P - Technische Universiteit Eindhoven

3. Crystallization under wetting/non-wetting conditions 33
of a non-wetting condition the contact lines between fluid and solid are free to move, and
as a result the droplet will shrink almost uniformly during drying. In this case it is found
that the concentration remains homogeneous throughout the droplet [64, 65].
In the case of a wetting surface the contact lines are pinned. As a result the contact
angle will change during drying and inhomogeneous drying will occur, since the evaporation
per unit fluid volume will be largest along the rim of the droplet. Consequently,
there will be salt transport by advection towards the rim. This is also known as the coffee
stain effect [65–67], where after complete evaporation of a droplet the edge of the stain
has a more pronounced color than the inner part.
In general, the transport of ions in a fluid can be described by a diffusion-advection
equation:
∂c
∂t = D∇2 c − ⃗ U · ∇c , (3.1)
where c is the concentration in molals [M = mole/kg], D [m 2 s −1 ] the ion diffusion coefficient,
and U the water velocity [m s −1 ]. The competition between advection and diffusion
can be defined in terms of the Peclet number [68]:
P e ≡ |⃗ U|L
D , (3.2)
where L [m] is the length scale of interest, e.g., the diameter of the droplet.
In the case of wetting conditions we expect that P e > 1 and hence concentration
gradients are present which could influence the crystallization dynamics [65]. On the
other hand, for non-wetting surfaces we expect P e < 1 and, consequently, a homogeneous
crystallization behavior throughout the droplet.
3.3 Experimental setup
We have used a Nuclear Magnetic Resonance (NMR) setup operating at a field of 0.78 T
in this study to measure the concentration in the droplet non-destructively while drying.
An extensive description of this setup can be found in [29, 40]. In order to create either
wetting or non-wetting conditions during drying of a droplet, two types of sample holders
were manufactured from transparent quartz-glass and Polymethyl Methacrylate (PMMA),
respectively. Each type of sample holder has a height of 15 mm, a diameter of 20 mm,
and a flat bottom.
A schematic representation of the experimental setup is given in figure 3.2. The
NMR setup is equipped with a Faraday shield to suppress the effect of changes of the
dielectric properties of the sample [29]. In addition, the quality factor of the LC circuit
containing the sample is chosen rather low (Q ≈ 40), to suppress the effects of the
(electrically conducting) sodium sulfate solution. The tuned circuit of the setup can be
toggled between 33 MHz for 1 H and 8.9 MHz for 23 Na, giving the possibility to quasisimultaneously
measure the Na and H content, and thereby the concentration.
At the echo times used in the experiments (T E = 750 µs for Na and 650 µs for H),
only the Na and H nuclei in the solution are measured and no signal is obtained from these
nuclei incorporated in sodium sulfate crystals. The magnetic field gradient was chosen