Inversion of a Laplace transform with applications from the study of the structure of material to the flow of Internet communications. Y. BAKOPOULOS and A.S.DRIGAS Department of Applied Technologies NCSR “DEMOKRITOS” Ag. Paraskevi GREECE Abstract: The percolation effect is studied through the inversion of a Laplace transformation. A universal law in the form of a stretched exponential is used to obtain information about the transition leading to percolation. As an example, the problem of the mathematical calculation of the pore size distribution of cement and cement – like materials showing a fractal pore construction is solved by the inversion of the Laplace transform which connects the pore size distribution function to the magnetization relaxation rate function calculated by Nuclear Magnetic Resonance experiments. Key-words: Laplace transform, percolation, stretched exponential, pore size, relaxation rate, cement, NMR 1. Introduction The percolation effect may appear in technological applications starting from the information flow in the Internet and proceeding to the pore structure of cement like materials. The problem of cement hydration is of strategic importance to the world construction technology and economy [1 – 7], [10], [12 – 14]. The possibility of the fractal geometry of the pore structure in cement – like materials, as well as in rocks, and its study by experimental methods is well known for years. The theory of fractals [8], [9], has applications in a diverse spectrum of physical problems (see for example [11] and [16]). The fractal geometry of the pores of cement has been studied by the use of various techniques and methods [1 – 7], [10], [12] (and references therein). In this work, some results having been obtained by Nuclear Magnetic Resonance (NMR) experiments are examined. The results are analyzed according to the theory developed in [7] and [10] and the resulting equation is regarded as a linear Fredholm equation of the first kind [12] The solution of the equation requires the inversion of the Laplace transformation of the relaxation rate distribution function [7]. Nuclear Magnetic Resonance is a standard method for studying cement hydration. The experiment is described by the fundamental equation for the calculation of the magnetization recovery function: (1): M0 −M M0 t α ) T1 AV where M0 is the maximum magnetization of the material when in relaxed condition in a static magnetic field, M is the magnetization measured at a specific time after the material has received the excitation of an exterior high radiofrequency pulse, t is the time after the excitation and T1AV is the average spin – lattice relaxation time measured from the bulk of the sample. Finally, α is a parameter related to the fractal dimension of the cement pore structure, taking values: o.5 ≤ α ≤ 1 [7]. The theoretical model describing cement hydration is based on two fundamental equations: (1): M0 −M M0 and (2): e 2. Description of the Experimental Method =e −( −( =e t α ) T1 AV −( t α ) T1 AV 1 − 1 = ∫ P ( )e T1 d ( ) T1 T1 0 ∞ t A third equation connects the pore size distribution with the relaxation rate distribution, permitting the modeling of the pore geometry, once the relaxation rate distribution function P( (3): g (ξ )d ξ = P ( 1 ) is known. T1 1 1 )d ( ) T1 T1 Here, T1 indicates the relaxation time of a specific pore, in any part of the cement bulk, since T1 depends primarily by the pore geometry and size. Equation (2) connects the local values of T1 to the average value T1AV, referring to the cement sample as a whole, during the experiment. In the next sextion, the solution of Equation (2) is given. For reasons of convenience, the following substitutions are used: 1 1 α = l, ( ) = λ, T1 AV T1 and equation (2) becomes: ∫ Ae ∞ (4): − lt P (l ) = A0e − λt Where q is a suitable normalization constant. By the calculation of the integral, the desired probability function will be obtained. To achieve the calculation, a more suitable path is chosen, such that, united with the straight line (c − i∞, c + i∞) it creates a closed contour in the complex numbers plane which contains no singularities. (Fig. 3). The definition of this path is as follows: Taking the origin as center, a circle of radius R tending to infinity is defined. As R goes on to infinity it will include the points c − i∞ , c + i∞ , as well as −∞ − iε , −∞ + iε , where ε tends to zero. The arcs from c − i∞ to −∞ − iε and −∞ + iε to c + i∞ are included in the new path. A second circle is defined, again having the origin as center but with a radius equal to , tending to zero. The arc between the two points D = – - iε and F = - + iε is excluded and the rest of the circle is included in the new path. Finally, the two straight line intervals, (- ∞ - iε, - – iε) and (- + iε, - ∞+ iε) are included to complete the closed contour A – C - D - E - F - H - B - A (Fig. 3), where A = c − i∞ , C = −∞ − iε , D = – -iε, Ε = , F = - +iε, Η = −∞ + iε , B = c + i∞ . The integral breaks down to the following parts: ∫e e α C I1 = 0 lt − λ tα dt which is an integral along the A 3. The Inversion of the Laplace Transformation in Equation (4) ∫ Ae The equation: ∞ (4): − lt P (l ) = A0e − λt α is in fact a Laplace transformation of the relaxation rate probability distribution P (l ) . The result of the transformation is the average magnetization recovery − λ tα function A0 e where A0 is the magnetization at t = 0, λ is the average relaxation rate raised to the α power. By inverting the transformation, we obtain the equation: (5): P (l ) = q ∫ c − i∞ ∫e e D I2 = 0 c + i∞ quadrant of a circle of radius R, tending to infinity, starting from A: c - i∞ and ending at C: - ∞ - iε. e − λt elt dt dt which is an integral along a straight C line parallel to the negative real semiaxis at a distance - ε, from C: - ∞ - iε to D: – – iε. lt − λ t ∫ e e dt which is an integral along the arc of a E I3 = α D circle with a radius D: – -iε, through E: tending to zero, from the point to F: - +iε. lt − λ t ∫ e e dt which is an integral along a straight H I4 = E α lt − λtα α line parallel to the negative real semiaxis at a distance ε, from F: - +iε to H: −∞ + iε lt − λ t ∫ e e dt which is an integral along the α B I5 = H quadrant of a circle of radius R, tending to infinity, starting from H: −∞ + iε to B: c + i∞ . I1 may be transformed as follows: t = Reiθ , I1 = ∫ π −π el Re e − λ R iθ − /2 ∫ π −π = − /2 ∫ π −π − /2 α iαθ e dt = Reiθ exp(lReiθ ) exp(−λ Rα eiαθ )dθ = exp(lR (cos(θ ) + i sin(θ )) exp(−λ Rα (cos(αθ ) + i sin(αθ )) dθ I5 = ∫ −π Reiθ exp{(lR(cos(θ ) + i sin(θ )) − (λ Rα (cos(αθ ) + i sin(αθ ))}dθ In both these integrals, the absolute value of the integrand is: exp(lR cos(θ ) − λ R a cos(αθ )) , where cos( ) and cos(α ) are negative quantities and α < 1. Therefore the integrant exponent, dominated by the term: lR cos(θ ) goes to - ∞ as R goes to ∞ and as a result the integrand goes to zero So the contribution of I1 and I5 is zero. Next, I3 = ∫ Re π −π iθ distribution P( exp{(lR(cos(θ ) + i sin(θ )) − (λ Rα (cos(αθ ) + i sin(αθ ))}dθ , where now R tends to zero. Again the contribution of I5 is zero. By reversing signs and integration limits and due to symmetry considerations, the contribution of I2 + I4 is: ∫ ∞ I = 2 elt e − λt dt . This is a real integral, easily α 0 calculated by numerical methods. 4. Conclusions 1 ) may be calculated. Then , by use of T1 the equation: P( I5 may be transformed in a similar manner: π /2 A method has been presented here for the calculation of the pore size distribution of Ordinary Portland Cement and other cementicious material, such as C3S, which show indications of a fractal pore structure, with fractal dimension substantially larger than 2, reaching 2.7 – 2.8. By calculating the parameters α and T1AV through NMR magnetization recovery experiments, inverting the Laplace transformation equation and calculating the ensuing integral, the relaxation rate 1 1 )d( ) = g(ξ)dξ [5], [6], [7] T1 T1 the pore size distribution may be calculated. This result is very useful in the theoretical and experimental studies for the understanding and improvement of the properties of cement and cement – like materials, which is of strategic value for the economy and the development of new materials and technologies. 5. Appendix on Proton NMR T1 Magnetization Recovery Measurements A sample containing a large percentage of protons as nuclei of hydrogen atoms will react by developing its own static magnetic field. This is due to the alignment of all proton spins either homo-parallel or antiparallel to the external field. The magnitude of the sample’s field is calculated from a Boltzman distribution function, depending on the external field. By the application of an suitable electromagnetic pulse, the sample magnetization along the external field may be driven to zero. This requires a suitable pulse, of frequency equal to the Larmor frequency of the sample and of such a strength and duration as to make the populations of homo-parallel and antiparallel spins equal in numbers. The sample regains equilibrium by relaxation and recovery of its magnetization, as the protons having been excited to an antiparallel direction relax back to their ground state, homo-parallel to the external field. The effect known as magnetization recovery, is, in its basic form, a simple exponential relaxation with a characteristic half-life’ time known as “spin- lattice relaxation time, T1”. In a more complex situation the magnetization recovery curve may be complex, ranging from the superposition of a small number of simple exponential functions to a full ‘stretched’ exponential indicating a continuous relaxation time distribution. 6. Aknowledgements The authors are grateful to Prof. R. 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