Nanomechanics of Hall–Petch relationship in nanocrystalline materials

Nanomechanics of Hall–P in nanocrystalline mater C.S. Pande, K.P. Cooper * Materials Science and Technology Division, Naval Res a r t i c l e i n f o profession and a genius by nature, blessed with a unique ability to treat everyone as his equal. During his later years he was very much interested in the mechanical properties of nanocrystalline materials. This review on that topic is our contribution to the spe- cial issue of Progress in Materials Science honoring him. Published by Elsevier Ltd. * Corresponding author. E-mail address: khershed.cooper@nrl.navy.mil (K.P. Cooper). Progress in Materials Science 54 (2009) 689–706 Contents lists available at ScienceDirect Progress in Materials Science journal homep 0079-6425/$ - see front matter Published by Elsevier Ltd. to be successful in explaining experimental results provided a real- istic grain size distribution is incorporated into the analysis. Masumura et al. [Masumura RA, Hazzledine PM, Pande CS. Acta Mater 1998;46:4527] were the first to show that the Hall–Petch plot for a wide range of materials and mean grain sizes could be divided into three distinct regimes and also the first to provide a detailed mathematical model of Hall–Petch relation of plastic deformation processes for any material including fine-grained nanocrystalline materials. Later developments of this and related models are briefly reviewed. Prof. Frank Nabarro was a physicist by training, a metallurgist by doi:10.1016/j.pmatsci.2009.03.008 etch relationship ials earch Laboratory, Washington, DC 20375-5343, USA a b s t r a c t Classical Hall–Petch relation for large grained polycrystals is usually derived using the model of dislocation pile-up first investi- gated mathematically by Nabarro and coworkers. In this paper the mechanical properties of nanocrystalline materials are reviewed, with emphasis on the fundamental physical mechanisms involved in determining yield stress. Special attention is paid to the abnor- mal or ‘inverse’ Hall–Petch relationship, which manifests itself as the softening of nanocrystalline materials of very small (less than 12 nm) mean grain sizes. It is emphasized that modeling the strength of nanocrystalline materials needs consideration of both dislocation interactions and grain-boundary sliding (presumably due to Coble creep) acting simultaneously. Such a model appears age: www.elsevier .com/locate / 1. ical underpinnings to the mechanical behavior of nanocrystalline materials. s ¼ s0 þ kd ð1Þ 2. Experimental background Mechanical behavior of nanocrystalline materials has been the theme of over 500 publications and several review articles [3,34–50]. These articles conclude that yield stress and microhardness of nano- crystalline materials can be 2–10 times higher than the corresponding coarse-grained polycrystalline materials with the same chemical composition. Similarly, some published values of microhardness of nanocrystalline composite coatings [13,14] are of the same order as microhardness (HV � 70–90 GPa) of diamond. In the range of grain sizes d above about 10 nm, the dependence of the yield stress s on d deviates little from the classical Hall–Petch relationship given by the formula, �1=2 with s0 and k being material constants [6–12]. Yield stress may also depend upon on the mode of pro- 2. Experimental background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690 3. Mechanisms of deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 4. Models using lattice dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 5. Role of Coble creep as a competing mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 6. A generalized expression for yield stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 7. Relationship between hardness and yield strength in metals and alloys . . . . . . . . . . . . . . . . . . . . . . . . 700 8. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 1. Introduction Nanocrystalline materials are polycrystalline materials consisting of grains in nanometer range. They have the potential to exhibit outstanding physical, mechanical and chemical properties, which could, in principle, lead to new applications and novel technologies (see Refs. [1–5]). There are now several examples of real applications even at present where the outstanding physical mechanical and chemical properties of nanomaterials are used for commercial products (e.g. soft magnets, coat- ings, structural repair, etc.). These outstanding properties can be due to interface and nano-scale ef- fects due to the high volume fraction of the interfacial phase (up to 50%), and smaller mean grain sizes (not exceeding 100 nm). Of special importance and subject of this review are the uniquemechan- ical properties of nanocrystalline materials, especially yield stress, which are essentially different from those of conventional coarse-grained polycrystalline materials. For example, bulk nanocrystalline materials and thin nanocrystalline coatings in some cases show superhardness, extremely high strength and good fatigue resistance [5–14], which are desired for many applications. The high strength however is often accompanied by low ductility at room temperature, which may limit their practical utility. (However, more recently, several researchers have claimed substantial strength as well as high tensile ductility in nanocrystalline materials [15–18].) It has been also surmised that some nanocrystalline ceramics and metallic alloys may even exhibit superplasticity at lower temper- atures and faster strain rates than their coarse-grained counterparts [19–29]. There is also experimen- tal evidence [30–33] that nanocrystalline materials may show anomalously fast diffusion, which may in turn explain their deformation behavior. These developments lead one to hope that nanocrystalline materials with unique combination of high strength and good ductility may provide new structural and functional applications in the future. The main aim of this paper is to provide a brief overview of the theoretical models of yield stress in nanocrystalline materials paying special attention to their microscopic mechanisms. Once established, these mechanisms are then expected to provide theoret- cessin Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690 Contents 690 C.S. Pande, K.P. Cooper / Progress in Materials Science 54 (2009) 689–706 g [51,52]. However, any further grain refinement may lead to lower yield stress. Thus, in the Fig. 1. Scaled yield stress as a function of (grain size)�1/2 for several materials [53]. C.S. Pande, K.P. Cooper / Progress in Materials Science 54 (2009) 689–706 691 range of smaller grain sizes, heat-treated materials exhibit the so-called ‘inverse’ Hall–Petch behavior (softening with further reduction of grain size). Masumura et al. [53] have plotted some of the avail- able data (till 1998) in a Hall–Petch plot (see Fig. 1). It is seen that the yield stress-grain size exponent for relatively large grains appears to be very close to �1/2, as in Eq. (1), and generally this trend con- tinues until the very fine grain regime (�100 nm) is reached. The large scatter of the data for grain sizes below 100 nm could be attributed to problems in preparing these materials or to differences in thermal treatments. With the advent of better prepared nanocrystalline materials whose grain sizes are of nanometer (nm) dimensions, the applicability and validity of Eq. (1) as well as the underlying mechanisms became of great interest. In addition, these nanocrystalline materials were found to exhibit, in general, low tensile ductility at room temperature [6–12]. However more recent results indicate that nanocrystalline materials with very low porosity [15] or with dendrite-like inclusions [17,18] or with bimodal grain size distributions (consisting of both nano- and micron-sized grains) [16] show better ductility. Some reports also indicate nanocrystalline materials with high-strain-rate (tensile) superplasticity [21–29]. As far as microstructures in these materials are concerned, mechanically loaded nanocrystalline materials are reported to show grain rotations [27,54], formation of shear bands [55–59], or emis- sion of (usually) partial lattice dislocations by grain boundaries into grain interiors [27,29,60]. Fig. 2 shows a Hall–Petch plot for copper using early data from various researchers. It also defines the three regions of the Hall–Petch plot. The limitation of the classical ideas of Hall–Petch plots is dra- matically demonstrated in Fig. 3 by plotting the yield strength data in terms of grain size instead of (grain size)�1/2. 3. Mechanisms of deformation As early as 1977, Armstrong and coworkers [61] noted the increase in yield stress on grain refine- ment up to the beginning of the nanocrystalline (<100 nm) regime. Much effort has been spent to theoretically describe the Hall–Petch relationship in nanocrystalline materials. Classically, high 692 C.S. Pande, K.P. Cooper / Progress in Materials Science 54 (2009) 689–706 values for yield stress were considered to be related to the effect of increased grain boundaries pro- viding additional obstacles for movement of lattice dislocations. In early theoretical studies, models of nanocrystalline materials were considered as two-phase composites consisting of nanograin interiors and grain-boundary regions (see, e.g. Refs. [55,62– 68]). Yield stress s is then accounted for using the so-called rule-of-mixture, yield stress s being given by some weighted sum of the yield stresses characterizing the grain-interior and grain-bound- ary phases. The ratio of the two phases, of course, strongly depends on the grain size d. In this calculation, the yield stress of the grain-boundary phase is assumed to be lower than that of the grain-interior phase, and with suitable adjustable parameters, the deviations from the conventional Hall–Petch relationship can be described roughly in accordance with experimental data. It is obvious that the ‘‘rule-of-mixture” approach is too approximate and arbitrary and sheds no light on the actual mechanisms [11]. Fig. 2. Compilation of yield stress data for pure copper from various publications [53]. Fig. 3. Plots showing limitation of standard Hall–Petch law at small grain sizes and existence of optimum grain size for yield strength [49,53]. (a) Schematic of hardness or strength as a function of normalized grain size shows the limitation dramatically. (b) Normalized yield strength plotted against (normalized grain size)�1/2. occur mode agreem fact that in nanocrystalline materials with small grain sizes, the number of dislocations in a pile-up within a grain cannot be very large. In the limit at still smaller grain sizes, this mechanism should cease when the grains are so small that there are only two dislocations in the pile-up. Mathematically, the model utilizes the fact that the length of the pile-up is no longer proportional to the number of dislocations in the pile-up if the pile-up is not large. Then, Pande and Masumura [74], by considering the conventional Hall–Petch model, showed that a dislocation theory for the Hall–Petch effect does not give a linear dependence of s on d�1/2 when grain sizes are in the nanometer range. When the number of dislocations in the pile-up falls to one, no fur- ther increase in the yield stress is possible by this mechanism and it saturates. As mentioned before, if the number of dislocations n in a pile-up is not too large, the length of the pile-up L is not linear in n. Chou [75] gives the relation between L and n as: L ffi A 2s 4 nþm� 1� 2i1 2n3 � �1=3 !����� �����; ð2Þ where i1 = 1.85575 and mb is the Burgers vector of the lead dislocation in the pile-up. (The lead dis- location could be in the grain-boundary itself and, hence, may have a Burgers vector different from plastic deformation mechanisms have been mentioned acting individually or in competition: (1) grain-boundary sliding, (2) grain-boundary diffusional creep, (3) triple junction diffusional creep, (4) rotational deformation (occurring through motion of grain-boundary disclinations) and (5) lattice dislocations. The experimental data is usually not precise enough to allow one to select a theoretical concept from a variety of theoretical models describing the same experimental data using various mechanisms. In this context, in next sections of this review article, we will pay special attention to theoretical models of plastic deformation mechanisms in nanocrystalline materials, that can account for some additional microstructural results (either experimental or computational) and provide math- ematical results rather than qualitative concepts. Needless to say, the subject is still a matter of some controversy. 4. Models using lattice dislocations The most obvious idea is to use conventional lattice dislocation slip model for nanocrystalline materials, but taking into account the influence of smaller grain sizes and high-density ensembles of grain boundaries on the formation of lattice dislocation pile-ups in grain interiors. Thus, this treat- ment extends the classical derivation but assumes that there are very few dislocations available in any one grain. For this purpose it is instructive to start with a brief discussion of the models describing the clas- sical Hall–Petch relationship (Eq. (1)) in coarse-grained polycrystals. Most of these models use the concept of dislocation pile-ups (see review by Li and Chou [69]). In deriving the Hall–Petch relation, grain boundaries here are considered as barriers to dislocation motion [70,71], causing stresses to con- centrate and activating dislocation sources in the neighboring grains, thus initiating the slip from grain to grain. In other type of models, though mentioned less often [72,73], the grain boundaries are re- garded as dislocation barriers limiting the mean free path of the dislocations, thereby increasing strain hardening and resulting in a Hall–Petch type relation. Several variations of these concepts are possible. It is also possible that several dislocation processes could compete or reinforce the deformation process. Pande and Masumura [74] were the first to extend mathematically the classical derivation of Hall– Petch relation to nanocrystalline materials. They assumed that the classical Hall–Petch dislocation pile-up model is still dominant with the sole exception that the analysis must take into account the the re ring in mechanically loaded nanocrystalline materials. At present, there are many theoretical ls of the abnormal Hall–Petch effect based on different deformation mechanisms and claiming ent with the corresponding experimental data from nanocrystalline materials. The following Our goal is to briefly review a more precise physical mechanism of plastic flow in nanocrystalline materials in terms of lattice dislocations, grain-boundary dislocations, vacancies and grain rotations C.S. Pande, K.P. Cooper / Progress in Materials Science 54 (2009) 689–706 693 st of the dislocations.) Pande and Masumura [74] give an improved expression, L ffi A 2s 2ðnþm� 1Þ1=2 � i1 þ e 121=3ðnþm� 1Þ1=6 i1 " # ; ð3Þ where e is a small correction term (e� 1) and can be neglected. Then, following the classical analysis, but using Eq. (3), they find that for small grain sizes there are additional terms to the Hall–Petch relation, s ¼ k�1=2 þ c1ðk�1=2Þ5=3 þ c2ðk�1=2Þ7=3; ð4Þ where s = s/[m/s*], c1 = �0.6881, c2 = 0.1339 and l = Lms*/2A. Eq. (4) is expected to be correct for all grain sizes, as long as dislocation pile-up mechanism is operating. This model thus recovers the clas- sical Hall–Petch at large grain sizes but for smaller grain sizes the s levels off. This mechanism there- fore cannot explain a drop in s. If, on the other hand, the yield stress is source limited, s � Gb/d, i.e., the yield stress should rise as d�1. Thus, from these arguments, at smaller grain sizes, either the yield stress should rises faster than d�1/2 or it should saturate, but it should not decrease. The Hall–Petch plot using Eq. (4) is given in Fig. 4 showing the leveling, but not the ‘inverse’ Hall–Petch curve. Several researchers [76–78] have developed models similar to that of Pande and Masumura [74]. 694 C.S. Pande, K.P. Cooper / Progress in Materials Science 54 (2009) 689–706 On the other hand, Malygin [79] has suggested that dislocations are absorbed in grain boundaries, the effect being larger in nanocrystalline materials. The assumption is that grain boundaries act pre- dominantly as sinks for dislocations (just the opposite to what was proposed by Li [80], who consid- ered grain boundaries as sources for dislocation generation). Malygin’s model is attractive as a dislocation mechanism and should be considered further. However, we point out two problems with the model. First, it is doubtful if the dislocations play the same role whether the grains are large or small. It is more likely that dislocations in ultrafine grains, if present at all, are confined to grain boundaries [81]. Second, in Malygin’s model [79], the stress calculated is a work-hardened flow stress rather than a yield stress and, as with Li [80], it merely assumes a relation between dislocation density and yield stress, instead of proving it. Lu and Sui [82] assume an enhancement of lattice dislocation penetration through the grain bound- aries and the corresponding softening of nanocrystalline materials. Scattergood and Koch [83] assume that the yield stress of fine-grained materials is controlled by the intersection of mobile lattice dislocations with the dislocation networks at grain boundaries. Zaichenko and Glezer [84] have pro- posed rotational defects (disclinations), formed at triple junctions, as sinks and sources of the lattice dislocations moving in grain interiors and causing the plastic flow in nanocrystalline materials (see also Ref. [85]). Fig. 4. Plot of Hall–Petch dislocation model for nanocrystalline materials [74]. interio crysta even a were mode in a si range an analytical expression for the Hall–Petch relation for the whole range of grain sizes. This model, based on the idea of competition between lattice dislocation slip and grain-boundary diffusional creep (Coble creep), is summarized below. In this model, it is assumed that polycrystals with a relatively large average grain size obey the classical Hall–Petch relation Eq. (1), and for very small grain sizes, it is assumed that Coble creep is active and that the s vs. d relationship is given by, sc ¼ A=dþ Bd3; ð6Þ where B is both temperature and strain-rate dependent. The additional term A/d (the threshold term) was added by Masumura et al. [53] on an ad hoc basis (see, however, the later part of this section for justification on physical grounds). Contribution from this term can be significant if d is in the nanome- ter range. For intermediate grain sizes, both mechanisms might be active simultaneously especially if the specimen has a grain size distribution as is usually the case. The model is illustrated graphically in Fig. 5. The presence of a distribution of the grain sizes in a polycrystal is taken into consideration by using an analysis similar to Kurzydlowski [91]. The volumes of the grains are assumed to be log-normally dis