Viscoelastic Properties of Organosilicon Fluid Interlayer at Low-Frequency Shear Deformations

1. Introduction

The shear elasticity of liquids is observed at high frequencies of 10¹⁰ - 10¹² Hz and is explained by the diffuse mobility of individual particles. The frequency of transfers of liquid particles from one equilibrium position to another has exactly this order of magnitude. Liquid molecules do not have time to react to external effects of such frequencies and higher, and the liquid shows properties characteristic of solid bodies.

The first studies of liquids with shear deformations were for strongly viscous liquids [1,2]. The early works of Mason [3] are known, which discovered the shear elasticity of viscous polymer liquids by impedance method. Many works [4,5,6,7,8] are devoted to the study of shear mechanical properties of viscous liquids at high (MHz) frequencies. At low frequencies, this phenomenon is observed primarily in viscous liquids or in liquids in a supercooled state. It is shown that for a number of pure liquids the frequency dependencies of the shear elasticity and viscosity are described by the Maxwell model. Low-frequency (10⁵ Hz) shear elasticity in low-viscosity fluids was first detected in work [9,10]. Study of low frequency viscoelastic behavior of fluids away from phase transitions was carried out in works [11,12,13,14,15,16,17,18]. These studies confirm that in fluids along with the high-frequency process there is a low-frequency viscoelastic relaxation process with a relaxation time exceeding classical estimates. It is likely that the collective effects caused by the interaction of large molecular groups play a key role in some cases. Therefore, the relaxation will require a coordinated movement of many particles, and the low-frequency shear elasticity of liquids may be related to their micro-heterogeneous structure.

This work investigates the viscoelastic characteristics of organosilicon fluids using an acoustic resonance technique. Polymer fluids exhibit more complex structural organization compared to simple liquids, where collective intermolecular interactions are expected to play a significant role. New experimental data on viscoelastic relaxation in organosilicon fluids would provide valuable insights into the underlying mechanisms of low-frequency viscoelastic relaxation in liquid systems.

Organosilicon fluids demonstrate remarkable thermal stability across broad temperature ranges, making them suitable for diverse applications including hydraulic and diffusion systems, heat transfer media, and liquid dielectrics, and their use in the biomedical field is known. Their implementation extends to instrument oils, lubricating greases, damping fluids, polishing compositions, paint and lacquer additives, and as base components for consistent lubricants [19,20,21,22,23,24,25,26,27] or polymer matrices for developing functional materials with tailored characteristics [28]. This extensive application spectrum underscores the importance of investigating the viscoelastic properties of these fluids.

2. Materials and Methods

Polyorganosiloxane fluids are characterized by low glazing temperatures, good dielectric and hydrophobic properties, significant compressibility and low surface tension coefficient. Their viscosity has low temperature dependence. The characteristics of the mechanics and rheology of these substances are due to the high flexibility of polymer siloxane chains [19]. Linear polymethylsiloxane fluids of PMS brand were used for the studies, which have a linear structure (CH3)₃Si–O–[Si(CH₃)₂O]_n–Si(CH₃)₃ and meet the requirements of GOST 13032-77 [29]. The numbers in the brand name indicate the viscosity of the PMS fluid in cSt.

The study of viscoelastic characteristics of liquid media is effectively carried out by means of dynamic methods, which provide for cyclic deformation of the liquid with a given frequency of shearing oscillations. When applying sinusoidal varying stress $\mathrm{\sigma }={\mathrm{\sigma }}_0\cos \mathrm{\omega }t$, with a frequency of ω = 2πf, the deformation will also sinusoidal varies $\mathrm{\epsilon }={\mathrm{\epsilon }}_0\cos \left(\mathrm{\omega }t-\mathrm{\theta }\right)$, where θ is the phase shift between stress and deformation. In this case, the complex shear modulus G* = G′ + iG″ is considered. For the rheological model of a viscoelastic material proposed by Maxwell, the complex shear modulus G* is expressed as follows:

$$G*=G_M{\displaystyle \frac{{\mathrm{\omega }}^2{{\mathrm{\tau }}_M}^2}{1+{\mathrm{\omega }}^2{{\mathrm{\tau }}_M}^2}}+\mathrm{i}\mathrm{\omega }{\displaystyle \frac{{\mathrm{\eta }}_M}{1+{\mathrm{\omega }}^2{{\mathrm{\tau }}_M}^2}}.$$

(1)

The real part of expression (1) $\mathrm{Re}G*=G^{\prime }(\mathrm{\omega })$ is called the dynamic modulus of elasticity for the frequency ω, or the storage modulus. $\mathrm{Im}G*=G^{″}\left(\mathrm{\omega }\right)$ is the loss modulus. The quantity $\mathrm{\eta }=G^{″}/\mathrm{\omega }$ plays the role of viscosity. Energy losses are characterized by the tangent of the angle of mechanical losses $\tan \mathrm{\theta }=G^{″}/G^{\prime }$. Therefore, the higher the dissipation, the greater the angle θ. Maximum energy loss corresponds to peak G″ when force exposure time coincides with stress relaxation time ${\mathrm{\tau }}_M$. From equation (1), it follows that at low frequencies (ωτ << 1) the medium behaves as a viscous liquid, and with increasing frequency the storage modulus begins to increase, and at very high frequencies (ωτ >> 1) the medium behaves like an elastic solid.

The low-frequency viscoelastic properties of polymethylsiloxane fluids are investigated by acoustic resonance method using a piezoelectric resonator [30,31,32]. This method covers a wide range of viscosities from 10^-3 to 10⁵ Pa∙s and is applicable to liquid layers 1-100 μm thick. The piezoelectric crystal is fixed in the points of the nodal line and oscillates at the main resonance frequency, creating tangential displacements (see fig. 1). The liquid is applied onto one end of the horizontal surface of the piezoquartz and covered with a cover-plate made of fused quartz. The cut of the X-18.5° crystal, which has a zero Poisson coefficient on the working surface, provides exclusively shear deformations in the liquid layer, forming in the layer standing shear waves propagating along the z-axis with wave equation:

$$\mathrm{\rho }{\displaystyle \frac{{\partial }^2\mathrm{\xi }(\mathrm{z},t)}{\partial t^2}}=G^{*}{\displaystyle \frac{{\partial }^2\mathrm{\xi }(\mathrm{z},t)}{\partial \mathrm{z}{}_2}},$$

(2)

where the displacement of liquid particles ξ(z, t) along the x-axis is described by the expression $\mathrm{\xi }(\mathrm{z},t)=\mathrm{A}\exp \left[\mathrm{i}(\mathrm{\omega }t-\mathrm{\kappa }\mathrm{z})\right].$

Figure 1. Piezoelectric quartz crystal with additional coupling.

Figure 1. Piezoelectric quartz crystal with additional coupling.

The influence of the liquid in the layer is manifested in the change of resonance characteristics of the system. An increase in the resonance frequency relative to the free quartz frequency indicates conservative forces in the liquid layer. While the dissipative forces of viscous friction would cause a reduction in the resonance frequency of the system.

It is assumed that the liquid layer has viscoelastic properties, represented by the complex shift module G* = G′ + iG″. Then the complex shift of the resonant frequency of the piezoelectric quartz crystal can be determined from the equality of the impedance of the liquid and the crystal and expressed as follows [30,31,32]:

$$\mathrm{\Delta }\mathrm{\omega }={\displaystyle \frac{2\mathrm{S}G*\mathrm{\kappa }}{M\mathrm{\omega }}}\cdot {\displaystyle \frac{1+\cos (2\mathrm{\kappa }H-\mathrm{\varphi })}{\sin (2\mathrm{\kappa }H-\mathrm{\varphi })}},$$

(3)

where S is the contact area of the liquid and the quartz, H is the thickness of the liquid layer, M is the mass of the quartz, φ = φ′ + iφ″ is the complex phase shift when the viscoelastic wave is reflected from the boundary of the liquid - cover plate, and κ = β - iα is the complex wave number of the liquid.

With the oscillation of the piezoquartz the cover-plate of sufficient mass remains virtually immobile due to weak coupling through the liquid layer. So, there is no phase shift (φ = 0), which corresponds to the complete reflection of the wave energy. As a result, the real and imaginary parts of the complex shift of the linear frequency ∆f * = ∆f ′ + i∆f ″ take the following form [27]:

$$\mathrm{\Delta }{f}^{\prime }={\displaystyle \frac{S{G}^{\prime }\mathrm{\beta }}{4{\mathrm{\pi }}^{2}M{f}_0\cos \mathrm{\theta }}}\cdot {\displaystyle \frac{\sin 2\mathrm{\beta }H-\tan \raisebox{1ex}{$\mathrm{\theta }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\sinh (2\mathrm{\beta }H\cdot \tan \raisebox{1ex}{$\mathrm{\theta }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)}{\cosh (2\mathrm{\beta }H\cdot \tan \raisebox{1ex}{$\mathrm{\theta }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)-\cos 2\mathrm{\beta }H}},$$

(4)

$$\mathrm{\Delta }{f}^{″}={\displaystyle \frac{S{G}^{\prime }\mathrm{\beta }}{4{\mathrm{\pi }}^{2}M{f}_0\cos \mathrm{\theta }}}\cdot {\displaystyle \frac{\sin 2\mathrm{\beta }H\cdot \tan \raisebox{1ex}{$\mathrm{\theta }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\sinh (2\mathrm{\beta }H\cdot \tan \raisebox{1ex}{$\mathrm{\theta }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)}{\cosh (2\mathrm{\beta }H\cdot \tan \raisebox{1ex}{$\mathrm{\theta }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)-\cos 2\mathrm{\beta }H}},$$

(5)

where f₀ is the intrinsic resonance frequency of the quartz, tan(θ/2) = α/β. These expressions show that for a viscoelastic fluid, the frequency shifts are functions of the layer thickness H, and with an increase in the latter, damped oscillations of the frequency shifts are observed.

According to expressions (4) and (5), when the thickness H is much smaller than the wave length λ established within the liquid (H << λ), the complex shear modulus can be determined as:

$$G*={\displaystyle \frac{4{\mathrm{\pi }}^{2}M{f}_{\mathrm{o}}\mathrm{\Delta }f*H}{S}}$$

(6)

So the resonance frequency shift of piezoelectric quartz is inversely proportional to the thickness of the liquid layer. The ∆f ″ is determined as half of the change in the resonance curve width (fig.1). The tangent of the loss angle is expressed as:

tgθ = G″/G′ = Δf ″/Δf ′.

(7)

Consequently, for determining the complex shear modulus of liquids using thin layers (H ≪ λ), it suffices to analyze the dependence of both real and imaginary frequency shifts on the reciprocal of layer thickness (1/H). In the present experimental study, we employed an X-18.5° cut quartz resonator with a fundamental resonance frequency of 73.18 kHz, having a mass of 6.82 g. The cover plate contact area was 0.2 cm². Liquid layer thickness was precisely controlled using optical measurement techniques. For reproducibility of the measurement results, it is particularly important to thoroughly clean the working surfaces of the quartz and cover-plate, which facilitate good wetting [32].

3. Results and Discussion

The shift of the oscillating system resonance frequency is influenced by an amplitude of the shear deformation [33,34]. When the amplitude of the oscillation increases, the resonance curves of the system deform. This indicates the non-linear nature of the shear elastic properties of the liquids [33,34,35,36]. Figure 2a shows normalized resonance curves for the free piezoquartz and the piezoquartz loaded with a PMS-100 fluid layer of thickness H = 2.1 μm at maximum amplitude A of 15 Å in both cases. The Fabry-Perot interferometer principle was used to determine the amplitude of the oscillation of piezoelectric quartz A. In this method, one of the mirrors is an optically polished end surface of a quartz resonator. The method is described in [35,36]. Analysis of the graphs shows that the resonance frequency of the loaded quartz is higher than that of the free piezoquartz, which corresponds to a positive shift in the resonance frequency. At the same time, the resonance curve broadens, showing an increase in the system’s damping. On the change of the width of the resonance curve it is possible to obtaine the imaginary shift of the resonance frequency Δf ″, defined as half the change in the width of the resonance curve (Fig. 2a). It should also be noted that the resonance curves of free and loaded piezoquartz are almost symmetrical, which indicates the lack of non-linearity of the shear properties of the liquid. Figure 2b shows resonance curves at different forcing force values corresponding to the maximum amplitude of 15 and 65 Å. For ease of comparison, the amplitude of the curves is also normalized to unit. As can be seen from the figure with increasing amplitude of oscillations up to 66 Å resonance curve deforms and acquires pronounced asymmetry.

The A/H ratio between oscillation amplitude and liquid interlayer thickness can be used to measure angular deformation. Figure 3 shows the dependences of the components of complex shear modulus G′ and G″ on the deformation magnitude (A/H)^1/2 for PMS-100 fluid. The figure shows that at small angles of deformation the dependences are linear, and the shear modulus remains constant. It indicates that linear elasticity is present. Assuming that the liquid has an equilibrium supramolecular structure with final strength and a relatively large relaxation period, it is obvious that at small angles of shift deformation the liquid structure remains unbroken. At a certain critical angle, the equilibrium structure will begin to break down, which will lead to changes in the viscoelastic properties of the liquid. Such behaviour of viscoelastic properties, which is typical for dispersed systems, has been observed for other low-viscosity liquids at low-frequency shear deformations [33,34,35,36].

Measurements of the complex shift modulus of polymethylsiloxane fluids at small thicknesses of liquid interlayer in [37,38,39] were performed at small amplitudes when the resonant curve is symmetrical and corresponds to the linear elasticity region. The values of the storage modulus G′ and the mechanical loss angle tanθ were calculated using formulas (6) and (7). Figure 4 present the dependencies of G′ and tanθ on the logarithm of viscosity lgη of PMS fluids. As can be seen from the figure, the shear modulus increases by more than an order as the viscosity of the fluid increases. The tangent of the mechanical loss angle passes through a maximum as viscosity increases, but remaining less than unity.

Assuming that the observed viscoelastic relaxation conforms to the Maxwell model, the frequency of the relaxation process f_M = f₀∙tanθ in PMS fluids should be below the frequency of the shear oscillations applied in the experiment. The relaxation frequency of PMS-100 fluid is equal to 37 kHz, and that of PMS-900 is 59.2 kHz. The relaxation frequency could have been directly determined experimentally at the given shear frequency for a fluid with a viscosity greater than 900 cSt but less than 5000 cSt, where tanθ would equal unity. Table 1 presents the calculated resonance frequency values f_M for all investigated PMS fluids with density ρ, which have the storage modulus G′ and mechanical losses tanθ, obtained at temperature t. The last column of the Table shows the effective viscosity values calculated according to Maxwell’s rheological model using the formula:

$${\mathrm{\eta }}_M={\displaystyle \frac{G^{\prime }(1+t{\mathrm{g}}^2\mathrm{\theta })}{2\mathrm{\pi }{f}_{\mathrm{o}}t\mathrm{g}\mathrm{\theta }}}$$

(8)

According to Table 1, the calculated viscosities η_M for low-viscous PMS fluids are significantly higher than nominal values. For instance, in the case of PMS-100 with a dynamic viscosity of 97 mPa∙s, the effective viscosity η_M calculated using the Maxwell formula (8) is more than three times higher and is equal to 0.32 Pa∙s. Figure 5, which plots the effective viscosity of PMS-100 against shear deformation correlating with the data of G* presented in Figure 3, illustrates a shear-thinning behavior. The graph evidences that viscosity decreases with increasing shear deformation and subsequently asymptotically approaches the standard (nominal) viscosity value. For high-viscosity PMS fluids, the rise in nominal viscosity accompanying increased molecular weight reduces molecular diffusion and energy dissipation and so the Maxwell viscosity computed from rheological model proves lower than the standard value. It should be noted that the single-relaxation-time Maxwell model provides an approximate description of real liquid behavior. However, it can be assumed that at oscillations with small shear deformation, the liquid exhibits an equilibrium supramolecular structure. This may explain the observed high effective viscosities for low-viscosity PMS fluids and anomalously long relaxation times, where collective interactions of molecules become significant. In contrast, the nominal viscosity of PMS likely corresponds to a liquid with a broken spatial structure, corresponding to high-frequency relaxation.

Low-frequency relaxation in liquids can be compared with the slow λ-relaxation process in amorphous polymers, which is observed above their glazing temperature and is explained by the breakdown and recovery of the micro-volume physical nodes of the molecular network [40,41]. The cluster model of liquids we are developing to explain the low frequency shear elasticity assumes the existence in liquids and amorphous materials of fluctuating dynamic structural micro-heterogeneity - clusters capable of formation and disintegrate over time [42,43,44]. Thus, the liquid represents a micro-heterogeneous media consisting of ordered and unordered areas. The lifetime of clusters is significant due to the large number of bound molecules in the cluster and corresponds to a long relaxation time. Considering the relaxation time to be the reciprocal of the relaxation frequency, which corresponds to the maximum mechanical losses, the study [43] established an exponential temperature dependence of the relaxation time and determined the activation energy of the low-frequency relaxation process for a polyethylsiloxane polymer fluid, U≈26 kJ/mol. Consequently, low-frequency viscoelastic relaxation in polymer (and possibly other) liquids can be classified as a low-activation process. It is noteworthy that the found activation energy of the low-frequency relaxation process in the polyethylsiloxane fluid coincides with the delocalization energy of the atom Δε ≈ (20–25) kJ/mol in silicate glasses and their melts with siloxane bonds [43,45,46] in line with generic conclusions of relaxation character in amorphous materials [47,48,49].

If the liquid possesses shear elasticity and the tangent of the angle of mechanical losses is less than unit, then the distance at which the shear wave decays can be equal to several wavelengths, which enables the study of their propagation [44,50,51]. In [30,31,37,38,39], the lengths of shear waves λ and viscoelastic parameters G′ and tanθ of some polymethylsiloxane and other liquids were experimentally determined using an ultrasonic interferometer on shear waves at a shear oscillation frequency of 74 kHz. The studies of shear wave propagation indicate that low-frequency shear elasticity is a property of the liquid in bulk and is not related to boundary phenomena. However, the method for determining viscoelastic parameters based on shear wave propagation is well suited for studying only viscous liquids. At thicknesses comparable to the wavelength, the normal component of piezoelectric quartz oscillations, which is always present in real crystals, begins to influence the experimental data. As the viscosity of the liquid increases, this influence decreases but does not completely disappear. Therefore, certain assumptions in the method yield shear modulus values based on the experimental shear wave length that are slightly lower than those presented in Table 1. Using the data in Table 1 for calculations obtained at H << λ will allow to obtain more accurate shear wave parameters.

From the theory of the acoustic resonance method for determining the complex shear modulus of liquids, the wavelength λ will be determined by the position of the maximum attenuation values [27,30,31]:

$$H={\displaystyle \frac{\mathrm{\lambda }}{2}}n.$$

(9)

Accordingly, the first maximum of attenuation will be observed at a thickness of the liquid layer equal to λ/2. Figure 6 shows the dependencies of the components of the resonance frequency shifts Δf ′ and Δf ″ on the layer thickness H for PMS-400, calculated using equations (4) and (5) using the experimentally obtained data G′ and tanθ from Table 1. According to expression (9), the shear wavelength λ in PMS-400 is determined from the curve dependence of Δf ″(H), it is equal to 180 μm, while experimental measurements give a value of 160 μm [38,39].

To refine the wavelength data for the homologous series of PMS fluids the Δf ″(H) dependences were calculated using viscoelastic parameters obtained by the method at H << λ. Figure 7 presents similar curves for the imaginary shift of the resonance frequency as a function of thickness for a series of PMS fluids. As can be seen from the figure, increasing thickness leads to damped oscillations of Δf ″, and the length of the shear waves increases with increasing fluid viscosity. Consequently, the wave penetration depth also increases, which is equal to the distance at which the wave amplitude decreases by a factor of e.

Table 2 presents the shear wave lengths for PMS fluids obtained using expression (9) from the positions of the first maximum of the Δf ″(H) dependences shown in Figure 7. It turned out that the obtained λ exceeded the experimental values in [30,31,37,38,39] by approximately 10-12%, which may be important for modeling the rheological properties of fluids and influence their subsequent practical application. The last column of Table 2 shows the attenuation coefficients α, calculated using expression (10) [50]. It is evident from the table that the attenuation coefficients decrease with increasing viscosity of PMS fluids.

$$\mathrm{\alpha }={\displaystyle \frac{2\mathrm{\pi }}{\mathrm{\lambda }}}\tan {\displaystyle \raisebox{1ex}{$\mathrm{\theta }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}.$$

(10)

With increasing molecular weight of the PMS fluid in the homologous series, the shear wavelength increases, while the attenuation coefficients decrease. This indicates a change in the liquid structure. The greater the molecular weight of the polymer, probably, the larger the sizes of the ordered areas in the liquid become. Larger associates are formed due to the increase in possible intermolecular and intramolecular interactions. The attenuation of the shear wave amplitude with increasing molecular weight is associated with energy dissipation and indicates that the elastic response begins to increasingly dominate the viscous response. This correlates with the effective viscosity data for high-viscosity PMS fluids compared to the nominal viscosity in Table 1. In addition, as shown in Figure 4a, the shear modulus also increases, and the fluid exhibits more pronounced elastic properties inherent to solids with increasing molecular weight.

4. Conclusions

In the work, a study of viscoelastic properties of polymethylsiloxane fluids was carried out in dynamic mode using an acoustic resonance method. It has been shown that the relaxation time of the observed viscoelastic relaxation process is much longer than the settled lifetime of individual liquid particles, and according to the cluster model of the fluid is due to microheterogeneity of structure. The change of their physical-mechanical and viscous properties depending on the amplitude of shear oscillations is shown. Values of shear wave lengths and attenuation factors are determined.

The results obtained are of considerable practical value due to the wide use of polymethylsiloxane fluids in modern technologies. The data on the viscoelastic properties of PMS can be used to develop science-based approaches for creating new organosilicon materials and composite materials based on them with nano-fillers for application in various fields of technology and technology, in particular, to create high-performance lubricants for precision instruments. Data on shear wave and attenuation parameters can be of practical importance for the selection of optimal polymethylsiloxane fluids used in measurement technology such as damping and working fluids, and taking into account their vibration and acoustic characteristics can contribute to improving the efficiency of devices.