Model droplet formation in extensional filament stretching within a Filament Extension Atomizer

The dominant type of deformation in filament stretching processes is extensional strain. Due to the nonbond surface and only one direction of forced deformation the strain in a filament can be approximately described as uniaxial (see Fig. 2) although that is not completely true at the outer regions. For a constant extensional rate \(\dot{\upvarepsilon }\), the length of the stretched object must increase exponentially (cf. Eq. (1)). At the same time, similar equations apply for the radius and surface at constant volume and radial symmetry (cf. Eq. (2) and (3)) (Mezger 2021).

Typically, polymers and polymer solutions show a strain hardening tendency under uniaxial strain that exceeds the effect of shear thickening by far (Pahl et al. 1995). The strain hardening is a result of the molecules interaction and depends strongly on the morphology of the molecules. It is widely accepted that branched polymer always show strain hardening behavior (Auhl 2006; Münstedt and Laun 1981). Even smallest amounts of branched polymers are sufficient to trigger this effect (Bin Wadud and Baird 2000). However, other research also shows that strain hardening can occur in linear polymers with a wide molar mass distribution (Ide and White 1978; Münstedt 1980; Takahashi et al. 1993; Wagner et al. 2000). But, according to Barnes (2000), linear polymers with low molar masses might even show strain softening under specific circumstances.

In special cases strain-hardening due to linear molecules and long chain branching can be described analytically with the Tube Model by Doi and Edwards (Doi and Edwards 1978a, b, c) and its extensions ((extended) Pom-Pom- (McLeish and Larson 1998; Verbeeten et al. 2001), (double) Rolie-Poly- (Abuga and Chinyoka 2020; Azahar et al. 2019; Boudara et al. 2019; Likhtman and Graham 2003), GLaMM- (Auhl et al. 2008; Graham et al. 2003; Kröger 2019) and Molecular Stress Function-(MSF)-model (Rolón-Garrido and Wagner 2007; Wagner et al. 2000, 2003)). On the other hand, the influence of the ‘Kinetic Models’, e.g. FENE-model (Bird et al. 1980; Chilcott and Rallison 1988) and Giesekus-model (Giesekus 1982), has increasingly diminished over the last decade (Azahar 2020). Tube models are based on the idea of polymer movement along defined tubes. Lateral movement to the main direction is not possible and relaxation of the polymers occurs only along the tube, since deviation by other polymers is prevented. For long-chain branched polymers, only the MSF- and Pom-Pom models are suitable. The MSF-model assumes that the tube can contract orthogonally to the direction of strain, thus creating a contraction stress on side chains (Mattes 2007). The transverse contraction of the tube and the tension of the side chains create additional resistance to further elongation. The (extended) Pom-Pom-model assumes idealized polymers. The polymers have a backbone and several branches at both ends. The backbone can move in a tube, whereas the branches are entangled with other polymers (McLeish and Larson 1998; Verbeeten et al. 2001). Despite the partially good results for certain cases, no model has emerged as universally applicable. Moreover, none of the models is suitable for investigating droplet formation and, with the prediction of rheological quantities, only reflects part of the problem (filament formation). Further research is necessary.

In general, the dissolving of polymers is separated into three different cases. In a poor solvent the chemical interaction between the polymer and the solvent is unfavorable, so polymer-polymer interactions are dominating (cf. Fig. 3). If the interactions between solvent and polymer are favorable, the polymers swell, resulting in a larger average distance between the ends of the polymer what can be described via the “Flory end-to-end”-distance R_F (cf. Fig. 3). Otherwise, the solution is called theta solution. Thermodynamically, the chemical potential between the solvent and the polymer is 0 and no excess volume is created by mixing them. The ‘end-to-end’ distance R_θ of the molecule can be exactly determined by the random walk of polymers (Rubinstein and Colby 2010).

The described behavior of the polymer in solution can be illustrated by means of the “excluded-volume” parameter \(\upnu\) (also called solvent quality exponent, 0,5 for theta solvents) which is used to calculate the intrinsic viscosity [η]. The rheological behavior of a polymer solution is therefore directly dependent on \(\upnu\). Dissolved polymers produce a noticeable increase in viscosity even at low concentrations (Colby 2010; Rubinstein and Colby 2010).

The rheological behavior of a polymer solution therefore directly depends on \(\upnu\). Dissolved polymers produce a noticeable increase in viscosity even at low concentrations. The concentration-dependent behavior of polymer solutions however is a function of the coil-overlap-concentration (COC). The COC describes the overlap of the imagined polymer tubes (cf. strain hardening models) and helps to divide solutions into three different domains (Graessley 1982; Ying and Chu 1987). In the diluted domain, the physical properties are dependent on the solvent-polymer interactions. The limit of this domain is the critical overlap concentration c* (Poinot et al. 2014). Above c*, the polymers begin to overlap and interact. The domain of c* < c < c_e is called semi-diluted and is limited by the ‘entanglement concentration’ c_e., which is 10-times the critical overlap concentration c* (cf. Eq. (4)) (Arnolds et al. 2010; Colby 2010). The critical overlap concentration is a function of the intrinsic viscosity: \({\mathrm{c}}^{*}\sim {\left[\upeta \right]}^{-1}\). Depending on the source, the proportionality factor of the previous equation differs, but is usually around 1 what is taken for granted within this publication (Graessley 1982; Tirtaatmadja et al. 2006). For simplified representation, the dimensionless parameter of the concentration ratio c_COC is defined in Eq. (4).

The dynamics of the droplet formation and the decay of fluid necking for Newtonian fluids dripping from a nozzle has been of interests since the mid-1800s. Initial investigations of droplet formation for inviscid fluids already revealed a clear correlation between gravity and surface tension but over time with the establishment of various experimental setups, further correlations for viscous fluids were found. Worth mentioning is the mathematical formulation of the idealized instability for cylindrical fluid threads of Plateau and Rayleigh (Plateau 1873; Rayleigh 1878, 1892a, b) and its further development of Tomotika (1935, 1936) for more precise application depending on fluid properties (viscosity and surface tension of the dripped/jetted fluid) and environmental conditions (viscosity and surface tension of the surrounding fluid; detailed description in ‘Plateau-Rayleigh instability’; Fig. 4). Another important variable is the Ohnesorge number (Eq. (8)). In his dissertation, Wolfgang von Ohnesorge presents the dimensionless number, which describes the flow of a liquid out of a capillary as a function of the velocity and the type of liquid (Ohnesorge 1936). He distinguished between four areas: dripping as a result of gravity; Decay of fluid jets due to the Plateau-Rayleigh instability; Decay of fluid jets due to wave-shaped interference, whereby this was experimentally discovered by Weber (Weber 1931) and theoretically confirmed by Haenlein (Haenlein 1931); Atomization into droplets (Ohnesorge 1936; Lefebvre 1989). In the 1970s, research began to characterize the behavior of polymer solutions. This was due to the emergence of new applications such as inkjet printing, in which individual ink droplets had to be created (Cooper-White et al. 2002). However, due to low viscous and strain-hardening effects of long-chain components of the ink, the production of individual ink droplets was only possible to a limited extent. At low viscosities, satellite droplets are formed in addition to the primary drops which lead to reduced quality in printing. The formation of satellite droplets was investigated in detail in the 1990s (Carrier et al. 2015; Chaudhary and Maxworthy 1980a, b; Chaudhary and Redekopp 1980; Tjahjadi et al. 1992; Vassallo and Asgriz 1991; van der Geld and Vermeer 1994; Wagner et al. 2005). The long-chain components cause the formation of filaments, which are also unwanted (inaccurate printing), but which are of importance for later droplet formation (Beads-on-a-string; detailed descriptions in ‘Filament formation and behavior’ and ‘Droplet formation’) (Anna and McKinley 2001; Clasen et al. 2006; McKinley and Tripathi 2000; Papageorgiou 1995). Two experimental standards have been established for the investigation of viscoelastic fluids, especially polymer solutions. On the one hand, the Plateau and Rayleigh experiments were used as a basis. Depending on the flow rate, the fluid flows through a capillary and the detachment of primary drops, the formation of satellite droplets and further droplets as well as the filament formation are investigated. Gravitational effects are studied with a vertical nozzle, whereas the decay from a jet is often studied with a horizontal orientation. Second there are filament stretching experiments (FSD/FBD/FISER) (Bazilevsky et al. 1990; Matta and Tytus 1990; Sridhar et al. 1991; Tirtaatmadja and Sridhar 1993) and ‘Capillary Breakup Extensional Rheometer’ (CaBER) (McKinley et al. 1999; McKinley and Sridhar 2002)). The experimental results could be accurately represented over the years by mathematical models with increasing computing power. Numerical experiments using different models have been performed for fluid spraying/jetting, thread formation and breakup (Fontelos and Li 2004; Forest and Wang 1990; Forest and Wang 1994; Renardy 2004; Renardy 1995), Plateau-Rayleigh instability, satellite droplet formation (van der Geld and Vermeer 1994; Vassallo and Asgriz 1991), BOAS structure (Bhat et al. 2010; Clasen et al. 2006; Li and Fontelos 2003) and strain hardening (e.g. Doi and Edwards 1978a; Gennes 1971; Wagner et al. 2000)). In this paper a semi-empirical model for droplet formation while dripping out of a nozzle by continuous stretching and Plateau-Rayleigh instability will be developed. Therefore, special attention is paid to the dripping of fluids from a nozzle. The relevant theory is described for Newtonian and viscoelastic fluids.

A Plateau-Rayleigh instability occurs in a system of a cylindrical fluid threads (length >> diameter) and is driven by capillary forces, which minimizes the surface area. For Newtonian fluids the instability was already investigated by Plateau and Rayleigh at the end of the 19th century. Tomotika extended the Plateau-Rayleigh instability by further cases. (Plateau-Rayleigh-instability = linear instability of inviscid liquids (Rayleigh 1878, 1892a, b); Rayleigh-Tomotika instability = linear instability of viscous liquids with fluid threads that are highly viscous and/or external fluid that is highly viscous (Tomotika 1935, 1936)). The explanation for the instability is self-enforcing perturbations in a thread, beginning with small random perturbations on the free surface. Growth of particular areas will occur, while other areas decrease, if the perturbations are sinusoidal with an appropriate wavelength Λ. A detailed description can be found in Fig. 4.

The mathematical correlation of the pressure gradients was established by Plateau and Rayleigh, and was extended by Tomotika (1935, 1936). They found a periodic decay of the cylindrical threads into droplets as a function of the unperturbed diameter of the threads and the viscosity. Plateau and Rayleigh assumed that only inertia and capillary forces had to be considered. This results in a maximum growth rate with the presented dependencies of Eq. (5) whereas R₀ is the initial radius and λ the wavelength (Rayleigh 1878).

Two effects result from the Plateau-Rayleigh instability: The formation of a capillary thread (filament) that breaks up at a time t and the formation of droplets. For both a distinction must be made between Newtonian and viscoelastic fluids.

Therefore, for the viscoelastic polymer solution, the filament behavior and the droplet formation depend on the surface tension, the viscosity, the temperature, the solvent, the dissolved polymer, its molecular weight and concentration and the relaxation time of the polymer (Anna and McKinley 2001; Bazilevsky et al. 1990; Dinic and Sharma 2019; Entov and Hinch 1997; Sachsenheimer et al. 2014; Sousa et al. 2017; Tirtaatmadja et al. 2006). Various (dimensionless) quantities can be derived from these variables, so that the dependencies can be displayed in a simplified way (see also c_COC (Eq. (4)). Important (dimensionless) quantities are the Weber number We, the Reynolds number Re, the Ohnesorge number Oh, the Rayleigh time t_R, the capillary time t_cap (Anna and McKinley 2001)(also called t_VC (Dinic and Sharma 2019) or viscous timescale t_V (Sachsenheimer et al. 2014)), the Weissenberg number Wi, the Deborah number De and the elasto-capillary number EC, the relaxation time λ and the disturbance frequency k (Eq. (5)). The detailed description of the used quantities can be found below whereas other combinations of these quantities can be obtained from various sources, although the definitions of the individual quantities are sometimes not consistent (e.g. (Anna and McKinley 2001; Christanti and Walker 2001, 2002; Dinic and Sharma 2019; Sachsenheimer et al. 2014; McKinley 2005)).

The Weber number describes the ratio of inertial force to surface tension, so inertia increases with increasing Weber number. It is often used to describe the outflow of fluids from a capillary and, together with the Reynolds number and the Ohnesorge number, establishes a classification for modes of disintegration (Rayleigh, varicose breakup, sinuous wave breakup, wave-like breakup with air friction, secondary atomization; according to Ohnesorge (1936)) (Lefebvre 1989).

The Reynolds number describes the ratio of inertia to viscous effects. The inertia increases with increasing Reynolds number. For small Reynolds numbers the viscous effects dominate, and dissipation occurs. Instabilities on a thread are continuously damped (Andrade et al. 2012).

The Ohnesorge number describes the relationship between the viscosity, inertia and capillary forces. It is the ratio between the capillary time (\({t}_{cap}=\frac{{\eta }_{0}{R}_{0}}{\sigma }\); breakup time of visco-capillary narrowing) and Rayleigh time (\({t}_{R}=\sqrt{\frac{\rho {R}_{0}^{3}}{\sigma }}\); breakup time of inertial-capillary narrowing) (Eggers and Villermaux 2008; Ohnesorge 1936).

The filament formation essentially depends on the viscosity, the dissolved substances (polymers) and the additional elastic stress of dissolved substances. A distinction between Newtonian and non-Newtonian viscoelastic flow behavior is necessary (e.g., Fig. 5).

The filament formation can be separated into four regimes: The inertia-capillary (IC) regime or viscous-capillary (VC) regime, the elasto-capillary (EC) regime and the terminal visco-elasto-capillary (TVEC) regime, whereby only the IC and the VC-regime is possible for Newtonian fluids (McKinley 2005). Depending on the viscosity, the dynamics of the thinning-process can either be a ‘potential flow’, inertia is dominating (IC-regime), or a ‘viscous thread’, viscosity is dominating (VC-regime) (Lister and Stone 1998; Cooper-White et al. 2002). The decay of the diameter in the IC-regime is proportional to t^2/3 (Eq. (9)), whereas it is linear depended on the time in the VC-regime (Eq. (10)). Dinic and Sharma (2019) also made an empirical approach for the transition of the regimes but only for an viscoelastic fluid. Their approach depends on the Ohnesorge number (Eq. (8)) but also on the dimensionless concentration c_COC. (Eq. (4)). The IC-regime is forming for Oh < 0,1 and c_COC < 4, whereas the VC regime develops at Oh > 4. In the intermediate area, there is a singularity of inertial and viscous forces, which was first proposed by Eggers as the ‘inertial-viscous’ regime with a linear decay over the time (Eggers 1997; Lister and Stone 1998). The ‘inertia-viscous’ regime also occurs just before the final breakup of the filament. Yildirim and Basaran (2001) made numerical simulations of the transition between the regimes and were able to predict the transition.

The filament development for the IC-regime is given by Eq. (9), whereby the time of the breakup is given by t_p (also called pinch-point), t_R is the Rayleigh timescale (cf. Eq. (8)) and X is a proportionality factor. Hence the IC regime only depends on the surface tension as the Rayleigh timescale is a function of it. Recent simulations and experiments showed that X is close to 0.4 (Deblais et al. 2018; Dinic and Sharma 2019), although earlier mathematical approaches and experiments resulted in values between 0.6 and 0.8 (Anna and McKinley 2001; Chen et al. 2002; McKinley 2005; Rodd et al. 2005).

The filament development for the VC-regime is given by Eq. (10). In addition to the surface tension γ, the decrease of the diameter depends on the viscosity η. The thinning process takes longer with higher viscosity (Cooper-White et al. 2002; Rodd et al. 2005). Papageorgiou (1995) used self-similarity with neglected inertial forces to determine the proportionality factor κ. The value κ = 0.0709 was later confirmed by McKinley and Sridhar for high viscous fluids (McKinley and Sridhar 2002). Eggers (1997) predicted κ = 0.0304 for the ‘inertial-viscous’ regime, which especially fits for the dynamics near the breakup of Newtonian fluids.

The transition from the IC/VC-regime to the EC-regime is to be expected when the viscoelastic timescale (relaxation time λ) is in the same order of magnitude as the visco-capillary timescale (capillary time t_cap) and the inertia-capillary timescale (Rayleigh time t_R).

The linear behavior of Newtonian systems can be completely described with the Eq. (9) and (10) until breakup. The nonlinear behavior of viscoelastic fluids can only be described up to shortly before the breakup. Further regimes have to be considered. During the elongation, the molecules are oriented, and the elastic stress of the molecules has to be considered. This regime is called EC-regime (elasto-capillary) (Dinic and Sharma 2019). The inertial and viscous effects are neglectable, and the elastic stress is in equilibrium with the capillary force (Tirtaatmadja et al. 2006). Perturbations in the cylindrical form cannot develop, since the Plateau-Rayleigh instability is counteracted by strain-hardening effects of the polymers (higher viscosity at necked regions due to greater extension rates). In the EC-regime, the radius decreases exponentially with time (Eq. (11)) (Cooper-White et al. 2002; Dinic and Sharma 2019). The thinning behavior depends on the elastic modulus (G = η_P/λ_c) with η_P as the contribution of the polymer to the solutions viscosity (η_S = η_P + η_solvent), the surface tension γ and the initial radius of the filament R₀. The proportionality factor is also known as the elasto-capillary-number of Anna and McKinley (EC = (G⋅R₀)/γ) (Anna and McKinley 2001).

The decrease is proportional to a characteristic relaxation time λ_c of the polymer solution (Anna and McKinley 2001; Anna et al. 2001; Arnolds et al. 2010; Christanti and Walker 2002). The characteristic relaxation time is usually the greatest relaxation time (Zhou and Doi 2018), but for diluted solutions it is often replaced by the model based Zimm-relaxation time. The Zimm-relaxation time can be directly calculated from the solution properties (Eq. (12)). In this equation, the Zimm-relaxation time depends on the pre-factor (1/ζ(3ν)), which depends on the ‘excluded-volume’ parameter \(\upnu\) (previously mentioned while discussing polymer solutions below Fig. 3), the intrinsic viscosity [η], which can be calculated with suitable reference books (e.g. (Brandrup 1999)), the molecular mass of the polymer M_w, the Avogadro constant N_A, the Boltzmann constant k_b and the temperature T (Anna et al. 2001; McKinley et al. 1999; McKinley 2005; Del Giudice et al. 2017; Zimm 1956).

Dinic noted, that even though the Entov-Hinch’s (1997) expression (Eq. (11)) is the most often cited description of the filament thinning behavior in the EC regime, adjustments have to be made because of striking differences between shear rheological measurements and strain behavior (strain hardening) (Dinic and Sharma 2019). The shear modulus should be replaced with the extensional elastic modulus (G_E = η_P/λ_E ≠ G) and the characteristic relaxation time with the longest extensional relaxation time λ_E.

Considering Eq. (2), it can be shown, that the strain rate is constant in the EC regime. The combination of the Eq. (2) and (11) results in a constant Weissenberg number Wi = 2/3 (Eq. (13)) (Bhattacharjee et al. 2003; Liang and Mackley 1994). In general, the viscosity of the unperturbed filament can always be determined from the ratio of capillary stress (surface tension) and the change of the diameter over the time (dR(t)/dt). The determination is simpler in the EC-regime (constant strain rate). It is possible to calculate the transient extensional viscosity from Eq. (14). Since the strain rate is constant, the transient viscosity increases monotonically until the polymers are orientated and completely stretched (Amarouchene et al. 2001).

In the final regime the polymer molecules are completely stretched. This regime is called TVEC regime (terminal visco-elastocapillary). The internal frictional force of the solution (viscosity), the elastic stress of the polymers and the surface tension are now in balance. There is a linear decrease of the radius over time until filament breakup (Eq. (15)), in which the factor \({\mathrm{t}}_{\mathrm{R}}\) (cf. Eq. (8)) and t_EC are the onset of the regime. Furthermore, the equation (15) depends on the Ohnesorge number (Eq. (8)) and the infinite Trouton-ratio (Tr_∞ = η_E,TVEC/η_S,TVEC (extensional and shear viscosity in the TVEC-regime). The polymer cannot further stabilize the instability of the capillary forces and small droplets will emerge from the filament. While still connected with the filament, it is called ‘beads-filament’- (Chang et al. 1999) or ‘Beads-on-a-string’(BOAS)-structure (Bhat et al. 2010; Oliveira and McKinley 2005). The effects of finite extensibility in this regime were broadly discussed via the FENE-P- and Giesekus (with inertia) -simulations by Renardy (1995, 2002, 2004). Renardy also subdivided the breakup into ‘with inertia’ (BOAS-structure) and ‘without inertia’ (no beads) (Renardy 2002).

Three different types of droplets—primary drops, satellite droplets and the Beads-on-a-string—are described in the literature. Concerning the performed dripping experiments, primary drops form directly on the outlet of a capillary until the gravity of the droplet predominates the surface tension (see Fig. 6). Individual primary drops can be identified for smaller Weber and Reynolds number (dripping). The size of the primary drops increases with the Weber number until there is no more distinction possible between the individual drops (chaotic dripping, jetting) (Lefebvre 1989). Furthermore, the primary drop size increases with increasing surface tension, which can be derived from the balance of forces at the capillary (Ambravaneswaran et al. 2002).

Satellite droplets, sometimes called secondary droplets, were already detected in the experiments of Plateau and Rayleigh. In principle, satellite droplets occur in both, Newtonian fluids and viscoelastic fluids in the IC/VC-regime near the breakup point as a result of nonlinear capillary instabilities. At this point, the three timescales Rayleigh time, capillary time and relaxation time (for viscoelastic fluids), i.e., inertia, viscosity and viscoelasticity, overlap. The ratio of the timescales can be expressed more simply by the Ohnesorge number (Eq. (8)) and Deborah number (Eq. (16)).

Lafrance (1975) as well as Rutland and Jameson (1970) investigated the ratio of primary drop size to satellite droplet diameter as a function of the perturbation wavelength in water. The exemplary description according to Bousfield et al. (1986) agrees with the experimentally found results. Tjahjadi et at. (1992) then investigated the influence of viscosity on the formation of satellite droplets for Newtonian fluids (change in the Ohnesorge number). For small viscosities it was found that the formation of satellite droplets becomes a self-similar problem. Between the first-generation satellite droplets and the primary drops, sub-satellites are formed (multiple satellite droplets, see Fig. 12). With increasing viscosity initially only one satellite droplet is formed, which diameter decreases with increasing viscosity. For sufficiently high viscosities, no more satellite droplets are formed. Typically, the limit is Oh > 1. Recent investigations have mostly dealt with viscoelastic solutions (polymer solutions). Tirtaatmadja et al. (2006) should be mentioned, who phenomenologically investigated the formation of satellite droplets at constant viscosity and thus at a constant Oh. It was found that the diameter of the satellite droplets decreases with increasing viscoelasticity until the relaxation time outweighs the Rayleigh time and no satellite droplet is formed. Bhat et al. numerically investigated the dependencies of inertia, viscosity and viscoelasticity and were able to define regions of multiple satellite droplets as well as single satellite droplets as a function of De and Oh ((Bhat et al. 2010) p.629 fig. 5). The location and size of the satellite droplets cannot be calculated from the fluid properties up to now.

The beads-on-a-string structure was first mentioned and numerical examined by Chang et al. (1999). The structure only occurs for polymer solutions if the polymers are oriented and completely extended. During the stretching process, the polymer molecules form a crystalline like structure and a phase separation occurs ((Sattler et al. 2012) p.24 fig. 15). The resulting droplets consist primarily of the solvent, while a polymeric thread is formed in between. When the polymers are fully stretched, periodic instabilities form according to the Plateau-Rayleigh instability. The shape of the periodic perturbation can be described according to the Eq. (17) (compare Fig. 4). It describes the spatial and temporal development of the shape which corresponds well with shape of droplets at the beginning of the experiment. In Equation (17) h₀(t) is the diameter of the filament, s(t)z is the adaptation to conical shape, a(t) is the amplitude of the perturbation and Λ(t) is the wavelength of the perturbation. The amplitude of the perturbation can be calculated from empirical proportionality factor a₁ and the growth rate θ, which is a function of the surface tension γ (assumed to be constant), the diameter of the filament h₀ and the effective Newtonian elongational viscosity η_eff (Fontelos and Li 2004). Sattler et al. (2012) pointed out, that the effective viscosity cannot be derived from Eq. (14). The effective viscosity is lower than the calculated viscosity since the solution differs from the solution of the EC regime because of the phase separation. A numerical calculation of the effective viscosity is necessary as well as a fitting of the empirical proportionality factor a₁.

The droplets (BOAS) are also called first generation droplets. After the formation of these first-generation droplets, second and subsequent generation of droplets can be formed (cf. Fig. 12). Chang et al. (1999) set up a general equation to describe the formation (Eq. (20)), whereby they found a power law exponent m = 3/2 for the Generations N ≥ 2. Later Sattler et al. (2008) and Oliveira and McKinley (2005) found a power law exponent closer to m = 2. In Eq. (20), the diameter of the resulting droplet D_N is related to the diameter of the undisturbed filament D^*. The power law exponent m represents the relation between the generations. The smallest undisturbed filament had a diameter of 44 µm, which corresponds well with the inner length scale l_v of Eggers (Eggers 1997; Haenlein 1931) (Eq. (21)) what is in good agreement with the results of Eggers (39 µm) for the same PEO-Water solution. The diameter of the resulting first-generation droplets corresponds well with the calculations from the Plateau-Rayleigh instability (Eq. (5) and (17)), but differs from the results by Arnolds et al. (2010). Arnolds et al. found a narrow range between 5 µm and 10 µm for the diameter of the droplets over a broad range of the viscosity.

After the droplets have completely formed, the description according to Eq. (17) is no longer possible. A potential description of the shape is discussed in Fig. 10. In addition, the droplets can move along the polymer filament depending on their size and environmental conditions, e.g. gravity.

All experiments were performed in a temperature-controlled laboratory at 22°C ± 0.5°C. The shear rheological measurements were performed using an Anton Paar Physica MCR 501 rheometer with a CC27 coaxial system. Two different experiments were performed to characterize the shear rheological properties of the solution: constant shear rate at 7.76 s^-1 and logarithmical shear rate ramp between 1 s^-1 and 100 s^-1. All relevant results are displayed in Table 2 (columns 5 and 6).

The setup for the extensional rheological investigation was inspired by filament stretching devices (Matta and Tytus 1990; Sridhar et al. 1991; Tirtaatmadja and Sridhar 1993) as well as the CaBER (McKinley et al. 1999; McKinley and Sridhar 2002) and is similar to the setup of Tirtaatmadja et al (2006). It resembles a slow stretching process in the FEA. A polymer solution is dripped at low flow rates (1ml/min) from a nozzle with an inner diameter of 2 mm using a syringe pump (KD Scientific Gemini 88 from fisher scientific). A drop forms at the frontal face of the nozzle (see Fig. 6; hereinafter referred to as ‘primary drop’), which separates from the nozzle when a certain volume is reached. The release point is a function of density and surface tension. Depending on the viscous and viscoelastic properties, satellite droplets and/or a filament are formed which connects the two primary drops (one falling down and one at the nozzle). The nozzle could be moved to different heights for image acquisition as the camera (Fastcam SA-X2 Type 1080k-m3 from Photron) was fixed. For better contrast and less noise, a backlight source was used.

Polyethylene oxide with different molecular weights from Sigma Aldrich (M_w = 6⋅10⁵ g/mol; M_w = 1⋅10⁶ g/mol; M_w = 4⋅10⁶ g/mol) was investigated in this study. The PEO powder was dissolved in a mixture of deionized water and Propane-1,2,3-triol (glycerol) (w_%,water = 0,64; η_S = 2,958 mPa⋅s). The solution was homogenized with a magnetic stirrer for at least two days. Concentrations have been varied between 0.14 wt% and 2.23 wt% according to Table 1. The density was constant at 1090 kg/m³ ± 0,2%. The surface tension was about 60 mN/m for every solution which corresponds well with other publications (Arnolds et al. 2010; Bhat et al. 2010; Tirtaatmadja et al. 2006; Cooper-White et al. 2002).

The idea behind the choice of these solutions is to compare the three parameters—mass concentration Φ, dimensionless coil-overlap-concentration c_COC and shear viscosity η_Sh,0—for each molecular weight. From Table 1 it is obvious that the mass concentration and dimensionless coil overlap concentration are the same in the different solutions. A shear viscosity of 0.1 Pas is prepared in different dilution levels from the stock solution, which is the highest concentration for each molecular weight. The relevant results of the shear rheological measurements are shown in Table 2.

To investigate the filament thinning behavior, the filament diameter was measured at a fixed position over time. The Reynolds and Weber numbers are so low that Rayleigh breakup occurs (Andrade et al. 2012; Lefebvre 1989). The filament was assumed to be circular (a quite good assumption), and its diameter was calculated directly from the images. For a better comparison, the area A(t) is normalized with the initial area (frontal face of the nozzle). The error resulting from the camera sensors pixels reaches 10% for D = 50 µm or a normalized area (A(t)/A₀) of 10^-3. An exemplary progression is shown in Fig. 7.

The qualitative temporal progression of the normalized area reflects the previously shown progressions in Fig. 5. For further evaluation, special attention is paid to the EC regime, since the relaxation time can be calculated directly from the strain rate via the constant Weissenberg number (combination of Eqs. (2), (11) and (13)).

The relaxation time is compared later with theoretical models (Table 3) and a mass concentration dependent equation of the relaxation time for each molecular mass was created (Eq. (29)). For the model presented subsequently (see “Modeling of the filament behavior and the droplet formation in the Filament Extension Atomizer”), the relaxation time plays an important role for the filament behavior during filament extension atomization and is significantly responsible for the maximum angular velocity of the FEA-cylinders (cf. Eq. (34)).

It is obvious (cf. Table 2) that the relaxation time increases with increasing concentration (constant molecular weight) and increasing molecular weight (constant concentration). This was expected since increasing concentration and/or increasing molecular weight raises the viscosity and the relaxation time coupled to the viscosity as well as the interactions of the polymers. It must be noted that the relaxation time at constant viscosity (η_Sh ≈ 0.1 Pa s) increases when the concentrations decrease with increasing molecular weight. This can be ascribed to the viscoelasticity and strain hardening effects which originate from the long chain branching of polymers with higher molecular weight.

The results also show that the relaxation time in the semi-diluted regime differs from the Zimm relaxation time (compare Tables 1 and 2) because the effective relaxation time depends on the concentration. An adjustment of the Zimm relaxation time described in Eq. (12) is necessary and leads to Eq. (22) (Tirtaatmadja et al. 2006).

However, Eq. (22) is only valid for diluted solutions. At higher concentrations, the viscosity of the solution and thus the relaxation time is strongly underestimated. A possible solution for this is the combination of the findings of Arnolds et al. ((2010) p.1212 fig. 1) and Tirtaatmadja (Tirtaatmadja et al. 2006). Arnolds et al. calculated the slope of zero-shear viscosity as a function of c_COC in the concentration domains (diluted, semi-diluted, entangled) of polymer solutions. Eq. (22) is extended by a further term for the semi-diluted regime (Eq. (23)). The exponents x_d (diluted solution) and x_s (semi-diluted solution) correspond to the power law of the slope of zero-shear viscosity with increasing concentration ((Arnolds et al. 2010) p.1212 fig. 1). c_COC,d represents the transition from diluted to semi-diluted solutions. Eq. (23) can be simplified to Eq. (24) with \({\mathrm{c}}_{\mathrm{COC},\mathrm{d}}=1\) as transition from a diluted to a semi-diluted system (see section ‘Polymer Solution’). This results in Eq. (25). Another possibility is the incorporation of real measured quantities like the zero-shear viscosity or the shear viscosity at \(\upeta (\dot{\upgamma }=\dot{\upepsilon })\) (cf. Eqs. (26) and (27)). All approaches (Eqs. 25-27) formally correspond to the power law approach for determining the strain relaxation time according to Rubinstein and Colby (Eq. (28)) (Rubinstein 2010).

All calculated relaxation times (cf. Table 3) are in the same order of magnitude as the effective relaxation times of the experiments. Irrespective of this, the model-based relaxation times show striking differences to reality, so that a complete theoretical prediction of the relaxation time is not possible at this stage, because the nonlinear strain hardening of the polymer molecules is overestimated. The power law of the approaches (Eqs. (25), (26) and (27)) is greater than in reality, and therefore an adjustment is necessary. Since a model-based determination of the relaxation time does not yield sufficient results, the relaxation time is calculated using the empirical approach of Rubinstein and Colby (Eq. (28)) for the later presented model. The fitting parameter ψ is used to fit the equation to experimental data. It results in Eq. (29).

From the recorded data of the shear and extensional rheological investigations, the Oh and De numbers can be calculated. In all experiments, with the exception of the lowest concentrations of \({\mathrm{M}}_{\mathrm{W}}=6\cdot {10}^{5}\mathrm{g}/\mathrm{mol}\) and \({\mathrm{M}}_{\mathrm{W}}=4\cdot {10}^{6}\mathrm{g}/\mathrm{mol}\), the Ohnesorge number is in the range of 0.1 to 1.134 and therefore they are in the ‘inertial-viscous’ regime (Eggers 1997). Regarding the Ohnesorge number, only at certain Deborah numbers the formation of satellite droplets would be expected. The requirements are only met at \({\mathrm{M}}_{\mathrm{W}}=6\cdot {10}^{5}\mathrm{g}/\mathrm{mol}\) (Fig. 8).

In most cases, droplet formation cannot be observed directly at the capillary outlet. For the observation, the imaging area is shifted downwards approx. 5 cm (Fig. 9).

According to the literature, two possible shapes occur during drop formation. Equation (17) and the simplification (30) explain the spatial and temporal development of the shape which corresponds well with the experimental findings, especially at beginning of the experiment.

The simplification is possible because the images clearly reveal that the filament is in fact cylindrical and not conical. Therefore, the adaptation to a conical shape is negligible (s(t)z = 0). With the assumption of rotational symmetry, the integration over the length z is possible and the volume (\({V}_{Sattler}=\pi \cdot {l}_{d}\cdot \frac{{h}_{0}^{2}+4\cdot {a}_{0}\cdot {h}_{0}+6\cdot {a}_{0}^{2}}{4}\); with \({a}_{0}=\frac{1}{2}\cdot \left(\frac{{w}_{d}}{2}-\frac{{h}_{0}}{2}\right)\); cf. Fig. 10) as well as the equivalent diameter of the droplets can be calculated. Unfortunately, it was not possible to investigate the influence of time with the used setup.

The computed shape corresponds quite well to the real shape of the droplets, especially at the beginning of the droplet formation process. At the end of the droplet formation the shape resembles more an ellipsoid (cf. Fig. 10: B1; \({V}_{elli}=\frac{4}{3}\cdot \pi \cdot {l}_{d}\cdot {w}_{d}^{2}\)). Regardless of the shape, the volume can be calculated in both cases. The difference, and thus the error, is at most 21.1% of the volume and thus less than 10% of the equivalent diameter. Based on this evaluation, the equivalent diameters of the resulting droplets can be calculated (cf. Table 4).

In general, the diameter of the droplets and the stable filament diameter decrease with increasing polymer concentration and increasing molecular mass, but the dependence of the molecular mass predominates. The quotient of the two diameters remains constant within small deviations. Based on these results, it is not possible to derive a functional relationship between the concentration and the droplet size or between the concentration and the stable filament diameter. Especially for the stable filament diameters of the molecular mass of 4000000 g/mol, the accuracy is already not better than about 20% of the measured value. Regarding \(\frac{\Lambda }{dsF}\), the spectrum of this ratio is the same as that found by Sattler et al. using a CaBER (4.5 < Λ/d_sF < 6.5) (Sattler et al. 2012). The lower limit (Λ/d_sF = 4.5) is connected to the Plateau-Rayleigh instability. This is surprising, since the Plateau-Rayleigh instability refers to Newtonian fluids and still seems to be applicable to viscoelastic fluids while being consistent with the results of other publications (Christanti and Walker 2001; Sattler et al. 2012). Although the size of the droplets is well matched, significantly fewer droplets are formed than expected. Usually the Plateau-Rayleigh instability infers regular, harmonically oscillating instabilities out of which many droplets should be formed. Referring to the calculated volume of the filament, approx. 100 droplets per filament should thus be produced which is far from the gained results. A solution to this deviation might be formation of individual instabilities that lead to a stabilization of the filament and to a hindrance of further instabilities. A similar behavior could also be shown numerically by Li and Fontelos ((2003) cf. Figs. 3, 13, and 14) and Mead-Hunter and King (2012). Additionally, some droplets are formed below the image area, which are not included in the evaluation and thus an exact absolute number cannot be given. Nevertheless, this has been checked qualitatively and the absolute number is still lower than expected. Another mechanism reducing the droplet number is coalescence (Fig. 11).

High velocity differences occur between single droplets when they were formed at different time steps. The polymer molecules are more or less immobilized in the center at high elongations (interactions with other polymer molecules and resulting strain hardening), but the solvent is not. When the instabilities develop, polymer molecules can therefore no longer pass into the droplets and remain in the filament. The droplets thus have a different composition and, in the case of PEO-water-glycerol mixtures, a significantly lower viscosity. Gier and Wagner (2012) performed Particle Image Velocimetry measurements in filaments and was able to show that relative movements occur. Combining this and the instability formation at different time steps, relative movements between the droplets and the filament as well as relative movements between droplets occur.

In addition to the droplets formed in the EC regime, another beads-on-a-string structure is formed later in the phase of final stretching of the polymer molecules. In the final phase, there is an absolute phase separation and the formation of droplets that are connected by a polymer filament. From the optical evaluation of the experiments, droplet sizes of up to approx. 30 µm could be determined (Fig. 12), which as a whole structure resembles the beads-on-a-string structures found in the literature before ((Sattler et al. 2012)) and could be reproduced in all experiments. An exact size determination was not possible due to the image quality. No dependence on concentration and molecular mass was found. This result is consistent with the results of Arnolds et al. (2010), who only found a minimal dependence on the dimensionless concentration c_COC. However, In contrast to the experimental results of this publication and the results of Sattler et al., the particle size of Arnolds et al. (2010) is between 5 µm and 10 µm.

Figure 9 shows exemplarily that the diameter of the filament is independent of the location. This could be shown by the superposition of the two curves and Fig. 10 as well as Fig. 11, excluding the droplets. In addition, there are a slight increase of the primary drop velocity at a second location with increasing concentration and molecular mass, which is not shown or discussed within this publication, but corresponds well with literature (Ambravaneswaran et al. 2002), and an increase of the filament volume with rising viscosity of the solution (Table 5). Likewise, the volume fraction of the filament to the total volume (primary drop and filament) increases. The total volume is almost constant (\(35.5 \mu l\pm 2 \mu l\)) regardless of the viscosity and molecular mass, but should theoretically decrease slightly with increasing concentration of polymer (equilibrium of forces at the nozzle). Tirtaatmadja et al (2006) was able to show that the surface tension is constant regardless of concentration and molecular weight. The density of the polymer solution, however, increases slightly from 1.089 g/cm³ (Φ = 0; water-glycerol mixture) to 1.093 g/cm³ (Φ = 2.23; maximum mass fraction PEO). For the volume fraction of the filament, similar to the relaxation time, a concentration-dependent empirical equation is derived Eq. (31), with Y and Z as empirical fitting parameters). The volume fraction was derived from the filament diameter in the second observing location while the total volume was known from the syringe pump.

The developed model is very complex and can be described as a system of eight equations what will be explained in the following paragraphs. Therefore, new formulaic correlations are developed and known formulaic correlations are adapted and combined in a new way. The system will be presented in three steps focusing on the filament formation, the droplet formation and the description of the extensional rate (Fig. 13).

As a starting point three main aspects have to be considered. First, a filament must be formed, which is stretched sufficiently until instabilities arise. From these instabilities, drops must form and at the same time the filament must not break before. Therefore, the strain rate times the relaxation time is limited to 2/3 in the EC-regime as shown above (Fig. 7).

It was shown in the experimental section that a large part of the solutions volume is bound in the primary drops (Figs. 7 and 8). Approximately these primary drops form a hemispherical shape on both rotating cylinders during filament stretching. To determine the filament volume at the beginning of the stretching process, it is assumed to be of cylindrical shape (volume of the primary droplets and the filament, Fig. 14 (1)). The total volume is therefore defined as the gap distance L₀ and the moistened surface of the cylinders (\({V}_{\mathrm{total}}={\mathrm{A}}_{0}\cdot {\mathrm{L}}_{0}=\frac{\uppi }{4}\cdot {\mathrm{D}}_{0}^{2}\cdot {\mathrm{L}}_{0}\)). In addition to Eq. (31), the zero length of the filament can be calculated as a function of the mass concentration (Eq. (32)).

The maximum length of the filament is limited by the radius of the FEA cylinders R_FEA, the gap distance L₀ and the volume fraction of the primary drops. According to the conducted experiments they will be handled as spherical droplets with the radius R₀. It is important that perturbations occur before the maximum length is reached, so that the formation of defined, spherical droplets is possible over time.

Under the premise of a cylindrical filament and a Gaussian distributed wavelength of the instabilities (Table 4), Eq. (30) is used for the temporal description of droplet formation. The amplitude of the instability grows exponentially with time (Eq. (18)), while the growth rate depends on the surface tension, the effective viscosity and the stable filament diameter (d_sF; cf. Table 4). As the predictive power of the Eq. (21) could not be confirmed on the basis of the experimental results, only experimental data of stable filament diameters are used as model input parameters. Furthermore, no growth rates of the droplets could be determined due to the static observation setup. However, a numerical theoretical determination is possible with a sufficient amount of measurement data. From Sattler (2012), various parameters of the growth can be identified and estimated. For example, the effective viscosity of the stable filament is assumed to be 5 Pas (cf. p.24 Table II) and the pre-factor a₁ of the growth rate is calculated as 1 µm. According to Eq. (19), this results in an exponential factor θ = 28.7 s^-1. The formation time of the droplets can be calculated from Eq. (19) with the given quantities (Eq. (33)).

In Eq. (18), the maximum amplitude a_max(t) is limited by the filament radius (h₀/2). The shape of the droplet can no longer be described by the Eq. (30). The result is a rounding of the droplets which is elliptical at first and spherical at the end.

It is assumed that exactly one droplet is created from one wavelength of the instability and that this droplet does not coalesce with other created droplets (which is more suitable for higher viscosities). The volume of the unstable section can be calculated from the theoretical consideration of Rayleigh (Λ/d_sF = 4.5) and the experimentally found ratio (Λ/d_sF = 4.5-6.5; Table 4 and (Christanti and Walker 2001)) as the maximum growth rate. From the maximum growth rate results a normal distribution of further instabilities, which for example in the case of a molecular mass of 1000000 g/mol showed a standard deviation of about 30 µm (Table 4). It has to be noted, however, that the standard deviation should correctly be concentration-dependent. According to other researchers it is assumed that the resulting polymer thread is infinitesimally small (cf. (Sattler et al. 2012)). This is based on the fact that the volume fraction of the polymer corresponds approximately to the mass fraction.

The strain rate profile is essential for the stretching process in the FEA and is given by Eq. (34) with l₀ as the zero length as a function of the volume fraction and therefore a function of the concentration (dependency is not shown in the equation), the radius of the FEA cylinders R_FEA and the angular velocity ω. It can be shown that the maximum strain rate increases with increasing angular velocity, decreasing zero length and increasing radius (see Fig. 15). To justify the assumption of Rayleigh decay, the maximum strain rate of the FEA (Eq. (35)) has to be set according the EC regime for the given solution. Via exemplary calculation of the strain relaxation time it is possible to derive the maximum strain rate for each solution. The point of the maximum strain rate is mathematically dependent on the angular velocity, the effective zero length, the radius of the FEA and the rotation velocity n (Eq. (36)). Substituting Eq. (36) into Eq. (34), for example, results in a maximum strain rate depending on the relaxation time of the polymers and the design parameters R_FEA and l₀ (Eq. (37)). The time and angular velocity dependent maximum strain rates are shown in Fig. 16. The process time t_pro can be calculated directly from the angular velocity (Eq. (38)).

The time dependent strain rate of the FEA is shown in Fig. 15. However, as mentioned before, the strain rate in the EC regime is constant and adjusts itself according to the relaxation time of the solutes. There is a decrease in the filament volume according to Eq. (39), whereas the temporal filament length is given by ’\(2\cdot {\mathrm{R}}_{\mathrm{FEA}}\cdot \left(1-\mathrm{cos}\left(\upomega \cdot \mathrm{t}\right)\right)+{\mathrm{l}}_{0}\)’ and the temporal cross-section is given by ‘\({A}_{0}\cdot exp(-{\dot{\varepsilon }}_{real}\cdot t)\)’. The initial area A₀ depends on the IC-/VC-EC-transition (Fig. 17 and 18; Table 6).

This Eq. (39) does not apply to the IC/VC regime. Even though the strain rate in the IC/VC regime is higher than in the EC regime it is acceptable to assume an EC regime – decrease of the diameter over both. In the course of this assumption the equation can be further simplified. Furthermore, l₀ can be substituted by the function of the volume fraction of the filament (Eq. (32)). The real strain rate can be substituted by the Weissenberg number and the relaxation time, which is a function of the concentration. A₀ can be substituted by the radius to implement the effect of a perforated surface (Eq. (40)).

At a time t_inst smaller than the maximum processing time of a single filament (cf. Eq. (42)), the diameter of the filament must be thinned enough to develop instabilities (see Table 4).