MATHEMATICAL MODEL OF CONDUCTING NANOPORE FOR MOLECULAR DYNAMICS SIMULATIONS

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Introduction. Nanopores represent a type o pores with a diameter of about several nanometers. Channels of such size can be both natural and artificially made objects. Currently researchers take great interest in the issues of studying and modeling of nanopore properties due to their broad application in science and technology [1; 2]. Artifi- cial nanopores can be used, for example, for imitation of biological nanopore functions. They also have attractive potential applications in nanofluid fields [3]. Among these applications are desalination, ionic sieves [4], sensors for biological agents and sequencing of DNA or RNA [5]. Various forms of artificial nanopores using different production technologies and materials, including poly- mers, inorganic substances, biotic and composite materi- als have been developed [6; 7]. Conducting nanopores attract considerable attention due to their important potential applications. In a number of works production methods of similar structures are given. For example, such nanopores can be made via in- tegration of carbon structures into the corresponding nanoporous material. A multilayered carbon nanotube was made inside a pore and on the surface of an anode aluminum oxidic membrane, applying the method of chemical sedimentation from a steam phase [8]. Strength- ened single-layer metal carbon nanotubes were grown on a silicon plate via chemical sedimentation from a steam phase [9]. Synthesis of conducting carbon tubules inside the pores of the anode aluminum and oxidic membrane was investigated [10]. Similar structures can be of interest in terms of production of selective membranes switched by electric field. In recent years mathematical and electrostatic models of ion transport in various types of nanopores and nano- channels have been vigorously developed, computer modeling of processes have been carried out as well [11-13]. At computer modeling of ion transport, applying In the work [15] a method of the conducting nanopore electric field in the presence of external charges (ions) modeling in relation to problems of molecular dynamics was offered. In the present work an alternative for further development of this method is being investigated. The results of simulation can be used when develop- ing various devices, including, for instance, a component of life support systems in piloted spacecrafts. Problem definition. The membrane containing a nanopore is simulated in the form of L wide uniform dielectric plate. The nanopore is a cylindrical hole with radius a and length L (fig. 1). Fig. 1. Model of nanopore. Dots denote fictitious electric charges Рис. 1. Модель нанопоры. Точками обозначены фиктивные электрические заряды We shall consider the problem of ion transport in the context of continual electrostatic model. We shall assume that the ion is located in random position r* . Electrostatic potential of j(r) in and outside the nanopore satisfies the Poisson's equation å n n 1 N * methods of molecular dynamics, the problem of electric field defining, created in the system, has to be solved at Dφ = q d(r - r ), 4πε0 n=1 (1) each step in time. In presence of electro-conductive ele- ments in the system, solving such a task with classical methods becomes rather resource-intensive. Therefore, problem solving methods of such electrostatic tasks should be fairly simple, although they are not as accurate as classical ways. In this regard there is a problem of such methods accuracy assessment in comparison with classi- cal ones. A rather typical assessment might be a compari- son of integrated electric parameters of the system, ob- tained by the offered methods, with classical decisions. In this work electric capacitance parameter will be compared with the data from the reference book [14]. The purpose of this work is to simulate the conducting nanopore electric field in presence of external charges (ions). This model is intended for use in fundamental re- search of charged particles transfer phenomenon (ions and charged proteinaceous residue) in conducting nanopores. where D - Laplasian operator; qn - ion charge; N - ion number; d(r - r* ) - delta-Dirac function. The equation (1) n has to be solved across all the space, except the area of membrane material with the potential set of boundary conditions on the tubulen surface. It is also necessary to demand the continuity of the membrane, normal to the surface, components of electric induction vector and, whenever possible, decrease of potential while being re- moved from the system. Under consideration of charged particles transport phenomenon, we are interested in the force affecting the ion from the conducting nanopore which is expressed as f = -q × grad (φ(r* )) . The solution of the equation (1) with the set boundary conditions is a difficult task. The problem becomes even more complicated if to consider not only cylindrical, but also other forms of nanopores. Further we will try to pre- sent simple and rather universal model of conducting nanopores which does not require solving the Poisson equation for the potential. Electrostatic model of conducting nanopore. We shall approximate electric field distributed on the body (tubulen) of the charge by the field of the dot charges lo- cated on the same surface. We shall place these fictitious charges in the chosen in advance surface dots. We will determine their size from the condition of maximum apnotes transposing operation of a vector column in a vector line; 1 = (1,1,1,…1) - made of units dimension vector K. Solution to variational problem (5), (6), on condition that nondegenerate matrix Aˆ , will be vector Aˆ -1 ´1 Q = q . (7) 1¢´ Aˆ -1 ´1 In case of nondegenerate matrics Aˆ , under condition that a0 - evector Aˆ , corresponding to zero evalue, nonproach to mechanical balance. Assuming that fictitious charges could move on a tubulen surface, then equality to zero projection on the turbulen surface of total force af- fecting each charge from other charges would be condiorthogonal 1, we obtain Q = q a0 . 1¢´ a0 (8) tion of balance. This case is equivalent to the requirement of equality to zero projections on the tubulen surface of total electric field strengths created by all other charges in the point of each fictitious charge. In our case as positions of charges are fixed, the above-stated requirement is gen- erally impracticable. Therefore, we shall demand the minimal sum of these projections squares. The result (8) can be easily summarized and in case of greater 1 dimension of the evector subspace, correspond- ing to zero evalue Aˆ : in this case a0 - any non orthogo- nal to 1 vector from that subspace. However, in case a0 is orthogonal to 1, the task should be reduced, while de- creasing Aˆ size and leaving behind only zero evalues. Thus, we shall place the fictitious charges Qk, k = 1, …, K, at the point with coordinates rk. Then we obtain electric-field vector Projection Eiτ at point ri: It should be noted that from the method Aˆ that Aˆ is symmetrical non-negative matrix. We shall consider the alternative where Aˆ it is clear is a degen- å E (r ) = 1 Q iτ i 4πε k (ri - rk ) - ni (ni *(ri - rk )), | r - r |3 (2) erate matrix. Generally this is true: we shall place nonzero dot charges of one sign on the tubulen surface and allow 0 k ¹i i k where ni - normal vector to the surface at the point ri; * is dot product of vectors. By summing squares (2) where all i = 1, …, K, we obtain åK E2 (r ) = 1 ´ them to move on a tubulen surface. It is obvious that such system has a stationary position of charges. If to arrange fictitious charges similarly, then the matrix Aˆ will de- generate as there are charge values at which vector pro- jection of electric field strength affecting each charge is equal to zero. Moreover, another stationary position, iτ i 16π2ε2 mostly continuously depending on charges, corresponds i=1 0 (ri - rk ) *(ri - rl ) - ´ ååå K Q Q - (ni *(ri - rk ))(ni *(ri - rl )). k l | r - r |3| r - r |3 (3) to a different set of value sizes which are freely moving on the tubulen surface. Therefore, we have a rather big area of coordinates of fictitious charges at which the ma- trix degenerate. At numerical calculations it will mean i=1 k ¹i l ¹i i k i l We shall look for the minimum (3) on condition of specified total charge q of conducting tubulen: Подпись: K q = åQi . (4) i=1 Considering that a set of unknown charges is vector Q = (Qk ) , than we have a standard problem of square form minimization of (3): åK E2 (r ) = 1 Q¢´ Aˆ ´ Q, that if we chose the “wrong” positions of fictitious charges, and the matrix is not degenerated, then, while increasing in number of fictitious charges, generally, con- ditionality ratio of this matrix will grow. The present work is devoted to preliminary assess- ment of the offered method practical applicability under molecular and dynamic calculations, therefore further cases of small number of fictitious charges at which in the reviewed examples matrix Aˆ is not degenerated are inves- tigated. The offered method is a further development of [15]. iτ i 16π2ε2 In the work [15] comparative analysis of the presented i=1 0 (5) method of electrostatic task solution with other methods of electrostatics is carried out. As a comparative criterion, å Aˆ = (ri - rk ) *(ri - rl ) - (ni *(ri - rk ))(ni *(ri - rl )) . kl | r - r |3| r - r |3 the results of electric capacitance calculation given in the reference book [14] were used. In this work for the above i¹k ,l i k i l Under condition (4): q = 1¢´ Q. (6) method applicability assessment we will use the results of electric capacitance of the conducting tubulen calculations and compare them with the results [14] and [15]. We shall continue the above suggested calculations in Here ´ denotes dot product of vectors of K dimension and vectors and dimensional matrix K ´ K ; stroke deorder to obtain estimates of conducting tubulen electric capacitance. We shall receive these estimates, comparing the energy of the system considered with the expression for the energy of the conducting body: same algorithm of fictitious charges arrangement and N = 80 are given. W = 1 å QiQj = q2 (9) Results presented in table and fig. 2 show that when L/a > 8, discrepancy amounts to less than 15 % already , 4πε0 i¹ j | ri - rj | 2C where C - conducting tubulen electric capacitance. Add- ing (7) to (9), finally obtain the expression for electric capacitance: (1¢´ Aˆ -1 ´1)2 for N = 20, and smoothly decreases under increase both L/a , and N, which is also an advantage of the suggested method. For comparison we must note that given in fig. 2 works [15] do not possess this property. From the practical point of view it might be expected that at L/a > 10, N = 30 parameters of charged particles interac- C = 2πε0 ¢ ˆ -1 ˆ ˆ -1 , tion with the conducting nanopore are defined to a preci- 1 ´ A ´ B ´ A ´1 sion of less than 10 %. Bˆ = ï| r - r | ì 1 , ï ij í i j î 0 i ¹ j; i = j. (10) Conclusion. One electrostatic model of conducting nanopore is presented in the work. The model is intended for use when modeling a nanopore, applying the method of molecular dynamics. This computing method is based It shall be noted that in [15] nondegenerate matrics Bˆ was proved Model validation. In order to validate the accuracy of the model suggested, we shall calculate the self- capacitance of the cylindrical hole of radius a and length L and compare it with the data given in the table [14]. We shall place the fictitious charges in points (zn, a, qm) of the cylindrical system of coordinates, where on the second law of Newton. In case force intensity op- erating on each of atoms is known, integration of the movement equations allows to obtain a trajectory which describes positions, velocities and accelerations of parti- cles over the time. If fictitious charges are not entered, then to find the force operating on charged particle from the conducting nanopore, at first it is necessary to solve the equation (1) in each timepoint. In [15] the method allowing a single zn = (n -1) L N -1 2π , n = 1,..., N , (11) invert of the matrix of equation system by fictitious charges introduction is offered. The method presented in this work is similar, but demonstrates a better accuracy as θm = (m -1) N -1 , m = 1,..., N. Then K = N2. Calculations are given in table for 5 £ L/a £ 20 и N = 10, 20, 30. Electrical capacitance value form the reference book [14] are marked C0, discrepancy between calculated and table-based [14] capacitance value d = (C - C0)/C0. In fig. 2 the obtained results are presented graphically, the best results from the work [15] obtained under the it is shown in table and fig. 2. It is obvious that the offered model can be easily applied to nanopores of any form. Thus, the offered method showed the sufficient accu- racy and predictability. Therefore, its further research and development is necessary for both large number of ficti- tious charges leading to significant growth of matrix con- ditionality and for the case of rather close approximation of ions to a nanopore surface. The results of calculations of the electrical capacitance of a conducting nanopore L/a, relative units. N = 10 N = 20 N = 30 C0/2pe0a, [14], relative units. C/2pe0a, relative units. d, % C/2pe0a, relative units. d, % C/2pe0a, relative units. d, % 5 2.294 35.58 2.129 25.83 2.073 22.52 1.692 6 2.457 30.90 2.283 21.63 2.223 18.43 1.877 7 2.604 28.34 2.418 19.17 2.354 16.02 2.029 8 2.740 25.75 2.543 16.70 2.475 13.58 2.179 9 2.869 23.45 2.660 14.46 2.588 11.36 2.324 10 2.993 21.47 2.772 12.50 2.696 9.42 2.464 11 3.113 19.68 2.879 10.69 2.799 7.61 2.601 12 3.229 18.06 2.983 9.07 2.899 6.00 2.735 13 3.342 16.65 3.083 7.61 2.995 4.54 2.865 14 3.451 15.30 3.180 6.25 3.088 3.17 2.993 15 3.557 14.08 3.276 5.07 3.179 1.96 3.118 16 3.659 12.90 3.369 3.95 3.268 0.83 3.241 17 3.759 11.84 3.461 2.98 3.355 -0.18 3.361 18 3.855 10.78 3.551 2.04 3.441 -1.12 3.480 19 3.948 9.73 3.639 1.14 3.524 -2.06 3.598 20 4.038 8.75 3.726 0.35 3.607 -2.85 3.713 Fig. 2. Relationship between capacitance of a nanopore and the value of L/a for various numbers of fictitious charges. Solid line - tabulated data on capacitance [14] Рис. 2. Зависимость емкости нанопоры от величины L/a при различном количестве фиктивных зарядов. Сплошная жирная линия - табличные значения емкости [14]

About the authors

V. E. Zalizniak

Siberian Federal University

79, Svobodny Av., Krasnoyarsk, 660041, Russian Federation

O. A. Zolotov

Siberian Federal University

79, Svobodny Av., Krasnoyarsk, 660041, Russian Federation

O. P. Zolotova

Reshetnev Siberian State University of Science and Technology

Email: zolotova@sibsau.ru
31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation

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