Area distribution of two-dimensional random walks on a square lattice

The algebraic area probability distribution of closed planar random walks of length N on a square lattice is considered. The generating function for the distribution satisfies a recurrence relation in which the combinatorics is encoded. A particular case generalizes the q-binomial theorem to the case of three addends. The distribution fits the L\'evy probability distribution for Brownian curves with its first-order 1/N correction quite well, even for N rather small.Comment: 8 pages, LaTeX 2e. Reformulated in terms of q-commutator

Microphase separation in cross-linked polymer blends: Efficient replica RPA post-processing of simulation data for homopolymer networks

We investigate the behaviour of randomly cross-linked (co)polymer blends using a combination of replica theory and large-scale molecular dynamics simulations. In particular, we derive the analogue of the random phase approximation for systems with quenched disorder and show how the required correlation functions can be calculated efficiently. By post-processing simulation data for homopolymer networks we are able to describe neutron scattering measurements in heterogeneous systems without resorting to microscopic detail and otherwise unphysical assumptions. We obtain structure function data which illustrate the expected microphase separation and contain system-specific information relating to the intrinsic length scales of our networks.Comment: 8 pages, 5 figure

Onset of entanglement

We have developed a theory of polymer entanglement using an extended Cahn-Hilliard functional, with two extra terms. One is a nonlocal attractive term, operating over mesoscales, which is interpreted as giving rise to entanglement, and the other a local repulsive term indicative of excluded volume interactions. We show how such a functional can be derived using notions from gauge theory. We go beyond the Gaussian approximation, to the one-loop level, to show that the system exhibits a crossover to a state of entanglement as the average chain length between points of entanglement decreases. This crossover is marked by critical slowing down, as the effective diffusion constant goes to zero. We have also computed the tensile modulus of the system, and we find a corresponding crossover to a regime of high modulus.Comment: 18 pages, with 4 figure

Screening of interactions in polymer blends

This paper discusses the screening of interactions in compatible binary polymer blends. It is shown that before the spinodal is reached a θ-like point exists, where the effective potential is essentially zero and the chain behaves ideally. A general expression for the radius of gyration of a single chain in the blend as a function of the concentration of the two components and the Flory interaction parameter is calculated by perturbation theory. This shows a progressive shrinking from a value above that of ideal Gaussian chains until the process reaches completion with the phase separation. This is of relevance in neutron scattering experiments, where the single chain structure factor of a single chain can be measured as a function of the temperature (χF-parameter) directly.Dans cet article, nous étudions l'écrantage d'intéractions de mélanges binaires de polymères compatibles. Nous montrons que, avant d'atteindre la spinodale, il y a un point du type θ pour lequel le potentiel effectif est pratiquement nul et la chaîne se comporte comme une chaîne idéale. Nous établissons une expression générale du rayon de giration d'une chaîne dans le mélange en fonction de la concentration des deux composants et nous calculons le paramètre d'intéraction de Flory par la théorie des perturbations. Nous observons une réduction progressive depuis la valeur au-dessus de celle des chaînes gaussiennes idéales, jusqu'à la séparation de phase. Cette évolution est intéressante pour des expériences de diffusion neutronique dans la mesure où le facteur de forme d'une chaîne pourra être déterminé directement en fonction de la température (paramètre χ F)

Theory of the Order-Disorder Phase Transition in Cross-Linked Polymer Blends

We consider the system of a network formed by crosslinking a polymer blend. In such a system there is a competition between monomer-monomer interactions and elastic energy which may result in a microphase separation. We first treat the problem by adapting a model due to de Gennes to include the effect of concentration fluctuations `frozen in' by the crosslinking process. This yields an expression for the scattering structure factor which gives finite scattering in the limit of low wavenumber (this has been observed in experiments on crosslinked blends). The expression also shows that the frozen-in fluctuations `seed' the phase transition, so that the structure factor diverges as $(\chi{\rm s} -\chi)^{-2}$ on approaching the spinodal. We then reconsider the problem at a molecular level. We model the network as a blend of interacting chains anchored at either end to fixed points in space. The system is treated using a variant of the Random Phase Approximation (RPA) which deals with the quenched chain-end variables but which does not resort to replica methods. The resulting structure factor has an identical form to that obtained by modifying the de Gennes model, but allows us to investigate the effect of varying composition, crosslink density, and applied strain. We find that the characteristic lengthscale for the early stages of microphase separation is controlled by the least concentrated component or the one with the shortest chain length between crosslinks, depending on which parameter shows the strongest difference between the two chain types. We also find that strain produces a phase separation with wavevectors in the direction(s) of least stretching (or greatest compression)

The 1H NMR visibility of intracellular lactate in Streptococcus faecalis

H NMR studies of glycolysis in washed cell suspensions of Streptococcus faecalis indicated that intracellular lactate is not H NMR visible. Evidence for this was gained from time course studies of glycolysis at increasing concentrations of glucose. A close correlation existed between the relative increase in the lactate integral and the enzymatically determined extracellular lactate concentration [Lo]. When ionophores which cause the collapse of the positive intracellular/extracellular lactate gradient were added to cell suspensions following fermentation of 5, 10 and 50 mM glucose, the increase in the lactate integral was proportional to the respective increase in [Lo]. A more direct method for determining the origin of the lactate signal involved centrifugation of a cell suspension after fermentation of 50 mM glucose and measurement of lactate in the extracellular and intracellular fluid. H spectra of the cell suspension, supernatant and sonicated pellet revealed that the lactate observed in the cell suspension was equivalent to the lactate in the supernatant alone. The intracellular lactate contained in the pellet represented 42% of the total lactate, indicating that only 58% of lactate is detected by in vivo H MRS of S. faecalis. This result is in contrast with the high percentage (70–90%) of in vitro lactate which is detected by in vivo H MRS of mammalian brain tissue (Williams S. R. et al. Magn. Res. Med. 7, 425–431, 1988). This may be due to a higher proportion of extracellular lactate in mammalian tissue or differences in the intracellular environments of bacterial and mammalian cells

P57 peripheral blood monocyte count as a prognostic marker in fibrotic interstitial lung disease (fILD): analysis from a single UK specialist centre

Background: predicting individual patient course and prognosis in fILD is challenging and there are no established prognostic biomarkers to aid clinical judgement. An association between peripheral blood monocyte (PBM) count and survival was recently proposed in patients with Idiopathic Pulmonary Fibrosis (IPF) (Scott et al. Lancet Respir Med 2019–7:497– 508). We investigated in a single UK centre whether monocyte count was an independent predictor of survival in cohorts of patients with an MDT diagnosis of IPF, chronic hypersensitivity pneumonitis (cHP) and unclassified ILD (uILD).Methods: single centre study of consecutive patients with an MDT diagnosis of IPF, cHP or uILD. Electronic records and blood results were reviewed. The PBM count nearest to the MDT diagnosis was imputed. Time to death/transplant was calculated by Kaplan-Meier analysis and cumulative risk of death/transplant quantified by multivariate Cox-regression analysis (IBM-SPSS®v25).Results: 385 patients (IPF n= 199, cHP n= 101, uILD n=85) were included. Baseline demographics, IPF-cHP-uILD respectively - mean (SD). Age (years): 72.7 (7.8), 66.6 (12.3), 71.7 (8.4). FVC% predicted (FVC%pred): 73.8% (19.8), 77.0% (20.0), 83.5% (20.8). DLCO%pred: 46.5% (15.1), 53.8% (15.3), 57.1% (18.1). Gender: (Males%): 76.1%, 36.0%, 61.7%.The IPF cohort had significantly higher absolute PBMs compared to cHP but not uILD (0.77x109/L vs 0.65x109/L vs. 0.74x109/L respectively IPFvs.HP p0.94x109/L (Mono>0.94) or Low≤0.94x109/L (Mono≤0.94) as Scott et al. 2019; in the IPF cohort Mono>0.94 was associated with significantly reduced time to death/transplant compared to Mono≤0.94 (171.7weeks (95%CI 132.6–210.8weeks) vs. 262.2weeks (95%CI 211.7–312.7weeks) p=0.035). Multivariate Cox-regression analysis (age, sex, FVC%pred and DLCO%pred) identified Mono>0.94 as an independent predictor of death/transplant in IPF; Hazard-Ratio 1.576 (95%CI 1.023–2.300) p=0.031. There was no association between absolute or stratified monocyte count and survival in the cHP or uILD cohorts.Conclusions: in this UK single-centre study a stratified PBM of >0.94x10*9/L was an independent risk factor for death/transplant in IPF but not in patients cHP or uILD. Prospective studies are required to confirm this observation