Persistence length, $L_p$, is a fundamental mechanical property that quantifies the bending stiffness of a polymer. It’s defined as the characteristic length scale over which the correlation of bond angles decays. This correlation is expressed as the average cosine of the angle, $\\theta$, between bonds separated by a distance $s$ along the chain: <\/p>
$\\langle\\cos(\\theta(s))\\rangle = \\exp(-sl\/L_p)$<\/p>
To calculate $L_p$, we need to determine $\\langle\\cos(\\theta(s))\\rangle$. For adjacent bonds, where $s=1$, this simplifies to:<\/p>
$\\langle\\cos(\\theta)\\rangle = \\exp(-l\/L_p)$<\/p>
In simulations, the bending stiffness is often modeled using a bending energy, $U_b$. When $U_b$ is large, and excluded volume effects are negligible, we can calculate $\\langle\\cos(\\theta)\\rangle$ as:<\/p>
$\\langle\\cos(\\theta)\\rangle=\\frac{\\int_0^\\pi \\cos(\\theta)\\sin(\\theta)\\exp(-\\beta U_b)\\mathrm{d}\\theta}{\\int_0^\\pi\\sin(\\theta)\\exp(-\\beta U_b)\\mathrm{d}\\theta}$<\/p>
The factor $\\sin(\\theta)$ represents a geometric weight. When two bonds form an angle $\\theta$, the number of possible bond configurations in 3D space is proportional to $\\sin(\\theta)$. <\/p>
A common example of bending energy is the harmonic potential:<\/p>
$U_b(\\theta)=-\\frac{1}{2}k \\theta^2$<\/p>
where $k$ is the stiffness constant in units of $k_BT$. For this potential, $\\langle\\cos(\\theta)\\rangle$ is given by:<\/p>
$\\frac{e^{\\frac{3}{4 k}} \\left(-2 \\text{erf}\\left(\\frac{1}{\\sqrt{k}}\\right)+\\text{erf}\\left(\\frac{1+i \\pi k}{\\sqrt{k}}\\right)+\\text{erf}\\left(\\frac{1-i \\pi k}{\\sqrt{k}}\\right)\\right)}{2 \\left(-2 \\text{erf}\\left(\\frac{1}{2 \\sqrt{k}}\\right)+\\text{erf}\\left(\\frac{1+2 i \\pi k}{2 \\sqrt{k}}\\right)+\\text{erf}\\left(\\frac{1-2 i \\pi k}{2 \\sqrt{k}}\\right)\\right)}$
<\/p>
For large values of $k$ ($k\\ge 6$), $\\langle\\cos(\\theta)\\rangle$ approaches the Langevin function, $L(k)$.<\/p> \n","protected":false},"excerpt":{"rendered":"
Persistence length, $L_p$, is a fundamental mechanical property that quantifies the bending stiffness of a polymer. It’s defined as the characteristic length scale over which the correlation of bond angles decays. This correlation is expressed as the average cosine of the angle, $\\theta$, between bonds separated by a distance $s$ along the chain: $\\langle\\cos(\\theta(s))\\rangle = […]<\/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-226","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\/226","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=226"}],"version-history":[{"count":1,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts\/226\/revisions"}],"predecessor-version":[{"id":227,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts\/226\/revisions\/227"}],"wp:attachment":[{"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/media?parent=226"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/categories?post=226"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/tags?post=226"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}