Diffusion Distortions in a Differential Mobility Analyzer with Inclined Electric Field

Abstract

An approach based on the calculation of the probabilities of the random displacements of particles around their regular trajectories enables us to derive analytical expressions for the normalized apparent mobility spectrum. A particular derivation is carried out for the case of a DMA with one collecting electrode, with inclined electric field, and variable electric field strength. Calculations show that an inclined electric field actually results in a sharper apparent spectrum, i.e., in a higher resolution than a transverse electric field does. A way to improve resolution by calculations is outlined.

INTRODUCTION

Differential Mobility Analyzer (DMA) has a wide use in the measurements of aerosol particle size distributions and, also, air ion mobility distributions. A comprehensive historical overview of this apparatus was composed by Flagan (1998). A systematical approach to DMA can be found in Tammet (1970). An overview of contemporary electrical aerosol size spectrometers was published by Intra and Tippayawong (2007, 2008). In general lines, a DMA has air inlet(s), air outlet(s), electrodes, and collector(s) of electric current or aerosol particles. Following the terminology of Tammet (1970), a DMA with a narrow inlet for aerosol and a narrow collector for aerosol particles can be called a second-order DMA. It is possible to design many different structures of DMA. For example, the DMA by Knutson and Whitby (1975) has a divided air inlet and one collector at the air outlet. The electrical aerosol spectrometer of the University of Tartu has a divided air inlet and multiple collecting electrodes (Mirme et al. 1984; Tammet et al. 2002). Salm (1995) proposed a DMA with a divided air inlet and multiple collectors at the air outlet; an analogous DMA was built by Box and Collings (2007).

The Brownian motion of particles and turbulence are two of the factors that cause distortions in the measurements of mobility distributions by means of DMA. These processes may also be called molecular diffusion and turbulent diffusion, respectively. The diffusion distortions can be divided into two groups: particle losses in the entrance region of the analyzer, and the distortion of the shape of the mobility distribution. We will confine ourselves to the second phenomenon. The most essential diffusion distortion is the smoothing of the mobility distribution and the relevant reduction of resolution. The historical survey of the problem up to 2000 is given in Salm (2000). Additional information to the survey can be found in the Introduction of the paper (Flagan 1999). The diffusion of charged particles in nonuniform force field and the corresponding DMA resolution have been studied in Alonso and Endo (2001) and Alonso (2002). Mamakos et al. (2007) by numerical calculations investigated the validity of the semianalytical theory of DMA resolution by Stolzenburg (1988). Stolzenburg and McMurry (2008) have presented a summary of equations governing the responses of single DMA and TDMA systems including diffusion broadening of the DMA transfer function. Diffusion distortions in a simple DMA have systematically been studied also in Salm (2000). Here we will demonstrate that the method developed in the above paper is applicable also for a more complicated DMA. In particular, we consider the design of a DMA with an inclined electric field, which has certain advantages with respect to the resolution (Loscertales 1998; Tammet 1998; 1999). The principle of inclined velocities was developed in an analyzer IGMA by Tammet (2002, 2003). In general, we will follow the methodology used in Salm (2000).

APPARENT MOBILITY SPECTRUM

In an ideal DMA, a charged particle with fixed mobility moves along a regular trajectory, and reaches a certain end point. In reality, the Brownian motion of the particles exists, turbulence may take place, and the actual trajectories disperse randomly around the ideal trajectory; charged particles of single mobility reach various dispersed end points, as if a continuous mobility spectrum of finite width exists. Then an ordinary data processing algorithm calculated for ideal conditions yields an apparent mobility spectrum of finite width in place of a discrete line.

If diffusion is lacking, then the current strength through one collector or the characteristic of the analyzer is expressed by the integral

where ψ is a variable operational parameter (electric field strength, length of the analyzer, flow rate, etc.); Z is the electrical mobility of particles; G(ψ, Z) is the transfer (apparatus) function; and ρ(Z) is the differential distribution of polar charge density of particles by their mobility, or the mobility spectrum. Integration is supposed over the entire range of the spectrum.

The characteristic I(ψ) is measured and a proper operator H _ψ is applied to the characteristic to solve the Equation (1) and to find the mobility spectrum ρ(Z):

The structure of the operator H _ψ depends on the particular type of the mobility analyzer and on the choice of the variable parameter ψ. The subscript ψ →Z denotes that the variable parameter should be expressed by mobility.

Without restricting generality, the particles of a single mobility Z ₁ (unimobile particles) may be considered only (the subscript 1 or the symbol Z ₁ denotes this presumption). The current through a collector is then I ₁. In the case of diffusion, Equations (1) and (2) can be expressed:

where ρ₁ is the total polar charge density of the unimobile particles at the entrance of the analyzer; the asterisk * denotes that diffusion takes place in the analyzer; and w*(Z, Z ₁) will be called the normalized apparent spectrum. The normalized apparent spectrum entirely describes the effect of diffusion in the DMA.

DMA WITH ONE COLLECTING ELECTRODE, WITH AN INCLINED ELECTRIC FIELD AND VARIABLE ELECTRIC FIELD STRENGTH

Description of the DMA

Let us consider a simplified second-order DMA, which is similar to that in Salm (2000). The main difference is that the traverse electric field strength E is replaced by two components E _x and E _y, where E _x is directed against the airflow u (Figure 1). Figure 1 represents the DMA schematically; the technical structure can be designed in several shapes and is not discussed here.

FIG. 1 Schematic representation of the DMA.

The regular velocity of a charged particle is in general

The horizontal velocity of a charged particle in our DMA is

where E _x is the absolute value of the field strength component.

We will consider here the case E _x∝E _y. Let us express E _x = kE _y, where k is the coefficient of proportionality. Equation for the characteristic (limiting) mobility is replaced by a modified equation

Charged particles get to the collector, if the characteristic (limiting) mobility equals to the particle mobility Z ₁.

Current Strength and Solving Algorithm

The following derivation of equations is quite similar to that in Salm (2000), with understandable replacements. Let the electric field strength E _y be the variable parameter. At first, the current strength through the collector I*₁(E _y) is to be calculated, and then the algorithm (4) is to be applied:

As in the above paper, the current strength I*₁(E_y ) through the collector, which is proportional to the current density through the same collector, is calculated by kinematical approach based on the consideration of the regular trajectories and of the random walk of particles.

Let us consider unipolarly charged particles entering the DMA at a point (0,0) at a moment t = 0, and drifting to the collector, in average (see Figure 1). The current strength of the entering particles:

where S_e is the cross-section area of the inlet slit for aerosols.

In general, the current density of particles is composed of a conduction component and of a diffusion component (Salm 2000). The current density of the particles through the collector is determined by the y-components of corresponding vectors and by the charge density of particles in the vicinity of the collector:

where ρ _c is the charge density of particles in the vicinity of the collector, D is the coefficient of diffusion. In the case of pure molecular diffusion, the Nernst-Townsend-Einstein equation could be taken into account. However, we will keep the coefficient of diffusion in equations, in order to consider also small-scale turbulent diffusion. For estimation of turbulent diffusion, the characteristics of turbulence should be measured at any particular set-up of measurements (Salm 1983).

The relative importance of the regular and random components of fluid motion may be expressed by the Peclet number

The Peclet number is an essential factor that determines the resolution of the DMA. We are considering the case of weak diffusion, i.e., the range of large Peclet numbers, Pe ≫ 1. In the case of weak diffusion, the deposition of particles on electrodes obviously has little effect upon the results of calculation (Salm 2000). In the theory of Brownian motion, the deposition of particles is called absorption. In the present paper, we will neglect the effect of absorption.

The two-dimensional density of probability for finding an entered particle at a point (x, y) at a later moment t is expressed by two-dimensional Gaussian law, if we neglect the absorption on electrodes:

If we take into account only the regular term Z ₁ E _y ρ _c in Equation (10) in the vicinity of the collecting electrode, then the probability that an entered particle hits upon the collector at any time is

where Δ _C is the width of the collecting electrode in the x direction.

The current strength through the collecting electrode:

The integration yields:

where K₀(ζ) is the Macdonald's function (a modified Bessel function).

In order to specify the general algorithm (2) for the present DMA, we use the same way as in Salm (2000). As a result, in the absence of diffusion, the algorithm (2) can be expressed as follows:

Normalized Apparent Spectrum

The normalized apparent spectrum can be expressed on the basis of Equations (8) and (16) as follows:

Substituting of I*₁(E_y ) from Equation (15) into Equation (17) gives the normalized apparent spectrum explicitly:

The shape of the normalized apparent mobility spectrum is illustrated in Figure 2. As it is seen in Figure 2, an inclined electric field actually results in a sharper apparent spectrum, i.e., in a higher resolution according to the concept by Loscertales (1998). The width of the spectrum essentially depends on the Peclet number. Figure 2 also clearly shows that the apparent spectrum is asymmetrical and that the peak is shifted towards lower mobility comparing with the actual mobility; this result was discovered by Salm (1970).

FIG. 2 The solid line represents the function w*(Z, Z ₁) in the case of Z ₁ = 1.0; d = L; Pe = 20 and E_x = E_y . The dashed line shows the function with the exception E_x = 0, as in Salm (2000).

Simplification of Equations

At weak diffusion, the argument ζ of the function K₀(ζ) is large, and we can use the asymptotic equation of this function:

Equation (19) is sufficiently accurate also at moderate diffusion: the error is less than 3%, if ζ>5. Equation (18) can asymptotically be expressed as follows:

Equation (20) represents an explicit analytical function composed of elementary functions. At the limit of weak diffusion, a subsequent simplification of the equation is possible. As in Salm (2000), the expression under square root symbol and the same under fourth root symbol can be transformed algebraically, and Equation (20) is then expressed as follows:

where

Equation (21) has the same shape as Equation (30) in Salm (2000). However, here the variable a has somewhat different expression, depending on the ratio k. In the limit k→0, the expression approaches to that in Salm (2000).

At weak diffusion only small deviations of Z around Z ₁ are essential. Then b≪a and by means of geometrical considerations as related to Figure 2 in (Salm 2000) it is possible to prove that

Then Equation (21) can be transformed as

Thus we have obtained a quite simple analytical equation for the description of the effect of diffusion on mobility spectrum measurements by a particular DMA. The normalized apparent spectrum according to Equation (24) has a certain affinity to the Gaussian curve, but shows asymmetry. The mode of mobility is shifted toward lower mobilities. The affinity to the Gaussian curve becomes better noticeable, if a new variable 1/Z is introduced. The exact Gaussian curve with respect to 1/Z is obtained, if we consider only small deviations of Z in the vicinity of Z ₁, i.e., Z≈Z ₁. Then

The mean value of the variable 1/Z is then 1/Z ₁, and the standard deviation is

The factor has appeared here in comparison with Equation (38) in Salm (2000). This factor expresses the effect of inclined electric field.

The standard deviation of the variable 1/Z can be used for the expression of the resolution of mobility spectrometers (Salm 2000). The resolution parameter can in general be expressed as follows Salm (1983, 2000) and Zhang and Flagan (1996):

where ΔZ _min is the least distinguishable interval in the spectrum at mobility Z. The interval ΔZ _min may be determined by means of the width of a normalized apparent spectrum. It is natural to use the standard deviation multiplied by two 2σ_1/Z for the least distinguishable interval with respect to the variable 1/Z. Since the function w*(Z,Z ₁) has a finite integral, it is justified to match corresponding quantiles for the variables 1/Z and Z, and to propose the following equation:

Equations (26) and (28) are in accordance with relevant estimations in Loscertales (1998) and Tammet (1999), which were derived on the basis of a relatively rough diffusion model. Here we have seen that several steps of simplification are necessary in order to obtain the estimations.

The knowledge of the analytical apparent spectrum opens a way to the improvement of resolution by calculations. This idea was first published by Salm (1970) and is summarized as follows. If the normalized apparent spectrum w*(Z,Z ₁) is known, then the apparent spectrum ρ*(Z) for any general case is expressed as

Here we consider only diffusion distortions. In many cases the apparent spectrum w*(Z,Z ₁) depends on the ratio Z/Z ₁. This is the case also in Equations (18), (20), and (24). Then, using a simple exponential transformation, it is possible to express Equation (29) in the shape of convolution, and to solve the convolution equation by means of the Fourier transform. Thus the mobility spectrum ρ(Z) can be found with an enhanced resolution.

CONCLUSIONS

An approach based on the calculation of the probabilities of the random displacements of particles around their regular trajectories enables to derive analytical expressions for the normalized apparent spectrum. A particular derivation is carried out for the case of a second-order DMA with one collecting electrode, an inclined electric field and variable electric field strength. Equation (18) represents the most exact explicit equation for the normalized apparent spectrum in this study. Analogously as in Salm (2000), a simplification of Equation (18), and the derivation of explicit analytical expressions of various approximation degrees for the normalized apparent spectrum are possible. Sufficiently simplified equations are in accordance with relevant estimations in Loscertales (1998) and Tammet (1999), which were derived on the basis of a relatively rough diffusion model. The knowledge of the analytical apparent spectrum opens a way to the improvement of resolution by calculations.

Acknowledgments

This research was in part supported by the Estonian Science Foundation through grants 6223 and 6988, and the Estonian Research Council Targeted Financing Project SF0180043s08.