Termodinamica de los materiales!

Samer Elatrache V-17810600 Estudiante de CRF


Thermodynamics is definite about events that are impossible. It will say, for
example, that reactions or processes are thermodynamically impossible. Thus,
gold films do not oxidize and atoms do not normally diffuse up a concentration
gradient. On the other hand, thermodynamics is noncommittal about permissi-
reactions and processes. Thus, even though reactions are thermodynamically
favored, they may not occur in practice. Films of silica glass should
revert to crystalline form at room temperature according to thermodynamics,
but the solid-state kinetics are so sluggish that for all practical purposes
amorphous SiO, is stable. A convenient measure of the extent of reaction
feasibility is the free-energy function G defined as
G = H - TS, (1-5)
where H is the enthalpy, S the entropy, and T the absolute temperature.
Thus, if a system changes from some initial (i) to final (9 state at constant
temperature due to a chemical reaction or physical process, a free-energy
change AG = G, - C, occurs given by
AG = AH - TAS,
where AH and AS are the corresponding enthalpy and entropy changes. A
consequence of the second law of thermodynamics is that spontaneous reactions
occur at constant temperature and pressure when AG < 0. This condition
implies that systems will naturally tend to minimize their free energy and
successively proceed from a value G, to a still lower, more negative value G,
until it is no longer possible to reduce G further. When this happens, AG = 0.
The system has achieved equilibrium, and there is no longer a driving force for
change.
On the other hand, for a process that cannot occur, AG > 0. Note that
neither the sign of AH nor of AS taken individually determines reaction
direction; rather it is the sign of the combined function AG that is crucial.
Thus, during condensation of a vapor to form a solid film, AS < 0 because
fewer atomic configurations exist in the solid. The decrease in enthalpy,
however, more than offsets that in entropy, and the net change in AG is
negative.
The concept of minimization of free energy as a criterion for both stability in
a system and forward change in a reaction or process is a central theme in
materials science. The following discussion will develop concepts of thermodynamics
used in the analysis of chemical reactions and phase diagrams. Subsequent
applications will be made to such topics as chemical vapor deposition,
interdiffusion, and reactions in thin films.
Chemical Reactions
The general chemical reaction involving substances A, B, and C in equilibrium
is
aA + bB * cC.
The free-energy change of the reaction is given by
AG = c G ~- uGA - bG,, (1-8)
where a, b, and c are the stoichiometric coefficients. It is customary to denote
the free energy of individual reactant or product atomic or molecular species
by
Gi = G,o + RTIn ai. (1-9)
G,o is the free energy of the species in its reference or standard state. For
solids this is usually the stable pure material at 1 atm at a given temperature.
The activity ai may be viewed as an effective thermodynamic concentration
and reflects the change in free energy of the species when it is not in its
standard state. Substitution of Eq. 1-9 into Eq. 1-8 yields
I
a;
a:.; '
AG = AG" -k RT In- (1-10)
where AG' = cGG - aGi - bGi. If the system is now in equilibrium,
AG = 0 and ai is the equilibrium value ai(eg)T. hus,
or
O = A G o + R T l n ( L1 4 ( e 4 l ) }
'A(,)
-AGO = RTln K,
(1-1 1)
(1-12)
where the equilibrium constant K is defined by the quantity in braces.
Equation 1-12 is one of the most frequently used equations in chemical
thermodynamics and will be helpful in analyzing CVD reactions.
Combining Eqs. 1-10 and 1-11 gives
Each term a, / ai(eg)r epresents a supersaturation of the species if it exceeds 1,
and a subsaturation if it is less than 1. Thus, if there is a supersaturation of
reactants and a subsaturation of products, AG < 0. The reaction proceeds
spontaneously as written with a driving force proportional to the magnitude of
AG. For many practical cases the ai differ little from the standard-state
activities, which are taken to be unity. Therefore, in such a case Eq. 1-10
yields
AG = AGO
Quantitative information on the feasibility of chemical reactions is thus
provided by values of AGO, and these are tabulated in standard references on
thermodynamic data. The reader should be aware that although much of the
data are the result of measurement, some values are inferred from various
connecting thermodynamic laws and relationships. Thus, even though the
vapor pressure of tungsten at room temperature cannot be directly measured,
its value is nevertheless "known." In addition, the data deal with equilibrium
conditions only, and many reactions are subject to overriding kinetic limitations
despite otherwise favorable free-energy considerations.
A particularly useful representation of AGO data for formation of metal
oxides as a function of temperature is shown in Fig. 1-10 and is known as an
Ellingham diagram. As an example of its use, consider two oxides of importance
in thin-film technology, SiO, and A,O, , with corresponding oxidation
reactions
Si + 0, + SiO, ; AGiio2, (1-15a)
(4/3)A + 0, + (2/3)A1,O3; AGi12~. , (1-15b)
Through elimination of 0, the reaction
(4/3)AI + S~O, + (2/3)~l,0, + Si (1-16)
results, where AGO = AGAz0, - AGiio2. Since the AGO- T curve for Al,O,
is more negative or lower than that for SiO,, the reaction is thermodynamically
favored as written. At 400 O C , for example, AG" for Eq. 1-16 is
- 233 - (- 180) = - 53 kcal/mole. Therefore, Al films tend to reduce SiO,
films, leaving free Si behind, a fact observed in early field effect transistor
structures. This was one reason for the replacement of Al film gate electrodes
by polycrystalline Si fdms. As a generalization then, the metal of an oxide that
has a more negative AGO than a second oxide will reduce the latter and be
oxidized in the process. Further consideration of Eqs. 1-12 and 1-15b indicates
that
)2/3 AG
( aAl,03 = exp - -. K =(aAYl 3pO2 RT

The A1,0, and A1 may be considered to exist in pure standard states with unity
activities while the activity of 0, is taken to be its partial pressure Po,.
Therefore, AGO = RT In Po,. If Al were evaporated from a crucible to
produce a film, then the value of Po, in equilibrium with both Al and A1203
can be calculated at any temperature when AGO is known. If the actual oxygen
partial pressure exceeds the equilibrium pressure, then A1 ought to oxidize. If
the reverse is true, AZO, would be reduced to Al. At lo00 'C, AGO = -202
Phase Diagrams
The most widespread method for representing the conditions of chemical
equilibrium for inorganic systems as a function of initial composition, temperature,
and pressure is through the use of phase diagrams. By phases we not only
mean the solid, liquid, and gaseous states of pure elements and compounds but
a material of variable yet homogeneous composition, such as an alloy, is also a
phase. Although phase diagrams generally contain a wealth of thermodynamic
information on systems in equilibrium, they can readily be interpreted without
resorting to complex thermodynamic laws, functions, or equations. They have
been experimentally determined for many systems by numerous investigators
over the years and provide an invaluable guide when synthesizing materials.
There are a few simple rules for analyzing phase diagrams. The most
celebrated of these is the Gibbs phase rule, which, though deceptively simple,
is arguably the most important linear algebraic equation in physical science. It
can be written as
f = n + 2 - 4, (1-18)
where n is the number of components (i.e., different atomic species), 4 is the
number of phases, and f is the number of degrees of freedom or variance in
the system. The number of intensive variables that can be independently varied
without changing the phase equilibrium is equal to f.
One-Component System. As an application to a one-component
system, consider the P- T diagram given for carbon in Fig. 1-1 1. Shown are
the regions of stability for different phases of carbon as a function of pressure
and temperature. Within the broad areas, the single phases diamond and
graphite are stable. Both P and T variables can be independently varied to a
greater or lesser extent without leaving the single-phase field. This is due to the
phase rule, which gives f = 1 + 2 - 1 = 2. Those states that lie on any of
the lines of the diagram represent two-phase equilibria. Now f = 1 + 2 - 2
= 1. This means, for example, that in order to change but maintain the
equilibrium along the diamond-graphite line, only one variable, either T or
P, can be independently varied; the corresponding variables P or T must
change in a dependent fashion. At a point where three phases coexist (not
shown), f = 0. Any change of T or P will destroy the three-phase equilib-
rium, leaving instead either one or two phases. The diagram suggests that
pressures between lo4 to lo5 bars (- 1O,O00-100,000 atm) are required to
transform graphite into diamond. In addition, excessively high temperatures
(- 2000 K) are required to make the reaction proceed at appreciable rates. It
is exciting, therefore, that diamond thin films have been deposited by decomposing
methane in a microwave plasma at low pressure and temperature, thus
avoiding the almost prohibitive pressure-temperature regime required for bulk
diamond synthesis.