Laser speckle probes of relaxation dynamics in soft porous media saturated by near-critical fluids Dmitry A. Zimnyakov,1,* Sergey P. Chekmasov,1 Olga V. Ushakova,1 Elena A. Isaeva,1 Victor N. Bagratashvili,2 and Sergey B. Yermolenko3 1 2

Department of Physics, Saratov State Technical University, Polytechnicheskaya Str. 77, Saratov 410054, Russia

Department of Atomic-Molecular Laser Technology, Institute on Laser and Information Technologies of the Russian Academy of Sciences, 2 Pionerskaya Str., Troitsk, Moscow Region 142092, Russia 3

Department of Correlation Optics, Chernivtsi National University, Kotsubinsky Str., 2, Chernivtsi 58012, Ukraine *Corresponding author: [email protected] Received 12 November 2013; revised 14 December 2013; accepted 16 December 2013; posted 18 December 2013 (Doc. ID 200769); published 3 February 2014

Speckle correlation analysis was applied to study the relaxation dynamics in soft porous media saturated by near-critical carbon dioxide. The relaxation of soft matrix deformation was caused by a stepwise change in the fluid pressure. It was found that the deformation rate in the course of relaxation and the relaxation time strongly depend on the temperature of the system. The values of relaxation time reach their maximal values in the vicinity of the critical point of saturating fluid. The contributions of hydrodynamic relaxation of the fluid density and viscoelastic relaxation of the porous matrix to its creeping are analyzed. © 2014 Optical Society of America OCIS codes: (030.6140) Speckle; (290.4210) Multiple scattering; (290.7050) Turbid media. http://dx.doi.org/10.1364/AO.53.000B12

1. Introduction

Numerous experimental and theoretical studies over the past three decades demonstrated high potential of quasi-elastic scattering (QELS) techniques in applications to the probes of multiply scattering systems with complex structure and dynamics. Various QELS methods based on correlation analysis of the speckle intensity fluctuations in multiply scattered light are very sensitive to microscopic movements of scattering sites in a probed medium. Consequently, they allow one to examine the peculiarities of complex dynamics of a probed system for a wide range of spatial and temporal scales (from a few nanometers to tens of micrometers and from submicroseconds up to thousands of seconds) [1–10]. Despite the abundance of names for these methods introduced in literature (diffusing light spectroscopy, 1559-128X/14/100B12-10$15.00/0 © 2014 Optical Society of America B12

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light correlation spectroscopy, speckle correlometry, etc.), they all come down to a few core principles. These principles include gathering of a stochastically modulated optical signal from an arbitrarily chosen unit coherence volume (a single speckle) in a multiple scattered optical field and subsequent time domain correlation analysis of the acquired signal. In addition, the ergodicity of processed dynamic speckle patterns can give additional advantages, such as the increasing robustness and the performance of data processing. These advantages occur due to the additional ensemble averaging in the spatial domain [9,11–13]. A vast variety of stochastic processes in multiple scattering random media probed with the use of speckle correlation can be classified as the relaxation transition of the medium from one stable structural state to another state (stable or metastable). Among them are drying and imbibition in porous media [14,15], aggregation in colloidal systems, and slow dynamics in glassy soft matter [16–19], thermally

induced modification of biological tissues [20,21], formation of domain structures in two-component mixtures due to phase separation [22], fluid dynamics [23], solid deformations [24–26], etc. In all these cases the correlation analysis of intensity fluctuations of speckle-modulated scattered light provides the comprehensive information concerning the behavior of relaxing systems in the wide range of time scales. In this paper we present the results of experimental study and phenomenological interpretation of laser light dynamic scattering by the system’s “soft porous medium—saturating near-critical fluid” in the course of an isothermal transition from one thermodynamically stable state to another. Such a transition, which is induced by a stepwise change in the pressure of the saturating fluid, is accompanied by various relaxation processes. These processes are the hydrodynamic relaxation of the saturating fluid density in the porous matrix and the viscoelastic relaxation (creeping) of the matrix. In turn, the relaxation-induced changes in the matrix structure should cause the expressed dynamic speckle modulation of laser light propagating in the system. The observed speckle dynamics should be very sensitive to the temperature variations because of the strong impact of the temperature on the fluid properties near the critical point. The motivation of this study is that the supercritical fluids are excellent carrying agents for molecular and permolecular mass transfer in dispersive systems [27–31]. Besides, the microfluidics in random capillary systems is the object of particular research interest in many interdisciplinary areas. This is due to its very significant influence on structural and functional properties of various materials and systems [32–35]. Therefore, the development of physically substantiated and easy-to-use speckle correlation techniques for analysis of diffusion-like and filtration-like fluid transfer in soft porous media is valuable for various technological and biomedical applications. It should be noted that the optical methods (for example, the correlation spectroscopy [36,37] or the fiberoptical reflectometry [38,39]) were traditionally

used to study the near-critical effects (in particular, the phenomenon of critical opalescence or the pressure and temperature dependencies of the fluid density in the vicinity of the critical point). However, the optical diagnostics was applied mainly to singlecomponent fluids and binary mixtures but not to spatially confined multiphase media with near-critical components such as fluid-saturated mesoporous systems and dense suspensions of micro- or nanoparticles. The only exceptions are a few works related to coherent anti-Stokes Raman scattering (CARS) spectroscopy of near-critical fluids in nanoporous glasses [40,41] but the major limitation of this approach is the necessity of using an almost transparent matrix porous substance. In this regard, the discussed approach can extend the possibilities of optical diagnostics of near-critical systems. 2. Experimental Technique

The dynamic speckle modulation of laser light multiply scattered by the system’s “soft porous layer—near-critical fluid” was analyzed in our experiments in the course of an isothermal transition of the system from one equilibrium state to another. The transition was induced by the abrupt partial depressurization of a near-critical fluid (carbon dioxide) filling a high-pressure optical cell with a porous layer inside. The scheme of the experimental setup is presented in Fig. 1. The temperature-stabilized highpressure optical cell with sapphire windows contained the porous layer placed normally to the cell axis. A beam of single-mode He–Ne laser (GN-5P, the product of JSC «PLASMA», Russia; 633 nm, 5 mW output, linear polarization, the long-term instability of output is not more than 2%) shined on the layer through the sapphire window; forward scattered light was detected by a CMOS camera without a lens (Thorlabs DCC1545M, the bit depth is 8 bits, the resolution is 1280 × 1024 pixels, the pixel size is 5.2 μm × 5.2 μm, the quantum efficiency and the threshold sensitivity of the sensor are equal to ≈0.65 and ≈2 nW∕ cm2 at 633 nm) axially aligned with the probe beam at a distance of 300 mm from the cell (this corresponds to the near-zone condition).

Fig. 1. Scheme of the experimental setup. 1—He–Ne laser; 2—foamed plastic jacket; 3—aluminum housing; 4—high-pressure cell; 5—sapphire window; 6—high-pressure capillary; 7—pressure sensor; 8—quartz oscillator; 9—heater; 10—thermocouple; 11—sample holder; 12—sample under study; 13—CMOS camera. 1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS

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Slow relaxation of the examined system after the depressurization caused gradually decaying lowfrequency speckle dynamics, which was captured as the bitmap frame sequences (the capture rate was 100 frames per second and the used length of sequences was adequately large for the relaxation analysis). The static character of captured speckle patterns before and at the final stage of transition between the equilibrium states allows us to conclude that the scattering by the fluid density fluctuations does not cause any remarkable contribution to the observed speckle dynamics, even in the vicinity of the critical point. The size of the captured subframes was equal to 100 × 100 pixels, and the area of interest typically covered 15–20 speckles (the average speckle size was about 30 μm). It was enough for the robust ensemble averaging in the course of data analysis. Filter paper (type F with density of 75 g∕m2 ) and Teflon films with fibrillar porous structure (PTFE thread sealing tape) were used as the porous layers. The thickness of the samples was equal to 100
5 μm. The temperature of the cell was stabilized with an accuracy of
0.02°K and was varied from 298.16 °K to 306.16 °K with a step of 0.5 °K from one experiment to another. In order to provide the appropriate temperature stabilization conditions, the optical cell made of stainless steel was placed inside a massive aluminum housing. The housing was provided with narrow windows and channels for the incoming laser beam, the outgoing scattered light, the high-pressure capillary, and the thermocouple placed inside the cell. The housing was coated by a thick foamedplastic jacket. It was heated by a complementary pair of high-power transistors placed on opposite sides. Temperature-dependent frequency deviation of a quartz oscillator placed on the cell inside the housing was used for the feedback control with PC. The oscillator data were synchronized with the thermocouple data for minimization of systematic errors in the temperature stabilization. The initial value of pressure in the cell was always set equal to 7.70
0.02 MPa and the pressure drop in the course of partial depressurization was chosen equal to 0.30
0.02 MPa; thus, the initial and final values of the pressure were slightly above the critical value of pC ≈ 7.3773 MPa for CO2 [42]. Typically, the stabilization time for the pressure in the cell after the drop was on the order of 0.2 s; therefore the recording of speckle patterns was started after pressure stabilization. The extra volume of saturating fluid flowing out from the porous layer in the course of relaxation appeared too small in comparison with the cell volume (2.1 cm3 ); rough upper estimates of the effluent fluid volume give the value of the order of 3∕5 × 10−4 cm3 . No pressure oscillations exceeding the absolute error limit of the used pressure sensor (0.02 MPa) were observed in the course of the speckle correlation experiments. Consequently, we can consider the effect of the fluctuations of external B14

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parameters (the pressure and the temperature) on the speckle dynamics as a negligible one. 3. Speckle Data Analysis: Techniques and Results

The time series of speckle intensity fluctuations recovered from the obtained subframe sequences were used to calculate the normalized correlation functions g2 τ at the initial stage of relaxation (approximately 0.5–3 s after the stepwise pressure change) with the use of the following procedure: g2 τ  g2 k · Δt (Pnn N−k

m ) M m m 1 Im 1 X nn1 nk − hI i I n − hI i :   Pnn1 N
m M m1 I n − hI m i 2 nn1 (1) Here, M is the number of statistically independent coherence areas (speckles) used for ensemble averaging, n1 corresponds to the first frame in the processed sequence containing N frames, hI m i  P 1 N m I n is the averaged brightness level 1∕N nn nn1 of mth pixel for the time interval from n1 · Δt to n1  N · Δt (the origin corresponds to the moment of stepwise change in the pressure; Δt is the sampling interval as the reciprocal of the frame rate), and I m n is the instantaneous brightness of mth pixel at the moment n · Δt. It should be noted that the procedure used (calculation of the normalized intensity correlation function for each coherence area with subsequent averaging over the ensemble of speckles) differs from the procedure usually adopted in multispeckle diffusing wave spectroscopy (the averaging of the unnormalized correlation functions over the speckle ensemble with the subsequent normalization by the average speckle intensity). This equation actually defines the normalized unbiased single-point temporal correlation function of speckle intensity fluctuations, which is averaged over the set of detection points located in the statistically independent coherence areas (speckles). Calculated in this way, the correlation function is equal to 1 at the origin, tends to zero at large time scales, and is related to the intensity correlation function calculated in the framework of the discrete scattering model (see Section 4). The value of M was taken as equal to 10. The maximal values of k were selected as substantially less than N in order to exclude the influence of edge effects on the calculated correlation functions (typically N∕20). The sequence length, N, was chosen as adequately small (350–400) to minimize the effect of nonstationarity of speckle intensity fluctuations occurring at large time scales. Figure 2 displays the typical correlation functions, g2 τ, for paper samples at the initial stage of relaxation, which were obtained for various values of the system temperature. The dependencies of the correlation time, τc , of intensity fluctuations at the initial stage of relaxation on the thermodynamic temperature of

with the initial point n  K. The introduced value of Σ reaches the maximal (unit) value at the initial stage of relaxation and asymptotically falls to zero when the system approaches the equilibrium state after the depressurization. In this case, the observed

speckle patterns go into a static state. The magnitude of Σ is calculated for a sampling interval significantly exceeding the speckle correlation time and characterizes the slow nonstationary changes in speckle dynamics. Thus, the intensity correlation function characterizes speckle dynamics at small time scales (Fig. 3), and the decay time of the normalized power of speckle intensity fluctuations characterizes the duration of the transition of speckle patterns from a dynamic to static state in the course of relaxation. Here. 2K  1 is the running window width and I~ m k is the fluctuating component of the mth pixel brightness obtained with the use of a detrending procedure. The latter one was provided by the raw data smoothing with the Savitsky–Golay algorithm and subsequent extraction of the smoothed values from raw data. Figure 4 displays the typical behavior of Σt with the thermodynamic temperature approaching its critical value. The dashed line marks the cutoff level used for estimates of the relaxation time; it was chosen equal to 1∕e. The values of window width for the Savitsky–Golay procedure and 2K  1 were chosen with the use of raw data preprocessing and preliminary analysis. Note that the increase in the relaxation time leads to the appearance of high-amplitude and low-frequency fluctuations of Σt (see curve 3 in Fig. 4), which are caused by the insufficient width of the running window in the case of very slow relaxation of the examined systems. Note that these fluctuations can also be caused by peculiarities of the relaxation process. In order to avoid the uncertainties in estimates of the total relaxation time caused by Σt, fluctuations in the values of trel were also averaged over a series of 5 independent experiments. It is remarkable that the decay in Σt can be approximated by the stretched exponential function Σt ∼ exp−t∕t0 χ  with the index χ remarkably exceeding 1. Figure 5 shows the values of the total relaxation time trel against the thermodynamic temperature of the examined systems. The relaxation

Fig. 3. Values of the correlation time of speckle intensity fluctuations at the initial stage of viscoelastic relaxation against the temperature of the system. 1—Teflon samples; 2—cellulose (paper) samples.

Fig. 4. Evolution of the normalized power of speckle intensity fluctuations in the course of viscoelastic relaxation of the porous matrices. Dashed line corresponds to 1∕e level. Samples: Teflon layers. The notations of the graphs are the same as in Fig. 3.

Fig. 2. Typical normalized correlation functions of speckle intensity fluctuations at the initial stage of viscoelastic relaxation. Dashed line corresponds to 1∕e level. Samples: paper layers; 1—T  298.16°K; 2—T  300.66°K; 3—T  303.66°K.

the examined systems are presented in Fig. 3. Each value of τc was obtained by averaging over a series of 5 independent experiments; the shown error bars correspond to the confidence level of 0.9. The remarkable decay in τc with T approaching its critical value should be mentioned. Another approach to dynamic speckle analysis was applied for estimates of the total relaxation time of the examined systems. The time domain running window was applied, in this case, for the evaluation of the ensemble-averaged normalized power of the fluctuating component of speckle intensity, (P ) M kK ~m 2 1 X k−K I nk  Σt  Σn · Δt  Pk2K1 ~ m 2 ; M m1 I k 

(2)

k0

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  2 k hΔ¯r2 τisi g1 τ; si  ≈ exp − ; 3l

Fig. 5. Values of total relaxation time against the temperature of the system. 1—the paper samples; 2—the Teflon samples. Error bars correspond to the confidence level of 0.9. Inset: the dependence of the fluid bulk modulus on the temperature (recovered on the base of the pressure-density data obtained with the use of Ref. [36].

time reaches its maximal value in the vicinity of the critical point, T C , and falls down with the increasing temperature detuning factor, jT − T C ∕T C j. 4. Phenomenological Basis for Speckle Correlation Analysis

We consider the classical discrete scattering model repeatedly described in literature (see, for example, Refs. [1,43]) in the phenomenological analysis of speckle pattern decorrelation due to the system relaxation. The multiply scattered light field in an arbitrarily chosen observation point can be considered in the framework of this approach as the result of the superposition of partial contributions (“plane waves”) propagating in a scattering medium along unique paths. Each path length, si , is the random quantity characterized by P the path length probability distribution, fPi g ( i Pi  1). The normalized single-path correlation function for an arbitrarily chosen partial contribution with the path length, si , can be introduced as g1 τ; si   hEt; si E t  τi∕hjEt; si j2 i;

(3)

where Et; si  is the complex amplitude of the electric field of the ith partial contribution at the moment, t,  denotes the complex conjugation, and the averaging is carried out over the ensemble of statistically independent partial contributions with the same values of si. This equation is applicable in the strict sense only in the case of ergodic and stationary scattering systems, but it can also be used with the acceptable accuracy for weak nonstationary scattering media. The necessary condition is that the time interval used for data gathering is adequately small. The analysis of the accumulation of random phase shifts in the sequence of scattering events (see, e.g., Ref. [43]) leads to the following expression for g1 τ; si : B16

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(4)

where k is the wavenumber of the propagating light in the medium, Δ¯rτ is the displacement vector for a randomly chosen scatterer during the time interval, τ, (the averaging of Δ¯r2 τ is carried out over the ensemble of scatterers), and l is the transport mean free path of light propagation in the medium [44]. Here, we dropped the factor exp−jωτ, which does not play any role in the following consideration (ω is the light frequency). The total field correlation function of scattered light in the observation point can be obtained as X g1 τ; si Pi ; (5) g1 τ ≈ where the single-path correlation functions are weighted by the corresponding statistical weights of partial contributions. We can obtain the expression for the field correlation function in the scalar approximation (without consideration of polarization effects in multiple scattering) with the use of transition from the discrete path length distribution to the continuous distribution and introducing the path length probability density ρs, Z g1 τ ≈

∞ 0

 2  k hΔ¯r2 τis exp − ρsds: 3l

(6)

We should specify the term hΔ¯r2 τi by taking into account peculiarities of the scatter dynamics in the soft porous matrix in the course of relaxation. The value of hΔ¯r2 τi as the function of τ can be estimated as follows: let us consider the squared displacements of elementary volumes inside the scattering layer, which are averaged along the most probable trajectory of propagating partial contributions (see Fig. 6). The most probable path length of diffusing light in the transmission mode is approximately equal to ~ 2 ∕l , where the factor K~ is determined by smod ≈ KL the boundary conditions for light diffusion approximation at the layer boundaries. The geometry of our experiments (planar deformable layers with the transverse dimensions substantially exceeding the layer thickness, which are embedded in the homogeneous isotropic medium) allows us to assume the predominant uniaxial deformation of the layers in a direction perpendicular to the layer boundaries (Fig. 6). In this case. the value of hΔ¯r2 τi can be expressed as hΔ¯r2 τi ≈ ≈

2

~ 2 ∕l  KL 2_ετ2 ~ 2 ∕l  KL

Z Z

~ 2 ∕2l KL 0 ~ 2 ∕2l KL 0

f_εl ξτxξg2 dξ fxξg2 dξ;

(7)

Fig. 6. Schematic presentation for the estimation of the averaged squared displacement of scattering sites hΔ¯r2 τi.

where ε_ l ξ  dεl ξ∕dt is the local deformation rate of a thin element of the layer (“sub-layer”) placed at the distance of x from the layer midplane (Fig. 6), ξ is the parametric variable corresponding to the distance from the midplane to the “sub-layer” along the most probable path, and ε_ is the characteristic averaged value of the deformation rate of the layer. ~ 2 ∕2l  Using the dimensionless variables, x~  x∕KL 2  ~ ∕2l , we obtain and ξ~  ξ∕KL  ~ 2 2 Z 1 KL ~ 2 dξ~ f~xξg  2l 0  ~ 2 2 KL ;  ℵ_ετ2 2l

hΔ¯r2 τi ≈ _ετ2

(8)

where ℵ is the numerical factor depending on the optical properties of the layer. The relationship between the field correlation function, g1 τ, and the normalized intensity correlation function, g2 τ  hfIt  τ − hIigfIt − hIigi∕ hfIt  τ − hIig2 i, follows from the Siegert relation (see, for example, Ref. [45]), g2 τ  jg1 τj2 :

(9)

Thus, we obtain, finally, Z  g2 τ ≈ 

∞ 0

2   2  k ℵ_ετ2 K~ 2 L4 s  : ρsds exp −   3 12l 

(10)

Figure 7 displays the calculated values of dimensionless parameter f_ετg1∕e, which correspond to 1∕e decay of g2 τ, in the dependence on the mean scattering free path, l, and the scattering anisotropy factor of the scattering layer. The layer thickness was taken as equal to 100 μm (as the thickness of the porous samples used in our experiments). The path length probability density and the ℵ and K parameters were calculated with the use of the Monte Carlo procedure (see, for example, Ref. [46]). It should be noted that the problem of light decorrelation in nonstationary media with the slab geometry was solved previously in the analytical form using the diffusion approximation (see for example Ref. [23]). But, in our case, the values of sample thickness occur too small for rigorous consideration of this problem in the framework of diffusion approximation (see Ref. [47]).

Fig. 7. Theoretical dependencies of the dimensionless parameter f_ετg1∕e on the mean scattering free path and the scattering anisotropy parameter for transilluminated nonabsorbing layers with a thickness of 100 μm. Open circles and squares correspond to the optical properties of examined paper (squares) and Teflon layers (circles). Error bars are related to measurement errors in the course of evaluation of l and g and correspond to the confidence level of 0.9.

Therefore, we have applied the Monte Carlo technique to mimic light transport in the examined samples. The effective refractive index of scattering layers was taken as equal to ≈1.3 (this value is on the order of the expected effective refractive indices of the examined porous Teflon and paper layers). The optical properties of the examined samples [namely, the mean transport free path (MTFP) l  l∕1 − g and the scattering anisotropy factor, g] were estimated with the use of integrating sphere measurements of the Rd ∕T t ratio (Rd is the diffuse reflectance and T t is the total transmission of the examined dry samples) in the wavelength range from 400 to 900 nm and subsequent processing of the obtained data with the use of the inverse Monte Carlo procedure. This allowed us to estimate the values of l and g at 633 nm, taking into account the larger values of the refractive index of saturating fluid with respect to air as equal to 23.0
1.6 μm and ≈0.65 for the Teflon layers and 14.9
1.1 μm and ≈0.35 for the paper layers (the corresponding points are marked in Fig. 7). It should be noted that the MTFP value for the Teflon sample appeared as significantly smaller than that usually reported for bulk Teflon samples in the visible range (see, for example, Ref. [48]). This discrepancy can be obviously explained by the expressed fibrillar structure of the Teflon films used in our experiments. Also, the obtained g value is smaller than the scattering anisotropy parameter usually reported for the bulk Teflon samples (g ≈ 0.9, see Ref. [48]). On the other hand, the obtained values for paper samples satisfactorily agree with previously reported data on the optical properties of filter paper [49]. The comparison of the obtained values of τc with the theoretically predicted values of f_ετg1∕e (Fig. 7) allowed us to estimate roughly the characteristic deformation rates in the 1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS

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Table 1. Estimates of the Deformation Rate at the Initial Stage of Relaxation for Temperatures (a) Far Enough from and (b) in the Vicinity of the Critical Point

Sample Teflon layer, f_ετc g1∕e ≈ 6.24 × 10−4 Paper layer, f_ετc g1∕e ≈ 4.23 × 10−4

T ≈ 298.16°K (a)

T ≈ 304.66°K (b)

6.9 × 10−3 s−1

3.47 × 10−2 s−1

3.0 × 10−3 s−1

1.28 × 10−2 s−1

vicinity of the critical point and far enough from it; these estimates are listed in Table 1. 5. Discussion of the Results

Two basic phenomena control the decaying deformation (creeping) of the examined system after the stepwise depressurization. These are the fast hydrodynamic relaxation of the fluid density inside the porous matrix and the slow viscoelastic relaxation of the porous matrix to a new equilibrium state. A.

11

where ρd x; t and vx; t are the fluid density and the velocity inside the layer, the x coordinate axis is directed normally to the layer surfaces, K~ is the permeability of the layer, η is the dynamic viscosity of the fluid, and p is the fluid pressure. We can obtain after some transformations the following equation:   ∂~ρd ξ; η ∂ ∂~ρ ξ; η  ρ~ d ξ; η d ; ∂η ∂ξ ∂ξ

(12)

with the following boundary conditions: 8 > < −1 < ξ < 1; η ≥ 0; ρ~ d −1; ηη>0  ρ~ d −1; ηη>0  1 − Δ; > : ρ~ ξ; 0  1: d

13

Characteristic Time Scale of Hydrodynamic Relaxation

Let us consider the equalization of the internal (inside the porous matrix) and external fluid density with the use of the following assumptions. 1. The temperature variations inside the matrix in the course of hydrodynamic relaxation are insufficient and the density equalization can be considered as the isothermal process. 2. The interrelation between local values of the fluid density and the pressure inside the matrix and in the surrounding space is determined by the same equation of the fluid state. This assumption is acceptable if the characteristic size of pores significantly exceeds the characteristic intermolecular distance in the fluid (i.e., the examined system belongs to the class of mesoporous and macroporous systems [50] with characteristic pore sizes from several tens of nanometers up to several micrometers). It becomes questionable because of the scale-dependent effects in the case of microporous systems with characteristic pore size of a few nanometers or less. 3. The fluid transfer inside the porous layer due to the pressure gradients is governed by the Darcy law. It should be noted that, despite the relatively low values of the dynamic viscosity of carbon dioxide in near-critical state [42], the characteristic values of the length of hydrodynamic interaction and the fluid discharge rate in the examined porous layers are small. Correspondingly, we can expect the low values of Reynolds number. 4. The geometry of porous systems (thin porous layers with relatively large lateral dimensions) allows us to consider the quasi-1D fluid transfer in the system. The spatial-temporal evolution of the local density of fluid in the porous layer can be described under these assumptions by the following system of equations: B18

8 ∂ρd x; t ∂fρd x; tvx; tg > >   0; < ∂t ∂x K~ ∂px; t > > : vx; t   0; η ∂x

APPLIED OPTICS / Vol. 53, No. 10 / 1 April 2014

Here, ρ~ d ξ; η  ρd ξ; η∕ρd0 is the dimensionless density of the fluid, ρd0 is the fluid density for the initial equilibrium state, Δ is the dimensionless density drop in the surrounding space caused by partial depressurization, ξ  2x∕L is the dimensionless spatial coordinate, L is the layer thickness, and η  ta is the dimensionless time. The value of a is de~ d0 ∕L2 η∂ρd ∕∂p and can be contermined as a  4Kρ sidered as the reciprocal of the relaxation time, τhg , for fluid density equalization in the layer. Here, we assume that we can neglect the dependence of ∂ρd ∕∂p on ξ in the case of Δ ≪ 1. Estimates of τhg for the examined systems give the values at microsecond time scales due to the small thickness of the porous layer and the low viscosity of the compressible saturating fluid. In particular, in the case of paper matrices with K~ ≈ 3 × 10−14 m2 − 6 × 10−14 m2 (this range of values was estimated with the use of the filtration rate probes for distilled water at the atmospheric pressure) we obtain τhg ≈ 1 × 10−6 s − 2 × 10−6 s for the near-critical parameters of carbon dioxide (η ≈ 5.0 × 10−5 Pa · s, ρd0 ≈ 651.1 kg∕m3 , ∂ρd ∕∂P ≈ 1.36 · 10−4 kg∕m3 Pa at T ≈ T C °K and p  7.7 MPa, see, e.g., Ref. [42]). The relaxation time, τhg , for the PTFE matrix is expected to be slightly larger because of the lower values of permeability. But it should be noted that the characteristic time of hydrodynamic relaxation is negligibly small in comparison with the decay time for scattered light fluctuations (Figs. 4 and 5). Thus, we can conclude that the observed speckle dynamics is caused by the viscoelastic relaxation of deformations in the soft porous matrix. Preliminary experiments with uniaxially loaded/unloaded Teflon and paper dry samples (strain–stress tests) have shown the relaxation times of the order of several seconds. This is comparable with the results on speckle

dynamics decay for temperatures far enough from the critical point. B. Temperature Dependencies of the Speckle Correlation Times τc at the Initial Stage of Viscoelastic Relaxation of the Soft Porous Matrices

The decrease in the speckle decorrelation time at the initial stage of viscoelastic relaxation with the temperature approaching the critical value, T C (Fig. 3) is presumably affected by the decreasing dynamic viscosity, η, of saturating carbon dioxide. A controversial point in this interpretation of the experimental data is that the relatively small changes in η (from η ≈ 6.4 × 10−5 Pa · s at T  298.16°K to η ≈ 5.0 × 10−5 Pa · s in the vicinity of the critical point) induce the dramatic decay in τc (from τc ≈ 140 ms to τc ≈ 29 ms for the paper layers and from τc ≈ 90 ms to τc ≈ 21 ms for the Teflon films). A relevant argument for the “viscosity-based” interpretation is related to the experimental data on the viscoelastic relaxation of creeping hard gel-like structures (cements) [51]. We can infer much slower relaxation dynamics in the case of more viscous saturating liquid at the time scales from ≈1 s up to ≈104 s on the basis of the experimental results of the relaxation kinetics for a cement rod in water or isopropanol. The values of η for water and isopropanol at room temperature differ approximately by 2.5 times (ηH2 O ≈ ×10−3 Pa · s against ηIPA ≈ 2.43 × 10−3 Pa · s) but the rough estimates of the relaxation rates from the relaxation history graphs [51] give the rate for the isopropanol-soaked sample approximately 7–10 times less than that for the water-soaked sample. It was noted that the viscoelastic relaxation function can be approximated by the stretched exponential function, Ψt  exp−t∕τVE b ; with the viscoelastic relaxation time, τVE , and the b index dependent on the structural and viscoelastic properties of a porous matrix and the viscosity of saturating liquid [51]. The fit of experimental data with the use of this formula showed the much larger value of τVE and lesser value of b for the isopropanolsoaked sample in comparison with the water-soaked sample. Of course, it should be noted that the comparison of such different porous systems as cement samples saturated by water or isopropanol and soft porous matrices saturated by liquid carbon dioxide in our case is very nonrigorous. Nevertheless, we can see the general tendency in both cases such as the strong nonlinear dependence of the viscoelastic relaxation rate on the viscosity of saturating fluid. C. Temperature Dependencies of the Total Relaxation Time

The values of the total relaxation time also exhibit a strong dependence on the temperature and reach their maxima near the critical point of the saturating fluid (Fig. 5). We can qualitatively characterize the

temperature dependence of τrel by considering the temperature impact on the instantaneous deformation, ε0 , of the fluid-saturated soft porous matrix at the initial stage of viscoelastic deformation of the matrix, which is induced by isothermal transition between two equilibrium thermodynamic states differing by the pressure values. The increase in ε0 should consequently cause the rise in the characteristic time of viscoelastic relaxation (τrel ∼ ε0 − ε∞ ∕_ε, where ε∞ is the final equilibrium deformation of the matrix determined for a large time interval after the transition. The value of ε0 − ε∞ is determined by the elastic properties of the matrix and the fluid compressibility as [52]  ε0 − ε∞ p~

1 − bλ 1 ; − Kp Ks

(14)

where K p is the bulk modulus of the drained porous matrix, K s is the bulk modulus of a solid phase, which forms the porous matrix, b is the Biot coefficient, which is equal to 1 − K p ∕K s , λ is equal to Mb∕K p  Mb2  (M is the Biot modulus defined as 1∕M  φ∕K L  1 − φ∕K s , where φ is the porosity of the matrix and K L is the bulk modulus of the saturating fluid, which is the reciprocal of its compressibility), and p is the final pressure in the system in the course of depressurization. We can simplify Eq. (14) by taking into account that the value of K s for the examined systems obeys the following relationship: K s ≫ K L , K p . The simplified form predicts the value of ε0 − ε∞, depending only on the compressibilities of the drained matrix and the fluidic phase,  ε0 − ε∞ ∼ p

φ 1 1 − : ≈ pφ φK p  K L K s φK p  K L

(15)

Note that the ratio ε0 ∕ε∞ is equal to 1 − bλ∕1 − b and can significantly exceed the unit value with b approaching 1 (this is expected in our case). Thus, the increase in the fluid compressibility in the vicinity of the critical point (see inset in Fig. 5) should cause the increasing value of ε0 − ε∞, and, correspondingly, the increase in the relaxation time, τrel , in the case of the comparable matrix and fluid compressibilities (Fig. 5). The dependence of the fluid bulk modulus on the thermodynamic temperature was recovered from the isothermal pressure-density data for carbon dioxide, which were calculated with the use of the online calculator of thermodynamic properties of fluids [42]. It is remarkable that the value of T ≈ 304.13°K, for which the maximal relaxation times occur (Fig. 5), is slightly shifted to higher temperatures with respect to the critical point. This value is in reasonable agreement with the calculated temperature corresponding to the minimum of the fluid bulk modulus (T ≈ 304.44°K). Besides, the expressed skewness of the empirical dependencies 1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS

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trel T (Fig. 5) can be explained in terms of the difference between the temperature coefficients of the fluid bulk modulus below and above the critical point. The bulk moduli of the drained soft porous matrices examined in our experiments can be roughly compared using the obtained values of ε_ and trel at T ≈ 304.66°K (near the maximum of the fluid TM compressibility). Thus, the ratio K PM P ∕K P ≈ TM PM TM PM ε0 − ε∞  ∕ε0 − ε∞  ≈ _εtrel  ∕_εtrel  (here “TM” denotes the Teflon matrix and “PM” denotes the paper matrix) can be roughly estimated as ≈4∕5. This estimate is significantly less than the ratio of Young moduli in the direction along the layers, which was obtained from the preliminary uniaxial stress-strain tests of the examined dry porous matrices. Nevertheless, it should be noticed that the bulk moduli can significantly differ from the results of uniaxial tests because of the expressed anisotropic fibrillar structure of the examined samples. Such anisotropy should cause the great difference between the viscoelastic properties along the layers and in the perpendicular directions. 6. Conclusions

Finally, the obtained results demonstrate convincingly that the speckle correlation technique has a high potential for analysis of the nonstationary relaxation in soft porous systems with saturating near-critical fluids. The time-dependent statistical and correlation properties of dynamic speckle patterns can be considered as the fingerprints of the current dynamic state of the studied system. The additional opportunity can be given by examination in the vicinity of the critical point of the saturating fluid, where its compressibility abruptly rises up. In this case, the influence of elastic properties of the fluid is minimal and viscoelastic relaxation of the porous matrix is mainly governed by the matrix properties (the bulk modulus in the drained state and the porosity). This opens the way for development of the measurement techniques for quantitative characterization of the structure and functional properties of examined soft porous systems. Further efforts will be directed toward the elaboration of the speckle-based quantitative approaches for the characterization of nonstationary dynamics of similar systems at the mesoscopic level. This work was supported by the RFBR grants #1302-00440, #13-02-12092, and #13-02-90468 from the Russian Foundation for Basic Research, and grant F53/103-2013 from the State Fund for Fundamental Research of the Ukraine. References 1. D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988). 2. A. G. Yodh, N. Georgiades, and D. J. Pine, “Diffusing-wave interferometry,” Opt. Commun. 83, 56–59 (1991). B20

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