J. Phys. Chem. C 2008, 112, 2359-2369
2359
Size Effect on the Thermodynamic Properties of Silver Nanoparticles Wenhua Luo,†,‡ Wangyu Hu,*,† and Shifang Xiao† Department of Applied Physics, Hunan UniVersity, Changsha 410082, People’s Republic of China, and Department of Physics and Electronic Information Science, Hunan Institute of Science and Technology, Yueyang 414000, China ReceiVed: August 31, 2007
The Gibbs free energy of silver nanoparticles has been obtained from the calculations of bulk free energy and surface free energy for both the solid and liquid phase. On the basis of the obtained Gibbs free energy of nanoparticles, thermodynamic properties of silver nanoparticles, such as melting temperature, molar heat of fusion, molar entropy of fusion, and temperature dependences of entropy and specific heat capacity have been investigated. Calculation results indicate that these thermodynamic properties can be divided into two parts: bulk quantity and surface quantity, and surface atoms are dominant for the size effect on the thermodynamic properties of nanoparticles. The method that the intersection of the free-energy curves for solid and liquid nanoparticles decide the melting point of nanoparticles demonstrates that the surface freeenergy difference between the solid and liquid phase is a decisive factor for the size-dependent melting of nanostructural materials.
1. Introduction
(
Tm(d) ) Tmb 1 -
Nanoparticle systems currently attract considerable interest from both academia and industry because of their interesting and diverse properties, which deviate from those of the bulk. Owing to the change of the properties, the fabrication of nanostructural materials and devices with unique properties in atomic scale has become an emerging interdisciplinary field involving solid-state physics, chemistry, biology, and materials science.1 Understanding and predicting the thermodynamics of nanoparticles is desired for fabricating the materials for practical applications.1 The most striking example of the deviation of the corresponding conventional bulk thermodynamic behavior is probably the depression of the melting point of small particles of metallic species. A relation between the radius of nanoparticles and melting temperature was first established by Pawlow,2 and the first experimental investigations of melting-temperature dependence on particle size was conducted more than 50 years ago.3 Further studies were performed by a great number of researchers.4-12 The results reveal that isolated nanoparticles and substrate-supported nanoparticles with relatively free surfaces usually exhibit a significant decrease in melting temperature as compared with the corresponding conventional bulk materials. The physical origin for this phenomenon is that the ratio of the number of surface-to-volume atoms is enormous, and the liquid/vapor interface energy is generally lower than the average solid/vapor interface energy.9 Therefore, as the particle size decreases, its surface-to-volume atom ratio increases and the melting temperature decreases as a consequence of the improved free energy at the particle surface. A lot of thermodynamic models of nanoparticles melting assume spherical particles with homogeneous surfaces and yield a linear or almost linear decreasing melting point with increasing the inverse of the cluster diameter2,6,10-12 * Corresponding author. E-mail:
[email protected]. † Hunan University. ‡ Hunan Institute of Science and Technology.
β D
)
(1)
where Tm and Tmb are the melting points of the cluster and bulk, respectively, D is the diameter of the cluster, and β is a material constant. Many phenomenological models have been proposed in order to determine constant β.2,6,10-12 However, the quantities involved in the expression for estimating β may not be easy to evaluate. Actually, the melting-phase transition is one of the most fundamental physical processes. The crystal and liquid phases of a substance can coexist in equilibrium at a certain temperature, at which the Gibbs free energies of these two phases become the same. The crystal phase has lower free energy at a temperature below the melting point and is the stable phase. As the temperature goes above the melting point, the free energy of the crystal phase becomes higher than that of the liquid phase and phase transition take places. The same holds true for nanoparticles. In the present paper, we have calculated the Gibbs free energies of solid and liquid phases for silver bulk material and its surface free energy using molecular dynamics with the modified analytic embedded-atom method (MAEAM). By representing the total Gibbs free energies of solid and liquid clusters as the sum of the central bulk and surface free energy,5,13,14 we can attain the free energies for the liquid and solid phase in spherical particles as a function of temperature. The melting temperature of nanoparticles is obtained from the intersection of these free-energy curves. This permits us to characterize the thermodynamic effect of the surface atoms on size-dependent melting of nanoparticles and go beyond the usual phenomenological modeling of the thermodynamics of melting processes in nanometer-sized systems. In addition to obtaining the results for size effect on the melting temperature of silver nanoparticles, we further calculate the molar heat of fusion, molar entropy of fusion, entropy, and specific heat capacity of silver nanoparticles. The size effect on the specific heat capacity has attracted a lot of attention recently,15,16 but many publications only concern the case at low temperature.16 Starting from the elastic continuum assumption, Wang et al.15 considered the different contributions of surface
10.1021/jp0770155 CCC: $40.75 © 2008 American Chemical Society Published on Web 01/29/2008
2360 J. Phys. Chem. C, Vol. 112, No. 7, 2008
Luo et al.
TABLE 1: Model Parameters for Aga F0
n
r1
R (×10 )
Fe
Pe
kc
1.6994
0.6760
2.8772
2.6179
12.7642
12.0435
0.75
-3
k-1
k0
k1
k2
k3
k4
43.0999
-20.2263
-15.1377
-1.3830
-1.2743
-5.2531
n, Fe, Pe, and kc are dimensionless; F0,R, and ki (i ) -1,0,1,2,3,4) in eV; r1 in Å. a
and interior atoms to the specific heat capacity of a nanoparticle and calculated the specific heat capacity of a nanoparticle with the Einstein and Debye models, respectively, to study the size and surface effects on the specific heat capacity of a nanoparticle. Different from the previous molds, we obtain the temperature and particle size dependence of the specific heat capacity based on the expression for the Gibbs free energy and find that the specific heat capacity of a nanoparticle exceeds the same values of the corresponding bulk materials. 2. Computational Scheme 2.1. Modified Analytic Embedded-Atom Method. Metallic bonding has long been understood to arise from the interaction of individual atoms with the electron gas, or Fermi sea, of the metal. Because the local electron density depends on the local arrangement of atoms, this bonding depends strongly on the local atomic density in the vicinity of an atom. The embeddedatom method17,18 is an attempt to model this dependence of the atomic interactions on the local environment and has met with considerable success in describing properties such as surface reconstruction, point-defect properties in pure metals, and surface segregation.18,19 Considering the hypothesis of linear superposition of atomic electronic density and inability to treat elements with negative Cauchy pressure, our group introduced a modified term in the atomic potential expression and reconstructed the pair potential function. These MAEAM potentials have already been applied successfully for the modeling of Ni clusters melting and microstructural evolution for Ag nanocrystalline and its amorphous phase.20-22 The basic equations are
Etot ) Ei ) F(Fi) +
1 2
∑ Ei
(2)
φ(rij) + M(Pi) ∑ j*i
(3)
where Etot and Ei are the total energies of the model system and a single atom, respectively, rij is the separation distance between atoms i and j, φ(rij) is the pair potential as a function of the distance rij, F(Fi) is the embedding energy to embed atom i in an electron density Fi, and M(Pi) is the modified term, which describes the energy change caused by the nonspherical distribution of atoms and deviation from the linear superposition of atomic density. The functions of φ(rij), F(Fi), and M(Pi) take the following forms
[
F(Fi) ) -F0 1 - n ln
( )]( ) Fi Fe
Fi Fe
n
(4)
4
φ(rij) ) k-1e((r1)/(rij)-1) + M(Pi) ) -
klel(1-(r )/(r )) ∑ l)0
RPe Pi (Pe + Pi )2
ij
1
(5) (6)
where r1 is the first-neighbor distance in the equilibrium state, Fi is the electron density induced at site i by all others in the system, and Pi is the second-order item of electron density. Fe and Pe correspond to their equilibrium values, respectively. The pair potential function is truncated at a specific cutoff distance rc) r3 + kc(r4 - r3); ri is ith nearest-neighbor distance for a perfect fcc crystal, and kc is an adjustable parameter.
Fi )
f(rij) ∑ j*i
(7)
Pi )
f 2(rij) ∑ j*i
(8)
where f(rij) is the electron density distribution function of an atom and is taken as
f(rij) ) fe
()( r1 rij
4.7
)
rce - rij rce - r1
2
(9)
where fe is taken to be 1.0. f(r) is truncated at rce, rce ) r4 + kc(r5 - r4). The model parameters are determined by fitting the physical properties of silver, such as cohesive energies, vacancy formation energy, and elastic constants. The calculated model parameters for silver are listed in Table 1. 2.2. Bulk Free-Energy Calculations. In recent times, several articles appeared showing procedures to evaluate the free energy numerically using empirical descriptions for the total energy, either in molecular-dynamics (MD) or Monte Carlo frames (MC).23-26 In this paper, the procedures used follow closely those described in refs 25 and 26, and all of the free-energy calculations were carried out at zero pressure. The Gibbs free energy per atom at a temperature T and zero pressure, g(T, P ) 0), is related to the internal energy per atom, u(T, P ) 0), and the specific entropy, s(T, P ) 0), by the thermodynamic relation:27
g(T, P ) 0) ) u(T, P ) 0) - Ts(T, P ) 0)
(10)
The entropy can be eliminated from eq 10 by expressing it in terms of the Gibbs free energy using the standard thermodynamic relation:27
s(T, P ) 0) ) -
|
∂g(T, P ) 0) ∂T
(11)
P)0
Substituting eq 11 into eq 10 yields a differential equation for the Gibbs free energy in terms of the internal energy. Upon solving this equation, one obtains an expression, valid for any system, for the Gibbs free energy at different temperatures
g(T, P ) 0) ) T
[
g(T0, P ) 0) T0
∫TT 0
u(τ, P ) 0) τ2
]
dτ
(12)
where T0 is a predetermined reference temperature. The internal energy is first computed as a function of temperature T using constant-pressure MD simulations, and it is fitted with a secondorder polynomial in T, which allows an analytic integration in eq 12. Equation 12 is the fundamental equation we will use in determining the free energy as a function of temperature. In order to use eq 12, the Gibbs free energy at the reference temperature must be obtained separately by thermodynamic integration from a suitable ideal reference state. To do this, we consider a system with Hamiltonian H ) (1 - λ)W + λU, where
Thermodynamic Properties of Silver Nanoparticles
J. Phys. Chem. C, Vol. 112, No. 7, 2008 2361
{
WL ) 0.1[φ(r) + |φmin |] dφ/dr e 0 dφ/dr > 0 0
(16)
where φmin is the minimum value of the pair potential part of the MAEAM energy. The repulsive potential cannot be very strong because the sample may crystallize, an irreversible process. As in the solid phase, the integration is carried over the coupling parameter λ, which varies between 0 and 1. The system is kept at the constant volume V0, which equilibrates the U Hamiltonian at temperature T0 and P ) 0. Therefore, the free-energy change due to the switch is given by ∆g1.
Figure 1. Pair potential and repulsive potential as functions of r/r1. The repulsive potential is indicated by the black solid curve.
U describes the actual system and W is the Hamiltonian of the reference system, with known free energy. The Gibbs free energy per atom relative to that of the reference system can then be obtained by thermodynamic integration
g(T0, P ) 0) ) gw(T0) + ∆g1 ∆g1 )
1 N
1 1 dλ ) ∫0 〈U - W〉λ,P)0 dλ ∫01 〈∂H ∂λ 〉λ,P)0 N
(13)
where gw(T0) is the free energy of the reference system at T0. The integration is carried over the coupling parameter λ, which varies between 0 and 1, and stands for the average over a constant NVT simulation. For the solid the reference system, W is a set of Einstein oscillators centered on the average positions of the atoms in the ensemble corresponding to the Hamiltonian U
1 Ws ) mω2 2
∑i (br i - br i 0)2
The second step is a reversible expansion of the repulsive gas, from V0 and high pressure, to reach the low-density limit (where it becomes identical to the ideal gas), followed by a reversible compression of the ideal gas, to recover the initial density or volume. The change in free energy due to both processes is
∆g2 ) kbT0
∫0F
0
[
]
P dF -1 FkbT0 F
(17)
where F0 ) N/V0 is the atom density. After the processes represented by eq 17 have taken place, we end up with an ideal gas at (T0, F0), whose free energy is calculated as follows
glw(T0) ) kbT0[ln(F0Λ3) - 1]
(18)
where Λ is the de Broglie thermal wavelength (Λ2 ) h2/ 2πmkbT0),28 Then the free energy of the liquid phase is calculated as the sum of these contributions:
gl(T0) ) ∆g1 + ∆g2 + glw(T0)
(19)
(14)
Equations 12-19 give the free energies of the solid and liquid phase as a function of temperature.
where {ri0} is the ideal crystal lattice vector, m is the mass of the atom, and ω is the frequency of oscillations, which to minimize the difference between the reference and actual system should be chosen to give similar radial distribution functions and mean-squared displacement for the atoms at the temperature of interest. The free energy of the Einstein oscillators can be calculated analytically:28
2.3. Surface Free-Energy Calculations. The surface free energy constitutes a large part of the total free energy of nanoparticles. The reversible work per unit area involved in forming a new surface of a substance (for instance by cleavage) is defined as the specific surface free energy γ (or simply called the surface free energy), and the reversible work per unit area required to elastically stretch a surface is the surface tension σ.29 For liquids, the configuration of a surface produced either by cleavage or by stretching is the same because the mobility of liquid molecules is high and the surface takes the equilibrium configuration of minimum energy. Thus, the work required for cleavage or surface stretching is the same, and the surface free energy equals the surface tension. For this reason, surface free energy is also called surface tension, and which is easily calculated directly from the interatomic interactions, that is, using the mechanical expressions for the surface tension.30,31 Thus, γ of the liquid is computed from
gsw(T0) ) -3kbT0 ln
() T0 TE
(15)
Here kb is the Boltzmann constant, TE is the Einstein temperature of the oscillators, and TE ) hω/2πkb, where h is the Planck constant. The reference system for the liquid phase is an ideal gas with the same temperature and density as the actual system. The process to switch from U to W involves an intermediate step to avoid atom overlap during integration. First, we compute the free-energy difference between the actual system with potential U (the MAEAM potential) and a system with a repulsive potential WL. In this work, we combine methods in refs 25 and 26. WL (illustrated in Figure 1) is determined by the following formula
1 γ l ) (σxx + σyy - 2σzz) 2
(20)
Here the σRβ is the component of the surface stress tensor, expressed in terms of MAEAM potential functions as follows32
2362 J. Phys. Chem. C, Vol. 112, No. 7, 2008
σRβ ) 1
∑ 2 j*i
1 As
∑i
{
Luo et al. AA(BB)
-miυRi υβi
+
[ ( ) ( ) ] } ∂φ
∂rij
+
∂F
+
∂Fi
∂F ∂f(rij)
∂Fj
+
∂rij
∂M
+
∂Pi
R β 2 ∂M ∂f (rij) rij rij
∂Pj
where (R,β) ≡ (x,y,z), As is the area of the surface, mi is the atomic mass, and υRi is the velocity component in the R direction of atom i. However, for solids, the atoms are relatively immobile and the work needed in cleavage and surface deformation (like stretching) is not the same; hence, usually surface tension does not equal the surface free energy in value,29 and they are related by the expression
σ ) γ + As
∂γ ∂As
(22)
Taking into account the difference between solids and liquids, the computational surface free-energy approach for solids is necessarily different from that normally used for liquids. Here we adopt two kind of methods to calculate solid surface free energies. The first strategy33 used for the calculation of solid surface free energies involves the evaluation of the bulk free energy and of the free energy of a slab system with two surfaces using eqs 12, 13, and 15. The surface free energy is then determined as
γarea ) γatom )
gslab - Ngbulk 2As
(23a)
gslab - Ngbulk 2Nsurf
(23b)
All
Fi (Fi) + 1
1 2
BB
AA
φ(rij) + ∑ i*j 1
AB
All
λφ(rij) + ∑ M(Pi) ∑ φ(rij) + 2 ∑ 2 i*j i*j i
(24)
(25)
AB
∑ i*j
f 2(rij) +
φAB rep )
λ f 2(rij) ∑ i*j
(26)
{
AB
(1 - λ)[φ(rij) + |φmin |] ∑ i,j
(27)
dφ/dr e 0 dφ/dr > 0
0
Differing from Grochola et al. ’s “blanket lambda” path, there is not any additional parameter to be determined in our repulsive potential. According to Grochola et al., the blanket lambda path is a two-part process. The first stage involves turning off the main A-B interactions as well as simultaneously turning on the repulsive potential. From standard λ-integration theory, the freeenergy change for this process can be shown as
∆F1 )
∫1
0
[〈 〉 〈 〉] ∂φAB ∂E(λ) rep + ∂λ ∂λ
dλ
(28)
where
∂E(λ)
All
)
∑i
∂F(Fi) ∂Fi
+
1 2
∂Fi ∂λ
∂Fi
AB
∑ i*j
∂Pi ∂λ
All
φ(rij) +
∑i
∂M(Pi) ∂Pi (29) ∂Pi
∂λ
AB
f(rij) ∑ i*j
)
∂λ
where represents surface free energy per area (per atom), gslab is the total free energy of the slab, N is the total number of atoms in the slab, Nsurf is the number of atoms on each surface, gbulk is the free energy per bulk atom, and the 1/2 accounts for the two free surfaces of the simulation cell; periodic boundary conditions are applied in the planar directions. The second strategy used for the calculation of solid surface free energies is application of the “simple lambda” and the “blanket lambda” path34,35 combined with our repulsive potential. The “simple lambda” path involves finding the work done in reversibly transforming a bulk into a slab. Briefly, a dividing plane perpendicular to the z axis placed in the center of the bulk cell is used to label atoms on one side A and on the other as B, and the interactions between A and B atoms at the center of the cell are “turned off” gradually. Note that the A-B interactions across the periodic boundary conditions are not affected or changed in any way so that instead of producing four surfaces we produce two, that is, one slab. The process of “turning off” A-B interactions is performed by changing λ from 1 to 0:
λ f(rij) ∑ i*j
Because at λ ) 0 the interactions between A and B atoms are “turned off” and atoms A and B may overlap making ∂E/ ∂λf ∞, we introduce our repulsive potential instead of the “blanket potential” and “corrugation potential”35 to keep atoms A and B separating. The repulsive potential is defined as follows:
∂λ
γarea(atom)
∑i
AB
f (rij) +
AA(BB)
Pi )
rij
∂rij
(21)
E(λ) )
∑ i*j
Fi )
(30)
AB
)
f 2(rij) ∑ i*j
(31)
In the second part, only repulsive potential exists between atoms A and B. To form two noninteracting surfaces A and B, we relax the z-axis periodic boundary conditions in a new λ-like integration expanding the A-B interface. The free-energy change for this step can be shown to be35
∆F2 )
∫L∞ 0 z
〈 〉 ∂φAB rep ∂Lz
dLz
(32)
Lz
where Lz, the z-axis cell length, is an integrating variable analogous to λ. L0z is the initial bulk cell length, and Lz is the z component force between surfaces A and B. Finally, the work required for the total process is the free energy necessary to create the slab. 2.4. Free Energy and Thermodynamic Properties of Nanoparticles. In order to explore the size effect on the thermodynamic properties of silver nanoparticles, we first write the total Gibbs free energy Gtotal of a nanoparticle as the sum of the volume free energy Gbulk and the surface free energy Gsurface
Thermodynamic Properties of Silver Nanoparticles
J. Phys. Chem. C, Vol. 112, No. 7, 2008 2363
Gtotal ) Gbulk + Gsurface ) Ng(T) + γ(T)As
(33)
Assuming a spherical particle leads to a specific surface area of5,10,36
As )
6NVat(T) D
(34)
where N is the total number of atoms in the particle, D is the radius of the particle, and Vat(T) is the volume per atom. Secondorder polynomials are adjusted to the simulation results of the internal energy for the solid and liquid phase shown in Figure 2. The Gibbs free energies per atom for the solid and liquid phase are written as
g(T) ) g(T0)
[
]
()
a 0 a0 T T - T a2(T - T0) + a1 ln - + (35) T0 T0 T T0
where ai are the polynomial coefficients, resulting from molecular dynamics (MD) simulations together with g(T0), which are summarized in Table 2. The surface free energy of a solid spherical particle may be determined by the average surface free energy of the crystallite facets and the Gibbs-Wulff relation37
∑ Aiγi ) minimum
n
γs )
µ
Aiγi ∑ γi ∑ i)1 γi i)1 )
n
∑ i)1
n
Ai
µ
∑ i)1 γ
i
γi(T0)
[
s (T,D) ) s
() ∂gs ∂T
)
2as2T
n
) n
(37)
as1
()
ln
+
()
al0
T
-
gs(T0)
T0
T0
-as2T0 +
2Vat(T)(γs)2
-
T0
D
3
∑ i)1
γ′i γ2i
(39)
where the primes denote derivatives with regard to temperature. The contribution from the derivative of atomic volume is trivial; it is reasonable to neglect. Using the relation between the specific heat capacity at constant pressure and the entropy, we can write the expression for the specific heat capacity per mole as
Csp(T, D) ) N0T
D
() { ∂ss ∂T
γs
1
∑ i)1 γ
+
as1
p
6N0TVat(T) 2
) 2N0as2T + N0as1 -
p
[(γ )′] + s
2
(γs)2 3
3
∑ i)1
[
γ′i′
-
]}
2(γ′i)2
γ2i
γ3i
(40)
where N0 is Avogadro’s number. The internal energy per atom for nanoparticles can be wrirren as5,10
i
where n is the number of facets under consideration. Each crystal has its own surface energy, and a crystal can be bound by an infinite number of surface types. Thus, we only consider three low index surfaces, (111), (100), and (110), because of their low surface energies, and the surface free energy γi of the facet i is calculated by eq 23 to be
γi(T) )
constant pressure conditions, and temperature and particle size dependence of the entropy per atom for solid nanoparticles can then be defined using the following expression
(36)
The equilibrium crystal form develops so that the crystal is bound by low surface energy faces in order to minimize the total surface free energy.38 For two surfaces i and j at equilibrium, Aiγi ) Ajγj ) µ, where µ is the excess chemical potential of surface atoms relative to interior atoms. A surface with higher surface free energy (γi) consequently has a smaller surface area (Ai), which is inversely proportional to the surface free energy. Accordingly, the average surface free energy of the crystal, weighted by the surface area, is n
Figure 2. Internal energy as a function of temperature for bulk material. Heating and cooling runs are indicated by the arrows and symbols.
]
b0i b0i T T - T b2i(T - T0) + b1i ln (38) + T0 T0 T T0
where bki (k ) 0,1,2) are the coefficients for the surface freeenergy calculation for facet i, and γi(T0) is surface free energy at the reference temperature T0, which are shown in Table 4. On the basis of the expression for the Gibbs free energy, general trends for thermodynamic properties may be deduced. For example, the melting temperature Tm for nanoparticles of diameter D can be obtained by equating the Gibbs free energy of solid and liquid spherical particles with the assumption of
hV,D(T, D) ) hV(T) +
6Vat(T) γ(T) D
(41)
where hV represents the internal energy per atom of bulk material. The molar heat of fusion and molar entropy of fusion for nanoparticles can be derived from the internal energy difference of solid and liquid nanoparticles easily. l s (Tm, D) - hV,D (Tm, D)] ) ∆Hm ) N0[hV,D
[
∆Hmb 1 -
(
)]
6Vsat(Tm) s Vlat(Tm) γ (Tm) - γl(Tm) s LD V (T )
[
at
∆Hmb 1 -
∆Sm )
∆Hm Tm
m
)
]
β(Tm) (42) D (43)
2364 J. Phys. Chem. C, Vol. 112, No. 7, 2008
Luo et al.
TABLE 2: Free-Energy Parameters at the Reference Temperature T0a solid
gs(T0)
∆g1
gsw
a2(×10-8)
a1(×10-4)
a0
T0 ) 300 K
-3.0029
-2.9318
-0.0711
4. 7345
2.21
-2.9395
liquid
gl(T0)
∆g1
∆g2
glw
a2(×10-8)
a1(×10-4)
a0
T0 ) 2150 K
-4.5987
-2.2718
0.2732
-2.6001
-6.4377
3. 7148
-2.9286
Energy changes of the Hamiltonian switch (∆g1) and reversible expansive (∆g2). Einstein crystal (gsw) and gas (glw) free energies. Coefficients ai of the free-energy calculations. (All entries are in units of eV/atom.) a
TABLE 3: Atomic Layer Numbers with the Corresponding Number of Atoms surface
x-axis layers
y-axis layers
z-axis layers
total atoms
(111) (100) (110)
24 16 16
36 16 24
14 16 22
2016 2048 2112
TABLE 4: Surface Free-Energy Parameters for the (111), (100), and (110) Facesa surface
γs(T0)
b2(×10-7)
b1(×10-4)
b0
(111) (100) (110)
0.8453 0.9129 1.0024
-1.8019 0.0874 3.0730
2.6493 0.3757 -2.5491
0.7951 0.9259 1.0817
a Coefficients bi of the surface free-energy calculation. (All entries are in units of J/m2.)
where ∆Hmb is the molar heat of fusion for bulk, and L is the latent heat of melting per atom. The superscript “s” and “l” represent solid phase and liquid phase, respectively. 3. Simulation Details In order to obtain the free-energy curves needed in eq 12 as a function of temperature, we implemented molecular dynamics (MD) simulations in a constant temperature and pressure ensemble (NPT) with temperature controlled by a Nose-Hoover thermostat39 and pressure kept at 0 Pa with a Parinello-Rahman scheme.40 The atomic motion equations are integrated numerically with a fourth-order Gear predictor-corrector algorithm and a time step of 2 fs (1 fs ) 10-15 s) for a system of 2048 atoms with periodic boundary conditions. The samples are heated in successive runs between 300 and 2200 K, and cooled down to 300 K, with a temperature interval of 50 K. In every run, the first 2 × 104 time steps are used to equilibrate the sample, the statistical average of the thermodynamic variable is obtained on an additional set of 2 × 104 time steps, and the internal energy per particle is obtained. The reference temperature for the calculation of the solid free energy is selected to be T0 ) 300 K, and 21 points on the interval of λ ) 0-1 have been used for the calculation of the λ integration of eq 13. Again, the calculations at a given λ are carried out in two successive runs each of 2 × 104 steps and averages are taken on the second of time steps. We fit these points with a sixth-order polynomial. Simulation for step 1 of the liquid free energy at a temperature of T0 ) 2150 K is performed in the same manner as for the crystal. The average volume obtained from the constant NPT simulation at 2150 K is used to generate a cubic sample. The cubic sample is taken as the starting sample for the calculation of the λ integration of eq 13. The switching parameter λ varies between 0 and 1, in intervals of 0.05. A sixth-order polynomial is fitted to the points, and the integral of eq 13 is solved analytically. We then expand the sample keeping the temperature constant and equal to T0. This is done in constant NPT
simulations reducing the pressure from 41 to 2 kbar, with a pressure interval of 2 kbar. Two more values are added to this data: the origin (P ) 0, F ) 0), corresponding to the ideal gas limit, which is obtained by an analytical calculation of the second virial coefficient,41 and the average pressure and density obtained for the case λ ) 0. The simulation for both steps of the liquid free-energy integrations are equilibrated for 3 × 104 steps and sampled for 3 × 104 steps at each value of the integrand. Again, instead of performing the numerical integration, we have fitted the integrand to a sixth-order polynomial and free-energy change ∆g2 is obtained by analytical integration of this polynomial. All of the calculations of the surface free energy are performed in the NVT ensemble. We have employed finite slabs with periodic boundary conditions in the lateral cells. When calculating liquid surface tensions, we first utilize the correct zero pressure cell sizes between 1250 and 2200 K, with a temperature interval of 50 K from NPT simulations. For each of these temperatures, the Ag(100) system, which has 5x, 5y, and 6z axis layers with a total of 300 atoms, is melted and equilibrated for 105 steps. Statistics for eqs 20 and 21 are taken over 105 steps. The solid slabs contain 1872, 2176, and 2208 atoms for the Ag(111), Ag(100), and Ag(110) systems and consist of 13, 17, and 23 atomic layers with zero pressure lattice constants from NPT simulations. An odd number of atomic layers is taken for the sake of symmetry of the upper and lower surface. The three systems are chosen to have approximately equal spatial dimensions, and the z axis is normal to the surface. In order to facilitate an accurate comparison, we performed the second scheme for the calculation of solid surface free energies at 300 and 750 K adopting Grochola et al.’s “blanket lambda” path.35 Table 3 summarizes the number of x, y, and z cell layers and the total number of atoms used. Initially, λ-integration runs are started from perfect fcc crystal configurations and equilibrated for 2 × 104 steps. Zero pressure lattice constants from NPT simulations are used. When changing λ, we use the atomic configurations from the previous run, which are then equilibrated to the new λ value. All thermodynamic integration is carried out by fitting polynomials up to sixth order to surface free energy. 4. Results and Discussion Figure 2 shows the behavior of the solid and liquid internal enthalpies as a function of temperature, and an abrupt jump in the internal energy during heating can be observed, but this step does not reflect the thermodynamic melting because periodic boundary condition calculations provide no heterogeneous nucleation site, such as free surface or the solid-liquid interface, for bulk material leading to an abrupt homogeneous melting transition at about 1500 K (experimental melting point 1234 K), as it is revealed that the confined lattice without free surfaces can be significantly superheated.42 The latent heat of fusion is 0.115 eV/atom, in good agreement with the experimental value of 0.124 eV/atom.43
Thermodynamic Properties of Silver Nanoparticles
Figure 3. Simulation results for the integrand λ appearing in the switching Hamiltonian method, eq 13, vs the switching parameter λ, corresponding to the solid (left axis) and liquid (right axis) phase.
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Figure 5. Gibbs free energy of the solid and liquid phase in units of eV/atom. The asterisks denote the experimental values from ref 43. The solid curve is the MAEAM solid free energy, and the dashed curve is the MAEAM liquid free energy. The temperature at which Gibbs free energy of the solid and liquid phase is identical is identified as the melting point.
Figure 4. Evaluation of the integrand of eq 17. The black dot indicates the initial density and volume corresponding to the pressure arrived at the end of the preceding switching process.
As for the calculation of the switch ∆g1 for the solid phase, it is necessary to take into account how to determine the Einstein temperature TE. If the path connecting the initial and final states is reversible, then the free-energy difference between the two states will only be a function of both states because the free energy of a system is a state function. In theory, all λ-integration paths between the two states in question are reversible, as long as the computation samples the total phase space for each λ value adequately. In practice, the process of choosing an appropriate path becomes a trial-and-error approach because of the difficulty some paths have in covering the entire phase space adequately in a finite computer simulation. Sturgeon and Laid propose the optimal value for the Einstein temperature is that giving a similar mean-squared displacement between the reference and actual system to reduce numerical errors and obtain a good free-energy difference estimate. According to the suggestion, the reference free energy of the solid Ag is calculated at T0 ) 300 K using the Einstein temperature TE ) 120 K. The integrand for the switching Hamiltonians, eq 13, is shown in Figure 3 for Ag in the solid and liquid phase. The shape of these curves satisfies the Gibbs-Bogoliubov inequality 〈∂2H(λ)/ ∂λ2〉 < 0,28 which is used to test the accuracy of the integrand. These curves are fitted with a sixth-order polynomial and integrated analytically. Figure 4 shows the second step of the liquid free-energy calculation, which includes a series of constant NPT simulations at decreasing pressures starting with the repulsive potential system from the conclusion of the first
Figure 6. Solid surface free energies vs temperature for the (111), (100), and (110) faces obtained using the thermodynamic integration technique and the lambda integration method. Also shown are L. Vitos et al.’s FCD results at 0 K.
step. The black solid circle indicates the density and volume corresponding to the pressure arrived at the end. The free-energy functions for the solid and liquid phases have been plotted in Figure 5. The melting temperature Tmb is obtained from the intersection of these curves. From Figure 5, two curves cross at Tmb ) 1243 K, which is in good agreement with the experimental melting point T exp ) 1234 K. The good agreement in melting point is consistent with accurate prediction of the Gibbs free energies. The calculation results of solid surface free energy for the (111), (100), and (110) surfaces with thermodynamic integration approach (TI)33 represented by eqs 23 and 35 is depicted in Figure 6. It can be seen that the free energies of the surfaces at low temperatures are ordered precisely as expected from packing of the atoms in the layers. The close-packed (111) surface has the lowest free energy, and loosely packed (110) the largest. As temperature increases, the anisotropy of the surface free energy becomes lower and lower because the crystal slowly disorders. For comparison, we also utilize Grochola et al’s “simple lambda” and “blanket lambda” path (BLP)34,35 to calculate the solid surface free energy for the three low-index faces. The results are in good agreement with the TI calculation for temperature from 300 to 750 K. As an example, the simulation results for the integrand λ + λ
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Luo et al.
Figure 7. Simulation results for the integrand λ + λ appearing in eq 28 for the (110) face at 750 K. Figure 9. Surface free energy of the solid and liquid phase in units of J/m2 as a function of temperature. The data for liquid surface free energy is fitted to a linear function of temperature T.
Figure 8. z density plots for the expansion part of Grochola et al.’s “blanket lambda” path at (Lz - L0z )/L0z ) 0, 0.045, and 0.08 applied to the (110) face at a temperature of 750 K.
appearing in eq 28 for the (110) face at 750 K is shown in Figure 7. It is obvious that the results are very smooth and completely reversible. In order to create the slab, we also show the expansion process using z-density plots for (Lz - L0z )/L0z ) 0, 0.045, and 0.08, in Figure 8. At (Lz - L0z )/L0z ) 0.08, the adatoms appearing between A and B sides can be seen. According to Grochola et al.,35 it indicates that the BLP samples the rare events more efficiently than the cleaving lambda method44 because the two surfaces interact via the adatoms when separated, as seen in Figure 8. These adatoms would tend to have greater fluctuations in the z direction interacting with each other than if they were interacting with a static cleaving potential. They should therefore be more likely to move onto other adatoms sites or displace atoms underneath them, which should result in better statistics. The work obtained from the system in this expansion is roughly 5% of the work put into the system in the first part. For comparison, ab initio calculation results at T ) 0 K performed by L.Vitos et al. adopting the FCD method45 is shown in Figure 6. According to eq 37, the average solid surface free energy can be obtained, and the calculation results are shown in Figure 9. Also shown are the liquid surface free energies and their linear fitting values, γL(Τ) ) 0.5773 - 2.3051 × 10-4(T - 1243). At melting point, we acquire the solid surface free energy and the liquid surface free-energy values of 0.793 J/m2 and 0.577 J/m2, respectively. The semitheoretical estimates of Tyson and
Miller46 for the solid surface energy at Tmb are 1.086 (J/m2), and the experimental value47 for the surface energy of the solid and the liquid states at Tmb are 1.205 and 0.903 (J/m2), respectively. It should be emphasized that surface free energies of crystalline metals are notoriously difficult to measure and the spread in experimental values for well-defined low-index orientations is substantial, as Bonzel et al.48 pointed out. It is obvious that because MAEAM is developed using only bulk experimental data, it underestimates surface free energy in both the solid and the liquid states as many EAM models do.49,50 Though there is the difference between the present results and experimental estimates, we note that the surface free-energy difference between the solid and liquid phase is 0.216 (J/m2) and is between Tyson and Miller’s result of 0.183 (J/m2) and the experimental value of 0.3 (J/m2). Furthermore, the average temperature coefficient of the solid and liquid phase surface free energy is 1.32 × 10-4 (J/m2K) and 2.3 × 10-4 (J/m2K), respectively. Such values compare reasonably well with Tyson and Miller’s estimate of 1.3 × 10-4 (J/m2K)46 for the solid and the experimental results of 1.6 × 10-4 (J/m2K)51 for the liquid. Therefore, we expect the model to be able to predict the melting points of nanoparticles by means of determining the intersection of free-energy curves. Because the liquid surface free energy is lower than the solid surface free energy, the solid and liquid free-energy curves of nanoparticles change differently when the size of the nanoparticle decreases so that the melting points of nanoparticles decrease with decreasing particle size, as is depicted by Figure 10. This indicates actually that the surface free-energy difference between the solid and liquid phase is a decisive factor for the size-dependent melting of nanostructural materials. In order to test our model, we plotted the results for the melting temperature versus inverse of the particle diameter in Figure 11. Because there is no experimental data available for the melting of Ag nanoparticles, the predictions of Nanda et al.10 and Yang et al.’s52 theoretical model are shown in Figure 11 for comparison. It can be seen that agreement between our model and Nanda et al.’s10 theoretical predictions for Ag nanoparticles is excellent. The nonlinear character of the calculated melting curve results from the temperature dependence of the surface free-energy difference between the solid and liquid phase, which is neglected in Nanda et al.’s10 model. Alternatively, Yang et al.’s52 theoretical predictions may overestimate the melting point depression of Ag nanoparticles.
Thermodynamic Properties of Silver Nanoparticles
Figure 10. Gibbs free energies of the solid and liquid phase in units of eV/atom for the bulk material and 5 nm nanoparticle.
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Figure 12. (a) Molar latent heats of fusion ∆Hm and (b) molar entropy of fusion ∆Sm of Ag nanoparticals as a function of particle diameter D.
Figure 11. Melting point vs the reciprocal of nanopartical diameter. The solid line is the fitting result. The dashed line is the result calculated from the thermodynamic model Tm ) Tmb(1 - β/d),10 (β ) 0.96564).
It is believed that understanding and predicting the melting temperature of nanocrystals is important. This is not only because their thermal stability against melting is increasingly becoming one of the major concerns in the upcoming technologies1,52,53 but also because many physical and chemical properties of nanocrystals follow the exact same dependence on the particle sizes as the melting temperature of nanocrystals does. For example, the size-dependent volume thermal expansion coefficient, the Debye temperature, the diffusion activation energy, the vacancy formation energy, and the critical ferromagnetic, ferroelectric, and superconductive transition temperature of nanocrystals can be modeled in a fashion similar to the size-dependent melting temperature.52,54,55 However, Lai et al.56 pointed out that in order to understand the thermodynamics of nanosized systems comprehensively an accurate experimental investigation of “the details of heat exchange during the melting process, in particular the latent heat of fusion” is required. Allen and co-workers developed a suitable experimental technique to study the calorimetry of the melting process in nanoparticles and found that both the melting temperature and the latent heat of fusion depend on the particle size.56-58 In the present work, we calculate the molar heat of fusion and molar entropy of fusion for Ag nanoparticles, and the results are shown in Figure 12. It can be seen that both the molar heat and entropy of fusion undergo a nonlinear decrease as the particle diameter D
Figure 13. Molar heat capacity as a function of temperature for Ag nanoparticles and bulk sample.
decreases. Actually, eq 42 discloses that the molar heat of fusion of nanoparticles is inversely proportional to the reciprocal of particle size. Taking Tm ) 1243 K, the calculated β(Tm) equals 1.09 nm and is close to 0.9656 nm for the temperature depression constant. In analogy with the melting point, Figure 12 shows that the system of smallest size possesses the lowest latent heat of fusion and entropy of fusion. In a particle with a diameter of 2.5 nm or smaller, all of the atoms should indeed suffer surface effects, and the latent heat of fusion and the entropy of fusion are correspondingly expected to vanish. It is also observed that the size effect on the thermodynamic properties of Ag nanoparticles is not really significant until the particle is less than about 20 nm. Figure 13 plots the molar heat capacities as a function of temperature for bulk material and nanoparticles. One can see that the molar heat capacity of nanoparticles increases with increasing temperature, as the bulk sample does. The temperature dependence of molar heat capacity qualitatively coincides with that observed experimentally. Figures 13 and 14 show that the molar heat capacity of bulk sample is lower compared to the molar heat capacity of the nanoparticles, and this difference increases with the decrease of particle size. The discrepancy in
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Luo et al. are in good agreement with experimental values.43 The molar entropy of nanoparticles is higher than that of the bulk sample, and this difference increases with the decrease of the particle size and increasing temperature. According to eq 39, the reduced molar entropy S/Sb (Sb denotes bulk entropy) also varies inversely with the particle diameter D, just as the heat capacity of a nanoparticle does. Because entropy is only related to the first derivatives of Gibbs free energy with regard to temperature, and we have obtained the average temperature coefficient of solid surface energy agreeing with the value in literature,46 it may be believed that Figure 15 rightly reveals the molar entropy of nanoparticles. 5. Conclusions
Figure 14. Dependence of heat capacities of Ag nanoparticles with different sizes relative to the bulk sample. Cb is the heat capacity of the bulk sample.
Figure 15. Molar entropy as a function of temperature for Ag nanoparticles and bulk material.
heat capacities of the nanoparticles and bulk samples is explained in terms of the surface free energy. Equation 40 indicates that the molar heat capacity of a nanoparticle consists of the contribution from the bulk and surface region, and the reduced heat capacity C/Cb (Cb denotes bulk heat capacity) varies inversely with the particle diameter D. Likhachev et al.59 point out that the major contribution to the heat capacity above ambient temperature is determined by the vibrational degrees of freedom, and it is the peculiarities of surface phonon spectra of nanoparticles that are responsible for the anomalous behavior of heat capacity. This is in accordance with our calculation. Recently, Li and Huang60 calculated the heat capacity of an Fe nanoparticle with a diameter around 2 nm by using MD simulation and obtained a value of 28J/mol-K, which is higher than the value of 25.1J/mol-K43 for the bulk solid. It might be a beneficial reference data for understanding the surface effect on the heat capacity of nanoparticles. The ratio C/Cb ) 1.1 they obtained for 2 nm Fe nanoparticles is comparative to our value of 1.08 for 2 nm Ag nanoparticles. Because we set up a spherical face by three special low-index surfaces, the molar heat capacity of nanoparticles calculated through eq 40 necessarily depends on the shape of the particle. The molar entropy as a function of temperature is shown in Figure 15. It can be seen that the calculated molar entropies
A method of calculating the thermodynamic properties of nanoparticles has been described in the present paper. The application of this method to siver nanoparticles reveals that the melting temperature, heat of fusion, reduced entropy, and reduced heat capacity are inversely proportional to the reciprocal of the particle size, and the results are in good agreement with the theoretical analysis from the liquid drop model. In particular, both the melting temperature and the molar heat of fusion decrease as the particle size decreases in the same way. Moreover, the heat of fusion, the entropy of fusion, the entropy, and the heat capacity of nanoparticles can be divided into two parts: bulk quantity and surface quantity. It is the surface atoms that decide the size effect on the thermodynamic properties of nanoparticles. This conclusion is further supported by the evidence that the specific heat of fusion of 2 nm Ag nanoparticles exceeds the value of the analogous bulk sample by 8%. The result is agreement with Li and Huang’s similar conclusion for 2 nm Fe nanoparticles. Finally, although the surface free energy of Ag is underestimated, the method that the intersection of the free-energy curves for solid and liquid nanoparticles decide the melting point of nanoparticles demonstrates that the surface free-energy difference between the solid and liquid phase is a decisive factor for the size-dependent melting of nanostructural materials. Acknowledgment. This work is financially supported by the NSFC (nos. 50371026, 50571036, and 50671035), the Hunan Pervincial Natural Science Foundation, and the High Performance Computing Center of Hunan University. References and Notes (1) Gleiter, H. Acta Mater. 2000, 48, 1. (2) Pawlow, P. Z. Phys. Chem. 1909, 65, 1. (3) Takagi, M. J. Phys. Soc. Jpn. 1954, 9, 359. (4) Ding, F.; Bolton, K.; Rose´n, A. J. Vac. Sci. Technol., A 2004, 22, 1471. (5) Qi, Y.; Cagin, T.; Johnson, W. L.; Goddard, W. A. J. Chem. Phys. 2001, 115, 385. (6) Buffat, P.; Borel, J.-P. Phys ReV. A 1976, 13, 2287. (7) Peters, K. F.; Cohen, J. B.; Chung, Y.-W. Phys. ReV. B 1998, 57, 13430. (8) Cleveland, C. L.; Luedtke, W. D.; Landman, U. Phys. ReV. Lett. 1998, 81, 2036. (9) Gu¨lseren, O.; Ercolessi, F.; Tosatti, E. Phys ReV. B 1995, 51, 7377. (10) Nanda, K. K.; Sahu, S. N.; Benera, S. N. Phys. ReV. A 2002, 66, 013208. (11) Safaei, A.; Shandiz, M. A.; Sanjabi, S.; Barber, Z. H. J. Phys.: Condens. Matter 2007, 19, 216216. (12) Reiss, H.; Mirabel, P.; Wetten, R. L. J. Phys. Chem. 1988, 92, 7241. (13) Lee, J.-G.; Mori, H. Philos. Mag. 2004, 84, 2675. (14) Tanaka, T.; Hara, S. Z. Metallkd. 2001, 92, 467. (15) Wang, B. X.; Zhou, L. P.; Peng, X. F. Int. J. Thermophys. 2006, 27, 139. (16) Comsa, G. H.; Heitkamp, D.; Ra¨de, H. S. Solid State Commun. 1977, 24, 547.
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