BROWN UNIVERSITY Solute strengthening from first-principles and applications to Al and Mg alloys by Gerard Paul Marcelo Leyson A dissertation submitted in partial fulfillment for the degree of Doctor of Philosophy in Solid Mechanics School of Engineering May 2013 c Copyright 2012 by Gerard Paul M. Leyson ii This dissertation by Gerard Paul M. Leyson is accepted in its present form by the School of Engineering as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date William A. Curtin, Director Recommended to the Graduate Council Date Allan F. Bower, Reader Date Pradeep Guduru, Reader Date Sharvan K. Kumar, Reader Approved by the Graduate Council Date Peter M. Weber, Dean of the Graduate School Vitæ Gerard Paul Marcelo Leyson was born on March 1, 1981 in the Marikina, Philippines. In 2001, he was awarded the Singapore Ministry of Foreign Affairs ASEAN1 scholar- ship and took an undergraduate degree in the National University of Singapore (NUS) in Materials Science. In that program, he finished with First Class Honours in 2005 and was awarded the Materials Research Society (Singapore) and the Lijen Industrial Development medals. From there, he pursued a Masters Degree in the same university under supervision of Prof. Chiu Cheng Hsin and graduated in 2007. He is currently pursuing a doctoral degree in Solid Mechanics at Brown University under the guidance ˆ of Prof. William A. Curtin. In September 2011, he came to Ecole Polytechnique F´ed´eral de Lausanne (EPFL) in Switzerland as a visiting doctural student in the Institute of Mechanical Engineering. 1 Association of South East Asian Nations iii “Every honest researcher I know admits he’s just a professional amateur. He’s doing whatever he’s doing for the first time. That makes him an amateur. He has sense enough to know that he’s going to have a lot of trouble, so that makes him a professional.” Charles Franklin Kettering (1876-1958) U. S. engineer and inventor. Abstract of “Solute strengthening from first-principles and applications to Al and Mg alloys” by Gerard Paul Leyson, Ph.D, Brown University, May 2013 In this dissertation, a quantitative, parameter free model to predict the flow stress as a function of temperature and strain rate of such alloys is presented. The model builds on analytic concepts developed by Labusch but introduces key innovations rectifying shortcomings of previous models. To accurately describe the solute/dislocation interaction energies in and around the dislocation core, density functional theory and a flexible-boundary-condition method are used. The model then predicts the zero temperature flow stress, the energy barrier for dislocation motion, and thus the finite temperature flow stresses. The model is first used to predict the flow stresses of various Al alloys. Excellent results are obtained for Al-Mg and Al-Mn. Al-Fe with ppm levels of Fe is not predicted well but, using experimental results for Fe, results for the quasi-binary Al-Cr-(Fe) and Al-Cu-(Fe) alloys agree well with experiments. The model is also consistent with the “stress equivalency” postulate of Basinski. The model is then applied to Mg-Al alloys undergoing basal slip. Due to the wide partial separation of the Mg basal edge dislocation, a smaller roughening is required to decorrelate the solute fluctuations in the partials compared to that required to decorrelate the fluctuations in the “far field”. As a consequence, the dislocation has two stable configurations. When these two configurations are taken into account, the model predictions are in good agreement with experiments over a wide range of solute concentrations and temperature. The model also explains the origins of the “plateau stress” observed in ex- periments. Finally, the model in conjunction with the standard Friedel strengthening model is used to study the transition between weak-pinning and strong-pinning at finite temperatures. The transition concentration is found to be a strong function of the dislocation core structure and can be significantly different from the zero-temperature transition concentration. It is also found that, for the case of Al-X and basal Mg-Al alloys, the weak-pinning mechanism is always dominant at temperatures and concentrations relevant to engineering applications. Acknowledgements First and foremost, I would like to express my heartfelt thanks to Prof. William A. Curtin. I am lucky that I have worked with a supervisor that is concerned both with the quality of the research as well as the personal growth of his students. His guidance over the past few years have been invaluable and I will always be grateful that I have had this opportunity to work with him. I would also like to thank General Motors for supporting my research, with special mention to Dr. Lou Hector Jr. At the dawn of my PhD studies, Dr. Hector took me under his wings and showed me the wonders of first-principles calculations. I would like to thank everyone who has helped me get to where I am today. One should not forget his roots lest he lost himself in the future. A special thanks to my parents, who have been a constant source of support and encouragement. I would also like to mention Prof. Augusto Hemorsilla, who is one of the major reasons I pursued a career in the Academia. It has been a long time, but I have not forgotten all you have taught me. Finally, I would thank my friends, near and far. I apologize for the general statement, but unfortunately, you are too numerous list here. That said, I’m sure you know who you are. Thank you for keeping me sane all these years. vi Contents Signature Page ii Vitæ iii Abstract v Acknowledgements vi List of Figures ix List of Tables xii 1 Introduction 1 1.1 Strong-pinning mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Weak-pinning mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Transition between strong- and weak-pinning regimes . . . . . . . . . . . . 7 1.4 Yield strength at finite temperature . . . . . . . . . . . . . . . . . . . . . 8 1.5 Limitations of existing models . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Numerical simulations of solute-strengthening . . . . . . . . . . . . . . . . 12 1.7 Overview of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Solute strengthening in Al alloys from first-principles 15 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Solute Strengthening Model . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3 Solute strengthening in Mg-Al alloys from first-principles 50 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 Solute Strengthening Model . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 Solute/dislocation Interaction Energies . . . . . . . . . . . . . . . . . . . . 56 3.4 Predictions and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 vii Contents viii 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4 Strong-/weak-pinning transition at finite temperature 68 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Finite-Temperature Friedel and Labusch-type Models . . . . . . . . . . . 72 4.3 Effect of core structure in the Friedel-Labusch transition . . . . . . . . . . 77 4.4 Finite temperature transition concentration using DFT energies . . . . . . 84 4.5 Discussion and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5 Summary and Conclusions 93 Bibliography 96 List of Figures 1.1 A schematic representation of (a) the Friedel strengthening mechanism, wherein solutes are treated as point pinning obstacles, and (b) the Labusch strengthening mechanism, wherein favorable fluctuations in solute concen- tration trap the dislocation segments. . . . . . . . . . . . . . . . . . . . . 4 2.1 Nye tensor for the (a) edge and (b) screw components of the dislocation core computed using DFT and the Lattice Green’s Function method. The x-axis is in units of Burgers vector. Peaks in the Nye tensor roughly correspond to the locations of the partial dislocations in the dissociated core. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Pressure versus concentration in fixed-volume supercells containing so- lutes atoms, for Mg, Si, Cu, Cr, Mn and Fe solutes, for both 27- and 64-atom supercells. Dashed lines are linear fits using data from both su- percells for each solute, and the slopes of these lines are used in Eq. (2.4) to compute the misfit volume. . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Solute/dislocation interaction energy Uij versus solute position position (xi , yj ) for Mg, Cr, Si, Cu, Mn and Fe solutes. Misfit volume ∆vm is shown for each solute. Negative energies indicate binding. Bold circles show positions at which first-principles interaction energies are calculated; “x” markers denote positions of the b/11 Burgers vector used to compute the analytic pressure field outside the immediate core sites (non-bold circles). 27 2.4 Schematic of a dislocation roughening into segments of length 2ζ with roughening amplitude w due to favorable solute fluctuations. . . . . . . . 28 2.5 Energy correlation function χ (w, yj ) for a solute initially at (0, yj ) as the dislocation moves a distance w, as computed for an Al-Mg alloy. . . . . . 32 2.6 Normalized total energy per unit length ∆Etot /Lc2/3 versus roughening amplitude w for an Al-Mg alloy, for various partial sums over solute sites yj demonstrating the dominance of the core region y−1 , y1 and conver- gence of the total energy with increasing range of |yj |. . . . . . . . . . . . 34 2.7 (a) Energy versus position of a straight dislocation segment of length ζ = 18.06 nm moving through a random 5 at.% Mg solute field; (b) Stress required to move the dislocation across the energy landscape. . . . 37 2.8 Experimental [1–3] versus predicted flow stress σy for 9 distinct alloys, showing overall good agreement. The dashed line has a slope of unity and indicates perfect agreement. . . . . . . . . . . . . . . . . . . . . . . . 44 ix List of Figures x 2.9 Experimental [3] versus predicted flow stress σy for 3 Al-Mn alloys at 78K, 198K and 263K, showing overall good agreement. The dashed line has a slope of unity and indicates perfect agreement. . . . . . . . . . . . 45 2.10 Normalized activation volume M V /b3 versus 78K shear yield stress τy for both experimental [1] and predicted results, in log-log form, demonstrat- ing stress equivalency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 Schematic of a dislocation roughening into segments of length 2ζ over length w due to favorable solute fluctuations. . . . . . . . . . . . . . . . . 53 3.2 (a) Solute/dislocation interaction energy map for Al solutes as a function of solute position. Circles with bold outline are from Ref. [4]; cicles with non-bold outline are from Eq. (3.12); cicles with no outline are interpo- lated using cubic splines. (b) A contour plot of the correlation function χ (w, yj ) for a solute initially (0, yj ) as the dislocation moves a distance w away, as computed for an Mg-Al alloy. . . . . . . . . . . . . . . . . . . 58 3.3 Normalized energy per unit length ∆Etot /Lc2/3 versus roughening ampli- tude w for an Mg-Al alloy. . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4 Comparison between the stress dependent energy barrier ∆E 0 (τ ) deter- mined using the direct stochastic simulation and that expected by the an- alytic model for a dislocation segment of length ζ = 16.6 nm and ζ = 56.4 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.5 (a) Experimental [5] and predicted finite temperature yield stress τy (T ) for three different Mg-Al alloys. The solid lines are τy,eff , the dashed lines are τy1 and the dashed-dot lines are τy2 . (b) Experimental versus predicted yield stress τy for the three alloys presented above. The dashed line has a slope of unity and indicates perfect agreement. . . . . . . . . . 65 4.1 (a) Burgers vector distribution ∆b (xk ) using Eqn. (4.17) for d = 10b and σ = 2b. (b) Solute dislocation interaction energy U as a function of solute position, for Mg solutes around an edge dislocation in Al; dashed line: stacking fault between two partial dislocations. (c) Solute dislocation interaction energy and resistive force f for solutes just above the glide plane (solutes highlighted in the gray bar). . . . . . . . . . . . . . . . . . 79 4.2 The Friedel-Labusch transition as a function of concentration and normal- ized temperature T /T◦F for various core structures. The interaction ener- gies are calculated using Eqs. (4.17)–(4.19), with d = 0–20b and σ = b–3b. The dashed lines indicate the transition between the two Labusch config- urations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3 The partial Burgers vector ∆b (xk ) derived from dislocation core struc- tures obtained from EAM calculations [6] for (a) basal and (b) pris- matic dislocations in Mg. Also shown are the solute/dislocation interac- tion energies U (xi , yj ) for (c) basal and (d) prismatic dislocations using Eqs. (4.18)–(4.19) and the ∆b (xk ) shown in (a) and (b). . . . . . . . . . 84 List of Figures xi 4.4 (a) The Friedel-Labusch transition as a function of normalized tempera- ture T /T◦F for Mg-Al alloys undergoing prismatic and basal slip using the continuum approximation (Eqn. (4.19)) in conjunction with the ∆b (xk ) derived from EAM dislocation structures. (b) The Friedel-Labusch transi- tion as a function of normalized temperature T /T◦F for Mg-Al undergoing basal slip using DFT-derived interaction energies for the core sites. The dashed line indicate the transition between the two Labusch configura- tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.5 (a) A contour plot of the yield stress τy for Mg solutes in Al as a function of concentration and actual/normalized temperature. The Friedel-Labusch transition is shown as a black line. (b) The Friedel-Labusch transition as a function of concentration and normalized temperature T /T◦F for Mg, Si, Cu, Cr, Mn and Fe solutes in Al. . . . . . . . . . . . . . . . . . . . . . 87 4.6 The normalized activation volume V /b3 as a function of finite temperature yield stress τy for Mg, Si, Cu, Cr, Mn and Fe at T = 78 K. Note that the Friedel model predicts that V = 0 for Mg, Si and Cu solutes. . . . . . . . 91 List of Tables 2.1 Pseudopotential valence configurations (USPP = ultrasoft pseudopoten- tial) and plane wave cutoff energies (PWCO) for atoms considered in this work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 The misfit volume ∆vm for Mg, Cr, Si, Cu, Mn and Fe solutes in Al. Also shown are the predicted T = 0 yield stress τy0 and energy barrier ∆Eb for each solute. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Predicted and experimental [1, 2] tensile yield stresses at T = 78K for various Al-X alloys. Solute concentrations are from Diak et al. [1, 2]. . . 43 2.4 Predicted and experimental [1, 2] tensile yield stresses at T = 78K for Al- Cr and Al-Cu alloys containing additional Fe solutes, using experimentally- derived parameters for Fe. . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1 Parameters for the two dislocation configurations in Mg-Al alloys, nor- malized by the appropriate scaling of the solute concentration. . . . . . . 61 4.1 Friedel and Labusch force, energy, length, and/or temperature parame- ters, as computed for Mg, Si, Cu, Cr, Mn and Fe solutes in Al. Also shown are the zero-temperature transition concentrations c◦ and the geometrical lower limit cmin for the Labusch model, for each alloy. . . . . . . . . . . . 86 xii Chapter 1 Introduction Five thousand years ago, Sn and Cu were mixed to form Bronze, ushering in a new era for mankind. Since that time, alchemists and scientists have developed many more materials through alloying of base metals. The results of their efforts surround us to- day in the form of cars, computer chips, airplanes, jet engines, and hip implants, that use advanced engineered materials ranging from lightweight Magnesium alloys to high- strength, high-temperature Ni-based superalloys. The evolution in material design has been achieved through a combination of experiment, physical insight based on mech- anistic ideas, and modeling of those mechanisms to verify basic concepts and scaling. In recent years, computational materials science has emerged as a powerful supplement to experiments, providing guidelines for materials development and supporting trends found experimentally. However, quantitative parameter-free computational prediction of material properties has yet to be achieved for properties as basic as the flow or yield stress in metals. 1 Chapter 1. Introduction 2 In this dissertation, we present a new approach for quantitative prediction of metal yield stress in solute-strengthened materials, with application to Aluminum (Al) and Magne- sium (Mg) alloys. Solute-strengthening arises in Al 2XXX, 3XXXX, 5XXX and 6XXX alloys, which represent important classes of lightweight alloys where solute strengthening is crucial and for which an understanding of strengthening via multiple types of solutes is valuable for green engineering/recycling. Magnesium alloys, with excellent material properties such as low density and high specific strength, have been enjoying increased interest in the recent years for similar reasons. However, wide-spread adoption of Mg alloys is held back by limitations in formability in these alloys. It is therefore impor- tant to develop theoretical and computational tools to gain insight in the strengthening mechanisms in these alloys and guide their development. Although applied specifically to Al and Mg alloys, the framework developed here is quite general and is applicable to other fcc metals, other solute atoms, and also various other systems such as basal slip in hcp alloys such as Ti and Zn. It has been long known that alloys containing substitutional solutes exhibit strengths higher than that of pure metals. Solutes interact with the dislocations and pin them in place so that a higher stress must be applied in order to overcome the solute pinning. The underlying physics is associated with the fact that, due to lattice mismatch, a substitutional solute acts as an eigenstrain that interacts with the stress field created by the dislocation. The interaction energy between a dislocation at ~r = 0 and a solute at ~r is given by Z     U (~r) = σij r~0 sij r~0 − ~r dr~0 (1.1) Ω where σij (r~0 ) is the stress field at r~0 due to the dislocation and sij (r~0 − ~r) is the strain Chapter 1. Introduction 3 field at r~0 due to the solute. If the solute is approximated as a point dilitational source, then sij (r~0 − ~r) = ∆vm δ(~r − r~0 ) and the solute-dislocation energy can be approximated by U (~r) ≈ −p (~r) ∆vm (1.2) where ∆vm is the misfit volume of the solute and p = −σkk /3 is the pressure field exerted by the dislocation at the location of the solute [7]. In other words, the energy is equal to the work required to insert a dilatational source of strength ∆vm against the dislocation pressure field. At finite concentrations, the solutes can modify the elastic modulus of the material and generate a second-order contribution to the energy [7]. Since the dislocation pressure field decreases as 1/r, there is a gradient in the interaction energy that ultimately leads to dislocation pinning phenomena. Solute strengthening is typically divided into two classes: strong-pinning and weak- pinning. These two classes gives rise to different scaling of the strengthening with concentration as well as different energy and length scales. 1.1 Strong-pinning mechanism In the case of strong-pinning, individual solutes pin the dislocation at the solute posi- tion. Applying a stress τ causes the dislocation to bow out between these pinning points with an angle θ, exerting a force f~ = −∇~r U (~r) on the pinned portion of the dislocation (schematically shown in Figure 1.1(a). Friedel pioneered the solution to this problem [8] Chapter 1. Introduction 4 Γ Γ θ ζF f (a) w ζL (b) Figure 1.1: A schematic representation of (a) the Friedel strengthening mechanism, wherein solutes are treated as point pinning obstacles, and (b) the Labusch strength- ening mechanism, wherein favorable fluctuations in solute concentration trap the dis- location segments. and a short summary of his results are presented here. Once force equilibrium is estab- lished, the line tension Γ balances out f = |f~|, f = 2Γ sin θ. (1.3) For small concentrations, the resistive force can also be expressed as f = τ bζF (1.4) where ζF is the average distance between obstacles in the dislocation line. When τ is under but close to the flow condition, the length of the pinned segments on average follows “Friedel sampling”,  1 ζF (τ ) 2Γ 3 = (1.5) λ τ bλ Chapter 1. Introduction 5 where λ ∼ c−1/2 is the average solute spacing on the slip plane and b is the Burgers vector. F , then corresponds to the The zero temperature yield stress for the Friedel mechanism, τy0 stress required to overcome the maximum resistive force fmax . Combining Eqns. (1.3)– F, (1.5), yields an expression for τy0  3   F fmax fmax 2 2Γ 1 τy0 =  = c2 . (1.6) F bζL τy0 2Γ b2 There have been refinements to the strong-pinning model [7, 9], but the general scal- ing behavior of the yield strength with concentration is identical for all strong-pinning models. 1.2 Weak-pinning mechanism Weak pinning solutes trap the dislocation in regions where the collective configuration of the solutes are relatively energetically favorable for the dislocation. Because of random fluctuations in solute concentration, any given segment of the dislocation may be able to lower its energy by bowing out toward nearby favorable solute configurations at the cost of increased elastic energy due to the bowing, shown schematically in Figure 1.1(b). For clarity, the discussion on this section follows Zaiser’s presentation of Labusch’s anal- ysis [10]. For a more detailed treatment of the derivations, readers are directed to this reference. The total energy of the dislocation is given by Etot = Ep + Eel (1.7) Chapter 1. Introduction 6 where Ep is the energy gain due to favorable solute fluctuations and Eel is the energy cost due to bowing out. In this treatment, the solute/dislocation interaction energy is assumed to have the form of Umax f (~r), where f (~r) is a non-dimensional function with a finite range w1 . Considering a dislocation segment of length X, the segment must at least bow out a distance w in order to take advantage of favorable solute fluctuations. The change in elastic energy is then given by   Γ w  ∆Eel = X. (1.8) 2 X2 The potential energy change, on the other hand, is proportional to the fluctuation of cw2 X/b3 within the interaction volume Xw2 , p the number of solutes ∂n = p ∆Ep = −Umax ∂n = Umax w cX/b3 . (1.9) When X is small, the cost of bowing ∆Eel is greater than the energy gain ∆Ep due to fluctuations. Therefore, there is a critical lengthscale ζL (referred to as the “Labusch length”) at which bowing out becomes favorable and the dislocation is free to adjust to the energy landscape. This critical lengthscale is obtained by substituting Eqns. (1.8) and (1.9) into Eqn. (1.7), then setting Etot to zero,  13 Γ2 w2  ζL ≈ 2 c b. (1.10) 4Umax 1 R∞ Specifically, f (0) = 1 and −∞ f (~r) d~r = w3 Chapter 1. Introduction 7 The characteristic energy scale EL associated with a “Labusch length” is then given by  13 Γw4 Umax 2 c  EL = (1.11) 2b3 L follows as and the zero temperature flow stress τy0  31 c2 Umax 4  L EL τy0 = = . (1.12) bwζL wΓb9 Regardless of the mechanism, the degree of strengthening is a function of solute con- centration c, some measure of the magnitude and/or gradient of the solute-dislocation interaction energies U (~r), the dislocation line tension Γ, the solute misfit volume ∆vm , the Burgers vector b, and shear modulus µ of the matrix. These mechanisms have been characterized in classical studies [9, 11–14]. Solutes that exhibit strong pinning are said to follow “Friedel statistics” with a yield stress that exhibits a concentration dependence of c1/2 , while weak pinning solute follow “Mott statistics” and a yield stress scaling as c2/3 . 1.3 Transition between strong- and weak-pinning regimes The transition between strong and weak pinning regimes has also been studied by Labusch [13, 15]. Labusch quoted the paramater, fmax b2 β= (1.13) 4Γcw2 Chapter 1. Introduction 8 which determines determines the dominant pinning mechanism of the material. Specif- ically, if β  1 then the strong-pinning mechanism dominates and if β  1, weak pinning dominates 2 . Equation (1.13) is can be derived by taking the ratio between the zero-temperature yield stresses of the Friedel and the Labusch strengths (given by Eqns. (1.6) and (1.12) respectively), and therefore is inherently a zero-temperature quan- tity. Regardless, this parameter is widely quoted in the literature [16–19] and is routinely applied to interpret finite-temperature data. The behavior of the strong-/weak-pinning transition with temperature has not been studied in the literature. 1.4 Yield strength at finite temperature At finite temperatures, the dislocation can be thermally activated over an energy barrier it encounters at stress less than the zero-temperature yield stress. Derivation of the finite temperature yield stress τy (T ) can be obtained through thermal activation theory [20, 21]. For concreteness, the τy (T ) of a dislocation experiencing strong-pinning is derived in this section. The dislocation is assumed to have a quadratic-cubic energy landscape characterized by a maximum resistive force fmax and energy barrier Emax . A resolved shear stress τ lowers the energy barrier by τ bζF x. The energy landscape of the dislocation with stress is therefore given by     x 3  x 2 E (x) = Emax 3 −2 − τ bζF (τ ) x (1.14) d d 2 The transition parameter given in Eqn. (1.13) is essentially the same β parameter was first quoted in Ref. [13], although derived through a different approach. However, in that paper, Labusch clearly states that his theory is valid when β  1. Despite this inconsistency, subsequent referral to this β parameter takes the weak-pinning mechanism to be applicable when β  1. Chapter 1. Introduction 9 3 where d = 2 Emax /fmax . The stress dependent energy barrier ∆E (τ ) is obtained by taking the difference between the two critical points in Eqn. (1.14). Using Eqn. (1.6) 1 F = τ /τ F 3 , ∆E (τ ) can be expressed as  and the fact that ζF (τ ) /ζF τy0 y0  ! 2  32 3 τ ∆E (τ ) = Emax 1 − F  . (1.15) τy0 For a quasistatic loading, such that the system is always in a state of metastable equilib- rium, the rate of escape ν by activation over the stress-dependent barrier height ∆E (τ ) can be approximated using transition state theory [21] as   ∆E(τ ) ν = ν◦ exp − (1.16) kT where ν◦ is the attempt frequency, k is the Boltzmann constant and T is the temperature. The microscopic dislocation escape rate ν can be related to the macroscopic strain rate by the well established model [12],   ∆E (τ ) ˙ = ˙◦ exp − , (1.17) kT where ˙◦ = ρbdν◦ , ρ is the dislocation density per unit area and d is the flight distance over which the dislocation moves after each escape. Equations (1.17) and (1.15) can be inverted to obtain the finite temperature stress required for dislocation flow as "   2 # 32 F kT ˙◦ 3 τy (T ) = τy0 1− ln . (1.18) Emax ˙ Chapter 1. Introduction 10 The procedure is quite general and can be applied straightforwardly to the weak-pinning model as well as other energy landscapes. In general, the stress dependent energy barrier can be approximated by [20]  q τ p   ∆E (τ ) = ∆E◦ 1 − (1.19) τy0 where ∆E◦ is the characteristic energy barrier for the strengthening mechanism, 0 < p ≤ 1 and 1 < q ≤ 2. For strong-pinning, p = 2/3 while for weak-pinning, p = 1. The exponent q depends on the force/energy profile assumed. The corresponding finite temperature yield stress takes the general form, "   1 # p1 kT ˙◦ q τy (T ) = τy0 1 − ln . (1.20) ∆E◦ ˙ 1.5 Limitations of existing models Both strong and weak pinning models, as presented in the existing literature, have features that are ill-defined, inhibiting predictability. For strong pinning, only the solutes along the glide plane are usually considered even though the interaction energies of solutes off the glide plane can remain substantial in the vicinity of the dislocation. Applicability of the point pinning model at concentrations typical of engineering alloys (on the order of 1%) is questionable because (i) the distance between point pinning solutes, even just in the glide plane, is only a few Burgers vectors and (ii) for fcc and some hcp metals, wherein the dislocation core is split into partials that are separated by a nanometer or more, multiple solutes can be interacting with the dislocation in the Chapter 1. Introduction 11 same region. Furthermore, the total interaction energy Utot of a dislocation line of length L lying along the z-direction is the sum of all individual solute/dislocation interactions. Since solutes with the same x- and y-coordinates have the same interaction energy, Utot is given by X Utot = nij U (xi , yj ) (1.21) ij where nij is the number of solutes with position (xi , yj ) perpendicular to the dislocation line. For random alloys, and for the 1/r scaling of the energy with distance, the total energy can be divergent (although angular terms can lead to cancellation). The mean interaction energy of a dislocation in a random distribution of solutes is X X hUtot i = hnij i U (xi , yj ) = cN U (xi , yj ) (1.22) ij ij where c is the concentration of the solute and N is the number of atomic sites along the dislocation line. The mean energy hUtot i is thus independent of dislocation position as the dislocation moves from one atomic position to the next equivalent one. The finite- difference measure of the configurational glide force fc = −∂ hUtot i /∂x due to the mean effect of the solutes,     2cN X b fc = U xi − , yj − U (xi , yj ) (1.23) b 2 ij is then zero. Therefore, it is only the random f luctuations in solute concentration around the dislocation that can provide strengthening, and this is the genesis of the weak-pinning models. Segments along an initially straight dislocation can be attracted to energetically-favorable fluctuations, and bow out towards them, at the cost of dislocation Chapter 1. Introduction 12 line energy, as discussed in the pioneering work of Labusch [13, 15]. Difficulties arise, however, because the scaling of the energy fluctuations is proportional to (p∆vm )2 , thus depending on distance as r−2 , for which the summation over all solutes still diverges as ln(R) where R is the system size. To circumvent this problem, previous works have either arbitrarily set a finite interaction range or considered only solutes along the glide plane. Predictions of such models depend on the ad-hoc finite range of the interactions, which is typically taken to be on the order of b. Hence, no insight yet exists into selecting any “correct” range or its possible dependence on alloy type or other material parameters. As noted above, neglect of solutes outside the glide plane is also not justified. Finally, none of the literature models accounts for solute interactions with the actual dislocation core structure. 1.6 Numerical simulations of solute-strengthening Computer simulations based on continuum analysis [22–25] have been developed to bet- ter understand the strengthening behavior of solute strengthened alloys. In these studies, the dislocation is modeled as a continuum line defect and often only the interactions be- tween the dislocation and solutes in the glide plane are considered. While providing insight into more-detailed behavior of the dislocations, such as temperature effects and dynamics of dislocation motion, these approaches have inherent limitations comparable to those associated with the corresponding analytic models. To address some of these limitations, molecular dynamics methods have recently been used to better describe the physics occurring in the core region [26–31] while also capturing the effects of solutes off the glide plane, and permitting some assessment of energy barriers to dislocation motion Chapter 1. Introduction 13 [26]. It is difficult, however, to capture the proper mesoscale dislocation configurations on the 10-100nm scale via MD. MD studies are also limited to systems where the semi- empirical interatomic potentials for the alloy constituents are well-established. Even then, such potentials are often not designed to capture solute/dislocation interactions and can give inaccurate predictions[32]. Recent advances in computational capabili- ties have enabled the use of first-principles calculations based upon Density Functional Theory (DFT) to capture both the mechanical and chemical processes occurring in the dislocation core [4, 33–35], providing improved input parameters for use in some exist- ing models. For instance, Yasi et al. [4] used a point-pinning model in order to predict basal-slip strengthening for various solutes in magnesium alloys and the predicted zero temperature strength of magnesium alloys compared favorably with experiments. Zan- der et al. [34, 35] used first-principles calculations to determine the size and modulus misfit parameters of solutes in aluminum alloys and used this information in conjunc- tion with the Labusch (weak-pinning) model to develop a one-parameter strengthening model. 1.7 Overview of the dissertation Despite of all the excellent progress made in this field, existing models are either non- quantitative, rely on fitting with existing experimental data or make predictions at zero-temperature. There is, therefore, a need for a robust and quantitative theory that predicts finite temperature yield behavior of arbitrary substitutional alloys. Chapter 1. Introduction 14 This dissertation is organized as follows. In Chapter 2, we develop a new weak-pinning theory that addresses the limitations of existing models and apply it to Al-X alloys (X = Mg, Si, Cu, Cr, Mn and Fe). The new theory is parameter-free and takes input from first-principles calculations. As a result, the model gives quantitative predictions of finite temperature yield stresses directly from first-principles. We make predictions to several alloys cited in the literature and assess the accuracy of the predictions. In Chapter 3, we apply the theory to Mg-Al alloys without modification. As with previous chapter, predictions of the finite temperature yield stresses are compared to experimental data available in the literature. In Chapter 4, we explore the transition between the strong- pinning and the weak-pinning regime at finite temperature. A parametric study is conducted to see how the dislocation core structure affects the transition. The transition is also studied using existing first-principles data. Finally, we conclude our findings in Chapter 5. Chapter 2 Solute strengthening in Al alloys from first-principles 2.1 Introduction To facilitate the development of advanced materials, computational materials science can be used to create guidelines for design as well as give physical insight to experimentally- derived trends. However, robust and predictive models are critical. Predictions of the macroscopic mechanical properties, such as flow stress, work hardening, and fatigue be- havior, in metals or other materials undergoing dislocation-mediated plastic flow, hold particular challenges because the behavior is controlled by interacting phenomena at multiple scales, from atomistic through mesoscale and microstructural scales, and is associated with the collective interactions among defects (solutes, dislocations, grain 15 Chapter 2. Solute strengthening in Al alloys from first-principles 16 boundaries, precipitates). In this chapter, we consider the problem of solute strength- ening, which combines the chemistry of solute interactions with dislocations in the host metal and the resulting configurations of the dislocation line. The fundamental interac- tions are at the Angstrom scale, but other characteristic scales from 10–100 nm control the overall material response. By capturing the phenomena at these scales, the model here provides a robust means for predicting the yield stress of solute-strengthened alloys, here with application to Al alloys, as a function of composition, temperature, and strain rate. Some basic aspects of solute strengthening mechanisms were established decades ago [11– 14, 36]. The general picture is that substitutional solute atoms interact with dislocations due to their misfit volume and elastic mismatch, leading to pinning of the dislocations and thus requiring higher applied stresses to move the dislocations. The standard models assume the applicability of elasticity theory, treat the dislocation core as a point, neglect the solute atoms nearest to the core, and ignore divergences in the total solute/dislo- cation interaction energy [21]. While trends such as the scaling of the strengthening with solute concentration and the temperature dependence of the strengthening have emerged from some models [11–14, 36], they are not quantitatively accurate. For in- stance, Argon reports a factor of 4 difference in model predictions and experiments for the zero-temperature strengths of Al and Au solutes in Cu, with no estimate of the energy barriers for motion. Here, we capture both elastic and chemical contributions to the dislocation/solute interaction energies, show that the solute atoms nearest the (dissociated) core of a dislocation provide the largest contribution to the strengthening, and predict the energy barriers for motion and finite-temperature flow stresses, with Chapter 2. Solute strengthening in Al alloys from first-principles 17 no adjustable parameters. The theory is also applicable to multi-component systems of dilute non-interacting solutes. In this chapter, we expound on a model first presented in Ref. [37] which addresses both the divergence of the fluctuations in the interaction energy and the interaction energy for solutes within the core of the dislocation for arbitrary solute additions to aluminum. The latter is addressed using DFT to compute the interaction energy of solutes in the immediate region of a core, the geometry of which is also computed fully within DFT using a flexible boundary condition method. The former is addressed by following the general strategy of the Labusch model but with a careful analysis of the concentration fluctuations that considers the spatial correlations in those fluctuations, which then leads to a parameter-free theory with no divergences and no ad-hoc restriction on the range of solute/dislocation interactions. The resulting theory is applied to edge dislocations in Al-X alloys (X = Mg, Si, Cu, Cr, Mn and Fe) to predict the finite-temperature yield stress. Screw dislocations can be evaluated similarly, but the interactions are limited to the immediate core due to the absence of a pressure field around the screw disloca- tion, and so we consider the behavior of the edge dislocations to control the flow stress. The predictions for binary alloys Al-Mg and Al-Mn are in very good agreement with experiments over a range of concentrations and temperature. The model is not accurate for Al containing very dilute (ppm) Fe. When the anomalous behavior associated with very dilute concentrations of Fe is included, the predictions for Al-Cr-(Fe) and Al-Cu- (Fe) are also in very good agreement with experiment. Predictions of the activation volume, a sensitive test of the thermal activation process, are also in general agreement with experiments and the overall model obeys Basinski’s “stress equivalency” principle. Chapter 2. Solute strengthening in Al alloys from first-principles 18 Although the analysis is carried out using Al as the host matrix, the approach is appli- cable to other metals with substitutional solutes, particularly fcc and hcp systems. The predictive success of the model demonstrates that computational materials science can provide quantitative predictions of material mechanical behavior including the effects of alloying, opening the path to computational alloy design. 2.2 Computational Methods Central to the solute strengthening mechanism is the interaction energy U (~r) between a solute at ~r and the dislocation structure centered at the origin ~r = 0. In order to make quantitative predictions, it is necessary to accurately determine these interactions over a wide range of solute atoms and for all possible solute locations. In particular, solutes that are immediately above and below the partial dislocations and stacking fault (hereafter called “the core”) are in a local environment much different from the bulk and therefore likely cannot be described by classical continuum elasticity. To accurately calculate the interaction energies of solutes in the core requires the use of a first-principles method to determine both the structure of the core and the energy changes when a solute atom substitutes for a host atom in the core region. Here, we use Density Functional Theory (DFT), implemented using the Vienna Ab-initio Simulation Package (VASP) [38, 39], Vanderbilt ultrasoft pseudopotentials [40], and the generalized gradient correction by Perdew and Wang [41, 42]. For bulk fcc Al, the method predicts a Burgers vector b = 2.86 A, a shear modulus µ = 27.4 GPa and a stacking fault energy σsf = 124 mJ m−2 , all in ˚ good agreement with experiments [43]. The pseudopotential valence configurations and plane wave cutoff energies used for Al and for all of the substitutional solutes considered Chapter 2. Solute strengthening in Al alloys from first-principles 19 here are summarized in Table 2.1. In calculating the relaxed core structure, a 2 × 2 × 8 Monkhorst-Pack kmesh [44] and Methfessel-Paxton energy broadening [45] of 1 eV are used. Table 2.1: Pseudopotential valence configurations (USPP = ultrasoft pseudopoten- tial) and plane wave cutoff energies (PWCO) for atoms considered in this work. Solute USPP PWCO (eV) Al [N e]3s2 3p1 129 Cr [Ar]d5 s1 227 Cu ([Ar]3d10 )p1 234 Fe [Ar]d7 s1 238 Mg [N e]3s2 106 Mn [Ar]d6 4s1 295 Si [N e]3s2 3p2 151 We first determine the equilibrium geometry of a pure Al a/2 h110i edge dislocation with line direction along the h1¯ 12i direction. We use the Lattice Green’s Function method of Woodward et al. [43, 46]. In this method, a central group of ions (region I) containing the core are allowed to relax fully in response to DFT forces. The forces caused on a surrounding group of ions (region II) are then relaxed by updating ion positions in regions I and II using the Lattice Green’s Function for a perfect infinite fcc crystal, and updating ion positions in an outer group of ions (region III) using the Continuum Elastic Green’s Functions. The entire collection of ions and their valence electrons are surrounded by a small layer of vacuum within a periodic rectangular simulation cell. Region III serves to isolate the inner two regions from spurious electronic effects emanating from the surrounding vacuum region. This overall approach accommodates the long-range elastic fields of the dislocation, eliminating boundary effects that can plague standard approaches. Details of the method are found in the literature and are not repeated here. Here, we use a computational cell comprised of 569 Al ions forming Chapter 2. Solute strengthening in Al alloys from first-principles 20 a prolate ellipse with major and minor axes dimensions of approximately 18 ˚ A and 14 ˚ A within an overall simulation box of dimensions 61˚ A×65˚ A×4.95˚ A and having periodic boundary conditions. Geometry optimizations were conducted until the forces on the ions in region I were 5 meV/˚ A or less. This ellipsoidal shape differs from the cylindrical shape used in Ref. [43] and provides more freedom for the splitting of the fcc core, and thus a better representation of the core structure than in the previous work. There are 50 ions in region I, 157 ions in region II, and 362 in region III. Nye tensor plots of the distribution of edge and screw components of the Burgers vector in the resulting relaxed dislocation core are shown in Figure 2.1. The equilibrium dislocation geometry shows that the core dissociates into Shockley partials separated by approximately 10.8 ˚ A, well within the 50-ion region I, and slightly larger than the value previously quoted in Ref. [43]. The solute-dislocation interaction energy as a function of solute position is computed by substituting a single solute atom into each of 22 distinct atomic core sites immediately above and below the slip plane. For Mn and Fe solutes, only 9 distinct sites around one partial were used and the symmetry of the core was then invoked to obtain the energies around the second partial, an approach validated by the results obtained on other solutes using the entire set of 22 distinct sites. For Mg and Cr, two additional sites in the middle of the core in the second rows above and below the slip plane were also investigated. The structure was then relaxed using DFT-derived forces in region I until the forces on all the ions in region I were 5 meV/˚ A or less. The Lattice Green Function optimization was not performed because continuum elasticity and molecular statics using embedded atom method (EAM) potentials for Al-Mg show that the difference in energies is negligible Chapter 2. Solute strengthening in Al alloys from first-principles 21 Note: I included Nye Tensor Plot for LGF-DFT a b/4 shift to 3b 0.06 match the Nye tensor plots for OFDFT-EAM 2b 0.04 Edge Component b 0.02 [111] → 0 0 !b !0.02 !2b !0.04 !3b !0.06 !5b !4b !3b !2b !b 0 b 2b 3b 4b 5b [ ¯110] → 3b 0.04 2b 0.02 Screw Component b [111] → 0 0 !b !0.02 !2b !3b !0.04 !5b !4b !3b !2b !b 0 b 2b 3b 4b 5b [ ¯110] → Figure 2.1: Nye tensor for the (a) edge and (b) screw components of the dislocation core computed using DFT and the Lattice Green’s Function method. The x-axis is in units of Burgers vector. Peaks in the Nye tensor roughly correspond to the locations of the partial dislocations in the dissociated core. (∼ 1 meV) when the radius of region I is much larger than the atomic radius, which is the case here. A set of ancillary DFT tests confirmed this conclusion. For magnetic solutes (Cr, Fe and Mn), spin polarized calculations were performed assuming an initial spin moment of 5 µB. The Fe and Mn solutes retained magnetic moments in all sites explored following VASP minimization, while Cr solutes did not. From the energy of the simulation cell containing the solute at (xi , yj ) and dislocation, Edisl+solute (xi , yj ), the interaction energy is computed as U (xi , yj ) = Edisl+solute (xi , yj ) − [Edisl only + Eref ] (2.1) Chapter 2. Solute strengthening in Al alloys from first-principles 22 where Edisl only is the energy of the pure Al core and Eref is the energy of a single solute in a bulk fcc Al crystal. Direct calculation of Eref using the same simulation cell but containing undeformed fcc Al is not accurate, presumably due to differences in the surfaces effects/energies between the two geometries. For consistency with the core calculations, Eref is estimated as the mean of the additional energy due to solutes in the 22 sites 1 XX Eref = (Edisl+solute (xi , yj ) − Edisl only ) (2.2) 22 i j where the summations are over the 22 sites in the dislocation core considered in the DFT calculations. Due to asymmetry in the magnitude of the solute energies just above and below the stacking fault, this estimate of Eref has some uncertainty. However, as will become clear later, the uncertainty in the predicted strength is second order in the uncertainty of Eref and is small. For instance, in the case of Al-Mg alloys, an uncertainty of as much as 0.05 eV (a very large value) would result in only a 7% change in the predicted strength. Our estimate for Eref also has the advantage of filtering out systematic errors in energy that are independent of solute position and further mitigates the boundary effects caused by not relaxing ions outside region I. The effect of solute-solute interactions along the h1¯12i direction was explored by doubling the periodicity of the simulation cell along the line direction from 4.95 ˚ A to 9.90 ˚ A, thereby increasing the solute-solute spacing by the same amount. These simulations required a simulation cell containing 1, 138 ions. The solute interaction energies of supercells containing one and two solutes along the dislocation line at a few sites in the dislocation partial cores were calculated. The differences in interaction energies were minimal (≈ 6%), consistent with EAM results on Al-Mg by Olmsted et al. [26] and Chapter 2. Solute strengthening in Al alloys from first-principles 23 with the continuum prediction of zero interaction energy between two separate point dilatational sources. It is not computationally feasible to calculate U (xi , yj ) on all sites using first-principles. However, away from the immediate core region, substitutional solutes are in a local fcc environment that is far less distorted than in the immediate core. Therefore, the continuum approximation of Eq. (1.2), U (xi , yj ) ≈ −p(xi , yj )∆vm , should provide a good approximation of the interaction energies, particularly with increasing distance from the core region. However, the pressure field p(xi , yj ) must be that appropriate for the real dissociated core structure. One approach to computing p(xi , yj ) is to use the Peierls-Nabarro model, where the Burgers vector of the dislocation is distributed along the stacking fault using the differential displacements obtained from the dislocation geometry. We have used a related but simpler approach by spreading the Burgers vector into n + 1 dislocations each of Burgers vector b/(n + 1) evenly spaced by b/2 over n + 1 = 11 interstitial sites along the slip plane in the core region. The pressure field is then computed as n/2 1+ν b X yj p(xi , yj ) = − µ , (2.3) 3π (1 − ν) n + 1 b 2 2   l=−n/2 x i − l 2 + y j where ν is the Poisson’s ratio of Al. This simple approximation was validated in EAM calculations of Al-Mg by Olmsted et al. [26]. Direct calculation of the misfit volume ∆vm through DFT volume relaxation calculations of perfect Al crystals containing a single solute atom shows inconsistent results between different supercell sizes. To obtain consistent results, we follow a more-complex method Chapter 2. Solute strengthening in Al alloys from first-principles 24 Table 2.2: The misfit volume ∆vm for Mg, Cr, Si, Cu, Mn and Fe solutes in Al. Also shown are the predicted T = 0 yield stress τy0 and energy barrier ∆Eb for each solute. Solute ∆vm (˚A3 ) τy0 /c2/3 (MPa) ∆Eb /c1/3 (eV) Mg 5.71 342 4.06 Cr -11.68 705 6.65 Si -2.65 137 2.58 Cu -5.57 348 4.10 Mn -13.31 711 7.53 Fe -16.44 1072 8.20 to calculate ∆vm proposed by Vannarat et al. [47]. Using both 27-atom supercell and 64-atom supercells, internal geometry optimizations are performed with one, two and three solute atoms, and the change in pressure in these supercells was computed via DFT in VASP. The dependence of the pressure with concentration is very linear for all solutes considered, as shown in Figure 2.2, consistent with results in Ref. [47]. The misfit volume is then calculated using the slope of the pressure versus solute concentration and the DFT-computed bulk modulus B of Al,     ∂∆v 1 ∂p ∆vm = c1sol = c1sol (2.4) ∂c B ∂c where c1sol is the concentration of 1 solute in the supercell. The misfit volumes calcu- lated using the two supercells differ by less than 15% across all the solutes considered, The misfit volume is taken as the average of the two derived values, and is shown in Ta- ble 2.2 for all solutes. The uncertainty in ∆vm causes uncertainties in the yield strength predictions of 10% or less. Using the DFT results in the core and the continuum analytic model outside the core leads to the solute/dislocation energies Uij ≡ U (xi , yj ) shown in Figure 2.3 for Mg, The quick brown fox jumps over the lazy dog. Chapter 2. Solute strengthening in Al alloys from first-principles 25 40 1 Figure 1 20 The quick brown fox jumps over the lazy dog. concentration (at.%) [111] → Mg 64 Si 64 0 Figure 2 pressure (kbar) Cu 64 (kbar) Cr 64 [¯ 110] → Mn 64 Mg 64 Fe 64 !20 pressure [111] → 110] → Si 64 Mg 27 Cu 64 Si 27 Cr 64 Cu 27 [¯ Mn 64 Cr 27 !40 Fe 64 2 Figure 2 Mn 27 2 Mg 27 Fe 27 Si 27 !60 Cu 27 Cr 27 The quick brown fox jumps over the lazy dog. Mn 27 Fe 27 !80 pressure (kbar) 0 2 4 6 8 10 12 concentration (at.%) concentration (at.%) Figure 2.2: Pressure versus concentration in fixed-volume supercells containing so- 6 8 10 lutes atoms,12for Mg, Si, Cu, Cr, Mn and Fe solutes, for both 27- and 64-atom supercells. oncentration (at.%) Dashed lines are linear fits using data from both supercells for each solute, and the slopes of these lines are used in Eq. (2.4) to compute the misfit volume. Si, Cr, Cu, Mn and Fe solutes. Not surprisingly, solutes that are smaller (∆vm < 0) than the host Al atom (Si, Cr, Cu, Mn and Fe) prefer to be on the compression side of the core while those that are larger (Mg) prefer to be on the tension side. The interaction energies are generally asymmetric between tension and compression in the layers immediately adjacent to the stacking fault. In most cases, there is a smooth transition between the DFT and the continuum regions, wherein the energies are highest in the immediate core region and decreases steadily with increasing distance from the core, The exceptions to this are Mn and Fe solutes, where the interaction energies in the second row are slightly higher than those in the core. In general, there are no sharp discontinuities in Uij despite the different methodologies employed in calculating Chapter 2. Solute strengthening in Al alloys from first-principles 26 the energies in the two regions, showing the consistency of the two approaches and the accuracy of U (xi , yj ) ≈ p(xi , yj )∆vm just outside the core region. The first-principles computation of solute/dislocation interaction in the dislocation core is the first main result of this chapter. While a significant computational challenge with various subtleties, the interaction energies by themselves are insufficient to predict strengthening and so in the next section we turn our attention to a detailed model of solute strengthening that uses the interaction energies computed here as input. 2.3 Solute Strengthening Model An initially straight dislocation of length L in a field of randomly-distributed solutes can lower its energy by bowing out into regions of the solid containing favorable fluctuations. The lateral length ζ of the favorable regions and their typical separation w along the slip direction must be determined by minimizing the total energy of the dislocation line. To make analytic progress, we simplify the possible dislocation configurations. In this simplified geometry, the dislocation is comprised of straight segments of length ζ lying in favorable local environments separated by roughening amplitude w along the slip direc- tion, as depicted in Figure 2.4. Since it is generally not possible to abruptly transition from one favorable local environment to another, straight segments are connected to each other via additional segments of length ζ. For the case of w  ζ, this geometry is close to a smooth piecewise sinusoidal dislocation configuration. In this geometry, the dislocation can be divided into L/2ζ unit cells, each with a segment of length ζ lying in an energetically favorable local environment. Cu The quick brown fox jumps over theMg lazy dog. Cr Cr Mg Mn 1 SiFigure 3 The quick brown fox jumps ove 1 Mn 3 1 Si Figure Fe Cu The quick brown fox jumps Mg over the lazy dog. CuFe eV Cr Chapter 2. SoluteThestrengthening quick browninfox Aljumps alloysoverfromthe Mg first-principles 1 Figure 3dog. lazy Si 27 CreV x (nm) ([111]) Mn 1 Figure 3 Mg Si x (nm) ([111]) The quick brown y (nm) ¯110]) over theCu fox ([jumps lazy dog. Mn Si Fe Cu The quick brown y (nm) fox jumps over the Mg ([¯110]) lazy dog. vm = 5.71 ˚ A3 Cr Fe Cu eV Cr Mg eVvm = 5.71 ˚A3 1 Si0.3 Figurevm3= 2.65 ˚A3 Mn 1 x (nm) ([111]) 0.3 11 Figure Cr 3 1 0.25 Mn ˚3 A3 ˚3 0.25 Fe 11.68 A 13.31 ˚ 16.44 A Si Cu A3 ˚3 vm = 2.65 A ˚ 3 vm = ¯ 5.57 A ˚ 3 2.65 ˚ 5.57 A 0.2 0.2 ˚3 x (nm) ([111]) y (nm) fox([jumps 110]) over the lazy 0.5 0.15 Fe vm = 5.71 A Mn The quick brown dog. x (nm) ([¯110]) (nm)([1([111]) 0.15 y (nm) ([1 1 1]) 1 1]) 0.5 TheCu quick brown vfox m =jumps 5.57 ˚ 3 Cr A the lazy over 0.1 dog. vvmm==5.71 11.68 ˚ A 3 eV 0.1 0 0.3 y (nm) ([ ¯110]) Mg 0.05 ˚ A3 0.050.30 0 Fe 0 0 eV vm = vm = vm = vm = vm = Cr Mn yy (nm) Mg v = 11.68 ˚ A 1 1 3 Figure −0.05 3 ˚ A 3 x −0.05 Mn v = 13.31 vm m= 5.71 ˚ Si m (nm) 0.25([111]) eV A3 v = 2.65 ˚ A3 0.25 Fe Cr −0.1 −0.1 −0.5 eV m −0.5 −0.15 x (nm) ([111]) −0.15 1−1 Mn SiFigurevm3 vm = 13.31 = 2.65 ˚ A3 Fe ˚ A3 −1 Cu −0.2 −0.25 The quick brown v vm = 16.44 m y (nm) = ¯ 5.57 fox([jumps ˚3A A ˚3 110]) over the lazy 0.2 −0.20.20 y (nm) −0.25 dog. ([¯ 110]) x (nm) ([111]) Fe A3 eV −0.3 −0.3 Cu −1 −0.5 0vm = 0.5 16.44 1˚3 ˚ −1 −0.5 Cr 0 0.5 1 3 vm0.150.15 1 = 5.71 ˚ A3 y (nm) ([1 1 1]) v = 5.57 A vm([−1 = 1 0]) 11.68 ˚ 3A x (nm) y m ([−1 (nm) 1 ([¯0]) 110]) The quick brown fox jumps over 1 eV 0.5 Mg x (nm) 0.3 dog. m the lazy v = 5.71 ˚ A 0.3 0.1 2.65 ˚A3 ˚3 A3 ˚3 1 x 0.25 (nm) ([111]) 0.250.10 11.68 A 13.31 ˚ 16.44 A Cr1 Figure Mn A3 ˚3 vm = 3 11.683 A ˚3 1 Figure 3 ˚ 3 2.65 ˚ 5.57 A v Si0.15 vm = 2.65 A 0.2 = 13.31 A v m = 0.2 ˚3 m ˚ Mg vm = 5.71 A ˚ 3 vm = 5.71 A x (nm) ([¯110]) (nm)([1([111]) 0.15 0.10.05 y (nm) ([1 1 1]) 1 1]) 0.5 x (nm) ([111]) 0.5 y (nm) ([¯ 110]) 0.05 Mn vm = 13.31 ˚ A 3 Fe Cu 0.1 v = 16.44 ˚ A 3 v0.05 m = 5.57 ˚A3 Si 0 y (nm) ([¯ The quick vm = 2.65 A ˚3 brown fox jumps0over 0 0v 0.05 The the lazy quick m vm = 5.57 A dog. brown ˚ ˚ 3 fox jumps over0the00.00 lazy dog. 110]) A3 vm = vm = vm = vm = vm = m = 5.71 eV yy (nm) Fe ˚ A 3 −0.05 −0.05 Mn v = 16.44 Cu vmm=3 5.57 ˚ A3 −0.5 Cr 11.68 ˚A3 eV vm = 11.68 ˚ A3 v−0.1 m = Fe Cr −0.1 −0.5 vm = Mg 5.71 A ˚ vm = −0.15Mg2.65 ˚A (nm) ([111]) x −0.2 3 −0.05 -0.05 −0.15 eV Cr ˚ 3 Mn ˚ 3 −0.2 = 13.31 ˚ A3 m Si −1 v = 2.65 −1(nm) −0.5 v m ˚3 0 = A 11.68 0.5 A −1 −0.25 1˚3−0.5 y−1(nm) vm = −0.3 −0.5 v m Si([¯110]) = 5.57 A 0 ˚ 3 0.5 13.31 A 1˚3 v −0.3−0.1 m-0.10 −0.25 Mn x ([111]) vm([−1 13.31 A =˚13 0]) Fe vm([−1 16.44 A = 1 30]) m−0.15 v0.3 = 16.44 ˚ -0.15 A3 1 vm =Cu x (nm) 5.57 A 0.3 vm = Cu x (nm) vm = 5.71 ˚ 1 0.25 11.68 A3 ˚ A ˚3 A3 ˚3 0.25 Fe 11.68 A 13.31 ˚ 16.44 A y (nm) ([¯110]) eV 0.2−0.2 A3 ˚3 vm = ˚316.44 ˚ A3 -0.20 2.65 ˚ 5.57 A 0.2 ˚3 vm =Cr 11.68 A vm = Cr 13.31˚A˚ 3 vm = 5.71 A x (nm) ([¯110]) ([111]) 0.15 3 0.15 1 1]) y (nm) ([1 1 1]) 0.5 0.5 vm = 2.65 A eV vm = 5.71 ˚ vm = 13.31 A Mn A3 ˚3 −1 0.1 (nm) ([111]) 3 x 0.05 vm = Mn5.5716.44˚A˚ 0.05−0.25 0.1 -0.25 (nm) ([1 0 0 0vm = A3 0 vm = vm = vm = vm = vm = −0.3 yy (nm) -0.30 x v(nm) = ([111]) ˚3 3 16.44A˚ 2.65 (nm) ([¯110]) y −0.05 −0.05 Mn m = eV vm A −0.1 −0.1 Fe Cr −0.5 Fe −0.5 −0.15 vm = Fe 11.68 ˚A 3 −0.15 −1 −0.5 0 0.5 1 y (nm) ([¯ 110]) ˚ 3 v −0.2= 5.71 ˚ A3 −0.2 1 vm = 5.57 A m −1 eV ˚3 −1 −0.25 vm = −0.3 eV13.31 A ˚ 3 −0.25 −0.3 x (nm) ([−1 1 0]) v m = −1vm −0.55.71 A 0 ˚ 3 −1vm −0.5 = 2.65 ˚ 3 0 A 0.5 1 = 11.68 A0.5 1 1 xx (nm) ([¯110]) (nm) ([−1 1 0]) vm = 16.44 xx (nm) (nm) ([−1 ˚ A130]) ([¯110]) vm = 2.65 ˚ A˚33 vm = 5.57 ˚ A3 vm = 13.31 A y (nm) ([111]) y (nm) ([111]) = 5.57 ˚ Figure 2.3:vmSolute/dislocationA33 interaction energy v = Uij versus 11.68 ˚ A3 solute position position (xi , yj ) for Mg, ˚ Si, Cu,A Mn and Fe solutes. Misfit volume ∆vm is shown for each vm =Cr, 16.44 m 1 ˚3 binding. Bold circles ˚ A3 solute. Negative vm = 5.71 vm =energies ˚3A 11.68 indicate A vm =show5.71 positions at which first- principles interaction energies are calculated; “x” v = 13.31 ˚ A 3 m markers denote positions 1 of the b/11 Burgers vector v used = to 2.65 compute ˚ A 3 the analytic pressure v = field 2.65 outside ˚ A 3the immediate core vm = m13.31 ˚ A 3 m sites (non-boldvm = 16.44 ˚ circles). A3 vm = vm16.44 = 5.57 A3 ˚ ˚ A3 vm = 5.57 ˚ A3 ˚3 ˚3 1 vm = 11.68 A vm = 11.68 A ˚3 ˚3 1 vm = 13.31 A vm = 13.31 A vm = 16.44 ˚ A3 vm = 16.44 ˚ A3 1 1 Chapter 2. Solute strengthening in Al alloys from first-principles 28 ζ w [¯110] 2ζ L [¯1¯12] [111] Figure 2.4: Schematic of a dislocation roughening into segments of length 2ζ with roughening amplitude w due to favorable solute fluctuations. The change in total energy as the dislocation moves from the straight configuration to the bowed out configuration is the sum of the potential energy decrease due to so- lute/dislocation binding, ∆Ep , and the elastic energy increase due to the increase in line length, ∆Eel , ∆Etot = ∆Ep + ∆Eel . (2.5) For an isotropic line tension model with line tension Γ, the elastic energy is h 1 i L 2 2 ∆Eel = Γ ζ +w 2 −ζ · (2.6) 2ζ  2  w L ≈ Γ (2.7) 2Γ 2ζ since w  ζ is expected. Calculation of the binding energy due to solute fluctuations is more subtle, as noted in the Introduction, due to divergences that arise when sim- ply summing the energies or fluctuation in energies. However, by recognizing that the total interaction energy as the dislocation segment moves between two positions is corre- lated over some distance, a finite, calculable solute binding energy and strength emerge Chapter 2. Solute strengthening in Al alloys from first-principles 29 naturally without any need to truncate the range of interaction energies, as described below. Consider a dislocation segment of length ζ in a random field of solutes. At each atomic position (xi , yj ) projected in the plane normal to the dislocation line, there are nij solutes √ in the length ζ, where nij is a random number between zero and N = ζ/ 3b (for a line direction of 110 in an fcc material). The energy of a segment at the origin is then X Utot (ζ) = nij Uij (2.8) ij Moving the dislocation a distance w from the origin in the slip direction is equivalent to moving the solutes from (xi , yj ) to (xi − w, yj ) while keeping the dislocation at the origin. The energy change, ∆Utot (ζ, w), of the dislocation segment of length ζ as it glides a distance w away from the origin can then be expressed as X ∆Utot (ζ, w) = nij ∆Uij (w) (2.9) ij where ∆Uij (w) = U (xi − w, yj ) − U (xi , yj ) and the summation in Eq. (2.8) extends to all in-plane sites. As noted earlier, the mean energy change h∆Utot (ζ, w)i vanishes for all values of w because hnij i = cN is constant, implying that the configurational force due to solute/dislocation interaction can only arise due to spatial fluctuations in solute concentration. Spatial fluctuations in solute concentration create a statistical distribution of total energies for a segment characterized by the standard deviation of Eq. (2.9), hD E i1 ∆Utot (ζ, w)2 − h∆Utot (ζ, w)i2 . 2 σ∆Utot (ζ, w) = (2.10) Chapter 2. Solute strengthening in Al alloys from first-principles 30 Some regions give rise to binding, σ∆Utot < 0, with characteristic energy −σ∆Utot , and others give rise to repulsion, σ∆Utot > 0 with characteristic energy σ∆Utot . Therefore, the characteristic potential energy decrease by allowing the dislocation segment to move into regions in the solid containing favorable solute fluctuations is L ∆Ep = −σ∆Utot · . (2.11) 2ζ To evaluate Eq. (2.10), we first express the first term on the right hand side of the equation as D E X ∆Utot (ζ, w)2 = n2ij ∆Uij (w)2 +hnij i hnlm i X X ∆Uij (w) ∆Ulm (w) (2.12) ij ij lm6=ij D E since only nij is the random variable in the equation. The quantity n2ij is given by the binomial theorem as N X N! n2ij = pn q 1−p n2 (2.13) n! (N − n)! n=0 where p is the probability of finding a solute at position (xi , yj ) which, for random solutes, is independent of position. Noting that n2 cn = (p · ∂/∂p)2 (pn ), nij ≤ N and that p = c, Eq. (2.13) reduces to n2 = (cN )2 + N c(1 − c) = hni2 + c (1 − c) N. (2.14) Chapter 2. Solute strengthening in Al alloys from first-principles 31 Substituting Eq. (2.14) into Eq. (2.12) yields D E h iX ∆Utot (ζ, w)2 = hni2 + c (1 − c) N ∆Uij (w)2 ij 2 X X + hni ∆Uij (w) ∆Ulm (w) , (2.15) ij lm6=ij ∆Uij (w)2 + h∆Utot (ζ, w)i2 . X = c (1 − c) N (2.16) ij Substituting Eq. (2.16) into Eq. (2.10) gives the expected fluctuation in the interaction energy when a segment of length ζ moves a distance w as ∆Uij (w)2 . X σ∆Utot (ζ, w) = c (1 − c) N (2.17) ij The total average binding energy of a dislocation due to solute/dislocation interaction is then obtained by substituting Eq. (2.17) into (2.11). For low concentrations c  1, " 1 #   cζ 2 ˜p (w) · L ∆Ep (ζ, w) = − √ ∆E (2.18) 3b 2ζ where  1 2 ˜p (w) =  2 X ∆E (U (xi − w, yj ) − U (xi , yj ))  . (2.19) ij Equation (2.19) can be rearranged into the form  1 2 ˜p (w) = 2 2 X ∆E (1 − χ (w, yj )) U (xi , yj ) (2.20) ij where P k U (xk − w, yj ) U (xk , yj ) χ (w, yj ) = P 2 . (2.21) k U (xk , yj ) The quick brown fox jumps over the lazy dog. The quick brown fox jumps over the lazy dog. The quick brown fox jumps over the lazy dog. Chapter 2. Solute strengthening in Al alloys from first-principles 32 ∆Etot /Lc2/3 (eV/nm) 2 Potential Energy (eV) 0.9 1.5 1 Figure 4 0.8 ([111]) 1 Figure 4 Figure 5 Figure 6 Figure 7 0.7 1 1]) 0.5 0.6 (nm) ([1 yjyj (nm) 0 2 Figure 5 0.5 w (nm) w (nm) !0.5 0.4 wc !1 0.3 The quick brown fox jumps 0.2 over the lazy dog. 1 2 3 4 !1.5 0.1 !2 yj (nm) ([111]) 0 1 2 3 ww (nm) (nm) Figure 2.5: Energy correlation function χ (w, yj ) for a solute initially at (0, yj ) as the dislocation moves a distance w, as computed for an Al-Mg alloy. 3 Figure 6 is the correlation function between the interaction energies of a solute at yj before (xi ) The quick and after (xi − w) the dislocation glides brown w away from fox jumps over its original the lazy position. dog. For any finite w, the correlation χ(w, yj ) increases ∆Eas |yj/Lc | increases, 2/3 i.e. the interaction energies (eV/nm) tot of a solute far from the dislocation are highly correlated, χ(w, yj ) → 1, as shown in w (nm) Figure 2.5. Such solutes therefore do not contribute significantly to the change in total wc ˜p (w) is thus finite and there is no need to introduce any energy. The quantity ∆E artificial truncation in the range of the solute/dislocation interactions. 4 Figure 7 Substituting Eq. (2.6) and Eq. (2.18) into Eq. (2.5) gives the total energy change as a function of ζ and w, for a dislocationThe taking on the quick configuration brown depicted fox jumps over in Figure the lazy 2.4, dog. as "  Potential 1Energy (eV) #   w2  cζ 2 ˜ L ∆Etot (ζ, w) = Γ − √ ∆Ep (w) · . (2.22) 2ζ Stress3b(MPa) 2ζ To obtain the equilibrium configuration of the dislocation, we minimize Eq. (2.22) with 5 Figure 8 The quick brown fox jumps over the lazy dog. experimental σy (MPa) Chapter 2. Solute strengthening in Al alloys from first-principles 33 respect to ζ and w. Analytic minimization with respect to ζ is straightforward, yielding the characteristic segment ζc as a function of the roughening amplitude w given by !1 √ Γ2 w4 b 3 ζc (w) = 4 3 . (2.23) ˜ 2 (w) c∆Ep Substituting this expression into Eq. (2.22), the total energy is now solely a function of the roughening amplitude w, !1 32/3 c2 ∆E ˜p2 (w) 3 ∆Etot (w, ζc (w)) = − 1/3 L. (2.24) 82 b2 w2 Γ Minimizing ∆Etot (w, ζc (w)) with respect to w yields the equilibrium energy and a char- acteristic amplitude wc . This minimization cannot be done analytically and so must be done numerically. Scaling out the concentration, the total energy per unit length ∆Etot /Lc2/3 versus w is shown in Figure 2.6. Figure 2.6 also shows the convergence of ∆Etot /Lc2/3 as a function of the upper limit of the summation over yj , which demon- strates that (i) the characteristic value wc is established quite well by the core sites only (y1 and y−1 ), (ii) the sum converges quickly with increasing number of layers of sites (increasing j), and (iii) the magnitude of the contributions from sites within the core (y1 ,y−1 ) and outside the core (yj , |j| > 1) are approximately equal, with significant con- tributions for |j| = 2, 3. Point (i) above demonstrates that the core structure is crucial for obtaining wc , justifying the computational expense of DFT in this region, but point (iii) shows that the region outside the core is also crucial for obtaining an accurate total energy. Calculations for other solutes yield the same general conclusions but with wc varying slightly (1.57 nm for Mg, Si and Cu; 1.71 nm for Cr and Fe; and 1.86 nm for 3 Figure 6 1 Figure 4 The quick brown fox jumps over the lazy dog. The quick brown fox jumps over the lazy dog. The quick brown fox jumps over the lazy dog. The quick brown fox jumps over the lazy dog. The quick brown fox jumps over the lazy dog. Chapter 2. Solute strengthening in Al alloys from first-principles 34 ∆Etot /Lc2/3 (eV/nm) 2 Figure 5 0 wThe(nm) quick brown fox jumps over the lazy dog. !0.02 !0.04 yj (nm) ([111]) wc experimental σy (MPa) (eV/nm) y j : "1 # j # 1 Potential Energy (eV) !0.06 w (nm) predicted σy (MPa) /L c2/3(eV/nm) !0.08 yj (nm) ([111]) Figure 6 Figure 7 Figure 8 y j : "3 # j # 3 !0.1 ! Stress (MPa) 2/3 ! E/Lc !0.12 4 3Figure Figure 67 y j : "10 # j # 10 tot w (nm) w (nm) ! ∆Etot !0.14 wc !0.16 ! y j : "50 # j # 50 The quick brown fox jumps over the lazy dog. 3 4 5 !0.18 The quick brown fox jumps over the lazy dog. !0.2 /Lc2/3 (eV/nm) ∆Etot ! 0 1 Potential 2 w (nm) Energy 3 (eV) 4 w (nm) Figure 2.6: Normalized total energyStress perw c (MPa) unit length ∆Etot /Lc2/3 versus roughening amplitude w for an Al-Mg alloy, for various partial sums over solute sites yj demon- strating the dominance of the core region y−1 , y1 and convergence of the total energy with increasing range of |yj |. 4 Figure 7 Mn) in spite of a large range of misfit volumes among these solutes. The values of wc 5 Figure 8 are approximately equal to 1.5 times theThe corequick brown width, foxwhen so that jumps theover the lazymoves dislocation dog. by wc , it encounters a new set of The solutesquick brown in Potential the core fox(eV) Energy that create jumps a binding over energythe lazy that is dog. Stress statistically uncorrelated with the energy at (MPa) the prior position. experimental σy (MPa) After minimization, we have a final total energy per unit length of ∆Etot (wc , ζc (wc )) /L. predicted σy (MPa) The relevant characteristic energy is the energy associated with each pinned segment of 5 Figure 8 length ζc and the surrounding length of ζc (see Figure 2.4), which is thus, from Eqs. (2.23) The quick brown fox jumps over the lazy dog. and (2.24), experimental σy (MPa) !1 2 35/6 σycw(MPa) ˜2 3 2ζc predicted c Γ∆Ep (wc ) ∆Ec = ∆Etot (wc , ζc (wc )) = − 5/3 . (2.25) L 2 b 1 Chapter 2. Solute strengthening in Al alloys from first-principles 35 Although ∆Ec is the characteristic energy, it is not the energy barrier for determining thermally activated motion of the dislocation. ∆Ec is the typical energy change on moving from one arbitrary point to another point wc away. The dislocation is pinned in a local minimum and must escape over the adjacent local maximum, in an energy landscape with a characteristic energy ∆Ec . Thus, the energy barrier ∆Eb is typically larger than ∆Ec , and is determined as follows. First we note that the elastic energy contribution to ∆Eb is the same as to ∆Ec . We thus concentrate on the potential energy contribution ∆Ep to ∆Eb . The potential energy landscape, i.e. the energy versus dislocation position (x, y=0) along the glide plane of the dislocation segment of length ζc , can locally be approximated as a sinusoid with some amplitude ∆Eb0 , i.e. ∆Eb0    πx E (x) = 1 − cos . (2.26) 2 wc where ∆Eb0 is the potential energy barrier. From our discussion above, ∆Ep is the standard deviation of the change in energy of the dislocation segment as it moves wc away,  2wc  12 1 1 Z 2 ∆Ep = (E (x − wc ) − E (x)) dx = √ ∆Eb0 . (2.27) 2wc 0 2 √ Thus, the typical potential energy barrier ∆Eb0 is 2 times larger than ∆Ep . The total energy barrier includes the elastic energy contribution, and so is given by √ ! ˜p2 (wc ) !1 4 2 − 1 35/6 cwc2 Γ∆E 3 ∆Eb = ∆Eb0 − ∆Eel = · 5/3 . (2.28) 3 2 b The sign convention in Eq. (2.28) is opposite of that in Eq. (2.25) because the latter refers to the energy change due to the bowing out process while the former refers to the Chapter 2. Solute strengthening in Al alloys from first-principles 36 magnitude of the energy barrier. √ We validate the above analysis showing that ∆Eb0 = 2∆Ep using a stochastic simulation of a straight dislocation moving through a random field of solutes. Remaining straight, only the potential energy ∆Ep is computed. A segment of length ζ = 18.06 nm is placed in a simulation box of size 1425nm x 5.12nm x 18.06nm containing a 5 at.% Mg solutes randomly distributed. This simulation box is large enough such that the relative potential energy differences with position within wc are converged. The total potential energy of the straight dislocation is calculated as a function of dislocation position along the glide plane in the simulation box, as shown in Figure 2.7(a). The expected standard deviation ±∆Ep as calculated analytically is shown in Figure 2.7(a) and agrees well with the simulated result; this is not surprising. Locally, the energies in Figure 2.7(a) are sinusoidal and the typical energy barrier, from a local minimum to a local maximum, is larger than ∆Ep . The energy barriers are best probed by examining the resisting stress exerted by the solutes on the dislocation, which is equal to the spatial derivative of ∆Ep with respect to x, τresist = − (1/bζ) (∂∆Ep /∂x), and is shown in Figure 2.7(b). The maximum and minimum values of τresist are related to the underlying energy barrier ∆Eb0 as max/min π ∆Eb0 τresist = . (2.29) 2 bζwc max/min √ The predicted values of τresist using ∆Eb0 = 2∆Ep are shown in Figure 2.7(b) and agree well with the simulated values, thus validating the analytic result. We have determined that dislocation segments of length ζc (Eq. (2.23)) are pinned in typical energy minima with barriers of height ∆Eb (Eq. (2.28)) over a length wc (e.g. Figure 2.6). We can now use standard analyses of thermally-activated dislocation motion (eV)quick brown fox jumps over the lazy dog. quick brown fox jumps over the lazy dog. 1 Chapter 2. Solute strengthening in Al alloys from first-principles 37 1 Figure 3 (MPa) The quick brown fox jumps over the lazy dog. The quick brown fox jumps over the lazy dog. Potential Energy (eV) (a) Potential Energy (eV) (MPa) 2 The quick brown fox jumps over the lazy dog. +ΔEp y Figure 7 Figure 8 Stress (MPa) (MPa) experimental Mg y 0 predicted −2 -ΔEp Si (MPa) wc Cu Potential EnergyThe The 1000 10 20 30 40 50 60 70 80 90 100 4 5 (b) x (nm) y Figure 7 Figure 8 +τp Cr (MPa) 50 experimental y Stress (MPa) predicted Mn 0 Stress −50 Fe -τp 4 5 −100 eV 0 10 20 30 40 50 60 70 80 90 100 x (nm) ([¯110]) Figure 2.7: (a) Energy versus position of a straight y (nm) ([111]) dislocation segment of length ζ = 18.06 nm moving through a random 5 at.% Mg solute field; (b) Stress required to move the dislocation across the energy3 landscape. vm = 5.71 ˚ A vm = ˚3 2.65 A to determine the finite-T and rate-dependent yield stress, as follows. When the material m v = 5.57 ˚ A3 is subjected to an applied resolved shear stress τ , the energy landscape E (τ, x) must be vm = stress, modified to include the work done by the applied ˚ A3 so becomes 11.68 and vm = 13.31 ˚ A3    ∆Eb πx E (τ, x) = 1 − cos −˚τ3ζc bx. (2.30) 2 vm = w16.44 c A . The stress-dependent barrier height ∆E(τ ) can be approximated (within 6%) by [21]    3 τ 2 ∆E (τ ) ≈ ∆Eb 1 − . (2.31) τy0 Chapter 2. Solute strengthening in Al alloys from first-principles 38 Here, τy0 is the zero-temperature yield strength, i.e. the stress at which the energy landscape exhibits no barrier for dislocation motion, !1 π ∆Eb c2 ∆E˜p4 (wc ) 3 τy0 = = 1.01 (2.32) 2 bζc (wc )wc Γb5 wc5 √ with 1.01 = π 4 2 − 1 /31/3 210/3 a constant containing all the numerical factors. At  finite temperature, the dislocation can be thermally activated over the barrier at stresses below τy0 . Using the same arguments as that stated in Section 1.4, the finite temperature stress required for dislocation flow is given by "  2 # kT ˙◦ 3 τy (T ) = τy0 1 − ln . (2.33) ∆Eb ˙ Equation (2.33) applies at low T but not at cryogenic T where dynamic effects can enter [21]. At high T, Labusch has shown that the characteristic length ζc effectively increases as the dislocation can explore other configurations, but with the behavior still controlled by the energy barrier ∆Eb . Over a wide range of temperature, 0.2 < kT ln (˙◦ /) ˙ /∆Eb < 0.6, this result of Labusch can be matched to Eq. (2.33) as   1 kT ˙◦ τy (T ) = τy0 exp − ln (2.34) 0.51 ∆Eb ˙ where the parameter 0.51 provides a good fit (< 7%) to both Eq. (2.33) and its derivative with respect to T . While nearly identical over a range of temperature, Eq. (2.34) does not reach zero at any finite temperature, and is thus more accurate at elevated temperatures. Eqs. (2.28), (2.32), (2.33)–(2.34) are the second main results of this chapter. Associated with thermally-activated flow is the activation volume V = −∂∆E/∂τ , which Chapter 2. Solute strengthening in Al alloys from first-principles 39 is the area swept out by the dislocation during the activation process multiplied by the Burgers vector. The activation volume can be measured experimentally using strain rate jump tests and other methods and is a sensitive measure of the response of the material that can provide insight into underlying mechanisms [1]. Taking the derivative of Eq. (2.31) with respect to τ yields  1 ∂E0 3 ∆Eb kT ˙0 3 V (T ) = − = ln . (2.35) ∂τ 2 τy0 ∆Eb ˙ Experimentally, a direct relationship between V and the finite temperature yield stress τy has been observed. This led Basinski to postulate the so-called “stress equivalency principle”: if two different alloys have the same τy at the same temperature T and strain rate , ˙ then they will also have the same V , regardless of the concentrations and types of solutes in the two different alloys [48]. “Stress equivalency” is satisfied by the present model under some weak constraints. Specifically, if the roughening amplitude wc varies ˜p2 controls the only mildly with the type of solute in the alloy, then the parameter c∆E values of both τy0 and ∆Eb , which in turn govern both V and τy . Equation (2.33) can then be rewritten as a function of V ,   2  3  12 bkT wc3 Γη   1 − 0.67  kT η   τy = 2.23  (2.36) V3 3     b3 kT wc Γ η 9 3  4 V where η = ln(˙0 /). ˙ The activation volume V and the finite temperature flow stress τy are thus related by an equation of the form τy = AV −2/3 (1 − BV −1/2 ), where A and B 2/3 are constants. If B  1, then V and τy are related by a simple power law, V ∼ τy , which was previously derived for the Labusch model by Nabarro [49]. For different Chapter 2. Solute strengthening in Al alloys from first-principles 40 solutes with the same wc , Basinski’s stress equivalency principle is thus strictly obeyed. Slight variations in wc will be manifested as slight rigid shifts preserving the functional form; for all materials studied in this chapter, these shifts are small. The activation 4 1 volume is also predicted to scale as c− 9 , which is close to the ∼ c− 2 trend reported in the literature [1]. The present model is easily generalized to consider more than one solute. For an arbi- trary number of random non-interacting solutes, Eq. (2.8) is replaced by X (1) (1) (2) (2) (3) (3)  ∆Utot (ζ, w) = nij ∆Uij (w) + nij ∆Uij (w) + nij ∆Uij (w) . . . (2.37) ij where the superscripts indicate the type of solutes. For the multisolute case, the analysis ˜ 2 replaced by above then applies with c∆E P ˜ (i)2 p i ci ∆Ep . For solutes with the same wc , the zero temperature strength and characteristic energy barrier are " #2 X 3 3 (i) 2 τy0 = τy0 , (2.38) i " #1 3 (i) 3 X  ∆Eb = ∆Eb , (2.39) i the former consistent with Labusch’s analysis. Numerical minimization must be used for solutes with different wc to determine the effective wc but the difference between direct minimization and Eq. (2.39) is small for all the solutes considered in this chapter. For instance, the differences in τy0 and ∆Eb between numerical minimization and Eq. (2.39) for a 1% Mg (wc = 1.57nm)/1% Mn (wc = 1.86nm) alloy are 5% and 4%, respectively. The theory presented in this section utilizes the fundamental solute/dislocation energies Chapter 2. Solute strengthening in Al alloys from first-principles 41 as input into a detailed model that culminates in the prediction of the characteristic energy barrier and length scales for dislocation motion through the material. Use of first-principles data overcomes the problem of lack of knowledge of solute interactions in the core of the dislocation. Careful analysis of the correlations in solute fluctuations over- comes the apparent problem of diverging energies that inhibited earlier approaches from being parameter-free. From these fundamental properties, the finite-T, rate-dependent yield stress follows naturally via thermal-activation theory, with no adjustable param- eters at any stage. The model is related conceptually to the classic Labusch model, predicting similar scalings with concentration, stress-equivalency, and generalized to multiple solute types, but has succeeded in eliminating ad-hoc features and fitting of parameters used in prior implementations of the Labusch concepts. We next turn to predictions of the model when applied to Al-X alloys to demonstrate the quantitative predictability of the model. 2.4 Results and Discussion Using the interaction energies calculated in Section 2.2 and the strengthening model presented in Section 2.3, the zero temperature stress scaled by concentration, τy0 /c2/3 , and the energy barrier scaled by concentration, ∆Eb /c1/3 , are presented in Table 2.2 for Mg, Si, Cu, Cr, Mn and Fe solutes in Al. The line tension used, Γ = 0.47 eV/˚ A, is based on new robust molecular statics calculations on EAM Al [50]. The quoted values in Table 2.2 have an estimated uncertainty of 5–10% due to uncertainties in the reference interaction energy, Eref , and the computed misfit volumes, ∆vm . Using Eqs. (2.38)–(2.39) and Eq. (2.33), the finite temperature flow stress of an alloy containing Chapter 2. Solute strengthening in Al alloys from first-principles 42 any number of these solutes can be predicted, with no parameters. Modifications to the detailed analysis in Section 2.3 could lead only to a shift by an overall constant factor independent of solute type or concentration. The experimental flow stress values of 12 distinct and well-characterized Al-X alloys have been measured at 78K and ˙ = 3 × 10−5 s−1 by Diak et al. [1–3]. Values for ˙◦ typically range from 104 –106 s−1 ; appearing in a logarithm, the precise value is not crucial and we use 104 s−1 . The flow stresses are measured using a 0.02% strain offset, corresponding to the onset of plastic flow associated with dislocation motion in tension tests on polycrystalline materials. The predicted shear flow stresses must therefore be multiplied by the Taylor factor M = 3 in order to compare with experiments. Table 2.3 shows the predicted and experimental flow stresses for all nominally binary Al- X alloys Predictions for the Al-Mg alloys are in very good agreement with experimental values, being only ∼ 15% lower. Predictions for the Al-Mn alloys are similarly close to the experimental values. In contrast, the Al-Fe alloys containing parts per million (ppm) of Fe solutes are drastically underestimated, with predictions of virtually zero strength and experiments showing strengths on the order of 10’s of MPa. It is clear that the strengthening due to dilute Fe solutes cannot be explained by the interaction energies calculated in Section 2.2 and the model of Section 2.3. The predicted strengths for the Al-Cr and Al-Cu alloys are rather smaller than experiments although the general trends are captured. For instance, the Al-Cu and Al-Cr alloys with similar solute concentra- tions have strengths differing by approximately a factor of two in both predictions and experiments. The difference between theory and experiment for these two alloys lies in the presence of ppm Fe in these nominally binary alloys. Chapter 2. Solute strengthening in Al alloys from first-principles 43 Table 2.3: Predicted and experimental [1, 2] tensile yield stresses at T = 78K for various Al-X alloys. Solute concentrations are from Diak et al. [1, 2]. Solute c (%) Tensile Yield Stress σy (MPa) Predicted (78K) Experiment (78K) Mg 0.444 18.2 20.6 Mg 0.810 28.9 34.2 Mn 0.123 17.1 20.8 Mn 0.246 28.8 28.9 Mn 0.494 48.1 44.0 Fe 7.7 × 10−4 0.1 11.0 Fe 16.9 × 10−4 0.5 16.7 Fe 43.5 × 10−4 1.6 33.4 Cr-(Fe) 0.073 (10 × 10−4 ) 11.0 23.7 Cr-(Fe) 0.302 (12 × 10−4 ) 32.5 50.2 Cu-(Fe) 0.090 (12 × 10−4 ) 5.3 12.3 Cu-(Fe) 1.650 (50 × 10−4 ) 51.1 86.6 It is beyond the scope of this chapter to address the Al-Fe system in more detail, which will be done in a subsequent publication. We can, however, account for the role of Fe in the nominally binary Al-Cr-(Fe) and Al-Cu-(Fe) alloys using the experimental results for Al-Fe, as follows. The experimental yield stresses for Al-Fe do scale reasonably well with c2/3 suggesting that Labusch scaling applies to the Fe solutes despite the anomalous magnitude. We thus use the experimental yield stresses at T=78K to back-calculate an ˜ 2 for Al-Fe, from which we then obtain experimentally-derived effective value for c∆Ep values of τy /c2/3 = 13, 238M P a and ∆Eb /c1/3 = 28.8eV . Predictions for the Al-Cr-(Fe) and Al-Cu-(Fe) alloys using the theory plus the experimentally-derived parameters for Fe are shown in Table 2.4, and good agreement with experiments is now obtained for all cases. Figure 2.8 shows the experimental yield stress versus the predicted value for all of the alloys except Al-Fe, graphically demonstrating consistent agreement across the entire range of alloys at T=78K. Diak et al. have also recently measured the yield stresses of The quick brown fox jumps over the lazy dog. ∆Etot /Lc2/3 (eV/nm) Chapter 2. Solute strengthening in Al alloys from first-principles 44 w (nm) Table 2.4: Predicted and experimental [1, 2] tensile yield stresses at T = 78K for Al-Cr and Al-Cu alloys containing additional Fe solutes, using experimentally-derived wparameters 1 c for Fe. Solute c (%) Tensile Yield Stress σy (MPa) Predicted (78K) Experiment (78K) Cr-(Fe) 0.073 (10 × 10−4 ) 19.4 23.7 The quick brown fox jumps over the lazy dog. Cr-(Fe) Cu-(Fe) 0.302 0.090 4 Figure 7 (12 × 10−4 ) (12 × 10−4 ) 39.8 16.7 50.2 12.3 Cu-(Fe) 1.650 (50 × 10−4 ) 75.8 86.6 The quick brown fox jumps over the lazy dog. 100 Al!Mg 90 Al!Mn Potential Energy (eV) Al!Cr!Fe 80 experimental σyy (MPa) Al!Cu!Fe exp=predStress (MPa) experimental ! (MPa) predicted σy (MPa) 70 60 Figure 8 Stress (MPa) 50 40 5 Figure 8 30 20 The quick brown fox jumps over the lazy dog. 5 10 experimental σy (MPa) 0 0 20 40 60 80 100 predicted !σ (MPa) predicted (MPa) yy Figure 2.8: Experimental [1–3] versus predicted flow stress σy for 9 distinct alloys, showing overall good agreement. The dashed line has a slope of unity and indicates perfect agreement. the Al-Mn alloys at several elevated temperatures [3]. Using the parameters in Table 2.2 and the theory of Eq. (2.34), predictions of the model are compared against experiment for the Al-Mn system at three temperatures in Figure 2.9. The agreement is again very good over the entire range although there is a small systematic deviation. The quick brown fox jumps over the lazy dog. 4 Figure 7 Chapter 2. Solute strengthening in Al alloys from first-principles 45 The quick brown fox jumps over the lazy dog. 60 78K 198K Potential Energy (eV) 50 263K y (MPa) exp=pred Stress (MPa) !y (MPa) 40 predicted σy (MPa) Figure 8 experimental σ Stress (MPa) experimental 30 5 Figure 8 20 The quick brown fox jumps over the lazy dog. 10 5 experimental σy (MPa) 0 0 10 20 30 40 50 60 predicted !σy (MPa) predicted y (MPa) Figure 2.9: Experimental [3] versus predicted flow stress σy for 3 Al-Mn alloys at 78K, 198K and 263K, showing overall good agreement. The dashed line has a slope of unity and indicates perfect agreement. Diak and colleagues measured the activation volumes of some of the alloys listed in Table 2.3 [1, 2]. Figure 2.10 shows the activation volume M V /b3 versus the T=78K shear yield stress τy = σy /M in log-log form to elucidate any power-law scaling and stress equivalency. The predicted V are of the same magnitude as the experimentally measured values but typically over-estimate the experiments by roughly a factor of two. We consider this parameter-free prediction to be in good agreement with experiments because the activation volume reflects the product bζc wc that involves the two key length scales emerging from the theory. The predicted values fall nearly along a single curve, reflecting the stress-equivalency principle with the small variations attributable to dif- ferences in wc among alloys. Both experiments and predictions are approximately linear, Chapter 2. Solute strengthening in Al alloys from first-principles 46 indicating power-law scaling, but the slopes indicate two different power-laws, V ∼ τy−0.6 is predicted while the experimental trend is V ∼ τy−1.0 . One possible source of the dis- crepancy between experimental and predicted activation volumes is associated with the value of the line tension Γ. The finite temperature stress τy (T ) is only weakly dependent on Γ because the zero temperature yield stress scales as Γ−1/3 and the energy barrier scales as Γ1/3 , and these opposite dependencies on Γ partially cancel. However, the activation volume depends on the ratio of these quantities, and scales as Γ5/9 . A line tension Γ smaller than the EAM-Al-derived value of Γ = 0.47eV would decrease the predicted activation volumes by a constant factor, with a rather smaller decrease in predicted yield stress. Additionally, non-solute related strengthening mechanisms, such as a Peierls barrier, can influence the slope of the experimentally-measured V without having too much effect on the yield stress itself. Overall, reasonable agreement in the values for V supports the model presented here. 2.5 Conclusion We have shown that first-principles calculations of the interaction energy coupled with a parameter-free solute strengthening model leads to predicted flow stresses that agree with experiments over a wide range of Al alloys. The model both resolves fundamen- tal difficulties encountered in existing literature models and avoids introducing any ad- justable parameters. Previous predictions using first-principles input were limited to zero temperature flow stresses and provided little insight into the temperature and strain-rate dependence of the strengthening. Chapter 2. Solute strengthening in Al alloys from first-principles 47 4000 1000 M V /b3 Mg pred Mn pred Cr pred Cu pred Mg exp Cr exp Cu exp Fe exp 100 1 10 50 τy (MPa) Figure 2.10: Normalized activation volume M V /b3 versus 78K shear yield stress τy for both experimental [1] and predicted results, in log-log form, demonstrating stress equivalency. While the results obtained from the model are promising, the model only applies to moderate solute concentrations where Mott/Labusch statistics apply. For extremely dilute concentrations, Friedel statistics govern the strengthening behavior or solutes, and a point pinning model such as the one presented in Ref. [4] may be more applicable. It is therefore important to better understand the transition between these two regimes. Previous literature has explored this transition but a quantitative determination of the transition is yet to be achieved. Possibly related to this issue is the anomalous behavior of dilute Fe in Al. The high strengthening of Fe, relative to predictions here and all other solute additions, has implications for alloy design and processing, and so must be better understood. Chapter 2. Solute strengthening in Al alloys from first-principles 48 In the absence of first-principles input, for other solutes or other host materials than those studied here, the analytic model here can be treated as a two-parameter model in terms of the zero temperature flow stress τy0 (Eq. (2.32)) and the energy barrier ∆Eb (Eq. (2.28)). This is just a consequence of thermal activation theory. More subtle is ˜ 2 and wc . If wc the fact that τy0 and ∆Eb depend on the fundamental parameters c∆Ep is taken as 1.5 times the core width for a given host, then the model here reduces to a one-parameter theory. Experimental data can then be compared to predictions in which ˜ 2 is used to fit temperature- and/or concentration-dependent data. c∆Ep An alternative simple approach also emerges by interpreting our results from a pure continuum elasticity perspective. While formally inadequate in the core, continuum 4/3 ˜p ∼ ∆vm , and thus that τy0 ∼ ∆vm and ∆Eb ∼ ∆vm . 2/3 elasticity suggests that ∆E Our full first-principles data, culminating in the results of Table 2.2, follows this scaling fairly closely: τy0 /c2/3 ≈ 31.1 ± 6.3 MPa/˚ A4 ∆vm 4/3  , (2.40) ∆Eb /c1/3 ≈ 1.31 ± 0.03 eV/˚ A2 ∆vm 2/3  . (2.41) Thus, it may be possible to use inexpensive first-principles methods to compute the surrogate material parameter ∆vm . For new host matrix materials, the coefficients above would be unknown, but fitting to limited experimental data on one solute might reveal the coefficient, which would then permit application to other solutes or combinations of solutes. However, the variations in Eq. (2.40) suggest that such an approach has limited accuracy, with predictions of the zero-temperature strength deviating by up to ±30%. With smaller variations in the energy barrier, the finite-T strength would thus deviate Chapter 2. Solute strengthening in Al alloys from first-principles 49 by a similar ±30%. The full predictions here are generally in much better agreement with experiment, although requiring far greater computational cost. We will report on results of such an analysis for the Cu-Mn and Cu-Ge systems, providing a different interpretation of the data in Ref. [17], in a future publication. Such a strategy is similar (i) in spirit to the work of Zander et al. [34, 35] who computed ∆vm for various solutes P 2/3 (i)4/3 using first-principles and then assumed σy = K i ci ∆vm , with K as a single fitting parameter independent of solute. However, our analysis here shows that the additivity assumed by Zander et al. should be replaced by Eq. (2.38). We provide additional insight into the energy barrier so that the results can be applied at finite temperature, although Eq. (2.40) shows such an approach to only have reasonable accuracy. The overall methodology presented here, including the full first-principles analysis of the solute/dislocation core interactions and validated for Al-X alloys, is robust and can readily be applied to other fcc alloys as well as basal slip in hcp alloys. We will report on application of the model to basal slip in Mg-Al in the next chapter. This model thus represents a significant advance in computational metallurgy that can be used to guide the design of technologically important multicomponent solute strengthened alloys. Chapter 3 Solute strengthening in Mg-Al alloys from first-principles 3.1 Introduction Interest in using magnesium alloys as structural materials has been increasing due to their excellent material properties, such as low density and high specific strength [51, 52]. In particular, applications of magnesium alloys to the automotive industry looks promis- ing due to the ever increasing demand for more light weight, fuel-efficient vehicles [53]. However, limitations in formability prevent the wide-spread adoption of magnesium al- loys. This limitation stems from the anisotropy in plasticity response associated with the various dislocation slip systems in the hexagonal close packed (hcp) structure. Pro- cessing and alloying strategies are aimed at overcoming these limitations, and these 50 Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 51 strategies can be informed through a detailed understanding of the deformation mech- anisms occurring in Mg and its alloys. Advancements in computational capabilities have enable the study of disocation struc- ture and mobility mechanisms. Molecular dynamics has been use to study dislocation [54–60] and twin [61–63] structures in magnesium. However, the predicted behavior depends strongly on the interatomic potentials used so that such studies have limited predictive capability, and the availability of potentials for alloy systems is limited. First- principles calculations have emerged to examine some dislocation core structures and energetics [4, 6, 64–66]. However, dislocations such as hc + ai have a large burgers vector and MD studies suggest that the cores are spread considerably, making first-principles modeling computationally challenging. Extending first-principles studies to direct mod- eling of dislocation motion or core structural changes upon alloying, for instance, remains prohibitive in most cases. Thus, despite notable advancements, predictions for impor- tant quantities such as the finite temperature yield stress of an alloy do not yet exist. In this chapter, we advance the field by using first-principles energetics of solutes in and around the basal dislocation core as input to an analytic model for solute strengthening, from which we make parameter-free predictions of the yield stress versus temperature for basal slip in Mg-Al alloys. Our predictions are in excellent agreement with literature experimental data over a range of Al concentration. The main feature of the work is our use of a new model for solute strengthening that has been successfully applied to Al-X alloys [37, 67] with the first-principles input coming from Yasi et al. [4]. While the quantitative success is impressive, a new physical feature of Mg-Al also emerges naturally from the theory. Specifically, because the dissociation of the basal edge core into partials Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 52 is wider than in Al, the theory predicts the existence of two possible energetically stable dislocation configurations. One configuration has a low energy barrier but a high zero- temperature flow stress while the other configuration has a high energy barrier but a low zero-temperature flow stress. Considering both configurations, the flow stress is controlled by the first configuration at low temperature and the second configuration at high temperature. The flow stress versus temperature thus shows a high-temperature domain that is relatively temperature-insensitive and appears similar to an “athermal” strength. Such an “athermal” regime is often observed in many materials [5, 17, 21, 68–75] and is usually attributed to unquantified “long-range” dislocation interactions. Our theory demonstrates that this apparent “athermal” regime remains quantitatively attributable to the solute strengthening and predicts the variation of this “athermal” regime as a function of Al solute content in good agreement with data. This chapter is limited to the study of Al solutes effects on basal slip in Mg, but the concepts are readily extended to other solutes and dislocation types. Moreover, our study demonstrates that a steady insistence on mechanistic details based on computable material quantities pays dividends in the form of both quantitative results and new physical insights. 3.2 Solute Strengthening Model The solute strengthening model is described in detail in Ref. [37, 67]. The model uses concepts first developed by Labusch [13, 15] but introduces key modifications that ren- der the model convergent and parameter-free. The key concept in the model is that an initially straight dislocation in a random field of solutes can lower its energy by seek- ing out regions in the solid containing favorable solute fluctuations. The dislocation Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 53 ζ w [1000] 2ζ L [2¯ 4¯ 10] [0001] Figure 3.1: Schematic of a dislocation roughening into segments of length 2ζ over length w due to favorable solute fluctuations. accomplishes this by bowing out towards these favorable fluctuations. While the bowed out segments lower their interaction energy with the solutes, the bowing out process in- creases the line energy of the dislocation. The total change in energy as the dislocation goes from the straight configuration to the bowed out configuration is therefore the sum of the contribution due to favorable solute fluctuations and the elastic energy cost of bowing out. We consider a positive basal edge dislocation of length L lying in the z−direction, with Burgers vector b in the x−direction and evaluate the total energy of a fluctuating config- uration shown schematically in Figure 3.1, where straight dislocation segments of length ζ lie in regions of favorable solute fluctuations, with bowing out of amplitude w and inter- mediate segments of length ζ connecting these segments to form the overall dislocation line. The interaction energies between solutes and the dislocation are obtained through first-principles computations. We minimize the total energy of the dislocation to find the characteristic scales ζc and wc as a function of the interaction energies, dislocation line tension, solute concentration, and other material parameters. We now sketch out the path from solute/dislocation energies to dislocation strength- ening; a full description is provided in Ref. [67]. A key quantity in the analysis is the Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 54 potential energy decrease due to favorable solute fluctuations, which is obtained by con- sidering the standard deviation of the total solute/dislocation interaction energy as the dislocation segment of length ζ moves a distance w from its original position, " 1 #   cζ 2 ˜p (w) · L ∆Ep (ζ, w) = − √ ∆E (3.1) 3b 2ζ where  1 2 ˜p (w) = 2 2 X ∆E (1 − χ (w, yj )) U (xi , yj ) , (3.2) ij Here, U (xi , yj ) is the solute/dislocation interaction energy of a solute at position (xi , yj , z) with a dislocation centered at the origin and P k U (xk − w, yj ) U (xk , yj ) χ (w, yj ) = P 2 . (3.3) k U (xk , yj ) is the correlation function of the solutes in the yjth layer when the dislocation segment moves a distance w. The total energy change as the dislocation goes from the straight configuration to the one depicted in Figure 3.1 is then " 1 #   w2   cζ 2 ˜p (w) · L . ∆Etot (ζ, w) = Γ − √ ∆E (3.4) 2ζ 3b 2ζ Minimizing Eq. (3.4) with respect to ζ yields the characteristic segment length ζc as a function of the roughening amplitude w, !1 √ Γ2 w4 b 3 ζc (w) = 4 3 , (3.5) ˜p2 (w) c∆E Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 55 and the total energy change ∆Etot !1 32/3 c2 ∆E ˜p4 (w) 3 ∆Etot (w, ζc (w)) = − 1/3 L. (3.6) 82 b2 w2 Γ as a function of only w. Numerical minimization of the total energy per unit length ∆Etot /L gives the characteristic roughening amplitude wc . The associated characteristic energy ∆Ec of each “unit cell” of dislocation line consisting of the straight segment of length ζc and the surrounding segment length of zetac is then given by !1 35/6 cwc2 Γ∆E˜ 2 (wc ) 3 p ∆Ec = 5/3 . (3.7) 2 b The zero-temperature energy barrier for motion of the dislocation is the average energy change as the dislocation segment moves from some favorable minimum to a maximum wc , which can be shown to be [37], √ ! 2 Γ∆E˜p2 (wc,i ) !1 4 2 − 1 35/6 cwc,i 3 ∆Eb,i = · 5/3 . (3.8) 3 2 b The finite-temperature flow behavior follows from standard thermal-activation theory with the energy barrier computed above plus considerations from Labusch regarding larger-scale fluctuations of the dislocation configuration that can arise with increasing temperature (corresponding to lower applied stresses). Under an applied resolved shear Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 56 stress τ , the energy barrier can be approximated by   τy0   0.51 ∆Eb log , τ /τy0 < 0.46   τ ∆E (τ ) = (3.9)  h  i 3 2 ∆Eb 1 − ττy0  , τ /τy0 ≥ 0.46   where τy0 is the zero temperature yield stress, !1 c2 ∆E˜p4 (wc ) 3 τy0 = 1.01 · . (3.10) Γb5 wc5 Once again, using the same arguments as that stated in Section 1.4, the finite tempera- ture stress required for dislocation flow is given by     1 kT ˙◦  τy0 exp − 0.51 ∆Eb ln ˙ τ /τy0 < 0.46   τy (T ) = (3.11)    2  kT ˙◦ 3  τy0 1 − ∆Eb ln ˙ τ /τy0 ≥ 0.46   3.3 Solute/dislocation Interaction Energies To evaluate Eq. (3.1), accurate determination of U (xi , yj ) is essential. Far from the dislocation core, the continuum approximation U (xi , yj ) ≈ −p (xi , yj ) ∆v (3.12) is accurate, where ∆v is the misfit volume of the solute and p = −σkk /3 is the pressure field exerted by the dislocation at the location of the solute [7]. The misfit volume is Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 57 calculated using Density Functional Theory (DFT), implemented through Vienna Ab- initio Simulations Package (VASP) using a methodology described in Ref. [37], and is ˚3 . The pressure field p (xi , yj ) is calculated using the determined to be ∆v = −7.89 A Peierls-Nabarro model, ∞ 1−ν µ b0 (x0 ) yj Z p (xi , yj ) = − dx0 (3.13) 1 + ν 3π −∞ (xi − x0 )2 + yj2 where ν is the Poisson ratio and µ is the shear modulus of magnesium. The distribution of the Burgers vector b0 (x0 ) = db/dx is calculated using the first-principles-computed dislocation core structure from Ref. [4] that is shown in Figure 3.2(a). This structure is obtained through the use of a DFT domain that is coupled with a Lattice Green’s Function method to simulate an isolated dislocation in an infinite solid. Near the dislocation partials, the deformation is very large and the continuum approxi- mation is not accurate, necessitating the first-principles computations. The DFT results of Yasi et al. in this core region are shown in Figure 3.2(a) and shows that the DFT results must be used for atomic sites in the two rows above the slip plane and in one row below the slip plane. The DFT data does not extend laterally outside the core and also does not include all sites within the core. We thus use the DFT data and cubic splines to interpolate the energies at sites along each atomic plane while enforcing convergence to the continuum results far to the left and right of the core. The solute/dislocation interaction energies so obtained are shown in Figure 3.2(a), matching the DFT data and smoothly merging into the continuum results outside the core. With the interaction energies established, we can compute the correlation function Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 58 (a) eV0.1 2 1.5 1 0.05 y (nm) ([0 0 0 1]) 0.5 0 0 -0.5 -1 -0.05 -1.5 -2 -0.1 -2 -1 0 1 2 x (nm) ([1 0 0 0]) (b) 1.0 2.5 0.9 2 0.8 1.5 yj (nm) ([0 0 0 1]) 0.7 1 0.5 0.6 0 0.5 -0.5 0.4 -1 0.3 -1.5 0.2 -2 0.1 -2.5 0.0 0 2 4 6 w (nm) Figure 3.2: (a) Solute/dislocation interaction energy map for Al solutes as a function of solute position. Circles with bold outline are from Ref. [4]; cicles with non-bold outline are from Eq. (3.12); cicles with no outline are interpolated using cubic splines. (b) A contour plot of the correlation function χ (w, yj ) for a solute initially (0, yj ) as the dislocation moves a distance w away, as computed for an Mg-Al alloy. Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 59 χ (w, yj ), which reflects how the interaction energy of a solute on a particular atomic plane parallel to the slip plane is correlated as the dislocation slides by on the slip plane. This correlation function is shown in Figure 3.2(b), and shows that the solutes immediately in the partials are decorrelated when the dislocation has moved by one-half the partial separation distance, and then are slightly more correlated as the dislocation moves by another one-half of the partial separation, followed by a rapid decay in the correlation. In contrast, solutes in atom planes further from the slip plane are highly cor- related over much longer distances, with even the first few planes reasonably correlated over distances of the order of 3nm. These differences in correlation of the solute energies will be shown below to have an important effect on the overall dislocation configurations and energetics, and subsequently the finite-temperature flow stress. 3.4 Predictions and Discussion Using the energies and correlation function shown in Figures 3.2(a) and 3.2(b), we can compute the dislocation energy per unit length as a function of the amplitude w of the fluctuations (Eq. (3.6)) and look for the value wc that minimizes the energy. In the calculations, we use the line tension Γ = 4.0 eV/nm [50]. Figure 3.3 demonstrates that for Mg-Al there are two minima, not just one as in Al, corresponding to two roughening amplitudes wc,1 and wc,2 , and two associated segment lengths ζc,1 and ζc,2 and characteristic energy barriers ∆Eb,1 and ∆Eb,2 . The two minima arise because of the nature of the correlations in the interaction energies. Due to the large partial separation in Mg basal edge dislocations, moving the dislocation Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 60 -0.007 -0.008 -0.009 wc,1 wc,2 ∆Etot /Lc2/3 (eV/nm) -0.010 -0.011 -0.012 -0.013 -0.014 -0.015 0 1 2 3 4 5 6 7 w (nm) Figure 3.3: Normalized energy per unit length ∆Etot /Lc2/3 versus roughening am- plitude w for an Mg-Al alloy. by wc,1 = 0.98 nm almost completely decorrelates the solute fluctuations within the partials of the dislocation (|yj | ≤ 2), resulting in the first minimum. However, this small roughening amplitude is not sufficient to the decorrelate solutes farther from the dislocation core (|yj | > 2). By increasing the roughening amplitude, the dislocation is able to access favorable fluctuations far away from the core, leading to the second minimum at wc,2 = 3.67 nm. In Al, which has a smaller partial dissociation distance, only one minimum exists. Thus, we see that the detailed structure of the dislocation core, which is accurately predicted only by first-principles methods, plays a key role in determining the characteristic length and energy scales for solute strengthening. The values for wc , ζc , and ∆Eb for the two minimizing configurations 1 and 2 are shown in Table 3.1. Configuration 1, associated with the near-core behavior, has a relatively Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 61 Table 3.1: Parameters for the two dislocation configurations in Mg-Al alloys, normal- ized by the appropriate scaling of the solute concentration. Normalized Parameter Config. 1 Config. 2 wc 1.12 nm 3.67 nm ζc · c1/3 6.04 nm 20.66 nm ∆Eb /c1/3 1.92 eV 6.07 eV τy0 /c2/3 224.4 MPa 63.0 MPa higher zero temperature yield stress but a relatively low energy barrier. Configuration 2, associated with the longer-range fluctuations, has a lower zero temperature yield stress but exhibits a higher energy barrier. The effect on the finite-temperature yield strength is that the dislocation can sample both configurations, and so must overcome both via thermal activation to achieve glide. The strain rate ˙ as a function of stress and temperature is thus given by      1 1 1 1 ∆E1 (τ ) ∆E2 (τ ) = + = exp + exp . (3.14) ˙ ˙1 ˙2 ˙◦ kT kT Inverting Eq. (3.14) gives the finite temperature yield stress τy (T, ) ˙ of the alloy. Phys- ically, the two configurations control different stress/temperature regimes of the flow behavior. At low stresses/high temperatures, since ∆E1 < ∆E2 then ˙2  ˙1 and ˙ ≈ ˙2 ; the temperature is sufficiently high such that the dislocation is easily thermally activated over the smaller ∆E1 barriers and the larger ∆E2 barrier controls the thermal activation. Conversely, at high stresses/low temperatures, ∆E2 < ∆E1 , and ˙1  ˙2 , so that ˙ ≈ ˙1 ; the stress is sufficiently high to suppress the longer-range barrier of configuration 2, leaving the barrier ∆E1 to control the thermal activation. The analytical prediction of the existence of two different minimum configurations, cor- responding to two different energy barriers at two different length scales, is unexpected. Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 62 To validate this result further, we thus performed direct stochastic simulations of straight dislocation segments moving through a random field of solutes. In this case the value of ζ is fixed but the presence of the solute correlations should still reveal the existence of two energy barriers that control the flow in different regimes of stress. The simulation is performed as follows. The total solute/dislocation interaction energy is calculated as a function of the dislocation segment glide position x. The dislocation is then placed at one of the many local energy minima and the largest energy barrier ahead of the dislo- cation is recorded, but restricting to a distance 2wc,2 to avoid very long-range barriers that are irrelevant. An increasing applied stress is then imposed on the system and the barrier height versus stress is measured, generating a simulated ∆E (τ ). This process is repeated 3000 times to get statistically significant results. The analytic model predicts two barriers that should vary as " #3 2 τ ∆Ei0 (τ, ζ) = 0 ∆Eb,i 1− 0 , (3.15) τy0,i (ζ) with 0 0 π ∆Eb,i (ζ) τy0,i (ζ) = . (3.16) 2 bζwc,i where the prime (’) indicates that these predictions are based only on the potential energy, since elastic effects of bowing are not included. The theory also shows that the energy barrier should scale as ζ 1/2 and the strength should scale as ζ −1/2 , so we plot ∆Eζ −1/2 vs. τ ζ 1/2 to obtain results independent of the actual ζ. The simulated energy barrier vs. stress and the predictions of the analytical model are shown in Figure 3.4(a), and show precisely the behavior anticipated in the discussion above. At low stresses, the longer-range fluctuations with the larger energy barrier ∆E20 persist and control the Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 63 0.4 ζ = 16.6 nm 0.35 ζ = 56.4 nm ∆E 0 /ζ 1/2 (eV/nm1/2 ) 0.3 ∆E10 /ζ 1/2 ∆E20 /ζ 1/2 0.25 0.2 0.15 0.1 0.05 0 0 50 100 150 1/2 1/2 τζ (MPa nm ) Figure 3.4: Comparison between the stress dependent energy barrier ∆E 0 (τ ) deter- mined using the direct stochastic simulation and that expected by the analytic model for a dislocation segment of length ζ = 16.6 nm and ζ = 56.4 nm. overall energy barrier. At high stresses the long-range barriers are defeated and the short- range barriers persist and control the overall energy barrier. The stochastic simulations thus validate the existence of two different energy barriers, with two different length scales and zero-temperature yield stresses, and show that two corresponding regimes of flow behavior exist. We now compare predictions of the model against experimental data. The critical re- M g−Al solved shear stress (CRSS) τy,expt for basal slip in a range of Mg-Al alloys over a range of temperatures was measured at a strain rate of ˙ = 1.66 × 10−4 s−1 by Akhtar and Teghtsoonian [5]. They also measured the CRSS tauM g y,expt for pure Mg, i.e. the Peierls barrier, at the same temperatures. Because the alloy strengths are not significantly larger than the Peierls stress for pure Mg, it is necessary to include the Peierls stress Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 64 along with the solute-strengthening prediction. Because the Peierls barrier is associated with slip over one burgers vector while the solute strengthening is associated with slip over the larger length scales wc , it is appropriate to add the two contributions to obtain the total strength. Our prediction is thus Mg M g−Al τy = τy,expt + τy,theory (3.17) M g−Al where τy,theory is computed using our theory. In the predictions, we use ˙◦ = 105 but the results depend very weakly on the precise since it appears in the logarithm in Eq. (3.11). Figure 3.5(a) shows the yield stress τy of three different Mg-Al alloys at various tem- peratures and as predicted by our model. The agreement is extremely good over the entire range of concentrations and temperatures, with no fitting or adjustable param- eters whatsoever. Figure 3.5(b) shows the experimental results versus the predictions; all of the materials cluster along a line of slope unity that indicates perfect agreement. We note that the predictions nicely capture the transition in strength with increasing temperature, where the experimental data appears to approach a “plateau”. Such a plateau, widely seen in solute-strengthened materials, is often ascribed to some long range, internal stresses that contribute some “athermal” strengthening due to some other mechanism, e.g. solute multiplets [17], elastic dislocation/dislocation interactions [76], or solute drag “friction” [21], but without quantitative analysis or prediction [11]. Other workers, attempting to explain the solute-concentration dependence of the plateau, have invoked the standard solute-strengthening theories themselves to predict the plateau, leaving open the origins of the lower-temperature strengthening [17, 69, 73, 74, 77, 78] and are still quantitatively lacking. Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 65 (a) 12 1.63% Al 0.89% Al 10 0.59% Al 8 τy (MPa) 6 4 2 0 100 200 300 400 Temperature (K) (b) 10 1.63% Al 0.89% Al experimental τy (MPa) 8 0.59% Al exp = pred 6 4 2 0 0 2 4 6 8 10 predicted τy Figure 3.5: (a) Experimental [5] and predicted finite temperature yield stress τy (T ) for three different Mg-Al alloys. The solid lines are τy,eff , the dashed lines are τy1 and the dashed-dot lines are τy2 . (b) Experimental versus predicted yield stress τy for the three alloys presented above. The dashed line has a slope of unity and indicates perfect agreement. Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 66 In contrast, the present theory, based on a sound mechanistic analysis and the most ac- curate computations currently possible, reveals a physical origin for this athermal stress. We find that the “plateau” is indeed directly connected to the solute strengthening it- self, but physically due to the wide dissociation of the dislocation core structure that leads to correlations in solute/dislocation interactions over a longer length scale than just the core itself. The higher energy barrier ∆Ec,2 over a longer range (wc,2 leads to a much weaker dependence of strength versus temperature in the higher temperature regime than in the low temperature regime. At the same time, the short-range solute correlations are responsible for the low-temperature strengthening. We postulate that the mechanism observed here for Mg is a general one that operates for materials with a large partial spacing, the latter arising in materials with a low stable stacking fault energy. And indeed, the material systems where a concentration-dependent “athermal” “plateau” stress is observed are those with low stacking fault energies: Cu[17, 21, 71–75], Ag[21, 70, 73], and Au[69] alloys. Thus, no new mechanisms are required to explain the high-temperature behavior in Mg and the explanation for Mg is qualitatively consistent with trends in other materials. 3.5 Conclusion We have applied the parameter-free solute strengthening model of Ref [37] to accurately predict the finite temperature basal flow stresses of Mg-Al alloys, using first-principles data on solute/dislocation interactions from the work of Yasi et al.[4]. This demonstrates that the model, previously applied to Al-X alloys, has broader applicability to materials where solute effects are limited to influencing the glide along the slip plane. In the Chapter 3. Solute strengthening in Mg-Al alloys from first-principles 67 case of Mg, the dissociated nature of basal edge dislocation has consequences for solute strengthening that are absent from the case of Al-X alloys. Specifically, the broader core spreading leads to two different possible minimum energy configurations for the dislocation, leading ultimately to a transition in the controlling configuration as the temperature is increased. This transition is manifested in an apparent plateau in the strength vs. temperature at higher temperatures. Overall, our results demonstrate that an insistence on rigorous mechanistic modeling, with accurate first-principles input, pays high dividends in achieving predictive models that, moreover, can reveal new mechanisms of material behavior that has previously evaded explanation. Chapter 4 Strong-/weak-pinning transition at finite temperature 4.1 Introduction Solute strengthened alloys are some of the most technologically important materials in industry. Understanding the strengthening mechanisms in solid solutions is therefore important for developing accurate constitutive models used to design components and, moreover, to design new alloys with improved processing. The fundamental physical origin of solute strengthening is the interaction between the dislocation deformation, e.g. the stress field, and the solute deformation, e.g. the distortional strain it creates when introduced into a substitional lattice site. Due to this interaction, dislocation motion under an applied shear stress is suppressed, leading to an increase in the alloy yield stress with increasing solute concentration. However, since the dislocation can interact 68 Chapter 4. Strong-/weak-pinning transition at finite temperature 69 with many solutes (on, above, and below) the glide plane and since the dislocation is a flexible line, the actual strengthening depends on the configurations the dislocation takes within the random field of solutes. There are two broad classes of theories for solute strengthening. The strong-pinning theory (Friedel [8], Fleischer [7, 9]) considers the solute atoms in the plane of the dis- location as independent point obstacles that pin the dislocation, which bows out in the regions between the solutes. With the parameter fmax characterizing the solute’s max- F, imum resistive force, the zero temperature yield stress for the Friedel mechanism, τy0 corresponds to the stress required to bow out the dislocation line in between the pinning solutes sufficiently to overcome their resistive force, leading to  3   F fmax fmax 2 2Γ 1 τy0 =  = c2 . (4.1) F bζF τy0 2Γ b2 where ζF is the Friedel sampling length and Γ is the dislocation line tension. The weak-pinning model (Mott [14] and Labusch [13, 15]) considers the collective in- teraction of many solutes around the dislocation, and attributes strengthening to the occurence of favorable statistical fluctuations in the solute configuration, when each indi- vidual solute alone would not be strong enough to pin the dislocation at the same stress level. Labusch first developed a model for considering solutes in the glide plane of the dislocation, although the concept is really three-dimensional in nature. Labusch derived a zero-temperature flow stress uses many of the same parameters as in the Friedel model plus a parameter w capturing the spatial range of interaction of the solutes with the Chapter 4. Strong-/weak-pinning transition at finite temperature 70 dislocation. The zero-temperature Labusch strength in this case is given by  31 c2 Umax 4  L τy0 = . (4.2) wΓb9 where Umax is the maximum solute/dislocation interaction energy. Using the zero-temperature results above, the transition between the two can be made by assuming that the strongest of the two mechanisms is operative. By taking the ratio of the Friedel strength to the Labusch strength, we obtain the condition F  16 τy0 fmax b2  ∝ . (4.3) τy0,L Γcw2 This ratio is essentially the β parameter first identified by Labusch, fmax b2 β= (4.4) 4Γcw2 although the analysis followed a different path. For a given solute, all material param- eters are unchanged, so that the transition occurs at a solute concentration c◦ given by fmax b2 c◦ = . (4.5) 4Γw2 Using typical material properties (see below for more details), the Friedel model should be applicable at concentrations below ∼ 10−2 . Thus, it is often assumed that the Friedel model is applicable for dilute alloys. However, this result is based on zero-temperature considerations and a restricted version of the Labusch concept for “weak” pinning. It is the flow stresses at finite temperatures and finite strain rates that are experimentally Chapter 4. Strong-/weak-pinning transition at finite temperature 71 relevant, and the transition should be based on a Labusch-type model that includes solutes off the glide plane. Since all of the necessary material parameters to assess the two models were not available, and neither model could quantitatively predict alloy strengthening (although trends and scaling could be identified), the domains of applicability of the two models were not of great importance. In recent years, the ability to obtain dislocation/solute interaction energies using first-principles quantum mechanical methods, and the creation of a new predictive model for the Labusch regime, has overcome these limitations. A clear de- lineation of the domains of dominance of the Labusch-like and Friedel pinning models, including at finite temperature, thus has new relevance for understanding and designing solute-strengthened metals. The purpose of this chapter is to examine this transition in detail, using parametric studies that span a wide range of possible dislocation structure and highly-accurate studies that use first-principles data for solutes in Al alloys. We show that over a wide range of dislocation structures, and specifically for Al alloys and basal/prism slip in Mg alloys, that the Labusch-type model controls that strength over a wide range of temperatures and at atomic solute concentrations higher than 10−4 . Thus, contrary to the conventional wisdom of the zero-temperature transition condition given by Eqn. (4.4), the Friedel model has very limited applicability, and the Labusch model should be the prevailing model for solute strengthening across a wide class of metals. The remainder of this chapter is organized as follows. In Section 4.2, we briefly discuss thermally-activated flow, leading to the finite-temperature, finite-strain rate flow stress, for the Friedel model. We then present the results of our recent predictive Labusch-type model for finite-temperature flow stress [37, 67]. In Section 4.3 we use an analytic Chapter 4. Strong-/weak-pinning transition at finite temperature 72 Peierls-Nabarro-type model to generate the dislocation/solute interaction energies for dislocation cores of varying degrees of dissociation and partial spreading, and examine the Friedel-Labusch transition quantitatively. In Section 4.4, we use first-principles dislocation/solute interaction energies for Al-X (X = Mg, Si, Cu, Cr, Mn and Fe) and Mg-Al alloys to make specific predictions for the Friedel-Labusch transition as a function of temperature and solute concentration. These results show that the Labusch-type model is dominant over nearly the entire practical range of temperatures (T > 78K) and solute concentrations (c > 10−5 ) in these materials systems. In Section 4.5 we discuss some additional considerations and summarize. 4.2 Finite-Temperature Friedel and Labusch-type Models At finite temperature, the dislocation can pass an individual solute via thermal activation over the energy barrier imposed by the solute atom at any applied stress τ less than the zero temperature yield stress τy0 [21]. The stress-dependent energy barrier ∆E F (τ ), including the effects of bowing that relate the applied stress to the force on the solute, follows the general form " !p #q F τ ∆E (τ ) = Emax 1 − F (4.6) τy0 where Emax is the maximum solute/dislocation interaction energy and the parameters 0 < p < 1 and 1 < q < 2 depend on the details of the force-distance profile of the solute obstacle [12]. A representative case is p = 2/3 and q = 3/2 corresponding to a Chapter 4. Strong-/weak-pinning transition at finite temperature 73 parabolic force-distance profile [20]. Other profiles can be considered [17], but this does not significantly change the conclusions of this chapter. For a quasistatic loading, the rate of escape ν by activation over the stress-dependent barrier height ∆E F (τ ) can be approximated using transition state theory [21] as ν = ν◦ exp(−∆E F (τ )/kT ) where ν◦ is the attempt frequency, T is the temperature, and k is Boltzmann’s constant. The microscopic dislocation escape rate ν can be related to the macroscopic strain rate ˙ by the well established model [12], ∆E F (τ )   ˙ = ˙◦ exp − (4.7) kT where ˙◦ = ρbdν◦ is the reference strain rate, ρ is the dislocation density per unit area and d is the flight distance over which the dislocation moves after each escape. This relation, along with Eq. (4.6), can be inverted to obtain the finite temperature stress required for dislocation flow as "   2 # 23 T 3 τyF (T, ) F ˙ = τy0 1− (4.8) T◦F where T◦F = Emax /k ln(˙◦ /) ˙ is the characteristic temperature for the Friedel mechanism. The Friedel model is thus characterized by two key features of the dislocation/solute interaction: the energy barrier Emax and the maximum force fmax , which corresponds to the maximum slope of the interaction energy versus solute/dislocatoin distance along the glide plane. In the Labusch-type model, an initially-straight dislocation is allowed to bow in the glide plane to seek out favorable local environments of the random solutes that minimize the Chapter 4. Strong-/weak-pinning transition at finite temperature 74 potential energy. The bowing has a cost in elastic energy, and so the typical configuration of the dislocation is that which minimizes the total energy as a function of the wavelength ζ and amplitude w of the bowing. We have recently developed a model based on this general physical picture in Refs. [37, 67] that includes the finite-temperature flow stress. The outcomes of our analysis are a characteristic wavelength ζc and amplitude wc , and L . These quantities an associated energy barrier ∆Eb and zero-temperature flow stress τy0 are given by !1 √ Γ2 wc4 b 3 ζc (w) = 4 3 , (4.9) ˜ 2 (wc ) c∆E p √ ! ˜ 2 (wc ) !1 4 2 − 1 35/6 cwc2 Γ∆E p 3 ∆Eb = · 5/3 . (4.10) 3 2 b and !1 ∆E˜ 4 (wc ) 3 2 L p τy0 = 1.01 · c3 . (4.11) Γb5 wc5 where  1 2 ˜p (wc ) = 2 2 X ∆E (1 − χ (wc , yj )) U (xi , yj ) , (4.12) ij and U (xi , yj ) is the interaction energy between a solute at position (xi , yj ) relative to a dislocation centered at the origin and P k U (xk − w, yj ) U (xk , yj ) χ (w, yj ) = P 2 . (4.13) k U (xk , yj ) is the correlation function of that interaction energy as the dislocation glides from one location to another location a distance w away. The characteristic quantity wc is com- puted by minimization of the total energy, and so is not expressible in an analytic form. We note that this formulation includes the dislocation/solute interactions throughout Chapter 4. Strong-/weak-pinning transition at finite temperature 75 all space. It is through the correlation function of this energy that the sums in the formulation become convergent, because U scales only as 1/r for large distances r, and so the formulation is rather different than the nominal Labusch model. Physically, the length wc seems similar to the “range of the solute/dislocation interaction w” in the original Labusch model. but we will see below that wc and w enter rather differently in the predicted transition from weak to strong pinning. The thermally-activated behavior for this Labusch-type model follows arguments similar to those for the Friedel model with the stress-dependent energy barrier taking the form of Eq. (4.6) with p = 1 and q = 3/2, " !# 3 2 L τ ∆E (τ ) = ∆Eb 1 − L , (4.14) τy0 At elevated temperatures, Eq. (4.14) actually transitions over to an logarithmic form, as discussed by Labusch [79]. This means the Labusch mechanism becomes more dominant than predicted by Eq. (4.14) with increasing temperature. However, the temperature range at which the logarithmic form is relevant in the Friedel-Labusch transition is narrow, occurs at very low concentrations (< 10−5 ), and Eq. (4.14) is assumed to apply for T < T◦F and this assumption has no important effects on our conclusions. The finite-temperature flow stress follows by the equivalent of Eq. (4.8) as "  2 # T 3 τyL (T, ) L ˙ = τy0 1− (4.15) T◦L where T◦L = ∆Eb /k ln(˙◦ /) ˙ is the characteristic temperature for the Labusch mecha- nism. Chapter 4. Strong-/weak-pinning transition at finite temperature 76 At a given applied stress, the motion of the dislocation should be controlled by the mechanism having the largest energy barrier. For instance, if the energy barrier for individual solute pinning exceeds that of the collective pinning, then after the dislocation can overcome the individual barrier it can rapidly pass over the collective barrier. In terms of the flow stress at finite temperature and imposed strain rate, the controlling mechanism is therefore the one with the highest flow stress. The transition between the two regimes can thus be taken to occur when τyF (T ) = τyL (T ), which naturally extends the zero-temperature result to finite temperatures. For any given solute, we determine the solute concentration at which the transition F = τL occurs. At zero temperature, this critical concentration follows by setting τy0 y0 using Eqs. (4.8) and (4.15) above to yield !8  wc2 fmax  wc fmax c◦ = 0.12 . (4.16) ∆E˜p (wc ) b2 Γ Comparing to the form in Eq. (4.5), something like the traditional β parameter seems to emerge but with (i) wc in the numerator instead of w in the denominator and (ii) an additional factor that is the ratio of the Friedel maximum force fmax to a characteristic ˜p /wc emerging from the collective solute energies in the Labusch-type force quantity ∆E model. The transition between the Friedel and Labusch regimes at finite temperature is calculated as the concentration at which τyF = τyL at the specified temperature. Since the finite temperature yield stress for the two mechanisms scale differently with temperature, the transition concentration can be significantly different from c◦ . From the formalism above, we can see that the transition between Friedel and Labusch Chapter 4. Strong-/weak-pinning transition at finite temperature 77 mechanisms is intimately connected to the solute/dislocation interaction energies, which are in turn tied to the structure of the dislocation core. To quantitatively compute the transition concentration at any temperature requires a specific model for these interac- tion energies. We address this in the next section. 4.3 Effect of core structure in the Friedel-Labusch transi- tion Both Friedel and Labusch models require solute-dislocation interaction energies U (xi , yj ), which in turn are highly dependent on the dislocation core structure. At the atomic scale, the dislocation is delocalized to some degree over the glide plane. Additionally, in materials with sufficiently small stacking faults, the dislocation can dissociate into two partials. Both the core spreading and partial dissociation affect the characteristic forces and energies associated with both strengthening mechanisms. In order to systematically study the effect of the core structure to the transition concentration, an analytic model for a generalized dissociated dislocation core is considered. To consider a general dislocation core, the dislocation is envisioned to have separated into two partials with separation distance d. In addition to this, each partial is delocal- ized around its center by a normal distribution with standard deviation σ. From this description, we generate a discrete Burgers vector distribution at positions xk along the Chapter 4. Strong-/weak-pinning transition at finite temperature 78 slip plane ∆b (xk ) as   2    2  1 xi −d/2 1 xi +d/2 exp − 2 σ + exp − 2 σ ∆b (xk ) = ∞   2   2  (4.17) b    exp − 12 xm −d/2 1 xm +d/2 P σ + exp − 2 σ m=−∞ Figure 4.1(a) shows b (xk ) versus x for the case d = 10b and σ = 2b. We then use the use continuum the continuun Peierls-Nabarro model to calculate the pressure field p (xi , yj ) around the dislocation. We use a discrete a discrete Peierls-Nabarro model ∞ X (1 + ν) ∆b (xk ) yj p (xi , yj ) = −µ (4.18) 3π (1 − ν) (xi − xk )2 + yj2 k=−∞ where µ is the shear modulus, ν is the Poisson ratio . Finally, the solute/dislocation interaction energy is given by the expression [80] U (xi , yj ) = −p (xi , yj ) ∆vm , (4.19) where p (xi , yj ) is the pressure field due to the dislocation at position (xi , yj ) and ∆vm is the misfit volume of the solute. From these energies, the maximum resistive force fmax = max|∇x U | due to solutes along the glide plane is calculated, which is the key material parameter in the Friedel model. These energies are also used in Eqs. (4.10)– (4.11) of our Labusch-type model. In order to systematically ascertain the effect of the dislocation core structure to the transition concentration, d and σ is varied from 0 to 20b and b to 3b respectively. The material parameters we choose for this set of calculations correspond to Al-Mg alloys; µ = 27.4 GPa, ν = 0.33, b = 2.85 ˚ A, Γ = 0.47 eV/˚ A and ∆vm = 5.7 ˚ A3 . Other metal Chapter 4. Strong-/weak-pinning transition at finite temperature 79 (a) d 0.1 ∆b/b 0.05 σ 0 (b) eV 0.10 4 2 0.05 y/b 0 0.00 -2 -0.05 -4 -0.10 (c) 0.2 fmax U f 0.10 Emax f (eV/nm) U (eV) 0.05 0 0.00 -0.2 -10 -5 0 5 10 x/b Figure 4.1: (a) Burgers vector distribution ∆b (xk ) using Eqn. (4.17) for d = 10b and σ = 2b. (b) Solute dislocation interaction energy U as a function of solute position, for Mg solutes around an edge dislocation in Al; dashed line: stacking fault between two partial dislocations. (c) Solute dislocation interaction energy and resistive force f for solutes just above the glide plane (solutes highlighted in the gray bar). Chapter 4. Strong-/weak-pinning transition at finite temperature 80 systems can be considered, but this does not change the general trends observed in this analysis since the energies and forces in both the Friedel and Labusch models both scale with the elastic modulus and misfit stress, while continuum models suggest the line ten- sion also scales with elastic modulus. Thus, when comparing Friedel and Labusch model predictions, the transition concentration versus temperature is not strongly dependent on the material properties. Where necessary, we use ln(˙◦ /) ˙ = 20 corresponding to typical experimental strain rates. The locus of (c, T ) points at which the transition from the Friedel mechanism to the Labusch mechanism occurs are shown in Figure 4.2, with T normalized by the concentration- independent Friedel temperature T◦F . For an undissociated dislocation (d = 0), the transition concentration increases monotonically with temperature, as shown in Fig- ures 4.2(a)–(c). The transition concentration decreases with the degree of dislocation delocalization σ for any given temperature. For dissociated dislocations d > 0, some core structures exhibit two minimum Labusch configurations [81]. This occurs when the interaction energies in the region between the partials are sufficiently low (i.e., when d/σ is sufficiently large). The first configuration, dominant at high T , is associated with decorrelating the interaction energies of the entire dislocation core by moving the dis- location approximately one dislocation core width away from its original position. The second configuration, dominant at low T , is associated with decorrelating the energies in the partials of the dislocation by moving about half the partial separation distance d. For cores exhibiting two Labusch minima, the transition concentration initially increases with T since both mechanisms are controlled by the partial cores, and is qualitatively similar to results for the undissociated core. However, at higher temperatures, the first Chapter 4. Strong-/weak-pinning transition at finite temperature 81 Labusch configuration becomes more favorable and is associated with much larger en- L ), so that the Labusch mechanism can compete ergy barriers (but with a smaller τy0 with the Friedel mechanism at these higher temperatures. As a result, the transition concentration decreases with temperature as observed in Figures 4.2(d)–(e),(g)–(i). The intermediate situation occurs when the core is dissociated into two partials but suffi- ciently to give two Labusch configurations (Figure 4.2(f)), the transition concentration initially decreases with T but can increase with T at higher temperatures. Examining the actual concentrations at which the transition occurs, Figure 4.2 indicates that, with the exception of the most localized and undissociated core (σ = 0; d = 0), the transition concentration is below c ∼ 4 × 10−4 at all temperatures. Furthermore, over wide portions of the domain of core structure and temperature, the transition con- centration is below 10−4 . Including the more accurate high-temperature Labusch model would suppress the transition further. We also note that the Friedel model predicts zero strength at T = T◦F , so that the actual strengths in this regime are becoming irrele- vant for engineering purposes. We therefore conclude that the Friedel model has a very narrow range of applicability, limited to very localized core structures and, even within such core structures, concentrations in the range of 10−3 or below. For a more realistic description of the dislocation core, the partial burgers vector ∆b (xk ) can be calculated directly from a numerically obtained dislocation core structure. In particular, edge basal and prism dislocations in Mg provide good examples of a dis- sociated and compact core, respectively, and have been studied using EAM and first principles methods [6]. The Burgers vector distribution ∆b (xk ) for the EAM-computed Chapter 4. Strong-/weak-pinning transition at finite temperature 82 σ=1 σ=2 σ=3 -2 (a) (b) (c) 10 Labusch concentration -3 10 Labusch d=0 Labusch -4 10 -5 Friedel 10 Friedel Friedel -6 10 -2 (d) (e) (f) 10 Labusch Labusch concentration -3 10 d = 5b Labusch -4 10 -5 10 Friedel Friedel -6 Friedel 10 -2 (g) (h) (i) 10 -3 Labusch concentration 10 Labusch Labusch d = 10b -4 10 10 -5 Friedel Friedel Friedel -6 10 0 0.5 1 0 0.5 1 0 0.5 1 T /T◦F T /T◦F T /T◦F Figure 4.2: The Friedel-Labusch transition as a function of concentration and nor- malized temperature T /T◦F for various core structures. The interaction energies are calculated using Eqs. (4.17)–(4.19), with d = 0–20b and σ = b–3b. The dashed lines indicate the transition between the two Labusch configurations. Chapter 4. Strong-/weak-pinning transition at finite temperature 83 cores, which agree well with those calculated by first principles, and the interaction en- ergies U (xi , yj ) calculated using Eqs. (4.18) and (4.19) are shown in Figure 4.3. The anisotropy in the elastic constants of Mg are small and for simplicity we use material parameters corresponding to Mg-Al alloys in the direction of basal slip: µ = 19.6 GPa, ν = 0.27, Γ = 0.40 eV/˚ A and ∆vm = −7.9 ˚ A3 . The predicted finite temperature Friedel-Labusch transition concentrations are shown in Figure 4.4(a). Note that using the continuum approximation for the basal interaction energies (Eq. (4.19)) yields one minimum energy configuration for the Labusch mechanism and not the two predicted using DFT-calculated energies [81]. As expected, the transition concentration for the compact prism dislocation core is higher than for the dissociated basal dislocation core. This implies that for complex crystal structures where multiple slip systems with varying core structures exist, such as in hcp, there are solute concentrations at which the Labusch mechanism controls strengthening on some slip systems (e.g. basal) while the Friedel mechanism controls strengthening on other slip systems (e.g. prism). However, the Labusch mechanism still controls strengthening for prism slip at concentrations greater than 10−4 over the entire range of temperatures. From these results, it can be seen that the general trends of the finite temperature Friedel-Labusch transition can very greatly with dislocation core structure. In general, the transition can occur at concentrations much different from (and usually much lower than) the zero temperature transition concentration c◦ . Moreover, with the exception of very highly localized and undissociated dislocations, the transition concentration oc- curs in the range c ∼ 10−4 –10−5 , below the range of typical solute-strengthening alloy additions and where the strengthening is quite small. Chapter 4. Strong-/weak-pinning transition at finite temperature 84 (a) (b) 0.16 0.05 0.14 0.12 0.04 0.1 b0 /b 0.03 0.08 0.02 0.06 0.04 0.01 -5 0 5 -5 0 5 (c) (d) eV 8 8 0.08 6 6 0.06 4 4 0.04 2 2 0.02 y/b 0 0 0 -2 -2 -0.02 -4 -4 -0.04 -6 -6 -0.06 -8 -8 -0.08 -5 0 5 -5 0 5 x/b x/b Figure 4.3: The partial Burgers vector ∆b (xk ) derived from dislocation core struc- tures obtained from EAM calculations [6] for (a) basal and (b) prismatic dislocations in Mg. Also shown are the solute/dislocation interaction energies U (xi , yj ) for (c) basal and (d) prismatic dislocations using Eqs. (4.18)–(4.19) and the ∆b (xk ) shown in (a) and (b). 4.4 Finite temperature transition concentration using DFT energies Continuum descriptions of the solute/dislocation interaction energies fail within the core, where the deformation and interaction energies are largest where explicit chemical effects may prevail. In previous work, we have thus used quantum-mechanical Density Func- tional Theory (DFT), in conjunction with a Greens Function method, to compute the Chapter 4. Strong-/weak-pinning transition at finite temperature 85 -3 (a) -3 (b) 10 10 10 -4 Labusch -4 Labusch concentration concentration 10 -5 10 -5 10 -6 10 prism Friedel -7 basal Friedel 10 -6 10 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 T /T◦F T /T◦F Figure 4.4: (a) The Friedel-Labusch transition as a function of normalized tempera- ture T /T◦F for Mg-Al alloys undergoing prismatic and basal slip using the continuum approximation (Eqn. (4.19)) in conjunction with the ∆b (xk ) derived from EAM dislo- cation structures. (b) The Friedel-Labusch transition as a function of normalized tem- perature T /T◦F for Mg-Al undergoing basal slip using DFT-derived interaction energies for the core sites. The dashed line indicate the transition between the two Labusch configurations. solute/dislocation interaction energies U (xi , yj ) in the dislocation core for Al-X alloys (X = Mg, Si, Cu, Cr, Mn and Fe) [67]. These energies also include the stacking fault in the dissociated dislocations typically of fcc metals, and are generally more accurate than the continuum approximation. Away from the core, a continuum interaction energy is cal- culated using the methods described in the previous section, and the results for U (xi , yj ) are consistent with the quantum calculations. Characteristic forces and energies for the Friedel and Labusch model are then calculated as described in the previous section. All quantities needed to assess the finite-temperature flow stress are shown in Table 4.1 for a range of solutes in Al. The predicted transition concentrations for the Friedel-Labusch transition are shown in Fig. 4.5. Fig. 4.5(a) shows the locus of (c, T ) transition points for Al-Mg along with contours of the associated finite temperature flow stress τy and with the temperature Chapter 4. Strong-/weak-pinning transition at finite temperature 86 Table 4.1: Friedel and Labusch force, energy, length, and/or temperature parameters, as computed for Mg, Si, Cu, Cr, Mn and Fe solutes in Al. Also shown are the zero- temperature transition concentrations c◦ and the geometrical lower limit cmin for the Labusch model, for each alloy. Friedel Parameters Labusch Parameters Transition Paramters Solute F /c1/2 L /c2/3 τy0 Emax τy0 ∆Eb /c1/3 c◦  T◦F cmin  (MPa) (eV) (MPa) (eV) ×10 3 (K) ×105 Mg 147 0.11 342 4.06 6.3 63.4 1.34 Si 97 0.07 137 2.58 123.4 40.1 0.67 Cu 199 0.13 348 4.10 35.7 72.8 1.36 Cr 269 0.26 705 6.65 3.1 152.7 1.86 Mn 271 0.20 710 7.52 3.1 117.8 1.59 Fe 372 0.26 1072 8.20 1.8 152.3 2.59 normalized by T◦F . Fig. 4.5(b) shows the locus of (c, T◦F ) points for all the different solutes considered, where normalization by T◦F collapses all the data nicely. The zero temperature transition concentration c◦ is on the order of 10−3 for most solutes, as shown in Table 4.1. For Si, the critical concentration is larger, 10−1 , but the Si is the weakest alloying element among all those considered here. In all cases, the transition concentration is a strong decreasing function of temperature, dropping rapidly from c◦ to very low values 10−5 as T approaches T◦F . Thus, with increasing temperature, the Labusch model rapidly become dominant over a very wide range of concentrations. The DFT core structure corresponds reasonably closely to our simple model with d = 5b and σ = 3b (Figure 4.2(f)). Compared to the continuum approximation for the energies, the Friedel-Labusch transition occurs at higher concentrations at any given temperature, with the increase being larger at low temperatures. This difference arises because the quantum interaction energies U (xi , yj ) are generally larger in magnitude due to chemical contributions that are not included in the purely mechanical model of Eqn. (4.19). Such local increases in local interaction energy favor the strong-pinning Friedel mechanism Chapter 4. Strong-/weak-pinning transition at finite temperature 87 Temperature (K) τy 100 (MPa) Mg −1 0 10 20 30 40 50 60 10 30 Si (a) 10−1 Cu Cr 10 Mn 10−2 Fe concentration 10−2 concentration 3 10−3 10−3 Labusch Labusch 1 10−4 Labusch 10−4 0.3 Friedel Friedel Friedel 10−5 0.1 10−5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 T /T◦F T /T◦F (a) (b) Figure 4.5: (a) A contour plot of the yield stress τy for Mg solutes in Al as a function of concentration and actual/normalized temperature. The Friedel-Labusch transition is shown as a black line. (b) The Friedel-Labusch transition as a function of concentration and normalized temperature T /T◦F for Mg, Si, Cu, Cr, Mn and Fe solutes in Al. more than weak-pinning Labusch mechanism where a significant contribution (∼ 50%) of the overall energy comes from solutes outside the core. Considering actual temperature ranges, for the alloys here T◦F < 156K (See Table 4.1). Therefore, at liquid nitrogen temperature T = 78K, T /T◦F > 0.5 and the transition concentration is typically c ∼ 10−4 , with corresponding strengths of τy ∼ 5 MPa or less. Temperatures relevant to many engineering applications are generally greater than T◦F , so that the Labusch mechanism is the only applicable mechanism. It is therefore unlikely that the point pinning model is ever relevant for engineering applications of these Al-X alloys. DFT-derived core energies for Mg-Al are available in the literature [4]. Using these ener- gies, we calculate the relevant Friedel and Labusch parameters as we did for Al-X alloys, as well as the finite-temperature Friedel-Labusch transitition. The transition concentra- tion as a function of normalize temperature T /T◦F is shown in Fig. 4.4(b). Similar to Chapter 4. Strong-/weak-pinning transition at finite temperature 88 Al-X, Mg-Al core energies near the partials are significantly larger in magnitude com- pared to the continuum approximation (see Figure 4.3). However, the core energies in between the partials are similar in magnitude to that of the continuum approximation. As a result, two minima are found during energy minimization, corresponding two sta- ble dislocation configurations. The higher magnitude of the DFT core energies makes the Friedel mechanism favorable at higher conecentration compared to the continuum model but the existence of second Labusch configurations makes the Labusch model dominant at lower concentrations when T /T◦F is small. Thus, the Labusch model con- trols strengthening for concentrations larger than ∼ 10−5 . We expect that materials like Cu will be similar to basal slip in hcp Mg. 4.5 Discussion and Summary There is physical statistical lower limit to the validity of the Labusch concept that does not emerge immediately in the analysis above. Specifically, when the total actual number of solutes around the characteristic dislocation segment of length ζc is 1, the statistical concept of a dislocation interacting with a random field of surrounding solutes breaks down. To calculate at what concentration this occurs, we consider the region around the dislocation within which the contributions to the interaction energy represent ∼ 80% of the total solute/dislocation interaction energy. This region encompasses the first 3 atomic rows of atoms above and below the glide plane of the dislocation and over the length wc . At concentration c, the total number N of host atoms within this dominant interaction volume is    2wc ζL,c N =6 √ . (4.20) b/2 3b Chapter 4. Strong-/weak-pinning transition at finite temperature 89 The average number of solutes in this region is cN and by setting cmin N = 1 and solving for the concentration we obtain the minimum concentration cmin for applicability of the Labusch model as  5 ˜p (wc ) ! b 2 ∆E cmin = 0.24 . (4.21) 4wc Γwc Values of cmin for solutes in Al considered here are summarized in Table 4.1 and are all ∼ 10−5 . If we consider the Labusch concept to fail when cmin N = 10, i.e. 10 solute atoms in the Labusch region, then this concentration would increase by 102/3 ∼ 30, putting the limit in the range of 4 × 10−4 . These statistical lower limit estimates are typically close to the transition concentrations computed using the two models directly, and so do not change the conclusion that the Friedel model only prevails at very low concentrations under most conditions of interest. Our analysis here assumes that fmax is determined by the greatest solute energy differ- ence between two neighboring solute sites in the material (see Figure 4.1). It remains possible that, for some solutes, there is an additional energy barrier, and hence a larger force, in between these two locations that must be overcome if the dislocation is to glide. Such a situation might arise if the solute forms strong covalent-type bonds with the matrix material, such that the breaking of these bonds during glide determines the maximum force. Such a situation would have little or no influence on the Labusch-type model predictions but would, naturally, increase the Friedel flow stress and energy bar- rier. This would drive the transition between Friedel and Labusch models to higher concentrations than calculated here. Such a situation can only be assessed by further detailed first-principles calculations of the glide process, akin to measuring the unstable stacking fault energy in the presence of a solute along the glide plane. While we have Chapter 4. Strong-/weak-pinning transition at finite temperature 90 previously applied our Labusch model with success to predict the strengths in various Al-X alloys, we were not able to predict the strength of Al-Fe at the 10–50 ppm level of Fe. The anomalous case of Fe may thus fall within the situation just above, and future studies are needed to gain a quantitative understanding of dilute Fe in Al. Finally, our conclusion about the broad irrelevance of the Friedel model for solute strengthening is generally supported by experimental measurements of the activation volume V = −∂∆E (τ ) /∂τ . Many solute-strengthened alloys exhibit stress equivalence [1, 2, 16, 21, 82], wherein the relationship between the activation volume V and the finite temperature yield stress τy is invariant to the type of solute in a given host material. It is well-known that the Friedel model does not show stress equivalence while the Labusch model does [16, 67]. For clarity, Figure 4.6 shows the predicted activation volumes ver- sus finite temperature flow stress for all the Al-X alloys considered here over a range of concentrations at T = 78 K. While the Friedel model predicts the experimentally observed scaling of V ∼ τy−1 [20], the prefactor in the Friedel scaling is solute-dependent and does not exhibit stress equivalence. Furthermore, at typical experimental conditions at which stress equivalence is observed (that is, T ≥ 78 K), the Friedel model predicts no strengthening for Mg, Si and Cu solutes in Al, as shown in Figure 4.6. On the other hand, the Labusch-type model gives a smaller power scaling of ∼ 2/3 but is consistent with stress equivalence [67]. Small deviations on the curve is due to small difference in wc between the solutes, which vary very slightly between all solute types. Experi- mental observations of stress equivalence down to solute concentrations at the level of 10−4 , and at low temperatures [16], suggest that the Labusch model applies at such low concentrations, consistent with our analyses here. Chapter 4. Strong-/weak-pinning transition at finite temperature 91 T = 78 K Friedel Labusch Mg 3 Si 10 Cu Cr Mn Fe V /b3 2 10 1 10 0 1 2 3 10 10 10 10 τy (MPa) Figure 4.6: The normalized activation volume V /b3 as a function of finite temperature yield stress τy for Mg, Si, Cu, Cr, Mn and Fe at T = 78 K. Note that the Friedel model predicts that V = 0 for Mg, Si and Cu solutes. In summary, the finite temperature transition between strong-pinning Friedel mechanism and the weak-pinning Labusch mechanism has been studied by evaluating the finite temperature yield stress as predicted by the Friedel model and our new Labusch-type model. A parametric study using a range of dislocation core structure showed that the Labusch model controls strengthening for most core structures and concentrations above 10−4 , over a wide range of temperature, but with the transition concentration increasing for very localized cores. The transition was also studied using DFT-calculated solute/dislocation interaction energies for Al-X (X = Mg, Si, Cu, Cr, Mn and Fe) and Mg-Al (basal) alloys. In all cases, the transition concentration are found to be a strong function of temperatures, with the Labusch-type model controlling the strength at most temperatures (T > 78K) and solute concentrations c > 10−4 of relevance to engineered Chapter 4. Strong-/weak-pinning transition at finite temperature 92 alloys. This conclusion is contrary to a conclusion obtained by considering only the zero- temperature limit, where the Labusch model dominates only for c ∼ 2 × 10−3 or higher. We conclude that applicability of the Friedel model is very limited in core structure, temperature, and solute concentration, so that use of the Labusch-type model is more appropriate for most materials and engineering applications. Chapter 5 Summary and Conclusions In the first part of this dissertation, we developed a new Labusch-type model for solute strengthening in general substitutional alloys that addresses some of the shortcomings inherent in previous weak-pinning models. The major feature of the model is that it takes into account the correlations between solute/dislocation interaction energies as the dislocation moves along the glide plane. By considering these correlations, the diver- gent nature of the dislocation field is taken into account without the need to introduce arbitrary cut-offs in the interaction energy. The resulting model is parameter-free, gives predictions for finite temperature yield stresses and is applicable to multi-solute alloys. The model takes as input the solute/dislocation interaction energy of a solute around the dislocation as a function of solute position. In positions immediately adjacent to the stacking fault, continuum elasticity breaks down and chemical effects are more signifi- cant. To obtain accurate energies on those positions, we use the flexible Greens Func- tions Flexible Boundary method is used in conjunction with state-of-the-art Density 93 Chapter 4. Strong-/weak-pinning transition at finite temperature 94 Functional Theory calculations. This methodology is used to compute the interaction energies of Al-X alloys (X = Mg, Si, Cu, Cr, Mn and Fe). Using the DFT-computed energies and the new parameter-free model, quantitative predictions of finite temper- ature yield stresses are made for several Al-based alloys characterized in experimental studies. The predictions are accurate for most of the solutes considered. The notable exceptions to this are for alloys containing Fe solutes, wherein the experiments show that alloys containing Fe solutes with c ∼ 10−6 have yield strengths on the order of 10 MPa. While this result is several orders of magnitude higher than what the model predicts, the experimental yield stresses for Al-Fe alloys obey the same scaling behavior as what the model predicts (that is, τy ∝ c2/3 ). The model is used to back calculate the parameters necessary to obtain the experimental results. When the anomalous behavior of Fe solutes is taken into account, the predictions made by the model are accurate for all the solutes types as well as all temperature ranges. We also rigorously showed that Basinski’s “stress equivalency principle” is obeyed by the model. We then applied the model without modification to basal slip in Mg-Al alloys using DFT-obtrained interaction energies existing in the literature [4]. Due to the large partial separation of basal edge dislocations in Mg, two stable configurations are found. The first configuration, dominant at high τ /low T conditions, is associated to decorrelating the solutes within the partials of the dislocation while the second, dominant at low τ /high T conditions, is associated with decorrelating the solutes in the far-field. The existence of these two configurations is confirmed by stochastic analysis. Due to the two configurations, the dependence of τy (T ) versus T in Mg-Al undergoing basal slip have features not found in Al-X alloys. In particular, the temperature dependence of Chapter 4. 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