# Modeling the Nucleation Kinetics of Aqueous NaCl with Modified Poisson Distribution

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## Principle of Droplet Generation in Microfluidics

The generation of discrete droplets which could serve as independent crystallizers is crucial in microfluidic experiments. Here, we will focus on droplet-based microfluidics based on the generation of monodispersed drops by mixing two immiscible liquids. These drops are isolated from each other by a continuous phase and can thus be considered as a closed system without contact with the outside and therefore can serve as real independent nanocrystallisers. A large number of drops can then be generated to allow the repetition of the experiments and to carry out statistical studies, which is particularly important in the study of nucleation as it is inherently a stochastic process. Ideally, each droplet must be of the same volume (monodispersed), does not coalesce (stable), and equally spaced between each other. To achieve these, it is important to understand the physical principles governing the droplet formation in microfluidics.

### Relevant Dimensionless Numbers

Although the physical laws of fluid mechanics are the same on a microscopic scale and on a large scale, some macroscopically negligible phenomena become preponderant at the microscopic scale, such as the capillarity force, while others like gravity become negligible. Dimensionless numbers are useful in analyzing the dominating or negligible forces in the system such as viscous forces, inertial forces, gravitational forces, and interfacial forces.
1. Bond Number This corresponds to the ratio of gravitational to interfacial forces Bo = 2 (2.41) where Δρ is the density difference of the fluids, g is the acceleration due to gravity, D is the hydrodynamic diameter, σ is the surface tension. The Bond number can be used to assess whether gravitational forces are insignificant (Bo << 1).
2. Capillary Number This describes the ratio of viscous forces to interfacial surface tension forces Ca = (2.42).
where µ is the dynamic viscosity of the continuous phase, σ is the surface tension, U is the average velocity of the two fluids, i.e. if qC and qD are the volumetric of the two fluids flowing in a channel with a cross section A, then U = (qC+qD)/A. In terms of droplet velocity vd and interfacial tension of the fluid pair γCD, the expression for Ca becomes Ca = (2.43) CD.
3. Reynolds Number This describes the ratio of inertial forces to viscous forces = (2.44). Where ρ is the density, U is the average velocity of the two fluids, D is the characteristic hydrodynamic diameter. Reynolds number characterizes the flow pattern as laminar (Re < 1000) or turbulent (Re > 2100).
4. Weber Number This corresponds to the ratio of inertial forces to interfacial forces and is defined as the product of Capillary number and Reynolds number. We = Ca × Re = 2 (2.45). The Weber number can be used to assess whether inertial forces are insignificant (We << 1).

#### Droplet Formation in T-junction

While there are different design structures for droplet generation, we will focus our attention on cross-flow system using T-junction. This is perhaps the most popular method because T-junctions are commercially available and can easily be integrated in plug-and-play setups. In T-junctions, two immiscible fluids meet at a 90o angle resulting in the formation of droplets. The stages of droplet formation are illustrated in Figure 2.7.

There has been tremendous progress in the use of microfluidics in crystallization as reviewed by Leng47, Shi50, and Candoni.57 Indeed, the choice of materials in the fabrication of microfluidc platform is an important consideration. The advantages and disadvantages of different materials (silicon, glass, ceramics, elastomers, hydrogels, thermoplastics, etc) in terms of mechanical properties, thermal properties, solvent resistance, optical transmissivity, biocompatibility and material cost have been reviewed by Tsao58 and Niculescu et al.59 In this section, I will focus on the advances developed in our laboratory to study crystallization fundamentals by taking advantage of cheap and commercially-available materials instead of employing sophisticated fabrication technologies.

Microfluidic Experiments in Transparent Capillary

As solvent incompatibility limits the diversity of the compounds that can be studied in microfluidics, Ildefonso et al.51 developed a microfluidic platform consisting of a T-junction made of polyether ether ketone (PEEK) and a Teflon tubing which has been found to be superior to both pure PDMS and mixed PDMS/Teflon device in terms of stability in various solvents namely ethanol, acetone, ethyl acetate, nitrobenzene and acetonitrile. Moreover, this platform enables droplet storage for several weeks without significant evaporation.
To develop a cheaper alternative to the sophisticated microfabrication technologies, Zhang et. al.52 developed a home-made microfluidic platform built entirely from commercially available modules as illustrated in Figure 2.9. This platform has been shown to enable generation of stable monodisperse droplets with uniform spacing and was successfully used to study the crystallization of lysozyme, isonicotinamide, gliclazide and paracetamol.

Evaporative Sessile Microdroplet Experiments

To study nucleation in picoliter to femtoliter volume ranges, Grossier et al63 developed a simple yet efficient method to generate such microdroplets without the need for surfactants (which can alter fluid-interface properties). This set-up is shown in Figure 2.11. The motorized microinjector can move in three direction by 16 nm increments while the glass microcapillary is connected to a pressure-control system. In contrast to microfluidic systems using capillaries, this system is called « open » as droplets are accessible. The size of the droplets (fL-pL) can be adjusted depending on the pressure-drop and the translation speed of the microcapillary across the surface.63 The setup has been successfully used by Hammadi et al19 to induce and localize primary nucleation events.

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Image Analysis and Application to Crystallization Studies

An important advantage of the setup in Figure 2.11 is that it allows simultaneous measurement of hundreds of droplets via image analysis. The basic concepts of image analysis and how it can be applied on crystallization studies is thus discussed in this section.
Digital images are made up of 2D array of pixels. For 8-bit gray-scale images, each pixel can have values ranging from 0 (black) to 255 (white). The distribution of pixel values can be analyzed using histograms. Dark images would have more pixels close to zero whereas bright images would have most of them close to 255. The general process of extracting the histogram from a gray-scale image is shown in Figure 2.12.

Affidavit
List of Publications and Conferences
Abstract
Résumé
Résumé étendu
Chapter 1 Introduction
1.1 Motivation
1.2 Background and Objective
1.3 Structure of the Thesis
Chapter 2 Literature Review
2.1 Fundamentals of Crystallization
2.1.1 Mechanisms of Nucleation
2.1.2 Classical Nucleation Theory
2.1.3 Two-step Nucleation Theory (2-SNT)
2.2 Nucleation Kinetics: Approaches on Data Acquisition and Treatment
2.2.1 Deterministic Approach
2.2.2 Stochastic Approach
2.3 Droplet-based Microfluidics in Crystallization Studies
2.3.1 Principle of Droplet Generation in Microfluidics
2.3.2 Recent Advances in Microfluidics Crystallization
2.4 Principle of Microdroplet Evaporation
2.4.1 Contact Line Behavior (CCA, CCR, SS mode)
2.4.2 Evaporation Rate Models
Chapter 3 Materials and Methods
3.1 Model Compounds
3.1.1 Para-Aminobenzoic acid (PABA)
3.1.2 Glutamic acid (GA)
3.1.3 Sodium chloride (NaCl)
3.2 Polymers used in the microfluidic set-up
3.2.1 Polyetheretherketone (PEEK)
3.2.2 Fluorinated ethylene propylene (FEP)
3.2.3 Polydimethylsiloxane (PDMS)
3.2.4 Polymethylmethacrylate (PMMA)
3.3 Process Analytical Tools (PATs)
3.3.1 Optical Reflectance Measurement (ORM)
3.3.2 In situ Raman spectroscopy
3.4 Experimental Setups
3.4.1 Setup for Liter Scale Experiments
3.4.2 Setup for mL Scale Experiments
3.4.3 Setup for μL Scale Experiments
3.4.4 Setup for Sessile Microdroplet Experiment
Chapter 4 Measuring Primary Nucleation Rates in Agitated Systems using Particle Count Approach
4.1 Introduction
4.2 Materials and Methods
4.2.1 Chemicals and Equipment
4.2.2 Solubility Measurement
4.2.3 Calibration of in-situ Raman Spectroscopy
4.2.4 Calibration of In-situ 3D ORM
4.2.5 Crystallization Process Monitoring
4.3 Results and Discussion
4.3.1 Identification of Polymorphs
4.3.2 Solubility Data
4.3.3 Calibration Curves of ORM
4.3.4 Validation of ORM Measurement by Raman Spectroscopy
4.3.5 Total Nucleation Rates from in-situ ORM
4.3.6 Estimation of Primary Nucleation Rate
4.4 Conclusion
Chapter 5 Nucleation Kinetics in Agitated Systems: Particle Counts vs Induction Time Approach
5.1 Introduction
5.2 Materials and Methods
5.2.1 Chemicals and Equipment
5.2.2 Induction Time Measurement
5.2.3 Extraction of Nucleation Rate from Induction Time
5.3 Results and Discussion
5.3.1 Nucleation Rates from Induction Time Probability Distribution
5.3.2 Confidence Intervals of Estimated Parameters
5.3.3 Comparing Nucleation Kinetic Parameters
5.4 Conclusion
Chapter 6 Quantifying Nucleation Kinetics: A Multi-scale Comparison
6.1 Introduction
6.2 Material and Methods
6.2.1 Liter Scale
6.2.2 Milliliter Scale
6.2.3 Submicroliter Scale
6.3 Results and Discussion
6.4 Conclusion
Chapter 7 Probing Nucleation in Microdroplets via Image Analysis: Effect of Diffusive Interactions
7.1 Introduction
7.2 Materials and Methods
7.2.1 Details of Instrumentation
7.2.2 Microdroplet Generation
7.2.3 Humidity Regulation
7.2.4 Numerical Detection of Oscillations
7.3 Results and Discussion
7.3.1 Effect of Diffusive Interactions
7.3.2 Eliminating Diffusive Interactions
7.4 Conclusion
Chapter 8 Nucleation in Sessile Microdroplets: Measuring Induction Time via Deliquescence-Efflorescence Cycle
8.1 Introduction
8.2 Materials and Methods
8.3 Results and Discussion
8.3.1 Analysis of σ-curves
8.3.2 Assessment of Reproducibility
8.3.3 Statistical Analysis
8.3.4 Checking for Possible Influence of Impurities
8.3.5 Nucleation Kinetic Parameter Estimation
8.4 Conclusion
Chapter 9 Modeling the Evaporation Dynamics of Sessile Saline Microdroplets
9.1 Introduction
9.2 Modeling
9.2.1 Influence of oil thickness on the evaporation rate
9.2.2 Considering the presence of neighboring droplet
9.2.3 Considering the evolution of droplet density as water evaporate
9.2.4 Dependence of water activity on solute concentration
9.2.5 Models for Contact Line Behavior
9.3 Materials and Methods
9.4 Results and Discussion
9.4.1 Model Predictions for Pure Microdroplets
9.4.2 Model Predictions for Saline Microdroplets
9.4.3 Implications on Crystallization Studies
9.5 Conclusion
Chapter 10 Modeling the Nucleation Kinetics of Aqueous NaCl with Modified Poisson Distribution
10.1 Introduction
10.2 Theory and Modeling
10.2.1 Classical Nucleation Theory for Ionic Systems
10.2.2 Modified Poisson Distribution Function
10.3 Results and Discussion
10.3.1 Kinetic Parameter Estimation
10.3.2 Comparison with Literature
10.3.3 Observing Confinement Effects
10.4 Conclusion
Chapter 11 Concluding Remarks and Perspective
11.1 Notable Findings
11.2 Perspective
11.2.1 Influence of interfering variables in Agitated Crystallizers
11.2.1 Evaporative microdroplet experiments
References

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