1. Introduction

It is well-known that the mean grain size and the corresponding grain shape and grain size distribution have a significant influence on the processing (e.g., sintering, annealing and forming) of polycrystalline metals and ceramics ^[1]. Further, it is well-established ^[2,3] that concurrent grain growth occurs during the plastic deformation of fine-grained materials (“dynamic”grain growth), and that the corresponding dynamic grain growth rate is appreciably greater than that which occurs during conventional annealing without an applied stress (“static”grain growth). Our understanding of the reason for the greater dynamic grain growth rate compared to static is still incomplete.

It is well-established that the plastic deformation of fine-grained materials generally occurs by grain boundary (GB) sliding ^[4]. Two physical mechanisms have been proposed for the greater dynamic grain growth due to the GB sliding; see ^[5,6]: (a) the sliding creates excess vacancies, which leads to an enhancement of the GB mobility and (b) in materials which contain solutes segregated at the GBs the sliding disrupts the segregated solute clusters which normally retard grain growth. Specific, quantitative details regarding the manner in which these models enhance the grain growth rate are however lacking.

A significant paper comparing dynamic and static grain growth rates in ceramics is that by Nich and Wadworth (N-W) ^[2] on 3 mole% yttria-stabilized zirconia (3Y-TZP) at 1550°C and 1650°C. They found that both the dynamic and static grain size obeyed the cubic time law, but that both the activation energy Q (520 vs 580 kJ/mole) and the pre-exponential A (~7.5×10¹⁰ vs~10¹² m/s) were lower for dynamic compared to static grain growth. They attributed the larger dynamic grain growth to an enhancement by the applied stress of atom migration across the GBs. However, no specific or quantitative details are given regarding the manner in which the stress enhances atom migration across the GBs.

To provide additional insight into the physical mechanism(s) by which GB sliding enhances the dynamic grain growth rate, we analyze the data by N-H ^[2] (and related results) employing a different approach than that used by these authors.

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2. Pertinent Models and Physical Mechanisms

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2.1. Static grain growth

Theoretical considerations ^{[7,8,9,10,11]} give that the mean grain growth rate $\dot d$ in polycrystals is the product of the driving force P and the GB mobility M, i.e.,

$\dot d={\text{PM}}$

(1)

In pure, unalloyed, and fully-annealed materials,

$P \cong 2{\gamma _b}/r$

(2)

where P is the capillary driving force, $\gamma _b$ is the GB energy, r=(3/2)$\bar d$^[12] is the local GB radius of curvature and $\bar d$ is the mean linear intercept grain size. The GB mobility is given by

${\text{M}}={{\text{M}}_0}{\text{exp}}(-\Delta {G^*}/kT)$

(3)

where ∆G^*=∆H^*-T∆S^* is the Gibbs free energy for an atomic jump across the GB to its nearest neighbor, and

${{\text{M}}_0}={\text{ }}{{\text{N}}_{\text{j}}}{\nu\text{ x}}*$

(4)

where ${{\text{N}}_{\text{j}}} \approx {{\text{A}}_{\text{b}}}/{\Omega ^{2/3}}$ is the number of atomic sites per unit volume from which an atom (or ion) can jump from one grain onto its neighbor, A_b=2/$\bar d$^[13,14] is the existing GB area per unit volume and $\Omega$ is the atomic volume, ν is the atomic vibration frequency (Debye frequency) and ${{\text{x}}^*} \approx {\Omega ^{1/3}}$ is the jump distance. Combining Eqs.1-4 one obtains for the static grain growth rate in pure, fully-annealed, single phase materials

$\dot d=[\frac{{A{\gamma _b}}}{{{d^2}{\Omega ^{1/3}}}}{\text{exp}}(\Delta {S^*}/R)]{\text{exp}}(-Q/RT)$

(5)

where R is the gas constant and $Q \equiv \Delta {H^*}$. To be noted regarding Eq.5 is that the existing grain size d has two effects on the concurrent grain growth rate: (a) on the magnitude of the driving force P and (b) on the number of sites available for atomic jumps from one grain to the next. The latter effect has generally been neglected in considerations of grain growth. It accounts for the cubic grain growth time law being generally observed, rather than a square time law.

The relationship for static grain growth rate in impure materials or those which contain solute additions is more complex than Eq.5. This is due to the segregation of the impurity or solute ions at the GBs, which thereby exert a drag on the GB mobility ^[8,9], and is generally referred to as Cahn’s solute drag theory. Vandermeer ^[15] has pointed out that according to Cahn’s solute drag theory the grain growth rate vs the driving force behavior is that shown in Figure 1. The slope of the logvs log P curve shown is unity at low (Regime I) and high (Regime III) values of the driving force, and appreciably larger in the transition (Regime II). Further, according to Vandermeer ^[15], the rate controlling mechanism in Regime I is the diffusion of solute atoms (or ions) perpendicular to the GB and that in Region III by the solvent atoms (ions) across the boundary. As pointed out by Cahn ^[8] any activation energy determined from the temperature dependence of in the transition regime has no physical significance.

For materials containing impurities or solute additions, Eq.5 becomes

$\dot d=\left[{\frac{A}{{d{\Omega ^{2/3}}}}{{\left({\frac{{{\gamma _b}}}{d}} \right)}^n}\frac{{{\text{exp}}(\Delta {S^*}/R)}}{C}} \right]{\text{exp}}(-Q/RT)$

(6)

where C is the solute concentration, and A, ${\Delta {S^*}}$, ${{\gamma _b}}$ and n can vary with the magnitude of the driving force. Moreover, ${{\gamma _b}}$ may vary with P depending on the atomic structure of the GB, and on the nature of the segregated solute. In the case of ceramics, the magnitude of ${{\gamma _b}}$ depends on the degree and nature of the solute segregation, which in turn depends on the valence difference, as well as the size difference between the solute and solvent ions ^[10].

In prior work, Wang and Conrad (W-C) ^[16] proposed that the GB energy ${{\gamma _b^0}}$ in 3 Y-TZP consists of three major components,

$\gamma _b^0=\gamma _b^\Sigma + \gamma _b^s + \gamma _b^e$

(7)

where $\gamma _b^\Sigma$ results from the crystallographic misorientation between neighboring grains, $\gamma _b^s$ from the ionic size difference between the yttrium ions and the solvent zirconium ions, and $\gamma _b^e$ is the electrostatic space charge component, which results from the difference in valence between the segregated solute and host ions. It was determined that $\gamma _b^e$ accounted for~60% of the total GB energy and the sum of the two other components about 40%. W-C ^[16] proposed that the reduction in $\gamma _b^0$ by an externally-applied electric field was mainly responsible for the observed reduction in P and the resulting retardation in the grain growth rate in 3 Y-TZP by an electric field.

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2.2. Dynamic grain growth and concurrent cavitation

It is well-established that the plastic deformation of fine-grained materials at high homologous temperatures occurs mainly by GB sliding ^[4]. The observed, concurrent, dynamic grain growth in ceramics is given by ^[17]

${d_d}={d_0}\exp \left({B \in } \right)$

(8)

where d₀ is the initial grain size and $\in$ the overall plastic strain. The parameter B depends on the temperature T, applied strain rate and the existing grain size d. A coupled GB sliding-migration model for dynamic grain growth in keeping with Eq.8 proposed by Wilkinson and Caceres ^[5] gives,

$B=ab$

(9)

where $\alpha={{\dot \in }_s}/\dot \in d=0.5-0.8$ is the fraction of the strain rate due to the GB sliding compared to the overall strain rate ${\dot \in }$. The parameter b=cf represents the average migration distance per unit sliding, c being the depth of deformation zone produced per unit sliding and f is a function which depends on the GB distribution. b has a value between 3.8 and 6.

It is expected that the GB sliding which occurs during plastic deformation at high temperatures affects the governing grain growth mechanism. Clark and Alden ^[18] proposed that the increase in grain growth rate due to GB sliding results from an increase in the concentration of vacancies, which leads to an increase in GB mobility. However, there is no direct evidence for this. Alternatively, Grey and Higgins ^[19] proposed that the enhanced dynamic grain growth rate resulted from a decrease in the GB pinning due to the segregated solute clusters, or by a decrease in immobile steps in the boundary itself. They suggested that support for the latter is provided by the observation that GB sliding and migration are observed sequential events in high purity Al, i.e. GB pinning can occur by the steps in the boundary without solute segregation.

Additional information on the mechanism for the enhanced dynamic grain growth in 3 Y-TZP is provided by the results obtained by Di Yang and Conrad (D-C) ^[20] in their investigation on the influence of an applied DC electric field on the two types of grain growth. These authors confirmed that the dynamic grain growth rate was greater than the static rate both without and with an applied field. Moreover, they found that the field retarded both the dynamic and static grain growth rates, but it had a greater effect on the former. This suggests that the disruption of the GB atomic structure by the GB sliding gave either an atomic arrangement at the GBs with an increase in the magnitude of the electronic component $\gamma _b^e$ in the GB energy (e.g. an increase in the space charge), or gave an atomic arrangement with which the electric field had a higher degree of interaction. Di Yang and Conrad ^[20] further found that the amount of GB cavitation was a unique function of the grain size, independent of the applied electric field, see Figure 2.

It is well-documented that concurrent with grain growth cavitation along the GBs generally occurs during the high temperature deformation of fine-grained materials ^[4,5,6,21]. Desirable is a knowledge of the relationship between the cavitation (which increases with plastic strain) and the existing grain size. The plastic deformation model developed by Stowell ^[22] gives for the effect of plastic strain $\in$ on the volume fraction $\phi$ of cavitation voids

$\phi={\phi _0}\exp \left({n \in } \right)$

(10)

where ${\phi _0}$ is the cavitation at zero strain and n is a parameter which usually has a value in the range of 2 to 3. The effect of $\in$ on $\phi$ in 3 Y-TZP (data from Schlisser et al. ^[6]) is shown in Figure 3. Included are measurements by Di Yang and Conrad ^[20] on 3 Y-TZP deformed at 1450°C, 1500°C and 1600°C with and without an applied DC electric field. The data from these two sources are in reasonable agreement, and in reasonable accord with Stowell’s model ^[22]. Furthermore, the parameter given by the slope of the line is relatively independent of the applied electric field. This indicates that the electrostatic component $\gamma _b^e$ of the GB energy (the space charge) is not a major factor in the relationship between the cavitation $\phi$ and the plastic strain $\in$.

The model developed by Chokshi and Langdon ^[23] for diffusion-controlled GB cavity growth rate gives

$\frac{{d{r_c}}}{{d \in }}=\frac{{45\Omega {\delta _b}{D_g}_b}}{{{d^2}kT}}\left({\frac{\sigma }{{\dot \in }}} \right)$

(11)

where r_c is the cavity radius, $\in$ is the total strain, Ω is the atomic volume, ${\delta _b}$ is the GB width, D_gb is the GB diffusion coefficient and d is the grain size. According to this model the cavity growth rate increases appreciably with decrease in GS, which is in keeping with experimental results based on the initial grain size, but not the existing GS. As mentioned above, in general the GS increases concurrently with straining (i.e., dynamic grain growth) and this needs to be taken into account.

Although it is generallyrecognized that grain growth occurs concurrently with cavitation during high temperature plastic deformation, very little quantitative information is available on the relationship between cavitation and the concurrent grain growth. Considering 3 Y-TZP, it is seen in Figure 4 that there occurs an appreciable (hyperbolic) increase in cavitation with increase in grain size, contrary to the prediction by Eq.11. This gives that the observed cavitation in 3 Y-TZP is not primarily diffusion-controlled.

Figure 5 shows that the hyperbolic increase in $\phi$ with the concurrent increase in d_a shown in Figure 4 approximates a straight line in a plot of ln$\phi$ vs d, which gives the empirical relationship

$ln\phi=ln{\alpha _\phi } + {\beta _\phi }{d_\alpha }$

(12)

where $ln{\alpha _\phi } \approx-2.4$ and ${\beta _\phi } \approx 3.4$. The value of ${\alpha _\phi }={10^{-1}}\%$ agrees with the relative density of fully-sintered 3Y-TZP specimens ^[16], and hence is the magnitude of $\phi _0$. In keeping with Stowell’s GB cavitation model (Eq.10), we will take ${\alpha _\phi }={\phi _0}$ and ${B _\phi }{d_\alpha }=n \in$ to give

$ln\left({\frac{\phi }{{{\phi _0}}}} \right)=n \in={\beta _\phi }{d_\alpha }$

(13)

where n≈2.5 ^[22] and ${B_\phi } \approx 3.4$ (Fig. 5), and in turn ${d_\alpha }=n \in /{B_\phi }=0.74 \in$. Further, Eq.13 specifies that the volume fraction of cavitation increases exponentially with the concurrent dynamic GS, in accord with the experiment. Furthermore, taking according to W-C ^[5] $\overline {{d_\alpha }} \approx \bar x/\left({0.7 \in } \right)$, where x is the average GB sliding distance, and inserting for ${d_\alpha }$ in Eq.13, one obtains

$ln\left({\frac{\phi }{{{\phi _0}}}} \right)={B_\phi }\bar x/\left({0.7 \in } \right)$

(13a)

i.e. the cavitation results from the GB sliding. Moreover, since ${d_\alpha } \approx \bar x/\left({0.7 \in } \right)$ ^[5]=$n \in /{B_\phi }$ (Eq.13), one obtains for the plastic strain dependence of x

$\bar x \approx \frac{{0.7 \in n \in }}{{{B_\phi }}} \approx 0.51{ \in ^2}$

(14)

Based on the above it is concluded that the W-C ^[5] coupled, sequential GB sliding-migration model is a good description of the strain dependence of the dynamic grain growth in 3Y-TZP, and moreover, the strain and grain size dependence of GB cavitation that results from the sliding.

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3. Analysis of Data by Nieh and Wadsworth ^[2]

In this section we consider the data in Figures 2 and 3 in the paper by N-W ^[2] in terms of the models and mechanisms reviewed above. To obtain the pertinent parameters we employ the following equations:

${d^3}-d_0^3=K(T)t$

(15)

$d={(Kt + d_0^3)^{1/3}}$

(16)

$\dot d=K/3{d^2}$

(17)

$t=({d^3}-d_0^3)/K$

(18)

$\in=\dot \in t$

(19)

where d is the mean grain size (GS), d₀ is the initial GS, K(T) is the time law constant, which is an exponential function of the temperatures, ${\dot d}$ is the grain growth rate, $\in$ is the overall strain, $\dot \in$ is the strain rate and t is the time.

Eq.6 in section 2 gives that ${\dot d}$ is proportional to d^-(n+1), where n is the capillary driving force exponent. The magnitude of the exponent (n + 1)=p can thus be obtained from the slope of a plot of log${\dot d}$ vs logd. Such slopes for the N-W data are presented in Figure 6. It is seen that the results for both dynamic and static grain growth fall on straight lines with slope equal to~2. This gives that the capillary driving force exponent n=1. The N-W data thus represents either Regime I or Regime III in Figure 1. Taking the GB energy $\gamma _b^0=1.215-0.358{\text{ }}{10^{-3}}T$ (J/m²) ^[24], and the range of values of in Figure 6, the magnitude of the capillary driving force in the N-W data lies in Regime I when compared with prior results ^[25,26]. It is therefore concluded that the N-W test results refer to Regime I.

Of further interest regarding the mechanism governing dynamic grain growth compared to static in 3 Y-TZP is the magnitudes of the activation energy Q in Eq.6 and of the pre-exponential A^* given by

${A^*}=(A{\gamma _b}/{d^2}{\Omega ^{1/3}})\left[{\frac{{\exp (\Delta {S^*}/R)}}{C}} \right]$

(20)

At constant A^* (constant d²) the magnitude of Q is given by

$Q = Rln(\frac{{{{\dot d}_2}}}{{{{\dot d}_1}}})/(T_1^{ - 1} - T_2^{ - 1})$

(21)

where ${{{\dot d}_2}}$ and ${{{\dot d}_1}}$ are the gain growth rates, respectively, at T₁=1550°C and T₂=1650°C. Having determined the values of Q, the magnitudes of A^* were obtained employing

${A^*}=(ln\dot d)\exp (Q/RT)$

(22)

To obtain the value of $\dot d$ at a constant d² for use in Eq.21, a log-log plot of $\dot d$ vs d² is presented in Figure 7. In accord with Eq.6 the datapoints for both dynamic and static grain growth fall on straight lines with a slope of −1.

The values of Q and A^* as a function of the grain size d determined from the N-W data employing Eqs.21 and 22 are given in Figure 8. To be noted is that Q for both dynamic and static grain growth increases slightly with d, while A^* decreases appreciably. Both parameters are however larger for dynamic grain growth compared to static over the entire grain size range considered (e.g., Q_d=506 kJ/mole vs Q_s=440 kJ/mole and $logA_d^*(m/s)=4$ vs $logA_s^*(m/s)=2$ determined for d²=1.0 (µm)². These larger values of Q and A^* for dynamic grain growth are in contrast to the smaller values for dynamic grain growth obtained by N-W ^[2] from their plots of K_d and K_s vs T⁻¹. The effect of grain size on the ratio Q_d/Q_s and the ratio $A_d^*/A_s^*$ are shown in Figure 9. It is seen that the ratio Q_d/Q_s decreases only slightly with d, while the ratio $A_d^*/A_s^*$ decreases appreciably.

The results in Figs. 8 and 9 thus give that the greater dynamic compared to static grain growth rate results mainly from the appreciably larger value of the pre-exponential parameter A^*. Desired is a knowledge of the parameterscomprising A^* which could account for the two orders of magnitude larger values of this factor for dynamic grain growth. Two parameters which could give a significant increase in A^* as a result of plastic deformation (GB sliding) are: (a) an increase in the number of atomic jump sites N_j due to an increase in the number of GB steps or ledges and (b) an increase in due to the increase in $\Delta {H^*}=Q$ ^[27] or to an otherwise disruption of the GB atomic structure ^[28]. Increases in the GB energy ${\gamma _b}$ and in the atomic jump distance x^* could conceivably also contribute to the increase in A^* as a result of the disruption of the GB structure by the GB sliding.

Regarding the effect of $\Delta {S^*}$ on $A_s^*$, the entropy of atomic migration $\Delta S_m^*$ in metals is related to the enthalpy of migration $\Delta {H^*}$ by ^[27]

$\Delta S_m^*=\beta _m^*\Delta H_m^*/{T_M}$

(23)

where T_M is the relating temperature and $\beta _m^*$ is a constant in the range of 0.25 to 0.45 for metals, but could conceivably be higher for oxide ceramics. Taking for 3 Y-TZP, T_M=2837K, ${\beta _m^*}=1.0$, $\Delta H_{(m, d)}^*=506{\text{kJ}}/{\text{mole}}$ and $\Delta H_{(m, s)}^*=440{\text{kJ}}/{\text{mole}}$, one obtains $\Delta S_{(m, d)}^*=176{\text{J}}/{\text{mole}}K$ and $\Delta S_{(m, s)}^*=153{\text{J}}/{\text{mole}}K$. These values in turn give for the contribution of the increase in entropy to the ratio of the pre-exponential $A_d^*/A_s^*=\exp \left[{\frac{{\Delta S_{m, d}^*-\Delta S_{m, s}^*}}{R}} \right]=15.9$, which accounts for about 1/10 of the magnitude of this ratio shown in Figs. 8 and 9. The remaining order of magnitude increase must therefore result from an increase in one or more of the parameters A_gb, ${\gamma _b}$ or x^* by the GB sliding. A determination of possible increases in these parameters requires further theoretical and experimental work.

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4. Conclusion

1. The overall physical mechanism governing dynamic grain growth rate is the same as that governing static grain growth.

2. An important parameter in the grain growth kinetics equation is existing grain size; i. e., the grain growth rate is proportional to d⁻².

3. The major factor which determines the higher dynamic compared to static grain growth rate in 3Y-TZP is the appreciably larger magnitude of the pre-exponential in the Arrhenius-type rate equation.

4. It is proposed that the slightly higher enthalpy which occurred in dynamic grain growth results from the disruption of the grain boundary atomic structure produced by the grain boundary sliding responsible for the high temperature plastic deformation.

5. A parameter which leads to the higher pre-exponential in dynamic grain growth is the increase in entropy associated with the governing atomic diffusion mechanism. It accounts for about 10% of the increase.

6. The Wilkinson-Caceres coupled, sequential grain boundary sliding-migration model provides a good description of dynamic grain growth rate.

7. The grain boundary cavitation that occurs during the plastic deformation at high temperatures increases with the concurrent grain size.

8. The grain boundary cavitation which occurred during the plastic deformation of 3Y-TZP at high temperatures resulted from the concurrent grain boundary sliding; the hyperbolic increase in cavitation which occurred with increase in grain size is due to the greater total sliding distance which occurs with a larger grain size.

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Acknowledgements

The senior author Hans Conrad wishes to acknowledge prior financial support by the U.S. Army Research Office and by the Materials Science Division of the National Science Foundation. Dr. Jun Wang would like to acknowledge prior financial support by the Materials Science Division of the National Science Foundation.

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Conflict of Interest

The authors declare no conflicts of interest regarding this paper.

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