Understanding the Concept of Entropic Force in Ideal Chains

[Sam Gralla]

In the article for "ideal chain", http://en.wikipedia.org/wiki/Ideal_chain, wikipedia states

"If the two free ends of an ideal chain are attached to some kind of micro-manipulation device, then the device experiences a force exerted by the polymer. The ideal chain's energy is constant, and thus its time-average, the internal energy, is also constant, which means that this force necessarily stems from a purely entropic effect."

Could somebody explain what it means for the force on the polymer to be "purely entropic"? The force is clearly electromagnetic: molecules in its bath are jostling it around, and eventually it ends up in a higher entropy state.

 

[alxm]

The change in energy with a change in coordinates (dE/dx) can be considered a force.

The jostling-around proceeds by means of the electromagnetic force, but the change in energy isn't from a change in electromagnetic potential, but entropy.

 

[DrDu]

Some sources tend to describe entropic forces as something misterious. However, they are always due to real forces due to collisions of molecules of the heat bath with the macromolecules.

Think of a child holding the end of a rope that is tied to the wall and other children throwing balls at the rope. The child will have to do work against the force of the balls which on the average will tend to shorten the length of the rope, so somehow it is statistical but nevertheless it is due to the concrete collisions of the rope with the balls.

On the other hand, any mechanical force excerted by a thermodynamic machine, e.g. a steam engine, can be thought of as being an entropic force.

 

[Mapes]

I've found it useful to visualize a chain in the bed of a pickup truck, or a necklace on a shaker table. Stretch it out into a straight line, and when you come back later you'll find that the ends have moved closer together. This is spring-like behavior. But instead of electromagnetic attraction between atoms pulling the ends together (an enthalpic spring), it's thermal energy (represented in the visualization by mechanical agitation). Thus, an entropic spring.

 

[Sam Gralla]

Thanks for the responses. This is all consistent with what I believed, but it still seems crazy to call such a force "purely entropic", since the force is electromagnetic. I wouldn't object if they were just calling it an "entropic" force, although it of course can be a bit misleading to a newbie if they don't discuss the microscopic origin. But what is the word "purely" doing in there, if not purposely trying to mask the microscopic origin?

It might help if somebody knows of a thermodynamic example where force is not purely (but only partially) entropic.

 

[Mapes]

...it still seems crazy to call such a force "purely entropic", since the force is electromagnetic.

Is it? Don't atoms repel each other in part due to Pauli exclusion?

Besides, even without collisions from other atoms, a chain at any finite temperature would still tend to become disorganized because of the Second Law and the tendency for increasing entropy. The common element is entropy, not collisions, and definitely not electromagnetism.

It might help if somebody knows of a thermodynamic example where force is not purely (but only partially) entropic.

Bond stretching is always a factor, it's just negligible compared to entropic effects in long chains. As the chain length decreases, the two effects act more evenly.

EDIT: Corrected "Heisenberg Uncertainty Principle" to "Pauli exclusion".

 

[Sam Gralla]

Is it? Don't atoms repel each other in part due to the Heisenberg Uncertainty Principle?

It's news to me if exclusion plays any role in molecular collisions. I thought it only mattered in extreme situations like neutron stars, but this is not my field.

Besides, even without collisions from other atoms, a chain at any finite temperature would still tend to become disorganized because of the Second Law and the tendency for increasing entropy. The common element is entropy, not collisions, and definitely not electromagnetism.

I don't think I'm contradicting you here, but to be clear, the second law is not some fundamental truth that gives rise to forces. Rather, the second law is derived from underlying force dynamics, based on the assumption that evolution won't prefer one microstate over another. I agree that any "finite temperature" chain would have this force on it, but you'd have to have some sort of interaction to make a chain "finite temperature". (How could an ideal chain in empty space be jiggling around?) But I guess you're saying that we could imagine in some strage situation that this interaction was (say) the weak interaction rather than electromagnetism, so it makes sense to call the force on a chain "entropic" regardless of what the actual underlying mechanism is. I guess I still dislike the word "purely" because you're not going to get a force if this chain is just sitting in empty space. You need the real forces on the chain.

Bond stretching is always a factor, it's just negligible compared to entropic effects in long chains. As the chain length decreases, the two effects act more evenly.

Okay, so if the chain had some effective elasticity, you'd say this was not an entropic force. So I guess the idea is that forces which represent the average of lots of little forces are entropic, whereas a force that always acts in one direction is "normal".

 

[Mapes]

It's news to me if exclusion plays any role in molecular collisions. I thought it only mattered in extreme situations like neutron stars, but this is not my field.

(Sorry, meant to write Pauli exclusion, as you must have figured out. Corrected above.)

I don't think I'm contradicting you here, but to be clear, the second law is not some fundamental truth that gives rise to forces.

But it is! Energy can be written in differential form as $dU=T\,dS+F\,dL$. (You may have seen this as $dU=T\,dS-P\,dV$, but let's assume a polymer chain in a vacuum here.) With no collisions and thus no energy exchange, $dU\approx 0$. Thus, the force $F$ to straighten the chain is $F\approx -T(dS/dL)$, which is a positive number because a straighter, longer chain has a lower entropy. This is covered in detail in texts on mechanics of polymers.

Intermediate thermo books (e.g., Callen) derive how entropy maximization (Second Law) implies energy minimization, which is the reason why electric charges flow under voltage, fluids flow under pressure, surfaces are minimized, and objects move under force. You can't show me a force that doesn't arise from energy minimization, entropy maximization, or a combination of the two (such as when the Gibbs free energy is minimized to make reactions proceed at constant temperature and pressure).

 

[DrDu]

Mapes, your argument, although found in many books is flawed. You cannot assume at the same time that the entropy depends only on the length of the chain and on the other hand express the differential of U in terms of both S and L, or, what is the same, S in terms of U and L. If the chain is massless and isolated, with L being the only thermodynamical degree of freedom, it is not regular. If it is a chain where each element has a mass, then the entropy will not only depend on the length but also on the amount of kinetic energy, i.e, entropy will depend not only on the length of the chain but on phase space volume. Another interpretation would be to consider the kinetic energy degrees of freedom as a heat bath (the children throwing balls in my example).

 

[Mapes]

No doubt the model is incomplete. What does regular mean?

 

[Sam Gralla]

But it is! Energy can be written in differential form as $dU=T\,dS+F\,dL$. (You may have seen this as $dU=T\,dS-P\,dV$, but let's assume a polymer chain in a vacuum here.) With no collisions and thus no energy exchange, $dU\approx 0$. Thus, the force $F$ to straighten the chain is $F\approx -T(dS/dL)$, which is a positive number because a straighter, longer chain has a lower entropy. This is covered in detail in texts on mechanics of polymers.

Intermediate thermo books (e.g., Callen) derive how entropy maximization (Second Law) implies energy minimization, which is the reason why electric charges flow under voltage, fluids flow under pressure, surfaces are minimized, and objects move under force. You can't show me a force that doesn't arise from energy minimization, entropy maximization, or a combination of the two (such as when the Gibbs free energy is minimized to make reactions proceed at constant temperature and pressure).

Wow, I have to say, I really think this is a totally absurd and untenable point of view! (All due respect . I suspected something like this was behind wikipedia's claim, which is one of the reasons I posted about it. I just can't see how entropy (classical entropy, to be safe)could possibly be viewed as fundamental. You can't even define entropy unless you have a collection of stuff. How are you going to derive the force between two electric charges from entropy? What could the entropy (or temperature) of this system possibly be? Your argument starts with an equation involving T and S, neither of which are defined outside of statistical physics. (And of course you can't do non-equilibrium stat mech with that equation, either, as far as I know.) I'd believe you if you said "entropy provides a really convenient, powerful, and unifying viewpoint for equilibrium statistical physics", but it's just impossible to view ordinary forces as arising from entropy.*

If your "finite temperature ideal chain in outer space" really made sense as an object and had a force on it, then you'd have an argument that entropy is fundamental. In this case, would you be writing the powers that be, telling them that there are actually five fundamental interactions: gravity, EM, weak, strong, and entropy? Did the physics community really overlook this fifth force which acts on ideal chains in outerspace? If you want to go down this path, please tell me the physical setup that keeps this ideal chain at finite temperature, and tell me why your first law of thermodynamics is valid for the system. I think you're going to have a problem setting this up without using one of the four fundamental interactions of nature :).

*disclaimer: there are people like Verlinde trying to get forces from entropy considerations. But this involves really vague arguments about "holographic screens" which are motivated from quantum considerations and black hole physics. Also, it doesn't work as far as I can tell :).

 

[Mapes]

Wow, I have to say, I really think this is a totally absurd and untenable point of view! (All due respect . I suspected something like this was behind wikipedia's claim, which is one of the reasons I posted about it. I just can't see how entropy (classical entropy, to be safe)could possibly be viewed as fundamental. You can't even define entropy unless you have a collection of stuff. How are you going to derive the force between two electric charges from entropy? What could the entropy (or temperature) of this system possibly be? Your argument starts with an equation involving T and S, neither of which are defined outside of statistical physics. (And of course you can't do non-equilibrium stat mech with that equation, either, as far as I know.) I'd believe you if you said "entropy provides a really convenient, powerful, and unifying viewpoint for equilibrium statistical physics", but it's just impossible to view ordinary forces as arising from entropy.*

Are we not talking about polymers---collections of atoms---tested under near-equilibrium conditions? Sorry, that's all I'm talking about: Force viewed as $d\Phi/dL$, where $\Phi$ is some potential function of energy and entropy, with the relative contributions being considerably different in elastomers vs. metals, for example. If it's not useful to you, OK. If you hate the terminology of "entropic spring," OK. I'm not interested in an argument about the relative importance of the Second Law vs. fundamental interactions.

 

[Sam Gralla]

Are we not talking about polymers---collections of atoms---tested under near-equilibrium conditions? Sorry, that's all I'm talking about: Force viewed as $d\Phi/dL$, where $\Phi$ is some potential function of energy and entropy, with the relative contributions being considerably different in elastomers vs. metals, for example. If it's not useful to you, OK. If you hate the terminology of "entropic spring," OK. I'm not interested in an argument about the relative importance of the Second Law vs. fundamental interactions.

I'm happy to back off arguing about what is fundamental, but for the record, unless I misunderstood what you meant by responding "but it is" to my statement that entropy is not fundamental and saying how entropy causes charge to flow in response to voltage, etc., you started it :).

But anyway there is content to my claim/belief that there is no sense in which an ideal chain (polymer if you like) can be sitting in vacuum and yet be at finite temperature. I'd like to settle this because if there really is a force on a polymer in outer space then I'd say we've discovered a new force of nature. I don't think the polymer can just fluctuate all on its own until it maximizes its entropy--there has to be a force on it causing the fluctuations. Do you see what I mean?

 

[Physics Monkey]

But anyway there is content to my claim/belief that there is no sense in which an ideal chain (polymer if you like) can be sitting in vacuum and yet be at finite temperature. I'd like to settle this because if there really is a force on a polymer in outer space then I'd say we've discovered a new force of nature. I don't think the polymer can just fluctuate all on its own until it maximizes its entropy--there has to be a force on it causing the fluctuations. Do you see what I mean?

It's absolutely true that a classical polymer chain sitting at rest in outer space with no forces on it or between the links does nothing. It just sits at a particular point in phase space and never moves (it's at rest with no forces hence no motion in phase space).

Then you include those magic words: "weakly coupled to a heat bath". Suddenly the polymer chain bunches up due to an "entropic force". But as must be, this motion is due to forces arising from the weak coupling to the bath that makes thermalization possible. Given this fact, the main interesting feature is that the resulting force at macroscopic scales is universal, independent of the detailed coupling and determined only by entropy or state counting for the ideal chain which we assume is essentially unchanged due to the weakness of the bath coupling.

 

[Mapes]

I'm happy to back off arguing about what is fundamental, but for the record, unless I misunderstood what you meant by responding "but it is" to my statement that entropy is not fundamental and saying how entropy causes charge to flow in response to voltage, etc., you started it :).

I stand by my claim that, at the macroscale, spontaneous processes can be traced back to the Second Law (and sure, mediated by interactions such as electromagnetism). I wasn't thinking about single particles but rather bulk materials, and I should have made this context clear before using a loaded word like "fundamental."

But anyway there is content to my claim/belief that there is no sense in which an ideal chain (polymer if you like) can be sitting in vacuum and yet be at finite temperature.

I'm not seeing what the problem is with the links swiveling and vibrating. What am I missing?

 

[Sam Gralla]

I stand by my claim that, at the macroscale, spontaneous processes can be traced back to the Second Law (and sure, mediated by interactions such as electromagnetism). I wasn't thinking about single particles but rather bulk materials, and I should have made this context clear before using a loaded word like "fundamental."

Okay, makes sense. I think we're in agreement, even if I have a bone to pick (and a bit of a temper) regarding the words "entropy" and "fundamental"!

I'm not seeing what the problem is with the links swiveling and vibrating. What am I missing?

The links could move around, but only inertially, not in a random walk as (I understand) normally assumed for the thermodynamic ideal chain. The evolution of the chain would depend very strongly on initial conditions, and while you might see higher entropy states more often, you might not (depending on initial conditions). It's not a thermodynamical setup. If you stuck a probe to one end, I think you'd probably measure a little something while the probe forces the end of the chain to slow down, and then you'd measure nothing once inertial motion begins with the new boundary condition of "one end fixed to the probe". In constrast, when the polymer is in a heat bath, the particles in the bath are constantly buffeting it, which transfers force to the probe.

 

[Sam Gralla]

Then you include those magic words: "weakly coupled to a heat bath". Suddenly the polymer chain bunches up due to an "entropic force". But as must be, this motion is due to forces arising from the weak coupling to the bath that makes thermalization possible. Given this fact, the main interesting feature is that the resulting force at macroscopic scales is universal, independent of the detailed coupling and determined only by entropy or state counting for the ideal chain which we assume is essentially unchanged due to the weakness of the bath coupling.

Yeah, okay, this makes sense, and is in line with what Mapes said earlier about entropy being the common element between various situations. I guess the point is that when the force is due to thermal fluctuations, its macroscopic effect is universal, so there isn't much point (except to satisfy onesself that nothing spooky/new is at work) in thinking about the force itself.

Thanks for all the help guys!

 

[Mapes]

Rethinking, I see I'm off base with the idea of a isolated polymer chain, as a deterministic system, evolving to maximum entropy. As Physics Monkey nicely pointed out in post #14, one needs the heat bath. Sorry about the slip-up.