Temperature Dependence of the Stress Due to Additives in KCl Single Crystals

Abstract

The influence of the state of additive cations on the various deformation characteristics was studied for KCl:Sr2+ single crystal at room temperature. This result gives the heat treatment suitable for the crystal immediately before deformation tests, such as compression and tension. Four kinds of single crystals (KCl: Mg2+, Ca2+, Sr2+ or Ba2+) were plastically deformed by compression at 77 to room temperature. The plasticity of the crystal depends on dislocation motion from a microscopic viewpoint. When a dislocation breaks away from the defect around the additive cation with the help of thermal activation on the slip plane in the crystal, the variation of effective stress with the temperature was investigated by the combination method of strain-rate cycling tests and ultrasonic oscillations. Furthermore, the critical temperature Tc at which the effective stress due to the additives is zero was estimated for each of the crystals. As a result, Tc value tends to be larger with the divalent cationic size.

1. Introduction

In alkali halide crystals doped with divalent cations (divalent impurities), the additive cations are expected to be paired with vacancies to conserve the electrical neutrality. They are often formed a divalent impurity-vacancy (I-V) dipole for the impure crystals quenched from a high temperature. Then, the asymmetrical distortions are produced around the I-V dipoles. Mobile dislocations on a slip plane interact strongly only with these defects lying within one atom spacing of the glide plane [1]. The solution hardening is named “rapid hardening,” which can be distinguished from “gradual hardening” due to the defects of cubic symmetry around the monovalent dopants in the crystals [2, 3, 4]. Effects of different defects on the hardness of some alkali halide crystals are listed in Table 1. The effects are expressed as an increase in flow stress per square root of concentration of point defects (i.e., Δτ /Δc^1/2) in terms of the shear modulus, μ. Despite the same matrix (see NaCl in Table 1), the hardening due to substitutional divalent additions is much larger than the case of monovalent ones. It has been well known for many years that aliovalent impurities (aliovalent cations) are a much more potent source of solution strengthening in ionic crystals than isovalent cations [5].

Crystal	Different types of point defects	Δτ /Δc^1/2 (×2 $\sqrt{c}$) ^a
NaCl	Monovalent substitutional impurities	< μ /30
KCl	F-centers, additive coloring	μ /2.5
NaCl	Divalent substitutional impurities	2 μ
KCl (irradiated)	Interstitial chlorine	18 μ
LiF	Divalent impurity clusters	6 μ
LiF	I-V dipoles (at 77 K)	10 μ
LiF (irradiated)	VK-centers (at 77 K)	25 μ
LiF (irradiated)	Interstitial fluorine	5 μ

Table 1.

Hardening due to various defects in alkali halide crystals. Defects concentration is below 10⁻⁴ [2, 3, 4].

^a

Δc represents the increment of the concentration of point defects and μ is the shear modulus. The measurements were made at room temperature unless otherwise noted. Δτ /Δc in Refs. [2, 3, 4] is replaced by Δτ/Δc^1/2.

In view of the different types of atomic defects, solution hardening may be divided into two classes: rapid hardening and gradual hardening by Fleischer and Hibbard [3] and Johnston et al. [4]. Roughly speaking, the value of Δτ /Δc^1/2 for rapid hardening is over several ten times as large as that for the gradual hardening as shown in Table 1.

It is well known that various characteristics of deformation are influenced by the state of impurities in a crystal. The following is concerned with it and presents the heat-treatment condition suitable for the deformation tests such as compression and tension for KCl:Sr²⁺ crystals. Furthermore, the influence of divalent cationic size on the deformation characteristics is reported analyzing the data obtained by the original method (strain-rate cycling tests associated with ultrasonic oscillation), which can separate the effective stress due to weak obstacles such as additive ions from that due to dislocation cuttings.

2. Experimental procedure

The initial dislocation density, the dielectric loss peak due to the I-V dipoles, and yield stress were measured for KCl:Sr²⁺ crystal as explained in Sections 2.2 to 2.4.

2.1 Preparation of specimens

KCl doped with SrCl₂ was grown from the melt of reagent-grade powders by the Kyropoulos method in air. The specimens, which were cloven out of single-crystalline ingots to the size of 5 × 5 × 15 mm³, were kept immediately at 973 K for 24 h, followed by cooling to room temperature at a rate of 40 Kh⁻¹. This treatment is because the density of dislocations is reduced as much as possible. Owing to the gradual cooling, the additive ions (Sr²⁺) are expected to aggregate in the crystal. Accordingly, the specimens were further kept at 373 to 873 K for 30 min, followed by quenching to room temperature to disperse the additive ions (Sr²⁺) into them.

2.2 Initial dislocation density ($\rho$)

Using an etch pits technique, the initial density ($\rho$) of dislocations in KCl:Sr²⁺ (0.3 mol.% in the melt) was detected with a corrosive liquid (saturated solution of PbCl₂ + ethyl alcohol added two drops of water). The etching was made at room temperature for 30 min. The measurement of dislocation density in a crystal was carried out by the etch pit technique.

2.3 Dielectric loss factor (tan $\delta$)

The dielectric loss factor tan $\delta$ as a function of frequency was measured for KCl:Sr²⁺ (1.0 mol.% in the melt) in a thermostatic bath at 300 to 873 K by using an Andoh electricity TR-10C model.

2.4 Yield stress (τ_y)

The values of yield stress τ_y were obtained at room temperature for KCl:Sr²⁺ (1.0 mol.% in the melt) compressed along the <100 > axis at the crosshead speed of 20 μm min⁻¹. The τ_y values were determined by the intersection of the tangent to the easy glide region in the stress-strain curve and the straight line extrapolated from the elastic region of the curve.

2.5 Combination method of strain-rate cycling tests and ultrasonic oscillation

Four kinds of single crystals (KCl:Mg²⁺ (0.035 mol.% in the melt), Ca²⁺ (0.050, 0.065 mol.% in the melt), Sr²⁺ (0.035, 0.050, 0.065 mol.% in the melt) or Ba²⁺ (0.050, 0.065 mol.% in the melt)) were prepared by cleaving the single crystalline ingots to the size of 5 × 5 × 15 mm³. The test pieces were kept immediately below the melting point (1043 K) for 24 h and were gradually cooled to room temperature at a rate of 40 Kh⁻¹. Further, they were held at 673 K for 30 min and were rapidly cooled by water-quenching immediately before the following tests.

The heat-treated test pieces were compressed along the <100> axis at 77 K to the room temperature and the ultrasonic oscillatory stress (τ_v) was intermittently superimposed in the same direction as the compression. The strain-rate cycling test off or on the ultrasonic oscillation (20 kHz) is illustrated in Figure 1. Superposing oscillatory stress, a stress drop (Δτ) is caused during plastic deformation. The strain-rate cycling between strain-rates of ${\dot{\epsilon }}_1$ (2.2 × 10⁻⁵ s⁻¹) and ${\dot{\epsilon }}_2$ (1.1 × 10⁻⁴ s⁻¹) was undertaken keeping the stress amplitude of τ_v constant. This led to the increase Δτ’ in stress due to the strain-rate cycling. The strain-rate sensitivity (Δτ’/Δln $\dot{\epsilon }$) of the flow stress, which is derived from Δτ’/1.609, was used as a measurement of the strain-rate sensitivity.

Figure 1.

Change in applied shear stress (τ_a) for the strain-rate cycling tests between the two strain rates, ${\dot{\epsilon }}_1$ (2.2 × 10⁻⁵ s⁻¹) and ${\dot{\epsilon }}_2$ (1.1 × 10⁻⁴ s⁻¹), off or on the ultrasonic oscillatory stress (τ_v) due to the oscillation (20 kHz).

3. Results and discussion

3.1 Deformation characteristics influenced by different heat treatments for KCl:Sr²⁺ crystals

Figure 2 shows the optical micrograph of the etch pits for KCl:Sr²⁺ (0.05 mol.% in the melt) at room temperature after the annealing at 973 K for 24 h. The position of the dislocation after the annealed treatment is marked by a pyramidal pit, and the position where a dislocation slipped out of the crystal after the treatment is marked by a flat-bottom pit.

Figure 2.

Dislocations on a (100) plane for KCl:Sr²⁺ (0.05 mol.%) after the heat treatment at 973 K for 24 h.

Although, it is difficult to resolve the individual etch pits for high dislocation density, the dislocation density on a (100) plane is found to be 1.27 × 10⁴ cm⁻² from this micrograph for the annealed specimen (i.e., KCl:Sr²⁺ (0.05 mol.% in the melt)).

The height of the loss peak is related to the concentration of the isolated I-V dipole (see Eq. (1)). The details are explained about KCl:Sr²⁺ (0.05 mol.% in the melt) below.

Dielectric absorption of an I-V dipole causes a peak on the relative curve of tan $\delta$-frequency. The Debye peak height is proportional to the concentration of I-V dipoles as expressed by the following Eq. (1) [6].

$$tan\delta =\frac{2\pi e^{2}c}{3{\epsilon }^{\prime }\mathit{akT}},\left(\text{maximum}\right)$$E1

where e is the elementary electric charge, c is the concentration of I-V dipoles, ${\epsilon }^{\prime }$ is the dielectric constant in the matrix, a is the lattice constant, k is the Boltzmann constant, and T is the absolute temperature. Figure 3 shows the tan $\delta$-frequency curves for KCl:Sr²⁺ at 393 K. The solid and dotted curves correspond to the quenched KCl:Sr²⁺ (0.05 mol.% in the melt). That is to say, the crystals were held within 673 K for 30 min, followed by quenching to room temperature. The dotted line shows Debye peak obtained by subtracting the d.c. part which is obtained by extrapolating the linear part of the solid curve in the low-frequency region to the high-frequency region. By introducing the peak height of the dotted curve into Eq. (1), the concentration of the isolated I-V dipoles was determined to be 98.3 ppm for the quenched crystal by dielectric loss measurement.

Figure 3.

Dielectric loss in KCl:Sr²⁺(0.05 mol.% in the melt) at 393 K. Dotted line (− − - -) shows the losses coming from the I-V dipoles.

As mentioned above, the dielectric loss factor tan $\delta$ is proportional to the concentration of the isolated I-V dipoles at a given temperature.

Figure 4 shows the variations in the initial dislocation density ($\rho$), the dielectric loss peak due to the I-V dipoles (tan $\delta$), and yield stress (τ_y) as against the temperature quenched KCl:Sr²⁺ (0.3 and 1.0 mol.% in the melt) single crystals [7]. The $\rho$ value is about $5\pm 1\times {10}^4{\mathrm{cm}}^{-2}$ independently of quenching temperature below 673 K, but it remarkably increases above 673 K. The τ_y value also remarkably increases for the crystals quenched at the temperature above 673 K as the variation in dislocation density. And then it becomes a constant value 29 MPa above 723 K. While the tan $\delta$ value does not vary and is almost constant by quenching from the temperature below 573 K or above 673 K. Its value becomes 0.3 $\times {10}^{-2}$ up to 0.9 $\times {10}^{-2}$ between the two quenching temperatures (i.e., 573 K and 673 K). The variation in tan $\delta$ value is similar to it in the yield stress within the temperature.

Figure 4.

Quenching temperature dependence of initial dislocation density (ρ), the dielectric loss peaks due to the I-V dipoles (tanδ), and yield stress (τ_y) for KCl:Sr²⁺ crystals (reproduced from Ref. [7]).

The difference in dislocation density is slight and the tan $\delta$ obviously becomes larger with a higher quenching temperature between 573 and 673 K as shown in Figure 4. The concentration of isolated I-V dipoles, which is proportional to the tan $\delta$ (see Eq. (1)), affects the yield stress, as reported in the papers [8, 9, 10, 11]. Therefore, the specimens are determined to be quenched from 673 K to room temperature immediately before deformation tests such as compression in this chapter.

3.2 Temperature dependence of τ_p1, τ_p2, and yield stress (τ_y)

The variation of the strain-rate sensitivity and the stress decrement with the shear strain is shown in Figure 5 for KCl:Sr²⁺ (0.050 mol.% in melt) single crystals at 200 K. Δτ′/Δln $\dot{\epsilon }$ tends to increase with shear strain and decrease with stress amplitude in Figure 5(a). Δτ does not change significantly with shear strain but increases with stress amplitude at a given temperature and shear strain in Figure 5(b).

Figure 5.

Shear strain ($\mathrm{\epsilon }$) dependence of (a) the strain-rate sensitivity (Δτ′/Δln $\dot{\epsilon }$) and (b) the stress decrement (Δτ) for KCl:Sr²⁺ (0.050 mol.%) at 200 K. τ_v (arb. units): (○) 0, (●) 10, (▲) 25, (▽) 35, (▼) 45, and (□) 50 (reproduced from Ref. [12] with permission from the publisher).

Δτ′/Δln $\dot{\epsilon }$ vs. Δτ curve is further obtained from Figure 5 at a given strain, which provides the relative curve for a fixed internal structure of the crystal and is shown by open squares in Figure 6 for KCl:Sr²⁺ (0.050 mol.% in melt) crystal at the shear strain of 10%. The details were described in the review article [14].

Figure 6.

Strain-rate sensitivity (Δτ′/Δln $\dot{\epsilon }$) vs. the stress decrement (Δτ) at strain of 10% for KCl:Sr²⁺ (0.050 mol.%) at temperatures of (○) 103 K, (Δ) 133 K, (□) 200 K, (⋄) 225 K (reproduced from Ref. [13] with permission from the publisher).

Relation between the strain-rate sensitivity and the stress decrement for KCl:Sr²⁺ (0.050 mol.% in melt) at the shear strain of 10% is shown by open symbols in Figure 6. The relative curve has a stair like shape. Figure 6 shows the influence of temperature on the relation between the strain-rate sensitivity, Δτ′/Δln $\dot{\epsilon }$, and the stress decrement, Δτ, for KCl single crystals doped with Sr²⁺ as weak obstacles. As the temperature is high, the Δτ value at first bending point, τ_p1, shifts in the direction of low stress decrement and does not appear up to 225 K. The first plateau region indicates that the average length of the dislocation segment remains constant in that region. This is because the strain-rate sensitivity of effective stress (τ^*) due to impurities is inversely proportional to the average length of the dislocation segment. That is to say, it is given by

$${\left(\frac{∆{\tau }^{\ast }}{∆ln\dot{\epsilon }}\right)}_T=\frac{\mathit{kT}}{\mathit{bLd}}$$E2

where b is the magnitude of the Burgers vector, L is the average length of dislocation segments, and d is the activation distance. Therefore, the application of oscillation with low-stress amplitude cannot influence the average length of the dislocation segment at low temperature, but even a low-stress amplitude can do so at a temperature of 225 K. Such a phenomenon was also observed for the other specimens: KCl doped with Mg²⁺, Ca²⁺ or Ba²⁺ separately.

Figure 7(a)–(c) shows the dependence of τ_p1, τ_p2, and yield stress (τ_y) on temperature for KCl:Sr²⁺ ((a) 0.065, (b) 0.050 and (c) 0.035 mol.%, respectively) crystals. τ_p2 is the Δτ value at the second bending point on the plots of Δτ vs. (Δτ′/Δln $\dot{\epsilon }$). It is clear from the figure that both τ_p1 and τ_p2 tend to increase with decreasing temperature as well as τ_y for the three crystals and the τ_y curve seems to approach a constant stress at high temperature. Two values of τ_p1 and τ_p2 increase with increasing Sr²⁺ concentration at a given temperature as shown in the figure. Similar results as the case of KCl:Sr²⁺ are also observed for the other crystals (i.e., KCl: Mg²⁺, Ca²⁺ or Ba²⁺).

Figure 7.

Temperature dependence of (○) τ_p1, (Δ) τ_p2, and (□) τ_y for KCl:Sr²⁺ ((a) 0.065, (b) 0.050, (c) 0.035 mol.% in the melt) (reproduced from Ref. [12]).

3.3 Critical temperature (T_c)

Figure 8(a)–(d) shows the dependence of τ_p1 on the temperature for KCl doped with Mg²⁺ (0.035 mol.% in melt), Ca²⁺ (0.065 mol.% in melt), Sr²⁺ (0.050 mol.% in melt) or Ba²⁺ (0.065 mol.% in melt). τ_p1 is considered the effective stress due to the weak obstacles (Mg²⁺, Ca²⁺, Sr²⁺ or Ba²⁺ ions in this Section 3.2) on the mobile dislocation during plastic deformation [13]. τ_p1 decreases with increasing temperature for the four kinds of crystals in the figure. The critical temperature (T_c) at which τ_p1 is zero can be determined from the intersection with the abscissa and is around 180, 220, 230 and 260 K for KCl:Mg²⁺, KCl:Ca²⁺, KCl:Sr²⁺, and KCl:Ba²⁺, respectively.

Figure 8.

Temperature dependence of τ_p1 for various crystals: (a) KCl:Mg²⁺ (0.035 mol.% in the melt), (b) KCl:Ca²⁺ (0.065 mol.% in the melt), (c) KCl:Sr²⁺ (0.050 mol.% in the melt), and (d) KCl:Ba²⁺ (0.065 mol.% in the melt). (Reproduced from Ref. [13] with permission from the publisher).

The tetragonal distortion resulting from the introduction of the divalent cations into alkali halides is generally formed in the lattice. A dislocation moves on a single slip plane and interacts strongly only with those defects. Then, the relation between the effective stress and the temperature can be approximated as the linear relationship of ${\tau }^{\ast 1/2}$ vs. T^1/2 (i.e., the Fleischer’s model [1]). ${\tau }^{\ast }$ is the effective stress due to the divalent cations. The critical temperature can be also determined from τ_p1^1/2 vs. T^1/2 for each specimen. The values of T_c are given in Table 2. When the divalent ionic size becomes closer to it of K⁺ from the small divalent cationic size side, T_c tends to increase. T_c is not influenced by the concentration of additives (i.e., Mg²⁺, Ca²⁺, Sr²⁺or Ba²⁺ here) [12, 15, 16] and is expressed by

Single crystal	Tc (K)	Ionic radius (Å)
KCl:Mg^2+	191	Mg^2+ 0.72
KCl:Ca^2+	221	Ca^2+ 1.00
KCl:Sr^2+	227	Sr^2+ 1.13
KCl:Ba^2+	277	Ba^2+ 1.36
		K^+ 1.38

Table 2.

T_c and ionic radius values for various crystals.

$$T_c=\left(∆G_0/k\right)ln\left[\dot{\epsilon }/\left\{{\rho }_{\mathrm{m}}{b}^2{\nu }_D{\left(L_0/L\right)}^2\right\}\right],$$E3

where $∆G_0$ is the Gibbs free energy for the breakaway of a dislocation from an impurity, ${\rho }_{\mathrm{m}}$ is the density of mobile dislocations, the ${\nu }_D$ is the Debye frequency, and L₀ is the average spacing of divalent cations on a slip plane. At the temperature of T_c, thermal fluctuations can provide the entire energy for breaking through the impurity. The additive ions act no longer as obstacles to dislocation motion.

4. Conclusions

The concentration of isolated I-V dipoles affects the τ_y values. The values of τ_y and $\rho$ remarkably increase with the quenching temperature above 673 K. As for tan $\delta$, it does not vary by quenching from the temperature below 573 K or above 673 K. Within 573 to 673 K, the difference in $\rho$ is slight and the values of tan $\delta$ and τ_y obviously become larger with a higher quenching temperature. Based on these results, KCl:Sr²⁺ single crystals are determined to be quenched from 673 K to room temperature immediately before deformation tests such as compression.

The following two points were mainly mentioned from the experimental results and the discussion based on the data τ_p1 of the first bending point on the plots of Δτ vs. (Δτ’/Δln $\dot{\epsilon }$).

1. The plots of Δτ vs. (Δτ’/Δln $\dot{\epsilon }$) have a stair like shape (two bending points and two plateau places) for the KCl doped with the divalent cations. The Δτ values at the first and second bending points, τ_p1 and τ_p2, become obviously larger at lower temperature as well as τ_y for the crystals within the temperature.

2. The values of T_c were derived from the relation between τ_p1^1/2 and T^1/2 with respect to the Fleischer’s model for KCl single crystals doped with Mg²⁺, Ca²⁺, Sr²⁺ or Ba²⁺ as divalent impurities. T_c tends to increase when the additive cationic size is increasingly close to the K⁺ ionic size from the smaller side than K⁺ size in the matrix crystal.

Conflict of interest

The author declares no conflict of interest.