Article
Polymer Science
Jiugeng Cheng, Zhenxing Zhong, Yuan Lin, Zhaohui Su, Chunyu Zhang, Xuequan Zhang
Summary: This work investigates the miscibility behavior of high molecular weight isotactic poly(1-butene) and isotactic polypropylene blends using AFM-IR technique. It is found that the two polymers partially miscible under certain conditions, and the content of polypropylene in each phase can be determined using a quantitative analysis method. The phase separation kinetics and phase diagram of the blends were studied, suggesting a lower critical solution temperature-type phase behavior of PB/PP blends.
Article
Materials Science, Composites
Sara Tarashi, Hossein Nazockdast, Milad Karbalaei-Bagher, Amir Rostami, Milad Mehranpour
Summary: The crystallization properties of polypropylene (PP) can be modified by adding ethylene octene copolymer (EOC) and nanoclay. The presence of EOC decreases the crystallinity and crystallization rate of PP due to partial miscibility. The effect of nanoclay on PP crystallization depends on its loading and distribution, serving as a nucleation agent at lower contents but hindering crystallization at higher contents.
POLYMER COMPOSITES
(2023)
Article
Polymer Science
Przemyslaw Sowinski, Sivanjineyulu Veluri, Ewa Piorkowska
Summary: This study prepared nanocomposites of isotactic polypropylene with fibrillated PTFE and investigated their crystallization behavior and resulting structure under high pressure. It was found that the addition of fibrillated PTFE increased the crystallization peak temperature of polypropylene and formed a fine-grain structure. This suggests that fibrillated PTFE can promote the crystallization of polypropylene under high pressure.
Article
Chemistry, Physical
Zhenxing Zhong, Li Peng, Juanxia Su, Zhongfu Luo, Charles C. C. Han, Xianbo Huang, Zhaohui Su
Summary: In this study, the distribution of antioxidant 1010 (AT1010) in blends of isotactic polypropylene (iPP) with an ethylene-octene copolymer (POE) was investigated using atomic force microscopy-infrared (AFM-IR) technique. The results showed that AT1010 preferred to stay in the POE disperse phase rather than in the iPP matrix, and the concentration ratio of AT1010 in the POE increased during the injection molding process. This study provides important insights into the application of AT1010 in plastic products.
Article
Polymer Science
Deyu Niu, Pengwu Xu, Jiaxuan Li, Weijun Yang, Tianxi Liu, Piming Ma
Summary: This work successfully obtains designable strength and ductility of PGA/PBAT films by tuning the drawing temperature. The films exhibit high tensile strength and remarkable strain hardening at low temperature, high ductility at moderate temperature, and excellent strength, stiffness, and ductility at high temperature.
Article
Polymer Science
Jingru Liu, Dexing Jiang, Zihan Li
Summary: This study introduces ABS into iPP and further annealing treatment to improve the impact toughness of iPP. The introduction of ABS and annealing treatment have a synergistic effect on toughening iPP.
POLYMERS FOR ADVANCED TECHNOLOGIES
(2022)
Article
Materials Science, Multidisciplinary
Wenjun He, Lin Ye, Phil Coates, Fin Caton-Rose, Xiaowen Zhao
Summary: By innovatively constructing tailored interfacial intermolecular interaction and entanglement between the two phases of the PLA/POE blend through reactive melt blending, the PLA/GPOE-MPOE blend exhibited enhanced complex viscosity/storage modulus, lower tan delta values, and significant strain-hardening behavior compared to neat PLA and directly blended PLA/POE blends. This enhancement was attributed to the increased molecular entanglement between the long-branched chains, leading to improved stretching ability during biaxial stretching of the blend, resulting in uniform films with higher tensile strength and elongation at break.
JOURNAL OF MATERIALS SCIENCE & TECHNOLOGY
(2022)
Article
Polymer Science
Dongyang Li, Yichong Chen, Shun Yao, Hong Zhang, Dongdong Hu, Ling Zhao
Summary: POE elastomers composition and molecular weight affect its foaming behaviors and dimensional stability. Lower octene content leads to higher melting point and crystallinity, resulting in a higher foaming temperature and larger cell diameter under high CO2 pressure.
Article
Polymer Science
E. Raghuram, Ajay Singh Panwar
Summary: Using molecular dynamics simulations, this study investigated chiral transitions in an isolated isotactic polypropylene chain on a nucleating agent substrate during the early stages of ordering. The results showed that selective stabilization of helical segments was more efficient when selecting helices from extended-coil conformations compared to coiled segments.
Article
Materials Science, Characterization & Testing
Qiang Wang, Shiqi Xie, Zhe Hua, Hui Niu
Summary: 4-Furfuryloxymethylstyrene (S-f) grafted isotactic polypropylene (iPP-g-S-f) was prepared by melt grafting reaction. The crystallization temperature of iPP after grafting increased and the crystallization rate was accelerated, resulting in smaller crystal morphology. The introduction of a small amount of S-f groups reduced the surface free energy of the crystals.
Article
Polymer Science
Yuan-Xia Wang, Cun-Ying Zou, Nan Bai, Qun-Feng Su, Li-Xin Song, Xian-Liang Li
Summary: Blending octene random copolymer (ORC) with other polymers, such as octene block copolymer (OBC) and high-density polyethylene (HDPE), has been studied to improve the mechanical properties of ORC. It was found that a small amount of OBC can enhance the crystallization ability of ORC, while the crystallization ability of ORC is significantly suppressed in ORC/HDPE blends. The tensile strength and elongation of ORC/HDPE blends are higher than those of ORC/OBC blends. Furthermore, cocrystallization can be achieved when ORC, OBC, and HDPE are blended at a ratio of 70:15:15.
Article
Polymer Science
Liyuan Liu, Wenhong Yang, Xinyan Chen, Ying Zhao, Xia Dong, Alejandro J. Mueller, Dujin Wang
Summary: We investigated the epitaxial crystallization of S-nucleated isotactic polypropylene with minor topographic modification at the interface caused by the inclusion of an ethylene comonomer. Experimental results showed that the amount of ethylene comonomer reduced the fraction of S-crystal (KS) due to the decreased S-nucleation ability and growth rate of the S-crystal. Molecular mechanics simulation revealed that the topographic modification weakened the intermolecular interactions, affecting the epitaxial nucleation and growth kinetics. This study provides insights into the control of polymorphism through the design of nucleation and growth interfaces.
Article
Materials Science, Composites
Zhongguo Zhao, Siyang Shen, Yapeng Li, Xin Zhang, Juqiao Su, Huan Li, Taotao Ai, Dahang Tang
Summary: The study found that the dispersion of multiwalled carbon nanotubes in PP/POE/MWCNTs composites significantly impacted their electrical conductivity, thermal stability, and toughness, while the introduction of POE phase improved resilience and stability. Additionally, stretching induced an increase in electrical conductivity in PP/POE nanocomposites.
POLYMER COMPOSITES
(2022)
Article
Polymer Science
Ziwei Lai, Nan Zheng, Ziying Liang, Yuanjie Wang, Hui Niu, Ming-Sheng Ji, Bo Ni, Xiang Huang, Xing Ouyang, Xiaoguang Li, Bernard Lotz, Yan Cao
Summary: The research demonstrates the presence of three-dimensional helical chain assembly in isotactic polypropylene under 2D confinement, with integrated assembly observed in both alpha- and gamma-phase crystals. Additionally, two unit-cell orientations of the gamma-phase crystal are shown under cylindrical confinement. The formation mechanism for the radial gamma-phase crystals is explained by the kinetic favorability of alignment with the rod long axis.
Article
Materials Science, Characterization & Testing
Xiang Liu, Lin Zou, Baobao Chang, Honghui Shi, Qingqing Yang, Kaichang Cheng, Taolin Li, Konrad Schneider, Gert Heinrich, Chuntai Liu, Changyu Shen
Summary: The crystallization behavior of oriented isotactic polypropylene during solid-state stretching was studied using in-situ wide-angle X-ray scattering. Results indicated that stretching induced crystallization is influenced by the stretching direction, Hencky strain, and amorphous phase orientation. The high Pearson correlation coefficient suggests that the crystallization process originates from the orientation of the chain network during stretching.
Article
Polymer Science
Jianming Shao, Xia Dong, Dujin Wang
Summary: In this work, a multiblock polyurethane (PU-Im) with imidazole dangling groups is demonstrated, which can coordinate with Ni2+ to form PU-Im-Ni complexes. By controlling the ligand content and metal-ligand stoichiometry ratio, PU-Im-Ni complexes with different mechanical behavior can be obtained. These complexes show excellent mechanical performance, thermal/aquatic self-healing ability, and unique damping properties, making them potential candidates for self-healing engineering plastic and cushion protection fields.
MACROMOLECULAR RAPID COMMUNICATIONS
(2023)
Article
Chemistry, Multidisciplinary
Bo Yang, Hui Zhao, Chunbo Zhang, Xiuqin Zhang, Xia Dong, Guoming Liu, Dujin Wang
Summary: The interplay between crosslinks and crystals in Eucommia ulmoides gum (EUG) was investigated in this study. It was found that crosslinks inhibit the crystallization of EUG, leading to a decrease in melting/crystallization temperature, crystallinity, modulus, and yield stress. The vulcanized EUG can gradually crystallize during annealing at room temperature, demonstrating a nonequilibrated state and slow crystallization kinetics.
Article
Chemistry, Multidisciplinary
Weijian Lu, Ping Zhu, Ying Zhao, Dujin Wang, Xia Dong
Summary: By incorporating isophoronediamine (IPDA), an alicyclic diamine monomer with asymmetric structure and steric hindrance, into the backbone of poly(ether-b-amide) multiblock copolymer, the mechanical properties and segmental mobility of copolymers can be modulated on a large scale. The resulting self-healing polyamide elastomers exhibited extraordinary tensile strength, excellent elongation at break, and record-high toughness due to the synergism between dynamic H-bonding networks and polymer chain diffusion. These copolymers showed great potential in protective coatings and soft electronics due to their adjustable mechanical performance, rapid scratch self-healing ability, and superior impact resistance.
CHEMISTRY-A EUROPEAN JOURNAL
(2023)
Article
Chemistry, Physical
Yunrui Long, Haochen Li, Rongbo Li, Xiaolei Guo, Zefan Wang, Xia Dong, Ping Zhu, Qian Xing
Summary: The phase morphology, crystallization behavior, and mechanical properties of PLA modified by PAE were investigated. PAE was found to be uniformly dispersed in the PLA matrix, forming a sea-island morphology. The addition of PAE promoted the crystallization of PLA and improved its elongation at break and impact strength. The sample with 20 wt% PAE showed the best rigidity-toughness balance and ductile failure, and the toughening mechanism was proposed based on the deformation of the impact fracture surfaces.
COLLOID AND POLYMER SCIENCE
(2023)
Article
Polymer Science
Liyuan Liu, Wenhong Yang, Xinyan Chen, Ying Zhao, Xia Dong, Alejandro J. Mueller, Dujin Wang
Summary: We investigated the epitaxial crystallization of S-nucleated isotactic polypropylene with minor topographic modification at the interface caused by the inclusion of an ethylene comonomer. Experimental results showed that the amount of ethylene comonomer reduced the fraction of S-crystal (KS) due to the decreased S-nucleation ability and growth rate of the S-crystal. Molecular mechanics simulation revealed that the topographic modification weakened the intermolecular interactions, affecting the epitaxial nucleation and growth kinetics. This study provides insights into the control of polymorphism through the design of nucleation and growth interfaces.
Article
Polymer Science
Tao Wen, Yuting Gao, Jiajia Zhou, Jie Qiu, Shuo Wang, Joachim Loos, Dujin Wang, Xia Dong
Summary: In this study, we introduce a simple method for fast fabrication of porous films and coatings of long-chain polyamides through a nonconventional evaporation induced phase separation. The amphiphilic nature of polyamide 12 allows it to be dissolved in a mixture of high-polarity and low-polarity solvents, leading to the formation of porous structures within 1 min through sequential and fast evaporation. We also investigate the dependence of pore structures on the composition of solutions and demonstrate the applicability of our approach to other long-chain polycondensates. Our findings provide valuable insights into the fabrication of porous materials using amphiphilic polymers.
Article
Engineering, Manufacturing
Shunxin Qi, Xia Gao, Yunlan Su, Guoming Liu, Ying Lu, Yong Zhou, Xia Dong, Dujin Wang
Summary: The crystalline structures and their spatial distributions of polyamide 12 (PA12) parts processed by material extrusion additive manufacturing (MEAM) were studied using microfocus wide-angle X-ray scattering combined with transmission electron microscopy. Two types of gradient structures were observed in the PA12 parts. One was found in the topmost layer, where the total crystallinity dropped gradually from the free surface to the core region, and the coexistence of alpha and gamma crystals was observed. The other gradient structure was related to the distance from the deposited layer to the platform, with the bottom layer having higher crystallinity than the top layer.
ADDITIVE MANUFACTURING
(2023)
Article
Materials Science, Multidisciplinary
Chenxu Zhou, Ping Zhu, Yu Wang, Qian Xing, Wen Tao, Xia Dong, Dujin Wang
Summary: The relaxation spectra of Long Chain Polyamides (LCPAs) were investigated using multiple testing methods, revealing the dynamic relaxation behaviors from macro- to micro-scale. The influences of zero-shear viscosity and hydrogen bond density on relaxation dynamics, crystallization behavior, and mechanical properties were systematically quantified.
MATERIALS TODAY COMMUNICATIONS
(2023)
Article
Polymer Science
Yuwen Wang, Rongbo Li, Yunrui Long, Haochen Li, Xia Dong, Qian Xing
Summary: The effect of styrene-glycidyl acrylate copolymer Joncryl ADR4368 (ADR) as a compatibilizer in immiscible blends of poly(lactic acid) (PLA) and poly(e-caprolactone) (PCL) was studied. The morphology, crystallization behavior, and mechanical properties were investigated. The addition of ADR improved the compatibility of PLA and PCL by reducing the size of PCL domains and delaying the cold crystallization ability of PLA. PLA/PCL/ADR obtained a more uniform microcrystalline dispersion during cooling and showed improved elongation at break and impact strength compared to PLA/PCL blends.
POLYMER-PLASTICS TECHNOLOGY AND MATERIALS
(2023)
Article
Materials Science, Multidisciplinary
Yang Liu, Li Peng, Jielong Lin, Yong Zhou, Dujin Wang, Charles C. Han, Nanbiao Ye, Xianbo Huang, Xia Dong
Summary: The mechanism of regulating hydrogen bond organization for polyamide 6,6 (PA 6,6) by nigrosine was investigated using temperature-dependent FTIR and in situ synchrotron X-ray diffraction. The study confirmed the intermolecular interaction between amide groups and nigrosine. Nigrosine promoted the formation of H-bonds in the initial stage of cooling, but hindered it at lower temperatures. The retardation effects on crystallization improved the flowability of PA 6,6 for high-speed injection molding applications.
ACS APPLIED POLYMER MATERIALS
(2023)
Article
Polymer Science
Yi-Lang Qin, Ping Zhu, Chen-Xi Ouyang, Xia Dong
Summary: This study investigates the effect of the chemical structure of chain extenders on the hydrogen bonding and mechanical properties of PCUs by synthesizing PCUs with different types of chain extenders. The results show that even minor differences in the chain extender's chemical structure can result in significant variations in phase separation, condensed morphologies, thermal and mechanical properties. The symmetry of the chain extender improves the ordering of the hydrogen bonding, and the coherence strength of the urethane/urea group also plays a key role in the polymer's properties.
CHINESE JOURNAL OF POLYMER SCIENCE
(2023)
Article
Engineering, Chemical
Runhao Zhu, Gongyi Wei, Peng Wang, Rongzhe Li, Xuan Li, Lei Wang, Xia Dong
Summary: For the first time, a segmented block copolyamide with a microphase-separated structure was used as a template to fabricate high-temperature proton exchange membranes (PEMs) with continuous porous proton-transport channels. The resulting membranes showed excellent cell performances, with an ultra-high PA doping and high in-plane conductivity. The efficient and simple in situ etching strategy is expected to become a routine method for preparing PA-doped HT-PEMs.
JOURNAL OF MEMBRANE SCIENCE
(2023)
Article
Polymer Science
Xuan Li, Yu Wang, Lihui Yuan, Yihan Yang, Dujin Wang, Alejandro J. Mueller, Xia Dong
Summary: The present study aims to generate different crystal forms in even-even polyamides by designing thermal protocols. The choice of low molecular weight (M (n)) Polyamide 1012 (PA1012) facilitates easy control of crystal forms. Different crystal forms can be obtained at room temperature by cooling the chosen PA1012 at different rates.
Article
Multidisciplinary Sciences
Xiansheng Zhang, Hongwei Yan, Chongzhi Xu, Xia Dong, Yu Wang, Aiping Fu, Hao Li, Jin Yong Lee, Sheng Zhang, Jiahua Ni, Min Gao, Jing Wang, Jinpeng Yu, Shuzhi Sam Ge, Ming Liang Jin, Lili Wang, Yanzhi Xia
Summary: A facile strategy using total amorphization of polymer is proposed to achieve multi-functionality of cryogels. The cryogels exhibit tissue-like ultrasoftness, stretchability, high transparency, self-adhesion, and instantaneous self-healing, along with superior ionic-conductivity, antifreezing, and water-retention abilities. This paves the way for the development of skin-like cryogel electronics.
NATURE COMMUNICATIONS
(2023)
Article
Polymer Science
Yang Liu, Li Peng, Jie-Long Lin, Yong Zhou, Du-Jin Wang, Charles C. Han, Xian-Bo Huang, Xia Dong
Summary: The crystallization behavior of polyamide 6,6 (PA66) under complex flow field was studied with the addition of poly(vinyl pyrrolidone) (PVP) as a crystallization modifying agent. The regulation of hydrogen bonds interaction by PVP was investigated, and the microstructure evolution under external injection field was revealed. The changes in crystallization and microstructure during processing led to improved processability without sacrificing mechanical properties.
CHINESE JOURNAL OF POLYMER SCIENCE
(2023)
Article
Polymer Science
Yuanzhang Jiang, Xuyi Wang, Yanting Han, Dakai Gong, Yingchun Gu, Lin Tan
Summary: In this study, FeCl3 and poly(acrylic acid) (PAA) were doped into waterborne polyurethanes (WPUs) to create multifunctional materials with self-healing and antimicrobial properties. WPU/Fe/PAA-5.5, containing 5.5 wt % PAA, exhibited excellent mechanical properties, strong self-healing capabilities, and an impressive antimicrobial rate of over 90%. These supramolecular films were also highly recyclable through hot-pressing.
Article
Polymer Science
Longyu Tian, Min Wang, Guangming Liao, Baoliang Liu, Sujuan Zhang, Yucheng Sun, Zhen Meng, Jintao Zhang, Zaijun Lu
Summary: Two kinds of benzoxazine monomers were synthesized and used to prepare polybenzoxazine anion exchange membranes (AEMs) with quaternary ammonium functionalization. The LQPBZ AEM exhibited better ion conductivity and alkaline stability.
Article
Polymer Science
Claire A. Lemarchand
Summary: This study investigates the shock behavior of three different polymers through molecular dynamics simulations. The results reveal specificities in the shock behavior of polymers, including deviations from the linear relation between shock velocity and particle velocity, as well as differences in shear stress relaxation behind the shock front. It is found that the deviation of the Hugoniot locus is related to the change in the relative contribution of bonding and non-bonding potential energies, while polymers with higher glass transition temperatures exhibit slower shear stress relaxation.
Article
Polymer Science
Zengquan Liu, Xiaochun Yin, He Zhang, Shuo Gao, Qinglin Kuang, Yanhong Feng
Summary: A powder solid-state extension (PSSE) technology was proposed to address the high melt viscosity issue of UHMWPE. By uniaxial extension and sintering in the solid state, a highly self-reinforced material was prepared. X-ray diffraction and scattering techniques were used to study the structural evolution during the PSSE process, and a solid-phase induced transformation model was established.
Article
Polymer Science
Zhike Li, Anyu Luo, Rui Zhou, Xin Li, Haiyan Li
Summary: A novel high temperature resistance IL@SiO2 nanocapsules were successfully prepared and their friction and wear properties in PA6 composites were studied. The addition of IL@SiO2 nanocapsules reduced the friction coefficient and wear rate without affecting the mechanical properties of PA6. The synergy between the IL core and SiO2 wall in the nanocapsules improved the self-lubricating performance of the PA6 composites.
Article
Polymer Science
Xin Liu, Xuhong Guo, Qi Liao
Summary: In this study, an algorithm for accurately estimating viscosity is developed using molecular dynamics simulations and the Green-Kubo formula. This algorithm can be applied to complex systems with long correlations, such as macromolecular and biological simulation systems.
Article
Polymer Science
Hao Pu, Yun-Lei Hou, Jing -Zhou Chen, Dong -Lin Zhao
Summary: The use of modified graphene improves the interfacial adherence of carbon fibers to epoxy resin, resulting in enhanced interfacial and bending properties of CF and matrix in CF/EP composites, as well as increased interlaminar shear strength and flexural strength.
Article
Polymer Science
Wei-Chung Ke, Jin-Wei Lin, Manohar Reddy Busireddy, Yueh-Hsing Lee, Jiun-Tai Chen, Chain-Shu Hsu
Summary: This study introduces a crosslinkable monomer, TAIC, to synthesize three crosslinked polyimide films, which exhibit improved thermal, mechanical, and dielectric properties. The introduction of TAIC enhances the tensile strength, reduces the dielectric constant and dielectric loss, and decreases the coefficient of thermal expansion of the polyimide films.
Article
Polymer Science
Alex Kwasi Kumi, Ruiling Fan, Ye Chen, Yumei Zhang
Summary: The difference in leaching of amylopectin from cellulose/amylopectin/1-butyl-3-methylimidazolium chloride ([Bmim][Cl]) blends during regeneration in water and aqueous ethanol has been studied. Molecular dynamics simulations showed that the dissolution and regeneration mechanisms of amylopectin in [Bmim][Cl] are similar to cellulose in ionic liquids. Water regeneration leads to weak electrostatic interactions, resulting in high leaching of amylopectin. In contrast, ethanol-water interactions enhance electrostatic interactions among amylopectin chains, limiting amylopectin leaching in aqueous ethanol.
Article
Polymer Science
Xi Zeng, Junwei Zhou, Junbiao Peng, Chunsheng Zhang, Danling Wang, Yihu Song, Qiang Zheng
Summary: This study utilizes an oligomer deep eutectic solvent (DES) based on polyethylene glycol to regulate the vulcanization kinetics of rubber. The results show that the DES can accelerate vulcanization at low temperatures without affecting the crosslinking density and Mullins effect.
Article
Polymer Science
Qianqian Yue, Yongfei Peng, Xingjian Liu, Aihua He, Huarong Nie
Summary: The addition of metal deactivators can effectively suppress the aging of TBIR, prolong its lifespan, and have no negative effects on its physical and processing properties. These research findings are of great importance for improving the stability and aging resistance of polymers.
Article
Polymer Science
J. Gomez-Caturla, J. Ivorra-Martinez, R. Tejada-Oliveros, V Moreno, D. Garcia-Garcia, R. Balart
Summary: This work focuses on the development of environmentally friendly PLA formulations by using different esters derived from geraniol as plasticizers. The results show that these esters have good compatibility with PLA, and they can effectively enhance the elongation and plasticity of PLA, reduce its glass transition temperature, and slightly improve its water absorption capabilities.
Article
Polymer Science
Dandan Li, Zhaohui Lu, Zhao Ke, Ke Xu, Fengna Dai, Youhai Yu, Guangtao Qian, Chunhai Chen
Summary: In this study, cross-linked PI aerogel membranes with low dielectric constant and high moisture resistance were prepared by co-polymerization and scraping coating technology. The incorporation of fluorinated blocks and benzimidazole ring structures resulted in novel PI aerogel membranes with fascinating dielectric properties and outstanding moisture resistance.
Article
Polymer Science
Zhaoyang Chu, Qing Zhang, Haihua Luo, Han Zhou, Fapei Zhang, Wei Chen, Wenhua Zhang
Summary: Multi-physical fields solution processing strategy is a universal and facile method for preparing various conjugated polymer films and high-performance devices. In this study, we employed a combined microfluidic flow and ultrasonication strategy (FU) for processing CP solutions, and found a pronounced synergetic effect in promoting the pre-ordering of chains in solution. The conformation order and anisotropy of the solution were revealed through various characterizations. A non-classical nucleation model for polymer crystallization in non-equilibrium solution processing was confirmed. The roles of microfluidic flow and ultrasonication in chain aggregation and crystallization were addressed through multi-physical simulations. Compared to pristine solutions, the FU strategy showed improved solution anisotropy and crystallization kinetics, resulting in higher crystallinity in films and increased mobilities in OFET devices. The FU processing strategy provides a universal approach for regulating chain conformation and aggregation in conjugated polymer solutions.
