Darcy's Law

Patrick E Trapa

Dost thou know of a mythical class,
In a land where beauty and truth are one.
Feebly the apprentices strive to pass
For the horrible trial soon will come.

Unmatched, the great pain and suffering endured,
So that the art of magic could be their master.
In toil, the tools of the wizards must be secured
Or else, May 25th will come in disaster.....

Not but the giants of Newton's fame declare:
"The Truth is Out There" somewhere.... Somewhere.

Footnote:If you want to keep it a true acrostic (and probably
 a bit better "poem"), you can change the last line to: 
No one has found the truth to set them free.

Clark Allred

The Kinetiwhacky (by Lewis and Clark)

'Twas springtime and reptating profs
Did preach and hand wave to the class.
All murky were the offered proofs:
A kinetical morass.
"Beware the Driving Forces, son!
The Gradiant Slide doth often trap!
Beware steady state's siren song
And all that Matano crap!"

He took his coarsened grain in hand:
Long time on random walk he mused --
Found small-angle tilt boundary,
And for a while self-diffused.
And, as Balluffish notes he read,
The Driving Forces fast emerged,
Equilibrating rapidly,
As their flux diverged!

One-D! Two-D! Source point, source plane:
The Driving Forces withdrew, as
With char'teristic length the grain
Parabolically grew.
"Now thou hast calmed the Driving Force,
With diffusion paths short-circuit,
Know vacancy mech'nisms, of course,
Leave your crystal too imperfect."
'Twas springtime and reptating profs
Did preach and hand wave to the class.
All murky were the offered proofs:
A kinetical morass.

Wendy Mao

Last Monday,
Taking a break from kinetics homework,
I took a walk down Memorial Drive toward the salt and pepper bridge
I paused after trying to make a moment
Of the peaking cherry trees and the daffodils as they gave out their
last dying sigh
I plucked one blushing blossom, to kill and keep in the cruel,
thoughtless way humans do
Admiring its beauty and mourning its fate-as it lay dry and dying on my desk
Then I wished I hadn't picked it and wished it wouldn't lose its beauty
as even those left on the tree would-defenseless against the rain and time
A gust of wind interrupted my dirge, snatching the blossom from my hand,
Lifting it up into the cloudless sky,
Letting it fall among the bushes
Lost there, it would remain forever pink and alive
Never to see decay
I thought about the injured pigeon my roommate had found the night
before She discovered it flailing wildly, helplessly at 2 AM
outside our house on Concord Ave.
A weary public works officer offered little consolation,
"I can't do anything as long as the bird was alive."
Probably to get rid of us, he took the dying bird,
Shoveling it into a blue plastic recycle bin,
reassuring us that he'd take care of it.
The bright orange hearse lumbered down the empty street,
And I was glad I didn't see the pigeon die,
glad the neighborhood cats were robbed of an easy meal,
And as the universe lurched forward,
toward death and equilibrium,
I was left a glimmer of hope in the face of inevitability
Perhaps the pigeon didn't die,
instead becoming one of a thousand perfectly opened flowers waiting for
the day.

Robert Bernstein: Two Semesters

May 25th is the day!!!
When we find out if it did pay.
Calculate that phase diagram,
Negative free energy, Ma'am.
Regular solution model or quasichemical approximation,
Ask Gibbs, and he'll tell you the derivation.
Where'd that free electron go,
Drude would know.
If there's a periodic potential,
A Bloch function is all.
How about some linear elastic fracture mechanics,
That's one way to get your kicks,
But wait, there's kinetics,
And that's where my record will get its nicks!
Remember the postulate of positive entropy production,
And don't confuse the three kinds of diffusion.
Darken and Kirkendall will show you how,
What frame its in, you better determine now!
More kinetics and my brain will be done,
Just look to Cahn for the solution.
He calculated all the simple cases,
Of course nothing real shows those faces.
Oh, my head,
I feel the diffusion of the dread!
Can I use a Laplace Transform?
Or maybe a variable separation is norm.
What was I thinking,
Maybe I should resort to drinking!

Larry Lee and Wendy Mao

Kinetics Limerick
There once was a man from Connecticut
Who wanted to study kinetics, but
He got stuck on diffusion
And lost in confusion
He ended up a schizophrenic nut.

W. Craig Carter (1999)

Portions of Diffusion

The value of diffusivity is not unfram'd,
but move as the gentle markers from thereon
Upon the underlying lattice; it is twice named;
It derives from Darken and from Kirkendahl taken:
'Tis self-diffusion in the unmoving frame: it shall be
interdiffusivity when measured, at last, in the laboratory.

Shall I compare thee to a growing grain?
Thou art more stable and of constant rate:
Stong quench does cause the darling buds to strain,
And fluctuations have too short a date:
Sometimes too quick the heat of temper do change,
And often are the structures misaligned;
And every phase from phase will soon arrange,
By chance or gamma's constant force aligned;
But thy eternal voume shall not fade,
Nor lose posession of that phase thou ow'st,
Nor shall Ostwald brag thy mass was unmade,
When in eternal lines to time thou growest;
So long as heat can flow like entropy,
So long lives this, and this gives size to thee.

Ashley, Douglas, Ullal, Wei Sp2000

Atoms to Mecca

Traveling paths random by one and trends by groups
they are pilgrims of fundamental belief
moving through lattice by grace of vacancy or over mountains,
filling valleys in their ride towards balance
Other concerns pale before their motion.
Always seeking the sphere.

Solar Olugebefola SP2000

The Atomistic Address (Address delivered at the closing of a group homework meeting)

2 score and 3 years ago, our forebearers brought forth upon this argument a new notion,
conceived in Probability and dedicated to the proposition that all jumps are created equal.
Now we are engaged in a long homework problem, testing whether that notion,
or any theory so conceived and so enumerated, can be endured.
We've been here for a great many hours for that end.
We have come to elucidate a portion of that model,
as a fruitful passing of knowledge for those who
dedicated their lives that others could model diffusion.
It is altogether expected that we do this.
But in a larger sense, we can not illustrate--
we can not explicate-
we can not expand-
this model.
The brilliant men, PhD's and ScD's, who struggled with this, have researched it,
far above our poor power to add or detract with a few hours of thought.
The TA's will little note, nor long remember what we pass in here,
but it can affect our grade because of the weight factor.
It is for us the students, rather, to explore the unfinished work in our theses
which they who theorized have thus far so nobly advanced.
It is rather for us to learn now to better prepare for the task before us-
that from these honored men we take increased devotion
to that cause which they gave their hours of lab time-
that we here highly resolve that these men shall not have researched in vain-
that this model, under our advisor, shall be worked into new equations-
and that models of diffusion, by diffusion, for diffusion, shall not produce errnoeous data.

Abraham Lincoln

A Vacancy Named Cheshire

there once lived a vacancy in the valley of silicon
who met many an atom via self diffusion.
the vacancy was known by the name of cheshire;
throughout his entire life he had but one desire-
migration to the surface, the land just beyond.
life in outside, he was told, was really swell;
"it's much better," he thought, "than life in this hell."
not a second had he here to claim as his own,
for in silicon, he was all too well known
by anxious atoms as a new place to dwell.
as their walks were random, cheshire was quite sure,
that even the atoms' thoughts were non sequitur.
"chaos and disorder, they act without rules,
i'm thrust to and fro in this deranged ship of fools."
and he made haste at once towards the exterior.
without further adieu he charted his route,
"i really must get out. out, out out!"
cheshire stayed up late and even missed the smurf's
to study kinetics and derive the erf(z).
"yes," he realized, "i can do it-no doubt."
befriending adam, his neighbor to the west,
cheshire embarked on an unheard of quest.
"the atom is dumb; he'll do what i say,
and before i know it, i'll be on my way."
there truly was a method to his madness.
"adam, do you see the gal to my right?
she says you are the smartest guy in sight."
adam hopped over with a slight prance
to talk to the girl and display his brilliance.
one step closer, cheshire's smile was bright.
with tricks like these, cheshire sailed steadily on,
until he met conrad, the interstitial atom.
"a defect no more will i be," conrad announced,
and before cheshire could move, conrad pounced
desiring a life of less stress and greater freedom.

Ellen J. Siem, 2001

Three Limericks

A Limerick on The First Year Experience

We first learned of systems at rest
But then there was more to digest
Kinetics was taught
With flux we all fought
'til o'er quals we'll have our conquest.

Limerick at Zero K

This limerick is at zero K
If it weren't it'd diffuse away
The letters would mix
There would be no fix
For entropy would rule the day.

The Life of an Atom

If only I was an atom
I'd be so up and at 'em
Hopping around
Making small bounds
Moving from top to bottom
The lattice would be my park
I'd wander from dusk to dark
My path a dance
I'm in a trance
Following a vacancy my trademark
My path is a random walk
I'd be very easy to stalk
On average quite dull
My displacement is null
You'd know where to stop by to talk

Garry Maskaly, 2001

The Drunk
Once upon a midnight dreary, as I wandered, drunk and weary,
Over many a quaint and winding street of ole Boston town--
While I roamed, with misdirection, suddenly came an intersection,
And after some small retrospection, one path I chose to follow down,
"'Tis the right way," I muttered, "to get me back uptown--
To my bed and ev'ning gown."
Not so clearly I recall--it was back in early fall--
That night was quite the free-for-all, when I got lost in dear Beantown,
Eagerly I wished to go home; in vain I hoped to end my slow roam,
But those drinks with heads of thick foam caused my brain to just meltdown,
Clouding my head were the pints of Ale (Nut Brown)
"I must get a-homeward bound."
And thus I wandered in a daze, my mental map became a maze,
And so in such an awful haze, I meandered through the old downtown--
Each crossroads I could not discern, and so with each I chose a turn,
And with no capacity to learn, nor with a 'T' to take outbound,
Thus was a random path mine to round,
"These streets do truly confound."
Deep into that darkness peering, long I walked there wondering, fearing,
While this circle-ish path I was clearing, I reached conclusions profound:
Though on the average I moved not, I was seldom at my starting spot,
Some mean square distance was my lot, as I wandered all 'bout town.
All I wanted was to nap in my bed of eiderdown
"Have I already walked by this mound?"
And then on further close inspection, I realized a deeper connection:
If I had a preferred direction, my average path would indeed move down,
Though I jumped not always in that way, my mean position would not stay,
And if I see the light of day, I'll be far from the bar I was at downtown,
These were my thoughts hours past sundown.
"I don't recognize this part of ground."
Then came such an epiphany--I was struck as I snuck down Newbury--
I like an atom in a lattice be, while I move wild as a circus clown;
Atoms move with thermal energy, like a drunk they move randomly,
Unless a driving force they do see, for then they go like a trail-sniffing hound
These wondrous thoughts in my head did circle round
"I recognize that building!" I did resound
Then it falls quite naturally that atoms move with a velocity
(it goes as root m under root k-T)--a motion such as described by Brown
And as I moved erratically, I thought of carbon in iron-BCC,
or atoms in an interstitialcy, "A King of Science I'll be crowned."
So after many hours my thoughts became sound:
"Is that my home? Thank God I'm found!"

Joshua Hertz, 2001

Based on Green Eggs and Ham by Dr. Seuss

That kinetics man! 
That kinetics man!

I do not like that kinetics man!

Would you like to random walk?

I want to make it down the block.
So I do not want to random walk.

But you could go here and there,
You could wander everywhere.

I would not like to here and there
I would really like to get somewhere.
I do not like to random walk
I want to make it down the block.

Would you like nowhere to go?
Would you like average displacement to be zero?

I have somewhere I'd like to go
I don't want average displacement to be zero.
I would not like to here and there
I would really like to get somewhere.
I do not like to random walk
I want to make it down the block.

Would you like to on a lattice?
Your center of mass will not change in status.

I would not like to on a lattice
I want my center of mass to change in status.
I have somewhere I'd like to go
I don't want average displacement to be zero.
I would not like to here and there
I would really like to get somewhere.
I do not like to random walk
I want to make it down the block.

But your steps would be independent,
Wouldn't you prefer an uncorrelated event.

I do not want my steps independent,
I would not prefer an uncorrelated event.
I would not like to on a lattice
I want my center of mass to change in status.
I have somewhere I'd like to go
I don't want average displacement to be zero
I would not like to here and there
I would really like to get somewhere.
I do not like to random walk
I want to make it down the block.

Would you like to with Einstein? 
You won't have to walk in a straight line. 

I do not want to with Einstein.
The way I plan to go is fine.
I do not want my steps independent,
I would not prefer an uncorrelated event.
I would not like to on a lattice
I want my center of mass to change in status.
I have somewhere I'd like to go
I don't want average displacement to be zero
I would not like to here and there
I would really like to get somewhere.
I do not like to random walk
I want to make it down the block.

But don't you love Brownian Motion
Isn't it the best way to undergo diffusion

I do not like Brownian motion
It isn't the best way to undergo diffusion.
I do not want to with Einstein.
The way I plan to go is fine.
I do not want my steps independent,
I would not prefer an uncorrelated event.
I would not like to on a lattice
I want my center of mass to change in status.
I have somewhere I'd like to go
I don't want average displacement to be zero
I would not like to here and there
I would really like to get somewhere.
I do not like to random walk
I want to make it down the block.

Elsa Olivetti, 2001

Competitive Nucleation

Ahh, finally I made it, critical size
Surface energy no longer a barrier
Being outside the spinodal, Arrhenius lent me a hand
Against probability I've made it this far
Collecting from around me I will expand
Straining for that lowest fr
ee energy
As long as I stay ahead of the curve, above average
I will survive.  My brothers around me I will savage
It is a grain eat grain crystal

Jennifer McKeehan 2002

From this state of unordered and fluctuating feelings
Generated by the perturbations of the first meetings
Atoms of concideration, attraction and admiration
Aggregate in this kinetic process called nucleation ,
Spinodal decomposition between the present and the past,
Leading to small nuclei of Love; But will they last ?
Under the classical and reasonnable assumption
That after this preliminary period of incubation|
The free energy of hearts decreases as loneliness ceases
And in the case where the surface energy of bodies
Provides enough attraction against any elastic constraint,
Even coarsening won't make their growth faint
And these initial nuclei will, from their critical point,
Build a stronger and stronger new love, that's the point

Gilles Benoit 2002

And so begins a single phase, all happy and true,
But then the temperature drops, so the atoms change too!
All quiet and still, yet some finite fluctuation.
Lo and behold, the beast of nucleation!

We start with grains, sub-critical nuclei,
But then alas, they grow, oh please, don't cry.
The atoms don't leave, they simply reform.
A new matrix, it grows, all they do is conform.

What type of growth, it must be better, oh!
Of course, due to surface defects, it's commonly hetero!
And then we come to regimes, there's only but four.
Incubation and steady-state, but wait, there's more!

Equilibrium and coarsening, these are the last two,
It's easy to remember, like cows going moo.
To incubate, there's a delay and we call it tow.
moving on to steady-state, whoa the particles increase, wow!

We slow down at the top it's called saturation,
Some call it equilibrium, and some just maturation.
And as all singers tire out, and their voices hoarsen.
We see in nucleation, the particles will coarsen.

And now let's step back, well don't get too cheery,
What we'll do now is form a simple theory.
So if we take free energy per unit volume and surface tension
We're obtaining free energy, being isotropic and spherical, not to mention.

Combine them, there's a hump that it must get over,
And unless you're thermally activated, it's like the Cliffs of Dover!
So once we find the radius just above critical
It's home-free to grow, and it feels, oh so, lyrical

Now you know in all this I'm not being whimsical
You can really observe this, it's true, it's empirical
So once again, it's all about free energy
it can be so simple, like reading a liturgy

This concludes my treatise, if you want to go farther,
Just ask our professors, Sam Allen and W.C. Carter!

Kuangshin Tai 2002

A Tribute to Nucleation and Growth

Free energy barrier is a major burden.
  Transformation cannot happen without aid.
The question is what can we do to make it happen.
  The answer is something has to nucleate.

In the beginning there was incubation.
  The question is "When will we get to steady state?"
Patience will lead to further fluctuation.
  Then nuclei will show and grow at a steady rate.

Rolling down the gradient slope.
  Gathering buddies along the way.
Which one of them will have hope?
  The truth is only the big ones will stay.

Such is the story of nucleation and growth.
  Nature has chosen this approach.

Preston LI 2002

Nucleation and Growth in Ice-Cream

Undercooling below freezing
Nucleates the ice-cream mix.
Heterogenous nucleation
Occurs from all the solute bits.
To get a smoother, creamier texture,
We need to make the crystals smaller.
So freeze it faster, stir it quicker,
And don't forget the stabilizer!
For ice-cream crystals are unstable,

Growing, fusing, becoming fewer,
Till all small crystals dissappear.
Behold, the coarse and icy texture
That all the ice-cream makers fear.
If you, by chance, let ice-cream melt,
And put it back into the freezer,
It's not as good as freshly-made
As you can't get it to re-nucleate.
The larger crystals just get bigger,
And refrozen water makes it icier.
So if you make ice-cream, remember
To freeze it fast, and mix it well
And always keep it very cold.

Shona Pek 2002

Nucleation is creation

Of a new phase in the old phase.
Homogeneous, het'rogeneous,
It's one by one or all at once.

Which will it be? Well it depends,
Free energy decides who wins.
Het'rogeneous is the favorite,
But so we thought of the rabbit.

He thought for sure he would have it,
Het'rogeneous mister rabbit,
But given the right conditions,
Turtle Homogeneous won!

Growth also comes from diffusion,
But it causes me confusion
By fighting with nucleation.
Why can't they both get along?

Hong Linh 2002

A short poem on the 4 regimes of nucleation and growth:

Nucleation begins with incubation,
as small nuclei struggle to rise in concentration.
When stable nuclei form at constant rate,
nucleation is at steady state.
Alas, supersaturation will ultimately decrease,
causing the nucleation rate to almost freeze.
Now particles will grow by coarsening,
which the smaller ones find frightening.

A haiku on nucleation and growth:

Metastable phase
Critical nuclei form
Grow as G declines

Tseh-Hwan Yong 2002

"The solace of homogeneity:  The Trials and Tribulations of a lonely 'A' Phase Atom"

Loneliness is but the faintest
hint of what I suffer;

I had another atom once--
she left me for another.

I had such aspirations then--
I strived toward different state,

But now my dreams have washed away--
statistically, I wait

And wonder: "Could there possibly,
a different order be?"

(Let's call it, for the sake of it,
the phase of nomen "B").

I'd anything to leave my life--
become a part of "B,"

But 'till then I'll incubate,
in favored energy.`

Adam J. Nolte 2002

The Lament

We are the subcritical nuclei,
Oh me, oh my!
If we could only grow, 
Think of where we would go!
A particle more or two,
Think of what we could do!
I know! We should join forces!
Of courses!!!
I really am trying to diffuse,
It's not a ruse!
Oh no- some atoms are leaving!
Time to start grieving!
I wish I could have grown,
I will never be a grain of my own!!!

Lara Abbaschian 2002


Where nuclei these are I think I know
They are in the matrix though
They will not see me stopping here
To study them how to grow

My mother must think it queer
How can I know it clear
Between the incubation and steady state
The most complex theory here

The smaller is becoming smaller
The bigger is getting bigger
Want to be a stable nuclei
Passing the critical size is much better

Bo Zhou 2002

something was happening
it was cooler
or everything was imperceptibly trembling
or both
i felt different uneasy

oddly i started getting closer to my neighbors
we had never been very close
but something was happening
maybe we needed solace
maybe just the knowledge that there were others
that this wasn't our own insanity

sometimes those i had rallied around
would be pulled apart
i never knew why
no one would ever say
sometimes an individual would drift off
and although this made more sense
they never gave reasons

we didn't talk much in general
i think the fact that we were moving together
worsened the internal disquiet we all felt
our need for one another screamed
that something was different

we always stood shaking in the dark
and as more joined those huddled masses
i felt their fear
their discomfort
and my own grew stronger
perhaps that's why we so often slipped away in the night
leaving in smaller groups or alone
to ease that tension

at some point
after i don't know how long
i was introduced to another gathering
probably the largest i'd seen
it was crowded and difficult to see everyone
they seemed to have drive
to know where they were going
my unease
my disquiet
and my fear were lifted
i don't know what happened

since then i have moved around a lot
i always ask but no one ever knows
we all seem closer now
not just the initial group i ran into
everyone i meet
and our relationships have changed
but i don't know what happened
no one does

Robert Damien Boyer 2002

Heterogeneous Haika (Heterogeneous Haiku Series)

Poor material!
You're full of imperfections -
Some dislocations.

Like leaves fall to earth
with energetic ease, you
Nucleate quickly.

Often beginning
Where the grain edges do meet;
Look! Lenticular!

At dislocations:
Incoherent to the core
Cahn is your daddy

As years have seasons
You pass through the four regions;
Now fewer, larger.

Mike Tarkanian 2002