The links given below provide MathCad files to compute the intrinsic viscosity and the hydrodynamic radius using the helical-worm model developed by Yamakawa and Yoshizaki (see Yamakawa, H., Helical Wormlike Chains in Polymer Solutions. Springer, New York, 1997).

The MathCad files were written in MathCad 6 on a MAC platform. They should work with that or more recent MathCad versions on either Window or MAC platforms.

The user should down load the files to a common folder (directory) for use, and then open and run the files via a MathCad application.


The user should consult the book cited above, or the original publications cited therein and listed below for definitions of the parameters L, d, k, t, etc., used in the calculaton.  The parameters k and t in the model, both bearing subscripts "o" in the nomenclature in the reference cited above, may be related to the pitch h and radius r of the helix in the model via Eqn 4.14 in the same reference.  All lengths given herein are the reduced lengths defined in those sources.  

The helical-worm model reduces to the wormlike chain model of Yamakawa and Fujii (see references cited below) for k = 0.

The calculated results are presented here as normally used dimensionless parameters (thereby avoiding the need to specify the absolute length scale).

    q   =  Rg/{Rg}L=Infinity

    F'  =  M[h]/Rg^3

    Z   =  Rh/Rg

    K   =  F'/(pNAVGZ)


HW-cylinder model

Z (contains q)

F' (contains q)

K (must have Z and F' open)

HW-touched bead model

Z (contains q)

F' (contains q)

K(must have Z and F' open)

The above must be downloaded before they are opened by MathCad:
Windows: Right click and "save as"
Mac: Hold down "option key" while clicking on the symbol.

The files for Z and F' run independently, but both must be open to run the file for K, with all three placed in the same folder (directory).
The user will then have to use the "Reference" option in the "Insert" dialog in MathCad to renew the links in the K file to connect to the Z and
F' files--the order these appear in the original K file must be preserved in that renewal.


The touched-bead model is most appropriate for essentially flexible chains, with some helical content.

The cylinder model is most appropriate for semiflexible chains with a substantial persistence length and some helical content. 



Original Literature:

Yamakawa, H.; Fujii, M., "Translational friction coefficient of wormlike chains"; Macromolecules 1973, 6; 407-15.

Yamakawa, H.; Fujii, M., "Intrinsic viscosity of wormlike chains. Determination of the shift factor"; Macromolecules 1974, 7; 128-35

Yamakawa, H.; Yoshizaki, T.; Fujii, M., "Transport coefficients of helical wormlike chains. 1. Characteristic helices"; Macromolecules 1977, 10; 934-43.

Yamakawa, H.; Yoshizaki, T., "Transport coefficients of helical wormlike chains. 2. Translational friction coefficient"; Macromolecules 1979, 12; 32-8.

Yamakawa, H.; Yoshizaki, T., "Transport coefficients of helical wormlike chains. 3. Intrinsic viscosity"; Macromolecules 1980, 13; 633-43.