[Paper Review] Crystallization Inhibitors: Explaining Experimental Data through Mathematical Models

Maria Paola Bracciale, Gabriella Bretti+5 more|arXiv (Cornell University)|Jan 21, 2015

Building materials and conservation29 references3 citations

TL;DR

This study presents a one-dimensional mathematical model that simulates salt crystallization in porous bricks, incorporating phosphocitrate (PC) as a crystallization inhibitor. The model captures inhibitor effects via two calibrated parameters—crystallization rate ($K_s$) and specific crystal volume ($\gamma$)—showing that PC increases nucleation rate while reducing crystal volume, thereby lowering damaging tensile stresses and preserving hydraulic continuity in the stone matrix.

ABSTRACT

In this paper we propose a new mathematical model describing the effect of phosphocitrate (PC) on sodium sulphate crystallization inside bricks. This model describes salt and water transport, and crystal formation in a one dimensional symmetry. This is the first study that takes into account mathematically the effects of inhibitors inside a porous stone. To this aim, we introduce two model parameters: the crystallization rate, which depends on the nucleation rate, and the specific volume of precipitated salt. These two parameters are determined by numerical calibration of our system model for both the treated and non treated case.

Motivation & Objective

• To develop a mathematical model explaining the action of crystallization inhibitors in porous building materials.
• To quantify how phosphocitrate (PC) alters salt crystallization dynamics in masonry bricks.
• To calibrate model parameters using experimental data from treated and untreated bricks.
• To assess the impact of inhibitors on crystal formation, porosity, and stress development in porous media.
• To provide a predictive simulation tool for evaluating crystallization modifiers in conservation science.

Proposed method

• A one-dimensional mathematical model simulates coupled water and salt transport, nucleation, and crystal growth in porous bricks.
• The model incorporates two key parameters: crystallization rate ($K_s$), influenced by nucleation, and specific crystal volume ($\gamma$), reflecting crystal habit changes.
• Numerical calibration is performed using experimental data from four brick sections (NT1–NT4 and PC1–PC4) under treated and untreated conditions.
• The model solves partial differential equations for liquid water content ($\theta_l$), salt concentration ($c_i$), and solid salt content ($c_s$) over time and height.
• Trapezoidal quadrature is used to compute average porosity ($n^{num}_i$) across brick segments.
• Model validation is performed by comparing simulated salt content ($q^{num}_i$) with measured values ($q_i$) across experimental samples.

Experimental results

Research questions

• RQ1 How does phosphocitrate (PC) alter the crystallization rate and crystal volume in sodium sulfate-affected porous bricks?
• RQ2 To what extent can a mathematical model reproduce experimental salt distribution and water content profiles in treated and untreated bricks?
• RQ3 Does the presence of PC lead to reduced crystal volume and increased nucleation rate, thereby minimizing structural damage?
• RQ4 How do changes in $K_s$ and $\gamma$ affect porosity and hydraulic continuity in the porous matrix?
• RQ5 Can the model predict the effectiveness of crystallization inhibitors based on calibrated parameters?

Key findings

• The model achieved a calibration error of 11.6% for the non-treated case using $K_s = 4.1 \times 10^{-5}\ \text{s}^{-1}$ and $\gamma = 0.6\ \text{cm}^3\text{g}^{-1}$.
• For the PC-treated case, the calibration error was 13.7% with $K_s = 6.0 \times 10^{-5}\ \text{s}^{-1}$ and $\gamma = 0.53\ \text{cm}^3\text{g}^{-1}$.
• The calibrated model shows that PC increases the crystallization rate ($K_s$) while decreasing the specific crystal volume ($\gamma$).
• Despite faster nucleation, PC-treated samples exhibit lower crystal volume, reducing tensile stress development in the porous matrix.
• The model predicts that PC preserves hydraulic continuity by minimizing pore clogging, as shown by stable porosity values across brick sections.
• Numerical simulations confirm experimental observations: higher salt content is observed in PC-treated bricks, but with reduced crystal volume and lower damage potential.