Radius of Gyration

Chapter 26 - RADIUS OF GYRATION CALCULATIONS The radius of gyration is a measure of the size of an object of arbitrary shape. It can be obtained directly from the Guinier plot [ln(I(Q)] vs Q2] for SANS data. The radius of gyration squared Rg2 is the second moment in 3D.

1. SIMPLE SHAPES First consider some simple shape objects. y

R

r φ r cos(φ)

x

Figure 1: Representation of the polar coordinate system for a disk. For an infinitely thin disk of radius R, Rg2 is given by the following integral using polar coordinates. 2π R1

2

Rgx =

∫ ∫r

2

cos (φ)rdrdφ

0 0

2π R1

∫ ∫ rdrdφ 0 0

Similarly for

Rgy2

R1

2π

∫ r dr ∫ cos 3

2

=

0

2

(φ)dφ

0

R1

2π

0

0

∫ rdr ∫

= dφ

R2 . 4

(1)

2 R2 2 2 2 R = . For an infinitely thin disk Rg = Rgx + Rgy = . 4 2

1

z

R

θ

r

y

φ

x

Figure 2: Representation of the spherical coordinate system for a sphere. In the case of a full sphere, the integration is performed with spherical coordinates.

Rg2 =

π

R

0

0

2 2 ∫ sin(θ)dθ∫ r dr r π

R

∫ sin(θ)dθ∫ r 0

2

dr

=

3R 2 . 5

(2)

0

The radius of gyration (squared) for the spherical shell of radii R1 and R2 is given by: Rg2 =

3

R2

∫ 4π(R 2 − R13 ) R1

=

3

4πr 4 dr

(3)

5 5 3 (R 2 − R 1 ) . 5 (R 2 3 − R 13 )

2

y

x

H

W Figure 3: Representation of the Cartesian coordinate system for a rectangular plate. For an infinitely thin rectangular object of sides W and H, the integration is performed in Cartesian coordinates. W/2

Rgx2 =

∫ dx x

− W/2 W/2

2

2

=

∫ dx

1⎛ W⎞ ⎜ ⎟ . 3⎝ 2 ⎠

(4)

− W/2

Similarly for

1 ⎡⎛ W ⎞ ⎛ H ⎞ . The sum gives Rg = ⎢⎜ ⎟ + ⎜ ⎟ 3 ⎢⎣⎝ 2 ⎠ ⎝ 2 ⎠ 3⎝ 2 ⎠

1⎛H⎞ Rgy2 = ⎜ ⎟

2

2

2

2

⎤ ⎥. ⎥⎦

Note that the moment of inertia I for a plate of width W, height H and mass M is also given by the second moment. I = Ixx + Iyy =

2 2 M ⎡⎛ W ⎞ ⎛ H ⎞ ⎤ + ⎢⎜ ⎟ ⎜ ⎟ ⎥ . 3 ⎣⎢⎝ 2 ⎠ ⎝ 2 ⎠ ⎦⎥

(5)

3

2. CIRCULAR ROD AND RECTANGULAR BEAM

Circular Rod Rectangular Beam

Figure 4: Representation of the cylindrical rod and rectangular beam. The radius of gyration for a cylindrical rod of length L and radius R is given by: 2

2

Rg =

2

R2 1⎛ L⎞ R2 L + ⎜ ⎟ = . + 2 3⎝ 2 ⎠ 2 12

(6)

The radius of gyration for a rectangular beam of width W, height H and length L is given by: Rg

2

2 2 2 1 ⎡⎛ W ⎞ ⎛ H ⎞ ⎛ L ⎞ ⎤ = ⎢⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ ⎥ . 3 ⎣⎢⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎦⎥

(7)

This formula holds for a straight “ribbon” where W = i, j

2

1 2n 2

n

∑< r

ij

2

>.

i, j

The vectorial notation has been dropped for simplicity.

9

(18)

i

r ri

r Si

r r Sij = rij

center of mass

r Sj j

r rj

Figure 10: Schematic representation of a Gaussian coil showing monomers i and j and r r their inter-distance rij. Note that Sij = rij in the notation used.

< Sij 2 >= a 2 | i − j | .

(19)

Here a is the statistical segment length, and is an average over monomers. The following formulae for the summation of arithmetic progressions are used: n (n + 1) 2 k =1 n n (n + 1)(2n + 1) k2 = . ∑ 6 k =1 n

∑k =

(20)

The radius of gyration squared becomes: Rg

2

a2 = 2 2n

a2 | i − j| = ∑ n i, j n

n

k

∑ (1 − n )k

(21)

k

a 2 (n 2 − 1) a 2 n = ≅ for n >> 1. 6 n 6 Note that taking the n >> 1 limit early on allows us to replace the summations by integrations. Using the variable x = k/n, one obtains: 1

2 ⎛ 1 1⎞ a n 2 . R g = a 2 n ∫ dx (1 − x ) x =a 2 n ⎜ − ⎟ = 6 ⎝ 2 3⎠ 0

(22)

Similarly, the end-to-end distance squared R1n2 for a Gaussian polymer coil is given by: 10

2

R 1n = a 2 n for n >> 1.

(23)

These results are for Gaussian coils that follow random walk statistics (Flory, 1969).

6. THE EXCLUDED VOLUME PARAMETER APPROACH

The Flory mean field theory of polymer solutions describes chain statistics as a random walk process along chain segments. For Gaussian chain statistics, the monomer-monomer inter-distance is proportional to the number of steps: < Sij 2 >= a 2 | i − j | 2ν .

(24)

Here a is the statistical segment length, ν is the excluded volume parameter, Sij represents an inter-segment distance and is an average over monomers. The radius of gyration squared for Gaussian chains is expressed as: 2

Rg =

1 2n 2 =

n

∑ < Sij > = 2

i, j

a2 n

a2 2n 2

n

∑| i − j |

2ν

(25)

i, j

k 2ν a2 (1 ) k n 2ν . − = ∑k n (2ν + 1)(2ν + 2) n

i and j are a pair of monomers and n is the number of chain segments per chain. Three cases are relevant: (1) Self-avoiding walk corresponds to swollen chains with ν = 3/5, for which 25 2 6 5 Rg2 = a n . 176 (2) Pure random walk corresponds to chains in theta conditions (where solvent-solvent, monomer-monomer and solvent-monomer interactions are equivalent) with ν = ½, for 1 which R g 2 = a 2 n . 2 (3) Self attracting walk corresponds to collapsed chains with ν = 1/3, for which 9 2 23 Rg2 = a n . 40 Note that the renormalization group estimate of the excluded volume parameter for the fully swollen chain is ν = 0.588 (instead of the 0.6 mean field value). Note also that the radius of gyration for a thin rigid rod can be recovered from this excluded volume approach by setting ν = 1 and defining the rod length as L = na.

11

2

Rg =

a2 a 2 n 2 L2 n 2ν = = . (2ν + 1)(2ν + 2) 12 12

(26)

This is the same result derived earlier for a thin rod.

REFERENCES

http://en.wikipedia.org/wiki/List_of_moments_of_inertia P.J. Flory, “Statistical Mechanics of Chain Molecules”, Interscience Publishers (1969)

QUESTIONS

1. How is the radius of gyration measured by SANS? 2. How is the center-of-mass of an object defined? 3. Why is the radius of gyration squared for an object related to the moment of inertia for that object? 4. Calculate Rg2 for a full sphere of radius R. Calculate Rg2 for a thin spherical shell of radius R. 5. What is the value of Rg2 for a Gaussian coil of segment length a and degree of polymerization n? How about the end-to-end distance? 6. What is the radius of gyration squared for a rod of length L and radius R?

ANSWERS

1. The radius of gyration is measured by performing a Guinier plot on SANS data. The slope of the linear variation of ln[I(Q)] vs Q2 is Rg2/3. 2. The center-of-mass of an object is defined as the spot where the first moment is zero. 3. The radius of gyration squared and the moment of inertia for that object are both expressed in terms of the second moment. 4. Rg2 for a full sphere of radius R is given by: R R ⎛π ⎞ ⎛π ⎞ 3R 2 Rg2 = ⎜⎜ ∫ sin(θ)dθ∫ r 2 drr 2 ⎟⎟ / ⎜⎜ ∫ sin(θ)dθ∫ r 2 dr ⎟⎟ = . Rg2 for a thin spherical shell is 5 0 0 ⎝0 ⎠ ⎝0 ⎠ 2 2 simply given by: Rg = R . 5. For a Gaussian coil of segment length a and degree of polymerization n, one can calculate the radius of gyration squared as Rg2 = a 2 n / 6 and the end-to-end distance squared as R1n2 = a 2 n . 6. The radius of gyration squared for a rod of length L and radius R is given by: Rg

2

2

R2 1⎛ L ⎞ = + ⎜ ⎟ . 2 3⎝ 2 ⎠

12