Free energies of vacancy complex formation in fcc metals with different characters of interatomic interaction forces

Journal of Nuclear North-Holland

Materials

journal of nuclear materials

186 (1992) 277-282

Free energies of vacancy complex formation in fee metals with different characters of interatomic interaction forces V.G. Chudinov and V.M. Dyadin Physical Technical Institute of the Ural Branch of the USSR Academy of Sciences, ul. Kirora 132, Izheusk, 426001, USSR Received

15 May 1990; accepted

6 August

1991

Free energies (FEs) of mono, di-, tri- and tetravacancies in face centered cubic lattices (fee), both with a small (Cu) and a large (Al) stacking fault energy (SFE), are calculated. It is shown that relative values of FE in these defects essentially differ. Particular differences are observed in the tetravacancy. For the case of a small SFE at low temperatures the most preferable is three-dimensional vacancy aggregation (tetrahedron); and for the case of high temperatures it is the dendrite. For the case of a large SFE the most advantageous is plane vacancy aggregation in the form of a rhombus in the plane of type (ill), which is in fact a Frank dislocation loop with the Burgers vector of (a, /3)( 111) type. Lower swelling under radiation of metals having a fee structure with a large SFE (Ni and Al) rather than a small one (Cu, Ag, Au, stainless steel. etc.) is associated with this difference. A mechanism of swelling suppression is discussed in materials with a small SFE due to preliminary cold deformation

1. Introduction The main problem that we encounter in applying structural materials in nuclear and thermonuclear reactors is their dimensional changes as a result of the irradiation environment (swelling, creep and radiation growth). Swelling is connected with void formation. The values of the rate of swelling for pure fee metals under identical conditions and at the same homologous temperatures (T/T,, T, is the melting temperature) can vary by orders of magnitude. The present representations based on the use of the chemical reaction rate theory cannot explain these differences, as the main constants describing the properties of defects and sinks under the above conditions are close to each other. The present experimental and theoretical knowledge testifies that there is a homogeneous character of void nucleation from vacancies. In any case nucleation begins at temperatures of 0.2T,, to 0.3T,,, at which vacancies become mobile. Besides, swelling is also observed in the cases when merely isolated Frenkel pairs are formed (for example, under radiation with * 1 MeV electrons). At present the theory describes well enough the evolution of voids or their aggregations under different conditions, but cannot explain the formation mecha0022-3115/92/$05.00

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nism of a nucleus they develop from. A nucleus of the void is usually considered to consist of less that 20-30 vacancies. The probability of nucleus formation due to fluctuations even in radiation fields of the highest intensity is negligible. In ref. [l] we obtained a possible mechanism of nucleus formation in copper by use of the molecular dynamics technique (MDT). In contrast with the well-known scheme of void formation (derived from the simplest model of broken bonds in which dilatation around defects is not taken into account) a vacancy-a divacancy-a closed trivacancy-a tetrahedron from vacancies, etc.-a void, we observed the following scheme: a vacancy-a divacancy-a trivacancy with 120 o and 180”-a dendrite-a void of a critical size. This scheme can be qualitatively understood by taking the entropy contribution to FE only into account, since its value for the aggregations of the given number of vacancies differs substantially, whether it is a void, a dislocation or a dendrite. In this article the quantitative estimations are given. Electron microscopy investigations permit one to determine a number of interesting regularities. Vacancy aggregations both in the form of voids and dislocation loops were found in fee metals [2-41. The latter could be formed in the following ways: 1) heterogeneous (in cascade regions (CR) under radiation with

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fast neutrons and heavy ions) and 2) homogeneous from isolated mobite vacancies. The possibility of the heterogeneous mode of loop formation was shown earlier [5]. The vacancy dislocation loop forms in some stages: at the cascade stage a depleted zone forms in the CR centre and a zone enriched with interstitials forms on the periphery. At the thermal stage, because of enormous temperature gradients, vacancies move towards the centrc of the CR, whiIe intcrstitials move towards the periphery. As a result, at sufficientIy large CRs (subsequentiy, large lifetimes of the thermal spike, too> an irregularly-shaped void is formed. It, in turn, can be transformed into a single vacancy type dislocation loop due to heat pressure. The homogeneous mechanism realization remains unclear, both from the point of view of the mechanism examined above and from the point of view of the preferential interstitial absorption phenomenon by edge dislocations [41. A three-dimensional aggregation (beginning from the simplest, namely, a tetravacancy) is ~ncr~~ticaI~y mure favourable than a plane one, since at adding each subsequent vacancy to the aggregate a smaller number of bonds (for a tetravacancy 9 and 10, respectively) is necessary to be truncated. Therefore, to understand this problem it is necessary to estimate the consriburion of the dilatation component and entropy to FE. No systematic investigations on the problem are avaitable. A more thorough analysis shows that in FCC metals the homogeneous mechanism of dislocation iloop nudeation is most probably realized in Al, Ni and their alloys with significant Al or Ni content uuder irradiation with electrons, the energy being 1 MeV [6-81. On the other hand, it is only these among fee metals that possess the maximum SFE. Earlier, the decrease of swelling with the SFE increase was emphasized in ref, PI. The present paper artide devoted to the investigation of the possibility of homogeneous nucleation of vacancy aggregation in fee metals on the basis of the FE thermodynamj~ analysis for the metals with large and small SFEs,

2. Formulation nique

of the problem and cakxdafkm

or due to self-clustering. In the case of a certain vacancy excess the uniting of vacancies into voids is possible in the presence of preferential sinks or if interstitials can unite into any type of aggregate ias a rule, into dislocation loops). Formation of dislocation loops is known to cease at comparatively low temperatures, N 0.4T, [lo]. Preferential interstitial sinks are usualfy edge dislocations, which are always contained in a material. The following questions is to be answered: what is the configuration of defect aggregations possessing a low FE at the given temperature? The FE value of the vacancy formation was calculated in the Mowing way [II]: F = (U,, - U&,) - TfS, + S,),

(0

where U, is the energy of broken bonds, Udi, is the dilatation energy connected with the atomic displacement around the defects; S, is the configuration entropy; and S, is the vibration entropy. The values (U,, - U&j were calculated by the program MOLDYN [IZ]. S, was calculated in the Einstein approximation according to the technique proposed in ref. [Ill. The authors proceeded from the approximate catculation according to the formula

where zjpj and pri are the frequencies of independent harmonic vibrations of perfect and nonperfect crystals, respectively. The product of squares of frequencies of atomic vibrations is equal to the matrix determinant of force constants. This fact has been used for the numerical calculations. Equilibrium atomic arrangements have been applied for the present calculations. The general tendency is that S, increases when the atomic displacements near the vacancies increase, that is when ihe dilatation entropy contribution increases. ‘ft is not difficult to show the values of the configuration entropy for various types of defects at constant conce.ntration of vacancies, proceed& from the wellknown technique Ill], if 1 B c z+ co:

tech-

It is known that vacancies and interstitials in equal. numbers form under the influence of fast neutrons and charged particles. If the defects are mobile, the system has a tendency to a low FE. Their number decreases either due to mutual annihilation, absorption by sinks

where c is the vacant number of independent from n vacancies (Z = 6 Z = 8 for the trivacancies

site concentration, Z is the ways of aggregate formation for the divacancies with n = 2; with closed bonds n = 3, etc.)

WI= Contributions to the entropy from electrons and static displacements can be neglected because of their

% G. Chudinoo, KM. Dyadin / Free energy of 1:acancy complex formation

279

Table 1 Calculated energy and entropy values for various vacancy aggregates

1. 2. 3. 4. 5. _ 6. 7. 8. 9. 10. 11. 12. 13.

14.

Aggregate type

n

z

vacancy divacancy I divacancy II divacancy III divacancy IV divacancy (II-IV) trivacancy 60 o trivacancy 90 o trivacancy 120 o trivacancy iS0 3 trivacancy (90-180 * 1 dendrites tetrahedron rhombus

1 2 2 2 2 2 3 3 3 3 3 4 4 4

1 6 3 12 6 21 8 12 24 6 42 461 2 12

small values *. Great difficulties arise when calculating S, for the defect aggregation. The problem is that conventional calculational techniques proceed from the broken bond model in which interaction of only nearest neighbours takes place. in the present case the defects interact with the more distant neighbours (in aiuminium up to the fourth). Therefore, it is impossible to consider these defects independently and they should be regarded as complexes, at least, those that have the binding energy B > k,T. The number of ways the complex formation increases is extremely large and can be calculated only in the case of divacancies. Equilibrium positions of atoms around the defects were calculated by MDT, where the correct reproduction of the compressibility value has the most significance during the calculations. We shall dwell on this question in detail. Compressibility in metals is almost completely determined by compressibility of the electronic subsystem [14]. The contribution from the crystal lattice for the metals located in the middle of the

* The estimations concerning the electronic contribution to the entropy were made in ref. [13]. The contribution from the static relaxation is equal [13] to 7 1-n where Y is Poisson’s coefficient; K is the modulus of compression; cx is the volume expansion coefficient; and AY is the change of the volume (caused by the defect formation). In the examined cases S, is approximately two orders of magnitude less than S, and 5,.

SC 5.61 3.7 3.35 4.04 3.7 4.32 2.56 2.7 2.93 2.47 3.12 2.93 I .57 2.02

Al

cu E/E,

S”

E/E,

S”

1

1.73 1.61

1 0.936 0.981 0.992 0.981 0.973 0.877 0.895 0.903 0.913 0.904 0.887 0.871 0.825

2.379 1.721 2.888 2.311 2.331 2.163 2.146 1.866 1.751 1.639 1.752

0.92 _ 0.83 0.886 0.886 0.886 0.886 0.875 0.75 0.795

1.48 1.57 1.58 1.57 1.57

1.55 1.34 1.41

1.744

1.844 2.186

Periodic Table is N 100 times less. The MDT cannot simulate the electronic subsystem, as the classic rather than quantum equations of motion are used. Thcrefore, applying the Cauchy pressure to the external borders will lead to significant atomic displacements (not observed experimentally), if the pair interaction potential (PIP) is calculated according to any electronic theory (for example, the pseudopotential theory 1153). Thus, we used the empirical potentia1 for aiuminium from the ref. 1161.It was adjusted to the experimental SFE (1.7 X 10m5 J/cm’), the compressibility modulus, the formation and migration energies of the point defects, the elastic constants, etc. The PIP was truncated between the fourth and fifth neighbours. The calculation according to the model of broken bonds for Cu corresponded to the extreme case of SFE N 0 J/cm’. The situation is not sufficiently adequate to the real one GFE in Cu is _ 50 x IO-’ J/cm’), but we applied the above PIP as this afforded us the opportunity to determine distinctively the difference, both in the mechanism of vacancy aggregation and the SFE influence upon the aggregates. The above mentioned PIP for Cu was used in ref. [I] during the MDT simulation. In MDT, it was impossible to reproduce the formation energy of the defect complex, as the electronic subsystem [ll] also contributed to this energy. This complication can be avoided if one is limited to the case I/= const. (The change of energy of the electronic subsystem is determined only by that of the volume.) The MDT calculations have been carried out by use of a relaxation technique on a crystallite consisting of

V. G. Chudinor,,

2x0

V. M. Dyadin

/ Free energy of Lucancy complex formation

Fig. 1. Free energy (vertical axis) dependence of different vacancy complexes versus temperature in Cu according to the model of broken bonds (normalized to the energy of single vacancy formation).

1724 atoms. The form of the crystallite was close to a spherical one. Approximately 500 atoms of the external layer were fixed. The calculations have a qualitative character and we choose to discuss the relative values only. Energy minimization was carried out with the help of a dynamical procedure, according to which atoms were permitted to relax easily after a defect had been produced. By their achieving the maximum kinetic energy, the rates went to zero. The atoms relaxed, repeatedly reaching a new maximum being lesser in value, and the rates went repeatedly to zero.

3. Calculation

results and discussions

In table 1 the energy the vacancy characteristics

0

200

and entropy values and also for the different aggregates

400

600

are shown. The calculation of the configuration entropy has been carried out in the approximation of the broken bond model. Figs. 1 and 2 present the dependence of the FE of a vacancy being in different aggregations on temperature with the concentration c = 0.01. The intersection with the ordinate is determined by the defect energy in the aggregation at T = 0 K, and the slope by the complete entropy. The intersecting points of the straight lines correspond to the temperature for changing one type of aggregate into another if, of course, there is no need to overcome an energetic barrier from any aggregate with a larger energy *. Numerically, the temperature is defined by the difference of the entropy and internal energy of the defect in various aggregations

where m, n are the number of defects in the aggregation. It should be noted that even slight changes in the absolute values (Um - U,) and (S,,, - S,l) can result in significant changes of T,,. In our estimations an error in T,, may reach 100%. Because of this the direct comparison of the experimental and the computational results is rather difficult. It is obvious that large aggrcgations during their growth from smaller sizes must pass through the small size aggregate formations with the most probability through those possessing a smaller FE at the given temperature. If any aggregate cannot exist because of annealing for isolated vacancies, then aggregates of a greater number of vacancies also do not form. The results connected with the tetravacancies arouse the most interest. In media with a small SFE the most energetically favoured is three-dimensional aggregation in the form of a tetrahedron, and in materials with a large SFE it is plane aggregation in the form of a rhombus. The latter, in principle, is a Frank dislocation loop of a minimum size. The main reason for this is associated with the following: in the second case the dilatation energy essentially contributes to the energy of defect formation, especially in the planar aggregates (both a trivacancy with 60 a and a rhomb with 60 ’ in the (111) plane). This is connected with the fact that the atom arranged over the plane of defects is displaced so that a tetrahedron forms. The atom, in turn,

TIK J

Fig 2. Free energy (vertical axis) dependence of different vacancy complexes versus temperature normalized to a vacancy in Al (normalized to the energy of single vacancy formation).

For example, upon transformation of a void into a dislocation it is necessary to pass the stage of a plane disk, the energy of which is larger than that of a void and a dislocation.

KG. Chudinoc, I/M. Dyadin / Free energy of r:acancy complex formation

moves to the centre of that tetrahedron. Upon forming a rhombus there are two such tetrahedra rotated 180 ‘. The analysis of the dependences presented in figs. 1 and 2 permits us to predict qualitatively the most obvious mode of vacancy aggregate formation. So, for example, in Al at T= 300 K (see fig. 2) the following mode must be realized: a vacancy-a divacancy-a trivacancy with closed bonds-a nucleus of a dislocation loop. At T = 400 K the probability of vacancy aggregation formation will begin to decrease, since the divacancy formation from vacancies becomes disadvantageous and may take place merely with overcoming the barrier. A different scheme of formation must be observed at the same temperature for the case presented in fig. 1: a vacancy-a trivacancy with closed bonds-a tetrahedron. The latter is a nucleus of a void under these conditions. At present, this is a generally adopted scheme. It denotes that in materials with a small SFE voids form preferentially and they must swell but in materials with a large SFE dislocation loops form and swelling must be lower. It is not difficult to imagine that at different T and for different relationships between the formation energy and the entropy, various other schemes also may be realized, for example, the scheme observed in ref. [l]: a vacancy-a divacancy-a trivacancy with disconnected bonds-a dendrite-a void of a critical size. After emerging a three-dimensional or plane nucleus, they can grow only into a void or a dislocation loop, respectively, even if the other aggregations are more advantageous at later stages of growth. This is caused by the fact that at the transformation it is necessary to overcome a significant energetic barrier related to the form change. Only a plane disk between the three-dimensional and plane dislocations is possible. According to the calculations [2] for aggregates consisting of 100 vacancies in copper, this barrier is equal to 24 eV and for aggregates consisting of four vacancies it equals 1 eV. It should be marked, however, that along with the most probable formation mechanism of a nucleus, other mechanisms may be realized but with less probability. It is not surprising, therefore, that in some materials both three-dimensional and plane aggregations are simultaneously observed. Such an approach may also be applied to investigate impurity formation, but this aspect is beyond the scope of the present paper article. The difference in nucleus formation mechanisms in materials with large and small SFEs has a decisive significance for understanding their behaviour under radiation. If the SFE is small, the materials will swell

281

unlimitedly after a void of a critical size or a tetrahedron forms. The decrease of swelling rate in this case is possible only due to a decrease of the vacancy supersaturation, that is, the delay of the nucleus formation. If, for example, a dislocation net of large density in the metal is created by cold deformation in a material, then at the initial stage it will play the role of a powerful sink for vacancies and interstitials. With increasing dose, the net density will naturally decrease. At a certain moment the nuclei of voids will begin to form and then swelling will occur. Stainless steel with a small SFE (_ 0.25 X 10-j J/cm*) is a typical example of such a behaviour [17]. If a nucleus of a vacancy dislocation loop is formed (a material with a large SFE), then under radiation it may grow only to a certain critical size. On the one hand clustering of vacancies is preferred for the loop and on the other hand, with increasing size, a preference to interstitial absorption must be apparent. As a result, the flows of vacancies and of interstitials are ultimately equivalent. That is the above vacancy loops are neutral sinks, where annihilation of vacancies and interstitials takes place. The critical size is determined both by the power of defect sources and of sinks in the system. It is obvious that with accumulation of radiation dose, their concentration will increase, but the rate of formation will decrease. Swelling of the material must also decrease and will stop upon reaching the equilibrium loop concentration. Such a mechanism is confirmed by the experimental results for pure Ni and Al (SFE according to different authors up to 1.50-2.00 X lo-” J/cm* [2,3]). At the initial stage of irradiation the linear dependence of dose is observed. The embryonic period is absent. In our opinion, it is also possible to relate small swelling levels of Al and high Ni alloys to the mechanism involved. Unfortunately, these alloys do not possess adequate properties on the other parameters: high-temperature strength, tendency to embrittlement, corrosion behaviour, adaptability to manufacture, etc. Within the framework of the examined mechanism, the temperature at which the formation of voids in materials with a small SFE stops is the decay temperature of any intermediate aggregation into isolated vacancies. In the case of a large SFE, the situation is more complicated. In principle, at certain ratios between U and S a critical temperature can exist at which the three-dimensional aggregations of four vacancies are energetically more advantageous than the plane aggregations. That is, vacancy void formation must start. A similar situation is experimentally observed in pure

282

KG. Chudinor: KM. Dyadin / Free energy of vacancy complex formation

molybdenum (a body-centered lattice, SFE - 7.5 X 10e5 J/cm*), where void formation takes place only at temperatures over 900 o C. At lower temperatures vacancy type dislocation loops with the Burgers vector (a,2)( 110) [18] are formed.

4. Conclusion The character of interatomic interaction forces affects decisively the defect structure of the materials under radiation. SFE, which is connected with the PIP, has an important significance. At small SFEs the material has a tendency to void formation and unlimited swelling. At large SFEs dislocation loops playing the role of neutral sinks are formed. This results in comparatively small swelling levels.

Acknowledgement The authors are indebted to E.I. Salamatov for helpful suggestions and help in calculations of values of the oscillated entropies.

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