Effect of Alkyl Side‐Chain Length on Photovoltaic Properties of Poly(3‐alkylthiophene)/PCBM Bulk Heterojunctions

The morphological, bipolar charge‐carrier transport, and photovoltaic characteristics of poly(3‐alkylthiophene) (P3AT):[6,6]‐phenyl‐C61‐butyric acid methyl ester (PCBM) blends are studied as a function of alkyl side‐chain length m, where m equals the number of alkyl carbon atoms. The P3ATs studied are poly(3‐butylthiophene) (P3BT, m = 4), poly(3‐pentylthiophene) (P3PT, m = 5), and poly(3‐hexylthiophene) (P3HT, m = 6). Solar cells with these blends deliver similar order of photo‐current yield (exceeding 10 mA cm^?2) irrespective of side‐chain length. Power conversion efficiencies of 3.2, 4.3, and 4.6% are within reach using solar cells with active layers of P3BT:PCBM (1:0.8), P3PT:PCBM (1:1), and P3HT:PCBM (1:1), respectively. A difference in fill factor values is found to be the main source of efficiency difference. Morphological studies reveal an increase in the degree of phase separation with increasing alkyl chain length. Moreover, while P3PT:PCBM and P3HT:PCBM films have similar hole mobility, measured by hole‐only diodes, the hole mobility in P3BT:PCBM lowers by nearly a factor of four. Bipolar measurements made by field‐effect transistor showed a decrease in the hole mobility and an increase in the electron mobility with increasing alkyl chain length. Balanced charge transport is only achieved in the P3HT:PCBM blend. This, together with better processing properties, explains the superior properties of P3HT as a solar cell material. P3PT is proved to be a potentially competitive material. The optoelectronic and charge transport properties observed in the different P3AT:PCBM bulk heterojunction (BHJ) blends provide useful information for understanding the physics of BHJ films and the working principles of the corresponding solar cells.     

Nanoscale Morphology of Conjugated Polymer/Fullerene‐Based Bulk‐ Heterojunction Solar Cells

The relation between the nanoscale morphology and associated device properties in conjugated polymer/fullerene bulk‐heterojunction “plastic solar cells” is investigated. We perform complementary measurements on solid‐state blends of poly[2‐methoxy‐5‐(3,7‐dimethyloctyloxy)]‐1,4‐phenylenevinylene (MDMO‐PPV) and the soluble fullerene C₆₀ derivative 1‐(3‐methoxycarbonyl) propyl‐1‐phenyl [6,6]C₆₁ (PCBM), spin‐cast from either toluene or chlorobenzene solutions. The characterization of the nanomorphology is carried out via scanning electron microscopy (SEM) and atomic force microscopy (AFM), while solar‐cell devices were characterized by means of current–voltage (I–V) and spectral photocurrent measurements. In addition, the morphology is manipulated via annealing, to increase the extent of phase separation in the thin‐film blends and to identify the distribution of materials. Photoluminescence measurements confirm the demixing of the materials under thermal treatment. Furthermore the photoluminescence of PCBM clusters with sizes of up to a few hundred nanometers indicates a photocurrent loss in films of the coarser phase‐separated blends cast from toluene. For toluene‐cast films the scale of phase separation depends strongly on the ratio of MDMO‐PPV to PCBM, as well as on the total concentration of the casting solution. Finally we observe small beads of 20–30 nm diameter, attributed to MDMO‐PPV, in blend films cast from both toluene and chlorobenzene.     

Mechanical Properties of Conjugated Polymers and Polymer‐Fullerene Composites as a Function of Molecular Structure

Despite the importance of mechanical compliance in most applications of semiconducting polymers, the effects of structural parameters of these materials on their mechanical properties are typically not emphasized. This paper examines the effect of length of the pendant group on the tensile modulus and brittleness for a series of regioregular poly(3‐alkylthiophenes) (P3ATs) and their blends with a soluble fullerene derivative, PCBM. The tensile modulus decreases with increasing length of the alkyl side‐chain, from 1.87 GPa for butyl side chains to 0.16 GPa for dodecyl chains. The moduli of P3AT:PCBM blends films are greater than those of the pure polymers by factors of 2–4. A theoretical model produces a trend in the effect of alkyl side chain on tensile modulus that follows closely to the experimental measurements. Tensile modulus correlates with brittleness, as the strain at which cracks appear is 6% for P3BT and >60% for P3OT. Adhesion of the P3AT film to a polydimethylsiloxane (PDMS) substrate is believed to play a role in an apparent increase in brittleness from P3OT to P3DDT. The additive 1,8‐Diiodooctane (DIO) reduces the modulus of P3HT:PCBM blend by a factor of 3. These results could enable mechanically robust, flexible, and stretchable electronics.     

Solvent‐Induced Morphology in Polymer‐Based Systems for Organic Photovoltaics

Studies on the influence of four different solvents on the morphology and photovoltaic performance of bulk‐heterojunction films made of poly(3‐hexylthiophene) (P3HT) and [6,6]‐phenyl‐C₆₁ butyric acid methyl ester (PCBM) via spin‐coating for photovoltaic applications are reported. Solvent‐dependent PCBM cluster formation and P3HT crystallization during thermal annealing are investigated with optical microscopy and grazing‐incidence wide‐angle X‐ray scattering (GIWAXS) and are found to be insufficient to explain the differences in device performance. A combination of atomic force microscopy (AFM), X‐ray reflectivity (XRR), and grazing‐incidence small‐angle X‐ray scattering (GISAXS) investigations results in detailed knowledge of the inner film morphology of P3HT:PCBM films. Vertical and lateral phase separation occurs during spin‐coating and annealing, depending on the solvent used. The findings are summarized in schematics and compared with the IV characteristics. The main influence on the photovoltaic performance arises from the vertical material composition and the existence of lateral phase separation fitting to the exciton diffusion length. Absorption and photoluminescence measurements complement the structural analysis.     

Effects of Thermal Annealing Upon the Morphology of Polymer–Fullerene Blends

Grazing incidence X‐ray scattering (GIXS) is used to characterize the morphology of poly(3‐hexylthiophene) (P3HT)–phenyl‐C61‐butyric acid methyl ester (PCBM) thin film bulk heterojunction (BHJ) blends as a function of thermal annealing temperature, from room temperature to 220 °C. A custom‐built heating chamber for in situ GIXS studies allows for the morphological characterization of thin films at elevated temperatures. Films annealed with a thermal gradient allow for the rapid investigation of the morphology over a range of temperatures that corroborate the results of the in situ experiments. Using these techniques the following are observed: the melting points of each component; an increase in the P3HT coherence length with annealing below the P3HT melting temperature; the formation of well‐oriented P3HT crystallites with the (100) plane parallel to the substrate, when cooled from the melt; and the cold crystallization of PCBM associated with the PCBM glass transition temperature. The incorporation of these materials into BHJ blends affects the nature of these transitions as a function of blend ratio. These results provide a deeper understanding of the physics of how thermal annealing affects the morphology of polymer–fullerene BHJ blends and provides tools to manipulate the blend morphology in order to develop high‐performance organic solar cell devices.     

Real‐Time Investigation of Crystallization and Phase‐Segregation Dynamics in P3HT:PCBM Solar Cells During Thermal Annealing

Crystallization and phase segregation during thermal annealing lead to the increase of power‐conversion efficiency in poly(3‐hexylthiophene) (P3HT):[6,6]‐phenyl C61‐butyric acid methyl ester (PCBM) bulk‐heterojunction solar cells. An understanding of the length and time scale on which crystallization and phase segregation occur is important to improve control of the nanomorphology. Crystallization is monitored by means of grazing incidence X‐ray diffraction in real time during thermal annealing. Furthermore, the change in film density is monitored by means of ellipsometry and the evolution of carrier mobilities by means of field effect transistors, both during annealing. From the combination of such measurements with those of device performance as a function of annealing time, it is concluded that the evolution of microstructure involves two important time windows: i) A first one of about 5 minutes duration wherein crystallization of the polymer correlates with a major increase of photocurrent; ii) a second window of about 30 minutes during which the aggregation of PCBM continues, accompanied by an increase in the fill factor.     

Evolution of Structure,Optoelectronic Properties,and Device Performance of Polythiophene:Fullerene Solar Cells During Thermal Annealing

We use spectroscopic ellipsometry to study the evolution of structure and optoelectronic properties of poly(3‐hexylthiophene) (P3HT) and [6,6]‐phenyl C61‐butyric acid methyl ester (PCBM) photovoltaic thin film blends upon thermal annealing. Four distinct processes are identified: the evaporation of residual solvent above the glass transition temperature of the blend, the relaxation of non‐equilibrium molecular conformation formed through spin‐casting, the crystallization of both P3HT and PCBM components, and the phase separation of the P3HT and PCBM domains. Devices annealed at 150 °C for between 10 and 60 min exhibit an average power conversion efficiency of around 4.0%. We find that the rate at which the P3HT/PCBM is returned to room temperature is more important in determining device efficiency than the duration of the isothermal annealing process. We conclude that the rapid quenching of a film from the annealing temperature to room temperature hampers the crystallization of the P3HT and can trap non‐equilibrium morphological states. Such states apparently impact on device short circuit current, fill factor and, thus, operational efficiency.     

Charge Transport and Photocurrent Generation in Poly(3‐hexylthiophene): Methanofullerene Bulk‐Heterojunction Solar Cells

The effect of controlled thermal annealing on charge transport and photogeneration in bulk‐heterojunction solar cells made from blend films of regioregular poly(3‐hexylthiophene) (P3HT) and methanofullerene (PCBM) has been studied. With respect to the charge transport, it is demonstrated that the electron mobility dominates the transport of the cell, varying from 10^–8 m² V^–1 s^–1 in as‐cast devices to ≈3 × 10^–7 m² V^–1 s^–1 after thermal annealing. The hole mobility in the P3HT phase of the blend is dramatically affected by thermal annealing. It increases by more than three orders of magnitude, to reach a value of up to ≈ 2 × 10^–8 m² V^–1 s^–1 after the annealing process, as a result of an improved crystallinity of the film. Moreover, upon annealing the absorption spectrum of P3HT:PCBM blends undergo a strong red‐shift, improving the spectral overlap with solar emission, which results in an increase of more than 60 % in the rate of charge‐carrier generation. Subsequently, the experimental electron and hole mobilities are used to study the photocurrent generation in P3HT:PCBM devices as a function of annealing temperature. The results indicate that the most important factor leading to a strong enhancement of the efficiency, compared with non‐annealed devices, is the increase of the hole mobility in the P3HT phase of the blend. Furthermore, numerical simulations indicate that under short‐circuit conditions the dissociation efficiency of bound electron–hole pairs at the donor/acceptor interface is close to 90 %, which explains the large quantum efficiencies measured in P3HT:PCBM blends.     

Measuring Domain Sizes and Compositional Heterogeneities in P3HT‐PCBM Bulk Heterojunction Thin Films with 1H Spin Diffusion NMR Spectroscopy

The application of ¹H spin diffusion nuclear magnetic resonance (NMR) is expanded to polymer‐fullerene blends for bulk heterojunction (BHJ) organic photovoltaics (OPV) by developing a new experimental methodology for measuring the thin films used in poly‐3‐hexylthiophene–phenyl C61‐butyric acid methyl ester (P3HT‐PCBM) OPV devices and by creating an analysis framework for estimating domain size distributions. It is shown that variations in common P3HT‐PCBM BHJ processing parameters such as spin‐coating speed and thermal annealing can significantly affect domain size distributions, which in turn affect power conversion efficiency. ¹H spin diffusion NMR analysis reveals that films spin‐cast at fast speeds in dichlorobenzene are primarily composed of small (<10 nm) domains of each component; these devices exhibit low power conversion efficiencies (η = 0.4%). Fast‐cast films improve substantially by thermal annealing, which causes nanometer‐scale coarsening leading to higher efficiency (η = 2.2%). Films spin‐cast at slow speeds and then slowly dried exhibit larger domains and even higher efficiencies (η = 2.6%), but do not benefit from thermal annealing. The ¹H spin diffusion NMR results show that a significant population of domains tens of nanometers in size is a common characteristic of samples with higher efficiencies.     

Precise Structural Development and its Correlation to Function in Conjugated Polymer: Fullerene Thin Films by Controlled Solvent Annealing

The impact of controlled solvent vapor exposure on the morphology, structural evolution, and function of solvent‐processed poly(3‐hexylthiophene):[6,6]‐phenyl‐C₆₁‐butyric acid methyl ester (P3HT:PCBM) bilayers is presented. Grazing incident wide angle X‐ray scattering (GIWAXS) shows that the crystallization of P3HT increases with solvent exposure, while neutron reflectivity shows that P3HT simultaneously diffuses into PCBM, indicating that an initial bilayer structure evolves into a bulk heterojunction structure. Small angle neutron scattering (SANS) shows the agglomeration of PCBM and the formation of a PCBM pure phase when solvent annealing for 90 min. The structural evolution can be described as occurring in two stages: the first stage combines the enhanced crystallization of P3HT and diffusion of PCBM into P3HT, while the second stage entails the agglomeration of PCBM and formation of a PCBM pure phase. The phase separation of PCBM from P3HT is not driven by P3HT crystallinity, but is due to the concentration of PCBM exceeding the miscibility limit of PCBM in P3HT. Correlation of the morphology to photovoltaic activity shows that device performance significantly improves with solvent annealing for 90 min, indicating that both sufficient P3HT crystallization and formation of a PCBM pure phase are crucial in the optimization of the morphology of the active layer.     

Photophysics and Photocurrent Generation in Polythiophene/Polyfluorene Copolymer Blends

Here, studies on the evolution of photophysics and device performance with annealing of blends of poly(3‐hexylthiophene) with the two polyfluorene copolymers poly((9,9‐dioctylfluorene)‐2,7‐diyl‐alt‐[4,7‐bis(3‐hexylthien‐5‐yl)‐2,1,3‐benzothiadiazole]‐2′,2′′‐diyl) (F8TBT) and poly(9,9‐dioctylfluorene‐co‐benzothiadiazole) (F8BT) are reported. In blends with F8TBT, P3HT is found to reorganize at low annealing temperatures (100 °C or below), evidenced by a redshift of both absorption and photoluminescence (PL), and by a decrease in PL lifetime. Annealing to 140 °C, however, is found to optimize device performance, accompanied by an increase in PL efficiency and lifetime. Grazing‐incidence small‐angle X‐ray scattering is also performed to study the evolution in film nanomorphology with annealing, with the 140 °C‐annealed film showing enhanced phase separation. It is concluded that reorganization of P3HT alone is not sufficient to optimize device performance but must also be accompanied by a coarsening of the morphology to promote charge separation. The shape of the photocurrent action spectra of P3HT:F8TBT devices is also studied, aided by optical modeling of the absorption spectrum of the blend in a device structure. Changes in the shape of the photocurrent action spectra with annealing are observed, and these are attributed to changes in the relative contribution of each polymer to photocurrent as morphology and polymer conformation evolve. In particular, in as‐spun films from xylene, photocurrent is preferentially generated from ordered P3HT segments attributed to the increased charge separation efficiency in ordered P3HT compared to disordered P3HT. For optimized devices, photocurrent is efficiently generated from both P3HT and F8TBT. In contrast to blends with F8TBT, P3HT is only found to reorganize in blends with F8BT at annealing temperatures of over 200 °C. The low efficiency of the P3HT:F8BT system can then be attributed to poor charge generation and separation efficiencies that result from the failure of P3HT to reorganize.     

Cover Picture: Tuning the Dimensions of C60‐Based Needlelike Crystals in Blended Thin Films (Adv. Funct. Mater. 6/2006)

A new ordered structure of the C₆₀ derivative PCBM is obtained in thin films based on the blend PCBM:P3HT, as detailed by Swinnen, Manca, and co‐workers on p. 760. Needlelike crystalline PCBM structures, whose dimensions and spatial distribution ca be tuned by adjusting the blend ratio and annealing conditions, are formed. In typical solar‐cell applications of these blended films, these results indicate that during long‐term operation under normal conditions (50–70 °C) morphology changes and a decrease in cell performance could occur. A new ordered structure of the C₆₀ derivative PCBM ([6‐6]‐phenyl C₆₁‐butyric acid methyl ester) is obtained in thin films based on the blend PCBM:regioregular P3HT (poly(3‐hexylthiophene)). Rapid formation of needlelike crystalline PCBM structures of a few micrometers up to 100 μm in size is demonstrated by submitting the blended thin films to an appropriate thermal treatment. These structures can grow out to a 2D network of PCBM needles and, in specific cases, to spectacular PCBM fans. Key parameters to tune the dimensions and spatial distribution of the PCBM needles are blend ratio and annealing conditions. The as‐obtained blended films and crystals are probed using atomic force microscopy, transmission electron microscopy, selected area electron diffraction, optical microscopy, and confocal fluorescence microscopy. Based on the analytical results, the growth mechanism of the PCBM structures within the film is described in terms of diffusion of PCBM towards the PCBM crystals, leaving highly crystalline P3HT behind in the surrounding matrix.     

Tuning the Dimensions of C60‐Based Needlelike Crystals in Blended Thin Films

A new ordered structure of the C₆₀ derivative PCBM ([6‐6]‐phenyl C₆₁‐butyric acid methyl ester) is obtained in thin films based on the blend PCBM:regioregular P3HT (poly(3‐hexylthiophene)). Rapid formation of needlelike crystalline PCBM structures of a few micrometers up to 100 μm in size is demonstrated by submitting the blended thin films to an appropriate thermal treatment. These structures can grow out to a 2D network of PCBM needles and, in specific cases, to spectacular PCBM fans. Key parameters to tune the dimensions and spatial distribution of the PCBM needles are blend ratio and annealing conditions. The as‐obtained blended films and crystals are probed using atomic force microscopy, transmission electron microscopy, selected area electron diffraction, optical microscopy, and confocal fluorescence microscopy. Based on the analytical results, the growth mechanism of the PCBM structures within the film is described in terms of diffusion of PCBM towards the PCBM crystals, leaving highly crystalline P3HT behind in the surrounding matrix.     

Effect of Trace Solvent on the Morphology of P3HT:PCBM Bulk Heterojunction Solar Cells

The performance of bulk‐heterojunction (BHJ) solar cells is strongly correlated with the nanoscale structure of the active layer. Various processing techniques have been explored to improve the nanoscale morphology of the BHJ layer, e.g., by varying the casting solvent, thermal annealing, solvent annealing, and solvent additives. This paper highlights the role of residual solvent in the “dried” BHJ layer, and the effect of residual solvents on PCBM diffusion and ultimately the stability of the morphology. We show that solvent is retained within the BHJ film despite prolonged heat treatment, leading to extensive phase separation, as demonstrated by the growth in the size and quantity of PCBM agglomerates. The addition of a small volume fraction of nitrobenzene to the casting solution inhibits the diffusion of PCBM in the dry film, resulting in smaller PCBM agglomerates, and improves the fill factor of the BHJ device to 0.61 without further tempering. The addition of nitrobenzene also increases the P3HT crystalline content, while increasing the onset temperature for melting of P3HT side chains and backbone. The melting temperature for PCBM is also higher with the nitrobenzene additive present.     

High‐Resolution Spectroscopic Mapping of the Chemical Contrast from Nanometer Domains in P3HT:PCBM Organic Blend Films for Solar‐Cell Applications

A high‐resolution near‐field spectroscopic mapping technique is successfully applied to investigate the influence of thermal annealing on the morphology of a poly(3‐hexylthiophene) and [6,6]‐penyl‐C₆₁ butyric acid methyl ester (P3HT:PCBM) blend film. Based on the simultaneously recorded morphological and spectroscopic information, the interplay among the blend film morphology, the local P3HT:PCBM molecular distribution, and the P3HT photoluminescence (PL) quenching efficiency are systematically discussed. The PL and Raman signals of the electron donor (P3HT) and acceptor (PCBM) are probed at an optical resolution of approximately 10 nm, which allows the chemical nature of the different domains to be identified directly. In addition, the local PL quenching efficiency, which is related to the electron transfer from P3HT to PCBM, is quantitatively revealed. From these experimental results, it is proposed that high‐resolution near‐field spectroscopic imaging is capable of mapping the local chemical composition and photophysics of the P3HT:PCBM blends on a scale of a few nanometers.     

Abrupt Morphology Change upon Thermal Annealing in Poly(3‐Hexylthiophene)/Soluble Fullerene Blend Films for Polymer Solar Cells

The in situ morphology change upon thermal annealing in bulk heterojunction blend films of regioregular poly(3‐hexylthiophene) (P3HT) and 1‐(3‐methoxycarbonyl)‐propyl‐1‐phenyl‐(6,6)C₆₁ (PCBM) is measured by a grazing incidence X‐ray diffraction (GIXD) method using a synchrotron radiation source. The results show that the film morphology—including the size and population of P3HT crystallites—abruptly changes at 140 °C between 5 and 30 min and is then stable up to 120 min. This trend is almost in good agreement with the performance change of polymer solar cells fabricated under the same conditions. The certain morphology change after 5 min annealing at 140 °C is assigned to the on‐going thermal transition of P3HT molecules in the presence of PCBM transition. Field‐emission scanning electron microscopy measurements show that the crack‐like surface of blend films becomes smaller after a very short annealing time, but does not change further with increasing annealing time. These findings indicate that the stability of P3HT:PCBM solar cells cannot be secured by short‐time annealing owing to the unsettled morphology, even though the resulting efficiency is high.     

Time‐Dependent Morphology Evolution by Annealing Processes on Polymer:Fullerene Blend Solar Cells

Changes in the nanoscale morphologies of the blend films of poly (3‐hexylthiophene) (P3HT) and [6,6]‐phenyl‐C₆₁‐butyric acid methyl ester (PCBM), for high‐performance bulk‐heterojunction (BHJ) solar cells, are compared and investigated for two annealing treatments with different morphology evolution time scales, having special consideration for the diffusion and aggregation of PCBM molecules. An annealing condition with relatively fast diffusion and aggregation of the PCBM molecules during P3HT crystallization results in poor BHJ morphology because of prevention of the formation of the more elongated P3HT crystals. However, an annealing condition, accelerating PCBM diffusion after the formation of a well‐ordered morphology, results in a relatively stable morphology with less destruction of crystalline P3HT. Based on these results, an effective strategy for determining an optimized annealing treatment is suggested that considers the effect of relative kinetics on the crystallization of the components for a blend film with a new BHJ materials pair, upon which BHJ solar cells are based.     

Poly(3-hexylthiophene) coated graphene oxide for improved performance of bulk heterojunction polymer solar cells

An effective method for preparing poly(3-hexylthiophene) (P3HT) coated graphene oxide (GO), (P-GO), based on an ethanol mediated mixing and solvent evaporation method is described. P-GO exhibits good dispersibility in the non-polar solvent o-dichlorobenzene (DCB) allowing the preparation of polymer blend composites. P-GO was doped into P3HT: PCBM blends by solution mixing and shown to facilitate phase separation of P3HT and PCBM in P3HT: PCBM blend films to achieve a more optimum morphology for polymer photovoltaic cells. Bulk heterojunction P3HT: PCBM solar cells exhibit ∼18% power conversion efficiency enhancement in the presence of P-GO.     

A new model for PCBM phase segregation in P3HT:PCBM blends

The phase segregation in P3HT:PCBM blend films has been investigated from an experimental and theoretical viewpoint. Optical microscopy, atomic force microscopy, scanning electron microscopy and X-ray diffraction show that thermal annealing of P3HT:PCBM blend films leads to the formation of PCBM aggregates. These aggregates are composed of dense randomly packed ∼50 nm PCBM crystallites with an overall aggregate density of ∼0.85 g cm⁻³. By applying the critical radius of nucleation for PCBM and the Stokes-Einstein equation for mobility of PCBM in a P3HT matrix, a model is developed which explains the formation of both crystallites and aggregates.     

Quantifying Interfacial Electric Fields and Local Crystallinity in Polymer–Fullerene Bulk‐Heterojunction Solar Cells

The challenges of experimentally probing the physical and electronic structures of the highly intermixed organic semiconductor blends that comprise active layers in high‐performance organic photovoltaic (OPV) cells ultimately limit the fundamental understanding of the device performance. We use Fourier‐transform IR (FTIR)‐absorption spectroscopy to quantitatively determine the interfacial electric field in blended poly(3‐hexylthiophene) (P3HT):phenyl‐ C61‐butyric acid methyl ester (PCBM) thin films. The interfacial electric field is ≈0.2 V nm^?1 in the as‐spun film and blends annealing at temperatures as high as 150 °C, which is the optimal annealing temperature in terms of OPV performance. The field decreases to a negligible value upon further annealing to 170 °C, at which temperature PCBM changes from amorphous to crystalline and the open‐circuit voltage of the solar cell decreases from 0.62 to 0.4 V. In addition, our measurements also allow determination of the absolute degree of crystallinity within the acceptor material. The roles of interfacial field and local crystallinity in OPV device performance are discussed.