## Abstract

In this brief communication, we identify intrinsic length scales of several physical properties at the nano-scale and show that, for nano-structures whose characteristic sizes are much larger than these scales, the properties obey a simple scaling law. The underlying cause of the size-dependence of these properties at the nano-scale is the competition between surface and bulk energies. This law provides a yardstick for checking the accuracy of experimentally measured or numerically computed properties of nano-structured materials over a broad size range and can thus help replace repeated and exhaustive testing by one or a few tests.

## 1. Introduction

The study of the variation of the properties of materials with their geometrical feature size has a long history because of its importance in many fields. The interest has been heightened recently at the nano-scale because nano-structures are pervasive in nature (Kamat *et al*. 2000; Gao *et al*. 2003) and in modern industry (Goldstein *et al*. 1992; Streitz *et al*. 1994; Bertsch 1997; Miller & Shenoy 2000; Miyata *et al*. 2004), and the large ratio of surface atoms to the bulk can have a profound effect on their properties. In physics and chemistry, the effect of particle size on melting and evaporation temperatures has been discussed since 1900s, and this effect is not restricted to any particular material; rather, it is observed in a variety of materials from metals and alloys to semiconductors and semi-crystalline lamellar polymers, and the typical size range over which the melting temperature undergoes a large change is 2–20 nm (Pawlow 1909; Buffat & Borel 1976; Couchman & Jesser 1977; Castro *et al*. 1990; Peters *et al*. 1998; Zhao *et al*. 2001; Dick *et al*. 2002; Nanda *et al*. 2002*b*; Sun *et al*. 2002; Jesser *et al*. 2004). Many phenomena in solid-state physics and materials science also exhibit size-dependence. For example, the elastic constants of Ag and Pb nano-wires of diameter 30 nm are nearly twice those of the bulk metals (Cuenot *et al*. 2004). Such increase in stiffness cannot be explained by structural modifications of the materials at the nano-scale (Cuenot *et al*. 2004). Reduction in the size of solids also results in a change of their failure mode. Thus, when the size of brittle calcium carbonate particles is reduced to a critical value 850 nm, comminuting by fracturing becomes impossible (Kendall 1978; Karihaloo 1979) and the particles behave as if they were ductile.

Properties of nano-structured materials are affected by the energy competition between the surface and bulk, and a common feature of many physical properties, such as those mentioned above, is that when the characteristic size of the object is very large, the physical property under consideration tends to that of the bulk material. By dimensional analysis, the ratio of a physical property at a small size *L*, denoted by *F*(*L*), to that of the bulk, denoted by *F*(∞), can be expressed as a function of some non-dimensional variables *X*_{j} (*j*=1, …, *M*) and a non-dimensional parameter *l*_{in}/*L* related to the size(1.1)where *l*_{in} is an intrinsic length scale related to the surface property, and *L* is the feature size of the object under consideration. We will confirm below that for many physical properties the function in equation (1.1) can be represented by the following expansion(1.2)where *O* (*l*_{in}/*L*)^{2} denotes the remainder to the order two, *at most*. For *l*_{in}/*L*≪1, this term can be neglected, leading to a simple relation(1.3)In what follows, we shall confirm that the scaling law (1.3) is obeyed by many properties at the nano-scale and we shall identify the corresponding intrinsic length scale *l*_{in}. We shall also provide indirect theoretical and computational evidence in support of this scaling law.

## 2. Melting temperature

Size-dependence of the melting temperature at nano-scale has enormous implications in the production of nano-crystals and thin films, and in the thermal stability of quantum dots (Goldstein *et al*. 1992). A large body of test data has been accumulated on this size-dependence, and a number of theoretical models have been proposed to explain it (Buffat & Borel 1976; Couchman & Jesser 1977; Castro *et al*. 1990; Peters *et al*. 1998; Zhao *et al*. 2001; Dick *et al*. 2002; Nanda *et al*. 2002*b*; Sun *et al*. 2002). Thermodynamically, the melting temperature of nano-particles has been described by three models (Peters *et al*. 1998; Sun *et al*. 2002): (i) the homogeneous melting and growth (HMG) model (Pawlow 1909; Buffat & Borel 1976); (ii) the liquid shell nucleation (LSN) model (Buffat & Borel 1976) and (iii) the liquid nucleation and growth (LNG) model (Couchman & Jesser 1977). All three models predict a size-dependent melting temperature *T*(*R*) that varies inversely with the radius *R* of spherical nano-particles, precisely according to the scaling law (1.3),(2.1)where(2.2)Equation (2.1) is actually the Gibbs–Thomson equation following from consideration of the relative thermodynamic contributions of surface and bulk energies. It lends theoretical support to the scaling law (1.3).

In equation (2.2), *H* is the latent heat of fusion, and *ρ* and *γ* are the mass density and the interface energy, respectively. Subscripts ‘s’, ‘l’ and ‘v’ represent the solid, liquid, and vapour phases, respectively. *δ*≪1 (Peters *et al*. 1998) is the liquid-layer thickness normalized by *R*. The LSN model assumes that a liquid layer of thickness *δ* is in equilibrium at the surface of the solid, implying that the surface of the solid melts prior to its core. Using the data from Castro *et al*. (1990) and letting *δ*=0.02 and *γ*_{sl}=*γ*_{sv}−*γ*_{lv} (which is true of most cubic metals; Peters *et al*. 1998; Zhao *et al*. 2001), the intrinsic length scales *l*_{in} in the above three models are in the order of 0.01–0.1 nm for Au and Ag nano-particles. This value of *l*_{in} is also true for many other metals. Therefore, the scaling law (1.3) can be expected to give good predictions for *L*>3–5 nm. We have confirmed this on numerous available test results (Buffat & Borel 1976; Couchman & Jesser 1977; Castro *et al*. 1990; Peters *et al*. 1998; Zhao *et al*. 2001; Dick *et al*. 2002; Nanda *et al*. 2002*b*; Sun *et al*. 2002; Jesser *et al*. 2004). Nanda *et al*. (2002*b*) proposed a scaling law similar to equation (2.1) based on empirical relations among cohesive energy, surface energy and melting temperature. The scaling law not only captures the size-dependent reduction in the melting temperature of unconstrained nano-particles, but also if they are embedded in a matrix. Moreover, it has been confirmed that the size-dependent evaporation temperature of nano-particles (Nanda *et al*. 2002*a*) also follows the scaling law (1.3). It is noted that if nano-structured materials are embedded in a matrix and the interfaces between them and the surrounding matrix are coherent or semi-coherent, then the melting temperature actually increases with decreasing nano-particle size (Zhao *et al*. 2001).

## 3. Surface effect on mechanical properties

The surface effect on the elastic deformation of nano-structures can be substantial. For homogeneous nano-structured materials, e.g. extension of nano-plates, nano-beams, nano-wires, etc., when the surface elasticity, characterized by the surface elastic constant *τ*, is taken into account, an intrinsic length scale automatically emerges (Miller & Shenoy 2000; Shenoy 2002),(3.1)where *E* is the Young modulus of the bulk material. Thus, the non-dimensional mechanical properties of nano-structures are governed by the scaling law (1.3) with *l*_{in} given by equation (3.1).

For metals and some other materials, *l*_{in} in equation (3.1) is typically in the order of 0.01–0.1 nm (Streitz *et al*. 1994; Miller & Shenoy 2000). Thus, the scaling law (1.3) can be expected to give good predictions for *L*>3–5 nm. *Ab initio* and molecular dynamic simulations, and experimental results show that the elastic constants of nano-plates, nano-beams and nano-wires obey the scaling law (1.3) almost exactly in and above the range 1–100 nm (Miller & Shenoy 2000; Shenoy 2002; Cuenot *et al*. 2004; Villain *et al*. 2004; Zhou & Huang 2004). The elastic constants of the nano-plates of copper and tungsten obtained from atomistic calculations by Villain *et al*. (2004) and Zhou & Huang (2004) are shown in figure 1. It is seen that the ratio *E*/*E*(∞)for these materials almost exactly obeys the scaling law in equation (1.3). Villain *et al*. (2004) and Zhou & Huang (2004) did not explicitly use the concept of surface elastic constant to explain the variation of the elastic constants with the thickness of the nano-plates, but, obviously, the surface effect is implicitly taken into account in the atomistic calculations. These computations provide a further objective justification of the scaling law (1.3). Moreover, had this scaling law been known, the numerical computations could have been significantly reduced, as only two points are needed to determine a linear function. This is particularly important in the problems of characterization of heterogeneous materials given below for which it may be very difficult and/or time-consuming to perform atomistic simulations.

For general heterogeneous nano-structured materials, the elasticity of an isotropic surface is characterized by two surface elastic constants *λ*_{s} and *μ*_{s} (Bottomley & Ogino 2001; Sharma *et al*. 2003; Duan *et al*. 2005*a*), giving rise to two intrinsic length scales *l*_{λ}=*λ* _{s}/*E* and *l*_{μ}=*μ*_{s}/*E* (Duan *et al*. 2005*a*,*b*). Thus, the size-dependence of non-dimensional physical properties associated with the deformation problems of heterogeneous nano-solids can be expected to follow a scaling law similar to equation (1.3), but with an intrinsic length scale which is a linear combination of these two scales:(3.2)Here, *α* and *β* are two non-dimensional parameters, *H*(*L*) is the property corresponding to a characteristic size *L* at nano-scale, and *H*(∞) denotes the same property when *L*→∞ or, equivalently, when the surface effect is vanishingly small. The scaling law (3.2) is applicable to a wide variety of properties, as evidenced by the following examples.

The maximum stress concentration factor *k*(*R*) at the boundary of a circular nano-pore of radius *R* in a plate under uniaxial tension is(3.3)where *k*(∞)=3 is the classical elasticity result. The transverse Young modulus *E*_{T}(*R*) of a nanochannel-array material containing parallel cylindrical nano-pores (Masuda & Fukuda 1995; Miyata *et al*. 2004) is(3.4)where *ν* is the Poisson ratio of the matrix material, and *f* is the porosity of the nanochannel-array material.

Formulas (3.3) and (3.4) have been obtained using the Eshelby tensor (Eshelby 1957) for inhomogeneities. This tensor is fundamental to the solution of many problems in materials science, solid-state physics and mechanics of composites. The classical Eshelby tensor depends on the shape of the inhomogeneity but not on its size or on the position of a material point within it. In contrast, for inhomogeneities at nano-scale when the interface stress effect is substantial (Cammarata 1997), the Eshelby tensor itself becomes a function of the size of the inhomogeneity and the position of a material point within it besides its shape (Duan *et al*. 2005*b*). For example, for circular or spherical inhomogeneities at nano-scale the Eshelby tensor (*R*, * x*) can be accurately expressed in the following general form (Duan

*et al*. 2005

*b*):(3.5)where

*is the position vector, and (∞) is the classical Eshelby tensor. Here,*

**x***α*(

*) and*

**x***β*(

*) are two position-dependent tensors. The scaling law (3.2) holds for many physical properties derived using the Eshelby tensor (3.5). For example, the effective moduli of heterogeneous solids obtained by various micromechanical schemes through the algebraic operations involving the Eshelby tensor will obey the scaling law (3.2). Two such examples were given above.*

**x**Again, for metals and some other materials, *l*_{λ} and *l*_{μ} in equation (3.2) are also typically in the order of 0.01–0.1 nm (Miller & Shenoy 2000; Sharma *et al*. 2003). Thus, like the one-dimensional nano-structural problems, we would expect the scaling law (3.2) to give good predictions for *L*>3–5 nm. This has been confirmed by the Eshelby tensors, stress concentration tensors and effective elastic constants of heterogeneous materials containing nano-inhomogeneities (Duan *et al*. 2005*a*,*b*).

The scaling law (3.2) shows that the surface effect can increase (*H*(*L*)/*H*(∞)>1) or decrease (*H*(*L*)/*H*(∞)<1) a property at the nano-scale relative to that at large feature size depending on the sign of the mixed intrinsic length parameter (*αl*_{λ}+*βl*_{μ}). Shenoy (2005) pointed out that the surface elastic modulus tensor need not be positive definite, but the total energy (bulk+surface) needs to satisfy the positive definiteness condition. Therefore, the signs of terms involving *α* and *β* will depend on the properties of the surfaces, as seen from figure 1 and from the theoretical calculations of Miller & Shenoy (2000) and Shenoy (2005). It should be mentioned that although the intrinsic lengths *l*_{in}, *l*_{λ} and *l*_{μ} are very small, they do have a significant and measurable effect on the properties of the nano-structured materials, as evidenced by the above examples.

It should be noted that *α* and *β* in equations (1.3) and (3.2) depend only on the properties of the materials and the geometry of the nano-structures but not the size of the latter. This is true of all the examples given above in which only tensile stress fields are present. If other stress fields are also present, e.g. those due to flexure, then *α* may additionally depend on the size of the test specimen (Cuenot *et al*. 2004), and there will be a coupling between the scaling and size effects.

Another intrinsic length scale essentially similar to that in equation (3.1) is the ratio of surface energy to the Young modulus of a bulk material(3.6)It can be traced back to the classical Griffith theory for brittle fracture.

Kendall (1978) and Karihaloo (1979) have shown that under compression small brittle particles cease to fail by cracking and appear to behave in a ductile manner when their size is reduced to *d*_{crit}=*A*(*E*/*σ*_{y})^{2}*l*_{in}, where *A* is a constant and *σ*_{y} the yield stress of the bulk material. In biology, Gao *et al*. (2003) showed that when the thickness of brittle mineral platelets in hard biological tissues reduces to the critical value *λ*_{crit}=*B*(*E*/*σ*_{th})^{2}*l*_{in}, where *B* is a constant, the platelets are no longer susceptible to brittle fracture but appear to approach the theoretical strength *σ*_{th} This again points to a transition from the brittle to a ductile mode of failure. Recently, Buehler *et al*. (2003) have found that an intrinsic length scale *Χ*=*C*(*E*/*σ*)^{2}*l*_{in}, where *C* is a constant and *σ*, the applied stress, is associated with the energy transport near a dynamically growing crack tip. This length scale plays a central role in understanding the effect of hyper-elasticity. Hyper-elasticity completely dominates crack dynamics if the size of the hyper-elastic zone in the vicinity of a crack tip approaches this intrinsic length scale. The known surface instability/stability transition phenomenon of solids is also found to scale with the intrinsic length scale (3.6). For a stressed solid, a surface undulation with a wavelength greater than a critical value *λ*_{crit}=*π*(*E*/*σ*)^{2}*l*_{in} is known to grow unstably, whereas an undulation with a wavelength smaller than *λ*_{crit} is smoothed out by surface diffusion (Srolovitz 1989).

## 4. Concluding remarks

As mentioned in the beginning, the common cause of the size-dependence in all the physical phenomena considered above is the competition between surface and bulk energies. Thus, in the size-dependence of the melting temperature of nano-particles, it is the competition between the surface energy and the bulk latent heat of fusion (as in the Gibbs–Thomson equation); that of the elastic deformation at the nano-scale, it is the competition between the surface energy and the strain energy in the bulk; and that of the brittle fracture and surface instability of solids, it is the competition between the energy consumed in creating new surfaces and the strain energy released by the bulk.

The scaling law (1.3) or (3.2) not only accurately predicts the scale-dependence of several properties of nano-structured materials, as shown above, but it also provides a benchmark for checking experimental measurements and numerical computations of nano-structural properties. Moreover, it can help replace multitude of experimental measurements and/or numerical computations needed to cover a wide range of sizes by one or a few tests on samples of one relatively large size, say, larger than 3 nm, to determine accurately the parameter *αl*_{in}. Then the properties at other nano sizes can be determined by simple scaling according to the scaling law in equation (1.3). This can be most useful, even when excellent theoretical models are available, because almost all models rely on some experimentally determined parameters, for example *ρ*_{s}, *ρ*_{l}, *γ*_{sv}, *γ*_{lv}, *γ*_{sl} and *H* in equation (2.2). The experimental techniques to determine these parameters at the nano-scale require highly specialized instruments and elaborate analysis routines which can be very costly, time-consuming and often difficult, if compounds, alloys (Jesser *et al*. 2004) and embedded nano-particles (Bergese *et al*. 2004) are considered. Surfaces/interfaces are pervasive in multi-phase materials containing nano-inhomogeneities such as nano-particles and nano-voids. Besides the elastic properties of the heterogeneous solids identified above, the effective thermal conductivities of nanofluids consisting of nano-scale solid particles dispersed in liquids (Eastman *et al*. 2004; Kumar *et al*. 2004) are also found to obey the scaling law in equation (1.3) due to the surface effects induced by the nano-particles (Kumar *et al*. 2004).

## Acknowledgments

H.L.D., J.W. and Z.P.H. are supported by the National Natural Science Foundation of China under grant no. 10525209.

## Footnotes

- Received June 2, 2005.
- Accepted December 5, 2005.

- © 2006 The Royal Society