Light scattering detectors and methods for the same
Methods for determining an assumption-free radius of gyration of particles in solution using a light scattering detector are disclosed. The method may include determining a first weighting factor based on a first form factor. The method may also include determining a second weighting factor based on a second form factor. The method may further include determining a first form factor contribution for the first form factor based on the first weighting factor and the second weighting factor. The method may also include determining a second form factor contribution for the second form factor based on the first weighting factor and the second weighting factor. The method may also include determining the assumption-free radius of gyration from the first and second form factor contributions.
1 . A method performed by a computer system for determining an assumption-free radius of gyration (Rg) of particles in solution using a light scattering detector, the method comprising:
obtaining, by the computer system, light scattering intensities of the particles in the solution from the light scattering detector, wherein the light scattering intensities comprise a first scattering intensity (I θ1 ), a second scattering intensity (I θ2 ), and a third scattering intensity (I θ3 ) of the particles in the solution at a first angle, a second angle, and a third angle, respectively, wherein the first angle, the second angle, and the third angle are determined relative to a beam of light of the light scattering detector;
determining, by the computer system, a first weighting factor (FW 1 ) based on a first form factor (FF 1 ) and the light scattering intensities, wherein determining the first weighting factor FW 1 ) comprises:
determining a ratio of observed light scattering of the particles at the first angle to observed light scattering of the particles at the third angle (R′ θ1/θ3 ) based on the first scattering intensity (I θ1 ) and the third scattering intensity (I θ3 );
determining a ratio of observed light scattering of the particles at the second angle to observed light scattering of the particles at the third angle (R′ θ2/θ3 ) based on the second scattering intensity (I θ2 ), and the third scattering intensity (I θ3 );
determining a ratio of calculated light scattering at the first angle to calculated light scattering at the third angle
(
R
″
θ
1
θ
3
Calculated
FF
1
)
based on the first form factor (FF 1 ), the first angle, and the third angle;
determining a ratio of calculated light scattering at the second angle to the calculated light scattering at the third angle
(
R
θ
2
θ
3
Calculated
FF
1
″
)
based on the first form factor (FF 1 ), the second angle, and the third angle;
determining a best fit of the ratio (R′ θ1/θ3 ) to the ratio
(
R
″
θ
1
θ
3
Calculated
FF
1
)
and the ratio (R′ θ2/θ3 ) to the ratio
(
R
θ
2
θ
3
Calculated
FF
1
″
)
based on the first form factor (FF 1 ); and
determining, by the computer system, the first weight factor (FW 1 ) based on the best fit of the ratio (R′ θ1/θ3 ) to the ratio
(
R
″
θ
1
θ
3
Calculated
FF
1
)
and the ratio (R′ θ2/θ3 ) to the ratio
(
R
″
θ
2
θ
3
Calculated
FF
1
)
;
determining, by the computer system, a second weighting factor (FW 2 ) based on a second form factor (FF 2 ) and the light scattering intensities, wherein determining the second weighting factor (FW 2 ) comprises:
determining a ratio of calculated light scattering at the first angle to calculated light scattering at the third angle
(
R
″
θ
1
θ
3
Calculated
FF
2
)
based on the second form factor (FF 2 ), the first angle, and the third angle;
determining a ratio of calculated light scattering of the second angle to calculated light scattering at the third angle
(
R
θ
2
θ
3
Calculated
FF
2
″
)
based on the second form factor (FF 2 ), the second angle, and the third angle;
determining a best fit of the ratio (R′ θ1/θ3 ) to the ratio
(
R
″
θ
1
θ
3
Calculated
FF
2
)
and the ratio (R′ θ2/θ3 ) to the ratio
(
R
θ
2
θ
3
Calculated
FF
2
″
)
based on the second form factor (FF 2 ); and
determining, by the computer system, the second weight factor (FW 2 ) based on the best fit of the ratio (R′ θ1/θ3 ) to the ratio
(
R
″
θ
1
θ
3
Calculated
FF
2
)
and the ratio (R′ θ2/θ3 ) to the ratio
(
R
θ
2
θ
3
Calculated
FF
2
″
)
;
determining, by the computer system, a first form factor contribution (FC 1 ) for the first form factor (FF 1 ) based on the first weighting factor (FW 1 ) and the second weighting factor (FW 2 ), wherein the first form factor contribution (FC 1 ) is determined according to Equation (1):
FC
1
=
FW
1
∑
FW
x
,
(
1
)
wherein ΣFW x is the sum of each of the weighting factors;
determining, by the computer system, a second form factor contribution (FC 2 ) for the second form factor (FF 2 ) based on the first weighting factor (FW 1 ) and the second weighting factor (FW 2 ), wherein the second form factor contribution (FC 2 ) is determined according to Equation (2):
FC
2
=
FW
2
∑
FW
x
;
(
2
)
determining, by the computer system, the assumption-free radius of gyration (Rg) of the particles based on the first and second form factor contributions (FC 1 , FC 2 ); and
normalizing the light scattering detector based on the assumption-free radius of gyration (Rg).
2 . The method of claim 1 , wherein the first scattering intensity (I θ1 ), the second scattering intensity (I θ2 ), and the third scattering intensity (I θ3 ) are excess scattering intensities, and wherein the respective excess scattering intensities of each of the first, second, and third angles are normalized for quantum efficiency, gains, geometric volume, or a combination thereof.
3 . The method of claim 1 , wherein determining the best fit comprises an iterative process.
4 . The method of claim 1 , wherein determining the first weighting factor (FW 1 ) further comprises determining a radius of gyration (Rg 1 ), based on the first form factor (FF 1 ), associated with the best fit of the ratio (R′ θ1/θ3 ) to the ratio
(
R
θ
1
θ
3
Calculated
FF
1
″
)
and the ratio (R′ θ2/θ3 ) to the ratio
(
R
″
θ
2
θ
3
Calculated
FF
1
)
.
5 . The method of claim 4 , wherein the determining the radius of gyration (Rg 1 ) associated with the best fit comprises determining the radius of gyration (Rg 1 ), based on the first form factor (FF 1 ), associated with a minimum sum of squares residual (SSR 1 ) of a difference between the ratio (R′ θ1/θ3 ) and the ratio
(
R
θ
1
θ
3
Calculated
FF
1
″
)
and a difference between the ratio (R′ θ2/θ3 ) and the ratio
(
R
″
θ
2
θ
3
Calculated
FF
1
)
.
6 . The method of claim 5 , wherein determining the second weighting factor (FW 2 ) further comprises determining a radius of gyration (Rg 2 ), based on the second form factor (FF 2 ), associated with the best fit of the ratio (R′ θ1/θ3 ) to the ratio
(
R
θ
1
θ
3
Calculated
FF
2
″
)
and the ratio (R′ θ2/θ3 ) to the ratio
(
R
″
θ
2
θ
3
Calculated
FF
2
)
.
7 . The method of claim 6 , wherein the determining the radius of gyration (Rg 2 ) associated with the best fit comprises determining the radius of gyration (Rg 2 ), based on the second form factor (FF 2 ), associated with a minimum sum of squares residual (SSR 2 ) of a difference between the ratio (R′ θ1/θ3 ) and the ratio
(
R
θ
1
θ
3
Calculated
FF
2
″
)
and a difference between the ratio (R′ θ2/θ3 ) and the ratio
(
R
″
θ
2
θ
3
Calculated
FF
2
)
.
8 . The method of claim 7 , wherein the first weighting factor (FW 1 ) is based on the minimum sum of squares residual (SSR 1 ), the second weighting factor (FW 2 ) is based on the minimum sum of squares residual (SSR 2 ), or a combination thereof.
9 . The method of claim 1 , wherein one or more of the first form factor (FF 1 ), the second form factor (FF 2 ), or a combination thereof is a form factor for a homogenous rigid particle, and wherein the form factor for the homogenous rigid particle is one of: a homogenous sphere, a spherical shell, a spherical concentric shell, particles of spherical subunits, an ellipsoid of revolution, a tri-axial ellipsoid, a cube and rectangular parallelepipedon, a truncated octahedra, a faceted sphere, a lens, a cube with terraces, a cylinder, a cylinder with an elliptical cross section, a cylinder with hemi-spherical endcaps, a cylinder with half lens end caps, a toroid, an infinitely thin rod, an infinitely thin circular disk, or a fractal aggregate.
10 . The method of claim 1 , wherein one or more of the first form factor (FF 1 ), the second form factor (FF 2 ), or a combination thereof is a form factor for a polymer model, and wherein the form factor for the polymer model is one of: a spherical particle form factor, a rodlike particle form factor, a random coil particle form factor, or a combination thereof.
11 . The method of claim 1 , wherein:
the light scattering detector comprises a multi-angle light scattering detector;
the particles in solution comprise monodispersed particles;
the light scattering detector comprises a goniometer configured to observe light scattering at two or more angles;
the particles in solution comprise polydispersed particles; and/or
the assumption-free radius of gyration (Rg) of the particles is from about 5 nm to about 500 nm, from about 30 to about 250 nm, or from about 50 to about 150 nm.
12 . The method of claim 1 , wherein the first angle is about 90° relative to the beam of light of the detector, the second angle is about 170° relative to the beam of light of the detector, and the third angle is about 10° relative to the beam of light of the detector.
13 . The method of claim 1 , further comprising:
determining a third weighting factor (FW 3 ) based on a third form factor (FF 3 );
determining a third form factor contribution (FC 3 ) for the third form factor (FF 3 ) based on the all the weighting factors (FW i ); and
determining the assumption-free radius of gyration (Rg) from the first, second, and third form factor contributions (FC 1 , FC 2 , FC 3 ).
14 . The method of claim 1 , wherein the method does not comprise one or more of:
determining a molecular weight of the particles in solution prior to determining the assumption-free radius of gyration (Rg); and
fitting a Zimm, Berry, Debye, or Guinier plot to a polynomial function.
15 . The method of claim 1 , further comprising implementing a rejection criteria to determine a corrected assumption-free radius of gyration, wherein the rejection criteria is based on one of: a respective minimum sum of squares residual, respective a minimum sum of squares squared, or respective a minimum sum of squares absolute for each of the form factors.
16 . A method for determining a molecular weight of the particles in solution of claim 1 , the method comprising determining the molecular weight of the particles in solution based on the assumption-free radius of gyration (Rg).
17 . A method performed by a computer system for determining one or more properties of composite particles in solution, the method comprising:
obtaining, by the computer system, light scattering intensities of the particles in the solution from the light scattering detector, wherein the light scattering intensities comprise a first scattering intensity (I θ1 ), a second scattering intensity (I θ2 ), and a third scattering intensity (I θ3 ) of the particles in the solution at a first angle, a second angle, and a third angle, respectively, wherein the first angle, the second angle, and the third angle are determined relative to a beam of light of the light scattering detector;
determining, by the computer system, a first weighting factor (FW 1 ) based on a first form factor (FF 1 ) and the light scattering intensities, wherein determining the first weighting factor (FW 1 ) comprises:
determining a ratio of observed light scattering of the particles at the first angle to observed light scattering of the particles at the third angle (R′ θ1/θ2 ) based on the first scattering intensity (I θ1 ) and the third scattering intensity (I θ3 );
determining a ratio of observed light scattering of the particles at the second angle to observed light scattering of the particles at the third angle (R′ θ2/θ3 ) based on the second scattering intensity (I θ2 ), and the third scattering intensity (I θ3 ); and
determining a ratio of calculated light scattering at the first angle to calculated light scattering at the third angle
(
R
θ
1
θ
3
Calculated
FF
1
″
)
,
based on the first form factor (FF 1 ), the first angle, and the third angle;
determining, by the computer system, a second weighting factor (FW 2 ) based on a second form factor (FF 2 ) and the light scattering intensities, wherein determining the second weighting factor (FW 2 ) comprises:
determining a ratio of calculated light scattering at the first angle to calculated light scattering at the third angle
(
R
θ
1
θ
3
Calculated
FF
2
″
)
based on the second form factor (FF 2 ), the first angle, and the third angle;
determining a ratio of calculated light scattering of the second angle to calculated light scattering at the third angle
(
R
θ
2
θ
3
Calculated
FF
2
″
)
based on the second form factor (FF 2 ), the second angle, and the third angle;
determining a best fit of the ratio (R′ θ1/θ3 ) to the ratio
(
R
θ
1
θ
3
Calculated
FF
2
″
)
and the ratio (R′ θ2/θ3 ) to the ratio
(
R
θ
2
θ
3
Calculated
FF
2
″
)
based on the second form factor (FF 2 );
determining the second weight factor (FW 2 ) based on the best fit of the ratio (R′ θ1/θ3 ) to the ratio
(
R
θ
1
θ
3
Calculated
FF
2
″
)
and the ratio (R′ θ2/θ3 ) to the ratio
(
R
″
θ
2
θ
3
Calculated
FF
2
)
;
determining, by the computer system, a first form factor contribution (FC 1 ) for the first form factor (FF 1 ) based on the first weighting factor (FW 1 ) and the second weighting factor (FW 2 ), wherein the first form factor contribution (FC 1 ) is determined according to Equation (1):
FC
1
=
FW
1
∑
FW
x
,
(
1
)
wherein ΣFW x is the sum of each of the weighting factors;
determining, by the computer system, a second form factor contribution (FC 2 ) for the second form factor (FF 2 ) based on the first weighting factor (FW 1 ) and the second weighting factor (FW 2 ), wherein the second form factor contribution (FC 2 ) is determined according to Equation (2):
FC
2
=
FW
2
∑
FW
x
;
(
2
)
determining, by the computer system, the one or more properties of the composite particles based on the first form factor contribution (FC 1 ) and the second form factor contribution (FC 2 ); and
generating an output, on a display, a readout, a report, or a disk storage of the computer system, based on the one or more properties of the composite particles.
18 . The method of claim 17 , wherein the one or more properties comprise structural features, substructures, or combinations thereof, and/or wherein the composite particles in solution comprise branched particles.
19 . A non-transitory computer-readable medium storing instructions that, when executed by one or more processors of a computing system, cause the computing system to perform operations for determining an assumption-free radius of gyration (Rg) of particles in solution using a light scattering detector, the operations comprising:
obtaining, by the computing system, light scattering intensities of the particles in the solution from the light scattering detector, wherein the light scattering intensities comprise a first scattering intensity (I θ1 ), a second scattering intensity (I θ2 ), and a third scattering intensity (I θ3 ) of the particles in the solution at a first angle, a second angle, and a third angle, respectively, wherein the first angle, the second angle, and the third angle are determined relative to a beam of light of the light scattering detector;
determining, by the computing system, a first weighting factor (FW 1 ) based on a first form factor (FF 1 ) and the light scattering intensities, wherein determining the first weighting factor (FW 1 ) comprises:
determining a ratio of observed light scattering of the particles at the first angle to observed light scattering of the particles at the third angle (R′ θ1/θ3 ) based on the first scattering intensity (I θ1 ) and the third scattering intensity (I θ3 );
determining a ratio of observed light scattering of the particles at the second angle to observed light scattering of the particles at the third angle (R′ θ2/θ3 ) based on the second scattering intensity (I θ2 ), and the third scattering intensity (I θ3 );
determining a ratio of calculated light scattering at the first angle to calculated light scattering at the third angle
(
R
θ1
θ3
Calculated
FF
1
″
)
based on the first form factor (FF 1 ), the first angle, and the third angle;
determining a ratio of calculated light scattering at the second angle to the calculated light scattering at the third angle
(
R
θ2
θ3
Calculated
FF
1
″
)
based on the first form factor (FF 1 ), the second angle, and the third angle;
determining a best fit of the ratio (R′ θ1/θ3 ) to the ratio
(
R
θ1
θ3
Calculated
FF
1
″
)
and the ratio (R′ θ2/θ3 ) to the ratio
(
R
θ2
θ3
Calculated
FF
1
″
)
based on the first form factor (FF 1 ); and
determining, by the computing system, the first weight factor (FW 1 ) based on the best fit of the ratio (R′ θ1/θ3 ) to the ratio
(
R
θ1
θ3
Calculated
FF
1
″
)
and the ratio (R′ θ2/θ3 ) to the ratio
(
R
θ2
θ3
Calculated
FF
1
″
)
;
determining, by the computing system, a second weighting factor (FW 2 ) based on a second form factor (FF 2 ) and the light scattering intensities, wherein determining the second weighting factor (FW 2 ) comprises:
determining a ratio of calculated light scattering at the first angle to calculated light scattering at the third angle
(
R
θ1
θ3
Calculated
FF
2
″
)
based on the second form factor (FF 2 ), the first angle, and the third angle;
determining a ratio of calculated light scattering of the second angle to calculated light scattering at the third angle
(
R
θ2
θ3
Calculated
FF
2
″
)
based on the form factor (FF 2 ), the second angle, and the third angle;
determining a best fit of the ratio (R′ θ1/θ3 ) to the ratio
(
R
θ1
θ3
Calculated
FF
2
″
)
and the ratio (R′ θ2/θ3 ) to the ratio
(
R
θ2
θ3
Calculated
FF
2
″
)
based on the second form factor (FF 2 ); and
determining, by the computing system, the second weight factor (FW 2 ) based on the best fit of the ratio (R′ θ1/θ3 ) to the ratio
(
R
θ1
θ3
Calculated
FF
2
″
)
and the ratio (R′ θ2/θ2 ) to the ratio
(
R
θ2
θ3
Calculated
FF
2
″
)
determining, by the computing system, a first form factor contribution (FC 1 ) for the first form factor (FF 1 ) based on the first weighting factor (FW 1 ) and the second weighting factor (FW 2 ), wherein the first form factor contribution (FC 1 ) is determined according to Equation (1):
FC
1
=
FW
1
∑
FW
x
,
(
1
)
wherein μFW x is the sum of each of the weighting factors;
determining, by the computing system, a second form factor contribution (FC 2 ) for the second form factor (FF 2 ) based on the first weighting factor (FW 1 ) and the second weighting factor
FC
2
=
FW
2
∑
FW
x
;
(
2
)
determining, by the computing system, the assumption-free radius of gyration (Rg) of the particles based on the first and second form factor contributions (FC 1 , FC 2 ); and
normalizing the light scattering detector based on the assumption-free radius of gyration (Rg).