polykin.kinetics.emulsion

nbar_Ugelstad

nbar_Ugelstad(
    alpha: float,
    m: float,
    *,
    tol: float = 1e-10,
    maxiter: int = 100
) -> float

Average number of radicals per particle according to the Ugelstad-Mørk exact quasi-steady-state solution.

\[ \bar{n} = \frac{1}{2} \frac{2 \alpha}{m + \frac{2 \alpha}{m + 1 + \frac{2 \alpha}{m + 2 + \frac{2 \alpha}{m + 3 + ...}}}} \]

The continued fraction is evaluated using Lentz's modified method.

References

• Ugelstad J., Mörk P.C. and Aasen J.O., Kinetics of emulsion polymerization. J. Polym. Sci. A-1 Polym. Chem., 1967; 5:2281-2288.

Note

Included mainly for historical completeness. Today, the Stockmayer-O'Toole solution is preferred: it is equally exact, simpler to evaluate, and numerically stable for large α and m.

PARAMETER	DESCRIPTION
alpha	Dimensionless entry frequency. TYPE: float
m	Dimensionless desorption frequency. TYPE: float
tol	Tolerance for convergence of the continued fraction evaluation. TYPE: float DEFAULT: 1e-10
maxiter	Maximum number of iterations for the continued fraction evaluation. TYPE: int DEFAULT: 100

RETURNS	DESCRIPTION
float	Average number of radicals per particle.

See Also

• nbar_Stockmayer_OToole: Preferred exact solution; simpler and numerically robust.
• nbar_Li_Brooks: Approximate solution; typically an order of magnitude faster.

Examples:

Evaluate the average number of radicals per particle for α=1e-2 and m=1e-4.

>>> from polykin.kinetics import nbar_Ugelstad
>>> nbar = nbar_Ugelstad(alpha=1e-2, m=1e-4)
>>> print(f"nbar = {nbar:.2e}")
nbar = 5.02e-01

Source code in src/polykin/kinetics/emulsion/smithewart.py

def nbar_Ugelstad(
    alpha: float,
    m: float,
    *,
    tol: float = 1e-10,
    maxiter: int = 100,
) -> float:
    r"""Average number of radicals per particle according to the Ugelstad-Mørk exact
    quasi-steady-state solution.

    $$ \bar{n} = \frac{1}{2}
      \frac{2 \alpha}{m +
      \frac{2 \alpha}{m + 1 +
      \frac{2 \alpha}{m + 2 + \frac{2 \alpha}{m + 3 + ...}}}} $$

    The continued fraction is evaluated using Lentz's modified method.

    **References**

    *   Ugelstad J., Mörk P.C. and Aasen J.O., Kinetics of emulsion polymerization.
        J. Polym. Sci. A-1 Polym. Chem., 1967; 5:2281-2288.

    Note
    ----
    Included mainly for historical completeness. Today, the Stockmayer-O'Toole solution is
    preferred: it is equally exact, simpler to evaluate, and numerically stable for large
    α and m.

    Parameters
    ----------
    alpha : float
        Dimensionless entry frequency.
    m : float
        Dimensionless desorption frequency.
    tol : float
        Tolerance for convergence of the continued fraction evaluation.
    maxiter : int
        Maximum number of iterations for the continued fraction evaluation.

    Returns
    -------
    float
        Average number of radicals per particle.

    See Also
    --------
    * [`nbar_Stockmayer_OToole`](nbar_Stockmayer_OToole.md):
      Preferred exact solution; simpler and numerically robust.
    * [`nbar_Li_Brooks`](nbar_Li_Brooks.md):
      Approximate solution; typically an order of magnitude faster.

    Examples
    --------
    Evaluate the average number of radicals per particle for α=1e-2 and m=1e-4.
    >>> from polykin.kinetics import nbar_Ugelstad
    >>> nbar = nbar_Ugelstad(alpha=1e-2, m=1e-4)
    >>> print(f"nbar = {nbar:.2e}")
    nbar = 5.02e-01
    """
    tiny = 1e-30
    a = 2.0 * alpha
    delta = 1e99

    # Initial denominator term (b0 = m)
    f = max(m, tiny)
    C = f
    D = 0.0

    for k in range(1, maxiter + 1):
        b = m + k

        D = 1.0 / max(b + a * D, tiny)
        C = max(b + a / C, tiny)

        delta = C * D
        f *= delta

        if abs(delta - 1.0) < tol:
            return alpha / f

    raise ConvergenceError(
        f"Lentz method failed to converge after {maxiter} iterations "
        f"(alpha={alpha}, m={m}, tol={tol}). "
        f"Final |delta-1|={abs(delta - 1):.3e}, F={f:.6e}"
    )