# Thermoreversible Gelation with Supramolecularly Polymerized Cross-Link Junctions

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Method

#### 2.1. Self-Association

#### 2.2. Linear Growth of the Cross-Link Junctions

#### 2.3. Chain/Ring Supramolecular Cross-Link Junctions

## 3. Metallo-Supramolecular Cross-Link Junctions

#### 3.1. Ladder Model

#### 3.2. Egg-Box Model

## 4. Discussion

## 5. Conclusions

- (1)
- Chain model: In addition to the sol–gel transition, there occurs a polymerization transition at a certain concentration just after the gel point is passed under a fixed temperature. The transition is not a true phase transition in the sense that it is not accompanied by any singularity in the physical properties. In particular, the average chain length grows to infinity only in the inaccessible limit of complete reaction. However, its variation becomes sharper and sharper with the cooperativity parameter, leading eventually to a singularity at finite reactivity. The increasing sharpness of the sol–gel transition with cooperativity parameter, in particular sharp rise of the gel fraction, makes the experimental detection of the gel point easier.
- (2)
- Chain/ring model: Under a certain simple condition on the association constants, a new phase transition occurs at a low temperature (large $\lambda $) deep in the postgel region, where the average length of rings goes to infinity. There appears a discontinuity in the physical properties at this condensation point of rings. The average molecular weight of the cross-linked polymers, the extinction probability, and the gel fraction all stay constant below this temperature. The transition is analogous to the Bose–Einstein condensation of an ideal Bose gas where a finite fraction of particles falls into the condensate of zero momentum.
- (3)
- Ladder model: A ladder is one of the simplest structures of multi-nuclear metal-coordinated complexes. As a function of the composition u of metal ions, there occur two transitions: one from sol to gel at a low value ${u}_{1}$, and the other from gel back to sol at a higher value ${u}_{2}$ (reentrant gel–sol transition). In the gel phase between them, there is a composition u at which the gel fraction reaches a maximum (optimal gel point). The average length of the ladder increases around this optimal gel point, but is limited within a finite value, and hence there is no polymerization transition. The ratio $\mu $ between the intra-layer association constant and the inter-layer one plays a role of the cooperativity parameter. The transitions become sharper with its decrease.
- (4)
- Egg-box model: Overall variation in physical observables is the same as the ladder model, although there are some quantitative differences. For instance, the gel fraction becomes asymmetric in the postgel region.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) A network of a tree type consisting of low molecular weight trifunctional ($f=3$) molecules with cross-link junctions of linear chains and rings. A chain of the length k (dotted line) is regarded as a connected cross-link junction of multiplicity k. Similarly, each ring of the length k is regarded as a cross-link junction of multiplicity k in the loop form. There are branching points where the primary reactive molecules have more than one reacted functional groups. The smallest ring has the size $k=3$. (

**b**) A network consisting of high molecular weight bifunctional ($f=2$) molecules (telechelic polymers) with coexisting cross-link junctions of linear chains and rings. Functional groups (low-mass gelators) are shown by the blue thick rods at the ends of molecules.

**Figure 2.**The reciprocal weight-average molecular weight (red solid lines) ${P}_{\mathrm{w}}^{-1}$ in the pregel region, and ${{P}_{\mathrm{w}}^{\left(s\right)}}^{-1}$ in the postgel region, the gel fraction ${W}_{\mathrm{gel}}$ (blue broken line), the extinction probability ${x}_{1}$ (red broken line), the reciprocal average chain length ${{\overline{\mu}}_{\mathrm{w}}}^{-1}$ (black line), and the fraction ${W}_{\mathrm{C}}$ of the reacted functional groups (green line) plotted against the volume fraction of the primary molecules for $f=3,n=6,\lambda =5.0$. The cooperativity parameter is fixed at $\sigma ={10}^{-3}$. The sol–gel transition is very sharp. There is a polymerization point just after the gel point is passed.

**Figure 3.**(

**a**) The reciprocal weight-average molecular weight (red solid lines) ${P}_{\mathrm{w}}^{-1}$ in the pregel region, and ${{P}_{\mathrm{w}}^{\left(s\right)}}^{-1}$ in the postgel region, and the gel fraction ${W}_{\mathrm{gel}}$ (blue broken lines) plotted against the volume fraction of the primary molecules. (

**b**) The reciprocal average chain length ${\overline{\mu}}_{\mathrm{w}}^{-1}$ (black lines), and the gel fraction ${W}_{\mathrm{gel}}$ (blue broken lines) plotted against the volume fraction of the primary molecules, both for $f=3,n=6,\lambda =5.0$. The cooperativity parameter is varied from curve to curve from $\sigma ={10}^{0}$ to $\sigma ={10}^{-5}$. Both the sol–gel transition and the polymerization transition become sharper and sharper with decrease in the cooperativity parameter.

**Figure 4.**Variation of physical properties characteristic to ring/chain competing TRG of telechelic polymers ($f=2,n=30$) plotted against the strength $\lambda $ of the association constant. The reciprocal of the weight-average molecular weight ${{P}_{\mathrm{w}}}^{-1}$ (red line) of the three-dimensional cross-linked polymers in the pregel region, that of the sol parts ${{P}_{\mathrm{w}}^{\left(s\right)}}^{-1}$ (red line) in the postgel region are shown. In the postgel region, we also plot gel fraction ${W}_{\mathrm{gel}}$ (blue broken line), and extinction probability ${x}_{1}$ (red broken line). The fraction of chain cross-links ${W}_{\mathrm{C}}$ (green line), and that of ring cross-links ${W}_{\mathrm{R}}$ (green broken line) are plotted in both regions. The fraction of infinite rings ${{W}_{\mathrm{R}}}_{\infty}$ (black line) start to appear at deep point inside the postgel region. The cooperativity parameters are fixed at ${\sigma}_{C}=3.00,{\sigma}_{R}=0.05$. In this model calculation, TRG occurs at $log\lambda =2.3$, while the second transition (BEC of rings) takes place at $log\lambda =4.6$, deep in the postgel region.

**Figure 5.**(

**a**) Network structure with cross-link junctions of ladder form made up of trifunctional ($f=3$) low-mass ($n=6$) molecules. The cross-linker (metal ion) is shown by a red sphere. The elementary unit of a cross-link is a sandwich complex with multiplicty index $(2,1)$. A network is made up of ladder cross-links and branch molecules [55] bearing more than one reacted functional groups. (

**b**) Ternary phase diagram for the ladder model of low-mass ($n=6$) trifunctional ($f=3$) molecules showing reentrant sol–gel–sol transition (red lines). The association constant $\lambda $ of the ladder unit is changed from curve to curve at a constant ratio $\mu =1.0$. For a given solute volume fraction $\varphi $, there are two composition ${u}_{1}$ and ${u}_{2}$ for the gel point; the former from sol to gel, and the latter from gel to sol.

**Figure 6.**Reentrant TRG with ladder cross-link junctions for trifunctional ($f=3$) low-mass (${n}_{A}=6$) molecules. (

**a**) $\mu =1.0,\lambda =8.0$, (

**b**) $\mu ={10}^{-4},\lambda =5.5\times {10}^{-3}$. There are a pregel region ($u<{u}_{1}$), a postgel region (${u}_{1}<u<{u}_{2}$), and a reentrant sol region (${u}_{1}<u$). The average molecular weight ${P}_{\mathrm{w}}^{-1}$ in the sol region, ${{P}_{\mathrm{w}}^{\left(s\right)}}^{-1}$ in the gel region, and the gel fraction ${W}_{\mathrm{gel}}$, the extinction probability ${x}_{1}$ of the functional group A, the average length ${{\overline{\mu}}_{\mathrm{w}}}^{-1}$ of the ladder cross-link junctions, all plotted as functions of the solute composition u. The total solute volume fraction is fixed at $\varphi =0.3$.

**Figure 7.**Networks formed by egg-box cross-link junctions made up of (

**a**) trifunctional low-mass ($f=3,{n}_{A}\sim 1$) molecules, (

**b**) telechelic polymers ($f=2,{n}_{A}>>1$). Cross-linkers (metal ions) are indicated by red spheres. The elementary unit of a cross-link is an egg-box complex with multiplicity index $(4,1)$. A network is made up of linear assembly of egg-boxes and branch molecules bearing more than one reacted functional groups A.

**Figure 8.**Rentrant TRG with egg-box cross-link junctions for telechelic polymers. (

**a**) $\mu =1.0,\lambda =40$ and (

**b**) $\mu ={10}^{-4},\lambda =1.9\times {10}^{3}$. The average molecular weight ${P}_{\mathrm{w}}^{-1}$ in the sol region (red lines), ${{P}_{\mathrm{w}}^{\left(s\right)}}^{-1}$ in the gel region (red line), and the gel fraction ${W}_{\mathrm{gel}}$ (blue broken line), the extinction probability ${x}_{1}$ of the functional group A (red broken line), the average length ${{\overline{\mu}}_{\mathrm{w}}}^{-1}$ of the egg-box cross-link junctions (green line), all plotted as functions of the solute composition u. The total solute volume fraction is fixed at $\varphi =0.3$.

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**MDPI and ACS Style**

Tanaka, F.
Thermoreversible Gelation with Supramolecularly Polymerized Cross-Link Junctions. *Gels* **2023**, *9*, 820.
https://doi.org/10.3390/gels9100820

**AMA Style**

Tanaka F.
Thermoreversible Gelation with Supramolecularly Polymerized Cross-Link Junctions. *Gels*. 2023; 9(10):820.
https://doi.org/10.3390/gels9100820

**Chicago/Turabian Style**

Tanaka, Fumihiko.
2023. "Thermoreversible Gelation with Supramolecularly Polymerized Cross-Link Junctions" *Gels* 9, no. 10: 820.
https://doi.org/10.3390/gels9100820