Physical Model for Rapid and Accurate Determination of Nanopore

Subscriber access provided by UNIVERSITY OF WISCONSIN-MILWAUKEE

Article

A physical model for rapid and accurate determination of nanopore size via conductance measurement Chenyu Wen, Zhen Zhang, and Shi-Li Zhang ACS Sens., Just Accepted Manuscript • DOI: 10.1021/acssensors.7b00576 • Publication Date (Web): 04 Oct 2017 Downloaded from http://pubs.acs.org on October 4, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

ACS Sensors is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Sensors

A physical model for rapid and accurate determination of nanopore size via conductance measurement Chenyu Wen,† Zhen Zhang,† Shi-Li Zhang†* † Solid-State Electronics, The Ångström Laboratory, Uppsala University, SE-751 21 Uppsala,

Sweden. KEYWORDS: nanopores, physical model, effective transport length, algebraic solution, conductance measurement in electrolyte.

ABSTRACT: Nanopores have been explored for various biochemical and nanoparticle analyses, primarily via characterizing the ionic current through the pores. At present, however, size determination

for

solid-state

nanopores

is

experimentally tedious

and

theoretically

unaccountable. Here, we establish a physical model by introducing an effective transport length, Leff, that measures, for a symmetric nanopore, twice the distance from the center of the nanopore where the electric field is the highest to the point along the nanopore axis where the electric field falls to e-1 of this maximum. By G = σ

S0 , a simple expression S0 = f (G,σ ,h, β ) is derived to Leff

algebraically correlate minimum nanopore cross-section area S0 to nanopore conductance G, electrolyte conductivity σ and membrane thickness h with β to denote pore shape that is

ACS Paragon Plus Environment

1

ACS Sensors

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 29

determined by the pore fabrication technique. The model agrees excellently with experimental results for nanopores in graphene, single-layer MoS2 and ultrathin SiNx films. The generality of the model is verified by applying it to micrometer-size pores.

Nanopore structures and nanopore-based devices hold great promise in application areas concerning health, energy, environment, etc. First used as an erythrocyte counter in 1950s,1 the nanopore-based sensor technology has lately attracted tremendous attention as it offers high sensitivity and versatility.2 Today, examples of nanopore applications span from biological analysis,3 gas separation,4,5 water desalination,6,7 ion selective filtering,8,9 power generation,10-13 to optical antenna.14 Experimental demonstrations of various types of nanopore sensors include detection and analysis of biological molecules (DNA,15-18 RNA19,20 and proteins21-23), chemical molecules,24 ions,25 polymers,26 and nanoparticles.27-32 For such a wide range of applications, two major classes of nanopores have been widely studied: biological and solid-state. The present work has a focus on the latter that are enabled by the mature micro/nanofabrication technology. Usually, smaller solid-state pores can be manufactured in thinner membranes33-36 in order to boost the resolution in sensing. The solid-state nanopore technology is particularly attractive for its potential in producing high-density nanopore arrays as well as in integrating with on-chip electronic circuitry for control and data processing. The general concept of nanopore-based sensing is based on monitoring and characterizing the variation of ionic current passing through a pore caused by the spatial occupation of target molecules during their translocation. The information extracted from the measured ionic current is critically dependent on the size of the pore (diameter or equivalent, length that is usually


2

Page 3 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Sensors

determined by membrane thickness and shape), in addition to electrical and chemical properties of the pore as well as of the target molecules and their host electrolyte.37 Knowledge of the pore size is, hence, highly desirable. At present, high-resolution electron microscopy is the predominantly used technique to directly and precisely observe the size of a nanopore. However, high-fidelity imaging at nanometer-scale is both costly and tedious requiring expensive equipment and highly skilled operators. An alternative is to extract the nanopore size from its (ionic) conductance information, which relies on an accurate relationship between the two parameters, i.e., size and conductance (or resistance). Such a relationship is yet to establish despite extensive studies available in the literature34,37-42. A major challenge lies in the treatment of the access resistance in the vicinity of the nanopore with respect to the pore resistance. For cylindrical pores (i.e., with vertical sidewalls, hereon denoted cylinder) that can be realized using reactive ion etching, the most widely used model considers the resistance from the pore volume and that from the access region separately.37 This simplified treatment leads to an algebraic relationship that expresses the pore conductance as a sum of two terms one quadratically and one linearly dependent on the reciprocal of nanopore diameter.38,43 However, this model can give rise to excessively large errors in determining the pore diameter for relatively large pores unless unrealistic access resistance is assumed. This deficiency undermines the effort of extracting the pore size from the conductance measurement and invalidates the predictive capability of the model. Moreover, this model is not applicable to pores of other shapes than cylindrical. For circular pores with hour-glass shape sidewalls (hereon denoted hour-glass), which constitute another commonly observed nanopore shape usually resulting from ion/electron beam milling processing,34,39,44 an analytical solution can be derived by invoking a hyperbolic function.40 Major shortcomings with this approach are elaborated mathematics involved and insufficient


3

ACS Sensors

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 29

accuracy in determining the pore size especially the minimum cross-section area of an hour-glass pore. Solutions to nanopores of irregular shapes can also be sought for by solving integral equations.41 In addition to circular shape pores, Liu et al. have introduced correction parameters in the existing model38,43 to describe the conductance of triangular shape nanopores form in hBN membranes.42 A common deficiency with all the theoretical attempts so far is their focus on the geometrical effects thereby overlooks the essence of ionic transport driven by an electric field. It is, hence, understandable of their inability to serve the purpose of directly determining the pore size using the conductance measurement data. In this work, we demonstrate a physical model that essentially applies for all kinds of nanopores. We note that the region of the highest electric field in the nanopore-electrolyte system is responsible for the largest voltage drop and thereby dictates the electrical behavior of the system, including current distribution, pore conductance/resistance, translocation characteristics,45,46 noise properties,47,48 etc. Thus, our physical model is established first by analyzing the electric field distribution in the nanopore-electrolyte system and then by introducing the concept of effective transport length, Leff, in a resistor model. Simple algebraic solutions are obtained and compared with literature results for two typical pore shapes. The physical model By referring to the column of star-like cross-section illustrated with the assistance of the vertical dash lines in the middle of Figure 1a, our model takes on the well-known relationship for pore conductance, G:

G =σ

S0 Leff


(1)

4

Page 5 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Sensors

where, S0 and Leff represent, respectively, the minimum cross-section area and the length of the resistor extending from the nanopore to the two sides into the electrolyte, while σ is the conductivity of the electrolyte in the extended nanopore. The thickness of the membrane that hosts the nanopore, h, goes in Leff. As h is usually known, the objective of this work is to find an analytical expression for S0 of nanopores of any cross-section shape in the form of:

S0 = f (G,σ ,h, β )

(2)

with β as the pore shape parameter that is usually known once the pore fabrication technique is specified. The definition of Leff is found in Figure 1b for a symmetric system with a cylinder pore and with all parameters identical on the two sides of the membrane for illustration, but the principle can be readily extended to more complex systems with asymmetric designs including pore geometry and electrolyte properties. With this definition, Leff is equal to twice the distance from the center of the nanopore where the electric field is at its maximum magnitude to the point along the nanopore axis where the electric field falls to e-1 of this maximum. This model, though extremely simple as expressed in Equation 1, compares almost perfectly with the numerical simulation result using COMSOL Multiphysics (see Methods), see Figure 1c for a cylinder nanopore of 10 nm diameter (dp=10 nm) in a 10 nm thick membrane (h=10 nm). The small error, 2.1%, arises from the departure of electric field from being ideally exponential with distance as shown in Figure 1d. By virtue of the definition of Leff, an exact solution is attained for the electric field of ideal exponential character. Also shown in Figure 1d is how the electric field varies with distance for an hour-glass nanopore of minimum dp=10 nm, h=10 nm, and half wedge angle 60 (see more later for its definition).


5

ACS Sensors

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 29

To find S0 expressed in Equation 2, it is imperative to relate Leff to h and S0 since Leff is not a measurable parameter itself although its physical significance is well defined. In an electrolyte of relatively high salt concentration under low electric field without consideration of surface charge, σ can be regarded as homogeneous in the system, which is also a prerequisite for the validity of Equation 1. Since the density of ionic current is linearly proportional to the magnitude of electric field in accordance to Ohm’s law, they share the same distribution pattern. As there is no source along the path of the ionic current, the total current passing through any equi-field surface (e.g., S1) is equal to that through S0, see Figure 1a. In other words, on the specific equifield surface S1 where S1 = e × S0 holds, the current density and the electric field on S1 are both e-1 of their counterparts on S0. Hence, Leff is twice the distance from the pore center to S1 for a symmetrical system and the electrical problem is then now converted to a geometrical problem that can be readily solved analytically. According to COMSOL simulation, S1 assumes a dome shape that can be viewed as a flattened hemisphere (see Figure 1b) whose solution would involve elliptic integrations that are obviously too complex to pursue for the objective of this work. Instead, S1 is approximated by a hemisphere with its top section being replaced by a flat cap that resembles the geometrical form of the pore cross-section as illustrated in Figure 1a. To elucidate the mathematical derivations with physics backing, a cylinder nanopore of diameter dp is considered below. Thus, this 2 acc

approximation makes S1 a hemisphere of diameter D = 2 L +

d p2 4

with a flat circular cap of

area S0 of diameter dp that is vertically distanced from the pore opening by Lacc. The following relationship can then be obtained:


6

Page 7 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Sensors

S1 π Lacc D + S0 = =e S0 S0

(3a)

The solution Lacc=0.35dp is found to lead to an underestimate of G in comparison with COMSOL simulation. For very small pore openings (small dp) and/or with S1 sufficiently distanced from the pore in comparison with the pore diameter (Lacc>dp), the S0 term in S1 can be neglected so that S1 becomes a hemisphere of diameter D=2Lacc. Hence, the relationship below holds: S1 2π L2acc = =e S0 S0

(3b)

The solution Lacc=0.58dp leads to an overestimate of G. The reality obviously lies between the two examined cases and the simplified expression below is shown by numerical simulation (Figure S1 of Supporting Information) to give an improved accuracy: S1 2π L2acc + S0 = =e S0 S0

(3c)

with the solution Lacc=0.46dp. The significance of Equation 3c is beyond the improved accuracy it brings about. Its simple mathematical representation leads to algebraic solutions to S0 essentially also for any columnlike nanopores of non-cylindrical cross-section shape illustrated in Figure 1a. Non-cylindrical or irregular shapes can be generated by special process designs or simply is a result of poor process controllability in lithography, etching or milling, etc. In what follows, we will proceed with Equation 3c for cylinder and hour-glass nanopores, the two most commonly encountered experimental pores in the literature. The cylinder pores can be formed using reactive ion etching in SiNx or SiO2 membranes. They can also represent pores formed in atomic-thick single-layer


7

ACS Sensors

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 29

graphene or MoS2 membranes. The hour-glass pores are often generated when ion/electron beam milling processing is used.

Algebraic solution for S0 The discussion above with the solution to Equation 3 leads a generalized simple relationship: (4)

Leff = b + kd p

with b=h and k=0.92 for cylinder pores that are presented in Table 1 along with the sketch of a cylinder pore. For an hour-glass pore, its geometrical profile can be well described by the hyperbolic function but it involves complex mathematics.41 Here, we use two head-to-head identical truncated cones (abbreviated as truncone) to approximate the hour-glass shape (see Table 1) in order to attain a simple relationship in accordance to Equation 2. Here, two different cases with Leff ≤ h and Leff ≥ h need be treated separately. Details about the mathematical derivation as well as error analysis are given in Supporting Information Note and Figure S2-S4, while the results for b and k in Equation 4 are shown in Table 1 where θ is half of the wedge angle between the two truncated cones.

Table 1 Parameters b and k in Equation 4 for cylinder and truncone pores. Pore type cylinder

Schematics

ܾ

݇

H

0.92


8

Page 9 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Sensors

Leff ≤ h

0

0.92

90 90 − θ

truncone Leff ≥ h

h (1-

90 − θ ) 90

0.92

By substituting Equation 4 in Equation 3, the solution to dp is obtained:

dp =

dp =

1.84G + 3.39G 2 + 4π hGσ

, for cylinder pores

(5a)

90 , for truncone pores with Leff ≤ h 90 − θ

(5b)

πσ

3.68G

πσ

1.84G + 3.39G 2 + 4π hGσ (1 − dp =

πσ

90 − θ ) 90

, for truncone pores with Leff ≥ h (5c)

The minimum cross-section area of the pores can now be calculated by S0 = π

d p2 4

, which

remains algebraic and simple. Finally, for a column-like nanopore of irregular cross-section shape, cf. Figure 1a, it can be shown that its S0 solution resembles that derived from Equation 5a for cylinder pores.

Results and discussion The model is first validated by comparing with the numerical solution using COMSOL simulation. The results shown in Figure 2a-c are the comparisons between modeling (based on the model) and simulation (using COMSOL) for cylinder nanopores, while those in Figure 2d are


9

ACS Sensors

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 29

for truncone nanopores. The linear relationship between Leff and dp in Equation 4 holds for dp

Physical Model for Rapid and Accurate Determination of Nanopore

Recommend Documents