Reaction conditions (temperature and pressure) in scCO2-assisted synthesis of nanoparticles.
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More than half of the publishers listed alongside IntechOpen (18 out of 30) are Social Science and Humanities publishers. IntechOpen is an exception to this as a leader in not only Open Access content but Open Access content across all scientific disciplines, including Physical Sciences, Engineering and Technology, Health Sciences, Life Science, and Social Sciences and Humanities.
\\n\\nOur breakdown of titles published demonstrates this with 47% PET, 31% HS, 18% LS, and 4% SSH books published.
\\n\\n“Even though ItechOpen has shown the potential of sci-tech books using an OA approach,” other publishers “have shown little interest in OA books.”
\\n\\nAdditionally, each book published by IntechOpen contains original content and research findings.
\\n\\nWe are honored to be among such prestigious publishers and we hope to continue to spearhead that growth in our quest to promote Open Access as a true pioneer in OA book publishing.
\\n\\n\\n\\n
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'
Simba Information has released its Open Access Book Publishing 2020 - 2024 report and has again identified IntechOpen as the world’s largest Open Access book publisher by title count.
\n\nSimba Information is a leading provider for market intelligence and forecasts in the media and publishing industry. The report, published every year, provides an overview and financial outlook for the global professional e-book publishing market.
\n\nIntechOpen, De Gruyter, and Frontiers are the largest OA book publishers by title count, with IntechOpen coming in at first place with 5,101 OA books published, a good 1,782 titles ahead of the nearest competitor.
\n\nSince the first Open Access Book Publishing report published in 2016, IntechOpen has held the top stop each year.
\n\n\n\nMore than half of the publishers listed alongside IntechOpen (18 out of 30) are Social Science and Humanities publishers. IntechOpen is an exception to this as a leader in not only Open Access content but Open Access content across all scientific disciplines, including Physical Sciences, Engineering and Technology, Health Sciences, Life Science, and Social Sciences and Humanities.
\n\nOur breakdown of titles published demonstrates this with 47% PET, 31% HS, 18% LS, and 4% SSH books published.
\n\n“Even though ItechOpen has shown the potential of sci-tech books using an OA approach,” other publishers “have shown little interest in OA books.”
\n\nAdditionally, each book published by IntechOpen contains original content and research findings.
\n\nWe are honored to be among such prestigious publishers and we hope to continue to spearhead that growth in our quest to promote Open Access as a true pioneer in OA book publishing.
\n\n\n\n
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The pharmaceutical industry has a major problem concerning the production of active pharmaceutical ingredients, which have a low water solubility and bioavailability. Therefore an appropriate technology for producing these active components is needed with certain properties like particle size (smaller than 1000 nm, typically under 500 nm), solubility, efficacy, state transition (polymorphism and crystallization), cost-effectiveness, etc.
In many nanomedical applications used in nanomedicine, processes based on supercritical fluids (SCFs) can be applied because they allow controlled fabrication of biological active nanostructured microparticles, nanoparticles, and nanoporous/nanostructured materials. Supercritical carbon dioxide (scCO2) as a green solvent that possesses many beneficial properties (it is nonflammable, nontoxic, biocompatible, cost-effective, and abundant) has gained huge interest in the food and pharmaceutical industries; it is considered environmentally benign and one of the few solvents not regulated as a volatile organic compound (VOC) by the US Environmental Protection Agency.
There are several methods for the manufacturing of solid particles scaled from micron to nanosize, divided into bottom-up, top-down, and combination approaches of these [1].
Bottom-up techniques produce nanosized particles by precipitation from a supersaturated drug solution. Precipitation by addition of liquid antisolvent is made by simple mixing methods (using a static mixer) or by modified mixing methods (sonoprecipitation or high gravity controlled precipitation). Other bottom-up techniques involve supercritical fluids, rapid expansion of supercritical solution (RESS), and supercritical antisolvent technique (SAS) or solvent removal by nano-spray dryer and spray freezing into liquid techniques. These methods have some disadvantages which include the size of particles that cannot be properly controlled in the non-sized range, but using of some additives (excipients, surfactants, etc.) can produce the stabilization nanocrystals or nanoparticles regarding the morphological properties and the crystallized polymorphic form [2]. Bottom-up techniques showed some advantages because they are low-energy processes and less expensive than the other methods, and obtained particles have narrow size distribution. In order to obtain smaller particles, these methods have been used in combination with top-down techniques [1].
Top-down techniques are used for particle size reduction of drugs to the nanometer size range by application of friction, involving high-energy processes such as media milling (wet bead milling) and high-pressure homogenization techniques (Dissocubes homogenization and NanoPure technology). These methods have some disadvantages which include a limited control of crystal size and surface properties and thermal or mechanical degradation generated by intensive energy of mixing [3]. These techniques are used in the last decade, but few nanocrystals under 100 nm have been obtained. Drug particles with smaller size than 100 nm have novel physical properties and better permeation through different biological barriers [4] and improved bioavailability of poorly aqueous soluble drugs, having different routes of administration such as oral, ocular, dermal, buccal, and pulmonary.
Supercritical fluids and their mixtures have specific properties like very fast mass transfer, near zero surface tension, and effective solvent elimination. The liquid-like and/or gas-like properties of SCFs and the possibility to modify several process parameters (temperature, pressure, and surface tension) can be benefits to produce several medical products at nanoscale. Several SCF-based processes are applied to nanomedicine applications: supercritical antisolvent precipitation (SAS), rapid expansion of supercritical solutions, supercritical emulsion extraction (SEE), supercritical assisted phase separation, supercritical gel drying, supercritical assisted liposome formation (SuperLip), supercritical assisted atomization (SAA), electrospinning in scCO2, supercritical assisted injection in a liquid antisolvent (SAILA), and depressurization of an expanded solution into aqueous media (DESAM) [5].
Numerous methods for nanoparticle fabrication and functionalization using supercritical carbon dioxide have been proposed, in various reaction conditions. In Table 1, some examples of temperature and pressure range frequently used for the preparation of nanoparticulated materials are summarized, with particular emphasis on drug or drug delivery systems.
scCO2 method | Nanoparticles | Temperature (K) | Pressure (MPa) | References |
---|---|---|---|---|
RESS | Drug micro−/nanoparticles Polymeric micro−/nanoparticles Drug encapsulated in polymeric microparticles | 313–373 | 8–25 | [6, 7] |
SAS | Drug micro−/nanoparticles Inorganic nanoparticles | 303–323 | 9–15 MPa | [8] |
SEE | Drug encapsulated in lipid or polymeric nanoparticles | 308–323 | 8–10 MPa | [5, 9, 10] |
SAA | Drug nanoparticles | 343–353 | 7–9 | [11] |
SAILA | Drug nanoparticles | 343–353 | 7–10 | [5] |
RESOLV | Drug nanoparticles Polymer-stabilized drug NPs | 308–318 | 10–20 | [12] |
Reaction conditions (temperature and pressure) in scCO2-assisted synthesis of nanoparticles.
The selection of the temperature and pressure condition is crucial for the particle size and drug encapsulation efficiency, and the specific values are chosen according to the characteristics of the active substance, such as solubility in scCO2 phase, thermal stability, etc.
In this chapter, we present several applications of supercritical carbon dioxide (scCO2) for preparation of polymeric nanoparticles as drug delivery systems, active principle encapsulation, water-in-scCO2 microemulsion in the synthesis of metal nanoparticles, and various methods for surface functionalization of nanoparticles in supercritical and subcritical CO2.
SCFs can bring their contributions in different fields for certain applications as chromatography, fluid extractions, and micro- and nanoparticle formation. There are several compounds that can be used as supercritical fluids. Among them, hydrocarbons are toxic and inflammable, water has high critical parameters, but carbon dioxide has appropriate critical temperature and pressure, being suitable to be used as a green solvent.
Methods using supercritical fluids (SCFs) can produce particles with narrow size distribution. Because of their special properties, SCFs can be applied to micronization of several types of compounds: drugs, biopolymers, polymers, food, coloring maters, explosives, etc. Pharmaceutical micronizations using supercritical fluids have some advantages due to the absence of organic solvent, and the particle size distribution could be controlled by process parameters. During the micronizations by SCFs methods, the dissolution rate is increased. The use of scCO2 provides several advantages in comparison with the previous conventional techniques. The scCO2 can be used as a solvent, antisolvent, and extracting agent for the organic phase of oil-in-water emulsions or at particle formation from gas saturated solution and improve the spraying process in different techniques [13].
The RESS process is based on the saturation of the supercritical medium with a solute (polymer) followed by a rapidly depressurization of the solution through a heated nozzle at high speed. During the pressure drop, the system passes from supercritical to atmospheric conditions, the solvent power decreases, and a fast nucleation of the solute in the form of very small particles with uniform size takes place. The properties of the obtained particles are influenced by its solubility in scCO2; state parameters from precipitation vessel (temperature and pressure); size, length, and shape of the nozzle; distance of the jet stream; and impact angle onto the surface. A schematic representation of this process is presented in Figure 1.
Schematic representations of the RESS process.
The RESS technique has several advantages like the simple control of process parameters, the absence of organic solvents, and easy implementation on lab-scale when a single nozzle is used. RESS has also disadvantages as difficulty in scaling-up, the possible particle aggregation and/or nozzle blockage, using an important amount of carbon dioxide, and low solubility of most pharmaceutical compounds in supercritical CO2. But, the scCO2 solvent power can be increased by using of a co-solvent.
RESOLV represents a variation of RESS. This technique can reduce the particle aggregation during the jet expansion. In this process the supercritical solution is depressurized through a nozzle into a collection chamber containing an aqueous solution at room temperature. Different types of water-soluble polymers or surfactants can added to the aqueous medium in order to stabilize the obtained nanoparticle suspension [12]. Biocompatible polymer nanoparticles composed of biodegradable polymers such as polylactic acid (PLA), poly(lactic-coglycolic) acid (PLGA), and poly(ε-caprolactone) (PCL) are drawing a considerable interest in the scientific community because they can be used in medicine as biodegradable support materials and drug delivery vehicles. There are studies focused on the generation of polysaccharide particles. Polysaccharides are biobased polymers used in a lot of domains such as nutrition, energy, health care, and materials science, with large applications in the industry. An example of this family is chitosan, an amino-polysaccharide derived from chitin, being the second most biosynthesized polymer after cellulose. Because chitosan is biocompatible and biodegradable (mucoadhesive with antibacterial and cytocompatible), it can be used in pharmaceutics and biomedical applications, cosmetics, food packaging, agriculture, water treatment, etc. Another polysaccharide is alginate that can be used in several areas like drug delivery, tissue engineering, and wound dressing. For these purposes and due to its essential functional groups (hydroxyl and carboxyl), alginates can be transformed to hydrogels, porous scaffolds, and micro- and nanoparticles [14].
In order to obtain microparticles, an experimental apparatus equipped with a high-pressure cell having a capillary nozzle was used. The scheme of this handmade experimental setup is presented in Figure 2.
Experimental setup for RESS technique.
Biopolymeric microparticles were prepared in our laboratory by rapid expansion of high-pressure CO2-chitosan solution in sodium bis-(2-ethylhexyl) sulfosuccinate solution. At pressures higher than 2 MPa, ultrafine particles were formed, while under this value, wires were obtained (Figure 3). The Chi/AOT ultrafine particles are instantaneously formed when 2 mL Chi solution 2% (wt/wt), preheated 1 hour at 40°C, is sprayed using CO2 at different pressures into 20 mL AOT 0.03 M aqueous solution through a stainless steel capillary nozzle of 30 mm length and 0.4 mm diameter. The pre-expansion pressure was 1–5 MPa, and the distance from the nozzle tip to surfactant solution interface was of about 20 mm. We observe that with increasing the spraying pressure of polymer, the size of the particles decreases. The microparticles obtained at high pressure are quasi-spherical in aqueous medium and irregular with many pores and a rough surface after freeze-drying. The morphology of synthesized particles recommends them for possible applications in adsorption of organic and inorganic substances from aqueous medium. The Chi/AOT microparticles were an effective adsorbent for removal of phenol and o-cresol from aqueous solution [15].
Optical images of CO2-chitosan (chi)/bis-(2-ethylhexyl) sulfosuccinate (AOT) microparticles and wires.
Porous alginate matrices were obtained using sub- and supercritical carbon dioxide. Calcium alginate matrices had uniform porous texture generated by high-pressure CO2 as foaming agent without co-solvents. Sodium alginate solutions were processed in high-pressure CO2, with freezing. After depressurization, the frozen samples were ionically cross-linked with calcium ions with and without glycerol. The effects of the presence of glycerol as plasticizer, carbon dioxide pressure, temperature, and processing time (20 minutes and 5 hours) on the structure of the obtained calcium alginate matrices were investigated. The porosity of alginate matrices increased with CO2 pressure, processing time, and glycerol adding. The plasticizer, glycerol, improves mechanical properties and texture for scaffolds.
Microparticles of poly(ε-caprolactam) were obtained by rapid expansion in water of the PCL polymeric solution (with a cosolvent, methanol or dimethylformamide) saturated with high-pressure carbon dioxide, in experimental setup presented in Figure 3. The solution was denoted by P1 – PCL in methanol (1 g/mL) and P2 – PCL in dimethylformamide (1 g/mL). The working temperature was 70°C and pressure 6.5 and 8.5 MPa. In Table 1 and Figure 4, values for pressure are expressed in bar. Depending on the nature of the cosolvent, temperature, and pressure, poly(ε-caprolactam) particles of spherical shape and variable dimensions (particles diameter 1.4–6.7 μm) were obtained.
FTIR spectra for PCL, P1, and P2 and different temperatures and pressures.
The morphological changes of PCL solutions in sub- and supercritical carbon dioxide and the addition of cosolvent were revealed by Boethius microscopy.
The interactions between CO2 and the carbonyl groups in the PCL molecules led to the lower melting temperature of the P1 and P2 samples treated with sub- and supercritical CO2 (Table 2).
System | Melting point (0C) | Observations |
---|---|---|
PCL | 60.3–61.9 | White crystals |
P1-25-1 | 55.8–57.5 | White crystals |
P2-25-1 | 57.1–58.9 | White crystals |
P1-70-85 | 55.5–57.6 | Amorphous |
P1-70-65 | 58.7–59.8 | White crystals |
P2-70-85 | 58.4–60.3 | White crystals |
P2-70-65 | 57.0–59.3 | Amorphous |
Melting point of high-pressure samples of PCL.
Structural changes of PCL were evidenced by the FTIR technique (Figure 4).
The presence of high-pressure CO2 influences the stretching vibration in the group C〓O (carbonyl). For P1, once the carbon dioxide pressure increases, the maximum absorption of the carbonyl group moves to higher values than P2 where the maximum absorption decreases with increasing pressure. This displacement reaches a limit at the highest pressures as a result of increased carbon dioxide mobility. The integral area of the carbonyl peak decreases linearly with increasing pressure for the P1 system (data consistent with those in the literature) and varies nonlinearly for the P2 system. It has been observed that with the decrease of carbon dioxide pressure, the frequency of the C-O-C stretching vibration moves to higher values. At higher temperatures (70°C) and higher pressures (6.5 and 8.5 MPa), the polymer melts and recrystallizes after expansion in a semicrystalline or amorphous state, resulting in melting temperatures below PCL. For the samples treated at high pressures, we observe a shift of the maximum attributed to the crystallinity to smaller wave numbers, so a decrease of the crystallinity produces a decrease of melting temperature for the PCL samples treated under high-temperature and high-pressure conditions.
In the last decade, drugs loaded in porous biodegradable polymeric foams have found to have important applications in tissue engineering and delivery systems. These polymeric porous scaffolds with an open-pore structure can ensure and increase seeding, attachment, growth of cells, extracellular matrix production, vascularization, and tissue growth. Supercritical CO2 is an excellent choice to produce impregnated polyester foams in a one-step process creating porosity, without residual solvent in the products. After polymer scaffold degradation, obtained tissue would not contain synthetic polymer. The rate degradation of the scaffold should be similar or slower than the rate of tissue formation; therefore, it is important to elucidate the mechanism of this degradation process.
Some examples of polymers used as porous scaffolds were poly(D,L-lactide) (PLA) and poly(D,L-lactide-co-glycolide) (PLGA), respectively, and poly(methyl methacrylate), PMMA, PMMA/poly(ε-caprolactone), PCL, etc. Such polymers are harmless to the growing cells and can be removed from the organism by normal metabolic pathways. They also can be used in other in vivo applications, such as resorbable sutures [16].
Encapsulation of drugs within colloidal-sized polymeric matrix is largely used to improve the sustained release, reduce the side effect of the drugs, and increase the bioavailability of the drug from the pharmaceutical formulations. Polymeric beads for the controlled release of the drugs are most often used as the best solution due to easy preparation procedure and high drug-loading efficiency.
The special properties of scCO2 made it a good transport vector for solid matrix impregnation. This process depends on the partition of interest substance between the supercritical fluid phase and matrix (such as porous polymers) used for impregnation. First the substance is mixed with high-pressure carbon dioxide and then is placed in contact with matrix used as support. Because scCO2 has a good diffusion capacity into porous scaffolds, inducing a swelling and/or plasticization, impregnation of a lot of materials could be possible. Among them were polymers as swellable matrices or silica as non-swellable matrices [17].
The controlled release of the very hydrophilic drugs is still a challenge due to the difficulties to encapsulate the drug in a suitable drug delivery system able to control the rate of the release and to ensure a minimum retention of the active substance inside the pharmaceutical vehicles, but with a good control of the particle size and surface properties, nanoparticles may be directed to specific sites for targeted drug delivery.
The encapsulation of active substances can be improved by impregnation in the presence of compressed fluids like carbon dioxide [18], at temperatures and pressures near or above critical point (CO2,
In a typical experiment in our laboratory, 2-pyridinealdoxime methochloride (PAM) was chosen as a hydrophilic drug model, with a high solubility in water. PAM is an acetylcholinesterase reactivator used as an antidote in poisoning with organophosphoric substances characterized by high solubility in water. Due to its hydrophilicity, PAM is rapidly eliminated from the body. Consequently, PAM is relatively short acting, and repeated doses may be needed. The encapsulation of PAM in alginate beads (used as sustained drug release system) was tested.
Alginate is a water-soluble linear polysaccharide extracted from brown seaweed, and it is composed of alternating blocks of 1–4 linked α-L-guluronic (G) and β-D-mannuronic (M) acid fragments. Alginate is a biocompatible and a hydrophilic biopolymer. These properties and its relatively low cost have recommended it for pharmaceutical applications [19]. The anionic biopolymer has the ability to bind multivalent cations, leading to the formation of insoluble hydrogels with “egg box”-type structure [19]. In the formation of water-insoluble gels, a specific interaction occurs between calcium ions (Ca2+) and -COO− and -OH groups of the guluronic acid fragments in a simple procedure of dripping a sodium alginate aqueous solution into a calcium chloride solution [20]. The drug can be encapsulated in alginate beads by two methods. A first method consists in dripping the aqueous solution of alginate/PAM mixture in calcium chloride solution. The deficiency of this method is the elimination of PAM outside the beads during the ionically cross-linking of alginate with calcium ions. A second method consists in immersing and soaking the polymeric beads into the solution of drug. In this case the impregnation efficiency is low.
A certain number of calcium alginate beads were immersed in the 3 × 10−3 wt% solution of PAM. The impregnation of PAM in calcium alginate beads was carried out in a high-pressure cell, in the presence of compressed carbon dioxide. The impregnation was performed at different pressures (2.5, 5.0, 7.5 MPa) and temperatures (20, 40, 60°C) and also at atmospheric conditions (0.1 MPa, 20°C). The impregnation time was 30 minutes. The samples are designated
The efficiency of PAM impregnation at various temperatures and pressures is presented in Figure 5.
Efficiency of encapsulation of PAM in alginate microbeads at different pressures and temperatures.
The results showed that the efficiency of PAM impregnation increases with temperature at 2.5 MPa and the interaction between the cationic drug and active sites of biopolymer are favored by high pressure. Both state parameters influence the encapsulation of the ionic drug in the polymeric beads. The encapsulation of PAM in calcium alginate beads at 2.5 MPa depends on temperature. Adsorption or/and absorption of PAM are the main processes which take place between 20 and 40°C. At 60°C, the dissociation constants of polyelectrolyte and ionic drug increase and electrostatic interactions between the polymer and drug molecules are promoted. At higher pressure (5.0 and 7.5 MPa), the impregnation efficiency varies in the same way like ratio of CO2 density to viscosity versus temperature and pressure.
The release curves for sample 1/20 and for samples with higher encapsulation efficiency are presented in Figure 6. The release curves for samples 0.1/20, 7.5/20, and 5.0/20 show a “burst effect”; the encapsulated PAM is released in the first 15 minutes. The sustained release of PAM is observed for samples 7.5/40 and 2.5/60. The maximum amount of released PAM is lower than 50% for samples 7.5/40, 2.5/60, and 5.0/20. The observed initial burst release profiles may be an indication that the impregnated drug was mainly located at the polymer surface for the sample impregnated with PAM at 20°C and different pressures. The kinetic data were fitted by first-order, Weibull, and Korsmeyer-Peppas kinetic equations. The experimental concentration values of the active substance released as a function of time were processed for establishing the release kinetics.
PAM release profiles from impregnated alginate beads; inset, the release curves in the first 30 minutes.
The kinetic parameters offer information about the encapsulation mechanism which can take place by adsorption on the surface of the beads or by interactions between drug and polymer inside the alginate beads. The PAM release curves show that the release mechanism depends on the drug impregnation conditions.
The encapsulation of PAM in alginate beads was improved by impregnation in the presence of compressed carbon dioxide at high temperature. The release of ionic drug from drug delivery beads obtained by impregnation of PAM in supercritical CO2 (7.5 MPa, 40°C) is controlled by diffusion of active substances. The alginate beads with encapsulated PAM in subcritical conditions (7.5 MPa, 40°C) release PAM according to swelling and erosion of an amorphous biopolymer.
The release of PAM from alginate beads can be controlled by changing the initial conditions of impregnation in the presence of CO2.
Nanoparticles with tunable size and high monodispersity present great interest for biomedical application, in particular to be used in bioimaging or as drug delivery systems. Synthesis of inorganic and organic NPs in scCO2 as green alternative to the physical and chemical traditional methods suffer from the same difficulties in controlling the nanoparticle growth step and to ensure narrow size distribution of the final product. One of the most efficient routes developed to fabricate monodisperse nanoparticles is the synthesis performed in heterogeneous media, i.e., in microemulsions or in blue emulsions, in order to ensure the nucleation and growth of nanoparticles in the restricted domain of liquid nanoreactors (oil or aqueous droplets). Microemulsions, as liquid-liquid colloidal systems consisting of nanometric droplets with size ranging from few nanometers to 100, stabilized with surfactant and cosurfactant mixture, are considered the most suitable reaction media for the synthesis of organic and inorganic nanoparticles with tunable size and morphology. Oil-in-water microemulsions are usually used for the fabrication of hydrophobic polymeric nanoparticles, while water-in-oil ones are used as reaction media for inorganic nanoparticles, especially for metal NPs obtained from reduction of metallic salt. All advantages of the controlled synthesis in water-in-oil microemulsions could be preserved using water-in scCO2 microemulsions, since the general structure of the colloidal system is the same (Figure 7).
The schematic view of structure and stabilization of water droplets in W/O and W/scCO2 microemulsions.
However, the stabilization of miniemulsions and the formation of thermodynamically stable microemulsions in supercritical CO2 are challenging tasks, due to the specific properties of carbon dioxide. Supercritical carbon dioxide possesses peculiar properties such as very high diffusivity and low viscosity, different from usual oily phases in the microemulsion composition; thus special requirements are needed for the surfactants that enable the formation of W-in-scCO2 microemulsion: (i) enhanced absorbability at water/scCO2 interface, (ii) enhanced surface activity in lowering water/scCO2 interfacial tension, and (iii) strong repulsive interaction (steric effect) between CO2-philic groups, which reduces the probability of droplet aggregation and leads to the stabilization of microemulsion [21].
Many researches have been performed addressing the problem of surfactant solubility in supercritical and dense CO2, in order to evidence their ability to stabilize water-in-CO2 microemulsions, and surfactants with special designed molecular structure were synthesized for this purpose [21].
The early studies [22] conclude that conventional theories in molecular design for surfactant with superior efficiency in lowering interfacial tension at W/O interfaces are inapplicable to the W-scCO2 ones, because CO2-philicity could not be directly compared with oleophilicity. Considerable efforts have been made to find the most favorable chemical structures able to ensure adequate solubility of the surfactant in supercritical CO2, together with better adsorption at W-scCO2 interface. Highly branched hydrocarbon surfactants were found to possess good solubility, but unfortunately they are not efficient enough as stabilizer, compared to AOT sodium salt, a commercially available cost-effective common surfactant widely used for the formulation of water-in-oil microemulsions. The typical surfactants exhibit very different Hildebrandt solubility parameters for carbon dioxide; thus their solubility in CO2 is very low, explaining their lack in stabilizing supercritical CO2 emulsions and microemulsions.
Based on the data evaluated [22], an interesting family of surfactants were synthesized as AOT analogues AO-VAc bearing vinyl-acetate oligomeric chains and AOK bearing
In order to increase the region of microemulsion in the phase diagram of the water-supercritical CO2-surfactant system, novel amphiphilic compounds were synthesized with a more flexible molecule due to the presence of linkers. A series of fluorocarbon derivatives having different fluorocarbon chain lengths F(CF2)n with n = 4,6,8, and 10 and oxyethylene groups (CH2CH2O)m/2 (where m = 2 and 4) introduced into the molecule as spacers were developed [23]. The analysis of phase diagram of quaternary systems water-scCO2-
These fluorinated surfactants with specific chemical structure are quite difficult to obtain, and their use is restricted in the “green chemistry” methods due to the high toxicity. In this respect, other classes of surfactants were investigated, in particular polymeric derivatives. Block copolymers have been extensively studied regarding their ability to act as stabilizer of water droplets in scCO2 since J. de Simone reports for the first time in 1994 the solubility of fluorinated polymers in liquid and scCO2. Chemicals such as fluoropolymers, polysiloxanes, and some poly(vinyl esters) derivatives prove to be CO2-philic polymers but with reduced surface activity. In the last decades, the progresses in the synthesis techniques allow the fabrication of a large variety of block copolymers with CO2-philic moieties and specially designed structures, such as amphiphilic block, amphiphilic comb copolymers, or gradient copolymers bearing linear or branched chains, such as poly(1,1,2,2-tetrahydroperfluorooctyl methacrylate)-based copolymers with either poly(N,N-dimethylaminoethyl methacrylate) (PDMAEMA) or poly(oligo(ethylene glycol) methacrylate) (POEGMA) as hydrophilic blocks or 1,1,2,2-tetrahydroperfluorodecyl tetrahydroperfluorodecyl acrylate and vinyl benzoic phosphonic acid [24]. A remarkable strategy to enhance the surfactant efficiency in scCO2 media is based on the synergistic effect obtained by ion-pairing cationic and anionic surfactants. Sagisaka et al. [25] report the properties of short tail, unbranched fluorinated surfactants with sulfate ionic groups and their cationic analogues synthesized by Verdia [26].
Individual compounds, for example, [C6F13S] and [C6F13mim], both exhibit very low solubility in CO2 and produce limited water solubility; even in water they have modest surface activity. The anionic and cationic pair of surfactants with similar hydrophobic tail used together in adequate molar ratio produces a spectacular synergistic effect, which enable the formulation of water-in-scCO2 microemulsion with significant water content (W0 max up to 50).
Metal nanoparticles (Ag, Au, and Cu) are particularly interesting for many industrial applications, due to their size- and shape-dependent unique optic and electric properties. In the last decades, applications of metal nanoparticles in nanomedicine, in emerging fields such as in vivo bioimaging, innovative cancer therapy, and diagnostics evolve in solution with enormous positive impact in the health-care system. The large-scale applicability of these nanomaterials is still restricted due to the concern about their intrinsic biocompatibility and the difficulties to remove toxic residual precursors and solvents during the synthesis, in particular regarding synthesis in W/O microemulsion as heterogeneous reaction media. Thus, new methods to synthesize metal or metal oxide nanoparticles using “green chemistry” approach as alternative to classic technologies have been investigated. The first reported synthesis of silver nanoparticles in CO2 microemulsion was reported in the late 1990s. The results were promising in terms of the quality of the nanopowder obtained, with additional advantage of fine-tuning the nanoparticle size. W/O microemulsions, as heterogeneous reaction media allow the synthesis of nanoparticles with desired average size through the specific mechanism of precipitation in arrested environment, are defined by the water droplets. The radius of the water nanoreactor could be tuned in a facile way by changing water/surfactant ratio (W0c) in the composition of microemulsion. Fernandez et al. [27] demonstrated that the use of scCO2 microemulsion as reaction media exhibits additional advantage in continuous tuning of the Ag nanoparticle size. The nanoparticles were prepared in scCO2 microemulsion with different water/surfactant molar ratios (W0c), that is, different water nanopool dimensions, stabilized with surfactant AOT fluorinated analogue at 10 mM concentration. The change of fluid phase density is obtained by simply adjusting temperature and pressure conditions. For the scCO2 microemulsion with lower water content (W = 6) very fine tuning was achieved, producing silver nanoparticles with average size ranging from 1.9 to 9.3 nm for a variation of density from 0.96 up to 0.80 g/mL.
The same strategy is used to obtain tunable size of quantum dots, cadmium sulfide and zinc sulfide [28]. Similarly, Cu nanoparticles have been obtained [29] by reduction of the copper precursor (copper nitrate) dissolved in the water droplets of microemulsion after injection of CO2-soluble reducing agents such as
Polymeric nanoparticles could also be obtained through polymerization processes in scCO2 microemulsion, using adapted strategy from emulsion/microemulsion polymerization with common oily phases. The preparation of polyamide nanoparticles was reported by Ohde et al. [30], using a scCO2 microemulsion stabilized with a mixture of commercially available hydrocarbon and fluorinated surfactants (AOT and perfluoropolyether-phosphate PFPE–PO4). The NPs were prepared by using polymerization of acrylamide monomer in the presence of potassium persulfate, both dissolved in the water core of scCO2 microemulsions with identic composition. The reaction is performed in homemade interconnected high-pressure vessels that allow the equilibration of separate scCO2 microemulsions containing monomer and initiator and mixing them at the desired time. The polymerization was completed very quickly, in less than 1 minutes at 25°C, 20 MPa, and in less than 30 s at 50°C, 15 MPa.
Metallic nanoparticles deposited on the surface of the graphenes or carbon nanotubes are hybrid material with particular interest in various domains, for example, catalysis, energy saving, modified electrodes in biosensing, etc. The classic methods employed suffer from various drawbacks, from high amount of organic reagents to the lack of dimension control. Deposition of metal nanoparticles on the carbon nanotubes using scCO2 microemulsion as reaction media was proposed as green alternative to wet chemistry or other technologies of deposition using physical methods.
Shimizu et al. [31] report the preparation of hybrid electrocatalyst consisting in Pt NPs deposited on multiwalled carbon nanotubes in three different media: water-in-supercritical CO2 microemulsion, supercritical CO2 fluid, and water-in-hexane microemulsion.
As it is shown in Figure 8, the reaction in scCO2 microemulsion leads to a better dispersion of Pt NPs on the surface of nanotubes that is obtained in homogeneous scCO2 phase and conventional W/O microemulsion.
TEM images of hybrid catalyst Pt-CNT synthesized in various media by (a) water-in-supercritical CO2 microemulsion, (b) water-in-oil microemulsions, (c) direct deposition in supercritical CO2 (adapted with permission from ref. [
Among the many techniques of processing and synthesis in fluid carbon dioxide phases, the use of water-in-scCO2 microemulsions is one of the most effective in terms of control on the size, size distribution, and shape of the final product. The stabilization of such reaction media with low amount of affordable and nontoxic surfactants remains the major bottleneck and limits the extension of this method.
The modification of nanoparticle surfaces, either in solutions or as layers deposited on various surfaces, is a key issue in many processes to fabricate functional materials, in many applications. Commonly used methods for the modification of nanoparticles such as quantum dots, graphene, metallic nanoparticles, etc. are based on solution processing involving organic solvents, which makes the whole process very difficult to implement as eco-friendly technology. For biomedical applications in particular, the proper functionalization of the surfaces in synthesis that involve green solvents is particularly encouraged. Industrial or large-scale processes for the modification of surfaces by vapor-related physical methods suffer from huge limitations due to the high-cost and energy-consuming devices. On the other hand, liquid-related chemical methods have an additional drawback: the solvent removal after the reaction is completed on the targeted surface. Due to the variety of roles that can be attributed to carbon dioxide, i.e., solvent, antisolvent, reaction medium, respectively, unique flexibility for surface engineering is obtained using scCO2.
A variation of RESS method can be applied for coating processes, with the condition that coating material dissolves in scCO2. The solution is introduced via a nozzle in another chamber that contains the solid support to be functionalized. During the rapid depressurization, the density of CO2 changes significantly, and consequently a supersaturation followed by precipitation of the material forming a coating occurs. For example, superhydrophobic coatings on paper were obtained from alkyl ketene dimer (AKD) dissolved in scCO2 by this modified RESS technique.
Surface modification of silica particles with organomodified silane derivatives for highly hydrophobic films has been reported by Purcar et al. [32]. The formation of silica nanoparticles using sol-gel synthesis in various conditions of temperature and pressure of CO2 was investigated, both in subcritical and supercritical phases. Tetraethoxysilane (TEOS) was used as the particle precursor and octyltriethoxysilane (OTES) as the co-precursor for coating. The dynamic light scattering measurements show that the higher CO2 pressure leads to larger diameters of coated particles. The agglomeration of the silica nanoparticles was found to be dependent on the temperature and pressure of the CO2 reaction media. The SEM images confirm the tendency of aggregation and differences in the size of silica nanoparticles coated with octyl-modified derivative (Figure 9).
SEM images of coated silica nanoparticles prepared in CO2 in various conditions of temperature and pressure. (1 = 25°C, 6 MPa; 2 = 40°C, 6 MPa; 3 = 60°C, 6 MPa; 4 = 25°C, 7.5 MPa; 5 = 40°C, 7.5 MPa; 6 = 60°C, 7.5 MPa). Reprinted with permission from Ref. [
The size of the resulting silica particles is around 100–250 nm when the CO2 pressure is 6 MPa and decreases up to 150 nm when the CO2 pressure increases to 7.5 MPa. It was observed that the higher CO2 pressure gives narrower size distributions. The thin films obtained by the as-prepared silica nanodispersions deposited on glass substrate show high contact angle values, ranging from 127 to 1450.
The NP processing using scCO2 as solvent leads not only to the adequate modification of the surface but, sometimes, also to the decrease of the aggregates and better dispersibility and interaction with polymeric matrix. Wei et al. [33] reported recently the silane modification of graphene nanoparticles for the fabrication of composite polymeric films with superior barrier properties. The obtained graphenes functionalized with silane exhibit a fluffy appearance with small size of sheets. The graphene surface was silanized through the hydrolysis and condensation of different organomodified silane derivatives: aminopropyl triethoxysilane, glycidyloxypropyltrimethoxysilane, and methacryloxy propyl trimethoxysilane. The reaction was conducted in scCO2 for 2 hours at 40
Nanoparticle surface functionalization with polymeric shells is an important issue in preparation of biocompatible colloidal vectors in drug delivery that allow further attachment of bioactive ligands. The transfer of such synthesis in scCO2 for the functionalization of nanoparticles used as drug delivery systems in pharmaceutical formulations benefits from the absence of many posttreatment steps that are generally required to remove remaining solvents or nondesirable by-products generated in wet chemistry methods. Marre et al. [34] demonstrated the feasibility of a versatile method for coating silica particles used as model substrates, with either a hydrophilic (polyethylene glycol PEG) or hydrophobic polymer (hydroxyl-terminated polybutadiene, HTPB) using a supercritical antisolvent process (Figure 10).
The variation of polymeric shell thicknesses on nanoparticles as a function of the initial mass fraction of polymer in the initial solution (reprinted with permission from ref
The principle of the precipitation from compressed antisolvent PCA techniques is allowing the contact of the polymeric solution with the scCO2 as antisolvent, when a succession of different phenomena governed by hydrodynamic, precipitation kinetics, thermodynamic, and mass-transfer effects result eventually in the precipitation of an organic shell on the nanoparticles dispersed previously in the polymeric solution. According to the polymer used (PEG or PBHT) and the reaction conditions, various sizes of the deposited polymeric shell have been obtained, from 3 to 28 nm, by simply modification of polymer concentration in the initial solution (Figure 10).
In order to effectively exploit the advantages of “green” processing in scCO2, Roy et al. describe the successful grafting of a model drug (salicylic acid, SA) on TiO2 nanoparticles [35]. The chosen active compound could react with the nanoparticle surfaces through the carboxylic and hydroxyl groups. The functionalization of the TiO2 surface was performed in mild conditions, at 40
In the last decades, methods to perform functionalization of nanoparticles using scCO2 were reported, as viable alternative to classical synthesis and showing remarkable advantage of reduced impact on the environment.
The recent works in supercritical fluids demonstrate the multiplicity of advantages in synthesis of nanomaterials, either organic or inorganic. Complex synthesis of hybrid materials consisting in nanoparticles deposited on other structures (carbon nanotubes) or core-shell nanoparticles was also developed in scCO2.
An important problem that requires further research, especially for fine-tuning the nanoparticle size, is design and fabrication of CO2-philic surfactants that enable the formation of water-in-scCO2 microemulsion as reaction media. The successful synthesis of nontoxic and cost-effective surfactants will open the opportunity to extend this method to many nanoparticles used as drug delivery vehicles, where the size and size distribution control are a key issue.
Another domain which is expected to benefit from further research in order to advance this technology is the development of novel strategy for surface functionalization, especially for scalable processes.
This work was supported by grants of the Romanian National Authority for Scientific Research and Innovation, CCCDI - UEFISCDI, project number PN-III-P1-1.2-PCCDI-2017-0743/P5, and PN-III-P1-1.2-PCCDI-2017-0428/P2 within PNCDI III.
The authors have no conflict of interest to declare.
pressure, MPa temperature, K time, s maximum water content in microemulsion, expressed as water-to-surfactant molar ratio [H2O]/[surfactant] polymeric shell thicknesses on nanoparticles, nm wavelength, cm−1 initial mass fraction of polymer, % critical alkyl ketene dimer sodium bis-(2-ethylhexyl) sulfosuccinate chitosan depressurization of an expanded solution into an aqueous medium dynamic light scattering Fourier-transform infrared spectroscopy graphene nanoparticles octyltriethoxysilane 2-Pyridinealdoxime methochloride polybutadiene hydroxy terminated poly(ε-caprolactone) poly(N,N-dimethylaminoethyl methacrylate) polyethylene glycol polylactic acid poly(lactic-coglycolic) poly(methyl methacrylate) poly(oligo(ethylene glycol) methacrylate) rapid expansion of supercritical solution into a liquid solvent rapid expansion of supercritical solution supercritical assisted atomization supercritical assisted injection in liquid antisolvent supercritical antisolvent technique supercritical carbon dioxide supercritical fluids supercritical emulsion extraction scanning electron microscopy tetraethoxysilane volatile organic compound Water-in-oil water-in-supercritical CO2
In recent years, many researchers have developed models to discuss the importance of lowering carbon emissions and its potential impact on society by examining economic growth, international trade, and health benefits. Khan et al. [1] examined the relationship between green logistics indices, economic, environmental, and social factors through the perspective of Asian emerging economies. By adopting a Fully Modified OLS (FMOLS) Model and Dynamic OLS (DOLS) they claimed that logistics operations, particularly the efficiency of customs clearance processes, quality of logistics services and trade and transport-related infrastructure are positively and significantly correlated with per capita income, manufacturing value added and trade openness, whereas greater logistics operations are negatively associated with social and environmental problems including, climate change, global warming, carbon emissions, and poisoning atmosphere. Khan et al. [2] examined the potential relationship between public health expenditures, logistics performance indices, renewable energy, and ecological sustainability in members of the Association of Southeast Asian Nations by applying the structural equation modeling approach. They showed that the use of renewable energy in logistics operations will improve environmental and economic performance to reduce emissions, whereas environmental performance is negatively correlated with public health expenditures, indicating that greater environmental sustainability can improve human health and economic growth. In [3], economic growth and environmental sustainability in the South Asian Association for Regional Cooperation using the data from the South Asian Association for Regional Cooperation (SAARC) member countries from 2005 to 2017 was examined. Adopting the panel autoregressive distributed lag technique to examine the hypotheses, they find that environmental sustainability is strongly and positively associated with national scale-level green practices, including renewable energy, regulatory pressure, eco-friendly policies, and the sustainable use of natural resources. In [4], the consumption of renewable energy with international trade and environmental quality in Nordic countries from 2001 to 2018 is investigated. Their findings concluded that renewable energy is strongly and positively associated with international trade in Nordic countries. Furthermore, [5] adopted multi-criteria-decision-making techniques to examine barriers in the sustainable supply chain management (SSCM) when firms are facing heavy pressure to adopt green practices in their supply chain (SC) operations to achieve better socio-environmental sustainability.
Around the world governments, businesses and individuals have committed to reducing carbon emission. As a result, the energy economy is highly exposed to these processes. As industries push for renewable energies, technology will need to step in to ensure reliability of the power supply. Therefore, there remains a need for exploiting the role of hybrid technology, its dynamics, limitations on the reduction of pollution levels and policy implementation within the wider carbon emissions debate. This is due to the vital role hybrid technology plays in energy production processes and the ability for the energy system to offer a better energy security. Development of such lower carbon emission policies has potential benefits to the environment and ecological sustainability to those economies. However, within many of these environmental policy models, technology is incorporated as an exogenous variable and limited attention is given to endogenous technology, other technological breakthroughs, potential government subsidies or collaborative innovations to integrate low carbon technology in environmental economics. Such interventions will promote the renewable energy sector to use natural resources and undertake public-private partnership investments to minimize dependence on fossil fuel derived energy.
To investigate the effects of hybrid-enabling technology when producing energy to meet consumption demand, we assume energy producing firms follow the Bertrand game paradigm. In the presence of government subsidy for the development and sustainability of renewable energy, tax on pollution created by energy producing firms will motivate them to undertake Research & Development (R&D) measures to improve hybrid enabling technologies to further reduce the level of carbon pollution. As a result, from an economic point of view it is an interesting question to examine the Bertrand-Nash equilibrium under such a dynamic environment. This chapter examines this concept via a Stochastic Differential game paradigm.
Many researchers have applied game theory to study carbon reduction behavior in electricity markets. In [6], the Cournot equilibria in an oligopolistic electricity market subject to a linear demand function is examined. In [7], the power suppliers bidding behavior is evaluated under the supply function equilibrium (SFE) paradigm, where the market power of an independent system operator (ISO) is modeled as a bi-level multi-objective problem. In [8], the equilibrium strategies in random-demand procurement auctions in the electricity market is obtained and presented a method for explicit calculation of the bid strategies is presented. [9] proposed a Nash bargaining game model to examine how governments can determine the taxes and subsidies in a competitive electricity market whilst achieving their environmental objectives. In [10, 11, 12, 13, 14], the role of government as a leading player who intervenes in competitive electricity markets to promote environmental protection is evaluated. In [15], the role of government, when managing environmental sustainability in a complete electricity market in a Stackelberg game paradigm is examined. In [16], a more robust trans-boundary industrial pollution reduction strategy for global emission collaborations is presented. The dynamics of each country’s quantity of pollution is modeled as a Brownian motion with Jumps to capture the systematic jumps caused by surprise effects arising from policy uncertainties within the economy. However, a crucial limitation within many of these environmental policy models, is that technological change is incorporated as an exogenous variable and does not consider the role of endogenous hybrid-enabling technology or other technological breakthroughs, hence limiting the dynamics of these models.
We quantify an optimal level of subsidy for the sharing of hybrid-enabling technology in an energy market under a Bertrand-duopoly game. We formulate a Stochastic Differential Game (SDG) to analyze the stability of the Stackelberg, Nash and cooperative equilibria via a feedback control strategy. We then adopt limit expectation and variance of the improvement degree to identify the influence of external environment limitations on the decision maker. We show that the game depends on its parameters and the equilibria chosen. We consider an electricity market composed of power plants I and II, with each one having the choice between fossil fuels
In this model, we consider a Bertrand duopoly game for two power plants under endogenous hybrid-enabling technology. In the first stage the matrix of prices
stochastic endogenous hybrid-enabling technology innovation is introduced into a two-player stochastic differential game with random interference factors, which capture uncertain external environment factors and the internal limitations within the shared hybrid-enabling technology decision process. In doing so, we provide a framework to quantify the impacts of market power on prices.
we show that both power plants invest in R&D measures and that the limit of expectation and variance of the improvement degree can be applied to identify the influence of random factors.
mathematically, we show that the issue of the game depends on the parameters of the game and the type of equilibrium one considers.
by applying the HJB equation we obtain the optimal effort level and the optimal level of subsidy for sharing hybrid-enabling technology via feedback equilibrium strategies whilst examining the Stackelberg equilibria, Nash equilibria and cooperative equilibria under Bertrand duopoly.
we reveal that for a given level of payoff distribution the Stackelberg equilibria under endogenous hybrid-technology innovation and the sharing paradigm dominate the Nash equilibria.
we show that in Stackelberg and Nash games, optimal hybrid-enabling technology innovation is proportional to the government subsidy, but the variance improvement degree of the Stackelberg game is different to the results of the Nash game.
our characterization of the short run price competition by strategic supplies for renewable and fossil resources, provides a more robust model than that presented by Bertrand-Edgworth, in which price competition with fixed (endogenous) capacities was used.
our model shows that robust cost-reducing R&D investments with effective hybrid-enabling technology innovation strengthens an innovator’s competitive position and the Stackelberg structure emerges as an equilibrium outcome, allowing each power plant to optimally use energy sources to produce electricity while maximizing their payoffs.
Therefore, under a Stochastic Differential Game (SDG) paradigm with uncertainty, each power plant can optimally use energy sources to produce electricity while maximizing their payoffs. Each power plant is capable of using fossil fuels
In recent years, researchers have incorporated the theory of SDGs, originated from [14, 17, 18, 19, 20] to analyzed environmental issues. Especially [21] analyzed (two player) zero-sum stochastic differential games in a rigorous way, and proved that the upper and lower value functions of such games satisfy the dynamic programming principle whilst being the unique viscosity solutions of their associated Hamilton-Jacobi-Bellman-Isaacs equations.
In Section 2, the proposed model and elements of evolutionary game theory are presented. In Section 3 by implementing the Stackelberg game we examine feedback Stackelberg equilibria, optimal level of subsidy for the shared hybrid-enabling technology from its counterpart and the limit of expectation and variance. In Section 4 by implementing a Nash game we examine feedback Nash equilibria and the limit of expectation and variance under hybrid-enabling technology. In Section 5 by implementing a cooperative game we examine feedback equilibria and the limit of expectation and variance under hybrid-enabling technology. In Section 6, comparative analysis of equilibrium results are described. Section 7 concludes the study. Appendix at the end of the chapter contains proofs.
We propose that the production process of electricity leads to emissions and is proportional to the power industry’s use of energy source. We assume that there are two power plants (Player I) and (Player II) in the energy market and each power plant is capable of using
Under the theory of evolutionary games, the production strategies in the
Then each iteration of an evolutionary game, where two matched power plants in accordance with Bertrand paradigm compete with each market and play a one-shot non-zero-sum game, represents the benchmark game of the population. If
We define the continuous demand function
where
The government’s tariff policy for the power plants with respect to their source of energy for long-time periods are transparent, and this information is available to the public. Therefore, it is assumed that the competitive power plants follow the government’s financial legislation, having the capability and technological skills to produce electricity from specific sources at any given time to meet energy demand. Then for each time-period the power plant will consider the tariff-rate quota or feed-in-tariff and adopt a pricing strategy for the selected energy source. Hence, we conclude that the production rate of the power plants is equal to the corresponding demand rates with a negligible internal consumption and waste rate.
To apply the Backward induction technique to investigate the equilibrium prices, demand, and profits, we define the profit function for each power plant as
where
Defining
Simultaneously solving Eq. (4), obtain
such that
where
Restricting ourselves to a two matched symmetric two-person bi- matrix game in random contest in a one-population evolutionary game, we define the payoff (utility) in Table 1.
Production Strategy | Fossil Fuel | Renewable Sources | |
Fossil Fuel | |||
Renewable Sources |
Bi matrix for two power plants by different energy sources.
Then via equation
Obviously the bimatrix of the power plant II, is given by:
The probability that player I choosing row 1 is
The probability that player II choosing row 1 is
Then the value of the game for Player I is
and the value of the game for Player II is
Suppose (
Otherwise Player I can change
Similarly, for
Otherwise Player II can change y slightly and do better. It follows that the unique Nash equilibrium (
Both players will undertake R&D measures on hybrid-enabling technology to ensure immediate reliability and affordability in energy production whilst reducing GHG-emissions. We assume that the strategic effects implemented by power plant I (Player I), has improved hybrid-enabling technology to generate energy and utilize energy sources in a much efficient way. This gives a superior advantage to power plant I overpower plant II (Player II) and both power plants are rational to maximize their profits. Although Power plant II has heterogeneous resources to hybrid-enabling technology, from a practical point of view it is logical for power plant I to share this technology with power plant II, because the price competition is typically characterized by a second-mover advantage. Many researchers have investigated the effects of these commitments in Cournot, Bertrand and Stackelberg setups. See [29, 30, 31]. Due to the government incentives, tariff-rate quota, feed-in-tariff and R&D incentive measures, the power companies will be competitive to improve their efficiency. Let
and
where
Let
Let
where
The first inequality says that each demand is downward sloping in own price, and the second that goods are substitutes (each demand increases with the price of the other good). [32] shows that in case of symmetric firms, there is a second-mover (first-mover) advantage for both players when each profit function is strictly concave in own action and strictly increasing (decreasing) in rival’s action, and reaction curves are upward (downward) sloping.
Then a sufficient condition on the super-modularity of the profit function is obtained via the profit function
where
We further assume that the total revenue is allocated between two players and
and
where
Theory of strong Stackelberg reasoning is an improved version of an earlier theory [35], which provides an explanation of coordination in all dyadic (two-player) common interest games. It provides an explanation of why players tend to choose strategies associated with a payoff-dominant Nash equilibrium. Its distinctive assumption is that players behave as though their co-players will anticipate any strategy choice and invariably choose a best reply to it. Stackelberg strategies resulting from this form of reasoning do not form Nash equilibria. The theory makes no predictions, because a non-equilibrium outcome is inherently unstable, leaving at least one player with a reason to choose differently and thereby achieve a better payoff. Strong Stackelberg reasoning is a simple theory, according to which players in dyadic games choose strategies that would maximize their own payoffs if their co-players could invariably anticipate their strategy choices and play counter-strategies that yield the maximum payoffs for themselves. The key assumption is relatively innocuous, first because game theory imposes no constraints on players’ beliefs, apart from consistency requirements, and second because the theory does not assume that players necessarily believe that their strategies will be anticipated, merely that they behave as though that is the case, as a heuristic aid to choosing the best strategy. Strong Stackelberg reasoning is, in fact, merely a generalization of the minorant and majorant models introduced by [36] and used to rationalize their solution of strictly competitive games.
To promote the sharing of hybrid-enabling technology, the Player II (the leader) determine an optimal sharing effort sharing level and an optimal subsidy scheme. Then the Player I (the follower) choose his/her optimal sharing level according to the optimal sharing effort level and subsidy. This leads to a Stackelberg equilibrium.
where
where
Via the first order conditions, we obtain the optimal values
where
Substituting the results of Eqs. (37) and (38) into Eq. (39), obtain
Via the first order conditions of (Eq. (40)), we obtain the optimal values
And the optimal value for
and
Hence, the solution of the HJB equation is an unary function with
and simplifying (Eq. (40)), obtain:
This implies that,
where
and
Substituting the results of
The payoff of Player I and Player II, under the Stackelberg game paradigm is related to the improvement degree of hybrid-enabling technology via Proposition 4. To analyze the limit of expectations and variance under Stackelberg game equilibrium rewrite (Eq. (19)) as follows.
where
and
Then
Solving the above non-homogeneous linear differential equation, will obtain the results.
Under Nash-non-cooperative game setting, Player I and Player II simultaneously and independently choose their optimal efforts levels of heterogeneous hybrid-enabling technology sharing concept to maximize their profits.
where
where
where
Poof of Proposition 8 is like the derivation of Proposition 6.
Under cooperative game paradigm, Player I and Player II will choose to collaborate/share their hybrid-enabling technology development knowledge while sharing the payoff function in order to maximize their total payoffs. As a result, hybrid-enabling technology can be improved through this effort as well.
and the optimal cooperative payoff function under hybrid-enabling technology on renewable sources and on fossil fuel, respectively.
where
Then the optimal revenue sharing function satisfies the following HJB equation
Via the first order conditions, now obtain the optimal values
Substituting the results of Eqs. (69) and (70), obtain
Hence, the solution of the HJB equation is an unary function with
Substituting the results of Eqs. (72) and (73), into
where
and
(ii) Player I will prefer to participate in a cooperative game over a non-cooperative game with Player II under the condition such that
and
(iii) The total payoff for Player I under a Stackelberg game exceeds the total payoff of Nash non-cooperative game with Player II under the condition such that
and
(iv). Player II will prefer to participate in a cooperative game over a non-cooperative game with Player I under the condition such that
and
and
In this chapter a complete study of an energy market by considering a Bertrand duopoly game with two power plants using endogenous hybrid-enabling technology was presented. Numerous game paradigms were articulated and defined including Stackelberg, Nash non-cooperative and cooperative games as well as their relevant equilibria via a feedback control strategy. Mathematically, the necessary conditions under which a power plant will move from taking part in a non-cooperative Nash game to participate as a leader in a Stackelberg game was derived. In doing so, this model allowed us to quantify the optimal level of subsidy for sharing the hybrid-enabling technology. We then adopted the concept of limit expectation and variance of the improvement degree to identify the influence of random factors of external environment and limitations of the decision maker. It is found that for a given level of payoff distribution the Stackelberg equilibria with technological enhancements, the knowledge sharing paradigm dominates the Nash equilibria. In both Stackelberg and Nash games, optimal technological enhancements for power plants were found to be proportional to the government subsidy, but the variance improvement degree of the Stackelberg game differed to the results of the Nash non-cooperative game due to the influence of random factors.
Furthermore, we have shown that due to optimal price reaction functions being upward sloping, the subsidy level plays a decisive role on the payoff function of power plant II as the leader in a Stackelberg game. This model shows that cost reducing R&D investments with efficient hybrid-enabling technology innovation/s strengthens one’s competitive bargaining position via the level of subsidy for Power Plant I to become a follower in the Stackelberg game. By analyzing this stochastic differential game model, we capture the government subsidy incentive as well as the subsidy that the leader (Power Plant II) pays the follower (Power Plant I) to share hybrid-enabling technology.
The proposed quantitative framework could assist policymakers when determining the appropriate R&D incentives for the development of hybrid-enabling technology within the energy market to achieve desired short and long-term environmental objectives with respect to budget limitations and environmental considerations.
Via the first order conditions, first obtain the optimal values
where
Substituting Eqs. (A.2) and (A.3) results to Eq. (A.4) and via the first order conditions, we obtain the optimal values
Hence, the solution of the HJB equation is an unary function with
where
and
Substituting
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\\n\\nOur services – The support we offer our authors and editors is second to none. Each book in our program receives the following:
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\n\nInterested? Contact Ana Pantar (book.idea@intechopen.com) for more information.
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