An ideal chain (or freely-jointed chain) is the simplest model in polymer chemistry to describe polymers, such as nucleic acids and proteins. It assumes that the monomers in a polymer are located at the steps of a hypothetical random walker that does not remember its previous steps. By neglecting interactions among monomers, this model assumes that two (or more) monomers can occupy the same location. Although it is simple, its generality gives insight about the physics of polymers.

In this model, monomers are rigid rods of a fixed length , and their orientation is completely independent of the orientations and positions of neighbouring monomers. 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, is the Kuhn length. For example, chromatin is modeled as a polymer in which each monomer is a segment approximately in length.

Model

N mers where N is the number of mers, and l is the length of each mer.

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.

the vectors corresponding to individual mers. Those random vectors have components in the three directions of space. Most of the expressions given in this article assume that the number of mers N is large, so that the central limit theorem applies. The figure below shows a sketch of a (short) ideal chain.

brackets will be used to denote the mean (of values taken over time) of a random variable or a random vector, as above.

are independent, it follows from the central limit theorem is distributed according to a normal distribution are distributed according to a normal distribution of mean 0 and of variance

. The end to end vector of the chain is distributed according to the following probability density function

A quantity frequently used in polymer physics is the radius of gyration

at the thermodynamic limit. This result is a general property of statistical systems.

is not as direct as it appears above: from the application of the usual (1D) central limit theorem are distributed according to a centered normal distribution are uncorrelated are also uncorrelated. Alternatively, this result can also be demonstrated by applying a multidimensional generalization of the central limit theorem, or through symmetry arguments.

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). Since the N mers are of a rigid, fixed length, the model also does not consider bond stretching, though it can be extended to do so.

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 pulled apart by some sort of device, then the device experiences a force exerted by the polymer. As the ideal chain is stretched, its energy remains constant, and its time-average, or internal energy, also remains 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.

of each individual monomer (with i varying from 1 to N). In its solvent, the ideal chain is constantly subject to shocks from moving solvent molecules, and each of these shocks sends the system from its current microscopic state to another, very similar microscopic state. For an ideal polymer, as will be shown below, there are more microscopic states compatible with a short end-to-end distance than there are microscopic states compatible with a large end-to-end distance. Thus, for an ideal chain, maximizing its entropy means reducing the distance between its two free ends. Consequently, a force that tends to collapse the chain is exerted by the ideal chain between its two free ends.

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

(micro-states are defined in the introduction to this section). Since the ideal chain's energy is constant, each of these micro-states is equally likely to occur. The entropy is the Boltzmann constant

The above expression gives the absolute (quantum) entropy associated with the end-to-end vector of the unconstrained. This leads to the following expression for the classical (relative) entropy the force exerted by the chain on the point to which its end is attached. From the above expression of the entropy, then the variation of internal energy of the chain is zero, since the energy

is defined as the elementary amount of mechanical work, the system passes through a series of thermodynamic equilibrium macro-states. This has two consequences:

1. first, the amount of heat received by the system during the transformation can be tied to the variation of its entropy where is the temperature of the chain.

2. second, in order for the transformation to remain infinitely slow, the mean force exerted by the operator on the end points of the chain must balance the mean

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.

For the arbitrary extensions the force-extension dependence can be approximated by: is the inverse Langevin function, is the number of bonds in the molecule (therefore if the molecule has bonds it has monomers making up the molecule.).

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

on the ideal chain. In this case the two ends of the polymer fluctuate around a mean.

.

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

Ideal chain under a constant force constraint – calculation

at its ends in a constant electric field axis as sketched in the figure on the right. If the direct Coulomb interaction at the two ends.

Thus, different chain conformation have different statistical Boltzmann factors.

The partition function

Every monomer in the spherical coordinate system. Therefore:

The Gibbs free energy

(analogous to the isothermal–isobaric ensemble, which has constant temperature and pressure).

This expression is the Langevin function, also mentioned in previous paragraphs: .

and follows Hooke's law

See also

• Polymer

• Worm-like chain, a more complex polymer model

• Kuhn length

• Coil–globule transition

References