Coagulative Transition of Gold Nanoparticle Spheroids into Monolithic

This paper deals with the structural changes of spherical aggregates composed of elemental gold nanoparticles, called spheroids. The spheroid diameter...
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Langmuir 2001, 17, 3863-3870

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Coagulative Transition of Gold Nanoparticle Spheroids into Monolithic Colloids: Structure, Lifetime, and Transition Model Eiki Adachi* L’Ore´ al Tsukuba Center, 5-5 Tokodai, Tsukuba, Ibaraki 300-2635, Japan Received December 27, 2000. In Final Form: April 3, 2001

This paper deals with the structural changes of spherical aggregates composed of elemental gold nanoparticles, called spheroids. The spheroid diameter is reduced to almost 70% of the original diameter by maintaining the spheroid suspension at a constant temperature ranging from 65 to 91 °C for 2-12 h (from 2 h at 91 °C to 12 h at 65 °C). The size-reduced spheroids become monolithic colloids of similar size at each temperature by the end of the reduction process. This is due to an irreversible transition, resulting from fusion among the nanocolloids in spheroids: coagulative transition. The 30% reduction of the diameter implies that one spheroid is initially composed of 67 nanocolloids at 0.26 volume fraction of the nanocolloids. Assuming that one spheroid gives one monolithic colloid (one-to-one coagulative transition), the lifetime of spheroids can be estimated from the dependence of the transition time on the suspension temperature, for example, 1 month at 22 °C and 6 months at 4 °C. A nonequilibrium thermodynamic model of the spheroid suspension describes the coagulative transition, suggesting that the exclusion and osmotic pressures against intercalated molecules in spheroids are additional factors keeping spheroids in water.

Introduction Metallic nanoparticles show characteristic physical properties originated in the nanometric volume, for example, nonlinear optical susceptibility,1-3 singleelectron tunneling,4-6 and abnormality of specific heat,7,8 differing from the bulk. These properties have been applied for catalysis9-11 and as sensor and optical devices12-14 by randomly depositing nanoparticles on solid substrates or dispersing them in media. The applications require each nanoparticle to be one functional unit. The number of functions is thus equal to the number of properties at most. Collective physical properties definitely play a major role in advanced nanoparticle-based devices. They originate in optical, electronic, and magnetic coupling among nanoparticles, which make it possible to obtain novel collective functions. The device applications are likely to be superior to the simple applications mentioned above. These devices are essentially made from regularly struc* Phone: +81-298-47-7984. Fax: +81-298-47-7985. E-mail: [email protected], [email protected]. (1) Ricard, D.; Roussignol, P.; Flytzanis, C. Opt. Lett. 1985, 10, 511. (2) Hache, F.; Ricard, D.; Flytzanis, C. J. Opt. Soc. Am. B 1986, 3, 1647. (3) Hache, F.; Ricard, D.; Flytzanis, C.; Kreibig, U. Appl. Phys. A 1988, 47, 347. (4) Devoret, M. H.; Grabert, H. In Single Charge Tunneling; Grabert, H., Devoret, M. H., Eds.; Plenum: New York, 1992; p 1. (5) Dorogi, M.; Gomez, J.; Osifchin, R.; Andres, R. P. Phys. Rev. B 1995, 52, 9071-9077. (6) Yau, S.-T.; Mulvaney, P.; Xu, W.; Spinks, G. M. Phys. Rev. B 1998, 57, 124-127. (7) Stewart, G. R. Phys. Rev. B 1997, 15, 1143. (8) Goll, G.; Lo¨neysen, H. v.; Kreibig, U.; Schmid, G. Z. Phys. D: At., Mol. Clusters 1991, 20, 329. (9) Vargaftik, M. N.; Zagorodnikov, V. P.; Stolarov, I. P.; Moiseev, I. I.; Kochubey, D. I.; Likholobov, V. A.; Chuvilin, A. L.; Zamaraev, K. I. J. Mol. Catal. 1989, 53, 315-348. (10) Schmid, G. Chem. Rev. 1992, 92, 1709-1727. (11) Toshima, N.; Yonezawa, T. New J. Chem. 1998, 22, 1179-1201. (12) Stietz, F.; Trager, F. Philos. Mag. B 1999, 79, 1281-1298. (13) Shipway, A. N.; Jatz, E.; Willner, I. ChemPhysChem 2000, 1, 18-52. (14) Reisfeld, R. Struct. Bonding 1996, 85, 101-143.

tured nanoparticles, which can be regarded as a collective functional unit. A two-dimensional (2D) array of nanoparticles is one of the typical regular functional structures made from stabilized nanoparticles (i.e., nanocolloids), for example, latex colloids for interferometric coloring,15,16 silver colloids for a tunable optical susceptibility,17 semiconductor colloids for a tunable optical absorption,18 and magnetic colloids for storage.19 One colloid has 6 nearest neighbors in the case of a 2D hexagonal close-packed monolayer. The number of nearest neighbors roughly determines the strength of coupling among nanocolloids. A larger number will make the structures more collective in functions. A three-dimensional (3D) structure is favorable with respect to coupling because the number can be larger than 6. Thus, dimensional expansion of regular structures, from 2D to 3D, is an appropriate direction to make the coupling stronger. However, the expansion may cause structural instability in ordinary environments, even though a 3D structure can fortunately be formed in a laboratory. Therefore, it is important to evaluate the stability of the 3D structure prior to further developments. A 3D assembly of elemental gold nanocolloids was recently synthesized by reducing an aurate solution in the presence of mercaptoacetate, because of subsequent dialysis of the nanocolloid suspension in water.20 This 3D assembly, a spheroid, is one example of the dimensional expansion and a promising 3D structure for nanoparticlebased devices. Spheroids seem to sustain most of their original form and size in water for several weeks at room temperature. However, temporal stability has not been quantitatively evaluated yet. (15) Dushkin, C. D.; Nagayama, K.; Miwa, T.; Kralchevsky, P. A. Langmuir 1993, 9, 3695-3701. (16) Dimitrov, A. S.; Nagayama, K. Langmuir 1996, 12, 1303-1311. (17) Collier, C. P.; Saykally, R. J.; Shiang, J. J.; Henrichs, S. E.; Heath, J. R. Science 1997, 277, 1978-1981. (18) Maenosono, S.; Dushkin, C. D.; Saita, S.; Yamaguchi, Y. Jpn. J. Appl. Phys. 2000, 39, 4006-4012. (19) Sum, S.; Murray, C. B.; Weller, D.; Folks, L.; Moser, A. Science 2000, 287, 1989-1992. (20) Adachi, E. Langmuir 2000, 16, 6460-6463.

10.1021/la001809s CCC: $20.00 © 2001 American Chemical Society Published on Web 05/18/2001

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Time-dependent reduction of the spheroid diameter is measured by holding the spheroid suspension at various constant temperatures using dynamic light scattering and a transmission electron microscope to evaluate the stability. In this report, structural features and the lifetime of spheroids are presented by analyzing reduction processes. A nonequilibrium thermodynamic model of spheroid suspension is introduced to describe the processes. Materials and Methods Spheroid Synthesis. Hydrogen tetrachloroaurate(III) tetrahydrate (HAuCl4‚4H2O, MW ) 411.8), sodium borohydride(NaBH4, MW ) 37.8), and sodium mercaptoacetate (HSCH2COONa, MW ) 114.1) were purchased from Wako Chemical Co., Ltd., Osaka, Japan. They were dissolved in pure water (MilliQPlus, Millipore Inc., Bedford, MA) to prepare 0.53 mM HAuCl4‚4H2O, 0.15 M NaBH4, and 20 mM HSCH2COONa aqueous solutions. One milliliter of the sodium mercaptoacetate solution was added to 94 mL of the aurate solution in a Pyrex beaker while stirring; 5 mL of NaBH4 solution was then immediately added. The solution quickly turned from transparent light yellow to brown, indicating the synthesis of elemental gold nanocolloids. The solution pH went from 3.5 to 5.0 at the start of reduction. The concentrations of HAuCl4‚4H2O, NaBH4, and HSCH2COONa eventually became 0.5, 7.5, and 0.2 mM, respectively. The total volume was 100 mL. The solution was first filtered using a syringe filter (0.2 µm pore, polyvinylidene fluoride, Whatman Inc., Clifton, NJ) to remove large contaminants. It was subsequently dialyzed overnight in 2 L of pure water to obtain the spheroid suspension, using a dialysis tube (Spectra/Por MWCO ) 6-8000, Spectrum Medical Industries Inc., Gardena, CA). The suspension pH finally became 6 by the end of the dialysis. Dynamic Light Scattering. Temporal changes of spheroid diameter were measured by dynamic light scattering (DLS) using the apparatus DLS/SLS-5000 with a 632.8 nm He-Ne laser (ALV Co., Ltd., Langen, Germany). The measurements were carried out in water at a constant temperature. The spheroid suspension was again filtered by using the syringe filter. A volume of the suspension, ∼3 mL, was then injected into a cylindrical Pyrex tube whose top was loosely capped, indicating that the suspension was in an isobaric state. The tube was set to the sample holder of the DLS apparatus whose temperature was adjusted in advance to remain constant. The measurement of the autocorrelation function was immediately started, with an appropriate interval of 30-240 s. The duration time for measuring each correlation function was 30 s. The functions were automatically saved for subsequent determination of diameter. Spheroid diameters were obtained from the decay time of autocorrelation functions using the Stokes-Einstein relationship (the fitting range: 2-64 µs). The size distribution was evaluated by the Laplace transform of the autocorrelation functions. Because the suspension temperature became constant in 10 min, the diameter change was reliable after 10 min. This measurement was independently conducted at 91, 83, 78, 74, 70, and 65 °C for 1, 3, 5, 7, 10, and 12 h, respectively. Evaporation of water reduced less than 10% of the suspension volume at all temperatures, indicating that the number density of spheroids was almost constant. Transmission Electron Microscopy. A volume of spheroid suspension was gently sampled from the tube during DLS measurements at specified time intervals. The sampled suspension was dried directly on grids for transmission electron microscope (TEM) observation. The grids were covered with collodion and carbon films in advance. The spheroids on grids were observed by using TEM with a liquid nitrogen trap at 80 kV acceleration voltage (Leo EM902, Leo Electron Microscopy Ltd., Oberkochen, Germany). The exposure time was restricted to less than 1 min to avoid fusion between colloids, caused by the electron beam irradiation. The images were digitized at a spatial resolution of 640 × 480 with 256 gray-scale.

Results The original spheroids are spherical aggregates of elemental gold nanocolloids (Figure 1a). The inset is an

Figure 1. (a) TEM image of spheroids sampled from their suspensions at 25 °C. Each spheroid is composed of black dots representing the elemental gold nanocolloids. (b) Distribution of decay time. This is the Laplace transform of the correlation function obtained from the DLS measurement at 25 °C. The single peak at ca. 0.1 ms corresponds to a spheroid diameter of 29.5 nm.

image of one spheroid in which small black dots represent the elemental gold nanocolloids. The average diameters of the spheroids and the elemental gold nanocolloids are 23.4 ( 5.6 nm and 3.0 ( 0.5 nm, respectively, according to the TEM images. The Laplace transform of the autocorrelation function shows a single peak at ca. 0.1 ms of decay time, corresponding to 29.5 nm of diameter at 25 °C (Figure 1b). The spheroid diameter obtained from DLS measurements 2RDLS falls from 29 to 20 nm in 4 h at 74 °C, then remaining almost constant (Figure 2a). The Laplace transforms in Figure 2b show single peaks for each labeled point in Figure 2a. The TEM images of spheroids at the labeled points give information about time-dependent changes in the internal structures of spheroids (Figure 3a-d). Images a-d correspond to the labeled points 1-4 in Figure 2a. Initially, the elemental gold nanocolloids constituting one spheroid can be clearly distinguished as black dots (Figure 3a). The nanocolloids then start to fuse with the neighboring nanocolloids in the spheroid (Figure 3b). The fusion decreases the spheroid size, although the constituents become larger (Figure 3c). Finally, the spheroids become the monolithic gold colloids (Figure 3d). The schematic drawing below the photographs

Transition of Gold Nanoparticle Spheroids

Figure 2. (a) The reduction process of the spheroid diameter in water at 74 °C. RDLS is the spheroid radius obtained from DLS measurements by the Stokes-Einstein relationship. A volume of the spheroid suspension was sampled for TEM observation at each labeled point 1-4. (b) Distributions of decay times at the labeled points.

illustrates these internal changes (Figure 3). The size reduction process is an irreversible transition of spheroids into monolithic colloids, because of the fusion. The spheroid diameters measured from the TEM images, 2RTEM, are always smaller than DLS diameters, 2RDLS, at the labeled points in Figure 2a (Figure 4). The same transitions are also observed at 65, 70, 78, 83, and 91 °C with different transition times (Figure 5). From the plots in Figure 5, the lifetime is defined as the time required for a 15% reduction of the diameter from its original value, that is, half of the total reduction. The relationship between lifetime tl [min] and the suspension temperature T [°C] can be fitted by the single-exponential function (tl ) 630 exp(-(T - 65)/10)) (Figure 6). The lifetime of a spheroid at a given temperature can be estimated from this relationship. Discussion Structural Features. According to the TEM images, the spheroids are slightly dispersed in size aggregates (Figure 1a). Free elemental gold nanocolloids are hardly found in the images. It indicates that the difference in chemical potentials of the nanocolloid in water and in the spheroids is large. Strictly speaking, the free nanocolloids and spheroids coexist at an equilibrium condition. However, the large difference makes it possible to regard the spheroid as a stationary structure. The single peak of decay time appears to support monodispersity of spheroids in water (Figure 1b). This is an illusion from the logarithmic axis of decay time. The width of the peak covers the size dispersion in the TEM image. Nevertheless, it is possible to consider spheroids as monodisperse in water in the sense that the Laplace transform shows a single peak. The peak does not split into multiple peaks and almost maintains the constant width throughout the reduction

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process (Figure 2a and b). The degree of size dispersion of spheroids and monolithic colloids is almost the same in the TEM images (Figure 3a-d). These two facts suggest that the spheroids become monolithic colloids, keeping same degree of dispersion. The transition is named “coagulative transition,” which means solidification of spheroid. Large aggregates were not found in the TEM images, which show that each spheroid either keeps its original form or collapses into small aggregates during the coagulative transition. Here, it is assumed that the spheroid number is constant, corresponding to the first case. This means that each spheroid becomes a monolithic colloid, the assumption of one-to-one coagulative transition. This assumption concludes that one spheroid is composed of almost 67 elemental gold nanocolloids, calculated as the 3rd power of the ratio between the diameters of nanocolloids in a spheroid (3.0 nm) and a monolithic colloid (12.2 nm). The gold density in colloids is supposed to be a constant. The volume fraction of nanocolloids in one spheroid is also calculated as 0.26 corresponding to the 3rd power of the ratio between the diameters of the monolithic colloid (12.2 nm) and the original spheroid (19.1 nm). The volume fraction is less than 40% of that of the 3D closest packing (0.74). Therefore, the spheroid is depicted as a spherical aggregate composed of 67 elemental gold nanocolloids with a 0.26 volume fraction based on the above calculations. Each diameter was taken from TEM images at 74 °C (Figures 3 and 4). In the case where spheroids collapse, a spheroid is thought to be composed of at least 67 nanocolloids with a volume fraction larger than 0.26. In general, a spheroid can be considered as a spherical aggregate of loose-packed elemental gold nanocolloids. The assumption of one-to-one coagulative transition also makes it possible to estimate the number density of spheroids as 5.4 × 1014/100 mL solution, when the amount of aurate salt is constant throughout the synthesis process. This density gives 0.57 µm of averaged distance among spheroids in water. The diffusion coefficient of a 29 nm spheroid is 1.69 × 10-11 m2 s-1 at 25 °C, giving a root-mean-square distance of 5.8 µm for 1 s. This implies the existence of collisions among the spheroids (0.57 µm < 5.8 µm). Despite the collisions, spheroids do not appear to disintegrate into small fragments or to form larger aggregates. Their degree of dispersion is almost conserved during the coagulative transition. There are two possible collision types that sustain the degree of dispersion: In one case, spheroids collide with others exchanging momentum only. In the other case, spheroids collide, exchanging momentum and nanocolloids. These are static and dynamic schemes, respectively. It is currently unknown which type of collision is dominant. Throughout the coagulative transition, the spheroid size in water is larger than that in a vacuum, according to DLS and TEM (Figure 4). This discrepancy probably results from two main reasons. The first one may come from the shrinkage of the spheroids in a vacuum, because they seem to be soft aggregates. The second reason comes from the electric double layer around spheroids and monolithic colloids in water, which should be charged; otherwise they would sediment. The diffusion coefficient of a charged spherical colloid is smaller than that of a noncharged colloid, because of the counteraction from the double layer around the moving colloids.21,22 The diameter (21) Ohshima, H.; Healy, T. W.; White, L. R.; O’Brien, R. W. J. Chem. Soc., Faraday Trans. 2 1984, 80, 1299-1317. (22) Ohshima, H.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1984, 80, 1643-1667.

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Figure 3. TEM images taken from the labeled points in Figure 2a. The spheroids gradually lose their internal structure as the size reduction progresses and finally become monolithic colloids. The drawing schematically reconstructs the stages of transition seen in the photographs as follows: (a) initial state of spheroids composed of elemental gold nanocolloids at a distance from each other; (b) start of transition due to fusion among nanocolloid neighbors; (c) progressing transition, where the spheroid size decreases and the size of elemental nanocolloids increases; (d) final state of transition into monolithic gold colloids.

Figure 4. Relationship between the spheroid diameters obtained from DLS (2RDLS) and from TEM images (2RTEM). The labels (1-4) are those in Figure 2a. The relationship is RTEM ) 0.68RDLS - 1.74 (R ) 0.9997).

of a charged colloid is thus always larger than that of a noncharged one, as far as the Stokes-Einstein relation-

ship is applied. Therefore, the size obtained from DLS is larger than that from TEM images. Lifetime of Spheroids. The linear relationship between RTEM and RDLS makes it possible to discuss the relative variation of the spheroid diameter. As shown, it is reduced to almost 70% of the original diameter at every temperature by the end of the process, according to DLS measurements (Figure 5). The respective time series of TEM images shows the disappearance of boundaries between the elemental gold nanocolloids in one spheroid, which indicates temperature-dependent coagulative transition. This implies that the spheroids are in a metastable state. The lifetime of spheroids at a given temperature can be estimated from the relationship between transition time and temperature, for example, 1 month at room temperature (22 °C) and 6 months in a refrigerator (4 °C) (Figure 6). Such a long lifetime indicates that a spheroid is actually a stable 3D structure in water. Model of Coagulative Transition. It is again supposed that each spheroid becomes one monolithic colloid

Transition of Gold Nanoparticle Spheroids

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Figure 5. Coagulative transitions at various temperatures: 65, 70, 74, 78, 83, and 91 °C. The size of the final monolithic colloids is almost the same, although the transition times are different.

Figure 7. (a) Mass and heat transports from inside to outside a spheroid. The large circles inside the dotted circle represent elemental gold nanocolloids; the small black dots at the end of the arrows represent the molecules that were originally intercalated among the nanocolloids. The dotted lines trace the path of mass transport, and the large arrows around the spheroid indicate the heat transport. (b) Entropy production in a spheroid suspension. The left scheme shows the spheroid ensemble before the coagulative transition; the right one shows the final state of the transition. The spheroids can distribute more randomly as they become smaller, thus increasing the entropy.

the three phenomena lead to the following expression describing the process (see Appendix). Figure 6. Lifetime of spheroids versus temperature. The black circles represent the experimental data. The solid line is the fit described in the text (tl). This shows the lifetimes of 34 days at 22 °C and 195 days at 4 °C.

via volume reduction. This is the assumption of one-toone coagulative transition. It is reasonable to assume that this transition is accompanied with three phenomena: two kinds of transports arising from the transition of each spheroid and an entropy effect arising from the volume reduction of all spheroids in the entire suspension. The first two phenomena are transports of heat and mass from inside to outside the spheroid (Figure 7a). The surfaces of nanocolloids in a spheroid disappear, because of fusion. The surface energy thus decreases and changes into heat, which is exhaled to the outside during the coagulative transition. Simultaneously, spheroids exclude intercalated molecules among nanocolloids, for example, water, mercaptoacetate ions, and others. The reduced volume of spheroids corresponds exactly to the volume of excluded molecules. Thus, heat and mass are transported from inside the spheroids to bulk water, as the spheroids reduce their volume. The second phenomenon is entropy production, occurring in the spheroid suspension (Figure 7b). The reduction of spheroid volume increases the suspension entropy, because the size-reduced spheroids can distribute more randomly in water. The three phenomena do not cause the coagulative transition but result from it. However, the thermodynamic relations between

(

t ) A(T) ln x

)

( ( ))

1 - xf x - xf + B(T) 1 x - xf 1 - xf

m

where x ) (RS/R0S)3 and xf ) (RN/R0S)3 (1 g x g xf). t, x, xf, and T are time, a variable, a constant, and the suspension temperature. RS, R0S, and RN are the respective radii of the spheroid at given time, the initial spheroid (t ) 0), and the monolithic colloid (t ) ∞). A and B are coefficients depending on T. m is supposed to be a constant during the transition, representing the complexity of internanocolloid space in a spheroid. This expression agrees well with the experimental curves of reduction processes at appropriate values of A, B, and m that were obtained from least-squares fittings (Figure 8). Here, the actual value of the averaged spheroid radius in water was supposed to be equal to RDLS. The second term on the right-hand side (tB in Figure 9) almost outlines the curve with the first term (tA) merely contributing to the curve tail at a given temperature. Therefore, the profile of coagulative transition is determined mainly by the second term. It is still unclear which of the phenomena described above is most responsible for the temperature dependence of the coagulative transition. According to the fittings, the coefficient A takes small positive values with large errors, which makes it difficult to recognize the temperature dependence. On the other hand, the coefficient B logarithmically increases as the temperature decreases. m slightly varies in the range of 3-4.5, as the tem-

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Figure 8. Theoretical fits (the solid curves) representing the experimental data for 2RDLS at 65, 70, 74, 78, 83, and 91 °C. The values of fitting parameters A, B, and m are obtained by the least-squares method. Figure 10. Temperature dependencies of A, B, and m. B is fitted by the empirical formula described in the text (the solid line).

Figure 9. Contributions of first (tA) and second (tB) terms to the theoretical fit (A ) 7.86, B ) 231, and m ) 4.38). RS and R0S are spheroid radii at a given time and at t ) 0, respectively. The dotted lines correspond to tA and tB; the solid line corresponds to tA + tB. The hollow circles represent experimental data at 74 °C for comparison.

perature increases (Figure 10). The first term of the expression represents the ratio between entropy production and heat transport (see Appendix). The entropy production is explicitly independent of the suspension temperature, as is the configuration number of spheroids. A weak temperature dependence of A in the specified range (65-91 °C) may result from the heat transport but not from the entropy production. Therefore, the heat transport is independent of temperature or only weakly dependent on it. The second term expresses the ratio between mass and heat transports (see Appendix). The logarithmic dependence of B on T suggests that the mass transport strongly depends on the temperature. In conclusion, the mass transport is the phenomenon that is most responsible for the observed temperature dependence of the coagulative transition in the frames of our model. B is fitted by t ) 636 exp(-(T - 65)/12)) where t is in min and T is in °C (Figure 10). Coincidentally, this is almost equal to tl (Figure 6). This agreement confirms

that the mass transport determines the lifetime of spheroids in water. However, this statement needs to be clarified, because the transport itself results from the coagulative transition. The lifetime of spheroids is determined by thermal motion of nanocolloids in a spheroid and their fusion induced by the motion. Thus, the molecules intercalated among fusing nanocolloids are excluded, causing the mass transport toward the outside of the spheroid. This leads to the hypothesis of exclusion pressure with respect to the intercalated molecules. Meanwhile, the osmotic pressure would probably resist the exclusion of the same molecular species. Although the balance between repulsion and attraction among nanocolloids basically forms a spheroid, this model indicates that the exclusion and osmotic pressures play an additional role in keeping the aggregates of nanocolloids stable. Two of the factors described above, exclusion and osmotic pressures, as well as the mass transport should be introduced to an advanced model for further understanding of the coagulative transition. Conclusions Spheroids composed of elemental gold nanocolloids are found to become monolithic colloids in water, which is named coagulative transition. Structural features and the lifetime of spheroids can be estimated from the coagulative transition. The transition time depends exponentially on the temperature, thus determining the practical lifetime of spheroids. This indicates that a spheroid is, in fact, a stable three-dimensional structure in water. A theoretical model including heat and mass transports and an entropy effect can describe the coagulative transition. The most important phenomenon for the transition is the mass transport from inside to outside the spheroids, resulting from thermal motion of nanocolloids and their fusion. This observation suggests that the exclusion and osmotic pressures are additional counterbalancing factors, keeping the aggregate of elemental gold nanocolloids spherical in water.

Transition of Gold Nanoparticle Spheroids

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and II. Because it should be zero for an isolated system, the differential is II

dH0+I+II )

dHj ) 0 ∑ j)0

(1)

where dH0+I+II ) 0, dH0 ) ∆0Q, dHI ) ∆IQ + ∆Iq + µI dNI, and dHII ) ∆IIq + µII dNII. Constant pressure is assumed in all regions (dPj ) 0), because the coagulative transition is an isobaric process. µI and µII are the chemical potentials of one water molecule in regions I and II, respectively. Equation 1 is a concise expression of energy conservation. Entropy Production. The entropy differential for the entire system dS0+I+II is the sum of the entropy differentials of all regions: II

dS0+I+II ) Figure 11. Thermodynamic model of the spheroid suspension. The numbers indicate different regions as follows: heat reservoir (region 0), water (region I), and spheroids (region II). Regions II, I + II, and 0 + I + II are open, closed, and isolated systems, respectively. TI is the temperature of region 0 and I; TII is that of region II. TII is supposed to be larger than TI for promotion of the heat transport from region II to region I. ∆qI and dNI are the amounts of heat and the number of water molecules that region I obtains from region II per unit time. ∆qII and dNII are the respective quantities lost from region II per unit time. ∆Q0 is the heat that region 0 obtains from region I. ∆QI is the heat loss from region I.

Acknowledgment. The author thanks Dr. K. Yase for his support in TEM (National Institute of Material and Chemical Research, 1-1 Higashi, Tsukuba, Ibaraki 305-8565, Japan). Appendix A spheroid suspension is divided into two regions: water (region I) and spheroids (region II). The suspension is in contact with a heat reservoir (region 0) whose temperature TI is constant (Figure 11). Hence, region I + II and region II are closed and open systems, respectively. The entire system, region 0 + I + II, is an isolated system. The temperature in region I is supposed to be equal to TI. An ideal sphere replaces the spheroid in order to efface the details of the fusion mechanism of nanocolloids inside. All spheres are supposed to exhale molecules and thus reduce the total volume VS () NSvS where NS and vS are the sphere number and the unit volume). The number of molecule compounds is set to 1 for simplicity, for example, only water molecules. The sphere temperature TII is supposed to be constant during the reduction, indicating that heat is constantly transported to region I from region II (TII > TI). Consequently, region I acquires an amount of heat ∆qI and a number of water dNI from region II. The latter simultaneously loses heat ∆qII and water dNII. The total amount of heat transport from region I + II to region 0 is ∆QI. The hypothesis TII > TI is essential in the subsequent derivation for the introduction of heat transport. The finite difference in temperature indicates that this model is a nonequilibrium thermodynamic model. “d” indicates an exact differential, and “∆” indicates that heat is not a function of state. Thermodynamic Energy Conservation. The enthalpy differential of the entire system dH0+I+II is the sum of the enthalpy differentials for each region dHj, j ) 0, I,

dSj ∑ j)0

(2)

dS0+I+II is actually the internal entropy production for the isolated system, because heat and mass are not transported across the system. Each entropy differential is divided into internal entropy production and entropy transport.

dSj ) diSj +

dHj - µj dNj Tj

(3)

The first and second terms on the right-hand side of eq 3 are the internal entropy production in region j and the entropy transport to the same region from other regions (subscript i means internal). Applying the assumption of II diSj ) 0 to eqs 2 and 3, one can write local equilibrium ∑j)0 II

dS0+I+II )

∑ j)0

dHj - µj dNj

)

Tj

dH0

+

dHI

TI

+

TI

dHII TII

-

µI dNI TI

-

µII dNII TII

(4)

Equation 4 is simplified by using dH0 + dHI ) -dHII (eq 1):

dS0+I+II )

(

)

µI dNI µII dNII 1 1 dHII (5) TII TI TI TII

Substituting dHII ) ∆IIq + µII dNII in eq 5 derives the Clausius definition for the entropy differential,

dS0+I+II )

(

)

µII - µI 1 1 ∆qII dNII TII TI TI

(6)

where the total number of water molecules in regions I and II is supposed to be a constant (dNI + dNII ) 0). The two terms on the right-hand side are the heat and mass transports, respectively. Equation 6 means that the entropy of the entire system is produced by the heat and mass transports from region II to region I. For further developments, dS0+I+II should be expressed by using known parameters. The spheres distribute more randomly in water as their volume VS is reduced. Consequently, the statistical entropy of the isolated system increases as the spheres become smaller. This increment can be calculated

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from the change in the configuration number of spheres in water, using a grid model (an isolated system is a microcanonical system). The statistical entropy differential of the entire system dSStat is equal to the differential of entropy of mixing d(∆MS) between water and spheres, because of the assumption of local equilibrium. This differential is

dSStat ) d(∆MS) ) -kBNS d

((

) (

)

)

VS V0 - VS V0 - VS ln + ln (7) VS V0 V0

where V0 and kB are the suspension volume and Boltzmann constant, respectively. Equation 7 is approximated as

dSStat ≈ -kB

NS dVS VS

)

(

)

NS µII - µI 1 1 ∆qII ) kB + dVS TI TII VS vWTI

(9)

dVS + vW dNII ) 0 is used because the volume reduction dVS matches the water exclusion vW dNII (vW is the effective volume of one water molecule in region II). The left term is the heat transport. The two terms in the right bracket are the entropy production and mass transport. Thermodynamics gives only the relation between ∆qII, µII - µI, and VS (eq 9). ∆qII and µII - µI should be defined as a function of VS based on special models in the following sections. Heat Loss in Region II. The fusion among nanocolloids in a spheroid generates heat, because nanocolloid surfaces disappear. The disappearance during a time period dt (t is time) is first thought to be proportional to the entire surface of nanocolloids in the spheroid. Consequently, heat generation would be proportional to the entire surface, that is, equivalent to the ratio between the internanocolloid space VS - VN and the average distance among the nanocolloid surfaces h. Here, VN is the total volume of the nanocolloids in the spheroids, which is equal to that of monolithic colloids. h is supposed to be a constant during the transition; this means that the chemical composition on the nanocolloid surface is constant and independent of temperature. It is also assumed that a spheroid keeps its temperature constant, despite the heat generation. This means that heat is continuously transported from region II to region I, and the heat loss becomes equal to

VS - VN dt h

(10)

a is the efficiency of surface energy conversion into heat that is transported to the outside, being independent of VS. Thus, ∆qII becomes an exact differential by specifying how heat depends on quantity of state (VS). Difference in the Chemical Potentials. The motion of a water molecule from region II to region I is caused by the chemical potential difference µII - µI. The respective work is proportional to the average path length of transportation. The length is nonlinearly proportional to the internanocolloid space, because packing of the nanocolloids would complicate the space in a spheroid. It is considered to be proportional to mth power of the internanocolloid space.

(8)

because VS , V0 (VS ∼ 2.1 × 10-10 m3 and V0 ∼ 3 × 10-6 m3, according to experiments). Here, the differential of Clausius entropy is supposed to be equal to that of the statistical entropy. The relationship between ∆qII and VS is obtained by comparing eqs 6 and 8 as

(

∆qII ) -a

µII - µI ) b(VS - VN)m

(11)

Here, b and m are constants independent of VS. Equation 11 means that the mass transport from a sphere stops when VS becomes equal to VN. Expression of Coagulative Transition. The relationship between t and VS is derived by substituting eqs 10 and 11 into eq 9 and integrating eq 9:

( (

)

VS V0S - VN h kBTITII NS ln t) a TII - TI VN VS - VN V0S b

)

(VS - VN)m - (V0S - VN)m (12) mvWkBTI

Here, V0S is VS at t ) 0. The quantities TII, a, and b cannot be obtained in the frames of the sphere model outlined here. Equation 12 can be written simply as

(

)

( ( ))

1 - xf x - xf + B(TI) 1 t ) A(TI) ln x x - xf 1 - xf

m

(13)

where x ) (RS/R0S)3 and xf ) (RN/R0S)3 (1 g x g xf). x is a variable, and xf is a constant. RS, R0S, and RN are the radii of the spheroid at a given time, the initial spheroid (t ) 0), and the monolithic colloid (t ) ∞). The radii are related to the respective volumes of species by V ) 4πR3/3. A, B, and m are coefficients generally depending on TI. The respective expressions of A and B do not give any information about their temperature dependencies, because of unknown parameters TII, a, and b. Assuming that the radii are equal to those obtained from DLS measurements, the dependencies of A, B, and m can be determined from the experiments. LA001809S