The key reason that chemists can determine molecular weights using measurements of osmotic pressure, or measurements of melting point depression, or boiling point elevation, is that these "colligative properties" depend only on the molar concentration of dissolved substances, completely independent of their chemical properties. I notice that the word colligative isn't in the spell checker!

Electro-osmosis refers to two or three related phenomena, that are "two sides of the same coin" but may seem different the first time you learn about them. If you have water in a glass tube, and create a voltage gradient down the length of the tube, then a small "DC" electrical current will flow through the water, even if it isn't salt water. [Be careful trying this experiment with voltages larger than dry cell batteries can produce. Do NOT try it with anything like an electrophoresis power supply, because that can easily kill you!] If you look with a microscope, you will be able to see a directional flow of a thin layer of water closest to the glass surface. The reason is that positive ions in the water (even if there aren't very many) become slightly more concentrated near the glass surface, because glass itself ionizes with a slightly larger number of negative groups, immobile in and on the glass. Because the glass has a tiny excess of negative groups, positive ions will become slightly more concentrated than negative ions in the water a few microns from the glass surface. This layer of water is pulled toward the negative electrode. In time-lapse films of cells being subjected to voltage gradients, this water flow looks like a hurricane. I apologize for not having digitized any of the thousands of hours of 16 mm movie film that Nancy Pryer, David Paydarfar and I took of this phenomenon. We were not studying the water flow itself, but rather the effects of electric fields on tissue culture cells. But you can't avoid the water flow, unless you have a culture substratum that either releases no ions, or releases exactly the same numbers of positive and negative ions. Even polystyrene and silicone rubber, not to mention plasma membranes of cells, produce quite strong electro-osmotic water flow.

Incidentally, there is also water flow in the opposite direction; but it is broadly distributed up in the water that is not close to solid surfaces. So it's very slow and not noticeable.

If you look up the word "electro-osmosis" in a real encyclopedia, on Wikipedia, or even in a physical chemistry textbook, this flow of water along surfaces will be the only meaning they mention. This is ironic, because cartilage is a very important part of our skeleton, and the stiffness and resistance of cartilage to compression is caused by another form of electro-osmosis. It is also the main reason why local voltages result from stretching or compressing parts of the skeleton, especially cartilage, but also bone. These electro-osmotic voltages are much bigger than the piezoelectric voltages, about which much more has been written.

Electric fields can restrict the dispersal of ions about as effectively as semi-permeable membranes can prevent the free dispersal of solutes. In other words, electrical attraction can restrict ions to a particular area, so that the concentration of dissolved ions is much larger inside a piece of cartilage, as compared with concentrations of solutes in water surrounding the cartilage. Water will therefore tend to dilute this larger concentration of ions, and produce an osmotic pressure proportional to the excess of the ion concentration. This is not because the charge on the ions is attracting the water molecules. It's a colligative effect, driven by entropy.

The main reason for the excess of ions dissolved in the water inside a piece of cartilage is the large number of sulfate groups covalently bound to chains of sugars attached to protein fibers. These sulfate groups dissociate, so that each one has a single unit of negative charge. Sodium ions, potassium ions, hydrogen ions, calcium ions and other cations therefore become obliged to be more concentrated near these sulfate groups. They are prevented from leaving almost as effectively as if held in by a semi-permeable membrane. One difference is that any individual cation can diffuse away, as long as some other cation takes its place. The excess concentration of cations is just as effective in producing an osmotic pressure, by inward diffusion of water, as if the same concentration of solute molecules were held in place by a semi-permeable membrane.

In fact, if there were some way to use magnets or magic wands to restrict solutes or ions from equalizing their concentrations, then osmotic pressures would be exerted on the magnets or the wands. Think of water as "wanting" to equalize its concentration, and able to push hard on anything that resists that equalization.

If you squeeze a piece of cartilage, then water will be forced to flow out of it, almost as if you were squeezing a sponge. A difference is that the water squeezed out of cartilage has a net excess of cations dissolved in it, and so you will produce a voltage difference, that can be as much as hundreds of volts. The water squeezed out will be positive and the cartilage will be negative. This voltage will only exist for a second or less. After all, salt water is a conductor. Any voltage difference you can produce in it, by any means, will "short out" by a redistribution of ion concentrations. Furthermore, a reverse voltage will be created (again for seconds or fractions of a second), when you release the pressure and let the water flow back into the sponge.

These voltage changes, caused by squeezing and stretching, will also occur for other materials, including even bone, to the extent it has an excess of cationic or anionic groups, plus some degree of flexibility.

Voltage generation (temporary voltage generation) can be considered to be the third meaning of the word "electro-osmosis". However, all three are results of the same basic principles. Unfortunately, you need to understand the three phenomena pretty well before the connections between them become clear. Consider "centrifugal force" and "inertia"; they are the same phenomenon, in two manifestations; but sometimes it helps to have different words.

Probably more scientists would understand electro-osmosis if two or three different words had been invented as names for its different manifestations. My guess is that the word must have originally been coined to refer to pressure like that of cartilage, and only later was applied to the directional flow of water along an ionized tube, driven by an imposed voltage. Otherwise, why would any version of "osmosis" be used to mean a lateral flow, rather than a pressure?

Medical applications of electro-osmosis fall into at least two categories: first, damage to cartilages, either articular cartilages in joints, or herniated inter-vertebral discs (slipped discs) which are made of fibrocartilage; and second, the possible use of induced voltages to control the strengthening of bone in proportion to mechanical loads.

For about 35 years, it has been widely believed (without good evidence) that bones detect imposed forces by means of voltages caused by piezoelectric effects of some crystals. People have been very credulous about this. Patients hospitalized with broken bones have had wires connected to areas of skin near the beaks, and special machines sold and used to apply small voltages, in the hope of stimulating bone growth. Belief has decreased, and the whole subject has become unfashionable. Partly, this was because electric currents don't stimulate much or any bone growth. Another thing is that calcium phosphate crystals are not piezoelectric.

If attraction were the cause of osmotic pressure, then the amount of pressure produced by a given concentration of solute would be larger for some chemicals than others, in proportion to how strongly they attract water molecules. Intuitively, that seems so reasonable, that students fall for it. Elementary textbook authors (who usually are professional writers, and not actual scientists) claim "It makes osmosis easier to understand". Yeah, and if the earth were flat, that would make Geography easier to understand. If you are ever in the position of having to choose a textbook to teach from, quickly check how they explain osmotic pressure. The cause of nerve resting potentials is another topic to check, and so is the origin of antibody binding sites.