1 Introduction
In this reference we outline many of the behavioral properties of twisting fibers. We survey
many of the general experimental techniques that have been either completed or commenced
in the course of these inveterate studies. In particular, we focus our efforts to explain the novel
measurements made in our experiments. In some cases, we compare these results to
theoretically predicted values. We note, however, that most of our work has not yet been
discussed in the literature. These studies, we claim, have ultimately helped us develop a
comprehensive picture of how fibers of various lengths, diamters, and mass densities, behave
when twisted. Finally, it is outlined below how we have attempted, in each of our studies, to
search for a respective 'master equation' that can be used to help physicists isolate and properly
interpret useful information about how fibers behave when twisted.
We begin with an overview of the central motivations that have guided our basic research.
Then we outline: how calibration techniques have been used to promote the veracity of our
work, how non-Hookian behavior has been measured in our research, and how relaxation times
have been measured and compared to known structural constants. Finally, a blue print for
future research is given so that we can precisely identify geometric features of fiber twisting
with the minimization of the free energy of the fiber.
OVERVIEW
2 Motivations
2.1
Natural Phenomena
2.2
Modeling And Interpolation
In our present studies we have two aspirations for modeling and interpolating the results of
our experiments which we address here. Unfortunately, we have come short in accomplishing
either of our primary goals. By no means, however, are we precluded from obtaining either of
these goals. Our main obstacles, so far, have included reconciling unmatched calibration data
with theory and following up 'loose-end' research which we generally consider,
comprehensively speaking, important though not central to our basic research programs.
Our first aspiration has been to identify micrscopic effects by measuring macroscopic
quantities. This is a simple research program, but is not without its problems. First, we have
been unsuccessful at finding direct quantitative links to theoretical models which would
corroborate our data with microscopic effects. In truth, very few theoretical calucaltions have
been done for elastic fibers, making our task all the more difficult to accomplish. Nevertheless,
we believe, in the long term, our research programs will succeed at finding a correlation
between the miscroscopic internal forces with with the macroscopic forces measured using our
apparatuses.
Our second aspriation has been to develop a set of 'master equations' that help us isolate and
properly interpret crucial information about the internal forces within the fiber. These 'master
equations' provide us the the ability to predict how a fiber will react in a specific manner when
twisted or relaxed or allowed to contract. In each case, we have attempted to isolate a number
of measurable parameters with one or two unknowns to be solved. The genesis for many of
these equations comes to us, primarily, by way of two methods. Directly, we can work out
basic mechanical equations, as is the case of torsion due to twisting fibers, and less directly
with dimensional analysis. In some cases we simply cannot work out, from first principles, the
dynamics of the fibers. In these cases, we look for relationships, like exponential decay, whose
familiar solutions help us identify fiber constants, like decay constants, with experimental
unknowns, like Young's modulus. Looking ahead, we will plan to build several other types of
experiments that exploit the construction of formulas whose constants can reveal important
experimental unknowns from measurable parameters. In this way, the techniques we use and
develop are equally as important as the data we retrieve in each experiment.
Together these aspirations form the basic connection with theory. In the long run, we hope
to establish direct connections with statistical and thermal physics. In particular, we hope to
develop a model or several models that, in the style of condensed matter research, would
relate the geometry of distortions in the fibers due to tristing, with the free energy of the fiber.
These are long range and amibitious goals, but we believe they are realizable. More
importantly, we believe that these sorts of theoretetical steps must necessarily be taken if we
are to accomplish our first aspiration.
There are two caveats, however, that we must state concerning our idealized aspirations.
First, the specific behavioral features we measure in many of our experiments are inherently
non-linear. That is to say, some of the simplest experiments we run cannot be expressed in
terms of first principles, but must be viewed as a perturbative expansion with appropriately
non-trivial solutions. Historically speaking, physicists have been prejudiced against nonlinear
systems, in that they have no simple solition nor can they be identified with any sort of
universal geometry. Recently, triumphant efforts have been made, certainly in the field of
chaos and non-linear dynamics, which have only modestly succeeded in demonstating
predictions for behavior of previously thought insoluble problems. Nevertheless, many of our
experiments run themselves very close to the point where these so-called non-linear effects are
simply too calculationally burdensome to overcome. On the other hand, we have been quite
successful at realizing when our simple experiments break down, and it is just as important, in
all cases, to know what the limits of our knowledge are. To quote, Mitchell J. Feigenbaum, the
modern father of chaos theory, "no real spring is linear". Touché.
Second, in generalizing microscopic details in the bulk of our macroscopic measurments
we are in danger of forbidding ourselves the pertinent information about the micrscopic
details we set out to recover. But this is partly a conundrum. The essential problem of
statistical themodyamics, to quote Erwin Schrödinger, is to discover the given amount of
energy E over N identical particles. By this we not only assume that interactions are weakly
coupled and can be disregarded, but that we cannot explicitely know anything about those
interactions. In other words, the private energies between microscopic particles is necessarily
lost. Does this remove the credulity in our first aspiration? Are we fixed only to describe
macroscopic features whose connections to microscopic details are necessarily cut? The
simple (and short) answer is: it might not be important. Our research programs may never
evolve to the stage where it can consider handling those types of questions anyhow.
Whatever the outcome of our attempts to manufacture, compare, or embellish theoretical
models, the process of experimentation will not go without its rewards. The primarily
motivation for this research is summarized here: we are measuring things that cannot be
calculated. This is not the first time, i.e., atomic spectra were known a whole generation before
their explanation was given, and this shall not be the last time experiment precedes theory.
We are simply exploring what we know exists.
RESEARCH DIRECTIONS
3 Experiments and their Ends
The completion of an experiment is a frightful thing. In many cases the experimenter does not
know a lump of gold from a lump of pyrite. It is the duty then of the experimentalist to search
his or her toolbox of intuition, savvy, and pragmatic thought in order to know imlications of
his or her knowledge of the experiment.
Below, we list the three primary directions that have helped guide our basic research from
falling into traps of ostentation and improbity. First, we describe callibration techniques used.
Calibration is the tool which physicsicts use to defend all of their observations. It is the
instrument by which all comparisons can be justifiably made. Second, we list a (incomplete)
selection of novel experiments performed. And lastly, we point out where our future research
is headed.
3.1
Calibration
There are many levels of corroboration between theory and exerperiment. In some cases, we
learn 'late-in-the-game' that the theory that was once thought applicable is no longer valid (or
even worse—never valid!). Or we learn, as proud humans, that our experimental set-up is
inadequately or unskillfully set-up . By calibrating our equipment we hope to remove the
possibility of either of these scenarios from frustrating our work's intention.
Calibration occurs during many points in the research program. First, we calibrate every
pertinent measuring device, tracable to the NIST (National Institute for Stantards and
Technology; formerly NBS, National Bureau of Standards). Second, if possible, we calibrate
(corroborate) a single measurement with a known or predicted theoretical value. We do both,
but have only been successful in limited cases.
Since only one strain guage is primarily used, it's calibration is paramount to our findings.
Indeed, it is well calibrated for the weight of objects given our experimental set-ups, i.e., it can
measure force to an accuracy of at least two decimal places. On the other hand, other
exerperiments have been run to determine the efficacy of our experimental set-ups or rather
how well torsional rod theory purports to predict the behavior of thin elastic fibers. These
results have proven less auspiscious. These experiments are run specifically to compare our
data directly with the results of calculations done for torsional rods. Here, we list two
pertinent results for both tension, T, and torsion, , measured from a fiber that is being twisted:
TENSION:
T = (1/2)·C·(Omega)^2
TORSION:
tau = C·Omega
where  = (2π·turns/unit length) and C is sometimes called the torsional rigidity. Assuming
these relations hold for our elastic thin fibers and given our experimental findings we should
be able to identify C. Unfortunately our findings are only sporadically succesful, with no clear
evidence that we have either negated the validity of this particular theoretical comparison or
that our set-up requires extensive renovations.
There are many other examples of calibration experiments that have been run, most of
them have to do with scaling or allometry. In general, we find that fibers whose lengths are
increased by integral amounts must also be 'twisted' by those same integral amounts in order
to obtain the same result. We call this the twist-per-unit-length-assumption or hypothesis: it
simply (and physically) means that fibers are homogenous and that each twist affects each unit
length the same. Though these sorts of studies may seem superfluous, they cannot go
undiscovered or left as assumptions without testing. If we are to make these assumptions they
must be justified.
3.2
Novel Experiments
In the course of experimentation we have run into many interesting artifacts that belittle our
attempts to make an experiment simple (remember the pyrite and the gold!). As
experimentalists, we discriminate between the voices in our heads that lead us to the fool's
gold or the argosy of apprehensible evidence. But the difference between an unexplained
artifact and one that is expained, is, in many cases, a superflusous distinction—or a semantic
distinction only.
Our novel experiments have followed these so-called gray areas of research—to what end?
Here there is danger (of losing a month's worth of experimental time!), but also the possibility
of immense reward. We choose to investigate these avenues largely because we have nothing
to lose (except time!) but also, as stated above, in many cases, only experiment can reveal any
quantitative features to what we see.
Present novel experiments include: hysteresis, relaxation, and observation of non-Hookian
behavior among others. Each of these studies is yet to be concluded and is reported below,
only in the form of a plot that depicts the general behavior of each experiment resptively.
3.3
Comparison's to Theory
Probably the most difficult task that lies ahead is the search for fundamental connections
between our experimental data and thermodynamic quantities, like the free energy of a fiber.
The discovery of a connection, if any, between free energy and the geometrical features
produced by the twisting process will be truly profound. It is fundamentally astonishing to us
that Nature has chosen to minimize the energy of a twisted fiber by means of transforming the
fiber from a line to a helical structure. (Even more astonishing is that Life has exploited this
helical structure to contain vast amounts of information in seemingly impossible small spaces!)
GENERAL BEHAVIOR
4 Gallery of Experiments
Below we have briefly described some the specific details of some of the major recent
experiments. Below these descriptions are plots of these experiments. In some cases, the plots
are taken from raw data, in other cases, the plots are either real data that has been analyzed or
an amalgamation of data which accurately depicts the specific features of a paricular
experiment. Choices for each plot are based on the need to illustrate overall experimental
findings, they are not meant to obfuscate or conceal experimental errors.
4.1
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4.2
Calibration
Turns/ Unit Length
In this experiment, we measured the number of turns just before a the fiber
would snap. In general, we performed the experiment several times with
different weights, only to find the outcome was unchanged. Here we plot turns
per length. The graph shows that this relationship is nearly perfectly linear—
hence our wist-per-unit-length-hypothesis is affirmed.
Torque/ (2π·Turns)
In this experiment, we measured the torque on a fiber due to twisting. From this
measurement we could determine a value for the torsional rigidity, C. Below, we
have plotted a typical value for a fiber of a given length.
Tension/(2π·Turns)^2
In this experiment, we measured the tension on a fiber due to twisting. From
this measurement we could determine yet another a value for the torsional
rigidity, C. For comparison, we have plotted below, a typical value for a fiber of
a given length (same as above).
Turns per Weight
4.3
Beads per Length
4.4
Relaxation
4.4
Non-Hookian Behavior
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