New model for randomly branched polycondensates

[gonzalezma]

An ILL colleague (Apostolos Vagias, working now in D22) would like to have the model described in an old paper: "ANGULAR DISTRIBUTION OF RAYLEIGH SCATTERING FROM RANDOMLY BRANCHED POLYCONDENSATES", by K. KAJIWARA, W. BURCHARD* and M. GORDON, Br. Polym. J., 1970, Vol. 2, 110-115). I guess this one can be hard to find, so I added it below.

He also provided the C code used in Sasfit (also attached), which seems very simple (a single line formula?)

pi.4980020203.pdf

sasfit_sq_p_ra.c.txt

[butlerpd]

Do we know which of the equations in the reference we are trying to implement? Equation 21a seems like the one but it does not seem to match the c code from SASfit. Does anybody know?

[pkienzle]

The sasfit docs for the model say:

• B = bond length of the polycondensates
• F = functionality
• NW = weight-average number of the complex units per polyelectrolyte complex

The paper gives the weight-averaged degree of polymerization as

• D Pw = (1 + α) / (1 - α (f - 1))

Solving for α and substituting into 21a I get a different set of equations from sasfit:

• numerator = 1 + (1-DPw)(f-1)/(2 - DPw)[1 - sin qb / qb]
• denominator = 1 - (1-DPw)/(2 - f DPw)[1 - sin qb / qb]

[pkienzle]

My algebra above was incorrect. Redoing with wolframalpha:

solve w = (1+a)/(1 - a(f-1)) for a
=> a = (w-1)/((f-1)*w+1)

substitute a into numerator and denominator terms, then simplify
num = 1 + a * (f-1)/(1 - a * (f-1))*t
den = 1 - a / (1 + a)*t

simplify (w-1)/((f-1)*w+1) * (f-1)/(1 - (w-1)/((f-1)*w+1) * (f-1))
=> ((f - 1) (w - 1))/f
simplify (w-1)/((f-1)*w+1) / (1 + (w-1)/((f-1)*w+1))
=> (w - 1)/(f w)

multiply numerator and denominator by (f w)

The following reproduces curve (b) from Fig 4a,4b in the paper:

def do(q, B, f, w):
    t = 1 - np.sin(q*B)/(q*B)
    num = f*w + w*(f-1)*(w-1)*t
    den = f*w - (w-1)*t
    Pinv = num/den
    plt.plot(q, Pinv)
    plt.grid(True)
do(np.linspace(0, 10, 200)[1:], 1, 2, 1499.5)
do(np.linspace(0, 10, 200)[1:], 1, 4, 1499.5)

Noting that the paper uses $P^{-1}$, we swap the numerator and denominator to get the same equations as SASfit.