Analizar los componentes identitarios y determinar su implicancia

In order to fully understand the response of SPM techniques it becomes necessary to use theoretical methods. In the work presented in this thesis, a combination of established analytical equations, and computational modelling techniques, specifically finite element method (FEM) simulations, are used to inform and quantify the observed results.

As outlined above, the nanopipette properties, both geometrical and chemical, influence strongly on the SICM response.5 Consequently, the characterisation of nanopipettes is an important task that first needs to be performed in order to allow for a meaningful understanding of experimental results. The assumption is commonly made that in the region of the nanopipette that contributes most significantly to the ionic current response, the geometry exhibited is conical. Whether this assumption is justified is explored in chapter 5 but what this assumption does allow is for analytical approaches to be applied. Through using equations for the resistance of a cone and knowledge about the electrolyte properties, in conditions where surface charge is not playing a role, and so the nanopipette exhibits an ohmic response, it becomes possible to predict the opening size of the nanopipette without the use of further microscopy tools. Furthermore, this assumption allows then for theoretical approach curves to insulating surfaces to be produced. For a well-defined, simple, conical geometry, there are analytical equations for how the access resistance would be expected to increase with a decreasing tip-substrate distance as in equation 1.3:3, 15

𝑅_!"# = 𝑅_!"#$ + ! !!"# !_! !_! !.!.! (1.3)

where ro and ri are the outer and inner radii of the nanopipette at the opening

respectively, d is the separation distance and κ is the solution conductivity. Such equations are useful approximations, but are limited in their applicability. If surface charge of either the nanopipette or substrate has a significant influence, as it would in aqueous environments of low ionic strength, or, if indeed the nanopipette geometry deviates greatly from the assumed conical geometry, such an approximation would no longer be valid.

FEM models provide an attractive alternative for giving a theoretical understanding to such systems. In FEM simulations, computational software, such as COMSOL Multiphysics, as used in the studies herein, use numerical methods to solve a specified set of partial differential equations across a defined geometry, subject to the initial conditions and boundary conditions chosen by the user. A typical FEM model first involves defining the desired geometry to closely match the experimental system and specifying the necessary properties of the materials that need to be captured such as viscosity, dielectric constants and density. This geometry can be in 1D, 2D or 3D with the higher dimensions often requiring more computational power to solve. Concentrations, charge numbers and diffusion coefficients are also required for each ionic species present in solution. Then the physical equations that are to be solved need to be chosen depending on the system that is being studied. In the case of the simulations performed herein, this includes the Nernst-Plank equation, to describe the flux of ions in solution:

J_! = −D_!∇c_!−z_! !!

!"Fc!∇ϕ+c!u (1.4)

where Ji is the total flux of species i, Di is the diffusion coefficient of species i, R is the

universal gas constant, T is the temperature and u is the fluid velocity, and the Poisson equation to describe the electric potential:

∇!_ϕ= !!

!!! !z!c! (1.5)

where F is the Faraday constant, ε is the relative permittivity of the solution,3 ε0 is

the vacuum permittivity, and zi is the charge on species i. Using the initial conditions

as a first approximation, FEM simulations then estimate subsequent solutions until a calculated error has been minimised, giving the final steady-state solution. It is also possible for time-dependent simulations to be performed to study the transport processes involved, as will be used in chapter 8. FEM models find application in a great deal of systems including the study of mixing problems, crystal growth and dissolution,185 heat transfer problems186 and the study of electrode materials.125

In the studies presented herein, FEM simulations are predominantly used as a means of exploring the effects of surface charge of the sample and nanopipette, as well as to help characterise the nanopipette properties and validating a new protocol for nanopipette characterisation. Additionally, time-dependent FEM simulations form the basis of understanding and quantifying the mixing timescales required for driving crystallisation in a nanopipette, which is explored in chapter 8. In other studies, FEM simulations have helped provide insights into the limitations of SICM including the possible resolution achievable.56-57 _{3D simulations performed of} the nanopipette over a cylindrical pipette showed that it was possible to clearly resolve features when they were separated by 3r where r is the radius of the nanopipette opening.57 _{Whilst there are limitations with these simulations in that} they assumed nanopipettes to exhibit a conical geometry, the simulation results help to understand the SICM response and the effect of various experimental parameters, such as the nanopipette size required to get the desired resolution and the separation distances that are needed to achieve this. There is some discrepancy between simulated and experimental studies of resolution in SICM,32 _{which have} suggested a resolution as low as 0.5r32, 58 is possible, and this could be due to inaccurate characterisation of nanopipettes as well as different definitions of resolution being used, which varies from study to study.32

FEM simulations can also help understand how the nanopipette response changes with the slope of the sample being probed.187 The change in resistance on

approach to a flat sample could be very different to that where the slope of the sample varies greatly over the scale of the nanopipette opening. Consequently, FEM simulations have been useful for implementing algorithms to deal with this and to give image-processing procedures to help correct for any discrepancy. This involves combining the initial topographical information extracted from SICM with FEM simulations of the nanopipette and using an iterative procedure to improve the final SICM topographical image based on this.187