  We present a theoretical model to study the effect of counterion size on the
effective charge, size, and thermodynamic behavior of a single, isolated, and
flexible polyelectrolyte (PE) chain. We analyze how altering counterion size
modifies the energy and entropy contributions to the system, including the
ion-pair free energy, excluded volume interactions, entropy of free and
condensed ions, and dipolar attraction among monomer-counterion pairs, which
result in competing effects challenging intuitive predictions. The PE self
energy is calculated using Edwards-Muthukumar Hamiltonian, considering a
Gaussian monomer distribution for the PE. The condensed ions are assumed
confined within a cylindrical volume around the PE backbone. The dipolar and
excluded volume interactions are described by the second and third virial
coefficients. Assumption of freely-rotating dipoles results in a first-order
coil-globule transition of the PE chain. A more realistic weaker dipolar
attraction, parameterized in our theory, shifts it to a second-order continuous
transition. We calculate the size scaling-exponent of the PE and find exponents
according to the relative dominance of the electrostatic, excluded volume, or
dipolar effects. We further identify the entropy- and energy-driven regimes of
the effective charge and conformation of the PE, highlighting the interplay of
free ion entropy and ion-pair energy with varying electrostatic strengths. The
crossover strength, dependent on the counterion size, indicates that
diminishing sizes favor counterion condensation at the expense of free ion
entropy. The predictions of the model are consistent with trends in
simulations, and generalize findings of the point-like counterion theories.
