Science in China Series E: Technological Sciences © 2009

www.scichina.com tech.scichina.com www.springerlink.com

SCIENCE IN CHINA PRESS

Springer

Theoretical progress in non-equilibrium grain-boundary segregation (II): Micro-mechanism of grain boundary anelastic relaxation and its analytical formula XU TingDong1†, WANG Kai1 & SONG ShenHua2 1 2

Central Iron and Steel Research Institute, Superalloy Department, Beijing 100081, China; Department of Materials Science and Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China

Finding the internal-friction peak of grain boundary anelastic relaxation was one of the important breakthroughs in the study of internal friction in the last century. But the micro-mechanism of grain boundary anelastic relaxations is still obscure. Based on the observations of the grain boundary segregation or depletion of solute induced by an applied stress, the following micro-mechanism was suggested: grain-boundaries will work as sources to emit vacancies when a compressive stress is exerted on them and as sinks to absorb vacancies when a tensile stress is exerted, inducing grainboundary depletion or segregation of solute, respectively. The equations of vacancy and solute concentrations at grain boundaries were established under the equilibrium of grain-boundary anelastic relaxation. With these the kinetic equations were established for grain boundary segregation and depletion during the grain boundary relaxation progress. grain boundary, segregation, anelastic relaxation, kinetics

The variation of micro-structure in materials induced by an applied stress has been an important study field in materials science and engineering. The slip, climb and interaction of dislocations under applied stresses induce the plastic deformation of materials, which are the main content in the research of mechanical properties. A great deal of engineering practice has confirmed that the degradation of properties, embrittlement, creep, fatigue and brittle fracture without any sign will occur in metallic materials during service to produce engineering accidents. Metal usually serves under an applied stress which is lower than its yield limit and at the same time the degradation of its mechanical properties occurs. Therefore, what kind of micro-structure variations will appear for polycrystalline materials under an elastic stress? How will these variations influence the mechanical properties and the performance in service? These have been the most urgent and vexing challenges

to the present materials scientists and engineers. In the 1940s, Chinese scientist Ke found the internal friction peak of grain boundary anelastic relaxation, which was one of the important achievements in the field of elastic stress to induce the variations of micro-structure in the last century. But the micro-mechanism of grain boundary anelastic relaxation was still obscure[1], and the situation gave rise to the argument whether or not the internal friction peak found by Ke was relative to the grain boundary anelastic relaxation[2]. Recently, we found that the main micro-structure variation of polycrystalline under an elastic stress was absorbing or emitting vacancies to produce grain boundary segregation or depletion of solutes during grain boundary anelastic re Received November 26, 2008; accepted January 22, 2009 doi: 10.1007/s11431-009-0143-z † Corresponding author (email: [email protected]) Supported by the National Natural Science Foundation of China (Grant No. 50771036)

Sci China Ser E-Tech Sci | Jun. 2009 | vol. 52 | no. 6 | 1679-1687

laxations[3 5]. Based on the findings, we established the thermodynamic and kinetic equations of grain boundary segregation and depletion under grain-boundary anelastic relaxations, which can systemically and analytically express the micro-mechanism of grain boundary anelastic relaxations. This paper serves as a review of the recent progress in terms of theories, research topics, and approaches on the above issues. ―

1 Experimental phenomena and the logic troubles In 1981, Shinoda and Nakamura[6] reported that a Cr low-alloy steel used in their experiments were first aged for 1000 h at 773 K to reach the equilibrium grain boundary concentration of phosphorus, then aged under an applied tensile stress and compressive stress of less than about 40 MPa, respectively. At the beginning of the tensile stress ageing, the grain boundary concentration of phosphorus increased first, reached a maximum, and then decreased to the original equilibrium grain-boundary concentration with the increasing ageing time. For the compressive stress ageing, the grain boundary concentration of phosphorus decreased first, reached a minimum, and then increased to the original equilibrium grain boundary concentration with increasing ageing time, as shown in Figure 1.

Figure 1 Changes in the mean Auger peak ratio of phosphorus to iron as a function of ageing time under an applied stress of 30 MPa at 773 K for smooth steel specimens [6]. 1680

In 1996, Misra[7] studied the effects of tensile stress ageing on the grain boundary segregation of sulphur in a 2.6Ni-Cr-Mo-V steel. His results supported the above effects, as shown in Figure 2[7]. Misra confirmed that the sulphur segregation concentration induced by tensile stress ageing would decrease and finally disappear with extending ageing time, reaching its equilibrium grainboundary concentration[7]. In 1996, Lee and Chiang[8] found that the segregation of bismuth at grain boundaries in ZnO could be almost completely suppressed under high hydrostatic pressures. Segregation and pressure-induced desegregation are highly reversible. This result is in line with the study by Shinoda and Nakamura[8]. As suggested by Hondros and Seah, the source of the solute is thought to be other grain boundaries rather than the lattice, since the activation energy for grain boundary diffusion is much lower than that for lattice diffusion[9]. In fact, the grain-boundary depletion of phosphorus observed by Shinoda and Nakamura[6] implies that there is only grain boundary depletion and no phosphorus enrichment in the specimens. Both grain boundary enrichment and depletion should occur in a specimen if the other grain boundaries were the source of the solute for segregation rather than the lattice. Therefore, these observations cannot show that the source of the solute is the grain boundary rather than the lattice, especially under high hydrostatic pressures. Similarly, they did not demonstrate either Herring-Nabarro or Coble creep as described by Misra[7]. Under these circumstances, the solute is transported from some grain boundaries to other boundaries. It may be concluded

Figure 2 Auger peak ratios of sulfur to iron at the grain-boundary recorded for a low-alloy steel under an applied tensile stress at 883 K[7].

XU TingDong et al. Sci China Ser E-Tech Sci | Jun. 2009 | vol. 52 | no. 6 | 1679-1687

that there must be diffusion of solute between the grain boundary and the lattice in the anelastic relaxation induced by the application of elastic stress. This diffusion can make the solute move to or depart from the grainboundary.

2 Micro-mechanism of grain boundary anelastic relaxation The experimental data by Shinoda and Nakamura[6] and by Misra[7] showed that an elastic stress can produce a critical time just as the grain boundary segregation induced by thermal cycle. A maximum concentration at the grain boundary occurs for the tensile stress and a minimum concentration occurs for a compressive stress. Based on the experimental phenomena above, Xu suggested the following hypotheses[4,5,10]: When a low applied tensile stress is exerted on a grain boundary at a high temperature, the vacancies near the grain boundary will be absorbed into the grain boundary. Because of the thermal equilibrium between vacancies, solute atoms and vacancy-solute complexes in the bulk, a decrease in vacancy concentration at the grain boundary causes the dissociation of the complexes into vacancies and solute atoms. Consequently, the concentration gradient of complexes formed drives the complexes to diffuse to the grain boundary, resulting in non-equilibrium grain boundary segregation of solute. When the equilibrium of anelastic relaxation is attained at the grain boundary the absorption of vacancies terminates, suggesting that the supply of complexes to the grain boundary is exhausted. At the same time, while the vacancy-solute complexes diffuse to the grain boundary, a concomitant but reverse diffusion of solute atoms away from the boundary takes place along the solute concentration gradient established. At the beginning of this process, the complex diffusion is dominant and decreases with increasing ageing time because only a certain quantity of vacancies are absorbed into one unit volume of grain boundary for a constant tensile stress, while the reverse diffusion of solute atoms increases. Accordingly, an ageing time must exist at which the solute diffusion balances the complex diffusion and the solute boundary concentration reaches the maximum value. This ageing time is called the critical time of the tensile stress ageing. The non-equilibrium segregation, in general, disappears as the ageing time approaches infinity to reach full equilibrium.

When a low compressive stress is exerted on a grain boundary at a high temperature, the grain boundary will emit vacancies into the bulk and thus oversaturated vacancies must exist near the grain boundary. These vacancies may combine with solute atoms to form vacancy-solute complexes owing to the equilibrium mentioned above. These complexes will diffuse away from the grain boundary along their concentration gradient between the boundary region and the interior and dissociate vacancies and solute atoms in the interior. The diffusion may cause a depletion region of solute near the grain boundary and a reverse solute concentration gradient between the grain boundary and the interior, making the diffusion of solute atoms to the boundary occur along the gradient of solute concentration. At the beginning of this process, the complex diffusion is dominant and decreases with increasing ageing time for a certain compressive stress, while the reverse diffusion of solute atoms increases. Therefore, an ageing time must exist at which the solute diffusion balances with the complex diffusion, and at this time a maximum extent of depletion is reached. This ageing time is called the critical time of the compressive stress ageing. Finally, the depletion region gradually attenuates and dies away with further increasing ageing time. This grain boundary depletion induced by an applied compressive stress is called the non-equilibrium grain boundary depletion. Xu[4,5,10] formulated the critical time for the stress induced non-equilibrium grain boundary segregation or depletion as follows tc (T ) = ⎡⎣ R 2 ln ( Dc / Di ) ⎦⎤ ⎣⎡δ ( Dc − Di ) ⎦⎤ ,

(1)

where Dc is the diffusion coefficient of complexes, Di is the diffusion coefficient of solute atoms, both of which are affected by the applied stress, R is the grain radius, and δ is a constant. Stress analysis in the grain boundary region. A high angle grain boundary is usually assumed to be a thin region and only several atomic diameters thick. It is also assumed that grain boundaries are elastically softer than grain interiors and therefore they are preferentially deformed when a tress is applied[11]. The atomic arrangement is irregular and many voids exist at grain boundaries[4,10]. The atomic structure of a grain boundary is known to be dependent on the orientation relationship between the two adjacent crystals and the boundary inclination. All grain boundaries of a material have dif-

XU TingDong et al. Sci China Ser E-Tech Sci | Jun. 2009 | vol. 52 | no. 6 | 1679-1687

1681

ferent atomic structures characterized by, for example, different interatomic distances because the adjacent crystals are oriented randomly[12,13]. The degree of fit at a grain boundary varies from point to point and this is also found in the bubble model[14]. Since the disorder is different from point to point at grain boundaries, the stresses vary locally under an applied elastic stress. Due to the disorder of atomic structure in the grain boundary region and the variation of the direction of the applied stress relative to the grain boundaries, there must be sites where the stress condition preferentially promotes vacancy emission, and at other sites the emission is suppressed. Therefore, it is almost impossible that an overall decrease or increase in interatomic spacing in the grain boundary region can be produced time-independently by an applied stress without ever emitting or absorbing any vacancies. Size factor analysis. The observed density reduction from 7.9 g/cm3 for polycrystalline iron to 6 g/cm3 for nanocrystalline iron results primarily from a lower density of the grain boundary component. If the grain boundaries are removed from nanocrystalline iron by the grain growth, the density increases to 7.9 g/cm3[15]. It may be stated that the average interatomic spacing in the grain boundary region is larger than that in the matrix. Therefore the vacancies in the grain boundary region have a larger average volume than those in the matrix. Given that the total number of vacancies is constant under a compressive stress, the volume of grain boundaries will decrease but the volume of the matrix will increase when the vacancies move from the grain boundary to the grain interior. As the vacancies take a larger volume in the grain boundary than in the matrix, the total volume (grain boundaries + bulk) would be decreased. Under a tensile stress, the reverse effect occurs. This results in such an anelastic deformation that the total volume of the material increases under a tensile stress and reduces under a compressive stress. Energy analysis. Transporting a vacancy between the grain boundary and the grain interior induces a change of energy equal to the difference of the vacancy formation energies between the grain boundary and the grain interior. Destroying a vacancy in the grain boundary or interior regions induces a change of energy equal to the vacancy formation energy in that region. In general, the former is less than the latter. Since all processes in nature generally follow the path of the lowest activa1682

tion energy, the anelastic deformation at the grain boundary under an applied compressive stress can preferentially emit vacancies into the matrix without ever destroying vacancies in the boundary region, and the anelastic deformation at grain boundaries under an applied tensile stress can preferentially absorb vacancies from the matrix without ever creating vacancies in the boundary region. In general, the boundaries are good sinks or sources of vacancies, as suggested by Balluffi et al.[16] and Gleiter et al.[17]. Therefore, it is suggested that the anelastic deformation in the grain boundary region under a compressive (tensile) stress occurs time-dependently by emitting (absorbing) vacancies, but not by an overall decrease (increase) in the interatomic spacing time-independently[18,19].

3 Equilibrium equations of grain boundary anelastic relaxation 3.1

Equations of vacancy concentration

When the stress equilibrium during anelastic relaxation is reached under a low applied compressive stress, the emission of vacancies stops and therefore a given number of vacancies will be emitted from a unit volume of grain boundary for a certain applied compressive stress. In this case, Hooke’s law is valid: σn = Egbεn, where σn is the compressive stress perpendicular to the grain boundary, Egb is the elastic or young’s modulus in the grainboundary region and εn is the strain induced by the compressive stress σn. The anelastic deformation energy, W, in a unit volume of grain boundary induced by the compressive stress σ, is given by W W = K 0σ 2 ( 2 Egb ) ,

(2)

where K0 is a geometric factor. The vacancy release process by the grain boundary under an applied compressive stress should be an adiabatic process during which there is no loss or gain of heat by the grain boundary. The entropy change, dS, involved in the process of removal of a vacancy from the boundary is equal to zero, i.e., TdS = 0,

(3)

where T is the absolute temperature. By the first law of thermodynamics, the internal energy change involved in the process of vacancy removal, ΔU, is equal to the work done by the external forces in a unit volume of boundary, i.e.,

XU TingDong et al. Sci China Ser E-Tech Sci | Jun. 2009 | vol. 52 | no. 6 | 1679-1687

ΔU = K 0σ 2 / ( 2 Egb ) .

(4)

The internal energy change of grain boundary is induced by the decrease of vacancies due to vacancy release. Therefore,

ΔU = nFv ,

(5)

where n is the number of vacancies decreased in a unit volume of boundary, Fv is the formation energy of a vacancy in the grain boundary region. Thus, nFv = K 0σ 2 / ( 2 Egb ) ,

(6)

i.e., n = K 0σ 2 / ( 2 Egb Fi.e., ) .

(7)

In fact, n is the decrement in the vacancy concentration of grain boundary. As a result, the grain boundary vacancy concentration Cv(′ σ =σ ) , induced by an applied compressive stress σ under anelastic stress equilibrium, can be expressed as[4,5] Cv′ ( σ =σ ) = Cv( σ =0 ) − K 0σ 2 ( 2 Egb Fv ) ,

(8)

where Cv(σ = 0) is the equilibrium vacancy concentration of grain boundary under no stress. It should be noted that the existence of vacancy sinks or sources such as dislocations within the diffusion zone near grain boundaries is neglected in this model[20]. In the same manner, when the stress equilibrium during anelastic relaxation is reached under a low applied tensile stress, the absorption of vacancies stops and therefore a certain number of vacancies will be absorbed into a unit volume of grain boundary for a certain applied tensile stress[4]. The grain boundary vacancy concentration Cv(σ = σ), induced by an applied tensile stress σ under anelastic stress equilibrium, can be expressed as Cv( σ =σ ) = Cv( σ =0 ) + ( K 0 / 2)σ 2 /( EFv ).

(9)

Clearly, this is the sixth method of creating and annihilating vacancies in solids besides the five methods mentioned by Smallman[21], namely rapid quenching, intermetallic compound deviation from stoichiometry, bombardment with high energy particles such as neutrons and protons, plastic deformation, and oxidization. 3.2

Equations of solute concentration

Owing to the fact that grain boundaries emit vacancies into the grain interior under a low applied compressive stress, supersaturated vacancies must exist near the grain boundaries. These vacancies combine with solute atoms

to form vacancy-solute atom complexes due to the thermal equilibrium between the vacancies, solute atoms and their complexes. These complexes diffuse away from the grain boundaries along their concentration gradient between the boundary region and the interior and then dissociate into vacancies and solute atoms in the interior to keep the above equilibrium. The following assumptions are proposed: i) One complex contains statistically one vacancy and one solute atom; ii) almost all the decrease of solute and vacancy in the grain boundary region is produced by the diffusion of complexes away from grain boundaries. This is because of the equilibrium between vacancies, solute atoms and complexes at the region of boundaries and the grain interior and the faster diffusion of complexes than that of solute atoms[22]. Therefore, the minimum concentration of solute under stress equilibrium in the grain boundary region, Cb(′ σ =σ ) , induced by an applied compressive stress σ, is derived from eq. (8) as Cb(′ σ =σ ) = Cb(σ =0 ) − K 0σ 2 (2 Egb Fv ),

(10)

where Cb(σ =0) is the grain boundary concentration of solute under no stress. In the same way, the maximum concentration of solute under stress equilibrium in the grain boundary region, Cb(σ = σ), induced by an applied tensile stress σ, is derived from eq. (9) as Cb( σ =σ ) = Cb( σ =0 ) + K 0σ 2 (2 Egb Fv ).

(11)

In order to confirm the existence of the physical processes above, Xu and Zheng[20] calculated the elastic modulus of the grain boundary region with eq. (10) using the observations by Misra[7]. The increased concentration Cb(σ = σ) − Cb(σ =0)＝0.20 at the grain boundary induced by an applied tensile stress of σ ＝ 3.43×108 Pa at 883 K was from the data for sulfur in 2.6Ni-Cr-Mo-V steel observed by Misra[7] (see Figure 2). Simonen et al.[23] evaluated defect kinetic parameters for radiation induced grain boundary segregation in austenitic stainless alloys and reported that the vacancy formation energy at the grain boundary was ~1.5 eV. In addition, using atomic resolution Z contrast STEM, Chisholm et al.[24] revealed preferential nucleation of electron beam induced damage in selected atomic columns of an Si tilt grain boundary. The results obtained by atomic simulations showed that the formation energies of isolated vacancies, isolated divacancies and chains of monovacan-

XU TingDong et al. Sci China Ser E-Tech Sci | Jun. 2009 | vol. 52 | no. 6 | 1679-1687

1683

cies were between 0.8 and 1.8 eV. In view of these results, Fv＝ 1.5 eV will be selected as the vacancy formation energy in the grain boundary region in the present paper. For polycrystalline material, the normal directions of grain boundaries are oriented randomly to the direction of applied stress over angles ranging from 0° to 90°. Here, the angle of 45° is selected as an average. Therefore, K0 is equal to cos45° = 0.707. Substituting the determined values into eq. (11) gives the grain boundary elastic modulus Egb = 2.03×107 Pa [20]. Up to now, there are no experimental data of the elastic modulus at grain boundaries. Kluge et al.[25] presented an approach for use in atomistic simulations to calculate the local elastic constants in terms of local stresses and strains. Furthermore, by employing an embedded atom method (EAM) potential fitted to Au, all the independent elastic constants C11, C12, C13, C33, C44 and C66 of a high angle (001) twist grain boundary were simulated based on an ideal defect-free bi-crystal. The simulation by Kluge et al. was for an ideal defect free grain boundary without taking the effect of the structural disorder into account. The structural disorder includes point defects, defect clusters due to impurity segregation or the presence of voids, the grain boundary geometry, and so on. These defects will greatly influence the elastic modulus at grain boundaries. Kluge concluded that the large reduction in C44 was a generic phenomenon in high angle grain boundary simulations. The experimentally determined grain boundary elastic modulus Egb should actually be an apparent value, which may be considered as being composed of contributions from C11, C12, C13, C33, C66 and C44. It is seen from comparing the measured results with the simulation ones that Egb is one order of magnitude lower than C44. It may be claimed that the elastic modulus at grain boundaries calculated with eq. (11) can represent the elastic property at grain boundaries. Eq. (11) is valid for the anelastic relaxation of grain boundaries. This could be a unique method of obtaining the elastic modulus of the grain boundary region experimentally.

4.1 Kinetic equations of solute depletion

The equilibrium equation, eq. (10), gives a boundary condition to solve the depletion equations with Gauss’s method and Carlslaw and Jaeger’s math method[26]. The kinetic equations of depletion induced by an applied compressive stress were detailed in refs. [10, 27]. For the depletion stage, when the stress ageing time t is shorter than the critical time tc, the kinetics are given by[5] ⎡⎣Cb ( t ) − Cg ⎤⎦ / ( Cb( σ =0 ) − Cg )

{

= erf ⎡⎣ d

16Dc t ⎤⎦ − erf ⎡⎣ −d

} 2.

16Dc t ⎤⎦

(12)

For the de-depletion stage, when the stress ageing time t is longer than the critical time tc, the kinetics are given by

( Cb ( t ) − Cb' (σ =σ ) ) =

{

2 2 2⎤ ⎡ ⎤ ⎡ ⎣ K 0σ ( 2 Egb Fv ) ⎦ 1 − exp ⎣ 4 Di ( t − tc ) α i +1d ⎦

}

⋅erfc ⎡⎣ 2 Di ( t − tc ) α i +1d ⎤⎦ .

(13)

Eqs. (12) and (13) present analytically the variation of grain boundary concentration of solute Cb(t) with the stress ageing time under an applied compressive stress. Figure 3 shows the simulation of Shinoda and Nakamura’s observations with eqs. (12) and (13). The simulations showed that the diffusion coefficient of vacancyphosphorus complexes is Dp-v = 2.7×10−20 m2/s. It is decreased by three orders of magnitude compared with the diffusion coefficient of complexes (Dp = 1.06×10−16 m2/s) under no stress. However, in contrast, the diffusion coefficient of phosphorus atoms is increased from

4 Kinetic equations of grain boundary anelastic relaxation The results by Shinoda et al.[6] and Misra[7] presented the kinetics of grain boundary segregation and depletion during the anelastic relaxations. The development of relative theories should describe these kinetic processes. 1684

Figure 3 Auger peak ratio (APR) curve represents phosphorus grain boundary segregation isotherms obtained under an applied compressive stress condition at 773 K. The Cb curve is the grain boundary concentration of phosphorus and represents a simulation of the APR curve using eqs. (12) and (13)[5].

XU TingDong et al. Sci China Ser E-Tech Sci | Jun. 2009 | vol. 52 | no. 6 | 1679-1687

7.4×10−20 m2/s under no stress to 1.06×10−14 m2/s, leading to an increase of about six orders of magnitude under the compressive stress. 4.2 Kinetic equations of segregation

The equilibrium equation, eq. (11), presents the interactions of grain-boundaries, vacancies and solute atoms but also gives a boundary condition to solve the segregation equations with Carlslaw and Jaeger’s math method and Gauss’s method[26]. The kinetic equations of segregation induced by an applied tensile stress were detailed in References [10, 27]. For the segregation stage, when the stress ageing time t is shorter than the critical time tc, the kinetics are given by[4] ⎡⎣Cb ( t ) − Cb(σ =0 ) ⎦⎤ ⎡⎣Cb(σ =σ ) − Cb(σ =0 ) ⎦⎤ 1/ 2 α i +1 d ⎦⎤ , = 1 − exp ( 4 Dc t α i2+1 d 2 ) ⋅ erfc ⎣⎡ 2 ( Dc t )

(14)

and for the de-segregation stage, when the stress ageing time t is longer than the critical time tc, the kinetics are given by ⎡⎣Cb ( t ) − Cb(σ =0 ) ⎦⎤ ⎡⎣Cb ( tc ) − Cb(σ =0 ) ⎤⎦ =

{erf ⎡⎣d

16 Di ( t − tc ) ⎤⎦ − erf ⎡⎣ −d

} 2,

16 Di ( t − tc ) ⎤⎦

(15) where Cb(t) is the boundary concentration at the stress ageing time t, d is the boundary enriched width, and α is the enrichment ratio, given by α = Cb(σ =σ)/Cg, where Cg is the matrix concentration. Eqs. (14) and (15) present analytically the variation of grain boundary concentration of solute, Cb(t), with the stress ageing time under an applied tensile stress. Xu[4,5] simulated the experiments of Shinoda and Nakamura with eqs. (14) and (15). The results were shown in Figure 4. The simulated results indicated that the diffusion coefficient of vacancy-phosphorus complexes is 3.14×10−14 m2/s during the segregation stage. The diffusion coefficient of phosphorus atoms is 7.59×10−23 m2/s during the de-segregation stage. Comparing the diffusion coefficient of vacancy-phosphorus complexes obtained here with the complex diffusion coefficient 9.0×10−17 m2/s under no stress[28] demonstrates that the applied tensile stress makes the diffusion coefficient of vacancy-phosphorus complexes increase by three orders of magnitude. Comparing the diffusion coefficient of phosphorus atoms obtained above with the phosphorus diffusion coefficient 7.4×10−20 m2/s under no stress[29]

Figure 4 APR-curve is phosphorus grain boundary segregation isotherms acquired in the tensile stress condition at 773 K. Cb curve in the grain boundary concentration of P is the simulation of APR-curve with eqs. (14) and (15) [4].

shows that the applied tensile stress makes the diffusion coefficient of phosphorus atoms during the de-segregation decrease by three orders of magnitude. Xu[4] simulated the experiments of Misra with eqs. (14) and (15), as shown in Figure 5. It was shown from the simulation results that the diffusion coefficient of vacancy-sulfur complexes (2.06×10−12 m2/s) under stress is larger than the diffusion coefficient of the complexes (1.35×10−15 m2/s) under no stress[30] by about three orders of magnitude. The diffusion coefficient of sulfur (1.86×10−22 m2/s) under stress is smaller than the diffusion coefficient of sulfur (1.24×10−17 m2/s) under no stress[30] by about five orders of magnitude. Such a great difference between the diffusivities of the diffusion processes towards and away from the grain boundaries implies that the diffusers towards the grain boundaries are much different from the diffusers away

Figure 5 Auger peak ratios of sulfur to iron at the grain-boundary recorded for a low-alloy steel under an applied tensile stress at 883 K in Figure 2[7]. The Cb curve is the simulation of Auger peak ratio curve with eqs. (14) and (15)[4].

XU TingDong et al. Sci China Ser E-Tech Sci | Jun. 2009 | vol. 52 | no. 6 | 1679-1687

1685

from the boundaries. Both processes produce a change only in the sulfur concentration at the grain boundaries and, therefore, they are both related to sulfur atoms. This confirms that the vacancy-sulfur complexes diffuse towards the grain boundaries and sulfur atoms diffuse away from the grain boundaries during ageing under an applied tensile stress. It is interesting to compare the variation of the diffusion coefficients under compressive stresses with those under tensile stresses. The diffusion coefficient of vacancy-phosphorus complexes is 3.14 × 10−14 m2/s under a 30 MPa tensile stress. This is increased by three orders of magnitude compared with the diffusion coefficient of the complexes of 9.0 × 10−17 m2/s under no stress, but in contrast the diffusion coefficient of phosphorus atoms is decreased from 7.4 × 10−20 m2/s under no stress to 7.59 × 10−23 m2/s, resulting in a decrement of about three orders of magnitude under tensile stress. Therefore, it may be concluded that an applied tensile stress enhances the diffusion coefficient of complexes and reduces that of solute atoms. Nevertheless, an applied compressive stress decreases the diffusion coefficient of complexes and increases that of solute atoms. It can be envisaged that the vacancy-phosphorus complex is such a cell (a region within the crystal lattice) that it is in the tensile stress condition and the phosphorus in lattice is such a cell that it is in the compressive stress condition in the matrix. When the complex moves from one site to another site in a tensile stress field, the stress condition in the new site will be closer to that of the complex and therefore it can move to the new site more easily than in a compressive stress field or in the zero stress field. When a solute atom in the lattice moves from one site to another site in a compressive stress field, the stress condition in the new site will be closer to that of the solute atom in the lattice and therefore it can move to the new site more easily than in the tensile stress field or in the zero stress field. This could be the reason why an applied tensile stress increases the diffu1

Ge T S. Contribution of internal-friction study on grain boundaries

2

Woirgard J, Amirault J P, Fouquet J. A dislocation model for grain

and dislocation substructures. Adv Sci China Phys, 1990, (3): 1―113 boundary peaks in pure face centered cubic metals. In: Proc

sivity of complexes and decreases the diffusivity of solute atoms, and an applied compressive stress decreases the diffusivity of complexes and increases the diffusivity of solute atoms.

5 Conclusions The grain boundary anelastic relaxation is an important question in the study of mechanical properties of materials. In the past it was studied mainly by the internal friction method. The variation of grain boundary concentration induced by an elastic stress confirms the existence of grain boundary anslastic relaxations experimentally and most undoubtedly. Based on this a micro-mechanism of grain-boundary anslastic relaxations was suggested. The equilibrium equations and kinetic equations were established to analytically describe the process of grain boundary anslastic relaxations. A new study direction in the field of the mechanical properties of materials is proposed. Some physical parameters which can not be obtained experimentally before can be gained by simulation of experimental data with the equations of grain boundary anslastic relaxations. The simulations may reveal some new physical concepts, such as the mechanism of complex diffusion for grain boundary segregation or depletion and the effects of stress on the diffusivity of complexes and solute atoms. A lot of engineering practices have shown that the property degradation, embrittlement and delaying fractures appear for metallic materials in service under an elastic stress. What is the micro-mechanism of these phenomena? It is shown in the present paper that due to the interactions of grain boundaries, vacancies and solute atoms, the grain boundaries can absorb or emit vacancies, producing non-equilibrium grain boundary segregation or depletion under an applied tensile or compressive elastic stress. This should be one of the main reasons why the property degradation of metallic materials often occur in service. 2002, 46(11): 759―763 4

Xu T. Creating and destroying vacancies in solids and non-equilib-

5

Xu T. Grain-boundary anelastic relaxation and non-equilibrium dilu-

rium grain-boundary segregation. Phil Mag, 2003, 83(7): 889―899

ICIFUAS-5. Berlin: Springer-Verlag, 1975. 392―400 3

tion induced by compressive stress and its kinetic simulation. Phil

Xu T. Kinetics of non-equilibrium grain boundary segregation induced by applied tensile stress and its computer simulation. Scr Mater,

1686

Mag, 2007, 87(10): 1581―1599 6

Shinoda T, Nakamura T. The effects of applied stress on the inter-

XU TingDong et al. Sci China Ser E-Tech Sci | Jun. 2009 | vol. 52 | no. 6 | 1679-1687

granular phosphorus segregation in a chromium steel. Acta Metall, 1981, 29: 1631―1637 7

boundaries in metals. Phys Rev, 1947, 71: 533―540 19

Ke T S. Stress relaxation across grain boundaries in metals. Phys Rev,

20

Xu T, Zheng L. The elastic modulus in the grain-boundary region of

21

Smallman R E. Modern Physical Metallurgy. London: Butterworth &

22

Ning C, Yu Z. Anomalously fast diffusion mechanism of solute atoms

23

Simonen E P, Bruemmer S M. Radiation-induced segregation: A mi-

Misra R D K. Issues concerning the effects of applied tensile stress on intergranular segregation in a low alloy steel. Acta Mater, 1996, 44(3):

1947, 72: 41―46

885―890 8

polycrystalline materials. Philo Mag Lett, 2004, 84(4): 225―233

Lee J R, Chiang Y M. Pressure-thermodynamic study of Bi segregation at ZnO grain boundaries. Mater Sci Forum, 1996, 207-209(1): 129―133

9

Co (Publishers) Limited, 1983. 258―259

Hondros E D, Seah M P. Interfacial and surface microchemistry. In: Cahn R W, Haasen P, eds. Physical Metallurgy. 4th revised and

in Pb. J Mater Sci Lett, 1995, 14(8): 557―560

enlarged ed. New York: Elsevier Science Publication, 1996. 894―895 10

crochemical gauge to quantify fundamental defect parameters. Mater

Xu T, Cheng B. Kinetics of non-equilibrium grain boundary segregation. Prog Mater Sci, 2004, 49(2): 109―208

11

Xu T. A model for intergranular segregation/dilution induced by ap-

12

Gleiter H. On the structure of grain boundaries in metals. Mater Sci

Res Soc Symp Proc, 1995, 373(1): 95―100 24

vacancy-induced structural transformation in Si grain boundaries.

plied stress. J Mater Sci, 2000, 35(22): 5621―5628

13

Mater Sci Forum, 1999, 294-296: 161―164 25

local elastic constants at grain boundaries by means of atomistic

Herr U, Jing J, Birringer R, et al. Investigation of nanocrystalline iron

simulation. J Appl Phys, 1990, 67(5): 2370―2375 26

472―475

28

Faulkner R G. Non-equilibrium grain-boundary segregation in aus-

29

Sebel G. Diffusion of S and P in solid Fe. Mem Sci Rev Met, 1964, 61:

30

Zhang Z L, Lin Q Y, Yu Z S. Non-equilibrium intergranular segregation

Ssegregation. Acta Metall, 1989, 37(9): 2499―2506 tenitic alloys. J Mater Sci, 1981, 16(2): 373―383

Gleiter H, Hornbogen E, Baro G. The mechanism of grain boundary glide. Acta Metall, 1968, 16: 1053―1060

18

Xu T, Song S. A kinetic model of non-equilibrium grain boundary

Balluffi R W, Chan J W. Mechanism for diffusion-induced grainboundary migration. Acta Metall, 1981, 29: 493―500

17

nealing. Acta Mater, 1999, 47(3): 893―903 27

Hort E. Diploma Thesis. Saarbrücken: Universitaet des Saarlandes, 1986. 1―55

16

Kameda J, Bloomer T E. Kinetics of grain-boundary segregation and desegregation of sulfur and phosphorus during post-irradiation an-

McLean D. Grain Boundaries in Metals. Oxford: Oxford University Press, 1957. 42―100

15

Kluge M D, Wolf D, Lutsko J F, et al. Formalism for the calculation of

Eng, 1982, 52(1): 91―93 materials by Mössbauer spectroscopy. Appl Phys Lett, 1987, 50(8): 14

Chisholm M F, Maiti A, Pennycook S J, et al. Vacancy formation and

Ke T S. Experimental evidence of the viscous behavior of grain

413―436 in ultra low carbon steel. Mater Sci Technol, 2000, 16(3): 305―308

XU TingDong et al. Sci China Ser E-Tech Sci | Jun. 2009 | vol. 52 | no. 6 | 1679-1687

1687