Methods of determining radius of gyration of a particle using light scattering detectors

1. A method for determining a radius of gyration (Rg) of a particle in a solution using a light scattering detector, the method comprising:

passing the particle in the solution through a flowpath in a sample cell, wherein the flowpath has a centerline aligned with a beam of light directed through a portion of the detector;

determining an angular normalization factor (N θ1 ) for a first angle of the detector, wherein the first angle is either about 90° or about 170° relative to the centerline;

obtaining a first scattering intensity (I θ1 ) of the particle in the solution at the first angle;

obtaining a second scattering intensity (I θ2 ) of the particle in the solution at a second angle, wherein the second angle is greater than 0° to less than 90°;

determining a first particle scattering factor (P 01 ) with the first scattering intensity (I θ1 ), the second scattering intensity (I θ2 ), and the angular normalization factor (N θ1 ) for the first angle;

plotting an angular dissymmetry plot, wherein the angular dissymmetry plot comprises the first particle scattering factor (P θ1 );

fitting a line to the angular dissymmetry plot;

determining a slope of the line at a selected location on the line;

determining the radius of gyration (Rg) of the particle in the solution from the slope of the line; and

outputting the radius of gyration.

2. The method of claim 1 , wherein determining the angular normalization factor (N θ1 ) for the first angle of the detector comprises:

passing each of a plurality of known particles in the solution through the flowpath of the sample cell;

obtaining scattering intensity values of each of the plurality of known particles in the solution at the first angle and the second angle; and

determining the angular normalization factor (N θ1 ) for the first angle with a plot of a ratio of the scattering intensity values of each of the plurality of known particles at the first angle to the scattering intensity values of each of the plurality of known particles at the second angle with respect to a respective weight average molecular weight of each of the plurality of known particles in the solution.

3. The method of claim 2 , wherein each of the plurality of known particles in the solution have a known molecular weight.

4. The method of claim 1 , wherein the first particle scattering factor (P θ1 ) is in the form

P

θ

⁢

⁢

1

=

(

I

θ

⁢

⁢

1

I

θ

⁢

⁢

2

)

N

θ

⁢

⁢

1

,

wherein:

I θ1 is the scattering intensity of the particle in the solution at the first angle;

I θ2 is the scattering intensity of the particle in the solution at the second angle; and

N θ1 is the angular normalization factor for the first angle.

5. The method of claim 1 , wherein plotting the angular dissymmetry plot comprises:

plotting a first point on a plane, the first point comprising a first coordinate and a second coordinate, wherein the first coordinate of the first point is the first particle scattering factor (Poi), and wherein the second coordinate of the first point is in the form

μ

θ

⁢

1

2

=

(

4

⁢

π

⁢

n

0

⁢

sin

⁢

θ

1

2

λ

)

2

,

wherein:

n 0 is a refractive index of the solution;

θ 1 is the first angle; and

λ is a wavelength of the beam of light.

6. The method of claim 1 , wherein the line of the angular dissymmetry plot is a straight line.

7. The method of claim 1 , wherein the radius of gyration (Rg) of the particle in the solution is less than 10 nm.

8. The method of claim 1 , further comprising:

obtaining an angular normalization factor (N θ2 ) of a third angle of the detector;

obtaining a third scattering intensity (I θ3 ) of the particle in the solution at the third angle; and

determining a second particle scattering factor (P θ2 ) with the third scattering intensity (I θ3 ), the second scattering intensity (I θ2 ), and the angular normalization factor (N θ2 ) for the third angle, wherein the angular dissymmetry plot further comprises the second particle scattering factor (P θ2 ).

9. The method of claim 8 , wherein the third angle is either about 90° or about 170° relative to the centerline, and wherein the third angle is different from the first angle.

10. The method of claim 9 , wherein determining the angular normalization factor (N θ2 ) of the third angle of the detector comprises:

obtaining scattering intensity values of each of the plurality of known particles in the solution at the third angle; and

determining the angular normalization factor (N θ2 ) for the third angle with a plot of a ratio of the scattering intensity values of each of the plurality of known particles at the third angle to the scattering intensity values of each of the plurality of known particles at the second angle with respect to a respective weight average molecular weight of each of the plurality of known particles in the solution.

11. The method of claim 10 , wherein the second particle scattering factor (P θ2 ) is in the form

P

θ

⁢

⁢

2

=

(

I

θ

⁢

⁢

3

I

θ

⁢

⁢

2

)

N

θ

⁢

⁢

2

,

wherein:

I θ3 is the scattering intensity of the particle in the solution at the third angle;

I θ2 is the scattering intensity of the particle in the solution at the second angle; and

N θ2 is the angular normalization factor for the third angle.

12. The method of claim 8 , wherein plotting the angular dissymmetry plot further comprises:

plotting a second point on the plane, the second point comprising a first coordinate and a second coordinate, wherein the first coordinate of the second point is the second particle scattering factor (P θ2 ), and wherein the second coordinate of the second point is in the form

μ

θ

⁢

2

2

=

(

4

⁢

π

⁢

n

0

⁢

sin

⁢

θ

3

2

λ

)

2

,

wherein:

n 0 is a refractive index of the solution;

θ 3 is the third angle; and

λ is the wavelength of the beam of light.

13. The method of claim 1 , wherein the line of the angular dissymmetry plot is a curved line.

14. The method of claim 1 , wherein the radius of gyration (Rg) of the particle in the solution is less than 100 nm.

15. A method for determining a radius of gyration (Rg) of a particle in a solution using a light scattering detector, the method comprising:

passing the particle in the solution through a flowpath in a sample cell, wherein the flowpath has a centerline aligned with a beam of light directed through a portion of the detector;

determining an angular normalization factor (N θ1 ) for a first angle of the detector relative to the centerline;

determining an angular normalization factor (N θ2 ) of a second angle of the detector relative to the centerline;

obtaining a first scattering intensity (I θ1 ) of the particle in the solution at the first angle;

obtaining a second scattering intensity (I θ2 ) of the particle in the solution at the second angle;

obtaining a third scattering intensity (I θ3 ) of the particle in the solution at a third angle of the detector relative to the centerline;

determining a first particle scattering factor (P θ1 ) with the first scattering intensity (I θ1 ), the third scattering intensity (I θ3 ), and the angular normalization factor (N θ1 ) for the first angle;

determining a second particle scattering factor (P θ2 ) with the second scattering intensity (I θ2 ), the third scattering intensity (I θ3 ), and the angular normalization factor (N θ2 ) for the second angle;

plotting an angular dissymmetry plot, wherein the angular dissymmetry plot comprises the first particle scattering factor (P θ1 ) and the second particle scattering factor (P θ2 );

fitting a line to the angular dissymmetry plot;

determining a slope of the line at a selected location on the line;

determining the radius of gyration (Rg) of the particle in solution from the slope of the line; and

outputting the radius of gyration (Rg).

16. The method of claim 15 , wherein determining the angular normalization factor (N θ1 ) for the first angle and the angular normalization factor (N θ2 ) of the second angle comprises:

passing each of a plurality of known particles in the solution through the flowpath of the sample cell;

obtaining scattering intensity values for each of the plurality of known particles in the solution at the first angle, at the second angle, and at the third angle;

determining the angular normalization factor (N θ1 ) for the first angle with a plot of a ratio of the scattering intensity values of each of the plurality of known particles at the first angle to the scattering intensity values of each of the plurality of known particles at the third angle;

determining the angular normalization factor (N θ2 ) for the second angle with a plot of a ratio of the scattering intensity values of each of the plurality of known particles at the second angle to the scattering intensity values of each of the plurality of known particles at the third angle.

17. The method of claim 16 , wherein each of the plurality of known particles in the solution have a known molecular weight.

18. The method of claim 15 , wherein the first particle scattering factor (P θ1 ) is in the form

P

θ

⁢

⁢

1

=

(

I

θ

⁢

⁢

1

I

θ

⁢

⁢

3

)

N

θ

⁢

⁢

1

,

where:

I θ2 is the scattering intensity of the particle in the solution at the first angle;

I θ3 is the scattering intensity of the particle in the solution at the third angle; and

N θ2 is the angular normalization factor for the first angle.

19. The method of claim 15 , wherein the second particle scattering factor (P θ2 ) is in the form

P

θ

⁢

⁢

2

=

(

I

θ

⁢

⁢

2

I

θ

⁢

⁢

3

)

N

θ

⁢

⁢

2

,

wherein:

I θ2 is the scattering intensity of the particle in the solution at the second angle;

I θ3 is the scattering intensity of the particle in the solution at the third angle; and

N θ2 is the angular normalization factor for the second angle.

20. The method of claim 15 , wherein plotting the angular dissymmetry plot comprises:

plotting a first point on a plane, the first point comprising a first coordinate and a second coordinate, wherein the first coordinate of the first point is the first particle scattering factor (P θ1 ), and wherein the second coordinate of the first point is in the form

μ

θ

⁢

1

2

=

(

4

⁢

π

⁢

n

0

⁢

sin

⁢

θ

1

2

λ

)

2

,

wherein:

n 0 is a refractive index of the solution;

θ 1 is the first angle; and

λ is a wavelength of the beam of light; and

plotting a second point on the plane, the second point comprising a first coordinate and a second coordinate, wherein the first coordinate of the second point is the second particle scattering factor (P θ2 ), and wherein the second coordinate of the second point is in the form

μ

θ

⁢

2

2

=

(

4

⁢

π

⁢

n

0

⁢

sin

⁢

θ

2

2

λ

)

2

,

wherein:

n 0 is a refractive index of the solution;

θ 2 is the second angle; and

λ is the wavelength of the beam of light.