Microrheological Study of Physical Gelation in Living Polymeric

Jun 8, 2016 - Microrheological Study of Physical Gelation in Living Polymeric Networks ... *E-mail: [email protected] Phone: 1-312-567-8844. Abstract. Ab...
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Microrheological Study of Physical Gelation in Living Polymeric Networks Tetsuharu Narita† and Tsutomu Indei*,‡ †

Laboratoire PPMD-SIMM, UPMC-ESPCI ParisTech-CNRS, UMR7615, 10 rue Vauquelin, 75005 Paris, France Department of Chemical and Biological Engineering, and Center for Molecular Study of Condensed Soft Matter, Illinois Institute of Technology, 3440 S. Dearborn Street, Suite 150, Chicago, Illinois 60616, United States

‡

S Supporting Information *

ABSTRACT: “Living” polymeric networks made by noncovalent reversible bonds exhibit gel-like elasticity at time scales shorter than the lifetime of the network bridge, but show sol-like fluidity at longer time scales. To explore sol−gel transition of such fluid living networks, we studied rheology of the poly(vinyl alcohol)-borax solutions and diffusion of Brownian particles dispersed in the solutions by using diffusing-wave spectroscopy microrheology over a wide range of frequency (approximately 0.1−105 rad/s). At a certain borax concentration Cb = C*b , the microrheologically estimated dynamic modulus exhibits a power-law behavior in terms of the frequency ω at ω above a terminal flow regime (>100 rad/ s). We developed a theory to describe the linear viscoelasticity of critical physical gels by extending a known theory of chemical gelation. The theoretically derived dynamic modulus agrees well with the experimental results. Also the time-curesuperposition is experimantally satisfied for the mean-square-displacement of the Brownian particles at borax concentrations around C*b . The shift factors to construct the master curves obey the power-law if plotted against the relative distance from this particular concentration ϵ ≔ (Cb−C*b )/C*b . All these facts indicate that a percolated network is formed at the borax concentrations above Cb*. We found an anomalous domain around the thus-estimated gel point in which the viscosity is nearly independent of the extent of cross-linking. We argue that the plateau viscosity is inherent in flowable weak physical gels. used as a criterion for gelation (WC criterion),2 and the underlying mechanism of gelation is percolation.1,2 Compared to the solid chemical gels, percolation in fluid (flowable) physical gels formed by breakable but reproducible cross-links due to noncovalent bond such as electrostatic attractions, hydrophobic interactions and hydrogen bonding4 has been little explored. In these flowable physical gels, the lifetime τ× of the network bridges or strands is smaller than the observation time. The network is living and transient over a time scale longer than τ× in the sense that the network structure continuously changes as time goes on (as if it is “living”). Thus, the network flows in accessible observation time even in the postgel regime. These “living” polymeric network63 exhibits solid-like mechanical properties only at the time scales shorter than τ×. Thus, study of the materials in the short-time scales t < τ× is inevitable to understand the sol−gel transition comprehensively. In many flowable physical gels, an upturn or inflection observed in the zero-shear viscosity curve is frequently

1. INTRODUCTION Many polymeric materials undergo a liquid−solid transition, or sol−gel transition, upon changing thermodynamic conditions such as temperature, pH, concentration of polymers or crosslinking agents. Conversion from a sol state to a gel state due to covalent chemical bonds (chemical gelation) is evidently observed in rheology as a divergence of the zero-shear viscosity at a certain thermodynamic condition.1,2 This critical behavior of the viscosity is attributed to a divergence of the molecular weight of branched molecular clusters that causes a formation of a material-spanning molecular network. Furthermore, the large clusters near the gel point are self-similar in space as well as in time t (and frequency ω) for several orders of magnitude, thereby exhibiting a unique power-law behavior in the viscoelastic relaxation spectrum near the gel point. That is, as first reported by Winter and Chambon (WC),3 the viscoelastic relaxation modulus shows G(t) ∝ t−n at a wide range of t. Equivalently, its counterpart in the frequency domain, dynamic modulus, exhibits G′(ω) = G″(ω)/tan(nπ/2)∝ ωn where G′(ω) and G″(ω) are the storage modulus and the loss modulus, respectively. Nowadays it is widely acknowledged for chemical gels that the appearance of a power-law behavior in these moduli over a wide range of time or frequency can be © XXXX American Chemical Society

Received: April 10, 2016 Revised: May 26, 2016

A

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origin of the near-single relaxation time is supposed to be the narrow lifetime distribution of the reversible didiol complexes formed by borate ion (borax is dissociated into borate ions and boric acids) and adjacent OH functions on the PVA chains.15,31 Also the time− temperature superposition is applicable for construction of a master curve of the dynamic modulus around terminal flow regime.15,18 Schultz and Myers reported that the activation energy (associated with the complex dissociation) estimated from the horizontal shift factors to make the master curve of the time−temperature superposition is 42 kJ/mol,15 whereas Koga et al. reported it as 80 kJ/mol.18 The difference might be due to different saponification of the PVA samples. Similarly we performed the time−temperature superposition of G*(ω) for our samples and estimated the activation energy from the horizontal shift factor as 42.4 kJ/mol. This is very close to the value Schultz and Myers reported. PVA (molecular weight, 89 000 g/mol; degree of deacetylation, 99%) and borax were purchased from Sigma-Aldrich. Polystyrene microspheres (stabilized with carboxylate groups at the surface) used as probe particles for DWS microrheology measurements were purchased from Micromod (Rostock, Germany). We use the particles of diameter 2R = 500 nm in most of our microrheology measurements. Some different size of particles are also used to investigate the size effects of the probe particles on the GSER-based microrheology. Stock aqueous solutions of PVA (dissolved at 95 °C) and borax (dissolved at room temperature) prepared in advance were mixed to obtain physically cross-linked PVA solutions/gels. At the studied conditions the mixtures were stable, no precipitation or phase separation were observed after 6 months. The dynamic correlation length of the PVA− borax mixtures was measured from cooperative diffusive mode (gel mode) of the system by using dynamic light scattering at the same conditions as the literature.31 For the mixture of PVA 5.5% and borax 1.7 mM, we found a dynamic correlation length of 14 nm. As previously reported, it depends little on the borax concentration. For DWS measurements, the polystyrene microspheres were also dispersed in the mixture. The probe concentration was 1% unless otherwise indicated. 2.2. Classical Macrorheology. Macroscopic rheological measurements of the PVA−borax solutions were performed with two rheometers depending on the conditions: a strain-controlled ARES rheometer (TA Instruments, USA) using a cone−plate geometry (diameter, 25 or 35 mm; angle, 2°) operated in the linear domain (strain: 1 or 10%) and/or a stress-controlled Haake RS600 (Thermo Scientific, USA) using a cone−plate geometry (diameter, 60 mm; angle, 2°) operated in the linear domain (stress: 9 Pa). Frequency sweep (between 0.01 and 100 rad/s) was performed at 25 °C. 2.3. Microrheology. DWS Microrheology. Microrheological measurements based on diffusing-wave spectroscopy (DWS) were conducted using a laboratory-made setup. A Spectra-Physics Cyan CDRH laser, operating at the wavelength λ = 488 nm with an output power of 50 mW, was used as coherent light source. The laser beam was expanded to approximately 1 cm in diameter with a beam expander. The sample cell thickness was L = 4 mm, thus a sampling volume of 0.3 cm3 containing 1011 particles was probed. The diffused light was collected by an optical fiber placed in the transmission geometry connected to a photon counter (ALV, Lanssen, Germany). The obtained intensity autocorrelation function g(2)(t) was converted into the field autocorrelation function g(1)(t) by the Siegert relation, g(2)(t) = β[1+g(1)(t)]2, then the MSD of the probe particles ⟨Δrb2(t)⟩eq was calculated by solving numerically the following equations32,33 for transmission geometry:

interpreted as a signature of a formation of the transient network.5−8 That is, the network formation is estimated in the long time terminal regime t > τ×. Typical examples of such flowable gels are poly(vinyl alcohol)−borax mixture,5 hydrophobically modified associating polymers, and so on.6−8 On the other hand, for “permanent” physical gels that do not flow on an experimentally accessible time range due to the long-lived junctions, the gel point can be estimated based on the WC criterion in the short time regime t < τ×.9−14 The junctions of these permanent physical gels are typically formed by partial crystallization,9−11 helix formation,12 cooperative binding that makes “egg-box” structure,13 hydrophobic interaction whose attraction energy is much higher than the thermal energy14 etc. In this paper, we comprehensively study both time regimes (t > τ× and t < τ×) for a representative flowable physical gel, and discuss how the gel point determined in the short time regime is related with that in the long time regime. As a model material, we employ aqueous solutions of poly(vinyl alcohol), PVA, physically cross-linked by borate ions.15−18 To achieve the short-time regime, or the corresponding high-frequency regime, we use passive microrheology19 based on diffusing-wave spectroscopy (DWS),20,21 which allows us to elucidate the transition behavior over a broad frequency range up to 105 rad/ s. On the other hand, conventional macrorheology is used to examine the long-time regime. Larsen and Furst22 are the first to conduct passive microrheological measurements of gelation for chemical gels and permanent physical gels based on the time−cure superposition.23 They employed multiple particle tracking microrheology to measure the ensemble-averaged mean square displacement (MSD) of the probe particles, and applied the time-cure superposition23 for the MSD curves to analyze material properties near the gel point. By applying their methods, gelation of milk protein β-lactoglobulin,24,25 gelation of Fmoc-derivative hydrogels,26 fluid-gel transition of geometrically confined polymer solutions27 etc have been studied. Very recently, Chen et al.28 reported reversible gel formation of lightly sulfonated polystyrene oligomer melts. They utilized the time−temperature superposition to construct master curves over almost 10 orders of magnitude of frequency. They found that the master curves of the dynamic modulus for the system close to the gel point exhibit the mean-field gelation region G′ ∼ G″∼ ω together with the critical percolation region G′ ∼ G″∼ ωn in addition to the Rouse region of the precursor chain as well as the terminal flow region. On the other hand, it seems unlikely that the time−temperature superposition is applicable for the PVA−borax solution at high frequencies/low temperature where the critical percolation region (and/or the meanfield region) may occur; the material would freeze at low temperature without showing the percolation regimes because of the relatively short lifetime of the cross-linking. Instead of using the time−temperature superposition, we apply the microrheology technique to examine the high-frequency percolation regime.

2. EXPERIMENTAL SECTION g(1)(t ) =

2.1. Materials. We use the aqueous solution of poly(vinyl alcohol) (PVA) and sodium tetraborate decahydrate (borax)5,15−18 as a model material. Its rheological as well as phase behaviors have been studied extensively. It is known that the viscoelastic properties of mixtures of polyhydroxy compound and borate ions such as this system are described approximately by the Maxwell model with a single relaxation time of order 0.1 s when the network is well developed.5,29,30 The

L l* z0 l*

×

+ +

4 3 2 3

z z 2 sinh⎡⎣ l *0 r (̃ t )⎤⎦ + 3 r (̃ t ) cosh⎡⎣ l *0 r (̃ t )⎤⎦ L L 4 2 1 + 9 r (̃ t )2 sinh⎡⎣ l * r (̃ t )⎤⎦ + 3 r (̃ t ) cosh⎡⎣ l * r (̃ t )⎤⎦

(

)

(1) B

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using eq 3, then G*(ω) can be obtained without approximation through the GSER. See Appendix A for details of the fitting method.

where r (̃ t ) ≔ (2π /λ)2 ⟨Δrb(t )2 ⟩eq is the root of the MSD nondimensionalized by the wavelength, l* is the sample transport mean free path of the scattered light determined from the values of transmission intensity of the sample and a reference sample (water) whose l* is known. For this system l* was found to be 220 μm. z0 is the distance the light must travel through the sample before becoming randomized, here it is set z0 = l*. The measured multiply scattered light signal was found to be ergodic for all the sample studied. With the experimental setup used, we can measure accurately the value of MSD up to 400 nm2. Thus, the probe particles probe the length scale ( 400 = 20 nm) much smaller than the size of the particles (500 nm in diameter). MSD Analysis. The dynamic modulus of the medium was estimated from the MSD of the probes by using the generalized Stokes−Einstein relation (GSER)19 G*(ω) =

3. THEORY In this section, we derive theoretical expressions for the dynamic modulus, zero-shear viscosity and relaxation time of the physical gel near its gel point. For this purpose we extend the theory for chemical gelation developed by Martin et al.41 so as to accommodate the finite lifetime of the molecular association and the cluster size. Prior to Martin et al., Cates42 developed a theory for selfsimilar macromlecules and used it for a polymer sol−gel system close to the critical point. The present approach is more directly related to the Martin et al.’s theory. 3.1. Dynamic Modulus. We consider aggregation of precursor polymer chains due to physical cross-linking. The dynamic modulus originating from the clusters formed by m chains (called m-cluster hereafter)65 is

kBT πRiωF ̅ {⟨Δrb2(t )⟩eq }

(2)

where kB is the Boltzmann constant, T is temperature, and F ̅ {···} indicates taking the one-sided Fourier (or Laplace) transformation. Particle inertia and medium inertia34−37 are not included in the GSER.64 In usual single-bead microrheological analysis, ln MSD is fit to a polynomial of ln t, and then an approximate expression is used for the one-sided Fourier (or Laplace) transformation of the polynomial function.38,39 Alternatively we used a more accurate and physically intuitive method proposed in ref 36. Our analysis method makes use of the power-law spectrum often used for the analysis of viscoelastic relaxation spectrum.40 We express the MSD in terms of the spectrum h(λ) and the two functions f (0) and f (1) as ⟨Δrb2(t )⟩eq =

∫0

∞

f (1) (t , λ)

h(λ) dλ + γ0 f (0) (t , λmax ) λ

m

Gm*(ω) = g0 ∑ j=1

j=1

Gm*(ω) ≃ g0

(3)

(4)

m

dj

iωτj , m 1 + iωτj , m

(8)

where 2F1 is the hypergeometric function. eq 8 describes a terminal flow behavior at ω ≪ 1/(m α τ 0 ), i.e., 1 − 2α

1−α

1−m 1−m Gm′ (ω) = α − 1 g0mα τ0ω and Gm″(ω) = 2α − 1 g0(mα τ0ω)2 . On the other hand, at 1/(mατ0) ≪ ω ≪ 1/τ0, eq 8 obeys power m 1 1 Γ 1 − g (iωτ0)1/ α , law in terms of ω: Gm*(ω) = Γ

f (0) (t , λmax ) ≔ e−t / λmax − 1 + t /λmax

α

(α) (

α

)0

where Γ is the gamma function. Thus, relaxation exponent for a monodisperse solution of the m-clusters is 1/α (corresponding to n in eq 1 of ref 43 or δ in ref 41). If hydrodynamic interaction is screened, the exponent α is given by41

(5)

where λmax ≔ λnmax. On the other hand, a weight function f(1) in front of the spectrum is defined by

α=

f (1) (t , λ) ≔ 1 − (1 + t /λ)e−t / λ 2 2 ⎧ ⎪ t /(2λ ) (for t ≪ λ) ≃⎨ ⎪ (for t ≫ λ) ⎩ 1

∫1

⎡ ⎛ 1 1 1 ⎞ = g0⎢m 2F1⎜1, ; 1 + ; − ⎟ ⎢⎣ α iωτ0 ⎠ ⎝ α ⎛ 1 1 1 ⎞⎤ − 2F1⎜1, ; 1 + ; − ⎟⎥ α iωτ0mα ⎠⎥⎦ ⎝ α

where H(λ) is a unit Heaviside function. Distribution of the relaxation time {λj}(λj > λj−1), amplitude {γj} and the exponent {αj} for each mode characterize the spectrum h(λ) . The total number of the modes is nmax. The weight γ1 of the first mode determines the rest due to a αk−αk+1 for 2 continuity for the weighting of the modes, i.e., γj = γ1∏j−1 k=1λk (0) ≤ j ≤ nmax. In eq 3, a function f is defined by

2 ⎧ 2 ⎪ t /(2λ max ) (for t ≪ λ max ) ≃⎨ ⎪ (for t ≫ λmax ) ⎩ t /λmax

(7)

where τj,m = (j/m) τ0 is the relaxation time of the jth mode of the m-cluster41 (τ0 ≔ τm,m is the relaxation time of the precursor chain, and α is explained below). By replacing the sum over j with an integral and by performing the integral, eq 7 becomes as

nmax

∑ γjλαjH(λj − λ)H(λ − λj − 1)

1 + iωτj , m

−α

The spectrum is given by

h(λ) =

iωτj , m

D+2 D

(9)

where D is fractal dimension. The dynamic modulus of the solution is obtained by taking a superposition of Gm* with a weight of the cluster size distribution Nm, as

(6)

mz

We impose a condition that both terms in the right-hand side of eq 3 are equal at t = λmax so that the MSD described by eq 3 shows the diffusive behavior ⟨Δrb2(t)⟩eq ∝ t at the longest time regime t ≫ λmax. This requirement relates γ0 with the other parameters by γ0 = max j γj(λαj j − λαj−1 )/αj. Therefore, the number of free adjustable ∑nj=1 parameters {α1, ..., αmax, λ0, .., λmax, γ1} is 2nmax + 2 in total for a given nmax. (In ref 36, γ0 was treated as an independent parameter.) We assumed nmax = 4. Equation 3 can be analytically Laplace transformed,36 so that once the MSD fitting is performed successfully by

G*(ω) =

∑ Gm*(ω)Nm ≃ ∫ m=1

1

mz

dm Gm*(ω)Nm

(10)

where the upper limit of the integral mz is the number of chains in the largest cluster. Since the chain−chain association is temporal, mz does not diverge even at gel point. Near the gel point, the cluster-size distribution is self-similar,1 so that the distribution function of the cluster size is C

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Macromolecules Nm = N0m−τF

(1 < m < mz )

(11)

where τF is called the Fisher exponent. As confirmed below (see eq 16), the relaxation exponent n for the dynamic modulus is related to the Fisher exponent as41 n=

τF − 1 α

(12)

Usually τF is slightly larger than 2, and hence n > 1/α. By putting eq 9 into eq 12, n is obtained as41,43

n=

D(τF − 1) D+2

(13)

Depending on the extent of screening of the excluded-volume interaction, D in eqs 13 and 9 takes some appropriate value between fractal dimension df if excluded-volume interaction is fully screened out and fractal dimension df with that effect unscreened.43 These two fractal dimensions are related as43,44 df =

Figure 1. Dynamic modulus of the PVA−borax aqueous gels. Borax concentrations are 2.2 mM (pink) and 3.0 mM (purple) for a fixed PVA concentration 5.5%. Solid (dashed) lines: storage (loss) modulus by microrheology. Closed (open) circles: storage (loss) modulus by strain-controlled macrorheometry.

2df ds + 2 − 2df

can be approximated as nΓ(n) Γ(1−n) (iωτ0)n at ω ≪ 1/τ0. Therefore, eq 15 becomes as

(14)

G*(ω) ≃ S Γ(1 − n)(iω)n

where ds(=3) is space dimension. Substituting eq 8 into eq 10, and then conducting the integral with respect to m, we obtain the expression for the dynamic modulus of the critical physical gel as G*(ω) =

(16)

(

where a prefactor S =

⎧ ⎛ G0 1 ⎞ ⎨ 2F1⎜1, n; 1 + n; − ⎟ iωτ0 ⎠ (nα − 1)nα ⎩ ⎝ ⎛ 1 1 1 ⎞ − nαmz1 − nα 2F1⎜1, ; 1 + ; − ⎟ α iωτ0 ⎠ ⎝ α ⎪

η ≔ lim

ω→0

⎪

⎪

(15)

=

where G0 ≔ g0 N0 is approximately the modulus at ω = 1/τ0. Equation 15 describes both the terminal flow behavior G′∼ ω2, G″∼ ω at low frequencies ω ≪ 1/(τ0mαz ) and the critical power-law behavior G′ ∝ G″∼ ωn at high frequencies 1/(τ0mαz ) ≪ω ≪ 1/τ0. 3.2. Chemical Gel Limit. Critical behavior can be derived explicitly by taking a chemical-gel limit mz → ∞ in eq 15 that removes the terminal flow regime. In this limit the first term in the curly brackets of eq 15 becomes dominant, and this term τz̅ ≔ lim

ω→0

(nα − 1)α

) is called the gel stiffness.

2

eq

G″ ω

mzα(1 − n)(nα − 1) + mz1 − nα (1 − n)α + 1 − α G0τ0 α(α − 1)(nα − 1)(1 − n) (17)

G0τ0 ≃mzα(1 − n) α(α − 1)(1 − n)

(18)

The last equation is the approximate expression of η when mz is sufficiently larger than 1.66 The weight-average relaxation time46 or the apparent longest relaxation time near the gel point can be derived from eq 15 as

[mzα (nα − 1) − mz(n − 1)α (2α − 1) + mz1 − α (2 − n)α](1 − n)(α − 1) (1 − n)(α − 1) G′ = τz τ0 ≃ n ( 1) α 1 α − − (2 − n)(2α − 1) G″ ω [(nα − 1) − mz (α − 1) + mz (1 − n)α](2 − n)(2α − 1)

where τz ≔ mαz τ0 is the relaxation time of the largest cluster, or the “true” longest relaxation time, near the gel point. The last equality is the approximation when mz is sufficiently large.67 Equation 19 indicates that τz is smaller than τz by a factor of (1 − n)(α − 1) (2 − n)(2α − 1)

G0τ0n Γ(n)

16 is a well-known expression for the dynamic modulus of the critical chemical gels.2,45 3.3. Terminal Behavior. For physical gels, mz is finite so that the terminal regime exists even at the gel point. As a result the zero-shear viscosity and the relaxation time are finite at the gel point. The zero-shear viscosity near the physical gel point is derived from eq 15 as

⎪

⎛ 1 ⎞ − mz−nα 2F1⎜1, n; 1 + n; − ⎟ iωτ0mzα ⎠ ⎝ ⎛ 1 1 1 ⎞⎫ + nαmz−nα 2F1⎜1, ; 1 + ; − ⎟⎬ α iωτ0mzα ⎠⎭ ⎝ α

(at mz → ∞ , ω ≪ 1/τ0)

(19)

the largest clusters, i.e., Nm = N0δm,mz, both time scales coincide: τz = τz.) At any rate, the dynamics of the material near its gel point is substantially governed by a single time constant τz.

4. RESULTS AND DISCUSSION 4.1. Dynamic Modulus. 4.1.1. Postgel Regime. Figure 1 shows the dynamic moduli of the PVA−borax aqueous gels for borax concentrations Cb = 2.2 mM and 3.0 mM with a fixed PVA concentration Cp = 5.5% measured by conventional macrorheology and DWS-based microrheology. Macro- and

which is approximately 0.1 in the present system.

Origin of this discrepancy is that the crossover domain of the critical power-law regime and the terminal flow regime in G*(ω) is broad because of the broad power-law distribution of the cluster size given by eq 11. (For monodisperse solution of D

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Macromolecules microrheological data satisfactorily superpose in the intermediate overlapping frequencies. With increasing the borax concentration, the dynamic modulus becomes close to that of the Maxwell model. The plateau in the storage modulus indicates a presence of a well-developed network of the PVA chains cross-linked by borate ions. 4.1.2. Near Gel Point. Figure 2 shows the dynamic modulus for the borax concentration Cb = 1.7 mM with the same PVA

n=

ds D+2

(20)

Therefore, if the hyper-scaling relation is assumed, fractal dimension is obtained from n by using eqs 20 (with D = df) as df =

ds n

− 2 , and hence df =

(2 + ds)(ds − 2n) 2(ds − n)

due to eq 14. Since

n = 0.59 in the present case, we obtain df = 1.89. This value is slightly (10%) larger than that obtained without assuming the hyper-scaling relation (df = 1.69). (Also if this value is used to fit G* through α, then the agreement gets worse.) The disagreement is probably caused by neglect of the hydrodynamic interaction in eq 9,41,43 experimental errors, etc. and does not necessarily mean a violation of the hyper-scaling relation. Shibayama et al.49 performed small-angle neutron scattering experiments for PVA−borax physical gels/solutions, and determined fractal dimension from the slope at the high-q region in double logarithmic plots of the scattered intensity I versus the scattering vector q. They reported thus-obtained fractal dimensions were 2.6 and 2.8 for PVA concentrations 12.5% and 16.7%, respectively. These values lie between df = 1.89 and df = 3.08 we obtained from n (assuming the hyperscaling relation). On the basis of a mean-field consideration, Rubinstein and Semenov predicted that G′ ∼ G″∝ ω at frequencies above a terminal zone (for pregel) or above a plateau zone (for postgel) near the gel point (Figure 4 of ref 50). Very recently, Chen et al. modified the Rubinstein and Semenov’s mean-field theory to include the critical percolation region.28 For lightly sulfonated unentangled polystyrene melts, they observed a transition between the mean-field region and the critical percolation region in master curves of G* constructed by utilizing the time−temperature superposition. The mean-field region is not observed in our PVA−borax systems. 4.1.3. Discussion about Justification of GSER near Gel Point of Fluid Gel. For chemical gels and permanent physical gels, the applicability of the GSER may be questioned near the gel point when the length scale of the material heterogeneities may exceed the probe size and the probes may exist in different microenvironment.24 In order to test the validity of the GSERbased microrheology near the gel point of this flowable physical gel, we measured the MSD of the larger particles (750 nm and 2 μm in diameter) in the same condition as in Figure 2 (Cp = 5.5%, Cb = 1.7 mM). The results are shown in Figure 3 where the products of the MSD and the particle radius ⟨Δrb2(t)⟩eqR are plotted for several different R. This plot provides an indication of any bead size dependence of the particle motion in the material,51,52 that is, the deviation from the master curve indicates the violation of the GSER-based microrheology. We can see in Figure 3 that the ⟨Δrb2(t)⟩eqR curves overlay each other, thereby indicating that in these length scales there is no probe-size dependence and the dynamic modulus can be measured by the GSER, despite the heterogeneous structure of percolated unrelaxed cluster having various characteristic length scales ξ up to the sample size. This is because the length scale that rheologically characterizes the material homogeneity is the size of the largest relaxed cluster, Rz, rather than ξ (size of unrelaxed clusters). If we assume D = df, then Rz is estimated as Rz ≃ m1/D z Rg ≃ 200 nm (Rg = 7.9 nm is the radius of gyration of a precursor PVA chain, Rg = Re/√6, where Re = 19 nm is the end-to-end distance of the PVA chain as estimated above). On the other hand, if we put D= df, then Rz ≃ 60 nm. Note that this

Figure 2. Dynamic modulus of the PVA−borax aqueous solution/gel. PVA concentration is 5.5% and borax concentration is 1.7 mM. Solid (dashed) lines: storage (loss) modulus by microrheology. Closed (open) circles: storage (loss) modulus by strain-controlled macrorheometry. Closed (open) squares: storage (loss) modulus by stresscontrolled macrorheometry. Dotted lines: theoretical prediction from eq 15.

concentration. For this sample with relatively low moduli, in order to cover a wide range of frequency, both a stresscontrolled and strain-controlled rheometers were used, as well as a probe concentration of 0.1% for microrheology (l* was estimated to be approximately 1700 μm). Again, macro- and microrheological data superpose well in the intermediate frequencies. A power-law behavior with an exponent n = 0.59 is observed in the microrheologically estimated G* at frequencies higher than 100 rad/s while a terminal zone exists at lower frequencies. The power law indicates a presence of large fractal self-similar structures.23,47 We consider that the appearance of the power-law in the dynamic modulus is one of the signatures that the system is on, or very close to, percolation threshold (or physical gel point) at this particular condition. The power-law and the terminal zone coexist at the gel point differently from the permanent networks3,28 because the cross-link formation due to diol complexation is temporal. As shown in Figure 2, eq 15 with n = 0.59, α = 1.95, mz = 550, G0 = 1.7 kPa, and τ0 = 5.9 × 10−7 s fits well with experimental data at Cb = 1.7 mM and Cp = 5.5% for a wide range of frequencies. (See ref 68 on how to estimate the values of τ0 and G0.) The good agreement indicates that the system is close to the physical gel point. The zero-shear viscosity and the apparent longest relaxation time estimated by using eqs 18 and 19 are η = 0.2 Pa·s and τz = 0.013 s, respectively. By using eq 9 with D= df, fractal dimension can be estimated from α as df = 2/ (α − 1), and therefore, with the help of eq 14, df = (ds + 2)/(α + 1) . Since α = 1.95 in the present case, we obtain df = 1.69. So far, hyperscaling relation τF = 1 + ds/D1,48 has not been assumed. If assumed, according to eq 13, the following relation is satisfied41,43 E

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results indicate that this latter method can also be microrheologically used for the transient gels to see the power-law behavior up to the longest time scale τz which is materialdependent (but measurement-independent). 4.1.4. Pregel Regime. Figure 4 shows the dynamic moduli at lower borax concentrations (pregel). In order to increase the

Figure 3. Mean-square-displacement of probe particles scaled by the particle radius R for varying size: 2R = 500 nm (red), 750 nm (green), and 2 μm (blue). The borax concentration is 1.7 mM and the PVA concentration is 5.5%.

is close to the value estimated from the relation ⟨Δrb2(t)⟩eqRz ≃ Rz3 that holds at the crossover time t ≃ τz. In both cases, our probe particles are always larger than Rz (i.e., R > Rz). Note that for both macro- and microrheology, inhomogeneity due to the large clusters (ξ > Rz) contribute to rheology only in the terminal flow regime (i.e., since these large clusters do not have time to fully relax, the length scale longer than Rz escapes the rheological measurements). Therefore, the GSER is applicable. We emphasize that, in this transient system, the spatial inhomogeneity is also dynamic, and the structure changes with time; thus, the structural inhomogeneity and the dynamical heterogeneity are associated with each other. Because of their short lifetime, the large clusters continuously break/reform their structure near the gel point, thus the spatial inhomogeneity probed by the particles is dynamically averaged over time, or an ensemble average of the material is measured by a time average. In fact, for all the DWS experiments shown in this article, the dynamic light scattering signals are ergodic, indicating that each probe particle can freely explore the entire space of the sample having very short-lived inhomogeneity. Thus, in the DWS measurements, the contribution from the distributions of the size and structure of clusters around the particle which determine the local viscoelasticity is spatially and temporally averaged out to the output MSD of the multiple probe particles (∼1011) differently from a single particle method53,54 that tracks a single probe placed in a part of the sample. We also underline here that with this microrheological approach, although we do not show a direct evidence of the network percolation at the length scale of the sample (much larger than R), but we show another evidence of the percolation by the power-law behavior of the viscoelasticity. That is, what the multiple particles “detect” in the DWS microrheology is the self-similar fractal nature of the material as a signature of the percolation. It is well accepted that the sol−gel transition can be evidenced not only by the divergence of the longest relaxation time, but also by all the shorter relaxation modes showing a distinct pattern, power-law dependence.2 The latter method is usually used for permanent gels because it is generally difficult to apply the former method to rheologically probe the longest relaxation of the percolated network due to the limit of experimentally accessible measurement time, whereas the shorter relaxation times are easier to access. Our

Figure 4. Dynamic modulus of the PVA−borax aqueous solutions (pregel). Borax concentrations are 0.8 mM (green) and 1.2 mM (blue) for a fixed PVA concentration 5.5%. Solid (dashed) lines: storage (loss) modulus by microrheology. Closed (open) circles: storage (loss) modulus by strain-controlled macrorheometry. Closed (open) squares: storage (loss) modulus by stress-controlled macrorheometry.

accuracy at the low frequency range, we performed measurements with higher l* value (about 1700 μm) by using lower probe concentration (0.1%). The values of G″ are sufficiently close to macrorheological data at narrow overlapping frequencies, while some differences between the macro- and microrheology are observed for G′. It should be noted that with microrheology, G* is calculated from the MSD of the probes, then it is decomposed into G′ and G″. Thus, one of the two moduli which is lower than the other is more sensitive to the experimental noises and artifacts than the other, and the value can be less precise. In other words, when the viscoelastic exponent n is close to 1 (purely viscous) or 0 (purely elastic), the precision of the G′ (for n = 1) or G″ (for n = 0) can be very low. Therefore, for these pregel samples having low viscoelasticity (n is close to 1 for lower frequency range), the values of G′ are not very accurate, while those of G″ are sufficiently close to macrorheological data.69 4.2. Uniqueness of Loss Tangent. The WC law in the critical regime G′ ∝ G″ ∼ ωn is equivalent to the observation that the viscoelastic loss tangent tan δ = G″/G′ is independent of frequency at the gel point.2 This property (uniqueness of tan δ) is often used to determine the gel point for chemical gels3 and also for permanent physical gels without terminal zone.10,55 We microrheologically estimated tan δ of PVA−borax solutions and performed the multifrequency plot of tan δ as a function of borax concentration as shown in Figure 5. The tan δ curves intersect at C*b = 1.70 ± 0.03 mM. Thus, we see that the material is very close to the physical gel point at this condition. Furthermore, the critical exponent n for G* can be estimated from the value of δ at the gel point δ* as n = 2δ*/π = 0.59, which is the same as that estimated from the slope of G′ and G″ (see Figure 2). Thus, the uniqueness of the tan δ method using the DWS-based microrheology is useful even for the flowable physical gels. Technical details on how to estimate the intersection of the tan δ curves is presented in Appendix B. F

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Figure 5. Microrheologically estimated viscoelastic loss tangent plotted against the borax concentration for several frequencies.

4.3. Time−Cure Superposition for MSD. Larsen and Furst showed that the time−cure superposition is satisfied for the particle MSD near gel point for both chemical gels and permanent physical gels.22 We applied their method to the present system to get a further understanding of the physical gelation of the living networks. It should be noted that in the reversible gels, frequency range of the superposition is narrow and limited because the lifetime of the transient bonds is finite and thus limits the size of the largest cluster that is rheologically detectable. Such a limitation tends to cause artifactual effects in the superposition process. To avoid the uncertainties as much as possible, we used the computational algorithm explained in Appendix C. Figure 6A shows the superposition of the MSD data for different borax concentrations made by shifting individual curves both horizontally and vertically. The vertical (time) and horizontal (MSD) shift-factors are plotted in Figure 6B as functions of the borax concentration Cb or of the relative distance from the gel point (extent of cross-linking) ϵ ≔ (Cb− Cb*)/Cb*. The MSD master-curve can be made because the Brownian motion of the tracer particles in the solution near the gel point is characterized by a single relaxation time, which is interpreted as a crossover time between the viscoelastic and viscous (for pregel) or elastic (for postgel) regimes τz.41 The presence of the MSD master-curves indicates that the time-cure superposition is satisfied for the relaxation modulus G(t) and the dynamic modulus G*(ω) near the gel point.22,23 The timecure superposition is violated at times larger than τz23 where a plateau begins to emerge in the MSD (or G*(ω)) in the postgel regime. In Figure 6A, two different master curves can be constructed for Cb < 1.7 mM and for Cb > 1.7 mM. For the lower borax concentrations (Cb < 1.7 mM), a master curve is obtained by shifting the MSD curves for higher Cb to lower left, thereby indicating that τz increases (and Gz decreases) with increasing Cb. This is a reasonable result; since the molecular weight of the clusters formed by the PVA chains increases with increasing the extent of cross-linking, the relaxation time of the clusters (represented by τz) increases with increasing Cb. On the other hand, for the higher borax concentrations (Cb > 1.7 mM), another master curve can be constructed by shifting the MSD curve to the opposite direction, thus τz decreases (and Gz increases) with increasing Cb. This opposite behavior can be explained by assuming a presence of the network in this higher concentration regime. That is, if τz is interpreted as the relaxation time of the network strands, then it decreases with

Figure 6. (A) MSD master curves of the particles. PVA concentration is Cp = 5.5%. Inset: MSD before the shift. (B) Shift factors for the MSD and the time as a function of borax concentration. Inset: shift factors plotted against the extent of cross-linking ϵ.

increasing Cb because the network strands shorten with increasing the extent of cross-linking. The sol−gel transition is characterized by the critical powerlaw behavior τz ∼ |ϵ|−y and Gz ∼ |ϵ|z (or ⟨Δrb2(τz)⟩eq ∼ |ϵ|−z).47 We found these power-law behaviors in the shift factors (inset of Figure 6B) and estimated the values of the critical exponents. See Appendix B for technical details of the evaluation of the exponents. When the PVA concentration is 5.5%, y− = 2.75 ± 0.17 and z− = 1.73 ± 0.11 in the pregel regime. On the other hand, the values in the postgel regime y+ = 3.69 ± 0.20 and z+ = 2.21 ± 0.10 are larger than those in the pregel regime y−, z−. It appears that the critical exponents slightly depend on the physical bond concentration (or distance from the gel point ϵ) probably due to the intrinsic, transient nature of the system. This dependence is averaged out in the above values of y±, z±. We performed the similar analysis for several different polymer concentrations Cp’s to see how the critical exponents n,y, and z depend on the polymer concentration. Exponents y and z were estimated from the shift factors of the MSD master curves up to Cp= 5.5%, whereas n was obtained by using the uniqueness of tan δ for Cp ≤ 8.25%. Results are shown in Figure 7. For all polymer concentrations surveyed, we found that the exponents y,z in postgel regime are larger than those in pregel regime, i.e., y+ > y−, z+ > z− (see Figure 7A,B). Only for the purpose of qualitative comparison are the values of the exponents for the electrical network model and Rouse model also shown in the figure. The values we found are rather close to those expected from the electrical percolation analogy (y ≃ 2.6, z ≃ 1.9).57,58 Scaling behavior expected from the electrical G

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symmetry is often assumed in experimental analysis of the gelation.2,24,26 As seen in Figure 7D, the values of n for our PVA−borax physical gels are close to those reported for PVA− glutaraldehyde chemical gels 56 over a range of PVA concentrations we surveyed. This fact also indicates that fractal dimensions (estimated from n) of these two systems are close to each other. For low PVA concentrations, the value of n is close to that expected from the electrical percolation analogy58 (≃0.72), and decreases with increasing the PVA concentration. The monotonic decrease of n with increasing cross-linker concentration has been also reported for end-linked poly(dimethylsiloxane) for the prepolymer with a medium chainlength.60 We emphasize the experimental fact that τz and Gz obey the power law only when these quantities are expressed as a function of ϵ = (Cb − C*b )/Cb* rather than the borax concentration Cb itself. This is a typical feature when the material undergoes percolation transition at Cb = Cb*,1 thereby strongly indicating that the present material is gelated at Cb*. It should be remarked that what we detected in the time-curesuperposition experiments is not a network formation due to entanglements of long supermolecules formed by the PVA chains, but a percolation caused by the local attractive polymer−polymer interactions. Actually, if we perform the similar experiments for the entangled polymer solutions, the power-law appears only when the shift factors are plotted against the polymer concentration Cp itself, rather than the relative distance (Cp − Coverlap )/Coverlap from the overlapping p p overlap concentration Cp above which entanglements occur (data are not shown here). By plotting the horizontal shift factor as a function of Cp, we see that the relaxation time of the concentration blob behaves as ∼ Cp−3ν/(3ν−1) as predicted from scaling argument for semidilute polymer solutions (ν is the Flory exponent).48 Thus, the difference between the physical gelation due to noncovalent bond and network formation due to entanglements can be clearly identified by plotting the shift factors as a function of the concentration itself or ϵ. Detailed studies of entangled polymer solutions based on the MSD master curve will be reported elsewhere. 4.4. Sol−Gel Phase Diagram. We constructed the sol−gel phase diagram of the PVA−borax aqueous system based on the gel points determined from the uniqueness of the loss tangent as explained in section 4.2. For polymer concentrations higher than 2.75%, a critical borax concentration Cb* was determined for each fixed polymer concentration. On the other hand, for lower polymer concentrations, we determined the critical polymer concentration C*p for each fixed borax concentration. The Cb-Cp phase diagram constructed in this way is presented in inset of Figure 8. We see that Cb* hits a bottom at Cp≃ 3.85%. The sol−gel phase diagram in terms of the PVA concentration and the mean functionality of the PVA chain (i.e., number of the association cites per chain) f due to the reversible didiol complexation is also presented in Figure 8. The mean functionality is given by f = nrepRc where Rc ≔ 2Cb/Cp is the cross-linking ratio defined as the molar ratio of borate ions to repeating units of the PVA chain, and nrep = 2023 is the total number of the repeat units per chain. In this plot the functionality of the chain is constant along the vertical axis, and therefore this diagram is more convenient to analyze than the Cb−Cp diagram (inset). For example, the sol−gel line shows a

Figure 7. (A) Time exponent, (B) modulus exponent, (C) ratio of the exponents between pregel (−) and postgel (+) regimes, and (D) relaxation exponent as a function of PVA concentration. In parts A, B, and D, dashed lines indicate the exponents for electrical networks (n = 0.72, y = 2.6, z = 1.9) and dotted lines show those for the Rouse model (n = 2/3, y = 4, z = 8/347). In part C, a horizontal gray line indicates the mean value of y+/y−and z+/z−(1.42). In part D, symbols are literature data for PVA gels chemically cross-linked by 9 mM (□), 13 mM (○), and 22 mM (△) glutaraldehyde.56

percolation analogy has been reported for physical gels of SnO2 colloidal suspensions containing PVA.59 It deserves to be mentioned that the ratios y+/y− and z+/z− are equal and independent of the polymer concentration within experimental errors (see Figure 7 (C)). A simple dimensional analysis derives a scaling relation between y,z and the relaxation exponent n as n = z/y.70 To confirm this relation experimentally for the present system, we plotted n together with z/y for both pregel and postgel regimes in Figure 7D. We see that approximately n = z−/y− = z+/y+. The second equality is consistent with the equality y+/y− = z+/z−(≃ 1.42) discussed above (Figure 7C). That is, although y+ ≠ y− and z+ ≠ z−, the ratio z/y cancels out the discrepancy giving the values of z/y close in both phases. Furthermore, these values of z/y almost agree with the value of n estimated from the uniqueness of tan δ. Summarizing, in spite of the asymmetry of the exponents y,z before and after the gel point, the expected scaling relation is satisfied for each phase (n− = z−/y−, n+ = z+/y+) and the relaxation exponent is symmetric (n− = n+). To our knowledge, there is no theoretical requirement for the exponents y and z to be symmetric before and after the gel point, although the H

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fitting the macrorheologically observed dynamic modulus with the single-mode Maxwell model. Thus, G× is the plateau modulus of the Maxwell model (there is no clear plateau in experimental G′ near the gel point). As seen in the right figure, the linear viscoelastic relation η = τ×G× is satisfied. As for the postgel regime, the exponent z = 1.7 for G× is close to the value of 1.9 predicted for the electrical networks, This value is smaller than z+ (=2.21 ± 0.10) for Gz estimated based on the timecure-superposition of the MSD curves. This fact indicates that the plateau modulus at the low frequency side (G×) increases more gradually than that at the high frequency side (Gz) with increasing the cross-linker concentration. On the other hand, the exponent x = 0.98 for τ× is close to 1.0 that Rubinstein and Semenov62 predicted based on the concept of the red bonds of the structures.71 Thus, the exponent k = 2.6 for η appears to be reasonable because this is close to the value of x + z. As for the pregel regime, the viscosity exponent k = 0.96 almost agrees with that estimated from the microrheological measurements (k = y−z = 1.02), indicating a consistency between the two independent measurements. As seen in Figure 9, the zero-shear viscosity levels off around the gel point −ϵ̃−< ϵ < ϵ̃+ with approximately ϵ̃− ≃ 0.05 (i.e., 1/ ϵ̃− ≃ 20) and ϵ+ ≃ 0.1. A plateau in the zero-shear viscosity near the physical gel point is a counterpart to the divergence of the viscosity at the chemical gel point. The appearance of the plateau viscosity may be explained by extending the argument by Rubinstein and Semenov.62 They assumed that the terminal relaxation time is flat inside a certain regime (we call it the RS regime) around the gel point where the lifetime of the largest cluster τlife (corresponding to τ× in our notation72) is smaller than its relaxation time τrelax (corresponding to τz) as shown in Figure 10 (B). We consider that the plateau domain of the viscosity |ϵ| < ϵ̃ corresponds to the RS regime because of the following reasons. At low cross-linker (borax) concentrations satisfying ϵ < −ϵ̃− (see (i) of Figure 10, parts A and B), clusters of the PVA chains are small, and therefore, the configurational relaxation time of the largest cluster τrelax is smaller than its lifetime τlife. All clusters can relax into equilibrium before they are dissociated into smaller clusters. With increasing cross-linker concentration, the clusters become large due to the associations with the other clusters, so that τrelax increases while τlife decreases.62 We consider that τlife becomes comparable with τrelax at ϵ = −ϵ̃−(ii). In the following, we denote the number of monomers forming the largest clusters at this concentration (ii) as M̃ (corresponding to mz in our notation). Inside the RS regime − ϵ̃− < ϵ < 0 (iii), the merged clusters that became larger than M̃ break apart before relaxing into equilibrium. Thus, there are two types of clusters in this regime; one is the relaxed cluster smaller than M̃ but long-lived, and the other is the unrelaxed cluster larger than M̃ but short-lived. Only the long-lived relaxed clusters (and the local relaxed segments of the large unrelaxed clusters) exclusively contribute to the viscoelasticity, but they cannot grow beyond M̃ . Even though they coalesce with the other clusters, they break into small clusters before contributing to the viscoelasticity as a whole. Therefore, the zero-shear viscosity is nearly independent of the cross-linker concentration in regime (iii) (see Figure 10C). Nevertheless, the large but unrelaxed short-lived clusters keep growing with increasing the cross-linker concentration due to the temporary association with the other clusters. At ϵ = 0 (iv), it is expected that the unrelaxed clusters form a network

Figure 8. Sol−gel phase diagram of PVA−borax aqueous solutions/ gels.

monotonic decrease differently from that in the Cb−Cp diagram. Mean-field theories for transient gelation of the system similar to the present PVA−borax solution/gel has been proposed by Ishida−Tanaka61 and Semenov−Rubinstein.50 But both theories assume that the stickers are covalently fixed to polymer chains differently from the present system where the cross-linkers are detachable and mobile. Semenov-Rubinstein also considered the case that the bond strength is very high.50 We made a comparison of the experimental data (Figure 8) with these theories despite the fact that the assumptions of the theories do not match perfectly with the experimental conditions. The results are discussed in Supporting Information. Briefly, the theories can explain the overall feature of the sol−gel boundary, but a discrepancy is observed at low polymer concentrations (i.e., at high mean functionalities). More detailed characterizations of the system and modification of the theory are required to achieve a better agreement, however, these are out of the scope of this paper. 4.5. Viscosity in the Vicinity of the Gel Point. Figure 9 shows the macrorheologically measured zero-shear viscosity η, the “plateau” modulus G× and the longest relaxation time τ× as a function of the extent of cross-linking ϵ = (Cb − C*b )/C*b , i.e., relative distance from the microrheologically determined gel point Cb*. G× and τ× for the postgel regime were estimated by

Figure 9. Zero-shear viscosity as a function of the extent of crosslinking ϵ (red symbols). For comparison, product of the plateau modulus (upper inset of the right figure) and the longest relaxation time (lower inset) is also plotted (blue symbols). PVA concentration is fixed at Cp = 5.5%. I

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determine a physical gel point (a borax concentration C*b ) at which percolation is expected to occur: (1) The dynamic modulus G* shows the critical power-law behavior near Cb*, and it agrees well with the theory that assumes the self-similar fractal structure of the material. (2) The characteristic time and the characteristic MSD (or G*) estimated from the MSD shift factors of the master curves obey the power-law near Cb*. These time and MSD (G*) scale as a function of the distance from the gel point ϵ = (Cb − C*b )/C*b rather than Cb itself, similar to the situation in chemical gelation. (3) The characteristic time becomes large at Cb < Cb* but small at Cb > Cb* with increasing the cross-linker concentration Cb. Thus, we can speculate that a network is formed near C*b . An appearance of the plateau modulus also indicates a presence of the network at Cb > C*b . (4) The values of the relaxation exponent n and the PVAconcentration dependence of n are similar to those for chemically cross-linked PVA gels. It is expected that the material in the postgel regime is always percolated at any instant in time but flows at the time scales longer than the lifetime of the network strands while changing the network architecture. We infer that the transient percolation is achieved by temporarily percolating unrelaxed long strands. At Cb = Cb*, the network is percolated transiently but not fully relaxed at any moment in time, so that the viscosity is finite. We experimentally showed for the first time that the viscosity is nearly independent of cross-link concentration inside the narrow regime around Cb* where the relaxation time of the cluster (for pregel) or strands (for postgel) is larger than its lifetime. The viscosity plateau is a counterpart of a divergence of the viscosity at the chemical gel point and highlights the difference between the physical gelation and chemical gelation. The largest relaxed clusters govern the rheological properties of this viscosity plateau domain, since there are larger clusters but they are unrelaxed (they break before relaxing), thus they do not contribute to the rheological meaurements. We estimated the size of the largest relaxed cluster, by fitting our theoretical model to the experiment, and found that it is smaller than the probe particle size. Thus, our microrheological measurements by GSER are valid. In fact we did not see probe size dependency for the MSD measurement. In the postgel regime, above a certain extent of cross-linking at which these two characteristic times meets, the network is fully relaxed and consequently the viscosity rises abruptly. This crossover point at which the zero-shear viscosity starts to increase rapidly is different from the gel point at Cb = Cb*. Though this crossover point is rheologically interesting, it appears that there is no critical phenomenon at this point, contrary to the geometrical gel point which resides in the viscosity plateau regime. The plateau regime of the viscosity is narrow in the present system, but it is expected that this regime would broaden in systems of weaker association strength. Investigations for the weaker physical gels would be necessary to understand more details inside the plateau regime.

Figure 10. (A) Schematic illustration of the gelation process of physical gels. Red coils: relaxed parts of the clusters (or strands). Gray coils: unrelaxed parts. In parts (i)−(iii), only the largest clusters are shown. (B) Sketch of the longest relaxation time reproduced from Figure 3 of ref 62. (C) Sketch of the zero-shear viscosity (from Figure 9).

that transiently percolates the solution. This is the physical gel point in terms of connectivity. We infer that the percolated network of the unrelaxed strands is almost continuously destroyed and reconstructed near ϵ = 0 as τlife can be very small. A similar argument holds true for the postgel regime where the crossover between τrelax and τlife of the network strands (rather than the clusters) determines ϵ̃+. That is, in the RS regime 0 < ϵ < ϵ̃+ (v), only the unrelaxed strands percolate the material. The viscosity is almost independent of the cross-linker concentration since the number of monomers in a local relaxed part of the unrelaxed strands is always less than M̃ . These unrelaxed strands become shorter as the cross-linker concentration increases and finally at ϵ = ϵ̃+ (vi), τlife becomes comparable to τrelax, i.e., all strands can relax before destroyed. In other words, the material is percolated by the relaxed strands. Above this regime ϵ > ϵ̃+ (vii), the zero-shear viscosity increases again because the relaxed strands of the percolated network become dense. In the present PVA−borax system, it seems that the width of the RS regime is not symmetric before and after the gel point. Probably this is attributed to the asymmetries in the exponents y for the relaxation time (discussed in section 4.3) and x for the lifetime (unconfirmed yet) as well as the asymmetry in the prefactor before and after the gelation.

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5. SUMMARY AND CONCLUSION A living network of poly(vinyl alcohol) cross-linked by borax in water is a typical flowable physical gel. We applied DWS-based microrheology complemented by macrorheology to this living network. On the basis of the following experimental observations, we conclude that our experiments allows us to

APPENDIX A. FITTING METHOD OF MSD

Values of the parameters {α1, ..., αnmax, τ0, ..., τmax, g1} were max (|ln MSDdata(tj) | − |ln determined so as to minimize χ2 ≔ ∑jj=1 2 MSDfit(tj)|) /|ln MSDdata(tj) | within a certain window of a parameter space, where MSDfit(t) is eq 3, MSDdata(t) is the MSD data at t, and jmax is the number of data used for the J

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fitting. We manually estimated the parameter window for a certain borax concentration near the gel point, and used the same window for all borax concentrations to find the gel point. It is very important to choose an appropriate window because the fitting curve eq 3 is a nonlinear function of these parameters and therefore practically the minimization of χ2 is attainable only locally in the parameter space. If our minimized χ2 value was much larger than the conventional one based on the leastsquares fitting of the polynomial function of ln t38 with order 7, we modified the parameter window so that our χ2 becomes smaller than or at least comparable to the conventional one. Then we calculated G* and tan δ = G″/G′ by using the GSER for the thus-obtained parameter values. Our MSD data set includes 27 points per order of magnitude of time (sec), and the observation time spans 3−4 orders. Usually it takes some time to fit all these data by our fitting function 3. For time-saving purpose, we first used the data for every 10 points to fit the MSD and to estimate tan δ, and repeated this procedure 10 times for different data sets for every 10 points, and finally took average of these 10 tan δ’s. The gel point was determined from the intersection of the thusobtained averaged tan δ (Figure 5).

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APPENDIX C. CONSTRUCTION OF THE MSD MASTER CURVE In constructing the MSD master curves, a domain at which the individual MSD curve changes its slope on a log−log scale are landmarks to superpose the curves appropriately. However, in the pregel regime, the MSD curves are monotonous and do not exhibit clear slope changes, thereby making it difficult to construct the master curves uniquely by hand. To ensure objectivity as well as reproducibility, we automatically constructed the master curves using the computational algorithm as follows. First, we introduce the slope functions g(t) for each MSD curve in a log−log scale by linearly interpolating the difference quotient of ln ⟨Δrb2(ln t)⟩eq. The horizontal time shift-factor a is determined so that the vertical distance between the nearest two slope functions is minimized inside the overlapped area on the time axis. Second we define a continuous MSD function f(t) just by linearly interpolating the MSD data. The vertical MSD shift-factor b is determined so that the vertical distance between the two MSD functions (one of which is already horizontally shifted and fixed) is minimized inside the overlapped area on the t axis. For high borax concentrations in the postgel regime (for example Cb ≥ 2.1 mM), MSD data at t > 0.001 were not used when constructing the mastercurve to exclude influences of the elastic plateau.

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APPENDIX B. ESTIMATION OF THE GEL POINT AND THE CRITICAL EXPONENTS

In the multifrequency plot of tan δ (Figure 5), the intersection domain of the multiple tan δ curves is usually not a single point but is comprised of a group of several intersection points. We derived Cb* and n for each intersection point and estimated the means and standard errors of these values. Usually the error in n is less than 1% and therefore negligible, but the error in C*b is a few percent and non-negligible. For example, in the case of Cp = 5.5%, the critical borax concentration is C*b = 1.7 ± 0.03 mM. The error in C*b affects the shape of the master curve and is propagated from the master curve to the critical exponents. In addition, the values of the critical exponents depend on the data range of the MSD to be used for the construction of the master curve because the data scatter at upper and lower boundaries of the accessible experimental range (see inset of Figure 6A, for example). To take these errors into account, we took averages and estimated standard errors of the exponents obtained from the master curves constructed for several different data range around the upper Cb,upper * (=1.73 mM) and lower Cb,lower * (=1.67 mM) limits of the uncertainty in Cb*. Parts A and B of Figure 6 show the results from one representative sample (C*b = C*b,upper, 7 × 10−6 μm2 ≲ MSD ≲ 2 × 10−4 μm2). In general, the pregel exponents are larger than the postgel exponents if the exponents are estimated around C*b,upper as seen in inset of Figure 6B. On the other hand, if the exponents are estimated around C*b,lower, the opposite tendencies are observed. In estimating the exponents y and z for Cp = 5.5% from the shift factors of the MSD master curves (see Figure 6B), MSD of the two borax concentrations closest to the gel point (Cb = 1.7 and 1.8 mM) were excluded because the MSD curves for these borax concentrations are almost linear and therefore the superposition of the MSD curves (and the shift factors) include large uncertainties. These data, Cb = 1.7 and 1.8 mM, are not included in insets of Figure 6B (but are included in the main figure of Figure 6B for reference).

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b00745. Comparison of experimental data (Figure 8) and theoretical predictions for the sol−gel phase diagram (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] Phone: 1-312-567-8844. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS T.N. thanks Pascal Hébraud, Costantino Creton, and Gregory B. McKenna for their helpful comments and Agence Nationale de Recherche of France (ANR JCJC 2010 Dynagel) for financial support. T.I. is grateful to Jay D. Schieber for useful and enlightening comments and the Army Research Office (Grants W911NF-08-2-0058, W911NF-09-1-0378, and W911NF-11-2-0018) for financial support.

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REFERENCES

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DOI: 10.1021/acs.macromol.6b00745 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

example, [Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: Oxford, U.K., 1986 ]) of thePVA chain is estimated to be 0.325 × ηsRe3/kBT = 5.9 × 10−7 s, where ηs = 1 mPa·s isthe viscosity of the solvent (water). We assume that this is τ0. On the other hand, an initial value of G0 to fit the data can be guessed from the extrapolateddata of the storage or loss modulus at ω = 1/τ0. Some adjustment of the initial value is required to fit the datawell. (69) Generally, such low modulus samples are better characterized by the (zero shear or complex) viscosity than the dynamic moduli even with macrorheology. For the zero shear modulus of PVA solutions without borax, a good agreement is reported. [ Narita, T. Macromolecules 2001, 34, 8224. ] (70) Putting G*(ω) ∼ (iω)n into the GSER, we get ⟨Δrb2(t)⟩eq ∼ tn. Therefore, ⟨Δrb2(τz)⟩eq∼ τzn ∼ |ϵ|−ny. On the other hand, the exponent z is defined by ⟨Δrb2(τz)⟩eq ∼ |ϵ|−z. By comparing these two equations, we find this relation. (71) τ× corresponds to the lifetime of network strands. The shorter the strand length, the longer the lifetime because of the smaller number of breakable points per strand. Therefore, as the cross-linker concentration increases and the network becomes dense (i.e., strands become short), τ× increases. (72) Below we use the same notations as those used in ref 62 except ϵ.

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DOI: 10.1021/acs.macromol.6b00745 Macromolecules XXXX, XXX, XXX−XXX