Experimental measurement of the nucleation landscape reveals classical and nonclassical features

Matthew A. Gebbiea,b, Hitoshi Ishiwatab,c, Patrick J. McQuadea,b, Vaclav Petrakd, Andrew Taylord, Christopher Freiwalde,f, Jeremy E. Dahlb, Robert M. K. Carlsonb, Andrey A. Foking,h, Peter R. Schreinerg, Zhi-Xun Shenb,i, Milos Nesladeke,f, and Nicholas A. Melosha,b,1

aDepartment of Materials Science and Engineering, Stanford University, Stanford, CA 94305; bStanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA 94025; cDepartment of Electrical and Electronic Engineering, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8550, Japan; dInstitute of Physics of the Czech Academy of Sciences, CZ-18221 Prague, Czech Republic; eInstitute of Materials Research, University of Hasselt, B-3590 Diepenbeek, Belgium; fInstitute for Materials Research in Microelectronics, Interuniversity Microelectronics Centre, B-3590 Diepenbeek, Belgium; gInstitute of Organic Chemistry, Justus Liebig University, D-35392 Giessen, Germany; hDepartment of Organic Chemistry, Igor Sikorsky Kiev Polytechnic Institute, 03056 Kiev, Ukraine; and iApplied Physics, Stanford University, Stanford, CA 94305

Edited by Catherine J. Murphy, University of Illinois at Urbana–Champaign, Urbana, IL, and approved July 9, 2018 (received for review March 1, 2018) Nucleation is a core scientific concept that describes the formation of pathways, additional thermodynamic barriers, and the existence new phases and materials. While classical nucleation theory is applied of distinct subcritical and postcritical regimes with differing scaling across wide-ranging fields, nucleation energy landscapes have never in nucleation rates and probabilities are key elements of these been directly measured at the atomic level, and experiments suggest updated theories. that nucleation rates often greatly exceed the predictions of classical However, there is still a large discrepancy between the theoretical nucleation theory. Multistep nucleation via metastable states could models and experimental measurements, which show that nucle- explain unexpectedly rapid nucleation in many contexts, yet exper- ation occurs much more rapidly than predicted by classical nucle- imental energy landscapes supporting such mechanisms are scarce, ation theory (8–11). Resolving the origin of such rapid nucleation particularly at nanoscale dimensions. In this work, we measured the rates remains an area of active investigation, often explained by nucleation energy landscape of diamond during chemical vapor metastable states that lower nucleation barriers (9–15) or multistep deposition, using a series of molecules as atomically pathways. One of the central challenges has been that these es- defined protonuclei. We find that 26- atom clusters, which do sential regimes have remained inaccessible experimentally. This is not contain a single bulk atom, are postcritical nuclei and measure particularly true for industrially important gas-phase reactions, such the nucleation barrier to be more than four orders of magnitude as plasma-enhanced chemical vapor deposition (PECVD) of di- smaller than prior bulk estimations. These data support both classical amond and (16–19). For example, diamond is used as a and nonclassical concepts for multistep nucleation and growth functional coating in many applications(16)andisanemerging during the gas-phase synthesis of diamond and other semiconduc- material for quantum information technologies (20) and advanced tors. More broadly, these measurements provide experimental biolabeling (21). While there is a rich history of studying diamond evidence that agrees with recent conceptual proposals of multistep growth, the extreme conditions of diamond synthesis have pre- nucleation pathways with metastable molecular precursors in di- vented the measurement of diamond nucleation energetics (17, 18). verse processes, ranging from cloud formation to protein crystalli- Nucleation energy landscapes have never been measured at zation, and nanoparticle synthesis. the atomic level due to the extreme challenges associated with observing the dynamic, few-atom clusters thought to participate nucleation | diamond | nanomaterials | thermodynamics | plasma synthesis Significance lassical nucleation theory (1–4) proposes that nucleation is a Cthermally activated process, where atomic clusters, called nu- Nucleation is the limiting step for thermodynamic phase transi- clei, stochastically evolve from metastable solution to form ther- tions. While classical models predict that nucleation should be modynamically stable phases and materials. In classical models, extremely rare, nucleation is surprisingly rapid in the gas-phase nuclei are often assumed to be microscopic clusters of the final bulk synthesis of diamond, silicon, and other industrial materials. We phase, and nucleation barriers originate in a competition between developed an approach for measuring nucleation landscapes us- unfavorable surface energies and favorable bulk free energies, ing atomically defined precursors and find that diamond critical which exhibit differing scaling with particle size. Below a certain nuclei contain no bulk atoms, which leads to a nucleation barrier size, called the critical size, the unfavorable surface energy is that is four orders of magnitude lower than prior bulk estima- dominant, biasing protonuclei toward dissolution. Random energy tions. Our findings suggest that metastable molecular precursors and concentration fluctuations allow the protonuclei to grow or play a key role in lowering nucleation barriers during materials shrink diffusively within the nucleation energy landscape, occa- synthesis and provide quantitative support for recent theoretical sionally overcoming the critical nucleation barrier, leading to rapid A proposals of multistep nucleation pathways with much lower particle growth (Fig. 1 ). barriers than the predictions of classical nucleation theory. The difference in scaling between surface and volumetric en- ergies leads to two key hallmarks of classical nucleation theory: Author contributions: Z.-X.S., M.N., and N.A.M. designed research; M.A.G., H.I., P.J.M., (i) an exponential relationship between subcritical nuclei surface V.P., A.T., and C.F. performed research; J.E.D., R.M.K.C., A.A.F., and P.R.S. contributed new areas and the probability of crossing the nucleation barrier, and reagents/analytic tools; M.A.G., M.N., and N.A.M. analyzed data; and M.A.G. and N.A.M. (ii) a critical size where relative nucleation probabilities abruptly wrote the paper. increase, indicating a boundary between the subcritical regime, The authors declare no conflict of interest. where nuclei are thermodynamically biased to dissolve, and the This article is a PNAS Direct Submission. postcritical regime, where nuclei are biased to grow. Published under the PNAS license. Recent theoretical advances (5–7) have extended classical 1To whom correspondence should be addressed. Email: [email protected]. nucleation theory, for example, by including both size and den- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. sity as order parameters to describe nucleation pathways (6) or 1073/pnas.1803654115/-/DCSupplemental. including size-dependent interfacial free energies (7). Multistep Published online August 1, 2018.

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dissoluon* growth J) r

-20 1 2 diamantane * ΔG * C H POCl C H POCl i ΔG 10 15 2 14 19 2 nucleaon

Energy (10 Energy barrier 0

10 14 18 22 26 3 4 Number of carbon atoms triamantane [1(2)3]tetramantane B C H POCl C H POCl

) 18 23 2 22 27 2 26 275 2 22 250 225 18 200 14 175

Carbon atoms 10 150 5 5 Surface area (A area Surface 12345 [12(1)3]pentamantane [1(2,3)4]pentamantane C H POCl C H POCl Diamondoid order, N 26 31 2 26 31 2

Fig. 1. Diamond nucleation landscape and diamondoid protonuclei. (A) Approach for measuring nanoscale thermodynamic energy landscapes during nu- cleation. Molecular nuclei of atomically defined sizes and shapes, (28), were used to map out the nucleation energy landscape during diamond PECVD by starting at precise locations on the thermodynamic nucleation energy landscape and measuring relative nucleation probabilities. (B) Diamondoid order, N, versus the number of carbon atoms in each diamondoid (Left) and diamondoid van der Waals surface areas (Right). N is defined as 1 for ada- mantane, and each subsequent addition of a four-carbon isobutyl substituent increases N by an integer value. (C) Molecular structures of the diamondoid protonuclei, ranging from 10 to 26 carbon atoms in size, are shown below the plot. The diamondoid order, N, is shown above each chemical name. Dia-

mondoids were functionalized with POCl2 self-assembly groups (29) for covalent attachment to PECVD substrates.

in nucleation processes. Recently, researchers have made strides hypothetical nucleation energy landscape in Fig. 1A. Assuming that toward visualizing nucleation in liquid–liquid and colloidal systems, all of the diamondoids are located along the same nucleation where optical imaging was used to study colloids (22, 23) and pathway, the measured density of fully formed diamond nano- proteins (12, 24), since large particle sizes permit direct observation. particles, ρ, should scale as the nucleation barrier passage proba- Emerging techniques, like liquid-state transmission electron mi- bility, P. As a result, by measuring the relative number of particles croscopy (25–27), enabled striking advances in visualizing nanoscale that nucleate from subcritical diamondoids, we can ascertain the growth. However, the putative dynamic nuclei, in the order of tens- relative nucleation energetics. of-atoms size, remain inaccessible, and questions surrounding the Interestingly, if one of the diamondoids is larger than r*, then CHEMISTRY influence of the probe beam remain. we should also observe an abrupt, superexponential increase in Rather than attempt to directly image few-atom clusters, we de- density, ρ, as nucleation and growth probabilities enter a dif- veloped a way to measure nucleation energy landscapes by creating ferent regime for protonuclei that exceed the critical size, since atomically defined putative protonuclei and quantifying the relative there is no thermodynamic barrier to growth above the critical densities of diamond nanoparticles, ρ (in particles per square meter), size. The atomically defined series of nuclei we studied were formed from monolayers of each protonuclei. Different-sized sub- formed using diamondoids (28), which are molecular fragments critical nuclei have different free energies with respect to the of diamond, ranging from 10 to 26 carbon atoms (Fig. 1C). critical nucleus (Fig. 1A) and grow or contract stochastically over time. If a nucleus reaches the critical size, r*(Fig.1A), the particle Results is taken to then grow deterministically. Hence, if seed protonuclei Nucleation Densities from Diamondoid Monolayers. To test this are larger than r*, then there should be an abrupt increase in the concept for diamond, a silicon substrate was functionalized with a observed particle density. monolayer of phosphonyl dichloride (POCl2)-modified diamondoid The probability, P, that a protonuclei crosses this nucleation protonuclei, which were covalently attached to the surface via barrier can then be calculated as a mean first-passage time (5), P–O–Si bonds (29). PECVD diamond growth was then per- which scales exponentially with the energy difference between a formed using standard conditions, resulting in different densi- nucleus of size i and the critical nucleus of size r*, as shown in ties of large diamond particles depending on the size of the Eq. 1: diamondoid seed (Fig. 2). The number of diamond particles per square millimeter was measured with scanning electron mi- −ΔGp i k T croscopy (SEM), representing the probability that the proto- P ∼ e B . [1] nuclei crossed the nucleation barrier before being etched away by the plasma. where P is the barrier passage probability; ΔGi* (in joules) is the Fig. 2C shows the measured diamond nanoparticle densities free energy difference between ΔGi (in joules), which is the free energy of a nucleus of a specific size i, and ΔG* (in joules), which grown from six different diamondoid monolayers, grown at 750 °C in a 99% H ,1%CH PECVD plasma at 23.5 torr and 350 W. A is the free energy maximum at the critical size r*; kB (in joules 2 4 per kelvin) is the Boltzmann constant; and T (in kelvin) is the vertical wafer configuration (30) enabled testing multiple conditions temperature. Eq. 1 is only rigorously applicable to nucleation in parallel, as the temperature of the wafer varies from 950 °C at pathways with a single reaction coordinate and where particles the top to 250 °C at the bottom. Each point corresponds to the grow deterministically upon reaching the critical size. More ad- average particle density from at least eight different independent vanced models have examined how this equation will change experiments for each diamondoid. The diamond particles exhibited under differing assumptions (6, 7). a faceted, “quasispherical” shape (Fig. 2A) with a large diamond– By starting with monodisperse monolayers of diamondoid silicon contact angle, so the substrate surface energy has minimal protonuclei, we initialize the system at specific points on the impact on the measured diamond nucleation energy barrier.

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300 μm 500 nm scale bars C 11

) 750 ˚C, 99% H2 , 1% CH4 -2 9

7

5 3 ln(Parcles mm ln(Parcles Density range on unseeded wafers 300 μm 1 10 14 18 22 26 Number of carbon atoms per diamondoid [1(2,3)4]pentamantane

Fig. 2. Diamondoid-seeded PECVD growth densities. (A) Photograph of PECVD growth using 10 × 6-mm silicon wafers in a 1% CH4,99%H2 plasma. Scanning electron microscopy (SEM) images of representative diamond particles. (White scale bars: 500 nm.) (B) Representative SEM images comparing nanoparticle growth densities on a wafer seeded with the largest precritical diamondoid [1(2)3]tetramantane (Top) to a wafer seeded with the postcritical diamondoid [1(2,3)4]pen- tamantane. The small white dots are faceted diamond nanoparticles, as shown in A.(C) Particle growth densities following vertical substrate growth at 750 °C with a

feed gas concentration of 99% H2 and 1% CH4. Each data point corresponds to at least eight independent experiments for the diamondoid pictured, with the diamondoid attachment orientation as shown. The error bars indicate the standard error of the mean. The 750 °C conditions occurred at a distance of 2 mm from the top edge of the seeded wafer, and other temperature conditions during the vertical growth are shown in SI Appendix,Fig.S1. Diamond growth densities were largely independent of temperature for the 100 °C range tested in the vertical configuration. The purple shading illustrates nucleation densities on unseeded vertical ori- entation substrates that were immersed in pure before growth.

To ensure that the measured growth densities map the underly- classical nucleation theory. First, the growth density depends ing thermodynamic nucleation energy landscape, the experiments exponentially on the protonuclei size from adamantane through reported in Fig. 2 and SI Appendix,Fig.S1were carried out under tetramantane, with the slope remaining strikingly consistent very low carbon supersaturation, where nucleation is quasi- over a large range of conditions. Second, the growth density reversible and governed by the underlying thermodynamic en- increases abruptly upon reaching pentamantane, exhibiting a ergy landscape (4–6). Diamond nanoparticles grown under the 1% pronounced break from the exponential relationship of the CH4, 99% H2 conditions shown in Fig. 2 have narrow size distri- smaller diamondoids. butions and preferentially exhibit the lowest surface energy (111) A nucleation barrier where pentamantane exceeds the critical facet of the diamond lattice. Further decreasing the CH4 content size is the most plausible explanation for these two observations. in the PECVD feed gas to less than 0.25% results in full etching of The nucleus size appears to be the dominant factor during nu- the diamondoid monolayers. These observations are consistent cleation, yet symmetry also plays a secondary role, as evidenced with nucleation and growth in a largely thermodynamically by the higher diamond density seeded by the symmetric [1(2,3)4] controlled regime. pentamantane , relative to the lower-symmetry [12(1)3] As shown in SI Appendix,Fig.S2, raising the supersaturation of pentamantane. Since both of pentamantane exceed the carbon in the plasma phase by increasing the CH content of the 4 critical size, these data suggest a critical nucleus size between 22 feedgasorreplacingH2 with Ar, which is inert, readily drives nucleation and growth into a kinetic regime. For example, under and 26 carbon atoms or about 0.8 nm in diameter. feed gas conditions of 1% CH in 99% Ar, selected to mimic the 4 Determination of Nucleation Energy Barrier and Interfacial Energy. A conditions of ultrananocrystalline diamond (UNCD) synthesis significant advantage of this approach is that relative nucleation (31), diamond growth densities are much higher, but particle size distributions are heterogeneous and diamond particles form rod- energy barriers and the effective surface energy of the diamondoid like shapes due to preferential growth of the higher surface energy protonuclei can be determined without needing to assume that (100) facet for kinetic reasons (SI Appendix,Fig.S2). Increasing the critical nuclei behave as small clusters of the bulk phase. The energy barriers separating each of the diamondoid molecules in concentration of CH4 in the feed gas above 10% resulted in the the subcritical regime were determined by normalizing the growth exclusive growth of graphitic carbon, even in the presence of H2. To assess the generality of our conclusions, diamond growth densities were determined at different heights on the substrate, Table 1. Nucleation barriers and calculated diamond–plasma representing a range of plasma conditions in the low supersatu- interfacial energies ration regime. Experiments were also repeated with a different PECVD chamber with a remote plasma source. As shown in Table Diamond–plasma 1andSI Appendix,Fig.S1, the nucleation barrier, critical nucleus Wafer Adamantane-[1(2)3]tetramantane interfacial energy, − − size, and diamond–plasma interfacial energy were in agreement temperature, °C energy barrier, 10 20 J mJ·m 2 across varying conditions, despite vastly different temperatures 750 1 ± 0.5 12 ± 6 and nucleation rates, showing the results are general and not 700 2 ± 0.5 14 ± 5 particular to one chamber or growth geometry. 650 1 ± 0.5 12 ± 6 In this low supersaturation regime, relative growth densities 250 1 ± 0.5 10 ± 6 as a function of particle size exhibit the two key elements of

8286 | www.pnas.org/cgi/doi/10.1073/pnas.1803654115 Gebbie et al. Downloaded by guest on October 1, 2021 3 nucleus of about 5,000 atoms (Fig. 4A). Within this previous theoretical framework, diamondoids should be incapable of seeding diamond growth. 2 These capillary calculations are two to four orders of magnitude − larger than the 10 20-J barrier and 26-atom critical nucleus mea- sured here. For prior classical, single-step models of diamond 1 nucleation to be consistent with the probabilities we measure, the driving chemical potential in the plasma would need to exceed · −1 0 4,500 kJ mol , which is not physically realistic (32). Therefore, while the nucleation energy landscape retains the size dependence Slope: γ = 12 ± 6 mJ m-2 of classical theory, bulk properties cannot be used to quantitatively -1 predict PECVD diamond nucleation energetics. 0 20 40 60 80 100 In addition, the postcritical nucleus, pentamantane, does not contain a single “bulk” atom, calling into question the concept that the driving force for nucleation is formation of the bulk phase. In Fig. 3. Estimation of plasma–nucleus interfacial energy calculated based on classical nucleation theory, the nucleation barrier results from Δ relative nucleation probabilities. The y axis ( Gx-tetra) is the proportion of the differing scaling between a favorable “volumetric” energy that nucleation energy barrier separating the subcritical diamondoids using “ Δ results from the creation of bulk atoms and an unfavorable sur- [1(2)3]tetramantane as a reference at 750 °C. Gx-tetra is determined using Eq. 3, ” and the error bars indicate the standard error of the mean. The x axis (ΔA)isthe face energy from the creation of surface atoms. Here, we find calculated van der Waals surface area differences between each of the sub- that all of the atoms in diamond critical nuclei are surface atoms − critical diamondoids and [1(2)3]tetramantane. The 12 ± 6mJ·m 2 slope is the and that the critical barrier is crossed before forming internal energy cost per unit surface area created during diamond nucleation, assuming atoms with volumetric scaling. Thus, our results suggest that the that each diamondoid is at different points along the nucleation pathway. presence of a nucleation barrier during diamond synthesis has a different molecular mechanism than in classical nucleation theory. We propose that the nucleation-driving force for these molec- densities of each of the subcritical diamondoid nuclei to that of ular clusters comes from the formation of favorable surface [1(2)3]tetramantane as a reference, using Eqs. 2 and 3: chemical bonds that scale as the nucleus surface area, which re- places the bulk volumetric driving force of classical nucleation −ΔGp i k T theory. The barrier still originates in an unfavorable entropic ρi ∼ e B , [2] contribution, which is captured by the effective molecular surface   energy determined in Fig. 3. Since both the creation of favorable ρ −ΔG i = i−tetra [3] surface bonds and the growth of unfavorable interface scale as the ln ρ k T . tetra B nucleus surface area, the abrupt change in nucleation probability shown in Fig. 2 cannot result purely from geometric scaling ar- i where corresponds to a specific subcritical diamondoid proto- guments. To be consistent within this model, at least one of the CHEMISTRY nuclei, ρi (in particles per square millimeter) is the density of two surface terms must exhibit an explicit size dependence to give ρ particles grown from the subcritical diamondoid, tetra (in parti- rise to the observed diamond critical nucleation behavior. cles per square millimeter) is the density of particles grown from A size-dependent interfacial energy is one plausible explana- ΔG [1(2)3]tetramantane, i-tetra (in joules) is the free energy dif- tion for the critical barrier. For example, a recent theoretical ference between each subcritical diamondoid and [1(2)3]tetra- mantane, kB (in joules per kelvin) is the Boltzmann constant, and T (in kelvin) is the temperature. When the relative energy barrier between each subcritical diamondoid and [1(2)3]tetramantane is ABCleavage barrier: Nucleaon barrier: plotted versus the difference in van der Waals surface areas > 1000 kBT several kBT between each subcritical diamondoid and [1(2)3]tetramantane μ (Fig. 3), the slope yields the plasma–nucleus interfacial energy. vapor μ Fig. 3 shows the measured subcritical energy landscape at 750 °C. surface −20 Measured The nucleation barrier is in the order of 10 J, which is comparable barrier:

to thermal energy at the PECVD temperatures (kBT), and the slope several k T Energy − B yields a diamond–plasma interfacial energy of 12 ± 6mJ·m 2 at 750 °C. As shown in Table 1, the nucleation barriers and interfacial μ vapor μ μ energies agree to within a factor of 3 across the conditions tested. solid bulk

Discussion Reacon coordinate Reacon coordinate Our results show that the probability of diamond nanoparticle growth exponentially depends on the size of the diamondoid seed. Fig. 4. Schematic of diamond nucleation reaction coordinates. (A) Nucleation of Furthermore, we observe an abrupt increase in diamond growth condensed carbon phases from supersaturated carbon vapor, where relative μ μ densities at a critical size between 22 and 26 carbon atoms (Fig. 2). chemical potentials are per carbon atom with two distinct states: vapor and solid. Estimating nucleation barriers from the bulk cleavage energy leads to a barrier The exponential scaling and the presence of a critical size are two exceeding 1,000 kBT under PECVD conditions; this approach inherently assumes key predictions of classical nucleation theory. From our results, that nucleation is a single-step process into the bulk crystalline diamond phase. In

the measured energy of the nucleation barrier is on the order of contrast, the measured nucleation barrier is in the order of several values in kBT.(B) −20 Two-step diamond growth mechanism as the simplest example of a multistep di- 10 J, which is several kBT under growth conditions. μ μ In contrast, capillary models of diamond nucleation (18, 32, 33) amond nucleation and growth pathway, with three distinct states: vapor, surface, μ using energetics calculated from the bulk diamond lattice and and bulk. The critical nucleus is composed entirely of surface atoms with diamond- γ – · −2 like bonding, requiring an additional bulk transformation step to form bulk di- surface energy, cleave, predict an interfacial energy of 1 3Jm , amond. The carbon supersaturation in the plasma drives nucleation, and the nu- −16 leading to an estimated nucleation barrier of 10 J. This exceeds cleation barrier is determined by the plasma–nucleus interfacial energy, which is 1,000 kBT under diamond growth conditions and implies a critical strongly influenced by the diamond surface termination (38).

Gebbie et al. PNAS | August 14, 2018 | vol. 115 | no. 33 | 8287 Downloaded by guest on October 1, 2021 study of nucleation using model “polycube” clusters shows that increases the probability for silicon nucleation during CVD by size-dependent surface energies may be a key element of nu- the tremendous factor of e1,000, at which point the prevalence cleation (7). Here, the critical increase in diamond nucleation of silicon nucleation during CVD is readily understood. probability could be explained if the nucleus–plasma interfacial Overall, this direct experimental measurement of a nanoscale energy abruptly decreases between a cluster size of 22 and 26 nucleation energy landscape provides a quantitative test of carbon atoms. Alternatively, the surface-bonding energy could classical nucleation theory at the molecular level. We find that also have a size dependence. Additionally, a recent extension to the diamond nucleation landscape exhibits both the size de- classical nucleation theory predicts that cluster distribution pendence and abrupt change in growth rate of classical nucleation functions for nuclei in the subcritical regime may exhibit de- theory (4). Unexpectedly, we also find that critical nuclei are partures from a diffusive exploration of phase space charac- diamond-like carbon clusters that do not contain a single bulk terized by the Boltzmann population distribution of classical atom, meaning that a critical size can be characteristic of a system nucleation theory in certain cases (6). The implications of this where there is no distinction between surface and bulk effects. finding remain under active investigation, and further theo- Thus, the nucleation barrier and critical size do not seem to retical progress along these lines may shed additional insight originate from a geometric scaling competition between surface into the data reported here. and volumetric energies, but instead appear to arise from either Consequently, our measurements show that two key tenets of an interfacial energy or surface bonding energy that changes with classical nucleation theory are observed at molecular scales, but particle size. These results highlight the importance of revising with origins that differ from the capillary assumption-based in- classical nucleation theory to include complex pathways with tuition of competing geometric scaling between nucleus surface metastable molecular precursors that cannot be understood as areas and volumetric energies. Accordingly, these measurements microscopic clusters with bulk properties (6–10, 34). support recent suggestions (8, 11, 34) to unify both classical and nonclassical concepts into a comprehensive framework. Materials and Methods One model that is consistent with our observations is that di- Diamondoid Purification and Functionalization. Adamantane, diamantane, amond growth occurs via a two-step pathway with distinct nucle- triamantane, [121]tetramantane, [1(2)3]tetramantane, [12(1)3]pentam- ation and bulk transformation steps, as shown in Fig. 4B.Critical antane (racemic), and [1(2,3)4]pentamantane were isolated and purified diamond nucleation occurs through the formation of metastable, from extracts using previously reported protocols (28). Mass hydrogen-terminated diamondoid clusters before forming the bulk spectrometry, gas chromatography, and NMR spectroscopy were used to phase. The nucleation step follows the expected exponential size ensure diamondoid purity. Diamondoid molecules were then selectively dependence of classical theory, but with an interfacial energy and functionalized via multistep synthesis (29) to enable covalent attachment of diamondoid self-assembled monolayers on oxidized silicon surfaces. Ada- nucleation energy barrier that is much lower than ideal classical mantane was also functionalized with a 2(trichlorosilyl)ethyl group (SiCl3)at estimates that rely on bulk properties. The second, bulk trans- the 1-position to assess the impact of attachment chemistry under the milder formation step then occurs as diamondoid nuclei atoms react into remote plasma CVD growth conditions. Diamond growth densities were

bulk diamond atoms. independent of attachment chemistry for POCl2 and SiCl3 functionalization, From our results, we hypothesize that crystalline diamond grows further confirming that carbon etching and addition primarily determine when the chemical potential of carbon in the plasma phase is diamondoid nucleation energetics. closely matched to the surface energy of the protonuclei, rather than the ideal classical case where crystalline diamond would only Formation of Diamondoid Self-Assembled Monolayers. Diamondoid mono- form when the chemical potential of carbon in the plasma is closely layers were attached to plasma-activated silicon surfaces via the formation of bidentate P–O bonds between POCl2 groups and Si–OH surface groups or matched to that of bulk diamond. This picture agrees with and – – provides explicit experimental evidence for thermodynamic en- tridentate Si O bonds between SiCl3 groups and Si OH groups. Control ex- periments utilizing nonfunctionalized diamondoids were ineffective for ergy landscapes that support recent proposals of multistep nu- seeding diamond growth, highlighting the importance of covalent attach- cleation in diverse contexts, ranging from cloud formation (34) ment. Additional control experiments with POCl2 functionalized non- to protein crystallization (35), biomineralization (9), and nano- diamondoid carbonaceous monolayers, including linear and aryl particle synthesis (10). substituents, were ineffective for seeding diamond growth, confirming the Furthermore, a 26-carbon critical nucleus resolves an out- importance of diamondoid seeds. Further details of monolayer self-assembly standing question surrounding growth of diamond versus graphite and characterization are provided in SI Appendix, section 1.1. in PECVD reactors. Prior estimates of carbon critical nuclei being composed of several thousand atoms (33) imply that diamond PECVD Diamond Growth. High-temperature vertical growths were carried out nuclei should not form thermodynamically relative to graphite using a Seki Technotron PECVD system with a 2.45-GHz direct plasma mi- (17). However, hydrogen-terminated diamond is more thermo- crowave source (see SI Appendix, section 1.2 for full details). Nucleation and dynamically stable than hydrogen-terminated graphite for carbon growth experiments in the low supersaturation thermodynamic regime were carried out under the following growth conditions: 23.5 torr, 300 clusters of less than about 100 carbon atoms in size (36). Our standard cubic centimeters per min (sccm) H2, 3 sccm CH4, and 350 W for the measurement of a 26-carbon critical nucleus resolves this paradox first 15 min of nucleation and growth and 23.5 torr, 300 sccm H , 1.5 sccm 3 2 by showing that sp diamond-like particles are the favored state CH4, and 350 W for the last 45 min of growth. CH4 concentrations were at the critical size, consistent with ultra-nanocrystalline dia- reduced from 1 to 0.5% after the first 15 min to suppress diamond renu- mond growth (31), where sub–5-nm crystalline domain sizes are cleation while diamond particles were grown to a large enough size to frequently observed. enable rapid determination of diamond growth densities via SEM. De- These results also have significant implications to other in- creasing the feed gas concentration of CH4 to below 0.25% resulted in dustrial vapor deposition processes. For example, homoge- complete etching of the diamondoid seeds. neous nucleation of silicon and other semiconductors impedes Nucleation and growth experiments in the high-supersaturation kinetic the chemical vapor deposition (CVD) production of crystalline regime were carried out under the following growth conditions: 15 torr, 90 films(37),butthecleavageenergyofbulksilicon(38)isaround sccm Ar, 1 sccm CH4, and 300 W for 1 h. Increasing the feed gas concentration − − of CH to above 10% resulted in the exclusive growth of graphitic carbon. 1–2J·m 2, leading to a calculated 10 16-J barrier for homo- 4 Control experiments were carried out where silicon wafers were immersed geneous nucleation. This implies that nucleation should almost in pure toluene for at least 48 h and then used for vertical growth experi- never occur, which is in contrast to experimental observations. ments. Diamond nanoparticles were sporadically observed on unseeded Since the chemical nature of hydrogenated silicon and diamond wafers, as shown by the shaded area in Fig. 2C. However, these densities surfaces are similar (38), multistep models would predict a ranged from no observable diamond nanoparticles to a maximum density that −20 nucleation barrier on the order of 10 J for silicon. This was significantly lower than for adamantane seeding. The baseline density of

8288 | www.pnas.org/cgi/doi/10.1073/pnas.1803654115 Gebbie et al. Downloaded by guest on October 1, 2021 diamond particles observed with no seeding likely results from the stochastic ACKNOWLEDGMENTS. This work was supported by the Department of deposition of diamond nanoparticles that homogeneously nucleate in the Energy, Office of Basic Energy Sciences, Division of Materials Sciences plasma phase. This implies that homogeneous nucleation provides some of the and Engineering, under Contract DE-AC02-76SF00515. M.A.G. acknowl- inherent experimental variability present in the measured nucleation densities. edges the funding provided by the Geballe Laboratory for Advanced Ma- terials Postdoctoral Fellowship program at Stanford University. C.F. and M.N. acknowledge support of the Flemish Funds for Scientific Research Determination of Diamond Growth Densities. The density of diamond particles (FWO), Project G0E7417N (Single-Atom Diamond Quantum Probe: Proof- was measured via SEM and counted following each growth. Diamond nano- of-Principle Molecular Engineering Methodology and ERANET Project particles were grown to diameters of ∼0.5–1 μm to enable particle densities to Nanobit). Part of this work was performed at the Stanford Nano Shared be automatically processed and counted via the NIH ImageJ analysis software. Facilities, supported by the National Science Foundation under Award Full details can be found in SI Appendix,section1.3. ECCS-1542152.

1. Gibbs JW, Smith AW (1878) Equilibrium of heterogeneous substances. Trans Conn 20. Wrachtrup J, Jelezko F (2006) Processing quantum information in diamond. J Phys Acad Arts Sci 3:343–524. Condens Matter 18:S807–S824. 2. Becker R, Doring W (1935) Kinetic treatment of germ formation in supersaturated 21. Mochalin VN, Shenderova O, Ho D, Gogotsi Y (2011) The properties and applications vapour. Ann Phys 24:719–752. of nanodiamonds. Nat Nanotechnol 7:11–23. 3. Turnbull D, Fisher JC (1949) Rate of nucleation in condensed systems. J Chem Phys 17:71–73. 22. Gasser U, Weeks ER, Schofield A, Pusey PN, Weitz DA (2001) Real-space imaging of 4. Kelton K, Greer AL (2010) Nucleation in Condensed Matter: Applications in Materials nucleation and growth in colloidal crystallization. Science 292:258–262. and Biology (Elsevier, Amsterdam). 23. Tan P, Xu N, Xu L (2014) Visualizing kinetic pathways of homogeneous nucleation in 5. Wedekind J, Reguera D (2008) Kinetic reconstruction of the free-energy landscape. colloidal crystallization. Nat Phys 10:73–79. J Phys Chem B 112:11060–11063. 24. Sleutel M, Lutsko J, Van Driessche AES, Durán-Olivencia MA, Maes D (2014) Observing 6. Lutsko JF, Durán-Olivencia MA (2015) A two-parameter extension of classical nucle- classical nucleation theory at work by monitoring phase transitions with molecular ation theory. J Phys Condens Matter 27:235101. precision. Nat Commun 5:5598. 7. Legg BA, De Yoreo JJ (2016) The energetics of prenucleation clusters in lattice solu- 25. Zheng H, et al. (2009) Observation of single colloidal platinum nanocrystal growth tions. J Chem Phys 145:211921. trajectories. Science 324:1309–1312. 8. De Yoreo JJ (2017) A holistic view of nucleation and self-assembly. MRS Bull 42: 26. Nielsen MH, et al. (2014) Investigating processes of nanocrystal formation and 525–536. transformation via liquid cell TEM. Microsc Microanal 20:425–436. 9. De Yoreo JJ, et al. (2015) Crystallization by particle attachment in synthetic, biogenic, 27. Ross FM (2015) Opportunities and challenges in liquid cell electron microscopy. and geologic environments. Science 349:aaa6760. Science 350:aaa9886. 10. Loh ND, et al. (2016) Multistep nucleation of nanocrystals in aqueous solution. Nat 28. Dahl JE, Liu SG, Carlson RMK (2003) Isolation and structure of higher diamondoids, Chem 9:77–82. nanometer-sized diamond molecules. Science 299:96–99. 11. Smeets PJ, et al. (2017) A classical view on nonclassical nucleation. Proc Natl Acad Sci 29. Fokin AA, et al. (2014) Selective preparation of diamondoid phosphonates. JOrg USA 114:E7882–E7890. Chem 79:5369–5373. 12. Vekilov PG (2010) The two-step mechanism of nucleation of crystals in solution. 30. Tzeng Y-K, et al. (2017) Vertical-substrate MPCVD epitaxial nanodiamond growth. Nanoscale 2:2346–2357. Nano Lett 17:1489–1495. 13. Sleutel M, Van Driessche AES (2014) Role of clusters in nonclassical nucleation and 31. Gruen DM (1999) Nanocrystalline diamond films. Annu Rev Mater Sci 29:211–259. growth of protein crystals. Proc Natl Acad Sci USA 111:E546–E553. 32. Hwang NM, Hahn JH, Yoon DY (1996) Chemical potential of carbon in the low 14. Schreiber RE, et al. (2017) Real-time molecular scale observation of crystal formation. pressure synthesis of diamond. J Cryst Growth 160:87–97. Nat Chem 9:369–373. 33. Wang CX, Yang GW (2005) Thermodynamics of metastable phase nucleation at the 15. Lee S, et al. (2016) Multiple pathways of crystal nucleation in an extremely super- nanoscale. Mater Sci Eng Rep 49:157–202. saturated aqueous potassium dihydrogen phosphate (KDP) solution droplet. Proc Natl 34. Lupi L, et al. (2017) Role of stacking disorder in ice nucleation. Nature 551:218–222. Acad Sci USA 113:13618–13623. 35. Yau ST, Vekilov PG (2000) Quasi-planar nucleus structure in apoferritin crystallization. 16. Williams OA (2011) Nanocrystalline diamond. Diam Relat Mater 20:621–640. Nature 406:494–497. CHEMISTRY 17. Derjaguin BV, Fedoseev DV (1977) The Growth of Diamond and Graphite from the 36. Badziag P, Verwoerd WS, Ellis WP, Greiner NR (1990) Nanometre-sized are Gas Phase (Nauka, Moscow). more stable than graphite. Nature 343:244–245. 18. Angus JC, Hayman CC (1988) Low-pressure, metastable growth of diamond and 37. Stenberg P, et al. (2017) Silicon chemistry in fluorinated chemical vapor deposition of “diamondlike” phases. Science 241:913–921. silicon carbide. J Phys Chem C 121:2711–2720. 19. Kruis FE, Schoonman J, Scarlett B (1994) Homogeneous nucleation of silicon. J Aerosol 38. Hong S, Chou MY (1998) Effect of hydrogen on the surface-energy anisotropy of Sci 25:1291–1304. diamond and silicon. Phys Rev B 57:6262–6265.

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