ISSN 10628738, Bulletin of the Russian Academy of Sciences: Physics, 2010, Vol. 74, No. 5, pp. 592–596. © Allerton Press, Inc., 2010. Original Russian Text © N.A. Koneva, N.A. Popova, E.V. Kozlov, 2010, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2010, Vol. 74, No. 5, pp. 630–634.

Critical Grain Sizes of Micro and Mezolevel Polycrystals N. A. Koneva, N. A. Popova, and E. V. Kozlov Tomsk State University of Architecture and Building, Solyanaya pl. 2, Tomsk, 634003 Russia email: [email protected] Abstract—The influence of size effects on the dislocation structure and parameter k of the Hall–Petch rela tion is considered. Three critical grain sizes, in the vicinity of which the mechanisms of polycrystal deforma tion and strengthening change, are shown to exist. These critical grain sizes are due mostly to the presence of geometrically essential dislocations. DOI: 10.3103/S1062873810050035

INTRODUCTION Almost all industrial materials are polycrystals; it is therefore essential to know the influence of the sizes of polycrystal grains on their mechanical properties. This is true for polycrystals of both the meso and the microlevel. Of particular importance are the critical grain sizes. Quantitative studies on grain structure and polycrystal properties allow us to distinguish three main critical grain sizes. These are the average grain sizes in whose vicinity considerable changes occur in the properties of a polycrystal aggregate. The present work considers the critical grain sizes of polycrystals obtained by samples of pure metals and solid solutions. EXPERIMENTAL First Critical Grain Size The Hall–Petch (H–P) relation σyp = σ0 + kd–1/2, (1) where σyp is the polycrystal yield point, σ0 is the single crystal deformation resistance, and d is the average grain size, determines the yield point of a polycrystal line aggregate. H–P coefficient k is an essential char acteristic of grain boundary strengthening. It deter mines the growth of the yield point as the grain size is reduced. The dependence of H–P coefficient k on the grain size was established in [1]. It is known that at some low value of the grain size, H–P coefficient k cr becomes equal to zero. The first critical grain size, d 1 , is the size at which the sign of the Hall–Petch coeffi cr cr cient k changes. At d > d 1 , k > 0; at d < d 1 , k < 0. The

tion. There is no grain boundary strengthening. The dependences of H–P coefficient k on average grain size 〈d〉 for a series of pure metals are presented in Fig. 1, from which it is obvious that the conversion of k into zero occurs in the proximity of 〈d〉 ≈ 10 nm. Curiously, this value does not depend on the type of metal. It is reasonable to suppose that 〈d〉 = 10 nm is the critical parameter in whose proximity the mecha nisms of nanopolycrystal deformation change and the role of grain boundary processes in deformation [2] increases. The diffusion processes at grain boundaries (GBs), the sliding of the lattice and grain boundary dislocations along GBs, the migration of GBs, and so on, are attributed to these processes. Second Critical Grain Size Three types of grain [3] that are usually observed after intensive plastic deformation can be distin

cr

value of d 1 for pure metals Al, Cu, Ni, Fe, and Ti is close to 10 nm. The change in coefficient k ’s sign sig nifies that the grainboundary strengthening is replaced by softening. In other words, further grinding results not in a rise in the yield point, but in its reduc 592

〈k〉, MPa m1/2 0.2

1

3

2 5

0.1 4

0

−0.1

10

102

103 〈d〉, nm

Fig. 1. Dependences of Hall–Pettch coefficient k on grain size for Fe (1), Ti (2), Ni (3), Cu (4), and Al(5).

CRITICAL GRAIN SIZES OF MICRO AND MEZOLEVEL POLYCRYSTALS

guished. These possess different dislocation structures: (a) dislocationfree grains; (b) grains with random dis location structure; and (c) grains containing disloca tion substructures (cells or fragments). The average size of each type of grain increases as we move from dislocationfree grains to grains with cells and frag ments. Such grain structures and size distributions are due to the method used in preparing the nanostruc tural polycrystalline aggregate [4]. cr

The second critical grain size, d 2 , is associated with the formation of dislocationfree grains [3]. The interaction between GBs and dislocations becomes so intense that dislocations stretch from the grain body and migrate to the GBs through GB stress fields, par ticularly through the stress fields formed by GB steps cr and triple joints. For pure metals, d 2 is about 100 nm [3, 5]. The formation of dislocationfree grains makes a submicropolycrystal stronger and leads to changes in the mechanisms of its deformation. Fig. 2 presents the dependence of the average scalar dislocation density (ρ) typical of various grain types on the average size (d) for grains of the same type. As the grain size decreases, the dislocation density is reduced. This first occurs with the averaged dislocation density visible in the dependence on the average grain size d (Fig. 2a). The dashed lines that extend these depen dences depend on the grain size d = 60 and 80 nm at the dislocation density of ρ ≈ 0 (1 and 2 in Fig. 2a). The second critical grain size is therefore around 50– 100 nm. This value of the critical grain size is reached upon averaging by all types of grain. Fig. 2b demon strates the dependences ρ = f(d) for grains of random dislocation structure (3) and grains of fragmented dis location structure (4). The nanpolycrystal samples were prepared by the method of twisting under hydro static pressure. The dashed lines are longer since grains with such structures do not transform into dis locationfree ones. The critical grain size can never theless be determined, and is close to 70–80 nm. Dis locationfree grains are observed when their sizes are smaller than the second critical grin size, which for cr pure metals is close to d 2 ≤ 100 nm (see Fig. 2). Greater dispersion turns out to be typical for this phe nomenon: σd ≈ 100 nm. The idea of critical grain size for the dislocationfree grains was first introduced in paper [6]. The results presented here confirm the pre cr viously published value of d 2 , which was close to 100 nm [3, 5, 6]. The range of sizes of dislocationfree grains was given in the same works. The range of dislo cationfree grain sizes most often extends from infini tesimal to d ≈ 200 nm, independent of the type of material for both Cu and Ni, testifying to the common nature of this phenomenon. Dislocationfree grains play a great role in the for mation of the mechanical properties of submicrocrys tals. Due to their small sizes, these grains make a poly

593

ρ, 10−14 m−2 6 (a)

4

2 2

1

0 8 (b)

3 4 4

0

100

200

300 d, nm

Fig. 2. Dependences of scalar dislocation density ρ on average grain size d (1 and 2), the size of grains with ran dom dislocation structure (3), and the size of grains with fragmented substructure (4). The dashed line indicates the critical grain sizes: 1, 3, and 4—Cu, twisting under hydro static pressure; 2—Ni, equal channel angular extrusion.

crystal stronger. The number of dislocationfree grains relative to the total number of grains varies from 0.1 to 0.4. The volume fraction of these grains does not exceed the values of 0.05–0.20, due to their small sizes. Geometrically Essential Dislocations and the Critical Size of Dislocation Fragments Dislocation structure is characterized by several parameters [7]. Along with scalar dislocation density (ρ), a great role is played by the density of statistically accumulated dislocations (SADs) (ρS) and the density of geometrically essential dislocations (GEDs) (ρG). GEDs, introduced many years ago by Eshbi [8], are now the focus of current interest [9, 10]. GEDs are the accumulated dislocations that are necessary to accom modate the crystal lattice curvature [8, 11] that appears due to the nonuniformity of plastic deforma tion, i.e. due to the presence of a deformation gradient [10, 12]. The value of ρG determines the nonunifor

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mity of deformation, its gradients, internal stress fields, and their shielding. This is particularly impor tant in the study of nanopolycrystals. GEDs arise upon the deformation of polycrystal line aggregates, in materials with deformed twins, in dispersionstrengthened materials, and in other cases where there are strong barriers to dislocation sliding. Nowadays, other geometrically essential defects are also considered and studied along with GEDs. These include geometrically essential boundaries (GEBs), geometrically essential twins (GETs), and other defects. The boundaries and subboundaries of the nature of deformation were first classified into two types in work of KulmannVilsdorf and Hansen [13], and were experimentally studied further by the PEM method in a series of works by Hansen et al. [14–16]. In particular, cell boundaries are denoted as random dislocation boundaries (RDBs) formed by the mutual crossing of dislocations, while the boundaries of cell blocks are denoted as geometrically essential bound aries (GEBs). Such a division had actually been intro duced several years before this by Rybin using the terms cell and knife boundaries [17]. These bound aries separate the element volumes in the dislocation structure that deform under the action of a combina tion of different types of sliding systems, accompanied by a deformation gradient. Various types of misori ented boundaries in cell substructures and the misori entations in them were studied in detail by the authors of the present work (see, e.g., [18, 19]), and Sevillano [20] later introduced the concept of geometrically essential twins (GETs). The entire spectrum of geo metrically essential defects is now used to analyze the deformation mechanisms and regularities in the for mation of dislocation structures. For many years, dislocation structure was com monly characterized by scalar dislocation density ρ. The development of dislocation science led to a divi sion of the value ρ into components differing in the physical sense [7, 21, 22]. Dislocation reproduction and reactions are random processes, so the disloca tions impeded in this way are called statistically accu mulated with density ρS (SAC) [8, 11]. Statistically accumulated dislocations are retarded by relatively weak barriers: other dislocations. If the material has stronger barriers (particles of the second phases and grain boundaries), gradients of plastic deformation appear. If such gradients are present, there is an accu mulation of geometrically essential dislocations (GEDs) with density ρS in addition to the density of dislocations ρG [8, 11]. In this case, ρ = ρS + ρG.

(2)

while their difference yields the excess dislocation density ρ± [24, 25]: ρ± = ρ+ – ρ–. (4) The excess dislocation density is directly associated with the curvature twisting of the crystalline lattice and the dislocation density ρG [7, 12, 24, 25]:

∂ϕ χ (5) ρ ± = 1 = = (rb)−1 , b ∂l b where b is a strength of dislocation (Burger’s vector magnitude), ϕ is the slope angle of the crystallographic layer, l is the distance along the layer, ∂ϕ / ∂ l is the gra dient of crystalline lattice curvature twisting, χ is the curvature twisting of the crystalline lattice, and r is the crystal curvature radius. The sizes of nanolevel grains (d) can be compared to the sizes of dislocation fragments at the mesolevel (df), and their ranges actually overlap. This can be seen from a comparison of Figs. 2 and 3. Fig. 3 shows the dependences of ρ, ρS, and ρG on the sizes of a fragment in deformed VCC steel. These dependences are quite interesting: all three components of the dislocation structure are reduced as the fragment sizes decrease. Extrapolation lines (the dashed lines in Fig. 3) indi cate the critical size of fragments equal to ~100 nm. This is of paramount interest, as it testifies to the intensive interaction of dislocations and fragment boundaries. The nature of the phenomena is undoubt edly the same in both Fig. 2 and Fig. 3. It should be noted that there are three types of fragments in the deformed steel: (a) dislocationfree, (b) with random dislocation structure, and (c) with a cell dislocation substructure. The structures of the dislocation frag ments and micrograins are undoubtedly almost identi cal. The dependences of the dislocation densities on the fragment and micrograin sizes also coincide. The critical grain sizes and the sizes of nanofragments turn out to be close. Doubtless both the size of grains at the microlevel and the size of the fragments lead to identi cal behavior in the parameters of their dislocation structures (compare Figs. 2 and 3). Third Critical Grain Size cr

The third critical grain size, d 3 , is associated with the change in the role of dislocation parameters. If d > cr d 3 , then statistically accumulated dislocations with a density of ρS predominate in the dislocation assembly. Their number is higher than that of geometrically cr essential dislocations ρG (ρS > ρG). The value of d 3 is close to 10 μm (Fig. 4), while ρS = ρG. Transition cr

Dislocations are divided into those positively (ρ+) and negatively (ρ ⎯ ) charged [23]. Their sum yields the total scalar dislocation density: ρ = ρ+ + ρ–,

(3)

through this grain size (d < d 3 ) changes the nature of most dislocations, the conditions for shielding dislo cations due to stress concentrators, and the level of internal stress fields. Under these conditions, the den sity of the geometrically essential dislocations (GEDs)

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CRITICAL GRAIN SIZES OF MICRO AND MEZOLEVEL POLYCRYSTALS ρ, ρG, ρS, 10−11 cm−2 2

1 2 1

3

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tion of a defective structure varies depending on their density. At higher GED densities, they form both dis location and disclination structures, as well as internal stress fields. Even at low values of GED density, how ever, their role is considerable, since they determine the type of the formed dislocation substructure [27, 28]. We may state that the role of grain sizes in the struc ture and mechanical properties of metallic aggregates is determined mostly by geometrically essential dislo cations and joint disclinations. REFERENCES

0

400

800 dΦ, nm

Fig. 3. Dependences of ρ] (1), ρS (2), and ρG (3) on frag ment size dФ in deformed VCC steel.

ρG/ρS 2 ρ G = ρS 1

0

0

10

20

30 d, μm

Fig. 4. Ratio of ρG/ρS depending on size (d) of grains and fragments. The vertical dashed line indicates the third crit ical size of the grain or fragment.

is higher than the density of the statistically accumu lated dislocations (SAD), so ρG > ρS. CONCLUSIONS The attainment of each critical grain size changes the mechanisms of polycrystal deformation strength ening, along with creating the physical difference between micro and mesolevels. The transition from the mesolevel to the microlevel over virtually the entire range of grain and fragment sizes can be described using GED. If ρG is 0.1–0.2 of the value of ρ at the mesolevel, then the dislocation structure is fully real ized by GED (ρ ≈ ρG) as we approach grain sizes of 200–300 nm. The gradient dislocation structure cre ated by the stress fields of disclinations situated at the GB and triple grain joints [3, 26], is formed. The role of geometrically essential dislocations in the forma

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