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Surface Properties

▶Nanostructures for Surface Functionalization and

Surface Properties

Surface Tension and Chemical Potentialat Nanoscale

▶ Surface Energy and Chemical Potential at Nanoscale

Surface Tension Effects of Nanostructures

Ya-Pu Zhao and Feng-Chao Wang

State Key Laboratory of Nonlinear Mechanics (LNM),

Institute of Mechanics, Chinese Academy of Sciences,

Beijing, China

Synonyms

Surface energy density; interface excess free energy

Definition

The surface tension is the reversible work per unit area

needed to elastically stretch/compress a preexisting

surface. In other words, it is a property of the surface

that characterizes the resistance to the external force.

Surface tension has the dimension of force per unit

length or of energy per unit area. For nanostructures

with a large surface to volume ratio, the surface tension

effects dominate the size-dependent mechanical

properties.

Overview

Nanostructures have a sizable surface to volume ratio

as compared to bulk materials, which leads its mechan-

ical properties to be quite different from those of bulk

materials [1]. The size-dependent mechanical proper-

ties of nanostructures are generally attributed to the

surface effects, in which surface tension is one of the

most predominant factors.

In the framework of the thermodynamics, the sur-

face (interface) between two phases is first modeled as

a bidimensional geometrical boundary of zero thick-

ness, that is, the mathematical surface, as shown in

Fig. 1a. The physical quantities between the two

phases are discontinuous, and the surface (interfacial)

Surface Tension Effects of Nanostructures 2599 S

S

http://dx.doi.org/10.1007/978-90-481-9751-4_234

http://dx.doi.org/10.1007/978-90-481-9751-4_234

http://dx.doi.org/10.1007/978-90-481-9751-4_276

http://dx.doi.org/10.1007/978-90-481-9751-4_100807

http://dx.doi.org/10.1007/978-90-481-9751-4_100827

tension is a jump in stress. This idealization was then

extended by Gibbs [2], who proposed that the physical

quantities should undergo a smooth transition at the

surface (interface) while the surface is still modeled as

an infinitesimal thin boundary layer, as shown in

Fig. 1b. In order to preserve the total physical proper-

ties of the system, the excess physical properties have

to be assigned to the geometrical surface. The surface

tension, which is defined based on the interface excess

free energy, can be written as

g ¼Z 10

wðyÞ � wA½ �dyþZ 0

�1wðyÞ � wB½ �dy (1)

where w is the free energy distribution of the actual

surface, wA and wB are free energy in the two phases of

A and B.

In some other theoretical models, the surface is

treated as an extended interfacial region with a nonzero

thickness. According to Cahn–Hilliard theory [3], the

surface tension can be derived as

g¼NV

Z 1�1

w0ðcÞþ k dc dy=ð Þ2� cmB� 1� cð ÞmAh i

dy;

(2)

in which NV is the number of atoms per unit volume, c

is one of the intensive scalar properties, such as com-

position or density, w0(c) is the free energy per atom of

a solution of uniform composition c, k reflects the

crystal symmetry, mA and mB are the chemical poten-

tials per atom in the A or B phase. The surface

(interfacial) thickness can be obtained by

l ¼ 2Dce

ffiffiffiffiffiffiffiffiffiffiffiffiffik

Dwmax

r; (3)

where 2Dce ¼ cB�cA is the difference of the uniform

composition in the two phases, Dwmax is the maximum

of the free energy referred to a standard state of an

equilibrium mixture. Cahn–Hilliard model gives

a finite thickness through Eq. 3, which has been used

in many applications [4].

From the standpoint of molecular theory, the sur-

face tension effects arise due to the difference of the

atomic interactions in the bulk and on the surface.

Atoms are energetically favorable to be surrounded

by others. At the surface, the atoms are only partially

surrounded by others and the number of the adjacent

atoms is smaller than in the bulk. Thus the atoms at the

surface are energetically unfavorable. If an atom

moves from the bulk to the surface, work has to be

done. With this view, the surface tension can be

interpreted as the energy required to bring atoms

from the bulk to the surface [5]. Therefore the term

“surface energy density” is often used to when the

surface tension is referred to. For the surface of solid,

Gibbs pointed out that surface tension and surface

energy density are not identical. The surface tension

is the reversible work per unit area needed to

Mathematicalinterface(surface)

Phase B

Phase A

Thickness l

WB

W(y)

WA

x

yy

x

a bSurface Tension Effects ofNanostructures, Fig. 1 An

illustration for the surface

(interface) models. (a) The

mathematical surface of zero

thickness, the physical

quantities are discontinuous.

(b) Gibbs’s surface with an

infinitesimal thickness, the

physical quantities are

continuous. Cahn–Hilliard

model for surface with a finite

thickness is also illustrated

S 2600 Surface Tension Effects of Nanostructures

elastically stretch/compress a preexisting surface. The

surface energy density is the reversible work per unit

area needed to create a new surface. The surface ten-

sion can be positive or negative, while the surface

energy density is usually positive. The surface tension

for liquid surface is a property that characterizes its

resistance to the external force, which is identical to

the surface energy density.

Basic Methodology

Since the surface to volume ratio increases as the

dimension scale decreases, surface tension effects of

nanostructures can be overwhelming. In the absence of

external loading, the surface tension effects would

induce a residual stress field in bulk materials. The

relations between surface stress and surface tension

for small deformations can be described by the

Shuttleworth-Herring equation [6, 7],

sij ¼ gdij þ@g@eij

; (4)

where g is the surface tension, dij is the Kronecher

delta, sij and eij are the surface stress tensor and the

surface strain tensor, respectively. The Shuttleworth-

Herring equation interprets that the difference between

the surface stress and surface tension is equal to the

variation of surface tension with respect to the elastic

strain of the surface.

With the development of computational materials

science, molecular dynamics (MD) simulations are

wildly performed to investigate the surface tension

effects on the mechanical properties of nanostructures.

Especially for the surface elastic constant, atomistic

simulation is almost the only way to get them up to

now. According to the Gibbs’s definition, the surface

tension of a solid is given by g ¼ ES � nEBð Þ A0= ,

where A0 is the total area of the surface considered,

ES is the total energy of a n-layer slab andEB is the bulk

energy per layer of an infinite solid. In cases where the

surfaces of the slab are polar, the electrostatic energy

of the slab contains an energy contribution, Epol, pro-

portional to the substrate thickness and the surface energy

needs corrections. Thus there is an alternative way to

calculate the surface tension g ¼ ES � nEB � Epol

�A0= ,

which does not rely on an exact knowledge of the lattice

or polar energy. MD simulations have identified that the

surface tension induced surface relaxation is proved to

be a dominate factor of the size-dependent mechanical

properties. When the surface stress is negative, the sur-

face relaxation is inward; otherwise, the relaxation is

outward.

Surface stress has been used as an effective molec-

ular recognition mechanism. Surface stresses due to

DNA hybridization and receptor-ligand binding

induce the deflection of a cantilever sensor [8]. The

curvature of bending beam under a surface stress is

governed by Stoney’s formula [9]. Stoney’s formula

serves as a cornerstone for curvature-based analysis

and a technique for the measurement of surface stress,

which is given as follows as a general form for a film/

substrate system,

s ¼ Et2s f

6 1� nð Þ (5)

in which E is the effective Young’s modulus, n is

Poisson’s ratio of the sensor material, ts is the substrate

thickness, f¼ 3Dz/2 L2 is the sensor curvature, L is the

length and Dz is the deflection. The applicability of theabove Stoney’s formula relies on several assumptions,

which are well summarized as the following six:

(1) both the film and substrate thicknesses are small

compared to the lateral dimensions; (2) the film thick-

ness is much less than the substrate thickness; (3) the

substrate material is homogeneous, isotropic, and lin-

early elastic, and the filmmaterial is isotropic; (4) edge

effect near the periphery of the substrate are inconse-

quential and all physical quantities are invariant under

change in position parallel to the interface; (5) all stress

components in the thickness direction vanish through-

out the material; and (6) the strains and rotations are

infinitesimally small. However, the one or several of

above six assumptions can be easily violated in reality,

which is to say that the Stoney’s formula needs to be

revised to fit in the real applications.

To summarize for the solid cases, the behaviors of

nanostructures can be affected significantly by either

of the two distinct parameters, surface tension and

surface stress. The relation between the two parame-

ters can be obtained by the Shuttleworth-Herring equa-

tion. For the surface stress induced deflection of

cantilever sensors, the curvature is by Stoney’s for-

mula. MD simulation is helpful to understand the sur-

face effects of nanostructures and partial results are

comparable to the experiments.


S

For liquid droplets in contact with the surface of

a nanostructure, surface tension is responsible for the

shape of liquid droplets in the equilibrium state, as well

as the dynamics response in wetting and dewetting.

There are several characteristic time which are related

to the surface tension effects, listed in Table 1. Dimen-

sionless number related to the surface tension effects

are listed in Table 2. Surface tension is dependent on

temperature T, concentration of surfactants c, and the

electric field V. The gradient of surface tension caused

by these factors can be described by [10]

dg ¼ @g@T

dT þ @g@c

dcþ @g@V

dV: (6)

To the first order, the dependency of the surface

tension on temperature is given by Guggenheim–

Katayama formula with power index n ¼ 1,

g ¼ g0 1� T=Tcð Þ. Here g0 is a constant for each liquidand Tc is the critical temperature. For the dependency

of the surface tension on concentration c, the surface

tension can be expressed as a linear function of the

concentration, g ¼ g0 1þ b c� c0ð Þ½ �. The solid–liquidsurface tension can be changed by applying a voltage

V, gSL ¼ g0SL � e0eD2d V2, where g0SL is the solid–liquid

surface tension in the absence of the applied voltage,

e0 is the permittivity of vacuum, eD is the relative

permittivity of the dielectric layer with a thickness d

separating the bottom electrode from the liquid.

The wetting properties of a solid surface can be

described by the introduction of the contact angle,

which is defined as the angle at which the liquid–

vapor interface meets the solid surface. The contact

angle is affected by various factors, including the sur-

face tension, the line tension, the applied voltage as

well as the molecular interactions between the liquid

and solid surfaces. It is proposed that the dependence of

the contact angle on these factors can be represented by

the generalized Young’s equation, which has a form of

cos y ¼ cos y0 �t

gLVRþ e0eD2dgLV

V2 þ A

12ph2gLV; (7)

where

cos y0 ¼gSV � gSL

gLV(8)

is the classical Young’s equation, y0 is the equilibriumcontact angle (also the Young contact angle); The sub-

scripts S, L, and V denote solid, liquid, and vapor,

respectively. The second term on the right-hand side

of Eq. 7 is related to the line tension. t is the line

tension and R is the radius of the contact area. For

a more generalized case, R can be replaced by 1/k, inwhich k is the geodesic curvature of the triple contact

line. �e ¼ e0eD2gLVd

V2 is defined as the dimensionless

electrowetting number. The last term in Eq. 7measures

the strength of the effective interaction energy com-

pared to surface tension. The interaction energy is

related to the disjoining pressure P(h), which can be

obtained by WðhÞ ¼R1h P h0ð Þdh0, where h is the film

thickness and A is the Hamaker constant [11, 12]. As

discussed in the following paragraphs, each term

would be illustrated in detail.

If only the classical Young’s equation is referred to,

the wetting properties of the surface can be obtained

directly if the three tensions are known. The spreading

parameter S determines the type of spreading, which is

defined as S ¼ gSV � gSL þ gLVð Þ. If S > 0, the liquid

spreads on the solid surface and forms a macroscopic

liquid layer covers the whole solid surface; it is the

case for complete wetting. If S < 0, the contact angle

0� < y0 < 180�, which means the liquid forms a droplet

with a finite contact angle; it is the case for speak of

partial wetting. The wettability of the solid surface

could be distinguished by the contact angle y0, asshown in Fig. 2. If 0� � y0 < 90�, the solid substrate

is hydrophilic. If 90� < y0� 180�, the solid substrate ishydrophobic. Especially, if 150� < y0� 180�, the solidsubstrate is superhydrophobic.

Surface Tension Effects of Nanostructures,Table 1 Characteristic time related to the surface tension effects

Name Expression Meaning

Capillary

characteristic

time

tc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffim gLV=

pCharacteristic time

derived from the droplet

mass and the liquid–

vapor surface tension

Lord Rayleigh’s

periodtp ¼

p4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffird30 gLV=

qThe period of a free

droplet in free oscillation

Lord Rayleigh’s

characteristic

time

tR �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirl3 gLV=

pCharacteristic time of

droplet dynamics

Viscous

characteristic

time

tvis � �l gLV= Characteristic time

related to the liquid

viscosity

m mass of the droplet, d0 diameter, r density, � viscosity,

l characteristic length.


The physics behind wettability is that, the solid

surfaces have been divided into high energy and low-

energy types [11, 12]. The relative energy of a solid has

to do with the bulk nature of the solid itself. (1) High-

energy surfaces such as metals, glasses, and ceramics

are bound by the strong chemical bonds, for example,

covalent, ionic, or metallic, for which the chemical

binding energy Ebinding is of the order of 1 eV.

The solid–liquid interface tension is given by

gSV � Ebinding a2�� 0:5� 5N m= , in which a2 is the

effective area per molecule. Most high-energy surfaces

are hydrophilic, some can permit complete wetting.

(2) For low-energy surfaces, such as weak molecular

crystals (bound by van der Waals forces or in some

special cases, by hydrogen bonds), the chemical bind-

ing energy is of the order of kBT. In this category, the

surface tension is gSV � kBT a2�� 0:01� 0:05N m= .

Depending on the type of liquid chosen, low-energy

surfaces can be either hydrophobic or hydrophilic.

When the liquid droplet comes to the nanoscale, the

classical Young’s equation seems to be not applicable,

since it has been derived for a triple line without

consideration of the interactions near the triple contact

line. The molecules close to the triple line experience

a different set of interactions than at the interface [10].

To take into account this effect, the “line tension” term

has been introduced in the generalized Young’s equa-

tion. A sketch of line tension is shown in Fig. 3.

Line tension was first introduced by Gibbs [2]. The

line tension t was introduced as an analogue of the

surface tension:

F¼ gLV

ZS0dS0 þ gLV �Wð Þ

ZS dS þ t

ZC dC ; (9)

where F, S’, S∗, and C∗ are the free energy, the free

surface, the adhering interface, and the contact line of

the droplet. W ¼ gLV þ gSL � gSV is the adhesion

potential. The line tension, depending on the radius R

of the contact line, should be connected to the classical

Young’s equation to include the interactions near the

triple contact line. In the three-phase equilibrium

Surface Tension Effects of Nanostructures, Table 2 Dimensionless number related to the surface tension effects

Name Expression Meaning

Adhesion number Na ¼ cos ya � cos yr ya and yr are the advancing and receding angle.

Bond number Bo ¼ rgl2 gLV= The capillary length lc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigLV rgð Þ=

pcan be determined

Capillary number Ca ¼ �v gLV= Ca ¼ On �ffiffiffiffiffiffiffiWep

; the capillary velocity v ¼ gLV �= can be obtained

Deborah number De ¼t tR= ¼ tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirl3 gLV=

p. t is the relaxation time, Lord Rayleigh characteristic time can be derived

Elasto-capillary number Ec ¼ tgLV �lð Þ= t is the relaxation time, viscous characteristic time can be obtained

Electrowetting number �e ¼ e0eDV2 2dgLVð Þ= The ratio of the electrostatic energy to the surface tension.

Laplace number La ¼ 1 On=ð Þ2 See Ohnersoge number

Marangoni numberMa ¼� dgLV

dT

L � DT�a

Thermal surface tension force divided by viscous force

Ohnersorge number On ¼ �ffiffiffiffiffiffiffiffiffiffiffirlgLV

p�Viscousforceffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Inertiaforce Surfacetensionp

Weber number We ¼ rv2l gLV= The inertia force compared to its surface tension

g gravitational acceleration, a thermal diffusivity, DT temperature difference.

Complete wetting

Vapor Vapor Vapor

Liquid

Liquid

Liquidq0 = 0°

q0 < 90° q0 > 90°gSV gSVgSL gSL

gLVgLV

Solid substrate

Partial wetting--Hydrophilic

Partial wetting--Hydrophobic

a b c

Surface Tension Effects of Nanostructures, Fig. 2 Wetting properties of the surface


S

systems when R decreases, the contact angle y will

increase at t > 0 and will decrease at t < 0. However,

in accordance to the mechanical equilibrium stability

conditions, while the surface tension can only be pos-

itive, the line tension can have either positive or neg-

ative values. The characteristic length of this problem

l ¼ tj j gLV= (|t| < 10�9 N, gLV � 0.1 N/m for water)

ranges from 10�8 to 10�6 m, which means small drop-

let (with typical dimension of |l|) should appreciate

the line tension effect. The line tension can be indi-

rectly measured through the contact angle, t �4d

ffiffiffiffiffiffiffiffiffiffiffiffiffigSVgLVp

cot y, in which d denotes the average dis-

tance between liquid and solid molecules. Thus the line

tension is negative for an obtuse contact angle, while it

is positive for an acute contact angle.

In the presence of a charged interface, which can be

achieved by applying a direct or alternating-current

electric field, the wetting properties of solid surface

will be modified. The physics describing the electric

forces on interfaces of conducting liquids and on triple

contact lines is called “electrowetting” [10]. In the year

of 1875, Gabriel Lippmann observed the capillary

depression of mercury in contact with an electrolyte

solution could be varied by applying a voltage between

the mercury and electrolyte. This phenomenon is

called electrocapillarity, which is the basis of modern

electrowetting. Then, the idea was developed to isolate

the liquid droplet from the substrate using a dielectric

layer in order to avoid electrolysis. This concept has

subsequently become known as electrowetting on

dielectric (EWOD) and involves applying a voltage

to modify the wetting behavior of a liquid in contact

with a hydrophobic, insulated electrode. When an

electric field was applied to the system (as shown in

Fig. 4), electric charges gather at the interface between

the conductive electrodes and the dielectric material;

the surface becomes increasingly hydrophilic (wetta-

ble), the contact angle is reduced and the contact line

moves. The change in contact angle over the buried

electrodes can be evaluated by the Lippmann–Young

equation cos y ¼ cos y0 þ e0eDV2 2dgLVð Þ= . The para-

bolic variation of cos y in the Lippmann–Young

Surface Tension Effects ofNanostructures,Fig. 3 Illustration of the line

tension t

BottomElectrode

+−

+−

Dielectric Layer

Electrode

a

b

Droplet

Surface Tension Effects of Nanostructures,Fig. 4 Illustration of EWOD. The external voltage is applied

between a thin electrode (the Pt wire) and the bottom electrode

(typically the indium-tin-oxide glass). Partially wetting liquid

droplet in the absence of an applied voltage (a) and after the

voltage is applied (b)


equation is only applicable when the applied voltage

lies below a threshold value. If the voltage is increased

above this threshold, then contact angle saturation

starts to occur and cos y eventually becomes indepen-

dent of the applied voltage.

The physics of why a liquid film wets or dewets is

found in the derivative of the effective interfacial

potential W ¼ gLV þ gSL � gSV with respect to film

thickness h, called the disjoining pressure P(h) [11,12]. PðhÞ ¼ dWðhÞ dh= , where the surface area

remains constant in the derivative. The effective inter-

face potential W(h), which arises from the interaction

energies of molecules in a film being different from

that in the bulk, is the excess free energy per unit area

of the film. If the interactions between the molecules in

the film and the solid substrate are more attractive than

the interactions between molecules in the bulk liquid,

W(h) > 0. Consequently, a liquid film with a thickness

in a range whereP(h)> 0 can lower its free energy by

becoming thicker in some areas while thinning in

others, that is, by dewetting. When P(h) < 0, wetting

or spreading occurs.

The van der Waals interaction wðrÞ / 1 r6�

includes all intermolecular dipole–dipole, dipole–

induced dipole, and induced dipole–induced dipole

interactions. Performing a volume integral over all

molecules present in the two half spaces bounding

the film one finds a corresponding decay P(h) �A/6ph3 [11, 12], where the Hamaker constant

A (�10�19 J) gives the amplitude of the interaction.

In the “attractive” case, in which the layer tends to thin,

A < 0. In the “repulsive” case, in which the layer tends

to thicken, A > 0. Because of the disjoining pressure,

there is a precursor film ahead of the nominal contact

line of the liquid droplet [13]. The thickness of the

precursor film can be defined by a molecular length

[11, 12] hPF �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 6pgLV=

p, which is on the order of

several A.

Key Research Findings

Elastic Models for the Nanostructures

Gurtin and Murdoch established the theoretical frame-

work of the surface elasticity under the classical theory

of membrane [14]. Recently, studies [15] have shown

that, even in the case of infinitesimal deformations, one

should distinguish between the reference and the cur-

rent configurations; otherwise the out-plane terms of

surface displacement gradient, associated with the sur-

face tension, may sometimes be overlooked in the

Eulerian descriptions, particularly for curved and

rotated surfaces. By combining elastic models for sur-

face and bulk, the size-dependent elastic and properties

of nanomaterials have been investigated. Usually, in

the absence of external mechanical or thermal load-

ings, the surfaces of a nanostructure will be subjected

to residual surface stresses, and an elastic field in the

bulk materials will be induced by such residual surface

stresses induce from the point of view of equilibrium

conditions. This self-equilibrium state without external

loadings is usually chosen as the reference configura-

tion, from which nanostructures will deform (see

Fig. 5). That is to say, the bulk will deform from the

residual stress states. However, in the prediction of

elastic and thermoelastic properties of nanostructures,

the elastic response of the bulk is usually described by

classical Hooke’s law, in which the aforementioned

residual stress was neglected in the existing literatures.

Considering a bulk material with the surface prop-

erties aforementioned, the surface tension would

induce a stress field in the bulk. According to

Young–Laplace equation, the surface tension will

result in a nonclassical boundary condition. The

boundary condition together with the equations of

classical elasticity forms a coupled system of field

equations to determine the stress distribution. To

solve a problem considering the surface properties,

the surface model and the bulk model are established

separately and using the Young–Laplace equation to

bridge the two models together.

Here, an example named surface elasticity would be

used to show how to establish a model combined the

surface and bulk together. Assuming the surface to be

isotropic and homogeneous, the constitutive relations

of the surface in the Lagrangian description can be

written as [15]

Ss ¼ g 0I0 þ g 0 þ g 1 �

trEsð ÞI0 � g 0ð�H0su0Þ

þ g1Es þ g 0FðoÞs ; (10)

where Ss is the first kind Piola–Kirchhoff stress of the

surface, I0 is the identity tensor on the tangent planes

of the surface in the reference configuration; the con-

stants g 0, g 1, and g1 are the surface tension and the

surface Lame moduli; Es, �H0su0, and FðoÞs denote,

respectively, the surface strain tensor, the in-plane


S

part of surface displacement gradient and the out-plane

term of surface deformation gradient. In view of the

importance of the linearization of the general constitu-

tive equations, the linear elastic constitutive relations

of the bulk with residual stresses can be written as

follows:

S ¼ TR þ uH � TR þ ltr Eð Þ1þ 2mE; (11)

where S is the first Piola–Kirchhoff stress, TR is the

residual stress in the reference configuration, uH is the

displacement gradient calculated from the reference

configuration, E is the infinitesimal strain, and l and

m are material elastic constants.

In the absence of external loading, surface tension

will induce a compressive residual stress field in the

bulk of the nanoplate and there may be self-equilibrium

states which correspond to plate self-buckling. The self-

instability of nanoplates is investigated and the critical

self-instability size of simplify supported rectangular

nanoplates is proposed. The critical size for self-

buckling is b ¼ phffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia�2A þ 1 �

Eh 24g 0 1� n2ð Þ� �q

,

where b and h are the width and thickness of the

nanoplate, respectively; E and ndenote the Young’s

modulus and Poisson’s ratio, aA ¼ l b= is the aspect

ratio [15].

Surface Tension Effects on the Mechanical

Properties of Nanostructures

Manymodels are developed to relax one or some of the

six above assumptions to extend Stoney’s formula to

a more generalized and realistic application. For exam-

ple, the effects such as axial force, the damaged/

nonideal interface effect and gradient stress, which

violates one or several of the above assumptions, are

analyzed and the extended/revised Stoney’s formulas

are given. Surface stress physically is a distributed one

and this characteristic is not emphasized in many stud-

ies. The analysis by Zhang et al. shows that the

Stoney’s formula is obtained by assuming the influ-

ence of surface stress as a concentrated moment

applied at the free end of a cantilever beam [16].

However, if the influence of surface stress is modeled

as a distributed axial load and bending moment, the

following nonlinear governing equation is obtained:

EId4z

dx4� sbðL� xÞ d

2z

dx2þ sb

dz

dx¼ 0; (12)

in which EI is the cantilever effective bending stiff-

ness; b, L, and z are the beam width, length, and

deflection, respectively. To solve the above nonlinear

equation with the (given) boundary conditions at the

two ends of the beam is a two-point-boundary value

Surface Tension Effects ofNanostructures, Fig. 5 The

choice of the reference

configuration: the stressed

state


problem [16], which is rather difficult. The semi-

analytical series solutions can more or less ease the

difficulty of solving Eq. 12. One implication of the

Stoney’s formula is that because the beam curvature

is constant, the beam deflection under a surface stress

is an arc of a circle (or a parabola if the approximate

curvature definition is used). There is no mechanism

to guarantee such kind of deflection. In general

the curvature of a beam under a surface stress is

not a constant, which has been verified in the

experiments.

The Stoney’s formula has been used to explain

recent experimental results, in which a hybrid device

based on a microcantilever interfaced with bacterio-

rhodopsin (bR), undergoes controllable and reversible

bending when the light-driven proton pump protein,

bR, on the microcantilever surface is activated by

visible light [17]. It should be pointed that the Young’s

modulus of a nanostructure is size-dependent, that is, it

would be enhanced or softened with decreasing the

size of the nanostructure, which is generally attributed

to the surface effects. Surface tension is one of the most

important factors that cause the size effects of the

Young’s modulus of a nanostructure. The surface ten-

sion can be introduced into mechanical model via

energy method. Using the relation of energy equilib-

rium, the effective elastic modulus of nanobeams are

dependent on the surface tension [18].

Surface Tension Effects Induced the Deformation

of Nanostructures

The classical Young’s equation describes the equilib-

rium of forces in the direction parallel to the solid

surface, while the vertical component of liquid–vapor

interfacial tension is ignored. That is, there is a net

force gLV sin y acting normal to the smooth solid sur-

face at the solid–liquid–vapor contact line. Due to the

unbalance force, there will be a surface deformation, as

shown in Fig. 6. Indeed, several decades ago, surface

deformation of semi-infinite solid was theoretically

analyzed with the physical assumption that the liq-

uid–vapor has a finite thickness (maybe at the order

of tens nanometers) and the liquid–vapor interfacial

tension acts uniformly in this region. Their research

suggests there is a wetting ridge at the three-phase

contact line. Later, Shanahan and Carre used dimen-

sional analysis to characterize the maximum height at

the order of gLV G= , where G is the shear modulus of

solid [19]. For the material widely used at that time

were very rigid (at the order of at least 100 GPa), such

a deformation is too small to be considered. However,

to meet with the rapid development of microelectro-

mechanical systems (MEMS) and nanoelectrome-

chanical systems (NEMS), polydimethylsiloxane

(PDMS) is widely fabricated to channels or membrane,

which has at least one dimension on the order of sub-

millimeters or even nanometers. The surface deforma-

tion might no longer be neglected. Moreover, it should

be noted that whether the theoretical solution for semi-

infinite case can be extended to the case of thin flexible

membrane. Recently, Yu and Zhao considered the

deformation of thin elastic membrane induced by ses-

sile droplet and gave a theoretical solution correspond-

ingly [20]. There are two important conclusions. The

first is that there exists a saturated membrane thickness

at the order of millimeter, if the solid is thicker than

this, it can be taken regard as semi-infinite; otherwise,

it is better to consider the effect of membrane thick-

ness. The second is that if the membrane has a very low

Young’s modulus (for example, on the order of MPa or

much less), the effect of membrane thickness will

become significant.

Apart from theoretical analysis, experimental

investigations on surface deformation induced by

droplet have also been reported. Because of the surface

resolving ratio, it is difficult to get the detailed

gLV

gLVcosq

gLVsinq

gSVgSL

q

ZY

X

Surface Tension Effects ofNanostructures,Fig. 6 Sketch of deformation

of PDMS membrane induced

by a water droplet


S

information of surface deformation at the contact line.

Moreover, there might be a highly stressed zone near

the contact line so that such a question should be

studied further when necessary.

Surface Tension Effects of Wetting at the

Three-Phase Contact Line

When considering the surface tension effects at the

three-phase contact line, there is a famous paradox

named Huh–Scriven paradox. It was first pointed out

by Huh and Scriven [21] that there is a conflict between

the moving contact line and the conventional no-slip

boundary condition between a liquid and a solid. The

interface meets the solid boundary at some finite con-

tact angle y. Owing to the no-slip condition, the fluid atthe bottom moves with constant velocity U and viscos-

ity �, while the flux through the cross section is zero.

The energy dissipation per unit time and unit length of

the contact line is obtained, _E � ��U2 ln L 0=ð Þ, whereL is an outer length scale like the radius of the spread-

ing droplet. Stresses are unbounded at the contact line,

and the force exerted by the liquid on the solid

becomes infinite. The energy dissipation is logarithmi-

cally diverging, “not even Herakles could sink a solid”

(Fig. 7)

In reality, dynamic wetting occurs at a finite rate

with changes in the wetted area and liquid shape. These

processes are thermodynamically irreversible and

therefore dissipative. But, the energy dissipation is

finite. There are two typical theories in identifying

the effective channel of energy dissipation for small

Capillary and Reynolds number. One of the two

approaches is the hydrodynamic theory emphasizes

energy dissipation caused by viscous flow within the

wedge of liquid near the moving contact line. The other

is the molecular kinetic theory emphasizes energy

dissipation caused by of attachment (or detachment)

of fluid molecules to (or from) the solid surface [22].

The Huh–Scriven paradox is raised from four ideal

assumptions summarized as: incompressible Newto-

nian fluid, smooth solid surface, impenetrable liquid/

solid interface, and no-slip boundary. Hence, the typ-

ical methods proposed to relieve the dynamical singu-

larity near the contact line are the precursor film,

surface roughness, diffuse interface, and nonlinear

slip boundary, aiming the four assumptions respec-

tively, as shown in Fig. 8.

Examples of Application

Nanostructures of Silicon Used as the Anode

Material of Lithium Ion Batteries

Rechargeable lithium ion batteries become the most

suitable energy carrier for portable electro-equipments,

electromobiles, and high performance computing, not

only for the high-energy density and low cost, but also

for the environmental needs for energy storage. Silicon

hU2

2h

qE~ –· L

rtrq

0

( c cos q – d sin q)

In

=

Moving Contact Line

Water

Surface Tension Effects ofNanostructures, Fig. 7 Huh

and Scriven’s paradox: Not

even Herakles could sink

a solid


is selected as a promising anode material of the lithium

ion batteries due to the high-energy density (about

4,200 mAhg�1). Nevertheless, a loss of electrical con-

tact due to the fracture and crack of the bulk silicon

induced by huge volume change (�400%) in charging

and discharging cycles hinders its applications.

In recent years, nanostructured silicon is investigated

as the material for electrode effectively circumvented

the fragmentation, since surface tension effects of

the nanostructured silicon on diffusion induced

stresses would have much effect on the stress distribu-

tion [23].

Surface Tension Effects Induced the Deflection of

the Cantilever Sensor

Surface stresses scale linearly with dimension and

surface to volume ratio increases as micro/nanostruc-

ture scale decreases. Therefore, surface stress can be

very important in microstructures in the size domain of

MEMS/NEMS. Surface stress has been used as an

effective molecular recognition mechanism. For

a microcantilever used in a bioactuator, some bioma-

terials and biomolecules, which are immobilized on

the microcantilever surface, are used to convert chem-

ical energy into mechanical energy. When excitations

Precursor film Diffuse layer Slippage

Roughness

Molecular kinetic theory Ridge induced by γ⊥

γ

ΔP

Trapped nanobubbles

Surface Tension Effects ofNanostructures,Fig. 8 Possible mechanisms

to solve the Huh and Scriven’s

paradox

Target Molecule

Probe Molecule

Cantilever Beam

Target Binding

Deflection, Δz

a

b

Surface Tension Effects ofNanostructures,Fig. 9 Schematic illustration

of the deflection of a cantilever

sensor due to the surface stress

induced by surface tension

effects [8, 17]. (a) The

cantilever is functionalized on

one side with the probe

molecules. (b) After the target

binding, the cantilever bends

and the deflection can be

measured


S

such as DNA hybridization and receptor-ligand bind-

ing are applied, the microcantilever generates a

nanomechanical deflection response due to the surface

tension effects, as shown in Fig. 9.

Surface Tension Effects Used in the Application of

Electrowetting

One particularly promising application area for

electrowetting is the manipulation of individual drop-

lets in digital microfluidic systems. Applications range

from “lab-on-a-chip” devices to adjustable lenses and

new kinds of electronic displays. Besides, electrowetting

has been used for the application to displays showing

video content since the switching speed is very high

(only a few milliseconds) [10].

Summary

The surface tension effects become particularly pre-

dominant in nanostructures since the surface to volume

ratio is sizable. In this entry, the surface tension effects

that lead to the size-dependent mechanical properties

of nanostructures are considered. Some key

research findings related to this issue were listed

and several examples of application were given,

which are expected to be helpful to readers. Under-

standing and controlling these surface tension effects is

a basic goal in the design and application of

nanodevices.

Cross-References

▶Disjoining Pressure and Capillary Adhesion

▶Electrowetting

▶Nanoscale Properties of Solid–Liquid Interfaces

▶ Surface Energy and Chemical Potential at

Nanoscale

▶Wetting Transitions

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surface tension and chemical potential at nanoscale

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