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Seiichi Inayama

Publications and source records attributed to Seiichi Inayama.

10 recordsLinked to original sources
arXiv2026

On the Determination of Gel Points

A critical composition of cross-linked polysiloxanes observed by Scanlan and Winter is reinvestigated in comparison with the theory of gelation. We assume, based on the Scott findings, the geometric distribution for one of the monomers, divinyl-terminated poly(dimethylsiloxane). Calculation results show that the two theories are in near-consistency, supporting the Scanlan-Winter estimation based on the linear viscoelastic theory. On the other hand, there is a disturbing result that calculation using the mean molecular weight, $M_{n}$, leads to exact agreement between the two theories, suggesting that the distribution is in effect monodisperse, contrary to the assumed geometric one and also the observed polydispersity, $M_w/M_n=2.1$. Further experimental studies employing monodisperse monomers would be highly valuable to consolidate the bridge between these two fundamental theories.

cond-mat.soft↗
arXiv2024

Critical Nature of the Size Exponent of Polymers

On the basis of the thermodynamic theory of the excluded volume effects, we show that the size exponent varies abruptly, depending on the change of the segment concentration. For linear polymers, the exponent changes discontinuously from $\nu=3/5$ for the isolated system ($\bar{\phi}=0$) in good solvents to $\nu=1/2$ in the finite concentration ($0<\bar{\phi}\le1$), while for branched polymers having $\nu_{0}=1/4$, the corresponding exponent varies from $\nu=1/2$ ($\bar{\phi}=0$) to $\nu\cong 1/3$ ($0<\bar{\phi}\le1$).

cond-mat.soft↗
arXiv2022

Size Exponents of Branched Polymers/ Extension of the Isaacson-Lubensky Formula and the Application to Lattice Trees

Branched polymers can be classified into two categories that obey the different formulae: \begin{equation} ν= \begin{cases} \hspace{1mm}\displaystyle\frac{2(1+ν_{0})}{d+2} & \hspace{3mm}\mbox{for polymers with}\hspace{2mm}\displaystyleν_{0}\ge\frac{1}{d+1}\hspace{10mm}\text{(I)}\\[3mm] \hspace{5mm}2ν_{0}& \hspace{3mm}\mbox{for polymers with}\hspace{2mm}\displaystyleν_{0}\le\frac{1}{d+1}\hspace{10mm}\text{(II)} \end{cases}\notag \end{equation} for the dilution limit in good solvents. The category II covers the exceptional polymers having fully expanded configurations. On the basis of these equalities, we discuss the size exponents of the nested architectures and the lattice trees. In particular, we compare our preceding result, $ν_{d=2}=1/2$, for the $z$=2 polymer having $ν_{0}=1/4$ with the numerical result, $ν_{d=2}\doteq 0.64115$, for the lattice trees generated on the 2-dimensional lattice. Our conjecture is that while both the conclusions in polymer physics and condensed matter physics are correct, the discrepancy arises from the fact that the lattice trees are constructed from less branched architectures than the branched polymers having $ν_{0} = 1/4$ in polymer physics. The present analysis suggests that the 2-dimensional lattice trees are the mixture of isomers having the mean ideal size exponent of $\barν_{0}\doteq0.32$.

cond-mat.soft↗
arXiv2022

Branched Polymers with Excluded Volume Effects/ Configurations of Comb Polymers in Two- and Three-dimensions

We investigate the excluded volume effects in good solvents for the isolated comb polymers having $ν_{0}=1/4$. In particular, we investigate the change of the size exponent, $ν$, defined by $\langle s_{N}^{2}\rangle\propto N^{2ν}$, for the various fully-expanded configurations. The results show that, given the fully stretched backbone and side chains, the exponent takes the value, $ν=1/2$, irrespective of the configurational isomerization of side chains; only the pre-exponential factor changes.

cond-mat.soft↗
arXiv2021

Branched Polymers with Excluded Volume Effects/ Relationship between Polymer Dimensions and Generation Number

We discuss the extension of the empirical equation: $\left\langle s_{N}^{2}\right\rangle_{0}\propto g\,l^{2}$, where the subscript 0 denotes the ideal value with no excluded volume and $g$ the generation number from the root to the youngest (outermost) generation. By analogy with the linear chain problem, we introduce the assumption that the scaling relation, $\left\langle s_{N}^{2}\right\rangle_{0}\propto g^{2λ}\,l^{2}$, exists for arbitrary polymeric architectures, where $λ$ is an exponent for the backbone structure. Then, making use of the relationship between $g$ and $N$ (monomer number), we can deduce the exponent, $ν$, for polymers with various architectures. The theory of the excluded volume effects impose the severe restriction on the quantities: $ν_{0}$, $ν$, and $λ$; for instance, the inequality, $ν_{0}\ge\frac{1}{d+1}$, must be satisfied for isolated polymers in good solvents. An intriguing question is whether or not there exists an actual molecule that violates this inequality. We take up two examples having $ν_{0}=1/4$ for $d=2$ and $ν_{0}=1/6$ for $d=3$, and discuss this question.

cond-mat.soft↗
arXiv2020

Segment Distribution around the Center of Gravity of a Triangular Polymer

The segment distribution around the center of gravity is investigated for a special comb polymer (triangular polymer) having the side chains of the same generation number, $g$, as the main backbone. Common to all the other polymers, the radial mass distribution is expressed as the sum of the distribution functions for the end-to-end vectors, $\{\vec{r}_{Gh}\}$, from the center of gravity to the monomers on the $h$th generation; the result being, for a large $g$, \begin{equation} φ_{\text{triang}}(s)=\frac{1}{N}\left\{\sum_{h=1}^{g}\left(\frac{d}{2π\left\langle r_{Gh}^{2}\right\rangle}\right)^{\frac{d}{2}}\text{Exp}\left(-\frac{d}{2\left\langle r_{Gh}^{2}\right\rangle}s^2\right)+\sum_{h=2}^{g}\sum_{j=1}^{g-h}\left(\frac{d}{2π\left\langle r_{Gh_{j}}^{2}\right\rangle}\right)^{\frac{d}{2}}\text{Exp}\left(-\frac{d}{2\left\langle r_{Gh_{j}}^{2}\right\rangle}s^2\right)\right\}\notag \end{equation} It is found that the mean square of the radius of gyration varies as $\left\langle s_{N}^{2}\right\rangle_{0}\doteq\frac{7}{15}\,g\,l^{2}$, as $g\rightarrow\infty$. Since $g\propto \sqrt{N}$ for the triangular polymer, this leads to $\left\langle s_{N}^{2}\right\rangle_{0}^{1/2}\propto N^{1/4}$, giving the same exponent as observed for the randomly branched polymer. On the basis of the present result, we put forth that all the known polymers obey the equality: $\left\langle s_{N}^{2}\right\rangle_{0}=A\, g\,l^{2}$, where $A$ is a polymer-species-dependent coefficient and also depends on the choice of the root monomer. We discuss the extension of this empirical equation.

cond-mat.soft↗
arXiv2020

Segment Distribution around the Center of Gravity of Branched Polymers

Mathematical expressions for mass distributions around the center of gravity are derived for branched polymers with the help of the Isihara formula. We introduce the Gaussian approximation for the end-to-end vector, $\vec{r}_{Gν_{i}}$, from the center of gravity to the $i$th mass point on the $ν$th arm. Then, for star polymers, the result is \begin{equation} φ_{star}(s)=\frac{1}{N}\sum_{ν=1}^{f}\sum_{i=1}^{N_ν}\left(\frac{d}{2π\left\langle r_{Gν_{i}}^{2}\right\rangle}\right)^{d/2}\exp\left(-\frac{d}{2\left\langle r_{Gν_{i}}^{2}\right\rangle}s^{2}\right)\notag \end{equation} for a sufficiently large $N$, where $f$ denotes the number of arms. It is found that the resultant $φ_{star}(s)$ is, unfortunately, not Gaussian. For dendrimers \begin{equation} φ_{dend}(s)=\sum_{h=1}^{g}ω_{h}\left(\frac{d}{2pi\left\langle r_{G_{h}}^{2}\right\rangle}\right)^{d/2}\exp\left(-\frac{d}{2\left\langle r_{G_{h}}^{2}\right\rangle}s^{2}\right)\notag \end{equation} where $ω_{h}$ denotes the weight fraction of masses in the $h$th generation on a dendrimer constructed from $g$ generations, so that $\sum_{h=1}^{g}ω_{h}=1$. To be specific, $ω_{1}=1/N$ and $ω_{h}=(f-1)^{h-2}/N$ for $h\ge 2$. These distributions can be described by the same grand sum of each Gaussian function for the end-to-end distance from the center of gravity to each mass point. Note that for a large $f$ and $g$, the statistical weight of younger generations becomes dominant. As a consequence, the mass distribution of unperturbed dendrimers approaches the Gaussian form in the limit of a large $f$ and $g$. It is shown that the radii of gyration of dendrimers increase logarithmically with $N$, which leading to the exponent, $ν_{0}=0$. An example of randomly branched polymers is also discussed.

cond-mat.soft↗