Upper critical dimension

An interesting and practically valuable result was obtained in [21] for PE + N2 melts, and in [43] for PS + N2 melts. The authors classified upper critical volumetric flow rate and pressure with reference to channel dimensions x Pfrerim y Qf"im-Depending on volume gas content

channel entrance (pressure of 1 stm., experimental temperature), x and y fall, in accordance with Eq. (24), to tp 0.85. At cp 0.80, in a very narrow interval of gas concentrations, x and y fall by several orders. The area of bubble flow is removed entirely. It appears that at this concentration of free gas, a phase reversal takes place as the polymer melt ceases to be a continuous phase (fails to form a continuous cluster , in flow theory terminology). The theoretical value of the critical concentration at which the continuous cluster is formed equals 16 vol. % (cf., for instance, Table 9.1 in [79] and [80]). An important practical conclusion ensues it is impossible to obtain extrudate with over 80 % of cells without special techniques. In other words, technology should be based on a volume con-... 

G. Biroli and J. P. Bouchaud, Diverging length scale and upper critical dimension in the mode coupling theory of the glass transition. Europhys. Lett. 67, 21 (2004). 

At the upper critical dimension d — 4 Eq. (7.29) shows that not only subdiagrams (M = lyL = 0) but also subdiagrams (Af = 1,L = 1), (M = 2. L = 0) axe divergent. Redefining all the parameters of the theory, we then can define a nontrivial form of the continuous chain limit, differing from... 

In the construction of the RGf dimension d = 4 plays a special role as upper critical dimension of the thebry. This for instance shows up in the estimate of the nonuniversal corrections to the theorem of renormalizability, or in the feature that the nontrivial fixed point u merges with the Gaussian fixed point for d — 4. It naturally leads to the e-expansion. However, the RG mapping constructed in minimal subtraction only trivially depends on e. Also results of renormalized perturbation theory do not necessarily ask for further expansion in e. Equation (12.25) gives an example. We should thus consider the practical implications of the -expansion in some more detail. 

These powers a, (3, 7, p and i/ are called the critical exponents. These exponents are observed to be universal in the sense that although Pc de-pends on the details of the models or lattice considered, these exponents depend the only on the lattice dimensionality (see Table 1.2). It is also observed that these exponent values converge to the mean field values (obtained for the loopless Bethe lattice) for lattice dimensions at and above six. This suggests the upper critical dimension for percolation to be six. 

The values of the critical exponents r and a and the cutoff functions /+ (N/N ) and/ (N/N ) depend only on the dimension of space in which gelation takes place. The percolation model has been solved analytically in one dimension (d=, see Sections 1.6.2 and 6.1.2) and critical exponents have been derived for two dimensions (d = 2). The mean-field model of gelation corresponds to percolation in spaces with dimension above the upper critical dimension (d>6). The cutoff function in the mean-field model [see Eq. (6.77)] is approximately a simple exponential function [Eq. (6.79)]. The exponents characterizing mean-field gelation are o — 1/2 and... 

Note that whenever a new exponent is defined, there is also a scaling relation that calculates this exponent from t and a. There are only two independent exponents that describe the distribution of molar masses near the gelation transition, with the other exponents determined from scaling relations. Table 6.4 summarizes the exponents in different dimensions that have been determined numerically, along with the exact results for 1, d=2, and d>6. It turns out that t/=6 is the upper critical dimension for percolation, and the mean-field theory applies for all dimensions d>6. 

This expression is expected to be valid for any statistical critical cluster in the presence of clnsters having a size distribution. The upper critical dimension for critical clusters can be determined assuming that in Equation (11.19), D=4. The result thus obtained, d =6, is in good agreement with the data for percolation [78]. 

Let US consider the limiting cases for h. At h=0 =-Q.5 or, as it follows from the Eq. (74), A =1.5. The last value defines the so-called permeable macromolecular coil, completely open for diffusion of various kinds of dififusates, including similar macromolecular coils [31], As it has been noted above, this case corresponds to the classical behavior (/t=0, Ar =const) [22], At /t=1.0 A =4, that is the upper critical dimension for separate macromolecular coil. In the last case the coil is Gaussian one (phantom one), that is, for it the excluded volume effects are abse c[6]. 

At d = 1 one has a completely stretched chain with ly = 1. At d = 2 the exact result v = 3/4) [13] is obtained. The upper critical dimension is d = 4, above which the polymer behaves as a random walker. The values of the universal exponents for SAWs on d - dimensional regular lattices have also been calculated by the methods of exact enumerations and Monte Carlo simulations. In particular, at the space dimension d = 3 in the frames of field-theoretical renormalization group approach one has (v = 0.5882 0.0011 [11]) and Monte Carlo simulation gives (i/ = 0.592 0.003 [12]), both values being in a good agreement. 

The field theory for SAWs on the percolation cluster developed in Ref. [22] supports an upper critical dimension d p = 6. The calculation of Up was presented to the first order of perturbation theory, however the numerical estimates obtained from this result are in poor agreement with the numbers observed by other means. In particular, they lead to estimate that i/p i/ in d = 3. Recently this investigation has been extended to the second order in perturbation theory [101], which leads to the qualitative estimates of critical exponents in good agreement with numerical studies and Flory-like theories. 

The critical exponent v at Gaussian fixed point reads z/ = 1/2, which corresponds to the fact that SAWs at dimensions above the upper critical one behave like RWs (simple random walks). There are essentially two ways to proceed in order to obtain the qualitative characteristics of the critical behavior of the model. The first is to substitute the loop integral in equations (94), (95) by its e-expansion ... 

In Ref. [39] it was argued that while the upper critical dimension for the problem of polj niers in disorder may be larger than dc = 4 one may analytically continue the e-expansion from d = 4 — ctoa4 + e-expansion. For the fixed points this has a dramatic effect (see Fig.3b). In the 4 + e case the unphysical fixed point (U) is mapped to U in the physical region. It is interpreted by these authors as a strong disorder fixed point. Also... 

This prediction iff = 5/3 in three dimensions is close to the actual value near 1.7 df = and 4/3 for d = and 2, respectively, is even exact. Note that we recover the ideal chain dimension for d = A. This is the upper critical dimension above which the excluded volume interaction becomes irrelevant, and the chain is ideal. For higher dimensions, the interaction with itself is negligible for the exponents, because space is sufficiently large that the polymer almost does not cross itself. Therefore, for d> d chains with or without interactions are equivalent. 

The dimension d = 3 corresponds to the upper critical dimension, at which logarithmic divergences appear due to the term t (y ) in the Lagrangian (or Hamiltonian). 

Capillary waves do not only broaden the width of the interface but they can also destroy the orientational order in highly swollen lamellar phases (see Fig. 1 for a phase diagram extracted from Monte Carlo Simulations). Those phases occur in mixtures of diblock-copolymers and homopolymers. The addition of homopolymers swells the distance between the lamellae, and the self-consistent field theory predicts that this distance diverges at Lifshitz points. However, general considerations show that mean-field approximations are bound to break down in the vicinity of lifshitz points [61]. (The upper critical dimension is du = 8). This can be quantified by a Ginzburg criterion. Fluctuations are important if... 

It seems plausible that fluctuations affect the Lifshitz point. If the lamellar distance is large enough that the interfaces between A and B sheets can bend around, the lamellae may rupture and form a globally disordered structure. A Ginzburg analysis reveals that the upper critical dimension of isotropic Lifshitz points is as high as 8 (see also Sect. 4.1). Unfortunately, the lower critical dimension of isotropic Lifshitz points is not known [ 102],... 

Matyjaszewski et al. [2] patented a novel and flexible method for the preparation of CNTs with predetermined morphology. Phase-separated copolymers/stabilized blends of polymers can be pyrolyzed to form the carbon tubular morphology. These materials are referred to as precursor materials. One of the comonomers that form the copolymers can be acrylonitrile, for example. Another material added along with the precursor material is called the sacrificial material. The sacrificial material is used to control the morphology, self-assembly, and distribution of the precursor phase. The primary source of carbon in the product is the precursor. The polymer blocks in the copolymers are immiscible at the micro scale. Free energy and entropic considerations can be used to derive the conditions for phase separation. Lower critical solution temperatures and upper critical solution temperatures (LCST and UCST) are also important considerations in the phase separation of polymers. But the polymers are covalently attached, thus preventing separation at the macro scale. Phase separation is limited to the nanoscale. The nanoscale dimensions typical of these structures range from 5-100 nm. The precursor phase pyrolyzes to form carbon nanostructures. The sacrificial phase is removed after pyrolysis. 

The question about the existence and nature of a spin-glass transition in d = 3 has remained controversial. The SK model does not answer the question since it is a mean-field theory, and the upper critical dimension d for spin glasses is = 6 or for certain exponents even d = 8 (Fisher and Sompolinsky 1985). 

Exponent relations involving the dimensionality d explicitly are called hyperscaling relations. They only hold below the upper critical dimension Above they are destroyed by dangerously irrelevant variables. 

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