Acta Polytechnica


doi:10.14311/AP.2014.54.0439
Acta Polytechnica 54(6):439–441, 2014 © Czech Technical University in Prague, 2014

available online at http://ojs.cvut.cz/ojs/index.php/ap

THE SHAPE OF A NUCLEUS GROWING ON THE STRONGLY
CURVED SURFACE OF A NANOFIBER

Alexey M. Sveshnikova, b, ∗, Pavel Demoa, b, Zdeněk Kožíšekb,
Petra Ticháa

a Faculty of Civil Engineering Czech Technical University in Prague
b Institute of Physics, Academy of Sciences of the Czech Republic
∗ corresponding author: sveshnik@fzu.cz

Abstract. The equilibrium shape of a critical nucleus has a strong impact on important parameters
of nucleation theory, such as the nucleation rate. While on a flat substrate the growing nucleus has the
shape of a spherical cap, the shape of a nucleus on the strongly curved surface of a nanofiber is more
complex. In the present paper, we propose a simple model for estimating the deviation of the shape of
the critical nucleus from spherical. It is shown that the nucleus extends more in the direction of the
axis of the nanofiber than in the perpendicular direction, and the deviation from a spherical shape is
stronger in the case of well wettable surfaces.

Keywords: nanofiber, nucleation, equilibrium shape.

1. Introduction
In recent years, nanotechnology has found a con-
stantly growing range of applications in civil engi-
neering. Nano-capsules filled with gel and placed in
the volume of concrete can provide a self-healing abil-
ity for the material; submicron optical fibers mixed
with Portland cement have enabled the creation of
translucent concrete; carbon nanotubes improve the
mechanical durability of concrete; titanium dioxide
nanoparticles break down organic pollutants; zinc ox-
ide nanoparticles improve the resistance of concrete
to water, etc. [1,2]. One of such possible applications
of nanotechnology in civil engineering is protection of
buildings from aggressive influence of the environment.
Many deteriorative phenomena start at the surface
of a building material and slowly find their way into
the volume, which makes protection of the surface a
very important task [3]. However, nanoparticles tend
to attract each other and to coagulate, which usually
makes it a non-trivial task to cover the surface of a
building material.

In recent times, nanotextiles have been proposed as
a potential way to solve this problem. Nanotextiles by
themselves have a number of useful properties, which
make them a promising candidate for protective lay-
ers. Their small thickness make them very flexible and
nearly transparent, which is important, for example,
for protection of historical building, which often have
strongly curved architectonic elements. Because a
nanotextile is a porous material, the air can relatively
freely pass through it. This porosity and extremely
small radius of curvature of nanofibers lead to very
high surface-to-volume ratio for this type of a mate-
rial in comparison with usual protective materials like
plasters. It means that nanotextiles interact with the
environment stronger than usual materials. Moreover,

if we manage to distribute nanoparticles with desired
properties uniformly over the surface of a nanotex-
tile, the resulting textile will combine good properties
of both types of nanomaterial. For example, silver
nanoparticles are known to have strong bactericide ef-
fect. The nanotextile imbued with silver nanoparticles
and attached to the surface of construction material
will inhibit a growth of molds and significantly im-
prove the longevity of building. Taking into account
huge amount of money paid every year for restoration
of surface of civil constructions every factor which
can reduce the rate of required treatment will lead to
appreciable savings.

One possible way to achieve a uniform distribution
of nanoparticles on the surface of a nanotextile is by
a nucleation process. The nanotextile is placed in a
chamber filled with a medium in a metastable state.
Due to the subsequent nucleation and growth process,
nanoparticles are formed on the surface of the nan-
otextile. Since the nucleation process is stochastic, a
generally uniform distribution of nanoparticles over
the surface can be expected. A better understanding
of the nucleation process on a highly curved nanotex-
tile surface will also be valuable in applications outside
civil engineering. For example, technology involving
this process has been suggested for improving the
efficiency of semiconductor devices [4].

2. Heterogeneous nucleation
From the thermodynamical point of view, the forma-
tion of nuclei on the surface of a nanofiber is controlled
by an excess of Gibbs free energy. In a metastable
medium, atoms have higher free energy than in the
form of a cluster. The expression for the change of free
energy during this transition consists of two terms:
the volume term, and the surface term. The volume

439

http://dx.doi.org/10.14311/AP.2014.54.0439
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A. M. Sveshnikov, P. Demo, Z. Kožíšek, P. Tichá Acta Polytechnica

Figure 1. Cross-sections of the nucleus on the nanofiber: a) parallel to the nanofiber axis; b) perpendicular to the
nanofiber axis.

term expresses the general tendency of a metastable
phase to become stable. This term makes a negative
contribution to the total free energy of the cluster.
The surface term is positive, because the atoms on the
interface have relatively high free energy. Thus, the
change in the total free energy of n atoms during their
transition from the metastable phase into a nucleus
can be written as follows:

∆G = −n∆µ + γσn2/3, (1)

where ∆µ is the difference of the chemical potentials
of the initial metastable phase and the cluste, σ repre-
sents the average excess surface energy, and γ stands
for the so-called shape factor. The growing nucleus
is in contact with the ambient metastable phase and
the supporting nanofiber. Both areas of contact are
proportional to n2/3, and the specific proportionality
coefficient leads to the shape factor γ.

Equation (1) represents the energy barrier that the
growing nuclei have to overcome, because for small n
the surface term in (1) prevails. The critical nucleus is
in non-stable equilibrium with the environment, and
its size can be found to be

n =
( 2γσ

3∆µ

)3
. (2)

The nucleation rate, i.e. the number of overcritical
nuclei appearing in the system per unit volume per
unit time, is proportional to the exponent of the free
energy of a critical nucleus:

I ∼ exp(−∆Gc). (3)

It can be seen from the above formulas that the nu-
cleation rate depends on the specific shape of the
critical nucleus, represented by its shape factor γ. In

order to describe the nucleation process on the sur-
face of nanofibers one needs to know the shape of the
corresponding nuclei [5,6].

In the standard theory of heterogeneous nucleation
it is assumed that the characteristic radius of curva-
ture of the substrate is much larger than the radius of
the nucleus. This assumption leads to the conclusion
that growing nuclei have the shape of a spherical cap.
However, this assumption is not valid for heteroge-
neous nucleation on a nanotextile.

3. Equilibrium shape of a nucleus
The equilibrium shape of a nucleus on an arbitrary
surface can be obtained by minimizing its free energy
under the condition of constant volume. The appli-
cation of this principle to the line of contact of the
nucleus with the substrate leads to the well-known
Young-Laplace equation for contact angle θ:

σsa − σsn − σna cos θ = 0, (4)

where σsa stands for the surface tension between sub-
strate and ambient phase, σsn stands for the surface
tension between substrate and nucleus, and σna stands
for the surface tension between nucleus and ambient
phase.

From (4) it follows that contact angle θ is constant
along the whole line of contact of the nucleus with
the nanofiber. Consequently, the equilibrium shape
of a nucleus on the cylindrical surface of a nanofiber
cannot be a spherical cap, because the contact angle
on the intersection of a sphere with a cylinder varies
along the contact line.
Some research has been done on the equilibrium

shape of droplets on fibers [7,8]. In these papers it
is usually assumed that the droplet already has a

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vol. 54 no. 6/2014 The Shape of a Nucleus Growing on the Strongly Curved Surface of a Nanofiber

macroscopic size. However, for applications in nucle-
ation theory one needs to consider very small droplets
smaller in size than or comparable in size with the
radius of the nanofiber. In the present paper, we pro-
pose a simple theoretical model allowing the shape
factor of very small nuclei to be estimated.

While the nucleus as a whole cannot be considered
as a spherical cap, one can assume that its cross-
sections by planes parallel and perpendicular to the
axis of nanofiber may be well approximated by circular
arcs. However, these arcs have a different diameter
and their centers do not coincide.
We denote the thickness of the nucleus (i.e., the

maximum distance of its surface from the surface of
the nanofiber) as h, the radius of a nanofiber as R,
and the radii of the circular arcs in parallel and per-
pendicular cross-sections as r1 and r2 (see Figure 1).
Then

h = r1(1 − cos θ) (5)

and
h = d + r2 − R, (6)

where d is the distance between the axis of the
nanofiber and the center of the circular arc in perpen-
dicular cross-section:

d2 = r22 + R
2 − 2r2R cos θ. (7)

From (6) and (7), we obtain the expression for the
radius of curvature of the surface of the nucleus in the
plane perpendicular to the nanofiber axis:

r2 =
h2 + 2hR

2(h + R − R cos θ)
. (8)

The deviation of the shape of the nucleus from spheri-
cal can be estimated by the ratio of its radii of curva-
ture r1 and r2:

r1
r2

=
2(x + 1 − cos θ)

(x + 2)(1 − cos θ)
, (9)

where x = h/R is the dimensionless thickness of the
nucleus. In Figure 2, the dependence of ratio (9)
on dimensionless thickness x is shown for different
values of the contact angle. It is clear that the more
wettable the surface of the nanofiber is, the larger
is the deviation of the shape of the nucleus from a
spherical cap.

4. Conclusion
It follows from (9) that the growing nucleus is al-
ways more extended in the direction parallel to the
nanofiber axis than in the perpendicular direction
(r1/r2 is always greater than 1). For a well-wettable
surface (θ ≈ 0, cos θ ≈ 1) this effect is stronger, be-
cause the area of contact of the nucleus with the fiber
is larger and the nucleus better “feels” the deviation
of the surface of the fiber from the flat surface.

Figure 2. Deviation of the equilibrium nucleus shape
from spherical as a function of nucleus relative thick-
ness x; r1 is the radius of curvature in the plane
parallel to the axis of the nanofiber; r2 is the radius
of curvature in the plane perpendicular to the axis of
the nanofiber; θ is a contact angle.

Acknowledgements
This work has been supported by CTU project
SGS14/111/OHK1/2T/11 and by GACR projects
P108/12/0891 and 14-04431P, and has been performed
within the framework of the Joint Laboratory of Nanofiber
Technology of the Institute of Physics ASCR and the
Faculty of Civil Engineering, CTU in Prague.

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441


	Acta Polytechnica 54(6):439–441, 2014
	1 Introduction
	2 Heterogeneous nucleation
	3 Equilibrium shape of a nucleus
	4 Conclusion
	Acknowledgements
	References