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Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2026,8,25]]},"abstract":"<jats:p>\n                    The flow of dense emulsions underlies applications from food and pharmaceutical processing to bioengineering, yet their rheology remains difficult to interpret under conventional volume-imposed conditions. Here, we change the control variable to osmotic-pressure using a recently developed instrument\u2014the Capillarytron. This approach reveals a unified rheological structure where the osmotic pressure\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"normal\">\u03a0<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    , by controlling droplet deformation, sets a\n                    <jats:italic toggle=\"yes\">pressure-dependent<\/jats:italic>\n                    jamming volume fraction. When expressed in terms of the distance to this jamming point, all rheological data\u2014spanning both\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"normal\">\u03a0<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    - and\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mi>\u03d5<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    -imposed measurements\u2014collapse onto a single power-law divergence, akin to granular suspensions. The resulting constitutive relations provide a predictive, parameter-free description of emulsion rheology across Newtonian, yielding, and shear-thinning regimes. Together with recent results on soft spheres, our findings point to a unifying paradigm: Soft amorphous materials\u2014from soft spheres to emulsions and likely foams\u2014obey the same\n                    <jats:italic toggle=\"yes\">hard rules<\/jats:italic>\n                    as granular suspensions, with softness entering through a pressure-dependent jamming point. This framework rationalizes Herschel\u2013Bulkley rheology, assigns its parameters microscopic meaning, and opens perspectives on rigidity transitions in soft, deformable systems, including biological tissues.\n                  <\/jats:p>","DOI":"10.1073\/pnas.2611013123","type":"journal-article","created":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T17:37:08Z","timestamp":1787161028000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":0,"title":["Soft matter, hard rules: Emulsions follow the laws of granular suspension rheology"],"prefix":"10.1073","volume":"123","author":[{"given":"Wenjun","family":"Chen","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0190ak572","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Physics, Center for Soft Matter Research, New York University","place":["New York"]},{"id":[{"id":"https:\/\/ror.org\/04bgbbh33","id-type":"ROR","asserted-by":"publisher"}],"name":"Aix Marseille Universit\u00e9, CNRS, Institut Universitaire des Syst\u00e9mes Thermiques Industriels","place":["Marseille, France"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eric","family":"De Giuli","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/05g13zd79","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Physics, Toronto Metropolitan University","place":["Toronto, Canada"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Matthieu","family":"Wyart","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00za53h95","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Physics and Astronomy, Johns Hopkins University","place":["Baltimore"]},{"id":[{"id":"https:\/\/ror.org\/02s376052","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute of Physics, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL)","place":["Lausanne, Switzerland"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6052-7291","authenticated-orcid":false,"given":"Yo\u00ebl","family":"Forterre","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04bgbbh33","id-type":"ROR","asserted-by":"publisher"}],"name":"Aix Marseille Universit\u00e9, CNRS, Institut Universitaire des Syst\u00e9mes Thermiques Industriels","place":["Marseille, France"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jasna","family":"Bruji\u0107","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0190ak572","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Physics, Center for Soft Matter Research, New York University","place":["New York"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3031-6543","authenticated-orcid":false,"given":"Bloen","family":"Metzger","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04bgbbh33","id-type":"ROR","asserted-by":"publisher"}],"name":"Aix Marseille Universit\u00e9, CNRS, Institut Universitaire des Syst\u00e9mes Thermiques Industriels","place":["Marseille, France"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2026,8,19]]},"reference":[{"key":"e_1_3_4_1_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00397-025-01488-w"},{"key":"e_1_3_4_2_2","doi-asserted-by":"publisher","DOI":"10.1021\/la00073a007"},{"key":"e_1_3_4_3_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.75.2051"},{"key":"e_1_3_4_4_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1017716108"},{"key":"e_1_3_4_5_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1201499109"},{"key":"e_1_3_4_6_2","doi-asserted-by":"crossref","unstructured":"Q. 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