The Boyle temperature, TB, for an n-segment polymer in solution is the temperature where the second osmotic virial coefficient, A2, is zero. This characteristic is of interest for its connection to the polymer condensation critical temperature, particularly for n. TB can be measured experimentally or computed for a given model macromolecule. For the latter, we present and examine two approaches, both based on the Mayer-sampling Monte Carlo (MSMC) method, to calculate Boyle temperatures as a function of model parameters. In one approach, we use MSMC calculations to search for TB, as guided by the evaluation of temperature derivatives of A2. The second approach involves numerical integration of an ordinary differential equation describing how TB varies with a model parameter, starting from a known TB. Unlike general MSMC calculations, these adaptations are appealing because they neither invoke a reference for the calculation nor use special averages needed to avoid bias when computing A2 directly. We demonstrate these methods by computing TB lines for off-lattice linear Lennard-Jones polymers as a function of chain stiffness, considering chains of length n ranging from 2 to 512 monomers. We additionally perform calculations of single-molecule radius of gyration Rg and determine the temperatures Tθ, where linear scaling of Rg2 with n is observed, as if the polymers were long random-walk chains. We find that Tθ and TB seem to differ by 6% in the n limit, which is beyond the statistical uncertainties of our computational methodology. However, we cannot rule out systematic error relating to our extrapolation procedure as being the source of this discrepancy.

1.
P. J.
Flory
,
Principles of Polymer Chemistry
(
Cornell University Press
,
Ithaca
,
1953
).
2.
Y. J.
Sheng
,
A. Z.
Panagiotopoulos
,
S. K.
Kumar
, and
I.
Szleifer
,
Macromolecules
27
,
400
(
1994
).
3.
A.
Rodriguez
,
J.
Freire
, and
A.
Horta
,
J. Phys. Chem.
96
,
3954
(
1992
).
4.
A. Z.
Panagiotopoulos
,
V.
Wong
, and
M. A.
Floriano
,
Macromolecules
31
,
912
(
1998
).
5.
G. L.
Dignon
,
W.
Zheng
,
R. B.
Best
,
Y. C.
Kim
, and
J.
Mittal
,
Proc. Natl. Acad. Sci. U. S. A.
115
,
9929
(
2018
).
6.
G. L.
Dignon
,
W.
Zheng
,
Y. C.
Kim
, and
J.
Mittal
,
ACS Cent. Sci.
5
,
821
(
2019
).
7.
J.
Mayer
and
M.
Mayer
,
Statistical Mechanics
, 1st ed. (
Wiley
,
New York
,
1940
).
8.
J.
Mayer
and
M.
Mayer
,
Statistical Mechanics
, 2nd ed. (
Wiley
,
New York
,
1977
).
9.
W. G.
McMillan
and
J. E.
Mayer
,
J. Chem. Phys.
13
,
276
(
1945
).
10.
D. J.
Ashton
and
N. B.
Wilding
,
J. Chem. Phys.
140
,
244118
(
2014
).
11.
M.
Janssens
and
A.
Bellemans
,
Macromolecules
9
,
303
(
1976
).
12.
13.
I.
Szleifer
,
E. M.
O’Toole
, and
A. Z.
Panagiotopoulos
,
J. Chem. Phys.
97
,
6802
(
1992
).
14.
P.
Grassberger
and
R.
Hegger
,
J. Chem. Phys.
102
,
6881
(
1995
).
15.
S.
Caracciolo
,
B.
Mognetti
, and
A.
Pelissetto
,
J. Chem. Phys.
125
,
094903
(
2006
).
16.
S.
Caracciolo
,
B. M.
Mognetti
, and
A.
Pelissetto
,
J. Chem. Phys.
128
,
065104
(
2008
).
17.
A.
Yethiraj
,
K. G.
Honnell
, and
C. K.
Hall
,
Macromolecules
25
,
3979
(
1992
).
18.
T.
Boublík
,
J. Chem. Phys.
119
,
7512
(
2003
).
19.
C.
Vega
,
C.
McBride
, and
L. G.
MacDowell
,
Phys. Chem. Chem. Phys.
4
,
853
(
2002
).
21.
L. G.
MacDowell
and
C.
Vega
,
J. Chem. Phys.
109
,
5670
(
1998
).
22.
C.
Vega
,
J. M.
Labaig
,
L. G.
MacDowell
, and
E.
Sanz
,
J. Chem. Phys.
113
,
10398
(
2000
).
23.
A.
Shultz
and
P.
Flory
,
J. Am. Chem. Soc.
74
,
4760
(
1952
).
24.
V. I.
Harismiadis
and
I.
Szleifer
,
Mol. Phys.
81
,
851
(
1994
).
25.
J. M.
Wichert
and
C. K.
Hall
,
Macromolecules
27
,
2744
(
1994
).
26.
C.
Vega
and
A.
López Rodríguez
,
J. Chem. Phys.
105
,
4223
(
1996
).
27.
A. M.
Rubio
and
J. J.
Freire
,
Macromolecules
29
,
6946
(
1996
).
28.
A. M.
Rubio
and
J. J.
Freire
,
J. Chem. Phys.
106
,
5638
(
1997
).
29.
C. W.
Yong
,
J. H. R.
Clarke
,
J. J.
Freire
, and
M.
Bishop
,
J. Chem. Phys.
105
,
9666
(
1996
).
30.
Y.
Chiew
and
V.
Sabesan
,
Fluid Phase Equilib.
155
,
75
(
1999
).
31.
I. M.
Withers
,
A. V.
Dobrynin
,
M. L.
Berkowitz
, and
M.
Rubinstein
,
J. Chem. Phys.
118
,
4721
(
2003
).
32.
D.
Ida
and
T.
Yoshizaki
,
Polym. J.
40
,
1074
(
2008
).
33.
A. J.
Schultz
and
D. A.
Kofke
,
J. Chem. Phys.
133
,
104101
(
2010
).
34.
A.
Mohammadi
,
A.
Ramazani Saadatabadi
, and
M.
Khanpour
,
Chem. Phys.
397
,
26
(
2012
).
35.
L.
Liu
,
C.
Duan
, and
R.
Wang
,
Polymer
258
,
125312
(
2022
).
36.
J. K.
Singh
and
D. A.
Kofke
,
Phys. Rev. Lett.
92
,
220601
(
2004
).
37.
E.
Mason
and
T.
Spurling
,
The Virial Equation of State
(
Pergamon Press
,
Oxford
,
1969
).
38.
J.-P.
Hansen
and
I.
McDonald
,
Theory of Simple Liquids
, 4th ed. (
Academic Press
,
London
,
2013
).
39.
K. M.
Benjamin
,
A. J.
Schultz
, and
D. A.
Kofke
,
Ind. Eng. Chem. Res.
45
,
5566
(
2006
).
40.
K. M.
Benjamin
,
J. K.
Singh
,
A. J.
Schultz
, and
D. A.
Kofke
,
J. Phys. Chem. B
111
,
11463
(
2007
).
41.
42.
A.
Bansal
,
A. J.
Schultz
,
J. F.
Douglas
, and
D. A.
Kofke
,
J. Chem. Phys.
157
,
224801
(
2022
).
43.
D. A.
Kofke
,
J. Chem. Phys.
98
,
4149
(
1993
).
45.
46.
D.
Sanchez
, “
Ordinary differential equations and stability theory
,” in
Dover Books on Mathematics
(
Dover Publications
,
2019
).
47.
F.
Vargas-Lara
,
F. W.
Starr
, and
J. F.
Douglas
,
Soft Matter
13
,
8309
(
2017
).
48.
A. J.
Schultz
and
D. A.
Kofke
,
J. Comput. Chem.
36
,
573
(
2015
).
49.
See https://www.etomica.org/ for more information about downloading and using etomica molecular simulation software,
2024
.
50.
N.
Metropolis
,
A. W.
Rosenbluth
,
M. N.
Rosenbluth
,
A. H.
Teller
, and
E.
Teller
,
J. Chem. Phys.
21
,
1087
(
1953
).
51.
D.
Frenkel
and
B.
Smit
,
Understanding Molecular Simulation: From Algorithms to Applications
, 3rd ed. (
Academic Press
,
San Diego
,
2023
).
52.
M. A.
Floriano
,
V.
Firetto
, and
A. Z.
Panagiotopoulos
,
Macromolecules
38
,
2475
(
2005
).
53.
Y.
Nakamura
,
N.
Inoue
,
T.
Norisuye
, and
A.
Teramoto
,
Macromolecules
30
,
631
(
1997
).
54.
A. M.
Rubio
,
J. J.
Freire
,
C. W.
Yong
, and
J. H. R.
Clarke
,
J. Chem. Phys.
111
,
1302
(
1999
).
55.
K. R. S.
Shaul
,
A. J.
Schultz
, and
D. A.
Kofke
,
J. Chem. Phys.
135
,
124101
(
2011
).
You do not currently have access to this content.