Ideal chain

 ( Polymer Chemistry ) 

An ideal chain (or freely-jointed chain) is the simplest Molecular model in polymer chemistry to describe polymers, such as nucleic acids and proteins. It assumes that the in a polymer are located at the steps of a hypothetical 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 Polymer physics.

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 peptide. 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.

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Model N Repeat unit form the polymer, whose total unfolded length is: $$L\; =\; N\; \backslash ,\; l,$$ where N is the number of mers.

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 $\backslash vec\; R$ the total end to end vector of an ideal chain and $\backslash vec\; r\_1,\backslash ldots\; ,\backslash vec\; r\_N$ 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.

The two ends of the chain are not coincident, but they fluctuate around each other, so that of course: $$\backslash left\backslash langle\; \backslash vec\{R\}\; \backslash right\backslash rangle\; =\; \backslash sum\_\{i=1\}^N\; \backslash left\backslash langle\; \backslash vec\; r\_i\; \backslash right\backslash rangle\; =\; \backslash vec\; 0~$$

Throughout the article the $\backslash langle\; \backslash rangle$ brackets will be used to denote the mean (of values taken over time) of a random variable or a random vector, as above.

Since $\backslash vec\; r\_1,\backslash ldots\; ,\backslash vec\; r\_N$ are independent, it follows from the central limit theorem that $\backslash vec\; R$ is distributed according to a normal distribution (or gaussian distribution): precisely, in 3D, $R\_x,\; R\_y,$ and $R\_z$ are distributed according to a normal distribution of mean 0 and of variance: $$\backslash sigma\; ^2\; =\; \backslash langle\; R\_x^2\; \backslash rangle\; -\; \backslash langle\; R\_x\; \backslash rangle\; ^2\; =\backslash langle\; R\_x^2\; \backslash rangle\; -0$$$$\backslash left\backslash langle\; R\_x^2\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; R\_y^2\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; R\_z^2\; \backslash right\backslash rangle\; =\; N\; \backslash ,\; \backslash frac\{l^2\}\{3\}$$

So that $\backslash langle\; \{R^2\}\; \backslash rangle\; =\; N\backslash ,\; l^2\; =\; L\; \backslash ,\; l~$. The end to end vector of the chain is distributed according to the following probability density function:$$P(\backslash vec\; R)\; =\; \backslash left\; (\; \backslash frac\{3\}\{2\; \backslash pi\; N\; l^2\}\; \backslash right\; )^\{3/2\}e^\{-\backslash frac\{3\; \backslash left|\backslash vec\; R\backslash right|^2\}\{2\; N\; l^2\}\}$$

The average end-to-end distance of the polymer is: $$\backslash sqrt\{\backslash left\backslash langle\; \{R^2\}\; \backslash right\backslash rangle\}\; =\; \backslash sqrt\; N\; \backslash ,\; l\; =\; \backslash sqrt\{L\; \backslash ,\; l\}~$$

A quantity frequently used in polymer physics is the radius of gyration:$$\backslash langle\backslash mathit\{R\}\_G\backslash rangle\; =\; \backslash frac\{\; \backslash sqrt\; N\backslash ,\; l\; \}\{\; \backslash sqrt\; 6\backslash \; \}$$

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 $N\backslash ,l$ at the thermodynamic limit. This result is a general property of statistical systems.

Mathematical remark: the rigorous demonstration of the expression of the density of probability $P(\backslash vec\; R)$ is not as direct as it appears above: from the application of the usual (1D) central limit theorem one can deduce that $R\_x$, $R\_y$ and $R\_z$ are distributed according to a centered normal distribution of variance $N\backslash ,\; l^2/3$. Then, the expression given above for $P(\backslash vec\; R)$ is not the only one that is compatible with such distribution for $R\_x$, $R\_y$ and $R\_z$. However, since the components of the vectors $\backslash vec\; r\_1,\backslash ldots\; ,\backslash vec\; r\_N$ are correlation for the random walk we are considering, it follows that $R\_x$, $R\_y$ and $R\_z$ are also correlation. This additional condition can only be fulfilled if $\backslash vec\; R$ is distributed according to $P(\backslash vec\; R)$. Alternatively, this result can also be demonstrated by applying a multidimensional generalization of the central limit theorem, or through symmetry arguments.

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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 length 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.

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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 entropy 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 entropy origin.

What is the microscopic origin of such an entropy 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 $\backslash vec\; r\_i$ 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.

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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 $\backslash vec\; R$ joining these two points characterizes the macroscopic state (or macro-state) of the ideal chain. Each macro-state corresponds a certain number of micro-states, that we will call $\backslash Omega(\backslash vec\; R)$ (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 associated to a macro-state is thus equal to: $$S(\backslash vec\; R)\; =\; k\_\backslash text\{B\}\; \backslash log\; (\backslash Omega(\backslash vec\; R)),$$ where $k\_\backslash text\{B\}$ is the Boltzmann constant

The above expression gives the absolute (quantum) entropy of the system. A precise determination of $\backslash Omega(\backslash vec\; R)$ would require a quantum model for the ideal chain, which is beyond the scope of this article. However, we have already calculated the probability density $P(\backslash vec\; R)$ associated with the end-to-end vector of the unconstrained ideal chain, above. Since all micro-states of the ideal chain are equally likely to occur, $P(\backslash vec\; R)$ is proportional to $\backslash Omega(\backslash vec\; R)$. This leads to the following expression for the classical (relative) entropy of the ideal chain: $$S(\backslash vec\; R)\; =\; k\_\backslash text\{B\}\; \backslash log\; (P(\backslash vec\; R))\; +\; C\_\{st\},$$ where $C\_\{st\}$ is a fixed constant. Let us call $\backslash vec\; F$ the force exerted by the chain on the point to which its end is attached. From the above expression of the entropy, we can deduce an expression of this force. Suppose that, instead of being fixed, the positions of the two ends of the ideal chain are now controlled by an operator. The operator controls the evolution of the end to end vector $\backslash vec\; R$. If the operator changes $\backslash vec\; R$ by a tiny amount $d\backslash vec\{R\}$, then the variation of internal energy of the chain is zero, since the energy of the chain is constant. This condition can be written as: $$0\; =\; dU\; =\; \backslash delta\; W\; +\; \backslash delta\; Q~$$

$\backslash delta\; W$ is defined as the elementary amount of mechanical work transferred by the operator to the ideal chain, and $\backslash delta\; Q$ is defined as the elementary amount of heat transferred by the solvent to the ideal chain. Now, if we assume that the transformation imposed by the operator on the system is quasistatic (i.e., infinitely slow), then the system's transformation will be time-reversible, and we can assume that during its passage from macro-state $\backslash vec\; R$ to macro-state $\backslash vec\; R\; +\; d\backslash vec\{R\}$, 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: $$\backslash delta\; Q\; =\; T\; \backslash ,\; dS,$$ 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 force exerted by the chain on its end points. Calling $\backslash vec\; f\_\backslash text\{op\}$ the force exerted by the operator and $\backslash vec\; f$ the force exerted by the chain, we have: $$\backslash delta\; W\; =\; \backslash langle\; \backslash vec\; f\_\backslash text\{op\}\; \backslash rangle\; \backslash cdot\; d\backslash vec\{R\}\; =\; -\; \backslash langle\; \backslash vec\; f\; \backslash rangle\; \backslash cdot\; d\; \backslash vec\; \{R\}$$

We are thus led to: $$\backslash langle\; \backslash vec\; f\; \backslash rangle\; =\; T\; \backslash frac\{dS\}\{d\backslash vec\; \{R\}\}\; =\; \backslash frac\{k\_\backslash text\{B\}\; T\}\{P(\; \backslash vec\; R)\}\backslash frac\{dP(\; \backslash vec\; R)\}\{d\backslash vec\; \{R\}\}$$ $$\backslash langle\; \backslash vec\; f\; \backslash rangle\; =\; -\; k\_\backslash text\{B\}\; T\; \backslash frac\{3\; \backslash vec\; R\}\{N\; l^2\}$$

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 $\backslash mathcal\{L\}$: $$\backslash frac\{R\}\{Nl\}\; =\; \backslash coth\backslash left(\backslash frac\{fl\}\{k\_\backslash text\{B\}T\}\backslash right)\; -\; \backslash frac\{k\_\backslash text\{B\}T\}\{fl\}\; =\; \backslash mathcal\{L\}\backslash left(\backslash frac\{fl\}\{k\_\backslash text\{B\}T\}\backslash right),$$ where the extension is $R=|\backslash vec\{R\}|$.

For the arbitrary extensions the force-extension dependence can be approximated by: $$\backslash frac\; \{fl\}\{k\_\backslash text\{B\}T\}\; =\; \backslash mathcal\{L\}^\{-1\}\{\backslash left(\backslash frac\; \{R\}\{Nl\}\; \backslash right)\}\; \backslash approx$$

 3\frac {R}{Nl}
 + \frac{1}{5} \left(\frac {R}{Nl}\right)^{2}\sin \left(\frac{7R}{2Nl}\right)
 + \frac{\left(\frac {R}{Nl}\right)^{3}}{1-\frac {R}{Nl}},
     

where $\backslash mathcal\{L\}^\{-1\}$ is the inverse Langevin function, is the number of bondsPolymer Physics , 76, Rubinstein 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.

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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 $\backslash vec\; f\_\backslash text\{op\}$ on the ideal chain. In this case the two ends of the polymer fluctuate around a mean position $\backslash langle\; \backslash vec\; R\; \backslash rangle$. The ideal chain reacts with a constant opposite force $\backslash vec\; f\; =\; -\backslash vec\; f\_\backslash text\{op\}$.

For an ideal chain exchanging length with a reservoir, a macro-state of the system is characterized by the vector $\backslash vec\; f$.

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 $\backslash vec\; R$ is now subject to fluctuations: $$\backslash vec\; f\; =\; -\; k\_\backslash text\{B\}\; T\; \backslash frac\{3\; \backslash langle\; \backslash vec\; R\; \backslash rangle\}\{N\; l^2\}~.$$

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Ideal chain under a constant force constraint – calculation Consider a freely jointed chain of $N$ bonds of length $l$ subject to a constant elongational force $f$ applied to its ends along the $z$ axis and an environment temperature $T$. An example could be a chain with two opposite charges $+q$ and $-q$ at its ends in a constant electric field $\backslash vec\{E\}$ applied along the $z$ axis as sketched in the figure on the right. If the direct Coulomb interaction between the charges is ignored, then there is a constant force $\backslash vec\{f\}$ at the two ends.

Different chain conformations are not equally likely, because they correspond to different energy of the chain in the external electric field. $$U\; =\; -q\backslash vec\{E\}\backslash cdot\; \backslash vec\{R\}=\; -\; \backslash vec\{f\}\backslash cdot\; \backslash vec\{R\}\; =\; -f\; R\_z$$

Thus, different chain conformation have different statistical Boltzmann factors $\backslash exp(-U/k\_\backslash text\{B\}T)$.

The partition function is: $$Z\; =\; \backslash sum\_\{\backslash text\{states\}\}\backslash exp(-U/k\_\backslash text\{B\}T)\; =\; \backslash sum\_\{\backslash text\{states\}\}\backslash exp\; \backslash left(\backslash frac\{f\; R\_z\}\{k\_\backslash text\{B\}\; T\}\backslash right)$$

Every monomer connection in the chain is characterized by a vector $\backslash vec\{r\}\_i$ of length $l$ and angles $\backslash theta\_i,\; \backslash varphi\_i$ in the spherical coordinate system. The end-to-end vector can be represented as: $R\_z\; =\; \backslash sum\_\{i=1\}^N\; l\; \backslash cos\backslash theta\_i$. Therefore:

The Gibbs free energy G can be directly calculated from the partition function: $$G(T,f,N)\; =\; -\; k\_\backslash text\{B\}\; T\; \backslash ,\; \backslash ln\; Z(T,f,N)=-N\; k\_\backslash text\{B\}\; T\; \backslash left\backslash ln\backslash left(4\backslash pi$$

The Gibbs free energy is used here because the ensemble of chains corresponds to constant temperature $T$ and constant force $f$ (analogous to the isothermal–isobaric ensemble, which has constant temperature and pressure).

The average end-to-end distance corresponding to a given force can be obtained as the derivative of the free energy: $$\backslash langle\; R\; \backslash rangle\; =\; -\; \backslash frac\{\backslash partial\; G\}\{\backslash partial\; f\}\; =\; N\; l\backslash left\backslash coth\backslash left(\backslash frac\{fl\}\{k\_\backslash text\{B\}$$

This expression is the Langevin function $\backslash mathcal\{L\}$, also mentioned in previous paragraphs: $$\backslash mathcal\{L\}(\backslash alpha)\; =\; \backslash coth(\backslash alpha)-\{1\; \backslash over\; \backslash alpha\}$$ where $\backslash alpha\; =\; \backslash frac\{fl\}\{k\_\backslash text\{B\}\; T\}$.

For small relative elongations ($\backslash langle\; R\; \backslash rangle\; \backslash ll\; R\_\backslash text\{max\}\; =\; lN$) the dependence is approximately linear, $$\backslash mathcal\{L\}(\backslash alpha)\backslash cong\; \backslash frac\{\backslash alpha\}\{3\}\; \backslash qquad\; \backslash text\{\; for\; \}\; \backslash alpha\; \backslash ll\; 1$$ and follows Hooke's law as shown in previous paragraphs: $$\backslash vec\{f\}\; =\; k\_\backslash text\{B\}\; T\; \backslash frac\{3\; \backslash langle\; \backslash vec\{R\}\; \backslash rangle\}\{Nl^2\}$$

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See also

• Polymer
• Worm-like chain, a more complex polymer model
• Kuhn length
• Coil–globule transition

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