Rouse Dynamics of Topological Polymers through Gaussian Random Embeddings and Comparison with Experiments

<p class=MsoNormal align=left style='text-align:left;line-height:14.0pt; mso-line-height-rule:exactly;mso-layout-grid-align:none;text-autospace:none'><span lang=EN-US style='font-size:11.0pt;mso-bidi-font-size:12.0pt;mso-fareast-font-family: "\FF2D\FF33 \660E\671D";mso-fareast-language:JA'>Recently, polymers of complex chemical connectivity expressed with graphs have been synthesized in experiments [1-3]. It is indeed marvelous that even such polymers expressed with K3<span class=GramE>,3 bipartite graph have been produced [2]. We call polymers with nontrivial structures in chemical connectivity <i style='mso-bidi-font-style:normal'>topological polymers. We also call polymers with nontrivial topology of spatial graphs as embeddings in three dimensions topological polymers [4]. <p class=MsoNormal align=left style='text-align:left;line-height:14.0pt; mso-line-height-rule:exactly;mso-layout-grid-align:none;text-autospace:none'><span lang=EN-US style='font-size:11.0pt;mso-bidi-font-size:12.0pt;mso-fareast-font-family: "\FF2D\FF33 \660E\671D";mso-fareast-language:JA'>The Rouse dynamics of a polymer plays an important role in the dynamical aspects of the polymer in dilute solution, and also in melts if the molecular weight is small [5]. In this talk, we formulate the Rouse dynamics of the topological polymer with a given graph. We derive several physical consequences of the model. In particular, we compare the experimental data of Size Exclusion Chromatography (SEC) of some topological polymers with their theoretical estimates of the mean-square radius of gyration obtained by the Gaussian method [6]. We argue physical backgrounds of SEC data and discuss how they are consistent. <p class=MsoNormal align=left style='text-align:left;line-height:14.0pt; mso-line-height-rule:exactly;mso-layout-grid-align:none;text-autospace:none'><span lang=EN-US style='font-size:11.0pt;mso-bidi-font-size:12.0pt;mso-fareast-font-family: "\FF2D\FF33 \660E\671D";mso-fareast-language:JA'>After reviewing the method for constructing Gaussian random configurations of a topological polymer, i.e. Gaussian random graph embeddings [6], we derive the normal coordinates and modes for the topological polymer. Here we remark that while the Moore-Penrose generalized inverse matrix has been addressed for general Gaussian molecules about three decades ago [7], it seems that important consequences were found later independently [8]. Moreover, it has not been known until quite recently how to generate Gaussian random configurations of a given topological polymer [6]. In fact, we can calculate any physical quantity at least numerically by taking the ensemble averages over generated configurations. It is quite nontrivial to generate such random walks that satisfy the constraints of all independent loops in the graph. <p class=MsoNormal align=left style='text-align:left;line-height:14.0pt; mso-line-height-rule:exactly;mso-layout-grid-align:none;text-autospace:none'><span lang=EN-US style='font-size:11.0pt;mso-bidi-font-size:12.0pt;mso-fareast-font-family: "\FF2D\FF33 \660E\671D";mso-fareast-language:JA'>The results of this talk are obtained in collaboration with Jason Cantarella, Clayton Shonkwiler, and Erica Uehara. <i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:11.0pt; mso-bidi-font-size:12.0pt;mso-fareast-font-family:"Times New Roman";mso-fareast-language: JA'>1)&nbsp;&nbsp;&nbsp; <span lang=EN-US style='font-size:11.0pt;mso-bidi-font-size:12.0pt;mso-fareast-font-family: "\FF2D\FF33 \660E\671D";mso-fareast-language:JA'>Topological Polymer Chemistry: Progress of cyclic polymers in synthesis, properties and <i style='mso-bidi-font-style: normal'><span lang=EN-US style='font-size:11.0pt;mso-bidi-font-size:12.0pt; mso-fareast-font-family:"\FF2D\FF33 \660E\671D";mso-fareast-language:JA'><span style="mso-spacerun:yes">&nbsp;functions<span lang=EN-US style='font-size:11.0pt;mso-bidi-font-size:12.0pt;mso-fareast-font-family: "\FF2D\FF33 \660E\671D";mso-fareast-language:JA'>, Y. Tezuka ed., World Scientific, Singapore, 2013.&nbsp; <span style="mso-spacerun:yes">&nbsp;&nbsp; <p class=MsoListParagraph style='margin-left:18.0pt;mso-para-margin-left:0gd; text-indent:-18.0pt;line-height:115%;mso-list:l3 level1 lfo8;text-autospace: none;word-break:break-all'><span lang=EN-US style='font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:115%;mso-fareast-font-family: "Times New Roman";mso-font-kerning:0pt;mso-fareast-language:JA'><span style='mso-list:Ignore'>2)&nbsp;&nbsp;&nbsp; <span lang=EN-US style='font-size:11.0pt; mso-bidi-font-size:12.0pt;line-height:115%;color:black;mso-font-kerning:0pt; mso-fareast-language:JA'>T. Suzuki, T. Yamamoto and Y. Tezuka. J. Am. Chem. Soc., 2014, 136, 10148&#8211;10155.<span lang=EN-US style='font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:115%; mso-font-kerning:0pt;mso-fareast-language:JA'> <p class=MsoListParagraph style='margin-left:18.0pt;mso-para-margin-left:0gd; text-indent:-18.0pt;line-height:115%;mso-list:l3 level1 lfo8;text-autospace: none;word-break:break-all'><span lang=EN-US style='font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:115%;mso-fareast-font-family: "Times New Roman";mso-font-kerning:0pt;mso-fareast-language:JA'><span style='mso-list:Ignore'>3)&nbsp;&nbsp;&nbsp; <span lang=EN-US style='font-size:11.0pt; mso-bidi-font-size:12.0pt;line-height:115%;color:black;mso-font-kerning:0pt; mso-fareast-language:JA'>Y. Tezuka. Acc. Chem. 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Edwards, The Theory of Polymer Dynamics, Oxford University Press, Oxford, 1986.<span style="mso-spacerun:yes">&nbsp; <p class=MsoListParagraph style='margin-left:18.0pt;mso-para-margin-left:0gd; text-indent:-18.0pt;line-height:115%;mso-list:l3 level1 lfo8;text-autospace: none;word-break:break-all'><span lang=EN-US style='font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:115%;mso-fareast-font-family: "Times New Roman";mso-font-kerning:0pt;mso-fareast-language:JA'><span style='mso-list:Ignore'>6)&nbsp;&nbsp;&nbsp; <span lang=EN-US style='font-size:11.0pt; mso-bidi-font-size:12.0pt;line-height:115%;mso-font-kerning:0pt;mso-fareast-language: JA'>J. Cantarella, T. Deguchi, C. Shonkwiler and E. 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