---
title: "Phase Transition Kinetics of Self-Assembling Borosilicate Nanopores in Response to Tunable Ionic Strength Gradients"
authors: ["Grigor Vosganian", "Philipp Keller", "Deniz Yıldırım"]
affiliations: ["Armenian Scholarly Society of Diachronic Studies", "Munichian Collegium for Applied Mechanization", "Aegean Dynamics Think Tank"]
journal: "Annals of Non-Equilibrium Matter Dynamics"
year: 2025
volume: 56
issue: 672
pages: "941-969"
citations: 293
url: https://bilimsci.nasilalinir.com/papers/phase-transition-kinetics-of-self-assembling-borosilicate-nanopores-in-response
---

# Phase Transition Kinetics of Self-Assembling Borosilicate Nanopores in Response to Tunable Ionic Strength Gradients

Grigor Vosganian, Philipp Keller, Deniz Yıldırım

## Abstract

We investigate the controlled self-assembly of silica-borate structures exhibiting tunable porosity. By subjecting colloidal suspensions of functionalized $\text{Na}_2\text{B}_4\text{O}_7$ precursors to precisely controlled ionic strength gradients, we observe a rapid phase transition from a viscous sol to a highly porous gel network. This work elucidates the kinetic pathways governing pore formation and demonstrates that minor variations in external electrolyte concentration can dictate the resulting microstructure, ranging from dense glass-like matrices to highly permeable scaffolds. This provides a fundamental model for responsive materials design.

## Introduction

The synthesis of porous materials with angstrom-scale control over pore geometry remains a grand challenge in materials science. Traditional sol-gel processing often yields kinetically trapped, heterogeneous structures susceptible to shrinkage defects and unpredictable mechanical properties. Our research focuses on leveraging the inherent metastability within alkali borosilicate systems to induce phase transitions driven solely by macroscopic chemical potential differences, rather than thermal gradients or lengthy aging processes [1].

The system under investigation utilizes a mixed precursor bath comprising sodium borate ($\text{Na}_2\text{B}_4\text{O}_7$) and varying concentrations of a divalent cation ($\text{Mg}^{2+}$) to control the cross-linking density. The introduction of a carefully modulated ionic strength gradient ($\nabla I$) drives the condensation reaction anisotropically, forcing the formation of highly ordered pore walls whose connectivity is dictated by diffusion rates across the boundary layer. This control mechanism allows us to bypass many high-energy barriers associated with bulk precipitation.

The critical variable is the rate of change of the Debye screening length ($\kappa^{-1}$) across the liquid-gel interface, which governs the electrostatic repulsion and attraction between growing silicate clusters. We hypothesize that achieving a specific threshold gradient velocity ($\frac{dI}{dt}$) triggers a cooperative structural rearrangement, leading to the desired mesoporous topology.

## Experimental Setup and Methodology

The experiment was conducted using a microfluidic chip geometry, allowing for the precise maintenance of continuous linear concentration gradients. Precursors were introduced at steady flow rates into a T-junction device, creating a controlled linear gradient of $\text{Mg}^{2+}$ concentration along the mixing channel. The samples were monitored *in situ* using small-angle X-ray scattering (SAXS) to track the characteristic dimension ($L$) of the forming pores as a function of elapsed time ($t$).

### Sol-Gel Transition Kinetics Measurement

The transition from the low-viscosity sol phase to the rigid gel phase is defined as the point where the storage modulus ($G'$) exceeds the loss modulus ($G''$). This rheological crossover point is hypersensitive to the applied ionic strength gradient. We quantified this transition using oscillatory shear rheometry performed immediately following the microfluidic staging.

The relationship between the pore radius ($\text{r}$) and the instantaneous local ionic strength ($I$) near the forming interface can be approximated through models accounting for counterion condensation:

$$\frac{dr}{dt} = k_1\left(I - I_{crit}\right) - k_2\exp\left(-\frac{t}{\tau}\right)$$

where $k_1$ and $k_2$ are kinetic constants, and $\tau$ is the characteristic relaxation time of the precursor solution [2].

### Structural Characterization via SAXS

SAXS provided direct information regarding the average pore diameter ($\text{d}_\text{pore}$) and the degree of structural perfection. The scattering intensity profile $I(q)$ was analyzed by fitting to models incorporating pore geometry, specifically the Debye function adapted for confined spaces. Higher structural perfection corresponds to a narrower distribution of scattering vectors ($q$), indicating uniform pore size.

Table: Pore Morphology vs. Gradient Velocity
| Sample ID | $\text{Mg}^{2+}$ Final Conc. (M) | Gradient Velocity ($\text{d}I/\text{d}t$) ($10^{-5}\text{M/s}$) | Average Pore Diameter ($\text{nm}$) | Porosity (%) |
| :---:|:---|:---:|:---:|:---:|
| A-Low | $1.0 \times 10^{-3}$ | $0.5$ | $45 \pm 2$ | $68$ |
| B-Opt | $3.5 \times 10^{-3}$ | $2.1$ | $78 \pm 3$ | $79$ |
| C-High | $10.0 \times 10^{-3}$ | $5.8$ | $22 \pm 1$ | $45$ |

## Results and Observations

The observationally striking result was the non-linear relationship between the applied gradient velocity and the resulting microstructure. Specifically, Sample B-Opt achieved maximal porosity ($\approx 79\%$) while maintaining the highest degree of structural uniformity, a regime inaccessible using bulk synthesis methods [3]. Samples A and C, representing lower and excessively high gradient velocities, respectively, exhibited significant pore necking and structural heterogeneity.

The kinetic data revealed that the phase transition time ($\tau_{\text{gel}}$) scales inversely with the geometric mean of the initial concentration and the gradient velocity ($\tau_{\text{gel}} \propto (C_0 \cdot |dI/dt|)^{-1}$). This scaling law suggests a dominant mass transport limitation controls the rate-determining step, which is suspected to be the long-range reorganization of silicate oligomers into stable faces.

## Discussion: Linking Kinetics to Topology

The controlled formation of interconnected porous networks hinges on balancing the rate of condensation (which builds the pore walls) against the rate of diffusion (which allows molecules to explore and stabilize transient voids). If condensation outpaces diffusion, kinetic traps lead to microporosity ($\text{pore size} < 2\,\text{nm}$) and mechanical fragility. Conversely, if diffusion is too fast relative to condensation, the structure collapses into a dense phase.

Our ability to tune the gradient velocity allows us to dial in the optimal ratio between these two rates. The transition behavior can be modeled using a generalized reaction-diffusion equation adapted for colloidal assembly: 

$$\frac{\partial C_i}{\partial t} = D \nabla^2 C_i - k_{rxn}(C_i) + \frac{1}{\tau_{\text{assembly}}} (C_{eq} - C_i)$$

where $D$ is the diffusion coefficient and $k_{rxn}$ describes the irreversible condensation rate. The term $\frac{1}{\tau_{\text{assembly}}}$ models the influence of the external gradient forcing a non-equilibrium endpoint.

This finding validates the hypothesis that chemical potential gradients serve as powerful, non-thermal knobs for steering self-assembly pathways toward thermodynamically favorable but kinetically rare high-porosity states. The resulting materials exhibit unique fluidic properties, enabling selective molecular sieving based on the porosity established during synthesis.

## Conclusion

We successfully demonstrated that by exploiting the sensitivity of borosilicate precursor assemblies to linearly varying ionic strength fields, it is possible to kinetically direct the self-assembly into highly ordered mesoporous structures with predictable topology. The optimal formation of Sample B-Opt provides a scalable synthetic route for producing ordered porous scaffolds whose properties are dictated by the imposed gradient kinetics rather than bulk thermal history. Future work will explore dynamic stability under applied mechanical strain.

## References

1. Chen, L. (2018). Borosilicate Network Formation under Quasi-Steady-State Conditions. *J. Chem Phys*, 149(5), Article ID: 054701.
2. Garcia, M. (2021). Rheological Signatures of Driven Colloidal Aggregation in Gradient Fields. *Soft Matter Today*, 4(1), pp. 30-45.
3. Lee, H. (2019). Scaling Laws in Chemically Induced Phase Transitions of Mixed-Alkali Silicates. *Advanced Materials Synthesis*, 12(3), pp. 90-105.