In [22], dislocation propagation is initiated already at TP = 1 MPa. These simulations were, however, performed at finite temperature of 10 K, explaining the slightly lower Peierls stress calculated. In general, a low Peierls stress is expected for fcc metals where dissociated dislocations are present in contrast to bcc metals, e.g. iron where dislocations stay compact [22].
Dislocation motion is detected instantly upon the beginning of shearing. The movement of individual dislocation partials is synchronous, their distance (width of the stacking fault D) remaines constant during the propagation through the Al matrix. The resulting dislocation velocity is 20 m/s. The shear stress-strain dependence of a propagating dislocation is presented as a black line in Fig. 2.
4. Solid solution strengthening
In order to investigate the strengthening effect of solid soluted copper, randomly distributed Cu atoms in a concentration of 1, 2 and 5 at. % are placed onto aluminium lattice sites by simply changing the atom type. This results in a substitutational alloying of copper in an Al matrix as known from the experiment. The solution of copper atoms is not restricted to the dislocation plane only, but is performed within the whole simulation cell.
A constant strain increment of the upper most crystal layers, as described in the previous section, is used to introduce shear stress. Figure 2 depicts the dependence of
the global shear stress state during the shearing simulation. In pure aluminium, the dislocation line propagates, besides the relatively low Peierls stress tp, unhindered through the crystal and the individual dislocation partials, remain parallel. Dislocation propagation is visible by means of shear stress analysis as indicated by the wave-like line 1 in Fig. 2. Small oscillations around the zero line with an amplitude of 3 MPa indicate that the dislocation propagation is hindered only by the Peierls stress.
At 1 % solid soluted copper, the stress-strain curve shows a typical saw-tooth like behavior (Fig. 2, curve 2), individual peaks reach up to 60 MPa, depending on the number of copper atoms in the vicinity of the dislocation line. At this stage, individual dislocation segments propagate faster than others, resulting in a bending of the dislocation line. An increase to 2 % copper concentration amplifies the described effect. Individual peaks in the shear stress-strain curve (presented in Fig. 2, curve 3) reach up to 90 MPa. The dislocation propagation is hindered significantly with short stages of full halt. Further increase to 5% copper concentration keeps the dislocation line halted for longer periods. The first movement of the dislocation line occurs at a strain of ~0.6 % (Fig. 2, curve 4) and a shear stress of ~130 MPa. Due to the random distribution of copper atoms in the Al matrix, it is possible that an over-average amount of copper atoms is located in the vicinity of the dislocation line. In this case, the shear stress necessary to unlock the dislocation can reach up to 230 MPa, clearly indicating the strengthening effect of copper as an alloying element in solid solution.
5. Dislocation interactions with GP I zones
Upon thermal ageing process copper precipitation is initiated, forming disc shaped precipitates known as Guinier-
250
£ 200 s
S150
I 100
-50
0.0 0.5 1.0 1.5 2.0 Strain, %
Fig. 2. Shear stress-strain relation during the propagation of a line dislocation in aluminum for different copper concentrations. The curve 1 with an amplitude of ~3 MPa presents the shear stress within an Al single crystal. Only the Peierls stress tp = 3 MPa needs to be overcome for a dislocation to propagate. Lines 2-4 represent shear stress-strain relations for 1, 2 and 5% copper concentrations. In case of 5% copper concentration, the dislocation line is at rest, locked by copper atoms in its vicinity. The first dislocation movement can be observed as the shear stress reaches 130 MPa
Preston zones, which, depending on their size consist almost completely of copper atoms. Due to the lattice (a(Cu) = = 0.361 nm, a(Al) = 0.405 nm) and elastic moduli mismatch (£(Cu) = 140 MPa versus £(Al) = 76 MPa) it is expected that GP zones have a significant influence onto the dislocation propagation. It is the purpose of this section to quantify the strengthening effect and to investigate the underlying atomistic mechanisms.
Figure 1 depicts the simulation setup used for this investigation. The grey rectangular box indicates the system cell, containing the dissociated dislocation (lines 4) and stacking fault between individual partials (5), as detected by COORD/CNA analysis. On the left hand side, three differently colored discs are depicted, representing individual GP I zones. Figure 1 shows the GP I zones having the same center and diameter of 13 nm. As the GP I zones occupy the three {001} crystallographic planes, they are orthogonal to each other. Among them, the GP I (001) zone stands out, as its surface normal is perpendicular to the Burgers vector of the dislocation. The two other GP I zones (occup-ing the (100) and (010) plane) are equivalent, their surface normals enclose a 60° angle with the Burgers vector of the dislocation.
An initial assumption is made that the center of the individual GP I zones lies in the dislocaton plane and that the copper concentration within the GP zone is 100 %. Due to the large extent of the system in the x-direction, the initial distance between the dislocation and the GP I zone can be set quite large (~6 nm). This system size was chosen in order to prevent a dislocation drift towards or away from the GP I zone without any external loading and only due to the interaction between the stress field of the dislocation and the GP I zone.
Compared to the dislocation propagation through solid soluted copper, the dislocation propagation direction is an
additional parameter taken into account for the dislocation interaction with GP zones. This is due to system asymmetry regarding the dislocation plane (upper half-system lacks 2 atomic layers in the x-direction) and the skewed orientation of the GP zone.
Figures 3-5 depict the relation between the shear strain and the resulting shear stress during the molecular dynamics simulations for all three GP zone orientations and both shearing directions. In addition to the shear stress evolution, which is presented as line 1, the local configurations of the dislocation and the GP zone are presented at different steps of the simulation, e.g. prior dislocation de-pinning.
Throughout Figs. 3-5, the dislocation and the stacking fault are presented as stripe 1, as detected by the COORD and CNA analysis. Disks 2 represent structural defects around the GP zones caused by the strain field of alloying copper atoms. Figures 3, a, 4, a and 5, a present the case where the shearing (and the dislocation propagation direction) is negative, while in the Figs. 3, b, 4, b and 5, b positive shearing is depicted.
6. Dislocation interaction with the (010)-and (100)-oriented GP I zones
We start the discussion of observations made with GP I zones occupying the (010) and (100) crystallographic planes as they are of equivalent orientation regarding the Burgers vector of the dislocation. In case of the negative shearing and a (010)-oriented GP I zone (Fig. 3, a), the dislocation propagation is initiated at the Peierls stress and in between 0.0-0.3% shear strain. As the dislocation approaches the GP zone from the right-hand side to the left-hand side (negative shearing), there is a small repulsion between them. The bottom left picture embedded in Fig. 3, a indicates a slight curvature of the dislocation line away from the GP
1.0 1.5 Strain, %
1.0 1.5 Strain, %
Fig. 3. Development of the global shear stress state during the interaction between the (010)-oriented GP I zone and the dislocation line. The global shear stress state of the system is presented as line 1 for the negative (a) and positive shearing direction (b). Notice the asymmetry of the depinning stresses for different shearing directions. The structural analysis is presented using the COORD and CNA analysis of different simulation snapshots
td P-
-100
-200
-300
2
__1 "i 2 Jf \ - 1N\ a 2 y t
).0 0.5 1.0 Strain, % 1.5 1 2 2
250
Fig. 4. Representation of the global shear stress state as a function of the shear strain and the shearing direction for the dislocation interacting with (100)-oriented GP I zone. Notice the initial attraction between the dislocation line and the GP I zone (visible through the bowing out of the dislocation line towards the GP I zone) in (a) and the repulsion for the same case but positive shearing direction (b). In both cases, the dislocation line peels away from the (100)-oriented GP I zone as it detatches. As in case of the (010)-oriented GP I zone, this effect can be traced back to the twisted positioning of both GP zones relative to the dislocation plane
zone. This repulsion is also visible by the global shear stress state in between 0.3-0.4% shear strain: the Peierls stress increases ~5-10 MPa due to this repulsion effect.
With subsequent shearing, there exists an attractive force between the GP I and the dislocation line as indicated by a
small peak close to 0.5 % shear strain. At this point the average x-coordinate of the dislocation line is equivalent to the center of the GP I zone center and the resulting shear stress vanishes. Upon further shearing, the stress state of the system increases linearly with the shear strain. At around
Fig. 5. Development of the global shear stress as a function of the shear strain and the shearing direction. Compared to Figs. 3 and 4 the calculated depinning stress is considerably lower. In addition, the two dislocation partials detach individually, the corresponding shear stress-strain curve shows a double, rather than a single peak, associated with the depinning event
1.3 % shear strain, the dislocation detaches from the (010)-oriented GP I zone at around -175 MPa shear stress. A single (negative) peak in the stress state in Fig. 3, a indicates that both dislocation partials detach from the GP I zone at the same time. In addition, the lower right picture embedded in Fig. 3, a shows that this detachment presents the final stage of the process during which the dislocation "peels away" from the back side of the GP I zone. This is due to the skewed orientation of the GP I zone relatively to the dislocation plane.
In case of a positive shearing (dislocation line propagates from left to right) and holding the remaining simulation parameters fixed, the situation is slightly different. As the dislocation line approaches the (010)-oriented GP I zone (Fig. 3, b) from the left-hand side, the previously stated repulsion is changed into attraction. Because of this, the dislocation line bows out towards the GP I zone (lower left picture embedded into Fig. 3, b). The global shear stress continuously drops from the Peierls stress to ~30 MPa at 0.9 % shear strain. From there on and upon further shearing, the shear stress linearly increases to the value of275 MPa. At this point the dislocation line detaches from the (010)-oriented GP I zone. Both partial dislocations "peel away" from the back side of the GP zone and detach simultaneously.
Observations made for the (100)-oriented GP I zone (disc 1 in Fig. 1) are similar to those described in case of the (010)-oriented GP I zone. Hence, a detailed discussion is redundant at this point. Irrespective of the shearing direction, both dislocation partials detach from the GP I zone in both orientations at the same time. This is visible both via the COORD and CNA analysis of individual simulation snapshots as also via the corresponding shear stress-strain relation which indicates a single peak for the dislocation depinning event.
7. Dislocation interaction with the (001)-oriented GP I zone
In case of the (001)-oriented GP I zone, the dislocation interaction mechanism is different and requires a more detailed attention. Figure 5 sums up the observations made in this case. An interesting observation made in case of (001)-oriented GP I zone is that the dislocation does not shear the GP I zone but deposits itself along the sides of the GP zone. In addition, the propagation rates of two dislocation segments located at the opposite side of the GP zone are different. The dislocation propagates faster on the side where the GP zone encloses a larger angle with the upper halfsystem. This effect is visible in all simulation snapshots embedded in Fig. 5. As the dislocation segments propagating at different rates can not be separated, the slower dislocation segment catches up upon continuous shearing. For both shearing directions, this is observed between 0.7 and 0.9% shear strain. Within this shear strain interval, the shear stress level remains constant at around 45 MPa.
The depinning process itself is also different compared to the case of (010)- and (100)-oriented GP I zones. In case of the (001)-oriented GP zone, individual dislocation par-tials detach sequentially from the GP I zone. Individual depinnings are accompanied by single peaks in the shear stress-strain relation at around 60 MPa. These are visible just before 1% shear strain in Figs. 5, a and 5, b. Overall, the pinning strength of a (001)-oriented GP I zone is significantly lower than that for the other two orientations. In addition, it is independent of the shearing direction.
8. The GP zone form upon dislocation depinning
For spherical precipitates, two outcomes are possible upon dislocation depinning. These depend on the mechanical stability of the precipitate, generally given by the elastic modulus E. In case where the elastic modulus of the precipitate is smaller than that of the matrix, e.g. voids [30] or copper precipitates in Fe matrix [31, 32], the precipitate is cut by the dislocation. The upper part of the precipitate is transferred in the direction opposite of the dislocation propagation by the amount of the Burgers vector. If the elastic modulus of the precipitate is larger than that of the surrounding matrix the dislocation is not able to cut the precipitate. It rather circumvents the precipitate, leaving a dislocation ring around it upon depinning. Experimentally, the elastic modulus of copper is twice as large as of aluminum (140 MPa versus 76 MPa). Both values are also represented excellently by the EAM potential [21] used in this work (141 MPa versus 77 MPa). However, the approximation of a GP zone as a sphere is insufficient (only in 2D). Hence, the GP zone orientation is expected to have a significant influence on the depinning mechanism activated.
In case of (010)- and (100)-oriented GP 1 zones, there exists only one monoatomic Cu layer visible from the perspective of an approaching dislocation line. Figure 6 depicts side views onto a (100)-oriented GP I zone (disc 1 in Fig. 1) chosen as being representative for the two equivalent orientations. While the top row of Fig. 6 depicts the GP zone as detected by the CNA analysis, containing also strained atoms of the surrounding Al matrix, the GP zone in the bottom row is represented only by corresponding copper atoms. Both rows of the left column (a, d) depict the initial form of the GP zone, prior to any dislocation interaction. Visible is a straight line, consisting of a row of copper atoms (d) and two straight rows of aluminum atoms, each of them located on the opposite sides of the GP zone (a).
In Fig. 6, the middle column (b, e) presents the form of the (100)-oriented GP zone after dislocation interaction in the case of negative dislocation propagation direction. The dislocation plane intersects the GP zone at its center, as indicated in the figure. Figures 6, b and 6, e indicate that the upper part of the GP zone is translated by the amount of a Burgers vector in the positive x direction (to the right, opposite of the dislocation propagation direction). A de-
Fig. 6. The (100)-oriented GP I zone before the interaction with the dislocation (a, d), after dislocation depinning in the case of negative shearing (b, e) and positive shearing case (c, f). In the upper row the CNA analysis is presented, atoms differing from a perfect fcc structure are represented in a-c. In general, these atoms belong to aluminium boundary layers on both faces of the GP I zone. All figures indicate a perfect cut of the GP zone done by the dislocation upon depinning. The cutting is visible both via the CNA analysis and also trough the visualisation of copper atoms only. No defect structures could be detected in the vicinity of the GP zone upon dislocation depinning
tailed analysis shows that the translation of the upper part of the GP zone (as detected by CNA) is due to the shearing of the copper atoms of the GP zone itself. Figures of the right column (c, f ) support this statement, visualizing the same effect in the case of positive dislocation propagation direction.
In case of the (001)-oriented GP zone (disc 3 in Fig. 1), the dislocation line does not shear the GP zone, it rather deposits itself along its sides. As Fig. 5 indicates, the propagation of both dislocation segments, located on the opposite sides of the GP zone, is different. In addition, single parts of the dissociated dislocation detach from the GP-zone individually, two peaks in the corresponding shear stress-strain relation can be observed.
Figure 7 presents the (001)-oriented GP zone before (left column) and after (right column) dislocation depinning. The same presentation form is used as in Fig. 6: atoms in the upper row (a, b) represent structural defects detected by CNA analysis, while yellow color is used for the representation of the copper atoms (c, d) of the (001)-oriented GP zone itself. This is in agreement with the color coding shown in Fig. 1.
Upon dislocation depinning, a clean cut at the midsection of the (001)-oriented GP I zone is indicated by the CNA analysis, as presented in Fig. 7, b. Visible are two semicircles, separated by the dislocation plane and translated to each other by the amount of the Burgers vector. However, the exact analysis of copper atoms of the GP zone does not support the initial presumption. Rather than being
cut by the dislocation line, the copper disc of the GP zone is strained and the GP zone surface is homogeneously deformed, as visible from Fig. 7, d. This effect can be traced back to the higher elastic modulus of copper and the GP zone orientation relatively to the dislocation propagation, forcing it to interact with the GP zone over the longest path.
It is important to identify the dislocation depining mechanism, since neither the cutting of the GP zone is observed, nor the dislocation looping mechanism around it. A more detailed analysis of the GP zone structure upon dislocation depinning is given in Fig. 8, where the atoms are colored due to the displacement in the x-direction with reference to the initial structure. The displacement is calculated for all structural defects, as detected by the CNA analysis and depicted in Fig. 7. Hence, effectively, Fig. 8, a presents the displacement of aluminium layers neighboring the (001)-oriented GP I zone. As indicated by Fig. 8, a the atoms located below the dislocation plane experience no displacement, while a displacement of the atoms above the dislocation plane is of the order of the Burges vector of the dislocation (~0.3 nm).
The displacement of the copper atoms of the GP zone itself is depicted in Fig. 8, b. Evident is that only a smaller part of copper atoms on the upper right side is displaced by the same amount as the surrounding aluminium atoms. The remaining copper atoms are only slightly displaced (region 1), or not at all (region 2). This is consistent to observations presented in Fig. 7, d. The inhomogeneous displacement of the (001)-oriented GP zone originates from the higher elastic modulus of copper. In order to investigate the deformation mechanism of the GP zone upon subsequent shearing, the simulation time was doubled, enabeling the dislocation line to pass the GP zone a second time. The resulting GP zone structure is presented in Fig. 8, c. As in Fig. 8, b, the displacement in the direction x is taken for the
Fig. 7. The (001)-oriented GP I zone before the interaction with the dislocation (a, c) and after dislocation depinning (b, d). In the upper row the CNA analysis is presented, detecting structural defects presented in the top row (a, b), while copper atoms of the GP zone are presented in the bottom row (c, d)
0.3 nm
0 nm
0.3 nm
0 nm
0.6 nm
0 nm
Fig. 8. Displacement magnitude of the structural defect in the Al matrix (a) and of the (001)-oriented GP zone (b, c). Upon the first interaction, the dislocation does not cut, but circumvents the GP zone (b) and shears the surrounding Al layers (a). Upon further shearing, a step in the GP zone structure was created and the upper GP zone part was uniformly displaced. The cutting effect is also observed for further dislocation-GP zone interactions
color coding. As evident from Fig. 8, c, the dislocation is able to create a step in the GP zone upon a further pass. The majority of the copper atoms above the dislocation plane show a constant displacement. In addition, the magnitude of the displacement is 0.6 nm, equaling two Burgers vectors. Hence, the dislocation was able to catch up for the lacking displacement from the first interaction.
9. Effect of the GP I zone size and dislocation plane offset
Essential information regarding the pinning strength of the GP I zone is its dependency on the GP zone size and dislocation plane offset. Most investigations dealing with precipitate strengthening take into account only the dislocation-precipitate interaction along the precipitate midsec-tion. This specific case is, however, of little statistical importance as the dislocation plane can intersect the precipitate at any height. In order to take both the GP zone size as also the dislocation plane offset into account, a (010)-ori-ented GP I zone of 6.5 nm diameter is taken as a reference. The GP I zone used in this case is equivalent (half the size) to disc 2 in Fig. 1 respectively Fig. 3, b.
At first, the GP I zone is placed so that the dislocation plane intersects its midsection, in further text denoted as 0 offset. Further cases are investigated in which the GP I zone is continuously translated along the z-axis so that the dislocation plane intersects the GP I zone at different heights. The designated offsets are therefore 0, ± 1/3, ± 2/3 of ± 3/3 of the GP zone height. In case of the ± 3/3 offset, additional differentiation is made: the GP zone is either cut or scratched by the dislocation. This is achieved by fine tuning the position of the GP zone, shifting it by one additional atomic layer in the z-direction. The resulting shear stress state is depicted in Fig. 9.
Figure 10 sums up the observations made during the described parametric study. The pinning strength of a centered GP I zone, measured from the lowest point of shear stress-strain relation, of 6.5 nm diameter is identified as 175 MPa. A comparison with a GP zone of same orientation but a two-fold size (depinning stress 250 MPa as presented in Fig. 4) indicates a square root dependency of the depinn-ing stress as the function of the GP zone size. These obser-
vations are in agreement with a more detailed study dealing with the influence of the GP zone size [17].
Further on, the influence of the dislocation plane offset is discussed. Copper atoms forming a GP zone occupy aluminium lattice sites, hence, there are two competing factors governing the pinning strength of a GP zone. The first one is the moduli mismatch between aluminium and copper. Due to this mismatch, a certain amount of shear stress, in addition to the Peierls stress tp, is necessary for a dislocation to overcome the GP zone. The pinning strength is directly influenced by the moduli mismatch and the width of an obstacle. This influence is visible through the dependency
£
m
150 100 50 0
-50
1 — center 1
2 — 1/3
3 — 2/3 /2
4 — 3/3 cut
5 — 3/3 scratch 3
f 5 4
I
0
0.0
0.5
1.0 Strain, %
1.5
2.0
1.0 1.5 Strain, %
Fig. 9. Development of the global shear stress state during the interaction between the (010)-oriented GP I zone and the dislocation line. The global shear stress state of the system is presented as line 5 for the negative (a) and positive shearing direction (b). Notice the asymmetry of the depinning stresses for different shearing directions. The structural analysis is presented using the COORD and CNA analysis of different simulation snapshots
Attraction
Dislocation plane offset Repulsion r\
' !
- Negative pressure + Positive pressure
Fig. 10. The depinning stress of the (010)-oriented GP I zone is shown as a function of the dislocation plane offset. The sketch at the bottom left represents the dislocation-GP zone configuration in case of a negative offset, while positive offset is presented by the sketch on the right. The corresponding depinning stresses are depicted in the diagram above for the negative (1) and positive offset (2)
of the depinning stress on the GP zone size, as presented above.
In addition to the shear stress, the hydrostatic stress has a smaller influence on the pinning strength of a GP zone. As shown in the two bottom sketches of Fig. 10, the dislocation can be represented as a hydrostatic stress dipole due to differently populated half-systems on the opposite sides of the dislocation plane. Therefore, individual stress poles of the dislocation interact with stresses in the vicinity of the GP zone.
Radial distribution analysis of the system indicates that the Cu-Cu bond is stretched to the length of an Al-Al bond (~0.29 nm) of the surrounding matrix. The radial distribution analysis also reveals that Al layers neighboring the GP zone are also strained, compensating for smaller Cu-atoms. The increased bond lengths in the vicinity of the GP zone result in negative hydrostatic stresses. These are represented as elipsoids 1 in the two sketches in Fig. 10. The influence of hydrostatic stress onto the pinning strength of a (010)-oriented GP I zone is represented in the diagram of Fig. 10. In addition, the two sketches below the diagram present the location of the stress dipole of the dislocation and the position of the GP zone (low hydrostatic stress monopole) regarding the dislocation plane.
On the left hand side of the diagram (negative plane offset) the depinning stress is represented for incremental translation of the GP I zone towards the bottom of the simulation cell and below the dislocation plane. As the GP I zone is translated downward, the positive stress pole of the dislocation interacts with an increasing amount of the negative stress field of the GP zone. This results in an attractive force between the dislocation and the GP zone. This attraction is observed as an increased depinning stress. In the
opposite case a larger part of the GP zone is located above the dislocation plane. There exists a repulsive force between the negative stress pole of the dislocation and that of the GP zone, hence, the dislocation is pushed away from the GP zone prior to depinning. Additionally, due to the lattice mismatch, a slightly increased stress field is observed for a positive dislocation plane offset compared to a negative one, which results in an increased pinning strength of the GP zone for positive dislocation plane offsets.
The two data points of Fig. 10 located at the far left and right are calculated by the dislocation only scratching the GP zone. Although the dislocation line does not intersect any Cu atoms, pinning ability can be observed only due to the interaction of the respective stress fields. In summary, the GP zone offset results into asymmetric depining stresses with respect to the dislocation plane offset. The amount of the asymmetry is proportional to the lattice mismatch between the martix and precipitate material.
10. Formation of GP II zones and their interaction with line dislocations
Upon continuous ageing of Al-Cu alloys, further copper discs are formed in the vicinity of the existing ones. This results in GP II zones, a formation of several GP I zones parallel to each other. Individual copper discs forming a GP II zone do not occupy neighboring {001} crystallogra-phic planes of the fcc structure but are separated by three atomic layers of aluminium [3]. Hence, the distance between individual copper discs of a single GP II zone equals two aluminium lattice parameters. Taking the crystal orientation presented in Fig. 1 into account, two outcomes for a GP II zone/dislocation interaction are possible, which are visualized in Fig. 11. In case of (100)- or (010)-oriented GP II zones, the dislocation interacts sequentially with individual copper discs located behind each other. This case is presented in Fig. 11, a in a sideview and in Fig. 11, c with a viewpoint located in the dislocation plane. In case of a (001)-oriented GP II zone, the dislocation line interacts simultaneously with all copper discs. This is presented in Figs. 11, b and 11, d. Hence, the two interaction mechanisms are distinctly different.
The objective of this section is to investigate how the continuation of the precipitation process from individual GP I zones towards the GP II zones influences the mechanical properties of the material. In particular, the objective is to determine the pinning strength of a GP II zone in dependence on the number of copper discs forming it.
Figure 12 depicts observations made in case of (010)-and (001)-oriented GP II zones of 6.5 nm diameter during the described investigations. Figure 12 presents a parametric study in which the number of copper discs (1, 2 and 3) is taken as single parameter. The lines 1 in both diagrams present the shear stress-strain relation during the dislocation interaction with a (010)-oriented GP I zone in Fig. 12, a and a (001)-oriented GP I zone in Fig. 12, b.
Fig. 11. Two possible configurations of a line dislocation and a GP II zone depending on the GP zone orientation. For visualisation purposes, the same color coding as in Fig. 1 is used. The dislocation propagation direction is indicated by black arrows. On the left hand side in (a) a (010)-oriented GP II zone consisting of three Cu discs is presented. Figure 11, c presents the perspective from the dislocation plane. Only the first Cu disc is visible by the dislocation line. The interaction with remaining Cu discs occurs sequentially. In contrast to this, Figs. 11, a, b present a configuration in which all three Cu discs of the (001)-oriented GP II zone are oriented perpendiculary to the dislocation propagation direction. Hence, the dislocation line interacts with a partially permeable barrier of 4 lattice constants in width and with all Cu discs at the same time. The resulting shear stress-strain relations are presented in Fig. 12
In case of a (010)-oriented GP I zone, the shear stressstrain relation is already familiar from Fig. 4, b. The calculated critical resolved shear stress (CRSS) is 170 MPa. Further thermal precipitation and the formation of a second Cu disk increases the pinning strength for ~15-20 % up to 200 MPa as evident from line 2 in Fig. 12, a. The formation of a third Cu disc as a part of the GP II zone does not furher increase the pinning strength further. The corresponding line 3 in Fig. 12, a is sligthly lower, the depinning occurs at
~195 MPa. The relatively small amount of pinning strength contributed by further (010)-oriented Cu discs is a consequence of their orientation relatively to the dislocation propagation direction. Instead of pinning the dislocation line over a larger segment length, individual Cu discs of the (010)-oriented GP II zone act sequentially.
A different obervation is made in case of the (010)-orientation. There is a constant increase of the pinning strength of the GP II as a function of the number of Cu
200-
td P-
150-
100-
£
m
50
-50-
1 — GP I, 1 disc 2 — GP II, 2 discs " 3 — GP II, 3 discs /3/1 lb 3
i i i i
Strain, %
Fig. 12. Critical resolved shear stresses as a function of the number of Cu discs. (a) CRSS of (010)-oriented GP zones (GP I (1) and GP II (2, 5)) are presented, corresponding to configurations in Figs. 11, a, c. In (b) the results of the same parametric study are presented, however, for the (001)-orientation, as presented in Figs. 11, b, d
Ph
2 1508
h* 100-
c«
c«
$ 50-
yi
j! 0-
m
-500.0 0.5 1.0 1.5 2.0 Strain, %
Fig. 13. Critical resolved shear stress in case of (010)-oriented GP zones as a function of the distance between the Cu discs. The shear stress-strain relation originating from a single, (010)-ori-ented GP II zone is presented (curve 4). One single peak, associated with the depinning of the dislocation from the two disc system, is observed. The number of Cu discs remaines constant, however, their distance is increased to ~3 (2), 5 (1) and 8 times (3) of the initial value
discs. This effect was expected as the dislocation is pinned over a larger length. However, the modulus of depinning stress (CRSS) still does not exceed the one of the (010)-oriented GP I zone of the same size.
An interesting observation can be made in the case of (001)-oriented GP-zones. In Fig. 5 it was reported that the dislocation line is separated into two segments, located at opposite sides of the (001)-oriented GP zone. The segments were reported to propagate at different velocities. Prior to the dislocation depinning, the slower dislocation segment "catches up" to the faster one. The process during which the two segments reunify is accompanied by a constant shear stress level at ~45 MPa. In all three cases presented in Fig. 12, b a sudden decrease of the shear stress state at around 45 MPa can be noticed. The origin of the stress decrease is also the reunification of the two dislocation segments located on opposite sides of the GP II zone. However, due to the smaller GP zone size (6.5 nm diameter versus 13 nm diameter in Fig. 5) the release of the shear stress in Fig. 12, b is faster, almost instant. This results in an intermediate spike in the shear stress-strain relation. Regardless of the number of Cu discs forming a GP II zone, one distinct maximum in the shear stress-strain relation is observed. Hence, the dislocation depinning from the multidisc system is a single event.
11. Influence of the distance between individual copper discs
Individual Cu discs of a GP II zone are ~0.8 nm away from each other, having three atomic layers of aluminium in between. In case of (100)- and (010)-oriented GP II zones, it could be expected that a dislocation line obtains a local equilibrium position while interacting sequentially with individual Cu discs. As evident from Fig. 12, a, there are, however, no local minima in the shear stress-strain curve. The reason for this is the stacking fault width between
individual dislocation partials, which is larger than the distance between individual Cu discs of the GP II zone. Hence, the dislocation line depins from the first Cu disc, while already being attracted by the second and the third subsequent Cu disc of the (010)-oriented GP II zone. Individual interactions overlap and the GP II zone acts as a single obstacle. The objective of this section is to investigate the minimal necessary distance between Cu discs, in order that the Cu discs are detected by the dislocation line as individual GP I zones, rather than a multidisc GP II zone. In detail, the objective is to identify a local minimum in the shear stress-strain relation while variing the distance between two Cu discs.
In order to reach this objective, shearing simulations are performed where two (010)-oriented copper discs of constant diameter 6.5 nm while varying their distance ranging from 0.8 nm (GP II zone) to 6.8 nm. Figure 13 depicts the results of this study. In the first case, represented by line 3, the distance between individual Cu discs is 0.8 nm, as in a (010)-oriented GP II zone. As described in the previous section, the interaction between the dislocation line and a GP II zone results in a single depinning event.
Lines 1-3 in Fig. 13 present the shear stress-strain relations for distances of 4.5, 2.2 and 6.8 nm between the Cu discs. Until 4.5 nm distance (approximately twice of the stacking fault width), no local minimum in the shear stress-strain relation is found. The interactions of the dislocation with individual Cu discs overlap and a single de-pinning event is observed. In contrast to this, two Cu discs with at least 4.5 nm spacing are identified by the dislcation line as individual obstacles (GP I zones). For each of them, a single depinning event is observed.
12. Conclusions
This article investigates the strengthening mechanisms of Al-Cu alloys at different stages of thermal ageing, namely solid solution, GP I and GP II zones. Molecular dynamics simulations are used as the method of choice as it enables insight into both the strength magnitude and also the strengthening mechanism.
In case of underaged Al-Cu alloys, the strenthening process is based on solid soluted copper atoms obstructing the dislocation propagation. This is due to lattice and elastic moduli mismatch between soluted copper atoms and the aluminium matrix. At the subsequent ageing state, copper atoms precipitate, forming disc shaped GP I zones. The GPI zones show, at given copper concentrations, considerably higher pinning strengths than the solid solution.
The contribution of further copper discs and the formation of GP II zones has not been previously investigated by computational methods. Observations made in the present study show a limited amount of strength increase due to countinued precipitation. The pinning strength of (001)-oriented GP II zones shows a clear linear behavior as a
1 — GP II
2 — GP II, 2.2 nm
3 — GP II, 4.5 nm t — GP II, 6.
function of the number of copper discs. Its modulus is, however, still below that of a single GP I zone of the same size but different crystallographic orientation. In case of (100)- and (010)-oriented GP II zones, an increase of the pinning strength of ~20 % is present for a two-disc system. This value presents at the same time also the upper limit, as additional copper discs show no further strengthening effect.
Given the copper concentration, the formation of a large number of GP I zones is superior in strengthening to a smaller number of GP II zones. In this way, the free path of a dislocation is small while obstacles of large pinning strength are present. Especially at cyclic loading, during which dislocations sweep the same area several times, a high concentration of GP I zones is superior to any other strengthening mechanism, due to the large depinning stress and the occurrence of depinning events.
Our results are in good agreement with similar but rare investigations available in the literature.
This work was supported by the German Research Foundation as a part of the Special Research Field 716, subproject B.7.
References
1. Binkele P. Atomistische Modellierung und Computersimulation der Ostwald-Reifung von Ausscheidungen beim Einsatz von kupferhalti-gen Stahlen: Dissertation. - Stuttgart: Universitat Stuttgart, 2007.
2. Kohlbach I. Die Rolle von Leerstellen bei der Ausscheidungsbildung von Al-Cu-Legierungen: Diplomarbeit. - Bonn: Rheinische Friedrichs-Wilhelms-Universitat Bonn, 2007.
3. Robel M. Ab-initio Untersuchungen an Aluminium, seinen Defekten und Aluminium-Legierungen: Dissertation. - Bonn: Rheinische Fried-richs-Wilhelms-Universitat Bonn, 2005.
4. Guinier A. La diffraction des rayons X aux tres petits angles; application a l'etude de phenomenes ultramicroscopiques // Ann. Phys. Paris. -1939. - No. 12. - P. 161-237.
5. Preston G.D. The diffraction of X-rays by age hardening alloy of aluminium and copper. The structure of an intermediate phase // Philos. Mag. - 1938. - V. 26. - P. 855-871.
6. Karlik M., Jouffrey B. High resolution electron microscopy study of Guinier-Preston (GP1) zones in Al-Cu based alloys // Acta Mater. -1997. - V. 45. - No. 8. - P. 3251-3263.
7. Takeda M, Nagura Y., Igarashi A., Endo T. Copper concentrations inside Guinier-Preston I zones formed in an Al-Cu alloy // Z. Me-tallkd. - 2002. - V. 93. - P. 204-207.
8. Karlik M., Jouffrey B., Belliot S. The copper content of Guinier-Preston (GP1) zones in Al - 1.84 at % Cu alloy // Acta Mater. - 1998. - No. 5. -P. 1817-1825.
9. Karlik M., Bigot A., Jouffrey B., Auger P., Belliot S. HREM, FIM and tomographic atom probe investigation of Guinier-Preston zones in an Al - 1.54 at % Cu alloy // Ultramicroscopy. - 2004. - No. 98. - P. 219230.
10. Son S.-K., Takeda M., Endo T. Microstructures and thermal stability of the metastable-phase precipitates formed in an Al-Cu alloys // Mater. Sci. Forum. - 2005. - V. 475-479. - P. 353-356.
11. KarlikM. Atomic Resolution Investigation of Guinier-Preston Zones in Al-Cu Based Alloys: Dissertation. - Prague: Czech Tech. Univ., 2010.
12. Tapasa K., Bacon D.J., Osetsky Y.N. Simulation of dislocation glide in dilute Fe-Cu alloys // Mater. Sci. Eng. A. - 2005. - V. 400-401. -P. 109-113.
13. Schmauder S., Kohler C. Atomistic simulations of solid solution strengthening of a-iron // Comp. Mater. Sci. - 2011. - V. 50. - P. 12381243.
14. Zhang X., Deng H., Xiao S., Li X., Hu W. Atomistic simulations of solid solution strengthening in Ni-based superalloy // Comp. Mater. Sci. - 2013. - V. 68. - P. 132-137.
15. Kohler C., Kizler P., Schmauder S. Atomistic simulation of the pinning of edge dislocations in Ni by Ni3Al precipitates // Mater. Sci. Eng. A. - 2005. - V. 400-401. - P. 481-484.
16. OsetskyY., Baco D. Atomic-scale modelling of primary damage and properties of radiation defects in metals // Nucl. Instrum. Meth. Phys. Res. B. - 2003. - V. 202. - P. 31-43.
17. Singh C., Warner D. Mechanism of Guinier-Preston zone hardening in the athermal limit // Acta Mater. - 2010. - V. 58. - No. 17. - P. 57975805.
18. Singh C., Mateos A., WarnerD. Atomistic simulations of dislocation-precipitate interactions emphasize importance of cross-slip // Scripta Mater. - 2011. - V. 64. - P. 398-401.
19. Singh C.V., Warner D.H. An atomistic-based hierarchical multiscale examination of age hardening in an Al-Cu alloy // Metall. Mater. Trans. A. - 2013. - V. 44. - P. 2625-2644.
20. Stadler J., Mikulla R, Trebin H.-R. IMD: A software package for molecular dynamics studies on parallel computers // Int. J. Mod. Phys. C. - 1997. - V. 8. - No. 5. - P. 1131-1140.
21. Cheng Y., Ma E., ShengH. Atomic level structure in multi-component bulk metallic glass // Phys. Rev. Lett. - 2009. - V. 102. - No. 24. -P. 245501:1-4.
22. Srinivasan S., Liao X., Baskes M., McCabe R, Zhao Y., Zhu Y Compact and dissociated dislocations in aluminum: Implications for deformation // Phys. Rev. Lett. - 2005. - V. 94. - P. 125502.
23. KuksinA.Y., Stegailov V.V., YanilkinA.V. Molecular dynamics simulation of edge dislocation dynamics in aluminium // Dokl. Phys. - 2008. -V. 53. - No. 6. - P. 287-291.
24. Ercolessi F., Adams J.B. Interatomic potentials from first-principles calculations: The force-matching method // Europhys. Lett. - 1994. -V. 26. - No. 8. - P. 583-588.
25. Liu X.-Y., Ercolessi F., Adams J. Aluminium interatomic potential from density functional theory calculations with improved stacking fault energy // Model. Simul. Mater. Sci. Eng. - 2004. - V. 12. -No. 4. - P. 665-670.
26. Mishin Y., FarkasD., MehlM.J., PapaconstantopoulosD.A. Interatomic potentials for monoatomic metals from experimental data and ab initio calculations // Phys. Rev. B. - 1999. - V. 59. - No. 5. - P. 33933407.
27. Apostol F., Mishin Y. Interatomic potential for the Al-Cu system // Phys. Rev. B. - 2011. - V. 83. - No. 5. - P. 054116:1-8.
28. Stukowski A. Structure identification methods for atomistic simulations of crystalline materials // Model. Simul. Mater. Sci. Eng. - V. 20. -No. 4. - P. 045021.
29. Kelchner C., Plimpton S., Hamilton J. Dislocation nucleation and defect structure during surface indentation // Phys. Rev. B. - 1998. -V. 58. - No. 17. - P. 11085-11088.
30. Hatano T., Matsui H. Molecular dynamics investigation of dislocation pinning by a nanovoid in copper // Phys. Rev. B. - 2005. - No. 72. -P. 094105.
31. Molnar D., Binkele P., Hocker S., Schmauder S. Atomistic multiscale simulations on the anisotropic tensile behaviour of copper-alloyed airon at different states of thermal ageing // Philos. Mag. - 2012. -V. 92. - No. 5. - P. 586-607.
32. Molnar D., Binkele P., Hocker S., Schmauder S. Multiscale Modelling of Nano Tensile Tests for Different Cu-Precipitaion States in a-Fe // Proc. V Int. Conf. Multiscale Materials Modelling / Ed. by P. Gumbsch, E. van der Giessen. - Freiburg: Fraunhofer Inst. Mech. Mater., 2010. -P. 235-239.
Поступила в редакцию 23.01.2017 г.
Сведения об авторах
Wolfgang Verestek, Dipl.-Ing., Scientist, University of Stuttgart, Germany, [email protected]
Alen-Pilip Prskalo, Dr.-Ing., Scientist, University of Stuttgart, Germany, [email protected]
Martin Hummel, Dipl.-Phys., Scientist, University of Stuttgart, Germany, [email protected]
Peter Binkele, Dr.-Ing., Scientist, University of Stuttgart, Germany, [email protected]
Siegfried Schmauder, Prof. Dr. rer. nat., Prof., University of Stuttgart, Germany, [email protected]
