**3. Relation between stress decrement (Δ***τ***) and strain-rate sensitivity (***λ***)**

**Figure 3** shows the influence of temperature on Δ*τ* vs. *λ* curve for the NaCl:Rb<sup>+</sup> (0.5 mol%) single crystals at strain 6%. The variation of *λ* with Δ*τ* has stair-like shape: two bending points and two plateau regions are on the each curve. That is to say, the first plateau region ranges below the first bending point at low Δ*τ* and the second one extends from the second bending point at high Δ*τ*. *λ* gradually decreases with increasing Δ*τ* between the two bending points. The length of Δ*τ* within the first plateau region is named *τ*p as denoted in **Figure 3**. *τ*p tends to be lower at higher temperature. Similar phenomena as **Figure 3** are observed for all the other NaCl single crystals contained with the monovalent impurities (i.e. Li<sup>+</sup> , K<sup>+</sup> , Cs<sup>+</sup> , F<sup>−</sup>, Br<sup>−</sup> or I<sup>−</sup> ions).

The relation between Δ*τ* and *λ* reflects the effect of ultrasonic oscillation on the dislocation motion on the slip plane containing many weak obstacles such as impurities and a few forest dislocations during plastic deformation [40]. Δ*τ* vs. *λ* curve is divided into three regions as shown in **Figure 3**. Within the first plateau region of relative curve (i.e. region 1 in **Figure 3**), the application of oscillation with low stress amplitude cannot influence the average length of dislocation segments (¯*l*) and ¯*l* is considered to remain constant. All weak obstacles act as impedimenta to the dislocation motion there. In region 2, the dislocation begins to break-away from the weak ones between the forest dislocations by applying oscillation with high stress amplitude. As a result, ¯*l* begins to increase and the *λ* of flow stress starts to decrease at the stress decrement Δ*τ* of *τ*p. This is because *λ* is inversely proportional to ¯*l* [57].

**93**

below *τ*0 (i.e. effective stress *τ*

*Study on Dislocation-Dopant Ions Interaction during Plastic Deformation by Combination…*

Some weak obstacles stop acting as impedimenta in the region. The weak obstacles

*strain 6% and various temperatures. The numbers besides each symbol represent the temperature (reproduced* 

*Relation between the stress decrement (Δ*τ*) and the strain-rate sensitivity (*λ*) for NaCl:Rb+*

not to be vacancies here, since the vacancies have low density as against the dopants in the specimen. When the specimens were plastically deformed, it is imagined that a dislocation begins to overcome from the dopants which lie on the dislocation with the help of thermal activation. Then, *τ*p is considered to represent the effective stress due to the ions. Accordingly, *τ*p is expected to decrease with increasing temperature. Δ*τ* vs. *λ* curves shown in **Figure 3** correspond to this. As the temperature becomes larger, *τ*p shifts in the direction of lower Δ*τ*. *τ*p depends on type and density of the weak obstacle [36, 38]. Applying still larger stress amplitude during plastic deformation of the specimens, the second plateau region within stage 3 becomes to appear on the relative curves in **Figure 3**. In stage 3, the dopants are no longer act as the impedimenta to mobile dislocations and the dislocations are hindered only by forest dislocations. Then, ¯*l* becomes constant again. This leads to the constant *λ* of

, K+ , Rb+

, Cs+

, F<sup>−</sup>, Br<sup>−</sup> or I<sup>−</sup> ions) and

 *(0.5 mol%) at* 

*DOI: http://dx.doi.org/10.5772/intechopen.92607*

are supposed to be monovalent dopants (Li+

*from Ref. [58] with permission from the publisher).*

**Figure 3.**

flow stress. *λ*p denoted in **Figure 3** is introduced later.

\*

**4. Model overcoming the thermal obstacle by a dislocation**

A dislocation will encounter a stress field illustrated schematically in **Figure 4** as it moves through on the slip plane containing many weak obstacles and a few strong ones. In the figure, the positive stress concerning axis of the ordinate opposes the flow stress (applied stress *τ*) and the negative stress assists it. Extrinsic resistance to the dislocation motion has two types: one is long-range obstacle (the order of 10 atomic diameters or greater) and the other short-range obstacle (less than about 10 atomic diameters). The former is considered to be forest dislocations, large precipitates or second-phase particles, and grain boundary, for instance, and the latter impurity atoms, isolated and clustered point defects, small precipitates, intersecting dislocations, etc. Overcoming the latter type of obstacles (byname, thermal obstacles) by a dislocation, thermal fluctuations play an important role in aid of the flow stress above the temperature of 0 K. Then the aid energy, Δ*G*, supplied by the thermal fluctuations is given by the shaded part in **Figure 4**. Thus the dislocation can move through

due to short-range obstacles and internal stress *τ*<sup>i</sup>

*Study on Dislocation-Dopant Ions Interaction during Plastic Deformation by Combination… DOI: http://dx.doi.org/10.5772/intechopen.92607*

#### **Figure 3.**

*Electron Crystallography*

50 μm min<sup>−</sup><sup>1</sup>

**Figure 2.**

1 *(2.2 × 10<sup>−</sup><sup>5</sup>*

 *s<sup>−</sup><sup>1</sup> ) and* ε̇

*rates,* ε̇

(1.1 × 10<sup>−</sup><sup>4</sup>

or I<sup>−</sup> ions).

s<sup>−</sup><sup>1</sup>

stress was monitored by the output voltage from the piezoelectric transducer set between a specimen and the support rod, which was observed by an a.c. voltmeter or an oscilloscope. Since the wavelength, which is 226 mm on the basis of calculating from the data of ref. [56], is 15 times as long as the length of specimen, the

*Explanatory diagram of a change in applied shear stress,* τa*, for the strain-rate cycling test between the strain* 

Strain-rate cycling tests made between the crosshead speeds of 10 and

associated with the ultrasonic oscillation is illustrated in **Figure 2**. Superposition of oscillatory stress (*τ*v) causes a stress drop (Δ*τ*) during plastic deformation.

variation of stress due to the strain-rate cycling is Δ*τ*'. The strain-rate sensitivity

**3. Relation between stress decrement (Δ***τ***) and strain-rate sensitivity (***λ***)**

**Figure 3** shows the influence of temperature on Δ*τ* vs. *λ* curve for the NaCl:Rb<sup>+</sup> (0.5 mol%) single crystals at strain 6%. The variation of *λ* with Δ*τ* has stair-like shape: two bending points and two plateau regions are on the each curve. That is to say, the first plateau region ranges below the first bending point at low Δ*τ* and the second one extends from the second bending point at high Δ*τ*. *λ* gradually decreases with increasing Δ*τ* between the two bending points. The length of Δ*τ* within the first plateau region is named *τ*p as denoted in **Figure 3**. *τ*p tends to be lower at higher temperature. Similar phenomena as **Figure 3** are observed for all the other NaCl

The relation between Δ*τ* and *λ* reflects the effect of ultrasonic oscillation on the dislocation motion on the slip plane containing many weak obstacles such as impurities and a few forest dislocations during plastic deformation [40]. Δ*τ* vs. *λ* curve is divided into three regions as shown in **Figure 3**. Within the first plateau region of relative curve (i.e. region 1 in **Figure 3**), the application of oscillation with low stress amplitude cannot influence the average length of dislocation segments (¯*l*) and ¯*l* is considered to remain constant. All weak obstacles act as impedimenta to the dislocation motion there. In region 2, the dislocation begins to break-away from the weak ones between the forest dislocations by applying oscillation with high stress amplitude. As a result, ¯*l* begins to increase and the *λ* of flow stress starts to decrease at the stress decrement Δ*τ* of *τ*p. This is because *λ* is inversely proportional to ¯*l* [57].

were performed within the temperatures. The strain-rate cycling test

) was carried out keeping the stress amplitude of *τ*v constant, the

 Δ*τ*'/1.609, was used as a measurement of the strain-rate sensitivity (*λ* = Δ*τ*'/Δlnε). Slip system for rock-salt structure ̇

1 (2.2 × 10<sup>−</sup><sup>5</sup>

*), under superposition of ultrasonic oscillatory shear stress,* τ*v.*

0> so that shear stress (*τ*) and shear strain (*ε*)

s<sup>−</sup><sup>1</sup>

, K<sup>+</sup> , Cs<sup>+</sup>

, F<sup>−</sup>, Br<sup>−</sup>

) and ε̇ 2

strain of specimen is supposed to be homogeneous.

2 *(1.1 × 10<sup>−</sup><sup>4</sup>*

 *s<sup>−</sup><sup>1</sup>*

When the strain-rate cycling between strain-rates of ε̇

(Δ*τ*'/Δlnε) of the flow stress, which is given by ̇

such as NaCl crystal is {110} <11\_

calculated for the slip system were used in this study.

single crystals contained with the monovalent impurities (i.e. Li<sup>+</sup>

**92**

*Relation between the stress decrement (Δ*τ*) and the strain-rate sensitivity (*λ*) for NaCl:Rb+ (0.5 mol%) at strain 6% and various temperatures. The numbers besides each symbol represent the temperature (reproduced from Ref. [58] with permission from the publisher).*

Some weak obstacles stop acting as impedimenta in the region. The weak obstacles are supposed to be monovalent dopants (Li+ , K+ , Rb+ , Cs+ , F<sup>−</sup>, Br<sup>−</sup> or I<sup>−</sup> ions) and not to be vacancies here, since the vacancies have low density as against the dopants in the specimen. When the specimens were plastically deformed, it is imagined that a dislocation begins to overcome from the dopants which lie on the dislocation with the help of thermal activation. Then, *τ*p is considered to represent the effective stress due to the ions. Accordingly, *τ*p is expected to decrease with increasing temperature. Δ*τ* vs. *λ* curves shown in **Figure 3** correspond to this. As the temperature becomes larger, *τ*p shifts in the direction of lower Δ*τ*. *τ*p depends on type and density of the weak obstacle [36, 38]. Applying still larger stress amplitude during plastic deformation of the specimens, the second plateau region within stage 3 becomes to appear on the relative curves in **Figure 3**. In stage 3, the dopants are no longer act as the impedimenta to mobile dislocations and the dislocations are hindered only by forest dislocations. Then, ¯*l* becomes constant again. This leads to the constant *λ* of flow stress. *λ*p denoted in **Figure 3** is introduced later.

#### **4. Model overcoming the thermal obstacle by a dislocation**

A dislocation will encounter a stress field illustrated schematically in **Figure 4** as it moves through on the slip plane containing many weak obstacles and a few strong ones. In the figure, the positive stress concerning axis of the ordinate opposes the flow stress (applied stress *τ*) and the negative stress assists it. Extrinsic resistance to the dislocation motion has two types: one is long-range obstacle (the order of 10 atomic diameters or greater) and the other short-range obstacle (less than about 10 atomic diameters). The former is considered to be forest dislocations, large precipitates or second-phase particles, and grain boundary, for instance, and the latter impurity atoms, isolated and clustered point defects, small precipitates, intersecting dislocations, etc. Overcoming the latter type of obstacles (byname, thermal obstacles) by a dislocation, thermal fluctuations play an important role in aid of the flow stress above the temperature of 0 K. Then the aid energy, Δ*G*, supplied by the thermal fluctuations is given by the shaded part in **Figure 4**. Thus the dislocation can move through below *τ*0 (i.e. effective stress *τ* \* due to short-range obstacles and internal stress *τ*<sup>i</sup>

**Figure 4.** *Stress fields encountered by a dislocation moving through the crystal lattice [57].*

#### **Figure 5.**

*The process for thermal activated overcoming of the short-range obstacle by a dislocation. Variation in (*a*) the Gibbs free energy of activation and (b) the force acted on the dislocation with the distance for a dislocation motion [57].*

due to long-range ones in **Figure 4**). *τ*0 is the value of *τ* at 0 K. As for the long-range obstacles (byname, athermal obstacles), the energy barrier is so large that the thermal fluctuations play no role in overcoming them within the temperature range.

The representation of **Figure 5** is concerned with a common type of thermal activation barrier. The free energy (*G*) varies with the distance (*x*) between a

**95**

**Figure 6.**

*permission from the publisher).*

*Relation between τp and activation volume (*V*) for NaCl:Rb+*

*Study on Dislocation-Dopant Ions Interaction during Plastic Deformation by Combination…*

dislocation and the obstacle as given in **Figure 5(a)**. When a dislocation overcomes the short-range obstacles, the free energy becomes high on account of the work (Δ*W*) done by the applied stress. Then the resistance (*F*), where it can be defined by the differentiation of free energy with respect to *x* (i.e. ∂*G*/∂*x*), to the dislocation motion is revealed as **Figure 5(b)** in accord with the abscissa of **Figure 5(a)**. *F* value is maximum at position *x*1. **Figure 5(b)** corresponds to typical force-distance curve for short-range obstacle among those in **Figure 4**. Shape of this curve represented by *F*(*x*) means the model overcoming the obstacle by a dislocation. *G*0, which is taken as the shaded area under *F*(*x*) between saddlepoint positions *x*0 and *x*2 in **Figure 5(b)**, is the Gibbs free energy of activation for the breakaway of the dislocation from the obstacle in the absence of an applied stress (in this case it is equivalent to the Helmholtz free energy for the

**5. Relation between the effective stress due to impurities on activation** 

When the dislocation breaks-away from the defects on a slip plane with the aid of thermal activation during plastic deformation, observations of *τ*p and *λ*p would provide information on the dislocation-defect interaction in the specimen. *λ*p is the difference between *λ* at first plateau place and at second one on Δ*τ* vs. *λ* curve as presented in **Figure 3**, which has been regarded as the component of strain-rate sensitivity due to dopant ions when a dislocation moves forward with the help of

**Figure 6** shows the relation between *τ*p and activation volume (*V*) for NaCl:Rb+

\_∂ ln *ε*̇ 

<sup>∂</sup>*<sup>τ</sup>* ) (1)

 *(0.5 mol%) (reproduced from Ref. [58] with* 

*V* = *kT*(

(0.5 mol%). The activation volume has been expressed as [57].

*DOI: http://dx.doi.org/10.5772/intechopen.92607*

dislocation motion).

**volume**

oscillation [40].

*Study on Dislocation-Dopant Ions Interaction during Plastic Deformation by Combination… DOI: http://dx.doi.org/10.5772/intechopen.92607*

dislocation and the obstacle as given in **Figure 5(a)**. When a dislocation overcomes the short-range obstacles, the free energy becomes high on account of the work (Δ*W*) done by the applied stress. Then the resistance (*F*), where it can be defined by the differentiation of free energy with respect to *x* (i.e. ∂*G*/∂*x*), to the dislocation motion is revealed as **Figure 5(b)** in accord with the abscissa of **Figure 5(a)**. *F* value is maximum at position *x*1. **Figure 5(b)** corresponds to typical force-distance curve for short-range obstacle among those in **Figure 4**. Shape of this curve represented by *F*(*x*) means the model overcoming the obstacle by a dislocation. *G*0, which is taken as the shaded area under *F*(*x*) between saddlepoint positions *x*0 and *x*2 in **Figure 5(b)**, is the Gibbs free energy of activation for the breakaway of the dislocation from the obstacle in the absence of an applied stress (in this case it is equivalent to the Helmholtz free energy for the dislocation motion).

## **5. Relation between the effective stress due to impurities on activation volume**

When the dislocation breaks-away from the defects on a slip plane with the aid of thermal activation during plastic deformation, observations of *τ*p and *λ*p would provide information on the dislocation-defect interaction in the specimen. *λ*p is the difference between *λ* at first plateau place and at second one on Δ*τ* vs. *λ* curve as presented in **Figure 3**, which has been regarded as the component of strain-rate sensitivity due to dopant ions when a dislocation moves forward with the help of oscillation [40].

**Figure 6** shows the relation between *τ*p and activation volume (*V*) for NaCl:Rb+ (0.5 mol%). The activation volume has been expressed as [57]. \_∂ ln *ε*̇

$$\begin{aligned} \text{(a)} \quad & \begin{pmatrix} \cos \alpha & \sin \alpha & \cos \alpha \\ \sin \alpha & \cos \alpha & \cos \alpha \end{pmatrix} \\ & \begin{pmatrix} \cos \alpha & \cos \alpha & \sin \alpha \\ \sin \alpha & \cos \alpha & \cos \alpha \end{pmatrix} \\ & \begin{pmatrix} \cos \alpha & \cos \alpha & \sin \alpha \\ \sin \alpha & \cos \alpha & \cos \alpha \end{pmatrix} \\ & \begin{pmatrix} \cos \alpha & \cos \alpha & \sin \alpha \\ \sin \alpha & \cos \alpha & \cos \alpha \end{pmatrix} \end{aligned}$$

$$W = kT \left(\frac{\partial \ln \dot{\varepsilon}}{\partial \tau}\right) \tag{1}$$

#### **Figure 6.**

*Relation between τp and activation volume (*V*) for NaCl:Rb+ (0.5 mol%) (reproduced from Ref. [58] with permission from the publisher).*

*Electron Crystallography*

**Figure 4.**

**94**

**Figure 5.**

*motion [57].*

due to long-range ones in **Figure 4**). *τ*0 is the value of *τ* at 0 K. As for the long-range obstacles (byname, athermal obstacles), the energy barrier is so large that the thermal

*The process for thermal activated overcoming of the short-range obstacle by a dislocation. Variation in (*

*Stress fields encountered by a dislocation moving through the crystal lattice [57].*

*Gibbs free energy of activation and (b) the force acted on the dislocation with the distance for a dislocation* 

The representation of **Figure 5** is concerned with a common type of thermal activation barrier. The free energy (*G*) varies with the distance (*x*) between a

a*) the* 

fluctuations play no role in overcoming them within the temperature range.


**Table 1.**

*Values of energy* G*0.*

#### **Figure 7.**

*Variation of the interaction energy (*G*0) between dislocation and the dopant ion with the defect size (reproduced from Ref. [58] with permission from the publisher).*

where *k* is the Boltzmann constant and *T* is the absolute temperature. Here, the ( \_∂ ln ε̇ ∂τ ) in Eq. (1) is obtained from *λ*p. Eq. (1) is namely replaced by *V* = *kT*/ λp. (2)

This dependence (*τ*p vs. *V*) also represents the force-distance profile between dislocation and Rb+ ion. The *τ*p vs. *V* curve gives the value of *G*0 for the specimen. The *G*0 values for the other specimens (i.e. NaCl: Li+ , K+ , Cs+ , F<sup>−</sup>, Br<sup>−</sup> or I<sup>−</sup>) are similarly estimated and are listed in **Table 1**.

**Figure 7** shows the obtained energies *G*0 with the isotropic defect size (), which is estimated from the difference between the lattice constants of host crystal and dopant, around ion doped in the each specimen. The ions beside each plot represent the dopants in NaCl single crystals. *G*0 values vary linearly with in the specimens. The intercept of the straight line is 0.36 eV, which is considered to be the interaction energy between dislocation and inherent obstacle of the host crystal because is zero.

### **6. Conclusions**

The following conclusions were derived from the data analyzed in terms of the Δ*τ* vs. *λ* curves for NaCl: Li+ , K+ , Rb+ , Cs+ , Br<sup>−</sup>, I<sup>−</sup>, F<sup>−</sup>, Br<sup>−</sup> or I<sup>−</sup> single crystals.

**97**

**Author details**

Yohichi Kohzuki

Japan

*Study on Dislocation-Dopant Ions Interaction during Plastic Deformation by Combination…*

1.The relation between Δ*τ* and *λ* has stair-like shape for the specimens at a given temperature and strain. There are two bending points and two plateau regions. *λ* decreases with Δ*τ* between the two bending points. The measurement of *τ*p and *V* calculated with *λ*p provides information on the interaction between mobile dislocation and the dopant ion in the specimens during

2.The Gibbs free energy *G*0 for the overcoming of dislocation from the dopant is obtained for each of the specimens and increases linearly with increasing the defect size . This result leads to the conclusion that the dopant ion as weak obstacle to dislocation motion becomes slightly stronger with larger defect size

Dr. T. Ohgaku, as well as S. Yamaguchi, M. Azuma, H. Teraji, E. Ogawa and Y. Yamanaka are acknowledged for his collaboration in the analysis on Δ*τ* and *λ*

Department of Mechanical Engineering, Saitama Institute of Technology, Fukaya,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: kohzuki@sit.ac.jp

provided the original work is properly cited.

*DOI: http://dx.doi.org/10.5772/intechopen.92607*

around the dopant in NaCl single crystal.

data, as well as for their experimental assistance.

The author declares no conflict of interest.

plastic deformation.

**Acknowledgements**

**Conflict of interest**

*Study on Dislocation-Dopant Ions Interaction during Plastic Deformation by Combination… DOI: http://dx.doi.org/10.5772/intechopen.92607*

