{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T13:42:28Z","timestamp":1783690948241,"version":"3.55.0"},"reference-count":41,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,2,7]],"date-time":"2021-02-07T00:00:00Z","timestamp":1612656000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Fundamental Research Funds for the Central Universities","award":["2018JBM071"],"award-info":[{"award-number":["2018JBM071"]}]},{"name":"the Fundamental Research Funds for the Central Universities","award":["2020YJS180"],"award-info":[{"award-number":["2020YJS180"]}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["2182057"],"award-info":[{"award-number":["2182057"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["21774013"],"award-info":[{"award-number":["21774013"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>The application of single chain mean-field theory (SCMFT) on semiflexible chain brushes is reviewed. The worm-like chain (WLC) model is the best mode of semiflexible chain that can continuously recover to the rigid rod model and Gaussian chain (GC) model in rigid and flexible limits, respectively. Compared with the commonly used GC model, SCMFT is more applicable to the WLC model because the algorithmic complexity of the WLC model is much higher than that of the GC model in self-consistent field theory (SCFT). On the contrary, the algorithmic complexity of both models in SCMFT are comparable. In SCMFT, the ensemble average of quantities is obtained by sampling the conformations of a single chain or multi-chains in the external auxiliary field instead of solving the modified diffuse equation (MDE) in SCFT. The precision of this calculation is controlled by the number of bonds Nm used to discretize the chain contour length L and the number of conformations M used in the ensemble average. The latter factor can be well controlled by metropolis Monte Carlo simulation. This approach can be easily generalized to solve problems with complex boundary conditions or in high-dimensional systems, which were once nightmares when solving MDEs in SCFT. Moreover, the calculations in SCMFT mainly relate to the assemble averages of chain conformations, for which a portion of conformations can be performed parallel on different computing cores using a message-passing interface (MPI).<\/jats:p>","DOI":"10.3390\/ma14040778","type":"journal-article","created":{"date-parts":[[2021,2,12]],"date-time":"2021-02-12T23:38:44Z","timestamp":1613173124000},"page":"778","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Single Chain Mean-Field Theory Study on Responsive Behavior of Semiflexible Polymer Brush"],"prefix":"10.3390","volume":"14","author":[{"given":"Yingli","family":"Niu","sequence":"first","affiliation":[{"name":"School of Science, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiangyu","family":"Bu","sequence":"additional","affiliation":[{"name":"School of Science, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8737-5312","authenticated-orcid":false,"given":"Xinghua","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Science, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/00018732.2013.771509","article-title":"Emergent complexity of the cytoskeleton: From single filaments to tissue","volume":"62","author":"Huber","year":"2013","journal-title":"Adv. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1038\/nphys260","article-title":"A bottom-up approach to cell mechanics","volume":"2","author":"Bausch","year":"2006","journal-title":"Nat. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1038\/nphoton.2009.170","article-title":"Electrically tunable liquid crystal optical microresonators","volume":"3","author":"Humar","year":"2009","journal-title":"Nat. Photon."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1411","DOI":"10.1021\/ma4020164","article-title":"Sub-5 nm domains in ordered poly(cyclohexylethylene)- block -poly(methyl methacrylate) block polymers for lithography","volume":"47","author":"Kennemur","year":"2014","journal-title":"Macromolecules"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"028103","DOI":"10.1103\/PhysRevLett.110.028103","article-title":"From flexible to stiff: Systematic analysis of structural phases for single semiflexible polymers","volume":"110","author":"Seaton","year":"2013","journal-title":"Phy. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"204903","DOI":"10.1063\/1.4936324","article-title":"Compression induced phase transition of nematic brush: A mean-field theory study","volume":"143","author":"Tang","year":"2015","journal-title":"J. Chem. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3811","DOI":"10.1063\/1.1680554","article-title":"Statistical mechanics of wormlike chains: Path integral and diagram methods","volume":"59","author":"Yamakawa","year":"1973","journal-title":"J. Chem. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"5814","DOI":"10.1021\/ma049958v","article-title":"Exact results for a semiflexible polymer chain in an aligning field","volume":"37","author":"Spakowitz","year":"2004","journal-title":"Macromolecules"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1209\/epl\/i2002-00253-5","article-title":"The distribution function of a semiflexible polymer and random walks with constraints","volume":"60","author":"Stepanow","year":"2002","journal-title":"EPL"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1103\/PhysRevLett.71.93","article-title":"Continuous isotropic-nematic transition of partially flexible polymers in two dimensions","volume":"71","author":"Chen","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2817","DOI":"10.1021\/ma00063a028","article-title":"Free energy functional for semiflexible polymer solutions and blend","volume":"26","author":"Liu","year":"1993","journal-title":"Macromolecules"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3235","DOI":"10.1103\/PhysRevLett.73.3235","article-title":"Semiflexible Polymers near Interfaces","volume":"73","author":"Morse","year":"1993","journal-title":"Phy. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6300","DOI":"10.1021\/ma9007412","article-title":"New numerical implementation of self-consistent field theory for semiflexible polymers","volume":"42","author":"Song","year":"2009","journal-title":"Macromolecules"},{"key":"ref_14","unstructured":"Fredrickson, G.H. (1993). The Equilibrium Theory of Inhomogeneous Polymers, Oxford University Press."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"10668","DOI":"10.1021\/ma1022814","article-title":"Isotropic-nematic interface in a lyotropic system of wormlike chains with the onsager interaction","volume":"43","author":"Jiang","year":"2010","journal-title":"Macromolecules"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1039\/C2SM26758F","article-title":"Non-lamellae structures of coil\u2013semiflexible diblock copolymers","volume":"9","author":"Gao","year":"2013","journal-title":"Soft Matter"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"138305","DOI":"10.1103\/PhysRevLett.110.138305","article-title":"Influence of Chain Rigidity on the Phase Behavior of Wormlike Diblock Copolymers","volume":"110","author":"Jiang","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"042603","DOI":"10.1103\/PhysRevE.88.042603","article-title":"Self-consistent field theory and numerical scheme for calculating the phase diagram of wormlike diblock copolymer","volume":"88","author":"Jiang","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"7769","DOI":"10.1021\/ma060871e","article-title":"Free energy of a wormlike polymer chain confined in a slit crossover between two scaling regimes","volume":"39","author":"Chen","year":"2006","journal-title":"Macromolecules"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"244910","DOI":"10.1063\/1.4811515","article-title":"Modified diffusion equation for the wormlike-chain statistics in curvilinear coordinates","volume":"138","author":"Liang","year":"2013","journal-title":"J. Chem. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3597","DOI":"10.1063\/1.449167","article-title":"Chain organization and thermodynamics in micelles and bilayers. I. Theory","volume":"83","author":"Szleifer","year":"1985","journal-title":"J. Chem. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3197","DOI":"10.1021\/ma00113a023","article-title":"On the structure and pressure of tethered polymer layers in good solvent","volume":"28","author":"Carignano","year":"1995","journal-title":"Macromolecules"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"8900","DOI":"10.1021\/ma960782+","article-title":"Ordered phases in rod-coil diblock copolymers","volume":"29","author":"Muller","year":"1996","journal-title":"Macromolecules"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"8323","DOI":"10.1021\/ma980052x","article-title":"Wetting and capillary condensation in symmetric polymer blends: A comparison between monte carlo simulations and self-consistent field calculations","volume":"31","author":"Muller","year":"1998","journal-title":"Macromolecules"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1124","DOI":"10.1021\/ma0343481","article-title":"Development of an importance sampling single chain mean field theory for polymer adsorption onto a flat wall","volume":"37","author":"Avalos","year":"2004","journal-title":"Macromolecules"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"104910","DOI":"10.1063\/1.1860558","article-title":"Prediction of the critical micelle concentration in a lattice model for amphiphiles using a single-chain mean-field theory","volume":"122","author":"Alanber","year":"2005","journal-title":"J. Chem. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2481","DOI":"10.1039\/C5SM02865E","article-title":"Microphase separation of short wormlike diblock copolymers with a finite interaction range","volume":"12","author":"Jiang","year":"2016","journal-title":"Soft Matter"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4168","DOI":"10.1021\/ma060014a","article-title":"Structure of adsorbed polymers on a colloid particle","volume":"39","author":"Yang","year":"2006","journal-title":"Macromolecules"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3409","DOI":"10.1016\/j.polymer.2012.05.017","article-title":"Structures and interactions between two colloidal particles in adsorptive polymer solutions","volume":"53","author":"Li","year":"2012","journal-title":"Polymer"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3455","DOI":"10.1021\/ma9023813","article-title":"Wormlike polymer brush: A self-consistent field treatmen","volume":"43","author":"Deng","year":"2010","journal-title":"Macromolecules"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1322","DOI":"10.1002\/adom.201600281","article-title":"Recent Progress in Photoswitchable Supramolecular Self-Assembling Systems","volume":"4","author":"Yao","year":"2016","journal-title":"Adv. Opt. Mater."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"164901","DOI":"10.1063\/1.4981914","article-title":"Light-responsive expansion-contraction of spherical nanoparticle grafted with azopolymers","volume":"146","author":"Fu","year":"2017","journal-title":"J. C hem. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"6664","DOI":"10.1021\/ja00094a023","article-title":"Thermal and Light Control of the Sol-Gel Phase Transition in Cholesterol-Based Organic Gels. Novel Helical Aggregation Modes As Detected by Circular Dichroism and Electron Microscopic Observation","volume":"116","author":"Murata","year":"1994","journal-title":"J. Am. Chem. Soc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"065702","DOI":"10.1103\/PhysRevLett.107.065702","article-title":"Glass Transitions in Quasi-Two-Dimensional Suspensions of Colloidal Ellipsoids","volume":"107","author":"Zheng","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"15185","DOI":"10.1021\/la202997r","article-title":"Photonic Control of Surface Anchoring on Solid Colloids Dispersed in Liquid Crystals","volume":"27","author":"Chandran","year":"2011","journal-title":"Langmuir"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1039\/B616752G","article-title":"From supramolecular chemistry towards constitutional dynamic chemistry and adaptive chemistry","volume":"36","author":"Lehn","year":"2007","journal-title":"Chem. Soc. Rev."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1603155","DOI":"10.1002\/smll.201603155","article-title":"Optimal reactivity and improved self-healing capability of structurally dynamic polymers grafted on janus nanoparticles governed by chain stiffness and spatial organization","volume":"13","author":"Xu","year":"2017","journal-title":"Small"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1799","DOI":"10.1002\/adma.201504394","article-title":"Rapid Reversible Photoswitching of Integrin-Mediated Adhesion at the Single-Cell Level","volume":"28","author":"Kadem","year":"2016","journal-title":"Adv. Mater."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"14141","DOI":"10.1021\/acs.langmuir.9b03009","article-title":"Theoretical Study of the Optimal Design of a UV-Controllable Smart Surface Decorated by a Hybrid Azobenzene-Containing Polymer Layer","volume":"35","author":"Bu","year":"2019","journal-title":"Langmuir"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"8117","DOI":"10.1038\/ncomms9117","article-title":"Sliding tethered ligands add topological interactions to the toolbox of ligand-receptor design","volume":"6","author":"Bauer","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"9418","DOI":"10.1039\/C9SM01825E","article-title":"Bistable adhesion behavior of an anisotropic particle on a sliding polymer brush","volume":"15","author":"Bu","year":"2019","journal-title":"Soft Matter"}],"container-title":["Materials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1996-1944\/14\/4\/778\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:20:46Z","timestamp":1760160046000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1996-1944\/14\/4\/778"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,7]]},"references-count":41,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["ma14040778"],"URL":"https:\/\/doi.org\/10.3390\/ma14040778","relation":{},"ISSN":["1996-1944"],"issn-type":[{"value":"1996-1944","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,2,7]]}}}