Grain boundary configurational entropy: a challenge

COMPUTATION & THEORY Grain boundary configurational entropy: a challenge Pavel Lejc ˇek 1,2, * and Andrea S ˇkola ´kova ´ 1 1 Institute of Physics, Czech Academy of Sciences, Na Slovance 2, 182 21 Prague 8, Czech Republic 2 Central European Institute of Technology, CEITEC BUT, Brno University of Technology, Purkyn ˇova 123, 612 69 Brno, Czech Republic Received: 17 April 2023 Accepted: 24 May 2023 Published online: 12 June 2023 ÓThe Author(s) 2023 ABSTRACT While the bulk of the high-entropy alloys is widely studied and characterized by their configurational entropy, there is a lack of general information regarding the configurational entropy of the grain boundaries. Here, we derived for the first time the basic relationships of this thermodynamic quantity related to the solute segregation at grain boundaries. Some examples of the appearance of the grain boundary configurational entropy are shown, and its effect on intergranular properties is discussed. It is stated that the role of grain boundary configurational entropy in interfacial properties is not completely clear and represents a challenge for future research. Introduction One of the new, dynamically developing fields of materials science, is complex concentrated systems, also called high-entropy alloys (HEAs) [1]. The concept of HEAs containing 5 or more principal elements has attracted scientific attention since Cantor [1] and Yeh [2] published their influential papers independently of each other. It is generally claimed that such complex concentrated systems exhibit high configurational entropy of the random mixing of elements leading to the formation of simple solid solutions or their mixture and preventing the formation of intermetallic phases. Therefore, the tendency for the formation of any clusters or precipitation of phases is lowered [3]. The value of configurational entropy thus mainly affects the stability of phases [4]. However, it is widely shown that the presence of intermetallic phases is highly likely [5–8]. In work [7] was shown that the formation of intermetallic phases is possible only at an intermediate temperature range. Such temperature range affects the role of entropy which starts to lose dominance and the formation of intermetallic phases is favored and triggered due to sufficient diffusion. The crystallographic structures of solid solutions are simple fcc [1,9], bcc [10–12], and hcp [13,14] types depending on the chemical composition of HEAs. The solid solutions are kinetically stable due to the sluggish diffusion of atoms; thereby, the possible precipitation of intermetallic phases is also suppressed. HEAs have become studied intensively due to their unique microstructure, phase composition, and mainly adjustable properties, and opened new possibilities and strategies in advanced alloy design. Among the Handling Editor: P. Nash. Address correspondence to E-mail: [email protected] https://doi.org/10.1007/s10853-023-08634-w J Mater Sci (2023) 58:10043–10057 Computation & theory exceptional properties belong high strength, hardness, excellent wear resistance and high-temperature strength, structure stability, and corrosion and oxidation resistance [15]. As was mentioned above, one of the general characteristics of these materials is the high value of the configurational entropy which seems to be responsible for the exceptional properties of the HEAs. However, the configurational entropy can also be applied to the grain boundaries. Recently, a concept of high-entropy grain boundaries was proposed [16,17]. The main advantage of these grain boundaries whose composition is affected by solute segregation, is high value of the configurational entropy which may become origin of exceptional properties such as suppression of interfacial precipitation and stabilization of nanocrystalline structures [16,18]. Nevertheless, it was argued in [19] that high mixing entropy may not be sufficient to prevent the segregation of the elements, although other authors claimed the opposite [2]. Grain boundaries generally act as barriers for the motion of dislocations resulting in the enhancing of the strength or hardness of alloys. This strengthening could be reduced at the intermediate temperatures as the boundaries are weakened by the segregation of solutes and/or impurities [20]. On the other hand, the ductility of alloys tested at high temperatures increased thanks to the easy motion of the boundaries. The pure metals with fcc structure as well as the alloys with low stacking fault energy (SFE) exhibit high ductility in a wide temperature range. The opposite trend could be observed in the case of the most studied HEAs, specifically Cantor alloy with equimolar chemical composition CoNiFeCrMn. As mentioned above, the ductility decreased at higher temperatures, especially at the intermediate temperatures [21], although the alloy is single fcc-phase solid solution, as well. This was attributed to the nanosegregation of one of the Ni, Cr, and Mn which resulted in the decohesion of the grain boundary [22,23]. Schuh et al. [24] emphasized that strengthening cannot be exclusively explained only by the consideration of solute segregation at grain boundaries. Further, the segregation could also negatively influence the initial stage of corrosion. The dissolution of Cantor alloy is accelerated when the Cr-enriched phases form due to the ongoing microsegregation [25]. However, grain boundary segregation and grain boundary engineering have not been studied and developed extensively in the case of HEAs so far. Therefore, there is a substantial lack of information. One of the unsolved fundamental problems in this respect is the relationship between the configurational entropy and the thermodynamic characteristics of grain boundary segregation. In this context, a general question arises what the configurational entropy of the grain boundaries is itself, and how is related to that of the bulk material as well as to the characteristics of the solute segregation not only in HEAs but also in diluted systems. This topic is discussed in this paper for the first time. Thermodynamic fundamentals The total configurational entropy, TSconf (in J K –1 ), in an n-component real system is defined as TSconf ¼X n i niS0;conf i¼RX n i niln Xi;ð1Þ where niand Xiare the number of moles and the mole fractions in the alloy, respectively, of component i,S0;conf i(in J mol –1 K –1 ) is the molar entropy of component I, and R= 8.3141 J mol –1 K –1 is the universal gas constant [26]. It is worth noting that the configurational entropy differs from the real mixing entropy, defined as TSmix ¼X n i niSi¼ THmix TRX n i niln ai;ð2Þ where ai¼ciXiis the activity of component iin the alloy, ciis the activity coefficient of component i, THmix (kJ mol –1 ), is the total mixing enthalpy and Siis the partial molar entropy of component i[26]. Equation (1) can also be rewritten to express the molar configurational entropy as Sconf ¼X n i XiS0;conf i¼RX n i Xiln Xi;ð3Þ Besides their application to the bulk, expressions (1) and (2) can also be applied to the grain boundary (GB), Sconf GB ¼X n i XGB iS0;conf;GB i¼RX n i XGB iln XGB i;ð4Þ where the index GB relates the variable to the grain boundary. Accepting 10044 J Mater Sci (2023) 58:10043–10057 XGB M¼1X n6¼M i XGB i;ð5Þ where Mis the host component, we can rewrite Eq. (4)as Sconf GB ¼RX n6¼M i XGB iln XGB i XGB M þln XGB M ! :ð6Þ In contrast to Sconf , the grain boundary configurational entropy, Sconf GB , is temperature dependent as the grain boundary concentrations change substantially with temperature due to the segregation effects which reflect the minimization of the Gibbs energy of the system. Under some assumptions (segregation of all solutes at the substitutional sites, full coverage of the grain boundaries by all solutes, monolayer segregation), the general segregation isotherm of the Langmuir–McLean type for a real system can be written as [27] aGB I aGB M ¼aI aM exp DG0 I RT  :ð7Þ Alternatively, XGB I 1Pj6¼MXGB j ¼XI 1Pj6¼MXj exp DGI RT  ;ð8aÞ i.e., XGB I¼XIexp DGI=RTðÞ 1Pj6¼MXj1exp DGj=RT  ;ð8bÞ In Eq. (7), DG0 I¼DH0 ITDS0 I;ð9Þ T(K) is the temperature, DG0 Iis the standard (ideal) Gibbs energy of segregation of solute Iat the grain boundaries, composed of the standard enthalpy of grain boundary segregation, DH0 I, and the standard entropy of grain boundary segregation, DS0 I.In Eqs. (8a) and (8b), DGI¼DHITDSI¼DG0 IþDGE I:ð10Þ In Eq. (10), DHIand DSIare the enthalpy and entropy of segregation, respectively, in a real system. DGE Iis the excess Gibbs energy of segregation representing the difference between the Gibbs energy of segregation and the standard Gibbs energy of segregation [27]. Its value is hardly measurable and therefore, it is frequently estimated using the binary (Fowler) and ternary (Guttmann) interaction coefficients [27,28]. As Eq. (8b) is identical with the Butler equation [29]ifDGI¼exp xðr0 Ir0 MÞ=RT  , where x is the molar grain boundary area and r0 I;Mare the molar energies of the components Iand M. Nevertheless, the nature of DGIcan further be extended based on, e.g., extended Butler equation [29,30] and Wynblatt model [28,31]. As the nature of DGIis not primarily important in the relationship with grain boundary configurational entropy, we will consider it here only in the sense of Eq. (10). To get primary insight into the relationship between the configurational entropy and the grain boundary segregation, we will consider a binary system for simplicity. As follows from Eq. (2), the partial molar entropy of component iin a binary alloy can be expressed as Sconf i¼oTSconf oni  T;nj ¼Rln Xi:ð11Þ This expression is valid for both the bulk and the grain boundary GB, as well as for solutes iand host metal M. Consequently, we can adopt Eq. (6) for an ideal binary system to get exp Sconf;GB ISconf;GB M  =R hi exp Sconf ISconf M  =R  ¼exp DGI RT  ;ð12Þ i.e., Sconf;GB ISconf;GB M  Sconf ISconf M  ¼oSconf GB oXGB I  T oSconf oXI  T ¼DGI T¼DHI TDSI ¼DH0 I TDS0 IþDGE I T:ð13Þ Differentiation of Sconf GB for binary alloy (Eq. (6)) by XGB Iat constant temperature results in oSconf GB oXGB I  T ¼Rln XGB I 1XGB I ¼Rln XI 1XI DGI RT  ; ð14Þ and differentiation by Tprovides dSconf GB dT ¼XGB I1XGB I  DH0 IþDGE IdDGE I=dT RT2:ð15Þ Using Eqs. (6) and (8b), we get J Mater Sci (2023) 58:10043–10057 10045 Sconf GB ¼RPj6¼MXjexp DGj=RT  ln Xj=1Pj6¼MXj  DGj=RT hi 1Pj6¼MXj1exp DGj=RT  ð16Þ Equation (16) provides the direct relationship between the segregation quantities and the configurational entropy in a real system. In principle, we can also apply more sophisticated approaches to describe the segregation isotherms, e.g., for multilayer segregation based on the BET (Brunauer–Emmett–Teller) approach [32]. Up to now, we treated the problem from the viewpoint of averaged grain boundary composition and corresponding averaged (= effective) thermodynamic quantities. However, theoretical calculations provide us with the energy of solute segregation for individual grain boundary sites which might result in the concentrations at these sites. To refine the above relationships, we can rewrite Eq. (8b) according to the White and Coghlan model [33] for a single grain boundary as XGB I¼X N k nkXGB I;k ¼X N k nk XIexp DGI;k=RT  1Pj6¼MXj1exp DGj;k=RT  ; ð17Þ with nkbeing a weight factor for individual grain boundary sites fulfilling the condition P N k nk¼1 and Nbeing the number of grain boundary sites. Accordingly, Sconf GB ¼RX N k nkX N k XGB k;iln XGB k;i ! :ð18Þ Under some assumptions, we can consider the segregation energy as a combination of the segregation energies of sites k,DEI;k,[34] DEI¼X N k¼1 nkDEI;k:ð19Þ Analogous expressions will hold for the other thermodynamic quantities of grain boundary segregation. Then, the above given formulas can be refined by using Eqs. (17)–(19). For simplicity, we will deal here with the averaged quantities. A further reason is that we do not have representative data calculated for individual grain boundary sites in complex multicomponent systems. Relations between configurational entropy and grain boundary segregation Note to high-entropy alloys It is evident from Eq. (14) that the maximum grain boundary configurational entropy in a binary alloy is obtained for its equimolar composition. It is worth noting that the condition of maximum configurational entropy also holds for equimolar composition in an n-component alloy. Due to the similarity of Eqs. (3) and (4), the same conclusion is also drawn for the equimolar crystal bulk supposing it represents the homogeneous solid solution. This result has an important consequence. If we have, e.g., 5-component single phase alloy with equimolar bulk composition (i.e., atomic concentration of each solute is 0.2), then Sconf ¼1:609 R.However, after annealing the solutes segregate to the grain boundary and thus, the composition of the grain boundary is no longer equimolar. An example representing this situation in a nearly equimolar FeMnNiCoCr alloy after annealing at 450 °C for various time periods [35] is given in Table 1. It is apparent that this annealing results in gradual solute segregation and consequently, in the reduction of the configurational entropy. In all cases, its values are lower than that in the bulk and it loses the character of a HEA region if Sconf GB \1:5R[36]. This situation can also be interpreted as an ordering of the grain boundary. In fact, this is a common feature of the HEAs. However, in the quinary system Fe–Mn–Ni–Co–Cr with non-equimolar compositions, we can find conditions for reaching a maximum grain boundary Table 1 Grain boundary composition (at%) of a quinary alloy [35] Fe Mn Ni Co Cr Sconf=R Bulk 18.9 19.9 20.6 20.1 20.5 1.609 GB 450 °C/6 h 15 25 34 15 11 1.525 GB 450 °C/18 h 2 38 53 4 3 1.016 GB 450 °C/48 h 1 41 54 3 1 0.896 10046 J Mater Sci (2023) 58:10043–10057 configurational entropy of 1.609 R. Accepting for simplicity a very rough assumption of the ideality of this system and accepting the values of the Gibbs energy of segregation in bcc iron at 800 K to be -3.8 kJ mol –1 for Co, -5.6 kJ mol –1 for Cr, -6.6 kJ mol –1 for Mn, and -6.6 kJ mol –1 for Ni (the absolute values of the Fowler coefficients are less than 3 kJ mol –1 )[28], the equimolar composition of a general grain boundary and thus Sconf GB = 1.609 Rcan be obtained for the bulk composition of this alloy being 9 at% Fe, 16 at% Co, 25 at% Cr, 25 at% Ni and 25 at% Mn exhibiting Sconf ¼1:550 R. Similarly, the maximum value of the grain boundary configurational entropy can be reached for slightly tuned bulk composition at different temperatures. Dilute binary alloys An opposite situation occurs in the case of grain boundary segregation in dilute alloys. Here, the composition of the grain boundaries is characterized by increased solute concentration and reduction of the concentration of the host metal. This fact contributes to an increase in the grain boundary configurational entropy compared to that of the bulk. We can demonstrate it by the example of diluted Fe–P alloys. The grain boundary segregation in P-doped bcc iron-based alloys has been studied rather frequently since the pioneering quantitative work of Erhart and Grabke on polycrystalline Fe–P-based alloys [37]. The measurements of its segregation in polycrystalline materials [37] as well as in well-characterized bicrystals [38] resulted in the evaluation of the segregation enthalpies and entropies [28]. Using the data for phosphorus segregation at a general boundary which is most frequently present in typical polycrystalline materials, DH P0 =-39 kJ mol –1 and DS P0 =?13 J mol –1 K –1 , we did calculate the grain boundary concentrations at a temperature range 700– 1100 K for three bulk concentrations, 0.1, 0.3, and 0.5 at% using Eq. (8b) with accounting for P–P interaction in Fe, a P(Fe) =?4.5 kJ mol -1 , and maximum grain boundary coverage X 0 = 2/3 [28]. These data together with the values of the grain boundary configurational entropy are listed in Table 2and shown in Fig. 1. As mentioned above, the condition for maximum configurational entropy, X PGB = 0.5, depends on the temperature and bulk composition of the alloy. This composition of the grain boundaries is reached at lower temperatures in the case of lower bulk concentrations compared to higher ones. Using Eq. (9), we may derive the value of the temperature, T MAX ,of maximum configurational entropy, 0.693 R, for the equimolar composition of the grain boundary. For an ideal binary system, we can write 1¼XI 1XI exp DGI RTMAX  ;ð20Þ and thus ln XI 1XI ¼DGI RTMAX ¼DH0 I RTMAX DS0 I RþDGE I RTMAX ð21Þ i.e., TMAX ¼DH0 IþDGE I Rln XI=1XI ðÞ½þDS0 I :ð22Þ It is evident that the temperature at which the maximum configurational entropy (TMAX) is reached, increases with increasing the bulk concentration of the solute (Fig. 2), Table 2 Grain boundary concentrations of phosphorus (at%) at a general grain boundary in an Fe–P alloy containing 0.01, 0.03, and 0.05 at% P, and corresponding values of Sconf GB =R. The values of Sconf=Rfor the bulk of individual alloys are listed in the right column X P T(K) X PGB Sconf GB =RS conf=R 0.001 700 0.530 0.691 0.0079 800 0.418 0.680 900 0.313 0.621 1000 0.223 0.531 1100 0.169 0.454 0.003 700 0.614 0.667 0.0204 800 0.556 0.687 900 0.483 0.693 1000 0.406 0.675 1100 0.333 0.636 0.005 700 0.634 0.657 0.0315 800 0.596 0.675 900 0.543 0.689 1000 0.482 0.692 1100 0.414 0.678 J Mater Sci (2023) 58:10043–10057 10047 oTMAX oXI ¼RTMAX ðÞ 2 DH0 IþDGE I 1 XI1XI ðÞ [0;ð23Þ with decreasing the value of the segregation enthalpy (i.e., increasing the absolute value of DH0 I) (Fig. 3a), oTMAX oDH0 I ¼TMAX DH0 IþDGE I \0;ð24Þ and with decreasing the value of the segregation entropy (Fig. 3b), oTMAX oDS0 I ¼ TMAX ðÞ 2 DH0 IþDGE I [0:ð25Þ Let us mention that in inequalities (23)–(25), DH0 I [DGE I ¼2aIðMÞ XGB I, where aIðMÞis the Fowler coefficient [28], Further, according to Eq. (19), the above formulas are also valid if the segregation quantities are considered for individual sites and Figs. 2and 3remain the same if these quantities are averaged according to Eq. (19). It is also worth noting that Sconf GB possesses the same value in binary alloys with interchanged values of XGB Iand XGB M. Then we must keep in mind that we consider two cases, segregation of Iin Mand segregation of Min I. due to different values of DGIand DGM. The same values of Sconf GB are then obtained for different values of bulk concentrations of particular systems (cf. Eq. (16)). Figure 1 Temperature dependence of the configurational entropy for grain boundary and bulk in Fe–P systems with various bulk concentrations of P (denoted by the values in the figure). Figure 2 Dependence of TMAX for Fe–P alloys with varied bulk concentrations of phosphorus, X P . Figure 3 Model dependence of the temperature of maximum configurational entropy, TMAX, for varied bulk concentrations of a solute, X I .afor varied segregation enthalpy (represented by the data in kJ mol –1 ); bfor varied segregation entropy (represented by the data in J mol –1 K –1 ). 10048 J Mater Sci (2023) 58:10043–10057 High-entropy grain boundaries As the maximum value of theconfigurational entropy of a considered equimolar HEA increases with the increasing number of components, Sconf ¼Rln n,thevalueof Sconf GB can reach the values corresponding to those, which are characteristic for HEAs even in more diluted nonHEAs. For a simple example, a maximum value Sconf GB ¼ 1:099Rshould be reached for a ternary system Fe–17 at%Cr–0.7at%Pat800Kasestimatedusingthe values DH0 P¼39 kJ mol -1 ,DS0 P¼þ13 J mol -1 K -1 , a P(Fe) =?4.5 kJ mol -1 ,X 0 =2/3,DH0 Cr ¼12 kJ mol -1 , DS0 Cr ¼8Jmol -1 K -1 ,a Cr(Fe) =?1.1 kJ mol -1 ,X 0 =1 [28], and a’ P-Cr(Fe) = –17 kJ mol -1 [39]. It is worth noting that the value of the configurational entropy in the bulk is Sconf ¼0:446 R, only. Although the value of 1.099 Rdoes not correspond yet to the HEA condition, it is evident that the alloy composition and suitable temperature produce grain boundaries characterized by high entropy as was shown already in Part Note to high-entropy alloys. Anisotropy of grain boundary configurational entropy Much larger differences between the values of the configurational entropy for the grain boundary and the bulk can be obtained in the case of quaternary alloys. A systematic study of the temperature dependence of the grain boundary segregation in an Fe–3.55 at% Si alloy containing 0.0089 at% P and 0.014 at% C (Sconf ¼0.121 R) using Auger electron spectroscopy [40] resulted in the evaluation of the averaged standard enthalpy and standard entropy of solute segregation at individual grain boundaries [41]. From the compositions of the grain boundaries, we can also determine the values of the grain boundary configurational entropy. The grain boundary concentrations are listed in Table 3and depicted in Fig. 4. The values of Sconf GB are not so high as was shown in the previous example. This is because the concentrations of the solutes do not reach the equimolar composition at the grain boundary. However, the values of Sconf GB are still high enough compared to that of the bulk (cf. Figure 4): the maximum shown values are nearly by one order of magnitude higher than Sconf of the bulk. It only confirms the fact that in diluted alloys the grain boundary configurational entropy is higher than that in the bulk. Even here, the character of the anisotropy of Sconf remains identical when the concentrations and thermodynamic quantities considered for individual sites at the grain boundary are averaged according to Eqs. (17) and (19), respectively. Table 3 Concentrations of phosphorus, silicon, and carbon (at%) at individual grain boundaries of an Fe–Si– P–C alloy. The average standard error of the measured grain boundary concentrations is ±0.5 at% for P, ±1.0 at% for C, and ±0.4 at% for Si [40] T(K) 773 873 973 1073 1173 Si P C Si P C Si P C Si P C Si P C {016} 4.1 4.3 19.4 2.5 4.3 10.1 3.6 3.4 6.0 2.5 2.3 3.2 2.3 1.9 2.4 {015} 2.2 7.8 14.4 2.4 6.9 9.2 2.5 6.1 5.9 2.4 5.1 3.5 2.4 4.6 2.4 {014} 1.1 13.3 20.8 3.2 7.3 13.2 3.7 5.6 7.2 3.9 3.6 4.6 3.8 2.7 3.1 {013} 1.0 13.6 16.2 1.5 11.8 9.7 1.9 10.0 6.6 2.1 8.9 4.6 2.5 7.7 3.1 {0kl}* 1.0 13.7 26.5 2.5 8.3 17.9 2.8 5.4 11.2 3.1 4.2 7.4 3.1 3.2 4.6 {0 7 15} 1.2 14.8 16.4 4.6 6.8 10.8 5.3 5.7 6.4 5.6 3.7 3.8 5.5 2.8 3.1 * 45°[100] {0kl} is an incommensurate interface with irrational indices kand l Figure 4 Orientation and temperature dependence of the configurational entropy at individual grain boundaries in bicrystals of and Fe–3.55 at% Si alloy containing 0.0089 at% P and 0.014 at% C. The horizontal line (bulk) represents the configurational entropy of the bulk. J Mater Sci (2023) 58:10043–10057 10049 Nevertheless, there is another interesting finding. The orientation dependence of Sconf GB follows the anisotropy of the absolute values of the segregation enthalpy [42] exhibiting minima at 22.6°[100] {015} and 36.9°[100] {013} special grain boundaries at 773 K (Fig. 4). However, it is apparent from Fig. 4that the differences among the values of Sconf GB of individual grain boundaries reduce and eventually reverse, thus exhibiting an opposite character of the anisotropy. This is also in agreement with the anisotropy of the values of the grain boundary concentrations [43] and was also observed in HEAs [44]. As shown previously [42], a specific relationship was found between the changes of the standard molar enthalpy and standard molar entropy, both of solute segregation at the grain boundaries, with changes in the grain boundary structure W, oDH0 I oW  T =oDS0 I oW  T ¼TCE;ð26Þ which is called enthalpyentropy compensation effect, and T CE is the compensation temperature. Accordingly, oDG0 I oW  T ¼oDH0 I oW  T 1TCE T  ¼oDH0 I oW  T TTCE T  :ð27Þ It was concluded [42,43] that the Gibbs energy of segregation is constant at T CE (K) and independent of the grain boundary structure. This fact has a consequence that oDG0 I=oW  Tchanges its sign and thus, the character of the anisotropy of grain boundary concentrations is reversed above and under T CE . As the grain boundary configurational entropy is composed of the grain boundary concentrations, we can expect similar dependence, which is also apparent from Fig. 4. To understand the changes of Sconf GB , we will consider a binary ideal alloy for simplicity. In this case according to Eqs. (14) and (21) oSconf GB oW  T ¼Rln XI 1XI  DG0 I RT 1 1XGB I  oXGB I oW  T DXGB I RT oDG0 I oW  T : ð28Þ As oXGB I oW  T ¼XGB I1XGB I  TTCE RT oDH0 I oW  T ;ð29Þ using Eq. (27) we can write oSconf GB oW  T ¼XGB I TTCE RT 1XGB I  ln XI 1XI DG0 I RT  oDH0 I oW  T : ð30Þ If we accept gTðÞ¼ XGB I RT 1XGB I  ln XI 1XI DG0 I RT  ¼XGB I1XGB I  RT ln XGB I 1XGB I \0ð31Þ for all temperatures and XGB I\0:5. Then, oSconf GB oW  T ¼gTðÞTTCE ðÞ oDH0 I oW  T :ð32Þ It is apparent from Eq. (17) that the character of the anisotropy of DSconf GB reverses by crossing T CE similarly to that of DG0 Iand XGB I. This change is also apparent from Fig. 4. It was established previously that the value of T CE = 900 K [41] is the temperature at which the reversion occurs. However, it seems from Fig. 4that the reversion of the anisotropy of Sconf GB takes place at a slightly higher temperature. This discrepancy results from the simplification of the mathematical treatment applied to only an ideal binary system as the shift in T CE can be affected by the real behavior of the system [45] characterized here mainly by strong repulsive interaction between Si and P atoms (Guttmann ternary interaction parameter a0 SiPðFeÞ¼92 kJ mol -1 ) [46]. Despite the compensation temperature being the same for all mentioned solutes and grain boundaries, the compensation effect splits into two branches, one for phosphorus and carbon, and the other one for silicon [41]. This reflects in somehow diffuse transition and apparent shift of T CE . However, Fig. 4 clearly demonstrates the reversed character of the anisotropy of Sconf GB at temperatures above and under T CE despite of the value of the compensation temperature. 10050 J Mater Sci (2023) 58:10043–10057 Consequences of grain boundary configurational entropy and future perspectives Materials properties are controlled by their Gibbs energy, G, and tend to reach the equilibrium characterized by a minimum of this thermodynamic quantity. The Gibbs energy of a system can be expressed as a combination of the ideal enthalpy, Hid i, entropy, Sid i, and mixing parameter, Gmix, G¼X i XiHid iTSid i  þDGmix ¼X i XiHid iTSid i  TSconf þDGE;ð33Þ where DGEis the excess Gibbs energy of the alloy [26]. It is apparent that besides the configurational entropy, there are other entropic contributions such as vibrational, harmonic, anharmonic, and magnetic entropies which are included in the terms Sid i.An expression analogous to Eq. (33) can be written for both the bulk and the grain boundary. Therefore, the properties of the material are affected by a synergistic influence of individual terms contributing to the total Gibbs energy of the system. However, here we will discuss the effect of one of these terms, i.e., of the grain boundary configurational entropy, despite that we are aware of the fact that observed behavior is not exclusively the result of the value of Sconf GB . It is widely accepted that entropy is a measure of the disorder of the system. In some cases, high entropy of grain boundaries can thus result in changing the structures of the grain boundaries up to an amorphous state [47]. The systems exhibiting high entropy may then exhibit exceptional properties [48]. If high configurational entropy results from solute segregation at grain boundaries, we may expect, for example, higher resistance to any type of clustering such as precipitation of second phases and intermetallic compounds [16]. It is also expected that the interfacial segregation will be reduced at lower temperatures in HEAs due to the opposite effects of the configurational entropy and driving force of the segregation process [49]. A similar effect might be expected in non-HEAs when the level of segregation induces high values of entropy. Maybe, it can also be connected to some tendency to short range ordering which has been observed in HEAs [50–52], and affecting, e.g., local distortion [50]. Consequently, the solid solubility of solutes increases at the grain boundaries, compared to the bulk materials. This is evident from the data for P and C listed in Table 3 which are by 1–2 orders of magnitude higher than the solid solubility of these solutes in bcc iron. We can also find other examples in the literature, e.g., Bi in Cu [53] and In in Ni [54]. Additionally, we may reach such a composition of the grain boundaries exhibiting the configurational entropy on the level of HEAs, and occurring high-entropy grain boundaries can then serve as stabilizers of nanocrystalline structures [16]. Such segregated grain boundaries can also undergo various transitions in multicomponent alloys which can result in changed width of the grain boundaries and their structure [48]. Similarly, convoluted grain boundaries form in HEAs. However, in the case of equimolar HEAs, the segregated grain boundaries exhibit lower configurational entropy than the bulk. It is question, how does this fact reflect in the properties of the grain boundaries? In this respect, we can expect similar effects as in the case of the above-mentioned interfaces. Nevertheless, an impact is expected on, e.g., electrical, thermal, and ionic conductivities, the coercivity of magnets, and the stability of batteries as summarized in [48]. However, another question arises whether the behavior of the grain boundaries is primarily controlled by configurational entropy or by atomic bonds at the interface. It is known that a high concentration of phosphorus at the grain boundaries causes both the loss of intergranular cohesion [39] and an increase of the configurational entropy [55]. However, in the case of diluted systems with several segregating elements which compete for the sites but have different effects on the cohesion such as phosphorus and carbon in steels, the situation is rather complicated. For example, grain boundary segregation of phosphorus in Fe–Cr-based alloys increases with increasing content of chromium in the alloy while that of carbon is decreasing [55] (Table 4). In both these alloys, the values of Sconf GB are nearly equal, but the alloys differ substantially in mechanical behavior as that with lower content of chromium possesses higher cohesion, while that with higher content of chromium exhibits intergranular brittle fracture [55]. This example suggests that chemical bonds play a dominant role in materials cohesion despite nearly identical values of the grain boundary configurational entropy. J Mater Sci (2023) 58:10043–10057 10051