Abstract
In this reference we outline many of the behavioral properties of twisting fibers. We
survey 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 known values found in the literature. 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, diameters, 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 our experimental setup and the central motivations
that have guided our basic research. Then we outline the following: how calibration
techniques have been used to promote the veracity of our work, how non-Hookean
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.
1
OVERVIEW
1 Introduction
Anyone who has ever twisted a fiber will notice many of the properties that we
investigate. Moreover, these properties, i.e., measurable quantities like torque, tension
and geometrical deformations, scale appropriately for nearly all fibers of any length,
diameter, or mass density. Electrical cables, elastic cords, musical instrument strings,
telephone wires, plant fibers, plastic fibers, and various hoses and tubes, all exhibit
similar properties; they differ only by specific scaling factors, like length, diameter, and
Young's modulus. The internal stresses and strains ultimately manifest themselves in
the geometrical deformations that one can actually see and feel when twisting a fiber.
For instance, if a fiber of length L is taken between the thumb and index fingers of
both hands as shown in [Fig. 1] and then twisted one will immediately notice a torque
and tension acting on the fingers. The torque will act in such a way to restore the fiber
into its natural, untwisted state. The tension will act in such a way to pull the ends of
the fiber together. If one shortens the fiber, i.e., relaxing the tension now felt by the
hands, then something remarkable happens. The twisting has caused internal strains in
the fiber, which report back to the twister's fingers a complex amalgamation of external
stresses and strains. Relaxing the tension causes the fiber to deform into, what we call,
a simple helix, as shown in [Fig. 2]. If the ends are brought closer together, the helix
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tightens and "snaps" into a twist or snap helix, as shown in [Fig. 3]. This snap helix
occurs rapidly and has a strong tendency to form perpendicularly to the initial fiber
axis.
Our experimental setup, as shown below [Fig. 4], is based on measuring what can be
felt and seen in the observation described above. Primarily, we measure the torque
(twisting induced strain perpendicular to fiber axis) and tension (twisting induced endto-end force along fiber axis) resulting from the twisting of a fiber through a certain
angle, , where  = 2·turns/unit length.
We also measure several dynamical
characteristics as a result of the twisting, including the tension and torque as a function
of change in length, L.
In quantifying what we know others see when reproducing our experiments, it is
necessary, as was done above, to define many of the crucial characteristics that we
notice when twisting fibers.
Other crucial terms, not defined above, but used
throughout this paper (and our research) include:
beads: these are the constituent twists that can be discretely counted in a snap
helix (see Fig. 3);
relaxation: this is a temporal measurement of the tendency of a fiber to return to
its original untwisted state; the relaxation of a fiber is usually measured
between or after twisting has occurred (see Plot 7 & Plot 8), but can also be
measured during twisting (see Plot 4).
Non-Hookean behavior: this behavior is measured when a pre-twisted, straight
fiber has its ends brought slowly together, so that a simple helix forms and
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then a snap helix. In the course of this evolution, it is noted that tension in
the fiber increases when it is expected to decrease--opposite of Hooke's
Law (see Plot 8).
With these definitions, we proceed to quantify what anyone can observe by twisting
a fiber. Each of our experiments is performed so that we can identify how fibers behave
and maybe learn a little bit about why they behave the way they do. Below we list the
motivations of why we think this problem is physically and mathematically important
to study.
2 Motivations
The motivation to study the behavior of twisting fibers is threefold. First, both natural
and artificial fibers are ubiquitous in our everyday observations. In particular, we are
forced to recognize that fibers, their interactions, and behavioral dynamics play a
crucial, if not fundamental, role in many of the physical mechanisms that govern the
natural selection processes exploited by biological organisms. The vast majority of
cellular material fall under the classification of fibers. That is, in principle, many of the
species of cellular and sub-cellular structures can be modeled as a single fiber or a set of
fibers. Dynamical models, of course, generally simplify the action of these fibers as
simple springs or harmonic oscillators or torsional rods. We use these same models to
manifestly simply our work so that reliable predictions can be obtained from a
minimum number of measurements. One of our ultimate goals is to find connections
between the macroscopic dynamics of fibers with those of microscopic strains and
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stresses now being measured with optical tweezers in materials like DNA and other
chromatin material.
Similarly, we also recognize that there are numerous basic research applications that
utilize artificially produced fibers. By learning some of the universal characteristics
about how these types of fibers behave when twisted, we can learn how to manufacture
fibers whose properties can be selected more precisely and more craftily to meet the
needs of the fibers’ uses.
Secondly, we recognize that in the course of our studies that many unexpected
results have impressed us enough to force us to think that more work needs to be done
to uncover how various parameters, like torsion, stress, or strain, should evolve in the
course of twisting fibers. These outstanding observations are presently the impetus for
several research programs which we are concurrently running experiments in order to
find different techniques that measure the same physical parameters, like the modulus
of elasticity or Young’s modulus.
And lastly, in performing our experiments, we recognize that the dynamics of a
twisting fiber resemble the behavior of living organisms. In particular, when elastic
fibers are twisted they display an uncanny resemblance to the reactions observed in
many biological organisms adapting to their environment. For comparison, elastic
fibers can react quite violently when heated—in essence they generally shorten, rather
than lengthen, and consequently, reduce their entropy. Observations like these have
provided abundant incentives to quantify the dynamics of these fibers so that we may,
in principle, understand some of the fundamental connections between physical and
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biological phenomena.
Using this sort of reductionist picture, there is a realistic
potential for substantial gains in our understanding of how certain microscopic
biological mechanisms can manifest themselves in the macroscopically observable
behavior that we measure.
Below, we outline a number of central tenets that have re-enforced our motivations
and directed our research program in beneficial ways.
2.1
Natural Phenomena
In life, as in nature, nothing is rigid. This universal result is not only demanded by
special relatively, but consequently, realized as a fundamental result of the atomic
hypothesis, which states that the stuff we are made of is (for our purposes) atoms and
atoms are, at short distances, essentially attractive but at even shorter distances they
become repulsive. Nature (and life) must exploit this crucial property of matter in order
to build macroscopic objects whose dynamical behavior simulate the same mathematics
of simple harmonic oscillators or sometimes called Hooke’s law. Natural selection,
then, must also perform its duty according to these physical laws. In this way, all
biological organisms should be studied, at least from this distinct point of view, as
though they are fundamentally composed of tiny harmonic oscillators. Our present
research is, in principle, not far removed from this discriminating picture life. In the lab
we observe elastic fibers under stress and strain and we model and interpret their
behavior as a complex collection of springs with modifiable parameters that can be used
to predict the behavior of all fibers, natural or unnatural.
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Since we do not deal directly with living tissues, we are at risk of losing this
ostensibly tenuous connection between our work and the dynamics of living organisms.
However, we are inspired by the results we have obtained and consider that much of
our work can and should be compared to behavior observed in biological systems.
From a pragmatic point of view, we have not experimented with natural or
biological fibers. Instead, we use artificial fibers for several reasons. First, artificial
fibers are easy to obtain from the standard industrial production of fish line and
trimmer line.
These types of fibers have fairly reliable properties.
Namely, their
diameters are usually consistent to within three thousands of an inch (or eight
thousands of a centimeter) and their mass density (per unit length) is consistent for a
given fiber.
What is not yet known is whether the fiber’s used have internally
consistent properties, i.e., it is not known if there are particular manufacturing
processes that would drastically alter the result of our measurements if the production
of the fibers were slightly or dramatically modified. It is our assumption, however, that
such an inconsistency would and should not be present in these fibers, since it is not a
beneficial property of the fibers to be made inhomogeneously. In fact, we argue that it
is difficult to construct fibers with inhomogeneous properties. Second, we use artificial
fibers precisely because of their assumed homogenous properties. Because we are
searching for relationships, that we believe, have not yet been quantified, we are using
very simple fibers whose properties are very well defined. Lastly, we do not use
natural fibers because they are generally more expensive and not readily available.
Natural fibers tend not to be homogenous and are also, in many cases, ‘predeformed’ in
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such ways that studying them becomes impossible. Similarly, we do not yet possess the
machinery to study these fibers: our set-up has been established to study the behavior
of specific types of fibers, whose diameters’ and lengths’ are macroscopic (5 –100 cm
and 150 microns – 2.5 mm, respectively), and whose elasticitys’ are comparatively
equal.
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 microscopic effects by measuring
macroscopic quantities. This is a simple research program but it 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 calculations 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 microscopic
internal forces with the macroscopic forces measured using our apparatuses.
8
Our second aspiration 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 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 twisting,
with the free energy of the fiber. These are long range goals are ambitious, but we believe
9
they are realizable. More importantly, we believe that these sorts of theoretical 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
solution 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 demonstrating 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 calculationaly 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
measurements we are in danger of forbidding ourselves the pertinent information
about the microscopic details we set out to recover. But this is partly a conundrum.
The essential problem of statistical thermodynamics, to quote Erwin Schrödinger, is to
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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 explicitly 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
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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 implications 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 calibration
techniques used. Calibration is the tool which physicists 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 experiment. 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, traceable to the NIST (National Institute for Standards
and Technology; formerly NBS, National Bureau of Standards). Second, if possible, we
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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 gauge is primarily used, its 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 experiments 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 auspicious.
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:
T
1
C 2
2
  C
where  = (2·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 successful, 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
13
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-lengthassumption 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 explained, is, in many cases, a superfluous
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 nonHookean behavior among others. Each of these studies is yet to be concluded and is
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reported below, only in the form of a plot that depicts the general behavior of each
experiment respectively.
3.3
Comparisons 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 and Results
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
15
of a particular 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
Calibration
•
Weight Calibration (Plot 1)
Here, we calibrate our strain gauge with a known weight (paper clips) to
show that there is no drift in our measuring apparatus. In this calibration
experiment, we ran two overlapping measurements, one that increased
weight and another that decreased the weight. We found that there was
no drift, as shown below, and that strain increased monotonically.
•
Turns/ Unit Length (Plot 2)
In this experiment, we measure the number of turns just before a fiber
snaps. We perform the experiment several times with different weights,
only to find the outcome is unchanged. Here we plot turns per length.
The graph below shows that this relationship is nearly exactly linear—
hence our twist-per-unit-length-hypothesis is affirmed.
•
Torque/ (2·Turns per Unit Length) (Plot 3)
In this experiment, we measure the torque on a fiber due to twisting.
From this measurement we determine a value for the torsional rigidity, C.
Below, we have plotted a typical value for a fiber of a given length. In
16
each experiment, we note that the torque depends linearly on turns. This
is exactly what theory predicts.
•
Tension/(2·Turns per Unit Length)^2 (Plot 4)
In this experiment, we measure the tension on a fiber due to twisting.
From this measurement we 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). In each experiment, we note that
the tension depends on the square of the turns. This is exactly what
theory predicts. (Note the relaxation in the fiber due to internal elastic
strain in the fiber. This relaxation is not detectable in small diameter
fibers, i.e., fibers with 0.03-inch diameters or less.)
4.2
Turns per Weight (Plot 5)
In this experiment, we measure the number of turns it takes before a fiber
snaps into a helical structure for a given weight. The length of the fibers
used is fixed, since we assume the twist-per-unit-length-hypothesis. Below
we have reported a typical plot of this fundamental relationship.
4.3
Beads per Length (Plot 6)
In this experiment, we measure the number of helical beads that appear
due to twisting fibers of various lengths with various weights. These socalled beads are the countable twists in the helical structure after
snapping.
It should be noted that this particular behavior is sharply
dependent on where a fiber decides to snap; i.e., the snap can be collet
17
constrained. In general, though, the short fibers tend to "snap" in the
center of the fiber, whereas, longer fibers tend to be effected by gravity
and are consequently forced to snap closer to the bottom collet.
4.4
Relaxation (Plot 7)
In this experiment, we measure the relaxation of a fiber after it has been
twisted as a function of time. In this particular experiment, only tension
has
been
measured.
Future
experiments
will
include
torque
measurements. Below is a typical plot of this behavior.
4.5
Non-Hookean Behavior (Plot 8)
In this experiment, we measured the tension on a fiber due to twisting as a
function of bringing the fiber ends closer together.
In this particular
experiment we were surprised to find that the tension did not obey
Hooke’s law. Below is a typical plot of this behavior. The history of the
dynamics are given below: (see Fig. 1-3 for references to L, L', and L*)
A
As twisting occurs, L = 0, i.e., the length is kept fixed, but
the tension increases quadratically (see Plot 4),
B
Twisting is stopped by the experimenter and a slight
relaxation in tension occurs (see Plot 7),
C
The distance, now L', between the ends of the fiber decreases
and the tension decreases and a simple helix develops
(see Fig. 2),
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D
At some length, L' < L, but greater than L*, the tension
increases again; a result which is opposite of Hooke's law,
E
As the length approaches L*, the tension increases and the
snap helix develops until the twist forms (first spike),
F
A second spike occurs due to the slipping of the twist;
consequently several other spikes occur for the creation of
each bead in the snap helix.
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