792 Y.Estrin,A.Vinogradov/Acta Materialia 61 (2013)782-817 case of evolving dislocation structure is a major assump- terms in the evolution equations represent possible disloca- tion.which definitely needs further substantiation. tion reactions involved.Their physical origin was explained The total dislocation density is considered to be given by in Refs.[153,154].For example,the loss of cell interior dis- the weighted sum of the dislocation densities in the cell locations to the walls where they get entrapped is repre- walls(subscript "w")and cell interiors(subscript "c") sented by the first term of Eq.(6)and the second term of p=fpw+(1-f)Pe, (3) Eq.(7).The last terms in both equations account for the dynamic recovery.The quantity io denotes a reference where fdenotes the volume fraction of the cell walls.Obvi- shear rate.In the low-temperature regime typical for SPD ously,the latter quantity is linked to the cell size d and the processing,i.e.below half the melting temperature,the wall thickness w through a relation that should account for exponent n in both equations can be taken to be inversely cell morphology.As this relation is not very sensitive to the proportional to the absolute temperature T,while the coef- particular shape of the cells,a simple expression ficient ko can be considered constant.This corresponds to f=d-(d-w)3 dynamic recovery being governed by dislocation cross-slip. (4) In Zehetbauer's model [146,155,156],it was assumed that d dynamic recovery in cell walls is controlled by dislocation corresponding to the idealized case of cube-shaped cells is a climb-a non-conservative process involving vacancy diffu- reasonably good one.In modelling SPD processes sion.This process is characterized by io,which is given by [161,162],it was assumed that the volume fraction de- an Arrhenius equation with the activation energy repre- creases from an initial value fo to a saturation value fo senting that for self-diffusion,the exponent n being a This approach of saturation is governed by a phenomeno- constant. logical equation: As vacancy formation is influenced by hydrostatic pres- f=foo+(fo-foo)exp(-7/i), (5) sure,this approach introduces a direct effect of hydrostatic pressure on the dislocation recovery kinetics [163].An which is supported by experiments on Cu.The parameter appropriate modification of the dynamic recovery term in represents the inverse of the rate of this variation with the Eq.(7)to include the hydrostatic pressure made it possible plastic shear strain y.The growth of the total dislocation to account for the effect of back-pressure on the ECAP density p and the concomitant decrease of the average cell processing of an Al alloy [164]. size d with strain suggests that a decrease of the cell volume The type of model described above proved to be a useful fraction f,which is implied by Eq.(5)for f<fo,is only descriptive and predictive tool for computer simulations of possible if the cell wall thickness decreases fast enough with ECAP processing of copper [157],aluminium [161],Al strain.This assumption was implicit in the original model alloy 6016 [164],interstitial-free steel [165],CP titanium [153,154]."Thinning"of cell walls with the progress of [158]and HPT processing of copper [53,166].The focus straining is a reasonable notion,as "geometrically unneces- of these simulations was the calculation of the strength sary"dislocations not contributing to misorientation be- characteristics of the ECAP-processed materials.However, tween neighbouring cells will recover with strain the model was also used for calculating the texture of cop- However,the phenomenological ansatz made in Eq.(5) per developing as a result of ECAP processing,albeit in a can be replaced by a more physical assumption of the var- very simplified way [167].We shall return to the issue of iation of the cell wall thickness.For instance.w can be de- texture below. fined in terms of a characteristic length controlling the The model of grain fragmentation presented above is decay of the stress field of the cell wall dislocations,which based on the idea that fine granularity is attained through should scale with 1/p.In this way,both the average cell dislocation cell formation with(tacitly assumed)accumula- size and the cell wall thickness would be expressed in terms tion of misorientations across the dislocation cell bound- of the dislocation densities. aries.In other words,gradual transformation of The evolution of the dislocation ensemble is captured in dislocation cell walls,or at least a large proportion of them, a set of coupled differential equations for the dislocation to high-angle grain boundaries is implied.Pantleon [168] densities in the cell interior and cell walls under the "Tay- and Estrin et al.[169]considered the increase of dislocation lor"-type assumption that the shear strain is the same in cell misorientations with progressing straining in terms of a both“phases'”: probabilistic approach.In both models,preferred storage de(V3(Dvp--ko 1/m of dislocations with a certain sign of the Burgers vector, dy bdf fb which occurs locally within a cell wall,gives rise to misori- entation between the cells separated by the wall.This (6 occurs stochastically,leading to a continual increase in 1r-g,6 the misorientation angle,which eventually saturates.The dy v3 b (7) results are consistent with the observed levels of misorien- tation angles (several degrees)associated with the so-called Here b is the magnitude of the Burgers vector and the coef- "incidental dislocation boundaries"(in the terminology of ficients a*and B are numerical constants.The various the Hansen school [170-172)),but cannot account for the
case of evolving dislocation structure is a major assumption, which definitely needs further substantiation. The total dislocation density is considered to be given by the weighted sum of the dislocation densities in the cell walls (subscript “w”) and cell interiors (subscript “c”) q ¼ f qw þ ð1 f Þqc; ð3Þ where f denotes the volume fraction of the cell walls. Obviously, the latter quantity is linked to the cell size d and the wall thickness w through a relation that should account for cell morphology. As this relation is not very sensitive to the particular shape of the cells, a simple expression f ¼ d3 ðd wÞ 3 d3 ; ð4Þ corresponding to the idealized case of cube-shaped cells is a reasonably good one. In modelling SPD processes [161,162], it was assumed that the volume fraction decreases from an initial value fo to a saturation value f1. This approach of saturation is governed by a phenomenological equation: f ¼ f1 þ ðfo f1Þ expðc=~cÞ; ð5Þ which is supported by experiments on Cu. The parameter ~c represents the inverse of the rate of this variation with the plastic shear strain c. The growth of the total dislocation density q and the concomitant decrease of the average cell size d with strain suggests that a decrease of the cell volume fraction f, which is implied by Eq. (5) for f1 < fo, is only possible if the cell wall thickness decreases fast enough with strain. This assumption was implicit in the original model [153,154]. “Thinning” of cell walls with the progress of straining is a reasonable notion, as “geometrically unnecessary” dislocations not contributing to misorientation between neighbouring cells will recover with strain. However, the phenomenological ansatz made in Eq. (5) can be replaced by a more physical assumption of the variation of the cell wall thickness. For instance, w can be de- fined in terms of a characteristic length controlling the decay of the stress field of the cell wall dislocations, which should scale with 1= ffiffiffiffiffi qw p . In this way, both the average cell size and the cell wall thickness would be expressed in terms of the dislocation densities. The evolution of the dislocation ensemble is captured in a set of coupled differential equations for the dislocation densities in the cell interior and cell walls under the “Taylor”-type assumption that the shear strain is the same in both “phases”: dqw dc ¼ 6b ð1 fÞ 2=3 bdf þ ffiffiffi 3 p b ð1 fÞ ffiffiffiffiffi qw p fb k0 c_ c_ 0 1=n qw; ð6Þ dqc dc ¼ a 1 ffiffiffi 3 p ffiffiffiffiffi qw p b b 6 bdð1 f Þ 1=3 k0 c_ c_ 0 1=n qc: ð7Þ Here b is the magnitude of the Burgers vector and the coef- ficients a and b are numerical constants. The various terms in the evolution equations represent possible dislocation reactions involved. Their physical origin was explained in Refs. [153,154]. For example, the loss of cell interior dislocations to the walls where they get entrapped is represented by the first term of Eq. (6) and the second term of Eq. (7). The last terms in both equations account for the dynamic recovery. The quantity c_ 0 denotes a reference shear rate. In the low-temperature regime typical for SPD processing, i.e. below half the melting temperature, the exponent n in both equations can be taken to be inversely proportional to the absolute temperature T, while the coef- ficient ko can be considered constant. This corresponds to dynamic recovery being governed by dislocation cross-slip. In Zehetbauer’s model [146,155,156], it was assumed that dynamic recovery in cell walls is controlled by dislocation climb—a non-conservative process involving vacancy diffusion. This process is characterized by c_ 0, which is given by an Arrhenius equation with the activation energy representing that for self-diffusion, the exponent n being a constant. As vacancy formation is influenced by hydrostatic pressure, this approach introduces a direct effect of hydrostatic pressure on the dislocation recovery kinetics [163]. An appropriate modification of the dynamic recovery term in Eq. (7) to include the hydrostatic pressure made it possible to account for the effect of back-pressure on the ECAP processing of an Al alloy [164]. The type of model described above proved to be a useful descriptive and predictive tool for computer simulations of ECAP processing of copper [157], aluminium [161], Al alloy 6016 [164], interstitial-free steel [165], CP titanium [158] and HPT processing of copper [53,166]. The focus of these simulations was the calculation of the strength characteristics of the ECAP-processed materials. However, the model was also used for calculating the texture of copper developing as a result of ECAP processing, albeit in a very simplified way [167]. We shall return to the issue of texture below. The model of grain fragmentation presented above is based on the idea that fine granularity is attained through dislocation cell formation with (tacitly assumed) accumulation of misorientations across the dislocation cell boundaries. In other words, gradual transformation of dislocation cell walls, or at least a large proportion of them, to high-angle grain boundaries is implied. Pantleon [168] and Estrin et al. [169] considered the increase of dislocation cell misorientations with progressing straining in terms of a probabilistic approach. In both models, preferred storage of dislocations with a certain sign of the Burgers vector, which occurs locally within a cell wall, gives rise to misorientation between the cells separated by the wall. This occurs stochastically, leading to a continual increase in the misorientation angle, which eventually saturates. The results are consistent with the observed levels of misorientation angles (several degrees) associated with the so-called “incidental dislocation boundaries” (in the terminology of the Hansen school [170–172]), but cannot account for the 792 Y. Estrin, A. Vinogradov / Acta Materialia 61 (2013) 782–817
Y.Estrin,A.Vinogradov/Acta Materialia 61 (2013)782-817 793 occurrence of a very significant fraction of high-angle grain thesis [1591 that the dislocation cell size and the misorienta- boundaries usually found experimentally in SPD-processed tion angle tend to saturation with the same rate and that materials,which are referred to as"geometrically necessary their product is independent of strain. boundaries"[170,171].Malygin,who also considered the It is pretty obvious that the mechanism of grain frag- evolution kinetics of cell size and misorientation angle mentation considered cannot lead to an arbitrarily small [173],associates the occurrence of the geometrically neces- grain size.Indeed,an estimate of the dislocation density sary boundaries with elastic bending of the crystal caused in Eq.(2)in terms of the applied stress a=Mr(where M by non-uniformity of plastic deformation and the atten- is the Taylor factor,which accounts for texture)leads to dant local distortions in the crystal bulk or the outer shape the following estimate for the smallest achievable grain size of the specimen.Obviously,such boundaries are not d accounted for by the models discussed. d￥KM G Although Eq.(2),which describes cell/grain size evolu- (8) tion with the accumulation of the dislocation density,was Gm introduced in a heuristic way,there are strong reasons to Here om is the highest possible stress,G is the shear modu- believe that it is a robust relation which can be used in cur- lus and a is a numerical constant,typically of the order of rent and future models.This can be expected based on both 0.5.With K of the order of 10 and M around 3 it is easily dimensional considerations and rudimentary modelling of seen that even for am close to the theoretical strength the dislocation pattern formation [171,174].Nevertheless, final grain size cannot be smaller than d~100b.That is developing a model which would provide an adequate to say,the average grain size cannot attain values in the description of the evolution of dislocation cell size and mis- "nano range",i.e.below 100 nm,if the assumed mecha- orientation angle distributions is as much a challenge as is nism of grain subdivision via dislocation cell formation the development of a full theory explaining the formation controls grain refinement. of a dislocation cell pattern in the first place. There is also a further limitation on the smallest grain Obviously,according to Eq.(2),the cell/grain size d has size achievable by this mechanism.As suggested in Ref. to saturate,asymptotically reaching a steady-state value [177],below a certain critical value of the grain size,de,dif- prescribed by the steady-state values of the dislocation den- fusive accommodation of dislocations in the walls is pre- sities governed by Egs.(6)and (7).(We note that the vol- dominant and their storage is negligible.In this regime, ume fraction of cell walls,f,also tends to saturation.) no strain hardening will occur.The critical grain size is Observations tell us that even when d approaches satura- given by: tion,e.g.with the number of ECAP passes or the rotation 1/3 angle in HPT,the misorientation angle still evolves visibly d. (9) (cf.[175)).Data for a Ni single crystal processed by HPT (Fig.4 [1761)show that the various characteristics of the where DGB is the grain boundary diffusivity.The critical microstructure,such as the mean length of small-and grain size de estimated for room temperature SPD of cop- high-angle boundaries,saturate at rates different from that per is about 250 nm.This is,indeed,consistent with the for the average cell/grain size.This is at variance with the average grain size found in copper processed by ECAP at 1.2 100 High-angle 0.8 60 0.6 0.4 0 Traction 02 10 15 20 25 30 35 10 15 20 25 30 eea oNickel <001>TOR single crystal ◆Cells2°-15° Nickel <111>TOR single crystal -Cell blocks >15 -Nickel single crystals ..High-angle boundary fraction Fig.4.Tendency to saturation with equivalent strain for various characteristics of the microstructure of monocrystalline Ni fragmented into a dislocation cell/grain structure.The data in the right-hand side figure represent the mean lengths of cell walls(misorientation angle below 15)and high-angle grain boundaries (misorientation angle above 15),as well as the fraction of the high-angle grain boundaries.After Ref.[176](reprinted with permission)
occurrence of a very significant fraction of high-angle grain boundaries usually found experimentally in SPD-processed materials, which are referred to as “geometrically necessary boundaries” [170,171]. Malygin, who also considered the evolution kinetics of cell size and misorientation angle [173], associates the occurrence of the geometrically necessary boundaries with elastic bending of the crystal caused by non-uniformity of plastic deformation and the attendant local distortions in the crystal bulk or the outer shape of the specimen. Obviously, such boundaries are not accounted for by the models discussed. Although Eq. (2), which describes cell/grain size evolution with the accumulation of the dislocation density, was introduced in a heuristic way, there are strong reasons to believe that it is a robust relation which can be used in current and future models. This can be expected based on both dimensional considerations and rudimentary modelling of dislocation pattern formation [171,174]. Nevertheless, developing a model which would provide an adequate description of the evolution of dislocation cell size and misorientation angle distributions is as much a challenge as is the development of a full theory explaining the formation of a dislocation cell pattern in the first place. Obviously, according to Eq. (2), the cell/grain size d has to saturate, asymptotically reaching a steady-state value prescribed by the steady-state values of the dislocation densities governed by Eqs. (6) and (7). (We note that the volume fraction of cell walls, f, also tends to saturation.) Observations tell us that even when d approaches saturation, e.g. with the number of ECAP passes or the rotation angle in HPT, the misorientation angle still evolves visibly (cf. [175]). Data for a Ni single crystal processed by HPT (Fig. 4 [176]) show that the various characteristics of the microstructure, such as the mean length of small- and high-angle boundaries, saturate at rates different from that for the average cell/grain size. This is at variance with the thesis [159] that the dislocation cell size and the misorientation angle tend to saturation with the same rate and that their product is independent of strain. It is pretty obvious that the mechanism of grain fragmentation considered cannot lead to an arbitrarily small grain size. Indeed, an estimate of the dislocation density in Eq. (2) in terms of the applied stress r ¼ Ms (where M is the Taylor factor, which accounts for texture) leads to the following estimate for the smallest achievable grain size ds: ds b ffi KMa G rm : ð8Þ Here rm is the highest possible stress, G is the shear modulus and a is a numerical constant, typically of the order of 0.5. With K of the order of 10 and M around 3 it is easily seen that even for rm close to the theoretical strength the final grain size cannot be smaller than ds 100b. That is to say, the average grain size cannot attain values in the “nano range”, i.e. below 100 nm, if the assumed mechanism of grain subdivision via dislocation cell formation controls grain refinement. There is also a further limitation on the smallest grain size achievable by this mechanism. As suggested in Ref. [177], below a certain critical value of the grain size, dc, diffusive accommodation of dislocations in the walls is predominant and their storage is negligible. In this regime, no strain hardening will occur. The critical grain size is given by: dc ¼ DGBb c_ 1=3 ; ð9Þ where DGB is the grain boundary diffusivity. The critical grain size dc estimated for room temperature SPD of copper is about 250 nm. This is, indeed, consistent with the average grain size found in copper processed by ECAP at Fig. 4. Tendency to saturation with equivalent strain for various characteristics of the microstructure of monocrystalline Ni fragmented into a dislocation cell/grain structure. The data in the right-hand side figure represent the mean lengths of cell walls (misorientation angle below 15) and high-angle grain boundaries (misorientation angle above 15), as well as the fraction of the high-angle grain boundaries. After Ref. [176] (reprinted with permission). Y. Estrin, A. Vinogradov / Acta Materialia 61 (2013) 782–817 793