# Difference between revisions of "Probing Surface Charge Fluctuations with Solid-State Nanopores"

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==Discussion== | ==Discussion== | ||

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+ | ''Effect of ionic concentration'' | ||

The change in behavior of the noise and conductance in the two regimes can be explained by DeBye screening. At high concentrations, the DeBye length <math>\lambda_D</math> ~ <math>1/\sqrt{c}</math> is smaller than the radius of the nanopore and the number of charge carriers that are affected by the fluctuation changes follows the equation: | The change in behavior of the noise and conductance in the two regimes can be explained by DeBye screening. At high concentrations, the DeBye length <math>\lambda_D</math> ~ <math>1/\sqrt{c}</math> is smaller than the radius of the nanopore and the number of charge carriers that are affected by the fluctuation changes follows the equation: | ||

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<math>c \cdot A_{ring}=c \cdot \pi\ \cdot (2 \cdot \lambda_D \cdot R- (\lambda_D)^2)</math> ~ <math>c \cdot \lambda_D</math> ~ <math>\sqrt{c}</math> | <math>c \cdot A_{ring}=c \cdot \pi\ \cdot (2 \cdot \lambda_D \cdot R- (\lambda_D)^2)</math> ~ <math>c \cdot \lambda_D</math> ~ <math>\sqrt{c}</math> | ||

− | The conductance is proportional to the concentration. The current fluctuation, <math>S_I</math>, and the conductance fluctuation, <math>S_G</math>, in an Ohmic system are related by <math>S_I = (S_G/G^2) I^2</math>, where G is the conductance and I is the current. | + | The conductance is proportional to the concentration. The current fluctuation, <math>S_I</math>, and the conductance fluctuation, <math>S_G</math>, in an Ohmic system are related by <math>S_I = (S_G/G^2) I^2</math>, where G is the conductance and I is the current. In this regime, the term <math>(S_G/G^2)</math> is approximately equal to <math>c^{-(1/2)}</math>. |

+ | |||

+ | On the other hand, in the low concentration regime, the DeBye length is comparable or larger than the radius of the pore, which means that the ions in the entire cross section of the pore are affected by the fluctuations and the local ion concentration and conductance are independent of the bulk concentration and proportional to the surface charge. However, since the surface charge slightly decreases with electrolyte dilution, the noise is bounded by the limit of constant surface charge (when the conductance and thus its fluctuation, <math>S_G</math>, are constant) and the limit of no surface charge effect (when the conductance and its fluctuations vary proportionally to c). This model indeed agrees with the experimental results. | ||

+ | |||

+ | ''Effect of pH'' | ||

+ | |||

+ | According to the data, the noise level peak is similar to that observed in protein ion channels, in which the previous studies attributed the peak to the protonization reactions of individual protein residues in the channel. The silicon nitride surface has several amphoteric silicon oxide groups which become active in the range of pH in the experiment. The minimal conductance occurs at the pH at which the densities of positive and negative surface groups balance, and the point is called ''point of zero charge''. Away from this point, the dominating charge species increases the local concentration of charge carriers and total conductance. For example, if the surface has net negative charge, hydrogen ions will be attracted to the nanopore walls, lowering the local pH. | ||

+ | |||

+ | The equations related to this model are as followed: | ||

+ | |||

+ | <math>SiO^- + H^+ \rightleftharpoons SiOH</math> when <math>k_R</math> is the association rate constant and <math>k_D</math> is the dissociation rate constant of this reaction. | ||

+ | |||

+ | <math>SiOH + H^+ \rightleftharpoons SiOH_2^+</math> when <math>l_R</math> is the association rate constant and <math>l_D</math> is the dissociation rate constant of this reaction. | ||

+ | |||

+ | <math>K=k_D/k_R=\frac{N_{SiO^-}[H^+]_0}{N_{SiOH}}=10^{-pK}</math> | ||

+ | |||

+ | <math>L=l_D/l_R=\frac{N_{SiOH}[H^+]_0}{N_{SiOH_2^+}}=10^{-pL}</math> when <math>[H^+]_0</math> is the hydrogen ion activity at the surface, <math>N_i</math> is the density of surface sites in state i. | ||

+ | |||

+ | <math>[H^+]_0=[H^+]_{bulk} e^{-\beta \mathit{e} \psi_0} </math> | ||

+ | when <math>[H^+]_{bulk}=10^{-pH}</math>, <math>\psi_0</math> is electrical potential at the pore surface, <math>\beta=(kT)^{-1}</math> is thermodynamic factor, <math>\mathit{e}</math> is the unit of elementary charge, and <math>\beta \mathit{e} \psi_0</math> is the energy required for a positively charged ion to get closer to the surface. | ||

+ | |||

+ | <math>\Gamma=sum N_i</math> is the total density of surface active sites, <math>\sigma</math> is the surface charge density. | ||

+ | |||

+ | These equation result in the generalized form of the Behrens-Grier equation | ||

+ | |||

+ | <math>10^{pL-pH} (\sigma-\mathit{e}\Gamma) e^{-2\beta \mathit{e} \psi_0}+\sigma e^{-\beta \mathit{e} \psi_0}+10^{pH-pK} (\sigma-\mathit{e} \Gamma)=0</math> | ||

+ | |||

+ | Consequently, the potential at the nanopore surface, <math>\psi_0</math>, can be related to the potential at the double layer, <math>\psi_D</math>, by the Stern capacitance, <math>C_s</math> in the Stern model: <math>\psi_0-\psi_D=\sigma/C_s</math> | ||

+ | |||

+ | Then, according to the Grahame equation, | ||

+ | |||

+ | <math>\sigma(\psi_D)=\frac{2\epsilon\epsilon_0}{\beta\mathit{e}\lambda_D}sinh(\frac{\beta\mathit{e}\psi_D}{2})</math> | ||

+ | |||

+ | The conductance can be calculated from the solution of Poisson-Boltzmann equation |

## Revision as of 22:19, 29 November 2011

*Entry by Pichet Adstamongkonkul, AP 225, Fall 2011*

*work in progress*

Reference:

**Title**: Probing Surface Charge Fluctuations with Solid-State Nanopores

**Authors**: David P. Hoogerheide, Slaven Garaj, and Jene A. Golovchenko

**Journal**: Physical Review Letters, 2009, Vol. 102, No. 25

## Contents

## Summary

Noise characteristics in nanopores are attributed to the dynamics of both pore and electrolyte, and often times the noise interferes with DNA and protein detections. The analysis in this study suggested that the current noise in solid-state nanopores in the range of 0.1-10 kHz may come from the surface charge fluctuations. The authors also proposed a model of protonization of surface functional groups and tested its validity. The method is quite sensitive; the local surface properties can be examined and single-molecule detection can be optimized.

## Methodology

The nanopores were fabricated to be single, hourglass-shaped channels within silicon nitride film, separating two compartments of KCl electrolyte. The measurement was done via Ag/AgCL electrodes positioned in each compartment.

## Results

The measurements indicated that the noise detected was intrinsic to the nanopore surface. From the Power Spectral Densities (PSDs), in the absence of applied voltage across the membrane, only thermal noise and high-frequency capacitive noise were observed, whereas, in the presence of the applied voltage, the conductance fluctuations, including <math>1/f</math> noise, appeared. In addition, frequency-independent noise between 0.1 kHz and RC filter cutoff at 20 kHz was also detected. It was found that the latter so-called 'white noise' resulted from the conductance fluctuations and was not from the electronics, electrodes, or analysis.

By varying the electrolyte concentration, the noise characteristics changes. The authors mentioned that at high concentrations, the noise varies as <math>c^-3/2</math> and the conductance, determined from the slope of the I-V curve, varies proportionally with c, the electrolyte concentrations. In contrast, at concentrations lower than 100mM KCl, both noise and conductance deviate from the high-concentration behaviors.

## Discussion

*Effect of ionic concentration*

The change in behavior of the noise and conductance in the two regimes can be explained by DeBye screening. At high concentrations, the DeBye length <math>\lambda_D</math> ~ <math>1/\sqrt{c}</math> is smaller than the radius of the nanopore and the number of charge carriers that are affected by the fluctuation changes follows the equation:

<math>c \cdot A_{ring}=c \cdot \pi\ \cdot (2 \cdot \lambda_D \cdot R- (\lambda_D)^2)</math> ~ <math>c \cdot \lambda_D</math> ~ <math>\sqrt{c}</math>

The conductance is proportional to the concentration. The current fluctuation, <math>S_I</math>, and the conductance fluctuation, <math>S_G</math>, in an Ohmic system are related by <math>S_I = (S_G/G^2) I^2</math>, where G is the conductance and I is the current. In this regime, the term <math>(S_G/G^2)</math> is approximately equal to <math>c^{-(1/2)}</math>.

On the other hand, in the low concentration regime, the DeBye length is comparable or larger than the radius of the pore, which means that the ions in the entire cross section of the pore are affected by the fluctuations and the local ion concentration and conductance are independent of the bulk concentration and proportional to the surface charge. However, since the surface charge slightly decreases with electrolyte dilution, the noise is bounded by the limit of constant surface charge (when the conductance and thus its fluctuation, <math>S_G</math>, are constant) and the limit of no surface charge effect (when the conductance and its fluctuations vary proportionally to c). This model indeed agrees with the experimental results.

*Effect of pH*

According to the data, the noise level peak is similar to that observed in protein ion channels, in which the previous studies attributed the peak to the protonization reactions of individual protein residues in the channel. The silicon nitride surface has several amphoteric silicon oxide groups which become active in the range of pH in the experiment. The minimal conductance occurs at the pH at which the densities of positive and negative surface groups balance, and the point is called *point of zero charge*. Away from this point, the dominating charge species increases the local concentration of charge carriers and total conductance. For example, if the surface has net negative charge, hydrogen ions will be attracted to the nanopore walls, lowering the local pH.

The equations related to this model are as followed:

<math>SiO^- + H^+ \rightleftharpoons SiOH</math> when <math>k_R</math> is the association rate constant and <math>k_D</math> is the dissociation rate constant of this reaction.

<math>SiOH + H^+ \rightleftharpoons SiOH_2^+</math> when <math>l_R</math> is the association rate constant and <math>l_D</math> is the dissociation rate constant of this reaction.

<math>K=k_D/k_R=\frac{N_{SiO^-}[H^+]_0}{N_{SiOH}}=10^{-pK}</math>

<math>L=l_D/l_R=\frac{N_{SiOH}[H^+]_0}{N_{SiOH_2^+}}=10^{-pL}</math> when <math>[H^+]_0</math> is the hydrogen ion activity at the surface, <math>N_i</math> is the density of surface sites in state i.

<math>[H^+]_0=[H^+]_{bulk} e^{-\beta \mathit{e} \psi_0} </math> when <math>[H^+]_{bulk}=10^{-pH}</math>, <math>\psi_0</math> is electrical potential at the pore surface, <math>\beta=(kT)^{-1}</math> is thermodynamic factor, <math>\mathit{e}</math> is the unit of elementary charge, and <math>\beta \mathit{e} \psi_0</math> is the energy required for a positively charged ion to get closer to the surface.

<math>\Gamma=sum N_i</math> is the total density of surface active sites, <math>\sigma</math> is the surface charge density.

These equation result in the generalized form of the Behrens-Grier equation

<math>10^{pL-pH} (\sigma-\mathit{e}\Gamma) e^{-2\beta \mathit{e} \psi_0}+\sigma e^{-\beta \mathit{e} \psi_0}+10^{pH-pK} (\sigma-\mathit{e} \Gamma)=0</math>

Consequently, the potential at the nanopore surface, <math>\psi_0</math>, can be related to the potential at the double layer, <math>\psi_D</math>, by the Stern capacitance, <math>C_s</math> in the Stern model: <math>\psi_0-\psi_D=\sigma/C_s</math>

Then, according to the Grahame equation,

<math>\sigma(\psi_D)=\frac{2\epsilon\epsilon_0}{\beta\mathit{e}\lambda_D}sinh(\frac{\beta\mathit{e}\psi_D}{2})</math>

The conductance can be calculated from the solution of Poisson-Boltzmann equation