Light scattering detector and method therefor

Abstract

A method for determining an assumption-free radius of gyration of a particle in a solution using a light scattering detector is disclosed. The method can include determining a first weighting factor based on a first shape factor. The method can also include determining a second weighting factor based on a second shape factor. The method can further include determining a first shape factor contribution to the first shape factor based on the first weighting factor and the second weighting factor. The method can also include determining a second shape factor contribution to the second shape factor based on the first weighting factor and the second weighting factor. The method can also include determining an assumption-free radius of gyration from the first and second shape factor contributions.

Description

[Background technology]

[0001] Conventional light scattering detectors are often used in conjunction with chromatography techniques to determine one or more physical attributes or properties of various particles (e.g., molecules or solutes) suspended in a solution. For example, light scattering detectors are often used in conjunction with gel permeation chromatography (GPC) or other liquid separation processes to determine the molecular weight (MW) and radius of gyration (Rg) of various particles, such as polymers. In a light scattering detector, a sample or eluate containing particles (e.g., molecules, polymers, etc.) is flowed through a sample cell from an inlet to an outlet. As the eluate flows through the sample cell, it is illuminated by a collimated light beam (e.g., a laser). The interaction of the light beam with the polymers in the eluate generates scattered light. The scattered light is then measured and analyzed for various attributes, such as intensity and angle, to determine the solution properties and / or physical characteristics of the particles.

[0002] While conventional light scattering detectors have proven effective in providing clearer insight into the solution and / or physical attributes of a wide range of particles, conventional methods for analyzing these particles have limitations. For example, conventional methods for analyzing particles with relatively high Rg potentially have significant errors because molecular weights determined by multi-angle light scattering (MALS) can be perturbed by extrapolating from multiple angles to zero angle. Furthermore, the determination of Rg by angle extrapolation is based on the slope of the angle extrapolation at zero angle. Therefore, if there is variation in the minimum angle, the resulting Rg can have significant errors if the minimum angle is not "small enough" or if an inappropriate extrapolation method is selected. In addition, extrapolation methods often rely on algorithms or equations that assume a single structural model or rely on polynomial "fit orders" to estimate or approximate the trend toward zero angle. However, if a particle does not fit or fits poorly to a single structural model or is not adequately represented by a polynomial, the analysis can have significant errors. Summary of the Invention [Problem to be solved by the invention]

[0003] Thus, what is needed are improved light scattering detectors and methods for analyzing or determining the behavior of particles in solution. [Means for solving the problem]

[0004] This summary is intended merely to introduce a brief overview of some aspects of one or more implementations of the present disclosure. Further areas of applicability of the present disclosure will become apparent from the detailed description provided hereinafter. This summary is not an extensive overview, and is not intended to identify key or critical elements of the present teachings or to delineate the scope of the present disclosure. Rather, its purpose is merely to present one or more concepts in a simplified form as a prelude to the detailed description that follows.

[0005] These and / or other aspects embodied in the present disclosure can be achieved by providing a method for determining an assumption-free radius of gyration (Rg) of a particle in a solution using a light scattering detector. The method can include determining a first weighting factor (FW1) based on a first shape factor (FF1). The method can also include determining a second weighting factor (FW2) based on a second shape factor (FF2). The method can further include determining a first shape factor contribution (FC1) to the first shape factor (FF1) based on the first weighting factor (FW1) and the second weighting factor (FW2). The method can also include determining a second shape factor contribution (FC2) to the second shape factor (FF2) based on the first weighting factor (FW1) and the second weighting factor (FW2). The method can also include determining the assumption-free radius of gyration (Rg) from the first and second shape factor contributions (FC1, FC2).

[0006] In at least one implementation, the method includes measuring a first scattering intensity (I) of particles in the solution at a first angle, a second angle, and a third angle relative to the light beam using a light scattering detector. θ1 ), the second scattering intensity (I θ2 ), and the third scattering intensity (I θ3 ) may be obtained.

[0007] In at least one implementation, the first scattering intensity (I θ1 ), the second scattering intensity (I θ2 ), and the third scattering intensity (I θ3 ) may be the excess scattering intensity. The respective excess scattering intensity for each of the first, second, and third angles may be normalized with respect to quantum efficiency, gain, geometric volume, or a combination thereof.

[0008] In at least one implementation, the method also includes determining a ratio of the observed light scattering of the particle at the first angle to the observed light scattering of the particle at the third angle (R' θ1 / θ3 ) and determining a ratio of the observed light scattering of the particle at the second angle to the observed light scattering of the particle at the third angle (R' θ2 / θ3 and determining a

[0009] In at least one implementation, determining the first weighting factor (FW1) further comprises determining a ratio of calculated light scattering at the first angle to calculated light scattering at a third angle based on the first shape factor (FF1). determining a ratio of calculated light scattering at the second angle to calculated light scattering at the third angle based on the first shape factor (FF1); TIFF0007749813000002.tif8150 and the ratio Ratio to TIFF0007749813000003.tif8150 (R' θ1 / θ3 ) and ratio Ratio (R') to TIFF0007749813000004.tif8150 θ2 / θ3) and determining the best fit.

[0010] In at least one implementation, determining the second weighting factor (FW2) comprises determining a ratio of calculated light scattering at the first angle to calculated light scattering at a third angle based on the second shape factor (FF2). determining a ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle; TIFF0007749813000006.tif8150 and the ratio Ratio (R') to TIFF0007749813000007.tif8150 θ1 / θ3 ) and ratio Ratio (R') to TIFF0007749813000008.tif8150 θ2 / θ3 ) and determining the best fit.

[0011] In at least one implementation, the step of determining the best fit disclosed above may involve an iterative process.

[0012] In at least one implementation, determining the first weighting factor (FW1) comprises determining a ratio based on the first shape factor (FF1). Ratio (R') to TIFF0007749813000009.tif8150 θ1 / θ3 ) and ratio Ratio (R') to TIFF0007749813000010.tif8150 θ2 / θ3 ) to determine a turning radius (Rg1) associated with a best fit.

[0013] In at least one implementation, determining the radius of gyration (Rg1) associated with the best fit comprises determining a ratio (R' θ1 / θ3 ) and Difference and ratio (R') between TIFF0007749813000011.tif8150 θ2 / θ3 ) and The method may include determining a radius of gyration (Rg1) associated with the smallest sum of squares residual (SSR1) of the differences between the two images.

[0014] In at least one implementation, the first weighting factor (FW1) may be based on the least sum of squares residual (SSR1).

[0015] In at least one implementation, determining the second weighting factor (FW2) comprises determining a ratio based on the second shape factor (FF2). Ratio (R') to TIFF0007749813000013.tif8150 θ1 / θ3 ) and ratio Ratio (R') to TIFF0007749813000014.tif8150 θ2 / θ3 ) can include determining a turning radius (Rg2) associated with a best fit.

[0016] In at least one implementation, determining the radius of gyration (Rg2) associated with the best fit comprises determining a ratio (R' θ1 / θ3 ) and Difference and ratio (R') between TIFF0007749813000015.tif8150 θ2 / θ3 ) and The method may include determining a radius of gyration (Rg2) associated with the smallest sum of squares residual (SSR2) of the differences between the two images.

[0017] In at least one implementation, the second weighting factor (FW2) may be based on the sum of squares residual (SSR2).

[0018] In at least one implementation, at least one of the first shape factor (FF1) or the second shape factor (FF2) can be a shape factor for a homogeneous rigid particle.

[0019] In at least one implementation, the shape factor for a homogeneous rigid particle can be one of a homogeneous sphere, a spherical shell, a spherical concentric shell, a particle of spherical subunits, a spheroid, a triaxial ellipsoid, a cube and rectangular prism, a truncated octahedron, a polyhedral sphere, a lens, a stepped cube, a cylinder, an elliptical cylinder, a hemispherical end cylinder, a semi-lens end cylinder, a torus, a thin rod, an ultrathin disk, or a fractal aggregate.

[0020] In at least one implementation, at least one of the first form factor (FF1) or the second form factor (FF2) can be a form factor for a polymer model.

[0021] In at least one implementation, the shape factor for the polymer model may be one of a spherical particle shape factor, a rod-like particle shape factor, a random coil particle shape factor, or a combination thereof.

[0022] In at least one implementation, the light scattering detector can include a multi-angle light scattering detector.

[0023] In at least one implementation, the particles in the solution may include monodisperse particles.

[0024] In at least one implementation, the particles in the solution may include polydisperse particles.

[0025] In at least one implementation, the particle's hypothetical radius of gyration (Rg) can be from about 5 nm to about 500 nm, from about 30 to about 250 nm, or from about 50 to about 150 nm.

[0026] In at least one implementation, the first angle can be approximately 90 degrees relative to the detector light beam.

[0027] In at least one implementation, the second angle can be approximately 170° relative to the detector light beam.

[0028] In at least one implementation, the third angle can be approximately 10 degrees relative to the detector light beam.

[0029] In at least one implementation, the method includes determining a third weighting factor (FW3) based on a third shape factor (FF3), and calculating all weighting factors (FW i ), and determining an assumption-free turning radius (Rg) from the first, second, and third shape factor contributions (FC1, FC2, FC3).

[0030] In at least one implementation, the method includes measuring a fourth scattering intensity (I) of particles in the solution using a light beam of a light scattering detector at a fourth angle relative to the light beam. θ4 )

[0031] In at least one implementation, the light scattering detector can include a goniometer configured to observe light scattering at two or more angles.

[0032] In at least one implementation, the method may not include determining the molecular weight of the particles in the solution prior to determining the assumption-free radius of gyration (Rg).

[0033] In at least one implementation, the method may not include fitting a Zimm, Berry, Debye, or Guinier plot to a polynomial function.

[0034] In at least one implementation, the method can include implementing exclusion criteria to determine the corrected assumption-free turning radius.

[0035] In at least one implementation, the exclusion criteria may be based on one of the following for each of the shape factors: a respective least sum of squares residual, a respective least sum of squares squared, or a respective least sum of squares absolute value.

[0036] In at least one implementation, the exclusion criteria may be implemented to determine a corrected least sum of squares residual, a corrected least sum of squares squared, or a least sum of squares absolute for each shape factor.

[0037] In at least one implementation, the rejection criteria may be based on the number of angles utilized by the light scattering detector.

[0038] In at least one implementation, the exclusion criteria may be based on the number of form factors utilized.

[0039] In at least one implementation, the exclusion criteria may be based on at least one of a first shape factor contribution (FC1) or a second shape factor contribution (FC2).

[0040] In at least one implementation, the rejection criteria may be based on the respective signal-to-noise at each of the angles.

[0041] These and / or other aspects and advantages embodied in the present disclosure can be achieved by providing a method for evaluating composite particles in a solution, the method including determining a first weighting factor (FW1) based on a first shape factor (FF1), determining a second weighting factor (FW2) based on a second shape factor (FF2), determining a first shape factor contribution (FC1) to the first shape factor (FF1) based on the first weighting factor (FW1) and the second weighting factor (FW2), determining a second shape factor contribution (FC2) to the second shape factor (FF2) based on the first weighting factor (FW1) and the second weighting factor (FW2), and determining one or more characteristics of the composite particles using the first shape factor contribution (FC1) and the second shape factor contribution (FC2).

[0042] In at least one implementation, one or more characteristics of the composite particles may include structural features, substructures, or a combination thereof.

[0043] In at least one implementation, the composite particles in solution may include branched particles.

[0044] Further areas of applicability of the present disclosure will become apparent from the detailed description provided hereinafter. It should be understood that the detailed description and specific examples, while indicating some typical aspects of the present disclosure, are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure.

[0045] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate various implementations of the present disclosure. These and / or other aspects and advantages of implementations of the present disclosure will become apparent and more readily appreciated from the following description of various implementations with reference to the accompanying drawings. It should be noted that some details of the drawings have been simplified and drawn to facilitate understanding of the present disclosure rather than maintaining strict structural accuracy, detail, and scale. These drawings / illustrations are intended to be illustrative rather than limiting. [Brief explanation of the drawings]

[0046] [Figure 1A] FIG. 1 is a schematic diagram of an exemplary light scattering detector including an exemplary sample cell according to one or more implementations of the present disclosure. [Figure 1B] FIG. 1B is a schematic diagram of the exemplary sample cell of FIG. 1A in accordance with one or more implementations of the present disclosure. [Figure 1C] 1B is a schematic diagram of the exemplary sample cell of FIG. 1A without analyte scattered light in accordance with one or more implementations of the present disclosure. [Figure 1D] 1D is an enlarged view of a portion of the sample cell indicated by the box labeled 1D in FIG. 1C in accordance with one or more implementations of the present disclosure. [Figure 2] FIG. 1 illustrates an exemplary computer system or electronic processor for receiving and / or analyzing data from a light scattering detector according to one or more implementations of the present disclosure. [Figure 3]FIG. 3 is a block diagram of the computer system or electronic processor of FIG. 2 in accordance with one or more implementations of the present disclosure. [Figure 4] FIG. 1 illustrates first, second, and third order polynomial fits of the Zimm plot of Example 3. [Figure 5] FIG. 10 shows a plot of the assumption-free Rg extrapolated using the angle data for each of the ranges disclosed in Table 15 of Example 4. [Figure 6] FIG. 10 illustrates a chromatogram of a sample of branched particles from Example 5. DETAILED DESCRIPTION OF THE INVENTION

[0047] DETAILED DESCRIPTION OF THE INVENTION The following description of various exemplary embodiments is merely exemplary in nature and is in no way intended to limit the present disclosure, its application, or uses.

[0048] As used throughout this disclosure, ranges are used as a shorthand for describing each and every value within that range. It is to be appreciated and understood that the description in range format is merely for convenience and brevity and should not be construed as an inflexible limitation on the scope of any embodiment or implementation disclosed herein. Accordingly, the scope of the present disclosure should not be construed as specifically disclosing all possible subranges and individual numerical values ​​within that range. Thus, any value within a range can be selected as an end value of the range. For example, a description of a range such as 1 to 5 should be considered to specifically disclose subranges such as 1.5 to 3, 1 to 4.5, 2 to 5, 3.1 to 5, etc., as well as individual numbers within that range, e.g., 1, 2, 3, 3.2, 4, 5, etc. This applies regardless of the breadth of the range.

[0049] Additionally, all numerical values ​​are "about" or "approximate" designations that take into account experimental error and variability that would be expected by one of ordinary skill in the art. It should be understood that all numerical values ​​and ranges disclosed herein are approximate values ​​and ranges, whether or not "about" is used in conjunction with them. It should also be understood that the term "about" used in conjunction with a numerical value herein can refer to ±0.01% (inclusive), ±0.1% (inclusive), ±0.5% (inclusive), ±1% (inclusive), ±2% (inclusive), ±3% (inclusive), ±5% (inclusive), ±10% (inclusive), or ±15% (inclusive) of that number. It should also be understood that when a numerical range is disclosed herein, any numerical value falling within that range is specifically disclosed.

[0050] All references cited herein are hereby incorporated by reference in their entirety. In the event of a conflict between the definitions of the present disclosure and those of the cited references, the present disclosure will control.

[0051] As used herein, the term or phrase "detector sensitivity" may refer to the signal-to-noise ratio. It should be recognized by those skilled in the art that increasing the laser power of a light scattering detector does not necessarily improve its sensitivity.

[0052] FIG. 1A illustrates a schematic diagram of an exemplary light scattering detector (LSD) 100 including a sample cell 102 according to one or more implementations. The LSD 100 can be operably coupled to a sample source or device 104 and can be capable of or configured to receive a sample or eluate therefrom. For example, as illustrated in FIG. 1A, the LSD 100 can be fluidly coupled to the sample source or device 104 via line 106 and configured to receive an eluate therefrom. The exemplary sample source or device 104 can include, but is not limited to, a chromatographic instrument capable of or configured to separate one or more analytes of a sample or eluate from one another. For example, the sample source or device 104 can be a liquid chromatographic instrument capable of or configured to separate analytes of an eluate from one another based on their respective charge (e.g., ion exchange chromatography), size (e.g., gel permeation chromatography), flow field fractionation (FFF), fluid force injection polymer analysis (FIPA), or the like. In an exemplary implementation, the LSD 100 is operably coupled to a liquid chromatography instrument configured to separate analytes from one another based on their respective sizes, for example, the LSD 100 is operably coupled to a liquid chromatography instrument that includes a gel permeation chromatography column.

[0053] The LSD 100 can include a sample cell 102, a collimated light beam source 108, such as a laser, and one or more detectors 110, 112, 114 (three shown) operably coupled to one another. The detectors 110, 112, 114 can be any suitable detector capable of or configured to receive sample-scattered light. For example, any one or more of the detectors 110, 112, 114 can be photodetectors, such as silicon photodetectors. The LSD 100 can include lenses 116, 118, 120, 122, 124 (five shown) capable of or configured to refract, focus, attenuate, and / or collect light transmitted therethrough, and one or more mirrors 126, 128 (two shown) capable of or configured to reflect or redirect light.

[0054] In at least one implementation, the first lens 116 and the second lens 118 are disposed on opposite sides or axial ends of the sample cell 102 and can be configured to refract, focus, attenuate, and / or collect light transmitted therethrough. In another implementation, the body 130 of the sample cell 102 can define recesses 132, 134 configured to receive the first and second lenses 116, 118. For example, as illustrated in FIG. 1A and further illustrated in detail in FIG. 1B, the body 130 of the sample cell 102 can define first and second recesses 132, 134 extending longitudinally or axially therethrough and configured to receive the first and second lenses 116, 118, respectively. 1A and 1B, each of the first and second lenses 116, 118 can define a convex surface along a respective first or outer end portion 136, 138. Although the first end portions 136, 138 of the first and second lenses 116, 118 are illustrated as defining a convex surface, the respective first end portions 136, 138 of the first and second lenses 116, 118 can alternatively define a flat surface. As further illustrated in FIG. 1A, each of the first and second lenses 116, 118 can define a flat surface along its respective second or inner end portion 140, 142. As described in further detail herein, the respective second end portions 140, 142 of the first and second lenses 116, 118 can seal and / or at least partially define a channel or flow path 144 extending through the sample cell 102.

[0055] The laser 108 may be any suitable laser capable of or configured to provide a light beam 146 having sufficient wavelength and / or power. For example, the laser 108 may be a diode laser, a semiconductor laser, or the like. The laser 108 may be configured to emit a light beam 146 that passes through the sample cell 102. For example, as illustrated in FIG. 1A , the lasers 108 may be arrayed or positioned around the LSD 100 such that the light beam 146 emitted therefrom is transmitted through the sample cell 102. As further illustrated in FIG. 1A , a third lens 120 may be interposed between the sample cell 102 and the laser 108 and configured to focus the light beam 146 directed to and through the sample cell 102.

[0056] In at least one implementation, at least one of the mirrors 126, 128 is associated with a respective detector 110, 112 and can be configured to reflect or redirect light (e.g., scattered light or analyte-scattered light) toward the respective detector 110, 112. For example, as illustrated in FIG. 1A , the first mirror 126 can be positioned proximate to the first lens 116 and configured to reflect at least a portion of the light from the first lens 116 toward the first detector 110. In another example, the second mirror 128 can be positioned proximate to the second lens 118 and / or interposed between the second lens 118 and the third lens 120 and configured to reflect at least a portion of the light from the second lens 118 toward the second detector 112. In at least one implementation, one or more lenses 122, 124 can be interposed between the first mirror 126 and the first detector 110 and between the second mirror 128 and the second detector 112 to focus, refract, or otherwise direct light from the mirrors 126, 128 onto the detectors 110, 112. For example, as illustrated in FIG. 1A , a fourth lens 122 can be interposed between the first detector 110 and the first mirror 126, and a fifth lens 124 can be interposed between the second detector 112 and the second mirror 128.

[0057] In at least one implementation, at least one of the detectors 110, 112, 114 can be configured to receive light (e.g., scattered light or analyte-scattered light) from the sample cell 102 without the aid of reflection from one of the mirrors 126, 128. For example, as illustrated in FIGS. A1 and 1B , the third detector 114 can be positioned adjacent to or coupled to the sample cell 102 and configured to receive light (e.g., scattered light or analyte-scattered light) from the sample cell 102 at an angle of approximately 90° relative to the light beam 146. As discussed in more detail herein, an optically transparent material or a sixth lens 186 can be configured to refract or direct the scattered light toward the third detector 114.

[0058] As illustrated in FIG. 1A , the sample cell 102, the first, second, and third lenses 116, 118, 120, and at least one of the first and second mirrors 126, 128 can be arranged parallel to, coaxial with, or otherwise aligned with one another along the direction of the light beam 146 emitted by the laser 108. As further illustrated in FIG. 1A , each of the first and second detectors 110, 112 can be arranged or positioned to receive light (e.g., scattered light or analyte-scattered light) from the respective mirror 126, 128 in a direction approximately perpendicular to the light beam 146 emitted by the laser 108. Each of the first and second mirrors 126, 128 can define a respective bore or passageway 150, 152 extending therethrough. For example, the first mirror 126 can define a bore 150 therethrough that extends parallel to, coaxial with, or otherwise aligned with the light beam 146. Similarly, the second mirror 128 may define a bore 152 extending therethrough in a direction parallel to, coaxial with, or otherwise aligned with the light beam 146. It should be appreciated that the bores 150, 152 extending through the respective mirrors 126, 128 may enable the light beam 146 emitted from the laser 108 to be transmitted through the first and second mirrors 126, 128, thereby preventing the light beam 146 from being reflected towards the first and second detectors 110, 112.

[0059] FIG. 1D illustrates an expanded view of a portion of an exemplary LSD 100 according to one or more implementations indicated by the box labeled 1D in FIG. 1C. As discussed above, the body 130 of the sample cell 102 can at least partially define a channel or flow path 144 extending therethrough. For example, as illustrated in FIG. 1D, an inner surface 154 of the body 130 can at least partially define the flow path 144 extending therethrough. The flow path 144 can define a volume of the sample cell 102. The flow path 144 can include a central axis or centerline 156 extending therethrough and configured to define a general orientation of the flow path 144. As illustrated in FIG. 1B, the flow path 144 and its central axis 156 can be aligned or coaxial with the light beam 146 emitted from the laser 108. The flow channel 144 of the sample cell 102 can be interposed between the first lens 116 and the second lens 118. In at least one implementation, the first lens 116 and the second lens 118 sealingly engage the body 130 of the sample cell 102 on opposite sides of the body 130, thereby interfacing the body 130 with the respective first and second lenses 116, 118 to prevent sample or eluate from exiting the flow channel 144. In another implementation, seals (e.g., gaskets, O-rings, etc.) (not shown) can be disposed between the body 130 and the first and second lenses 116, 118 to provide a fluid-tight seal therebetween.

[0060] The flow channel 144 can include an inner section 158 and two outer sections 160, 162 disposed along a centerline 156 of the flow channel 144. As illustrated in FIG. 1D, the inner section 158 can be interposed between the two outer sections 160 and 162. The inner section 158 can be configured to fluidly couple with the sample source 104 and receive sample or eluate therefrom. For example, with continued reference to FIG. 1A and as illustrated in FIG. 1D, the body 130 of the sample cell 102 can define an inlet 164 extending therethrough and configured to fluidly couple the sample source 104 with the inner section 158 via the line 106. In a preferred implementation, the inlet 164 is configured such that sample from the sample source 104 is directed toward the center or center of the flow channel 144 or its inner section 158.

[0061] In at least one implementation, the inner section 158 can be cylindrical or define a cylindrical volume and can have a circular cross-sectional profile. However, it should be appreciated that the cross-sectional profile can be represented by any suitable shape and / or size. For example, the cross-sectional profile can be oval, rectangular such as rounded, square, or the like. The inner section 158 can have any suitable dimensions. In at least one implementation, the inner section 158 can have a length extending between the two outer sections 160 and 162 of about 4 mm to about 12 mm or more. For example, the inner section 158 can have a length of about 4 mm, about 5 mm, about 6 mm, about 7 mm, or about 7.5 mm to about 8.5 mm, about 9 mm, about 10 mm, about 11 mm, about 12 mm, or more. In another example, the inner section 158 can have a length of about 4 mm to about 12 mm, about 5 mm to about 11 mm, about 6 mm to about 10 mm, about 7 mm to about 9 mm, or about 7.5 mm to about 8.5 mm. In a preferred implementation, the inner section 158 can have a length of about 7 mm to about 9 mm, preferably about 7.5 mm to about 8.5 mm, and more preferably about 8 mm. In at least one implementation, the inner section 158 can have a diameter of about 1.2 mm to about 2.0 mm or more. For example, the inner section 158 can have a diameter of about 1.2 mm, about 1.3 mm, about 1.4 mm, about 1.5 mm, or about 1.55 mm to about 1.65 mm, about 1.7 mm, about 1.8 mm, about 1.9 mm, about 2.0 mm, or more. In other examples, inner section 158 can have a diameter of about 1.2 mm to about 2.0 mm, about 1.3 mm to about 1.9 mm, about 1.4 mm to about 1.8 mm, about 1.5 mm to about 1.7 mm, or about 1.55 mm to about 1.65 mm. In preferred implementations, inner section 158 can have a diameter of about 1.5 mm to about 1.7 mm, preferably about 1.55 mm to about 1.65 mm, and more preferably about 1.6 mm.

[0062] The outer sections 160, 162 of the flow channel 144 can be fluidly coupled to the inner section 158 and configured to receive a sample or eluate therefrom. In at least one implementation, at least one of the first and second outer sections 160, 162 can be cylindrical or define a cylindrical volume and have a circular cross-sectional profile. For example, at least one of the first and second outer sections 160, 162 can be sized and shaped similarly to the inner section 158 of FIG. 1D . In another implementation, at least one of the first and second outer sections 160, 162 can have a relatively smaller cross-sectional area at its respective first end portion or inlet 166, 168 than at its respective second end portion or outlet 170, 172. In a preferred implementation, the first outer section 160 and the second outer section 162 can both be frusto-conical or define a truncated cone, with respective first end portions or inlets 166, 168 configured to receive sample from the inner section 158 and respective second end portions or outlets 170, 172 configured to deliver the sample to a waste line 174 (see FIG. 1A).

[0063] The inner surface 154 of the body 130 can at least partially define the taper angles (T1, T2) of the first outer section 160 and the second outer section 162, respectively. For example, as illustrated in FIG. 1D , the portion of the inner surface 154 that defines or forms the first outer section 160 of the flow channel 144 and the centerline 156 of the flow channel 144 can define the taper angle (T1) of the first outer section 160. In another example, the portion of the inner surface 154 that defines or forms the second outer section 162 of the flow channel 144 and the centerline 156 of the flow channel 144 can define the taper angle (T2) of the second outer section 162. The first and second outer sections 160, 162 can have any taper angle (T1, T2) that is functional or configured to enable the LSD 100 and its detectors 110, 112, 114 to receive scattered light at any desired angle. 1D illustrates the taper angles (T1, T2) of the first and second outer sections 160, 162 as being relatively equal to one another, it should be appreciated that one of the taper angles (T1, T2) can be relatively greater than the other. It should be further appreciated that any one or more attributes (e.g., length, taper angle, diameter, shape, size, etc.) of the first and second outer sections 160, 162 can be different. In a preferred implementation, the attributes (e.g., length, taper angle, diameter, shape, size, etc.) of the first outer section 160 and the attributes (e.g., length, taper angle, diameter, shape, size, etc.) of the second outer section 162 are the same or substantially the same.

[0064] Each of the outer sections 160, 162 can be fluidly coupled to a waste line 174 (see FIG. 1A ). For example, as illustrated in FIGS. 1A and 1D , the body 130 can define a first outlet 176 and a second outlet 178 extending therethrough configured to fluidly couple the first outer section 160 and the second outer section 162 to the waste line 174 via a first outlet line 180 and a second outlet line 182, respectively. As illustrated in FIG. 1D , the first and second outlets 176, 178 can be fluidly coupled to second end portions 170, 172 of the outer sections 160, 162, respectively. It should be appreciated that the orientation (e.g., circumferential orientation) or location of the inlet 164 and the first and second outlets 176, 178 can be varied. For example, the inlet 164 can be circumferentially aligned with at least one of the first and second outlets 176, 178. In another example, the inlet 164 can be circumferentially offset from at least one of the first and second outlets 176, 178. In yet another example, the first outlet 176 and the second outlet 178 can be circumferentially aligned with one another or circumferentially offset from one another.

[0065] As illustrated in FIG. 1D , the body 130 of the sample cell 102 can define an opening 184 extending through at least a portion thereof and configured to allow light (e.g., scattered light) from the inner section 158 to be directed or transmitted to the third detector 114. The opening 184 can be sealed with an optically transparent material 186, such as quartz, thereby allowing light from the inner section 158 to be directed to the third detector 114. In the exemplary implementation illustrated in FIGS. 1B and 1D , the optically transparent material 186 can be shaped to refract a portion of the light toward the third detector 114. For example, the optically transparent material 186 can be a sixth lens (e.g., a ball lens) configured to seal the opening 184 and at least partially refract the light toward the third detector 114.

[0066] The body 130 can include or be made of any suitable material. The body 130 can be configured so that its inner surface 154 attenuates light reflection. For example, the body 130 can be made of a non-reflective material. In another example, the body 130 can be at least partially made of a reflective material and at least partially coated with a non-reflective material. In at least one implementation, the sample cell 102 can be made of quartz, such as black quartz. In an exemplary implementation, the body 130 can include or be made of a polymer. Exemplary polymers can be or include, but are not limited to, polyolefin-based polymers, acrylic-based polymers, polyurethane-based polymers, ether-based polymers, polyester-based polymers, polyamide-based polymers, formaldehyde-based polymers, silicone-based polymers, copolymers of any of these, or any combination thereof.For example, the polymer may be poly(ether ether ketone) (PEEK), TORLON®, polyamide-imide, polyethylene (PE), polyvinyl fluoride (PVF), polyvinyl chloride (PVC), polyvinylidene fluoride (PVDF), polyvinylidene chloride (PVDC), polychlorotrifluoroethylene (PCTFE), polytetrafluoroethylene (PTFE), polypropylene (PP), poly(1-butene), poly(4-methylpentene), polystyrene, polyvinylpyridine, polybutadiene, polyisoprene, polychloroprene, styrene-acrylonitrile copolymer, acrylonitrile-butadiene-styrene terpolymer, ethylene-methacrylic acid copolymer, styrene ... The polymer may include, but is not limited to, ethylene-butadiene rubber, tetrafluoroethylene copolymer, polyacrylate, polymethacrylate, polyacrylamide, polyvinyl acetate, polyvinyl alcohol, polyvinyl butyral, polyvinyl ether, polyvinyl pyrrolidone, polyvinyl carbazole, polyurethane, polyacetal, polyethylene glycol, polypropylene glycol, epoxy resin, polyphenylene oxide, polyethylene terephthalate, polybutylene terephthalate, polydihydroxymethylcyclohexyl terephthalate, cellulose ester, polycarbonate, polyamide, polyimide, copolymer of any of these, or any combination thereof. The polymer may be or include, but is not limited to, an elastomer or elastic material, synthetic rubber, or the like. Exemplary elastomeric materials and synthetic rubbers may include, but are not limited to, VITON®, nitrile, polybutadiene, acrylonitrile, polyisoprene, neoprene, butyl rubber, chloroprene, polysiloxane, styrene-butadiene rubber, hydrin rubber, silicone rubber, ethylene-propylene-diene terpolymer, copolymers of any of these, or any combination thereof.

[0067] 1A-1D , in an exemplary operation of the LSD 100, a sample source 104 (e.g., a liquid chromatograph including a gel permeation chromatography column) can inject or direct a sample or eluent (e.g., a diluted particle and / or polymer solution) through line 106 and inlet 164 to and through the flow path 144 of the sample cell 102. As illustrated in FIG. 1D , the sample from the sample source 104 can be directed to the center or middle of the flow path 144 and / or to the inner section 158 of the sample cell 102. As the sample flows to the center of the inner section 158, the sample flow can be split so that a first portion of the sample flows toward a first outer section 160 and a second portion of the sample flows toward a second outer section 162. Portions of these samples within the first and second outer sections 160, 162 may then be directed from the sample cell 102 to first and second outlets 176, 178, respectively, and through first and second outlet lines 180, 182 to a waste line 174.

[0068] The flow rates of the sample through the first outer section 160 and the second outer section 162 can be modified or adjusted (i.e., increased or decreased) by adjusting the lengths of the first outlet line 180 and the second outlet line 182, respectively. In at least one implementation, the flow rates of the first portion of the sample through the first outer section 160 and the second portion of the sample through the second outer section 162 can be the same or substantially the same. For example, the flow rate of the first portion of the sample through the first outer section 160 is the same or substantially the same as the flow rate of the second portion of the sample through the second outer section 162. In another implementation, the flow rates of the first portion of the sample through the first outer section 160 and the second portion of the sample through the second outer section 162 can be different. However, it should be appreciated that a time correction can be applied when the flow rates through the first outer section 160 and the second outer section 162 are different.

[0069] Concurrently with sample flowing through the flow path 144 of the sample cell 102, the laser 108 can emit a light beam 146 through the bore 152 of the second mirror 128 and through the flow path 144 along a centerline 156 of the flow path 144. In at least one implementation, as illustrated in FIG. 1A , the light beam 146 can be transmitted through a third lens 120 that can at least partially focus the light beam 146 along the centerline 156 of the flow path 144. In another implementation, the third lens 120 can be omitted. In at least one implementation, an optional screen or diaphragm 188 can be disposed between the laser 108 and the sample cell 102 and configured to “screen out,” screen out, or otherwise filter stray light (e.g., halos) from the light beam 146. For example, the diaphragm 188 can define a hole or aperture (e.g., an adjustable aperture / iris) capable of or configured to filter stray light from the light beam 146.

[0070] At least a portion of the light beam 146 may travel or be transmitted from the laser 108 to and through the sample cell 102, the first lens 116, the bore 152 of the second mirror 128, and / or the diaphragm 196. For example, at least a portion of the light beam 146 may be transmitted from the laser 108 to and through the sample cell 102, the first lens 116, the bore 152 of the second mirror 128, and / or the optional diaphragm 188 without being obstructed by or interacting with any of the analytes in the sample. The remaining portion of the light beam 146 transmitted through the flow path 144 may interact with or otherwise contact analytes suspended, dispersed, or otherwise contained in the sample and / or flowing through the sample cell 102.

[0071] Contact between the light beam 146 and the analytes in the sample may generate or induce scattered light or analyte scattered beams 190, 192, 194 (see FIGS. 1A and 1B). For example, contact between the light beam 146 and the analytes contained in the sample or flowing through the flow path 144 of the sample cell 102 may generate forward or reverse analyte scattered beams 190, 192. The forward analyte scattered beam 190 may be referred to as low-angle light scatter (LALS). The reverse analyte scattered beam 192 may be referred to as high-angle light scatter (HALS). In another example, contact between the light beam 146 and the analytes contained in the sample or flowing through the flow path 144 of the sample cell 102 may generate a scattered beam 194 at a right angle (e.g., about 90° relative to the centerline 156) in a direction approximately perpendicular to the light beam 146. The right-angle scattered beam 194 may be referred to as right-angle light scatter (RALS).

[0072] It should be appreciated that flow of sample through inlet 164 to the center of channel 144 allows the sample to interact with light beam 146 instantaneously, thereby minimizing peak broadening. For example, flowing sample directly into the center of channel 144 allows the sample to interact with light beam 146 without flowing (laterally or axially) through at least half the length or volume of sample cell 102 and its channel 144. Flowing sample directly into the center of channel 144 also minimizes the amount of time required for the sample to interact with light beam 146 and generate analyte scattered beams 190, 192, 194. It should further be appreciated that one or more components of LSD 100 are configured such that only light scattered from the center of channel 144 is collected by detectors 110, 112, 114. For example, at least one of the first lens 116, the first mirror, and the fourth lens 122 can be configured to distinguish forward light scatter 190 emanating from the center of the flow channel 144 from forward light scatter 190 emanating from other regions of the flow channel 144, such that the first detector 110 receives only forward light scatter 190 emanating from the center of the flow channel 144. Similarly, at least one of the second lens 116, the second mirror 128, and the fifth lens 124 can be configured to distinguish backward light scatter 192 emanating from the center of the flow channel 144 from backward light scatter 192 emanating from other regions of the flow channel 144, such that the second detector 112 receives only backward light scatter 192 emanating from the center of the flow channel 144.

[0073] It should be appreciated that in at least one implementation, the sample can flow through the flow path 144 of the sample cell 102 in the "flow mode" described above, and in another embodiment, can be evaluated or analyzed in a "batch mode." In at least one exemplary operation of the LSD 100 operating in a "batch mode," the sample can be placed into a sample cell, such as a cuvette, without an inlet and / or outlet stream. In another exemplary operation of the LSD 100 operating in a "batch mode," the sample can be placed into the sample cell 102 without an inlet and / or outlet stream (e.g., by a pipette or other manual device). In yet another exemplary operation, the sample can be placed into the inlet 164 of the sample cell 102 such that the sample is not separated. For example, the sample can be placed directly into the inlet 164 without separation by a column or using a non-separation column (e.g., through fluid injection polymer analysis). In each of the aforementioned operations, the laser 108 can emit a light beam 146 along a centerline 156 of the sample cell (e.g., a cuvette) and through the sample cell. In at least one implementation, the light beam 146 can be transmitted through a third lens 120 that can at least partially focus it. In another implementation, the third lens 120 can be omitted. In at least one implementation, an optional screen or diaphragm 188 can be disposed between the laser 108 and the sample cell (e.g., a cuvette) and configured to "screen out," filter out, or otherwise filter stray light (e.g., halos) from the light beam 146. For example, the diaphragm 188 can define a hole or aperture (e.g., an adjustable aperture / iris) that functions or is configured to filter stray light from the light beam 146.

[0074] 1A , the forward analyte scattered beam, or forward scattered light 190, can be directed through a first lens 116, a first mirror 126, and a fourth lens 122 toward the first detector 110. At least a portion of the forward scattered light 190 can be at least partially refracted by a convex surface defined along a first end portion 136 of the first lens 116. As illustrated in FIG. 1A , the forward scattered light 190 can be refracted by this convex surface toward the first mirror 126, which can reflect the forward scattered light 190 through the fourth lens 122 toward the first detector 110. The fourth lens 122 can collect the forward scattered light 190 and direct and / or focus the forward scattered light 190 toward the first detector 110.

[0075] The forward scattered light 190 (LALS) may be scattered at various angles greater than 0° and less than 90° relative to the light beam 146 emitted from the laser 108 and / or the centerline 156 of the flow channel 144. For example, the forward scattered light 190 may be scattered at an angle greater than 0°, about 5°, about 10°, about 15°, about 20°, about 25°, about 30°, about 35°, about 40°, or about 45° to about 50°, about 55°, about 60°, about 65°, about 70°, about 75°, about 80°, about 85°, or any angle less than 90°. In another example, the forward scattered light 190 may be scattered at any angle of about 5°, about 6°, about 7°, about 8°, about 9°, or about 9.5° to about 10.5°, about 11°, about 12°, about 13°, about 14°, or about 15° relative to the light beam 146 emitted from the laser 108 and / or the centerline 156 of the flow channel 144. In yet another example, the forward scattered light 190 may be scattered at an angle of about 5° to about 15°, about 6° to about 14°, about 7° to about 13°, about 8° to about 12°, about 9° to about 11°, or about 9.5° to about 10.5°. It should be appreciated that the LSD 100 and any of its components may be configured to receive forward scattered light 190 scattered at any angle greater than 0° and less than 90°. For example, any attribute (e.g., shape, location, orientation, etc.) of one or more of the first detector 110, first lens 116, first mirror 126, fourth lens 122, and / or any additional optional diaphragms can be adjusted, modified, or otherwise configured to enable the first detector 110 to receive any of the forward scattered light 190. In a preferred implementation, the LSD 100 and its first detector 110 are configured to receive or collect the forward scattered light 190 at an angle of about 9° to about 11°, preferably about 9.5° to about 10.5°, and more preferably about 10°, relative to the light beam 146 and / or the centerline 156 of the flow channel 144. It should be appreciated that the forward scattered light 190 can be scattered at 360° minus any of the aforementioned angles.

[0076] 1A , the reverse analyte scattered beam or reverse light scatter 192 can be directed through the second lens 118, the second mirror 128, and the fifth lens 124 toward the second detector 112. At least a portion of the reverse scattered light 192 can be at least partially refracted by the convex surface of the second lens 118. As illustrated in FIG. 1A , the reverse scattered light 192 can be refracted by this convex surface toward the second mirror 128, which can reflect the reverse scattered light 192 through the fifth lens 124 toward the second detector 112. The fifth lens 124 can collect and / or direct and / or focus the reverse scattered light 192 toward the second detector 112.

[0077] The backward scattered light 192 (HALS) may be scattered at various angles greater than 90° and less than 180° relative to the light beam 146 emitted from the laser 108 and / or the centerline 156 of the flow channel 144. For example, the backward scattered light 192 may be scattered at any angle from greater than 90°, about 95°, about 100°, about 105°, about 110°, about 115°, about 120°, about 125°, about 130°, or about 135° to about 140°, about 145°, about 150°, about 155°, about 160°, about 165°, about 170°, about 175°, or less than 180°. In another example, the backscattered light 192 may be scattered at an angle of about 165°, about 166°, about 167°, about 168°, about 169°, or about 169.5° to about 170.5°, about 171°, about 172°, about 173°, about 174°, or about 175° relative to the light beam 146 emitted from the laser 108 and / or the centerline 156 of the flow channel 144. In yet another example, the backscattered light 192 may be scattered at an angle of about 165° to about 175°, about 166° to about 174°, about 167° to about 173°, about 168° to about 172°, about 169° to about 171°, or about 169.5° to about 170.5°. It should be appreciated that the LSD 100 and any of its components can be configured to receive back-scattered light 192 scattered at any angle greater than 90° and less than 180°. For example, any attribute (e.g., shape, location, orientation, etc.) of one or more of the second detector 112, second lens 118, second mirror 128, fifth lens 124, and / or any additional optional diaphragms can be adjusted, modified, or otherwise configured to enable the second detector 112 to receive any of the back-scattered light 192. In a preferred implementation, the LSD 100 and its second detector 112 are configured to receive or collect the forward-scattered light 190 at an angle of about 169° to about 171°, preferably about 169.5° to about 170.5°, and more preferably about 170°, relative to the light beam 146 and / or the centerline 156 of the flow channel 144. It should be appreciated that the backscattered light 192 can be 360° minus (-) scattered at any of the aforementioned angles.

[0078] As illustrated in FIG. 1D , the right-angle analyte scattered beam, or right-angle scattered light 194, can be directed toward the third detector 114 through an opening 184 extending between the third detector 114 and the inner section 158 of the flow channel 144. In at least one implementation, the third detector 114 can be positioned within the opening 184 adjacent to the inner section 158. In another implementation illustrated in FIG. 1D , an optically transparent material 186 can be disposed within the opening 184 to seal the inner section 158 of the flow channel 144. The optically transparent material 186 can be any suitable material capable of allowing the right-angle scattered light 194 to be transmitted to the third detector 114. The optically transparent material 186 can be shaped to refract at least a portion of the right-angle scattered light 194 toward the third detector 114. For example, as previously described, the optically transparent material 186 can be a ball lens shaped to refract the right-angle scattered light 194 toward the third detector 114.

[0079] The right angle scattered light 194 (RALS) may be scattered in a direction approximately perpendicular to the light beam 146 and / or the centerline 156 of the flow channel 144. For example, the right angle scattered light 194 may be scattered at an angle of about 87°, about 88°, about 89°, about 89.5°, or about 90° to about 90.5°, about 91°, about 92°, or about 93°. In other examples, the right angle scattered light 194 may be scattered at an angle of about 87° to about 93°, about 88° to about 92°, about 89° to about 91°, or about 89.5° to about 90.5°. It should be appreciated that the LSD 100 and any of its components may be configured to receive right angle scattered light 194 scattered in a direction approximately perpendicular to the light beam 146 and / or the centerline 156 of the flow channel 144. For example, the shape, location, orientation, or any other attribute of the optically transparent material 186 (e.g., the sixth lens) and / or the third detector 114 can be adjusted, modified, or otherwise configured to enable the third detector 114 to receive any of the right-angle scattered light 194. In a preferred implementation, the LSD 100 and its third detector 114 are configured to receive or collect the right-angle scattered light 194 at an angle of about 89° to about 91°, preferably about 89.5° to about 90.5°, and more preferably about 90°, relative to the centerline 156 of the light beam 146 and / or the flow channel 144. It should be appreciated that the right-angle scattered light 194 can be scattered at any of the aforementioned angles plus (+) 180°.

[0080] In at least one implementation, the LSD 100 is capable of receiving, analyzing, collecting, and / or otherwise measuring light scattered at one or more of the following angles: 12°, 20°, 22.5°, 28°, 32°, 36°, 38°, 44°, 50°, 52°, 57°, 60°, 64°, 68°, 72°, 76°, 81°, 84°, 90°, 99°, 100°, 108°, 116°, 117°, 124°, 126°, 132°, 134°, 140°, 141°, 147°, 148°, 156°, 164°, or any combination thereof. It should be appreciated that any one or more of the angles disclosed herein, such as the foregoing angles, can be adjusted and / or modified. It should further be appreciated that the adjustment and / or modification of one or more angles may depend, at least in part, on the refractive index of the cell, the refractive index of the solvent, or a combination thereof. In at least one implementation, no adjustment or modification of angles is required.

[0081] Method for determining turning radius or no-assumption turning radius

[0082] The present disclosure can provide a method for determining one or more properties of one or more particles in a solution using a light scattering detector. For example, the present disclosure can provide a method for determining the radius of gyration (Rg) of one or more particles (e.g., nanoparticles, microparticles, etc.) in a solution using a light scattering detector such as the LSD100 disclosed herein. For example, the present disclosure can provide a method for determining the assumption-free radius of gyration (Rg) of particles in a solution by analyzing (e.g., by an electronic processor or computer system) data from a light scattering detector (e.g., the LSD100). AF ) can be provided.

[0083] While reference may be made to the LSD 100 and its components described herein, it should be appreciated that the method for determining the assumption-free radius of gyration (Rg) can be performed or implemented using any suitable light scattering detector. For example, any multi-angle light scattering detector or any light scattering detector capable of or configured to observe light scattering over at least two or more angles (e.g., at least two, three, or four or more angles) can utilize the methods disclosed herein. For example, a light scattering detector capable of or configured to observe light scattering over or using at least three angles, such as the LSD 100 described above, can be utilized with the methods disclosed herein. In another example, a light scattering detector including a goniometer capable of or configured to observe light scattering over at least two or more angles can utilize the methods disclosed herein. The methods disclosed herein are also not limited to measuring each of two or more angles simultaneously or substantially simultaneously. For example, a light scattering detector capable of or configured to measure a first angle and then measure a second or subsequent angle (e.g., with a goniometer) can utilize the methods disclosed herein. In addition to the foregoing, while the methods disclosed herein may be described with reference to three angles (θ1, θ2, θ3), it should be appreciated that the use of additional angles is contemplated and within the scope of the present disclosure. Furthermore, those skilled in the art should appreciate that increasing the number of angles may improve the accuracy of the results. However, it should be appreciated that increasing the number of angles will only result in a marginal improvement in accuracy. For example, the accuracy of the results may depend at least in part on the respective quality of each of the angles. The quality of each of the angles may depend at least in part on one or more variables, including, but not limited to, signal-to-noise, any respective angles, or a combination thereof.

[0084] As used herein, the term or phrase "assumption-free radius of gyration (Rg)" or the like can refer to a radius of gyration (Rg) derived or determined from light scattering (e.g., multi-angle light scattering) and two or more shape factors. For example, the term or phrase "assumption-free radius of gyration (Rg)" can refer to a radius of gyration (Rg) based on the slope at zero angle of a multi-angle light scattering measurement, such as a constrained extrapolation of normalized excess scattering, or a radius of gyration (Rg) derived from the slope as a composite result of two or more shape factors combined (mathematically combined) to determine a composite average molecular shape and / or molecular size that can be represented by one or more of the respective shape factors. In another example, the "assumption-free radius of gyration (Rg)" can refer to a radius of gyration (Rg) derived from multi-angle light scattering and two or more shape factors using weighting coefficients to generate a simultaneous composite extrapolation to molecular weight (MW), radius of gyration at zero angle (Rg), or a combination thereof. The assumption-free radius of gyration (Rg) can refer to the radius of gyration (Rg) and / or molecular weight (MW) derived from multi-angle light scattering and two or more shape factors without fitting the Zimm, Berry, Debye, or Guinier plot to a polynomial function. For example, the assumption-free radius of gyration (Rg) can be determined without fitting the Zimm plot, Berry plot, Debye plot, Guinier plot, or a combination thereof to a polynomial function.

[0085] A method for determining Rg of particles in a solution using a light scattering detector can include placing the particles in solution in a sample cell, such as the sample cell 102 of the LSD 100. For example, the method can include advancing or flowing the particles in solution in the sample cell 102 of the LSD 100 through a channel 144 having a centerline 156 aligned with the light beam 146 of the LSD 100. In another example, the particles in solution can be placed in a sample cell (e.g., a cuvette) and not flowed through a channel in the sample cell.

[0086] The one or more particles in the solution can be or include, but are not limited to, nanoparticles, microparticles, macroparticles, macromolecules, cross-linked molecules, natural particles, synthetic particles, polymeric particles, particles with multicomponent structures, proteins, antibodies, virus-like particles (VLPs), particles or molecules comprising structures or features that can be at least partially represented by or are rod-shaped, spherical, hollow, donut-shaped, disc-shaped, cylindrical, worm-shaped, or spiral-shaped rigid chains distributed regularly and / or randomly throughout each of the particles or molecules, or the like, or combinations and / or aggregates thereof. The one or more particles can be or include rigid particles, semi-rigid particles, or combinations thereof. Exemplary multicomponent structures can be or include, but are not limited to, copolymers with random or ordered incorporation, comb-branched polymers, or polymer blends having a mixture of ring-, coil-, and rod-like structures, individual molecules containing a mixture of these structural elements, or combinations thereof.

[0087] The one or more particles can have one or more radii of gyration (Rg) of about 1 nm to about 1 μm, respectively. For example, the one or more particles can have a respective radius of gyration (Rg) of about 1 nm, about 5 nm, about 20 nm, about 30 nm, about 50 nm, or about 100 nm to about 150 nm, about 200 nm, about 250 nm, about 300 nm, about 400 nm, about 500 nm, about 600 nm, about 800 nm, or about 1 μm, respectively. In exemplary implementations, the one or more particles can have a respective radius of gyration (Rg) of about 5 nm to about 500 nm, about 30 to about 250 nm, or about 50 to about 150 nm.

[0088] The one or more particles in the solution can be dissolved in the solution, suspended in the solution, or otherwise contained in the solution. In at least one implementation, the particles in the solution can be disposed in a mobile phase. The particles in the solution can be monodisperse particles. For example, the particles in the solution can have substantially the same molecular weight. The particles in the solution can also be polydisperse particles. For example, the particles in the solution can have substantially different molecular weights. In at least one implementation, the particles in the solution can be homogeneous. For example, the particles in the solution can be substantially the same with respect to one or more characteristics. In another implementation, the particles in the solution can be heterogeneous. For example, the particles in the solution can be substantially different with respect to one or more characteristics. Exemplary particle characteristics can be or include, but are not limited to, molecular weight, including statistically averaged characteristics, structure, chemical composition, shape, molecular conformation, size, viscosity, diffusion rate, or the like, or a combination thereof.

[0089] In at least one implementation, one or more particles in solution can be provided by one or more liquid separation processes or devices capable of or configured to receive a bulk liquid phase and separate one or more particles or analytes in the bulk phase from one another. For example, one or more particles in solution can be provided by a sample source or device 104, illustrated in FIG. 1A, capable of or configured to perform a liquid separation process. Exemplary liquid separation processes can be or include, but are not limited to, size exclusion chromatography (SEC), gel permeation chromatography (GPC), flow field fractionation (FFF), temperature gradient interaction chromatography (TGIC), temperature rising elution fractionation (TREF), crystallization elution fractionation (CEF), two-dimensional liquid chromatography (2D-LC), hydrodynamic chromatography (HDC), capillary electrokinetic chromatography (CEC), supercritical fluid chromatography (SFC), crystallization fractionation (CRYSTAF), critical conditions liquid chromatography (LCCC), gradient polymer elution chromatography (GPEC), phase fluctuation chromatography (PFC), or the like, or a combination thereof.

[0090] The method measures the respective scattering intensities (I) of particles by a light beam of a light scattering detector at each of two or more angles (θ). θ For example, the method may include acquiring a first scattering intensity (I) of particles in the solution by a light beam of a light scattering detector. θ1 ), the second scattering intensity (I θ2 ), and the third scattering intensity (I θ3 ) at a first angle (θ1), a second angle (θ2), and a third angle (θ3), respectively, where each of the angles (θ1, θ2, θ3) is an angle relative to the light beam 146. θ1 ), the second scattering intensity (I θ2 ), and the third scattering intensity (I θ3It should be appreciated that the respective excess scattering intensities at each of the first, second, and third angles can be normalized for quantum efficiency, gain, geometric volume, or a combination thereof. For example, the method can include normalizing one or more angles of the light scattering detector or determining an angle normalization factor for one or more angles of the light scattering detector. Determining an angle normalization factor for one or more angles of the LSD 100 can be performed to account for differences in scattering volume of the LSD 100 or different sensitivities of any one or more of the detectors of the LSD, such as detectors 110, 112, 114 of the LSD 100.

[0091] In at least one implementation, the respective scattering intensities (I) of particles in solution by the light beam of a light scattering detector at each of two or more angles (θ) are measured. θ ) may be acquired by measuring the respective scattering intensities (I) at two or more of the angles 22.5°, 28.0°, 32.0°, 38.0°, 44.0°, 50.0°, 57.0°, 64.0°, 72.0°, 81.0°, 90.0°, 99.0°, 108.0°, 117.0°, 126.0°, 134.0°, 141.0°, 147.0°, or any combination thereof. θ ) can be included. It should be appreciated that any one or more of the angles disclosed herein, such as the angles described above, can be adjusted and / or modified. It should further be appreciated that the adjustment and / or modification of one or more angles can depend, at least in part, on the refractive index of the cell, the refractive index of the solvent, or a combination thereof. In at least one implementation, no adjustment or modification of the angles is required.

[0092] A method for determining Rg or hypothesis-free Rg of a particle in a solution can include utilizing two shape factors (FF) to determine at least two respective weighting factors (FW). For example, the method can include determining a first weighting factor (FW1) based on a first shape factor (FF1) and determining a second weighting factor (FW2) based on a second shape factor (FF2). It should be appreciated that the method can include utilizing any number of shape factors (FF) and determining a respective weighting factor (FW) for each of these shape factors. For example, the method can include determining a first weighting factor (FW1), a second weighting factor (FW2), and a third weighting factor (FW3) based on a first shape factor (FF1), a second shape factor (FF2), and a third shape factor (FF3). It should further be appreciated that increasing the number of shape factors utilized in the method can at least partially improve the determined Rg or hypothesis-free Rg. For example, increasing the number of shape factors utilized in the method can at least partially improve one or more, or a combination thereof, of accuracy, precision, significance, robustness, convergence, robustness, specificity, selectivity between particle structural characteristics (e.g., shape, branching), or the like. The method can also include determining a respective shape factor contribution (FC) for each of the shape factors (FF) based on a respective weighting factor (FW). For example, the method can include determining a first shape factor contribution (FC1) and a second shape factor contribution (FC2) for a first shape factor (FF1) and a second shape factor (FF2) based on a first weighting factor (FW1) and a second weighting factor (FW2), respectively. The method can also include determining an assumption-free radius of gyration (Rg) using each of the shape factor contributions (FC). For example, the method can include determining an assumption-free radius of gyration (Rg) using the first and second shape factor contributions (FC1, FC2).

[0093] In an exemplary implementation, a method for determining Rg or assumption-free Rg of a particle in a solution can include determining a first weighting factor (FW1) based on a first shape factor (FF1), determining a second weighting factor (FW2) based on a second shape factor (FF2), determining a first shape factor contribution (FC1) and a second shape factor contribution (FC2) to the first shape factor (FF1) and the second shape factor (FF2) based on the first weighting factor (FW1) and the second weighting factor (FW2), respectively, and determining an assumption-free radius of gyration (Rg) using the first and second shape factor contributions (FC1, FC2).

[0094] As used herein, the term or phrase "shape factor" can refer to scattering resulting from one or more characteristics or structural properties of one or more particles. For example, the term or phrase "shape factor" can refer to a normalized excess scattering pattern predicted (e.g., from models proposed in the literature or by mathematical derivation) as a function of observation angle resulting from one or more characteristics of one or more particles. The one or more characteristics can be or include, but are not limited to, shape, conformation, size, molecular weight, or any combination thereof. The shape factor can be represented by an expression or formula. The expression or formula representing the shape factor can be a function of any one or more characteristics of one or more particles. It should be appreciated that any shape factor known to those skilled in the art can be utilized for the methods disclosed herein. It should be further appreciated that any shape factor that may be developed or determined in the future is contemplated for use in the methods disclosed herein. Specifically, it should be appreciated that the methods disclosed herein can utilize any particular shape factor and are not limited thereby.

[0095] Exemplary shape factors can be or include, but are not limited to, each of the following shape factors: rigid particles, semi-rigid particles, polymers, anisotropic particles with local planar geometries, anisotropic particles with local cylindrical geometries, spheres, rods or rods, worm-like geometries, rigid-chain structures, helical structures, brush-like structures, bottle brushes, ellipsoids, core-shell particles, prolate ellipsoids with a shell of constant thickness, cylinders, hollow cylinders, glucagon fibrils, particles with arbitrary shapes, dissolved polymer chains, closed-bottom cylinders, block copolymer micelles, discs, lenticular discs, star polymers, multi-branched star polymers, ribbon-like ribbons in solution, hollow cylinders, amino acids, helical nanostructures, random coils, functional representations of scattered light from charge-induced condensation or elongation, flow effects or steric effects, or the like, or combinations thereof. Any one or more of these shape factors can represent homogeneous particles, heterogeneous particles, monodisperse particles, and / or polydisperse particles.

[0096] Shape factors for polymers include spheres, rods or rod-like particles, Gaussian particles, random coils, flexible polymers with Gaussian distribution, polydisperse flexible polymers with Gaussian distribution, flexible cyclic polymers with Gaussian distribution, flexible self-avoiding polymers, polydisperse flexible self-avoiding polymers, semi-flexible polymers without self-avoiding, semi-flexible polymers with self-avoiding, polyelectrolyte semi-flexible polymers with self-avoiding, star polymers with Gaussian distribution, polydisperse star polymers with Gaussian distribution, ordered starburst polymers (dendrimers) with Gaussian distribution, polycondensation polymerized structures including combinations of various monomers, structural modifications of preformed polymers, e.g. The shape factors can be or include, but are not limited to, regular comb polymers exhibiting a Gaussian distribution, such as polysaccharides, proteins, or peptides, arbitrarily branched polymers exhibiting a Gaussian distribution, arbitrarily branched semi-maximally flexible polymers, arbitrarily branched self-avoiding polymers, Gaussian chain-attached spheres, Gaussian chain-attached ellipsoids, Gaussian chain-attached cylinders, polydisperse thin-walled cylinders with polydisperse Gaussian chains attached to their ends, semi-flexible interactions of corona chains, spheres with a corona of self-avoiding chains, functional representations of scattered light due to branching of long or short chains, due to aggregation, from condensation due to crosslinking, or from stretching due to charge, flow effects, or steric effects, or the like, or combinations thereof.

[0097] Shape factors for homogeneous rigid particles can be or include, but are not limited to, each of the following shape factors: homogeneous spheres, spherical shells, spherical concentric shells, particles of spherical subunits, spheroids, triaxial ellipsoids, cubes and rectangular prisms, truncated octahedrons, polyhedral spheres, lenses, stepped cubes, cylinders, elliptical cylinders, hemispherical end cylinders, "half-lens" end cylinders, tori, thin rods, ultra-thin disks, fractal aggregates, or the like, or combinations thereof.

[0098] Shape factors for anisotropic particles with local planar geometry can be or include, but are not limited to, shape factors of two ultrathin planes, a layered centrosymmetric cross section, and / or a homogeneous cross section such as a Gaussian chain attached to a surface, an ultrathin spherical shell, an ellipsoidal shell, a cylindrical shell, an ultrathin disk, or the like, or combinations thereof.

[0099] Shape factors for anisotropic particles with local cylindrical geometry can be or include, but are not limited to, shape factors for a homogeneous circular cross section, concentric circular shells, elliptical homogeneous cross section, elliptical concentric shells, Gaussian chains attached to a surface, thin rods, semi-flexible polymer chains with or without excluded volume, or the like.

[0100] In an exemplary implementation, determining the R or assumption-free R of a particle in solution includes determining the form factor for random coil (FF ランダムコイル ), the shape factor for rod-shaped particles (FF Rod ), the form factor for a sphere (FF Sphere ), or any combination thereof. ランダムコイル ), the shape factor for rod-shaped particles (FF Rod ), and the form factor for a sphere (FF Sphere ) can be expressed by the following equations (1), (2), and (3), respectively. TIFF0007749813000017.tif10150(1) where: TIFF0007749813000018.tif17150n0 is the refractive index of the solvent containing the particles, θ is the respective angle (e.g., about 90° or about 170°), λ0 is the wavelength of the light beam. TIFF0007749813000019.tif11150(2) where: TIFF0007749813000020.tif16150n0 is the refractive index of the solvent containing the particles, θ is the respective angle (e.g., about 90° or about 170°), λ0 is the wavelength of the light beam. TIFF0007749813000021.tif9150(3) where: TIFF0007749813000022.tif16150n0 is the refractive index of the solution containing the particles, θ is the respective angle (e.g., about 90° or about 170°), λ is the wavelength of the light beam, R = radius of the sphere The file is TIFF0007749813000023.tif6150.

[0101] As used herein, P シータ (P θ The term or expression P theta (P ) can refer to the ratio between the actual light scattering and the scattering that is believed to occur from a particle. θ The term or expression θ can refer to the ratio between the actual light scattering and the scattering that would occur from a particle at zero angle or angle theta.

[0102] Determination of respective weighting factors (FW) based on each of the form factors (FF)

[0103] As discussed above, a method for determining the Rg or hypothesis-free Rg of a particle in a solution can include determining at least two respective weighting factors (FW) utilizing at least two shape factors (FF). For example, the method can include determining a first weighting factor (FW1) based on a first shape factor (FF1) and determining a second weighting factor (FW2) based on a second shape factor (FF2).

[0104] As used herein, the term or phrase "weighting factor" or "FW" may refer to a value or quantity that is proportional to the overall "goodness of fit" and / or inversely proportional to the residual error or a function thereof. It should be appreciated that a weighting factor (FW) with a relatively stronger correlation may indicate or represent a higher selectivity or normalized multiplier compared to a relatively weaker correlation.

[0105] In at least one implementation, determining the first weighting factor (FW1) includes determining a theoretical or calculated ratio of light scattering at a first angle to calculated light scattering at a third angle based on the first shape factor (FF1). TIFF0007749813000024.tif8150 and the ratio of calculated light scattering at the second angle to calculated light scattering at the third angle determining a ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle; TIFF0007749813000026.tif8150 or the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle The step of determining TIFF0007749813000027.tif8150 includes determining a calculated particle scattering factor (P) for the first angle. θ1計算値FF1 ) and the calculated particle scattering factor for the second angle (P θ2計算値FF1 ) and the calculated particle scattering factor for the third angle (P θ3計算値FF1 ) each based on the first shape factor (FF1).

[0106] In at least one implementation, determining the first weighting factor (FW1) comprises determining a ratio of the observed light scattering of the particle at the first angle to the observed light scattering of the particle at the third angle (R' θ1 / θ3 ) and the ratio of the observed light scattering of the particle at the second angle to the observed light scattering of the particle at the third angle (R' θ2 / θ3 ) and determining a ratio (R') of the observed light scattering of the particle at the first angle to the observed light scattering of the particle at the third angle. θ1 / θ3) and the ratio of the observed light scattering of the particle at the second angle to the observed light scattering of the particle at the third angle (R' θ2 / θ3 ) is the first scattering intensity (I) of a particle in solution measured at a first angle (θ1), a second angle (θ2), and a third angle (θ3) using the LSD, respectively. θ1 ), the second scattering intensity (I θ2 ), and the third scattering intensity (I θ3 ) can be determined using

[0107] In at least one implementation, determining the first weighting factor (FW1) comprises determining a ratio of calculated light scatter at a first angle to calculated light scatter at a third angle. The ratio of the observed light scattering at the first angle to the observed light scattering at the third angle (R' θ1 / θ3 ) the best fit at or between the calculated light scattering at the second angle to the calculated light scattering at the third angle. The ratio of the observed light scattering at the second angle to the observed light scattering at the third angle (R' θ2 / θ3 ) or the best fit therebetween. Ratio (R') to TIFF0007749813000030.tif8150 θ1 / θ3 ) and ratio Ratio (R') to TIFF0007749813000031.tif8150 θ2 / θ3 The step of determining the best fit of the ratio (R' θ1 / θ3 ) and TIFF0007749813000032.tif8150 and / or ratio (R' θ2 / θ3 ) and TIFF0007749813000033.tif8150. For example, the optimization function may be a function of optimizing the fit between the ratio (R' θ1 / θ3 ) and TIFF0007749813000034.tif8150 and / or ratio (R' θ2 / θ3 ) and TIFF0007749813000035.tif8150 and then determine the minimum difference or discrepancy therebetween. In another example, the optimization function may be any function configured or capable of measuring the difference or discrepancy between the two images (R' θ1 / θ3 ) and TIFF0007749813000036.tif8150 and the difference and ratio (R' θ2 / θ3 ) and TIFF0007749813000037.tif8150, and then determining the minimum difference therebetween (e.g., a subtraction function). Any optimization function suitable for, capable of, or configured to determine the best fit can be utilized. Exemplary optimization functions can be or include, but are not limited to, minimum sum of squares residual, minimum absolute sum of residuals, custom optimizers, convergence obtained from commercially available optimizers such as the Microsoft Excel® Solver, grid search, Newton's method or gradient method, or the like, or combinations thereof. It should be appreciated that the residuals can be normalized by a function proportional to the sampled values.

[0108] In at least one implementation, the ratio (R' θ1 / θ3 ) and TIFF0007749813000038.tif8150 and ratio (R' θ2 / θ3 ) and Determining the best fit of TIFF0007749813000039.tif8150 may involve an iterative process. For example, an optimization function may be used to determine the ratio (R' θ1 / θ3 ) and TIFF0007749813000040.tif8150 and the ratio (R' θ2 / θ3 ) and Determining the best fit between the image and the image may involve an iterative process. For example, the ratio (R' θ1 / θ3 ) and TIFF0007749813000042.tif8150 and ratio (R' θ2 / θ3 ) and Determining the best fit of TIFF0007749813000043.tif8150 can include iterating through the first shape factor (FF1) using a series of theoretical Rg values. The series of theoretical Rg values ​​can be or include incremental values, decremental values, random values, or combinations thereof. In at least one implementation, the ratio (R' θ1 / θ3 ) and TIFF0007749813000044.tif8150 and ratio (R' θ2 / θ3 ) and Determining the best fit of TIFF0007749813000045.tif8150 can include iterating through the value of the first form factor (FF1) while incrementing or decrementing a predetermined or random theoretical Rg value, for example, a theoretical Rg value starting at about 1 nm. The theoretical Rg value can be increased or decreased in increments of about 0.1 nm or about 0.01 nm. g The value is the ratio (R' θ1 / θ3 ) and The difference and / or ratio (R') between θ2 / θ3 ) and TIFF0007749813000047.tif8150 is minimized or minimized, indicating a best fit between them (in increments of about 0.01 nm).

[0109] In at least one implementation, determining the first weighting factor (FW1) comprises determining a ratio (R' θ1 / θ3 ) and TIFF0007749813000048.tif8150 and ratio (R' θ2 / θ3 ) and The method may further include determining a radius of gyration (Rg1) based on a first shape factor (FF1) associated with the best fit of the TIFF0007749813000049.tif8150. θ1 / θ3 ) and TIFF0007749813000050.tif8150 and ratio (R' θ2 / θ3 ) and Determining Rg1 based on a first shape factor (FF1) associated with the best fit of TIFF0007749813000051.tif8150 can include determining Rg1 based on a first shape factor (FF1) associated with an optimization function. For example, in at least one example implementation, determining the radius of gyration (Rg1) associated with the best fit can include determining a ratio of observed light scatter at a first angle to observed light scatter at a third angle (R' θ1 / θ3 ) and the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle TIFF0007749813000052.tif8150, and the ratio of the observed light scatter at the second angle to the observed light scatter at the third angle (R' θ2 / θ3 ) and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle The method may include determining Rg1 based on a first shape factor (FF1) associated with a smallest sum of squares residual (SSR1) of the differences between the first and second images.

[0110] In at least one implementation, determining the first weighting factor (FW1) comprises determining a ratio (R' θ1 / θ3 ) and TIFF0007749813000054.tif8150 and ratio (R' θ2 / θ3 ) and The method may include utilizing variables or values ​​associated with the best fit of the first weighting factor (FW1). For example, the first weighting factor (FW1) may be determined using any suitable calculation, equation, or expression incorporating variables or values ​​associated with the best fit. In at least one implementation, as discussed above, the best fit may be determined using a ratio of the observed light scatter at the first angle to the observed light scatter at the third angle (R' θ1 / θ3 ) and the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle TIFF0007749813000056.tif8150, and the ratio of the observed light scatter at the second angle to the observed light scatter at the third angle (R' θ2 / θ3 ) and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle TIFF0007749813000057.tif8150. Accordingly, the first weighting factor (FW1) can be determined by any suitable calculation, equation, or representation utilizing a residual, such as the minimum sum of squares residual. In an exemplary implementation, the first weighting factor (FW1) is determined as the reciprocal of the sum of squares residual (SSR1) according to the following equation (4a): 2 can be determined as the reciprocal of FW1=1 / SSR1(4a) FW1=1 / SAR1(4b) FW1=1 / SAR1 2 (4c) Here, SSR1 may be the smallest sum of squares residual associated with the best fit with the first shape factor (FF1), where SSR1 may be normalized to the ratio of observed or calculated scatter by a function. For example, SSR1 may be normalized to the ratio of observed or calculated excess scatter by a function.

[0111] In at least one implementation, determining the second weighting factor (FW2) includes determining a theoretical or calculated ratio of light scattering at a first angle to calculated light scattering at a third angle based on the second shape factor (FF2). TIFF0007749813000058.tif8150 and the ratio of calculated light scattering at the second angle to calculated light scattering at the third angle determining a ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle; TIFF0007749813000060.tif8150 or the ratio of calculated light scattering at the second angle to calculated light scattering at the third angle The step of determining TIFF0007749813000061.tif8150 includes determining a calculated particle scattering factor (P) for the first angle. θ1計算値FF2 ) and the calculated particle scattering factor for the second angle (P θ2計算値FF2 ) and the calculated particle scattering factor for the third angle (P θ3計算値FF2 ) each based on the second shape factor (FF2).

[0112] In at least one implementation, determining the second weighting factor (FW2) comprises determining a ratio of the calculated light scatter at the first angle to the calculated light scatter at the third angle. The ratio of the observed light scattering at the first angle to the observed light scattering at the third angle (R' θ1 / θ3 ) the best fit at or between the calculated light scattering at the second angle to the calculated light scattering at the third angle. The ratio of the observed light scattering at the second angle to the observed light scattering at the third angle (R' θ2 / θ3 ) or the best fit therebetween. Ratio (R') to TIFF0007749813000064.tif8150 θ1 / θ3 ) and ratio Ratio (R') to TIFF0007749813000065.tif8150 θ2 / θ3 The step of determining the best fit of the ratio (R' θ1 / θ3 ) and TIFF0007749813000066.tif8150 and / or ratio (R' θ2 / θ3 ) and TIFF0007749813000067.tif8150. For example, the optimization function may be a ratio (R' θ1 / θ3 ) and TIFF0007749813000068.tif8150 and / or ratio (R' θ2 / θ3 ) and TIFF0007749813000069.tif8150 and then determine the minimum difference or discrepancy therebetween. As discussed above with respect to the first weighting factor (FW1), any optimization function suitable, capable of, or configured to determine the best fit may be utilized.

[0113] In at least one implementation, the ratio (R' θ1 / θ3 ) and TIFF0007749813000070.tif8150 and ratio (R' θ2 / θ3 ) and Determining the best fit of TIFF0007749813000071.tif8150 may involve an iterative process. For example, an optimization function may be used to determine the ratio (R' θ1 / θ3 ) and TIFF0007749813000072.tif8150 and the ratio (R' θ2 / θ3 ) and Determining the best fit between the image and the image may involve an iterative process. For example, the ratio (R' θ1 / θ3 ) and TIFF0007749813000074.tif8150 and ratio (R' θ2 / θ3 ) and Determining the best fit of TIFF0007749813000075.tif8150 can include iterating through a second shape factor (FF2) using a series of theoretical Rg values. The series of theoretical Rg values ​​can be or include incremental values, decremental values, random values, or combinations thereof. In at least one implementation, the ratio (R' θ1 / θ3 ) and TIFF0007749813000076.tif8150 and ratio (R' θ2 / θ3 ) and Determining the best fit of TIFF0007749813000077.tif8150 can include iterating the second shape factor (FF2) while incrementing or decrementing a predetermined or random theoretical Rg value, for example, a theoretical Rg value starting at about 1 nm. The theoretical Rg value can be increased or decreased in increments of about 0.1 nm or about 0.01 nm. g The value is the ratio (R' θ1 / θ3 ) and The difference and / or ratio (R') between θ2 / θ3 ) and TIFF0007749813000079.tif8150 is minimized or minimized, indicating the best fit between them.

[0114] In at least one implementation, determining the second weighting factor (FW2) comprises determining a ratio (R' θ1 / θ3 ) and TIFF0007749813000080.tif8150 and ratio (R' θ2 / θ3 ) and The method may further include determining a radius of gyration (Rg2) based on a second shape factor (FF2) associated with the best fit of the TIFF0007749813000081.tif8150. θ1 / θ3 ) and TIFF0007749813000082.tif8150 and ratio (R' θ2 / θ3 ) and Determining Rg2 based on a second shape factor (FF2) associated with the best fit of TIFF0007749813000083.tif8150 can include determining Rg2 based on a second shape factor (FF2) associated with the optimization function. For example, in at least one example implementation, determining the radius of gyration (Rg2) associated with the best fit can include determining a ratio of observed light scatter at a first angle to observed light scatter at a third angle (R' θ1 / θ3 ) and the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle TIFF0007749813000084.tif8150, and the ratio of the observed light scatter at the second angle to the observed light scatter at the third angle (R' θ2 / θ3 ) and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle The method may include determining Rg2 based on a second shape factor (FF2) associated with a smallest sum of squares residual (SSR2) of the differences between the two images.

[0115] In at least one implementation, determining the second weighting factor (FW2) comprises determining a ratio (R' θ1 / θ3 ) and TIFF0007749813000086.tif8150 and ratio (R' θ2 / θ3 ) and The method may include utilizing a variable or value associated with the best fit of the second weighting factor (FW2). For example, the second weighting factor (FW2) may be determined using any suitable calculation, equation, or expression incorporating a variable or value associated with the best fit. In at least one implementation, as discussed above, the best fit may be determined using a ratio of the observed light scatter at the first angle to the observed light scatter at the third angle (R' θ1 / θ3 ) and the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle TIFF0007749813000088.tif8150, and the ratio of the observed light scatter at the second angle to the observed light scatter at the third angle (R' θ2 / θ3 ) and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle TIFF0007749813000089.tif8150. Thus, the second weighting factor (FW2) may be determined by any suitable calculation, equation, or expression that utilizes, for example, the smallest sum of squares residual (SSR2). In an exemplary implementation, the second weighting factor (FW2) may be determined according to the following equation (5a), (5b), or (5c): FW2=1 / SSR2(5a) FW2=1 / SAR2(5b) FW2=1 / SAR2 2 (5c) SSR2 here can be the smallest sum of squares residual associated with the best fit with the second form factor (FF2).

[0116] Determination of each form factor contribution

[0117] As discussed above, a method for determining the Rg or hypothesis-free Rg of a particle in a solution can include determining a respective form factor contribution (FC) for each of the shape factors (FF) based on one or more of the respective weighting factors (FW). For example, a method for determining the Rg or hypothesis-free Rg of a particle in a solution can include determining a first form factor contribution (FC1) for a first shape factor (FF1) based on all of the weighting factors (FW) and determining a second form factor contribution (FC2) for a second shape factor (FF2) based on all of the weighting factors (FW) (e.g., a sum of each of the weighting factors). In another example, a method for determining Rg or hypothesis-free Rg of a particle in solution can include determining a first shape factor contribution (FC1) to a first shape factor (FF1) based on a first weighting factor (FW1) and a second weighting factor (FW2), and determining a second shape factor contribution (FC2) to a second shape factor (FF2) based on the first weighting factor (FW1) and the second weighting factor (FW2). In this alternative method, the method for determining Rg or hypothesis-free Rg of a particle in solution can include determining the first shape factor contribution (FC1) and the second shape factor contribution (FC2) to the first shape factor (FF1) and the second shape factor (FF2) based on the first weighting factor (FW1) and the second weighting factor (FW2), respectively. As used herein, the term or phrase "form factor contribution" or "FC" can refer to a quantity or value that represents an apportioned or determined fraction (FW) of a scattering observation represented by an individual component or respective shape factor compared to the sum of all of the shape factor observations (e.g., normalized excess scattering observations). The form factor contribution (FC) can represent a physical weight percentage or a best fit percentage (e.g., specificity) from a given convergence criterion for a weighting factor. The form factor contribution (FC) quantity or value can be a normalized value between 0 and 1, inclusive, that can represent a range from 0% composition to 100% composition of a given form factor structure in a composite scattering observation.

[0118] At relatively low Rg values ​​(e.g., less than about 25 nm), the assumption-free Rg can remain accurate, but it must be recognized that at such Rg values, most shape factors may converge to a similar linear slope when plotted versus angle, so that the selectivity of each shape factor contribution or the amount of a particular or individual shape factor(s) exhibits only low significance. For example, a mixture of amounts of 0.50:0.50 or 0.33:0.33:0.34 may represent a lack of ability to distinguish between preferred structures, but at the same time may still provide or produce a fit that can be more clearly constrained than any conceivable polynomial, which may give an indeterminate Rg value from an improperly extrapolated slope at zero angle.

[0119] Determining the first shape factor contribution (FC1) and the second shape factor contribution (FC2) for the first shape factor (FF1) and the second shape factor (FF2), respectively, based on the first weighting factor (FW1) and the second weighting factor (FW2) may include determining a respective weight, amount, quantity, or percentage that the first and second shape factors (FF1, FF2) contribute to Rg or hypothesis-free Rg. The respective shape factor contributions (FC1, FC2) may be determined using any suitable calculation, equation, or expression that utilizes or incorporates the first weighting factor (FW1) and / or the second weighting factor (FW2). In an exemplary implementation, the first shape factor contribution (FC1) and the second shape factor contribution (FC2) for the first shape factor (FF1) and the second shape factor (FF2) may be determined based on the first weighting factor (FW1) and the second weighting factor (FW2). For example, the first and second shape factor contributions (FC1, FC2) can be determined based on respective values ​​corresponding to the first weighting factor (FW1) and the second weighting factor (FW2). For example, the first and second shape factor contributions (FC1, FC2) can be determined based on respective least sum of squares residuals (SSR1, SSR2) corresponding to the first and second weighting factors (FW1, FW2) as discussed above with respect to equations (4a)-(4c) and (5a)-(5c). In an exemplary implementation, the first shape factor contribution (FC1) and the second shape factor contribution (FC2) to the first shape factor (FF1) and the second shape factor (FF2) can be determined according to the following equations (6) and (7): TIFF0007749813000090.tif10150(6) TIFF0007749813000091.tif10150(7) Here, ΣFW x is the sum of each of the weighting coefficients. For example, when two shape factors are used, the first shape factor contribution (FC1) and the second shape factor contribution (FC2) can be expressed through the following equations (8) and (9), respectively. TIFF0007749813000092.tif10150(8) TIFF0007749813000093.tif10150(9)

[0120] In at least one implementation, the respective shape factor contribution (FC) of each of the shape factors can be evaluated or utilized to determine one or more characteristics, structures, subdivisions of structural features, substructures, or combinations thereof, of particles in solution. For example, many particles cannot be accurately or completely represented by a single or pure shape factor. Instead, many particles can be at least partially represented by at least two or more shape factors. For example, multi-branched particles such as dextran contain a combination of structural features or substructures, often including various molecular weights and / or sizes. Therefore, determining Rg based on a single shape factor can introduce inaccuracies and / or provide incomplete information about the branched particle. In at least one implementation, the respective shape factor contribution (FC) of each of the shape factors can be utilized to at least partially or more accurately determine the structural features of the particle. For example, a relatively large shape factor contribution for a first shape factor compared to a second shape factor can indicate that the particle's structure can be more closely represented by the first shape factor. Similarly, a relatively small shape factor contribution for a first shape factor compared to a second shape factor can indicate that the particle's structure cannot be closely represented by the first shape factor. In yet another example, a particle having relatively similar or substantially equal shape factor contributions for a first shape factor and a second shape factor can indicate that the particle's structure can be substantially represented by both the first shape factor and the second shape factor. It should be appreciated that more than two shape factors and shape factor contributions can be evaluated.

[0121] In at least one implementation, the shape factor contribution (FC) can be used as or provide a relative "weight" for selecting and / or deselecting one or more shape factors according to predetermined criteria. For example, a first shape factor contribution (FC1) can be compared to a second shape factor contribution (FC2) or any additional shape factor contributions to determine whether each of these shape factors has a relatively strong or weak correlation. In at least one implementation, one or more of the shape factors can be deselected when a weaker correlation is determined. A weaker correlation can be represented by a shape factor contribution (FC) of about 33% or less, about 30% or less, about 25% or less, about 20% or less, about 15% or less, about 10% or less, about 5% or less, or about 3% or less. Similarly, one or more shape factors can be selected when a stronger correlation is determined. A strong correlation can be represented by a form factor contribution (FC) of about 50% or more, about 60% or more, about 70% or more, about 80% or more, about 90% or more, or about 95% or more.

[0122] Determining the turning radius (Rg) without assumptions

[0123] As discussed above, a method for determining Rg or assumption-free Rg of a particle in solution can include determining an assumption-free radius of gyration (Rg) using each of the form factor contributions (FC). For example, the method can determine an assumption-free radius of gyration (Rg) using first and second form factor contributions (FC1, FC2). 無仮定 The method can include determining the respective radii of gyration (Rg) associated with the best fit of each form factor (FF). xFor example, the method may include utilizing a turning radius (Rg1) determined using a first shape factor (FF1) associated with a respective optimization function (e.g., smallest sum of squares residual SSR1). In another example, the method may include utilizing a turning radius (Rg2) determined using a second shape factor (FF2) associated with a respective optimization function (e.g., smallest sum of squares residual SSR2). In an exemplary implementation, the method may include utilizing a turning radius (Rg2) determined using a shape factor (FF x ) and the corresponding form factor contribution (FC) and the respective radii of gyration (Rg x For example, the method may include determining an assumption-free turning radius (Rg) using first and second shape factor contributions (FC1, FC2), each turning radius (Rg1, Rg2) being determined using each of the shape factors (FF1, FF2).

[0124] The assumption-free Rg is the form factor (FF i ) corresponding to each of the form factor contributions (FC i ) and / or their respective turning radii (Rg i It should be appreciated that the assumption-free Rg can be determined using any suitable calculation, equation, or expression incorporating (Rg). In an exemplary implementation, the assumption-free Rg can be determined according to the following equation (10): TIFF0007749813000094.tif8150 formula (10) Here, Σ(FC i ×Rg i ) are the respective form factors (FF i ) associated with the best fit of the form factor contribution (FC i ) for each turning radius (Rg i For example, when two form factors are used, the assumption-free Rg can be determined according to the following equation (11): TIFF0007749813000095.tif6150(11)

[0125] A method for determining the assumption-free radius of gyration (Rg) of one or more particles in a solution can also include outputting or displaying the radius of gyration (Rg). For example, the method can include outputting the radius of gyration (Rg) on ​​a display (e.g., a computer display), readout, report, or disk storage of a computing system, such as the computing systems described herein.

[0126] In at least one implementation, the method can include normalizing the detector at one or more angles. Normalizing the detector at one or more angles can be performed according to the manufacturer of the instrument or its detector, or by any one or more methods known in the art. In a preferred implementation, normalizing the detector at one or more angles can include using either a standard spherical coil or a standard random coil, such as a standard spherical coil or a standard random coil with a known Rg of less than about 10 nm. The method can also include utilizing an appropriate shape factor. For example, a spherical coil shape factor or a random coil shape factor is utilized for a spherical coil or a random coil, respectively. In this case, utilizing the respective or appropriate shape factor allows for direct calculation of the angular dependence, which is independent of both the refractive index section (dn / dc) and the molecular weight. P シータ The (Pθ) ratio is the reference angle (also known as P θ of scattering (e.g., excess scattering) observed in シータ It is possible to calibrate from the scattering (eg, excess scattering) at each of the observation angles by a single constant to the (Pθ) ratio.

[0127] How to Determine Molecular Weight

[0128] In at least one implementation, the methods disclosed herein can be used to determine or estimate the molecular weight or estimated molecular weight of a particle. For example, the methods disclosed herein can include determining the molecular weight of a particle using two or more shape factors. In another example, the methods disclosed herein can include determining the molecular weight of a particle using one or more shape factor contributions (FC) of each shape factor (FF). The molecular weight value is calculated from the above calibration and the observed peak area or excess scattering height, e.g., chromatographic elution excess scattering height or batch mode (e.g., cuvette sample cell) excess scattering height, and further using the appropriate P at each angle. θ It should be appreciated by those skilled in the art that the molecular weight at zero (0) angle is corrected by P θ However, the methods disclosed herein may include determining P from two or more shape factors. θ The method includes determining:

[0129] A method for determining the molecular weight of a particle in a chromatographic mode can include determining a calibration constant (K) using known standards. The calibration constant can be determined according to equation (12a) or (12b). In at least one implementation, the determination of the calibration constant can depend at least on the operating mode (e.g., flow mode or batch mode). For example, when operating in flow mode, the calibration constant (K) can be determined according to equation (12a) or (12b). In another example, when operating in batch mode, the calibration constant (K) can be determined according to equation (12b). TIFF0007749813000096.tif14150(12a) TIFF0007749813000097.tif14150(12b) In equations (12a) and (12b), P θ can be determined from the respective shape factors, such as the random coil shape factor, relative to known standards. For example, P θis the random coil form factor (FF) using known standard values. ランダムコイル ) determined from the predicted P θ The LS area can be or refer to the area of ​​the light scattering chromatogram of excess light scattering. The RI or concentration detector area can be or refer to the area under the peak of the refractive index chromatogram of the same injection and may depend at least in part on the total mass of polymer injected into the detector. For example, the RI area may be proportional to the mass injected into the LSD 100. The molecular weight (MW) can be a known standard or the molecular weight of a known sample. It should be recognized by those skilled in the art that the calibration constant (K) can be factored into or adjusted for any one or more or combination of dn / dc, absolute mass, and geometric volume, other chromatographic and detector factors, or the like.

[0130] The method for determining the molecular weight of a particle may also include determining the molecular weight of the particle at one or more of each angle (i.e., θ1, θ2, θ3) using a calibration constant (K) according to the following equation (13): TIFF0007749813000098.tif7170(13) where θ is any angle disclosed herein and K is a calibration constant. For example, θ can be any angle between 0° and 180°. In at least one exemplary implementation, P θ is P θ1 , P θ2 , and P θ3 In at least one implementation, P θ1 , P θ2 , and P θ3 are P RALS , P HALS , and P LALS For example, P θ1 , P θ2 , and P θ3 is P 90 , P 170 , and P10 It can be expressed as or by these. P θ is the respective P from each of the form factors (FF) θ The form factor contribution (FC) of each of these form factors can be determined from the form factor values.

[0131] P θ3 can be expressed through the following equation (14). TIFF0007749813000099.tif6170(14) where: P (θ3)FF1計算値 is calculated from the first form factor (FF1) and is the P associated with the best fit at the third angle. θ The value can be P (θ3)FF2計算値 is calculated from the second form factor (FF2) and is the P associated with the best fit at the third angle. θ The value can be P (θ3)FF3計算値 is calculated from the third form factor (FF3) and is the P associated with the best fit at the third angle. θ It can be a value.

[0132] P θ1 can be expressed through the following equation (15). TIFF0007749813000100.tif6170(15) where: P (θ1)FF1計算値 is calculated from the first form factor (FF1) and is the P associated with the best fit at the first angle. θ The value can be P (θ1)FF2計算値 is calculated from the second form factor (FF2) and is the P associated with the best fit at the first angle. θ The value can be P (θ1)FF3計算値 is calculated from the third form factor (FF3) and is the P associated with the best fit at the first angle. θ It can be a value.

[0133] P θ1 can be expressed through the following equation (16). TIFF0007749813000101.tif6170(16) where: P (θ2)FF1計算値 is calculated from the first form factor (FF1) and is the P associated with the best fit at the second angle. θ The value can be P (θ2)FF2計算値 is calculated from the second form factor (FF2) and is the P associated with the best fit at the second angle. θ The value can be P (θ2)FF3計算値 is calculated from the third form factor (FF3) and is the P associated with the best fit at the second angle. θ It can be a value.

[0134] In each of equations (14), (15), and (16), FC1, FC2, and FC3 refer to the form factor contributions corresponding to the first form factor (FF1), the second form factor (FF2), and the third form factor (FF3), respectively.

[0135] The molecular weight (MW) is P θ1 , P θ2 , P θ3 For example, the molecular weight (MW) can be determined by using one or more of the following equations: LALS , P RALS , P HALS The molecular weight (MW) can be determined through equation (13) using one or more of the following: θ1 Only, P θ2 Only or P θ3 Similarly, molecular weight (MW) can be determined using only P θ1 , P θ2 , and P θ3 In at least one implementation, the molecular weight (MW) can be determined using two or more of P θ1 , P θ2 , and P θ3For example, the molecular weight (MW) is determined using two or more of the following: LALS , P RALS , and P HALS For example, the molecular weight (MW) of each of the θ3 or P LALS , P θ1 or P RALS , and P θ2 or P HALS The molecular weight (MW) is determined through equation (13) for each of the P θ It should be appreciated that the determination may be made using a number of values ​​(e.g., more than three angles). In at least one implementation, P θ3 or P LALS , P θ1 or P RALS , and P θ2 or P HALS Each molecular weight determined via Equation (13) using each of the above can be averaged to determine a standard deviation. This standard deviation can be used to determine the accuracy of the molecular weight (MW). This standard deviation can also be used to determine or facilitate the appropriateness of selecting a shape factor to use. For example, the standard deviation can be used as an exclusion criterion for selecting or deselecting one or more shape factors.

[0136] The molecular weight (MW) (determined from extrapolation to zero angle as known in the art) is the sum of the molecular weights of each P θ It should be appreciated by those skilled in the art that the function can be at least partially improved by expressing it in terms of a form factor (FF) and a corresponding form factor contribution (FC). In view of the above, improving Rg (e.g., its accuracy, precision, etc.) toward zero angle by the assumption-free Rg method disclosed herein will simultaneously result in improved molecular weight determination.

[0137] Exclusion criteria for improving assumption-free Rg

[0138] In at least one implementation, the method can include implementing an exclusion criterion to improve the determined assumption-free turning radius (Rg). For example, the exclusion criterion can improve the accuracy, consistency, and / or precision of the assumption-free Rg determined in the manner herein. The exclusion criterion can include utilizing a respective minimum sum of squares residual, minimum sum of squares squared, or minimum sum of squares absolute for each of the shape factors, the number of angles utilized by the detector, the number of angle ratios, the number of shape factors, or a combination thereof. The exclusion criterion can utilize a respective minimum sum of squares residual, minimum sum of squares squared, or minimum sum of squares absolute for each of the shape factors, the number of angles utilized by the detector, the number of angle ratios, the number of shape factors, or a combination thereof, thereby improving the corrected minimum sum of squares residual (SSR) for any one or more of the shape factors. 補正値 ), corrected least sum of squares (SSR 2 補正値 ), or Corrected Absolute Sum of Squares (SAR 補正値 ) for any one or more of the shape factors. 補正値 ), corrected least sum of squares (SSR 2 補正値 ), or Corrected Absolute Sum of Squares (SAR 補正値 ) can be used to improve the assumption-free turning radius or to provide a corrected assumption-free turning radius.

[0139] In at least one implementation, the exclusion criterion includes utilizing the respective minimum sum of squares residual or the respective minimum sum of squares absolute value and the number of angle ratios. For example, the exclusion criterion can include dividing the respective minimum sum of squares residual or the respective minimum sum of squares absolute value by the number of angle ratios according to Equations (17a, 17b). Dividing by the number of angle ratios can provide an SSR 2 It must be recognized that the same can be corrected. TIFF0007749813000102.tif11150(17a) TIFF0007749813000103.tif11150(17b)

[0140] In at least one implementation, the exclusion criteria can include utilizing a respective least sum of squares residual, least sum of squares squared, or least sum of squares absolute for each of the shape factors, the number of angles utilized by the detector, the number of angle ratios, the number of shape factors, the respective shape factor contributions (FC) for each of the shape factors, or a combination thereof. The exclusion criteria can include utilizing a corrected least sum of squares residual (SSR) for any one or more of the shape factors. 補正値 ), corrected least sum of squares (SSR 2 補正値 ), or Corrected Absolute Sum of Squares (SAR 補正値 ) can be included.

[0141] In at least one implementation, the exclusion criteria can include utilizing a respective minimum sum of squares residual, minimum sum of squares squared, or minimum sum of squares absolute for each of the shape factors, and a combination of the number of angle ratios, the number of shape factors, and the respective shape factor contribution (FC) for each of the shape factors, as shown through the following equations (18a, 18b): TIFF0007749813000104.tif11150(18a) TIFF0007749813000105.tif11150(18b)

[0142] In at least one implementation, as shown through equations (19a, 19b), the exclusion criteria can include utilizing a respective minimum sum of squares residual, minimum sum of squares squared, or minimum sum of squares absolute for each of the shape factors, and a combination of the number of angle ratios and a respective shape factor contribution (FC) for each of the shape factors. TIFF0007749813000106.tif11150(19a) TIFF0007749813000107.tif11150(19b)

[0143] In at least one implementation, as shown through equations (20a, 20b), the exclusion criteria can include utilizing a respective minimum sum of squares residual, minimum sum of squares squared, or minimum sum of squares absolute for each of the shape factors, and a combination of the number of shape factors and a respective shape factor contribution (FC) for each of the shape factors. TIFF0007749813000108.tif11150(20a) TIFF0007749813000109.tif11150(20b)

[0144] As discussed above, the shape factor contributions (FC) can be used as or provide relative "weights" for selecting and / or deselecting one or more shape factors according to predetermined criteria, such as exclusion criteria. For example, a first shape factor contribution (FC1) can be compared to a second shape factor contribution (FC2) or any additional shape factor contributions to determine whether the respective shape factors have a relatively strong or weak correlation. Thus, in at least one implementation, the exclusion criteria can include using one or more of the shape factor contributions (FC) to deselect one or more of the respective shape factors. The exclusion criteria can be determined by a weak correlation, such as a shape factor contribution (FC) of about 33% or less, about 30% or less, about 25% or less, about 20% or less, about 15% or less, about 10% or less, about 5% or less, or about 3% or less. Similarly, one or more shape factors can be selected when a stronger correlation is determined. A strong correlation can be expressed through a form factor contribution (FC) of about 50% or more, about 60% or more, about 70% or more, about 80% or more, about 90% or more, or about 95% or more.

[0145] In at least one implementation, the exclusion criteria can include utilizing respective signal-to-noise ratios at one or more angles and / or signal bias (e.g., insufficient normalization, baseline settings, etc.). For example, the respective signal-to-noise ratios at one or more angles can be utilized to eliminate one or more of the angles utilized to determine the hypothesis-free Rg. For example, the exclusion criteria can include determining a signal-to-noise ratio at a first angle and a signal-to-noise ratio at a second angle, and eliminating, excluding, or otherwise filtering out the first angle or the second angle having or associated with a relatively low signal-to-noise ratio.

[0146] Assumption-free Rg for particles with complex shapes

[0147] In at least one implementation, the methods disclosed herein can be used to describe or characterize one or more particles or composite particles having complex molecular shapes, such as branched molecules. For example, the methods disclosed herein can be used to determine one or more characteristics or properties of the composite particles. Exemplary characteristics or properties can be or include, but are not limited to, shape factors of each of the branches, structural units, structural features, substructures, subdivisions of structural features, branching frequency, long-chain branching frequency, or the like, or combinations thereof.

[0148] It should be recognized that branched particles often exhibit more compact structures, subdivisions of structural features, or substructures than particles that can be represented by a single structure, such as a random coil. For example, as the molecular weight of a branched particle increases, the number of branch points packed into the volume of a hydrodynamic sphere correspondingly increases. In multibranched particles or samples, branch points may also form on existing branches. For example, dextran particles may have a relatively large amount of long-chain branches. Thus, complex particles, such as branched particles, are not adequately described or represented by a single shape factor. The methods disclosed herein can utilize a combination of shape factors that more accurately represent these complex particles than conventional methods.

[0149] A method for evaluating composite particles can include determining a respective shape factor contribution (FC) for each of the shape factors. The method can also include determining a respective percentage (%) of each of the shape factor contributions relative to all of the shape factor contributions (FC). The method can further include determining each shape factor contribution (FC) and / or its respective percentage (%).

[0150] In at least one implementation, the respective shape factor contributions (FC) can be utilized to determine the relative branching level or degree (e.g., long-chain branching frequency [LCBf]). For example, branched particles can be represented by a combination of shape factors, such as a random coil shape factor and a spherical shape factor. In at least one implementation, increased branching of a branched particle can be represented by a corresponding increase in the shape factor contribution (FC) relative to the spherical shape factor. In another implementation, decreased branching of a branched particle can be represented by a corresponding decrease in the shape factor contribution (FC) relative to the spherical shape factor and / or an increase in the shape factor contribution (FC) relative to the random coil shape factor. It should be appreciated that these methods of evaluating branched particles can be applied to homopolymers, heterogeneous copolymers (e.g., random or regular heterogeneous copolymers), or combinations thereof.

[0151] In at least one implementation, the respective shape factor contributions (FC) can be used to represent, observe, or evaluate coil expansion or contraction, which may be due to steric hindrance in the helical backbone, polyelectrolyte effects, solubility parameters, or a combination thereof.

[0152] Data acquisition setting values

[0153] 2 illustrates a computer system or electronic processor 200 for receiving and / or analyzing data from the LSD 100 or any LSD according to one or more implementations. The computer system or electronic processor 200 may be a general-purpose computer and may enable a user or chromatographer to process data, analyze data, interpret data, store data, retrieve data, display data, display results, interpret results, store results, or any combination thereof. The results may be in graphical and / or tabular format. While the electronic processor 200 is shown operatively and / or communicatively coupled to the LSD 100 of FIG. 1A, it should be appreciated that the electronic processor 200 may be operatively and / or communicatively coupled to any suitable light scattering detector known in the art.

[0154] The computer system or electronic processor 200 may be capable of or configured to operate, communicate with (e.g., send and receive data from), modify, modulate, or otherwise operate any one or more components of a light scattering detector, such as the LSD 100. For example, the electronic processor 200 may be operatively and / or communicatively coupled to the pump (not shown), the laser 108, the sample source 104, any one or more of the detectors 110, 112, 114, or any other components of the LSD 100, and may be capable of or configured to operate, communicate with, modify, modulate, or otherwise operate these components.

[0155] In at least one implementation illustrated in FIG. 2 , electronic processor 200 can be operatively and / or communicatively coupled to detectors 110, 112, 114 and can be capable of or configured to transmit and / or receive signals and / or data 202 from these detectors. Data 202 from one or more detectors 110, 112, 114 can be or include analog data, such as a varying analog voltage. In at least one implementation, electronic processor 200 can be capable of or configured to convert the analog data to digital data. For example, electronic processor 200 can include an analog-to-digital converter (not shown). In another implementation, an analog-to-digital converter can be interposed between LSD 100 or its detectors 110, 112, 114 and electronic processor 200.

[0156] The electronic processor 200 may be capable of or configured to receive, collect, record, and / or store data 202 from any one or more components of the LSD 100. For example, as illustrated in Figure 2, the electronic processor 200 may receive data 202 from one or more detectors 110, 112, 114 of the LSD 100 and record and / or store the data 202 in computer memory, such as a local drive or a network drive (e.g., a cloud drive).

[0157] The electronic processor 200 may be capable of or configured to analyze, process, display, and / or output the data 202. For example, the electronic processor 200 may include software capable of or configured to analyze, process, display, and / or output the data 202. The software may also be capable of or configured to process the data 202 and output or display it on a workstation or display 204. The software may include any one or more of the algorithms, equations, methods, steps, processes, or formulas disclosed herein. The electronic processor 200 may process and / or extract information from the data 202 to prepare results and provide the data 202 and / or results, such as in a report or on the display 204. The electronic processor 200 may include a graphical user interface (GUI) that allows a user or chromatography technician to interact with all systems, subsystems, and / or components of the electronic processor 200 and / or LSD 100.

[0158] 3 illustrates a block diagram of the computer system or electronic processor 200 of FIG. 2 that can be used in conjunction with one or more light scattering detectors, including LSD 100, and / or one or more methods disclosed herein. For example, computing system 200 (or system, server, computing device, or device) can represent any of the devices or systems described herein that perform any of the processes, operations, or methods of the present disclosure. It should be noted that while computing system 200 illustrates various components, computing system 200 is not intended to be limited to such details, as any particular architecture or manner of interconnecting these components is not germane to the present disclosure. It will also be appreciated that other types of systems having fewer or more components than those illustrated can be used in conjunction with the present disclosure.

[0159] As shown, computing system 200 may include a bus 302 that may be coupled to a processor 304, a ROM (read-only memory) 308, a RAM (or volatile memory) 310, and a storage (or non-volatile memory) 312. Processor 304 may store data 202 (see FIG. 2) in one or more of memories 308, 310, and 312. Processor 304 may also retrieve data from one or more of memories 308, 310, and 312. One or more of memories 308, 310, and 312 may store the software disclosed herein, which may contain instructions for performing any one or more of the processes, operations, or methods described herein. Processor 304 may also retrieve the stored software or instructions thereof from one or more of memories 308, 310, and 312 and execute those instructions to perform any one or more of the processes, operations, or methods described herein. These memories represent examples of non-transitory computer-readable media (or machine-readable media) or storage containing instructions that, when executed through processor 304 (or system or computing system), cause processor 304 to perform any one or more of the processes, operations, or methods described herein. RAM 310 may be implemented, for example, as dynamic RAM (DRAM) or other types of memory that require constant power to refresh or maintain data in memory. Storage 312 may include, for example, magnetic, semiconductor, tape, optical, removable, non-removable, and / or other types of storage that retain data even after power is removed from computer system 200. It should be appreciated that storage 312 may be remote from system 200 (e.g., accessible via a network).

[0160] A display controller 314 may be coupled to bus 302 for receiving data to be displayed on display 204, which may display any of the user interface features or implementations described herein and which may be local or remote display device 204. Computing system 200 may also include one or more input / output (I / O) components 316, including a mouse, keyboard, touch screen, network interface, printer, speakers, and other devices. Typically, input / output components 316 are coupled to system 200 through an input / output controller 318.

[0161] Module 320 (or program code, instructions, components, subsystems, units, functions, or logic) may represent any of the instructions, subsystems, steps, methods, equations, calculations, plots, or engines described above. Module 320 may reside, completely or at least partially, in a memory (e.g., a non-transitory computer-readable medium) described above, or in processor 304 during execution of module 320 by computing system 200. In addition, module 320 may be implemented as software, firmware, or functional circuitry within computing system 200, or as a combination thereof.

[0162] Example The examples and other implementations described herein are exemplary and are not intended to be limiting as describing the full scope of the present disclosure's structures and methods. Equivalent changes, modifications, and variations of specific implementations, materials, structures, and methods may be made within the scope of the present disclosure with substantially similar results. [Example]

[0163] The assumption-free Rg of a narrow polystyrene sample with a nominal molecular weight of approximately 5,500,000 Da in tetrahydrofuran solution was determined using a system including an automated liquid chromatograph coupled with a light scattering detector (LSD).

[0164] To determine the assumption-free Rg, the system was normalized and calibrated using known polystyrene standards. The light scattering detector angle was normalized with respect to LALS. Specifically, the light scattering detector was calibrated at a low-angle light scattering angle (LALS) of 10 degrees (θ3) and normalized for relative excess scattering at a right-angle light scattering angle (RALS) of 90 degrees (θ1) and a high-angle light scattering angle (HALS) of 170 degrees (θ2). The theoretical form factor ratios between these angles were determined based on the random coil model. The normalized ratios and MW constants are summarized in Table 1.

[0165] Measure the first scattering intensity (I) of a polystyrene sample (approximately 5,500,000 Da) using the system at each of the aforementioned angles (θ1, θ2, θ3). θ1 ), the second scattering intensity (I θ2 ), and the third scattering intensity (I θ3 ) were acquired. The raw chromatograms were integrated to determine the respective areas. The measured areas for LALS (θ3), RALS (θ1), and HALS (θ2) are summarized in Table 1.

[0166] (Table 1) TIFF0007749813000110.tif52140

[0167] The area correction value was determined by dividing the measured area by a normalized ratio. Using the LALS signal or the observed light scattering at the third angle (θ3) as a reference, the observed light scattering of the particle at the first angle relative to the third angle (R' θ1 / θ3 ) and the observed light scattering (R') of the particle at the second angle relative to the third angle θ2 / θ3 ) and the normalized observed ratios (R') of the observed ratios (R') were determined. These ratios were determined according to equations (21a), (21b), and (21c). The calculated observed ratios R' are summarized in Table 2. R LALS or (R'θ3 / θ3 )=R θ3 / R θ3 (21a) R RALS or (R' θ1 / θ3 )=R θ1 / R θ3 (21b) R HALS or (R' θ2 / θ3 )=R θ2 / R θ3 (21c)

[0168] (Table 2) TIFF0007749813000111.tif31139

[0169] It should be appreciated that the ratios may alternatively be determined using any representative scattering value, for example, the ratios may instead be determined using the height and / or partial area of ​​each chromatographic data slice, as opposed to an area integral, and may be expressed as a quantity proportional to a calibrated or observed value of scattering intensity.

[0170] Determining theoretical or calculated ratios of light scattering at the first angle to the third angle and at the second angle to the third angle based on the first shape factor.

[0171] The ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle based on the first form factor (FF1) TIFF0007749813000112.tif11157 and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle The Rg values ​​were determined for various Rg values. Specifically, a random coil form factor was selected as the first form factor (FF1), and the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle was calculated using increasing Rg values. TIFF0007749813000114.tif11157 and the ratio of calculated light scattering at the second angle to calculated light scattering at the third angle More specifically, the random coil form factor (FF1) was used to determine the LALS (P θ3 ), RALS(P θ1 ), and HALS(P θ2 ) theoretical or calculated values ​​of light scattering (e.g., P θ particle scattering factor values) and these LALS (P θ3 ), RALS(P θ1 ), and HALS(P θ2 ) using the theoretical or calculated values ​​of light scattering, the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle TIFF0007749813000116.tif11157 and the ratio of calculated light scattering at the second angle to calculated light scattering at the third angle The ratio of LALS (P) at each increasing Rg value was determined. θ3 ), RALS(P θ1 ), and HALS(P θ2 ) and the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle based on the first form factor (FF1) at each increasing Rg value. TIFF0007749813000118.tif11157 and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle TIFF0007749813000119.tif11157 are shown in Table 3 as abbreviations.

[0172] TIFF0007749813000120.tif147166

[0173] Determining the best fit of observed ratios to calculated or theoretical ratios

[0174] Ratio of calculated light scattering at the first angle to calculated light scattering at the third angle The ratio of the observed light scattering at the first angle to the observed light scattering at the third angle (R' θ1 / θ3) and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle. The ratio of the observed light scattering at the second angle to the observed light scattering at the third angle (R' θ2 / θ3 ) was determined. The best fit utilized an iterative process. The best fit was determined as the ratio of the calculated light scatter at the first angle to the calculated light scatter at the third angle. The ratio of the observed light scattering at the first angle to the observed light scattering at the third angle (R' θ1 / θ3 ) and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle. The ratio of the observed light scattering at the second angle to the observed light scattering at the third angle (R' θ2 / θ3 ) and the convergence of the residuals (R'). These convergences were determined using the least sum of squares residual (SSR1) and the least sum of absolute residual (SAR1). As summarized in Table 2, θ1 / θ3 ) and (R' θ2 / θ3 ) were 0.19109 and 0.10160, respectively. For each increasing Rg value, the ratio of the observed light scatter at the first angle to the observed light scatter at the third angle (R' θ1 / θ3 ) is the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle TIFF0007749813000125.tif11157 and the ratio of the observed light scattering at the second angle to the observed light scattering at the third angle (R' θ2 / θ3 ) is the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle The smallest sum of squares residual (SSR1) and smallest absolute sum residual (SAR1) between the TIFF0007749813000126.tif11157 and the contrast are presented in Table 4 in abbreviated form.

[0175] (Table 4) TIFF0007749813000127.tif140149

[0176] As illustrated in Table 4, Rg1 based on the first form factor (FF1) associated with the best fit was 127.2 nm. Specifically, Rg1 at 127.2 nm corresponded to the smallest deviation or difference between the observed ratio and the calculated ratio. More specifically, Rg1 at 127.2 nm corresponded to the ratio of the observed light scatter at the first angle to the observed light scatter at the third angle (R' θ1 / θ3 ) and the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle TIFF0007749813000128.tif11157, and the ratio of the observed light scatter at the second angle to the observed light scatter at the third angle (R' θ2 / θ3 ) and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle TIFF0007749813000129.tif11157 corresponded to the smallest sum of squares residual (SSR1) of the difference between

[0177] Next, the first weighting factor (FW1) was determined based on the shape factor (FF1) of the first or random coil using the least sum of squares residual (SSR1). Specifically, the first weighting factor (FW1) was determined using the following equation (22). TIFF0007749813000130.tif11157(22) where SSR1 is the smallest sum of squares residual associated with the best fit with the first shape factor (FF1). Using equation (22), we set the first weighting factor (FW1) to 1.8859x10 8 It was decided that:

[0178] Determining theoretical or calculated ratios of light scattering at the first angle to the third angle and at the second angle to the third angle based on the second shape factor.

[0179] The ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle based on the second form factor (FF2) TIFF0007749813000131.tif11157 and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle The Rg values ​​were determined for various Rg values. Specifically, a rod-like shape factor was selected as the second shape factor (FF2), and the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle was calculated using increasing Rg values. TIFF0007749813000133.tif11157 and the ratio of calculated light scattering at the second angle to calculated light scattering at the third angle More specifically, the rod-like shape factor (FF2) was used to determine the LALS (P θ3 ), RALS(P θ1 ), and HALS(P θ2 ) and determine the theoretical or calculated scattering values ​​of these LALS(P θ3 ), RALS(P θ1 ), and HALS(P θ2 ) using theoretical or calculated scattering values ​​of the first angle to the third angle, TIFF0007749813000135.tif11157 and the ratio of calculated light scattering at the second angle to calculated light scattering at the third angle TIFF0007749813000136.tif11157 and LALS (P θ3 ), RALS(P θ1 ), and HALS(P θ2 ) and the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle based on the theoretical scattering value of Rg and the second form factor (FF2) at each increasing Rg value. TIFF0007749813000137.tif11157 and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle TIFF0007749813000138.tif11157 are shown in Table 5 as abbreviations.

[0180] (Table 5) TIFF0007749813000139.tif166146

[0181] Determining the best fit of observed ratios to calculated or theoretical ratios

[0182] Ratio of calculated light scattering at the first angle to calculated light scattering at the third angle The ratio of the observed light scattering at the first angle to the observed light scattering at the third angle (R' θ1 / θ3 ) and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle. The ratio of the observed light scattering at the second angle to the observed light scattering at the third angle (R' θ2 / θ3 ) was determined. The best fit utilized an iterative process similar to that for the random coil discussed above. This best fit was determined as the ratio of the calculated light scatter at the first angle to the calculated light scatter at the third angle. The ratio of the observed light scattering at the first angle to the observed light scattering at the third angle (R' θ1 / θ3 ) and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle. The ratio of the observed light scattering at the second angle to the observed light scattering at the third angle (R' θ2 / θ3 These convergences were determined using the least sum of squares residual (SSR2) and the least absolute sum residual (SAR2). For each increasing Rg value, the ratio of the observed light scatter at the first angle to the observed light scatter at the third angle (R' θ1 / θ3 ) is the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle TIFF0007749813000144.tif11157 and the ratio of the observed light scattering at the second angle to the observed light scattering at the third angle (R' θ2 / θ3 ) is the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle The least sum of squares residual (SSR2) and least absolute sum residual (SAR2) between the TIFF0007749813000145.tif11157 and the contrast are presented in Table 6 in abbreviated form.

[0183] TIFF0007749813000146.tif179170

[0184] As illustrated in Table 6, Rg2 based on the second form factor (FF2) associated with the best fit was 243.1 nm. Specifically, Rg2 at 243.1 nm corresponded to the smallest deviation between the observed and calculated ratios. More specifically, Rg2 at 243.1 nm was the ratio of the observed light scatter at the first angle to the observed light scatter at the third angle (R' θ1 / θ3 ) and the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle TIFF0007749813000147.tif11157, and the ratio of the observed light scatter at the second angle to the observed light scatter at the third angle (R' θ2 / θ3 ) and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle TIFF0007749813000148.tif11157 corresponded to the smallest sum of squares residual (SSR2) of the difference between

[0185] Next, the second weighting factor (FW2) was determined based on the second or rod-like shape factor (FF2) using the least sum of squares residual (SSR2). Specifically, the second weighting factor (FW2) was determined using the following equation (23): FW2=1 / SSR2(23) where SSR2 is the smallest sum of squares residual associated with the best fit with the second shape factor (FF2). Using equation (23), the second weighting factor (FW2) was determined to be 1218.36.

[0186] Determining the theoretical or calculated ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle based on the third shape factor and the calculated light scattering at the second angle to the calculated light scattering at the third angle.

[0187] The ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle based on the third form factor (FF3) TIFF0007749813000149.tif11157 and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle The Rg values ​​were determined for various Rg values. Specifically, a spherical shape factor was selected as the third shape factor (FF3), and the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle was calculated using increasing Rg values. TIFF0007749813000151.tif11157 and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle More specifically, the spherical shape factor (FF3) was used to determine the LALS (P θ3 ), RALS(P θ1 ), and HALS(P θ2 ) and calculate the scattering values ​​of these LALS (P θ3 ), RALS(P θ1 ), and HALS(P θ2 ) is used to calculate the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle. TIFF0007749813000153.tif11157 and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle TIFF0007749813000154.tif11157 was determined. The LALS(P θ3 ), RALS(P θ1 ), and HALS(P θ2 ) and the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle based on the theoretical scattering value of Rg and the third form factor (FF3) at each increasing Rg value. TIFF0007749813000155.tif11157 and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle TIFF0007749813000156.tif11157 are shown in Table 7 in their abbreviations.

[0188] TIFF0007749813000157.tif151157

[0189] Determining the best fit of observed ratios to calculated or theoretical ratios

[0190] Ratio of calculated light scattering at the first angle to calculated light scattering at the third angle The ratio of the observed light scattering at the first angle to the observed light scattering at the third angle (R' θ1 / θ3 ) and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle. The ratio of the observed light scattering at the second angle to the observed light scattering at the third angle (R' θ2 / θ3 ) was determined. The best fit utilized an iterative process similar to that for the random coil and rod shape factors discussed above. This best fit was determined as the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle. The ratio of the observed light scattering at the first angle to the observed light scattering at the third angle (R' θ1 / θ3 ) and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle. The ratio of the observed light scattering at the second angle to the observed light scattering at the third angle (R' θ2 / θ3 ) and the convergence of the least sum of squares residual (SSR3) and the least sum of absolute residual (SAR3).

[0191] The Rg3 based on the third form factor (FF3) associated with the best fit was 81.9 nm. Specifically, Rg3 at 81.9 nm corresponded to the smallest deviation between the observed and calculated ratios. More specifically, Rg3 at 81.9 nm corresponded to the ratio of the observed light scatter at the first angle to the observed light scatter at the third angle (R' θ1 / θ3 ) and the ratio of the calculated light scattering at the first angle to the calculated light scattering at the third angle TIFF0007749813000162.tif11157, and the ratio of the observed light scatter at the second angle to the observed light scatter at the third angle (R' θ2 / θ3 ) and the ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle TIFF0007749813000163.tif11157 corresponded to the smallest sum of squares residual (SSR3) of the difference between

[0192] Next, the third weighting factor (FW3) was determined based on the third or spherical shape factor (FF3) using the least sum of squares residual (SSR3). Specifically, the third weighting factor (FW3) was determined using the following equation (24). FW3=1 / (SSR3) (24) where SSR3 is the smallest sum of squares residual associated with the best fit with the third shape factor (FF3). Using equation (24), the third weighting factor (FW3) was determined to be 151.964.

[0193] Determination of the respective form factor contributions to each of the form factors

[0194] After determining the first, second, and third weighting coefficients (FW1, FW2, FW3), the form factor contributions (FC) for each of the form factors (FF1, FF2, FF3) were determined based on these weighting coefficients (FW1, FW2, FW3). Specifically, the first form factor contribution (FC1), the second form factor contribution (FC2), and the third form factor contribution (FC3) for the first form factor (FF1), the second form factor (FF2), and the third form factor (FF3) were determined based on the first weighting coefficient (FW1), the second weighting coefficient (FW2), and the third weighting coefficient (FW3), respectively. The first, second, and third form factor contributions (FC1, FC2, FC3) were determined according to equations (25a, 25b, 25c). TIFF0007749813000164.tif11157(25a) TIFF0007749813000165.tif11157(25b) TIFF0007749813000166.tif11157(25c) Here, ΣFW iis the sum of each of the weighting coefficients. The calculated form factor contributions were FC1 = 0.999993, FC2 = 0.000006, and FC3 = 0.000001 for the random coil form factor (FF1), rod form factor (FF2), and spherical form factor (FF3), respectively.

[0195] Assumption-free radius of gyration determination using shape factor contributions.

[0196] After determining the first, second, and third shape factor contributions (FC1, FC2, FC3), the assumption-free radius of gyration (Rg) was determined. Specifically, the assumption-free Rg was determined by utilizing the respective radius of gyration (Rg) associated with the best fit of each shape factor (FF). Tables 8, 9, and 10 summarize the determined or calculated values ​​for the random coil form factor (FF1), rod-like form factor (FF2), and spherical form factor (FF3), respectively.

[0197] (Table 8) TIFF0007749813000167.tif47142

[0198] (Table 9) TIFF0007749813000168.tif41138

[0199] (Table 10) TIFF0007749813000169.tif36127

[0200] As shown above in Tables 8-10, the radius of gyration associated with the best fit of the random coil form factor (FF1) was 127.2 nm, the radius of gyration associated with the best fit of the rod-like form factor (FF2) was 243.1 nm, and the radius of gyration associated with the best fit of the spherical form factor (FF3) was 81.9 nm. The assumption-free radius of gyration (Rg) was then determined using the following equation (26): TIFF0007749813000170.tif11157(26) ΣFC here i ×RGi is the form factor contribution (FC i ) to the respective radius of gyration (Rg i ) is the sum of the multiplications of the three form factors (FF1, FF2, FF3). Therefore, the following equations (27a, 27b) are given by using the three form factors (FF1, FF2, FF3). TIFF0007749813000171.tif11157(27a) TIFF0007749813000172.tif6157(27b) Calculation no assumptions turning radius (Rg 無仮定 ) was determined to be about 127.2 nm. It should be noted that in Example 1, the iterations were increased to about 0.1 nm intervals before and after the area of ​​minimum residual. It should be recognized that iterating at smaller intervals, such as about 0.01 nm, would result in a more accurate and / or precise assumption-free Rg of 127.18 nm.

[0201] As determined above, the shape factor contribution (FC1) of 0.999993 for the random coil shape factor (FF1) was several orders of magnitude greater than the shape factor contributions (FC2, FC3) for the rod and sphere shape factors (FF2, FF3), and was nearly exactly 1, or 100%, of the formulation. Therefore, the particles can be determined to be substantially similar to random coils, as confirmed by the assumption-free Rg. It should be noted that when the iterative process was extended to an accuracy of 0.01 nm, the FC1 value increased to 0.999997, revealing that iterative processing with smaller intervals results in more robust or improved structural discrimination. It should be acknowledged that high accuracy may require significant digitization resolution and attention to the baseline setting of the light scattering signal. [Example]

[0202] The molecular weight of the sample in Example 1 was determined according to the method disclosed herein. To determine the molecular weight, the molecular weight constant or calibration constant (K) was determined according to Equation (12a). As shown in Table 1, the calibration constant (K) was determined to be 18,183 Da / (LALS / RI). After determining the calibration constant (K), the P of the sample at the third angle (θ3) was θ The weighted average of the values ​​was determined according to equation (14) using the form factor contributions (FC1, FC2, FC3) from each of the form factors (FF1, FF2, FF3). TIFF0007749813000173.tif11157(14) TIFF0007749813000174.tif11157

[0203] The molecular weight of the particle as determined at the third angle (θ3) was then determined according to Eq. (13). TIFF0007749813000175.tif11157(13) TIFF0007749813000176.tif11157

[0204] The molecular weight of the particle as determined at the first angle (θ1) and the second angle (θ2) was similarly determined according to Equation (13). Specifically, to determine the molecular weight of the particle at the first angle (θ1), P θ1 or P RALS Furthermore, to determine the molecular weight of the particle at the second angle (θ2), P θ2 or P HALS I used P RALS and P HALS The values ​​of P were determined to be 0.182291 and 0.096996, respectively. RALS and P HALS Using the values ​​of θ, the molecular weights of each of the particles determined using equation (13) at the first angle (θ) and the second angle (θ) were 5,912,954 Da and 5,910,143 Da, respectively. These results are summarized in Table 11.

[0205] Independent weighted Pθ The average molecular weight and standard deviation were also determined by averaging the molecular weights given by each of the values. As shown in Table 11, the average molecular weight was 5,911,576 Da and the standard deviation was 1,406.

[0206] (Table 11) TIFF0007749813000177.tif47138

[0207] It was surprisingly and unexpectedly discovered that molecular weights with improved accuracy and precision were obtained by the methods disclosed herein. In particular, the calculated standard deviation revealed a relatively high degree of precision for determining molecular weight. Even more surprisingly, it was also unexpectedly discovered that the methods disclosed herein can provide such standard deviations that facilitate or assess the accuracy and / or precision of the calculated molecular weight. [Example]

[0208] The assumption-free Rg was determined for a narrow polystyrene sample with a nominal molecular weight of approximately 4,500,000 Da. The system was calibrated using known narrow polystyrene standards. Samples and calibrants were dissolved in THF, and a wavelength of 660 nm was used for the incident light scattering beam. Angular data between 12 and 164 degrees were collected in 8-degree increments or intervals.

[0209] For 4,500,000 polystyrene, after calibration, significant distortion (downward bending) was observed at high angles by extrapolation of the Zimm plot, illustrated in Figure 4. Figure 4 illustrates first, second, and third order polynomial fits of the Zimm plot. Without being bound by theory, it is believed that the possible source of distortion in the Zimm plot can be attributed, at least in part, to back reflections within the cell. This distortion is due to the slopes at the intercepts for the first, second, and third order polynomials, respectively, of 7.21x10 -7 , 9.59x10 -7 , and 5.51x10 -7These intercept values ​​are of concern because they yield Rg values ​​of 117 nm, 134 nm, and 101 nm, respectively. These intercept values ​​yielded MW estimates of 4,475,000 Da, 5,187,000 Da, and 4,565,000 Da, respectively. The wide range or distribution of values ​​for each slope used in determining Rg contributed to imprecision or lack of precision. It should be recognized by those skilled in the art that if the sample is a broadly distributed sample, as opposed to the narrowly distributed sample used, the conventional fit (e.g., polynomial fit order according to Zimm, Berry) may need to be adjusted as a function of molecular weight to maintain accuracy, or high angles may need to be excluded to accurately determine Rg. However, the methods disclosed herein do not rely on any particular polynomial fit from Zimm plots, Debye, Berry, Guinier, and the like. Thus, undesired distortions in polynomial fits are largely eliminated by the methods disclosed herein. Therefore, the shape factor model is likely to produce an inappropriate gradient estimate (or Rg 2 The assumption-free Rg also improves accuracy in measuring low Rg values, even when favorable shape factor contributions (FCs) are insufficient. Because weighting coefficients can be determined for each chromatographic slice, independent slice Rg measurements can be made without the need to modify the fit function by this method.

[0210] In accordance with the methods disclosed herein, one of the measured angles was utilized as a reference angle. Specifically, the lowest angle at 12° was utilized as the reference angle to determine a series of R" ratios for the random coil, rod, and spherical form factors (FF1, FF2, FF3). Table 12 summarizes the respective observed R' ratios and respective calculated R" ratios as a function of angle for each of the random coil, rod, and spherical form factors at the Rg corresponding to the best fit (i.e., minimum sum of squares residual (SSR)).

[0211] The respective Rg values ​​corresponding to the smallest sum of squares residual (SSR) for the random coil, rod, and sphere shape factors (FF1, FF2, FF3) were 106 nm, 117 nm, and 83 nm, respectively, as shown in Table 12. The shape factor contributions (FC) from SSR and the no-assumption Rg values ​​obtained from SAR and SSR are also shown in Table 12. It should be noted that the methods for determining the respective Rg values ​​associated with the best fit, the shape factor contributions (FC), and the no-assumption Rg values ​​obtained from SAR and SSR, were described above with respect to Example 1. The variability between the no-assumption Rg values ​​from the polynomial fits of the Zimm plots and the observed Rg values ​​may be at least partially attributed to a relatively poor fit or distortion at higher angles due to unexpected curvature at high angles. Based on the shape factor contributions (FC), the random coil shape factor was selected as the most preferred structure.

[0212] The mean calculated shape factor P is calculated over all angles for both the sum of absolute residuals (SAR) and the squared sum of squared residuals (SSR) and through an optimization criterion. θ The molecular weights corrected for contributions are shown in Table 12.

[0213] TIFF0007749813000178.tif234170

[0214] To improve the determined assumption-free Rg, an exclusion criteria method or process was implemented. Specifically, the two lowest angles (i.e., 12° and 20°) and the six highest angles (i.e., 124°, 132°, 140°, 148°, 156°, and 164°) were not utilized, thereby implementing angle selection from 28° to 116°. The assumption-free radius of gyration (Rg) obtained from the SAR and SSR after implementing the exclusion criteria are summarized in Table 13.

[0215] TIFF0007749813000179.tif115153

[0216] Surprisingly, an unexpected finding was that the determined assumption-free Rg was consistent. Even more surprisingly, the assumption-free Rg yielded consistent P values ​​without the need for fitting orders or angle exclusion, as is commonly practiced in multi-angle light scattering. θ It was an unexpected discovery that the results yielded corrected molecular weight extrapolations. However, it should be noted that the specificity of the shape factor contribution depends on ideal light scattering behavior. For example, the light scattering detector utilized in Example 1 was specifically designed to eliminate back reflections into the cell, along with improved sensitivity, and determinations using three angles yielded particularly accurate molecular weight determinations, in this case showing high shape factor contribution (FC) sensitivity compared to those determined in Example 3. However, the inventors were able to detect an improved fit through the shape factor contribution (FC) of the shape factor that most closely resembled the polymer when the two lowest angles and the six highest angles were eliminated, resulting in poor or distorted signal-to-noise.

[0217] It should be appreciated that the inventors have provided a straightforward means for determining the suitability of MALS calibration and calculations by examining the specificity of the form factor contribution (FC) using particles of molecular structure corresponding to known form factors as input. These examples also demonstrate that the disclosed and utilized methods are applicable to MALS detectors with approximate angular ranges (e.g., maximum minus minimum angle is less than 160° and can be utilized for angles of 90° or less). Nevertheless, it should be further appreciated that even when the form factor contribution has some uncertainty, the no-assumption Rg will still be accurate because all of the utilized form factor models converge to a straight line at low Rg. [Example]

[0218] To improve the assumption-free Rg determined in Example 3, an exclusion criterion method or process was implemented. The exclusion criterion method involved utilizing the respective minimum sum of squares residual or the respective minimum sum of squares absolute value for each of the shape factors and angle ratios or number of angles utilized by the detector. Specifically, the exclusion criterion was implemented by dividing the respective minimum sum of squares residual or the respective minimum sum of squares absolute value by the number of angle ratios. In Example 3, a total of 20 angles were utilized for the detector, resulting in a total of 19 angle ratios. The respective minimum sum of squares residual and the respective minimum sum of squares absolute value for each of the shape factors after implementing the exclusion criterion are summarized in Table 14.

[0219] (Table 14) TIFF0007749813000180.tif47158

[0220] That is, we can propose to include and / or exclude angles in the sum of squares and observe their significance or non-significance. As exemplified in Table 14, the SSR for the random coil form factor 補正値 and SAR 補正値 decreases when observing angles excluded from the widest range (i.e., 12° to 164°) to a narrower range (i.e., 28° to 116°). Surprisingly, this unexpectedly demonstrates improved accuracy and precision when implementing the exclusion criteria. However, as further illustrated in Table 14, the SSR for the random coil form factor 補正値 and SAR 補正値 Neither of these improved when higher angles (i.e., greater than 90°) were eliminated alone. It must be acknowledged that Example 4 evaluates the SAR and SSR of only the dominant form factor (i.e., the random coil form factor).

[0221] FIG. 5 illustrates a plot of assumption-free Rg extrapolated using the angle data for each of the ranges disclosed in Table 14. It should be noted that high angles that were not consistent with light scattering theory were automatically excluded from the fit using exclusion criteria. However, in conventional methods utilizing Zimm analysis, an operator would manually exclude points or fit this plot to a higher-order polynomial, which would introduce inaccuracies and / or inconsistencies in generating a stable slope at zero angle. Notably, the respective y-intercepts and slopes thereat of the plot in FIG. 5 are relatively more consistent and accurate compared to the respective y-intercepts and slopes thereat of the Zimm plot in FIG. 4. [Example]

[0222] The methods disclosed herein were used to evaluate branched particles in comparison with conventional methods. In particular, the unassumed Rg and molecular weight of the branched particles were determined using the methods disclosed herein and compared with conventional methods for determining Rg and molecular weight. The branched particles were broad dextran with a nominal molecular weight of approximately 990,000 Da. The dextran particles were evaluated according to the methods disclosed herein to determine the unassumed Rg and molecular weight, thereby describing the complex molecular shape of these particles.

[0223] The system of Example 1 was utilized. The light scattering detector was aligned and normalized with a known standard polymer.

[0224] A wide range of dextran samples from American Polymer Standards (D990K) were injected into the system, and the detector signal was integrated to calculate the molecular weight results by LALS. The chromatogram of the sample is shown in Figure 6. As shown in Figure 6, the sample contained a combination of high and low molecular weight species. This combination of high and low molecular weight species was as expected.

[0225] A conventional calculation method was used to generate an integration range around or around the lower molecular weight peak position. Specifically, the integration range from about 42 minutes to about 43 minutes, representing the lower molecular weight species or sample components, was evaluated by the conventional method. The conventional light scattering method gave an Rg of about 23.0 and an average molecular weight of about 477,400 Da. The Rg and weight average molecular weight are summarized in Table 15.

[0226] SSR using the exemplary method disclosed herein (i.e., the no-assumption method) 2 By minimizing (equations (4C) and (5C)), P θ The values ​​were determined and the weighting factor (FW) was determined as in Example 1. The determined unassumed Rg was also determined to be 23.4 nm for the low molecular weight species. As in Example 2, the shape factor contribution (FC) was used to determine the estimated molecular weight. The unassumed Rg and weight average molecular weight are summarized in Table 15.

[0227] (Table 15) TIFF0007749813000181.tif78150

[0228] As shown in Table 15, the exemplary method disclosed herein yielded Rg values ​​consistent with conventional methods. Specifically, the assumption-free Rg (Rg AF = 23.4 nm) agreed with the Rg determined by the conventional method (Rg = 23.0 nm).

[0229] After demonstrating agreement between the Rg determined by the conventional light scattering method and the assumption-free Rg, the integration range near or around the higher or highly branched, high molecular weight region was evaluated. Specifically, the integration range from about 37 minutes to about 38 minutes, representing higher molecular weight species or sample components, was evaluated by the conventional method. The conventional method gave an Rg of about 41.9 nm, which resulted in a weight average molecular weight of about 8,437,000 Da. The Rg and weight average molecular weight are summarized in Table 16.

[0230] SSR using the exemplary methods (e.g., no-assumption methods) disclosed herein 2By minimizing P θ The values ​​were determined and the weighting factor (FW) was determined as in Example 1. The unassumed Rg was also determined to be 43.2 nm for the high molecular weight species. As in Example 2, the shape factor contribution (FC) was used to determine the estimated molecular weight. The unassumed Rg and weight average molecular weight are summarized in Table 16.

[0231] (Table 16) TIFF0007749813000182.tif78150

[0232] As shown in Table 16, the exemplary method disclosed herein yielded Rg values ​​consistent with conventional methods. Specifically, the assumption-free Rg (Rg AF = 43.2 nm) was in agreement with the Rg determined by conventional light scattering method (Rg = 41.9 nm).

[0233] After demonstrating agreement between the Rg determined by conventional light scattering methods and the assumption-free Rg for the high molecular weight region, the shape factor contributions for the random coil, rod, and globular form factors were determined similarly as described in Example 1. The determined respective shape factor contributions (FC) for the random coil, rod, and globular form factors are summarized in Table 17.

[0234] (Table 17) TIFF0007749813000183.tif21132

[0235] As shown in Table 17, the dextran samples were not fully represented by a single shape factor. Specifically, the shape factors for random coil and sphere were emphasized. Therefore, the shape factor contributions in Table 17 revealed that the dextran samples were best represented by a combination of random coil and spherical shape factors. It was further concluded that the dextran samples were not adequately represented by a rod-like shape factor. Without being bound by theory, it is believed that the dextran samples are best described when they have approximately 40% random coil characteristics, traits, and / or substructures and approximately 60% spherical characteristics, traits, and / or substructures. Therefore, from a density perspective, this structure was determined to be somewhere between random coil and sphere. It should be appreciated that the more highly branched a dextran sample, the more likely the particle or dextran is to be represented by a spherical shape factor. Furthermore, the less branched a dextran sample, the more likely the particle or dextran is to be represented by a random coil shape factor. Therefore, without being bound by theory, it is believed that the shape factor contribution (FC) of each of the random coil and globular shape factors can be used to determine the relative branching level or degree. It must be recognized that as the sample Rg increases, the specificity of the shape factor contribution (FC) will increase, as shape factors branch more at higher Rg values.

[0236] Overall, the exemplary methods disclosed herein have been shown to be capable of determining assumption-free Rg that is at least consistent with, matches, or more accurate than conventional methods. Furthermore, the exemplary methods disclosed herein have been shown to provide additional insight or information regarding particle structure through evaluation of the respective shape factor contributions of each of the shape factors. Furthermore, unlike conventional branching calculations in size exclusion chromatography (SEC), the exemplary methods disclosed herein do not require a linear reference comparator and can be performed on very low polydispersity ranges as well as polydisperse samples to demonstrate reference-independent structural heterogeneity. Additionally, it should be appreciated that the shape factor contribution is a function of Rg, and thus, determination of mass, dn / dc, concentration profile, and / or molecular weight is not required.

[0237] The present disclosure has been described with reference to exemplary implementations. While only a limited number of implementations have been shown and described, those skilled in the art will recognize that modifications can be made to these implementations without departing from the principles and spirit of the above detailed description. It is intended that the present disclosure be construed as including all such modifications and alterations insofar as they come within the scope of the appended claims and equivalents thereof. [Explanation of symbols]

[0238] 100 Light Scattering Detector (LSD) 104 Sample Source or Device 110, 112, 114 detectors 188 Screen or diaphragm 190, 192, 194 Specimen scattered beam

Claims

1. 1. A method for determining the assumption-free radius of gyration (Rg) of particles in solution using a light scattering detector, comprising: First Form Factor (FF 1 ) based on which a first weighting factor (FW 1 ) determining Second Form Factor (FF 2 ) based on the second weighting factor (FW 2 ) determining The first weighting factor (FW 1 ) and the second weighting factor (FW 2 ) based on the first form factor (FF 1 ) to the first form factor contribution (FC 1 ) determining The first weighting factor (FW 1 ) and the second weighting factor (FW 2 ) based on the second shape factor (FF 2 ) with the second form factor contribution (FC 2 ) determining The first and second form factor contributions (FC 1 , F.C. 2 determining the assumption-free turning radius (Rg) from A method comprising:

2. The light beam of the light scattering detector is used to measure a first scattering intensity (I) of the particles in the solution at a first angle, a second angle, and a third angle relative to the light beam, respectively. θ1 ), the second scattering intensity (I θ2 ), and the third scattering intensity (I θ3 10. The method of claim 1, further comprising the step of obtaining a

3. The first scattering intensity (I θ1 ), the second scattering intensity (I θ2 ), and the third scattering intensity (I θ3 ) is the excess scattering intensity, the excess scattering intensity for each of the first, second, and third angles is normalized with respect to quantum efficiency, gain, geometric volume, or a combination thereof; The method of claim 2.

4. the ratio of the observed light scattering of the particle at the first angle to the observed light scattering of the particle at the third angle (R' θ1 / θ3 ) determining the ratio of the observed light scattering of the particle at the second angle to the observed light scattering of the particle at the third angle (R' θ2 / θ3 ) determining The method of any one of claims 1 to 3, further comprising:

5. The first weighting factor (FW 1 ) is determined by The first form factor (FF 1 a ratio of calculated light scattering at the first angle to calculated light scattering at the third angle based on and determining The first form factor (FF 1 a ratio of the calculated light scattering at the second angle to the calculated light scattering at the third angle based on and determining The ratio The ratio (R' θ1 / θ3 ) and the ratio The ratio (R' θ2 / θ3 ) determining the best fit; Further comprising: The method of claim 4.

6. The second weighting factor (FW 2 ) is determined by The second form factor (FF 2 a ratio of calculated light scattering at the first angle to calculated light scattering at the third angle based on and determining The second form factor (FF 2 a ratio of calculated light scattering at the second angle to calculated light scattering at the third angle based on and determining The ratio The ratio (R' θ1 / θ3 ) and the ratio The ratio (R' θ2 / θ3 ) determining the best fit; Including, The method of claim 5.

7. 7. A method according to claim 5 or claim 6, wherein the step of determining the best fit comprises an iterative process.

8. The first weighting factor (FW 1 The step of determining the first shape factor (FF 1 ) based on the ratio The ratio (R' θ1 / θ3 ) and the ratio The ratio (R' θ2 / θ3 ) associated with the best fit of the turning radius (Rg 1 7. The method of claim 5 or claim 6, further comprising the step of determining:

9. The turning radius (Rg 1 ) is determined by determining the first form factor (FF 1 ) based on the ratio (R' θ1 / θ3 ) and the ratio and the ratio (R' θ2 / θ3 ) and the ratio The smallest sum of squares residual (SSR) of the difference between 1 ) associated with the turning radius (Rg 1 9. The method of claim 8, further comprising determining

10. The first weighting factor (FW 1 ) is the minimum sum of squares residual (SSR 1 10. The method of claim 9, wherein the method is based on

11. The second weighting factor (FW 2 The step of determining the second shape factor (FF 2 ) based on the ratio The ratio (R' θ1 / θ3 ) and the ratio The ratio (R' θ2 / θ3 ) associated with the best fit of the turning radius (Rg 2 11. The method of claim 10, further comprising determining:

12. The turning radius (Rg 2 ) is determined by determining the second form factor (FF 2 ) based on the ratio (R' θ1 / θ3 ) and the ratio and the ratio (R' θ2 / θ3 ) and the ratio The smallest sum of squares residual (SSR) of the difference between 2 ) associated with the turning radius (Rg 2 12. The method of claim 11, comprising determining

13. The second weighting factor (FW 2 ) is the minimum sum of squares residual (SSR 2 13. The method of claim 12, wherein the method is based on

14. The first form factor (FF 1 ) or the second form factor (FF 2 14. The method of claim 1, wherein at least one of the factors is a shape factor for a homogeneous rigid particle.

15. 15. The method of claim 14, wherein the shape factor for the homogeneous rigid particle is one of a homogeneous sphere, a spherical shell, a spherical concentric shell, a particle of spherical subunits, a spheroid, a triaxial ellipsoid, a cube and a rectangular prism, a truncated octahedron, a polyhedral sphere, a lens, a stepped cube, a cylinder, an elliptical cylinder, a hemispherical end cylinder, a semi-lens end cylinder, a torus, a thin rod, an ultrathin disk, or a fractal aggregate.

16. The first form factor (FF 1 ) or the second form factor (FF 2 16. The method of claim 1, wherein at least one of the parameters is a shape factor for the polymer model.

17. 17. The method of claim 16, wherein the shape factor for the polymer model is one of a spherical particle shape factor, a rod-like particle shape factor, a random coil particle shape factor, or a combination thereof.

18. 18. The method of any of claims 1 to 17, wherein the light scattering detector comprises a multi-angle light scattering detector.

19. 19. The method of any of claims 1 to 18, wherein the particles in solution comprise monodisperse particles or polydisperse particles.

20. 20. The method of any of claims 1 to 19, wherein the hypothetical radius of gyration (Rg) of the particle is from about 5 nm to about 500 nm, from about 30 to about 250 nm, or from about 50 to about 150 nm.

21. 14. The method of claim 2, wherein the first angle is approximately 90° relative to the light beam of the light scattering detector, the second angle is approximately 170° relative to the light beam of the light scattering detector, and the third angle is approximately 10° relative to the light beam of the light scattering detector.

22. The third form factor (FF 3 ) based on the third weighting factor (FW 3 ) determining All the weighting factors (FW i ) based on the third form factor (FF 3 ) with the third form factor contribution (FC 3 ) determining The first, second, and third form factor contributions (FC 1 , F.C. 2 , F.C. 3 determining the assumption-free turning radius (Rg) from 22. The method of any of claims 1 to 21, further comprising:

23. a fourth scattering intensity (I) of the particles in solution using the light beam of the light scattering detector at a fourth angle relative to the light beam of the light scattering detector; θ4 23. The method of claim 2, further comprising the step of obtaining a

24. 24. A method according to any preceding claim, wherein the light scattering detector comprises a goniometer configured to observe light scattering at two or more angles.

25. 25. The method of any one of claims 1 to 24, which does not include determining the molecular weight of the particle in solution prior to determining the assumption-free radius of gyration (Rg), or fitting a Zimm, Berry, Debye, or Guinier plot to a polynomial function.

26. 26. The method of any of claims 1 to 25, further comprising implementing exclusion criteria to determine a corrected assumption-free turning radius.

27. 27. The method of claim 26, wherein the exclusion criterion is based on one of a respective least sum of squares residual, a respective least sum of squares squared, or a respective least sum of squares absolute value for each of the first and second shape factors.

28. 28. The method of claim 27, wherein the exclusion criterion is implemented to determine a corrected least sum of squares residual, a corrected least sum of squares squared, or a corrected least sum of squares absolute for each of the first and second shape factors.

29. The exclusion criteria include the number of angles utilized by the light scattering detector, the number of shape factors utilized, the first shape factor contribution (FC 1 ) or the second shape factor contribution (FC 2 27. The method of claim 26, wherein the method is based on at least one of: the signal to noise ratio at each of the angles;

30. 1. A method for determining the molecular weight of particles in a solution, comprising: Determining the assumption-free radius of gyration (Rg) of the particles in the solution of any of claims 1 to 29; determining the molecular weight of the particle in solution using the assumption-free radius of gyration (Rg); A method comprising:

31. 1. A method for evaluating composite particles in a solution, comprising: First Form Factor (FF 1 ) based on which a first weighting factor (FW 1 ) determining Second Form Factor (FF 2 ) based on the second weighting factor (FW 2 ) determining The first weighting factor (FW 1 ) and the second weighting factor (FW 2 ) based on the first form factor (FF 1 ) to the first form factor contribution (FC 1 ) determining The first weighting factor (FW 1 ) and the second weighting factor (FW 2 ) based on the second shape factor (FF 2 ) with the second form factor contribution (FC 2 ) determining The first form factor contribution (FC 1 ) and the second form factor contribution (FC 2 ) determining one or more characteristics of the composite particles; A method comprising:

32. 32. The method of claim 31, wherein the one or more characteristics include a structural feature, a substructure, or a combination thereof.

33. 33. The method of claim 31 or claim 32, wherein the composite particles in solution comprise branched particles.

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