Atomistic studies of stress effects on self-interstitial diffusion in α-titanium

Transcription

1 Journal of Computer-Aided Materials Design, 7: , KLUWER/ESCOM 2000 Kluwer Academic Publishers. Printed in the Netherlands. Atomistic studies of stress effects on self-interstitial diffusion in α-titanium M. WEN a,b,c.h.woo a, and HANCHEN HUANG a a Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong b Department of Engineering Mechanics, Tsinghua University, Beijing, People s Republic of China Received 1 December 1999; Accepted 25 January 2000 Abstract. The diffusion anisotropy of intrinsic point defects is an important factor governing the behavior of the HCP metals bombarded by energetic particles. The effects of stress on the diffusion and its anisotropy, although known to be important, have not been well understood. In this paper, we use a combination of molecular dynamics and molecular statics methods to investigate energy states of a self-interstitial in α-titanium, a typical HCP metal. Our calculation shows that the most stable configuration of the self-interstitial is the basal-split dumbbell configuration on the basal plane. Compression along the [0001] or the [1 100] directions leads to an insignificant change in the migration energies, while compression along the [11 20] direction leads to a larger migration energy. A significant change of the diffusion anisotropy is observed when a uni-axial compressive stress of 200 MPa is applied along the [11 20] direction. Similar stress along the other two directions does not produce substantial changes of the anisotropy. We also show that an applied hydrostatic stress can significantly change the diffusion anisotropy of HCP metals and alloys. Thus, under irradiation, a hydrostatic stress can produce a significant creep-like deformation (i.e., with a deviatoric strain rate) through a stress-dependent change of the growth rate. Keywords: Diffusion anisotropy, Molecular dynamics, Self-interstitial, Stress effects, Titanium 1. Introduction The effects of displacement damage on the physical and mechanical properties of metals and alloys, caused by the bombardment of energetic particles, have been under active investigation for many years. Besides the obvious technical and industrial implications, an important motive of such investigations is to understand the factors that differentiate the response of different metals under different irradiation conditions. Recently, much interest has been shown in the possible role of the crystal lattice structure on variations in the damage accumulation behavior of metals and alloys. From a historical perspective, the understanding of macroscopic behavior of materials due to damage accumulation was first established based on reaction kinetics governed by the isotropic migration of vacancies and interstitials. The studies were primarily directed towards cubic metals, in modeling the swelling behavior due to the growth of irradiation-induced voids. Because of crystallographic anisotropy, diffusion of the lattice defects in HCP metals and alloys (simply called HCPs in the rest of the paper) can be expected to be anisotropic. Article based on a presentation at the Fifth International Conference on Advanced Materials (International Union of Materials Research Societies), Symposium on Multiscale Materials Modeling (JJ), Beijing, China, June 13 18, See the Special Issue of Journal of Computer-Aided Materials Design, Vol. 6, nos. 2 3, 1999 for the Symposium Proceedings. To whom correspondence should be addressed. mmchwoo@polyu.edu.hk

2 98 M. Wen et al. In addition, dislocations with different Burgers vectors are non-equivalent in their motion and the reactions with the point defects and their clusters. The geometrical arrangements of sinks relative to the anisotropy of the diffusion then become important factors that dictate the reaction kinetics and damage accumulation behavior in these materials. This is expected to lead to a damage accumulation behavior much different from that of the cubic metals. Indeed, in cubic metals, it is well known that the irradiation-induced climb of dislocations and growth of interstitial loops cause a concomitant volume increase due to the growth of voids. In the HCPs, the anisotropy of both the diffusion of the intrinsic point-defects and the evolving dislocation structure necessarily produce a deviatoric straining in addition to volumetric swelling, even in the absence of an external stress. Indeed, irradiation growth is the name given to the volume-conserved shape deformation that occurs to HCPs under irradiation in the absence of an applied stress. The best known examples of irradiation growth are found in graphite [1 4], uranium [5], zirconium and its alloys [6 9]. The deformation behavior of the HCPs during irradiation is best understood in terms of the DAD (Diffusional Anisotropy Difference) theory [10]. Once generated under irradiation, the intrinsic point defects will either recombine or undergo long-range migration and annihilate preferentially at sinks, such as dislocations and grain boundaries. DAD produces a large difference in their annihilation rates at sinks, producing a strong sink bias. The anisotropic diffusion of point defects can either be intrinsic, i.e., due to the crystallographic anisotropy, or stress-induced. The HCPs have a crystal structure that requires the simultaneous consideration of both components. The intrinsic component contributes to irradiation growth [11]. The stress-induced component gives rise to a stress-induced preferred absorption (SIPA) effect that contributes to irradiation creep [10,12]. Although SIPA due to DAD has also been studied extensively in cubic metals after its introduction by Woo [12,13], due to the geometric complexity, a similar analysis in the HCPs has yet to be achieved. A clear understanding of the behavior of the HCPs during irradiation requires information on the intrinsic defects thus generated, particularly the anisotropy of the migration of both the vacancies and the interstitials. Much work has been done to obtain information, both experimentally and using computer simulation, concerning the configuration and energetics of these defects in the HCPs. The computer simulation results have been reviewed by Bacon [14]. Relevant experimental data have been comprehensively reviewed and analyzed with reference to the simulation results by Frank [15,16] and Seeger [17]. The migration of the vacancy does not show any evidence of a strong anisotropy in computer simulation studies. From experimental data for six metals, Hood [18] suggested that vacancies might diffuse faster on the basal plane than along the c-direction. The first comprehensive study of energy and stability of interstitials in HCPs was performed by Johnson and Beeler [19]. Based on symmetry considerations, the basic interstitial configurations that have been considered are as follows (Fig. 1): the tetrahedral (T), octahedral (O), basal tetrahedral (B T ), basal octahedral (B O ), basal crowdion (C B ), non-basal crowdion (C N ), c-dumbbell (D C ), and the basal dumbbell (D B ). All computer simulation results indicate the possible existence of multiple stable and metastable configurations, with transition barriers between them well within reach by thermal activation. The relative stability of various possible configurations depends on the c/a ratio. In general, the basal configurations are the preferred stable configurations when c/a is less than ideal (i.e., c/a < 8/3). On increasing the value of c/a from below to above the ideal value, it is expected that the formation energies and stability of the T, OandC N are the least affected. At the same time, B T,B O,C B and D B are expected to become less stable with an increase in their formation energy, and D C to become more stable with a

3 Atomic studies of stress effects on self-interstitial diffusion in α-titanium 99 Figure 1. Some possible self-interstitial configurations in the HCP structure. decrease in its formation energy [16]. The migration takes place via in-plane jumps through the configurations B O,B T,D B or B C, or out-of-plane jumps through the configurations T, O, C N,D C. The anisotropy of the interstitial diffusion depends on the diffusivities of the two types of jumps, and is expected to depend on the c/a ratio, being weakest for those closest to the ideal ratio. Little attention has been paid to the effect of stress on the diffusion properties of the intrinsic defects in the HCPs. In this connection we note the numerical calculation of the irradiation-induced deformation in Zr [20] due to stress-induced changes in the saddle-point energies. α-titanium is a typical HCP metal with a c/a ratio that is less than ideal, with irradiation damage properties much like α-zirconium. An investigation of the intrinsic defects of this metal is interesting in a systematic study of the HCPs. Various interatomic potentials, including pair potentials [14,21] and many-body potentials [22,23], have been used to perform atomistic simulation of intrinsic point defects in this metal. More recently, an interatomic potential of α-titanium, based on the Finnis-Sinclair approach, was developed by Ackland [24]. Using this potential, Wooding and Bacon studied the production and evolution of displacement cascades in α-titanium [25]. So far, all studies indicate that the diffusion of the self-interstitial atom is highly anisotropic [21]. However, no consensus has been reached among various calculations based on the different potentials [14,20,21,26 28]. Furthermore, none of these calculations so far consider the effect of an external stress. This work aims at investigating the effects of external stress on the diffusion anisotropy of the self-interstitial of α-titanium using a combination of the molecular dynamics (MD) and molecular statics (MS) methods. In Section 2, we present details of the simulation method. In Section 3, we present and discuss the simulation results. And finally we conclude our studies in Section 4.

4 100 M. Wen et al. Table 1. Self-interstitial formation energy for cases without stress or applying 200 MPa compression along the [1 100], [0001], and [11 20] directions Configuration Formation energy (ev) Stress free [11 20] [1 100] [0001] C B D B B O O C N Simulation method We use Ackland s interatomic potential for α-titanium. A simulation cell of Å, Å, and Å along the [11 20], [1 100] and [0001] directions is used with the periodic boundary condition. A self-interstitial is generated by adding an extra titanium atom in the simulation cell. The formation energy of the self-interstitial atom is calculated according to E f = E N+1 E N E sub, (1) where E N is the energy of the reference system without the extra atom, E N+1 is that of the defected system containing the self-interstitial atom in its fully relaxed configuration, and E sub is the sublimation energy of the titanium, which is 4.86 ev according to Ackland s potential. The relaxation of the crystal lattice is achieved by using a combination of annealing and quenching. The temperature is first raised to several hundred degrees Kelvin, and then gradually decreased to about 5 K. This temperature is further brought down to 10 5 Kusing the conventional quenching technique. During the annealing process, Langevin forces are used to simulate a thermal bath, and the frictional coefficient is chosen so that temperature transition takes 1 ps. Such a choice is made to optimize the computational efficiency and minimize the occurrence of artifacts caused by the random force. Because of the complex crystal structure of the HCP metals, several initial temperatures were used in the annealing of each configuration to find the state of minimum energy. To ensure the independence of the results on the size of the computational cell, we repeat the calculations by doubling the linear dimension of the computational cell. Our calculations indicate that results using a computational cell of Å Å Å are not affected by further increase of its dimension. The formation energies for both cases, with or without external applied stresses, are calculated under constant volume. The lattice constant in the basal plane and that along the c-axis, under various external stresses, are calculated using the Parrinello Rahman algorithm [29]. The calculated formation energies arepresented in Table 1. Temperature effects are eliminated by quenching the simulated system to 10 5 K. To investigate the relative stability of the self-interstitial configuration on the basal plane, we obtain the relaxed configurations and calculate the formation energies, with the selfinterstitial atom constrained on straight lines on the basal plane, oriented at various angles θ to

5 Atomic studies of stress effects on self-interstitial diffusion in α-titanium 101 Figure 2. The nodal points before (open diamond) and after (solid diamond) the relaxation in calculating the formation energy of the self-interstitial atom. The position of each nodal point is represented by the angle, θ. The solid circles represent the lattice atoms in the same basal plane as the self-interstitial atom, and the open circles represent the lattice atoms in the neighboring basal planes. the [10 10] direction (Fig. 2). The positions before (open diamond) and after (solid diamond) the relaxation are indicated. The formation energy as a function of θ is then plotted, as shown in Fig. 3. To calculate the energy barrier for migration of the self-interstitial atom from one D B position to another, the self-interstitial atom is constrained to move on the plane perpendicular to the straight line connecting the two D B positions during relaxation (Fig. 4). For the migration barrier out of the basal plane, the same procedure is used. These calculations not only locate the most stable configuration of the self-interstitial atom, they also help identify the likely diffusion path. If the relaxed coordinates of the self-interstitial atom constitute a continuous path, this may be taken as the diffusion path. However, it happens sometimes that the path may appear to be discontinuous, as shown in Fig. 2. In this case, we would have to introduce more nodal points at the discontinuities ; this is the case if the barrier of rotation is sought. 3. Results and discussions We first consider the formation energy of the self-interstitial atom in the basal plane in the absence of external stress. Because of the symmetry, we only need to consider an angular range of 60, as shown in Fig. 2. Formation energies of the self-interstitial atom, corresponding to the relaxed positions on various straight lines making an angle θ with the [10 10] direction, are determined. As can be seen from Fig. 3, the D B configuration, corresponding to θ 45, is the most stable one in the absence of an external stress. This is in agreement with the results of Bacon [21]. The B O configuration has a formation energy that is only 0.07 ev higher, and is therefore also thermodynamically probable.

6 102 M. Wen et al. Figure 3. The formation energy of the self-interstitialatom as a function of θ in the basal plane. The solid squares represent calculated data points, and the solid line is a linear interpolation between neighboring points. D B1 and D B2 stand for D B configuration with different orentations. Figure 4. The nodal points before (open diamond) and after (solid diamond) the relaxation in calculating the migration energy of the self-interstitial atom. The position of each nodal point is represented by the distance from a lattice site, d. The solid circles represent the lattice atoms in the same basal plane as the self-interstitial atom, and the open circles represent the lattice atoms in the neighboring basal planes.

7 Atomic studies of stress effects on self-interstitial diffusion in α-titanium 103 Figure 5. The formation energy of the self-interstitial atom as a function of distance from one lattice site, d, as shown in Fig. 4. The energies under zero stress, compression along the [1 100], the [0001], and the [11 20] directions are represented by solid circles, solid squares, open diamonds, and solid triangles, respectively. The solid line is a linear interpolation between neighboring points. To study the stress effects, we consider three stress conditions. In each case, we apply a uniaxial compressive stress of 200 MPa along the [11 20], the [1 100], or the [0001] directions; as a first study of the stress effects, we take 200 MPa as a typical value of stress. In all three cases, the D B configuration is found to be the most stable. The formation energy of the selfinterstitial atom is increased by 0.11 ev, 0.11 ev, and 0.13 ev, for the three stress states, respectively. The change of the formation energy can be attributed to the change of nearest neighbor distance of the self-interstitial atom. For example, the compressive stress along the [11 20] directions pushes the nearest neighbors closer to the self-interstitial atom, and therefore increases the formation energy. We next consider the migration of the self-interstitial. The formation energies as a function of position between one D B configuration and another is shown in Fig. 5 and 6, for the inplane and out-of-plane jumps, respectively. Four cases, under zero stress and under the three different compressive stresses, are shown for comparison. The self-interstitial atom migrates on the basal plane through a replacement mechanism. One of the D B atoms moves toward a nearby lattice atom and forms a new D B on the same basal plane, leaving the other one of the D B atoms occupying a lattice site. The migration barrier of this replacement process is found to be 0.05 ev, which is close to the 0.07 ev obtained by Foster (unpublished work reviewed in Reference 14) but is substantially smaller than the 0.37 ev obtained by Johnson [30]. For jumps that involve a change of orientation, a rotation of the dumbbell axis has to be included. Utilizing the molecular static method, the rotation barrier is 0.11 ev for the stress-free case, which is larger than the migration barrier of 0.05 ev. For the out-of-plane jumps, one of the D B atoms moves from one basal plane to the other, leaving the other D B atom occupying one lattice site. In these cases, the method we adopted can only follow the path with minimum energy barrier. Between other equivalent sites that involve a change of the dumbbell axis, we have to restrict the self-interstitial atom on the nodal points along the straight line connecting the initial and the final sites. With this scheme, we obtained a migration barrier of 0.27 ev

8 104 M. Wen et al. Figure 6. The formation energy of the self-interstitial atom as a function of the distance from a basal plane, h.the symbols are the same as in Fig. 6. for the out-of-plane jumps, which is larger than the 0.15 ev obtained by Foster (unpublished work reviewed in Reference 14) and the 0.12 ev obtained by Johnson [30]. The defect diffusivity in Zr, as well as the stress effects, have been studied in Reference 20. Using the diffusivity tensor components in the basal plane D a and out of the basal plane D c, the anisotropy factor η can be written as η = D c /D a For the stress-free case, the diffusivity tensor is defined by [31]: (2) D kl = N n 0 n ij j=1 n=1 n ij =1 ν 0ij 2n 0 exp [ E m (j,n,n ij )/kt ] s ij k sij l (3) where N j=1 is the sum over N neighbour sites equivalent to i that can be reached by a thermally activated single diffusion jump with jump vector s ij. n 0 n=1 is the sum over the possible orientations of the defect at equilibrium. n ij labels the different paths from i to j with different configuration symmetries at the saddle point, characterized by a migration energy E m (j, n, n ij ). ν 0 stands for the attempt frequency of the diffusion jump. For the D B configuration the possible thermal jumps are either in the basal plane or out of the basal plane. The diffusivity tensor results in Da 0 = ν 0 2 a2 [2exp( Em a /kt) + exp( Ea rot /kt)]+ ν 0 2 a2 [2exp( Em c 1 /kt) + exp( Em c 2 /kt)] (4) Dc 0 = ν 0 2 c2 [ 7 2 exp( Ec m 1 /kt) + exp( Em c 2 /kt)], where Em a and Ec rot stand for energy barriers to diffusion jumps, with and without a rotation, in the basal plane, and Em c 1 and Em c 2 for jumps out of the basal plane along different paths,

9 Atomic studies of stress effects on self-interstitial diffusion in α-titanium 105 Figure 7. Configuration changes during out-of-plane jumps of the self interstitial. (a) One of the atoms in the dumbbell moves along the empty channels in the HCP lattice to the next basal plane and forms a new dumbbell (jump barrier = E c m 1 ); (b) one of the atoms in the dumbbell moves through the C N position in the HCP lattice to the next basal plane and forms a new dumbbell (jump barrier = E c m 2 ). Table 2. Calculated and experimental energy barriers for self-interstitial diffusion in the basal plane Em a, Ea rot (ev), and out of the basal plane Em c 1, Em c 2 (ev). Erot a, Ec m 1 and Em c 2 result from a change of the orientation of the self-interstitial configuration during diffusion Energy barrier (ev) Calculation for α-ti Mikhin et al. for α-zr Em a Erot a 0.11 Em c Em c respectively (see Figs. 7a and 7b for the detailed configuration changes during the two kinds of jumps). The calculated values are tabulated in Table 2. The anisotropy factor p = (D c /D a ) 1/6 for interstitial diffusion in Zr has been determined experimentally by fitting to HVEM measured loop growth rates [32], assuming that the vacancy migration is isotropic. Expressed in an effective single mechanism form, p can be expressed by p = 1.38 exp( 0.026/kT) (5) Mikhin et al. [28], using a pair potential for α-zr constructed in terms of the generalized pseudo-potential theory, obtained a significant anisotropy for interstitial migration: a B O - D B -B O basal plane migration energy was one order of magnitude lower than for B O -C N -B O migration out of the basal plane. Using Equation 3 we may write the diffusion coefficients as follows: Da 0 = 3ν 0 2 a2 exp( Em a /kt) + 3ν 0c 2 exp( Em c /kt) Dc 0 = 3ν (6) 0 2 c2 exp( Em c /kt) from which p can be calculated and expressed in the effective single mechanism form Equation 5 using an Arrhenius fit (Fig. 8). The pre-exponential and the activation energy of p are

10 106 M. Wen et al. Figure 8. Arrhenius plot of the diffusion anisotropy factor (D c /D a ) 1/6 for self-interstitial under compression along the [1 100], the [11 20], the [0001] directions and under zero stress (from top to bottom) in α-ti. Table 3. Calculated and experimental activation energies E (ev) and the pre-exponential factor D of the anisotropy factor p = (D c /D a ) 1/6 = p 0 exp( E/kT) Calculation for α-ti Experiment for α-zr Mikhin et al. for α-zr p E listed in Table 3. Listed in comparison are the results of α-ti fitted to the diffusion anisotropy calculated from Equations 2 and 5 using the effective single mechanism approximation. The three sets of data are roughly consistent, indicating moderate anisotropic diffusion that is faster in the basal plane than along the c-axis. We note that the pre-exponential is very much a characteristic of the interstitial configuration. In this regard, the relatively large difference between the experimental and the calculated pre-exponential factors in the α-zr case may indicate that the equilibrium interstitial configuration of α-zr is more likely to be D B,as presently obtained for α-ti, instead of B O originally calculated in Reference 28. Application of a uniaxial compressive stress along the [11 20] direction leads to a negligible change of the migration barrier in the basal plane, and a decrease of 0.04 ev for the energy barrier of the out-of-plane jumps. If the entropy of the diffusion is not sensitive to external stresses, application of the stress reduces the diffusion anisotropy. Under the compression, the c/a ratio is changed from to 1.595, and the nearest neighbors of the self-interstitial atom are compressed on the basal plane, although the inter-plane spacing is in tension. The selfinterstitial atom being more easily accommodated in tension, its formation energy between the two basal planes is reduced under the [11 20] compression, causing the reduction of the diffusion anisotropy.

11 Atomic studies of stress effects on self-interstitial diffusion in α-titanium 107 Table 4. Energy barriers for self-interstitial diffusion in the basal plane Em a (ev), and out of the basal plane Em c (ev) for cases without stress or applying 200 MPa compression along the [11 20], [1 100] and [0001] directions, respectively. Stress free [11 20] [1 100] [0001] Em a Erot a Em c Em c Table 5. Under a uniaxial compressive stress along [1 100], the c/a ratio is changed from to The effect of the stress on both the in-plane and out-of-plane migration energies is small. When the uniaxial compressive stress is along [0001], the c/a ratio is reduced from to There is little change in the migration energies for the in-plane jumps, and a decrease of 0.02 ev for out-of-plane jumps, resulting in a small reduction of the diffusion anisotropy. The change of the migration barrier due to the application of the three uniaxial stresses is summarized in Table 4. The corresponding parameters fitted to the effective one-mechanism approximation are summarized in Table 5. The values of p at kt = 0.05 ev are also given for comparison. It can be seen that only the uniaxial stress along the [11 20] direction is effective. The application of this stress mode may change the growth rate by almost 20%. Taking into account the difference in the induced change in the dislocation structure, this effect can be further magnified by several times through the SIG (SIPA Induced Growth) effect [33]. In comparison, the effects of the uniaxial stresses along the other directions are negligible. In terms of the creep compliance, the specimen is much more compliant along the a-direction than along the c-direction, when subjected to uniaxial stresses. This is in agreement with the experimental observation [34]. Table 6. Calculated activation energies E (ev) and the pre-exponential factor p 0 of the anisotropy factor p = (D c /D a ) 1/6 = p 0 exp( E/kT) for cases without stress or applying 200 MPa compression along the [11 20], [1 100] and [0001] directions, respectively Stress free [11 20] [1 100] [0001] p E p (kt = 0.05eV)

12 108 M. Wen et al. Following the suggestion of Leibfried and Breuer [35] it is convenient to represent the applied stress, σ, using the six orthonormalized basis tensors b (λ),where, σ = 6 σ (λ) b (λ) λ=1 In Cartesian coordination b (λ) can be written as b (1) = b (2) = b (4) = b (5) = b (3) = 1 2 b (6) = (7) (8) In terms of the compression along the [11 20], [1 100], and the [0001] directions the unit basis stresses can be represented as ˆσ (1) = 1 [ σx 6 σ x + σy σ y 2 σ ] z σ z ˆσ (2) = 1 [ σx 2 σ x + σy ] (9) σ y ˆσ (6) = 1 [ σx 3 σ x + σy σ y + σ ] z σ z where σ x, σ y and σ z represent the uniaxial compression along the [11 20], [1 100], and [0001] directions, respectively. The corresponding relative changes in the anisotropy per unit stress due to stress states (1) (pyramidal), (2) (prismatic) and (6) (hydrostatic) are then given by δη (1) = (δη x + δη y 2δη z )/ 6 δη (2) = (δη x + δη y )/ 2 (10) δη (6) = (δη x + δη y + δη z )/ 3 Using Table 5, at kt = 0.05 ev, we obtain δη (1) = MPa 1, δη (2) = MPa 1 and δη (6) = MPa 1.Hereδη (1) is the change of η due to ˆσ (1) relative to the stress-free η. From these results, it can be seen that an applied hydrostatic stress σ (6) may change the diffusion anisotropy by a significant amount (36% for 1000 MPa). Indeed, in the present case, a hydrostatic stress affects the diffusion anisotropy as much as a prismatic shear stress, and more significantly than a pyramidal shear stress. Thus, under irradiation, a hydrostatic stress can produce a significant creep-like deformation (i.e., with a deviatoric strain rate) through a stress-dependent change of the growth rate. This possibility has to be, but so far has not been, considered in analyzing the irradiation creep and growth of HCPs.

13 4. Conclusions Atomic studies of stress effects on self-interstitial diffusion in α-titanium 109 Our calculations show that in the absence of an external stress the D B configuration is the most stable one. It remains the most stable configuration under the three orthogonal uniaxial compressive stress conditions that are investigated in this work. The migration energies of both in-plane and out-of-plane diffusion jumps are calculated for various possible paths as a function of external applied stress. Under a compressive uniaxial stress along the [11 20] direction, the diffusion anisotropy is significantly reduced. However, when a similar uniaxial compressive stress is applied along the [1 100] or [0001] direction, the diffusion anisotropy has an effect that is an order of magnitude smaller. The implication of this to irradiation creep and growth in α-ti is discussed. The present analysis shows that an applied hydrostatic stress can significantly change the diffusion anisotropy of HCPs. Thus, under irradiation, a hydrostatic stress may produce a significant creep-like deformation (i.e., with a deviatoric strain rate) through a stress-dependent change of the growth rate. More detailed calculations need to be performed to consider this possibility quantitatively. Acknowledgements The work described in this paper was fully supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region (PolyU 5156/97E and PolyU 5123/98E). References 1. Nettley, P.T., Bridge, H. and Simmons, J.H.S., J. Brit. Nucl. Energy Soc., 2 (1963) Kelly, B.T., In Second Conference on Industry Carbon and Graphite, Society of Chemical Industry, London, Nettley, P.T., Brocklehurst, J.E., Martin, W.H., and Simmons, J.H.S., Advanced and High-Temperature Gas Cooled Reactors, IAEA, Vienna, 1969 p Kelly, B.T., Gray, B.S., Martin, W.H., Howard, V.C. and Jenkins, M.J., Society of Chemical Industry, London, 1966, p Proceedings of the International Conference on Peaceful Uses of Atomic Energy, Geneva, 1955, Vol. 7, United Nations 1956 (whole volume). 6. Buckley, S.N., Properties of Reactor Materials and Effects of Irradiation Damage, Butterworth, London, 1962, p Carpenter, G.J.C., Coleman, C.E. and MacEwen, S.R., (Eds.), Fundamental Mechanisms of Radiation Induced Creep and Growth, J. Nucl. Mater., Vol ASTM STP 633, 1977 (whole volume). 9. Woo, C.H. and McElroy, R.J., (Eds.), Fundamental Mechanisms of Radiation Induced Creep and Growth, J. Nucl. Mater., Vol. 159, Woo, C.H., ASTM STP, 955 (1987) Woo, C.H. and Goesele, U., J. Nucl. Mater., 119 (1983) Woo, C.H., J. Nucl. Mater., 120 (1984) Woo, C.H. and Garner, F.A., J. Nucl. Mater., (1992) Bacon, D.J., J. Nucl. Mater., 159 (1988) Frank, W., Phil. Mag. A, 63 (1991) Frank, W., J. Nucl. Mater., 159 (1988) Seeger, A., Phil. Mag. A, 64 (1991) Hood, G.M., Zhou, H., Gupta, D. and Shultz, R.J., J. Nucl. Mater., 233 (1995) 122.

14 110 M. Wen et al. 19. Johnson, R.A. and Beeler, J.R., In Lee, J.K. (Ed.), Interatomic Potentials and Crystalline Defects, The Metallurgical Society of AIME, Pennsylvania, PA, 1981, p Fernandez, J.R., Monti, A.M., Sarce, A. and Smetniansky-De Grande, N., J. Nucl. Mater., 210 (1994) Bacon, D.J., J. Nucl. Mater., 206 (1993) Igarashi, M., Khantha, M. and Vitek, V., Phil. Mag. B, 63 (1991) Oh, D.J. and Johnson, R.A., J. Mater. Res., 3 (1988) Ackland, G.J., Phil. Mag. A, 66 (1992) Wooding, S.J. and Bacon, D.J., Radiation Effects and Defects in Solids, (1994) Fernandez, J.R., Monti, A.M. and Pasianot, R.C., J. Nucl. Mater., 220 (1996) Fuse, M., J. Nucl. Mater., 136 (1985) Mikhin, A.G., Osetsky, Yu. N. and Kapinos, V.G., Phil. Mag. A, 70 (1994) Parrinello, M. and Rahman, A., J. Appl. Phys., 52 (1981) Johnson, R.A., Phil. Mag. A, 63 (1991) Savino, E.J. and Smetniansky-De Grande N., Phys. Rev. B, 35 (1987) Woo, C.H., Radiation Effects Defects in Solids, 144 (1998) Woo, C.H., Phil. Mag. A, 42 (1980) Woo, C.H., J. Nucl. Mater. 276 (2000) Leibfried, G. and Breuer, N., Point Defects in Metals I: Introduction to the Theory, Springer-Verlag, Berlin, 1978.