Accelerated Nucleation Due to Trace Additives: A Fluctuating

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Accelerated Nucleation due to Trace Additives: A Fluctuating Coverage Model Geoffrey G. Poon, and Baron Peters J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b08510 • Publication Date (Web): 20 Oct 2015 Downloaded from http://pubs.acs.org on October 20, 2015

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The Journal of Physical Chemistry

Accelerated Nucleation Due to Trace Additives: A Fluctuating Coverage Model Geoffrey G. Poon and Baron Peters∗ Department of Chemical Engineering, University of California, Santa Barbara, CA 93106 E-mail: [email protected] Phone: (805) 284-8293

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Abstract We develop a theory to account for variable coverage of trace additives that lower the interfacial free energy for nucleation. The free energy landscape is based on classical nucleation theory and a statistical mechanical model for Langmuir adsorption. Dynamics are modeled by diffusion-controlled attachment and detachment of solutes and adsorbing additives. We compare the mechanism and kinetics from a mean-field model, a projection of the dynamics and free energy surface onto nucleus size, and a full two-dimensional calculation using Kramers-Langer-Berezhkovskii-Szabo theory. The fluctuating coverage model predicts rates more accurately than mean-field models of the same process primarily because it more accurately estimates of the potential of mean force along the size coordinate.

Introduction Additives are used to control nucleation and growth in many natural and industrial processes, including crystallization, 1–9 biomineralization, 10–16 and material synthesis. 17–19 Currently most experimental and computational studies focus on additives that influence growth. 20–26 Additives that influence nucleation have only been examined recently. Several experimental studies have investigated the effects of salts, acids, and surfactants on induction times and polymorph selection. 27–32 Clouet et al. revealed that precipitates in Al-Zr-Sc alloys have a Zr-rich external shell and Sc-rich core, which may explain the enhanced nucleation rate of Al-Zr-Sc alloys compared with Al-Sc alloys. 33 Anwar et al. studied the effect of amphiphilic Leonard-Jones (LJ) particles on nucleation in an otherwise binary LJ mixture. 34 Duff et al. computed the effects of NaCl on the interfacial free energy of glycine polymorphs using alchemical transformations. 35 In a recent paper, we suggested a general theoretical model to predict how additive properties and concentration modulate nucleation rates. We modified classical nucleation theory to account for the reduction in surface tension with increasing additive adsorption onto the 2 ACS Paragon Plus Environment

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nucleus surface. 36 van Dongen and coworkers have used similar models to analyze the effects of gas adsorption on water condensation. 37–39 Their model improved upon ours by including configurational degeneracies of adsorbed additives. Both models take a mean-field approach: neglecting fluctuations in additive coverage. Fluctuations in coverage may be important for two reasons. First, potent additives should work at low coverages and concentrations where deviations from the equilibrium average coverage are statistically relevant. Secondly, the dynamics of additive attachment to a nucleus may be too slow to assume equilibrium adsorption at each nucleus size. These concerns about current mean-field models were expressed at a recent Faraday Discussion. 40 This work derives the two-dimensional free energy surface and dynamical model for nucleation in the presence of fluctuating additive coverage. We compare barriers and rates from the fluctuating coverage model to those from mean-field models and classical nucleation theory.

Free energy landscape We decompose the reversible work to grow a nucleus of n solutes with m adsorbed additives into a two stage process. First, we use classical nucleation theory (CNT) to model the free energy of a bare nucleus of size n in solution: 41–46

FCNT (n) = −n∆µ + γ0 aφn2/3 ,

(1)

where ∆µ is the chemical potential difference of the solute in the metastable solution and in the precipitate, γ0 is the interfacial tension of an uncovered nucleus, φ is a shape factor such that φn2/3 is the number of adsorption sites, and a is the area per adsorption site. With the Girshick-Chiu correction, 47,48 the reversible work to grow a nucleus from monomers (i.e. n = 1) is ∆FCNT (n) ≡ FCNT (n) − FCNT (1). 3 ACS Paragon Plus Environment

(2)


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If P (m, n) is the relative population of nuclei of size n with m adsorbed additives, then ∆FCNT (n) = −kB T ln P (0, n)/P (0, 1). Second, we use a statistical mechanical model for Langmuir adsorption to model the work to bind m additives from solution to a nucleus of size n. Before outlining the model, we show how it will be used. The free energy of adsorption ∆Fa (m, n) is defined as

∆Fa (m, n) ≡ −kB T ln [P (m, n)/P (0, n)] .

(3)

The sum ∆F (m, n) ≡ ∆FCNT (n) + ∆Fa (m, n) is therefore

∆F (m, n) = −kB T ln [P (m, n)/P (0, 1)] ,

(4)

which defines a free energy landscape corresponding to the concentration of nuclei of different sizes and coverages relative to the isolated solute concentration. The nucleus of size n has Ns = φn2/3 binding sites. If each adsorption site is identical and independent then

P (m|Ns ) =

Ns ! hθim (1 − hθi)Ns −m , m!(Ns − m)!

(5)

where P (m|Ns ) is the fraction of nuclei of size n with m adsorbed additives, θ = m/Ns is the fractional coverage, and hθi is its equilibrium average. Using the Stirling approximation and P (m|Ns )/P (0|Ns ) = P (m, Ns )/P (0, Ns ), the adsorption free energy is approximately

∆Fa (m, Ns ) = −Ns Φ(θ) + m∆µa (θ),

(6)

where Φ/a is the surface or spreading pressure, 49

Φ(θ) ≡ −

∂∆Fa = −kB T ln(1 − θ), ∂Ns


(7)

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and ∆µa is the chemical potential difference between adsorbed additives and additives in solution, ∂∆Fa = kB T ln ∆µa (θ) ≡ ∂m

θ 1−θ

1 − hθi hθi

.

(8)

Equation 6 is a continuous representation of the free energy, so a Jacobian factor is needed to map the adsorption work from (m, Ns ) to (m, n) coordinates:

∆Fa (m, n) = ∆Fa (m, Ns (n)) +

kB T ln n. 3

(9)

If the barrier is large, the Jacobian factor has a negligible effect on the free energy surface near the saddle point and will be ignored from now on. For equilibrium adsorption onto a macroscopic surface, the chemical potential of additives on the surface is equal to the chemical potential of additives in solution (i.e. ∆µa = 0 and θ = hθi) and is related to additive concentration: 49

ln

hθi µ◦ (T, P ) = a + ln xa , (1 − hθi)q kB T

(10)

where q is the partition function for an adsorbed additive, µ◦a is the reference chemical potential of the additive in solution, and xa is the additive mole fraction in solution. Equation 10 can be rearranged to produce the Langmuir isotherm: 49

hθi =

Kxa , 1 + Kxa

◦

K ≡ qeµa /kB T ,

(11)

where K is the Langmuir constant. For small additive concentrations, Equation 11 simplifies to Henry’s law (i.e. hθi = Kxa ). We can divide ∆Fa by the number of sites Ns = φn2/3 and see that increasing additive



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coverage on the surface reduces the surface tension: ∆Fa (m, n) φn2/3

= γ(θ) − γ0 1 = − [Φ(θ) − θ∆µa (θ)], a

(12)

which for small additive coverages is approximately kB T θ γ(θ) − γ0 = − θ 1 − ln . a hθi

(13)

Therefore, the free energy of a nucleus with m adsorbed additives is (ignoring the Jacobian factor) ∆F (m, n) = −[n − 1]∆µ + aφ γ(θ)n2/3 − γ0 ,

(14)

which is shown in Figure 1.

Figure 1: Free energy landscape as a function of number of absorbed additives (m) and nucleus size (n). The parameters used in Equation 14 are: ∆µ/kB T = 4/3, γ0 a/kB T = 4/3, φ = 6, and hθi = 0.05. Mean-field models assume nucleating trajectories follow the θ = hθi contour (solid white line). No-additive trajectories follow classical nucleation theory (dashed white line). The saddle point is approximately at m‡ = hθi φn2/3 and n‡ = (2aφγ(hθi)/3∆µ)3 .


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One-dimensional models Mean-field models Mean-field models simplify the full two-dimensional landscape to a one-dimensional one along the size coordinate by ignoring fluctuations in coverage (i.e. θ = hθi). 36,37 In the mean-field approximation, the free energy is

∆FMF (n) = − [n − 1] ∆µ + γ(hθi)aφ n2/3 − 1 .

(15)

This predicts an approximately linear drop in barrier heights with increasing additive concentration at low coverages: " # ‡ ‡ ‡ ∆FCNT FCNT ∆FMF − =− 3 − φ Kxa + · · · , kB T kB T γ0 a

(16)

where the ‡ superscript represents the free energy evaluated at the critical nucleus size n‡ . Equation 16 differs from our previous mean-field model which predicts a different slope when the drop in barrier is plotted as function of Kxa . 36 This is because our previous study did not account for the configurational degenercies of adsorbed additives that are now included in the spreading pressure.

Projection onto nucleus size We improve upon mean-field predictions by using the potential of mean force (PMF) along the size coordinate. For simulations, this is equivalent to computing the PMF along the n-coordinate using umbrella sampling or other free energy methods. The PMF ∆FPMF (n) is estimated by using a quadratic expansion of ∆F (m, n) around the m = hθi φn2/3 contour and integrating out the m-dependence. For small coverages, the predicted drop in barrier



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using the PMF is ‡ ‡ ∆FPMF ∆FCNT − = kB T kB T

"

# ‡ FCNT − 3 − φ Kxa γ0 a q   2/3 1 + erf φn‡ Kxa /2  + ··· , p − ln  1 + erf φKxa /2 (17)

which includes an additional term not found in Equation 16. Sometimes this term is comparable to the predicted effect of additives using the mean-field model. This is especially √ true at very low coverages, where the extra term approximately scales with Kxa and is often larger than the mean-field term that only scales with Kxa . Therefore the sum over fluctuations should be included if accurate barriers and rates are required. Figure 2 shows that Equation 17 successfully models data from our previous work, 36 including the expected data collapse when barrier reductions is plotted against coverage. Mean-field models that lack the additional term in Equation 17 cannot capture the sudden drop in barrier at very small coverages. This suggests that Equation 17 should be used by simulations to predict the effect of additives on barriers. We can estimate the effect of additives on the rate JPMF using Equation 17 and the Zeldovich-Frenkel equation: J = cs Mn‡ Z exp[∆F ‡ /kB T ] where Z is the Zeldovich factor (2πkB T Z 2 = (∂ 2 ∆F/∂n2 )‡ ). For low coverages, the ratio of Zeldovich factors is

ln

3 kB T ZPMF = Kxa + · · · , ZCNT 2 γ0 a

(18)

where ZPMF and ZCNT is the Zeldovich factor when additives are and are not added. The ratio of rates is " # ‡ ‡ JPMF ZPMF ∆FPMF ∆FCNT ln = ln − − , JCNT ZCNT kB T kB T

(19)

where JCNT is the rate when no additives are added. It is important to note that even small


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Figure 2: Measured drop in barrier (∆FPMF −∆FCNT )/kB T as a function of additive coverage Kxa from PMF calulations using additives with different adsorption constants K. Data is from Poon et al. 36 Solid lines are fits using Equation 17 with K as the only varying parameter. The inset shows the drop in barrier as a function of additive mole fraction xa . errors in predicted barriers are exponentiated in rate calculations. This causes mean-field models to yield large errors in predicted rates (see Figure 3). However, Equation 18 and Figure 3 show that changes in the Zeldovich factor have a much less pronounced effect on ‡ the rate and can be ignored if the barrier is very large (i.e. ∆FCNT kB T ).

Rates and reaction coordinates Equation 19 predicts the rate by assuming nucleus size is the reaction coordinate and projecting the free energy and dynamics onto the n-coordinate. Many studies of nucleation in other systems have suggested that nucleus size alone is an adequate reaction coordinate. 50–59 However, these were analyses of single component nucleation without additives. It is unclear whether the extremely slow adsorption of trace additives needs to be explicity accounted for to properly model the dynamics. Here, we model the effect of additives on the rate without assuming nucleus size is the reaction coordinate a priori. The rate can be calculated directly from the two-dimensional landscape using KramersLanger theory. 60–62 The theory requires a quadratic expansion of the free energy surface at



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Figure 3: Predicted rate relative to the no-additive rate as a function of average additive coverage (hθi = Kxa ) using three levels of theory of increasing accuracy: the mean-field free energy (blue), the PMF along the size coordinate (red), and the KLBS theory result that uses PMF with Zeldovich and reaction coordinate corrections (green). We use the same parameters used in Figure 1 except for hθi. Mean field models underpredict the effect of additives on the rate. the saddle point (n‡ , m‡ ): T

 ∆F (n, m) ≈ ∆F (n‡ , m‡ ) +





1  δn   δn  ,   H‡  2 δm δm

(20)

where δn ≡ n − n‡ , δm ≡ m − m‡ , and H‡ is the second derivative matrix. Kramers-Langer theory also requires the mobility tensor near the saddle point M‡ to describe the diffusive dynamics along each coordinate. The mobilities along the m and n-coordinates are the attachment frequencies of additives and solutes onto the nucleus. If we assume uncoupled diffusion-controlled attachment, the mobility tensor is  M‡ = 4πR‡2

D s cs  1  R‡ 0

 0 Da ca /Ds cs

 ,

(21)

where Da and Ds are the diffusion constants of the additive and solute, ca and cs are the concentration of additive and solute, and 4πR‡ Ds cs is the attachment frequency of solutes


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for compact spherical nuclei with a critical radii of R‡ . The absolute value of the attachment frequency is unimportant for our anyalsis but can be estimated using simulations. 59,63–66 The steady-state nucleation rate is then −1/2 H‡ ∆F (n‡ , m‡ ) cs J= λ det exp − . 2π 2πkB T kB T

(22)

where −λ is the negative eigenvalue that corresponds to the lone unstable eigenvector of M‡ H‡ . For low additive concentrations (i.e. Da ca /Ds cs 1),

λ

4πR‡ Ds cs ∂ 2 ∆F ≈− kB T ∂n2 ‡ ! 4πR‡ Ds cs 3kB T 2aφγ0 ≈ 1− Kxa + · · · . 4/3 kB T aγ0 9n‡

(23)

Equation 22 is slightly different from Langer’s result. Langer’s rate expression is for the decay of the metastable state which includes the entire reactant basin (or all small nuclei commonly found in the metastable solution). It is often used when the free energy landscape has largest nucleus size as a coordinate. However, our landscape is built using CNT which uses population ratios of nuclei of different sizes. By redefining the reactant state as (m, n) = (0, 1), J in Equation 22 is the rate that uncovered solute monomers grow to post-critical nuclei (see Supporting Information for details). In the spirit of variational transition theory (a theory which owes much to the work of Bruce Garrett), Berezhkovskii and Szabo demonstrated that the rate calculated using the PMF along the coordinate that minimizes the rate is identical to the Langer result in Equation 22. 62 That coordinate is both the eigenvector u of H‡ M‡ and the true reaction coordinate. For small coverages, " u≈

aγ0 φ 1/3

3kB T n‡

#T 3kB T 1− 1+ Kxa + · · · , 1 . aγ0


(24)


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The reaction coordinate deviates from the minimum free energy path and tilts toward the slow coordinate (i.e. number of adsorbed additives). Estimating rates using the PMF along an arbitrary coordinate e will either overestimate the true rate or equal it if e = u. This begs an important question: 36 can rates be reasonably estimated if only a PMF along the size coordinate (not u) is used? The ratio of the true rate and the estimated rate is calculated using Berezhkovskii-Szabo theory and the size PMF: J JPMF

=λ

nT M‡ n nT H−1 n ‡

!−1 ,

(25)

where n = [1, 0]T points along the size coordinate. When additive coverage is small, the ratio simplifies to J JPMF

=1−

2kB T Kxa + · · · . aγ0

(26)

Equation 26 shows that reasonably accurate rate predictions can be obtained from a onedimensional analysis using the size PMF as long as additive coverage is low and the barrier ‡ is high (i.e. ∆FCNT kB T ). This is also demonstrated in Figure 3. This suggests that

simulations likely do not have to explicitly account for the dynamics of additive adsorption when investigating the effect of additives, which will be both simpler and less expensive than a full two-dimensional study. It is interesting to note that Berezhkovskii-Szabo theory still holds in this example of severe anisotropic mobility. Berezhkovskii and Zitserman showed that, in the anomalous regime where mobility is severely anisotropic, dynamical trajectories typically avoid the saddle because the slow coordinate is ”frozen.” 67,68 However, in this special case, both the position of the saddle and the relative mobility are proportional to the concentration of additives. So as the relative mobility along the m-direction approaches zero, the minimum free energy path becomes more like the no additive case where trajectories moving only in the n-direction pass through the saddle.


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Conclusion This work develops a model for the kinetics of nucleation in the presence of additives that bind to the surface of nuclei. We combined elements of classical nucleation theory with a statistical model for Langmuir adsorbtion to derive the free energy as a function of nucleus size and additive coverage. We also developed a simple dynamical model for diffusion controlled attachment of additives and solutes to nuclei. The model accounts for both the size and dynamics of fluctuations in coverage during the nucleation process. To examine the importance of fluctuations and dynamical effects, we compared rates and free energy barriers computed using several approximations to those from the full multidimensional Langer theory. We revise a previous mean-field model in which the coverage is assumed to always be the Langmuir average. We also compute free energy barriers and rates by projecting of the free energy and dynamics onto the nucleus size coordinate. This calculation essentially ignores dynamics of the additive adsorption process. Finally, we use Kramers-Langer-Berezhkovskii-Szabo theory to compute rates and the ideal reaction coordinate. All of these results are also compared to the kinetics and free energy barriers obtained from classical nucleation theory. Simple and convenient mean-field models capture most of the trends, and they may help devise experiments to screen for potent nucleants. However, the mean-field calculation ignores the effects of fluctuating coverage, and these effects are significant when the mean coverage is small. The potential of mean force includes the effects of additive coverage fluctuations, and thus barriers and rates from the potential of mean force are more accurate for sparse additive coverages. The full two-dimensional rate calculation captures the dynamics of additive adsorption, but it gives results which are similar to the potential of mean force calculation for small additive concentrations. We show that the potential of mean force model accurately collapses all simulation results from previous work by Poon et al. 36 whereas their previous mean field model did not. The results suggest that the new model may help interpret the kinetics of nucleation in the presence of additives that bind to the precipitate 13 ACS Paragon Plus Environment


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surfaces.

Acknowledgement The authors thank Peter Vekilov, Joop ter Horst, Richard Sear, Wenhao Sun, Marco Mazzotti, James J. De Yoreo, and Valeria Molinero for helpful comments and discussions. G.G.P. thanks the National Science Foundation (NSF) for support through the Graduate Research Fellowship under DGE 1144085. B.P. was supported by a Camille Dreyfus Teacher-Scholar award.

Supporting Information Available Derivation of the Jacobian in Equation 9, the second term in Equation 17, and the modified reference free energy used in the Langer-like multidimensional rate expression in Equation 22. This material is available free of charge via the Internet at http://pubs.acs.org/.

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1D projections of the free energy and dynamics onto the nucleus size coordinate capture the effect of adsorbing trace additives on the nucleation rate better than mean-field models can.


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Accelerated Nucleation Due to Trace Additives: A Fluctuating

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