Sodium chloride precipitation reaction coefficient from crystallization experiment in a microfluidic device

Highlights

• NaCl crystal growth is analyzed from high speed imaging.

• Value of NaCl precipitation reaction coefficient (PRC) is determined.

• The value of PRC in previous experiments was underestimated.

Abstract

The crystal growth of sodium chloride from an aqueous solution is studied from evaporation experiments in microfluidic channels in conjunction with analytical and numerical computations. The crystal growth kinetics is recorded using a high speed camera in order to determine the intrinsic precipitation reaction coefficient. The study reveals that the crystal growth rates determined in previous studies are all affected by the ions transport phenomena in the solution and thus not representative of the precipitation reaction. It is suggested that accurate estimate of sodium chloride precipitation reaction coefficient presented here offers new opportunities for a better understanding of important issues involved in the damages of porous materials induced by the salt crystallization.

Introduction

The crystallization of sodium chloride from an aqueous solution is a key phenomenon in relation with evaporation from porous media [1], the generation of damages in buildings and monuments [2], or the injection of CO₂ in underground formations [3], to name only a few. The crystallization process is generally decomposed into two main steps: the nucleation step and the growth step. In this respect, it is important to distinguish the crystal growth kinetics [4] from the nucleation kinetics, which involves the induction time between the application of a supersaturation state and the appearance of the first crystals [5]. In the literature, they can both be found under the expression of “crystallization kinetics”. In this paper, we focus on the crystal growth kinetics.

The crystal growth is studied within the framework of the diffusion reaction theory [6] (where other crystal growth theories: surface energy theory, adsorption layer theory and kinematic theory, are also presented). Crystal growth starts only once a stable nucleus, large enough to be stable, appears in the metastable solution. It relies on two coupled steps: an ion diffusion process from the solution to the crystal surface, followed by a reaction process where ions fit in the crystal lattice. These processes have been highlighted in Refs. [7], [8].

As illustrated in Fig. 1, the two steps occur in series and three zones can be defined. The first zone corresponds to a stagnant film (or adsorption layer) at the crystal-liquid interface. Far from the crystal, there is the bulk solution with a constant concentration. The concentration increases following a diffusion law in the intermediate zone of size d between the crystal and the bulk.

The crystal growth rate, J (kg/m²·s) in the adsorption zone and diffusion zone can be modelled as:

$$J_D=\frac{1}{A}\frac{\mathit{dM}}{\mathit{dt}}=k_D(c_b-c_i)\text{,}$$

$$J_R=\frac{1}{A}\frac{\mathit{dM}}{\mathit{dt}}=k_R{(c_i-c_{\mathit{eq}})}^n\text{,}$$

where M (kg) is the mass of the crystal, A is the crystal total surface, c_b (kg/m³) is the bulk salt concentration of the solution, c_i is the salt concentration at the liquid crystal interface, and c_eq is the ion concentration at equilibrium; n is the order of the reaction and k_D and k_R (m/s) are the coefficients of mass transfer by diffusion and reaction, respectively. k_D can be seen as the ratio of the salt molecular diffusion coefficient D_s to the diffusion length d.

Considering the diffusion zone and the adsorption layer as two mass transfer resistances in series (as sketched in Fig. 1) and for a first order reaction (n = 1, which is the case for sodium chloride), an equation combining these two steps can be obtained,

$$J_G=\frac{1}{A}\frac{\mathit{dM}}{\mathit{dt}}=k_G(c_b-c_{\mathit{eq}})$$

with

$$k_G=\frac{1}{\frac{1}{k_D}+\frac{1}{k_R}}=\frac{k_D{k}_R}{k_D+k_R}\text{,}$$

where k_G is referred to as the overall growth rate parameter. The equivalent of the mass transfer resistance is equal to the inverse of the growth rate parameter. In order to characterize the phenomenon driving the crystallization process, Garside [9] suggests to define the effectiveness factor for crystal growth η_r as the ratio between the overall growth rate and the growth rate obtained when the crystal surface is exposed to the bulk concentration:

$${\eta }_r=\frac{J_G}{k_R(c_b-c_{\mathit{eq}})}$$

In other words, it is the ratio between the overall growth rate and the growth rate obtained when the crystallization is limited only by reaction, with an infinitely fast diffusion; η_r can be expressed as:

$${\eta }_r=(1-\mathit{Da}{\eta }_r)$$

where $\mathit{Da}=\frac{k_R}{k_D}$, is the Damkhöler number, which represents the ratio between the reaction flux and the mass transport flux. Thus

$${\eta }_r=\frac{1}{1+\mathit{Da}}\text{.}$$

Therefore, the process is controlled by diffusion when Da is large and η_r is low. On the contrary, it is controlled by reaction (the controlling process is the slowest one) when Da is small and η_r is large. Moreover, because NaCl crystal has a cubic shape, the mass precipitation rate can be related to the mean linear velocity of its faces w_cr (m/s) by (see Appendix A):

$$w_{\mathit{cr}}=\frac{\mathit{dr}}{\mathit{dt}}=\frac{J_G}{{\rho }_c}=\frac{k_G}{{\rho }_c}(c_b-c_{\mathit{eq}})$$

where r is the half length of the side of a cubic crystal (m) and ρ_c is the crystal density (kg/m³). In case of a spherical crystal or a growth in 1 dimension on both sides, Eq. (8) remains valid with r as the sphere radius or the crystal half length.

The above considerations clearly show that the crystal growth kinetics depends on both the local concentration c_i at the interface and coefficients k_D and k_R. The experiments typically allow determining w_cr (references are given below in the section on the results). Since both c_i and k_R are unknowns, it is clearly difficult to determine k_R from the experimental data. Also, as stated in [9], k_R is difficult to measure because it is hard to separate the reaction step from the diffusion one. In this context, the main objective of the paper is precisely to provide an accurate estimate of k_R.

It should be mentioned that correct values of k_R are of the uttermost importance for correctly evaluating the crystallization pressure, which is the key concept in relation with the damages caused by the salt crystallization in porous materials [10]. As explained in [11], what matters for evaluating the crystallization pressure is the salt concentration at the crystal surface when it becomes confined between the pore walls. This concentration is highly dependent on k_R (see [11] for more details).