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Pinning force from multiple second-phase particles in grain growth
Computational Materials Science 93 (2014) 81–85
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Pinning force from multiple second-phase particles in grain growth Nan Wang a,⇑, Youhai Wen b, Long-Qing Chen a a b
Department of Materials Science and Engineering, Pennsylvania State University, University Park, PA 16802, USA National Energy Technology Laboratory, Albany, OR 97321, USA
a r t i c l e
i n f o
Article history: Received 13 January 2014 Received in revised form 16 June 2014 Accepted 17 June 2014
Keywords: Grain boundary Particle pinning Phase-field Grain growth
a b s t r a c t A factor that can reduce particle pinning force significantly in grain growth is found when the grain-boundary is pinned by multiple particles. The pinning force, in this case, is a function of particle radius over inter-particle distance. A previously proposed phase-field model for particle pinning is used to validate this predicted pinning force reduction in two and three dimensions. When applied to coherent pinning particles, the same effect is observed in simulations. It is shown that, at application relevant high particle volume fraction, the average grain size is affected by this reduction of pinning force. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Study of the particle pinning in grain growth has been an important topic for a long time since introducing second-phase precipitates is one of the most commonly used method to inhibit grain growth and achieve better mechanical properties through Hall–Petch relation in polycrystalline materials. The pinning force, as formulated by Zener, comes from grain-boundary (GB) which forms a bow-out toward the migration direction during its interaction with a second-phase particle [1]. Therefore, the GB shape near the pinning particle is critical to pinning force evaluation [2,3]. When multiple particles are considered, the total pinning pressure is assessed using a summation of pinning force from individual particles [4–6]. While both experimental and computational studies have shown good agreements with this theory, they also suggested a deviation from the prediction at high particle volume fraction [5]. Different corrections have been proposed in this case based on the break-down of random particle distribution [10,12], particle overlaps [20] or the so called Louat effect where the pulling force of particles in front of an advancing boundary is considered [7,8]. However, an important factor, the change of near-particle GB shape at high particle volume fraction has been ignored. In this work, we address this issue by calculating the GB shape between nearby particles exploiting a previously ignored factor in Ashby’s work [2], and demonstrate, using both analytics and numerical method, how the change of GB shape reduces the ⇑ Corresponding author. Tel.: +1 814 880 0997. E-mail address: [email protected] (N. Wang). http://dx.doi.org/10.1016/j.commatsci.2014.06.030 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.
pinning force and affects the average grain size at high particle volume fraction. 2. Particle pinning force in grain growth In a configuration where two pinning particles are close to each other (which occurs at high volume fraction, or in low volume fraction case with highly inhomogeneous distribution of second-phase particles), the GB shape needs be determined using Ashby’s method [2] since the near-particle boundary shape in Hellman’s approach [3] is only a zero driving force approximation. The axi-symmetric curvature equation for a single particle in cylindrical coordinates is a second order ordinary differential equation (ODE) with driving force P as a free parameter [2].
zR 1 zRR P qffiffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ : 2 3=2 R 1 þ z2 c 1 þ z R R
ð1Þ
Here c is the GB energy, R and z are defined in Fig. 1, zR and zRR are the 1st and 2nd derivative of z with respect to R. Ashby et al. solved this equation by integrating from the GB-particle contact point with a given position angle b (as defined in Fig. 1) and an interface balance condition cos a ¼ ðc1 c2 Þ=c where c1 is the surface energy for particle-grain1 interface, c2 is the surface energy for particle-grain2 interface [2]. These two boundary conditions specify the coordinates and the GB slope at the contact point. With a prescribed driving force P, GB energy c and two particle-grain interfacial energies c1 and c2, the GB shape is completely determined by solving Eq. (1). However, Ashby imposed another condition that requires a flat boundary at the midway (L/2) of
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pinning becomes less effective when the inter-particle distance becomes comparable with the particle size. Although derived in 2D, Eq. (3) clearly demonstrates that the maximum pinning force is given by not only the local contact condition (angle a) but also the relative size of particle over inter-particle distance. 2.2. Pinning theory in 3D Fig. 1. A boundary seperating grain1 (lower grain) and grain2 (upper grain) is pinned by multiple second-phase particles of radius R0 and inter-particle distance L. Only one pinning particle is shown in this window of size L to demonstrate the geometry, other nearby particles are located at distance L from this one. Window boundary lines (the two vertical solid lines) are drawn exactly at the mid-point of two adjacent pinning particles. The GB surface is a function of (R, Z) coordinates. Position angle b gives the GB-particle contact point. Angle a which is defined between GB line and the particle surface tangent is given by Young’s force balancing condition. Surface energies are not labeled here. A modification angle a is needed due to particle coherency. For an incoherent particle, a = p/2 since c1 = c2. For a coherent particle, a < p/2.
two particles, and argued that such a condition gave a relationship between driving force P and angle b (note: this is an L dependent relation). This argument seems reasonable but is inconsistent with their theory on the critical pinning angle presented earlier in the same paper [2] as we will see. The pinning force in Fig. 1 configuration is simply F z ¼ ¼ 2pR0 c cos b cos ða bÞ. The maximum pinning force is then F max z
pR0 cð1 þ cos aÞ with the critical pinning angle given by bcrit ¼ a=2. However, according to Ashby’s flat midpoint argument, the angle bcrit should be associated with the maximum driving force Pmax with inter-particle distance L as a parameter. For b > bcrit, a stable pinning configuration can exist only under a decreased driving force P < Pmax. Now, it is easy to see an inconsistency between the two bcrit relations since one depends on the inter-particle distance L while the other is a simple function of a. To resolve this controversy, the maximum pinning force will be derived analytically in two dimensions (2D) with the inter-particle distance L as a parameter, and generalized to three dimensions (3D) numerically. 2.1. Pinning theory in 2D While the particle pinning theory in 2D is very different from its 3D cousin, it is a simpler demonstration of the basic physics. We choose to present our L dependent pinning force theory in 2D first because our key argument can be presented analytically in this case. In 2D, Fig. 1 is still a good illustration of the basic geometry. Instead of solving for 3D GB shape from Eq. (1), the 2D GB shape is a simple circle with radius q given by q = c/P. The flat midpoint condition can be easily expressed as the following:
L c ¼ R0 cos b þ cosða bÞ: 2 P
ð2Þ
This configuration is stable as long as the driving force is smaller than Pmax beyond which the grain-boundary will detach from the pinning particle. One can see that the condition for maximum driving force is sin (a b) = 2 sin aR0/L by rearranging Eq. (2) and using oP/ob = 0. Since the pinning force is given by 2c cos(a b), by eliminating the cosine factor with the maximum driving force condition, the maximum pinning force is then:
F max z
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2R0 2 ¼ 2c 1 sin a: L
ð3Þ
For incoherent particles, the maximum pinning force is reached at b = cos1(2R0/L) rather than the simple 2D maximum pinning force condition b = p/2. Comparing with the classic pinning force theory that gives F max ¼ 2c, this equation indicates that particle z
To formulate the same L-dependent particle pinning theory in 3D, one needs to solve for the GB shape from Eq. (1) numerically (we use simple 4th order Runge–Kutta). Integration of the curvature equation (Eq. (1)) starts from the midway (L/2) with flat boundary condition dz/dR = 0 and z = C1. On the other end, the GB should join the pinning particle at the correct angle such that the surface energy balance condition cos a = (c1 c2)/c is satisfied. Assuming a spherical particle of radius R0 is centered at the origin, this joining angle condition can be expressed as:
dz ¼ cotða bÞ: dRR¼R0 cos b
ð4Þ
By tuning the initial condition C1, the GB line can be moved up and down such that the point with correct joining angle can be positioned on the particle sphere. The maximum driving force Pmax and the associated position angle bmax are then found by increasing P continuously until there is no correct solution for the GB shape. Since the pinning force is given by F z ¼ 2pR0 c cos b cos ða bÞ, the maximum pinning force can be directly evaluated with bmax. In the regime where the particle radius is much smaller than the inter-particle distance, we found that the GB-particle joining angle bmax for the maximum pinning force configuration is actually very close to the critical value bcrit = a/2. This is expected as this condition is very similar to the single particle configuration Ashby used to derive the relation. As the inter-particle distance becomes comparable to the particle size, bmax becomes smaller than bcrit. Solution of GB shape for b = bcrit in this condition still exists but only for a smaller driving force P and is not dynamically accessible by surface energy minimization in the phase-field method we use to validate the theory in the next section. 3. PF model for quantitative evaluation of particle pinning force To validate the pinning force theory proposed in the previous section numerically, we employ a phase-field (PF) model to evaluate the particle pinning force. As demonstrated in previous works [9,19], this method can quantitatively reproduce the particle pinning force at single particle level in grain growth. Here the model is briefly described and more details can be found in Ref. [9]. The phase-field free energy functional is:
# Z " 2 1X 2 F¼ n Kðrgi Þ þ Phðg1 ; g2 ; g3 Þ dv ; f0 ðg1 ; g2 ; g3 Þ þ 2 i¼1 i
ð5Þ
where the gradient energy coefficient is a product of numerical parameter ni and dimensional constant K, and
" f0 ¼ Df a
# X 2 3 X 3 X Y g2 g4 i þ i þ cij g2i g2j þ b g2i ; 2 4 i¼1 i¼1 j
ð6Þ
with dimensional constant Df, numerical parameter a, b and cij. Here, the first two terms are commonly seen in PF grain growth model, while the last term in f0 is added to account for the triple junction. A P term coupled with function h ¼ g31 ð10 15g1 þ 6g21 Þ is also added to account for the grain growth driving force in this twograin-one-particle simple geometry. The simple grain growth in Fig. 1 is modeled with g1 for grain1, g2 for grain2 and g3 for pinning
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particle. The time evolution of gi follows the Allen–Cahn phase-field dynamics [18].
@g dF i ¼ Li ; dgi @t
ð7Þ
with time t and kinetic coefficient Li. This equation above applies to
g1 and g2 only, while g3 is fixed since we assume the coarsening of pinning particles is a slow process at the grain growth time scale. Dimensionless form of Eq. (7) is:
@ gi ~ @ð f0 =Df Þ ~ @h ¼ Li þP w2 ni r2 gi ; @ gi @ gi @~t
ð8Þ
~ ¼ P=Df with characteristic length scale ~t ¼ t=s, ~Li ¼ Df sLi , P where pffiffiffiffiffiffiffiffiffiffiffi ffi w ¼ K=Df and time scale s. In case of incoherent particle pinning, the three surface energies involved are two incoherent particle-grain surface energies (c1 and c2) and one grain-boundary energy (c). Since the nature of those interfaces are rather similar, we choose to use model parameters a = 0.1, n1 = n2 = 0.2, c13 = c23 = 2.0 and c12 = 0.0993 which give c1 = c2 = c = 0.6 J m2. In case of coherent particle pinning, the surface energy relations are more complicated. Since the particle was initially formed coherently with grain2, a typical coherent surface energy should be used for c2 while the choice of incoherent surface (c1) and grain-boundary (c) should be similar to the case of incoherent particle. To be specific, we use a = 0.1, n1 = 0.5, n2 = 0.1, c13 = 0.386, c23 = 0.0849, c12 = 0.1428 which correspond to c1 = c = 0.75 J m2 and c2 = 0.15 J m2. In both cases, the triple junction parameter b is chosen to be 50. Numerical integration of Eq. (8)is implemented on a simple square mesh with Dx/w = 0.4. Standard central difference and forward Euler scheme are used. By increasing P gradually, a series of stable configuration where the grain growth from grain1 to grain2 is completely pinned by the particle can be achieved until it reaches Pmax beyond which a stable configuration no longer exists. The maximum pinning force can be numerically evaluated using the distortion of grain-boundary [19].
F z ¼ Pmax
Z
^ ^z dA; n
ð9Þ
Correction factor
^ is the grain-boundary surface normal, dA is the surface elewhere n ment, ^z is the unit vector in z direction and the maximum driving force Pmax for a pinned grain-boundary.
1
Based on the pinning theory we proposed in Section 2 and the PF numerical method in Section 3, the deviation of maximum pinning force from classic single particle theory is shown in Fig. 2 for an incoherent particle with c1 = c2 = c in 2D and 3D. As shown in Fig. 2, the PF calculated maximum pinning force for incoherent particles agrees well with our theory in the regime where the inter-particle distance L is comparable with the particle size. It is clear that the effect of small inter-particle distance is noticeable in 2D even for 2R0/L 0.2, while the same effect in 3D is insignificant until 2R0/L reaches 0.6. This dimension-dependent result is expected since the 3D GB shape is given by the balance of 2 principal curvatures (the curvature in R–z plane and the curvature in R–h plane) while the 2D GB shape is only given by a single curvature. Such a small deviation at fairly large 2R0/L in 3D could be part of the reason that this effect was ignored in previous works. As mentioned before, the GB-particle joining angle is related to the particle coherency. The angle a between GB line and particle surface tangent is given by the balance of surface tensions at the joining point. For a coherent second-phase particle, this angle is quite small since c2 c1 and c c1. It is well known that a coherent second-phase particle provides larger pinning force comparing with an incoherent particle of the same size. As outlined in the PF model part, with a different set of model parameters, the PF numerics for incoherent pinning particle can be used to study the pinning force from a coherent particle. The same pinning force reduction is also observed for coherent particles as shown in Fig. 3. Here, instead of looking at the correction to previous single particle theory as a function of relative particle size in Fig. 2, this plot relates the pinning force reduction to a more experiment relevant measure, the particle volume fraction. Zener’s theory suggested a maximum pinning pressure Pz = 3fvc/(4R0) which is an average of maximum pinning force from all particles within a thin layer (R0) of GB over the total GB area [1,3]. The particle volume fraction fv is given by 4pnR30 =3 with n being the particle density per volume. For coherent particles, all arguments above apply except for the maximum pinning force which should be multiplied by a factor 1 + cos a [2]. As can be seen from the comparison with PF simulations, the old single particle theory agrees well with PF results for particle volume fraction smaller than 0.1. After that, a deviation from the theory becomes noticeable. Applying our L-dependent theory gives the same result for small volume fraction (fv < 0.1), while it predicts a reduction of pinning force at high particle volume fraction that agrees well with PF simulations. While the coherent pinning particle maintains its advantage over the incoherent particle at all volume fractions, the same pinning force saturation effect is observed regardless of particle coherency. Here,
0.8 theory
0.6
2D theory 2D PF 3D theory 3D PF
0.4 0.2
0
0.2
correction
0.02
PF
0.4
0.6
0.8
Pzw/γ
4. Results and discussions
0.01
2R 0 /L Fig. 2. Comparison of inter-particle distance (L) dependent pinning theory with previous single particle theory. The result is then validated by numerically simulation with phase-field model for incoherent second-phase particles. Y-axis is the ratio of maximum pinning force predicted with L-dependent theory and classic single particle theory. Red line shows the deviation of pinning force from classic theory in 2D based on Eq. (3). Green line shows the pinning force deviation in 3D based on numerical integration of Eq. (1) using simple Runge–Kutta method. Red dots and green dots are the maximum pinning force evaluated using phasefield method in 2D and 3D. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
0
0
0.2
0.4
0.6
Volume fraction f v Fig. 3. Comparison of inter-particle distance (L) dependent pinning theory (dot line) with previous single particle theory (solid line) at various particle volume fraction. The numerical results from phase-field model are shown as squares. Y-axis is the pinning pressure Pz normalized by c/w. X-axis is the particle volume fraction. The upper branch is for coherent pinning particle while the lower branch is for incoherent pinning particle.
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Table 1 Previous results on the relation between average grain size and particle volume fraction. Theory
Simulation
[3] Gave exponential 1 With GB bow-out, [4] gave exponential 0.92 [10] Refined [3] gave exponential 0.93 With Louat effect, [11] gave exponential 0.87 High volume fraction, [12] gave exponential 0.5 High volume fraction, [10] gave exponential 0.33
MC in [16] gave exponential 0.31 MC in [15] gave exponential 1 MC in [21] gave exponential jump from 1 at low volume fraction to 0.33 at high volume fraction MC in [17] gave exponential 1, and results depend on particle size PF in [13] gave exponential 0.87 PF in [14] gave exponential 0.93
3 PF results -1
log (D/R 0)
we need to point out that the demonstrated pinning pressure reduction in Fig. 3 is a mere extrapolation of single particle results in Fig. 2, and is not based on simulations of polycrystalline grain growth with multiple pinning particles. Volume fraction in the single particle setting is measured by the product of pinning particle size and average particle density in the simulation box. So far, we have demonstrated the effect of inter-particle distance on the pinning force for both incoherent and coherent particles. Its effect on the average grain size in 3D is discussed here. In the work of Hellman and Hillert [3], the average grain size (D) comes in as the curvature radius of a macroscopic outer boundary. A simple relation D=R0 fve derived from this approach with e = 1 becomes the foundation of several later works [4,10–12] and was used to benchmark many simulation results [13–17]. Nes et al. derived a correction to Hellman’s expression by taking into account the bow-out of GB near the pinning particle and showed a similar expression with a slightly modified exponential D=R0 fv0:92 [4]. Later, Hillert refined his theory and gave a similar expression D=R0 fv0:93 [10]. By carefully considering Louat effect [7,8], Hunderi et al. gave a volume fraction exponential e = 0.87 [11]. At higher particle volume fraction (fv > 0.1), the relation features a different exponential as suggested by Anand (e = 0.5) [12] and Hillert (e = 1/3) [10]. This is primarily due to the breakdown of particle distribution randomness at small grain size since the pinning particles are more likely to be found at the grainboundary rather than grain interior in this case. Early simulation result from Monte Carlo (MC) method [16] suggested an exponential e = 0.31 for the volume fraction range from 0.005 to 0.16. Later, another group reported a different exponential e = 1 for volume fraction from 0.025 to 0.15 [15] also using Monte Carlo method. In the work of Hazzledine et al., a discontinuous exponential with a jump from 1 at low volume fraction to 0.33 at high volume fraction was observed [21]. A recent work by Di Prinzio et al. indicated that the exponential from MC method depend on the pinning particle size and e = 1 was reached only by using larger pinning particles [17]. Phase-field method was also used to investigate the dependence of average grain size on pinning particle volume fraction. The first result reported by Suwa et al. gave e = 0.87 [13]. Later, Vanherpe et al. demonstrated an exponential e = 0.93 for both spherical and spheroid particle shape [14]. In Table 1 we summarize the results from previous works on this topic. Since the pinning force depends on the inter-particle distance as we see in previous parts of this work, Hellman’s average grain size relation should be modified at large particle volume fraction where the reduction of pinning force is significant. Using the results of coherent particles in Fig. 3 as an example, the dependence of grain size on the particle volume fraction is shown below. Clearly, from Fig. 4a fv1 behavior is observed at small volume fraction as expected since the effect of inter-particle distance on the pinning force is negligible there. At high volume fraction, a slower exponential emerges as a result of the reduced pinning force. This change of exponential behavior at high volume fraction is different from the one stated by Hillert [10] and Anand [12] since it comes from simple reduction of pinning force per particle rather
fv -0.3 fv
2 1 0
-3
-2
-1
0
log (f v) Fig. 4. Grain size as a function of pinning particle volume fraction. Grain size (red dot) is calculated based on the pinning force evaluated in PF model in Fig. 3 following Hellman’s work. The two lines with different exponential are simply presented for eye guidance. Similar to Fig. 3, the volume fraction here is simply derived from the single particle simulation setting used in Fig. 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
than the breakdown of particle distribution. Also, the physics that stabilizes the average grain size in the non-random distribution case described in Refs. [10,12] is completely different from the curvature relation presented in Eq. (1), and is not captured in our single particle PF simulation. 5. Conclusions In summary, by considering the constraint of inter-particle grain boundary shape, a previously ignored pinning force reduction factor is found to be related to the ratio of particle size over interparticle distance. This effect is particularly important when the volume fraction of second-phase pinning particles becomes large and applies both to coherent and incoherent pinning particles. A slower exponential relation between average grain size and particle volume fraction is shown to rise from this pinning force reduction effect. Acknowledgments The authors would like to acknowledge the Strategic Center for Coal, NETL, for supporting this activity through the Innovative Process Technologies Program, and in particular Robert Romanosky as Technology Manager, Patricia Rawls as Project Manager and David Alman as ORD Technical Team Coordinator. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
C.S. Smith, Trans. Metall. Soc. AIME 175 (1948) 15. M.F. Ashby, J. Harper, J. Lewis, Trans. Metall. Soc. AIME 245 (1969) 413. P. Hellman, M. Hillert, Scand. J. Metall. 4 (1975) 211. E. Nes, N. Ryum, O. Hunderi, Acta Metall. 33 (1985) 11. P.A. Manohar, M. Ferry, T. Chandra, ISIJ Int. 38 (1998) 913. Y. Liu, B.R. Patterson, Acta Metall. 44 (1996) 4327. N. Louat, Acta Metall. 30 (1982) 1291. N. Louat, Philos. Mag. 47 (1983) 903. N. Wang, Y.H. Wen, L. Chen, submitted. M. Hillbert, Acta metall. 36 (1998) 3177. O. Hunderi, E. Nes, N. Ryum, Acta Metall. 37 (1989) 129. L. Anand, J. Gurland, Metall. Trans. A 6 (1975) 928.
N. Wang et al. / Computational Materials Science 93 (2014) 81–85 [13] Y. Suwa, Y. Saito, H. Onodera, Scr. Mater. 55 (2006) 407. [14] L. Vanherpe, N. Moelans, B. Blanpain, S. Vandewalle, Comput. Mater. Sci. 49 (2010) 340. [15] M. Miodownik, E.A. Holm, G.N. Hassold, Scr. Mater. 42 (2000) 1173. [16] M.P. Anderson, G.S. Grest, R.D. Doherty, Kang Li, D.J. Srolovitz, Scr. Metall. 23 (1989) 753.
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[17] C.L. Di Prinzio, E. Druetta, O.B. Nasello, Modell. Simul. Mater. Sci. Eng. 21 (2013) 025007. [18] L. Chen, W. Yang, Phys. Rev. B 50 (1994) 15752. [19] K. Chang, L. Chen, Modell. Simul. Mater. Sci. Eng. 20 (2012) 055004. [20] E. Flores, J.M. Cabrera, J.M. Prado, Metall. Mater. Trans. A 35 (2004) 1097. [21] M.P. Hazzledine, R.D.J. Oldershaw, Philos. Mag. A61 (1990) 579.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Pinning force from multiple second-phase particles in grain growth Nan Wang a,⇑, Youhai Wen b, Long-Qing Chen a a b
Department of Materials Science and Engineering, Pennsylvania State University, University Park, PA 16802, USA National Energy Technology Laboratory, Albany, OR 97321, USA
a r t i c l e
i n f o
Article history: Received 13 January 2014 Received in revised form 16 June 2014 Accepted 17 June 2014
Keywords: Grain boundary Particle pinning Phase-field Grain growth
a b s t r a c t A factor that can reduce particle pinning force significantly in grain growth is found when the grain-boundary is pinned by multiple particles. The pinning force, in this case, is a function of particle radius over inter-particle distance. A previously proposed phase-field model for particle pinning is used to validate this predicted pinning force reduction in two and three dimensions. When applied to coherent pinning particles, the same effect is observed in simulations. It is shown that, at application relevant high particle volume fraction, the average grain size is affected by this reduction of pinning force. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Study of the particle pinning in grain growth has been an important topic for a long time since introducing second-phase precipitates is one of the most commonly used method to inhibit grain growth and achieve better mechanical properties through Hall–Petch relation in polycrystalline materials. The pinning force, as formulated by Zener, comes from grain-boundary (GB) which forms a bow-out toward the migration direction during its interaction with a second-phase particle [1]. Therefore, the GB shape near the pinning particle is critical to pinning force evaluation [2,3]. When multiple particles are considered, the total pinning pressure is assessed using a summation of pinning force from individual particles [4–6]. While both experimental and computational studies have shown good agreements with this theory, they also suggested a deviation from the prediction at high particle volume fraction [5]. Different corrections have been proposed in this case based on the break-down of random particle distribution [10,12], particle overlaps [20] or the so called Louat effect where the pulling force of particles in front of an advancing boundary is considered [7,8]. However, an important factor, the change of near-particle GB shape at high particle volume fraction has been ignored. In this work, we address this issue by calculating the GB shape between nearby particles exploiting a previously ignored factor in Ashby’s work [2], and demonstrate, using both analytics and numerical method, how the change of GB shape reduces the ⇑ Corresponding author. Tel.: +1 814 880 0997. E-mail address: [email protected] (N. Wang). http://dx.doi.org/10.1016/j.commatsci.2014.06.030 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.
pinning force and affects the average grain size at high particle volume fraction. 2. Particle pinning force in grain growth In a configuration where two pinning particles are close to each other (which occurs at high volume fraction, or in low volume fraction case with highly inhomogeneous distribution of second-phase particles), the GB shape needs be determined using Ashby’s method [2] since the near-particle boundary shape in Hellman’s approach [3] is only a zero driving force approximation. The axi-symmetric curvature equation for a single particle in cylindrical coordinates is a second order ordinary differential equation (ODE) with driving force P as a free parameter [2].
zR 1 zRR P qffiffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ : 2 3=2 R 1 þ z2 c 1 þ z R R
ð1Þ
Here c is the GB energy, R and z are defined in Fig. 1, zR and zRR are the 1st and 2nd derivative of z with respect to R. Ashby et al. solved this equation by integrating from the GB-particle contact point with a given position angle b (as defined in Fig. 1) and an interface balance condition cos a ¼ ðc1 c2 Þ=c where c1 is the surface energy for particle-grain1 interface, c2 is the surface energy for particle-grain2 interface [2]. These two boundary conditions specify the coordinates and the GB slope at the contact point. With a prescribed driving force P, GB energy c and two particle-grain interfacial energies c1 and c2, the GB shape is completely determined by solving Eq. (1). However, Ashby imposed another condition that requires a flat boundary at the midway (L/2) of
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pinning becomes less effective when the inter-particle distance becomes comparable with the particle size. Although derived in 2D, Eq. (3) clearly demonstrates that the maximum pinning force is given by not only the local contact condition (angle a) but also the relative size of particle over inter-particle distance. 2.2. Pinning theory in 3D Fig. 1. A boundary seperating grain1 (lower grain) and grain2 (upper grain) is pinned by multiple second-phase particles of radius R0 and inter-particle distance L. Only one pinning particle is shown in this window of size L to demonstrate the geometry, other nearby particles are located at distance L from this one. Window boundary lines (the two vertical solid lines) are drawn exactly at the mid-point of two adjacent pinning particles. The GB surface is a function of (R, Z) coordinates. Position angle b gives the GB-particle contact point. Angle a which is defined between GB line and the particle surface tangent is given by Young’s force balancing condition. Surface energies are not labeled here. A modification angle a is needed due to particle coherency. For an incoherent particle, a = p/2 since c1 = c2. For a coherent particle, a < p/2.
two particles, and argued that such a condition gave a relationship between driving force P and angle b (note: this is an L dependent relation). This argument seems reasonable but is inconsistent with their theory on the critical pinning angle presented earlier in the same paper [2] as we will see. The pinning force in Fig. 1 configuration is simply F z ¼ ¼ 2pR0 c cos b cos ða bÞ. The maximum pinning force is then F max z
pR0 cð1 þ cos aÞ with the critical pinning angle given by bcrit ¼ a=2. However, according to Ashby’s flat midpoint argument, the angle bcrit should be associated with the maximum driving force Pmax with inter-particle distance L as a parameter. For b > bcrit, a stable pinning configuration can exist only under a decreased driving force P < Pmax. Now, it is easy to see an inconsistency between the two bcrit relations since one depends on the inter-particle distance L while the other is a simple function of a. To resolve this controversy, the maximum pinning force will be derived analytically in two dimensions (2D) with the inter-particle distance L as a parameter, and generalized to three dimensions (3D) numerically. 2.1. Pinning theory in 2D While the particle pinning theory in 2D is very different from its 3D cousin, it is a simpler demonstration of the basic physics. We choose to present our L dependent pinning force theory in 2D first because our key argument can be presented analytically in this case. In 2D, Fig. 1 is still a good illustration of the basic geometry. Instead of solving for 3D GB shape from Eq. (1), the 2D GB shape is a simple circle with radius q given by q = c/P. The flat midpoint condition can be easily expressed as the following:
L c ¼ R0 cos b þ cosða bÞ: 2 P
ð2Þ
This configuration is stable as long as the driving force is smaller than Pmax beyond which the grain-boundary will detach from the pinning particle. One can see that the condition for maximum driving force is sin (a b) = 2 sin aR0/L by rearranging Eq. (2) and using oP/ob = 0. Since the pinning force is given by 2c cos(a b), by eliminating the cosine factor with the maximum driving force condition, the maximum pinning force is then:
F max z
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2R0 2 ¼ 2c 1 sin a: L
ð3Þ
For incoherent particles, the maximum pinning force is reached at b = cos1(2R0/L) rather than the simple 2D maximum pinning force condition b = p/2. Comparing with the classic pinning force theory that gives F max ¼ 2c, this equation indicates that particle z
To formulate the same L-dependent particle pinning theory in 3D, one needs to solve for the GB shape from Eq. (1) numerically (we use simple 4th order Runge–Kutta). Integration of the curvature equation (Eq. (1)) starts from the midway (L/2) with flat boundary condition dz/dR = 0 and z = C1. On the other end, the GB should join the pinning particle at the correct angle such that the surface energy balance condition cos a = (c1 c2)/c is satisfied. Assuming a spherical particle of radius R0 is centered at the origin, this joining angle condition can be expressed as:
dz ¼ cotða bÞ: dRR¼R0 cos b
ð4Þ
By tuning the initial condition C1, the GB line can be moved up and down such that the point with correct joining angle can be positioned on the particle sphere. The maximum driving force Pmax and the associated position angle bmax are then found by increasing P continuously until there is no correct solution for the GB shape. Since the pinning force is given by F z ¼ 2pR0 c cos b cos ða bÞ, the maximum pinning force can be directly evaluated with bmax. In the regime where the particle radius is much smaller than the inter-particle distance, we found that the GB-particle joining angle bmax for the maximum pinning force configuration is actually very close to the critical value bcrit = a/2. This is expected as this condition is very similar to the single particle configuration Ashby used to derive the relation. As the inter-particle distance becomes comparable to the particle size, bmax becomes smaller than bcrit. Solution of GB shape for b = bcrit in this condition still exists but only for a smaller driving force P and is not dynamically accessible by surface energy minimization in the phase-field method we use to validate the theory in the next section. 3. PF model for quantitative evaluation of particle pinning force To validate the pinning force theory proposed in the previous section numerically, we employ a phase-field (PF) model to evaluate the particle pinning force. As demonstrated in previous works [9,19], this method can quantitatively reproduce the particle pinning force at single particle level in grain growth. Here the model is briefly described and more details can be found in Ref. [9]. The phase-field free energy functional is:
# Z " 2 1X 2 F¼ n Kðrgi Þ þ Phðg1 ; g2 ; g3 Þ dv ; f0 ðg1 ; g2 ; g3 Þ þ 2 i¼1 i
ð5Þ
where the gradient energy coefficient is a product of numerical parameter ni and dimensional constant K, and
" f0 ¼ Df a
# X 2 3 X 3 X Y g2 g4 i þ i þ cij g2i g2j þ b g2i ; 2 4 i¼1 i¼1 j
ð6Þ
with dimensional constant Df, numerical parameter a, b and cij. Here, the first two terms are commonly seen in PF grain growth model, while the last term in f0 is added to account for the triple junction. A P term coupled with function h ¼ g31 ð10 15g1 þ 6g21 Þ is also added to account for the grain growth driving force in this twograin-one-particle simple geometry. The simple grain growth in Fig. 1 is modeled with g1 for grain1, g2 for grain2 and g3 for pinning
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particle. The time evolution of gi follows the Allen–Cahn phase-field dynamics [18].
@g dF i ¼ Li ; dgi @t
ð7Þ
with time t and kinetic coefficient Li. This equation above applies to
g1 and g2 only, while g3 is fixed since we assume the coarsening of pinning particles is a slow process at the grain growth time scale. Dimensionless form of Eq. (7) is:
@ gi ~ @ð f0 =Df Þ ~ @h ¼ Li þP w2 ni r2 gi ; @ gi @ gi @~t
ð8Þ
~ ¼ P=Df with characteristic length scale ~t ¼ t=s, ~Li ¼ Df sLi , P where pffiffiffiffiffiffiffiffiffiffiffi ffi w ¼ K=Df and time scale s. In case of incoherent particle pinning, the three surface energies involved are two incoherent particle-grain surface energies (c1 and c2) and one grain-boundary energy (c). Since the nature of those interfaces are rather similar, we choose to use model parameters a = 0.1, n1 = n2 = 0.2, c13 = c23 = 2.0 and c12 = 0.0993 which give c1 = c2 = c = 0.6 J m2. In case of coherent particle pinning, the surface energy relations are more complicated. Since the particle was initially formed coherently with grain2, a typical coherent surface energy should be used for c2 while the choice of incoherent surface (c1) and grain-boundary (c) should be similar to the case of incoherent particle. To be specific, we use a = 0.1, n1 = 0.5, n2 = 0.1, c13 = 0.386, c23 = 0.0849, c12 = 0.1428 which correspond to c1 = c = 0.75 J m2 and c2 = 0.15 J m2. In both cases, the triple junction parameter b is chosen to be 50. Numerical integration of Eq. (8)is implemented on a simple square mesh with Dx/w = 0.4. Standard central difference and forward Euler scheme are used. By increasing P gradually, a series of stable configuration where the grain growth from grain1 to grain2 is completely pinned by the particle can be achieved until it reaches Pmax beyond which a stable configuration no longer exists. The maximum pinning force can be numerically evaluated using the distortion of grain-boundary [19].
F z ¼ Pmax
Z
^ ^z dA; n
ð9Þ
Correction factor
^ is the grain-boundary surface normal, dA is the surface elewhere n ment, ^z is the unit vector in z direction and the maximum driving force Pmax for a pinned grain-boundary.
1
Based on the pinning theory we proposed in Section 2 and the PF numerical method in Section 3, the deviation of maximum pinning force from classic single particle theory is shown in Fig. 2 for an incoherent particle with c1 = c2 = c in 2D and 3D. As shown in Fig. 2, the PF calculated maximum pinning force for incoherent particles agrees well with our theory in the regime where the inter-particle distance L is comparable with the particle size. It is clear that the effect of small inter-particle distance is noticeable in 2D even for 2R0/L 0.2, while the same effect in 3D is insignificant until 2R0/L reaches 0.6. This dimension-dependent result is expected since the 3D GB shape is given by the balance of 2 principal curvatures (the curvature in R–z plane and the curvature in R–h plane) while the 2D GB shape is only given by a single curvature. Such a small deviation at fairly large 2R0/L in 3D could be part of the reason that this effect was ignored in previous works. As mentioned before, the GB-particle joining angle is related to the particle coherency. The angle a between GB line and particle surface tangent is given by the balance of surface tensions at the joining point. For a coherent second-phase particle, this angle is quite small since c2 c1 and c c1. It is well known that a coherent second-phase particle provides larger pinning force comparing with an incoherent particle of the same size. As outlined in the PF model part, with a different set of model parameters, the PF numerics for incoherent pinning particle can be used to study the pinning force from a coherent particle. The same pinning force reduction is also observed for coherent particles as shown in Fig. 3. Here, instead of looking at the correction to previous single particle theory as a function of relative particle size in Fig. 2, this plot relates the pinning force reduction to a more experiment relevant measure, the particle volume fraction. Zener’s theory suggested a maximum pinning pressure Pz = 3fvc/(4R0) which is an average of maximum pinning force from all particles within a thin layer (R0) of GB over the total GB area [1,3]. The particle volume fraction fv is given by 4pnR30 =3 with n being the particle density per volume. For coherent particles, all arguments above apply except for the maximum pinning force which should be multiplied by a factor 1 + cos a [2]. As can be seen from the comparison with PF simulations, the old single particle theory agrees well with PF results for particle volume fraction smaller than 0.1. After that, a deviation from the theory becomes noticeable. Applying our L-dependent theory gives the same result for small volume fraction (fv < 0.1), while it predicts a reduction of pinning force at high particle volume fraction that agrees well with PF simulations. While the coherent pinning particle maintains its advantage over the incoherent particle at all volume fractions, the same pinning force saturation effect is observed regardless of particle coherency. Here,
0.8 theory
0.6
2D theory 2D PF 3D theory 3D PF
0.4 0.2
0
0.2
correction
0.02
PF
0.4
0.6
0.8
Pzw/γ
4. Results and discussions
0.01
2R 0 /L Fig. 2. Comparison of inter-particle distance (L) dependent pinning theory with previous single particle theory. The result is then validated by numerically simulation with phase-field model for incoherent second-phase particles. Y-axis is the ratio of maximum pinning force predicted with L-dependent theory and classic single particle theory. Red line shows the deviation of pinning force from classic theory in 2D based on Eq. (3). Green line shows the pinning force deviation in 3D based on numerical integration of Eq. (1) using simple Runge–Kutta method. Red dots and green dots are the maximum pinning force evaluated using phasefield method in 2D and 3D. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
0
0
0.2
0.4
0.6
Volume fraction f v Fig. 3. Comparison of inter-particle distance (L) dependent pinning theory (dot line) with previous single particle theory (solid line) at various particle volume fraction. The numerical results from phase-field model are shown as squares. Y-axis is the pinning pressure Pz normalized by c/w. X-axis is the particle volume fraction. The upper branch is for coherent pinning particle while the lower branch is for incoherent pinning particle.
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N. Wang et al. / Computational Materials Science 93 (2014) 81–85
Table 1 Previous results on the relation between average grain size and particle volume fraction. Theory
Simulation
[3] Gave exponential 1 With GB bow-out, [4] gave exponential 0.92 [10] Refined [3] gave exponential 0.93 With Louat effect, [11] gave exponential 0.87 High volume fraction, [12] gave exponential 0.5 High volume fraction, [10] gave exponential 0.33
MC in [16] gave exponential 0.31 MC in [15] gave exponential 1 MC in [21] gave exponential jump from 1 at low volume fraction to 0.33 at high volume fraction MC in [17] gave exponential 1, and results depend on particle size PF in [13] gave exponential 0.87 PF in [14] gave exponential 0.93
3 PF results -1
log (D/R 0)
we need to point out that the demonstrated pinning pressure reduction in Fig. 3 is a mere extrapolation of single particle results in Fig. 2, and is not based on simulations of polycrystalline grain growth with multiple pinning particles. Volume fraction in the single particle setting is measured by the product of pinning particle size and average particle density in the simulation box. So far, we have demonstrated the effect of inter-particle distance on the pinning force for both incoherent and coherent particles. Its effect on the average grain size in 3D is discussed here. In the work of Hellman and Hillert [3], the average grain size (D) comes in as the curvature radius of a macroscopic outer boundary. A simple relation D=R0 fve derived from this approach with e = 1 becomes the foundation of several later works [4,10–12] and was used to benchmark many simulation results [13–17]. Nes et al. derived a correction to Hellman’s expression by taking into account the bow-out of GB near the pinning particle and showed a similar expression with a slightly modified exponential D=R0 fv0:92 [4]. Later, Hillert refined his theory and gave a similar expression D=R0 fv0:93 [10]. By carefully considering Louat effect [7,8], Hunderi et al. gave a volume fraction exponential e = 0.87 [11]. At higher particle volume fraction (fv > 0.1), the relation features a different exponential as suggested by Anand (e = 0.5) [12] and Hillert (e = 1/3) [10]. This is primarily due to the breakdown of particle distribution randomness at small grain size since the pinning particles are more likely to be found at the grainboundary rather than grain interior in this case. Early simulation result from Monte Carlo (MC) method [16] suggested an exponential e = 0.31 for the volume fraction range from 0.005 to 0.16. Later, another group reported a different exponential e = 1 for volume fraction from 0.025 to 0.15 [15] also using Monte Carlo method. In the work of Hazzledine et al., a discontinuous exponential with a jump from 1 at low volume fraction to 0.33 at high volume fraction was observed [21]. A recent work by Di Prinzio et al. indicated that the exponential from MC method depend on the pinning particle size and e = 1 was reached only by using larger pinning particles [17]. Phase-field method was also used to investigate the dependence of average grain size on pinning particle volume fraction. The first result reported by Suwa et al. gave e = 0.87 [13]. Later, Vanherpe et al. demonstrated an exponential e = 0.93 for both spherical and spheroid particle shape [14]. In Table 1 we summarize the results from previous works on this topic. Since the pinning force depends on the inter-particle distance as we see in previous parts of this work, Hellman’s average grain size relation should be modified at large particle volume fraction where the reduction of pinning force is significant. Using the results of coherent particles in Fig. 3 as an example, the dependence of grain size on the particle volume fraction is shown below. Clearly, from Fig. 4a fv1 behavior is observed at small volume fraction as expected since the effect of inter-particle distance on the pinning force is negligible there. At high volume fraction, a slower exponential emerges as a result of the reduced pinning force. This change of exponential behavior at high volume fraction is different from the one stated by Hillert [10] and Anand [12] since it comes from simple reduction of pinning force per particle rather
fv -0.3 fv
2 1 0
-3
-2
-1
0
log (f v) Fig. 4. Grain size as a function of pinning particle volume fraction. Grain size (red dot) is calculated based on the pinning force evaluated in PF model in Fig. 3 following Hellman’s work. The two lines with different exponential are simply presented for eye guidance. Similar to Fig. 3, the volume fraction here is simply derived from the single particle simulation setting used in Fig. 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
than the breakdown of particle distribution. Also, the physics that stabilizes the average grain size in the non-random distribution case described in Refs. [10,12] is completely different from the curvature relation presented in Eq. (1), and is not captured in our single particle PF simulation. 5. Conclusions In summary, by considering the constraint of inter-particle grain boundary shape, a previously ignored pinning force reduction factor is found to be related to the ratio of particle size over interparticle distance. This effect is particularly important when the volume fraction of second-phase pinning particles becomes large and applies both to coherent and incoherent pinning particles. A slower exponential relation between average grain size and particle volume fraction is shown to rise from this pinning force reduction effect. Acknowledgments The authors would like to acknowledge the Strategic Center for Coal, NETL, for supporting this activity through the Innovative Process Technologies Program, and in particular Robert Romanosky as Technology Manager, Patricia Rawls as Project Manager and David Alman as ORD Technical Team Coordinator. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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