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# (p.257) Appendix E Kink-pair Models

Source:
Dislocations, Mesoscale Simulations and Plastic Flow
Publisher:
Oxford University Press

# E.1 Dislocations and Peierls potentials

This section outlines methods for estimating in elastic terms the energetics of the kink-pair mechanism for a dislocation moving over a periodic Peierls barrier in metallic materials. More detailed accounts can be found in Kocks et al. (1975) and Caillard and Martin (2003). These calculations make use of various potential energy profiles, sinusoidal, anti-parabolic, camel-hump or more complicated ones. For example, the sinusoidal potential is written

(E.1)
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where $E o$ is the energy at the bottom of a Peierls valley, y is the direction normal to the Peierls relief in the slip plane (Fig. E.1) and $h o$ is the periodicity in the same direction.

Fig. E.1. Top view of a periodic Peierls energy relief with energy minima (thin lines) and a maximum (thin dashed line). (a) Two critical bulge configurations with curvature radii that decrease with increasing temperature and decreasing stress. (b) Geometric low-stress, high-temperature critical configuration.

A straight dislocation experiences a resistive stress $τ P = − d E ( y ) / b d y$, which is maximum when $y = h o / 4$. Hence, the Peierls stress is

(E.2)
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Other potentials yield similar results. Under an applied stress, $τ$, a straight dislocation moves upwards on the Peierls profile and reaches an equilibrium position $y o$ given by $τ b = − [ d E ( y ) / d y ] y o$. Its energy is then $E = E ( y o )$.

# (p.258) E.2 High-stress solutions

We now consider high-stress critical configurations, which form bulges like those shown in configurations $( a )$ of Fig. E.1. The length of an element dx of line is $d x [ 1 + ( d y / d x ) 2 ] 1 / 2$. The energy fluctuation $Δ U$ required for going from the initially straight, infinite configuration at $y = y o$ to a bowed configuration is obtained by integration, taking into account the work of the applied stress $τ$

(E.3)
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This expression can only be solved numerically. It is often simplified by assuming that $d y / d x$ ≪ 1, as done in the vibrating string model (Section 2.2.2.3). This leads to

(E.4)
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Denoting by $Φ ( y ) = E ( y )$$E ( y o )$$τ b ( y$$y o )$ the effective potential under stress, eqn. E.4 is rewritten

(E.5)
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The saddle point configuration is among all possible solutions of eqns. E.3 or E.4 the one that minimizes the activation energy. It can be obtained from a variational argument as a solution of an Euler–Lagrange differential equation. In the present case, a simple integral solution exists, which is given by the Beltrami identity. If $f ( y , y x )$ is the functional form to be integrated, this solution is written f$( d y / d x ) [ ∂ f / ∂ ( d y / d x ) ] = c o n s t .$ (see Weisstein 2002 for a demonstration). Applying this result to eqn. E.5, one obtains

(E.6)
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where the constant vanished upon integration.

An equivalent approximate result can be obtained by using the concept of line tension. Energy minimization is then equivalent to a condition of equilibrium between the local forces on the dislocation, specifically the line tension, the applied force and the resistive Peierls force

(E.7)
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(p.259) In this equation, the curvature radius appears in the approximated form of a second derivative, which is valid when $d y / d x$ ≪ 1. Integration with respect to y, assuming a constant line tension and with $2 d 2 y / d x 2 = d ( d y / d x ) 2 / d y$, yields

(E.8)
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The second equality is valid in the line tension approximation if one assumes that the fluctuations of the Peierls potential are small compared to the energy at rest of the dislocation. Then, $E ( y )$ $≈$ $E o$ and the constant line tension has same formal expression as the line energy per unit line. Simple relations are obtained by inserting the second equality of eqn. E.8 into eqn. E.5. The integration variable is taken as the height y of each elementary segment, which varies between the initial and critical values $y o$ and $y c$. With $d x = ( d x / d y ) d y = ( E o / 2 Φ ) 1 / 2 d y$ and $E ( y )$ $≈$ $E o$, one obtains

(E.9)
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The factor of two arises from the fact that the integration is carried out twice, on the left side and on the right side of the configuration. The energy of a single kink under zero stress, $U k$, follows by setting $τ = 0$ and integrating over one period $h o$ of the Peierls potential. Equation E.9 reduces to

(E.10)
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Upon integration with various simple potentials, one always finds

(E.11)
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The stress dependence of the activation energy, $Δ U ( τ )$, is obtained by integrating eqn. E.9 for successive values of $y c$. For example, with the antiparabolic potential one obtains

(E.12)
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(p.260) Other Peierls potentials yield similar forms with different exponents or more complex forms. At low stresses, eqn. E.12 yields $Δ U$ $≈$ $2 U k$, which is the energy of two non-interacting kinks. This type of solution, in which the interaction between kinks is neglected as a result of the line tension approximation, is not valid for small kink separations (Section E.3).

To bypass the problem of fitting Peierls potentials to experimental results, Kocks, Argon and Ashby (1975) proposed a general parametric form for the activation enthalpy $Δ H ( τ )$

(E.13)
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where $Δ H o$ is the total activation enthalpy. The two constants p and q are usually such that 0 $〈$ $p ≤$ 1 and 1 ≤ q ≤ 2. Within these ranges of values, it is possible to reasonably fit both the high and low stress parts of the activation energy for the interaction of a dislocation with localized obstacles.

# E.3 Kink-pairs at low stresses

When the critical configuration consists of a geometric kink-pair, as in configuration $( b )$ of Fig. E.1, the interaction between the two kinks has to be taken into account for small kink-pair widths. Such small stress configurations can be treated easily (Seeger and Schiller 1962). We consider a kink-pair of width w larger than its height $h o$ and the width of a single kink. The total formation energy is the sum of the energies of the two kinks and of their interaction energy. It is written here as an activation enthalpy to keep same notations as in Section 3.2.4.2. As the two kinks are of atomic dimension, they cannot be treated like two interacting dislocation segments. The interaction energy is inversely proportional to the distance w, as for two electrical point charges of magnitude $h o$ and of opposite sign, and

(E.14)
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The factor $μ b 2$ is introduced for the sake of dimensionality and k $≈$ 1$/ 8 π$ (Hirth and Lothe, 1992). The activation enthalpy under stress, $Δ H ( τ )$, is obtained by subtracting the work of the resolved stress, $Δ W$ $≈$ $τ b h o w$, during the formation of this configuration

(E.15)
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(p.261) The critical configuration is again obtained by minimizing the activation energy with respect to the unknown kink-pair width w. A straightforward calculation yields

(E.16)
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The activation energy is eventually written

(E.17)
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These relations are confronted to atomistic results in Section 3.2.4.2. As they are no longer valid for critical bulge configurations, the question was raised of a possible transition between low and high stress kink-pair regimes (Section 3.2.4.4). However, this question cannot be answered within a framework based on empirical Peierls potentials.

# E.4 The kink-diffusion model

This section discusses the steady-state motion of a perfect dislocation by the kink-pair mechanism, from which the existence of two distinct regimes is predicted. Although the demonstration is quite general, it is of particular interest in the presence of a strong secondary Peierls force as in covalent materials. In such a case, the velocity of the expanding kinks is much lower than in metals.

The lateral expansion of single kinks is limited to a maximum distance $X / 2$, which is governed either by their annihilation with kinks of opposite sign or by fixed obstacles on the dislocation lines (Fig. E.2).

Fig. E.2. The expansion and mean free path, X, of kink-pairs of height $h o$.

A single kink has a life-time $t k$, during which it propagates along a distance $X / 2$. Its velocity is given by

(E.18)
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(p.262) In steady-state, the motion of a dislocation line with a constant velocity implies that every time a pair of moving kinks is blocked or annihilated, a nucleation event occurs. If J is the nucleation rate of kink-pairs per unit time and length of dislocation line, the steady-state condition requires that one kink-pair is nucleated during the time interval $t k$, taking into account that the length available for kink-pair nucleation is in average half the mean free path X. Thus,

(E.19)
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From eqns. E.18 and E.19, we obtain

(E.20)
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The dislocation velocity $v d$ is of the form $v d = h o / τ n u c l$, where $τ n u c l$ is the time interval between two successive kink-pair nucleations. It takes a time interval $t k$ to create a new dislocation segment of length X, but during this time interval a new kink-pair is generated on the expanding segment, in average at a time $t k / 2$ after it was generated. Thus, the periodicity of kink-pair nucleation events is $t k / 2$. According to eqn. E.19, $τ n u c l = t k / 2 = 1 / J X$ and the dislocation velocity is given by

(E.21)
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When the mean free path of kinks is governed by strong obstacles with spacing L, one has $L = X$ and, according to eqn. E.21, the dislocation velocity is written

(E.22)
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When the mean free path of kinks is governed by their annihilation with kinks of opposite sign, the combination of eqns. E.20 and E.21 yields an expression for the dislocation velocity that does not contain the free-flight distance of the kinks

(E.23)
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Thus, two different regimes are defined, a length-dependent regime and a kink-collision regime. This last regime prevails for long segments or large kink densities.

## E.4.2 Dislocation velocities

Owing to the high strength of the covalent bond, the critical kink-pairs can be assumed to have geometric shapes (Fig. E.2). Critical bulges may only form under very high stresses, because the lines are restricted to stay in the deep and narrow (p.263) minima of the primary and secondary Peierls valleys. Thus, an insight into the formation energy of geometric kink-pairs can be obtained without assuming any shape for the Peierls potential. The kink-diffusion model by Hirth and Lothe (1992) is based on a diffusive treatment of the kink velocity over the secondary Peierls relief. Kinks of atomic height are treated like point defects undergoing thermally activated motion with a migration energy $W m$$k B T$.15 This allows defining the jump frequency of a kink over the secondary Peierls barrier by

(E.24)
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The velocity of a kink submitted to a small external force $F = τ b$ is calculated using Einstein’s derivation for point defect drift in the linear regime. The mechanical work per atomic step of motion of a kink of height $h o$ is $τ b h o a / 2$, assuming that the saddle point configuration is situated at one-half of the barrier width a. The latter is taken as the reconstructed period along the line (Section 3.4.3). One defines activation energies for jumps in the direction of the applied stress ($+$) or in the reverse direction (−) by $W m ± = W m ± τ b h o a / 2$. If $f +$ and $f −$ are the corresponding jump frequencies, the forward and reverse kink velocities are given by $v k ± = a f ±$ and the net kink velocity is $v k = ( v k +$$v k − )$. The exponential term in the rate equation (eqn. E.24) can be expanded with respect to stress when $τ b h o a / 2$$k B T$. As a result, one obtains a linear stress dependence of the net kink velocity

(E.25)
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where $D k$ is the kink diffusivity.

The nucleation rate of kink-pairs can be estimated in an approximate manner and within specific conditions for the two particular cases of low and high stresses. What is called high stress in this context is in fact a regime of interacting geometric kinks similar to the low stress regime in metals (Section E.3). In the present case, however, it is necessary to account for the fact that kink-pairs are nucleated in the presence of the secondary Peierls potential. At low stresses, the formation energy of kink pairs is derived from the equilibrium concentration of non-interacting single kinks under zero stress. Identical expressions for the dislocation velocities are obtained in the two cases and we restrict ourselves here to the low stress regime. A derivation of the velocities in the high stress regime is given by Hirth and Lothe (1992) and Caillard and Martin (2003).

(p.264) In thermal equilibrium and under zero stress, the concentration of kinks of a given sign, c, is estimated by a traditional derivation for point defect concentrations. Denoting by $F k$ the (free) energy for kink formation, one has

(E.26)
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At small stresses and for a sufficiently large critical kink-pair width, the kink–kink interaction can be neglected. The concentration of kink-pairs, $c k p$, is half the value given above for single kinks. At or near thermal equilibrium, the average distance between kink-pairs, X, is given by $X = 1 / c k p = 2 / c$ and

(E.27)
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The kink-pair nucleation rate follows from eqn. E.20

(E.28)
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In the length-dependent regime, the dislocation velocity is obtained by inserting into eqn. E.28 the value of $v k$ given by the last equality in eqn. E.25 and importing the result into eqn. E.22

(E.29)
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Taking into account that the kink diffusivity involves the migration energy (eqn. E.25), the total activation energy in this regime is $2 F k$ $+$ $W m$, that is, the sum of the formation energy of a kink-pair and the migration energy of a single kink.

In the kink-collision regime, the dislocation velocity is obtained by combining eqns. E.25, E.28 and E.23

(E.30)
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In this regime, the total activation energy, $F k$ $+$ $W m$, is the sum of the formation and migration energies of a single kink. The full expressions for the two velocities are given by eqns. 3.11 and 3.12.

(p.265) The transition between the two regimes is obtained by setting $X = L$ in eqn. E.27 for the average distance between kink-pairs in thermal equilibrium

(E.31)
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where $T ∗$ is the crossover temperature and $L ∗$ the corresponding segment length. Thus, with increasing segment length, the velocity should linearly increase in the length-dependent regime and saturate to a constant value when the kink-collision regime is reached. In parallel, the total activation energy should decrease by $F k$ at this transition. Actually, whereas the first prediction is confirmed by experimental results, the second one is not (Sections 3.4.7 and 3.4.8).

In strongly covalent materials, the domain of validity of the kink-diffusion model is assumed to extend up to very high stresses, typically a fraction of the Peierls stress, beyond which the kink-pairs have to be treated as small bulges. Although the Peierls stress is not accurately known in silicon, experimental results show that the total activation energy becomes stress dependent typically around $10 − 3 μ$, well below the high stress limit. This is interpreted as arising from elastic interactions between kinks in narrow critical kink-pairs. As a consequence, the formation energy of kink-pairs should preferably be taken in the stress-dependent form proposed by Seeger and Schiller (eqn. E.17).

Alternative models to the kink-diffusion model make use of Peierls potentials. They also account for the secondary Peierls barrier and the kink migration energy (Indenbom et al. 1992; Iunin and Nikitenko 2001). The possible presence of point defects along the dislocations lines and their interaction with moving kinks was examined by several authors (Celli et al. 1963, Rybin and Orlov 1970, Iunin and Nikitenko 2001).

## Notes:

(15) Strictly speaking, the formation and migration energies of kinks are free energies. Entropy terms are, however, not explicitly included in the kink-diffusion model (see Section 3.4.6).