RADIUS OF GYRATION CALCULATIONS - PDF document

1 RADIUS OF GYRATION CALCULATIONS The radius of gyration is a measure of the size obtained directly from the Guinier plot [ln(I(Q)] is the second moment in 3D. First consider some simple shape objects. 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. = 201R02021R02drdrrdrd)(cosr = 11R020R02023drdr)d(cosdrr = . (1) Similarly for R 4R2. For an infinitely thin disk Rg= Rgx+ Rgy= 2R2. y cal coordinate system for a sphere. In the case of a full sphere, the integration is performed with spherical coordinates. 2 = 0R020R022drrd)sin(r drrd)sin( = . (2) (3) )RR()RR(5331325152. y 3 Figure 3: Representation of the Cartesian coordinate system for a rectangular plate. e integration is performed in = W/2W/2W/2W/22dxdx x = . (4) Similarly for R . The sum gives R 222H2W31 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 = I . (5) y 4 2. CIRCULAR ROD AND RECTANGULAR BEAM drical rod and rectangular beam. The radius of gyration for a cylindrical rod of length L and radius R is given by: . (6) The radius of gyration for a rectangular beam of width W, height H R. (7) This formula holds for a stra for a cylindrical rod with radius R = 10 (diameter D = 20) and length L = 58.3. This value is to be compared with the case of a rectangular beam with sides W = L = 20 and length L = 10 for which RThe radius of gyration squared can be calculated for other more complicated shapes as the second moment for each of the symmetry direction. Note that R for a horizontal strip is the same as that for the whole square plate R = 22W31 is independent of the height of the object. Of course Rgy depends of the height but not on the width. Circular Rod Rectangular Beam . This is the same value for an infinitely thin spherical shell of radius R. simple case of a rigid helical of a rigid twisted rithickness. Helical Wire origin of the Cartesian coordinate system at the center-of mass of the twisted wire. The z L/2. The parametric equation of y y X = R cos() (8) Y = R sin( Z = pHere p is the helix pitch and is the azimuthal angle in the horizontal plane. The wire center-of-mass, the average of this vector is null, Figure 7: Schematic representation of the twisted wire. is defined as follows: . (9) . The azimuthal angle integration is readily performed to give: 2 = R2 + . (10) Twisted Wire x y Note that this is the same result as for a cylindrical shell of radius R not surprising since a cylinder could be built by a number of twisted wires stacked Thin Twisted Ribbon al ribbon of width W can be worked out similarly using a two-variable parametric notation r is the azimuthal angle and z is the vertical ribbon width with –W/2 z W/2. Figure 8: Schematic representation of the thin twisted ribbon. Here, the variable Z is replaced by Z+z. The radius of gyration (squared) is therefore ,z) = (11) . The integrations can here also be readily performed to give: Twisted Ribbon x y 8 2 = R2 + 22L31 . (12) These involve contributions from &#xZ000;. The cross term gives no contribution because it involves the nu&#xz000;ll average Thick Twisted Ribbon The calculation of the second moment proceeds as before: X = ) (13) Y = Z = pvariable in the horizontal plane with limits: R-T/2 R+T/2. In this case r where z is the same parameter as before. R� + (+z) 2/TR2/TR2022/TR2/TR20dd dd = 22442TR2TR212TR2TR41 = 222TR Thick Twisted Ribbon To View R T (14) ontal and vertical) averages is: 2 = R2 + 22T 2L31 . (15) Note that all terms add up in quadrature since all cross terms (first moments) average to 5. GAUSSIAN POLYMER COIL for a polymer coil is defined as: R. (16) refers to the position of monomer i with respect to the center-of-mass of the polymer coil and n is the total number of monomers per coil. The inter-distance vector between two monomers within the same macromolecule is defined as . Consider the . (17) The last summation is null since by definition of the center-of- mass d) is therefore simplified as: . (18) een dropped for simplicity. Figure 10: Schematic representation of a Gaussian coil showing monomers i and j and their inter-distance r. Note that in the notation used. . (19) Here a is the statistical segment length, &#x…000;and e over monomers. The following formulae for the summation of arithmetic progressions are used: (20) The radius of gyration squared becomes: R = (21) = 6nan)1n(6a222�� for n 1. Note that ta�king the n � 1 limit early on allows us to replace the summations by integrations. Using the variable x = k/n, one obtains: . (22) Similarly, the end-to-e for a Gaussian polymer coil is given by: iS ijijrS center of mass i j r jr S for n �� 1. (23) coils that follow random walk 6. THE EXCLUDED VOLUME PARAMETER APPROACH The Flory mean field theory of polymer soluwalk process along chain segments. For Gaussian chain statistics, the monomer-monomer inter-distance is proportional to the number of steps: . 24 Here a is the statistical segment length, is the excluded volume parameter, S represents an inter-segme&#x…000;nt distance and erage over monomers. The radius of gyration R (25) i and j are a pair of monomers and n is the number of chain segments per chain. Three (1) Self-avoiding walk corresponds to swollen chains with = 3/5, for which 5622gna17625R (2) Pure random walk corresponds to chains monomer-monomer and solvent-monomer interactions are equivalent) with a21R22g (3) Self attracting walk corresponds to collapsed chains with = 1/3, for which 3222gna409R Note that the renormalization group estimate of the excluded volume parameter for the fully swollen chain is 6 mean field value). Note also that the radius ofcan be recovered from this excluded volume approach by setting . (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 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? for a thin spherical shell of 5. What is the value of R 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 ng a Guinier plot on SANS data. The ()vs Q2 is Rg2/3. 2. The center-of-mass of an object is defined as the spot where the first moment is zero. 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: Rg2 = 0R020R022drrd)sin(/drrrd)sin( = for a thin spherical shell is simply given by: R 5. For a Gaussian coil of segment length a and degree of polymerization n, one can calculate the radius of and the end-to-end distance 6. The radius of gyration squared for a rod of length L and radius R is given by: 222g2L312RR