Bogumiła Pilarczyk
email: bogumila.pilarczyk@zut.edu.pl

Diana Hendzel

Renata Pilarczyk

Agnieszka Tomza-Marciniak

Barbara Błaszczyk

Marta Dąbrowska-Wieczorek

Małgorzata Bąkowska

Ewa Adamowicz

Tomasz Bujak

Liver and kidney concentrations of selenium in wild boars (Sus scrofa) from northwestern Poland

Keywords: Selenium, Trace elements, Wild boar, Sus scrofa, Kidney, Liver, Poland

The aim of this study was to determine liver and kidney concentrations of selenium in wild boars from the northwest part of Poland, depending on season of the year, age, sex, and body weight. Altogether, samples of livers and kidneys from 172 wild boars that were shot in 2005-2008 were investigated. Liver and kidney concentrations of selenium were determined using spectrofluorometric method. In all the animals studied, selenium concentration was several times lower in the liver than in the kidneys. Selenium concentration averaged 0.19 μg/g wet weight (w.w.) in the liver and 1.20 μg/g w.w. in kidneys. The present study showed that season (P≤0.05), age (P≤0.01), and body weight (P≤ 0.01) have a significant effect on selenium concentration in the liver of wild boars. Liver selenium concentration was the highest in spring (0.23 μg/g w.w.) and the lowest in autumn (0.16 μg/g w.w). Young animals (up to 1 year of age) and those with the lowest body weight (up to 20 kg) were characterized by a slightly lower selenium concentra-tion in the liver compared to older and heavier animals. No significant differences were found in organ selenium concentration between males and females. According to biochemical criteria for the diagnosis of selenium deficiency in pig liver, which were used to evaluate selenium concentration in the liver of wild boars, no individuals were found to have optimal levels. Considering that in Se deficiency higher selenium concentrations are found in kidneys than in the liver, it can be presumed that the wild boars had Se deficiency. However, this is difficult to state conclusively because there are no reference values for this species.

Introduction

Selenium is an essential trace element that determines the normal growth and development of humans and animals. The biological role of selenium stems mainly from its presence in selenium-dependent enzymes (glutathione peroxidase, iodothyronine deiodinase, selenoproteins P and W, thioredoxin reductase) which play a key role in many metabolic processes [START_REF] Musik | Effect of organic selenium compounds on the activity of glutathione peroxidase and superoxide dismutase in selected mouse tissues[END_REF]Kohrle 2004;[START_REF] Floriańczyk | Selenium and selenoproteins in health and disease[END_REF].

In nature, selenium occurs irregularly, with some areas rich in selenium (considerable areas of North and South America and part of China) and some deficient in this element (a considerable part of Europe, including some regions of Poland, several provinces of China, and New Zealand; [START_REF] Kabata-Pendias | Biogeochemistry of trace elements (in Polish). PWN, Warszawa Kohrle J (2004) Selenium in biology and medicine-further progress and increasing interest[END_REF][START_REF] Zust | Assessment of selenium and vitamin E deficiencies in dairy herds and clinical disease in calves[END_REF]Pilarczyk et al. 2008a[START_REF] Pilarczyk | Selenium concentration in liver and kidney of free living animals (roe and red deer) from West Pomerania (Poland)[END_REF].

Until recently, research on Se levels in animals and the health consequences of its deficiency was limited to farm animals only. Polish research in this area [START_REF] Jaśkowski | O diagnozowaniu i ocenie niedoborów miedzi, selenu kobaltu i manganu u bydła i owiec[END_REF][START_REF] Balicka-Ramisz | Effects of selenium administration on blood serum Se content and on selected reproductive characteristics of sheep[END_REF][START_REF] Pilarczyk | Selengehalte im Blutserum von Kühen, Schafen und Schweinen in der Westukraine und Westpommern[END_REF]Pilarczyk et al. , 2008b) ) demonstrated low serum concentrations of selenium in animals, possibly suggesting that these animals are raised in selenium-deficient areas.

Thanks to a full integration with their habitat, free-living animals are a good indicator of environmental selenium concentrations throughout their lives. Game animals have been examined relatively seldom [START_REF] Mcdowell | Selenium status of white-tailed red deer in southern Florida[END_REF][START_REF] Vikøren | Liver concentrations of copper, cobalt, and selenium in wild Norwegian Red deer (Cervus elaphus)[END_REF]Pilarczyk et al. 2008a[START_REF] Pilarczyk | Selenium concentration in liver and kidney of free living animals (roe and red deer) from West Pomerania (Poland)[END_REF]. Selenium deficiency in free-living animals is strictly associated with the amount and availability of this element in the soil. Selenium is taken up from the soil environment by plants, from which it reaches animals. The amount of selenium absorbed by plants depends on a number of factors such as soil pH, oxidation and reduction conditions of the soil environment, and degree of selenium oxidation. As a result of low amounts of available selenium in the soil, it is low in plants and deficient in animals (Kabata-Pendias and Pendias 1993).

The wild boar (Sus scrofa) is an omnivorous animal that ingests over 90% of its food from plants and around 10% from animals. Wild boars consume acorns, beech nuts, nuts, berries and other fruits, some grasses and herbs, rootlets and rhizomes, imagoes, their pupae and larvae, small rodents, frogs, eggs and chicks of birds nesting on the ground, young mammals that let themselves be caught, and carrion [START_REF] Genov | Food composition of wild boar in north-eastern and western Poland[END_REF][START_REF] Pinna | Vegetable and animal ford sorts fund in the gastric content of Sardinian Wild boar (Sus scrofa meridionalis)[END_REF].

At present, selenium is acknowledged to be a microelement that is vital for the normal height and development of animals and humans. Selenium deficiency in farm and freeliving animals may cause large economic losses due to its associated white muscle disease, mulberry heart disease, and hepatosis dietetica in hogs and wild boars; in females, it can lead to the death of fetuses due to their arrested development, retained placenta after delivery, inflammation of the genital tract, the MMA syndrome, and a decrease in the breeding abilities of males [START_REF] Grela | Organic selenium compounds in animal nutrition (in Polish)[END_REF][START_REF] Radostits | Veterinary medicine: a textbook of the diseases of cattle, sheep, pigs, goats and horses[END_REF]. Although there is no official information about the incidence of selenium deficiency diseases in wild boars from the northwest part of Poland, hunters reported seeing smaller litters from time to time. [START_REF] Jacyno | Reproductive performance of young boars receiving during their rearing inorganic or organic selenium + vitamin E in diets[END_REF] observed that animals receiving higher dietary amounts of Se were characterized by better libido, higher concentration and total number of spermatozoa in ejaculate, and lower percentage of spermatozoa with morphological defects. Likewise, Reglero et al. (2009a) found semen quality to be strictly related to selenium concentration in testes. This shows that selenium deficiency may be one of the reasons for a decline in the wild boar population.

The aim of this study was to determine selenium concentrations in the liver and kidneys of wild boars from the northwest part of Poland, depending on the season, sex, and body weight.

Materials and methods

Samples

Research material included liver and kidney samples obtained from 172 wild boars originating from northwestern Poland. Wild boars were shot during the hunting seasons of 2005-2008 in compliance with the hunting limits set. Thirty-seven, 56, and 49 wild boars were harvested during the 2005/2006, 2006/2007, and 2007/ 2008 hunting seasons, respectively. After the shooting, carcasses and organs were transported to a wild game collection center, where the carcasses were marked and the age, sex, and weight of animals were determined. The age of animals was estimated using fur characteristics [START_REF] Lochman | Age determination in game animals[END_REF]. After these procedures, the prepared organs (n liver =172, n kidney =172) were transported to a laboratory and stored at -20°C until chemical analyses. Whole livers and kidneys were homogenized to obtain a uniform material, from which samples were collected for selenium determination.

Reagents

All chemicals used were of analytical reagent grade. Most of the chemicals were obtained from Chempur® except 2,3diaminonaphtalene, which was obtained from Sigma Aldrich. Certified reference material BCR-185R (bovine liver; European Commission Joint Research Center Institute for Reference Materials and Measurements) was obtained from LGC Standards GmbH, Germany.

Analysis

Concentrations of selenium in liver and kidneys were determined using spectrofluorometric method [START_REF] Grzebuła | The determination of selenium trace levels in biological materials with fluorometric method. Selenium determination in tissues and bodily fluids (in Polish)[END_REF]. The tissues were digested in HNO 3 at 230°C for 180 min and in HClO 4 at 310°C for 20 min. Then, the samples were hydrolyzed with 9% HCl. Selenium was derivatized with 2,3-diaminonaphtalene (Sigma Aldrich) under the conditions of controlled pH (pH 1-2) with the formation of selenadiazole complex. This complex was extracted into cyclohexane. EDTA and hydroxylamine hydrochlorine were used as masking agents. Se concentration was determined fluorimetrically using RF-5001 PC Shimadzu spectrophotofluorimeter. The excitation wavelength was 376 nm; the fluorescence emission wavelength was 518 nm.

The accuracy of the analyses was verified using certified reference material BCR-185R (bovine liver). A reference sample was analyzed in triplicate. The Se concentration obtained ranged between 89.4% and 101.7% of the reference values (mean 95.5%±5.9%). The detection limit was 0.003 μg/g wet weight.

Statistical data analyses

Statistical calculations were performed using procedures of STATISTICA PL 7.1. All data are expressed throughout as an arithmetic mean ± standard deviation and also geometric mean and median. The concentrations of the Se were logtransformed to attain or approach a normal distribution of the data. The effects of season, sex, age, and body weight on selenium concentration in the liver and kidneys of wild boars were analyzed using generalized linear models. Differences were considered as significant at the level of P<0.05 and P<0.01. Relationships between concentrations of selenium in the liver and in the kidneys were evaluated by calculating the coefficients of correlation. Pearson's correlation coefficient (r x,y ) was counted for every season separately and together for all data. Statistical significance of coefficients of correlation was tested at the level P<0.05 and P<0.01.

Results

The present study showed that season (P≤0.05), age (P≤ 0.01), and body weight (P≤0.01) have a significant effect on selenium concentration in the liver of wild boars, while kidney selenium concentration was significantly influenced only by season (P≤0.05; Table 1). In all the animals studied, selenium concentration was higher in kidneys than in livers (Table 2). Liver selenium concentration averaged 0.19 μg/g w.w. in the liver and was several times higher in kidneys (1.20 μg/g w.w.). The lowest selenium concentration in the liver was found in the autumn period and the highest in the spring and summer periods, with statistically significant differences (P≤0.05). In kidneys, selenium concentration was also the lowest in the autumn. In the summer period, a statistically significant (P≤0.05) positive mean correlation of r=0.36 was found between selenium concentration in the liver and in kidneys, with no such correlations observed in the other periods.

Our results showed that the sex of animals had no significant effect on Se concentration in the organs. Mean selenium concentration in kidneys and liver was 1.23 and 0.19 μg/g w.w. in females and 1.14 and 0.20 μg/g w.w. in males.

Young animals up to 1 year of age (Table 2) were characterized by lower selenium concentration in the liver (P ≤ 0.01) compared to older animals. In addition, a statistically significant (P≤0.05) positive mean correlation of r=0.36 was found between liver and kidney selenium concentration of young animals.

The analysis of the effect of body weight on selenium concentration in the liver of wild boars revealed similar relationships as for the effect of age. Animals weighing up to 20 kg were characterized by significantly (P≤0.01) lower selenium concentration in the liver (0.15 μg/g w.w. on average) compared to animals weighing above 40 kg. In animals weighing between 41 and 60 kg, there was a statistically significant (P≤0.05) positive mean correlation of r=0.30 between selenium concentration in the liver and kidneys.

Discussion

At present, no standards are available for selenium concentrations in the liver and kidneys of wild boars. The results of environmental studies should not be compared with the results obtained for pigs because of major differences in the feeding of these animals, which translates directly into Se concentration in their bodies. Because there are no reference values concerning selenium content in the organs of wild boars, we used the standards designed for pigs. According to [START_REF] Puls | Mineral levels in animal health: diagnostic data, 2nd edn[END_REF], biochemical criteria used in the diagnosis of selenium deficiency in the liver of pigs are as follows: below 0.11 µg/g w.w., deficiency; 0.12-0.39 µg/g w.w., marginal level; above 0.40 µg/g w.w., suitable (optimal) level for animals. Using these criteria to evaluate selenium concentration in the liver of wild boars, no individuals were found to have optimal levels. Se deficiency was observed in 31.0% of animals in autumn, 20.9% in summer, 15.2% in winter, and 12.0% in spring. Marginal levels were found in the other animals.

In the examined wild boars, liver selenium concentration ranged from 0.05 to 0.58 µg/g w.w. The highest mean concentration of selenium in the liver (0.23 µg/g w.w.) was found in spring, during the period when the lowest proportion of animals had Se deficiency. Our study has shown that, in all seasons of the year, mean selenium concentrations in the wild boars reached only marginal levels. The animals' habitat has long been considered deficient in selenium. [START_REF] Zabłocki | Selenium in soils and plants of West Pomerania[END_REF] reported soil selenium concentrations in northwest part of Poland to range from 0.026 to 0.293 mg/kg d.w., while even a threefold higher concentration of this element was found in the soils of south Poland [START_REF] Piotrowska | Concentration of selenium in cultivated soil of Poland (in Polish)[END_REF]. Research in the northwest part of Poland on Se concentration in wild ruminants showed Se deficiency (based on Se status criteria for ruminants; Puls 1994) in more than half of the animals studied (Pilarczyk et al. 2008a[START_REF] Pilarczyk | Selenium concentration in liver and kidney of free living animals (roe and red deer) from West Pomerania (Poland)[END_REF].

We found that season of the year has a significant effect of selenium concentration in the organs of wild boars, especially in their livers. The highest concentration of this element was found in the spring and summer period and the lowest in the autumn. This phenomenon is probably due to seasonal variations in wild boar diets [START_REF] Giménez-Anaya | Food habits of wild boars (Sus scrofa) in a Mediterranean coastal wetland[END_REF][START_REF] Baubet | Diet of the wild boar in the French alps[END_REF]). [START_REF] Baubet | Diet of the wild boar in the French alps[END_REF] observed that green parts of plants are dominant in the diet of European wild boars in the spring. [START_REF] Galeas | Seasonal fluctuations of selenium and sulfur accumulation in selenium hyperaccumulators and related nonaccumulators[END_REF] reported that research on seasonal fluctuations in Se content of plants reveal that the highest concentrations of this element in leaves are found during the spring period. According to [START_REF] Baubet | Diet of the wild boar in the French alps[END_REF], plant roots, seeds, and fruits form a considerable proportion of the diet in spring. Compared to spring and summer, an increased proportion of underground parts of plants (roots and bulbs) in the diet is observed in autumn. During this period, Se returns to the roots but in smaller amounts [START_REF] Galeas | Seasonal fluctuations of selenium and sulfur accumulation in selenium hyperaccumulators and related nonaccumulators[END_REF] and, what is more, volatile forms of Se are formed in roots as a result of increased metabolism [START_REF] Wójcik | Phytoremediation-a method to clean up the environment (in Polish)[END_REF]. In summer, in addition to forest and meadow plants, wild boars eagerly use agricultural crops such as maize, cereals, and potatoes The same lower case letters denote statistically significant differences at p<0.05; the same upper case letters denote statistically significant differences at p<0.01 GM geometric mean,no correlation (non significant correlation) *p<0.05 (statistically significant coefficient of correlation); **p<0.01 (statistically significant coefficient of correlation) [START_REF] Genov | Food composition of wild boar in north-eastern and western Poland[END_REF]. [START_REF] Schley | Diet of wild boar Sus scrofa in Western Europe, with particular reference to consumption of agricultural crops[END_REF] report that cultivated plants are often part of the diet and are consumed in large amounts.

It is also worth noting that, in the autumn (October and November), wild boars show increased sexual activity accompanied by no food intake [START_REF] Weiler | Influence of the photoperiod and a light reverse program on metabolically active hormones and food intake in domestic pigs compared with a wild boar[END_REF]. In addition to seasonal changes in the availability of seleniumrich foods, this phenomenon probably influences selenium concentration in the organs of these animals.

The mean Se concentration in the liver (geometric mean ≈0.576 µg/g d.w., taking into account the mean d.w. of liver 31.2%) of wild boars, obtained in our study, was comparable with the findings of Reglero et al. (2009b), who reported that the liver of wild boars from south Spain contained 0.453 µg Se/g d.w. (Sierra Madrona) and 0.589 µg Se/g d.w. (Montes de Ciudad Real and eastern Sierra Morena). However, these results are lower compared to other species of animals. In roe deer from Western Pomerania (Poland), Pilarczyk et al. (2008a) observed that the mean selenium concentration was 0.71±0.64 μg/g d.w. in the liver and 3.09±1.73 μg/g d.w. in the kidneys. [START_REF] Mcdowell | Selenium status of white-tailed red deer in southern Florida[END_REF] found that average Se concentration in the liver of white-tailed red deer was 0.676 μg/g d.w. [START_REF] Millán | Levels of heavy metals and metalloids in critically endangered Iberian lynx and other wild carnivores from Southern Spain[END_REF] reported that the liver selenium concentration in red foxes and badgers from Spain averaged 1.72 and 4.88 μg/g d.w., respectively Statistical analysis showed that young animals up to 1 year of age were characterized by significantly (P≤0.01) lower concentration of selenium in the liver compared to older animals. A similar relationship in other animal species was observed by [START_REF] Rush | Geographic distribution of selected elements in the liver of polar bears from Greenland, Canada and the United States[END_REF] and [START_REF] Conover | Concentrations of selenium and mercury in eared grebes (Podiceps nigricollis) from Utah's Great Salt Lake, USA[END_REF]. We believe that such a low concentration of Se in young wild boars could result from more rapid metabolism in these animals. The increased metabolic rate may result in a release of greater amounts of free radicals, thus increasing the body's requirement for substances with antioxidant properties, including glutathione peroxidase that contains selenium, which is found in the liver. Based on results of experimental studies with rats, [START_REF] Sunde | Dietary selenium requirements based on tissue selenium concentration and glutathione peroxidase activities in old female rats[END_REF] concluded that selenium requirement in young animals can be half as large as in adult animals.

In our study, we observed that liver Se concentrations in males and females were similar. No correlation between Se concentration and sex was also reported by [START_REF] Nicpoń | Selenium level of dog sera from Lower Silesia[END_REF] in dogs, [START_REF] Conover | Concentrations of selenium and mercury in eared grebes (Podiceps nigricollis) from Utah's Great Salt Lake, USA[END_REF] in eared grebes (Podiceps nigricollis), and [START_REF] Rush | Geographic distribution of selected elements in the liver of polar bears from Greenland, Canada and the United States[END_REF] in polar bears. Meanwhile, [START_REF] Millán | Levels of heavy metals and metalloids in critically endangered Iberian lynx and other wild carnivores from Southern Spain[END_REF] found that the liver of female red foxes and mongooses had higher Se concentrations compared to the males. Perhaps, such differences are observed when there are distinct differences in feeding habits between sexes. [START_REF] Pollock | Trace elements status of white-tailed red deer (Odocoileus virginianus) and moose (Alces alces) in Nova Scotia. Wildlife Damage Management[END_REF] holds that selenium concentrations in the liver are a better indicator of Se status than the kidneys.

However, considering the lack of proper reference values, knowledge of the content of this element also in kidneys can help diagnose Se deficiency in animals. [START_REF] Oh | Glutathione peroxidase response to selenium intake in lambs fed a torula yeast-based, artificial milk[END_REF] observed that kidney selenium concentrations in animals receiving selenium-deficient feeds were much higher than in the liver and found an opposite situation in animals receiving selenium-rich feeds. In our study, we observed that liver Se concentration was several times lower than in kidneys, possibly suggesting that the dietary content of this element was too low or the form of Se was poorly available.

Comparison of the Se concentrations in wild boar kidneys, obtained in the present study, with the criteria used in the diagnosis of selenium deficiency in the kidneys of swine [START_REF] Puls | Mineral levels in animal health: diagnostic data, 2nd edn[END_REF], shows that optimal concentrations in wild boar kidneys were shown only in 4.0% of animals in spring, 37.2% in summer, 5.2% in autumn, and 8.7% in winter.

Conclusion

Because of limited literature data on selenium concentration in the organs of wild boars and the lack of reference values, it is difficult to state conclusively if the content of this element fell within the range that could be considered appropriate. The selenium deficiency could be evidenced by the fact that Se concentration in the liver of wild boars was several times lower than in kidneys. It seems necessary to conduct further research in this area while analyzing the health status and reproductive parameters of wild boars to determine the reference values for this species.

Table 1

 1 Effect of season, sex, age, and body weight on selenium concentration in the liver and kidneys of wild boars

	Treatment	Liver	Kidneys
		P values	P values
	Season	0.02	0.03
	Sex	-	-
	Age	<0.001	-
	Body weight	<0.001	-

no correlation (non significant correlation)

Table 2

 2 Selenium concentration

		N	Se concentration (µg/g w.w.)
			Liver					Kidney	Value of the coefficient
									of correlation (r x,y ) between
			Mean	SD	Range	GM Median Mean SD	Range	GM Median	Se concentrations in the
									liver and kidney
	Season
	Winter	46 0.17	0.07 0.08-0.38 0.16	0.16	1.20	0.26 0.70-2.03 1.17	1.21	-
	Spring	25 0.23a	0.11 0.10-0.40 0.21	0.21	1.16	0.32 0.14-1.56 1.08	1.22	-
	Summer	43 0.21b	0.09 0.09-0.40 0.19	0.18	1.39a 0.36 0.22-1.98 1.32	1.41	0.36*
	Autumn	58 0.16ab	0.07 0.05-0.36 0.15	0.16	1.11a 0.31 0.24-1.62 1.04	1.21	-
	Total	172 0.19	0.09 0.05-0.58 0.17	0.17	1.21	0.33 0.14-2.03 1.15	1.22	0.21**
	Sex
	Male	83 0.20	0.10 0.05-0.40 0.17	0.19	1.14	0.36 0.14-1.97 1.09	1.22	0.29**
	Female	89 0.19	0.08 0.06-0.39 0.17	0.18	1.23	0.32 0.24-2.03 1.19	1.22	-
	Total	172 0.19	0.09 0.05-0.58 0.17	0.17	1.21	0.33 0.14-2.03 1.15	1.22	0.21**
	Age [year]
	<1	49 0.14AB 0.06 0.05-0.39 0.13	0.13	1.19	0.35 0.30-2.03 1.13	1.22	0.36*
	1-2 years	99 0.20A	0.09 0.06-0.40 0.18	0.19	1.24	0.30 0.22-1.97 1.19	1.22	-
	>2	24 0.24B	0.08 0.09-0.39 0.22	0.23	1.13	0.40 0.14-1.80 1.01	1.17	-
	Total	172 0.19	0.09 0.05-0.58 0.17	0.17	1.21	0.33 0.14-2.03 1.15	1.22	0.21**
	Weight (kg)
	≤20	29 0.15AB 0.07 0.06-0.39 0.13	0.13	1.23	0.36 0.53-2.03 1.21	1.21	-
	21-40	41 0.18	0.07 0.05-0.38 0.16	0.15	1.15	0.36 0.24-1.97 1.11	1.22	-
	41-60	54 0.21A	0.08 0.07-0.40 0.19	0.20	1.24	0.33 0.22-1.97 1.22	1.22	0.30*
	>60	48 0.21B	0.10 0.06-0.40 0.17	0.20	1.14	0.36 0.14-1.80 1.07	1.20	-
	Total	172 0.19	0.09 0.05-0.58 0.17	0.17	1.21	0.33 0.14-2.03 1.15	1.22	0.21**

Ethical standards

The authors declare that the experiment complies with the current laws of Poland.

Conflict of interest

The authors declare that they have no conflict of interest.


T Kovanen

T Tarhasaari

L Kettunen

J Korppi-Tommola

Modelling chemical composition in electric systems -implications to the dynamics of dye-sensitised solar cells

Keywords: 82, 47 Applied electrochemistry -72, 40 Photoconduction and photovoltaic effects -73, 63 Electronic transport

Classical electromagnetism provides limited means to model electric generators. To extend the classical theory in this respect, additional information on microscopic processes is required. In semiconductor devices and electrochemical generators such information may be obtained by modelling chemical composition. Here we use this approach for the modelling of dye-sensitised solar cells. We simulate the steady-state current-voltage characteristics of such a cell, as well as its transient response. Dynamic simulations show optoelectronic hysteresis in these cells under transient light pulse illumination.

Introduction

In certain physical systems the main process is the generation of electric power by chemical reactions. By itself, the classical (phenomenological) electromagnetic theory provides limited means to the modelling of these systems.

Consequently, they have drawn a lot of attention from the physics and chemistry research communities, and addia E-mail: tuomas.kovanen@tut.fi, Fax: +358331152160 tional information on microscopic processes has been combined with the classical theory for a more detailed model.

When employing the classical theory, generation and dissipation are usually confined to distinct parts of space.

In the dissipative part, the electric field intensity e of a given charge distribution is determined by the Poisson equation of electrostatics together with the charge conservation law and the classical Ohm's law. In the generator part of space, on the other hand, the product of e and the current density j needs to be made negative. To accomplish this, the Ohm's law could be modified by an additional source current term assumed to be known in advance.

For some purposes, a more detailed model than provided by the classical theory can be useful. Such a model necessarily involves an alternative for Ohm's law to be used in the description of current generation. In addition to the current density, also charge density ρ is not known in advance but need to be obtained as a part of the problem solution. Here we employ the modelling of chemical composition for the above mentioned tasks. Since we have been unable to find a treatment where the physicalchemical setting is introduced in logical order, a detailed formalisation is given in the first part of the paper.

Examples in which the classical approach is usually adjusted in this way include semiconductor devices and various types of solar cells [1][2][3][4][5][6]. A particular example is a dye-sensitised solar cell in which generation occurs throughout a film of macroscopic thickness [7,8]. In the latter part of the paper, we demostrate the method by generating a model for dye-sensitised solar cells. From the point of view of these cells, we focus here on their dynamic behaviour under transient light pulse illumination.

Formalisation

Let us consider a physical system as being composed of different subsystems. In constructing and superposing these subsystems, quantities for which addition and scalar multiplication make physical sense are of particular importance for modelling purposes. These two elementary oper-ations open the way to efficiently employ mathematics in a modelling process, that is, in the translation of physical assumptions and laws into form accessible by experimentation. The value of mathematics is thus in assisting us with practical problems, and, on the other hand, a request for such assistance makes its use indispensable.

The chemical subsystem

To describe chemical composition in the spatial domain, we assume the precence of appropriate energy carriers called substances. These can be, for instance, certain molecules, electrons or photons. By composition we refer to quantities that indicate amounts of substances per unit volume at each mesoscopic point of space. 1 Formally, a proper quantity for each element s of a set of substances is the substance density n s , see Appendix A for details.

Since each point of space can be considered as an open system, transport of substances exists. In addition, substances are generated and regenerated locally by chemical reactions. Proper quantities to take these processes into account are substance fluxes j s and generation rates r s . Also, energy is stored and released in the spatial domain, 1 At each mesoscopic point of space there is statistical abundance of quanta of energy carriers for their density to be a meaningful concept. Also, the length scale of the largest spatial inhomogenities smoothed out in the model determines the smallest length scale under explicit consideration. Length scales much smaller than this limit are then referred microscopic. Unless otherwise stated, all points of space referred in this paper are mesoscopic. and quantities that characterise the response of the free energy density to changes in substance densities will be of crucial importance. Such quantities are named with hindsight as electrochemical potentials µ s , the detailed definition of which is given in Appendix B.

To obtain starting points (axioms) for modelling, physical and chemical knowledge of relationships between these quantities is utilised. Such laws serve as predicates for the quantities and may be used to define relations between them, see Appendix A for details. A conjunction of carefully chosen laws (expressed as the intersection of relations) will then uniquely determine these quantities.

A type of conservation law is that the substance flowing out of each space point per unit time must be generated at the point or the substance density at this point must change accordingly. This idea is employed in the model by introducing continuity relations

{(n s , r s , j s ) : divj s + ∂ t n s = r s } (1)
for each substance s. In addition, we need to express the behaviour of background materials in an appropriate way.

Relying on the second law of thermodynamics we require that substance is transported to the direction of lower electrochemical potential until equilibrium is reached [START_REF] Bockris | Modern Electrochemistry[END_REF][START_REF] Morrison | Electrochemistry at Semiconductor and Oxidized Metal Electrodes[END_REF][START_REF] Atkins | Physical Chemistry[END_REF].

This is done by introducing pointwise mappings L s , such that j s = -L s (gradµ s ). Using these mappings, the idea can be employed by introducing relations

{(µ s , j s ) : j s = -L s (gradµ s )} (2)
for each substance s. Denoting the array of all substance densities as n S , we express the dependence of generation rates on composition by introducing pointwise mappings F s , such that r s = F s (n S ), and further relations

{(n S , r s ) : r s = F s (n S )}, (3)
which are referred in chemistry as rate laws [START_REF] Atkins | Physical Chemistry[END_REF]. To obtain the substance densities as a part of the solution, we would still need relations between µ s 's and n s 's. This need will drive us to decompose electrochemical potentials into two parts. For this, we turn our focus to the electric subsystem.

Connection to the electric subsystem

Informally, the charge density ρ is a sum of contributions by different charged substances. To make use of the idea, we introduce relation

{(n S , ρ) : ρ = s q s n s }, (4)
where each coefficient q s is the charge per unit amount of substance s. The same idea applies for the current density j. Therefore, we introduce relation

{(j S , j) : j = s q s j s }. (5)
By considering (1), ( 4) and ( 5) together, it can be seen that the charge conservation divj +∂ t ρ = 0 further implies a relation for generation rates. That is, we have

{(r S ) : s q s r s = 0}, (6)
which is imposed in the construction of the rate laws.

Let us next introduce quantities that characterise the response of the electric energy density to changes in substance densities. These quantities are named here substance potentials µ el s , and they are constructed in view of the correspondence µ el s = q s φ el with the electric potential φ el , see Appendix B for details. It is notable that the substance potentials need not coincide with the experimentally relevant electrochemical potentials introduced above. To explain this difference (as is done for electrolytes in [START_REF] Bockris | Modern Electrochemistry[END_REF]), microscopic processes are called for.

At this point we decompose the electrochemical potentials into microscopic (or chemical) and mesoscopic (or electric) parts by introducing the chemical potential of substance s as the difference µ ch s = µ s -µ el s . Since µ el s and φ el are related the electrochemical potential can be written as µ s = µ ch s + q s φ el . The substance fluxes are also decomposed according to the above decomposition of potentials. For this, mappings L s are required to be additive, and relations (2) are replaced with relations {(φ el , µ ch s j s ) : j s = -L s (gradµ ch s ) -L s (q s gradφ el )}, (7) which describe the chemical and electric influences to the individual substance fluxes in separate parts. In the above relations the mappings L s are further required to be odd, so the model includes the idea that zero flux is obtained by balancing the two influences, which is the content of the (generalised) Einstein relation. To express the pointwise dependence of µ ch s 's on n s 's, we introduce pointwise mappings H s , such that µ ch s = H s (n s ), and relations

{(n s , µ ch s ) : µ ch s = H s (n s )} (8)
for each substance s.

By taking the intersection of (1), ( 3), ( 7) and ( 8), we

get 2 {(n S , φ el ) : -div(L s (gradH s (n s )) + L s (q s gradφ el )) + ∂ t n s = F s (n S )}, (9)
for each substance s. By further taking the intersection of (4), [START_REF] Bockris | Modern Electrochemistry[END_REF] and the relation of electrostatics,

{(ρ, φ el ) : -div(ǫgradφ el ) = ρ}, (10)
we obtain a unique element (up to boundary conditions)

(n S , ρ, φ el ) which satisfies all the requirements of the modelling. Furthermore, according to ( 5) and ( 7), the current density is given as

j = - s q s L s (gradµ ch s ) - s q s L s (q s gradφ el ), (11)
which can be interpreted as an extended Ohm's law. Relations ( 9) and [START_REF] Morrison | Electrochemistry at Semiconductor and Oxidized Metal Electrodes[END_REF] form what is called the coupled driftdiffusion-reaction problem, in which the substance diffusivity is defined as D s = L s • ∂µ ch s /∂n s (pointwise derivative, not the Fréchet derivative of Appendix B), and the substance conductivity is defined as

σ s = q s L s • q s .
Finally, in view of a concrete instance such as the dyesensitised solar cell, the set of substances, the chemical reactions present and the constitutive relations (3), (7) and ( 8) are left to be determined.

2 Strictly speaking, the relations need to be considered as subsets of the same set (the linear space of elements (nS , φ el , µ ch S , rS , j S ) in this case) to take the intersection. This should be kept in mind throughout this paper.

Application

Detailed modelling of dye-sensitised solar cells (referred by dye cells below) under steady-state operation have been performed elsewhere [START_REF] Ferber | [END_REF]. With an eye for these results

we aim to generate a model with some predictive power concerning the dynamic behaviour of the cell, and simulate the steady-state operation mainly for illustration purposes. To the authors' knowledge, such dynamic modelling have been restricted to a model that includes only one substance [14]. Alternatively, one has considered timeharmonic situations [15][16][17][18][19][20].

Relevant substances

Keeping in mind the preceding ideas, we begin with specifying the set of substances. A typical electrolyte contains at least iodide/triiodide redox pair (I -/I - 3 ), lithium ions (Li + ) and tert-butylpyridine (tBP ) [18,22]. In the semiconductor nanoparticles, free electrons can be trapped to localised energy states [8]. Hence, in the active electrode region two types of electrons are of interest, trapped electrons e - t and conduction electrons e - c , in addition to the electrolyte species. This is not the case in the counter electrode where only conduction electrons need to be considered. Finally, by including photons hν, the relevant substances can be collected into

{hν, e - c , e - t , I -, I - 3 , Li + , tBP }. ( 12
)
This selection of substances means that the dye, the electrolyte solvent and the semiconductor crystal structure are taken into account only by the constitutive relations.

Often, in modelling the dye cell, also the effect of photons, ions and molecules on the transport and generation of the modelled substances is dubbed into the constitutive relations. 3 Accordingly, the resulting set of substances consists of only conduction and trapped electrons. It is typically thought that the larger the set of substances, the deeper the physical and/or chemical understanding. Here we consider conduction and trapped electrons explicitly to obtain a dynamic model for the cell.

Chemical reactions

Chemical reactions occur in each point of the active electrode (anode) and at the surface of the counter electrode (cathode). The reactions may be written using chemical formalism as follows [8,23]:

Anode: 2hν + 3I -→ I - 3 + 2e - c , e - c → e - t e - t → e - c I - 3 + 2e - c → 3I -, I - 3 + 2e - t → 3I -, I - 3 + e - c + e - t → 3I -.
Cathode:

I - 3 + 2e - c → 3I -.
Here we have written the net photoreaction at the anode which does not explicitly state the precence of the dye. Notice that the cathode reaction occurs also at the anode. This recombination reaction is considered to be a significant degrading factor of the cell performance, and it largely motivates the mesoscopic modelling of the cell.

Constitutive relations

There is a controversy of what should be done with the constitutive relations of field theories in general. One conception defends their determination strictly by measurement to confine any uncertainty into such measurements, while others try to derive them from the fundamental assumptions of microscopic theories. In connection with dye cells both of the approaches have gained interest, and here we will also employ both of them.

"Flux laws" (L s )

In accordance with the band theory of solids, relation (7) for conduction electrons is given as

{(φ el , µ ch e - c , j e - c ) : j e - c = - n e - c u e - c |q e - c | (gradµ ch e - c + q e - c gradφ el )}, (13)
where u e - c is the mobility [START_REF] Blakemore | Solid State Physics[END_REF], which holds information on the microstructure of each mesoscopic point of Fig. 1.

Here we have identified µ e - c with the difference of the quasi-Fermi level of electrons and the Fermi-level at equilibrium. The trap states are localised in space so for trapped electrons the mobility is zero. Therefore, the trapped electron flux vanishes so relation (7) is

{(φ el , µ ch e - t , j e - t ) : j e - t = 0}. (14)

"Potential laws" (H s )

By treating conduction electrons as a non-degenerate (classical) gas [START_REF] Blakemore | Solid State Physics[END_REF], relation (8) for conduction electrons is given as

{(n e - c , µ ch e - c ) : µ ch e - c = k B T ln( n e - c n eq c )}, ( 15
)
where n eq c is the equilibrium conduction electron density in the dark, k B is Boltzmann's constant, and T is temperature. Here we have further identified q e - c φ el with the deviation of the conduction band edge from its equilibrium level. As a consequence of the predicate in relation (14), the relation (8) for trapped electrons will not be needed.

"Rate laws" (F s )

The rate law of generation of conduction electrons is composed of contributions of the different reactions at the anode. That is, we have

{(n hν , n e - c , n e - t , r e - c ) : r e - c =F p e - c (n hν ) + F t e - c (n e - c , n e - t ) + F cb e - c (n e - c )}, ( 16
)
where the generation rate of conduction electrons by photons is

F p e - c (n hν ) = -F hν (n hν ) (17)
and the generation rate of conduction electrons by trapping and detrapping can be given as

F t e - c (n e - c , n e - t ) = k detrap n e - t -k trap (1 - n e - t N trap )n e - c , (18)
where k detrap is the rate constant of detrapping, k trap is the rate constant of trapping, and N trap is the density of traps [15,19]. Equation ( 18) is a simplification of theories where the trapped electrons are distributed in energy [26,27]. To employ such theories it would be necessary to determine the trapped electron density (per unit volume and energy) also in "energy space". Finally, the generation rate of conduction electrons by recombination (negative quantity) is taken as first order in excess conduction electron density [15,20]. That is

F cb e - c (n e - c ) = -k cb (n e - c -n eq c ), (19)
where k cb is the rate constant of recombination from the conduction band. The above rate law is a simplification of experimental evidence suggesting that recombination is second order in electron density [18]. To obtain an expression for F p e - c (n hν ) in ( 16), the generation rate of photons (negative quantity) F hν (n hν ) appearing on the right hand side of ( 17) is usually given a priori as a function of position and time. This corresponds to the assumption that the photon density is independent of the densities of other substances in the cell.

For the rate law of generation of trapped electrons, we have

{(n e - c , n e - t , r e - t ) : r e - t = F t e - t (n e - c , n e - t ) + F tb e - t (n e - t )}, ( 20
)
where the generation rate of trapped electrons by trapping and detrapping is

F t e - t (n e - c , n e - t ) = -F t e - c (n e - c , n e - t ), (21)
and the generation rate of trapped electrons by recombination with electrolyte (negative quantity) is taken as first order in excess trapped electron density [28], that is

F tb e - t (n e - t ) = -k tb (n e - t -n eq t ), (22)
where k tb is the rate constant of recombination of the trapped electrons, and n eq t is the equilibrium trapped electron density in the dark.

Finally, we emphasise that the present formulation of the rate laws excludes the effect of the density variations of ions and molecules on the generation rates. The use of such rate laws is justified here due to the large density of electolyte species which is only slightly perturbed under cell operation according to earlier modelling results [START_REF] Ferber | [END_REF].

Formulation

The axioms necessary for mathematical modelling have now been collected into relations (1), ( 13), ( 14), ( 15), (16) and (20). By taking the intersection of these relations, we get the coupled drift-diffusion problem with trapping and recombination expressed by relations

{(n e - c , n e - t , φ el ) : -div(D e - c gradn e - c + n e - c u e - c gradφ el ) + ∂ t n e - c = -F hν (n hν ) + k detrap n e - t -k trap (1 - n e - t N trap )n e - c -k cb (n e - c -n eq c )}, (23)
and

{(n e - c , n e - t ) : ∂ t n e - t = -k detrap n e - t + k trap (1 - n e - t N trap )n e - c -k tb (n e - t -n eq t )}, (24)
and further relation

{(n e - c , n e - t , n I -, n I - 3 , n Li + , φ el ) : -div(ǫgradφ el ) = q e - c n e - c + q e - t n e - t + s∈{I -,I - 3 ,Li + } q s n s }. (25)
Notice that although the ions are not considered explicitly in the chosen modelling they need to be taken into account in (4), and hence in [START_REF] Vanmaekelbergh | [END_REF]. Clearly, we have taken a shortcut when leaving them out of the model. However, this was done with the knowledge of an earlier modelling result [START_REF] Ferber | [END_REF]. According to this result, the local charge neutrality is maintained (to a good approximation) in the active electrode under cell operation, corresponding to a vanishing electric field according to relation [START_REF] Vanmaekelbergh | [END_REF]. Relying on this result, we leave out relation [START_REF] Vanmaekelbergh | [END_REF] and the electric field term in (23).

Current and voltage

We need the global electric quantities, the cell current and the cell voltage, in terms of conduction electron density.

As stated in Appendix C, the problem is reduced to one dimension, and the spatial domain consisting of the active electrode is parametrised by x ∈ [0, d], such that x = 0 is the substrate interface and x = d is the electrolyte interface of the active electrode. As a result, the cell current (per unit area) I is given directly by ( 5), ( 13) and (15) as

I(t) = -j(0, t) • n(0) = - s∈{e - c ,I -,I - 3 ,Li + } q s j s (0, t) • n(0) = -q e - c D e - c ∂n e - c (x, t) ∂x |x=0 , (26)
since the current due to the ions vanish at x = 0. The cell voltage, on the other hand, needs some argumentation. Let us assume for a while that the counter electrode is part of the spatial domain. Then, excluding the ohmic voltage drop caused by the conducting substrate, the (positive) cell voltage V is given as

V (t) = µ e - c (counter electrode, t) q e - c - µ e - c (0, t) q e - c . (27)
Typically, it is assumed that the counter electrode is in its equilibrium state of the dark, so that the electrochemical potential vanishes there. This leads to the widely employed expression

V (t) = - µ e - c (0, t) q e - c = - k B T q e - c ln( n e - c (0, t) n eq c (0) ), (28)
where the latter equality follows from relation (15) with the assumption that µ e - c = µ ch e - c .

Boundary conditions

To specify the electron densities uniquely, boundary and initial conditions are still required. For clarification we review here three types of conditions. To begin with, the current-voltage characteristics of the cell -the macroscopic counterpart of (11) -is obtained for steady state by successively adjusting the cell voltage through the voltage range of interest and solving for the current. As the cell voltage is related to conduction electron density by (28) this can be done by introducing relation {n e - c : n e - c (0) = n eq c (0)e -q e - c V /(kBT ) and dn e - c (x)

dx |x=d = 0}, (29)
where the latter part of the predicate expresses the zero electron flux to the electrolyte. The procedure of finding the unique element of the intersection of ( 23), ( 24) and ( 29) is given in Appendix C for light illumination from the substrate side of the cell. The current-voltage curves thus obtained are given in the results section.

The above method, however, cannot be used in the modelling of transient responses with a fixed load resistance since the voltage corresponding to the fixed resistance depends on light intensity (and possibly on the history of light intensity), and it cannot be known in advance.

To model transient responses with an arbitrary load resistance R, the Ohm's law V = RI need to be imposed.

This corresponds to a nonlinear boundary condition, and it results in relation

{(n e - c , n e - t ) : - k B T q e - c ln( n e - c (0, •) n eq c (0) ) = -Rq e - c D e - c ∂n e - c (x, •) ∂x |x=0 , ∂n e - c (x, •) ∂x |x=d = 0, n e - c (•, 0) = n ref c ,
and

n e - t (•, 0) = n ref t }, (30)
where

{(n e - c , n e - t ) : D c ∂n c (x, •) ∂x |x=0 = k ext (n c (0, •) -n eq c (0)), ∂n e - c (x, •) ∂x |x=d = 0, n e - c (•, 0) = n ref c and n e - t (•, 0) = n ref t }. (31)
The open-circuit case when negligible amount of electrons is extracted to the circuits is obtained by setting k ext = 0, and the short-circuit situation when n c (0) = n eq c (0) is (formally) obtained in the limit k ext = ∞. It is notable that a finite resistance situation cannot be modelled by a finite value of k ext since this would contradict the Ohm's law V = RI. The method to take the intersection of ( 23), ( 24) and ( 31) is given in Appendix C.

Parameter values

The parameter values of the constitutive relations used in the modelling were taken from literature and are listed in Table 1. The order of magnitude values for the first order recombination rate constants, as well as for the rate constants of trapping and detrapping, were estimated based on references [18,19], and further adjusted to obtain typical steady-state current-voltage characteristics for various light intensities. Other parameter values were taken directly from the references indicated.

Results

Simulated

R 2 defined by the laws, that is R 1 ∩ R 2 .

Appendix B: Definitions

For the definition of the electrochemical potential, we first consider the density of free energy at each point in space and time, g(x, t), as a function of composition. 4 Electrochemical potentials are constructed to express the response of the free energy density to variations in substance densitites. Since substance density at a point y can affect to the free energy density at another point x (Coulomb interactions are included), the electrochemical potential µ s is defined for each substance s at each point (x, t) as the "derivative of g(x, t) at composition n S to the direction n s ", and is denoted as

∂ nS g(n s ) (the argument (x, t)
is dropped for notational simplicity). The operator ∂ nS g 4 To make a connection to the thermal subsystem, we identify this energy with the (electrochemical) Gibbs free energy by assuming constant temperature and pressure [START_REF] Bockris | Modern Electrochemistry[END_REF]. Hence, all non-thermal and non-mechanical energy exchange is included.

(bounded and linear) is the Fréchet derivative at n S , defined by

lim h→0 |g(n S + h) -g(n S ) -∂ nS g(h)| h = 0,
where • is a norm given for the substance densities.

It is noticed that the potentials constructed in this way need not be continuous in space. However, they are usually required to be continuous in certain parts of space called regularity regions. In practice, these regions are specified a priori based on the qualitative understanding of the situation. In this paper, the derivatives are restricted to the regularity regions of the potential in question.

For the definition of the substance potential, the electric energy density g el is considered. Parallelling the construction above, substance potentials are defined as µ el s = ∂ nS g el (n s ). For the correspondence of µ el s and the electric potential φ el , we express (4) by a linear pointwise mapping qS , such that ρ = qS (n S ), and we further give g el as a composition mapping g el = v el • qS by using a new mapping v el . Then, by the chain rule, it follows that

µ el s = ∂ nS g el (n s ) = ∂ ρ v el (ρ) ∂ qS ∂n s (n S ) = φ el q s (B.1)
for each substance s such that q s = 0, where the latter derivative is taken pointwisely.

the resulting spatial domain by x ∈ [0, d], such that x = 0 is the substrate interface and x = d is the electrolyte interface of the active electrode. We further consider illumination from the substrate side of the cell to give the driving term for the problem using photon flux at the boundary,

I hν (t) = -j hν (0, t) • n(0) for all t ∈ [0, T ], as -F hν (n hν ) = gI hν , (C.1)
where g describes the a priori fixed spatial variation. The function g is typically given using the absorption coefficient α as g(x) = αe -αx [15,19]. Before proceeding, we clarify our notation somewhat. That is, we set

n c = n e - c , (C.2)
n t = n e - t , (C.3)
D = D e - c , (C.4)
N = n eq c (0)e -q e - c V /(kBT ) , (C.5)

c = k trap (1 - n t N trap + k cb , (C.6) f = gI hν + k detrap n t + k cb n eq c . (C.7)

Steady-state situation

With the above notation, for a given n t , the conduction electron density n c that belongs to the intersection of ( 23), For n c given, on the other hand, the trapped electron density of the intersection satisfies

-k detrap n t + k trap (1 - n t N trap )n c -k tb (n t -n eq t ) = 0 in [0, d]. (C.9)
To solve the combined problem of (C.8) and (C.9) we employ the finite element method and a fixed point iteration procedure to solve the nonlinearity caused by the used trapping rate law.

It is noticed that the chemical subsystem is constructed in such a way that the pairs (µ s , n s ), (µ s , r s ) and (gradµ s , j s ) are related to energies or powers. As a consequence, the existence of the constitutive relations suggests us to search such solutions n c and n t that are square integrable in the domain together with their gradients -the square integrability guarantees the finiteness of energy. 5 In the case of (C.8), we end up in searching for an approximation to its variational formulation from the following subspace of the linear space C 0 ([0, d]) of continuous functions in [0, d]:

S N h ={χ ∈ C 0 ([0, d]) : χ affine in each interval [x j-1 , x j ], where {x i } is a partition of [0, d] such that 0 = x 0 < x 1 < • • • < x M = d, and
χ(0) = N }. (C.10) (Affine function is a function of the form f (x) = αx + β.)
Since this is also a subspace of an appropriate Sobolev space [START_REF] Larsson | Partial Differential Equations with Numerical Methods[END_REF], a relevant problem (the solution of which, if 5 This is the case if there are strictly positive coefficients representing the constitutive relations that are bounded below and above [START_REF] Bossavit | Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements[END_REF]. Intuitively, the "fixed point" means that for a given n c , (C.9) results in some n t , and by (C.11) we have n c again, which must be the same we started from. To prove that the iteration procedure self-consistently converges to a unique fixed point for all initial guesses, it would be sufficient to show that the associated mapping (which maps the array of nodal values N j to new nodal values) is a contraction mapping. This is bypassed here but in practice the procedure does converge.

Dynamic situation

Parallelling the above strategy, for a given n t , the conduction electron density n c that belongs to the intersection of ( 23), ( 24) and ( 31) satisfies

∂n c ∂t - ∂ ∂x D ∂n c ∂x + cn c = f in (0, d) × (0, T ], (C.14a) D ∂n c ∂x -k ext (n c -n eq c ) = 0 on {0} × [0, T ], (C.14b) ∂n c ∂x = 0 on {d} × [0, T ], (C.14c) n c = n ref c on [0, d] × {0}, (C.14d)
whereas, for n c given, the trapped electron density of the intersection satisfies This time we employ the function space

∂n t ∂t = -k detrap n t + k trap (1 - n t N trap )n c -k tb (n t -n eq t ) in [0, d] × (
S h ={χ ∈ C 0 ([0, d]) : χ affine in each interval [x j-1 , x j ], where {x i } is a partition of [0, d] such that 0 = x 0 < x 1 < • • • < x M = d}. (C.16)
We further employ the time discretisation that n k c ∈ S h (resp. n k t ∈ S h ) is the approximation of n c (t) (resp. n t (t)) at t = t k = k△t, where △t is the time step and k ∈ N.

Then, an approximation to the variational formulation of the time discretised version of (C.

△t -1 (n k t -n k-1 t ) + k detrap 2 (n k t + n k-1 t ) -k trap (1 - 1 2N trap (n k t + n k-1 t )) 1 2 (n k c + n k-1 c ) + k tb 2 (n k t + n k-1 t
) -k tb n eq t = 0, (C.23a)

n 0 t = I h n ref t . (C.23b)
The basis for S h is given by hat functions {λ i } M i=0 ⊂ S h , λ i (x j ) = δ ij . To obtain finite element equations n k c and n k t are expressed in this basis to write (C.17) and (C.23) in matrix form. Finally, the solution procedure can be stated as follows:

1. Set initial conditions n 0 c and n 0 t .

2. Set k = 1.

3. Solve n k c and n k t by fixed point iteration from (C.17 The method is close to the Crank-Nicholson-Galerkin method (see [START_REF] Larsson | Partial Differential Equations with Numerical Methods[END_REF]) equipped with the fixed point iteration procedure. Here, however, the bilinear form a(•, •) depends on the time step which is not considered in the analysis of the standard method. We skip further mathematical analysis, and are satisfied with the practical functioning of the algorithm.

3. 1

 1 photoactive dye on the surface where the photochemical reactions take place. The structure of the cell is shown in Fig.1. The continuous performance of the cell is made possible by the electrolyte which reduces the photoexcited dye while the electrons released to the semiconductor are transported through the nanoparticle network, the conducting substrate and the external load to the counter electrode and again back to the electrolyte[7,21].

4 Conclusions

 4 steady-state current-voltage characteristics are shown in Fig.2for two different light intensities. As the curves are typical for these cells[START_REF] Ferber | [END_REF], we may hope for that also dynamic simulations show reasonable results. We focus on two dynamic relations predicted by the model, one between the light intensity I hν and the short-circuit current I SC , and the other between I hν and the open-circuit voltage V OC . Fig.3and Fig. 4 show the simulation results for short-circuit and open-circuit situations under a transient light pulse of five milliseconds in duration. It is clearly shown that the electric quantities lag the light exitation under such short light pulses. This is in contrast to the normal operating conditions of a solar cell under which the cell performance can be considered as successive steady states.The fast transient simulations under short-circuit conditions elucidate the cell performance under the effect trapping-detrapping process. Typically, the process is taken into account only partly by the use of an "effective" diffusion coefficient (see, for instance,[31]). It is known that a dynamic model involving only conduction electrons is restricted to such a quasistatic regime under which the effective diffusion coefficient is sufficient to describe the cell dynamics[31]. The model presented here allows the simulation of the time evolution of short-circuit current and open-circuit voltage even beyond the quasistatic regime.With the parameter values used, the short-circuit hysteresis behaviour is clearly visible when the cell is illuminated with light pulses of duration of milliseconds or shorter. It was noticed that at short-circuit conditions the detrapping rate constant in particular is a sensitive parameter that controls the extent of hysteresis. For instance, if the rate constant is reduced by two orders of magnitude then to observe hysteresis light pulses of a few hundreds of milliseconds are needed.Even though the trapping-detrapping process need not be included explicitly in the model to describe the cell dynamics at typical working conditions of a solar cell, it may be important to study its role in the hysteresis effect. Especially, to describe the cell dynamics beyond the quasistatic regime a detailed account of the process is required. In particular, this may be necessary in other devices that use similar hybrid material structures, and where reproducible and fast transient behaviour is of importance. For instance, hysteresis behaviour of a TiO 2 -dye-electrolyte based photoconductor has been observed recently[32].Limitations in using classicalelectromagnetism in modelling local generation of electric power can be circumvented by considering chemical composition in detail. The modelling of chemical composition allows one to incorporate information on microscopic processes in the classical description. This can be done by decomposing the influence quantity (electrochemical potential) into microscopic (labeled chemical) and mesoscopic (labeled electric) parts. A key step in the approach is to establish the constitutive relations of the chemical subsystem -this step essentially determines the uncertainty of modelling. The dye-sensitised solar cell appears as an interesting example in which the microscopic part of the influence quantity seems to play the dominant role. In this connection, we point out assumptions that are often made to construct a model for the cell. To solve the steadystate and dynamic boundary value problems, we employ spaces of square integrable functions in methods based on finite elements. Steady-state simulations resulted in current-voltage characteristics closely resembling those typically obtained in experiments. Dynamic simulations were performed to evaluate open-circuit voltage and short-circuit current dependence on the illumination intensity of the cell under a transient light pulse illumination. This is the first time that a model is presented that is capable of describing the time evolution of these quantities under the full influence of trapping and detrapping of electrons. A question is raised on the role of this electronic process in the optoelectronic hysteresis effect observed under fast light pulse transients. functions of the same space and multiply them by a scalar to obtain a new function of this space. Relations between physical quantities are subspaces of cartesian products of the corresponding function spaces. As an example, if quantities a and b are functions of the spaces A and B, and F is a mapping from A to B, a physical law b = F (a) serves as a predicate for a and b, and it defines a relation R = {(a, b) ∈ A × B : b = F (a)} (a subspace of A × B). In this paper, we denote this as {(a, b) : b = F (a)} for short. A conjuction of two laws is expressed as the intersection of the two relations R 1 and

  c = f in (0, d), (C.8a)n c = N on {0}, (C.8b) dn c dx = 0 on {d}. (C.8c)

  smooth enough, satisfies (C.8)) can be set to find n c ∈ S N h such thata(n c , χ) = (f, χ) for all χ ∈ S 0 h , functions {λ i } M i=1 ⊂ S 0 h , λ i (x j ) = δ ij(the Kronecker delta), as a basis for S 0 h , we can write n c ∈ S N h by using a finite number of degrees of freedom asn c = N λ 0 + M j=1 N j λ j ,where the hat function N λ 0 is an extension of the boundary value N . By using the hat functions {λ i } M i=1 also as the weight functions χ in (C.11), we obtain M equations for the unknowns N j , j = 1, . . . , M , which are solved at each step of the fixed point iteration procedure. The algorithm goes as 1. Guess N j for j = 1, . . . , M, 2. Solve the coefficients for n t from (C.9), 3. Solve new coefficients N j for j = 1, . . . , M, from (C.11), 4. Go to step 2 until converged.

  0, T ], (C.15a) n t = n ref t on [0, d] × {0}. (C.15b) To solve the problem the finite element method is equipped with time discretisation, and the fixed point iteration procedure is performed at each time step. The initial condition was specified to the steady-state situation of low illumination (approximately 10 -3 sun) by first solving the corresponding steady-state problem for n ref c and n ref t .

1 c), χ) + a k- 1 (n k t + n k- 1 t)

 111 14) is defined by the problem: Find n k c ∈ S h recursively for k > 0 such that(△t -1 (n k c -n k-k-1/2 uv)dx + k ext u(0)v(0), (C.19) b(f k-1/2 , v) = d 0 f k-1/2 vdx + k ext n eq c (0)v(0), (C.20) and c k-1/2 = k trap (1 -1 2N trap (n k t + n k-1 t )) + k cb , (C.21) f k-1/2 = gI hν (t k-1/2 ) + k detrap 1 2 + k cb n eq c ,(C.22)andI h n ref c is the interpolant of n ref c ∈ C 0 ([0, d]) in S h .The same time stepping is used for (C.15), so that n k t is defined recursively for k > 0 by

4 .

 4 ) and (C.23). Set k = k + 1, go to the previous step.

Fig. 1 . 2 )Fig. 2 .

 122 photoactive dye molecules

Fig. 3 .Fig. 4 .

 34 Fig. 3. Short-circuit simulation under fast transient light pulse. Light intensity versus time (top), short-circuit current versus time (center), and the corresponding I hν I SC -curve (bottom). The arrows show the direction of time evolution.

Table 1 .

 1 Parameter values used in the modelling, and their

	sources.
	parameter	value	source
	d	10 -5 m	[18]
	T	300K	[19]
	Dc	10 -6 m 2 s -1	[19]
	k cb	10s -1	estimated [18, 19]
	k tb	10s -1	estimated [18, 19]
	k trap	10 7 s -1	estimated [18, 19]
	k detrap	10 6 s -1	estimated [18, 19]
	α	2.5 • 10 5 m -1	[18]
	N trap	2.3 • 10 24 m -3	[19]
	n eq c	10 11 m -3	[19]

This is not justified a priori, but it requires certain special properties from the situation. Such properties are discussed in sections 3.4 and 3.5.

Kovanen et.al.: Modelling chemical composition in electric systems

Acknowledgement Scholarship (TK) from FinNano research program of the Academy of Finland under FUNANO consortium (contract No 118040) is greatfully acknowledged. T.K. acknowledges Finnish Cultural Foundation for additional financial support.

Appendix A: Mathematical background

Physical quantities, like the substance density, are considered as functions in space and time. For instance, by considering the space and time as a cartesian product of Ω and Γ , the substance density is denoted as function n s : Ω × Γ → R, whereas the image of a point (x, t) by n s would be called n s (x, t). In many cases, the set Ω can be considered as a subset of the three dimensional Euclidean space, and Γ is the closed interval [0, T ] of R. By recalling the idea of addition and multiplication mentioned earlier, a prerequisite for the physical quantities considered in this paper is that each quantity belongs to the (abstract) linear space of appropriate functions. That is, one can add

Appendix C: Solution procedures

The structure of the dye cell has enough symmetry to reduce the problem to one spatial dimension. We parametrise