NationMaster - Encyclopedia: Ideal chain

An ideal chain (or freely-jointed chain) is the simplest model to describe a polymer. It only assumes a polymer as a random walk and neglects any kind of interactions among monomers. Although it is simple, its generality gives us some insights about the physics of polymers. Polymer is a generic term used to describe a very long molecule consisting of structural units and repeating units connected by covalent chemical bonds. ... In mathematics and physics, a random walk is a formalization of the intuitive idea of taking successive steps, each in a random direction. ... In chemistry, a monomer (from Greek mono one and meros part) is a small molecule that may become chemically bonded to other monomers to form a polymer. ... A Superconductor demonstrating the Meissner Effect. ...

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.

N monomers form the polymer, whose total unfolded length is:

$L = N,l$ , where N is the number of monomers.

In this very simple approach where no interactions between monomers are considered, the energy of the polymer is taken 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. In thermodynamics, a thermodynamic system is in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium. ... The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. ...

Let us call $vec R$ the total end to end vector of an ideal chain and $vec r_1,ldots ,vec r_N$ the vectors corresponding to individual monomers. Those random vectors have components in the three directions of space. Most of the expressions given in this article assume that the number of monomers N is large, so that the central limit theorem applies. The figure below shows a sketch of a (short) ideal chain. In chemistry, a monomer (from Greek mono one and meros part) is a small molecule that may become chemically bonded to other monomers to form a polymer. ... Central limit theorems are a set of weak-convergence results in probability theory. ...

The two ends of the chain are not coincident, but they fluctuate around each other, so that of course: Image File history File linksMetadata Ideal_chain_random_walk. ...

$langle vec{R} rangle = Sigma_{i=1}^N langle vec r_irangle = vec 0~$

Throughout the article the $langle rangle$ brackets will be used to notice the mean (of values taken over time) of a random variable or a random vector, like above. In statistics, mean has two related meanings: the average in ordinary English, which is more correctly called the arithmetic mean, to distinguish it from geometric mean or harmonic mean. ...

Since $vec r_1,ldots ,vec r_N$ are independent, it follows from the Central limit theorem that $vec R$ 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: Central limit theorems are a set of weak-convergence results in probability theory. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... In statistics, mean has two related meanings: the average in ordinary English, which is more correctly called the arithmetic mean, to distinguish it from geometric mean or harmonic mean. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ...

$sigma ^2 = langle R_x^2 rangle - langle R_x rangle ^2 =langle R_x^2 rangle -0$

$langle R_x^2rangle =langle R_y^2 rangle = langle R_z^2rangle = N,frac{l^2}{3}$

So that $langle vec{R^2} rangle = N, l^2 = L , l~$ . The end to end vector of the chain is distributed according to the following probability density function: In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

$P(vec R ) = left ( frac{3}{2 Pi N l^2} right )^{3/2}e^{-frac{3 vec R^2}{2 N l^2}}$

The average end-to-end distance of the polymer is:

$sqrt{langle vec{R^2} rangle} = sqrt N , l = sqrt{L , l}~$

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 in front of the total unfolded length of the polymer $N,l$ at the thermodynamic limit. This result is a general property of statistical systems. In physics, the thermodynamic limit is the statistical mechanical limit described by a system in which the number of particles approaches infinity. ...

mathematical remark: the rigorous demonstration of the expression of the density of probability $P(vec R)$ 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 $N, l^2/3$ . Then, the expression given above for $P(vec R)$ 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 $vec r_1,ldots ,vec r_N$ are uncorrelated for the random walk we are considering, it follows that $R x$ , $R y$ and $R z$ are also uncorrelated. This additional condition can only be fulfilled if $vec R$ is distributed according to $P(vec R)$ . Alternatively, this result can also be demonstrated by applying a multidimensional generalization of the central limit theorem, or through symmetry arguments. Central limit theorems are a set of weak-convergence results in probability theory. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... In probability theory and statistics, correlation, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random variables. ... In probability theory and statistics, correlation, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random variables. ... Central limit theorems are a set of weak-convergence results in probability theory. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

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 solvant (in that case, the interactions between monomer and monomer, solvant molecule and solvant molecule, and between monomer and solvant are identical, and the system's energy can be considered constant, validating the hypothesis 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). Steric effects are the interaction of molecules dictated by their shape and/or spatial relationships. ...

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 segment 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. The worm-like chain (WLC) model in polymer physics is used to describe the behavior of semi-flexible polymers; it is sometimes referred to as the Kratky-Porod worm-like chain model. ... In physics, the thermodynamic limit is the statistical mechanical limit described by a system in which the number of particles approaches infinity. ...

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 a constant, which means that this force necessarly stems from a purely entropic effect. The internal energy of a system (abbreviated E or U) is the total kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the total potential energy associated with the vibrational and electric energy of atoms within molecules or crystals. ... Ice melting - a classic example of entropy increasing In thermodynamics, thermodynamic entropy (or simply entropy) S is an important state function of a thermodynamic system: that is, a property depending only on the current state of the system, independent of how that state came to be achieved. ...

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 that it is not an energy effect that tends to increase the volume of the box like gas pressure does. This means the pressure of an ideal gas has a purely entropic origin. An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of negligible volume, with no intermolecular forces. ... The internal energy of a system (abbreviated E or U) is the total kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the total potential energy associated with the vibrational and electric energy of atoms within molecules or crystals. ... An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of negligible volume, with no intermolecular forces. ... Pressure (symbol: p) is the force per unit area acting on a surface in a direction perpendicular to that surface. ... Pressure (symbol: p) is the force per unit area acting on a surface in a direction perpendicular to that surface. ... Ice melting - a classic example of entropy increasing In thermodynamics, thermodynamic entropy (or simply entropy) S is an important state function of a thermodynamic system: that is, a property depending only on the current state of the system, independent of how that state came to be achieved. ...

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 word, thermal fluctuations tend to bring a system toward its macroscopic state of maximum entropy. Ice melting - a classic example of entropy increasing In thermodynamics, thermodynamic entropy (or simply entropy) S is an important state function of a thermodynamic system: that is, a property depending only on the current state of the system, independent of how that state came to be achieved. ... Ice melting - a classic example of entropy increasing In thermodynamics, thermodynamic entropy (or simply entropy) S is an important state function of a thermodynamic system: that is, a property depending only on the current state of the system, independent of how that state came to be achieved. ...

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 $vec r_i$ of each individual monomer (with i varying from 1 to N). In its solvant, the ideal chain is constantly subject to shocks with fluctuating solvant molecules, and each of these shocks sends the system from its current microscopic state to another, very close microscopic state. Now, for the ideal polymer, and as will be shown below, there are more microscopic states compatible with a short end-to-end distance than there are of 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. Ice melting - a classic example of entropy increasing In thermodynamics, thermodynamic entropy (or simply entropy) S is an important state function of a thermodynamic system: that is, a property depending only on the current state of the system, independent of how that state came to be achieved. ...

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. In statistics, mean has two related meanings: the average in ordinary English, which is more correctly called the arithmetic mean, to distinguish it from geometric mean or harmonic mean. ... In physics, the thermodynamic limit is the statistical mechanical limit described by a system in which the number of particles approaches infinity. ...

Ideal chain under length constraint

The case of an ideal whose two ends are attached to fixed points will be considered in

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Entropic elasticity of an ideal chain

Ideal chain under length constraint

Generality of the model