= Solution
With polymer degree $N_A$ and single-site solvent molecules, <Flory–Huggins solution theory> gives $f/(k_BT)=(\phi/N_A)\ln\phi+(1-\phi)\ln(1-\phi)+\chi\phi(1-\phi)$. Expanding the solvent entropy at low concentration gives the <dilute-polymer free-energy expansion>
$$
\boxed{\frac f{k_BT}=\frac\phi{N_A}\ln\phi+(\chi-1)\phi+\left(\frac12-\chi\right)\phi^2+\frac{\phi^3}{6}+O(\phi^4).}
$$
The linear term sets a reference chemical potential. The quadratic term controls the leading effective two-segment <excluded volume>, $v_{\rm eff}=v_0(1-2\chi)$ for site volume $v_0$.
In a <good solvent>, $\chi<1/2$ and positive <excluded volume> swells a dilute chain: the <Flory approximation> gives $R\sim bN_A^{3/5}$ in three dimensions. In a <poor solvent>, $\chi>1/2$ favors segment contacts and a sufficiently long isolated chain forms a <polymer globule>, with compact scaling $R\sim bN_A^{1/3}$ up to density-dependent factors.
At the <theta temperature>, the leading two-segment interaction cancels, $\chi=1/2$ in this model, and the long-chain conformation is approximately an <ideal chain>, $R\sim b\sqrt{N_A}$. The positive cubic term remains. The <critical point of a Flory–Huggins polymer solution> at finite $N_A$ has $\chi_c=\tfrac12(1+N_A^{-1/2})^2$; its limiting value, rather than every finite-chain critical point, is the theta condition.
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