Article pubs.acs.org/JPCB
Osmotic Bulk Modulus of Charged Colloids Measured by Ensemble Optical Trapping Joseph Junio,† Joel A. Cohen,‡ and H. Daniel Ou-Yang*,† †
Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015, United States Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, United States
‡
ABSTRACT: The optical-bottle technique is used to measure osmotic bulk moduli of colloid suspensions. The bulk modulus is determined by optically trapping an ensemble of nanoparticles and invoking a steady-state force balance between confining optical-gradient forces and repulsive osmotic-pressure forces. Osmotic bulk moduli are reported for aqueous suspensions of charged polystyrene particles in NaCl solutions as a function of particle concentration and ionic strength, and are compared to those determined by turbidity measurements under the same conditions. Effective particle charges are calculated from the bulk moduli and are found to increase as a function of ionic strength, consistent with previously reported results.
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INTRODUCTION Colloidal interactions can be investigated by direct methods, such as surface-force measurements,1 atomic force microscopy,2 and optical trapping,3,4 or alternatively, by quantifying the bulk properties of a colloidal suspension with thermodynamic measurements. In contrast to direct interparticle measurements, ensemble parameters such as osmotic pressure and isothermal bulk modulus provide a noninvasive macroscopic link to the microscopic physics. The bulk modulus of a colloidal suspension is an ensemble measurement of the excess osmotic pressure generated by either electrostatic5 or entropic interactions,6 which can be measured directly by osmotic stress7 or inferred from the static structure factor of small-angle light scattering.8 The experimental method described here employs an optical bottle, which operates similarly to optical tweezers. For dielectric particles having a refractive index higher than that of the surrounding medium, a tightly focused laser beam can manipulate the particles through light-scattering and electromagnetic-field gradient forces. For sufficiently high laser intensities, the gradient force is strong enough to overcome the randomizing effect of Brownian motion, thus locally enhancing the particle concentration near the focus. As the intensity of the trapping laser is increased, more particles are pulled toward the focus. Enrichment of the colloid concentration due to the trapping laser increases the colloid osmotic pressure, and a steady state is reached when the osmotic and gradient forces are balanced. Thus, the particle concentration in the trap is a function of both the particle coupling to the laser and interactions between the particles themselves. Once the particle−laser coupling is known, the particle concentration in the trap as a function of laser intensity can be used to infer the bulk modulus and particle interactions. Thus, the optical-bottle method is analogous to the application of osmotic stress, where instead of a tightly focused laser, a neutral polymer suspension © 2016 American Chemical Society
squeezes a colloid suspension confined by a semipermeable flexible membrane (Figure 1). Alternatively, in an optical trap, in which particles can diffuse into and out of the trap, it is the average particle number density that increases with laser intensity. Whereas for osmotic stress the particles are totally confined by the membrane, in our experiment individual particles are only transiently confined, so we measure their time-averaged concentration in the trap, as shown in Figure 1. Previously, we used optical bottles to determine particle− laser coupling for different particle sizes.9,10 In this work, we vary the salt concentration and show that the optical bottle can be used to measure the osmotic bulk modulus directly. To confirm the accuracy of this method, we also determine the osmotic bulk moduli of identically prepared samples by turbidity measurements. In this article, we review the theories for the optical-bottle and turbidity techniques and report bulkmodulus results from both methods for suspensions of charged colloids at different ionic strengths and volume fractions. Finally, we calculate the corresponding effective charges and report their dependence on ionic strength.
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THEORY
Optical Bottle. We first consider the pressure exerted on particles in the focal volume due to single-particle opticalgradient forces. For particles whose diameters 2a are small compared with the laser wavelength (2a ≲ λlaser/10), the optical-gradient force is much larger than the optical-scattering force,12 which thus may be neglected. Moreover, since the gradient force is conservative,13−15 the trapping region can be defined in terms of an optical potential well. The maximum trapping energy per particle U0 at the focus depends on the Received: June 3, 2016 Revised: June 27, 2016 Published: June 27, 2016 9187
DOI: 10.1021/acs.jpcb.6b05608 J. Phys. Chem. B 2016, 120, 9187−9194
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The Journal of Physical Chemistry B
Figure 1. (a) In an osmotic stress experiment, the colloids are confined by a semipermeable flexible membrane that exchanges water and small ions with a reservoir containing a neutral polymer whose osmotic pressure is determined by its concentration. (b) For the optical-bottle experiment, the particle concentration n(r) in the detection region (green ring) is higher than that in the reservoir due to the laser trap (red arrows). Although the particle concentration profile of the trap n(r) can be spatially resolved,11 here we measure the average particle concentration in the detection region, n = ∫ n(r) dr⃗/V, where the integral is over the detection volume V. At steady state, n is determined by a balance between particle confinement produced by the trapping laser and entropic forces that resist confinement.
κT = [n(1 + 2B2 n + 3B3n2 + ...)kBT ]−1
volume polarizability of the particle and is insensitive to interparticle interactions.10 For simplicity, we assume the gradient force to be isotropic. We have previously shown10 that the optical gradient force exerts a confining pressure on the particle ensemble, ΔP ≈ 1 nU0,16 where n is the average particle 2 number density in the trapping volume (initially equal to the reservoir particle number density) and the prefactor 1 is 2 numerically calculated for noninteracting particles in a tightly focused Gaussian beam when Δn/n ≲ 0.2.10 For a beam of fixed numerical aperture, U0 is linearly related to the laser intensity, U0 = βIlaser.10,11 The experimental determination of β is described below. The trap causes the time-averaged local particle concentration to increase until a steady-state balance is reached between the colloid osmotic pressure and the confining pressure produced by the optical gradient force. The isothermal compressibility of a substance of volume V subject to applied pressure P is defined as: κT ≡ −
1 ∂V V ∂P
where the first term is the ideal gas expression and Bi are the osmotic virial coefficients. Experimentally (details below), we measure Δn/n as a function of laser intensity at different reservoir particle concentrations n. Defining α ≡ the initial slope of Δn/n versus Ilaser, eq 3 gives:
α(n) = κT(nβ /2)
1 ∂n Δn 1 ∼ n ∂P n ΔP
α(n) = β[(1 + 2B2 n + 3B3n2 + ...)(2kBT )]−1
(6)
Defining γ ≡ (2α)−1 gives: γ(n) = (1 + 2B2 n + 3B3n2 + ...)(kBT /β )
(7)
Experimental plots of γ(n) versus n are shown below, but first we note that in the limit of infinite dilution (n → 0), the yintercept is γ(0) = kBT/β, which provides an experimental determination of β independent of n and of virial coefficients and thus calibrates the single-particle gradient force of the laser trap. Once β is known, initial slopes α of eq 3 yield the compressibilities, or bulk moduli, via eq 5. Once the compressibility is known for a given n under given conditions (such as ionic strength), the virial coefficients can be found from eq 4. The virial coefficients contain information about the particle interactions. Turbidity. The osmotic compressibility κT can also be recovered from light-scattering experiments in the small-q limit of the static structure factor. However, static light scattering is limited to dilute solutions because of errors introduced by multiple scattering at higher concentrations. It has been shown that turbidimetry is a suitable technique for studying particle interactions in concentrated colloidal suspensions because it provides a means of obtaining the long-wavelength limit of the structure factor, which is equivalent to the low-q limit.17 Turbidity measurements are insensitive to multiple scattering and are used here to validate the optical-bottle measurements at the same particle concentrations and ionic strengths. Turbidity τ is defined as the ratio of the transmitted scattered light intensity It to the incident intensity I0 through a sample of optical path length l:
(1)
(2)
at constant V, where ΔP is the confining pressure generated by the gradient force of the optical trap.10 Using eq 2 and the maximum single-particle trapping energy U0, we can express the relative enhancement of the particle number density Δn/n in the detection region as a function of the laser intensity:10 ⎛1 ⎞ ⎛1 ⎞ Δn ∼ κT ΔP ≈ κT⎜ nU0⎟ = κT⎜ nβ ⎟Ilaser ⎝ ⎠ ⎝2 ⎠ n 2
(5)
where β = U0/Ilaser. From the expression for κT in eq 4:
For an N-particle gas whose particle concentration (number density) is n ≡ N/V, an applied pressure ΔP causes a decrease in volume, ΔV < 0, hence an increase in n such that Δn/n = −ΔV/V (to first order in ΔV/V and Δn/n). However, in the present experiment, the volume V of the trap detection region remains constant, and the confined particle number N (thus the number density n) increases due to the confining forces of the trap. From the above relation between Δn/n and ΔV/V, we rewrite eq 1 for the present case as: κT =
(4)
(3)
Considering the colloids as a weakly interacting gas, the isothermal compressibility in eq 3 may be expanded as: 9188
DOI: 10.1021/acs.jpcb.6b05608 J. Phys. Chem. B 2016, 120, 9187−9194
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The Journal of Physical Chemistry B τ ≡ ln(It /I0)/l
(Sigma Aldrich). After dialysis, the samples were suspended in fresh solutions. The optical-trapping sample chamber was a microcapillary (2 mm × 50 mm × 100 μm; Vitrocom) attached to a microscope glass slide and sealed with vacuum grease. Each microcapillary held 10 μL of sample. Optical trapping was achieved with a variable-power laser (λtrap = 1064 nm) coupled into an oil-immersion objective lens (NA = 1.3, 100×, PlanFluor; Olympus). A second laser (λfluor = 532 nm) for fluorescence excitation was focused by the same lens to coincide with the trapping laser. Fluorescent signals emanating from the focal region common to both beams were optically band-passed to a pinhole at a location conjugate to the common focal region in order to ensure confocal detection of particle number densities. An optical chopper with a lock-in amplifier enhanced the signal-to-noise ratio.9 The particle number density in the detection volume was calibrated with samples of known number densities. The operating principle of the measurement is as follows: the gradient force of the trapping laser pulls particles toward the focal region, increasing the local number density, which causes an increase in fluorescence emission from the detection volume. Laser intensities were varied by changing the laser power. To rule out effects other than changes in particle number density from contributing to the fluorescence signal, we mention five possible sources of error: (1) Optical binding forces. For particles of size a < λ, the ratio of the opticalscattering force to the optical-gradient force scales as (a/λ)3, which for 100 nm particles and a 1064 nm laser is of order 10−4. Therefore, optical binding forces resulting from optical scattering are expected to be insignificant. (2) Local temperature changes. The rise of temperature due to heating by the focused IR laser is expected to be negligibly small due to the low absorbance of water for light at 1064 nm, our maximum laser power of 32 mW, and the good thermal conductivity of water. Under conditions similar to ours, a temperature change of only a fraction of a degree was reported by other authors.20 (3) Thermophoresis. Thermal gradients created by the trapping laser, which could contribute to particle migration, are expected to be negligibly small due to the minimal heating mentioned above. (4) Leaching of dye or photobleaching products. The ThermoFisher Firefli Red fluorescent dye in our particles is hydrophobic and has been incorporated into the internal matrix of the polystyrene spheres by a solvent swelling/deswelling process. Although leaching has not been studied, the manufacturer expects the dye and photobleaching products to be stably contained when the particles are suspended in aqueous solvent. (5) Perturbation of particle charge by the dye. Because of its hydrophobic and electroneutral molecular structure (similar to that of Nile Red) and its incorporation into the polystyrene matrix, the dye is not expected to contribute to the particle’s charge or ζ potential. Turbidity spectra of identically prepared particles (but without fluorescent dye) were recorded by a spectrophotometer (Lambda 9; PerkinElmer) with adjustable incident optical wavelengths in the range λ = 400−750 nm. The samples were contained in quartz cells (Starna) having 0.20, 0.50, and 1.0 mm optical path lengths corresponding to sample volumes of 60, 150, and 310 μL, respectively. The polystyrene particle mass density ρp = 1.054 g/cm3 was used to determine the mass concentrations in each turbidity experiment. Independent static light-scattering measurements of the particles yielded Rg = 40.3 nm, giving a mean particle diameter 2a = 2(5/3)1/2Rg = 104 nm.
(8)
In the absence of absorption by the colloids themselves, turbidity τ is related to the scattered light intensity by: τ = 2π
∫0
π
R(q) sin θ dθ
(9)
where R(q) is the Rayleigh ratio for unpolarized light at each scattering angle, which can be factored into a form factor P(q) related to scatterer size and shape and a structure factor S(q) related to scatterer interactions. The scattering vector q is related to the scattering angle θ by q = (4πnw/λ) sin(θ/2), where nw is the refractive index of water at each wavelength. Since turbidity τ is the scattered light integrated over all angles, it can be factored into analogous integrated form and structure factors, Q and Z, respectively, as:
τ ∝ CQ (λ 2)Z(λ 2 , C)
(10)
where C is the mass concentration of the sample, and the proportionality constant is given by eq 15 in the Appendix.18 In turbidity experiments the incident optical wavelengths λ are varied to change the scattering vector q for probing the sample. We measure the normalized turbidity τ/C: τ / C ∝ Q ( λ 2 ) Z (λ 2 , C )
(11a)
In the limit of low concentrations (infinite dilution) interparticle interferences are negligible, so Z → 1, thus:
(τ /C)C → 0 ∝ Q (λ 2)
(11b)
where the proportionality constant is the same as that in eq 11a. Dividing eq 11a by eq 11b: (τ / C ) = Z (λ 2 , C ) (τ /C)C → 0
(11c)
Note that eq 11c is an equality, as the proportionality constants in eqs 11a and 11b cancel out. From eq 11c, the y-intercepts of plots of τ/C versus C may thus be used to determine Z(λ2, C) as a function of wavelength at each particle concentration.17 As q ∝1/λ, we can experimentally probe the region of zero q by determining the infinite wavelength limit of Z(λ2, C). Plots are made of Z(λ2, C) as a function of 1/λ2, and as seen from eq 13, the y-intercepts at 1/λ2 = 0 (thus at q = 0) are the structure factors S(0) at each concentration. These values determine the isothermal compressibility, and equivalently the bulk modulus, through the relation:19
κT−1 =
nkBT S(0)
(12)
Turbidity measurements can thus be used to determine the bulk modulus of an ensemble of interacting particles for direct comparison with the bulk modulus measured by the optical bottle.
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EXPERIMENTAL SECTION Materials and Methods. We trapped 100 nm-diameter negatively charged polystyrene spheres internally dyed with Firefli Fluorescent Red, a lipophilic electroneutral dye similar to Nile Red (RO100B; ThermoFisher). All samples were first dialyzed against NaCl solutions of varying ionic strengths, diluted from a stock made with deionized water (Easypure RF; Barnstead), 100 mM NaCl (Fisher), 10 mM MOPS sodium salt (Sigma Aldrich), and 1 mM ethylenediaminetetraacetic acid 9189
DOI: 10.1021/acs.jpcb.6b05608 J. Phys. Chem. B 2016, 120, 9187−9194
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The Journal of Physical Chemistry B Results Obtained with Optical Bottle. We first measured the initial slope of Δn/n versus Ilaser at different reservoir particle concentrations and ionic strengths. Figure 2 shows Δn/n versus Ilaser for a 3% particle volume fraction. As the ionic strength is increased, reduced electrostatic repulsions permit more particles to be trapped at the same laser intensity, shown by the increasing slopes in the direction of the black arrow. In Figure 3, we plot γ(n) ≡ (2α)−1 = (2 × initial slope in Figure 2)−1 as a function of reservoir number density n. From eq 7, the y-intercepts at n = 0 determine β ≡ U0/Ilaser. The intercepts give β = (1.52 ± 0.24) × 10−10kBT m2/W, yielding U0 = (0.19 ± 0.03)kBT/mW. From β, the osmotic bulk moduli are calculated using eq 3 and plotted in Figure 4 for each ionic strength. For reference, Figure 4 also shows the bulk modulus calculated for the limiting case of infinite ionic strength where electrostatic interactions are absent. Results Obtained with Turbidity. The data in Figures 5 and 6 show τ/C as a function of C at each wavelength for 10 and 1 mM salt, respectively (other ionic strengths not shown). The data are fitted with either a line or a polynomial. The yintercepts (τ/C)C→0 are proportional to the experimental integrated form factors Q(λ2) as previously described. The integrated structure factors Z(λ2, C) are determined from eq 11c by dividing the experimental τ/C versus C data by (τ/C)C→0. From eq 13, the intercept of Z(λ2, C) as 1/λ → 0 (thus q → 0) is equal to S(0), and the bulk modulus is obtained from eq 12. To analyze the turbidity data and determine the intercepts for S(0), we fit the integrated structure factors shown in Figures 7 and 8 by performing numerical calculations of Z(λ2, C) using eq 13. To do this, expressions for P(q) and S(q) are required. As the particle size parameter 2πanw/λ is < 1 for 2a ∼ 100 nm, the form factor P(q) can be obtained from Rayleigh−Gans− Debye theory (eq 14). As the colloid volume fractions used in these experiments were turbidity Zeff
Figure 10. Average Zeff at each salt concentration as a function of κa, where κ−1 is the Debye screening length and a is the particle radius. Open red circles are from optical-bottle bulk-modulus measurements using eq 21. Open blue squares are from turbidity structure-factor fits using the RPA via eq 20. Symbols are average Zeff values for six particle volume fractions at each value of κa (see the legends of Figures 7 and 8); error bars are standard errors of the mean. Linear fits for the optical bottle: slope = 68.6 ± 6.2, intercept = 55 ± 61. For turbidity: slope = 46.0 ± 1.2, intercept = 7 ± 12.
2
means the approximate prefactor 1 in the ΔP expression is too 2 small. A better estimate of this number would produce better agreement. In future work, we propose using the turbidity results to calibrate the optical-bottle prefactor. Both methods produce a linearly increasing dependence of Zeff on κa, consistent with results obtained by other investigations of electrostatically interacting colloids, and in qualitative accord with charge renormalization theory for saturated colloid effective charges. We propose the optical bottle as a novel technique to measure colloidal ensemble bulk moduli, offering more direct and faster determinations, far smaller samples, and more convenience than classical turbidimetric methods.
with charge renormalization theory for saturated effective charges.24,28−32
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CONCLUSIONS In this article, we have used turbidity spectra to measure the integrated structure factor Z(λ2, C) as a function of q, which can be extrapolated to find S(0) and the bulk modulus. Alternatively, the optical bottle can measure the change in particle number density directly in response to an externally applied optical-trapping pressure,9,10 thus providing a straightforward measure of the bulk modulus. To test our methods, we modified the colloid−colloid interaction strength and range by
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APPENDIX
Form and Structure Factors
The integrated structure factor Z(λ2,C) for monodisperse particles is related to the form and structure factors P(q) and S(q), respectively: 9192
DOI: 10.1021/acs.jpcb.6b05608 J. Phys. Chem. B 2016, 120, 9187−9194
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The Journal of Physical Chemistry B π
2
Z (λ , C ) =
Using eq 12, eq 20 can be expressed directly in terms of the isothermal bulk modulus, providing an analytical relation between bulk modulus and Zeff:
∫0 P(q)S(q)(1 + cos2 θ) sin θ dθ π
∫0 P(q)(1 + cos2 θ) sin θ dθ
(13)
⎡ ⎛ 2 + ϕ ⎞⎤ κT−1 = nkBT ⎢1 + 4ϕ⎜ ⎟⎥ ⎢⎣ ⎝ (1 − ϕ)2 ⎠⎥⎦
where S(q) depends implicitly on concentration. Assuming that the attenuation of light through the cell is not due to absorption by the particles themselves and that the particles are small (2πanw/λ < 1), the form factor for low 2qa can be expressed by Rayleigh−Gans−Debye theory:33
P(q) = 1 −
⎡ ⎛ ⎞⎤ 1 + 2κa 2 ⎛ lB ⎞ ⎢1 + 12πϕZeff ⎜ ⎟⎜ ⎟⎥ ⎝ a ⎠⎝ (κa)2 (1 + κa)2 ⎠⎥⎦ ⎢⎣
(qR g)2 3
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(14)
where Rg is the optical radius of gyration, which we determine by static light scattering to be 40.3 nm. The complete equation for τ (see eq 10) includes the particle radius, the index of refraction of water, and the relative refractive index of the particles:18 ⎛ 2πan w ⎞3 16π ⎟ CQ (λ 2)Z(λ 2 , C) τ = K *⎜ ⎝ λ ⎠ 3 3n w ⎛ m2 − 1 ⎞ ⎟ ⎜ 4λρp ⎝ m2 + 2 ⎠
*E-mail:
[email protected] Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This research was supported in part by NSF-DMR: 0923299. The authors thank Prof. Ivan Biaggio for his time and instrumentation support in recording the turbidity spectra.
(15)
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ADDITIONAL NOTE This paper was originally invited and accepted for the “William M. Gelbart Festschrift”, published as the July 7, 2016 issue of J. Phys. Chem. B (Vol. 120, No. 26).
(16)
and m ≡ np/nw. Random Phase Approximation
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In this work we assume the particles to interact through the purely repulsive part of the DLVO potential, involving only hard-sphere and electrostatic repulsions. In the RPA, the structure factor S(q) is expressed using perturbation potentials of the hard-sphere system. The total structure factor is given by:22 S(q) =
SHS(q) n 1 + k T VC(q) B
VC(q) =
(17)
(18)
2 2 4πZeff e [κ sin(2qa) + q cos(2qa)]
ε(1 + κa)2 q(q2 + κ 2)
(19)
The q → 0 limit of the RPA structure factor is found by inserting the q → 0 limits of eqs 18 and 19 into eq 17 and using the relation lim j1 (x)/x = 1/3. Thus: x→0
⎡ ⎛ 2 + ϕ ⎞⎤ ⎡ 2 ⎛ lB ⎞ ⎜ ⎟ S(0)−1 = ⎢1 + 4ϕ⎜ ⎟⎥ ⎢1 + 12πϕZeff 2 ⎝a⎠ ⎢⎣ ⎝ (1 − ϕ) ⎠⎥⎦ ⎢⎣ ⎞⎤ ⎛ 1 + 2κa ⎟⎥ ⎜ ⎝ (κa)2 (1 + κa)2 ⎠⎥⎦
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where SHS is the structure factor for a hard-sphere system given by eq 18, VC(q) is the Fourier transform of the screened Coulomb potential given by eq 19, ϕ is the particle volume fraction calculated by dividing particle mass concentration C by the particle mass density 1054 g/L, j1 is the first-order spherical Bessel function, ε is the dielectric constant of the medium, and ϕ is related to particle number density n by ϕ = (4πa3/3)n. ⎛ 2 + ϕ ⎞ j1 (2qa) SHS(q)−1 = 1 + 12ϕ⎜ ⎟ ⎝ (1 − ϕ)2 ⎠ (2qa)
AUTHOR INFORMATION
Corresponding Author
where K* =
(21)
(20)
where lB is the Bjerrum length. 9193
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