Logarithmic Size-Dependent Melting Temperature of Nanoparticles

Article pubs.acs.org/JPCC

Logarithmic Size-Dependent Melting Temperature of Nanoparticles Zhiyuan Liu,† Xiaohong Sui,† Kai Kang,‡ and Shaojing Qin*,† †

State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China Science and Technology on Surface Physics and Chemistry Laboratory, P.O. Box 718-35, Mianyang 621907, Sichuan, China

‡

ABSTRACT: This work describes the logarithmic term in the sizedependent melting temperature of nanoparticles. The Lindemann melting criterion and the thermal phonon contribution are intended primarily as a clear approach for our study of melting temperature. The ratio of the melting temperature (Tmn) of a nanoparticle to the bulk melting temperature (Tmb) is numerically evaluated up to the order of the inverse of the size L of a nanoparticle: Tmn/Tmb = 1 − A/L. We show that the coefficient A is not a constant but has a logarithmic part. The logarithmic term is the dominate term in the factor A(L). Our investigation suggests the behavior of the size-dependent melting is Tmn/Tmb = 1 − (B + C ln L)/(L − 2ls) where B and C are constant factors and ls is the number dimension of the surface layers reconstructed or premelted. The results present a challenge to reach a higher resolution in future thermal experiments in order to distinguish ln L/L behavior from the previously accepted pure 1/L behavior.

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considering the free boundary conditions.24 As the size of the crystal becomes small, the surface becomes a substantial part.13,14,17 Different surface properties correspond to different boundary conditions for the crystal. Following Sui’s work, Xu et al. studied the effect of boundary conditions and discussed the boundary scattering phase shift of phonons.25 They found that the upper bound of melting temperature is set by the fixed boundary condition. In this paper, we present that the relation between the ratio of melting temperature of nanoparticles (Tmn) to the bulk melting temperature (Tmb) and the inverse of particle size (1/ L) is not simply linear, Tmn/Tmb = 1 − A/L; rather, A is logarithmic size-dependent. On the basis of the Lindemann criterion and the thermal phonon contribution, we find the relation to be Tmn/Tmb = 1 − (B + C ln L)/(L − 2ls), where ls is the characteristic length for atoms all on the surface. The expression is general for different boundary conditions and crystals and is compared with the experimental data.

INTRODUCTION The study of nanoparticle melting is of great importance because of its possible application as a source of new materials. Since the size-dependent melting temperature was found by Takagi by means of TEM,1 extensive experiments have been performed.2−11 While the depression of melting temperature with the decrease of particle size was found in some experiments,2−7 the superheating phenomenon was also reported for some nanoparticles embedded in matrices.8−10 Theoretical study began in the early 1900s when Pawlow predicted the size-dependent melting temperature for nanoparticles.12 Different theoretical models explain the phenomenon based on different assumptions,12−21 and it is still difficult to decide which model is the best. The linear relation between the ratio of melting temperature of nanoparticles (Tmn) to bulk melting temperature (Tmb) and inverse of particle size (1/L) has been derived as follows: Tmn/Tmb = 1 − A/L, where A is a constant. The Lindemann melting criterion is simple but effective for the melting phenomenon: a solid melts when the ratio of the mean displacement of atoms (u) to the lattice constant (a) reaches the Lindemann critical value (Lc).22,23 In the rootmean-square displacement model,17−19 it was used as the criterion for the melting of nanoparticles. For an ideal crystal, as the size of the lattice decreases, the phonon contribution becomes important. The surface-phonon instability model considered the intrinsic defect contribution to phonon modes.13 The quantum size effect had not been involved before Sui et al. carefully investigated the effect of discrete thermal phonons contributing to the melting of nanoparticles, and they found the lower boundary for the melting temperature of nanoparticles © XXXX American Chemical Society

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THERMAL PHONON EQUATION The discrete phonon energy level equations are introduced first. Then, the melting temperature is evaluated in the next section. The detailed derivation and discussion could be found in refs 24−26. For a simple cubic lattice with one atom at each lattice point, the thermal phonon contribution gives thermal displacement of the atoms. The equation for the melting temperature (Tmb) of a Received: February 5, 2015 Revised: April 30, 2015

A

DOI: 10.1021/acs.jpcc.5b01188 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C 1 N

bulk material can be calculated from the thermal phonon population:

∑

2v

′

⎡ aC Tmbω(k) ⎤ aC0ω(k) tanh⎣⎢ 02vT ⎦⎥

kx , k y , k z

π /a

Lc 2 =

3ℏa 2π 3M

∫ 0

mn

d3k

π /a

1 ⎡ ℏω(k) ⎤ ω(k) tanh⎣⎢ 2k T ⎦⎥ B mb

= (1)

⎛ a ⎞3 ⎜ ⎟ ⎝π ⎠

∫

d3k

0

in which Lc is the critical ratio in Lindemann criterion, M the mass of the atom, a the lattice constant, and ω(k) the energy dispersion relation of the phonon at wave vector k = (kx, ky, kz) . The melting temperature of nanoparticles depends on the reflective phase change of low-energy phonon modes at the nanoparticle surface. In this paper, nanoparticles under free boundary condition (free-BC) and fixed boundary condition (fixed-BC) are investigated, which are of zero and π reflective phase shifts, respectively. We also evaluate the melting temperature for the ideal periodic boundary condition (PBC). Wave vectors of phonon modes are simple and completely determined under these boundary conditions. Actually, the PBC is an ideal boundary condition but is useful in the study of ideal crystals. Realization of absolute free-BC and fixed-BC is also impossible; however, the approximation is probable. In experiments, the nanoparticles are always substrate-supported. When the nanoparticles are approximately free from the substrate, we could treat the boundary condition as free-BC, and for particles embedded in rigid matrices, the boundary condition may be estimated as fixed-BC. We introduce a parameter C0 = ℏv/akBTmb for the parent bulk crystal with Tmb already measured and v the bulk sound speed. In Figure 1, the parameter C0 for more than 60 crystals is

2v ⎡ aC ω(k) ⎤ aC0ω(k) tanh⎣ 02v ⎦

(2)

where ∑′ sums all modes except the k = 0 zero mode. Equation 2 is also the equation for the melting temperature (Tmn) for free-BC. The difference is in the discrete wave-vector lattice points in summation, which is kα = nαπ/La with nα = 0,···, L − 1 for free-BC. For fixed-BC, the melting temperature (Tmn) is determined by ⎡ 1⎢ 2v ⎢ ∑ N ⎢ kx , ky , kz aC ω(k) tanh⎡ aC0Tmbω(k) ⎤ ⎢⎣ 2vTmn ⎥⎦ 0 ⎣

∑

−

kxe , k ye , kze

⎤ ⎥ ⎥ e kαa 2 ⎡ aC Tmbω(k) ⎤ ⎥ ∏ aC0ω(k) tanh⎣⎢ 02vT [ L sin ] ⎦⎥ α 2 ⎦ mn 16v

π /a

=

⎛a⎞ ⎟ ⎝π ⎠

⎜

3

∫ 0

d3k

2v ⎡ aC ω(k) ⎤ aC0ω(k) tanh⎣ 02v ⎦

(3)

in which kα = nαπ/La with nα = 1,···, L and = (2nα + 1)π/La with nα = 0,···, L/2 − 1. The discrete phonon mode summation will not be transformed into continuous integration in numerical evaluation, and 1/L terms will be analyzed in the next section. keα

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LOGARITHMIC SIZE-DEPENDENT TERM The equations in the previous section are solved numerically for Tmn/Tmb at each size L for a typical magnitude of C0. The phonon dispersion is approximated by ω(k) =

k ya ka ka 2v sin 2 x + sin 2 + sin 2 z a 2 2 2

(4)

Although C0 changes for different parent crystals, the emphasis is on the size-dependent behavior of melting temperature at the order of 1/L: Tmn/Tmb = 1 − A/L. For a typical magnitude C0 = 0.05 (e.g., elemental crystals of Ca, Ge, In, and Bi etc. have C0 ∼ 0.05) in Figure 1, we calculate melting temperatures of nanoparticles at different sizes and under different boundary conditions. The size of one nanoparticle (L) is from 2 to 1000, where we assume L is large from 100−1000 and small around 10. Actually, the L should be larger than the 2ls ∼ 6, in which 2ls is the character length for the nanoparticle in which all the atoms are located on the surface. When L ≤ 2ls, the crystal will be unstable17,21 and the thermal phonon equations fail. High-precision Tmn/Tmb values are computed numerically from eq 2 and eq 3. Size-dependent behavior up to the order of 1/L is plotted as A(L) versus L in logarithmic scale in Figure 2. A pure linear quantum finite size effect term in the inverse of the size for PBC is demonstrated. This pure linear term has been obtained in a previous study24 from the difference

Figure 1. Parameter C0 for monatomic crystals of different elements, listed in increasing order of C0. The horizontal axis represents the order number.

listed, and each crystal is made of one kind of element. In the following, Tmn is the melting temperature for a nanoparticle of size La, in which L is the number size of the nanoparticle in units of lattice constant. The total number of atoms in a simple cubic lattice is N = L3. For PBC, phonon wave vector takes discrete values: kα = 2nαπ/La with nα = −L/2 + 1,···, L/2, where α is x, y, or z. The melting of a nanoparticle is estimated by the same critical ratio (Lc) for bulk material in eq 1. The melting temperature Tmn for PBC will be calculated from the equation u2(Tmn) = u2(Tmb) without the appearance of Lc: B

DOI: 10.1021/acs.jpcc.5b01188 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Figure 2. Size-dependent melting temperature for nanoparticles under different boundary conditions.

Figure 3. Parameter A at size L = 100 for nanoparticles consisting of different parent materials characterized by different C0 and under different boundary conditions.

between the phonon zero-mode volume in a finite size particle and in the bulk material. The line becomes flat for PBC in the large L limit (Figure 2). This behavior indicates the factor A in Tmn/Tmb = 1 − A/L is a constant. We also see straight lines with finite slopes in the large L limit for free-BC and fixed-BC, which means that A is not a constant but A(L) = B + C ln L. C is the slope of the line in the logarithmic scaled figure. The factor B is relatively small while C is big. Therefore, the logarithmic term is dominant at the order of 1/L for both situations. After the investigation on the logarithmic term for nanoparticles with a monatomic parent bulk parameter C0 = 0.05, we study the logarithmic term for nanoparticles of various compositions and under general boundary conditions. For the phonon states on the top of the acoustic phonon band, their energy can be expressed through the Debye frequency (ωD). In a bulk material, the phonon population on this mode is of the order of 1/[2 tanh ((ℏωD)/(2kBTmb))], which is represented by 1/(2C0) through our parameter C0. For free-BC and fixedBC, we evaluated Tmn/Tmb for 0 < C0 < 1/2 and found the logarithmic term is the dominant term at the order of 1/L: A(L) ∼ ± ln L. This property is almost independent of the magnitude of C0, which is shown in Figure 3, i.e., the magnitude of A(L) is almost the same for different kinds of parent bulks characterized by different magnitudes of C0. Next, we analyze how the logarithmic term dominates the variation of melting temperature of nanoparticles under general boundary conditions. A reflective phase change (δkα) of the phonon wave on the surface of a nanoparticle25 shifts the lattice of k-points inside k-space. The reflective phase change for a small wave vector can be expanded as δkα = δ + a1kα, where δ is a constant phase shift and a1 is the expansion coefficient in the first order of small kα. This shift changes the magnitude of the function on each point in the discrete summation in the main term of eqs 2 and 3. The function in summation is in the form of 1/k2 when k is small, and the change is big for them. The most important contribution to the change is the integration on surfaces with kα ∼ 0 in k-space, which is dominated by the logarithmic term (δ/π − 1/2)( ln L/L). Such a dominant change leads to the dominant logarithmic term in Tmn/Tmb − 1 ∼ (δ/π − 1/2)(ln L/L). Thus, the boundary condition plays an important role in changing the melting temperature of nanoparticles.

Now we discuss the logarithmic term for nanoparticles of different composition and lattice types. For any lattice type and composition that a nanoparticle can have, small wave-vector acoustic phonons contribute the most to the total magnitude change of the discrete summation in eqs 2 and 3. The ln L/L term will be dominant whenever ω(k) ∼ vk is a good approximation for small wave vectors. Because ω(k) ∼ vk generally holds for bulk materials of any composition and lattice type, we have a dominant logarithmic term in Tmn/Tmb − 1 ∼ ln L/L for nanoparticles of general composition and general lattice type. Now we reach the result of the investigation in this section: the logarithmic term dominates the melting temperature change in the order of 1/L; the sizedependent melting temperature for nanoparticles behaves according to Tmn/Tmb = 1 − (B + C ln L)/L.

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DISCUSSION AND CONCLUSION It is a challenge for new experimental techniques to distinguish ln L/L from 1/L behavior. For example, for Tmb ∼ 1000 K in the parent bulk material, a 50 K drop of melting temperature in experiment for free-standing nanoparticles has been measured at the size around 20 nm (L ∼ 100). Then a constant A fitting with experimental data for size around 20 nm will be 5 and Tmn/Tmb ∼ 1 − 5/L, which predicts a melting temperature drop of 1 K for nanoparticles at the size of 1 μm and L ∼ 5000. While a fitting for A dominated by the logarithmic term will be Tmn/Tmb ∼ 1 − (5/ln 100) (ln L/L), which suggests a drop of 1.8 K at the size of L ∼ 5000. A very high-resolution 0.4 K in melting temperature measurement is required at the size of 1 μm particles in order to distinguish the difference between ln L/L and 1/L behavior. Such a resolution is difficult and challenging in thermal measurements. The equation Tmn/Tmb = 1 − (B + C ln L)/L is expected to be correct for large sized particles in 1/L → 0. It makes no claim to being a reference for small L where higher-order terms in 1/L cannot be ignored. When the size of almost freestanding nanoparticles is small, L ∼ 10, the melting temperature falls sharply and error in measurement is large.3 The size-dependent melting temperature of small nanoparticles must be represented in a practical fitting or estimation. The estimation 1/(L − 6) has been successively applied to describe this sharp fall in several previous studies under some profound reasoning.17 We also include 1/(L − 2ls) in the following C

DOI: 10.1021/acs.jpcc.5b01188 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C Notes

formula for the behavior of size-dependent melting of small nanoparticles: Tmn B + C ln L =1− Tmb L − 2ls

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Yuanyuan Xu and visitors to KITPC for discussions. This work was supported by NNSF11421063 of China.

(5)

If we move the effective boundary of the nanoparticle several layers toward the center, we leave those surface atoms not on lattice points as the environment outside the effective boundary. lsa and ls can be understood as the thickness of the surface layers outside the effective boundary and the number dimension of the thickness, respectively. When L − 2ls approaches one, the small nanoparticle can be accessed only by ab initio simulation27 case-by-case for particular clusters and under particular environments. The ls introduced into eq 5 is close to 3.17 We read out the experimental melting temperature Tmn and size for gold nanoparticles in Figure 6 in ref 3. With known Tmb = 1337.33 K and a = 4.07/(2)1/2 Å for gold, in Figure 4 we list

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Figure 4. Enveloping the experimental data taken from ref 3 according to eq 5.

the experimental data points by Tmn/Tmb versus the inverse of the number size L. We show that the points for small size particles can be enveloped by two lines with ls = 3 ± 0.3. Whether ls is close to 3 for most of free-standing nanoparticles is important in the understanding of the size-dependent melting of small nanoparticles. In summary, the size-dependent effect of melting temperature of nanoparticles was investigated based on the thermal phonon equations and Lindemann criterion. We studied the melting temperature of nanoparticles under different boundary conditions. Through numerical extrapolation, we found a new relation between the melting temperature and particle size, presented by eq 5. In this equation, the logarithmic term of particle size L was contained in parameter A because of the quantum size effect. This interesting relation has not been predicted by other models before. This new relation is in reasonable agreement with the experimental data of Au nanoparticles. Future accurate experiments in large particle size are expected to resolve the difference between our (B + C ln L)/(L − 2ls) relation and the previously accepted A/L relation.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

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DOI: 10.1021/acs.jpcc.5b01188 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C (24) Sui, X. H.; Wang, Z. G.; Kang, K.; Qin, S.; Wang, C. Can Nanoparticle Melt below the Melting Temperature of Its Free Surface Partner? Commun. Theor. Phys. 2015, 63, 249−254. (25) Xu, Y. Y.; Kang, K.; Qin, S. J. Can Nano-particle Stand above the Melting Temperature of Its Fixed Surface Partner? 2015, arXiv:1501.01735. arXiv Preprint. (26) Kang, K.; Qin, S. J.; Wang, C. L. Size-dependent Melting: Numerical Calculations of the Phonon Spectrum. Phys. E (Amsterdam, Neth.) 2009, 41, 817−821. (27) Tartaglino, U.; Zykova-Timan, T.; Ercolessi, F.; Tosatti, E. Melting and Nonmelting of Solid Surfaces and Nanosystems. Phys. Rep. 2005, 411, 291−321.

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DOI: 10.1021/acs.jpcc.5b01188 J. Phys. Chem. C XXXX, XXX, XXX−XXX