Thermodynamic Stability of Nanograined Alloys Against Grain Coarsening and Precipitation of Macroscopic Phases
Abstract
Thermodynamic conditions are derived here for binary alloys to have their grain boundary (GB) energies negative, ensuring the stability of some nanograined (NG) alloys. All binary alloys are found to belong to one of the following three types. Type 1 is the unstable NG alloy both against grain coarsening and precipitation of a macrophase. Type 2 is the partly stable NG alloy, stable against coarsening but not against precipitation. Type 3 is the fully stable NG alloy, both against coarsening and precipitation. Alloys type 1 have negative, or lowpositive interaction energies between the components. Alloys type 2 have mediumpositive interaction energies, while alloys type 3 have highpositive interaction energies. Equations are derived for critical interaction energies separating alloys type 1 from type 2 and those from type 3, being functions of the molar excess GB energy of the solute, temperature (T) and composition of the alloy. The criterion to form a stable NG alloy is formulated through a new dimensionless number (Ng), defined as the ratio of the interaction energy to the excess molar GB energy of the solute, both taken at zero Kelvin. Systems with Ng number below 0.6 belong to alloy type 1, systems with Ng number between 0.6 and 1 belong to alloy type 2, while systems with Ng number above 1 belong to alloy type 3, at least at T = 0 K. The larger is the Ng number, the higher is the maximum T of stability of the NG alloy. By gradually increasing temperature alloys type 3 convert first into type 2 and further into type 1. The Ng number is used here to evaluate 16 binary tungstenbased (WB) alloys. At T = 0 K type 3 NG alloys are formed with B = Cu, Ag, Mn, Ce, Y, Sc, Cr; type 2 is formed in the WTi system, while type 1 alloys are formed with B = Al, Ni, Co, Fe, Zr, Nb, Mo and Ta. For the WAg system the region of stability of the NG alloys is shown on a calculated phase diagram, indicating also the stable grain size.
Nomenclature
 \( A \)
The total interface area of grain boundaries (m^{2})
 \( A_{\text{sp}} \)
The average specific interfacial area of the grain boundaries (1/m)
 \( f \)
A geometric parameter in Eqs. [8a] through [8b] (\( \cong 1. 2 5 \) for bcc grain) (–)
 \( f_{\text{b}} \)
The bulk packing fraction within the grain (= 0.68 for bcc grain (–)
 \( f_{\text{gb}} \)
The 2dimensional packing fraction along the GB (= 0.75) (–)
 \( G_{\text{A}}^{ \circ } \)
The Gibbs energy of pure component A with GBs (J)
 \( G_{\text{b,A}}^{ \circ } \)
The bulk Gibbs energy of pure component A without GBs (J)
 \( G_{\text{m,A}}^{ \circ } \)
The molar Gibbs energy of pure component A with GBs (J/mol)
 \( G_{\text{m,b,A}}^{ \circ } \)
The molar Gibbs energy of pure component A without GBs (J/mol)
 \( G_{\text{m,Ag,L}}^{ \circ } \)
The standard molar Gibbs energy of pure silver in liquid state (J/mol)
 \( G_{\text{m,Ag,V}}^{ \circ } \)
The standard molar Gibbs energy of pure silver in vapor state (J/mol)
 \( G_{\text{m,b,B}}^{ \circ } \)
The molar Gibbs energy of pure component B without GBs (J/mol)
 \( G_{\text{m,mac}} \)
The molar Gibbs energy of equilibrium mixture of two macroscopic solid solutions (J/mol)
 \( G_{\text{m,nano}} \)
The molar Gibbs energy of the NG alloy (J/mol)
 \( G_{\text{m,nano,b}} \)
The bulk molar Gibbs energy of the NG alloy (J/mol)
 \( k \)
A semiempirical constant of Eq. [4], (\( k \cong 3. 3 6 \)) (–)
 \( n_{\text{A}}^{ \circ } \)
The amount of matter in the pure phase A (mole)
 \( n_{\text{b}} \)
The amount of matter within the bulk of one average grain (mole)
 \( n_{\text{gb}} \)
The amount of matter in the halfGB (mole)
 \( n_{\text{B,gb}} \)
The amount of matter of component B in the halfGB (mole)
 \( N_{\text{Av}} \)
The Avogadro number (= 6.02 × 10^{23}) (1/mole)
 \( Ng \)
The dimensionless Ngnumber defined by Eq. [28] (–)
 \( p \)
Pressure in the system (Pa)
 \( r \)
The average effective grain radius (for a spherical grain) (m)
 \( r_{\text{a}} \)
The atomic radius (m)
 \( r_{ \hbox{min} } \)
The minimum size of the grain hat ensures the GB is fully covered by component B (m)
 \( R \)
The universal gas constant (= 8.3145 J/molK)
 \( T \)
Absolute temperature (K)
 \( V \)
The total volume of the NG metal (m^{3})
 \( V_{\text{b}} \)
The volume of the bulk of one average grain (m^{3})
 \( V_{\text{m,b}} \)
The molar volume of the bulk of the grain (m^{3}/mol)
 \( V_{\text{m,A}}^{ \circ } \)
The molar volume of pure component A (m^{3}/mol)
 \( V_{\text{m,B}}^{ \circ } \)
The molar volume of pure component B (m^{3}/mol)
 \( V_{\text{m,L,B}}^{ \circ } \)
The molar volume of pure liquid component B (m^{3}/mol)
 \( Z \)
The dimensionless number defined by Eq. [29] (–)
 \( x \)
The average mole fraction of component B in the alloy (–)
 \( x_{\text{b}} \)
The mole fraction of component B in the bulk of the grain (–)
 \( x_{\text{gb}} \)
The mole fraction of component B in the GB (–)
 \( x_{\text{e}} \)
\( x_{\text{e}} \) and \( 1  x_{\text{e}} \) are the equilibrium mole fractions of B in case of phase separation of bulk alloy A–B (–)
 \( x_{ \hbox{min} } \)
The minimum average mole fraction of component B, needed to stabilize the grain (–)
 A
The first component (subscript) (–)
 a
Atomic (subscript) (–)
 b
Bulk (subscript) (–)
 B
The second component (subscript) (–)
 L
Liquid (subscript) (–)
 m
Molar (subscript) (–)
 mac
Macroscopic (subscript) (–)
 min
Minimum (subscript) (–)
 nano
Nanosized (–)
 NG
Polycrystalline nanograined alloy (–)
 nost
Nostability (subscript) (–)
 stab
Stability (subscript) (–)
 \( \beta \)
The ratio of bonds in the GB to the same in the bulk of the grain (–)
 \( \sigma \)
The grain boundary energy of alloy A–B (J/m^{2})
 \( \sigma_{\text{A}} \)
The partial grain boundary energy of component A in alloy A–B (J/m^{2})
 \( \sigma_{\text{A}}^{ \circ } \)
The grain boundary (free) energy in a pure A crystal (J/m^{2})
 \( \sigma_{\text{B}} \)
The partial grain boundary energy of component B in alloy A–B (J/m^{2})
 \( \sigma_{\text{B}}^{ \circ } \)
The grain boundary (free) energy in a pure B crystal (J/m^{2})
 \( \omega_{\text{A}}^{ \circ } \)
The molar GB interfacial area in pure crystal A (m^{2}/mol)
 \( \omega_{\text{B}}^{ \circ } \)
The molar GB interfacial area in pure crystal B (m^{2}/mol)
 \( {{\varOmega }} \)
The bulk interaction energy between components A and B (J/mol)
 \( {{\varOmega }}_{\text{gb}} \)
The special \( {{\varOmega }} \) value ensuring \( \sigma = \, 0 \) (J/mol)
 \( {{\varOmega }}_{\text{cr}} \)
The critical \( {{\varOmega }} \) value above which phase separation takes place (J/mol)
 \( {{\varOmega }}_{\text{stab}} \)
The minimum interaction energy needed to fully stabilize the NG alloy (J/mol)
 \( {{\varOmega }}_{\text{no  st}} \)
The largest value of the interaction energy below which there is even no partial stability of the NG alloy (J/mol)
1 Introduction
Nanomaterials play an increasing scientific and social role.[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] In the past, nanomaterials were simplified to single nanoparticles. However, it has been clear for some decades that the real use of nanomaterials is expected if they form macroscopic articles with nanostructure inside. One class of such materials is the polycrystalline nanograined (NG) alloys. It should be admitted that producing such NG alloys is easier than to ensure their longterm stability, especially at hightemperatures when diffusion is fast enough to drive materials towards their equilibrium state within reasonable times.[20, 21, 22, 23, 24, 25, 26, 27, 28] That is why the purpose of this paper is to develop a model for thermodynamic stability of such NG alloys. Although there is plenty of previous literature on both the synthesis[29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47] and on modeling the stabilization of NG alloys[48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88]) (see also reviews[89, 90, 91, 92, 93]), the present paper is novel as it puts this question into a wider framework of the nanoCalphad method,[94,95] i.e., into the thermodynamic framework originally developed by Gibbs.[96]
All previous models on GB stability apply the simplest Langmuir–McLean model[97,98] (see also Reference 99) for modeling grain boundary (GB) energy. Since the pioneering works of Weismuller[48,49] it is known that for the stabilization of PCNG alloys strong repulsion between the components is needed in the bulk alloy. Thus, there is an inner contradiction in the previous sentences, as the original Langmuir model[97] treated the surface layer as an ideal solution, and so this modeling framework is not suited to describe strongly interacting systems. The novelty of the present paper is that here the extended Butler equation is applied to describe the GB energy.[100] As the Butler equation[101] was originally designed for strongly interacting systems, it provides a more natural framework to describe the stability of strongly interacting NG alloys.
The interplay between thermodynamic and kinetic reasons of the stability of NG alloys has been also discussed[102, 103, 104, 105, 106] (see also References 107 through 111). However, in the present paper only thermodynamic aspects will be discussed in a novel way.
2 On Thermodynamic Instability of OneComponent NG Metals
Simplified Parameters Used for Model Calculations
Quantity  Unit  Value  Source 

\( G_{\text{m,b,A}}^{ \circ } \)  J/mol  0  Eq. [6] 
\( G_{\text{m,b,B}}^{ \circ } \)  J/mol  0  Eq. [6] 
\( \sigma_{\text{A}}^{ \circ } \)  J/m^{2}  1.00  arbitrary 
\( \sigma_{\text{B}}^{ \circ } \)  J/m^{2}  0.30  \( \sigma_{\text{B}}^{ \circ } < \sigma_{\text{A}}^{ \circ } \) 
\( V_{\text{m,A}}^{ \circ } \)  cm^{3}/mol  10.0  arbitrary 
\( V_{\text{m,B}}^{ \circ } = V_{\text{m,L,B}}^{ \circ } \)  cm^{3}/mol  15.0  arbitrary 
\( \beta \)  —  0.90  approximated 
\( f \)  —  1.25  for bcc grains 
\( \omega_{\text{A}}^{ \circ } \)  m^{2}/mol  4.90 10^{4}  Eq. [8a] 
\( \omega_{\text{B}}^{ \circ } \)  m^{2}/mol  6.42 10^{4}  Eq. [8b] 
\( r_{\text{a}} \)  nm  0.157  Eq. [19] 
\( r_{ \hbox{min} } \)  nm  4.3  Eq. [23a] 
Similar to pure metals, the specific surface area and the molar volume have always positive values even for alloys. However, contrary to pure metals, the GB energy can be also negative for alloys under special conditions.[48] If this is indeed the case, then NG alloys of special compositions can be more stable compared to the corresponding bulk single crystals, at least in a limited temperature range. In the next sections, the thermodynamic conditions of the stability of NG alloys will be discussed.
3 Selection of Models and Parameters to Demonstrate the Stability of NG Alloys
In the present paper, the essential conditions for the thermodynamic stability of NG alloys are demonstrated. For this purpose, the simplest possible set of models and model parameters will be applied, necessary for stabilization. All those details will be neglected which are not necessarily needed for this demonstration. All those neglected details would provide only small quantitative changes for the final result, without any qualitative influence. Parameter values for demonstrational purposes are given in Table I.
As was shown above, stability is not possible for onecomponent metals. The minimum requirement to ensure stable NG alloys is to have a twocomponent alloy, so this is our choice here. Let us mention that if the number of components in the alloy is increased further, the configurational entropy of the macrosolid solution will increase, stabilizing the macroalloy vs the NG alloy (see the extrastability of highentropy macroalloys). Thus, multicomponent alloys are not preferred vs binary alloys to stabilize NG alloys.
The polycrystalline alloy modeled here is not limited in its total size, thus it contains virtually an infinite number of nanograins. However, for modeling purposes each nanograin is considered equal. In real life, it means that we consider an average nanograin with average properties. An average nanograin will be characterized by its effective radius (r, m) of its bulk not including the thickness of the GB surrounding this grain. The volume of the grain is calculated as if the grain was spherical, but its surface area is calculated by Eqs. [3] and [4] with the average value of k = 3.36 (see above).
For simplicity, we suppose that the two solid components A and B of the binary system have in equilibrium the same crystal structure, and Eq. [6] is valid for both of them. We also presume for simplicity that the thermodynamics of the solid solution between components A and B can be described by the simplest regular solution model. However, the exponential Tdependence of the interaction energy will be taken into account below; in this way, implicitly the excess molar entropy is taken into account. We take this Tdependence into account, as it has a much more significant influence on the maximum stability of the NG alloys compared to the neglected Tdependence of \( \omega_{\text{B}}^{ \circ } \cdot \sigma_{\text{B}}^{ \circ } \) (see below).
It is also presumed here for simplicity that although the pure components A and B have different molar volumes (denoted as \( V_{\text{m,A}}^{ \circ } \) and \( V_{\text{m,B}}^{ \circ } \), both in m^{3}/mol), their temperature dependence has a negligible effect on the stability of NG alloys. Note that in Table I, the ratio \( V_{\text{m,B}}^{ \circ } /V_{\text{m,A}}^{ \circ } = 1.5 \) is selected, as it is in the middle of the interval of actual systems considered in Table II with values of \( V_{\text{m,B}}^{ \circ } /V_{\text{m,A}}^{ \circ } \) ranging from 0.7 to 2. In addition, we will suppose for simplicity that the molar volume of the A–B solid solution will be a linear combination of \( V_{\text{m,A}}^{ \circ } \) and \( V_{\text{m,B}}^{ \circ } \) along the mole fraction of component B, i.e., the excess molar volume of the alloy is also neglected for simplicity. For simplicity, it is also supposed that the molar volume of pure solid and pure liquid B is the same: \( V_{\text{m,B}}^{ \circ }=V_{\text{m,L,B}}^{ \circ } \).
We also presume for simplicity that although the pure components A and B have different GB energies, their temperature dependence and their orientation dependence are neglected. Thus, for simplicity only highangle GBs are taken into account, with an average GB (free) energy, denoted as \( \sigma_{\text{A}}^{ \circ } \) and \( \sigma_{\text{B}}^{ \circ } \) (both in J/m^{2}). Other (low angle) GBs are neglected for simplicity, as they have smaller GB energies. In addition, the outer surface area of the alloy is neglected as it is usually much smaller compared to the total GB interfacial area within the NG alloy; moreover, its contribution is identical for the NG alloy and for the reference macroalloy, so this term falls out when the two are compared. However, the GB energy of the alloy will be not a linear combination of \( \sigma_{\text{A}}^{ \circ } \) and \( \sigma_{\text{B}}^{ \circ } \) along the average mole fraction of the alloy, as preferential segregation of the component with smaller GB energy will be considered. Component B will be selected as the GBactive component, so in this paper: \( \sigma_{\text{A}}^{ \circ } > \sigma_{\text{B}}^{ \circ } \).
Three state parameters will be considered in this paper: the average mole fraction of component B in the alloy (denoted by x, dimensionless), absolute temperature (T, K) and the average radius of the bulk of the grain (r, m). The radius of the grain is an independent state parameter here as the concentration and temperature dependences of the molar volumes of the components are neglected. Pressure will be kept at standard constant value of 1 bar. As condensed phases are studied here, all results will be identical below 100 bar.
This paper is written using mostly the methods of chemical thermodynamics, which is a statistical science. Therefore, care will be taken to make sure that there are at least 1000 atoms in each nanograin. Taking into account the average molar volume of 10 cm^{3}/mol of metals and supposing for simplicity that the grains are spherical, the effective radius of each nanograin studied in this paper will be larger than r = 1.6 nm. Therefore, the results of the present paper will be valid for NG alloys with average grain diameters above 3.2 nm.
4 A Model for the Concentration Dependence of the GB Energy
In this paper, the GB energy will be modeled using the extended Butler model.[100] The original Butler model was developed for surface tension of a liquid/gas surface[101] and it was criticized in the literature for using undefined partial surface tensions of the components and for being not consistent with thermodynamics of Gibbs. Recently, the partial surface tensions of the components were defined[112] in agreement with the thermodynamics of Gibbs and it was proven that even in this case the original Butler equations follow. Additionally, it was proven that the Butler equations can also be derived from the requirement that a solution phase should have a minimum Gibbs energy including its surface term.[113] This question was reviewed recently.[95]
To perform calculations with Eqs. [7] through [9] the following initial information must be known for the given A–B system: \( \sigma_{\text{A}}^{ \circ } \), \( V_{\text{m,A}}^{ \circ } \), \( \sigma_{\text{B}}^{ \circ } \), \( V_{\text{m,B}}^{ \circ } \), \( {{\varOmega }} \). Additionally, the model parameters \( \beta \) and \( f \) should be known. Finally, the state parameters should be given: \( x \) and \( T \) (note: p = 1 bar is fixed above for the whole paper). If the values of all these parameters are given, then, first Eqs. [7a] and [7b] should be substituted into the righthand side of Eq. [9] (\( \sigma_{\text{A}} = \sigma_{\text{B}} \)) and from here the equilibrium mole fraction of component B in the GB region is found (\( x_{\text{gb}} \)). Then, substituting this latter value back into Eqs. [7a] and [7b], both \( \sigma_{\text{A}} \) and \( \sigma_{\text{B}} \) are calculated. These two later values must be equal, and according to Eq. [9] their values also equal the GB energy of the alloy (= solid solution).
Substituting the values from above and from Table I into Eq. [12]: \( \, \varOmega_{\text{gb}} \cong 37.6\;{\text{kJ/mol}} \), which is in agreement with Figure 3(b). As follows from Eq. [12], at x = 0 and x = 1: \( \varOmega_{\text{gb}} = \infty \). As this is an unrealistic requirement, for pure metals \( \sigma \le 0 \) is indeed impossible.
Substituting parameters x = 0.15 and T = 500 K into Eq. [13]: \( \, \varOmega_{\text{cr}} = 10.3\;{\text{kJ/mol}} \). Compared to the above value of \( \, \varOmega_{\text{gb}} \cong 37.6\;{\text{kJ/mol}} \) it follows that \( \varOmega_{\text{cr}} < \varOmega_{\text{gb}} \) (see also Figure 3(b)). It can be generally proven that this inequality is obeyed at any reasonable values of \( \omega_{\text{B}}^{ \circ } \cdot \sigma_{\text{B}}^{ \circ } \), x and T.[116] It means that when the A–B system is in equilibrium as a 1phase macroscopic solid solution, then \( \sigma > 0 \), which is a usual boundary condition for any interfacial energy. On the other hand, \( \sigma < 0 \) takes place only, when the A–B system is in equilibrium as a mixture of two macroscopic solutions of different compositions; however, in this case the Butler equation is not valid as applied above, thus \( \sigma < 0 \) is not a real result for macroscopic equilibrium systems. On the other hand, for nanograins, the mutual solubility of the components is increased,[117] thus for NG alloys the case of \( \sigma < 0 \) is possible also in equilibrium (see below).
Equation [11a] is one of the most essential equations of this paper. Let us note that in the literature different equations are published instead of Eq. [11a]. However, their detailed critical analysis goes out of the framework of the present paper.
5 The Model for the Molar Gibbs Energy of Macroscopic Alloys
Substituting the above values of T = 500 K and \( \varOmega \) = 50 kJ/mol into Eq. [15b]: \( x_{\text{e}} \cong 5.9795 \cdot 10^{  6} \) is obtained. When the two above solutions to Eqs. [15a] and [15b] are substituted into Eq. [14], the two results are: \( G_{\text{m,mac}} =  \,0.0249 \) and − 0.0254 J/mol, the difference being below 0.001 J/mol. This difference is insignificant, so Eqs. [14] and [15b] can be used in the first approximation to calculate the reference value of the molar Gibbs energy.
The NG alloy will be thermodynamically fully stable, if the minimum value of its molar Gibbs energy vs its radius will be more negative than the reference value calculated by Eqs. [14] and [15b]. This reference value will be valid at any temperature below the eutectic temperature of the eutectic type and eutectic + monotectic type phase diagrams. On the other hand, the same will be valid at any temperature below the melting point of the lower melting component for the peritectic type and the peritectic + monotectic type phase diagrams.
6 The Material Balance for NG Alloys
The material balance should be separately discussed as an average grain shown in Figure 1 should be divided into its crystalline bulk and into its amorphous halfGB. All physical parameters will be denoted by subscript “b” for the bulk of the grain and by subscript “gb” for the halfGB (note: any GB belongs to two grains, so a halfGB belongs to each grain). For example, the average mole fraction of component B in the alloy and in the average grain (denoted as x) will be divided into a value x_{b} (defined as the mole fraction of component B in the bulk of the grain) and x_{gb} (defined as the mole fraction of component B in the GB). For the case when the NG alloys are expected to be stable, i.e., the GB energy is expected to have a negative value, the approximation of x_{gb} ≅ 1 is valid (see Figure 3(a)), i.e., the GB is composed almost entirely of component B, due to its segregation to the GB (and due to the selection made above: \( \sigma_{\text{A}}^{ \circ } > \sigma_{\text{B}}^{ \circ } \)).
Substituting the above parameters and r = 1.6 nm into Eq. [24], the value of x_{min} ≅ 0.25 is found. From the numerical solution of Eq. [23a] x_{min} ≅ 0.29 is found. Thus, the present model will be thermodynamically valid below about x = 0.3 (see also Figure 7(b) below).
7 The Molar Gibbs Energy of NG Alloys; Demonstration of the Three Alloy Types
Note that Eq. [26] is similar to Eq. [14] with x_{e} replaced by x_{b}, while Eq. [27] is similar to Eq. [11a] with x replaced by x_{b}. In both Eqs. [26] and [27] x_{b} is calculated by Eq. [22]. In the second term of Eq. [25] the outer radius of the grain (\( r + 2 \cdot r_{\text{a}} \)) is used instead of \( r \), as the latter is only the radius of the bulk of the grain, while the former is the radius of the whole grain including the double thickness of the halfGB.

Type 1 NG alloy (with \( \, \varOmega \, = 20\;{\text{kJ/mol}} \) as an example in Figure 5(a)) is unstable both against grain coarsening and precipitation of a macrophase; its molar Gibbs energy does not pass through a minimum as function of grain size and it is more positive at any grain size than that of the corresponding macroalloy.

Type 2 NG alloy (with \( \varOmega \, = 35\;{\text{kJ/mol}} \) as an example in Figure 5(b)) is partly stable: it is stable against grain coarsening, but not against precipitation of a macrophase; its molar Gibbs energy passes through a minimum as function of grain size, but the minimum molar Gibbs energy of the NG alloy is more positive than that of the corresponding macroalloy at any grain size.

Type 3 NG alloy (with \( \, \varOmega \, = 50\;{\text{kJ/mol}} \) as an example in Figure 5(c)) is fully stable both against grain coarsening and precipitation of a macrophase; its molar Gibbs energy passes through a minimum as function of grain size, and the minimum molar Gibbs energy of the NG alloy is more negative than that of the corresponding macroalloy. In Figure 5(c) the equilibrium grain radius corresponding to this minimum point is: r_{eq} = 4.6 nm, being slightly larger than the value of r_{min} = 4.3 nm found above.
The evolution of the stability of NG alloys with increasing values of the interaction energy is similar to Figures 5(a) through (c) even if different parameter combinations are applied. Thus, type 1 alloys appear with interaction energies having negative or zero or lowpositive values, type 2 alloys appear with interaction energies having mediumpositive values, while type 3 alloys appear with interaction energies having largepositive values.
8 On a Stability Criterion and Stability Diagrams to Select Stable NG Alloys
As was shown above, even in the framework of the simplest model, the stability of NG alloys depends on seven independent parameters (\( V_{\text{m,A}}^{ \circ } \), \( V_{\text{m,B}}^{ \circ } \), \( \sigma_{\text{A}}^{ \circ } \), \( \sigma_{\text{B}}^{ \circ } \), \( \varOmega \), \( T \), x). If a more complex solution model, or the excess volume or the Tdependence of physical quantities are taken into account, the number of parameters can be easily doubled, at least (see for example the 24 parameters below for the real WAg system). Thus, the identification of a single stability criterion or at least a 2dimensional stability graph seems elusive. Nevertheless, for fast screening of stable NG systems such a criterion or diagram is desirable.
Coordinates for the DataPoints Given in Figs. 6(a) to (c) for W–B Systems (\( \omega_{\text{B}}^{ \circ } \cdot \sigma_{\text{B}}^{ \circ } \) and \( {{\varOmega }} \) are valid at T = 0 K, see Appendix A)
Component B  \( \omega_{\text{B}}^{ \circ } \cdot \sigma_{\text{B}}^{ \circ } \) (kJ/mol)  \( {{\varOmega }} \) (kJ/mol)  Ng  NG Stability  Alloy Type 

Ag  20.6  160  7.8  strong  3 
Al  21.6  − 14  − 0.65  none  1 
Ce  22.9  144  6.3  strong  3 
Co  30.6  − 5  − 0.16  none  1 
Cr  29.1  45  1.5  weak  3 
Cu  22.7  200  8.8  strong  3 
Fe  31.8  0  0  none  1 
Mn  21.0  160  7.6  strong  3 
Mo  43.1  − 1  − 0.02  none  1 
Nb  43.0  − 33  − 0.77  none  1 
Ni  30.1  − 12  − 0.42  none  1 
Sc  26.9  44  1.6  weak  3 
Ta  46.6  0  0  none  1 
Ti  37.8  33  0.87  partial  2 
W  49.8  —  —  —  — 
Y  29.6  112  2.8  medium  3 
Zr  40.9  − 36  − 0.88  none  1 
For an average value of x = 0.15 ± 0.05, Eq. [29] leads to \( Z \cong 0.022 \cdot T \) (where T is substituted in K, and Z is obtained in kJ/mol). Expressing T from this equation and substituting it into Eq. [31], the first exponential term of Eq. [30] is obtained, which is a correction factor for the Tdependence of the interaction energy. Note that \( \omega_{\text{B}}^{ \circ } \) slightly increases, while \( \sigma_{\text{B}}^{ \circ } \) slightly decreases with temperature, and thus no correction is used in Eq. [30] for the Tdependence of \( \omega_{\text{B}}^{ \circ } \cdot \sigma_{\text{B}}^{ \circ } \).
 1.
if for a given A–B system Ng ≤ 0.6, then this system is unstable as an NG alloy at any temperature and composition (i.e., it is alloy type 1),
 2.
if for a given A–B system 0.6 < Ng ≤ 1, then this system is partly stable as an NG alloy (i.e., it is alloy type 2), at least in a limited temperature and concentration range.
 3.
if for a given A–B system Ng > 1, then this system is fully stable as an NG alloy (i.e., it is alloy type 3), at least in a limited temperature and concentration range.
 1.
stable NG alloys are expected in the WCu, WAg, WMn, WCe, WY, WCr and WSc systems,
 2.
the WTi alloy is expected to be a partly stable NG alloy,
 3.
unstable NG alloys are expected to form in the WAl, WCo, WFe, WMo, WNb, WNi, WTa and WZr systems.
 i.
the lower regions of Figures 6(a) through (c) correspond to systems of stable macroscopic alloys (= type 1 alloy);
 ii.
the middle regions of Figure 6(a) through (c) correspond to systems of partly stable NG alloys (= type 2 alloys);
 iii.
the upper regions of Figures 6(a) through (c) correspond to systems of fully stable NG alloys (= type 3 alloys).
As follows from Figures 6(a) through (c), by increasing the value of Z (i.e., increasing T at fixed x) the demarcation lines are shifted towards higher Ω values. Thus, increasing temperature type 3 alloys become first type 2 alloys and then type 1 alloys. This is due to the effect of entropy, stabilizing macrosolutions and destabilizing GB segregation as a 2D ordered state.
9 On the Stable Region of NG Alloys in the WAg Phase Diagram
As was shown above, in some A–B systems, the NG alloy is found fully stable in finite temperature and composition ranges. This theoretical prediction is of primary importance. However, the next reasonable question is about the theoretical prediction on the temperature and compositional borders of this stability range. The usual way to show the stability of different states in materials science is to present stable phases/states in equilibrium phase diagrams. For binary macroscopic systems phase diagrams are routinely constructed in T vs x diagrams at fixed p = 1 bar.[120] Thus, for engineering purposes the stability of NG alloys should be shown in the same type of phase diagrams. In this section an example will be shown for the WAg system, found stable till a relatively high temperature (see Table II and Figures 6(a) through (c)).
Thermodynamic Properties of the W–Ag System (T in K)
Quantity  Unit  Equation  TRange (K)  Source 

\( G_{\text{m,W,bcc}}^{ \circ } \)  J/mol  0  0 … 3695  
\( G_{\text{m,Ag,bcc}}^{ \circ } \)  J/mol  \( 3400  1.05 \cdot T \)  0 … 3000  
\( G_{\text{m,Ag,fcc}}^{ \circ } \)  J/mol  0  0 … 1235  
\( G_{\text{m,Ag,L}}^{ \circ } \)  J/mol  \( 11508.141  9.301748 \cdot T \)  1235 … 2433  
\( G_{\text{m,Ag,V}}^{ \circ } \)  J/mol  \( 292992  224.04 \cdot T + 12.686 \cdot T \cdot \ln T + R \cdot T \cdot \ln \left( {\frac{p}{{p^{ \circ } }}} \right) \)  T ≥ 2433 K at p = 1 bar)  
\( V_{\text{m,W}}^{ \circ } \)  cm^{3}/mol  \( 9.47 \cdot \left( {1 + 2.04 \cdot 10^{  5} \cdot T} \right) \)  0 … 3000  
\( V_{\text{m,Ag}}^{ \circ } \)  cm^{3}/mol  \( 10.146 + 7.077 \cdot 10^{  5} \cdot T^{1.314} \)  0 … 1235  
\( 9.889 + 8.694 \cdot 10^{  4} \cdot T \)  1235 … 3000  
\( V_{\text{m,L,Ag}}^{ \circ } \)  cm^{3}/mol  \( 10.490 + 9.648 \cdot 10^{  5} \cdot T^{1.314} \)  0 … 1235  
\( 10.140 + 1.185 \cdot 10^{  3} \cdot T \)  1235 … 3000  
\( \sigma_{\text{W}}^{ \circ } \)  J/m^{2}  \( 1.054  4.55 \cdot 10^{  5} \cdot T \)  0 … 3000  Appendix A 
\( \sigma_{\text{Ag}}^{ \circ } \)  J/m^{2}  \( 0.393  3.05 \cdot 10^{  5} \cdot T \)  0 … 3000  Appendix A 
\( {{\varOmega }} \)  kJ/mol  \( 160 \cdot { \exp }\left( {  \frac{T}{4900}} \right) \)  0 … 3000  Appendix A 
As the bulk of the grains are Wrich bcccrystals, the molar grain boundary areas of both components are calculated using parameter f = 1.25 (see above). Let us note that in this case different molar volumes are used for solid and liquid Ag and also their Tdependencies are taken into account (see Table III).
The calculation procedure includes the construction of molar Gibbs energy diagrams similar to Figures 5(a) through (c) using different values of temperatures and average mole fractions of Ag. The first conclusion from each such diagram is whether at any grain size the molar Gibbs energy of the NG alloy calculated by Eqs. [22], [25], [27] and [34] is more negative than the most negative of the molar Gibbs energies of the corresponding macroscopic states calculated by Eqs. [33a] through [33c]. If the answer is “yes”, then at the given T–x point of the WAg phase diagram the NG alloy is found stable. If the answer is “no”, then at the given T–x point of the WAg phase diagram the original stable state shown in Figure 7(a) is found stable. In this way the contours of the full stability of the NG alloy can be drawn in the phase diagram, as shown in left bottom part of Figure 7(b).
Additionally, for the case of stable NG alloy, the equilibrium grain radius corresponding to the minimum of the molar Gibbs energy of the NG alloy can be found. Using these data, the isograinradius lines were drawn as dotted, almost vertical lines in Figure 7(b). The stability range of the NG alloys is terminated when the equilibrium grain size reaches the smallest allowed value (r = 1.6 nm, see above), as at a higher Agcontent the equilibrium grain size would be smaller than this value and for such small grains the validity of the present model becomes questionable. Therefore, this line in Figure 7(b) is shown as a dotted line, as the choice of 1.6 nm above was somewhat arbitrary. The stability region of the NG alloy in the Wrich side of the phase diagram Figure 7(b) starts at T = 0 K at x = 0 and terminates at x = 0.33 … 0.34 at T = 0 … 2130 K.
It should be noted that the stability region of NG alloys in Figure 7(b) extends above the melting point of Ag. This might mean GB melting. However, in this model the GB is supposed to be amorphous / quasiliquid anyway, and so this change is not reflected in Figure 7(b). However, it might have an influence on properties of the NG alloys.
Similar extended phase diagrams can be calculated for any other system with its Ng number being higher than 1. Such extended phase diagrams are considered useful tools for NG alloy design. Let us note that for all systems with the Ng number not exceeding 1 the existing macroscopic phase diagrams remain valid,[120] so for them the production of fully stable NG alloys (type 3) is hopeless.
10 Conclusions
 1.
It is shown that onecomponent polycrystalline nanograined (NG) metals are prone to grain coarsening due to their always positive GB energies.
 2.
Using the extended Butler equation, thermodynamic conditions are found for negative GB energy of binary nanoalloys (note: for macroalloys only positive GB energies exist). This finding is built into a general thermodynamic model describing the molar Gibbs energy of NG alloys. Three types of alloys are considered: (i). the Type 1 NG alloy is unstable, as its molar Gibbs energy has no minimum as function of grain size, moreover, its molar Gibbs energy is more positive than that of the most stable macroalloy (“unstable” here means it is not stable against grain coarsening and precipitation of a macrophase); (ii). the Type 2 NG alloy is partly stable, as its molar Gibbs energy passes through a minimum as function of grain size, but its value is more positive than that of the most stable macroalloy (“partly stable” here means it is stable against grain coarsening but not stable against precipitation of a macrophase); (iii). the Type 3 NG alloy is fully stable, as its molar Gibbs energy passes through a minimum as function of grain size, and this molar Gibbs energy is more negative than that of the most stable macroalloy (“fully stable” here means the NG alloys is stable both against grain coarsening and the precipitation of a macrophase).
 3.
A new dimensionless number Ng is defined as the ratio of the bulk interaction energy between the A–B components to the molar excess GB energy of the solute component B at zero Kelvin. Based on the value of this Ng number all A–B systems can be categorized, at least at T = 0 K. When the Ng number is smaller then 0.6, the A–B system is type 1. The A–B system is a type 2 alloy when the Ng number is between 0.6 and 1 and it is a type 3 alloy if the Ng number is larger than 1. Higher is the Ng number for the given A–B system, the wider is the temperature—composition space of the A–B alloy that is stable as an NG alloy. With increasing temperature systems gradually transform from type 3 to type 2 and further to type 1.
 4.
General equations are worked out for the critical values of the interaction energy as function of temperature, composition and excess molar GB energy of component B. These special interaction energy values serve as demarcation values/lines to separate alloys type 1 from alloys type 2 and those from alloys type 3. The method is demonstrated on 16 binary Wbased alloys (see Table II and Figures 6(a) through (c)).
 5.
The temperature and concentration ranges of stability of NG alloys are calculated for the WAg system as an example, and the findings are presented in the binary equilibrium WAg phase diagram, indicating also isograinsize lines (see Figure 7(b)). The same method can be applied to calculate the stability ranges of other NG alloys with Ng > 1.
Notes
Acknowledgments
Open access funding provided by University of Miskolc (ME). This work was partly financed by the ICARUS project which has received funding from the European Union’s Horizon 2020 research and innovation program under Grant Agreement No 713514 and partly by the nanoGinop Project GINOP2.3.215201600027 in the framework of the Szechenyi 2020 program, supported by the European Union. The author is grateful to the following individuals for their helpful discussions during this work: prof. Santiago CuestaLópez and prof. Nicolas A. Cordero of the University of Burgos (Spain), dr. Antonio Locci and prof. Francesco Delogu of the University of Calgary (Italy), Dr Tomas Polcar of Advamant Ltd (UK), prof. Antonio Rinaldi of ENEA (Italy), Andras Dezso, prof. Andras Roósz, prof. Zoltan Gacsi and dr. Greta Gergely of the University of Miskolc (Hungary), prof. Spiros Pantelakis of the University of Patras (Greece), prof Pal Barczy and Tamas Barczy of Admatis Ltd (Hungary). The author is also grateful for Anna Vasileva and Koppány Juhász for their help in the figures to this paper.
References
 1.A. Táborosi, R.K. Szilagyi, B. Zsirka, O. Fónagy, E. Horváth, J. Kristóf.: Inorg. Chem., 2018, 57, pp. 7151–7167.CrossRefGoogle Scholar
 2.E. Illés, M. Szekeres, I.Y. Tóth, Á. Szabó, B. Iván, R. Turcu, L. Vékás, I. Zupkó, G. Jaics, E. Tombácz, J Magnetism Magnetic Mater, 2018, 451, pp. 710720.CrossRefGoogle Scholar
 3.K. Molnar, C. Voniatis, D. Feher, A. Ferencz, G. Weber, M. Zrinyi, A. JedlovszkyHajdu, Byophys J, 2018, 114, pp. 363aCrossRefGoogle Scholar
 4.Z. Fogarassy, N. Oláh, I. Cora, Z.E. Horváth, T. Csanádi, A. Sulyok, K. Balázsi, J Eur Ceram Soc, 2018, 38, pp. 28862892.CrossRefGoogle Scholar
 5.M. Weszl, K.L. Tóth, I. Kientzl, P. Nagy, D. Pammer, L. Pelyhe, N. E. Vrana, D. Scharnweber, C. WolfBrandstetter, J.F. Árpád, E. Bognár, Mater Sci Eng C, 2018, 78, pp. 6978.CrossRefGoogle Scholar
 6.É. Fazakas, Z. MátyásKarácsony, R. Bak, L.K. Varga, Am. J. Analyt. Chem, 2017, 8, pp. 171179.CrossRefGoogle Scholar
 7.G.F. Samu, Á. Veres, S. P. Tallósy, L. Janovák, I. Dékány, A. Yepez, R. Luque, C. Janáky, Catalysis Today, 2017, 284, pp. 310.CrossRefGoogle Scholar
 8.M. Czagány, P. Baumli, G. Kaptay, Appl Surf Sci, 2017, 423 pp. 160169.CrossRefGoogle Scholar
 9.O. Pitkänen, T. Järvinen, H. Cheng, G. S. Lorite, A. Dombovari, L. Rieppo, S. Talapatra, H. M. Duong, G. Tóth, K. L. Juhász, Z. Kónya, A. Kukovecz, P. M. Ajayan, R. Vajtai, K. Kordás: Sci Reports, 2017, 7, 16594.CrossRefGoogle Scholar
 10.F. Bíró, C. Dücső, G.Z. Radnóczi, Z. Baji, M. Takács, I. Bársony, Sensors Actuators B, 2017, 247, pp. 617625.CrossRefGoogle Scholar
 11.V. Takáts, J. Hakl, A. Csík, H.F. Bereczki, G. Lévai, M. Godzsák, T.I. Török, G. Kaptay, K. Vad, Surf Coating Technol, 2017, 326, pp. 121125.CrossRefGoogle Scholar
 12.O. Tapasztó, J. Balko, V. Puchy, P. Kun, G. Dobrik, Z. Fogarassy, Z.E. Horváth, J. Dusza, K. Balázsi, C. Balázsi, L. Tapasztó: Sci Rep. 2017, 7, 10087.CrossRefGoogle Scholar
 13.W. Rao, D. Wang, T. Kups, E. Baradács, B. Parditka, Z. Erdélyi, P. Schaaf, ACS Appl. Mater. Interfaces, 2017, 9, pp. 6273–6281.CrossRefGoogle Scholar
 14.G. Czel, K. Tomolya, M. Sveda, A. Sycheva, F. Kristaly, A. Roosz, D. Janovszky. J Noncryst Solids, 2017, 458, pp. 4151.CrossRefGoogle Scholar
 15.N. Oláh, Z. Fogarassy, A. Sulyok, M. Veres, G. Kaptay, K. Balázsi: Surf. Coat. Technol. 2016, 302, pp. 410419.CrossRefGoogle Scholar
 16.G. Levai, M. Godzsák, T.I. Török, J. Hakl, V. Takáts, A. Csik, K. Vad, G. Kaptay, Metall Mater Trans A, 2016, 47A, pp. 3580 – 3596.CrossRefGoogle Scholar
 17.A. Lekatou, A.E. Karantzalis, A. Evangelou, V. Gousia, G. Kaptay, Z. Gacsi, P. Baumli, A. Simon, Mater Design, 2015, 65, pp. 11211135.CrossRefGoogle Scholar
 18.O.L. Galkina, A. Sycheva, A. Blagodatskiy, G. Kaptay, V.L. Katanaev, G.A. Seisenbaeva, V.G. Kessler, A.V. Agafonov, Surf Coat Technol, 2014, 253, pp. 171179.CrossRefGoogle Scholar
 19.E. Sohn, X. Li, W.Y. He, S. Jiang, Z. Wang, K. Kang, J.H. Park, H. Berger, L. Forró, K.T.Law, J.Shan, K.F. Mak, Nature Mater, 2018, 17, 504508.CrossRefGoogle Scholar
 20.H. Gleiter, Acta Mater, 2000, 48, pp. 129.CrossRefGoogle Scholar
 21.R. Valiev, Nature Mater, 2004, 3, pp. 511516.CrossRefGoogle Scholar
 22.M. A. Meyers, A. Mishra, D. J. Benson, Prog. Mater Sci, 2006, 51, pp. 427556.CrossRefGoogle Scholar
 23.M. Dao, L. Lu, R.J. Asaro, J.T.M. De Hosson, E. Ma, Acta Mater, 2007, 55, pp. 40414065.CrossRefGoogle Scholar
 24.R.H.R. Castro, Mater Letters, 2013, 96, pp. 4556.CrossRefGoogle Scholar
 25.K. Lu: Nat. Rev. Mater. 2016, 1, 16019.CrossRefGoogle Scholar
 26.R.H.R.Castro, D.Gouvea, J Amer Ceram Soc, 2016, 99, pp. 11051121.CrossRefGoogle Scholar
 27.T.J.Rupert, Curr Opinion Solid State Mater Sci, 2016, 20, pp. 257267.CrossRefGoogle Scholar
 28.I.A.Ovidko, R.Z.Valiev, Y.T.Zhu, Progr Mater Sci, 2018, 94, pp. 462540.CrossRefGoogle Scholar
 29.P.K.Sahoo, S.S.K. Kamal, M.Premkumar, B.Sreedhar, S.K.Srivastava, L.Durai. Int J Refr Met Hard Mater, 2011, 29, 547554.CrossRefGoogle Scholar
 30.G.T.P.Azar, H.R. Rezaie, B. Gogari, H.Razavizadeh, J Alloys Compds, 2013, 574, pp. 432436.CrossRefGoogle Scholar
 31.C.Ren, M.Koopman, Z.Z. Fang, H.Zhang, B. van Devener, Int J Refract Met Hard Mater, 2016, 61, pp. 273278.CrossRefGoogle Scholar
 32.W.T.Qiu, Y.Pang, Z.Xiao, Z.Li, Int J Refr Met Hard Mater, 2016, 61, pp. 9197.CrossRefGoogle Scholar
 33.C.Ren, Z.Z.Fang, H.Zhang, M.Koopman, Int J Refr Met Hard Mater, 2016, 61, pp. 273278.CrossRefGoogle Scholar
 34.M.Kapoor, T.Kaub, K.A.Darling, B.L.Boyce, G.B.Thompson, Acta Mater, 2017, 126, pp. 564575.CrossRefGoogle Scholar
 35.J.T.Zhao, J.Y.Zhang, L.F.Cao, Y.Q.Wang, P.Zhang, K.Wu, G.Liu, J.Sun, Acta Mater, 2017, 132, pp. 550564.CrossRefGoogle Scholar
 36.Z.L.Zhang, J.M.Guo, G.Dehm, R.Pippan, Acta Mater, 2017, 138, pp. 4251.CrossRefGoogle Scholar
 37.Chakraverty, K. Sikdarm, S.S. Singh, D. Roy, C.C. Koch: J. Alloys Compds., 2017, 716: 197203.CrossRefGoogle Scholar
 38.G.Csiszár, A.Makvandi, E.J.Mittemeijer, J Appl Cryst, 2017, 50, pp. 152171.CrossRefGoogle Scholar
 39.R.Raghavan, T.P.Harzer, S.Djaziri, S.W.Hieke, C.Kirchlechner, G.Dehm, J Mater Sci, 2017, 52, 913920.CrossRefGoogle Scholar
 40.S.S.V.Tatiparti, F.Ebrahimi, J Alloys Compds, 2017, 694, pp. 632635.CrossRefGoogle Scholar
 41.P.S. Roodposthi, M. Saber, C. Koch, R.Scattergood, S.Shahbazmohamadi, J Alloys Compds, 2017, 720, pp. 510520.CrossRefGoogle Scholar
 42.O.K.Donaldson, K.Hattar, T.Kaub, G.B.Thomson, J Mater Res, 2018, 33, pp. 6880.CrossRefGoogle Scholar
 43.J.Hu, Y.N.Shi, K.Lu, Scr mater, 2018, 154, pp. 182185.CrossRefGoogle Scholar
 44.H. Kotan, J Alloys Compds, 2018, 749, pp. 948954.CrossRefGoogle Scholar
 45.X.G.Li, L.F.Cao, J.Y.Zhang, J.Li, J.T.Zhao, X.B.Feng, Y.Q.Wang, K.Wu, P.Zhang, G.Liu, J.Sun, Acta Mater, 2018, 151, pp. 8799.CrossRefGoogle Scholar
 46.J.F. Curry, T.F. Babuska, T.A. Furnish et al.: Adv. Mater. 2018, 30, 1802026.CrossRefGoogle Scholar
 47.Y.Z.Chen, K.Wang, G.B.Shan, A.V.Ceguerra, L.K.Huang, H.Dong, L.F.Cao, S.P.Ringer, F.Liu, Acta Mater, 2018, 158, pp. 340353.CrossRefGoogle Scholar
 48.J. Weissmuller, Nanostruct. Mater, 1993, 3, pp. 261272.CrossRefGoogle Scholar
 49.J. Weissmuller, J. Mater. Res., 1994, 9, pp. 47.CrossRefGoogle Scholar
 50.D.L. Beke, C. Cserháti, I.A. Szabó: Nanostr. Mater., 1997, 9, pp. 665668.CrossRefGoogle Scholar
 51.R.Kirchheim, Acta Mater, 2002, 40, pp. 413419.CrossRefGoogle Scholar
 52.D.L.Beke, C.Cserháti, I.A.Szabó, J Appl Phys, 2004, 95, pp. 49965001.CrossRefGoogle Scholar
 53.F. Liu, R. Kirchheim, J Cryst Growth, 2004, 264, pp. 385391.CrossRefGoogle Scholar
 54.F. Liu, R. Kirchheim, Scripta Mater, 2004, 51, pp. 521525.CrossRefGoogle Scholar
 55.P. C.Millett, R. P. Selvam, S. Bansal,A. Saxena, Acta Mater, 2005, 53, pp. 36713678.CrossRefGoogle Scholar
 56.L.S.Shvindlerman, G.Gottstein, Scripta Mater, 2006, 54, pp. 10411045.CrossRefGoogle Scholar
 57.R. Kirchheim, Scripta Mater, 2006, 55, pp. 963964.CrossRefGoogle Scholar
 58.R. Kirchheim, Acta Mater, 2007, 55, pp. 51295138.CrossRefGoogle Scholar
 59.R. Kirchheim, Acta Mater, 2007, 55, pp. 51395148.CrossRefGoogle Scholar
 60.P. C. Millett, R. P. Selvam, A. Saxena, Acta Mater, 2007, 55, pp. 23292336.CrossRefGoogle Scholar
 61.J. R. Trelewicz, C. A. Schuh: Phys. Rev B, 2009, 79:094112.CrossRefGoogle Scholar
 62.F.Liu, Z.Chen, W.Yang, C.L.Yang, H.F.Wang, G.C.Yang, Mater Sci Eng A, 2010, 457, pp. 1317.CrossRefGoogle Scholar
 63.Y. Purohit, L. Sun, D.L. Irving, R.O. Scattergood, D.W. Brenner: Mater. Sci. Eng. A 2010, 527, 17691775.CrossRefGoogle Scholar
 64.K.A.Darling, M.A.Tschopp, B.K.VanLeeuwen, M.A.Atwater, Z.K.Liu, Comput Mater Sci, 2014, 84, pp. 255266.CrossRefGoogle Scholar
 65.N. X. Zhou, J. Luo, Mater Lett, 2014, 115, pp. 268271.CrossRefGoogle Scholar
 66.T. Chookajorn, H.A.Murdoch, C.A.Schuh, Science, 2012, 337, pp. 951954.CrossRefGoogle Scholar
 67.H.A. Murdoch, C.A. Schuh, Acta Mater, 2013, 61, pp. 21212132.CrossRefGoogle Scholar
 68.H.A. Murdoch, C.A. Schuh, J Mater Res, 2013, 28, pp. 21542163.CrossRefGoogle Scholar
 69.M. Saber, H. Kotan, C.C. Koch, R.O. Scattergood: J. Appl. Phys., 2013, 113, 063515.CrossRefGoogle Scholar
 70.T. Chookajorn, C.A. Schuh: Phys. Rev. B, 2014, 89, 064102.CrossRefGoogle Scholar
 71.T. Chookajorn, C.A. Schuh, Acta Mater, 2014, 73, 128138.CrossRefGoogle Scholar
 72.A.R.Kalidindi, T.Chookajoorn, C.A.Schuh, JOM, 2015, 67, 28342843.CrossRefGoogle Scholar
 73.F. Abdeljawad, S.M. Foiles, Acta Mater, 2015, 101, 159171.CrossRefGoogle Scholar
 74.N. Zhou, T. Hu, J. Huang, J. Luo, Scr Mater, 2016, 124, pp. 160163.CrossRefGoogle Scholar
 75.T. Liang, Z. Chen, X. Yang, J. Zhang, P. Zhang, Int J Mater Res, 2017, 108, pp. 435440.CrossRefGoogle Scholar
 76.O. Waseda, H. Goldenstein, G.F.B. LenzSilva, A. Neiva, P. Chantrenne, J. Morthomas, M. Perez, C.S. Becquart, R.G.A. Veiga: Model. Simul. Mater. Sci. Eng., 2017, 25, 075005.CrossRefGoogle Scholar
 77.D.A.Aksyonov, A.G.Lipnitskii, Comp Mater Sci, 2017, 137, pp. 266272.CrossRefGoogle Scholar
 78.F. Abdeljawad, P. Lu, N. Argibay, B.G. Clarke, B.L. Boyce, S.M. Foiles, Acta Mater, 2017, 126, pp. 528539.CrossRefGoogle Scholar
 79.A.R.Kalidindi, C.A.Schuh, Acta Mater, 2017, 132, pp. 128137.CrossRefGoogle Scholar
 80.Y.Zhang, G.J.Tucker, J.R.Trelevicz, Acta Mater, 2017, 131, pp. 3947.CrossRefGoogle Scholar
 81.A.R.Kalidindi, C.A.Schuh, J Mater Res, 2017, 32, 19932000.CrossRefGoogle Scholar
 82.J.D.Schuler, T.J.Rupert, Acta Mater, 2017, 140, pp. 196205.CrossRefGoogle Scholar
 83.S.B. Kadambi, S. Patala: Phys. Rev. Mater., 2017, 1, 043604.CrossRefGoogle Scholar
 84.F.D.Fischer, G.A.Zickler, J.Svoboda, Phil Mag, 2017, 97, 19631977.CrossRefGoogle Scholar
 85.W.T.Xing, A.R.Kalidindi, C.A.Schuh, Scr Mater, 2017, 127, 136140.CrossRefGoogle Scholar
 86.Y.J. Zhao, J.Q. Zhou, J Nanopart Res, 2017, 19, 406.CrossRefGoogle Scholar
 87.C.J.O Brien, C.M.Barr, P.M.Price, K.Hattar, S.M.Foiles, J Mater Sci, 2018, 53, pp. 29112927.CrossRefGoogle Scholar
 88.K.Graetz, J.S.Paras, C.A.Schuh, Materialia, 2018, 1, pp. 8998.CrossRefGoogle Scholar
 89.M. A. Tschopp, H. A. Murdoch, L. J. Kecskes, K. A. Darling, JOM, 2014, 66, pp. 1000 – 1019.CrossRefGoogle Scholar
 90.R. A. Andrievski, J Mater Sci, 2014, 49, pp. 14491460.CrossRefGoogle Scholar
 91.M. Saber, C.C. Koch, and R.O. Scattergood, Mater Res Lett, 2015, 3, pp. 6575.CrossRefGoogle Scholar
 92.H.R.Peng, M.M.Gong, Y.Z.Chen, F.Liu, Int Mater Rev, 2017, 62, pp. 303333.CrossRefGoogle Scholar
 93.P.Lejcek, M.Vsianska, M.Sob, J Mater Res, 2018, 33, pp. 26472660.CrossRefGoogle Scholar
 94.G.Kaptay, J Mater Sci, 2012, 47, 83208335.CrossRefGoogle Scholar
 95.G.Kaptay, Adv Colloid Interface Sci, 2018, 256, pp. 163192.CrossRefGoogle Scholar
 96.J.W. Gibbs: Trans. Conn. Acad. Arts Sci. 1875–1878, vol. 3, pp. 108–248 and 343–524.Google Scholar
 97.I.Langmuir, J Am Chem Soc, 1918, 40, pp. 1361–1403.CrossRefGoogle Scholar
 98.D. McLean. Grain boundaries in metals. Clarendon Press, Oxford, 1957.Google Scholar
 99.P. Lejcek. Grain boundary segregation in metals, vol. 136 Springer Series Maters Sci, Springer, Berlin, 2010.Google Scholar
 100.G.Kaptay, J Mater Sci, 2016, 51, pp. 17381755.CrossRefGoogle Scholar
 101.J.A.V.Butler, Proc Roy Soc A, 1932, 135, pp. 348375.CrossRefGoogle Scholar
 102.C. C. Koch, R. O. Scattergood, K.A. Darling, J. E. Semones, J. Mater. Sci, 2008, 43, pp. 72647272.CrossRefGoogle Scholar
 103.C.C.Koch, R.O.Scattergood, M.Saber, H.Kotan, J Mater Res, 2013, 28, 17851791.CrossRefGoogle Scholar
 104.B.G.Clark, K.Hattar, M.T.Marshall, T.Chookajorn, B.L.Boyce, C.A.Schuh, JOM, 2016, 68, pp. 16251633.CrossRefGoogle Scholar
 105.D.Amram, C.A.Schuh, Acta Mater, 2018, 144, pp. 447458.CrossRefGoogle Scholar
 106.H.R.Peng, L.K.Huang, F.Liu, Mater Letters, 2018, 219, pp. 276279.CrossRefGoogle Scholar
 107.F.D.Fischer, J.Svoboda, P.Fratzl, Phil Mag, 2003, 83, pp. 10751093.CrossRefGoogle Scholar
 108.J.Li, J.Wang, G.Yang, Scr Mater, 2009, 60, 945948.CrossRefGoogle Scholar
 109.M.M.Gong, F.Liu, K.Zhang, Scr Mater, 2010, 63, pp. 989992.CrossRefGoogle Scholar
 110.Z. Chen, F. Liu, X.Q. Yang, C.J. Shen, and Y.M. Zhao, J. Alloys Compd, 2014, 608, pp. 338342.CrossRefGoogle Scholar
 111.D.L.Beke, Yu. Kaganovskii, G.L.Katona, Progr Mater Sci, 2018, 98, pp. 625674.CrossRefGoogle Scholar
 112.G. Kaptay, Langmuir, 2015, 31, pp. 57965804.CrossRefGoogle Scholar
 113.J.Korozs, G.Kaptay, Coll Surf A, 2017, 433, pp. 296301.CrossRefGoogle Scholar
 114.A.S.Skapski. J. Chem. Phys. 1948, 16, pp. 389393CrossRefGoogle Scholar
 115.G.Kaptay, Mater Sci Eng A, 2008, 495, pp. 1926.CrossRefGoogle Scholar
 116.G.Kaptay, Langmuir, 2017, 33, 10550–10560.CrossRefGoogle Scholar
 117.G.Kaptay, Int J Pharmaceutics, 2012, 430, pp. 253257.CrossRefGoogle Scholar
 118.G.Kaptay, Calphad, 2004, 28, pp. 115124.CrossRefGoogle Scholar
 119.G.Kaptay, Calphad, 2017, 56, pp. 169184.CrossRefGoogle Scholar
 120.T.B. Massalski, ed.: Binary Alloy Phase Diagrams, second ed., vol. 3, ASM International, New York, 1990.Google Scholar
 121.A.T.Dinsdale, Calphad, 1991, 15, pp. 317425.CrossRefGoogle Scholar
 122.I. Barin: Thermochemical Properties of Pure Substances, VCh, 1993, in 2 parts.Google Scholar
 123.J.Emsley: The Elements. Clarendon Press, Oxford, 1989.Google Scholar
 124.Y.S.Touloukian, R.K.Kirby, R.E.Taylor, T.Y.R.Lee: Thermal Expansion, IFI/Plenum, NY, 1977.CrossRefGoogle Scholar
 125.G.Kaptay, J. Mater. Sci., 2015, 50, pp. 678687.CrossRefGoogle Scholar
 126.T.Iida, R.I.L.Guthrie: The Physical Properties of Liquid Metals, Clarendon Press, Oxford, 1993.Google Scholar
 127.G. Guisbiers, M. JoseYacaman: in Reference Module in Chemistry, Molecular Sciences and Chemical Engineering, Elsevier, Amsterdam, 2017Google Scholar
 128.D.Schreiber, V.I.Razumovskiy, P.Rusching, R.Pippan, L.Romaner. Acta Mater, 2015, 88, pp. 180189.CrossRefGoogle Scholar
 129.D. Schreiber, R. Pippan, P. Ruschnig, L. Romaner. Model. Simul. Mater. Sci. Eng. 2016, 24, 035013.CrossRefGoogle Scholar
 130.D. Schreiber, R. Pippan, P. Ruschnig, L. Romaner. Model. Simul. Mater. Sci. Eng. 2016, 24: 085009.CrossRefGoogle Scholar
 131.B. Predel: Phase Equilibria, Crystallographic and Thermodynamic Data of Binary Alloys, vol. 5 of group IV of LandoltBörnstein Handbook, Springer, Berlin, 1991–1997.Google Scholar
 132.S.V.Meschel, O.J.Kleppa, J Alloys Compds, 1993, 197, pp. 7581.CrossRefGoogle Scholar
 133.L.Kaufman, H.Nesor, Calphad, 1978, 2, pp. 5580.CrossRefGoogle Scholar
 134.L.Kaufman, H.Nesor, Calphad, 1978, 2, pp. 80108.Google Scholar
 135.J.L.Murray, Bull Alloy Phase Diagr, 1981, 2, pp. 192196.CrossRefGoogle Scholar
 136.J.O.Andersson, P.Gustaffson, Calphad, 1983, 7, pp. 317326.CrossRefGoogle Scholar
 137.S.K.Lee, D.N.Lee, Calphad, 1986, 10, pp. 6176.CrossRefGoogle Scholar
 138.P Gustaffson, Metall Mater Trans A, 1988, 19A, pp. 25312546.CrossRefGoogle Scholar
 139.M.Vijayakumar, A.M.Siriamamutry, S.V.N. Naidi, Calphad, 1988, 12, pp. 177184.CrossRefGoogle Scholar
 140.K.Frisk, P.Gustaffson, Calphad, 1989, 12, pp. 247254.CrossRefGoogle Scholar
 141.L.Kaufman, Calphad, 1991, 15, pp. 243259.CrossRefGoogle Scholar
 142.N.Dupin, B.Sundman, Metallurgy, 2001, 30, pp. 184192.Google Scholar
 143.L.Kaufman, P.E.A. Turchi, W.Huang, Z.K.Liu, Calphad, 2001, 25, pp. 419433.CrossRefGoogle Scholar
 144.P.E.A. Turchi, V. Drchal, J. Kudrnovsky, C. Colinet, L. Kaufman, Z.K. Liu: Phys. Rev. B, 2005, 71, 094206.CrossRefGoogle Scholar
 145.J.Popovic, P.Broz, J.Bursik, Intermetallics, 2008, 16, pp. 884888.CrossRefGoogle Scholar
 146.Y.F.Cui, X.Zhang, G.L. Xu, W.J.Zhu, H.S.Liu, Z.P.Jun, J Mater Sci, 2011, 46, pp. 26112621.CrossRefGoogle Scholar
 147.S.Y. Yang, M.Jiang, H.X.Li, L. Wang, Trans Nonferr Met Soc China, 2011, 21, pp. 22702275.CrossRefGoogle Scholar
 148.P.Zhou, Y.Peng, Y.Du, S.Wang, G.Wen, Int J Refr Metal Hard Mater, 2015, 50, pp. 274281.CrossRefGoogle Scholar
 149.S.L.Wang, L.J.Zhou, Y.Ye, X.H.Wang, Y.H.Lin, C.P.Wang, X.J.Liu, J Phase Equil Diff, 2015, 36, pp. 39.CrossRefGoogle Scholar
 150.H.W.Yao, J.W.Qiao, M.C.Gao, J.A.Hawk, S.G.Ma, H.F.Zhou, Y.Zhang, Mater Sci Eng A, 2016, 674, pp. 203211.CrossRefGoogle Scholar
 151.F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema: Cohesion in Metals, NorthHolland, Amsterdam, 1988Google Scholar
 152.G.Kaptay, Metall Mater Trans A, 2012, 43, pp. 531543.CrossRefGoogle Scholar
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