Fyzikální principy tvorby nanovláken Teoretické základy (2) Povrchové napětí a elektrická dvojvrstva D. Lukáš 2010

Surface tension

Electrospun nanofibres are not the only “nano-scale objects” in electrospinning. Surfaces of physical bodies, including liquids and polymeric solution, within a depth of several tens or units of nanometres, embody properties quite different from that in the bulk. Even without any external electrostatic field a liquid surface exhibits surface tension that is the consequence of short range intermolecular forces, and, hence, this phenomenon itself is bound to a liquid surface layer, whose thickness is comparable with the reach / range of intermolecular forces. If, moreover, on surface of a liquid, under an external electrostatic field, charges tend to distribute in a way to shield the field in the bulk, an ‘electric bi-layer’ is formed.

Intermolecular forces 0.3-0.5 nm 0.1 nm

Electrospinning is commonly described as a tug of war between electric and capillary forces. So, a brief introduction to the nature of capillary phenomena is meaningful. The phenomenon of surface tension will be explained for the sake of brevity, using a lattice model at zero temperature limits. Details about this simple approach to surface tension can be found in Lukas et al. [26]. capillary forces electric forces [26] D. Lukas, E. Kostakova, and A. Sarkar, Computer simulation of moisture transport in fibrous materials, in Thermal and moisture transport in fibrous materials, N. Pan and P. Gibson, Woodhead Publishing Limited, Cambridge, 2006, pp. 469 – 541.

A regular cubic lattice of cells in a three dimensional space may be imagined as in (Figure 3.1). Each cell may be considered to be occupied by one of the two species of fluid particles (!!!not cells!!!), out of which one is assigned with a value 1 and depicted as white, the second kind is allocated a value 2 and black colour. There are various interaction energies between various pairs of particles. These interactions represent short range molecular interactions and hence they appear in-between neighbouring cells only. Only cells having a common wall are counted towards as neighbours.

Lennard-Jones potential From Wikipedia, the free encyclopedia The Lennard-Jones potential is a mathematically simple model that describes the interaction between a pair of neutral atoms or molecules. A form of the potential was first proposed in 1924 by John Lennard-Jones. r o 1 A=0.1 nm

The most common expression of the L-J potential is Where: ε is the depth of the potential well, σ is the (finite) distance at which the inter-particle potential is zero, r is the distance between the particles. The r−12 term describes Pauli repulsion at short ranges due to overlapping electron orbitals. The r−6 term describes attraction at long ranges (van der Waals force, or dispersion force).

r r0 Intermolecular forces: - Dispersive forces Van der Waals forces Hydrogen bonds System Energy Pauli repulsion r Attractive van der Waals force, or dispersion force Bond Energy r0

As a rule, particles of the same nature attract each other more than different ones. So, interaction energies E between neighbouring particles of the same kind of fluids have to be modelled smaller. Figure 3.1. Lattice model of binary mixture of fluids: At the zero temperature limit, fluids separate completely, since the system minimizes its total energy. There is an extra energy bonded to the interface between liquids, whose surface density is called surface energy.

Systém Energy per an interaxting pair of particles. System Energy r Bond Energy r0

The lattice model of this binary mixture of fluids tends naturally to minimize its total / free energy. At the zero temperature limit the fluids separate completely, since the system minimizes its total energy, as shown in (Figure 3.1). Unlike bulks of both fluids there is an extra energy belonging to the interface between them. In bulks. the interaction energy per a bond is equal either to 1 e.u. or to 2 e.u., while cells on the interface create at least one bond which energy content is higher and equal to 3 e.u. The effect of the increment of total energy due to interaction energy between different species of fluid particles is associated to the interface in a depth comparable with the order of short range inter-molecular forces and, hence, the phenomena of surface energy belongs to a superficial layer having thickness in nanometre-scale. The same effect appears at each boundary independent on physical state of the material.

Energy z

Creation of a new area A of a rectangular interface of a width w needs some force F acting on a side of the length w. The work, W, done by the displacement of the side w through a distance l is This work has to be equal to the surface energy, WA , of an elementary area of interface having an area of Surface energy, WA, according to its aforementioned definition, is equal to The equation defines also surface tension  as force acting on a unit length of the triplet line.

If one divides an ideally spherical liquid droplet of radius r into two mirror parts, cutting it along its equator, the linear force  of surface tension acts along the perimeter given by which creates a total capillary force, that attracts and tends to attach the hemispheres together. This force causes a capillary / Laplace pressure if the droplet is again constructed from its two parts,

Two generalised relationships for above-cited brief introduction to capillary phenomena, as cited above, will be used further. First of them is the expression for capillary force, given by (3.8) where θ represents the contact angle between the vector, representing the surface tension and the plain of perimeter.

The other well-known Laplace -Young formula represents the generalized capillary pressure pc caused by arbitrarily curved liquid surface as a multiple of the surface tension and a sum of two principal curvatures K1 and K2. (3.9) In the case of sphere of radius R, both principal curvatures K1 and K2 are of the same value and are equal to in agreement with the previously derived formula for capillary pressure,

L S R

More information about surface tension and capillary pressure are given in: Adamson and Gast [27] and Shchukin et al. [28].