Water dielectric and microwave radiationa

The water dipoleb attempts to continuously reorient in electromagnetic radiation's oscillating electric field (see external applet). Dependent on the frequency the dipole may move in time to the field, lag behind it or remain apparently unaffected. When the dipole lags behind the field then interactions between the dipole and the field leads to an energy loss by heating, the extent of which is dependent on the phase difference of these fields. The ease of the movement depends on the viscosity and the mobility of the electron clouds. In water these, in turn, depend on the strength and extent of the hydrogen bonded network.

The applied field potential (E, volts) of electromagnetic radiation is given by;

E = Emax.cos(wt)

where Emax is the amplitude of the potential, w is the angular frequency in radians.second-1 and t is the time (seconds). If the polarization lags behind the field by the phase (d, radians) then the polarization (P, coulombs) varies as

P = Pmax.cos(w t - d)

where Pmax is the maximum value of the polarization.

Hence the current (I, amperes) varies as

I = (dP/dt) = -w Pmax.sin(wt - d)

The power (P, watts) given out as heat is the average value of (current x potential). This is zero if there is no lag (i.e. if d = 0), otherwise

P = 0.5 PmaxEmax w.sin(d)

It is convenient to express the dielectric constant in terms of a complex number (e r*, = i) (dielectric permittivity) defined as:

e r* = e r´ - iLf

Where e r´ is the ability of the material to be polarized by the external electric field, Lf (the loss factor) quantifies the efficiency with which the electromagnetic energy is converted to heat and i = Ö(-1). This equation may be visualised by considering the total current is the vector sum of the charging current and the loss current; the angle d as the phase difference (lag) between the electric field and the resultant (orientation) polarisation of the material (see similar treatment in rheology).

tan(d) = loss current/charging current = Lf/(e r´ - 1) (see derivation)

The terms (e r*, e r´, Lf ) are all affected by the frequency of radiation; the relative permittivity (e r´, dielectric constant) at low frequencies (e S, static region) and at high ( visible) frequencies the (e¥, optical permittivity) are the limiting values. The relative permittivity changes with the wavelength (and hence frequency):

where e S is the relative permittivity at low frequencies (static region), and l S is the critical frequency (maximum dielectric loss).

(see complex dielectric permittivity)

where t is the relaxation time (a measure of the time required for water to rotate (where r is the molecular radius, k is the Boltzman constant and h is the viscosity), also considered as the delay for the particles to respond to the field change, or for reversion after disorientation. The maximum loss occurs when w = 1/t c. For water at 25°C, t is about 8 ps and r is half the (diffraction-determined) inter-oxygen distance (1.4 Å).

Figure 1. Dielectric permittivity and dielectric loss of water between 0°C and 100°C, the arrows showing the effect of increasing temperature (data is indicative only but based on [64, 135]). As the temperature increases, the strength and extent of the hydrogen bonding both decrease. This (1) lowers both the static and optical dielectric permittivities, (2) lessens the difficulty for the movement dipole and so allows the water molecule to oscillate at higher frequencies, and (3) reduces the drag to the rotation of the water molecules, so reducing the friction and hence the dielectric loss. Most of the dielectric loss is within the microwave range of electromagnetic radiation (~1 - ~300 GHz). The frequency for maximum dielectric loss lies higher than the 2.45 GHz produced by most microwave ovens. This is so that the radiation is not totally adsorbed by the first layer of water it encounters and may penetrate further into the foodstuff, heating it more evenly; unabsorbed radiation passing through is mostly reflected back, due to the design of the microwave oven, and absorbed on later passes.

The dielectric loss factor (Lf) increases to a maximum at the critical frequency.

Dissolved salt depresses the dielectric constant dependent on its concentration (C ) and the average hydration number of the individual ions (HN)

Figure 2. Dielectric and dielectric loss of 0.15 M NaCl between 0°C and 100°C, the arrows showing the effect of increasing temperature. The salt decreases the natural structuring of the water so reducing the static dielectric permittivity, in a similar manner to increased temperature. At the lower frequencies the ions are able to respond and move with the changing potential so producing frictional heat and increasing the loss factor (Lf).

The dielectric loss is increased by a factor that depends on the conductivity (L, S cm2 mol-1; S = siemens = mho), concentration and frequency. It increases with rise in temperature and decreasing frequency.

Ensuring that all units are SI, the 1000 factor in the denominator goes. This 1000 is a conversion between SI and cm, mol, L units. This emphasises that careful consideration must be given to the units used in the microwave literature.

The rate of temperature increase is proportional to the loss factor and inversely proportional to the density times specific heat. The heating time = specific heat x temperature increase x weight/power input

Bound water and ice have critical frequencies (l S) at about 10 MHz (t about 0.1 m). At the much higher frequency of microwave ovens such water has a low dielectric permittivity (e.g. ice-1h, e¥ = 3.1; c.f. ice-1h, e S = 97.5 [94]; e S water (0°C) = 87.9), and is almost transparent, absorbing little energy. Thus whereas water becomes a poorer microwave absorber with rising temperature, a lossy salty food such as salt meat becomes a better microwave absorber with rising temperature. This is particularly noticed on thawing. Therefore for balanced heating the thermal effects must be balanced.

The electromagnetic penetration is infinite in a perfectly transparent substance and zero in reflective material (e.g. metals). At the microwave oven frequency (2.45 GHz), most energy is absorbed by water. The attenuation (a) is given by:

This equation may be approximated where the attenuation is (approximately) directly proportional to the loss factor and inversely proportional to the wavelength times the square root of the relative dielectric constant:

For a plane wave, incident microwaves decrease to 1/e (0.36788; i.e. 63% absorbed) in a penetration distance Dp given approximately by::

At 2.45 GHz:

The amount of power (P, in watts m-3) that is absorbed is given by:

P = 2 p f e 0LfE2

where e 0 = 8.854x10-12 F m-1, f is the frequency (Hz, = w/2 p) and E is the potential gradient (V m-1).

a Background theory and definitions are given on another page [Top]

b Hydroxyl groups in sugars and polysaccharides behave similarly, creating a high shear environment. Fats exhibit a lesser effect but their lower specific heat gives rise to rapid heating. [Back]

c Note the relaxation time is the reciprocal of the frequency in radians per second whereas the electromagnetic frequency is commonly reported in cycles per second (Hz). [Back]

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This page was last updated by Martin Chaplin
on 26 September 2001