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Calculate Branching Frequency from Zimm-Stockmayer Theory

Source: R/polymer-analysis.R
measure_branching_frequency.Rd

Calculates the number of branch points per molecule using Zimm-Stockmayer theory, which relates the branching index (g) to the branching frequency.

Usage

measure_branching_frequency(
  g,
  architecture = c("random", "star_3", "star_4", "star_f", "comb"),
  arms = NULL,
  mw = NULL
)

Arguments

g

Numeric vector of branching ratios g = Rg^2(branched) / Rg^2(linear).

architecture

Branching architecture model:

  • "random": Random trifunctional branching (default)

  • "star_3": 3-arm star polymer

  • "star_4": 4-arm star polymer

  • "star_f": f-arm star (requires arms parameter)

  • "comb": Comb-like branching

arms

Number of arms for star polymers (required for "star_f").

mw

Optional molecular weight vector (same length as g) for calculating total branches in sample.

Value

A data frame of class branching_frequency containing:

g

Input branching ratio

branches_per_molecule

Estimated branch points per molecule

branch_density

Branches per unit MW (if mw provided)

Details

Zimm-Stockmayer Theory:

For random trifunctional branching, the relationship between g and the average number of branch points (n_b) is:

$$g = \left[\left(1 + \frac{n_b}{7}\right)^{1/2} + \frac{4n_b}{9\pi}\right]^{-1/2}$$

For star polymers with f arms: $$g = \frac{3f - 2}{f^2}$$

Interpretation:

  • g = 1.0: Linear polymer (no branching)

  • g ~ 0.5-0.8: Lightly branched

  • g ~ 0.3-0.5: Moderately branched

  • g < 0.3: Highly branched/hyperbranched

References

Zimm, B.H. and Stockmayer, W.H. (1949). "The Dimensions of Chain Molecules Containing Branches and Rings." J. Chem. Phys., 17, 1301-1314.

See also

Other sec-polymer: measure_branching_index(), measure_branching_model_comparison(), measure_conformation_data(), measure_mh_parameters(), measure_rg_mw_scaling()

Examples

# Calculate branching for random trifunctional polymer
g_values <- c(0.9, 0.7, 0.5, 0.3)
bf <- measure_branching_frequency(g_values)
print(bf)
#> Branching Frequency Analysis (Zimm-Stockmayer)
#> ================================================== 
#> 
#> Architecture: random
#> N samples: 4
#> 
#> g ratio range: 0.300 - 0.900
#> Branches/molecule: 1.12 - 57.14 (mean: 19.74)
#> 
#> # A tibble: 4 × 2
#>       g branches_per_molecule
#>   <dbl>                 <dbl>
#> 1   0.9                  1.12
#> 2   0.7                  5.12
#> 3   0.5                 15.6 
#> 4   0.3                 57.1 

# For star polymer with 4 arms
bf_star <- measure_branching_frequency(0.625, architecture = "star_4")