  The negative internal energetic contribution to the elastic modulus (negative
energetic elasticity) has been recently observed in polymer gels. This finding
challenges the conventional notion that the elastic moduli of rubberlike
materials are determined mainly by entropic elasticity. However, the
microscopic origin of negative energetic elasticity has not yet been clarified.
Here, we consider the $n$-step interacting self-avoiding walk on a cubic
lattice as a model of a single polymer chain (a subchain of a network in a
polymer gel) in a solvent. We theoretically demonstrate the emergence of
negative energetic elasticity based on an exact enumeration up to $n=20$ and
analytic expressions for arbitrary $n$ in special cases. Furthermore, we
demonstrate that the negative energetic elasticity of this model originates
from the attractive polymer--solvent interaction, which locally stiffens the
chain and conversely softens the stiffness of the entire chain. This model
qualitatively reproduces the temperature dependence of negative energetic
elasticity observed in the polymer-gel experiments, indicating that the
analysis of a single chain can explain the properties of negative energetic
elasticity in polymer gels.
