Scale-free power-law distribution of nonlinear polymers formed in a homeostatic system

e-Polymers ◽  
2004 ◽  
Vol 4 (1) ◽  
Author(s):  
Hidetaka Tobita

Abstract A power-law distribution of molecular weight is found for low-density polyethylene formed in a one-zone autoclave reactor. This reactor could be modelled as a continuously stirred tank reactor (CSTR) operated under homeostatic conditions where the concentration of each component as well as temperature and pressure are kept constant. A simple stochastic model is used to investigate the molecular weight distribution (MWD) formed in free-radical polymerization involving chain transfer to polymer. The MWD profiles are dominated by the Pb value, which represents the probability that the chain end is connected to a backbone chain. By increasing Pb, MWD changes from an exponential to a powerlaw distribution. A scale-free power-law distribution is found for Pb > 0.5. Because the power of the weight fraction distribution, W(M) ~ M -γ, is 1 < γ < 2, the number average molecular weight will reach a stationary-state value but the weightaverage may continue to increase unlimitedly under ideal CSTR conditions.

e-Polymers ◽  
2004 ◽  
Vol 4 (1) ◽  
Author(s):  
Hidetaka Tobita

Abstract The high-molecular-weight tail of the molecular weight distribution (MWD) of branched polymers formed via free-radical polymerization involving polymer transfer reactions conducted in a continuously stirred tank reactor (CSTR) follows a power-law distribution. The exponent, α, of the weight fraction distribution W(M) ~ M-α is determined by the Pb value, which represents the probability that the chain end is connected to a backbone chain, with the simple relationship, α = 1/Pb. Since α > 1, the number-average molecular weight will always reach a stationary-state value but the weight-average and higher-order averages of molecular weights may continue to increase without limit under ideal CSTR conditions. Because it takes an infinitely long time for the second or a higher moment of the MWD to go to infinity, the kinetic behaviour is significantly different from usual gelation phenomena. The i-th moment of the MWD goes to infinity for Pb ≥ 1/i.


2007 ◽  
Vol 7 (5) ◽  
pp. 13805-13838 ◽  
Author(s):  
M. T. Latif ◽  
P. Brimblecombe

Abstract. Surfactants in atmospheric aerosols determined as methylene blue active substances (MBAS) and ethyl violet active substances (EVAS). The MBAS and EVAS concentrations can be correlated with surface tension as determined by pendant drop analysis. The effect of surface tension was more clearly indicated in fine mode aerosol extracts. The concentration of MBAS and EVAS was determined before and after ultrafiltration analysis using AMICON centrifuge tubes that define a 5000 Da (5 K Da) nominal molecular weight fraction. Overall, MBAS and to a greater extent EVAS predominates in fraction with molecular weight below 5 K Da. In case of aerosols collected in Malaysia the higher molecular fractions tended to be a more predominant. The MBAS and EVAS are correlated with yellow to brown colours in aerosol extracts. Further experiments showed possible sources of surfactants (e.g. petrol soot, diesel soot) in atmospheric aerosols to yield material having molecular size below 5 K Da except for humic acid. The concentration of surfactants from these sources increased after ozone exposure and for humic acids it also general included smaller molecular weight surfactants.


1955 ◽  
Vol 28 (2) ◽  
pp. 504-507
Author(s):  
G. W. Drake

Abstract Fractionation of the rubber hydrocarbon in temperate climates has usually resulted in high molecular-weight fractions, with a molecular weight of the order of one million. Bloomfield has shown that fresh latex contains a considerable proportion of hydrocarbon having an intrinsic viscosity (η) of 10 or over and, therefore, a molecular weight of well over 106. The fractionation technique used by Bloomfield in Malaya has now been applied by the writer to smoked sheet and to F rubber, working in the United Kingdom. No very high molecular-weight fractions were found in the smoked sheet, but the F rubber yielded a fraction of (η)=7.3 and a number average molecular weight 6×106, determined osmometrically. The average molecular weight of natural rubber when freshly prepared is probably well over a million, and includes a substantial portion having a molecular weight of several millions. By the time smoked sheet has reached temperate climates, the high molecular-weight portion has probably been converted to gel. F rubber, presumably because of its different method of preparation, retains the major part of its high molecular-weight material during prolonged storage.


2013 ◽  
Vol 753-755 ◽  
pp. 2959-2962
Author(s):  
Jun Tao Yang ◽  
Hui Wen Deng

Assigning the value of interest to each node in the network, we give a scale-free network model. The value of interest is related to the fitness and the degree of the node. Experimental results show that the interest model not only has the characteristics of the BA scale-free model but also has the characteristics of fitness model, and the network has a power-law distribution property.


1999 ◽  
Vol 09 (12) ◽  
pp. 2249-2255 ◽  
Author(s):  
S. HAINZL ◽  
G. ZÖLLER ◽  
J. KURTHS

We introduce a crust relaxation process in a continuous cellular automaton version of the Burridge–Knopoff model. Analogously to the original model, our model displays a robust power law distribution of event sizes (Gutenberg–Richter law). The principal new result obtained with our model is the spatiotemporal clustering of events exhibiting several characteristics of earthquakes in nature. Large events are accompanied by a precursory quiescence and by localized fore- and aftershocks. The increase of foreshock activity as well as the decrease of aftershock activity follows a power law (Omori law) with similar exponents p and q. All empirically observed power law exponents, the Richter B-value, p and q and their variability can be reproduced simultaneously by our model, which depends mainly on the level of conservation and the relaxation time.


2012 ◽  
Vol 229-231 ◽  
pp. 1854-1857
Author(s):  
Xin Yi Chen

Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a power-law distribution. This feature was found to be a consequence of three generic mechanisms: (i) networks expand continuously by the addition of new vertices, (ii) new vertex with priority selected different edges of weighted selected that connected to different vertices in the system, and (iii) by the fitness probability that a new vertices attach preferentially to sites that are already well connected. A model based on these ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena. Experiment results show that the model is more close to the actual Internet network.


2007 ◽  
Vol 17 (07) ◽  
pp. 2419-2434 ◽  
Author(s):  
FRANCESCO SORRENTINO ◽  
MARIO DI BERNARDO ◽  
FRANCO GAROFALO

We study the synchronizability and the synchronization dynamics of networks of nonlinear oscillators. We investigate how the synchronization of the network is influenced by some of its topological features such as variations of the power law degree distribution exponent γ and the degree correlation coefficient r. Using an appropriate construction algorithm based on clustering the network vertices in p classes according to their degrees, we construct networks with an assigned power law distribution but changing degree correlation properties. We find that the network synchronizability improves when the network becomes disassortative, i.e. when nodes with low degree are more likely to be connected to nodes with higher degree. We consider the case of both weighed and unweighed networks. The analytical results reported in the paper are then confirmed by a set of numerical observations obtained on weighed and unweighed networks of nonlinear Rössler oscillators. Using a nonlinear optimization strategy we also show that negative degree correlation is an emerging property of networks when synchronizability is to be optimized. This suggests that negative degree correlation observed experimentally in a number of physical and biological networks might be motivated by their need to synchronize better.


2006 ◽  
Vol 43 (3) ◽  
pp. 665-677 ◽  
Author(s):  
J. E. Yukich

We consider a family of long-range percolation models (Gp)p>0on ℤdthat allow dependence between edges and have the following connectivity properties forp∈ (1/d, ∞): (i) the degree distribution of vertices inGphas a power-law distribution; (ii) the graph distance between pointsxandyis bounded by a multiple of logpdlogpd|x-y| with probability 1 -o(1); and (iii) an adversary can delete a relatively small number of nodes fromGp(ℤd∩ [0,n]d), resulting in two large, disconnected subgraphs.


2011 ◽  
Vol 50-51 ◽  
pp. 166-170 ◽  
Author(s):  
Wen Jun Xiao ◽  
Shi Zhong Jiang ◽  
Guan Rong Chen

It is now well known that many large-sized complex networks obey a scale-free power-law vertex-degree distribution. Here, we show that when the vertex degrees of a large-sized network follow a scale-free power-law distribution with exponent  2, the number of degree-1 vertices, if nonzero, is of order N and the average degree is of order lower than log N, where N is the size of the network. Furthermore, we show that the number of degree-1 vertices is divisible by the least common multiple of , , . . ., , and l is less than log N, where l = < is the vertex-degree sequence of the network. The method we developed here relies only on a static condition, which can be easily verified, and we have verified it by a large number of real complex networks.


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