Grain size evolution during discontinuous dynamic recrystallisation E. I. Galindo-Nava1,2 and P. E. J. Rivera-D´ıaz-del-Castillo1 1 Department 2 Department

of Materials Science and Metallurgy, University of Cambridge, United Kingdom

of Materials Science and Engineering, Delft University of Technology, Netherlands Abstract

Grain size behaviour during discontinuous dynamic recrystallisation in alloys is characterised by a theoretical approach. It combines a novel thermostatistics framework, with classical grain nucleation and growth formulations. Configurational effects from dislocation migration paths control microstructure variation via an entropic effect. An alternative approach for accounting for solute–drag effects is proposed. It is shown that the drag atmosphere linearly amounts to the atomic radius of solute atoms. The approach is validated for various deformation conditions in 20 single–phase metals.

Keywords: Dynamic recrystallization; modeling; statistical mechanics; grain boundary migration; dislocation theory. Discontinuous dynamic recrystallisation (DRX) occurs during the hot deformation of low to medium stacking fault energy alloys. New grains are formed by strain–induced grain ∗ is reached [1]. Dislocation–free grains boundary motion once a critical shear strain γDRX

then grow as deformation continues, by bulging into their surroundings and consuming the deformed regions [2]. Once steady state is achieved, a constant average grain size Dss is reached; Dss is shown to be independent of the average initial size D0 [3]. However, Dss is shown to be highly sensitive to the material’s purity and solute concentration [4, 5, 6]. The objective of this work is to present a novel approach for describing grain size behaviour during discontinuous DRX at high temperatures in single-phase FCC alloys. Moderate strains and strain rates are considered only. These assumptions are introduced to avoid the potential

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occurrence of other mechanisms such as continuous DRX. A new statistical thermodynamics– based theory is combined with well established formulations for grain boundary nucleation and migration. The steady state grain size is firstly obtained, once the driving pressure for grain growth reaches equilibrium. This is followed by deriving an evolution equation for describing the average grain size D (containing deformed and recrystallised grains) and the recrystallised (DDRX ) grain size evolution with strain. A thermodynamic framework has been derived by the authors for predicting the dynamic recovery rate and average dislocation density ρ evolution at various temperatures and strain rates [7, 8]. The evolution of the average dislocation subgrain size dc has also been modelled by performing a balance between the energy produced by a dislocation forest and by the
T ∆S √1 , where 1 + ordered material in the form of dislocation cells [9]: dc = √κcρ = 12π(1−ν) 3 (2+ν) µb ρ ν is the Poisson ratio, µ is the shear modulus, b is the magnitude of the Burgers vector, and ∆S is the entropy associated to the possible dislocation migration paths. The latter has been introduced to account for the energy loss due to the different dislocation velocity configurations. The entropy equals:  ∆S = kB ln

 ε˙0 + ϑ , ε˙

(1)

where ε˙ is the axial strain rate1 ; ε˙0 = cbρY is a constant related to the speed of sound c,
2 and the ρY is the dislocation density consistent with the yield point (ρY = 0.9σY /µb [8]);
ϑ = 1013 exp − ERTm is the vacancy atomic jump frequency, and Em is the vacancy migration energy. Details on ∆S derivation and application can be found elsewhere [7, 8, 9]. Once recovery and subgrain evolution were obtained, the critical conditions to onset DRX have been described [8]: the strain energy to nucleate dislocation–free grains (being proportional to 21 µb2 , around the initial boundary length) equals the stored energy at subgrain 1

The shear (γ) and axial (ε) strain are directly related by the Taylor factor γ = M ε, where M = 3 for the tested materials [8].

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boundaries ( 21 µb3 ), being reduced by the dissipation (entropy) effects when grain boundary bulging occurs (T ∆S 2 ), and from the fraction of dislocations in the subgrain interiors moving ∗ towards the walls ( κ1c T ∆S [8]). From this balance, γDRX is obtained [8]:

∗ γDRX =

1 µb3 2

− 1+

1 κc




T ∆S

1 µb3 2

.

(2)

This equation represents the ratio between the effective activation energy to ignite grain boundary bulging, and the stored energy in the material. Thus, the activation energy for grain nucleation Qnuc should be proportional to the numerator in the previous equation:

Qnuc

    1 1 3 1 = µb − 1 + T ∆S , 8 2 κc

(3)

where 8 is a geometric factor that accounts for the effective boundary sites where grain bulging occurs; this constant amounts to the subgrain surface area per unit volume of a subgrain, and has been considered to be in the range 1–10 [10, 11]. By combining the previous results, the average dislocation density evolution and flow stress response (via ρ and dc ) during DRX have been described for several multicomponent systems [8]. The average grain size at a given shear strain (γ) is obtained by adding the size of the deformed (DjDef ) and recrystallised (DkRex ) grains over the total number of grains N : PN D=

i=1

N

Di

1 = N

X N0 j=1

DjDef

+

N X

DkRex

 ,

(4)

k=N0 +1

where N0 is the number of deformed grains prior to recrystallisation. N and Di (i = 1, ..., N ) P ∗ are dependent on γ; also D = D0 and N = N0 for γ ≤ γDRX . (N −1 DjDef ) represents an P average deformed grain size, whereas (N −1 DkRex ) defines an average recrystallised grain 2

Grain boundaries are considered to be formed by a dislocation arrangement, hence configurational effects from moving boundaries equal T ∆S [8].

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size DDRX . Dss is obtained before deriving the equations for D and DDRX . A grain boundary moves in response to a difference in the free energy between adjacent grains, allowing atoms to move towards a preferred configuration [1, 12]. Such is promoted by P a net driving pressure P = i Pi induced on the boundary, where Pi accounts for different acting pressures. The average velocity v for moving boundaries can be expressed by the product of the driving pressure and its mobility M > 0 [13]: v = M P . Steady state is achieved when grains stop growing and v is null, which occurs when P = 0. The driving force for grains to grow during DRX accounts for the energy variations as the grain size evolves. Such is composed by the contributions of : 1) the pressure available for grains to grow, P1 , being this term proportional to the stored energy in dislocations 12 µb2 ρ [1]; 2) capillary pressure effects −P2 , due to grain size variations

χGB D

[13, 14], where χGB is the grain boundary

energy; and 3) solute and/or impurity drag pressure −P3 , preventing further growth [12]. ρ and D account for regions containing both deformed and recrystallised grains; however, P should include effects from recrystallised grains only. A term multiplying

1 µb2 ρ 2

in 1) is
2 /D , introduced to account for the regions on the freshly nucleated grains: β1 d2c exp − QkBnuc T
represents the effective area consuming dislocations by new grains, where β1 d2c exp − QkBnuc T and β1 is a constant related to the number of effective nucleation sites around subgrain boundaries. Another term multiplies

χGB D

in item 2) to account for the effective length where

capillary effects can take place; this equals the ratio between the grain boundary perimeter of P PN Rex recrystallised grains and the total grain boundary perimeter3 : N j=N0 +1 πDk / i=1 πDi = DDRX /D (equation 4). On the other hand, classical solute–drag models are based on the assumption that a moving boundary drags a solute atmosphere that exerts a retarding force on it [12]. This force is defined in terms of the boundary velocity and solute concentration. Although they provide good qualitative agreement with experimental observations, it has been shown that they are limited in making real quantitative predictions [1]. An alternative 3

The average grain size is estimated from two dimensional micrographs, hence the analysis defined in two dimensions.

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approach is proposed by estimating the probability for a moving boundary to drag a number of impurity/solute atom being randomly located on its surroundings. This probability depends on the distortion atmosphere and the binding energy between impurity atoms and grain boundaries. Boundary–solute interactions will exhaust a fraction of the pressure available for further growth (P1 ). If it is assumed that the boundary–drag atmosphere is αsol b metre per solute atom, where αsol is a dimensionless constant related to the distortion field induced around solute atoms and to the solute–boundary binding energy, the probability p that during recrystallisation, all boundaries will drag xsol solute atoms during the whole process, where xsol is the solute atom fraction, corresponds to an exponential distribution [15]: p = 1/3

1/3

1 − exp(−αsol b/Λsol ) = 1 − exp(−αsol xsol ), where Λsol = b/xsol is the average solute spacing [16]. Then, P3 equals pP1 . Finally, the driving pressure for grain growth during DRX becomes (P = P1 − P2 − P3 = (1 − p)P1 − P2 ):      Qnuc χGB DDRX 1 2 β1 d2c 1/3
− exp − αsol xsol − . P = µb ρ 2 exp 2 kB T D D D

(5)

1/3

If additional solute elements are incorporated into the alloy, then αsol xsol is replaced by P

s

1/3

αs xs , to account for the total drag effect from different elements. An impurity–drag term 1/3

αimp ximp is included in the previous summation. When steady state is achieved, equation 5 is null, and the average and recrystallised grain sizes are constant D = DDRX = Dss : X
κ2 µb2 Dss = 340 c exp − αs xs1/3 exp 2χGB s



 Qnuc − , kB T

(6)

where β1 = 340 was found for all (20) modelled materials. An important aspect of this equation is that the entropy term in Qnuc is the principal contribution for temperature and strain rate variations in Dss . The model results for Dss are compared against experimental measurements obtained from the literature for various metallic systems. Table 1 shows the deformation conditions and impurity concentrations for the modelled materials. It also shows 5

the value of αimp for each alloy family. 7N Cu, 6N Cu and 4N Cu stand for copper with the corresponding impurity concentrations, Ni-imp stands for pure nickel with impurities, and Steel A–D stand for austenitic stainless steels (their compositions are reported in the following paragraphs). Table 1: Deformation Material 7N Cu 6N Cu 4N Cu Ni-imp Ni(0.05-20)Fe Ni30Fe Ni30Fe Ni(0-1)Nb Ni20Cr Steel A Steel B Steel C Steel D

conditions and impurity concentration for the tested alloys. Ref. T (K) ε˙ (s−1 ) Imp. (at) αimp [4] 573−773 0.0002−0.2 3×10−8 16 [4] 723 0.0002−0.2 3 × 10−7 16 [4] 573−773 0.0002−0.2 3 × 10−4 16 [17] 1073−1273 0.01−10 4 × 10−4 3 −4 [5] 1035−1380 0.0006−1.32 6 × 10 3 −4 [18] 1073−1173 0.01−0.1 6 × 10 3 [19] 1273−1473 1 3 × 10−4 3 −5 [20] 1073−1273 0.1 5 × 10 3 −4 −2 [21] 973−1223 7×10 4.5 × 10 3 [22] 1123−1323 10−3 3 × 10−5 0.1 [22] 1123−1323 10−3 6 × 10−5 0.1 −4 [23] 1073−1173 2×10 2.5 × 10−2 0.1 [24] 1073−1173 0.1 3.4 × 10−2 0.1

Additionally, αs values were obtained for Ni alloys and steels. A linear relationship was found between αs and the atomic radius rs (in nm) of the respective solute elements. Additional elements were considered as impurities. Table 2 shows αs values and the formulae for Ni and Fe (in FCC phase) alloys. This illustrates how the distortion atmosphere from different elements affects the dragging force. Solute-boundary binding effects are related to the coefficients in the relationships displayed in Table 2; a future analysis can be made to identify their explicit dependence. Table 2: Solute-drag atmosphere constants. Solvent Solute (αs ) αs in terms of rs Ni Fe (3.75), Cr (4.8), Nb (15) 562.6rs − 67.15 Fe Cr (3.97), Ni (3.71), C(0.2) 6.52rs − 4.37

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Figure 1(a) shows the model comparison for 7N Cu, 6N Cu and 4N Cu and for steels A and B with concentrations Fe18.1Cr12.1Ni0.0005C and Fe18.2Cr12.2Ni0.0015C (wt%), respectively. In order to show temperature and strain rate variations simultaneously, the Qself
, and Qself is the activation horizontal axis is shown in terms of log Z, where Z = ε˙ exp RT energy for self diffusion of Cu and Fe, respectively. The values for µ, b, Em and c were obtained from the literature for all tested materials and can be found elsewhere [8]. χGB = 0.625, 0.8 and 0.76 for Cu, Ni and steels respectively [1]; these values were assumed constant for multicomponent systems, as their compositional variation was not found in the literature. To verify solute–drag effects on Dss , Figure 1(b) shows the model results for various Ni alloys containing Nb and Fe (concentrations in wt%), at several deformation conditions. The impurity/solute–drag effect is described with good accuracy for wide concentration ranges, and for up to three orders of magnitude in Dss . The model also shows excellent agreement on the temperature and strain rate variations.

Dss (µm)

Exp - 7N Cu Exp - 6N Cu Exp - 4N Cu Mod - 7N Cu Mod - 6N Cu Mod - 4N Cu

Exp - Steel A Exp - Steel B Mod - Steel A Mod - Steel B

3

2

10

(b) 10

3

Mod - Dss (µm)

(a) 10

2

10

Ni0.05Fe Ni5Fe Ni10Fe Ni20Fe Ni30Fe Ni30Fe

Ni-imp Ni0Nb Ni0.01Nb Ni0.1Nb Ni1Nb

1

10

1

10

6

9

12

15

18

log(Z)

1

10

2

10

Exp - Dss (µm)

3

10

Figure 1: Dss variation for (a) Cu with different purity, two austenitic stainless steels, and (b) for Ni and its alloys. D evolution with strain can be prescribed by deriving equation 4 with respect to γ and

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applying the chain rule: N N dD 1 X dN 1 X dDi DN dN =− 2 Di , + + dγ N i=1 dγ N i=1 dγ N dγ | {z } | {z } | {z } A)

B)

(7)

C)

where DN is the grain size of the last term in the summation. Term A) can be simplified to (equation 4)

1 dN D. N dγ

For moderate strains, it can be considered that no spatial contraction or

expansion occurs at the microstructural scale [25]; therefore, the overall grain size variation P 0 dDjDef PN PN dDi dDkRex with γ should be null, i.e.: N + = j=1 k=N0 +1 dγ i=1 dγ = 0, as the former and dγ latter are negative and positive, respectively. Thus, term B) in equation 7 vanishes. Due to the fact that D and

dD dγ

are independent of term permutation inside the second

summation in equation 4, term C) in equation 7 should be independent of DN choice; also ∗ 0 ≤ DN ≤ Dss . On the other hand, for γ = γDRX , N = N0 and DN = 0, and for large strains

(when steady state has been reached), N >> N0 and DN ≈ Dss , as most recrystallised grains are expected to have reached Dss . Thus, a linear interpolation in DN is proposed to describe
such behaviour: DN = Dss 1 − NN0 . Combining the previous expressions, the evolution equation is simplified to:     1 dN dD N0 = Dss 1 − −D . dγ N dγ N dN dγ

(8)

represents the grain number variation with strain, which is directly related to the grain

nucleation rate. Following the classical nucleation theory, the grain nucleation rate J depends on the potential number of new grains formed per second n˙ nuc [26]. For a constant shear strain rate

dγ , dt

n˙ nuc is represented by

dγ dN . dt dγ

J equals [26, 12]:

dγ dN exp J= dt dγ

8



 Qnuc − . kB T

(9)

On the other hand, it is well established that the grain nucleation rate increases when the initial grain size decreases [1]. Moreover, the nucleation rate should be limited by the number and size of grains at the steady state, hence the ratio Dss /D0 can approximate the fraction of nucleated grains per grain. An alternative expression for J can be defined by adopting a ∗ linear interpolation in γ, from zero (at γ = γDRX ) up to Dss /D0 :

∗ ) dγ Dss (γ − γDRX N, J = β2 dt D0

(10)

where β2 is a constant. The introduction of the linear increase is equivalent to consider a linear grain growth behaviour. This assumption is consistent with classical recrystallisation approaches, such as the use of JMAK statistics [1]. Although the previous equation considers an unbounded increase in γ, a limiting value of the nucleation rate is displayed once steady state is achieved. Combining equations 9 and 10, leads to: 1 dN ∗ = β2 aD (γ − γDRX ), N dγ where aD =

Dss D0

exp

Qnuc kB T




(11)

∗ . N can then be obtained by setting N = N0 for γ ≤ γDRX . The

∗ is: solution of equation 8 with D = D0 at γ = γDRX



∗ ∗ ∗ D = D0 exp − aD (γ − γDRX )2 + Dss (1 − (1 + aD (γ − γDRX )2 ) exp − aD (γ − γDRX )2 ) | {z } | {z } (A)

(B)

(12) where β2 = 2 for all modelled materials. Figure 2 shows the average grain size evolution with axial strain for Ni and Cu, and their comparison against experimental data obtained from [27] and [28], respectively. The impurity concentration for Ni equals 3 × 10−4 (at), whereas for Cu it was fixed to 2 × 10−3 (at), as no specific concentration was provided. The model is able to reproduce the experimental measurements in all cases for various loading conditions during straining. A decrease in D before reaching steady state is experimentally 9

and numerically observed, due to a higher nucleation rate when D0 is small [27]. Ni - Exp - D0=60 µm

70

Ni - Exp - D0=30 µm

60

-4

D (µm)

50 40 30 20 10

673 K

D0= 24 µm

0 0.0

0.2

-1

Cu - Exp - 5x10 s -3 -1 Cu - Exp - 5x10 s -2 -1 Cu - Exp - 5x10 s -2 -1 Cu - Mod - 5x10 s -3 -1 Cu - Mod - 5x10 s -4 -1 Cu - Mod - 5x10 s

923 K -3 -1 2x10 s

0.4

0.6

ε

0.8

1.0

1.2

Figure 2: D variation with strain for Ni and Cu. Term (A) in equation 12 decreases with strain upon eventually vanishing, whereas term (B) is initially null and increases with strain upon reaching Dss . Thus, it can be inferred that the average deformed and recrystallised (DDRX ) grain size equal terms (A) and (B), respectively. Figure 3 shows DDRX evolution for a Ni20Cr (wt%) alloy and Steel C, of composition Fe18Cr8.35Ni0.058C (wt%), and their comparison against experimental measurements. The model shows good agreement; it is also shown that a finer initial structure leads to faster grain growth kinetics.

DDRX (µm)

10 8 6 4

Exp - 1223 K Ni20Cr Exp - 1173 K D0=80 µm Exp - 1073 K -4 -1 Exp - 973 K 7x10 s Model Exp - 1173 K Steel C Exp - 1123 K D =25 µm 0 Exp - 1073 K -4 -1 2x10 s Model

2 0 0.0

0.2

0.4

0.6

ε

0.8

1.0

1.2

Figure 3: DDRX evolution for Ni20Cr and Steel C. To verify that the nucleation rate (via N ) proposed in this work (equation 10) is in agreement with experimental observations, Figure 4 shows a comparison in D and N for Steel D, with 10

composition Fe18.35Cr8.96Ni0.064C (wt%). N was estimated by direct measurement of the grain number from EBSD imaging; the error bars display the stereological error from the 2D measurements [29]. An initial grain population (per unit area) N0 was estimated to be 2.3 × 10−5 grains/µm2 . The model shows good agreement, confirming that the nucleation rate is consistent with experiments. 0

10 D - Exp D - Mod

30

N - Exp N - Mod

D (µm)

20

-2

10

0

-3

10

Steel D 0.0

0.2

2

10

N (grains / µm )

-1

10

1323 K -1 0.1 s

-4

0.4

0.6

0.8

10

ε

Figure 4: D and N evolution with strain for Steel D. This work provides a fundamental description on grain size behaviour during discontinuous DRX: the average, recrystallised and steady state grain size are predicted. A remarkable feature is that an evolution equation for D is obtained from the definition of arithmetic mean and taking its variation with strain (equation 8). It is shown that temperature and strain rate variations in Dss , D and DDRX (via the migration rates) are mainly controlled by the entropy term T ∆S in Qnuc (equation 3). This allows to preserve the same expression for the activation energy in all tested materials; this is the first theory to account for moving dislocations’s configurational effects into Qnuc . Alternative approaches for impurity/solute–drag effects and grain nucleation rate were postulated. A linear relationship between the drag atmosphere and the solute radius was found. Thus, a finer grain structure may be tailored during DRX, by increasing the concentration of elements with larger atomic radii. The theory was validated against experimental measurements for 20 single–phase metals. Additional work can be made to characterise particle pinning effects. 11

Acknowledgements The authors are grateful to Prof. A.L. Greer for the provision of laboratory facilities. E.I. Galindo-Nava is grateful to CONACYT and the Roberto Rocca Education Program for the provision of financial support.

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