Possible induced enhancement of dispersion forces by cellular phones

We derive the dispersion forces between objects in the presence of a non-thermal radiation field.

We apply the formalism to a model system representing two human blood cells in blood.

By focusing the same radiation density, as in room-temperature thermal radiation, in the microwave region we find a huge enhancement of the attractive force.

Related effects are predicted to occur also in other types of biological tissue.

The quantitative results should not be taken at face value, since the model is crude.

The effects are so large though, that further investigation is motivated.

The origin of the effects lies in the variation of water and/or ion content.

In the microwave region of the spectrum both the dipolar contributions from the water-molecules, and the mobile-ion contributions are important parts of the dielectric function, and cause the enhancement.


The dispersion forces between two objects can be expressed in terms of the electromagnetic normal modes of the system.1

These forces are van der Waals forces2 at intermediate separation and Casimir forces3,4 at large separations.

We know how the forces behave at zero and finite temperatures but we do not fully understand the behavior in non-equilibrium situations.

It is a well-known fact that gradients of optical fields can produce forces on microscopic dielectric objects.5

This effect is utilized in so-called optical tweezers.6

Optical binding between two dielectric objects in the presence of a strong optical field was discussed and demonstrated by Burns et al.7

Other related studies have been performed investigating the van der Waals force involving excited state atoms.8–10

The work by Burns et al. is an example of virtual excitations of the normal modes; the other studies involve real electronic excitations within one of the two interacting objects, not excitations of the normal modes.

In the present work11 we study the effects of real excitations of the normal modes of the system.

Thus the present work complements the previous studies.

The outline of the article is the following.

In section II we derive the force between two objects at zero temperature in terms of the electromagnetic normal modes; in this case the modes are not occupied.

In section III we extend the derivation to finite temperatures where the modes are in equilibrium with the thermal radiation field; in this case the modes are occupied according to the distribution function for mass-less bosons at the given temperature.

In section IV we use a Lorentzian distribution function.

In section V we concentrate on spherical objects.

In section VI we let the Lorentzian distribution function be peaked in the microwave region and apply the theory to the force between two water drops, between two air bubbles in water and finally between two human blood cells in blood.

Section VII is a summary and conclusion section.

Forces at zero temperature

At zero temperature the force, fo(r), between two objects as a function of separation, r, is given by the derivative of the energy, E: where the energy is the sum of the zero-point energies of the modes,

The modes are of two types: one type is localized to the surfaces of the objects and gives rise to the van der Waals interaction; one type extends between the objects and gives rise to the Casimir interaction (see ref. 1).

In a simple system with a small number of well-defined modes this summation can be performed directly.

In the general case this is not so.

Then we can rely on an extension of the so-called argument principle to find the results.

Let us study a region in the complex frequency plane.

We have two functions defined in this region; one, φ(z), is analytic in the whole region; one, f(z), has poles and zeros inside the region.

The following relation holds for an integration path around the region: where z0 and z are the zeros and poles, respectively, of function f(z).

In the argument principle the function φ is replaced by unity and the right hand side then equals the number of zeros minus the number of poles for function f(z) inside the integration path.

If we choose the function f(z) to be the function in the defining equation for the normal modes of the system,f(ωi) = 0,the function φ (z) to beand let the contour enclose all the zeros of the function f(z), eqn. (3) produces the energy of eqn. (2).

Using this theorem we end up integrating along a closed contour in the complex frequency plane.

In most cases it is fruitful to choose the contour shown in Fig. 1.

We have the freedom to multiply the function f (z) by an arbitrary constant without changing the result on the right hand side of eqn. (3).

If we choose it carefully we can make the contribution from the curved part of the contour vanish and we are only left with an integration along the imaginary frequency axis: where the last line was obtained from an integration by parts.

This needs some clarification.

The function f(z) contains polarizabilities or other time-correlation functions representing the response of the system one is treating.

These functions approach zero as the argument goes towards infinity since there is an upper limit for the frequency the system can respond to.

Consequently f(z) approaches a constant and after we have rescaled the function it approaches unity and the value of the logarithm approaches zero.

One further property of f(z) we should mention is that on the imaginary axis it is an even function of its argument; this is a general property of the time-correlation functions contained in f(z).

We will refer to this property below.

Forces at thermal equilibrium

The contribution to the internal energy from the interaction between two objects separated by the distance r can in general be written as:where the index i runs over all electromagnetic modes of the system and ni and εi are the occupation number and energy, respectively, of the mode number i.

Let us now concentrate on one of the modes and omit the indexNote that the occupation number n(r) is a function of energy or frequency and becomes a function of r because the mode energy depends on r.

An infinitesimal change in r modifies E both because of the change in mode energy and because of the change in occupation number The first term is due to mechanical work and the second to heat exchange with the surroundings.

The free energy Θ = E − Φ is the quantity that determines the force.

Let us now determine Φ and Θ where on the last line we have omitted the constant.

The energy functions are relative to some arbitrary reference level.

We usually choose this to be the value at infinite separation.

Thus the free energy is where on the last line we have identified the result as the Helmholtz free energy.

The force is

This was the contribution from one of the modes.

The total contribution to the Helmholtz free energy is We make the observation that we may use the argument principle but now with ln[2sinh(βħz/2)]/β instead of ħz/2 for φ(z) in the integrand.

There is however one complication.

This new function has poles of its own in the complex frequency plane.

We have to choose our contour so that it includes all poles and zeros of the function f(z) but excludes those of φ(z).

The poles of function φ(z) all are on the imaginary frequency axis.

We use the same contour as in Fig. 1., but now let the straight part of the contour lie just to the right of, and infinitesimally close to, the imaginary axis.

We have The coth function has poles on the imaginary z-axis and they should not be inside the contour.

The poles are at and all residues are the same, equal to 2/ħβ.

We integrate along the imaginary axis and deform the path along small semicircles around each pole.

The integration path is illustrated in Fig. 2.

The integration along the axis results in zero since the integrand is odd with respect to ω.

The only surviving contributions are the ones from the small semicircles.

The result is Since the summand is even in n (see comment at the end of Section II) we can write this as where the prime on the summation sign indicates that the n = 0 term should be multiplied by a factor of one half.

This factor of one half is because there is only one term with |n| = 0 in the original summation but two for all other integers.

When the temperature goes to zero the spacing between the discrete frequencies goes to zero and the summation may be replaced by an integration: and we regain the contribution to the internal energy from the interactions, the change in zero-point energy of the modes, as given in eqn. (6).

Forces at non-thermal equilibrium

In the absence of thermal equilibrium the occupation numbers are different from the Bose–Einstein ones for mass-less bosons and the summation along the imaginary frequency axis will no longer be the same.

The summation points came from the poles of the distribution function.

We want a narrow distribution function that is peaked in the microwave region of the spectrum where the cellular phones operate.

We want a distribution function that is peaked in a narrow energy range and has poles that give rise to feasible calculations.

A Gaussian distribution function would seem like a good candidate but it does not fulfil these criteria; it has an infinite number of poles on the imaginary axis.

We instead choose a Lorentzian distribution function.

Let us now specify our choice of n(ω): n(ω) = iA[(ω − ω0 + iB)−1 − (ω − ω0 − iB)−1 + (ω + ω0 + iB)−1 − (ω + ω0 − iB)−1]This function is real-valued on the real frequency axis and is a sum of two Lorentzian functions.

This function has the proper analytic properties and symmetries of a distribution function.

The energy or frequency where the distribution is focused is given by ω0 and the width of the distribution by B.

The function has two poles in the right half plane; it is an even analytic function on the imaginary frequency axis.

Our integration contour will now consist of the contour in Fig. 1., and two small circles clockwise around the two poles of the distribution function.

The integrand will be derived next.

Using eqns. (7), (10) and (11) gives

We first assume that we have an explicit expression for the second term of the free energy:

Then we may write: and we may use the argument principle again and get one contribution from the integration along the contour of Fig. 1. and one from the contribution of the small circles around the poles of the distribution function: Θ(r) = Θ1(r) + Θ2(r), and Θ2(r) = iħA[ln f(ω0 − iB) − ln f(ω0 + iB)]Thus in eqn. (24) we have a similar expression to that for the equilibrium case for T = 0 in eqn. (6), but now with a factor n + 1/2 instead of just 1/2.

Note that the occupation number is evaluated on the imaginary frequency axis.

Forces between spherical objects

We will now limit the treatment to the interaction between two spherical objects at large and intermediate separations where the dipole-dipole interaction is dominant.

We limit the treatment to the van der Waals region.

Then the mode defining function is (see ref. 1)f(z) = [1 − α1(z)α2(z)/r6]2[1 − 4α1(z)α2(z)/r6]and the free energy is

In eqns. (27) and (28) we have expanded the logarithm for small arguments.

Now the polarizability for the spherical object i in some ambient medium, a, is where Ri is the radius of the object and εa(ω) the dielectric function of the ambient medium.

Numerical results

We now apply the theory to some systems where the objects have a water content differing from that of the ambient medium.

Water is strongly active in the microwave region due to the permanent dipole moments of the water molecule.

This property is the basis of the microwave oven.

The field induces rotations of the molecules.

The rotational energy is transformed into heat, due to the internal friction of water.

We study two water drops, two air bubbles in water, and two blood cells in blood.

The dielectric properties of water have been studied extensively and are available in the literature.12

We need these data both on the imaginary frequency axis for eqn. (27) and at general points of the complex frequency plane for eqn. (28).

The measured values are of course on the real frequency axis.

The values on the imaginary axis can be obtained from the knowledge of the imaginary part, ε2(ω) on the real frequency axis from a generalization of the Kramers–Kronig dispersion relations, derived from general properties of analytical functions.

The relation is For a general point, ω = a + ib, in the complex frequency plane the relations for the real part ε1 (ω) and imaginary part, ε2(ω), are

In the case of water drops the dielectric function for the ambient medium in eqn. (29) is unity and εi(ω) is the dielectric function of water.

Note that the index i here runs over the two drops and does not represent real and imaginary parts.

In the case of air bubbles in water the ambient dielectric function is that of water and εi(ω) is unity.

The results for two water drops are shown in Fig. 3 and the corresponding results for two bubbles in water are shown in Fig. 4.

In this work the main objective is the force between two human red blood cells in blood.

To be able to perform a reliable, quantitative, calculation of the effects on the force between objects one needs to know the dielectric function in the full spectral range, both for the objects and for the ambient medium.

Since the information we have is limited we are restricted to rather crude model calculations.

The dielectric properties of blood in the range of 10 Hz to 100 GHz are known.13

The data for red blood cells14 are sparser, though.

We model the cell as a spherical object of radius R enclosed by a membrane; the radius of a human red blood cell is 4.5 μm and the thickness of the cell membrane is 8 nm.

We assume that the membrane has the same dielectric parameters as the cell membrane of a white blood cell of B-type.15

The interior is treated as homogeneous and we adjust the water and ion content of the cytoplasm so that the real and imaginary parts of the dielectric function for the cell agree with the few available, measured values;14 we end up with a water content of 75% and obtain the value 0.525 S m−1 for the conductivity from the ions; these values seem reasonable.

We should be more specific with the dielectric functions used for the blood and blood cells.

For the blood we use the expression in eqn. (4) of .ref. 13

It is valid for frequencies up to 100 GHz and consists of four terms; the first is the high-frequency asymptote; the second is the rotational contribution from water molecules; the third is the rotational contribution from larger entities; the fourth is the contribution from all mobile ions.

The authors of ref. 13 adjusted the eight parameters to give a good fit to the experimental data.

The red blood cell consists of a membrane enclosing the cytoplasm.

We thus need the dielectric properties of these two materials.

Here we use just two of the terms discussed above, the first and the fourth, just as in eqn. (3) of ref. 15 with parameter values given in that reference.

For the cytoplasm we keep in principle the first, second and fourth terms.

The two first terms we replace by 1 + F[εw(ω) − 1], where F is the fraction of water and εw(ω) is the dielectric function of water.

This gives us two parameters to vary, viz., F and the parameter of the fourth term of the dielectric function, the static conductivity of the mobile ions.

The effective dielectric function of the blood cell we determine along the lines of eqns. (4)–(6) of ref. 15, where we remove the effects from the nucleus.

The red blood cells have no nucleus.

We then vary the two parameters, to get a good fit to the experimental imaginary and real parts of the dielectric function of a red blood cell.

The fit is seen in Fig. 5.

We have probably got good enough estimates of the dielectric properties of blood and cells for the spectral range up to 100 GHz, but we need the properties also for higher frequencies.

We lack this information.

We know the dielectric function of water16 in the whole spectral range.

The most characteristic feature of this dielectric function is the large contribution from Debye1,17 rotational relaxation in the microwave region.

We model the dielectric function of the cell and of blood by letting them be equal to the function for water for high frequencies.

The functions used are displayed in Fig. 5.

The solid, dashed and dotted curves are the imaginary parts of the dielectric function for the cell, for blood and for water, respectively.

The circles are the experimental values for the cell from .ref. 14

The short piece of solid curve is our real part of the dielectric function for the cell compared to the experimental values from ref. 14, indicated by the squares.

The energy density at the surface of a black-body radiator at 300 K is approximately 6 × 10−6 J m−3.

The corresponding radiative power is 46 mW cm−2.

We choose a Lorentzian function peaked at the photon energy 3.5 × 10−6 eV and with a FWHM of 5% of the peak energy.

The results are found to be insensitive to the peak width.

We have let the strength of the distribution be such that the energy density in the radiation field is 1 × 10−6 J m−3.

The corresponding radiative power is 7.5 mW cm−2.

Thus we have chosen a slightly weaker radiation than that of the black-body radiator.

The numerical values used for ω0, B and A are 5.32 × 109 s−1, 1.33 × 108 s−1 and 2.95 × 1024 s−1, respectively.

In Fig. 6 we show the result, the upper solid curve, for the interaction potential in the microwave field.

For comparison we have included the room temperature result, middle curve, and one in the absence of radiation, lower curve.

To give the reader a feeling for the strength of the potential we have added two reference potentials: The first is the dash-dotted curve.

It is the unscreened potential between two unit charges; in a colloid the particles become charged and typically attain a charge of the order of 100 unit charges, but the resulting repulsive potential between the particles becomes screened by the counterions in the solution and the net potential is strongly reduced and also short range.

Thus, a comparison between the repulsive Coulomb potential and the attractive dispersion potential, induced by the microwave field, should be made towards the left in the figure.

The second reference potential is the short horizontal line in the figure.

It represents the energy 6kBT for room temperature.

This number is an estimate used to determine if a potential barrier is large enough to prevent the particles from overcoming it through the Brownian motion.

The force between the cells is obtained by taking the derivative of the potential with respect to separation.

Doing this we find that the force ranges from 0.15 μN at the small-separation end of the figure to 1.5 zN at the large-separation end.

The potential and force resulting from the microwave radiation can not be considered negligible in comparison with other forces experienced by the blood cells.

The lowest curve in the figure is the result without radiation.

This potential comes entirely from the zero-point energy of the normal modes.

It is only of academic interest since it corresponds to absolute zero temperature.

The thermal radiation enhances the interaction.

At room temperature the radiation causes an enhancement of the potential and force at intermediate separations of several orders of magnitude.

The enhancement at microwave radiation is much more impressive.

It gives an additional enhancement of ten orders of magnitude.

The photon energy where the microwave radiation is concentrated is indicated by a bar at the bottom of Fig. 5.

We see that the energy or frequency is here low enough for the mobile ions to contribute to the screening.

The ion contribution is of the Drude-tail type and varies as ω−1.

The enhancement of the force in the microwave region is probably due to both ion contributions and contributions from rotations of the water molecules.

In the calculations for water drops and for bubbles in water we found strong enhancement of the force but not as strong as the result in Fig. 6.

In these two cases the ion contributions are absent but the rotational contributions are stronger and there are contributions from the whole spectral range including the infrared and ultraviolet regions.

Summary and conclusions

To summarize, we have gone beyond the approximation of treating cellular phones as black-body radiators, which is usually done in setting the radiation standards.

We have focused the radiation power to a peak in the microwave region.

We have studied the effect of this radiation on the attractive part of the force between two blood cells in blood using a simplified model for the dielectric properties of the system.

We found an enhancement of the force by ten orders of magnitude when changing the radiation spectrum from thermal to one that is peaked in the microwave region.

Since the results depend on the water and ion content in the blood and blood cells and since these vary between individuals and also depend on the person's health status the effects can vary from person to person.

Another possible result from the radiation could be that thin blood vessels contract and thereby limit the flow of blood through them; the surface energy of the blood vessel might increase drastically making a contraction energetically favorable.

Still other unwanted effects could be growth of precipitates in tissue and organs, in the eye for example.

At this stage it is only speculation.

A word of caution: The present work should not be considered as a proof of that cellular phones are harmful.

It shows that there may be effects not considered before.

The weakness of the model calculations lies in the incomplete input data for the dielectric properties of biological tissue and in the description of the radiation field.

We have not taken into account that the field may be polarized; in a polarized field the forces between the two cells also depend on their relative orientation with respect to the polarization direction; the force may be repulsive for some orientations and even more strongly attractive for others.

In general, the force induced by a radiation field has attractive and repulsive contributions from different parts of the spectral range.

The origin of the enhancement, we have found, for the interaction between the cells is that the energy density in the radiation field has been concentrated in a spectral range where the contributions are attractive.

The result is not sensitive to the particular choice of distribution as long as it is peaked at the same spectral position.

We have used a Lorentzian distribution function because it has only two poles in the right half of the complex frequency plane, giving rise to the two terms in eqn. (28).

The Bose–Einstein distribution function for thermal radiation has an infinite number of poles giving rise to the infinite number of terms in eqn. (17).

Had we used a Gaussian distribution function we would also have ended up with an infinite number of terms.

The final result would have been similar but the mathematics more cumbersome and the treatment less transparent.

Hopefully this study will stimulate efforts to measure the dielectric properties of all components in biological tissue over the full spectral range, making more realistic studies feasible.