# Theoretical minimum temperature required to melt any material

Given a pressure level (like 1 atm) and a sufficiently hot temperature, I have the intuition that no material stays solid, and turns to plasma if hot enough.

So here's the question: According to modern physics models, what is the lowest known temperature beyond which we can guarantee that any material will be past its melting point? We can consider an arbitrary material sample being heated under isobar conditions at 1 bar.

Can we in theory make a material that remains solid at 1 bar and 4500K? 6000K? 20000K?

• Do you mean 101.3 kPa? I.e. 1 atm pressure? Nov 2 '21 at 9:41
• Very hard question, needs a lot of guess-work, approximations, etc. But as for upper-bound I'm pretty sure that no solid material can maintain it's solid state under quark-gluon plasma temperature of $10^{12}~\text{K}$, which is about $100\,000 \times T_{\odot}$. Nov 2 '21 at 13:59

Using the Debye model leads to the Lindemann melting formula for the melting Temperature: see reference), for p = 1 bar there is an upper limit for a given material structure.

$$T_m = \frac{4\pi^2 A\, r_0^2 k_B \eta^2 }{9N_Ah^2}\Theta_D^2\,$$ in K with A atomic mass, $$r_0$$ interatomic distance, $$\eta$$ Lindemann factor = 0,2 - 0,25 and Debye temperature $$\Theta_D$$.

In the reference the highest calculated value $$T_m$$ is for the element Tungsten W with 3955 K. The only variables A, $$r_0$$ and $$\Theta_D$$ can be altered, but you don't know them for the "theoretical melting temperature of any material", but only for a specific one. Moreover the whole Debye theory is an approximation.

• Is this model for compounds or does it only apply to pure elements? Nov 2 '21 at 13:27
• The formular given here is for pure elements, but the original idea of the model can be applied to every structure, computationally only that's the point. You will find different approaches on Google. For compounds even taking averages of the involved elements should give you an indication. So to find the absolute maximum, you would have to calculate Millions of different structures, a heavy search problem. Nov 2 '21 at 13:40
• I think this answer goes towards answering the question (Which seems harder to answer than it looked at first), but as you say it's for pure elements. The article I linked in the question related to a meting point of 4400K for a new compound. This means that 3955K at 1 bar is a very high temperature, but it's not the upper bound I'm looking for. Nov 2 '21 at 18:07

The outer part of a neutron star is considered solid and its temperature can reach $$10^6$$ K. This is probably the highest temperature that a solid can reach.

• I thought the surface of a neutron star was more like frozen plasma than a typical solid. Nov 3 '21 at 2:09
• If the crust is a solidified plasma then it is solid. See physics.stackexchange.com/q/219264 Nov 4 '21 at 21:07
• While this is a simplistic answer/doesn’t really answer the question specifically, I’d guess it’s probably closer to an accurate maximum melting point (but under\low) than the quark-gluon plasma comment on the question (that [seems to be] over/higher). Great addition, for background/real world reference outside theory if nothing else Nov 4 '21 at 23:05

Currently, the best possible answer, although not a verifiably correct answer, is "over 4400K" or whatever the exact value determined in the research mentioned from the article you provided is. It actually does describe the "Theoretical minimum temperature required to melt any material” that you're looking for, at least as best as any human currently can.

In theory there is a temperature you could prove no bonds can be maintained between multiple elements, which we already can prove is higher than the melting point of any individual element. However, that temperature may also be far higher than the minimum needed to melt any compound that could exist (or, a maximum melting point).

There also (sh)/(c)ould be undiscovered compounds that take a higher temperature to melt than this newly("newly", 6 yrs ago) discovered material you linked. (and I have not researched to confirm this isn't the case) - but this is an NP-Hard problem. Anyone here that can solve this, I suggest you check out the Clay Mathematics Institute before you post anything...

So, the question can't be answered any better than confirmation from the scientific community of a newly discovered compound with a new record for melting point, until we, as a species, figure out timely solutions to NP-hard problems. This doesn’t preclude studying known high melting point substances to narrow the possible fields and more rapidly develop a new compound with a higher melting point - but it is still not possible to prove it’s the highest melting point.

That any one compound has the maximum melting point cannot (currently) be proven, this makes it impossible to define the maximum melting point.

• The question is whether we can use the known laws of physics to analytically determine a theoretical maximum melting point. This isn't really a question that befits a frame challenge - it's both clear and answerable. A frame challenge suggests that the original question was misguided, or was rooted in a misguided approach. I don't think that's the case here.
– J...
Nov 2 '21 at 20:01
• @J... The question is answerable but only as an NP-hard problem. So yes the answer exists, but it is currently unanswerable by any human in existence. Maybe a frame challenge is the wrong word choice, but the goal is to express: "This question isn't currently, definitively, answerable by any known possible method, although the answer does clearly exist" - the best answer anyone can give now it what's provided in that article. Nov 2 '21 at 20:26
• @J... As you phrase the question, the answer is "Yes, we can use the known laws of physics to analytically determine a theoretical maximum melting point." but the funny thing is, no one can post that answer here, because no one can compute it before they're dead. (currently, I'm hopeful this changes in our life times) Nov 2 '21 at 20:35
• @TCooper computing the exact $T$ where the last material melts might be (NP)-hard, but that doesn't mean there can't be narrower upper bounds than infinity. It's just about finding an upper bound for the binding energy in crystals, and require thermal fluctuations bigger than that (so that in a Landau free energy analysis, entropy wins over energy). Nov 3 '21 at 7:09
• @Wouter If you can prove some specific temperature under infinity is guaranteed to melt anything, I’d love to read that answer (I see what you’re saying, personally I’m too ignorant to get there). While I think that’d be great to add to this QA, I can’t see how this question, as asked, is looking for anything other than the exact 𝑇 where the last material melts. Nov 3 '21 at 14:00