Method for calculating two-body distribution functions, method for calculating two-body potentials, program for calculating two-body distribution functions, program for calculating two-body potentials, and particle observation device.

Abstract

This pair distribution function calculation method comprises: an observation step for observing a two-dimensional image of fine particles using a fine particle observation device; a first calculation step for calculating an apparent pair distribution function from the two-dimensional image observed in the observation step; and a second calculation step for inversely calculating a true pair distribution function from the apparent pair distribution function calculated in the first calculation step. This pair potential calculation method comprises: a third calculation step for calculating a total correlation function from the true pair distribution function calculated by the pair distribution function calculation method; a fourth calculation step for calculating a direct correlation function from the total correlation function calculated in the third calculation step; and a fifth calculation step for calculating a pair potential from the direct correlation function calculated in the fourth calculation step.

Description

[Technical Field]

[0001] The present invention relates to a method for calculating a two-body distribution function, a method for calculating a two-body potential, a program for calculating a two-body distribution function, a program for calculating a two-body potential, and a microparticle observation device. [Background technology]

[0002] For example, if fine particles of about 1 nm to 1,000 nm are dispersed in a dispersion medium (gas or liquid), the two-body distribution function and two-body potential can be determined from the scattering data using small-angle X-ray scattering (SAXS), small-angle neutron scattering (SANS), ultra-SAXS (USAXS), or ultra-SANS (USANS). For example, the method for analyzing the two-body distribution function and two-body potential disclosed in Non-Patent Literature 1 is applicable not only to SAXS but also to SANS, USAXS, and USANS. [Prior art documents] [Patent Documents]

[0003] [Non-Patent Document 1] Ken-ichi Amano, Ryosuke Sawazumi, Hiroshi Imamura, Tomonari Sumi, Kota Hashimoto, Kazuhiro Fukami, Haru Kitaoka, Naoya Nishi, and Tetsuo Sakka, "An Improved Model-potential-free Analysis of the Structure Factor Obtained from a Small-Angle Scattering: Acquisitions of the Pair Distribution Function and the Pair Potential", Chemistry Letters, 49 (2020) 1017-1021. [Overview of the project] [Problems that the invention aims to solve]

[0004] SAXS, SANS, USAXS, and USANS can be used to determine the two-body distribution function and two-body potential between microparticles dispersed in a dispersion medium. However, these instruments are relatively large and expensive, making them difficult to install and use in a typical research laboratory. Furthermore, even to use SAXS, SANS, USAXS, and USANS, which are available for shared use, it is necessary to apply for use, travel to the location where the instrument is located, and perform measurements within a limited number of days, which is relatively time-consuming. Moreover, when using shared instruments, it is difficult to conduct a large number of trial-and-error experiments due to the limitations on the number of days of use. This is inconvenient, for example, in the development of pharmaceuticals and functional materials, which require a lot of trial and error.

[0005] SAXS, SANS, USAXS, and USAXS each have limitations in their measurement range regarding the diameter of the microparticles. For example, the measurement range of SAXS and SANS is said to be around a few Å to 100 nm. Therefore, it is difficult to determine the two-body distribution function and two-body potential of microparticles with a diameter exceeding 100 nm. To avoid this problem, USAXS and USAXS are used. Using these instruments, the measurement range can be extended to approximately 1000 nm. However, the camera length of these instruments is very long, several tens of meters, and the number of instruments is limited, making them more difficult to use than SAXS and SANS. Furthermore, even with USAXS and USAXS, it is difficult to determine the two-body distribution function and two-body potential of microparticles with a diameter exceeding 1000 nm. Moreover, when the microparticles are mainly composed of organic molecules, the problem of measurement limits becomes even more pronounced. For example, if the diameter of microparticles composed of organic molecules is approximately 50 nm or more, measurement by small-angle scattering becomes extremely difficult.

[0006] SAXS, SANS, USAXS, and USANS can be used to determine the two-body distribution function and two-body potential of particles within the above-mentioned measurement range. When determining the two-body distribution function and two-body potential, the process involves deriving the structure factor from the scattering data (intensity factor), and then deriving the two-body distribution function and two-body potential from the structure factor. However, only the low-wavenumber structure factor can be used; in other words, the high-wavenumber structure factor cannot be used. Therefore, analyzing the two-body distribution function and two-body potential from SAXS, SANS, USAXS, and USANS scattering data is a poorly designed problem. For this reason, a model two-body potential is generally used when determining the two-body distribution function and two-body potential. However, the actual two-body potential between particles cannot be strictly represented by the model two-body potential. Therefore, the model two-body potential-free method is sometimes used in the analysis. However, if sufficient high-wavenumber structure factors are not obtained, the calculation convergence is unstable. Thus, there are various problems with calculating the two-body distribution function and two-body potential using SAXS, SANS, USAXS, and USANS.

[0007] This disclosure has been made in view of the above circumstances, and one of its purposes is to provide a method for calculating a two-body distribution function, a method for calculating a two-body potential that includes the same, a two-body distribution function calculation program, a two-body potential calculation program, and a microparticle observation device that can calculate two-body distribution functions and two-body potentials relatively easily. [Means for solving the problem]

[0008] A method for calculating a two-body distribution function according to one aspect of the present invention is a method for calculating a two-body distribution function of fine particles in a dispersion medium, comprising: an observation step of observing a two-dimensional image of the fine particles using a fine particle observation device; and calculating the apparent two-body distribution function from the two-dimensional image observed in the observation step. By computer A first calculation step to calculate, and the true two-body distribution function obtained from the apparent two-body distribution function calculated in the first calculation step. By the aforementioned computer It has a second calculation step that performs the reverse calculation.

[0009] In the above calculation method, the first calculation step may include: a reference image generation step of creating an artificial two-dimensional image in which a plurality of artificial particles that are fine particles of an ideal gas are randomly arranged on a screen having the same area as the two-dimensional image; an artificial particle density function generation step of extracting the distance between the center points of the artificial particles from the artificial two-dimensional image and generating a probability density function of the artificial particles (hereinafter referred to as the artificial probability density function); a real particle density function generation step of extracting the distance between the center points of the two-dimensional image and generating a probability density function of the fine particles (hereinafter referred to as the real probability density function); and a distribution function calculation step of dividing the real probability density function by the artificial probability density function to obtain the apparent two-body distribution function.

[0010] In the above calculation method, the second calculation step includes a step of preparing a forward calculation formula to derive a forward calculation formula for calculating the apparent two-body distribution function from the true two-body distribution function, wherein the two-dimensional image has a thickness of l in the observation direction of the two-dimensional image. z When the three-dimensional space is a planar image viewed from the observation direction, the distance from one of the microparticles (hereinafter referred to as "reference microparticles") located at a predetermined position on the two-dimensional image to a predetermined position v2 is defined as the two-dimensional distance r, and the predetermined position in the three-dimensional space corresponding to the predetermined position v2 on the two-dimensional image is defined as v3, the distance from the reference microparticle to the predetermined position v3 in the three-dimensional space is defined as the actual distance r0, the axis extending in the observation direction at the predetermined position v3 in the three-dimensional space is defined as the Z' axis, and the relative distance on the Z' axis from the reference microparticle to the predetermined position v3 is defined as z', then the apparent two-body distribution function is a function g of the two-dimensional distance r. a (r) is such that the true two-body distribution function is a function of the real distance r0 g(r0), and by geometrical relationship the real distance r0 is √(r 2 +z' 2 Since it can be expressed as ), the true two-body distribution function is g(√(r 2 +z' 2 The above forward calculation formula is expressed as the true two-body distribution function g(√(r 2 +z' 2 Regarding the variable z' in ), the thickness l along the Z' axis directionz It may include a single integral integrated over.

[0011] In the above calculation method, the second calculation step may further include an inverse calculation formula derivation step of deriving an inverse calculation formula for calculating the true two-body distribution function from the apparent two-body distribution function based on the forward calculation formula.

[0012] In the above calculation method, when it is assumed that the reference particle is arranged at a position (hereinafter, variable) z on the Z-axis parallel to the Z'-axis, the forward calculation formula includes the double integral integrated over the thickness l along the Z-axis direction with respect to the variable z z It may include a double integral integrated over.

[0013] In the above calculation method, the second calculation step includes an actual average residence time acquisition step of referring to the two-dimensional image and acquiring an actual average residence time which is an average time that a to-be-observed particle serving as a predetermined particle stays in the two-dimensional image, a diffusion coefficient calculation step of observing Brownian motion in the two-dimensional image and obtaining a diffusion coefficient of the particle, a virtual average residence time acquisition step of performing Brownian motion simulation of particles (hereinafter, simulation particles) inside a virtual three-dimensional space with a thickness (hereinafter, virtual thickness) s and obtaining a virtual average residence time which is an average time that the simulation particles stay in the virtual three-dimensional space by using the diffusion coefficient, a virtual thickness selection step of extracting the virtual thickness s in the virtual three-dimensional space for achieving the virtual average residence time that coincides with the actual average residence time, and a thickness substitution step of substituting the virtual thickness s extracted in the virtual thickness extraction step into the thickness (hereinafter, actual thickness) l in the forward calculation formula. z It may further include a thickness substitution step of substituting the virtual thickness s extracted in the virtual thickness extraction step into the thickness l.

[0014] The above calculation method may further include a light irradiation step of irradiating light from a light source toward the dispersion medium in a direction intersecting with an observation direction of the two-dimensional image before the observation step.

[0015] A method for calculating a two-body potential according to one aspect of the present invention comprises the above method for calculating a two-body distribution function and the total correlation function obtained from the true two-body distribution function calculated by the above method for calculating a two-body distribution function. By the aforementioned computer A third calculation step to perform calculations, and a direct correlation function from the total correlation function calculated in the third calculation step By the aforementioned computer A fourth calculation step involves calculating the two-body potential from the direct correlation function calculated in the fourth calculation step. By the aforementioned computer It comprises a fifth calculation step of performing calculations.

[0016] A two-body distribution function calculation program according to one aspect of the present invention causes a computer to function as a first calculation means for calculating an apparent two-body distribution function from a two-dimensional image of the fine particles in a dispersion medium, and a second calculation means for inversely calculating the true two-body distribution function from the apparent two-body distribution function calculated by the first calculation means.

[0017] A two-body potential calculation program according to one aspect of the present invention causes a computer to function as a first calculation means for calculating an apparent two-body distribution function from a two-dimensional image of the fine particles in a dispersion medium; a second calculation means for inversely calculating the true two-body distribution function from the apparent two-body distribution function calculated by the first calculation means; a third calculation means for calculating the total correlation function from the true two-body distribution function calculated by the second calculation means; a fourth calculation means for calculating a direct correlation function from the total correlation function calculated by the third calculation means; and a fifth calculation means for calculating the two-body potential from the direct correlation function calculated by the fourth calculation means.

[0018] A microparticle observation apparatus according to one aspect of the present invention comprises a light source that irradiates a dispersion medium containing microparticles with light, a two-dimensional image acquisition unit that observes and acquires a two-dimensional image of the microparticles from a direction intersecting the direction of light irradiation, and a calculation unit that calculates a two-body distribution function of the microparticles in the dispersion medium based on the two-dimensional image, wherein the calculation unit includes a first calculation unit that calculates an apparent two-body distribution function from the two-dimensional image, and a second calculation unit that inversely calculates the true two-body distribution function from the apparent two-body distribution function calculated by the first calculation unit.

[0019] In the above-described particle observation apparatus, the calculation unit may further include a third calculation unit that calculates the total correlation function from the true two-body distribution function calculated by the second calculation unit, a fourth calculation unit that calculates the direct correlation function from the total correlation function calculated by the third calculation unit, and a fifth calculation unit that calculates the two-body potential from the direct correlation function calculated by the fourth calculation unit. [Effects of the Invention]

[0020] According to the method for calculating a two-body distribution function, the method for calculating a two-body potential comprising the same, the two-body distribution function calculation program, the two-body potential calculation program, and the microparticle observation device described above, the two-body distribution function and the two-body potential can be calculated relatively easily. [Brief explanation of the drawing]

[0021] [Figure 1] This is a schematic diagram of the three-dimensional space that forms the basis of the two-dimensional image observed by the microparticle observation device using the calculation method of this embodiment. [Figure 2] This figure shows the two-dimensional image described above. [Figure 3] This figure shows the procedure for calculating the apparent two-body distribution function from the above two-dimensional image. [Figure 4] This diagram shows the procedure for deriving a sequential calculation formula from the above two-dimensional image, and shows a cylindrical space and a cylindrical microspace where the reference particle is located on the central axis. [Figure 5]This figure shows the procedure for deriving the above sequential calculation formula, and represents a small space that is a part of the cylindrical small space shown in Figure 4, specifically a part in the circumferential direction and the Z' axis direction. [Figure 6] This diagram shows the procedure for deriving the above sequential calculation formula, and illustrates a small space that is a part of the circumferential direction of the cylindrical small space shown in Figure 4. [Figure 7] This figure shows the XZ plane (cross-sectional view) where the center point of a reference particle on the Z axis in the three-dimensional space of Figure 1 exists. [Figure 8] Figure 1 shows the XZ plane (cross-sectional view) where the center point of a reference particle on the Z axis in three-dimensional space exists, and is a diagram used to explain the approximate sequential calculation formula. [Figure 9] This flowchart shows the procedure for determining the thickness lz of the three-dimensional space that forms the basis of the above two-dimensional image in the second calculation step. [Figure 10] This figure shows a schematic diagram of a particle observation device incorporating a program that performs the calculation method described above. [Figure 11] This is a diagram of an apparent two-body distribution function with a general shape. [Figure 12] This figure shows a true two-body distribution function with a general shape, and also demonstrates the accuracy of the inverse calculation. [Modes for carrying out the invention]

[0022] Preferred embodiments of the present invention will be described. The method for calculating the two-body distribution function of the present invention and Embodiment 1, which embodies the method for calculating the two-body distribution function, will be described with reference to the drawings.

[0023] <Embodiment 1> First, we will explain the procedure for determining the true two-body distribution function g(r0) between microparticles. Here, g(r0) is the radial distribution function, and since it is an isotropic function, the target function we are looking for can also be considered as g(r). Therefore, the true two-body distribution function will be denoted as g(r0) or g(r) as appropriate. Hereinafter, r is the distance between the centers of the microparticles (hereinafter referred to as the two-dimensional distance (see Figure 1)). Furthermore, g(r) is the distribution function normalized by the (number) density ρ of the microparticles in the bulk liquid, and ρg(r), which is simply the product of ρ and g(r), is the two-body density distribution function. Therefore, determining g(r) and determining ρg(r) are technically synonymous here. First, prepare a two-dimensional image (two-dimensional video or two-dimensional image) of the microparticles observed with a microparticle observation device (for example, an optical microscope, a liquid-phase electron microscope, an interference light microscope, a microscope that can observe scattered light or fluorescence from particles irradiated with laser light, etc.). If the two-dimensional image is a two-dimensional image, it is preferable to prepare multiple copies. Observation of a two-dimensional image of fine particles using a fine particle observation device corresponds to the "observation step" of the present invention. In this case, the fine particle observation device takes a certain thickness l z Assume that microparticles are observed in a space S (see Figure 1) with a certain thickness (actual thickness) l. z A device capable of observing microparticles existing in a space with a specific configuration is defined as a microparticle observation device. The incident light L (e.g., laser light) from the microparticle observation device is irradiated from the light source towards the bulk liquid (dispersion medium) in the direction of the X-axis (left-right direction in Figure 1), which is perpendicular to the observation direction D (light irradiation step). The axis perpendicular to the X-axis and parallel to the observation direction D is defined as the Z-axis, and the axis perpendicular to both the X-axis and Z-axis is defined as the Y-axis. The width dimension of the above space S in the X-axis direction is l x The depth dimension in the Y-axis direction is l y This means that, as shown in Figure 2, the two-dimensional image 10 is a planar image of the above space S as viewed from the observation direction D.

[0024] Using the two-dimensional image (two-dimensional video or two-dimensional image) 10 obtained from the microparticle observation device, the central position (center coordinates) of each microparticle can be determined, and the distance between the center points of each microparticle pair can be calculated. Using the distance between the center points of each microparticle pair, the apparent two-body distribution function g can be calculated. a(r) can be calculated. Here, the subscript 'a' means apparent. The apparent two-body distribution function g can be obtained from the two-dimensional image observed in the above observation step. a Calculating (r) corresponds to the "first calculation step" of the present invention. Then, in the calculation formula of the present invention, g a (r) and l z If you input this, you can find the true two-body distribution function g(r). Therefore, the "apparent two-body distribution function g" a (r)" refers to an extremely thin thickness l z This function represents the normalized number density (number density at the point of interest divided by the number density of particles in the bulk dispersion) of particles existing on the circumference of a circular disk of radius r when viewed from a planar direction D, and does not represent the probability of particle existence in a perfect two-dimensional space. Furthermore, the "true two-body distribution function g(r)" is a function that represents the normalized number density of particles existing on a spherical surface of radius r.

[0025] In the first calculation step described above, the following steps are specifically performed. First, the reference image generation step is performed. In this step, as shown in Figure 3, artificial particles P, which become ideal gas particles, are placed on a screen with the same area as the two-dimensional image 10 obtained from the microparticle observation device, with a screen of the same area as the two-dimensional image 10 obtained from the microparticle observation device. r Multiple artificial two-dimensional images 20 are created by randomly arranging artificial particles P. r The subscript 'r' signifies randomness.

[0026] Next, the artificial particle density function generation step is performed. In this step, artificial particles P are preferably involved. r From multiple artificial two-dimensional images 20 with different distributions, artificial particles P r The distance between the centers of the two artificial particles (distance between the centers of the artificial particles) r is extracted, and the artificial particle P r The probability density function of (hereinafter referred to as the artificial probability density function) q r (r) is generated. Then, the real particle density function generation step is performed to extract the distance between the center points of the microparticles P (two-dimensional center point distance) r from the two-dimensional image 10 and generate the probability density function of the microparticles P (hereinafter referred to as the real probability density function) q(r).

[0027] Finally, the distribution function calculation step is performed, and the real probability density function is normalized by dividing it by the artificial probability density function q(r) / q r (r) is the apparent two-body distribution function g a Let (r) be the case. The reason for normalizing the real probability density function is to take into account that the probability of finding particles P approaches zero at the edges of the two-dimensional image 10.

[0028] By the way, the apparent two-body distribution function g calculated in the first calculation step above a Inversely calculating the true two-body distribution function g(r) from (r) corresponds to the "second calculation step" of the present invention. The details of the calculation method of the present invention are described below.

[0029] Using a particle observation device, the apparent two-body distribution function g a (r) is obtained, and the apparent two-body distribution function g is already found. a Assuming (r) is available, the apparent two-body distribution function g a The method for obtaining the true two-body distribution function, the radial distribution function g(r), from (r) is explained below. From g(r) g a I have independently discovered that the formula (1) for finding (r) (which I call a sequential calculation formula) can be expressed as follows.

[0030]

number

[0031] By the way, the sequential calculation formula shown in equation (1) above can be derived, for example, by the following procedure. Returning to Figure 1, we assume that a single microparticle (hereinafter referred to as a reference particle) Pc located at a predetermined position in a three-dimensional space S is situated on the Z-axis, and we consider a cylindrical space Sc with radius r (i.e., two-dimensional distance r) centered on this Z-axis. Then, as shown in Figure 4, we focus on the number of particles dN(r) present in the cylindrical microspace Sco, which is located dr away from a predetermined position v2 at a distance r from the reference particle Pc on the two-dimensional image 10. Note that the predetermined position v2 at a distance r from the reference particle Pc on the two-dimensional image 10 corresponds to the predetermined position v3 at a distance r0 from the reference particle Pc in the three-dimensional space S.

[0032] To obtain the particle number dN(r), we first calculate the number of particles in a microspace Ss, which is part of a microspace Sco located outside the cylindrical space Sc with radius r, as shown in Figure 5. This microspace Ss is located outside the portion of the cylindrical space Sc that occupies an angle dφ around the Z axis. When a Z' axis parallel to the Z axis is defined at a distance r from the reference particle Pc, the height in the Z-axis direction of the microspace Ss is dz', and the volume of the microspace Ss is "dz'·dr·dφ·r[m 3 ]」. The number of particles present in this minute space Ss is the normalized number density g(r0) and the number density per unit volume in bulk liquid ρ0 [particles / m³]. 3 By multiplying by ], it is obtained as "dz'·dr·dφ·r·ρ0·g(r0)[particles]". And when the reference particle Pc is assumed to be at position z on the Z axis (hereinafter referred to as variable z) in the three-dimensional space S (see Figure 1) which is the basis of the two-dimensional image 10, then for variable z', from -z to l z Integrating down to -z gives the number of particles dN present in the space Ssz shown in Figure 6. R This was found, and this particle number dN R It can be expressed by the following equation (1-2).

number

[0033] And in equation (1-2) above, with respect to the radius r, as shown in Figure 7, r0 2 =r 2 +z' 2Since the above equation (1-2) holds true, the above equation (1-3) can be expressed by the following equation (1-3).

number

[0034] Then, assuming that the position z of the reference particle Pc can take any position on the Z axis, we take the average of the above equations (1-3) to obtain the following equation (1-4), which represents the average number of particles in the above space Ssz.

number

[0035] Next, in order to calculate the number of particles dN(r) in the cylindrical microspace Sco (see Figure 4), we integrate equation (1-4) over one full rotation with respect to the central angle φ to obtain the following equation (1-5).

number

[0036] On the other hand, the number of particles dN(r) in the microspace Sco (see Figure 4) can also be expressed by the following equation (1-6).

number

[0037] In equation (1-6), ``dφ·r·l'' z ·dr is the volume [m³] of space Ssz shown in Figure 6. 3 ] and "ρ0·g a (r) is the number density in the bulk, g a (r) times the original amount [pieces / m 3 ]

[0038] Then, from equations (1-5) and (1-6) above, the apparent two-body distribution function g(r0) is obtained from the true two-body distribution function g(r0). a The sequential calculation formula for equation (1) to calculate (r) is obtained.

[0039] By the way, as shown in Figure 8, the reference particle Pc is l z If we assume that it is located at position / 2, then it becomes unnecessary to integrate with respect to the above variable z in the reference particle Pc, and the forward calculation formula can be approximately (simplified) shown in the following equation (1-7). The approximate forward calculation formula shown in equation (1-7) is the true two-body distribution function g(r0), i.e., g(√(r 2 +z' 2 This includes the true two-body distribution function g(√(r 2 +z' 2 )) with respect to z', thickness l along the Z' axis. z -l z / 2 to +l z The single integral is performed up to / 2. Also, in equation (1-7), the right-hand side has "l z -1 The inclusion of "" is intended to eliminate dimensions.

number

[0040] Furthermore, the formulas obtained by multiplying the above formulas (1) and (1-7) by, for example, a predetermined correction coefficient may be used as the sequential calculation formulas.

[0041] If this sequential calculation formula (formula (1)) is correct, then the thickness l z When g is infinitesimally thin (almost 0), a The equation (r) = g(r) is obtained. In fact, l z If we consider it to be infinitesimally thin, then g aSince it is confirmed that (r) = g(r), this forward calculation formula (Equation (1)) is correct. In other words, when the thickness l of the two-dimensional image 10 actually observed by the particle observation device is sufficiently thin, g z can be approximated as g a (r) ≈ g(r). By the way, the goal (invention) here is not to obtain g a (r) from g(r), but to obtain g(r) from g a (r). Therefore, consider dividing the width from 0 to l z into n parts and performing the integration by the trapezoidal rule. Also, define r i = i(l z / n) (where i is an integer). As a result, the inventor found that by multiplying both sides of the following matrix equation (Equation (2)) by the inverse matrix of the n×n square matrix in the matrix equation, g a (r) can be obtained from g(r).

[0042]

Number

[0043] Here, f(i, j) is a function represented by the following case division. i and j are integers from 1 to n, and f(i, j) is represented by Equation (3) when i = j and i ≠ n, Equation (4) when j = n and i ≠ n, Equation (5) when i = j = n, and Equation (6) when i ≠ j and i ≠ n and j ≠ n and i < j.

[0044]

Number

[0045]

Number

[0046]

Number

[0047]

Number

[0048] The notation in equations (2) through (6) is merely one example of an inverse calculation formula, and the notation may differ slightly depending on how the strips are positioned in the Riemann sum method and how the area of ​​the strips is calculated. Note that if the inverse calculation formula is correct, it is possible to repeat the forward and inverse calculations. That is, input g(r) into the forward calculation formula (equation (1)) and g a (r) obtained, and g a If inputting (r) into the inverse calculation formula yields the original g(r), then the inverse calculation formula can be said to be correct. This has already been verified, and the results of the verification will be shown in the example described later.

[0049] Furthermore, the "second calculation step" of the present invention, that is, the apparent two-body distribution function g calculated in the first calculation step described above. a Inversely calculating the true two-body distribution function g(r) from (r) includes methods that combine the forward calculation formula in equation (1) with repeated trial and error by a computer, numerical optimization methods, machine learning, etc., without using the method of obtaining the inverse analytical solution from equation (2) to equation (6). For example, substituting the "trial function of g(r)" into equation (1) gives "g derived from the trial function". a (r) is calculated, and that is the g derived from the microparticle observation obtained in the first calculation step. a If it matches (r), the "trial function of g(r)" obtained by substitution can also be taken as a solution.

[0050] By the way, let's assume that the thickness of the incident light L (laser light, etc.) used in the particle observation device is the thickness of the three-dimensional space mentioned above. z If it is unknown, in the forward and reverse calculation formulas above, l z Since the thickness l in three-dimensional space remains an unknown, it is not possible to calculate the true two-body distribution function. Therefore, in such cases, the thickness l in three-dimensional space can be calculated using the following method. z The thickness l is determined. That is, in the second calculation step described above, the following steps are performed as shown in Figure 9, and the thickness l z To decide.

[0051] First, a two-dimensional image preparation step ST1 is performed, in which multiple two-dimensional images 10 are prepared at predetermined time intervals using a microparticle observation device. In this step ST1, for example, a still image two-dimensional image 10 is prepared at predetermined time intervals, or multiple two-dimensional images 10 are prepared by extracting frames from a video at predetermined time intervals. Then, a real average residence time acquisition step ST2 is performed, in which the real average residence time t, which is the average time that multiple microparticles (hereinafter referred to as observed microparticles) stay in the two-dimensional image 10, is obtained by referring to the multiple two-dimensional images 10. That is, the average time from when the observed microparticles appear on the two-dimensional image 10 until they disappear is obtained. The state of "the observed microparticles disappearing from the two-dimensional image 10" represents the state of "the observed microparticles moving outside of three-dimensional space in the Z-axis direction." If it is assumed that the observed microparticles have escaped to the outer frame of the observation screen, those observed microparticles are not used in the calculation of the real average residence time. Then, a diffusion coefficient calculation step ST3 is performed, in which the Brownian motion in the two-dimensional images 10 is observed by referring to the multiple two-dimensional images 10, and the diffusion coefficient of the microparticles is determined.

[0052] Next, step ST4 is performed to obtain the virtual average residence time using the calculated diffusion coefficient. In step ST4, Brownian motion simulation of microparticles (hereinafter referred to as simulation particles) is performed inside a virtual three-dimensional space of a predetermined thickness s (a specific numerical value such as 1 μm or 2 μm), and the virtual average residence time t', which is the average time that these simulation particles stay in the virtual three-dimensional space, is obtained. Hereafter, this thickness s will be defined as the virtual thickness s.

[0053] Then, a virtual thickness selection step ST5 is executed to extract a virtual thickness s in a virtual three-dimensional space that achieves a virtual average stay time t' that matches the actual average stay time t. In the virtual thickness selection step ST5 of this embodiment, a determination step ST51 is executed to determine whether the actual average stay time t and the virtual average stay time t' match. If the actual average stay time t and the virtual average stay time t' match, the process proceeds to the thickness substitution step ST6. On the other hand, if the actual average stay time t and the virtual average stay time t' do not match in the determination step ST51, a virtual thickness change step ST52 is executed to change the set value of the virtual thickness s, and then the process returns to the virtual average stay time acquisition step ST4. In the thickness substitution step ST6, the thickness l of the forward calculation formula is used. z Substitute the virtual thickness s into the equation. That is, the virtual average residence time t' in the virtual three-dimensional space for the virtual thickness s assumed in the simulation is equal to the actual thickness l z If the virtual thickness s in the virtual three-dimensional space matches the actual average dwell time t in the three-dimensional space, then the virtual thickness l in the virtual three-dimensional space is equal to the actual thickness l in the three-dimensional space. z It can be assumed that this matches, and in the thickness determination step, the thickness l of the sequential calculation formula z This process is performed because it is possible to substitute a virtual thickness s into it.

[0054] By the way, the virtual thickness selection step ST5 is not limited to the present embodiment. For example, the results of simulations with various combinations of diffusion coefficient and virtual thickness s may be prepared in advance as data such as a matrix table, and the virtual thickness s that achieve a virtual average stay time t' that matches the actual average stay time t may be extracted based on this data. Alternatively, the above data may be used to train artificial intelligence to estimate the virtual thickness s that achieve a virtual average stay time t' that matches the actual average stay time t.

[0055] Next, the method for determining the two-body potential (average force potential) u(r) between microparticles is shown below. Note that the "method for calculating the two-body potential" of the present invention is applicable only when u(r) is calculated using the two-body distribution function g(r) obtained from the "method for calculating the two-body distribution function" of the present invention. First, prepare g(r) using the method for calculating the two-body potential of the present invention. Then, find the total correlation function h(r) using the following equation (7). Calculating the total correlation function h(r) from the true two-body distribution function g(r) calculated using the above method for calculating the two-body distribution function corresponds to the "third calculation step" of the present invention.

[0056]

number

[0057] Then, the direct correlation function c(r) is calculated using the following equation (8) (a variation of the single-component Ornstein-Zernike equation [JPHansen and IRMcDonald, Theory of Simple Liquids (4th edition), Academic Press]). Calculating the direct correlation function c(r) from the total correlation function h(r) calculated in the third calculation step above corresponds to the "fourth calculation step" of this invention.

[0058]

number

[0059] Here, F +1 This is the Fourier transform in spherical polar coordinates, F -1 This represents the inverse Fourier transform in spherical polar coordinates. Finally, u(r) is obtained from the following equation (9) (an equation combining the hypernetted-chain equation and the bridge function b(h(r),c(r))). Calculating the two-body potential u(r) from the direct correlation function c(r) calculated in the fourth calculation step above corresponds to the "fifth calculation step" of the present invention.

[0060]

Number

[0061] Here, k B is the Boltzmann constant, and T is the absolute temperature. When the particles are sufficiently dilute, in Equation (9), setting h - c + b = 0 allows for a simple calculation of the two-body potential. Additionally, if b = 0 in Equation (9), the equation is called the Hypernetted-chain equation. Generally, if there is no particular requirement, b = 0 can be assumed. Note that several bridge functions have been proposed and published, and one should determine which bridge function to use according to the conditions of the dispersion system. For example, when the particles can be approximated as rigid spheres, the Verlet bridge function shown in Equation (10) [L. Verlet, Mol. Phys., 41(1980)183] can be used.

[0062]

Number

[0063] The calculation method of the two-body potential of the present invention is applicable only when u(r) is calculated using the two-body distribution function g(r) obtained from the "calculation method of the two-body distribution function" of the present invention.<>

[0064] The present invention is not limited to the embodiments described above and shown in the drawings. For example, the following embodiments are also included in the technical scope of the present invention. Recommended embodiments are also described below.

[0065] When obtaining a two-dimensional image (two-dimensional video or two-dimensional image) using a particle observation device, when changes in the state of the sample such as evaporation or deterioration of the observation sample can be ignored, the longer the time of the two-dimensional video, the better, and the more the number of two-dimensional images, the better.

[0066] The diameter of the particles observed using the particle observation device is approximately within the range specified or favored by the particle observation device itself.

[0067] It is preferable that the type of microparticle observed using the microparticle observation device be of a single type. However, when observing fluorescent microparticles, the type of fluorescent microparticle observed can be intentionally selected by choosing the wavelength of the irradiated laser light; therefore, in a colloidal dispersion containing fluorescent microparticles, it is not necessary for the type of microparticle to be single.

[0068] It is preferable that the diameter of the microparticles observed using the microparticle observation device is monodisperse. If there are multiple types of microparticles, it is preferable that the diameter of each type of microparticle is monodisperse. However, if there is only one type of microparticle being observed, the diameter may be polydisperse. When the diameter of the microparticles is polydisperse, the present invention is used to obtain a single two-body distribution function by averaging the two-body distribution functions of various diameter pairs (there are multiple two-body distribution functions for each pair).

[0069] The dispersion medium is preferably a gas or liquid. However, the dispersion medium may be a gel, glass, or solid. Even if the dispersion medium is a gel, glass, or solid, as long as the particle observation device can observe the dispersed particles, the apparent two-body distribution function g a (r) can be found, and its g a It is possible to input (r) into the calculation formula for the two-body distribution function of the present invention to obtain the true two-body distribution function (radial distribution function) g(r) in gels, glasses, and solids. For example, by using a cryo-electron microscope, fine particles dispersed in glass can be observed, and g can be obtained from the observation results. a It is thought that g(r) can be obtained by inputting (r) into the calculation formula of the present invention. However, when the two-body potential u(r) is determined from the g(r) of fine particles obtained in gel, glass, or solid, a meaningful u(r) is not always obtained.

[0070] Next, we will briefly describe the configuration of the particle observation device 100, which incorporates a two-body distribution function calculation program that performs the calculation of the two-body distribution function described above, and a two-body potential calculation program that performs the calculation of the two-body potential. As shown in FIG. 10, the fine particle observation apparatus 100 has a thickness l in the Z-axis direction, which is the height direction, of a dispersion medium containing fine particles z of the light source 105 that irradiates the light L, the two-dimensional image acquisition unit (camera) 110 that observes and acquires the two-dimensional image of the fine particles, and the operation unit 120 that calculates the pair distribution function of the fine particles P in the dispersion medium M based on the two-dimensional image 10 obtained by the two-dimensional image acquisition unit 110. The operation unit 120 is composed of a processor or the like, and includes a first calculation unit 121 that calculates the apparent pair distribution function g a (r), and a second calculation unit 122 that inversely calculates the true pair distribution function g(r) from the apparent pair distribution function g a (r) calculated by the first calculation unit 121. Here, the operation unit 120 may further include a third calculation unit 123 that calculates the total correlation function h(r) from the true pair distribution function g(r) calculated by the second calculation unit 122, a fourth calculation unit 124 that calculates the direct correlation function c(r) from the total correlation function h(r) calculated by the third calculation unit 123, and a fifth calculation unit 125 that calculates the pair potential u(r) from the direct correlation function c(r) calculated by the fourth calculation unit 124. By incorporating the pair distribution function calculation program and the pair potential calculation program into the fine particle observation apparatus 100 in this way, the pair distribution function g(r) and the pair potential u(r) can be obtained only from the fine particle observation apparatus 100.

Example

[0071] Next, an example of the forward calculation and the inverse calculation of the present invention will be described with reference to the drawings. FIG. 11 shows the apparent pair distribution function g that is generally assumed to be obtained from a fine particle observation apparatus a(r) is obtained by substituting the Benchmark curve (two-body distribution function g(r) with a general shape) shown in Figure 12 into the forward calculation formula (Equation (1)). Figure 12 also shows the two-body distribution function obtained using the calculation method of the two-body distribution function of the present invention, which is shown as "Result from the inverse calculation". Since the Benchmark curve and the curve shown as "Result from the inverse calculation" are identical, it can be confirmed that the calculation method of the two-body distribution function of the present invention is correctly performing the inverse calculation (functioning).

[0072] Note that when the distance between the centers of the particles is less than the diameter of the particles, it means that the particles overlap, and therefore, the two-body distribution function is generally zero when the distance between the centers is less than the diameter of the particles. However, in Figure 11, which represents the apparent two-body distribution function, there is a significant g value that is not zero even in the near-field region (around zero on the horizontal axis in Figure 11) where the distance between the centers is less than the diameter. a The value of (r) can be seen on the vertical axis. The reason for this is easier to understand by looking at Figure 1. When observing the microparticles from the observation direction shown in Figure 1, the microparticles may appear to overlap. This is also true when viewing from directly below, opposite to the observation direction shown in Figure 1. As a result, in Figure 11, a significant value that is not zero can be seen on the vertical axis even in the short-range region less than the diameter. The same thing can be seen in Figure 1(b) of the non-patent document [A.Toyotama, T.Okuzono, J.Yamanaka, Scientific Reports, 6(2016)23292].

[0073] It should be noted that the embodiments disclosed herein are illustrative and not restrictive in all respects. The scope of the present invention is not limited to the embodiments disclosed herein, and is intended to include all modifications within the scope set forth in the claims or equivalents thereof. [Industrial applicability]

[0074] According to the present invention, the method for calculating a two-body distribution function, the method for calculating a two-body potential incorporating the same, the two-body distribution function calculation program, the two-body potential calculation program, and the microparticle observation device, the two-body distribution function and the two-body potential can be calculated relatively easily. [Explanation of Symbols]

[0075] 10 2D image 20 Artificial two-dimensional image Pc fine particles (reference fine particles) L (incident) light 100 Particle Observation Device 105 Light source 110 2D image acquisition unit 120 Arithmetic section 121 1st calculation section 122 2nd calculation section 123 3rd Calculation Department 124 4th calculation section 125 5th Calculation Department

Claims

1. A method for calculating the two-body distribution function of fine particles in a dispersion medium, An observation step of observing a two-dimensional image of the fine particles using a fine particle observation device, A first calculation step involves calculating the apparent two-body distribution function from the two-dimensional image observed in the observation step using a computer. A second calculation step involves the computer inversely calculating the true two-body distribution function from the apparent two-body distribution function calculated in the first calculation step, A method for calculating the two-body distribution function having [a certain property].

2. The first calculation step is, A reference image generation step involves creating an artificial two-dimensional image in which multiple artificial particles, which are fine particles of an ideal gas, are randomly arranged on a screen with the same area as the aforementioned two-dimensional image. An artificial particle density function generation step involves extracting the distance between the center points of the artificial particles from the artificial two-dimensional image, which is the distance between the center points of the artificial particles, and generating a probability density function of the artificial particles (hereinafter referred to as the artificial probability density function), A real particle density function generation step involves extracting the two-dimensional distance between the center points of the fine particles from the two-dimensional image and generating a probability density function of the fine particles (hereinafter referred to as the real probability density function), A distribution function calculation step in which the apparent two-body distribution function is obtained by dividing the real probability density function by the artificial probability density function, A method for calculating the two-body distribution function according to claim 1, comprising:

3. The second calculation step described above is: The process includes a step of preparing a sequential formula to derive a sequential formula for calculating the apparent two-body distribution function from the true two-body distribution function, The two-dimensional image has a thickness of l in the observation direction of the two-dimensional image. ｚ In the case where the three-dimensional space is a planar image viewed from the direction of observation, From one of the aforementioned microparticles (hereinafter referred to as the reference microparticle) located at a predetermined position on the two-dimensional image, to a predetermined position v ２ Let the distance to be the two-dimensional distance r. The predetermined position v on the two-dimensional image ２ v ３ When defining this, the distance from the reference particle in the three-dimensional space to the predetermined position v ３ The distance to the actual distance r ０ year, In the three-dimensional space, the predetermined position v ３ The axis extending in the observation direction is defined as the Z' axis, and the reference particle is located at the predetermined position v ３ When z' is the relative distance along the Z' axis up to the aforementioned point, The apparent two-body distribution function is a function g ａ (r) of the two-dimensional distance r, and The true two-body distribution function is given by the actual distance r. ０ The function g(r ０ ) and The actual distance r is determined by the geometric relationship. ０ is √(r ２ +z' ２ Since it can be expressed as ), the true two-body distribution function is g(√(r ２ +z' ２ It is represented as )) and The aforementioned forward calculation formula is the true two-body distribution function g(√(r) ２ +z' ２ Regarding the variable z' in ), the thickness l along the Z' axis direction ｚ A method for calculating a two-body distribution function according to claim 1, which includes a single integral integrated over a certain distance.

4. The method for calculating a two-body distribution function according to claim 3, wherein the second calculation step further comprises a step of deriving an inverse formula for calculating the true two-body distribution function from the apparent two-body distribution function based on the forward calculation formula.

5. Assuming that the reference particles are positioned on the Z-axis parallel to the Z' axis (hereinafter referred to as variable) z, the forward calculation formula is the single integral of the thickness l along the Z-axis direction with respect to variable z. ｚ A method for calculating a two-body distribution function according to claim 3, which includes a double integral integrated over 2.

6. The second calculation step described above is: A step of obtaining the actual average residence time, which is the average time that the observed particles, which are predetermined fine particles, stay in the two-dimensional image by referring to the two-dimensional image, A diffusion coefficient calculation step involves observing the Brownian motion in the two-dimensional image and determining the diffusion coefficient of the fine particles, A virtual average residence time acquisition step involves performing a Brownian motion simulation of microparticles (hereinafter referred to as simulation particles) inside a virtual three-dimensional space of thickness (hereinafter referred to as virtual thickness) s using the aforementioned diffusion coefficient, and obtaining a virtual average residence time, which is the average time the simulation particles stay in the virtual three-dimensional space. A virtual thickness selection step for extracting the virtual thickness s in the virtual three-dimensional space in order to achieve the virtual average stay time which matches the actual average stay time, The thickness (hereinafter referred to as actual thickness) l in the above sequential calculation formula ｚ The process includes a thickness substitution step in which the virtual thickness s extracted in the virtual thickness extraction step is substituted, A method for calculating a two-body distribution function according to any one of claims 3 to 5, further comprising the above.

7. A method for calculating a two-body distribution function according to any one of claims 1 to 5, further comprising a light irradiation step of irradiating the dispersion medium with light from a light source in a direction intersecting the observation direction of the two-dimensional image, prior to the observation step.

8. A method for calculating a two-body distribution function according to any one of claims 1 to 5, A third calculation step in which the computer calculates the total correlation function from the true two-body distribution function calculated using the method for calculating the two-body distribution function, A fourth calculation step in which the computer directly calculates a correlation function from the total correlation function calculated in the third calculation step, A fifth calculation step involves calculating the two-body potential using the computer from the direct correlation function calculated in the fourth calculation step, A method for calculating two-body potentials that include [a specific feature / condition].

9. To calculate the two-body distribution function of fine particles in a dispersion medium, a computer is used. A first calculation means for calculating an apparent two-body distribution function from a two-dimensional image of the fine particles, and A second calculation means that inversely calculates the true two-body distribution function from the apparent two-body distribution function calculated by the first calculation means, A program for calculating two-body distribution functions to function as such.

10. To calculate the two-body potential of fine particles in a dispersion medium, a computer was used. A first calculation means for calculating an apparent two-body distribution function from a two-dimensional image of the fine particles, and A second calculation means that inversely calculates the true two-body distribution function from the apparent two-body distribution function calculated by the first calculation means, A third calculation means calculates the total correlation function from the true two-body distribution function calculated by the second calculation means. A fourth calculation means that directly calculates the correlation function from the total correlation function using the third calculation means, and A fifth calculation means calculates the two-body potential from the direct correlation function calculated by the fourth calculation means. A two-body potential calculation program designed to function as such.

11. A light source that irradiates light onto a dispersion medium containing fine particles, A two-dimensional image acquisition unit that observes and acquires a two-dimensional image of the fine particles from a direction intersecting the direction of light irradiation, A calculation unit that calculates the two-body distribution function of the fine particles in the dispersion medium based on the two-dimensional image, Equipped with, The aforementioned arithmetic unit, A first calculation unit that calculates the apparent two-body distribution function from the two-dimensional image, A second calculation unit inversely calculates the true two-body distribution function from the apparent two-body distribution function calculated by the first calculation unit, A microparticle observation device having the following features.

12. The aforementioned arithmetic unit, A third calculation unit calculates the total correlation function from the true two-body distribution function calculated by the second calculation unit, A fourth calculation unit calculates a direct correlation function from the total correlation function calculated by the third calculation unit, A fifth calculation unit calculates the two-body potential from the direct correlation function calculated by the fourth calculation unit, The particulate matter observation apparatus according to claim 11, further comprising:

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