Prediction of stability and lifetime of carbyne, carbyne–graphene and similar low-dimensional nanostructures

Abstract

The approach is offered and the analytical dependences are obtained which enable to predict with sufficient accuracy the lifetime of carbyne–graphene and similar low-dimensional nanostructures under the simultaneous action of temperature and force field. Ideas on atomistic of the force field influence on the fluctuation-induced atomic bond break is developed. Existence of two components of above effect is ascertained, namely (i) decrease in the energy barrier under the force action, and (ii) reduction of the energy cost for the bond break due to release of the accumulated energy of elastic deformations. On the example of carbyne–graphene nanoelement, it is shown that the effect of force field can cause a drop of lifetime by tens of orders of magnitude. This is a manifestation of the synergy effect of temperature and force field on the stability and lifetime of nanostructures. The obtained dependences are a convenient tool for predicting the lifetime of low-dimensional nanostructures, in particular, of straintronics elements. On the whole, the approach proposed may be considered as a generalization of Arrhenius’ theory of reactions in the case of force field action.

Appendices

Appendix A

Derivation of the approximation to calculate the integrals (2), (8), (11) and (12)

Calculation of the approximate values of integrals in general form.

• 1. Absence of the IZ, i.e., the work of internal forces is not taking into account (formula 2)

  The sub-integral function is expanded into a series in the vicinity of $\delta = \delta_{C}$ and $\delta = 0$, limiting expansion by the first two series members:

  $$\begin{gathered} \varepsilon \left( \delta \right) \approx \varepsilon \left( 0 \right) + \varepsilon^{/} \left( 0 \right) \cdot \delta = \varepsilon^{/} \left( 0 \right) \cdot \delta \hfill \\ \varepsilon_{C} \left( \delta \right) \approx \varepsilon \left( {\delta_{C} } \right) + \varepsilon^{/} \left( {\delta_{C} } \right) \cdot \left( {\delta - \delta_{C} } \right). \hfill \\ \end{gathered}$$

  (A1)

  Then, the value of the corresponding integral in (2) is:

  $$\begin{gathered} \int\limits_{{\delta_{C} }}^{{\delta_{br} }} {\exp \left[ { - \beta \varepsilon \left( \delta \right)} \right]} {\text{d}}\delta \approx \int\limits_{{\delta_{C} }}^{{\delta_{br} }} {\exp \left\{ { - \beta \left[ {\varepsilon \left( {\delta_{C} } \right) + \varepsilon^{/} \left( {\delta_{C} } \right) \cdot \left( {\delta - \delta_{C} } \right)} \right]} \right\}} {\text{d}}\delta \hfill \\ = - \frac{1}{{\beta \varepsilon^{/} \left( {\delta_{C} } \right)}} \cdot \exp \left[ { - \beta \varepsilon \left( {\delta_{C} } \right)} \right] \cdot \exp \left[ { - \beta \varepsilon^{/} \left( {\delta_{C} } \right) \cdot \left( {\delta - \delta_{C} } \right)} \right] \, \left| {_{{\delta_{C} }}^{{\delta_{br} }} } \right. \hfill \\ = \left| {\exp \left[ { - \beta \varepsilon^{/} \left( {\delta_{C} } \right) \cdot \left( {\delta_{br} - \delta_{C} } \right)} \right] < < \exp \left[ 0 \right]} \right| \approx \frac{1}{{\beta \varepsilon^{/} \left( {\delta_{C} } \right)}} \cdot \exp \left[ { - \beta \varepsilon \left( {\delta_{C} } \right)} \right]. \hfill \\ \end{gathered}$$

  (A2)

  The value of the statistical sum (Eq. 3) is:

  $$\begin{gathered} Z = \int\limits_{0}^{{\delta_{{{\text{br}}}} }} {\exp \left[ { - \beta \varepsilon \left( \delta \right)} \right]} {\text{d}}\delta \approx \int\limits_{0}^{{\delta_{{{\text{br}}}} }} {\exp \left[ { - \beta \varepsilon^{/} \left( 0 \right) \cdot \delta } \right]} {\text{d}}\delta = \hfill \\ = - \frac{1}{{\beta \varepsilon^{/} \left( 0 \right)}} \cdot \exp \left[ { - \beta \varepsilon^{/} \left( 0 \right) \cdot \delta } \right] \, \left| {_{0}^{{\delta_{{{\text{br}}}} }} } \right. = \left| {\exp \left[ { - \beta \varepsilon^{/} \left( 0 \right) \cdot \delta_{{{\text{br}}}} } \right] < < \exp \left[ 0 \right]} \right| \approx \frac{1}{{\beta \varepsilon^{/} \left( 0 \right)}}. \hfill \\ \end{gathered}$$

  (A3)

  Accordingly, from the formula (2) we obtain the following approximate expression for the probability of contact bond breaking:

  $$\begin{aligned}P\left( {\delta \ge \delta_{C} } \right) & \approx \frac{1}{{\beta \varepsilon^{/} \left( {\delta_{C} } \right)}} \cdot \exp \left[ { - \beta \varepsilon \left( {\delta_{C} } \right)} \right] \times \beta \varepsilon^{/} \left( 0 \right) \\ & = \frac{{\beta \varepsilon^{/} \left( 0 \right)}}{{\beta \varepsilon^{/} \left( {\delta_{C} } \right)}} \cdot \exp \left[ { - \beta \varepsilon \left( {\delta_{C} } \right)} \right]. \end{aligned}$$

  (A4)

  From the condition of the constant applied force $\varepsilon^{/} \left( {\delta_{C} } \right) \equiv \varepsilon^{/} \left( 0 \right) = F$ we have:

  $$P\left( {\delta \ge \delta_{C} } \right) \approx \exp \left[ { - \beta \varepsilon \left( {\delta_{C} } \right)} \right].$$

  (A5)

• 2. Existence of the IZ—“High-energy” (the 1st) mechanism (formula 8)

  All considerations related to formula (2) are also valid for formula (8), if $\varepsilon \left( \delta \right)$ is replaced by $\varepsilon_{IZ} \left( \delta \right) = \varepsilon \left( \delta \right) - A_{R} \left( F \right)$ in it, which takes into account the work of internal forces. That is, the value of integral in (8) will be the following:

  $$\int\limits_{{\delta_{C} }}^{{\delta_{{{\text{br}}}} }} {\exp \left[ { - \beta \varepsilon_{IZ} \left( \delta \right)} \right]} {\text{d}}\delta \approx \frac{1}{{\beta \varepsilon_{IZ}^{/} \left( {\delta_{C} } \right)}} \cdot \exp \left[ { - \beta \varepsilon_{{_{IZ} }} \left( {\delta_{C} } \right)} \right].$$

  (A6)

• 3. Existence of the IZ—“Low-energy” (the 2nd) mechanism (formula 11)

  The sub-integral function is expanded into a series in the vicinity of $\delta = \delta_{R}$. Similarly to (A1):

  $$\varepsilon_{R} \left( \delta \right) \approx \varepsilon \left( {\delta_{R} } \right) + \varepsilon^{/} \left( {\delta_{R} } \right) \cdot \left( {\delta - \delta_{R} } \right).$$

  (A7)

  So, the value of the corresponding integral in (11) is:

  $$\begin{gathered} \int\limits_{{\delta_{R} }}^{{\delta_{{{\text{br}}}} }} {\exp \left[ { - \beta \varepsilon_{R} \left( \delta \right)} \right]} {\text{d}}\delta \approx \int\limits_{{\delta_{C} }}^{{\delta_{{{\text{br}}}} }} {\exp \left\{ { - \beta \left[ {\varepsilon \left( {\delta_{R} } \right) + \varepsilon^{/} \left( {\delta_{R} } \right) \cdot \left( {\delta - \delta_{R} } \right)} \right]} \right\}} {\text{d}}\delta \hfill \\ \quad = - \frac{1}{{\beta \varepsilon^{/} \left( {\delta_{R} } \right)}} \cdot \exp \left[ { - \beta \varepsilon \left( {\delta_{R} } \right)} \right] \cdot \exp \left[ { - \beta \varepsilon^{/} \left( {\delta_{R} } \right) \cdot \left( {\delta - \delta_{R} } \right)} \right] \, \left| {_{{\delta_{R} }}^{{\delta_{{{\text{br}}}} }} } \right. = \hfill \\ \quad = \left| {\exp \left[ { - \beta \varepsilon^{/} \left( {\delta_{R} } \right) \cdot \left( {\delta_{{{\text{br}}}} - \delta_{R} } \right)} \right] < < \exp \left[ 0 \right]} \right| \approx \frac{1}{{\beta \varepsilon^{/} \left( {\delta_{R} } \right)}} \cdot \exp \left[ { - \beta \varepsilon \left( {\delta_{R} } \right)} \right]. \hfill \\ \end{gathered}$$

  (A8)

  Since $\varepsilon^{/} \left( {\delta_{R} } \right) = F_{R}$, then:

  $$\int\limits_{{\delta_{R} }}^{{\delta_{{{\text{br}}}} }} {\exp \left[ { - \beta \varepsilon_{R} \left( \delta \right)} \right]} {\text{d}}\delta \approx \frac{1}{{\beta F_{R} }} \cdot \exp \left[ { - \beta \varepsilon \left( {\delta_{R} } \right)} \right].$$

  (A9)

• 4. Existence of the IZ—the statistical sum (formula 12)

  Accounting for (A1), (A3) and (A8), we obtain:

  $$Z \approx \frac{1}{\beta F} + \left( {\delta_{R} - \delta_{un} } \right) \cdot \exp \left[ { - \beta \varepsilon \left( {\delta_{un} } \right)} \right] + \frac{1}{{\beta F_{R} }} \cdot \exp \left[ { - \beta \varepsilon \left( {\delta_{R} } \right)} \right].$$

  (A10)

Direct numerical integration in calculating the statistical sum indicates that the second and third terms are smaller than the first one by many orders of magnitude (at a temperature of 600 K: the second term—by about 20 orders of magnitude, the third term—by 13 orders), therefore, they may be neglected:

$$Z \approx \frac{1}{\beta F}.$$

(A11)

The probability of breaking the contact bond at the 1st mechanism is:

$$P_{{\text{I}}} = P\left( {\delta \ge \delta_{C} } \right) \approx \exp \left[ { - \beta \varepsilon_{IZ} \left( {\delta_{C} } \right)} \right] = \exp \left\{ { - \beta \left[ {\varepsilon \left( {\delta_{C} } \right) - A_{R} \left( F \right)} \right]} \right\}.$$

(A12)

The probability of breaking the contact bond at the 2nd mechanism is:

$$P_{{{\text{II}}}} = P\left( {\delta \ge \delta_{C} } \right) \approx \frac{\beta F}{{\beta F_{R} }} \cdot \exp \left[ { - \beta \varepsilon_{IZ} \left( {\delta_{R} } \right)} \right] = \frac{F}{{F_{R} }} \cdot \exp \left\{ { - \beta \left[ {\varepsilon \left( {\delta_{R} } \right) - A_{R} \left( F \right)} \right]} \right\},$$

(A13)

where $F_{R}$ is determined by the formula (1).

Appendix B

Lifetime formulae in an explicit form, as functions of relative load

To obtain the formulae for lifetime (19)–(21) explicitly as functions of the force field, first, one need to derive the expressions both for the length of critical fluctuation of the contact bond required to break it and for the corresponding fluctuations of energy, as well as the work of internal forces included in above formulae. Morse potential in differential form (24) was utilized for appropriate calculations.

To do this, the equation was solved relatively to the displacement value $u$:

$$E^{/} (u) = F\left( u \right) = F,$$

(B1)

and we obtain two roots—$u_{1} \left( F \right)$ and $u_{2} \left( F \right)$:

$$u_{1,2} \left( F \right) = \left( { - \frac{1}{b}} \right) \cdot \ln \left( {0.5 \pm 0.5 \cdot \sqrt {1 - \overline{F}} } \right),$$

(B2)

where

$$\overline{F} = \frac{2F}{{bE_{0} }} = \frac{F}{{F_{{{\text{un}}}} }},$$

(B3)

since the Morse potential parameters and the magnitude of the instability force are related as follows (from the condition of equality to 0 of the second derivative of the potential $\frac{{d^{2} E\left( u \right)}}{{du^{2} }} = \frac{dF\left( u \right)}{{du}} = 0$):

$$F_{{{\text{un}}}} = F_{\max } = \frac{{E_{0} \cdot b}}{2}.$$

(B4)

Accordingly, the value of critical fluctuation is:

$$\delta_{C} \left( F \right) = u_{2} \left( F \right) - u_{1} \left( F \right) = \frac{2}{b} \cdot \tanh^{ - 1} \left( {\sqrt {1 - \frac{2F}{{bE_{0} }}} } \right) = \frac{2}{b} \cdot \tanh^{ - 1} \left( {\sqrt {1 - \overline{F}} } \right).$$

(B5)

General expression for the magnitude of energy fluctuation is:

$$\varepsilon \left( F \right) = E_{0} \cdot \left( {\left\{ {\exp \left[ { - b\left( {u_{f} + \delta } \right)} \right] - 1} \right\}^{2} - \left\{ {\exp \left( { - bu_{f} } \right) - 1} \right\}^{2} } \right).$$

(B6)

Critical energy fluctuation (energy barrier) is:

$$\varepsilon_{C} \left( F \right) = E\left( {u_{f} + \delta } \right) - E\left( {u_{f} } \right) = E_{0} \cdot \sqrt {1 - \frac{2F}{{bE_{0} }}} = E_{0} \cdot \sqrt {1 - \frac{F}{{F_{{{\text{un}}}} }}} .$$

(B7)

For similar considering, we find

$$u_{R} \left( F \right) = - \frac{1}{b} \cdot \ln \left( {0.5 - 0.5 \cdot \sqrt {1 - \sqrt {1 - \alpha \cdot \overline{F}^{2} } } } \right),$$

(B8)

$$\delta_{R} \left( F \right) = u_{R} - u_{f} = \left( { - \frac{1}{b}} \right) \cdot \ln \frac{{1 - \sqrt {1 - \sqrt {1 - \alpha \cdot \overline{F}^{2} } } }}{{1 + \sqrt {1 - \overline{F}} }},$$

(B9)

$$E_{R} \left( F \right) = E\left( {u_{R} } \right) = E_{0} \cdot \left( {0.5 + 0.5 \cdot \sqrt {1 - \sqrt {1 - \alpha \cdot \overline{F}^{2} } } } \right)^{2} ,$$

(B10)

$$\varepsilon_{R} \left( F \right) = E\left( {u_{R} } \right) - E\left( {u_{f} } \right) = \frac{1}{4} \cdot E_{0} \cdot \left[ {\left( {1 + \sqrt {1 - \overline{F}_{R} } } \right)^{2} - \left( {1 - \sqrt {1 - \overline{F}} } \right)^{2} } \right].$$

(B11)

The elongation magnitude at the instability moment is:

$$u_{un} = \frac{\ln 2}{b},$$

(B12)

$$\delta_{un} \left( F \right) = u_{un} - u_{f} = \frac{{\ln \left( {1 + \sqrt {1 - \overline{F}} } \right)}}{b},$$

(B13)

$$E_{un} = E\left( {u_{un} } \right) = \frac{1}{4}E_{0} .$$

(B14)

Work of internal forces is:

$$A_{R} \left( F \right) = E_{R} - E_{un} = \frac{1}{4}E_{0} \cdot \left[ {\left( {1 + \sqrt {1 - \sqrt {1 - \alpha \cdot \overline{F}^{2} } } } \right)^{2} - 1} \right],$$

(B15)

or

$$A_{R} = \frac{1}{4} \cdot E_{0} \cdot \left[ {\left( {1 + \sqrt {1 - \overline{F}_{R} } } \right)^{2} - 1} \right],$$

(B16)

where

$$\overline{F}_{R} = \sqrt {1 - \alpha \cdot \overline{F}^{2} } .$$

(B17)

Substituting the values (B7), (B11) and (B15) into formulae (19)–(21), we obtain the corresponding formulae of lifetime in the explicit form.

For the case without the IZ:

$$\ln \frac{\tau }{{\tau_{0} }} = \frac{{E_{0} }}{{K_{B} T}} \cdot \sqrt {1 - \overline{F}} .$$

(B18)

For the 1st mechanism:

$$\ln \frac{\tau }{{\tau_{0} }} = \frac{{E_{0} }}{{K_{B} T}} \cdot \left\{ {\sqrt {1 - \overline{F}} - \frac{1}{4} \cdot \left[ {\left( {1 + \sqrt {1 - \sqrt {1 - \alpha \cdot \overline{F}^{2} } } } \right)^{2} - 1} \right]} \right\}.$$

(B19)

For the 2nd mechanism:

$$\ln \frac{\tau }{{\tau_{0} }} = \ln \frac{{\sqrt {1 - \alpha \overline{F}^{2} } }}{{\overline{F}}} + \frac{1}{4} \cdot \frac{{E_{0} }}{{K_{B} T}} \cdot \left[ {1 - \left( {1 - \sqrt {1 - \overline{F}} } \right)^{2} } \right].$$

(B20)