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## Fundamentals of drop breakup

The flow of an incompressible fluid (i.e. ~r· ~v = 0) is governed by the Navier- Stokes equation, that includes a time-dependent domain in the case of drop breakup: ⇢ &~v &t + ⇢(~v · ~r)~v = −~rP + ⌘~r2~v + ~F (2.4).

where ⇢ is the density, ~v the vector field of the fluid velocity, P the pressure and ⌘ the dynamic fluid viscosity. ~F resumes all present body forces [8].

The assignment of a Navier-Stokes dynamics inherently entails a large divergence between space and time dimensions. As a consequence, the characteristic length scale of the solution deviates by orders of magnitude from the length scale of the physical system it is supposed to describe. This is illustrated in Fig. 2.3 and 2.4: the finite time singularity that occurs during drop fission is assumed to be characterizable by a single local length scale, the minimum fluid neck diameter dmin, that will go to zero as the fluid filament becomes thinner and the instant of breakup is approximated. At the same time, other physical quantities such as the newly created surface A blow up in space illustrating that the whole pinchoff process is driven by capillary forces: formation of a (spherical) drop from a cylindrical fluid filament minimizes potential energy and the rate at which this occurs is primarily dependent on the physical parameters of the fluid, i.e. surface tension, viscosity and density [9].

### Proton diffusion in bulk water

Protons (and hydroxides) were found to exhibit anomalously high diffusion coefficients (DH+ ⇠ 10−9m2s−1) in bulk aqueous solutions, which seems rather surprising when considering their radius/charge ratios that are very similar to alkali or alkaline earth metal ions (D ⇠ 10−10m2s−1). What causes the proton to stand out to such a noticeable extent with respect to other chemically very similar ions such as Li+, Na+ or Mg2+? First of all, unlike metal cations, free protons per se do not exist in water, rather they immediately form hydronium ions (H3O+) with surrounding water molecules. This outstanding property was further investigated by Eigen [28] and Zundel [29] who proposed the existence of two hydration complexes, referred to as ’Eigen’ (H9O + 4 ) and ’Zundel’ (H5O + 2 ) ion respectively. Ultrafast proton diffusion is enabled through continuous interconversion between these two limiting structures, which is driven by fluctuations in the solvation shell of the hydrated ions. The resulting ’proton hopping’ therefore does not involve a net transport of mass, but

merely of positive charge (Fig. 2.9). This so-called ’structural diffusion’ mechanism is highly dependent on the unhindered rotation of the water molecules in close proximity to the diffusing proton. Recent experimental evidence suggests that ⇠ 10 water molecules need to reorientate/ are involved in the transfer of a single proton charge [30]. More recent literature suggests the existence of even higher-order structures such as unique H13O + 6 entities [31]. After all, the full dynamics of a diffusing proton in a hydrogen bond network are yet to be completely understood and lie beyond the scope of this thesis.

#### Determination of proton diffusion coefficients using a microfluidic setup

Proton diffusion is known to be extremely fast and quantitative derivation of accurate diffusion coefficients is challenging. Known methods for determining proton diffusion coefficients such as conductivity measurements [21] proved to be unsuitable for our purpose due to the abundance of salts and buffers in the CMS with respect to the low physiological proton concentrations. We used the well-controlled reaction environment of a micrometric Y-shaped glass channel (45 μm width and 20 μm height, Micronit) that allows for purely diffusive mixing of two inlet streams (Fig. 3.2B and C). We dyed all solutions with pH-sensitive fluorophore (Fluorescein, 25 μM) the fluorescence of which rapidly decreases below its first pKa of 6.4. For each experiment two different pH values of the same solution were injected into the two channel inlets by the help of a syringe pump (PhD 2000, Havard Apparatus). The resulting proton gradient induces a diffusive proton flux from the low pH stream (non-fluorescent) to the high pH stream (fluorescent). As the pH of the high pH stream decreases, an increasing fraction of the Fluorescein molecules is quenched which allows to directly follow the progression of the protons by confocal microscopy. At the same time, the microfluidic setup decomposes this movement into a time (t) and a space component (y) and thus the whole proton diffusion process can be grasped within a single image.

From these binary fluorescence microscopy recordings the mean proton diffusion length « d can be determined as the distance between the center of the channel y0 and the fluorescent front y1 (Fig. 3.2C). This switching point corresponds to the position where the local pH has reached the value of the second pKa of Fluorescein, which is equal to pH = 6.4.

As the two inlet fluids mix by diffusion only, the square of this mean travelled distance « d2 is linearly dependent on the lapsed contact time t of the two fluids and the medium-specific diffusion coefficient D: « d2 ⇡ Dt.

**Assessment of water dynamics using ultrafast IR spectroscopy**

In our experiments we sought to assess the effect of biological components on water dynamics that occur on a picosecond time scale. For such extremely short times, so-called femtosecond time resolved infrared spectroscopy (fsTRIR) has been succesfully employed in the past [22, 23, 24]. The corresponding pumpprobe setup is depicted in Fig. 3.3. For each measurement, the sample is first excited by a so-called ’pump’ pulse and the resulting excited state is then probed by a second laser. The impact of this ’probe’ pulse on the excited sample is a function of both the wavelength ! and the time t after excitation which results in a transient absorption signal « A(!, t): « A(!, t) = Absorbanceprobe − Absorbancepump (3.3).

Since the dipole moment of water has a very strong IR absorption, exciting the OH stretch vibration would result in spectral saturation, unless an extremely thin sample is used. For this reason, fsTRIR usually probes the equivalent stretch vibration of partially deuterated water instead (6 % HDO in H2O or CMS respectively in our experiments) which occurs at 2500 cm−1. The deuteration does not affect the properties of the other water molecules and consequently leaves the hydrogen bonding network of the sample unchanged. At the same time.

**Proton diffusion coefficients in bulk water and cytosolic mimic solutions**

For the determination of proton diffusion coefficients in our cytosolic mimic solution we used a microfluidic confocal microscopy setup (Fig. 3.2). We first performed a control experiment where we derived diffusion coefficients for D+ in deuterated bulk water and for H+ in (normal) bulk water in order to check whether our method was capable of resolving the expected isotope effect. For the proton diffusion coefficient in bulk water we found a value in the order of ⇠ 10−9 m2 s−1, which is consistent with previous findings [27, 28]. In contrast, D+ diffusion in D2O was observed to be retarded by a factor of p2 with respect to H+ diffusion in H2O (Fig. 3.6), which is in line with theoretical predictions assuming a Grotthus-type transport mechanism [29].

**Table of contents :**

Summary

Résumé

Samenvatting

**1 Introduction **

1.1 Preface: water

1.1.1 Hydrogen bonding, collectivity and provocative thoughts .

1.1.2 Surfaces and interfaces

1.2 Scope of this thesis

**2 Theory **

2.1 Surface tension and capillarity

2.2 Drop breakup

2.2.1 Fundamentals of drop breakup

2.2.2 Inviscid pinch-off

2.3 Diffusion

2.3.1 Principles of diffusion

2.3.2 Proton diffusion in bulk water

2.4 Microfluidics

**3 Proton mobility in cellular environments **

3.1 Introduction

3.2 Materials and Methods

3.2.1 Cellular mimic

3.2.2 Determination of proton diffusion coefficients using a microfluidic setup

3.2.3 Assessment of water dynamics using ultrafast IR spectroscopy

3.3 Results

3.3.1 Proton diffusion coefficients in bulk water and cytosolic mimic solutions

3.3.2 Origin of proton diffusion retardation in cytosolic mimic solutions

Role of water reorientation dynamics

Role of cytosolic buffers

Role of viscosity

3.4 Discussion

Role of cytosolic buffers

Role of water reorientation dynamics

Role of viscosity

Other macromolecular effects

Interplay of different factors

3.5 Conclusion

**4 The dynamic surface tension of water **

4.1 Introduction

4.2 Materials and Methods

4.2.1 Surface tension measurements

4.2.2 Ultrarapid imaging of drop breakup

4.2.3 Derivation of breakup dynamics from movies

4.3 Results

4.3.1 Determination of the universal prefactor

4.3.2 Effect of pH

4.3.3 Effect of salt concentration

4.4 Discussion & Conclusion

**5 Drop breakup dynamics on the nanosecond time scale **

5.1 Introduction

5.2 Materials and Methods

5.2.1 Electrical measurements

5.2.2 Data analysis

5.2.3 MATLAB simulations

5.3 Results

5.3.1 Breakup dynamics of mercury

Ultrarapid imaging

Electrical measurements

Comparison ultrarapid imaging versus electrical measurements

5.3.2 MATLAB simulation of the electrical pinch-off

5.3.3 The pinch-off behaviour of liquid gallium

Ultrafast imaging

Electrical measurements

5.4 Discussion

Noise from surface evaporation

Noise from surface oxidation

Noise from thermal capillary waves and other noise sources

5.5 Conclusion

**6 Oil-water displacement in rough pore microstructures **

6.1 Introduction

6.2 Materials and Methods

6.2.1 Design of rough microdevices

Design of rough single pore structures

Design of rough network structures

6.2.2 Water flooding experiments

6.3 Results and Discussion

6.3.1 Effects of roughness and viscosity on oil recovery

6.3.2 Effect of the flowrate on trapped droplet sizes

6.3.3 Towards a generic scaling law incorporating capillarity and roughness effects

6.3.4 Oil-water displacement patterns in complex porous networks

6.4 Conclusion

Acknowledgements

**A Simulation of mercury pinch-off MATLAB code **