User:Its the economy stupid/Dynamic Force Spectroscopy

Dynamic Force Spectroscopy (DFS) is an analytical technique used to determine the kinetic and energetic characteristics of meta-stable states. The technique is applicable to meta-stable systems that, given enough time, will stochastically change configuration by thermal activation. By applying a steadily increasing force field, the activation energy barrier is lowered with time until 'escape' occurs. The rate of thermal activation thus becomes a function of time, and the typical force at which escape occurs depends on how quickly the force is increased. Hence the term Dynamic Force Spectroscopy was coined to describe the spectrum of typical escape force versus the rate, or frequency, of increasing force.

Examples of applicable systems include ligand-receptor bonds, weak chemical bonds, magnetic polarization, macroscopic quantum tunneling...

Infinite-Barrier Model

Infinite-Barrier Statistics

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Mean Force Spectrum

Parameters $\scriptstyle {\dot {f}}=df/dt$ : loading rate [N/s] $\scriptstyle x^{\ddagger }\,\!$ : distance to transition state [m] $\scriptstyle k(0)\,\!$ : force-free transition rate [s-1] Scaled Parameters $\scriptstyle R={\dot {f}}x^{\ddagger }/(k(0)k_{B}T)$ : dimensionless loading rate $\scriptstyle f_{\beta }=k_{B}T/x^{\ddagger }\,\!$ : thermal force scale [N] Probability density function (pdf) ${\frac {1}{f_{\beta }R}}\mathrm {exp} {\Bigg [}R^{-1}{\Big (}1-\mathrm {e} ^{f/f_{\beta }}{\Big )}+{\frac {f}{f_{\beta }}}{\Bigg ]}$ Cumulative distribution function (cdf) $1-\mathrm {exp} {\Big [}R^{-1}{\Big (}1-\mathrm {e} ^{f/f_{\beta }}{\Big )}{\Big ]}$ Survival function $\mathrm {exp} {\Big [}R^{-1}{\Big (}1-\mathrm {e} ^{f/f_{\beta }}{\Big )}{\Big ]}$ Mean $f_{\beta }\mathrm {e} ^{R^{-1}}{\rm {E}}_{1}(R^{-1})$ Mode $f_{\beta }\mathrm {ln} (R)\mathrm {,} ~~R\geq 1$ Variance $\sim [f_{\beta }/(1+R^{-1})]^{2}\,\!$

The infinite-barrier (or phenomenological) model assumes that the activation energy barrier to escape is sufficiently larger than the thermal energy $k_{B}T$ (Boltzmann's constant times the temperature) such that any distortion of the barrier shape or location by the application of force can be neglected. This is not to say that the barrier is actually infinite. It is referred to as this simply because the derivation including a finite barrier (see below) gives this result in the limit of an infinitely high barrier. The barrier is in fact finite, however, it is large enough that the shape of the potential is not altered by the applied force. Therefore the effect of force acts only to lower the intrinsic activation energy barrier height, $E_{a}$, by $\scriptstyle -fx^{\ddagger }$ where $\scriptstyle f={\dot {f}}t$ is the load increasing linearly with time, and $\scriptstyle x^{\ddagger }$ is the distance from the potential minimum to the barrier maximum (or transition state). The force-dependent energy barrier takes the form:

$U(f)=E_{a}-fx^{\ddagger }\,\!$

The kinetics of thermally-activated escape are assumed to follow the Arrhenius form,

$k(f)=A\mathrm {e} ^{-(E_{a}-fx^{\ddagger })/k_{B}T}$

where $A$ is the attempt frequency (or pre-exponential factor).

The time-dependent probability of the meta-stable state surviving up to a time $t$ is assumed to follow a first-order kinetic process such that the probability of re-entry to the state is zero,

${\frac {dS(t)}{dt}}=-k(t)S(t)$,

where $S(t)$ is the survival function, with initial conditions $S(t=0)=1$. Typically it is more intuitive to solve the above differential equation in terms of force with $\scriptstyle dt=df/{\dot {f}}$. Some statistical solutions for this model are shown in the table on the right. The derivations are left to the reader or can be found in the cited references.

Finite-Barrier Model

Finite-Barrier Statistics

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Mean Force Spectrum The dashed line is the infinite-barrier model

Parameters $\scriptstyle {\dot {f}}=df/dt$ : loading rate [N/s] $\scriptstyle x^{\ddagger }\,\!$ : distance to transition state [m] $\scriptstyle k(0)\,\!$ : force-free transition rate [s-1] $\scriptstyle E_{a}\,\!$ : activation energy barrier [J] $\scriptstyle b=\,\!$ 3/2 (linear-cubic), 1/2 (harmonic), 1 (infinite-barrier) Scaled Parameters $\scriptstyle R={\dot {f}}x_{1}/(k(0)k_{B}T)$ : dimensionless loading rate $\scriptstyle f_{\beta }=k_{B}T/x^{\ddagger }\,\!$ : thermal force scale [N] $\scriptstyle f_{c}=bE_{a}/x^{\ddagger }\,\!$ : critical force [N] $\scriptstyle \epsilon =1-f/f_{c}\,\!$ : reduced force Probability density function (pdf) ${\frac {\epsilon ^{b-1}}{f_{\beta }R}}\mathrm {exp} {\Bigg [}{\frac {E_{a}}{k_{B}T}}(1-\epsilon ^{b})+R^{-1}{\Big (}1-\mathrm {e} ^{{\frac {E_{a}}{k_{B}T}}(1-\epsilon ^{b})}{\Big )}{\Bigg ]}$ Cumulative distribution function (cdf) $1-\mathrm {exp} {\Big [}R^{-1}{\Big (}1-\mathrm {e} ^{{\frac {E_{a}}{k_{B}T}}(1-\epsilon ^{b})}{\Big )}{\Big ]}$ Survival function $\mathrm {exp} {\Big [}R^{-1}{\Big (}1-\mathrm {e} ^{{\frac {E_{a}}{k_{B}T}}(1-\epsilon ^{b})}{\Big )}{\Big ]}$ Mean $f_{c}{\Bigg \{}1-{\Big [}1-{\frac {k_{B}T}{E_{a}}}\mathrm {e} ^{R^{-1}}{\rm {E}}_{1}(R^{-1}){\Big ]}^{1/b}{\Bigg \}}$ Mode $f_{c}{\Bigg \{}1-{\Big [}1-{\frac {k_{B}T}{E_{a}}}\mathrm {ln} (R){\Big ]}^{1/b}{\Bigg \}}\mathrm {,} ~~R\geq 1$ Variance {{{variance}}}

... given by,

$k(f)=\nu (1-f/f_{c})^{b-1}\mathrm {e} ^{-E_{a}(1-f/f_{c})^{b}/k_{B}T}$

System-Dependent Parameters

Mechanical	Magnetic	Electric
$U_{0}$	$H_{0}$	$V_{0}$
$x_{0}$	row 2, cell 2	row 2, cell 3
$k_{0}$	row 3, cell 2	row 3, cell 3