Modulated dilatometry as a tool for simultaneous study of vacancy formation and migration

Roland Würschum; Robert Weitenhüller; Robert Enzinger; Wolfgang Sprengel

doi:10.1515/ijmr-2022-0022

Article Publicly Available

Modulated dilatometry as a tool for simultaneous study of vacancy formation and migration

Roland Würschum , Robert Weitenhüller , Robert Enzinger and Wolfgang Sprengel

Published/Copyright: July 21, 2022

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From the journal International Journal of Materials Research Volume 113 Issue 8

Abstract

A model is presented to derive both vacancy formation and migration characteristics from length change measurements upon modulated time-linear heating. The length variation with linear heating yields access to the equilibrium concentration of thermal vacancies. The modulation amplitude and the phase shift between modulated temperature and length change is determined by the ratio of equilibration rate and modulation frequency which yields access to the vacancy migration characteristics. The contribution from thermal lattice expansion is obtained from a reference measurement at high modulation frequencies. Compared to static isothermal equilibration measurements after temperature jumps, the processes are monitored under quasi-equilibrium conditions avoiding obstacles associated with fast temperature changes. Furthermore, in contrast to the static isothermal case where the equilibration rate is obtained from the time-exponential decay, its determination from the amplitude and phase shift of modulation offers higher precision. The method is suitable for materials with high thermal vacancy concentrations and low vacancy diffusivities, among which is the important class of intermetallic compounds with B2-structure.

Keywords: Kinetic model; Modulated dilatomtry; Thermal vacancy formation; Vacancy equilibration; Vacancy migration

1 Introduction

The principle of dilatometry, i.e., of measuring volume changes caused by physical or chemical processes, dates back to the nineteenth century [1]. Meanwhile, dilatometry has developed to a powerful toolbox with a number of measurement routines applicable to various phenomena in materials science. A major application field of dilatometry is the study of transformation and precipitation processes in alloys [2], [3], [4], [5], [6], [7]. Owing to recent progress in sensitivity and long-term stability by using a specially designed highly stable non-contact dilatometer [8], this application route could be extended to the most application-relevant class of Al alloys with low amounts of Mg and Si [9, 10] (for review see [11]).

Dilatometry is further successfully applied to study defect annealing phenomena in sub-microcrystalline metals after severe plastic deformation yielding access to grain boundary excess volume [12, 13]. In the same manner the concentration of deformation-induced excess vacancies became measurable by dilatometry for ultrafine-grained Ni [14], [15], [16]. With dilatometry as a function of orientation, even anisotropic behaviour of vacancy annealing, which arises from structural anisotropy in this material, could be measured allowing conclusions regarding the vacancy formation and relaxation volume [14].

The fundamental idea of detecting thermal vacancies in metals by macroscopic length change measurements goes back to Feder and Nowick [17]. They compared the length change obtained from the macroscopic measurement with that from X-ray diffraction in order to subtract the contribution from thermal lattice expansion for estimating the contribution from thermal vacancies to the overall macroscopic length change. Based on this principle, but measuring macroscopic length change and X-ray diffraction simultaneously on the same sample, Simmons and Balluffi [18] could successfully determine thermal vacancy concentrations, which was later further refined by Hehenkamp et al. [19].

For a high vacancy concentration and low vacancy diffusivity as observed, e.g., in various intermetallic compounds, both the vacancy formation and migration can be studied by isothermally monitoring the length change after a temperature jump [20, 21]. From the time dependence of the length change, the vacancy equilibration time constant, and from its amplitude the difference in thermal vacancy concentration upon temperature jump can be deduced. The inherent advantage of the method is that isothermal monitoring of the length change is not affected by thermal lattice expansion. Disadvantages, however, associated with the required temperature jump are potential defect losses during cooling and transient phemonena affecting the measuring accuracy right after sudden temperature change.

These disadvantages can be avoided by monitoring the vacancy processes by temperature-modulated time-linear heating, which is the concept proposed in the present work. Upon time-linear heating, the temperature-dependence of the thermal vacancy concentration can be monitored in thermal equilibrium, rather than upon step-like temperature change. The superimposed temperature modulation allows monitoring of the temperature-dependence of the vacancy equilibration under conditions near to equilibrium. As will become clear in the forthcoming, the modulation is also the essential key for determining the contribution from thermal lattice expansion without the necessity of another technique.

The modulation model presented in the following makes dilatometry a powerful tool for simultaneous study of vacancy formation and migration for a wide spectrum of alloys exhibiting high vacancy concentrations and low vacancy diffusivities. Although the model is elaborated for dilatometry, the concept is not restricted to length change measurements, but applicable to other defect-sensitive techniques as well.

Dilatometric measurement techniques and temperature modulated thermal analysis including their various applications are reviewed by James et al. [22] and by Kraftmakher [23]. The latter also discusses temperature modulated dilatometry, however, mainly describing various experimental setups for assessing instantaneous thermal expansion coefficients. Aspects of analysing kinetic processes using temperature modulated dilatometry have so far only been addressed scarcely, e.g., in specific applications related to second order phase transitions [24], to processes in polymers [25], or to coating/substrate interaction [26]. To our knowledge there have been no applications up to now of temperature modulated dilatometry for the study of vacancy formation and migration processes as proposed in the present article.

2 Model

We consider vacancies in thermal equilibrium with the concentration C_V(T) as function of temperature T

(1) C V,eq ( T ) = exp S V F k B exp − H V F k B T

where k_B denotes the Boltzmann constant, and H V F and S V F the vacancy formation enthalpy and entropy, respectively. Any deviation ΔC_V = C_V − C_V,eq from the equilibrium concentration C_V,eq = C_V,eq(T₀) at a given temperature T₀ gives rise to equilibration, which can be described by the rate equation

(2) d C V ( t ) d t + C V ( t ) − C V,eq τ = 0

with the equilibration rate [21]

(3) 1 τ = 1 τ 0 exp − H V M k B T

where H V M denotes the vacancy migration enthalpy and τ 0 − 1 a prefactor characterizing the density of vacancy sources or sinks. The solution of Equation (2) for an initial deviation Δ C V ( t = 0 ) = Δ C V ini = C V ini − C V,eq with respect to the equilibrium concentration C_V,eq(T₀) reads

(4) C V ( t ) = C V,eq + C V ini − C V,eq exp − t τ

In the following, at first equilibration of vacancy concentration associated with modulation with respect to a constant temperature will be considered (Subsection 2.1) and subsequently the case of modulation upon time-linear heating (Subsection 2.2).

2.1 Equilibration of vacancy concentration associated with modulation

In order to describe vacancy equilibration associated with temperature modulation, we consider a periodic small oscillation of the temperature with amplitude Δ T ̂ and angular frequency ω

(5) T ( t ) = T 0 + Δ T ̂ sin ( ω t )

with respect to a constant temperature T₀. Associated with the small temperature variation is a small variation in the equilibrium vacancy concentration C_V,eq(T(t)) with respect to C_V,eq(T₀):

(6) C V,eq ( T ) = C V,eq ( T 0 ) + d C V,eq d T T 0 ( T − T 0 )

By means of the temperature-derivative of the vacancy concentration (Equation (1)).

(7) d C V,eq d T = C V,eq H V F k B T 2

C_V,eq(T) (Equation (6)) can be written for T(t) according to Equation (5) as

(8) C V,eq ( T ( t ) ) = C V,eq ( T 0 ) + C V,eq ( T 0 ) H V F k B T 0 Δ T ̂ T 0 sin ( ω t )

For a small temperature variation we can further neglect a variation in the equilibration rate (Equation (3)) within the modulation amplitude Δ T ̂ . This is justified for

(9) d 1 τ d T Δ T ̂ ≪ 1 τ i . e . H V M k B T Δ T ̂ T ≪ 1

The time behaviour of the vacancy concentration C_V(t) associated with equilibration in the wake of the periodic modulation C_V,eq(T(t)) is determined by the differential equation (Equation (2)) after insertion of Equation (8):

(10) d C V ( t ) d t + 1 τ C V ( t ) − C V,eq ( T 0 ) − C V,eq ( T 0 ) H V F k B T 0 Δ T ̂ T 0 sin ( ω t ) = 0

or with ΔC_V(t) = C_V(t) − C_V,eq(T₀) and dC_V,eq(T₀)/dt = 0:

(11) d Δ C V ( t ) d t + Δ C V ( t ) τ = 1 τ C V,eq ( T 0 ) H V F k B T 0 Δ T ̂ T 0 sin ( ω t )

Equation (11) represents an inhomogenous differential rate equation where the right-hand side defines the driving force for concentration changes. One should emphasize that ΔC_V(t) denotes the deviation from the unmodulated value C_V,eq(T₀) rather than from C_V,eq(T). The inhomogeneous differential equation (Equation (11)) can be solved by the Ansatz ΔC_V(t) = b sin(ωt) + c cos(ωt), leading to the solution:

(12) Δ C V ( t ) = C V,eq ( T 0 ) H V F k B T 0 1 1 + ω 2 τ 2 Δ T ̂ T 0 sin ( ω t + φ )

with the phase shift

(13) φ = − arctan ( ω τ )

An essential characteristic of the solution is the phase shift between the modulation of the temperature (Equation (5)) and of the vacancy concentration (Equation (12)). This retardation is caused by the equilibration which also damps the modulation amplitude. For high equilibration rates τ⁻¹, the phase shift and the damping of the amplitude vanishes.

The analogue of the inhomogeneous differential equation (Equation (11)) in electronic circuit engineering is the series connection of a resistance R and a capacitance C with the time constant τ = R C , where the vacancy concentration C_V(t) corresponds to the charge Q(t) and where the driving force on the right-hand side is given by C Δ U ̂ sin ( ω t ) with the modulation amplitude Δ U ̂ of the voltage with respect to a voltage U₀. The solution of this electronic analogue fully corresponds to that given by Equations (12) and (13).^[1]

2.2 Length change upon modulated time-linear heating

Based on the solution for vacancy equilibration upon modulation with respect to a constant temperature T₀ (Equation (12)), in the following the length change upon modulation during time-linear heating will be considered.

The length change associated with time-linear heating (LH) with constant heating rate R starting from temperature T_start:

(14) T ( t ) = T start + R t

is composed of two parts, i.e., lattice expansion (Lat) and change of thermal vacancy concentration (Vac) (Equation (1)). The latter reads.

(15) Δ L L 0 LH,Vac = 1 3 ( 1 − r ) C V,eq ( T ) − C V,eq ( T start )

where r denotes the vacancy relaxation. Here, the heating rate is considered sufficiently small compared to the vacancy equilibration rate to ensure that C_V,eq(T) is in equilibrium with respect to linear heating.

The length change due to lattice expansion reads

(16) Δ L L 0 LH,Lat = α 0 R t + 1 2 α 1 ( R t ) 2

assuming a linear temperature dependence of the thermal coefficient of lattice expansion

(17) α ( T ) = α 0 + α 1 T

where the cofficients α_0,1 refer to T_start. We note that the proper choice of the temperature dependence of α is irrelevant for the analysis method I of primary choice, presented below; the same is true for method III.

Next, the temperature modulation (mod) (Equation (5)) is superimposed to linear heating, which adds two further terms to the length change; that of lattice expansion and, the most relevant one for the present study, that arising from vacancies. The modulation part from lattice expansion reads

(18) Δ L L 0 Mod,Lat = α ( T ) Δ T ̂ sin ( ω t )

For the length change associated with the change ΔC_V(t) of vacancy concentration upon modulation, one can apply the solution above (Equation (12)), provided that the linear temperature increase within one period is small compared to the modulation amplitude, i.e.,

(19) R 2 π ω ≪ T ̂

This vacancy-associated modulated length change then reads:

(20) Δ L L 0 Mod,Vac = 1 3 ( 1 − r ) C V,eq ( T ) H V F k B T 1 1 + ω 2 τ 2 × Δ T ̂ T sin ( ω t + φ )

with T = T_start + Rt. The total relative length change upon modulated time-linear heating is given by the sum of the four contributions (Equations (16)–(20)):

(21) Δ L L 0 = Δ L L 0 LH,Lat + Δ L L 0 LH,Vac + Δ L L 0 Mod,Lat + Δ L L 0 Mod,Vac

This equation (Equation (21)) respresents the basis for deducing the characteristics of both thermal vacancy formation and equilibration from length measurements upon modulated time-linear heating, as will be pointed out in detail in the next section.

3 Analysis and discussion

3.1 Method I: high- and low-frequency measurement

In order to derive the vacancy characteristics from the measured length change (Equation (21)) upon modulated time-linear heating, the contribution from lattice expansion has to be subtracted. This is best done by means of a reference measurement at a sufficiently high frequency (HF), where the modulation is only determined by the lattice expansion, but not by vacancy equilibration, i.e., where the part ( Δ L / L 0 ) HF Mod,Vac (Equation (20)) is neglibly small. This allows determination of the lattice expansion coefficient α(T) from the HF-modulated length change ( Δ L / L 0 ) HF Mod,Lat (Equation (18)) and, in this way, subtraction of the lattice expansion contribution from the length change measured at low frequency, where modulation is both affected by thermal expansion and vacancy equilibration.

When the entire frequency-independent length change arising from thermal expansion, i.e., the sum of the linear heating Equation (16)) and the modulation part (Equation (18)), obtained from the HF measurement, is subtracted from a second length change ( Δ L / L 0 ) LF measured at low frequency (LF), the vacancy-associated length change arising from linear heating and modulation is obtained:

(22) Δ L L 0 I = Δ L L 0 LF − Δ L L 0 HF LH,Lat − Δ L L 0 HF Mod,Lat = Δ L L 0 LF LH,Vac + Δ L L 0 LF Mod,Vac

where the index I denotes the method I. The part of vacancy-associated length change from linear heating is

(23) Δ L L 0 LF LH,Vac = 1 3 ( 1 − r ) C V,eq ( T ) − C V,eq ( T start )

according to Equation (15) and that from modulation

(24) Δ L L 0 LF Mod,Vac = 1 3 ( 1 − r ) C V,eq ( T ) H V F k B T 1 ω 2 τ 2 + 1 × Δ T ̂ T sin ( ω t + φ )

according to Equation (20) with T = T_start + Rt.

From Equation (22) the vacancy characteristics can be deduced as follows. The linear heating part (Equation (23)) of ( Δ L / L 0 ) I directly yields the temperature variation of C_V,eq(T), from which H V F can be deduced as well as the temperature-independent prefactor containing the product of (1 − r) and exp S V F / k B . Using the data on vacancy formation from the linear part, from the temperature-dependent modulated part the characteristics on vacancy migration follow. Both the amplitude (Equation (24)) and the phase angle (Equation (13)) of the modulated part depend on H V M and τ₀, the amplitude in addition on H V F , S V F , r.

For considering the amplitude A ̂ I of the modulated part (Equation (24)), we introduce the dimensionless inverse temperature x and the dimensionless coefficients β and γ

(25) x = H V F k B T , β = H V M H V F , γ = ω τ 0

yielding

(26) A ̂ I = C x 2 ⁡ exp ( − x ) γ 2 ⁡ exp ( 2 β x ) + 1

with the temperature-independent dimensionless coefficient

(27) C = 1 3 ( 1 − r ) exp S V F k B k B T ̂ H V F

3.1.1 Analysis of Method I

The modulation amplitude A ̂ I according to Equation (26) is shown in Figure 1a in dependence of the dimensionless temperature. The amplitude increases with temperature as both the vacancy concentration and the vacancy equilibration rate are increasing. For a given temperature, A ̂ I increases with decreasing γ, e.g., with decreasing ω, since the vacancies can equilibrate to greater extent. Correspondingly, a given amplitude is attained at lower temperature with decreasing γ. Increasing β, e.g. H V M , has the reverse effect, i.e., the amplitude for a given temperature decreases or else a given amplitude is reached at higher temperature.

$Figure 1: Modulation amplitude A ̂ I,II ${\hat{A}}_{\text{I,II}}$ according to Equation (26) (part a, method I) and Equation (39) (part b, method II) in dependence of dimensionless temperature x − 1 = ( k B T ) / H V F ${x}^{-1}=({k}_{\text{B}}T)/{H}_{\text{V}}^{\text{F}}$ for dimensionless parameters C = 1, β = H V M / H V F = 1.5 $\beta ={H}_{\text{V}}^{\text{M}}/{H}_{\text{V}}^{\text{F}}=1.5$ and γ = ωτ0 = 6.25 × 10−8 (dotted line), γ = 2.5 × 10−8 (full line), γ = 2.5 × 10−9 (dashed line), as well as for β = 1.55 and γ = 2.5 × 10−8 (red dashed-dotted line).$

Figure 1:

Modulation amplitude A ̂ I,II according to Equation (26) (part a, method I) and Equation (39) (part b, method II) in dependence of dimensionless temperature x − 1 = ( k B T ) / H V F for dimensionless parameters C = 1, β = H V M / H V F = 1.5 and γ = ωτ₀ = 6.25 × 10⁻⁸ (dotted line), γ = 2.5 × 10⁻⁸ (full line), γ = 2.5 × 10⁻⁹ (dashed line), as well as for β = 1.55 and γ = 2.5 × 10⁻⁸ (red dashed-dotted line).

Next, we address the choice of appropriate modulation frequency and linear heating rate.

Appropriate modulation frequency

In order that the high-frequency (HF) modulation is not affected by vacancy equilibration, which is the condition for this method, the modulation amplitude of Equation (20) associated with vacancy equilibration, for the highest applied temperature, has to be smaller than the measurement uncertainty Δ(ΔL/L₀), i.e.,

(28) 1 3 ( 1 − r ) C V,eq ( T ) H V F k B T 1 1 + ω 2 τ 2 Δ T ̂ T < Δ Δ L L 0

which sets a lower limit for the high frequency. Correspondingly, for the measurements at low frequency, the opposite condition is required, i.e., the amplitude has to exceed the uncertainty for the lowest applied temperature.

Appropriate linear heating rate

In order to ensure that the vacancy state is equilibrated with respect to linear heating, the associated length change ( Δ L / L 0 ) LH,Vac ( Δ t = τ ) within the characteristic time constant τ of equilibration may not exceed the measurement uncertainty Δ(ΔL/L₀). The vacancy-associated relative length change upon time-linear heating ΔT = RΔt with a rate R in a time interval Δt can be obtained from the temperature-derivative of Equation (15) multiplied by ΔT, yielding:

(29) Δ L L 0 LH,Vac ( Δ t ) = 1 3 ( 1 − r ) C V,eq ( T ) H V F k B T R Δ t T

This sets an upper limit for the heating rate R, i.e., ( Δ L / L 0 ) LH,Vac ( Δ t = τ ) ≃ Δ ( Δ L / L 0 ) . The upper limit must be fullfilled for the lowest temperature, where τ is highest. The condition ( Δ L / L 0 ) LH,Vac ( Δ t = τ ) < Δ ( Δ L / L 0 ) is then fullfilled throughout the subsequent course of linear heating.

A lower heating rate R than that defined in this way, on the other hand, yield no gain in information, since the associated relative length change is below the detection limit, even in the low temperature regime where τ is high.

Case example

For the sake of exemplary illustration, we consider CuZn, which exhibits a low vacancy formation enthalpy in combination with a high vacancy migration enthalpy and which, therefore, should be suited for the proposed kind of measurement. Table 1 shows the temperature T and phase shift φ (Equation (13)) for given values of amplitude A of ( Δ L / L 0 ) Mod,Vac (Equation (20)) and frequencies ν of modulation based on characteristic values for vacancy formation and migration enthalpy (see [27]) and pre-exponential factor [28]. As expected, with increasing modulation frequency, the phase shift increases as well as the temperature for a given modulation amplitude. The higher the temperature, the higher is the modulation amplitude and the lower is the phase shift.

Table 1:

Temperature T and phase shift φ (Equation (13)) for given values of amplitude A ( Δ L / L 0 ) Mod,Vac (Equation (20)) and frequency ν = ω/(2π) of modulation. Parameters: Vacancy formation enthalpy H V F = 0.42 eV and vacancy migration enthalpy H V M = 1.02 eV for CuZn (see [27]), pre-exponential factor τ 0 − 1 = 1 0 8 s⁻¹ [28], vacancy formation entropy S V F = 2 k_B, vacancy relaxation r = 0, temperature modulation amplitude T ̂ = 10 K.

ντ ₀	2.5 × 10⁻¹⁰		10^–11		10^–12
ν (mHz)	25		1		0.1
A	T (K)	−φ (°)	T (K)	−φ (°)	T (K)	−φ (°)
10^–6	501	90.0	455	90.0	427	90.0
10^–5	542	90.0	488	89.9	459	89.8
10^–4	610	89.5	602	79.7	602	28.7
5 × 10⁻⁴	835	29.5	835	1.3	835	0.1

With the data set of the given example (see caption of Table 1), the minimum linear heating rate as outlined above can be derived. For R = 10⁻³ K s⁻¹ = 3.6 K h⁻¹ an amplitude ( Δ L / L 0 ) LH,Vac = 1 0 − 6 is obtained from Equation (29) for T = 480 K, i.e., this value R represents the lower limit, if 10⁻⁶ is considered as experimental uncertainty and 480 K the minimum temperature. This value of R well fullfills the condition (Equation (19))

(30) R 2 π ω = R 1 ν = 1 K ≪ T ̂ = 10 K

of the example for the frequency of ν = 1 mHz. Also the condition according to Equation (9) is fairly well fullfilled. For T = 500 K, a value of

(31) H V M k B T Δ T ̂ T = 0.47

is obtained and correspondingly lower values for higher temperatures.

3.2 Method II: measurement without high-frequency approach

When the HF-approach as used for method I is not taken into consideration, the length change associated with lattice expansion α cannot be obtained from the modulation. An alternative way in this case is to determine an expansion coefficient

(32) α LH ( T ) = d d T Δ L L 0 LH

from the length change part ( Δ L / L 0 ) LH of linear heating and to obtain a difference curve corresponding to Equation (22), but for the same frequency.

However, the linear heating part and, therefore, α_LH(T) is not simply given by the lattice expansion coefficient

(33) d d T Δ L L 0 LH,Lat = α ( T ) = α 0 + α 1 T

but is the sum of Equation (33) and the temperature-derivative ( d / d T ) ( Δ L / L 0 ) LH,Vac from the vacancy contribution (Equation (15)). Subtracting the length change α LH ( T ) ( R t + T ̂ sin ( ω t ) ) deduced in this way from the measured length change (ΔL/L₀) yields the difference curve for this method II:

(34) Δ L L 0 II = − 1 2 α 1 ( R t ) 2 + Δ L L 0 Mod, Vac − d d T Δ L L 0 LH, Vac Δ T ̂ sin ( ω t )

where the summand −1/2α₁(Rt)² arises from

(35) d d T Δ L L 0 LH,Lat R t = Δ L L 0 LH,Lat + 1 2 α 1 ( R t ) 2

For ( Δ L / L 0 ) II (Equation (34)) follows with Equation (15):

(36) Δ L L 0 II = − 1 2 α 1 ( R t ) 2 + 1 3 ( 1 − r ) C V,eq ( T ) H V F k B T Δ T ̂ T × 1 1 + ω 2 τ 2 sin ( ω t + φ ) − sin ( ω t )

and after trigonometric transformations

(37) Δ L L 0 II = − 1 2 α 1 ( R t ) 2 − 1 3 ( 1 − r ) C V,eq ( T ) H V F k B T × ω 2 τ 2 1 + ω 2 τ 2 Δ T ̂ T sin ( ω t + ϕ )

with the phase shift

(38) ϕ = arctan 1 ω τ

and with T = T_start + Rt.

Note that in this case of method II, the difference (Equation (34)) is obtained from one and the same heating curve, in contrast to method I where the difference of two curves measured for low and high frequency is taken. However, the result (Equation (37)) of method II also contains a contribution from the lattice expansion (−1/2α₁(Rt)²), in contrast to method I. Since this is a non-oscillating contribution it can easily be subtracted from the oscillating part and, therefore, will not be considered in the following.

Rewriting the amplitude A ̂ II of Equation (37) in dimensionless quantities (Equations (25) and (27)) as for method I (Equation (26)), yields

(39) A ̂ II = C x 2 ⁡ exp ( − x ) γ 2 ⁡ exp ( 2 β x ) γ 2 ⁡ exp ( 2 β x ) + 1

The temperature variation of the amplitude of Equation (37) exhibits an extremum which can be deduced from the implicit equation:

(40) T ext = 1 2 H V F k B − H V M k B 1 ω 2 τ 2 ( T ext ) + 1

For graphical solution, Equation (40) can be rewritten with the dimensionless quantities (Equation (25)) as follows:

(41) 1 β − 2 β x = 1 γ 2 ⁡ exp ( 2 β x ) + 1

From Equation (37) the vacancy characteristics ( H V F , S V F , r, H V M , and τ₀) can be deduced. As for method I, both the amplitude and the phase angle (Equation (38)) of the modulated part depend on H V M and τ₀, the amplitude in addition on H V F , S V F , r. However, in contrast to method I (Equation (23)), here the part from vacancy formation cannot be separately obtained from the linear heating part. Therefore, although Equation (37) for a given ω principally contains the entire information, in practice, measurements for at least two frequencies are necessary.

3.2.1 Analysis of Method II

The modulation amplitude according to Equation (39) is shown in Figure 1b in dependence of the dimensionless temperature. Striking differences compared to the amplitude of method I are the occurence of a maximum and the reversed variation with the parameters β and γ (compare Figure 1a and b). This is due to the different square-root term of A ̂ II compared to A ̂ I (Figure 2).

$Figure 2: Square root term of A ̂ I ${\hat{A}}_{\text{I}}$ according to Equation (26) ( 1 / ( γ 2 ⁡ exp ( 2 β x ) + 1 ) 1 / 2 ) $(1/{({\gamma }^{2}\mathrm{exp}(2\beta x)+1)}^{1/2})$ (dashed line), of A ̂ II ${\hat{A}}_{\text{II}}$ according to Equation (39) γ 2 ⁡ exp ( 2 β x ) / { γ 2 ⁡ exp ( 2 β x ) + 1 } 1 / 2 $\left({\left[{\gamma }^{2}\mathrm{exp}(2\beta x)/\left\{{\gamma }^{2}\mathrm{exp}(2\beta x)+1\left.\right)\right\}\right]}^{1/2}\right)$ (dotted line) and ratio A ̂ II / A ̂ I ( γ ⁡ exp ( β x ) ) ${\hat{A}}_{\text{II}}/{\hat{A}}_{\text{I}}\ (\gamma \mathrm{exp}(\beta x))$ (full line) in dependence of dimensionless temperature x − 1 = ( k B T ) / H V F ${x}^{-1}=({k}_{\text{B}}T)/{H}_{\text{V}}^{\text{F}}$ for dimensionless parameters β = H V M / H V F = 1.5 $\beta ={H}_{\text{V}}^{\text{M}}/{H}_{\text{V}}^{\text{F}}=1.5$ , γ = ωτ0 = 2.5 × 10−8, and C = 1.$

Figure 2:

Square root term of A ̂ I according to Equation (26) ( 1 / ( γ 2 ⁡ exp ( 2 β x ) + 1 ) 1 / 2 ) (dashed line), of A ̂ II according to Equation (39) γ 2 ⁡ exp ( 2 β x ) / { γ 2 ⁡ exp ( 2 β x ) + 1 } 1 / 2 (dotted line) and ratio A ̂ II / A ̂ I ( γ ⁡ exp ( β x ) ) (full line) in dependence of dimensionless temperature x − 1 = ( k B T ) / H V F for dimensionless parameters β = H V M / H V F = 1.5 , γ = ωτ₀ = 2.5 × 10⁻⁸, and C = 1.

Towards low temperatures A ̂ II decreases for the same reason as for method I, namely the decreasing vacancy concentration C_V. A maximum value arises for method II, since in that case the difference ( Δ L / L 0 ) II (Equation (37)) also goes to zero for high temperatures. This is the case for ωτ → 0, i.e., ω ≪ τ⁻¹, when the equilibration rate is high, i.e., when C_V is always in equilibrium, or either when ω is low. Correspondingly, A ̂ II decreases in this regime with increasing temperature (i.e., increasing equilibration rate) and, for given T, with decreasing γ (e.g., ω). The lower γ is, the lower is the temperature where the decrease of A ̂ II sets in with increasing T (see shift of maximum), which also becomes evident from Equation (40). Lowering of β, e.g., of H V M , causes a similar trend as a decrease of ω, due to the concomitant increase in the equilibration rate. Figure 2 underlines that the high-temperature decrease in A ̂ II arises from its square-root factor, which in contrast to method I, decreases with increasing T.

A further remarkable difference to method I is the behaviour for large values of ωτ, i.e., for ω ≫ τ⁻¹, when either ω is high or the equilibration occurs slowly. Here, for method I the amplitude (Equation (24)), which is the difference to the high-frequency part, goes to zero. For method II, the square-root term (ω²τ²)/(1 + ω²τ²) in Equation (37) approaches one and the phase angle ϕ (Equation (38)) zero. In that case only the second sine-function (−sin(ωt)) (Equation (36)) remains for ( Δ L / L 0 ) II . This is because α_LH(T), contains only the unmodulated contribution from vacancies. Therefore, the function −sin(ωt) in the difference, where α_LH (T) T(t) is subtracted (Equation (36)), takes into account the deviation of C_V from the unmodulated value upon modulation.

In summary, one may state that although method II yields access to the characteristics of vacancy formation and equilibration to the same extent as method I, the latter high-frequency approach, if applicable, should be preferred as the less laborious way.

3.3 Method III: Modulation after step-like temperature change

The application of the modulation technique is not restricted to time-linear heating, but may also be used to monitor isothermal vacancy equilibration after step-like temperature change.

The standard solution according to Equation (4) gives the isothermal length change as answer to an initial step-like change of the equilibrium vacancy concentration

(42) Δ C V ini = Δ C V ( t = 0 ) = C V,eq ( T ini ) − C V,eq ( T 0 )

that is caused by temperature jump from T_ini to T₀. The solution with temperature modulation T = T 0 + Δ T ̂ sin ( ω t ) after step-like temperature change is given by simply adding the inhomogeneous solution (Equation (12)), which yields

(43) Δ C V ( t ) = C V,eq ( T ini ) − C V,eq ( T 0 ) exp − t τ + C V ( T 0 ) H V F k B T 0 1 1 + ω 2 τ 2 Δ T ̂ T 0 sin ( ω t + φ )

with the corresponding length change

(44) Δ L L 0 = 1 3 ( 1 − r ) Δ C V ( t ) + α ( T 0 ) Δ T ̂ sin ( ω t )

where φ denotes the phase shift (Equation (13)) and α the thermal expansion coefficient of the lattice.

As for method I, the contribution from thermal lattice expansion can be separately obtained from a measurement at high frequency, where vacancy equilibration does not contribute to modulation. Compared to the static isothermal case (Equation (4)), in the modulation case (Equation (43)) the characteristic vacancy equilibration rate τ⁻¹ cannot just be derived from the exponential decay but also from the modulation amplitude and the phase shift which improves the accuracy.

4 Concluding remarks

The major advantages of the presented modulation technique compared to static isothermal measurements after temperature jumps can be summarized as follows:

Thermal vacancy formation and equilibration are monitored under quasi-equilibrium conditions avoiding obstacles associated with fast temperature changes.
Principally the entire vacancy characteristics are included in one (ΔL/L₀) (t)-curve measured upon a single modulated linear-heating sequence provided heating rate and modulation frequency are estimated appropriately in advance. The contribution from thermal lattice expansion can unambiguously be determined from a reference measurement at high modulation frequencies.
In contrast to the static isothermal case, the modulation yields access to the phase shift (Equation (13)) between (ΔL/L₀) (t) and T(t). This yields additionally most direct information on the equilibration time constant τ, which can be determined in this way more reliably compared to the static case where this is obtained from the time-exponential decay (Equation (4)). It should be mentioned, however, that after temperature jump the equilibration of a high excess concentration of vacancies can be monitored which, on the other hand, can be favourable compared to the present method in the range where the equilibrium vacancy concentration is still low.

Best suited to exploring the potentials of the proposed method appear to be intermetallic alloys with B2-type structure, which exhibit low H V F - and high H V M -values. Among those are the ones listed in the following table: FeAl, NiAl [21], PdIn [29, 30], and CuZn (see [27]) (Table 2).

Table 2:

Vacancy formation enthalpy H V F and vacancy migration enthalpy H V M in B2 intermetallic alloys. The activation energy of self-diffusion Q^SD is given, when H V M is deduced from Q SD − H V F .

	H V F (eV)	H V M (eV)	Q^SD (eV)
Fe₅₅Al₄₅	1.0 [21]	1.5 [21]
Ni₄₇Al₅₃	1.5 [21]	1.8 [21]
PdIn	0.41 [29]	1.9	2.3 [30]
CuZn	0.42 (see [27])	1.02	1.56 (see [27])

Corresponding author: Roland Würschum, Institute of Materials Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria, E-mail: wuerschum@tugraz.at

Acknowledgements

This work was performed in the framework of the interuniversity cooperation of TU Graz and Univ. Graz on natural sciences (NAWI Graz). The authors are indebted to Peter Schlosser, Institute of Applied Mathematics, TU Graz, for fruitful discussion.

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Received: 2022-01-19

Revised: 2022-06-22

Accepted: 2022-04-01

Published Online: 2022-07-21

Published in Print: 2022-08-26

Articles in the same Issue

https://doi.org/10.1515/ijmr-2022-0022

Keywords for this article

Kinetic model; Modulated dilatomtry; Thermal vacancy formation; Vacancy equilibration; Vacancy migration