  When compressed in a slit of width $D$, a $\Theta$-chain that displays the
scaling of size $R_0$ (diameter) with respect to the number of monomers $N$,
$R_0\sim aN^{1/2}$, expands in the lateral direction as $R_{\parallel}\sim a
N^{\nu}(a/D)^{2\nu-1}$. Provided that the $\Theta$ condition is strictly
maintained throughout the compression, the well-known scaling exponent of
$\Theta$-chain in 2 dimensions, $\nu=4/7$, is anticipated in a perfect
confinement. However, numerics shows that upon increasing compression from
$R_0/D<1$ to $R_0/D\gg 1$, $\nu$ gradually deviates from $\nu=1/2$ and plateaus
at $\nu=3/4$, the exponent associated with the self-avoiding walk in two
dimensions. Using both theoretical considerations and numerics, we argue that
it is highly nontrivial to maintain the $\Theta$ condition under confinement
because of two major effects. First, as the dimension is reduced from 3 to 2
dimensions, the contributions of higher order virial terms, which can be
ignored in 3 dimensions at large $N$, become significant. Second and more
importantly, the geometrical confinement, which is regarded as an applied
external field, alters the second virial coefficient ($B_2$) changes from
$B_2=0$ ($\Theta$ condition) in free space to $B_2>0$ (good-solvent condition)
in confinement. Our study provides practical insight into how confinement
affects the conformation of a single polymer chain.
