Two Particles X And Y Are 4m Apart

[Everardo Frost]

Iwas wondering why atoms, when pulled apart and then released, attract together, but I notice that there is a special distance at which this attractive force doesn't seem to act but rather those atoms repel each other when pushed even closer.

The poles face opposite of each other and thus attract, no matter how far or close I place the magnet apart, as long as they're opposite poles they will indeed attract. So obviously, this isn't the right analogy, at least to my knowledgeable extent, to represent my solution.

Another thing I notice is using the Spring analogy as the interactive force between these particles, but for the sake of this question, please try not to include that analogy or anything similar as personally I find it a circular argument. In other words, if you were to say the atoms/molecules have a spring-like mechanism and act just like a regular spring you might find at a lab, I'll be asking, similarly, "How do the atoms in that laboratory spring work at a molecular level?" And you'll be answering, again, "just like the one at the lab" This is what I meant by the "question loop".

So if this was the case. Why do atoms attract each other at farther distances instead of continuously getting repelled by one another? And vice versa: Why do atoms repel each other at shorter distances instead of continuously getting attracted by one another?

I see that you are going deeper into the rabbit hole and this might be a continuation of our previous disscusion on why solid matter behaves in the way rigid body dynamics suggests from a microscopic perspective.

Since in your previous question you said you where in highschool I'm not going to jump to the maths or abstract concepts behind this, but I'm going to try a more intuitive approach. The problem here is that in our previous discussion it was easy to keep us focused on classical mechanics, where physics is more or less intuitive (we all have some knowledge of what a force is and how a spring works), but here we have to deal with concepts that are completely out of any conceivable "common sense intuition"; concepts that have to be developed throughout many years at university, concepts that require heavy math and a fair amounts of easiness handling simplier but still complex concepts that constitute the foundations, concepts on quantum mechanics that are so out of touch with our everyday macroscopic experience of reality that many physicists just avoid trying to understand them in a conceptual manner and just adhere to the "Shut up and calculate!" doctrine. I'm saying this because what I'm about to tell you is not only an extreme oversimplification but I'm also going to need to use metaphors and other toy models that might be closer to our daily experience to explain this, and in doing so I want you to realize that they are just that, models, and all of what I'm going to say has to be taken with a grain of salt. The truth is that in mathematical language things are clear but when trying to use words things might get even to the absurd (since human languages have evolved to accomodate our everyday experience of the world and not this realm of reality) so please rememeber that everything has a correct and more rigorous explanation.

In classical mechanics we learn about potential energy. This is usually shown as some sort / "latent" form of the kinetic energy of an object, an energy that can be potentially unleashed and awaits to be converted into real kinetic energy. Here we have our first example of intuitive but imprecise approach to a physical concept.

The important thing is that potential energy is different in different contexts (situations in terms of space and time), depending on whether or not this energy has more or less potential to become kinetic energy. For example, a rock on the floor is usually regarded as having zero potentiall energy since there are no means to make it move (to see it gain kinetic energy) until someone acts on it. But a rock on the top of the empire state building has the potentiality to archive extreme amounts of kinetic energy as soon as it starts falling. As you can see my explanation suggest some kind of subjectivity to what it really means to have the potential to move but the reality is that potential energy is a well defined quantity in physics that has some notion of relativism in terms of where you place "the zero of the potential". I'm not going to expand further on this since it is not so usefull and it is some of the basics of classical mechanics.

My interest here is to explain that this dependence of the potential energy in terms of location, moment in time and other contextual parameters suggests a view of the world where potential energy constitutes a "landscape". Since force is the change in motion of an object and motion is associated to kinetic energy we can see that this "landscape" is in fact related to the behaviour of force. We tend to think about it like this: the slope on the terrain of the potential energy "landscape" tells us how much force (change in motion) is going to be at play at any time, the slope is in fact the rate at which potentiall energy might be converted to kinetic. Again this is oversimplified and can be missleading if you don't state assumptions, but for us this is enought. So, there's (usually) a direct relationship between force and this potential energy "landscape" in classical mechanics that is expressed as

The interesting thing is that these potential energy "landscapes" are very usefull in terms of giving some intuition on the evolution of the system. You can immagine a ball rolling down hill on this "lanscape" and relate that to the changing state of the system. In our case the ball could roll downhill (as we move throught the $x$ axis of the plot) and then continue uphill until the force brings it back downhill to the other side. This will make the so called harmonic oscillator, and springs are like that: if you contract them (low $x$) they are going to expand (to high $x$) and after expansion they are going to contract again in a never ending oscillation. In real life there are frictional interactions that would make this system to damp, so the ball finally gets to a static situation on the equilibrium point (the lower part of the valley). This equilibrium point is in fact reached when the spring is relaxed (no need for contraction and no need for expansion), or mathematically, when $x = x_0$ (which means $F=0$).

This is called the Lennard-Jones potential and as you can see is a bit more complex than the potential associated with a pring. They both share some qualitative commonalities; if you expand the spring (if you separate the atoms a little) it would try to contract (the atoms would atract each other) and if you try to contract the spring (get the atoms closer toghether) then it would try to expand (the atoms will repel each other). This is because both potentials look like a valley.

In fact both systems are oscillators (but the Lennard-Jones potential is not for a simple harmonic oscillator is just a bit different), this means that if you separate the atoms they are going to pull each other closer and by inertia they are going to surpass the equilibirum point and get to close toghether. In fact enought to start repeling each other and expand again in a periodic fashion. In fact the bonds between atoms in a molecule are generally oscillating, this vibrations inside molecules explain a lot of stuff in physics (why is the sky blue or how is temperature defined from a microscopic perspective). But if some dissipative process occurs (just like friction for the spring) the oscillation between the atoms in molecules and solids damps until the atoms reach an equilibrium distance (as the relaxation length of the spring).

But chemical bonds are not springs and in fact there are some key differences. As you can see the Hooke's potential is symetric but the Lennard-Jones is not. You can immagine it as a spring that reacts much more violently to a compression that to an expansion. You can also see (if you immagine a ball rolling down hill), that of you put the atoms very close toghether they are going to repel each other so much that they are going to get infinitely far (the bond can be broken by this method). This doesn't happen for the spring case, you can conpress it as much as you whish and after expanding it is going to return back. So in atoms there is a minumum energy to allow the complete disruption of the connected system of a molecule but this doesn't happen on springs. All of this can be seen just by the shape of this potential.

So, we know the potential of an harmonic oscillator (the Hooke's potential) comes from a mechanical force related to elastic tension on a spring and described by Hooke's Law. But what is the nature of the force that generates the Lennard-Jones potential? Well, the asymetric nature of it suggests that maybe there are two different causes (one force might explain the resistance to expansion of the bonds and the other, completely different in nature, might explain the resistance to contraction of the bonds), and indeed this is the case. So let's get into each cause separately.

This looks very similar to gravitational interaction but the key difference is that the "gravitational charge", mass, is always positive while electrical charge can be of two kinds. This means that you can concentrate charges in such a way as to shield each other to make the entire ensemble neutral (without charge). You can't do this with mass since there is no way to add mass to an object and stop being attracted to it.

Atoms in a solid are generally neutral, This is because the electron charge is the same as the proton charge and thus any atom with the same number of electrons and protons is neutral overall. There is no Coulombian reason in sight as to why they should atract each other.

But the thing is that things are more subtle. There are two phenomena at play; Van der Waals forces and London dispersion forces. Both are rooted in Coulumb's law in fact. The thing is that atoms, even if neutral, might get polarized. This means that the charges inside them might be displaced when another charge gets close to the atom. Suppose an electron getting closer to an Hydrogen atom (one proton and one electron), this incoming electron will repel the electron inside the atom and atract the proton on the nucleus, the force is tiny (since the binding force between the electron and proton in the atom is huge due to their vicinity) so the atom wouldn't shatter into pieces, but is enought to create an imbalance in the disposition of the charge inside the atom. This doesn't mean that the electron on the atom gets to the opposite side of the atom as the other electron comes closer, it means, that on average, the electron will be located more on the opposite side of the atom. This polarity thing is what generates the Van der Waals forces and surprisingly, if you do the math, you can see that this force (which is an emergent property rooted in Coulumb's interaction) is not dependant on $x^-2$ as Coulumb's interaction is, but it turns out it is dependant on $x^-6$!

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