Front Cover
 Table of Contents
 University of Arizona
 Meet your students: 2. Susan and...
 Edwin N. Lightfoot of Wisconsi...
 From molecular theory to thermodynamic...
 Chemical engineering in the spectrum...
 Chemical processing of electrons...
 Book reviews and books receive...
 Thermal oxidation of silicon
 Working in the integrated circuit...
 Chemical vapor deposition epitaxy...
 The impedance response of semiconductors:...
 Stochastic modeling of chemical...
 Back Cover


Chemical engineering education
http://cee.che.ufl.edu/ ( Journal Site )
Full Citation
Permanent Link: http://ufdc.ufl.edu/AA00000383/00105
 Material Information
Title: Chemical engineering education
Alternate Title: CEE
Abbreviated Title: Chem. eng. educ.
Physical Description: v. : ill. ; 22-28 cm.
Language: English
Creator: American Society for Engineering Education -- Chemical Engineering Division
Publisher: Chemical Engineering Division, American Society for Engineering Education
Place of Publication: Storrs, Conn
Publication Date: Winter 1990
Frequency: quarterly[1962-]
annual[ former 1960-1961]
Subjects / Keywords: Chemical engineering -- Study and teaching -- Periodicals   ( lcsh )
Genre: periodical   ( marcgt )
serial   ( sobekcm )
Citation/Reference: Chemical abstracts
Additional Physical Form: Also issued online.
Dates or Sequential Designation: 1960-June 1964 ; v. 1, no. 1 (Oct. 1965)-
Numbering Peculiarities: Publication suspended briefly: issue designated v. 1, no. 4 (June 1966) published Nov. 1967.
General Note: Title from cover.
General Note: Place of publication varies: Rochester, N.Y., 1965-1967; Gainesville, Fla., 1968-
 Record Information
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: oclc - 01151209
lccn - 70013732
issn - 0009-2479
Classification: lcc - TP165 .C18
ddc - 660/.2/071
System ID: AA00000383:00105

Table of Contents
    Front Cover
        Front Cover 1
        Front Cover 2
    Table of Contents
        Page 1
    University of Arizona
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
    Meet your students: 2. Susan and Glenda
        Page 7
    Edwin N. Lightfoot of Wisconsin
        Page 8
        Page 9
        Page 10
        Page 11
    From molecular theory to thermodynamic models: Part 1. Pure fluids
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
    Chemical engineering in the spectrum of knowledge
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
    Chemical processing of electrons and holes
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
    Book reviews and books received
        Page 33
    Thermal oxidation of silicon
        Page 34
        Page 35
        Page 36
        Page 37
    Working in the integrated circuit industry
        Page 38
        Page 39
        Page 40
        Page 41
    Chemical vapor deposition epitaxy on patternless and patterned substrates
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
    The impedance response of semiconductors: An electrochemical engineering perspective
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
    Stochastic modeling of chemical process systems: Part 1. Introduction
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
    Back Cover
        Back Cover 1
        Back Cover 2
Full Text

chemica en gi e e* e dc

Powerful New Materials Introduce Health, Safety and Loss Prevention Concepts
in Undergraduate Engineering Courses

ABET Criteria require accredited schools to teach an understanding of the ethical, social and safety
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* vapor releases

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NOW AVAILABLE Instructor's guide, complete with problems, solutions and notes, list price $50*,
608 pp est., at no cost for those using the problems in their courses, Write, on university letterhead, to ask
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Chemical Engineering Education
Department of Chemical Engineering
University of Florida
Gainesville, FL 32611

EDITOR: Ray W. Fahien (904) 392-0857
MANAGING EDITOR: Carole Yocum (904) 392-0861


E. Dendy Sloan, Jr.
Colorado School of Mines

Gary Poehlein
Georgia Institute of Technology

Lee C. Eagleton
Pennsylvania State University

Richard M. Felder
North Carolina State University

Jack R. Hopper
Lamar University

Donald R. Paul
University of Texas

James Fair
University of Texas

J. S. Dranoff
Northwestern University

Frederick H. Shair
California Institute of Technology

Alexis T. Bell
University of California, Berkeley

Angelo J. Perna
New Jersey Institute of Technology

Stuart W. Churchill
University of Pennsylvania

Raymond Baddour
Massachusetts Institute of Technology

Charles Sleicher
University of Washington
Leslie W. Shemilt
McMaster University

Library Representative
Thomas W. Weber
State University of New York


Chemical Engineering Education


2 University of Arizona,
G. K. Patterson


8 Edwin N. Lightfoot,
by his Colleagues


12 From Molecular Theory to Thermodynamic Models:
Part 1. Pure Fluids,
Stanley I. Sandler


20 Chemical Engineering in the Spectrum of Knowledge,
Davor P. Sutija, John M. Prausnitz


26 Chemical Processing of Electrons and Holes,
Timothy J. Anderson

34 Thermal Oxidation of Silicon,
Dennis W. Hess

38 Working in the Integrated Circuit Industry,
Carol M. McConica

42 Chemical Vapor Deposition Epitaxy on Patternless and
Patterned Substrates,
Christos G. Takoudis

48 The Impedance Response of Semiconductors: An
Electrochemical Engineering Perspective,
Mark E. Orazem


56 Stochastic Modeling of Chemical Process Systems:
Part 1. Introduction,
R.O. Fox, L.T. Fan


7 Meet Your Students: 2. Susan and Glenda,
Richard M. Felder

33 Book Reviews
33 Books Received

CHEMICAL ENGINEERING EDUCATION (ISSN 0009-2479) is published quarterly by the Chemical Engi-
neering Division, American Society for Engineering Education and is edited at the University of Florida. Cor-
respondence regarding editorial matter, circulation, and changes of address should be sent to CEE, Chemical
Engineering Department. University of Florida, Gainesville, FL 32611. Advertising material may be sent di-
rectly to E.O. Painter Printing Co., PO Box 877, DeLeon Springs, FL 32130. Copyright @ 1990 by the
Chemical Engineering Division, American Society for Engineering Education. The statements and opinions
expressed in this periodical are those of the writers and not necessarily those of the ChE Division, ASEE,
which body assumes no responsibility for them. Defective copies replaced if notified within 120 days of pub-
lication. Write for information on subscription costs and for back copy costs and availability. POSTMAS-
TER: Send address changes to C, Chem. Engineering Dept., University of Florida, Gainesville, FL 32611.

department I




G. K. Patterson
University of Arizona
Tucson, AZ 85721

THE UNIVERSITY OF Arizona has come a long way
in the 103 years since it was a one-room school
house in the middle of the desert. The original 32 stu-
dents have grown to more than 34,000 and the six
charter members of the UA faculty now number about
2,200 (which does not include the more than 1,000
graduate assistants who aid in teaching and research).
In order to administer such a large educational enter-
prise, The University of Arizona requires more than
7,500 staff personnel. Their work is carried out in 131
buildings spread across 540 acres in the heart of Tuc-
The University of Arizona is a very strong re-
search-oriented university. In 1988, it was ranked
12th among U.S. public universities for research fund-
ing. Engineering at The University of Arizona ranked
17th in research funding, and the Chemical Engineer-
ing Department ranked 25th in research funding.
Chemical engineering was proposed for The Uni-
versity of Arizona in the mid-1950s. The dean of engi-
neering at that time saw no place for chemical engi-
neering in his college, so the College of Mines began
development of the department. The first courses
were taught by the Metallurgical Engineering Depart-
The Chemical Engineering Department first took
its place among the other engineering curricula and
was duly proclaimed in print in the 1957-58 UA
catalog. It is interesting to note that while it listed
thirteen courses, it listed NO faculty-a problem
quickly remedied, even though those who were inter-
viewed had to pay their own travel expenses. A strong
faculty was assembled by the first department head,
Copyright ChE Dimsion ASEE 1990

View of Tucson, home of the University of Arizona.

Aerial view of the campus.



Don H. White, and the department quickly gained rec-
Due to strong leadership during those initial years,
the department gained the reputation as a research
and education leader with both industrial and
academic sectors of the profession. The early em-
phasis on computer utilization, continued stress on
design, and a strong foundation in chemistry have
brought the department into the present where it en-
joys international respect in the crystallization, com-
bustion, pollution control, biomedical, biochemical,
and turbulent flow areas.
Enrollment has averaged 150 undergraduates and
40 graduate students for the last ten years. An in-

Laser doppler anemometry.

creasing emphasis on doctoral research has changed
the composition of the graduate enrollment toward a
majority pursuing the PhD. These students benefit
from working at a highly technological campus with
well-equipped laboratories.
The University provides several outstanding
facilities of interest to students and faculty in chemical
engineering. The Main and Science libraries contain
over 2,500,000 volumes, and the combination ranks
21st nationally. The University Computer Center
houses a CYBER 175, an IBM 4381, and three VAX
11-780's, one VAX-8650, one SCS-40 mini-supercom-
puter, and a large microcomputer laboratory. An IBM
3090 is used for financial and student records. Addi-
tionally, the University is one of eight member institu-
tions in the Consortium for Scientific Computing at
the John von Neumann Supercomputer Center in
Princeton, New Jersey. Direct satellite link to this
center provides interactive capabilities. In addition to
those campus-wide facilities, the department has a
multitude of student-accessible personal computers,
both for undergraduate education and graduate re-
search, including data acquisition, and numerous
minicomputers serving similar purposes. The College
of Engineering and Mines has several computers on a
network which are accessible to chemical engineering
faculty and students. Some of those are MV10000 and
MV8000 Data General computers (the 10000 has an
array processor), three VAX 11/750's, one VAX 11/
780 and several DEC minicomputers of various sizes.
All computer systems are accessible through interac-
tive terminals and microcomputers located in the
chemical engineering building. All computers on both
the engineering and the university networks can send
and accept files to and from the others.
The departments of mathematics, physics, chemis-
try, biochemistry, microbiology, pharmacy, electrical
081MY&MIa~a~ ra.- lp-ddw ~~a~a

Coal combustion research lab.

Two-story combustion facility.

The graduate program is designed to provide advanced work with core courses in
transport phenomena, thermodynamics, and reaction engineering, with additional selected work
in mass transfer, heat transfer, fluid dynamics, control theory, and process simulation.

and computer engineering, and aerospace and
mechanical engineering supply chemical engineering
with the support necessary for good undergraduate
and graduate research programs. Projects are done
jointly by undergraduate and graduate students and
faculty in chemical engineering, and in many cases
other departments provide service to and/or coopera-
tion with chemical engineering. A high degree of
cooperative research and course-sharing exists be-
tween departments on the campus, assuring a rich
and diverse atmosphere for research and study.

The original faculty member, Don H. White, was
called upon by the Dean of the College of Mines to
organize a chemical engineering department in 1958.
He arrived in Tucson in September of 1958 to find
eager junior and sophomore classes of chemical en-
gineers expecting to be taught, but empty chemical
engineering laboratory rooms and no other faculty.
He quickly organized a small group of faculty to begin
teaching the students who had been admitted in the
fall of 1956. This first faculty consisted of Robert
Damon, fresh out of Montana State, Richard Edwards
from Mallinkrodt Corporation in St. Louis, and James
Carley, engineering editor of Modern Plastics in New
York City. Also here, for a short time, were James
Hall and Ray Richardson.
Other early faculty included Edward Freeh in
1962, Neil Cox in 1962, Alan Randolph in 1968,
Thomas Rehm in 1966, and Richard Williams and Wil-
liam Cosart in 1968. A second round of "early faculty"
included Jost Wendt in 1972, Joseph Gross in 1972,
and James White in 1971. At that time a new young
university president (John Schaefer) led the univer-
sity in its push to become a research university.
Not all early faculty stayed in the teaching profes-
sion. Robert Damon is now with Crown Zellerbach,
James Carley is with Lawrence Livermore Labs, and
James White is vice president of his own successful
computer company (in partnership with a UA alum-
nus, who is president). Richard Edwards retired in
Tucson after becoming a vice president of the Univer-
sity of Arizona. Neil Cox went to the Idaho Nuclear
Test Facility, and Richard Williams ended up at
Exxon Research and Development. All the others are
still active faculty in our chemical engineering group.
William Cosart is Associate Dean of our College of
Engineering and Mines.

Faculty: University of Arizona
MILAN BIER Fordham University
protein purification
HERIBERTO CABEZAS University of Florida
solution thermodynamics
WILLIAM P COSART Oregon State University
Associate Dean
EDWARD J. FREEH Ohio State University
control (adjunct)
JOSEPH F. GROSS Purdue University
biomedical transport processes; department head 1975-1981
ROBERTO GUZMAN North Carolina State University
GARY K. PATTERSON University of Missouri-Rolla
fluid mechanics; department head 1984-1990
THOMAS W. PETERSON California Institute of Technology
ALAN D. RANDOLPH Iowa State University
THOMAS R. REHM University of Washington
process design
FARHANG SHADMAN University of California, Berkeley
chemical reaction engineering
JOST O.L. WENDT Johns Hopkins University
DON H. WHITE Ohio State University
polymer processes
mixing; (visiting half years)

As can be quickly discerned from this short his-
tory, the faculty in this department have been re-
markably loyal. Undoubtedly, part of the reason is
the persuasiveness of Don White, who was depart-
ment head until 1974. Other factors contributing to
such a stable faculty are the unduplicated climate and
quality of life in Tucson and the excitement of being
part of a rapidly growing university.
More recent faculty are Thomas Petersen,
Farhang Shadman, Gary Patterson, Milan Bier,
Heriberto Cabezas, and Roberto Guzman. The pres-
ent faculty, their origins, and their main professional
interests are shown in Table 1.


The University of Arizona is located in the heart
of Tucson, a metropolitan area of over 700,000 people.
Tucson is located sixty miles north of Mexico, at an


elevation 2,400 feet and in a valley surrounded by rug-
ged mountains. Not much more than a century ago
Tucson was a small, dusty Mexican outpost-a pueblo.
Life in Tucson places emphasis on the out-of-doors.
The sun shines more than 300 days each year, making
outdoor recreation a way of life for many Tucsonans
(exemplified by the thirty-one golf courses in the
area). Tennis, golf, and swimming are obvious favor-
ites, but Tucson offers much more than that. Within
an hour of the campus are the ski resort of Mount
Lemmon, the internationally acclaimed Arizona-Son-
ora Desert Museum, and one of the world's largest
accumulations of telescopes at Kitt Peak National Ob-
servatory. Backpacking and off-road vehicle trails are
abundant and easily accessible. Camping equipment
can even be rented from the campus Hiking, Outing,
and Travel Center. For longer treks there are the ski
resorts of the White Mountains and the incredible
Grand Canyon. Tucson city parks and golf courses also
provide varied recreation. For those who like to stay
close to home, on campus there are several restau-
rants, a game room, three swimming pools, handball/
racquetball courts, a weight room, a gymnasium, and
a movie theater. A new comprehensive student recre-
ation center is being constructed to add to these

The undergraduate program in chemical engineer-
ing at the University of Arizona has developed as one
of strength in fundamental chemical engineering
utilizing basic science, requiring thirteen semester
units of math, eight semester units of physics, twenty-
three units of chemistry, and three units of any other
advanced-level science. The math strength is consoli-
dated by six units of applied math taught within the
department. Altogether, thirty-seven units of courses
are required within the department, along with fifteen
units of other engineering outside the department.
The addition of six units of freshman composition and
sixteen units of humanities and social sciences leaves
room for ten units of technical electives. The technical
electives must be chosen from a list of specialties such
that depth of study is achieved in some field.
Such a program of study is rigorous and attracts
only the most motivated students. Even with a heavy
study load, most students find time to participate in
the student AIChE chapter, which has Friday after-
noon picnics several times each year. The chapter also
organizes the annual open house for high school stu-
dents and sends members to talk to high school classes
about chemical engineering. Each year the chapter
sends a delegation to the regional student meeting.

The graduates of the undergraduate program have
been very successful when entering industry or
graduate programs at other universities. This is prob-
ably because they can rely on the strong fundamental
science base that the curriculum provided-they are
prepared to learn any specialized field of chemical en-
A relatively recent trend, which is of great benefit
to the undergraduates, is their participation in the
various graduate research programs. All the students
seem to be highly motivated by their inclusion in the
laboratory research of the graduate programs. An
added benefit is that acquainting them with the joys
of research and the graduate student culture undoubt-
edly results in more of them pursuing graduate work

Altogether, thirty-seven units of courses are
required within the department, along with fifteen
units of other engineering outside the department. ...
Such a program is rigorous and attracts only
the most motivated students.

before taking a job. Additionally, contact with the
other undergraduates and discussions of their re-
search work convinces them of the value of the re-
search projects. Such knowledge breeds understand-

The Department of Chemical Engineering has a
strong and diverse graduate program. The Master of
Science degree has been offered for many years (the
first degree was awarded in 1961) and the first Doctor
of Philosophy was awarded in 1964. Since that time
144 MS degrees and 23 PhD degrees have been
granted. The department is heavily involved in a
transition from a primarily MS graduate program to
a primarily PhD graduate program. The ratio of PhD
candidates to MS students has increased in the last
four years from less than 0.3 to greater than 1.0.
Graduate studies in chemical engineering are ad-
ministered by a graduate studies committee of the fac-
ulty under the general direction of the Graduate Col-
lege. Work toward a master of science degree consists
of a mix of course work and research leading to a
thesis. The doctor of philosophy degree requires some
additional course work (including a minor in a related
area) and extensive research leading to a dissertation.
Along the way a qualifying examination, a prelimi-
nary examination, and a final oral examination must
be passed.
All graduate students must take, or have equiva-
lent credit for, three basic courses: Advanced Trans-


port Phenomena, Advanced Thermodynamics, and
Advanced Chemical Reaction Engineering.
The graduate program is designed to provide ad-
vanced work with core courses in transport
phenomena, thermodynamics, and reaction engineer-
ing with additional selected work in mass transfer,
heat transfer, fluid dynamics, control theory, and pro-
cess simulation. The following interdisciplinary op-
tions with several courses in each are also available:
biomedical engineering, bioprocess engineering,
energy systems engineering, and materials engineer-


Each department of chemical engineering develops
its own style and philosophy of research. At Arizona
neither pure engineering science nor applied process
development is emphasized at the expense of the
other. Rather, the approach is to apply fundamental
engineering science to the solution of process-relevant

Departmental Research Topics

Aerosol Dynamics: Physical and chemical mechanisms governing
particle formation, transport, and transformation application to
atmospheric problems, combustion and clean rooms.
Biochemical and Biomedical Engineering: Transport phenomena
in the microcirculation development of physiological pharma-
cokinetic models thermodynamic properties of protein solutions
protein separation andpurification techniques dynamic model-
ing of kidney-stone formation bioreactor design and scale-up *
transport in fermenters
Catalysis and Kinetics: Sulfur oxide emissions control via catalyzed
lime reactions catalytic gasification of coal
Combustion: Control ofNOx/SOx during coal combustion; returning
as a method of NOx control mechanisms of fly ash formation in
pulverized coal combustion model characterization of flame
Computer-Aided Design: Development of unit operation modules
for undergraduate education
Crystallization: Use of chemical inhibitors as nucleation/growth rate
modifiers laboratory modeling of kidney stone formation real-
time control of crystal population dynamics use of supercritical
fluids as crystallization media crystallization in microgravity
Fuels: Fuel combustion efficiency and relationship to pollutant
formation coal gasification and liquefaction high-pressure
liquefaction of biomass
Polymers: Viscoelastic properties of polymers shear degradation of
polymer molecular weights polymer extrusion methods
Separations: Large scale free-flow electrophoresis as a means of
protein separation and purification
Solar Energy: Solar desalination of sea water and brackish solutions
solar desiccation methods for air conditioning
Surface Analysis: Scanning auger spectroscopy of fly ash particles
and catalysts
Thermodynamics: Modeling of multicomponent and polyelectrolyte
properties in solution
Turbulence: Effect of mixing on reactor yields laser-doppler
measurement of turbulence properties in mixing computer simu-
lation of turbulent mixing and reaction

problems of major engineering concern. This is indeed
a formidable task and requires good relationships and
communication with the industrial world where such
process-relevant problems are identified. About half
of the chemical engineering faculty members at
Arizona have had several years experience in industry
and are encouraged to maintain close industrial ties
through consulting arrangements.
Our department plays a significant role in the
energy and biotechnology fields. In energy-related
work, numerous projects emphasize both the
economic and environmental aspects of current energy
concerns. These projects combine practical experi-
mental and theoretical studies with the latest in
analytical tools.
Biochemical engineering studies are being con-
ducted in a number of new and exciting areas. In fer-
mentation and bioreactor studies, microorganisms and
enzymes are used to generate useful products. In the
bioseparation area, mixed products are separated and
purified. Our bioseparation group is one of the pre-
mier groups in the world, particularly in the area of
electrophoretic methods of separation. Other separa-
tion methods, notably two-phase aqueous extraction
and chromatography, are under development.
An interdisciplinary program emphasizing mate-
rials processing is available in cooperation with the
Materials Science and Engineering Department. Simi-
larly, courses of study are also designed to include
electronic materials processing through the Electrical
and Computer Engineering Department. Current re-
search topics in the department are described in Table 2.
Research funding from outside agencies and com-
panies has grown dramatically in the last five years,
particularly since inclusion of the Center for Separa-
tion Science. Total funding is now about $1.5 million
for a permanent faculty of eleven. Growth in the
biotechnology area was recently spurred by a univer-
sity program which provided new capital and support

The department has a rigorous undergraduate pro-
gram which has produced graduates who have contri-
buted to the chemical engineering profession since its
first graduating class in 1960. Many entered graduate
studies at other universities and did well in their pro-
grams, as indicated earlier.
From the beginning, the department has worked
hard at building its graduate program, with the aim
of having a PhD program of national impact. That
capability grew sporatically at first, but it was marked
Continued on page 60.


Random Thoughts ...


2. Susan and Glenda

North Carolina State University
Raleigh, NC 27695

Susan and Glenda are seniors in chemical en-
gineering at a private northeastern university. They are
both bright and personable. They like to study with
friends and enjoy the lengthy bull sessions that the
study sessions sometimes turn into. They both have a
hard time saying no to requests for help with class-
work, even if they don't have the time for it. Neither one
cares for laboratory courses. They have almost identical
grade point averages-about 3.2/4.0.
The resemblance ends there, however. Susan was
an outstanding student in junior high and high school,
and in college she has gotten B's in almost all of her
courses, with an occasional A. Her instructors have an
easy time grading her homework and test papers: the
solutions are neatly laid out, with each step clearly fol-
lowing the preceding one, and she gets a great deal of
credit even when her answers are incorrect.
Glenda is another story. Her transcript is a mix-
ture of A's and C's. She usually starts out in a class by
doing poorly on the homework and failing the first quiz,
and she may spend the rest of the semester trying to
catch up. Her problem solutions are jumbles of appar-
ently unrelated numbers and equations with the
answer magically appearing at the end; she rarely gets
much partial credit, and if anyone asks her to explain
what she did, she has an extremely difficult time doing
Sometimes, however, Glenda seems to undergo a
transformation. She begins to solve homework and test
problems with ease, occasionally using methods that
were not taught in class. She may then go on to get an
easy A in the course, or, if the class moves on to com-
pletely new material, she may revert to her previous
performance level and struggle until either another
breakthrough is achieved or the semester ends. Even
after she makes a breakthrough, her problem solutions
are frequently incomprehensible to anyone else; the
difference is that the answer that suddenly appears at
the end is correct. She has been hurt on several occa-
sions by instructors who implied that she had cheated,
although no one ever had any proof. (In fact, she never
Susan is a sequential learner, Glenda is a global
learner (1}. Sequential learners tend to gain under-
standing in a linear fashion, with each new piece of in-

formation building logically from previous pieces. They
tend to solve problems the way they learn-in a linear,
stepwise fashion-and their solutions make sense to
others. They generally have little trouble in school be-
cause of their sequential way of learning and solving
problems: their courses, books, and teachers are all
geared to their style.
Global learners function in a much more all-or-
nothing fashion. They absorb information almost ran-
domly, in no apparent logical sequence. In conse-
quence, when they are first learning a subject nothing
may make sense to them, and they may be incapable of
solving trivially simple problems. But then at some
point a key piece of data is taken in, a critical
connection is made, the light bulb goes on, and they "get
it." They may be fuzzy about details after that, but they
see the big picture in a way that most sequential
learners never achieve. Thereafter, when presented
with new material that they can fit into this picture,
they may appear to assimilate it instantly, and when
solving problems they may leap directly to the solution
without seeming to go through the required
intermediate steps. They may also see surprising
connections between newly-learned material and
material from other subjects and disciplines.
Strongly global learners often have difficulty in
school. Before they make their mental breakthrough in
a given subject, their struggle to solve problems that
their sequential counterparts handle with ease makes
them feel stupid. Even after they make breakthroughs,
their inability to explain their problem-solving pro-
cesses can get them into trouble, as when Glenda was
suspected of cheating. These difficulties-which most of
them experience from the first grade on-are truly un-
fortunate, since global learners collectively constitute
one of society's most valuable and underutilized re-
sources. If they are allowed to progress in their seem-
ingly disjointed manner, some of them will go on to be-
come our most creative researchers, our systems
analysts-our global thinkers.
Felder and Silverman [1] suggest ways that en-
gineering instructors can accommodate the learning
styles of global learners. Most of these suggestions in-
volve providing a broad perspective on the course mate-
rial, relating it to material in other courses and disci-
plines and to the students' prior experience. Perhaps
the best thing we can do for these individuals, however,
Continued on page 11.

( Copyright ChE Division ASEE 1990



Edwin N.


... of Wsconsin

University of Wisconsin
Madison, WI 53706

ED LIGHTFOOT is an extraordinary individual
driven by enormous energy and curiosity. He is
an engineer in Olaf Hougen's sense of the word:
Successful solutions to industrial problems de-
pend upon engineering judgement and experi-
ment with the unknown and undocumented
science as well as with the principles that have
already been well established. This is the prin-
cipal distinction between the scientist and the

The broad interdisciplinary nature of Ed's profes-
sional interests has led to significant contributions in
a number of fields. He is sought and respected as a
collaborator by physicians, biologists, physiologists,
mathematicians, engineers, and businesspeople. He
has produced forty PhD students, published two
books, and written nearly two hundred research pub-

Honors: E. N. Lightfoot

* 1962
* 1972
* 1975
* 1979
* 1979
* 1980
* 1984
* 1985
* 1985
* 1986
* 1986
* 1987
* 1988
* 1988

Fulbright Scholar Norway
Erskine Fellow New Zealand
Wm. Walker Award AIChE
Elected to National Academy of Engineering
Food, Pharmaceutical and Bioengineering Award AIChE
Hilldale Professor University of Wisconsin
Lacey Lectureship Caltech
Elected to Norwegian Society of Sciences and Letters
Honorary Doctorate Norway
Stanley Katz Lecturer City College of New York
Goff Smith Lecturer University of Michigan
Kloor Memorial Lecturer Indian Institute of Sciences
Reilly Lecturer University ofNotre Dame
Benjamin Smith Reynolds Award for Teaching Excellence

locations appearing in AIChE Journal, Journal of the
American Chemical Society, Industrial and En-
gineering Chemistry, Chemical Engineering Science,
Journal of Chemical Engineering, Journal of Physi-
cal Chemistry, Journal of the Electrochemical Soci-
ety, Applied Microbiology, Transactions of the Fara-
day Society, International Journal of Heat and Mass
Transfer, Chimie et Industrie-Genie Chimique, Jour-
nal of Biomedical Materials Research, Biophysics
Journal, Physics of Fluids, Journal of Undersea
Biomedical Research, Annals of Biomedical Re-
search, Science, Separation Science, Journal of Food
Science, Progress in Water Technology, Journal of
Applied Physiology, Respiration Physiology, Jour-
nal of Theoretical Biology, Laboratory Animal Sci-
ence, Water Resources Research, Biotechnology and
Bioengineering, Environmental Research, Protein
Purification, Environmental Toxins and Chemicals,
Israel Journal of Chemistry, Chemical Engineering

Copyright ChE Division ASEE 1990


His incredible record of productivity carries over to the classroom, and his students
are the beneficiaries of his efforts to bring new ideas and challenges to his courses. He is an
extremely enthusiastic lecturer whose courses are at the frontier of knowledge.

Lightfoot Students

A.D. Di Benedetto
G. B. Wills
D. W. Hubbard
E. L. Cussler
D. O. Cooney
R. E. Safford
K. A. Solen
H. O. Ozbelge
T. A. Hatton
B. O. Palsson
A. M. Lenhoff

Who are Currently Teaching

University of Connecticut
Virginia Polytechnic Institute
Michigan Technical University
University of Minnesota
University of Wyoming
Mayo Clinic
Brigham Young University
Middle East Technical University
Massachusetts Institute of Technology
University of Michigan
University of Delaware

Education, Critical Reviews in Food Engineering,
Recent Developments in Separation Science, Annual
Reviews of Fluid Mechanics, and Journal of Phar-
maceutical Science.
His incredible record of productivity carries over
to the classroom, and his students are the bene-

ficiaries of his efforts to bring new ideas and chal-
lenges to his courses. He is an extremely enthusiastic
lecturer whose courses are at the frontier of knowl-
edge. They are progress reports on the continual
evolution of his thoughts on science and pedagogy. Of
particular value to his graduate students is his skill in
the one-on-one interaction necessary in innovative re-
search at the doctoral level.
Ed is the recipient of the 1989 Benjamin Smith
Reynolds Award for excellence in teaching in the Col-
lege of Engineering at the University of Wisconsin.

As everyone knows, Ed's gregarious
and talks very fast (it's hilarious);
with such animation
and hyperventilation
he nearly blacks out It's precarious.

on Ed's 60th birthday

Ed's love of being on the go is apparent by his
travel schedule. The East Coast, West Coast, Rocky
Mountains, India, Norway, New Zealand, Finland,

Warren Stewart, Bob Bird, and Ed Lightfoot, authors of Transport Phenomena.


Sweden, Taiwan, Australia, Canada, Switzerland,
Denmark, Czechoslovakia, Mexico, and Spain are all
on his itinerary. He was elected to the Norwegian
Society of Sciences and Letters, received an Honorary
Doctorate at the Norwegian Technical University at
Trondheim, and was an Erskine Fellow at the Univer-
sity of Christchurch in New Zealand. At a different
pace are his visits, with his wife Lila and five children
(together with their dog, Rascal), to his cabin in the
woods north of Madison, and his bicycle rides and
cross-country ski tours through the lovely Wisconsin
Ed was born in Wisconsin 64 years ago and re-
ceived his college education at Cornell University. A
young John Prausnitz was taught mechanical drawing

at Cornell by an equally young Ed Lightfoot. After
several years at Chas. Pfizer and Company, where he
developed a patented process for vitamin B12 produc-
tion and purification, Ed joined the University of Wis-
consin in 1953 to lead one of the first bioengineering
programs in the United States.
At that time Olaf Hougen was promoting the de-
velopment of educational material in transport
phenomena. Ed joined Bob Bird and Warren Stewart
in an enormously productive period of several years
to produce the classic book Transport Phenomena.
Much of the writing was done in a small cabin Bob
Bird rented on the north shore of Lake Mendota to
which Ed often commuted by canoe and sailboat. That
book has gone through over forty printings and has
been translated into Italian, Czech, Spanish, and Rus-

Lightfoot: "My Most Satisfying Research Projects."

/ E. N. Lightfoot and R. J. Taylor
"Recovery and Purification of Vitamin B12"
U. S. Patent 2,787,578 (1954)
This represents a major engineering effort at Chas. Pfizer
and Co., which resulted in a full-scale production process
for crystalline vitamin B,2 Ed found this type of process
development work particularly attractive and has returned
to this first love in his current research on the production of
chemicals from agricultural wastes.

/ J. B. Angelo and E. N. Lightfoot
"Generalization of the Penetration Theory for Surface Stretch:
Application to Forming and Oscillating Drops"
AIChE Journal, 12, 751-760 (1966)
This work showed the applicability of asymptotic analysis in
complex process equipment, as well as providing a mass-
transfer model which has proven useful for applications as
widely different as direct-contact heat transfer and oxy-
genation of sewage in U-tube aerators.

/ E. M. Scattergood and E. N. Lightfoot
"Diffusional Interaction in an Ion-Exchange Membrane"
Transactions of The Faraday Society, 64, 1135-1146 (1968)
This was the first measurement of all of the multicom-
ponent diffusivities in a membrane system and the first
comprehensive treatment of boundary-layer polarization in
test equipment.

/ V. Ludviksson and E. N. Lightfoot
"The Dynamics of Thin Films in the Presence of Surface-Tension
AIChE Journal, 17, 1166-1173 (1971)
This is one of a series of papers describing the effects of
surface tension gradients on the dynamics of thin sup-
ported liquid films. These are important in a variety of sys-
tems, including electrochemical cells and secondary
recovery in oil wells.

/ E.N. Lightfoot, A. Baz, E.H. Lanphier, E.P. Kindwall,
and A. Seirig
"Role of Bubble Growth Kinetics in Decompression"
VI Symposium on Underwater Physiology, San Diego,
California (1975)
This work predicted that the existing basis of decompres-

sion tables for divers and tunnel workers was unsound, a
fact borne out by experimental studies reported at the
same symposium.

/ J.F.G. Reis, P.T. Noble, A.S. Chaing, and E. N. Lightfoot
"Chromatography in Beds of Spheres"
Separation Science and Technology, 14 367-394 (1979)
This research developed among the most practical
descriptions of gradient elution chromatography, the
mainstay of commercial protein fractionation.
/ A.M. Lenhoff and E.N. Lightfoot
'The Effects of Axial Diffusion and Permeability Barriers on the
Transient Response of Tissue Cylinders"
J. of Theoretical Biology,,106, 207-238 (1984)
This work provides the basis for an understanding of
transient mass transfer processes in the microcirculation,
applied to such clinical situations as the identification of
abnormal metabolism in the brain.
/ B.O. Palsson, H. Palsson, and E.N. Lightfoot
"Mathematical Modelling of Dynamics and Control in Metabolic
J. of Theoretical Biology, 113, 279-298 (1985)
This is the latest in a series of pioneering papers which
helped to open the field of the dynamics and control of
metabolic networks of chemical reactions to systematic
/ E.N. Lightfoot and M.C.M. Cockrem
"What Are Dilute Solutions?"
Separation Science and Technology, 22, 165-189 (1987)
This was a first major step in continuing efforts to speed the
development of separations equipment and processes by
relating economic goals to the underlying physico-chemi-
cal principles.
/ J.C. Liao and E.N. Lightfoot
"Applications of Characteristic Reaction Paths: Rate Limiting
Capability of Phosphofructolkinase in Yeast Fermentation"
Biotechnology and Bioengineering, 126, 253-273 (1988)
This work shows how to locate bottlenecks in metabolic
networks by using systems analysis techniques. It points
out the importance of systems analysis in commercial


sian. It has been a regular "citation classic" for being
referenced in the literature with a frequency compar-
able to such classics as Abramowitz and Stegun
(Handbook of Mathematical Functions) and Carslaw
and Jeager (Conduction of Heat in Solids). It is recog-
nized as the most important textbook in the profession
in the last quarter century.
In 1974 Ed published the pioneering text Trans-
port Phenomena and Living Systems which showed
how to use the art and science of chemical engineering
to solve important bioengineering problems by includ-
ing details of the physiological and pharmacological
Recently he has devoted his efforts to the develop-
ment of modern biotechnology, with special emphasis
on the engineering of metabolic pathways and mate-
rials separations. He was the driving force in the or-
ganization of the Bioprocess and Metabolic Engineer-
ing Consortium at the University of Wisconsin. With
the support of Abbott Labs, Agracetus, APV Crepaco
Inc., Becton Dickson, Bio-Technical Resources, Du-
Pont, Kraft, New Brunswick Scientific Co., Procter
and Gamble, Promega Biotech, Universal Foods and
Xylan, the consortium promotes the use of biological
organisms and biochemical processes to produce spe-
cialty chemical products.
Just this year Ed led a University/Industry/State-
of-Wisconsin team in the development and design of
an industrial process to produce high purity lactic acid
from waste cheese whey. This industry seeks to pro-
duce valuable chemicals and jobs from a particularly
troublesome waste product of one of the state's largest
industries. This service to industry and state follows
Ed's successful approach to research in combining the
science and practice of engineering. OE

Continued from page 7.
is to watch for them, and when we find them (which we
will), explain and affirm their learning process to
them. They probably already know all about the draw-
backs of their style, but it usually comes as a revelation
to them that they also have advantages-that their
creativity and breadth of vision can be exceptionally
valuable to future employers and to society. Any en-
couragement we provide could substantially increase
the likelihood that they will succeed in school and go on
to apply their unique abilities after they graduate.

Postscript: 10 years later

Susan graduated and went on to get a masters de-
gree in chemical engineering, got a number of good job

offers, and went to work in the process design division
of a large petrochemical company. She did extremely
well and is now making rapid progress up the technical
management ladder. Glenda went through a lengthy
job search when she graduated-all those C's on her
transcript worried prospective employers-and finally
found a position with a small firm of design consul-
tants. Her first project involved designing and
installing process simulation software for a
pharmaceuticals manufacturer. She did almost
nothing on the project for months, despite increasing
pressure from her supervisor. Then she came up with a
package that not only did the required simulation but
also used it to schedule production, manage inventory,
and determine production bottlenecks and the best
methods of eliminating them. The company estimated
that the program led to savings of two million dollars in
its first year of use. Glenda now gets the problems too
difficult for anyone else in the firm to solve. Sometimes
long periods go by without any apparent results, but no
one pressures her any more. O

11] R.M. Felder and L.K. Silverman, "Learning and Teaching Styles in
Engineering Education," Engineering Education, 78(7), p.674 (1988).
Susan is a representative sequential learner and Glenda is a
representative global learner, but not all sequential are just like Susan
and not all global are just like Glenda These labels simply denote ten-
dencies that may be strong or weak in any given individual, and
everyone exhibits characteristics of both types to different degrees.

I E NEEN EDUI ATION 1 0 919|0 0 9/.....8
i uaY yTrl 4 see Attach.d 9Rt.e
CHEICAL GINEERING EDUCA TIONf 9od 31', chemical Egineerlng 03a3rtent
UnesiLty of Floida, .n2v0 le1, lhua, Flo.. ida 32611 2
Ceical Enqineerlnq D9Visfon Ar0ca society for Ag=En n Educaton.

ASEE he0i0cal Enqlnlrnq OlvlsIon, 11 D3pont Circle, Washlnqton, C 20030
ay W. Flen, ~~e .Ical glne-n rl G Dep 1tmnt. o. o 319,

*Cao 3__ h,8 0tcall o3inloia.ng .ap.n.t. 7 3 17,
Un- rs ty of Flo da., G -neille, FL 32611

1 ff or3 a- 1r.stedt. h.u. uld 9

o., 23l73 .... ,... ___ .0._________

G- ..--l...,. 3r 73


:,~"~""='~'~..c---.c~=~.~.~.,.~., i -uau~,.lm*ma~*.la ~n,.M
I_ rull*~ i Mn ~n. ~ru

* AwardLecture



Part I. Pure Fluids

University of Delaware
Newark, DE 19716

THERMODYNAMICS AND physical properties are
central to the practice of chemical engineering.
This is evident from the fact that 70 to 90% of both
investment and energy operating costs in a typical
chemical plant involves the separations and purifica-
tions equipment which are designed largely on the
basis of phase equilibrium. Further, the complete flow
sheet of a chemical plant may depend on whether an
azeotrope or two liquid phases are formed somewhere
in the process. With the availability of modern process
simulators it is usually the uncertainty in ther-
modynamic behavior, rather than the design al-
gorithms or calculational complexity, which presents
the biggest difficulty in accurate process design.
Because of their importance in process design,
many thermodynamic and physical properties models
have been developed. Indeed, there are more than
100 variations of the van der Waals equation of state
in addition to numerous other equations of state and
activity coefficient models. A problem that arises in
teaching thermodynamics to chemical engineering stu-
dents is providing a coherent scientific (rather than

Stanley I. Sandier earned a
i..-.B BChE degree from the City College
of New York in 1962 and his PhD
from the University of Minnesota in
1966. His research at Minnesota
was on the kinetic theory of gases.
He was a NSF postdoctoral fellow at
the Institute for Molecular Physics
at the University of Maryland for the
1966-67 academic year. He joined
the faculty of the University of
SDelaware in 1967 and was its chair
from 1982 to 1986.
His research has mainly been in the field of thermody-
namics and fluid properties, with specializations in molecular
theory, computer methods, and the experimental measure-

merely empirical) basis for these models. A related
problem is introducing students to the use of molecu-
lar theory for the development of thermodynamic
models. This paper presents a framework which al-
lows one to identify the molecular level assumptions
underlying many thermodynamic models. We then
continue on to test these assumptions using theory
and computer simulation and to show how we can
make better assumptions which lead to improved mod-
els. Here we consider only pure fluids and their equa-
tions of state; in Part 2 (to be published in the next
issue of CEE) we will consider mixtures and activity
coefficient models.

The Generalized van der Waals Partition Function
The molecular theory from which one can derive
thermodynamic models is statistical mechanics. For
the case in which the temperature T, volume V, and
number of particles N are the independent variables,
the canonical partition function
Q(N,V,T)= I e-E'(NV)/kT

ment of phase equilibrium. He is the author of 135 papers,
the editor of five conference proceedings books, and author
of the textbook Chemical and Engineering Thermodyanmics,
which has been translated into Spanish and Chinese. The
second edition has just appeared.
He was cochair of the Chemical Engineering Faculty
Summer School of the ASEE in 1982, and was originator and
chair of the 1988 Chemical Engineering Education in a
Changing Environment Conference, held for the purpose of
introducing new technology examples into standard under-
graduate courses. He has served on the Editorial Advisory
Board of the AIChE Journal, Industrial and Engineering
Chemistry Fundamentals, and the University of Delaware
Press. He is an ABET/AIChE chemical engineering
accreditation visitor and is on the AIChE subcommittee for
New Technology Educational Materials.

V Copyright ChE Division ASEE 1990


This paper presents a framework which allows one to identify the molecular level assumptions underlying
many thermodynamic models. We then continue on to test these assumptions using theory and computer
simulation and to show how we can make better assumptions which lead to improved models. Here we consider
only pure fluids and their equations of state; part 2 .. will consider mixtures and activity coefficient models.

is the starting point for our work [1]. Here the sum is
over all the quantum states of N molecules in a volume
V, k is Boltzmann's constant, T is temperature, and
Ei is the energy of the system in the ith quantum state.
Once we know the partition function, all other ther-
modynamic properties can be computed as follows
A(N, V,T)= kTinQ(N, V,T) (2)

P=kT (InQ)
V T,

a- T v,N

S= kT -klnQ (4a)

SdlT V,N
Eq. 2, which relates the Helmholtz free energy to
N, V, and T, is one of the fundamental equations of
state in the sense of Gibbs; from it all other ther-
modynamic properties of a fluid can be obtained with-
out any other information, as is evident from the equa-
tions above.
Identifying each quantum state of an assembly of
molecules is at present an impossible task except for
special cases such as the ideal gas. For the case of
relatively simple molecules (for the moment excluding
long chain hydrocarbons or polymers) the total energy
of an assembly of molecules can be separated into
translational (t), rotational (r), vibrational (v), elec-
tronic (e), and interaction (i) energies, each of which
is independent of the others. Further, except for the
interaction energy term, each of the contributions is
a sum of the energies of the individual molecules.
Therefore, for a pure fluid of N identical molecules we
Q(N,V,T)= e-(E +Er+E,+E.+fE)/kT

1 N ))N N N Z(N,V,T)
= R.(q,(T)) (qr(T)) (q,(T)) (q.(T)) VN
Here q,, qv, and qe are the single particle rotational,
vibrational, and electronic partition functions which
are only a function of temperature. Also, qt =
(2lrrmkT/h2)3/2 V is the single particle translational par-
tition function where m is the particle mass and h is

Planck's constant.
Of special interest is the last term, the configura-
tion integral, which arises from the interactions be-
tween molecules. For spherical molecules in a volume
element of macroscopic dimensions, classical mechan-
ics can be used (thereby replacing summations with
Z=f...fe-u(r' rN)/kT dr, dr2... drN (6)

where u(rl,r2,...rN) is the interaction energy when a
molecule is located between position vectors r, and r1
+ dr1, a second molecule between position vectors r2
and r2 + dr2, etc., and the integrals are over all values
of the position vectors within the volume V. It is only
the configurational integral Z which depends upon the
interactions among the molecules and therefore, from
Eqs. 3 and 4, it is the derivative of Z with respect to
temperature that gives information about the average
interaction energy among the molecules. We refer to
this average total interaction energy as the configura-
tional energy EONF.
To proceed further we need to make some state-
ment about the interactions between the molecules.
We will assume that the interaction energy for the
assembly of molecules in any particular configuration
can be computed as the sum of the interaction energies
between all possible pairs of molecules (i.e., the pair-
wise additivity assumption) so that
u(r,r2,...rN)= u(rU) (7)


f r

FIGURE 1. The square-well potential with an unpene-
trable hard wall at r = r.


and for the purpose of illustration here, we will con-
sider two molecules to interact with the square-well
potential of Figure 1

r u(r)= -e acr 0 Ra
though other potential models may be used [2]. This
very simplified model does have the essential features
of a real interaction; it has a repulsive region (r an attractive region (o-rs Rcr), and vanishes at large
The average total interaction or configurational
energy, ECONF, for a fluid of square-well molecules can
be gotten from a simple analysis. If Ne(p,T) represents
the coordination number, that is the average number
of molecules in the well of a central molecule at the
density p and temperature T, then the interaction
energy of that molecule with all others is Nc(p,T)E.
Since there are N choices for the central molecule, the
total interaction energy is

ECONF -NNc(p,T)E (9)
where the factor of 2 accounts for the fact that each
interaction is counted twice as each member of the
interacting pair is considered to be the central
To proceed further it is useful to relate the config-
urational energy to the configuration integral using
Eqs. 3 and 5 as follows
inZ(p,T)=tnZ(p,T =_o)+ i kT E dT
Z(p T) VN(p) ep N (10)
yN ex 2kT)

Here, for convenience, we have defined Z(p,T = 0)
= V(p), where Z(p,T = ) is the configurational in-
tegral at infinite temperature when only hard core





0 (o' #O

FIGURE 2. Free volume as total volume less the volume
around each molecule from which the center of another
molecule is excluded: (a) low density; (b) high density
including overlap of excluded volume regions.

forces are important; Vf is referred to as the free vol-
ume. For the square-well fluid Z(p,T = co) is the con-
figuration integral for hard spheres since only the in-
finite repulsive energy, not the finite attractive
energy, is important at T = o-. The second term

9= kT dT
-2kT i ECONF
which for the square-well fluid is

T= I/T=-
T=- 1/T=-

is the free energy change accompanying a change from
T = co to the temperature of interest, T. We will refer
to < as the mean potential. Combining all of the above,
we have

Q= (qrq,qvq,)N Z(N,V,T)
= -(qrqqe) )(qt V exp(-0 /2kT)) = (qi. N (q ()N
N( t ((qN! )* (13)

where we have grouped the short wavelength rota-
tional and vibrational motions and the electronic
energy term into the internal partition function qint,
and the long wavelength translational motions into an
external partition function qext. Eq. (13), in which the
partition function has been separated into an internal
part, a hard-core part (Vf or Z), and an interaction
part ((), will be referred to as the generalized van der
Waals partition function [3].
In Eq. (13) the internal partition function, qnt, is
a function of temperature but not volume and as such
does not affect the equation of state, though it is im-
portant when computing values of the ideal gas
energy, entropy, and heat capacity. The hard core
part, Vf or Z, will lead to a repulsive or configurational
term in the equation of state, while the mean potential
( will lead to the interaction or residual term, as will
be seen shortly.

Application of Generalized van der Waals Theory
to Equations of State
With this background, we can examine the equa-
tions of state commonly used by engineers in terms of
the assumptions that have been made about the free
volume Vf and the mean potential (. For example,
though not explicitly stated this way, van der Waals
used the literal interpretation of the free volume as
the volume accessible to the center of mass of a new
molecule of diameter a when put into a volume V oc-
cupied by N similar molecules. As shown in Figure
2a, this results in Vf = V NP with p = 2rr(T/3,


where the excluded volume Np is equal to one-half
the volume of N spheres of radius u. (The factor of
one-half arises from considering each molecule of the
pair to contribute one-half of the excluded volume.) In
essence, van der Waals also assumed that the coordi-
nation number is a linear function of density and inde-
pendent of temperature, i.e., Nc = cp, where c is a
constant. We then have that 4 = -N, = -cpe, and
ZIN. =V N" NeNNE' ['V N 'e~ ( c N
Z(N, V, T)= (V Np) exp = [(V N) exp( (14a)

from which we obtain

PkT(/anQ) NkT CEN2 RT a (14b)
P= kT =--+'"- -= (14b)
Sav V-Nb 2V V-b V2

where V = V/(N/Na) is the molar volume, b = NaP,
a = -cN e/2 and Na is Avogadro's number. Con-
sequently, we can now understand the molecular basis
for the van der Waals equation of state in terms of the
assumptions made about the coordination number and
the free volume. Further, we can also relate the

parameters in this equation of state to the inter-
molecular potential function parameters.
Other equations of state can be analyzed in a simi-
lar manner. Table 1 contains the free volume and coor-
dination number models imbedded in some other equa-
tions of state. Clearly different assumptions have been
made for the free volume and coordination number in
each of the equations in the table, and many others
are possible. We can now ask which, if any, of the
models in Table 1 is correct?

Use of Theory and Computer Simulation
to Test Molecular Assumptions
We now need to answer the question raised at the
end of the last section. From statistical mechanics we
know quite a bit about the free volume; the simple van
der Waals model is correct only in one dimension. In
three dimensions it underpredicts the free volume at
moderate and high densities because of the overlap-
ping of the excluded volume regions shown in Figure
2b. However, the Carnahan-Starling expression [9]

Free Volume and Coordination Number Approximations
in Several Equations of State

Equation of State

" van der Waals

* Redlich-Kwong [4]

*Redlich-Kwong-Soave [5]

* Peng-Robinson [6]

* Widom, et al. [7]

* Alder, et al. [8]

* Lattice Gas Models

V-b V2
RT a/l
-b +b)
V-b V(V+b)

RT a(T)
V-b V(V+b)

SRT a(T)
V-b V(V+b)+b(V-b)

SRT+ 1 +T12 -r3] a(T)
PY (1-)3 v2





V exp (3"1- 4)
v (1- )" J

T Ln(1+pp)

C (T) n (1+ Pp)

C,(T) 1+ 1+ )Pp
CTn1+ (1-[2)0p


P RT= 1++ 12 3 f- A \fV ex 2FnA3- 1 -nl /
P RT[ ) LL+ m kT1- jA V exp [n(3l- 4) 2 E

PV [i+ni+ -3] NmVo(e'/2 -1)
RT (1-1 )3 V (+V (e/2" -1)

=[rN+0.77bl NmVo(e'2kT -l)
N-0.42b V[V+Vo(e"/2-1)]

V exp (31-1)

(V 0.42 Nb)28333

NmVo e-z/2
V + Vo[e/2kT -1]

NmV+Vo e2kT
v + v O l e k T- i]


_ -=exp[3(3-4)
V (I _,1)2

with 7, = pp/4 is in almost exact agreement with com-
puter simulation data for hard spheres, while, as can
be seen from Figure 3, the simpler equation of Kim,
Lin, and Chao [10]

V, (V-0.42NP' .8333
Vl V )

is in very good agreement with Eq. (15) and has the
advantage of still producing cubic equations of state
when combined with some coordination number mod-
Choosing among the coordination number models
is more difficult. At low density the exact result for
thPe snnar well fluid is

so t



Absolute Average Deviation in the Compressibility
Factor of the Square-Well Fluid Predicted by
Various Equations of State

Eauatlon of State

van der Waals
Alder, et al
Aim-Nezbeda [13]
Aim-Nezbeda + 3 body
Ponce-Renon [14]
Equation (19)

aad Z


though we can obtain such information from computer
Lm N 4x 3 ekT R3 (17) simulation methods such as Monte Carlo or molecular
p-0 3 dynamics [11]. In brief, by considering many different
hat at low density the coordination number is a configurations of molecules in a volume element which
ar function of density, as is predicted by all the exist only in the memory of a computer, these simula-
els in Table 1, though none has the same temper- tion methods can be used to obtain average values for
'e dependence. At higher densities, we do not have all mechanical variables such as energy, pressure, and
dination number information from theory, al- the coordination number for any chosen intermolecu-
lar potential. Coordination number values so obtained
0 I I I I for the square-well fluid [12] are plotted in Figure 4a
as a function of dimensionless temperature e/kT and
density po3. In Figure 4b we have drawn curves for
some of the coordination number models of Table 1.
There are a number of things to be seen from these
Figures. First, unlike the van der Waals assumption,
Sthe coordination number is a function of both temper-
ature and density, and the density dependence is non-
linear. Second, the density dependence is smooth, ex-
\ cept at the lowest simulation temperature. When we
examined the location of the molecules at these condi-
tions we found that the fluid had separated into re-
gions of high density and others of low density; that
\ is, a phase separation had occurred. (Since we did not
5 impose a gravitational field in our simulations, the
Separation was not of a low density vapor above a
high density liquid, but rather of vapor and liquid re-
gions interdispersed as would occur in a phase separa-
I I I I I I I tion on the space shuttle.) The last and most important
0 2 6 8 10 12 14 16 observation is that none of the coordination number
Sp models in common equations of state are in agreement
with the simulation data.
IRE 3. Free volume as a function of reduced density: Thus, we find that the equations of state commonly
represents the Carnahan and Starling equation used in chemical engineering are reasonably satisfac-
15) and the result of computer simulation; --- is tory, not because they are fundamentally correct but
van der Woals model; and-- is the Kim-Lin-Chao rather as a result of a cancellation of errors between
It (Eq. 16). the free volume (repulsive) and mean potential


(largely attractive) terms. Further, since the expres-
sions used for the mean potential or residual term in
the common cubic equations have been empirically
chosen to give reasonably accurate results when com-
bined with the van der Waals free volume term, this
also means that it would not be very productive to try
to develop better equations of state by improving only
the free volume term (i.e., replacing the van der
Waals term with the Carnahan-Starling or Kim-Lin-
Chao expressions) while leaving the mean potential
term unchanged, or vice versa. Both need to be im-
The coordination number behavior found in our
simulations (except within the two-phase region) can
be described by a simple lattice gas model in which
the likelihood of two neighboring sites being occupied
is proportional to the Boltzmann factor of E/2kT which
leads to [12]

N m Vo eEf/2kT (18)
N V e I (18)
V + V0 (e"kT _1-)

where Vo = N V3//2 is the close-packed volume and
Nm is the coordination number at close packing (18
when R in the square-well potential is equal to 1.5).
The success of this simple, theoretically-based model
in describing the square-well fluid is evident from Fig-
ure 4b.
Using Eq. (18) in the generalized van der Waals
partition function together with the Carnahan-Star-

ling free volume expression (Eq. 15) leads to the rela-
tively simple equation of state
PV 1 +1+12- q NmVo(e/2kT-)
RT (I _)2 [v+Vo (e'/2kT -
In Table 2 the results of this and other equations of
state for the square-well fluid are compared. We see
from that table that the empirical equations (vdW,
PR, and RK) are, in fact, not very good for describing
this fluid. Better is the twenty-three term Alder et al.
[6] equation which is a double power series expansion
in temperature and density, with parameters that had
been fit to their simulation data. The best equation,
however, is Eq. (19) which has no adjustable paramet-
ers! That is, once the parameters in the potential
model have been fixed, there is nothing left to adjust
in Eq. (19) to fit the simulation data. The success of
this relatively simple, theoretically-based equation
over the empirical equations of state is the first exam-
ple of the advantage of using the generalized van der

0 o -

00 01 0.2 0.3 04 05 06 07 08 0.9

0.2 0.4 0.6 0.8

FIGURE 4. Coordination number of the square-well fluid as a function of reduced temperature and density. In all
cases the points are the result of Monte Carlo simulation and the solid lines are the result of Eq. (18).

(a) All simulation results including the two-phase region
at the lowest temperature (e/kT = 1); extent of two
phase region is indicated by the dotted line;

(b) predictions of various equations including the van
der Waals (...), Redlich-Kwong (---) and Peng-Robin-
son (---) models.


Waals theory as a basis for developing thermodynamic
Of course engineers are interested in the equation
of state for real fluids, not merely models such as the
square-well fluid. We show in Figures 5a and 5b how
well Eq. (19) does in describing the phase behavior of
argon and methane. Of even more concern to chemical
engineers is the behavior of more complicated
molecules which are not spherical, and chain molecules
such as hydrocarbons and polymers.

103 L

100 10-1 10-2 10-3

10 1 : I ilJ l iI I I I 1111 I



FIGURE 5. The compressibility of (a) argon and (b)
methane. The points are experimental data for the two-
phase or saturation envelope, and the line results from
Eq. (19).

Extension to Chain Molecules
While, in principle, the generalization of the dis-
cussion above to nonspherical, and especially to chain
molecules, is very difficult, a very clever approximate
formulation was presented by Prausnitz and co-work-
ers [15,16] more than a decade ago in the form of the
perturbed hard chain theory (PHCT). In brief, this
model considers a chain molecule to behave like a
chain of m spherical beads, each of which interacts
with the square-well potential. A difficulty in evaluat-
ing the partition function of a chain molecule is that
some of its rotations and vibrations are unaffected by
the presence of neighboring beads, and can therefore
be treated as in Eq. (5), while others (the long wave
length motions) are hindered. Following a suggestion
of Prigogine [17], these latter degrees of freedom are
assumed to have the same density dependence as the
translational degrees of freedom. Letting C be the
external degree of freedom parameter (which is unity
for atomic fluids) we have

S(p, [ eT) ) (20)

That is, a chain molecule is considered to have (C-1)/3
rotational or vibrational modes which are behaving as
3-dimensional translations, where C is taken to be an
adjustable parameter.
The free volume for this fluid of chains is described


0 0.2 0.4 0.6 0.8 1.0

FIGURE 6. Bubble points of mixtures of methane and
hexadecane at 300 K. The points are the experimental
data of ref. 16, the solid line is the result of the
simplified perturbed hard chain theory and the dashed
line results from the Soave-Redlich-Kwong equation. The
calculations, reported in ref. 16, are predictions in that
no adjustable parameters were fit to the experimental


by the Canahan-Starling term with iq = mpp/4; here
we will replace the 23-term Alder expansion of Table
1 used in the original PHCT with our new single term
expression of Eq. (18). The resulting equation for this
simplified perturbed hard chain theory [18] is
PV 1+(4C-3)y1+(3-2C)12-113 CNmVo(e/2kT-1)
RT (1-t)3 V+Vo (e/2kT -1) (21)

This relatively simple, three-parameter (e, C, and b
or Vo) equation of state has been remarkably success-
ful in describing the properties of pure fields, espe-
cially for large molecules, as shown in reference 18,
and even more successful in predicting the properties
of nonpolar mixtures of molecules of very different
size. This is shown in Figure 6, for the mixture of CH4
+ C16H34 [16] where the predictions (no adjusted
parameters) of the simplified perturbed hard chain
theory are found to be more accurate than those of the
Soave-Redlich-Kwong equation [5].
The success of the simplified perturbed hard chain
equation is another example of the value of developing
thermodynamic models from theory, rather than
merely choosing algebraic functions or a power series
expansion with parameters fit to experimental data or
using power series expansions. Note that if we use
the simpler Kim-Lin-Chao expression (Eqn. 16) for
the free volume, we obtain an even more simplified
perturbed hard chain equation

PV V+b(1.19C-0.42) CNmVo(e/2kT-1)
RT V-0.42b V+V0 (e/2kT 1)
The properties of this three-parameter cubic equation
of state have not yet been thoroughly studied.


We leave the reader first with some new equations
of state to explore. More importantly, however, we
also leave him or her with a formulation which allows
one to understand the molecular level assumptions in
the equations of state now being used and a proper
theoretical basis for developing new ones. We have
also shown that many of the equations of state now in
use do not have a good basis in theory. In fact, each
consists of repulsive and interaction (or configura-
tional and residual) terms which are incorrect, but
which have been empirically chosen so that when they
are combined, reasonable results are obtained. Thus
there is much room for improvement and further re-
In the next paper we will consider the extension
of the generalized van der Waals partition function to
mixtures, which allows us to understand and test the

basis for activity coefficient models and equation of
state mixing rules.


This work was supported by Grant No. DE-FG-
85ER13436 from the United States Department of
Energy to the University of Delaware.

1. See, for example, T. Hill, Introduction to Statistical
Mechanics, Addison-Wesley, Reading, MA (1960)
2. Dodd, L.R., and S.I. Sandler, "A Monte Carlo Study of
the Buckingham Exponential-Six Fluid," Molecular
Simulation, 2, 15 (1989)
3. Sandler, S.I., "The Generalized van der Waals Parti-
tion Function. I. Basic Theory," Fluid Phase Eq., 19,
4. Redlich, O., and J.N.S. Kwong, "On the Thermody-
namics of Solutions. V. An Equation of State. Fugaci-
ties of Gaseous Solution," Chem. Rev., 44, 233 (1949)
5. Soave, G., "Equilibrium Constants From a Modified
Redlich-Kwong Equation of State," Chem. Eng. Sci.,
6. Peng, D.-Y., and D.B. Robinson, "A New Two-Con-
stant Equation of State," IEC Fund., 15, 49 (1976)
7. Lonquet-Higgins, H.C., and B. Widom, Molec. Phys.,
8. Alder, B.J., D.A. Young, and M.A. Mark, J. Chem.
Phys., 56, 3013 (1972)
9. Carnahan, N.F., and K.E. Starling, "Equation of State
for Nonattracting Rigid Spheres," J. Chem. Phys., 51,
10. Kim, H., H.M. Lin, and K.C. Chao, "Cubic Chain-of-
Rotators Equation of State," IEC Fund., 25, 75 (1986)
11. Allen, M.P., and D.J. Tildesley, Computer Simulation
of Liquids, Oxford Science Publications, Oxford (1987)
12. Lee, K.-H., M. Lombardo, and S.I. Sandier, "The Gen-
eralized van der Waals Parition Function. II. Appli-
cation to the Square-Well Fluid," Fluid Phase Eq., 21,
13. Aim, K., and I. Nezbeda, "Perturbed Hard Sphere
Equations of State of Real Fluids. I. Examination of a
Simple Equation of the Second Order," Fluid Phase Eq.,
12,235 (1983)
14. Ponce, L., and H. Renon, J. Chem. Phys., 64, 638 (1976)
15. Beret, S., and J.M. Prausnitz, "Perturbed Hard Chain
Theory: An Equation of State for Fluids Containing
Small or Large Molecules," AIChE J., 21, 1123 (1975)
16. Donohue, M.D., and J.M. Prausnitz, "Perturbed Hard
Chain Theory for Mixtures: Thermodynamic Proper-
ties of Mixtures in Natural Gas and Petroleum Refin-
ing," AIChEJ., 24, 849 (1978)
17. Prigogine, I., The Molecular Theory of Solutions,
North-Holland, Amsterdam (1957)
18. Kim, C.-H., P. Vimalchand, M.D. Donohue, and S.I.
Sandler, "Local Composition Model for Chain-Like
Molecules: A New Simplified Version of the Perturbed
Hard Chain Theory," AIChE J., 32, 1726 (1986)
19. Peters, C.J., J. deSwaan Arons, J.M.H. Levelt Sengers,
and J.S. Gallagher, "Global Phase Behavior of Mix-
tures of Short and Long n-Alkanes,"AIChE J., 34, 834






University of California
Berkeley, CA 94720

or at least two generations, chemical engineers
have claimed the ability to "do anything." Because
their education has been so broad, they had both the
basic tools and the self-confidence needed to tackle
nearly any problem. This versatility has remained
even as chemical engineers have shifted their atten-
tion from commodity chemicals to biotechnology or
materials processing. Unfortunately, undergraduate
education has not yet sufficiently changed to meet the
new challenges. The prevailing repertoire of home-
work problems and classroom examples does not
adequately reflect the advent of new fields and,

FIGURE 1. Woman Before a Mirror, Pico (1930).
FIGURE 1. Woman Before a Mirror, Picasso (1930).

perhaps more serious, has failed to show the growing
interconnections between chemical engineering and
societal concerns.
The dichotomy between current chemical en-
gineering practice and what is commonly presented to
undergraduates may be illustrated by analogy to
Picasso's painting, Woman Before a Mirror (1930),
shown in Figure 1. A woman gazing at herself in a
mirror sees a distorted profile rather than an accurate
representation of her face and figure. The painting
symbolizes her inability or unwillingness to see herself
as a complete, integrated whole; she can only see a
part of herself. Similarly, by confining illustrative
examples in undergraduate chemical engineering to
traditional topics, we fail to reflect properly how
"real," contemporary chemical engineering is prac-
ticed, and how intimately our branch of knowledge is
related to issues of wider scope.
In practice, chemical engineering does not exist in
a vacuum. As a field of knowledge, it is closely con-
nected with many other disciplines. Therefore, mod-
ern chemical engineering education must transcend
the compartmentalization of academic subjects. Stu-
dents must be shown that what they learn in the class-
room relates to the world outside. This relationship is
best established through illustrative classroom exam-
ples and homework problems.
We present here three such problems, drawn from
current societal concerns; these problems link chemi-
cal engineering with broad policy issues.

The depletion of the ozone layer
A nuclear-winter scenario
Air pollution by chemical solvents

In these examples, we show the student how ther-
modynamics, fluid flow, and chemical kinetics can sup-
ply partial answers to significant social questions. At
the same time, these problems serve to expose the
student to issues which do not have a unique solution,
where competing claims require consideration, and
where chemical engineering skills must be integrated
Copyright ChE Division ASEE 1990


S. undergraduate education has not .. changed to meet the new challenges. The prevailing repertoire of
homework problems and classroom examples does not . reflect the advent of new fields and, perhaps more
serious, has failed to show the growing interconnections between chemical engineering and societal concerns.

with insights from other perspectives to arrive at a
comprehensive solution.
We do not claim to give unique answers. Rather,
the solutions presented should be viewed as best esti-
mates. It is the procedure, rather than the numerical
outcome, which we hope to stress.

The first problem (suggested by Professor H. S.
Johnston) analyzes alternative strategies for coun-
teracting the depletion of the stratospheric ozone
layer by chloro-fluorinated hydrocarbons (CFCs). The
production of CFCs has increased markedly over the
last decade, to about one billion kilograms per year
[1]. CFCs are used as refrigerants, propellants, and
as foaming agents in the production of polystyrene
and polyurethane packing materials. Distinguished by
their lack of toxicity and chemical stability at sea
level, CFCs become photoactive and deplete ozone in
the stratosphere at altitudes above fifteen kilometers.
Ozone decomposition is dangerous because ozone
shields the earth from harmful ultraviolet radiation.
Even a ten percent reduction in stratospheric ozone
concentration would lead to a significant increase in
skin cancer and cause a drastic increase in the number
of cataracts. As a result of recent activity by the
United Nations, a landmark international treaty has
been negotiated which would cut CFC emissions by
fifty percent by the year 1999 [2].
Rising CFCs enter the stratosphere at an altitude

studying electrodeposition with Prof. C.W.
Tobias in the department of chemical engi-
neering at the University of California,
Berkeley. He received his BSE in chemical
engineering at the University of Pennsyl-
vania in 1983. After spending a year at
Cambridge University, he entered Berkeley
and received his MS in December, 1986.

John Prausnitz is a professor of
chemical engineering at the University of
California, Berkeley. He is a member of both
the National Academy of Science and the
National Academy of Engineering, and was
most recently elected to the American
Academy of Arts and Sciences. His inter-
ests include the history and philosophy of

of approximately eleven kilometers. Below this layer,
air temperature drops linearly with height, with a gra-
dient close to the adiabatic limit (9.9"C per km) [3].
An inverted temperature gradient occurs in the
stratosphere, however, caused by the absorption of
incident solar radiation by oxygen and ozone, espe-
cially in the ultraviolet portion of the electromagnetic
spectrum. It is this absorption which blocks harmful
radiation from reaching the earth's surface.
In the stratosphere, oxygen radicals react with an
oxygen molecule forming ozone [1]

02 +hv(X < 242nm)-> 20 (1)
02 + 0.- 03 (2)
h = Planck's constant
v = frequency
S= wavelength
Ozone absorbs ultraviolet radiation and converts it
into heat by

03 + hv (UV, visible) 02 + 0 (3)

0 +02-03 +heat (4)

The heat released warms the stratospheric air mass
so that the temperature rises with height. This rise
creates hydrodynamic conditions where free convec-
tion is almost completely suppressed, and a stagnant
layer of air results. Thus, CFCs travel through this
quiescent layer by molecular diffusion alone. At this
height, they become photo-active and form chlorine
radicals. For example, if the CFC is fluoro-

CFCl3 +hv ( <280nm) CFC12 *+CI.

The chlorine radical reacts either directly with ozone
or with an oxygen radical [1]. The chlorine radical acts
as a reaction intermediate, continuously depleting the
stratosphere of ozone:
C1l+03- 02 + C10 (6)
C1-+0. CIO (7)
C10-+03- Cl1+202 (8)
The result of this reaction chain is a perturbation
of the steady-state ozone concentration. While simula-
tion of the complex set of reactions lies beyond the
scope of an undergraduate course, it is educationally


useful and relevant to inquire what can be done to
return the stratospheric ozone concentration to the
level prior to the introduction of man-made chemicals.
Students are told that under present conditions a
ten-percent reduction in the steady-state ozone con-
centration may occur by the year 2000. To counteract
this reduction, one alternative might be to augment
the ozone production rate artificially to achieve the
desired 03 concentration, that is, to produce ozone on
the earth and to send it to the stratosphere.
Students are given data on natural 03 rates of pro-
duction from the literature [4]. While these rates vary
with height, latitude and season, an average rate of

1x106 molecules
cm sec

may be used to estimate the energy required to re-
place the "lost" ozone. As a homework problem, stu-
dents are asked to calculate the minimum amount of
energy required to replenish the ozone layer and to
compare that to annual U.S. energy production.
To establish a solution, students must first deter-
mine the volume of the stratosphere and then calcu-
late a global production rate. The free energy of for-
mation of ozone gives the approximate minimum
energy required.
The calculated global production rate is 1.2 x 107
moles of ozone per second. Since the free energy of
formation of ozone is positive

AGf = 39.06 (9)

energy must be supplied for the reaction to proceed.
The minimum energy needed to augment the 03 pro-
duction rate by ten percent would be
4.7 x10" cal
Converting this into more conventional units, and
comparing it to the annual U.S. production of usable
energy, we see that the energy required to increase
the ozone production rate is prohibitive [5]. It would
require a tripling of our annual energy production:

Energy to replace lost 03:

5.77x 1016 BTU (57.7 Quads/ yr)
1983 U.S. energy production:

2.71 x 1016 BTU (27.1 Quads/ yr)
This simple analysis shows that, if continued un-
checked, the problem of ozone depletion could be
beyond our direct control well before the year 2000.

To forestall serious and lasting damage to the environ-
ment, it is therefore necessary to address the issue of
CFC emissions and to consider alternate chemical ma-
terials to replace CFCs in current chemical technol-

We now turn to another form of energy release:
the detonation of nuclear weapons. Until recently, sci-
entists believed that, however devastating, the effects
of nuclear war on global climate due to blast, heat,
and radio-activity would be slight. Recent research
has brought this conclusion into question [6], noting
that detonation of a nuclear arsenal would cause large-
scale forest fires which would blow huge quantities of
dust into the stratosphere. This dust prevents sun-
light from reaching the earth, triggering nuclear
winter. As proposed by Professor M. C. Williams, stu-
dents are asked to estimate the settling time for the
stratospheric dust.
The students are told to model the dust as spheri-
cal particles ten microns in diameter, forming a dilute
dispersion in the stagnant stratospheric layer. They
are asked to calculate the settling time from a height
of fifty kilometers, and are given a hint that Stokes'
law may apply.
In this context, the condition of diluteness implies
that the particles have a nearest approach of 100 diam-
eters, or one millimeter. Even as a dilute dispersion,
this layer of dust would be quite opaque, since it would
be twenty kilometers thick. The incident solar radiation
would be completely blocked from reaching the earth's
surface. The earth would be engulfed in darkness and
there would be no light for photosynthesis.
To apply Stokes' law, we assume a stagnant layer
of air; this is an optimistic assumption, since any winds
or natural convection would tend to keep particles air-
borne longer. It may also be argued that large parti-
cles scavenge some small dust particles. However,
once the large particles settled, it is plausible to as-
sume that there would still be a dilute dispersion of
small-diameter dust.
First, the student calculates the terminal velocity,
Vt, from a force balance:
6 xp.RV = 4nR3( Pair)g (10)
g = gravitational constant
R = particle radius
Vt = terminal velocity
p. = viscosity of air
ps = particle density
Pair = density of air
Assuming that ps > pair, and substituting known val-


ues, we obtain Vt for dust particles ten microns in

(5x10o-4cm)2(2.0 -g 980 cJ
2 cm sec2 2
9 2.0 x10- g
cm sec

S= 0.55 cm (12)
The settling time, T, is the maximum height (50
kilometers) divided by the terminal velocity. For ten-
micron spheres, -= 9.09 x 106 sec., or about 105
days. This magnitude of T is great enough to demon-
strate the possibility of nuclear winter. If the initial
dust content in the stratosphere were large enough to
cool the earth to winter-like conditions, then at least
one harvest, and perhaps two, would be destroyed.
More serious, all summertime vegetation would
perish, severely affecting wildlife dependent on such
vegetation for food.
Settling time T is inversely related to the square
of particle diameter. Thus, T becomes very large for
very small particle diameters. For one-micron
spheres, the settling time is 29 years!
This analysis does not consider the effects of nu-
clear explosions staggered in time. Staggering would
delay the settling process considerably by re-injecting
the stratosphere with dust. The consequences of nu-
clear war could cause climatic damage for a period of
Similar calculations may also be used to consider
the effect on the atmosphere of a large meteorite im-
pact or prolonged volcanic activity. It has been pro-
posed that either of these mechanisms may have
caused the extinction of dinosaurs at the Cretaceous-
Tertiary boundary [7].
For our final example, we examine a common air-
pollution problem: smog caused by the evaporation of
solvents in lacquers and paints. This example differs
from the previous two because it concerns a response
to existing legislation rather than evaluating a need
for political or social action.
Solvents vaporize and are subject to photochemical
reaction with ozone, forming smog. In some geo-
graphic areas, local legislation has been enacted for
controlling the emission of volatile materials. Los
Angeles was the first major metropolitan area to enact
such legislation, in conjunction with the Environmen-
tal Protection Agency. Los Angeles' Rule 66 limits
both the type and amount of solvents which may be
used in paint formulations. To meet Rule 66 limita-
tions, typical paint and lacquer solvents have to be

This example concerns the cost of choosing a per-
missible solvent mixture for cellulose nitrate, which is
widely used as a lacquer for textiles and furniture.
Cellulose nitrate has been used for such applications
for over a century, due to its low cost and the durabil-
ity of nitrocellulose films. For use as a coating mate-
rial, cellulose nitrate is dissolved in a mixture of sol-
vents. The active (and relatively expensive) solvent is
a polar liquid having functional groups containing oxy-
gen: aliphatic esters of acetic acid, ketones, and glycol
ethers are the most common solvents. Co-solvents and
diluents may also be used to reduce cost. However,
these diluents tend to be smog-forming aromatic hy-
drocarbons. Rule 66 limits both the aggregate volume
fraction in the paint mixture of these diluents, as well
as individual volume fractions of certain types of di-
luents. Olefins are limited to five percent by volume,
eight-carbon aromatics are limited to eight percent,
while toluene, trichlorethane, and branched ketones
are also subject to the aggregate limit of twenty vol-
ume percent.
The students are introduced to a method of design-
ing solvent blends based on a 2-dimensional map of
Gardon's fractional polarity versus solubility parame-
ter, shown in Figure 2. Fractional cohesion paramet-

Gardon's Solubility Map
Ir I I I I I I

0.9 -









0.0 1 I I I I I I
7 8 9 10 II 12 13 14
Solubility Parameter ( cal) i
A = Acetone {6 = 10.0, p = 0.695}
B = Solvent Naphtha (6 = 7.6, p = 0.001}

FIGURE 2. A solubility map for cellulose nitrate.


ers are used to represent the solubility behavior of
polymer-solvent systems. Gardon has shown that a
good solvent matches both the solubility parameter
and the fractional polarity of the solute [8, 9].
Students are given solubility data and current
prices for various categories of solvents, as listed in
Table 1. They are asked to compare the cost of the
cheapest acceptable solvent mixture to that of the
least expensive mixture which also satisfies Rule 66.
Several additional constraints must also be met to
achieve an acceptable solvent for cellulose nitrate:

The volume fraction of the active solvent should be three
times that of the co-solvent (this insures solubility).
High-boiling solvents should not exceed ten percent of the
total solvent volume. (Required for proper drying charac-
The volume fraction of the diluent must not exceed three
times that of the slow-evaporating, high-boiling compo-
nent. (Required for even flow and uniform coating.)
Proper flow and blush resistance requires that low-boiling
active solvents do not exceed twenty percent of the total
volume. (Nitrocellulose lacquers tend to precipitate resins
if the temperature is lowered by rapidly evaporating sol-

Table 2 shows both the original inexpensive sol-
vent mixture and the mixture which satisfies Rule 66.

Suggested Solvents and Prices (Sept. 1986)

Low boiling solvents:
methyl ethyl ketone
Medium boiling solvents:
methyl isobutyl ketone
n-butyl acetate
iso-butyl acetate
High boiling solvents:
methyl methoxy pentanone
methyl amyl acetate


$ 0.27/lb
$ 0.235/lb

$ 0.38/lb
$ 0.52/lb
$ 0.45/lb

$ 0.50/lb
$ 0.52/lb

$ 1.31/gal
$ 0.34/lb
$ 1.81/gal

$ 0.73/gal
$ 0.80/gal
$ 0.85/gal
$ 1.30/gal

iso-propyl alcohol
n-butyl alcohol
ethyl alcohol

solvent naphtha

The cost of the original solvent mixture is $1.48/gal-
lon, while that of the environmentally benign, rede-
signed mixture is fifty percent higher, $2.21/gallon.
This example shows students that chemical en-
gineers have a role in establishing a cost-effective re-
sponse which adheres to legislated concerns.


The examples presented here have been drawn
from environmental issues. While forming a coherent
set, they illustrate an important point: chemical en-
gineering science offers useful contributions toward a
better understanding of broad topics. Just as we com-
bine kinetics, thermodynamics, and fluid mechanics
into an engineering curriculum, so must we integrate
chemical engineering with knowledge and concerns
from other academic areas. Educational experience is
enriched when its relevance is clearly demonstrated
to the student.
We plead for integrating chemical engineering
education with the world around us. The goal of such
integration can be illustrated by looking at another
Picasso portrait, Dora Maar (1937), in Figure 3. In
contrast to the first painting, we now have an integ-
rated portrait of the woman, that is, a representation
which shows the subject in many dimensions. The art-
ist combines both full-face and profile images to give

Cost Comparison Between Solvent Mixtures

Standard Solvent (fre-Rule 66)

Active Solvents
methyl ethyl ketone
methyl isobutyl ketone
methyl methoxy pentanone
Latent Solvent
isopropyl alcohol
Total Cost Per Gallon

Environmentally Benign Solvent
Active Solvents
n-butyl acetate
methyl amyl acetate
Latent Solvent
ethyl alcohol
includes solvent naphtha
Total Cost Per Gallon

$ 0.235/lb
$ 0.38/lb
$ 0.50/lb

$ 1.31/gal

$ 0.73/gal

$ 0.275/1b
$ 0.52/lb
$ 0.52/lb

$ 1.81/gal

$ 1.30/gal
$ 2.21


FIGURE 3. Dora Maar, Picasso (1937)

us a more complete description of his subject. By anal-
ogy, showing students how to link their technical skills
with contemporary problems gives them a more com-
plete image of what chemical engineering is and what
it can do. Versatility means to establish connections,
to practice a form of "networking" by building bridges
between a variety of intellectual domains. By helping
our students to become more versatile, they will see
chemical engineering as a component in the spectrum
of knowledge.

Viscosities at High Altitudes
The calculations presented in the preceding prob-
lems are meant to be estimates rather than exact val-
ues. They incorporate enough precision to allow the
examination of possible environmental scenarios. It
has been suggested, for example, that the estimates
presented in the nuclear winter problem are sensitive
to the value of viscosity used in the model. While vis-

cosities are only a function of temperature at low pres-
sures, it may be instructive to analyze the variation
of calculated viscosities as a function of height. Ac-
cording to the kinetic theory of gases, as modified by
Chapman and Enskog (1906), the viscosity of air de-
pends only on the square root of temperature. Thus,
the viscosity varies as the particle falls through
warmer regions of air. The range of values is rela-
tively small, however. At the extreme of 200K, the
viscosity is 1.30 cp, only 35% lower than the quantity
used in the present work.
A second objection concerns the inapplicability of
using bulk-viscosity values in regions of very low pres-
sure. A simple calculation shows that the mean free
path of a dust particle is less than one diameter up to
heights of 20 25 kilometers. Even at 30 kilometers,
the particle experiences over 2000 collisions per sec-
ond. It would seem appropriate, given the statistically
large number of collisions, to use ordinary viscosities
to predict the particle's terminal velocity. The model
is here applied to a particle falling from a height of 50
kilometers; if 30 kilometers were substituted for 50,
there would be no change in the qualitative conclusion
that injection of sufficient dust into the upper stratos-
phere may portend grave environmental damage.

We are grateful to Professors H. S. Johnston and
M. C. Williams for guidance and helpful suggestions,
and to Juan de Pablo for preparing the problem on
chemical solvents. Davor Sutija thankfully acknowl-
edges fellowship support from the Fannie and John
Hertz Foundation.


1. Johnston, H. S., "Human Effects on the Global Atmo-
sphere," Ann. Rev. Phys. Chem., 35, 481 (1984)
2. Crawford, M., Science, 234; 927 (1986)
3. Sychev, V. V., Complex Thermodynamic Systems,
Plenum Press, New York, 186 (1983)
4. Solomon, S., et al., "Instantaneous Global Ozone Bal-
ance Including Nitrogen Dioxide," Pageoph, 118, 58
5. Annual Energy Review/Energy Information Adminis-
tration, Office of Energy Markets and End Use: U.S.
Department of Energy, Washington, D.C. (1984).
6. Turco, R. P., et al, Science, 222,1283(1983)
7. Alvarez, L. W., W. Alvarez, F. Asaro, H. V. Michel,
Science, 208, 1095 (1980)
8. Kumar, R., J. M. Prausnitz, "Solvents in Chemical
Technology," in Weissberger, Techniques of Chem-
istry, Vol. 8, Part 1, ed. by M. R. J. Dack, Wiley, New
York (1975).
9. Gardon, J. L., J. Paint Technol., 38, 43 (1966) C





University of Florida
Gainesville, FL 32611

N THE EARLY twentieth century the four engineer-
ing disciplines of chemical, civil, electrical, and me-
chanical were founded. In a very short period of time,
each discipline evolved along relatively independent
paths to produce, in part, quite different curricular
contents. These differences are best exemplified by
chemical and electrical engineering. Chemical en-
gineering has developed into the most general of the
founding engineering disciplines and is characterized
by an isolated and rigid curriculum with an emphasis
on the engineering sciences, particularly those which
involve chemical change. Textbooks in our discipline
tend to experience longevity, time as a variable is not
emphasized, and mature technologies are often
graduated (e.g., nuclear engineering, environmental
engineering, petroleum engineering, polymer en-
gineering, metallurgical engineering, and biomedical
engineering). In contrast, the electrical engineering
curriculum is more option oriented, reflecting a his-
torical retention of developed technologies (e.g.,
power engineering, solid-state electronics, computer
architecture, optical engineering). The increased
technological content of the curriculum also translates
to short textbook lifetimes.
Mother Nature, however, is totally unaware of our
somewhat arbitrary partitioning of her behavioral pat-
terns. As a result, the foundations of chemical en-
gineering prepare the student to understand a variety

Tim Anderson joined the faculty at the
University of Florida in 1978 after receiving
degrees in chemical engineering from Iowa
State University (BS) and the University of
California, Berkeley (MS,PhD). He has been a
visiting scientist at RADC Air Force laboratory
and a Fullbright Scholar at the University of
Grenoble. He has an active research program
in bulk crystal growth and epitaxy of compound
semiconductors and ceramic superconduc-
Copyright Che Division ASEE 1990

FIGURE 1. Mechanical analogy of electron energy state
of topics included in other curricula once the terminol-
ogy and nomenclature are translated. One such exam-
ple is the operation of solid state electronic devices in
an integrated circuit. Presented below is a synopsis of
four lectures given in an elective senior-level elec-
tronic materials processing course which introduces
the student to solid state electronics. The terminology
of chemical engineering is used primarily with the
equivalent electrical engineering terminology con-
tained in brackets.

In order to understand the behavior of electrons
in the solid state, the student must first appreciate
the concept of electron energy states. This is intro-
duced by comparing the free electron, for which all
energy values are accessible; the hydrogen atom, for
which only discrete states exist; and a large collection
of H atoms in the solid state, for which the collection
of electron states becomes so closely spaced in energy
that we speak of bands of energy states (e.g., Is band,
2s band). The splitting of electron energy states when
atoms are brought together is illustrated by recalling
from quantum mechanics the bonding and anti-bond-
ing states between two hydrogen atoms and also by
the mechanical analogy shown in Figure 1. This figure
illustrates two identical balls suspended from two


By their senior year, students have already received the foundations
of chemical engineering. One of the objectives of these lectures is to convince
seniors that the digital integrated circuit is nothing more than a chemical processing plant.

identical springs. Neglecting frictional losses, an ini-
tial displacement of each uncoupled ball will result in
a single natural vibrational frequency. If the two balls
are permitted to interact through the third spring
shown in this figure, two natural vibrational states
are possible for a given displacement of each ball: a
low frequency state when each ball is initially dis-
placed in the same direction and a higher frequency
state when displaced by the same amount but in oppo-
site directions.
The concept of energy bands is next applied to
semiconductors by showing a plot of energy versus
the density of electron states for both the conduction
and the valence bands, pointing out the gap in energy
for which no intrinsic states exist. This plot is then
compared with the Fermi-Dirac distribution function
which gives the probability of finding an electron at a
certain energy value. Integration of the product of
the density of states and the probability of a state
being occupied for energies above the conduction band
minimum gives the number of electrons in the conduc-
tion band. These electrons are essentially free because
the vast majority of conduction band states are not
occupied. A similar integration over unoccupied states
in the valence band gives the concentration of holes
(empty electron states). The Fermi-Dirac distribution
contains a parameter called the Fermi energy that is
a function of temperature, pressure, and concentra-
tion. The Fermi energy is equivalent to the elec-
trochemical potential of electrons, a quantity that is
understandable to chemical engineers.
The process of doping a semiconductor is next de-
scribed by illustrating the incorporation of B and P in
Si. The group III dopant B introduces an electron
state with energy level located just above the highest
valence band energy, while the group V dopant P in-
troduces a state just below the conduction band
minimum energy. A doped semiconductor material
under equilibrium conditions is a good example of
chemical equilibrium. Consider the following three
reactions at equilibrium:

These three chemical reactions involve the chemi-
cal species free electron (e-), free hole (h+) neutral
donor (D), ionized donor (D+), neutral acceptor (A),
and ionized acceptor (A ) with corresponding equilib-
rium concentrations of n, p, [D], [D ], [A], and [A ].
The chemical species 0 represents an electron com-
bined with a hole in the valence band (normal state)
with a large, nearly constant concentration.
Equations (1-3) are further constrained by the con-
dition of charge neutrality

P+[D+]=n+[A-] (4)

Since the electron and hole are highly mobile in the
semiconductor, even at room temperature, the mate-
rial reaches equilibrium very rapidly. These four
equations contain six concentration variables. The
total donor dose, ND = [D+] + [D], and acceptor
dose, NA = [A-] + [A], however, are usually
specified. Solution of Equations (1-4) in terms of ND
and NA involves finding the roots of a 4th order
polynomial in the variable n. If the donor and acceptor
ionization energies are small compared to the bandgap
energy, Eg, and comparable or smaller than RT, then
reactions (2) and (3) are shifted to right and

ND [D+]

NA [A-]

This simplification leads to a quadratic equation with
meaningful root

n= ND-NA+ (N NA)2+4KI (5)

This example is easily understood by the senior chem-
ical engineering student and is translated into electri-
cal engineering terminology according to the relation-

KI = exp(ASO / R) exp(-AH / RT)= NcNv exp(-Eg / RT) = n

(1) In this equation N, and Nv are conduction and valence
band effective density of states

Ni=2( mkT/h'3
Ni = 2(2cm? kT / h2)

m7 = effective mass of an electron or hole

0=e- +h+



K1 =np

K2= [D]

K3 [A]


and are related to the entropy of reaction (1). The
quantity ni is the intrinsic carrier concentration and
represents the electron or hole concentration in the
undoped semiconductor (n = p in the intrinsic material
since a hole is created for every electron promoted to
the conduction band).


As in operational chemical plants, functioning
semiconductor devices operate under non-equilibrium
conditions by the action of external influences (e.g.,
electric field, magnetic field, optical excitation, elec-
tron bombardment). The basic equations that describe
transport of electrons and holes include species mate-
rial balances, a statement of species flux in terms of
available potential gradients, and Maxwell's equations
since these two species are charged. The principles of
basic device operation can be illustrated with a
simplified set of these equations. Considering only low
electric fields and one-dimensional transport in the ab-
sence of magnetic fields for an ideal (dilute) solution
of electrons and holes, the species material balances
[continuity equations] are:



an 1 WJ
-=- +R (electrons)
at q ax

ap -1 Jp
pt =--x +Rp (holes)
at q ax

q = magnitude of electric charge
t = time
Ji = flux of positive charge [current
Ri = net rate of production of species i (number/cm3s)
by homogeneous reaction [carrier recombination,
optical excitation, impact ionization]

The charge fluxes [current densities] are given by

vice physics is the Poisson equation which relates the
electric field gradient to the net charge distribution,
p, according to

a p
ax e (11
c = semiconductor permittivity (F/cm)

The senior chemical engineering student who has com-
pleted courses in transport phenomena and introduc-
tory physics can easily understand the significance of
these equations; the only new term in the transport
equations is migration due to an electric field.
A simple and useful example of these equations is
illustrated in Figure 2. A uniformly n-type doped
semiconductor slab is illuminated on one side with
light (Figure 2a). The photon energy is chosen so that

photons semiconductor


_h+ diffusion

S e diffusion
h+ drift

e drift

Jn = q rnn+ qDna

Jp = qpp-qup (10)
pi = mobility of species i (cm2V.s)
S= electric field (V/cm)
Di = concentration independent diffusion coefficient
of species i (cm2/s)

Electrons and holes can migrate in response to both
an electric field [drift] and concentration gradient [dif-
fusion]. The most important Maxwell equation to de-

FIGURE 2. Surface absorption of photons in an n-type
semiconductor; (a) schematic, (b) sketch of relative mag-
nitude of hole and electron currents due to diffusion and
drift, (c) carrier concentration at equilibrium (no illumi-
nation), (d) steady state carrier concentration with il-




absorption creates electron-hole pairs near the surface
only (photon energy greater than the bandgap energy
and large absorption coefficient). Before illumination
the slab is at equilibrium (Figure 2c) and the concen-
tration of electrons [majority carrier] greatly exceeds
the concentration of holes [minority carrier]; for exam-
ple, Si at room temperature and doped at

n = 1017 cm-

gives p = 2.1 x 103 cm-

Upon illumination electron/hole pairs are generated
at the surface and are transported into the slab by
diffusion. The diffusion coefficient of an electron, how-
ever, is normally greater than that for a hole (by a
factor of 3 for Si) and a small electric field is estab-
lished. The direction of the field is such that the elec-
tron flux is reduced and the hole flux is enhanced (Fig-
ure 2b). Realizing there is no net current in the slab
(Jn = Jp) and the carrier concentration gradients are
nearly identical at a specified position [electroneutral-
ity approximation], it can be shown that the minority
carrier transports almost exclusively by diffusion pro-
vided the quantity

k4.p/pn-1)pn/nnl << 1
This condition is satisfied if the photon flux is not too
large (pn n,). The same conclusion is not found for
the electron [majority carrier], since there is a large
population of electrons to respond to the electric field
(Figure 2b).
As the carriers diffuse into the slab, they attempt
to return to their equilibrium concentrations through
homogeneous reaction recombinationn]. The steady
state minority carrier concentration profile can be de-
termined by solution of Equation (10), with only the
diffusion term, and Equation (8):

0=Dp +Rp (12)

The applicable boundary conditions are

pn(x=0) equal to a constant (due to steady illumination)

p( (x= oo)= pO (semi-infinite slab)

An expression for the net rate of generation of holes,
R,, is required. The simplest homogeneous reaction
mechanism is direct recombination of a conduction
band electron with a valence band hole [band to band

0 -- e- ++ h(13)

The rate of production of h I by this reversible reaction
Rp = k, -k_, np, (14)
The rate of the forward reaction is pseudo-zero order
since the concentration of electrons in the valence
band and holes in the conduction band are not signifi-
cantly changed by the reaction at low doping levels.
The rate constant k, can be eliminated in favor of the
known equilibrium constant
K1 = n 0 p
to give
Rp =k_ (n p -n,) kin (p pn) (11

The rate at which holes disappear by this particular
mechanism is thus pseudo-first order since the equilib-
rium majority carrier concentration is barely dis-
turbed at low illumination. In device physics texts the
k_, n
is termed the minority carrier lifetime, Tp. The solu-
tion to Equation (12) with the recombination rate
given above is:
pn(x)-pn _e-i(Dp/ (16)
pn(x= )-p

The term (D, Tp))' is called the diffusion length for
obvious reasons.
In chemical engineering terms, the above problem
is simply one of diffusion with first-order homogene-
ous reaction into a semi-infinite slab from a constant
composition source. The simplicity of this example al-
lows students to make the connection between
semiconductor physics and their own background in
chemical engineering. The problem is also useful since
it is the basis for understanding minority carrier injec-
tion necessary to describe the operation of p-n junc-
tion devices (diode, bipolar transistor). Useful home-
work problems include analysis of: carrier concentra-
tion decay with steady and uniform photoexcitation
(batch reactor), transient and steady-state transport
of carriers generated by localized illumination with/
without an electric field (transient and steady-state
dispersion of a line source with plug flow/no flow),
recombination rates for materials having mid-gap
states (homogeneous catalysis), and surface recombi-
nation with uniform and steady illumination
(homogeneous and heterogeneous reaction in a semi-
infinite stagnant liquid).


The p-n junction is the basic building block of many
solid state devices, including the junction diode and
bipolar transistor. The lecture material begins with
the equilibrium p-n junction and then examines the
junction under non-equilibrium conditions with both a
positive and a negative applied potential. Finally, the
behavior of two p-n junctions (bipolar transistor)
under non-equilibrium conditions is described.
When a piece of n-type semiconductor is metallur-
gically joined to a piece of p-type semiconductor, the
hole and highly mobile electron species diffuse in di-
rections of lower chemical potential. Electrons, in ex-
cess in the n-type material, will diffuse into the p-type
material, where their concentration is extremely
small, and holes will diffuse in the opposite direction.
If immobile donor and acceptor ions were not present,
this process would continue until the electron and hole
chemical potentials were the same in both materials.
As diffusion occurs, however, a positive space charge
density in the n-type material and a negative charge
density in the p-type material are "uncovered." The
resulting charge distribution produces a diffusion po-
tential [built-in voltage] that opposes further diffu-
sion. When the spatial variation of the chemical po-
tential is just balanced by the variation in electric po-
tential, the joined semiconductors are in equilibrium
and the electrochemical potentials [Fermi level] are
constant. The discussion of the equilibrium p-n junc-
tion continues with a numerical illustration of a Si ab-
rupt junction in which the charge distribution is ap-
proximated as a step function [depletion approxima-
tion]. The details of this example are given in most
device textbooks [1-6] and it invokes our first use of
Poisson's equation.
The steady state operation of a p-n junction with
an applied external voltage is treated next. The equi-
librium junction described above is dynamic, repre-
senting a balance between drift and diffusion currents.
To understand the operation of the junction with a
positive voltage applied to the n-type material [re-
verse bias], different sources of carriers are
examined. If I place myself at the metallurgical junc-
tion and count the electrons and holes which cross the
junction, I will see three sources of carriers. The first
source is homogeneous reaction in the depletion region
[generation]. The rate of electron production equals
that of hole production as given by Eq. (15) pn K1
since carriers are assumed to be depleted in this re-
gion. The electric field in the depletion zone will sweep
an equal number of generated electrons and holes in
opposite directions towards material of the same type.
Therefore, at the metallurgical plane I can count the

holes coming from the n-type material side of the de-
pletion region and the electrons originating from the
p-type material side. The total charge crossing the
plane is equal to the rate of hole production in the
entire depletion region, or equivalent electron produc-
tion, times the total width of the depletion region, W
J=qk_ n2W (17)
The second source of carriers crossing the junction
plane are produced by diffusion of minority carriers to
the boundary between the neutral and depletion re-
gions, where they are swept across the depletion zone
by the electric field. This problem is similar to the
surface illumination problem treated earlier, except
that the minority carrier concentration is reduced [de-
pleted] at the boundary instead of elevated to a con-
stant value by the photon absorption. The flux of
minority carriers at the edges of the depletion region
can be determined from Fick's first law of diffusion
and Equation (16) with pn(x=0) = 0. The currents
arising from extraction of minority carriers are given

J, =qp(D,/,)2 (18

The currents given by Equations (17-19) have the
same sign and at equilibrium are just balanced by the
third source of carriers; majority carriers from the
neutral regions capable of overcoming the built-in po-
tential. Application of a positive voltage, VR, to the
n-type material increases the potential which majority
carriers must overcome, thus decreasing the majority
carrier diffusion current (proportional to exp[qVR/
kT]). The minority carrier extraction currents given
by Equations (18) and (19), however, are independent
of voltage while the current due to homogeneous reac-
tion (Equation (17)) actually increases since the deple-
tion region widens with increasing VR, enlarging the
reactor volume. Therefore, with increasing VR, the
majority current rapidly becomes small and the re-
verse bias current is given by the sum of currents in
Equations (16-18). This current is small since the

W, p,, and nP

are small.
If the sign of the applied potential is reversed [for-
ward bias], the potential barrier decreases and the
number of majority carriers capable of overcoming the
decreased potential barrier dramatically increases


J = qqn(D, / ,)1/2

(proportional to exp[qVF/kT]). Development of the
current equations for the forward bias condition is
similar to the reverse bias case, requiring only the use
of Equations (15) and (16). Instead of extracting
minority carriers from each side of the depletion re-
gion, they are injected, and instead of carrier genera-
tion in the depletion region, they recombine (pn >
pno). The p-n junction device is thus shown to operate
as a leaky check valve, permitting a large current to
flow under forward bias and only a very small current
to flow under reverse bias. An interesting homework
problem is the analysis of a p-n junction under uniform
illumination (solar cell, photodetector).
With a background in p-n junction behavior the
operation of a p -n-p bipolar (both electrons and holes
participate) transistor (transfer resistor) is next dis-
cussed. This transistor consists of three slabs of
semiconductors joined in the series p+-n-p and electri-
cal contacts made to each slab. The transistor is biased
such that the p+ (heavily doped) -n junction is forward
biased and the second n-p junction is reverse biased.






n+ depletion +


FIGURE 3. Metal oxide semiconductor (MOS) transistor;
(a) cross section view of the device; (b) schematic of the
gate voltage induced n-channel.

The diffusion current at the forward biased p+-n junc-
tion is largely due to holes because of the heavy doping
in the p+ slab [hole emitter junction]. These holes then
diffuse as a minority species across the n-type slab
[base]. If the width of this region is kept small com-
pared to the diffusion length, (Dp/Tp)12, most of the
holes reach the depletion region of the reverse biased
junction and are swept across this junction [hole col-
lector junction] by the favorable electric field. These
holes are now a majority carrier in the p-type slab and
appear as the collector current. The electrons that
enter the base contact and are extracted from the re-
verse biased junction either participate in a small dif-
fusion current at the forward biased junction or react
homogeneously with holes in the base region. With
proper transistor design the collector current can be
significantly greater than the base current [amplifica-
tion], thanks to the "pumping" action of the emitter.
The bipolar transistor can act as either a pump
[amplification] or an on-off valve [switch].

Though the bipolar transistor can be made to act
as a fast switch, the power requirements can be fairly
high. In order to decrease the base current, an insulat-
ing oxide layer is sandwiched between the base and
the metal contact as illustrated in Figure 3. The MOS
transistor is a three-terminal device with a source (S),
a gate (G), and a drain (D). In this particular config-
uration, the source and drain lead wires are connected
to "pockets" of n-type material which are isolated by
a p-type region. Application of a potential between
the drain and source contacts will not produce any
significant current since one of the junctions is reverse
biased. The application of a positive voltage to the
gate contact attracts electrons and repels holes in the
semiconductor near the oxide interface, uncovering
immobile ions. For a sufficiently large applied gate
voltage [threshold voltage], the population of elec-
trons near the interface will exceed that of the holes
[inversion], and a continuous n-type channel forms be-
tween the source and the gate that permits a current
to flow. Of course, the n-type channel and p-type ma-
terial are separated by a depletion region. A further
increase in the applied voltage will increase the n-type
channel "pipe" diameter to produce a larger "flow
rate." The device can be operated as either an "on-off
valve" or "gate valve."
The lecture presentation includes a graphical rep-
resentation of the band diagrams in the equilibrium,
accumulation (negative gate voltage), depletion, and
inversion regimes. The operation of a functional




f F--

capacitor is also analyzed. The p-n junction has a small
capacitance since the depletion width changes with
applied voltage. Similarly, the MOS structure can be
used as a capacitor. Actually, this structure has two
capacitors connected in series; a parallel-plate-like
capacitor with the oxide as the dielectric material and
the accumulation/depletion regions of the semiconduc-
tor. The equations which describe the operation of an
ideal MOS capacitor are relatively straightforward to
develop [1-6] and are presented in the course.


By their senior year, students have already re-
ceived the foundations of chemical engineering. One
of the objectives of these lectures is to convince
seniors that the digital integrated circuit is nothing
more than a chemical processing plant. Integrated cir-
cuits contain only a few types of devices (resistors,
capacitors, transistors, and diodes). These four lec-

Comparison Between a Chemical Processing
Plant and an Integrated Circuit

Chemical Plant

Integrated Circuit

many but depleting electrical ground

2 (electron, hole)

pipe (10 inch O.D.)

103 moles/s
10 hp
tanks (106 moles)
check valve
on-off valve
gate valve

Reactions many
Unit operation 104/mi2
Cost $108 ($1
Waste disposal problem
Diffusion 10-2- 10-



106 1/moles s

Metal interconnect
(10-5 inch O.D.)
10-11 moles/s
10-9 hp (bipolar transistor)
capacitor (10-10 moles)
field effect transistor

$10 ($109/mi2)
electrical ground
10 103 cm2/s

1016 1/moles s

tures demonstrate their operation in terms that, for
the most part, are understandable by chemical en-
A comparison between a large scale chemical pro-
cessing plant and an integrated circuit is given in
Table 1. A typical chemical plant processes hundreds
of species, while the integrated circuit processes only
two charged species, the electron and the hole. Power-
ful pumps move fluids through large diameter pipes
at high flowrates in a chemical plant, while power re-
quirements, dimensions, and flowrates are orders of
magnitude lower in an integrated circuit. A high per-
centage of the land area at a plant site can be devoted
to storage of raw materials and products. In contrast,
charge storage in a p-n junction or MOS structure is
very limited in an integrated circuit. As discussed
above, control valves have analogs in an integrated
circuit. Indeed, a diode is used to protect the circuit
against excessive voltages, just as a check valve pro-
tects against excessive pressures. Relief in the inte-
grated circuit is accomplished by simply dumping the
current to ground. The E.P.A., however, does not
permit this luxury at a chemical plant site. One of the
major difficulties in the simulation of a chemical pro-
cess is the large number of chemical reactions, often
coupled and with unknown rate constants. In the elec-
tron-hole plant the reactions involve only recombina-
tion and generation, often of reduced order. Due to
the size difference in the basic unit operations, the
densities are dramatically different, though the costs
per unit area are similar. As in chemical plants, the
rates of most processes are limited by either reaction
or diffusion. The diffusion coefficient and reaction rate
constants for electrons and holes are very high. Com-
bining these properties with the small dimensions
found in an integrated circuit gives an extremely rapid
response time to input parameter changes in the cir-


1. Adler, R.B., A.C. Smith, and R.L. Longini, Introduction
to Semiconductor Physics, Wiley, New York (1964)
2. Gray, P.E., D. DeWitt, A.R. Boothroyd, and J.F. Gibbons,
Physical Electronics and Circuit Models of Transistors,
Wiley, New York (1964)
3. Grove, A.S., Physics and Technology of Semiconductor
Devices, Wiley, New York (1967)
4. Streetman, B.G., Solid State Electronic Devices, 2nd ed.,
Prentice Hall, Englewood Cliffs, NJ (1980)
5. Sze, S.M., Physics of Semiconductor Devices, 2nd ed.,
Wiley, New York (1981)
6. Sze, S.M., Semiconducting Devices: Physics and Tech-
nology, Wiley, New York (1985) 0


Raw material
Number of

Flow rates

Reaction rate
(1st order)

book reviews

Edited by Naim Afgan
Hemisphere Publishing Corporation, 79 Madison
Ave., New York, NY 10016; 466 pages, $95, (1989)

Cesar C. Santana and Judit Z. Halasz
State University of Campinas
Campinas, SP, BRAZIL

The purpose of this book is to present selected
contributions to the scientific meetings organized by
the International Center for Heat and Mass
Transfer between 1968 and 1987. It includes forty
papers on fundamentals and applications ranging
from boundary layers to high temperature heat
According to the editorial preface, the aim of the
book is to select contributions representative of the
state-of-the-art in each category which had the most
impact on each field during a time-span of twenty
years. This aim has been achieved. Additionally, a
very important and complete list of references is
available for each topic.
In the reviewers' opinion, some of the papers
had lost their up-to-date importance and new
selections could probably have been considered.
Considering the book as a whole, it can serve as
a good reference source for several subjects in heat
and mass transfer research. O

Technology of Resists
by Arnost Reiser
John Wiley & Sons, NY; (1989) $49.95

David S. Soane
University of California, Berkeley

Photoreactive Polymers covers a broad range of
subjects, including a brief history of resists, nega-
tive photoresists, photophysics and photochemistry
in solid polymers, photoinitiated polymerization,
positive resists based on diazonaphthoquinones, the
rudiments of imaging science, deep-UV lithogra-
phy, electron beam lithography, X-ray and ion beam
lithographies, and finally multilayer resists and
plasma processing. The presentation of these sub-
jects parallels approximately the chronological ap-
pearance of the resists and their associated tech-
nologies. Each topic is dealt with in the space of one
chapter. Taken as a whole, this book provides a
truly comprehensive overview of the science of
photoreactive polymers.
Chapter One is unique in that no other mono-
graphs seem to have given such a thorough cover-

age of the dawning days of photoreactive polymers.
This degree of care and research dedication has
permeated throughout the book, and the author has
achieved a rather unbiased treatment of all the sub-
ject areas of the book. I find that practically all the
important issues and major developments have
been described.
Chemistry, such as explicit details of chemical
reactions, chemical and physical photoevents, pro-
posed mechanisms, and the wide varieties of resists
and their structures, is the strong suit of the book.
For chemical engineering students who have not
been exposed to much organic chemistry, especially
photoreactive polymer chemistry, this book is an es-
sential tool. It will undoubtedly save the readers
much library time and provide the necessary back-
ground for advanced reading of current literature.
Comparatively, this book devotes less to processes
that are much more familiar to chemical engi-
neers, i.e., processes that involve basic transport
theories and polymer dynamics. Fortunately, these
are the exact places where traditional chemical en-
gineers may grasp the concepts most readily and
further contribute to the advancement of the science
and technology of resists. Even though an in-depth
treatment of these areas has not been given, the ba-
sics of these processes and related research prob-
lems have been prominently identified. Adequate
references have also been cited for beginners.
In short, this book is quite useful for chemical
engineers who are interested in the field of photore-
active polymers. O

books received

Corrosion: For Students of Science and Engineering, by K. R.
Trethewey, J. Chamberlain; John Wiley & Sons, Inc., 1
Wiley Drive., Somerset, NJ 08875-1272 (1988) 382 pages, $38.95
Fundamentals of Chemistry With Qualitative Analysis,
Third Edition, by Brady and Holum; John Wiley & Sons,
Inc., 1 Wiley Drive, Somerset, NJ 08875-1272 (1988) 1112+
pages, $51.50
Concepts in Biochemistry, Third Edition, by William K.
Stephenson; John Wiley & Sons, Inc., 1 Wiley Drive,
Somerset, NJ 08875-1272 (1988) 229 pages $19.40
Industrial Energy Management and Utilization, by L. C.
Witte, P. S. Schmidt, and D. R. Brown; Hemisphere
Publishing Co., 79 Madison Ave., New York, NY 10016; 666
pages (1988) $40
Kinetic Aspects of Analytical Chemistry, by H. A. Mottola;
Wiley-Interscience, 605 Third Ave., New York, NY 10158-
0012; 285 pages
New Polymer Technology for Auto Body Exteriors, Schmeal
and Purcell (eds); AIChE, 345 East 47 St., New York, NY
10017; 92 pages, $15 members, $30 non-members
Heat Transfer in Tube Banks in Crossflow, A. Zukauskas
and R. Ulinskas; Hemisphere Publishing Co., 79 Madison
Ave., New York, NY 10016-7892; (1988) 199 pages, $69.50




University of California
Berkeley, CA 94803

OR NEARLY THIRTY years, silicon has been the
semiconductor material of choice for the fabrica-
tion of microelectronic devices and integrated circuits
(ICs). This situation has arisen and continues today
despite the fact that silicon is not the best semiconduc-
tor material from the standpoint of device speed (i.e.,
the electron mobility is not as high as in materials
such as gallium arsenide and indium antimonide).
However, in order to fabricate solid state devices and
ICs in the surface of a semiconductor, it is necessary
to greatly reduce the number of unsatisfied orbitals
("dangling bonds" or surface electronic states); other-
wise, the electron (or hole) concentration at the semi-
conductor surface cannot be reproducibly established
and controlled. A reduction in "dangling bond" density
was demonstrated in the late 1950s by merely expos-
ing the silicon surface to air so that a thin "native
oxide" layer formed [1]. Subsequent studies showed
that a further reduction could be achieved if inten-
tional oxidation of the silicon surface was performed
at high (> 600C) temperature [2]. Currently, no other
semiconductor/insulator solid state structure can
achieve the low level of surface or interface states
that is obtained in the Si/Si02 interface system. Fur-
thermore, other formation methods (such as chemical
vapor deposition) for Si02 do not yield the excellent
interfacial properties that exist in the thermal growth
of Si02 on Si. Finally, amorphous Si02 films thermally
grown on Si are unparalleled in their dielectric proper-
ties, and can serve as diffusion barriers for common
dopants (e.g., boron, phosphorus, arsenic) in silicon
IC process technology [3]. These facts have led to the
extensive use of thermal Si02 in device components,
device isolation, and as a passive insulator and a
mechanical and chemical protection (passivation)
layer. As a result, a large number of silicon oxidation
studies have been performed since the early 1960s [4-
7]. The investigations have yielded a reasonable de-
scription of the kinetics of silicon oxidation. However,
a detailed atomistic model is still lacking. Therefore,
fundamental research efforts in silicon oxidation con-
tinue [8-10].
O Copyright (hE Division ASEE 1990

Dennis W. Hess is professor and vice
chairman of the chemical engineering depart-
ment at the University of California, Berkeley.
He received his BS in chemistry from Albright
College and his MS and PhD in physical
chemistry from Lehigh University. Prior to
joining the Berkeley faculty in 1977 he was a
member of the research staff and manager of
process development at Fairchild Semicon-
ductor. His research efforts involve thin film
science and technology and rf glow discharge
(plasma) processes, as applied to the fabrica-
tion of electronic materials and microelectronic

The fabrication of silicon ICs consists of a number
of individual steps ("unit operations") that are care-
fully sequenced to yield an overall process. For in-
stance, since ICs are built up of layers of thin films,
various means of forming thin film materials (e.g.,
chemical vapor deposition, sputtering, evaporation,
oxidation) are needed. In addition, precise patterns
must be established in these layers (lithography) and
selective regions of the silicon doped (solid state diffu-
sion) to control the resistivity level and type (n or p).
Of these various process steps, silicon oxidation has
probably been the most extensively studied. Further-
more, since the chemistry and chemical engineering
principles behind silicon oxidation have been covered
by the time a materials and energy balance course has
been completed, this "unit operation" can serve as an
elementary example of a process step in a non-tradi-
tional field.

Silicon is oxidized by exposure to oxygen or water
vapor at elevated (> 700C) temperatures. For these
oxidants, the overall oxidation reactions to form
amorphous SiO2 can be written
Si(s)+ 02(g) Si02(s) (1)
Si(s) + 2H20(g) SiO2(s) + 2H2(g) (2)

Oxidant species diffuse through the growing SiO2 film
to the Si/SiO' interface where they react with Si.
Therefore, Si is consumed and the Si/Si02 interface
moves into the bulk Si as oxidation proceeds. It can
be shown from the densities and molecular weights of
Si and Si02, that if a thickness of SiO2, Xo, is formed,
0.45 Xo silicon is consumed.


Thermal oxidation of Si is generally performed in
a tubular quartz reactor contained in a resistance
heated furnace. Silicon substrates are placed upright
in slotted quartz carriers or boats, and pushed into
the reactor. The oxide thickness is established by pre-
cise control of temperature, oxidant partial pressure,
oxidation ambient, and oxidation time.

A general kinetic relationship describing the oxida-
tion of silicon was proposed over twenty years ago
[12]. Although the mechanistic details of the oxidation
process have not been firmly established, the overall
form of the rate expression that results from this
model can assimilate data generated by numerous in-
vestigators over a wide range of temperature, silicon
crystal orientation, oxide thickness, and oxidation am-
bient. As described in Figure 1, the approach to model
formulation for silicon oxidation considers three fluxes
that could control oxidation rate [12]. Oxidant (gener-
ally 02, H20, or both) is transported (Fi) to the sur-
face of the growing SiO2 film and is subsequently in-
corporated. Since nearly two orders of magnitude
change in gas flow rate has no effect on silicon oxidation
rate, these steps are considered rapid and thus are not
rate limiting under normal conditions. Oxidant species
then diffuse across the growing oxide (F2) to the Si02/
Si interface, where they react with Si (F3) to form
Si02. The overall oxidation rate can be derived by
writing analytical expressions for each flux, F, equat-
ing them, since steady state conditions apply, and de-
termining oxide thickness as a function of time. The
following formulation of this problem parallels the







... since the ... chemical engineering principles behind
silicon oxidation have been covered by the time a
materials and energy balance course has been
completed, this "unit operation" can serve
as an elementary example of a process
step in a non-traditional field.

original derivation of an expression for the oxidation
rate of silicon [12, 13].
Referring to Figure 1, the gas phase flux F1 is
assumed to be proportional to the difference between
the oxidant concentration in the bulk gas (Cg), and
that near the oxide surface (Cs). The proportionality
constant is defined as the gas phase mass transfer
coefficient, hg
F, = h,(C,-C.) (3)

In order to estimate the concentration of oxidant in
the oxide (solid) surface, we assume that Henry's Law
holds. Thus
Co = kLP, (4)

where the concentration of oxidant in the outer sur-
face of the oxide, Co, is proportional to the partial
pressure of the oxidant next to the oxide surface, Ps,
and the proportionality constant is Henry's Law con-
stant, kHL. Finally, the oxidant concentration, C*,
that would be in equilibrium with the partial pressure
in the bulk gas, Pg, can be written

C* = kHPg

If ideal gas behavior is assumed, the concentration of
oxidant in the bulk gas and near the oxide surface can
be written
C =g (6)
g kT
C- T(7)

Combining Eqs. (3) to (7),
F= h(C*-Co)

where h = hg/(kHLkT), and represents a gas phase
mass transfer coefficient written in terms of oxidant
concentration in the solid. Thus, Eq. (8) defines the
flux of oxidant from the gas to the oxide surface.
The flux of oxidant across the growing oxide layer
is given by Fick's First Law
F2 = -Deff (9)
where the effective diffusion coefficient, Deff, is used


because at present it is not clear what the diffusing
species is (probably 02, but 02, 0-, and 0 have also
been proposed), and x represents the distance into
the oxide film from the SiO2 surface. If quasi-steady
state oxidation is assumed (i.e., no accumulation of
oxidant in the oxide), F, must be the same at any
point in the oxide layer, so that dF,/dx = 0. There-
fore, Eq. (9) can be written

F2 = Deff o- (10)

where X0 represents the oxide thickness.
Finally, the flux of oxidant due to the oxidation
reaction at the SiOg/Si interface is assumed to be pro-
portional to the concentration of oxidant at the inter-
face, Ci. The proportionality constant is the surface
reaction rate coefficient for oxidation, ks

F3 = kCi (11)

Under steady state conditions, Fi = F2 = F3 =
F; therefore, we can develop an expression for the
concentration of oxidant reaching the silicon surface.
The flux is

F3=F= k X (12)
1+ k + k
h Deff

The growth rate can now be described if the number
of oxidant molecules incorporated into a unit volume
of oxide is known. If this quantity is defined by 0,
then the oxidation rate is
dX_ F3 kC* / 0 (13)
dt 0 k k. X (sX
1+ -+ -
h Deff
This differential equation can be solved if an initial
condition is specified. To formulate the initial condi-
tion, it is useful to consider X0 consisting of two parts:
an initial oxide layer Xi that might have been present
on the silicon surface prior to the present oxidation
step, and the additional oxide grown during the oxida-
tion cycle. Such an approach makes the model general
to multiple oxidations. The initial condition used is
therefore Xo = Xi at t = 0.
Solution of Eq. (13) yields the general relationship
for the thermal oxidation of silicon

X2 + AX = B(t+T)

A 2 Dffk +

B 2 Def



Xi2 +AXi


where 7 is a constant (time units) that corrects for the
presence of an initial oxide layer, Xi, or for an initial
"rapid oxidation rate" in dry oxygen [6-12]. Eq. (14)
can also be solved for X0 as a function of oxidation
time, t.

X- (1 t+g 2
X0 1+ )-1 (15)
A/2 ( A24B (15)

It is useful to consider this expression in two limiting
forms. At relatively long oxidation times or thick
oxides, Eq. (15) reduces to

Xo2 =Bt

This represents the parabolic oxidation regime
wherein the oxidation rate depends upon diffusion of
oxidant through the growing oxide; B is the parabolic
rate coefficient. For relatively short oxidation times
or thin oxides, Eq. (15) becomes

X0 B-(t+,)

Eq. (17) describes the linear or surface reaction rate
controlled regime; B/A is the linear rate coefficient.
As a result, Eq. (14) is often referred to as a linear-
parabolic oxidation law.
Using the definitions (14a) and (14b), the
semiquantitative dependence of the rate coefficients
B and B/A on temperature (through h, ks, and Deff)
and pressure (through C*) can be considered. Fur-
thermore, the activation energy for the linear rate
coefficient (B/A) at temperatures of 1000C and above
is -2.0 eV for both dry 02 and steam oxidation [1, 6].
This value is approximately equal to the Si-Si bond
energy, which is consistent with the linear kinetics


Oxide Thickness (um)
Oxidation Time (hr) (100)Si (111)Si

1 0.0490 0.0700
2 0.0780 0.1050
4 0.1235 0.1540
7 0.1800 0.2120
16 0.2980 0.3390


regime controlling the oxidation by breaking a Si-Si
bond on the silicon surface. By comparison, the activa-
tion energy for the parabolic rate coefficient B [1, 6]
is higher for dry 02 (-1.2 eV) than for steam (-0.8
eV). These results are consistent with values reported
for diffusion of 02 and H20 through fused silica and
suggest that the rate controlling step in the parabolic
oxidation regime is diffusion of oxidant through the
oxide film.
Although the above model is extremely useful for
most oxidation regimes. it appears inadequate to de-
scribe the initial "rapid" oxidation rate observed in
dry 02 and the curvature of Arrhenius plots at tem-
peratures below 10000C. As a result, new models are
being formulated, and additional experimental data
are being generated [7-11).


A simple example can be used as a homework prob-
lem or can be incorporated into lectures or discussion
sections to demonstrate the use of the general re-
lationship for the thermal oxidation of silicon (Eq. 14).
Silicon wafers are thermally oxidized in dry oxy-
gen at 1000C, and the kinetic data shown in Table 1
are obtained by measuring the SiO, thickness grown
on (100) and (111) crystal orientations of silicon.

a. From the data in Table 1, determine the
parabolic (B) and linear (B/A) rate constants via
a graphical method for (100) and (111) silicon
b. Discuss the comparison of the rate constants for
these two orientations of silicon.

a. Dividing Eq. (14) by Xo and rearranging yields

Xo =B( t+ -A (18)

This is the equation of a straight line; thus if Xo is
plotted versus (t + T)/Xo, the slope of the line is B and
the intercept is -A. The parabolic (B) and linear (B/A)
rate constants can therefore be determined by linear
regression analysis. First, however, we need a value
for T, a correction factor for the initial "rapid oxidation
rate" in dry 02. Evaluation of 7 is performed by ex-
trapolating a plot of Xo versus t to zero oxide thick-
ness. For these data, the extrapolation crosses the
time axis at -0.35 hr, so that 7 is defined as 0.35 hr
(this value can be given to the student as a constant).
Linear regression analysis of the oxidation rate
data in the form of Eq. (18) gives

For (100) orientation
A = 0.196 pm, B = 9.07 x 10-3 gm2/hr
so that
B = 0.0091 gm2/hr and B/A = 0.0463 gm/hr

For (111) orientation
A = 0.105 im, B = 9.19 x 10-3 gm2/hr
so that
B = 0.0092 gm2/hr and B/A = 0.0874 gtm/hr

b. The two orientations of Si have essentially the
same parabolic rate constant. Since B relates to the
diffusion of oxidant through the amorphous SiO2
layer, there should be no effect of B on silicon surface
orientation provided that the oxide is the same in both
The linear rate constant is larger for (111) than for
(100) orientation. This observation correlates with the
higher packing density of the Si (111) plane. Since the
linear rate constant appears to be controlled by the
reaction of oxidant with the Si surface, such differ-
ences are consistent with the atom density in the dif-
ferent planes.
Thermal oxidation of silicon is an important step
in the manufacture of silicon devices and integrated
circuits. A general relationship describing the thermal
oxidation process can be derived easily by considering
fundamental chemical engineering principles. This ex-
pression can be used as an example to demonstrate
the reduction of kinetic data obtained for silicon oxi-
dation in undergraduate core chemical engineering
1. Atalla, M.M., E. Tannenbaum, and E.J. Scheibner, Bell Sys.
Tech. J., 56, 749 (1959)
2. Ligenza, J.R., and W.G. Spitzer, J. Phys. Chem. Solids, 14, 131
3. Tsai, J.C.C., in VSLI Technology, ed. by S.M. Sze, McGraw-Hill
Publishing Co., New York, p. 169 (1983)
4. Pliskin, W.A., and R.A. Gdula, in Handbook on Semiconductors,
ed. by T.S. Moss, p. 641, Vol. 3 of Materials, Properties, and
Preparation, ed. by S.P. Keller, North Holland Publishing Co.,
Amsterdam (1980)
5. Nicollian, E.H., and J.R. Brews, MOS Physics and Technology,
John Wiley & Sons, New York (1981)
6. Deal, B.E., in "Proceedings of the Tutorial Symposium on Semi-
Conductor Technology," ed. by D.A. Doane, D.B. Fraser, and
D.W. Hess, The Electrochemical Society Inc., p. 15 (1982)
7. Murali, V., and S.P. Murarka, J. Appl. Phys., 60, 2106 (1986)
8. Irene, E.A., J. Appl. Phys., 54, 5416 (1983)
9. Massoud, H.Z., and J.D. Plummer, J. Appl. Phys., 62, 3416
10. Irene, E.A., and G. Ghez, App. Surf Sci., 30, 1(1987)
11. Irene, E.A., Crit. Rev. Solid State Matl. Sci., 14, 175 (1988)
12. Deal, B.E., and A.S. Grove, J. Appl. Phys., 36, 3770 (1965)
13. Grove, A.S., Physics and Technology of Semiconductor De-
vices, John Wiley, New York, Chap. 2 (1967) O





Colorado State University
Ft. Collins, CO 80523

THE INTENT OF THIS article is to introduce you to
the work environment and the culture in the IC
(integrated circuit) industry. This introduction to the
work environment is necessary to adequately prepare
you for making a career choice. In each of the para-
graphs below, a statement of fact about the industry
will be given and then discussed in terms of how it
creates a culture or a unique work environment. A
design process typical of the industry will be con-
trasted with chemical engineering design so that you
can grasp the level of sophistication of the manufactur-
ing environment. Lastly, the advantages of working
in this industry will be highlighted.
In the IC industry, the product has an electrical
engineering application, and thus the management
and the majority of the employees are electrical en-
gineers. Because electrical properties (film resistivity,
electromigration properties, contact resistance, leak-
age current, dielectric constant, breakdown voltage,
etc.) of materials directly relate to the material prop-
erties (grain size, contamination, stress, adhesion,
alloy type, etc.), the second most frequently encoun-
tered employee is a material scientist. Only rarely
does one find a chemical engineer in the IC industry.
This distribution of employees results in a culture
which considers process development and reactor de-
sign as extraneous to the primary function of the com-
pany. The most enlightened form of the industry rec-
ognizes that reactor conditions during film deposition
contribute to the film properties, and the cause and
effect relationships are correlated through orthogonal
matrices. Because the average material scientist or

The product . is not just one chemical
which has been modified through a series of
processes. It is a layered structure resulting from at
least 100 sequential operations, each one with the
potential to influence an earlier layer or a later layer.

Copyright ChE Dwision ASEE 1990

CAROL M. McCONICA received her
PhD (1982) in chemical engineering from
Stanford University. She spent three years
with Hewlett Packard (1979-1982) developing
state-of-the-art deposition/etching processes
for their 128Kb RAM and 640Kb ROM, all
fabricated with 1 micron NMOS double-layer
metal technology. She is currently an
associate professor at CSU leading a graduate
program in IC processing.

electrical engineer has very little, if any, education in
heat, mass, or momentum transport, in gas-solid kine-
tics or in reactor design, the whole problem of process
design takes on the mystique of being a black art. A
properly-educated chemical engineer in this environ-
ment can become an instant hero, if the stage is prop-
erly set, by applying his/her knowledge of reactor de-
sign to the manufacturing processes. But beware-
any great opportunity comes with an equally great
The product which is being manufactured is not
just one chemical which has been modified through a
series of processes. It is a layered structure resulting
from at least 100 sequential operations, each one with
the potential to influence an earlier layer or a later
layer. You will be working with 50-100 other en-
gineers, each responsible for a different set of steps
in the process of building an integrated circuit. Most
of these people will be trained in chip design and fail-
ure analysis, but will generally be undereducated in
process design. Compound this with the fact that the
product you make is not visible to the naked eye. The
best analogy is of one hundred blindfolded sculptors
trying to recreate Michelangelo's statue of David
while it sits on a rotating table. Success is determined
by an artist at the end of the process deciding if it
looks like the original. Failure analysis is a process of
trying to determine which piece was sculpted poorly
many layers and many rotations ago.
Because the chip yield is the only true measure of
the viability of a new process, all proposed processes
must be demonstrated upon a real product. In a devel-
opment facility there may be only ten lots (25 wafers


each) every eight weeks that are available to all 100
process engineers for development purposes. This
means that any one engineer may receive only two to
five patterned test wafers every two months on which
to test his/her ideas. Because process interactions are
so strong, the best process developed on unpatterned
wafers may yield horribly on the real patterned test
wafers. The best process engineers strive hard to un-
derstand the interactions within their own process,
whether it be an etch step, a deposition step, or photo-
lithography, before receiving those very few and very
precious test wafers. The clever process engineer also
gets his/her hands dirty and learns the details of all
the steps in creating films which will be contacting his
or her film. This is a job best suited to experimen-
talists, and if you went through school attached only
to a computer, you will find your work overwhelming.
You better know when vacuum pump oil can contami-
nate wafers and how to run a scanning electron micro-
An interesting result of the test wafer starved en-
vironment is an underground black market for test
wafers within any facility. Imagine a product market
in the 1200s where the people bartered their goods
one at a time, and after a full day on the town square
they had enough to live on for the next day or two.
Any IC industry is a culture of 50-100 individuals net-
working furiously among themselves to obtain a few
test wafers so that they can get their own jobs done
and become heros. Some individuals may operate
through intimidation, others through bribing their
friends with weekends on a family catamaran. Stu-
dents simply won't need their speech classes or formal
presentation skills-a much better background is in
interpersonal relationships and in the art of negotia-
tion or coercion 101.
Another interesting result from 50-100 people de-
veloping one invisible product with very strong pro-
cess interactions is the "hot potato" syndrome. Imag-
ine the case of shorts between metal lines on the chip's
interconnect. Let's investigate where this failure
mode might originate. It could be that conductive con-
tamination was left during processing prior to metal
deposition; or the metal itself may have an improper
composition, making the etch difficult; or the photo-
lithography process could be failing, leaving photo-
resist where it should not be; or the etch reactor may
have changed its performance and is no longer clear-
ing out the metal between the lines. These are the
obvious possibilities. It could also be that the engineer
in charge of oxide deposition (several layers ago)
changed the process slightly, creating steeper
sidewalls, and thus the etch can no longer clear metal

from under the new geometries. Who owns the prob-
lem? Clearly, the lab manager owns the problem, just
like he/she owns all of the other problems in the clean
room. It is too often the case that the lab manager is
faced with a group of engineers, each claiming the
problem belongs to another process group-thus the
"hot potato" syndrome.
IC fabrication lines are very expensive to build
and operate. Therefore, companies are forced to de-
velop new processes on equipment producing the cur-
rent product. The consequences of this constraint are
profound. Hardware modifications are nearly always
forbidden because they may interfere with current
product yield. Remember that the hardware/process
interactions are so poorly understood that any

Not only is there the obvious problem of
making an inadequate tool (today's batch reactor) to
create a new product (tomorrow's chip set), but there
is the added problem of getting time on the
production machines to run tests.

hardware change is viewed as potentially dangerous.
The chip manufacturers do not consider themselves in
the business of inventing processes and would rather
work with a low-yielding piece of hardware than to
modify it to optimize its performance. Imagine a
sculptor making statues out of soap when the manage-
ment decides to make statues out of metal. The
sculptor now has to figure out how to sculpt metal
with the same old tools because the production of soap
statues must not be interrupted.
Not only is there the obvious problem of making
an inadequate tool (today's batch reactor) to create a
new product (tomorrow's chip set), but there is the
added problem of getting time on the production
machines to run tests. Today's product represents
today's profit, and therefore production wafers take
precedence over test wafers. Engineers have to find
time for development without interfering with produc-
tion. This creates another interesting situation: the
art of bribing the machine operator off of the reactor
for a while. (Once a chip set goes to production, it is
handled almost exclusively by operators, not en-
gineers.) Common tactics range from encouraging the
operator to take long coffee breaks to designating the
machine as "down," implying that it has hardware
problems and is not suitable for production runs.
Either method allows the engineer to process a batch
of test wafers, and he/she works diligently trying to
make the old tools fit the new need. Sometimes they
are successful, but usually the IC industry is forced


to wait for new tools to be developed by equipment
Because the chip facilities view process develop-
ment as the job of the equipment vendor, and the
equipment vendor has no idea what the next genera-
tion of chips will look like, there exists a mismatch
between the time a process is needed (now) and when
the vendor can make a machine to fulfill this need
(two years from now). The equipment companies are
expected to produce the very best IC batch reactors
with almost no capital investment from the IC indus-
tries. Not only is there no monetary support, but the
IC fabrication engineers generally distrust equipment
vendors because the last generation of batch reactors
did not come with an adequate recipe for processing
wafers. The vendors cannot find that optimum recipe
because they do not have access to patterned wafers,
which are the ultimate test of a process. In fact the
IC fabs are very reluctant to help the vendor create
the process because they don't want any information
to leak out about the next generation of chips. As a
consequence of this nearly adversarial vendor-IC fab-
rication facility relationship, the process becomes an
orphan. The IC chip fabrication engineer is then faced
with spending $1.5 million on a piece of hardware (a
batch reactor useful for one step of the 100-step pro-
cess) with only a vague idea of the best operating con-
ditions for his/her chip set.
There is usually a honeymoon period, albeit a short
one, where the new machine belongs to a development
engineer and not yet to production. In this window of
one to three months, the development engineer is
given carte blanche on the machine and some priority
on obtaining test wafers. This is where six to eight
years of chemical engineering buys leverage. The ad-
vantage is best understood by contrasting the
textbook chemical engineering methods of process de-
velopment with the methods currently used in the in-
For the purpose of discussion, film properties and
device behavior are termed "level 1" variables in this
document, for they are the important surface proper-
ties dependent upon the local chemical environment
present during growth (Figure 1). These level 1 prop-
erties are most directly related to the surface compo-
sition during growth, which in most cases is un-
measurable and only roughly predictable. The only ex-
ception to this is the measurement of surface composi-
tion during growth with Raman spectroscopy or in
situ low energy electron spectroscopy, a technique re-
quiring vacuum capabilities in the 10-10 torr range. The
measurable variables which most directly affect film
properties are the local gas composition, the wafer

dep rate
FILM PROPERTIES encroachment
nucleation rate
contact resistance
leakage current
I _







FIGURE 1. Variable chart

pretreatment, and the surface temperature-the local
gas composition being that within a few mean free
paths of the wafer surface. These variables are refer-
red to as independent level 2 variables, for they con-
trol the level 1 dependent variables. Ultimately the
level 2 variables are determined by the gas flow rates,
pressure, reactor hardware, species residence times,
reactor materials, transience times in the reactor,
heating method, and other "reactor knobs." It has
been the tradition in the IC industry to turn the level
3 reactor knobs in an orthogonal manner in order to
determine the best conditions for running a piece of
hardware. The level 1 properties are related to level
3 knobs by statistical correlations without any funda-
mental understanding of the processes involved. It is
a perfect tool for engineers with an educational mis-
match to the product they are expected to build. It is,
for instance, how I would go about building a strong
bridge since I am not trained in stress analysis.
This method of orthogonal experimental design is
very well suited to processes where nothing is known
about the behavior of the process and the hardware.
Plasma processes, for instance, are so complex that
they have traditionally qualified for this category. For
processes which are dominated by chemical reactions
with known kinetics (classical chemical vapor deposi-
tion), this method is not optimal. With a few heat and


mass balances, calculations of dimensionless groups,
and an understanding of kinetics a skillful chemical
engineer can often solve in an afternoon what a pro-
cess engineer has been statistically correlating for
For demonstration purposes, let us contrast chem-
ical engineering reactor design to orthogonal matrix
design for a semi-batch reactor with no heat and mass
transfer limitations (the most primitive case). As a
chemical engineer, you are aware that the film proper-
ties are dominated by the local reactant concentra-
tions and temperature. Any reactor design text will
give the appropriate design equation, depending upon
the Peclet number, for the reactor. Knowing the de-
sign equation, the kinetics, and the stoichiometry, the
growth rate is perfectly predictable. Level 1 proper-
ties can then be related to the actual deposition envi-
ronment given by the calculated level 2 variables.
If the kinetics are unknown, a chemical engineer
is aware that the variables critical to film deposition
are the local concentrations and surface temperatures.
At this point the engineer can choose to determine the
basic kinetics and create a predictive model, or more
realistically he/she will be in an industrial situation
that does not allow fundamental studies and will de-
sign an experiment based upon orthogonal matrices.
The key in being efficient is in realizing that the en-
gineering staff will have a more fundamental under-
standing of which variables actually control material
properties if the matrix is built around level 2 vari-
ables (concentrations) rather than level 3 knobs (flow
rates). This approach differs dramatically from the
blind approach of most process engineers in the IC
industry today.
Figure 2 is a plot of the operating conditions for
performing an orthogonal matrix in flow rate space,
the industry standard, for reactants A, B, and C. Fig-

A (scem)

FIGURE 2. Orthogonal,matrix in flowrate space for reac-
tants A, B, and C.

ure 3 shows those same operating points plotted in
concentration space when given a fixed surface area
for deposition. As can be seen, in concentration space
(the only one that really matters) the matrix is far
from orthogonal. Interpretation of the experiments in
terms of fundamentals is all but impossible. In fact it
is no mystery at all that the industry has so much
difficulty in converging on optimal processes.
The US IC industry is battling for survival against
foreign competition. If it is to survive, it will have to
develop a new strategy. We are already witnessing
cooperation among competing industries and vendors
in the form of Sematech. To really become an interna-
tional leader, the US industry will have to launch itself
out of the mode of empirical correlations and base its
manufacturing processes upon science. This is the
very heart of chemical engineering, and chemical en-
gineering should therefore become the very center of
the future of the IC industry. It is a waste of time to
have fine circuit designers attempting to derive heat
transfer relationships given in junior level chemical
engineering textbooks, or attempting to define kine-
tics based upon pseudo-orthogonal matrices. It is our
responsibility to use our knowledge and our tools to
solve these problems efficiently and scientifically.
For chemical engineers, the IC industry is a gold
mine of opportunity. Compared with the petroleum
industry, the technical problems are easy. The pro-
cesses operate at low pressure and moderate temper-
atures. Reactors often behave as mixed flow reactors
because the diffusivities are so large. Pressure drops
never exist. Processes are just as likely to be kinet-
ically limited as mass transfer is limited. The reactions
are often inorganic. One is usually limited to two
Continued on page 60.

A (torr)
FIGURE 3. Operating points in Figure 2 plotted in partial
pressure space, which is identical to concentration space
when divided by the constant RT.





Purdue University
West Lafayette, IN 47907

ulum at Purdue consists primarily of a one-semes-
ter course. The objective of this course is to provide
chemical engineering students with the basic princi-
ples and practical aspects of the most advanced state
of electronics processing. The main emphasis of the
course is on fundamental processes especially useful
for Very Large Scale Integration (VLSI) schemes [1].
About five weeks are devoted to epitaxy, which is
a process by which material is deposited onto a crys-
talline substrate or seed, and the crystalline config-
uration is maintained. Two and one-half weeks are
devoted to Vapor Phase Epitaxy (VPE) on patternless
substrates, one week to VPE on patterned substrates
typically known as Selective Epitaxial Growth (SEG),
one week to doping profiles in epitaxially grown thin
films, and one-half week to other kinds of epitaxy
(e.g., Molecular Beam Epitaxy (MBE), Plasma As-
sisted Chemical Vapor Deposition (PACVD)). The
purpose of this paper is to focus on Chemical Vapor
Deposition (CVD) epitaxy on patternless and pat-
terned substrates in the manner in which it has been
developed in our course over the past five years.
Journal articles play a very important role in many
educational aspects of the CVD epitaxy on patternless
and patterned substrates. A list of such journal arti-
cles typically used in class is presented at the end of
this paper [7-23]. The rapid developments in the field
preclude adequate discussion in a book, and in general
there is very little discussion in books, if any, about
CVD on patterned substrates.
First, the process of epitaxy is introduced and

About five weeks are devoted to epitaxy,
which is a process by which material is deposited onto
a crystalline substrate or seed, and the crystalline
configuration is maintained.

Copyright ChE Division ASEE 1990

0 0
0 0



FIGURE 1. Schematic representation of (a) vertical, (b)
horizonal, (c) barrel, (d) pancake, and (e) low pressure
chemical vapor deposition epitaxial reactors.
classified into types (e.g., VPE, MBE, PACVD, Solid
Phase Epitaxy (SPE)) [2, 3, 6], and important features
of epitaxy are briefly discussed. At the same time,
some potential problems of epitaxy are briefly pre-
sented. Such problems with VPE, for example, can be
autodoping, pattern shift, and pattern washout [2-4,

An introduction to different basic types of reactors


Christos G. Takoudis is an associate
professor at Purdue University. He received
his Diploma in 1977 from the National Techni-
cal University of Athens, Greece, and his PhD
in chemical engineering in 1982 from the Uni-
versity of Minnesota. He joined the faculty at
Purdue in December of 1981. His research in-
terests are in the areas of electronic materials,
catalysis of new materials, and reaction

used in the epitaxial thin film growth of electronic
materials forms the first stage of our course. For
VPE, five types of CVD reactors are discussed [2, 3,
6-11, 22] along with some recent reactor designs such
as the Vapor Levitation Epitaxial (VLE) system [24]
and the Epsilon One system [25]. They are the vertical
(typically used in Metal Organic Chemical Vapor De-
position (MOCVD)), the horizontal, the barrel, the
pancake, and the Low Pressure Chemical Vapor De-
position (LPCVD) reactors (see Figure 1). The
LPCVD reactor has been increasingly used in reduced
pressure epitaxy since problems associated with VPE,
such as pattern shift, washout, and autodoping, have
been remedied to a significant degree with low pres-
sure epitaxy [11, 13, 14, 19].
The Epsilon One system is a one-wafer horizontal
reactor with very low contact times between gas mix-
tures and substrates [25]. VLE uses growth vapors
and carrier gas not only to provide reactants to a
wafer surface but also to lift the wafer and keep it
suspended above the growth apparatus during the epi-
taxial growth process [24]. PACVD reactor systems
are also briefly presented. Throughout this section the
emphasis is on discussing main features and potential
advantages and disadvantages of the various systems
used in electronic materials CVD.

Sources typically used in silicon (Si) or gallium ar-
senide (GaAs) epitaxy are presented next. Si and
GaAs are the base semiconducting materials studied
in the Purdue course on microelectronics processing,
other materials being conceptually presented as
rather simple extensions of these two base ones. Mass
spectrometry and other gas phase analytical tools
along with in situ substrate surface analysis tech-
niques are shown to provide a means of understanding
some of the chemical reactions that may occur during
epitaxy. On the other hand, important chemical reac-
tions during the pretreatment and preparation of a
substrate surface are also discussed in detail. One
example from silicon epitaxy is the process of remov-
ing all native oxide just before an epitaxial growth

step since device quality epitaxial thin film is,
perhaps, the main objective of any epitaxial process.
In the context of SEG, only silicon epitaxy on pat-
terned substrates is covered (see Figure 2). There has
to be a higher degree of supersaturation for the nucle-
ation of silicon on Si02 and Si3N4 as compared to that
for nucleation on silicon surfaces. Thus, by keeping
the supersaturation below a critical value it is possible
to selectively deposit silicon on Si substrates masked
by either silicon nitride or silicon oxide (e.g., Figure
2). Crystal growth theories as discussed, for example,
by Bennema and van Leeuwen [26] explain the initia-
tion of growth by the adsorption of silicon at the
growth interface. Adsorbed atoms form little clusters
which are thermodynamically unstable until they
reach a certain critical size. Thereafter it is energeti-
cally more favorable for them to remain in the solid
phase than in the vapor phase [27]. The adsorption
energy on foreign substrates is generally higher than
that for Si. Thus it becomes possible to operate at a
point where the nucleus size on the foreign material
is held below the critical value, while nuclei of over-
critical size can form on the silicon-growth interface.
The process is a delicate balance between reasonable
growth rates and polynucleation on the masking mate-
rial, most often Si02. The onset of nucleation on the
mask is a function of temperature, pressure, mask ma-
terial, and the Cl/Si ratio in the vapor phase.
Doping profiles in epitaxially grown thin films are
presented from two points of view. First, an epitaxial
Mask Material

FIGURE 2. Selective epitaxial growth and epitaxial lat-
eral overgrowth schematic cross sections.


layer doped with a desired element can be obtained
with a cofeed of a dopant source along with other
species of interest. In this case, additional chemical
reactions, which include dopant species, have to be
accounted for; these additional reactions can signifi-
cantly affect the quality of growing doped epitaxial
layers. Second, intrinsic epitaxial growth of, say, sili-
con on substrates with buried layers raises questions
of doping a growing thin film with impurities coming
from these buried layers through autodoping or out-
diffusion [2, 3].

Thermodynamic calculations of a CVD reaction
system are discussed next. Such an analysis may pro-
vide important insights into several aspects of the sys-
tem. Starting with a nonequilibrium inlet state, chem-
ical equilibrium calculations can provide boundary val-
ues of operating parameters necessary for successful
thin film growth and provide information on the re-
sponse of the process to changes in operating condi-
tions [7, 28]. Furthermore, the computation of equilib-
rium compositions with intentionally limited reactants
may assist in the assessment of reaction mechanisms.
In the course, students are presented with a computer
program that allows quick equilibrium calculations of
CVD reaction systems.
Thermodynamics calculations are also helpful for
the pretreatment and preparation steps of substrates
as well as for the growth of thin films doped with a
desired impurity. However, in our course on micro-
electronics processing, the need for chemical equilib-
rium calculations is emphasized even more in the SEG
of silicon on patterned substrates [29]. Nucleation on
Si02 (or SisN4) during SEG, silicon oxide (or nitride)
degradation during SEG, and undesired impurities in
Si02 (or Si3N4) films used for the patterning of a sub-
strate are some of the many additional problems one
does not have to worry about in CVD on a patternless
substrate. Such issues are addressed in detail for the
CVD of epitaxial silicon on patterned wafers.

The chemistry of CVD systems follows their chem-
ical equilibrium calculations. Such chemistry is com-
plex and usually involves surface and gas phase reac-
tions [7]. With respect to gas phase reactions, two
approaches are discussed. The first one includes ex-
perimental data from studies on individual gas phase
reactions. The second approach is theoretical. Start-
ing from basic data of all conceivable species and reac-
tions in a given CVD reaction environment, one can

estimate rate constants from kinetic theory. Once this
is done, dominating gas phase reactions can be deter-
mined at any set of CVD reactor conditions. It is em-
phasized in class that, typically, a combination of both
approaches seems to be best. With such a conceptual
understanding of gas phase reactions, particle forma-
tion (for example, during Si deposition from silane)
can be reasonably well predicted. Thus, because parti-
cle formation in the gas phase can be detrimental to
the quality of growing epitaxial thin films, the impor-
tance of gas phase chemistry and kinetics becomes
On the other hand, it is pointed out that the role
of gas phase reactions is becoming less important with
decreasing CVD temperatures and partial pressures
of the reactants. Therefore, in silicon SEG, which is
typically carried out at reduced pressures and temper-
atures, many gas phase reactions are not expected to
play an important role. Yet, both approaches dis-
cussed previously are also presented as thorough
ways of accounting for gas phase reactions in CVD on
patterned substrates.
When it comes to substrate surface reactions in
CVD, it is pointed out that little is known even for
epitaxial silicon deposition, which is perhaps the reac-
tion system studied the most. Several difficulties in
the understanding of CVD surface reactions are dis-
cussed. These are the typically unknown extent of gas
phase reactions, the typically significant role of trans-
port phenomena in the neighborhood of a substrate in
particular (with the only exception perhaps of LPCVD
epitaxial reactors), the potentially high levels of unde-
sired impurities in the feed gases or in the reactor
itself, the potentially high conversions of key reac-
tants, and the possibility that some reactions may be
very near their chemical equilibrium. It is mentioned
that in a few studies, spectroscopic techniques have
been utilized in CVD so that some surface reactions
can be monitored. Although helpful, such studies are
shown to provide more questions than answers.
In spite of our incomplete understanding of CVD
surface reactions, a few reaction mechanisms for Si
and GaAs growth are discussed in detail. Fairly
widely acceptable gas phase and surface intermediates
are used.
Naturally, the role of surface reactions in CVD on
patterned wafers is presented as even more complex.
In silicon SEG, there are two kinds of surfaces that
any species is in contact with: the silicon seed windows
area and the Si02 (or Si3N4) area. It is indicated to
the students that, typically, silicon SEG in the seed
windows is assumed to be similar to growth on pat-
ternless wafers; that is, the only potential contribution


coming from the oxide (or nitride) surface is assumed
to be just surface diffusion close to the SiO2-Si inter-
face. Yet, recent developments are shown to suggest
that the oxide or nitride surface seems to participate
to a much greater extent in the overall surface reac-
tion scheme than thought before.
Also, what is usually called "kinetics of epitaxial
growth" in several books and some journal articles
[2-6] is discussed at the end. The above term includes
mass transport in series with a truly kinetic step, the
rate expression of which is assumed to be linear.
Therefore, the final growth rate expression obtained
involves an overall mass transfer coefficient along
with a kinetic rate constant. Although such a growth
rate expression may help in the understanding of dif-
fusion controlled and surface reaction controlled de-
position, it is emphasized that such an analysis is not
the intrinsic kinetics of epitaxial thin film growth and
that it simply provides an elementary, though clear,
conceptual understanding of kinetically or mass trans-
fer controlled processes in CVD systems.

Today more than 70% of all fabricated integrated
circuits employ epitaxy in one way or another. The
requirements made on the quality of the epitaxially
grown layers are stringent: less than 5% thickness
variation over a wafer and from wafer to wafer, less
than + 5% doping nonuniformity and high growth
rates to suppress dopant redistribution.
Selective epitaxy is even more sensitive to the
variation of parameters than is epitaxy on patternless
wafers; one has only a limited operating range in
which nucleation on SiO2 (or Si3N4) does not occur.
Also, local depletion effects can significantly alter
growth rates.
The basic continuity, momentum, energy, and
mass equations of a comprehensive model of a CVD
reactor are covered through the detailed modeling of
a pancake reactor. Such equations in their general
form apply to any type of reactor, the main variations
being related to entrance effects and to whether natu-
ral convection plays an important role in a given CVD
reactor system. Also, the special geometry and inlet
and exhaust configurations of the reactor used have
to be accounted for in a CVD reactor model. Gas phase
chemistry is shown to be included in these modeling
equations, whereas surface chemistry is accounted for
through appropriate boundary conditions for a chosen
CVD system. Important differences between cold-
wall and hot-wall reactors are discussed in detail.
Results from the detailed model of a pancake reac-
tor are presented in detail for patternless and pat-

turned substrates [30]. It is readily seen that one re-
sult of the solution of a detailed reactor model is an
understanding of velocity, temperature, and species
mass fraction profiles throughout the reactor of in-
terest. Another result is predictions of the growth
rate profiles on substrates.
Because of our incomplete understanding of CVD
kinetics, it is emphasized that any CVD gas phase and
surface chemistry should be tested in at least two dif-
ferent types of reactors. Furthemore, even within
each reactor, wide regimes of operating parameters
such as substrate temperature and reactor pressure
are suggested for testing. However, even if such a
model is able to predict all trends of thin film growth
rate profiles quantitatively, it may not be able to pre-
dict other features such as defect density, surface re-
sistivity, and quality of Si/SiO, interfaces that can
characterize the quality of a grown thin film. The qual-
ity of a thin film grown in an epitaxial CVD system is
also shown to be a fairly strong function of the quality
of the substrate used, the purity of gases or other
materials used, the impurities of the reactor itself,
and the predeposition treatment. Film characteriza-
tion after an epitaxial processing step is presented as
an essential integrated part of any CVD on pattern-
less or patterned substrates [2, 3, 6].

Specific focus on VPE on patterned substrates fol-
lows. SEG of silicon is presented as being most often
carried out by employing an SiH2C12-HC1 system at
reduced pressure and temperatures of about 800-
10000C [31-35]. SEG leads to structures exhibiting dis-
tinct faceting which depends on substrate orientation
and seed window alignment relative to crystal planes.
It is pointed out that (100) substrates and pattern
alignment along [100] directions seem to give the best
results for application purposes [36]. At reactor pres-
sures greater than about 20 torr, SEG rates appear
to depend rather strongly on the amount of exposed
silicon area. However, a reduction of pressure below
20 torr or an increase of the reactor inlet ratio Cl/Si
is shown to decrease such a loading effect. In a fabri-
cation line it is indicated that both these remedies may
be undesirable because they result in smaller growth
If the film is grown longer than necessary to fill
the void created by the etching of the SiO2 mask, it
will not only grow vertically but horizontally as well
(Epitaxial Lateral Overgrowth (ELO), Figure 2). This
leads to a Silicon-on-Insulator (SOI) type structure
which is very desirable from a device application point
of view. Typical ratios of horizontal growth rates over


vertical ones, i.e., aspect ratios, are pointed out to be
about 1:1.
A different technique for growing epitaxial silicon
over Si02 is also discussed [37]. Jastrzebski, et al. [37]
report almost nucleation-free growth by growing
without any HCI for a short time and then etching
with HC1 for about the same amount of time. These
steps are repeated until the desired film thickness is
achieved. The aspect ratio is about 1:1.
Large aspect ratios (much greater than 1:1) are
shown to be of great interest for advanced dielectric
isolation and the design of new three dimensional in-
tegrated circuits. One promising avenue for such a
high aspect ratio is pointed out to be Lateral SEG
(LSEG), which is depicted in Figure 3 [38]. The top
structure is a cavity with prepared wing layers of dif-
ferent etched-rate materials and with a seed hole deep
inside. In the center structure, selective growth ex-
tends up into the cavity and is technically ELO at this
stage. As the top of the ELO meets the cavity ceiling,
growth is now constrained to proceed only laterally,
as shown in the bottom structure. This lateral growth
is referred to as LSEG.
The importance of selective epitaxial growth in
VLSI is stressed because it allows for novel device
isolation techniques with higher densities as well as

L l' -,licon

Lateral selective epitaxial growth of silicon [38).

Titles of Final Projects in CVD Epitaxy on
Patternless and Patterned Substrates

* Silicon on Insulators: A Focus on Epitaxial Lateral Overgrowth
* Solid Phase Epitaxy of Silicon
* GaAs Contacts: Theory and Practice
* Kinetics in the Vapor Phase Epitaxy of GaAs
* Recent Studies on the Kinetics of Epitaxial Silicon Growth
* Metalorganic Chemical Vapor Deposition of III-V Compounds
* Chemical Vapor Deposition of ll-VI Materials
* Recent SOI Technologies
* Increasing the Throughput of High Electron Mobility Transistors
Grown by III-V Molecular Beam and Chemical Beam Epitaxy
* Plasma Enhanced Chemical Vapor Deposition
* Silicon Epitaxial Growth Research at Purdue University: An Overview
* Silicon on Insulator Technologies

new device structures such as silicon on insulator ar-
rangements [13, 14, 31-34, 38].


Two potential problems associated with the doping
profiles in such epitaxial thin films are addressed as
being very important: autodoping (etchback) and solid
state diffusion. Typically, a lightly doped epitaxial
layer may have to be deposited on a heavily doped
substrate with the same kind of dopant, or vice versa
(e.g., n- or n on n-, p- or p on pi). Also, for the forma-
tion of a pn junction, a p-doped epitaxial layer has to
be deposited on an n-doped substrate or vice versa.
Autodoping is discussed first in detail. Etchback is
shown to result in sharper transitions from the dopant
concentration level in a substrate to the dopant level
in the growing epitaxial layer, as substrate tempera-
ture or reactor pressure decreases. Simple semi-em-
pirical models are developed for autodoping. These
models are shown to be able to predict trends like the
ones just mentioned as well as a shift in the position
of the pn junction delineated by the two layers [2].
Comprehensive models of autodoping are briefly pre-
sented. Furthermore, although increased etchback is
pointed out to be technologically undesirable, it is
demonstrated that autodoping may be a very useful
tool in determining velocity profiles just above a sus-
ceptor in some CVD reactors (e.g., a pancake reactor).
Solid state diffusion is presented next. Although
redistribution of the dopants because of diffusion dur-
ing epitaxial growth of a (doped) thin film may not be
very important compared with the redistribution that
takes place during subsequent processing, a simple




model for solid state diffusion is discussed. This model
is shown to result in a graded junction between sub-
strate and epitaxial layer and in a shift of the pn junc-
tion delineated by the two layers. This shift, though,
seems to compensate for the junction lag due to the
autodoping effect. The intensity of solid state diffusion
effects is shown to depend on the substrate tempera-
ture during epitaxy, the duration of this step, and the
magnitude of solid state diffusivities at standard con-
ditions. Also, a brief discussion of redistribution of
dopants due to diffusion during subsequent processing
is presented.


After a brief coverage of other kinds of CVD
epitaxy, such as PACVD, the last stage is a final term
paper. Each student works on his/her own project
after choosing a topic. Within such projects, students
are expected to critically review any existing litera-
ture and to present their own "innovative ideas" in
improving or developing various CVD epitaxial pro-
Topics in the chemical vapor deposition epitaxy on
patternless and patterned substrates covered in the
past four years are listed in Table 1.


1. Takoudis, C. G., "Fundamentals of Microelectronics Processing,"
Chem. Eng. Ed., 21,170 (1987)
2. Ghandhi, S.K., VLSI Fabrication Principles, Wiley, New York
3. Sze, S.M., VLSI Technology, 2nd ed., McGraw-Hill, New York
4. Sze, S.M., Semiconductor Devices: Physics and Technology,
Wiley, New York (1985)
5. Till, W.C., and J.T. Luxon, Integrated Circuits: Materials, Devices
and Fabrication, Prentice Hall, Englewood Cliffs (1982)
6. Wolf, S., and R.N. Tauber, Silicon Processing for the VLSI Era-
Process Technology, Lattice Press, Sunset Beach (1986)
7. Hess, D.W., K.F. Jensen, and T.J. Anderson, "Chemical Vapor
Desposition: A Chemical Engineering Perspective," Reviews in
Chem. Eng., 3, 97 (1985)
8. Sherman, A., "Modeling of Chemical Vapor Deposition Reactors,"
J. Elec. Mat., 17, 413(1988)
9. Ban, V.S., "Novel Reactor for High Volume Low Cost Silicon Epi-
taxy," J. Crystal Growth, 45,97 (1978)
10. Juza, J., and J. Cermak, "Phenomenological Model of the CVD
Epitaxial Reactor," J. Electrochem. Soc., 129, 1627 (1982)
11. Cullen, G.W., and J.F. Corboy, "Reduced Pressure Silicon Epi-
taxy: A Review," J. Crystal Growth, 70, 230 (1984)
12. Arnaud D'Avitaya, F., S. Delage, and E. Rosencher, "Silicon MBE:
Recent Developments," Surf. Sci., 168, 483 (1986)
13. Jastrzebski, L., "SOI by CVD: Epitaxial Lateral-Overgrowth
Process-Review," J. Crystal Growth, 63, 493 (1983)
14. Jastrzebski, L., "Silicon on Insulators: Different Approaches A
Review," J. Crystal Growth, 70, 253 (1984)
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ical Vapor Deposition," Chapter 3 in VLSI Electronics: Mi-
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Vapour Phase: Confrontation of Theory with Experiment," J.
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on SiO2 and Si3N4 Substrates, J. Electrochem. Soc., 127, 194
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Reduced Pressure in a Hot-Wall Reactor: Equilibrium and Ki-
netics," J. Crystal Growth, 56, 659 (1982)
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Temperature Silicon Epitaxy Due to Water Vapor and Oxygen in
the Growth Ambient," Appl. Phys. Lett., 53, 2543 (1988)
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Neudeck, "On the Gas Flow: Temperature Profile and Epitaxial
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89-1, 333 (1989)
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Through SEG Techniques," Solid State Tech., Aug., 141, (1985)
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Matsumato, Solid State Tech, January, 111(1988)
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Biennial University/Government/Industry Microelectronics
Symposium, Westborough, MA (1989) 0





An Electrochemical Engineering Perspective

University of Florida
Gainesville, Florida 32611

C HEMICAL ENGINEERS working in the field of
electronic materials are not normally concerned with
processes taking place within the semiconductor. Most direct
application of chemical engineering principle is seen in the
analysis of the growth of semiconductors in the gas phase
(CVD or MOCVD) or in the liquid phase (crystallization,
Czochralski crystal growth, and Bridgman growth). Applica-
tion of chemical engineering principles to these processes is
not easy but is direct because the species of concern are not
electrically charged. In contrast, the species within the
semiconductor (e.g., electrons, holes, ionized electron donors
or acceptors) are charged, and proper analysis of processes
taking place within the semiconductor requires that this elec-
trical charge bee treated.
Since ions in electrolytic solutions are also charged, the
principles learned in the application of transport phenomena,
reaction engineering, and thermodynamics to electrochemi-
cal systems can be applied almost directly to the study of
semiconductor devices. Here, these principles are applied to
interpret the impedance response of semiconducting elec-
Impedance techniques can be applied to semiconductors
to identify the electronic structure, i.e., the distribution of
states within the semiconductor bandgap. A simplified
schematic representation of the band structure is shown in
Figure 1. Electrons can be excited from the valence or bond-
ing orbitals to the conduction band by receiving thermal or
electromagnetic (illumination) energy. The species formed
by this excitation are electrons (in the conduction band) and
holes (absence of an electron in the valence band). Both
species are charged (electrons have a negative charge and
holes have a positive charge) and can move in response to
concentration or potential gradients.
The minimum energy required to excite an electron from
the valence band to the conduction band is the bandgap
energy. In the ideal semiconductor, electrons cannot exist
at energy levels between the valence and conduction ener-
gies. In real materials, electronic states within the band gap
can exist due to the presence of impurities (carbon, oxygen,

Mark Orazem is associate professor of
chemical engineering at the University of
Florida, where he contributes to Microfabritech
(a center for study of electronic materials). He
holds BS and MS degrees from Kansas State
University and a PhD from UC Berkeley. His
research interests include applications of
impedance techniques to electrochemical sys-
tems, corrosion, and semiconductors.


10 @

2 D 0

FIGURE 1. Generalized reaction scheme showing electronic
transitions between the conduction hand edge with energy E,
the valence band edge with energy E,, and a defect level with
energy E,.
and chromium are examples) or of dislocations, vacancies,
or other lattice defects. These states can be electron donors
or electron acceptors. Donor species are those which become
positively charged when an electron is released, while accep-
tors become negatively charged when an electron is added.
Because these species are charged, the distribution of elec-
trical potential can be affected. Inter-band electronic states
can be undesirable since they facilitate electronic transitions
which can reduce the efficiency of electronic devices. In some
cases, inter-band states are intentionally added when the
added reaction pathways for electrons result in desired ef-
fects. Electroluminescent devices, for example, rely on emis-
sion of photons which takes place when electrons are trans-
ferred from the conduction band to an inter-band state in a
large-bandgap semiconductor. The energy level of the states
caused by introduction of the impurity determines the color
of the emitted light. The impact of these states can be signif-
icant, even in concentrations that would seem to be very low
by normal chemical engineering standards. There is, there-
fore, a need for developing new ways to evaluate the concen-
tration, energy, and distribution of such electronic states.
A variety of techniques have been developed to study
semiconductors which are based on impedance spectros-
copy. We wish to focus here on a variant of electrochemical
photoccapacitance spectroscopy [1-5 in which the capacity of
a reverse-biased electrode is measured as a function of the
wavelength of incident sub-bandgap light. Let us note here
that we really do not measure a capacity. Instead, we meas-
ure a periodic cell potential in response to a periodic current
(or vice-versa) from which we calculate aan impedance which
has real and imaginary components. If we assume that this
system behaves like an electrical circuit consisting of a
capacitor and a resistor in series, we can, through regres-
sion techniques, obtain a value for a capacity and a resist-
ance. The capacity obtained in this way is usually em-
phasized in this type of work since it can be easily related
to the charge held in the semiconductor.
Since light of energy sufficient to cause an electronic
transition will change the amount of charge held in a given
Copyright ChE Division ASEE 1990


state, changes in capacity at a given photon energy indicate
the presence of states that allow transitions requiring that
amount of energy. From this type of data we can obtain the
energy levels of electronic states. The problem in this is
that the largest contribution to the capacity is due to shallow
level electronic states that are usually intentionally intro-
duced as dopants. In fact, the change in capacity seen under
illumination is (at best) proportional to the square root of
the ratio of the defect concentration to the dopant concentra-
tion. This means that the technique of Haak and Tench [1-4]
can be applied to semiconductors with a large defect concen-
tration as compared to dopant concentration, but provides
an unacceptable low signal to noise ratio when the dopant
concentration is moderately large. On the other hand, the
real part of the impedance, normally ignored since it is so
difficult to relate to physical parameters, is very sensitive
to these defects as low frequencies. We wish to focus here
on the application of electrochemical principles to the prob-
lem of identifying the relationship between the real part of
the impedance response and the energy, concentration, and
distribution of defects. We can do this through development
of a mathematical model based on the principles used in
analysis of electrochemical systems. The treatment pre-
sented here follows a qualitative description of the experi-
mental technique and the methods usually used in its


Impedance techniques involve perturbation of a steady-
state condition by a sinusoidal current or applied potential

S40 0 A

10 --

0 10 20 30 40 5(
Zr, Ohms
I 11II I 1 111 I IIIII 1 1 n 1111 1 111 i ing II IuIi 111111

zr [wl

0 I llllli 1 1111I 1 11111 lln l IIII 1111111 1 11111 1 lll 11 llll 1l _25
10-3 10-2 10-1 100 101 102 10 104 105
Frequency. Hz

FIGURE 2. Impedance data for a system consisting of a resistor
(with no capacitive component): a) impedance plane plots
with frequency as a parameter; b) Bode plots for real and
imaginary components of impedance.

of low magnitude. A typical amplitude for an applied poten-
tial perturbation might be 10 mV, and the resulting sinusoi-
dal current should have the same frequency, but may be
shifted in phase. Thus the impedance, obtained by dividing
potential by current, can be described as having real and
imaginary components, i.e.,
Z=Z, +j (1)

A typical way to analyze impedance experiments is to com-
pare the results to the impedance of simplified "equivalent"
electrical circuits.

Equivalent Circuit Representations of Simple Systems

Electrochemists commonly present the resulting data in
the form of an impedance plane plot (-Zj as a function of Zr
with frequency as a parameter). An impedance plane plot is
given in Figure 2 for an electrical circuit consisting of a
resistor. This is, of course, a very simple case. A Bode plot
for this system (see Figure 2b) shows that the real part of
the impedance is constant for all frequencies, and, since
there is no phase shift, the imaginary part of the impedance
is equal to zero. Thus, Zr = R, and Zj = 0.
The impedance data for a resistor and capacitor in series
are given in Figure 3. The real part of the impedance is
independent of potential, and the magnitude of the imagi-
nary part is inversely proportional to frequency, i.e., the
highest values are seen at low frequencies. For this case: Zr
= R1, and Zj = -1/oC1.

0 I

R =250

C, =20,.F

10 20 30 40 50
Zr, Ohms
....... ...... ....... .... .. ..... .. ...... ..... ... 1 0 '

10 .

. I,

10-3 10-2 10-1 100 101 102 103 104 105
Frequency, Hz
FIGURE 3. Impedance data for a system consisting of a resistor
and a capacitor in series: a) impedance plane plots with fre-
quency as a parameter; b) Bode plots for real and imaginary
components of impedance.


S -um -I

Z,- \

I I I~1111 4 II 1 \II III1111111III


Equivalent Circuit Representations for
Electrochemical Systems

Simple electrochemical reactions at an electrode surface
are often modeled in terms of the circuit shown in Figure 4.
The resistance Rs is associated with the Ohmic resistance of
the cell, the capacity is associated with the double layer
capacity, and the resistance R, is related to the rate con-
stant for the surface reaction. The impedance plane plot for
this case is in the shape of a semicircle with the high fre-
quency asymptote shifted from the origin by an amount
equal to the solution resistance. Additional elements can be
added to account for reactions proceeding in parallel or in
series. A perfect semicircle is usually not observed experi-
mentally, and a number of factors have been used to explain
the observed depression of the semicircle. Roughening of
the surface or growth of films during the course of an exper-
iment can, in some cases, account for these observations.
Mass transfer effects are also often important. These are
treated by adding a Warburg element (see Figure 5). The
impedance response of a Warburg element is a function of
frequency and is derived by solving the convective diffusion
equation for a given geometry to obtain the frequency de-
pendent concentrations of reactants at the electrode surface.
See reference 6 and chapter 9 in reference 7 for more discus-
sion on the application of impedance techniques to typical
electrochemical systems.





100 -

7 T r 1 7T
C, = 20 F
R, 250 [-

R =4250


0 100 200 300
Zr, Ohms

300 -

too -
- 200


An Equivalent Circuit Representation for
Defects in Semiconductors

The fifth case considered here is that of a second resistor
and capacitor in series added in parallel to the capacitor of
Figure 3. The resulting impedance data are shown in Figure
6. The magnitude of the imaginary part of the impedance is
largest at lower frequencies, and the impact of the added
circuit components is seen at lower frequencies. The real
and imaginary components of impedance, based on the equi-
valent circuit given in Figure 6, are

,=R4 R,2 (2)
S (C,+C,)2 +w(C2C,R,)2

Cz + C,2+2CC2R, (3)
Wj(C,+C2) + (C1CR)2
If the experimental system behaves like a given electri-
cal circuit, nonlinear regression techniques could be used to
obtain values for the resistor and capacitor components in
that circuit. If the electrical circuit chosen does not account
for all aspects of the data, e.g., if the circuit of Figure 3 is
used to model the data shown in Figure 6, the circuit compo-
nents will be functions of frequency. Note that the circuits
given in Figures 3 and 6 do not allow passage of direct cur-






Z". Ohms



t )0



-0 2

10 10 100 10' 102 103 104
Frequency, Hz

10-3 10-2 10- 100 10 10 103 10
Frequency. Hz




FIGURE 4. Impedance data for a system consisting of a resistor
in series with the parallel combination of a capacitor and a
resistor: a) impedance plane plots with frequency as a
parameter; b) Bode plots for real and imaginary components
of impedance.

FIGURE 5. Impedance data for a system consisting of a resistor
in series with the parallel combination of a capacitor and a
resistor and Warburg element in series: a) impedance plane
plots with frequency as a parameter; b) Bode plots for real
and imaginary components of impedance.


400 500

rent. This corresponds to an ideally polarized or completely
blocking electrode. To allow passage of direct current, a
resistor in parallel to the other elements would be added as
was done in Figures 4 and 5.
The electrical circuit given in Figure 6 is especially relev-
ant to our system because it describes the behavior of an
ideally polarized semiconductor electrode that contains a
reasonable concentration of inter-band defects. In the high
frequency limit,
Z, = R2 (4)
Z =-1- (5)
This behavior is more easily seen in a logarithmic impedance
plane plot as shown in Figure 7. This type of plot emphasizes
the high frequency data at the expense of the low frequency
asymptote. The high frequency limit obscures the influence
of the defects and yields the same result as would be ob-
tained for a resistor and capacitor in series. For this reason,
experimental data are frequently taken at high frequencies
(greater than 10 kHz is usually sufficient). The defects, rep-
resented by C1 and Ri, have a major influence at low fre-
quencies, i.e.,

Z,=R2+ R (6)
(C, +C,)2

n lOu
L 105

C IlnF
- R= 0000
R,= 100n

C, 1 nF

C, +C2
So(C, +C,)2

The imaginary part of the impedance tends toward -oc while
the real part of the impedance is shifted from the bulk resist-
ance by a constant which includes the time constant as-
sociated with the defects R1C1 and and averaged capacity
(C1 + C2)2/C1. This interpretation of the circuit elements is
based, to a large extent, on the results of the mathematical
model presented in subsequent sections.
We can compare these idealized cases to experimental
results. Impedance plane plots are presented in Figure 8
with potential as a parameter for an n-GaAs electrode in
contact with a mercury pool [8]. The logarithmic plot was
used to emphasize the behavior at the high frequency limit.
Linear regression of these data with Eqs. (2) and (3) yields
frequency-independent values of circuit components which
correspond to the solid line. The component values do vary
with applied potential, and, if illumination had been used,
the component values would vary with the photon energy
of the illumination. The problem we face is how to tie these
component values to physical characteristics of the semicon-
ductor. One way to gain this intuition is to develop models
for the system based on treatment of transport phenomena



E 107



05 10 1.5 20 25
Zr, Ohmsx107

E-f i llTiIRi 1iT 711nT r i nlli I 111111 1 I 11



-z ) 106



F I l.ili I E 11111in I 1111i I i It I1i 1111 1 i 1111 1 1111 103
1072 10-' 100 101 12 103 104 105
Frequency, Hz

FIGURE 6. Impedance data for a system consisting of a resistor
in series with the parallel combination of a capacitor and a
resistor and capacitor in series: a) impedance plane plots with
frequency as a parameter; b) Bode plots for real and imag-
inary components of impedance.

I 111 11 I 11111111 I II hi I I lillll I i II I

103 104 105 106 107 100
Zr, Ohms

FIGURE 7. Impedance data for the system of Figure 6 consist-
ing of a resistor in series with the parallel combination of a
capacitor and a resistor and capacitor in series.

N 105

1- rTMF r" Tl--n 1ITT TTI T Im ilf "-T"nT

1 o3 104 10 1 06 1 7 IC

Zr, Ohms
FIGURE 8. Impedance plane plot for a semi-insulating n-GaAs
electrode in contact with a mercury pool [8].


and reaction kinetics and to compare the results of these
models to those from the equivalent electrical circuits.


Development of mathematical models for the impedance
response of semiconducting systems generally takes place
in two steps: development of a steady-state model followed
by development of a model treating the sinusoidal perturba-
tion of voltage or current about the steady-state values.
Since the species of interest have a charge associated with
them, we need to include treatment of electrical potential
as well as concentrations. Thus, the electrostatic potential
and the concentrations of electrons, holes, and ionized defect
states become dependent variables for this system. The
shallow-level doping species are usually assumed to be com-
pletely ionized at room temperatures and thus contribute to
a fixed concentration of charge. Parts of the development
presented here are given in references 9, 10, 11, and 12.
References 13 and 14 provide good background to general
aspects of semiconductor physics, and 15 provides a good
mathematical foundation for electrochemical engineering.

Mass Transport Expressions

The electrochemical potential L, of a given species i can
arbitrarily be separated into terms representing a second-
ary reference state I4, a chemical contribution, and an elec-
trical contribution, i.e.,
Ii = Wi +RTtn(cf )+ziF (8)
where ci is the volumetric concentration of species i, fi is the
activity coefficient, z is the charge number, and (F is a po-
tential which characterizes the electrical state of the system
and can be defined in many ways. This treatment is entirely
analogous to the definition of chemical potentials as used for
electrically neutral systems. In fact, the usual chemical po-
tential is recovered for the case where z is equal to zero.
The flux Ni of species i is governed by the gradient of
the electrochemical potential, given in one dimension by

Ni =--uicigi (9)
where ui is the mobility of species i. If the semiconductor is
nondegenerate, the electron and hole activity coefficients fi
can be considered to be constant, and Eq. (8) can be substi-
tuted into Eq. (9) to give the dilute solution transport ex-
Ni =-Di d -uziFcd (10)

where the transport properties Di and ui are related through
the Nernst-Einstein equation; i.e.,
Di = RTu, (11)

From Eq. (10), the fluxes of electrons and holes are driven
by concentration and potential gradients. This distinction is
a result of the separation of the chemical and electrical con-
tributions given in Eq. (8). If desired, degenerate semicon-
ductor conditions can be modeled by calculating the value of
the activity coefficients fi for electrons and holes (e.g., [16]
and [17]). The flux expression for species i is constrained by
the equation of continuity, i.e.,

2Ei JN +Gi (12)
at ay

Usually inter-band defect states are considered to be im-
mobile; the rate of change of the concentration of ionized
inter-band states is equal to their (position-dependent) rate
of production, Gi.
For most electrochemical systems, the separation of
charge associated with interfacial regions can be treated
simply as contributing to rate constants associated with
electrode kinetics. This is not appropriate for a semiconduc-
tor because this separation of charge is integral to the oper-
ation of electronic devices. Poisson's equation,
a20 F
=---[p-n+Nd-N ] (13)

can be used to relate the electrostatic potential (P to the
charge held within the semiconductor. The scaling length
for this system, found by making the governing equations
dimensionless, is given by the Debye length,

,RT (14)
S F2(N, -N) j)

The term (Nd Na) includes the charge associated with par-
tially ionized mid-bandgap acceptors (which may be a func-
tion of applied potential) as well as the completely ionized
dopant species (which may have an arbitrary distribution,
but is usually assumed to be independent of operating condi-

Kinetic Expressions for Electronic Transitions

Calculation of a rate expression for Gi requires the choice
of a kinetic framework. In this work, electrons are allowed
to pass between the conduction band (with energy E,), the
valence band (with energy E,), and the inter-band species
(with energy E,). A general scheme for the various electron
transitions associated with this approach are shown in Fig-
ure 1. With these representations, the rates of the electronic
transitions between the various energy levels can be de-
scribed by applying mass action principles (e.g., [13]) to give
r, = k,c (15)
r2=k2(ct --C )p (16)
r, =kc- c;) (17)
r4 = k4 cn (18)
rs =k5 (19)
r6= k6np (20)
where ki is the rate constant of reaction i, cj is the concen-
tration of positively charged, inter-band donor species, c4 is
the total concentration of inter-band donors, n is the elec-
tron concentration, and p is the hole concentration.
In the absence of inter-band states, generation of elec-
trons and holes occurs through band-to-band mechanisms.
The rate of electron generation is given by

G =k( n?-np)

where the two righthand terms represent thermal genera-
tion (k5 = keni) and recombination, respectively, and ni is
termed the "intrinsic concentration" (a physical property
equal to the concentration of electrons and holes in the
"ideal" undoped semiconductor). The constraint that the
rates of generation and recombination are equal provides
that np = n under equilibrium conditions. In the presence
of inter-band states, the net rate of production for electrons
(and holes) is given by


G- = k kk4p (n2 -np) (22)
=k, [1+ k6(k,+k2p) ) (22)

Again, at equilibrium, the rates of generation and recombi-
nation are equal and np = nf.
Use of the above six kinetic expressions requires selec-
tion of the six rate constants (or three rate constants and
three equilibrium constants) associated with these expres-
sions. This apparently arbitrary selection can be approached
by deriving equilibrium expressions to relate the rate con-
stants for the reversible, homogeneous reaction pairs
explicitly in terms of the energy differences between the
valence band, inter-band species, and the conduction band,

=N,gexp F(E, -E,) (23)
k2 RT

E, N, exF(E -E,) (24)
k4 g RT
Eex = p F(E, -E,) (25)
k6 RT

where EK is the equilibrium constant for reaction pair ij, g
is the degeneracy associated with the inter-band state, Nc
is the conduction band density of states, and N, is the val-
ence band density of states. These expressions were derived
by assuming thermal equilibrium and substituting standard
statistical expressions for electron, hole, and defect concen-
tration in terms of energy level. The numerical value for g
is determined by the electronic character of the state, e.g.,
g = 4 for electron acceptors and g = 2 for electron donors
Parameter variation studies can be further simplified by
the assumption that the rate constants are interrelated such
that, given energy levels for the electronic states, all rate
constants can be obtained from a single rate constant. For
example, the relationship,

k4 =k2(f2 (26)

was obtained by assuming that changes in the free energy
of reaction associated with varying the energy of an elec-
tronic state are distributed equally between the activation
energies for the forward and the reverse directions. This is
similar to the standard approach used to separate the free
energy of an electrochemical reaction into chemical and elec-
trical terms. The symmetry factor in this application is as-
sumed to have a value of 1/2 (e.g., [15]).
Similar expressions can be developed for band-to-band
recombination, i.e.,

k2=k E52 (27)

The use of Eq. (27) to relate the homogeneous, band-to-band
rate constant k6 to the corresponding inter-band constants
k2 (and k4) is equivalent to assuming that the reaction cross
section is the same for recombination through trap sites as
it is for direct band-to-band recombination. This assumption
could easily be relaxed to account for enhanced rates of re-
combination through trap sites.
In the case where solar illumination is applied to the
semiconductor, the expression for the optical generation of
electrons under solar illumination is

Ge_.. = qom exp(-my)

where y is the fraction of incident photons with energy
greater than the bandgap Eg, m is the band-to-band absorp-
tion coefficient, and qo is the solar flux. Similar expressions
apply for sub-bandgap illumination; however, treatment of
optical excitation by light with photon energies smaller than
the bandgap requires expressions for the effective absorp-
tion coefficient. Such expressions can be found in the litera-
ture (e.g., [18]) for the absorption coefficient m correspond-
ing to the transition of electrons from inter-band acceptor
states to the conduction band. This absorption coefficient is
a function of the inter-band state energy, the photon energy,
and the concentration of ionized states. Absorption of sub-
bandgap illumination is negligible for the usual values of
semiconductor thickness, inter-band species density, and
absorption coefficients. This allows the effects of sub-
bandgap illumination to be included as a modification of the
rate constants in the expressions for r, and r3.

Impedance Modeling
A system whose time response y(t) to a perturbation x(t)
can be described by the expression

b dy(t)+b d (t).. +by(t)
dt" dt ..by(t)
dmx(t) dm-'x(t)
=a dtm +a dtm-- ... +amx(t) (29)

is defined as a linear system. One characteristic of such a
system is that a perturbation of the form x(t) = cos(wt) will
result in a response of the form y(t) = cos(ot + 0). This
behavior is also observed for nonlinear systems if the
amplitude of the perturbation is small enough that a first-
order Taylor series expansion about the steady state is ap-
Experimental impedance measurements are evaluated
using this theory since the current response to a sinusoidal
applied potential is also sinusoidal. The important restric-
tions are that the system be stationary, that the system
response be driven by the imposed signal, and that the im-
posed voltage perturbation be sufficiently small that the sys-
tem can be described by Eq. (29). If these conditions are not
violated, all variables of the system will take the form
x=x+(x, +jii) exp(jot) (30)
where x, Rr, and i are functions of position, but are inde-
pendent of time. This means that impedance measurements
are usually made in the region where the voltage perturba-
tion is small enough for the system to be linear, yet large
enough to give a significant signal to noise ratio. For a cur-
rent density given by

i=I+i, exp(jet) (31)

the concentration of electrons is given by

n =D+(i, +jfij)exp(jcot)

Similar expressions are used for potential and the concentra-
tions of holes, ionized electron acceptors, and ionized elec-
tron donors. In the above equations, an overbar represents
the steady-state value, and a tilde represents the perturba-
tion value. The actual concentration or potential at a given
point in time and space is given by the real part of the
expressions given above. The approach described here has


been used to model the impedance response of semiconduc-
tors in the absence of inter-band states [19, 20] and in the
analysis of electrochemical systems (e.g., [21-24]).
The above expressions are substituted into the govern-
ing equations which are solved sequentially for the steady-
state and the sinusoidal steady-state portions, respectively.
The impedance can be resolved from the calculated potential
variation across the space charge region into real and imagi-
nary components according to

Z, = (33)



Z =---

Steady State Boundary Conditions

The governing equations are initially solved under the
steady-state condition, subject to the boundary conditions

NP = 0, = 0, and i=0
at the semiconductor-current collector interface ohmicc con-
tact), and
de q
N,=0, 0=0, and d=-
dy E,

at the semiconductor-electrolyte interface (ideally polariza-
ble contact). These conditions are appropriate for a semicon-
ductor-mercury contact or for a semiconductor-electrolyte
contact where the electrolyte is chosen so that no chemical
reaction occurs.

Sinusoidal Steady State Boundary Conditions

The time-dependent equations are solved for the re-
sponse to a superimposed sinusoidal current by introducing
expressions for the dependent variables (such as Eq. (32))
into the governing equations and linearizing around the
steady state solution obtained in the previous step. Appro-
priate boundary conditions for the impedance calculations
are given by

= =N,, = pj=, =0, and i j= ,=

at the semiconductor-current-collector interface, and by

N =Nr=0, 6 = = _0,0 = -L, and dr =0
dy e,o' dy
at the semiconductor-electrolyte interface. Again, these
conditions are consistent with an ideally polarized electrode
where the superimposed current acts only as a charging

Numerical Method for Solution

The solution of the coupled differential equations is non-
trivial, and a complete solution requires use of a computer.
The results of this type of numerical solution are presented
elsewhere [11, 12]. The point here is to emphasize that the
apparently complex behavior associated with transport and
reaction processes within the semiconductor in response to
a sinusoidal perturbation of current or applied potential can

be described by a straightforward application of principles
learned in the study of electrochemical systems.

Analytic Expressions Used for Analysis of
Experimental Data

Analytic solutions to the above equations have been de-
veloped that are valid in the high frequency limit. These
solutions are based on integration of Poisson's equation
coupled with assumption of equilibrium concentration distri-
butions. The relationship between the applied potential and
the R-C series capacitance was derived by Mott and
Schottky (see, e.g. Joffe [25]) in the late 1930's to be (for an
n-type semiconductor)

2 V+-i
I F(35)
C2 cF(N, -N,)

This is the well-known Mott-Schottky relationship.
Deviations from straight lines in Mott-Schottky plots,
are frequently attributed to the influence of potential depen-
dent charging of surface or bulk states. While deviations
can also be attributed to non-uniform dopant concentrations,
this interpretation is supported by analytic calculations of
the contribution of defects to the space charge as a function
of applied potential (i.e., [26-27]).


The principles learned in the study of mass transport,
thermodynamics, and heterogeneous and homogeneous
kinetics associated with electrochemical systems can be
applied directly to the transport and reaction processes that
take place within a semiconductor. The theory of dilute sol-
utions is generally appropriate, and values for needed
parameters can be obtained through application of statistical

This material is based upon work supported by the Na-
tional Science Foundation under Grant No. EET-8617057
and on work supported by DARPA under the Optoelec-
tronics program of the Florida Initiative in Advanced Micro-
electronics and Materials.

Roman Characters
ci concentration of species i, cm-3
C space charge capacitance calculated from an R-C
series circuit, F/cm2
AC Change in C from a chosen reference level, F/cm2
Di diffusivity of species i, cm2/s
Ea inter-band acceptor energy, eV
Ec conduction band energy, eV
Ed inter-band donor energy, eV
Ef Fermi energy, eV
Eg bandgap energy, Ec Ev, eV
Ejk equilibrium constant for reversible reactions j and k
Et Energy of generalized inter-band trap species, eV
Ey valence bandedge energy, eV
fi activity coefficient for generalized species i
F Faraday's constant, 96487 C/equiv.
g degeneracy of inter-band species
i current density, mA
j 4=-


kj rate constant for species j
m absorption coefficient, cm-1
n electron concentration, cm-3
ni intrinsic carrier concentration, cm-3
Nc effective density of conduction band states, cm"3
Nd doping concentration, cm-3
Nv effective density of valence band states, cm-3
Nyi molar flux of species i, mol/m2*s
ri rate of reaction of species i, mol/cm3* s
R universal gas constant, 8.314 J/mol*K
R resistance associated with a given electrical circuit, Q
t time, s
T absolute temperature, K
ui mobility of species i, m2/V*s
V applied potential, referenced to flatband, V
I steady state symbol for variable x
ir real component of the perturbation in variable x
xj imaginary component of the perturbation in variable x
y distance from interface, cm
zi charge number for species, i
Z complex impedance, L2*cm2
Greek Characters
e permittivity, Farad/cm
0 phase angle, rad
X Debye length, cm
pi electrochemical potential of species i, J/mol
p. reference electrochemical potential of species i, J/mol
D electrostatic potential, V
AD change in the real or imaginary portion of the potential
across the semiconductor sample, V
o frequency, s-1

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Part I: Introduction

R. O. FOX and L. T. FAN
Kansas State University
Manhattan, KS 66506

A STOCHASTIC SYSTEM is a system evolving ac-
cording to probabilistic laws as opposed to de-
terministic laws. In practical terms this implies that
given a system in a certain measurable state, the
evolution of the system through other possible states
can only be predicted in terms of a probability. We
are thoroughly familiar with deterministic systems
whereby, for example, knowledge of the initial posi-
tion and momentum will allow us to exactly determine
the future position.
Imagine a system for which knowledge of the ini-
tial conditions only allows us to predict the future pos-
ition with a certain probability. Such a system would
seem to go against the scientific belief of strict deter-
minism. For our purposes, however, we can assume
that although in principle it may be possible to make
strict deterministic statements about the behavior of

Rodney O. Fox's doctoral research with
Professor Fan was supported by a NSF
Graduate Fellowship and involved the analysis
of non-linear stochastic processes and other
probabilistic models applied to coalescence
and breakage phenomena. Prior to his doc-
toral studies, he was a Fulbright Scholar at the
Federal Institute of Technology in Zurich, H
Switzerland. He also worked at the Laboratoire
des Sciences du Genie Chimique in Nancy,
France, as a NATO Postdoctoral Fellow.

L.T. Fan has been on the faculty at
Kansas State University since 1958 and has
been department head since 1968. He re-
ceived his BS from the National Taiwan
University, his MS from Kansas State Univer- A
sity, and his PhD from West Virginia University,
all in chemical engineering. He holds eight
patents and has published numerous techni-
cal articles and six books, the most recent be-
ing Controlled Release: A Quantitative Treat-
ment, published by Springer Verlag.

any physical system, such statements would require
exact and complete knowledge about the initial condi-
tions and the external forces acting on the system.
Since such exact knowledge is often beyond us, the
reality as we perceive it may be represented best by
stochastic models. This philosophy is in line with cur-
rent theories involving deterministic chaos where a
small error in the value of the initial conditions pro-
duces an enormous error in later predictions about
the process (see, e.g., reference 1).
Consider, for example, a bubbling fluidized bed.
Theoretically, it is possible to exactly predict the sizes
and positions of bubbles at each moment in time. How-
ever, the prediction would be dependent upon the ini-
tial conditions since the bubbles do not occur with
exactly the same positions and sizes each time a
fluidized bed is started up. Such a system appears to
us to be stochastic, and thus we speak of the random
coalescence and movement of the bubbles. This is
equivalent to stating that although in principle we
may be able to understand the mechanism of coales-
cence for two or three isolated bubbles in a deter-
ministic manner, we are unable to extend the deter-
ministic model to accurately predict the behavior of a
large swarm of bubbles. Therefore, we resort to a
model involving random movement and coalescence.
Nevertheless, it is important to note that neither the
deterministic nor the probabilistic mode of modeling
excludes or negates the utility of the other. Indeed,
while the deterministic model may be intractable for
large complicated systems, the basic knowledge it pro-
vides about the dependence of the rate constants ap-
pearing in the probabilistic model on system parame-
ters is invaluable. Both modes of modeling should be
seen as working hand-in-hand, providing complemen-
tary understanding of complicated systems. An exam-
ple of this aspect can be found in the recent work of
Muralidhar and Ramkrishna [2] in modeling coales-
cence efficiencies.

Copyright ChE Division ASEE 1990


Numerous chemical process systems lend themselves to a stochastic description due to their inherent
complexity and fluctuating nature. Examples of such systems can be found in dispersed phase
flow, turbulence, solids mixing, and in many other chemical engineering fields of study.

At this point we wish to carefully distinguish be-
tween the deterministic models mentioned above,
which allow an exact determination of the behavior of
the system, and macroscopic models, which are also
deterministic but are volume-averaged over the exact
deterministic equations. Macroscopic models, there-
fore, are deterministic models involving variables
such as overall temperature and concentration. In con-
trast, the exact deterministic or so-called microscopic
models deal with the position and momentum of indi-
vidual molecules. The exact relationship between
these two domains is the subject of study of statistical
mechanics. Although the stochastic models considered
in this paper are less detailed than microscopic mod-
els, they are more detailed than the macroscopic mod-
els describing only the average behavior of a system.
Thus, our desire to arrive at an accurate formulation
of the stochastic model necessitates a close scrutiny of
the mechanisms underlying the kinetic behavior of the
process. In fact, a multitude of stochastic models cor-
responds to any given macroscopic model. Hence, the
ad hoc addition of fluctuating terms to a macroscopic
model is of relatively limited value if we wish to pre-
dict the effect of changing operating conditions on the
higher moments of the probability distribution of the
random variables.

Numerous chemical process systems lend them-
selves to a stochastic description due to their inherent
complexity and fluctuating nature. Examples of such
systems can be found in dispersed phase flow, turbu-
lence, solids mixing, and in many other chemical en-
gineering fields of study. Research efforts in these
areas have been reported extensively. For example,
by using probabilistic methods, coalescence and
breakage in dispersed phase systems have been
studied by Valentas and Amundson [3], Ramkrishna
and Shah [4], Ramkrishna [5], and Bajpai, Ram-
krishna and Prokop [6], among others. Stochastic
modeling of mixing and chemical reactions has been
reported by Krambeck, Katz, and Shinnar [7], King
[8], Pell and Aris [9], Mann and O'Leary [10], and
Nauman [11], as well as work done by Fan and co-
workers [12-14], and others. A fluidized-bed reactor
is a notable example of a stochastic system with the
random generation and coalescence of bubbles leading
to pressure and density fluctuations. Stochastic mod-

els for fluidized beds have been discussed by Bukur,
et al. [15], Shah, et al., [16], Ligon and Amundson [17,
18], and recently by Fox and Fan [19].

The incorporation of stochastic analysis and model-
ing into the repertoire of our profession is a matter of
great urgency. Indeed, the need for a monograph or
textbook on this subject is noted in a list compiled by
Bird [20] and published in this journal. Devising ap-
propriate stochastic models for chemical process sys-
tems, however, can be difficult. Construction of valid
models requires the proper determination of the
source of fluctuations and the mechanisms by which
they evolve. The fact that relatively little interest has
been shown for stochastic analysis and modeling of
chemically reacting systems rests most likely with the
nature of the internal fluctuations; such systems con-
tain roughly the Avogadro number of molecules. A
well-known result of statistical mechanics states that
the number of density fluctuations are of the order of
N where N equals the total number of molecules in
the system. The implication is that, in terms of con-
centration, the fluctuations are negligible with respect
to the mean value equations and thus will be of little
concern in the macroscopic description of chemical
reactions. This result may be satisfying to the physi-
cist who wishes to build a unified theory of matter
based on molecular dynamics, but it is usually of little
practical value to the chemical engineer modeling an
actual chemically reacting system.
Visible or detectable fluctuations do exist in count-
less process systems, but their roots are not to be
sought at the molecular level. A fluidized bed, for
example, often fluctuates violently. These fluctuations
obviously do not stem from the transfer of individual
molecules among different phases in the bed; they
stem from the transfer of relatively large entities,
e.g., clusters of particles and bubbles. It is well known
that the bubbles can be modeled as entities which ran-
domly enter into the bed, coalesce in it, and leave
from it. Thus, the importance of properly identifying
the source of fluctuations for a successful description
of their impact on the system is obvious. Stochastic
models based on independent molecular processes will
show that the fluctuations are negligible in large sys-
tems, while a stochastic model based on mechanisms


involving, for example, bubble interactions will yield
significant fluctuations.
From the stochastic model of a chemically reacting
system, the more familiar kinetic expressions found in
the chemical reaction engineering literature can be de-
rived by calculating the average numbers of molecules
of each species and expressing these in terms of con-
tinuous variables. The latter is of course possible and
quite accurate since the number of molecules in any
system is usually very large-it is on the order of the
Avogadro number. As noted earlier, the variance of
the numbers of molecules of each species will be of the
order of the mean number of molecules. Con-
sequently, when working in terms of molar concentra-
tion, the standard deviation will be several orders of
magnitude smaller than the mean concentration. The

From the discussion . it should be clear that the
stochastic model is more fundamental in nature
than the deterministic rate equations of chemical
kinetics or, in general, macroscopic models.

probability distribution of the random variables will
then approach a delta function centered at the mean
or average concentration for a system containing a
large number of independent particles. In the statisti-
cal physics literature, this limit is often referred to as
the thermodynamic limit. In this limit it is possible to
describe the system in terms of the thermodynamic
variables of chemical concentration and temperature
instead of more fundamental quantities such as posi-
tion and momentum.
From the discussion presented thus far it should
be clear that the stochastic model is more fundamental
in nature than the deterministic rate equations of
chemical kinetics or, in general, macroscopic models.
However, we are justified in using the deterministic
rate equations when the number of molecules in the
system is extremely large. In general, we can say that
stochastic population balances for large numbers of
independent entities almost always reduce to the de-
terministic mean value rate expressions. Neverthe-
less, in all cases, the stochastic model represents a
fundamentally more basic description of the physical
behavior of the system. It recognizes the existence of
the individual members of the population and their
ability to undergo change at random times.
For relatively small populations the random nature
of the changes in the population numbers can be quite
significant. For example, the change in the number of
bubbles of each size in a fluidized bed takes place
rather quickly, resulting in the widely fluctuating be-
havior of this system. A detailed stochastic model of

the fluidized bed might include a stochastic population
balance for the bubble phase from which other physi-
cally important quantities, e.g., the total surface area
of the bubble phase, could be derived and their ran-
dom nature quantified [19, 21]. These observations, of
course, carry over to dispersed phase systems in gen-
eral where deterministic population balances are
widely used (see, e.g., Ramkrishna [22]).

An appropriate stochastic model should depict the
details of the internal mechanisms generating the fluc-
tuations and can be solved by means of a rational ap-
proximation technique when the resultant equations
are non-linear or be amenable to numerical simulation.
A general formulation possessing both of these qual-
ities is known in the modern literature as the master
equation (see, e.g., van Kampern [23] and Gardiner
[24]). The master equation was first introduced into
the statistical chemistry literature as a method of de-
riving statistical mechanics from molecular dynamics
(see, e.g., Cohen [25]). In the ensuing years much
work has been done to understand the nature of the
solutions to the master equation. Numerous approxi-
mation schemes have been devised to solve nonlinear
master equations [24]. Perhaps the most successful of
these has been the system size expansion [23].
A stochastic formulation based on the Janossy den-
sity function can be found elsewhere [26, 27]. How-
ever, we prefer to work with the master equation for-
mulation for several important reasons: (1) the master
equation uses as random variables the numbers of en-
tities or particles that are the natural variables when
considering a population balance; (2) although the
Janossy density function and the joint probability dis-
tribution in the master equatipn are theoretically in-
terchangable through a correct change of variables,
the master equation is easier to formulate once the
fundamental events that change the values of the
numbers of entities in each state are known; (3) in
contrast to the Janossy density function, there is a
vast body of literature pertaining to the master equa-
tion wherein numerous solutions are discussed, ra-
tional approximation techniques are introduced, and
statistics such as the first passage time and the prob-
ability of large fluctuations are derived (see, e.g., van
Kampen [23] and Gardiner [24] for partial lists of ref-
erences and basic derivations, solutions, and approxi-
mation techniques); (4) the rates of transition for each
possible event appearing in the master equation are
exactly the quantities needed when performing a


Monte-Carlo simulation of the system; (5) the proce-
dure to go from the master equation to a multivariate
Fokker-Planck equation or to a stochastic differential
equation is straightforward, thus opening the possibil-
ity of applying the large body of literature in these
fields to problems involving the master equation; and
(6) except for the limited work carried out with the
Janossy density function in the chemical engineering
literature, the master equation formulation is perhaps
the most commonly used stochastic formulation for
population balance problems in the current scientific
Numerous physico-chemical systems have been
studied through formulation of their master equa-
tions. In particular, various chemically-reacting sys-
tems have been thoroughly studied and numerous
examples are available in the literature [23, 24, 28,
29]. Nicolis and Prigogine [28] discuss stochastic
methods for reaction-diffusion systems and non-
equilibrium statistical mechanics with an emphasis on
self-organization in nonequilibrium conditions. Op-
penhiem, et al. [29] present an interesting and useful
compilation of basic papers on stochastic methods in
chemical physics. Van Kampen [23] discusses in detail
the effects of internal and external fluctuations in
chemically reacting systems, while Gardiner [24] has
collected many examples of nonlinear chemical reac-
tions in both lumped and distributed systems. These
authors and others have also dealt with the effects of
fluctuations on the so-called "critical slowing down" in
chemical systems and with other random effects and
have presented methods for the stochastic treatment
of mean passage time in bistable systems. While these
systems are well documented in the statistical physics
literature, the results have made little headway into
chemical engineering.
Many chemical process systems are governed by
nonlinear equations; this, in turn, implies that the
stochastic model should also be nonlinear. This compli-
cation naturally leads to a coupling between the mo-
ment equations describing the population. It is then
no longer possible to find the moments of the probabil-
ity distribution of the random variables by solving an
independent equation for each moment. To solve these
equations, approximation techniques need to be intro-
duced. Common ad hoc assumptions of independence
between random variables or formulation of the
higher-order moments as products of lower-order ones
are of limited value. Instead, a rational expansion
technique where the magnitude of higher-order terms
can be controlled is clearly preferable. This technique
will allow us to uncouple and solve the equations for
lower order moments and then to use them in the

Part II of this series will be concerned with the
derivation and solution of the master equation.
The System Size Expansion will also be outlined . .

coefficients of the equations for higher order moments
while minimizing the error introduced through the ap-
proximation procedure. The System Size Expansion
is such an approximation technique for the master
equation [23].
Part II of this series will be concerned with the
derivation and solution of the master equation. The
System Size Expansion will also be outlined and used
to find approximations for the moments and correla-
tion functions of the random variables. For illustration
the master equation will be applied to the modeling of
a chemically-reacting system in the final part, Part
III. It will be demonstrated that fluctuations in a large
population are extremely small compared to the mean
value and thus can often be ignored.


This material is mainly based upon work supported
under a National Science Foundation Graduate Fel-
lowship awarded to the first author.


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Continued from page 6.
with a tendency for high quality. Many of the early
PhD graduates, for instance, went on to become pro-
fessors and administrators at major universities.

Since the majority of graduate students in a de-
veloping program were in the MS degree program,
they were very closely supervised and generally pro-
duced publication-quality work. Many have gone on to
important positions: three are company vice presi-
dents; one is a director of overseas development; and
several are heads of company divisions of various
types. Several others pursued PhDs at other univer-
sities and have entered academia or research and de-
As mentioned before, the PhD/MS student ratio
has recently increased to a level that will ensure a
high rate of PhD graduates in future years. It appears
that the department is beginning to achieve its early
objectives for the graduate program. In terms of doc-
toral students, the department has been long on qual-
ity but short on quantity. Now that graduate enroll-
ment has reached the desired level, we are focusing
our efforts on maintaining quality in both graduate
and undergraduate programs. E

Continued from page 41.
phases: gas and solid. Because of problems with par-
ticulates, liquids have been all but eliminated from
the clean room. The reactors are small, and a batch of
product can be held in one hand. Reaction times are
on the order of minutes rather than days, so the turn-
around is fast. Process control is simply a matter of
using in situ diagnostics to predict the endpoint of an
etch or deposition step. Compared with the difficulties
of death, mutation, complex organic chemistry, living
membranes, and mass transfer limitation typical of
bio-engineering, the challenges of the IC industry are
controllable. The problems are straightforward, but
they generally require experimental solutions. There
is enough work to be done to keep surface scientists
occupied for several decades. Not only do the prob-
lems require experimental solutions, but the chemical
engineer who lacks knowledge of device physics is just
as handicapped as the electrical engineer with his/her
ignorance of continuum mechanics. The need for a
cross-disciplinary education cannot be overem-
In conclusion, if you have the people skills to run
for congress, the patience to spend a day in a junior
high school, the perseverance to climb Mt. McKinley,
the hands-on skills to keep dual Weber carburetors
perfectly tuned on a 1960 Porsche, and the desire to
help an industry which is vital to our national security
and economy, consider obtaining a graduate degree in
IC processing and joining a US IC company. [


Process Sensing and Diagnostics

Edited by: Jaromir J. Ulbrecht

AIChE Symposium Series Vol. 85, No. 267

These papers, presented at the AIChE Annual Meeting in Washington, D.C. in 1988, analyze various
aspects of integrating sensors with microprocessors and include presentations reporting the exploitation
of optical, electrical, electrochemical, electromagnetic, and other phenomena to monitor a wide range
of physical, chemical and biochemical variables.

Contents: Chemical Sensing on the Factory Floor.
Electrochemical Sensors Produced by Microelectronic Fabrication Techniques.
Micron and Submicron Electrochemical Sensors.
Integrated Microsensors.
Feedback Strategies in Multiple Sensor Systems.
Synergism Between Applied Statistics and Sensor/Microcomputer Technology.
Development of a Process Diagnosis Scheme Using Al Techniques.
pH Sensors Based on Iridium Oxide.
Applying Capacitance Sensors.
Model Studies of Tin Oxide-Based Gas Sensors.
Determination of Solids Fraction in Slurries by Radio Frequency Diagnostic Techniques.
A System Approach to Ingestible Temperature Monitoring.
Thin Film Thermocouples for Heat Engines.
Spectroscopic Sensors for Bioprocessing.
On-Line Process Monitoring for Optical Density in Fermentations.
Rapid Microbe Detection Through Membrane Mediated Fluorescence.
Biosensors for Bioprocessing.
A New Sensing System for Continuous Specific Gravity Measurement in Fermenters.
Transient Measurements and Analyses of Biosensors.

1989 106 pp. ISBN 0-8169-0463-4 LC89-405 (Softcover)
Pub # S-267 AIChE Members $18 Others $36 Foreign extra $6

Send orders to: AIChE Publications Sales, 345 East 47 Street, New York NY 10017. Prepayment in U.S. currency
is required (check, VISA, MasterCard, international money order, bank draft drawn on a foreign bank with a New
York office). Members may order only one copy at member price and must indicate membership number when ordering.
Credit card customers: Please indicate "VISA" or "MasterCard" and include card number, expiration date, printed name
of cardholder and signature. (Europe, Middle East & Africa: Contact Clarke Associates-Europe Ltd, 13a Small Street,
Bristol BSI 1DE England.)


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We offer a stimulating environment for personal and
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If interested, send your resume, including country
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