http://www.sklogwiki.org/SklogWiki/index.php?action=history&feed=atom&title=Rouse_modelRouse model - Revision history2026-05-01T03:18:48ZRevision history for this page on the wikiMediaWiki 1.41.0http://www.sklogwiki.org/SklogWiki/index.php?title=Rouse_model&diff=19422&oldid=prevCarl McBride: Slight tidy of references2016-12-01T10:50:07Z<p>Slight tidy of references</p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:50, 1 December 2016</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Gaussian distribution]]. Therefore, the [[polymers | polymer]] chain can be described by a </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Gaussian distribution]]. Therefore, the [[polymers | polymer]] chain can be described by a </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[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 ([[Harmonic spring approximation |Hookean springs]]). The Rouse model <del style="font-weight: bold; text-decoration: none;">(Ref</del>. 1) describes a polymer chain as a set of N coupled [[Harmonic spring approximation | 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>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[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 ([[Harmonic spring approximation |Hookean springs]]). The Rouse model <ins style="font-weight: bold; text-decoration: none;"><ref>[http://dx.doi.org/10</ins>.<ins style="font-weight: bold; text-decoration: none;">1063/</ins>1<ins style="font-weight: bold; text-decoration: none;">.1699180 Prince E. Rouse, Jr. "A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers", Journal of Chemical Physics '''21''' pp. 1272-1280 (1953</ins>)<ins style="font-weight: bold; text-decoration: none;">]</ref> </ins>describes a polymer chain as a set of N coupled [[Harmonic spring approximation | 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>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In dilute solution, the relaxation times obtained from the Rouse model are not correct due to the presence of hydrodynamic interactions between units, introduced by Zimm (<del style="font-weight: bold; text-decoration: none;">Ref. 2</del>). 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 solvent | theta]]) solvents. For solvent of good quality, excluded volume interactions have also to be accounted. However, the scheme of normal modes is maintained (<del style="font-weight: bold; text-decoration: none;">Ref. 3</del>).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In dilute solution, the relaxation times obtained from the Rouse model are not correct due to the presence of hydrodynamic interactions between units, introduced by Zimm <ins style="font-weight: bold; text-decoration: none;"><ref>[http://dx.doi.org/10.1063/1.1742462 Bruno H. Zimm "Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss", Journal of Chemical Physics '''24''' pp. 269-278 </ins>(<ins style="font-weight: bold; text-decoration: none;">1956</ins>)<ins style="font-weight: bold; text-decoration: none;">]</ref></ins>. 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 solvent | theta]]) solvents. For solvent of good quality, excluded volume interactions have also to be accounted. However, the scheme of normal modes is maintained <ins style="font-weight: bold; text-decoration: none;"><ref>[http://www.oup.com/uk/catalogue/?ci=9780198520337 M. Doi and S. F. Edwards "The Theory of Polymer Dynamics" Oxford University Press </ins>(<ins style="font-weight: bold; text-decoration: none;">1988</ins>)<ins style="font-weight: bold; text-decoration: none;">]</ref></ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#[http:</del>/<del style="font-weight: bold; text-decoration: none;">/dx.doi.org/10.1063/1.1699180 Prince E. Rouse, Jr. "A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers", Journal of Chemical Physics '''21''' pp. 1272-1280 (1953)]</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><references</ins>/<ins style="font-weight: bold; text-decoration: none;">></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#[http://dx.doi.org/10.1063/1.1742462 Bruno H. Zimm "Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss", Journal of Chemical Physics '''24''' pp. 269-278 (1956)]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#[http://www.oup.com/uk/catalogue/?ci=9780198520337 M. Doi and S. F. Edwards "The Theory of Polymer Dynamics" Oxford University Press (1988)]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Models]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Models]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: polymers]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: polymers]]</div></td></tr>
</table>Carl McBridehttp://www.sklogwiki.org/SklogWiki/index.php?title=Rouse_model&diff=3477&oldid=prevNice and Tidy at 14:48, 30 July 20072007-07-30T14:48:26Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:48, 30 July 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Gaussian distribution]]. Therefore, the [[polymers | polymer]] chain can be described by a </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Gaussian distribution]]. Therefore, the [[polymers | polymer]] chain can be described by a </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[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 ([[Harmonic spring approximation | Hookean springs]]). The Rouse model (Ref. 1) describes a polymer chain as a set of N coupled [[Harmonic spring approximation | 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>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[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 ([[Harmonic spring approximation |Hookean springs]]). The Rouse model (Ref. 1) describes a polymer chain as a set of N coupled [[Harmonic spring approximation | 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-<ins style="font-weight: bold; text-decoration: none;">''</ins>th<ins style="font-weight: bold; text-decoration: none;">'' </ins>Rouse relaxation time is proportional to <math>1/p^2</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In dilute solution, the relaxation times obtained from the Rouse model <del style="font-weight: bold; text-decoration: none;">is </del>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 solvent | 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).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In dilute solution, the relaxation times obtained from the Rouse model <ins style="font-weight: bold; text-decoration: none;">are </ins>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 solvent | 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).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
</table>Nice and Tidyhttp://www.sklogwiki.org/SklogWiki/index.php?title=Rouse_model&diff=3475&oldid=prevCarl McBride at 14:40, 30 July 20072007-07-30T14:40:46Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:40, 30 July 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Gaussian distribution]]. Therefore, the [[polymers | polymer]] chain can be described by a </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Gaussian distribution]]. Therefore, the [[polymers | polymer]] chain can be described by a </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[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>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[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 (<ins style="font-weight: bold; text-decoration: none;">[[Harmonic spring approximation | </ins>Hookean springs<ins style="font-weight: bold; text-decoration: none;">]]</ins>). The Rouse model (Ref. 1) describes a polymer chain as a set of N coupled <ins style="font-weight: bold; text-decoration: none;">[[Harmonic spring approximation | </ins>harmonic springs<ins style="font-weight: bold; text-decoration: none;">]]</ins>. 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>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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 solvent | 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).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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 solvent | 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).</div></td></tr>
</table>Carl McBridehttp://www.sklogwiki.org/SklogWiki/index.php?title=Rouse_model&diff=3470&oldid=prevCarl McBride at 14:23, 30 July 20072007-07-30T14:23:31Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[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>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[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>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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 <ins style="font-weight: bold; text-decoration: none;">[[Theta solvent | </ins>theta<ins style="font-weight: bold; text-decoration: none;">]]</ins>) solvents. For solvent of good quality, excluded volume interactions have also to be accounted. However, the scheme of normal modes is maintained (Ref. 3).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
</table>Carl McBridehttp://www.sklogwiki.org/SklogWiki/index.php?title=Rouse_model&diff=3469&oldid=prevCarl McBride: /* References */2007-07-30T14:17:41Z<p><span dir="auto"><span class="autocomment">References</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:17, 30 July 2007</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">#[http://dx.doi.org/10.1063/1.1699180 Prince E. Rouse, Jr. "A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers", Journal of Chemical Physics '''21''' pp. 1272-1280 (1953)]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">#[http://dx.doi.org/10.1063/1.1742462 Bruno H. Zimm "Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss", Journal of Chemical Physics '''24''' pp. 269-278 (1956)]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">#[http://www.oup.com/uk/catalogue/?ci=9780198520337 M. Doi and S. F. Edwards "The Theory of Polymer Dynamics" Oxford University Press (1988)]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Models]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Models]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: polymers]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: polymers]]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#[http://dx.doi.org/10.1063/1.1699180 Prince E. Rouse, Jr. "A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers", Journal of Chemical Physics '''21''' pp. 1272-1280 (1953)]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#[http://dx.doi.org/10.1063/1.1742462 Bruno H. Zimm "Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss", Journal of Chemical Physics '''24''' pp. 269-278 (1956)]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"># M. Doi, S. F. Edwards "The Theory of Polymer Dynamcis", Clarendon, Oxford, 1986.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
</table>Carl McBridehttp://www.sklogwiki.org/SklogWiki/index.php?title=Rouse_model&diff=3468&oldid=prevCarl McBride: /* References */2007-07-30T14:14:01Z<p><span dir="auto"><span class="autocomment">References</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Models]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Models]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: polymers]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: polymers]]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div># <del style="font-weight: bold; text-decoration: none;">P</del>. E. Rouse, <del style="font-weight: bold; text-decoration: none;">J</del>. <del style="font-weight: bold; text-decoration: none;">Chem</del>. <del style="font-weight: bold; text-decoration: none;">Phys. 21, </del>1272 (1953)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#<ins style="font-weight: bold; text-decoration: none;">[http://dx</ins>.<ins style="font-weight: bold; text-decoration: none;">doi.org/10.1063/1.1699180 Prince </ins>E. Rouse, <ins style="font-weight: bold; text-decoration: none;">Jr</ins>. <ins style="font-weight: bold; text-decoration: none;">"A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers", Journal of Chemical Physics '''21''' pp</ins>. 1272<ins style="font-weight: bold; text-decoration: none;">-1280 </ins>(1953)<ins style="font-weight: bold; text-decoration: none;">]</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div># <del style="font-weight: bold; text-decoration: none;">B</del>. H.Zimm, <del style="font-weight: bold; text-decoration: none;">J</del>. <del style="font-weight: bold; text-decoration: none;">Chem. Phys. 24, </del>269 (1956)<del style="font-weight: bold; text-decoration: none;">.</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#<ins style="font-weight: bold; text-decoration: none;">[http://dx.doi.org/10</ins>.<ins style="font-weight: bold; text-decoration: none;">1063/1.1742462 Bruno </ins>H. Zimm <ins style="font-weight: bold; text-decoration: none;">"Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity</ins>, <ins style="font-weight: bold; text-decoration: none;">Flow Birefringence and Dielectric Loss", Journal of Chemical Physics '''24''' pp</ins>. 269<ins style="font-weight: bold; text-decoration: none;">-278 </ins>(1956)<ins style="font-weight: bold; text-decoration: none;">]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># M. Doi, S. F. Edwards "The Theory of Polymer Dynamcis", Clarendon, Oxford, 1986.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># M. Doi, S. F. Edwards "The Theory of Polymer Dynamcis", Clarendon, Oxford, 1986.</div></td></tr>
</table>Carl McBridehttp://www.sklogwiki.org/SklogWiki/index.php?title=Rouse_model&diff=3465&oldid=prev62.204.202.244 at 14:02, 30 July 20072007-07-30T14:02:11Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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).</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In <del style="font-weight: bold; text-decoration: none;">semi</del>-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.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In <ins style="font-weight: bold; text-decoration: none;">non</ins>-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.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Models]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Models]]</div></td></tr>
</table>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Rouse_model&diff=3464&oldid=prev62.204.202.244 at 14:00, 30 July 20072007-07-30T14:00:42Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:00, 30 July 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l3">Line 3:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[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>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[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>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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^{<del style="font-weight: bold; text-decoration: none;">(</del>3/2<del style="font-weight: bold; text-decoration: none;">)</del>}</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).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
</table>62.204.202.244http://www.sklogwiki.org/SklogWiki/index.php?title=Rouse_model&diff=3463&oldid=prevCarl McBride at 14:00, 30 July 20072007-07-30T14:00:16Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<col class="diff-content" />
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<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:00, 30 July 2007</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Models]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Models]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: polymers]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: polymers]]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">1. </del>P. E. Rouse, J. Chem. Phys. 21, 1272 (1953)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"># </ins>P. E. Rouse, J. Chem. Phys. 21, 1272 (1953)</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">2. </del>B. H.Zimm, J. Chem. Phys. 24, 269 (1956).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"># </ins>B. H.Zimm, J. Chem. Phys. 24, 269 (1956).</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">3. </del>M. Doi, S. F. Edwards "The Theory of Polymer Dynamcis", Clarendon, Oxford, 1986.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"># </ins>M. Doi, S. F. Edwards "The Theory of Polymer Dynamcis", Clarendon, Oxford, 1986.</div></td></tr>
</table>Carl McBridehttp://www.sklogwiki.org/SklogWiki/index.php?title=Rouse_model&diff=3462&oldid=prev62.204.202.244 at 13:59, 30 July 20072007-07-30T13:59:03Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:59, 30 July 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Gaussian distribution]]. Therefore, the [[polymers | polymer]] chain can be described by a </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Gaussian distribution]]. Therefore, the [[polymers | polymer]] chain can be described by a </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[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 forces between units jointed by the Gaussian segments <del style="font-weight: bold; text-decoration: none;">that </del>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 <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>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[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 <ins style="font-weight: bold; text-decoration: none;">that the </ins>forces between units jointed by the Gaussian segments are proportional to their distances (Hookean springs). The Rouse model <ins style="font-weight: bold; text-decoration: none;">(Ref. 1) </ins>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>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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 <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.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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 <ins style="font-weight: bold; text-decoration: none;">(Ref. 2)</ins>. 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 <ins style="font-weight: bold; text-decoration: none;">(Ref. 3)</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Models]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Models]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: polymers]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: polymers]]</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">1. P. E. Rouse, J. Chem. Phys. 21, 1272 (1953)</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">2. B. H.Zimm, J. Chem. Phys. 24, 269 (1956).</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">3. M. Doi, S. F. Edwards "The Theory of Polymer Dynamcis", Clarendon, Oxford, 1986.</ins></div></td></tr>
</table>62.204.202.244