Article pubs.acs.org/JPCC
Diffusional Nanoimpacts: The Stochastic Limit Shaltiel Eloul, Enno Kaẗ elhön, Christopher Batchelor-McAuley, Kristina Tschulik, and Richard G. Compton* Department of Chemistry, Physical & Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom S Supporting Information *
ABSTRACT: The probability expressions for the average number of diffusional impact events on a surface are established using Fick’s diffusion in the limit of a continuum flux. The number and the corresponding variance are calculated for the case of nanoparticles impacting on an electrode at which they are annihilated. The calculations show the dependency on concentration in the limit of noncontinuous media and small electrode sizes for the cases of linear diffusion to a macroelectrode and of convergent diffusion to a small sphere. Using random walk simulations, we confirm that the variance follows a Poisson distribution for ultradilute and dilute solutions. We also present an average “first passage time” for the ultradilute solutions expression that directly relates to the lower limit of detection in ultradilute solutions as a function of the electrode size. The analytical expressions provide a straightforward way to predict the stochastics of impacts in a “nanoimpact” experiment by using Fick’s second law and assuming a continuum dilute flux. Therefore, the study’s results are applicable to practical electrochemical systems where the number of particles is very small but much larger than one. Moreover, the presented analytical expression for the variance can be utilized to identify effects of particle inhomogeneity in the solution and is of general interest in all studies of diffusion processes toward an absorbing wall in the stochastic limit.
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INTRODUCTION The detection of nanoparticles is a rapidly growing area with implications for environmental, technological, and fundamental studies. The extensive use of nanoparticles in medicine, electronics, and energy production as well as in the food and textile industries has vastly increased the demand for tracking and identifying the inevitable release of nanoparticle materials into the environment, and it is especially important because concerns have been raised regarding the possible toxicity of various nanoparticles toward humans.1,2 Beyond analytical use, in both fundamental research and materials synthesis, understanding mechanisms that involve nanoparticles and their properties is critically required to sense and analyze them.3−8 The electrochemical “nanoimpact” detection methodology has significant advantages over other techniques, including the ability to directly measure nanoparticles in situ without the need for drying or modifying the solution.9 In an electrochemical cell, nanoparticles diffuse freely and even at ultralow concentrations can be detected via their random impacts on the electrode surface facilitated through Brownian motion.10 When an impact occurs, an electrochemical reaction with a corresponding Faradaic current is measured, providing direct information about the individual nanoparticle size. The statistical average number of impacts/hits as a function of time can be described by Fick’s diffusion equation, allowing the concentration of nanoparticles in the sample to be inferred.11 (Often, the terminology for events of meeting a boundary/ target due to diffusion is referred to as a “hit” or “passage”. © 2015 American Chemical Society
Therefore, in the following, we use the term hit rather than “impact” in most cases. However, when directly referring to the nanoimpact method, the term impact is also mentioned.) With the increased accessibility of more sensitive potentiostats (high signal-to-noise ratio) and better understanding of the nanoparticle role in electrochemical systems, the nanoimpact method has proved to be a valuable practical detection method for a range of particle sizes and various types of nanoparticles,11−13 organic molecules,14,15 and biomolecules.5,16 Additionally, it provides fundamental understanding of chemical mechanisms,8,17,18 information about aggregation and agglomeration,8,19,20 and, lately, the possibility of nanoparticle detection at ultradilute concentrations, in the subpicomolar region.21 The lower limit of detection in ultradilute nanoparticle solutions is experimentally constrained to situations in which a tiny number of impacts on the electrode are detected over an experimentally viable time scale. Moreover, experiments with small data sizes exhibit deviations from the statistical average, as described by Fick’s diffusion equation. The magnitudes of these deviations are dependent upon the cell setup, the sample concentration, the shape or size of the detecting electrode, and the nanoparticle of interest.21 Physically, this deviation from Fickian behavior occurs by virtue of the breakdown of Received: April 2, 2015 Revised: May 25, 2015 Published: June 3, 2015 14400
DOI: 10.1021/acs.jpcc.5b03210 J. Phys. Chem. C 2015, 119, 14400−14410
Article
The Journal of Physical Chemistry C
microsphere. The macroelectrode represents the linear diffusion regime that is inherent to any large electrode or short time scale of electrochemical experiments. The microsphere is a representative symmetric case, where a full convergent (radial) diffusion is obtained that results in enhanced mass transport as compared to the linear case. We explain the difference between the cases of linear and convergent (radial) diffusion and conduct random walk simulations to support our analytical expressions for the average number of hits in the case of the linear diffusion regime.
continuous theory approximation inherent in Fick’s second law. We expand the discussion on the application of Fick’s second law in the transition between continuous and noncontinuous media in the Supporting Information. In ultradilute media, the solution no longer behaves as a continuous medium, and instead of describing the space through an average concentration or concentration profile, probability density functions need to be introduced.22,23 Evaluating expressions for the probability of an impact allows the determination of the expected deviation in the impact frequency and is thus essential for the interpretation of analytical results in the nanoimpact method at various experimental setups. Additionally, comparisons between the theoretically expected variance and experimentally obtained measurements can be utilized to estimate the uncertainty in the concentration or the inhomogeneity of the nanoparticle solution. The nanoimpact method is a diffusion process toward an absorbing wall and might be modeled with a continuous media approximation (see also the Supporting Information), where it is reasonable to describe the system with a concentration or as a noncontinuous medium at ultralow concentrations, where stochastic processes occur and probability density needs to be introduced. Hence, the present study is important for understanding this system. Furthermore, there is a more general interest in such fundamental studies of the flux in Fickian diffusional processes. Fick’s diffusion equation provides a practical way to mathematically describe the average number of many hits on an absorbing wall in the case of free diffusion. Therefore, Fick’s second law is widely applicable to many systems, even for very small diffusional flux. Among others, it is used to study the electrochemistry of small amounts of molecules,24−28 the growth of small particles,29,30 and diffusion toward a biological target.31 However, because the obtained results are sensitive to the size of the absorbing wall and to the concentration, understanding the limitations of and the deviations from Fick’s second law is critical. Moreover, using analytical expressions for the determination of the stochastic number of hits on an absorbing wall has an important practical advantage over computational stochastic methods such as random walk simulations. The analytical assessment provides a straightforward way to analyze and predict experimental results, while random walk simulation is time-consuming and computationally costly, especially in two or more dimensional coordinates.23 Previous works in electrochemistry that focused on small amounts of molecules compared stochastic and statistical processes for single molecules25 and for small current responses.26,32 Recently, Boika and Bard33 showed numerical simulation of first passage times of micro-electrodes in ultradilute solutions. However, for the nanoimpact method, it is required to find a practical method to derive probability expressions and the corresponding variance in order to study the influence of particle concentration and electrode shape or size on the analysis. In this work, we investigate the stochastics of ultradilute solutions (pM−fM), where a very small but sufficient number of particles (N0 ≫ 1) are randomly distributed and independently diffuse in the cell. We use Fick’s second law of diffusion to derive general expressions for the average number of impacts/hits and for the deviation from this average. These expressions are explicitly evaluated for the two representative and important cases of a macroelectrode and for diffusion to a
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THEORY The study is divided into the cases of (i) linear and (ii) convergent diffusion regimes. In both cases, we start with Fick’s diffusion equation ∂c = D∇2 c ∂t
(1)
where c is the concentration and D is the diffusion coefficient, and use the analytical solution in one-dimensional linear and radial space.34,35 Linear diffusion represents the case of semiinfinite diffusion and corresponds to a planar electrode in an infinite-sized cell. Radial diffusion represents the case of diffusion toward a finite-sized spherical electrode. We provide comparison calculations of the impact/hit probability for different electrode sizes and concentrations. Additionally, in order to support the presented probability function and its variance, we show results of one-dimensional random walk simulations. Linear Diffusion: Macrosized Absorbing Wall. For full and irreversible absorption, the continuous one-dimensional solution of Fick’s second law gives the time-dependent concentration profile as a function of the distance from the absorbing wall ⎛ x ⎞ ⎟ c(x , t ) = c* erf⎜ ⎝ 2 Dt ⎠
(2)
which solves eq 1 with respect to the boundary conditions c(x = 0, t ≥ 0) = 0
c(all x , t < 0) = c*
c(x = ∞ , t ≥ 0) = c*
where c* is the bulk concentration. For simplicity, the number concentration (entities/m3) is used along this study. The flux toward the absorbing wall, which can for instance be a planar electrode, is then found by differentiation with respect to x at x =0 j = −D
∂c ∂x
= x=0
c* D πt
(3) 36
which is the Cottrell equation. The conventional representation of Fick’s second law as being applied to concentrations is meaningful in most chemical systems. However, the concentration, which is defined as the number of entities per volume, can also be interpreted as the probability of finding an entity in space.37 The underlying principle that allows such probability studies is the fact that the Brownian motions of different molecules or particles are independent of each other as their diffusion is only caused by collisions with the fluid.37 This principle enables us to find the probability of an individual particle hitting the absorbing wall, which can also be interpreted as an impact on an electrode 14401
DOI: 10.1021/acs.jpcc.5b03210 J. Phys. Chem. C 2015, 119, 14400−14410
Article
The Journal of Physical Chemistry C
Figure 1. Illustration of the two representations of Fick’s solution. (a) The probability density functions along x shows the surviving fraction and the absorbed fraction of particles. (b) The flux probability density toward a fully absorbing wall (i.e., an electrode with the boundary condition, c = 0 at x = 0).
surface. Using the summing rule of probabilities, the probability of N0 particles hitting the absorbing wall as a function of time is obtained. We define the probability density (p) on all space (x = 0−∞) to be the probability of finding unity particle density in a onedimensional space with length L and at the initial time (t0) 1 p*(t0) = L
x=0
D πt
(5)
phit (t ) =
∫0
∞
1 1 ⎛ x ⎞ ⎟ dx − erf⎜ L L ⎝ 2 Dt ⎠
1 1 2 Dt erfc(z) dz = L π L
2 Dt πL
(9)
∫0
t
−D
∂p dt = ∂x
∫0
t
1 L
2 Dt D dt = πt πL
(10)
where 1/L is assumed to be constant over time, as is discussed later. It is observed that the integration of the flux over time gives the same result as the integration of the complementary CDF of the probability density over space. The identity that we obtained
(6)
In the following, we investigate these two representations of the probability density function (eq 5), as illustrated in Figure 1a, and the probability density flux (eq 6), as illustrated in Figure 1b, in order to obtain an expression for the probability of a single particle hitting the absorbing wall. For the first representation, the probability density function is illustrated in Figure 1a. Integration of p(x,t) over x yields the cumulative function density (CFD), which at t0 equals the integration over L and, therefore, equals 1. At t > 0, the probability density along x provides the fraction of the probability density that “survived” at the time t. Thus, the complementary function of the CFD presents the fraction of particle that absorbed, as illustrated in Figure 1a. The hit probability as a function of time is hence equal to phit (t ) =
2 Dt
The same conclusion can be achieved by using the Cottrell equation, which is a representation of the flux at the boundary of the absorbing wall. Equation 6 describes the flux of the probability density at this boundary as a function of time. This provides the “amount of probability” that dissipates at the absorbing wall per time, as plotted in Figure 1b. Hence, integrating over time equally gives the hit probability (in dimensionless units)
x
