\r\n\t
\r\n\tWhile two-dimensional (2D) thin crystals limit the physical phenomena into a plane, in a one-dimensional (1D) quantum structure (nanowires, nanotubes, and nanoribbons (NRs)), charge carriers and excitations have only one degree of freedom. These crystal structures have been the focus of interest due to their unique properties such as the very high electronic density of states, enhanced exciton binding energy, diameter-dependent bandgap, increased surface scattering for electrons and phonons, and chirality-dependent electronic band structure.
\r\n\tNanoribbons (NRs), made of single- or few-atom-thick lamellar crystals, are novel forms of 1D nanoscale materials and are ideal systems for investigation of the size and dimensionality dependence of the fundamental properties. After the successful synthesis of many 2D monolayer materials, their 1D NR form came into prominence due to their necessity in nanoscale applications. In this context, this book will cover the synthesis techniques, characterization methods, fundamental properties, and state-of-the-art applications on NRs of recent 1D/2D materials such as graphene, transition metal dichalcogenides (TMDs) (MoS2, WS2, ReS2, and TiSe2), mono-chalcogenides (GaS, GaSe, ZnSe, and SnSe), tri-chalcogenides (TiS3 and ZrS3), black phosphorus, group-IV, III–V binary compounds, superstructures and so on. The proposed book is intended for academia, professionals, scientists and Graduate & Undergraduate students without any geographical limitations.
\r\n\t
Catenary arches are one of the main features of Art Nouveau architecture in Spain. Their shape is based on the modern theory of masonry arches. This theory was developed during the nineteenth century and claimed the work of Antoni Gaudí (1852–1926) as its main exponent [1]. Architect Cèsar Martinell i Brunet (1888–1973) built the so-called wine cathedrals (1918–1924) which were built by the Commonwealth of Catalonia (1907–1925).
\nThese buildings in the nineteenth-century style named as Noucentisme may be regarded as the last ig cluster of constructions featuring Catalan masonry [2]. The catenary arches designed and built by Cèsar Martinell belong to the Catalan modernist architecture Antoni Gaudí i Cornet (1852–1926). Thus, in a hanging chain, any inward pulling force is matched by an equal outward pushing force. Martinell knew Gaudi’s work and inherited his techniques. This is evidenced by his many writings on Gaudí [3], whom he met during a visit to the Sagrada Familia church in 1915, when Martinell was about to complete his studies at the Barcelona School of Architecture. From then on, Martinell joined an exclusive group of disciples who learned a way of doing architecture aside from university teachings [4].
\nFor Martinell, Gaudí was a much more interesting lesson of life and architecture than most of the teachings given at university. Gaudí’s words became architectural when the very statement of the scientific truth and the procedures he himself invented were able to explain problems and geometric concepts which remained unclear in the classrooms of the school of architecture. Antoni Gaudí’s theory of structures relies on the strength of geometry, in particular on the strength of the parabolic and catenary shapes. The construction technique used by Cèsar Martinell stems from the methods applied by Gaudí to calculate the geometric shapes of vaults and arches [5] (Figure 1). Other architects have written about this view; see, for instance, the lecture entitled
Cesar Martinell sketch of transversal section El Pinell de Brai [COAC H101I-6-Reg 2502].
Otherwise, these concepts were introduced in the formation of architects through the Escuela Especial de Arquitectura de Madrid (1844). The work
The theory of the chain, in the shape of a hanging collar, was proposed by Robert Hooke (1635–1703) at the end of his treatise
Simon Stevin (1548–1620), in
The mathematic equation of the catenary would be formulated some years later by David Gregory (1659–1708) and published in the
In Spain, the development and application of this theory take place in the context of the Mathematics Academy of Barcelona (1720). The work of Bernard Forest de Bélidor (1698–1761) is the main reference of the curve of equilibrium theory. In
On 13 January 1710, King Philip V appointed Jorge Prosper Verboom (1665–1744) as engineer in chief. Verboom was a disciple of Sebastián Fernández de Medrano. Together with Alejandro de Retz (c. 1660–c. 1732), chief engineer of the Catalan region, Verboom was the link with the former academy in Brussels. Mateo Calabro was appointed head of the Mathematics Academy of Barcelona (1720–1738), and he had profound disagreements with Jorge Prosper Verboom with regard to the training program. Therefore, in 1738 Verboom offered the head position of the academy to Pedro de Lucuze y Ponce (1692–1779), who held that position until his death. Among other duties, the academy had to build a collection of scientific works which would be used as reference texts for military training. This bibliographic interest led Vicente García de la Huerta (1734–1787) to publish
Some of the most relevant texts available for military engineers in the libraries are
La science des ingénieurs (1729) of Bernard Forest de Bélidor.
Another reference work for engineers is
Yet another work of reference in the libraries of the military engineers was
The
However, the main references for the Spanish military engineers were definetly the works of Bernard Forest of Bélidor (1698–1761): the
Military architecture treatises of José Cassani (1673–1750), developed at the end of the seventeenth and eighteenth centuries, make reference to the construction of these warehouses, especially if they have an element of high resistance, such as having been made bombproof or having been constructed underground (whether in manmade excavations or in caves) [34].
\nThe principal work of reference is
In the Spanish treatise
In
The work of Bélidor is translated into English by John Müller (1699–1784) and published under the title
Tratado de fortificación, ó Arte de construir los edificios militares, y civiles (1769), John Müller.
Gunpowder magazine projects made by the Spanish military engineers of the eighteenth century are based on previous military architecture treatises. Therefore, most of these ancillary constructions are built bombproof by shielding the roof. A distinction is made between two types of designs: vaults and wooden structures. The latter are protected by elastic components capable of cushioning the impact of a pyroballistic weapon. From a morphological point of view, gunpowder magazines can be classified depending on their protecting enclosures. From a formal point of view, a distinction is made between the following three morphologies:
\nGunpowder magazines having a simple construction body, with the outer wall directly exposed to hostile fire. This is the case of the gunpowder magazines built in Zaragoza (1729) [MPD, 39, 041] (Figure 4), Cádiz (1749) [MPD, 56, 029], San Sebastián (1738) [MPD, 27, 092], or Peñíscola (1739) [MPD, 18, 262].
\nZaragoza (1729) [MPD, 39, 041].
Gunpowder magazines having an outer protection enclosure and a simple enclosure for storing the gunpowder. This is the most common type of magazine in the military treatises. Examples that stand out are the gunpowder magazines in Cardona (1718) [MPD, 19, 028] (Figure 5), San Sebastián (1722) [MPD, 28, 034], Ceuta (1724) [MPD, 39, 083], Málaga (1724) [MPD, 59, 046], and Gerona (1738) [MPD, 01, 018].
\nCardona (1718) [MPD, 19, 028].
Lastly, gunpowder magazines having an outer protection enclosure and a two-element central construction body (where the inner wall is at ground level and the main magazine is above ground level). This is the case of the gunpowder magazines built in Tortosa (1721) [MPD, 64, 019], Málaga (1721) [MPD, 59, 044] (Figure 6), Barcelona (1726) [MPD, 10, 060], Cádiz (1728) [MPD, 08, 236], or Zaragoza (1729) [MPD, 28, 010].
\nMálaga (1721) [MPD, 59, 044].
These gunpowder magazines are built on load-bearing walls which are parallel to the vault’s longitudinal axis, with a perpendicular framework of wooden beams. Where the width is small (until about three toises), the project is built with a single span. Examples that stand out are the gunpowder magazines in Tortosa (1721) [MPD, 64, 019] (Figure 7), Málaga (1721) [MPD, 59, 044], Barcelona (1726) [MPD, 10, 060], Cádiz (1728) [MPD, 08, 236], or Zaragoza (1729) [MPD, 28, 010]. Where the width is greater, there are two structural spans. The latter projects are divided into three types: firstly, projects where the central body has load-bearing walls and the two vaulted spans are connected by small doors, such as in Cádiz (1728) [MPD, 08, 237] and Cartagena (1745) [MPD, 18, 258] or Alicante (1750) [MPD, 65, 088 and MPD, 65, 092]; secondly, projects where the two spans are separated by pillars and the roof is supported by wooden main beams, such as the gunpowder magazines in Hondarribia (1733) [MPD, 65, 044], Cartagena (1745) [MPD, 18, 257], and Tortosa (1798); and thirdly, projects where the roof is supported by masonry arches, such as in the gunpowder magazines of Ceuta (1735) [MPD, 07, 179] or Lleida (1739) [MPD, 07, 001].
\nTortosa (1721) [MPD, 64, 019].
Military architecture treatises require the outer walls of the gunpowder magazines to be reinforced with buttresses. Thus, buttresses were used very often as the abutment of diaphragm arches. These types of design emerged later, and structures thus generated may have a single span. This is the case of the gunpowder magazines in Benimàmet (1751) [MPD, 06, 169] (Figure 4), Valencia (1756) [MPD, 07, 028] (where the arch abutment is built outwards), and A Coruña (1774) [MPD, 28, 027] (where the abutment is concealed in the interior space, as Müller set out in his treatise (1769)). Other larger powder magazines feature two parallel vaulted spans, for instance, the one in Barcelona (1761) [MPD, 20, 031], with a central pillar between each pair of arches.
\nThese are gunpowder magazines having a pitched roof and a timber framing. In some instances a joggle-truss is used, i.e., in the projects for Cádiz (1718) [MPD, 64, 020] and Gerona (1755) [MPD, 10, 073]. In other instances a collar-beam truss is used, i.e., in Barcelona (1731) [MPD, 18, 100 and MPD, 18, 101]. But mainly these magazines are built using the Spanish double-framed roof. This is the case of the gunpowder magazines in Zaragoza (1729) [MPD, 28, 009], Zamora (1734) [MPD, 65, 042], El Ferrol (1772) [MPD, 04, 089], or Cartagena (1795) [MPD, 46, 051 and MPD, 46, 052]. Sometimes they are Spanish double roofs with small variations affecting the inclined tie beams, i.e., in Pamplona (1723) [MPD, 64, 023] and El Ferrol (1738) [MPD, 47, 094]. Other projects feature scissors trusses, i.e., in Monzón (1740) [MPD, 54, 049], Palma (1748) [MPD, 65, 047], and Valencia (1754) [MPD, 06, 170]. The project for Barcelona (1796) [MPD, 46, 035] features a double-framed roof but with two horizontal rafters as tie beams. In other cases, such as the project for Málaga (1721) [MPD, 64, 022] (Figure 8), there is a mansard truss with two horizontal rafters (this structure is more similar to the models used in the upper body of military barracks).
\nMálaga (1721) [MPD, 64, 022].
Masonry barrel vaults are the most commonly used type of bombproof vaults. The gunpowder magazines are the direct result of military architecture treatises. Because it is such a common construction technique, most of the engineers do not even mention the material used for the vault of the magazine. This is what happened in the projects for Ceuta (1724) [MPD, 39, 083], Longone (1725) [MPD, 12, 222], A Coruña (1738) [MPD, 17, 058], San Sebastián (1738) [MPD, 27, 092], Peñíscola (1739) [MPD, 18, 262 and MPD, 18, 263], Játiva (1748) [MPD, 54, 012], Palma (1748) [MPD, 65, 048], San Sebastián (1750) [MPD, 27, 093], Viveiro (1778) [MPD, 19, 241], and Santa Cruz de Tenerife (1792) [MPD, 05, 033]. With regard to the projects which do specify the material used for the construction of the barrel vault, a distinction must be made between those which use stone for the first layer of the vault and those which use ceramic bricks.
\nThe main projects which use stone masonry include those for Barcelona (1715) [MPD, 18, 097] and Dénia (1748) [MPD, 65, 085]. There is a variant which uses an ashlar arch with wooden beams on top, such as the project for Santa Cruz de Tenerife (1758) [MPD, 18, 050] (this variant was already used by Fernández de Medrano in the previous century). In other cases, ashlar masonry is substituted by brick masonry with a lime and pebble covering, as in the case of San Sebastián (1722) [MPD, 28, 034] (Figure 9).
\nSan Sebastián (1722) [MPD, 28, 034].
As for the projects which use ceramic bricks, a distinction must be made between four different types: firstly, those which feature a single-layer ceramic vault, like the ones for Zamora (1738) [MPD, 13, 113] or Ciudad Rodrigo (1739) [MPD, 12, 154]; secondly, those which feature a bottom layer of ceramic bricks, an intermediate layer of stone, and a top layer of lime and pebble, like the gunpowder magazine in Cardona (1718) [MPD, 18, 028]; thirdly, vaults designed with a double layer of ceramic bricks and a top fill of lime and pebble, like in the projects for Pamplona (1718) [MPD, 31, 032; MPD, 31, 033; MPD, 31, 034; MPD, 31, 035; MPD, 31, 036]; and lastly, projects which feature three layers of ceramic bricks, for instance, in Longone, Italy (1728) [MPD, 12, 221], or even four layers, like in Badajoz (1749) [MPD, 65, 045].
\nPointed vault structures were described by Bélidor (1729), and Müller (1769) said they are less resistant to bomb impacts than barrel vaults. They belong to the Gothic building tradition, and they need a smaller abutment than barrel vaults, even though magazine walls in military treatises are dimensioned depending on the impact of pyroballistic weapons (those employing gunpowder) and not on the basis of masonry mechanic criteria. This is the case of the gunpowder magazines in Zaragoza (1729) [MPD, 39, 041] and San Fernando de Cádiz (1749) [MPD, 56, 029], which have only one enclosure. It is the same construction type as the projects for Gerona (1738) [MPD, 01, 018] and San Sebastián (1749) [MPD, 27, 094], but these two magazines are protected by an encircling wall, and therefore they have two enclosures. The gunpowder magazine projects for Gibralfaro Castle in Málaga (1724) [MPD, 59, 046 and MPD, 59, 047] (Figure 10) and Ceuta (1737) [MPD, 07, 180] feature two parallel vaulted vans which are separated by square pillars supporting round arches, thus forming the central valley of the roof.
\nGibralfaro Castle in Málaga (1724).
In the construction of gunpowder warehouses, barrel and pointed vaults are generally used, although there are some examples with elliptical vaults, such as that one built in 1694 by Hércules Torelli in Pamplona. This construction was remodelled by Francisco Larrando de Mauleón (1718) [MPD, 31,031] (Figure 11) [43]. Mauleón was professor at the Mathematics School of Barcelona and Zaragoza and authored the
Warehouse of Pamplona, Francisco Larrando de Mauleón (1718) [MPD, 31,031].
The simple vault of the warehouse of Montjuïc mountain in Barcelona (1731) [MPD, 07, 057], a project attributed to Miguel Marín (Figure 12), is not generated through an arch of circumference. The geometric study reveals that the vault has a length of 16 feet in toise, a rise of 11.5 feet, a width of 3 feet, and a buttress of 7 feet. A geometric element having the shape of a catenary can be drawn running through the springing points and the key of the vault (Figure 13). The shape of this element, which has the same span and rise as the vault, is very similar to the shape drawn in the project.
\nGunpowder warehouses: Marín (1731), Marín (1733), and Feriére y Valentín (1736).
Metrology of the projects for the gunpowder magazines.
Other projects, such as the project from Miguel Marín for Tortosa (1733) [MPD, 13, 035] (Figure 12), have a span of 21 feet in toise, a rise of 14 feet, a width of 3.5 feet, and a buttress of 7 feet (Figure 13). Another similar project is the simple warehouse layout by Juan de la Feriére y Valentín in A Coruña (1736) [MPD, 17, 057] (Figure 12), which contains a span of 22 feet in toise, a rise of 14 feet, a width of 3 feet, and a buttress of 7 feet (Figure 13).
\nThe design of the pointed vault is initially compared with the catenary, as obtained with a chain over a reproduction of the plans on a larger scale (Figure 14). Thus, the arch described by the chain is very similar but not coincidental to the profile of the vault because there are small deviations near the springline of the vault. This deviation is because it is not possible to lay out the catenary with traditional drawing tools, such as rulers and compasses.
\nChaînette method applied to the projects for the gunpowder magazines.
The assessment of the original section drawing of the warehouse reveals three compass marks. One point is made over the vertical axis of symmetry of the figure, while the other two are made over the perpendicular axis, slightly below the springline of the vault. An oval was drawn on each project using these compass marks, and the obtained curves were coincident with the curves of the projects. Thus, to draw the projects of the warehouses, both Miguel Marín and Juan de Feriére y Valentín used the geometrical solution of an oval. Therefore, the curves drawn in the three projects are oval, but the major axes of the oval are higher than the springline of the arch. As a consequence, the curves are not tangential at the springing. So, the military engineers drawn the curve of the vault as an arch
Oval method applied to the projects for the gunpowder magazines.
The geometric layout of these vaults, based on ovals, was well known by the eighteenth-century military engineers. They began from the essential feature that oval vaults are tangential to the springline of these building elements. When the springline is higher than the axis, a non-tangential curve is obtained, which is a feature of the catenary definitions given by Frézier (1738). Concurrently, Bélidor (1729) specifies the method to lay out the true shape of the catenary vault. By knowing the rise and the span of the vault, the architectonic shape is determined with a hanging chain. Thus, a scale model can be built and can easily be taken to the construction site. By contrast, the layout of the catenary in military engineers’ projects is more complex, because it needs the use of an approximation of the catenary through the geometrical shape of a lowered oval.
\nThe ovals are derived from the centres of the circumferences (the compass center points on the paper). They are referred to (O1) for Barcelona (1731) [MPD, 07, 057], (O2) for Tortosa (1733) [MPD, 13, 035], and (O3) for A Coruña (1736) [MPD, 17, 057]. They are consistent with three different types of ovals, and they all share the common feature that the origin of the vertical tangent to the minor axis of the oval is located 1 foot below the impost line (Figure 16).
\nGeometrical analysis of the vaults of the projects for the gunpowder magazines.
The main ovals’ geometric data are shown in
e1 describes the clear span.
e2 describes the rise.
a1 describes the distance between centres of the minor axis.
a2 describes the distance between the centre of the minor arc and the minor axis.
d1 describes the ratio between the length of the semimajor axis and the vertical distance from the semiminor axis to the springline (feet).
d2 describes the ratio between the length of the minor axis and the distance from the center of the major arc to the point of tangency between the major arc and the minor axis.
d3 describes the ratio between the length of the semimajor axis and the radius of the oval’s minor arc.
p1 describes the ratio between the semimajor axis and the minor axis.
p2 describes the ratio between the center-to-center distance on the minor axis and the distance from the center of the semimajor axis to the minor axis.
The ovals used in the layout of the gunpowder magazines are thus used as a reference for purposes of comparison with the cellar’s layout. The layout of [O1, O2, O3] is based on a ratio between d3 and e2 of [0.39:0.36:0.50].
\nIn addition, the layout of each oval is compared with a catenary that has the same rise and span, which is drawn using InnerSoft software. According to the results, the inner surface defined between the corresponding geometric shape and the springline is different (1.33 m2 vs. 0.98 m2). Furthermore, the ratio between the maximum distance between geometric shapes and the arch’s span ranges from 2.14 to 3.44%. From these data, we can conclude that the approximation made by the engineers by drawing ovals in the three projects considered is sufficiently precise for the drawing scale used, between E: 1:90 and E: 1:70. Finally, the curves are compared with an elipse with the same rise and span. The obtained shape is clearly not coincident with the curves of the projects.
\nAfter the Bourbon dynasty’s ascendancy to the Spanish throne (1700), Catholic diplomatic and military families of Irish and Scottish origin emigrated under royal protection, preserving their status. O’Connor family was installed in Benicarló in the eighteenth century, and they associated with the McDonnells in the wine export business.
\nThe Bourbon dynasty, which established itself in Spain in 1700 with King Philip V (1683–1746), created the Army Corps of Engineers by the Royal Decree of 17 April 1711. Several Irish families moved to eastern Spain in the mid-seventeenth century. Patrick White Limerick, a trader in agricultural products and wine, and the O’Connor family settled in Benicarló in 1749. Gaspar White and the O’Gorman family were based in Alicante, whereas the Lilikells, the Tuppers, and Henry O’Shea lived in Valencia. Against this backdrop, the O’Connor family built a Carlón wine cellar in Benicarló in 1757 [45] (Figure 17).
\nFloor plan and cross section of the O’Connor cellar in Benicarló (1757).
The wine cellar’s construction, using diaphragm arches, is very similar to the gunpowder magazine projects built by the Army Corps of Engineers in a neighbouring geographical area. These include the project by Carlos Beranger [MPD, 06, 169] for Benimàmet (1751) and the one by Juan Bautista French (1756) for Peñíscola [MPD, 07, 208]. In these projects, as opposed to the O’Connor cellar, the arch abutment is on the outside of the building. Nonetheless, there is a subsequent project by Antonio López Sopeña [MPD, 28, 027] for A Coruña (1774), in which he uses diaphragm arches with inside abutments similars to those used in the Benicarló’s wine cellar.
\nThe O’Connors family Benicarló’s building was built in 1757. The geometric study of the cellar arches is based on the topographical survey conducted with a laser scanner. The Carlón wine cellar has a rectangular floor plan; its inside measures are 12.42 m in width and 43.01 m in length. In the nave, there are eight diaphragm arches, each having a single two-piece offset-jointed ring and an average depth of 0.60 m. The arches are made of solid ceramic bricks (measuring 0.37 × 0.18 × 0.04 m), and they rest on a limestone base that was brought from Santa Magdalena de Polpís. The top of this base determines the springline of the ceramic arch. The arch’s abutment and the outside walls are made of ordinary uneven masonry. The formal characteristics of the arches are different: arch a1 has a clear span of e1a1 = 9.65 m and a rise of e2a1 = 5.82 m, whereas the other seven arches can be grouped together. Their span is within a range of e1a(2–8) = [9.76–9.69 m], similar to arch a1, but their rise significantly differs from the first arch, within a range of e2a(2–8) = [5.46–5.45 m]. All of the arches share a special feature: they do not have a vertical tangent on the stone base. The angle of incidence (α) of these arches with respect to both vertical sides, left (αa)and right (αb), has the following values: αa.a1 = 13.94° and αb.a1 = 8.97° in arch a1 and αa.a(2–8) = [4.49°–1.80°] and αb.a(2–8) = [7.38°–2.58°] in arches a(2–8) (Figure 8). By statistically analysing the parameters, for arches a(2–8) the average span calculated is e1a(2–8) = 9.72 m, and the average rise is e2a(2–8) = 5.46 m (Figure 18).
\nTransversal section of the O’Connor cellar in Benicarló (1757).
It seems that the measurement units used for the construction of the cellar were the toise (194.90 cm) and the toise foot (32.48 cm). Arches a(1–8) have an average span of 29.92 toise feet (9.71 m), with an error of 0.02 m for 30 feet (5 toises, 9.74 m). The rise of arch a1 is 17.92 feet (5.82 m), i.e., there is an error of 0.02 m in 18 feet (5.84 m), which are 3 toises (Figure 19). Arches a(2–8) have a rise of 16.80 toise feet (5.46 m), i.e., there is an error of 0.06 m in 17 feet (5.52 m). The arches are 0.60 m in width (1 + 10/12 feet). Regarding the outside measurements, the nave is 41.50 feet (13.48 m) wide and 92.36 feet (30 m) long, and the arches’ abutments are structures measuring 5 + 9/12 feet. The inside length of the cellar is 43.01 m, i.e., 132 + 5/12 toise feet. The enclosure wall on the façade is 2 feet thick; thus, the span-to-arch ratio is 5.75/30 feet (Figure 20).
\nGeometrical analysis of the arch 1 in the O’Connor cellar (1757).
Geometrical analysis of the arches 2–8 in the O’Connor cellar (1757).
A metrological analysis of the arches in Benicarló’s cellar reveals that the eight arches show the same metric relations, i.e., a 5 toise span and a 3 toise rise. The dimension of the catenary arch a1 are 30 × 18 feet (exactly 5 × 3 toises). The dimensions of the elliptical or oval-shaped arches a(2–8) are 30 × 17 feet. If we follow the hypothesis that the minor axis (either the ellipse minor axis or the oval minor axis) is 1 foot below the impost, then the geometric relation of arches a(2–8) is also 30 × 18 toise feet.
\nA statistical analysis is now performed on each of the eight arches a(1–8) to determine the difference between the shape of the arches built and the shapes of reference: ellipse (E), catenary (C), and ovals [O1, O2, O3]. The following values are calculated for 29 points on each cellar arch:
The average and maximum deviation of these 29 points
The angle of incidence on the springline
The mean deviation has a spread of only 0.03 m, which is approximately 0.31% of the arch’s span, making it difficult to conclude whether it is a catenary or an oval. The determining feature is that arch a1 has an angle of incidence on the springline [αa.a1 = 13.94°, αb.a1 = 8.97°]. Because the catenary’s angle of incidence is 19.38°, the geometric shape that most closely resembles the arch is the catenary.
\nConversely, the statistical analysis of the remaining seven arches a(2–8) shows that the geometric shape that they most resemble is the ellipse, with an average deviation ranging between 0.001 and 0.015 m. The range for oval-shaped arches is 0.006–0.186 m (
Thus, arch a1 resembles a catenary arch, whereas the other seven arches a2–8 tend to be ellipses. These seven arches do not have a vertical tangent on the springline because their horizontal axis has been moved 1 foot below the arch’s springline. As defined by Frézier (1738), the shape of the catenary has the following essential property: the vertical line which is tangent to the curve at the springline does not form a right angle with the horizontal plane. Therefore, geometrically, the catenary can be understood as any curve that does not have a vertical tangent at its springline. This is what happens in the springline of St. Paul’s dome in London [11], which was designed by Christopher Wren in collaboration with Robert Hooke [46]. Otherwise, it should be noted that from a mechanical perspective, catenary arches are an optimal solution to build masonry arches, since the material has very low tensile strength.
\nFinally, from the construction point of view, the catenary shape can be approximated using other geometric forms such as ovals or ellipses, under the condition that there is not a vertical tangent at the springline. The catenary shape forms a barycentric axis, which minimizes the tensions on a linear element that is subject to only vertical loads. In the arch, the inverted catenary shape prevents the appearance of stresses other than compression stresses.
\nThus, there are two hypotheses regarding the construction of the wine cellar. The first one is that the construction work was started from the inside toward the façade; thus, arches a(2–8) were constructed before the catenary arch a1. The second hypothesis is that the construction work began with arch a1. According to the second hypothesis, there is also a difference between both types of arches: on the first brick courses from the springline of arch a1 (the first 9 courses on 1 side and the first 17 courses on the other side), the ring is 0.36 m wide. On the remaining seven arches, the ring is 0.60 m wide (just like the arch’s depth). It is clear that less ceramic material is necessary for the construction of arch a1 than for the other seven elliptical arches a(2–8) (Figure 21).
\nSpringing of arches no. 1 and no. 2 in the O’Connor cellar.
The assessment of some drawings of gunpowder warehouses, found in the collection of
This paper addresses the introduction of the concept of the catenary arch in Spain before the nineteenth century. After an exhaustive review of the theoretical framework, some cases are assessed. The aim of the research is to find out if the mechanical concept of the chain was used by the Spanish military engineers and by the exiled English engineers, who built several wine cellars in Spain. Thus, we intend to determine whether there is any geometrical relationship between the layout of several gunpowder magazines made by Spanish military engineers in the 1730s and the construction of a civil building—the Carlón wine cellar in Benicarló (1757)—in which catenary arches may have been used.
\nThe assessments of the gunpowder warehouses by Miguel Marín for Barcelona (1731) and Tortosa (1733) and by Juan de la Feriére y Valentín in A Coruña (1736) are only a mere 4.05% of the projects analysed. However, they prove the intention to lay out the vault as a catenary. These authors knew that in a catenary the tensility in the shape of a hanging chain has the same compression values in the inverted geometrical figure. These engineers had a vast knowledge of the mechanical principles of the modern theory for masonry. From a scientific perspective, catenary vaults are the most interesting because they introduce the principles established by Hooke (1676). Both the arches of gunpowder magazines and the arches a(2–8) of Benicarló were laid out using the geometrical construction of an oval. Otherwise, the location of the horizontal axis of the ovals under the springline reveals the application of one of the characteristics of the catenary. This causes that the vertical line which is tangent to the curve in the springing does not form a right angle with the horizontal, so they used the chain’s theory in the layout of the projects.
\nFormally, if the distance between the axes and the springline of the arch is small, then the angle of incidence has a minimum influence on the thrust and the line of pressure. Otherwise, the location of the axis under the springline reveals the intention to minimize stresses in this point and in the neighbouring areas, even though the final mechanical influence is small.
\nAlthough there is no evidence of the construction of the gunpowder warehouses, it is possible to confirm the use of catenary arches in the construction of the Carlón cellars of the O’Connor in Benicarló (1757). There are significant differences between the measures of the arches of the gunpowder magazines (maximum span: 22 feet; maximum rise: 14 feet) and the arches of the Benicarló cellar (span: 30 feet; rise: between 17 and 18 “toise” feet, until the springline). In addition, the span-to-rise ratio of the oval arches in the gunpowder magazine studies is [1.39:1.57], whereas in Benicarló, this ratio is [1.67:1.76]. It can be concluded that arch a1 is a catenary arch, whereas arches a(2–8) tend to be elliptical. Arches a(2–8) show the special feature that their (x) axis is located below the springline; therefore, the tangent of the curve on the springline does not form a right angle with the horizontal. This is a feature of the definition of the catenary.
\nThe theory of the equilibrium curve, followed by most of the British engineers, became known to the Bourbon military engineers through the academy of mathematics in the eighteenth century, and it was used by some immigrants of English origin, such as the O’Connors, a century before the modernist architecture of Antoni Gaudí.
\nThe author thanks the members of the PatriARQ group research: PhD Agustí Costa Jover, PhD Sergio Coll Pla, and Arch. Mónica López Piquer.
\nLight harvesting for generation of electric energy is one of the most important research topics in applied sciences. First, for an efficient harvesting one needs a material with a broad light absorption window having a strong overlap with the sunlight spectrum. Second, one needs an efficient conversion of photoexcited carriers into produced current or voltage which can be used for applied purposes. The maximum light conversion coefficient in semiconductor systems is designated by so called Shockley-Queisser law, which is around 32% for an optimal bandgap value of 1.2–1.3 eV. However the efficiency may be increased using a solutions based on semiconductor nano materials such as quantum dots. Solar cells based on such a structures are included in the group of 3rd generation solar cell. 3rd generation solar cell encompasses multiple materials as a base of cell, such as: perovskite, organic, polymers and biomimetics. The most promising and in the same time most discussed are quantum dots and perovskite. Both material has a potential to revolutionize the solar cell industry due to their wide absorption range and high conversion coefficient. Nonetheless before the most common used material in photovoltaic namely silicon is replace one must overcome few major issues such as: stability and lifetime for at least 5 to 10 years or more, manufacturing process for a large surfaces and low production cost as well as recycling after the time of optimal use.
In this chapter we focus on two most promising material for photovoltaic application. The basic overview of organometallic properties of perovskites and quantum dots from the point of view of photovoltaics and formulation description of the electronic structure in the form of a simplified effective Hamiltonian as an approximation of a tight tie will be presented. The electronic structure plays a key role in the photovoltaic effect and is responsible for the high efficiency of the effect. Additionally perovskites or quantum dots show the spin-orbit coupling in the general form, this coupling can increase the carrier’s lifetime - the quantity important for solar cell applications.
Some perovskite-structured oxides have an internal electrical field, which plays an important role as it leads to the separation of electrons and holes generated in the process of light absorption. These oxides have the general structure of the ABO3 type. In general, there are quite a few different materials called perovskites, but the crystalline structure for all perovskites is similar. Perovskite oxides and, above all, organometallic halogen perovskites play an important role for photoelectronics and photovoltaics. Nonetheless perovskite oxides turned out to be inefficient in terms of photovoltaics. The interest in perovskite materials increased significantly towards the end of the last year a decade, when a series of works appeared showing the possibility of increasing efficiency in organometallic perovskites [1]. It turned out that there was a fairly broad class organometallic halide perovskites of the type CH3NH3PbX3 (X = I, Br, Cl), which show promising properties from the photovoltaic point of view. Although the first results gave relatively low photovoltaic efficiency, however this efficiency is quite fast it grew with new research. Besides, the conducted research did not show any significant restriction on the upper limit of the photovoltaic efficiency organometallic perovskites, which now reaches over 20%, which in turn gives hope for its further growth. The main advantages of organometallic halide perovskites are their relatively low levels price and relatively simple technology, which makes these materials competitive. Recent research results show that the efficiency of the laudatory prototypes of perovskite solar cell are already equalled and even exceeded the silicon based solar cell. Hence the great interest these materials from the point of view of application in photovoltaic cells [2, 3]. Of course, these materials also have weaknesses. One of these weaknesses is the lead toxicity they contain. The second is quite rapid degradation resulting from the sensitivity of photovoltaic cells based on them on humidity and the effect of ultraviolet radiation to which they are exposed. Therefore, the main lines of current research are not only aimed at further increasing efficiency photovoltaic, but also removing these undesirable weaknesses. As mentioned for photovoltaic the most interesting and promising are halide perovskites, the crystal structure of these materials has the general form ABX3, where A is the cation of the methylammonium group CH3NH3 for organometallic halide perovskites (metal cation for oxides), B is the metal cation Pb or Sn (the smaller metal cation in the case of oxides), while X is a Cl, Br or I anion for halide perovskites (O for oxides). The unit cell of the ABX3 perovskite crystal in the cubic phase is shown in Figure 1.
Perovskite crystal unit cell, a - large cation (methylammonium group CH3NH3), B - smaller cation (Pb or Sn), X - anion (I, Br or Cl).
One of the most promising materials is a perovskite with the chemical composition CH3NH3PbI3, because in this case the photovoltaic efficiency turned out to be the highest in this class of materials. It is worth noting, however, that the class of organometallic perovskites is in fact quite rich and contains many elements, which allows the use not only of single perovskites, but their more complex structures, e.g. double perovskites or systems composed of various materials [4]. The high photoelectric efficiency of organometallic perovskites is related to their electronic properties. This material is a semiconductor with a band gap width of about 1.6 eV. The light absorption coefficient is very high while energy losses associated with the possibility of non-radiative electron processes (e.g., electron–hole recombination by phonons) are relatively low. Moreover, the mobility of the carriers (electrons and holes) in these perovskite materials is quite high due to the low effective mass of the carriers. All these properties underlie high photovoltaic efficiency. On the other hand, the physical mechanisms underlying these properties are not yet fully researched and elucidated.
The excellent photovoltaic properties of perovskites are related to their electronic structure, in particular to the quantum states of electrons and holes in the conduction and valence bands, respectively. In the case of organometallic halide perovskites these properties are related to the organic CH3NH3 positive ion and its orientation with respect to the crystallographic axes [5].
Even better results using perovskite material from energy harvesting point of view may be achieved using hybrid structure. Recent discovery by the group of Prof. Miyasaka of a highly efficient light-to-voltage conversion in hybrid organic–inorganic perovskites [6] made these material promising elements for photovoltaics, especially taking into account simple low-cost fabrication technology. The basic structure of the perovskite-based solar cell is presented in Figure 2.
Schematic picture of a hybrid organic–inorganic perovskite solar cell. (figure source: USA). Department of Energy website.
The first two compounds of this family investigated by Prof. Miyasaka group, that is CH3NH3PbBr3 and CH3NH3PbI3, deposited on the TiO2 surface, demonstrated a good power conversion efficiency η higher than 3% (now reaching as high as 20%, similar to that in best conventional semiconductor-based solar cell) accompanied by the open-circle voltage Voc higher that 0.6 V and generated current density Jsc higher than 5.5 mA/cm2. These results of Prof. Miyasaka group attracted a great deal of attention and caused a tide of research in this field.
These compounds belong to the family of perovskite structures, similar to the high- temperature superconductors, where the main element is represented by Cu-O octahedrons. Although some structure elements of these groups of materials are similar, their physical properties are very different. In general, all the perovskites are known for formation of different structures and a variety of temperature-induced structural transitions.
Due to a large variety of the organic cations, the entire family of hybrid organic- inorganic perovskites potentially contains more than 1000 members [7], all different in their properties. Structure-wise, the main element of these compounds as presented in two-dimensional projection in Figure 3 is an octahedron built by metal and halogen ions, these elements are surrounded by organic layers.
Typical crystal structure of hybrid organic–inorganic perovskite compounds.
Despite several years of extensive research efforts, many microscopic properties, which can help in the understanding of the high photovoltaic efficiency in these compounds, remain unknown. This holds true even for CH3NH3PbI3 and HC(NH2)2PbI3 - the leaders of the race for the low-cost high photovoltaic efficiency. Electrical properties such as conductivity of these compounds can strongly depend on the temperature since due to a relatively soft lattice, various structural phase transitions occur in the temperature range of the order of 100 K - the property common for all perovskite structures, as mentioned above. Here we review and analyze properties of these materials in the context of their applicability for photovoltaics and connect these properties to the processes related to their possibly high efficiency.
Typical hybrid perovskite structure has the known form of a vertex-sharing networks of BX6 octahedrons as shown in Figure 3, which can be modeled as negatively charged PbI−3 elements. The bandstructure and optical properties are due mainly to the metal and halogen orbitals. Mutual overlap of these orbitals determines the matrix elements of interatomic hopping and, in turn, the band structure, corresponding to direct-band semiconductors with the bandgap Eg. This gap can be evaluated by different techniques. The overlap of the orbitals forms a relatively small effective mass of the carriers as well as the optical properties of these systems.
The sunlight has the following physical properties of our interest. Its spectrum corresponds to maximum in the green light region at photon energies
The light-induced transitions produce electron–hole pairs in the energy interval E > Eg, where Eg is the fundamental gap at the R − point of the Brillouin zone. The fundamental gap can be understood from the Coulomb energy arguments for the energy necessary to transfer electron from halogen to the IV-group heavy ion.
A qualitative plot of injected distribution of electrons in the energy representation is presented in Figure 4. This strongly nonequilibrium distribution then relaxes to the quasi equilibrium which, as we will see below, determines the performance of the photovoltaic devices. The relaxation process is mainly attributed to the multiple emissions of phonons.
Interband transitions caused by different photons, and electron distribution over the energy states, as injected. The behavior of the distribution at energies close to the minimum of the conduction band Ec is due to small density of states ∼
The energy relaxation processes are understood even less sufficiently than the origin of the carrier’s finite mobility. Indeed, due to a complex unit cell structure, crystals possess a large variety of phonon branches (of the order of 100) of different nature and symmetry. Here we propose a simple picture of the relaxation due to electron–phonon coupling. The analysis done in [9] shows that coupling to acoustic phonons (with the frequency linear in the momentum) would lead to high mobilities of the order of 103 cm2 V−1 s−1 and, therefore, this coupling is not the limiting factor for the observed low mobilities. Therefore, we concentrate on the relevant coupling to optical phonons. The coupling is due to the asymmetry of the field and change of the hopping integrals due to change in the interatomic distances. The value of the deformation potential constant is attributed to two main effects. First effect is the change in the site energy, corresponding to atomic displacement in the crystal field formed by its interaction with surrounding ions. Second effect is the changing in the overlap transfer integrals between the iodine and the lead orbitals, contributing to the electron energy as well.
The energy relaxation of the photoexcited electrons due to electron–phonon coupling with optical phonons, is relatively fast and occurs on the time scale of the order of 10 ps. This fast relaxation demonstrates that a thermalized room-temperature energy distribution is quickly produced. As a result, the performance of the photovoltaic elements with typical involved time scales of the order of 1–10 ns, is determined by the thermalized distributions, where the static local defects, either charged or not, structural disorder, and low-frequency optical phonons can play a role for the kinetics of the carriers distributions. The relation of these energy relaxation processes to the photovoltaic performance of real solar cells needs experimental studies and remains to be investigated.
The light absorption is efficient due to the band structure of perovskite materials having a direct bandgap close to 1.5 eV in the vertex point of the Brillouin zone. As a result, almost the entire sunlight spectrum can be absorbed. The efficiency of the absorption, in addition, is enhanced by relatively large momentum matrix elements between the group-IV heavy metal and halogen atoms resulting from their spatial overlap what makes perovskite material promising material for III generation of photovoltaic.
Quantum dots are one of the most interesting objects in modern fundamental and applied solid state physics, including applications for photovoltaics systems [10]. Typical sizes of the quantum dots of the order of 10 nanometers determine the majority of their physical properties, including the spectra of light absorption and properties of light-injected carries. These spectra determine the application of quantum dots in photogalvanics and photovoltaics systems. In contrast to bulk materials, where free electron–hole pairs can be produced optically, a strong confinement of carriers in quantum dots and resulting interaction between them leads to formation of exciton-like states. This effect qualitatively modifies optical properties of quantum dots and can make them useful for solar cell elements applications [10].
Various types of quantum dots can be used in photovoltaics: semiconductor polycrystalline and granular materials, quantum dots obtained by epitaxial methods or from colloidal solutions, nanoparticles of organic dyes. There are also a number of possibilities for the architecture of photovoltaic cells. Their common feature is that the phenomenon of multiple exciton excitation in dots a is used the energized charges (electrons and holes) are conducted to the electrodes in various ways, ensuring, however, their spatial separation. One possibility is to use scatter dots.
in a conductive material (e.g. in organic polymers). With the appropriate concentration of the dots, the discharge of the charge from the dots to the electrodes can be accomplished by coupling between the quantum dots. For regular networks of dots (single, double, or three-dimensional), discrete states of dots are transformed into mini-electron bands, ensuring charge transport. Photovoltaic cells with the use of regular quantum dot networks and their electronic mini-band structure (also called intermediate bands) have become one of the important directions in the development of photovoltaics. The essence of this type of solution is the fact that in the area between the electrodes in the p-n junction there is a layer containing dots quantum between which the distance is so small that an intermediate band is created in this area during the energy gap. This allows the use of low-energy photons (with energy lower than the width of the output semiconductor gap) to generate electrons in the conduction band and holes in the valence band. This is due to optical transitions from the valence band to the intermediate band and from the intermediate band to the conduction band. An important element is also that the recombination processes are in the case of the intermediate band much less likely than in the case of isolated quantum dots. In this case, it is enough for the wave functions of the dots to be quite delocalized. This can be achieved in systems with complexes quantum suppositories instead of regular networks.
Semiconductor quantum dots are usually produced of two types of binary materials. First type is usually referred to as III-V semiconductors, where one of the elements belongs to the III group of the periodic table of elements and the other belongs to the V group such as GaAs, InAs, InSb, and similar ones. The other group, named in the same way, is the II-VI semiconductors such as ZnS, ZnSe, CdS, and similar ones. In addition, coated quantum dots, where the core and the coating layer are made of different materials of the same (usually II-VI) type can be produced and used for different applications. Conventionally produced quantum dots show a variety of sizes and shapes. On one hand, this variety of quantum dot geometries extends the ability to use them for various applications, including photovoltaics. On the other hand, this variety hampers controllability of their applications. This circumstance should be taken into account in the analysis of all applications quantum dots.
Direct calculations of properties of quantum dots are very difficult, simple, but still highly efficient, approach relies on employing of the effective mass approximation with the Hamiltonian
where
where
One-dimensional parabolic potential modeling a simple quantum dot. Schematic plots of the ground and first excited states with opposite spins (marked by up- and down-arrows) are presented. Spin-orbit coupling couples these two spatial states with opposite spins and, as result, leads to the spin-position entanglement and modification of the charge densities.
Another form of the potential is given by:
and determines the quantum dot shape. Usual model shapes of quantum dot are ellipsoidal (in simple realizations, spherical) or coin-like cylindrical with the radius much larger than the width, as shown in Figure 6.
(a) Typical spherical quantum dot with a coating layer. Typical value of the radius R is of the order of 10 nm. (b) Typical coin-like cylindrical model of a quantum dot. The width d is of the order or less than 10 nm.
It is well-known that in semiconductors, although the band electron velocities, being of the order of v ∼ 108 cm/s, are much smaller than the speed of light c = 3 × 1010 cm/s, the relativistic effects, dependent on the v/c ratio, should be taken into account. In both types of semiconductors these relativistic effects lead to a coupling between electron momentum and electron spin, which appears due to the effect of the electric field in the unit cell of a binary semiconductor without inversion symmetry.
The Hamiltonian describing the spin-orbit coupling in bulk III-V materials has the form
presenting the Dresselhaus realization of the spin-orbit coupling [11]. In this Hamiltonian
with the cyclic permutation of the indices defining the other two components. The coupling constant
For II-VI compounds the bulk Hamiltonian has the form
usually referred to as the” Rashba Hamiltonian”.
The same form of the Hamiltonian describes the spin-orbit coupling in two-dimensional systems with a structural inversion asymmetry. Nonzero values of acan be achieved, in addition, by applying an electric field across the two-dimensional structure or a quantum dot. Usually both (Rashba and Dresselhaus) terms are present in two-dimensional electron systems and quantum dots.
The corresponding spectrum of the Rashba Hamiltonian is given by:
as shown in Figure 7.
Two branches of spectrum of two-dimensional carriers in the presence of the Rashba spin-orbit coupling with
In the absence of an external magnetic field the presented states are double-degenerate as dictated by the time-reversal symmetry of the spin-orbit coupled Hamiltonian. In the presence of such a field, the spin-orbit coupling terms should be augmented by the Zeeman coupling for the interaction between electron spin and external magnetic field in the form:
Here B is the magnetic field applied to the quantum dot, and g is the g-(Lande) factor.
Being a relativistic effect, the g − factor is strongly related to the spin-orbit coupling and the Zeeman term plays an important role in the physics of the quantum dot. The values of g- factor can vary by two orders of magnitude, e.g. between −0.45 for GaAs and approximately −50 in InSb. The corresponding spin splitting can reach the values of the order of 10 meV at realistic fields of the order of 10 T. We mention here that while the spin-orbit coupling for a given quantum dot is fixed by its material and shape [12], the Zeeman coupling can easily be modified by applying a magnetic field. Thus, the absorption spectra and optics-related properties of the quantum dot can be modified as well.
The geometry of a quantum dot plays the crucial role for its spectrum and spin-orbit coupling, and, therefore in the light absorption and photovoltaics effects. A qualitative effect of the spin-orbit coupling in quantum dots is the quantum entanglement of particle spin and its position, where the particle wavefunction cannot be presented as a product of the spin and coordinate states. This entanglement determines several processes in quantum dots.
Since in the optical absorption electrons and holes are produced, similar spin-orbit coupling and Zeeman Hamiltonians should be defined for the holes as well. The Coulomb interaction between electrons and holes plays the crucial role in the light absorption in quantum dots. The spectrum of holes is described by 4 × 4 matrices due to degeneracy of the spectrum consisting of “light” and “heavy” hole branches. Dependent on the material and shape of the quantum dot, spin-orbit coupling and the Zeeman coupling for holes can be much stronger than that for electrons. In the presence of spin-orbit coupling, spin-defined states are prone to relaxation and loosing well-defined spin values. In bulk systems, where electron momentum is a well-defined quantum number, spins relax mainly due to the celebrated Dyakonov-Perel mechanism [13] with the spin relaxation rate of the order of
We mention here that a new very large class of very promising for the photovoltaics materials such as perovskites or quantum dots show the spin-orbit coupling in the general form of the Rashba and Dresselhaus terms. Therefore, the spin-orbit coupling-related aspects of their applications can be treated in general terms similarly to semiconductors. This coupling can increase the carrier’s lifetime - the quantity important for solar cell applications. However, these rather complex compounds, demonstrating a great variety of different properties, are not yet reliably functionalized in the form of quantum dots. At the same time, whether the combined effects of the spin-orbit coupling lead to an increase or to a decrease in the efficiency of the light-to-voltage conversion in solar cells, is not yet clear even on the qualitative scale.
This work was supported by the National Centre for Research and Development under the project No. POIR.01.02.00-00-0265/17-00.
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