n-Hexanol Enhances the Cetyltrimethylammonium Bromide

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n-Hexanol Enhances the Cetyltrimethylammonium Bromide Stabilization of Small Gold Nanoparticles and Promotes the Growth of Gold Nanorods Tilo Schmutzler, Torben Schindler, Tobias Zech, Sebastian Lages, Martin Thoma, Marie-Sousai Appavou, Wolfgang Peukert, Erdmann Spiecker, and Tobias Unruh ACS Appl. Nano Mater., Just Accepted Manuscript • DOI: 10.1021/acsanm.9b00510 • Publication Date (Web): 01 May 2019 Downloaded from http://pubs.acs.org on May 2, 2019

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ACS Applied Nano Materials

n-Hexanol

Enhances the

Cetyltrimethylammonium Bromide Stabilization of Small Gold Nanoparticles and Promotes the Growth of Gold Nanorods †

Tilo Schmutzler,

Thoma,

‡

Torben Schindler,

Marie-Sousai Appavou,

¶

†

Tobias Zech,

†

Wolfgang Peukert,

Tobias Unruh

†,k

Sebastian Lages,

‡

Martin

Erdmann Spiecker,

§

and

∗,†

†Institute for Crystallography and Structural Physics (ICSP),

Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstrasse 3, 91058 Erlangen, Germany. ‡Institute of Particle Technology (LFG), Friedrich-Alexander-Universität

Erlangen-Nürnberg (FAU), Cauerstr. 4, 91058 Erlangen, Germany. ¶Forschungszentrum Jülich GmbH, Jülich Centre for Neutron Science (JCNS), Outstation

at MLZ, 85747 Garching, Germany. §Institute of Micro- and Nanostructure Research, Friedrich-Alexander-Universität

Erlangen-Nürnberg, 91058 Erlangen, Germany. kMAX IV Laboratory, Lund University, P.O. Box 118, SE-22100 Lund, Sweden. E-mail: [email protected] Phone: +49 91318524189. Fax: +49 91318524182

1

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Abstract Gold nanorods (AuNRs) are of interest for many applications, since their absorption in the regime of visible light can easily be tuned by their exact shape. To produce these AuNRs a two-step synthesis, that starts from small seed particles, is used. These seed particles are stabilized by cetyltrimethylammonium bromide (CTAB), which forms micelles at the used concentration (0.1 mol L ). In this work, the inuence of the micelle morphology on the stabilization of these seed particles and the consequences on the formation of AuNRs is reported. The elongation of CTAB micelles by the addition of n-hexanol leads to much more stable seed particle dispersions and thus less polydisperse

AuNRs. In contrast, a higher number of micelles compared to pure CTAB dispersions results from the addition of n-pentanol. This promotes the formation of larger seed particles and leads to lower yields of AuNRs. The gold nanoparticles are characterized by UV-Vis-NIR absorption spectroscopy, transmission electron microscopy and smallangle X-ray scattering (SAXS). The morphology of the micelles has been determined by a combination of SAXS and small-angle neutron scattering (SANS). The experimental results were used to calculate the collision kinetics of seed particles using an improved approach of classical coagulation theory to consider the anisotropy of the micelles. The combination of these experiments with the calculations strongly supports the mechanistic model, that these gold seed particles are not stabilized by a CTAB bilayer but by the micelles itself. For the rst time, the inuence of the micellar size and shape on the stabilization mechanism of noble metal nanoparticles could be claried. Theses ndings contribute to the development of targeted design routes for distinct nanoparticle morphologies by the use of suitable dispersions.

Keywords gold nanoparticles, coagulation, stabilization, micelles, CTAB, SAXS, SANS TEM

2


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1

Introduction

Gold nanoparticles have been a focus in the scientic community for the past decades, especially because of their unique optical properties. plasmonic behavior and therefore their size and shape.

These are closely linked to their

1,2

Whereas isotropic particles like

gold nanospheres exhibit one distinct surface plasmon resonance (SPR) absorption band, anisotropic particles - like nanorods - typically absorb two dierent energy bands in the UVVis and NIR range of light.

3

These bands are characteristic for the existence of transversal

SPRs (TSPRs) and longitudinal SPRs (LSPRs) and can be inuenced by morphological changes of the particles.

4,5

The synthesis of well dened colloidal solutions with a narrow

distribution of the nanoparticle size is a fundamental criterion for various applications in plasmonics, photonics, drug delivery, bio(chemical) sensing and imaging.

69

Further applica-

tions in catalysis, cancer diagnostics and therapy are discussed within the scientic community.

1013

Figure 1: Schematic representation of the synthesis of gold nanorods (AuNRs) starting with the precursor HAuCl4 solved in water to form small seed particles after reduction with the reducing agent NaBH4 in the presence of cetyltrimethylammonium bromide (CTAB). The subsequent reduction of HAuCl4 using ascorbic acid (AscA) in the presence of AgNO3, CTAB and the seed particles leads to the formation of single crystalline AuNRs.

In this

work the stabilization of gold nanoparticles via attached CTAB micelles is discussed which is in contradiction to the previous assumption of a closed CTAB bilayer (bottom row).

3



Page 4 of 46

The common procedure to synthesize gold nanorods (AuNRs) in high yields, is the socalled seed-mediated growth approach, which is schematically depicted in Figure 1.

14

In this

synthesis, the precursor compound HAuCl4 is solved in water and chemically reduced using NaBH4 to form seed particles.

15,16

Afterwards, these seed particles are used to synthesize

AuNRs by anisotropic growth in the presence of additional HAuCl4.

17,18

Both reaction steps

involve the cationic surfactant cetyltrimethylammonium bromide (CTAB, C19H42BrN), which stabilizes the nanoparticles and inuences their shape.

17

Among others the nal AuNR

morphology depends on the size and shape of the seed particles used in the synthesis.

19

known that small seed particles lead to higher aspect ratios of the AuNRs and vice versa.

It is

20,21

Since the seed particles tend to age rapidly and thus get more and more undened in terms of their average size and shape, they need to be used within 2 hours after preparation to produce AuNRs in signicant yield and quality.

22

This aging seems to be driven by the coalescence

of seed particles leading to the formation of large and polydisperse gold nanoparticles which are no longer suitable for a controlled AuNR preparation.

23

It therefore would be a major

advantage to improve the stability of these seed particles, without impacting their potency for AuNR formation. Stabilizer molecules like CTAB can strongly aect the gold nanoparticle formation, growth and aging.

2426

However, the formation and stabilization mechanism of both the

seed particles and the AuNRs and especially the role of CTAB is not yet fully understood. Open questions are for instance: Why is the CTAB concentration needed to produce highquality AuNR dispersions that high? Why is a concentration of 0.1 stabilize the seed particles for a few days whereas 0.001

mol CTAB needed to L

mol 27 is enough to stabilize AuNRs? L

Does the structure and the amount of CTAB micelles inuence the seed particle formation or the subsequent AuNR growth? How does CTAB exactly stabilizes the gold nanoparticles, via the formation of bilayers

2731

or via the adsorption of micelles

3235

? The formation of

CTAB bilayers would lead to a partial transformation of CTAB micelles into bilayers. Another explanation would be that bromide ions adsorb at the AuNR surface in a rst step

4


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which has been predicted by theory.

32,34,35

In a second step the positively charged micelles

attach selectively to the negatively charged interface of the gold nanoparticles via Coulomb interactions.This might cause the transformation of the adsorbed micelles to bilayers at high

− 36 Br concentrations and at certain facets. This stabilization mechanism has recently been suggested for silver nanoparticles by a combination of small angle X-ray (SAXS) and neutron scattering (SANS).

33

As demonstrated by theoretical works,

32,34

the preferred adsorption of

bromide ions on {100} and {110} facets of gold nanoparticles might lead to an anisotropic coverage of the AuNRs by CTAB micelles.

Correspondingly, a dierent stabilization e-

ciency for these facets by the micelles might occur and thus aect the anisotropic growth of the AuNRs. Thus, it is prudent to say that also size and shape of the CTAB micelles are capable to inuence the formation and stabilization of gold nanoparticles. In this work, the impact of

n -alcohols on the formation and stabilization of seed particles

for the seed-mediated growth of AuNRs is presented with respect to the structural inuence of these alcohols on CTAB micelles. The most distinct eect could be observed for the addition of

n -pentanol (PeOH) and n -hexanol (HexOH) during the synthesis of seed particles.

alcohols strongly alter the structure of CTAB micelles

3741

the nucleation behavior and stability of gold seed particles.

These

which could be correlated to These alcohol modied seed

particle dispersions are well suited to control the yield of AuNRs among other morphologies in a subsequent gold nanorod synthesis. In recent studies, it has been observed that PeOH increases the number of CTAB micelles while not signicantly altering their shape.

42

The

incorporation of HexOH supports a strong elongation and thus a distinct anisotropy of these micelles.

43

In this work, the experimental results were completed by calculations about

the collision kinetics of seed particles and micelles to quantify the aggregation behavior. The classical model was therefore improved by a simplied consideration of the eect of anisotropic micelles on the aggregation of seed particles.

In this respect the calculations

describe the experimental ndings very well. Using a combination of UV-Vis-NIR absorption spectroscopy, transmission electron mi-

5



Page 6 of 46

croscopy (TEM), SAXS, SANS and calculations about the collision kinetics in these dispersions, we were able to correlate the structure of

n -pentanol

and

n -hexanol

micelles with the formation and stabilization of gold seed particles.

modied CTAB

Using this approach

we were able to strongly enhance the seed particle stability which enables a reliable AuNR synthesis. Furthermore, by this routine an advanced selectivity for the formation of AuNRs could be achieved.

2

Results and Discussion

2.1 Aging of gold seed particles CTAB stabilized seed particles were used to prepare AuNRs according to the seed-mediated growth procedure.

14,16,21

The idea that gold particles are not stabilized by CTAB bilayers but

assemblies of micelles implies that: the stabilization capability of the dispersion depends on the micellar structure and thus might be controlled by the morphology of the CTAB micelles. Co-surfactants are well known to aect the morphology of CTAB micelles whereby are probably the most common ones.

3741

Accordingly, the

hexanol were used in concentrations of 0.1

n-alcohols

n-alcohols

from methanol to

n-

mol to prepare gold seed particle dispersions. L

UV-Vis-NIR absorption spectroscopy has been used to evaluate the aging of these seed particles as a function of storing time. Spectra were recorded 2, 8, 24, 32 and 48 hours after

◦ preparation and storage at 30 C and are presented in gure 2. The aging of seed particles and thus the formation of larger particles (≥ 10 nm) can be detected by the formation of a distinct absorption band in the UV-Vis spectra around a wavelength of 530 nm,

44,45

which is accompanied by a decreased extinction at lower wavelengths

related to interband transitions

46,47

(c.f. arrows in Figure 2a). Both the decreasing absorp-

tion at small wavelengths and the formation of a distinct absorption band is characteristic for gold nanoparticles larger than approximately 10 nm.

48

All alcohols seem to inuence the

stability of seed particles. However, this behavior was not found to systematically depend

6


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Figure 2: UV-Vis-NIR spectra illustrating the temporal evolution of the absorption of seed particle dispersions (a, c, d) stabilized with CTAB (a), CTAB/PeOH (c) and CTAB/HexOH (d) micelles after 2 h (black), 8h (red), 24 h (green), 32 h (blue), 48 h (magenta) and 16 days (for HexOH only, orange). b) Summary of seed particle dispersions in CTAB solution

n-propanol n-hexanol (solid,

(solid, black) modied with methanol (dashed, dark blue), ethanol (dashed, grey), (dashed, yellow),

n-butanol

(dashed, brown),

n-pentanol

(solid, red) and

green) after 48 hours of storage.

on the chain length of the alcohol.

n-propanol, n-butanol

49

Furthermore, for short

n-alcohols

methanol, ethanol,

and unmodied CTAB solutions the aging of the seed particles has

been found to be hardly reproducible from batch to batch. Compared to the other

n-pentanol (PeOH) seems to support the aging of seed particles (c.f. n-hexanol

n-alcohols

Figure 2b). In contrast,

(HexOH) addition leads to deceleration of seed particle aging and to perfectly re-

producible reaction results.

Hence, this work focuses on these two alcohols.

The UV-Vis

spectra of unmodied CTAB stabilized seed particle dispersions exhibit a clear absorption band at about 530 nm after a storage time of 48 hours. This is linked to the consumption of the initial seed particles due to their coalescence to larger particles. This absorption band is weak and broad for all alcohol modied dispersions and almost absent in HexOH modied

7



solutions even after 16 days of storage (c.f.

Figure 2d).

Page 8 of 46

These results indicate that the

tendency of CTAB stabilized seed particles to almost completely change size within 48 hours can be tremendously decelerated by the addition of

n-alcohols.

In conclusion drastically

decelerated aging could be observed for both CTAB/PeOH and CTAB/HexOH dispersions, with the eect being considerably more pronounced for the latter mixture.

2.2 Inuence of aged seed particles on AuNRs Analysis by UV-Vis-NIR spectroscopy.

Seed particles characterized in the previous

section were used for a subsequent AuNR synthesis to study the inuence of the seed particle aging-state on the resulting AuNR morphology.

AuNRs without substantial amounts of

byproducts exhibit two absorption peaks in UV-Vis-NIR spectra: the transversal surface plasmon resonance (TSPR) band is situated at a wavelength of roughly

λT SP R = 520

whereas the longitudinal surface plasmon resonance (LSPR) band is located around

800

nm depending on the exact aspect ratio of the AuNRs.

50

nm,

λT SP R =

If particles with non rod-like

morphology are formed in distinct quantities, a third SPR absorption band around 580 nm can be observed which is superimposed with the TSPR of the AuNRs and is characteristic for nanocubes or nanospheres (c.f. Figure 3a,b).

5,51

If signicantly aged seed particles are used for the AuNR synthesis without alcohol, the

Figure 3:

UV-Vis-spectra of AuNR dispersions prepared using the seed particles characmol mol mol in aqueous 0.1 CTAB (a), 0.1 CTAB/PeOH (b) and 0.1 L L L HexOH/CTAB solution (c). terized in Fig.2:

8


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LSPR of the AuNRs is shifted towards smaller wavelengths, which is depicted in Figure 3a. Additionally, an increase of the TSPR can be observed indicating the formation of non rod-like particles. This behavior is very pronounced if conventional (CTAB-stabilized) seed particle dispersions were used for the AuNR synthesis. In contrast, seed particle dispersions with PeOH modied CTAB micelles lead to well dened AuNR dispersions for aging times up to 48 hours (c.f. Figure 3b). However, the peak at 520 nm is clearly a combination of the TSPR and the absorption band of non-rod-like byproducts in signicant quantities. The HexOH modied CTAB stabilized seed particles lead to a comparably low blue-shift of the LSPR and negligible amounts of byproducts even when stored for 16 days (c.f. Figure 3c). From the results of the UV-Vis-NIR absorption spectroscopy it can be concluded that the structure and structural changes of seed particles during aging strongly inuence the subsequent synthesis of AuNRs. Conventionally stabilized seed particles using CTAB drastically age within 32 hours and were not suitable to produce high quality AuNR dispersions. The addition of

n-pentanol

to the seed particle synthesis leads to more stable seed particles

but the preparation of AuNRs results in a lower yield of AuNRs and an increased amount of non rod-like byproducts.

The addition of

n-hexanol

leads to seed particle dispersions

with drastically enhanced stability, which can be used for AuNR synthesis even after several weeks of storing without compromising the yield and quality of rod-like particles. The full systematic study using all

n-alcohols up to n-hexanol has been performed and is summarized

in the supplementary information (c.f. Figure S1). Only for in seed particle stability could be observed. Longer

n-hexanol

n-alcohols

a signicant increase

were not tested in detail due

to the limited miscibility with water.

TEM analysis of AuNRs and byproducts.

AuNRs prepared with seed particles stored

for two hours in CTAB, CTAB/PeOH and CTAB/HexOH were found to have similar diameters (≈ 11 nm) and lengths (≈ 38 nm) which was quantied analyzing the TEM images(c.f. Figure 4). The quantitative analysis was performed with the imageJ software

9


52

and is pre-


Page 10 of 46

sented in detail in the supplementary information (c.f. Figure S2-S5 and Table S1-S3). A summary of these results is visualized in Figure 4b. Next to the AuNRs in high amounts two minority byproduct fractions known from literature could be observed in frequencies usually below 10%: single crystalline nanocubes (≈ 22 nm) and multi-twinned sphere-like particles (≈ 35 nm).

51

The use of CTAB stabilized seed particles leads to 84% rod-like par-

ticles and almost equal amounts for both byproduct fractions in the nal AuNR dispersions, which is in agreement to the results from UV-Vis-NIR spectroscopy. The use of

n-hexanol

modied seed particles seems to almost completely suppress the formation of the larger, spherical fraction by achieving a yield of 89% of AuNRs. In contrast,

n-pentanol

strongly

promotes the formation of multiple-twinned nanospheres at the cost of rod-like particles

Figure 4:

TEM images of AuNRs prepared using CTAB (a),

CTAB/PeOH (c) and

CTAB/HexOH (d) stabilized seed particles. The presented scale bars are 100 nm. Quantitative results for rod diameter, length and byproduct diameter (b): CTAB (blue circles), CTAB/PeOH (red squares) and CTAB/HexOH (green diamonds). Error bars indicate the standard deviation of Gaussian size distributions. The non rod-like particle fraction turned out to consist of two fraction: single crystalline cubes and multi-twinned spheres.

10


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with a sharply decreased yield of 69% compared to the other dispersions. That implies that the use of CTAB/PeOH stabilized seed particles lowers the morphological selectivity of the seed-mediated growth synthesis. This is clearly visible in the graphical summary in Figure 4b. The tendency to support the formation of spherical byproducts increases systematically with a longer chain length of the alcohol used during seed particle synthesis up to

n-pentanol

(c.f. supplementary information Figure S2h). CTAB/HexOH stabilized seed particles suppress the formation of these aggregates almost completely.

Additionally, the analysis of

signicantly aged CTAB stabilized particles indicates, that aging of the seed particles also promotes the formation of these multiple-twinned spheres (c.f. supplementary information Figure S3).

Whereas these spheres seem to be the product of aged seed particles, single-

crystalline cubic byproducts might be the result of the isotropic growth of the initial seed particles. The observed inuence of the use of

n-alcohols

during the seed particle synthesis on the

yield in the subsequent AuNR synthesis is extremely valuable to understand the inuence of aged seed particles on AuNR formation. In order to understand the underlying stabilization mechanism of the seed particles a detailed analysis of their size is needed.

In addition,

classical coagulation theory can be applied to describe the collision probability of these particles and the interaction with CTAB and alcohol modied CTAB micelles. Hence, the following section is about the characterization of the seed particles, CTAB, CTAB/PeOH and CTAB/HexOH micelles in dispersion.

2.3 Stabilization of seed particles studied by SAXS and SANS Small-angle scattering with X-rays (SAXS) and neutrons (SANS) is especially suited to study size and shape of structures in the nanometer-regime. To characterize gold nanoparticles, SAXS is the preferred method since the high electron density of gold provides a higher X-ray scattering contrast to the aqueous dispersion medium. If a deuterated dispersion medium

11



Page 12 of 46

Figure 5: a) SAXS (squares) and SANS (circles) data of CTAB micelles (black), (red) and

n-hexanol

n-pentanol

(green) modied CTAB micelles resulted in the morphological changed

micelles illustrated in d): the modication of CTAB micelles with PeOH does not aect the morphology of the CTAB micelles but the amount of formed micelles whereas the addition of HexOH results in strongly elongated micelles. Using these models, SAXS curves of gold seed containing micellar solutions were analyzed after 2 (b) and 36 (c) hours of storage at ◦ 30 C. The example of n-pentanol modied seed particle dispersions is depicted in b and c. In addition to the micelles one small (e) and one large seed particle fraction (f ) were detected and analyzed in dependence of storing time (2 h: solid, 36 h: dashed).

can be used, SANS is a powerful tool to study hydrogen containing compounds assembled in structures like CTAB micelles. Accordingly, the combination of SAXS and SANS is well suited to characterize nanoparticles stabilized by organic molecules.

24,27

This combination

has been chosen to reveal the mechanism behind the contrary behavior of CTAB/PeOH and CTAB/HexOH stabilized seed particle dispersions to characterize CTAB, alcohol modied CTAB micelles and gold seed particles. The purpose was to correlate the structure of these micelles with the morphology and stabilization of the gold seed particles. In Figure 5a the SAXS and SANS data of aqueous 0.1

mol CTAB, CTAB/PeOH and L

CTAB/HexOH dispersions, respectively, are depicted. These curves can be described by a tting model that assumes prolate core-shell ellipsoids with the semi-principal axis

12


a,

the

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equatorial semi-axis

b (b = 17.6 Å) and the shell thickness t (t = 7.0 Å) which is schematically

depicted in Figure 5d.

33.0

43

The semi-principal axis

Å) and CTAB/PeOH micelles (aCT AB/P eOH

a

is almost equal for CTAB (aCT AB

= 32.7

approximately 1.6 times more micelles are formed.

42

=

Å) but in the latter dispersion

In the CTAB/HexOH dispersion the

number of micelles is reduced to 80 %, however, a drastically elongated semi-principal axis

aCT AB/HexOH = 61.5

Å was formed (c.f. Figure 5d).

43

Since the number of micelles within the seed particle dispersions is roughly one thousand times higher than the number of seed particles, especially at high

Q the SAXS data is signi-

cantly determined by the micellar scattering pattern. This can be observed by comparing the SAXS data in Figure 5a with 5b and 5c. Seed particle dispersions in CTAB, CTAB/PeOH and CTAB/HexOH were measured with SAXS after 2 and 36 hours of storage, respectively. After considering the scattering of background and the micelles two spherical gold particle fractions could be detected in all tested seed particle dispersions.

The small fraction

was 2.2 nm in diameter 2 hours after preparation and storage in CTAB, CTAB/PeOH and CTAB/HexOH dispersion, respectively, with the lowest number of seed particles in CTAB dispersion (c.f. Figure 5e). Relative to this number 1.6 times more seeds in CTAB/HexOH and 2.1 times more seeds could be observed in CTAB/PeOH dispersions. The large particle fraction was around 30 nm in diameter and posses a volume fraction below 10 vol-% which results in roughly 2% of surface area of the large fraction compared to the small particles. According the low quantity of the large fraction, these particles will be neglected in the further discussion. After 36 hours of storage the smaller particles of CTAB and CTAB/PeOH stabilized seed dispersions aggregated to larger particles with a diameter of around 4.4 nm. In contrast, CTAB/HexOH stabilized seed particles did not change size nor particle number signicantly.

13



Page 14 of 46

2.4 Collision kinetics of seed particles Regarding the stabilization of seed particles by CTAB, it has been assumed for a long time that CTAB forms bilayers at the interface between gold nanoparticles and water.

2730

In the following part it will be shown, that a CTAB bilayer is not necessarily needed to explain a high temporal stability of gold nanoparticles. In this discussion it will be proposed that those are directly stabilized by CTAB micelles in agreement with collision kinetics. The experimental observations of the previous sections are going to be used to develop a model that describes the interaction of gold seed particles with CTAB, CTAB/PeOH and CTAB/HexOH micelles. For this purpose classical coagulation theory will be utilized to estimate the stability of gold seed particles by evaluating their collision rate including their collision probability which is determined by attractive and repulsive forces of the seed particles but also the micelles. The theory of coagulation kinetics as well as the respective equations are presented in the experimental section.

In the following, the formation of

clusters of gold nanoparticles and CTAB micelles will be considered as the main stabilizing eect as it has been proposed for silver nanoparticles.

33

A change of the micelle morphology

due to the adsorption at the gold nanoparticle surface as it has been proposed in literature for gold nanorods could not be detected.

36

Accordingly, size and shape of both micelles and

seed particles were taken into account as described in section 2.3. It can be assumed that the collision of two seed particles leads to coalescence and thus the formation of larger particles and the consumption of the small particles, which is in accordance to the SAXS data. Attachment of CTAB micelles at the bromide ion covered gold nanoparticle surface will drastically decrease the probability of a collision of two particles. The collision rate

Jr

for rapid coagulation can be described mathematically by:

Jr = −

with

n

4kB T n2 dn = dt 3η

,

denoting the particle concentration as a function of the time

14


(1)

t,

Temperature

T,

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Boltzmann constant

kB

η . Jr

and dynamic viscosity

in equation 1 is mainly depending on

the squared number density of the particles and the viscosity of the dispersion medium. The viscosity of the medium is strongly inuenced by the dispersed nanoparticles and thus mainly dependent on the amount of micelles in the seed particle dispersion.

Hence, the

existence of a high number of micelles in these dispersions naturally enhances the stability of seed particles. The relation between volume fraction medium with the viscosity

η0

φ

of the particles within the dispersion

has been described by Einstein for spherical particles:

53,54

η = 1 + 2.5φ... η0 for

φ below 0.1.

However, this equation cannot be applied to anisotropic particles like the el-

lipsoidal CTAB micelles. Hence, the viscosity perimentally.

55

(2)

CTAB dispersions (cCT AB

η of the micellar dispersions was determined ex-

= 0.1

mol ) possess a viscosity L

ηCT AB = 1.02

mPas

close to the value of water. Due to the increased volume fraction of micelles in CTAB/PeOH dispersions the value is slightly increased (ηCT AB/P eOH

= 1.23

mPas).

The anisotropy in

combination with the enhanced volume fraction of CTAB/HexOH micelles results in a distinct increase of the viscosity of these dispersions (ηCT AB/HexOH

= 2.58

mPas). It already

can be concluded from equation 1 that the stability of seed particles follows the trend CTAB < CTAB/PeOH < CTAB/HexOH due to the respective viscosity, which limits the mobility of the seed particles. To quantify the collision behavior of seed particles in these micellar dispersions the interaction potentials of the dierent particles need to be considered. The adsorption of dissolved bromide ions from CTAB on the seed particle surface would lead to negatively charged surfaces, which limit the particle collision possibility due to repulsive forces between the electrochemical double layers around the particles.

56

The bromide ions from CTAB are dissolved

in water and can adsorb to the surface of gold nanoparticles as it is described in literature. The resulting surface charge of the gold nanoparticles is approximately -1.4

15


e · nm−2

32

due to


Figure 6: Calculated total potential energy to-surface distance

H0

Vt

Page 16 of 46

kB T

divided per

and center-to-center distance

r

as a function of the surface-

between a) 2 hours (black) and 36

hours (red in CTAB/PeOH and blue in CTAB dispersion) stored gold seed particles and b) CTAB (blue), CTAB/PeOH (red) and CTAB/HexOH (green) micelles. In b) dashed lines

γH 0

according to

(grey),

r = 2(a + t)

represent the collision probability of two approaching ellipsoidal micelles the right axis of the plot. Vertical dashed lines indicate

r = 2(a + t) dependency of H0

r = 2(b + t)

for CTAB, CTAB/PeOH (red) and

for CTAB/HexOH micelles (green).

c)

Schematic representation of the

from the micellar orientation for CTAB

and CTAB/PeOH (left) and CTAB/HexOH micelles. (right).

the adsorption of bromide ions.

32

Also the micelles electrostatically repulse each other due

to their charged headgroups within the micellar shell. Van der Waals interactions counteracting the repulsive potential energy are especially high for gold and low for CTAB and alcohol modied CTAB micelles due to the values of the Hamaker coecient. to-surface distance

H0

57

The resulting total potential energies as a function of the surface-

of seed particles and micelles are depicted in Figure 6. The potential

energy of the 36 hours stored, 4.4 nm sized seed particles is much higher compared to the potential energy of the initial small particles.

The reason is the higher surface potential

of larger particles (210 mV) compared to smaller ones (166 mV) which leads to stronger repulsion.

58

The van der Waals attraction dominates the potential energy below

To evaluate the potential energy of micelles

Vt ,

account. As schematically depicted in Figure 6c

H0

Å.

their anisotropy needs to be taken into

H0

strongly depends on the orientation of

the ellipsoidal micelles at the same center-to-center distance function of

H0 = 3

r.

The potential energy as a

is very similar for all three micellar systems (c.f. supplementary information

Figure S15a). CTAB/PeOH micelles obtain a slightly lower surface charge density and thus a

16


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


smaller repulsive potential energy. For the determination of of the micelles was included via the collision probability

Vt

(c.f Figure 6b) the anisotropy

γH0 (r)

of two approaching micelles

according to:

r 2(a+t) b+t 2(a+t)

1− γH0 (r) =

1−

,

γH0 [r < 2(b + t)] = 1

(3)

γH0 [r > 2(a + t)] = 0

and the respective discussion in the experimental section (equation 32). as a function of

r

This probability

is also depicted in Figure 6b and illustrates that the repulsive potential

of the strongly anisotropic CTAB/HexOH micelles (aspect ratio:

Aell = 2.8)

starts already

at a center-to-center distance of around 12 nm. The reason is the enhanced probability of an orientation where

H0

is small and thus the repulsive potential energy comparably high.

In comparison, for CTAB and CTAB/PeOH micelles (Aell (CTAB, CTAB/PeOH) distinct interactions can be observed at around 9.5 nm which is slightly above

8.0nm

r = 2(a + t) =

representing a longitudinal orientation of these spheroids.

The total potential energies

Vt

depicted in Figure 6a,b were used to calculate the ratio

between the rate constant for rapid and slow coagulation for seed particles

W11 ,

micelles

W22

W =2

∞

exp

0

s

W.

This value has been calculated

and seed particles with micelles

Z

where

= 1.6)

ds Vt (s) kB T (s + 2)2

W12

according to:

,

(4)

denotes a dimensionless distance parameter that is described in detail in the exper-

imental section (c.f. equation 38). The values are summarized in Table 1 and can be used to already discuss some general observations.

W11

of the initial seed particles is equal in all

tested micellar dispersions due to the same size and surface potential of the formed seeds . The aging to larger particles leads to an enhanced surface potential and

Vt

(c.f. Figure

6) and thus a strong increase in the value of

W11 ,

probability of the aged particles. The value

1 of micellar collisions is very low for CTAB W22

resulting in a strongly decreased collision

and CTAB/HexOH micelles and higher for CTAB/PeOH micelles due to the lower surface

17



potential of the latter.

W12

Page 18 of 46

is below one in all calculations, which is a result of the attrac-

tive interaction between micelles and seed particles that increases the probability of these particles to collide.

Hence, the spontaneous formation of clusters between seed particles

and micelles can be assumed and the existence of isolated seed particles will be neglected in further calculations. Furthermore,

W11

is not assumed to reect the real collision probability

of seed particles since these values cover the interaction between isolated particles. The subsequent considerations were made by assuming that the aggregation behavior of seed particles is mainly determined by the interaction of seed particle-micelle-clusters. Hence, the potential energy between two clusters is a combination of the seed particle interaction and the micellar repulsion with a strong dependency on the relative orientation of these clusters. Distinct repulsion between two clusters can be expected at a center-to-center distance of

r = 2(a + t),

if the longitudinal alignment of the micellar part of the clusters is considered.

In this respect it is a useful simplication to think of these clusters as particles with the radius

a+t

and a certain collision probability of seeds and micelles within this interaction

volume. The volume fraction of seed particles volume

Vc

φ11

with radius

Rs

inside the clusters with

can be calculated:

φ11 =

Rs3 Vs = Vc (a + t)3

,

(5)

Figure 7: a) Temporal evolution of the relative seed particle number due to the collision of seed particle-micelle-clusters of 2 hours stored (solid lines) and 36 hours stored (dashed lines) seed particles stabilized with CTAB (black), CTAB/PeOH (red) and CTAB/HexOH (green) micelles.

b) Schematic illustration of the orientation dependency of the clusters

on the collision kinetics and the inuence of the anisotropy of the micelles. c) For better n comparison was normalized to the value of CTAB/PeOH stabilized seed particles. n0

18


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


where

Vs

denotes the volume of one seed particle. The interaction probability

dened that corrects the value of

φ211

can be

1 for the interaction of clusters compared to isolated W11

particles. The resulting equation for the collision rate of seed particles

Js =

4πKB T φ211 2 n = ks n2 3ηW11

Js :

(6)

was used to obtain the temporal evolution of the seed particle concentration relative to the initial particle concentration

n depicted in Figure 7. While the number of CTAB and n0

CTAB/PeOH stabilized seed particles drastically drops within a few days, CTAB/HexOH stabilized seed particles exhibit a strongly enhanced stability. The larger, aged particles in CTAB and CTAB/PeOH dispersions are even more stable which has already been discussed. A useful value to estimate the stability of particles is their half-life which was calculated to be roughly 6 days for CTAB, 3 days for CTAB/PeOH and 236 days for CTAB/HexOH stabilized seed particles.

As discussed earlier the synthesis in CTAB/PeOH yields the highest seed

particle number. According to equation 6 the collision of particles depends on the particle number squared.

Hence, the relative reduction of initial seed particles in CTAB/PeOH

dispersions is faster compared to CTAB dispersions but the overall number of initial particles is still higher which can be seen in Figure 7c. In this image the temporal evolution of the seed particle number relative to the initial number of CTAB/PeOH stabilized seed particles is depicted. The collision probability of seed particles between two clusters

Ws,11 was calculated

using the following equation:

φ2 1 = 11 Ws,11 W11

(7)

The calculated values are summarized in Table 1 and are very similar for CTAB and CTAB/PeOH stabilized seeds. value for

Ws,11

CTAB/HexOH stabilized seeds exhibit a 20 times larger

which underlines the eect of micellar anisotropy on the stabilization capa-

bility of the micelles. These calculations can be used to describe the experimental observations from UV-Vis-

19



Table 1:

Viscosity

η

Page 20 of 46

of CTAB, CTAB/PeOH and CTAB/HexOH dispersions and calcu-

ndim per volume and the volume φ22 within these dimers. Potential energy limited collision probability of gold seed particles W11 , CTAB micelles W22 and both with each other W12 and the collision probability of seed particles from two clusters Ws,11 . The latter one has been used to calculate the collision rate Js,11 of micelle-stabilized seed particles with the half-life t1/2 for slow coagulation. lated values for the number of seed particle-micelle-clusters

fraction of seed particles

parameter

−3

ndim / m η / mPas φ11 φ22 W11 W22 W12 Ws,11 Js,11 / m−3 s−1 t1/2 / s t1/2 / days

φ11

and micelles

CTAB

CTAB/PeOH

CTAB/HexOH

seeds 2 h 20 2.13·10

seeds 36 h 19 2.33·10

seeds 2 h 20 4.41·10

seeds 36 h 19 8.56·10

seeds 2 h 20 3.37·10

1.02

1.02

1.23

1.23

2.58

0.021

0.133

0.021

0.108

0.004

0.378

0.378

0.387

0.387

0.130

2.55·10

5

4.15·10

20

2.55·10

5

2.15·10

19

2.55·10

5

3.67·10

11

3.67·10

11

1.12·10

8

1.12·10

8

3.56·10

12

0.27 5.89·10 4.20·10

0.25

8

2.36·10

14 5

5.06·10 6

0.28

0.27

22

5.63·10

8

1.85·10

1.26·10

−1

1.85·10 15 2·10

20

1.57·10

15

2.81·10

5

3

0.26

21

1.49·10

10

1.80·10

1

1.66·10

13

4.76·10 13 6·10

18

2.04·10

7

236

NIR absorption spectroscopy very well. Additionally the results from SAXS can be explained, where 36 hours after preparation the scattering of CTAB and CTAB/PeOH dispersions was dominated by the aged, larger seed particles. On the other hand the scattering from CTAB/HexOH stabilized seed particles was not signicantly altered.

Especially the role

of the anisotropy of the micelles has been found to tremendously enhance the seed particle stability. In this respect the morphology of the micelles conrmed by SAXS and SANS could be successfully used for the calculation of the seed particle stability. In addition to understand the experimental ndings the proposed kinetic model can be used to predict the stability of seed particles in CTAB solutions with varying process parameters like HexOH concentration and temperature. The anisotropy of CTAB/HexOH micelles increases with the HexOH concentration. than 0.05

42

Above HexOH concentrations larger

mol the aspect ratio of the CTAB/HexOH micelles exceeds two. After the proposed L

model this dramatically increases the half life of the seed particles from a few days to many

20


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


months with increasing HexOH concentration (the results of these calculations are presented in the supplementary information in Table S12 and Figure S22). The stability of seed particles is expected to increase with decreasing storing temperature since the collision rate of the seed particles decreases according to equation 24 and 25. Additionally, more elongated micelles of CTAB and CTAB/HexOH are formed at lower temperatures, which can be studied using SAXS and SANS. The results of these structural investigations are presented in the supplementary information (c.f.

Figure S23, S24 and

Table S13). The structural information of micelles at dierent temperatures between 20 and 55

◦

◦

C

C were used to evaluate the half life of seed particles at dierent storing temperatures.

As a result in addition to the reduced collision rate of seed particles at low temperatures, the increased anisotropy of the CTAB and CTAB/HexOH micelles signicantly increases the stability of seed particles at temperatures below 30 are given in the supplementary information (c.f.

◦

C.

The results for these results

Table S13).

This is in accordance to

time-resolved UV-Vis-NIR absorption measurements of seed particle dispersions at dierent storage temperatures (c.f. supplementary information Figure S1). Hence, ideal results for preparing gold nanorods via the seed-mediated growth procedure can be achieved by using seed particles synthesized in a 0.1

mol solution of CTAB and HexOH and a further storage L

◦ at low temperature like at 6 C in a conventional laboratory fridge. Summarizing the mean statement of the previous discussion, one single CTAB or

n-

alcohol modied micelle is able to suciently stabilize one seed particle against coalescence as soon as this micelle posses a pronounced anisotropy like the CTAB/HexOH micelles. The model of adsorbed micelles stabilizing the seed particles rather than the existence of a closed bilayer of CTAB can be used to explain the subsequent growth to gold nanorods (AuNRs):

36

the anisotropic growth starts with the seed symmetry breaking, which is initiated by the preferred adsorption of CTAB micelles at the {100} and {110} facets of the gold crystals compared to the {111} facets.

32,35

A dense packing of the micelles at the {100} and {110}

facets diminishes the transport of gold precursor at these surfaces compared to the {111}

21



facets and thus the growth of the latter is preferred.

32,35,36

stabilized at a rather low CTAB concentration of 0.001 of 0.1

Page 22 of 46

The resulting AuNRs can be

mol compared to the concentration L

mol , which is needed to synthesis both seed particles and AuNRs. The reason is the L

repulsive potential between CTAB micelles adsorbed at the AuNR surface which is covered by negatively charged bromide ions compensating the van der Waals attraction of the gold particles.

The presented calculations indicate that this is already the case for aged seed

particles with a size of approximately 5 nm. The stability even increases for larger particles since more bromide ions and CTAB micelles can be adsorbed at the particle surface.

3

Conclusions

In this publication, a signicant increase in the seed particle stability for AuNR preparation has been demonstrated for the rst time. alcohols to the CTAB dispersion.

This was accomplished by the addition of

n-

It has been proven, that long-term stable seed particle

dispersions lead to more reliable synthetic conditions in the seed-mediated growth approach that enhance the morphological selectivity of the AuNRs preparation up to 89%. The combination of SAXS and calculations about the collision kinetics including the potential energies between seed particles and micelles, respectively, could be successfully used to quantify the aggregation behavior of CTAB stabilized seed particles. The calculated potential energy between gold seed particles and micelles indicates the formation of temporarily stable seed particle-micelle-clusters. The surface potential of larger particles (aged seed particles) is higher compared to small gold nanoparticles (initial seed particles) and thus the collision probability is reduced, since this repulsive potential is large enough to compensate the van der Waals attraction. Thus, AuNRs in a CTAB dispersion with concentration of 0.001

mol are stable for several months, L

while a comparably high CTAB concentration (c

mol ) is needed to synthesize and L

= 0.1

stabilize the gold seed particles.

22


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


It has been demonstrated that the incorporation of

n-alcohols

into CTAB micelles leads

to a change of the structure of the micelles with direct consequences on the stabilization capability of these dispersions. The use of CTAB/HexOH micelles that posses a relatively high aspect ratio of 2.8 strongly decelerate the coalescence of seed particles by the interaction of the seed particle-micelle-clusters.

The anisotropy of the micelles drastically limits the

probability to obtain two clusters in a certain relative orientation that is needed to bring the seed particles close enough to aggregate. In this work a detailed analysis of the aging behavior of gold seed particles stabilized by CTAB micelles has been used to understand the improved synthesis of these particles by the addition of

n-hexanol.

For the rst time theoretical considerations about the collision

kinetics of these seed particle-micelle-clusters were successfully applied to quantify the aggregation behavior and understand the stabilization mechanism by the interaction of seeds with micelles. The enhanced anisotropy of CTAB/HexOH micelles has a drastic eect on this stabilization. These ndings shed light on the general mechanism of structure formation and stabilization of nanoparticles in dispersions of anisotropic micelles.

4

Experimental

4.1 Particle Synthesis Chloroauric acid (HAuCl4 · 3H2O, 99.99%, metal basis), silver nitrate (AgNO3, > 99.9%, metal basis) and ascorbic acid (> 99.9%) were obtained from Alfa Aesar and used as received. Cetyltrimethylammonium bromide (CTAB, C19H42BrN, dride (NaBH4,

≥

>

99

%,

Sigma), sodiumboron hy-

98%, Merck KGaA), methanol (MeOH, > 99.9%, Roth), ethanol (EtOH,

99.98%, VWR Germany), 99.5%, Carl Roth),

n -propanol

n -pentanol

(PrOH, > 99.5%, Carl Roth),

(PeOH,

≥

99%, Sigma-Aldrich) and

99%, Sigma-Aldrich) were used without further purications.

n -butanol

n -hexanol

23

(HexOH,

≥

For the preparation of the

aqueous dispersions deionized water (Direct-Q 3 UV, Millipore) was used.


(BuOH, >

All gold seed


particle dispersions were prepared by the standard synthesis:

Page 24 of 46

19

of HAuCl4 · 3H2O in water were mixed with 7.5 mL of a 0.1 in these solutions, the

n-alcohol

concentration was also 0.1

0.2 mL of a 0.01

mol solution L

mol CTAB solution. If present L mol . L

The CTAB solution was

◦ heated to 30 C to completely dissolve the added CTAB and the reaction was started by the fast addition of 0.6 mL of a freshly prepared 0.01

mol aqueous NaBH4 solution. L

gold nanorod dispersions were prepared by the seed-mediated growth synthesis. approach 7.2 mL of an aqueous 0.1 HAuCl4 · 3H2O, 0.045 mL of 0.01

14

The

For this

mol mol CTAB solution was mixed with 0.225 mL of 0.01 L L

mol mol AgNO3 and 0.03 mL of 0.1 ascorbic acid all disL L

solved in water. The ascorbic acid solution was prepared freshly right before use. All stored solutions were kept in dark containers. The reaction started after the addition of 0.015 mL

◦ ◦ of the seed particle solution at 30 C. The AuNR dispersions were kept at 30 C over night and centrifuged at 9000 r/min and the sediment was re-dispersed with an aqueous 0.001

mol L

CTAB solution. This cleaning procedure was repeated twice.

4.2 UV-Vis-NIR Spectroscopy Absorbance spectra were recorded at 303 K in the wavelength range between 190 nm and 1000 nm using a TIDAS S 500 /MCS UV/NIR 1910 UV-Vis-NIR spectrometer (J & M Analytik AG, Germany). The samples were measured in silica glass cuvettes with 10 mm optical path length (Hellma Analytics, Germany).

4.3 Transmission Electron Microscopy (TEM) TEM investigations of AuNRs were carried out at room temperature using a Philips CM300UT operated at an acceleration voltage of 300 kV. To record the TEM images, a 2k x 2k CCD camera (Tietz FastScan-214) was used. Samples were prepared by drop-casting the colloidal dispersion on a carbon lm covered copper grid. After waiting until the solvent was completely evaporated this procedure was repeated twice. To quantify the particle morphology (length and diameter) the ImageJ software was used.

24

52

Plotting and tting of size distribu-


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


tions was done using QtiPlot. To achieve reasonable statistics, at least 200 particles were counted for each sample. The experimental data was tted by rod, sphere and cube models, respectively, assuming Gaussian distributions at the corresponding lengths and diameters:

(x−xc )2 A f (X) = √ e− 2σ2 σ 2π

where

A

denotes the area under the curve,

σ

(8)

the standard deviation and

xc

the location

parameter.

4.4 Small-angle X-ray Scattering (SAXS) SAXS images were collected at the SAXS beamline I911-4 at the MAXII synchrotron (MAX IV laboratory, Lund, Sweden) which was equipped with a bent Si (111) crystal (horizontally focusing) and a multilayer mirror in the vertical (R with a photon ux of

≈ 5·1010

= 400 m). 59,60

photons/s at the sample position was selected. The sample to

detector distance (SDD) was 2350 mm providing access to a The

Q

A wavelength of 0.91 Å

scale was calibrated using a silver behenate standard.

Q-range 61

from 0.01 to 0.3 Å

−1

.

A Pilatus 1M detector (Dec-

tris AG, Baden, Switzerland) was used for data acquisition. The setup was equipped with a ow through silica glass capillary with a mean diameter of 2 mm. All measurements were

◦ performed at 30 C. The scattering data was calibrated to the absolute scale of the dierential cross section using a water measurement standard.

62,63

The data has been corrected

for capillary thickness, acquisition time, absorption and ux. It was also corrected for background (where applicable) by the subtraction of a water measurement. Data reduction was performed using FIT2D.

64

All SAXS data tting was done using SASt.

65

4.5 Small-angle Neutron Scattering (SANS) SANS experiments were performed at the KWS-2 instrument of the Jülich Center for Neutron Science (JCNS) at the Heinz-Maier-Leibnitz Zentrum (MLZ) in Garching, Germany.

25


66

A


Page 26 of 46

2 beam of 8 x 8 mm at the sample position was chosen. Measurements were recorded at sample to detector distances (SDD) of 1400 mm and 8000 mm (collimation length: 8000 mm) to gather a of

∆λ λ

Q-range

= 20%.

from 0.006 to 0.469 Å

−1

. The wavelength was set to 4.54 Å with a spread

The data was corrected for background in the neutron guide hall using a

measurement with the beam being blocked by a boron carbide slab.

Detector sensitivity

correction and calibration to the absolute scale of the dierential scattering cross section was performed by using a polymethylmethacrylat (Plexiglas) standard sample. The data was reduced using QtiKWS.

67

SAXS and SANS data were tted simultaneously using SASt.

65

4.6 Analysis of SAXS and SANS data Because CTAB micelles and seed particles both contributing to the total scattering signal, the scattering contribution of CTAB micelles has to be described diligently to reliably evaluate the seed particle morphology. To consider this scattering contribution, the respective micellar dispersions (without seed particles) were measured as reference samples and subtracted by a measurement of pure water. In this respect the description of the residual seed particle scattering could be done more precisely.

Using the tting parameters from the reference

samples and a polynomial for the residual background scattering the scattering of the gold seed particles could be described.

The models that were used to calculate the respective

scattering contribution will be explained in the following. The modulus of the scattering vector and neutron scattering. the scattered wave.

Q

ki

Q

is dened as

Q = |Q| = |kf − ki |

is the wave vector of the incident wave and

kf

for both x-ray

the wave vector of

can be calculated by:

Q2 = |kf − ki |2 = kf2 + ki2 − 2kf ki cos (2θ) .

26


(9)

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


For elastic scattering we nd

k = kf = ki =

Q=

where

λ

2θ

2π : λ

4π 2π sin (θ) = λ d

denotes the scattering angle,

d

,

(10)

the real space distance corresponding to

Q

and

the wavelength of the neutrons and the x-rays, respectively. By detecting the scattered

neutrons and x-rays per second into the solid angle segment by the two angles section

χ

and

ϕ

dΩ

at a certain direction given

of a spherical coordinate system, the dierential scattering cross

dσ can be determined experimentally by normalizing this value by the incident ux dΩ

of the probe particles and the solid angle

dΩ.

The detector is located at a distance to the

sample so that all distances within the sample are small compared to the sample-to-detector distance (far eld approximation). The macroscopic scattering cross section

dΣ is given by: dΩ

dΣ 1 dσ = dΩ V dΩ and is obtained in terms of the scattering length density

(11)

ρb

at the position

Z 2 dΣ 1 dσ 1 iQ·ri 3 (Q) = = ρb (r)e d r dΩ V dΩ V V where

V

,

(12)

denotes the illuminated sample volume. The distribution of

to the atomic, molecular and mesoscopic structure of the sample.

69

ri : 68

ρb

is directly linked

Hence, equation 12 is

directly linked to the structure of the sample through the Fourier transform of the SLD distribution within the sample, where the obtainable length scale depends on the experimentally available

Q-range.

The scattering of neutrons and x-rays by a single, compact nanoparticle with well dened size, shape and internally constant SLD can be expressed in terms of the form amplitude

27



F (Q)

according to equation 12:

dΣ 1 (Q) = |F (Q)|2 dΩ V where

Page 28 of 46

|F (Q)|2

,

(13)

is called the form factor of this particle. In a dispersion of nanoparticles the

interference of the scattering of the single particles can be described by the static structure factor

S(Q): 70,71 dΣ 1 (Q) = h|F (Q)|2 iS(Q) . dΩ V

The brackets cles.

25,68

(14)

h...i denote the ensemble average over the size distribution of polydisperse parti-

For compact spherical particles with constant SLD, polydispersity can be respected

via the distribution function

f (Rs )

of the radius

2

Z∞

h|F (Q, Rs )| i =

Rs

of the spheres:

|F (Q, Rs )|2 f (Rs ) dRs

72

.

(15)

0

The expression

f (Rs ) dRs

within the range length density

ρpb

describes the probability to nd the radius of an analyzed particle

f (Rs ) + dRs . 68

The form amplitude of a spherical particle with scattering

dispersed in a medium with SLD

ρm b

is given by:

72

4 3 p m sin(QRs ) − QRs cos(QRs ) F (Q, Rs ) = πRs (ρb −ρb ) 3 = Vp (Rs ) ∆ρb K(Q, Rs ) , 3 (QRs )3 where

Vp

(16)

denotes the volume of one particle. Accordingly, the macroscopic scattering cross

section for poydisperse spherical particles can be described by:

1 dΣ (Q) = (∆ρb )2 S(Q) dΩ V

Z∞

71

Vp2 (Rs ) |K(Q, Rs )|2 f (Rs ) dRs

.

(17)

0

We applied a bimodal particle distribution for seed particles to match the experimental data. The fraction of larger particles could be described by a Gaussian distribution, whereas

28


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


for the smaller fraction a lognormal distribution was used:

65

2

Rs

ln( µ ) N · e− 2σ2 f (Rs , µ, σ) = cLN · Rs

√ σ2 cLN = 2π · σ · e 2

(18)

.

(19)

σ denotes the width parameter and µ the location parameter. cLN R∞ f (Rs , µ, σ) dRs = N , where N represents the amount of scatterers. 0

is chosen such that The mean radius

R0

of the particles can be calculated by:

R0 = µ · e−σ

2

(20)

CTAB micelles were assumed to posses a prolate, two-axial core-shell ellipsoidal shape with a rotational half-axis literature.

73,74

∆ρout b

main half-axis

b (b = c)

and shell thickness

The resulting form factor includes an expression

of scattering contrast contrast

a,

∆ρin b

between core with SLD

ρcore b

as described in

which describes the ratio

and shell with SLD

between shell and dispersion medium with SLD

µb =

µb

t

ρcore − ρshell ∆ρin b b b = shell sol ∆ρout ρ − ρ b b b

ρshell b

and the

ρsol b :

,

that is used to calculate the form amplitude of core-shell ellipsoids:

(21)

75

F (Q, µb ) = (ρcore − ρshell ) Vc K(xc ) + (ρshell − ρsol b b b b ) Vt K(xt )

xc = Q

p

a2 µ2b + b2 (1 − µ2b )

(22)

p xt = Q (a + t)2 µ2b + (b + t)2 (1 − µ2b ) , where

Vc

denotes the core volume and

Vt

the total volume of one micelles including core and

29



shell of thickness

Page 30 of 46

t: 4 4 Vc = πab2 , Vt = π(a + t)(b + t)2 3 3

In all ts the main half-axis result of our previous work.

42,43

b (b = 17.6

Å) and

t (t = 7.0

(23)

Å) were kept constant as a

A Hayter-Penfold structure factor was used to describe the

correlation peak in the scattering signal of CTAB micelles.

76,77

about the chosen tting model is given in earlier publications.

A more thorough discussion

42,43

4.7 Coagulation Theory 4.7.1

Rapid Coagulation

If each collision of particles results in aggregation, the collision rate spheres with radius

Rs

J = − dn dt

can be described by the theory of Smoluchowski.

78

of similar

If the collisions

are completely driven by the diusion of the particles, the following equation can be used to describe rapid coagulation:

53,79

Jr = −

D

denotes the diusion coecient

n

dn = 8πDRs n2 = kr n2 dt

(24)

the number of particles per volume and

kr

the rate

constant for rapid coagulation. Considering two identical spheres with their radius diusion coecient

D

can be calculated using the Stokes-Einstein relation:

D=

where

kB

is the Boltzmann constant (kB

Rs ,

53

kB T 6πηRs

= 1.3807 · 10−12 KJ ), T

the

(25)

the temperature and

η

the

dynamic viscosity. The combination of 24 and 25 leads to:

Jr = −

dn 4kB T n2 = dt 3η

30


(26)

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


4.7.2

Slow Coagulation

Equation 24 does not consider the interaction potential between two particles, which can strongly aect the aggregation kinetic by inuencing the collision possibility. potential energy

Vt

is the sum of all attractive energies

VA

and repulsive energies

The total

VR :

Vt = VA + V R

where

VA

(27)

is mainly described by van der Waals interactions and

sion. The extensive evaluation of

Vt

VR

by electrostatic repul-

and some assumptions that were made to consider the

ellipsoidal shape of the micelles are discussed in detail in the supplementary information. The collision rate

Js

for slow coagulation of two particles can be expressed by:

Js =

where

W

4πkB T 2 n = ks n2 3ηW

depends on the total interaction potential

center distance of the particles

On the other hand

the particles and hence

W

ks :

(29)

Vt (r)

exp

Rs

Jr .

and slow coagulation

kr W

∞

W = Rs

A repulsive potential leads to

kr

which is a function of the the center-to-

r: Z

to

(28)

is the ratio between the rate constants for rapid

ks =

W

53,7981

Vt (r) dr kB T r2

W > 1 that result in a decrease of Js

will be between

0 2(a + t)] = 0

the micelles were assumed to act like spherical particles with equivalent surface

Aell o : Aell o

and radius

H0 = 0

which is explained in detail in the supplementary information:

1−

For large

plus

The rotational spheres of two approaching micelles start to overlap if

surface area of the spheres. A probability to have with

a

√ 2 a − b2 a2 arcsin = 2πb · b + √ a a2 − b2

(33)

RA : r RA =

32

Aell o 4π


(34)

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


At

r ≤ 2(a + t)

the eective radius

Ref f (r)

depends on

r

and can be expressed via:

r Ref f (r) = γH0 · [(1 − γH0 )(b + t − RA )] + γH0 + (1 − γH0 )RA 2

(35)

and equation 31 needs to be expressed by:

H0 = r − 2Ref f

With

Ref f

a dimensionless distance parameter

s

can be dened according to equation 36,

which considers the anisotropy of the micelles in the evaluation of

can be expressed with respect to

W:

H0 (r) r = −2 Ref f (r) Ref f (r)

s(r) =

W

(36)

s: 53,79 Z

∞

W =2

exp

0

In this way it is possible to determine

W

Vt (s) ds kB T (s + 2)2

Js

(38)

for the collision of seed particles with seed particles

W11 , micelles with micelles W22 and seed particles with micelles W12 . rates

(37)

are not correct for the following reason:

J11

The respective collision

only allows for collisions of seed particles

with seed particles by ignoring the interactions with the micelles. The same holds for and

J22

J12 .

4.7.3 Since

Final evaluation of the collision rate

Vt

for the interaction between seed particles and micelles is completely attractive (c.f.

supplementary information), it is valid to assume the seeds and micelles to spontaneously form dimeric clusters. As demonstrated in the supplementary information, the attachement of another micelle at these cluster to form trimers is also likely for CTAB and CTAB/PeOH micelles but thermodynamically unfavored for CTAB/HexOH micelles.

33


In the following


calculations these trimers are not further considered. trimers on the aggregation kinetics is not expected.

Page 34 of 46

A signicant impact of the formed For the collision of a trimer with a

dimer or another trimer the high mobility of the seed particles at the micelle surface would still lead to seed particle collision. Hence, the formation of higher order clusters might even increase aggregation probability through the locally enhanced concentration of seed particles. In contrast CTAB/HexOH micelles and seeds will not form higher order clusters through the stronger repulsion of the micelles at comparable distances. The probability of a collision of two seed particles depends on the mobility and the orientation of the dimeric clusters. fraction

φ11

This probability is described by the squared volume

of the seeds inside the interaction volume of the cluster. This interaction volume

can be varied to reect the interaction dependency on, among other parameters, micelle size, anisotropy and moment of inertia and is discussed in further detail in the supplementary information. A convenient model for the interaction volume is to consider the cluster as part of the rotational sphere volume of the micelles with radius

Rs = a + t.

An aggregation of

seeds will take place between two approaching clusters with a certain probability depends on the volume fraction

φ11

φ211

which

of the seeds in the interaction volume of the clusters.

The potential energy between two seeds and two micelles is a function of

H0

and thus can

occur with a certain probability depending on the orientation of the clusters. To consider this, an eective collision probability

1 79 can be dened: Ws,11

1 φ2 φ2 φ2 1 1 1 = 11 + 22 + 12 = + + Ws W11 W22 W12 Ws,11 Ws,22 Ws,33 where

φ11

(39)

denotes the volume fraction of the seed particle in one dimer and thus

statistical probability of seed particle collision between two dimers. Accordingly, probability of micelle collisions and micelles.

Ws,12

φ212

φ211

φ222

the

is the

the probability of collisions of seed particles with

represents the attraction of two dimers via the seed particle of one and the

micelle of the other dimer. From these three collision scenarios

34


Ws,11

is the probability of

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


seed particle collision. Hence, equation 28 can be expressed as:

Js =

The temporal evolution of

n

4πKB T φ211 2 n = ks n2 3ηW11

(40)

can be determined:

n(t) 1 = t n0 (1 + t1/2 ) where

t1/2

denotes the half-life of the seed particles. It can be evaluated:

t1/2 =

where

n0

(41)

53

1 ks n0

is the initial particle number per volume. Because the highest

(42)

n0

has been observed

in CTAB/PeOH stabilized dispersions (n0 (CT AB/P eOH)), this value was used to normalize all other curves for better comparison:

n(t) n0

= norm

n0 1 · t n0 (CT AB/P eOH) (1 + t1/2 )

(43)

Acknowledgement The authors acknowledge the funding of the Deutsche Forschungsgemeinschaft (DFG) through the Cluster of Excellence Engineering of Advanced Materials (EAM), the research training group 1896 In Situ Microscopy with Electrons, X-rays and Scanning Probes, the funding for instruments (INST 90/825-1 FUGG, INST 90/751-1 FUGG, INST 90/827-1 FUGG) and the BMBF for the nancial support via the SAXS/SANS project 05K16WE1.

Supporting Information Available The following les are available free of charge.

35



Page 36 of 46

SI-alcseeds-SAS_reviewed.pdf: all relevant measurements, tting parameters and supporting calculations

This material is available free of charge via the Internet at

http://pubs.acs.org/.

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