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<hr />
<div>The '''Flory-Huggins theory''' for [[solutions]] of [[polymers]] was developed by [[Maurice L. Huggins ]] (Ref. 1) and [[Paul J. Flory]] (Ref. 2). 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, i, 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,<br />
<br />
:<math>\Delta S_m = -R \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B \right]</math><br />
<br />
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 <br />
<br />
:<math>\Delta H_m = RT \chi \phi_A \phi_B</math><br />
<br />
where <math>T</math> is the absolute [[temperature]].<br />
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<br />
<br />
:<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><br />
<br />
where <math>\chi</math> is the Flory-Huggins binary interaction parameter, which can be expressed as:<br />
<br />
:<math>\chi \propto \epsilon_{ij} - \frac{(\epsilon_{ii}-\epsilon_{jj})}{2} </math><br />
<br />
where <math>\epsilon_{ij}</math> is the net energy associated with two neighbouring lattice sites of the different<br />
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.<br />
<br />
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<br />
<br />
:<math>\chi_{cr} = \frac{1}{2} \left[ \frac{1}{\sqrt{n_A}} + \frac{1}{\sqrt{n_B}} \right]^2</math><br />
<br />
Therefore:<br />
*Positive values of <math>\chi</math> necessarily lead to immiscibility for polymer mixtures of high molecular weight.<br />
*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. Mixcibility 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.<br />
*For a polymer solution, <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. <br />
<br />
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><br />
<br />
:<math>\chi(T) = a + \frac{b}{T}</math><br />
<br />
where <math>a</math> and <math>b</math> are assumed to be constants, but can actually depend on density,<br />
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.<br />
<br />
<br />
==References==<br />
#[http://dx.doi.org/10.1063/1.1750930 Maurice L. Huggins "Solutions of Long Chain Compounds", Journal of Chemical Physics '''9''' p. 440 (1941)]<br />
#[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)]<br />
[[Category: Polymers]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Flory-Huggins_theory&diff=7047Flory-Huggins theory2008-08-07T11:46:58Z<p>62.204.202.244: </p>
<hr />
<div>The '''Flory-Huggins theory''' for [[solutions]] of [[polymers]] was developed by [[Maurice L. Huggins ]] (Ref. 1) and [[Paul J. Flory]] (Ref. 2). 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, i, 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,<br />
<br />
:<math>\Delta S_m = -R \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B \right]</math><br />
<br />
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 <br />
<br />
:<math>\Delta H_m = RT \chi \phi_A \phi_B</math><br />
<br />
where <math>T</math> is the absolute [[temperature]].<br />
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<br />
<br />
:<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><br />
<br />
where <math>\chi</math> is the Flory-Huggins binary interaction parameter, which can be expressed as:<br />
<br />
:<math>\chi \propto \epsilon_{ij} - \frac{(\epsilon_{ii}-\epsilon_{jj})}{2} </math><br />
<br />
where <math>\epsilon_{ij}</math> is the net energy associated with two neighbouring lattice sites of the different<br />
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.<br />
<br />
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<br />
<br />
:<math>\chi_{cr} = \frac{1}{2} \left[ \frac{1}{\sqrt{n_A}} + \frac{1}{\sqrt{n_B}} \right]^2</math><br />
<br />
Therefore:<br />
*Positive values of <math>\chi</math> necessarily lead to incompatibility for polymer mixtures of high molecular weight.<br />
*Polymer mixing always take place if the <math>\chi</math> parameter is negative. Compatible polymer mixtures with negative <math>\chi</math> are due to specific interactions between given polymer segments. 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.<br />
*For a polymer solution, <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. <br />
<br />
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><br />
<br />
:<math>\chi(T) = a + \frac{b}{T}</math><br />
<br />
where <math>a</math> and <math>b</math> are assumed to be constants, but can actually depend on density,<br />
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.<br />
<br />
<br />
==References==<br />
#[http://dx.doi.org/10.1063/1.1750930 Maurice L. Huggins "Solutions of Long Chain Compounds", Journal of Chemical Physics '''9''' p. 440 (1941)]<br />
#[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)]<br />
[[Category: Polymers]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Flory-Huggins_theory&diff=7045Flory-Huggins theory2008-08-07T11:42:58Z<p>62.204.202.244: </p>
<hr />
<div>The '''Flory-Huggins theory''' for [[solutions]] of [[polymers]] was developed by [[Maurice L. Huggins ]] (Ref. 1) and [[Paul J. Flory]] (Ref. 2). The descrition 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, i, 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,<br />
<br />
:<math>\Delta S_m = -R \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B \right]</math><br />
<br />
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 <br />
<br />
:<math>\Delta H_m = RT \chi \phi_A \phi_B</math><br />
<br />
where <math>T</math> is the absolute [[temperature]].<br />
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<br />
<br />
:<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><br />
<br />
where <math>\chi</math> is the Flory-Huggins binary interaction parameter, which can be expressed as:<br />
<br />
:<math>\chi \propto \epsilon_{ij} - \frac{(\epsilon_{ii}-\epsilon_{jj})}{2} </math><br />
<br />
where <math>\epsilon_{ij}</math> is the net energy associated with two neighbouring lattice sites of the different<br />
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.<br />
<br />
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<br />
<br />
:<math>\chi_{cr} = \frac{1}{2} \left[ \frac{1}{\sqrt{n_A}} + \frac{1}{\sqrt{n_B}} \right]^2</math><br />
<br />
Therefore:<br />
*Positive values of <math>\chi</math> necessarily lead to incompatibility for polymer mixtures of high molecular weight.<br />
*Polymer mixing always take place if the <math>\chi</math> parameter is negative. Compatible polymer mixtures with negative <math>\chi</math> are due to specific interactions between given polymer segments. 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.<br />
*For a polymer solution, <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. <br />
<br />
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><br />
<br />
:<math>\chi(T) = a + \frac{b}{T}</math><br />
<br />
where <math>a</math> and <math>b</math> are assumed to be constants, but can actually depend on density,<br />
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.<br />
<br />
<br />
==References==<br />
#[http://dx.doi.org/10.1063/1.1750930 Maurice L. Huggins "Solutions of Long Chain Compounds", Journal of Chemical Physics '''9''' p. 440 (1941)]<br />
#[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)]<br />
[[Category: Polymers]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Flory-Huggins_theory&diff=7042Flory-Huggins theory2008-08-07T11:13:44Z<p>62.204.202.244: </p>
<hr />
<div>The '''Flory-Huggins theory''' for [[solutions]] of [[polymers]] was developed by [[Maurice L. Huggins ]] (Ref. 1) and [[Paul J. Flory]] (Ref. 2). The descrition 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, i, 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,<br />
<br />
:<math>\Delta S_m = -R \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B \right]</math><br />
<br />
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 <br />
<br />
:<math>\Delta H_m = RT \chi \phi_A \phi_B</math><br />
<br />
where <math>T</math> is the absolute [[temperature]].<br />
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<br />
<br />
:<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><br />
<br />
where <math>\chi</math> is the Flory-Huggins binary interaction parameter, which can be expressed as:<br />
<br />
:<math>\chi \propto \epsilon_{ij} - \frac{(\epsilon_{ii}-\epsilon_{jj})}{2} </math><br />
<br />
where <math>\epsilon_{ij}</math> is the net energy associated with two neighbouring lattice sites of the different<br />
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.<br />
<br />
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<br />
<br />
:<math>\chi_{cr} = \frac{1}{2} \left[ \frac{1}{\sqrt{n_A}} + \frac{1}{\sqrt{n_B}} \right]^2</math><br />
<br />
Therefore:<br />
*Positive values of <math>\chi</math> necessarily lead to incompatibility for polymer mixtures of high molecular weight.<br />
*Polymer mixing always take place if the <math>\chi</math> parameter is negative. Compatible polymer mixtures with negative <math>\chi</math> are due to specific interactions between given polymer segments. 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.<br />
*For a polymer solution, <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 point of the polymer-solvent system. <br />
<br />
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><br />
<br />
:<math>\chi(T) = a + \frac{b}{T}</math><br />
<br />
where <math>a</math> and <math>b</math> are assumed to be constants, but can actually depend on density,<br />
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.<br />
<br />
<br />
==References==<br />
#[http://dx.doi.org/10.1063/1.1750930 Maurice L. Huggins "Solutions of Long Chain Compounds", Journal of Chemical Physics '''9''' p. 440 (1941)]<br />
#[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)]<br />
[[Category: Polymers]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Flory-Huggins_theory&diff=7041Flory-Huggins theory2008-08-07T11:10:46Z<p>62.204.202.244: </p>
<hr />
<div>The '''Flory-Huggins theory''' for [[solutions]] of [[polymers]] was developed by [[Maurice L. Huggins ]] (Ref. 1) and [[Paul J. Flory]] (Ref. 2). The descrition 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, i, 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,<br />
<br />
:<math>\Delta S_m = -R \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B \right]</math><br />
<br />
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 <br />
<br />
:<math>\Delta H_m = RT \chi \phi_A \phi_B</math><br />
<br />
where <math>T</math> is the absolute [[temperature]].<br />
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<br />
<br />
:<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><br />
<br />
where <math>\chi</math> is the Flory-Huggins binary interaction parameter, which can be expressed as:<br />
<br />
:<math>\chi \propto \epsilon_{ij} - \frac{(\epsilon_{ii}-\epsilon_{jj})}{2} </math><br />
<br />
where <math>\epsilon_{ij}</math> is the net energy associated with two neighbouring lattice sites of the different<br />
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.<br />
<br />
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<br />
<br />
:<math>\chi_{cr} = \frac{1}{2} \left[ \frac{1}{\sqrt{n_A}} + \frac{1}{\sqrt{n_B}} \right]^2</math><br />
<br />
Therefore:<br />
*Positive values of <math>\chi</math> necessarily lead to incompatibility for polymer mixtures of high molecular weight.<br />
*Polymer mixing always take place if the <math>\chi</math> parameter is negative. Compatible polymer mixtures with negative <math>\chi</math> are due to specific interactions between given polymer segments. 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.<br />
*For a polymer solution, <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 point of the polymer-solvent system. <br />
<br />
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><br />
<br />
:<math>\chi(T) = a + \frac{b}{T}</math><br />
<br />
where <math>a</math> and <math>b</math> are assumed to be constants, but can actually depend on density,<br />
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 systems 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.<br />
<br />
<br />
==References==<br />
#[http://dx.doi.org/10.1063/1.1750930 Maurice L. Huggins "Solutions of Long Chain Compounds", Journal of Chemical Physics '''9''' p. 440 (1941)]<br />
#[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)]<br />
[[Category: Polymers]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Flory-Huggins_theory&diff=7040Flory-Huggins theory2008-08-07T11:03:41Z<p>62.204.202.244: </p>
<hr />
<div>The '''Flory-Huggins theory''' for [[solutions]] of [[polymers]] was developed by [[Maurice L. Huggins ]] (Ref. 1) and [[Paul J. Flory]] (Ref. 2). The descrition 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, i, 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,<br />
<br />
:<math>\Delta S_m = -R \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B \right]</math><br />
<br />
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 <br />
<br />
:<math>\Delta H_m = RT \chi \phi_A \phi_B</math><br />
<br />
where <math>T</math> is the absolute [[temperature]].<br />
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<br />
<br />
:<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><br />
<br />
where <math>\chi</math> is the Flory-Huggins binary interaction parameter, which can be expressed as:<br />
<br />
:<math>\chi \propto \epsilon_{ij} - \frac{(\epsilon_{ii}-\epsilon_{jj})}{2} </math><br />
<br />
where <math>\epsilon_{ij}</math> is the net energy associated with two neighbouring lattice sites of the different<br />
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.<br />
<br />
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<br />
<br />
:<math>\chi_{cr} = \frac{1}{2} \left[ \frac{1}{\sqrt{n_A}} + \frac{1}{\sqrt{n_B}} \right]^2</math><br />
<br />
Therefore:<br />
*Positive values of <math>\chi</math> necessarily lead to incompatibility for polymer mixtures of high molecular weight.<br />
*Polymer mixing always take place if the <math>\chi</math> parameter is negative. Compatible polymer mixtures with negative <math>\chi</math> are due to specific interactions between given polymer segments. Compatibility can be induced by introducing opposite charges in the different polymers or by the presence of a copolymer containing A and B segments.<br />
*For a polymer solution, <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 point of the polymer-solvent system. <br />
<br />
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><br />
<br />
:<math>\chi(T) = A + \frac{B}{T}</math><br />
<br />
where <math>A</math> and <math>B</math> are assumed to be constants, but can actually depend on density,<br />
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 systems 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.<br />
<br />
<br />
==References==<br />
#[http://dx.doi.org/10.1063/1.1750930 Maurice L. Huggins "Solutions of Long Chain Compounds", Journal of Chemical Physics '''9''' p. 440 (1941)]<br />
#[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)]<br />
[[Category: Polymers]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Flory-Huggins_theory&diff=7035Flory-Huggins theory2008-08-06T13:43:13Z<p>62.204.202.244: </p>
<hr />
<div>The '''Flory-Huggins theory''' for [[solutions]] of [[polymers]] was developed by [[Maurice L. Huggins ]] (Ref. 1) and [[Paul J. Flory]] (Ref. 2). The Flory-Huggins theory defines the volume of a [[polymers |polymer system]] as a lattice which is divided into small subspaces (called sites) of the same volume. In the case of polymer solutions, the solvent is assumed to occupy single sites, while each polymer chain occupies <math>n</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 of two polymers, 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,<br />
<br />
:<math>\Delta S_m = -R \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B \right]</math><br />
<br />
where <math>\phi_i</math> is the [[volume fraction]] of the component <math>i</math> and <math>n_i</math> is the number of segments in each type of polymer chain, 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 <br />
<br />
:<math>\Delta H_m = RT \chi \phi_A \phi_B</math><br />
<br />
where <math>T</math> is the absolute [[temperature]].<br />
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<br />
<br />
:<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><br />
<br />
where <math>\chi</math> is the Flory-Huggins binary interaction parameter, defined as:<br />
<br />
:<math>\chi \approx \epsilon_{ij} - \frac{(\epsilon_{ii}-\epsilon_{jj})}{2} </math><br />
<br />
where <math>\epsilon_{ij}</math> is the net energy associated with two neighbouring lattice sites of the different<br />
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.<br />
<br />
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<br />
<br />
:<math>\chi_{cr} = \frac{1}{2} \left[ \frac{1}{\sqrt{n_A}} + \frac{1}{\sqrt{n_B}} \right]^2</math><br />
<br />
Therefore:<br />
*Positive values of <math>\chi</math> necessarily lead to incompatibility for polymer mixtures of high molecular weight.<br />
*Polymer mixing always take place if the <math>\chi</math> parameter is negative. Compatible polymer mixtures with negative <math>\chi</math> are due to specific interactions between given polymer segments. Compatibility can be induced by introducing opposite charges in the different polymers or by the presence of a copolymer containing A and B segments.<br />
*For a polymer solution, the critical Flory-Huggins parameter is close to <math>1/2</math>.<br />
<br />
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><br />
<br />
:<math>\chi(T) = A + \frac{B}{T}</math><br />
<br />
where <math>A</math> and <math>B</math> are assumed to be constants, but can actually depend on density,<br />
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 systems 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.<br />
<br />
<br />
==References==<br />
#[http://dx.doi.org/10.1063/1.1750930 Maurice L. Huggins "Solutions of Long Chain Compounds", Journal of Chemical Physics '''9''' p. 440 (1941)]<br />
#[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)]<br />
[[Category: Polymers]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Flory-Huggins_theory&diff=7034Flory-Huggins theory2008-08-06T13:37:44Z<p>62.204.202.244: </p>
<hr />
<div>The '''Flory-Huggins theory''' for [[solutions]] of [[polymers]] was developed by [[Maurice L. Huggins ]] (Ref. 1) and [[Paul J. Flory]] (Ref. 2). The Flory-Huggins theory defines the volume of a [[polymers |polymer system]] as a lattice which is divided into small subspaces (called sites) of the same volume. In the case of polymer solutions, the solvent is assumed to occupy single sites, while each polymer chain occupies <math>n</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 of two polymers, 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,<br />
<br />
:<math>\Delta S_m = -R \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B \right]</math><br />
<br />
where <math>\phi_i</math> is the [[volume fraction]] of the component <math>i</math> and <math>n_i</math> is the number of segments in each type of polymer chain, 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 <br />
<br />
:<math>\Delta H_m = RT \chi \phi_A \phi_B</math><br />
<br />
where <math>T</math> is the absolute [[temperature]].<br />
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<br />
<br />
:<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><br />
<br />
where <math>\chi</math> is the Flory-Huggins binary interaction parameter, defined as:<br />
<br />
:<math>\chi \approx \epsilon_{ij} - \frac{(\epsilon_{ii}-\epsilon_{jj})}{2} </math><br />
<br />
where <math>\epsilon_{ij}</math> is the net energy associated with two neighbouring lattice sites of the different<br />
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.<br />
<br />
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<br />
<br />
:<math>\chi_{cr} = \frac{1}{2} \left[ \frac{1}{\sqrt{n_A}} + \frac{1}{\sqrt{n_B}} \right]^2</math><br />
<br />
Therefore:<br />
*Positive values of <math>\chi</math> necessarily lead to incompatibility for polymers of high molecular weight.<br />
*Mixing always take place if the <math>\chi</math> parameter is negative. Compatible mixtures with negative <math>\chi</math> are due to specific interactions between given polymer segments. Compatibility can be induced by introducing opposite charges or by the presence of a copolymer containing A and B segements.<br />
*For a polymer solution, the critical Flory-Huggins parameter is close to <math>1/2</math>.<br />
<br />
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><br />
<br />
:<math>\chi(T) = A + \frac{B}{T}</math><br />
<br />
where <math>A</math> and <math>B</math> are assumed to be constants, but can actually depend on density,<br />
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 systems 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.<br />
<br />
<br />
==References==<br />
#[http://dx.doi.org/10.1063/1.1750930 Maurice L. Huggins "Solutions of Long Chain Compounds", Journal of Chemical Physics '''9''' p. 440 (1941)]<br />
#[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)]<br />
[[Category: Polymers]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Flory-Huggins_theory&diff=7026Flory-Huggins theory2008-08-05T13:08:46Z<p>62.204.202.244: </p>
<hr />
<div>The '''Flory-Huggins theory''' defines the volume of [[polymers |polymer system]] as a lattice which is divided by small subspaces (called sites) of the same volume. In the case of polymer solutions, the solvent is assumed to occupy single sites, while each polymer chain occupies <math>n</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 of two polymers, 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,<br />
<br />
:<math>\Delta S_m = -R \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B \right]</math><br />
<br />
<br />
where <math>\phi_i</math> is the volume fraction of the component <math>i</math> and <math>n_i</math> is the number of segments in each type of polymer chain, <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 <br />
<br />
:<math>\Delta H_m = RT \chi \phi_A \phi_B</math><br />
<br />
where <math>T</math> is the absolute [[temperature]].<br />
<br />
According to the preceding equations, the Flory-Huggins equation can be expressed in terms of the free-energy of mixing for a binary system<br />
<br />
:<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><br />
<br />
where <math>\chi</math> is the Flory-Huggins binary interaction parameter, defined as:<br />
<br />
:<math>\chi \approx \epsilon_{ij} - \frac{(\epsilon_{ii}-\epsilon_{jj})}{2} </math><br />
<br />
where <math>\epsilon_{ij}</math> is the net energy associated with two neighbouring lattice sites of the different<br />
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 the theoretical point of view it may also depend on model parameters as the coordination number of the lattice and segment length. The <math>\chi</math> parameter is somehow similar to a [[second virial coefficient]] expressing binary interaction between molecules and, therefore, it usually shows a linear dependence of <math>1/T</math><br />
<br />
:<math>\chi(T) = A + \frac{B}{T}</math><br />
<br />
<br />
<math>A</math> and <math>B</math> are assumed to be constants but can actually depend on density,<br />
concentration, molecular weight, etc. A usual interpretation is that <math>A</math> represents an enthalpic quantity and <math>B</math> an [[entropy | entropic]] contribution, although both <math>A</math> and <math>B</math> are actually empirical parameters.<br />
<br />
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. 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<br />
<br />
:<math>\chi_{cr} = \frac{1}{2} \left[ \frac{1}{\sqrt{n_A}} + \frac{1}{\sqrt{n_B}} \right]^2</math><br />
<br />
Therefore:<br />
*Positive values of <math>\chi</math> necessarily lead to incompatibility for polymers of high molecular weight.<br />
*Mixing always take place if the <math>\chi</math> parameter is negative.<br />
*For a polymer solution, the critical Flory-Huggins parameter is close to <math>1/2</math>.<br />
==References==<br />
[[Category: Polymers]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Theta_solvent&diff=3705Theta solvent2007-08-09T11:03:38Z<p>62.204.202.244: </p>
<hr />
<div>A '''theta solvent''' is the name for a condition (sometimes known as the [[Paul J. Flory |Flory]] condition) rather than an actual solvent. At the ''theta point'', in the words of Paul Flory: "''excluded volume interactions are neutralized''."<br />
An excluded volume of zero connotes a [[second virial coefficient]] of zero. The theta state also corresponds to the highest upper critical temperature of a given polymer-solvent system.<br />
==References==<br />
#[http://nobelprize.org/nobel_prizes/chemistry/laureates/1974/flory-lecture.html Paul J. Flory Nobel Lecture]<br />
[[category: polymers]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Rouse_model&diff=3465Rouse model2007-07-30T14:02:11Z<p>62.204.202.244: </p>
<hr />
<div>The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a <br />
[[Gaussian distribution]]. Therefore, the [[polymers | polymer]] chain can be described by a <br />
[[Coarse graining | coarse-grained model]] of N successive Gaussian segments. A dynamical model is consistent with this description and also with [[Boltzmann distribution | Boltzmann's distribution of energies]] if it assumes that the forces between units jointed by the Gaussian segments are proportional to their distances (Hookean springs). The Rouse model (Ref. 1) describes a polymer chain as a set of N coupled harmonic springs. The mathematical treatment of this model decomposes the global motion in a set of N-1 independent "normal modes". The [[time-correlation function]] of each one of these modes decays as a single exponential, characterized by a "Rouse relaxation time". The Rouse relaxation times are proportional to <math>N^2</math> (N is proportional to the chain length or the polymer molecular weight). The first Rouse mode represents the slowest internal motion of the chain. The p-th Rouse relaxation time is proportional to <math>1/p^2</math>.<br />
<br />
In dilute solution, the relaxation times obtained from the Rouse model is not correct due to the presence of hydrodynamic interactions between units, introduced by Zimm (Ref. 2). With this correction, the Zimm-Rouse relaxation times are proportional to <math>N^{3/2}</math>. This result is only valid for unperturbed (or theta) solvents. For solvent of good quality, excluded volume interactions have also to be accounted. However, the scheme of normal modes is maintained (Ref. 3).<br />
In non-dilute solutions and melts hydrodynamic interactions and excluded volume effects are screened out and the Rouse model is correct, though only in the scale of short times and distances.<br />
==References==<br />
[[Category: Models]]<br />
[[category: polymers]]<br />
# P. E. Rouse, J. Chem. Phys. 21, 1272 (1953)<br />
# B. H.Zimm, J. Chem. Phys. 24, 269 (1956).<br />
# M. Doi, S. F. Edwards "The Theory of Polymer Dynamcis", Clarendon, Oxford, 1986.</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Rouse_model&diff=3464Rouse model2007-07-30T14:00:42Z<p>62.204.202.244: </p>
<hr />
<div>The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a <br />
[[Gaussian distribution]]. Therefore, the [[polymers | polymer]] chain can be described by a <br />
[[Coarse graining | coarse-grained model]] of N successive Gaussian segments. A dynamical model is consistent with this description and also with [[Boltzmann distribution | Boltzmann's distribution of energies]] if it assumes that the forces between units jointed by the Gaussian segments are proportional to their distances (Hookean springs). The Rouse model (Ref. 1) describes a polymer chain as a set of N coupled harmonic springs. The mathematical treatment of this model decomposes the global motion in a set of N-1 independent "normal modes". The [[time-correlation function]] of each one of these modes decays as a single exponential, characterized by a "Rouse relaxation time". The Rouse relaxation times are proportional to <math>N^2</math> (N is proportional to the chain length or the polymer molecular weight). The first Rouse mode represents the slowest internal motion of the chain. The p-th Rouse relaxation time is proportional to <math>1/p^2</math>.<br />
<br />
In dilute solution, the relaxation times obtained from the Rouse model is not correct due to the presence of hydrodynamic interactions between units, introduced by Zimm (Ref. 2). With this correction, the Zimm-Rouse relaxation times are proportional to <math>N^{3/2}</math>. This result is only valid for unperturbed (or theta) solvents. For solvent of good quality, excluded volume interactions have also to be accounted. However, the scheme of normal modes is maintained (Ref. 3).<br />
In semi-dilute solutions and melts hydrodynamic interactions and excluded volume effects are screened out and the Rouse model is correct, though only in the scale of short times and distances.<br />
==References==<br />
[[Category: Models]]<br />
[[category: polymers]]<br />
# P. E. Rouse, J. Chem. Phys. 21, 1272 (1953)<br />
# B. H.Zimm, J. Chem. Phys. 24, 269 (1956).<br />
# M. Doi, S. F. Edwards "The Theory of Polymer Dynamcis", Clarendon, Oxford, 1986.</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Rouse_model&diff=3462Rouse model2007-07-30T13:59:03Z<p>62.204.202.244: </p>
<hr />
<div>The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a <br />
[[Gaussian distribution]]. Therefore, the [[polymers | polymer]] chain can be described by a <br />
[[Coarse graining | coarse-grained model]] of N successive Gaussian segments. A dynamical model is consistent with this description and also with [[Boltzmann distribution | Boltzmann's distribution of energies]] if it assumes that the forces between units jointed by the Gaussian segments are proportional to their distances (Hookean springs). The Rouse model (Ref. 1) describes a polymer chain as a set of N coupled harmonic springs. The mathematical treatment of this model decomposes the global motion in a set of N-1 independent "normal modes". The [[time-correlation function]] of each one of these modes decays as a single exponential, characterized by a "Rouse relaxation time". The Rouse relaxation times are proportional to <math>N^2</math> (N is proportional to the chain length or the polymer molecular weight). The first Rouse mode represents the slowest internal motion of the chain. The p-th Rouse relaxation time is proportional to <math>1/p^2</math>.<br />
<br />
In dilute solution, the relaxation times obtained from the Rouse model is not correct due to the presence of hydrodynamic interactions between units, introduced by Zimm (Ref. 2). With this correction, the Zimm-Rouse relaxation times are proportional to <math>N^{(3/2)}</math>. This result is only valid for unperturbed (or theta) solvents. For solvent of good quality, excluded volume interactions have also to be accounted. However, the scheme of normal modes is maintained (Ref. 3).<br />
In semi-dilute solutions and melts hydrodynamic interactions and excluded volume effects are screened out and the Rouse model is correct, though only in the scale of short times and distances.<br />
==References==<br />
[[Category: Models]]<br />
[[category: polymers]]<br />
1. P. E. Rouse, J. Chem. Phys. 21, 1272 (1953)<br />
2. B. H.Zimm, J. Chem. Phys. 24, 269 (1956).<br />
3. M. Doi, S. F. Edwards "The Theory of Polymer Dynamcis", Clarendon, Oxford, 1986.</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Rouse_model&diff=3457Rouse model2007-07-30T13:41:05Z<p>62.204.202.244: New page: The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a Gaussian distribution. Therefore, the chain can be described by a course-grai...</p>
<hr />
<div>The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a Gaussian distribution. Therefore, the chain can be described by a course-grained model of N successive Gaussian segments. A dynamical model is consistent with this description and also with Boltzmann's distribution of energies if it assumes forces between units jointed by the Gaussian segments that are proportional to their distances (Hookean springs). The Rouse model describes a polymer chain as a set of N coupled harmonic springs. The mathematical treatment of this model decomposes the global motion in a set of N-1 independent "normal modes". The time-correlation function of each one of these modes decays as a single exponential, characterized by a "Rouse relaxation time". The Rouse relaxation times are proportional to N^2 (N is proportional to the chain length or the polymer molecular weight). The first Rouse mode represents the slowest internal motion of the chain. The p-th Rouse relaxation time is proportional to 1/p^2.<br />
In dilute solution, the relaxation times obtained from the Rouse model is not correct due to the presence of hydrodynamic interactions between units, introduced by Zimm. With this correction, the Zimm-Rouse relaxation times are proportional to N^(3/2). This result is only valid for unperturbed (or theta) solvents. For solvent of good quality, excluded volume interactions have also to be accounted. However, the scheme of normal modes is maintained.<br />
In semidilute solutions and melts hydrodynamic interactions and excluded volume effects are screened out and the Rouse model is correct, though only in the scale of short times and distances.</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Lattice_simulations_(Polymers)&diff=3137Lattice simulations (Polymers)2007-06-27T10:59:31Z<p>62.204.202.244: </p>
<hr />
<div>[[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. <br />
Earlier simulations were performed with the [[Building up a simple cubic lattice | simple cubic]] or [[tetrahedral lattices]], employing different algorithms. Tetrahedral lattice models are mainly used because of their closer similarity of their bond angles with those of the [[aliphatic chains]]. <br />
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 (Ref. 1). A combinations of bends and crankshaft was shown to be nearly ergodic and reproduce adequately the [[Rouse dynamics]] of single chain systems in a cubic latice (Ref. 2). More efficient pivot moves have been devised to explore the equilibrium properties of very long single chains (Ref. 3). 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 (Ref. 4). Also, specific algorithms to perform ''NpT'' simulations have been designed (Ref. 5). <br />
The [[bond fluctuation model]] has been proposed to combine the advantages of lattice and off-lattice models.<br />
==References==<br />
#[http://dx.doi.org/10.1063/1.1732301 P. H. Verdier and W. H. Stockmayer "Monte Carlo Calculations on the Dynamics of Polymers in Dilute Solution", Journal of Chemical Physics '''36''' pp. 227-235 (1962)]<br />
#[http://dx.doi.org/10.1063/1.431297 H. J. Hilhorst and J. M. Deutch "Analysis of Monte Carlo results on the kinetics of lattice polymer chains with excluded volume", Journal of Chemical Physics '''63''' pp. 5153-5161 (1975)]<br />
#[http://dx.doi.org/10.1007/BF01022990 Neal Madras and Alan D. Sokal "The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk", Journal of Statistical Physics '''50''' pp. 109-186 (1988)]<br />
#[http://dx.doi.org/10.1021/ma00169a036 T. Pakula and S. Geyler "Cooperative relaxations in condensed macromolecular systems. 1. A model for computer simulation", Macromolecules '''20''' pp. 679 - 682 (1987)]<br />
#[http://dx.doi.org/10.1063/1.469450 A. D. Mackie, A. Z. Panagiotopoulos, S. K. Kumar "Monte Carlo simulations of phase equilibria for a lattice homopolymer model", Journal of Chemical Physics '''102''' pp. 1014-1023 (1995)]<br />
[[category: Computer simulation techniques]]<br />
[[category: Monte Carlo]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Lattice_simulations_(Polymers)&diff=3136Lattice simulations (Polymers)2007-06-27T10:56:31Z<p>62.204.202.244: /* References */</p>
<hr />
<div>[[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. <br />
Earlier simulations were performed with the [[Building up a simple cubic lattice | simple cubic]] or [[tetrahedral lattices]], employing different algorithms. Tetrahedral lattice models are mainly used because of their closer similarity of their bond angles with those of the [[aliphatic chains]]. <br />
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 combinations of bends and crankshaft was shown to be nearly ergodic and reproduce adequately the [[Rouse dynamics]] of single chain systems in a cubic latice (Ref. 1). More efficient pivot moves have been devised to explore the equilibrium properties of very long single chains (Ref. 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 (Ref. 3). Also, specific algorithms to perform ''NpT'' simulations have been designed (Ref. 4). <br />
The [[bond fluctuation model]] has been proposed to combine the advantages of lattice and off-lattice models.<br />
==References==<br />
#[http://dx.doi.org/10.1063/1.1732301 P. H. Verdier and W. H. Stockmayer "Monte Carlo Calculations on the Dynamics of Polymers in Dilute Solution", Journal of Chemical Physics '''36''' pp. 227-235 (1962)]<br />
#[http://dx.doi.org/10.1063/1.431297 H. J. Hilhorst and J. M. Deutch "Analysis of Monte Carlo results on the kinetics of lattice polymer chains with excluded volume", Journal of Chemical Physics '''63''' pp. 5153-5161 (1975)]<br />
#[http://dx.doi.org/10.1007/BF01022990 Neal Madras and Alan D. Sokal "The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk", Journal of Statistical Physics '''50''' pp. 109-186 (1988)]<br />
#[http://dx.doi.org/10.1021/ma00169a036 T. Pakula and S. Geyler "Cooperative relaxations in condensed macromolecular systems. 1. A model for computer simulation", Macromolecules '''20''' pp. 679 - 682 (1987)]<br />
#[http://dx.doi.org/10.1063/1.469450 A. D. Mackie, A. Z. Panagiotopoulos, S. K. Kumar "Monte Carlo simulations of phase equilibria for a lattice homopolymer model", Journal of Chemical Physics '''102''' pp. 1014-1023 (1995)]<br />
[[category: Computer simulation techniques]]<br />
[[category: Monte Carlo]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Lattice_simulations_(Polymers)&diff=2548Lattice simulations (Polymers)2007-05-31T10:56:20Z<p>62.204.202.244: </p>
<hr />
<div>[[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. <br />
Earlier simulations were performed with the [[Building up a simple cubic lattice | simple cubic]] or [[tetrahedral lattices]], employing different algorithms. Tetrahedral lattice models are mainly used because of their closer similarity of their bond angles with those of the [[aliphatic chains]]. <br />
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 combinations of bends and crankshaft was shown to be nearly ergodic and reproduce adequately the [[Rouse dynamics]] of single chain systems in a cubic latice (Ref. 1). More efficient pivot moves have been devised to explore the equilibrium properties of very long single chains (Ref. 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 (Ref. 3). Also, specific algorithms to perform ''NpT'' simulations have been designed (Ref. 4). <br />
The [[bond fluctuation model]] has been proposed to combine the advantages of lattice and off-lattice models.<br />
==References==<br />
#[http://dx.doi.org/10.1063/1.1732301 P. H. Verdier and W. H. Stockmayer "Monte Carlo Calculations on the Dynamics of Polymers in Dilute Solution", Journal of Chemical Physics '''36''' pp. 227-235 (1962)]<br />
#[http://dx.doi.org/10.1007/BF01022990 Neal Madras and Alan D. Sokal "The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk", Journal of Statistical Physics '''50''' pp. 109-186 (1988)]<br />
#[http://dx.doi.org/10.1021/ma00169a036 T. Pakula and S. Geyler "Cooperative relaxations in condensed macromolecular systems. 1. A model for computer simulation", Macromolecules '''20''' pp. 679 - 682 (1987)]<br />
#[http://dx.doi.org/10.1063/1.469450 A. D. Mackie, A. Z. Panagiotopoulos, S. K. Kumar "Monte Carlo simulations of phase equilibria for a lattice homopolymer model", Journal of Chemical Physics '''102''' pp. 1014-1023 (1995)]<br />
[[category: Computer simulation techniques]]<br />
[[category: Monte Carlo]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Models&diff=2492Models2007-05-30T15:31:13Z<p>62.204.202.244: </p>
<hr />
<div>__NOTOC__<br />
See also: [[force fields]].<br />
==Lattice models==<br />
*[[Hard hexagons]]<br />
*[[Heisenberg model]]<br />
*[[Ising Models]]<br />
*[[Lattice gas]]<br />
*[[Potts model]]<br />
*[[Bond fluctuation model]]<br />
<br />
=='Hard' models==<br />
*[[Hard rods]]<br />
*[[Hard disks]]<br />
*[[hard sphere model | Hard sphere]]<br />
*[[Widom-Rowlinson model]]<br />
*[[Two-dimensional hard dumbbells]]<br />
*[[Three-dimensional hard dumbbells]]<br />
*[[Tangent linear hard sphere chains]]<br />
*[[Flexible hard sphere chains]] (aka. pearl-necklace model)<br />
*[[Branched hard sphere chains]]<br />
*[[Fused hard sphere chains]]<br />
*[[Hard ellipsoids]]<br />
*[[Hard spherocylinders]]<br />
*[[Hard core Yukawa]]<br />
<br />
==Piecewise continuous models==<br />
*[[Square well]]<br />
*[[Square shoulder]]<br />
*[[Square shoulder + square well]]<br />
*[[Ramp model]]<br />
<br />
=='Soft' models==<br />
*[[Gaussian overlap model]]<br />
*[[Gay-Berne model]]<br />
*[[Kihara potential]]<br />
*[[Lennard-Jones model]]<br />
*[[9-3 Lennard-Jones potential]]<br />
*[[United-atom model]]<br />
*[[Intermolecular Interactions]]<br />
*[[Flexible molecules|Flexible molecules (intramolecular interactions)]]<br />
*[[Confined systems]]<br />
<br />
=='Charged' models==<br />
*[[Restricted primitive model]]<br />
*[[Charged hard dumbbells]]<br />
<br />
== Metals ==<br />
*[[Sutton-Chen]]<br />
*[[Embedded atom model]]<br />
*[[Finnis-Sinclair]]<br />
*[[Gupta potential]]<br />
[[category:models]]<br />
[[category:Computer simulation techniques]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Models&diff=2491Models2007-05-30T15:30:17Z<p>62.204.202.244: </p>
<hr />
<div>__NOTOC__<br />
See also: [[force fields]].<br />
==Lattice models==<br />
*[[Bond fluctuation model]]<br />
*[[Hard hexagons]]<br />
*[[Heisenberg model]]<br />
*[[Ising Models]]<br />
*[[Lattice gas]]<br />
*[[Potts model]]<br />
<br />
=='Hard' models==<br />
*[[Hard rods]]<br />
*[[Hard disks]]<br />
*[[hard sphere model | Hard sphere]]<br />
*[[Widom-Rowlinson model]]<br />
*[[Two-dimensional hard dumbbells]]<br />
*[[Three-dimensional hard dumbbells]]<br />
*[[Tangent linear hard sphere chains]]<br />
*[[Flexible hard sphere chains]] (aka. pearl-necklace model)<br />
*[[Branched hard sphere chains]]<br />
*[[Fused hard sphere chains]]<br />
*[[Hard ellipsoids]]<br />
*[[Hard spherocylinders]]<br />
*[[Hard core Yukawa]]<br />
<br />
==Piecewise continuous models==<br />
*[[Square well]]<br />
*[[Square shoulder]]<br />
*[[Square shoulder + square well]]<br />
*[[Ramp model]]<br />
<br />
=='Soft' models==<br />
*[[Gaussian overlap model]]<br />
*[[Gay-Berne model]]<br />
*[[Kihara potential]]<br />
*[[Lennard-Jones model]]<br />
*[[9-3 Lennard-Jones potential]]<br />
*[[United-atom model]]<br />
*[[Intermolecular Interactions]]<br />
*[[Flexible molecules|Flexible molecules (intramolecular interactions)]]<br />
*[[Confined systems]]<br />
<br />
=='Charged' models==<br />
*[[Restricted primitive model]]<br />
*[[Charged hard dumbbells]]<br />
<br />
== Metals ==<br />
*[[Sutton-Chen]]<br />
*[[Embedded atom model]]<br />
*[[Finnis-Sinclair]]<br />
*[[Gupta potential]]<br />
[[category:models]]<br />
[[category:Computer simulation techniques]]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Bond_fluctuation_model&diff=2490Bond fluctuation model2007-05-30T15:29:35Z<p>62.204.202.244: </p>
<hr />
<div>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.<br />
<br />
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.<br />
<br />
[[Image:fig1.redu.png|thumb|right]]<br />
<br />
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]]: <br />
*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]]. <br />
*A variety of bond lengths alleviate the restrictions due to the lattice constraints and, therefore, the model is more similar to the continuum behavior. <br />
*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]]. <br />
* The elementary jump moves can directly used in polymers with branch points ([[stars]], [[combs]], [[dendrimers]]).<br />
==References==<br />
[[Category: Models]]<br />
<br />
#[http://www.oup.com/uk/catalogue/?ci=9780195094381 K. Binder (ed.) “Monte Carlo and Molecular Dynamics Simulations in Polymer Science”, Oxford University Press (1995)]<br />
#[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)]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Bond_fluctuation_model&diff=2478Bond fluctuation model2007-05-30T14:35:32Z<p>62.204.202.244: </p>
<hr />
<div><br />
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.<br />
<br />
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>, but bonds vectors of the type (2,2,0) are excluded to avoid bond crossing during the simulation. <br />
<br />
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: <br />
*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]]. <br />
*A variety of bond lengths alleviate the restrictions due to the lattice constraints and, therefore, the model is more similar to the continuum behavior. <br />
*It is possible to perform simulations by using a single type of elementary bead jump. 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]]. <br />
* The elementary jump moves can directly used in polymers with branch points ([[stars]], [[combs]], [[dendrimers]]).<br />
==References==<br />
[[Category: Models]]<br />
<br />
#[http://www.oup.com/uk/catalogue/?ci=9780195094381 K. Binder (ed.) “Monte Carlo and Molecular Dynamics Simulations in Polymer Science”, Oxford University Press (1995)]<br />
#[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)]</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Bond_fluctuation_model&diff=2469Bond fluctuation model2007-05-30T12:18:00Z<p>62.204.202.244: </p>
<hr />
<div><p><br />
The bond fluctuation model has been used in the recent past to simulate a great variety of polymer systems.[1-2] It represents non-fixed bond lengths and 8-site non-overlapping beads placed in a simple cubic lattice. Bonds linking the beads may have lengths ranging between 2 and sqrt(10), but bonds vectors of the type (2,2,0) are excluded to avoid bond crossing during the simulation. </p><br />
<p><br />
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 in a simple cubic lattice: 1)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. 2) A variety of bond lengths alleviate the restrictions due to the lattice constraints and, therefore, the model is more similar to the continuum behavior. 3)It is possible to perform simulations by using a single type of elementary bead jump. 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. 4) The elementary jump moves can directly used in polymers with branch points (stars, combs, dendrimers). </p><br />
==References==<br />
[[Category: Models]]<br />
<br />
[1] K. Binder (ed.) “Monte Carlo and Molecular Dynamics Simulations in Polymer Science”, Oxford University Press, New York, 1995.<p><br />
<br />
[2] I. Carmesin, K. Kremer, Macromolecules 21, 2819 (1988).</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Bond_fluctuation_model&diff=2468Bond fluctuation model2007-05-30T12:03:51Z<p>62.204.202.244: /* References */</p>
<hr />
<div><p><br />
The bond fluctuation model has been used in the recent past to simulate a great variety of polymer systems.[1-2] It represents non-fixed bond lengths and 8-site non-overlapping beads placed in a simple cubic lattice. Bonds linking the beads may have lengths ranging between 2 and sqrt(10), but bonds vectors of the type (2,2,0) are excluded to avoid bond crossing during the simulation. </p><br />
<p><br />
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 in a simple cubic lattice: 1)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. 2) A variety of bond lengths alleviate the restrictions due to the lattice constraints and, therefore, the model is more similar to the continuum behavior. 3)It is possible to perform simulations by using a single type of elementary bead jump. 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. 4) The elementary jump moves can directly used in polymers with branch points (stars, combs, dendrimers). </p><br />
==References==<br />
[[Category: Models]]<br />
<br />
[1] K. Binder (ed.) “Monte Carlo and Molecular Dynamics Simulations in Polymer Science”, Oxford University Press, New York, 1995.<p><br />
<br />
[2] I. Carmesin, K. Kremer, Macromolecules 1988, 21, 2819.</div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Bond_fluctuation_model&diff=2467Bond fluctuation model2007-05-30T12:03:02Z<p>62.204.202.244: /* References */</p>
<hr />
<div>==References==<br />
[[Category: Models]]<br />
<p><br />
The bond fluctuation model has been used in the recent past to simulate a great variety of polymer systems.[1-2] It represents non-fixed bond lengths and 8-site non-overlapping beads placed in a simple cubic lattice. Bonds linking the beads may have lengths ranging between 2 and sqrt(10), but bonds vectors of the type (2,2,0) are excluded to avoid bond crossing during the simulation. </p><br />
<p><br />
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 in a simple cubic lattice: 1)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. 2) A variety of bond lengths alleviate the restrictions due to the lattice constraints and, therefore, the model is more similar to the continuum behavior. 3)It is possible to perform simulations by using a single type of elementary bead jump. 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. 4) The elementary jump moves can directly used in polymers with branch points (stars, combs, dendrimers). </p><br />
<br />
[1] K. Binder (ed.) “Monte Carlo and Molecular Dynamics Simulations in Polymer Science”, Oxford University Press, New York, 1995.<p><br />
<br />
[2] I. Carmesin, K. Kremer, Macromolecules 1988, 21, 2819.<p></div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Bond_fluctuation_model&diff=2466Bond fluctuation model2007-05-30T12:02:20Z<p>62.204.202.244: /* References */</p>
<hr />
<div>==References==<br />
[[Category: Models]]<br />
<br />
The bond fluctuation model has been used in the recent past to simulate a great variety of polymer systems.[1-2] It represents non-fixed bond lengths and 8-site non-overlapping beads placed in a simple cubic lattice. Bonds linking the beads may have lengths ranging between 2 and sqrt(10), but bonds vectors of the type (2,2,0) are excluded to avoid bond crossing during the simulation. <p><br />
<br />
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 in a simple cubic lattice: 1)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. 2) A variety of bond lengths alleviate the restrictions due to the lattice constraints and, therefore, the model is more similar to the continuum behavior. 3)It is possible to perform simulations by using a single type of elementary bead jump. 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. 4) The elementary jump moves can directly used in polymers with branch points (stars, combs, dendrimers). <p><br />
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[1] K. Binder (ed.) “Monte Carlo and Molecular Dynamics Simulations in Polymer Science”, Oxford University Press, New York, 1995.<p><br />
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[2] I. Carmesin, K. Kremer, Macromolecules 1988, 21, 2819.<p></div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Bond_fluctuation_model&diff=2465Bond fluctuation model2007-05-30T12:01:54Z<p>62.204.202.244: /* References */</p>
<hr />
<div>==References==<br />
[[Category: Models]]<br />
<br />
The bond fluctuation model has been used in the recent past to simulate a great variety of polymer systems.[1-2] It represents non-fixed bond lengths and 8-site non-overlapping beads placed in a simple cubic lattice. Bonds linking the beads may have lengths ranging between 2 and sqrt(10), but bonds vectors of the type (2,2,0) are excluded to avoid bond crossing during the simulation. <p><br />
<br />
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 in a simple cubic lattice: 1)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. 2) A variety of bond lengths alleviate the restrictions due to the lattice constraints and, therefore, the model is more similar to the continuum behavior. 3)It is possible to perform simulations by using a single type of elementary bead jump. 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. 4) The elementary jump moves can directly used in polymers with branch points (stars, combs, dendrimers). <p><br />
<br />
[1] K. Binder (ed.) “Monte Carlo and Molecular Dynamics Simulations in Polymer Science”, Oxford University Press, New York, 1995.<p><br />
[2] I. Carmesin, K. Kremer, Macromolecules 1988, 21, 2819.<p></div>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Bond_fluctuation_model&diff=2464Bond fluctuation model2007-05-30T12:00:57Z<p>62.204.202.244: </p>
<hr />
<div>==References==<br />
[[Category: Models]]<br />
<br />
The bond fluctuation model has been used in the recent past to simulate a great variety of polymer systems.[1-2] It represents non-fixed bond lengths and 8-site non-overlapping beads placed in a simple cubic lattice. Bonds linking the beads may have lengths ranging between 2 and sqrt(10), but bonds vectors of the type (2,2,0) are excluded to avoid bond crossing during the simulation.<br />
<br />
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 in a simple cubic lattice: 1)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. 2) A variety of bond lengths alleviate the restrictions due to the lattice constraints and, therefore, the model is more similar to the continuum behavior. 3)It is possible to perform simulations by using a single type of elementary bead jump. 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. 4) The elementary jump moves can directly used in polymers with branch points (stars, combs, dendrimers).<br />
<br />
[1] K. Binder (ed.) “Monte Carlo and Molecular Dynamics Simulations in Polymer Science”, Oxford University Press, New York, 1995.<br />
[2] I. Carmesin, K. Kremer, Macromolecules 1988, 21, 2819.</div>62.204.202.244