Chaîne idéale

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Ideal chain

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Une freely-jointed chain est le modèle le plus simple utilisé pour décrire les polymères, comme les acides nucléiques et les protéines. Le polymère est considéré comme une suite aléatoire et néglige toute interaction entre monomères. Bien que ce modèle soit élémentaire, il donne un aperçu de la physique des polymères.

Dans ce modèle, les monomères sont des tiges rigides de longueur fixée l, et leur orientation est complètement indépendante des positions et orientations de leurs voisins, dans la mesure où deux polymères peuvent coexister à la même place. Dans certains cas, le monomère peut être interprété physiquement, comme un amino-acide ou un polypeptide. Dans d'autres cas, un monomère est simplement un segment de polymère, représenté comme une unité libre, discrète et donc, si l est la longueur de Kuhn. Par exemple, la chromatine est modélisée comme un polymère pour lequel chaque segment mesure environ 14 à 46 paires de kilo bases de long.

Contenu

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• 1Le modèle
• 2Généralités du modèle
• 3Elasticité entropique d'une chaîne idéale
  • 3.1Chaîne idéale sous une contrainte en longueur
  • 3.2Polymère idéal échangeant de la longueur avec un réservoir
• 4Voir aussi
• 5Liens externes
• 6Références

Le modèle[edit source]

Soit un polymère constitué de N monomères, dont la longueur totale une fois déplié est :

$L=Nl$, avec N le nombre de monomères.

Dans cette approche simple, où l'on considère que les monomères ne sont pas en interaction, l'énergie du polymère est choisie de telle sorte qu'elle soit indépendante de sa forme. Ceci qui signifie qu'à l'équilibre thermodynamique, toutes les configurations du polymère ont autant de chances les unes que les autres de s'établir lors du mouvement du polymère, relativement à la distribution de Maxwell–Boltzmann.

Soit ${\vec {R}}$ le vecteur d'un bout à l'autre de la chaîne idéale, et ${\vec {r_{1}}},...,{\vec {r_{N}}}$ les vecteurs correspondant à chaque monomère. Ces vecteurs aléatoires ont des composantes dans les trois directions de l'espace. La plupart des expressions de cet article suivent l'hypothèse que le nombre de monomères $N$ est grand, c'est pourquoi le théorème de la limite centrale s'applique. La figure ci-dessous représente le schéma d'une chaîne idéale (courte).

IMAGE

Les deux extrémités de la chaîne ne coïncident pas, mais fluctuent l'une autour de l'autre, d'où :

$\langle {\vec {R}}\rangle =\Sigma _{i=1}^{N}\langle {\vec {r_{i}}}\rangle ={\vec {0}}$.

Tout au long de cet article, les symboles $\langle \rangle$ désigneront la valeur moyenne (par rapport au temps) de variables ou vecteurs aléatoires, comme ci-dessus.

Comme ${\vec {r_{1}}},...,{\vec {r_{N}}}$ sont indépendants, il vient du théorème de la limite centrale que ${\vec {R}}$ suit une distribution normale (ou distribution de Gauss) : précisément, en 3D, $R_{x},R_{y}$ et $R_{z}$ suivent une distribution normale de moyenne $0$ et de variance :

$\sigma ^{2}=\langle R_{x}^{2}\rangle -\langle R_{x}\rangle ^{2}=\langle R_{x}^{2}\rangle -0$.

$\langle R_{x}^{2}\rangle =\langle R_{y}^{2}\rangle =\langle R_{z}^{2}\rangle ={\frac {Nl^{2}}{3}}$.

Et ainsi : $\langle {\vec {R^{2}}}\rangle =Nl^{2}=Ll$. Le vecteur d'un bout à l'autre de la chaîne est distribué selon la fonction de densité de probabilité suivante :

$P({\vec {R}})=\left({\frac {3}{2\pi Nl^{2}}}\right)^{3/2}e^{-{\frac {3{\vec {R}}^{2}}{2Nl^{2}}}}$.

La distance moyenne d'une extrémité à l'autre du polymère est donnée par :

${\sqrt {\langle {\vec {R}}^{2}\rangle }}={\sqrt {N}}l={\sqrt {Ll}}$.

Une quantité fréquemment utilisée dans la physique des polymères est le rayon de giration, qui peut s'exprimer comme suit :

$R_{G}={\frac {{\sqrt {N}}l}{\sqrt {6}}}$.

Il est important de noter que la distance moyenne d'une extrémité à l'autre du polymère donnée ci-dessus, qui dans le cas de notre modèle simplifié représente typiquement l'amplitude des fluctuations du système, devient négligeable comparée à la longueur totale du polymère déplié à la limite thermodynamique. Ce résultat est une propriété générale des systèmes statistiques.

Remarque mathématique : la démonstration rigoureuse de l'expression de la densité de probabilité n'est pas si évidente qu'elle peut le sembler précédemment : de l'application du théorème de la limite centrale, il apparaît $R_{x},R_{y}$ et $R_{z}$ suivent une loi de distribution centrée normale de la variance.

Mathematical remark: the rigorous demonstration of the expression of the density of probability  is not as direct as it appears above: from the application of the usual (1D) central limit theorem one can deduce that ,  and  are distributed according to a centered normal distribution of variance . Then, the expression given above for  is not the only one that is compatible with such distribution for ,  and . However, since the components of the vectors  are uncorrelated for the random walk we are considering, it follows that ,  and  are also uncorrelated. This additional condition can only be fulfilled if  is distributed according to . 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[edit source]

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[edit source]

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  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[edit source]

The case of an ideal chain whose two ends are attached to fixed points will be considered in this sub-section. The vector  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  (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:

, where  is Boltzmann's constant

The above expression gives the absolute (quantum) entropy of the system. A precise determination of  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 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,  is proportional to . This leads to the following expression for the classical (relative) entropy of the ideal chain:

,

where  is a fixed constant. Let us call  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 . If the operator changes  by a tiny amount , 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:

is defined as the elementary amount of mechanical work transferred by the operator to the ideal chain, and  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  to macro-state , the system passes through a series of thermodynamic equilibriummacro-states. This has two consequences:

• first, the amount of heat received by the system during the transformation can be tied to the variation of its entropy:

, where T is the temperature of the chain.

• 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  the force exerted by the operator and  the force exerted by the chain, we have:

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  is the inverse Langevin function, N is the number of bonds in the molecule (therefore if the molecule has N bonds it has N+1 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[edit source]

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  on the ideal chain. In this case the two ends of the polymer fluctuate around a mean position . The ideal chain reacts with a constant opposite 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)^{[citation needed]}. 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  is now subject to fluctuations:

.