Elastic properties of vanadium pentoxide aggregates and topological defects
L. V. Elnikova
A. I. Alikhanov Institute for Theoretical and Experimental Physics,
25, B. Cheremushkinskaya st., 117218 Moscow, Russia
(Dated: September 12, 2021)
We study the aqueous solution of vanadium pentoxide by using topology methods. The exper-
iments by Zocher, Kaznacheev, and Dogic exhibited, that in the sol phases of V2O5 − H2O, the
tactoid droplets of V2O5 can coalesce. In the magnetic field, this effect is associated with a gauge
field action, viz. we consider coalescence (in the topologically more convenient term, ”junction”)
of droplets as annihilation of topological defects, concerning with the tactoid geometry. We have
shown, that in the magnetic field, the tactoid junction is mainly caused by non-Abelian monopoles
(vortons), whereas the Abelian defects almost do not annihilate. Taking into account this annihila-
tion mechanism, the estimations of time-aging of the V2O5 −H2O sols may be specified.
PACS numbers:
I. INTRODUCTION
The tactoid sol phase of the V2O5 −H2O system has
been discovered at the 20-th years of the last century
by Zocher (see references in [1, 2]). At the beginning of
our century, the tactoid drops (tactoids) have been in-
vestigated on the optical experiments by Kaznacheev [2],
Lavrentovich [3], Dogic (see [4] and references in [5]), and
their coworkers. The tactoid phase is chemically classi-
fied as the lyotropic inorganic nematic [1]. The tactoids
coexist with the isotropic liquid phase at the mass con-
centration of V2O5, amounting 0.3-2.1 percents, and un-
der other standard conditions [2].
The thermodynamic parameters and pHcause the dy-
namics of their formation, in particular, the junction.
The tactoid geometry is evolved complicatedly (and
mutually inversely) in depending on time-aging of the
sols [2].
Due to the de Gennes’s theory [6], the tactoid shape
stabilization is defined by competition between the elastic
energy of the nematic phase, the surface energy, and the
anchoring energy [2]. The minimum of the tactoid free
energy provides an equilibrium shape of a droplet. The
measured macroscopic elastic moduli are in a very large
ratio (K3
> 100), that distinguishes V2O5 − H2O from
other lyotropic liquid crystals (LC), whose typical values
of K3
are in order of ten.
In the magnetic field, the prolate droplets are aligned
by their long axes parallel to the field. Then the special
case of the junction of tactoid poles may be observed [1,
Remarkably, that the sol phases of V2O5 −H2O were
conditionally sorted on a shape polarity and a nematic
director field [5] as of a homogeneous and a non-uniform
field, and of the spherical and the bispherical [2] drops
with boojums. Strikingly simultaneously, these phases
have been parsed (see [5, 7, 8] and references therein)
basing on the experiments by Dogic (references in [5]),
performed independently of Kaznacheev.
In this paper, we study the mesomorphism of the
V2O5−H2O system during the tactoid junction and spec-
ify the character of the mesomorphic consequence there.
Our goal is to define the influence of junction onto dy-
namic parameters of the sol system, including time-aging
of the sols. In addition, aging of these sols in water is an
applied problem of ecology, since V2O5 contains in coal
impurities, generated in result of work of thermal power
stations.
From a topological standpoint, poles of a tactoid are
the point defects, boojums. As will readily be observed,
we have to do with a quantum phase transition, the anal-
ogous topological singularities of two poles (each admit-
ting a flux) were announced by Haldane [9] for the quan-
tum Hall semiconductors. Also, there is a convenient
analogy with the boojum formalism for the superfluid
phases of 3He and 4He [10], however their varied topol-
ogy descriptions does not allow to explain the case of the
tactoid coalescence.
II. FORMALISM
Geometry of the droplets obeys the local nematic order
parameter n, which is oriented relatively to a droplet
surface (Fig. 1).
The free energy functional of a tactoid in the magnetic
field is summed up from the Frank elastic energy Fel and
the magnetic energy Fm [2, 5]:
F = Fel + Fm, (1)
Fel =
(∇·n)2+K2
(n·∇×n)2+K3
[n×(∇×n)]2−K24∇·[n·∇·n+n×(∇×n)]2].
The magnetic energy density has the form −χa
(n ·H)2,
(where χa is the anisotropy of magnetic susceptibility,
and H is the magnetic field).
The terms at K1, K2, and K3 elastic constants in 2
mean splay, twist, and bend deformations of a bulk ne-
matic respectively, n is the coordinate dependent nematic
director. The term at K24 relates to saddle-splay defor-
mation mode [5]. In this continuum, the tactoid boojums
http://arxiv.org/abs/0704.1893v7
FIG. 1: The director field on the tactoid surface, taken over
[12]. Ri and α are the geometric parameters, γ = (
tan(α1/2
tan(α/2)
0 ≤ γ ≤ 1, the vectors ei i = ϕkazn, ξ, ηkazn denote the
bispherical coordinates.
were revealed by Kaznacheev [2] and by van der Schoot
[5] practically identically, independently of one another.
The final result of tactoid classification is the existence of
four regimes of form is possible, which depend on anchor-
ing between the local director and the tactoid surface,
and also on the total tactoid volume [11]. Only at the
week-coupled limit (γ = 0), Kaznacheev found an equi-
librium shape of a tactoid [2], [13], a fortiori at Fm = 0
and without the terms of K24-s in (2). At the limit (Fig.
1), the free energy (1) is the almost non-analytical func-
tion on f(α, γ) [2, 13]:
4π(sinα− α cosα) + π(3 sinα− 3α sinα− α2 sinα) +
π sin3 α
sin θ
(cosh ηkazn + cosα)
dηkazn +
[sinα(20 + cosα)− 3α cosα(7 + 2 sin2 α)] (3)
here θ is the parameter with the too long dependence of
α, γ, ηkazn [13], the last term of (4) corresponds to the
magnetic energy at γ → 1. For γ, see Fig.1.
Nematic surface defects of the tactoids [12] are of the
homotopic group π2(R, R̃) = P ×Q, the defects of the P
group are living only at the surface (P group is the kernel
of the homomorphism π1(R̃) → π1(R) and consists of
integers [3]), andQ’s defects are arrived from the interior.
(Here R and R̃ denote the space of degenerate states in
the volume and the non-vanishing states on the surface,
which are arrived from the interior, respectively). The
interior may be inhabited by hedgehogs. All of these
point defects keep within the exact homotopic sequence
[12]:
π2(R̃) −→ π2(R) −→ π1(R̃) −→ π1(R). (4)
Boojums are characterized by topological charges m and
n [3], which depend on a configuration of a nematic di-
rector’s field. Annihilation of the boojums of the ad-
jacent tactoids does not mean an influence of the rais-
ing hedgehog’s (in topology, they are not arbitrary float-
ing to the tactoid surface). Kurik and Lavrentovich [14]
have mentioned about some strings, connecting opposite
boojums via a hedgehog in nematic droplets, however,
non-triviality of π1 group hampered the revealing of the
droplet junction without the disclination concept. How-
ever, in our case we reasonably ignore lacking disclina-
tions (see the conclusions by Balachandran et al. [15]).
Interaction scales are the ’dipole length’ Ldip, and the
’correlation length’ Lξ [16], which are characterized an
action of the group of the order parameter. We assume
Ldip is in connection with a long-axis of a tactoid.
In the Cartesian coordinates (x, y, z), the director field
has the configuration n = n(0, 0, 1−cosh ηkazn cos ξ
cosh ηkazn−cos ξ
), where
ηkazn, ξ are the bispherical coordinates [2].
Quite evidently, that tactoid system is provided by
a gauge field [17] (and a field with SU(2) symmetry).
Concerning an universality class of the system, take the
V2O5 droplet surface as belonging to SO(3) group of ro-
tations of the two-dimensional sphere (here ’tactoid’) S2
[16]. U(1) will a group of rotations around a droplet axis,
which is agree closely with the magnetic phase group of
3He − A [18]. U(1)’s winding is realized of non-trivial
topology of tactoids.
The SO(3) and SU(2) groups are locally isomorphic
(as their Lie algebras) and are connected by the homo-
morphism, SO(3) ∼ SU(2)/Z2, where our Z2 is the boo-
jum’s boundary condition.
In our standpoint, at the bulk junction, the group
SO(3)n ×U(1)2n broken down to SO(3)n−1 ×U(1)2n−1,
where n is a number of tactoids.
A model of the sol should involve the monopole solu-
tions, according to the theorem [16] about requirement
of their existence (π2(G/H) −→ π1(H)).
On the other hand, inasmuch as π1(H) =
Z ⊗ Z ⊗ Z ⊗ ...⊗ Z︸ ︷︷ ︸
, the V2O5 − H2O sols are of the
group G. The tactoid annihilation may be described ei-
ther by non-Abelian or Abelian theory in depending on
the global field SU(2). Besides, we have to expect ap-
pearance of a compensative vector field [19].
Here, an each tactoid, in correspondence to two poles
(boojums) on a tactoid surface, may contain two vortons
with their tails (the wide and ”over-Witten’s” definition
for vortons see in [20, 21, 22], this is a kind of monopoles
with the definite pair of topological charges, vortex and
azimuthal windings). Just as vortices, they appear, if the
order parameter has extra degrees of freedom besides of
the overall phase [23, 24]. In the tactoid free energy, the
terms of twisted deformations [8] may play a role in these
excitations. By introducing a necessary parametrization,
the free energy equation, analogous to [2], was proposed
in [8], where the free parameters permit to be the non-
commutative relations in the droplet symmetry. Let us
note, that we use the factor-space CP 1 in accordance to
a chiral (gauge) field (2) [17].
Though, due to the electromagnetic (no topological)
reasons, the sol tactoids can survive coalescence owing
to the Coulomb attraction in water. But from topology
[14], we do not yet know about appearance of a physical
field from the configuration of defects. We have to note,
that because of in-homogeneity of a system, we have a
wide class of string models for a prototype.
III. ANNIHILATION OF TOPOLOGICAL
DEFECTS
So, a junction of droplets means, that the surface point
defect (boojum) configuration may be unstable (γ 6= 0).
We discuss the Abelian and non-Abelian string config-
urations [18, 25, 26, 28, 29], which support the sols of
tactoid nematics. Their combinations and interactions
are expected to define of the junction of tactoids.
A. Abelian space
The Abelian character of pair boojums and monopoles,
and also their integer charge were proven [10]. Boojums
of charge N = ±1 live at Lξ ≪ L ≪ Ldip [16, 20]. But
from the surface field phenomenology [2] of a solitary tac-
toid, one can not define a flux number k [16], concerning
an each boojum, only what k = 1 is preferable for their
pairing configuration, and k = 2 for a unit singularity.
In this scenario, annihilation of charge-opposite (topo-
logical) ’particles’ is possible.
Abelian monopoles may be associated with locations
of boojums, but, due to the topological properties of our
G, we ignore them. Let us consider only vortons of the
Abelian gauge. They are unstable [26], and appear to-
gether with the neutral strings. The open question is
which velocity will greater: of the tactoid coalescence or
the vorton decay.
In the U(1) gauge, the loop-radius dependent criterion
of the vorton stability was found and analyzed numer-
ically in the case of the potential expressed in the el-
liptic ansatz [27], as well as in the well-known Witten’s
U(1) × U(1) case (see review [21), that is an analogous
phase transition from U(1)×U(1) to U(1) for two neigh-
bouring randomly oriented tactoids, in absent of mag-
netic field.
B. Non-Abelian space
Usual Lagrangians of non-Abelian theories are often
linearized into the Bogomolny-Prasad-Sommerfeld (BPS)
equations [30]. A number of applications corresponding
to similar strings were considered, for example, in [15,
21, 22, 24, 25, 28, 29, 31, 32].
In the phase diagram [8], the regions of twist states
were indicated. If the tactoid junction carry out there,
for spherical and prolate droplets, one may make an
analogy between the non-Abelian vortons and ”rotation”
of the nematic order parameter, in spite of the ansatz
(α(η) = α0 sin η [8]) condition, labeled one of the topo-
logical invariants.
Let us formulate the string model with the boson La-
grangian density (due to [21, 22])
L = −1
µν − 1
µν −Dµ~φ† ·Dµ~φ− V (~φ). (5)
Fµν = ∂µAν − ∂νAµ (6)
are the Abelian field strengths. The global curvature is
Gµν = ∂µHν − ∂νHµ + gHµ ×Hν . (7)
The gauge covariant derivatives of vacuums are:
Dµ(~φ) = ∂µ~φ− ieAµ~φ+ gHµ × ~φ, (8)
In the formulas (5) - (8), µ and ν are indices of the gauge
field A and of the metrics g. Hµ and φ are the three-
dimensional vectors in the SU(2) Lie algebra. The field
potential V (φ) is expressing from (2). Due to [13]
x = a
sin ξ cosϕ
cosh η − cos ξ
, y = a
sin ξ sinϕ
cosh η − cos ξ
, z = a
sinh η
coshη − cos ξ
the bulk elastic energy [13] of a tactoid equals to
dηkazn
4K1 sinh
2 ηkazn sin ξ +K3 sin
(cosh ηkazn − cos ξ)3
The corresponding vector potential is
V (~φ) =
λ+ (~φ+ · ~φ− 1
ζ2)2 +
k|~φ · ~φ|2. (11)
At the parameter k > 0, the vacuum is characterize by
~φ · ~φ, ~φ† · ~φ = 1
− sin ξ sinh ηkazn cosϕkazn
cosh ηkazn−cos ξ
− sin ξ sinh ηkazn sinϕkazn
cosh ηkazn−cos ξ
1−coshkazn cos ξ
cosh ηkazn−cos ξ
 . (12)
The generators of SU(2) are denoted as Ti=1,2,3. T0 is
the generator of U(1). −iT1(~φj) = −ǫijk~φj , −iT0(~φ)j =
−~φj . Q = T2 + T0 is the annihilation condition. The
string generator (TS = T3) does not commutate with
the charge generator: [TS , Q] = [T3, T2] = −iT1 [22].
Here ~φ(α) = e−iαT3 ~φ are also meaning the generators.
Between the vacuums, the angular dependence is estab-
lished Q(θ) = e−iθTsQeiθTs [29]. Tactoid vortices revolve
SU(2). R, α are introduced to describe the tactoid ge-
ometry (Fig.1).
Further, we need to solve the next equations of motion:
gFµν = jα = je[~φ† ×Dν ~φ− ~φ · (Dν ~φ)†], (13)
gGµν = Jα = g[~φ† ×Dν ~φ+ ~φ× (Dν ~φ)†], (14)
g~φ =
. (15)
To confirm the existence of vortons, labeled by vac-
uum, and estimate the energy T2, the first-order Bogo-
mol’ny’s equations are usually applied. the first-order
Bogomolny’s equations are usually applied. For example,
in the sigma-model limit of the Lagrangian of the type
(5), the non-Abelian votrons with the (1, 1)-, (1, 2)- and
other pairs of winding numbers in SU(2) were numeri-
cally revealed by Radu and Volkov [21] just lately; to be
solvable, their model has included four free parameters
in the potential (Fig. 2).
There was numerically proven with help of Gauss-
Tschebuchev algorithm, that in the U(1) gauge, the sta-
ble vortons may appear [27], whereas in SO(3) it is not
so [26]. The stability criterion includes the radius R
of the vortex loop, which may be compared with the
Kaznacheev-van der Schoot theoretical analysis [2], [7],
and with the lattice Monte Carlo simulations, performed
by Bates [11].
IV. DYNAMICS AND ESTIMATIONS FOR
TIME-AGING OF THE TACTOID SOLS
Along with these assertions on the configurations sup-
plied with Non-Abelian gauge fields, the approximate
methods of analysis exist for quite attainable numer-
ical simulations of vorton states. One of there is so
called Abelian projection [33]. So, following the Maxi-
mal Abelian (MaA) projection approach, we fix SU(2)
gauge and leave the winding group U(1) unfixed. In ap-
plied numerical tasks, Abelian approximations of (11) are
yet acceptable.
For example, whether is an analogous Abelian projec-
tion of the V2O5 −H2O tactoid configuration realized in
the 2D ferromagnetic systems and thin films [34], if there
FIG. 2: The energy density of n=m=1 vortons [21], plotted by
Radu and Volkov numerically at four free parameters, where
z and ? are the polar coordinates.
FIG. 3: Competition between the magnetic Φm and the elastic
and the surface energies Φel +Φs of a tactoid, divided by σ =
10−3 erg/cm, errors are not indicated; as this is a qualitative
view of (3) and experiments [2], [12]; the data at γ → 1, from
which α ≈ 32o.
are defined the same topological invariants? This simpli-
fication is useful to estimate the case of annihilating par-
ticles with whole unit opposite charges [3]. One may ex-
press the vorton dynamics by the Landau-Lifshitz equa-
tion (LLE), including dissipation (labeled by the constant
ad [34]), and write:
= m× f− αdm× (m× f), f ≡ △m−Qmzêz, (16)
where m is the magnetization vector, Q is the free pa-
rameter, êz is the unit vector in the z magnetization
direction. According to the definition [3], the topologi-
cal invariant N connected with the topological density n
is N = 1
nǫµνr
3, ǫµν is the asymmetric tensor with
(µ, ν) = (1, 2), V is a tactoid volume, and the vector r
denotes its space.
The magnetic field stretches large tactoids (a in-
creases), whereas to annihilate, the tactoid shape should
become more oblate [12], Fig. 3. Therefore, the equilib-
rium angle a, corresponding to the large tactoid shape,
exists also for coalescence in the magnetic field. To de-
fine a is not difficult from the next simple algebra with
(1) and (2), by using the definitions [3, 10-11]. From
(1) - (3) and (12), dynamics characteristics of a solitary
tactoid may be expressed as:
(χH)4
)2 ∼ α2d(
1− cosh ηkazn cos ξ
cosh ηkazn − cos ξ
)2. (17)
E. g. stretch of a tactoid in z-axis direction increases its
magnetic energy, and the magnetic field is precipitating
for annihilation of droplets, as a free volume decreases.
On the experiment [14], the next parameters are mea-
sured: Ci =
, i = 1, 3, Ki are modulii of (2), and σ is
the surface tension. C3-s order is hundreds micrometers.
For C1-s, these are about unit. Both of they are drop-
down with time, but according to (1)-(2), have not affect
on the magnetic term.
V. CONCLUSIONS
We composed the topological classification of sols
V2O5 − H2O, owing to which, the qualitative practical
predictions for thermodynamic states of these sols may be
performed. The cosmological theory of superconductive
strings supposes that the nematic tactoids in V2O5−H2O
annihilate in accordance with non-Abelian statistics.
This process, carried out in magnetic field, increases a
time-aging of the sols, but does not yield to direct exact
estimations, since its nature is principally Non-Abelian.
One may connect an actual electromagnetic interaction
in the V2O5 − H2O solution via pH value and discuss
questions on the tactoid junction in frames of chemistry,
which we have wittingly ignored in favor of the impor-
tant topological role. The process of tactoid junction in
magnetic field leads to rise of the additional electromag-
netic field changing pH of water around tactoids and, for
one’s part, time-aging [2, 13]. These observations may
be important for ecology, as long as vanadium pentox-
ide is contained in impurities of coal soles, which are the
components of wastes of thermoelectric power stations
and are included in the impurity parameters at the back-
ground control for radiation.
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  We study the aqueous solution of vanadium pentoxide by using topology
methods. The experiments by Zocher, Kaznacheev, and Dogic exhibited, that in
the sol phases of $V_2O_5-H_2O$, the tactoid droplets of $V_2O_5$ can coalesce.
In the magnetic field, this effect is associated with a gauge field action,
viz. we consider coalescence (in the topologically more convenient term,
"junction") of droplets as annihilation of topological defects, concerning with
the tactoid geometry. We have shown, that in the magnetic field, the tactoid
junction is mainly caused by non-Abelian monopoles (vortons), whereas the
Abelian defects almost do not annihilate. Taking into account this annihilation
mechanism, the estimations of time-aging of the $V_2O_5-H_2O$ sols may be
specified

Elastic properties of vanadium pentoxide aggregates and topological defects
L. V. Elnikova
A. I. Alikhanov Institute for Theoretical and Experimental Physics,
25, B. Cheremushkinskaya st., 117218 Moscow, Russia
(Dated: September 12, 2021)
We study the aqueous solution of vanadium pentoxide by using topology methods. The exper-
iments by Zocher, Kaznacheev, and Dogic exhibited, that in the sol phases of V2O5 − H2O, the
tactoid droplets of V2O5 can coalesce. In the magnetic field, this effect is associated with a gauge
field action, viz. we consider coalescence (in the topologically more convenient term, ”junction”)
of droplets as annihilation of topological defects, concerning with the tactoid geometry. We have
shown, that in the magnetic field, the tactoid junction is mainly caused by non-Abelian monopoles
(vortons), whereas the Abelian defects almost do not annihilate. Taking into account this annihila-
tion mechanism, the estimations of time-aging of the V2O5 −H2O sols may be specified.
PACS numbers:
I. INTRODUCTION
The tactoid sol phase of the V2O5 −H2O system has
been discovered at the 20-th years of the last century
by Zocher (see references in [1, 2]). At the beginning of
our century, the tactoid drops (tactoids) have been in-
vestigated on the optical experiments by Kaznacheev [2],
Lavrentovich [3], Dogic (see [4] and references in [5]), and
their coworkers. The tactoid phase is chemically classi-
fied as the lyotropic inorganic nematic [1]. The tactoids
coexist with the isotropic liquid phase at the mass con-
centration of V2O5, amounting 0.3-2.1 percents, and un-
der other standard conditions [2].
The thermodynamic parameters and pHcause the dy-
namics of their formation, in particular, the junction.
The tactoid geometry is evolved complicatedly (and
mutually inversely) in depending on time-aging of the
sols [2].
Due to the de Gennes’s theory [6], the tactoid shape
stabilization is defined by competition between the elastic
energy of the nematic phase, the surface energy, and the
anchoring energy [2]. The minimum of the tactoid free
energy provides an equilibrium shape of a droplet. The
measured macroscopic elastic moduli are in a very large
ratio (K3
> 100), that distinguishes V2O5 − H2O from
other lyotropic liquid crystals (LC), whose typical values
of K3
are in order of ten.
In the magnetic field, the prolate droplets are aligned
by their long axes parallel to the field. Then the special
case of the junction of tactoid poles may be observed [1,
Remarkably, that the sol phases of V2O5 −H2O were
conditionally sorted on a shape polarity and a nematic
director field [5] as of a homogeneous and a non-uniform
field, and of the spherical and the bispherical [2] drops
with boojums. Strikingly simultaneously, these phases
have been parsed (see [5, 7, 8] and references therein)
basing on the experiments by Dogic (references in [5]),
performed independently of Kaznacheev.
In this paper, we study the mesomorphism of the
V2O5−H2O system during the tactoid junction and spec-
ify the character of the mesomorphic consequence there.
Our goal is to define the influence of junction onto dy-
namic parameters of the sol system, including time-aging
of the sols. In addition, aging of these sols in water is an
applied problem of ecology, since V2O5 contains in coal
impurities, generated in result of work of thermal power
stations.
From a topological standpoint, poles of a tactoid are
the point defects, boojums. As will readily be observed,
we have to do with a quantum phase transition, the anal-
ogous topological singularities of two poles (each admit-
ting a flux) were announced by Haldane [9] for the quan-
tum Hall semiconductors. Also, there is a convenient
analogy with the boojum formalism for the superfluid
phases of 3He and 4He [10], however their varied topol-
ogy descriptions does not allow to explain the case of the
tactoid coalescence.
II. FORMALISM
Geometry of the droplets obeys the local nematic order
parameter n, which is oriented relatively to a droplet
surface (Fig. 1).
The free energy functional of a tactoid in the magnetic
field is summed up from the Frank elastic energy Fel and
the magnetic energy Fm [2, 5]:
F = Fel + Fm, (1)
Fel =
(∇·n)2+K2
(n·∇×n)2+K3
[n×(∇×n)]2−K24∇·[n·∇·n+n×(∇×n)]2].
The magnetic energy density has the form −χa
(n ·H)2,
(where χa is the anisotropy of magnetic susceptibility,
and H is the magnetic field).
The terms at K1, K2, and K3 elastic constants in 2
mean splay, twist, and bend deformations of a bulk ne-
matic respectively, n is the coordinate dependent nematic
director. The term at K24 relates to saddle-splay defor-
mation mode [5]. In this continuum, the tactoid boojums
http://arxiv.org/abs/0704.1893v7
FIG. 1: The director field on the tactoid surface, taken over
[12]. Ri and α are the geometric parameters, γ = (
tan(α1/2
tan(α/2)
0 ≤ γ ≤ 1, the vectors ei i = ϕkazn, ξ, ηkazn denote the
bispherical coordinates.
were revealed by Kaznacheev [2] and by van der Schoot
[5] practically identically, independently of one another.
The final result of tactoid classification is the existence of
four regimes of form is possible, which depend on anchor-
ing between the local director and the tactoid surface,
and also on the total tactoid volume [11]. Only at the
week-coupled limit (γ = 0), Kaznacheev found an equi-
librium shape of a tactoid [2], [13], a fortiori at Fm = 0
and without the terms of K24-s in (2). At the limit (Fig.
1), the free energy (1) is the almost non-analytical func-
tion on f(α, γ) [2, 13]:
4π(sinα− α cosα) + π(3 sinα− 3α sinα− α2 sinα) +
π sin3 α
sin θ
(cosh ηkazn + cosα)
dηkazn +
[sinα(20 + cosα)− 3α cosα(7 + 2 sin2 α)] (3)
here θ is the parameter with the too long dependence of
α, γ, ηkazn [13], the last term of (4) corresponds to the
magnetic energy at γ → 1. For γ, see Fig.1.
Nematic surface defects of the tactoids [12] are of the
homotopic group π2(R, R̃) = P ×Q, the defects of the P
group are living only at the surface (P group is the kernel
of the homomorphism π1(R̃) → π1(R) and consists of
integers [3]), andQ’s defects are arrived from the interior.
(Here R and R̃ denote the space of degenerate states in
the volume and the non-vanishing states on the surface,
which are arrived from the interior, respectively). The
interior may be inhabited by hedgehogs. All of these
point defects keep within the exact homotopic sequence
[12]:
π2(R̃) −→ π2(R) −→ π1(R̃) −→ π1(R). (4)
Boojums are characterized by topological charges m and
n [3], which depend on a configuration of a nematic di-
rector’s field. Annihilation of the boojums of the ad-
jacent tactoids does not mean an influence of the rais-
ing hedgehog’s (in topology, they are not arbitrary float-
ing to the tactoid surface). Kurik and Lavrentovich [14]
have mentioned about some strings, connecting opposite
boojums via a hedgehog in nematic droplets, however,
non-triviality of π1 group hampered the revealing of the
droplet junction without the disclination concept. How-
ever, in our case we reasonably ignore lacking disclina-
tions (see the conclusions by Balachandran et al. [15]).
Interaction scales are the ’dipole length’ Ldip, and the
’correlation length’ Lξ [16], which are characterized an
action of the group of the order parameter. We assume
Ldip is in connection with a long-axis of a tactoid.
In the Cartesian coordinates (x, y, z), the director field
has the configuration n = n(0, 0, 1−cosh ηkazn cos ξ
cosh ηkazn−cos ξ
), where
ηkazn, ξ are the bispherical coordinates [2].
Quite evidently, that tactoid system is provided by
a gauge field [17] (and a field with SU(2) symmetry).
Concerning an universality class of the system, take the
V2O5 droplet surface as belonging to SO(3) group of ro-
tations of the two-dimensional sphere (here ’tactoid’) S2
[16]. U(1) will a group of rotations around a droplet axis,
which is agree closely with the magnetic phase group of
3He − A [18]. U(1)’s winding is realized of non-trivial
topology of tactoids.
The SO(3) and SU(2) groups are locally isomorphic
(as their Lie algebras) and are connected by the homo-
morphism, SO(3) ∼ SU(2)/Z2, where our Z2 is the boo-
jum’s boundary condition.
In our standpoint, at the bulk junction, the group
SO(3)n ×U(1)2n broken down to SO(3)n−1 ×U(1)2n−1,
where n is a number of tactoids.
A model of the sol should involve the monopole solu-
tions, according to the theorem [16] about requirement
of their existence (π2(G/H) −→ π1(H)).
On the other hand, inasmuch as π1(H) =
Z ⊗ Z ⊗ Z ⊗ ...⊗ Z︸ ︷︷ ︸
, the V2O5 − H2O sols are of the
group G. The tactoid annihilation may be described ei-
ther by non-Abelian or Abelian theory in depending on
the global field SU(2). Besides, we have to expect ap-
pearance of a compensative vector field [19].
Here, an each tactoid, in correspondence to two poles
(boojums) on a tactoid surface, may contain two vortons
with their tails (the wide and ”over-Witten’s” definition
for vortons see in [20, 21, 22], this is a kind of monopoles
with the definite pair of topological charges, vortex and
azimuthal windings). Just as vortices, they appear, if the
order parameter has extra degrees of freedom besides of
the overall phase [23, 24]. In the tactoid free energy, the
terms of twisted deformations [8] may play a role in these
excitations. By introducing a necessary parametrization,
the free energy equation, analogous to [2], was proposed
in [8], where the free parameters permit to be the non-
commutative relations in the droplet symmetry. Let us
note, that we use the factor-space CP 1 in accordance to
a chiral (gauge) field (2) [17].
Though, due to the electromagnetic (no topological)
reasons, the sol tactoids can survive coalescence owing
to the Coulomb attraction in water. But from topology
[14], we do not yet know about appearance of a physical
field from the configuration of defects. We have to note,
that because of in-homogeneity of a system, we have a
wide class of string models for a prototype.
III. ANNIHILATION OF TOPOLOGICAL
DEFECTS
So, a junction of droplets means, that the surface point
defect (boojum) configuration may be unstable (γ 6= 0).
We discuss the Abelian and non-Abelian string config-
urations [18, 25, 26, 28, 29], which support the sols of
tactoid nematics. Their combinations and interactions
are expected to define of the junction of tactoids.
A. Abelian space
The Abelian character of pair boojums and monopoles,
and also their integer charge were proven [10]. Boojums
of charge N = ±1 live at Lξ ≪ L ≪ Ldip [16, 20]. But
from the surface field phenomenology [2] of a solitary tac-
toid, one can not define a flux number k [16], concerning
an each boojum, only what k = 1 is preferable for their
pairing configuration, and k = 2 for a unit singularity.
In this scenario, annihilation of charge-opposite (topo-
logical) ’particles’ is possible.
Abelian monopoles may be associated with locations
of boojums, but, due to the topological properties of our
G, we ignore them. Let us consider only vortons of the
Abelian gauge. They are unstable [26], and appear to-
gether with the neutral strings. The open question is
which velocity will greater: of the tactoid coalescence or
the vorton decay.
In the U(1) gauge, the loop-radius dependent criterion
of the vorton stability was found and analyzed numer-
ically in the case of the potential expressed in the el-
liptic ansatz [27], as well as in the well-known Witten’s
U(1) × U(1) case (see review [21), that is an analogous
phase transition from U(1)×U(1) to U(1) for two neigh-
bouring randomly oriented tactoids, in absent of mag-
netic field.
B. Non-Abelian space
Usual Lagrangians of non-Abelian theories are often
linearized into the Bogomolny-Prasad-Sommerfeld (BPS)
equations [30]. A number of applications corresponding
to similar strings were considered, for example, in [15,
21, 22, 24, 25, 28, 29, 31, 32].
In the phase diagram [8], the regions of twist states
were indicated. If the tactoid junction carry out there,
for spherical and prolate droplets, one may make an
analogy between the non-Abelian vortons and ”rotation”
of the nematic order parameter, in spite of the ansatz
(α(η) = α0 sin η [8]) condition, labeled one of the topo-
logical invariants.
Let us formulate the string model with the boson La-
grangian density (due to [21, 22])
L = −1
µν − 1
µν −Dµ~φ† ·Dµ~φ− V (~φ). (5)
Fµν = ∂µAν − ∂νAµ (6)
are the Abelian field strengths. The global curvature is
Gµν = ∂µHν − ∂νHµ + gHµ ×Hν . (7)
The gauge covariant derivatives of vacuums are:
Dµ(~φ) = ∂µ~φ− ieAµ~φ+ gHµ × ~φ, (8)
In the formulas (5) - (8), µ and ν are indices of the gauge
field A and of the metrics g. Hµ and φ are the three-
dimensional vectors in the SU(2) Lie algebra. The field
potential V (φ) is expressing from (2). Due to [13]
x = a
sin ξ cosϕ
cosh η − cos ξ
, y = a
sin ξ sinϕ
cosh η − cos ξ
, z = a
sinh η
coshη − cos ξ
the bulk elastic energy [13] of a tactoid equals to
dηkazn
4K1 sinh
2 ηkazn sin ξ +K3 sin
(cosh ηkazn − cos ξ)3
The corresponding vector potential is
V (~φ) =
λ+ (~φ+ · ~φ− 1
ζ2)2 +
k|~φ · ~φ|2. (11)
At the parameter k > 0, the vacuum is characterize by
~φ · ~φ, ~φ† · ~φ = 1
− sin ξ sinh ηkazn cosϕkazn
cosh ηkazn−cos ξ
− sin ξ sinh ηkazn sinϕkazn
cosh ηkazn−cos ξ
1−coshkazn cos ξ
cosh ηkazn−cos ξ
 . (12)
The generators of SU(2) are denoted as Ti=1,2,3. T0 is
the generator of U(1). −iT1(~φj) = −ǫijk~φj , −iT0(~φ)j =
−~φj . Q = T2 + T0 is the annihilation condition. The
string generator (TS = T3) does not commutate with
the charge generator: [TS , Q] = [T3, T2] = −iT1 [22].
Here ~φ(α) = e−iαT3 ~φ are also meaning the generators.
Between the vacuums, the angular dependence is estab-
lished Q(θ) = e−iθTsQeiθTs [29]. Tactoid vortices revolve
SU(2). R, α are introduced to describe the tactoid ge-
ometry (Fig.1).
Further, we need to solve the next equations of motion:
gFµν = jα = je[~φ† ×Dν ~φ− ~φ · (Dν ~φ)†], (13)
gGµν = Jα = g[~φ† ×Dν ~φ+ ~φ× (Dν ~φ)†], (14)
g~φ =
. (15)
To confirm the existence of vortons, labeled by vac-
uum, and estimate the energy T2, the first-order Bogo-
mol’ny’s equations are usually applied. the first-order
Bogomolny’s equations are usually applied. For example,
in the sigma-model limit of the Lagrangian of the type
(5), the non-Abelian votrons with the (1, 1)-, (1, 2)- and
other pairs of winding numbers in SU(2) were numeri-
cally revealed by Radu and Volkov [21] just lately; to be
solvable, their model has included four free parameters
in the potential (Fig. 2).
There was numerically proven with help of Gauss-
Tschebuchev algorithm, that in the U(1) gauge, the sta-
ble vortons may appear [27], whereas in SO(3) it is not
so [26]. The stability criterion includes the radius R
of the vortex loop, which may be compared with the
Kaznacheev-van der Schoot theoretical analysis [2], [7],
and with the lattice Monte Carlo simulations, performed
by Bates [11].
IV. DYNAMICS AND ESTIMATIONS FOR
TIME-AGING OF THE TACTOID SOLS
Along with these assertions on the configurations sup-
plied with Non-Abelian gauge fields, the approximate
methods of analysis exist for quite attainable numer-
ical simulations of vorton states. One of there is so
called Abelian projection [33]. So, following the Maxi-
mal Abelian (MaA) projection approach, we fix SU(2)
gauge and leave the winding group U(1) unfixed. In ap-
plied numerical tasks, Abelian approximations of (11) are
yet acceptable.
For example, whether is an analogous Abelian projec-
tion of the V2O5 −H2O tactoid configuration realized in
the 2D ferromagnetic systems and thin films [34], if there
FIG. 2: The energy density of n=m=1 vortons [21], plotted by
Radu and Volkov numerically at four free parameters, where
z and ? are the polar coordinates.
FIG. 3: Competition between the magnetic Φm and the elastic
and the surface energies Φel +Φs of a tactoid, divided by σ =
10−3 erg/cm, errors are not indicated; as this is a qualitative
view of (3) and experiments [2], [12]; the data at γ → 1, from
which α ≈ 32o.
are defined the same topological invariants? This simpli-
fication is useful to estimate the case of annihilating par-
ticles with whole unit opposite charges [3]. One may ex-
press the vorton dynamics by the Landau-Lifshitz equa-
tion (LLE), including dissipation (labeled by the constant
ad [34]), and write:
= m× f− αdm× (m× f), f ≡ △m−Qmzêz, (16)
where m is the magnetization vector, Q is the free pa-
rameter, êz is the unit vector in the z magnetization
direction. According to the definition [3], the topologi-
cal invariant N connected with the topological density n
is N = 1
nǫµνr
3, ǫµν is the asymmetric tensor with
(µ, ν) = (1, 2), V is a tactoid volume, and the vector r
denotes its space.
The magnetic field stretches large tactoids (a in-
creases), whereas to annihilate, the tactoid shape should
become more oblate [12], Fig. 3. Therefore, the equilib-
rium angle a, corresponding to the large tactoid shape,
exists also for coalescence in the magnetic field. To de-
fine a is not difficult from the next simple algebra with
(1) and (2), by using the definitions [3, 10-11]. From
(1) - (3) and (12), dynamics characteristics of a solitary
tactoid may be expressed as:
(χH)4
)2 ∼ α2d(
1− cosh ηkazn cos ξ
cosh ηkazn − cos ξ
)2. (17)
E. g. stretch of a tactoid in z-axis direction increases its
magnetic energy, and the magnetic field is precipitating
for annihilation of droplets, as a free volume decreases.
On the experiment [14], the next parameters are mea-
sured: Ci =
, i = 1, 3, Ki are modulii of (2), and σ is
the surface tension. C3-s order is hundreds micrometers.
For C1-s, these are about unit. Both of they are drop-
down with time, but according to (1)-(2), have not affect
on the magnetic term.
V. CONCLUSIONS
We composed the topological classification of sols
V2O5 − H2O, owing to which, the qualitative practical
predictions for thermodynamic states of these sols may be
performed. The cosmological theory of superconductive
strings supposes that the nematic tactoids in V2O5−H2O
annihilate in accordance with non-Abelian statistics.
This process, carried out in magnetic field, increases a
time-aging of the sols, but does not yield to direct exact
estimations, since its nature is principally Non-Abelian.
One may connect an actual electromagnetic interaction
in the V2O5 − H2O solution via pH value and discuss
questions on the tactoid junction in frames of chemistry,
which we have wittingly ignored in favor of the impor-
tant topological role. The process of tactoid junction in
magnetic field leads to rise of the additional electromag-
netic field changing pH of water around tactoids and, for
one’s part, time-aging [2, 13]. These observations may
be important for ecology, as long as vanadium pentox-
ide is contained in impurities of coal soles, which are the
components of wastes of thermoelectric power stations
and are included in the impurity parameters at the back-
ground control for radiation.
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