Spatial correlations in a self‐repelling ring polymer

The scattering structure function SN(q) of a large flexible ring polymer with N monomers in a good solvent is calculated as a universal expansion in the variable q2<RG2≳, where <RG2≳ denotes the mean square radius of gyration of the ring. To solve the interacting polymer problem, the methods of renormalized field theory are applied directly to Fixman’s cluster expansion. The results are obtained using the approximation of expanding in ϵ≡4−(dimension of space) and indicate that the self‐repulsion has a greater influence on the large distance behavior of a ring polymer when compared with a linear polymer. In particular, we find that the radius of gyration ratio <RG2≳ring/<RG2≳ line is a universal quantity and to first order in ϵ is equal to 0.568. For ideal polymers, the ratio is known to be 0.5.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70385/2/JCPSA6-76-3-1574-1.pd

Renormalisation theory of the self-avoiding Levy flight

The self-avoiding Levy flight (SALF) in d dimensions with Levy exponent mu is formulated as a geometrical equilibrium statistical mechanical problem. A direct renormalisation theory, based on modern field theoretic techniques, is used to derive the critical exponents and the end-to-end distance probability function through first order in epsilon =2 mu -d. The non-perturbative structure of the probability function is characterised by a universal scaling function. The SALF represents a simple many-body system that can assume a continuum of values of epsilon near zero.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48802/2/jav18i14pL833.pd

The interpenetration of two chain polymers in a good solvent

The interpenetration of two excluded‐volume chain molecules of different size in dilute solution is studied via scaling and renormalization methods. The chains are found to interpenetrate much more strongly than smoothed‐density models suggest, in accordance with recent work by Khokhlov. The pair correlation funtion g(r) goes to zero at the origin only as a weak power of r. This power is related to Des Cloizeaux’s exponents ϑi describing intrachain correlations. The power is also related to the scaling exponents of star polymers. The mutual excluded volume MSL of two chains with greatly different length is proportional to the volume of the smaller chain and to the mass of the larger. Thus MSL is much smaller than a smoothed density model would predict. We discuss which chain correlations give rise to this small MSL. The universal coefficient relating MSL to the radius of gyration of the smaller chain is strongly dependent on the dimension d of space, according to our second‐order expansion in 4‐d. The interpenetration behavior predicted here affects measurable thermodynamic, scattering, and physical–chemical properties of the solution.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70233/2/JCPSA6-77-8-4247-1.pd

Renormalization theory of self-avoiding walks which cross a square

The renormalization group theory is used to calculate the critical behaviour of self-avoiding walks which cross a square. This problem, which has been proposed recently, is especially well suited for a renormalization group analysis. The fixed-endpoint, diagonal-span trademark of the square-crossing walks leads naturally to a well defined renormalization scheme. Unlike other finite-lattice renormalization schemes for self-avoiding walks, this square-crossing renormalization is exact in the sense that the finite lattice (square) is uniquely defined, the spanning rule is unambiguous, and the end-to-end correlations are exactly preserved. The results for the critical point are in excellent agreement with series analysis estimates and support a conjecture on its exact value.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48825/2/ja912121.pd

Universality of node-avoiding and path-avoiding Levy flights

The real space renormalisation group theory is applied to the self-avoiding Levy flight problem with Levy exponent mu on a two-dimensional square lattice. A finite-lattice renormalisation transformation is used to derive the critical behaviour exhibited by two classes of self-avoiding Levy processes: the node-avoiding Levy flight (NALF); the path-avoiding Levy flight (PALF). It is found that the NALF and the PALF belong to the same universality class.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48809/2/jav19i3pL161.pd

Self-avoiding random walks on multifractal lattices

A renormalisation theory is developed to study the critical behaviour of self-avoiding random walks on multifractals. Critical exponents and connectivity constants are calculated for walks on a class of square multifractal lattices using a finite lattice renormalisation. The effect of the multifractal disorder is considered for both annealed and quenched disorder.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48818/2/jav22i8p1101.pd

Rectangular renormalization

A generalized real-space renormalization scheme is developed for geometrical critical phenomena. The renormalization group is parametrized by the standard length-scaling factor and a new rectangular area-fraction factor. This rectangular renormalization scheme utilizes relatively small rectangular sublattices to effectively renormalize large square lattices. With the area-fraction factor, one can systematically study rectangular generalizations of the conventional square-cell renormalization theories. Application to self-avoiding random walks yields critical descriptors that are comparable to, and in most cases better than previous results obtained from more complex renormalization schemes.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48832/2/ja951904.pd

Optimal real-space renormalisation

Using a continuous class of real-space renormalisation transformations the authors study the critical behaviour of the Ising model in a variety of two-dimensional lattices. They use the simplest possible cluster, with two cells, and a single renormalised coupling. Certain of these transformations-well separated from the majority spin rule-produce sharply more accurate critical properties than the rest. From this study they note an optimal set of conditions which characterise the successful transformations. The optimal transformations appear to have a certain 'locality' property: renormalised couplings beyond nearest neighbour are especially small.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48799/2/jav14i2p447.pd

Sterically encumbered tin and phosphorus peri-substituted acenaphthenes

The work in this project was supported by the Engineering and Physical Sciences Research Council (EPSRC), EaStCHEM and the University of St Andrews.A group of sterically encumbered peri-substituted acenaphthenes have been prepared, containing tin moieties at the 5,6-positions in 1 – 3 ([Acenap(SnR3)2], Acenap = acenaphthene-5,6-diyl; R3 = Ph3 ( 1 ), Me3 ( 2 ); [(Acenap)2(SnMe2)2] ( 3 )) and phosphorus functional groups at the proximal peri-positions in 4 and 5 ([Acenap(PR2)(PiPr2)] R2 = Ph2 ( 4 ), Ph(iPr) ( 5 )). Bis(stannane) structures 1 – 3 are dominated by repulsive interactions between the bulky tin groups, leading to peri-distances approaching the sum of van der Waals radii. Conversely, the quasi-linear CPh-P···P three-body fragments found in bis(phosphine) 4 suggest the presence of a lp(P)−σ*(P–C) donor–acceptor 3c-4e type interaction, supported by a notably short intramolecular P···P distance and notably large JPP through-space coupling (180 Hz). Severely strained bis(sulfides) 4-S and 5-S , experiencing pronounced in-plane and out-of-plane displacements of the exocyclic peri-bonds, have also been isolated following treatment of 4 and 5 with sulfur. The resulting nonbonded intramolecular P···P distances, ∼4.05 Å and ∼12% longer than twice the van der Waals radii of P (3.60 Å), are among the largest ever reported peri-separations, independent of the heteroatoms involved, and comparable to the distance found in 1 containing the larger Sn atoms (4.07 Å). In addition we report two metal complexes with square planar [( 4 )PtCl2] ( 4-Pt ) and octahedral cis-[( 4 )Mo(CO)4] ( 4-Mo ) geometries. In both complexes the bis(phosphine) backbone is distorted, but notably less so than in bis(sulfide) 4-S . All compounds were fully characterized, and except for bis(phosphine) 5 , crystal structures were determined.PostprintPostprintPeer reviewe

Genetic bottlenecks in time and space: reconstructing invasions from contemporary and historical collections

Herbarium accession data offer a useful historical botanical perspective and have been used to track the spread of plant invasions through time and space. Nevertheless, few studies have utilised this resource for genetic analysis to reconstruct a more complete picture of historical invasion dynamics, including the occurrence of separate introduction events. In this study, we combined nuclear and chloroplast microsatellite analyses of contemporary and historical collections of Senecio madagascariensis, a globally invasive weed first introduced to Australia c. 1918 from its native South Africa. Analysis of nuclear microsatellites, together with temporal spread data and simulations of herbarium voucher sampling, revealed distinct introductions to south-eastern Australia and mid-eastern Australia. Genetic diversity of the south-eastern invasive population was lower than in the native range, but higher than in the mid-eastern invasion. In the invasive range, despite its low resolution, our chloroplast microsatellite data revealed the occurrence of new haplotypes over time, probably as the result of subsequent introduction(s) to Australia from the native range during the latter half of the 20th century. Our work demonstrates how molecular studies of contemporary and historical field collections can be combined to reconstruct a more complete picture of the invasion history of introduced taxa. Further, our study indicates that a survey of contemporary samples only (as undertaken for the majority of invasive species studies) would be insufficient to identify potential source populations and occurrence of multiple introductions.Eleanor E. Dormontt, Michael G. Gardner, Martin F. Breed, James G. Rodger, Peter J. Prentis, Andrew J. Low