1. Introduction

Sodium polystyrene sulfonate (NaPSS) is widely used in both industrial and medical applications [1]. It is used as an ion-exchanger or a cement superplastifier in industry [2], and also as a medication for patients with hyperkalemia [3]. Topical antimicrobial effect had also been reported [4]. Characteristics of the polymer solution have been described by several physical parameters, such as ionic strength [5], molecular weight [6], polymer concentration [7], and mechanical properties [8]. The mechanical properties to describe the dynamic structure of polymer solutions under different shear rate are often described by the complex viscoelastic modulus $G*(\omega )=G^{\prime }(\omega )+i{G}^{″}(\omega )$. The real part G’(ω) is the elastic modulus, which characterizes the ability of the material to store energy (as mechanical potential energy), and the imaginary part G”(ω) is the viscous modulus which characterizes the ability to dissipate energy (often in the form of heat) [9,10]. The viscous modulus G”(ω) of NaPSS polymer solutions has been measured by small-angle neutron scattering [11], dynamic light scattering [12], and conventional micro-rheometer [1]. However, the elastic modulus G’(ω) of NaPSS has not been well characterized, and the dynamic structure of NaPSS polymer solution is not yet well understood.

In general, the viscoelastic properties of NaPSS solutions depend on NaPSS molecular weight and concentration as well as the ionic strength of the solution. Conventional rheometry generally requires large sample volume (~several milli-liters); thus any method that requires only a smaller amount of solvent and chemicals is of interest. We decided to check if the rheological properties of NaPSS could be obtained with sample volume on the order of microliter and to apply a microrhelogical analysis to the understanding of the dynamic structure of NaPSS polymer solutions.

In this study, we investigated the microrheological properties of NaPSS (Mw = 70kDa) by oscillatory optical tweezers approach with frequencies ranging from 6rad/s to 6000rad/s. We studied the mechanical properties of NaPSS solutions with different polymer concentrations ranging from below to above the overlap concentration as well as the Critical Micelle Concentration (CMC). That includes both the dilute and the semi-dilute regimes [1,13,14]. We compared our experimental results with different theoretical models including the Maxwell and the Rouse models, and discussed the structural changes in the polymer solutions at different concentrations.

2. Experimental materials and methods

2.1 Sodium polystyrene sulfonate polymer

Sodium polystyrene sulfonate (NaPSS) is an anionic polyelectrolyte, with a linear chemical formula given by (C₈H₇SO₃ ^-Na⁺)_n., and the chemical structure shown in Fig. 1 . The polymer samples used in our experiments have a molecular weight of 70kDa with a polydispersity index Mw/Mn < 1.2 (product number 243051 from Sigma Aldrich). As NaPSS solutions rheology is very sensitive to the presence of ions in solutions, as for any type of polyelectrolyte solutions, de-ionized water was used as solvent in all our samples. The polymer solution structure depends on its concentration. The Critical Micelle Concentration is the polymer concentration where the molecules start to form micelles and alter the solution behavior [15]. The Critical Micelle Concentration (CMC) of NaPSS (Mw~70kDa), determined by surface tension measurement, was reported to be equal to 0.06mM [16]. The polymer overlapping concentration (C*) is the concentration where isolated polymer chains start to overlap and form transient network. At this concentration, the viscosity at zero shear rate increases significantly with a moderate increment in polymer concentration. The polymer overlapping concentration of NaPSS (Mw~70kDa) has been estimated to be 0.7mM [1].

 

Fig. 1 Chemical structure of Sodium polystyrene sulfonate (NaPSS).

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Average mesh size of NaPSS was estimated to be 5.89nm for 0.06mM NaPSS polymer solution (Mw ~82kDa) [17]. To characterize the network properties of the polymer solution via a continuum model [9], we used 1.5μm polystyrene beads as a probe to measure the mechanical properties of the polymer solution by active oscillation of the bead (embedded in the polymer solution) with oscillatory optical tweezers. As the size of probe particle is several times larger than the mesh size of the polymer, the network intrinsic inhomogeniety should not affect the average micromechanical properties sensed by the probe. As a consequence, characterization of the mechanical properties of the polymer solution via the continuum model is justified. In addition, this particular bead size (of 1.5μm) and (polystyrene) material were chosen for several reasons including sufficiently small spatial resolution (~1μm), relatively large trapping force (~tens of pico-Newton), and our familiarity with its characteristics as an optical probe. The possible effects of probe size and material on our results are discussed in more detail in Section 3 (Experimental results and discussion).

2.2 Optical tweezers setup

The main components of our experimental setup are schematically illustrated in Fig. 2 . A linearly polarized laser beam (wavelength = 1064nm) from an Nd: YVO₄ laser was expanded three times by a beam expander (BE₁) to fill the back aperture of an oil immersion objective (NA = 1.3, 100X), to serve as the trapping beam. The trapping laser power was controlled by a half-wave plate (λ/2) in conjunction with a polarized beam splitter (PBS). A mirror was mounted on a PZT-stage (PZT Mirror) which can be driven by a sinusoidal signal from a lock-in amplifier to generate an oscillating trapping beam. The telescopic relay lens (telescope) imaged the plane of the oscillating mirror onto the entrance aperture of the microscope objective so that there was no beam walk-off during the beam oscillation.

 

Fig. 2 A schematic diagram of the experimental setup. A linearly polarized laser beam (λ = 1064nm) was used to trap and oscillate a polystyrene bead; another laser beam (λ = 980nm) was used to track the position of the bead. A sinusoidal voltage from a lock-in amplifier was applied to a PZT mirror to generate the oscillating trapping beam. λ/2: half-wave plate; PBS: polarizing beam splitter; BE: beam expander; DM: dichroic mirror; QPD: quadrant photodiode; M₁, M₂,: mirrors.

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A second laser beam (from a cw laser; λ = 980nm) was used to track the position of the sample particle (a polystyrene bead) trapped and oscillated by the optical tweezers. The tracking beam was expanded by a beam expander (BE₂), and combined with the trapping beam via a dichroic mirror (DM₁) into the oil immersion objective. A QPD (Quadrant Photo Diode) was used to track the lateral position of the trapped particle. The displacement of the trapped particle on the xy-plane, perpendicular to the optical beam axis, can be deduced directly from the QPD’s output voltages via a standard calibration procedure [18]. The time response of our 2-d particle-tracking system is on the order of millisecond. We used the lock-in amplifier to measure the phase delay of the displacement of the probe particle relative to that of the focal spot of optical trapping beam; the latter was scanned by the reference signal provided by the lock-in amplifier to the PZT-mirror. The force constant in the linear regime of our optical tweezers for a polystyrene bead (with a diameter of 1.5μm) trapped in de-ionized water (with refractive n = 1.330) was determined to be approximately 15pN/μm at 6mW trapping power [19]. Variation in optical force constant (k_OT) due to the change in refractive index from 1.331 (for NaPSS polymer solution with 0.001mM concentration) to 1.414 (for NaPSS polymer solution with 10mM concentration) was calibrated according to the recipe given by Brau et al. [9].

3. Experimental results and discussion

We used 1.5μm polystyrene beads as probe, and we suspended them in NaPSS solutions with different concentrations. In the experiment, a single bead was trapped and oscillated with optical tweezers; a sinusoidal voltage with a peak-to-peak value of 0.1 voltage and an angular frequency ranging from 6 to 6000rad/s was applied to a PZT-stage with a mirror to scan the trapping laser beam (λ = 1064nm) such that the focal spot oscillated with an amplitude of approximately 36nm, while another laser beam (λ = 980nm) was used to track the amplitude and the phase delay of the particle relative to that of the focal spot of the trapping beam. The elastic modulus and the viscous modulus (G’, G”) as a function of the angular frequency (ω) were determined by the following equations [19,20]

The elastic modulus and the viscous modulus (G’, G”) of NaPSS solution as a function of the angular frequency (ω), at different polymer concentrations, are shown in Fig. 3 . All the data in Fig. 3 represent the mean value obtained by averaging over 6 repeated measurements under identical condition. The typical standard deviation is approximately ± 50% for G’, and ± 15% for G”. For the sake of clarity, error bars representing the standard deviation are not shown in Fig. 3. In contrast to the loss modulus G”(ω), which is proportional to ω for all polymer concentrations (in the range of 0.001mM to 10mM) and throughout the whole frequency range from 6 to 6000rad/s, the frequency response of the elastic modulus G’(ω) is quite different. In the low frequency range (~6 to 100rad/s), G’ is essentially independent of frequency for all polymer concentrations. In the higher frequency range (~100 to 6000rad/s), the value of G’ increases with frequency which can be approximated by a power law dependence. The characteristic exponent of this power law depends on polymer concentrations, varying from G’~ω^0.5 at low polymer concentration (~C < 1mM), to G’~ω^1.0 at higher polymer concentration (~C > 1mM).

 

Fig. 3 (a) The elastic modulus (G’), and (b) the viscous modulus (G”) as a function of angular frequency (ω) for eleven different polymer concentrations. In (a) the green solid line below the experimental data represents a power law dependence characterized [G’(ω) ~ω ^0.5] and the red dashed line above the experimental data represents a power law dependence characterized by [G’(ω) ~ω]; likewise, in (b), the red dashed line above the experimental data represents a power law dependence characterized by [G”(ω) ~ω].

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Since the loss modulus G”(ω) is linearly proportional to frequency (ω), the viscosity η, defined by η = G”/ω, is approximately constant, independent of frequency. Hence the zero-shear-rate viscosity defined by (${\eta }_0=\underset{\omega \to 0}{\lim }{G}^{″}/\omega$), can be approximated by the constant slope of G”(ω) in a linear plot of our experimental data associated with Fig. 3(b). The zero-shear-rate viscosity (η ₀) as a function of polymer concentration is shown in Fig. 4 . Obviously, the polymer solution viscosity cannot be lower than the viscosity of the pure solvent (i.e., water in this case). The viscosity of water at 27 °C is taken to be 0.0008513N*s/m² which is indicated by the purple dash line in Fig. 4.

 

Fig. 4 Zero-shear-rate viscosity as a function of concentration; the purple dashed straight line, with a constant value of 0.0008513Ns/m², represents the zero-shear-rate viscosity of pure water at 27°C.

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For NaPSS polymer concentrations (C) in the range of 0.001mM to 10mM investigated in this work, the polymer structure can be approximately divided into 3 regimes, each characterized by a different exponent in the power law which expresses the zero-shear-rate viscosity as a function of polymer concentration (confer Fig. 4). For polymer concentration C< 0.05mM, η ₀ ~C^0.04; for 0.05mM < C <1.0mM, η ₀ ~C^0.3 ; for C > 1.0mM, η ₀ ~C³ . The transition concentration (of ~0.05mM) from the 1st to the 2nd regimes is fairly consistent with the Critical Micelle Concentration (of 0.06mM reported in the literature [16]); and that from the 2nd to the 3rd regimes at the concentration (of ~1.0mM) is also consistent with the polymer-overlapping concentration (of ~0.7mM [1]). The polymer-overlapping concentration (C*) is revealed as the polymer concentration at which η ₀ increases drastically with a moderate increment in polymer concentration. These results indicate that microstructural changes in polymer solutions such as micelles formation and transient network formation can be revealed via this active microrheology approach.

While the scaling of loss modulus with polymer concentration, deduced from our experimental data, is compatible with the expectation from polymer theory [22,23], our experimental results for the elastic modulus G’(ω) deviate from the trend predicted by the transient network theory (Maxwell model) [22]. Maxwell model, which is a theory often used to describe polymer behavior, predicts that G’(ω) ~ω ² in the lower frequency regime (where ωτ_Μ << 1) and G’(ω) ~ω ⁰, i.e., independent of frequency (ω), in the higher frequency regime (where ωτ_Μ >> 1), where τ_Μ is the characteristic time of a relatively slow relaxation process of the transient polymer network. However our experimental data show the opposite trend with G’(ω) ~ω ⁰ in the lower frequency regime (~6 to 100rad/s), and G’(ω) ~ω ^0.5 to G’(ω) ~ω ¹ in the higher frequency regime (~100 to 6000 rad/s). The Rouse model [20,23] represents another theoretical model which describes the behavior of polymer solutions as micelles connected by elastic springs. According to the Rouse model, both G’ and G” are proportional to ω^0.5 at high frequency (where ωτ_Μ >> 1). At low polymer concentrations (~0.001mM to 1mM), our experimental data in the higher frequency range (~100rad/s to 6000rad/s) follow approximately the power law [G’ ~ω^0.5] predicted by the Rouse model. As the concentration increases, the exponent in the power law also increases, approaching G’ ~ω at the polymer concentration ~10mM. We speculate that such a change is due to the polymer entanglements and formation of transient network. Hence, our results indicate that the rheological behavior of NaPSS solutions may be a combination of the transient network effect and the micelle bridging effect, leading to an exponent of 0 to 1.0 in frequency response, in between the values of 0 and 2.0 predicted by the Maxwell model and the values of 0.5 to 2.0 predicted by the Rouse model. Quantitative interpretation of the details of the power law dependence requires further studies.

Two other parameters often used for the polymer behavior characterization are the crossing frequency (ω _c) and the transition frequency (ω _τ), which are discussed below in this section. The crossing frequency (ω _c), is defined as the frequency where G’ and G” crossed over. It is shown in the inset in the lower left of Fig. 5(a) . The inverse of the crossing frequency is often referred to as the polymer relaxation time (${\tau }_{\text{R}}=\text{1}/{\omega }_{\text{c}}$). In the regime where G” is larger than G’, the polymer is expected to behave more like a (dissipative) liquid than an elastic solid; in contrast, in the regime where G” is smaller than G’, the polymer is expected to behave more like an elastic solid than a dissipative liquid.

 

Fig. 5 (a) Crossing frequency of G’ and G”, and (b) Transition frequency of G’ as a function of polymer concentration. The inset figure of Fig. 5(a) shows that the crossing frequency (ω _c) is defined as the frequency where G’ and G” crossed over, and the inset figure of Fig. 5(b) shows that the transition frequency (ω _τ) is defined as the frequency where G’(ω) changes from a constant value to a power law dependence on frequency.

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The crossing frequency as a function of polymer concentration is shown in Fig. 5(a). In general, the crossing frequency decreases (and the relaxation time increases) as the polymer concentration increases due to increasing polymer entanglement effect. Similar effect was observed as the molecular weight of the polymer increases (experimental data not shown). For micelles structure formation (at a polymer concentration of C ~0.06mM), the relaxation time is often related to the lifetime of a hydrophobe in a micelle [20]. In the transient network theory, it is related to the elasticity of the polymer structure [20]. Previous studies on telechelic polymers have also indicated the possibility of two-step relaxation, where the Rouse model is applied to describe the relaxation mechanism of micelles structure at high frequencies, and the transient network theory accounts for the slow relaxation time of the polymer network [20]. Depending on specific parameters such as the molecular weight, the concentration, and the temperature of the polymer solution, a generic frequency dependence of G’(ω) and G”(ω) for a polymer solutions can have several crossing frequencies [24]. In the frequency range (~6.0rad/s to 6000rad/s) studied in our experiments, only one crossing frequency, where G’ > G” below the crossing frequency and G’ < G” above the crossing frequency, was observed as shown the inset figure of Fig. 5(a), except for samples with 5mM and 10mM polymer concentrations where the crossing frequency was apparently beyond our frequency range. We speculate that for our NaPSS samples (with Mw = 70kDa), the longer relaxation time (i.e., the lower crossing frequency) associated with the micelles structure is probably beyond our frequency range; and hence, not revealed in this study.

The transition frequency (ω _τ), defined as the frequency where G’(ω) changes from a constant value (independent of frequency) to a power law dependence on frequency (with a positive exponent), is shown in the inset in the lower left in Fig. 5(b), and determined by fitting the experimental data with a power law dependence in each of the two regimes, and subsequently the crossing point of the two fitted lines. We fit our experimental data from 6rad/s to 30rad/s with $G^{\prime }(\omega )=a{\omega }^0$ to define the constant value regime, and those from 2000rad/s to 6000rad/s with $G^{\prime }(\omega )=a{\omega }^b$ to determine the exponent in the power-law regime. The transition frequency (ω _τ) as a function of polymer concentration is shown in Fig. 5(b). Although the general trend of the dependence of the crossing frequency (ω _c) and the transition frequency (ω _τ) on polymer concentration, as shown in Figs. 5(a) and 5(b), respectively, looks somewhat similar, the physical meaning of latter is not clear and needs further investigation.

4. Summary and conclusions

In this study, we investigated the microrheological properties of NaPSS (Mw = 70kDa) solution with oscillatory optical tweezers to probe its viscoelasticity, in terms of its elastic modulus G’ and loss modulus G” as a function of angular frequency (ω). For polymer concentrations ranging from 0.001mM to 10mM, from dilute to semi-dilute regimes with or without micelles and polymer network formation. The zero-shear-rate viscosity, the crossing frequency, and the transition frequency were defined and determined from the experimental data as a function of polymer concentration. The power law dependence of the zero-shear-rate viscosity on the polymer concentration clearly reveals 3 different concentration regimes, each with a distinct exponent, which is consistent with the formation of micelles structure and polymer network reported in the literatures. Quantitatively, the Critical Micelle Concentration and the polymer overlapping concentration deduced from these results are also comparable to the corresponding values reported in literature. Our experimental results indicate that the rheological behavior of NaPSS solution may be a combination of the transient network and the micelle bridging effect, predicted by the Maxwell model and the Rouse model, respectively. Although a deeper understanding of the physics and the chemistry of the polymer behavior requires further studies, we have clearly demonstrated that subtle structural changes in polymer solutions can be probed by active microrheology with a single particle trapped and oscillated by oscillatory optical tweezers. Potential biological applications in microrheology of bio-fluid samples, available only in very small quantity, such as cerebral-spinal fluid, blood plasma, vitreous and synovial fluids, as well as studies of a large diversity of polyelectrolyte/medium combinations using small amount of solvent and chemicals look very promising.