Conjugated homopolymers based on six-member rings,e.g.,polyfluorene,always exhibit blue emission and conjugated homopolymers based on five-member rings,e.g.,polythiophene,can give red emission with low efficiency.In t...Conjugated homopolymers based on six-member rings,e.g.,polyfluorene,always exhibit blue emission and conjugated homopolymers based on five-member rings,e.g.,polythiophene,can give red emission with low efficiency.In this work,we report a series of new conjugated homopolymers based on six-member rings with high-efficiency deep-red emission.The repeating units of the red light emitting homopolymers are double B←N bridged bipyridine(BNBP)with the boron atoms functionalized with diphenyl,borafluorene,and 2,7-di-tert-butyl-borafluorene groups,respectively.The relationship between the chemical structures and the opto-electronic properties of the monomers and the homopolymers has been systematically studied.The three polymers emit pure red light(λ_(max)=656 nm)or deep red light(λ_(max)=693 nm)with fluorescence quantum efficiency in solution higher than 60%.The polymers can be used as the emitters in solution-processed organic light-emitting diodes with red emission and decent device performance.This work indicates a new strategy to design highly efficient light emitting conjugated polymers.展开更多
Polymer science encompasses a different range of materials critical to industries spanning from packaging to biomedicine. Understanding the synthesis, characterization, and applications of common homopolymers and copo...Polymer science encompasses a different range of materials critical to industries spanning from packaging to biomedicine. Understanding the synthesis, characterization, and applications of common homopolymers and copolymers is fundamental to advancing polymer research and development. In this comprehensive review, we explore various preparation methods, including free radical, anionic, and cationic polymerization, utilized for synthesizing homopolymers and copolymers. Furthermore, we investigate solvent choices commonly employed for polymer characterization, ranging from neat conditions, polar protic and polar aprotic solvents. We also explored characterization techniques, including Fourier Transform Infrared Spectroscopy (FTIR), Nuclear Magnetic Resonance (NMR), Atomic Force Microscopy (AFM), Differential Scanning Calorimetry (DSC), and Thermogravimetric Analysis (TGA). In addition to industrial applications, we highlight the diverse biological applications of homopolymers, poly(2-hydroxyethyl methacrylate) (pHEMA) and polystyrene, which find its extensive use in biomedicine. By synthesizing and analyzing this wealth of information, this review aims to provide a comprehensive understanding of the synthesis, characterization, and applications of homopolymers and copolymers, with a particular focus on their biological applications. This holistic approach not only contributes to advancements in polymer science and technology but also fosters innovation in biomedicine, ultimately benefiting human health and well-being.展开更多
Gold nanoparticles (GNs) are prepared through in situ reduction using NaBH4 in the presence of homopolymer PDMAEMA. The sizes of the GNs can be adjusted by alternating the molar ratio of gold to DMAEMA. Pure PDMAEMA a...Gold nanoparticles (GNs) are prepared through in situ reduction using NaBH4 in the presence of homopolymer PDMAEMA. The sizes of the GNs can be adjusted by alternating the molar ratio of gold to DMAEMA. Pure PDMAEMA aqueous solution shows a phase-transition at 50°C at pH 10 and 25°C at pH 14, while PDMAEMA-supported GNs aqueous solution shows a phase-transition at 47°C at pH 10 because of the increasing hydrophobic property resulting from GNs. Due to the pH and temperature-responsible characteristics of PDMAEMA, the resulting PDMAEMA-supported GNs exhibit pH adjustable temperature-responsive characteristics in optic and catalytic aspects. Under an acidic condition (pH 2), the GNs show unchanged surface Plasmon absorbance with a peak of 518 nm in a temperature range from 20 to 65°C. Under a basic condition (pH 10), the GNs first show the same absorbance with a peak at 518 nm in a temperature range from 20 to 40°C, and then the absorbance red shifts from 518 to 545 nm as temperature increases from 40 to 65°C. When the GNs are used as catalysts to catalyze the reduction of p-nitrophenol, the catalytic activity can be adjusted by changing the permeation of reactants in the PDMAEMA layer at low and high temperatures, respectively.展开更多
Nonin vasive ultraso und is a more effective strategy for on-demand drug delivery of polymeric nano particles than many other stimuli.However,the preparation of ultrasound-responsive homopolymer nanoparticles is still...Nonin vasive ultraso und is a more effective strategy for on-demand drug delivery of polymeric nano particles than many other stimuli.However,the preparation of ultrasound-responsive homopolymer nanoparticles is still very challenging.In this study,we disclose the regulating factors of ultrasound responsiveness of homopolymer nanoparticles and the disaggregation behavior of homopolymer nanoparticle aggregates.Homopolymer nanoparticles such as vesicles and large compound micelles(LCMs)are self-assembled from poly(methoxyethyl methacrylate)(PMEMA)and poly(amic acid)(PAA),respectively.The ultrasound responsiveness of PAA vesicles at metastable state could be regulated by tuning the self-assembly temperature(7^),and was optimized when Ts is around the glass transition temperature(7g)of PAA.However,the PMEMA LCMs did not respond to ultrasound as they are at stable state.On the other hand,poly(2-(2-ethoxyethoxy)ethyl acrylate)(PEEA)could self-assemble into vesicle aggregates or complex micelle aggregates,which were dissociated upon sonication.Overall,the above findings provide us with a fresh in sight for designing ultrasound-responsive polymeric nanoparticles.展开更多
The liquid-crystal assembly of semiflexible-coil diblock copolymers with coil or semiflexible homopolymers is studied by dissipative particle dynamics simulation. Phase diagrams of the blends and orientation ordering ...The liquid-crystal assembly of semiflexible-coil diblock copolymers with coil or semiflexible homopolymers is studied by dissipative particle dynamics simulation. Phase diagrams of the blends and orientation ordering parameters among semiflexible blocks are constructed as a function of chain stiffness and homopolymer volume fraction. For semiflexible-coil/coil blends with varying stiffness of semiflexible blocks, we display the rich phase behaviors of the system transited from coil-coil/coil to rod-coil/coil blends. The disorder- lamellae or lamellae-liquid crystalline transition and ＂dry brush＂ phenomenon induced by coil homopolymers are observed. For semiflexible-coil/semiflexible blends, adding semiflexible homopolymers also leads to a disorder-order transition and even a transition between monolayer and bilayer smectic-A phase. The results demonstrate that blending homopolymers into semiflexible copolymers can induce liquid-crystal assembly and even improve the orientation ordering of semiflexible blocks effectively.展开更多
The solid forms of drugs play a central role in controlling their physicochemical properties and consequently the bioavailability. Multiple types of drug solid forms have been developed to achieve the desirable pharma...The solid forms of drugs play a central role in controlling their physicochemical properties and consequently the bioavailability. Multiple types of drug solid forms have been developed to achieve the desirable pharmaceutical profiles, but new solid forms will provide more options for the solid-state property optimization and hence are highly desirable. This review focuses on a new pharmaceutical solid form, drug-polymer inclusion complexes （ICs）, and summarizes their structural features, structure- property relationships, as well as potential pharmaceutical applications展开更多
Polymer electron acceptors for all-polymer solar cells （all-PSCs） are usually conjugated copolymers, which contain alternating electron-rich units and electron-deficient units. In this manuscript, we report a conjug...Polymer electron acceptors for all-polymer solar cells （all-PSCs） are usually conjugated copolymers, which contain alternating electron-rich units and electron-deficient units. In this manuscript, we report a conjugated homopolymer （P-BNBP） based on an electron-deficient unit of double B,--N bridged bipyridine, which can be used as electron acceptor for all-polymer solar cells. P-BNBP shows low-lying LUMO energy level of -3.59eV, high absorption coefficient of 1.6 ×10^5Lmo1^-1 cm^-1 at 626nm and moderate electron mobility of 4.37 ×10^-6cm^2V^-1s^-1. AII-PSC devices exhibit power conversion efficiencies of 2.44%-3.04%. These results demonstrate that conjugated homopolymers are promising as electron acceptor materials for alI-PSCs.展开更多
Dented n anospheres show promisi ng potential in drug delivery,nano motors,etc.However,it is still challe nging to prepare them by homopolymer self-assembly because of the strict structural requirements of the homopol...Dented n anospheres show promisi ng potential in drug delivery,nano motors,etc.However,it is still challe nging to prepare them by homopolymer self-assembly because of the strict structural requirements of the homopolymer.Herein,we propose a strategy for preparing dented nano spheres from homopolymers by co-assembly with a short peptide.They were co-assembled from poly(2-hydroxy-3-((4-(ethoxycarbonyl)phenyl)amino)propyl methacrylate)(PHBzoMA59)and(S)-2-((S)-2-((((9H-fluoren-9-yl)methoxy)carbonyl)amino)-3-phenylpro-panamido)-3-phenyIpropanoic acid(Fmoc-FF-OH).PHBzoMA homopolymers can only self-assemble into nanospheres without dent,and the addition of a short peptide in troduced hydroge n bondi ng and complementary π-π stacki ng interactions led to the final dented nano sphere morphology.The weight fracti ons of the short peptide can be adjusted to regulate the final morphology.It was con firmed that the radius of curvature of the dent on the surface was related to the organic bubble inside the protospheres prepared at critical aggregation concentration(CAC).The orga nic bubble can be adjusted by altering the kind of orga nic solve nt and solution pH,which allowed control over the dented nano sphere dimension.The use of differe nt orga nic solve nts with various polarities allows adjustment of the interfacial tension,and hence the denting degree.This degree can also be controlled by manipulating the solution pH to(de)protonate the short peptide and homopolymer.Furthermore,the versatility of this method was highlighted by using a differe nt homopolymer and the applicability of the resulting den ted nanospheres was demonstrated by decoration with gold nanoparticles.Overall,this study provided important insights and a new simple strategy to prepare den ted n anospheres in a con trolled fashion.展开更多
Dynamic self-consistent field theory(DSCFT)is a fruitful approach for modeling the structural evolution and collective kinetics for a wide variety of multicomponent polymers.However,solving a set of DSCFT equations re...Dynamic self-consistent field theory(DSCFT)is a fruitful approach for modeling the structural evolution and collective kinetics for a wide variety of multicomponent polymers.However,solving a set of DSCFT equations remains daunting because of high computational demand.Herein,a machine learning method,integrating low-dimensional representations of microstructures and long short-term memory neural networks,is used to accelerate the predictions of structural evolution of multicomponent polymers.It is definitively demonstrated that the neural-network-trained surrogate model has the capability to accurately forecast the structural evolution of homopolymer blends as well as diblock copolymers,without the requirement of“on-the-fly”solution of DSCFT equations.Importantly,the data-driven method can also infer the latent growth laws of phase-separated microstructures of multicomponent polymers through simply using a few of time sequences from their past,without the prior knowledge of the governing dynamics.Our study exemplifies how the machine-learning-accelerated method can be applied to understand and discover the physics of structural evolution in the complex polymer systems.展开更多
The morphologies of triblock copolymer/homopolymer blend films, ABA/A and ABAIB, confined between two neutral hard walls were studied via Monte Carlo （MC） simulation on a simple .cubic lattice. The effects of φh （...The morphologies of triblock copolymer/homopolymer blend films, ABA/A and ABAIB, confined between two neutral hard walls were studied via Monte Carlo （MC） simulation on a simple .cubic lattice. The effects of φh （the volume fraction of homopolymer） and Md/Mb （the molecular weight of homopolymer in relation to that of the corresponding blocks in the copolymer） on the morphologies were investigated in detail.展开更多
We begin with the reference measure P0 induced by simple, symmetric nearest neighbor continuous time random walk on Zd starting at 0 with jump rate 2d and then define, for β≥0, t > 0, the Gibbs probability measur...We begin with the reference measure P0 induced by simple, symmetric nearest neighbor continuous time random walk on Zd starting at 0 with jump rate 2d and then define, for β≥0, t > 0, the Gibbs probability measure Pβ,t by specifying its density with respect to P0 as dPβ,t/dP0= Zβ,t（0）(-1eβ∫0tδ0（xs）ds),（0.1）where Zβ,t（0）≡E0[eβ∫<sup>t0δ0（xs）ds]. This Gibbs probability measure provides a simple model for a homopolymer with an attractive potential at the origin. In a previous paper（Cranston and Molchanov, 2007）, we showed that for dimensions d≥3 there is a phase transition in the behavior of these paths from the diffusive behavior for β below a critical parameter to the positive recurrent behavior for β above this critical value. The critical value was determined by means of the spectral properties of the operator ? + βδ0, where ? is the discrete Laplacian on Zd. This corresponds to a transition from a diffusive or stretched-out phase to a globular phase for the polymer. In this paper we give a description of the polymer at the critical value where the phase transition takes place. The behavior at the critical parameter is dimension-dependent.展开更多
基金financially supported by the National Natural Science Foundation of China(Nos.22135007 and 52073281)。
文摘Conjugated homopolymers based on six-member rings,e.g.,polyfluorene,always exhibit blue emission and conjugated homopolymers based on five-member rings,e.g.,polythiophene,can give red emission with low efficiency.In this work,we report a series of new conjugated homopolymers based on six-member rings with high-efficiency deep-red emission.The repeating units of the red light emitting homopolymers are double B←N bridged bipyridine(BNBP)with the boron atoms functionalized with diphenyl,borafluorene,and 2,7-di-tert-butyl-borafluorene groups,respectively.The relationship between the chemical structures and the opto-electronic properties of the monomers and the homopolymers has been systematically studied.The three polymers emit pure red light(λ_(max)=656 nm)or deep red light(λ_(max)=693 nm)with fluorescence quantum efficiency in solution higher than 60%.The polymers can be used as the emitters in solution-processed organic light-emitting diodes with red emission and decent device performance.This work indicates a new strategy to design highly efficient light emitting conjugated polymers.
文摘Polymer science encompasses a different range of materials critical to industries spanning from packaging to biomedicine. Understanding the synthesis, characterization, and applications of common homopolymers and copolymers is fundamental to advancing polymer research and development. In this comprehensive review, we explore various preparation methods, including free radical, anionic, and cationic polymerization, utilized for synthesizing homopolymers and copolymers. Furthermore, we investigate solvent choices commonly employed for polymer characterization, ranging from neat conditions, polar protic and polar aprotic solvents. We also explored characterization techniques, including Fourier Transform Infrared Spectroscopy (FTIR), Nuclear Magnetic Resonance (NMR), Atomic Force Microscopy (AFM), Differential Scanning Calorimetry (DSC), and Thermogravimetric Analysis (TGA). In addition to industrial applications, we highlight the diverse biological applications of homopolymers, poly(2-hydroxyethyl methacrylate) (pHEMA) and polystyrene, which find its extensive use in biomedicine. By synthesizing and analyzing this wealth of information, this review aims to provide a comprehensive understanding of the synthesis, characterization, and applications of homopolymers and copolymers, with a particular focus on their biological applications. This holistic approach not only contributes to advancements in polymer science and technology but also fosters innovation in biomedicine, ultimately benefiting human health and well-being.
基金Supported by the National Natural Science Foundation of China (Grant No. 20774051)Program for New Century Excellent Talents in Universitiesthe Outstanding Youth Fund (Grant No. 50625310)
文摘Gold nanoparticles (GNs) are prepared through in situ reduction using NaBH4 in the presence of homopolymer PDMAEMA. The sizes of the GNs can be adjusted by alternating the molar ratio of gold to DMAEMA. Pure PDMAEMA aqueous solution shows a phase-transition at 50°C at pH 10 and 25°C at pH 14, while PDMAEMA-supported GNs aqueous solution shows a phase-transition at 47°C at pH 10 because of the increasing hydrophobic property resulting from GNs. Due to the pH and temperature-responsible characteristics of PDMAEMA, the resulting PDMAEMA-supported GNs exhibit pH adjustable temperature-responsive characteristics in optic and catalytic aspects. Under an acidic condition (pH 2), the GNs show unchanged surface Plasmon absorbance with a peak of 518 nm in a temperature range from 20 to 65°C. Under a basic condition (pH 10), the GNs first show the same absorbance with a peak at 518 nm in a temperature range from 20 to 40°C, and then the absorbance red shifts from 518 to 545 nm as temperature increases from 40 to 65°C. When the GNs are used as catalysts to catalyze the reduction of p-nitrophenol, the catalytic activity can be adjusted by changing the permeation of reactants in the PDMAEMA layer at low and high temperatures, respectively.
基金supported by the National Natural Science Foundation of China(No.21674081)Fundamental Research Funds for the Central Universities(No.22120180109)。
文摘Nonin vasive ultraso und is a more effective strategy for on-demand drug delivery of polymeric nano particles than many other stimuli.However,the preparation of ultrasound-responsive homopolymer nanoparticles is still very challenging.In this study,we disclose the regulating factors of ultrasound responsiveness of homopolymer nanoparticles and the disaggregation behavior of homopolymer nanoparticle aggregates.Homopolymer nanoparticles such as vesicles and large compound micelles(LCMs)are self-assembled from poly(methoxyethyl methacrylate)(PMEMA)and poly(amic acid)(PAA),respectively.The ultrasound responsiveness of PAA vesicles at metastable state could be regulated by tuning the self-assembly temperature(7^),and was optimized when Ts is around the glass transition temperature(7g)of PAA.However,the PMEMA LCMs did not respond to ultrasound as they are at stable state.On the other hand,poly(2-(2-ethoxyethoxy)ethyl acrylate)(PEEA)could self-assemble into vesicle aggregates or complex micelle aggregates,which were dissociated upon sonication.Overall,the above findings provide us with a fresh in sight for designing ultrasound-responsive polymeric nanoparticles.
基金financially supported by the National Natural Science Foundation of China(No.21674082)
文摘The liquid-crystal assembly of semiflexible-coil diblock copolymers with coil or semiflexible homopolymers is studied by dissipative particle dynamics simulation. Phase diagrams of the blends and orientation ordering parameters among semiflexible blocks are constructed as a function of chain stiffness and homopolymer volume fraction. For semiflexible-coil/coil blends with varying stiffness of semiflexible blocks, we display the rich phase behaviors of the system transited from coil-coil/coil to rod-coil/coil blends. The disorder- lamellae or lamellae-liquid crystalline transition and ＂dry brush＂ phenomenon induced by coil homopolymers are observed. For semiflexible-coil/semiflexible blends, adding semiflexible homopolymers also leads to a disorder-order transition and even a transition between monolayer and bilayer smectic-A phase. The results demonstrate that blending homopolymers into semiflexible copolymers can induce liquid-crystal assembly and even improve the orientation ordering of semiflexible blocks effectively.
基金supported by the National Natural Science Foundation of China (No. 21434008)
文摘The solid forms of drugs play a central role in controlling their physicochemical properties and consequently the bioavailability. Multiple types of drug solid forms have been developed to achieve the desirable pharmaceutical profiles, but new solid forms will provide more options for the solid-state property optimization and hence are highly desirable. This review focuses on a new pharmaceutical solid form, drug-polymer inclusion complexes （ICs）, and summarizes their structural features, structure- property relationships, as well as potential pharmaceutical applications
基金supported by the National Natural Science Foundation of China (Nos.21625403,21574129)Strategic Priority Research Program of Chinese Academy of Sciences (No.XDB12010200)+2 种基金Jilin Scientific and Technological Development Program (No.20170519003JH)Youth Innovation Promotion Association of Chinese Academy of Sciences (No.2017265)Open Project (No.sklssm201803) of the State Key Laboratory of Supramolecular Structure and Materials in Jilin University of China
文摘Polymer electron acceptors for all-polymer solar cells （all-PSCs） are usually conjugated copolymers, which contain alternating electron-rich units and electron-deficient units. In this manuscript, we report a conjugated homopolymer （P-BNBP） based on an electron-deficient unit of double B,--N bridged bipyridine, which can be used as electron acceptor for all-polymer solar cells. P-BNBP shows low-lying LUMO energy level of -3.59eV, high absorption coefficient of 1.6 ×10^5Lmo1^-1 cm^-1 at 626nm and moderate electron mobility of 4.37 ×10^-6cm^2V^-1s^-1. AII-PSC devices exhibit power conversion efficiencies of 2.44%-3.04%. These results demonstrate that conjugated homopolymers are promising as electron acceptor materials for alI-PSCs.
基金by the National Science Fund for Distinguished Young Scholars,NSFC(No.21925505)Z.F.is supported by the National Natural Science Foundation of China(Nos.51803152,22075212)+2 种基金the program for professor of special appointment(Eastern Scholar)at Shanghai institutions of higher learning and Natural Science Foundation of Shanghai(19ZR1478800)Y.Z.is supported by the National Natural Science Foundation of China(No.51903190)and Shanghai Pujiang Program(No.19PJ140960)EJ.C.is supported by the China Postdoctoral Science Foundation(No.2020M671197).
文摘Dented n anospheres show promisi ng potential in drug delivery,nano motors,etc.However,it is still challe nging to prepare them by homopolymer self-assembly because of the strict structural requirements of the homopolymer.Herein,we propose a strategy for preparing dented nano spheres from homopolymers by co-assembly with a short peptide.They were co-assembled from poly(2-hydroxy-3-((4-(ethoxycarbonyl)phenyl)amino)propyl methacrylate)(PHBzoMA59)and(S)-2-((S)-2-((((9H-fluoren-9-yl)methoxy)carbonyl)amino)-3-phenylpro-panamido)-3-phenyIpropanoic acid(Fmoc-FF-OH).PHBzoMA homopolymers can only self-assemble into nanospheres without dent,and the addition of a short peptide in troduced hydroge n bondi ng and complementary π-π stacki ng interactions led to the final dented nano sphere morphology.The weight fracti ons of the short peptide can be adjusted to regulate the final morphology.It was con firmed that the radius of curvature of the dent on the surface was related to the organic bubble inside the protospheres prepared at critical aggregation concentration(CAC).The orga nic bubble can be adjusted by altering the kind of orga nic solve nt and solution pH,which allowed control over the dented nano sphere dimension.The use of differe nt orga nic solve nts with various polarities allows adjustment of the interfacial tension,and hence the denting degree.This degree can also be controlled by manipulating the solution pH to(de)protonate the short peptide and homopolymer.Furthermore,the versatility of this method was highlighted by using a differe nt homopolymer and the applicability of the resulting den ted nanospheres was demonstrated by decoration with gold nanoparticles.Overall,this study provided important insights and a new simple strategy to prepare den ted n anospheres in a con trolled fashion.
基金financially supported by the National Natural Science Foundation of China(Nos.22073028,21873029 and 22073004)the Fundamental Research Funds for the Central Universities。
文摘Dynamic self-consistent field theory(DSCFT)is a fruitful approach for modeling the structural evolution and collective kinetics for a wide variety of multicomponent polymers.However,solving a set of DSCFT equations remains daunting because of high computational demand.Herein,a machine learning method,integrating low-dimensional representations of microstructures and long short-term memory neural networks,is used to accelerate the predictions of structural evolution of multicomponent polymers.It is definitively demonstrated that the neural-network-trained surrogate model has the capability to accurately forecast the structural evolution of homopolymer blends as well as diblock copolymers,without the requirement of“on-the-fly”solution of DSCFT equations.Importantly,the data-driven method can also infer the latent growth laws of phase-separated microstructures of multicomponent polymers through simply using a few of time sequences from their past,without the prior knowledge of the governing dynamics.Our study exemplifies how the machine-learning-accelerated method can be applied to understand and discover the physics of structural evolution in the complex polymer systems.
基金Acknowledgements This work is supported by the National Natural Science Foundation of China （Projects No. 20236010. 20476025, 20490200）. E-Institute of Shanghai High Institution Grid （No.200303） and Shanghai Municipal Education Commission of China.
文摘The morphologies of triblock copolymer/homopolymer blend films, ABA/A and ABAIB, confined between two neutral hard walls were studied via Monte Carlo （MC） simulation on a simple .cubic lattice. The effects of φh （the volume fraction of homopolymer） and Md/Mb （the molecular weight of homopolymer in relation to that of the corresponding blocks in the copolymer） on the morphologies were investigated in detail.
基金supported by National Science Foundation of USA (Grant Nos. DMS1007176 and DMS-0706928)
文摘We begin with the reference measure P0 induced by simple, symmetric nearest neighbor continuous time random walk on Zd starting at 0 with jump rate 2d and then define, for β≥0, t > 0, the Gibbs probability measure Pβ,t by specifying its density with respect to P0 as dPβ,t/dP0= Zβ,t（0）(-1eβ∫0tδ0（xs）ds),（0.1）where Zβ,t（0）≡E0[eβ∫<sup>t0δ0（xs）ds]. This Gibbs probability measure provides a simple model for a homopolymer with an attractive potential at the origin. In a previous paper（Cranston and Molchanov, 2007）, we showed that for dimensions d≥3 there is a phase transition in the behavior of these paths from the diffusive behavior for β below a critical parameter to the positive recurrent behavior for β above this critical value. The critical value was determined by means of the spectral properties of the operator ? + βδ0, where ? is the discrete Laplacian on Zd. This corresponds to a transition from a diffusive or stretched-out phase to a globular phase for the polymer. In this paper we give a description of the polymer at the critical value where the phase transition takes place. The behavior at the critical parameter is dimension-dependent.
