ArticlePDF Available
Journal of Modern Physics, 2018, 9, 2657-2724
http://www.scirp.org/journal/jmp
ISSN Online: 2153-120X
ISSN Print: 2153-1196
Water Memory Due to Chains of Nano-Pearls
Auguste Meessen
Université Catholique de Louvain (UCL), Louvain-la-Neuve, Belgium
Abstract
Biologically active molecules create substitutes in liquid water by forming
single-domain ferroelectric crystallites.
These nanoparticles are spherical and
constitute growing chains. The dipoles are aligned, but can be set in oscilla-
tion at
the frequency of vibration of the charged part of active molecules.
They are then automatically trimmed and become information carriers.
Moreover, they produce an oscillating electric field, causing autocatalytic
multiplication of identical chains in the c
ourse of successive dilutions. Active
molecules are thus only required to initiate this process. Normally, they excite
their specific receptors by resonance, but trimmed chains have the same ef-
fect. This theory is confirmed by many measurements.
Keywords
Water Memory, Extra-High Dilutions, Ferroelectric Particles, Chains of
Nanoparticles, Water Bridges, Molecular Interactions
1. Introduction
The concept of water memory is based on experimental results of measurements,
published 30 years ago in the prestigious scientific journal Nature [1]. This asto-
nishing phenomenon had been discovered by
Jacques Benveniste
and his colla-
borators, but one month later, the same journal declared that it was a delusion
[2]. Even the publication of the discovery was already accompanied by an as-
tounding editorial reservation [3]. The editor in chief,
John Maddox
, declared
indeed that “there is no physical basis for such an activity” and announced even
that independent investigators would “observe repetitions of the experiments”.
When they arrived at Benveniste’s laboratory, it turned out that John Maddox
was accompanied by the professional magician
James Randi
and the debunker of
scientific fraud
Walter Steward.
The objective was thus to detect errors or fraud.
During their stay in Paris, the first experiments confirmed the published result,
How to cite this paper:
Meessen, A. (2018
)
Water Memory Due to Chains of N
a-
no
-Pearls.
Journal of Modern Physics
,
9,
2657
-2724.
https://doi.org/10.4236/jmp.2018.914165
Received:
November 20, 2018
Accepted:
December 26, 2018
Published:
December 29, 2018
Copyright ©
2018 by author and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
DOI: 10.4236/jmp.2018.914165 Dec. 29, 2018 2657 Journal of Modern Physics
A. Meessen
but not later ones. The inquisitors did immediately publish a devastating con-
clusion: the reported results are not reproducible and merely due to imagination
[2]. Randi stated even that they should be compared to the sensational claim of
having seen a unicorn, where there was merely a goat.
The load of the accusation fell on Jacques Benveniste (1935-2004). He was a
medical doctor, who had practiced several years before opting for research. In
California, he discovered the platelet-activating factor and determined its role in
immunology [4] [5]. He returned to France in 1973 and joined INSERM (Na-
tional Institute of Health and Medical Research). Since 1980, he directed Unit
200, devoted to research in immunology, allergy and inflammation. He discov-
ered there a very sensitive method to detect allergens. It required careful count-
ing of discolored cells, but
Elisabeth Davanas
succeeded very well [1]. The young
medical doctor
Bernard Poitevain,
who joined the team to prepare a thesis,
asked somewhat later if he could test the efficiency of homeopathic dilutions by
means of this method. Benveniste answered: “try if you want, but there will be
no effect; high dilutions are merely water” ([6], p. 45). Nevertheless, the results
were positive and confirmed by other members of the group.
This phenomenon was totally unexpected, since successive dilutions of bio-
logically active molecules in pure water do necessarily lead to their complete
elimination. Was it really possible to create substitutes that are only constituted
of water molecules? Further experimentation did prove that these structures
should even be able to mimic active molecules of different types. No one knew
how this might be achieved. Benveniste verified therefore if these hypothetical
structures could be destroyed. It appeared that after heating extra-high dilutions
(EHDs) of histamine during one hour at 70˚C, they had completely lost their bi-
ological efficiency. Exposition to ultrasound had the same effect ([6], pp. 53-54).
The puzzling phenomenon of “water memory” had even a characteristic proper-
ty: the biological efficiency of EHDs decreased at first, but increased again and
reached a high value after about the 9th decimal dilution. Then it dropped and
continued to vary in a quasi-periodic way during successive dilutions.
Since the reality of this phenomenon was tested many times for different sub-
stances, Benveniste thought that these facts had to be published, although they
were unexplained. He insisted on the observed quasi-periodic variations, by pro-
viding two figures [1]. However, Maddox was convinced that the reported re-
sults cannot be real. He required independent confirmations. They were pro-
vided by other laboratories in Italy, Canada and Israel. Eventually, after two-year
long discussions ([6], p. 51), Maddox accepted to publish the article if Benve-
niste did agree that a team of experts could come to “verify the quality of the ex-
perimentation”. This was more than bizarre, but Benveniste had nothing to hide.
The outcome was the terrible accusation that we mentioned [2]. Benveniste was
allowed to add a reply, but could only describe how the investigators had pro-
ceeded. They created a climate of “intense and constant suspicion”. Actually,
Benveniste and his collaborators were treated like “criminals”.
DOI: 10.4236/jmp.2018.914165 2658 Journal of Modern Physics
A. Meessen
In their report, the group of inquisitors insisted on the variability of the peaks
of activity and claimed that measurements had been treated in “disregard of sta-
tistical principles”. They declared even that “the laboratory has fostered and then
cherished a delusion about the interpretation of this data”.
Melinda Baldwin,
lecturer on History of Science at Harvard University, identified the actual cause
of this grave incident. Maddox considered that scientific journals are shaping
science by controlling its quality [7]. This implies enormous power, but also the
danger to defend orthodoxy by condemning any deviation. Baldwin mentioned
the quasi-periodic variations of the reported biological efficiency. They were
strange, but could have an intrinsic cause, deserving further research.
Benveniste considered that his duty was only to establish the reality of these
“unbelievable and fear-provoking” facts, since their meaning and the underlying
mechanism could be studied later on ([6], p. 61). This is not at all unusual in
science. After the events of 1988, the experiments were repeated many times
with the statistician
Spira
, but the previous results were validated. In the course
of further tests, Benveniste made a second discovery. He realized in 1993 that
two EHDs were able to perturb one another and thought that this was due to
electromagnetic fields. He suspected that these “signals” could play an essential
role in the constitution of water memory and did prove that they can be de-
tected, amplified, recorded and transmitted. They seemed to be noise in the fre-
quency domain of audible sound waves (lower that 20 kHz). However, when
pure water was exposed to them, it acquired the same properties as EHDs of the
initially dissolved active molecules. This
information transfer
confirmed that
water memory is real, but INSERM did not renew the contract for Unit 200 in
1995. Benveniste pursued his research in a room on the parking of his former
institute, with a caravan for storing materials. He focused now his efforts on de-
veloping “digital biology”.
Alain Kaufmann
presented in 1994 basic facts and an analysis of the sociolog-
ical context [8].
Michel Schiff
published the same year a more detailed descrip-
tion of the experiments and denounced the dangers of censorship [9]. Benve-
niste, who had reluctantly accepted to write the preface of this book, attributed
there the radical rejection of available evidence to the following reasons: “It
cannot be,
since if it were true it would have been found two hundred years ago”
and “there is
no theory
behind it.” We can add that the discovery of an anomaly
may lead at first to incredulity, but well-documented facts should induce a
search of their cause. It might be necessary, indeed, to correct some previous as-
sumptions. Peer evaluation is necessary and useful, but not infallible. Schiff in-
sisted that even scientists can “suppress unwanted knowledge” when it would
“shatter their current beliefs”. Actually, “the long history of scientific dogmatism
shows that today’s heresy could well become tomorrow’s scientific truth”.
Francis Beauvais
, a former collaborator of Benveniste, provided in 2007 much
more details on events at that time [10].
Yolène Thomas
[11] continued research
on water memory, but it had to be camouflaged as concerning properties of
EHDs. Even the French virologist
Luc Montagnier,
who received in 1988 the
DOI: 10.4236/jmp.2018.914165 2659 Journal of Modern Physics
A. Meessen
Nobel Prize for his contribution to the discovery of the HIV virus, was violently
attacked when he resumed the experimental study of water memory. He was
motivated by scientific curiosity and the constant need of improved medical
treatments, while his opponents negated
a priori
that EHDs in pure liquid water
could modify this solvent. Montagnier presented his experimental method in a
documentary, realized in 2014 by French TV [12]. It is also available in English
[13] and does clearly demonstrate that water memory involves detectable signals
at frequencies like those of audible sound waves (20 - 20,000 Hz). Montagnier
published in 2009 and 2010 important results concerning water memory [14]
[15]. He discovered even that viral DNA sequences can be reproduced by means
of transmitted signals when the building blocks are available in pure water.
These signals had thus to provide the required master plan and this fact might
account for strange resurgences of some sicknesses. The conference of Montag-
nier at UNESCO in 2014 stressed this fact and other medical applications [16].
Visceral opposition to the concept of water memory was often motivated by
fear that it could justify homeopathy. The aim of the present study is merely to
find out if water memory is real or not. This has to be viewed as a basic problem
for condensed matter physics, since bonds between water molecules are con-
stantly broken by thermal agitation in liquid water at the time scale of picose-
conds.
Martin Chaplin,
specialist of properties of water molecules, proposed
therefore in 2007 that water memory could result from creating statistically sta-
ble clusters of water molecules [17]. Individual molecules would there be re-
placed by other ones without affecting the global structure. This hypothesis was
the most plausible one, but Martin Chaplin added that much research work
remains to be undertaken if these real and observable facts are to be completely
understood”.
The structure of this article results from the itinerary that we followed. In Sec-
tion
2, we examine the internal structure of water molecules and their possible
interactions. This leads to the concept of “water pearls”. In Section
3, we explain
why biologically active molecules can create chains of these nano-pearls and why
they account for water memory. Section
4 presents more observational evidence
concerning these chains. It is diverse, detailed and very remarkable, but the con-
cept of water pearls accounts for known facts, while the alternative concept of
Coherent Domains does not. In Section
5, we insist on the most important con-
sequence of water memory: molecular interactions are not only possible by
means of the “key and slot” model of chemical reactions. Intermolecular com-
munications can also result from oscillating electric fields and resonance effects.
It will appear once more that “Nature is written in Lingua Mathematica”, as Ga-
lilei stated already, but we endeavor to be understandable by non-specialists.
2. Interactions of Water Molecules
2.1. Their Structure and the Dipole Approximation
Martin Chaplin provides detailed information about the internal structure of
DOI: 10.4236/jmp.2018.914165 2660 Journal of Modern Physics
A. Meessen
water molecules [18] and various models that have been proposed [19]. The
usual “stick and ball model” insists on the chemical composition (H2O), but the
protons of both light atoms are deeply embedded in the common electron cloud
of the oxygen and hydrogen atoms. Water molecules are thus practically spheri-
cal, but at close range, the protons are surrounded by a spherically symmetric
excess density of electrons. They are thus equivalent to point-like charges
q
e
/3. The core of the oxygen atom and the remaining part of the electron cloud
are equivalent to a central point-like charge −2
q
.
The kinetic diameter (for collisions) of water molecules in the terrestrial at-
mosphere is 0.265 nm. H2O molecules are thus smaller than O2, N2, CO2 and H2
[20]. Their size is slightly greater in the liquid state, because of interactions with
surrounding water molecules. The average separation of two oxygen atoms is
there measurable by means of x-ray diffraction. The resulting diameter is
d
=
0.275 nm. In the gaseous state, the angle HOH is 104.5˚, but it is close to 106˚ in
the liquid state. Vibrational and rotational spectra of water molecules disclosed
that the length of OH bonds is
δ
≈ 0.095 nm. Thus, δ/d ≈ 1/3. Since water mole-
cules behave in the liquid state like hard spheres that can easily roll on one
another, we adopt the model of Figure 1. To simplify later calculations, we chose
natural units, where
δ
= 1 and
q
= 1. The diameter of a water molecule in the
liquid state is then
d
≈ 2.90. Since the angle
φ
= 53˚, the distance
a
= cos
φ
≈ 0.60
and
b
= sin
φ
≈ 0.80. It follows that a/d = 1/5.
Because of their internal point-like charges, water molecules are tripoles, but it
is customary to replace them by dipoles. They are constituted by the central
charge −2
q
and a single charge +2
q
, situated in the middle between the charges
+
q
. This dipole is represented in Figure 2 by a red arrow. By definition, the di-
pole moment is then
p
= 2
qa
. The limited validity of this approximation appears
when we calculate the electrostatic potential
V
(
r
,
θ
) at large distances
r
from the
center of the effective dipole. The angle
θ
specifies the chosen direction with re-
spect to the axis of the dipole.
The test charge +1 does then “see” the charges along parallel lines, but their
distances are slightly different. They are indicated by thin red lines. Adopting
also natural units (
e
2/
δεo
= 1) for electrostatic potentials, their sum is
Figure 1. Model of water molecules.
DOI: 10.4236/jmp.2018.914165 2661 Journal of Modern Physics
A. Meessen
Figure 2. The dipole approximation.
( )
()
( )
222
23
2
,
22
o
oo
qq q
Vr rrr
qq
rr
θδδδ
δδδ δδδ
+−
+− +
=++
+−+
= +−+−+++
The first and second order approximations result from
( )
12
11x xx
+ =−+
when
1x
. Since
( )
cos
o
δ δ δ ϕθ
+
+= −
and
( )
cos
o
δ δ δ ϕθ
−= +
, the
value of
, where
δ
cos
φ
=
a
. Thus,
( )
23 3
2
, cos , cos , sin
r
ppp
Vr E E
rr r
θ
θθ θ θ
≈≈ ≈
(1)
The radial and angular components of the electric field
E
at the observation
point result from partial derivatives of
V
(
r
,
θ
). At closer range, there are inevita-
ble corrections. In liquid water, it is also necessary to account for intermediate
water molecules, since they are easily reoriented by an applied electric field. The
potential
V
(
r
,
θ
) is then reduced by the factor 1/εr, where the relative dielectric
constant εr ≈ 80. When neighboring water molecules are subjected to an electric
field, their effective dipoles will be aligned. These
molecular chains
are broken
by thermal agitation when the electric field is extinguished, but they could also
be stabilized by association, like sticks in a bundle.
This possibility merits attention, since it is known that electric field lines can
be visualized by means of neutral particles, like grains of semolina or short plas-
tic filaments dispersed on oil. The applied electric field does merely polarize
these particles, but the induced dipoles tend then to align one another. Could
similar chains be formed by means of water molecules? To answer this question,
we have to examine all possible types of interactions between water molecules.
2.2. The Origin of Van der Waals Forces
Even electrically neutral molecules attract one another in gases, because of Cou-
lomb forces and quantum-mechanical effects. Indeed, if such a particle were
subjected to an electric field, it would displace all weakly bound electrons inside
this particle. This produces surface charges that create a secondary electric field
inside the particle. It opposes displacements of the electrons and would restore
neutrality when the applied field is switched off. For oscillating electric fields,
this force leads to a resonance for the ensemble of oscillating electrons. That ex-
DOI: 10.4236/jmp.2018.914165 2662 Journal of Modern Physics
A. Meessen
plains the appearance of colors and peculiar optical properties of thin granular
metal films. They were said to be “anomalous” until they could be explained in
terms of collective oscillations of nearly free electrons [21]. Since oscillations of
electrons inside neutral particles do also create an oscillating electric field out-
side these particles, two neutral ones can interact with one another.
Nearly free electrons will be set in coupled oscillations inside neighboring par-
ticles. Their resonance frequency is then reduced, but in quantum mechanics
(QM), the lowest possible energy of an oscillator is proportional to its resonance
frequency. The (zero-point) energy of two neutral particles is thus reduced when
they come close enough to one another. This effect can be interpreted as result-
ing from an attractive force. The existence of this short-range force was discov-
ered by
Van der Waals
in 1873, since a dense gas does not behave like an ideal
one. It corresponds to a model, where velocities are only randomized by colli-
sions of point-like particles, but neutral particles attract already one another at
some small distance. The physical origin of this force could only be understood
after the development of QM.
Since Van der Waals forces are proportional to the volume of the polarizable
particles, they are negligible with respect to other forces for water molecules in
the liquid state. However, small metal particles that are suspended in liquid wa-
ter contain nearly free electrons. They are very polarizable and big enough to at-
tract one another by Van der Waals forces. In liquid water, small metal particles
attract thus one another and constitute chains. These “necklaces of pearls” are
observable by optical microscopy [22] and attract now much attention, because
of expected applications. Similar chains might be relevant for water memory.
2.3. Hydrogen Bonds and Exchange Effects
The concept of so-called “hydrogen bonds” was already introduced in 1920,
since some quantum effects could be treated in a semi-classical way [23], but
simplified models can lead to confusions. The Lewis model, proposed in 1916,
was merely based on the fact that many atoms are more stable when they con-
tain 2 or 8 electrons. Because of Bohr’s semi-classical model of atomic struc-
tures and Pauli’s exclusion principle, these values correspond to closed shells.
Hydrogen atoms contain only 1 electron, while C, N, O, and F atoms do respec-
tively have 4, 5, 6 and 7 electrons in their external shells instead of 8. Molecules
like CH4, NH3, OH2 and FH would thus result from the “tendency to complete…
the octet of electrons”. This lowers the total energy, but H2O molecules are spe-
cial. The left part of Figure 3 represents closed shells of the oxygen atom and the
hydrogen atoms by blue rings. The shared electrons are represented by dots, but
the oxygen atom is then surrounded by 4 pairs of electrons. There are 2 bound
pairs and 2 free pairs.
It was therefore proposed that the negative charge density of the free pairs
“might be able to exert sufficient force” on two neighboring oxygen atoms. This
would account for mutual attraction of water molecules that allows for structur-
ing of liquid and frozen water. The right part of Figure 3 represents these
DOI: 10.4236/jmp.2018.914165 2663 Journal of Modern Physics
A. Meessen
Figure 3. Semi-classical concepts of hydrogen bonds.
“hydrogen bonds” by means of red lines, when neighboring water molecules are
assumed to be situated in the same plane. QM revealed that electrons behave
according to laws that apply to waves. Every oxygen atom contains two strongly
bound electrons and four external electrons in (2s12p3) states. Superposition of
these wave functions leads to interference effects and the charge distribution of
the 4 external electrons acquires then tetragonal symmetry. We might thus think
that hydrogen bonds are merely due to stronger electrostatic attraction, but
modern biochemistry states that “in a hydrogen bond, a hydrogen atom is
shared by two other atoms” [24]. The right part of Figure 3 can then be inter-
preted in terms of two donor sites of protons and two acceptor sites. This de-
scription implies that the intermediate proton might change its position. Hy-
drogen bonds would then be due to
exchange effects
, which are also known for
nuclear forces.
Figure 4(a) represents the effects of hydrogen bonds between water molecules
in 3D space. The red dots define average positions of the cores of neighboring
water molecules in liquid water, while the red lines correspond in a schematic
way to electron pairs, but also to possible exchanges of protons between pairs of
oxygen atoms. This configuration requires a modification of the internal struc-
ture of water molecules, since the normal angular separation of two protons was
there 2
φ
= 106˚ (Figure 1). Here we get 4 equal angles 2
μ
≈ 110˚. They are de-
fined by joining the center of the cubic cell to four equally separated vertices.
The value of
cos 1 3hc
µ
= =
, since (2
c
)2 = 3(2
h
)2. It follows that
μ
= 54.74˚.
Figure 4(a) accounts only for one possible lattice structure of ice, since im-
posed temperatures and pressures can yield different phases for frozen water
[25]. In liquid water, adjacent molecules are moving around, since all bonds are
constantly broken and reconstituted at an extremely rapid pace [26]. Neverthe-
less, small-scale order is statistically preserved, while large-scale order is lost. At
an intermediate scale, we get the extended lattice-structure of Figure 4(b).
Body-centered and empty cubes are alternating. This yields many voids, which
will often be filled at higher temperatures. This fact explains the existence of
low-density and high density liquid water, as well as analogous amorphous states
for frozen water.
It is important to be aware of the quantum-mechanical nature of exchange ef-
fects. They are also possible between two X and Y atoms, when an intermediate
proton could belong to X or Y. Both possibilities are expressed by the notation
XH… Y or X… HY
.
QM accounts indeed for limited knowledge. The probability
distribution for possible positions of
electrons
is defined by means of their wave
functions. Exchange effects are then due to “tunneling” through an intermediate
DOI: 10.4236/jmp.2018.914165 2664 Journal of Modern Physics
A. Meessen
Figure 4. (a) Idealized model for the relative positions of oxygen atoms in neighboring
molecules for ice and liquid water; (b) Extended lattice structure of this type.
potential barrier. This is relevant for bonds between atoms inside molecules and
in particular for
2
H
+
, where one electron allows for H-H+ or H+-H. Although
protons have a greater mass than electrons, they are also subjected to quan-
tum-mechanical laws. The probability distribution for being at different places
in space is then not smeared out, but reduced to needle-like (delta) functions.
When a proton has two possible positions, it can be said to be
delocalized,
but
possible exchanges are then not due to tunneling. They result from the fact that
the energy of any physical system cannot be precisely determined during short
time intervals Δ
t
. There is always an irreducible uncertainty Δ
E
h
t
. In
semi-classical terms, the proton is able to “jump” over the intermediate potential
barrier, when this happens rapidly enough.
2.4. Coulomb Forces and Exchange Effects for Water Dimers
To analyze possible effects of protons for interacting water molecules, we begin
with the simplest case. Figure 5 represents the cores of two oxygen atoms by
open dots, separated by the distance d. The left part of this figure corresponds to
purely classical concepts. The intermediate proton has a well-defined position,
indicated by a black dot. Since the measured length of OH bonds is
δ
, we can
calculate the total potential energy U1 that results from electrostatic interactions
between two point-like particles of charge −2 and one charge +1. However, the
concept of
hydrogen bonds
means that the intermediate proton has two possible
positions. They are represented by gray dots in the upper right part of Figure 5.
When they are occupied with equal probabilities, we have to attribute an average
charge +1/2 to these positions. However, even when a particle is delocalized, it
cannot exert forces on itself. We will calculate the resulting electrostatic poten-
tial energy
U
2. The third configuration would be obtained if it were possible to
account for hydrogen bonds in a semi-classical way, by assuming that the proton
has only one well-defined position, situated in the middle. This position is
represented by a black dot and the potential energy would then be
U
3.
Using natural units for charges, distances and energies, we get
12
42 2 1.67UU
dd
δδ
=−− =− =
and
3
42 0.69Udd
=−=
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A. Meessen
Figure 5. Three conceivable models for O-H-O bonds.
Since
U
2 =
U
1, a strictly classical description is sufficient, although the proton
is delocalized. This would even be true (in the present case) if the uncertainty
did allow for any partition
q'
and (1
q'
) of the charge +1. The third configura-
tion is very unstable and does not account for quantum-mechanical exchange
effects. We are now ready to calculate the total potential energies
V
1 to
V
4 for
different configurations of dimers, represented in Figure 6. They result from the
fact that two water molecules can easily be rotated with respect to one another,
but this modifies the potential energy of the interacting tripoles. The usually as-
sumed configuration of (H2O)2 is close to the upper left one of Figure 6. The
tripole of one molecule is situated in the plane of the drawing and one of its two
protons is precisely oriented towards the core of the neighboring molecule. The
other tripole is perpendicular to this plane and seen in profile. We will calculate
the potential energy
V
1 for the indicated configuration.
V
2 corresponds to aligned effective dipoles. This configuration would be pre-
ferred if water molecules did only contain dipoles, but could even be privileged
for effective dipoles, when the dimer is subjected to an electric field. We want
thus to see if
V
2 is already close to
V
1 in the absence of an applied electric field.
The tripoles should then be orthogonal to one another to minimize repulsion
between protons in neighboring molecules.
V
3 is the potential energy for any
pair of molecules in a nearly linear chain, where all intermediate protons are
ideally situated between two oxygen atoms. The resulting zigzagging configura-
tion is planar and in conformity with a classical representation.
An applied electric field could even allow for a perfectly linear chain, because
of
intramolecular
exchange effects, although this was unknown. For clarity, we
consider here two coplanar tripoles. One proton is always situated as close as
possible to the core of the neighboring oxygen atom. The other proton has two
equally probable positions, above and below the symmetry axis. This would yield
the energy
V
4 for any pair of water molecules. Alternatively orthogonal tripoles
would reduce repulsion between the delocalized protons. The resulting potential
energy is then
V
5 <
V
4.
To facilitate this type of calculations, we note that the total potential energies
result always from adding the Coulomb potentials (
V
=
qq'
/Δ) for pairs of
point-like charges
q
and
q'
, separated by a distance Δ. We define thus a function
S
(
x
,
y
,
z
), where
x
,
y
and
z
are differences of Cartesian coordinates, respectively
measured towards the right, rear and top. Thus,
( )
,,V qq S x y z
=
where
( )
( )
12
2 22
,,S xyz x y z
= ++
(2)
The values of
V
1 depends on the angle
φ
= 106˚ and the complementary angle
ϕ
= 74˚. Since
δ
= 1 in natural units, the distance
a
1 = cos
ϕ
= 0.28 and
b
1 = sin
ϕ
DOI: 10.4236/jmp.2018.914165 2666 Journal of Modern Physics
A. Meessen
Figure 6. Conceivable structures of water dimers.
= 0.96. However,
a
2 =
aa
1 and
b
2 =
ab
1, where
a
= cos
φ
= 0.60 and
b
= sin
φ
=
0.80. Thus,
() ( ) ( )
( ) ( ) ( )
1 2 2 12 12
21 11
4 ,0,0 2 1 , , 2 , ,
2 1,0,0 4 ,0, 2 ,0,
0.116
V Sd Sd a bb Sd a a bb b
Sd Sd a b Sd a b
= +−+ +++ +
−−−+ −+
= −
( ) ( ) ( ) ( )
2
4 ,0, 0 4 , , 4 , , 0 4 , 0, 0.093V Sd Sdbb Dd ab Dd a b= + −+ −− =
V
1 can be slightly lowered when the repulsion between the nearest protons is
reduced by a small rotation of the left molecule around its center. An angle of
1.4˚ is sufficient to reach the minimal potential energy [27]. Since
V
1 <
V
2, it has
been assumed that more than 2 water molecules should always be assembled ac-
cording to the same rule as for the most strongly bound dimer. Water molecules
could then only constitute rings or clusters of limited size, but
linear polymeri-
zation
is not excluded for water molecules. It would even be preferred for
n
2 wa-
ter molecules, compared to clusters of
n
1 molecules, when (
n
2 1)
V
2 < (
n
1
1)
V
1. It is thus sufficient that
n
2 > 1.3
n
1. Of course, long chains of water mole-
cules would be too fragile to resist thermal agitation in liquid water, but we will
show (in Section 2.7) that chains of water molecules with aligned effective di-
poles can be stabilized.
Moreover,
V
3 = −0.111, which is quite close to
V
1 = −0.116. Since water mo-
lecules can easily be rotated, they are aligned according to this pattern inside
very narrow pores [28]. They are then said to form wires or filaments. Perfectly
linear polymerization with intramolecular exchange effects would yield
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
5 11 11 1 1
11 1 1
5 ,0,0 , , 1 , ,0 1 ,0,
2 1,0,0 2 1,0,0 2 , ,0 2 ,0,
0.084
V Sd Sdbb Sd a b Sd a b
Sd Sd Sd ab Sd a b
= + +−− +++
−− −+ −− −+
= −
An applied electric field does not only align water molecules, but also polarize
water molecules in such a chain. This yields stronger bonds, since all positive
charges of one molecule come closer to the central negative charge of the neigh-
boring molecule. Polarization of water molecules is possible [29] and favors thus
linear polarization by means of intramolecular exchange effects.
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2.5. Initial Evidence of Molecular Chains
When the young
Theodor von Grotthuss
was experimenting in1806 with a Volta
pile, he discovered that pure water has a much higher electric conductivity than
other liquids. The chemical structure of water molecules was not yet known.
[John Dalton asserted in 1808 that it is HO, while Avogadro proposed in 1811
that it could be H2O. This hypothesis was disregarded, since it resulted from the
assumption that all types of particles occupy the same volume in gases, whether
they are molecules or atoms. This was only accepted at about 1860, after the de-
velopment of the kinetic theory of gases.] However, Von Grotthuss knew that
water molecules are composed of positive and negative parts, since they can be
separated by electrolysis. This had already been proven in 1800.
Von Grotthuss thought therefore that water molecules are held together in
liquid water by mutual attraction of positive and negative parts. An electric field
should align them. The high electric conductivity of liquid water could then be
explained, if water molecules did contain tiny charge carriers that move more
easily inside these chains than outside them [30]. This hypothesis may have been
suggested by the method of fire-fighting of that time. People were standing in a
row and passed buckets from hand to hand, but the existence of protons was not
yet known. Nevertheless, this explanation was appealing and successful. Now, we
can justify this model in term of
intramolecular
exchange effects. The left part of
Figure 7 represents two water molecules in a linear chain, subjected to an elec-
tric field
E
. One molecule contains an additional proton. We indicate displace-
ments of charged particles by red lines, but the applied electric field does also
modify the intermediate potential barrier. It becomes dissymmetric and facili-
tates “hoping” over the potential barrier for the proton in best position. The blue
arrow represents a jump of this proton towards one of the nearby potential wells.
The right part of Figure 7 shows the result (without polarization effects). The
H3O+ ion became a normal H2O molecule, while the neighboring H2O molecule
was converted into an H3O+ ion. The proton did advance without being deviated
by collisions. [Ohm’s law is still valid for relatively small electric fields, as for io-
nic conduction in solid state physics]. We could equally well consider H2O mo-
lecules with one delocalized proton on the left side and an adjacent HO ion. It
contains only one proton, attracted toward the center of the next molecule. The
electric field
E
would then cause a jump of the intermediate proton towards one
of the two empty places. This would be equivalent to opposite motion of a pro-
ton hole. The essential result is that
intramolecular
exchange effects are realistic,
since they explain the high electric conductivity of liquid water in a more de-
tailed way.
Figure 7. Explanation of the von Grotthuss mechanism.
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2.6. Evidence of 2D Polymerization of Water Molecules
The story of this discovery is similar to that of water memory. It began with an
unexpected observation, made by the Russian chemist
Nikolai Fedyakin.
He
condensed water in thin capillary quartz tubes and found that its physical prop-
erties are different from those of ordinary water. This phenomenon was totally
unexpected, but arose at first much attention and scientific curiosity.
Lippincot
and Stromberg
combined, for instance, routine infra-red spectroscopy with an
improved method for producing this type of water. They confirmed that it has
peculiar properties and proposed an explanation [31]. They assumed that water
molecules can be bound to one another inside layers like that of Figure 8. The
nature of hydrogen bonds was misunderstood (see Section 2.4), but this confi-
guration was simple and heuristically useful. Since the layers can slide on one
another, water remains liquid, but is viscous near substrates that favor this con-
figuration.
2D polymerization of water molecules would thus yield a “new state” of liquid
water, since it is partially crystallized. However, other persons declared that this
is impossible, since they preferred to stick to customary ideas. The media prop-
agated the slogan of “bad science”, which had great impact. It led even to total
prohibition of research on “polywater”. Academic careers would have been bro-
ken for anyone who might dare to be involved in such pathologicalscience.
Even Stromberg, interviewed some 40 years later, accepted that researchers
might be misled by unconscious bias [32]. However, he added that “most mista-
ken hypotheses in science aren’t entirely wrong; they just have to be modified a
bit.”
Actually, it is well-known in material science that crystallization can be influ-
enced by the substrate, because of local attractions.
Rostrum Ray
proposed that
this phenomenon of epitaxy might explain water memory, because of the “ex-
treme structural flexibility” of water molecules [33]. Their internal structure and
relative positions could be modified, but this model would require molding and
stability of detached structures and even multiplication of positive and negative
molds in the course of successive EHDs. These hypotheses are not plausible
enough, but Ray tried at least to explain water memory, instead of simply pre-
tending that it is impossible.
Figure 8. Polywater with hexagonal cells.
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2D polymerization of water molecules was rediscovered by the bioengineer
Gerald Pollack.
He wondered why particles that imitate red blood cells can easily
move through narrow capillaries. Trying to understand this fact, he realized that
some materials create an “exclusion zone” near their surface [34]. Indeed, water
molecules can be so strongly bound to the substrate and to one another in suc-
cessive layers that the presence of red blood cells, for instance, becomes there
impossible. Pollack adopted the model of Figure 8 to explain the formation of
2D lattices. He deduced from this model that every oxygen atom is surrounded
by 3 half-hydrogen atoms. In natural units, this would yield a charge (3/2) − 2 =
−1/2 per molecule. Exclusion zones should thus be electrically charged and this
was proven to be true. Contact of pure water with some materials is sufficient to
constitute a battery.
Pollack’s discovery and empirical investigations were outstanding achieve-
ments, but Figure 8 can be replaced by Figure 9. Delocalized protons have then
3 equally probable positions inside any water molecule. This model combines
intramolecular with intermolecular exchange effects. It is then not necessary to
assume strong distortions of tripoles in all water molecules, since the normal an-
gles of 106˚ can be preserved when the third angle is 148˚. Regular hexagons
would merely be transformed into elongated ones. The upper and lower rows of
Figure 9 are even zigzagging chains, like those of Figure 6.
2.7. Formation and Stabilization of Molecular Chains
In interstellar space, there are ions that attract water molecules and align their
effective dipoles, since the configuration
V
2 of Figure 6 is sufficient for moderate
electric fields. Figure 10 represents such a molecular chain in 3D. The tripoles
are alternatively orthogonal to one another and the aligned effective dipoles are
represented by red arrows. On the average, the chain is axially symmetric. In
outer space, ions would thus collect water molecules and become radially
“haired”. This does facilitate the participation of water molecules in the forma-
tion of planetary systems.
Figure 9. Polywater with elongated cells.
Figure 10. A single chain of water molecules.
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Although positive and negative ions are strongly bound to one another in io-
nic crystals, they are easily dissolved in liquid water, since the small water mole-
cules are more attracted. They penetrate inside ionic crystals and dissolve them.
In liquid water, ions will thus usually be isolated and surrounded by a “hydra-
tion sphere”, where the effective dipoles of water molecules are oriented towards
the central charge. This polarization decreases further away, because of thermal
agitation. However, molecular chains could also be formed and rapidly stabilized
by attracting one another. Figure 11 represents two possibilities by means of
three water molecules that belong to parallel chains.
According to the dipole approximation (1), the electric field is −
p
/
r
3, when
θ
=
90˚. Real dipoles would thus be antiparallel in lateral positions, but we have to
consider chains of tripoles. Two parallel molecular chains will thus be shifted by
the distance
a
with respect to one another. This allows for parallel
or
antiparallel
effective dipoles. We expect that parallel ones are preferred, since the two pro-
tons of the lower molecule are then closer to the negative center of the upper
right molecule. However, it is useful to verify if this leads to significant differ-
ences for the resulting potential energies
Va
and
Vb
. Since the potential energy of
the upper pair is
V
2 = −0.093 in natural units, we get
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
2
4 , 0, 2 , , 4 , 0,
2 , 0, 2 , 2 , 2 0, 0,
4 2,, 4 ,, 4 2,,
0.28
a aa a
a aa
aa a
V V Sa d Sabd b SD a d
S da d S da bd S d b
S abd S dbd S d abd
++
=+ + ±+ −
+ + −±
− −−
= −
( ) ( )( )
() ( ) ( )
( ) ( )
2
4 , 0, 2 , , 4 , 0,
2 , 2 , 2 , 0, 2 0,0,
4 2 , , 8 , ,
0.09
b aa a
a aa
aa
V V Sa d Sabd b Sd a d
Sd a bd Sd a d S d b
S a b d S d bd
=+ + ±+ −
++ ++ − ±
−−
= −
The ± signs mean here that we have to sum two different terms. It appears
that
Va
is 3 times lower than
Vb
. Agglomerations of
parallel
chains of water mo-
lecules are thus preferred. Moreover, the effective dipoles are already oriented in
nearly the same way by the electric field of the ion. Molecular chains with paral-
lel effective dipoles get spontaneously assembled and stabilized.
It is very important to realize that biologically active molecules contain electr-
ically charged parts. They explain why these molecules are easily dissolved in
liquid water and require at least contact with saliva. Figure 12 shows some typi-
cal examples. We see that the charged parts are even situated on a branch, where
Figure 11. Mutual attraction of molecules on parallel chains.
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Figure 12. Biologically active molecules contain charged parts that can vibrate.
they can be set in oscillation by thermal agitation of water molecules in the sur-
rounding liquid. They
resonate
at a particular frequency, which is characteristic
of the active molecule. Its value is much lower than for vibrations of strongly
bound charged particles inside molecules of any type. For active molecules, the
resonance frequency is determined by the effective mass of its charged part and a
weak restoring force. It results from deformations of the soft cocoon of polarized
water molecules. These ideas are essential to unravel the puzzle of water memory.
2.8. Single-Domain Ferroelectric Crystallites
When the electric field of a biologically active molecule has started to assemble
water molecules, it becomes a germ of ongoing crystallization. More and more
water molecules are attracted and align their effective dipoles. This yields closely
packed molecular chains, like that of Figure 10. A frontal view of the alterna-
tively orthogonal tripoles yields a square lattice. Ideal alignment of all effective
dipoles is only achieved in “single-domain ferroelectric crystallites”. Water mo-
lecules are there more closely packed than in the surrounding water. We will
prove that these crystallites contain many water molecules and are thus spheri-
cal. We will call them
water pearls
(WPs) and prove that they have the same size.
Since they contain a great number
N
of water molecules, their equally oriented
effective dipole moment
p
yields a very great total dipole moment
P
=
Np
for
every WP. Each one of them creates thus an electric field, which is able to as-
semble other water molecules. This process creates an adjacent WP and even a
spontaneously growing chain of WPs.
To discard unnecessary objections, we mention that the solid state physicist
Kittel
realized already in 1946 that molecules of magnetite (Fe3O4) create sin-
gle-domain ferromagnetic crystallites [35]. Since they are ideal magnets, their
discovery led to important applications, like magnetic recording. Blackmore
discovered in 1975 that some species of aquatic bacteria collect Fe3O4 or Fe3S4
molecules. They are then spontaneously assembled and constitute single-domain
ferromagnetic crystallites [36]. They form chains of about a dozen beads. Mag-
netotactic bacteria produce these chains to remain in deeper layers of shallow
waters and are able to sense the orientation of their internal compass needles in
the inclined geomagnetic field of the Earth. They developed this stratagem to
survive, since the oxygen content would be too high for them near the surface.
This required the formation of specialized genes [37].
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To determine the radius
R
single-domain ferroelectric crystallites is a tricky
problem. It has been tackled for the most common ferroelectric material (Ba-
TiO3) by means of the theory of phase transitions [38]. Since this approach does
not apply to water molecules in the liquid state, we adopt another one. It does
simply result from a transposition of the method that is used in electrostatics to
determine the polarization of a homogeneous spherical particle. Figure 13(a)
represents such a WP in pale blue color. Because of its surface charge, it orients
water molecules in the surrounding liquid, but only in a limited region. The po-
larization of the surrounding water molecules is there progressively decreasing
because of thermal agitation. This region is represented in a different color.
We use polar coordinates (
r
,
θ
) with axial symmetry. On the average, the WP
is electrically neutral, because of the closeness of the charges ±2
q
inside all water
molecules. The internal surface charge density results from the charges ±2
q
at
the extremities of every molecular chain. It occupies the surface
d
2 in planes that
are perpendicular to these chains, but the surface of the sphere is inclined. The
intersected surface is thus increased and the internal surface charge is
σi
(
θ
) =
(2
q
/
d
2)cos
θ
at the positive side. The external surface charge density
σe
(
θ
) is low-
er, but proportional to the internal one. The total surface charge density is thus
σ
(
θ
) =
σo
cos
θ
. Positive and negative surface charges on opposite sides of a WP
create a homogeneous electric field
Ei
inside this sphere, as if it were composed
of many very thin condensers. The electrostatic potential inside the WP is
therefore
ϕi
(
r
,
θ
) =
Eir
cos
θ
.
Figure 13(b) defines the potential
ϕ
(
r
) = (Q/r)exp(−
r
/
λo
) that would be pro-
duced outside the sphere by the total charge
Q
=
N
2
q
of all effective positive
poles of water molecules, if this charge were situated at the center of the sphere.
The usual Coulomb potential (Q/r) is modified by
screening effects,
resulting
from positive and negative charges in the polarized region of the liquid water.
The radial decrease is characterized by the
Debye length λo
. Figure 13(c) defines
the actual potential
ϕe
(
r
,
θ
) in the external medium. It is due to the charges ±
Q
of all positive poles and negative poles, separated by the distance
a.
The external
potential depends then on Δr = (
a
/2)cos
θ
, since
Figure 13. (a) A water pearl and the external domain of oriented water molecules; (b) A
single charge
Q
situated at the center of the sphere would generate an electric potential
ϕ
(
r
). (c) The actual external potential
ϕe
(
r
,
θ
) results from two charges ±
Q.
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( ) ( ) ( )
,
e
r rr rr
φθφ φ
= −∆ − +∆
for
rR
ϕi
(r, θ) and
ϕe
(r, θ) are subjected to boundary conditions, which determine
the values of
R
and
Ei
. This more technical problem is solved in the short appen-
dix, but all required physical concepts have been explained here and the result is
that
R
≈ 10
λo
. The value of the Debye length
λo
depends on the concentration of
ions in the surrounding water. It is also proportional to the square root of the
absolute temperature T, but this factor is nearly constant between 20 and 30˚C.
The value of
λo
has been measured at 25˚C for water with different concentra-
tions of dissolved NaCl [39]. It appeared that
λo
≈ 1 nm at 0.1 M (mol/liter), but
increases for lower and higher concentrations. Actually,
λo
≈ 3 nm at 0.01 M and
5 M, which is only slightly higher than for Dead Sea water. For pure water,
λo
would depend on the concentration of H+ and OH ions. The radius
R
of WPs is
then somewhat smaller than 10 nm.
2.9. Properties of WPs and Their Poles
When the particle physicist
Shui-Yin Lo
was visiting professor at the famous
California Institute of Technology in 1996, he adopted a research project con-
cerning properties of liquid water. He was surprised to discover that EHDs of
HCl, NaOH or HNO3 molecules in very pure water led to the formation of “nov-
el stable structures”. Lo thought that they result from crystallization of hydration
spheres [40], while we attribute their existence to the formation of ferroelectric
crystallites. This explains the dipolar nature of these particles and their great sta-
bility. They subsisted when all ions had been removed by successive dilutions
and these structures were even multiplied in the course of successive dilutions,
but only when they were followed by vigorous agitation.
When S. Y. Lo determined the sizes of various types of structures by means
self-interference of scattered laser light, he found 3 distinct groups. The smallest
particles had a diameter
D
≈ 15 nm with very low dispersion. We interpret this
result as meaning that
D
is the diameter of WPs in pure water. Thus,
R
≈ 7.5 nm
and the Debye length
λo
≈ 0.75 nm. Since the volume occupied by every water
molecule is d3, where d = 0.275 nm, WPs contain
N
≈ 85,000
molecules.
This
huge number justifies the assumption that they are spherical. Nevertheless, WPs
are nanoparticles, since water molecules are very small.
The second group of structures, discovered by S.Y. Lo, had a size of about 300
nm. We consider that this group corresponds to the length
L
=
ND
of chains of
WPs, containing
N
≈ 20 water pearls. We will explain (in Section 3.3), why their
length L has to be limited. Its value depends on the mutual attraction between
positive and negative surface charges on adjacent hemispheres. It is thus useful
to replace the distributed surface charges of WPs by point-like poles. They are
situated inside the sphere, like those of magnetized steel balls, but we can be
more explicit. The total charge
Qo
on the surface of the positive hemisphere, is
the integral of 2π
(
θ
)
r
d
θ
, where
r
=
R
sin(
θ
) and
σ
(
θ
) =
σo
cos
θ
, while the angle
θ
varies from −π/2 to + π/2. This yields
Qo
= (/3)R2
σo
. We can also calculate the
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electrostatic potential V(x) for a test charge +1 that is situated on the symmetry
axis at the distance
x
from the center O of a WP. Figure 14 shows that V(x) is
the integral of 2π
σ
(
θ
)
r
2d
θ
divided by the distance
R'
. This allows us to define the
effective charge Q(x) of the positive pole, if it were situated at the distance
R
/2
from the center O. This pole is represented by a black dot and
( ) ( )
( )
( )
2
π2
0.5
π222
3sin cos d
22
2 cos
o
Qx
Q
Vx xR
R x Rx
θ θθ
θ
+
= =

+−

The result of numerical integration is shown in Figure 15. It appears that
Q(x) =
Qo
when
x
> 3
R
, but when the test charge is close to the surface of the
WP, it does mainly interact with the nearest surface charges. This reduces the
value of Q(x). On the surface, Q(R) = 0.68
Qo
. We neglected all screening effects,
but it is only important that neighboring poles are separated by the distance
R
and carry charges ±
Q
.
3. The Mechanism of Water Memory
3.1. Small Oscillatory Rotations of Water Pearls
Because of the rapidly decreasing Coulomb forces, it is sufficient to consider the
mutual attraction of neighboring positive and negative poles. At rest, they are
aligned and their poles are separated everywhere by the same distance
R
, but
small oscillatory rotations of WPs around their center will lead to transverse dis-
placements of the poles. They are represented in Figure 16 by red arrows. For
clarity, we exaggerated their magnitude. The essential point is, indeed, that
neighboring poles remain in tangential contact and that the transverse displace-
ments of the poles can vary along the chain. We characterize the instantaneous
rotation of the nth water pearl by the variable
un
(
t
).
Figure 17 shows that when u is the relative displacement of two neighboring
poles, they attract one another by the force
F
. In natural units, its magnitude is
Figure 14. Definition of the charge
Q
(
x
).
Figure 15. Calculated value of
Q
(
x
).
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Figure 16. Small rotations of water pearls in a chain are specified by dis-
placements of poles.
Figure 17. The restoring force is the transverse component of the force
F
.
Q
2/(
u
2 +
R
2). The transverse component
F
is reduced by the factor
u
/
R
. It follows
that when
uR
, the
restoring force
is
( )
F u Ku= −
where
2
3
Q
KR
=
(5)
This force is proportional to the relative displacement
u
, as for any elastic sys-
tem. When
M
is the effective inertial mass of poles, the equation of motion for
the nth water pearl is
( ) ( )
11n n n nn
Mu Ku u Ku u
+−
= −− −

(6)
Every dot stands for a time derivative. Since this equation is identical for all
WPs, it describes the behavior of the whole chain. It can be simplified when the
displacements
un
are smoothly varying along the chain, which is equivalent to
saying that the diameter
D
of WPs is small compared to the distance where the
relative displacements
un
are notably varying. We can then replace
un
(
t
) by
u
(
x
,
t
), where the coordinate
x
is treated as if it were a continuous variable. Actually,
2
1
2
nn
u u Du D u
±
′ ′′
=±+
, where
u
and
u′′
designate first and second order
partial derivatives with respect to
x
. Equation (6) is then reduced to
2
u vu
′′
=

where
22
v DKM
=
(7)
This is the usual wave equation for vibrating strings. An infinite chain would
allow for
( ) ( )
,uxt ux vt= ±
. This corresponds to a function of any shape, moving
at the velocity
v
towards the right or the left. Possible attenuations of oscillatory
rotations have been neglected in (6) and (7), but will be discussed later on.
3.2. Standing Waves on Chains of Water Pearls
For a chain of finite length
L
, we have to know the boundary conditions at
x
= 0
and
x
=
L
. When both ends are free, the first and last pearls are not subjected to
any force. Thus,
( )
,0u xt
=
for
x
= 0 and
x
=
L
. In other words,
u
(
x
,
t
) has to
reach maximal values at both extremities. This allows for a particular solution of
well-defined frequency f and well-defined wavelength
λ
:
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() ( ) ( )
, cos sinu x t A kx t
ω
=
where
2f
ω
= π
and
2πk
λ
=
(8)
Since
( )
,u xt
is proportional to sin(
kx
), the boundary condition
( )
0, 0ut
=
is satisfied, but
( )
,0
u Lt
=
requires that
kL
=
s
π, where
1, 2, 3,s=
. It follows
that
L
=
/2 and because of (7), that the spectrum of possible frequencies is de-
fined by
2
s
vv
f sf
L
λ
= = =
where
1, 2, 3,s=
(9)
The only possible frequencies are thus integer multiples of the fundamental
frequency
f
1 =
v
/2
L
. For sound waves, any pair of such frequencies would pro-
duce an impression of harmony. The spectrum
fs
=
sf
1 is therefore said to be a
“harmonic” one. These properties are well-known in physics, but everyone
should see why a chain of WPs with free ends does only allow for standing
waves. This means that for any particular solution (8), all WPs are set in oscilla-
tory rotations at the same frequency
f
, but the amplitude of these oscillations va-
ries along the chain.
However, the approximation (7) is of limited validity, since it requires that
DL
. The measurements of Lo imply that chains of WPs contain a relatively
small number of WPs (
N
20). To see how far the approximation (9) is realistic,
we have to solve the general Equation (6). This is easy when we use complex no-
tations, since standing waves are then defined by
( )
( )
en
i kx t
n
ut A
ω
=
where
( )
2ee 2
ikD ikD
K
M
ω
= +−
Thus,
( ) ( )
sin 2
2
kD
kvD
ω
=
and
( ) ( )
, sin π
π
v
fD D
D
λλ
=
(10)
The function
f
(
λ
,
D
) is represented by the dark curve in Figure 18. It reveals
that the domain of possible frequencies is limited, as well as the domain of poss-
ible wavelength. It is only physically possible that
λ
/2 ≥
D
. The approximation
f
=
v
/
λ
corresponds to the red line, which would allow for arbitrarily high fre-
quencies, but (10) imposes an upper limit (
f
max =
v
D
). Nevertheless, the linear
approximation is valid for a relatively larger domain of low frequencies. [Indeed,
( )
3
sin 6x xx x
=−≈
, when
2.5x
]. Actually, the spectrum of possible fre-
quencies is
fs
=
sf
1 when
sN
.
Figure 18. The general function f(λ) for chains of WPs.
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3.3. Automatic Trimming of Chains of Water Pearls
Figure 19 summarizes the essential steps that explain why chains of water pearls
are the
information carriers
of water memory. The upper line represents a bio-
logically active molecule, where the charged part oscillates at some very low fre-
quency
f
. It cannot communicate these oscillations to the WPs, since the length
of the chain
L
<
λ
/2, where the wavelength
λ
is determined by
f
=
v
/
λ
. The
second line represents what happens as soon as the length of the chain of WPs
reaches the value
L
=
λ
/2 =
v
/2
f
. The electric field of the oscillating charge does
then excite a standing wave for oscillatory rotations of WPs.
The last WP of the growing chain does suddenly start to oscillate when its
length
L
=
v
/2
f
. The amplitude of this oscillation is the same as for the first WP
of the chain, which can also be set in forced oscillation by the active molecule. It
remains attached to it, but can now communicate its motion to other WPs of the
chain. However, the rotation of the last pearl of the chain prevents the formation
and attachment of an additional WP. The growing chain is thus
automatically
trimmed.
Information that is characteristic of the type of active molecules has
been encoded by means of the length
L
of the chain. It depends indeed on the
frequency
f
.
The third line of Figure 19 shows that when the trimmed chain is detached
from its generator, it does still allow for a standing wave for the same length
L
=
λ
/2. The liberated chain of WPs conserves the acquired information. Moreover,
it produces itself an
oscillating electric field.
Its frequency
f
is adequate to create
more equally trimmed chains of WPs. They are thus multiplied by an autocata-
lytic process. This provides the key for a rational justification of water memory.
It resulted from a systematic examination of all possible interactions between
water molecules and their logical consequences.
We might object that oscillatory rotations of WPs will be damped by friction,
exerted by surrounding water molecules. This is true, but trimmed chains are
also subjected to local impacts of water molecules. Although the impacts are
random, the chain does pick-up energy when it allows for resonances at any
frequency
fs
for possible standing waves. Since it is sufficient that the free ends of
the chain can oscillate with maximal amplitude, standing waves of smaller wave-
lengths and higher frequencies can also be excited. Excitation of a standing wave
at a higher frequency can easily be demonstrated with a flute, since “overblow-
ing” is sufficient to double the frequency for standing waves, without modifying
the effective length for longitudinal oscillations. Oscillatory rotations of WPs at
Figure 19. Creation of trimmed chains of water pearls, resonating at the frequency
f
of
the active molecules for chains of length
L
. This remains true for detached chains.
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higher frequencies imply more rapid motions and thus greater kinetic energies
and more violent local impacts. Available energies depend on the statistical dis-
tribution of kinetic energies of water molecules in the liquid state. Although a
chain of WPs of given length allows for a superposition of different modes of os-
cillations, those of increasingly higher frequencies will thus be excited with de-
creasing amplitudes. They are byproducts of random re-excitation, but the low-
est possible frequency
f
1 remains the predominant one.
3.4. Negation of Water Memory Was Based on False Assumptions
Benveniste’s experimental proof of water memory was categorically rejected be-
cause of prevailing beliefs. They resulted from four erroneous assumptions:
1) Biologically active molecule can have no effects any more, when all of them
have been eliminated by successive dilutions.
2) Even if biologically active molecules could create substitutes, made of water
molecules, they would have to be adaptable. Such aggregates are unknown and
can thus not exist.
3) Biologically active molecules can only act on their specific receptors by means
of chemical affinities. Local structuring of liquid water would be unable to mim-
ic these processes. This is particularly implausible for various types of molecules,
since that would require an extraordinary capacity of adaptable imitation.
4) Extra high dilutions are also used for homeopathy, which is inefficient. The
preparation of EHDs does even involve shaking by vigorous “successions”. This
ritual is a sign of charlatanism.
We have already shown that the two first objections are contradicted by the
formation of trimmed chains of WPs. The third objection concerns the
funda-
mental
problem of molecular interactions. Modifications of the state of motion
can result from direct contact (collisions), but also from actions at a distance
(due to attractive or repulsive forces). We are accustomed to the idea that struc-
tural changes (combinations or dissociations) at molecular level result from
chemical reactions, requiring direct contact, chemical affinities and configura-
tional conformity. However, internal modifications can also result from energy
transfer (excitation or disexcitation) by means of force fields.
Figure 20 summarizes the required restructuring of our ideas, because of the
concept of ferroelectric water pearls. First of all, we have to realize that biologi-
cally active molecules contain an electrically charged part (Figure 12) that has a
resonance frequency
f
. It creates thus an
electric field
that oscillates at this fre-
quency. The upper line of Figure 20 represents the normal process, where this
electric field acts on a molecular receptor, which has also a charged part that can
oscillate. Specific receptors of a particular type of active molecules contain also a
charged part that can oscillate. It resonates at a frequency
fr
. It is thus sufficient
that
fr
f
to allow active molecules to stimulate their specific receptors. There is
some tolerance, since the probability of interaction by resonance corresponds to
a peak that has some width.
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Figure 20. Water memory reveals a new type of molecular interactions.
The assumption that molecular interactions are only possible according to the
“key and slot model” of chemical reactions is not correct. Biologically active
molecules can also interact with their specific receptors by means of
oscillating
electric fields and resonances.
This allow for a bypass, represented by the second
line of Figure 20, since the oscillating electric field of active molecules can also
create trimmed chains of WPs. As long as they are attached to their generators,
they resonate at a frequency
f
1
f
. This remains true when these chains are de-
tached, but reactivation of their oscillations by thermal agitation in liquid water
leads to a harmonic spectrum of possible frequencies (
fs
=
sf
1, where
1, 2, 3,s=
). The fundamental frequency
f
1 remains dominant, however.
Standing waves on trimmed chains of WPs do produce an oscillating electric
field of frequency
f
1
fr
and can thus stimulate the same receptors.
The collective electric field, generated by all trimmed chains of WPs, has even
the capacity to create more and more equally trimmed chains. Their number is
increased and the oscillating electric field is amplified by an autocatalytic
process. The possibility that molecular interactions can result from oscillating
electric fields and resonance effects had been overlooked. The discovery of water
memory did thus reveal the existence of a mechanism that is of fundamental
importance and even very efficient.
The fourth erroneous assumption concerns homeopathy. Since the underlying
mechanism was not understood, it was believed that its efficiency can only result
from placebo effects. We wonder how they can be justified for animals and small
children. Our purpose is not to defend homeopathy, but to restore truth, also in
this regard. It is therefore instructive to examine the argumentation advanced by
those who would like to eliminate homeopathy. The
Australian National Health
and Medical Research Council
published in 2015 a study on “Evidence on the
effectiveness of homeopathy” [41]. This report was sponsored by the Australian
Government, but was not based on a scientific study of underlying physical and
physiological process. It was merely a collection of 176 articles, supporting the
claim that “there are no health conditions for which there is reliable evidence
that homeopathy is effective.”
These evaluations are essentially dependent on subjective appreciations. It was
recognized that the general conclusion of their report was “based on all the evi-
dence considered”. Other evidence was discarded. The first report of 2012 had
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A. Meessen
even been concealed [42], since it was not sufficiently selective in this regard.
The published report stated that the aim of this study was “to assist people in
making health care choices”, but it was addressed to political deciders. The au-
thors mentioned, indeed, that they wanted to “influence policy” and to get “in-
creased funding for such research” ([41], pp. 6, 16 and 4). This report was widely
publicized and is equivalent to lobbying.
3.4. The Standard Procedure for Extra-High Dilutions
It is even necessary to clarify the origin of homeopathy, which has often been
misrepresented to denigrate it. The basic idea was due to
Samuel Hahnemann
(1755-1843). He was a regular medical doctor. After acquiring his diploma at the
age of 24, he practiced during 5 years, but decided then to cease. He had realized,
indeed, that it would have been better for some of his patients not to be treated
according to the “art of healing” of his time. He was even horrified that he might
“murder” suffering people, instead of helping them. This was an exceptionally
honest attitude, justified by recognizing the cause of this horrible situation. Nei-
ther the chosen substances, nor the doses were determined in a rational way, al-
though Paracelsus wrote already in 1543 that “only the right dose differentiates a
poison from a remedy”.
Hahnemann’s linguistic gifts made it preferable for him to translate books and
to search there for possible improvements of medical practice. In one of these
books, it was claimed that the bark of a Peruvian tree was able to treat malaria. It
is known today that the bark of cinchonatrees contains
quinine.
Most efficient
medicines were actually discovered by trial and error. It was already known in
Antiquity, for instance, that leaves of willow trees can stop pain. A chemist dis-
covered in 1853 that the active molecule is C9H8O4, which became famous as
as-
pirin.
Even elephants, apes and other animals know how to cure or avoid ail-
ments [43]. The textbook attributed the beneficial effects of cinchona
powder to
its taste, but Dr. Hahnemann could not believe this claim. Nevertheless, he de-
cided to verify if there were any detectable effects and was amazed that it pro-
duced malaria-like symptoms.
By experimenting with other substances, he realized that medicines could be
discovered in a more rational way, by adopting the “law of similars”. This was
merely an empirical rule, but such rules were often followed before understand-
ing why they are valid. [Even Newton’s law of gravity was expressed in terms of
actions at a distance. It did account for observed phenomena, but the real cause
is a gravitational field, which corresponds even to modifications of the metric of
space and time.] Since Hahnemann tried to discover medications by means of
tests, performed on healthy persons, he had to use the lowest possible doses. He
adopted thus the method of successive dilutions. If the result was beneficial, such
an EHD could also be administered to patients in a secure way. We recall that
Hahnemann was a learned medical doctor and was thus able to verify if a prepa-
ration is helpful of not.
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As an example, we mention
Apis mellifica
. Until recently, it was customary in
medicine to use Latin, also for anatomy, to overcome language barriers. The
European honey bee is called “
Apis mellifera
” and the main component of its
venom is
mellitine.
This molecule has also anti-inflammatory properties and
honey bees do even protect their larvae from infections by means of very effi-
cient substances. Hahnemann presented his discovery already in 1796 in a Ger-
man medical journal, by formulating the rule that “like cures like”. Objections
that are based on the finite divisibility matter are anachronistic. [The ancient
concept of atoms had been reintroduced by Boyle in 1661 and elaborated by the
chemist John Dalton in 1804, but the atomic theory was only accepted at about
1860, since the kinetic theory of gases did prove that Avogadro’s hypothesis was
correct. Nevertheless, Mendeleev did not yet dare to use the concept of atoms in
1869.]
Hahnemann could thus assume that even when a substance has been diluted
many times, there remains something of this substance. In 1810, he presented a
first collection of results and one year before his death, the 6th edition of his
“Organon of the Rational Art of Healing” was ready for publication. It is easily
available [44]. It should also be obvious that successive dilutions required always
homogenization before the next step. Hahnemann did this by holding the vessel
in his hand and stroking it with vigor on a semi-elastic surface, like leather.
Modern chemists use mechanical vortexing to insure homogenization of mix-
tures. This is merely a simpler method. We will show in the following section
that these “successions do also have another effect. Although Hahnemann was
only concerned with practical medicine, he discovered already the bypass of
Figure 20. Neither he nor his detractors were aware of this fact. Even when
Benveniste did empirically prove the biological efficiency of EHDs, it was cate-
gorically declared to be impossible.
3.5. The Quasi-Periodic Variations of Biological Efficiency
Sir John Maddox, long-term editor of Nature (1966-73 and 1980-95) accused
Benveniste of self-delusion, although his article contained two figures, displaying
results of measurements [1]. Figure 21 reproduces one of them. It is undistorted
Figure 21. Measured quasi-periodic variations of the biological efficiency of EHDs, pub-
lished by Benveniste
et al.
[1].
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A. Meessen
and was extracted from a publication in Japanese. We see 9 peaks. The first one
is higher than the following ones. Since every peak did result from several mea-
surements, the investigators should have realized that the quasi-periodic varia-
tions cannot result from “disregard of statistical principles” and “sampling er-
rors”. The investigators proclaimed even that Benveniste’s experimental results
were merely due to self-delusion [2]. What would result from the theory of water
pearls? We begin with a description of the underlying processes in usual lan-
guage by means of Figure 22. The first frame (a) shows some of the initially di-
luted active molecules, their charged parts (in red) and attached trimmed chains
of WPs. We know that their length
L
=
λ
/2 and depends on the frequency f of
the oscillating electric field of the active molecules. The second frame (b) illu-
strates the situation immediately after the first dilution and vigorous agitation.
The concentration of active molecules has been reduced and agitation lasted
long enough to detach all chains from the remaining ones, but some chains of
WPs were broken.
The third frame (c) shows that after a relatively short time interval, the re-
maining active molecules had again formed trimmed chains of WPs. Broken de-
tached chains did grow and new ones were generated by the global oscillating
electric field. These chains have the same characteristic length
L
, allowing for
standing waves at the frequency
f
1 as well as harmonics (
11
2,3,ff
). Some
chains may have reached the length 2
L
. It allows for a mode of oscillation where
2
L
=
λ
, which is equivalent to
L
=
λ
/2 and allows for the frequency
f
1. There did
also appear some “associated chains”, resulting from mutual attraction of
trimmed chains. Figure 23 shows how two parallel chains of WPs will be at-
tached to one another, but more than two chains are also possible. However, as-
sociated chains do not resonate at the same frequency or not at all.
Figure 22. Decoding the standard procedure for preparing EHDs. (a) Active molecules
and attached trimmed chains of WPs; (b) Detached and broken chains, immediately after
dilution and agitation; (c) Reconstituted chains; (d) Excess of associated chains.
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Figure 23. A simple associated chain of water pearls.
The fourth frame (d) of Figure 22 represents a state where nearly all single
chains of WPs got bound to one another. The biological efficiency of the result-
ing EHD is then much reduced, but associated chains can be broken by vigorous
agitation. Single chains can then be multiplied again during successive dilutions,
until there are so much of them that association will be favored. The global re-
sult is that the biological efficiency of EHDs has to vary in a quasi-periodic way.
They are not perfectly periodic, since these processes allow for statistical fluctua-
tions.
3.6. Kinetics of Water Memory
It is useful to express these ideas by means of equations, since they allow for log-
ical deductions. Let
Xo
be the initial number of active molecules, dissolved in a
given volume of twice distilled water. This concentration is reduced by succes-
sive dilutions, where the same fraction of the homogenized solution is eliminat-
ed at every step. Usually, this fraction is 9/10 or 99/100. It is replaced by pure
water to get always the same volume. When successive dilutions follow one
another at identical short time intervals Δ
t
, the concentration of active molecules
becomes a function
X
(
t
) that decreases step-wise, since
( ) ( ) ( )
Xt t Xt tXt
α
+∆ = − ∆
The value of
α
Δ
t
= 0.9 or 0.99. For smalltime intervals,
X
(
t
) can be treated as if
it were a continuous function. It decreases then according to the equation:
XX
α
= −
so that
( ) ( )
exp
o
Xt X t
α
= −
(11)
The exponential decrease does necessarily end up with
X
(
t
) = 0 when
1t
α
, but this does not prove that the biological efficiency of EHDs has to
vanish. Active molecules are able to generate trimmed chains of WPs with a
probability
g
per unit time and they do generate more of them with a probability
β
par unit time. The concentration
Y
(
t
) of trimmed chains of WPs increases thus
according the equation:
Y gX Y
β
= +
(12)
When the sequence of EHDs starts without previously formed trimmed
chains, the initial value
Y
(0) = 0. Because of (11) and (12), we get then
()
ee
tt
A
Yt
βα
αβ

= −

+
where
o
A gX=
(13)
It appears that
Y
(
t
) =
At
when
t
→ 0. The initial increase of
Y
(
t
) is thus linear
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