Effects of amine fluoride cleaning chemistry on metallic aluminum IC films: II. Determining causal chemistry of OCPs by a time-dependent free energy relationship

Article abstract of DOI:10.1149/1.1836152

A mathematical expression has been developed describing the change in thermodynamic free energy for a chemical system as a function of time to aid the interpretation of experimental time-dependent energy curves, such as the open circuit potential (OCP) plots, generated in corrosion studies. Accurate results of chemical potentials and reaction rates, one pair of constants for each causal chemical reaction, were found. Reaction rate constants were determined for OCPs of Fe(CN)₆^4- + 1/2I₂ → Fe(CN) ₆^3- + I^- at room temperature of ω₁ = 0.011 ± 0.002/s, for oxidation of Fe(CN) ₆^4- to Fe(CN)₆^3- and ω₂ = 0.108 + 0.003/s for reduction of I₂ to I ^-, and the known half-cell potentials were reproduced. Experimental aluminum dissolution OCP data was fit using regression analysis describing a four to ten chemical reaction model. The formalism was useful in describing results of OCP plots of integrated circuits (IC) interconnect metals, cleaned by fluoride-based silicon wafer remover formulas, in terms of identifying the causal corrosion chemistry.

Full text of DOI:10.1149/1.1836152

B30
Journal of The Electrochemical Society, 152 ͑1͒ B30-B38 ͑2005͒
0013-4651/2004/152͑1͒/B30/9/$7.00 © The Electrochemical Society, Inc.
Effects of Amine Fluoride Cleaning Chemistry on Metallic
Aluminum IC Films
II. Determining Causal Chemistry of OCPs by a Time-Dependent Free
Energy Relationship
Melvin Keith Carterz
DuPont Electronics Technologies, EKC Technology, Hayward, California 94545, USA
A mathematical expression has been developed describing the change in thermodynamic free energy for a chemical system as a
function of time to aid the interpretation of experimental time-dependent energy curves, such as the open circuit potential ͑OCP͒
plots, generated in corrosion studies. Accurate results of chemical potentials and reaction rates, one pair of constants for each
causal chemical reaction, were found. Reaction rate constants were determined for OCPs of Fe͑CN)46Ϫ ϩ 1/2I2 → Fe͑CN)36Ϫ
ϩ IϪ at room temperature of ␻ ϭ 0.011 Ϯ 0.002/s, for oxidation of Fe͑CN)46Ϫ to Fe͑CN)36Ϫ and ␻ ϭ 0.108 ϩ 0.003/s for
1
2
reduction of I2 to IϪ, and the known half-cell potentials were reproduced. Experimental aluminum dissolution OCP data was fit
using regression analysis describing a four to ten chemical reaction model. The formalism was useful in describing results of OCP
plots of integrated circuits ͑IC͒ interconnect metals, cleaned by fluoride-based silicon wafer remover formulas, in terms of
identifying the causal corrosion chemistry.
© 2004 The Electrochemical Society. ͓DOI: 10.1149/1.1836152͔ All rights reserved.
Manuscript submitted March 30, 2004; revised manuscript received June 3, 2004. Available electronically December 7, 2004.
Integrated circuits ͑ICs͒ with submicrometer aluminum intercon-
between thermodynamic equilibrium and kinetic systems such that
they may not be defined at precisely the same conditions.
nects are commonly cleaned using liquid chemical products de-
signed specifically to remove postplasma etch residues with minimal
corrosion damage of the metal surfaces. These products are typically
water/solvent formulations, containing active fluoride additives and
pH regulators, designed to clean high-feature density silicon wafers.
In Part I, Experimental Measurements and Chemical Modeling,1 it
was shown that formula FA removed surface residues and passiva-
tion layers from aluminum-coated silicon wafer surfaces. A number
of electrochemical experiments were conducted to illuminate the
variables of the chemical cleaning process in an attempt to discover
the chemical mechanism of aluminum dissolution ͑corrosion͒. A
process model was established and an indicated chemical model was
proposed consisting of a sequence of time-dependent reactions de-
scribing a multisloped open circuit electrochemical potential ͑OCP͒,
with a corresponding current flow of 6 ϫ 10Ϫ9 A, presented here as
Fig. 4a. Ideally, detailed chemical analysis could be conducted to
measure chemical concentration changes associated with each step
of the OCP measurement, but such analyses are impractical at ul-
tralow concentration levels. The purpose of this work is to develop a
mathematical formalism to relate experimental OCP data curves to
their underlying causal chemistries so that the desired chemical po-
tential and reaction rate constant might be revealed for each contrib-
uting chemical reaction in a set of reactions.
Thermodynamics addresses energy changes associated with
chemical reactions based on a definition of a system as being at rest
(⌬F Ӎ 0). The potential Eؠ of an electric cell is given at that po-
tential representing the final oxidation state of an element at a con-
dition of rest so no current flows between the electrodes. In reality a
small amount of current must flow to establish a working cell and to
allow measurement of the cell potential. Thus, real systems are not
exactly at equilibrium but approach equilibrium. It is precisely this
gray area as systems approach equilibrium in which kinetics is also
often described.
Kinetic reaction rates are commonly expressed in terms of dif-
ferential equations4 such as dCa /dt ϭ ϪkaCa that describe the
change in concentration with time near equilibrium as the rate ka of
the instantaneous change in the concentration of species a ignoring
the other simultaneous changes that must be occurring. Changes in
other components are tacitly ignored and are assumed to be small, or
independent thereof, compared to the changes in a. It is apparent
that real thermodynamic and kinetic systems approach equilibrium
rather than operating at equilibrium. Nonetheless, kinetic modeling
methods have produced considerable useful information, although
often approximate. A relation has been expressed5 between reaction
rates and an exponential function of free energy of such a system,
close to equilibrium, that has been extended linearly to nonequilib-
rium conditions.6 Here electrochemical potential may be considered
to be the same as the chemical potential as discussed in footnote a.a
The goal of this investigation is to develop a mathematical for-
malism that facilitates interpretation of kinetic rates and changes in
thermodynamic free energies for an energy-time curve such that it
a Electrochemical potential measured in a two electrode cell may be expressed7 as
⌬V ϭ V1(electrode1) Ϫ V2(electrode2) ϭ (E1 Ϫ ␺ewf) Ϫ (E2 Ϫ ␺ewf) where ␺
ewf
is the energy required to remove an electron from an electrode surface to a vacuum; this
is also known as the electron work function and has a value of Ϫ4.6 Ϯ 0.1 eV. The
electron work function ͑EWF or absolute͒ may be considered to be a reference energy.
Electrochemists prefer to use a normal hydrogen electrode ͑NHE͒, a calomel electrode,
or a silver chloride electrode as a reference potential. The NHE value is taken to be 0.00
V, and is actually a relative reference-4.6 V different from the EWF value, and is
applied to all measured electrochemical potential values. Thus, the electrochemical
potential of aluminum of Ϫ1.662 V is relative to the standard NHE or ϩ3.062 V
absolute. The electrochemical potential of an electron in a metal is its Fermi energy
while the Fermi energy of a semiconductor may be the average of the valence band
energy and the conduction band energy. Once equilibrium has been established the
electrochemical potentialb may be considered to be the same as the chemical potential.
b It is necessary to consider the relationship between thermodynamic chemical po-
tential and electrochemical potential. The energy state of an ion in solution is primarily
composed of the sum of short range ionic interactions and a long range energy field
required to establish the potential charge of the ion.6 Thus, the electrochemical poten-
tial, ␮i , is a sum of the standard chemical potential, ␮i , and the potential resulting
from local ionic species, ␸, as ␮ ϭ ␮ ϩ ziF␸ where ␮ ϭ (ץF/ץni)T,P,nj , F is Fara-
A kinetic system is a chemical organization in unidirectional
motion2 while a thermodynamic system is normally at or approach-
ing equilibrium. Energy is conserved so a kinetic system can comply
with the first law of thermodynamics. The term kinetics, however,
implies that a system is not at rest and, therefore, not precisely in
thermodynamic equilibrium ͑not in strict compliance with the sec-
ond law of thermodynamics͒.3 Thus, there is an apparent chasm
i
i
i
day’s constant, and zi is the ionic charge on component i. For a pure phase at unit
activity the electrochemical potential ␮ ϭ ␮i , the chemical potential. For equilibrium
i
␣
i
␤
i
species i between phases ␣ and ␤, ␮ ϭ ␮ implying that the terms containing poten-
tial ␸ become equal making a null contribution to the equation. Thus, for a state of
equilibrium especially for dilute solutions, a chemical potential may be considered to be
the same as an electrochemical potential. Constant open circuit potentials ͑OCPs͒ mea-
sured at iOCP ϭ 6 ϫ 10Ϫ9 A, as reported in the previous article, are considered to be
near equilibrium chemical potentials since the current, which is a measure of change in
charge or concentration with time, is small.
z E-mail: mcarter@ekctech.com
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Journal of The Electrochemical Society, 152 ͑1͒ B30-B38 ͑2005͒
B31
reveals the underlying chemistry. It becomes apparent that each
chemical reaction equation in the system is represented by one re-
action rate and one chemical potential. These values may be consid-
ered to be equation constants once all mutually interdependent reac-
tion information is accounted for, that is, when the system
approaches equilibrium.c Such a system may be considered to be in
compliance with the second law of thermodynamics3 when current
flow associated with an OCP is low and approaches zero.
where the total mass of the system is constrained. Inclusion of, for
example, one component of solvent and i Ϫ 1 components of solute
results in a total mass of unity defined as
Mini ϭ 1
͓3͔
͚
i
where ni is the number of moles of the ith component per 1000 g
and Mi is the molecular weight of the ith component.d The masses
are determined by the chemical equations derived from single-
electron electrochemical reactions. For example, the chemical equa-
tion for iodine oxidation of ferrocyanide may be written as two
single electron half cell reactions, namely Fe͑CN)46Ϫ → Fe͑CN)63Ϫ
ϩ eϪ and 1/2I2 ϩ eϪ → IϪ. The masses are the molecular weights
of the products as shown.
Mathematical Considerations
Detailed chemical and electrochemical disclosure was made in
the previous article1 showing an OCP curve for aluminum immersed
in IC remover chemistry that consisted of three separate and distinct
functions modeled as a set of sequential chemical reaction equa-
tions. They are ͑i͒ dissolution of the native surface oxide, (ii) reac-
tion of aluminum metal with water, and (iii) repassivation of the
exposed aluminum metal surface. This electrochemical process may
be treated as a system. In addition, it has become clear that some of
the reactions in the system were interdependent on other reactions in
the set so as to produce sufficient concentrations of critical interme-
diate chemical components in order for the whole process to proceed
to a final steady state over time.
The differential free energy equation expresses interaction be-
tween one molal component, i, and the others, j. This relation enters
naturally as a partial differential expression in
dF ϭ
ץF/ץn ͒ ץn /ץn ͒dn ϭ F
␮Ј ץn /ץn ͒dn ͓4͔
͑
i j j
i
͑
͑
i
i
j
j
͚
͚
iрj
iрj
An OCP graphical figure represents changes in potential caused
by combining reactive components. For example, oxidation of alu-
minum metal by water produces an OCP of Ϫ1.66 V. This circuit
potential is a direct result of the chemical changes that occur natu-
rally. Thus, this chemical system exhibits a negative change in free
energy, a changing potential for chemical rebalance as the system
proceeds toward equilibrium. Time-dependent changes of free en-
ergy indicate changing chemical composition. Development of a
time-dependent free energy equation, for times ␶ long compared to
the time of electron transport approximated by tt ϭ (d/c) between
electrodes ͑times longer than 1–2 s, for electrode spacing d and
velocity of light c, such that ␶ ӷ tt for a system of chemical reac-
tions proceeding toward thermodynamic equilibrium, follows.
Consider a dilute chemical system of sequential reactions that
proceed slowly in time ͑requiring more than a few seconds͒. A sys-
tem consisting of two well-mixed dilute solutions or a dilute solu-
tion exposed to a few atomic layers of a chemically active surface
proceed toward a state of thermodynamic equilibrium. The partial
differential molal free energy of such a system is given in terms of
the thermodynamic3 equation of state as
where the sum is taken for i being 1 to ᐉ and j being from i to ᐉ so
duplication of indexes is avoided. The concentration of chemical
component i may be represented by a negative exponential function
of time as
n ϭ ␩ a exp Ϫ␻ t͒
͓5͔
͑
i
i
ij
where ␩ is the initial molal concentration of component i ͑at t
i
Ͻ
Ͻ
ᐉ), a is the constant pre-
ϭ 0) for an ᐉ component set (1
i
exponential factor and ␻ is a frequency factor. Real-time-based
ij
asymptotic functions, like Bessel functions that appear in a wide
variety of physical problems,9 represent the rate functionality that
accompanies chemical potential. Interaction between two such func-
tions of ni and nj may be represented by the binary expression
n /n ͒ ϭ ␩ a exp Ϫ␻ t͒ / ␩ a exp Ϫ␻ t͒
͔ ͓ ͔
ji
͓6͔
͑
͓
͑
͑
i
j
i
ij
j
The change in free energy of the system with time may be expressed
by the differential equation
dF/dt ϭ F
iрj
␮Ј ץn /ץn ͒dn /dt
͓7͔
͑
i
j
j
͚
i
dF ϭ ץF/ץT͒dT ϩ ץF/ץP͒dP ϩ
ץF/ץn ͒dn
͑
͓1͔
͓2͔
͑
͑
i
i
͚
i
Equation 6 is expressed as a differential equation in time in terms of
the binary molal concentration function as
ϭ ϪSdT ϩ VdP ϩ
␮idni
͚
i
ץn /ץn ͒ ϭ ץn /ץt͒/ ץn /ץt͒ ϭ ␩ ␻ /␩ ␻ ͒exp ␻ Ϫ ␻ ͒t
͑
͑
͑
͑
͑
i
j
i
j
i
ij
j
ji
ji
ij
͓8͔
where ␮i is the chemical potential ͑Fermi level8͒ of component i. At
the normal conditions of constant T and P the differential expression
becomes dF ϭ ͚ ␮idni or dF ϭ F͚ ␮Јdni where F is Faraday’s
and
i
i
i
dn /dt͒ ϭ Ϫ␩ a␻ exp Ϫ␻ t͒
͓9͔
͑
͑
j
i
ji
ji
constant and ␮ ϭ F␮Ј .
i
i
A state of equilibrium is expressed by ⌬F Ӎ 0, corresponding to
zero applied potential, and becomes a boundary condition for math-
ematical treatment. In addition, it is required that (dF/dt) approach
zero. Conservation of mass is chosen as another limiting condition
The time-based free-energy expression, substituting expressions 8
and 9 into Eq. 7, becomes
dF/dt ϭ F
␮Ј ץn /ץn ͒dn /dt
͑
i j j
i
͚
iрj
c The possibility that the closed electrochemical system treated in this work may be
an irreversible thermodynamic system has been considered. Initial components, their
concentrations and thus their equilibrium constants, are interrelated and are not defined
as independent entities with restricted phase space. In addition, experimental data dem-
onstrated the interrelationship of the products. Thus, the chemical system may be con-
sidered as deterministic and not probabilistic. Furthermore, closed electrochemical sys-
tems may be shown to be chemically reversible systems allowing for a consumption of
energy and an increase in entropy. From a chemical viewpoint the system is considered
to be reversible but from an energy viewpoint it is considered to be irreversible as, of
course, real world thermodynamic systems are.
ϭ ϪF
␮Ј␩ a␻ exp Ϫ␻ t͒
͓10͔
͑
i
ij
ij
͚
i
iрj
d A more precise definition of the conservation of mass is ͚iMini ϭ b where b is a
constant and b ϭ 1 for a total mass of 1000 g s. However, by redefining the units of ni
to be other values, such as moles per 1 g or moles per 10 gs, then the constant is still
b ϭ 1.
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B32
Journal of The Electrochemical Society, 152 ͑1͒ B30-B38 ͑2005͒
through 13 contain time-based expressions, however, the reaction
order is ultimately determined by extracting the sum or product of
equilibrium constants related to chemical reaction model equations
and are typically constant during the time of reaction.
The constant a is readily evaluated from Eq. 5. By setting t ϭ 0 it
may be observed that ␩ ϭ ni so a ϭ 1. The constant term, a, is
i
also determined by the conservation of mass condition,
l
͚
iрjMi␩ia exp(Ϫ␻ijt) ϭ 1. At t ϭ 0 then ͚liMi␩ia ϭ 1 and the
Rate constants and accurate chemical potentials for industrial
reactions in a condensed state are often unavailable. Numerous mea-
sured values have been reported determining kinetic values for se-
lected gas-phase reactions11 and reaction rates can be estimated for
gases from thermodynamic data. For other cases where the rate con-
stants or chemical potentials are not available, that is when the num-
ber of unknowns exceeds ᐉ, a semiempirical approach, as guided by
application of Eq. 13, may be possible. However, the total number
of unknown values is limited to less than the number of experimen-
tal data points. Reasonable results may be obtained by an iterative
minimization of deviations ͑regression analysis͒ approach to match
Eq. 13 to experimental curves. Semiempirical computations are
readily facilitated for Eq. 13 expressed as
constant may be expressed as
a ϭ
1/M ␩ ͒
͓11͔
͑
i
i
͚
i
but ͚iMi␩ also sums to 1, thus a ϭ 1 and the free energy expres-
i
sion 10 becomes
dF/dt ϭ ϪF
␩ ␮Ј␻ ͒exp Ϫ␻ t͒
͓12͔
͑
͑
i
ij
ij
͚
i
iрj
Each expression in i contains all of the rate-related terms. This equa-
tion can be simplified by letting ␬ ϭ ͚ ␻i , a sum of constants, so
i
dF/dt ϭ Ϫ␬F͚j(␩ ␮Ј)exp(Ϫ␻j)t. Integration produces an equa-
j
j
⌬F t͒ ϭ F
␻ ϩ ␻ ϩ ... ϩ ␻ ͒ ␩ ␮Ј/␻ ͒exp Ϫ␻ ͒t
͑ ͑
1 2 l r r r
r
͑
͑
tion for limited change in free energy of the system in a region near
equilibrium as
͚
r
ϭ F
␬␩ ␮Ј/␻ ͒exp Ϫ␻ ͒t ϭ
c /␻ ͒exp Ϫ␻ ͒t
͑
r
͑
͑
͑
r
r
r
r
r
͚
r
͚
⌬F t͒ ϭ Ϫ␬F
␩ ␮Ј exp Ϫ␻ ͒tdt
͑
͑
͵
i
i
͚
i
r
r
i
͓16͔
t
0
ϭ ␬F
␩ ␮Ј 1/␻ ͒exp Ϫ␻ t͒
͓͑
͑
͔
i
i
i
͚
i
i
where
cr ϭ F␬␩ ␮Ј
͓17͔
ϭ ␬F
␩ ␮Ј 1/␻ ͒exp Ϫ␻ t͒ Ϫ 1/␻ ͒
͓͑
͓13͔
͑
͔
͑
͕
͖
r
r
i
i
i
i
͚
i
i
and where the index i has been replaced by the index r to identify
reaction r that contains chemical component i. Active redox reac-
tions usually drive the generation of experimental OCP curves.
where the constant Ϫ͚i(1/␻i) ϭ Ϫ(1/␬) is ignored as small. The
integral form of free energy equation 13 meets the equilibrium con-
dition as time approaches the time of the measurement when the
change in ⌬F(t) with time approaches zero. The index i is the
number of chemical reaction components in the system represented
by one equation per chemical reaction and a solution is accom-
plished for a set of such equations as one equation per unknown.
Thus, one unknown is allowed per component, either the reaction
However, it is recognized that the constants ␮Ј and ␻r also relate to
r
active chemical component i those minimally active and ion ex-
change species necessary to complete the reaction set but whose
chemical potentials are usually minimal ͑close to zero͒. Thus, ⌬F(t)
produces a pair of constants, cr ͑a function of chemical potential͒
and ␻ ͑reaction rate͒, for each underlying chemical reaction upon
r
rate, ␻j , or the chemical potential, ␮Ј . The equation ⌬F(t) repre-
j
fitting an experimental curve. These pairs of constants identify the
type of chemical reaction acting and individual reactions may be
specified through knowledge of the chemicals present. Should the
model chemical reactions not be rationalized with the constants gen-
erated by the curve fitting procedure then another set of model
chemical reactions should be found until agreement is attained.
sents a set of linear equations which if solved as a matrix would be
solved simultaneously. Instead, Eq. 13 is solved as a sum of sequen-
tial equations to facilitate determination of the constants by fitting to
a specific experimental curve.
Consider ⌬F(t) to be a real function of rate ␻ and chemical
j
potential ␮Ј . If this function is able to be differented at t ϭ 0 and
j
in a small region about t ϭ 0 such that, if no singularity exists for
d⌬F(t)/dt, then ⌬F(t) is analytic10 at t ϭ 0. A smooth, continuous
field of such points may be considered in a well-defined region
about t such that, if no singularity exists for d⌬F(t)/dr, then ⌬F(t)
is analytic throughout this region. A solution of the sum of sequen-
tial equations is sought by determining the unknown constants upon
fitting the experimental OCP curve.
The frequency factors, ␻j , present in (dnj /dt) expressions are
inherently first-order rate constants but may be related to apparent
second-order rate constants. Here a reaction rate expression may be
stated as dCa /dt ϭ ϪkaCa or expressed in terms of molal concen-
tration (dnj /dt) ϭ Ϫkjnj where nj ϭ ␩ja exp(Ϫ␻jt) and (dnj /dt)
ϭ Ϫ␩ja␻j exp(Ϫ␻jt) as before. Solving for the rate constant lets
Experimental Measurements
OCP measurements were conducted to record changes in poten-
tial with time for oxidation of ferrocyanide ion by iodine from a KI3
complex. To 13 mL of 18.2 mM solution of K4Fe͑CN)6 stirred at
500 rpm was injected 50 ␮L of 55 mM KI3 solution. Fresh
K4Fe͑CN)6 solutions were prepared often and kept sealed between
uses. Three sets of measurements were conducted at 22 Ϯ 2°C and
atmospheric pressure for this reaction but the concentration of the
ferrocyanide ion was difficult to control due to its propensity for
rapid air oxidation. Measurements were conducted using
a
computer-controlled Solartron 1287 electrochemical interface, using
Scribner software, for a closed glass cell consisting of two nitric
acid-cleaned platinum foil electrodes, a saturated Ag/AgCl reference
electrode, a micromagnetic stir bar, and a nitrogen gas bubbler. Each
solution was stirred at 500 rpm under nitrogen gas for less than 3
min while the electrochemical baseline stabilized and 50 ␮L of 10
mM KI3 ͑prepared by dissolving 0.2538 g of I2 in 150 mM KI
solution and diluting to 100 mL͒ was injected just above the liquid
surfacee to initiate the reaction. Perturbation due to the injection
kj ϭ ␻
͓14͔
j
where rate constants may be expressed in units of 1/s. An apparent
second-order rate constant may be expressed as
k ϭ ␻ M /1000␳͒
͓15͔
͑
j
j
j
e The electrochemical cell could become grounded if contacted by the needle
thereby invalidating the measurement.
in units of L/mol-s where Mj is the molecular weight of the jth
component and ␳ is the density of the solution. Mathematical Eq. 5
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Journal of The Electrochemical Society, 152 ͑1͒ B30-B38 ͑2005͒
B33
Figure 1. Computed curve to match mathematical table eϪx values.
Figure 2. ͑a, top͒ OCP of K4Fe͑CN)6 ϩ I2 . ͑b, bottom͒ Computed OCP of
K4Fe͑CN)6 ϩ I2 curve and data points.
caused an increase in potential lasting 2 s followed by a smooth
decline in potential that was completed in less than 3 min. Refer to
Fig. 2a for a sample OCP curve.
OCP measurements were conducted for aqueous formula FB that
contained 70 wt % N-dimethylformamide and 30 wt % of an aque-
ous solution of 27 mmol of diisopropylethylquaternaryammonium
fluoride adjusted to pH 9 with an alkylamine by immersing an alu-
minum working electrode 1 cm from a platinum counter electrode
and immediately initiating the measurement. Results of potential
were recorded as a function of time for 400 s where the resultant
current drain was 6 Ϯ 1 nA/cm2. Refer to Fig. 3a. Removal ͑corro-
sion͒ of the 570 nm aluminum film continued at Ϫ1.66 V until the
film was completely dissolved.
sultant current drain was 6 Ϯ 1 nA/cm2. Refer to Fig. 4a. Removal
͑corrosion͒ of the thin aluminum film was abated as the film became
passivated.
Table IIA. Sample data points from OCP of K4Fe„CN…6 ϩ I2 .
OCP measurements were also conducted for formula FA ͑similar
to formula FB except ammonium fluoride replaced the quateramine͒
by immersing an aluminum working electrode 1 cm from a platinum
counter electrode and immediately initiating the measurement. The
initial potential of Ϫ1.3 V changed in some 40 to 50 s to Ϫ1.66 V
where it remained until an elapsed time of 100 s and then abruptly
changed approaching Ϫ0.7 V after some 400 s. Here again the re-
Time
͑s͒
Potential
͑V͒
Time
͑s͒
Potential
͑V͒
Time
͑s͒
Potential
͑V͒
7.1
11.1
15.1
19.1
23.1
0.153
0.139
0.129
0.122
0.117
31.1
39.1
47.1
55.1
67.1
0.109
0.102
0.0971
0.0927
0.0869
79.1
91.1
103.1
115.1
127.1
0.0823
0.0782
0.0747
0.0715
0.0684
Table IIB. Computed chemical potentials and reaction rates for
Table I. Data points from mathematical table eÀx values.
OCP of K4Fe„CN…6 ϩ I2 .
Chemical
potential ␮Ј
kr
͑L/mol
s͒
Time
͑s͒
Potential
͑V͒
Time
͑s͒
Potential
͑V͒
cr
␻
r
͑l/s͒
r
ᐉ
r
NLREG
͑V͒
0.5
1.0
1.5
2.0
2.5
0.60653
0.36788
0.22313
0.13534
0.08208
3.0
3.5
4.0
4.5
5.0
0.04979
0.03020
0.01832
0.01111
0.00674
5.637 ϫ 10Ϫ3
4.052 ϫ 10Ϫ2
0.0112 Ϯ 0.0016
0.1084 Ϯ 0.0026
2
2
1
2
0.358
0.538
0.00237
0.0137
Here 1 represents Fe͑CN)46Ϫ and 2 represents IϪ.
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B34
Journal of The Electrochemical Society, 152 ͑1͒ B30-B38 ͑2005͒
Table IIIA. Data points from OCP of formulation FB.
Time
Potential
Time
Potential
Time
Potential
͑s͒
͑V͒
͑s͒
͑V͒
͑s͒
͑V͒
1
10
25
1.32
1.46
1.61
50
100
150
1.76
1.85
1.85
200
300
400
1.85
1.85
1.85
Table IIIB. Computed chemical potentials and reaction rates for
OCP model of formulation FB.
Chemical
potential
Reaction
no.
cr
␻
͑l/s͒
kr
r
l
r
NLREG
␮Ј ͑V͒
͑L/mol s͒
r
5.57 ϫ 10Ϫ8
4 1
4 2
4 3
4 4
19
20
21
0.0001
396.
14.7
Ϫ1.304 ϫ 10Ϫ7Ϫ7.27 ϫ 10Ϫ113.20 ϫ 10Ϫ5 1.44 ϫ 10Ϫ6
4.479 ϫ 10Ϫ5 2.50 ϫ 10Ϫ8
8.97 ϫ 10Ϫ4
0.0321
17.1
22 Ϫ2.953.
Ϫ1.65
0.460
Assumed values may be adjusted until the regression analysis re-
turns computed values that provide the best fit to experimental
curves.
The degree of accuracy provided by the NLREG computational
routine ͑available on the web͒ was determined by fitting constants cr
and ␻ of Eq. 16 to a set of ten five place eϪx values from a math-
r
ematical table12 (cr , ␻r , and ␩r were set to 1.000000 initially͒, refer
to Fig. 1. The algorithm was run on a Sony notebook computer
using Eq. 16 until a minimization of the sum of squared deviations
or residuals of the function was achieved. It returned the acceptable
values of ␻ ϭ 0.999992 and c1 ϭ 1.00001 indicating five figure
1
accuracy. Refer to data Table I for the data points used.
OCP curves were recorded for a specific redox reaction chemis-
try specified by the two chemical half-cell reactions Fe͑CN)64Ϫ
→ Fe͑CN)36Ϫ ϩ eϪ and 1/2I2 ϩ eϪ → IϪ resulting in chemical Eq.
18 as
Fe͑CN)46Ϫ ϩ 1/2I2 → Fe͑CN)63Ϫ ϩ IϪ
͓18͔
Figure 3. ͑a, top͒ OCP of quateramine formulation FB. ͑b, bottom͒ Com-
puted OCP of quateramine formulation FB curve and data points.
where iron ͑II͒ became oxidized to iron ͑III͒ by iodine, refer to Fig.
2a and Table IIA for the average of three sets of experimentally
measured data. Regression analysis was conducted using starting
chemical potentials of ␮Ј ϭ 0.358 V and ␮Ј ϭ 0.538 V, that
matched the data of Fig. 2a for ᐉ ϭ 2, refer to Fig. 2b for the
matched data curve. The X symbols on the figure represent mea-
sured data points, and the solid line was the computed curve for
constants presented in Table IIB. The concentrations, ␩r , were taken
from the prepared solutions presented in the experimental section.
1
2
Discussion
The goal of this effort was to interpret experimental OCP curves
in terms of a causal chemical reaction model by a computational
method consisting of three logical steps. First, Eq. 16 was solved for
an OCP figure by means of a regression analysis curve fitting routine
adjusting the number of terms that represent the minimum number
of chemical reactions acting and to compute a pair of constants,
chemical potential, and reaction rate, for each reaction. Second, a
model set of reactions was expressed based on knowledge of the
chemical compounds present in an effort to explain the chemical
basis for the OCP. Finally, the chemical reaction model was com-
pared with the computed constants to determine if the model was in
reasonable agreement with the mathematical analysis of the data.
For example, a computed chemical potential of Ϫ1.66 V would
confirm dissolution of aluminum metal, a chemical potential of
Ϫ0.45 or Ϫ0.037 V might indicate that iron metal was active and
ϩ0.34 V could indicate that copper was active. For those more
complex chemical situations where insufficient data was available a
semiempirical approach may be employed by estimating starting
values for some reaction rate constants ␻r , remembering that solv-
ing for r equations allows for not more than r unknown values.
The resulting computed rate constants were ␻ ϭ 0.011
1
Ϯ 0.002/s, for oxidation of Fe͑CN)46Ϫ to Fe͑CN)63Ϫ and ␻
2
ϭ 0.108 ϩ 0.003/s for reduction of I2 to IϪ. The values of cr
ϭ F␬␩ ␮Ј returned yielded values of ␮Ј ϭ 0.358 V and ␮Ј
r
r
1
2
ϭ 0.538 V in agreement with the literature values13 of 0.358 V for
the oxidation potential of the Fe͑CN)46Ϫ ion and 0.536 V for the
oxidation potential of the I2 . Rate constants14 of 0.014 Ϯ 0.002
cm/s and 0.011 Ϯ 0.002 cm/s have been reported15 for reduction of
Fe͑CN)46Ϫ ion using a microcylindrical carbon electrode and system-
atic variations of electrodes, solvents, and temperature on the redox
potential have also been reported.16-21 The derived value of constant
␻ was in agreement with these reported values. Limited I2 reaction
1
rate data reported a recombination rate22 for • I2Ϫ of 3.2 ϫ 109
L/mol s for a free radical version of this reactant but a directly
comparable rate value was not found. Results of the experimental
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Journal of The Electrochemical Society, 152 ͑1͒ B30-B38 ͑2005͒
B35
Figure 4. ͑a, top left͒ OCP of ammonium fluoride formulation FA. ͑b, top right͒ Computed OCP of 0 to 100 s of ammonium fluoride formulation FA curve and
data points. ͑c, bottom left͒ Computed OCP of 100 to 400 s of ammonium fluoride formulation FA curve and data points. ͑d, bottom right͒ Computed OCP of
ammonium fluoride formulation FA curve and data points.
method of reacting Fe͑CN)46Ϫ with I2 were sufficiently repeatable to
produce the reported chemical potentials and the one reported reac-
tion rate suggesting the time-dependent free energy formalism to be
useful.
A series of OCPs were recorded for immersion of a 570 nm thick
aluminum metal film on a silicon wafer coupon in a solution of
formula FB, refer to Fig. 3a. The NLREG was conducted for the
OCP of Fig. 3a ͑data Table IIIA͒, for ᐉ ϭ 3, with c1 set to 0.0001,
c2 set to 0.38, and the chemical potential ␮Ј (␮Ј was omitted͒
4
3
initially constrained to Ϫ1.662 V, but no quantitative solution was
found. Refer to chemical reaction Eq. 19 through 22. Increasing the
equation set to ᐉ ϭ 4 and assuming all concentration distributions
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B36
Journal of The Electrochemical Society, 152 ͑1͒ B30-B38 ͑2005͒
␩ to be equal to 4.5 ϫ 10Ϫ5 for this very dilute chemistry,f setting
a reasonable curve fit, refer to the results in Table IVB and Fig. 4c.
The chemistry of the complete OCP ͑0 to 400 s͒ was then modeled
by incorporating the major values of the previous two truncated data
sets as starting values in a new computation for ᐉ ϭ 8 and above
maintaining ␩r at 4.5 ϫ 10Ϫ5 as before. Approximate computational
solutions were found using eight and ten sets of variables. Refer to
Fig. 4d and results in Table IVB for the OCP curve and computed
results. The ᐉ ϭ 8 parameter set did approximate the curve of Fig.
4a, but the limited set produced a zero slope beyond 200 s instead of
the non-zero slope as shown in Fig. 4a. This also indicated that more
than eight parametric equations were required to produce a better
curve fit. Variations in the quality of the curve fitting were affected
by changes in the constant cr but the frequency factors, ␻r , changed
only slightly. Pursuit of acceptable results for the ᐉ ϭ 8 and
ᐉ ϭ 10 data sets required numerous attempts via small changes in
starting values for the NLREG algorithm before an approximate fit
was achieved, yet the NLREG software readily computed acceptable
results for ᐉ ϭ 2 to 4 with a wide range of starting data. This
indicated a weakness in the regression analysis approach for
r
starting chemical potentials as above, ␻ to near 400, c1 to 0.0001 V
1
and c2 to 0.38 where ␮Ј was set to Ϫ1.662 V produced the antici-
4
pated results; refer to X marks in Fig. 3b and values in Table IIIB.
The aluminum dissolution chemistry has been modeled by the
four chemical reactions identified below as Eqs. 19-22
NH4F ꢀ NHϩ4 ϩ FϪ r ϭ 1
͓19͔
͓20͔
͓21͔
13 Al͑OH)3 ϩ FϪ
ꢀ
13 AlF3 ϩ OHϪ r ϭ 2
surface- 13 AlF3 ꢀ surface ϩ 31 AlF3 r ϭ 3
13 Al ϩ OHϪ
→
13 Al͑OH)3 ϩ eϪ,
E ϭ Ϫ1.662 V,13
r ϭ 4
͓22͔
,g
Reaction 19 generated an equilibrium concentration of fluoride
ion, Reaction 20 replaced surface metal hydroxide by fluoride, Re-
action 21 allowed for dissolution of the aluminum fluoride surface
so oxidation of the aluminum metal surface, Reaction 22, could
proceed. Since no metal passivation mechanism was active this re-
action continued until the aluminum film became completely dis-
solved.
Ͼ
ᐉ
4.
Immersion of aluminum metal into remover chemistry formula
FA, that produced the OCP of Fig. 4a, was modeled by a ten chemi-
cal reaction equation set of Eq. 19 through 28
13 Al͑OH)3 ϩ OHϪ
→
13 Al͑OH)Ϫ4 r ϭ 5
͓23͔
͓24͔
͓25͔
Computed results for the ᐉ ϭ 4 equation set not only fit the OCP
curve but a correct value of c4 , the critical driving Reaction 22, was
returned; refer to data presented in Table IIIB. Computed values for
NH4OH ꢀ NH3 ϩ H2O r ϭ 6
an apparent equilibration reaction were ␮Ј ϭ 5.57 ϫ 10Ϫ8 V with a
1
NH3 ϩ 1/2 O2 → NH2OH r ϭ 7
reaction rate of ␻ ϭ 396/s (k1 ϭ 14.7 L/mol s͒. Chemical poten-
1
tials of ␮Ј of Ϫ7.27 ϫ 10Ϫ11 V and ␮Ј of 2.50 ϫ 10Ϫ8 V were
12 NH2OH ϩ OHϪ
→
14 N2O ϩ 4 H2O ϩ eϪ
5
2
3
insignificant compared to the value of ␮Ј4 , as were the reaction rates
of ␻ ϭ 3.20 ϫ 10Ϫ5/s (k2 ϭ 1.44 ϫ 10Ϫ6 L/mol s͒ and ␻
r ϭ 8 E ϭ 1.05 V
͓26͔
2
3
ϭ 3.21 ϫ 10Ϫ1/s (k3 ϭ 8.97 ϫ 10Ϫ4 L/mol s͒. Chemical Eq. 22
1
12 N2O ϩ OHϪ → NO ϩ 2 H2O ϩ eϪ r ϭ 9 E ϭ 0.76 V
was represented by a chemical potential of ␮Ј of Ϫ1.65 V13 and a
4
͓27͔
͓28͔
rate constant of ␻ ϭ 17.1/s (k4 ϭ 0.460 L/mol s͒. This data con-
4
13 Al ϩ NO → 61 Al2O3 ϩ 12 N2O r ϭ 10
firms chemical Eq. 22 as the driving reaction. Nonetheless, this ap-
plication required four sets of reaction constants to match the ex-
perimental OCP curve, and the set of model Eq. 19, 20, 21, and 22
matched that requirement.
Dissolution of the active aluminum surface, 22, generated a negative
potential ͑cathode͒ so the oxidation steps Eq. 26 and 27 may have
occurred at the counter electrode ͑anode͒. Table IVB, data set for
ᐉ ϭ 10, indicated chemical Eq. 19 could fit the chemical potential
and rate-constant parameters of ␮Ј ϭ 1.14 ϫ 10Ϫ8 V and ␻
The mathematical formalism provided results for four pairs of
constants for aluminum immersed in formula FB, but it was not
known if this mathematical analysis process could successfully
model a more complex OCP. Equation 16 was applied to the con-
voluted OCP of aluminum immersed in formula FA, see Fig. 4a,
using a semiempirical approach and data points of Table IVA. The
initial curve analysis was conducted in two distinct parts for partial
time segments to estimate the chemical potential and reaction rate
constants. These estimated values of chemical potentials and fre-
quency factors were applied as initial values for a new analysis of
the entire 400 s curve. First Eq. 16 was applied to the 0 to 100 s
segment, up to the plateau of the OCP, using a four-parameter pro-
cess (ᐉ ϭ 4 for Reactions 19, 20, 21, and 22͒ starting with results
of Table IIIB and setting ␩r to 4.5 ϫ 10Ϫ5 M as before. Refer to the
results in Table IVB and Fig. 4b. A three-variable set did not pro-
duce an acceptable solution, but a four-variable set did optimize the
degree of curve fit and provided the expected value of ␮Ј4 . Next Eq.
16 was applied to the 100 to 400 s segment of the curve, from the
plateau to the end of the OCP curve of Fig. 4a using the data of
Table IVA as initial values for a four-parameter process (ᐉ ϭ 4 for
Reactions 25, 26, 27, and 28͒. Again a four-variable set did produce
1
1
ϭ 1.19 ϫ 104/s different from the data of Table IIIB and ᐉ ϭ 8
sets in Table IVB ͑negative rate values imply a reversed reaction͒.
Here a near-zero chemical potential would indicate equilibrium and
a high value of the frequency factor would indicate rapid attainment
of equilibrium expected for Reaction 19. A similar set of values was
Ϫ8
computed for chemical Eq. 24 where ␮ ϭ 1.44 ϫ 10 V and
Ј
6
␻ ϭ 1.20 ϫ 104/s. Here the equilibration rates and the chemical
6
potentials were similar to those for r ϭ 1. The initial chemical po-
tential of Eq. 22 was set to Ϫ1.662 V for the dissolution ͑corrosion͒
of aluminum metal in a mildly alkaline solution and a value of
Ϫ1.662 V was returned while an apparent rate constant of ␻
4
ϭ 1.90 ϫ 10Ϫ5/s was computed. Chemical Eq. 26 for electro-
chemical conversion of hydroxylamine to nitrous oxide was repre-
sented by the values of ␮Ј ϭ 1.07 V and ␻ ϭ 2.78 ϫ 10Ϫ3/s
8
8
close to the estimated chemical potential of 1.05 V. Chemical Eq.
27, representing reoxidation of nitrous oxide to nitric oxide, pro-
duced a value of ␮Ј ϭ 0.77 V with a rate constant of ␻ ϭ 5.40
9
9
ϫ 10Ϫ3/s close to a reported redox potential of 0.76 V.11 Chemical
f Since ⌬F Ӎ 0 and since all concentration factors ␩ ϭ 1, the computed values
r
Eq. 20 may be viewed as an exchange reaction with a ␮Ј ϭ 1.20
2
will be essentially the same as for ␩r ϭ 10Ϫ3 or 10Ϫ6
.
ϫ 10Ϫ2 V and ␻ ϭ 4.97 ϫ 10Ϫ2/s. Chemical Eq. 21 represents
g The half reaction listed in Ref. 12, 1/3Alؠ ꢀ 1/3Al3ϩ ϩ e, is suspect. Two half
reactions are expected to add together to form one complete chemical reaction as
1/3Alؠ ϩ OHϪ ꢀ 1/3Al͑OH)3 ϩ eϪ and Hϩ ϩ eϪ ꢀ 1/2H2 to yield ͑as multiplied
by 3͒ Alؠ ϩ 3H2O ꢀ Al͑OH)3 ϩ 3/2H2 . Since water is a necessary part of the alu-
minum reaction sequence the later aluminum half reaction has been expressed in this
work.
2
dissolution of the passivated surface with a ␮Ј ϭ 3.90 ϫ 10Ϫ2
V
3
and ␻ ϭ 4.66 ϫ 10Ϫ2/s. Chemical Eq. 23 established an alterna-
3
tive condition for potential dissolution of the passivating film with
␮Ј ϭ Ϫ5.20 ϫ 10Ϫ2 V and ␻ ϭ 4.74 ϫ 10Ϫ2/s. Chemical Eq. 25
5
5
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Journal of The Electrochemical Society, 152 ͑1͒ B30-B38 ͑2005͒
B37
constant. However, it was shown that each partial curve fitting re-
quired a minimum of four parameters for the 0 to 100 s region and
at least four parameters for the 100 to 400 s region so that a mini-
mum set of eight chemical reaction equations were required to sat-
isfy the aluminum surface-cleaning model responsible for produc-
tion of the OCP of Fig. 4a. Furthermore, a zero slope computed for
the OCP in the 200 to 400 s region for ᐉ ϭ 8 indicated that ᐉ was
greater than 8. The additional equations with smaller constants serve
to refine the chemical model.
Table IVA. Data points from OCP of formulation FA.
Time
Potential
Time
Potential
Time
Potential
͑V͒
͑s͒
͑V͒
͑s͒
͑V͒
͑s͒
0
10
20
30
40
50
60
70
80
90
1.40
1.47
1.61
1.66
1.69
1.69
1.68
1.67
1.66
1.65
1.63
1.59
1.53
105
107
108
110
113
117
120
130
140
150
160
170
1.47
1.42
1.37
1.34
1.29
1.24
1.20
1.14
1.10
1.08
1.06
1.04
180
190
200
210
230
250
280
300
330
350
380
400
1.03
1.01
1.00
0.99
0.97
0.95
0.92
0.91
0.89
0.88
0.87
0.86
A quite limited number of chemicals were present initially1 in the
experiments performed. They consisted of a passivated aluminum
metal surface, water, a water soluble organic solvent such as dim-
ethylformamide ͑DMF͒, less than two percent ammonium fluoride
and a fractional percent concentration of an amine to adjust pH. This
picture became somewhat more complex as Al became exposed and
corroded, AlF3 formed, and some NH3 became vaporized in an
equilibrium shift. In addition, a minor amount of ammonia may have
been oxidized and low concentrations of impurities could have par-
ticipated.
Reactions capable of producing a potential significantly different
from zero involve redox, that is reduction-oxidation chemistry usu-
ally associated with a low level of electrical current for reactions
close to equilibrium. Once a potential field has been established,
chemical reaction equations that modify the potential as a function
of time may also participate. For example, exposure of a passivated
aluminum metal surface to an aqueous solution might result in a
measured potential of Ϫ1.3 to Ϫ1.4 V where water can react with
aluminum via a resistive, porous native oxide layer. Should a sec-
ondary chemical reaction proceed to remove this passivation layer
then the measured potential would be anticipated to change to
Ϫ1.66 V as observed, the redox potential for aluminum in water, as
seen in Fig. 3a and 4a. In this case chemical reactions directing both
removal of the passivating film and oxidation of the aluminum were
100
104
104.5
represents the possible reaction of ammonia with a ␮Ј ϭ Ϫ0.614 V
7
and ␻ ϭ 5.67 ϫ 10Ϫ3/s while chemical Eq. 28 represents forma-
7
tion of an thin film of insoluble aluminum oxide with a ␮Ј
10
ϭ Ϫ1.18 ϫ 10Ϫ2 V and ␻ ϭ 3.06 ϫ 10Ϫ3/s. Figure 4d showed
10
the computed curve did approximate the measured data points but
not overlay them. Computations conducted for all values con-
strained to those values believed to be correct did cause the compu-
tation to halt prior to completion. Possibly a more quantitative mini-
mization algorithm could improve the curve fit computation for
Ͼ
ᐉ
4. Nonetheless, the computed constants did approximately rep-
resent the chemical reaction equations indicating chemical Eq. 19
through 28 represented a reasonable model of aluminum surface
corrosion in remover FA.
The mathematical results of Table IVB did show there were at
least five main chemical equations, one for each large value of a
Table IVB. Computed chemical potentials and reaction rates for OCP of formulation FA.
Chemical
potential
kr
͑L/mol
s͒
cr
␻
͑l/s͒
Mr
r
l
r
NLREG
␮Ј ͑V͒
͑Daltons͒
Comments
r
5.57 ϫ 10Ϫ8
Ϫ7.27 ϫ 10Ϫ11
2.50 ϫ 10Ϫ8
Ϫ1.65
4
4
4
4
1
0.0001
396.
14.7
37.04
44.98
27.93
26.91
Fit to 0–100 s
of Fig. 4a
Ϫ1.30 ϫ 10Ϫ7
4.48 ϫ 10Ϫ5
Ϫ2.953
3.20 ϫ 10Ϫ5
1.44 ϫ 10Ϫ6
8.97 ϫ 10Ϫ4
0.460
2
3
4
0.0321
17.1
5.57 ϫ 10Ϫ8
Ϫ1.10 ϫ 10Ϫ10
1.38
4
4
4
4
5
6
7
8
0.0001
396.
13.9
35.04
33.55
39.01
38.99
Fit to 100–400
s of Fig. 4a
Ϫ4.22 ϫ 10Ϫ7
2.474
2.45 ϫ 10Ϫ4
0.328
8.24 ϫ 10Ϫ6
1.28 ϫ 10Ϫ2
3.69 ϫ 10Ϫ3
Ϫ1.28 ϫ 10Ϫ3
Ϫ2.30
0.0946
2.94 ϫ 10Ϫ8
2.94 ϫ 10Ϫ4
Ϫ1.74 ϫ 10Ϫ4
Ϫ1.70
8
8
8
8
8
8
8
8
1
2
3
4
5
6
7
8
0.0001
1.000
Ϫ0.590
Ϫ5.762
0.0001
Ϫ1.000
394.
14.6
37.04
44.98
27.93
26.91
35.04
33.55
39.01
38.99
Fit to 0–400 s
of Fig. 4a
4.93 ϫ 10Ϫ4
Ϫ1.24 ϫ 10Ϫ4
5.40 ϫ 10Ϫ5
13.6
0.0110
Ϫ0.00446
0.00201
388.
0.0110
0.00201
Ϫ0.00446
Fig. 4a
Fig. 4a
Fig. 4a
Fig. 4a
Fig. 4a
Fig. 4a
2.94 ϫ 10Ϫ8
Ϫ2.94 ϫ 10Ϫ4
1.70
3.68 ϫ 10Ϫ4
7.82 ϫ 10Ϫ5
5762
1.74 ϫ 10Ϫ4
Ϫ1.74 ϫ 10Ϫ4
0.590
5.183 ϫ 10Ϫ5
1.137 ϫ 10Ϫ8
1.188 ϫ 104
10
1
441.
37.04
Fit to 0–400 s
of Fig. 4a
2.23 ϫ 10Ϫ3
1.30 ϫ 10Ϫ3
5.14 ϫ 10Ϫ7
1.54 ϫ 10Ϫ3
420.
1.87 ϫ 10Ϫ4
9.35 ϫ 10Ϫ5
2.11 ϫ 10Ϫ4
1.19 ϫ 10Ϫ4
10
10
10
10
10
10
10
10
10
2
3
4
5
6
7
8
9
10
40.01
129.0
Ϫ5.442.
0.01200
0.03900
Ϫ1.662
0.04973
0.04662
44.98
27.93
26.91
32.58
35.04
33.03
33.55
39.01
38.99
Fig. 4a
Fig. 4a
Fig. 4a
Fig. 4a
Fig. 4a
Fig. 4a
Fig. 4a
Fig. 4a
1.904 ϫ 10Ϫ5
Ϫ169.0
Ϫ0.05200
0.04744
5.252 ϫ 10Ϫ5
Ϫ2.010.
3.496.
1.438 ϫ 10Ϫ8
Ϫ0.6140
1.068
1.120 ϫ 104
0.005668
0.002780
0.005402
0.003065
2.524.
Ϫ3.875.
0.7710
Ϫ1.183
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B38
Journal of The Electrochemical Society, 152 ͑1͒ B30-B38 ͑2005͒
indicated. The actual potential established could also be a function
of the pH of the acting solution. pH values of 8 to 9 resulted in an
aluminum redox potential of Ϫ1.66 V11 while higher pH values may
result in slightly different redox potentials.
a chemical potential and a reaction rate could be associated with
every chemical reaction equation.
Results of these analyses provided a conceptual chemical model
for dissolution ͑corrosion͒ of aluminum metal similar to that pro-
posed in the previous article,1 although it has become apparent that
variations in the chemical model set for more than eight chemical
reaction equations was indicated. This approach to analysis of OCP
curves, specifically corrosion of materials as represented by an en-
ergy vs. time curve, demonstrated the applicability of the mathemati-
cal formalism for simultaneously extracting details of both the ther-
modynamic process and kinetic reaction rates. While there were
difficulties in analysis of convoluted experimental curves represent-
ing more than four sets of chemical reactions, the analysis process
has been informative.
A rate of reaction for fluoride ion replacing hydroxide ion has
been reported23 in a pH region of 4 to 5 as K1 ϭ 303 molϪ2 sϪ1
,
K2 ϭ 5 molϪ1 sϪ1, and K3 ϭ 1.03 ϫ 105 molϪ2 sϪ1. These values
are higher than the value of k2 presented in Table IVB. Reaction
rates measured in liquids assume no boundaries ͑barriers͒ to diffu-
sion along reaction coordinates so molecular motion proceeds as
though in an infinite liquid. Reactions that proceed on a reactive
surface, such as this example, are anticipated to be slower since half
of the free volume is missing ͑semi-infinite model͒, diffusion near a
solid surface may be slower and formation of interfacial barriers
͑and double layers͒ may further restrict molecular motion. The ex-
perimental data indicated this reaction rate to be ␻ ϭ 4.97
2
Conclusions
ϫ 10Ϫ2/s. Higher rates produced in the presence of selected
ligands,23 shown to reversibly displace fluoride ion at pH 9 as a
subsequent reaction, also indicated this step may be slow.
Development of a mathematical formalism expressing changes in
thermodynamic free energy of a chemical system as a function of
time, for description of experimental time-dependent energy curves
such as the OCP metal corrosion plots, produced results in terms of
chemical potentials and reaction rates, one pair of constants for each
causal chemical reaction. Experimental OCP data was presented in
terms of aluminum surface removal ͑corrosion͒ models consisting of
sets of four to ten chemical reactions. Furthermore, derived reaction
rates and chemical potentials for the known Fe͑II͒ ϩ I2 reaction
were found to be in good agreement with known values. The for-
malism has been helpful in describing results of OCP metal corro-
sion plots of IC interconnects cleaned by silicon wafer remover
products. It has also enabled modified chemistry to be developed
͑formula FA compared to formula FB͒ for reduction of the amount
of surface material removed ͑corroded͒ during the cleaning process.
The rate of dissolution of Al͑OH)3 has also been measured24
under mildly acidic conditions indicating two distinct, apparently
age-dependent, rates. Amorphous Al͑OH)3 yielded a dissolution rate
of kЈa ϭ 0.45/s such that C(t) ϭ C0eϪ0.45t while aged or possibly
less hydrated Al͑OH)3 yielded a dissolution rate of kЈ ϭ 0.0095/s
c
so C(t) ϭ C0eϪ0.0095t. These constants are identified as frequency
factors in the present discussion so the apparent equivalent second-
order rate constants would be ka ϭ 3.5 ϫ 10Ϫ2 L/mol s and kc
ϭ 7.4 ϫ 10Ϫ4 L/mol s. Here the rates of dissolution of a suspended
solid and that of a hydrated aqueous gel differ by nearly two orders
of magnitude. The lower value was still orders of magnitude differ-
ent from that of a liquid-solid interface anticipated for dissolution of
a passivated aluminum surface. The data for Eq. 22 in Table IVB,
DuPont Electronics Technologies, EKC Technology, assisted in meeting
the publication costs of this article.
also of a solid-liquid interface, indicates the reaction rate to be ␻
4
ϭ 1.90 ϫ 10Ϫ5/s still slower than that of kЈ ϭ 9.5 ϫ 10Ϫ3/s
c
above. The small value seems reasonable considering dissolution of
aluminum occurs only at the solid-liquid interface at a measured rate
of Ͻ10 nm/min.1
References
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͑2003͒.
Constants for Eq. 20 indicate the small reaction rate of ␻
2. S. W. Benson, The Foundations of Chemical Kinetics, Chap. III, McGraw-Hill
Book Co., Inc., New York ͑1960͒.
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court Academic Press, New York ͑2001͒.
2
ϭ 4.97 ϫ 10Ϫ2/s to be orders of magnitude higher than for disso-
lution of surface aluminum metal. Dissolution of aluminum fluoride,
Eq. 21 might be anticipated to be similar to ␻ with a value of ␻
2
3
ϭ 4.66 ϫ 10Ϫ2/s. The chemical potential for this dissolution was
shown to be ␮Ј ϭ 3.90 ϫ 10Ϫ2 V.
3
The chemical potential constant derived for the more complex
reversible ammonium fluoride reaction in formulation FA, ␮Ј
1
ϭ 1.14 ϫ 10Ϫ8 V, was similar to the equilibrated or reversible
chemical reaction FB of ␮Ј ϭ 5.57 ϫ 10Ϫ8 V yet the equilibration
1
rate constant for formulation FA of ␻ ϭ 1.19 ϫ 104/s was nearly
1
two orders of magnitude higher than that for formulation FB of
␻ ϭ 3.96 ϫ 102/s, refer to data of Table IIIB. The values of ␮Ј
10. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, p. 401, Har-
court Academic Press, New York ͑2001͒.
1
2
ϭ Ϫ7.27 ϫ 10Ϫ11 V and ␻ ϭ 3.20 ϫ 10Ϫ5/s for the FB formu-
2
11. C. J. Marek, Technical Support Package, Physical Sciences, ͑no. 176͒, Lewis Re-
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81st ed., 8–21, Petr Vanysek, Section Editor, CRC Press, New York ͑2000-2001͒.
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lation and the values of ␮Ј ϭ 1.20 ϫ 10Ϫ2 V and ␻ ϭ 4.97
2
2
ϫ 10Ϫ2/s for the FA formulation indicate a shift in the chemical
potentials and reaction rates. The values for the dissolution of alu-
minum also show significant differences between the two different
types of chemistry. For formulation FB values of ␮Ј ϭ Ϫ1.65 V
4
and ␻ ϭ 1.71 ϫ 101/s indicated a substantial value for the rate
4
constant while the rate constant for formulation FA, of ␻ ϭ 1.90
4
ϫ 10Ϫ5/s was orders of magnitude lower as anticipated for a repas-
sivated surface. This result correlates well with dissolution of Al in
formula FB and passivation of Al in formula FA.
The very small apparent chemical potentials for chemical equi-
librium Eq. 19 and 24 imply some energy, though negligible, is
associated with these balances. Nonetheless, non-zero values of
chemical potentials must be associated with these chemistries. Ap-
plication of the time-dependent free energy formalism indicated that
22. P. Neta, R. E. Huie, and A. B. Ross, J. Phys. Chem. Ref. Data, 17, 1208 ͑1988͒.
23. K. Srinivasan and G. A. Rechnitz, Anal. Chem., 40, 1818 ͑1968͒.
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͑1991͒.
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