Consider a polymer chain dissolved in a solvent, where its behavior is influenced by the Flory parameter, denoted by $\\chi$. As the interaction parameter $\\chi$ varies, the chain undergoes several conformational transitions:<\/p>\n\n\n\n
\u2022\u2003For $\\chi > \\chi_c$, the polymer collapses into a dense, globular structure with a radius of gyration scaling as $R_g \\sim N^{1\/3}$.<\/p>\n\n\n\n
\u2022\u2003At the critical point\u2014commonly known as the $\\theta$ point\u2014where $\\chi \\sim \\chi_c$, the chain behaves like an ideal random coil, characterized by $R_g \\sim N^{1\/2}$.<\/p>\n\n\n\n
\u2022\u2003In a good solvent, when $\\chi < \\chi_c$, the polymer chain is swollen and stretched, with $R_g \\sim N^{3\/5}$.<\/p>\n\n\n\n
A “universal” expression that captures how the radius of gyration $R_g$ changes with $\\chi$ is given by<\/p>\n\n\n\n
\u2003\u2003$R_g(\\chi) = R_g^\\mathrm{glob} + \\frac{R_g^\\mathrm{coil}-R_g^\\mathrm{glob}}{1 + \\exp[(\\chi-\\chi_c)\/\\Delta \\chi]}$,<\/p>\n\n\n\n
where $\\Delta \\chi$ quantifies the width of the conformational transition region and $\\chi_c$ locates the transition.<\/p>\n\n\n\n
To estimate the window width $\\Delta \\chi$, we first define an order parameter that measures deviations of the polymer\u2019s size from its value at the $\\theta$ point. We write<\/p>\n\n\n\n
\u2003\u2003$R = R_0 (1 + m)$,<\/p>\n\n\n\n
with $R_0 \\sim N^{1\/2}b$, where $b$ is the monomer size and $m$ represents the fractional deviation from the ideal size.<\/p>\n\n\n\n
Next, we expand the free energy $F(m)$ in powers of $m$. Close to the $\\theta$ point, the expansion takes the form<\/p>\n\n\n\n
\u2003\u2003$\\frac{F(m)}{k_BT} \\simeq C_1 N (\\chi-\\chi_c) m^2 + C_2 N m^4 + \\cdots$,<\/p>\n\n\n\n
where $C_1$ and $C_2$ are constants, and $k_BT$ is the thermal energy.<\/p>\n\n\n\n
In the mean-field (infinite-system) limit, the phase transition is sharp. However, for finite systems, fluctuations smear the transition, leading to a rounded behavior characterized by a finite width $\\Delta \\chi$ defined by $|\\chi-\\chi_c|$.<\/p>\n\n\n\n
At the edge of the transition, the quadratic and quartic terms in the free energy become comparable. Equating these contributions gives<\/p>\n\n\n\n
\u2003\u2003$C_1 N\\,\\Delta\\chi\\,m^2 \\sim C_2 N\\,m^4,$<\/p>\n\n\n\n
which implies<\/p>\n\n\n\n
\u2003\u2003$m^2 \\sim \\frac{C_1}{C_2}\\,\\Delta\\chi$.<\/p>\n\n\n\n
Furthermore, the transition is significantly broadened when the free energy barrier, estimated by the quartic term, is of the order of unity (in units of $k_BT$). That is, when<\/p>\n\n\n\n
\u2003\u2003$N C_2\\, m^4 \\sim O(1).$<\/p>\n\n\n\n
Substituting our earlier estimate for $m^2$ into this condition, we obtain<\/p>\n\n\n\n
\u2003\u2003$N C_2 \\left(\\frac{C_1}{C_2}\\Delta\\chi\\right)^2 \\sim O(1),$<\/p>\n\n\n\n
which simplifies to<\/p>\n\n\n\n
\u2003\u2003$\\Delta\\chi^2 \\sim \\frac{1}{N}.$<\/p>\n\n\n\n
Thus, the rounding of the transition in terms of $\\chi$ scales as<\/p>\n\n\n\n
\u2003\u2003$\\Delta\\chi \\sim \\frac{1}{\\sqrt{N}}.$<\/p>\n","protected":false},"excerpt":{"rendered":"
Consider a polymer chain dissolved in a solvent, where its behavior is influenced by the Flory parameter, denoted by $\\chi$. As the interaction parameter $\\chi$ varies, the chain undergoes several conformational transitions: \u2022\u2003For $\\chi > \\chi_c$, the polymer collapses into a dense, globular structure with a radius of gyration scaling as $R_g \\sim N^{1\/3}$. \u2022\u2003At […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[29],"class_list":["post-313","post","type-post","status-publish","format-standard","hentry","category-notes","tag-polymer-physics"],"_links":{"self":[{"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts\/313","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/comments?post=313"}],"version-history":[{"count":10,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts\/313\/revisions"}],"predecessor-version":[{"id":324,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts\/313\/revisions\/324"}],"wp:attachment":[{"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/media?parent=313"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/categories?post=313"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/tags?post=313"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}