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

Appendix ADerivation 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 \(\delta = \delta_\) and \(\delta = 0\), limiting expansion by the first two series members:

$$\begin \varepsilon \left( \delta \right) \approx \varepsilon \left( 0 \right) + \varepsilon^ \left( 0 \right) \cdot \delta = \varepsilon^ \left( 0 \right) \cdot \delta \hfill \\ \varepsilon_ \left( \delta \right) \approx \varepsilon \left( } \right) + \varepsilon^ \left( } \right) \cdot \left( } \right). \hfill \\ \end$$

(A1)

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

$$\begin \int\limits_ }}^ }} \right]} }\delta \approx \int\limits_ }}^ }} } \right) + \varepsilon^ \left( } \right) \cdot \left( } \right)} \right]} \right\}} }\delta \hfill \\ = - \frac \left( } \right)}} \cdot \exp \left[ } \right)} \right] \cdot \exp \left[ \left( } \right) \cdot \left( } \right)} \right] \, \left| }}^ }} } \right. \hfill \\ = \left| \left( } \right) \cdot \left( - \delta_ } \right)} \right] < < \exp \left[ 0 \right]} \right| \approx \frac \left( } \right)}} \cdot \exp \left[ } \right)} \right]. \hfill \\ \end$$

(A2)

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

$$\begin Z = \int\limits_^}}} }} \right]} }\delta \approx \int\limits_^}}} }} \left( 0 \right) \cdot \delta } \right]} }\delta = \hfill \\ = - \frac \left( 0 \right)}} \cdot \exp \left[ \left( 0 \right) \cdot \delta } \right] \, \left| ^}}} }} } \right. = \left| \left( 0 \right) \cdot \delta_}}} } \right] < < \exp \left[ 0 \right]} \right| \approx \frac \left( 0 \right)}}. \hfill \\ \end$$

(A3)

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

$$\beginP\left( } \right) & \approx \frac \left( } \right)}} \cdot \exp \left[ } \right)} \right] \times \beta \varepsilon^ \left( 0 \right) \\ & = \frac \left( 0 \right)}} \left( } \right)}} \cdot \exp \left[ } \right)} \right]. \end$$

(A4)

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

$$P\left( } \right) \approx \exp \left[ } \right)} \right].$$

(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 \(\varepsilon_ \left( \delta \right) = \varepsilon \left( \delta \right) - A_ \left( F \right)\) in it, which takes into account the work of internal forces. That is, the value of integral in (8) will be the following:

$$\int\limits_ }}^}}} }} \left( \delta \right)} \right]} }\delta \approx \frac^ \left( } \right)}} \cdot \exp \left[ }} \left( } \right)} \right].$$

(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 \(\delta = \delta_\). Similarly to (A1):

$$\varepsilon_ \left( \delta \right) \approx \varepsilon \left( } \right) + \varepsilon^ \left( } \right) \cdot \left( } \right).$$

(A7)

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

$$\begin \int\limits_ }}^}}} }} \left( \delta \right)} \right]} }\delta \approx \int\limits_ }}^}}} }} } \right) + \varepsilon^ \left( } \right) \cdot \left( } \right)} \right]} \right\}} }\delta \hfill \\ \quad = - \frac \left( } \right)}} \cdot \exp \left[ } \right)} \right] \cdot \exp \left[ \left( } \right) \cdot \left( } \right)} \right] \, \left| }}^}}} }} } \right. = \hfill \\ \quad = \left| \left( } \right) \cdot \left( }}} - \delta_ } \right)} \right] < < \exp \left[ 0 \right]} \right| \approx \frac \left( } \right)}} \cdot \exp \left[ } \right)} \right]. \hfill \\ \end$$

(A8)

Since \(\varepsilon^ \left( } \right) = F_\), then:

$$\int\limits_ }}^}}} }} \left( \delta \right)} \right]} }\delta \approx \frac }} \cdot \exp \left[ } \right)} \right].$$

(A9)

4.

Existence of the IZ—the statistical sum (formula 12)

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

$$Z \approx \frac + \left( - \delta_ } \right) \cdot \exp \left[ } \right)} \right] + \frac }} \cdot \exp \left[ } \right)} \right].$$

(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:

$$Z \approx \frac.$$

(A11)

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

$$P_}} = P\left( } \right) \approx \exp \left[ \left( } \right)} \right] = \exp \left\ } \right) - A_ \left( F \right)} \right]} \right\}.$$

(A12)

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

$$P_}}} = P\left( } \right) \approx \frac }} \cdot \exp \left[ \left( } \right)} \right] = \frac }} \cdot \exp \left\ } \right) - A_ \left( F \right)} \right]} \right\},$$

(A13)

where \(F_\) is determined by the formula (1).

Appendix BLifetime 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—\(u_ \left( F \right)\) and \(u_ \left( F \right)\):

$$u_ \left( F \right) = \left( } \right) \cdot \ln \left( } } \right),$$

(B2)

where

$$\overline = \frac }} = \frac}}} }},$$

(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 \(\frac E\left( u \right)}} }} = \frac} = 0\)):

$$F_}}} = F_ = \frac \cdot b}}.$$

(B4)

Accordingly, the value of critical fluctuation is:

$$\delta_ \left( F \right) = u_ \left( F \right) - u_ \left( F \right) = \frac \cdot \tanh^ \left( }}} } \right) = \frac \cdot \tanh^ \left( } } \right).$$

(B5)

General expression for the magnitude of energy fluctuation is:

$$\varepsilon \left( F \right) = E_ \cdot \left( + \delta } \right)} \right] - 1} \right\}^ - \left\ } \right) - 1} \right\}^ } \right).$$

(B6)

Critical energy fluctuation (energy barrier) is:

$$\varepsilon_ \left( F \right) = E\left( + \delta } \right) - E\left( } \right) = E_ \cdot \sqrt }}} = E_ \cdot \sqrt }}} }}} .$$

(B7)

For similar considering, we find

$$u_ \left( F \right) = - \frac \cdot \ln \left( ^ } } } \right),$$

(B8)

$$\delta_ \left( F \right) = u_ - u_ = \left( } \right) \cdot \ln \frac^ } } }}} }},$$

(B9)

$$E_ \left( F \right) = E\left( } \right) = E_ \cdot \left( ^ } } } \right)^ ,$$

(B10)

$$\varepsilon_ \left( F \right) = E\left( } \right) - E\left( } \right) = \frac \cdot E_ \cdot \left[ _ } } \right)^ - \left( } } \right)^ } \right].$$

(B11)

The elongation magnitude at the instability moment is:

$$\delta_ \left( F \right) = u_ - u_ = \frac} } \right)}},$$

(B13)

$$E_ = E\left( } \right) = \fracE_ .$$

(B14)

Work of internal forces is:

$$A_ \left( F \right) = E_ - E_ = \fracE_ \cdot \left[ ^ } } } \right)^ - 1} \right],$$

(B15)

or

$$A_ = \frac \cdot E_ \cdot \left[ _ } } \right)^ - 1} \right],$$

(B16)

where

$$\overline_ = \sqrt ^ } .$$

(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:

$$\ln \frac }} = \frac }} T}} \cdot \sqrt } .$$

(B18)

For the 1st mechanism:

$$\ln \frac }} = \frac }} T}} \cdot \left\} - \frac \cdot \left[ ^ } } } \right)^ - 1} \right]} \right\}.$$

(B19)

For the 2nd mechanism:

$$\ln \frac }} = \ln \frac^ } }}}} + \frac \cdot \frac }} T}} \cdot \left[ } } \right)^ } \right].$$

(B20)

留言 (0)

沒有登入
gif