Surface Energy and Its Effects on Nanomaterials

Synonyms

Effective modulus; Nanocomposites; Surface tension

Definitions

Atoms at a free surface experience a different local environment than do atoms in the bulk of a material. As a result, the energy associated with these atoms will, in general, be different from that of the atoms in the bulk. The excess energy associated with surface atoms is called surface free energy. In traditional continuum mechanics, such surface free energy is typically neglected because it is associated with only a few layers of atoms near the surface and the ratio of the volume occupied by the surface atoms and the total volume of material of interest is extremely small. However, for nano-sized particles, wires, and films, the surface-to-volume ratio becomes significant and so does the effect of surface free energy. Consequently, the effective modulus of nano-sized structural elements should be considered, which by definition becomes size dependent.

Introduction

The elastic behavior of a material is...

Appendices

Appendix A: Coordinate Transformation

Consider the ellipsoid Ω shown in Fig. 3. When the ellipsoid is subjected to a uniform strain field, ε _ij, the surface of the ellipsoid deforms accordingly. Let the two-dimensional surface strain tensor, ${\varepsilon}_{\alpha \beta}^s$, be defined in a local coordinate system (i ₁, i ₂, i ₃), where i ₁ and i ₂ are tangent to the surface and i ₃ is normal to the surface. Clearly, the choice of i ₁ and i ₂ is not unique. The following approach is taken to uniquely define the local coordinate system on the ellipsoidal surface.

In the spherical coordinate system,

$$ \begin{aligned}{x}_1&=r\cos \theta \sin \phi, {x}_2=r\sin \theta \sin \phi,\\ {x}_3&=r\cos \phi, 0\le \theta \le 2\pi, 0\le \phi \le \pi,\end{aligned} $$

(A1)

a point on the surface of the ellipsoid can be represented by the vector

$$\begin{aligned} \mathbf{R}\left(\theta, \phi \right)&=a\cos \theta \sin \phi {\mathbf{I}}_1+b\sin \theta \sin \phi {\mathbf{I}}_2\\ &\quad+c\cos \phi {\mathbf{I}}_3\end{aligned} $$

(A2)

A local coordinate system at this point may be introduced by the following three unit vectors:

$$\begin{aligned} {\mathbf{i}}_3&=\frac{1}{d_1}\left(\cos \theta \sin \phi {\mathbf{I}}_1+\frac{a}{b}\sin \theta \sin \phi {\mathbf{I}}_2\right.\\ &\left.\quad+\frac{a}{c}\cos \phi {\mathbf{I}}_3\right),\end{aligned} $$

(A3)

$$ {\mathbf{i}}_2=\frac{\partial \mathbf{R}}{\partial \theta }/\left\Vert \frac{\partial \mathbf{R}}{\partial \theta}\right\Vert =\frac{1}{d_2}\left(-\frac{a}{b}\sin \theta {\mathbf{I}}_1+\cos \theta {\mathbf{I}}_2\right), $$

(A4)

$$\begin{aligned} {\mathbf{i}}_1&={\mathbf{i}}_2\times {\mathbf{i}}_3=\frac{a}{cd_1{d}_2}\cos \theta \cos \phi {\mathbf{I}}_1\\ &\quad+\frac{a^2}{bcd_1{d}_2}\sin \theta \cos \phi {\mathbf{I}}_2-\frac{d_2}{d_1}\sin \phi {\mathbf{I}}_3,\end{aligned} $$

(A5)

where

$$ {d}_1{=}\sqrt{\cos^2\theta {\sin}^2\phi \,{+}\,\frac{a^2}{b^2}{\sin}^2\theta {\sin}^2\phi \,{+}\,\frac{a^2}{c^2}{\cos}^2\phi }, $$

(A6)

$$ {d}_2=\sqrt{\frac{a^2}{b^2}{\sin}^2\theta +{\cos}^2\theta }, $$

(A7)

The transformation matrix between the global (I ₁, I ₂, I ₃) and the local (i ₁, i ₂, i ₃) coordinate systems is thus given by

$$ \begin{aligned}&\left[{t}_{ij}\right]\\ &\quad =\left[\begin{array}{ccc}\frac{a}{cd_1{d}_2}\cos \theta \cos \phi & \frac{a^2}{bcd_1{d}_2}\sin \theta \cos \phi & -\frac{d_2}{d_1}\sin \phi \\ {}-\frac{a}{bd_2}\sin \theta & \frac{1}{d_2}\cos \theta & 0\\ {}\frac{1}{d_1}\cos \theta \sin \phi & \frac{a}{bd_1}\sin \theta \sin \phi & \frac{a}{cd_1}\cos \phi \end{array}\right].\end{aligned} $$

(A8)

Therefore, according to the tensor transformation rule, the surface strain in the local coordinate system can be written as

$$ {\varepsilon}_{\alpha \beta}^s={t}_{\alpha i}{t}_{\beta j}{\varepsilon}_{ij}. $$

(A9)

For a spherical particle (a = b = c), the transformation matrix reduces to

$$ \left[{t}_{ij}\right]=\left[\begin{array}{ccc}\cos \theta \cos \phi & \sin \theta \cos \phi & -\sin \phi \\ {}-\sin \theta & \cos \theta & 0\\ {}\sin \phi \cos \theta & \sin \phi \sin \theta & \cos \phi \end{array}\right]. $$

(A10)

Bulk and Surface Elasticity Tensors

When subjected to a strain field ε _ij, the strain energy of an elastic body can be written as

$$ \Phi {=}\frac{1}{2}{C}_{ij kl}{\varepsilon}_{ij}{\varepsilon}_{kl}{+}\frac{1}{6}{C}_{ij kl mn}^{(3)}{\varepsilon}_{ij}{\varepsilon}_{kl}{\varepsilon}_{mn}{+}\cdots \cdots $$

(B1)

where C _ijkl is a fourth-order tensor consists of (second-order) elastic constants and ${C}_{ijkl}^{(3)}$ is a sixth-order tensor consisting of the third-order elastic constants of the solid. It can be easily shown that the following symmetry conditions must be met by these tensors:

$$ {C}_{ijkl}={C}_{jikl}={C}_{klij}, $$

(B2)

$$ \begin{aligned}{C}_{ijklmn}^{(3)}&={C}_{jiklmn}^{(3)}={C}_{klmnij}^{(3)}={C}_{mnijkl}^{(3)}\\ &={C}_{ijmnkl}^{(3)}={C}_{mnklij}^{(3)}={C}_{klijmn}^{(3)}.\end{aligned} $$

(B3)

Instead of the tensorial notation, it is convenient in certain cases to use the Voigt (contracted) notation for these tensors. For example, C ₁₁ is used for C ₁₁₁₁, C ₁₂ is used for C ₁₁₂₂, C ₁₂₃ is used for C ₁₁₂₂₃₃, etc. The general rules to contract the indices are (11) → (1), (22) → (2), (33) → (3), (12) → (6), (13) → (5), and (23) → (4). Of course, the symmetry properties of the elasticity tensor remain in their contracted form, e.g., C ₁₂ = C ₂₁ and C ₁₂₃ = C ₃₁₂.

For solids with cubic symmetry, there are three independent non-zero second-order elastic constants for C _ijkl:

$$ \begin{aligned}{C}_{11}&={C}_{22}={C}_{33},{C}_{12}={C}_{13}={C}_{23},\\ {C}_{44}&={C}_{55}={C}_{66}\end{aligned} $$

(B4)

and six independent non-zero third-order elastic constants for ${C}_{ijklmn}^{(3)}$:

$$ {C}_{111}={C}_{222}={C}_{333},{C}_{144}={C}_{255}={C}_{366}, $$

(B5)

$$ {C}_{112}={C}_{113}={C}_{122}={C}_{133}={C}_{223}={C}_{233}, $$

(B6)

$$ {C}_{155}={C}_{166}={C}_{244}={C}_{266}={C}_{344}={C}_{355}, $$

(B7)

$$ {C}_{123},{C}_{456} $$

For isotropic solids, the number of independent elastic constants is further reduced. For C _ijkl, there are only two independent ones. They are

$$ \begin{aligned}{C}_{11}&={C}_{22}={C}_{33}=K+\frac{4\mu }{3},\\ {C}_{12}&={C}_{13}={C}_{23}=K-\frac{2\mu }{3},\end{aligned} $$

(B8)

$$ {C}_{44}={C}_{55}={C}_{66}=\mu, $$

(B9)

where K is called the bulk modulus and μ the shear modulus.

For isotropic solids, ${C}_{ijklmn}^{(3)}$ has three independent non-zero constants L, M, and N. They are related to C _ijk by

$$ {C}_{111}={C}_{222}={C}_{333}=L+6M+8N, $$

(B10)

$$ {C}_{144}={C}_{255}={C}_{366}=M, $$

(B11)

$$ \begin{aligned}{C}_{112}&={C}_{113}={C}_{122}={C}_{133}={C}_{223}\\ &={C}_{233}=L+2M,\end{aligned} $$

(B12)

$$\begin{aligned} {C}_{155}&={C}_{166}={C}_{244}={C}_{266}={C}_{344}\\ &={C}_{355}=M+2N,\end{aligned} $$

(B13)

$$ {C}_{123}=L,{C}_{456}=N. $$

(B14)

In terms of the Kronecker delta δ _ij, these elasticity tensors can be written conveniently as

$$\begin{aligned} {C}_{ij kl}&=K{\delta}_{ij}{\delta}_{kl}\\ &\quad+\mu \left({\delta}_{ik}{\delta}_{jl}+{\delta}_{il}{\delta}_{jk}-\frac{2}{3}{\delta}_{ij}{\delta}_{kl}\right),\end{aligned} $$

(B15)

$$ {\displaystyle \begin{aligned}{C}_{ij klmn}^{(3)}&=L{\delta}_{ij}{\delta}_{kl}{\delta}_{mn}\\ &\quad+M\left({\delta}_{ij}{\delta}_{km}{\delta}_{ln}+{\delta}_{ij}{\delta}_{kn}{\delta}_{lm}\right.\\ &\quad+{\delta}_{im}{\delta}_{jn}{\delta}_{kl}+{\delta}_{in}{\delta}_{jm}{\delta}_{kl}\\ &\quad\left.+{\delta}_{ik}{\delta}_{jl}{\delta}_{mn}+{\delta}_{il}{\delta}_{jk}{\delta}_{mn}\right)\\ &\quad +N\left({\delta}_{ik}{\delta}_{jm}{\delta}_{ln}+{\delta}_{im}{\delta}_{jk}{\delta}_{ln}\right.\\ &\quad+{\delta}_{il}{\delta}_{jm}{\delta}_{kn}+{\delta}_{im}{\delta}_{jl}{\delta}_{kn}\\ &\quad+{\delta}_{ik}{\delta}_{jn}{\delta}_{lm}+{\delta}_{in}{\delta}_{jk}{\delta}_{lm}\\ &\left.\quad +{\delta}_{il}{\delta}_{jn}{\delta}_{km}+{\delta}_{in}{\delta}_{jl}{\delta}_{km}\right).\end{aligned}} $$

(B16)

Next, consider the surface elasticity tensors ${\Gamma}_{\alpha \beta}^{(1)}$ and ${\Gamma}_{\alpha \beta \kappa \lambda}^{(2)}$. Again, it follows from the definition (16) that certain symmetry conditions must be met:

$$ {\Gamma}_{\alpha \beta}^{(1)}={\Gamma}_{\beta \alpha}^{(1)},{\Gamma}_{\alpha \beta \kappa \lambda}^{(2)}={\Gamma}_{\kappa \lambda \alpha \beta}^{(2)}={\Gamma}_{\beta \alpha \kappa \lambda}^{(2)}. $$

(B17)

In general, ${\Gamma}_{\alpha \beta}^{(1)}$ and ${\Gamma}_{\alpha \beta \kappa \lambda}^{(2)}$ can be anisotropic in the surface (where they are defined). For isotropic surfaces, both ${\Gamma}_{\alpha \beta}^{(1)}$ and ${\Gamma}_{\alpha \beta \kappa \lambda}^{(2)}$ should be isotropic. It can be shown (Aris 1962) that ${\Gamma}_{\alpha \beta}^{(1)}$ is isotropic if and only ${\Gamma}_{12}^{(1)}={\Gamma}_{21}^{(1)}=0$ and ${\Gamma}_{11}^{(1)}={\Gamma}_{22}^{(1)}$ and ${\Gamma}_{\alpha \beta \kappa \lambda}^{(2)}$ is isotropic if and only ${\Gamma}_{1112}^{(2)}={\Gamma}_{1222}^{(2)}=0$ and ${\Gamma}_{1111}^{(2)}={\Gamma}_{2222}^{(2)}={\Gamma}_{1122}^{(2)}+2{\Gamma}_{1212}^{(2)}$. This is the case if the surface has a rotation axis of threefold or higher symmetry (Buerger 1963). Therefore, for a {111} surface, which has threefold symmetry, and for a {100} surface, which has fourfold symmetry, the surface stiffness tensors can be written as

$$\begin{aligned} {\Gamma}_{\alpha \beta}^{(1)}&={\Gamma}_{11}{\delta}_{\alpha \beta},{\Gamma}_{\alpha \beta \kappa \lambda}^{(2)}={K}^s{\delta}_{\alpha \beta}{\delta}_{\kappa \lambda}\\ &\quad+{\mu}^s\left({\delta}_{\alpha \kappa}{\delta}_{\beta \lambda}+{\delta}_{\alpha \lambda}{\delta}_{\beta \kappa}-{\delta}_{\alpha \beta}{\delta}_{\kappa \lambda}\right).\end{aligned} $$

(B18)

Special Cases

Films

For the film shown in Fig. 4, the integrals in Eq. 27 can be written as integrals on the top and bottom surfaces of the film. On these surfaces, the integrands in both integrals are constants. Thus, they can be easily carried out to yields Eqs. 41 and 42. Consequently, the non-zero components of the fourth-order tensor ${R}_{ijkl}={C}_{ijkl mn}^{(3)}{M}_{mnpq}{\tau}_{pq}$ are obtained as

$$\begin{aligned} {R}_{1111}&=2{\Gamma}_{11}\eta \left(\frac{C_{112}}{C_{12}}-\frac{C_{111}}{C_{11}}\right),\\ {R}_{1122}&={R}_{1133}{=}{\Gamma}_{11}\eta \left(\frac{C_{123}{+}{C}_{112}}{C_{12}}{-}\frac{2{C}_{112}}{C_{11}}\right),\end{aligned} $$

(C1)

$$\begin{aligned} {R}_{2222}&={R}_{3333}{=}{\Gamma}_{11}\eta \left(\frac{C_{111}{+}{C}_{112}}{C_{12}}{-}\frac{2{C}_{112}}{C_{11}}\right),\\ {R}_{2233}&=2{\Gamma}_{11}\eta \left(\frac{C_{112}}{C_{12}}-\frac{C_{123}}{C_{11}}\right),\end{aligned} $$

(C2)

$$ \begin{aligned}{R}_{2323}&=2{\Gamma}_{11}\eta \left(\frac{C_{155}}{C_{12}}-\frac{C_{144}}{C_{11}}\right),\\ {R}_{1313}&={R}_{1212}{=}{\Gamma}_{11}\eta \left(\frac{C_{144}{+}{C}_{155}}{C_{12}}{-}\frac{2{C}_{155}}{C_{11}}\right),\end{aligned} $$

(C3)

where C _ijk are related to their third-order elastic constants as indicated in Appendix B and η is defined by Eq. C10. The non-zero components of the effective elasticity tensor for the thin film in terms of the Voigt notation can then be obtained from Eq. 40:

$$ {\overline{C}}_{11}={C}_{11}+\frac{ 2{\Gamma}_{11}\eta }{a}\left(\frac{C_{111}}{C_{11}}-\frac{C_{112}}{C_{12}}\right), $$

(C4)

$$\begin{aligned} {\overline{C}}_{12}&={\overline{C}}_{13}{=}{C}_{12}\\ &\quad+\frac{\Gamma_{11}\eta }{a}\left(\frac{2{C}_{112}}{C_{11}}{-}\frac{C_{123}{+}{C}_{112}}{C_{12}}\right),\end{aligned} $$

(C5)

$$\begin{aligned} {\overline{C}}_{22}&={\overline{C}}_{33}={C}_{11}+\frac{1}{a}\bigg[\left({K}^s+{\mu}^s\right)\\ &\left.\quad+{\Gamma}_{11}\eta \left(\frac{2{C}_{112}}{C_{11}}-\frac{C_{111}+{C}_{112}}{C_{12}}\right)\right],\end{aligned} $$

(C6)

$$ \begin{aligned}{\overline{C}}_{23}&={C}_{12}+\frac{1}{a}\bigg[\left({K}^s-{\mu}^s\right)\\ &\quad\left.+2{\Gamma}_{11}\eta \left(\frac{C_{123}}{C_{11}}-\frac{C_{112}}{C_{12}}\right)\right],\end{aligned} $$

(C7)

$$ {\overline{C}}_{44}\,{=}\,{C}_{44}\,{+}\,\frac{1}{a}\left[{\mu}^s\,{+}\,2{\Gamma}_{11}\eta \left(\frac{C_{144}}{C_{11}}\,{-}\,\frac{C_{155}}{C_{12}}\right)\right], $$

(C8)

$$ \begin{aligned}{\overline{C}}_{55}&={\overline{C}}_{66}={C}_{44}+\frac{\Gamma_{11}\eta }{a}\left(\frac{2{C}_{155}}{C_{11}}\right.\\ &\quad\left.-\frac{C_{144}+{C}_{155}}{C_{12}}\right),\end{aligned} $$

(C9)

where η is a nondimensional constant given by

$$ \eta =\frac{C_{11}{C}_{12}}{\left({C}_{11}+2{C}_{12}\right)\left({C}_{11}-{C}_{12}\right)}. $$

(C10)

Note that the positive definiteness of the strain energy density requires C ₁₁ > |C ₁₂|. Thus, η ≥ 0 if C ₁₂ ≥ 0.

Wires

For the wire shown in Fig. 5, the integrals in Eq. 27 can be written as integrals on the lateral surfaces of the wire. On these surfaces, the integrands in both integrals are constants. Thus, they can be easily carried out to yield

$$ {\tau}_{11}={\tau}_{22}={\Gamma}_{11},{\tau}_{33}=2{\Gamma}_{11}, $$

(C11)

$$ {Q}_{1111}{=}{Q}_{2222}{=}{K}^s{+}{\mu}^s,{Q}_{3333}{=}2\left({K}^s{+}{\mu}^s\right), $$

(C12)

$$ {Q}_{1133}{=}{Q}_{2233}{=}{K}^s{-}{\mu}^s,{Q}_{2323}{=}{Q}_{1313}{=}{\mu}^s. $$

(C13)

Consequently, the non-zero components of the forth-order tensor ${R}_{ijkl}={C}_{ijkl mn}^{(3)}{M}_{mnpq}{\tau}_{pq}$ are obtained as

$$ \begin{aligned}{R}_{1111}&={R}_{2222}={\Gamma}_{11}\eta \left(\frac{C_{111}+3{C}_{112}}{C_{12}}\right.\\ &\quad\left.-\frac{2\left({C}_{111}+{C}_{112}\right)}{C_{11}}\right),\end{aligned} $$

(C14)

$$ {R}_{3333}=2{\Gamma}_{11}\eta \left(\frac{C_{111}+{C}_{112}}{C_{12}}-\frac{2{C}_{112}}{C_{11}}\right), $$

(C15)

$$ {R}_{1122}=2{\Gamma}_{11}\eta \left(\frac{C_{112}+{C}_{123}}{C_{12}}-\frac{2{C}_{112}}{C_{11}}\right), $$

(C16)

$$\begin{aligned} {R}_{1133}&={R}_{2233}={\Gamma}_{11}\eta \left(\frac{C_{123}+3{C}_{112}}{C_{12}}\right.\\&\quad\left.-\frac{2\left({C}_{112}+{C}_{123}\right)}{C_{11}}\right),\end{aligned} $$

(C17)

$$ \begin{aligned}{R}_{2323}&={R}_{1313}={\Gamma}_{11}\eta \left(\frac{C_{144}+3{C}_{155}}{C_{12}}\right.\\&\quad\left.-\frac{2\left({C}_{144}+{C}_{155}\right)}{C_{11}}\right),\end{aligned} $$

(C18)

$$ {R}_{1212}=2{\Gamma}_{11}\eta \left(\frac{C_{144}+{C}_{155}}{C_{12}}-\frac{2{C}_{155}}{C_{11}}\right). $$

(C19)

The non-zero components of the corresponding effective elasticity tensor are thus given by

$$\begin{aligned} {\overline{C}}_{11}&={\overline{C}}_{22}={C}_{11}+\frac{1}{a}\bigg[\left({K}^s+{\mu}^s\right)\\ &\quad+{\Gamma}_{11}\eta \left(\frac{2\left({C}_{111}+{C}_{112}\right)}{C_{11}}\right.\\ &\quad\left.\left.-\frac{C_{111}+3{C}_{112}}{C_{12}}\right)\right],\end{aligned} $$

(C20)

$$\begin{aligned} {\overline{C}}_{33}&={C}_{11}+\frac{1}{a}\bigg[2\left({K}^s+{\mu}^s\right)\\ &\quad\left.\,+2{\Gamma}_{11}\eta \left(\frac{2{C}_{112}}{C_{11}}-\frac{C_{111}+{C}_{112}}{C_{12}}\right)\right],\end{aligned} $$

(C21)

$$ {\overline{C}}_{12}{=}{C}_{12}+\frac{2{\Gamma}_{11}\eta }{a}\left(\frac{2{C}_{112}}{C_{11}}{-}\frac{C_{123}+{C}_{112}}{C_{12}}\right), $$

(C22)

$$\begin{aligned} {\overline{C}}_{13}&={\overline{C}}_{23}={C}_{12}+\frac{1}{a}\bigg[\left({K}^s-{\mu}^s\right)\\ &\quad+{\Gamma}_{11}\eta \left(\frac{2\left({C}_{112}+{C}_{123}\right)}{C_{11}}\right.\\ &\quad\left.\left.-\frac{3{C}_{112}+{C}_{123}}{C_{12}}\right)\right],\end{aligned} $$

(C23)

$$\begin{aligned} {\overline{C}}_{44}&={\overline{C}}_{55}={C}_{44}+\frac{1}{a}\bigg[{\mu}^s\\ &\quad+{\Gamma}_{11}\eta \left(\frac{2\left({C}_{144}+{C}_{155}\right)}{C_{11}}\right.\\ &\quad\left.\left.-\frac{C_{144}+3{C}_{155}}{C_{12}}\right)\right],\end{aligned} $$

(C24)

$$ {\overline{C}}_{66}\,{=}\,{C}_{44}{+}\frac{2{\Gamma}_{11}\eta }{a}\left(\frac{2{C}_{155}}{C_{11}}{-}\frac{C_{144}+{C}_{155}}{C_{12}}\right). $$

(C25)

Spherical Particles

$$\begin{aligned} {Q}_{ij kl}&=\frac{4}{3}{K}^s{\delta}_{ij}{\delta}_{kl}+\frac{1}{5}\left({K}^s+6{\mu}^s\right)\bigg({\delta}_{ik}{\delta}_{jl}\\ &\quad\left.+\,{\delta}_{il}{\delta}_{jk}-\frac{2}{3}{\delta}_{ij}{\delta}_{kl}\right),\end{aligned} $$

(C26)

$$ {\tau}_{ij}=2{\Gamma}_{11}{\delta}_{ij}, $$

(C27)

$$\begin{aligned} {R}_{ij kl}&=\frac{2{\Gamma}_{11}}{3K}\left(3L+6M+\frac{8}{3}N\right){\delta}_{ij}{\delta}_{kl}\\ &\quad+\frac{2{\Gamma}_{11}}{3K}\left(3M+4N\right)\bigg({\delta}_{ik}{\delta}_{jl}\\ &\quad\left.+\,{\delta}_{il}{\delta}_{jk}-\frac{2}{3}{\delta}_{ij}{\delta}_{kl}\right).\end{aligned} $$

(C28)