1
Electrochemical transport properties of a cone-shaped nanopore: revisited

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

2
There are reports in the literature that a single cone-shaped nanopore generated in a polymer foil separating two equally concentrated dilute aqueous potassium chloride solutions can reach high and low stationary electrical conductivity states respectively depending on the sign of the applied electrical potential.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

3
On the basis of published data it has been argued (D. Woermann, Phys. Chem. Chem. Phys., 2003, 5, 1853) that this phenomenon can be understood in terms of a well established model describing the electrochemical transport properties of polyelectrolyte membranes (“model of the membrane with narrow pores”).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

4
In the present contribution experimental evidence is presented which gives strong support to these arguments using a model system.

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa1

5
Based on the “model of the membrane with narrow pores” a composite membrane is constructed mimicking the structure and electrochemical function of an ensemble of conical nanopores.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met1

6
It is found that the characteristic electrochemical transport property of the composite membrane is that of a cone-shaped nanopore.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con1

Introduction

7
In recent publications observations of a non-ohmic (diode-like) current voltage characteristic of membranes carrying a single cone-shaped nanopore in contact with equally concentrated dilute aqueous potassium chloride solutions have been reported.1–4

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

8
The nanopore is produced by an advanced track etch technique.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

9
The following values characterizing the pore geometry are given in the literature: length of the pore, δ ≈ 10 μm; diameter of the base of the conical pore, 2rbase≈500 nm; tip of the conical pore 2rtip ≈ 2 nm–10 nm.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

10
It has to be kept in mind that the diameter of the tip of the cone-shaped pore is not well defined.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

11
It cannot be smaller than the diameter of the latent ion track in the membrane matrix.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

12
For polymeric organic materials an estimated value of the diameter of a latent ion track of about 10 nm is given.5

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

13
In ref. 6 the transport properties of single cone-shaped pores described in the literature are compared with predictions of the “model of the membrane with narrow pores”.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

14
It is concluded that these properties can be rationalised on the basis of this model.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

15
The arguments in ref. 6 and ref. 7 are based on three assumptions:8,9 (a) The internal surface of a cone-shaped pore carries negative charged groups (–COO− groups) which are generated by the etching process.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

16
The fixed charges are compensated by mobile counterions (e.g. K+, Na+ ions) (b) In the narrow tip region of the cone-shaped pore there exists a homogeneous positive electrical charge density ρtip which is formed by the thermal motion of the mobile counterions (ρtip = −FωXtip; ωXtip = −(c+ − c−)tip: F, Faraday number; ω, sign of the fixed charges, here ω = −1; Xtip, fixed ion concentration in the tip region ([mol cm−3]); (ci)tip, concentration of mobile on species i in the tip region).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

17
The concentrations of the mobile ions in the tip region determine the electrical conductivity of the pore fluid in that region.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

18
It is independent of the direction of the electrical current flow across the tip region.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

19
(c) In the wide region of the cone-shaped pore the positive space charge density does not extend into the lumen of the wide region.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

20
It is limited to a narrow region near the internal pore wall.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

21
The state of the pore fluid away from the region close to the pore wall is not influenced by the presence of the positive space charge.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

22
Under electrical current flow across the pore the stationary electrical conductivity of the pore fluid (and the stationary the electrolyte concentration) in the wide region of the cone-shaped pore will depend on the direction of the flow of electrical current.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

23
This is caused by the fact that value of electrical transference number t+,tip of the counterions in the tip region is larger than the corresponding transference number t+,wide in the wide region of the pore (e.g.cbulk/Xtip ≪1,t+,tip ≈ 1; t+,wide ≈ 0.5; free solution; cbulk, electrolyte concentration in the bulk phase of the composite membrane).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

24
In this contribution results of experiments are reported which are carried out with a composite track etched membrane mimicking the structure and function of a conical pore.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met2

25
The composite consists of two membranes carrying straight track-etched pores with the same pore diameter (e.g. ≈100 nm) arranged in series (see insert in Fig. 1 and Fig. 2).

Type: Method | Advantage: None | Novelty: New | ConceptID: Met2

26
The track-etched pores of the (ω = −1)-membrane (membrane m2) are filled with a polyelectrolyte gel carrying negatively charged fixed ionic groups (- SO3−groups).

Type: Method | Advantage: None | Novelty: New | ConceptID: Met2

27
This membrane is assumed to model the electrochemical transport properties of tip region of a cone-shaped pore.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met2

28
The pores of the (ω = 0)-membrane (membrane m1) are not treated.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met2

29
That membrane is assumed to model the electrochemical transport properties of the wide region of a cone-shaped pore.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met2

30
The influence of the fixed -COO− charges at the internal surface of the wide pores of membrane m1 on the state of the pore fluid is neglected.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met2

Experimental section

31
The matrix of the membranes used in this study consists of a poly(ethylene terephthathalate) foil (abbreviated PET) with a thickness of about δ ≈ 10μm.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

32
Track etched membranes with straight pores are produced from this foil.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

33
Three sets of membranes with different pore diameters are used (effective pore diameters: 50 nm, 100nm, and 200 nm; pore density n ≈ (2.5 ± 0.5) × 108 cm−2).

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

34
Several membranes are cut from stocks of irradiated and etched PET foils.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

35
The stocks (charge number, 586) have been produced by Apel (Felerov Laboratory of Nuclear Reactions, JINR, 141980, Russia).

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

36
Both sides of several track membranes with the same pore diameter are treated with a solution of perfluorinated ion exchange powder Nafion (5 wt.% in a isopropanol/water mixture; Sigma Aldrich Com).

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

37
Nafion carries -SO3− groups and is insoluble in water.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

38
This treatment has the purpose to fill the pores of the membrane with a gel carrying negatively charged ionic groups.10

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

39
The success of this treatment is checked by measuring the electrical potential difference Δφ0 (=φ′ − φ ″)0 across a treated membrane [(ω = −1- membrane] separating two bulk phases formed by two aqueous NaCl solutions of different compositions using two identical Ag/AgCl /KCl (aq)// electrodes with NaCl salt bridges ((′), left bulk phase; (″), right bulk phase).

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

40
The NaCl concentration c′s in the left bulk phase is lower by a factor of 10 than the corresponding NaCl concentration c″s in the right bulk phase.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

41
The mean electrolyte concentration of the solution 〈cs〉 is small compared with the estimated fixed ion concentration X in the NAFION gel plugging up the pores of the track etched membrane gel.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

42
Values of Δφ0 in the range 45 mV<Δφ0<52 mV are found (c′s = 5 × 10−3c+; c″s = 5 × 10−2c+; c+ = 1 mol dm−3).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

43
This indicates that from an electrochemical point of view the track etched (ω = −1)-membranes have the properties of a cation ion exchange membrane predominantly permeable to counterions (Na+ ions)(Δφ0 = (2.303RT/F)log(c″s/c′s); (T ≈ 300 K; R, universal gas constant; F, Faraday number; T, thermodynamic temperature; c″s/c′s = 10; Δφ0 ≈ +58 mV).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

44
Under electrical current flow the fraction t+ ( = I+/I) of the electrical current I carried by the Na+ ions across the (ω = −1)-membrane is expected to be close to t+ ≈ 1.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

45
Independent experiments show that the I–Δφ curve of the track-etched (ω = −1)-membranes prepared in a manner just described are symmetric (linear) with respect to a change of the direction of an electrical current flow.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

46
Corresponding measurements of the membrane potential Δφ0 of untreated track etched (ω = 0)-membranes show that the observed electrical potential differences Δφ0 is small (Δφ0<5 mV).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs2

47
It is identified with a diffusion potential.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

48
It is concluded that the electrochemical state of the electrolyte solution filling the pores of the untreated membranes is not influence by the presence of the fixed -COO− charges at the pore wall.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con2

49
Under electrical current flow the fraction of the electrical current I carried by the Na+ across (ω = 0)-membranes is expected to be close to that in free solution (for c′s = c″s, t+ ≈ 0.5).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

50
A composite track etched membrane is formed by placing an untreated (ω = 0) -track etched membrane and a treated (ω = −1)-track etched membrane wetted by distilled water with their flat sides on top of each other.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

51
Care is taken to prevent air bubbles to be trapped between the surfaces of the two membranes.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

52
The composite membrane is placed between two equally concentrated and mechanically stirred NaCl solutions (c′s = c″s = 5 × 10−2c+; Plexiglass cells with rubber gaskets; volume of each bulk phase, 160 cm3; effective membrane area a = 2.5 cm2).

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

53
Two titanium grid electrodes with a layer of platinum deposited onto the surface of the grid are immersed into the bulk phases.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

54
These electrodes are used to apply an electrical current under galvanostatic conditions.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

55
The total net amount of electrical charge transported across a membrane during the measurements of one stationary current- voltage curve is so small that a possible influence of pH changes in the bulk phases caused by the reactions at the electrodes on the electrochemical measurements can be neglected.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

56
Two identical Ag/AgCl/KCl(aq)// electrodes in contact with Luggin capillaries filled with the solution contained in the bulk phase are used to measure the electrical potential difference across the membrane.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

57
The tips of the two Luggin capillaries are positioned reproducibly opposite to each other at a fixed distance from the surface of the composite membrane.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

58
A typically stationary I–Δφ curve of a composite membrane with a pore diameter of 100 nm is shown in Fig. 1.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs3

59
The non-ohmic character of the I–Δφ curve is clearly visible.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res4

60
The insert in Fig. 1 shows schematically the experimental set up.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

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It also shows schematically the corresponding orientation of a single cone-shaped pore for which a similarly shaped I–Δφ is found.1–3

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

62
The shape of corresponding I–Δφ curves of composite track etched membranes with diameters of 50 nm and 200 nm is very similar to that of a composite membrane with a pore diameter of 100 nm.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res5

63
These curves are not shown.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs4

Discussion

64
The stationary I–Δφ curve shown in Fig. 1 indicates that a composite tack-etched membrane reach high conductivity states when the electrical potential difference Δφ (=(φ′ − φ″)) across the membrane has negative values.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

65
Positive charge carriers (Na+ ions) are transported from the right bulk phase into the left bulk phase.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

66
In the pores of the (ω = −1)-membrane the electrical current is transported predominantly by Na+ ions (cbulk/Xtip ≪1; c+,tip ≫ c−,tip).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

67
They are the counterions of the fixed charges of the polyelectrolyte gel filling the pores (t+(ω = −1)-membrane ≈ 1).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

68
This leads to an increase of the concentration of NaCl in the pores of the (ω = 0)-membrane.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

69
Within the pores of the (ω = 0)-membrane the electrical current is carried by Na+ ions as well as by Cl− ions (t+(ω = 0)-membrane ≈ 0.5).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

70
A stationary high conductivity state of the composite membrane is reached when the concentration profile of NaCl in the (ω = 0)-membrane has reached a time independent value.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

71
In this state the mean NaCl concentration in the pores of the (ω = 0)-membrane is higher than the NaCl concentration in the bulk phases.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs5

72
This is shown schematically in Fig. 2.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res8

73
The composite membrane reaches a low conductivity state when the electrical potential difference Δφ = (φ′ − φ″) across the membrane has positive values and positive charge carries (Na+ ions) are transported from the left into the right bulk phase.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res9

74
Since the value of transference number of the Na+ions within the (ω = − 1)- membrane is higher than that within the (ω = 0)-membrane the NaCl concentration in the pores of the (ω = 0)-membrane decreases.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res9

75
A stationary low conductivity state of the composite membrane is established when the concentration profile of NaCl in the (ω = 0)-membrane has reached a time independent value.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs6

76
In this state the mean NaCl concentration in the (ω = 0)-membrane is lower than the NaCl concentration in the bulk phases (see Fig. 2).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs6

77
In summary, the transition from a high conductivity state to a low conductivity state is caused mainly by changes of the mean electrolyte concentration and consequently by the electrical conductivity in the pores of the (ω = 0)-membrane.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con3

78
The (ω = 0)-membrane and the (ω = −1)-membrane are not bound together permanently but are separated by a thin film of an aqueous solution.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met3

79
Furthermore, most pores of he composite membrane do not run across both membranes uninterruptedly.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met3

80
These complications can be taken into account by assigning an effective thickness to the (ω = 0)-membrane which is larger than its geometrical thickness.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met3

81
This does not influence the interpretation the stationary I–Δφ curve just described.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met3

82
Under electrical current flow an electroosmotic volume flow Jv across the (ω = −1)-membrane could take place (Δφ > 0, Jv directed from left to right; (Δφ < 0, Jv directed from right to left).

Type: Method | Advantage: None | Novelty: New | ConceptID: Met4

83
This would not change the interpretation given because stationary I–Δφ values are considered.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met4

84
Under electrical current flow a change of concentration of NaCl will occur in the region to the right of the phase boundary (ω = − 1)-membrane/ bulk phase (″) at the mouth of the pores filled with the polyelectrolyte gel plugs.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met5

85
This concentration change is caused by the change of the electrical transference number of the Na+ ions at this phase boundary.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met5

86
It is argued that the influence of this concentration polarisation on the shape of the I-Δφ curve can be expected to be small.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

87
The solid angle for the transport of NaCl toward the mouths of the pores at Δφ < 0 and away from the mouths of the pores for Δφ > 0 is large compared with the corresponding solid angle within the (ω = −1)-membrane.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

88
This minimises the possible influence of a concentration polarisation at the interface (ω = −1)-membrane/ bulk phase (″).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

Conclusions

89
It has been argued in refs. 6 and 7 that the wide region and the tip region of a single cone-shaped nanopore have different electrochemical transport properties and that the combination of both properties generate the typical non-ohmic electrical current/voltage characteristic of a pore.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

90
It has been argued further that the “model of the membrane with narrow pores” can be used to describe thee transport properties of the two regions.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

91
To test this concept experimentally a composite membrane with two membranes (m1 and m2) arranged in series is constructed.

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa2

92
Membrane m1 is prepared to have the electrochemical properties of the wide region of the cone-shaped pore.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met6

93
Membrane m2 is prepared to have the electrochemical properties of the wide region of the cone-shaped pore.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met6

94
When the composite membrane separates two equally concentrated dilute aqueous potassium chloride solutions it reaches high and low stationary electrical conductivity states respectively depending on the sign of the electrical potential difference applied across the composite membrane.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res10

95
This experimental finding strongly supports the arguments presented in ref. 6 and ref. 7: In deed, the electrochemical transport properties of a cone-shaped nanopore reported in refs. 1–4 can be explained on the basis of the “model of the membrane with narrow pores”.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con4