#### 3.2. Magnetic Fields

In order to be able to read the magnetic field scan images properly, the analysis of a cylindrical bar magnet is presented first.

Figure 7 shows the field of such a magnet scanned with the device described in the method section in an area of 14 × 14 cm

^{2}, 3 cm above the magnet.

Because for the present analysis the sign (N or S) is not important, the field can also be displayed in terms of absolute intensity, and the three components can then be translated to the three primary colours, red (

x), green (

y) and blue (

z). These images are then combined to a composite RGB image. This procedure is shown in

Figure 8.

With the method describe above, composite images of magnetic fields of two different WCMs (serial numbers 62083545 and 62081992) are shown in

Figure 9. The dark lines are sudden changes in magnetic field strengths resembling strong gradients. Gradients in

x direction have been calculated and are shown in

Figure 10. Here a scale from white to the primary colours is used in order to make a clear distinction from the composite images in

Figure 9.

As can be seen from

Figure 10, the magnetic field gradients of the WCMs are in the order of 770~G·m

^{−1} (WCM 62081992) and 740~G·m

^{−1} (WCM 62083545) or 0.077 T·m

^{−1} (WCM 62081992) and 0.074 T·m

^{−1} (WCM 62083545). Moreover, all gradients also seem to have a finer structure embedded. Because the 3-axis probe of the VGM magnetometer has its three sensors placed within less than 1.5 mm (according to the manual) of the probe’s end, a spatial resolution beyond that cannot be recorded. The presented gradients with a step size of 1 mm are probably already spatial averages, and the actual gradients present are probably even higher than 0.077 T·m

^{−1} (WCM 62081992) and 0.074 T·m

^{−1} (WCM 62083545). A higher gradient would not change the results of this investigation, nevertheless the authors plan to measure the field with a different system providing a higher resolution in the future.

As mentioned in the introduction, Coey [

21] derived an inequality based on which the effectiveness of a magnetic device can be evaluated. For the sake of clarity and because of its importance, this inequality (2) is repeated here,

where

C is the criterion,

L the length of the magnetic device,

v the velocity of the DOLLOPs,

f_{P} the Larmor frequency of a proton, a the spin separation (0.25 nm) and ∇

B the magnetic field gradient. If

C ≥ 1, then the magnetic device can effectively influence the crystallisation of calcium carbonate. For the experiments presented, the magnetic device is larger than the volume of water treated, so the beaker diameter, 6 cm is used instead. One might argue at this point that for WCMs the actual width of the inhomogeneities, which is sometimes only a few mm, should be used, and not the measures of the device. Doing that does reduce

C, but does not change the result (

C ≥ 1, see end of this chapter) down to

L values of 8 mm.

The velocity of the DOLLOPs can be calculated from the Brownian motion in a liquid using Fick’s law,

$v=\frac{Dmg}{{\mathrm{k}}_{\mathrm{B}}T}$. In a dynamic equilibrium velocity is equal to the settling speed according to the Stokes-Einstein equation,

where

D is the diffusion coefficient, k

_{B} the Boltzmann constant,

T the absolute temperature,

µ and

m are the mass and mobility of the DOLLOP,

g is the acceleration due to gravity, ξ the drag coefficient, η the viscosity of the medium, and r the radius of a (assumingly) spherical DOLLOP. To know the mass of a 100 nm DOLLOP sphere, first its density must be calculated. Since a DOLLOP consists of both water (ρ

_{25°C} = 1.00 g·cm

^{−3}) and aragonite (ρ = 2.93 g·cm

^{−3}), we assess its density as an equal mixture of both, so ρ

_{DOLLOP,25°C} = 1.97 g·cm

^{−3}, resulting in a DOLLOP mass of 1.03 × 10

^{−15} g. Taking water’s dynamic viscosity η

_{25°C} = 0.89 × 10

^{−4} Pa s, the resultant DOLLOP velocity due to Brownian motion is 0.012 mm·s

^{−1}. The magnetic field gradients of the WCMs are in the order of 770~G·m

^{−1} (WCM 62081992) and 740~G·m

^{−1} (WCM 62083545) or 0.077 T·m

^{−1} (WCM 62081992) and 0.074 T·m

^{−1} (WCM 62083545) (see

Figure 8), resulting in a Coey criterion of 0.077 T·m

^{−1} (WCM 62081992) and 0.074 T·m

^{−1} (WCM 62083545), thus clearly ≥ 1. The real gradients are probably higher as mentioned above, which would further increase the value of

C and allow for higher fluid velocities. Smaller DOLLOPs than 100 nm are slower (the decrease of their mass in equal. 5 outweighs the increase of their mobility) and will therefore also further increase the value of

C (velocity in Equation (4) is in the denominator).

The velocity of a DOLLOP also allows an estimation of the time required for the treatment to be effective. A straight travel from one side of the beaker to the other (6 cm) would take a 100 nm DOLLOP 1.4 h. During such a journey, a DOLLOP would encounter a number of magnetic field gradients. One should, however, also consider that Brownian motion is not unidirectional, the DOLLOP might be smaller and thus slower, and so the time should be multiplied by a factor of 10 or more in order to guarantee a successful treatment. This estimation matches with the observed time frames for a measureable treatment effect of about 24–48 h.

#### 3.4. Complex Impedance

The proposed DOLLOP formation was tested in 16 independent experiments with at least 12 measurements per experiment, one measurement comprising the complex impedance at 65 frequencies, measuring two parameters (phase and impedance) per frequency. In each experiment at least two treated and two untreated samples were compared, and every sample was measured at least three times.

t-test results for the treated samples compared to blank values showed statistically highly significant differences in 15 of 16 experiments (binominal exact probability < 0.001) either for phase (φ), for impedance (

Z) or for both parameters. One experiment with 48 h treatment time did not show significant differences. In 9 of 16 experiments, Z showed frequency dependent variations which were highly different from the blank. In the case of differences a minimum of 4 and a maximum of 65 frequencies showed a significant difference, highly different results were obtained for 4 to 65 frequencies. In 15 of 16 experiments, φ showed frequency dependent variations which were highly different from the blank. Statistically significant differences in frequencies for a given φ occurred in a minimum of 2 and a maximum of 59 cases, highly significant results were calculated for 1 to 56 frequencies. In sum Z and φ showed statistically significant differences in a minimum of 3 and a maximum of 110 cases, highly different results in 2 to 106 frequencies. The exact binominal probability for an experiment to show a statistically highly different behaviour of treated water compared to untreated water is

p < 0.001 (with 15 out of 16 cases positive for significant statistical values). These results are summarized in

Table 1.

Figure 11 shows exemplary impedance and phase changes of treated compared to untreated tap water. In addition to impedance (a,d) and phase (b,e), a Nyquist-plot of the data is shown (c,f). In this plot both the real and imaginary part of the impedance, |

Z|·cos(φ) and -|

Z|·sin(φ), are plotted against each other. On a Nyquist plot the impedance is a vector with characteristic length and angle, therefore a mandatory precondition is equal scales of the ordinate and abscissa axes. Such a scaling would, however, be unfavorable for the details of the spectra presented in this work, thus, we chose to omit this precondition in favor of a better representation of spectral details knowing well that the from such depictions the impedance can no longer be read out directly.Whereas a Nyquist plot does not give frequency information it is better in showing differences between spectra over the same frequency range since it combines both impedance and phase in one curve.

The curves depicted in

Figure 11 were fitted using the model shown in

Figure 1. The parameters of the circuit elements are given in

Table 2 (case

a) and

Table 3 (case

b), respectively. Next to that, average curves of all 16 measurements were fitted with this model. Although sample composition and treatment time were not identical for all experiments as described in the experimental section, the most important model parameter, the conductivity of the water (

R_{aq}), shifted consistently depending on the case (

a or

b). This result is shown in

Table 4.

The direction in which this change was observed depended on the constituents of the original tap water:

When no micro precipitation was present in the beakers before and after the treatment, the impedance of the treated sample increased at high frequencies (case

a, see

Figure 11a–c, increase of

R_{aq} in

Table 2). Because fewer ions are available in the solution due to DOLLOP formation, there is less electrode polarisation at low frequencies (smaller phase shift in

Figure 7b at low frequencies and decrease of

A_{w} (ion diffusion constant) and

C in

Table 2).

As mentioned in

Section 3.1 sometimes the initial tap water contained tiny crystals. In some cases, precipitation would form during the treatment time in the reference beaker. Both of these cases are summarized in this work as case

b:

If there was visible micro precipitate in the reference beakers after the treatment or both beakers before treatment, the impedance of the treated sample was lower at high frequencies than the reference meaning that there were more ions in solution (case

b, see

Figure 11d–f, decrease of

R_{aq} in

Table 3), and higher at low frequencies due to increased electrode polarisation caused by these ions (larger phase shift in

Figure 11e at low frequencies and increase of

A_{w} and

C in

Table 2). Also here DOLLOPs are formed as described in case a, but the solubility product allows additional ions to dissolve from the precipitate. The solution is thus in a dynamic equilibrium between the formation of DOLLOPs and the solvation of the µm sized particles (see

Figure 12). The impedance increase observed suggests that the magnetic field gradient does not only facilitate the formation of DOLLOPs but also the dissolution of micro crystals. The authors plan to investigate this hypothesis in subsequent work.

In no case was precipitation found in the treated beaker after treatment, corroborating the DOLLOP formation hypothesis. In general, the shift in case b was smaller than in case a; and in the one case where we did not see a significant effect of the treatment, a trend towards case b was observable.

Next to ion diffusion constant, capacity and resistivity; parameters of the constant phase element (CPE) show large differences between the fits of the spectra from treated and untreated samples, again in opposite direction for cases a and b like the other parameters. The physical meaning of a CPE is an ongoing discussion in general; however, for the purpose of this work it is sufficient to say that, according to the model, together with the Warburg impedance it represents electrode polarisation and ion migration. Differences in the mobility of the ionic content of the solution due to the DOLLOP formation are also reflected in different parameters of the CPE.

A simplified sketch of this mechanism is given in

Figure 12.

These findings are in line with the many observations reported in theliterature [

4,

6,

13,

15,

16,

38,

39,

40] and most importantly, they agree with the model of Coey [

21]: the strong local gradients act on the mechanism of precipitation and induce DOLLOP formation. In case a, the ions form many small nuclei, DOLLOPs, which form a colloid and are thus no longer able to follow the alternating electric field during the impedance measurement.