# Civilization Springs, 3/5

### Part 3. Spring Civilization

^{[Image credit: By Lothar Spurzem - Own work, CC BY-SA 2.0 de, commons.wikimedia.org/w/index.php?curid=39574590 ] }

Previous part . Summary of the previous series.

So, what other ways are there to store energy, besides chemical fuels? Even if not for rockets, but in general?

Let's start with the electric battery. Here at least lithium-ion. Where does energy come from?

It's simple, there goes

^{[ 210 ] the}electrochemical reaction:

**LiC**

_{6}+

**CoO**

_{2}<->

**C**

_{6}+

**LiCoO**

_{2}

Going to the left - the

Of course, you guessed it. Since we know the limit of the energy intensity of a chemical reaction (≈20–30 MJ / kg), the maximum energy density for

**any**battery / accumulator is the same . Although lead, even nickel-cadmium, even sulfur-sodium. A simple look at the characteristics of different types of batteries on Wikipedia

^{[ 340 ] is}enough to confirm this conjecture. And yet to see: even the best batteries in terms of energy content (1-3 MJ / kg) to the theoretical limit do not yet reach the order. The battery in joules per kilogram does not beat gasoline and will never beat it - but it still has a lot to develop.

OK, let's try something radically different. Not at all on the battery is not like. Well, at least a spring. How is energy stored in it?

A load is applied to the material. The load shifts the atoms relative to each other. Because of the displacement, the electrical clouds of external, valence electrons are redistributed and slightly change their shape ... Stop! "

*It seems ... today I already said this ...*"

^{[Image Credit: Election Day film [ 630 ]]}

Yes, exactly. The energy of elasticity is stored mainly in the electric field of external electrons. And this means that it has the same limit: ≈20-30 MJ / kg, or 3-4 eV per atom, corresponding to the binding energy of either valence electrons with an atom (in covalent and ionic lattices), or an atom with electron “liquid” distributed electrons (in metal, where everything is actually more complicated and I cut a couple of corners here, but this did not radically affect the answer).

How to check this output? With fuels easy, the heat of combustion is in any directory. And what is the physical parameter for the material of the spring is a measure of the maximum stored energy?

**A bit of formulas**

**a**, площадь

**S**. Объём

**V**=

**aS**. Его растягивают. Делать это можно, пока давление (на растяжение) внутри бруска не достигнет предела прочности, обозначаемого буковкой

**σ**; попытка дальнейшего растяжения приведёт к необратимой деформации без запасания энергии. Сколько энергии в этом бруске? Работа

**A**= ∫

**F***

**dx**от нуля до

**ε***

**a**, где

**ε**— относительное удлинение, при котором деформация ещё обратима. За исключением резиноподобных материалов (где работает другой механизм)

**ε**<< 1 всегда — ибо атомы не могут поменять взаимное расположение на расстояния, сопоставимые с исходными, без радикальной перестройки всей решётки. Следовательно, удлинение мало, а силу сопротивления можно полагать линейно растущей с удлинением:

**F**≈ (

**x**/

**aε**)*

**S***

**σ**. Интегрируем, получаем работу

**A**=

**x***

^{2}**S***

**σ**/2

**aε**, подстановка от 0 до

**aε**, итого

**A**=

**Saσε**/2 =

**σVε**/2. А сколько энергии получается на единицу массы? Надо разделить

**A**на

**m**=

**ρV**. Будет

**w**=

**εσ**/2

**ρ.**

Величину

**ε**можно оценить как

**ε**≈

**σ**/

**E**, где

**E**— модуль Юнга для вещества. Но мы поступим проще. Поскольку для абсолютно подавляющего большинства конструкционных материалов

**ε**меньше

^{[358]}(и даже существенно меньше) единицы, мы просто загрубим оценку, записав

**w**<

**σ**/2

**ρ**. Как выяснится в дальнейшем, этой точности вполне хватит для понимания картины.

Answer: the limiting density of the elastic energy per kilogram does not exceed

**w**≈

**σ**/ 2

**ρ**, where

**σ**is the limiting pressure maintained by the material without irreversible deformation, and

**ρ**is its density. And if our understanding at the molecular level is at least approximately true, then this ratio should be no more than ≈30 MJ / kg. We look at the strengths

^{[ 350 ] [ 355 ] of}materials, compare:

Material | Ultimate tensile load σ, GPa (yield strength) | Плотность, кг/м^{3} | w = σ/2ρ, МДж на килограмм |

Нержавеющая сталь | 0.505 | 8000 | 0.031 |

Титановый сплав Beta C | 1.25 | 4810 | 0.13 |

Бериллий | 0.345 | 1840 | 0.19 |

Мартенситно-стареющая сталь [2800 Maraging steel] | 2.617 | 8000 | 0.33 |

Алмаз | 1.6 | 2800 | 0.57 |

Кевлар | 3.62 | 2514 | 1.25 |

Углеродное волокно Toray T1100G | 7.0 | 1790 | 2.96 |

It's like that. Moreover, the majority of construction materials do not reach the limit of 1-3 orders of magnitude. For real materials in crystal lattices always have numerous defects that do not allow them to attain even the strength that their atoms and molecules in principle are capable of. And real springs, in their turn, do not even reach the limit due to defects - because they “float” even with very small relative deformations.

And graphene

^{[ 95 ]}, you ask? As graphene, with the stated characteristics

^{[ 355 ]}at 65 MJ / kg? And all sorts of "colossal nanotubes"? We will talk about them in the fourth part. In the meantime, we will limit ourselves to the statement that, with a couple of very specific exception exceptions, the limit of the elastic energy capacity of solid matter does not really exceed ≈30 MJ / kg.

^{Article written for the site https://habr.com . When copying please refer to the source. The author of the article is Evgeny Bobukh . }

But maybe the problem with the spring is that it can not be compressed above the tensile strength of the material? However, this can be done with gases! What if you store energy in compressed gas?

So, it is given: a spherical balloon of radius

**r**made of metal with strength

**σ**with a thin wall. It is pumped gas under pressure

**p**. How thick should the wall be so that the balloon does not tear? The simplest calculation shows that this thickness is

**δ**= (

**r**/ 2) * (

**p**/

**σ**). How much does such a balloon weigh?

**m**=

**ρ**

**V**=

**ρ*** 4π

**r**

^{2 }

**δ**= 2π

**ρ**

**r**

^{3 }

**p**/

**σ.**How much energy is stored in it?

**E**≈

**pV**= 4π

**r**

^{3 }

**p**/ 3. We ignore the mass of the gas itself. Expansion losses too. How many joules per kilogram? Divide

**E**by the mass of the balloon

**m**, we get ...

**w**= 2

**σ**/ 3

**ρ**

The same spring. With the same fundamental Spring Limit

**independent of the pressure in the cylinder**. Of course, at the expense of tricky geometry or thick walls from this, you can probably squeeze a couple more times. But certainly not a couple of hundred ...

Flywheel? Its limit is determined by the ability of the material to resist the load created by

^{[ 640 ]}the same

**σ**/

**ρ**with an accuracy of a couple of times due to the geometry. However, in practice, this limit

**does not depend**on the flywheel.from relative elongation before failure, and, therefore, is (almost) fully achieved, unlike the spring.

Let's drop the mechanics. There is more modern electricity, let's store energy in it?

Suppose a vacuum condenser. The simplest: two plates, the electric field between them. As is known

^{[360, p. 106]}, each cubic centimeter of the electric field stores

**E**

^{2}/ 8π energy units (in the GHS, I used to count electricity in it). How much will it be per kilogram? And kilograms arise inevitably, because the capacitor needs strength. Plates are attracted to each other. They are attracted as if they were experiencing a negative electric field pressure. Which is equal to

^{[360]}the same

**E**

^{2}/ 8π! That is, this task is equivalent to the problem of a gas cylinder with negative pressure, which is kept from destruction by the strength of the walls. And we just solved this task. The answer is well known: all the same unfortunate

**σ**/

**ρ**plus or minus a couple of times.

And if the capacitor is not vacuum? If you fill the dielectric? He will take on part of the load. And it will increase the volume energy density

**ε**times, because in the dielectric it is equal to

^{[ 650 ] }

**ED**/ 8π =

**ε**

**E**/ 8π. It would seem, here it is, happiness? But alas, the compressive pressure on the capacitor also increases

^{2}**ε**times with a fixed internal

**E**and it turns out that way. But we still neglected electrical breakdown. The probability of which increases catastrophically as soon as the field

**E**becomes comparable to the interatomic fields created by external valence electrons. That is, here everything rests on the Spring Limit.

Then how about super-capacitors

^{[ 220 ]}with crazy capacities up to hundreds of farads? Alas, also nothing. By the principle of action, they are divided into two classes. Electrochemicals are actually redox batteries that store energy in a chemical form, simply very quickly. And electrostatic, more similar to capacitors in the conventional sense, only with a very thin gap between the "electrodes", several molecules wide. At the first the stock of energy, obviously, rests against chemistry. The second - in the magnitude of the breakdown electric field. Which can not significantly exceed the strength of the interatomic electric fields that hold matter in integrity. And these are the same units of eV on the size of the atom. Thus, supercapacitors are also limited in energy storage of ≈30 MJ / kg. Wikipedia testifies

^{[ 22 3 ]}: None of these devices even comes close in energy density to this limit. And, based on our understanding, will not work.

In the last attempt to jump over this limit with electrostatics, let's take a look at the spherical capacitor in a vacuum:

Take an ideally smooth metal sphere of radius

**r**. We cool it to (almost) absolute zero. We are taking far, far away into an infinitely deep vacuum. And bombarded with an electron beam, very far away. Electrons, falling onto the sphere, it will add the charge

**q**and (as we can calculate) the total energy

**W**=

**q**/2

^{2}**r**. Like, independent of the mass of the sphere. It???

Alas, such a capacitor can not be charged indefinitely. But only until the electric field created by him near the surface becomes comparable in strength with the electric fields between the atoms. If you approach this value by a negative charge, wild electron emission will begin ([ 390 , p. 13], [ 400 ]) and the charge will fly to the surrounding vacuum in a couple of minutes. If positive - the crystal lattice of the capacitor loses its strength, the substance will “evaporate” or simply crumble.

And if the field is not electric, but magnetic? Well, they took a ring from a superconductor of radius

**R**, wire thickness

**2r**, they started electric current

**I with it**, cooled it - and please: it runs in a circle forever current, energy in a magnetic field is waiting to be consumed. What is not a perfect battery?

But let us remember that oppositely directed currents repel each other. Therefore, a breaking force will act on the ring. For the opposition of which it is necessary to have a certain mass and elasticity. Having freed the readers from the details of the calculation, I will inform you that here the energy stored in the ring is approximately equal to all the same ratio

**σ**/

**ρ**.

Here, knowledgeable people will probably think: “Powerless configuration! But what about the powerless configuration ?! ”There is

^{[ 380 ]}such a thing. It is with a magnetic field that tricky geometry is possible, in which the field turns out

**to**be

**parallel**current in the system - and, thus, does not exert any force on this current. In the simplest embodiment, such a configuration, current is wrapped spirally field wrapped in the same direction and the force on the wire (almost) not valid:

^{[Image credit: Szabolcs Rembeczki, Design and Optimization of Force-Reduced High Field Magnets, [ 370 ]]}

Such a construction , at first glance, finally brings ordinary matter out of the role of a spring and jumps over the Spring Limit. However, it was not there. Careful and careful analysis

^{[ 370 ]}shows that, firstly, the force-free state is possible only in certain points of space - but not in the whole space; and secondly, the force-free

**finite**systemsize still requires external backups for its existence. Moreover, Szabolcs Rembeczki gives the exact result of another author (GE March) from 1996, where the total energy supply in such a system is compared with the elastic energy of these supports:

^{[Image credit: Szabolcs Rembeczki, Design and Optimization of Force-Reduced High Field Magnets, [ 370 ]]}

Slightly rewriting the last expression, we get:

**E**/

**M**≤

**σ**/

**ρ**. That is, the energy to mass still does not exceed the Spring Limit.

Finally, let's briefly touch the molten salt, because this topic is popular. How much energy can a kilogram of melt store? Obviously, this is the energy required for heating to the melting temperature, plus the specific heat of this melting itself. The first of the two is negligible: since 1 eV at this is 11600 degrees, then obviously no solid body can contain more than ≈0.4 eV / heat atom. The second is determined by the binding energy of a solid lattice and for this reason does not exceed units of eV per atom. For example, in salt

**NaCl**(a substance close to total ionicity and relatively harmless), the heat of fusion is

^{[ 660 ]}0.52 MJ / kg, or about 0.3 eV per atom. What is this topic can be closed.

The result is sad and a little funny.

Despite the millennial engineering progress; Despite the enormous variety of ways of storing energy, most of these methods rely on the same principle. The principle underlying the device, known to us for hundreds of years.

This device is a spring:

We are a civilization of springs.

Our rockets are expensive and heavy, because, in fact, the spring keeps their driving energy, the density of which is barely enough to overcome the earth's gravitational well. The Spring Limit determines the mechanical strength of the rockets, opposing the mass of the cocked chemical fuel spring. The spring limit dictates the limiting height of our buildings, the span length of bridges, the capacity of batteries, the thickness of truck bodies.

All that stores energy in the redistribution of the electric fields of external, valence electrons of ordinary matter, rests on the Spring Limit: 3-4 eV per atom, or 20-30 MJ / kg. The matter we use every day is like a greedy broker. All transactions go strictly through it: energy => matter => electric fields => matter => energy. But the broker prohibits storing more than 3-4 electronvolts per atom on one account, and delays a colossal commission in the form of a mass of a heavy atom for each account.

And although the internal electrons of the atom have binding energies of hundreds and thousands of electron volts, and the nuclei have millions and billions, we are barely able to work with these forces. So far we have learned well to manipulate only the thin outer shell of the atom. In it, in the form of electric field strength, almost all energy reserves of our civilization are stored.

Some Martians, you see, from the realization of this would have long since dropped the

Continued .

**References and literature**

[95] Про графен: https://en.m.wikipedia.org/wiki/Graphene

[210] Литиево-ионная батарейка, электрохимия: https://en.wikipedia.org/wiki/Lithium-ion_battery#Electrochemistry

[220] Суперконденсаторы: https://en.wikipedia.org/wiki/Supercapacitor

[223] https://en.wikipedia.org/wiki/Supercapacitor#Comparison_with_other_storage_technologies

[340] Параметры разных типов батарей: https://en.wikipedia.org/wiki/Comparison_of_commercial_battery_types

[350] Прочности материалов: https://en.wikipedia.org/wiki/Specific_strength#frb-inline

[355] Прочности материалов: https://en.wikipedia.org/wiki/Ultimate_tensile_strength#Typical_tensile_strengths

[358] Лекция по теории прочности и пределы разрушения материалов: https://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/MIT3_11F99_ss.pdf

[360] Л. Д. Ландау, Е. М. Лифшиц. Теория поля. Издания пятое, исправленное и дополненное. Издательство «Наука», Главная Редакция физико-математической литературы, Москва, 1967.

[370] Szabolcs Rembeczki, Design and Optimization of Force-Reduced High Field Magnets: https://research.fit.edu/media/site-specific/researchfitedu/hep/heplaba/documents/theses/DissertationSzabolcsRembeczki.pdf

[380] Википедия про бессиловые поля: https://en.wikipedia.org/wiki/Force-free_magnetic_field

[390] Формулы для расчёта полевой электронной эмиссии: http://www2.avs.org/conferences/IVNC/pdfs/Forbes-Tutorial.pdf

[400] Физическое описание полевой электронной эмиссии (с битыми формулами явно из [390], но пояснениями в предыдущих параграфах, позволяющими таки понять механизм): https://en.wikipedia.org/wiki/Field_electron_emission

[630] Режиссёр Олег Фомин, фильм День выборов: https://ru.wikipedia.org/wiki/День_выборов_(фильм)

[640] Про маховики и предел их плотности энергии: https://en.wikipedia.org/wiki/Flywheel_energy_storage#Specific_energy

[650] Лекция про электростатику диэлектриков: http://physics.usask.ca/~hirose/EP464/ch4-09.pdf (вообще-то, всё это во множестве совестких учебников ещё написано, у того же Тамма или Ландау).

[660] Теплота плавления поваренной соли: https://en.wikipedia.org/wiki/Sodium_chloride_(data_page)