# Ion Current Rectification in Extra-Long Nanofunnels
## 1 Introduction  
Nanofluidics has been attracting the attention of a wide scientific community for a long time [1, 2]. Nevertheless, its potential has not yet been fully exploited [3]. This interest is motivated by the fact that nanofluidic systems offer functionalities in terms of control of ionic and molecular transport that occur only at a nanoscale [4, 5]. Solid-state nanopores [6] and nanochannels [7] mimicking biological counterparts [8] have been investigated and exploited in many applications [9], e.g., the detection of molecules of biomedical interest, such as proteins [10], nucleic acids [11], viruses [12, 13, 14], and nanovesicles [15] or for developing new components of ionic circuits [16], such as transistors [17], diodes [18, 19], and ion pumps [20, 21, 22]. A large number of nanofluidic devices exploit electric fields for handling fluids and nano-objects near or through functional nanostructures; for this reason, understanding electrokinetic phenomena occurring at the nanoscale is of paramount importance. Many researchers focused their efforts on modeling, either by analytical or numerical methods, the electrokinetic behavior of fluids in nanoconfinement conditions or at micro-nanointerfaces [23, 24], but a comprehensive understanding of all the phenomena taking place and how they influence each other is still far off. This lack of understanding is due to the complexity of the problem, which involves coupled nonlinear fluid flow, charged species transport, and a dynamic evolution of the electric field in a multi-scale space spanning from nanometers to centimeters, as stated by [PERSON] and coworkers [25]. In addition to symmetric geometries, nanopores and nanochannels with asymmetric characteristics have also been studied due to their interesting functionalities [26, 27], especially when electric fields are applied. In particular, electrostatic interactions between molecules and ions in aqueous solutions, and the electric double layer(EDL), which forms on a nanostructure's walls, can be exploited for creating ion-selective devices [28]. Concentration polarization (CP) [29] was observed at micro-nano interfaces [30], in nanochannels [17], and in ion current rectification (ICR) in nanostructures [31] with asymmetric surface charge distribution [32; 33], geometry [20; 26; 27; 33; 34; 35; 36; 37; 38; 39; 40], or both [41]. Concerning conical nanochannels, many different parameters, such as length, cone angle, distribution of surface charge, size and shape, were analyzed to understand how they affect CP and ICR [34; 42; 43; 44]. In particular, CP has been exploited to concentrate diluted analytes, such as proteins [29; 45], to improve the sensitivity of biosensors. The challenging idea of combining accumulation capabilities with biological and chemical selectivity boosted the research of new methods and techniques for functionalizing the surface and/or immobilizing biomolecules. Nanochannel/nanopore surface functionalization was used for tuning the ion/molecule selectivity of the structure [46] and also for varying the rectification characteristics [46; 47]. For example, [PERSON] et al. [48] anchored biomolecules onto a conical nanopore to fabricate a biosensor for avidin, streptavidin, and immunoassay. Although research activities focused on the exploitation of CP for biosensing have been very intensive in recent years, as demonstrated by three recent reviews [29; 45; 49], the main disadvantage of CP-based accumulation systems is still the inability to precisely control the location of the preconcentrated biomolecule plug, as stated by [PERSON] and [PERSON] [50]. In fact, by controlling the position of the plug it would be possible to overlap target biomolecules with probe molecules anchored on the surface to enhance the detection sensitivity and promote the recognition mechanisms. With this idea in mind, we developed an immunosensor [51] to accumulate target biomolecules (e.g., antigens) inside a long funnel functionalized with probe molecules (e.g., antibodies).