[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-290666-105":59,"doc-detail-290666-en":135},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":128,"head_meta":130,"extra_data":132,"updated_unix":134},105,"en","the-orientational-structure-of-a-model-patchy-particle-fluid-simulations-integral-equations-density-functional-theory-and-machine-learning","The orientational structure of a model patchy particle fluid - simulations, integral equations, density functional theory and machine learning","","The work studies the orientational properties of a homogeneous and inhomogeneous tetrahedral 4-patch fluid using the Bol–Kern–Frenkel model. Pair and direct correlation functions are resolved with integral-equation methods, including HNC and a modified HNC scheme with simulation input. For inhomogeneous cases, two density-functional routes are developed: molecular DFT using the full direct correlation function and an isotropic hard-sphere bridge, and a machine-learning functional split into isotropic reference and mean-field orientational parts.",{"@graph":69,"@context":127},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/the-orientational-structure-of-a-model-patchy-particle-fluid-simulations-integral-equations-density-functional-theory-and-machine-learning/290666/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/the-orientational-structure-of-a-model-patchy-particle-fluid-simulations-integral-equations-density-functional-theory-and-machine-learning/290666.png","ImageObject",300,407,{"name":92,"@type":93},"Levi","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-12","2026-09-17",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",14,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119,123],{"name":110,"@type":111,"acceptedAnswer":112},"What fluid model is used to study orientational structure?","Question",{"text":113,"@type":114},"A homogeneous and inhomogeneous tetrahedral 4-patch fluid is investigated using the Bol–Kern–Frenkel model.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"Which integral-equation methods are applied to correlation functions?",{"text":118,"@type":114},"The full orientational dependence of pair and direct correlation functions is determined using HNC and a modified HNC scheme with simulation input.",{"name":120,"@type":111,"acceptedAnswer":121},"How are density functionals constructed for the inhomogeneous problem?",{"text":122,"@type":114},"One approach uses molecular density functional theory with the full direct correlation function and an isotropic hard-sphere bridge functional. The second uses machine learning by splitting the functional into an isotropic reference part and a mean-field orientational part.",{"name":124,"@type":111,"acceptedAnswer":125},"How do the proposed functionals compare with simulation benchmarks?",{"text":126,"@type":114},"Comparisons to simulation data at hard walls and around hard tracers show similar performance, and machine-learning strategies are discussed to reduce residual differences for general anisotropic fluids.","https://schema.org",{"og:url":83,"og:type":129,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":131,"canonical":83},"index,follow",{"doc_id":133,"site_id":62},290666,1789642318,{"code":4,"msg":5,"data":136},{"doc_id":133,"user_id":137,"nickname":92,"user_avatar":138,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":139,"file_id":140,"file_url":141,"file_type":142,"file_size":143,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":144,"language":145,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":146,"faqs":147,"seo_title":148,"seo_description":67,"update_tm":134,"read_time":149,"preview_page_count":4},7971461740909,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","arXiv :2411 .06973v2 [ cond-mat .soft] 6 Jan 2025  \nThe orientational structure of a model patchy particle fluid: simulations, integral equations, density functional theory and machine learning  \nAlessandro Simon,1, a) Luc Belloni,2 Daniel Borgis,3, 4 and Martin Oettel1, b)  \n1) Institute for Applied Physics, University of T¨ubingen, Auf der Morgenstelle 10, 72076 T¨ubingen, Germany  \n2) LIONS, NIMBE, CEA, CNRS, Universit´e Paris-Saclay, 91191 Gif-sur-Yvette, France  \n3) Maison de la Simulation, USR 3441 CNRS-CEA-Universit´e Paris-Saclay, 91191 Gif-sur-Yvette, France  \n4) PASTEUR, D´epartement de chimie, ´Ecole normale sup´erieure, PSL University, Sorbonne Universit´e, CNRS, 75005 Paris, France  \nWe investigate the orientational properties of a homogeneous and inhomogeneous tetrahedral 4-patch fluid (Bol–Kern–Frenkel model) . Using integral equations, either (i) HNC or (ii) a modified HNC scheme with simulation input, the full orientational dependence of pair and direct correlation functions is determined. Density functionals for the inhomogeneous problem are constructed via two different methods. The first, molecular density functional theory, utilizes the full direct correlation function and an isotropic hard-sphere bridge functional. The second method, a machine learning approach, uses a decomposition of the functional into an isotropic reference part and a mean-field orientational part, where both parts are improved by machine learning techniques. Comparison to simulation data at hard walls and around hard tracers show a similar performance of the two functionals. Machine learning strategies are discussed to eliminate residual differences, with the goal of obtaining machine-learning enhanced functionals for the general anisotropic fluid.  \nCONTENTS  \nI. Introduction 1  \nII. Model definitions and simulations 2  \nA. Bol–Kern–Frenkel model 2  \nB. Orientational expansions 3  \nC. Simulations 4  \nIII. Integral equations 5  \nA. Theory 5  \nB. Results 5  \nIV. Density functional theory 7  \nA. Molecular DFT 7  \nB. Machine learning approach 8  \n1. Splitting of the density functional 8  \n2. Training the weight functions for the  \nreference part 10  \n3. Direct correlation functions 10  \nC. Results 12  \n1. Contact density at hard wall and equation  \nof state 12  \n2. Averaged density n (z) near the hard wall 12  \n3. Orientational moment profiles α(i)(z) near the hard wall 12  \n4. Hard spherical tracers 13  \nV. Summary and conclusions 15  \nVI. Supplementary Material 15  \na) Electronic mail: [alessandro-rodolfo.simon@uni-tuebingen.de](alessandro-rodolfo.simon@uni-tuebingen.de)  \nb) [Electronic mail:](Electronic mail: martin.oettel@uni-tuebingen.de)[ martin.oettel@uni-tuebingen.de](Electronic mail: martin.oettel@uni-tuebingen.de)  \nAcknowledgments 15  \nReferences 16  \nI. INTRODUCTION  \nClassical fluids made of particles with strong directional bonding (associating fluids) are an important class of systems relevant for the understanding of polymerization, self-assembly in molecular and colloidal systems and phase behavior of proteins. A simple model system for associating fluids are spherical patchy particles made of hard spheres with Np localized spots (patches) on their surface, which lead to strong attraction when they overlap.  \nSpecifically, patchy particles with Np = 4 patches are interesting model systems for tetrahedrally coordinating particles and thus the system has also some relevance to molecular fluids such as water 1 . Patchy particles are becoming increasingly popular as colloidal models to explain certain features of protein solutions. An example is the phase and adsorption behavior of bovine serum albumine (BSA) in solutions with trivalent cations2–4 . In these systems, the proteins bind at specific sites with the cations acting as a bridge, and can be described as ion-activated patchy particles. Regarding bulk thermodynamic behaviour we note that the equation of state and consequently the phase diagram of these patchy partic","cbCaisAqYRAdp7Nb","https://ap.wps.com/l/cbCaisAqYRAdp7Nb","pdf",2253603,23,"English","# Contents\n## I. Introduction\n## II. Model definitions and simulations\n## III. Integral equations\n## IV. Density functional theory\n## V. Summary and conclusions\n## VI. Supplementary Material","[{\"question\":\"What fluid model is used to study orientational structure?\",\"answer\":\"A homogeneous and inhomogeneous tetrahedral 4-patch fluid is investigated using the Bol–Kern–Frenkel model.\"},{\"question\":\"Which integral-equation methods are applied to correlation functions?\",\"answer\":\"The full orientational dependence of pair and direct correlation functions is determined using HNC and a modified HNC scheme with simulation input.\"},{\"question\":\"How are density functionals constructed for the inhomogeneous problem?\",\"answer\":\"One approach uses molecular density functional theory with the full direct correlation function and an isotropic hard-sphere bridge functional. The second uses machine learning by splitting the functional into an isotropic reference part and a mean-field orientational part.\"},{\"question\":\"How do the proposed functionals compare with simulation benchmarks?\",\"answer\":\"Comparisons to simulation data at hard walls and around hard tracers show similar performance, and machine-learning strategies are discussed to reduce residual differences for general anisotropic fluids.\"}]","The orientational structure of a model patchy particle fluid - simulations, integral equations, density functional theory and machine learning | PDF",58]