Influence of Magnetic Field on Water and Aqueous Solutions

Abstract

Influences of an applied magnetic field and an applied field gradient are explored in pure water and in aqueous solutions. Effects such as diamagnetic levitation require a large magnetic field gradient force, and very large inhomogeneous fields. The weak effects on surface tension are best observed in compensated zero-susceptibility solutions of paramagnetic ions. Much larger effects of magnetic field on evaporation rate, in the range 10–100% may be of either sign. They are related to behaviour of the ortho and para nuclear isomers of water as quasi-independent gasses in the vapor, which are released from the surface of evaporating water in a ratio that is far from the expected equilibrium 3:1 triplet/singlet ratio in the ambient atmosphere. For pure water the ratio is found to be 39:61 The equilibrium is slow to be established by intermolecular collisions, but the ratio can be changed rapidly in the vapor phase, for example by a magnetic field gradient due to Larmour precession, which in turn alters the evaporation rate. The field is most effective in when the vapor is confined in a sheltered space such as a microchannel.

1 Introduction

The influence of a magnetic field B (measured in Tesla) on condensed matter depends on the nature of the field, whether homogeneous or inhomogeneous, static or dynamic. The simultaneous application of an electric field E (measured in Vm⁻¹) creates a dielectric polarization or excites an electric current that can modify the magnetic response. In this chapter, our focus is on the response of water and aqueous solutions to homogeneous and inhomogeneous static magnetic fields. B is the fundamental divergenceless magnetic field with no sources or sinks. The magnetic response of condensed matter however is determined by a different field, the local magnetic field strength H. The two are related by the equation,

$$ B = \,\mu_{0} \left( {H + M} \right) $$

(8.1)

where µ₀ is the magnetic constant 4π 10^–7 TmA⁻¹, and the magnetization M is the magnetic moment per unit volume of condensed matter. Units of H and M are both Am⁻¹; an equivalent unit for M is JT⁻¹ m⁻³, based on the expression for the energy of a magnetic moment m (units Am²) in a field of B, E = − m.B JT⁻¹.

1.1 Susceptibility

The basic response of water or aqueous solution to a field H is the appearance of an induced magnetization, proportional to the magnetic susceptibility χ of the liquid, defined by

$$ M = \chi H $$

(8.2)

M is isotropic and initially linear in field. Thus defined, susceptibility is a pure number, with no dimensions. The value for water is – 9.0 × 10⁻⁶, so water is diamagnetic with induced magnetization directed opposite to the applied field H₀. The local magnetic field strength is related to H₀ via a ‘demagnetizing’ factor that depends only on the shape of the sample with limits $0 \le {\mathcal{N}} \le 1$.

$$ H = H_{0} - {\mathcal{N}}M $$

(8.3)

Dividing Eq. 8.3 by H, it can be seen that H is actually bigger that H₀ when χ is negative, but the difference for water is very small, and it can usually be neglected.

Other definitions of susceptibility are possible, where it is not dimensionless. Sometimes H₀ is replaced by B₀. Also cgs units (emu, Oersted and Gauss) may be encountered. The dimensionless cgs susceptibility is smaller than its SI counterpart by a factor 4π because µ₀ = 1GOe⁻¹ in the cgs system. Table 8.1 lists values for the susceptibility of water, together with its units, for different definitions to ease confusion when reading literature that does not use Eq. 8.1 and SI units. Besides the magnetic moment per unit volume used in our definition of χ, the magnetic moment per unit mass or per mole may be used instead. There is little temperature dependence for water; the susceptibility is found to increase by 1% in the range 1–80 ℃ due to a slight weakening of the hydrogen bonds [12, 48].

Table 8.1 Summary of different ways of defining the susceptibility of water, with corresponding units in brackets

There are two notable magnetic consequences of the diamagnetic susceptibility of pure water in a non-uniform field. One is the Moses Effect [5], where a field is applied to region of water of depth d in an open bath. When a long bath of water is placed between the poles of an electromagnet, a small depression δ of the surface in the field is observed due to the magnetic pressure

$$ P_{{\text{m}}} = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \mu_{0} \chi H^{{2}} $$

(8.4)

exerted on the water. Equating this to the change in hydrostatic pressure δρg. The value of δ for µ₀H = 1 T is just 360 µm. In 10 T, the effect rises to 3.6 cm. (The red sea is 3 km deep)

The other effect is diamagnetic levitation. A pendant droplet in a vertical magnetic field gradient ∇_zB experiences an apparent change of weight due to the Kelvin force density, also known as the ‘magnetic field gradient force’ density F_K = − (1/2µ₀)χ ∇_zB ². The force will counterbalance the weight when − (1/µ₀)B∇_zB = − ρg, where ρ is the density of water. The numerical condition for levitation of water is therefore

$$ B\nabla_{{\text{z}}} B = - {136}0\,{\text{T}}^{{2}} {\text{m}}^{{ - {1}}} . $$

(8.5)

It is possible to suspend water droplets and other objects composed mainly of water (frogs, strawberries ….) in stable equilibrium close to the top of an open bore of a Bitter magnet or superconducting solenoid where the field is nonuniform and ~ 20 T [56].

A further effect on a pendant or suspended water droplet is due to Maxwell stress. An applied field H₀ induces a small uniform magnetization M in droplets of isotropic paramagnetic or diamagnetic liquids, as indicated in Fig. 8.1. The ‘demagnetizing’ effect can usually be neglected in water on account of its small susceptibility, although this is not the case in strongly paramagnetic ferrofluids with χ ~ 1, which minimize their total energy in the magnetic field by adopting spectacular spiked surfaces [54]. The pressure is not continuous at an interface between a liquid magnetized in a uniform field and vacuum, where a ‘magnetic normal traction’

$$ P_{{\text{n}}} = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \,\mu_{0} M_{ \bot }^{{2}} $$

(8.6)

Fig. 8.1

Effects of a magnetic field on water. a Moses effect in a localized field b levitation in a large vertical field gradient and c Maxwell stress on a spherical droplet in a uniform field

acts perpendicular to the interface [16, 54].

2 Pendant Droplets

Fitting the shape of pendant liquid droplets captured in an optical contact analyser is a commonly used method to determine the surface tension of liquids. The pear-shape of the droplet is a consequence of equilibrium between gravitational and surface tension forces. Commercial optical droplet analysers fit the shape of the outline of a digital photographic image of the pendant droplet to a parametric equation from which surface tension γ is determined, given the liquid density ρ. Both vary with temperature, which should remain constant over the course of the measurement. To use the method to investigate magnetic field effects, permanent magnets are preferred to electromagnets, which have the drawbacks of heating and restricting access.

There are three ways a magnetic field could influence the shape of a pendant droplet.

• (1) Maxwell stress

• (2) A change in the surface tension of the liquid

• (3) Kelvin force in the direction of the field gradient when the field is non-uniform.

The first two effects can be produced by a uniform applied field, the third depends on the field gradient to provide partial magnetic levitation.

Maxwell stress. A uniform field H₀ applied to a droplet of isotropic paramagnetic or diamagnetic liquid, induces a small uniform magnetization M = χH₀, as indicated in Fig. 8.1c. The ‘demagnetizing’ effect can usually be neglected in water and dilute solutions of paramagnetic ions on account of their small susceptibility, although this is not the case in strongly paramagnetic ferrofluids with χ ~ 1, which minimize their total energy in the magnetic field by adopting the characteristic spiked surfaces described by Rosensweig. The pressure is not continuous at an interface between magnetized liquid and vacuum, where the magnetic normal traction of Eq. 8.6 appears in a direction perpendicular to the interface [54]. There is also a tangential component S_θ of the Maxwell stress in the Lorentz formulation, and the components as given by Datsyuk and Pavlyniuk [16] when the susceptibility is small are

$$ \begin{aligned} P_{{\text{r}}} & = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \mu_{0} M^{{2}} {\text{cos}}^{{2}} \theta \\ S_{\theta } & = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \mu_{0} MH{\text{sin2}}\theta . \\ \end{aligned} $$

(8.7)

The tangential component for water is the larger by five orders of magnitude (1/χ). Integrating the component of the normal stress in the z-direction of the magnetic field over a hemisphere the stress in the z-direction is (1/4)µ₀M². A consequence of the deformation of the magnetized droplet due to Maxwell stress is an increase in its surface area. The deformation δd/d of the in the direction of the field, where d is the droplet diameter is a function of the dimensionless quantity µ₀M²d/4γ, where γ is the surface tension of the liquid and for small deformations of a pendant droplet they should be roughly equal. Hence, we can estimate that the deformation of a 4 mm pendant water droplet due to the normal Maxwell stress in 1 T as approximately 4.5 × 10^–7. However, if we take account of the z-component of the tangential Maxwell stress (1/3)µ₀MH, the estimated deformation in 1 T is 4%, or 1% in 0.5T.

A similar result is obtained by relating the Maxwell stress to the magnetic Bond number B_m, the dimensionless ratio of magnetic to surface energy, defined as B_m = χB²d/2µ₀γ. This ratio is 0.2 for a 4 mm droplet of water in 1 T and one or two orders of magnitude greater in solutions of paramagnetic ions. The deformation of the droplet in a uniform applied field is proportional to the square of the field, the susceptibility of the liquid and the size of the droplet itself [21, 22].

The field produced by a single permanent magnet is inherently non-uniform in magnitude and direction. To achieve an approximately-uniform field over a droplet, the magnets must have dimensions that are several times that of the drop. A simple configuration uses two identical rectangular magnet blocks magnetized the same direction, with the drop in the airgap between them. The permanent magnet configurations in Fig. 8.2a and b are assemblies of small magnet cubes designed apply a vertical magnetic field gradient ∇B to a pendant droplet.

Fig. 8.2

Magnet arrays that produce a a vertical field with a vertical gradient and b a horizontal field with a vertical gradient. Black arrows show direction of magnetization of the green permanent magnets. The variation of B and ∇_zB with height above the centre of each magnet array is shown. The measurement positions in the vertical and horizontal fields are noted

Measurements of the influence of magnetic field on the shape of a pendant drop are conveniently made by first observing the drop in zero field in an optical contact analyser, then raising the permanent magnet array so that the drop is immersed in the field and after several minutes of observation lowering the magnets and observing the drop again in zero field. In this way it is possible to correct for any drift. An example of a measurement of water in the nonuniform field produced by the magnet arrays of Fig. 8.2a and b is shown in Fig. 8.3. All three effects listed above are simultaneously present, but the Kelvin force is dominant, and it tends to partially levitate a drop of pure water since B∇_zB at the centre is − 43 T²m⁻¹, a fraction of that given by Eq. 8.5. The points on the figure correspond to fitted images, with the normal value of g, the acceleration due to gravity, and a change in ‘apparent surface tension’ γ_eff that best fits the shape.

Fig. 8.3

Left. Typical data on a drop of diamagnetic water in no field and in a vertical gradient field. The droplet shape in terms of an ‘apparent surface tension’ as the magnets are raised to influence the drop, and then lowered again to establish any baseline drift

The deformation due to the magnetic field gradient effect is the greatest whenever it is present, but it is eliminated when the field is uniform. The influence of a uniform 450 mT field on γ_eff of deionized water is an increase of 0.19 ± 0.21 mNm⁻¹_. The error is the standard deviation on the mean of 37 measurements. The contribution of Maxwell stress to the deformation of a pure water droplet is negligible.

3 Surface Tension

3.1 Static Surface Tension

Returning to the question of surface tension, it is possible to eliminate any direct influence of magnetic forces on a liquid that has exactly zero susceptibility. This is achieved in dilute solutions of the right concentration of a paramagnetic ion to cancel the diamagnetic susceptibility of water, giving a net zero-susceptibility aqueous solution. Any effect of a magnetic field on surface tension should then reflect its influence on the chemical bonding in the water. The zero-susceptibility molar concentration of paramagnetic ions is calculated by setting to zero the sum of the diamagnetic contribution of water (dimensionless susceptibility, χ_w = − 9 × 10^–6).

and the positive Curie law susceptibility the of 3d or 4f ions

$$ \chi = \mu_{0} {\text{ng}}^{{2}} {\text{p}}_{eff}^{{2}} \mu_{{\text{B}}}^{{2}} /{\text{3k}}_{{\text{B}}} T. $$

(8.8)

were n is the is the number of ions per unit volume, g is the Landé g-factor and the effective Bohr magneton number p_eff is (S(S + 1))^1/2 for 3d ions and (J(J + 1))^1/2 for 4f ions where S and J are the spin quantum number and the total angular momentum quantum number, respectively [13]. The effective ionic moment is p_effµ_B where µ_B is the Bohr magneton. The numerical expression for the susceptibility of a mole of ions is χ_mol = 1.571 × 10^–6 p_eff²/T. The susceptibility of an aqueous ionic solution of molarity x in the dilute limit is therefore

$$ \chi = \chi_{{\text{w}}} /{1}000\, + \,{1}.{571}\,{1}0^{{ - {6}}} x{\text{p}}_{{{\text{eff}}}}^{{2}} /T. $$

(8.9)

where the first term is the susceptibility of a litre of water. Hence, we can calculate the molar concentration x₀ of any paramagnetic ion where the net susceptibility of the solution is zero. Table 8.2 lists the molar susceptibility of Cu²⁺, Mn²⁺ and Dy³⁺_, the three ions we will consider here, as well as the molarity x₀ = − 9 × 10^–9/χ_mol of the zero-susceptibility solution where χ_mol is the molar susceptibility of the ions, given by Eq. 8.6 with n = N₀ and using a Landé g-factor of 2 for 3d ions and 4/3 for Dy³⁺. Results for the three ions at T = 295 K are shown in Table 8.2. Also included are the experimental values of x₀^exp measured by SQUID magnetometry, which are a little different.

Table 8.2 Calculated molar susceptibility at 295 K, and calculated and measured zero-susceptibility concentrations

From measurement of the zero-susceptibility solutions, we deduce that the change is surface tension in a 0.5 T gradient field is − 0.40mNm⁻¹ for 0.47 M Cu, − 0.48mNm⁻¹ for 0.051 M Mn and − 0.30mNm⁻¹ for 0.017 M Dy. Similar results, − 0.11mNm⁻¹ for 0.47 M Cu, − 0.51mNm⁻¹ for 0.051 M Mn and − 0.50mNm⁻¹ for 0.017 M Dy are obtained in a 0.4 T uniform field because the Maxwell stress is then negligible [51]. These are all small changes, barely outside experimental error of zero. A superconducting magnet and a different means of measuring surface tension is needed for a more significant result. Such measurements have been done by Fujimura and Iino who find an increase of just 1.83 mNm⁻¹ in 10 T for pure water [26] or 0.09 mNm⁻¹ in 0.5T.

3.2 Dynamic Surface Tension

Considering the influence of magnetic fields on surface tension in a dynamic context provides an interesting insight into their effects on water. Water has been suggested to have a dynamic surface tension that can be inferred by examining the process of droplet pinch-off, a highly surface tension-driven event [29]. When a drop detaches from an orifice, a thin filament is first formed as the drop pulls away under gravity, which elongates while decreasing in diameter until the filament breaks and the drop detaches. At times close to the filament breaking or pinch-off, the thinning behaviour of an inviscid fluid can be described by a universal scaling law

$$ D_{{{\text{min}}}} = A\left( {g/r} \right)^{{{1}/{3}}} t^{{{2}/{3}}} $$

(8.10)

where D_min is the minimum filament diameter, A is a universal prefactor and t is the time from detachment (t-t₀) [37]. Some evidence has been reported that shows viscous effects still have an effect for low viscosity fluids such as water even for filament diameters much greater than the viscous length scale, l_v = µ²/γρ, for dynamic viscosity µ [17]. Pinch-off in this scaling regime is independent of gravity as the filament diameters close to pinch-off are much lower than the capillary length κ⁻¹ = (γ/ρg)^1/2 (2.72 mm for water) (Fig. 8.4).

Fig. 8.4

Minimum filament diameter of water as a function of dimensionless time t*, in which the time to pinch-off is normalised by the capillary time t_c = (ρD₀³/γ)^½. Data for water in no magnetic field as well as three different gradient fields superpose. Inset: diameters at times close to pinch-off in which the proposed scaling law should apply. No effect of the magnetic field can be observed

Applying a gradient field to a detaching droplet has the effect of modifying the overall body force the fluid filament experiences due to the Kelvin force. This is evident when observing the pinch-off of a fluid such as 0.11M DyCl₃ which is more magnetically susceptible than water. Figure 8.5b shows the fluid filament of a 0.11M DyCl₃ drop pulled towards a permanent magnet placed to the right of the drop resulting in an angle of tilt of 8°, while a much smaller tilt is observed in water in Fig. 8.5a. The modified body force results in the effective gravity the drop experiences changing by a small amount. However, as the filament diameters close to pinch off are much smaller than the capillary length of 2.72 mm for water, the small change in the effective gravitational force due to the presence of the gradient field does not affect the pinch-off dynamics, even in the case of 0.11M DyCl₃ after correcting the image for tilt. Using high speed photography, we have shown this to be case as no measurable effect on the pinching dynamics by a gradient field was observed [51].

Fig. 8.5

Pinching filaments of a water and b 0.11M DyCl₃ in the presence of a gradient magnetic field due to the presence of a permanent magnet placed to the right of the filament in each image. A greater degree of tilt is observed for the more highly susceptible DyCl₃

4 Evaporation Rate

A possible influence of magnetic field on evaporation rate of water has attracted the attention of experimentalists in recent years, and a body of often-contradictory experimental results has accumulated. Some of the reports of persistent magnetic field enhancement of water evaporation rate [11, 24, 28, 40, 46, 55] been reviewed by Chibowski and Szcses [10]. Here we address the questions of if and why a static or intermittent magnetic field can change the evaporation rate? Whenever possible, control experiments where relevant environmental factors such as airflow, relative humidity and temperature are carefully controlled, or continuously monitored as two samples, one in magnetic field and the other a no-field reference are simultaneously subject to otherwise identical, but possibly time-varying ambient conditions.

Various types of experiments have been carried out: First, are those where the weight of an open vessel of water is measured while it is exposed to a static field B and compared to a no-field reference [2, 4, 11, 40, 55]. These sometimes involve interrupting the experiment at different times to weigh the vessel and the remaining water. A better approach is to monitor an in-field sample and a no-field control simultaneously. An alternative protocol is to measure the loss of weight by evaporation in zero field at different times after a single brief exposure to a static field [32], and compared with an unexposed sample in the same conditions. Here, a persistent memory effect of magnetic field exposure on the water is sought. The field is commonly produced by a permanent magnet, rather than an electromagnet, to avoid any heating but the field is inevitably nonuniform to some extent in magnitude and direction over the evaporating water surface unless the surface area is very much smaller than the airgap of the magnet. Fields used range from tens to hundreds of millitesla.

Second are those types of experiment where the water is exposed to a nonuniform field, by continuous flow at a velocity ≳ 1 cms⁻¹ around a circuit where a permanent magnet surrounds a section of the pipe [2, 23, 58, 60]. The experimental setup resembles that used for magnetic treatment of hard water to control limescale, except that the water is usually circulated not just once but repeatedly through the inhomogeneous magnetic field, which is equivalent to periodically exposing the water to ~ 20 ms magnetic field pulses at frequency of about 1 Hz. After a fixed circulation period with one or many pulses, the magnetically-treated water is removed and its rate of evaporation is tracked by weighing. Remarkably, both types of experiments give qualitatively similar results—a modest increase of evaporation rate is usually reported. They have been combined in a two-stage experiment to increase the effect [62].

There is little consistency in the experimental protocols adopted, and results depend on humidity, temperature, magnetic field and time in different ways. Reported increases in evaporation rate associated with the magnetic field range from a few percent to more than 30% [4, 8, 11, 23, 32, 40, 55]. The magnetic field was thought to modify somehow the network of molecular hydrogen bonding in water, but there is no agreed explanation of how this might occur. It must be a subtle phenomenon—the direct decrease in energy of ½χ_molB²/µ₀ of water with molar susceptibility χ_mol = − 1.6 10^–10 [13] in a field of 1 T is—64 µJ/mol, seven orders of magnitude less than the energy of a hydrogen bond in water [57].

The third type of experiment less controversial. Here water was exposed to an intense magnetic field of order 10 T in a superconducting solenoid, with a large horizontal gradient ∇_xB so that the product B∇_xB is 320 T²m⁻¹. The explanation here [46] lies in the paramagnetic susceptibility of atmospheric oxygen, which is displaced by the evaporating water vapor. The magnetic forces are 17% of the buoyancy forces on air, which leads to a field-induced modification of gaseous convection and hence the water evaporation rate.

4.1 Droplets

There are reports in the literature of the evaporation of water in sessile and pendant droplets. Notable are interferometric measurements based on Newtons rings from a sessile droplet by Verma and Singh [59] who reported an increase of evaporation rate of 28% in the presence of a 100 mT magnetic field. This method is rapid and accurate with a precision of 5 nm for the droplet height.

Evaporation rates of water can be modified by dissolved ions that have the propensity to make (kosmotropes) or break (chaotropes) the hydrogen-bonded structure of liquid water and thereby reduce or increase the evaporation rate compared with pure water. This influence is reflected in the Hofmeister series of dissolved cations or anions [53]. The evaporation rate of droplets on hydrophilic surfaces increases with salt content, and Marangoni convection of the solute, which is due to different evaporation rates at the centre and edge of the droplet that leads to a radial thermal and surface-tension gradients, is the dominant cause of advective flows in the solutions [36]. Flow in the liquid entrains a convective flow in the adjacent air that modifies the evaporation rate.

Similar flows in pendant droplets are modified by uniform magnetic fields in the 100 mT range acting on conducting ~ 0.1 M solutions of FeCl₃ CoCl₂ and NiSO₄, with significant enhancement of the evaporation rate [35]. These are essentially magnetohydrodynamic damping effects the Lorentz force

$$ F_{{\text{L}}} = {\sigma}(v{\times}B){\times}B $$

(8.11)

acting on the Marangoni flow in a liquid with conductivity σ, an explanation that does not depend on the magnetism of the dissolved cation. No increase of surface tension was found by Jaiswal et al.

4.2 Confined Water and Aqueous Solutions

Our own work has focussed on simultaneous measurements of twin samples in the same Perspex enclosure to restrict airflow, one exposed to a quasi-uniform 300–500 mT field, the other a no-field control. Temperature and relative humidity are monitored automatically for periods of up to 60 h. Evaporation of water or aqueous solutions was investigated either from open beakers or from tiny drops centred in microchannels. The twin magnetic setups are illustrated in Figs. 8.6 and 8.12.

4.2.1 Open Beakers

Two 100 ml beakers are half-filled with water or aqueous solution and placed on separate balances in the Perspex enclosure. One is surrounded by a large Halbach ring magnet producing a quasi-uniform magnetic field over the liquid surface close to 500 mT, with a field gradient reaching 3 Tmm⁻¹ at the edge. (Fig. 8.6). Evaporation is proportional to the surface area, and the weights of the two beakers are monitored for periods from 1 to 60 h. Some short-time runs were also made using a single beaker on a normal laboratory balance with doors shut to exclude draughts [52] for reasons that will become clear.

Deionized water. Most work, including 36 runs of 16 or 60 h, was done with Millipore deionized water. In every case, evaporation was greater in the presence of the magnetic field. The average enhancement was 12 ± 7% [52] (Fig. 8.6).

Fig. 8.6

Experimental arrangement where in-field and no-field evaporation of water is monitored continuously by measuring the time-dependence the mass of water in beakers. Magnetic field profiles are shown on the right

Fig. 8.7

Extended 60 h run of evaporation of water versus time for a 50 mL sample of water in 500 mT (red) and the no-field reference (black, grey). a relative weight loss by evaporation b evaporation rates versus time and c histograms of the evaporation rates, where the average values are marked by blue arrows [52]

The evaporation rate averaged over all runs for which RH = 0.73 ± 0.06 is 0.0297 ± 0.0081 kgm⁻² h⁻¹ without field and 0.0331 ± 0.0088 kgm⁻² h⁻¹ in the 500 mT field with the field gradient. The rate often fluctuated by 10% or more In the course of the experiments because of slow variations of in ambient temperature and relative humidity. An extended, 60 h run is analysed in Fig. 8.7. Fluctuations due to changes of the ambient temperature and humidity, as well as local variations of evaporation rate in the beakers lead to the fluctuations during the run illustrated in Fig. 8.7b, but it should be noted that at almost any instant, the evaporation rate of the sample in the magnetic field is greater than that of the reference. The net weight loss in the magnetic field in this case was 14% greater than that of the reference at the end of the run.

The evaporation rate varies non-linearly with temperature T which determines the capacity c_a of dry air to absorb water vapor plotted in Fig. 8.8a; the variation around room temperature is 6% per K. It also varies with relative humidity RH. Although we did not control it, the value was often in the range 60–70%. Sometimes the rate was quite steady over a 16-h period, but on other occasions it changed by 5–20%. Evaporation rates of water from half-filled 100 mL beakers versus (100—RH) with RH in % are plotted in Fig. 8.8b. The data are taken at different times of the year and corrected to 23 ℃. A star marks the average value of g obtained from the 36 extended runs where the average humidity was 72.8%. The data suggest that the evaporation rate is proportional to (100—RH) and they extrapolate to 0.115 (10) kgm⁻² h⁻¹ for dry air.

Fig. 8.8

a Variation of c_a the capacity of dry air to absorb water vapor in g/kg as a function of temperature [25]. b Variation of evaporation rate g with relative humidity RH, measured in the beaker experiments [50]

An empirical formula for the evaporation rate of water in open air, used by engineers is [25]

$$ g = \Theta c_{{\text{a}}} \left( {{1}{-}RH} \right) $$

(8.12)

The prefactor Θ is numerically equal to (25 + 17v) kgm⁻² h⁻¹, where v is the speed of the surface airflow in ms⁻¹. In still, dry air at 23 °C, the predicted value of g at 50% RH is 0.49 kgm⁻² h⁻¹. The water in the beaker evaporates 45 times more slowly, Fig 8.8b, extrapolated to 100%.

6 M urea solution. Next, we consider a different liquid where 26.5 w/w% of the water has been replaced by urea, a non-volatile liquid that has no appreciable vapor pressure and is expected to disrupt the hydrogen bonding when in solution in the host water. Here we find that the evaporation rate is 0.0422(23) kgm⁻² h⁻¹ at a relative humidity of 49%, which extrapolates to 0.0867 kgm⁻² h⁻¹ in dry air. This is appreciably less than the value for pure water. In urea solution, the magnetic field decreases the rate of evaporation significantly, by 28 ± 6% to 0.330 (41) kgm⁻² h⁻¹ [52].

Salt solutions. Results obtained on ionic salt solutions reveal a rich variety of behaviour both as regards dry evaporation rate and the influence of magnetic field. For example, the evaporation rate of water from lithium, sodium and potassium chloride solutions decrease monotonically with increasing molarity, and it is 50% lower in 4 M NaCl compared to pure water (Fig. 8.9). The effect of magnetic field is to increase the evaporation rate for most concentrations around 1 M. The evaporation rate of a sessile droplet of water in open air is 0.91 kgm⁻² h⁻¹, and it is not influenced by a magnetic field. However, when the droplet is surrounded by a quartz cuvette partially open at the base so that the droplet evaporates into its own vapor, Fig. 8.10b, the rate is reduced by a factor 20 but errors are large, as seen in Fig. 8.10c.

Fig. 8.9

Evaporation rates of water in beakers from salt solutions of different molarity. The trends with and without an applied field are indicated by red and black dashed lines, respectively

Fig. 8.10

a A sessile droplet of water evaporating in an inverted cuvette, b Effect of cuvette on the evaporation rate and c Comparison of the evaporation rates of water and 1 M NaCl in the cuvette

Time dependence

At this point another experimental result helps to cast light on what is controlling the evaporation of water from beakers. Water is simply poured into a 100 mL beaker, which is placed on a chemical balance with the doors shut to exclude drafts while weight loss is monitored for the first two hours. The beaker again is roughly half full. No magnetic field is applied. The initial transient is quite different from the steady rate reached after about an hour. (This is a reason for using long runs used to measure evaporation rates). Results for deionized water are shown in Fig. 8.11, where data were fitted to the equation

$$ m\left( {\text{t}} \right) = m\left( 0 \right){-}({1} - {\text{e}}^{ - r/t} )(g_{0} {-}g{\text{t}}) $$

(8.13)

Fig. 8.11

Short-term evaporation of Millipore water in stagnant conditions, showing the steady-state and initial transient regimes, measured in a chemical balance. The initial transient, which decays exponentially with a time constant of 9 min is followed by uniform steady-state evaporation after about 30 min

Fig. 8.12

a Experimental arrangement where in-field and no-field evaporation of water is monitored continuously by measuring the volume of water in two PMMA microchannels; b Dimensions of the evaporating water sample at the centre of the channel, with a schematic illustration of Marangoni flow

The mass of water at the start of an experiment is m(0), τ is the decay time of the initial transient, and g₀ and g are the initial and steady state evaporation rates in kgs⁻¹.

The average of eight one-hour runs for water gave an initial evaporation rate g₀ = 0.134 ± 0.030 kgm⁻² h⁻¹ and an average τ of 14 ± 3 min. The long-time evaporation rate was only half as great 0.068 ± 0.010 kgm⁻² h⁻¹, Table 8.3.

These key difference between the early and later stages of evaporation is that initially water is evaporating into ambient air whereas later it is evaporating into air in the beaker where most the water vapor originally in ambient air has been replaced by vapor from the evaporating liquid. How they differ is discussed in §6. When deionized water is replaced by heavy water, the evaporation rate is less than half as great (0.013 kgm⁻²h⁻¹) and no initial transient is observed (Table 8.3).

Table 8.3 Initial and steady-state water evaporation rates and initial decay time in no field

From the typical evaporation rates, which depend on both temperature and relative humidity, and a density of air of 1.2 kgm³, the time taken to evaporate enough fresh water vapor to create a relative humidity of 50% in the empty half of the beaker of order 20 min. The average time taken for a water molecule escaping from the surface with a root mean square velocity deduced from equipartition of energy of 630 ms⁻¹ to diffuse out of the beaker is of order a minute or two. The evaporation at the surface is controlled by a mass balance of freshly escaping water vapor and recondensing molecules from region a within a few mean free paths of the surface, where the freshly evaporated molecules become the majority after a time τ.

When evaporation is measured from aqueous solutions as a function of time it is important to establish thermal equilibrium at ambient temperature before beginning measurements in the balance. Some salts like LiCl are exothermic when mixed with water, but urea is endothermic. In these cases any initial transient may be masked by the effects of temperature changes on the evaporation rate.

4.2.2 Water in Microfluidic Channels

Following from the idea that confinement of the fresh water vapor may be necessary to observe a magnetic field effect on the evaporation rate, we have performed experiments in microchannels analogous to those described in Sect. 8.4.2.1 [50]. The channels are made from three layers of poly(methyl methacrylate) (PMMA) assembled by thermo-lamination after cutting a 1 mm wide channel, 54 mm long in the 0.38 mm thick middle layer with a CO₂ laser. The experimental arrangement is illustrated in Fig. 8.12. No magnetic field is applied to water in one channel while water in the other is exposed to a 300 mT field perpendicular to the channel, produced by rectangular 50 × 20 × 10 mm³ Nd-Fe-B magnets. Evaporation is monitored by simultaneously imaging 0.4 µL droplets of Millipore water positioned at the centres of the two 50 mm channels with PCE800mm USB cameras The ends of each microchannel are open to ambient air, and the evolution of the shapes of the drops are recorded as they shrink down to a membrane after several hours and rupture shortly afterwards. The twin setup was enclosed in a perspex box and the ambient temperature (26 ℃) and relative humidity RH (42%) in the laboratory were controlled throughout.

Menisci appear at the edges of the drops with contact angles θ of about 30°, as seen in Fig. 8.12. Water normally exhibits a contact angle < 90° on PMMA [63], but the menisci form because of the periodic variation of the channel width due to cutting with the CO₂ laser. The variation is 10% of the 1 mm channel width with a period of 300 µm. Just like a magnetic domain wall, the water seeks to minimize its surface area and form a circular meniscus because of surface tension. As the water evaporates, the meniscus jumps to a neighbouring pinning point in a stick–slip process. The volume of water in the confined drop is deduced from its approximately symmetric shape parameterized by L, the average of the contact lengths at the two edges and l, the distance between the centers of the two menisci, as shown in Fig. 8.12b. The volume V of liquid, calculated from L and l knowing the width w and the depth d of the channel, is

$$ V \approx wd(L - {2}w\theta /{3}) $$

(8.14)

where contact angle θ is tan⁻¹ [(2R-L + l)/w] [50].

Some results are illustrated in Fig. 8.13; Figure 8.13a captures the last moments of a sample of water that shrinks to a membrane, and then ruptures, while Fig. 8.13b shows a typical sample that takes almost four hours to evaporate. Sessile droplets of recondensed water vapor can be seen growing in the channel beyond the menisci, which indicate that the air in the channel is saturated with water vapor. The areas of the menisci from which evaporation is occurring remains roughly constant and equal to 2wd.

Fig. 8.13

a Evolution of 0.12 μL drop of water in a microfluidic channel. After 39 min it has shrunk to membrane that ruptures 30 s later. b A typical example of evaporation of a 0.5 μL drop. Sessile droplets of recondensed water vapor grow in the channel beyond the menisci, indicating that relative humidity there is 100% [50]

Data for two representative runs are shown in Fig. 8.14. Initial values of L and l are normalized to 1. The runs, like those in the beakers, showed significant variability. The average evaporation rate from the control channel over ten runs was 0.13 ± 0.03 kgm⁻² h⁻¹, which is four times that in the beakers. The magnetic field enhancement was much greater, ranging up to 140%, with an average of 61 ± 42%. The stable sessile droplets of water with diameters of 30–150 µm growing slowly in the channel throughout a run were observed in most of the measurements when a magnetic field was applied. They appear at distances greater than about 100 µm from the menisci in Fig. 8.2b, which means that the air in the microchannel at 299 K is be saturated with water vapor, and they may be a result of the enhanced evaporation rates shown in Fig. 8.14.

Fig. 8.14

Variation of L and l, normalized to the values at t = 0, during two evaporation experiments; a and b are for experiment 1. c and d are for experiment 2. Data in black circles are for the no-field channel and data in red squares are for the channel in a 300 mT magnetic field

The flow dynamics of water in the microchannel and in the beaker must be quite different. The flow regime in the two cases may be characterized by the inverse Bond number, the dimensionless ratio − (∂γ/∂T)/βρgd² of Marangoni to Rayleigh numbers, which reflects the relative importance of the surface tension and gravitational forces that influence the flow. Here, β = 2.1 × 10^–4 K⁻¹ is the thermal expansion coefficient of water and d is the depth. The ratio is 316,000 for the microchannels and 90 for the beakers. Surface tension dominates flow in the channel. The faster evaporation rate of water in the microchannel compared with the beaker is attributed to Marangoni convection. Thermocapillary flow with the vortex pattern sketched in Fig. 8.12b, like that observed at a meniscus in a capillary [6, 7, 20], will increase evaporation in the microchannel. Magnetic field is known to influence the flow in droplets of ferrofluids and conducting fluids [19, 35] (Eq. 8.9), but the deionized water we use is neither conducting nor magnetic. We need a different explanation.

The vapor close to the meniscus is composed of freshly evaporated molecules, which need high kinetic energy and well-timed making and breaking of at least three hydrogen bonds at the interface in to break loose [45]. The isomer ratio of such water vapor may differ from that in liquid water and its effective temperature will be higher than ambient in the channel, allowing it to evaporate without immediately recondensing. The Knudsen layer where there is a temperature gradient normal to the water surface [34] should be wide enough to allow evaporation to proceed into the channel at the ambient temperature of 26 ℃.

5 An Explanation

5.1 Isomers of Water

We saw in the discussion of Fig. 8.11 that water vapor freshly evaporated at a liquid surface seems to be different from that in equilibrium in the atmosphere. Not all molecules of H₂O are the same. Water exists as one of two nuclear isomers, according to whether the two proton spins are aligned parallel with net nuclear spin I = 1, or antiparallel with net nuclear spin I = 0, as shown in Fig. 8.15. Molecules with a nuclear spin triplet or singlet are known (somewhat confusingly) as ortho or para water respectively and every molecule is in one state or the other. To conserve angular momentum, the ortho isomer has an angular momentum of $\hbar$ in its ground state and the para isomer, which has none, is lower in energy by 2.95 meV (34.3 K). There are three possible spin substates with I = 1, namely I_z = 1, 0 and -1, but only one, I_z = 0, with I = 0. The spin substates are respectively symmetric |↑↑〉, |↑↓〉 + |↓↑〉, |↓↓〉 and antisymmetric |↑↓〉  −  |↓↑〉. The water vapor in equilibrium at ambient temperature is expected to exhibit an ortho:para ratio of 3:1, and this has been confirmed by terahertz spectroscopy of the vibrational energy levels [44] in the 0–3 THz range. However, the ratio in liquid water, which will be influenced by hydrogen bonding, is thought to be close to 1:1 [47], and the ratio in the escaping water vapor, which has to be involved in a three-body collision in order to break free [45] will be different again.

Fig. 8.15

Para (left) and ortho (right) water molecules. The spin and orbital moments of each are listed and the vibrational energy levels are shown in the centre

Unlike the corresponding isomers of hydrogen gas, H₂, which differ in energy by 15 meV in their ground states and can be separated in the gas phase just above the boiling point and then used for dynamic nuclear polarization to enhance the sensitivity of nmr, it has proved very difficult to separate ortho and para water by physical methods. Separation rates of 1 pL per day have been achieved in molecular beams, but the separated liquid isomers have half-lives of about an hour at ambient temperature [33] and 3:1 equilibrium is soon re-established. However, the molecular isomers in the vapor phase are much longer-lived and equilibrium can take weeks to establish [9].

Nuclear singlet–triplet transitions are strongly forbidden [1], so ortho and para water vapor may be regarded as separate molecular species—two quasi-independent gasses. This hypothesis allows us to rationalize the observed magnitudes of the evaporation rates with and without a magnetic field in terms of the ortho:para ratio f_v^o:f_v^p of the escaping water vapor.

5.2 Analysis

Data analysis is based on the hypothesis that ortho and para water vapor behave as independent gasses on the timescale of the experiments. The ratio in freshly-evaporated liquid f_L is evaluated from the data, assuming the re-condensation rate is constant and isomer-independent, and the effect of the magnetic field is to modify the isomeric ratio in the vapor. The main difference between the beaker, microchannel or confined sessile droplet and water in open air is that the water will be evaporating into its own vapor in a steady state, with an isomer ratio f^o_L:f^p_L whereas in open air water is evaporating into ambient water vapor with an isomer ratio f^o_V: f^p_V of 3:1. Water evaporating in a sheltered space the originally filled with ambient air, will gradually replace the ambient air by air with a the isomer ratio f^o_L:f^p_L of freshly evaporated water. If the space is not sheltered but open to air currents, the ratio will remain 3:1, and magnetic field has no influence on the evaporation rate. This is the case, for example, for unshielded sessile droplets.

The hypothesis is expressed as the ansatz that the evaporation rate is proportional to the dimensionless quantity

$$ g_{0} = \left[ {f^{{\text{o}}}_{{\text{L}}} \left( {{1} - f^{{\text{o}}}_{{\text{V}}} } \right) + p^{{\text{p}}}_{{\text{L}}} \left( {{1} - f^{{\text{p}}}_{{\text{V}}} } \right)} \right] $$

(8.15)

The plot of g₀ as a function of f^o_L = (1 − f^p_L) in Fig. 8.16 illustrates the evaporation rate for three different conditions of the surrounding vapor. The grey line f^o_V = 0.75 represents the 3:1 equilibrium ratio in open air. The green parabola represents the situation where the water is evaporating into its own vapor in a confined space. The horizontal red line is for the case where the ratio is 1:1. The horizontal dashed line represents the recombination rate c, which is supposed to be a constant independent of f^o_L; the net evaporation is measured as the height of the grey, red or green line above the dash line. The net value of g₀ is multiplied by a factor Θ_exp to give the actual mass evaporation rate in kgm⁻² h⁻¹.

Fig. 8.16

Evaporation diagram based on Eq. 8.15. See text for explanation

The effect of water vapor escaping from a from a liquid and accumulating in half-filled beaker is often to reduce the initial rate from the gray line to that given by the green parabola as time goes on. This means that f^o_L for water lies in zone A₁ or A₂. There is no effect for f^o_L = 0.5 or 0.75. When the initial rate increases with time, as it does for 6 M urea or 1 M NaCl, f^o_L lies in Zone B. The values of f^o_L and c can be determined more precisely from the no-field to in-field evaporation ratio, which is < 1 in almost all cases.

Magnetic field can alter the isomer ratio in the vapor, and hence the evaporation rate, in two ways [52]. One is via Larmor precession at 43 MHz/T, which tends to de-phase the two protons when a field gradient is present, making f^o_V = f^p_V (red line), The other is via Lorentz stress on the electric charge dipole, which tends to increase angular momentum of the H₂O. In Zone A₁ the field effect is always positive. In Zone B it may be negative.

Applying these ideas to pure water we find f^o_L = 39 ± 1% [52]. In the 2 M and 4 M Li and K solutions the value increases to 0.46 for Li and 0.44 for K, assuming c remains 0.4. Na is different, 2 M is similar to water and for 4 M the ortho ratio has increased to 0.46, but for 6 M urea has shifted into Zone B, with a negative field effect and f^o_L ≈ 0.6.

6 Crystallization from Solution

Finally, we summarize some information on a related topic—how the crystallization of ionic crystals from supersaturated aqueous solution can be influenced by a static magnetic field, either to promote single-crystal growth or to selectively inhibit the growth of an unwanted crystal polymorph in favour of another. Explanations of both effects involve proton dimers formed during crystal growth.

6.1 Ionic Crystals

The effect of a magnetic field on the crystallization of a selection of carbonates phosphates, sulphates and oxalates of calcium, magnesium or zinc, as well as the divalent magnetic ions Mn, Fe or Co were studied by Lundager Madsen [41,42,43]. He found that when these sparingly-soluble salts are crystallized from solution in a magnetic field of 270 mT, the rates of nucleation and crystal growth are enhanced in the diamagnetic salts of weak acids but there is no effect at high pH, or when the cations are magnetic. A key observation was that the field had no effect when heavy water is used as the solvent. In D₂O, hydrogen is replaced by deuterium, where the nucleus is composed of a proton and a neutron. Both have spin ½ and they couple to give I = 0 making the hydrogen a boson rather than a fermion. The growth of nuclei in solution is assumed to involve the creation of doubly protonated anions on the growing surface, which is possible with no energy penalty for D₂ or H₂ in an I = 0 singlet para state, but the triply degenerate I = 1 ortho state has a ground state with orbital angular momentum and corresponding vibrational energy that reduces the rate of the process. Nuclear spin relaxation for H₂ tends to equalize the ortho and para populations and thereby increase the growth rate [43]. There is no such effect for D₂.

6.2 Magnetic Water Treatment

Treatment of hard water to avoid precipitation of hard limescale on heated surfaces in domestic and commercial water heaters, boilers and heat exchangers has a long history of mixed practical success and widespread scientific scepticism [3]. The basic claim that is amenable to investigation is that by passing hard water once through a nonuniform magnetic field, typically generated by an array of permanent magnets, it is possible to influence the subsequent precipitation of calcium carbonate when the water is heated. Aragonite tends to precipitate rather than calcite, and the carbonate then has an acicular morphology that does not form hard scale [15, 30, 31, 38, 39]. The water somehow retains a memory of its magnetic exposure that persists for days. The dilemma of reconciling these observations with the rapidly-changing molecular structure of liquid water was lifted after it was discovered that the calcium carbonate dissolved in water was not entirely in the form of separate Ca²⁺ cations and HCO₃ anions, but partly in the form of amorphous polymeric nanoscale prenucleation clusters [27, 49], subsequently named DOLLOPS (Dynamically Ordered Liquid-Like Oxyanion Polymers) [18]. This has led to a new theory of nucleation [27] and also an idea of how magnetic water treatment may work [14]. The requirement is a durable modification of the pre-nucleation clusters during the fleeting exposure of water to the magnetic field. This could be provided by the magnetic field gradient acting on a layer of bicarbonate anions at the surface of the DOLLOP. Adding a Ca²⁺ ion to grow the cluster displaces a proton to form a hydrogen dimer in H₂CO₃ or its stable product H₂O + CO₂. Dimerization of the protons is a way to achieve a long-lived modification of the cluster by scrambling the proton singlet and triplet states as their spins are dephased in the magnetic field gradient, where their Larmor precession frequencies (42.6 MHzT⁻¹) are slightly different. The field gradient produced by the magnet arrays is comparable to that needed to dephase the protons by π in the time they take to pass through the inhomogeneous magnetic field [14].

7 Conclusions

Magnetic fields have little direct effect on liquid water. Because of its weak diamagnetic susceptibility, exceptionally large magnetic fields and magnetic field gradients are required to produce effects such a magnetic levitation. Compensated solutions of paramagnetic ions with zero susceptibility can be used to avoid extraneous effects of the Kelvin force density on properties such as surface tension. Changes in surface tension measured in pendant droplets are of order − 1% per tesla.

Much larger magnetic effects on the evaporation rate of pure water or aqueous solutions are found when the water is evaporating into its own vapor in a confined space sheltered, from external airflow. Increases of 10–100% are observed for deionized water, depending on confinement, but for aqueous solutions the evaporation rate may increase (many salt solutions) or decrease (6 M urea, 1 M NaCl) in the magnetic field. The sign of the field effect on evaporation appears to be associated with the sign of an initial transient in the evaporation rate observed in no field.

The remarkable magnetic field effects on the evaporation rate have been discussed in terms of a two-vapor model, where the two nuclear isomers of water behave as independent gasses. The ortho:para, triplet to singlet ratio is 3:1 in equilibrium in ambient air, but in freshly-evaporated water vapor it may be quite different. From the magnitude and sign of the magnetic field enhancement and the shape of the zero-field transient evaporation, it is possible to fix both the isomeric ratio of fresh vapor and the recombination rate to within a few percent. The ratio is 39:61 in Millipore water. The effect of a magnetic field is to modify this ratio in the vapor and thereby modify the rate of evaporation from the liquid surface. Two mechanisms are proposed, one is Larmour precession of the two protons on a single water molecule. A magnetic field gradient will dephase their precession and tend to equalize the two populations. Lorentz torque could augment the angular momentum.

Magnetic field effects on crystal growth from saturated aqueous solutions and magnetic water treatment are similarly related to the nuclear spin of proton dimers. We have not observed magnetic memory effects in pure water.

Surprisingly, the average evaporation rate of water in a microfluidic channel where the relative humidity exceeds 100% is notably greater than it is in an open beaker where the vapor is unsaturated. This is likely due thermocapillary flow in the channel. Future work should aim to visualize thermocapilliary flow in microchannels by particle image velocimetry in a field; evidence of a Lorentz force would reveal any related charge flow. The proposed isomer ratios in fresh water vapor should be verified by terahertz spectroscopy.

Our results suggest that the conventional treatment of evaporating water vapor as a single gas needs to be reconsidered in confined spaces where advection is limited. There is a prospect of relating the ortho-para ratio of fresh vapor to hydrogen bonding in different aqueous solutions [53], thereby obtaining new insight into the role of kosmotropic and chaotropic ions in the Hofmeister series. Furthermore, magnetic field may prove to be useful for applications where it is desirable to increase the evaporation rate of water in microscale porous media without raising the temperature [61].