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Modeling the interfacial energy of embedded metallic nanoparticles
Journal Pre-proof Modeling the interfacial energy of embedded metallic nanoparticles Mohadeseh Davari, Mohammad Amin Jabbareh PII:
S0022-3697(19)31473-8
DOI:
https://doi.org/10.1016/j.jpcs.2019.109261
Reference:
PCS 109261
To appear in:
Journal of Physics and Chemistry of Solids
Received Date: 23 June 2019 Revised Date:
3 November 2019
Accepted Date: 4 November 2019
Please cite this article as: M. Davari, M.A. Jabbareh, Modeling the interfacial energy of embedded metallic nanoparticles, Journal of Physics and Chemistry of Solids (2019), doi: https://doi.org/10.1016/ j.jpcs.2019.109261. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Modeling the interfacial energy of embedded metallic nanoparticles Mohadeseh Davari, Mohammad Amin Jabbareh* Department of Materials Engineering, Faculty of Engineering, Hakim Sabzevari University, Sabzevar, Iran
*E-mail address: [email protected] Tel: +(98) 51 44012776; fax: +(98) 51 44012773 Address: Hakim Sabzevari University, Towhid Shahr, Sabzevar, Iran, P.O. Box: 9617976487
Abstract The surface/interface energy is a key parameter for determining the properties of nanomaterials. Many studies have investigated the surface energy for free standing nanoparticles but the interface energy of embedded nanoparticles requires further analysis. In this study, we developed a model of the size-dependent interfacial energy for embedded metallic nanoparticles in terms of the size-dependent cohesive energy. The model was applied to Pb-Cu, Pb-Al, and Ag-Ni systems as three examples, and the results were compared with those obtained using previously developed models and the available experimental data. Depending on the system considered, the interface energy could increase or decrease as the particle size decreased. Similarly, the dependence of the interface energy on the temperature was also related to the system considered, where the interface energy could decrease or increase as the temperature increased. The results also showed that there was a critical value for interface coherency where the effect of size on the interface energy was reversed. Keywords: A. Interfaces; A. Nanostructures; D. Surface properties. 1. Introduction
1
It is well known that the physicochemical properties of nanoparticles, such as the melting temperature, melting enthalpy, Debye temperature, and thermal conductivity, differ from those of the corresponding bulk materials [1]. These special characteristics of nanoparticles are related to an increased surface to volume ratio, and thus the surface/interface energy is a key parameter for understanding the special properties of nanoparticles. Experimental measurements of the surface/interface energy are very difficult to obtain at the nanoscale, so mathematical models and computer simulations are essential for investigating the surface/interface energies of nanomaterials. Atomistic computational methods such as density functional theory (DFT) and molecular dynamics (MD) simulations have been used widely for calculating the surface/interface energies of nanomaterials. However, it is often difficult to reproduce the surface and interface energies in metals using the DFT [2], and the results generally deviate greatly from the experimental values [3]. By contrast, MD simulations are capable of obtaining more accurate estimations of the surface/interface energy [3]. Both methods are extremely computationally intensive, so the computations are usually limited to very small nanoparticles [4]. Recently, the interface force field method has been successfully employed to overcome the limitations of DFT and MD simulations [5,6]. Large-scale methods such as the liquid drop model [7], bond-order-length-strength model [8], and thermodynamic approaches [9] have also been used to study the size-dependent surface energy of nanoparticles. These investigations mainly focused on the surface energy of free standing nanoparticles [7–16]. Increases in the surface energy as the particle size decreases have been observed [17] and predicted [18,19] in some studies, but most have shown that the surface energy decreases as the particle size decreases. In general, it is accepted that the decrease in the surface energy with the particle size can be evaluated by: ( )=
(1 − / ), where
is the surface energy of a free standing nanoparticle, 2
is the surface energy of the corresponding bulk material,
is the nanoparticle diameter, and
is the material constant. In contrast to the comprehensive studies of the surface energies of free standing nanoparticles, few models are available for calculating the interfacial energy of an embedded nanoparticle. Guisbiers and Wautelet [20] assumed that the interface energy of a solid particle and substrate can be estimated based on the mean value of the solid surface energy of the particle and the solid surface energy of the substrate. Similarly Ouyang et al. [21] assumed that the solid–solid interfacial energy of a binary metallic multilayer at the nanoscale is the average value of the grain boundary energies of the two components. Luo et al. [22] used this approach and developed a Gibbs free energy function for embedded nanoparticles to study the size-dependent melting temperature. Xiong et al. [23] regarded an embedded nanoparticle as comprising a free nanoparticle and a cavity with the same size, and proposed that the size-dependent interface energy can be calculated as: = where
1−
,
−
,
1+
,
(1)
is the interfacial energy of the embedded nanoparticle,
,
and
,
are the
surface energies of the nanoparticle and nanocavity (matrix) in the bulk state, respectively, and
is a material constant, where denotes the nanoparticle or the matrix. Using the
bond-order-length-strength method, they suggested that
= 1.45 ℎ , where ℎ indicates
the atomic diameter of the material. However, other studies suggested different values for , e.g., Lu and Jiang [24] proposed that determining
= 4ℎ. Many other models have been proposed for
[25–27], where each assigns different values to
thereby leading to different predictions of the interfacial energy.
3
for the same material,
In addition to the particle size, the shape of the nanoparticle, temperature, and atomic structure of the interface (e.g., atomic relaxation, atomic reconstruction, and coherency of the interface) can affect the interfacial energy, but they are not considered in the models mentioned above. It has been shown that there is a linear relationship between the sizedependent cohesive energy of a nanoparticle and its surface energy [10]. In the present study, by calculating the cohesive energy of an embedded nanoparticle, we propose a simple model for determining the size-dependent interfacial energy of a metallic nanoparticle embedded in a metallic matrix. The proposed model can explain the effects on the interfacial energy of the size and shape of a nanoparticle, the temperature of the system, and atomic configuration at the interface. 2. Model According to a previous study [10], the surface energy of a free standing metallic nanoparticle, =
!
!
, can be obtained as follows:
,
where "
(2) and " are the cohesive energies of the nanoparticle and corresponding bulk
material, respectively, and
is the surface energy of the material in the bulk state. For
embedded nanoparticles, Eq. (2) can be rewritten as: #
#
=
!$
!
where
,
(3)
is the interface energy between the nanoparticle and the matrix,
interface energy of the corresponding bulk materials, and "!
is the
denotes the cohesive energy
of the embedded nanoparticle. The cohesive energy of an embedded nanoparticle can be written as: 4
"!
= "% + " ,
(4)
where "% and " are the cohesive energies of the interior and interface atoms in [Joule], respectively. The cohesive energy of the interior atoms can be expressed as: &
"% = ( () − )* )+, , '
(5)
where ) and )* are the numbers of total atoms and interface atoms for the nanoparticle, respectively, and thus ) − )* indicates the number of interior atoms; ( is the number of atomic bonds in each interior atom, and +, is the bond energy of the nanoparticle atoms in [J/atom]. The factor of 1/2 is employed because half of the bond energy belongs to each atom. The cohesive energy of the interface atoms can be given by: &
&
" = ' (* )* +, + ' -(( − (* ))* +,./ ,
(6)
where (* is the number of atomic bonds between a surface atom and the interior atoms, ( − (* indicates the number of atomic bonds between a surface atom and the matrix atoms, and - is a constant used to describe the degree of coherency of the interface. This coefficient is actually a correction factor that considers the effects of coherency changes when calculating the cohesive energy of embedded nanoparticles [25]. This coefficient does not indicate the actual degree of coherency but instead it is related to the coherency of the interface. Thus, - = 1 denotes that the interface is fully coherent [28,29], whereas - = 0 represents the free standing nanoparticle [25,30]. Semi-coherent and incoherent interfaces have values between 0 to 1. +,./ denotes the bond energy between a surface atom and the matrix. Liu et al. [31] showed that in binary alloys, charge transfer will occur if the difference in the electronegativity of the two metals is significant and this generates polar bonding, which increases the cohesion energy. However, if the difference in the 5
electronegativity is negligible (as found for the systems considered in this study), +,./ can be estimated as: &
+,./ = ' 1+, + +/ 2,
(7)
where +/ is the bond energy of the matrix atoms. By substituting Eqs (5), (6), and (7) into Eq. (4) and rearranging the equation, the cohesive energy of an embedded nanoparticle can be derived as follows: "!
= " 31 −
4 .45 4
5
&
61 − ' - 31 +
78 79
:;:
(8)
where " = )( +, /2 is the cohesive energy of the nanoparticle in the bulk state. The number of total atoms in the nanoparticle, ), can be determined as: ) = = >, />? , where >, and >? are the volumes of the nanoparticle and each atom, respectively, and = is the atomic packing fraction. The number of interface atoms is given by: )* = =* @, /@? , where =* is the planar packing fraction, @, is the nanoparticle’s surface area, and @? is the effective surface area of each atom, which indicates the contribution of each atom to the surface area of the nanoparticle. Without interface relaxation, half of the surface of each interface atom is in the nanoparticle, so the effective surface area of each atom is equal to: A(ℎ/2)' , where ℎ denotes the atomic diameter. However, interface relaxation is expected for embedded nanoparticles [32,33], so the effective surface area becomes smaller than A(ℎ/2)' . Thus, )* can be written as: )* = =* @, ⁄B A (ℎ/2)' , where the parameter B is a constant used to describe the effect of interface relaxation. B is greater than zero and smaller than or equal to unity (B = 1 indicates no relaxation). Considering that >? = AℎD /6, the ratio of the surface to total atoms in the nanoparticle can be determined as: )* ⁄) = (2/3)(=* /= )(ℎ/ B)(@, />, ). For spherical nanoparticles, @, />, = 6/ , where 6
is the nanoparticle’s
diameter. Qi et al. [34] introduced a shape factor parameter, G, to describe the difference in shape between spherical and non-spherical nanoparticles. The shape factor concept and the method used for its calculation were described by Qi et al. [34]. For non-spherical nanoparticles, @, />, = 6G/ . Thus, the ratio of the surface to total atoms can be expressed as Eq. (9). It should be noted that for regular polyhedral nanoparticles, 1 ≤ G ≤ 1.49 [34]. 5
=
JK L
M5
N
M
(9)
By inserting Eq. (9) into Eq. (8), the cohesive energy of a nanoparticle embedded in a matrix can be expressed as follows, !$
!
Let
= 1− 4 .45 4
4 .45
M5
4
M
= O and
M5
M
JK L
N
&
61 − ' - 31 +
78 79
:;
(10)
= P, and considering that the bond energy of a material, +, is
proportional to its cohesive energy, ", then the cohesive energy of an embedded nanoparticle can be written as: !$
!
= 1−
Q R L
4
K N
&
61 − ' - 31 +
!8 !9
:; ,
(11)
where "/ and ", are the cohesive energies of the matrix and the nanoparticle in the bulk state, respectively, and O, P, and B are parameters related to the atomic arrangements at the interface. For a close packed crystal structure (i.e., fcc or hcp) ( = 12 and = = 0.74. If the interface is also a close packed atomic plane, (* = 9 and =* = 0.91; hence, O = 0.25, and P = 1.23, and thus OP = 0.3. However, surface reconstruction leads to changes in (* and =* , so OP can deviate from 0.3 due to the interface reconstruction. Theoretically, 1 ≤ (* ≤ 11 and 0 < =* ≤ 0.91, and thus 0.08 ≤ O ≤ 0.92 and 0 < P ≤ 1.23, so OP must be greater than zero and smaller than 1.13. It should be noted that O and P are related to 7
each other such that an increase in O causes a decrease in P and vice versa. Thus, the maximum value of OP, i.e., 1.13, will never be achieved. It has also been reported that ( and consequently = are size-dependent parameters [8], thereby affecting the value of OP. Interface relaxation can also affect the cohesive energy. As mentioned above, the interface relaxation parameter, B, must be greater than zero and smaller than or equal to unity. Assuming that the minimum acceptable value of B is 0.1 (i.e., 0.1 ≤ B ≤ 1), the range of OP/B varies from zero to 11.3. For a system with no relaxation and no reconstruction, OP/B is equal to 0.3. It should be noted that if OP/B = 3/4 or OP/B = 1/2, the model will convert into the models proposed previously by [35] and [28] for the cohesive energy of embedded nanoparticles, respectively. Let OP/B = V and according to Eq. (3), the interface energy of an embedded nanoparticle can be expressed as follows. =
31 − V 4G
N
&
61 − ' - 31 +
!8 !9
:;:
(12)
Kaptay [36] proposed a simple model for estimating the coherent interface energy between metals A and B in the bulk state,
, as follows:
' Q W
=X
(13)
Y/Z
where Ω is the interaction parameter for the A/B alloy in the solid state, which is generally a function of the temperature, \, and composition, ]. For simplicity, the nanoparticle/matrix interface is assumed to be a sharp interface in the present study. If the interface is assumed to be fully coherent, each atom at the surface of the nanoparticle will be bonded with one atom in the matrix. Therefore, the interface atoms contain bonds similar to the A (particle)/B (matrix) alloy where: ]^ = ]_ = 0.5. According to Eq. (13), the energy of this interface is proportional to the A/B alloy formation energy ( ∆a = ]^ ]_ Ω ). Thus, the 8
interfacial energy can even be negative if the formation energy of this alloy is negative. Negative interface energies have been reported for multi-component systems in other studies [37]. b^/_ is the molar interface area and O = (( − (* )/( . The molar interface area can be given by: b^/_ = √b^ b_ ,
(14)
where b is the molar surface area of pure component , which can be calculated as: b = 1.1()d )&/D (> )'/D,
(15)
where )d is Avogadro’s number and > is the molar volume of the pure component in the solid state. Various models have been developed for estimating > [38,39], but we use the model proposed by Tanaka and Hara [39]: > = (1 + ef\ − \/ g)> / /(1 + h) ,
(16)
where e is the volumetric thermal expansion coefficient, \/ represents the melting temperature, > / is the molar volume of the liquid at its melting point, and h indicates the ratio of the volume change due to fusion. According to Eq. (12) and Eq. (13), the interfacial energy of the embedded nanoparticle can be obtained as follows. ' Q W
=X
Y/Z
31 − V 4G
N
&
61 − ' - 31 +
!8 !9
:;:
(17)
It should be noted that the interface energy is usually considered to comprise a chemical part and a structural part. The chemical part is related to the atomic bond energy and the structural part originates from the elastic strain energy at the interface. This strain energy is caused by the lattice mismatch of two crystals [40]. Lattice mismatch leads to 9
rearrangement of the atoms at the interface to achieve a stable atomic configuration [5]. Different types of defects such as Shockley dislocations, perfect dislocations, and twins are correlated with the lattice misfit at the interface. Hao and Lau [41] indicated that the interface becomes rougher and the strain energy becomes larger as the lattice mismatch increases. However, Ouyang et al. [21] showed that the structural part of the interface energy is one or two orders smaller than that of the chemical part at several nanometer scale sizes. Therefore, we neglect the structural part of the interface energy as a first order approximation in the present study.
We applied Eq. (17) to Pb-Cu, Pb-Al, and Ag-Ni systems, and determined the effects of the particle size and shape, degree of coherency, and temperature on the values of the interface energy. Table 1 shows the thermodynamic and physical properties used in the calculations. The interaction parameters for the Pb-Cu, Pb-Al, and Ag-Ni systems are listed in Table 2. Table 1. Thermodynamic and physical parameters used in the calculations. Parameter Cohesive energy (kJ/mol) Melting point (K) Molar volume at the melting point 3 (m /mol) – Volumetric thermal expansion (K 1 ) Ratio of the volume change due to fusion (%) Atomic diameter (nm)
Symbol " \/ >/ e h ℎ
Element Ag 284 1235 11.6 × –6 10 0.98 × –4 10 3.51 0.32
Ref. Al 327 933.5 11.3 × –6 10 – 1.5 × 10
Cu 336 1358 7.94 × –6 10 – 1.0 × 10 3.96
Ni 428 1728 7.43 × –6 10 1.51 × –4 10 6.3
4
4
6.9 0.286
3.81
[44]
0.27
0.27
0.36
[7]
Table 2. Interaction parameters of Pb-Cu, Pb-Al and Ag-Ni systems. System Pb-Cu Pb-Al Ag-Ni
Ω [J/mol] 111040.719 + 53.356 T 100000 54620.4 + 3.1 T +2800 (1–2XNi)
Ref. [45] [46] [47]
10
Pb 196 600.7 19.42 × –6 10 – 1.24 × 10
[42] [42] [43] [43]
4
3. Results and discussion Figure 1 shows the interface energy calculated as a function of the particle size for different A/B systems, where A represents the nanoparticle and B is the matrix. Spherical nanoparticles with fully coherent interfaces and no relaxation or reconstruction were considered. In all cases, the temperature was set to the half the melting temperature of the nanoparticle in the bulk state. The results demonstrate that decreasing the particle size in the Pb/Cu, Pb/Al, and Ag/Ni systems led to increases in the interface energy. By contrast decreasing the particle size in the Cu/Pb, Al/Pb, and Ni/Ag systems decreased the interface energy. According to Table 1, the cohesive energy of the matrix was greater than that of the nanoparticle in the Pb/Cu, Pb/Al, and Ag/Ni systems, so Em/Ep was greater than unity in these systems. According to Eq. (17), for fully coherent interfaces, this condition means that ⁄
is greater than unity. Thus, the interface energy of the embedded nanoparticle was
greater than the bulk interface energy under these conditions. When the nanoparticle and matrix were replaced by each other, Em/Ep was smaller than unity and the energy of the nanoparticle interface was smaller than that in the bulk state. Clearly, the size dependency of the interface energy varied among the different systems. According to Eq. (9), G, B, =* , and = were similar in all cases, so the difference between the size dependencies was due to the differences in the atomic diameters of the systems. Increasing the particle size allowed the value of the interface energy to approach that of the bulk interface energy in all cases. It is well known that the melting temperature of a substance is directly related to its surface energy. Hence, the increases in the interface energy after decreasing the nanoparticle diameter in the Pb/Cu, Pb/Al, and Ag/Ni systems indicate that the melting points of the
11
nanoparticles in these systems must have increased after decreasing the particle size. These results are consistent with the previously reported experimental melting temperatures determined for the embedded nanoparticles [48–51].
Figure 1. Size-dependent interfacial energies calculated for A/B systems (A represents the nanoparticle and B indicates the matrix). Other parameters were set as: c = 1, i = 0.3, α = 1, and T = Tm,A/2.
Figure 2 shows the interface energies calculated for the Pb/Cu, Pb/Al, and Ag/Ni systems with different values of V (i.e., different interface conditions). The results are also compared with the interface energies calculated using Eq. (1), where the different values of proposed by Xiong et al. [23] (i.e.,
= 1.45ℎ) and Lu and Jiang [24] (i.e.,
= 4ℎ) were
considered. The circles in Figure 2 indicate experimental data converted from melting temperature data using the relationship:
j8.
j8,
=
#
#
. As the value of V increased, the
calculated interface energies increased for all diameters. For the Pb/Cu and Pb/Al systems, V = 2.5 obtained better agreement with the experimental results, whereas 1.2 was the most appropriate value of V for the Ag/Ni system. The lattice constant mismatches for the Ag/Ni, Pb/Al, and Pb/Cu systems were 0.15, 0.22, and 0.36 respectively, which indicate that increasing the lattice constant mismatch increased the value of V. This is reasonable 12
because mismatched lattice constants lead to increased stress at the interfaces and rearrangements of the atoms can reduce this stress. It should be noted that the interfaces were assumed to be fully coherent in these calculations, and even for the Pb/Cu system with a relatively large lattice mismatch. Previous experimental studies [52] have shown that this assumption can be true for small particle sizes. In addition to the surface rearrangement of the atoms, changing the interface coherency can reduce the interface stress. Thus, the degree of coherency can be reduced by increasing the particle size. The effects of coherency on the interface energy of an embedded nanoparticle are discussed in the following. Figure 2 also shows that when using the value of interface energy, whereas the value of
from Lu’s model, Eq. (1) overestimated the
value from Xiong’s model obtained better
agreement with the experimental values.
Figure 2. Interfacial energies calculated for the embedded nanoparticles compared with the experimental values [48–51] and the results calculated using Eq. (1). The other parameters were set as: c = 1 and α = 1. (a) Pb/Cu, (b) Pb/Al, and (c) Ag/Ni.
13
Figure 3 shows the interface energy in the Pb/Cu system as a function of the interface coherency for three different particle sizes, where no relaxation/reconstruction was assumed. The results show that for a specific particle size, the interface energy increased after increasing the coherency of the interface. It should be noted that the interface energy was considered to comprise a chemical part and a structural part. After decreasing the coherency of the interface, the structural part of the interface energy increased due to the increases in structural defects such as dislocations at the interface. By contrast, the chemical part of the interface energy increased after increasing the coherency of the interface due to the increase in the atomic bonds at the interface. However, the net effect was an increase in the interface energy as the interface coherency decreased. We completely neglected the structural part of the interface energy, so the interface energy comprised the chemical part of the interface energy in the present study. Thus, the increase in the interface energy (chemical part) as the coherency increased was predicted by the model as expected. We also found that there was a critical interface coherency, c*, at which the value of the nanoparticle interface energy was equal to the interface energy for the bulk system. When the coherency of the interface exceeded the critical value, the interface energy increased after decreasing the particle size, whereas when the coherency was smaller than c*, decreasing the particle size decreased the interface energy. Zhao et al. [53] showed that the superheating of embedded nanoparticles with coherent interfaces is dominated by the chemical parameters and that the structural pressure has a small effect, whereas the effects of the structural parameters are important for nanoparticles with incoherent interfaces. Zhao et al. [53] only considered fully coherent interfaces but it is reasonable to assume that their results are also valid for semi-coherent interfaces. Therefore, we could investigate the melting behavior of embedded nanoparticles with coherent and semi-coherent interfaces 14
based only on the chemical part of the interface energy. It is considered that if "/ /", > 1, then the melting temperature of embedded nanoparticles will become greater than that of the bulk [54]. Considering the direct relationship between the chemical part of the interface energy and the melting temperature of an embedded nanoparticle, our results indicate that for superheating, in addition to the ratio of the cohesive energies, the coherency of the interface was also greater than the critical value. By contrast, when the interface coherency was smaller than the critical value, the melting behavior of the embedded nanoparticles was similar to that of the free standing nanoparticles, i.e., the melting temperature decreased as the particle size decreased. According to Eq. (17), - ∗ = 2", /("/ + ", ), which shows that if "/ ≤ ", , then - ∗ ≥ 1, whereas - > 1 has no physical meaning, thereby demonstrating that under these conditions, the interface energy will always be smaller than the bulk value, such as those for the Cu/Pb, Al/Pb and Ni/Ag systems shown in Figure 1. By contrast, if "/ > ", then - ∗ < 1 and the interface energy can be greater or smaller than the bulk value according to the degree of interface coherency.
15
Figure 3. Effect of interface coherency on the interfacial energy of embedded Pb nanoparticles in Cu matrix. The other parameters were set as: i = 0.3 and α = 1.
The shape of a particle is another factor that can affect the interface energy [12,15]. By changing the shape of the nanoparticle from spherical to polyhedral, some interior atoms move to the surface and the number of surface atoms, )* , increases. Thus, polyhedral nanoparticles have more particle–matrix bonds, +,./ , than spherical nanoparticles. Therefore, according to Eqs (4 – 7) the contribution of the cohesive energy of the interface atoms, " , to the total cohesive energy, "!
, will increase. Hence, it is expected that if
+,./ > +, (i.e., "/ > ", ), then the interface energy of polyhedral nanoparticles will become greater than that of the spherical nanoparticles. Conversely, if +,./ < +, (i.e., "/ < ", ), then it is expected that the interface energy of polyhedral nanoparticles will become smaller than that of the spherical nanoparticles. As mentioned above, we considered the effect of the shape of nanoparticles by introducing the shape factor, G. Increasing the shape factor indicates an increase in the number of surface atoms. Figure 4 shows the interface energies calculated for the Pb/Al and Al/Pb systems with different particle shapes. Clearly, when "/ > ", (i.e., Pb/Al system), the interface energy increased as the shape factor increased, whereas when "/ < ", (i.e., Al/Pb system), increasing the shape factor decreased the interface energy. In both cases, the shape effects were more evident after decreasing the particle size because decreasing the particle size increased the )* /) ratio, especially for nanoparticles with diameters smaller than 10 nm [34]. It should be noted that from an atomistic viewpoint, the interface contains several types of atoms, including face, edge, and vertex atoms as well as temporary adatoms [55]. Each of these atom types form different numbers of particle–matrix bonds, which can affect the interface
16
energy. The ratio of edge and vertex atoms relative to the total interface atoms also depends on the size and shape of the nanoparticles [55]. However, the contributions of edge and vertex atoms to the surface energy can be ignored when the particle diameter is larger than 5 nm [56]. Thus, we ignored the effects of edge and vertex atoms, and all of the interface atoms were assumed to be face atoms. This simplification may have led to some errors when calculating the interface energies of the small nanoparticles.
Figure 4. Effects of nanoparticle shape on the interfacial energies of the embedded nanoparticles. The other parameters were set as: i = 0.3 and c = 1, and T = Tm,A/2.
17
Figure 5. (a) Interfacial energies calculated for the embedded nanoparticles as a function of temperature. (b) Variations in n/op/q as a function of temperature.
Figure 5(a) shows the interface energies calculated for the Pb/Cu, Pb/Al, and Ag/Ni systems as a function of temperature. Spherical nanoparticles with parameters comprising d = 5 nm, c = 1, and V = 0.3 were considered. Decreases in the interface energy as the temperature 18
increased were determined for the Pb/Al and Ag/Ni systems. However, in the Pb/Cu system, the interface energy increased as the temperature increased, which contrasted with the temperature dependence of the surface free energy, which decreased as the temperature increased. Few data are available regarding the dependence of the solid–solid interface energy on the temperature. In particular, Luo et al. [22] used a thermodynamic model to investigate the effects of temperature on the solid–solid interfacial energy for a Cu-Fe system and showed that the interface energy increased as the temperature increased. In addition, studies of the solid–liquid interface energy showed that the interface energy increased as the temperature increased. For example, Bai and Li [57] conducted MD simulations to study the interfacial energy for the solid–liquid interface of a pure metal, where they found that the interfacial free energy increased as the temperature increased, and the positive temperature coefficient of the interfacial energy was in qualitative agreement with the analysis by Spaepen [58] and the empirical estimate of Turnbull [59]. Sato et al. [60] also reported a positive temperature coefficient for the solid–liquid interface energy in a Cu-Zr alloy. However, other studies obtained different results, e.g., Granasy and Tegze [61] used a broken bond model to estimate the solid–liquid interface energy for some metals and alloys, as well as providing experimental data, including the temperaturedependent interface energy for a Pb-Cu alloy. The experiments and model showed that the interface energy decreased as the temperature increased. According to the model, we found that the temperature dependence of the interface energy was related to the Ω/b^/_ ratio, which was an increasing function of the temperature for the Pb/Cu system but a descending function of the temperature for the other cases (Figure 5(b)). Therefore, in contrast to the Ag/Ni and Pb/Al interfaces, we found that the interface energy increased in the Pb/Cu system as the temperature increased. In the present study, we calculated Ω 19
based on the CALPHAD approach, but some studies have shown that the CALPHAD assessment is not a reliable method for determining the temperature dependence of the interaction energies in solid solutions [63]. Thus, according to previous research, determining the dependence of the interfacial energy on temperature requires further theoretical and experimental studies. Conclusion In this study, we developed a model for determining the size-dependent interface energy of an embedded metallic nanoparticle in a metallic matrix in terms of the size-dependent cohesive energy. The model considers the effects on the interface energy of embedded nanoparticles of the particle size, shape, and temperature as well as the interface coherency and atomic relaxation/reconstruction. The results showed that depending on the system considered, the value of the interface energy could decrease or increase as the particle size decreased. The temperature dependence of the interface energy was also related to the system considered, where the interface energy was an increasing or decreasing function of the temperature. Furthermore, the shape of the nanoparticle affected the interfacial energy of the embedded nanoparticle. The shape effect was more evident when the nanoparticle size was smaller than 5 nm. We also identified a critical value for the interface coherency at which the size dependency of the interface energy becomes reversed, i.e., if the coherency is greater than the critical value, the interface energy will be greater than that for the bulk, whereas if the interface coherency is below the critical value, the interface energy will be smaller than that for the bulk. The proposed model can be used to study various interfacedependent properties of nanoparticles, such as the melting behavior of embedded and core–shell nanoparticles as well as multilayers at the nanometer scale.
20
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28
Highlights •
Model of size-dependent interfacial energy of embedded metallic nanoparticles.
•
Effects of size, shape, temperature, and interface coherency on interface energy.
•
Proposed model agrees better with experimental data than previous models.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
M.A. Jabbareh & M. Davari
S0022-3697(19)31473-8
DOI:
https://doi.org/10.1016/j.jpcs.2019.109261
Reference:
PCS 109261
To appear in:
Journal of Physics and Chemistry of Solids
Received Date: 23 June 2019 Revised Date:
3 November 2019
Accepted Date: 4 November 2019
Please cite this article as: M. Davari, M.A. Jabbareh, Modeling the interfacial energy of embedded metallic nanoparticles, Journal of Physics and Chemistry of Solids (2019), doi: https://doi.org/10.1016/ j.jpcs.2019.109261. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Modeling the interfacial energy of embedded metallic nanoparticles Mohadeseh Davari, Mohammad Amin Jabbareh* Department of Materials Engineering, Faculty of Engineering, Hakim Sabzevari University, Sabzevar, Iran
*E-mail address: [email protected] Tel: +(98) 51 44012776; fax: +(98) 51 44012773 Address: Hakim Sabzevari University, Towhid Shahr, Sabzevar, Iran, P.O. Box: 9617976487
Abstract The surface/interface energy is a key parameter for determining the properties of nanomaterials. Many studies have investigated the surface energy for free standing nanoparticles but the interface energy of embedded nanoparticles requires further analysis. In this study, we developed a model of the size-dependent interfacial energy for embedded metallic nanoparticles in terms of the size-dependent cohesive energy. The model was applied to Pb-Cu, Pb-Al, and Ag-Ni systems as three examples, and the results were compared with those obtained using previously developed models and the available experimental data. Depending on the system considered, the interface energy could increase or decrease as the particle size decreased. Similarly, the dependence of the interface energy on the temperature was also related to the system considered, where the interface energy could decrease or increase as the temperature increased. The results also showed that there was a critical value for interface coherency where the effect of size on the interface energy was reversed. Keywords: A. Interfaces; A. Nanostructures; D. Surface properties. 1. Introduction
1
It is well known that the physicochemical properties of nanoparticles, such as the melting temperature, melting enthalpy, Debye temperature, and thermal conductivity, differ from those of the corresponding bulk materials [1]. These special characteristics of nanoparticles are related to an increased surface to volume ratio, and thus the surface/interface energy is a key parameter for understanding the special properties of nanoparticles. Experimental measurements of the surface/interface energy are very difficult to obtain at the nanoscale, so mathematical models and computer simulations are essential for investigating the surface/interface energies of nanomaterials. Atomistic computational methods such as density functional theory (DFT) and molecular dynamics (MD) simulations have been used widely for calculating the surface/interface energies of nanomaterials. However, it is often difficult to reproduce the surface and interface energies in metals using the DFT [2], and the results generally deviate greatly from the experimental values [3]. By contrast, MD simulations are capable of obtaining more accurate estimations of the surface/interface energy [3]. Both methods are extremely computationally intensive, so the computations are usually limited to very small nanoparticles [4]. Recently, the interface force field method has been successfully employed to overcome the limitations of DFT and MD simulations [5,6]. Large-scale methods such as the liquid drop model [7], bond-order-length-strength model [8], and thermodynamic approaches [9] have also been used to study the size-dependent surface energy of nanoparticles. These investigations mainly focused on the surface energy of free standing nanoparticles [7–16]. Increases in the surface energy as the particle size decreases have been observed [17] and predicted [18,19] in some studies, but most have shown that the surface energy decreases as the particle size decreases. In general, it is accepted that the decrease in the surface energy with the particle size can be evaluated by: ( )=
(1 − / ), where
is the surface energy of a free standing nanoparticle, 2
is the surface energy of the corresponding bulk material,
is the nanoparticle diameter, and
is the material constant. In contrast to the comprehensive studies of the surface energies of free standing nanoparticles, few models are available for calculating the interfacial energy of an embedded nanoparticle. Guisbiers and Wautelet [20] assumed that the interface energy of a solid particle and substrate can be estimated based on the mean value of the solid surface energy of the particle and the solid surface energy of the substrate. Similarly Ouyang et al. [21] assumed that the solid–solid interfacial energy of a binary metallic multilayer at the nanoscale is the average value of the grain boundary energies of the two components. Luo et al. [22] used this approach and developed a Gibbs free energy function for embedded nanoparticles to study the size-dependent melting temperature. Xiong et al. [23] regarded an embedded nanoparticle as comprising a free nanoparticle and a cavity with the same size, and proposed that the size-dependent interface energy can be calculated as: = where
1−
,
−
,
1+
,
(1)
is the interfacial energy of the embedded nanoparticle,
,
and
,
are the
surface energies of the nanoparticle and nanocavity (matrix) in the bulk state, respectively, and
is a material constant, where denotes the nanoparticle or the matrix. Using the
bond-order-length-strength method, they suggested that
= 1.45 ℎ , where ℎ indicates
the atomic diameter of the material. However, other studies suggested different values for , e.g., Lu and Jiang [24] proposed that determining
= 4ℎ. Many other models have been proposed for
[25–27], where each assigns different values to
thereby leading to different predictions of the interfacial energy.
3
for the same material,
In addition to the particle size, the shape of the nanoparticle, temperature, and atomic structure of the interface (e.g., atomic relaxation, atomic reconstruction, and coherency of the interface) can affect the interfacial energy, but they are not considered in the models mentioned above. It has been shown that there is a linear relationship between the sizedependent cohesive energy of a nanoparticle and its surface energy [10]. In the present study, by calculating the cohesive energy of an embedded nanoparticle, we propose a simple model for determining the size-dependent interfacial energy of a metallic nanoparticle embedded in a metallic matrix. The proposed model can explain the effects on the interfacial energy of the size and shape of a nanoparticle, the temperature of the system, and atomic configuration at the interface. 2. Model According to a previous study [10], the surface energy of a free standing metallic nanoparticle, =
!
!
, can be obtained as follows:
,
where "
(2) and " are the cohesive energies of the nanoparticle and corresponding bulk
material, respectively, and
is the surface energy of the material in the bulk state. For
embedded nanoparticles, Eq. (2) can be rewritten as: #
#
=
!$
!
where
,
(3)
is the interface energy between the nanoparticle and the matrix,
interface energy of the corresponding bulk materials, and "!
is the
denotes the cohesive energy
of the embedded nanoparticle. The cohesive energy of an embedded nanoparticle can be written as: 4
"!
= "% + " ,
(4)
where "% and " are the cohesive energies of the interior and interface atoms in [Joule], respectively. The cohesive energy of the interior atoms can be expressed as: &
"% = ( () − )* )+, , '
(5)
where ) and )* are the numbers of total atoms and interface atoms for the nanoparticle, respectively, and thus ) − )* indicates the number of interior atoms; ( is the number of atomic bonds in each interior atom, and +, is the bond energy of the nanoparticle atoms in [J/atom]. The factor of 1/2 is employed because half of the bond energy belongs to each atom. The cohesive energy of the interface atoms can be given by: &
&
" = ' (* )* +, + ' -(( − (* ))* +,./ ,
(6)
where (* is the number of atomic bonds between a surface atom and the interior atoms, ( − (* indicates the number of atomic bonds between a surface atom and the matrix atoms, and - is a constant used to describe the degree of coherency of the interface. This coefficient is actually a correction factor that considers the effects of coherency changes when calculating the cohesive energy of embedded nanoparticles [25]. This coefficient does not indicate the actual degree of coherency but instead it is related to the coherency of the interface. Thus, - = 1 denotes that the interface is fully coherent [28,29], whereas - = 0 represents the free standing nanoparticle [25,30]. Semi-coherent and incoherent interfaces have values between 0 to 1. +,./ denotes the bond energy between a surface atom and the matrix. Liu et al. [31] showed that in binary alloys, charge transfer will occur if the difference in the electronegativity of the two metals is significant and this generates polar bonding, which increases the cohesion energy. However, if the difference in the 5
electronegativity is negligible (as found for the systems considered in this study), +,./ can be estimated as: &
+,./ = ' 1+, + +/ 2,
(7)
where +/ is the bond energy of the matrix atoms. By substituting Eqs (5), (6), and (7) into Eq. (4) and rearranging the equation, the cohesive energy of an embedded nanoparticle can be derived as follows: "!
= " 31 −
4 .45 4
5
&
61 − ' - 31 +
78 79
:;:
(8)
where " = )( +, /2 is the cohesive energy of the nanoparticle in the bulk state. The number of total atoms in the nanoparticle, ), can be determined as: ) = = >, />? , where >, and >? are the volumes of the nanoparticle and each atom, respectively, and = is the atomic packing fraction. The number of interface atoms is given by: )* = =* @, /@? , where =* is the planar packing fraction, @, is the nanoparticle’s surface area, and @? is the effective surface area of each atom, which indicates the contribution of each atom to the surface area of the nanoparticle. Without interface relaxation, half of the surface of each interface atom is in the nanoparticle, so the effective surface area of each atom is equal to: A(ℎ/2)' , where ℎ denotes the atomic diameter. However, interface relaxation is expected for embedded nanoparticles [32,33], so the effective surface area becomes smaller than A(ℎ/2)' . Thus, )* can be written as: )* = =* @, ⁄B A (ℎ/2)' , where the parameter B is a constant used to describe the effect of interface relaxation. B is greater than zero and smaller than or equal to unity (B = 1 indicates no relaxation). Considering that >? = AℎD /6, the ratio of the surface to total atoms in the nanoparticle can be determined as: )* ⁄) = (2/3)(=* /= )(ℎ/ B)(@, />, ). For spherical nanoparticles, @, />, = 6/ , where 6
is the nanoparticle’s
diameter. Qi et al. [34] introduced a shape factor parameter, G, to describe the difference in shape between spherical and non-spherical nanoparticles. The shape factor concept and the method used for its calculation were described by Qi et al. [34]. For non-spherical nanoparticles, @, />, = 6G/ . Thus, the ratio of the surface to total atoms can be expressed as Eq. (9). It should be noted that for regular polyhedral nanoparticles, 1 ≤ G ≤ 1.49 [34]. 5
=
JK L
M5
N
M
(9)
By inserting Eq. (9) into Eq. (8), the cohesive energy of a nanoparticle embedded in a matrix can be expressed as follows, !$
!
Let
= 1− 4 .45 4
4 .45
M5
4
M
= O and
M5
M
JK L
N
&
61 − ' - 31 +
78 79
:;
(10)
= P, and considering that the bond energy of a material, +, is
proportional to its cohesive energy, ", then the cohesive energy of an embedded nanoparticle can be written as: !$
!
= 1−
Q R L
4
K N
&
61 − ' - 31 +
!8 !9
:; ,
(11)
where "/ and ", are the cohesive energies of the matrix and the nanoparticle in the bulk state, respectively, and O, P, and B are parameters related to the atomic arrangements at the interface. For a close packed crystal structure (i.e., fcc or hcp) ( = 12 and = = 0.74. If the interface is also a close packed atomic plane, (* = 9 and =* = 0.91; hence, O = 0.25, and P = 1.23, and thus OP = 0.3. However, surface reconstruction leads to changes in (* and =* , so OP can deviate from 0.3 due to the interface reconstruction. Theoretically, 1 ≤ (* ≤ 11 and 0 < =* ≤ 0.91, and thus 0.08 ≤ O ≤ 0.92 and 0 < P ≤ 1.23, so OP must be greater than zero and smaller than 1.13. It should be noted that O and P are related to 7
each other such that an increase in O causes a decrease in P and vice versa. Thus, the maximum value of OP, i.e., 1.13, will never be achieved. It has also been reported that ( and consequently = are size-dependent parameters [8], thereby affecting the value of OP. Interface relaxation can also affect the cohesive energy. As mentioned above, the interface relaxation parameter, B, must be greater than zero and smaller than or equal to unity. Assuming that the minimum acceptable value of B is 0.1 (i.e., 0.1 ≤ B ≤ 1), the range of OP/B varies from zero to 11.3. For a system with no relaxation and no reconstruction, OP/B is equal to 0.3. It should be noted that if OP/B = 3/4 or OP/B = 1/2, the model will convert into the models proposed previously by [35] and [28] for the cohesive energy of embedded nanoparticles, respectively. Let OP/B = V and according to Eq. (3), the interface energy of an embedded nanoparticle can be expressed as follows. =
31 − V 4G
N
&
61 − ' - 31 +
!8 !9
:;:
(12)
Kaptay [36] proposed a simple model for estimating the coherent interface energy between metals A and B in the bulk state,
, as follows:
' Q W
=X
(13)
Y/Z
where Ω is the interaction parameter for the A/B alloy in the solid state, which is generally a function of the temperature, \, and composition, ]. For simplicity, the nanoparticle/matrix interface is assumed to be a sharp interface in the present study. If the interface is assumed to be fully coherent, each atom at the surface of the nanoparticle will be bonded with one atom in the matrix. Therefore, the interface atoms contain bonds similar to the A (particle)/B (matrix) alloy where: ]^ = ]_ = 0.5. According to Eq. (13), the energy of this interface is proportional to the A/B alloy formation energy ( ∆a = ]^ ]_ Ω ). Thus, the 8
interfacial energy can even be negative if the formation energy of this alloy is negative. Negative interface energies have been reported for multi-component systems in other studies [37]. b^/_ is the molar interface area and O = (( − (* )/( . The molar interface area can be given by: b^/_ = √b^ b_ ,
(14)
where b is the molar surface area of pure component , which can be calculated as: b = 1.1()d )&/D (> )'/D,
(15)
where )d is Avogadro’s number and > is the molar volume of the pure component in the solid state. Various models have been developed for estimating > [38,39], but we use the model proposed by Tanaka and Hara [39]: > = (1 + ef\ − \/ g)> / /(1 + h) ,
(16)
where e is the volumetric thermal expansion coefficient, \/ represents the melting temperature, > / is the molar volume of the liquid at its melting point, and h indicates the ratio of the volume change due to fusion. According to Eq. (12) and Eq. (13), the interfacial energy of the embedded nanoparticle can be obtained as follows. ' Q W
=X
Y/Z
31 − V 4G
N
&
61 − ' - 31 +
!8 !9
:;:
(17)
It should be noted that the interface energy is usually considered to comprise a chemical part and a structural part. The chemical part is related to the atomic bond energy and the structural part originates from the elastic strain energy at the interface. This strain energy is caused by the lattice mismatch of two crystals [40]. Lattice mismatch leads to 9
rearrangement of the atoms at the interface to achieve a stable atomic configuration [5]. Different types of defects such as Shockley dislocations, perfect dislocations, and twins are correlated with the lattice misfit at the interface. Hao and Lau [41] indicated that the interface becomes rougher and the strain energy becomes larger as the lattice mismatch increases. However, Ouyang et al. [21] showed that the structural part of the interface energy is one or two orders smaller than that of the chemical part at several nanometer scale sizes. Therefore, we neglect the structural part of the interface energy as a first order approximation in the present study.
We applied Eq. (17) to Pb-Cu, Pb-Al, and Ag-Ni systems, and determined the effects of the particle size and shape, degree of coherency, and temperature on the values of the interface energy. Table 1 shows the thermodynamic and physical properties used in the calculations. The interaction parameters for the Pb-Cu, Pb-Al, and Ag-Ni systems are listed in Table 2. Table 1. Thermodynamic and physical parameters used in the calculations. Parameter Cohesive energy (kJ/mol) Melting point (K) Molar volume at the melting point 3 (m /mol) – Volumetric thermal expansion (K 1 ) Ratio of the volume change due to fusion (%) Atomic diameter (nm)
Symbol " \/ >/ e h ℎ
Element Ag 284 1235 11.6 × –6 10 0.98 × –4 10 3.51 0.32
Ref. Al 327 933.5 11.3 × –6 10 – 1.5 × 10
Cu 336 1358 7.94 × –6 10 – 1.0 × 10 3.96
Ni 428 1728 7.43 × –6 10 1.51 × –4 10 6.3
4
4
6.9 0.286
3.81
[44]
0.27
0.27
0.36
[7]
Table 2. Interaction parameters of Pb-Cu, Pb-Al and Ag-Ni systems. System Pb-Cu Pb-Al Ag-Ni
Ω [J/mol] 111040.719 + 53.356 T 100000 54620.4 + 3.1 T +2800 (1–2XNi)
Ref. [45] [46] [47]
10
Pb 196 600.7 19.42 × –6 10 – 1.24 × 10
[42] [42] [43] [43]
4
3. Results and discussion Figure 1 shows the interface energy calculated as a function of the particle size for different A/B systems, where A represents the nanoparticle and B is the matrix. Spherical nanoparticles with fully coherent interfaces and no relaxation or reconstruction were considered. In all cases, the temperature was set to the half the melting temperature of the nanoparticle in the bulk state. The results demonstrate that decreasing the particle size in the Pb/Cu, Pb/Al, and Ag/Ni systems led to increases in the interface energy. By contrast decreasing the particle size in the Cu/Pb, Al/Pb, and Ni/Ag systems decreased the interface energy. According to Table 1, the cohesive energy of the matrix was greater than that of the nanoparticle in the Pb/Cu, Pb/Al, and Ag/Ni systems, so Em/Ep was greater than unity in these systems. According to Eq. (17), for fully coherent interfaces, this condition means that ⁄
is greater than unity. Thus, the interface energy of the embedded nanoparticle was
greater than the bulk interface energy under these conditions. When the nanoparticle and matrix were replaced by each other, Em/Ep was smaller than unity and the energy of the nanoparticle interface was smaller than that in the bulk state. Clearly, the size dependency of the interface energy varied among the different systems. According to Eq. (9), G, B, =* , and = were similar in all cases, so the difference between the size dependencies was due to the differences in the atomic diameters of the systems. Increasing the particle size allowed the value of the interface energy to approach that of the bulk interface energy in all cases. It is well known that the melting temperature of a substance is directly related to its surface energy. Hence, the increases in the interface energy after decreasing the nanoparticle diameter in the Pb/Cu, Pb/Al, and Ag/Ni systems indicate that the melting points of the
11
nanoparticles in these systems must have increased after decreasing the particle size. These results are consistent with the previously reported experimental melting temperatures determined for the embedded nanoparticles [48–51].
Figure 1. Size-dependent interfacial energies calculated for A/B systems (A represents the nanoparticle and B indicates the matrix). Other parameters were set as: c = 1, i = 0.3, α = 1, and T = Tm,A/2.
Figure 2 shows the interface energies calculated for the Pb/Cu, Pb/Al, and Ag/Ni systems with different values of V (i.e., different interface conditions). The results are also compared with the interface energies calculated using Eq. (1), where the different values of proposed by Xiong et al. [23] (i.e.,
= 1.45ℎ) and Lu and Jiang [24] (i.e.,
= 4ℎ) were
considered. The circles in Figure 2 indicate experimental data converted from melting temperature data using the relationship:
j8.
j8,
=
#
#
. As the value of V increased, the
calculated interface energies increased for all diameters. For the Pb/Cu and Pb/Al systems, V = 2.5 obtained better agreement with the experimental results, whereas 1.2 was the most appropriate value of V for the Ag/Ni system. The lattice constant mismatches for the Ag/Ni, Pb/Al, and Pb/Cu systems were 0.15, 0.22, and 0.36 respectively, which indicate that increasing the lattice constant mismatch increased the value of V. This is reasonable 12
because mismatched lattice constants lead to increased stress at the interfaces and rearrangements of the atoms can reduce this stress. It should be noted that the interfaces were assumed to be fully coherent in these calculations, and even for the Pb/Cu system with a relatively large lattice mismatch. Previous experimental studies [52] have shown that this assumption can be true for small particle sizes. In addition to the surface rearrangement of the atoms, changing the interface coherency can reduce the interface stress. Thus, the degree of coherency can be reduced by increasing the particle size. The effects of coherency on the interface energy of an embedded nanoparticle are discussed in the following. Figure 2 also shows that when using the value of interface energy, whereas the value of
from Lu’s model, Eq. (1) overestimated the
value from Xiong’s model obtained better
agreement with the experimental values.
Figure 2. Interfacial energies calculated for the embedded nanoparticles compared with the experimental values [48–51] and the results calculated using Eq. (1). The other parameters were set as: c = 1 and α = 1. (a) Pb/Cu, (b) Pb/Al, and (c) Ag/Ni.
13
Figure 3 shows the interface energy in the Pb/Cu system as a function of the interface coherency for three different particle sizes, where no relaxation/reconstruction was assumed. The results show that for a specific particle size, the interface energy increased after increasing the coherency of the interface. It should be noted that the interface energy was considered to comprise a chemical part and a structural part. After decreasing the coherency of the interface, the structural part of the interface energy increased due to the increases in structural defects such as dislocations at the interface. By contrast, the chemical part of the interface energy increased after increasing the coherency of the interface due to the increase in the atomic bonds at the interface. However, the net effect was an increase in the interface energy as the interface coherency decreased. We completely neglected the structural part of the interface energy, so the interface energy comprised the chemical part of the interface energy in the present study. Thus, the increase in the interface energy (chemical part) as the coherency increased was predicted by the model as expected. We also found that there was a critical interface coherency, c*, at which the value of the nanoparticle interface energy was equal to the interface energy for the bulk system. When the coherency of the interface exceeded the critical value, the interface energy increased after decreasing the particle size, whereas when the coherency was smaller than c*, decreasing the particle size decreased the interface energy. Zhao et al. [53] showed that the superheating of embedded nanoparticles with coherent interfaces is dominated by the chemical parameters and that the structural pressure has a small effect, whereas the effects of the structural parameters are important for nanoparticles with incoherent interfaces. Zhao et al. [53] only considered fully coherent interfaces but it is reasonable to assume that their results are also valid for semi-coherent interfaces. Therefore, we could investigate the melting behavior of embedded nanoparticles with coherent and semi-coherent interfaces 14
based only on the chemical part of the interface energy. It is considered that if "/ /", > 1, then the melting temperature of embedded nanoparticles will become greater than that of the bulk [54]. Considering the direct relationship between the chemical part of the interface energy and the melting temperature of an embedded nanoparticle, our results indicate that for superheating, in addition to the ratio of the cohesive energies, the coherency of the interface was also greater than the critical value. By contrast, when the interface coherency was smaller than the critical value, the melting behavior of the embedded nanoparticles was similar to that of the free standing nanoparticles, i.e., the melting temperature decreased as the particle size decreased. According to Eq. (17), - ∗ = 2", /("/ + ", ), which shows that if "/ ≤ ", , then - ∗ ≥ 1, whereas - > 1 has no physical meaning, thereby demonstrating that under these conditions, the interface energy will always be smaller than the bulk value, such as those for the Cu/Pb, Al/Pb and Ni/Ag systems shown in Figure 1. By contrast, if "/ > ", then - ∗ < 1 and the interface energy can be greater or smaller than the bulk value according to the degree of interface coherency.
15
Figure 3. Effect of interface coherency on the interfacial energy of embedded Pb nanoparticles in Cu matrix. The other parameters were set as: i = 0.3 and α = 1.
The shape of a particle is another factor that can affect the interface energy [12,15]. By changing the shape of the nanoparticle from spherical to polyhedral, some interior atoms move to the surface and the number of surface atoms, )* , increases. Thus, polyhedral nanoparticles have more particle–matrix bonds, +,./ , than spherical nanoparticles. Therefore, according to Eqs (4 – 7) the contribution of the cohesive energy of the interface atoms, " , to the total cohesive energy, "!
, will increase. Hence, it is expected that if
+,./ > +, (i.e., "/ > ", ), then the interface energy of polyhedral nanoparticles will become greater than that of the spherical nanoparticles. Conversely, if +,./ < +, (i.e., "/ < ", ), then it is expected that the interface energy of polyhedral nanoparticles will become smaller than that of the spherical nanoparticles. As mentioned above, we considered the effect of the shape of nanoparticles by introducing the shape factor, G. Increasing the shape factor indicates an increase in the number of surface atoms. Figure 4 shows the interface energies calculated for the Pb/Al and Al/Pb systems with different particle shapes. Clearly, when "/ > ", (i.e., Pb/Al system), the interface energy increased as the shape factor increased, whereas when "/ < ", (i.e., Al/Pb system), increasing the shape factor decreased the interface energy. In both cases, the shape effects were more evident after decreasing the particle size because decreasing the particle size increased the )* /) ratio, especially for nanoparticles with diameters smaller than 10 nm [34]. It should be noted that from an atomistic viewpoint, the interface contains several types of atoms, including face, edge, and vertex atoms as well as temporary adatoms [55]. Each of these atom types form different numbers of particle–matrix bonds, which can affect the interface
16
energy. The ratio of edge and vertex atoms relative to the total interface atoms also depends on the size and shape of the nanoparticles [55]. However, the contributions of edge and vertex atoms to the surface energy can be ignored when the particle diameter is larger than 5 nm [56]. Thus, we ignored the effects of edge and vertex atoms, and all of the interface atoms were assumed to be face atoms. This simplification may have led to some errors when calculating the interface energies of the small nanoparticles.
Figure 4. Effects of nanoparticle shape on the interfacial energies of the embedded nanoparticles. The other parameters were set as: i = 0.3 and c = 1, and T = Tm,A/2.
17
Figure 5. (a) Interfacial energies calculated for the embedded nanoparticles as a function of temperature. (b) Variations in n/op/q as a function of temperature.
Figure 5(a) shows the interface energies calculated for the Pb/Cu, Pb/Al, and Ag/Ni systems as a function of temperature. Spherical nanoparticles with parameters comprising d = 5 nm, c = 1, and V = 0.3 were considered. Decreases in the interface energy as the temperature 18
increased were determined for the Pb/Al and Ag/Ni systems. However, in the Pb/Cu system, the interface energy increased as the temperature increased, which contrasted with the temperature dependence of the surface free energy, which decreased as the temperature increased. Few data are available regarding the dependence of the solid–solid interface energy on the temperature. In particular, Luo et al. [22] used a thermodynamic model to investigate the effects of temperature on the solid–solid interfacial energy for a Cu-Fe system and showed that the interface energy increased as the temperature increased. In addition, studies of the solid–liquid interface energy showed that the interface energy increased as the temperature increased. For example, Bai and Li [57] conducted MD simulations to study the interfacial energy for the solid–liquid interface of a pure metal, where they found that the interfacial free energy increased as the temperature increased, and the positive temperature coefficient of the interfacial energy was in qualitative agreement with the analysis by Spaepen [58] and the empirical estimate of Turnbull [59]. Sato et al. [60] also reported a positive temperature coefficient for the solid–liquid interface energy in a Cu-Zr alloy. However, other studies obtained different results, e.g., Granasy and Tegze [61] used a broken bond model to estimate the solid–liquid interface energy for some metals and alloys, as well as providing experimental data, including the temperaturedependent interface energy for a Pb-Cu alloy. The experiments and model showed that the interface energy decreased as the temperature increased. According to the model, we found that the temperature dependence of the interface energy was related to the Ω/b^/_ ratio, which was an increasing function of the temperature for the Pb/Cu system but a descending function of the temperature for the other cases (Figure 5(b)). Therefore, in contrast to the Ag/Ni and Pb/Al interfaces, we found that the interface energy increased in the Pb/Cu system as the temperature increased. In the present study, we calculated Ω 19
based on the CALPHAD approach, but some studies have shown that the CALPHAD assessment is not a reliable method for determining the temperature dependence of the interaction energies in solid solutions [63]. Thus, according to previous research, determining the dependence of the interfacial energy on temperature requires further theoretical and experimental studies. Conclusion In this study, we developed a model for determining the size-dependent interface energy of an embedded metallic nanoparticle in a metallic matrix in terms of the size-dependent cohesive energy. The model considers the effects on the interface energy of embedded nanoparticles of the particle size, shape, and temperature as well as the interface coherency and atomic relaxation/reconstruction. The results showed that depending on the system considered, the value of the interface energy could decrease or increase as the particle size decreased. The temperature dependence of the interface energy was also related to the system considered, where the interface energy was an increasing or decreasing function of the temperature. Furthermore, the shape of the nanoparticle affected the interfacial energy of the embedded nanoparticle. The shape effect was more evident when the nanoparticle size was smaller than 5 nm. We also identified a critical value for the interface coherency at which the size dependency of the interface energy becomes reversed, i.e., if the coherency is greater than the critical value, the interface energy will be greater than that for the bulk, whereas if the interface coherency is below the critical value, the interface energy will be smaller than that for the bulk. The proposed model can be used to study various interfacedependent properties of nanoparticles, such as the melting behavior of embedded and core–shell nanoparticles as well as multilayers at the nanometer scale.
20
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Highlights •
Model of size-dependent interfacial energy of embedded metallic nanoparticles.
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Effects of size, shape, temperature, and interface coherency on interface energy.
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Proposed model agrees better with experimental data than previous models.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
M.A. Jabbareh & M. Davari