### Abstract

- Top of page
- Abstract
- INTRODUCTION
- POLYMERIC SYSTEMS AT LOW DENSITIES
- NONLINEAR DYNAMICS FOR LOW DIMENSIONAL SYSTEMS
- DAMPING FOR VACUUM POLYMERS
- INTERNAL FRICTION
- FUTURE DIRECTIONS
- CONCLUSION
- REFERENCES AND NOTES
- Biographical Information

Some important recent advances in polymer science involve situations where molecular inertia plays an important role, for example, in the mass spectroscopy of large macromolecules. Only recently have we begun to understand how to analyze dynamics in such situations and what effects inertia has. In this review, we describe what is known about such effects and how nonlinearity plays a crucial role in generating damping, which is of a very different nature than is normally assumed. Not only is such analysis important in understanding polymers in low density environments but also in understanding the internal friction of a polymer chain in solution. The understanding of such situations is still in its infancy and we close by propounding a set of diverse questions for further study. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012

### INTRODUCTION

- Top of page
- Abstract
- INTRODUCTION
- POLYMERIC SYSTEMS AT LOW DENSITIES
- NONLINEAR DYNAMICS FOR LOW DIMENSIONAL SYSTEMS
- DAMPING FOR VACUUM POLYMERS
- INTERNAL FRICTION
- FUTURE DIRECTIONS
- CONCLUSION
- REFERENCES AND NOTES
- Biographical Information

The experimental and theoretical understanding of polymers is vast but confined almost exclusively to situations where they are in liquid or solid environments. However the advent of some recent technological advances has expanded their domain to the other environments as well, such as in a vacuum or a gas. Here, we shall review recent studies of this unusual situation.1–8 We shall see how polymers in a vacuum are expected to be fundamentally different in their characteristics than in the more common liquid environment. This will also have the benefit of elucidating the role inertia plays even in a liquid phase. In the following, I will review some systems in the low density state and why we expect them to contain a wealth of new physics.

Theoretical treatments of polymer dynamics generally start with the same basic assumption, that motion at the smallest scale is heavily overdamped so that inertia can be neglected.9, 10 For analyzing many experiments this is an excellent approximation. However there are some experimental systems where this approximation does not work and these situations are becoming increasingly important.

Theoretical treatments of polymer dynamics in solution usually start with a Langevin equation or something more approximate,9 such as the Metropolis algorithm, often used in Monte Carlo simulations.11 The Langevin equation can include hydrodynamic interactions and forces between atoms, and random thermal forces. However inertial effects are almost always neglected. Such massless models lead to an enormous wealth of phenomena and the highly successful prediction of important physical quantities such as viscoelastic behavior for a wide variety systems. By utilizing many theoretical techniques devised to understand the basic equations, such as scaling and the renormalization group, one can often see universal behavior spanning decades in concentration and chain length. This has made polymer dynamics one of the best understood areas in condensed matter physics.

First, let us briefly review the justification for neglecting inertia in the description of polymeric systems. In order for this to be justified, we have to be considering dynamics on large enough timescales. Most polymeric systems are studied in solutions or melts, and properties of experimental interest, such as visco-elastic properties are normally probed at timescales greater than a picosecond and length scales greater than a nanometer. When considering the dynamics of a macromolecule of mass *m*, there are inertial effects giving a force *m* where **r** is the center of mass position, and the dots represent time derivatives. The local drag force can be written *b***ṙ**, with the Stokes drag coefficient *b* = 6πη*R*, with η denoting the viscosity and *R* an average linear dimension. Comparing these terms dimensionally, gives a cross over timescale τ_{c} = *m*/*b*. Let us first estimate this value for something the size of a typical monomer, using the drag on a sphere of radius 10 Å in water, and *m* = 100 Da. This gives τ_{1} of about 10^{−14} s. For timescales much greater than this, inertia can confidently neglected.

Now let us look at what happens on larger length scales, say for a linear polymer of *N* units. The mass *m* ∝ *N* and the size of the molecule *R* ∝ *N*^{ν}. Then the crossover time will be τ_{N} ∼ *N*^{1 − ν}. ν for a compact chain is 1/3 and ∼ 3/5 for a polymer in good solvent.10 Therefore the cross over time increases as a power of chain length. However the overall relaxation time including only damping increases with a larger exponent, τ_{rel} ∝ *N*^{3ν}, or greater if the system is dense. Therefore inertial effects, in this case are not important on the relaxation time scale for a chain. However internal modes for high enough wave number will overlap with this inertial timescale. This is a crude argument, but it suggests that inertia will contribute to motion on intermediate timescales and it has indeed been possible to measure inertial effects in polymer dynamics, as we will see later, by a more detailed analysis of this problem.

The above analysis shows that although inertial effects do not dominate polymer dynamics in solutions, it suggests that effects on internal modes should be measurable. To understand this better, a useful question to ask is if there are real systems where the mass term is dominant. Therefore, we will first turn our attention to situations where damping is small and inertial effects determine the systems behavior. We will see that this case is highly unusual and exhibits very interesting behavior. Then we explain the crucial role the nonlinearity plays in low dimensional systems in the absence of an interacting bath. With this understanding, we will explain how this manifests itself in the case of polymer chains and can shed light on the internal friction in situations such as mentioned earlier. The consideration of polymeric systems with inertia leads to a large number of interesting question, most of which are still quite unexplored. There are also unusual features of the equilibrium properties of isolated polymers which are brought about by additional conservation laws in this situation but will not be described here in detail. The interested reader can consult the relevant literature.2, 3, 6, 7

### POLYMERIC SYSTEMS AT LOW DENSITIES

- Top of page
- Abstract
- INTRODUCTION
- POLYMERIC SYSTEMS AT LOW DENSITIES
- NONLINEAR DYNAMICS FOR LOW DIMENSIONAL SYSTEMS
- DAMPING FOR VACUUM POLYMERS
- INTERNAL FRICTION
- FUTURE DIRECTIONS
- CONCLUSION
- REFERENCES AND NOTES
- Biographical Information

Mass spectroscopy is a technique that is normally used on small molecules or atoms to determine their chemical composition. A surprising discovery leading to the 2003 Nobel prize in chemistry, was that mass spectroscopy can be performed on large macromolecules consisting of thousands of atoms. Techniques were developed to do this such as matrix-assisted laser desorption/ionization (MALDI) or electrospray ionization (ESI).12 Using these techniques produces ionized single polymer molecules in a vacuum where they are accelerated by applying a high voltage. The time of flight can then be used to estimate the mass of a molecule to very high accuracy. These have had an enormous number of applications and is now used frequently to characterize the composition of polypeptide chains with applications to many fields.12

The use of a laser to vaporize the substrate produces a hot plume of material which would ordinarily heat the molecules to such a high temperature as to completely degrade their chemical composition. However special matrix material is incorporated into a surface layer of the specimen. The ablated material contains matrix molecules which assist in vaporization and in keeping the temperature of the polymer low enough to maintain its structural integrity. The physics of this desorption process is still not completely understood. However the end result are single charged polymers at high temperatures in a gaseous phase where inertia effects will dominate their dynamics. So far, only the mass of the polymers is being measured but to great accuracy using time of flight measurements.

Although MALDI is an extremely useful technique, knowing the total mass of a macromolecule does not completely determine its composition as many sequences can have the same mass. Therefore extra information could be used to further identify the molecule. At the moment, the internal dynamical properties are not being probed. These should be very sensitive to the local mass inside the molecule. This should provide extra information to help in determining chemical composition.

One way, in principle, to probe the internal modes in this situation would be by measuring the absorption of radio frequency electromagnetic fields. Because these molecules are charged, they should couple strongly to external fields in the right frequency range. The absorption characteristics as a function of frequency should depend on the detailed structure of the chain. This signature, combined with the time of flight mass measurement has the potential for identifying sequences with much greater specificity. This would also require a better theoretical understanding of the chain's dynamics and what absorption characteristics are independent of the position of the charges.

The rapid recent growth in the ability to fabricate devices at the nanoscale has led to many potential situation where inertial effects are important, in particular the situation where macromolecules are suspended over surfaces in a vacuum. As an example, recent work has fabricated a bridge of a single DNA molecule between two contacts13 in solution. By evaporating the solvent molecules they are left with a DNA molecule in a vacuum. As far as the author is aware, only electronic properties of such systems have been examined but in principle mechanical properties could be as well. As an example where mechanical properties have been probed, recent work has fabricated Graphene sheets suspended over trenches of silicon dioxide.14 The mechanical response was probed at different frequencies to characterize the eigenmodes of oscillation. The ability to fabricate complicated nano-structures is improving rapidly and therefore it is expected that probing the properties of single polymers will correspondingly, become increasingly relevant. Because mass is important to the dynamics of such situations, this has the potential to allow for probing properties based on inertia. For example, a DNA molecule may have proteins bound to it, and the mechanical response will be greatly influenced by their presence, in contrast to solution, where their effects on dynamics will be much weaker.

### NONLINEAR DYNAMICS FOR LOW DIMENSIONAL SYSTEMS

- Top of page
- Abstract
- INTRODUCTION
- POLYMERIC SYSTEMS AT LOW DENSITIES
- NONLINEAR DYNAMICS FOR LOW DIMENSIONAL SYSTEMS
- DAMPING FOR VACUUM POLYMERS
- INTERNAL FRICTION
- FUTURE DIRECTIONS
- CONCLUSION
- REFERENCES AND NOTES
- Biographical Information

In the above situations, where there is no solvent, the effects of inertia dominate the dynamics and the massless Langevin equation formulation of polymer dynamics has to be abandoned. The physics of these situation are quite subtle for a number of reasons that will be discussed later.

We first consider a greatly simplified system: a one-dimensional (1D) chain of atoms bound together by nearest neighbor forces that act as springs. We assume that it is completely isolated, that is, has no contact to a dissipative medium This is certainly an inadequate description of a polymer in a vacuum as the atoms are confined to a single dimension. However this is a well studied problem and many of the same issues arise as in the case of a polymer. For this 1D chain, energy is exchanged between neighboring atoms but globally remains conserved. The dynamics depend crucially on if the system is integrable. For integrable systems, there are an infinite number of additional invariants aside from momentum and energy conservation. The simplest example of this kind of integrable system is where the forces between atoms are linear. In this case the system decouples into separate modes and energy between these cannot be exchanged. A system starting in a single mode will remain in it and the system will never reach thermal equilibrium.

If we now introduce nonlinearity into the springs, this allows for the exchange of energy between modes. The important physical question is whether this leads to thermal equilibrium for sufficiently long times. It is known by the theorem of Kolmogorov, Arnold, and Moser (KAM) that the addition of a sufficiently small nonlinearity does not lead to a complete exploration of phase space, and that the motion remains confined to a much smaller region, for most initial conditions of the system. It is not easy to see in general how large the nonlinearity must be in order to give rise to thermal relaxation. In particular does the nonlinearity needed for thermalization become arbitrarily small, as the number of degrees of freedom goes to infinity? This theorem does not answer this question and further analysis is needed to understanding the dynamics of long chains.

Because of these unanswered theoretical questions, the problem of nonlinear chains was one of the first scientific uses of a digital computer, studied numerically more than 50 years ago by Fermi et al.15 (FPU). The work used an additional nonlinear term in the forces between the springs. This model is to this day, still not well understood.16 Although there is exchange of energy between different modes, it does not relax monotonically to a thermal distribution of energies. Instead energy between different modes oscillates, so that energy originally put in a single mode will first leave it and subsequently return, a large number of times. Such recurrences depend on the whole spectrum of different modes present.

Note that in the usual Langevin formulation of damped systems, none of these issues arise. Energy will leave a mode without any recurrences and integrability does not prevent thermal relaxation. It is the lack of a bath that leads to these new issues, where relaxation of the energy depends on the exact form of the spring force.

In reality, physical systems are expected to be sufficiently nonlinear that the system will eventually thermalize. However the dynamics are still not expected to be the same as that of a simple Langevin model where damping and noise are explicitly included. There have been a number of recent studies characterizing the way that energy is transfered down a 1D chain of this kind, where energy and momentum are both conserved.17, 18 By analytic arguments and by simulations, the relaxation of many quantities, such as density, energy, and momentum follow power-law scaling between space and time.

To understand this situation, where local thermal equilibrium can be attained but there is still momentum and energy conservation, consider the propagation of a sound wave in such a system. It will travel in both directions at constant velocity. If we consider the propagation of a localized pulse, there is also dispersion, which causes the width of the pulse to increase with time. The width δ*x* scales with time *t* as *t* ∝ δ*x*^{3/2} for large *t*. Note that this is quite different than diffusional behavior which appears in a damped system. In that case the relaxation is diffusional, *t* ∝ δ*x*^{2}. For a system of finite length with reflecting boundaries, sound waves will bounce back and forth slowly dispersing. Without dispersion, the system will never equilibrate. It is therefore the dispersive timescale that determines the relaxation time *t*_{rel} implying that *t*_{rel} ∝ *L*^{3/2}. In a system that is diffusional, say based on Langevin equation dynamics, *t*_{rel} ∝ *L*^{2}. Therefore in the isolated system, the relaxation is asymptotically much faster than what is usual observed, and this difference can be understood as a result of 1D momentum conservation. This leads to interesting physical effects, such as thermal conductivity coefficient that diverges as a power law with system size. This anomalous power law behavior is a result of momentum conservation and has a similar explanation to long time tails found for liquids in higher dimensions19 which has been confirmed experimentally.20 It is also important to note that this power law scaling in one dimension has been shown to be quite universal, occurring in chaotic hard spheres,21 and analytic nonlinear potentials.22

### DAMPING FOR VACUUM POLYMERS

- Top of page
- Abstract
- INTRODUCTION
- POLYMERIC SYSTEMS AT LOW DENSITIES
- NONLINEAR DYNAMICS FOR LOW DIMENSIONAL SYSTEMS
- DAMPING FOR VACUUM POLYMERS
- INTERNAL FRICTION
- FUTURE DIRECTIONS
- CONCLUSION
- REFERENCES AND NOTES
- Biographical Information

How does this analysis carry over to polymers in a vacuum? In this case, the polymer is not along a 1D line, unless it has been stretched. If we ignore excluded volume interactions and consider only nearest neighbor forces, we run into the same issues that were discussed earlier, namely that without nonlinear spring forces binding nearest neighbor monomers, the system will not thermalize. Therefore in order for a model to be physically meaningful, these forces must include nonlinear terms. If these terms are not sufficiently strong, similar behavior to that seen in FPU chains is expected. In general, the forces between neighboring atoms are far from being linear, however this does not mean that transfer of energy between modes will be as rapid as it is for polymers in solvent.

It is useful for the purposes of this discussion to think of the relationship between a polymer in this situation and the usual one in a solvent. In the latter case, the solvent acts as a “heat bath” and has two important physical effects: (i) to provide transfer of the polymers kinetic energy into solvent degrees of freedom, leading to a dissipative term in the equation of motion and (ii) the transfer of kinetic energy out of the solvent and into the polymer, leading to a noise term. This is the basic physics behind the Langevin equation. Often the form of the drag is taken to be proportional to the local velocity with a viscosity taken from empirical observation. The requirement that this model is in accord with the predictions of equilibrium statistical mechanics leads to the fluctuation dissipation theorem23 which determines the form of noise acting on the polymer. This is the physics that is behind the well known Rouse equation.24 Let us call the position of the *i*th monomer in a polymer chain **r**_{i}. The drag force generated by the bath is then *b***ṙ**_{i}. This is balanced against the force due to neighboring monomers which, assuming linear springs of spring constant *k*, is *k*(**r**_{i + 1} − 2**r**_{i} − **r**_{i − 1}) and thermal noise from the heat bath **ξ**_{i}. As is often done when treating long wavelength properties, we will take the continuum limit, replacing the monomer index *i* with a continuous arclength variable *s* so that this becomes

- (1)

Relaxation in this case has a diffusive character. Consider a perturbation in the form of a pulse of width δ*s* along the backbone. This does not propagate ballistically as we saw it did for long wavelength disturbances of a 1D nonlinear chain. Instead, on average it spreads with time as *t* ∝ δ*s*^{2}.

In the case of a polymer in a vacuum, there is no bath. To continue to make an analogy to the solvent case, we are forced to think of the polymer chain as acting as its own heat bath. However it is no longer clear what form the dissipation should take. As mentioned earlier, in the case of linear springs, there is no dissipation and the polymer will oscillate indefinitely without thermalizing, so that the nonlinearity plays a key role in the damping. Aside from the strength of the nonlinearity, the form of the damping term should be quite different as we can see from the following argument.

If we consider a polymer chain in vacuum, the overall center of mass velocity makes no difference to its internal dynamics due to Galilean invariance. Therefore a local damping force **f**_{d} acting on this monomer such as **f**_{d} ∝ **ṙ**_{i} will not work in this case as it will slow down the center of mass of the entire chain. The simplest modification of the damping term is to consider the relative motion of adjacent monomers **f**_{d} ∝ **ṙ**_{i + 1} − **ṙ**_{i}. However this is unphysical because it depends on the direction that the monomers have been ordered. Relabeling the monomers to start at the chain's other end, leads to the same kind of term but with the opposite sign. Therefore we need to go to one higher order in the difference to get a physically viable damping

- (2)

which for long wavelengths can be written

- (3)

where *C* is a parameter which depends on the local form of the spring force and is zero for linear springs. This is sometimes referred to as “Kelvin Damping”.25 It is strikingly different in behavior from the usual case of damping seen in solutions as is exemplified by the Rouse equation, eq 1. This form was obtained by MacInnes for a model system26 based on a detailed theoretical analysis done by him. It was later proposed as the origin of Cerf friction27 where it was argued that this form was compatible with experiments characterizing internal friction28 in solutions, although other explanations have been proposed.29

It should be emphasized that although a damping term has been introduced, the underlying dynamics are still energy conserving. As discussed at the beginning of this section, such considerations lead to the fluctuation-dissipation theorem.23 The different modes are coupled together through nonlinear interactions and energy is thereby transfered. An unsolved problem is how to obtain the *k*-dependent damping analytically by such considerations. The corresponding problem has been understood for nonlinear 1D chains by a renormalization procedure involving the successive integrating out of high wave number,17 but it is harder to do for the polymeric case. However the form of the damping can be obtained by computer simulations. As we shall see, the simpler Kelvin damping, eq. 3, describes qualitatively the phenomenon seen in computer simulations, although its actual form is slightly different.

Because this form of damping is translationally invariant in both space and time, it will have sinusoidal modes with wave vector *k* and with amplitudes *r̂*(*k*, ω) that will satisfy the equation of a damped harmonic oscillator

- (4)

*M* is the mass per unit length, ν is a damping constant according to eq. 3 is ν = *Ck*^{2} and *K* = κ*k*^{2} where κ is related to the spring constant of a chain and κ = 3*k*_{B}*T*, with *T* denoting the temperature.10 The Rouse equation gives modes of the same form except for the absence of the inertial term. ν is also *k* independent and the modes will exponentially relax towards zero.

In contrast, to Rouse relaxation,24 eq 4 implies that each mode will damp out with a rate that depends strongly on wavelength, and the motion becomes underdamped for long wavelength modes therefore showing many oscillations as it slowly decays. The time for a mode to decay goes to infinity rapidly as the mode number *k* 0. One therefore expects to see strong oscillations in the time autocorrelation function for a chain that die out slowly for long chains.

To investigate these predictions, a model of a polymer with rigid links was used.4 The distance between monomers is constrained to a constant value, and the polymer is evolved according to Newton's laws, obeying conservation of momentum and energy. Time averages of the freely hinged model agree with exact statistical mechanical results that assume ergodicity.2, 3, 6, 7

Figure 1(a) shows the autocorrelation function of monomer positions **r**_{i} averaged over every monomer *i* and over 4000 independent runs, which can be written as 〈|**r**_{i}(*t*) − **r**_{i}(0)|^{2}〉. The curve is highly non-sinusoidal with cusp-like minima at every oscillation. This can be explained from understanding the individual behavior of the individual modes *r*_{k}.2 Figure 1(b) shows the first mode for the time autocorrelation function 〈*r*_{k}(*t*)*r*_{k}(0)〉 for single mode amplitudes *r*_{k} for the lowest wave vector *k*. What is striking about this result is the strength of the oscillations which are completely absent in the usual theory of polymers in solution. This data fits almost perfectly to the damped harmonic oscillator, eq. 4. Fitting the higher modes to this equation allows one to study how the frictional and mass terms depend on the wave-vector of a mode, and this was studied for a range of dihedral angle potentials4 as will be discussed briefly later.

A more realistic model of polymer forces should include a strong dependence on bond and dihedral angle. However even when such forces are quite strong, for long enough chains, the low *k* modes still show underdamped motion. The higher k-modes are more strongly dependent on the local angular potential, and the description in terms of single modes becomes quite poor. The dynamic correlations relax in an overdamped manner but no longer can be described in terms of as a damped spring. This interesting result is still not well understood.

It is also interesting to note that the damping coefficient only fits the Kelvin prediction qualitatively. The damped harmonic oscillator equation, eq. 4, has a damping coefficient ν = *Ck*^{2} but in fact the value of *C* is *k* dependent. The same is true for the inertial term *M*. It appears that either *C* ∝ *k*^{−α} where α is small or there are logarithmic correction, for example, *C* ∝ 1/log(*k*). This decrease in *C* occurs for a wide range of interatomic potentials employed, for small wave numbers.

Therefore for long wavelengths, there appears to be a wide degree in universality in the dynamics. There is a *k*-dependent damping term that leads to oscillatory behavior in the dynamics. This is very different than what is seen in polymer solutions. Even with potentials comparable to those for polyethylene,30, 31 for long enough chains, the same universal wave-vector dependence of the frictional term is seen. As mentioned earlier, for higher modes, the relaxation ceases to be well described by a single damped oscillator and shows interesting collective relaxation that deserves further study.4

The addition of excluded volume interactions changes the time autocorrelation functions considerably because collisions between monomers of distant backbone locations will speed up relaxation. The relaxation time τ_{rel} as a function of chain length *L* scales as τ_{rel} ∝ *L*^{p}, where *p* as we have shown above is close to 2 for no excluded volume interactions. However simulations including excluded volume show a large decrease of this exponent to *p* ≈ 1.15 ± 0.05. This is substantially smaller than what is seen in a solvent where *p* ≈ 1.8.10

Aside from autocorrelation functions, there are other interesting measures of a system's dynamics. It is possible to probe the chaotic nature of dynamics using Lyapunov exponents, and this was done for vacuum polymers with a variety of potentials.1 The systems studied included monomer–monomer interactions modeled with Leonard Jones potentials and nonlinear springs. It was shown that the largest Lyapunov exponent has an interesting dependence on the phase of the polymer, that is, whether it was extended or collapsed.

### INTERNAL FRICTION

- Top of page
- Abstract
- INTRODUCTION
- POLYMERIC SYSTEMS AT LOW DENSITIES
- NONLINEAR DYNAMICS FOR LOW DIMENSIONAL SYSTEMS
- DAMPING FOR VACUUM POLYMERS
- INTERNAL FRICTION
- FUTURE DIRECTIONS
- CONCLUSION
- REFERENCES AND NOTES
- Biographical Information

In de Gennes' book on Polymer Physics,10 he divides friction experienced by a chain into three parts, (i) solvent friction due to the hydrodynamic coupling of the chain to the solvent, (ii) barrier friction, due to the internal energy barriers impeding transitions between “trans” and “cis” conformations, and (iii) what he calls “Cerf friction.” This latter effect is the internal friction of the chain independent of solvent viscosity. At the time of that book's writing as is the case today, there is very little reference to the work of Cerf and Leray on this problem27, 28 who carried out experimental investigations to determine this third form of friction. In those experiments, they measured relaxation times in a variety of solvents and attempted to extrapolate these results to zero solvent viscosity. The origin of these findings was interpreted by Cerf27 in terms of a Kelvin damping term derived by MacInnes.26 This is in fact a model of a polymer in a zero viscosity solvent, as has been analyzed above. The size of the damping coefficient ν increases considerably with increasing rotational barriers,4 for example, for low modes, it increases in magnitude by a factor of approximately 15, from a freely hinged chain to a chain with a dihedral barrier height of 6*k*_{B}*T*. If we follow the argument of Cerf,27 and well explained by de Gennes,10 the argument can be revised slightly by what we have learned above. The nth Rouse mode in a viscous solution have a relaxation time τ_{n}

- (5)

and *k* = π*n*/*N*, from which it is apparent that the relaxation time diminishes rapidly with increasing *n*. For the point of illustration we can include the effects of internal friction by the simple approximation of Kelvin damping, so that eq. 3 this becomes

- (6)

This means that as we increase *k* the relaxation time plateaus instead of going to zero. The position of the plateau will depend on the value of *C* and the mode *k*. Because *C* can be large, this should substantially alter the chain's dynamical properties.

Other explanations may be possible10 for this effect, however it is clear that one does expect internal damping to have a form similar to the simple Kelvin damping model and so this effect should be seen experimentally.