# Energy for Expanding Bubbles

When I see soap bubbles or sea foam bubbles, it seems like they’re all expanding to the point where they pop.

Why do these bubbles almost always expand instead of contract or stay the same size?

Where does the energy to make them expand come from?

I believe the bubbles break because of evaporation of the material forming the film.

Don’t confuse energy with force. They expand because of the pressure of the gas inside. If the film of the bubble were infinitely elastic and stable with no forces of it’s own squeezing gas inside the bubble would only expand until it equalled outside pressure then stop.

Maybe I need a basic physics refresher, but how can a force be exerted without the expenditure of energy? Isn’t force = mass * acceleration?

Let’s take the example of a person blowing a soap bubble… is the expansion of the soap bubble caused by the energy the person expended blowing air into the bubble? If so, what about soap bubbles in lather where no blowing has occurred?

Bubbles merging?

More properly, force is the time-derivative of momentum. That has nothing (directly) to do with energy, though, and I’m not sure why you think it does. What is true is that force is the negative of the space-derivative of potential energy. A force causing change comes from an “attempt” to reach a minimum of potential energy (quotes because I shouldn’t really anthropomorphize nature like that).

What’s could be happening is that the thickness of the soap film is decreasing, since that will lead to a lower-energy state. The same amount of soap in a thinner layer means more area, so the bubble must expand. Of course, this doesn’t manage to handle how more air gets into the bubble…

I thought you couldn’t accelerate anything without an input of energy. Perhaps I was wrong. I’m a biologist, and my physics is a bit rusty to say the least.

OK, with ya so far!

OK, so the soap film would have to have energy input in order to maintain say, the hydrophilic portions and hydrophobic portions of a soap molecule close together, so they tend to create a thinner film of soap, which causes the bubble to expand without regard to the air inside of it?

The Earth moving in its (near-)circular orbit around the sun is constantly accelerating, but no energy goes in or out of the orbital system.

Well, as I mentioned before, the question of the volume inside increasing might be a weak point of my guess. Maybe the film is somewhat porous, maybe some reaction creates more gas (this is certainly happening in carbonated beverages, but there the production of CO[sub]2[/sub] is actually driving the expansion of the bubbles. Maybe the pressure actually drops inside the bubble.

So, break my first answer into two parts. (a) energy and force are not the same (b) a guess at what’s actually going on.

No more air is needed.
If you watch the colors on a large airborne bubble, you’ll see that they often form latitudinal rings about the bottom of the bubble. Gravity draws soap bi-layer down towards the bottom of the bubble. This thins the bilayer at the bubble’s top, and speeds evaporation of water. You can see the top become transparent, almost invisible, before the bubble pops. This thinning bubble wall exerts less surface tension, thus inward force, against the contained gas, so the bubble expands a bit.

Is there light shining on the bubble? Could it be the gas/air inside the freshly made bubble gets warmed by sunlight and expands?

Yeah, I was about to say this, mostly because if you get a nice big bubble and there’s good light, you can see the soap drift downward. It’s no doubt evaporating, and that no doubt does the same thing: less surface tension means less pressure on the gas inside. The gas inside a bubble is under slight pressure from the soap solution, which pulls together due to surface tension. As the pressure decreases, the equilibrium changes and the gas inside expands. It’s purely mechanical.

Conceptual nitpick:

These facts about forces are true under certain assumptions, but they don’t serve as definitions of force. Force is a primitive quantity of mechanics, given a priori, like mass. There is no way to define it in terms of other quantities like momentum and energy without making it contingent on some assumptions that may or may not be true.

Actually, from everything I see “force” really drops out of the picture once you move beyond Newtonian mechanics, and ever after all reference to force is as such a derivative.

Yeah. Isn’t what we sense as the “force” of gravity really that we are constantly being accelerated against the local curvature of space by 1 g. Or have I got it wrong again?

Well, that’s more or less true – it’s more that the Earth is pushing us from free-fall world-lines than that it’s “pulling us down”. What I meant was that Newtonian mechanics have long since been overthrown by Lagrangian and Hamiltonian mechanics.

Newtonian mechanics basically say what everyone knows from high school physics. It uses “force” and Newton’s law “F = dp/dt” (commonly misquoted as F = ma) and then relates force to position and time and sets up (generally second-order) differential equations.

Lagrangian mechanics sets out an “action” for each parametrized path in configuration space. That is, take a set of all configurations of the system (like the circle, for a rigid pendulum) and for every function from some time-interval to this set a value S. Then it says that the paths that actually occur are local extrema of this action. This is sort of like minimizing functions in basic calculus, except that instead of just one degree of freedom (as in calc 1) or two or three (in calc 3), there are infinitely many “nearby paths”. It turns out that from certain standard actions you get back exactly the differential equations from Newtonian mechanics, and without reference to any force.

Hamiltonian mechanics sets out a “phase space” of “states” of a system, which generally includes “position” and “momentum” coordinates q and p. Note that these momenta are not considered to be the mv from Newtonian mechanics. It also posits a Hamiltonian function H on this space. Then dq/dt = dH/dp and dp/dt = -dH/dq (partial derivatives of H, and the signs may be flipped). Again, Newtonian mechanics can easily be recovered as a special case. Again, there is no “force”.

Actually, tossing this out makes me unsure where force comes up other than in Newtonian mechanics, where the axioms clearly state that F = dp/dt. Hyperelastic, are you saying there are situations where this doesn’t hold? I mean, to the extent that the force function in a Newtonian system is part of the data of the system you’re right, and I may have been better off to state before that force is equal to the time-derivative of momentum, rather than that it is. Still, I think that’s an exceptionally tiny nit, especially given how useless force is in the long run.

Sure; F in general doesn’t equal dp/dt. It only equals dp/dt in the special case where the deformation is negligible, i.e. a rigid body. For instance, a force is required to statically deform a spring. Since nothing is moving, and we’re talking about a fixed quantity of matter, there’s no change in momentum; hence dp/dt is zero. Obviously, the force isn’t zero, so F doesn’t equal dp/dt. Even if you’re talking about the motion of a spring under a time-varying load, you need a more general expression that includes both inertial (m d[sup]2[/sup]x/dt[sup]2[/sup]) and material response (kx) terms.

For a real expanding soap bubble, the motion is so slow and the density so small that inertia probably plays no practical role, unless maybe you’re interested in oscillations after a pressure change. In any case, to understand the bubble, one needs to characterize the response of the soap film to stress. That corresponds to the “kx” part in the above discussion.

I am aware that in “fundamental” physics, it is attempted to explain everything in terms of little hard particles continually bouncing off one another, in which case the momentum is always changing and one can ignore the internals of the particles. I don’t profess to know much about “fundamental” physics except that any attempt to calculate the behavior of a soap bubble from that perspective is doomed to failure.

I see the problem. You’re omitting momenta. As I see it, if something is being deformed, the by definition something is moving. I’m talking about the entire configuration space here. That means that the shape matters, not just the position.

No, I’m talking about a static deformation. You push on the spring with a constant force and wait until all the points on the spring stop moving. You’re still exerting a force, but there’s no momentum.

There’s another way the force issue is commonly looked at - one can regard the “kx” part as an internal force, and then still say that F=dp/dt, where F is understood to include internal forces. There are some problems with this view. First, except in the very simplest cases, you need to introduce the concept of stress when deformation is important. Then you have to write your balance law in terms of stresses, so external influences have to be interpreted as surface tractions, not forces. Another problem is that for inelastic materials, you can’t get the dynamical equations simply by adding an “inertia” term to the statical equations. In some cases you can separate out all the terms involving density and call the leftovers “internal forces”, but I know of no proof that this can always be done, and it’s of dubious utility anyway.

In fundamental physics, the concept of momentum is preserved by associating momentum to a field. For instance, if you move a charge against an electrical potential, you have to keep applying a force even after the charge stops, or else it’ll go back in the other direction. Physicists want to be able to keep saying that momentum is preserved, so there’s an equation that assigns momentum to the field. I won’t say more for fear of making a fool of myself, because this is not my area, but I think the same thing is done with all fields in physics.

As for the OP, you’re never going to calculate the bubble’s motion from fundamental physics. So we adopt a continuum theory which explicitly disavows any explanations in terms of internal workings of the material. Dumping momentum into a field is right out. But, the theory is mathematically consistent (something that isn’t clear about the fundamental theories) and, just as important, it works.

Well, of course you’ve got to include internal forces. You’re not properly describing the system if you don’t include all forces along with all momenta. Also, of course you have to include stress since that’s just another form of the “same thing” as energy and momentum.

There’s actually a much better reason to say the Faraday field (rather than unnaturally splitting into electric and magnetic parts any more than we’d unnaturally split forces into internal or external parts) has a momentum: fields really aren’t well described in Newtonian mechanics, so we have to move to Lagrangian or Hamiltonian mechanics anyway. In the Hamiltonian formalism the momentum is part and parcel of the theory, while in the Lagrangian it’s a certain functional derivative of the Lagrangian function.

As for the OP, the whole point of this tangent was to break the disillusion that force and energy are the same thing, not to apply directly to the bubble problem. Read carefully.

What I’m saying is that the notion of internal forces has never been clearly defined, and we don’t need it in our theory, so why bother with it? Let me give yet another example: materials with internal constraints, like incompressibility. An internal constraint gives rise to a response that can only be deduced from boundary conditions, not from the deformation of the material. Is that response an internal force? It “comes from” the material, in that it wouldn’t exist if there were no material, but it is totally determined by the boundary conditions. Is it an internal force? You be the judge.

The Lagrangian and Hamiltonian formalisms contain no new physics so far as I can tell. They are simply mathematical expedients. The macroscopic behavior of fluids, liquid crystals and solids is governed by the classical field theory of continuous media, originated by Euler, Cauchy, and the Bernoullis, in which energy plays an occasional but secondary role.

Mathochist, I know you know what you’re talking about, but the way you phrase things suggests that you have a picture of classical mechanics as nothing but a series of deductions from Newton’s Laws. That is a common view among physicists and mathematicians, because most of them are taught by people who don’t know or care about anything in classical mechanics except occasionally mass-point mechanics. As you may have surmised, I like to be slightly argumentative on this issue. I wouldn’t bother the layman with these points, but I bother you with them because you’ll probably soon be in a position of influence with other scientists. If you’re interested in these matters, I highly recommend the writings of Truesdell, especially his articles in the Handbuch der Physik.