# How is potential energy actually stored in a steel spring at the atomic level?

Elasticity is one the most intriguing phenomena, wiki gives a summary explanation of what happens in a steel spring:

the atomic lattice changes size and shape when forces are applied (energy is added to the system). When forces are removed, the lattice goes back to the original lower energy state. Could you explain in detail how potential energy is actually stored in a steel spring and why the material never surrenders to the bending forces taking a new shape?

Update: it would be interesting if someone might add some technical details:

• is the quantity of energy transformed into heat during the compression $H_C$ equal to that lost during the release $H_R$
• is the heat loss during the two phases due to the exactly same causes, process
• how do you minimize heat loss, one factor is surely the mass, what is the max value of the ratio between energy spent in the compression and the energy retrieved ($c, \gamma$)? I read it can be: $E_R/E > 0.99$
• what determines the ratio between energy retrieved and velocity of rebound (CR)
• what is the max CR achieved to date in a spring, and with what material
• choosing the right material, can the speed of relaxation ever increase with the deflection in such a way to allow an elastic collisions

You are asking two questions really

1)

How is PE actually stored in a steel spring at the atomic level?

The explanation for this lies in quantum mechanics

2)

Could you explain in detail how/where potential energy is actually stored in a steel spring and why the material never surrenders to the bending forces taking a new shape?

Replying to 1) one has to think quantum mechanically. All solids form lattices. These lattices are formed because even though the individual atoms and molecules are neutral, there are electrical spill over fields due to the fact that the orbitals for the electrons are not spherical, except for l=0 angular momentum states, but have shapes. One can think of them as LEGO shapes that fit, positive spill over to negative spill over forces, forming new quantum mechanical solutions at a lower energy level, releasing energy into soft photons in the process ( that is why precipitation releases energy/heat to the environment). Thus the lattice is balanced at a lower quantum mechanical level than the free atoms/molecules themselves.

The state function of the lattice is a collective one of all the molecules/atoms that compose it, and acts as a higher level "particle", responds collectively quantum mechanically as long as the energies it interacts with are within the energy levels of the collective state. You must remember that in quantum mechanical solutions the lowest energy levels are filled but the higher ones exist and are available if energy is supplied.

What happens if an external force is imposed on the lattice? The collective state function, solution of the quantum mechanical state of the lattice, by absorbing soft photons collectively, moves to a higher lattice energy level. When the pressure is removed, it falls back to its lower energy level by releasing soft photons. These are non destructive deformations. If the energy is too high the lattice bonds will be broken and the lattice solutions will no longer be valid.

Elasticity means that some materials have enough energy levels to the lattice to be able to be deformed and fall back to ground state without being destroyed. The energy is stored in the higher energy level attained by the interaction "lattice forrce". So the steel spring lattice at the atomic level is at a higher energy level . Evidently steel lattices can take a lot of deformation so must have many higher than ground state energies.

Now for the second question, the models are continuum models with parameters that do depend on the quantum mechanical underlying framework but are emergent, from the many body problem. This seems to be still a research problem as it has to do with materials useful for industry and life in general. Models are proposed which use the spring modeling of the other answer at the lattice level, avoiding the quantum mechanical complications by approximations.

In a lattice spring model (LSM), the material is discretised into particles linked by springs. However, LSMs always adopt linear springs, which results in a stiff approximation of the corresponding elastic solution. In this work, a high order LSM is proposed to overcome this limitation by introducing additional degrees of freedoms (DOFs) to the particles.

In this case the internal quantum mechanics dynamics are approximated by springs.

• Hi, anna, thanks for the nice answer, it would become an excellent post if you could add some technical details on springs: 1) what is the max CR a spring can reach (made of what material) 2) supposing it's 97%, does the rest 3% all go into heat? 3) what happens to that heat when the spring extends itself does it increase or decrease? 4) if it decreases does it go into KE?, 5) does the restituted energy value (97%) correspond to the value of PE stored in the spring , or the stored PE is greater (say:98%) and some is lost in the release of the spring. If it is not clear I'll update my question – bobie Dec 5 '14 at 7:28
• @bobie I am not working in this engineering type field so I cannot provide you with such numbers. My answer is qualitative, not quantitative, based on general quantum mehanical systems.let us say you supply energy to the spring by pulling it . – anna v Dec 5 '14 at 11:40
• Transitions between lattice levels absorb photons or emit photons. The energy levels may well be vibrational energy levels,where the photons involved are very low frequency, black body radiation type. Suppose I pull a spring whose center is at rest at x=0 and attached on the other side at x=-a. \I pull it to x=a – anna v Dec 5 '14 at 12:15
• Pulling means that photons from the spill over coulomb forces in my hand transfer energy to the lattice. When I let go, some of the energy is transferred, again with photons in the opposite direction and compresses the spring . This vibration ends/dampens with dissipation into thermal photons. That is why there are spring like models which simplify the complexity. – anna v Dec 5 '14 at 12:21
• I updated my post, just in case some queries might find an answer. Thanks :) – bobie Dec 8 '14 at 14:01

Firstly, you can deform material permanently..spring is no exception. On the atomic level, you are working against Coulomb forces that bind the material id est, that form the lattice. One primitive cell is well defined by the conditions of minimal energy. You can describe this potential as a quadratic, so you get harmonic forces, but it is not truly quadratic, of course. In approximation, you get Hooks law. It is a good approx. for small displacements, but when you make large deformations, you enter a nonlinear zone in force or non-quadratic zone in terms of energy/work done for the displacement. Spring it self has an interesting geometrical structure which makes it more susceptible to forces so it can be as strong as you want it to be. A metal stick will vibrate in the same fashion but it is so stiff that you can not easily see these vibrations with small forces applied. Restoring force is Coulomb force, with repulsive and attractive forces balanced to form nice potential minimum.

http://en.m.wikipedia.org/wiki/File:HookesLawForSpring-English.png. I think this is also because of the spring constant which is I think is the gap present between the spring when it is coiled where the energy or the potential energy is stored and I don't think the atoms get affected

What you observe as mechanical deformation of a steel spring is an actual displacement (motion) of the atoms constituting the spring. In places, atoms will be slightly closer to their neighbors (compression) and in some other places actually futher apart (tension). The combination of compression on one side and compression on the other side of a beam or a wire causes it to bend to the side of compression. A coiled wire is the most stereotypical kind of spring, but all forms of springs will involve some compression or tension and usually both.

The (quantum) chemistry of your atoms determines the usual spacing between atoms. In metals, that usually creates a highly regular lattice, a crystal, but not a monocrystal: You have many crystalline domains, inidividual crystalls bordering on each other. When forces become too great, these crystalls can grow or shrink along their border, and they can slide against each other. This is the cause of permanent deformations and other changes (especially work hardening and embrittlement).

Your "technical" details are good questions to confirm one's understanding of the subject! Let me try to address them in order but rephrased:

1. Is the energy transformed into heat during the compression $H_c$ exactly equal to the mechanically lost energy (the bit of the compressive energy not recovered upon release)?

Your assumption is almost true. The lost energy is lost because it is dissipated (converted into heat). But it is (somewhat theoretically) possible that some of the heat temporarily generated during the compression gets adiabatically converted back into mechanical motion. This occurs for fast motion on not too small length scales (for the speed). It matters for details such as the spectrum of thermally driven, sub-microscopic fluctuations of metal surfaces (yes, there is Brownian motion everywhere).

2. Is the heat loss during compression and tension due to exactly the same processes?

Essentially, yes. You basically introduce a minus sign almost everywhere. For example, the adiabatic temperature change (compression heats things, expansion cools things) causes thermal conduction to go the other way but otherwise dissipate in just the same way because most things that matter tend to be linear in temperature change and hence much the same. Obviously there are minor exceptions: To address another quantitatively rather unimportant example, dissipation by thermal radiation is not quite symmetric: When you compress and hence heat, it increases more than it decreases for the same tension and hence cooling. Perhaps more relevant (but further outside my personal expertise), there are probably other heating mechanisms, such as heat released or absorbed in changes of the crystal domains, that could break the symmetry between compression and tension.

3. How does one minimize the loss of mechanical energy when cycling a spring?

Using (spring) steel is a good start; especially for very fast cycling (think steel balls bouncing into each other), it has a very high coefficient of restitution. You can do orders of magnitude better by using certain crystals or glasses (fused silica, quartz crytsalls, sapphire, silicon crystalls can reach 0.01 parts-per-million loss when operated in a vacuum). But there are lots of very unconvenient effects especially at surfaces and contact points. Quartz crystals for watches, tiny tuning forks, have Q factors (the inverse of the fractional energy loss per cycle) limited to about ten thousand despite coming in a vacuum package, because of losses at the mechanical mount points and electric contacts. If you open their vacuum package, you risk reducing the Q factor by an order of magnitude simply due to air friction and surface contamination. Must I even mention that glasses and (nonmetallic) crystals tend to have certain disadvantages such as being extremely brittle?

4. What determines the ratio between energy retrieved and the kinetic energy (squared speed, normalized by spring's effective mass for this purpose) at which the unloaded spring's end rebounds?

The short answer is that this ratio is one. Please excuse me for simplifying the question by pushing the bit that acts as effective mass of your spring's end (the entire mass weighted by how much is contributes to the total kinetic energy) into the question. The geometry and composition of your spring determines it. It is easy to calculate in principle and hence not very enlightening to consider in detail.

It is very helpful to consider the process as cyclically repeating. Then you have an oscillation of the same frequency of all relevant quantities, and the ratios between their amplitudes have special names. For example, the ratio between speed and force is called the (mechanical) impedance. An engineer might more naturally base an answer on this impedance than my personal (but accurate) concept of an effective mass for your purpose.

5. What is the maximum coefficient of restitution?

I don't know, and I won't accept any one value as it simply depends on what you still accept as a spring. If you consider the "restitution" of sub-microscopic indentations of solid quartz or fused silica test masses for gravitational wave detectors, the lost energy per deformation energy can be less than 0.01 parts per million.

6. Can there be essentially ellastic collisions?

Yes. Use steel balls for their springy behavior.

horizontal spring exerts a force $F = (−kx, 0, 0)$ that is proportional to its deflection in the $x$ direction. The work of this spring on a body moving along the space curve $s(t) = (x(t), y(t), z(t))$, is calculated using its velocity, $v = (vx, vy, vz)$, to obtain

$$W=\int_0^t\mathbf{F}\cdot\mathbf{v}\mathrm\,{d}t =-\int_0^t kx v_x \mathrm\,{d}t = -\frac{1}{2}kx^2$$

For convenience, consider contact with the spring occurs at $t = 0$, then the integral of the product of the distance x and the x-velocity, xvx, is $x_2/2$.

The function

$$U(x) = \frac{1}{2}kx^2,$$

is called the potential energy of a linear spring.

Elastic potential energy is the potential energy of an elastic object (for example a bow or a catapult) that is deformed under tension or compression (or stressed in formal terminology). It arises as a consequence of a force that tries to restore the object to its original shape, which is most often the electromagnetic force between the atoms and molecules that constitute the object. If the stretch is released, the energy is transformed into kinetic energy. As far as I can tell, based on General and special relativity, mass has a certain amount of energy inherent to it's existence. this energy regulates the matter's rate of time flow. as the mass increases in speed, it's kinetic energy increase while it's rate of time slows. So I think that until we get a better idea of how to conceptualise the matter/energy duality, the "potential energy" should be re-defined as "time rate energy".