Continuum Limits of the Becker-Döring Equations

Thomas Staudt

Georg-August University Göttingen Max-Planck Institute for Dynamics and Self-Organization

2016/08/17

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Overview

Lifshitz-Slyozov-Wagner (LSW) Theory of Ostwald Ripening
Becker-Döring (BD) Theory and its Continuum Limits
Discussion of the Continuum Limits

Lifshitz-Slyozov-Wagner Theory

Ostwald Ripening

Ripening: Phase separation of two-phase mixtures

Ostwald (1897): Ripening of salt crystals under constant thermodynamic parameters

Lifshitz-Slyozov-Wagner Theory

The LSW Model

Lifshitz, Slyozov, and Wagner (1961): Theory of Ostwald ripening for small droplet volume fraction
Deterministic, mean field

$$\partial_t n(v, t) = \partial_v \big(\dot{v}(t)\,n(v, t)\big)$$ $$\dot{v}(t) = \frac{v^{1/3}}{r^\mathrm{c}(t)} - 1$$

Critical radius $r^\mathrm{c}$ obtained by volume conservation

$$0 = \frac{\mathrm{d}}{\mathrm{d} t}V_0 = \int_0^\infty \!\!\!\!\!v~\partial_t n(v, t)\,\mathrm{d}v \quad~ \Longrightarrow\quad~ \quad $$ $$ r^\mathrm{c} = \langle r \rangle $$

Lifshitz-Slyozov-Wagner Theory

LSW Predictions

LSW predicted convergence to universal, self-similar solution

$$ \begin{aligned} N_t &\sim t^{-1} \\ \langle r \rangle, \sigma(r) &\sim t^{1/3} \end{aligned} $$

Experimental tests: Scaling holds, but distribution deviates

$\rho = r / \langle r \rangle$

$F(\rho) = 1 - \frac{\exp{\rho / (\frac{3}{2} - \rho)}}{\big(1 - \frac{2}{3}\rho\big)^{5/3}~\big(1 + \frac{1}{3}\rho\big)^{4/3}}$

Lifshitz-Slyozov-Wagner Theory

Other Solutions and Selection Rules

Niethammer (1999): Family $F_p$, $0 < p \le \infty$, of self-similar solutions (LSW: $~p=\infty$)
Dynamics: Zoom in tip of the distribution
Non-self-similar behaviour possible

Lifshitz-Slyozov-Wagner Theory

Emerging Questions

How universal is the LSW solution ($p=\infty$) in a physical sense?
Which corrections to the LSW theory make it more realistic?
Switch perspective: Can a microscopic theory help us to understand these questions better?
What happens in ripening systems with drift (temperature, volume, ...)? This has important applications in the synthesis of nanocrystals and the formation of cloud droplets.

Continuum Limits of the Becker-Döring Theory

The Becker-Döring Model

picture showing becker döring idea and notation

$~~ c_1 + c_l \rightleftharpoons c_{l+1}\,,\qquad\footnotesize l\in\mathbb{N}$

$~$

$c_l$: $~$Concentrations

$a_l$: $~$Coagulation rates

$b_l$: $~$Fragmentation rates

Becker and Döring (1935): Cluster growth by attaching or detaching single monomers, no coalescence
Used to study nucleation, metastability, and domain coarsening

$\scriptsize a_l$

$\scriptsize b_l$

Continuum Limits of the Becker-Döring Theory

Deterministic Description

Concentrations	$~~~c_l$, $z := c_1$
Net currents	$~~~J_l = a_l\,c_l\,z - b_{l+1}\,c_{l+1}$		$\qquad\qquad~~~~ c_1+c_l\rightleftharpoons c_{l+1}$
Density	$~~~\rho = \sum_{l=1}^{\infty} l\,c_l $
$~$
EOM $~$	$~~~\begin{aligned}\dot{c}_l &= -J_l + J_{l-1} \qquad\qquad\qquad\quad~~~\text{(for} \quad l > 1 \text{)}\\ \dot{z} &= \dot{\rho} - 2\,J_1 - \sum_{l=2}^\infty J_l \end{aligned}$
$~$
Case A	$~~~z ~ $ constant $\quad\qquad$	chemostated
Case B	$~~~\rho ~$ constant $\quad\qquad$	closed
Case C	$~~~\rho(t) = \rho_0 + \xi\,t$	monomer influx, drift

$\scriptsize a_l$

$\scriptsize b_l$

Continuum Limits of the Becker-Döring Theory

Basic Properties

Equilibrium:	$~~~ c^{\mathrm{eq}}_l(z) = Q_l\,z^l,$	$~~~~~Q_l = \frac{a_{l-1} \cdot \,\ldots\, \cdot a_1}{b_l \cdot \,\ldots\, \cdot b_2} \quad$ (case A,B)
Critical density:	$~~~ \rho_c := \sum_{l=1}^\infty l\,Q_l\,z_c^l,$	$~~~~~z_c~$ convergence radius

$ ~ $

Phase transition: $z_c$ saturation concentration, $\quad \rho_c$ critical density

Ball, Carr, and Penrose (1986): For constant $ \rho $ (case B)

$ \rho \le \rho_c \quad \Longrightarrow\quad c \longrightarrow c^\mathrm{eq}(z) \quad~ $ (strong convergence)

$ \rho > \rho_c \quad \Longrightarrow\quad c \longrightarrow c^\mathrm{eq}( z_c) \quad $ (weak convergence)

Ripening for $ \rho > \rho_c \,$: The excess density $ \rho - \rho_c $ is transported to larger and larger clusters

Continuum Limits of the Becker-Döring Theory

Monomer Influx

Numerical results suggest: Weak convergence also holds for monomer influx (case C)
Similar power-law decay as in case B observed

Continuum Limits of the Becker-Döring Theory

Towards the Continuum

$ x $	$ = $	$ \epsilon\,l $	$ \qquad $	$\tau $	$ = $	$ \epsilon\,t$
$ \alpha(x) $	$ = $	$ a_l $	$ \qquad $	$ \beta(x) $	$ = $	$ b_l $
$ n(x, \tau) $	$ = $	$ c_l(t) / \epsilon^2 $	$ \qquad $	$ I(x, \tau) $	$ = $	$ J_l(t) / \epsilon^2 $

$\epsilon:~$ scaling parameter

Density	$ ~~~~ \rho$	$~=~$	$ \epsilon \,\sum_{l=1}^\infty x\,n $ $~ =~ $ $ \int_0^\infty x\, n\,\mathrm{d}x $ $\,+ O(\epsilon) $ $ \quad \big(\approx V \big) $
$~$
Current	$ ~~~~ I $	$~=~$	$\big( \alpha\, n\,z - \beta\,n\big) - \big(\beta(x+\epsilon)\,n(x+\epsilon) - \beta(x)\,n(x)\big) $
$~$	$ ~$	$~=~$	$\big(\alpha\,z - \beta\big)\,n$ $\,- \epsilon\,\partial_x (\beta\,n)$ $\,+ O\big(\epsilon^2\big) $
$~$
EOM	$ ~~~~ \partial_\tau n $	$~=~$	$\frac{I(x - \epsilon) - I(x)}{\epsilon}~=\,$ $- \partial_x I$ $~~+ ~~ O(\epsilon) $

Continuum Limits of the Becker-Döring Theory

Continuum Limits

Density	$ ~~~~ \rho$	$~=~$	$ \int_0^\infty x\, n(x, \tau)\,\mathrm{d}x $	$~~ +~~ O(\epsilon) $
Current	$ ~~~~ I $	$~=~$	$\big(\alpha\,z - \beta\big)\,n $	$~~ +~~ O\big(\epsilon\big) $
EOM	$ ~~~~ \partial_\tau n $	$~=~$	$- \partial_x I$	$~~+ ~~ O(\epsilon) $

$~$

Choice of coefficients	$~~~~a_l = a_1 \, l^{1/3}$,	$~~~~b_l = a_1\,(z_s\,l^{1/3} + q)$
$~$
First order CL	$~~~~t := a_1\,q~\tau$,	$~~~~r_c(t) := q\,\epsilon^{1/3} / \big(z(t) - z_s\big) $

$~$

$ \partial_t n $	$=$	$ - \partial_x \big( ( x^{1/3} / r^c - 1)~n\big) $
$ V(t) $	$=$	$ \int_0^\infty x\, n(x, t)\,\mathrm{d}x $

Continuum Limits of the Becker-Döring Theory

Continuum Limits: Cases

$ \partial_t n $	$=$	$ - \partial_x \big( ( x^{1/3} / r^c - 1)~n\big) $
$ V(t) $	$=$	$ \int_0^\infty x\, n(x, t)\,\mathrm{d}x $

Case A	$~~~~ z ~~\text{constant} \quad\Longrightarrow \quad r^c = q\,\epsilon^{1/3}/ (z - z_s) ~~\text{constant}$
$~$	$~~~~$EOM analytically solvable
$~$
Case B	$~~~~V ~~\text{constant} \quad\Longrightarrow \quad r^c = \langle r \rangle $
$~$	$~~~~$LSW model recovered
$~$
Case C	$~~~~V(t) = V_0 + \xi\,t \quad \Longrightarrow \quad r^c = \langle r \rangle / k ~~$ with $~~ k = 1 + \xi/N $
$~$	$~~~~$Ripening model with drift

Continuum Limits of the Becker-Döring Theory

Observations and Remarks

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Becker-Döring model suggests correction terms for LSW theory
Diffusion smooths distributions: Only LSW solution $(p=\infty)$ reasonable
Change of scaling − Implications?
CL far easier to analyze / handle
CL can't describe effects at small cluster sizes: Dynamics not precise in situations where nucleation is important
Little research on the influence of stochasticity in these models

Discussion of the Continuum Limits

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Analytical treatment of model A
Properties of model C

Discussion of the Continuum Limits

Chemostated System: Method of Characteristics

EOM	$\quad \partial_t n = - \partial_x\big( (x^{1/3}/r^c - 1)\, n \big)$	$\quad~~ r^c~~\text{constant} $
Reduced radius	$\quad \rho := x^{1/3}/r^c ~~\Leftrightarrow ~~ x = (r^c\rho)^3 $	$\quad~~\frac{\mathrm{d}\,x}{\mathrm{d}\,\rho} = 3\,(r^c)^3\,\rho^2$
$~$
Change of variables	$\quad$$n^\rho(\rho):=\frac{\mathrm{d}\,x}{\mathrm{d}\,\rho}\,n(x)$	$\quad~~\hat{t} := 3\,(r^c)^3~t$

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Method of characteristics

$\partial_{\hat{t}} n^\rho$	$=$	$ - \partial_\rho \big( \frac{\rho - 1}{\rho^2}\, n^\rho \big)$	$\qquad f := \frac{\rho - 1}{\rho^2}\,n^\rho$
$\partial_{\hat{t}} f$	$=$	$ - \partial_{\hat{\rho}} f$	$\qquad\hat{\rho}(\rho) : ~\frac{\mathrm{d}\hat{\rho}}{\mathrm{d}\rho}= \frac{\rho - 1}{\rho^2}$
$f(\hat{\rho}, \hat{t})$	$=$	$f(\hat{\rho} - \hat{t})$

Discussion of the Continuum Limits

Chemostated System: Solutions

For given $ n_0(\rho) $ find $ f $ with

$ f( \hat{\rho} ) = \frac{\rho-1}{\rho^2}\,n_0(\rho) $

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Then the solutions are

$ n(\rho, t) = \frac{\rho^2}{\rho - 1}\,f\big(\hat{\rho} - \hat{t}\big) $

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For large times

$ \langle r \rangle$	$\sim$	$~t^{1/2} $	$~$
$ \langle r^n \rangle~$	$\sim$	$~\int (y + t)^{n/2}\,f(y)\,\mathrm{d}y $
$ N $	$\sim$	$~t^0 $

Discussion of the Continuum Limits

$~$

Analytical treatment of model A
Properties of model C

Discussion of the Continuum Limits

Ripening with Drift: Overview

EOM

$\quad \partial_t n = - \partial_x\big( (x^{1/3}/r^c - 1)\, n \big)$

$\quad~~ r^c = \langle r \rangle / k, $ $~~ k = 1 + \xi/N$

Vollmer, Papke, Rohloff (2014):

$ k$ always ends up larger than $3/2$
Droplet number $N$ approaches finite value
Monodispersity emerges: $ \quad \sigma(r) / \langle r \rangle \longrightarrow 0 $

Discussion of the Continuum Limits

Ripening with Drift: High Growth Rates

Analytics	$~~~~$Convergence of cdf of $~~I(r) = \big(r^2 - \langle r^2\rangle\big) / \langle r \rangle^{1/(k-1)} $
Simulations	$~~~~$This cdf is even constant for high $k$

Discussion of the Continuum Limits

Ripening with Drift: Small Growth Rates

Crossover between case B and C: Small growth rates, $ ~k_0 - 1 \ll 1$
End up with final $k_{\mathrm{f}} \approx 1.584$

Discussion of the Continuum Limits

Dependence of $\rho$ on $k$

Reduced radius EOM	$ \quad\dot{\rho} ~$ $=$ $~\frac{\mathrm{d}}{\mathrm{d}t}\,\frac{r}{\langle r \rangle}$ $~$ $\approx~$ $\frac{-3}{\langle r \rangle^3}~\frac{(\alpha\,k - \beta)\,\rho^3 - k\,\rho + 1}{\rho^2} $
$~$
Critical $k$	$~~~~$Solutions diverge if $~~k \approx k_{\mathrm{crit}}\approx 1.1$
Critical time	$~~~~k_{\mathrm{crit}} - 1 \sim \xi/N_{\mathrm{crit}} \sim \big(k_0 - 1\big)\,t_\mathrm{crit}$

$~$

Discussion of the Continuum Limits

Conclusion / Recap

Introduced Becker-Döring model of nucleation and derived continuum limit via separation of scales
Distinguished between three cases of boundary conditions, and derived results for the corresponding continuum limits

A	B	C
$ z$ constant	$\quad$$ V$ constant	$\quad$ drift $ \xi $ in $ V $
$ \langle r \rangle \sim t^{1/2}$	$\quad$$ \langle r \rangle \sim t^{1/3} $	$\quad$ $ \langle r \rangle \sim t^{1/3} $
$ \sigma(r) \sim t^{0}$	$\quad$ $ \sigma(r) \sim t^{1/3} $	$\quad$ $ \sigma(r) \sim t^{\frac{1}{3}\,\big(1/(k-1) - 1\big)} $

Crossover B to C: LSW theory unstable under volume influx, critical time $t_{\mathrm{crit}}\sim 1/\big(k_0 - 1\big)$

References

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Lifshitz, Slyozov	The Kinetics of Precipitation from Supersaturated Solid Solutions (1961)
Wagner	Theorie der Alterung von Niederschlägen durch Umlösen (1961)
Niethammer, Pego	Non-Self-Similar Behavior in the LSW Theory of Ostwald Ripening (1998)
$~$
Becker, Döring	Kinetische Behandlung der Keimbildung in übersättigten Dämpfen (1935)
Ball, Carr, Penrose	The Becker-Döring Cluster Equations (1986)
Penrose	The Becker-Döring Equations at Large Times and their Connection with the LSW Theory of Coarsening (1997)
$~$
Vollmer, Papke, Rohloff	Ripening and Focusing of Aggregate Size Distributions with Overall Volume Growth (2014)

Concentrations	\(~~~c_l\), \(z := c_1\)
Net currents	\(~~~J_l = a_l\,c_l\,z - b_{l+1}\,c_{l+1}\)		\(\qquad\qquad~~~~ c_1+c_l\rightleftharpoons c_{l+1}\)
Density	\(~~~\rho = \sum_{l=1}^{\infty} l\,c_l \)
\(~\)
EOM \(~\)	\(~~~\begin{aligned}\dot{c}_l &= -J_l + J_{l-1} \qquad\qquad\qquad\quad~~~\text{(for} \quad l > 1 \text{)}\\ \dot{z} &= \dot{\rho} - 2\,J_1 - \sum_{l=2}^\infty J_l \end{aligned}\)
\(~\)
Case A	\(~~~z ~ \) constant \(\quad\qquad\)	chemostated
Case B	\(~~~\rho ~\) constant \(\quad\qquad\)	closed
Case C	\(~~~\rho(t) = \rho_0 + \xi\,t\)	monomer influx, drift

Equilibrium:	\(~~~ c^{\mathrm{eq}}_l(z) = Q_l\,z^l,\)	\(~~~~~Q_l = \frac{a_{l-1} \cdot \,\ldots\, \cdot a_1}{b_l \cdot \,\ldots\, \cdot b_2} \quad\) (case A,B)
Critical density:	\(~~~ \rho_c := \sum_{l=1}^\infty l\,Q_l\,z_c^l,\)	\(~~~~~z_c~\) convergence radius

\( x \)	\( = \)	\( \epsilon\,l \)	\( \qquad \)	\(\tau \)	\( = \)	\( \epsilon\,t\)
\( \alpha(x) \)	\( = \)	\( a_l \)	\( \qquad \)	\( \beta(x) \)	\( = \)	\( b_l \)
\( n(x, \tau) \)	\( = \)	\( c_l(t) / \epsilon^2 \)	\( \qquad \)	\( I(x, \tau) \)	\( = \)	\( J_l(t) / \epsilon^2 \)

Density	\( ~~~~ \rho\)	\(~=~\)	\( \epsilon \,\sum_{l=1}^\infty x\,n \) \(~ =~ \) \( \int_0^\infty x\, n\,\mathrm{d}x \) \(\,+ O(\epsilon) \) \( \quad \big(\approx V \big) \)
\(~\)
Current	\( ~~~~ I \)	\(~=~\)	\(\big( \alpha\, n\,z - \beta\,n\big) - \big(\beta(x+\epsilon)\,n(x+\epsilon) - \beta(x)\,n(x)\big) \)
\(~\)	\( ~\)	\(~=~\)	\(\big(\alpha\,z - \beta\big)\,n\) \(\,- \epsilon\,\partial_x (\beta\,n)\) \(\,+ O\big(\epsilon^2\big) \)
\(~\)
EOM	\( ~~~~ \partial_\tau n \)	\(~=~\)	\(\frac{I(x - \epsilon) - I(x)}{\epsilon}~=\,\) \(- \partial_x I\) \(~~+ ~~ O(\epsilon) \)

Density	\( ~~~~ \rho\)	\(~=~\)	\( \int_0^\infty x\, n(x, \tau)\,\mathrm{d}x \)	\(~~ +~~ O(\epsilon) \)
Current	\( ~~~~ I \)	\(~=~\)	\(\big(\alpha\,z - \beta\big)\,n \)	\(~~ +~~ O\big(\epsilon\big) \)
EOM	\( ~~~~ \partial_\tau n \)	\(~=~\)	\(- \partial_x I\)	\(~~+ ~~ O(\epsilon) \)

Choice of coefficients	\(~~~~a_l = a_1 \, l^{1/3}\),	\(~~~~b_l = a_1\,(z_s\,l^{1/3} + q)\)
\(~\)
First order CL	\(~~~~t := a_1\,q~\tau\),	\(~~~~r_c(t) := q\,\epsilon^{1/3} / \big(z(t) - z_s\big) \)

\( \partial_t n \)	\(=\)	\( - \partial_x \big( ( x^{1/3} / r^c - 1)~n\big) \)
\( V(t) \)	\(=\)	\( \int_0^\infty x\, n(x, t)\,\mathrm{d}x \)

Case A	\(~~~~ z ~~\text{constant} \quad\Longrightarrow \quad r^c = q\,\epsilon^{1/3}/ (z - z_s) ~~\text{constant}\)
\(~\)	\(~~~~\)EOM analytically solvable
\(~\)
Case B	\(~~~~V ~~\text{constant} \quad\Longrightarrow \quad r^c = \langle r \rangle \)
\(~\)	\(~~~~\)LSW model recovered
\(~\)
Case C	\(~~~~V(t) = V_0 + \xi\,t \quad \Longrightarrow \quad r^c = \langle r \rangle / k ~~\) with \(~~ k = 1 + \xi/N \)
\(~\)	\(~~~~\)Ripening model with drift

CL far easier to analyze / handle
CL can't describe effects at small cluster sizes: Dynamics not precise in situations where nucleation is important

EOM	\(\quad \partial_t n = - \partial_x\big( (x^{1/3}/r^c - 1)\, n \big)\)	\(\quad~~ r^c~~\text{constant} \)
Reduced radius	\(\quad \rho := x^{1/3}/r^c ~~\Leftrightarrow ~~ x = (r^c\rho)^3 \)	\(\quad~~\frac{\mathrm{d}\,x}{\mathrm{d}\,\rho} = 3\,(r^c)^3\,\rho^2\)
\(~\)
Change of variables	\(\quad\)\(n^\rho(\rho):=\frac{\mathrm{d}\,x}{\mathrm{d}\,\rho}\,n(x)\)	\(\quad~~\hat{t} := 3\,(r^c)^3~t\)

\(\partial_{\hat{t}} n^\rho\)	\(=\)	\( - \partial_\rho \big( \frac{\rho - 1}{\rho^2}\, n^\rho \big)\)	\(\qquad f := \frac{\rho - 1}{\rho^2}\,n^\rho\)
\(\partial_{\hat{t}} f\)	\(=\)	\( - \partial_{\hat{\rho}} f\)	\(\qquad\hat{\rho}(\rho) : ~\frac{\mathrm{d}\hat{\rho}}{\mathrm{d}\rho}= \frac{\rho - 1}{\rho^2}\)
\(f(\hat{\rho}, \hat{t})\)	\(=\)	\(f(\hat{\rho} - \hat{t})\)

\( \langle r \rangle\)	\(\sim\)	\(~t^{1/2} \)	\(~\)
\( \langle r^n \rangle~\)	\(\sim\)	\(~\int (y + t)^{n/2}\,f(y)\,\mathrm{d}y \)
\( N \)	\(\sim\)	\(~t^0 \)

Analytics	\(~~~~\)Convergence of cdf of \(~~I(r) = \big(r^2 - \langle r^2\rangle\big) / \langle r \rangle^{1/(k-1)} \)
Simulations	\(~~~~\)This cdf is even constant for high \(k\)

Reduced radius EOM	\( \quad\dot{\rho} ~\) \(=\) \(~\frac{\mathrm{d}}{\mathrm{d}t}\,\frac{r}{\langle r \rangle}\) \(~\) \(\approx~\) \(\frac{-3}{\langle r \rangle^3}~\frac{(\alpha\,k - \beta)\,\rho^3 - k\,\rho + 1}{\rho^2} \)
\(~\)
Critical \(k\)	\(~~~~\)Solutions diverge if \(~~k \approx k_{\mathrm{crit}}\approx 1.1\)
Critical time	\(~~~~k_{\mathrm{crit}} - 1 \sim \xi/N_{\mathrm{crit}} \sim \big(k_0 - 1\big)\,t_\mathrm{crit}\)

A	B	C
\( z\) constant	\(\quad\)\( V\) constant	\(\quad\) drift \( \xi \) in \( V \)
\( \langle r \rangle \sim t^{1/2}\)	\(\quad\)\( \langle r \rangle \sim t^{1/3} \)	\(\quad\) \( \langle r \rangle \sim t^{1/3} \)
\( \sigma(r) \sim t^{0}\)	\(\quad\) \( \sigma(r) \sim t^{1/3} \)	\(\quad\) \( \sigma(r) \sim t^{\frac{1}{3}\,\big(1/(k-1) - 1\big)} \)

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