Double-Layer formation, ionic transport, charge accumulation

When light impinges the dye, the emitted electrons eventually reach the cathode after a time τ_e, charging it and producing an electric field and a voltage V_e; this field is proportional to the charge accumulated in a very thin layer on the electrode surface of the order of λ_e-e < 0.1 Å; here, n_eq_e = en_e, n_e, q_e, ε and A are the number and charge of the electrons, the dielectric permittivity of acetonitrile and the surface area; the thickness of the electron layer on the electrode λ_e-e is very small because of the tiny size of electrons and the fact that they can be transported out of the electrode through the interface by the redox couple (faradaic transport) leaving only a single layer on the metal surface and preventing the formation of a multi-layer structure. The field E_e moves the ions in the electrolyte toward the electrode of opposite charge and accumulates near it forming, in a time τ_i, a diffuse charge layer of thickness λ; this accumulation of ions in EDL produces the field and a voltage V_i; the field E_i is proportional to the accumulated charge n_iq_i (= n_iz_ie), where n_i, q_i, and z_i are the number, charge, and valence of the ions; because the fields and voltages are in opposition, the output voltage V_o is giving by V_o = V_e—V_i; according to this equation, V_e increases the output voltage, while V_i reduces it.

When the ions start moving towards EDL, the first layer that begins to fill is the Stern layer which is the layer closest to the interface [9]. The charge accumulation in the Stern layer involves high internal energies due to intense repulsive forces between charges of the same sign; there is a strong restriction imposed to the charge separation distance: there is a minimum approach distance d_min between charges, which limits the maximum concentration of charge that can be accumulated in a finite space; then, the filling of the Stern layer ends when the distance between ions reaches the value d_min; the rest of ions go to the next second layer, which is filled also respecting d_min; this process continues filling successive layers as close as possible to the interface; it is in this way that the EDL is formed: the Stern layer and a few adjacent layers form the EDL which has a thickness λ and is in contact with the interface; the restriction imposed by d_min produces non-uniform ions distribution with higher density near the charged surface, and with a diffuse structure (Bockris-Devanathan-Mueller model). The restriction imposed by d_min and the strong and attractive force between charged surfaces at both sides of the interface have interesting effects: a) produce a tight and orderly packing of ions in the diffuse layer, and b) the strong interfacial attractive force between charged surfaces stabilizes the whole EDL structure. Even when both phases are in physical contact with each other, the interface itself prevents the transport of ions through it (a non-faradaic process), avoiding positive and negative charges from mixing and canceling each other, but rather remaining separate forming a system of two charged surfaces separated by a dielectric; this structure resembles a capacitor, a particular type of capacitor called double-layer capacitor DLC or Helmholtz double-layer capacitor HDLC. This process produces a double-layer formation on both sides of the interface: on one side, an electrode containing a thin layer of electrons with a thickness λ_e-e, while on the other side, an electrolyte containing a diffuse layer of ions with thickness λ and in contact with the interface; the diffuse layer is made up of several layers, the first one and closest to the interface is the Stern layer which consists of ions strongly adsorbed on the interface due to strong, attractive electrostatic forces with the electrons in the electrode; in this layer, there are also solvent molecules which solvate the ions. The second layer, adjacent to the Stern layer, is composed of solvated ions and counterions and attracted electrostatically to the charge of the electrode with the corresponding screening effect produced by the Stern layer; in contract, the ions in the first layer are firmly anchored to the interface, in the second one these are loosely associated with the electrode and they can move in the fluid under the influence of electric interactions and thermal effects.

Shapes of the output voltage profiles

One of the most notable characteristics of the output voltage profiles is their initial shape; this is important because the cells are connected to different devices and the characteristics of the output signals can affect these; sometimes the shape is a rising exponential, while others times it is a decaying. This is produced by a competition between the arrival times of electrons τ_e and ions τ_i to their corresponding destinations. The arrival time τ_e is practically constant because the charges are always electrons at the same concentration at constant illumination, while the time τ_i depends on the size, charge, and concentration of various types of added ions introduced into the cell by the additives (mordants and brighteners). The added ions can be divided into two categories according to their physical characteristics: slow and fast ions. The slow ions are those that have a large size, low charge, and high concentration; these move at a small velocity (τ_i > τ_e), producing a considerable delay in the formation of EDL without any affectation to the deposition rate of electrons to the electrode, which increases the output voltage resulting in a rising profile. On the other hand, fast ions have a small size, high charge, and low concentration; they move rapidly (τ_i < τ_e) forming in short times EDL, which reduces the output voltage, resulting in a decaying profile. This variety of rising and decaying profiles can be seen in Fig 1A–1D; for example, in Fig 1D the dye is brazilwood and the brightener sodium metasilicate; from this set, two profiles are rising and three decaying; consequently, this effect can only be produced by concentration. From this discussion, it is possible to see that τ_i depends on the ratio R_i/q_i, where R_i is related to the viscous force and q_i to the electric force; it is necessary to include the dependence of τ_i with the concertation n_i. This is a peculiar characteristic of the ionic transport; due to the large size of the ions, their transport is modeled as a series of jumps to neighboring sites of the same size; however, when there are a large number of ions close to the ion that is jumping, they block the jump reducing the ionic transport; this is called “obstruction effect”; this means that the ionic conduction, i.e., τ_i, depends on the concentration n_i; this effect can be taken into account by introducing a factor n_i in the numerator in the expression for τ_i; then, τ_i ~ n_iR_i/q_i; because the ions were introduced at concentrations C1-C5: (1)

With this equation, it is possible to determine the arrival times of different types of ions to their destinations, and to predict the shape of the output profiles. In a time-concentration space, this equation corresponds to a straight line passing through the origin with a slope of R_i/q_i: for each type of ion, there is a straight line, as shown in Fig 2.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 1.

Voltage profiles of a) cochineal-metasilicate, b) cochineal-alum, c) cochineal-CT, d) brazilwood-metasilicate; the profiles were shifted for clarity.

https://doi.org/10.1371/journal.pone.0298776.g001

Download:

• PNG
  larger image
• TIFF
  original image

Fig 2. Plot of the equation τ_i ~ (R_i/q_i)C_i for four cations (Al³⁺, Sn²⁺, Na⁺, K⁺) at different concentrations C1-C5.

Fast ions (Al³⁺, Sn²⁺) produce decaying profiles (red); slow ions (Na⁺, K⁺) can produce either rising (blue) or decaying (red) profiles depending on concentration.

https://doi.org/10.1371/journal.pone.0298776.g002

A typical capacitor is a device that stores energy as electric field in a finite region of space; this is generally composed of two metal plates that delimit the electric field and are separated by a dielectric material. This device is governed by the relationship Q = CV, where Q, V, and C are charge, potential, and capacitance; the charge in a capacitor consists of an excess of electrons in one plate and a deficiency of them in the other and a polarized dielectric in between. On the other hand, the DLC has no metallic borders that delimit the electric field, but in its place is the double-layer structure which, by itself, plays that role: the double-layer does not produce a well-defined edge but a diffuse one; DLC is not a typical electrostatic capacitor. The double layer capacitor has received different names; double-layer capacitor DLC, Helmholtz double-layer capacitor HDLC or supercapacitor. The charge is accumulated at different speeds on the electrode and EDL: fast at the beginning and slower at the end; this is because the traveling charges have to overcome the field produced by the charges already accumulated. The rate at which the charges are being deposited depends on the charges already deposited: dq/dt ∞ -q; the solution of this differential equation assuming the q = 0 at t = 0 is q = q_o(1-e^-t/τ); because the charge and the voltage are proportional for electrons and ions: (2A, 2B)

These equations describe the filling kinetics of electrons at the electrode and ions at EDL. V_e and V_i are part of the output voltage V_o = V_e−V_i. There are two possible situations in the filling kinetics: for slow ions (τ_i > τ_e), they move so slowly that there is a delay in the EDL formation leaving unaffected the rate at which electrons are deposited on the electrode, which increases the output voltage; then, V_o can be approximated by V ≈ V_e, and from Eq 2A, resulting in a rising profile: (3A)

For fast ions (τ_i < τ_e), they form the EDL rapidly; however, a threshold voltage V_oe is required to initiate the ionic transport and once initiated, they quickly reach EDL while V_oe remains practically constant; then, using Eq 2B ; in this case, the output profile is decaying: (3B)

Eqs 3A and 3B correctly describe the shape of the output profiles; however, two exponentials are required to fit the experimental data due to the presence of different types of added ions: (4A) (4B) where τ₁, τ₂ are the relaxation times, V_o1, V_o2 the voltage amplitudes, and BL the baseline. It is necessary to include in these equations a term to fit the oscillations. The times τ₁ and τ₂ are related, but not equal, to the arrival times τ_e and τ_i.

Numerical estimation of DSSC parameters

It has been mentioned that the double-layer structure can store large amounts of energy and charge at a high electric field; to support this assertion, the values of important characteristic parameters associated with the performance of DSSC will be numerically estimated: stored energy, stored charge, electric field, and capacitance; the numerical estimation of these parameters was based on reported values of some basic quantities characteristic of the double-layer capacitor structure like: double-layer thickness λ, Stern layer thickness λ_SL, the distance between charged particles d_min in a charge accumulation. The separation distance δ between charged surfaces is δ = λ_SL + λ_e-e ≈ λ_SL; this distance is essentially given by the Stern distance λ_SL which is of the order of λ_SL ≈ 0.2 nm ≈ 2.0x10^-10 m [12]; this very small distance produces an intense attractive force F_I; this intense, interfacial force stabilizes the DLC structure causing accommodating ions in a tight, orderly manner very near to saturation conditions.

The basic quantities λ, λ_SL and d_min are reported in the literature for a wide variety of systems and different experimental conditions and have been grouped, in all cases, as ranges of numerical values associated; some of these values have been measured experimentally, while others have been obtained based on some theoretical model like the Debye-Huckel model [10]. From each of these ranges, a value was selected approximately in the middle range and assigned to the corresponding basic quantity; in this way, this value is representative of the basic quantity in question [10]. The estimated values of DSSC parameters obtained by this methodology provide a clearer idea of the potential applications of DLC. It is possible to mention, in advance, that the DLCs have a very appropriate structure to significantly increase the charge and energy that can be stored in a DL without losing its stability; an additional advantage of this type of structure is its extraordinarily small size which is of the order of Aλ where λ is the Debye length. These characteristics open up many interesting potential applications in different fields of science and technology: in the design of new, very high-performance batteries, in the analysis of body piezoelectric signals, in the development of a new generation of medical equipment, etc.

As mentioned, a charged surface in contact with an electrolyte forms a DLC where the charge, energy, electric field, and capacitance are confined to a small volume Aλ, being λ the double-layer thickness: (5)

The values of the basic quantities have a direct effect on the performance of the cell: the distance between charged surfaces δ ≈ λ_SL provides information with respect to the stability of the whole DLC structure; the thickness λ of DL is important because electric field, charge, energy, are limited to this distance, the area A of the charged surfaces contribute to the force between them. Once numerical values have been assigned to the basic quantities, the values of the parameters can be calculated. These parameters allow to estimate how much energy and charge can be stored in DLC; typical values of standard capacitors are reported in Table 1 for comparison purposes.

Download:

• PNG
  larger image
• TIFF
  original image

Table 1. Typical values for double layer and electrostatic capacitors.

https://doi.org/10.1371/journal.pone.0298776.t001

Numerical values will be assigned to the basic quantities. It has been reported values of λ and κ (= λ^-1) for different electrolytes at different concentrations and valences in aqueous solutions at 25°C [11]; from this published data, the values of λ are in the range λ ∈ [2.5x10^-10, 1.0x10^-8] m, and for its inverse κ = λ^-1, κ ∈ [1.0x10⁸, 4.0x10⁹] m^-1; then, values in the middle range were selected for λ and κ: λ = 10⁻⁹ m and κ = 10⁹ m^-1. Regarding the cell geometry, typical values were chosen for the area and separation distance: a typical area of 2x2 cm² was chosen, A = 2x2 cm² = 4x10^-4 m² and a distance d = 1 mm = 10⁻³ m; the electric permittivity of acetonitrile ε_acetonitrile = 3.32 x 10⁻¹⁰ Fm^-1. For the Stern layer, because this is built from solvated ions, the thickness of this layer has an average value [12] of 0.2 nm, i.e., λ_SL = 2.0x10^-10 m. The Stern layer is the layer that contributes the most to the interfacial force between the charged surfaces.

The accumulation of electrons in the electrode, although they are subject to the criterion of the minimum distance between charges, d_min, they generally form only a very thin single layer with a thickness of around λ_e-e ≤ 0.1 Å. The charges accumulated in the electrode and the electrolyte (Stern layer) are closely packed due mainly to the intense attraction force between electrons and ions; the ions in the Stern layer behave much like a two-dimensional ideal gas, accommodating an ion in an area of 10 nm²; this corresponds to a surface charge density σ_DL of [13]; the charges in this monolayer are separated by a distance d_min = √(10 nm²) = 3.16 nm. Then, there are two characteristic distances: ion-ion (d_min = 3.16 nm) and electron-ion (λ_SL = 0.5 nm); the ion-ion distance is approximately equal to electron-electron for monovalent ions.

Table 1 reports typical values for double layer and electrostatic capacitors; as can be seen, the energy and charge densities are several orders of magnitude higher with respect to the electrostatic one.

The electric potential Ψ and the electric field E are possibly the more measured and calculated quantities from theoretical models. The electric field in DL has been reported for different potentials and concentrations, and it is in the range E_DL ∈ [6.36x10⁶–1.51x10⁹] V/m, and for monovalent ions is 2.3x10⁷ V/m; choosing a value in the middle of this range, E_DL takes the value E_DL = 1.0x10⁸ V/m; even though this value of the electric field can be used to determine secondary parameters such as the surface charge density σ = E/ε, the potential Ψ_o will be used to calculate the secondary parameters since Ψ is the quantity more experimentally measured. Based on the Debye-Huckel theory, the potential at 25°C for monovalent ions Ψ_o = kT/e = 25.7 mV has been reported, and the potential drop occurs at 4.4 nm. In many situations, the potential Ψ_o varies in the range Ψ_o ∈ [10, 200] mV [14]; then, the value in the middle of the range is: Ψ_oDL = 100 mV = 0.1 V for DL and the potential Ψ_elec can be obtained by evaluating in x = λ: Ψ_oelec = Ψ_o(x = λ) = Ψ_oDL/e = 0.04 V.

The first secondary quantity that will be estimated is the force between the two charged surfaces, i.e., the interfacial force F_I; the charges on these surfaces are on both sides of the electrode/electrolyte interface separated by a small distance of the order of λ_SL; due to the small distance between charges of opposite signs, the attractive interfacial force F_I is very strong, stabilizing the entire DLC structure and producing, as mentioned, tight and orderly packing of charges near the saturation conditions; the intense force F_I allows to increase the charge accumulation in DL without losing stability, with a concomitant increment in the stored energy. There are several different ways to determine the interfacial force F_I. The force between two charged surfaces depends on the electric field between the layers . Several aspects must be considered about the validity of this expression for the double-layer structure under consideration; in a cell, there are in total four charged surfaces: two couples (charged electrode)-(diffuse layer) which are in contact through the interface, i.e., these charged surfaces are in two different phases, and the electric fields in each of these phases are different; then, it is not clear the validity of this expression for calculating F_I. Due to this, F_I will be calculated using the Coulomb law for the electrostatic attraction force between the electrons layer in the electrode and the ions layer in the EDL. The distance between the charged surfaces is δ = λ_SL + λ_e-e ≈ λ_SL = 0.2 nm; this distance is significantly smaller with respect to the distance d_min = √(10) = 3.16 nm between ions in the Stern layer, i.e., d_min it is six times larger with respect to λ_SL.

At such small distances, the charged surfaces cannot be considered as continuous charge distributions; then, to calculate the force F_I between charged surfaces, the Coulomb Law was applied to each electron-ion pair multiplied by the number of pairs to get the total force F_I; because d_min >> λ_SL, only the electron-ion pairs separated by a distance λ_SL (first neighbors) will be considered; the reason for this is because the charge in the Stern layer produces strong shielding effect on the electric field inside the double layer reducing the contribution of layers far away from the interface; due this, the pairs separated by a distance √(λ_SL² + d_min²) (second neighbors) or greater contribute with 2–3%. F_I can be approximated as where there are ions in area A; the minus sign is because the force is attractive; this expression provides the minimum value for F_I because it only takes into account the interaction between close pairs, i.e., first neighbors; using this equation it is obtained: F_I = 909 N; however, taking into account the cross terms F_I = 1,020 N. This strong, attractive force stabilizes both charged surfaces, the whole DLC structure, and it is responsible for the very tight and orderly packing of ions in DL. This intense attraction force allows to infer the possibility of increasing the charge stored in DL without sacrificing its stability.

Even though the electric field has been experimentally measured and reported for different systems, it is possible to determine it based on the potential Ψ; this has the advantage that Ψ has been directly measured experimentally and widely reported. The electric fields E_DL and E_elec can be obtained based on the potential Ψ. Because the electric field depends on the position, their average values will be determined in DL and electrostatic regions: [0, λ] for E_DL and [λ, d-λ] for E_elec; because there are two interfaces, and there is a double contribution in the middle region, the electric fields calculated from has to be duplicated: = 1.368x10⁸ V/m; similarly, E_elec can be calculated: = 0.368x10⁸ V/m; these values are in the range experimentally reported for the electric field; as can be observed, 80% of the electric field is in DL (in a volume of 4x10^-13 m³) and 20% in elec (in a volume of 4x10^-7 m³). Based on these results, it is possible to obtain the charge and charge density in DL and elec. Near a charged surface, the electric field E and surface charge density σ are related by E = σ/ε; this is a general expression for values of E close to the charged surface; then, σ_DL = εE_DL; and ρ_oDL = σ_DL/λ = εκE_DL = 4.54x10⁷ C/m³; for elec σ_elec = εE_elec and ρ_oelec = σ_elec/d = εE_elec/10⁻³ = 12.2 C/m³. From these values, it is possible to obtain the electric charge: Q_DL = ρ_oAλ = 1.82x10^-5 C and Q_elec = ρ_oAd = 4.88x10^-6 C; as in the electric field case, 93% of the charge is in DL and 7% in elec.

The energy stored in the double-layers was determined using , where the integration limits were taken in the corresponding regions for E_DL and E_elec; because E and Ψ decay rapidly with the distance, to evaluate the integral, the integrand (Ψ/x)² was approximated by (Ψ/λ)²; the energy stored in two double-layer capacitors is: = 3.89x10^-7 J, and the corresponding energy density is u_DL = U_DL/Aλ = 0.97 x 10⁶ J/m³. For = 7.67x10^-9 J, and energy density u_elec = U_elec/Ad = 1.92x10^-2 J/m³; the energy density in DL is seven orders of magnitude bigger with respect to elec. In a cell, three capacitors are formed: two C_DL at the interfaces and an electrostatic capacitor in the middle region; these capacitances are: C_DL = εA/2λ, and C_elec = εA/a = εA/(d– 2λ) ≈ εA/d. The total capacitance C_T is: (6)

The capacitances in DL and elec have the values: = εAκ = 1.33x10^-4 F and = 1.33x10^-10 F; these capacitances differ by six orders of magnitude. This big difference is due to the low values of the Debye length λ: all these parameters depend linearly with κ = λ^-1 a factor in the range [10⁸, 4x10⁹]. Because Q = CV, when C increases, Q increases, that is, the amount of charge that can be stored in DL is significantly greater than in a standard capacitor: the small distance between charged surfaces, i.e., the intense attraction force F_I allows to store a lot of charge on DL without losing stability. These results can be compared with a standard capacitor with the same area and distance used in the previous calculations: A = 4x10^-4 m² and a separation distance of 10⁻³ m; for this standard capacitor: U = 1.66x10^-13 J, u = 4.15x10^-7 J/m³, E = 50 V/m, Q = 1.0x10^-6 C, ρ_o = 16.6 C/m³, C = 1.33x10^-10 F; in all cases, the parameters in the double-layer were improved by a factor of 10⁶ or higher with respect to standard capacitor. Fig 3 shows a histogram plot in the log scale of the parameter for DL and elec.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 3. Comparative histogram of numerical values of parameters in DL and elec regions.

https://doi.org/10.1371/journal.pone.0298776.g003

These numerical values support the mentioned that a large amount of charge can be stored in double-layer capacitors. The advantage of these estimations is to visualize potential important applications of this type of structure: DLC structures with thickness in the range of nanometers are the basic units through which it is possible to build systems to store large amounts of energy and charge; this can be achieved by connecting in series a large number of these basic capacitors. The obtained values of the parameters in DL will be discussed in relation to those obtained for the elec capacitor:

• Energy: the energy stored in a volume Vol_DL = Aλ = 4x10^-13 m³ is U_DL = 3.89ex10^-7 J; this is two orders of magnitude greater than the energy stored in volume Vol_elec = Ad = 4x10^-7 m³: U_elec = 7.67x10^-9 J; Vol_DL is 10⁶ times smaller than Vol_elec; then, the energy densities are: u_DL = 0.97x10⁶ J/m³ and u_elec = 1.92x10^-2 J/m³; the difference is a factor of 10⁸. According to these values, the DLC can store large amounts of energy.
• Electric Charge: The electrical charges are similar in both regions; Q_DL = 1.82x10^-5 C and Q_elec = 4.88x10^-6 C; however, the charge densities are significantly higher in DL with respect to elec: ρ_oDL = 4.54x10⁷ C/m³ and ρ_oelec = 12.2 C/m³, i.e., the difference is a factor of 10⁶. The DL structure is ideal to be used in the manufacture of high charge storage batteries.
• Electric Field: the electric field in DL is E_DL = 1.37x10⁸ V/m, while in elec is E_elec = 0.37x10⁸ V/m; this means that 80% of the electric field is in DL region: the DLC is where the electric field is stored.
• Interfacial Force: this is an important quantity because this attractive force holds together the two charged surfaces, providing stability to the DL structure; this interfacial force is F_I = 1,020 N, which is an intense force with respect to the small size of the system 4x10^-13 m³. this value is considerable larger than that previously obtained, 112.5 N; the large difference is due to the small thickness of the Stern layer.
• Capacitance: because the thickness in the DL region is of the order of 10⁻⁹ m, the capacitance C_DL is significantly larger with respect to C_elec: C_DL = 1.33x10^-4 F and C_elec = 1.33x10^-10 F. For a capacitor Q = CV, a large capacitance allows storing high amounts of charge.
• Volume: the DL volume is Vol_DL = 4x10^-13 m³, while the volume for the middle region is Vol_elec = 4x10^-7 m³.

In all determinations of double-layer parameters appear, in a natural way, the main characteristic of the DL, its thickness λ, or equivalently κ = λ^-1; κ depends linearly on the valence and concentration Eq (5): if the valence z increases, κ increases; then, the initial assignment to basic quantities must take into account this dependence: for monovalent ions (Na⁺, K⁺) κ = κ_i, for divalent (Sn²⁺) κ = √2κ_i, for trivalent (Al³⁺) κ = √3κ_i; these corrections have to be made depending on the ions present in the system.

Based on the results obtained for energy, charge, electric field, and capacitance, it is clear that DLC has important technological applications; it is possible to design devices with high output voltages and with large amounts of stored energy and charge all in a reduced space, by staking in series a considerable number of DLCs. However, this configuration brings several technological problems; even when DLC can store, by itself, large amounts of energy and charge and maintain intense electric fields and voltages, the energy must be recoverable within a specific charge-discharge voltage window; this voltage window depends on the decomposition voltage of the electrolyte produced by the electrolysis; the electrolysis voltage depends on the medium in which the ions are found in the electrolyte: for organic solvents is around 3 V.

To have a supercapacitor with high operating voltages and/or large amounts of stored energy, as required at a commercial or industrial level, a bank of many small and identical DLCs stacked in series is required. For n identical DLC capacitors C_i connected in series, the total voltage is V_T = nV_i, and the total capacitance is ; from these expressions, it is possible to obtain a total stored energy: U_T = nU_i. Both quantities, energy and voltage, increase linearly with the number of capacitors in the stack; then, with n large enough, it is possible to design devices that satisfy the voltage and/or energy requirements. However, even when this seems feasible, several technological problems must be solved. The most difficult of these is related to making and stacking n small identical DLCs: these have to be identical, with the same capacitance and internal resistance. When a stack of DLCs is overloaded, capacitors with high resistance and low capacitance are at risk of decomposition and gas production, seriously affecting the performance of the entire stack. Regardless, this is just a reproducibility issue that the industry always faces. The battery industry faces similar problems because these are made by stacking voltage cells; then, in the near future, these types of DLC devices will be quotidian.

Chemical potential μ and chemical capacitance C_μ

A system formed by a charged surface in contact with an electrolyte gives rise to DLC; the first theoretical model to address this problem was the Debye-Huckel, providing an analytical expression for the electric potential and charge density and where, from the Poisson equation, ρ_o = εκ²Ψ_o. The DLC provides the main contribution to C_μ and, since this capacitance depends on the voltage V, this allows obtaining the relationship between C_μ and V. To achieve this goal, it is necessary to determine the chemical potential μ of an ions’ system in the double-layer; these ions follow the Boltzmann Distribution Law: . Due to long-range interaction, this system cannot be considered ideal; because the potential and charge density decay rapidly with the distance x, this is measured from the interface. The change in chemical potential Δμ_j of an ion j surrounded by n_i ions is given by [10]; solving for κ: (7A, 7B) where a = d_min, z_j, the valence, and q_j the charge of the ions, and β = z_jq_j²/4πε.

At the end of the Chemical Potential μ and Chemical Capacitance C_μ section, an analytical expression will be obtained between the chemical capacitance C_μ and the output voltage and the mordant concentration [%Al³⁺]; cochineal and brazilwood were used as dyes and alum as mordant. The chemical capacitance C_μ is defined as where μ_i is the internal chemical potential. Using the Boltzmann Distribution Law and the definition of C_μ, it is easy to obtain: ; C_μ depends quadratically on the more important parameter of the double layer: κ. Using Eqs (7A and 7B), defining the output voltage V = Ψ_o−Ψ measured from the interface and assuming, as in the Debye-Huckel approximation, that the potential Ψ is small, zeΨ << kT, i.e., Ψ = Ψ_o[1 - κx + …], the Boltzmann Distribution Law can be written as: (8) which is more appropriate for this calculation. From the definition of C_μ, it is possible to write ; the quantity can be obtained using Eq (7B): , while the factor can be obtained using Eq (8) . Using these expressions, the final form for C_μ is: (9A, 9B) where , a = d_min, , m = q_e/kT. Eq (9A) predicts that the chemical capacitance depends exponentially on the voltage; this equation is the same as that reported by Bisquert & Fabregat, [15]. The experimental results report that the logarithm of C_μ vs V is a sigmoid function where the middle section, of such sigmoid function, is a straight line identical to Eq (9B) with a positive slope: in this region C_μ dominates the capacitance. The dependence of C_μ on temperature has been experimentally measured in the range 0 to 60°C and reported by Bisquert & Fabregat, [16]; it coincides with that obtained here: m = q/kT.

Using Fig 1B, it was possible to obtain the dependence of the output voltage as a function of the [%Al³⁺]; Fig 4 shows, in red a linear dependence of voltage with [%Al]: V = 0.021–0.0013[%Al³⁺] with R = 0.97978; based on this equation, it was possible to obtain the dependence of Ln (C_μ) as a function of [%Al³⁺] for cochineal-alum samples as shown in blue; using this equation with Eq (9A) is obtained (10)

Download:

• PNG
  larger image
• TIFF
  original image

Fig 4. Dependence of: In red) the output voltage V, and in blue) Ln(C_μ/C_μo), as a function of [%Al], obtained from Eq (10) for cochineal-alum samples.

https://doi.org/10.1371/journal.pone.0298776.g004

In a semi-log space this equation is a straight line shown in blue (Fig 4). Unlike the graph reported by Bisquert et al. [16], where the relationship Ln(C_μ) vs V has a positive slope, in this case, the slope is negative. As mentioned, the Al³⁺ ions are very small with high charge and move rapidly, producing decaying profiles in all cases. These profiles reach, in short times, their final voltage values; then, if the concentration of Al³⁺ ions increases, V_i increases, and V decreases, i.e., they have an inverse relationship: the slope is negative for the line shown in red in Fig 4.

Mathematical model of unstable oscillations

Because the cell was suddenly illuminated, the output voltages show unstable oscillations on top of the signals; due to this, a theoretical model based on thermodynamics was developed to fit these oscillations; in addition to providing fit to the experimental data, this model provides information regarding the mechanisms involved in these unstable oscillations. It has been reported a mathematical model to describe the unstable oscillations found on top of the output voltage profile when the photocell is exposed to an abrupt dark-to-light transition [17]. This model is based on thermodynamics where there are two forces working in opposition. During the dye-photon interaction, electrons are released from the dye, leaving it with a net positive charge; these positive charges produce stretching of the dye molecules, which has an elastic recovery force f_elas. The stretched molecules can overlap each other, and this overlapping results in a local increment in the segment concentration given place to an osmotic pressure f_osmotic; this force tries to separate them to reduce the concentration gradient; the forces f_elas and f_osmotic are responsible for the oscillations in the output voltage; finally, the released electrons are recovered when the circuit is completed and the cell backs to its initial stage. The model presented here is based on the Flory-Krigbaum theory [18], and it was developed to study the stability of colloids. This model allows obtaining an appropriate expression to fit the oscillations present on the profiles: x(t) = x_o/[1+a_xsin²(ω_xt+δ_x)]; this term has to be included in Eqs (4A and 4B): (11A) (11B) where Eq (11A) corresponds to a decaying profile and Eq (11B) to a rising one.

The shape of the output signal od DSSC is important when these cells are connected to non-linear electrical charges, for example to inverter electronic circuits DC-AC [19]. When this DSSC voltage has harmonic distortion or its shape is unpredictable, it could have high content of harmonics of different orders that could affect the operation and performance of DSSCs and the connected non-linear charges. It is also possible that due to variations in the operating temperature of the DSSC, its performance is affected, since due to the type of ions present in the electrolyte, the equivalent resistive and capacitive circuits inside the cells they would be changing their conductance or capacitance and would modify the output impedance or resistance of the DSSC [20], as well as the charge constant (τ) of the double layer capacitors and charge transfer resistances, that form within this type of cells and could influence to transfer the maximum generated power by the DSSC itself to an electrical charge connected to its output. In this work the tests on the DSSC were carried out at a temperature of 25°C.

Materials, fabrication and characterization

Solar cells were manufactured as in Ref [21] using cochineal and brazilwood as natural dyes; additionally, four additives were included: two mordant, alum and Sn, and two brighteners, sodium metasilicate and cremor tartar. Five different concentrations of these four additives were prepared: C1 = 2.5, C2 = 7.5, C3 = 9.4, C4 = 14.0, and C5 = 19.0, all in mg/mL. The fabrication was done according to Ref [21], adjusting the thickness of the mesoporous to 34.7 microns. The electric characterization was done using a multimeter connected to the computer.