DOI: 10.1063/1.1499718. Simplified Onsager theory for isotropic–nematic phase equilibria of length polydisperse hard rods

Simplified Onsager theory for isotropic–nematic phase equilibria of length polydisperse hard rods

10.1063/1.1499718

Crossref journal-article

AIP Publishing

The Journal of Chemical Physics (317)

Abstract

Polydispersity is believed to have important effects on the formation of liquid crystal phases in suspensions of rodlike particles. To understand such effects, we analyze the phase behavior of thin hard rods with length polydispersity. Our treatment is based on a simplified Onsager theory, obtained by truncating the series expansion of the angular dependence of the excluded volume. We describe the model and give the full phase equilibrium equations; these are then solved numerically using the moment free energy method which reduces the problem from one with an infinite number of conserved densities to one with a finite number of effective densities that are moments of the full density distribution. The method yields exactly the onset of nematic ordering. Beyond this, results are approximate but we show that they can be made essentially arbitrarily precise by adding adaptively chosen extra moments, while still avoiding the numerical complications of a direct solution of the full phase equilibrium conditions. We investigate in detail the phase behavior of systems with three different length distributions: a (unimodal) Schulz distribution, a bidisperse distribution, and a bimodal mixture of two Schulz distributions which interpolates between these two cases. A three-phase isotropic-nematic-nematic coexistence region is shown to exist for the bimodal and bidisperse length distributions if the ratio of long and short rod lengths is sufficiently large, but not for the unimodal one. We systematically explore the topology of the phase diagram as a function of the width of the length distribution and of the rod length ratio in the bidisperse and bimodal cases.

Bibliography

Speranza, A., & Sollich, P. (2002). Simplified Onsager theory for isotropicânematic phase equilibria of length polydisperse hard rods. The Journal of Chemical Physics, 117(11), 5421â5436.

Authors 2

Alessandro Speranza (first)
Peter Sollich (additional)

References 53 Referenced 89

10.1039/tf9373300348 / Trans. Faraday Soc. (1937)
10.1063/1.1750183 / J. Chem. Phys. (1938)
10.1002/bip.1968.360060708 / Biopolymers (1968)
10.1002/zaac.19251470111 / Z. Anorg. Allg. Chem. (1925)
{'key': '2024021017073197300_r5', 'first-page': '274', 'volume': '123', 'year': '1937', 'journal-title': 'Proc. R. Soc. London, Ser. A'} / Proc. R. Soc. London, Ser. A (1937)
{'key': '2024021017073197300_r6', 'first-page': '513', 'volume': '141', 'year': '1938', 'journal-title': 'Nature (London)'} / Nature (London) (1938)
10.1038/1381051a0 / Nature (London) (1936)
10.1038/139923a0 / Nature (London) (1937)
10.1515/zna-1958-0716 / Z. Naturforsch. A (1958)
{'key': '2024021017073197300_r10'}
10.1111/j.1749-6632.1949.tb27296.x / Ann. N.Y. Acad. Sci. (1949)
10.1063/1.447098 / J. Chem. Phys. (1984)
10.1063/1.1673914 / J. Chem. Phys. (1970)
{'key': '2024021017073197300_r14', 'first-page': '2067', 'volume': '17', 'year': '1977', 'journal-title': 'Phys. Rev. A'} / Phys. Rev. A (1977)
10.1063/1.1748493 / J. Chem. Phys. (1951)
10.1098/rspa.1956.0016 / Proc. R. Soc. London Ser. A (1956)
10.1063/1.1731986 / J. Chem. Phys. (1961)
10.1080/15421407108083541 / Mol. Cryst. Liq. Cryst. (1971)
{'key': '2024021017073197300_r19', 'first-page': '316', 'volume': '30', 'year': '1986', 'journal-title': 'Polym. Sci. U.S.S.R.'} / Polym. Sci. U.S.S.R. (1986)
{'key': '2024021017073197300_r20', 'first-page': '234', 'volume': '4', 'year': '1956', 'journal-title': 'Proc. R. Soc. London, Ser. A'} / Proc. R. Soc. London, Ser. A (1956)
10.1039/tf9565200571 / Trans. Faraday Soc. (1956)
10.1016/0095-8522(62)90028-4 / J. Colloid Sci. (1962)
10.1016/0009-2509(93)80026-M / Chem. Eng. Sci. (1993)
10.1016/0166-6622(92)80161-T / Colloids Surf. (1992)
10.1016/0378-4371(81)90148-5 / Physica A (1981)
{'key': '2024021017073197300_r26', 'first-page': '605', 'volume': '112', 'year': '1981', 'journal-title': 'Physica A'} / Physica A (1981)
10.1063/1.434786 / J. Chem. Phys. (1977)
10.1063/1.434044 / J. Chem. Phys. (1977)
10.1021/j100146a027 / J. Phys. Chem. (1993)
{'key': '2024021017073197300_r30', 'first-page': '9451', 'volume': '8', 'year': '1996', 'journal-title': 'J. Phys.: Condens. Matter'} / J. Phys.: Condens. Matter (1996)
10.1103/PhysRevE.50.2849 / Phys. Rev. E (1994)
10.1080/02678298908027327 / Liq. Cryst. (1989)
10.1021/j100256a058 / J. Phys. Chem. (1985)
10.1063/1.473006 / J. Chem. Phys. (1996)
10.1021/j100116a026 / J. Phys. Chem. (1993)
10.1021/ma60066a012 / Macromolecules (1978)
10.1021/ma60066a011 / Macromolecules (1978)
10.1021/ma60066a013 / Macromolecules (1978)
10.1021/ma60066a014 / Macromolecules (1978)
10.1016/0032-3861(82)90095-7 / Polymer (1982)
10.1063/1.1734518 / J. Chem. Phys. (1963)
{'key': '2024021017073197300_r42', 'first-page': '1841', 'volume': '2', 'year': '1992', 'journal-title': 'J. Phys. I'} / J. Phys. I (1992)
10.1063/1.1290473 / J. Chem. Phys. (2000)
10.1103/PhysRevLett.80.1365 / Phys. Rev. Lett. (1998)
{'key': '2024021017073197300_r45', 'first-page': '265', 'volume': '116', 'year': '2001', 'journal-title': 'Adv. Chem. Phys.'} / Adv. Chem. Phys. (2001)
10.1103/PhysRevLett.80.1369 / Phys. Rev. Lett. (1998)
10.1088/0034-4885/55/8/003 / Rep. Prog. Phys. (1992)
{'key': '2024021017073197300_r48', 'first-page': '405', 'volume': '130', 'year': '1997', 'journal-title': 'Colloids Surf., A'} / Colloids Surf., A (1997)
{'key': '2024021017073197300_r49'}
{'key': '2024021017073197300_r50'}
10.1021/ma60017a607 / Macromolecules (1970)
10.1021/ma60048a025 / Macromolecules (1975)
{'key': '2024021017073197300_r53'}

Dates

Type	When
Created	23 years ago (Aug. 30, 2002, 1:02 p.m.)
Deposited	1 year, 6 months ago (Feb. 10, 2024, 12:52 p.m.)
Indexed	4 days, 15 hours ago (Aug. 29, 2025, 5:57 a.m.)
Issued	22 years, 11 months ago (Sept. 15, 2002)
Published	22 years, 11 months ago (Sept. 15, 2002)
Published Print	22 years, 11 months ago (Sept. 15, 2002)

Funders 0

None

BibTeX

@article{Speranza_2002, title={Simplified Onsager theory for isotropic–nematic phase equilibria of length polydisperse hard rods}, volume={117}, ISSN={1089-7690}, url={http://dx.doi.org/10.1063/1.1499718}, DOI={10.1063/1.1499718}, number={11}, journal={The Journal of Chemical Physics}, publisher={AIP Publishing}, author={Speranza, Alessandro and Sollich, Peter}, year={2002}, month=sep, pages={5421–5436} }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        8,
        29
      ]
    ],
    "date-time": "2025-08-29T09:57:06Z",
    "timestamp": 1756461426399
  },
  "reference-count": 53,
  "publisher": "AIP Publishing",
  "issue": "11",
  "content-domain": {
    "domain": [
      "pubs.aip.org"
    ],
    "crossmark-restriction": true
  },
  "published-print": {
    "date-parts": [
      [
        2002,
        9,
        15
      ]
    ]
  },
  "abstract": "<jats:p>Polydispersity is believed to have important effects on the formation of liquid crystal phases in suspensions of rodlike particles. To understand such effects, we analyze the phase behavior of thin hard rods with length polydispersity. Our treatment is based on a simplified Onsager theory, obtained by truncating the series expansion of the angular dependence of the excluded volume. We describe the model and give the full phase equilibrium equations; these are then solved numerically using the moment free energy method which reduces the problem from one with an infinite number of conserved densities to one with a finite number of effective densities that are moments of the full density distribution. The method yields exactly the onset of nematic ordering. Beyond this, results are approximate but we show that they can be made essentially arbitrarily precise by adding adaptively chosen extra moments, while still avoiding the numerical complications of a direct solution of the full phase equilibrium conditions. We investigate in detail the phase behavior of systems with three different length distributions: a (unimodal) Schulz distribution, a bidisperse distribution, and a bimodal mixture of two Schulz distributions which interpolates between these two cases. A three-phase isotropic-nematic-nematic coexistence region is shown to exist for the bimodal and bidisperse length distributions if the ratio of long and short rod lengths is sufficiently large, but not for the unimodal one. We systematically explore the topology of the phase diagram as a function of the width of the length distribution and of the rod length ratio in the bidisperse and bimodal cases.</jats:p>",
  "DOI": "10.1063/1.1499718",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2002,
        8,
        30
      ]
    ],
    "date-time": "2002-08-30T17:02:52Z",
    "timestamp": 1030726972000
  },
  "page": "5421-5436",
  "update-policy": "http://dx.doi.org/10.1063/aip-crossmark-policy-page",
  "source": "Crossref",
  "is-referenced-by-count": 89,
  "title": "Simplified Onsager theory for isotropic\u2013nematic phase equilibria of length polydisperse hard rods",
  "prefix": "10.1063",
  "volume": "117",
  "author": [
    {
      "given": "Alessandro",
      "family": "Speranza",
      "sequence": "first",
      "affiliation": [
        {
          "name": "Department of Mathematics, King\u2019s College London, Strand, London WC2R 2LS, United Kingdom"
        }
      ]
    },
    {
      "given": "Peter",
      "family": "Sollich",
      "sequence": "additional",
      "affiliation": [
        {
          "name": "Department of Mathematics, King\u2019s College London, Strand, London WC2R 2LS, United Kingdom"
        }
      ]
    }
  ],
  "member": "317",
  "reference": [
    {
      "key": "2024021017073197300_r1",
      "doi-asserted-by": "crossref",
      "first-page": "348",
      "DOI": "10.1039/tf9373300348",
      "volume": "33",
      "year": "1937",
      "journal-title": "Trans. Faraday Soc."
    },
    {
      "key": "2024021017073197300_r2",
      "doi-asserted-by": "crossref",
      "first-page": "873",
      "DOI": "10.1063/1.1750183",
      "volume": "6",
      "year": "1938",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2024021017073197300_r3",
      "doi-asserted-by": "crossref",
      "first-page": "973",
      "DOI": "10.1002/bip.1968.360060708",
      "volume": "6",
      "year": "1968",
      "journal-title": "Biopolymers"
    },
    {
      "key": "2024021017073197300_r4",
      "doi-asserted-by": "crossref",
      "first-page": "91",
      "DOI": "10.1002/zaac.19251470111",
      "volume": "147",
      "year": "1925",
      "journal-title": "Z. Anorg. Allg. Chem."
    },
    {
      "key": "2024021017073197300_r5",
      "first-page": "274",
      "volume": "123",
      "year": "1937",
      "journal-title": "Proc. R. Soc. London, Ser. A"
    },
    {
      "key": "2024021017073197300_r6",
      "first-page": "513",
      "volume": "141",
      "year": "1938",
      "journal-title": "Nature (London)"
    },
    {
      "key": "2024021017073197300_r7",
      "doi-asserted-by": "crossref",
      "first-page": "1051",
      "DOI": "10.1038/1381051a0",
      "volume": "138",
      "year": "1936",
      "journal-title": "Nature (London)"
    },
    {
      "key": "2024021017073197300_r8",
      "doi-asserted-by": "crossref",
      "first-page": "923",
      "DOI": "10.1038/139923a0",
      "volume": "139",
      "year": "1937",
      "journal-title": "Nature (London)"
    },
    {
      "key": "2024021017073197300_r9",
      "doi-asserted-by": "crossref",
      "first-page": "564",
      "DOI": "10.1515/zna-1958-0716",
      "volume": "13",
      "year": "1958",
      "journal-title": "Z. Naturforsch. A"
    },
    {
      "key": "2024021017073197300_r10"
    },
    {
      "key": "2024021017073197300_r11",
      "doi-asserted-by": "crossref",
      "first-page": "627",
      "DOI": "10.1111/j.1749-6632.1949.tb27296.x",
      "volume": "51",
      "year": "1949",
      "journal-title": "Ann. N.Y. Acad. Sci."
    },
    {
      "key": "2024021017073197300_r12",
      "doi-asserted-by": "crossref",
      "first-page": "3427",
      "DOI": "10.1063/1.447098",
      "volume": "80",
      "year": "1984",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2024021017073197300_r13",
      "doi-asserted-by": "crossref",
      "first-page": "4141",
      "DOI": "10.1063/1.1673914",
      "volume": "53",
      "year": "1970",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2024021017073197300_r14",
      "first-page": "2067",
      "volume": "17",
      "year": "1977",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2024021017073197300_r15",
      "doi-asserted-by": "crossref",
      "first-page": "1142",
      "DOI": "10.1063/1.1748493",
      "volume": "19",
      "year": "1951",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2024021017073197300_r16",
      "doi-asserted-by": "crossref",
      "first-page": "73",
      "DOI": "10.1098/rspa.1956.0016",
      "volume": "234",
      "year": "1956",
      "journal-title": "Proc. R. Soc. London Ser. A"
    },
    {
      "key": "2024021017073197300_r17",
      "doi-asserted-by": "crossref",
      "first-page": "658",
      "DOI": "10.1063/1.1731986",
      "volume": "35",
      "year": "1961",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2024021017073197300_r18",
      "doi-asserted-by": "crossref",
      "first-page": "193",
      "DOI": "10.1080/15421407108083541",
      "volume": "13",
      "year": "1971",
      "journal-title": "Mol. Cryst. Liq. Cryst."
    },
    {
      "key": "2024021017073197300_r19",
      "first-page": "316",
      "volume": "30",
      "year": "1986",
      "journal-title": "Polym. Sci. U.S.S.R."
    },
    {
      "key": "2024021017073197300_r20",
      "first-page": "234",
      "volume": "4",
      "year": "1956",
      "journal-title": "Proc. R. Soc. London, Ser. A"
    },
    {
      "key": "2024021017073197300_r21",
      "doi-asserted-by": "crossref",
      "first-page": "571",
      "DOI": "10.1039/tf9565200571",
      "volume": "52",
      "year": "1956",
      "journal-title": "Trans. Faraday Soc."
    },
    {
      "key": "2024021017073197300_r22",
      "doi-asserted-by": "crossref",
      "first-page": "638",
      "DOI": "10.1016/0095-8522(62)90028-4",
      "volume": "17",
      "year": "1962",
      "journal-title": "J. Colloid Sci."
    },
    {
      "key": "2024021017073197300_r23",
      "doi-asserted-by": "crossref",
      "first-page": "411",
      "DOI": "10.1016/0009-2509(93)80026-M",
      "volume": "48",
      "year": "1993",
      "journal-title": "Chem. Eng. Sci."
    },
    {
      "key": "2024021017073197300_r24",
      "doi-asserted-by": "crossref",
      "first-page": "47",
      "DOI": "10.1016/0166-6622(92)80161-T",
      "volume": "64",
      "year": "1992",
      "journal-title": "Colloids Surf."
    },
    {
      "key": "2024021017073197300_r25",
      "doi-asserted-by": "crossref",
      "first-page": "546",
      "DOI": "10.1016/0378-4371(81)90148-5",
      "volume": "108",
      "year": "1981",
      "journal-title": "Physica A"
    },
    {
      "key": "2024021017073197300_r26",
      "first-page": "605",
      "volume": "112",
      "year": "1981",
      "journal-title": "Physica A"
    },
    {
      "key": "2024021017073197300_r27",
      "doi-asserted-by": "crossref",
      "first-page": "5795",
      "DOI": "10.1063/1.434786",
      "volume": "67",
      "year": "1977",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2024021017073197300_r28",
      "doi-asserted-by": "crossref",
      "first-page": "1098",
      "DOI": "10.1063/1.434044",
      "volume": "66",
      "year": "1977",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2024021017073197300_r29",
      "doi-asserted-by": "crossref",
      "first-page": "11510",
      "DOI": "10.1021/j100146a027",
      "volume": "97",
      "year": "1993",
      "journal-title": "J. Phys. Chem."
    },
    {
      "key": "2024021017073197300_r30",
      "first-page": "9451",
      "volume": "8",
      "year": "1996",
      "journal-title": "J. Phys.: Condens. Matter"
    },
    {
      "key": "2024021017073197300_r31",
      "doi-asserted-by": "crossref",
      "first-page": "2849",
      "DOI": "10.1103/PhysRevE.50.2849",
      "volume": "50",
      "year": "1994",
      "journal-title": "Phys. Rev. E"
    },
    {
      "key": "2024021017073197300_r32",
      "doi-asserted-by": "crossref",
      "first-page": "111",
      "DOI": "10.1080/02678298908027327",
      "volume": "6",
      "year": "1989",
      "journal-title": "Liq. Cryst."
    },
    {
      "key": "2024021017073197300_r33",
      "doi-asserted-by": "crossref",
      "first-page": "2090",
      "DOI": "10.1021/j100256a058",
      "volume": "89",
      "year": "1985",
      "journal-title": "J. Phys. Chem."
    },
    {
      "key": "2024021017073197300_r34",
      "doi-asserted-by": "crossref",
      "first-page": "11237",
      "DOI": "10.1063/1.473006",
      "volume": "105",
      "year": "1996",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2024021017073197300_r35",
      "doi-asserted-by": "crossref",
      "first-page": "3601",
      "DOI": "10.1021/j100116a026",
      "volume": "97",
      "year": "1993",
      "journal-title": "J. Phys. Chem."
    },
    {
      "key": "2024021017073197300_r36",
      "doi-asserted-by": "crossref",
      "first-page": "1122",
      "DOI": "10.1021/ma60066a012",
      "volume": "11",
      "year": "1978",
      "journal-title": "Macromolecules"
    },
    {
      "key": "2024021017073197300_r37",
      "doi-asserted-by": "crossref",
      "first-page": "1119",
      "DOI": "10.1021/ma60066a011",
      "volume": "11",
      "year": "1978",
      "journal-title": "Macromolecules"
    },
    {
      "key": "2024021017073197300_r38",
      "doi-asserted-by": "crossref",
      "first-page": "1126",
      "DOI": "10.1021/ma60066a013",
      "volume": "11",
      "year": "1978",
      "journal-title": "Macromolecules"
    },
    {
      "key": "2024021017073197300_r39",
      "doi-asserted-by": "crossref",
      "first-page": "1134",
      "DOI": "10.1021/ma60066a014",
      "volume": "11",
      "year": "1978",
      "journal-title": "Macromolecules"
    },
    {
      "key": "2024021017073197300_r40",
      "doi-asserted-by": "crossref",
      "first-page": "558",
      "DOI": "10.1016/0032-3861(82)90095-7",
      "volume": "23",
      "year": "1982",
      "journal-title": "Polymer"
    },
    {
      "key": "2024021017073197300_r41",
      "doi-asserted-by": "crossref",
      "first-page": "1714",
      "DOI": "10.1063/1.1734518",
      "volume": "39",
      "year": "1963",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2024021017073197300_r42",
      "first-page": "1841",
      "volume": "2",
      "year": "1992",
      "journal-title": "J. Phys. I"
    },
    {
      "key": "2024021017073197300_r43",
      "doi-asserted-by": "crossref",
      "first-page": "5817",
      "DOI": "10.1063/1.1290473",
      "volume": "113",
      "year": "2000",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2024021017073197300_r44",
      "doi-asserted-by": "crossref",
      "first-page": "1365",
      "DOI": "10.1103/PhysRevLett.80.1365",
      "volume": "80",
      "year": "1998",
      "journal-title": "Phys. Rev. Lett."
    },
    {
      "key": "2024021017073197300_r45",
      "first-page": "265",
      "volume": "116",
      "year": "2001",
      "journal-title": "Adv. Chem. Phys."
    },
    {
      "key": "2024021017073197300_r46",
      "doi-asserted-by": "crossref",
      "first-page": "1369",
      "DOI": "10.1103/PhysRevLett.80.1369",
      "volume": "80",
      "year": "1998",
      "journal-title": "Phys. Rev. Lett."
    },
    {
      "key": "2024021017073197300_r47",
      "doi-asserted-by": "crossref",
      "first-page": "1241",
      "DOI": "10.1088/0034-4885/55/8/003",
      "volume": "55",
      "year": "1992",
      "journal-title": "Rep. Prog. Phys."
    },
    {
      "key": "2024021017073197300_r48",
      "first-page": "405",
      "volume": "130",
      "year": "1997",
      "journal-title": "Colloids Surf., A"
    },
    {
      "key": "2024021017073197300_r49"
    },
    {
      "key": "2024021017073197300_r50"
    },
    {
      "key": "2024021017073197300_r51",
      "doi-asserted-by": "crossref",
      "first-page": "665",
      "DOI": "10.1021/ma60017a607",
      "volume": "3",
      "year": "1970",
      "journal-title": "Macromolecules"
    },
    {
      "key": "2024021017073197300_r52",
      "doi-asserted-by": "crossref",
      "first-page": "819",
      "DOI": "10.1021/ma60048a025",
      "volume": "8",
      "year": "1975",
      "journal-title": "Macromolecules"
    },
    {
      "key": "2024021017073197300_r53"
    }
  ],
  "container-title": "The Journal of Chemical Physics",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "https://pubs.aip.org/aip/jcp/article-pdf/117/11/5421/19017994/5421_1_online.pdf",
      "content-type": "application/pdf",
      "content-version": "vor",
      "intended-application": "syndication"
    },
    {
      "URL": "https://pubs.aip.org/aip/jcp/article-pdf/117/11/5421/19017994/5421_1_online.pdf",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2024,
        2,
        10
      ]
    ],
    "date-time": "2024-02-10T17:52:06Z",
    "timestamp": 1707587526000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://pubs.aip.org/jcp/article/117/11/5421/184912/Simplified-Onsager-theory-for-isotropic-nematic"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        2002,
        9,
        15
      ]
    ]
  },
  "references-count": 53,
  "journal-issue": {
    "issue": "11",
    "published-print": {
      "date-parts": [
        [
          2002,
          9,
          15
        ]
      ]
    }
  },
  "URL": "http://dx.doi.org/10.1063/1.1499718",
  "relation": {},
  "ISSN": [
    "0021-9606",
    "1089-7690"
  ],
  "subject": [],
  "published-other": {
    "date-parts": [
      [
        2002,
        9,
        15
      ]
    ]
  },
  "published": {
    "date-parts": [
      [
        2002,
        9,
        15
      ]
    ]
  }
}