List of dialysers used for in vitro testing
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Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'IntechOpen is proud to announce that 179 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\nThroughout the years, the list has named a total of 252 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\nReleased this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
\n'}],latestNews:[{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"},{slug:"all-intechopen-books-available-on-perlego-20201215",title:"All IntechOpen Books Available on Perlego"},{slug:"oiv-awards-recognizes-intechopen-s-editors-20201127",title:"OIV Awards Recognizes IntechOpen's Editors"},{slug:"intechopen-joins-crossref-s-initiative-for-open-abstracts-i4oa-to-boost-the-discovery-of-research-20201005",title:"IntechOpen joins Crossref's Initiative for Open Abstracts (I4OA) to Boost the Discovery of Research"},{slug:"intechopen-hits-milestone-5-000-open-access-books-published-20200908",title:"IntechOpen hits milestone: 5,000 Open Access books published!"},{slug:"intechopen-books-hosted-on-the-mathworks-book-program-20200819",title:"IntechOpen Books Hosted on the MathWorks Book Program"}]},book:{item:{type:"book",id:"2283",leadTitle:null,fullTitle:"Advances in Crystallization Processes",title:"Advances in Crystallization Processes",subtitle:null,reviewType:"peer-reviewed",abstract:"Crystallization is used at some stage in nearly all process industries as a method of production, purification or recovery of solid materials. In recent years, a number of new applications have also come to rely on crystallization processes such as the crystallization of nano and amorphous materials. The articles for this book have been contributed by the most respected researchers in this area and cover the frontier areas of research and developments in crystallization processes. Divided into five parts this book provides the latest research developments in many aspects of crystallization including: chiral crystallization, crystallization of nanomaterials and the crystallization of amorphous and glassy materials. 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Lembrikov is a senior lecturer at the Faculty of Electronics, Electrical and Communication Engineering of the Holon Institute of Technology (HIT), Holon, Israel. B. I. Lembrikov received his Ph.D. in Nonlinear Optics at the Technion – Israel Institute of Technology in 1996. Since then he was an invited researcher at the Haifa University, at the Max Planck Institute High Magnetic Field Laboratory at Grenoble, France, at the Technion, Haifa, Israel. Dr. B. I. Lembrikov is an author of the book \\Electrodynamics of Magnetoactive Media\\, a number of chapters in scientific books, a large number of papers in international peer reviewed journals and reports delivered at the international scientific conferences. He actively participated in a number of research projects concerning optics of nanoparticles, optical communications, UWB communications. The main research fields of interest of Dr. B. I. 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The FMC-Genius®-Therapy System, which uses a double-sided peristaltic pump for both the patient\'s blood and the dialysis fluid (50…350 ml/min), even allows a ratio of QB=QD.
The relationship between the effectiveness of haemodialysis on the one hand, and the flows of blood and dialysate on the other, is guided by physical principles. These principles have been known for some time and have been widely discussed in the literature [1]. However, the fact that dialysis machines are completely automated makes it easy to neglect the underlying principles, so much so that knowledge of these may at times be lacking. There are also financial pressures that result in a desire to economise on water and dialysate concentrate, and a desire to work with as low a dialysate flow rate as possible.
The following sections will provide a brief summary of the theory behind the principles involved. In vitro and in vivo testing results from some of the more commonly available dialysers are used to provide a clearer picture of the theoretical principles involved. These measurements show that in spite of modern dialysis membranes and structural changes, it is not in fact possible to reduce the dialysate flow rate without also reducing the dialysis dose.
The connection between the dialyser\'s technical parameters and the flow rates involved (QB, QD in the counter current flow) is based on the following relationship [1]:
For QB=QD the following applies:
where Ko is the "overall mass transfer coefficient“ and A is the dialyser membrane surface area.
In dialysis log sheets, the KoA-value is usually recorded for urea, thus allowing the calculation of urea-clearance in relation to blood and dialysate flow rates.
Dependence of clearance K on the flow rates of blood QB and dialysate QD (KoA=1000 ml/min)
Fig. 1 shows how clearance rate K, derived from the KoA value for a low molecular substance (urea), changes in relation to blood flow rates, with the relationship demonstrated at different dialysate flow rates, and for a high-flux dialyser. At low dialysate flow rates, the dialysate compartment soon becomes saturated, leading to a large reduction in the concentration gradient. This in turn means that an increase in blood flow will no longer produce a substantial improvement in the rate of clearance. It is only when the dialysate flow rate is high enough for the substance in question to be removed quickly from the dialysate chamber that a higher blood flow rate can produce substantial improvements in the rate of clearance. This is effectively only ever the case when the dialysate flow rate QD is at least 1.5…2.0 times as high as the blood flow rate, QB.
This theoretical relationship requires the KoA value to be constant for all dialysate and blood flow rates. This does of course not apply in practice in cases where the total surface area of the dialyser\'s fibre bundle is not completely bathed in dialysate. This may happen when low dialysate flow rates lead to preferential channels being formed (please also refer to Sections 3 and 4). In the presence of relatively low dialysate flow rates, therefore, clearance values which may seem possible in theory are in fact unachievable in practice.
It must be emphasised, however, that these observations only apply to transport by diffusion, the most important transport mechanism for low molecular weight substances such as urea, creatinine and phosphate. As far as larger molecules are concerned, it is convective solute transport that becomes increasingly important as the molecular weight of the solute increases. With regard to the manner in which the overall clearance KT depends on both of these mechanisms, only an incomplete explanation can be provided. Werynski\'s equation [2] offers a reasonable approach:
Tr: transmittance coefficient; S: sieving coefficient; QF: ultrafiltration rate
As the value for clearance by diffusion, Kdiff in relation to blood flow rate, QB, decreases, the transmittance coefficient approaches the sieving coefficient, and the overall clearance KT is effectively determined by the ultrafiltration rate QF. It follows that the dialysate flow rate QD has less significance with regard to larger molecules than it does with regard to smaller ones.
Fig. 2 shows an enhanced ultrasound image depicting an unfortunate case of channelling observed in a test dialyser during the inflow of dialysate.
Enhanced ultrasound image of inflowing dialysate at QD=500 ml/min (Dialyser Altair 12G, US-machine Logic 7, GE Medical Systems, contrast agent: Optison®)
In order to avoid, or at least reduce, this channelling phenomenon, different structural alterations are commonly applied:
Flat sheet arrangement of hollow fibres
Wave-shaped hollow fibres (Moiré structure)
Spacer yarns
Pinnacle structure in the dialysate inflow and outflow areas.
Although a number of manufacturers at one point utilised flat-plate arrangements of interweaved hollow fibres in their dialysers (e.g. HFD 1.0 by MLW), this method has since been abandoned-probably due to cost reasons. Almost all manufacturers prefer the Moiré structure of fibres in order to obtain adequate spacing. In some instances-and sometimes in addition to the Moiré structure-spacing filaments (spacer yarns) are added (e.g. Asahi PAN650SF, MTP VitaPES). Although these measures ensure improved dialysate flow distribution within the cross-sectional area of the dialyser, in vitro testing using CT imaging has revealed that some preferential channelling through peripheral areas of the dialysate compartment remains [3, 4, 5]. With its FX series, Fresenius Medical Care followed a different path. Via a pinnacle structure in the inflow and outflow tracts, the dialysate is forced into even distribution across the entire surface area [6]. In the early days, dialyser technology included a number of devices whose inlet and outlet headers were fitted diagonally to improve the distribution of dialysate across the fibre bundle (e.g. EMC TriEx). For reasons unknown, this very simple solution did not prevail.
A dialysis machine (FMC 4008) was used to perform in vitro measurements of clearance rates in accordance with standard ISO 8637. A batch of 7.0 L stirred and thermostatically warmed dialysis liquid served as “blood” with dissolved urea and vitamin B12 as test substances. Measurements were carried out at different dialysate and blood flow rates, with each dialysate flow rate (QD=300, 500, 800 ml/min) being measured at blood flow rate settings of QB=100, 200, 300, 400 ml/min. These measurements were repeated for the 3 different types of dialysers.
\n\t\t\t\tDialyser\n\t\t\t | \n\t\t\t\n\t\t\t\tMembrane\n\t\t\t | \n\t\t\t\n\t\t\t\tManufacturer\n\t\t\t | \n\t\t\t\n\t\t\t\tKoA urea [ml/min]\n\t\t\t | \n\t\t
FDY-150 GW | \n\t\t\tPEPA®\n\t\t\t | \n\t\t\tNikkiso | \n\t\t\t874 | \n\t\t
FX 60 | \n\t\t\tHelixone®\n\t\t\t | \n\t\t\tFMC | \n\t\t\t967 | \n\t\t
VitaPES 150HF | \n\t\t\tPurema®\n\t\t\t | \n\t\t\tMTP Pirna | \n\t\t\t1167 | \n\t\t
List of dialysers used for in vitro testing
Clearance data were obtained for one small molecule (urea, relative molecular weight=60) and one larger molecule (vitamin B12, relative molecular weight=1357). KoA values were then calculated according to GI. 1a, b and compared with the values provided by the manufacturers.
Fig. 3 shows that the measured characteristics generally followed the theoretical model (refer to Fig. 1). However, the clearance rates calculated using the catalogue KOA values were not achieved at low dialysate flow rates. It was not until QD=800 ml/min that the values obtained managed to improve slightly on the ones provided. Similar curves were obtained for the other types of dialysers. The KoA values calculated from clearance, QB and QD were dependent upon both blood flow rate of blood and dialysate flow rate. In an ideal scenario, this kind of relationship should not exist. The overall mass transfer coefficient, Ko, is a physical characteristic of the fibre bundle and does not vary with the flow rates that might exist on the blood-side or the dialysate-side. Different KoA values in the presence of different flow rates can be easily explained provided if the effective surface area A, which is involved in solute transport, fails to remain constant. Values that may be theoretically possible will of course not be achieved in situations where a low dialysate flow rate results in sections of the fibre bundle not being immersed in dialysate. Fig. 4 shows that the KoA value calculated for the dialyser FDY150GW depends upon the dialysate flow rate.
Comparison of measured urea clearance values and urea clearance values derived from KoA data provided by the manufacturers.
KoA values calculated based on clearance, QD and QB, using the dialyser FDY150GW as an example
\n\t\t\t\t\n\t\t\t\t\tQD/dialyser type\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tFDY150GW\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tFX60\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tVitaPES150HF\n\t\t\t\t\n\t\t\t | \n\t\t
QD = 300 ml/min | \n\t\t\t740 ± 21 | \n\t\t\t728 ± 26 | \n\t\t\t792 ± 64 | \n\t\t
QD = 500 ml/min | \n\t\t\t753 ± 49 | \n\t\t\t751 ± 42 | \n\t\t\t936 ± 34 | \n\t\t
QD = 800 ml/min | \n\t\t\t817 ± 104 | \n\t\t\t756 ± 70 | \n\t\t\t1063 ± 106 | \n\t\t
Catalogue value | \n\t\t\t874 | \n\t\t\t937 | \n\t\t\t1167 | \n\t\t
In vitro KoA values [ml/min] for urea at different dialysate flow rates QD (mean values for different blood flow rates QB)
The dialyser FX60 showed the lowest degree of dialysate flow rate dependence of the KoA value. Although this was likely to be due to improved dialysate flow, the device still failed to achieve the manufacturer\'s catalogue value.
In summary, it has to be concluded that even modern high flux dialysers providing structural changes to optimise dialysate flow do not manage to negate the underlying relationship between the flow rates of blood and dialysate. Furthermore, full use of the fibre bundle\'s effective surface area A can only be guaranteed at high dialysate flow rates of around 800ml/min. Whilst this mirrors the results obtained by Leypolt [7], it contradicts the assumptions made by Golper and Ward, who concluded that structural improvements have made high dialysate flow rates unnecessary.
Naturally, in vitro investigations involving dialysers have to be treated with caution, with results not directly transferable to the clinical dialysis setting. Due to its viscosity and non-Newtonian flow properties, blood is a much more complex substance than an aqueous test solution. Also, unlike dialyser clearance rates, in vivo clearance rates depend upon physiological factors such as compartmentalisation of the blood and the uraemic toxins contained therein, fistula recirculation, cardiopulmonary recirculation, individual differences in haematocrit, as well as differences in protein and lipid concentrations. In spite of this, when looking at a sufficiently large number of patients and dialyses treatments, the basic principles discussed above should be mirrored in the clinical dialysis setting.
(a): Dialysate flow rate dependence of the spKt/V value in dialyser FDY150GW (b): Dialysate flow rate dependence of the spKt/V value in dialyser FX60 (c): Dialysate flow rate dependence of the spKt/V value in dialyser VitaPES150HF
At a constant blood flow rate (QB=300 ml/min), a group of 6 patients received three treatments each at QD=300, 500 und 800 ml/min, with each set repeated for each of the 3 types of dialysers tested in vitro. In order to assess the effectiveness of treatment, the urea reduction ratio, single pool Kt/V (spKt/V) and equilibrated Kt/V (eqKt/V) were obtained (Daugirdas formula). In addition, OCM data were obtained for the dialysis machine FMC 5008 (Eff.Kt, Eff.Kt/V).
The figures show (as do the urea reduction ratios and eqKt/V results) that each step increase in dialysate flow can significantly improve the effectiveness of haemodialysis.
The interesting question for dialysis patients is of course the exact impact this may have on treatment time. A sample calculation using results from the FX60 dialyser will demonstrate this (see Table 3).
\n\t\t\t\t\n\t\t\t\t\tPatient\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tBWopt\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tVcalc\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tK300\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tK500\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tK800\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tKtV300\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tt300\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\ttcalc500\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\ttcalc800\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tt300-t500\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tt300-t800\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t | \n\t\t
\n\t\t\t\t\n\t\t\t\t\tNo.\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tkg\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tL\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tml/min\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tml/min\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tml/min\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t | \n\t\t\t\t\n\t\t\t\t\tmin\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tmin\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tmin\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tmin\n\t\t\t\t\n\t\t\t | \n\t\t\t\n\t\t\t\t\n\t\t\t\t\tmin\n\t\t\t\t\n\t\t\t | \n\t\t
1 | \n\t\t\t75.0 | \n\t\t\t37.1 | \n\t\t\t186 | \n\t\t\t199 | \n\t\t\t209 | \n\t\t\t1.18 | \n\t\t\t240 | \n\t\t\t220 | \n\t\t\t209 | \n\t\t\t\n\t\t\t\t20\n\t\t\t | \n\t\t\t\n\t\t\t\t31\n\t\t\t | \n\t\t
2 | \n\t\t\t60.0 | \n\t\t\t26.9 | \n\t\t\t181 | \n\t\t\t197 | \n\t\t\t207 | \n\t\t\t1.57 | \n\t\t\t240 | \n\t\t\t215 | \n\t\t\t204 | \n\t\t\t\n\t\t\t\t25\n\t\t\t | \n\t\t\t\n\t\t\t\t36\n\t\t\t | \n\t\t
3 | \n\t\t\t66.0 | \n\t\t\t23.0 | \n\t\t\t158 | \n\t\t\t190 | \n\t\t\t202 | \n\t\t\t1.35 | \n\t\t\t180 | \n\t\t\t163 | \n\t\t\t154 | \n\t\t\t\n\t\t\t\t17\n\t\t\t | \n\t\t\t\n\t\t\t\t26\n\t\t\t | \n\t\t
4 | \n\t\t\t75.5 | \n\t\t\t33.6 | \n\t\t\t184 | \n\t\t\t202 | \n\t\t\t216 | \n\t\t\t1.71 | \n\t\t\t300 | \n\t\t\t285 | \n\t\t\t267 | \n\t\t\t\n\t\t\t\t15\n\t\t\t | \n\t\t\t\n\t\t\t\t33\n\t\t\t | \n\t\t
5 | \n\t\t\t69.0 | \n\t\t\t31.8 | \n\t\t\t173 | \n\t\t\t199 | \n\t\t\t207 | \n\t\t\t1.36 | \n\t\t\t240 | \n\t\t\t217 | \n\t\t\t208 | \n\t\t\t\n\t\t\t\t23\n\t\t\t | \n\t\t\t\n\t\t\t\t32\n\t\t\t | \n\t\t
6 | \n\t\t\t69.5 | \n\t\t\t27.9 | \n\t\t\t178 | \n\t\t\t188 | \n\t\t\t201 | \n\t\t\t1.93 | \n\t\t\t300 | \n\t\t\t287 | \n\t\t\t268 | \n\t\t\t\n\t\t\t\t13\n\t\t\t | \n\t\t\t\n\t\t\t\t32\n\t\t\t | \n\t\t
Time saved by using a dialysate flow rate of QD=500 ml/min or QD=800 ml/min instead of a dialysate flow rate of QD=300 ml/min
(Vcalc from OCM EffKt and measured Kt/V; tcalc=Kt/V(QD300)*V/K)
The same Kt/V value achieved at a dialysate flow rate QD=300 ml/min can be achieved at a dialysate flow rate of QD=500 ml/min, whilst treatment time can be reduced by between 13 and 25 minutes, depending on the individual patient\'s body mass and clearance. If a dialysate flow rate of QD=800 ml/min is used, treatment time will be reduced accordingly, namely by 26-36 min. Treatment time reduction of this order is certainly of interest to patients and should be able to outweigh the slight increase in costs incurred by higher dialysate flow rates. Naturally, if treatment time remains unchanged, a higher dialysate flow rate can also be used to improve the Kt/V value-as depicted in Fig. 5.
In vitro measurements obtained from a number of different modern high flux dialysers show that the characteristics "Clearance (QB, QD)“ manage to closely match theoretical scenarios of expected interdependence. However, in spite of a number of different structural improvements, the KoA value remains dependent upon the dialysate flow rate. A higher dialysate flow rate of around 800 ml/min ensures that the fibre bundle is bathed more evenly in dialysate, thus improving the effectiveness of dialysis.
In vivo investigations confirm that increases in dialysate flow rates are likely to produce significantly improved Kt/V values.
The density functional theory (DFT) is commonly used to study the crystal structure, lattice energy, the equation of state, the electronic bandgap, and vibration spectra properties [1]. Based on the kinetic energy density functional of Thomas [2] and the exchange-correlation effects of Dirac [3], DFT has been greatly developed by Kohn and Sham (KS) [4], who have established the fundamental approximation theorem on the functional status to describe real systems by electronic structure calculations. The eigenvalues of KS equations have no physical meaning, and the ionization energy is in the opposite state direction [5]. Moreover, one proposed approach is to introduce the eigenstates to calculate multi-body (many-body calculation) on the basis of Monte Carlo calculations [6] and perturbation theory [7]. The calculation of elastic constants is preceded by full geometry optimization and the stress tensor calculation of a number of distorted structures at the atomic scale. Polycrystalline structure constituted by a single crystal structure contains a variety of information (e.g., orientation) and the properties of a single crystal, such as anisotropy. Within the mechanics of typical crystals structures, the transition from the micro- to the meso-scale (homogenization) and vice versa (localization) can be estimated. Homogenization is an idealized description of a statistical distribution inside the actual heterogeneous material. Once the continuity model is admitted, the concept of homogeneity is deduced from it [8]. For quasi brittle materials, Zhu et al. [9] have formulated the anisotropic model in the framework of Eshelby-based homogenization methods. X-rays diffraction measurement is one of the stress assay test methods in physics field, of which the stress is actually determined by the strain [10]. Diffraction-based stress analysis depends critically on the use of the correct diffraction elastic constants [11]. X-ray method to test the material stress and to obtain elastic constants [12] is commonly based on the Reuss model [13]. Elasticity of single crystal and mechanical properties of polycrystalline material have been closely integrated. Various calculations methods are compared to determine homogenized moduli of the polycrystalline material composed of a single crystal, for example, the certain stress of Reuss model [13] and the certain strain of Voigt model [14].
DFT as a first-principles theory and a solid band theory in quantum mechanics has own a great success in linking physical properties and molecular structure, the calculation with exact accuracy but for low computational efficiency for macromolecular structure, which can be used to calculate elastic constants of anisotropic crystals, the monoclinic gypsum, and tobermorite crystals, for example. The chemical formula of gypsum is CaSO4•2H2O, which is an evaporite mineral and a kind of hydration product of anhydrite (chemical formula: CaSO4). Moreover, the 11 Å tobermorite model (chemical formula: Ca4Si6O14(OH)4·2H2O) as an initial configuration of C-S-H structure is commonly used. Since Young’s modulus parameters of gypsum and C-S-H are important to the multi-scale model [15], elastic constants of the gypsum crystal are investigated. The crystal is monoclinic, with 13 independent constants. For the homogenization of elastic deformation, especially for polycrystalline structures, the traditional Reuss-Voigt-Hill method is used to calculate the elastic moduli of monoclinic structures. Based on the ab initio plane-wave pseudopotential density functional theory method mentioned earlier, we focus on the monoclinic crystals to estimate their homogenized elastic moduli.
Despite the above advantages of DFT, however, the resolution of a system by Kohn-Sham equations involves difficulties due to an infinite number of electrons. These electrons maybe changed under an effective potential generated by an infinite number of cores or ions.
From a microscopic point of view, Schrödinger equation describing a periodic crystal system composed of atomic nuclei n in mutual interaction and electron spin σi is positioned
Hamiltonian, in simple cases, consists of five terms: the kinetic energy of the electrons and nuclei, and the various interactions between them.
The possible analytical representation and resolution of such a problem become a difficult task due to the limited memory of the computer tools. However, it is possible to reformulate the problem using appropriate theorems and approximations.
The fundamental principle approaches of mean field theory, in particular the DFT, are that any properties of an interacting particle system can be considered as a functional density in the ground state of the system n0(r). Besides, the scalar function of the position n0(r) essentially determines the wave functions of the system at the ground state and the excited states. Electronic and mechanical properties of a periodic crystal refer to solid state physics, quantum mechanics, and crystallography.
The crystalline ion movement of the electron is as
According to the Born-Oppenheimer or adiabatic approximation [16], the dynamics of the system (electrons and nuclei) is described. The electrons are assumed to react instantly to ionic motion. In electronic coordinates, the nucleus positions are considered as immobile external parameters.
where the last term of the Hamiltonian is constant and has been introduced in order to preserve the neutrality of the system and avoid the divergence of the eigenvalues. Clean the ground state of the system for fixed nuclear positions, total energy is given by the formula:
This energy has a surface in the space coordinates that is said to be ionic Born-Oppenheimer surface. The ions move according to the effective potential energy, including Coulomb repulsion and the anchoring effect of the electron, which are as follows:
The dissociation degrees of freedom of electrons from those of nucleons, obtained through the adiabatic approximation, are very important, because if the electrons must be treated by quantum mechanics, degrees of freedom of ions in most cases are processed in a conventional manner.
This theorem/approach of Hohenberg and Kohn tries to make an exact DFT theory for many-body systems. This formulation applies to any system of mutually interacting particles in an external potential
DFT and its founding principle are summarized in two theorems, first introduced by Hohenberg and Kohn [17], which refer to the set of potential
The total energy of the ground state of a system for interacting electrons is functional (unknown) of the single electron density
As a result, the density n0(r) minimizing the energy associated with the Hamiltonian (9) is obtained and used to evaluate the energy of the ground state of the system.
The principle established in the second theorem of Hohenberg and Kohn specifies that the density that minimizes the energy is the energy of the ground state
Because the ground state is concerned, it is possible to replace the wave system function by the electron charge density, which therefore becomes the fundamental quantity of the problem. In principle, the problem boils down to minimize the total energy of the system in accordance with the variations in the density governed by the constraint on the number of particles
The approach of Kohn-Sham system substitutes the interacting particles, which obeys the Hamiltonian in Eq. (3), by a less complex system easily solved. This approach assumes that the density in the ground state of the system is equal to that in some systems composed of non-interacting particles. This involves independent particle equations for the non-interacting system, gathering all the terms complicated and difficult to assess, in a functional exchange-correlation
T is the kinetic energy of a system of particles (electrons) independently (non-interacting) embedded in an effective potential which is no other than the real system,
The Hartree energy or energy of interaction is associated with the Coulomb interaction of the self-defined electron density.
Solving the auxiliary Kohn and Sham system for the ground state can be seen as a minimization problem while respecting the density n(r). Apart from orbital function TS, all other terms depend on the density. Therefore, it is possible to vary the functions of the wave and to derive the variational equation:
With the constraint of orthonormalization
Eqs. (16)–(18) are known equations of Kohn-Sham, the density n(r) and the resulting total energy EKS. These equations are independent of any approximation on the functional EXC(n), resolution provides the exact values of the density and the energy of the ground state of the interacting system, provided that EXC(n) is exactly known. The latter can be described in terms of Hohenberg Kohn function in Eq. (8)
or more precisely,
This energy is related to potential exchange-correlation
For the exchange-correlation functional, the only ambiguity in the approach of Kohn and Sham (KS) is the exchange-correlation term. It is subject to functional approximations of local or near local order of density that said energy EKS can be written as
where εxc([n], r) is the exchange-correlation energy per electron at point r, it depends on n(r) in the vicinity of r. These approximations have made enormous progress in the field.
The approximation of the local density (LDA)
The use of the local density approximation (LDA) in which the exchange-correlation energy
The exchange term
This approximation has been particularly checked to deal with non-homogeneous systems.
The generalized gradient approximation (GGA)
The generalized gradient approximation (GGA) involves the local density approximation providing a substantial improvement and better adaptation to the systems. This approximation is equal to the exchange-correlation term only as a function of the density. A first approach (GEA) was introduced by Kohn and Sham then used by the authors of Herman et al. [20].
This notion of GGA is the choice of functions, which allows us a better adaptation to wide variations so as to maintain the desired properties. The energy is written in its general form [21]:
where
Different states of the Schrödinger equation for an independent particle in a system. By Kohn and Sham equations, responding to eigenvalue equation is as:
where the electrons are immersed in an effective potential
The effective potential has the periodicity of the crystal and may be expressed using Fourier series, in a periodic system:
Gm is the reciprocal lattice vector:
where
As the translational symmetry, it is that states are orthogonal and conditioned by the limits of the crystal (infinite volume). In this case, the Eigen functions of KS are governed by the Bloch theorem: they have two quantum numbers: the wave vector k in the Brillouin zone (BZ) and the band index i, and this can be expressed by a product of a plane wave exp.(ik, r) and a periodic function:
where R is the vector of direct space defined by ai with
Solving Eq. (24) is equivalent to increase the periodic function
where
where each point is a set of k eigenstates, the label having i = 1, 2, … obtained by diagonalization of the Hamiltonian in Eq. (29).
It is necessary to integrate the points k in the Brillouin zone. For a function fi(k) where i defines the band index, the average value is
According to the crystal theory [28], any crystal lattice system contains six independent variables, namely the cell side length a, b, and c; unit cell angle α, β, and γ. Generally, the crystal under a certain deformation, temperature, and pressure can be described by the corresponding six-dimensional deformation tensor. The temperature and pressure will cause cell-deformed configuration tensor as
where
A multi-particle electronic structure satisfies the Schrödinger equation. As in [29], Kohn-Sham equation as an approximation to simplify Schrödinger equation is described. For crystal composed by vibrator with the vibration frequency wi, the total Helmholtz free energy is
Helmholtz free energy can be calculated for all the thermodynamic quantities. DFT-QHA (quasi-harmonic approximation) is a precise calculation method to calculate thermodynamic properties of solid materials elastic constants and Debye temperature with the accurate predictions.
According to the theory of elasticity, under the isothermal strain, the elastic modulus of Helmholtz free energy can be described by the form of the Taylor expansion, of which the coefficients of the polynomial are the elastic coefficient:
where
The components of the stress tensor can be extracted by
The second-order elastic coefficients can be obtained by the coefficient of the second-order Taylor expansion of Helmholtz free energy with the strain,
Here, strain and thermodynamics deformation are symmetric. There is only six independent deformation tensor in the nine-dimensional deformation tensor. LCEC is a second-order linear combination of independent elastic coefficients corresponding to Helmholtz free energy coefficient under some deformation mode [30, 31]. For all directions under monoclinic crystals, if a strain is added, the corresponding simultaneous equations can be solved to determine all elastic coefficients.
Deformation tensors to calculate independent Cij constants of monoclinic crystal are listed in Table 1.
Deformation tensor | ΔE-V relation of LCEC | LCEC |
---|---|---|
c11+ c22 + 2c12 | ||
c22 + c33 + 2c23 | ||
c11 + c33 + 2c13 | ||
c44 + c55 + 2c45 | ||
c11 + c66 + 2c16 | ||
c22 + c66 + 2c26 | ||
c33 + c66 + 2c36 | ||
c11 + c22 + c33 + 2c12 + 2c13 + 2c23 | ||
c11 + c22 − 2c12 | ||
c11 + c33 − 2c13 | ||
c22 + c33 − 2c23 | ||
4c44 | ||
c11 + 4c55 + 4c15 | ||
c11 + 4c55 − 4c15 | ||
c22 + 4c55 + 4c25 | ||
c33 + 4c55 + 4c35 |
For monoclinic crystal, elastic constants include C11, C22, C33, Cl2, C13, C23, C44, C55, C66 Cl5, C25, C35, and C46; the strain energy-volume relation and elastic moduli of monoclinic symmetry based on E-V method can be obtained. The calculated E-δ points are fitted to second-order polynomials E(V, δ). For all strains, different strain forms δ are taken to calculate the total energies E for the strained crystal structure. By applying a series of δ strain amplitude, the independent elastic constants of monoclinic crystal by these simultaneous ΔE-δ equations can be obtained.
Stress-strain relation in an orthotropic monoclinic crystal can be defined by the independent elastic stiffness parameters [32]:
where σ represents the normal stress and shear stress in each direction (unit: nN/nm2); ε and γ are the normal strain and shear strain in each direction, respectively.
The homogenized elastic properties of polycrystals can be calculated, of which elastic moduli and Poisson’s ratio can be obtained by calculating Voigt and Reuss bounds and averaging term as [32]
For monoclinic crystal structure, elastic constants include C11, C22, C33, Cl2, C13, C23, C44, C55, C66 Cl5, C25, C35, and C46. The criteria for mechanical stability are given by Wu [32]:
Young’s modulus and Poisson’s ratio can be rewritten based on the Voigt-Reuss-Hill approximation [33]. In terms of the Voigt-Reuss-Hill approximations [34], MH = (1/2)(MR + MV), M refers to B or G. Thus, Young’s modulus E and Possion’s ratio μ are obtained as
Then, Voigt-Reuss-Hill average [32] will be determined, and Young’s modulus can be calculated.
The gypsum morphology is monoclinic, and the initial lattice is as a = 5.677Å, b = 15.207Å, c = 6.528Å, α = β = 90°, and γ = 118.49°, its structure is monoclinic with space group I 2/a [35].
In Figure 1, the gypsum crystal can be summarized as follows: (1) the two hydrogen atoms of water molecules formed weak hydrogen bonds with the O atoms of Ca and S polyhedra; (2) a stacking sequence of CaO8 and SO4 chains in the (010) plane alternates with water layers along the b-axis; and (3) in (010) plane, the sulfate tetrahedra and CaO8 polyhedra alternate to form edge-sharing chains along [100] and zigzag chains along [001] direction [36] (Table 2).
Modeling of gypsum crystal. (a) Gypsum structure [36] along [001]; (b) the real cell; (c) in x-direction; (d) in y-direction; and (e) in z-direction.
Atom | x | y | z | Occupancy rate | Uiso or Ueq |
---|---|---|---|---|---|
Ca | 0.5000 | 0.0786 | 0.2500 | 1.00 | 1.00 |
S | 0.0000 | 0.0787 | 0.7500 | 1.00 | 1.00 |
O1 | −0.0384 | 0.1326 | 0.5512 | 1.00 | 1.00 |
O2 | 0.2429 | 0.0215 | 0.8347 | 1.00 | 1.00 |
Ow | 0.3784 | 0.1825 | 0.4554 | 1.00 | 1.00 |
H1 | 0.2504 | 0.1615 | 0.5009 | 1.00 | 1.00 |
H2 | 0.4022 | 0.2435 | 0.4900 | 1.00 | 1.00 |
Atomic coordinates and displacement parameters of gypsum [36].
Hamid model [37] as the 11 Å tobermorite (formula: Ca4Si6O14(OH)4·2H2O) as an initial configuration is commonly used. The morphology is monoclinic, and the initial lattice is [37]: a = 6.69 Å, b = 7.39 Å, c = 22.779 Å, α = β = 90°, and γ = 123.49°, space group P21. Modeling of 11 Å tobermorite is shown in Figure 2.
Modeling of 11 Å tobermorite crystal. Silicate chains, calcium octahedral, and oxygen atoms are shown as yellow tetrahedra, green spheres, and red spheres. (a) 11 Å Tob monoclinic crystal; (b) in x-direction; (c) in y-direction; and (d) in z-direction.
In Figure 2(a), the 11 Å tobermorite crystal can be summarized as follows: (1) the structure is basically a layered structure. (2) The central part is a Ca-O sheet (with an empirical formula: CaO2, of which the oxygen in CaO2 also includes that of the silicate tetrahedron part). (3) Silicate chains envelope the Ca-O sheet on both sides. (4) Ca2+ and H2O are filled between individual layers to balance the charges. The infinite layers of calcium polyhedra are parallel to (001), with tetrahedral chains of wollastonite-type along b and the composite layers stacked along c and connected through the formation of double tetrahedral chains [38]. Atomic coordinates and displacement parameters are seen in Table 3.
Atomic species | X | Y | Z | Occupancy rate | Uiso or Ueq | Atomic species | X | Y | Z | Occupancy rate | Uiso or Ueq |
---|---|---|---|---|---|---|---|---|---|---|---|
Si1 | 0.7710 | 0.3830 | 0.1578 | 1 | 0.031 | O8 | 0.7690 | 0.8430 | 0.0953 | 1 | 0.027 |
Si2 | 0.9250 | 0.7500 | 0.0721 | 1 | 0.030 | O9 | 0.5370 | 0.7980 | 0.1968 | 1 | 0.036 |
Si3 | 0.7720 | 0.9620 | 0.1596 | 1 | 0.015 | O10 | 0.0040 | 0.0420 | 0.2008 | 1 | 0.034 |
O1 | 0.7740 | 0.4950 | 0.0932 | 1 | 0.039 | O11 | 0.4330 | 0.2230 | −0.0250 | 0.5 | 0.072 |
O2 | 0.7620 | 0.1690 | 0.1305 | 1 | 0.019 | O12 | 0.9490 | 0.2560 | 0.0000 | 1 | 0.080 |
O3 | 0.0020 | 0.5270 | 0.2000 | 1 | 0.032 | O13 | 0.4300 | 0.7700 | −0.0220 | 0.5 | 0.090 |
O4 | 0.5360 | 0.3040 | 0.1926 | 1 | 0.035 | Cal | 0.2770 | 0.4257 | 0.2083 | 1 | 0.024 |
O5 | 0.9100 | 0.7470 | 0.0000 | 1 | 0.034 | Ca2 | 0.7630 | 0.9160 | 0.2951 | 1 | 0.027 |
O6 | 0.2020 | 0.8870 | 0.0942 | 1 | 0.053 | Ca3 | 0.5620 | 0.0640 | 0.0450 | 0.25 | 0.038 |
O7 | 0.2890 | 0.4360 | 0.0940 | 1 | 0.076 | — | — | — | — | — | — |
Atomic coordinates and displacement parameters of 11 Å tobermorite [38].
The initial conditions are as follows: the pressure region of 0–1 GPa is used. Besides, a plane-wave basis set and ultrasoft pseudopotentials using GGA are used with a plane-wave cutoff energy of 400 eV. Brillouin zone is 6 × 6 × 4. Self-consistent convergence of the total energy per atom is chosen as 10−4 eV. Elastic constants of monoclinic gypsum crystal under 0–1.0 GPa are shown in Figure 3.
Gypsum monoclinic crystal under pressure 0–1.0 GPa by DFT. (a) Relative change of a, b, c, and V and (b) elastic constants.
From Figure 3, elastic constants at 0 GPa are given as c11 = 82.464 GPa, c12 = 34.751 GPa, c13 = 33.643 GPa, c15 = −1.987 GPa, c22 = 63.046 GPa, c23 = 34.920 GPa, c25 = −8.071 GPa, c33 = 57.549 GPa, c35 = −3.054 GPa, c44 = 20.863 GPa, c46 = −4.688 GPa, c55 = 28.062 GPa, and c66 = 28.556 GPa. It is found that the oxygen atom of the water molecule did not change its position or occupancy under pressure conditions. A simple pressure increase at an ambient temperature cannot induce dehydration because of the unchange of water molecular in the gypsum structure within pressure range [36].
Elastic constants of gypsum crystal model based on DFT are calculated, and parameters are detailed in Table 4.
P | C11 | C12 | C13 | C15 | C22 | C23 | C25 | C33 | C35 | C44 | C46 | C55 | C66 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10–4[36] | — | — | — | — | — | — | — | — | — | — | — | — | — |
0.0 | 82.46 | 34.75 | 33.64 | −1.99 | 63.05 | 34.92 | −8.07 | 57.55 | −3.05 | 20.86 | −4.69 | 28.06 | 28.56 |
0.1 | 79.82 | 32.64 | 29.2 | 1.8 | 71.04 | 29.61 | −7.54 | 61.88 | −3.22 | 20.13 | −3.06 | 26.19 | 27.7 |
0.2 | 82.93 | 37.75 | 34.59 | 1.09 | 63.62 | 32.42 | −7.35 | 50.64 | −4.37 | 21.32 | −1.1 | 25.8 | 17.8 |
0.3 | 82.82 | 39.77 | 32.81 | 0.17 | 65.64 | 29.61 | −7.23 | 57.31 | −4.45 | 26.43 | −5.57 | 23.17 | 23.39 |
0.4 | 84.47 | 38.6 | 32.25 | 1.27 | 69.03 | 32.31 | −8.51 | 53.41 | −2.21 | 20.8 | −2.03 | 28.41 | 22.34 |
0.5 | 75.84 | 43.68 | 29.39 | 0.57 | 68.7 | 33.18 | −8.36 | 56.08 | −2.52 | 29.7 | −3.88 | 27.35 | 22.4 |
0.6 | 74.22 | 43.11 | 28.77 | 2.22 | 69.52 | 28.87 | −7.81 | 53.19 | −2.68 | 28.97 | −1.49 | 23.24 | 15.53 |
0.7 | 88.37 | 41.74 | 32.85 | 2.25 | 70.09 | 32.28 | −9.14 | 55.48 | −4.28 | 24.66 | −2.76 | 27.25 | 22.58 |
0.8 | 88.53 | 39.65 | 35.29 | 2.96 | 73.28 | 33.84 | −8.02 | 62.22 | −3.73 | 24.73 | −3.44 | 26.37 | 24.39 |
0.9 | 88.7 | 45.09 | 37.97 | 4.54 | 66.78 | 36.02 | −10.4 | 61.98 | −1.2 | 25.15 | −4.92 | 28.93 | 26.82 |
1.0 | 90.12 | 39.79 | 34.63 | 2.7 | 75.99 | 34.92 | −8.74 | 68.31 | −3.46 | 26.32 | −5.83 | 28.15 | 30.19 |
Elastic coefficient Cij (GPa) of gypsum by DFT.
Initial conditions of tobermorite are quite the same with that of gypsum crystal. Elastic constants of 11 Å tobermorite crystal under 0–1.0 GPa are shown in Figure 4. Elastic constants are shown in Table 5.
11 Å tobermorite monoclinic crystal under pressure 0–1.0 GPa by DFT. (a) Relative change of a, b, c, and V and (b) elastic constants.
P/GPa | C11 | C12 | C13 | C15 | C22 | C23 | C25 | C33 | C35 | C44 | C46 | C55 | C66 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
SHA[39] | 102.65 | 41.68 | 27.70 | 1.25 | 125.05 | 18.83 | −4.10 | 83.80 | −3.38 | 22.90 | −11.93 | 23.25 | 50.20 |
0.0 | 106.63 | 50.37 | 41.09 | −3.50 | 131.67 | 22.78 | −0.78 | 71.45 | −0.83 | 26.03 | −0.02 | 27.61 | 45.26 |
0.1 | 118.37 | 45.40 | 35.91 | −3.52 | 129.18 | 17.19 | 0.11 | 67.84 | −0.55 | 32.51 | 3.90 | 32.74 | 40.07 |
0.2 | 109.13 | 45.84 | 35.63 | −3.22 | 136.79 | 23.05 | 0.03 | 82.75 | 0.06 | 28.88 | 1.21 | 22.40 | 45.69 |
0.3 | 115.53 | 46.36 | 40.17 | −4.46 | 142.59 | 27.65 | −0.04 | 95.03 | 0.02 | 31.08 | 0.49 | 32.38 | 50.57 |
0.4 | 102.65 | 35.38 | 38.73 | −6.32 | 123.43 | 18.11 | −1.92 | 74.28 | 0.05 | 18.14 | −0.83 | 22.38 | 40.44 |
0.5 | 100.08 | 42.58 | 36.10 | −4.52 | 137.56 | 21.68 | −0.26 | 90.87 | 0.36 | 29.92 | −0.46 | 29.66 | 51.82 |
0.6 | 97.87 | 44.09 | 28.76 | −5.85 | 162.17 | 25.77 | 0.19 | 93.71 | −0.14 | 24.89 | −1.20 | 26.63 | 40.26 |
0.7 | 108.73 | 48.60 | 34.07 | −4.55 | 147.09 | 26.78 | −0.14 | 92.64 | −0.01 | 21.56 | 2.06 | 44.25 | 41.23 |
0.8 | 122.87 | 55.30 | 40.62 | −4.05 | 155.75 | 29.54 | −0.25 | 103.3 | −0.57 | 24.90 | 0.72 | 33.31 | 42.67 |
0.9 | 120.77 | 44.19 | 45.41 | −4.82 | 139.59 | 13.68 | 0.09 | 88.25 | −0.19 | 27.18 | −0.35 | 26.85 | 53.22 |
1.0 | 127.01 | 41.78 | 45.00 | −4.47 | 143.72 | 23.65 | −0.02 | 98.30 | −0.12 | 29.98 | 0.71 | 32.08 | 45.68 |
Elastic coefficient Cij (GPa) of 11 Å tobermorite by DFT.
A comparisonal results of Shahsavari [39] are provided. Elastic constants at 0 GPa are as follows: c11 = 106.63 GPa, c12 = 50.37 GPa, c13 = 41.09 GPa, c15 = −3.50 GPa, c22 = 131.67 GPa, c23 = 22.78 GPa, c25 = −0.78 GPa, c33 = 71.45 GPa, c35 = −0.83 GPa, c44 = 26.03 GPa, c46 = −0.02 GPa, c55 = 27.61 GPa, and c66 = 45.26 GPa. Thus, elastic modulus can be homogenized to compare with the results of LD C-S-H phase in nano-indentation test by Vandamme and Ulm [40].
Based on elastic constants, the elastic moduli of gypsum at 0 GPa are verified and averaged in Figure 5.
Elastic moduli of gypsum crystal under pressure 0–1.0 GPa.
As gypsum shows anisotropic compressibility along three crystallographic axes with b > c > a below 5 GPa [44], the pressure region of 0–1.0 GPa is used to verify whether the performance of model under low pressure is stable. Mechanical moduli of gypsum polycrystalline are listed in Table 6.
Pressure (GPa) | Gv (GPa) | Bv (GPa) | Gr (GPa) | Br (GPa) | B (GPa) | G (GPa) | E (GPa) | μ |
---|---|---|---|---|---|---|---|---|
Reference [43] | 26.5333 | 39.2556 | 24.8077 | 39.2381 | 25.6705 | 39.2469 | 63.2265 | 0.2315 |
0.0 | 22.1459 | 45.5208 | 19.7054 | 43.8224 | 20.9257 | 44.6716 | 54.2985 | 0.2974 |
0.1 | 22.8896 | 43.9624 | 21.7569 | 42.9250 | 22.3233 | 43.4437 | 57.1766 | 0.2806 |
0.2 | 19.1472 | 45.1906 | 17.4395 | 41.9064 | 18.2934 | 43.5485 | 48.1394 | 0.3158 |
0.3 | 21.5041 | 45.5718 | 19.3501 | 43.6675 | 20.4271 | 44.6197 | 53.1678 | 0.3014 |
0.4 | 21.2263 | 45.9146 | 19.5324 | 43.2463 | 20.3794 | 44.5805 | 53.0537 | 0.3017 |
0.5 | 22.1809 | 45.9026 | 19.4980 | 43.7566 | 20.8395 | 44.8296 | 54.1306 | 0.2988 |
0.6 | 19.9615 | 44.2709 | 17.7554 | 41.5818 | 18.8585 | 42.9264 | 49.3486 | 0.3084 |
0.7 | 22.0356 | 47.5189 | 20.1833 | 44.0623 | 21.1095 | 45.7906 | 54.8931 | 0.3002 |
0.8 | 22.7817 | 49.0659 | 21.3745 | 47.0681 | 22.0781 | 48.0670 | 57.4399 | 0.3008 |
0.9 | 22.7399 | 50.6242 | 19.4094 | 48.1542 | 21.0747 | 49.3892 | 55.3510 | 0.3132 |
1.0 | 25.2707 | 50.3430 | 23.4785 | 48.9327 | 24.3746 | 49.6379 | 62.8382 | 0.2890 |
Mechanical moduli of gypsum polycrystalline by different methods.
As an acoustic method [41] and mechanical properties [42] have been investigated, according to elastic constants of gypsum crystal [43], elastic moduli by experiment can be calculated, as shown in Table 6. Elastic moduli are as follows: Gv = 22.146 GPa, Gr = 19.705 GPa, Bv = 45.521 GPa, Br = 43.822 GPa, B = 44.672 GPa, G = 20.926 GPa, E = 54.299 GPa, and μ = 0.2974. These results are close to the plane-strain value of Young’s modulus by reference [44] E = 50 GPa, μ = 0.45. By comparison of gypsum crystal and CH crystal, axial moduli of gypsum in x, y, and z directions are 57.75, 37.22, and 34.91 GPa, while axial moduli of Ca(OH)2 in x, y, and z directions are 93.75, 93.75, and 42.39 GPa, showing that gypsum crystal is much less anisotropic than hydrogen-bonded layered Ca(OH)2 structure [42].
Based on elastic constants of 11 Å tobermorite crystal using GGA calculation method by DFT, bulk modulus B and shear modulus G are separately calculated by Eqs. (37)–(50) (Figure 6).
Elastic moduli of 11 Å tobermorite crystal under pressure 0–1.0 GPa.
Elastic moduli at 0 GPa are verified and averaged as Gv = 32.815 GPa, Bv = 59.803 GPa, Gr = 29.908 GPa, Br = 54.276 GPa, E = 79.512 GPa, and μ = 0.268. Young’s modulus is about 79.512 GPa by Reuss-Voigt-Hill estimation, which is close to the simulation result of 89 GPa [45] by Pellenq and result of 78.939 GPa [39] by Shahsavari. Mechanical moduli by different methods are listed in Table 7.
Pressure (GPa) | Bv (GPa) | Br (GPa) | Gv (GPa) | Gr (GPa) | B (GPa) | G (GPa) | E (GPa) | μ |
---|---|---|---|---|---|---|---|---|
Reference [191] | 54.2133 | 51.6976 | 34.1560 | 28.9168 | 52.9555 | 31.5364 | 78.9391 | 0.2516 |
0.0 | 59.8066 | 54.2778 | 32.8140 | 29.9063 | 57.0399 | 31.3615 | 79.5121 | 0.2677 |
0.1 | 56.9306 | 50.1535 | 35.5205 | 33.4468 | 53.5438 | 34.4843 | 85.1689 | 0.2349 |
0.2 | 59.7454 | 56.4236 | 34.3364 | 31.4164 | 58.0846 | 32.8763 | 82.9743 | 0.2619 |
0.3 | 64.6138 | 62.4671 | 38.7385 | 36.7029 | 63.5388 | 37.7202 | 94.4669 | 0.2522 |
0.4 | 53.8665 | 50.9999 | 30.0678 | 26.1841 | 52.4333 | 28.1257 | 71.5786 | 0.2725 |
0.5 | 58.8039 | 56.9635 | 37.4898 | 34.7400 | 57.8831 | 36.1145 | 89.6903 | 0.2417 |
0.6 | 61.2236 | 57.0694 | 35.3648 | 32.2837 | 59.1442 | 33.8242 | 85.2259 | 0.2598 |
0.7 | 63.0391 | 60.0785 | 37.3432 | 34.0050 | 61.5598 | 35.6744 | 89.6966 | 0.2572 |
0.8 | 70.3213 | 67.4998 | 37.2739 | 34.9015 | 68.9022 | 36.0849 | 92.1653 | 0.2771 |
0.9 | 61.6837 | 57.8325 | 37.8067 | 33.5164 | 59.7599 | 35.6606 | 89.2325 | 0.2511 |
1.0 | 65.5442 | 63.4872 | 38.6855 | 36.6712 | 64.5147 | 37.7292 | 94.7226 | 0.2553 |
Mechanical moduli of 11Å tobermorite polycrystalline by different methods.
However, these values considering the ordered Si-chain at a long range are far away from the nano-indentation experiment performed on the C-S-H phase [40, 46]. It confirms the absence of order at a long range in this phase and that the up-scaling to polycrystals cannot be done with the tobermorite model. Modeling of C-S-H structure with disordered Si chain should be fairly considered.
Elastic constants of gypsum and tobermorite structures under a certain pressure region are calculated by DFT method, which has a certain value for both application and reference. Results are as follows:
For monoclinic gypsum and tobermorite crystals, elastic coefficients are obtained in 0–1-GPa pressure range to verify the reliability of the model by comparing other literatures.
Elastic constants of gypsum single crystal at 0 GPa are given as follows: c11 = 82.464 GPa, c12 = 34.751 GPa, c13 = 33.643 GPa, c15 = −1.987 GPa, c22 = 63.046 GPa, c23 = 34.920 GPa, c25 = −8.071 GPa, c33 = 57.549 GPa, c35 = −3.054 GPa, c44 = 20.863 GPa, c46 = −4.688 GPa, c55 = 28.062 GPa, and c66 = 28.556 GPa.
Elastic constants of 11Å tobermorite single crystal at 0 GPa are as follows: c11 = 106.63 GPa, c12 = 50.37 GPa, c13 = 41.09 GPa, c15 = −3.50 GPa, c22 = 131.67 GPa, c23 = 22.78 GPa, c25 = −0.78 GPa, c33 = 71.45 GPa, c35 = −0.83 GPa, c44 = 26.03 GPa, c46 = −0.02 GPa, c55 = 27.61 GPa, and c66 = 45.26 GPa.
Young’s modulus of gypsum is about 54.299 GPa. Elastic moduli at 0 GPa are as follows: Gv = 22.146 GPa, Gr = 19.705 GPa, Bv = 45.521 GPa, Br = 43.822 GPa, E = 54.299 GPa, and μ = 0.297.
Young’s modulus of 11Å tobermorite is about 79.512 GPa. Elastic moduli at 0 GPa are as follows: Gv = 32.815 GPa, Bv = 59.803 GPa, Gr = 29.908 GPa, Br = 54.276 GPa, E = 79.512 GPa, and μ = 0.268.
Structural, elastic properties of monoclinic crystals are investigated, and Cij determination is given by DFT method. Reuss-Voigt-Hill estimation has been used for polycrystal structures and can be seen as an intermediate step in the homogenization of elastic properties.
The authors greatly acknowledge the financial support for this work provided by the China Scholarship Council (CSC) and the support of start-up foundation of Xi’an Shiyou University. Thanks to Qiufeng Wang for her proofreading.
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\\n\\nDISCLAIMER: Neither the CC BY 3.0 license, nor any other license IntechOpen currently uses or has used before, applies to figures and tables reproduced from other works, as they may be subject to different terms of reuse. In such cases, if the copyright holder is not noted in the source of a figure or table, it is the responsibility of the User to investigate and determine the exact copyright status of any information utilised. Users requiring assistance in that regard are welcome to send an inquiry to permissions@intechopen.com.
\\n\\nAll rights to Books and all other compilations published on the IntechOpen platform and in print are reserved by IntechOpen.
\\n\\nThe copyright to Books and other compilations is subject to separate copyright from those that exist in the included Works.
\\n\\nAll Long Form Monographs/Compacts are licensed under the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) license granted to all others.
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\\n\\nPolicy last updated: 2016-06-08
\\n"}]'},components:[{type:"htmlEditorComponent",content:'Copyright is the term used to describe the rights related to the publication and distribution of original Works. Most importantly from a publisher's perspective, copyright governs how Authors, publishers and the general public can use, publish, and distribute publications.
\n\nIntechOpen only publishes manuscripts for which it has publishing rights. This is governed by a publication agreement between the Author and IntechOpen. This agreement is accepted by the Author when the manuscript is submitted and deals with both the rights of the publisher and Author, as well as any obligations concerning a particular manuscript. However, in accepting this agreement, Authors continue to retain significant rights to use and share their publications.
\n\nHOW COPYRIGHT WORKS WITH OPEN ACCESS LICENSES?
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\n\nWork - a Chapter, including Conference Papers, and any and all text, graphics, images and/or other materials forming part of or accompanying the Chapter/Conference Paper.
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\n\nIntechOpen - Registered publisher with office at 5 Princes Gate Court, London, SW7 2QJ - UNITED KINGDOM
\n\nIntechOpen platform - IntechOpen website www.intechopen.com whose main purpose is to host Monographs in the format of Book Chapters, Long Form Monographs, Compacts, Conference Proceedings and Videos.
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\n\nTERMS
\n\nAll Works published on the IntechOpen platform and in print are licensed under a Creative Commons Attribution 3.0 Unported License, a license which allows for the broadest possible reuse of published material.
\n\nCopyright on the individual Works belongs to the specific Author, subject to an agreement with IntechOpen. The Creative Common license is granted to all others to:
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\n\nAll Works are published under the CC BY 3.0 license. However, please note that book Chapters may fall under a different CC license, depending on their publication date as indicated in the table below:
\n\n\n\n
LICENSE | \n\t\t\tUSED FROM - | \n\t\t\tUP TO - | \n\t\t
\n\t\t\t Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0) \n\t\t\t | \n\t\t\t\n\t\t\t 1 July 2005 (2005-07-01) \n\t\t\t | \n\t\t\t\n\t\t\t 3 October 2011 (2011-10-03) \n\t\t\t | \n\t\t
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The CC BY 3.0 license permits Works to be freely shared in any medium or format, as well as the reuse and adaptation of the original contents of Works (e.g. figures and tables created by the Authors), as long as the source Work is cited and its Authors are acknowledged in the following manner:
\n\nContent reuse:
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\n\nReposting & sharing:
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\n\nRepublishing – More about Attribution Policy can be found here.
\n\nThe same principles apply to Works published under the CC BY-NC-SA 3.0 license, with the caveats that (1) the content may not be used for commercial purposes, and (2) derivative works building on this content must be distributed under the same license. The restrictions contained in these license terms may, however, be waived by the copyright holder(s). Users wishing to circumvent any of the license terms are required to obtain explicit permission to do so from the copyright holder(s).
\n\nDISCLAIMER: Neither the CC BY 3.0 license, nor any other license IntechOpen currently uses or has used before, applies to figures and tables reproduced from other works, as they may be subject to different terms of reuse. In such cases, if the copyright holder is not noted in the source of a figure or table, it is the responsibility of the User to investigate and determine the exact copyright status of any information utilised. Users requiring assistance in that regard are welcome to send an inquiry to permissions@intechopen.com.
\n\nAll rights to Books and all other compilations published on the IntechOpen platform and in print are reserved by IntechOpen.
\n\nThe copyright to Books and other compilations is subject to separate copyright from those that exist in the included Works.
\n\nAll Long Form Monographs/Compacts are licensed under the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) license granted to all others.
\n\nCopyright to the individual Works (Chapters) belongs to their specific Authors, subject to an agreement with IntechOpen and the Creative Common license granted to all others to:
\n\nUnder the following terms:
\n\nThere must be an Attribution, giving appropriate credit, provision of a link to the license, and indication if any changes were made.
\n\nNonCommercial - The use of the material for commercial purposes is prohibited. Commercial rights are reserved to IntechOpen or its licensees.
\n\nNo additional restrictions that apply legal terms or technological measures that restrict others from doing anything the license permits are allowed.
\n\nThe CC BY-NC 4.0 license permits Works to be freely shared in any medium or format, as well as reuse and adaptation of the original contents of Works (e.g. figures and tables created by the Authors), as long as it is not used for commercial purposes. The source Work must be cited and its Authors acknowledged in the following manner:
\n\nContent reuse:
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\n\nReposting & sharing:
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\n\nAll Book cover design elements, as well as Video image graphics are subject to copyright by IntechOpen.
\n\nEvery reproduction of a front cover image must be accompanied by an appropriate Copyright Notice displayed adjacent to the image. The exact Copyright Notice depends on who the Author of a particular cover image is. Users wishing to reproduce cover images should contact permissions@intechopen.com.
\n\nAll Video Lectures under IntechOpen's production are subject to copyright and are property of IntechOpen, unless defined otherwise, and are licensed under the Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license. This grants all others the right to:
\n\nShare — copy and redistribute the material in any medium or format
\n\nUnder the following terms:
\n\nUsers wishing to repost and share the Video Lectures are welcome to do so as long as they acknowledge the source in the following manner:
\n\n© {year} IntechOpen. Published under CC BY-NC-ND 4.0 license. Available from: {DOI}
\n\nUsers wishing to reuse, modify, or adapt the Video Lectures in a way not permitted by the license are welcome to contact us at permissions@intechopen.com to discuss waiving particular license terms.
\n\nAll software used on the IntechOpen platform, any used during the publishing process, and the copyright in the code constituting such software, is the property of IntechOpen or its software suppliers. As such, it may not be downloaded or copied without permission.
\n\nUnless otherwise indicated, all IntechOpen websites are the property of IntechOpen.
\n\nAll content included on IntechOpen Websites not forming part of contributed materials (such as text, images, logos, graphics, design elements, videos, sounds, pictures, trademarks, etc.), are subject to copyright and are property of, or licensed to, IntechOpen. Any other use, including the reproduction, modification, distribution, transmission, republication, display, or performance of the content on this site is strictly prohibited.
\n\nPolicy last updated: 2016-06-08
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