An Electron in Water

What happens when a fast-moving electron hits water?

This question is important for understanding the effects of ionizing radiation, but it’s also just cool. I learned the answer here:

• H. Haberland and K. H. Bowen, Solvated electron clusters, Springer Series in Chemical Physics 56 (1994), 134–153.

There are four stages. To get a feel for these, it helps to remember that a femtosecond is 10-15 seconds. This is roughly the time it takes for light to travel a third of a micron—about 5000 times the radius of a hydrogen atom.

1. Thermalization

In 10 to 20 femtoseconds, our electron slows down due to its interaction with water molecules. It transfers energy to these molecules by knocking their electrons to higher energy levels or even knocking them off entirely. It does this until its energy has dropped to about 5 electron volts.

This is still quite energetic: for comparison, a lone electron at room temperature would have an energy of about 0.037 electron volts. Still, at this point we say the electron is thermalized. From now on it exchanges energy with water mainly by exciting motions of the water molecules.

2. Localization

Metals have a conduction band, a range of energies such that electrons with these energies can move freely. But materials called dielectrics, which do not conduct electricity well, also have a conduction band! The difference is that for metals, the conduction band has an energy low enough that electrons can easily jump into it. For dielectrics, the conduction band has an energy so high that it’s usually unoccupied.

Water is a dielectric. So, its conduction band is mostly empty. But even after our fast-moving electron is thermalized, it still has enough energy to occupy the conduction band! So, it moves along through the water!

It does this for about 110 to 180 femtoseconds until it finds a more localized state with about the same energy it has. You see, an electron in the conduction band has a wavefunction that’s very spread out. But there are also some states of about the same energy where the wavefunction is not so spread out. These are called shallow trap states. ‘Trap state’ means that the electron is stuck in one place. ‘Shallow’ means that the energy is still fairly high—there are also ‘deep trap states’.

All this sounds a bit abstract. What are these shallow trap states actually like? I think the electron finds itself a home in a little region that momentarily happens not to contain water molecules.

3. Solvation

Next, the water molecules around the newly trapped electron start reacting to its electric field. As this happens, the energy of the electron decreases. People say the electron “digs its own trap”. I like that metaphor!

This takes about 240 femtoseconds. When this is done, we have what’s called a solvated electron:

This picture is by Michael Tauber. The electron looks huge, but that’s just because its wavefunction is fairly spread out. He says this picture shows the ‘first and second coordination shells’ around the solvated electron.

4. Recombination

In the final stage, the electron combines with a positively charged ion. How long this takes depends radically on how many positively charged ions happen to be around.

For example, even in pure water there are some lone protons floating around. Not many! At 25 degrees Celsius, there is one lone proton per half billion molecules of water. But eventually the electron may combine with one of those and form a hydrogen atom.

In reality, it’s not quite that simple. A proton floating around in water will almost surely have already attached itself to a water molecule, forming a hydronium ion, H3O+:

And hydronium is still positively charged, so it will attract electrons in other water molecules. It will stick on to them in various various ways, the two most famous being the Eigen cation H₉O₄⁺:

and the Zundel cation H₅O₂⁺:

These in turn are still positively charged, so they attract more water molecules, making bigger and bigger structures, which I discussed in detail here:

Water, Azimuth, 29 November 2013.

Here’s a picture of one, called the protonated 21-water cluster:

Presumably when our lone electron meets these structures, they fall apart like a house of cards! I don’t know.

8 Responses to An Electron in Water

  1. Simon Burton says:

    What is the timescale for recombination?

    • John Baez says:

      The article did not say, perhaps because it depends on the concentration of ions that the electron can react with. The article is discussing electrons in a variety of liquids and even solids, not just pure water. It should be possible to determine a time scale for recombination in pure water at room temperature… but I can’t do it.

      All I can do is estimate the density of positive ions in pure water.

      In reality the pH of water is 7 only at 25 °C. As it gets hotter, the pH drops because the water jostles around more and there are more free H⁺ ions around. By 100 °C its pH is just 6.14.

      So, at 25 °C there are 10-7 moles of H+ per liter of water, while at 100 °C there are 10-6.14 = 7.24 × 10-7.

      The density of water also changes with temperature. At 25 °C it’s 0.997047 kilograms per liter, while at 100 °C, right before it boils, it’s 0.958366.

      A mole of water is 18.02 grams, so are 55.49 moles of water per kilogram. So, at 25 °C there are 55.33 moles of water per liter, but at 100 °C there are just 53.18.

      So, at 25 °C there are 10-7 moles of H+ per liter of water, or 55.33 × 107 molecules of water for each H+ ion. That’s 553.3 million molecules of water for each H+ ion.

      But at 100 °C there are 7.24 × 10-7 moles of H+ per liter of water, or 53.18 / (7.24 × 10-7) molecules of water for each H+ ion. That’s 73.4 million molecules of water for each H+ ion!

  2. Where’s the Cerenkov radiation in this story?

    • John Baez says:

      The paper I was reading didn’t mention that. That would happen in the initial slowdown phase, mainly before the story here even starts. I guess they (and I) were mainly interested in the later phases: the birth of a ‘solvated electron’.

  3. Erik Nelson says:

    I understand that the overall process, is one of progressively increasing LOCALIZATION. First, an incident, de-localized electron wave-function propagates into the matrix of water molecules. It progressively sheds energy, until it spatially contracts INTO, and conducts THROUGH, the molecular matrix. Thus, the electron wave-function PARTIALLY localizes into the regions NEAR positively (+) charged atomic nuclei, and shies away from inter-molecular voids, far from nuclei.

    If the electron wave-function resides WITHIN the molecular matrix, then does the electron EXIT that matrix, to Solvate in an inter-molecular void, between water molecules? Or could Solvation occur, when the electron wave-function propagates INTO a protonated water cluster? Are Zundel, Eigen, and 21-water clusters the LOCALIZED sites of Solvation into Shallow Trap States?

    If so, then perhaps pH values affect the physics? By judiciously manipulating the concentrations of protonated (acidic) and de-protonated (alkaline) water clusters, electron penetration distances and Recombination times might be affected.

    • John Baez says:

      I don’t completely understand your first question, but solvation refers to the process of an already localized electron affecting the orientation of nearby water molecules. When solvation is done, these water molecules are arranged in ‘coordination shells’ around the electron as shown in this picture by Michael Tauber:

      Of course everything wiggles around a lot, so this is just an ‘typical’ picture of what you might see.

      Solvation takes just 240 femtoseconds: about a quarter of a nanosecond!

      Are Zundel, Eigen, and 21-water clusters the LOCALIZED sites of Solvation into Shallow Trap States?

      No, these would play a role in the much slower process of recombination: the process where the solvated electron finds a positive ion to combine with. But I believe that pH can dramatically affect the rate of that process, since pH controls the density of positive ions.

      I don’t know how long recombination takes for water, but for liquid ammonia, which has a very high pH and thus very few positive ions, solvated electrons can last years before recombining! That’s what this article says:

      • H. Haberland and K. H. Bowen, Solvated electron clusters, Springer Series in Chemical Physics 56 (1994), 134–153.

      By the way, this articles is full of fascinating stuff and definitely worth reading.

      • Erik Nelson says:

        Thanks, that’s much more clear 😊

        The electric potential in water is vaguely like an egg carton, with peaks near the negatively charged oxygen atoms, and troughs near and between the positively charged hydrogens

        Sounds like thermalized electrons’ wave-functions propagate through the water like water trickling through an egg carton (on an incline). The electron clouds move through the positively charged regions BETWEEN molecules, presumably exerting electrical tugs on all the hydrogen atoms, as they propagate through. By a QM correspondence to Newton’s laws, the hydrogen atoms tug back on the electron, slowing it down (like lowering the incline), until the electron cloud ☁ finally pools in one egg carton trough, in BETWEEN a few molecules (egg carton on a flat surface).

        Then the water is shallowly trapped and soon solvated, but not yet recombined, and the situation looks like the Tauber picture.

        What is a characteristic value of the residual energy of solvated electrons, prior to Recombination?

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