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An ideal chain (or freely-jointed chain) is the simplest model to describe polymers, such as nucleic acids and proteins. It only assumes a polymer as a random walk and neglects any kind of interactions among monomers. Although it is simple, its generality gives insight about the physics of polymers.
Contents
- The model
- Generality of the model
- Entropic elasticity of an ideal chain
- Ideal chain under length constraint
- Ideal polymer exchanging length with a reservoir
- References
In this model, monomers are rigid rods of a fixed length l, and their orientation is completely independent of the orientations and positions of neighbouring monomers, to the extent that two monomers can co-exist at the same place. In some cases, the monomer has a physical interpretation, such as an amino acid in a polypeptide. In other cases, a monomer is simply a segment of the polymer that can be modeled as behaving as a discrete, freely jointed unit. If so, l is the Kuhn length. For example, chromatin is modeled as a polymer in which each monomer is a segment approximately 14-46 kbp in length.
The model
N mers form the polymer, whose total unfolded length is:
In this very simple approach where no interactions between mers are considered, the energy of the polymer is taken to be independent of its shape, which means that at thermodynamic equilibrium, all of its shape configurations are equally likely to occur as the polymer fluctuates in time, according to the Maxwell–Boltzmann distribution.
Let us call
The two ends of the chain are not coincident, but they fluctuate around each other, so that of course:
Throughout the article the
Since
So that
The average end-to-end distance of the polymer is:
A quantity frequently used in polymer physics is the radius of gyration:
It is worth noting that the above average end-to-end distance, which in the case of this simple model is also the typical amplitude of the system's fluctuations, becomes negligible compared to the total unfolded length of the polymer
Mathematical remark: the rigorous demonstration of the expression of the density of probability
Generality of the model
While the elementary model described above is totally unadapted to the description of real-world polymers at the microscopic scale, it does show some relevance at the macroscopic scale in the case of a polymer in solution whose monomers form an ideal mix with the solvent (in which case, the interactions between monomer and monomer, solvent molecule and solvent molecule, and between monomer and solvent are identical, and the system's energy can be considered constant, validating the hypotheses of the model).
The relevancy of the model is, however, limited, even at the macroscopic scale, by the fact that it does not consider any excluded volume for monomers (or, to speak in chemical terms, that it neglects steric effects).
Other fluctuating polymer models that consider no interaction between monomers and no excluded volume, like the worm-like chain model, are all asymptotically convergent toward this model at the thermodynamic limit. For purpose of this analogy a Kuhn segment is introduced, corresponding to the equivalent monomer length to be considered in the analogous ideal chain. The number of Kuhn segments to be considered in the analogous ideal chain is equal to the total unfolded length of the polymer divided by the length of a Kuhn segment.
Entropic elasticity of an ideal chain
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.
This entropic force is very similar to the pressure experienced by the walls of a box containing an ideal gas. The internal energy of an ideal gas depends only on its temperature, and not on the volume of its containing box, so it is not an energy effect that tends to increase the volume of the box like gas pressure does. This implies that the pressure of an ideal gas has a purely entropic origin.
What is the microscopic origin of such an entropic force or pressure? The most general answer is that the effect of thermal fluctuations tends to bring a thermodynamic system toward a macroscopic state that corresponds to a maximum in the number of microscopic states (or micro-states) that are compatible with this macroscopic state. In other words, thermal fluctuations tend to bring a system toward its macroscopic state of maximum entropy.
What does this mean in the case of the ideal chain? First, for our ideal chain, a microscopic state is characterized by the superposition of the states
In this section, the mean of this force will be derived. The generality of the expression obtained at the thermodynamic limit will then be discussed.
Ideal chain under length constraint
The case of an ideal chain whose two ends are attached to fixed points will be considered in this sub-section. The vector
The above expression gives the absolute (quantum) entropy of the system. A precise determination of
where
We are thus led to:
The above equation is the equation of state of the ideal chain. Since the expression depends on the central limit theorem, it is only exact in the limit of polymers containing a large number of monomers (that is, the thermodynamic limit). It is also only valid for small end-to-end distances, relative to the overall polymer contour length, where the behavior is like a hookean spring. Behavior over larger force ranges can be modeled using a canonical ensemble treatment identical to magnetization of paramagnetic spins. For the arbitrary forces the extension-force dependence will be given by Langevin function
where the extension is
For the arbitrary extensions the force-extension dependence can be approximated by:
where
Finally, the model can be extended to even larger force ranges by inclusion of a stretch modulus along the polymer contour length. That is, by allowing the length of each unit of the chain to respond elastically to the applied force.
Ideal polymer exchanging length with a reservoir
Throughout this sub-section, as in the previous one, the two ends of the polymer are attached to a micro-manipulation device. This time, however, the device does not maintain the two ends of the ideal chain in a fixed position, but rather it maintains a constant pulling force
For an ideal chain exchanging length with a reservoir, a macro-state of the system is characterized by the vector
The change between an ideal chain of fixed length and an ideal chain in contact with a length reservoir is very much akin to the change between the micro-canonical ensemble and the canonical ensemble (see the Statistical mechanics article about this). The change is from a state where a fixed value is imposed on a certain parameter, to a state where the system is left free to exchange this parameter with the outside. The parameter in question is energy for the microcanonical and canonical descriptions, whereas in the case of the ideal chain the parameter is the length of the ideal chain.
As in the micro-canonical and canonical ensembles, the two descriptions of the ideal chain differ only in the way they treat the system's fluctuations. They are thus equivalent at the thermodynamic limit. The equation of state of the ideal chain remains the same, except that