Ginzburg Number Research Articles

Dynamic Crossover of Critical Behavior in Polymer Blend Solutions

By the use of quasi-elastic light scattering, dynamic critical behavior was studied for symmetrical blend solutions of polystyrene/poly(methyl methacrylate)/deuterated benzene and polystyrene/poly(2-chlorostyrene)/deuterated benzene having different polymeric indices N and polymer concentrations c. The decay rate Γ I of polymer-composition fluctuations showed the dynamic crossover from the diffusive to the mode-mode-coupled hydrodynamic behavior in all solutions. The dynamic-crossover number τ*, which represents the distance between the stability limit and the location of the dynamic crossover, was comparable to that theoretically expected for the dynamically-unentangled system in the present ranges ofN and c including a chain-entangled region. The blend solutions with entanglements exhibited a wider mode-mode-coupled regime than theoretical prediction. The dynamic-crossover number τ* was larger than the Ginzburg number Gi, which characterizes the location of the static crossover from the mean-field to the nonclassical Ising behavior, and the mode-mode-coupled dynamics covered a wider region than the statical Ising behavior.

Static Crossover of Critical Behavior in Polymer Blend Solutions: Scaling Analysis of the Ginzburg Number Volume 28, Number 13, June 19, 1995, pp 4433−4440

ADVERTISEMENT RETURN TO ISSUEPREVAddition/CorrectionNEXTStatic Crossover of Critical Behavior in Polymer Blend Solutions: Scaling Analysis of the Ginzburg Number Volume 28, Number 13, June 19, 1995, pp 4433−4440Naoshi Miyashita and Takuhei NoseCite this: Macromolecules 1996, 29, 24, 8024Publication Date (Web):November 18, 1996Publication History Published online18 November 1996Published inissue 1 January 1996https://doi.org/10.1021/ma952017lCopyright © 1996 American Chemical SocietyRIGHTS & PERMISSIONSArticle Views115Altmetric-Citations-LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit Get e-Alerts Get e-Alerts

Ginzburg criterion for the mean-field to three-dimensional Ising crossover in polymer blends.

Composition fluctuations within the mean-field and three-dimensional Ising range were measured in a homogeneous binary polymer blend by small angle neutron scattering as a function of temperature and pressure. The experimental data were analyzed in terms of the crossover function of Belyakov and Kiselev [Physica A 190, 75 (1992)]. It is shown that the reduced-crossover-temperature, the Ginzburg number Gi, decreases with pressure sensitively, in accordance with the prediction of Belyakov and Kiselev. On the other hand, de Gennes' crossover criterion for polymer blends predicts an increase of Gi with pressure which contradicts our results.

Critical crossover phenomena in compatible polymer blends studied with SANS

Polymer blends show a much larger 3d-Ising regime, e.g. a much larger Ginzburg number Gi than predicted by the Ginzburg criterion. This discrepancy is supposed to be explained by the compressibility or the free volume of the blend. In this paper we present and discuss the Gi number of polymer blends, different in chemistry, molecular weight, and microstructure. One sample was also studied at a different pressure. Gi was determined by describing the susceptibility obtained from SANS with the crossover function from Belyakov et al. The essential results are that Gi depends sensitively on monomeric microstructure and on pressure. This clearly shows that Gi is not a universal function. The observed strong decrease of Gi with pressure is a clear experimental proof that the critical crossover behaviour in polymer blends is indeed strongly influenced by the compressibility or free volume of the blend.

Static Crossover of Critical Behavior in Polymer Blend Solutions: Scaling Analysis of the Ginzburg Number

By means of light scattering, the static critical behavior was investigated for symmetrical polymer blend solutions of polystyrene/poly(methyl methacrylate)/deuterated benzene and polystyrene/poly(2-chlorostyrene)/deuterated benzene with different polymeric indices N and polymer concentrations c. The temperature dependence of the isothermal susceptibility S T and the correlation length ξ t near the stability limit showed the static crossover from the mean-field to the three-dimensional Ising behavior in all solutions. The Ginzburg number Gi BK was evaluated with the aid of the crossover function of Belyakov and Kiselev. As c increases from the semidilute region to the bulk limit, the scaled Ginzburg number gi BK = Gi BK N decreased, with the power α of gi BK ∞ c α being -1.25 to -2 as expected by the theory based on the blob concept. The values of gi BK (bare) evaluated by the bare interactions were smaller than experimental values.

Microstructural effects on the Ginzburg number and the crossover behavior in d-PB/PS blends.

The critical behavior of three equimolar deuterated polybutadiene/polystyrene ($d\ensuremath{-}\frac{\mathrm{PB}}{\mathrm{PS}}$) blends was measured by small-angle neutron scattering. The three blends differ by the content of the 1,4 and 1,2 units of the $d\ensuremath{-}\mathrm{PB}$ component. It was found that the Ginzburg number increases with 1,2 unit content which even leads to a crossover like the "Fisher renormalized" critical behavior. The larger content of 1,2 units is supposed to increase the free volume change upon mixing which influences the Ginzburg number. Thus the nonuniversality of the Ginzburg number is clearly demonstrated.

Vortices in high-temperature superconductors

With the high-temperature superconductors a qualitatively new regime in the phenomenology of type-II superconductivity can be accessed. The key elements governing the statistical mechanics and the dynamics of the vortex system are (dynamic) thermal and quantum fluctuations and (static) quenched disorder. The importance of these three sources of disorder can be quantified by the Ginzburg number $Gi=\frac{{(\frac{{T}_{c}}{{H}_{c}^{2}}\ensuremath{\varepsilon}{\ensuremath{\xi}}^{3})}^{2}}{2}$, the quantum resistance $Qu=(\frac{{e}^{2}}{\ensuremath{\hbar}})(\frac{{\ensuremath{\rho}}_{n}}{\ensuremath{\varepsilon}\ensuremath{\xi}})$, and the critical current-density ratio $\frac{{j}_{c}}{{j}_{o}}$, with ${j}_{c}$ and ${j}_{o}$ denoting the depinning and depairing current densities, respectively (${\ensuremath{\rho}}_{n}$ is the normal-state resistivity and ${\ensuremath{\varepsilon}}^{2}=\frac{m}{M}l1$ denotes the anisotropy parameter). The material parameters of the oxides conspire to produce a large Ginzburg number $\mathrm{Gi}\ensuremath{\sim}{10}^{\ensuremath{-}2}$ and a large quantum resistance $\mathrm{Qu}\ensuremath{\sim}{10}^{\ensuremath{-}1}$, values which are by orders of magnitude larger than in conventional superconductors, leading to interesting effects such as the melting of the vortex lattice, the creation of new vortex-liquid phases, and the appearance of macroscopic quantum phenomena. Introducing quenched disorder into the system turns the Abrikosov lattice into a vortex glass, whereas the vortex remains a liquid. The terms and liquid are defined in a dynamic sense, with a sublinear response $\ensuremath{\rho}={\frac{\ensuremath{\partial}E}{\ensuremath{\partial}j}|}_{j\ensuremath{\rightarrow}0}$ characterizing the truly superconducting vortex glass and a finite resistivity $\ensuremath{\rho}(j\ensuremath{\rightarrow}0)g0$ being the signature of the phase. The smallness of $\frac{{j}_{c}}{{j}_{o}}$ allows one to discuss the influence of quenched disorder in terms of the weak collective pinning theory. Supplementing the traditional theory of weak collective pinning to take into account thermal and quantum fluctuations, as well as the new scaling concepts for elastic media subject to a random potential, this modern version of the weak collective pinning theory consistently accounts for a large number of novel phenomena, such as the broad resistive transition, thermally assisted flux flow, giant and quantum creep, and the glassiness of the solid state. The strong layering of the oxides introduces additional new features into the thermodynamic phase diagram, such as a layer decoupling transition, and modifies the mechanism of pinning and creep in various ways. The presence of strong (correlated) disorder in the form of twin boundaries or columnar defects not only is technologically relevant but also provides the framework for the physical realization of novel thermodynamic phases such as the Bose glass. On a macroscopic scale the vortex system exhibits self-organized criticality, with both the spatial and the temporal scale accessible to experimental investigations.

On the N-scaling of the Ginzburg number and the critical amplitudes in various compatible polymer blends

The susceptibility related to composition fluctuations in polymer blends can be described by a crossover function derived by Belyakov et al. [2, 3] near and far from the critical temperature Tc. This has quite recently been demonstrated by us using neutron and light scattering data [1]. In this paper we discuss consequences of this analysis with respect to the accomplishment of the mean-field approximation far from Tc and the N scaling (N is the degree of polymerization) of the critical amplitudes of the bare correlation length and susceptibility respectively and the Ginzburg number, Gi. A quite unexpected N dependence of Gi was found : polymer blends with N < 30 give the same Gi as low molecular liquids and at about N = 30 an increase of Gi of more than one order of magnitude which is synonymous to an increase of the 3d-Ising regime is observed. For $N \gtrsim 30$ a Gi ∝1/N2 scaling is found in a range up to $N \cong 400$. The observed N scaling of Gi can be explained by the large entropic contribution of the Flory-Huggins parameter Γσ related to the compressibility or free volume of the system in comparison with the combinatorial entropic term, which was obtained from the analysis of the scattering results with the crossover function.

Limits of validity for mean field description of compressible binary polymer blends

We examine phase diagrams of model binary polymer blends and estimate the temperature range where mean field theory is applicable. The computations use both compressible and incompressible Flory–Huggins theory and the corresponding random phase approximations. The size of the nonclassical regime is estimated in terms of the Ginzburg number Gi for both the one and two phase regions. Model calculations indicate that compressibility (and thus pressure) can significantly affect widths of the nonclassical region and critical temperatures (Tcrit) for phase separations. The qualitative scaling of Tcrit and Gi (and hence the widths of the nonclassical regime) also differs between compressible and incompressible blends. Incompressible symmetric blends with polymerization index N yield constant values for both the product NGi and the ratio Tcrit/N, while compressible polymer mixtures produce NGi as nonconstant and even possibly as varying nonmonotonically with chain length. Pressure and entropic interaction variations are shown to change miscibility limits and strongly to affect Ginzburg numbers for some blends. Model calculations provide insight into common experimental methods for determining the onset of the nonclassical domain.

Fluctuations and disorder in high temperature superconductors

The special material parameters of the oxide superconductors lead to a dramatic increase of the importance of thermal and quantum fluctuations. The latter can be quantified by the Ginzburg number Gi = [T c/ H 2 c(0)ϵξ 3(0)] 2/2 and the quantum resistance Q u = (e 2/ h ̵ ) [ϱN/ ϵξ(0)] , where H c(0), ξ(0), and ϱ N denote the thermodynamic critical field, the planar coherence length (both linearly extrapolated to zero), and the planar normal resistivity. ϵ 2 = m/ M < 1 is the anisotropy parameter. In the high T c 's (specifically for YBCO) we have Gi ≅ 10 −2 and Qu ≅ 1 and thus these parameters are by orders of magnitude larger than in conventional low- T c superconductors. The large fluctuations lead to the melting of the vortex lattice well below the upper critical field line. The inclusion of quenched disorder as parametrized by the critical current density ratio j c / j o drastically changes the dynamic behavior of the vortex system ( j c and j o denote the depinning and depairing current densities). We discuss the equilibrium statistical mechanics (vortex lattice melting) and the dynamic behavior (creep) of the vortex system with a particular emphasis on the role of quantum fluctuations.

How far is far from critical point in polymer blends? Lattice cluster theory computations for structured monomer, compressible systems

Although the lattice cluster theory (LCT) incorporates many features which are essential in describing real polymer blends, such as compressibility, monomer structures, local correlations, chain connectivity, and polymer–polymer interactions, it still remains a mean field theory and is therefore not applicable in the vicinity of the critical point where critical fluctuations become large. The LCT, however, permits formulating the Ginzburg criterion, which roughly specifies the temperature range in which mean field applies. The present treatment abandons the conventional assumptions of incompressibility and of composition and the molecular weight independent effective interaction parameter χeff upon which all prior analyses of the Ginzburg criterion are based. Blend compressibility, monomer structure, and local correlations are found to exert profound influences on the blend phase diagram and other critical properties and, thus, exhibit a significant impact on the estimate of the size of the nonclassical region. The LCT is also used to test various methods which employ available experimental data in computations of the Ginzburg number Gi. The reduced temperature τ=‖T−Tc‖/T defining the range of the validity of mean field theory (τ≳τMF) and the onset of the Ising-type scaling regime (τ≳τcrit) are quite different, and renormalization group estimates of τMF and τcrit are presented as a function of Gi to more precisely specify these scaling regimes.

Penetration depth and Ginzburg number in Rb 3C 60

The penetration depth λ, the Ginzburg-Landau parameter κ and the Ginzburg number Gi in the superconducting fulleride Rb 3C 60 are estimated. To determine these quantities the results of recent magnetization measurements performed by Politis et al. on the fine powder specimen are analyzed. A finiteness of superconducting grain sizes is taken into account and shown to influence considerably the calculated values of λ and κ. The experimental data are found to be consistent with the assumption that λ( T) obeys the Ginzburg-Landau theory within the whole temperature range studied and result in λ(0)=2100 Å, κ ≈90. The estimated critical region in Rb 3C 60 turns out to be narrow enough, Gi ≈10 -4, indicating that superconducting fluctuations in fullerides are weaker than in high- T c oxide superconductors.

Crossover behavior of the susceptibility and the specific heat near a second-order phase transition

Simple crossover equations for the susceptibility and the specific heat in zero field have been obtained on the basis of the renormalization-group method and ε-expansion. The equations contain the Ginzburg number as a parameter. At temperatures near the critical temperature, scaling behavior including the first Wegner corrections is reproduced. At temperatures far away from the critical temperature the classical Landau expansion with square-root corrections is recovered. For small values of the Ginzburg number the crossover equations approach a universal form. The equations are applied to represent experimental specific heat data for CH 4, C 2H 6, Ar, O 2 and CO 2 along the critical isochore in a universal form.

Specific heat anomalies in high temperature superconductors: Critical behavior or gaussian fluctuations?

The width of critical region and 2D→3D crossover temperature for high- T c superconductors of yttrium, bismuth and thallium families are estimated. For Bi- and Tl-based compounds the Ginzburg number Gi turns out to be 0.02-0.05 while for Y-123 Gi≈0.003. In Bi-2212 and Tl-2212 phases superconducting fluctuations appear to be two-dimensional up to ¦ T– T c¦∼0.3 –0.1 K. The results of recent high resolution measurements of the specific heat of the above superconductors [1–3] are discussed in detail. The experimentally accessible range is shown to cover the region of gaussian fluctuations and, for Bi and Tl compounds, the crossover region. It is argued that the logarithmic temperature dependence of fluctuation specific heat obtained would not be considered as an evidence of true 3D critical behaviour since it is found outside 3D critical region and the critical amplitude ratios measured [2,3] turn out to be essentially non-universal.