The spatial structure of jets from steady Taylor cone-jets

Abstract

This work presents a comprehensive analysis of steady cone-jet electrospray (SCJ-ES) that captures the full range of its steady jet scales within the Taylor-cone electric field. We identify three fundamental regions, each governed by distinct scaling laws and dominant physical mechanisms: (i) the transition region, characterised by the balances that fix the emitted current; (ii) the charge convection-dominated region, where surface charge transport dominates total charge transport and the Taylor field drives jet acceleration; and (iii) the ballistic region, where the jet attains a fixed cylindrical scale before undergoing Rayleigh breakup into charged droplets. This refined theoretical framework harmonises existing models, particularly those using the Taylor–Melcher leaky dielectric model as an electrokinetic approximation for SCJ-ES. Notably, our newly proposed spatial scales achieve a remarkable collapse of published experimental SCJ-ES jet profiles. We also apply this framework to study the charge of resulting droplets using extensive literature data, observing significant differences between weak and strong electrolytes, consistent with recent findings.

1. Introduction

Among the different working modes of electrospray (ES) (Zeleny 1917; Cloupeau & Prunet-Foch 1994; Jaworek & Krupa 1999a ), the so-called steady cone-jet mode (SCJ-ES) can be regarded as a form of complex lensing of liquid-phase matter: in addition to its function in reducing matter to micrometre or nanometre scales (that is, reducing matter to micro- and nanodroplets), SCJ-ES controllably provides an electrical charge close to the physical limit load per unit volume (Rayleigh 1881). This feature makes SCJ-ES an exceptional electromechanical converter from electrical potential energy into surface and kinetic energy involving an amazing combination of electrohydrodynamic, electrokinetic and electrochemical processes, and endows it with optimal features for widely known applications (Dole et al. 1968; Yamashita & Fenn 1984a , b ; Whitehouse et al. 1985; Fenn et al. 1989; Wilm & Mann 1996; Takáts et al. 2004; Jaworek 2007, 2008; Zhang et al. 2021; Kim & Velásquez-García 2025). The physical richness of the phenomenon has made SCJ-ES a subject of in-depth analysis (a few scattered works are Zeleny 1914, 1917; Smith 1986; Hayati, Bailey & Tadros 1986, 1987b , a ; de la Mora 1992; Fernandez de la Mora & Loscertales 1994; Gañán-Calvo 1997a ; Gamero-Castaño & de la Mora 2000).

The approaches to this subject can be considered to belong to a diffuse zone between electrostatics and electrohydrodynamics, attributed to the inherent presence of interdependent, yet disparate-dimensional scales of different nature, from the size of the emitter to the diameter of the jet and the thickness of diffuse charge layers. It encompasses a domain that can be classified as nearly electrostatic, characterised by the quasi-conical meniscus, and another zone of much smaller scale, characterised by its quasi-unidirectional motion at high velocity: the microjet emerging from the cone apex (Fernandez de la Mora & Loscertales 1994; Gañán-Calvo 1997b ; Jaworek & Krupa 1999b ; Gamero-Castaño & Hruby 2001, 2002). The latter is responsible for the steady ejection of the characteristic highly electrically charged aerosol of electrospray, such that the conversion from the available energy potential for this process (a voltage difference between the liquid meniscus and a counterelectrode) into surface energy is among the highest in nature. In contrast, the much larger-scale quasi-electrostatic conical meniscus encapsulates an underlying purely electrostatic balance between surface forces due to electric charges and surface tension. This balance is ultimately responsible for the focusing action of the SCJ-ES. The solution to this balance, discovered by Taylor (Taylor 1964; de la Mora 1992; Pantano, Gañán-Calvo & Barrero 1994; Gañán-Calvo 1997a ), is physically unattainable without disrupting the conical tip on a certain small local scale fundamentally determined by the liquid properties (Rosell-Llompart & de la Mora 1994; Fernandez de la Mora & Loscertales 1994; Gañán-Calvo et al. 1993, 1997; Gañán-Calvo 1997a , 1999b ; Gamero-Castaño & Hruby 2001, 2002; Gañán-Calvo 2004; Fernández de la Mora 2007; Jaworek & Sobczyk 2008; Rosell-Llompart, Grifoll & Loscertales 2018; Borra 2018; Gañán-Calvo et al. 2018). However, in contrast to other related unsteady ejection processes, the disruption leading to the emergence and subsequent breakup of a steady jet in SCJ-ES is governed not by an abrupt, catastrophic instability, but by a finely tuned, continuous progression of force equilibria. In this regime, a complex interplay of capillary forces, viscous stresses, inertial effects, electric forces and charge diffusion occurs, resulting in a delicate balance that undergoes gradual shifts as the system evolves (Gañán-Calvo 1999b ; Gañán-Calvo et al. 2018).

The complex balance mediating the transition from a stable quasi-electrostatic meniscus terminated in a conical tip to a dynamic jet has been shown to span a remarkable spectrum of physics, drawing upon classical fluid dynamics of free liquid surfaces (Eggers 1993, 1997) and electrodynamics (e.g. electrohydrodynamics) (Taylor 1966; Melcher & Taylor 1969; Melcher & Warren 1971; Saville 1997), while also sharing the complexities of interfacial chemistry encountered in natural systems (electrokinetics and electrochemistry) (Bazant et al. 2004, 2011; López-Herrera et al. 2015; Schnitzer & Yariv 2015; McEldrew et al. 2018; López-Herrera et al. 2023). The existing wealth of studies underscores that SCJ-ES is not merely a fluid dynamical curiosity but a microcosm of multifaceted physical interactions that continues to inspire both theoretical inquiry and technical innovation in areas ranging from mass spectrometry of complex biomolecules (Dole et al. 1968; Yamashita & Fenn 1984a , b ; Whitehouse et al. 1985; Fenn et al. 1989; Wilm & Mann 1996) and drug discovery (Koehn & Carter 2005; Cooks et al. 2006) to nanoengineering (Loscertales et al. 2002; Jaworek & Sobczyk 2008).

Although some outlier observations may not be fully consistent with our current understanding and models, the occurrence of the SCJ-ES phenomenon and its general logic is a physical puzzle that has been gradually solved. However, a comprehensive revision of the variety of physical scales and a general spatial structure of the jet is still pending, in particular when the rather general conditions of the leaky dielectric model (LDM) are applicable to the SCJ-ES. To accomplish this, a detailed revision of the certainties already achieved in the description of the SCJ-ES will be helpful.

Figure 1. Intermediate scale found in a Taylor SCJ-ES and sketch of the cone-jet transition region, here indicated by a characteristic length $R$ (assimilated to the radial coordinate $R$ of a spherical coordinates system centred at the apex of an underlying virtual Taylor cone). The variables used in the analysis are indicated $I$ is the emitted electric current, $\Delta V$ is the characteristic voltage decay along the jet, $K$ is the liquid electrical conductivity, $\sigma$ the surface tension, $\varepsilon_o$ the permittivity of vacuum, and $E_n$ and $E_s$ the normal and tangential electric fields on the jet surface, respectively. The inner diameter of the cone-ended emitter capillary is 0.5 mm (the emitter begins at the less glossy part of the cone).

1.1. The local solution at the tip, at intermediate scale

For introductory purposes, in this work we assume the existence of an intermediate scale (figure 1) between the jet diameter and the size of the emitter such that Taylor’s solution of the electric potential $\phi$ outside the cone, given by

(1.1) \begin{equation} \varPhi _T(R,\theta )=A_T R^{1/2} {Q}_{1/2}(\theta ) \end{equation}

can be assumed as a first local approximation to the general solution of the cone-jet in spherical polar coordinates $\{R,\theta \}$ centred at the tip of the cone, where $\theta$ is measured from its inner axis, $R$ has units of length, ${Q}_{1/2}(\theta )$ is the second-kind Legendre function of order 1/2 (usually expressed as ${Q}_{1/2}(x)={Q}_{1/2}(\cos (\theta ))$ as well), and the constant $A_T$ must be

(1.2) \begin{equation} A_T=A \left (\frac {\sigma }{\varepsilon _o}\right )^{1/2},\quad A=\frac {-2^{3/2} \sin (\alpha _T)}{3 \sqrt {\tan (\alpha _T)} {Q}_{\frac {3}{2}}(\cos (\alpha _T))}=0.856844\ldots , \end{equation}

where $\alpha _T$ is the well-known Taylor angle $\alpha _T=49.29\ldots ^\circ$ , corresponding to the zero of ${Q}_{1/2} (\cos (\theta ) )$ , $\sigma$ is the liquid surface tension, $\varepsilon _o$ is the permittivity of a vacuum, and $Q_{\frac{3}{2}}(\cos(\alpha_T))$ is the second-kind Legendre function of order 3/2.

Taylor’s electric field on the outer axis of the cone is

(1.3) \begin{equation} \{E_\theta,E_R \} = \left\{\frac {\pi A}{4} \left (\frac {\sigma }{\varepsilon _o}\right )^{1/2} R^{-1/2},0 \right \}. \end{equation}

The self-similarity of Taylor’s solution is altered by the liquid emission from the tip of the cone. Under stable and steady conditions with a constant emitted flow rate $Q$ , at the spatial scale of the jet prior to breakup (jet length) the jet occupies a marginal region of space surrounding the outer axis of the cone, such that the jet radius $R_J$ is

(1.4) \begin{equation} R_J=\sin \left (\theta (R)\right ) R = \sin \left (\pi - \epsilon _J(R) \right ) R \cong \epsilon _J(R) R, \end{equation}

with

(1.5) \begin{equation} \epsilon_{J} (R)\ll 1 \Rightarrow R_J \ll R, \end{equation}

where $\epsilon_J(R)$ is the small angle defined by the jet surface, consistent with the intrinsic geometric slenderness condition of the jet, which means that for $R_J \ll R$ to be true along distances of the order of $R$ , $R_J$ must have a sufficiently weak dependence on $R$ . This requires an estimation of the scale of $R_J$ (and the subsequent perturbation of the Taylor solution) from a slender model of the jet. It should be noted that this scale, $R_G$ , may not be equivalent to the local scale of the non-slender transition from the cone to the jet. However, this study determines the power-law dependencies of each region of the structured jet, particularly the larger power-law slope of the jet profile near the transition. The latter provides the precise form of how the jet transitions from the cone.

1.2. The energy balance for the jet’s scale

Assuming that the jet is sufficiently thin to facilitate a nearly perfect viscous diffusion of momentum from the stress exerted on the charged surface by the applied electric field $E_s$ , a basic mass balance relating the radius of the jet, denoted by $R_J$ , with its local velocity $v$ , leads to

(1.6) \begin{equation} Q=\pi R_J^2 v. \end{equation}

One of the fundamental consequences of the total viscous diffusion of momentum across the jet, even for relatively low-viscosity liquid and if the jet is thin, is that the kinetic energy of the liquid eventually emerges as the predominant mechanical product of the driving force applied on the jet surface. In this case, the flux of mechanical energy per unit time, or power, $W_m$ anywhere along the jet is as follows:

(1.7) \begin{equation} W_m=\frac {\pi }{2}\rho v^3 R_J^2, \end{equation}

where $\rho$ is the liquid density. The hypotheses that result in the relatively trivial outcomes (1.6) and (1.7) are not inconsequential. In general, to ensure that the applied surface stress is sufficiently strong and robust to sustain the extremely fast steady jets observed in this phenomenon, it is necessary for the jet surface to be highly charged, albeit at levels below the Rayleigh limit corresponding to a cylinder. Consequently, the charge diffusion should be present along the jet, from its inception at the cone-jet transition region. Thus, the total power $W_J$ made along the jet by the electric field on the emitted charges, part of which is the Joule effect (Gamero-Castaño & Hruby 2002; Gañán-Calvo et al. 2006; Gamero-Castaño 2010), must be given by

(1.8) \begin{equation} W_J=I\Delta \varPhi, \end{equation}

where $I$ is the steady electric current issued by the jet, and $\Delta \varPhi$ is the voltage decay at distances $R$ along the jet from the cone-jet transition. In this regard, Misra, Magnani & Gamero-Castaño (2025) have recently demonstrated that a steady Taylor cone-jet can be realised without the electrical relaxation time ( $\varepsilon \varepsilon_o /K$ ) being small compared with the residence time $R_J^3/Q$ , at least while an intermediate region like that assumed in this work exists ( $\varepsilon$ is the electrical permittivity of the liquid).

From (1.3), one has $\Delta \varPhi \cong ({\sigma }/{\varepsilon _o} )^{1/2} R^{1/2}$ . This is true as long as $R$ is sufficiently large compared with the local jet radius. On the other hand, the electric current $I$ must be $I\sim K R_J^2 E_s$ , at least along a certain range of $R$ up to the point where both charge conduction and convection become comparable. Assuming that the Ohmic dissipation is never dominant, the fluxes $W_m$ and $W_J$ must be comparable. This can be shown using the scaling of the Ohmic dissipation obtained by Magnani & Gamero-Castaño (2024):

(1.9) \begin{equation} W_\varOmega =\frac {7.74}{\delta _\mu ^{0.11}} W_o (Q/Q_o)^{0.89}, \end{equation}

where $Q_o= ({\sigma \varepsilon _o}/({\rho K}))$ and $W_o = ({\sigma ^5 \varepsilon _o}/{(\rho ^2 K)} )^{1/3}$ are the reference flow rate and power values, respectively, and $\delta _\mu$ is the electrohydrodynamic Reynolds number,

(1.10) \begin{equation} \delta _\mu =\left (\frac {\rho \varepsilon _o \sigma ^2}{\mu ^3 K}\right )^{1/3}, \end{equation}

where $\mu$ is the liquid viscosity. If $W_\varOmega$ is comparable to $W_J$ or smaller, from ((1.6)–(1.8)) one has (Gañán-Calvo 1997a ; Gañán-Calvo & Montanero 2009; Gañán-Calvo et al. 2018):

(1.11) \begin{eqnarray} \rho \frac {Q^3}{R_J^4} \sim K R_J^2 \left (\frac {\sigma }{\varepsilon _o R}\right )^{1/2} \left (\frac {\sigma R}{\varepsilon _o}\right )^{1/2} & = \frac {K \sigma R_J^2 }{\varepsilon _o} \Longrightarrow \nonumber \\ R_J \sim R_G &\equiv \left (\frac {\rho \varepsilon _o Q^3}{\sigma K}\right )^{1/6}, \end{eqnarray}

which, defining $d_o= ({\sigma \varepsilon _o^2}/({\rho K^2}) )^{1/3}$ as the reference length (Gañán-Calvo et al. 1997; Gañán-Calvo 1999b , 2004), one has $R_G=d_o (Q/Q_o )^{1/2}$ .

In (1.11), recall that $R$ refers to any length scale along the axis of the Taylor cone, which nicely cancels out and makes the balance independent of that scale. Remarkably, from (1.6), the jet’s velocity scales as

(1.12) \begin{equation} v\sim Q R_G^{-2}=\left (\frac {\sigma K}{\rho \varepsilon _o}\right )^{1/3}\equiv v_o. \end{equation}

This is the universal electrohydrodynamic velocity that determines the self-similar collapse of a meniscus into a conical shape, too (Gañán-Calvo et al. 2016a ).

1.2.1. The Ohmic dissipation is not larger than mechanical power

To verify the assumption of $W_\varOmega \sim W_J$ or smaller, using (1.11) and (1.12) and including numeric prefactors, one has

(1.13) \begin{equation} W_m = \frac {1}{2}\rho v^3 R_J^2 = \frac {1}{2\pi ^2} \rho Q^3 R_J^{-4} = \frac {(6 \lambda )^{4/3}}{2\pi ^2 k_d^4} W_o (Q/Q_o) \simeq 8.8 W_o (Q/Q_o), \end{equation}

where $\lambda \simeq 8$ is the breakup wavelength of the jet into droplets, and according to Gañán-Calvo & Montanero (2009); Gañán-Calvo et al. (2018), the diameter of the droplets is $d_g=k_d (Q/Q_o)^{1/2} d_o$ with $k_d \simeq 1$ . Hence, according to (1.9), one has

(1.14) \begin{equation} \frac {W_\varOmega }{W_m}=\eta \left (\frac {Q_o}{\delta _\mu Q}\right )^{0.11}, \end{equation}

where $\eta ={7.74(2\pi ^2)}/{(6 \lambda )^{4/3}}$ varies between 0.75 for small flow rates, with $\lambda \simeq 9$ (see Cloupeau & Prunet-Foch 1989) and 0.88 for large flow rates, with $\lambda \simeq 8$ . The latter is obtained making the conservative assumption that the charge of the jet portion of length $\lambda R_J$ has the same charge as the droplet of diameter $d_g$ , and both are at the Rayleigh limit. Consequently, unless $\delta _\mu$ is smaller than 0.07, not only $W_\varOmega$ is comparable to $W_m$ for $(Q/Q_o)\sim O(1)$ , but also the former can be smaller than the later for large flow rates compared with $Q_o$ , which justifies the assumption.

1.3. Motivation of the analysis

When problems involving multiple stages along a dominant dimension – such as gravitational jets or electrospray jets – are analysed globally with a single axial scale, as in Gañán-Calvo (1999b , 2004), the distinct sequence of regions can be obscured because the governing mechanisms weigh differently at different axial positions. Experimental electrospray profiles may indeed be collapsed under a global axial scale if the radial scale ( $R_G$ ) is correctly captured (Gañán-Calvo 1999b , 2004), but only under restrictive conditions: (i) the two electric transport mechanisms must be of comparable order, and (ii) both the normal and tangential electric fields, $E_n$ and $E_s$ , must vary uniformly along that scale. This explains the partial success of the scaling collapse in Gañán-Calvo (1999b ), where figure 2 reveals only a limited axial range over which these assumptions hold. Later, Gamero-Castaño and coworkers Gamero-Castaño (2010); Gamero-Castaño & Magnani (2019) demonstrated that the characteristic length $R_G$ provides the correct axial scaling of the jet, too, contradicting the alternative scaling proposed in Gañán-Calvo (1999b , 2004). A structured, region-by-region analysis along the jet axis therefore remained necessary.

A primary finding of this study is that the non-uniform scaling of conduction and convection currents, and of the electric fields $E_n$ and $E_s$ , can be significantly obscured – or effectively ‘averaged out’ – when a global axial scale is used, while $R_G$ emerges as the correct length for both axial and radial scales of the jet, consistent with the arguments of Gamero-Castaño. This work reveals a three-region structure with distinct power-law dependencies and field scalings: (i) a post cone-jet transition region where conduction and convection are comparable and the electric current is fixed, (ii) a subsequent acceleration region governed by the axial decay of the field from the conical meniscus and (iii) a ballistic region where the jet radius becomes constant, capillary instabilities develop, and breakup occurs. For longer jets that extend beyond the influence of the Taylor cone into the weaker emitter field, a residual acceleration of the cylindrical jet may be discerned, as observed numerically by López-Herrera et al. (2023) among others. Here, however, we restrict attention to jets short enough to remain entirely under Taylor’s field (figure 1), the regime most relevant to electrospray ionisation mass spectrometry (ESI-MS) and space micro-thrusters.

2. A slender jet model of the SCJ-ES

A classical slender model for the local average jet momentum can be written as (Eggers 1993; Gañán-Calvo 1997b , 1999b ; Gamero-Castaño & Hruby 2002)

(2.1) \begin{eqnarray} \frac {d}{{\rm d}R}\left (\frac {\sigma }{R_J}+\frac {1}{2\pi ^2}\frac {\rho Q^2}{R_J^4} -\frac {\varepsilon _o}{2} \big (E_n^2+(\varepsilon -1)E_s^2\big )\right )\nonumber \\ = \frac {2 \varepsilon _o E_s E_n}{R_J}-\frac {6 \mu Q}{\pi R_J^2}\frac {d}{{\rm d}R}\left (\frac {R^{\prime}_{\!J}}{R_J}\right ), \end{eqnarray}

where $\varepsilon$ is the relative permittivity of the liquid, and $R^{\prime}_{\!J}={\rm d}R_J/{\rm d}R$ , together with the charge continuity equation, Gañán-Calvo (1997a , 1999b ),

(2.2) \begin{equation} I=\pi \textit{KR}_J^2 E_s + 2 \sigma _s \textit{QR}_J^{-1}, \end{equation}

where $I$ is the total emitted electric current, and $\sigma _s\cong \varepsilon _o E_n$ is the surface charge on the jet.

To seek for an analytical approximate solution to (2.1), we can now define a generic electric potential in the presence of the jet as

(2.3) \begin{equation} \varPhi =\left (\frac {\sigma R_G}{\varepsilon _o}\right )^{1/2}\Big (A\, (R/R_G)^{1/2} Q_{\frac {1}{2}}(\theta )+(R/R_G)^{\alpha } \left (B_1 Q_{\alpha }(\theta )-B_2 P_{\alpha }(\theta )\right )\Big ) \end{equation}

which is an exact solution of ${\nabla} ^2 \varPhi = 0$ in spherical coordinates $\{R,\theta ,\phi \}$ using Legendre functions of the first and second kind $P_\alpha$ and ${Q}_\alpha$ , respectively. Here, $B_1$ , $B_2$ and the power $\alpha$ are unknown constants, which can be resolved by coupling (2.1) with the electrostatic equilibrium of the slightly deformed cone in the presence of the jet (Gañán-Calvo 1997a ). Using solution (2.3), $E_n$ and $E_s$ are expressed as

(2.4) \begin{eqnarray} E_n=&E_\theta \cos (\theta -R_J/R) + E_R \sin (\theta -R_J/R)\nonumber, \\ E_s=&-E_R \cos (\theta -R_J/R) + E_\theta \sin (\theta -R_J/R), \end{eqnarray}

where $E_R$ and $E_\theta$ are the $\{R,\theta \}$ components of the electric field. For slenderness requirements, $R_J$ is a slowly varying function of $R$ , and for geometric reasons $\theta \rightarrow \pi$ since the origin of the spherical coordinates is at the virtual apex of Taylor’s cone. Thus, defining

(2.5) \begin{equation} x=R/R_G, \, R_J = R_G fx^{-\lambda },\, e_{n,s}=E_{n,s} \left (\frac {\sigma }{\varepsilon _o R_G}\right )^{-1/2},\, I_o= \left (\frac {\sigma ^2\varepsilon _o}{\rho }\right )^{1/2}, \end{equation}

where $\lambda$ is here an unknown exponent with $0 \leqslant \lambda \lt 1$ , $\alpha \leqslant 1/2$ , $f$ and $k_I$ are unknown constants, the equations ((2.1)–(2.4)) lead to the following.

(i) Multiplying (2.1) by $R_G^2/\sigma$ , one obtains

(2.6) \begin{eqnarray} \frac {\mathrm d}{\mathrm dx}\left ( \frac {1}{f}x^\lambda + \frac {1}{2\pi ^2}\left (\frac {Q}{Q_o}\right )^{1/2} \frac {x^{4\lambda }}{f^4}-\frac {1}{2}\left (e_n^2+(\varepsilon -1)e_s^2\right )\right )\nonumber \\ =\frac {2 e_n e_s x^\lambda }{f}-\frac {6\lambda }{\pi f^2} \delta _\mu ^{-1} x^{2\lambda -2}. \end{eqnarray}

(ii) Under slenderness requirements, one has $x \gg 1$ . Thus, from (2.3), (2.4) and (2.5) one has to leading order

(2.7) \begin{equation} e_s=\frac {\pi A}{4} x^{-1/2},\quad e_n=\frac {B_1 \varphi }{f} x^{\alpha +\lambda }, \end{equation}

where

(2.8) \begin{equation} \varphi =\lim _{\theta \rightarrow \pi } \left (Q_{\alpha }(\theta )-\frac {B_2}{B_1} P_{\alpha }(\theta )\right )\sim O(1). \end{equation}

Equations (2.7)–(2.8) are valid for $\theta =(\pi - x^{-1+\lambda } f)\rightarrow \pi$ , as far as $\lambda \lt 1$ with $x\gg 1$ . Thus, in terms of the flow rate $Q$ and its reference value $Q_o=({\sigma \varepsilon _o}/({\rho K}))$ , (2.2) reads

(2.9) \begin{equation} I/I_o=\left (\frac {Q}{Q_o}\right )^{3/4} \frac {\pi ^2 A}{4}f^2 x^{-2\lambda -1/2}+\left (\frac {Q}{Q_o}\right )^{1/4} \frac {2 B_1 \varphi }{f^2} x^{\alpha +2\lambda }. \end{equation}

(iii) Combining (2.6) and (2.7), one arrives at

(2.10) \begin{eqnarray} \frac {\lambda \,x^{\lambda -1}}{f} & + & \frac {2 \lambda x^{4\lambda -1}}{\pi ^2 f^4}\left (\frac {Q}{Q_o}\right )^{1/2} + \frac {(\alpha +\lambda )\varphi ^2 B_1^2 x^{2(\alpha +\lambda )-1}}{f^2} + \frac {\varepsilon -1}{2}\left (\frac {\pi A}{4\,x}\right )^2\nonumber \\ & = & \frac {\pi A \, \varphi B_1 \, x^{-1/2+\alpha +2\lambda }}{2 f^2}-\frac {6\lambda }{\pi f^2} \delta _\mu ^{-1} x^{2\lambda -2}. \end{eqnarray}

Equations (2.9) and (2.10) determine the dependency of $f$ with $x$ along different scales of $x \gg 1$ . A careful weight of the different terms of (2.9) and (2.10) allows the identification of three main regions.

2.1. The three regions of the SCJ-ES’s jet

An approach to obtain the jet shape, scaling of the electric current, and eventual droplet size is here outlined from a variable-scaling analysis of equations (2.6)–(2.10). Introducing the renormalisation

(2.11) \begin{equation} f=\xi \left (\frac {Q}{Q_o}\right )^{\beta _1}, \quad x=\chi \left (\frac {Q}{Q_o}\right )^{\beta _2}, \quad B_1=b_1 \left (\frac {Q}{Q_o}\right )^{\beta _3}, \end{equation}

with $\xi$ and $b_1$ constants of the order unity, $\beta_{1,2,3}$ unknown exponents, and $\chi$ a variable of the order unity as well (in principle), one can identify the following three possibilities.

2.1.1. Region (1): transition region from charge conduction to charge convection

First, from the balance of the two terms in (2.9) and the definitions (2.11), one must have

(2.12) \begin{eqnarray} f^4 x^{-4\lambda -\alpha -1/2} B_1^{-1} \left (\frac {Q}{Q_o}\right )^{1/2} \sim O(1)\nonumber \\ \Longrightarrow 4\beta _1-(4\lambda +\alpha +1/2)\beta _2-\beta _3 = -1/2. \end{eqnarray}

In this region, if a balance between the inertia and the tangential and normal electric stresses is achieved, from (2.10) the following must be true:

(i) The powers of $x$ must be equal. Thus,

(2.13) \begin{equation} 4\lambda -1 = 2(\alpha +\lambda )-1=\alpha +2\lambda -1/2. \end{equation}

(ii) The order of magnitude of the three terms must be equal. Thus,

(2.14) \begin{equation} 1/2-4\beta _1=-2\beta _1+\beta _3=-2\beta _1+2\beta _3. \end{equation}

Equations (2.12) to (2.14) yield

(2.15) \begin{equation} \lambda =1/2,\, \alpha =1/2,\, \beta _1=1/4,\,\beta _2=1/2,\,\beta _3=0. \end{equation}

This solution shows that, in the case of steady emission, there is a natural extension of the Taylor electrostatic solution to include the solution of order $R^{1/2}$ with a singularity at $\theta =\pi$ , i.e. $P_{1/2}(\theta )$ , the Legendre function of the first kind and order 1/2 (see (2.3)). From a completely different approach, this solution was considered early by Fernández de la Mora (1992) in the same framework of self-similarity as Taylor’s solution, assuming that the space charge generated by the charged aerosol has a conical shape with the same origin as the conical meniscus (i.e. assuming the jet has a negligible length compared with the macroscopic scale). How the local solution obtained here corresponds to the outer solution proposed by Fernández de la Mora (1992) is an open question worth investigating in future studies.

In this region, the renormalised inertia, surface normal and tangential stresses terms scale as $\chi ^1$ . In particular, the polarisation stress term scales as $(\varepsilon -1)\chi ^{-2}$ . In addition, the viscous term scales as $ ({Q}/{Q_o} )^{-3/4}\delta _\mu ^{-1}\chi ^{-1}$ . These two latter terms are subdominant compared with the former three for $\chi \gt (\varepsilon -1)^{1/3}$ and $\chi \gt \delta _\mu ^{-1/2} ({Q}/{Q_o} )^{-3/8}$ , respectively. Interestingly, using definitions (2.12) and the reference scale $d_o= ({\sigma \varepsilon _o^2}/({\rho K^2}) )^{1/3}$ , the scaling of the jet radius $R_J$ in this region is

(2.16) \begin{equation} R_J \sim R_G \left (\frac {Q}{Q_o}\right )^{\beta _1-\lambda \beta _2=0} \chi ^{-1/2} \sim d_o \left (\frac {Q}{Q_o}\right )^{1/2} \chi ^{-1/2}, \end{equation}

which coincides with the scaling proposed by Gañán-Calvo (1999b ), Hartman et al. (1999), and Gañán-Calvo (2004).

The fundamental outcome from this region is that the electric current gets fixed. In effect, from (2.15) and (2.9) one finally and unequivocally arrives at

(2.17) \begin{equation} I / I_o = k_I \left (\frac {Q}{Q_o}\right )^{1/2} \end{equation}

as anticipated by Gañán-Calvo et al. (1993), Gañán-Calvo (1999b ), Hartman et al. (1999), and Gañán-Calvo (2004): note that $\delta _\mu$ and $\varepsilon$ are absent in (2.17) since the axial decay of the corresponding terms is faster than the others, at least as far as the jet is under the influence of a conical meniscus, as in figure 1. In this case, $k_I$ is a constant of the order unity that can be obtained from (2.9) and the solution of the electrostatic coupling of the cone-jet described in Gañán-Calvo (1997a). For long jets that exceed the characteristic length of the conical intermediate region, $k_I$ can be slightly dependent on the global applied voltage on the meniscus, as well as on the other parameters $\{\delta _\mu ,\varepsilon ,Q/Q_o\}$ . This dependence would arise from the external coupling of the electric potential between this region and the global geometry of the meniscus and its support, but a generally valid value is simply $k_I=2.5$ .

2.1.2. Region (2): region of dominance of surface charge convection

In this region, from the scaling of the electric current (2.17) already fixed in region (1), one must have

(2.18) \begin{equation} f^{-2} x^{\alpha + 2\lambda } B_1 \left (\frac {Q}{Q_o}\right )^{1/4} \sim \left (\frac {Q}{Q_o}\right )^{1/2}. \end{equation}

Thus, one must have

(2.19) \begin{equation} \alpha +2\lambda =0,\quad 1/4 -2\beta _1 +\beta _3=1/2. \end{equation}

In addition, the balance of the inertia and the tangential surface stress requires, as in region (1),

(2.20) \begin{equation} 4\lambda -1=\alpha +2\lambda -1/2,\quad 1/2-4\beta _1=-2\beta _1+\beta _3. \end{equation}

Equations (2.19) and (2.20) yield:

(2.21) \begin{equation} \lambda =1/8, \quad \alpha =-1/4, \quad \beta _1=1/16, \quad \beta _3=3/8. \end{equation}

Observe that $\beta _2$ cannot be fixed in this region, and therefore one just has that the jet shape scales as

(2.22) \begin{equation} R_J \sim R_G \left (\frac {Q}{Q_o}\right )^{1/16} x^{-1/8}\sim d_o \left (\frac {Q}{Q_o}\right )^{9/16} x^{-1/8}. \end{equation}

This jet profile has the same power decay with $x$ as the solution first proposed in Gañán-Calvo (1997a ). Note that this power decay is slower than that of region (1), but still indicates an acceleration of the liquid along the axial coordinate due to the action of the resultant of the tangential surface stresses.

2.1.3. Region (3): final ballistic jet region and Rayleigh breakup

When the resultant of the tangential surface stress decays below those of the inertia, normal stress and surface tension, the jet reaches a ballistic state with a constant radius. In effect, in this region the balance of inertia and normal electric stress demands

(2.23) \begin{equation} 2\alpha -2\lambda =0, \quad 1/2-4\beta _1=-2\beta _1+2\beta _3. \end{equation}

These equations and the dominance of charge convection, (2.19), consistently yield

(2.24) \begin{equation} \lambda =0, \quad \alpha =0, \quad \beta _1=0, \quad \beta _3=1/4. \end{equation}

These exponents reflect the strictly ballistic nature of this region, whose fundamental result is to fix the jet radius and the eventual size of the emitted droplets. According to (2.24), one has

(2.25) \begin{equation} R_J=k_J R_G= k_J d_o \left (\frac {Q}{Q_o}\right )^{1/2}, \end{equation}

where $k_J$ is a constant. Assuming that the range of Weber numbers (Gañán-Calvo & Montanero 2009; Li, Gañán-Calvo & López-Herrera 2011) is below that which promotes the asymmetric instability and that the diameter of the droplet $d_d$ is proportional to the radius of the jet, one obtains a result anticipated in Gañán-Calvo (1997a , 1999a); Hartman et al. (1999) and extensively verified in Gañán-Calvo & Montanero (2009); Gañán-Calvo et al. (2018), i.e. $d_d=k_{d}R_G$ with $k_d\simeq 1$ . The eventual charge of the emitted droplets after the jet breakup is an overall consequence of the electric current fixed in the first region and the jet size fixed in the ballistic region.

3. The charge of the emitted droplets

A review of the break up of charged jets can be found in Gañán-Calvo et al. (2018). In summary, the droplet charge $q$ over the Rayleigh limit charge $q_R$ ratio, given by the relationship

(3.1) \begin{equation} \frac {q}{q_R}=\frac {\frac {I\,\pi d_d^3}{6Q}}{\left (8 \pi ^2 \varepsilon _o\sigma d_d^3\right )^{1/2}}\simeq \frac {k_I}{6} \left (\frac {k_d^3}{8}\right )^{1/2} \left (\frac {Q}{Q_o}\right )^{1/4}, \end{equation}

anticipates that, for $k_d\simeq 1$ , $k_I=2.5$ and $Q/Q_o$ above $2\times 10^3$ , the droplet charge would be above the Rayleigh limit and would undergo fission (Rayleigh 1882; Taflin, Ward & Davis 1989; Gomez & Tang 1994; Davis & Bridges 1994; Duft et al. 2003; López-Herrera & Gañán-Calvo 2004; Nemes, Marginean & Vertes 2007; Arscott et al. 2012).

On the other hand, the ratio of the charge $q_J$ of a segment of the jet with length $\lambda R_J$ over the limit charge $q_{RJ}$ of that segment, given by

(3.2) \begin{equation} \frac {q_J}{q_{RJ}}=\frac {\frac {I \pi R_J^3 \lambda }{Q}}{(6 \pi ^2 \sigma \varepsilon _o R_J^3 \lambda ^2)^{1/2}}= k_I \left (\frac {k_J^3}{6}\right )^{1/2}\left (\frac {Q}{Q_o}\right )^{1/4}, \end{equation}

where $\lambda$ is the most probable wavelength of the varicose breakup mode, indicates that the limit would be reached for $Q/Q_o$ above $k_I^{-4} ({k_J^3}/{6} )^{-2}$ . Which of them would go first into electrically driven instability can be measured by the following ratio:

(3.3) \begin{equation} \varOmega =\frac {q_J/q_{RJ}}{q/q_{R}}=\sqrt {48}\left (\frac {k_J}{k_d}\right )^{3/2}\left (\frac {Q}{Q_o}\right )^{1/4}. \end{equation}

Droplets (jets) would do it first for $\varOmega \gt (\lt ) 1$ . Since $k_d$ and $k_J$ have been found experimentally to be approximately 1 and 0.3, respectively, then $\varOmega \simeq 1.14 \gt 1$ , which means that droplets would undergo fission at flow rates slightly below those that cause jet branching, consistent with many observations. However, since $k_J=(6\lambda )^{-1/3}k_d$ , $\lambda$ values of approximately 9 (or $k_J$ values as close to 0.3 as 0.26 for $k_d\simeq 1$ ), which are also found experimentally depending on $\delta _\mu$ (see § 4 and Gañán-Calvo 1999a), would cause $\varOmega \simeq 0.92$ , causing the jet to electrically branch before the resulting droplet would undergo fission. This is an indication of how close the two instabilities are in realisations of SCJ-ES. Now, assuming more precisely $\varOmega \simeq 1$ and defining the Weber number as in Gañán-Calvo & Montanero (2009),

(3.4) \begin{equation} We=\frac {\rho Q^2}{\pi ^2\sigma R_J^3}=\frac {6\lambda }{\pi ^2 k_d^3}\left (\frac {Q}{Q_o}\right )^{1/2}, \end{equation}

it is found that, assuming that the raise of asymmetric jet perturbations (azimuthal number $m=1$ ) over the classical Rayleigh varicose instability ( $m=0$ ) is caused by electrical instability, they appear for $We \simeq 20$ , which is consistent with more precise values of $k_d\simeq 1.35$ and the classical value $\lambda \simeq 9$ of Rayleigh capillary breakup, for a wide range of $\delta _\mu$ values (see Gañán-Calvo et al. 2018 for an extensive literature on this topic).

4. Experimental verification

4.1. Experimental verification of the jet scaling and profile

In this section we use the collection of experimental results obtained in 1998 and published in Gañán-Calvo (1999a). We use here 50 electrospray runs where the shapes of the cone-jets and the emitted electric current were collected and digitised. Their relevant physical properties are given in table 1.

Table 1. Liquids used from Gañán-Calvo (1999a) and some of their physical properties at 24.5 $^{\circ}$ C (Propyl. glycol) or 30 $^{\circ}$ C (Dodecanol) ( $K$ , S $\textrm {m}^{-1}$ ; $\rho$ , kg $\textrm {m}^{-3}$ ; $\sigma$ , N m⁻¹; $\mu$ , centipoise). An additional liquid (propylene glycol 2) from these early experiments has been included. Also given, the values of $\delta _\mu$ (note the different definition in Gañán-Calvo (1999a); also, the conductivity of dodecanol (2) has been corrected: 1.3 instead of 2.3.

Figure 2 shows the non-dimensional measured emitted electric current $I/I_o$ vs the liquid flow rate $Q/Q_o$ . The agreement with (2.9) for $k_I=2.5$ (Gañán-Calvo et al. 1993; Gañán-Calvo 1999a, 2004; Gañán-Calvo et al. 2018) is excellent.

Figure 2. Electric current vs flow rate. Dashed line is $I/I_o= 2.5 (Q/Q_o)^{1/2}$ .

Figure 3 shows the cone-jet digitised profiles published in Gañán-Calvo (1999b ), obtained at the 50 % grey level locations.

Figure 3. Raw digitised profiles of experimental steady electrospray cone-jets using the liquids of table 1, where $z$ and $z_0$ are the generic axial coordinate and the axial position of the origin of the jet, respectively.

To verify the prediction of the analytical jet profile from (2.5) and (2.21)–(2.22), and the proposed multi-scale description of the jet and its successive regions in the axial direction, the profiles are first made dimensionless with $R_G(Q/Q_o)^{1/16}$ as suggested by the scaling of the more extensive region 2. The profiles should collapse and show the spatial structure proposed once the non-dimensional axial position of the origin $x_{0,2}$ is correctly determined for each profile. This involves the following two steps.

(1) Select any profile as reference. Since region 1 has a small axial length compared with that of region 2, the profiles should reasonably collapse by setting an appropriate position of the origin. This is indeed what is found: once the origin of a first profile is arbitrarily fixed, the others can be easily collapsed onto the first one by finding their corresponding origin, showing a very good global collapse.

(2) A final shift of the axial origin of the profile collection to reveal the slope $-1/2$ in log–log coordinates establishes the final position of the origin $x_{0,2}$ . This corresponds to the initial arbitrarily selected position of the origin plus the final global shift. At this point, it should be noted that the origin of the underlying first-order Taylor field is essentially undetermined. Hence, a final axial shift is necessary to find the most accurate origin of the underlying cone-jet’s virtual first-order quasi-electrostatic meniscus with a Taylor-like end point for each real cone-jet shape (see Pantano et al. 1994). The only experimental guidance available to the researcher is to determine it by using the most accurate theoretical description of the jet’s structure as a template in the experiments. Naturally, the collapse in region 1, with slope $-1/2$ , cannot be perfect since the radius does not exactly scale as $R_G(Q/Q_o)^{1/16}$ , but as $R_G$ .

Figure 4. Non-dimensional raw profiles from figure 3. The yellow and black dashed lines are functions $0.73 x^{-1/8}$ and $1.8 x^{-1/2}$ , respectively (used as a guide to the eye). The jet radius is made dimensionless with (a) $(Q/Q_o)^{1/16} R_G$ and (b) $R_G$ to highlight the scaling of each jet’s region. The axial coordinate is made dimensionless with $R_G$ , but its origin is different in (a) and (b) ( $x_{0,2}$ and $x_{0,1}$ , respectively).

Figure 4(a) shows in logarithmic scale the profiles of figure 2 made dimensionless with $R_G(Q/Q_o)^{1/16}$ , as a function of the axial coordinate made dimensionless with $R_G$ , as in region 2. The predicted power laws, $x^{-1/2}$ and $x^{-1/8}$ , are also shown for reference. One can observe that the collapse is very good; nevertheless, the collapse in region 1 can be slightly improved by making $R_J$ dimensionless with $R_G$ , as suggested by the scaling in this region, and making minor adjustments in the axial position of the origin, i.e. $x_{0,1}$ . This is shown in figure 4(b). As a result of this improvement, however, the collapse in region 2 worsens slightly to the naked eye, which is revealed by the statistical regression but is not shown here for brevity. This illustrates that, despite the inherent impossibility of producing a perfect homogeneous collapse of the profiles due to the non-homogeneous scaling of the jet radius along the axial length, the power of $1/16$ is sufficiently small to preserve the clear appearance of the (larger) power-law dependencies of each region.

In summary, the excellent collapse found with each slightly different scaling in regions 1 and 2 provides a solid support for the proposed analytical model and scaling laws, again under the general hypothesis made in this work (i.e. the applicability of the Taylor–Melcher LDM to the SCJ-ES mode).

4.2. Experimental determination of the droplet charge

In this section, the comprehensive data set collected in Gañán-Calvo et al. (2018) is used. This study encompasses $\gt 1000$ data points from the existing literature, which includes all relevant liquid properties, droplet size and electric current. Using this extensive data set, the ratio $q/q_R$ is calculated.

Figure 5. Droplet charge to Rayleigh limit ratio $q/q_R$ calculated from the data collected in Gañán-Calvo et al. (2018) from the literature. Data labelled ‘R-M 2013’ refer to the data in Rebollo-Muñoz et al. (2013). The data in Gamero-Castaño (2008) have been excluded from the collection in Gañán-Calvo et al. (2018) because the droplet size is not directly measured. The black dashed line is (3.1) for $k_I=2.5$ and $k_d=1$ .

As shown in figure 5, this leads to two distinct groups of data: one that closely follows the predicted scaling law (3.1) for constant $k_I$ and $k_d$ , and another group that remains a nearly constant fraction of the Rayleigh limit over more than two orders of magnitude in the flow rate range, which would imply a certain dependence of prefactors $k_I$ and $k_d$ with $Q/Q_o$ . Interestingly, this latter group of data corresponds to non-polar solvents (heptane, dioxane) with the addition of a highly polar liquid, such as formamide or water, or a soluble additive that functions as a relatively strong electrolyte. In contrast, liquids that more closely follow the scaling law are pure liquids with certain degree of dissociation to be electrosprayed (weak electrolytes). In Lopez-Herrera et al. (2023), the authors explored the electrokinetics of the SCJ-ES in detail associated with the strength of the electrolyte. They discovered that while weak electrolytes follow closely the LDM, which is the backbone of our scaling laws, strong electrolytes do not. The authors suggested that a surface adsorption mechanism leading to a high-charge packaging at the surface could be missing in the electrokinetic model. However, the sheer local change of conductivity associated with the rise of volumetric concentration of the ions in the jet due to the well-described electrokinetic mechanisms could be enough to alter the prediction of the scaling laws. As illustrated in figure 5, the findings indicate that, regardless of the flow rate, the spatial ion concentration in the jet would reach levels commensurate with the resulting electric field (dominated by Taylor and, therefore, relatively constant), thus inducing a nearly constant droplet charge that aligns with the concentration, resulting in a substantial proportion (approximately 60 %–80 %) of the Rayleigh limit for all values of $Q/Q_o$ under stable SCJ-ES conditions. For $Q/Q_o$ larger than approximately 500 (corresponding to Weber numbers in the range 20–40, according to (3.4) and consistent with Gañán-Calvo & Montanero 2009), the ratio $q/q_R$ stabilises at approximately 60 %–80 %, aligning with the proximity of the Rayleigh limit and the intensification of asymmetric instabilities.

These findings provide experimental evidence in support of SCJ-ES outcomes that involve not only strong electrolytes but also complex electrokinetic mechanisms with varying dissociation levels in response to field strengths and ion concentrations, which would result in a modification of the composition of the charged ejecta with respect to the bulk liquid, as suggested in the recent literature. This should be taken into account during the sample introduction step in ESI-MS in the future.

5. Concluding remarks

The description of the SCJ-ES classically relies, as it should, on the fundamental mechanism by which the external field acts on the liquid: the electrokinetic relaxation of free charges in surface layers. This phenomenon is associated with a well-known characteristic time scale, allowing a conceptual distinction between inherently unsteady processes, such as the formation of the electrified meniscus and the onset of ejection, and the steady-state process represented by SCJ-ES. However, this distinction is rather formal, as both processes share electrohydrodynamic characteristics, with the former essentially providing the physical pathway for the establishment of the latter. Central to this approach is the identification of the universal electrohydrodynamic velocity, $v_o$ , governing the self-similar collapse of the charged liquid meniscus and the subsequent steady jet formation (Gañán-Calvo et al. 2016b ), providing a coherent basis for predicting the fundamental jet scale $R_G$ once the emitted liquid flow rate is imposed. These two primary scales, $R_G$ and $v_o$ , serve as the foundation for subsequent stages of analysis.

By identifying an intermediate spatial scale near the emitting tip of the SCJ-ES and employing a refined one-dimensional model spanning the full jet length under the local Taylor-cone electric field, the analysis reveals three fundamental regions: the transition region, where the emitted current is determined; the convection-dominated region, where surface charge transport governs charge transfer while jet acceleration is driven by the Taylor-cone electric field; and the ballistic region, where the jet acquires a cylindrical shape and undergoes Rayleigh breakup. The existence of this intermediate scale assumes that the jet length remains smaller than the radius of the feeding capillary, a condition typically met at low emitted liquid flow rates and, more generally, with sufficiently high liquid conductivities (de la Mora 1992). However, this is not an intrinsic limitation of the three-region theoretical model presented here. In fact, although not explicitly developed in this work, the effective externally applied electric field along the jet may exhibit a spatial distribution that deviates from Taylor’s solution. Nevertheless, the local scaling of this field would invariably sustain the fundamental balance between surface tension and normal electric stress revealed by Taylor. And more importantly, once the electric current is set by the electrohydrodynamic mechanisms in region 1, the action of the external electric field tends to vanish along the axial coordinate, the final scaling the jet is fixed as $R_G$ in the final ballistic region.

Despite these advances, the study also highlights persisting gaps in understanding, as revealed by the experimentally measured charge on the emitted droplets. The influence of electrokinetic and electrochemical phenomena, such as ion dissociation, possible surface ion adsorption, and dynamics of diffuse surface charge layers with sizes comparable to those of the ejected jets, warrants further investigation (López-Herrera et al. 2023). Observed deviations in strong electrolytes, coupled with the complex interplay between electrokinetic and electrohydrodynamic processes, and the resulting local conductivity enhancements suggest that existing models may require extensions to incorporate more intricate electrokinetic effects. The impact of future investigations in the fundamental field of ESI-MS for organic chemistry would be significant.

Thus, future research should focus on elucidating the microphysical mechanisms governing charge relaxation in SCJ-ES, exploring the role of local liquid composition and temperature, and refining models to capture the precise balance between electrohydrodynamic forces and surface phenomena. The continued development of high-fidelity numerical simulations from both molecular dynamics (MD) (Aschi 2019; Nuwal et al. 2021) and continuous models including phase field approaches (Misra & Gamero-Castaño 2022), combined with well-controlled experiments, will be crucial for deepening our understanding and expanding the predictive capabilities of SCJ-ES models.