1 Physical principles of nanofiber production 6. Electric pressure and liquid body disintegration D.Lukáš 2010

Electric pressure is another basic concept in electrospinning besides surface tension and electric bi-layer. Its analysis will start with the derivation of the field strength at the surface of a charged conductive liquid, as given by Smith [35]

Two points, A and B, at the vicinity of the surface of a charged conductive liquid drop. Field strengths E 1 and E 2 are contributions to the total field strength E. E 1 generated from charges that resides on surface element of drop, denoted by, dS, and E 2 is the field strength contribution of charges from the rest of the liquid and from all other charges in space. 3

It is worth to mention that the orientation of the total field strength, E, has to be perpendicular to the surface of the conductor, and the same is presupposed for electrostatic intensities, E 1 and E 2, since the charge distribution is considered to be in equilibrium. It is obvious that any tangential component of field strength with respect to the liquid surface should violate the equilibrium since then the charge will move along the liquid surface and the system cannot be considered as a one in equilibrium. E has to be zero in the liquid bulk, otherwise, a charge there should move too. Thus, the induced charge on the liquid drop, causing the field strength value E 1, shields the external field inside the drop, as has been shown in the article about the electric bi-layer. So the analysis may be carried out with the values of field strengths instead of their vector nature. 4

+ Conductive body (liquid body) Intensity inside conductive body is zero. Intensity vectors (on the surface of conductive body) are perpendicular to body surface. Surface of body is equipotentials. 5

6

As, inside the conductive liquid the total electric field, E(B) strength, is zero, two equations can be constructed for the total field strength at the points A and B. At point B, the following relation holds true. (3.14) and at the point A holds (3.15) 7

Due to infinitesimally small distance between points A and B the absolute values of E 1 and E 2 at these points may virtually be considered unchanged. On the other hand, since E 1 is generated by the surface charge and E 2 by the charge in the rest of the liquid sphere and elsewhere in the space, the mutual orientation of these electric intensities varies at points A and B, as has been expressed by equations (3.14) and (3.15). The direct consequence of these Equations is given as: (3.16) 8

Application of Gauss theorem of electrostatics, introduced as Equation (3.4), at the vicinity of the surface element, dS, results in the following relation, as depicted in (Figure 3.3). (3.17) 9

(3.19) 10

The onset of electrospinning appears under the condition that electric pressure p e exceeds the capillary pressure, p c, i.e.. This condition for the electrospinning onset will be commonly used further in the text. 11

Disintegration of liquid bodies 12

Disintegration of charged liquid conductive bodies to nano- scale matter can be illustrated through a single droplet. The related experimental physics, directly connected to disintegration of water drop under electric field, originated through Zeleny’s [10], Doyle et al. [36] and Berg and George [37] works. The stability analysis of charged liquid bodies, as carried out by Rayleigh [15], will be presented here in a simplified version to show the limiting charge, q, for spherical droplet disintegration. 13

Suppose now that the charged droplet, embedded in a space without any other external charges, is a perfect sphere with radius r. The liquid sphere has uniform surface charge density, and is considered as conductive. Thus, for the whole sphere, having radius r, the following relation is obtained directly from Gauss theorem of electrostatics, total net charge on the liquid sphere 14 (3.20) Gauss theorem Nabytá kapka v nulovém vnějším poli.

15 According to the statement at the end of the following article, the spherical droplet dissociates under the condition capillary pressure From the inequality, and from Equation (3.20) follows static disintegration criterion. Electric pressure (3.21)

16 The more advanced theoretical foundations for analyzing the dynamic stability of charged droplets were developed by Rayleigh [15]. Rayleigh has shown in the work that capillary wave instability on the droplet surface is responsible for this phenomenon. He derived the following condition for the onset of destabilization of a perfectly conductive spherical droplet (3.22) (3.21)

17 The integer, n, belongs to various vibration modes of the liquid droplet. The zero mode, n=0, corresponds to radial oscillations, that are unacceptable for incompressible fluids. The first mode, n=1, represents the reciprocating translational droplet motion. Hence, the smallest possible mode number is n=2.

18 The droplet instability can be observed visually as the ejection of a fine jet of highly dispersed daughter droplets whose charge / mass ratios are higher than for the original droplet, as mentioned in Grigor’ev [38]. Freely charged liquid droplets are, in principle, unstable since they elongate to reach the shape of spheroids with the major and minor axes as a and b, respectively, as was showed by Taylor [16]. As the ratio increases, the critical value of Q decreases since on the highly curved spheroid apexes is the charge density significantly greater then on the surface of original spherical droplet. Thus, electric pressure p e grows in these places more rapidly than the capillary one, p c.

Nano- particles Macro-particles

20 Consequently, it may be stated that charged spheroids are always unstable and therefore, disintegration proceeds inexorably as has been mentioned earlier in the introductory part of this subsection. It was found that the instability led to creation of daughter droplets that were approximately ten times smaller than the original one. Daughter droplets and their offsprings obey the same phenomenon too. Hence such cascade of droplet disintegrations leads finally to nanoparicles, i.e. nanodroplets. Analogous, but a complex self-similar process leads to creation of nanofibres from macroscopic liquid jets in the area of electrospinning. Equation (3.22) represents qualitative explanation of procedures that lead to creation of still tinier objects, made by charged liquid bodies. Electrospraying! (3.22)