# Entropy-Driven Grain Boundary Segregation: Prediction of the Phenomenon

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## Abstract

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## 1. Introduction

_{I}, which is formed by the segregation enthalpy, ΔH

_{I}; and by the product of temperature, T, and the segregation entropy, ΔS

_{I},

_{I}(0 K) = ΔH

_{I}, and the value of the segregation entropy cannot be obtained. It is noteworthy that the segregation enthalpy is related to the segregation energy, $\Delta {E}_{I},$ by $\Delta {H}_{I}=\Delta {E}_{I}+P\Delta {V}_{I}.$ However, the value of the product $P\Delta {V}_{I}$ is extremely low at normal pressure and thus negligible [5], so that practically $\Delta {H}_{I}\cong \Delta {E}_{I}$, and the Gibbs energy of segregation is equal to the Helmholtz energy of segregation. Accordingly, in the following, we consider $\Delta {H}_{I}$ and $\Delta {E}_{I}$ to be identical. By the term “segregation entropy”, we consider all its contributions, such as vibrational, anharmonic and multiplicity, except mixing entropy [6]. Without segregation entropy, it is impossible to realistically quantify the temperature dependence of grain boundary composition, given by [7]

_{I}is the bulk concentration, and ${\theta}_{I}^{GB}$ is the grain boundary (GB) occupation ratio, i.e., the ratio of the grain boundary concentration, ${X}_{I}^{GB}$, and the maximum reachable concentration of solute I at the grain boundary (site) in host M, ${X}_{I}^{*,GB}$,

_{I}, is larger (in absolute values) than the enthalpy, ΔH

_{I}, i.e.,

_{I}if the values of the product $T\Delta {S}_{I}>\Delta {H}_{I}$? This would mean that a solute could additionally occupy an “anti-segregation” site due to the effect of the segregation entropy.

## 2. Proposal of Entropy-Driven Grain Boundary Segregation

^{−1}. Numerous data, mainly for the segregation of Sb, Sn, P, Si and V, fit Equation (6) under the condition $\Delta {H}_{I}^{0}\le 0$.

_{CE}is the compensation temperature and $\Delta {S}^{\prime}$ is the integration constant of entropy character, as explained in detail in [11]. The dependence (7) is split into two branches, one for substitutional segregants, and the other for interstitially segregated solutes. In bcc iron-based systems, the values of $\Delta {S}^{\prime}$ are $\Delta {S}_{interstitial}^{\prime}$ = 56 J mol

^{−1}K

^{−1}and $\Delta {S}_{substitutional}^{\prime}$ = 5 J mol

^{−1}K

^{−1}, and T

_{CE}= 900 K [10]. The linear relationship (7) for the values of $\Delta {H}_{I}^{0}$ and $\Delta {S}_{I}^{0}$ reports on the compensation of the changes in the standard enthalpy of grain boundary segregation due to the changed grain boundary structure by respective changes in the standard segregation entropy, and this phenomenon is generally called the enthalpy–entropy compensation effect [11].

_{I}and positive and negative values of ΔS

_{I}in principle (Figure 1). In the case of ΔH

_{I}> 0 and ΔS

_{I}< 0, according to Equation (1), ΔG

_{I}> 0 at any temperature (cf. Equation (2)) and no segregation occurs. However, if the segregation entropy is positive, condition (6) can be fulfilled for the pairs of segregation enthalpy and entropy, thus resulting in ΔG

_{I}< 0 due to the prevailing value of the entropy term. The region of principal validity of ΔG

_{I}< 0 is shown by the horizontally hatched area in Figure 1. Due to the “anti-segregation” tendency of the segregation energy, the segregation in this area would be exclusively controlled by the entropy term, which we refer to as entropy-driven grain boundary segregation. It is noteworthy that this statement is only based on the fact that Equation (6) is fulfilled in a specific part of the plot $\Delta {H}_{I}^{0}$ vs. $\Delta {S}_{I}^{0}$. As is apparent from Figure 1, entropy-driven grain boundary segregation should be an extreme part of a more general phenomenon—entropy-dominated grain boundary segregation.

## 3. Indirect Support of Entropy-Driven Grain Boundary Segregation

_{H}

_{=TS}, at which the absolute values of the energy and entropy terms are equal, are given for individual data in Table 1. It is noteworthy that the values of the standard enthalpy and entropy were applied to estimate T

_{H}

_{=TS}in Equation (6) so that some deviations can be expected in real systems. However, as the values of ${\alpha}_{I}$ are positive, they reduce the value of ΔH

_{I}and, consequently, reduce the values of T

_{H}

_{=TS}compared to those shown in Table 1. Thus, the interaction strengthens the grain boundary segregation tendency of a particular solute.

_{C}= +1.3 kJ mol

^{−1}calculated by Hendy et al. [24]. According to Equation (7) for interstitial segregation ($\Delta {S}_{interstitial}^{\prime}$ = 56 J mol

^{−1}K

^{−1}, T

_{CE}= 900 K [10]), ΔS

_{C}= +57.4 J mol

^{−1}K

^{−1}. Using Equation (2) and these input data, we calculated the temperature dependence of C segregation at this boundary for X

_{C}= 0.0001 using the interaction parameter ${\alpha}_{C}$ = +7.7 kJ mol

^{−1}(Table 1) in Equations (2)–(5). This dependence is shown in Figure 3. It is apparent that the course of this dependence is reversed to the usual course, which is characterized by a negative value of ΔH

_{C}(with maximum ${\theta}_{C}$ = 1 at 0 K). it is worth mentioning that with increasing temperature the value of ${\theta}_{C}$ gradually approaches the maximum, i.e., the saturation value of segregation, given by ${X}_{C}\mathrm{exp}\left(-\Delta {S}_{I}/R\right)$. To assess the maximum extent of segregation, the grain boundary enrichment ratio was obtained as ${\theta}_{C}/{X}_{C}\approx \mathrm{exp}\left(-\Delta {S}_{C}/R\right)$ = 996. This suggests that grain boundary segregation can occur principally at high temperatures, even if the calculated segregation energy at 0 K is positive.

_{V}= 0.1). We used the data calculated by Kholtobina et al. for the {111} grain boundary [18]. The values of ΔH

_{V}are different for different sites of the {111} grain boundary shown in [18] (see also inset in Figure 4), i.e., +6.8 kJ mol

^{−1}(sites +1 and −1 as marked in [18]) listed in Table 1 but also −16.4 kJ mol

^{−1}(site 0) and −12.5 kJ mol

^{−1}(sites +2 and −2). For these energies, the values of the segregation entropy were determined for substitutional segregation according to Equation (7) as +12.5 J mol

^{−1}K

^{−1}(sites +1 and −1), −13.2 J mol

^{−1}K

^{−1}(site 0) and −8.9 J mol

^{−1}K

^{−1}(sites +2 and −2). The temperature dependence of the V concentration at individual sites is shown in Figure 4.

_{V}and ΔS

_{V}are responsible for different temperature dependences of solute segregation at individual sites. It is worth noting that the site characterized by positive segregation energy also exhibits segregation at temperatures above 600 K (note that the temperature of practical applications is 723 K), and it even dominates the segregation behavior at temperatures above 900 K. This reversion is a further consequence of the enthalpy–entropy compensation effect [6]. We can also calculate the average concentration of V at this grain boundary, ${\theta}_{V}^{GB}$, according to a proposal of White and Coghlan [28] and of Nowicki et al. [29],

_{i}is the weight of the site i, $\sum {f}_{i}=1$. At the {111} grain boundary, 5 sites, −2, −1, 0, +1 and +2, contribute to the segregation [18]. If we accept the equal probability of participation of all sites i, f

_{i}= 0.2, the average temperature dependence of the grain boundary concentration of V can be determined, which is marked as AVE in Figure 4. It can be described by the values ΔH

_{V}= −7.3 kJ mol

^{−1}and ΔS

_{V}= −2.9 J mol

^{−1}K

^{−1}. Although there are no suitable experimental data for comparison, we can model the temperature dependence using our recent prediction [25]. An excellent quantitative agreement was obtained for the prediction using the values corresponding to vanadium segregation at a general grain boundary [25], $\Delta {H}_{V}^{0}$ = −7.9 kJ mol

^{−1}, $\Delta {S}_{V}^{0}$ = −3.8 J mol

^{−1}K

^{−1}, and ${\alpha}_{V}$ = +0.9 kJ mol

^{−1}determined according to Equation (7). This dependence is represented in Figure 4 by solid circles. It is noteworthy that this excellent agreement was obtained using (i) all values of ΔH

_{V}, i.e., including its positive values, and (ii) the values of ΔS

_{V}estimated from the extrapolated compensation effect. This finding has two important consequences. First, the “anti-segregation” sites contribute equally to the average grain boundary concentration, thus supporting the existence of entropy-driven grain boundary segregation. Indeed, at low temperatures, the “anti-segregation” character of sites ±1 prevails, resulting in a reduced concentration of the segregant at the site compared to the bulk (line 1 in Figure 4). However, at higher temperatures, when the entropy term prevails over the positive segregation energy, thus resulting in negative value of the Gibbs energy of segregation, solute segregation occurs. This is exclusively caused by the reverse of segregation tendency due to the change in the sign of the Gibbs energy of segregation. Second, it also justifies the extrapolation of the compensation effect in as performed in this study.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Regions of entropy-dominated grain boundary segregation (horizontal hatching, $\Delta {H}_{I}^{0}\le 0$) and entropy-driven grain boundary segregation (vertical hatching, $\Delta {H}_{I}^{0}>0$) for binary iron-based systems with the limiting temperature T = 723 K. Full lines marked as “interstitial” and “substitutional” represent the branches of the enthalpy–entropy compensation effect, Equation (7) from [10], and the dashed lines represent their respective extrapolations (the slope of these lines is the reciprocal value of T

_{CE}, 1/900 K

^{−1}in both cases). Various symbols represent the experimental values of segregation of numerous solutes at individual grain boundaries, data from [7,10].

**Figure 3.**Tempe rature dependence of C segregation at the {112} grain boundary of bcc Fe calculated according to Equations (2)–(5), for X

_{C}= 0.0001. The value of ΔH

_{C}= +1.3 kJ mol

^{−1}was taken from Reference [24], and the value of $\Delta {S}_{C}^{0}$ = +57.4 J mol

^{−1}K

^{−1}was determined using Equation (7) from [25]. The value ${\alpha}_{I}$ = 7.7 kJ mol

^{−1}from [25] was applied.

**Figure 4.**Temperature dependence of V segregation at individual sites 0, ±1 and ±2 at the {111} grain boundary of bcc iron calculated according to Equation (2) for the values of ΔH

_{V}published in [18] (see the inserted Figure) and the respective values of ΔS

_{V}determined by Equation (7) from [25]. Individual dependences are depicted by the dashed, dotted and dash-dotted lines. The full line corresponds to the arithmetic average of the grain boundary concentrations over all sites (AVE), the solid symbols represent the data obtained by means of the predicted values $\Delta {H}_{V}^{0}$ = −7.9 kJ mol

^{−1}, $\Delta {S}_{V}^{0}$ = −3.8 J mol

^{−1}K

^{−1}, and ${\alpha}_{V}$ = +0.9 kJ mol

^{−1}; data from [25].

**Table 1.**Some candidates of the grain boundary sites and solutes to exhibit entropy-driven grain boundary segregation in α-iron. The values of the segregation enthalpy (energy), $\Delta {H}_{I}^{0}$, were calculated by DFT methods in the respective references. Values of the entropy, ΔS

_{I}, were determined according to Equation (7) from [25]. ${\alpha}_{I}$ is the binary (Fowler) interaction coefficient estimated according to data from [25]. T

_{H}

_{=TS}is the temperature for the equal contribution of enthalpy and entropy terms, T

_{H}

_{=TS}= $\Delta {H}_{I}^{0}$/$\Delta {S}_{I}^{0}$. Above this temperature, entropy terms dominate.

Solute | Grain Boundary | $\mathbf{\Delta}{\mathit{H}}_{\mathit{I}}^{0}$(kJ mol^{−1}) | Ref. | $\mathbf{\Delta}{\mathit{S}}_{\mathit{I}}^{0}$(J mol^{−1} K^{−1}) | ${\mathit{\alpha}}_{\mathit{I}}$(kJ mol^{−1}) [25] | T_{H}_{=TS}(K) |
---|---|---|---|---|---|---|

C | {112} | +1.3 | [24] | +57.4 | 7.7 | 2 |

B | {111} | +3.9 | [18] | +60.3 | 13.3 | 65 |

+13.5 | [18] | +71.0 | 190 | |||

+4.2 | [19] | +60.7 | 7 | |||

+8.3 | [19] | +65.2 | 127 | |||

N | {111} | +14.5 | [18] | +72.1 | 5.6 | 201 |

+11.6 | [18] | +68.9 | 168 | |||

{111} | +9.7 | [22] | +66.7 | 145 | ||

{210} | +12.6 | [22] | +70.0 | 180 | ||

O | {210} | +73.5 | [22] | +137.7 | 15.5 | 536 |

+78.3 | [22] | +143.0 | 548 | |||

S | {111} | +35.8 | [22] | +95.8 | 10.8 | 374 |

{210} | +26.1 | [22] | +85.0 | 307 | ||

P | {210} | +4.8 | [22] | +61.3 | 4.5 | 78 |

V | {111} | +6.8 | [18] | +12.5 | 0.9 | 544 |

+7.3 | [20] | +13.1 | 557 | |||

+3.3 | [23] | +8.6 | 384 | |||

{210} | +0.5 | [26] | +5.6 | 89 | ||

Mo | {111} | +13.8 | [20] | +20.3 | 2.3 | 680 |

Re | {111} | +1.9 | [18] | +7.1 | 1.5 | 268 |

Tc | {111} | +11.6 | [18] | +17.9 | 4.1 | 648 |

Cr | {111} | +2.8 | [21] | +8.1 | 1.1 | 346 |

+4.8 | [18] | +10.4 | 462 | |||

{210} | +8.6 | [23] | +14.6 | 589 | ||

{332} | +5.6 | [23] | +11.2 | 500 | ||

Sc | {111} | +18.2 | [23] | +25.2 | 7.0 | 722 |

C | {210} | +7.7 | [22] | +13.6 | 5.1 | 566 |

Al | {111} | +4.8 | [27] | +10.4 | 1.0 | 462 |

N | {210} | +6.8 | [24] | +12.6 | 3.7 | 540 |

S | {210} | +6.8 | [24] | +12.6 | 7.2 | 540 |

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Lejček, P.; Hofmann, S. Entropy-Driven Grain Boundary Segregation: Prediction of the Phenomenon. *Metals* **2021**, *11*, 1331.
https://doi.org/10.3390/met11081331

**AMA Style**

Lejček P, Hofmann S. Entropy-Driven Grain Boundary Segregation: Prediction of the Phenomenon. *Metals*. 2021; 11(8):1331.
https://doi.org/10.3390/met11081331

**Chicago/Turabian Style**

Lejček, Pavel, and Siegfried Hofmann. 2021. "Entropy-Driven Grain Boundary Segregation: Prediction of the Phenomenon" *Metals* 11, no. 8: 1331.
https://doi.org/10.3390/met11081331