Pair distribution function calculation method, pair potential calculation method, pair distribution function calculation program, pair potential calculation program, and fine 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

Method for calculating pair distribution function, method for calculating pair potential, program for calculating pair distribution function, program for calculating pair potential, and particle observation device

[0001] The present invention relates to a method for calculating a pair distribution function, a method for calculating a pair potential, a pair distribution function calculation program, a pair potential calculation program, and a particle observation apparatus.

[0002] For example, when fine particles of about 1 nm to 1,000 nm are dispersed in a dispersion medium (gas or liquid), the pair distribution function and pair potential can be obtained 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 analysis method of the pair distribution function and pair potential disclosed in Non-Patent Document 1 can be applied not only to SAXS but also to SANS, USAXS, and USANS.

[0003] 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.

[0004] SAXS, SANS, USAXS, and USANS can be used to determine the pair distribution function and pair potential between particles dispersed in a dispersion medium. However, these instruments are relatively large and expensive, making them difficult to install and use in a typical laboratory. Furthermore, even if you use a shared SAXS, SANS, USAXS, or USANS instrument, you must apply for use, travel to the instrument's location, and perform measurements within a limited number of days, which is relatively time-consuming. Furthermore, when using a shared instrument, the limited number of days of use makes it difficult to conduct a large number of trial-and-error experiments. This is inconvenient, for example, for the development of pharmaceuticals and functional materials, which require a lot of trial and error.

[0005] SAXS, SANS, USAXS, and USANS each have a measurement range limitation related to the diameter of particles. For example, the measurement range of SAXS and SANS is said to be from a few angstroms to approximately 100 nm. Therefore, it is difficult to calculate the pair distribution function and pair potential of particles with diameters exceeding 100 nm. To avoid this problem, USAXS and USANS are used. These instruments can extend the measurement range to approximately 1000 nm. However, the camera length of these instruments is very long, at 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 USANS, it is difficult to calculate the pair distribution function and pair potential of particles with diameters exceeding 1000 nm. Furthermore, the measurement limit problem becomes even more pronounced when the particles are primarily composed of organic molecules. For example, if the diameter of a particle made of organic molecules is approximately 50 nm or more, measurement using small-angle scattering is extremely difficult.

[0006] Using SAXS, SANS, USAXS, and USANS, it is possible to calculate the pair distribution function and pair potential for particles with diameters within the above measurement range. When calculating the pair distribution function and pair potential, a process is used in which structure factors are calculated from scattering data (intensity factors), and then the pair distribution function and pair potential are calculated from the structure factors. However, only structure factors on the low-wavenumber side can be used; in other words, structure factors on the high-wavenumber side cannot be used. In other words, the analysis of calculating the pair distribution function and pair potential from scattering data using SAXS, SANS, USAXS, and USANS is an ill-posed problem. Therefore, when calculating the pair distribution function and pair potential, a model pair potential is generally used. However, the pair potential between actual particles cannot be strictly expressed by the model pair potential. Therefore, the model pair potential-free method is sometimes used for analysis. However, if sufficient structure factors on the high-wavenumber side are not obtained, the convergence of the calculation is unstable. As described above, there are various problems in the calculation of pair distribution functions and pair potentials using SAXS, SANS, USAXS, and USANS.

[0007] The present disclosure has been made in consideration of the above-described circumstances, and has as an object to provide a method for calculating a pair distribution function that allows for relatively easy calculation of a pair distribution function or a pair potential, a method for calculating a pair potential that includes the same, a pair distribution function calculation program, a pair potential calculation program, and a particle observation apparatus.

[0008] A method for calculating a pair distribution function according to one aspect of the present invention is a method for calculating a pair distribution function of a particle in a dispersion medium, and includes an observation step of observing a two-dimensional image of the particle using a particle observation device, a first calculation step of calculating an apparent pair distribution function from the two-dimensional image observed in the observation step, and a second calculation step of inversely calculating a true pair distribution function from the apparent pair distribution function calculated in the first calculation step.

[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 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 an artificial particle center-to-center distance that is the distance between the centers of the artificial particles from the artificial two-dimensional image and generating a probability density function of the artificial particles (hereinafter referred to as artificial probability density function); a real particle density function generation step of extracting a two-dimensional center-to-center distance that is the distance between the centers of the particles from the two-dimensional image and generating a probability density function of the particles (hereinafter referred to as 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 pair distribution function.

[0010] In the above calculation method, the second calculation step includes a forward calculation formula preparation step of deriving a forward calculation formula for calculating the apparent pair distribution function from the true pair distribution function, and 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, a predetermined position v is measured from one of the particles (hereinafter, a reference particle) present at a predetermined position on the two-dimensional image. ２ The distance to the predetermined position v on the two-dimensional image is defined as a two-dimensional distance r. ２ The predetermined position in the three-dimensional space corresponding to v ３ When defining the above, the predetermined position v ３ The distance to the actual distance r ０ and the predetermined position v ３ The axis extending in the observation direction is defined as the Z' axis, and the distance from the reference particle to the predetermined position v ３ When the relative distance on the Z' axis to ａ (r), and the true pair distribution function is ０ Function g(r ０ ) and the actual distance r ０ is √(r ２ +z´ ２ ), the true pair distribution function can be expressed as g(√(r２ +z´ ２ )), and the forward calculation formula is expressed as the true pair distribution function g(√(r ２ +z´ ２ ) for the variable z' in the Z'-axis direction, ｚ 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 pair distribution function from the apparent pair distribution function based on the forward calculation formula.

[0012] In the above calculation method, when the reference particle is arranged at a position (hereinafter, variable) z on the Z axis parallel to the Z' axis, the forward calculation formula is calculated by multiplying the single integral by the thickness l along the Z axis direction with respect to the variable z. ｚ 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 the average time that a particle to be observed, which will be the predetermined particle, stays in the two-dimensional image; a diffusion coefficient calculation step of observing Brownian motion in the two-dimensional image and determining a diffusion coefficient of the particle; a virtual average residence time acquisition step of performing Brownian motion simulation of a particle (hereinafter, simulated particle) inside a virtual three-dimensional space of thickness (hereinafter, virtual thickness) s using the diffusion coefficient to acquire a virtual average residence time, which is the average time that the simulated particle stays in the virtual three-dimensional space; a virtual thickness selection step of extracting the virtual thickness s in the virtual three-dimensional space to achieve the virtual average residence time that coincides with the actual average residence time; and a method of calculating the thickness (hereinafter, actual thickness) l in the forward calculation formula. ｚ and a thickness substituting step of substituting the virtual thickness s extracted in the virtual thickness extracting step into the above-mentioned thickness substituting step.

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

[0015] A method for calculating a pair-body potential according to one aspect of the present invention includes the above-described method for calculating a pair distribution function, a third calculation step of calculating a total correlation function from the true pair-body distribution function calculated by the method for calculating the pair distribution function, a fourth calculation step of calculating a direct correlation function from the total correlation function calculated in the third calculation step, and a fifth calculation step of calculating a pair-body potential from the direct correlation function calculated in the fourth calculation step.

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

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

[0018] A particle observation device according to one aspect of the present invention includes a light source that irradiates light onto a dispersion medium containing particles; a two-dimensional image acquisition unit that observes and acquires a two-dimensional image of the particles from a direction intersecting the direction of the light irradiation; and a calculation unit that calculates a pair distribution function of the particles in the dispersion medium based on the two-dimensional image, wherein the calculation unit has a first calculation unit that calculates an apparent pair distribution function from the two-dimensional image, and a second calculation unit that inversely calculates a true pair distribution function from the apparent pair 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 a total correlation function from the true pair-body distribution function calculated by the second calculation unit, a fourth calculation unit that calculates a direct correlation function from the total correlation function calculated by the third calculation unit, and a fifth calculation unit that calculates a pair-body potential from the direct correlation function calculated by the fourth calculation unit.

[0020] According to the above-described aspects of the method for calculating a pair distribution function, the method for calculating a pair potential including the same, the pair distribution function calculation program, the pair potential calculation program, and the particle observation device, it is possible to calculate the pair distribution function and the pair potential relatively easily.

[0021] 4 is a schematic diagram of a three-dimensional space that is the basis of a two-dimensional image observed by a particle observation device using the calculation method of this embodiment. FIG. 4 is a diagram showing the two-dimensional image. FIG. 5 is a diagram showing a procedure for calculating an apparent pair distribution function from the two-dimensional image. FIG. 6 is a diagram showing a procedure for deriving a forward calculation formula from the two-dimensional image, showing a columnar space and a cylindrical minute space in which a reference particle is located on the central axis. FIG. 7 is a diagram showing a procedure for deriving the forward calculation formula, showing a minute space that is a part of the cylindrical minute space shown in FIG. 4 in the circumferential direction and in the Z'-axis direction. FIG. 8 is a diagram showing a procedure for deriving the forward calculation formula, showing a minute space that is a part of the cylindrical minute space shown in FIG. 4 in the circumferential direction. FIG. 9 is a diagram showing an XZ plane (cross-sectional image) in which the center point of a reference particle on the Z axis in the three-dimensional space of FIG. 1 exists. FIG. 10 is a diagram showing an XZ plane (cross-sectional image) in which the center point of a reference particle on the Z axis in the three-dimensional space of FIG. 1 exists, and is a diagram for explaining an approximate forward calculation formula. ｚ Fig. 1 is a flowchart showing a procedure for determining the above-mentioned calculation method. Fig. 2 is a diagram showing an outline of a particle observation device incorporating a program for executing the above-mentioned calculation method. Fig. 3 is a diagram showing an apparent pair distribution function with a general shape. Fig. 4 is a diagram showing a true pair distribution function with a general shape, which also shows the accuracy of the inverse calculation.

[0022] A preferred embodiment of the present invention will be described below. A method for calculating a pair distribution function and a first embodiment of the method for calculating a pair distribution function of the present invention will be described with reference to the drawings.

[0023] First, the true particle-to-particle pair distribution function g(r ０ Here, we will explain the procedure for finding g(r ０ ) is a radial distribution function, and is an isotropic function, so the target function to be sought can also be considered as g(r). Therefore, the true pair distribution function is g(r ０ ) or g(r) as appropriate. Here, r is the center-to-center distance of the microparticles (hereinafter referred to as two-dimensional distance (see Figure 1)). Furthermore, g(r) is a distribution function normalized by the (number) density ρ of the microparticles in the bulk liquid, and ρg(r), which is the simple multiplication of ρ and g(r), is the two-body density distribution function. Therefore, here, finding g(r) and finding ρg(r) are technically equivalent. First, a two-dimensional image (two-dimensional video or two-dimensional image) of a microparticle observed with a microparticle observation device (for example, an optical microscope, a submerged electron microscope, an interference microscope, a microscope capable of observing scattered light or fluorescence from particles irradiated with laser light, etc.) is prepared. When the two-dimensional image is a two-dimensional image, it is preferable to prepare multiple images. Observing the two-dimensional image of a microparticle using a microparticle observation device corresponds to the "observation step" of the present invention. In this case, the microparticle observation device is used to obtain a particle having a certain thickness l ｚ In other words, a particle existing in a space S (see Figure 1) with a certain thickness (actual thickness) l is observed. ｚ A device capable of observing particles present in a space having a given shape is defined as a particle observation device. Incident light L (e.g., laser light) from the particle observation device is irradiated from a light source toward a bulk liquid (dispersion medium) in the direction of the X axis (left-right direction on the paper in FIG. 1) that intersects (orthogonal) with the observation direction D (light irradiation step). The axis orthogonal to the X axis and parallel to the observation direction D is defined as the Z axis, and the axis orthogonal to the X and Z axes is defined as the Y axis. The width dimension of the space S in the X axis direction is l ｘ , the depth dimension in the Y-axis direction is l ｙ That is, the two-dimensional image 10 is a planar image of the space S as viewed from the observation direction D, as shown in FIG.

[0024] By using a two-dimensional image (two-dimensional video or two-dimensional image) 10 obtained from the particle observation device to determine the center position (center coordinates) of each particle, the distance between the center points of each particle pair can be calculated. Using the distance between the center points of each particle pair, the apparent pair distribution function g ａ (r) can be calculated. Here, the subscript a means apparent. The apparent pair distribution function g ａ Calculating (r) corresponds to the "first calculation step" of the present invention. ａ (r) and l ｚ If you input the above, you can find the true pair distribution function g(r). Therefore, the "apparent pair distribution function g ａ (r)" means an extremely thin thickness l ｚ When a disk-shaped three-dimensional space of radius r is viewed in plan from the observation direction D, it is a function that indicates the normalized number density (the number density at the position of interest divided by the number density of the particles in the bulk dispersion) of particles existing on the circumference of radius r, and does not indicate the probability of particles existing in a completely two-dimensional space. Also, the "true pair distribution function g(r)" is a function that indicates the normalized number density of particles existing on a spherical surface of radius r.

[0025] Specifically, the first calculation step involves the following steps: First, a reference image generation step is carried out. In this step, as shown in FIG. 3, an artificial particle P, which is a particle of an ideal gas, is generated on a screen having the same area as the two-dimensional image 10 obtained from the particle observation device. ｒ A plurality of artificial two-dimensional images 20 are created by randomly arranging the artificial particles P ｒ The subscript r in stands for random.

[0026] Next, an artificial particle density function generation step is performed. In this step, preferably, the artificial particle P ｒ From a plurality of artificial two-dimensional images 20 with different distributions of artificial particles P ｒ The distance between the centers of the artificial particles (the distance between the centers of the artificial particles) r is extracted, and the artificial particle P ｒ The probability density function (hereinafter referred to as the artificial probability density function) q ｒThen, a real particle density function generation step is executed to extract the distance between the centers of the particles P (two-dimensional distance between the centers) r from the two-dimensional image 10, and generate a probability density function q(r) of the particles P (hereinafter referred to as real probability density function).

[0027] Finally, a distribution function calculation step is performed to normalize the real probability density function by dividing it by the artificial probability density function, q(r) / q ｒ (r) is the apparent pair distribution function g ａ The reason for normalizing the real probability density function is to take into consideration that the probability of the particle P existing at the edge of the two-dimensional image 10 approaches zero.

[0028] By the way, the apparent pair distribution function g calculated in the first calculation step ａ Inverse calculation of the true pair distribution function g(r) from (r) corresponds to the "second calculation step" of the present invention. Details of the calculation method of the present invention are given below.

[0029] Using a particle observation device, the apparent pair distribution function g ａ Since (r) can be obtained, the apparent pair distribution function g ａ (r) is available, and the apparent pair distribution function g ａ The method for obtaining the radial distribution function g(r), which is the true pair distribution function, from g(r) is explained below. ａ We have independently discovered that the formula (1) for finding (r) (called the forward calculation formula) can be expressed as follows:

[0030]

[0031] The forward calculation formula shown in the above formula (1) can be derived, for example, by the following procedure. Returning to FIG. 1, let us assume that a particle (hereinafter referred to as the reference particle) Pc existing at a predetermined position in the three-dimensional space S is positioned on the Z axis, and consider a cylindrical space Sc having a radius r (i.e., a two-dimensional distance r) centered on the Z axis. Then, as shown in FIG. 4, a predetermined position v ２The number of particles dN(r) present in the cylindrical minute space Sco further away from the reference particle Pc by a distance dr is dN(r). ２ is the distance r from the reference particle Pc in the three-dimensional space S. ０ At a predetermined position v ３ Corresponds to.

[0032] To obtain the particle number dN(r), first, the number of particles in a minute space Ss, which is a part of the minute space Sco located outside the cylindrical space Sc of radius r, is calculated as shown in FIG. 5. This minute space Ss is a space located outside the part of the cylindrical space Sc that occupies an angle dφ around the Z axis. When the Z' axis parallel to the Z axis is defined at a position at a distance r from the reference particle Pc, the height in the Z axis direction in the minute space Ss is dz', and the volume of the minute space Ss is "dz' dr dφ r [m ３ The number of particles present in this minute space Ss is expressed as the normalized number density g(r ０ ) and the number density per unit volume in the bulk liquid ρ ０ [pcs / m ３ ] and is obtained by multiplying "dz' dr dφ r ρ ０ ・g(r ０ ) [pieces]. When the reference particle Pc is assumed to exist at a position z (hereinafter, variable z) on the Z axis in the three-dimensional space S (see FIG. 1) that is the source of the two-dimensional image 10, the variable z' is ｚ When integrated up to -z, the number of particles dN existing in the space Ssz shown in FIG. Ｒ is known, and the number of particles dN Ｒ is expressed by the following formula (1-2).

[0033] In the above formula (1-2), the radius r is geometrically expressed as r as shown in FIG. ０ ２ =r ２ +z´ ２ Since the above formula (1-2) is expressed by the following formula (1-3):

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

[0035] Next, to calculate the number of particles dN(r) in the cylindrical minute space Sco (see FIG. 4), the following equation (1-5) is obtained by integrating the above equation (1-4) for one revolution around the central angle φ.

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

[0037] "dφ r l" in equation (1-6) ｚ ・dr” is the volume [m ３ ] and "ρ ０ ・g ａ (r)" is the number density in the bulk, g ａ (r) times [pieces / m ３ ].

[0038] Then, from the above equations (1-5) and (1-6), the true pair distribution function g(r ０ ) to obtain the apparent pair distribution function g ａ The forward calculation formula (1) for calculating (r) is obtained.

[0039] As shown in FIG. 8, when the reference particle Pc is l ｚ / 2, it is not necessary to integrate the variable z in the reference particle Pc, and the forward calculation formula can be approximated (simplified) and expressed as the following formula (1-7). The approximate forward calculation formula shown in formula (1-7) can be expressed by the true pair distribution function g(r ０ ), that is, g(√(r ２ +z´ ２ )) and the true pair distribution function g(√(r ２ +z´ ２ )) with respect to z', the thickness l along the Z'-axis direction ｚ across -l ｚ / 2 to +lｚ / 2. Also, in equation (1-7), the right side of "l ｚ －１ " is included to make it dimensionless. Here, if the position z on the Z axis of the quasi-fine particle Pc is different, the calculation result of the above single integral will be different. Therefore, in order to avoid this, the forward calculation formula shown in the above formula (1) is used to calculate the above single integral with respect to z, ｚ from 0 to l ｚ In the forward calculation formula shown in formula (1), the right side contains "l ｚ －２ " is included to make it dimensionless.

[0040] The formulas (1) and (1-7) may be multiplied by a predetermined correction coefficient to obtain the forward calculation formula.

[0041] If this forward calculation formula (formula (1)) is correct, the thickness l ｚ When is infinitesimally thin (almost 0), g ａ (r) = g(r). ｚ If we make g infinitesimally thin, ａ Since it is confirmed that (r) = g(r), this forward calculation formula (formula (1)) is correct. In other words, the thickness l of the two-dimensional image 10 actually observed by the particle observation device ｚ When is thin enough, g ａ We can approximate it as (r) ≒ g(r). By the way, the goal (invention) here is to convert g(r) to g ａ It is not about seeking (r), but about g ａ The goal is to find g(r) from (r). ｚ Divide the width of into n parts and perform the integration by the quadrature method. ｉ = i(l ｚ / n) (i is an integer). By this, the inventors have determined that if both sides of the following matrix equation (Equation (2)) are multiplied by the inverse matrix of the n × n square matrix in the matrix equation, g ａ He independently discovered that g(r) can be calculated from (r).

[0042]

[0043] Here, f(i, j) is a function that can be expressed by the following cases: i and j are integers from 1 to n, and f(i, j) is expressed by formula (3) when i = j and i ≠ n, formula (4) when j = n and i ≠ n, formula (5) when i = j = n, and formula (6) when i ≠ j, i ≠ n, j ≠ n and i < j.

[0044]

[0045]

[0046]

[0047]

[0048] The notation of formulas (2) to (6) is merely an example of an inverse calculation formula, and the notation will differ slightly depending on how the rectangular rectangles are positioned in the area calculation method by division and how the rectangular rectangles' areas are calculated. If the inverse calculation formula is correct, it is possible to repeat the forward and inverse calculations. In other words, input g(r) into the forward calculation formula (formula (1)) and get g ａ (r) and its g ａ If the original g(r) can be obtained by inputting (r) into the inverse calculation formula, 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 examples below.

[0049] In the "second calculation step" of the present invention, i.e., the apparent pair distribution function g calculated in the first calculation step ａ Inverse calculation of the true pair distribution function g(r) from (r) includes a method that combines the forward calculation formula of formula (1) with repeated trial and error on a computer, numerical optimization methods, machine learning, etc., without using the method of obtaining an inverse analytical solution of formula (6) from formula (2). For example, by substituting the "trial function of g(r)" into formula (1) and obtaining "g derived from the trial function" ａ (r)" is calculated, and this is the g ａ If it matches (r), the substituted "trial function of g(r)" can be taken as the solution.

[0050] Now, let us suppose that the thickness l of the three-dimensional space, which is the thickness of the incident light L (laser light, etc.) used in the particle observation device, isｚ If is unknown, in the forward and inverse calculation formulas above, l ｚ becomes an unknown quantity, and the true pair distribution function cannot be calculated. Therefore, in such a case, the thickness l of the three-dimensional space can be calculated by the following method. ｚ That is, in the second calculation step, the following steps are carried out as shown in FIG. ｚ Determine.

[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 particle observation device. In this step ST1, for example, two-dimensional images 10 are prepared as still images at predetermined time intervals, or multiple two-dimensional images 10 are prepared by cutting out a video at predetermined time intervals. Then, an actual average residence time acquisition step ST2 is performed in which the multiple two-dimensional images 10 are referenced to acquire an actual average residence time t, which is the average time that multiple particles (hereinafter, "particles to be observed") stay within the two-dimensional image 10. That is, the average time from when the particle to be observed appears on the two-dimensional image 10 until it disappears is acquired. The state in which the particle to be observed disappears from the two-dimensional image 10 represents a state in which the particle to be observed has moved outside of three-dimensional space in the Z-axis direction. Note that if the particle to be observed is assumed to have escaped the outer frame of the observation screen, the particle to be observed is not used in calculating the actual average residence time. Then, a diffusion coefficient calculation step ST3 is performed in which Brownian motion in the two-dimensional image 10 is observed by reference to the multiple two-dimensional images 10, and the diffusion coefficient of the particle is calculated.

[0052] Next, the calculated diffusion coefficient is used to execute a virtual mean residence time acquisition step ST4. In this step ST4, a Brownian motion simulation of particles (hereinafter referred to as simulation particles) is performed inside a virtual three-dimensional space of thickness s (a predetermined thickness: a specific value such as 1 μm or 2 μm), and a virtual mean residence time t', which is the average time that the simulation particles stay in the virtual three-dimensional space, is acquired. Hereinafter, this thickness s is defined as the virtual thickness s.

[0053] Then, a virtual thickness selection step ST5 is executed to extract a virtual thickness s in the virtual three-dimensional space to achieve a virtual average residence time t' that matches the actual average residence time t. In the virtual thickness selection step ST5 of this embodiment, a judgment step ST51 is executed to determine whether the actual average residence time t and the virtual average residence time t' match, and if the actual average residence time t and the virtual average residence time t' match, the process proceeds to a thickness substitution step ST6. On the other hand, if the actual average residence time t and the virtual average residence time t' do not match in the judgment step ST51, a virtual thickness change step ST52 is executed to change the value of the set virtual thickness s, and then the process returns to the virtual average residence time acquisition step ST4. In the thickness substitution step ST6, the thickness l in the forward calculation formula is ｚ In other words, the virtual average residence time t' in the virtual three-dimensional space of the virtual thickness s assumed in the simulation is the actual thickness l ｚ If the virtual thickness s of the virtual three-dimensional space coincides with the actual average stay time t in the three-dimensional space, the virtual thickness s of the virtual three-dimensional space is the thickness l in the actual three-dimensional space. ｚ It can be assumed that the thickness l of the forward calculation formula is the same as ｚ Since it is possible to substitute the virtual thickness s into the equation, such processing is performed.

[0054] The virtual thickness selection step ST5 is not limited to the present embodiment. For example, the results of simulations performed by varying the combinations of the diffusion coefficient and the virtual thickness s may be prepared in advance as data such as a matrix table, and the virtual thickness s for achieving a virtual average residence time t' that coincides with the actual average residence time t may be extracted based on this data. Alternatively, the above data may be trained by an artificial intelligence to estimate the virtual thickness s for achieving a virtual average residence time t' that coincides with the actual average residence time t.

[0055] Next, a method for calculating the pair-body potential (potential of mean force) u(r) between particle-particles will be described below. Note that the "method for calculating pair-body potential" of the present invention is applicable only when u(r) is calculated using the pair distribution function g(r) obtained from the "method for calculating pair distribution function" of the present invention. First, g(r) is prepared using the method for calculating pair-body potential of the present invention. Then, the total correlation function h(r) is calculated using the following equation (7). Calculating the total correlation function h(r) from the true pair distribution function g(r) calculated using the above method for calculating pair distribution function corresponds to the "third calculation step" of the present invention.

[0056]

[0057] Then, the direct correlation function c(r) is calculated using the following equation (8) (a modified version of the single-component Ornstein-Zernike equation [J.P. Hansen and I.R. McDonald, 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 corresponds to the "fourth calculation step" of the present invention.

[0058]

[0059] Here, F +１ is the Fourier transform in spherical polar coordinates, F －１ means the inverse Fourier transform in spherical polar coordinates. Finally, u(r) is found 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 above fourth calculation step corresponds to the "fifth calculation step" of the present invention.

[0060]

[0061] Here, k Ｂis the Boltzmann constant, and T is the absolute temperature. If the particles are sufficiently sparse, the two-body potential can be calculated simply by setting h-c+b=0 in equation (9). Alternatively, if b=0 in equation (9), the equation is called a hypernetted-chain equation. Unless there are particular requirements, it is generally sufficient to set b=0. Several bridge functions have been proposed and published, and it is sufficient to determine which bridge function to use based on the conditions of the dispersion system. For example, if the particles can be approximated as hard spheres, the bridge function by Verlet shown in equation (10) [L. Verlet, Mol. Phys., 41 (1980) 183] may be used.

[0062]

[0063] The method for calculating a pair-body potential of the present invention is applicable only when u(r) is calculated using the pair-body distribution function g(r) obtained from the "method for calculating a pair-body distribution function" of the present invention.

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

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

[0066] The diameter of the particles observed using the particle observation device is assumed to be approximately within the range specified or suitable for the particle observation device itself.

[0067] It is preferable that the type of particles observed using a particle observation device is a single type. However, when observing fluorescent particles, the type of fluorescent particles to be observed can be intentionally selected by selecting the wavelength of the irradiated laser light, so the type of particles does not need to be a single type in a colloidal dispersion containing fluorescent particles.

[0068] It is preferable that the diameters of the microparticles observed using a microparticle observation device are monodisperse. When there are multiple types of microparticles, it is preferable that the diameters of each type of microparticle are monodisperse. However, when the type of microparticles observed is a single type, the diameters may be polydisperse. When the microparticle diameters are polydisperse, using the present invention will determine a single pair distribution function by averaging the pair distribution functions of various diameter pairs (there are multiple pair distribution functions for the number of pairs).

[0069] The dispersion medium is preferably a gas or a 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 particles, which are the dispersoid, the apparent pair distribution function g ａ (r) can be found and its g ａ It is possible to input (r) into the formula for calculating the pair distribution function of the present invention to obtain the true pair distribution function (radial distribution function) g(r) in gels, glass, and solids. For example, by using a cryo-electron microscope, it is possible to observe particles dispersed in glass, and g can be obtained from the observation results. ａ It is thought that g(r) can be calculated by inputting (r) into the calculation formula of the present invention. However, when calculating the two-body potential u(r) from g(r) of a particle calculated in a gel, glass, or solid, it is not necessarily the case that a meaningful u(r) is obtained.

[0070] Next, a brief description will be given of the configuration of a particle observation device 100 incorporating the pair distribution function calculation program for calculating the pair distribution function and the pair potential calculation program for calculating the pair potential. As shown in FIG. 10, the particle observation device 100 is configured to have a thickness l in the Z-axis direction, which is the height direction, of a dispersion medium containing particles. ｚ The apparatus includes a light source 105 that irradiates light L of the particle, a two-dimensional image acquisition unit (camera) 110 that observes and acquires the two-dimensional image of the particle, and a calculation unit 120 that calculates the pair distribution function of the particle P in the dispersion medium M based on the two-dimensional image 10 obtained by the two-dimensional image acquisition unit 110. The calculation unit 120 is configured with a processor or the like, and calculates the apparent pair distribution function g from the two-dimensional image 10. ａA first calculation unit 121 that calculates (r), and an apparent pair distribution function g calculated by the first calculation unit 121 ａ and a second calculation unit 122 that inversely calculates a true pair distribution function g(r) from the true pair distribution function g(r) calculated by the second calculation unit 122. Here, the calculation unit 120 may further include a third calculation unit 123 that calculates a 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 a 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 a pair potential u(r) from the direct correlation function c(r) calculated by the fourth calculation unit 124. By incorporating a pair distribution function calculation program and a pair potential calculation program into the particle observation device 100 in this way, the pair distribution function g(r) and the pair potential u(r) can be obtained only from the particle observation device 100.

[0071] Next, an embodiment of the forward calculation and the inverse calculation of the present invention will be described with reference to the drawings. ａ (r). This was obtained by substituting the Benchmark curve in FIG. 12 (a pair distribution function g(r) having a general shape) into the forward calculation formula (Equation (1)). FIG. 12 also shows the pair distribution function obtained using the method for calculating the pair distribution function of the present invention, shown as Result from the inverse calculation. Since the Benchmark curve and the curve shown as Result from the inverse calculation are consistent, it can be confirmed that the method for calculating the pair distribution function of the present invention is capable of performing the inverse calculation correctly (is functioning).

[0072] In addition, when the distance between the centers of particles is less than the diameter of the particles, it means that the particles overlap with each other. Therefore, when the distance between the centers is less than the diameter of the particles, the pair distribution function is generally zero. However, in Figure 11, which shows the apparent pair distribution function, there is a significant g that is not zero even in the short distance region less than the diameter (near the horizontal axis of Figure 11 is zero). ａThe value of (r) can be seen on the vertical axis. The reason for this can be easily understood by considering Figure 1. When observing particles from the observation direction shown in Figure 1, the particles may appear to overlap. This also occurs when viewed from directly below, opposite to the observation direction shown in Figure 1. For this reason, in Figure 11, significant values ​​other than 0 can be seen on the vertical axis even in the short distance region less than the diameter. The same 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 should be considered to be illustrative in all respects and not restrictive. The scope of the present invention is not limited to the embodiments disclosed herein, but is intended to include all modifications within the scope indicated by the claims or the scope equivalent to the claims.

[0074] According to the method for calculating a pair distribution function, the method for calculating a pair potential including the same, the pair distribution function calculation program, the pair potential calculation program, and the particle observation device of the present invention, it is possible to calculate the pair distribution function and the pair potential relatively easily.

[0075] REFERENCE SIGNS LIST 10 Two-dimensional image 20 Artificial two-dimensional image Pc Particle (reference particle) L (incident) light 100 Particle observation device 105 Light source 110 Two-dimensional image acquisition unit 120 Calculation unit 121 First calculation unit 122 Second calculation unit 123 Third calculation unit 124 Fourth calculation unit 125 Fifth calculation unit

Claims

1. A method for calculating a pair distribution function of a particle in a dispersion medium, comprising: an observation step of observing a two-dimensional image of the particle using a particle observation device; a first calculation step of calculating an apparent pair distribution function from the two-dimensional image observed in the observation step; and a second calculation step of inversely calculating a true pair distribution function from the apparent pair distribution function calculated in the first calculation step.

2. A method for calculating a pair distribution function according to claim 1, wherein the first calculation step comprises: a reference image generation step of creating an artificial two-dimensional image in which a plurality of artificial particles which are 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 an artificial particle center-to-center distance which is the distance between the centers of the artificial particles from the artificial two-dimensional image and generating a probability density function of the artificial particles (hereinafter referred to as artificial probability density function); a real particle density function generation step of extracting a two-dimensional center-to-center distance which is the distance between the centers of the particles from the two-dimensional image and generating a probability density function of the particles (hereinafter referred to as 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 pair distribution function.

3. The second calculation step includes a forward calculation formula preparation step of deriving a forward calculation formula for calculating the apparent pair distribution function from the true pair distribution function, and 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, a predetermined position v from one of the particles (hereinafter, a reference particle) existing at a predetermined position on the two-dimensional image is ２ The distance to the predetermined position v on the two-dimensional image is defined as a two-dimensional distance r. ２ A predetermined position in the three-dimensional space corresponding to v ３ When defining the above, the predetermined position v ３ The distance to the actual distance r ０ 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 distance from the reference particle to the predetermined position v ３ When the relative distance on the Z' axis to is z', the apparent pair distribution function is a function g of the two-dimensional distance r. ａ (r), and the true pair distribution function is ０ Function g(r ０ ) and the actual distance r ０ is √(r ２ +z' ２ ), the true pair distribution function can be expressed as g(√(r ２ +z' ２ )) and the forward calculation formula is expressed as the true pair distribution function g (√(r ２ +z' ２ For the variable z' in the Z'-axis direction, the thickness l ｚ 2. The method of claim 1, further comprising a single integral integrated over 4. The method for calculating a pair distribution function according to claim 3, wherein said second calculation step further comprises an inverse calculation formula derivation step of deriving an inverse calculation formula for calculating said true pair distribution function from said apparent pair distribution function based on said forward calculation formula.

5. When the reference particle is disposed at a position (hereinafter, variable) z on the Z axis parallel to the Z' axis, the forward calculation formula is expressed as follows: ｚ 4. The method of claim 3, further comprising a double integral integrated over 6. 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 the average time that a particle to be observed, which is the predetermined particle, stays in the two-dimensional image; a diffusion coefficient calculation step of observing Brownian motion in the two-dimensional image and calculating a diffusion coefficient of the particle; a virtual average residence time acquisition step of performing a Brownian motion simulation of a particle (hereinafter, a simulation particle) inside a virtual three-dimensional space of a thickness (hereinafter, a virtual thickness) s using the diffusion coefficient, and acquiring a virtual average residence time, which is the average time that the simulation particle stays in the virtual three-dimensional space; a virtual thickness selection step of extracting the virtual thickness s in the virtual three-dimensional space to achieve the virtual average residence time that coincides with the actual average residence time; and a calculation step of calculating the thickness (hereinafter, an actual thickness) l in the forward calculation formula. ｚ The method for calculating a pair distribution function according to claim 3 , further comprising: a thickness substitution step of substituting the virtual thickness s extracted in the virtual thickness extraction step into 7. A method for calculating a pair distribution function according to any one of claims 1 to 5, further comprising, prior to said observation step, a light irradiation step of irradiating light from a light source toward said dispersion medium in a direction intersecting the observation direction of said two-dimensional image.

8. A method for calculating a pair-body potential comprising: a method for calculating a pair-body distribution function according to any one of claims 1 to 5; a third calculation step for calculating a total correlation function from the true pair-body distribution function calculated by the method for calculating a pair-body distribution function; 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-body potential from the direct correlation function calculated in the fourth calculation step.

9. A pair distribution function calculation program for causing a computer to function as: a first calculation means for calculating an apparent pair distribution function from a two-dimensional image of a particle; and a second calculation means for inversely calculating a true pair distribution function from the apparent pair distribution function calculated by the first calculation means, in order to calculate the pair distribution function of a particle in a dispersion medium.

10. A two-body potential calculation program for causing a computer to function as: a first calculation means for calculating an apparent pair-body distribution function from a two-dimensional image of the particle; a second calculation means for inversely calculating a true pair-body distribution function from the apparent pair-body distribution function calculated by the first calculation means; a third calculation means for calculating a total correlation function from the true pair-body distribution function calculated by the second calculation means; a fourth calculation means for calculating a direct correlation function from the total correlation function by the third calculation means; and a fifth calculation means for calculating a two-body potential from the direct correlation function calculated by the fourth calculation means, in order to calculate a pair-body potential of a particle in a dispersion medium.

11. A particle observation device comprising: a light source which irradiates light onto a dispersion medium containing particles; a two-dimensional image acquisition unit which observes and acquires a two-dimensional image of the particles from a direction intersecting the direction of the light irradiation; and a calculation unit which calculates a pair-body distribution function of the particles in the dispersion medium based on the two-dimensional image, wherein the calculation unit has: a first calculation unit which calculates an apparent pair-body distribution function from the two-dimensional image; and a second calculation unit which inversely calculates a true pair-body distribution function from the apparent pair-body distribution function calculated by the first calculation unit.

12. The particle observation apparatus according to claim 11, wherein the calculation unit further comprises: a third calculation unit that calculates a total correlation function from the true pair-body distribution function calculated by the second calculation unit; a fourth calculation unit that calculates a direct correlation function from the total correlation function calculated by the third calculation unit; and a fifth calculation unit that calculates a pair-body potential from the direct correlation function calculated by the fourth calculation unit.

Images