Translational Anisotropy and Rotational Diffusion of Gold Nanorods in

Jul 26, 2015 - We studied the translational and rotational diffusion of gold nanorods within a rod/sphere composite liquid using fluctuation correlati...
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Translational Anisotropy and Rotational Diffusion of Gold Nanorods in Colloidal Sphere Solutions Sharmine Alam and Ashis Mukhopadhyay* Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, United States S Supporting Information *

ABSTRACT: We studied the translational and rotational diffusion of gold nanorods within a rod/sphere composite liquid using fluctuation correlation spectroscopy (FCS). The nanorods of length L ∼ 60 nm and diameter d ∼ 17 nm were used at a fixed concentration of ∼1 pM. The concentration of colloidal Ludox spheres (size ∼ 26 nm) was varied up to a volume fraction of ϕ = 0.3 or approximately 7 spheres/L3. Our experiments showed significant translational anisotropy as the sphere concentration was increased. The translational diffusion parallel to the rod axis (DT∥) followed very close to the bulk viscosity of the solution. However, the diffusion normal to the rod axis (DT⊥) experienced a significantly higher frictional force. For volume fraction ϕ > 0.1 a slightly modified caging theory by Pecora and Deutch could explain the rotational diffusion of the rods very well. At low volume fraction the agreement is poor, which we interpreted as modification of the local ordering of the spheres, which can affect the rod rotation.

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concentration of the particles, which has a unit of m−3, and thus nL3 is unit less. Within a highly concentrated solution, they argued that the neighbors create a “cage” which surrounds the probe and completely restricts its rotational motion. The rotation can happen only after the cage is “dissolved” due to the translational diffusion of the rod along its axis (DT∥). For ultrathin rods, L ≫ d, they derived9 DR = DR0β(nL3)−2, where DR0 is the rotational diffusion coefficient in neat solvent and β is a numerical factor, whose value is predicted to be ≈1. The subsequent experiments, however, showed significant discrepancies with the DE theory.10−12 By using rod-like polymer poly-γ-benzyl-L-glutamate (PBLG), the experiments had shown the rotational diffusion coefficient (DR) to be much higher so that β ≈ 1000 is needed to match with the DE theory.11 There is also a concentration offset effect, and the inverse-square dependence is not obtained until a much higher value of nL3. Eventually, sophisticated advances were made to refine the DE theory. According to Fixman,13,14 the rod rotation can be due to two mechanisms: cage renewal as described by the DE theory and slight transverse translation and rotation of both the caging particles and the confined rod. The model showed an angular jump (√⟨θ2⟩) of the rods is proportional to n−1/2, which yielded DR ∼ n−1 and, thus, was in qualitative disagreement with DE theory. Keep and Pecora developed a geometric model,15 which argued that predicted DR ∼ (nL3)−2 dependence will be obtained for nL3 > 500 and

INTRODUCTION The dynamics of nonspherical colloidal particles through complex fluids is a subject of great interest in diverse areas. For example, anisotropic particles such as gold nanorods (AuNRs) have shown promise for use in diagnostic and therapeutic purposes.1,2 The bottom-up approach of fabricating nanostructures using self-assembly can make use of anisotropic building blocks, which can provide a wider range of possibilities. Understanding the dynamics of such objects is important as the thermal energy which dictates the motion of the particles is comparable to the interaction energy at the nanoscale in many situations. One common approach for studying the dynamics is the probe diffusion method, where a particle (the probe) is monitored as it diffuses through a suspension of other particles (the matrix). Many combinations of probe and matrix had been studied using both hard (e.g., silica, gold, and so on) and soft (e.g., polymer, proteins, and so on) objects.3−5 Here, our interest is the study of gold nanorods within a matrix of Ludox spheres in the semidilute concentration regime, which is very relevant for the previously mentioned applications. The initial theoretical interest in studying these systems originated to understand the entanglement and caging in polymers, where rod-like molecules can behave as one of the model systems. Doi and Edwards (DE) provided the basis of understanding the dynamical behavior of rigid rods within semidilute or concentrated solutions of similar rods.6−8 In the semidilute solutions, cages are transient, appearing and disappearing with time. As a result, theoretically it is a very challenging problem. DE theory considered the concentration range of 1 ≪ nL3 ≪ L/d, where L and d are the length and diameter of the rod, respectively, and where n is the number © 2015 American Chemical Society

Received: May 7, 2015 Revised: June 30, 2015 Published: July 26, 2015 8780

DOI: 10.1021/acs.langmuir.5b01682 Langmuir 2015, 31, 8780−8785

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showed the diameter of the particles, 2Ro ≈ 26 ± 3 nm (Supporting Information Figure S1). The particles are negatively charged with ζ potential of −35 ± 5 mV. Cetyltrimethylammonium bromide (CTAB) stabilized AuNRs with length 60 nm and diameter 17 nm were also obtained from Sigma-Aldrich (Figures S2 and S3). The concentration of NRs in the stock solution was ≈2−3 nM, which was diluted a thousand times to prepare the solution used in our experiments. Distilled deionized water was used as a solvent for all experiments. Ludox solution with volume fraction ϕ = 0−0.3 and nanorods in water were prepared in a home-built cylindrical glass cell with the top and bottom plate made with a borosilicate cover glass of thickness ∼ 0.17 mm. The cell was sealed to prevent evaporation during measurements. Freshly prepared samples were used in all experiments. The viscosity (ηb) of solutions as a function of volume fraction was measured using a falling ball viscometer (Fish-Schurman Corp., New Rochelle, NY, USA). All measurements were performed at 20 °C. A major obstacle in determining the rod dynamics within a rod/ sphere mixture is the strong scattering from the spheres, which can easily overwhelm the signal arising from rods. We took advantage of the two-photon excitation of the gold nanorods. Experiments were performed on a Zeiss inverted microscope (Axiovert S200TV, Carl Zeiss, Inc.) and the sample cell was placed on the mechanical stage attached with the microscope. Near-infrared light from a femtosecond Ti:sapphire laser (Mai Tai, Spectra Physics) of wavelength 800 nm with pulses of width 150 fs at a repetition rate of 80 MHz was focused through a long working distance oil immersion objective (100×; numerical aperture (NA) = 1.25) into the sample. The laser excites the transverse surface plasmon resonance (TSPR) near 510 nm. The TSPR peak depends weakly on the diameter of the rod and is independent of the aspect ratio. We use a short-pass filter in front of the detector, which collects light of wavelength above 650 nm. The laser power was kept below 1 mW to avoid photothermal conversion, which can induce a heating effect in the sample. The fluctuations in photon counts as the gold NPs rotate and move into or out of the laser focus are detected by two single photon counting moduli (Hamamatsu, Inc.). A polarized beam splitter (Thorlabs, Inc.) is placed in front of the detectors. The digital output is recorded and analyzed by an integrated FCS data acquisition system (ISS, Champaign, IL, USA). The cross-correlation function (CCF), G(τ), of the fluctuation is measured for 5−10 min. At least 10 FCS trials were performed for each concentration. The CCF showed clearly two stage decays well-separated in time indicating both translational and rotational diffusion of the rods (Figure 1). We used Gtran(τ) = G(0)/[1

significant caging is not expected unless nL3 > 50. Farther improvements of the DE theories were made by considering that only a part of the rod needs to be translated for cage renewal, 10,11 smaller rods can relax away faster for a polydisperse system,16 and so on. We found only one experimental study on the diffusion of rod-like molecules within a sphere mixture. Phalakornkul et al.17 have studied PBLG molecules of three different molecular weights of aspect ratios (L/d) varying ≈30−90. The experiments probed the bulk anisotropy of the sample using transient electric birefringence (TEB) decay method. The matrix consisted of silica spheres with diameter 2Ro ≈ 124 nm with volume fractions between 0 and 0.08. For L > 2Ro they found that DR ∼ L−2.52 and not according to scaling prediction of L−4. The diffusion is also decoupled from the matrix viscosity as DR decreased much faster compared to the increase of the matrix viscosity. According to their analysis, the rotational relaxation of the rods is dominated by the collective diffusion of the spheres at the length scale of L, instead of the self-diffusion coefficient. It modified the concentration (ϕ) dependence, showing that DR ∼ ϕ−1 instead of ϕ−2/3. But, the shortest rod (L = 68 nm) did not show any concentration dependence of DR, so the results are not conclusive. The TEB decay method, which they used, had few issues. Because it is a bulk technique, a high concentration of rods (10 rods/L3) is needed, so the caging can be affected by the spheres and by the rods as well. The method is also perturbative. Because the silica particles are charged, the external electric field can affect the sphere motion and their ordering around the rods. The bending mode of the polymer can complicate interpretation of the data as Odjik18,19 had shown that even slight flexibility can have a drastic effect on the rod confinement. In addition, the measurements cannot provide the absolute values of the rotational diffusion coefficients but give only the ratio DR/DR0. So a quantitative comparison with theories is not possible. In this work, we used gold nanorods (AuNRs), which are truly a rigid rod within a semidilute concentration of spheres using a nonperturbative optical technique of fluctuation correlation spectroscopy (FCS). By using polarized optics, we were able to determine both the translational and rotational diffusion coefficients from the same experiment. We kept the rod concentration at nL3 ≈ 10−7 so that any correlations among them are negligible and true self-diffusion coefficient is measured. The rod has an aspect ratio ≈ 4, which is at least an order of magnitude smaller compared to polymer molecules. The size of the crowding spheres is comparable to the dimensions of the rods, and their concentration is varied up to ϕ = 0.3 or approximately 7 spheres/L3. These size and concentration regime are appropriate for understanding the effects of crowding on rod diffusion in biological systems, where gold NRs with aspect ratio ∼ 3−4 have shown promise for diagnostic and therapeutic applications.20 Our results will also be important for self-assembly of a mixture of spherical and nonspherical particles, where depletion interaction among the nonspherical particles due to the presence of similar sized spheres could create structures with anisotropic photonic or electrical properties.

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Figure 1. Cross-correlation function, G(τ), with fitting as described in the text showing both the translational and rotational diffusion of the gold nanorods in water (main panel). Rotational (inset bottom) and translational (inset top) cross-correlation curves are shown separately for different volume fractions of Ludox as indicated in the legends. The nanorods concentration was kept at about ∼1 pM. The imperfect fit of the translation curves are described later.

EXPERIMENTAL SECTION

Colloidal silica (Ludox TM-50, 50 wt % suspension in water) sphere was purchased from Sigma-Aldrich, Inc. Both dynamic light scattering (DLS) and transmission electron microscopy (TEM) measurements 8781

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Langmuir + (8DTτ/ω02)(1 + (8DTτ/z02))1/2] and Grot(τ) = Gr exp(−6DRτ) to fit the CCFs, which give translation (DT) and rotation (DR) diffusion coefficients, respectively.21,22 Here, G(0) is the cross-correlation magnitude at time lag τ = 0 and is inversely proportional to the number of particles within the laser focus. The amplitude of the translational correlation function Gtran(τ) depends upon the average number of the particles within the laser focus. The amplitude of rotational correlation function Grot(τ) of the rod depends upon the aspect ratio as well as the excitation and detected polarization state. Because the observation volume is a three-dimensional Gaussian intensity profile, the 1/e2 radius is given by ω0, whereas it is z0 in the axial direction. The values of ω0 ≈ 0.4 μm and z0 ≈ 2 μm were determined through calibration experiments in water. Parak et al.23 have used a rough approximation for measuring the focus radius in solution which has a different refractive index compared to the solution used for the calibration experiment. According to the approximation, ωLudox = ωwater(ηwater/ηLudox). In our experiment, ωwater = 0.4 μm, ηwater = 1.33, and ηLudox = 1.45, which gives ωLudox = 0.37. The maximum error determining the beam width is about 7.5%. So, the change in D is ∼15% as D ∼ ωo2. The statistical deviation in measuring diffusion coefficient is more than 20%. So the change in calibration volume can be neglected.

Figure 2. Imperfect fit with one-component translation diffusion coefficient. The solution is with Ludox volume fraction ≈ 0.3. The dashed line through the points is a one-component fit (DT ≈ 0.3 μm2/ s), and the solid line is a two-component fit (DT∥ ≈ 5 μm2/s and DT⊥ ≈ 0.1 μm2/s). The corresponding residuals are shown in inset a. Inset b shows one-component (dashed) and two-component (solid) fits of the correlation function with residuals in water, which showed no significant differences.

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RESULTS AND DISCUSSION Figure 1 shows a typical cross-correlation function of AuNRs in water. Following the convention used in ref 21, we used a configuration of [YXY], where the first letter corresponds to excitation polarization and the next two letters correspond to detected polarizations. Z is the propagation direction of the incident light. From the analysis as described earlier, we obtained DT0 = 14.6 μm2/s and DR0 = 33556 s−1, where the subscript “0” indicates that the measurements were performed in neat solvent. According to various theories of rod diffusion, DT ∼ L−1 log(L/d) and DR ∼ L−3 log(L/d),21,24 which indicates extreme sensitivity of DR on the length (L) of the rod. We used Tirado and de la Torre’s (TT) theory,25,26 which is most appropriate for shorter rods (i.e., low aspect ratio) to calculate the expected values for rod diffusion from the measured L and d, respectively, by using TEM. It gave DR0 ≈ 17900−26000 s−1 and DT0 ≈ 14−16.5 μm2/s depending upon the assumed thickness of the CTAB coating (∼1−3 nm).27 In the insets of Figure 1 we showed CCFs for translational and rotational diffusion separately for a few different concentrations of Ludox spheres. As shown, clearly the time scale of diffusion increases with the increase of volume fraction consistent with the overall increase of the viscosity of the solution. Translational Diffusion. A rod can translate along its axis (∥ or “easy” direction) or perpendicular to it (⊥ or “hard” direction). In neat solvent an isotropic diffusion is expected, i.e., DT∥/DT⊥ = 1, but diffusion anisotropy can be extremely large within a semidilute or concentrated solution.29 DE theory assumed an extreme situation, where diffusion along the hard direction is completely quenched (DT⊥ ≈ 0). Along the easy direction, DT∥ is unaffected by the presence of the other particles. Therefore, DT∥ ≈ DT∥0. These results, however, are strictly valid for ultrathin rods and within highly concentrated solution. The analysis of our data in neat solvent did not reveal any significant difference between fitting with one-component diffusion, Gtran(τ) = G(0)/[1 + (8DTavτ/ω02)] or twocomponent diffusion, Gtran(τ) = {[G1(0)/[1 + (8DT∥τ/ω02)]] + [G2(0)/[1 + (8DT⊥τ/ω20)]]} as shown in the Figure 2 inset. But, clearly one-component fitting is not satisfactory with the increase of concentration as shown by the systematic variation of the residuals (Figure 2). The CCF can be fitted very well

with two-component diffusion, giving a fast (DT∥) and slow component (DT⊥). In Figure 3 (top) we plotted the two components as a function of the Ludox volume fraction. Both DT⊥ and DT∥ decrease strongly beyond a concentration of ϕ = 0.1. Over the whole concentration regime, DT∥ decreases by a factor of 10, while DT⊥ decreases by a factor of ∼100. So both components of diffusion showed strong concentration dependence with the diffusion along the hard direction affected more dramatically. The diffusion anisotropy, the ratio of the fast component to slow component (DT∥/DT⊥), increased by a factor of ∼10 as the volume fraction of the spheres was increased from 0 to ∼7 spheres/L3. These results can be compared with Brownian dynamics simulation of rigid rods (aspect ratio = 50) within a mixture of similar rods.30 For a concentration change between 5 and 50 rods/L3, observed DT∥ decreases by about 20%, while DT⊥ decreases by a factor of ∼5. Obviously, our results showed more significant dependence of DT∥ and DT⊥ on the volume fraction. The average value of translational diffusion coefficients, DTav was calculated by fitting the CCF’s with 2D Gaussian model with two photon excitation: Gtran(τ) = G(0)/[1 + (8DTavτ/ω20)]. These coefficients are comparable with the theoretical value of average translational diffusion coefficients obtained from Stick theory: DTav = (1/3)(DT∥ + 2DT⊥). We can determine the nanoviscosity which is also known as nanofriction, ηc(ϕ), that the particles experienced from the measured diffusion coefficients.23,31 We used TT theory as described earlier because it provided good agreement with the measured DT in the neat solvent. Using this theory, we calculated ηc(ϕ) experienced by the rods along the easy and hard directions. A small numerical factor is included in the analysis to match the nanoviscosity with the bulk water viscosity at ϕ = 0 and to get isotropic diffusion in neat solvent. The comparison with ηb(ϕ) is shown in Figure 3 (bottom). It is clear that the diffusion along the easy direction closely followed the bulk viscosity, while the nanoviscosity experienced by the rod in the hard direction is much higher compared to the bulk viscosity. The diameter of the rod (≈17 nm) is slightly smaller compared to the Ludox spheres size (≈26 nm). The motion along this direction, therefore, involves rearrangement of 8782

DOI: 10.1021/acs.langmuir.5b01682 Langmuir 2015, 31, 8780−8785

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Figure 4. Rotational diffusion coefficient vs volume fraction of Ludox. The dashed line is fit with use of Pecora−Deutch theory with bulk viscosity, ηb(ϕ). The solid line is generated using the same theory, but with nanoviscosity that the rods experienced along the “easy” direction (ηT∥). (Inset) Normalized diffusion coefficients and normalized bulk viscosity showing that both DT∥ (circles) and DR (squares) followed the bulk viscosity (), while DT⊥ (stars) decreased more sharply with respect to the volume fraction of Ludox.

who extended the DE theory to a semidilute solution of rod (probe)/sphere (matrix) mixture. In their simplified theory, which is an extension of DE theory, the rod is trapped by neighboring spheres and its rotational motion is limited to a space between the rod and the nearest spheres. It naturally gives DR ∼ n−2/3. The restricted rotational diffusion of the confined rods is relaxed by the translation diffusion of the spheres away from the cage, giving the results, Dsphr ∝ (kBT/ R ηb(ϕ)L4 Roη2/3), where kB is Boltzmann’s constant, T is absolute temperature, ηb(ϕ) is the solution viscosity, and Ro is the radius of a sphere. Their result is valid in the concentration range such that the sphere concentration is high enough to restrict the rod relaxation to 2π(Ln1/3) > 1, but not so high that there is a notable overlap among the spheres, nRo3 < 1. This theory is also valid for a system where Ro ≤ L, so that the cage relaxation is dominated by the motion of the spheres. We estimated that in our experiments 2π(Ln1/3) ≈ 3−6 and nRo3 ≈ 0.01−0.07, so we expect the caging idea should be valid. In our experiments, L ≈ 3Ro, so the rod rotation is dominated by the translational diffusion of the spheres away from the rod. A comparison with this theory is shown in Figure 4 with a numerical prefactor = 1, which showed a stronger dependence on ϕ and higher disagreement at lower concentration compared to the expectation from caging theory. The theory assumed that the rotational relaxation is dominated by the bulk viscosity of the matrix, which governs the self-diffusion of the spheres. A better agreement with theory is obtained if we assume that the translational diffusion of the rods along the easy direction breaks up the cage. This is justified as this motion is the fastest as shown in Figure 3 and will, therefore, be the most dominant mechanism of cage relaxation. Substituting the corresponding nanoviscosity gives a better agreement with the rotational diffusion. The two lowest volume fractions still showed significant deviation from the theory. In fact, diffusion is found to be much slower compared to the expectation from either the bulk viscosity or nanoviscosity. For these two lowest concentrations studied, there are expected to be only 1−3 particles/L3 present. So caging is not expected to be significant. But we have not considered yet the ordering of the Ludox spheres around the rods. The spheres are negatively charged as the ζ-potential measurements showed. The bare gold nanorods are negatively charged as well, but there is a CTAB coating on the top of the

Figure 3. (Top) Plots of translational diffusion coefficients along the “easy” axis (DT∥, open squares) and “hard” axis (DT⊥, open circles) as a function of Ludox volume fraction (ϕ). The graph also shows the average translation diffusion coefficient of the rods (open triangle). The solid and dashed lines have slopes of −2.0 and −2.7, respectively, showing the different concentration dependences of DT∥ and DT⊥. (Bottom) Viscosity (η) extracted from translation diffusion showing rod motion along the easy direction (open squares) followed closely by the bulk viscosity, ηb (closed stars) while the motion along the hard direction (open circles) followed much higher nanoviscosity. The nanoviscosity experienced by spherical gold NPs of the similar size of the diameter of the rod also experienced the same nanoviscosity (open stars) as in the easy direction.

particles at the length scale of about 20 nm. To verify this statement, we conducted experiments with gold nanospheres (AuNSs) of size ∼ 20 nm within the matrix of the same Ludox spheres, which was used for nanorod experiments. Our results showed that the AuNSs experience the nanoviscosity very close to the viscosity experienced by the AuNRs in the easy direction, supporting this scenario. The long-time diffusion along the hard direction, however, is much slower as observed in Figure 3. The motion along this direction requires collective diffusion of a few particles at the length scale of L. It depends both upon hydrodynamic and interparticle interaction. The hydrodynamic interaction generally slows down the particle mobility with an increase of sphere concentration at most length scales. This is also manifested in the concentration dependence of nanoviscosity that we showed in Figure 3. A power-law fitting indicated that ,for ϕ > 0.1, DT∥ ∼ ϕ−2 and DT⊥ ∼ ϕ−2.7. The significance of these exponents is not clear at present, but it supports the general picture that rod motion is affected by different mechanisms in easy and hard directions. Rotational Diffusion. The rotation of the rod along its own axis does not change the relative orientation of the absorption and emission dipoles. So, the experiments are sensitive only to the rotation perpendicular to the major axis and only one rotational diffusion coefficient is measured (DR). The measured diffusion coefficients as a function of sphere concentration are shown in Figure 4. The most appropriate for the theoretical analysis is the study by Pecora and Deutch,32 8783

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Langmuir particles, which formed during their synthesis. It is believed that the coating consisted of a bilayer, which if perfect, will expose positively charged amino groups at the surface.33 Even if the bilayer is not perfect,34 it will have patches of positively charged regions with some exposed hydrophobic groups at the surface. The presence of positively charged regions can enhance the ordering of the spheres around the rods and increase the effective sphere density near the particles. Because of the local nature of the rotational motion, this might cause a slowdown of the rotational diffusion compared to the expectation from caging theory. In spite of caging there is no aggregation between the spheres and rods as we verified by UV spectra in the presence of Ludox spheres (Figure S4). In addition, the adsorption of even one layer of Ludox particles into the surface of the rods would cause dramatic slowing down of diffusion even at the lowest volume fraction studied (ϕ = 0.05), which was not observed in our experiments. Figure 4 (inset) summarizes the data, which showed a normalized plot of diffusion coefficients and viscosity vs volume fraction. The ratio DT∥/DT∥0 followed the same concentration dependence of normalized bulk viscosity, ηb0/ηb. The diffusion along the hard axis has stronger concentration dependence than the bulk viscosity, while the ratio DR/DR0 can be best explained by considering that cage relaxation occurs through the rod motion along the easy axis.

ACKNOWLEDGMENTS

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REFERENCES

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CONCLUSION We investigated a poorly understood problem of anisotropic particle diffusion within a mixture of spherical particles. This situation arises in diverse areas ranging from self-assembly to bioengineering. For anisotropic particles we used short gold nanorods of aspect ratio ∼ 4 and two-photon excitation to study their dynamics within a crowded solution of Ludox spheres. We found significant diffusional anisotropy for translational motion of the rod. Though the motion along the rod axis followed closely to the bulk viscosity, the motion perpendicular to the rod axis is significantly retarded. These features had been predicted for ultrathin rods both in theories and simulations. Our experimental results provided a direct quantitative test of these predictions. The rotation of the rods closely followed the modified Doi−Edward caging theory as developed by Pecora and Deutch considering the fastest relaxation mechanism that breaks the cage. ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.5b01682.

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Acknowledgements are made to the Donors of the American Chemical Society Petroleum Research fund (PRF No. 51694ND10) for support of this research.

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Three figures including the TEM images of gold NRs, histogram of length and diameter, and TEM and DLS characterizations of Ludox spheres (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: 313-577-2775. Fax: 313-5773932. Notes

The authors declare no competing financial interest. 8784

DOI: 10.1021/acs.langmuir.5b01682 Langmuir 2015, 31, 8780−8785

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DOI: 10.1021/acs.langmuir.5b01682 Langmuir 2015, 31, 8780−8785