A Hypothesis on Homeopathy, Part 4

OK, back to the exercise on the three states of matter. (If joining late, first go through my last post [^].)

First of all, take out your sketches. Then, do a Google search on “states of matter.” Click open the Google link to “Images for states of matter,” and browse further, esp. the Encyclopaedia Britannica entry. Also, see the Wiki entry on States of Matter [^] (version, today’s!!). Compare your sketches with these diagrams.

I can bet a coin of (Marathi) “chaar aaNe” (i.e. in Hindi, “chaar aanaa” or “chawanni”) that for most of you, the sketches you made would look very similar to those reported by the Wiki. Also, by the Britannica Encyclopaedia.

To make the matters somewhat more convenient, I also made a small diagram myself, using MS Paint. It appears below, and shows, from top to bottom, a solid, a liquid, and a gas. This diagram is not my idea of how they ought to be drawn, but, as per my aforementioned bet, it does show all yours’!!:

States of Matter, as Usually Shown

States of Matter, as Usually Shown

Now, on to the main point I said you all were going to miss (well, almost all of you, anyway!). To illustrate it, let’s take an example.

Take any metal, say, iron. When a piece of solid iron is heated, it melts into liquid iron. Does the above sketch adequately represent this change of state? Think about it.

If you know materials science, you would know that iron exists as a BCC (body-centered cubic) crystal at the room temperature and as an FCC (face-centered cubic) crystal at intermediately high temperatures. Etc. In contrast, my diagram shows iron as a SCC (simple cubic structure) crystal. Point granted. But that’s not the point I have in mind here.

So, my question still is: Do you find anything else wrong with that picture, esp. the sketches for the solid and liquid states?

In case you didn’t, here’s a hint.

What is the density of solid iron at room temperature? (Did at least that question give you a hint, now?) The density is, say, 7800 kg/m^3. OK. Now, what is the density of liquid iron, at a temperature a little beyond its melting point, say, at 1725 C? It is, say, 6900 kg/m^3. Given these data, suppose, you begin with one kg of iron, and melt it, and further heat the liquid iron to 1725 C. What do you think is the overall volume expansion, in percentage terms for such a solid to liquid change of state? (Note, volume and density are inversely related.) Say, roughly, 10 percent. That is, on the volume basis.

How does a volume expansion translate into lineal terms? For a unit cube to have a 10 percent increase in volume, what must be the increase in the length of each of its sides? I will provide some details for the benefit of some of my readers. Does it go like: \sqrt[3]{0.1} \approx 0.464 \approx 46 \% ? Or does it go like this: (1.0+e)^3 = 1.1 where e is the per-unit expansion (in fractional terms), and hence, 1.0 + e = \sqrt[3]{1.1} \approx 1.03228 , implying that e = 0.03228 \approx 3.2 \% ? 🙂

The second answer is correct. In lineal terms, the expansion accompanying the solid-to-liquid change is just about 3.2 %. Thus, when a cube of solid iron melts, if the molten liquid is to be held in a cube-shaped container, the container will have sides barely 3.2 % longer. That’s all!

For a viewpoint more convenient for us here: If you keep the volume of the liquid the same as that of the solid, how many atoms from the original solid would you expect to find in that same volume? The answer is: In volume terms, about 91% (because, the inverse of 1.1 is 0.909090…). Hence, in lineal terms, 97%. Hence, in areal terms, about 94%.

What we have drawn above is a 2D representation, and hence, areal terms apply. In the sketch for the solid state, there are exactly 100 atoms (10 rows and 10 columns). Hence, in the corresponding sketch for the liquid state, there should be 94 atoms.

Now, go, refer to the above sketch, and count the number of atoms actually drawn for the liquid state. I did. There are 63 of them.

The question is: How do you squeeze the extra 31 atoms—or approximately 50% more atoms—in the same space? Doesn’t it look impossible?

One way to look at it is this. Why not start with the sketch of those end-to-end stacked atoms in the solid state, and simply draw the area they would be permitted to play in, once the state of the matter changes from solid to liquid? That is precisely what I have done in the sketch below. The side of the square in which the solid atoms fit, was 480 pixels. At about 3.2 % increase, it becomes 496 pixels. I have drawn a blue square of 496 pixels to show you the room allowed to the 100 atoms, once they get into their liquid state.

Extra Space Available to a Solid When It Becomes Liquid

Extra Space Available to a Solid When It Becomes Liquid

Conclusion: Contrary to what (i) your high-school teacher, (ii) Wikipaedia, (iii) Encyclopaedia Britannica, (iv) etc. told you, (and contrary to what their high-school teachers, in turn, had told them) atoms simply don’t find as much room to roam around as the usual sketches show, when they go from the solid state to the liquid state. Indeed, in a great contrast to gases, both liquid and solid states are almost similar in terms of how tightly packed their atoms are.

Now, let’s try to integrate what that conclusion means, assuming the neat hypothetical substance we have sketched above. (Don’t rush to apply this discussion directly to water. We will cover water in some future post in this series. Water shows too much of anomalous behavior. The hypothetical substance we took here is based on the hard-spheres assumption, and as such, for some purposes, it might provide a good abstraction for metals—but not for salts, minerals, silicates, glasses, plastics, liquid crystals, etc., and, above all, in most important respects, not for water either.)

Is our model for the liquid state consistent with the fact that liquids flow readily? What really happens at the atomic level when liquids flow?

If the questions includes: “at the atomic level,” the answer is: “not much.” While you have been taught, perhaps right from your high-school science, that both liquids and gases are fluids, there is a great difference between how the flow processes must be occurring in them at the atomic level. Let me explain.

Here, you must first understand that, in the general sense of the term, a flow cannot be generated by simple direct pulls or pushes, i.e. by applying the normal load. Pressure does not result in transport of matter from one region of space to another. Pressure differences might. But not pressure by itself (whether it is positive or negative). Flow, in general, involves transport of a quantity of matter from one spatial region to another, together with change of shape, too—not just a local change of volume. For flow in this sense to occur, it is shear loading—and not normal loading—which must get applied. Liquids can flow only when the loading type is shear.

Do a physical experiment. Construct a large rectangular frame out of four wooden battons that are hinged at each of its four corners. It should also be possible to easily dis-assemble the sides, if necessary. Place the frame on a flat glass top, so as to make a rectangular tray. The frame should be large enough to carry a great number of (hundreds of) small glass marbles forming a monolayer on the surface of the table-top. Place the marbles in the tray following different systematic ways. For instance, first, let all blue marbles be together and all red marbles be together. Next, create one or two circular clusters of blue marbles in the other red marbles. Etc.

If the lengths of the frame-sides are such that they exceed the dense packing of marbles by a few percentage , the setup will show a liquid state. If the marbles instead fit snugly, it will show a solid state. (For representing the solid state, ideally, the hexagonal packing should be considered, not the square-packing shown above. (Exercise: Find out the lineal expansion factor e when a hexagonal solid becomes liquid.))

Starting with a setup for the liquid state (i.e. with, say, 3% extra length), first, shake the frame horizontally for some time. Observe how the marbles move. In particular, mark a few marbles and observe how they move with respect to their neighbouring marbles. Do the initial neighbours remain neighbours? (To CS guys: The issue here is not primarily the number of connected elements; it is: whether each connection keeps the same neighbouring element all throughout, or not.)

Notice, even if in this experiment, you have not allowed for interatomic bonds between marbles (which is an extremely important point), you would still find something interesting. Unlike in gases, thermal energy (or vibrations) does not serve to break “neighbourhoods.” For most atoms, even over very very long times of shaking, and even when there is no attractive interatomic potential present, the originally directly neighbouring atoms continue to remain direct neighbours. Even in the liquid state, not just the solid one. This is an important point, and though it is so easy to see, it is also very easily possible that you read it here first. Science is like that.

Next, take a fresh setup of liquid-like marbles in a frame. Now, completely remove two opposite sides of the frame, and, using a couple of extra battons, slowly apply shear to the marbles so that the “liquid” deforms in shear in such a way that two large chunks of marbles are made to merely slide against each other. What do you observe?

Obviously, even in such a shear deformation, most atoms remain as closely connected as ever. Only very few atoms, i.e. precisely those which happen to lie on the line of the shear-slip, slide against each other. Only these few atoms acquire new individual neighbours. The rest of the overwhelming majority of neighbourhoods remain intact.

Lesson: When liquids flow, they do not do so via independent motions of individual atoms as happens in gases, but via sliding motions of large blocks of atoms. There simply is not enough room for liquid atoms to behave in a neat conformance to your high school teachers’ imagination. Consequently, the mechanism of each atom moving independent of others is ruled out. And so, the only deformation mode left to explain flow is: movement of large blocks of neighbourhood-preserving atoms, sliding against each other in shear.

Suppose you argue that this is not the only shear deformation possible. Suppose, you advise this course of action: Keep all the four sides of the frame intact and simply apply shear to the overall frame, so as to deform the whole frame from a rectangular shape to a general parallelogram shape. Now, all atoms would move, albeit in a sliding mode. Accordingly, suppose that the battons that are originally alighed to the y-axis become, during shear-deformation, oblique-angled—their angle with the x-axis changes from 90 degrees to, say, 30 degrees. In this case, horizontal rows of atoms slide, and yet, all atoms move. If such a shear movement is carried out to a sufficiently large extent (large, as necessary to explain magnitudes experiences in real flows), then all atoms would have changed all their direct neighbours.

Why shouldn’t a liquid flow in this manner? You may try to buttress your argument by making a reference to the diagram they show while teaching Newton’s law of viscosity: The liquid at the ground surface is shown to remain stationary throughout, whereas planes of liquid are shown sliding at ever incresing velocity as you go away from the surface. Why not apply the suggested idea to the atomic model, you may ask.

To find the answer, just try it. Try it with actual marbles. See whether producing a shear this way is easier or more difficult. You will find that the mode I suggested—two blocks sliding on only one plane—is easier. And, the rows-sliding mode becomes plane-sliding mode for a 3D fluid. Between the block-sliding mode and the rows-sliding mode, the former requires far less energy, and hence, Nature would pursue it.

Thus, real liquids do not flow in the rows-sliding mode, despite the nice abstract illustration they use while teaching viscosity. (The illustration accompanying viscosity assumes a continuum, just in case you missed it!)

In fact, even the description of two blocks shearing away, the way we presented it above, is not an exact description. In reality, the boundary conditions are such that liquids can split into 3D regions—call them globules or cells or whatever. Each of the 3D globules might itself suffer deformation, but the point is, it would do so only as an independent block. Thus, the point is, within each globule/3D region, it’s not just planes of atoms but entire 3D blocks that remain together. Thus, when liquid flow occurs, for most atoms, there are blocks within which neighbouring atoms are never replaced by other atoms.

The idea of blocks makes it necessary to think in terms of scales. At what scale does separation via sliding occurs? How? etc.

It’s perhaps possible that real liquids flow with not just micro- but also meso- and even macro-scale structures.

However, even if you take only nano-scale blocks, precisely because the Avogadro number is so mind-bogglingly large, it still means that literally billions of atoms remain together even in extremely turbulent motion. Not convinced? Do a simple calculation. Take a reasonable 100 pm (1 angstrom or 0.1 nm) as the reasonable effective size for an atom while in the liquid state. For a 50 nm cubic block, it means: 500^3 \approx 125 million atoms. For a 100 nm cubic block, it means: 1 billion atoms.

Atoms this many (hundreds of millions) move together even in the most turbulent motion. Even if there are nasty mechanisms like generation of gas bubbles and their collapse. (If not convinced, estimate the overall volume fraction of gas bubbles in the given liquid, and the size of the tiniest bubble possible in it.) (BTW, in case you didn’t know, gas-bubbles are more disruptive of a liquid’s fabric than the motions of second phase particles such as undissolved solids, colloidally dispersed particles, etc.)

Now, keep aside the hard-spheres assumption for a while and think what it means for hundreds of millions (or even billions) of atoms to move together i.e. without destroying their neighbourhoods. (Even in the worst turbulent motion.)

It means: their quantum mechanical wavefunction involves sufficiently large number of particles that the configuration space is sufficiently large that it would be capable of supporting an incredibly greater number of at least metastable states brought about by whatever external means. In highly simplified terms, what it in turn means are two things: (i) structures undestroyed by most turbulent flows are possible in liquids in general (let alone in water), (ii) the structures can be of an incredibly great variety. In other words, it is possible that liquids can carry “memory,” at a suitable scale.

Notice, we made an extremely pathetic (i.e. unrealistic) assumption in the development of our argument above: a complete absence of interatomic forces (which allows for great slideability). Real liquids are not like that.

Further, to stay close to what could be happening in homeopathic preparation process, we thought of only nano-scale blocks. Despite this worst-case scenario, we found that we still have millions of atoms staying together, even during the most violent liquid motion, and within a space hardly different from the most rigidly structured solids.

Structure and liquid can only be rather thick friends.

Not surprising. Though liquids don’t keep their shape, they do keep their volume—they don’t fill the container they are poured in. (Philosophers must be thankful for this circumstance; it allows them the debate as to whether the glass is half-full or half-empty.)

The structures formed in liquids may be different in several respects from those in solids. The structures may not be sufficiently periodic. So, it may be impossible to catch them via diffraction techniques (like XRD/TEM). Yet, based even on the simplest of considerations as those touched upon above, a counter-question naturally arises: How can there not be structures—mers/linkages, rings, “fabrics,” clusters, etc. Involving not just hundreds or thousands, but millions and billions of atoms per block of those structures.

Your high-school teachers gave you an incredibly wrong idea when they drew those 60% volume filled representations for the liquid state—which, in turn, augmented in your mind the idea that liquids are structureless.

And, mind you, the “liquids” we considered here were very neat ones. Very hypothetical—without interatomic forces, and with all atoms spherical in shape, i.e., isotropic in the charge distribution. On both these counts, real water differs. A lot. Think how strong then can the evidence that 100% pure water, too, could carry structural imprints.

[PS: Might streamline this a bit or correct any minor mistakes still left, within a few days. If there is any major change, will note it as such.]

Links to my earlier posts on this topic:

A comment on homeopathy [^]
A Hypothesis on Homeopathy, Part 1 [^]
A Hypothesis on Homeopathy, Part 2 [^]
A Hypothesis on Homeopathy, Part 3 [^]

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A Song I Like:

(Marathi) “Tap Tap paDati, angaavarati, praajaktaachi phule…”
Lyrics: Mangesh Padgaonkar
Music: Shrinivas Khale
Singer: ?? (Shruti Sadolikar ??)

[E&OE]

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