A model is proposed for directly determining the volume of lattice vacancies by means of dilatometric measurements of the anisotropic irreversible length change which occurs during annealing of lattice vacancies at grain boundaries of shape-anisotropic crystallites. The model is tested using nanocrystalline Ni after the high-pressure torsion deformation which exhibits excess concentration of lattice vacancies and elongated crystallite shape. Different length changes upon annealing parallel and perpendicular to the elongation axis occur from which a vacancy volume can be derived.

Lattice vacancies are one of the most important defect types in solids as they serve in most cases as vehicle for self-diffusion and diffusion of substitutional foreign atoms.^{1} With a few exceptions, the lattice vacancy is the single defect type which is formed in thermodynamic equilibrium. In addition to the enthalpy and entropy of vacancy formation, the vacancy volume is also a fundamental key parameter, since due to lattice relaxation it is generally reduced in size compared to an atomic volume.

Measuring the vacancy volume is not a straightforward easy task.^{1} One way is via the pressure dependence of thermal vacancy formation which can be measured by positron annihilation.^{2,3} Another approach, the pressure dependence of self-diffusion yields the sum of vacancy formation and migration volumes.^{4}

A powerful tool for measuring the volume associated with lattice vacancies is dilatometry. An overview on the various measuring principles of dilatometry is for instance given by Sprengel *et al*.^{5} In the classical technique of differential dilatometry, the change of macroscopic length Δ*L*/*L* is measured simultaneously with the change of the lattice constant Δ*a*/*a* upon thermal formation of vacancies at high temperatures.^{1,6} However, since vacancy relaxation affects the macroscopic length and the lattice constant in the same way,^{7} the vacancy concentration *C _{V}* determined from the difference ($CV=1/3(\Delta L/L\u2212\Delta a/a)$) is independent from the vacancy relaxation.

Therefore, either Δ*L*/*L*- or Δ*a*/*a*-measurements have to be combined with another technique in order to determine the vacancy volume. Besides combining Δ*a*/*a*-measurements with diffuse x-ray scattering,^{8} the combination of Δ*L*/*L*-measurements and resistometry was applied to study the annealing out of quenched-in vacancies.^{6,9,10} Isotropic annealing of vacancies gives rise to a relative length contraction

where the relaxation factor,

characterizes the relaxation of the vacancy volume,

(Ω: atomic volume). In the pioneering studies,^{6,9,10} the vacancy relaxation *r* could be indirectly deduced from a comparison of the relative length contraction and the decrease of the electrical resistivity making reasonable assumptions for the resistivity increase with vacancy concentration.

In this letter, we suggest a direct way to determine the vacancy relaxation *r* by dilatometry on an absolute scale avoiding the indirect way of comparison with a technique, such as resistometry, which is sensitive to relative concentrations only. In contrast to the situation described above, the presented approach is concerned with the *anisotropic* annealing out of relaxed lattice vacancies in excess concentration. In this case, the annealing-induced relative length change depends on the measuring direction, enabling direct experimental access to the vacancy relaxation.

For a quantitative description, we consider the model structure as shown in Fig. 1, where anisotropic vacancy annealing occurs due to the elongated shape of the crystallites. The simple notion is adopted that vacancies exclusively anneal out at grain boundaries which are enclosing the crystallites along their elongation axis owing to the reduced diffusion length required for reaching the sink (see solid arrows in Fig. 1). Annealing out of vacancies perpendicular to the measuring direction (Fig. 1, left) causes a length increase in measuring direction since upon annealing *relaxed* vacancies are replaced by atoms. Owing to the used test example presented below, this measuring direction is called *tangential* in the following. The relative expansion in tangential measuring direction is given by

When the plane of preferential vacancy annealing lies parallel to the measuring direction (so-called axial direction, Fig. 1, right), the relative length change is given by the same elongation due to the lattice expansion, less the shrinkage due to the annealing of the relaxed vacancies

The factor 1/2 is due to the annealing of the vacancies in two directions perpendicular to the elongation axis (Fig. 1). Calculating the ratio *M* of the tangential and axial relative length change, the vacancy concentration *C _{V}* cancels out, which yields

From Eq. (6), the relaxation factor,

can be deduced. According to this model of anisotropic vacancy annealing, measurements of the length the change in both directions ($\Delta L/L0|tang.,\u2009\Delta L/L0|axial$) not only yield the absolute concentration *C _{V}* of vacancies but also the relaxation factor

*r*can be directly determined from the ratio

*M*(Eq. (7)).

An experimental test of the above presented model by means of dilatometric measurements requires samples (i) of macroscopic dimensions containing, (ii) shape-anisotropic elongated crystallites with preferred elongation axis, and (iii) an abundant concentration of lattice vacancies. To ensure that vacancy annealing predominantly occurs at grain boundaries enclosing the anisotropic crystallites, furthermore, (iv) the crystallite size has to be sufficiently small. A model system which fulfills these requirements is given with bulk nanocrystalline metals prepared by high-pressure torsion (HPT).^{11} HPT-Ni is particularly well-suited owing to the following aspects: Besides a macroscopic sample dimensions (condition (i)) and ultrafine crystallite sizes in the regime of 100 to 200 nm (condition (iv)), HPT-processing of Ni yields elongated crystallites with elongation axis lying in tangential direction of the HPT-disk (condition (ii), see Figs. 2 and 1)^{12–14}). Samples of HPT-Ni also contain a high abundant concentration of lattice vacancies of several 10^{−4} (condition (iii)).^{15} Moreover, lattice vacancies in Ni are stable at ambient temperature and become mobile at temperatures around 360 K^{16} below the recrystallization stage which starts at about 470 K.^{12} Therefore, the length change associated with the annealing out of vacancies is well separated from the length change which occurs due to the removal of grain boundaries upon crystallite growth. This allows a test of the model of anisotropic vacancy annealing.

The HPT-samples were prepared from high-purity nickel (99.99+%). For grain refinement by high-pressure torsion,^{11} 5 revolutions with a quasi-hydrostatic pressure of 2.2 GPa were applied yielding disks with a diameter of 30 mm and a height of ca. 8 mm (Fig. 2). Prism-shaped samples for dilatometry with a size of 3 mm × 3 mm × 7 mm were cut out from the disk at radii larger than 5 mm from the centre in order to guarantee a homogeneous microstructure independent of the radius and strain. The aspect ratio of the elongated crystallites is in the range of 1.8 as revealed by scanning electron microscopy. In the samples cut in tangential direction from the HPT-disk (Fig. 2), the elongation axis is parallel to the measuring direction of the length change (Fig. 1, left); in the axially cut samples the elongation axis is perpendicular to the measuring direction (Fig. 1, right). For measuring the length change upon vacancy annealing, a high-precision vertical difference dilatometer (Linseis L75VD500LT) was employed which allows simultaneous measurements of the test sample and of a reference sample. The relative length change Δ*L*/*L*_{0} due to vacancy annealing is obtained from the difference of the relative length changes in the HPT test sample and the reference sample. As reference served a well annealed, coarse-grained and defect-free nickel sample which was prepared from the same disk. The measurements were performed under pure argon (99.999%) gas flow.

Three measuring series were performed with in total 24 tangential samples and 18 axial samples cut from three identically HPT-processed disks. The results obtained from the dilatometer measurements are well reproducible. Fig. 3 exemplary shows the relative length change measured upon time-linear heating (3 K/s) in tangential (red full line) and axial directions (black dashed line). Two major regimes can be discerned. In regime (B) at higher temperatures starting at ca. 470 K, a contraction occurs in both measuring directions, which is due to grain growth. From the length contraction in this stage, values for the excess volume of the grain boundaries can be deduced as reported earlier.^{12} Relevant for the present study, however, is the regime (A) where lattice vacancies are mobile and anneal out. In tangential direction a length increase and in axial direction a length decrease occur in accordance with the above presented model of anisotropic annealing of vacancies. From the relative change in stage (A) in tangential ($\Delta L/L0|tang.$) and axial directions ($\Delta L/L0|axial$), length change ratios *M* (Eq. (6)) between −0.32 and −0.37 are derived for the three measuring series, corresponding to a value of the lattice vacancy relaxation *r* (Eqs. (2) and (7)) between 0.36 and 0.40, or a value for the vacancy volume *V _{V}* (Eq. (3)) between 0.60 Ω and 0.64 Ω (see Table I). The vacancy concentration

*C*as determined from these experiments is in the range of ca. (4–7) × 10

_{V}^{−4}(see Table I).

Disk . | Number of samples . | Measuring direction . | ΔL/L_{0} [10^{−4}]
. | C [10_{V}^{−4}]
. | M . | r . | V (Ω)
. _{V} |
---|---|---|---|---|---|---|---|

1 | 13 | Axial | −1.69 | ||||

3 | Tangential | +0.53 | 4.4 | −0.32 | 0.36 | 0.64 | |

2 | 1 | Axial | −1.75 | ||||

1 | Tangential | +0.62 | 4.8 | −0.35 | 0.39 | 0.61 | |

3 | 4 | Axial | −2.44 | ||||

20 | Tangential | +0.90 | 6.8 | −0.37 | 0.40 | 0.60 |

Disk . | Number of samples . | Measuring direction . | ΔL/L_{0} [10^{−4}]
. | C [10_{V}^{−4}]
. | M . | r . | V (Ω)
. _{V} |
---|---|---|---|---|---|---|---|

1 | 13 | Axial | −1.69 | ||||

3 | Tangential | +0.53 | 4.4 | −0.32 | 0.36 | 0.64 | |

2 | 1 | Axial | −1.75 | ||||

1 | Tangential | +0.62 | 4.8 | −0.35 | 0.39 | 0.61 | |

3 | 4 | Axial | −2.44 | ||||

20 | Tangential | +0.90 | 6.8 | −0.37 | 0.40 | 0.60 |

For testing the model, dilatometric length change measurements were also performed in a radial direction of the HPT disk (not shown). In this case, the elongation axis of crystallites is perpendicular to the dilatometric measuring direction as for the axial samples (Fig. 1, right). Similar to the axial orientation, a length contraction is observed in the regime of vacancy annealing for the radial orientation, supporting the model according to which vacancies anneal out perpendicular to the elongation axis (Eq. (5)).

The vacancy relaxation of 0.36–0.40 is in good agreement with most recent *ab initio* calculations of vacancy properties in Ni by Metsue *et al*.^{17} For the temperature range of 400 K of the present measurement, vacancy relaxation factors of 0.33 or 0.36 were obtained by local density approximation or generalized gradient approximation, respectively.^{17} A lower value of 0.22 was found earlier for Ni at 6 K using diffusive x-ray scattering.^{18} Garcia Ortega *et al*. computed a factor of 0.17 for Ni at 0 K by means of molecular dynamic simulations.^{19}

The model presented above can be extended to the case that a fraction *f _{tang}* of vacancies anneals out in the direction of the elongated axis of the crystallites, i.e., in measuring direction of the tangential sample (Fig. 1). In this case, the length increase in tangential direction (Eq. (4)) is reduced by

*f*and the fraction of vacancies annealing in axial direction in Eq. (5) reads

_{tang}C_{V}*f*= 1/2(1−

_{axial}*f*) instead of 1/2, yielding a relaxation factor

_{tang}*r*= 3(

*Mf*−

_{axial}*f*)/(

_{tang}*M*−1) instead of

*r*= 3

*M*/2(

*M*− 1) (Eq. (7)). The fractions

*f*,

_{tang}*f*sensitively depend on the vacancy distribution inside the crystallites and on the details of the diffusion kinetics, in particular, on the total fraction of vacancies which anneal out during linear heating prior to recrystallization. A simple random walk test for the given aspect ratio shows that the fraction

_{axial}*f*rapidly decreases under the reasonable assumption that a layer within the crystallites next to the grain-boundary sinks is initially free of vacancies. The relaxation factor

_{tang}*r*given above, neglecting the fraction

*f*, therefore, has to be considered as a lower limit.

_{tang}An assessment has to be given whether the observed anisotropic length change (Fig. 3) may be caused by other effects than anisotropic vacancy annealing. Residual stress may be ruled out as major source of the observed length change anisotropy. Preliminary residual stress analysis by x-ray diffraction performed on the dilatometer samples yield no evidence of macroscopic residual stress.^{20} Moreover, the length change anisotropy is not affected by uniaxial tensile straining which for the purpose of testing has been applied after HPT-processing in order to relieve potential residual strain anisotropies. The observed length change anisotropy is also considered as evidence that the grain boundaries act as major vacancy sinks rather than dislocations. Dislocations should exhibit a preferred orientation of the Burgers vector in the plane of the HPT-disk. The observed sign of the ratio *M* (Eq. (6)) appears to be incompatible with a climbing of such dislocations by vacancy annealing.

This supports our conclusion that the observed length change anisotropy arises from the anisotropic vacancy annealing rather than from any of the above mentioned other potential sources. Nevertheless, these other sources in the structurally complex HPT-material may affect the length change to a certain extent. Therefore, the test system of HPT-Ni should in the first instant be considered as a proof of principle for our model to measure vacancy volumes, rather than a way to deduce high-precision values for *V _{V}.* To exclude any other effect besides ansiotropic vacancy annealing, dilatometric studies after quenching of thermal vacancies in materials with ultrafine, shape-anisotropic crystallites are desirable.

This work was financially supported by the Austrian Science Fund (FWF): P25628-N20.