Soluble Oligomeric Nucleants: Simulations of Chain Length, Binding

Letter pubs.acs.org/JPCL

Cite This: J. Phys. Chem. Lett. 2017, 8, 5815-5820

Soluble Oligomeric Nucleants: Simulations of Chain Length, Binding Strength, and Volume Fraction Effects Geoffrey G. Poon,† Tobias Lemke,‡ Christine Peter,‡ Valeria Molinero,¶ and Baron Peters*,† †

Department of Chemical Engineering, University of California, Santa Barbara, California 93106, United States Department of Chemistry, University of Konstanz, Konstanz 78457, Germany ¶ Department of Chemistry, The University of Utah, Salt Lake City, Utah 84112, United States ‡

S Supporting Information *

ABSTRACT: Recent theories and simulations suggest that molecular additives can bind to the surfaces of nuclei, lower the surface energy, and accelerate nucleation. Experiments have shown that oligomeric and polymeric additives can also modify nucleation rates of proteins, ice, and minerals; however, general design principles for oligomeric or polymeric promoters do not yet exist. Here we investigate oligomeric additives for which each segment of the oligomer can bind to surfaces of nuclei. We use semigrand canonical Monte Carlo simulations in a Potts lattice gas model to study the effects of oligomer chain length, volume fraction, and binding strength. We find that increasing each of those parameters lowers the nucleation barrier. At extremely low oligomer concentrations, the nucleation kinetics can be modeled as though each oligomer is a heterogeneous nucleation site in solution.

S

adsorbing chain can bind to unoccupied surface sites on the nuclei. The nucleation rate J is

olute precipitate nucleation is often influenced by species other than the solute and the solvent, including small molecules,1−5 salts,6,7 surfactants,8−11 peptides,12 and impurities.13,14 Polymeric additives (e.g., PEG, PVP, HPMC, and PVA) have also been experimentally shown to enhance or inhibit the nucleation of proteins and small molecule drugs,15−17 lower or raise the freezing temperature of water,18,19 and selectively control polymorph selection during CaCO3 nucleation and crystallization.20,21 An understanding of the mechanisms by which these species influence nucleation rates might guide the development of tailor designed additives to modify nucleation rates or to selectively promote nucleation of certain polymorphs.1,15,22 Efforts to model the effects of additives began with Sangwal’s semiempirical models of metastable zone widths.23,24 Anwar et al. discovered that additive size and interaction strength with solutes are important design variables for a promoter and inhibitor.25 Their simulations of ternary Lennard-Jones systems showed that amphiphilic additives with a strong attraction to the nuclei surface decreased the waiting time for solute cluster formation. Poon et al.26,27 and van Dongen et al.28,29 combined Langmuir adsorption and classical nucleation theory (CNT) to show how adsorbing additives lower the surface free energy and accelerate nucleation. These theoretical models showed that stronger surface-binding additives are more effective promoters and predicted specific trends in free-energy barriers. The model from our previous work27 is not applicable to oligomeric nucleants because their adsorption onto surfaces does not follow a Langmuir isotherm. In this work, we consider the effects of oligomeric additives where each segment of the © XXXX American Chemical Society

J = A exp[−β ΔF ⧧]

(1)

where A is a pre-exponential factor, β = 1/kBT, and ΔF⧧ is the maximum of the free energy as a function of nucleus size. According to CNT, A is proportional to the attachment frequency. ΔF⧧ depends on the thermodynamic driving force for nucleation and on the interfacial free energy of the nuclei.30 Therefore, eq 1 suggests that additives can influence nucleation rates in three ways: by changing the thermodynamic driving force, by changing the interfacial free energies of nuclei, or by changing the attachment frequency.23,31,32 In general, additives must be present in large amounts to alter the thermodynamic driving force (e.g., by changing solubility limits or freezing points), although in some cases they can promote nucleation in very dilute solutions by nonideal mixing effects.33 In this work, we focus on trace oligomeric additives that lower the interfacial free energy of nuclei by adsorbing to their surfaces. More specifically, we investigate how different conditions (i.e., chain length, volume fraction, and surface attraction) affect ΔF⧧ and therefore nucleation rates. We do not consider the effects of additives on solute attachment (i.e., prefactors) because coverages on critical nuclei do not closely approach unity for any conditions examined. Received: October 6, 2017 Accepted: November 8, 2017 Published: November 8, 2017 5815

DOI: 10.1021/acs.jpclett.7b02651 J. Phys. Chem. Lett. 2017, 8, 5815−5820

Letter

The Journal of Physical Chemistry Letters Following previous works,26,34 we use a ternary Potts lattice gas (PLG) model to reveal generic design principles. The PLG in this work represents a solution with solutes, solvents, and dissolved oligomeric additives. Each site i on the 32 × 32 × 32 cubic lattice has two properties: a type mi and an orientation si. mi is either 0, 1, or 2, which indicates that the site is occupied by an oligomer segment, solute, or solvent, respectively. Oligomers occupy a series of N connected lattice sites where N is the oligomer chain length. In contrast, solutes and solvents occupy only one site. For each occupant, si can be one of 24 possible orientations (the number of distinct orientations of a cube), allowing similarly oriented solutes to be more attracted to each other.35 It also allows nonisotropic oligomer−solute interactions to ensure chains adsorb onto the cluster surface without becoming inclusions in the solute precipitate. The total energy E is the sum of pairwise interactions Ui,j between nearest-neighbor sites i and j: βE =

canonical moves are also used to control the chemical potential driving force for nucleation. In a small closed simulation, the formation of a nucleus artificially depletes the dissolved solute concentration and lowers the driving force, even while nuclei are precritical.40−43 Therefore, solute precipitate nucleation is best simulated using open, grand, or semigrand ensembles.35,44−46 We use Kofke−Glandt47 semigrand moves to swap solutes and solvents and Müller−Binder48 semigrand moves to swap oligomers with either solutes or solvents. These moves maintain the chemical potential differences between all three species. We use a rare event strategy to sample the distribution of solute clusters from precritical to postcritical sizes with varying number of bound oligomers. Specifically, we use umbrella sampling49 and the multistate Bennett acceptance ratio (MBAR)50 to compute the probability P(ñ, m, X) that the largest solute cluster in the simulation is size ñ and has m adsorbed oligomers with X segments in contact with the cluster surface. Clusters are contiguous groups of neighboring solutes. Adsorbed oligomers have at least one segment that contacts the cluster. Sampling was biased along the ñ and X coordinates with harmonic biasing potentials: Ub(n) = kn(ñ− ñ′)2/2 + kx (X − X′)2/2, where β(kn, kx) = (0.1, 0.03) are the restraining spring constants and (ñ′, X′) is where the biasing potential is centered. Each window is sampled for at least 500 000 additional MC sweeps after equilibrating for 10 000 MC sweeps, where a sweep is a series of 323 (i.e., number of lattice points) MC moves. The Supporting Information contains additional details about the simulation methods and justification of our quasiequilibrium approach. To visualize the nucleation barrier, we project P(ñ, m, X) onto the two-dimensional Landau free-energy surfaces FL(ñ, X) and FL(ñ, m):

∑ Ui ,j (2)

⟨i , j⟩

The pair interactions are defined by species-specific and orientation-specific parameters Amimj and Bmimj, respectively. ⎧ A 00 ⎪ ⎪ A11 + δs , s B11 i j ⎪ ⎪ ⎪ A 22 βUi , j = ⎨ ⎪ ξijB01 ⎪ ⎪ A 02 ⎪ ⎪ ⎩0

if mi = mj = 0 if mi = mj = 1 if mi = mj = 2 if mi = 0, mj = 1 if mi = 0, mj = 2 otherwise

(3)

where interactions (in order) are between oligomer segments (bonded and nonbonded), between solutes, between solvents, between segments and solutes, and between segments and solvents. δh,k is the Kronecker delta. See Table 1 for interaction

A11

B11

A22

A00

B01

A02

−1.25

−1.25

−1.25

−2.5 or −3.0

−1.25

(4)

βFL(n ̃, m) = −ln[P(n ̃, m)/P(1, 0)]

(5)

where P(ñ, X) = ∑mP(ñ, m, X) and P(ñ, m) = ∑XP(ñ, m, X). Figure 1a shows that increasing N in the range of 10 ≤ N ≤ 50 at constant oligomer volume fraction increases the average surface coverage θ on solute clusters. A long chain can make more of the contacts that drive adsorption,51,52 while losing only marginally more entropy than a short chain. Therefore, increasing chain length results in more surface contacts, larger binding affinity, and lower surface energy. Multiple chains can adsorb to the nucleus when N is sufficiently large to drive binding and when free sites remain available on the nucleus surface (see Figure 1b,c). For N = 10, the minimum free-energy path (MFEP) follows the traditional CNT mechanism: growth of a cluster unassisted by an oligomer. However, a small (but not negligible) fraction of critical nuclei have one adsorbed chain. For N = 30 and 50, the MFEP shows that nucleation is typically assisted by one or two oligomer(s). Note that a single adsorbed N = 50 chain covers a large fraction of the available surface sites (see Figure 1a). High coverage from the first chain hinders the adsorption of the second chain and results in a large fraction of the critical nuclei with m = 1 (see Figure 1c). For larger chains (N ≫ 50), we expect all transition states to have m = 1 as a single oligomer covers the surface and excludes additional chains from binding. The impact of additive adsorption on the rate is estimated by defining a cluster free energy F(n) analogous to that used in CNT:

Table 1. Pairwise Interaction Parameter Values −1.25

βFL(n ̃, X ) = −ln[P(n ̃, X )/P(1, 0)]

parameter values. All other interactions are zero. The absence of solute−solvent interactions leads to a nearly pure solute precipitate with poor solubility.35 Oligomer segments interact only with solutes they are oriented toward and therefore are weakly soluble in the bulk solute precipitate. We do not expect oligomers to be incorporated in nuclei unless growth is fast and far from equilibrium.36 ξij = 1 for the 4 orientations that “point” to neighbor j; otherwise, ξij = 0. We restrict our study to parameters where the solvent is a good/athermal solvent for the oligomer by setting A00 = A22 (i.e., Flory parameter χ is zero).37,38 We also consider cases where B01 is large enough that the chains adsorb onto a solute surface. If oligomer segments are not attracted enough to the surface, an oligomer depletion layer forms because the entropic cost to confine the chain to the surface outweighs the enthalpy of adsorption. We sample the configuration space by accepting or rejecting Monte Carlo (MC) moves according to the Metropolis criteria. Moves include orientation flips, local and nonlocal swaps between solutes and solvents, and four local polymer moves:39 end rotation, kink jump, crankshaft, and snake. Semigrand5816


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The Journal of Physical Chemistry Letters

Figure 1. (Rows top to bottom) Free energies and histograms for chain lengths N = 10, 30, and 50 at comparable volume fractions (φ × 105 = 1.47, 1.02, and 1.26, respectively). The segment−solute interaction parameter is B01 = −3.0. (a) β FL(ñ, X). The gray region above the dashed line is where X exceeds the number of available surface sites (i.e., θ > 1) on a compact cubic cluster of size ñ. (b) β FL(ñ, m). ●, ▲, and ■ correspond to transition states for motion along ñ at fixed values of m = 0, 1, and 2, respectively. The black line is the minimum free-energy path. (c) Histogram of m in the transition-state ensemble. Examples of the most common critical nuclei (blue) are shown with adsorbed chains (red and yellow) .

βF(n) = −ln[5(n)/5(1)]

not lower the barrier (within errors), but increasing N to 10, 30, and then 50 reduced the barrier by as much as 17 kBT. Figure 1b shows the MFEP on the FL(ñ, m) landscape for each chain length N. For N = 30 and 50, the value of m along the MFEP jumps discontinuously from 0 to 1, from 1 to 2, etc. Each jump from m to m + 1 occurs at a point along ñ where FL(ñ, m) crosses FL(ñ, m + 1), somewhat like the diabat crossings in a nonadiabatic reaction.54 Based on eq 1, the enhanced rate J due to oligomeric additives should be approximately

(6)

where 5(n) is the average number of clusters of size n and 5(1) is that of dissolved solute monomers. In contrast to P(ñ) = ∑m∑XP(ñ, m, X), 5(n)/5(1) is independent of system size.45,53 The free-energy barrier ΔF⧧ is defined as ΔF⧧ = F(n⧧), where the critical nucleus size n⧧ maximizes βF(n). Figure 2 shows that ΔF⧧ is strongly dependent on the length N of the oligomeric chain when oligomer volume fractions are held approximately constant. Monomeric additives (N = 1) did

J = J0 exp[−β ΔΔF ⧧]

(7)

where J0 = A exp[−βΔF⧧(N, 0)] is the rate without oligomers and ΔΔF⧧ = ΔF⧧(N, φ) − ΔF⧧(N, 0) is the difference between the barrier with oligomers of length N at volume fraction φ and the barrier without oligomers. ΔΔF⧧ also depends on the strength of chain segment−solute interactions B01. For example, the barrier is reduced by about 8.4 kBT when N = 30, B01 = −3.0, and φ = 1.0 × 10−5, but the change in barrier is negligible when B01 = −2.5. For N = 50, Figure 3 shows the additive still reduces the barrier when B01 = −2.5 but much less effectively than when B01 = −3.0. Relative to our previous models of molecular surfactants,26,27 the dependence on binding strength is exaggerated for oligomers because of cooperative binding of covalently linked segments.

Figure 2. Cluster free energy (see eq 6) for nucleation with and without (black) oligomeric additives of different chain lengths (N = 10, 30, and 50) at the same conditions in Figure 1. 5817


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chains φ/vsN, where vs is the volume of an oligomer segment. If the probability of a nucleus interacting with two or more chains is also small, we can estimate the apparent rate J as a volumeweighted average of nucleation rates in cells with zero or one chain: ⎡ vφ ⎤ vφ J = J0 ⎢1 − exp[β ΔF0⧧ − β ΔF1⧧] ⎥ + J0 vsN ⎦ vsN ⎣

(8)

where the first and second terms are the HON and HEN contributions, respectively. ΔF⧧0 − ΔF⧧1 is the difference in nucleation barriers in cells with zero or one chain. To compute ΔF⧧1 , the nucleation barrier is calculated for a system initiated with a single chain and simulated without Müller−Binder moves. Our simulations agree with the model in eq 8 at very low φ. However, significant deviations occur at larger φ when the predominant critical nuclei in the fully open system have more than one adsorbed chain. Therefore, the model of oligomers as isolated HEN sites works only at very dilute oligomer concentrations. Because ΔF⧧0 − ΔF⧧1 increases with increasing N, the HEN barrier may become vanishingly small for very long chains. HEN events would occur almost instantly on each chain, making the HON contribution negligible in eq 8. Beyond some chain length (not reached in our simulations), any further increase of N at constant volume fraction only decreases the number of cells where nucleation can be promoted, and the rate enhancement J/J0 becomes proportional to vφ/vsN. Equation 8 does not account for the possibility of multiple nuclei forming on a single chain. However, those nuclei would likely agglomerate to form a single particle. Therefore, our simulation results and eq 8 suggest the existence of an optimal oligomer size at constant oligomer volume fraction. Our simulations show that soluble oligomeric additives are more potent nucleants than their monomeric counterparts, sometimes by several orders of magnitude. Long oligomers can make more energetically favorable contacts, with only marginally more entropy losses than short oligomers. Increases in binding strength, chain length, and volume fraction of oligomers all tend to lower the nucleation barrier and therefore accelerate nucleation. Unlike our simulation results for molecular-scale nucleants,26,27 the effects of oligomeric nucleants are highly nonlinear in concentration. Very small concentrations have a large effect, but only modest additional rate enhancements are obtained from further concentration increases. We have provided a mixed homogeneous/heterogeneous nucleation (HON+HEN) model where oligomers act as heterogeneous nucleation sites. The mixed HON+HEN model explains the nonlinear dependence of the free-energy barriers on oligomer concentration, and it predicts an optimal oligomer chain length at constant oligomer volume fraction. Further studies are needed to verfiy the optimal chain length prediction and investigate behavior at high volume fraction where critical nuclei with multiple adsorbed additives become important.

Figure 3. Barrier reduction −β ΔΔF⧧ ≈ ln(J/J0) as a function of φ at two chain segment−solute interaction strengths B01. Results are for chain length N = 50. Dashed lines are mixed HON+HEN model predictions (eq 8) .

For oligomers with B01 = −3.0, we vary additive volume fraction φ by changing the additive segment-to-solvent fugacity ratios that control the acceptance probability of oligomer insertion and deletion. As expected from previous work,26,27 higher φ more strongly promotes nucleation. However, Figure 3 and 4b show that there are two distinct regimes that describe the φ-dependence of the nucleation barrier. −ΔΔF⧧ rises rapidly at very low φ but levels off at larger φ.

Figure 4. (a) Diagram of mixed HON+HEN model. (b) (left) Barrier reduction −βΔΔF⧧ ≈ ln(J/J0) as a function of φ for different chain lengths N for B01 = −3.0. (right) Same data on a semilog scale. Dashed lines are mixed HON+HEN model predictions (eq 8).

This qualitative behavior can be understood by modeling chains as heterogeneous nucleation sites. We call this model the mixed homogeneous/heterogeneous nucleation (HON+HEN) model (see Figure 4a). Imagine dividing the solution into small cells where the probability distribution for the number of oligomer chains in each cell follows a Poisson distribution. For very small φ, the probability of two or more chains residing in a cell is prohibitively small, so each cell contains zero or one chain. The fraction of cells with a chain is then the product of the volume of the cell v and the concentration of oligomer

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b02651. Simulation details and proof of detailed balance (PDF) 5818


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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Geoffrey G. Poon: 0000-0003-0237-6557 Tobias Lemke: 0000-0002-0593-2304 Christine Peter: 0000-0002-1471-5440 Valeria Molinero: 0000-0002-8577-4675 Baron Peters: 0000-0003-1935-6085 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

The authors thank Joop ter Horst and Richard Sear for stimulating discussions. G.G.P. was supported by the NSF Graduate Research Fellowship under DGE 1144085. T.L. and C.P. were supported by the German Research Foundation through the Collaborative Research Center 1214. V.M. was supported by NSF Center for Aerosols Impacts on Climate and the Environment under CHE-1305427. B.P. was supported by a Camille Dreyfus Teacher Scholar award.

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Soluble Oligomeric Nucleants: Simulations of Chain Length, Binding

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