20846 | Phys. Chem. Chem. Phys., 2015, 17, 20846--20852 This journal is© the Owner Societies 2015

Cite this:Phys.Chem.Chem.Phys.,

2015, 17, 20846

A simple model of burst nucleation†‡

Alexandr Baronov, Kevin Bufkin, Dan W. Shaw, Brad L. Johnson* andDavid L. Patrick*

We introduce a comprehensive quantitative treatment for burst nucleation (BN)—a kinetic pathway

toward self-assembly or crystallization defined by an extended post-supersaturation induction period,

followed by a burst of nucleation, and finally the growth of existing stable assemblages absent the

formation of new ones—based on a hybrid mean field rate equation model incorporating thermodynamic

treatment of the saturated solvent from classical nucleation theory. A key element is the inclusion of a

concentration-dependent critical nucleus size, determined self-consistently along with the subcritical

cluster population density. The model is applied to an example experimental study of crystallization in

tetracene films prepared by organic vapor–liquid–solid deposition, where good agreement is observed

with several aspects of the experiment using a single, physically well-defined adjustable parameter. The

model predicts many important features of the experiment, and can be generalized to describe other

self-organizing systems exhibiting BN kinetics.

Many self-organizing systems provided with an ongoing supplyof fresh growth units are found to undergo burst nucleation(BN), characterized by an extensive post-supersaturation induc-tion time, followed by a transient burst of nucleation, thencontinued growth of existing nuclei absent the formation ofmany new ones. BN has been observed in situations as diverseas setting concrete,1 infection-induced cellular lysis,2 the pre-paration of synthetic nanoparticles,3,4 and vapor–liquid–soliddeposition.5,6 In systems exhibiting BN kinetics, nucleation andgrowth are separated in time, and hence can be performedunder different experimental conditions; this distinguishes BNfrom other modes of monomer aggregation,7 and can be exploited,for example, to produce nanostructures incorporating hetero-interfaces.8 Since nucleation is limited to a relatively shortperiod of time, BN is perhaps the most widely invoked mecha-nism for understanding narrow size distributions in the pre-paration of monodisperse nanoparticles.9–11

The underlying mechanisms responsible for BN may bevaried, and system-dependent. For example, in a closed systemcontaining a finite number of heterogeneous nucleation sites,such as foreign particles, heterogeneous nucleation may betransitory, lasting only until all such sites have been titrated bynuclei.12 In systems subjected to mechanical agitation, so-calledsecondary nucleation can result from the growth of daughter

fragments detached from parent crystals, a mechanism whichcan lead to a rapid but brief increase in the apparent nucleationrate.13,14 Another mechanism, proposed for certain metal nano-particle syntheses,15 postulates a two-step process combiningslow, continuous nucleation with fast autocatalytic surfacegrowth. The large difference in rate constants for the two stepsleads to BN.16

In contrast, here we are interested in homogeneous nuclea-tion within a system fed by a continuous supply of fresh mono-mers with negligible heterogeneous or secondary nucleation,perhaps the simplest situation producing BN behavior. Such ascenario was originally studied by LaMer and Dinegar (LD),17

whose classic explanation for BN kinetics remains by far the mostwidely cited and studied one. LD postulated a time-dependent riseand fall of the monomer concentration n, where a flux F of newmonomers is continuously provided from an external reservoir:F(t) 4 0. The main features of the LD picture are illustrated inFig. 1. During the induction regime n increases as a result of theflux, rising until a critical supersaturation n* is reached andcrystallization commences. Drivers of n can include the chemicaldecomposition of monomer-generating precursors in solutionas studied originally by LaMer and Dinegar, the flux of growthunits from the vapor onto a surface as studied here, or acontinuous change of temperature in a closed system, suchas in a batch crystallizer, where it is the monomer chemicalpotential, rather than concentration, that is driven. The forma-tion and growth of stable nuclei in the subsequent nucleationregime rapidly outstrips the flux, depleting the monomerconcentration until a steady-state condition within the meta-stable zone nsat o n o n* is reached, where nsat is the monomer

Advanced Materials Science and Engineering Center, Western Washington

University, Bellingham, WA 98225, USA. E-mail: brad.johnson@wwu.edu,

david.patrick@wwu.edu

† 68.55.A-, 81.05.Fb, 81.10.Aj, 81.15.Aa‡ Electronic supplementary information (ESI) available. See DOI: 10.1039/c5cp01745a

Received 25th March 2015,Accepted 16th July 2015

DOI: 10.1039/c5cp01745a

www.rsc.org/pccp

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concentration at saturation; this halts further nucleation butpermits growth of existing nuclei. The LD model applies to early-stage growth, excluding coarsening processes such as Ostwaldripening or further aggregation among supercritical clusters.18

Despite its important role in understanding the origins ofBN in a wide range of systems, few quantitative formulations ofthe LD mechanism have been proposed, and direct comparisonto experiments under unambiguous LD conditions are rare.19

Recently, Privman and coworkers introduced a model combin-ing classical nucleation theory (CNT) with a rate equationtreatment for monomer and supercritical cluster aggregationto study burst nucleation and cluster size distributions in theformation of monodisperse colloids.20,21 However, the modelbegins with a large supersaturation, masking details of theinduction regime, and the total number of monomers in thesystem is conserved, i.e. F(t) = 0.

In this paper we introduce a semi-empirical hybrid kinetic-thermodynamic approach to modeling BN under LD conditions(F(t) 4 0, and no subsequent coarsening processes) wherein atraditional rate equation structure is modified to include theeffects of cluster surface energy and monomer concentration indynamically determining the critical nucleus size and alsoexplicitly incorporating the rates for attachment and detach-ment of subcritical clusters via exchange with a surroundingsolvent. The purpose is to provide a simple, effective quantita-tive description of BN that explicitly elucidates the key globalmechanisms applicable to a range of systems and growth habits(i.e. independent of crystal shape/morphology and diffusion-,reaction-, or attachment-limited growth kinetics; these detailscan be included in the present model, but we focus instead onthe foundational elements of BN22).

The key new feature of the present treatment is the self-consistent incorporation of a monomer-concentration-dependentcritical nucleus size. We demonstrate that the critical size playsa catalytic role, activating the nucleation mechanism (explicitlygoverned by the coupled rate equations), and after nucleation isinitiated, the subsequent reduction of monomer concentrationcauses the critical size to abruptly increase, terminating nucleation

on short time scales. As formulated, the model is the simplestembodiment of the LD theory capturing all microscopic pro-cesses, and the first quantitative model of the full LD mecha-nism. It reduces exactly to a standard mean field rate equation(MFRE) model in the (fixed) small critical nucleus size limit butproduces good agreement with classical nucleation theory (CNT)in the prediction of the critical nucleus size, thereby bridging thetwo paradigmatic frameworks. We apply our model to analyzeearly stage nucleation and growth in tetracene films prepared byorganic vapor–liquid–solid (VLS) deposition, under conditionswherein Ostwald ripening and supercritical cluster aggregationare both negligible, producing archetypal LD kinetics. We findgood quantitative agreement using a single, physically well-defined adjustable parameter.

The proposed model is constructed around a simplified MFREset incorporating terms accounting for the elementary processesof monomer flux, diffusion, and aggregation.23,24 The treatmentis limited to two bulk populations:25,26 the concentration ofsub-critical clusters n, which is dominated by monomers, andsupercritical (thermodynamically stable) clusters N.27 The two-population approximation reduces the model to the simplestcomponents capable of encompassing the essential mecha-nisms in a mathematically tractable and complete fashion-thermodynamically-driven critical nucleation and (mean-field)population dynamics. The kinetics of the two populations aregoverned by coupled rate equations, vis.

dn

dt¼ F � KP i�; nð Þn� KnN (1)

dN

dt¼ KP i�; nð Þn (2)

Here K is a collision and capture kernel for diffusing speciesand P(i*,n) is the concentration of aggregates (at monomerconcentration n) of size i*, with i* defined as one monomer lessthan the critical stable cluster size. The mathematical forms ofeqn (1) and (2) represent the processes of monomer addition,loss of subcritical clusters via the collision and capture of amonomer with/by a cluster of size i* (with a corresponding gainin the stable cluster density via the same process) and the loss ofsubcritical density via the collision (and possible capture) of sub-critical clusters with stable clusters. The capture of a monomerby a stable island is assumed irreversible. Most important, theconnection of the kinetics to the thermodynamics is accountedfor in the construction of the functions K and P.

To construct the distribution P(i*,n) we begin with the freeenergy of a cluster of size i at concentration n:28–30

DGiðnÞ ¼ �ði � 1ÞkT lnn

nsat

� �þ 4psa2 i2=3 � 1

� �(3)

Here nsat is the monomer equilibrium (saturation) concen-tration, k is the Boltzmann constant, T is the absolute tempera-ture, a is the monomer radius (assuming spherical monomers),and s is the cluster surface energy. The first term derives fromthe entropy of dispersed monomers in the solvent. The secondterm is the surface energy associated with the creation of acluster of size i, which is assumed for the sake of simplicity

Fig. 1 The LaMer–Dinegar model of burst nucleation. I = inductionregime, N = nucleation regime, G = growth regime. n* and nsat are themonomer concentrations where nucleation begins and at saturation,respectively. MSZ is the metastable zone where growth of existing stableclusters, absent nucleation, occurs.

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to be spherical with a size-independent surface energy.31

Assuming rapid attachment/detachment kinetics, the concen-tration of sub-critical clusters is a thermalized distribution, vis.

Pði; nÞ ¼ n exp�DGiðnÞ

kT

� �and therefore

Pði; nÞ ¼ nn

no

� �i�1exp

�4psa2 i2=3 � 1� kT

" #(4)

which then leads immediately to the relationship for P(i*,n)(eqn (4) has been constructed directly from the main ideas ofCNT, and is closely related to the so-called Walton relation.32

The difference, in this case, is centered around the monomerconcentration-dependent parameter i*, which is developed below).

We adopt the form of the collision kernel K as a modifiedSmoluchowski function33 for the rate of collision betweendiffusing monomers and spherical clusters of size i*: K Eg4pai*1/3D. Here D is the monomer diffusion coefficient andg is the condensation coefficient, equal to the monomer captureprobability given a collision has occurred. This (mean-field)parameter incorporates the microscopic rate constants govern-ing the association and dissociation of monomers to and fromthe cluster surface.

Lastly, the critical size i* itself is determined via maximiza-tion of the free energy (3) with respect to cluster size, and in thiscase we find

i�ðnÞ ¼ 8psa2

3kT ln n=nsatð Þ

� �3

(5)

The resulting model predicts the signature features of BN: in theinitial stages of flux delivery, the monomer concentration is onlyslightly greater than the equilibrium concentration, and thereforei* is very large (see eqn (5)). Large i* leads to the conditionP(i*,n) E 0, and the rate equations reduce to dN/dt E 0, (nonucleation, and thus N E 0) and dn/dt E F, giving n(t) = Ft.Together these conditions describe the induction regime with itslinear increase in monomer concentration. Similarly, as mentionedabove, following nucleation of stable clusters the monomer concen-tration drops rapidly, as diffusing monomers are significantly morelikely to encounter a stable cluster than to randomly aggregate into acluster of size i* � 1, and therefore the critical size will againincrease sharply and inhibit the terms involving P(i*,n) E 0, andnucleation ceases. Following nucleation, then, the stable clusterdensity is constant (N = Nq), and the rate equation for the subcriticaldensity is given by dn/dt E F � KNqn. If we make the simplifyingassumption that the critical nucleus size is a large constant(large enough to warrant the approximation P(i*,n) E 0), thenK = Kq (a constant), and the rate equation admits solutions given by

nðtÞ ¼ F

KqNqþ ce�KqNqt (6)

where c is an integration constant. Eqn (6) is the predicted formof the LD monomer curve in the growth regime, which asympto-tically approaches F/(KqNq) in the metastable zone.

We utilize the assumption that the size distribution ofsubcritical clusters is dominated by the monomers; this then

assumes the formation/dissolution rate of subcritical clustersis high, and is supported by the form of the concentrationdistribution given by eqn (4). In addition, for i* = 1, P = ndirectly, in agreement with established MFRE models for i* = 1.26

Hence the rate eqn (1) and (2) reduce to the familiar form for theirreversible deposition-diffusion-aggregation model in the limiti* = 1, with modifications of the diffusion coefficient to includecapture probability via the Smoluchowski equation.

We apply the model to study BN in the formation of mm-sizedcrystals of the organic semiconductor tetracene, grown in asolvent of bis(2-ethylhexyl)sebecate (BES). A steady rate of mono-mer addition was achieved by supplying growth units to a thinsolvent layer from the vapor, which resulted in the formation ofsmall crystals whose appearance and growth were quantified viadirect, in situ observation using videomicroscopy. The method isan all-organic analog to the VLS technique of crystal growthintroduced by Wagner and Ellis for inorganic materials inliquid metal alloy droplets,34–37 but employs an organic,38–41

ionic liquid,42–44 or liquid crystalline45 solvent combined withan organic precursor delivered via the vapor phase. As thesolvent layer becomes saturated, crystals form in a quasi-two-dimensional liquid environment, but they tend to stay in fixedpositions, similar to island growth on bare surfaces. Film for-mation thus combines elements of physical vapor deposition andsolution-phase crystal growth.

Tetracene vapor was deposited into a 1.9 � 0.3 mm thicklayer of BES spread onto a supporting indium-tin-oxide-coated(ITO) glass substrate. BES is a clear and colorless hydrophobicliquid with a low vapor pressure at room temperature and goodspreading properties on ITO-coated glass. Solvent layer thick-ness was determined by interferometry. Deposition was per-formed in a sealed chamber under 1 atm of N2. Tetracene vaporgenerated in a heated crucible was swept into flowing nitrogenand directed at the substrate by a tapered nozzle, producing alaminar axisymmetric impinging flow giving nearly uniformdeposition over an area E1 cm2 in size (Fig. 2). A movable

Fig. 2 Schematic of the experimental apparatus. Sublimate exited aheated source cell through a pinhole opening and was carried by a streamof N2 out a tapered nozzle toward the substrate. A shutter between thenozzle and substrate (not shown) controlled deposition. Film growth wasmonitored in situ through a transparent window with polarized opticalmicroscopy. The geometry of the impinging jet produces axisymmetricstagnation point flow.

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shutter was used to start the experiment. The gas flow serves toenhance deposition efficiency by increasing mass transportacross the static boundary layer at the gas–liquid interface.A tapered constriction at the nozzle’s terminus flattened thevelocity profile from Poiselle-like to plug-like flow, so that thevariation in F over the analysis region was less than 10%. Videos offilm growth were captured in situ with polarized optical micro-scopy. Further details are provided in ref. 47.

Fig. 3 presents micrographs showing the development of atetracene film. Fig. 3A, acquired t = 5.3 min after the start ofdeposition, shows the beginning stages of nucleation, where eachvisible bright object eventually grows into a distinct crystal. Fig. 3Band C correspond to the middle of the nucleation regime and thegrowth regime, respectively. Crystals were generally compact, withpartially branched habits (Fig. 3D). Post-deposition high resolu-tion polarized optical micrographs (Fig. S4, ESI‡) demonstratethat each object in Fig. 3 is a single monocrystal, and not anaggregate of smaller crystals. Growth was observed to be quasi-twodimensional, with lateral (in-plane) sizes approaching 100 mm bythe end of the analysis period, but crystal thickness, measured byatomic force microscopy after deposition was complete, beingonly E75 nm. This thickness was found to be about the same forall crystals measured, regardless of their size (Fig. S1, ESI‡).Crystals grew with random in-plane, but uniform out-of-planeorientation, with the (001) plane parallel to the liquid surface,as determined by X-ray diffraction (Fig. S2, ESI‡). Occasionally,very small crystals would be observed to move if the solventlayer was disturbed, but once they had grown to several tens ofmicrons in size they became essentially immobile. This sug-gests that crystals were not initially attached to the substrate,i.e. growth did not initiate with heterogeneous nucleation at theITO/solvent interface. Crystals were also fully wetted by BES andtherefore completely submerged in the solvent. Thus crystals

appeared to form in the solvent, as free-floating nuclei. Oncecrystals had grown to a certain size, capillary forces would havepressed them against the substrate, making them immobile.Halting deposition by closing the shutter led to the cessation ofgrowth without further changes in crystal size or shape over thetimescale of the experiment. Over a period of many hoursOstwald ripening was observed to gradually occur, but the rateof this process appeared negligible on the timescale of the analysis.Together these results support the assumption that processes ofsupercritical cluster aggregation and coarsening can be neglected,leading to archetypal La Mer–Dinegar BN kinetics.

The time evolution of the crystal concentration N(t) deter-mined by counting the number of visible crystals per unitvolume of solvent is shown in Fig. 4A. Three distinct regimesmay be identified: (i) an induction regime between the onset ofdeposition and the first appearance of crystals. Its durationdepended on the deposition rate, being shorter for higher rates.It could also be made shorter (but not equal to zero) by usingBES that was pre-saturated with tetracene46 prior to the start

Fig. 3 (A–C) Polarized optical micrographs at various stages of filmdevelopment. (A) t = 5.3 min after the start of deposition, near thebeginning of the nucleation regime; (B) t = 6.8 min, near the end of thenucleation regime; (C) t = 16.3 min, after nucleation had ceased andthe growth regime was well established. (D) Atomic force microscopyimage of a representative crystal.

Fig. 4 (A) The concentration of precursors and stable crystals as a function oftime since the onset of deposition. Open and solid points from experiment.Solid lines are model results (see text for parameters). (B) The model predic-tions for the critical nucleus size i* and the precursor distribution functionP(i*,n). Note the demarkation of the nucleation regime corresponding tosimultaneous minimum in i* and onset/peak/decline of P(i*,n).

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of deposition, demonstrating that the solution had to becomesignificantly supersaturated before crystals started to form.Since the smallest crystal which could be detected in the digitalmicroscope images was E10 mm2, and during the nucleationregime crystals grew in size at an average rate of E250 mm2 min�1,the induction time was essentially equivalent to the first appear-ance of crystals in the video images, which were acquired in 15 s.intervals. (ii) The induction regime was followed by a nucleationregime during which new crystals formed and existing crystalsgrew. The peak nucleation rate occurred at t = 8 min and droppednearly to zero less than two minutes later. In terms of 2D coverage,the projected fractional surface area occupied by crystals was0.006 at the peak nucleation rate, increasing to 0.03 by the timenucleation ceased. Crystals were thus widely separated relative totheir initial sizes, consistent with independent nucleation. (iii) Thenucleation regime was followed by a growth regime during whichcrystals increased in size but no new crystals formed.

To enable quantitative comparison to the proposed model,the time evolution of both the stable crystal density and thetetracene concentration are required. The latter may be obtainedas follows. Assuming uniform mixing,48 the concentration

CðtÞ ¼Pi�i¼1

iPði; nÞ of dissolved tetracene in the solvent layer is

found from the mass balance between the integrated flux andthe number of molecules incorporated into crystals:

CðtÞ ¼ðt0

Fdt� AðtÞdcrc=ds (7)

where A(t) is the total crystal area per unit substrate area,dc is the crystal thickness, ds is the solvent thickness, andrc = 3.42 � 1027 m�3 is the monomer number density ofcrystalline tetracene.49 The volumetric flux F was determined bymatching the arrival rate of new tetracene from the gas phasewith the incorporation rate of tetracene into growing crystalsduring the steady-state conditions occurring after nucleation hadceased and the growth regime was well established: F = Gdcrc/ds,where G = dA/dt is the total areal crystal growth rate per unitsubstrate area (in s�1) (in the present case, G = 2.67 � 10�5 s�1).Solvent layer and average crystal thickness were measured byinterferometry and atomic force microscopy, respectively, andfound to be ds = 1.9 mm, and dc = 75 nm (Fig. S1, ESI‡). This gaveF = 9.5� 1021 m�3 s�1. The monomer concentration computed inthis way is plotted in Fig. 4A (filled circles) along with theresulting density of stable crystals (open circles).

Overlaid in Fig. 4A (solid lines) are the model results obtainedby using parameters appropriate to the tetracene experiment,vis.: no = Csat, along with a = 4.11 � 10�10 m,49 s = 25 mJ m�2,50

and D = kT/6pea = 5.34� 10�11 m2 s�1 (e = 0.01 Pas is the solventviscosity, T = 300 K is the temperature and k is Boltzmann’sconstant). The only unknown parameter is the condensationcoefficient, which was determined by simultaneously fitting theexperimental results in Fig. 4A for both n(t) and N(t), givingg = 7 � 10�8. This value of the condensation coefficient isreasonable, based on its mean-field kinematic role as averagingthe stochastic rates of association and dissociation of clusters

of all sizes (including stable islands, which also suffer dissocia-tion, but have a net average growth). In the current application,nucleation and growth occur in a solvent solution, greatlynarrowing the gap between the attachment and dissociationrate constants. When the rate constants for attachment (ka) anddissociation (kd) obey ka c kd, the condensation coefficient isroughly equal to the areal growth of stable islands per time perarea; when dissociation rates are appreciable, as in the presentcase, the condensation coefficient is much smaller, as flux ratesand precursor concentrations greatly exceed areal growth rates.51

The model provides remarkable agreement with most aspectsof the experiment, including the existence of the three signatureregimes of BN, i.e. induction, nucleation, and growth regimes,a high monomer concentration relative to the stable clusterconcentration (n c N), as well as quantitative agreement in thenucleation onset time, peak monomer concentration, andconcentration of stable clusters. The main discrepancy is anunderestimation of the rate of decline in the monomer concen-tration at the beginning of the growth regime. We speculatethis may be caused in part by shear thinning of the solvent layerduring deposition under the impinging gas flow. Solventthinning-for a constant areal flux of monomers-results in ahigher volumetric-flux rate and a corresponding larger mono-mer concentration in the experiment than would be computedfrom the relationship C(t) above. The critical concentration(shown in Fig. 4A) is found to be C* = 3.6 � 1024 m�3, and thecritical supersaturation ratio S* = C*/Csat = 9, based on the equili-brium saturation concentration Csat = (4.2 � 0.6) � 1023 m�3,measured separately by UV-Vis absorption spectroscopy of a bulkequilibrated BES solution saturated in tetracence. The saturationratio determined here is similar to that reported for VLS growthin some inorganic systems,52 and further demonstrates thatsignificant supersaturation is required for crystal nucleation tooccur. The model prediction for n* via the value of n(t) at theonset of nucleation gives n* = 3.8 � 1024 m�3, in excellentagreement with the observed C*.

The central significance of the model, and the primary resultof this paper, is illuminated via self-consistent calculation ofthe critical size i* over the course of the experiment. In Fig. 4B,we show the model predictions for the critical nucleus size i*(t)and the corresponding distribution P(i*,n) for the parametersof the experiment. The important features of the model areevident: at the onset of deposition the monomer concentrationis small, so i* is initially large. As the flux of new growth unitsfrom the vapor increases n(t), i* decreases, reaching a minimumiN* E 53 in the nucleation regime. Nucleation occurs when thecritical size decreases sufficiently such that the distributionfunction P(i*,n) becomes appreciable, triggering the couplingof eqn (1) and (2), which is the mechanism for the initiation ofnucleation. Thereafter monomers are increasingly consumed bygrowing crystals, causing i* to rise again, and further nucleationto cease. Also note in Fig. 4B that the function P(i*,n), which, aspointed out, becomes appreciable only in the nucleation regime,serves to define both the onset and terminus of the nucleationregime. Together these results highlight the importance of thechanging critical nucleus size in producing BN behavior.

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The value of iN* given by the model can be compared to thecritical nucleus size predicted from classical nucleation theory.For a spherical nucleus, the latter is given by rc = (8/3)psa3/Dm* =1.63 nm, where the Gibbs potential energy difference Dm* betweena monomer in solution at concentration C* and in the crystallinestate is Dm* = kT ln(C*/Csat) = 8.9 � 10�21 J at T = 300 K. Thiscorresponds to iCNT* = 62 monomers, in good agreement with thevalue iN* reported above.

In addition, the model rate equations may be employed toanalyze the scaling of the equilibrium crystal density with theflux. It has been shown previously that two-component rateequation models with fixed i* predict that the equilibriumcrystal density should scale with flux, according to N(F) E Fw,where w = i*/(i* + 2).53 More recently, it has been shown that inthe presence of an energy barrier inhibiting attachment ofmonomers at crystal edges, the scaling of crystal density withflux may be significantly altered, i.e. wE 2i*/(i* + 3).54 Therefore,in the limit of large i*, the two predictions converge to wE 1 andw E 2, respectively. In order to compare the current model –which assumes an effective attachment barrier (small condensationcoefficient, via appreciable attachment/detachment rates) – werun a numerical experiment for the equilibrium crystal densityas a function of flux, using the parameters from the tetracene/BES experiment as a baseline. Fig. 5 shows the model predic-tion for the scaling of stable crystal density with flux. The plotdepicts the log of crystal density vs. the log of the volumetricflux into the liquid solvent; the resulting linear fit indicates ascaling exponent of w = 0.98. If we adopt the critical nucleus sizeof iN* = 53 as above, we expect, in the absence of an attachmentbarrier, w = 53/55 = 0.96. The agreement would indicate that therapid onset and rise of nucleation associated with BN, in thecurrent model, compensates for the detachment rate effects ofthe solvent. It is worth noting that a similar numerical result forthe scaling of the equilibrium crystal density with condensationcoefficient predicts a scaling of N E g�0.98, or roughly inversescaling – a surprising result.55 Lastly, we note that the modelpredicts a power-law scaling of the induction time with thecondensation coefficient, vis. tind E g�0.035, such that long

induction times are predicted when association and dissociationrate constants are comparable, resulting in a low probability ofmonomer sticking per collision.

In summary, we have studied the general phenomenon ofburst nucleation within a unified thermodynamic/kinetic meanfield rate equation framework, wherein the combined effects ofa monomer driving source (here provided by a vapor-phase flux)and a solvent (forcing the monomer concentration to signifi-cantly exceed the saturation concentration in order for nuclea-tion to occur) modify the rate equations, giving qualitativelydifferent nucleation and growth behavior relative to standardMFRE approaches. A key feature is the self-consistent inclusionof a monomer-concentration-dependent critical nucleus size,which controls the onset and duration of the nucleation regimeby activating the monomer capture terms in the rate equations.This leads to a substantial induction regime, a sharply-definednucleation regime followed by pure growth, as well as a largedifference in the equilibrium monomer and stable-crystal con-centrations. The model predicts a critical nucleus size consistentwith classical nucleation theory, but reduces to the standardMFRE result in the limit i* = 1. As the first comprehensivequantitative formulation of LD kinetics, we note the potential toextend the model to include different drivers of the monomerconcentration, reaction- or diffusion-limited growth, and size-and shape-dependent surface energy.

This work was supported by the National Science Founda-tion (DMR-0705908 and DMR-1207338). The authors thankB. Ricker for his contributions.

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Fig. 5 The log of the equilibrium crystal density vs. the log of the volumetricflux as predicted by the model, utilizing parameters from the experimental fitof Fig. 4. The lines is a linear fit, giving a scaling exponent of 0.98.

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20852 | Phys. Chem. Chem. Phys., 2015, 17, 20846--20852 This journal is© the Owner Societies 2015

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