薄膜资料(Thin film Material)连续送第5部分!

xx_912 · 发表于 2006-5-5 19:56:50

1.2.4 Example: Thermal barrier coatings
The thermal barrier coating (TBC) system is a multilayer arrangement introduced
to thermally insulate metallic structural components from the combustion gases in
gas turbine engines. The design of TBCs along with appropriate internal cooling
of the high-temperature metallic components has facilitated the operation of gas
turbine engines at gas temperatures well in excess of the melting temperature of
the turbine blade alloy. This thermal protection system, which reduces the surface
temperature of the alloy by as much as 300 ±C, leads to better engine e±ciency,
performance, durability and environmental characteristics.
The performance requirements for TBCs are stringent. The selection of
materials for TBCs and the design of the layered coating structure inevitably requires
consideration of highly complicated interactions among such phenomena and
processes as phase transformation, microstructural stability, thermal conduction,
di®usion, oxidation, thermal expansion mismatch between adjoining materials, radiation,
as well as damage and failure arising from interface delamination, ¯lm
buckling, subcritical fracture, foreign-object impact, erosion, thermal and mechanical
fatigue, inelastic deformation and creep. Summaries and reviews of such issues
for TBCs have been reported by Evans et al. (2001) and Padture et al. (2002).
A representative TBC system for a gas turbine engine comprises four layers:
(a) a metallic substrate, that is, the turbine blade itself, (b) a metallic interlayer or
bondcoat, (c) a thermally grown oxide (TGO), and (d) a ceramic outerlayer or topcoat.
A schematic of the turbine blade along with a scanning electron micrograph
of the cross-section of a layered TBC coating is shown in Figure 1.7.
The turbine blade is commonly made of a nickel-base or cobalt-base super1.2
Film deposition methods 15
Fig. 1.7. Illustration of the thermal-barrier coated turbine blade which is air-cooled
internally along hollow channels. The outer surface of the blade is coated for
thermal protection from the hot gases so that there exists a temperature gradient
through the cross-section of the TBC. Also shown is a scanning electron micrograph
of the cross-sectional view of the coating layers which comprise a ceramic topcoat
deposited by electron-beam PVD, an alumina TGO layer, and a NiCrAlY bondcoat
on a nickel-base superalloy substrate. Reprinted with permission from Padture et
al. (2002).)
alloy which is investment-cast as a single crystal or polycrystal. The bondcoat,
typically 75 to 150 ¹m in thickness and made of an oxidation-resistant alloy of
NiCrAlY or NiCoCrAlY, is deposited onto the substrate using plasma spray or
electron-beam PVD. In some cases, the bondcoats are deposited by electroplating
along with either di®usion-aluminizing or CVD with layers of Ni and Pt aluminides.
Occasionally, the bondcoats consist of sublayers of di®erent phases or compositions.
The bondcoat is a critical component of the TBC system that determines the spallation
resistance of the TBC.
The oxidation of the bondcoat at an operating temperature as high as 700 ±C
results in a 1 to 10 ¹m thick TGO layer between the bondcoat and the topcoat.
This oxidation process is aided by the transport of oxygen from the surrounding hot
gases in the engine environment through the porous TBC top layer. The TGO layer
is also commonly engineered to serve as a di®usion barrier so as to suppress further
oxidation of the bondcoat; for this purpose, the in-situ formation of a uniform,
defect-free ®-Al2O3 TGO interlayer is facilitated by controlling the composition of
the bondcoat.
Y2O3-stabilized Zr2O3 (YSZ), typically with a Y2O3 concentration of 7 to 8
wt%, is the common material of choice for the ceramic topcoat in light of its many
desirable properties as a TBC (Padture et al. 2002):
¡ the thermal expansion coe±cient of YSZ has a high value of approximately
16 Introduction and Overview
11 £10¡6=±C, which is closer to that of the metallic layer beneath it (» 14 £ 10¡6=±C) than to that of most ceramic materials. Consequently, the
stresses generated by thermal expansion mismatch with the underlying layers
during the thermal cycles generated by the operation of the turbine engine
would be minimized;
¡ YSZ has a very low thermal conductivity of approximately 2.3 W/m¢K at
1000 ±C when fully dense;
¡ the low density of YSZ, typically about 6:4 £ 103 kg/m3, improves the performance
of the rotating engine component in which it is used as a coating;
¡ the high melting temperature of approximately 2700 ±C makes YSZ a desirable
thermal barrier material;
¡ with a myriad of available processing methods, the YSZ topcoat can be
deposited with controlled pore content and distribution in a such a way that
it can be made compliant with an elastic modulus of approximately 50 GPa;
and
¡ YSZ has a hardness of roughly 14 GPa which renders it resistant to damage
from foreign object impact and erosion.
The two most common methods of producing the TBC topcoat entail air
plasma spray (see Section 1.2.3) and electron-beam physical vapor deposition, EBPVD
(see Section 1.2.1). The former deposition method produces 15 to 25% porosity
whereby low values of thermal conductivity and elastic modulus result. The
spray coating, typically 300 to 600 ¹m thick, contains pancake-shaped splats with
a diameter of 200{400 ¹m and thickness of 1{5 ¹m. The pores and cracks along
the inter-splat boundaries are generally oriented parallel to the interface with the
bondcoat and normal to the direction of heat °ow, a consequence of which is the low
thermal conductivity of 0.8{1.7 W/m¢K. The air plasma spray method provides an
economical means for the large-scale production of TBCs. However, the structural
defects inherent in this process and the rough interface between the sprayed coating
and the material beneath it make this deposition technique more suitable for
less critical parts. The EB-PVD deposited TBCs, on the other hand, are typically
125 ¹m thick and are more durable and costly compared to the plasma-sprayed
coatings. They are engineered to comprise the following microstructural features:
(a) an equiaxed structure of YSZ grains with a diameter of 0.5{1.0 ¹m near the interface
with the bondcoat, (b) columnar grains of YSZ, 2{10 ¹m in diameter which
extend from the equiaxed grains to the top surface of the coating, with the boundaries
between the columnar grains amenable to easy separation to accommodate
thermal stresses that develop during service, and (c) nanometer-size pores inside
the columnar grains.
1.3 Modes of film growth by vapor deposition 17
1.3 Modes of film growth by vapor deposition
There is enormous variation in the microstructures of ¯lms formed by deposition
of atoms on the surfaces of substrates from vapors. Final structures
can range from single crystal ¯lms, through polycrystalline ¯lms with
columnar or equiaxed grains, to largely amorphous ¯lms. Some materials
can be deposited in ways that yield any of these structures, with the ¯nal
microstructure depending on the materials involved, the deposition method
used and the environmental constraints imposed. The purpose in this section
is to discuss some general ideas in ¯lm growth by vapor deposition that
transcend issues of ¯nal microstructure. The discussion is largely descriptive,
and it is couched in the terminology of thermodynamics. The principles
of thermodynamics provide powerful tools for deciding whether or not some
particular change in a material system can occur. On the other hand, in
those cases in which the change considered can indeed take place, thermodynamics
is silent on whether or not it does take place and, if it does occur,
on how it proceeds. Nonetheless, the thermodynamic framework provides
a basis for establishing connections between deposition circumstances and
¯lm formation. Progress toward understanding physical processes of ¯lm
growth are discussed in depth by Tsao (1993), Pimpinelli and Villain (1998)
and Venables (2000).
1.3.1 From vapor to adatoms
In this section, the factors that control the very early stages of growth of a
thin ¯lm on a substrate are described in atomistic terms. The process begins
with a clean surface of the substrate material, which is at temperature Ts,
exposed to a vapor of a chemically compatible ¯lm material, which is at the
temperature Tv. To form a single crystal ¯lm, atoms of the ¯lm material in
the vapor must arrive at the substrate surface, adhere to it, and settle into
possible equilibrium positions before structural defects are left behind the
growth front. To form an amorphous ¯lm, on the other hand, atoms must
be prevented from seeking stable equilibrium positions once they arrive at
the growth surface. In either case, this must happen in more or less the
same way over a very large area of the substrate surface for the structure
to develop. At ¯rst sight, this outcome might seem unlikely, but such ¯lms
are produced routinely.
Atoms in the vapor come into contact with the substrate surface where
18 Introduction and Overview
















Fig. 1.8. Schematic showing the atomistics of ¯lm formation on substrates.
they form chemical bonds with atoms in the substrate. The temperature of
the substrate must be low enough so that the vapor phase is supersaturated
in some sense with respect to the substrate, an idea that will be made more
concrete below. There is a reduction in energy due to formation of the bonds
during attachment. Some fraction of the attached atoms, which are called
adatoms, may return to the vapor by evaporation if their energies due to
thermal °uctuations are su±cient to occasionally overcome the energy of
attachment, as suggested in the schematic diagram in Figure 1.8. To make
the discussion a bit more speci¯c, a simple hexagonal close-packed crystal
structure is assumed for convenience in counting bonds.
It has already been recognized that, for ¯lm growth to be possible, it is
necessary that the vapor in contact with the growth surface is supersaturated
with respect to the substrate at its temperature Ts. For a homogeneous
crystal at some temperature that is in contact with its own vapor at the
same temperature, the equilibrium vapor pressure pe of the system is de¯ned
as the pressure at which condensation of vapor atoms onto the solid surface
and evaporation of atoms from the surface proceed at the same rate. At
equilibrium, the entropic free energy per atom in the vapor equals the free
energy per atom in the interior of the crystal. The lower internal energy of
atoms in the interior of the crystal compared to those in the vapor, due to
chemical bonding, is o®set by the lower entropic energy within the crystal.
For net deposition on the substrate surface, it is essential that the pressure
p in the vapor exceeds the equilibrium vapor pressure pe at the substrate
temperature, that is, the vapor must be supersaturated. For the pressure
p, the entropic free energy per atom of the vapor, over and above the free
energy at pressure pe, is estimated as the work needed per atom to increase
the vapor pressure from pe to p at constant temperature. According to the
ideal gas law, the result is kTv ln(p=pe) where k = 1:38£10−23 J/K= 8:617£
1.3 Modes of film growth by vapor deposition 19
10−5 eV/K is the Boltzmann constant and Tv is the absolute temperature
of the vapor. If the vapor becomes supersaturated, a free energy di®erence
between the vapor and the interior of the crystal exists, providing a chemical
potential for driving the advance of the interface toward the vapor. As the
interface advances in a self-similar way, a remote layer of vapor of some mass
is converted into an interior layer of crystal of the same mass. The interface
does not advance or retreat when these energies are identical.
In ¯lm deposition, the situation is often complicated by the fact that
the vapor and the substrate are not phases of the same material, and by
the fact that the temperature of the substrate is usually lower than that of
the vapor. The de¯nition of equilibrium vapor pressure is not so clear in
this case. However, in most cases there will be some level of vapor pressure
below which deposition of ¯lm material onto the growth surface will not
occur; this serves as an operational de¯nition of pe for ¯lm growth.
Once adatoms become attached to the substrate, their entropic free
energy is reduced from that of the vapor. The adatoms form a distribution
on the substrate surface having the character of a two-dimensional vapor.
The deposited material it is believed to thermalize quickly and to take on the
temperature Ts. On a crystal surface, there is some density ½ad of adatoms
that is in equilibrium with a straight surface step or ledge bordering a partial
monolayer of ¯lm atoms, that is, for which the step neither advances nor
recedes. The step is a boundary between two phases in a homogeneous
material system. Consequently, the notion of equilibrium vapor pressure
or equilibrium vapor density ½e
ad can be invoked in an analogous way. For
¯lm growth to occur by condensation, the free energy per atom of the twodimensional
gas must exceed the free energy of a fully entrained surface
atom by the amount Eph = kTs ln(½ad=½e
ad).
1.3.2 From adatoms to film growth
Each adatom presumably resides within an equilibrium energy well on the
surface most of the time, and this well is separated from adjacent energy
wells by a barrier of height Ed > 0 with respect to the equilibrium position.
The atoms oscillate in their wells due to thermal activation and,
occasionally, they acquire su±cient energy to hop into adjacent equilibrium
wells. Surface di®usion results from such jumps. If the surface is spatially
uniform, di®usion occurs randomly with no net mass transport on a macroscopic
scale. The hopping rate of any given atom increases with increasing
substrate temperature. Occasionally, attached atoms may also change positions
with substrate atoms, but this possibility is not pursued further in
20 Introduction and Overview
this discussion. If the surface is not uniform, perhaps as a result of a strain
gradient or a structural gradient, then surface di®usion can be directionally
biased, resulting in a net mass transport along the surface on a macroscopic
scale. Consequences of such transport are considered in Chapters 8 and 9. If
the temperature is very low or if the di®usion barrier is very high, adatoms
stick on the growth surface where they arrive, and the ¯lm tends to grow
with an amorphous or very ¯ne-grained polycrystalline structure.
The growth surface invariably has some distribution of surface defects
{ crystallographic steps, grain boundary traces and dislocation line terminations,
for example { which provide sites of relatively easy attachment for
adatoms. The spacing between defects represents a length scale for comparison
to the extent of random walk di®usion paths. If the di®usion distance is
large compared to the defect spacing, then adatoms tend to encounter these
defects and become attached to them, giving up some free energy in the
process. This is the case of heterogeneous nucleation and growth of ¯lms.
General statements about such processes are also di±cult to make. In some
cases, surface steps are relatively transparent to migrating adatoms, that is,
adatoms often pass over the step in either the up direction or the down direction
without attachment. For other materials, virtually every encounter of
an adatom with the step results in attachment. In yet other cases, adatoms
easily bypass steps in the up direction but not in the down direction, a manifestation
of the so-called Schwoebel barrier (Schwoebel 1969). If the spacing
of defects is large compared to the di®usion distance, on the other hand,
then migrating adatoms have the potential for lowering the energy of the
system by binding together upon mutual encounters to form clusters. The
case of formation of such stable clusters is called homogeneous nucleation
of ¯lm growth. That there is some minimum cluster size necessary for formation
of a stable nucleus can be demonstrated by appeal to the following
argument based on classical nucleation theory.
It was noted in Section 1.3.1 that the free energy of an adatom in the
supersaturated distribution on the surface is Eph with respect to its state
once entrained within the surface layer. In other words, this is the amount
of energy reduction per atom associated with the phase change from a two
dimensional gas of adatoms on the surface to a completely condensed surface
layer. Suppose that a planar cluster of n atoms is formed on the surface.
Will it tend to grow into a larger cluster, and eventually into a ¯lm, or will
the cluster tend to disperse? If all n atoms are fully entrained within the
cluster then the free energy reduction due to cluster formation would be
¡nEph. However, those atoms on the periphery of the cluster are not fully
incorporated. They possess an excess free energy compared to those that
1.3 Modes of film growth by vapor deposition 21
are fully entrained; this energy can be estimated to be roughly 4p¼nEf for
an equiaxed cluster, if n is fairly large compared to unity. This quantity has
the character of a surface step energy. The free energy change associated
with cluster formation is then
¢F = ¡nEph + 4p¼nEf : (1.1)
A graph of F versus n for ¯xed ¢Eph and Ef shows a maximum value for
a cluster size of n∗ = 4¼(Ef=Eph)2. The value of ¢F at n∗, which is the
activation energy for cluster formation, is ¢F∗ = 4¼E2
f =Eph. The implication
is that clusters smaller than the size n∗ are unstable and that they tend to
disperse, whereas clusters larger than this size tend to grow, driven by a
corresponding reduction in free energy. It is likely that many clusters form
and disperse for each one that evolves into an island. In fact, the classical
nucleation theory says nothing about such processes, other than that they
are possible. In any case, for a ¯lm to form on the substrate surface, it is
necessary that either nuclei formed by such homogeneous processes are able
to grow or that a su±cient number of surface defects are available to serve
as sites of heterogeneous nucleation.
The mode of ¯lm formation is determined by the relative values of the
various energies involved in the process, and this mode largely determines
the eventual structure of the ¯lm. There are two main comparisons to be
considered. One of these contrasts the height of the di®usion barrier Ed to
the background thermal energy. If Ed is large compared to the background
thermal energy then surface mobility of adatoms is very low. Under such
conditions, adatoms more or less stick where they arrive on the substrate
surface.
For growth of crystalline ¯lms, it is important that Ed be less than
the background thermal energy so that adatoms are able to seek out and
occupy virtually all available equilibrium sites in the ¯lm crystal lattice
as it grows. This requires the substrate temperature and/or the degree of
supersaturation of the vapor to be high enough to insure such mobility.
Suppose that this is indeed so and that the adatoms are able to migrate
over the surface.
The other important energy comparison concerns the propensity for
atoms of ¯lm material to bond to the substrate. This is represented by the
magnitude of Efs, relative to their tendency to bond to other, less well-bound,
atoms of ¯lm material, as represented by Ef . Two kinds of growth processes
can be distinguished, one with Efs larger in magnitude than Ef, and a second
with the relative magnitudes reversed. If Efs is the larger of the two energy
changes in magnitude, then ¯lm growth tends to proceed in a layer by layer
22 Introduction and Overview
mode, as indicated in the schematic diagram in Figure 1.8. Adatoms are
more likely to attach to the substrate surface than to other ¯lm material
surfaces. Once small stable clusters of adatoms form on the surface, other
adatoms tend to attach to the cluster at its periphery where they can bond
with both substrate and ¯lm atoms, thereby continuing the planar growth,
as indicated in Figure 1.8. This layer-by-layer ¯lm growth mode is often
called the Frank—van der Merwe growth mode or FM mode, according to a
categorization of growth modes proposed by Bauer (1958) on the basis of
more macroscopic considerations of surface energy. This alternate point of
view will be considered in Section 1.3.5.
On the other hand, if Ef is larger in magnitude than Efs, then it is
energetically favorable for adatoms to form three-dimensional clusters or
islands on the surface of the substrate. Film growth proceeds by the growth
of islands until they coalescence; this type of growth is commonly called the
Volmer—Weber growth mode or the VW mode; see Figure 1.8.
A third type of growth, which combines features of both the Frank{van
der Merwe and the Volmer{Weber modes, is called the Stranski—Krastanov
growth mode or SK mode. In this mode, the ¯lm material tends to prefer
attachment to the growth surface rather than the formation of clusters on
the growth surface; that is, Efs is greater in magnitude than Ef. However,
after a few monolayers of ¯lm material are formed and after the structure of
the ¯lm becomes better de¯ned as a crystal in conformity with the substrate,
the tendency is reversed. In other words, once the planar growth surface
becomes established as ¯lm material, subsequent adatoms tend more to
gather into clusters than to continue planar growth. The magnitude of Efs
appears to depend on the thickness of the ¯lm in the early stages of growth,
decreasing from values larger than the magnitude of Ef to values that are
smaller. The occurrence of this mode is most likely when the ¯rst few layers
of ¯lm material are heavily strained due to the constraint of the substrate.
This issue is revisited in Section 1.3.5.
1.3.3 Energy density of a free surface or an interface
In the preceding discussion, the early stages in the growth of a ¯lm from a
vapor were considered in a qualitative way in terms of the behavior of atoms.
Many of the inferences drawn can also be made in terms of surface energy
and interface energy of solid materials. These quantities are macroscopic
measures attributable to the discreteness of the material. However, they
represent only ensemble averages of behavior at the atomistic level and do
not incorporate discreteness of the material in any direct way. In many
1.3 Modes of film growth by vapor deposition 23
cases of microstructure evolution, knowledge of these quantities provides an
adequate basis for understanding the ¯lm growth process. In this section,
these quantities are discussed in general terms, and implications for ¯lm
growth are considered.
Surface energy is an important concept in considering the evolution
of microstructure in small-scale material systems. The free energy of the
bounding surface of a crystal is a macroscopic quantity representing, in some
sense, the net work that had to be done to create that surface. This energy
is distributed over a speci¯c mathematical surface, which has no thickness,
and which approximates the physical boundary between the crystal and its
surroundings. From this de¯nition of surface energy it is clear that the
reference level for energy of a free surface is the state of the material on that
same crystallographic surface when it is embedded deep within a perfect
crystal. Presumably, creation of the surface was accomplished by dividing
a larger crystal into two parts by separating these parts along a common
bounding surface. The total work done per unit area is assumed to be
evenly divided between the two bounding surfaces created. The concept of
speci¯c surface energy, or energy of cohesion, appears to have its origins in
a theory of °uid surfaces due to Young (1805) and Rayleigh (1890). The
concept was subsequently extended to crystal surfaces, where it was given a
central role in the work of Gibbs (1878). A useful geometrical interpretation
was provided subsequently by Wul® (1901). Later important contributions
to the equilibrium theory of crystal surfaces and to the description of the
energetics of surface evolution were provided by Herring (1953).
In general, there is a free energy reduction (increase) associated with
forming (breaking) a chemical bond. It follows that there is an increase in
free energy associated with creating a free surface that is more or less proportional
to the area of surface created or the number of chemical bonds per
unit area that are directly involved. Once the surface is created, the corresponding
free energy is, ¯rst and foremost, in the form of chemical bonding
potential, but other physical factors can contribute to surface energy as
well. At the surface of a coherent crystal, the atoms retain the general
arrangement that de¯nes the crystal structure. However, the spacings in
this arrangement are altered once bonds are broken; this is necessary to
restore equilibrium following the `disappearance' of interactions across the
surface. Through this change in spacing, atoms at or very near the surface
are able to relax to energy states somewhat lower than those in the natural
lattice sites in the interior of the crystal; this change in positional energy
may reduce the surface energy from the level predicted by bond counting
alone. The e®ect is due primarily to the longer range interactions between
24 Introduction and Overview
atoms in the crystal. If only nearest neighbor interactions are taken into
account, then interactions forces are typically zero at equilibrium. However,
if next nearest neighbor interactions are also taken into account, then nearest
neighbor interaction forces are compressive and next nearest neighbor
forces are tensile at equilibrium. Consequently, if some of the interactions
are eliminated to form a surface, unbalanced forces remain in the con¯guration
unless rearrangements occur.
There is yet another factor that comes into play in considering the
energies of surfaces, namely, surface reconstruction or rebonding. Chemical
bonds that are broken in creating a free surface are free or `dangling'
in the process described up to this point. Atoms in surfaces with bonding
potential may seek out other atoms within the surface with the potential
to form bonds. Bond formation is likely to occur in such cases if it results
in a net lowering of the energy of the system. Usually, more than one
such reconstruction is possible for a given crystallographic surface. Each
reconstructed con¯guration represents a local minimum of free energy with
respect to that con¯guration. Presumably, the system seeks out the absolute
minimum from among these as the actual reconstructed con¯guration; the
choice might depend on temperature, lattice strain or other factors. Reconstructed
surfaces are typically periodic, but with lattice vectors that are very
di®erent from the those of the underlying crystal. Reconstruction may also
result in surface anisotropy that is not representative of the lattice itself.
In summary, surface free energy represents the net energy increase of
the system due to work that must be done to overcome chemical bonds in
creating the surface, reduced by any relaxation of atomic positions of lattice
sites adjacent to the surface, and further reduced by surface reconstructions
which take place to match up available bonds in the surface. It is a macroscopic
quantity represented mathematically by a scalar function of position
over a material surface. The value of surface energy per unit area of a given
crystallographic surface orientation is determined by the ¯ne scale structure
of that surface. The ¯ne scale structure of the 'perfect' crystal may
include steps only one atom spacing in height separating terraces, combinations
of facets, various rebounding con¯gurations and other features unique
to surface structure.
Consider an isolated small crystal of some particular initial shape.
Knowledge of the dependence of surface energy on surface orientation is
su±cient to predict a minimum energy shape for the small crystal when
surface energy is the only free energy contribution that changes when shape
changes. This shape is most readily determined by means of the Wul®
construction, as illustrated schematically in two dimensions in Figure 1.9.
1.3 Modes of film growth by vapor deposition 25
Fig. 1.9. Representative polar plot of surface free energy in a symmetry plane of a
crystal; the free energy of a surface at a certain orientation is the radial distance
from the origin to the lobed curve in the direction of the surface normal. The Wul®
construction is illustrated by the family of lighter weight lines, with one member
of the family and its corresponding perpendicular radial line shown in a heavier
weight. The constructed equilibrium shape of the crystal is shown by the dashed
line.
In this diagram, the solid lobed curve represents the variation of surface
free energy with orientation; the radial distance in some direction from the
origin of the polar plot to the solid curve is the free energy per unit area
of a planar surface with normal vector in the direction of the radial line. A
representative radial line is shown in the ¯gure. To form the surface shape,
a line perpendicular to the radial line is drawn through the point where the
radial line intersects the curve of surface energy. Several examples of such
lines are shown in the ¯gure, with the particular line perpendicular to the
radial line shown in heavier weight. The low temperature equilibrium shape
of the small crystal is then geometrically similar to the region completely
enclosed by the totality of these perpendicular lines as the angle of the radial
line varies continuously over its full range. This equilibrium shape is shown
as the heavy dashed line in the ¯gure. A proof that this construction leads
to the shape with minimum surface energy is outlined by Herring (1953).
The construction is readily generalized to three space dimensions.
Many additional observations can be made on the basis of the same
construction (Herring 1951). Among the most useful of these concerns the
possibility of a macroscopically °at surface assuming a corrugated or peak26
Introduction and Overview
and-valley shape in order to reduce its free energy. For example, if a macroscopically
°at surface has the orientation of one of the °at surfaces in the
equilibrium shape, then no corrugated surface composed microscopically of
other orientations can be more stable. Conversely, if a given macroscopically
°at surface of a crystal does not coincide in orientation with some portion
of the boundary of the equilibrium shape, then there will always be a corrugated
structure, with the same average orientation as the initial °at surface,
which has a lower free energy than the °at surface; this corrugated structure
will be made up of segments of surface coinciding in orientation with the
°at faces of the equilibrium shape. The scale of the surface corrugations
is indeterminate unless an energy is associated with the edges where the
di®erent facets of the con¯guration intersect.
Like free surfaces, interface surfaces at which materials are joined can
also be represented macroscopically as discrete surfaces and can be characterized
by an interface surface energy per unit area. The physical origin
of this energy is essentially the same as that of a free surface. The energy
density of an interface is presumably less than the sum of the free surface
energies of the two materials joined at that interface, and greater than the
energy densities within either of the materials at interior points remote from
the interface.
Suppose that a small cluster or island of ¯lm material of volume V is
deposited onto the surface of the substrate. With this deposition, there is
a decrease in the area of substrate free surface, a change in the area of free
surface of the ¯lm material, and an increase in the area of a shared interface.
There is a surface free energy density associated with both the substrate
and ¯lm materials, say °s and °f , respectively. Likewise, there is also a
characteristic interface free energy per unit area, say °fs, associated with
the interface. What e®ect does the change in free energy associated with
these surfaces have on the equilibrium shape of the island, assuming that
the substrate surface remains °at? The equilibrium shape is that particular
shape that minimizes the free energy for given island volume.
To illustrate the idea, assume that °f is independent of orientation of
the island surface. Furthermore, assume that the shape of the island is a
spherical cap of constant volume. Two geometrical parameters are needed
to specify the shape of the island. Denote the radius of the circular contact
line where the island surface meets the substrate surface by R, and denote
the angle between the substrate surface and the tangent plane to the island
surface at the contact line by µ. These parameters are indicated in the
sketch in Figure 1.10. In terms of R and µ, the island volume V and the
1.3 Modes of film growth by vapor deposition 27
Fig. 1.10. Schematic diargram of a spherical cap island on a substrate. The radius
of the circular contact line is R and the angle between the substrate surface and
the tangent plane to the island surface at the contact line is µ.
island surface area A are
V =
¼R3
3 µ2 + cos µ
1 + cos µ¶µ1 ¡ cos µ
sin µ ¶; A=
2¼R2
1 + cos µ
: (1.2)
Up to an additive of constant, the free energy change associated with island
formation is
F(V; µ) = A°f + ¼R2(°fs ¡ °s) (1.3)
where R is understood to depend on V and µ according to (1.2)1. The
requirement that µ must take on a value that minimizes the free energy F,
subject to the constraint that V is constant, leads to the condition that
°f cos µ + °fs ¡ °s = 0 ) cos µ =
°s ¡ °fs
°f
: (1.4)
This result is frequently said to follow from a `force balance' among surface
energies at the contact line, but the correctness of the result in only
fortuitous; the basis for the result is constrained energy minimization. The
expression in (1.4) is known commonly as Young's equation for the wetting
angle of a °uid droplet on a rigid solid surface (Young 1805); the dependence
of the wetting angle on the surface energies does not hinge on the assumption
of a spherical shape. If additional forms of energy depend on island
shape, the argument must be altered accordingly and the form of the edge
condition may be a®ected. An example arises in the next section through
the in°uence of elastic energy.

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