Correlation of Solubility with the Metastable Limit of Nucleation Using

Subscriber access provided by UNIVERSITY OF ADELAIDE LIBRARIES

Article

Correlation of the Solubility with the Metastable Limit of Nucleation Using Gauge-Cell Monte Carlo Simulations Michael David Clark, Kenneth R. Morris, and Maria Silvina Tomassone Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b01939 • Publication Date (Web): 16 Aug 2017 Downloaded from http://pubs.acs.org on August 17, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Correlation of the Solubility with the Metastable Limit of Nucleation Using Gauge-Cell Monte Carlo Simulations Michael D. Clark1, Kenneth R. Morris† Maria Silvina Tomassone2* 1,2

Department of Chemical & Biochemical Engineering, Rutgers University, Piscataway, NJ 08854

† College of Pharmacy and Health Sciences, Long Island University, Brooklyn, NY 11201 (*) Corresponding Author KEYWORDS: nucleation, metastable limit, solubility, surface tension, simulation, gauge-cell Monte Carlo We present a novel simulation-based investigation of the nucleation of nanodroplets from solution and from vapor. Nucleation is difficult to measure or model accurately, and predicting when nucleation should occur remains an open problem. Of specific interest is the "metastable limit", the observed concentration at which nucleation occurs spontaneously, which cannot currently be estimated a priori. To investigate the nucleation process, we employ Gauge Cell Monte Carlo simulations to target spontaneous nucleation and measure thermodynamic properties of the system at nucleation. Our results reveal a widespread correlation over 5 orders of magnitude of solubilities, in which the metastable limit depends exclusively on solubility and the number density of generated nuclei. This three-way correlation is independent of other parameters including intermolecular interactions, temperature, molecular structure, system composition, and the structure of the formed nuclei. Our results have great potential to further the prediction of nucleation events using easily measurable solute properties alone and open new doors for further investigation.

ACS Paragon Plus Environment

1

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

I. Introduction One important step in the manufacturing of many specialty chemicals, pharmaceuticals, food ingredients, and nanomaterials is the crystallization or condensation of a product1, 2. In many crystallizing systems, tiny new crystallites or new droplets will spontaneously form from a supersaturated state through the process of primary homogeneous nucleation. Differences in process conditions (e.g. the degree of or the rate of supersaturation, solvent, etc) can have a significant impact on nucleation and on the resultant particle properties such as their number, size, shape, uniformity, etc., which in turns alters the performance of the product in its applications 2-6. In multiple experimental studies, the crystallization community has shown particular interest in the "metastable limit" (MSL) of nucleation. The metastable limit is defined as the concentration of a supersaturated solution (or a supercooled vapor) at which nucleation will proceed spontaneously and rapidly.2, 7-9 In general, any supersaturated state can be described as "metastable": in the presence of any seed (a droplet, a crystallite, a surface with a defect), a solute which is supersaturated will precipitate uninhibited onto the seed; but in the absence of a seed, the solute cannot precipitate. At high enough supersaturations, the solute goes from "metastable" to "unstable" and is capable of generating its own seeds in the process of nucleation. The "metastable limit" is therefore the concentration at which the solute ceases to remain dissolved and instead begins to nucleate. A metastable state can also be described this way: it is stable (remains dispersed) in the presence of small fluctuations but unstable (and generates nuclei) in the presence of large enough fluctuations. Introducing a seed represents a large fluctuation. Generating a nucleus is also a large fluctuation, but it is a very rare event because there is a high free energy barrier to dissolved solute molecules spontaneously forming a large enough nucleus.10 According to several theories of nucleation, the free energy barrier of generating a nucleus decreases with supersaturation, reducing the size of the fluctuations necessary to cause nucleation.10, 11 At the "metastable limit" the free energy barrier goes to zero and nucleation occurs spontaneously. Predicting this metastable limit from other physical properties of the system, however, is so far an open problem. Solving it will greatly enhance our ability to control the nucleation process and the properties of crystallized products. However, the nucleation process is currently so poorly understood that the onset of nucleation is very difficult to be predicted a priori2, 7, 10 from other properties of the system. Only a few cases have been


2

Page 2 of 24

Page 3 of 24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

reported in the literature such as the onset of nucleation of methane hydrate that can be predicted according to the so-called cage adsorption hypothesis12. Experimental studies of nucleation have also offered some insight, but there are still inherent limitations to observing a nucleation process. Indirect experimental measurements such as spectroscopy and scattering methods cannot examine or locate individual nuclei, while more direct methods like electron microscopy or mass spectrometry will perturb the system before measurement takes place, altering the results sometimes substantially13, 14. Available theories of nucleation processes, ones based on first principles, have also had limited success: often, such models either fail to reproduce experimental data or provide different results depending on the author's choices within the model (such as in density functional theories).10, 11, 15, 16 In addition, there have been a myriad of molecular simulations on the nucleation of crystals17-24. However, the topic of nucleation is still not completely understood. In some cases, while molecular simulations have been used to study nucleation events, they are usually constrained to short times and small volumes and accurately representing the massive sample of an experiment is extremely time consuming24-31. Efforts to make more tractable simulations often resort to unphysical assumptions, such as starting with an unrealistically high supersaturation or biasing the dynamics to examine the free energy of a single cluster.32-35 Empirical models have also been proposed.2, 8, 36-38, which have widespread use in manufacturing. These models sometimes require largely heuristic assumptions to work, employing fitting parameters with no physical meaning which vary between systems, such that these models cannot predict nucleation in a new setting or offer insight into its mechanics.

Our work was motivated by previous work by Morris et al9 in which they showed that the solubility curve and the metastable zone width have the same functionality using acetaminophen as a model drug. Their studies suggest that there could be a strong correlation between the metastable limit and the solubilty for the prediction of MSL with solubility. The authors also proposed that there should be new studies to test the notion that the metastable limit is inversely proportional to solubility. In this work we propose a new investigation based on the Gauge-Cell Monte Carlo techniques and thermodynamics to shed light on this open question and on the nucleation process in general, which permits us to connect it directly to the system's physical properties (solubility) with the metastable limit of nucleation in a completely consistent way. The main goal is to elucidate whether it is possible to find a correlation between the metastable


3

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

limit of nucleation and a measurable physical property of the system such as the solubility. We will focus on 20 different systems with 68 distinct nucleating configurations at different temperatures, molecular structures, and compositions with the goal to obtain the isotherms, the metastable limit of nucleation and their solubilities and a correlation between them. To the best of our knowledge this has never been done before in the literature. Section II describes the methodology, section III includes results and discussion, and section IV is devoted to the conclusions. We add also a Supporting Information Section where we describe the details of the model and the algorithms for the intermediate calculations to get the fitting of the isotherms.

II. Methodology The Gauge-Cell MC (gcMC) simulation technique introduced by Neimark and Vishnyakov39, 40 is particularly amenable to this type of study in that it restricts density fluctuations, allowing for the simulation of metastable states such as crystal nuclei which are below their critical size. A full description of the Gauge-Cell Monte Carlo (gcMC) simulation technique can be found elsewhere

39-41

, but the most important feature of gcMC simulations is the ability to evaluate

directly the chemical potential ( µ ) of stable, metastable, and unstable states. Figure 1 shows a schematic of the algorithm for Gauge-Cell Monte Carlo. The gcMC method contains a main cell of interest (where in which nucleation takes place) that is in equilibrium with several implicit gauge cells, one for each component. The gauge cell is a finite-sized reservoir for the solute, but not for the solvent. It is the solute that we want to restrict the density fluctuations of, whereas we want the solvent to mimic bulk water. So, the solvent has an infinite gauge cell, which is equivalent to grand canonical Monte Carlo. This works well because the method was developed without any restriction of gauge-cell size. The finite size of the reservoir acts to restrict density fluctuations in the system, and artificially stabilize the system such that we are able to observe crystal nuclei below their critical size.

The trial moves in a gcMC simulation are as follows:

translation of particles within the main cell, removal of particles from the main cell to the gaugecell and insertion of particles from the gauge-cell into the main cell. A given gauge-cell MC simulation, results in a histogram of the solute chemical potential around a certain target value.


4

Page 4 of 24

Page 5 of 24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Multiple simulations around different target values allows one to generate an isotherm, the loading (N) versus the chemical potential. FIGURES

Figure 1: Schematic of the Gauge-Cell Monte Carlo algorithm. The method provides a means to measure the chemical potential. The trial moves are: translation, removal (main cell to a gauge cell); insertion (gauge cell to the main cell) such that the total number of molecules from the main cell and gauge cell remains constant.

Chemical potential, µ , is the thermodynamic parameter best suited to studying phase separation; at equilibrium, µ is equal in all phases, and any differences in µ will matter to flow from one phase to another. We will therefore use µ to study the steps of a single nucleation event as it evolves from a metastable supersaturated solution to an unstable nucleus to a stable droplet.42,

43

For any particular system of volume V at temperature T, gcMC simulations

determine the chemical potential µ for a system with N solute atoms in the system (corresponding to bulk concentrations N/V), and the collection of all µ data at various N for a


5

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 24

given V and T is called an "N- µ isotherm". Table 1 provides a list of all the systems for which an N- µ isotherm of a nucleation event was simulated. In total 68 systems marked with an (x) in Table 1 are considered. The systems employ different temperatures, interaction energies, molecular structures, solvent compositions, and simulation-box volumes. Analysis of an N- µ isotherm can provide the metastable limit as well as several thermodynamic properties of the simulated solute, without additional simulations. Finally, we quantitatively correlate the simulated measurements to establish a relationship between the metastable limit of nucleation and the solubility. The description of the simulation details and the method to calculate the chemical potential are included in the Supporting Information Section, section I, page S1 and the method to calculate the solubility is included in section II, page S3.

Table 1: List of gcMC simulation parameters used to generate isotherms. DD

T*

DS a SS

System I

System II

System III

System IV

System V

System VI

System VII

System VIII System IX

1.00

0.50 1.00

1.20

0.50 1.00

1.50

0.50 1.00

1.00

0.70 0.75 0.80 0.85 0.90 0.95 1.00 ×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

15³, ½(15)³

×

×

×

×

×

15³

×

×

×

×

×

15³

×

×

×

0.10

0.30 1.00

1.50

0.60 1.00

1.20

0.10 1.00

1.20

0.60 1.00

0.76

³

×

1.00 1.20

Simulation Box Volumes /

0.55

12³, 13³, 14³, 15³, 16³, 17³, 20³ 15³, ½(15)³, ¼(15)³

15³

×

15³

×

15³


6

15³, ½(15)³, 20³

Page 7 of 24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

1.00 1.06

System X

1.00 2.00

System XI

1.00

(no solvent) System XIII

System XIV

1.20

(two solvents) System XVI c (bonded dimers)

System XVII c (bonded dimers)

System XVIII c (bonded dimers)

System XIX c (bonded dimers)

System XX c (bonded dimers)

— —

1.50

(no solvent) b

— —

(no solvent)

a

1.00 1.00

System XII

System XV

0.66

— —

1.20

×

15³

×

17³

×

×

×

×

×

×

10³, 15³

×

×

×

×

×

×

15³

×

×

×

×

×

×

9³, 12³, 15³, 18³, 21³, 24³, 27³, 30³

b

0.1/0.8

×

1.00 1.0-1.0c — —

×

×

×

1.0-0.8c — — 1.0-0.5c — — 0.8-1.0c — — 0.5-1.0c — —

15³

×

×

×

15³, 30³

×

15³

×

15³

×

15³

×

15³

Interaction parameters for solute-solute(D-D), solute-solvent (D-S), and solvent-solvent (S-S)

pair potentials, all of which are Lennard-Jones 6-12 potentials with σ = 1. b

System XV has a solute, A, with two solvents, B and D. The A-B interaction is 0.10 while the

A-D interaction is 0.80. All solvents interact identically, εBB = εBD = εDD = 1.00. Solvent mixtures were simulated at 5%, 10%, 20%, 30%, 70%, 80%, 90%, and 95% B, thus 8 separate isotherms were generated for each "x" in System XV. c

This system's solute is a bonded diatomic Lennard-Jones molecule, simulated in vacuum. The

bond is a simple harmonic bond with energy k(d – d0)², where k = 100 and d0 = σ. The bonded dimers are treated as A-C molecules, where the AA and CC interactions are equal, but the AC interaction can be different. The notation "1.0-0.8" indicates εAA = 1.0, εCC = 1.0, and εAC = 0.8.


7

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

III. Results and Discussion Figure 2 shows a representative N − µ isotherm generated by the gcMC simulations. All isotherms for nucleating systems exhibit three distinct behaviors: one at low concentrations or the small-N regime, one at high concentrations or the large-N regime, and an intermediate regime. Simulation snapshots in Fig. 2a show that at low concentrations, the solute is completely dispersed either as a pure vapor or in a homogeneous solution. At high concentrations, spherical droplets form which coexist with dispersed solute molecules. The intermediate regime shows reversible clustering, where clusters form, dissipate, and reappear throughout the simulation. These behaviors are observed regardless of whether solvent is present (i.e. for nucleation from solution as well as for nucleation from vapor). The shape of the isotherm provides the definition for the metastable limit: it is the highest concentration NMSL/V at which the system remains dispersed and, more rigorously, the metastable limit is also the value of the concentration NMSL/V for which the NMSL value corresponds to a maximum in the chemical potential µ . Any N larger than NMSL will possess a lower, more favorable µ due to the formation of clusters.


8

Page 8 of 24

Page 9 of 24

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment

9

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 24

Figure 2. (a) An example isotherm generated by gcMC simulations, of System XIV at T = 0.85 and V = (15σ)³. Simulation snapshots (above) show the progression from a dispersed state, to reversible clustering, to a spherical droplet. Simulation data as well as the Kelvin equation and the regular solution model are shown. The horizontal dashed line represents µ bulk = µ which intersects the data at the solubility, N = N0. The metastable limit (MSL) is the concentration NMSL/V for which N = NMSL is defined where µ is a maximum. (b) Several isotherms for different values of the temperature T=0.70, 0.75, 0.80, 0.85, 0.90 and 0.95, illustrating how the reversibleclustering concentration changes with temperature.


10

Page 11 of 24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Each isotherm was analyzed to determine the solubility and the metastable limit of the solute in the simulated system. The metastable limit as stated above is the value of the concentration NMSL/V where NMSL is the number of particles corresponding to the maximum µ value. The solubility N0/V, however, must be calculated from the isotherm. (For clarity in the text, the term "solubility" is used interchangeably for solute-in-solvent systems and for solute-in-vacuum systems, even though in the latter case we are technically using the vapor density at the equilibrium vapor pressure.) To obtain the solubility, we employ macroscopic thermodynamic theories to portions of the isotherm. The small-N regime exhibits only dispersed molecules that are well described by regular solution theory.44 In regular solution theory, the chemical potential is the sum of the ideal-gas entropy plus a first-order energy term to account for the random interactions between two solute molecules: reg sol'n = + ln

+

(1)

where kB is Boltzmann's constant, T is the temperature, V is the volume of the simulation box,

µ1 is the chemical potential of a single dispersed solute atom, and ω is an average intermolecular attraction between two solute atoms that are close together. (For solute in vacuum, we define µ1 = 0 and Eq. (1) is related to the van der Waals equation of state for a pure substance.45) Meanwhile, the large-N regime is characterized by the existence of a spherical droplet in equilibrium with dispersed solute atoms. The chemical potential µ of a spherical droplet is given by the Kelvin equation:46 Kelvin = +

(2)

Here, R is the radius of the droplet, v and µ bulk are the molecular volume of the bulk-solute phase and the chemical potential respectively, and γ is the solute's surface tension.

Using linear regression as described in the Supporting Information Section, section II, page S3

µ data from the small-N and the large-N portions of the isotherm are fit to Eq. (1) and Eq. (2) respectively to determine the fit parameters ω, µ1, µ bulk and γν . Example curve fits are shown in Figure 2a. The thermodynamic definition of solubility is that it is the concentration at equilibrium with a bulk phase. Therefore, the chemical potential of the dispersed solute at N0/V


11

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 24

and the chemical potential µ bulk of the bulk-solute phase must be equal. N0/V can thus be determined from Eq. (1) and the fit parameters listed above:

 N 0 +1  ω N 0 = µ bulk . +  V  V

µ reg sol'n (N 0 / V ) = µ1 + kBT ln 

(3)

The regression analyses also provide the standard error of each fit parameter, which is used to estimate the error of the N0 estimation. Across all runs, the median error on N0 was 2% and the average was 3%.

The purpose of this study is to connect the metastable limit, NMSL/V, for each system to the solubility of the solute. To that end, primary results for the solubility and metastable limit are plotted in Figure 3 as a function of temperature for several systems as listed in Table 1. The raw data is in Figure 3(a). In Figure 3(b) and 3(c), the data is compared to scaling laws derived from regular solution theory. The method to do this comparison using regular solution theory is explained in section II page S3 of the Supporting Information Section. Several examples of graphs of isotherms can be seen in Section III, page S5 in the Supporting Information Section.

Figure 3: Plots of calculated solubilities, N0/V, and metastable limits, NMSL/V, vs. temperature for 3 simulated system. (a) Solubility (filled symbols) and metastable limit (open symbols) for Systems I, II, and III as listed in Table 1. (b) Plot of solubility on a log10 scale vs. 1/T, with dashed lines from linear regression. (c) Plot of inverse-metastable limit, V/NMSL vs. 1/T as per our scaling law, with dashed lines from linear regression.


12

Page 13 of 24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

The definition of the solubility in Eq. (3) suggests that for low solubility N0/V NMSL. The size of the droplet therefore varies with the simulation box size. Furthermore, a maximum of one nucleus per box means that the simulation box volume V determines the number-density of nuclei, *+, =

nucleus

, and the box length L represents the

average distance between nuclei at nucleation. Figure 5 illustrates this effect explicitly. When V is increased, the simulated nucleus is "farther away" from its own periodic image, and this is equivalent to a macroscopic solution in which nuclei are farther apart, i.e. at a lower number density. We therefore expect that different box volume V will produce a different N − µ isotherm with a differential metastable limit, even for an otherwise identical system.

L

Figure 5: Depiction of a nucleus in a simulation box of length L with periodic boundary conditions. One droplet in periodic boundary conditions is equivalent to a populations of droplets, each a distance L away from each other in every direction, and the number density of nuclei is therefore (1 droplet) / (volume of the box, L3).


15

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Two systems from Table 1 (System II with solvent and System XIV in vacuum) were simulated at T = 0.85 inside several differently-sized simulation boxes. The resulting isotherms, solubilities, and metastable limits are displayed in Figure 6. Isotherms are shown in Figures 6(a) and 6 (d).

Figure 6: Dependence of results on the simulation box size. (a) Isotherms of System XIV (no solvent) at T* = 0.85 simulated in cubic boxes with different box lengths L, thus V = L3. (b) Comparison of solubilities, N0/V vs. L. There is no significant variation (error bars, calculated as part of the linear regression described in Supporting Information, are smaller than the symbols). (c) Variation of metastable limit, showing NMSL/V is directly proportional to 1/L. Plots (d-f) are similar to (a-c) but for isotherms from System II (with solvent) at various L. Again, N0/V does not vary significantly but NMSL/V is directly proportional to 1/L.


16

Page 16 of 24

Page 17 of 24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

The solubility is a homogeneous property and should be constant with L, which Figure 6(b) and 6(e) show to be true within the error bars. However, NMSL/V is observed to increase for smaller box volumes. In other words, a higher supersaturation is required for nucleation to produce a higher number of nuclei. This is intuitively reasonable, as the system must be further from equilibrium to generate more densely-packed nuclei. It also agrees with the definition of metastability as stable against small fluctuations but unstable against large ones: in a small simulation box, only small fluctuations are allowed, which means a larger range of supersaturation will remain stable (dispersed) in a smaller volume. A higher supersaturated therefore is necessary for the formation of nucleus to become spontaneous in a small simulation box. Figure 6(c) and 6(f) indicate that the variation of the metastable limit is approximately NMSL/V ∝ 1/L. Note that the dependence is apparently on L and not on V. We wished to test the generality of this L-dependence across a wider array of simulated systems. Additional isotherms were constructed in different box volumes, as listed in the last column of Table 1. All solubilities N0/V and metastable limits NMSL/V were collected and plotted as V/NMSL vs. ln(N0/V) in Figure 7(a). To include the L-dependence, Figure 7(b) plots the same data as (V/NMSL)(1/L) vs. ln(N0/V), again for all systems at all volumes listed in Table 1.

(a)


17

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 24

(b)

Figure 7: Plot of all solubilities and metastable limits collected from all isotherms listed in Table 1. All temperatures, all solvent compositions, all molecular structures, all intermolecular interactions, and all simulation box volumes are included. (a) The y-axis is 1/(NMSL/V), and for each box-length L there is an apparently linear relationship between log(solubility) and 1/MSL, as implied by Figures 3 and 4. (b) The y-axis is changed to 1 / (NMSL/V) / L, as implied by Figure 6(c) and 6(f), and a new linear relationship emerges which does not depend on any other simulation parameter. Symbol shapes indicate Systems I-XX and symbol color indicates box lengths L = 10-30σ.

When all data is divided by L, all curves collapse into one. This 3-way correlation remains valid across 68 distinct nucleating systems with different temperatures, molecular structures, and compositions, suggesting a possibly universal description of nucleation revolving solely around


18

Page 19 of 24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

solubility and internuclear distance. The ordinary least squares correlation coefficient is R2 = 0.930, and using weighted least squares a value of R2 = 0.950 is obtained. Many studies of nucleation have shown that kinetic parameters can affect the supersaturation achieved before nucleation.2, 8, 36 It has also been observed that higher supersaturation rates lead to a greater number of nuclei2, 48 and that nucleation at smaller supersaturations leads to larger particles.49 It has been conjectured so far that this is due to a chemical reaction-like mechanism, where nuclei begin to form at some "threshold" supersaturation, and the number of nuclei depends only on how long the concentration remains above the threshold. However, our results present another new hypothesis, representing the first quantitative evidence in the literature that the metastable limit of nucleation depends thermodynamically on the distance between newly-forming nuclei and thus on nuclei concentration, and moreover, that this is not exclusively a kinetic effect. Figure 6 illustrates this. Our data provides a lower-bound to nucleation: no system can generate stable nuclei, which do not redissolve, at a density as high as 1/V if the solute concentration is lower than NMSL/V. Additionally, a lower supersaturation corresponds to a unique value of NMSL/V associated with larger V. Existing models do not address that this effect is possible.7, 8, 10, 16, 37 Experimental studies of nucleation also have not witnessed this effect, since most studies do not measure particle number-density or internuclear distance at all. Among those few studies that have reported the number-concentration of particles (for example, quantum dots48), none have correlated the number density with solubility or concentration at nucleation. Furthermore, the correlation seems independent of the surface tension, the Tolman length50, 51, and the equation of state of the dispersed phase, contrary to oftcited theoretical assessments of nucleation.52-54 This streamlined description of nucleation behavior using only two parameters, at the exclusion of other parameters and with one of the parameters with a heretofore unexamined role in nucleation, is the primary novel result of this study as embodied in Figure 7. To place any experimental data points onto Figure 7, all three relevant parameters must be known: solubility, metastable limit, and either number-density of (or distance between) nuclei. While many studies measure two of these quantities,48, 49, 55-60 no studies are currently found in literature which measure all three. An attempt was made by the authors to estimate the required data from the information in two literature sources,37, 61 but the error on the estimate was too


19

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 24

severe. New experimental data must be collected to make any comparison with our results in Figure 7. Our results also show that given only the solubility, the metastable limit and the distance between nuclei are intrinsically linked as shown in Figure 7. Figure 7 as a model, can predict these linked parameters together, but neither parameter can be determined individually from our study. To have full closure, one additional correlation or equation would be required to constrain the nucleation event and predict its onset. Despite this lack of closure, our simulation results predict a very strong linear correlation between the metastable limit of nucleation and the logarithm of the solubility of the system, which has never been reported before, and suggest that all nucleation processes might be described by a unifying, global pattern that is based on elementary physical properties and which scientists and engineers could apply to any phase separating system.

IV. Conclusions We presented a novel simulation-based investigation of the nucleation of nanodroplets based on thermodynamics theory. We found a strong linear correlation between the logarithm of the solubility, the metastable limit of nucleation and the inter-nuclear distance, a correlation which is valid across five orders of magnitude for solubility, calculated for 68 distinct nucleating systems with different temperatures, molecular structures, and compositions. This suggests the possibility of a universal description of nucleation revolving solely around solubility and inter-nuclear distance. This investigation opens new doors for future research.

Supporting Information Available: A description of the input parameters for gcMC simulations, graphs of multiple N − µ isotherms, and an explanation of the regression analysis is included in Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org. Corresponding Author E-mail: [email protected] Acknowledgments: The authors would like to acknowledge the funding of the National Science Foundation through the Engineering Research Center on Structured Organic Particulate Systems, funded by Grant NSF-ECC 0540855. The authors wish to thank Professor Alex


20

Page 21 of 24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Neimark for useful discussions and Franklin E. Bettencourt for assistance running the simulations.

REFERENCES 1.

2.

3.

4. 5.

6.

7. 8.

9.

10. 11. 12.

13. 14.

Sadati, M., Ramezani-Dakhel, H., Bu, W., Sevgen, E., Liang, Z., Erol , C., Rahimi, M., Taheri Qazvini, N., Lin, B., Abbott, N. L., Roux, B., Schlossman, M. L. & de Pablo, J. J. Molecular Structure of Canonical Liquid Crystal Interfaces. J. Am. Chem. Soc. 139, 3841 (2017). Nagy, Z.K., Fujiwara, M., Woo, X.Y. & Braatz, R.D. Determination of the Kinetic Parameters for the Crystallization of Paracetamol from Water Using Metastable Zone Width Experiments. Industrial & Engineering Chemistry Research 47, 1245-1252 (2008). Sugimoto, T., Shiba, F., Sekiguchi, T. & Itoh, H. Spontaneous nucleation of monodisperse silver halide particles from homogeneous gelatin solution I: silver chloride. Colloids and Surfaces a-Physicochemical and Engineering Aspects 164, 183-203 (2000). Garekani, H.e.a. Crystal Habit Modifications of Ibuprofen and Their Physicomechanical Characteristics Drug Development Industrial Pharmacy 27, 803 (2001). Owen, J.S., Chan, E.M., Liu, H. & Alivisatos, A.P. Precursor Conversion Kinetics and the Nucleation of Cadmium Selenide Nanocrystals. Journal of the American Chemical Society doi101021ja106777j 132, 18206-18213 (2010). Li, T., Li, B. & Tomassone, M.S. Surface Characterization of the Aspirin Crystal Planes using Molecular Dynamics Simulations Chemical Engineering Science 61 5159-5169 (2006). Kim, K.J. & Mersmann, A. Estimation of metastable zone width in different nucleation processes. Chemical Engineering Science 56, 2315-2324 (2001). Sangwal, K. A novel self-consistent Nývlt-like equation for metastable zone width determined by the polythermal method. Crystal Research and Technology 44, 231-247 (2009). Laird, A., Chougule, M., Hamad, M. & Morris, K. Thermodynamics Associated with Monitoring Pre-nucleation Aggregation at High Supersaturation. Int. J. Pharm. Sci. Rev. Res., 18, 6-12 (2013). Cahn, J.W. & Hilliard, J.E. Free Energy of a Nonuniform System .3. Nucleation in a 2Component Incompressible Fluid. Journal of Chemical Physics 31, 688-699 (1959). Oxtoby, D.W. HOMOGENEOUS NUCLEATION - THEORY AND EXPERIMENT. Journal of Physics-Condensed Matter 4, 7627-7650 (1992). Guo, G.J. & Rodger, P.M. Solubility of Aqueous Methane under Metastable Conditions: Implications for Gas Hydrate Nucleation. Journal of Physical Chemistry B 117, 64986504 (2013). Browning, N.D.e.a. Recent developments in dynamic transmission electron microscopy. Current Opinion in Solid State and Materials Science 16, 23-30 (2011). Sellberg, J.A., Huang, C., McQueen, T. A., Loh, N. D., Laksmono, H., Schlesinger, D., Sierra, R. G., Nordlund, D., Hampton, C. Y., Starodub, D., DePonte, D. P., Beye, M ,


21

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

15.

16. 17.

18.

19.

20.

21. 22. 23. 24.

25. 26.

27. 28. 29.

30. 31.

Chen, C., Martin, A. V., Barty, A., Wikfeldt, K. T., Weiss, T. M., Caronna, C., Feldkamp, J., Skinner, L. B., Seibert, M. M., Messerschmidt, M., Williams, G. J., Boutet, S., Pettersson, L. G. M., Bogan, M. J., Nilsson, A.; Nature 510, 381-384 (2014). Laaksonen, A., Talanquer, V. & Oxtoby, D.W. NUCLEATION - MEASUREMENTS, THEORY, AND ATMOSPHERIC APPLICATIONS. Annual Review of Physical Chemistry 46, 489-524 (1995). Uline, M.J. & Corti, D.S. Activated instability of homogeneous droplet nucleation and growth. J Chem Phys 129, 234507 (2008). Karayiannis, N.C., Malshe, R., Kroger, M., de Pablo, J.J. & Laso, M. Evolution of fivefold local symmetry during crystal nucleation and growth in dense hard-sphere packings. Soft Matter 8, 844-858 (2012). He, Z.J., Gupta, K.M., Linga, P. & Jiang, J.W. Molecular Insights into the Nucleation and Growth of CH4 and CO2 Mixed Hydrates from Microsecond Simulations. Journal of Physical Chemistry C 120, 25225-25236 (2016). Li, H.J., Yan, D., Cai, H. Q., Yi, H. B., Min, X. B., Xia, F. F.; Insights into watermediated ion clustering in aqueous CaSO4 solutions: pre-nucleation cluster characteristics studied by ab initio calculations and molecular dynamics simulations. Physical Chemistry Chemical Physics 19, 11390-11403 (2017). Liu, S., van Duin, A.C.T., van Duin, D.M., Liu, B. & Edgart, J.H. Atomistic Insights into Nucleation and Formation of Hexagonal Boron Nitride on Nickel from First-PrinciplesBased Reactive Molecular Dynamics Simulations. Acs Nano 11, 3585-3596 (2017). Anwar, M. & Schilling, T. Crystallization of polyethylene: A molecular dynamics simulation study of the nucleation and growth mechanisms. Polymer 76, 307-312 (2015). Bourque, A., Locker, C.R. & Rutledge, G.C. Molecular Dynamics Simulation of Surface Nucleation during Growth of an Alkane Crystal. Macromolecules 49, 3619-3629 (2016). Nicholson, D.A. & Rutledge, G.C. Analysis of nucleation using mean first-passage time data from molecular dynamics simulation. Journal of Chemical Physics 144 (2016). Anwar, J. & Zahn, D. Uncovering Molecular Processes in Crystal Nucleation and Growth by Using Molecular Simulation. Angewandte Chemie-International Edition 50, 19962013 (2011). Laaksonen, A., Talanquer, V. & Oxtoby, D.W. Annual Review of Physical Chemistry 46, 489-524 (1995). Chushak, Y. & Bartell, L.S. Crystal nucleation and growth in large clusters of SeF6 from molecular dynamics simulations. Journal of Physical Chemistry A 104, 9328-9336 (2000). Norman, G.E. & Stegailov, V.V. Homogeneous nucleation in a superheated crystal. Molecular-dynamic simulation. Doklady Physics 47, 667-671 (2002). Leyssale, J.M., Delhommelle, J. & Millot, C. Molecular simulation of the homogeneous crystal nucleation of carbon dioxide. Journal of Chemical Physics 122 (2005). Peng, L.J., Morris, J.R. & Lo, Y.C. Temperature-dependent mechanisms of homogeneous crystal nucleation in quenched Lennard-Jones liquids: Molecular dynamics simulations. Physical Review B 78 (2008). Yi, P., Locker, C.R. & Rutledge, G.C. Molecular Dynamics Simulation of Homogeneous Crystal Nucleation in Polyethylene. Macromolecules 46, 4723-4733 (2013). Sosso, G.C. , Chen, J., Cox, S. J., Fitzner, M., Pedevilla, P., Zen, A., Michaelides, A.; Crystal Nucleation in Liquids: Open Questions and Future Challenges in Molecular


22

Page 22 of 24

Page 23 of 24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

32.

33.

34. 35.

36.

37. 38.

39. 40. 41. 42. 43. 44. 45. 46. 47. 48.

49.

Dynamics Simulations. Chemical Reviews 116, 7078-7116 (2016). Laasonen, K., Wonczak, S., Strey, R. & Laaksonen, A. Molecular dynamics simulations of gas--liquid nucleation of Lennard-Jones fluid. The Journal of Chemical Physics 113, 9741-9747 (2000). Tanaka, K.K., Tanaka, H., Yamamoto, T. & Kawamura, K. Molecular dynamics simulations of nucleation from vapor to solid composed of Lennard-Jones molecules. The Journal of Chemical Physics 134, 204313-13 (2011). Toxvaerd, S. Molecular-dynamics simulation of homogeneous nucleation in the vapor phase. The Journal of Chemical Physics 115, 8913-8920 (2001). Milek, T. & Zahn, D. Molecular Simulation of Ag Nanoparticle Nucleation from Solution: Redox-Reactions Direct the Evolution of Shape and Structure. Nano Letters 14, 4913-4917 (2014). Kashchiev, D., Borissova, A., Hammond, R.B. & Roberts, K.J. Effect of cooling rate on the critical undercooling for crystallization. Journal of Crystal Growth 312, 698-704 (2010). Nývlt, J. Kinetics of nucleation in solutions. Journal of Crystal Growth 3–4, 377-383 (1968). Sugimoto, T., Shiba, F., Sekiguchi, T. & Itoh, H. Spontaneous nucleation of monodisperse silver halide particles from homogeneous gelatin solution I: silver chloride. Colloids and Surfaces A: Physicochemical and Engineering Aspects 164, 183-203 (2000). Neimark, A.V., Vishnyakov, A. & Review, E. Gauge cell method for simulation studies of phase transitions in confined systems. Physical Review E 62, 4611-4622 (2000). Neimark, A.V. & Vishnyakov, A. Monte Carlo simulation study of droplet nucleation. Journal of Chemical Physics 122 (2005). Vishnyakov, A. & Neimark, A.V. Multicomponent gauge cell method. Journal of Chemical Physics 130 (2009). Neimark, A.V. & Vishnyakov, A. Gauge cell method for simulation studies of phase transitions in confined systems. Physical Review E 62, 4611-4622 (2000). Neimark, A.V. & Vishnyakov, A. Monte Carlo simulation study of droplet nucleation. Journal of Chemical Physics 122 (2005). Hildebrand, J.H. & Wood, S.E. The derivation of equations for regular solutions. Journal of Chemical Physics 1, 817-822 (1933). Kontogeorgis, G.M. & Folas, G.K. Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories (Wiley, 2009). Thomson, W. On the Equilibrium of Vapour at a Curved Surface of Liquid. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science 52 (1871). Frenkel, D. & Smit, B. Understanding Molecular Simulation: From Algorithms to Applications (2002). Owen, J.S., Chan, E.M., Liu, H. & Alivisatos, A.P. Precursor Conversion Kinetics and the Nucleation of Cadmium Selenide Nanocrystals. Journal of the American Chemical Society 132, 18206-18213 (2010). Jia, J., Tian, J., Mi, W., Tian, W., Liu, X., Dai, J., Wang, X.; Growth kinetics of CdSe nanocrystals synthesized in liquid paraffin via one-pot method. Journal of Nanoparticle Research 15, 1-12 (2013).


23

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

50. 51. 52.

53. 54. 55. 56.

57.

58.

59.

60.

61.

Iwamatsu, M. THE SURFACE-TENSION AND TOLMAN LENGTH OF A DROP. Journal of Physics-Condensed Matter 6, L173-L177 (1994). Blokhuis, E.M. & Kuipers, J. Thermodynamic expressions for the Tolman length. Journal of Chemical Physics 124 (2006). Dillmann, A. & Meier, G.E.A. A refined droplet approach to the problem of homogeneous nucleation from the vapor phase. The Journal of Chemical Physics 94, 3872-3884 (1991). Kalikmanov, V.I. Generalized Kelvin equation and pseudospinodal in nucleation theory. J Chem Phys 129, 044510 (2008). Moody, M.P. & Attard, P. Curvature-dependent surface tension of a growing droplet. Physical Review Letters 91 (2003). Adams, G.W., Schmitt, J.L. & Zalabsky, R.A. THE HOMOGENEOUS NUCLEATION OF NONANE. Journal of Chemical Physics 81, 5074-5078 (1984). Chianese, A., Karel, M. & Mazzarotta, B. NUCLEATION KINETICS OF PENTAERYTHRITOL. Chemical Engineering Journal and the Biochemical Engineering Journal 58, 209-214 (1995). Chan, E.M. et al. Reproducible, High-Throughput Synthesis of Colloidal Nanocrystals for Optimization in Multidimensional Parameter Space. Nano Letters 10, 1874-1885 (2010). Koerner, H., MacCuspie, R.I., Park, K. & Vaia, R.A. In Situ UV/Vis, SAXS, and TEM Study of Single-Phase Gold Nanoparticle Growth. Chemistry of Materials 24, 981-995 (2012). Zhang, X., Qian, G. & Zhou, X. Effects of Different Organic Acids on Solubility and Metastable Zone Width of Zinc Lactate. Journal of Chemical & Engineering Data 57, 2963-2970 (2012). Valiulin, S.V., Karasev, V.V., Vosel, S.V. & Onischuk, A.A. A study of sulfur homogeneous nucleation from supersaturated vapor. Determination of surface tension of sulfur nanoparticles. Colloid Journal 75, 14-25 (2013). Woehl, T.J., Evans, J.E., Arslan, L., Ristenpart, W.D. & Browning, N.D. Direct in Situ Determination of the Mechanisms Controlling Nanoparticle Nucleation and Growth. Acs Nano 6, 8599-8610 (2012).


24

Page 24 of 24

Correlation of Solubility with the Metastable Limit of Nucleation Using

Recommend Documents