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Flory-Huggins theory

2012-08-29T10:10:58Z

Juan:

The '''Flory-Huggins theory''' (although chronologically speaking it should be known as the Huggins-Flory theory <ref>Paul J. Flory in [http://garfield.library.upenn.edu/classics1985/A1985AFW2600001.pdf Citation Classic '''18''' p. 18 May 6 (1985)]</ref>) for [[solutions]] of [[polymers]] was developed by [[Maurice L. Huggins ]] <ref>[http://dx.doi.org/10.1063/1.1750930 Maurice L. Huggins "Solutions of Long Chain Compounds", Journal of Chemical Physics '''9''' p. 440 (1941)]</ref> and [[Paul J. Flory]] <ref>[http://dx.doi.org/10.1063/1.1750971 Paul J. Flory "Thermodynamics of High Polymer Solutions", Journal of Chemical Physics '''9''' pp. 660-661 (1941)]</ref><ref>[http://dx.doi.org/10.1063/1.1723621 Paul J. Flory "Thermodynamics of High Polymer Solutions", Journal of Chemical Physics '''10''' pp. 51-61 (1942)]</ref>, following the work by Kurt H. Meyer <ref>[http://dx.doi.org/10.1002/hlca.194002301130 Kurt H. Meyer "Propriétés de polymères en solution XVI. Interprétation statistique des propriétés thermodynamiques de systèmes binaires liquides", Helvetica Chimica Acta '''23''' pp. pp. 1063-1070 (1940)]</ref>. The description can be easily generalized to the case of polymer mixtures. The Flory-Huggins theory defines the volume of a [[polymers |polymer system]] as a lattice which is divided into microscopic subspaces (called sites) of the same volume. In the case of polymer solutions, the solvent molecules are assumed to occupy single sites, while a polymer chain of a given type, <math>i</math>, occupies <math>n_i</math> sites. The repulsive forces in the system are modelled by requiring each lattice site to be occupied by only a single segment. Attractive interactions between non-bonded sites are included at the lattice neighbour level. Assuming random and ideal mixing, i.e. mixing volume <math>\Delta V_m = 0</math>, it is possible to obtain the well-known expression for the combinatorial [[entropy]] of mixing <math>\Delta S_m</math> per site of the Flory-Huggins theory for the general case of a mixture of two components, A (polymer or solvent) and polymer B (Eq. 10.1 of Ref. 3):

:<math>\Delta S_m = -R \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B \right]</math>

where <math>\phi_i</math> is the [[volume fraction]] of the component <math>i</math>, and <math>R</math> is the [[molar gas constant]]. Applying the concept of regular solutions and assuming all pair interactions in the framework of a mean-field theory yields for the [[enthalpy]] of mixing (Eq. 20 of Chapter XII in <ref>Paul J. Flory "Principles of Polymer Chemistry" Cornell University Press (1953) ISBN 0801401348</ref>):

:<math>\Delta H_m = RT \chi \phi_A \phi_B</math>

where <math>T</math> is the absolute [[temperature]].
According to the preceding equations, the Flory-Huggins equation can be expressed in terms of the [[Gibbs energy function]] of mixing for a binary system <ref>[http://dx.doi.org/10.1063/1.1724018 Robert L. Scott and Michael Magat "The Thermodynamics of High‐Polymer Solutions: I. The Free Energy of Mixing of Solvents and Polymers of Heterogeneous Distribution", Journal of Chemical Physics '''13''' pp. 172-177 (1945)]</ref>:

:<math>\Delta G_m = RT \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B + \chi \phi_A \phi_B\right]</math>

where <math>\chi</math> is the dimensionless Flory-Huggins binary interaction parameter (similar to the [[Johannis Jacobus van Laar |van Laar]] [[heat of mixing]] <ref>[http://www.dwc.knaw.nl/DL/publications/PU00013947.pdf J. J. van Laar "On the latent heat of mixing for associating solvents", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''7''' pp. 174-177 (1905)]</ref>), which can be expressed as (Eq. 21 of Chapter XII):

:<math>\chi = \frac{z\Delta w_{AB}}{RT}</math>

where <math>z</math> is the coordination number and (Eq. 17 of Chapter XII)

:<math>\Delta w_{AB} = w_{AB} - \frac{(w_{AA}+w_{BB})}{2} </math>

where <math>w_{AB}</math> is the net energy associated with two neighbouring lattice sites of the different
polymer segments for the same type or for the different types of polymer chains. Although the theory considers <math>\chi</math> as a fixed parameter, experimental data reveal that actually <math>\chi</math> depends on such quantities as temperature, concentration, [[pressure]], molar mass, molar mass distribution. From a theoretical point of view it may also depend on model parameters as the coordination number of the lattice and segment length.

For polymers of high molecular weight (i.e. <math>n_i \rightarrow \infty</math>) the entropic contribution is very small and the miscibility or immiscibility of the system mainly depends on the value of the enthalpy of mixing. In this case, miscibility can only be achieved when <math>\chi</math> is negative. Assuming a <math>\chi</math> temperature-dependent parameter, T vs. <math>\phi_i</math> phase-separation diagrams can be constructed. For long polymers, miscibility can only be achieved when <math>\chi < \chi_{cr}</math>. The <math>\chi</math> parameter at the [[critical points |critical point]] <math>\chi_{cr}</math> can be obtained from the definition of the critical point and the Flory-Huggins expression for the free-energy of mixing. The result is

:<math>\chi_{cr} = \frac{1}{2} \left[ \frac{1}{\sqrt{n_A}} + \frac{1}{\sqrt{n_B}} \right]^2</math>

Therefore:
*Positive values of <math>\chi</math> necessarily lead to immiscibility for polymer mixtures of high molecular weight.
*Polymer mixing always take place if the <math>\chi</math> parameter is negative. Miscible polymer mixtures with negative <math>\chi</math> exist due to specific interactions between given polymer segments. Miscibility or compatibility can be induced by several methods. For instance, introducing opposite charges in the different polymers or adding a copolymer containing A and B segments.
*For polymer solutions (whose sites have the volume of a solvent molecule, <math>n_A</math>=1), the critical Flory-Huggins parameter is close to <math>1/2</math>. The temperature corresponding to this value <math>\chi</math>=<math>1/2</math> would be the critical temperature if the polymer is infinitely long and defines the [[theta solvent | theta temperature]] of the polymer-solvent system. Good solvent systems show significantly smaller positive values of <math>\chi</math>.
*For polymer mixtures, <math>\chi</math> should be referred to the arbitrarily chosen microscopic volume defined as a site, e.g. 100 Angstroms. <math>\chi</math> values and can be positive or negative and they are usually very small <ref>[ N. P. Balsara "Thermodynamics of Polymer Blends", in J. E. Mark, editor, “Physical Properties of Polymers Handbook” AIP Press, pp. 257-268, (1996) ISBN 1563962950] </ref>

The <math>\chi</math> parameter is somewhat similar to a [[second virial coefficient]] expressing binary interactions between molecules and, therefore, it usually shows a linear dependence of <math>1/T</math>

:<math>\chi(T) = a + \frac{b}{T}</math>

where <math>a</math> and <math>b</math> are assumed to be constants, but can actually depend on density,
concentration, molecular weight, etc. A usual interpretation is that <math>a</math> represents an [[enthalpy |enthalpic]] quantity and <math>b</math> an [[entropy | entropic]] contribution, although both <math>a</math> and <math>b</math> are actually empirical parameters. According to this description, the systems should show an upper critical temperature. Many polymer mixtures and solutions, however, show an increase of <math>\chi</math> for increasing temperatures (negative entropic contribution) what implies the existence of a lower critical temperature.

==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1146/annurev.pc.02.100151.002123 P. J. Flory, and W. R. Krigbaum "Thermodynamics of High Polymer Solutions", Annual Review of Physical Chemistry '''2''' pp. 383-402 (1951)]
[[Category: Polymers]]

Flory-Huggins theory

2012-08-29T10:00:21Z

Juan:

The '''Flory-Huggins theory''' (although chronologically speaking it should be known as the Huggins-Flory theory <ref>Paul J. Flory in [http://garfield.library.upenn.edu/classics1985/A1985AFW2600001.pdf Citation Classic '''18''' p. 18 May 6 (1985)]</ref>) for [[solutions]] of [[polymers]] was developed by [[Maurice L. Huggins ]] <ref>[http://dx.doi.org/10.1063/1.1750930 Maurice L. Huggins "Solutions of Long Chain Compounds", Journal of Chemical Physics '''9''' p. 440 (1941)]</ref> and [[Paul J. Flory]] <ref>[http://dx.doi.org/10.1063/1.1750971 Paul J. Flory "Thermodynamics of High Polymer Solutions", Journal of Chemical Physics '''9''' pp. 660-661 (1941)]</ref><ref>[http://dx.doi.org/10.1063/1.1723621 Paul J. Flory "Thermodynamics of High Polymer Solutions", Journal of Chemical Physics '''10''' pp. 51-61 (1942)]</ref>, following the work by Kurt H. Meyer <ref>[http://dx.doi.org/10.1002/hlca.194002301130 Kurt H. Meyer "Propriétés de polymères en solution XVI. Interprétation statistique des propriétés thermodynamiques de systèmes binaires liquides", Helvetica Chimica Acta '''23''' pp. pp. 1063-1070 (1940)]</ref>. The description can be easily generalized to the case of polymer mixtures. The Flory-Huggins theory defines the volume of a [[polymers |polymer system]] as a lattice which is divided into microscopic subspaces (called sites) of the same volume. In the case of polymer solutions, the solvent molecules are assumed to occupy single sites, while a polymer chain of a given type, <math>i</math>, occupies <math>n_i</math> sites. The repulsive forces in the system are modelled by requiring each lattice site to be occupied by only a single segment. Attractive interactions between non-bonded sites are included at the lattice neighbour level. Assuming random and ideal mixing, i.e. mixing volume <math>\Delta V_m = 0</math>, it is possible to obtain the well-known expression for the combinatorial [[entropy]] of mixing <math>\Delta S_m</math> per site of the Flory-Huggins theory for the general case of a mixture of two components, A (polymer or solvent) and polymer B (Eq. 10.1 of Ref. 3):

:<math>\Delta S_m = -R \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B \right]</math>

where <math>\phi_i</math> is the [[volume fraction]] of the component <math>i</math>, and <math>R</math> is the [[molar gas constant]]. Applying the concept of regular solutions and assuming all pair interactions in the framework of a mean-field theory yields for the [[enthalpy]] of mixing (Eq. 20 of Chapter XII in <ref>Paul J. Flory "Principles of Polymer Chemistry" Cornell University Press (1953) ISBN 0801401348</ref>):

:<math>\Delta H_m = RT \chi \phi_A \phi_B</math>

where <math>T</math> is the absolute [[temperature]].
According to the preceding equations, the Flory-Huggins equation can be expressed in terms of the [[Gibbs energy function]] of mixing for a binary system <ref>[http://dx.doi.org/10.1063/1.1724018 Robert L. Scott and Michael Magat "The Thermodynamics of High‐Polymer Solutions: I. The Free Energy of Mixing of Solvents and Polymers of Heterogeneous Distribution", Journal of Chemical Physics '''13''' pp. 172-177 (1945)]</ref>:

:<math>\Delta G_m = RT \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B + \chi \phi_A \phi_B\right]</math>

where <math>\chi</math> is the dimensionless Flory-Huggins binary interaction parameter (similar to the [[Johannis Jacobus van Laar |van Laar]] [[heat of mixing]] <ref>[http://www.dwc.knaw.nl/DL/publications/PU00013947.pdf J. J. van Laar "On the latent heat of mixing for associating solvents", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''7''' pp. 174-177 (1905)]</ref>), which can be expressed as (Eq. 21 of Chapter XII):

:<math>\chi = \frac{z\Delta w_{AB}}{RT}</math>

where <math>z</math> is the coordination number and (Eq. 17 of Chapter XII)

:<math>\Delta w_{AB} = w_{AB} - \frac{(w_{AA}+w_{BB})}{2} </math>

where <math>w_{AB}</math> is the net energy associated with two neighbouring lattice sites of the different
polymer segments for the same type or for the different types of polymer chains. Although the theory considers <math>\chi</math> as a fixed parameter, experimental data reveal that actually <math>\chi</math> depends on such quantities as temperature, concentration, [[pressure]], molar mass, molar mass distribution. From a theoretical point of view it may also depend on model parameters as the coordination number of the lattice and segment length.

For polymers of high molecular weight (i.e. <math>n_i \rightarrow \infty</math>) the entropic contribution is very small and the miscibility or immiscibility of the system mainly depends on the value of the enthalpy of mixing. In this case, miscibility can only be achieved when <math>\chi</math> is negative. Assuming a <math>\chi</math> temperature-dependent parameter, T vs. <math>\phi_i</math> phase-separation diagrams can be constructed. For long polymers, miscibility can only be achieved when <math>\chi < \chi_{cr}</math>. The <math>\chi</math> parameter at the [[critical points |critical point]] <math>\chi_{cr}</math> can be obtained from the definition of the critical point and the Flory-Huggins expression for the free-energy of mixing. The result is

:<math>\chi_{cr} = \frac{1}{2} \left[ \frac{1}{\sqrt{n_A}} + \frac{1}{\sqrt{n_B}} \right]^2</math>

Therefore:
*Positive values of <math>\chi</math> necessarily lead to immiscibility for polymer mixtures of high molecular weight.
*Polymer mixing always take place if the <math>\chi</math> parameter is negative. Miscible polymer mixtures with negative <math>\chi</math> exist due to specific interactions between given polymer segments. Miscibility or compatibility can be induced by several methods. For instance, introducing opposite charges in the different polymers or adding a copolymer containing A and B segments.
*For polymer solutions (whose sites have the volume of a solvent molecule, <math>n_A</math>=1), the critical Flory-Huggins parameter is close to <math>1/2</math>. The temperature corresponding to this value <math>\chi</math>=<math>1/2</math> would be the critical temperature if the polymer is infinitely long and defines the [[theta solvent | theta temperature]] of the polymer-solvent system.
*For polymer mixtures, <math>\chi</math> should be referred to the arbitrarily chosen microscopic volume defined as a site, e.g. 100 Angstroms. Usual <math>\chi</math> values are very small <ref>[ N. P. Balsara "Thermodynamics of Polymer Blends", in J. E. Mark, editor, “Physical Properties of Polymers Handbook” AIP Press, pp. 257-268, (1996) ISBN 1563962950] </ref>

The <math>\chi</math> parameter is somewhat similar to a [[second virial coefficient]] expressing binary interactions between molecules and, therefore, it usually shows a linear dependence of <math>1/T</math>

:<math>\chi(T) = a + \frac{b}{T}</math>

where <math>a</math> and <math>b</math> are assumed to be constants, but can actually depend on density,
concentration, molecular weight, etc. A usual interpretation is that <math>a</math> represents an [[enthalpy |enthalpic]] quantity and <math>b</math> an [[entropy | entropic]] contribution, although both <math>a</math> and <math>b</math> are actually empirical parameters. According to this description, the systems should show an upper critical temperature. Many polymer mixtures and solutions, however, show an increase of <math>\chi</math> for increasing temperatures (negative entropic contribution) what implies the existence of a lower critical temperature.

==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1146/annurev.pc.02.100151.002123 P. J. Flory, and W. R. Krigbaum "Thermodynamics of High Polymer Solutions", Annual Review of Physical Chemistry '''2''' pp. 383-402 (1951)]
[[Category: Polymers]]

Flory-Huggins theory

2012-08-29T09:57:59Z

Juan:

The '''Flory-Huggins theory''' (although chronologically speaking it should be known as the Huggins-Flory theory <ref>Paul J. Flory in [http://garfield.library.upenn.edu/classics1985/A1985AFW2600001.pdf Citation Classic '''18''' p. 18 May 6 (1985)]</ref>) for [[solutions]] of [[polymers]] was developed by [[Maurice L. Huggins ]] <ref>[http://dx.doi.org/10.1063/1.1750930 Maurice L. Huggins "Solutions of Long Chain Compounds", Journal of Chemical Physics '''9''' p. 440 (1941)]</ref> and [[Paul J. Flory]] <ref>[http://dx.doi.org/10.1063/1.1750971 Paul J. Flory "Thermodynamics of High Polymer Solutions", Journal of Chemical Physics '''9''' pp. 660-661 (1941)]</ref><ref>[http://dx.doi.org/10.1063/1.1723621 Paul J. Flory "Thermodynamics of High Polymer Solutions", Journal of Chemical Physics '''10''' pp. 51-61 (1942)]</ref>, following the work by Kurt H. Meyer <ref>[http://dx.doi.org/10.1002/hlca.194002301130 Kurt H. Meyer "Propriétés de polymères en solution XVI. Interprétation statistique des propriétés thermodynamiques de systèmes binaires liquides", Helvetica Chimica Acta '''23''' pp. pp. 1063-1070 (1940)]</ref>. The description can be easily generalized to the case of polymer mixtures. The Flory-Huggins theory defines the volume of a [[polymers |polymer system]] as a lattice which is divided into microscopic subspaces (called sites) of the same volume. In the case of polymer solutions, the solvent molecules are assumed to occupy single sites, while a polymer chain of a given type, <math>i</math>, occupies <math>n_i</math> sites. The repulsive forces in the system are modelled by requiring each lattice site to be occupied by only a single segment. Attractive interactions between non-bonded sites are included at the lattice neighbour level. Assuming random and ideal mixing, i.e. mixing volume <math>\Delta V_m = 0</math>, it is possible to obtain the well-known expression for the combinatorial [[entropy]] of mixing <math>\Delta S_m</math> per site of the Flory-Huggins theory for the general case of a mixture of two components, A (polymer or solvent) and polymer B (Eq. 10.1 of Ref. 3):

:<math>\Delta S_m = -R \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B \right]</math>

where <math>\phi_i</math> is the [[volume fraction]] of the component <math>i</math>, and <math>R</math> is the [[molar gas constant]]. Applying the concept of regular solutions and assuming all pair interactions in the framework of a mean-field theory yields for the [[enthalpy]] of mixing (Eq. 20 of Chapter XII in <ref>Paul J. Flory "Principles of Polymer Chemistry" Cornell University Press (1953) ISBN 0801401348</ref>):

:<math>\Delta H_m = RT \chi \phi_A \phi_B</math>

where <math>T</math> is the absolute [[temperature]].
According to the preceding equations, the Flory-Huggins equation can be expressed in terms of the [[Gibbs energy function]] of mixing for a binary system <ref>[http://dx.doi.org/10.1063/1.1724018 Robert L. Scott and Michael Magat "The Thermodynamics of High‐Polymer Solutions: I. The Free Energy of Mixing of Solvents and Polymers of Heterogeneous Distribution", Journal of Chemical Physics '''13''' pp. 172-177 (1945)]</ref>:

:<math>\Delta G_m = RT \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B + \chi \phi_A \phi_B\right]</math>

where <math>\chi</math> is the dimensionless Flory-Huggins binary interaction parameter (similar to the [[Johannis Jacobus van Laar |van Laar]] [[heat of mixing]] <ref>[http://www.dwc.knaw.nl/DL/publications/PU00013947.pdf J. J. van Laar "On the latent heat of mixing for associating solvents", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''7''' pp. 174-177 (1905)]</ref>), which can be expressed as (Eq. 21 of Chapter XII):

:<math>\chi = \frac{z\Delta w_{AB}}{RT}</math>

where <math>z</math> is the coordination number and (Eq. 17 of Chapter XII)

:<math>\Delta w_{AB} = w_{AB} - \frac{(w_{AA}+w_{BB})}{2} </math>

where <math>w_{AB}</math> is the net energy associated with two neighbouring lattice sites of the different
polymer segments for the same type or for the different types of polymer chains. Although the theory considers <math>\chi</math> as a fixed parameter, experimental data reveal that actually <math>\chi</math> depends on such quantities as temperature, concentration, [[pressure]], molar mass, molar mass distribution. From a theoretical point of view it may also depend on model parameters as the coordination number of the lattice and segment length.

For polymers of high molecular weight (i.e. <math>n_i \rightarrow \infty</math>) the entropic contribution is very small and the miscibility or immiscibility of the system mainly depends on the value of the enthalpy of mixing. In this case, miscibility can only be achieved when <math>\chi</math> is negative. Assuming a <math>\chi</math> temperature-dependent parameter, T vs. <math>\phi_i</math> phase-separation diagrams can be constructed. For long polymers, miscibility can only be achieved when <math>\chi < \chi_{cr}</math>. The <math>\chi</math> parameter at the [[critical points |critical point]] <math>\chi_{cr}</math> can be obtained from the definition of the critical point and the Flory-Huggins expression for the free-energy of mixing. The result is

:<math>\chi_{cr} = \frac{1}{2} \left[ \frac{1}{\sqrt{n_A}} + \frac{1}{\sqrt{n_B}} \right]^2</math>

Therefore:
*Positive values of <math>\chi</math> necessarily lead to immiscibility for polymer mixtures of high molecular weight.
*Polymer mixing always take place if the <math>\chi</math> parameter is negative. Miscible polymer mixtures with negative <math>\chi</math> exist due to specific interactions between given polymer segments. Miscibility or compatibility can be induced by several methods. For instance, introducing opposite charges in the different polymers or adding a copolymer containing A and B segments.
*For polymer solutions (whose sites have the volume of a solvent molecule, <math>n_A</math>=1), the critical Flory-Huggins parameter is close to <math>1/2</math>. The temperature corresponding to this value <math>\chi</math>=<math>1/2</math> would be the critical temperature if the polymer is infinitely long and defines the [[theta solvent | theta temperature]] of the polymer-solvent system.
*For polymer mixtures, <math>\chi</math> should be referred to the arbitrarily chosen microscopic volume defined as a site, e.g. 100 Angstroms. Usual values are close to zero <ref>[ N. P. Balsara "Thermodynamics of Polymer Blends", in J. E. Mark, editor, “Physical Properties of Polymers Handbook” AIP Press, pp. 257-268, (1996) ISBN 1563962950] </ref>

The <math>\chi</math> parameter is somewhat similar to a [[second virial coefficient]] expressing binary interactions between molecules and, therefore, it usually shows a linear dependence of <math>1/T</math>

:<math>\chi(T) = a + \frac{b}{T}</math>

where <math>a</math> and <math>b</math> are assumed to be constants, but can actually depend on density,
concentration, molecular weight, etc. A usual interpretation is that <math>a</math> represents an [[enthalpy |enthalpic]] quantity and <math>b</math> an [[entropy | entropic]] contribution, although both <math>a</math> and <math>b</math> are actually empirical parameters. According to this description, the systems should show an upper critical temperature. Many polymer mixtures and solutions, however, show an increase of <math>\chi</math> for increasing temperatures (negative entropic contribution) what implies the existence of a lower critical temperature.

==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1146/annurev.pc.02.100151.002123 P. J. Flory, and W. R. Krigbaum "Thermodynamics of High Polymer Solutions", Annual Review of Physical Chemistry '''2''' pp. 383-402 (1951)]
[[Category: Polymers]]

Configurational bias Monte Carlo

2007-06-13T16:16:42Z

Juan:

It is usual that many of the accessible configurations have a small probability and only a few ones are probable. In these cases, the simulation is more efficient if the probabilities of the different configurations are previously considered. With this end, the new position for a unit is randomly chosen between a discrete number of possibilities (the neighboring sites in lattice models or a randomly chosen set of positions in other cases), taking into account their Boltzmann probabilities. In the case of polymers, an entirely new part of a chain up to an end can be generated following a path of easily accessible positions. This introduces a bias which should be compensated by considering a weight factor for each new position chosen (or a product of these factors for a new chain). A similar weight corresponding to reconstructing the old configuration from the new one has also to be calculated. The probability ratios are corrected by introducing the ratio between the new and the old configurational weight factors.

==References==

1. I. Siepman and D. Frenkel, Mol. Phys. 75, 59 (1992).

Configurational bias Monte Carlo

2007-06-13T16:14:16Z

Juan: New page: It is usual that many of the accessible configurations have a small probability and only a few ones are probable. In these cases, the simulation is more efficient if the probabilities of t...

Gibbs ensemble Monte Carlo

2007-05-31T10:33:35Z

Juan:

Phase separation is one of the topics for which simulation techniques have preferentially been focused in the recent past. Different procedures have been used for this purpose. Thus, for the particular case of chain systems, we can employ simulations in the [[Semi-grand ensembles | semi-grand canonical ensemble]], [[histogram re-weighting]], or characterization of the [[spinodal curve]] from the study of computed [[collective scattering function]].
The '''Gibbs ensemble Monte Carlo''' method has been specifically designed to characterize [[phase transitions]]. It was mainly developed by Panagiotopoulos (Ref. 1) to avoid the problem of finite size interfacial effects. In this method, an ''NVT'' (or ''NpT'')ensemble containing two (or more) species is divided into two (or more) boxes. In addition to the usual particle moves in each one of the boxes, the algorithm includes moves steps to change the volume and composition of the boxes at mechanical and chemical equilibrium. Transferring a chain molecule from a box to the other requires the use of an efficient method to insert chains. The [[Configurational bias Monte Carlo | configurational bias method]], is specially recommended for this purpose.
==References==
#[http://dx.doi.org/10.1080/00268978700101491 Athanassios Panagiotopoulos "Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble", Molecular Physics '''61''' pp. 813-826 (1987)]
[[category: Computer simulation techniques]]

Lattice simulations (Polymers)

2007-05-31T10:29:35Z

Juan:

Polymers have many interesting mesoscopic properties that can adequately represented through [[Coarse graining | coarse-grained]] models. [[Models | Lattice models]] are particularly useful since they employ integer coordinates that can be quickly processed. Another advantage is the possibility of checking the occupancy of the discrete number of sites and store it in an array. Therefore, overlapping between beads can be easily avoided.
Earlier simulations were performed with the simple cubic or tetrahedrical lattices, employing different algorithms. Tetrahedrical lattice models are mainly used because of their closer similarity of their bond angles with those of aliphatic chains.
Algorithms useful for lattice models may have special difficulties to comply with the ergodicity and microscopic reversibility conditions. Elementary bead jumps are commonly used to try to mimic dynamics. A combination of bents and crankshaft was shown to be nearly ergodic and reproduce adequately the Rouse dynamics of single chain systems in a cubic lattice [1]. More efficient Pivot moves have been devised to explore the equilibrium properties of very long single chains. [2] Although the lattice simulations can effectively represent melts, most algorithms employ a certain amount of voids to allow specific local motions. An exception is the cooperative motion algorithm that considers totally occupied systems [3]. Also, specific algorithms to perform NPT simulations have been designed. [4]
The bond fluctuation model has been proposed to combine the advantages of lattice and off-lattice models.
==References==
[1] P. H. Verdier, W. H. Stockmayer, J. Chem. Phys. 36, 227 (1962).
[2] N. Madras and A. D. Sokal, J. Stat. Phys. 50, 109 (1988).
[3] T. Pakula and S. Geyler, Macromolecules 20, 2909 (1987).
[4] A. D. Mackie, A. Z. Panagiotopoulos, S. K. Kumar, J. Chem. Phys. 102, 1014 (1995).

Lattice simulations (Polymers)

2007-05-31T10:29:16Z

Juan:

Polymers have many interesting mesoscopic properties that can adequately represented through [[Coarse graining | coarse-grained]] models. [[Models | Lattice models]] are particularly useful since they employ integer coordinates that can be quickly processed. Another advantage is the possibility of checking the occupancy of the discrete number of sites and store it in an array. Therefore, overlapping between beads can be easily avoided.
Earlier simulations were performed with the simple cubic or tetrahedrical lattices, employing different algorithms. Tetrahedrical lattice models are mainly used because of their closer similarity of their bond angles with those of the aliphatic chains.
Algorithms useful for lattice models may have special difficulties to comply with the ergodicity and microscopic reversibility conditions. Elementary bead jumps are commonly used to try to mimic dynamics. A combination of bents and crankshaft was shown to be nearly ergodic and reproduce adequately the Rouse dynamics of single chain systems in a cubic lattice [1]. More efficient Pivot moves have been devised to explore the equilibrium properties of very long single chains. [2] Although the lattice simulations can effectively represent melts, most algorithms employ a certain amount of voids to allow specific local motions. An exception is the cooperative motion algorithm that considers totally occupied systems [3]. Also, specific algorithms to perform NPT simulations have been designed. [4]
The bond fluctuation model has been proposed to combine the advantages of lattice and off-lattice models.
==References==
[1] P. H. Verdier, W. H. Stockmayer, J. Chem. Phys. 36, 227 (1962).
[2] N. Madras and A. D. Sokal, J. Stat. Phys. 50, 109 (1988).
[3] T. Pakula and S. Geyler, Macromolecules 20, 2909 (1987).
[4] A. D. Mackie, A. Z. Panagiotopoulos, S. K. Kumar, J. Chem. Phys. 102, 1014 (1995).

Lattice simulations (Polymers)

2007-05-31T10:29:01Z

Juan:

Polymers have many interesting mesoscopic properties that can adequately represented through [[Coarse graining | coarse-grained]] models. [[Models | Lattice models]] are particularly useful since they employ integer coordinates that can be quickly processed. Another advantage is the possibility of checking the occupancy of the discrete number of sites and store it in an array. Therefore, overlapping between beads can be easily avoided.
Earlier simulations were performed with the simple cubic or tetrahedrical lattices, employing different algorithms. Tetrahedrical lattice models are mainly used because of their closer similarity of their bond angles with those of the aliphatic chains.
Algorithms useful for lattice models may have special difficulties to comply with the ergodicity and microscopic reversibility conditions. Elementary bead jumps are commonly used to try to mimic dynamics. A combination of bents and crankshaft was shown to be nearly ergodic and reproduce adequately the Rouse dynamics of single chain systems in a cubic lattice [1]. More efficient Pivot moves have been devised to explore the equilibrium properties of very long single chains. [2] Although the lattice simulations can effectively represent melts, most algorithms employ a certain amount of voids to allow specific local motions. An exception is the cooperative motion algorithm that considers totally occupied systems [3]. Also, specific algorithms to perform NPT simulations have been designed. [4]
The bond fluctuation model has been proposed to combine the advantages of lattice and off-lattice models.
==References==
[1] P. H. Verdier, W. H. Stockmayer, J. Chem. Phys. 36, 227 (1962).
[2] N. Madras and A. D. Sokal, J. Stat. Phys. 50, 109 (1988).
[3] T. Pakula and S. Geyler, Macromolecules 20, 2909 (1987).
[4] A. D. Mackie, A. Z. Panagiotopoulos, S. K. Kumar, J. Chem. Phys. 102, 1014 (1995).

Lattice simulations (Polymers)

2007-05-31T10:28:36Z

Juan:

Polymers have many interesting mesoscopic properties that can adequately represented through [[Coarse graining | coarse-grained]] models. [[Models | Lattice models]] are particularly useful since they employ integer coordinates that can be quickly processed. Another advantage is the possibility of checking the occupancy of the discrete number of sites and store it in an array. Therefore, overlapping between beads can be easily avoided.
Earlier simulations were performed with the simple cubic or tetrahedrical lattices, employing different algorithms. Tetrahedrical lattice models are mainly used because of their closer similarity of their bond angles with those of the aliphatic chains.
Algorithms useful for lattice models may have special difficulties to comply with the ergodicity and microscopic reversibility conditions. Elementary bead jumps are commonly used to try to mimic dynamics. A combination of bents and crankshaft was shown to be nearly ergodic and reproduce adequately the Rouse dynamics of single chain systems in a cubic lattice [1]. More efficient Pivot moves have been devised to explore the equilibrium properties of very long single chains. [2] Although the lattice simulations can effectively represent melts, most algorithms employ a certain amount of voids to allow specific local motions. An exception is the cooperative motion algorithm that considers totally occupied systems [3]. Also, specific algorithms to perform NPT simulations have been designed. [4]
The bond fluctuation model has been proposed to combine the advantages of lattice and off-lattice models.
==References==
[1] P. H. Verdier, W. H. Stockmayer, J. Chem. Phys. 36, 227 (1962).
[2] N. Madras and A. D. Sokal, J. Stat. Phys. 50, 109 (1988).
[3] T. Pakula and S. Geyler, Macromolecules 20, 2909 (1987).
[4] A. D. Mackie, A. Z. Panagiotopoulos, S. K. Kumar, J. Chem. Phys. 102, 1014 (1995).

Lattice simulations (Polymers)

2007-05-31T10:28:13Z

Juan:

Polymers have many interesting mesoscopic properties that can adequately represented through [[Coarse graining | coarse-grained]] models. [[Models | Lattice models]] are particularly useful since they employ integer coordinates that can be quickly processed. Another advantage is the possibility of checking the occupancy of the discrete number of sites and store it in an array. Therefore, overlapping between beads can be easily avoided.
Earlier simulations were performed with the simple cubic or tetrahedrical lattices, employing different algorithms. Tetrahedrical lattice models are mainly used because of their closer similarity of their bond angles with those of the aliphatic chains.
Algorithms useful for lattice models may have special difficulties to comply with the ergodicity and microscopic reversibility conditions. Elementary bead jumps are commonly used to try to mimic dynamics. A combination of bents and crankshaft was shown to be nearly ergodic and reproduce adequately the Rouse dynamics of single chain systems in a cubic lattice [1]. More efficient Pivot moves have been devised to explore the equilibrium properties of very long single chains. [2] Although the lattice simulations can effectively represent melts, most algorithms employ a certain amount of voids to allow specific local motions. An exception is the cooperative motion algorithm that considers totally occupied systems [3]. Also, specific algorithms to perform NPT simulations have been designed. [4]
The bond fluctuation model has been proposed to combine the advantages of lattice and off-lattice models.
==References==
[1] P. H. Verdier, W. H. Stockmayer, J. Chem. Phys. 36, 227 (1962).
[2] N. Madras and A. D. Sokal, J. Stat. Phys. 50, 109 (1988).
[3] T. Pakula and S. Geyler, Macromolecules 20, 2909 (1987).
[4] A. D. Mackie, A. Z. Panagiotopoulos, S. K. Kumar, J. Chem. Phys. 102, 1014 (1995).

Lattice simulations (Polymers)

2007-05-31T10:27:29Z

Juan:

Polymers have many interesting mesoscopic properties that can adequately represented through [[Coarse graining | coarse-grained]] models. [[Models | Lattice models]] are particularly useful since they employ integer coordinates that can be quickly processed. Another advantage is the possibility of checking the occupancy of the discrete number of sites and store it in an array. Therefore, overlapping between beads can be easily avoided. 
Earlier simulations were performed with the simple cubic or tetrahedrical lattices, employing different algorithms. Tetrahedrical lattice models are mainly used because of their closer similarity of their bond angles with those of the aliphatic chains. 
Algorithms useful for lattice models may have special difficulties to comply with the ergodicity and microscopic reversibility conditions. Elementary bead jumps are commonly used to try to mimic dynamics. A combination of bents and crankshaft was shown to be nearly ergodic and reproduce adequately the Rouse dynamics of single chain systems in a cubic lattice [1]. More efficient Pivot moves have been devised to explore the equilibrium properties of very long single chains. [2] Although the lattice simulations can effectively represent melts, most algorithms employ a certain amount of voids to allow specific local motions. An exception is the cooperative motion algorithm that considers totally occupied systems [3]. Also, specific algorithms to perform NPT simulations have been designed. [4]
The bond fluctuation model has been proposed to combine the advantages of lattice and off-lattice models.
==References==
[1] P. H. Verdier, W. H. Stockmayer, J. Chem. Phys. 36, 227 (1962).
[2] N. Madras and A. D. Sokal, J. Stat. Phys. 50, 109 (1988).
[3] T. Pakula and S. Geyler, Macromolecules 20, 2909 (1987).
[4] A. D. Mackie, A. Z. Panagiotopoulos, S. K. Kumar, J. Chem. Phys. 102, 1014 (1995).

Lattice simulations (Polymers)

2007-05-31T10:27:03Z

Juan:

Lattice simulations (Polymers)

2007-05-31T10:11:52Z

Juan:

Bond fluctuation model

2007-05-31T09:46:40Z

Juan:

The '''bond fluctuation model''' has been used in the recent past to simulate a great variety of [[Polymers |polymer]] systems (see Ref.s 1 and 2). It represents 8-site non-overlapping beads placed in a [[Building up a simple cubic lattice | simple cubic lattice]]. Bonds linking the beads may have lengths ranging between 2 and <math>\sqrt 10</math> lattice length units but bonds vectors of the type (2,2,0) are excluded to avoid bond crossing during the simulation.

[[Image:BFM 1.png|thumb|right]]

This model exhibits some advantages with respect to more conventional representations of the polymer in a lattice, such as fixed-bond self-avoiding walk chains on a simple cubic or [[tetrahedral lattice]]:
*It is somehow more consistent with the theoretical description of the chain through Gaussianly distributed statistical segments. This description has been employed in the past to obtain useful analytical expression for equilibrium properties and its dynamical version corresponds to the [[Rouse model]].
*A variety of bond lengths alleviate the restrictions due to the lattice constraints and, therefore, the model is more similar to the continuum behavior.
*It is possible to perform simulations by using a single type elementary bead jumps. A bead jump is a simple translation of the bead to one of the six contiguous lattice units along the x, y or z axis.This is a difference with respect to the simple cubic lattice model where a combination of bent and crankshaft elementary moves involving two or three beads has to be used as the simplest local level. The implication is that one can generate dynamic trajectories with the bond fluctuation model by using a simple and natural type of bead jumps, which intuitively resembles the stochastic behavior of [[Brownian particles]].
* The elementary jump moves can directly used in polymers with branch points ([[stars]], [[combs]], [[dendrimers]]).
==References==
[[Category: Models]]

#[http://www.oup.com/uk/catalogue/?ci=9780195094381 K. Binder (ed.) “Monte Carlo and Molecular Dynamics Simulations in Polymer Science”, Oxford University Press (1995)]
#[http://dx.doi.org/10.1021/ma00187a030 I. Carmesin and Kurt Kremer "The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions", Macromolecules pp. 2819 - 2823 (1988)]

Lattice simulations (Polymers)

2007-05-31T09:45:54Z

Juan: New page: Polymers have many interesting mesoscopic properties that can adequately represented through coarse-grained models. Lattice models are particularly usefu...

Bond fluctuation model

2007-05-31T09:40:30Z

Juan:

Gibbs ensemble Monte Carlo

2007-05-31T09:27:13Z

Juan: New page: Phase separation is one of the topics for which simulation techniques have preferentially been focused in the recent past. Different procedures have been used for this purpose. Thus, for t...

Phase separation is one of the topics for which simulation techniques have preferentially been focused in the recent past. Different procedures have been used for this purpose. Thus, for the particular case of chain systems, we can employ simulations in the semi-grand canonical ensemble, histogram reweighting, or characterization of the spinodal curve from the study of computed collective scattering function. 
The Gibbs ensemble Monte Carlo method has been specificly designed to characterize phase transitions. It was mainly developed by Panagiotopoulos [1] to avoid the problem of finite size interfacial effects. In this method, a NVT (or NPT)ensemble containing two (or more) species is divided into two (or) boxes. In addition to the usual particle moves in each one of the boxes, the algorithm includes moves steps to change the volume and composition of the boxes at mechanical and chemical equilibrium. Transferring a chain molecule from a box to the other requires the use of an efficient method to insert chains. The configurational bias method, is specially recommended for this purpose.

==References==
[1] A. Z. Panagiotopoulos, Mol Phys. 61, 813 (1987).

Gibbs ensemble

2007-05-31T09:24:14Z

Juan: Removing all content from page

Gibbs ensemble

2007-05-31T09:21:53Z

Juan:

Phase separation is one of the topics for which simulation techniques have preferentially been focused in the recent past. Different procedures have been used for this purpose. Thus, for the particular case of chain systems, we can employ simulations in the semi-grand canonical ensemble, histogram reweighting, or characterization of the spinodal curve from the study of computed collective scattering function. 
The Gibbs ensemble Monte Carlo method has been specificly designed to characterize phase transitions. It was mainly developed by Panagiotopoulos [1] to avoid the problem of finite size interfacial effects. In this method, a NVT (or NPT)ensemble containing two (or more) species is divided into two (or) boxes. In addition to the usual particle moves in each one of the boxes, the algorithm includes moves steps to change the volume and composition of the boxes at mechanical and chemical equilibrium. Transferring a chain molecule from a box to the other requires the use of an efficient method to insert chains. The configurational bias method, is specially recommended for this purpose.

==References==
[1] A. Z. Panagiotopoulos, Mol Phys. 61, 813 (1987).

Gibbs ensemble

2007-05-31T09:21:39Z

Juan:

Phase separation is one of the topics for which simulation techniques have preferentially been focused in the recent past. Different procedures have been used for this purpose. Thus, for the particular case of chain systems, we can employ simulations in the semi-grand canonical ensemble, histogram reweighting, or characterization of the spinodal curve from the study of computed collective scattering function. 
The Gibbs ensemble Monte Carlo method has been specificly designed to characterize phase transitions. It was mainly developed by Panagiotopoulos [1] to avoid the problem of finite size interfacial effects. In this method, a NVT (or NPT)ensemble containing two (or more) species is divided into two (or) boxes. In addition to the usual particle moves in each one of the boxes, the algorithm includes moves steps to change the volume and composition of the boxes at mechanical and chemical equilibrium. Transferring a chain molecule from a box to the other requires the use of an efficient method to insert chains. The configurational bias method, is specially recommended for this purpose.

==References==
[1]A. Z. Panagiotopoulos, Mol Phys. 61, 813 (1987).

Gibbs ensemble

2007-05-31T09:15:33Z

Juan:

Phase separation is one of the topics for which simulation techniques have preferentially been focused in the recent past. Different procedures have been used for this purpose. Thus, for the particular case of chain systems, we can perform simulations in the semi-grand canonical ensemble, histogram reweighting, or characterization of the spinodal curve from the study of computed collective scattering function. 
The Gibbs ensemble Monte Carlo method has been specificly designed to characterize phase transitions. It was mainly developed by Panagiotopoulos to avoid the problem of finite size interfacial effects. In this method, a NVT (or NPT)ensemble containing two (or more) species is divided into two (or) boxes. In addition to the usual particle moves in each one of the boxes, the algorithm includes moves steps to change the volume and composition of the boxes at mechanical and chemical equilibrium. Transferring a chain molecule from a box to the other requires the use of an efficient method to insert chains. The configurational bias method, is specially recommended for this purpose.

Gibbs ensemble

2007-05-31T09:10:46Z

Bond fluctuation model

2007-05-30T15:45:40Z

Juan:

Polymers have many interesting mesoscopic properties that can adequately represented through [[Coarse graining | coarse-grained]] models. [[Models | Lattice models]] are particularly useful since they employ integer coordinates that can be quickly processed. Another advantage is the possibility of checking the occupancy of the discrete number of sites and store it in an array. Therefore, overlapping between beads can be easily avoided.

The '''bond fluctuation model''' has been used in the recent past to simulate a great variety of [[Polymers |polymer]] systems (see Ref.s 1 and 2). It represents 8-site non-overlapping beads placed in a [[Building up a simple cubic lattice | simple cubic lattice]]. Bonds linking the beads may have lengths ranging between 2 and <math>\sqrt 10</math> lattice length units but bonds vectors of the type (2,2,0) are excluded to avoid bond crossing during the simulation.

[[Image:BFM 1.png|thumb|right]]

This model exhibits some advantages with respect to more conventional representations of the polymer in a lattice, such as fixed-bond self-avoiding walk chains on a simple cubic or [[tetrahedral lattice]]:
*It is somehow more consistent with the theoretical description of the chain through Gaussianly distributed statistical segments. This description has been employed in the past to obtain useful analytical expression for equilibrium properties and its dynamical version corresponds to the [[Rouse model]].
*A variety of bond lengths alleviate the restrictions due to the lattice constraints and, therefore, the model is more similar to the continuum behavior.
*It is possible to perform simulations by using a single type elementary bead jumps. A bead jump is a simple translation of the bead to one of the six contiguous lattice units along the x, y or z axis.This is a difference with respect to the simple cubic lattice model where a combination of bent and crankshaft elementary moves involving two or three beads has to be used as the simplest local level. The implication is that one can generate dynamic trajectories with the bond fluctuation model by using a simple and natural type of bead jumps, which intuitively resembles the stochastic behavior of [[Brownian particles]].
* The elementary jump moves can directly used in polymers with branch points ([[stars]], [[combs]], [[dendrimers]]).
==References==
[[Category: Models]]

#[http://www.oup.com/uk/catalogue/?ci=9780195094381 K. Binder (ed.) “Monte Carlo and Molecular Dynamics Simulations in Polymer Science”, Oxford University Press (1995)]
#[http://dx.doi.org/10.1021/ma00187a030 I. Carmesin and Kurt Kremer "The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions", Macromolecules pp. 2819 - 2823 (1988)]

File:BFM 1.png

2007-05-30T15:44:41Z

Juan:

Bond fluctuation model

2007-05-30T15:21:03Z

Juan:

Polymers have many interesting mesoscopic properties that can adequately represented through [[Coarse graining | coarse-grained]] models. [[Models | Lattice models]] are particularly useful since they employ integer coordinates that can be quickly processed. Another advantage is the possibility of checking the occupancy of the discrete number of sites and store it in an array. Therefore, overlapping between beads can be easily avoided.

The '''bond fluctuation model''' has been used in the recent past to simulate a great variety of [[Polymers |polymer]] systems (see Ref.s 1 and 2). It represents 8-site non-overlapping beads placed in a [[Building up a simple cubic lattice | simple cubic lattice]]. Bonds linking the beads may have lengths ranging between 2 and <math>\sqrt 10</math> lattice length units but bonds vectors of the type (2,2,0) are excluded to avoid bond crossing during the simulation.

[[Image:fig1.redu.png|thumb|right]]

This model exhibits some advantages with respect to more conventional representations of the polymer in a lattice, such as the fixed-bond [[self-avoiding walk chain model]] on a simple cubic or [[tetrahedral lattice]]:
*It is somehow more consistent with the theoretical description of the chain through Gaussianly distributed statistical segments. This description has been employed in the past to obtain useful analytical expression for equilibrium properties and its dynamical version corresponds to the [[Rouse model]].
*A variety of bond lengths alleviate the restrictions due to the lattice constraints and, therefore, the model is more similar to the continuum behavior.
*It is possible to perform simulations by using a single type elementary bead jumps. A bead jump is a simple translation of the bead to one of the six contiguous lattice units along the x, y or z axis.This is a difference with respect to the simple cubic lattice model where a combination of bent and crankshaft elementary moves involving two or three beads has to be used as the simplest local level. The implication is that one can generate dynamic trajectories with the bond fluctuation model by using a simple and natural type of bead jumps, which intuitively resembles the stochastic behavior of [[Brownian particles]].
* The elementary jump moves can directly used in polymers with branch points ([[stars]], [[combs]], [[dendrimers]]).
==References==
[[Category: Models]]

#[http://www.oup.com/uk/catalogue/?ci=9780195094381 K. Binder (ed.) “Monte Carlo and Molecular Dynamics Simulations in Polymer Science”, Oxford University Press (1995)]
#[http://dx.doi.org/10.1021/ma00187a030 I. Carmesin and Kurt Kremer "The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions", Macromolecules pp. 2819 - 2823 (1988)]

Bond fluctuation model

2007-05-30T15:08:28Z

Juan:

Polymers have many interesting mesoscopic properties that can adequately represented through [[Coarse graining | coarse-grained]] models. [[Models | Lattice models]] are particularly useful since they employ integer coordinates that can be quickly processed. Another advantage is the possibility of checking the occupancy of the discrete number of sites and store it in an array. Therefore, overlapping between beads can be easily avoided.

The '''bond fluctuation model''' has been used in the recent past to simulate a great variety of [[Polymers |polymer]] systems (see Ref.s 1 and 2). It represents 8-site non-overlapping beads placed in a [[Building up a simple cubic lattice | simple cubic lattice]]. Bonds linking the beads may have lengths ranging between 2 and <math>\sqrt 10</math> lattice length units but bonds vectors of the type (2,2,0) are excluded to avoid bond crossing during the simulation.

[[Image:fig1.png|thumb|right]]

This model exhibits some advantages with respect to more conventional representations of the polymer in a lattice, such as the fixed-bond [[self-avoiding walk chain model]] on a simple cubic or [[tetrahedral lattice]]:
*It is somehow more consistent with the theoretical description of the chain through Gaussianly distributed statistical segments. This description has been employed in the past to obtain useful analytical expression for equilibrium properties and its dynamical version corresponds to the [[Rouse model]].
*A variety of bond lengths alleviate the restrictions due to the lattice constraints and, therefore, the model is more similar to the continuum behavior.
*It is possible to perform simulations by using a single type elementary bead jumps. A bead jump is a simple translation of the bead to one of the six contiguous lattice units along the x, y or z axis.This is a difference with respect to the simple cubic lattice model where a combination of bent and crankshaft elementary moves involving two or three beads has to be used as the simplest local level. The implication is that one can generate dynamic trajectories with the bond fluctuation model by using a simple and natural type of bead jumps, which intuitively resembles the stochastic behavior of [[Brownian particles]].
* The elementary jump moves can directly used in polymers with branch points ([[stars]], [[combs]], [[dendrimers]]).
==References==
[[Category: Models]]

#[http://www.oup.com/uk/catalogue/?ci=9780195094381 K. Binder (ed.) “Monte Carlo and Molecular Dynamics Simulations in Polymer Science”, Oxford University Press (1995)]
#[http://dx.doi.org/10.1021/ma00187a030 I. Carmesin and Kurt Kremer "The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions", Macromolecules pp. 2819 - 2823 (1988)]

Bond fluctuation model

2007-05-30T14:49:07Z

Juan:

Polymers have many interesting mesoscopic properties that can adequately represented through coarse-grained models. Lattice models are particularly useful since they employ integer coordinates that can be quickly processed. Another advantage is the possibility of checking the occupancy of the discrete number of sites and store it in an array. Therefore, overlapping between beads can be easily avoided.

The '''bond fluctuation model''' has been used in the recent past to simulate a great variety of [[Polymers |polymer]] systems (see Ref.s 1 and 2). It represents 8-site non-overlapping beads placed in a [[Building up a simple cubic lattice | simple cubic lattice]]. Bonds linking the beads may have lengths ranging between 2 and <math>\sqrt 10</math> lattice length units but bonds vectors of the type (2,2,0) are excluded to avoid bond crossing during the simulation.

This model exhibits some advantages with respect to more conventional representations of the polymer in a lattice, such as the fixed-bond [[self-avoiding walk chain model]] on a simple cubic or tetrahedrical lattice:
*It is somehow more consistent with the theoretical description of the chain through Gaussianly distributed statistical segments. This description has been employed in the past to obtain useful analytical expression for equilibrium properties and its dynamical version corresponds to the [[Rouse model]].
*A variety of bond lengths alleviate the restrictions due to the lattice constraints and, therefore, the model is more similar to the continuum behavior.
*It is possible to perform simulations by using a single type elementary bead jumps. A bead jump is a simple translation of the bead to one of the six contiguous lattice units along the x, y or z axis.This is a difference with respect to the simple cubic lattice model where a combination of bent and crankshaft elementary moves involving two or three beads has to be used as the simplest local level. The implication is that one can generate dynamic trajectories with the bond fluctuation model by using a simple and natural type of bead jumps, which intuitively resembles the stochastic behavior of [[Brownian particles]].
* The elementary jump moves can directly used in polymers with branch points ([[stars]], [[combs]], [[dendrimers]]).
==References==
[[Category: Models]]

#[http://www.oup.com/uk/catalogue/?ci=9780195094381 K. Binder (ed.) “Monte Carlo and Molecular Dynamics Simulations in Polymer Science”, Oxford University Press (1995)]
#[http://dx.doi.org/10.1021/ma00187a030 I. Carmesin and Kurt Kremer "The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions", Macromolecules pp. 2819 - 2823 (1988)]

Bond fluctuation model

2007-05-30T14:48:13Z

Juan:

Polymers have many interesting mesoscopic properties that can adequately represented through coarse-grained models. Lattice models are particularly useful since they employ integer coordinates that can be quickly processed. Another advantage is the possibility of checking the occupancy of the discrete number of sites and store it in an array. Therefore, overlapping between beads can be easily avoided.

The '''bond fluctuation model''' has been used in the recent past to simulate a great variety of [[Polymers |polymer]] systems (see Ref.s 1 and 2). It represents 8-site non-overlapping beads placed in a [[Building up a simple cubic lattice | simple cubic lattice]]. Bonds linking the beads may have lengths ranging between 2 and <math>\sqrt 10</math> lattice length units but bonds vectors of the type (2,2,0) are excluded to avoid bond crossing during the simulation.

This model exhibits some advantages with respect to more conventional representations of the polymer in a lattice, such as the fixed-bond [[self-avoiding walk chain model]] on a simple cubic or tetrahedrical lattice:
*It is somehow more consistent with the theoretical description of the chain through Gaussianly distributed statistical segments. This description has been employed in the past to obtain useful analytical expression for equilibrium properties and its dynamical version corresponds to the [[Rouse model]].
*A variety of bond lengths alleviate the restrictions due to the lattice constraints and, therefore, the model is more similar to the continuum behavior.
*It is possible to perform simulations by using a single type elementary bead jumps. A bead jump is a simple translation of the bead to one of the six contiguous lattice units along the x, y or z axis.This is a difference with respect to the simple cubic lattice model where a combination of bent and crankshaft elementary moves has to be used as the simplest local level. The implication is that one can generate dynamic trajectories with the bond fluctuation model by using a simple and natural type of bead jumps, which intuitively resembles the stochastic behavior of [[Brownian particles]].
* The elementary jump moves can directly used in polymers with branch points ([[stars]], [[combs]], [[dendrimers]]).
==References==
[[Category: Models]]

#[http://www.oup.com/uk/catalogue/?ci=9780195094381 K. Binder (ed.) “Monte Carlo and Molecular Dynamics Simulations in Polymer Science”, Oxford University Press (1995)]
#[http://dx.doi.org/10.1021/ma00187a030 I. Carmesin and Kurt Kremer "The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions", Macromolecules pp. 2819 - 2823 (1988)]