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Second-order thermodynamic perturbation theory for the inverse patchy colloids
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Second-order thermodynamic perturbation theory for the inverse patchy colloids O. Stepanenko a , T. Urbic b , Y. Kalyuzhnyi a, * a b
Institute for Condensed Matter Physics NASU, Lviv, Ukraine University of Ljubljana, Department of Chemistry and Chemical Technology, Chair of Physical Chemistry, Veˇcna pot 113, Ljubljana SI-1000, Slovenia
A R T I C L E
I N F O
Article history: Received 2 August 2016 Received in revised form 21 September 2016 Accepted 23 September 2016 Available online xxxx Keywords: Inverse patchy colloids Association Thermodynamic perturbation theory
A B S T R A C T In this paper we propose extension of the second-order thermodynamic perturbation theory (TPT2) for the inverse patchy colloids (IPC) with arbitrary number of patches. The theory is used to study thermodynamical properties and liquid-gas phase behavior of the IPC model with one, two and three patches. To validate the accuracy of the TPT2 we compare theoretical predictions against corresponding results obtained by computer simulations. The theory is accurate for the one-patch version of the model at all values of the temperature and density studied and less accurate for two- and three-patch versions at lower temperature and higher density. Theoretical predictions for the critical temperature and density of the two- and three-patch IPC models are relatively accurate, however the overall shape of the theoretical phase diagram appears to be too narrow. No liquid-gas phase coexistence for the one-patch IPC model was found. © 2016 Published by Elsevier B.V.
1. Introduction Inverse patchy colloidal (IPC) model [1] is a coarse-grained version of the model used to describe the properties of charged colloidal particles with nonuniform distribution of the charge on the surface. The model was developed to represent negatively charged colloids with several positively charged star polyelectrolytes adsorbed on its surface [2]. Coarse-graining procedure, which was used in Ref. [1], is based on the application of Debye-Hückel theory. The final potential acting between the two particle consists of the hard-sphere potential with additional short-ranged spherically symmetric soft repulsive potential and short-ranged orientationally dependent potential. The latter potential appears due to the presence of the patches, which are introduced to mimic adsorbed polyelectrolyte stars. Thus, in contrast to the usual patchy colloidal models, in the case of the IPC model interaction between patches is repulsive, attractive interaction is valid between patches and colloidal center. These features of the model is reflected in a very rich and unusual phase behavior (see Refs. [3,4] and references therein), which can be used to generate novel self-assembled materials with desired properties. The properties of the IPC model were studied using computer simulation methods in a number of publications [4–9]. More recently multidensity integral equation approach of Wertheim [10] has been
* Corresponding author.
extended [9] and applied [9,11] to describe different versions of the IPC models theoretically. Comparison of the theoretical and computer simulation results show good performance of the theory developed. However application of the integral equation theory in general requires application of the numerical methods of solution. Although recently an analytical method for the solution of the associative Percus-Yevick approximation has been developed [11], it is restricted to the case of the IPC model with interparticle pair potential represented by the combination of hard-sphere and sticky interactions. In the present paper we propose extension of the second-order thermodynamic perturbation theory (TPT2) of Wertheim [12] for the IPC model. Important advantage of the theories based on the TPT approach for associating fluids is due to their simplicity and flexibility in application and also due to the possibility of using analytical methods of description. We consider here slightly simplified version of the model, which allows us to formulate an analytical version of the theory. To determine the accuracy of the theoretical predictions we generate a set of the computer simulation data for thermodynamics and phase behavior of the model at hand. The paper is organized as follows. In Section 2, we present the model, continue with description of the second-order TPT specialized for the model at hand in Section 3 and in Section 4, we discuss details of the computer simulations. Our numerical results and discussion are presented on Section 5 and the paper is finished with concluding remarks in Section 6.
http://dx.doi.org/10.1016/j.molliq.2016.09.081 0167-7322/© 2016 Published by Elsevier B.V.
Please cite this article as: O. Stepanenko et al., Second-order thermodynamic perturbation theory for the inverse patchy colloids, Journal of Molecular Liquids (2016), http://dx.doi.org/10.1016/j.molliq.2016.09.081
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Here the reference system is represented by the system with the following pair potential
2. The model The model particles are represented as the hard-sphere fluid with additional orientational dependent attractive square-well and repulsive hard-sphere potentials, which appear due to ns patches. Interparticle pair potential U(1, 2) can be written as, U(1, 2) = Uhs (r) +
ns
[Upi c (1, 2) + Ucpi (1, 2)] +
i=1
ns
Upi pj (1, 2) =
0,
(1)
ij=1
for hpi < h0 and r < D + y otherwise, (1)
(2)
(2)
for hpi , hpj < h0 and r < D + y otherwise,
(3)
(1)
where hpi is the angle between the line connecting the centers of the two particles and the line connecting the center of the particle 1 and the center of its patch pi , y is the width of the patch-center and patch-patch potentials and D is the hard-sphere diameter. The width of the patch-center square-well potential is chosen to be narrow enough, so that one-bond per patch restriction is satisfied. At the same time due to the hard-sphere patch-patch repulsion of the similar width y double bonding of the two particles is blocked. In what follows we will consider one-, two- and three-patch versions of the model with symmetric location of the patches, i.e. the angle between the lines connecting the center of the particle and the centers of its patches is equal to 180◦ in the two-patch case and to 120◦ in the three-patch case.
ca =
∂ Dc(0) . ∂ sC−a
(0)
a⊂C(a=0)
ca sC−a −
Dc(0) , V
(4)
where b = 1/kB T, V is the system volume, q is the number density, C denotes the set of all attractive sites (all patches and particle center), a denotes subset of C, s a is the density parameter, which is equal to the sum of the densities of the particles with all possible subset of sites c from the set a bonded, i.e
c⊂a
qc .
(5)
(0)
(8)
where (0)
Dc1s = ns sC−c sC−p I1 ,
(0)
Dc2d =
(0)
Dc(0) = Dc1s + Dc2s + Dc2d ,
Important feature of our IPC model, which to a substantial degree defines its properties, is due to substantial asymmetry in bonding abilities of the patches and particle center. While each patch can be bonded only once, the particle center can bond up to twelve patches, each belonging to different particles. In addition, due to patch-patch repulsion, formation of the double patch-center and center-patch bond between two particles is not possible. To account for these features of the model we will use appropriately modified second order thermodynamic perturbation theory (TPT2) for associating fluid, proposed by Wertheim [12]. According to Wertheim [13] Helmholtz free energy of the system in excess to its reference system value DA = A − Aref can be written as follows:
(7)
Exact expression for Dc(0) contains the infinite sum of diagrams: to proceed one have to adopt certain approximation. We will follow Wertheim [12] and utilize here TPT2, retaining only the graphs with a chain of Fpi c -bonds on up to three points. To account for the blocking effects, which appear due to the patch-patch repulsion, we will omit the diagrams with double bonds between two particles. In addition we will assume that the properties of the reference system can be described using corresponding hard-sphere system. The latter approximation is expected to be sufficiently accurate, since the major contribution due to the patch-patch interaction is already taken into account. For the model at hand we have:
Dc2s = ns sC−c
sa =
(6)
Dc(0) = c(0) −cref , where c(0) is the fundamental sum of all possible irreducible diagrams with field points connected by fref (1, 2) = exp[−bUref (1, 2)] − 1 bonds and Fpi c (1, 2) = [fref (1, 2) + 1]fpi c (1, 2) (0) bonds (where fpi c (1, 2) = exp[−bUpi c (1, 2)] − 1) and cref is the sum of diagrams with the only bond fref (1, 2) [13]. Each field point carries the factor s C −a , where a is a subset of bonded sites. Finally for ca we have [13]:
3. Second-order thermodynamic perturbation theory for inverse patchy colloids
bDA s0 = q ln + V q
Upi pj (1, 2),
ij=1
(1)
4 < 0,
0, ∞,
ns
(0)
Upi pj (1, 2),
where Uhs (r) is hard-sphere potential, Upi c (1, 2) and Ucpj (1, 2) are patch-center and center-patch potentials, respectively and Upi pj (1, 2) is patch-patch potential. Here 1 and 2 stand for the position and orientation of the two particles and the lower indices pi and c denote the patch of the type i and particle center, respectively. For the patch-center and patch-patch potentials we have:
Upi c (1, 2) =
Uref (1, 2) = Uhs (r) +
(0)
1 (ns − 1)sC−c sC−p−p + ns sC−c−p sC−p I2 , 2
1 2 2 n sC−c sC−p J2 , 2 s
(9) (10) (11)
I1 =
d(2)fpc (1, 2)ghs (1, 2),
(12)
d(2)d(3)fcp (1, 2) fcp (1, 3) (ghs (1, 2, 3) − ghs (1, 2)ghs (1, 3)),
(13)
d(2)d(3) fcp (1, 2) fcp (1, 3) ghs (1, 2, 3),
(14)
I2 = J2 =
ghs (1, 2) and ghs (1, 2, 3) are two- and three-particle hard-sphere distribution functions and we have used the symmetry of the model which appears due to equivalence of the patches. Here we omit the lower indices, which denote the type of the patches and use the following notation: C − p − p ≡ C − pi − pj (i = j). Here (0) (0) the terms Dc1s and Dc2s represent contributions to Helmholtz free energy from the diagrams with both center and patches singly bonded. These terms are similar to those, which appear in the stan(0) dard TPT2 of Wertheim [12]. The term Dc2d describes contribution due to doubly bonded center and singly bonded patches. Note that in the framework of the present version of the TPT2 contribution from the diagrams with colloidal center bonded more then twice is neglected. The integrals I1 , I2 , J2 , which appear in the expression for Dc(0) (Eq. (8)), can be calculated using the scheme developed in Marshall et al. [14].
Please cite this article as: O. Stepanenko et al., Second-order thermodynamic perturbation theory for the inverse patchy colloids, Journal of Molecular Liquids (2016), http://dx.doi.org/10.1016/j.molliq.2016.09.081
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I1 = N(1) D,
(15)
J2 = N(2) Dd,
(16)
I2 = J2 − I12 .
(17)
Here D = fas Fyhs (D), fas = e −b4 − 1 - the value of Mayer function (1−cos h0 ) - the part of particle where the potential is non-zero, F = 2 2+g surface which allows bonding, yhs (D) = 2(1−g)2 - the contact value of
3
the pair distribution function in Percus-Yevick approximation, g = p 3 6 qD - the packing fraction. N(1) =
4 3 p rc − D3 , 3
N(2) =
N(1)
2
2
2 + p2 rc3 − D2 rc ,
1
d = (1 + 4k)− 2
(18)
1+
3 √ 1 + 4k , 2
(19)
(20)
where k ≈ 0.2336g + 0.1067g2 . Using expression for Dc(0) and relation (7) we have: cc = ns (ns − 1)sC−c sC−p−p J2 + n2s sC−c−p sC−p J2 + ns sC−p I1 +
Fig. 1. Internal energy U∗ = U/(|4|N) (upper panel), pressure P∗ = PD3 /|4| (middle panel) and chemical potential l ∗ = l /|4| (lower panel) vs density q∗ = qD3 for IPC with ns = 1 at T∗ = 0.3 (red lines and circles) and T∗ = 0.2 (blue lines and triangles). The lines denote results of the theory and symbols stand for computer simulation results. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
n2s 2 s I2 2 C−p (21)
cp = ns sC−c sC−c−p J2 + sC−c I1 + ns sC−c sC−p I2
(22)
ccp = ns sC−c sC−p J2
(23)
2 J2 cpp = sC−c
(24)
Fig. 2. Fraction of free particles x0 (upper panel) and fraction of particles with one bond x1 (lower panel) vs density q∗ for IPC with ns = 1 at T∗ = 0.3 (red lines and circles) and T∗ = 0.2 (blue lines and triangles). The lines denote results of the theory and symbols stand for computer simulation results. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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This result enable us to recast expression for Dc(0) in terms of the functions ca , i.e. Dc(0) ns (ns − 1) = cc sC−c − cpp sC−p−p . V 2
(25)
Now we are in a position to derive the so-called mass action equations, which can be used to calculate the density parameters s a . We recall that the densities qa and the quantities ca are connected by the following relation [13]: qa = q0
cc ,
where P(a) denotes the partitioning of the set of the sites a into subset c. According to the expressions for ca (Eq. (7)) and Dc(0) (Eq. (8)) the only nonzero functions ca are those with a = p, c, pp, cp. Using relation (26), we have:
(26)
P(a)={c} c
cc =
sc −1 s0
(27)
cp =
sp −1 s0
(28)
ccp =
scp sc sp − s0 s02
(29)
cpp =
sp2 spp − 2. s0 s0
(30)
Combining obtained expressions for ca Eqs. (21)–(24) and Eqs. (27)– (30) we obtain the set of nonlinear equations for the density parameters s a . For ns = 1, 2 and 3 these equations are presented in the Appendix explicitly. Solution of these sets of equations has to be carried out numerically. Using Eqs. (4), (25) and (27)–(30) we obtain the following final expression for Helmholtz free energy in terms of the density parameters s a
scp sp sc sp sC−c−p − 1 sC−p + ns − s0 s0 s02
sp2 spp + ns (ns − 1) − 2 sC−p−p . (31) s0 s0
s0 1 bDA = ln + N q q
Fig. 3. The same as in Fig. 1 for IPC with ns = 2 at T∗ = 0.3 (red lines and circles) and T∗ = 0.25 (blue lines and triangles). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
ns
Fig. 4. The same as in Fig. 2 for IPC with ns = 2 at T∗ = 0.3 (red lines and circles) and T∗ = 0.25 (blue lines and triangles). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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All the rest thermodynamical properties can be obtained using the standard relations. For the pressure, chemical potential and internal energy we have:
∂ P = qkB T
A V
∂q
,
(32)
l = A + P /q,
U /N = −T
2
∂
(33) A N
∂T
.
(34)
5
4. Computer simulation To test the theory we performed Monte Carlo simulations in grand canonical ensemble (constant l, V, and T ). The colloids interact with the potential given by Eq. (1). We used periodic boundary conditions and the minimum image convention to mimic an infinite system of particles. Starting configurations were selected at random. In each move we randomly tried to translate or rotate random particle or insert new or remove random particle. In one cycle we on average tried to translate and rotate each particle and to make as many insertions or removals as were averaged number of particles in system. The simulations were allowed to equilibrate for 100,000 cycles and averages were taken for 20 series each consisted for another 100,000 cycles to obtain well converged results. In the system we had from 100 to 1000 particles depending on density of the system. Thermodynamic quantities such as energy were calculated as statistical averages over the course of the simulations [15]. Increasing the number of particles had no significant effect on the simulated quantities. 5. Results and discussion In this section we present and discuss numerical results for thermodynamical properties of three versions of the IPC model, calculated using our TPT2 approach and MC computer simulation method. We consider the models with one, two and three patches with the width of the patch-center square-well potential y = 0.1D and limiting angle h0 = 27◦ . The patches are symmetrically located
Fig. 5. The same as in Figs. 1 and 3 for IPC with ns = 3 at T∗ = 0.42 (red lines and circles) and T∗ = 0.3 (blue lines and triangles). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. The same as in Figs. 2 and 4 for IPC with ns = 3 at T∗ = 0.42 (red lines and circles) and T∗ = 0.3 (blue lines and triangles). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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on the hard-sphere surface of each particle. We calculate internal energy, pressure, chemical potentia and fractions of the free x0 and singly x1 bonded particles at different temperatures and densities. We also calculate the liquid-gas phase diagrams for two- and threepatch versions of the model. In what follows we are using reduce quantities, i.e. q∗ = qD3 , T∗ = kB T/|4|, U∗ = U/(|4|N), P∗ = PD3 /|4| and l ∗ = l /|4|. To determine the accuracy of the theoretical predictions we compare them against corresponding results from computer simulations. This comparison is presented in Figs. 1–7. In Figs. 1 and 2 we show our results for one-patch version of the model (ns = 1) at T∗ = 0.2 and T∗ = 0.3. Here agreement between theory and simulation for thermodynamical properties (Fig. 1) is very good for both values of the temperature and all values of the density. Predictions for the fraction of the free particles x0 are also very accurate (Fig. 2, upper panel); slightly less accurate are theoretical results for the fraction of singly bonded particles x1 (Fig. 2, lower panel). The model with two patches (ns = 2) was studied at T∗ = 0.25 and T∗ = 0.3 (Figs. 4 and 5). Here TPT2 results are accurate for thermodynamics at higher temperature; at lower temperature and higher density theoretical predictions for the energy and pressure (Fig. 3, upper and intermediate panels) U∗ and P∗ are less accurate. At the same time theoretical results for the chemical potential (Fig. 3, lower panel) at low temperature are in a good agreement with corresponding MC simulation results. Similar as in the case of the one-patch model predictions for the fraction of free particles for two-patch model are
accurate at all values of the temperature (and density) and prediction for the fraction of singly bonded particle becomes only qualitatively correct. The accuracy of the theoretical results for the model with three patches (Figs. 5 and 6), which was investigated at T∗ = 0.3 and T∗ = 0.42 is similar to that of the two-patch model, they are reasonably accurate at higher temperature and much less accurate at lower temperature. This decrease of the accuracy can be related to the increase of the number of particles with more than two bonds per center and to the ring formation. Both effects, which are neglected in the present version of the theory, become important with the temperature decrease and/or increase of the number of patches. Finally in Fig. 7 we show our results for the liquid-gas coexistence curve. Only results for two- and three-patch versions of the model are presented here, since for one-patch version of the model we have not been able to observe the liquid-gas phase transition. Comparison of the TPT2 and MC simulation results shows that while theoretical predictions for the critical temperature and density are relatively accurate, the overall shape of the corresponding phase diagram is much too narrow.
6. Conclusions We extended the second order thermodynamic perturbation theory (TPT2) of Wertheim [12] and applied it to study the equilibrium properties of the inverse patchy colloids (IPC) with up to three patches. We also generated a set of new computer simulation data for thermodynamical properties and liquid-gas phase diagram of the IPC model and compare against them corresponding theoretical predictions. The theory appears to be accurate for the one-patch version of the model at all values of the temperature and density studied and less accurate for two- and three-patch versions at lower temperature and higher density. TPT2 predictions for the critical temperature and density are relatively accurate, however the overall shape of the theoretical phase diagram appears to be too narrow.
Acknowledgments O. S. and Y. K. acknowledge financial support of the Ministry of Education and Science of Ukraine in the framework of joint Ukrainian-Austrian research project M/186-2016. T. U. is grateful for the support of the NIH (GM063592) and Slovenian Research Agency (P1 0103-0201, N1-0042) and the National Research, Development and Innovation Office of Hungary (SNN 116198). Appendix A. Equations for the density parameters s a Combining expressions for ca Eqs. (21)–(24) and Eqs. (27)–(30) we obtain the set of nonlinear equations for the density parameters s a . For ns = 1, 2 and 3 we have ⎧ 2 ˜ ⎪ ⎪ ⎨xp xc − x0 + x0 xp xc I2 = 0 2 ˜ x0 − xp + x0 xp I2 + x0 xp ˜I1 + x0 xp xc ˜J2 = 0 ⎪ ⎪ ⎩x − x + x2 x ˜I + x x ˜I + 1 x x2 ˜J = 0, c c 2 0 0 c 1 0 2 0
Fig. 7. The curve of liquid-gas coexistence for temperature T∗ vs density q∗ for IPC with ns = 2 (upper panel) and ns = 3 (lower panel). Blue lines denote results of the theory and red symbols stand for computer simulation results. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
2
(35)
c
⎧ 2 2 ⎪ ⎪ ⎪2x0 xp xcp − 2xp xc + x0 xpp xc − x0 = 0 ⎪ ⎪ ˜ ˜ ˜ ⎪ ⎪ ⎨x0 − xp + 2x0 xp xpp I2 + x0 xpp I1 + 2x0 xpp xcp J2 = 0 2 2 ˜ 2 xp − x0 xpp + x0 xpp I2 = 0 ⎪ ⎪ ⎪ ⎪ x0 − xc + 2x0 xpp xc ˜I2 + 4x0 xp xcp ˜I2 + 2x0 xcp ˜I1 + 2x0 x2cp ˜J2 = 0 ⎪ ⎪ ⎪ ⎩x x − x x + 2x2 x x ˜I = 0, p c
0 cp
0 pp cp 2
(36)
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and ⎧ ⎪ 3x20 xpp xcp − 2x3p xc − x30 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ x0 − xp + 3x0 xpp xppp ˜I2 + x0 xppp ˜I1 + 3x0 xppp xcpp ˜J2 = 0 ⎪ ⎪ ⎪ ⎪ 2 2 ˜ 2 ⎪ ⎪ ⎨xp − x0 xpp + x0 xppp I2 = 0
3x0 xp xpp − 2x3p − x20 xppp = 0 ⎪ ⎪ ⎪ ⎪ x − xc + 6x0 xppp xcp ˜I2 + 9x0 xpp xcpp ˜I2 + 3x0 xcpp ˜I1 + 4.5x0 x2cpp ˜J2 = 0 ⎪ ⎪ 0 ⎪ ⎪ 2 ˜ ⎪ ⎪ ⎪xp xc − x0 xcp + 3x0 xppp xcpp I2 = 0 ⎪ ⎩ 2 2x0 xp xcp − 2xp xc + x0 xpp xc − x20 xcpp = 0, (37) respectively. Here xa =
sa q
7
[4] E. Bianchi, C.N. Likos, G. Kahl, Nano Lett. 14 (2014)3412. [5] E. Bianchi, C.N. Likos, G. Kahl, ACS Nano 7 (2013)4657. [6] E.G. Noya, I. Kolovos, G. Doppelbauer, G. Kahl, E. Bianchi, Soft Matt. 10 (2014)8464. [7] E.G. Noya, E. Bianchi, J.Phys.: Condens.Matter 27 (2015)234103. [8] S. Ferrari, E. Bianchi, G. Kahl, Y.V. Kalyuzhnyi, J.Phys.: Condens.Matter 27 (2015)234104. [9] Y.V. Kalyuzhnyi, E. Bianchi, S. Ferrari, G. Kahl, J.Chem.Phys. 142 (2015)114108. [10] W.S. Wertheim, J.Chem.Phys. 88 (1988)1145. [11] Y.V. Kalyuzhnyi, O.A. Vasilyev, P.T. Cummings, J.Chemi.Phys. 143 (2015)044904. [12] M.S. Wertheim, J.Chem.Phys. 87 (1987)7323. [13] M.S. Wertheim, J.Stat.Phys. 42 (1986)459. [14] Chapman, D. Bennett, G. Walter, J.Chem.Phys. 139 (2013)104904. [15] D. Frenkel, B. Smit, Molecular Simulation: From Algorithms to Applications, Academic Press, New York, 2000.
, ˜I1 = qI1 , ˜J2 = q2 J2 , ˜I2 = q2 I2 .
References [1] E. Bianchi, G. Kahl, C.N. Likos, Soft Matt. 7 (2011)8313. [2] C.N. Likos, R. Blaak, A. Wynveen, J.Phys.: Condens.Matter 20 (2008)494221. [3] E.G. Noya, I. Kolovos, G. Doppelbauer, G. Kahl, E. Bianchi, Soft Matt. 10 (2014)8464.
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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
Second-order thermodynamic perturbation theory for the inverse patchy colloids O. Stepanenko a , T. Urbic b , Y. Kalyuzhnyi a, * a b
Institute for Condensed Matter Physics NASU, Lviv, Ukraine University of Ljubljana, Department of Chemistry and Chemical Technology, Chair of Physical Chemistry, Veˇcna pot 113, Ljubljana SI-1000, Slovenia
A R T I C L E
I N F O
Article history: Received 2 August 2016 Received in revised form 21 September 2016 Accepted 23 September 2016 Available online xxxx Keywords: Inverse patchy colloids Association Thermodynamic perturbation theory
A B S T R A C T In this paper we propose extension of the second-order thermodynamic perturbation theory (TPT2) for the inverse patchy colloids (IPC) with arbitrary number of patches. The theory is used to study thermodynamical properties and liquid-gas phase behavior of the IPC model with one, two and three patches. To validate the accuracy of the TPT2 we compare theoretical predictions against corresponding results obtained by computer simulations. The theory is accurate for the one-patch version of the model at all values of the temperature and density studied and less accurate for two- and three-patch versions at lower temperature and higher density. Theoretical predictions for the critical temperature and density of the two- and three-patch IPC models are relatively accurate, however the overall shape of the theoretical phase diagram appears to be too narrow. No liquid-gas phase coexistence for the one-patch IPC model was found. © 2016 Published by Elsevier B.V.
1. Introduction Inverse patchy colloidal (IPC) model [1] is a coarse-grained version of the model used to describe the properties of charged colloidal particles with nonuniform distribution of the charge on the surface. The model was developed to represent negatively charged colloids with several positively charged star polyelectrolytes adsorbed on its surface [2]. Coarse-graining procedure, which was used in Ref. [1], is based on the application of Debye-Hückel theory. The final potential acting between the two particle consists of the hard-sphere potential with additional short-ranged spherically symmetric soft repulsive potential and short-ranged orientationally dependent potential. The latter potential appears due to the presence of the patches, which are introduced to mimic adsorbed polyelectrolyte stars. Thus, in contrast to the usual patchy colloidal models, in the case of the IPC model interaction between patches is repulsive, attractive interaction is valid between patches and colloidal center. These features of the model is reflected in a very rich and unusual phase behavior (see Refs. [3,4] and references therein), which can be used to generate novel self-assembled materials with desired properties. The properties of the IPC model were studied using computer simulation methods in a number of publications [4–9]. More recently multidensity integral equation approach of Wertheim [10] has been
* Corresponding author.
extended [9] and applied [9,11] to describe different versions of the IPC models theoretically. Comparison of the theoretical and computer simulation results show good performance of the theory developed. However application of the integral equation theory in general requires application of the numerical methods of solution. Although recently an analytical method for the solution of the associative Percus-Yevick approximation has been developed [11], it is restricted to the case of the IPC model with interparticle pair potential represented by the combination of hard-sphere and sticky interactions. In the present paper we propose extension of the second-order thermodynamic perturbation theory (TPT2) of Wertheim [12] for the IPC model. Important advantage of the theories based on the TPT approach for associating fluids is due to their simplicity and flexibility in application and also due to the possibility of using analytical methods of description. We consider here slightly simplified version of the model, which allows us to formulate an analytical version of the theory. To determine the accuracy of the theoretical predictions we generate a set of the computer simulation data for thermodynamics and phase behavior of the model at hand. The paper is organized as follows. In Section 2, we present the model, continue with description of the second-order TPT specialized for the model at hand in Section 3 and in Section 4, we discuss details of the computer simulations. Our numerical results and discussion are presented on Section 5 and the paper is finished with concluding remarks in Section 6.
http://dx.doi.org/10.1016/j.molliq.2016.09.081 0167-7322/© 2016 Published by Elsevier B.V.
Please cite this article as: O. Stepanenko et al., Second-order thermodynamic perturbation theory for the inverse patchy colloids, Journal of Molecular Liquids (2016), http://dx.doi.org/10.1016/j.molliq.2016.09.081
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Here the reference system is represented by the system with the following pair potential
2. The model The model particles are represented as the hard-sphere fluid with additional orientational dependent attractive square-well and repulsive hard-sphere potentials, which appear due to ns patches. Interparticle pair potential U(1, 2) can be written as, U(1, 2) = Uhs (r) +
ns
[Upi c (1, 2) + Ucpi (1, 2)] +
i=1
ns
Upi pj (1, 2) =
0,
(1)
ij=1
for hpi < h0 and r < D + y otherwise, (1)
(2)
(2)
for hpi , hpj < h0 and r < D + y otherwise,
(3)
(1)
where hpi is the angle between the line connecting the centers of the two particles and the line connecting the center of the particle 1 and the center of its patch pi , y is the width of the patch-center and patch-patch potentials and D is the hard-sphere diameter. The width of the patch-center square-well potential is chosen to be narrow enough, so that one-bond per patch restriction is satisfied. At the same time due to the hard-sphere patch-patch repulsion of the similar width y double bonding of the two particles is blocked. In what follows we will consider one-, two- and three-patch versions of the model with symmetric location of the patches, i.e. the angle between the lines connecting the center of the particle and the centers of its patches is equal to 180◦ in the two-patch case and to 120◦ in the three-patch case.
ca =
∂ Dc(0) . ∂ sC−a
(0)
a⊂C(a=0)
ca sC−a −
Dc(0) , V
(4)
where b = 1/kB T, V is the system volume, q is the number density, C denotes the set of all attractive sites (all patches and particle center), a denotes subset of C, s a is the density parameter, which is equal to the sum of the densities of the particles with all possible subset of sites c from the set a bonded, i.e
c⊂a
qc .
(5)
(0)
(8)
where (0)
Dc1s = ns sC−c sC−p I1 ,
(0)
Dc2d =
(0)
Dc(0) = Dc1s + Dc2s + Dc2d ,
Important feature of our IPC model, which to a substantial degree defines its properties, is due to substantial asymmetry in bonding abilities of the patches and particle center. While each patch can be bonded only once, the particle center can bond up to twelve patches, each belonging to different particles. In addition, due to patch-patch repulsion, formation of the double patch-center and center-patch bond between two particles is not possible. To account for these features of the model we will use appropriately modified second order thermodynamic perturbation theory (TPT2) for associating fluid, proposed by Wertheim [12]. According to Wertheim [13] Helmholtz free energy of the system in excess to its reference system value DA = A − Aref can be written as follows:
(7)
Exact expression for Dc(0) contains the infinite sum of diagrams: to proceed one have to adopt certain approximation. We will follow Wertheim [12] and utilize here TPT2, retaining only the graphs with a chain of Fpi c -bonds on up to three points. To account for the blocking effects, which appear due to the patch-patch repulsion, we will omit the diagrams with double bonds between two particles. In addition we will assume that the properties of the reference system can be described using corresponding hard-sphere system. The latter approximation is expected to be sufficiently accurate, since the major contribution due to the patch-patch interaction is already taken into account. For the model at hand we have:
Dc2s = ns sC−c
sa =
(6)
Dc(0) = c(0) −cref , where c(0) is the fundamental sum of all possible irreducible diagrams with field points connected by fref (1, 2) = exp[−bUref (1, 2)] − 1 bonds and Fpi c (1, 2) = [fref (1, 2) + 1]fpi c (1, 2) (0) bonds (where fpi c (1, 2) = exp[−bUpi c (1, 2)] − 1) and cref is the sum of diagrams with the only bond fref (1, 2) [13]. Each field point carries the factor s C −a , where a is a subset of bonded sites. Finally for ca we have [13]:
3. Second-order thermodynamic perturbation theory for inverse patchy colloids
bDA s0 = q ln + V q
Upi pj (1, 2),
ij=1
(1)
4 < 0,
0, ∞,
ns
(0)
Upi pj (1, 2),
where Uhs (r) is hard-sphere potential, Upi c (1, 2) and Ucpj (1, 2) are patch-center and center-patch potentials, respectively and Upi pj (1, 2) is patch-patch potential. Here 1 and 2 stand for the position and orientation of the two particles and the lower indices pi and c denote the patch of the type i and particle center, respectively. For the patch-center and patch-patch potentials we have:
Upi c (1, 2) =
Uref (1, 2) = Uhs (r) +
(0)
1 (ns − 1)sC−c sC−p−p + ns sC−c−p sC−p I2 , 2
1 2 2 n sC−c sC−p J2 , 2 s
(9) (10) (11)
I1 =
d(2)fpc (1, 2)ghs (1, 2),
(12)
d(2)d(3)fcp (1, 2) fcp (1, 3) (ghs (1, 2, 3) − ghs (1, 2)ghs (1, 3)),
(13)
d(2)d(3) fcp (1, 2) fcp (1, 3) ghs (1, 2, 3),
(14)
I2 = J2 =
ghs (1, 2) and ghs (1, 2, 3) are two- and three-particle hard-sphere distribution functions and we have used the symmetry of the model which appears due to equivalence of the patches. Here we omit the lower indices, which denote the type of the patches and use the following notation: C − p − p ≡ C − pi − pj (i = j). Here (0) (0) the terms Dc1s and Dc2s represent contributions to Helmholtz free energy from the diagrams with both center and patches singly bonded. These terms are similar to those, which appear in the stan(0) dard TPT2 of Wertheim [12]. The term Dc2d describes contribution due to doubly bonded center and singly bonded patches. Note that in the framework of the present version of the TPT2 contribution from the diagrams with colloidal center bonded more then twice is neglected. The integrals I1 , I2 , J2 , which appear in the expression for Dc(0) (Eq. (8)), can be calculated using the scheme developed in Marshall et al. [14].
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I1 = N(1) D,
(15)
J2 = N(2) Dd,
(16)
I2 = J2 − I12 .
(17)
Here D = fas Fyhs (D), fas = e −b4 − 1 - the value of Mayer function (1−cos h0 ) - the part of particle where the potential is non-zero, F = 2 2+g surface which allows bonding, yhs (D) = 2(1−g)2 - the contact value of
3
the pair distribution function in Percus-Yevick approximation, g = p 3 6 qD - the packing fraction. N(1) =
4 3 p rc − D3 , 3
N(2) =
N(1)
2
2
2 + p2 rc3 − D2 rc ,
1
d = (1 + 4k)− 2
(18)
1+
3 √ 1 + 4k , 2
(19)
(20)
where k ≈ 0.2336g + 0.1067g2 . Using expression for Dc(0) and relation (7) we have: cc = ns (ns − 1)sC−c sC−p−p J2 + n2s sC−c−p sC−p J2 + ns sC−p I1 +
Fig. 1. Internal energy U∗ = U/(|4|N) (upper panel), pressure P∗ = PD3 /|4| (middle panel) and chemical potential l ∗ = l /|4| (lower panel) vs density q∗ = qD3 for IPC with ns = 1 at T∗ = 0.3 (red lines and circles) and T∗ = 0.2 (blue lines and triangles). The lines denote results of the theory and symbols stand for computer simulation results. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
n2s 2 s I2 2 C−p (21)
cp = ns sC−c sC−c−p J2 + sC−c I1 + ns sC−c sC−p I2
(22)
ccp = ns sC−c sC−p J2
(23)
2 J2 cpp = sC−c
(24)
Fig. 2. Fraction of free particles x0 (upper panel) and fraction of particles with one bond x1 (lower panel) vs density q∗ for IPC with ns = 1 at T∗ = 0.3 (red lines and circles) and T∗ = 0.2 (blue lines and triangles). The lines denote results of the theory and symbols stand for computer simulation results. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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This result enable us to recast expression for Dc(0) in terms of the functions ca , i.e. Dc(0) ns (ns − 1) = cc sC−c − cpp sC−p−p . V 2
(25)
Now we are in a position to derive the so-called mass action equations, which can be used to calculate the density parameters s a . We recall that the densities qa and the quantities ca are connected by the following relation [13]: qa = q0
cc ,
where P(a) denotes the partitioning of the set of the sites a into subset c. According to the expressions for ca (Eq. (7)) and Dc(0) (Eq. (8)) the only nonzero functions ca are those with a = p, c, pp, cp. Using relation (26), we have:
(26)
P(a)={c} c
cc =
sc −1 s0
(27)
cp =
sp −1 s0
(28)
ccp =
scp sc sp − s0 s02
(29)
cpp =
sp2 spp − 2. s0 s0
(30)
Combining obtained expressions for ca Eqs. (21)–(24) and Eqs. (27)– (30) we obtain the set of nonlinear equations for the density parameters s a . For ns = 1, 2 and 3 these equations are presented in the Appendix explicitly. Solution of these sets of equations has to be carried out numerically. Using Eqs. (4), (25) and (27)–(30) we obtain the following final expression for Helmholtz free energy in terms of the density parameters s a
scp sp sc sp sC−c−p − 1 sC−p + ns − s0 s0 s02
sp2 spp + ns (ns − 1) − 2 sC−p−p . (31) s0 s0
s0 1 bDA = ln + N q q
Fig. 3. The same as in Fig. 1 for IPC with ns = 2 at T∗ = 0.3 (red lines and circles) and T∗ = 0.25 (blue lines and triangles). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
ns
Fig. 4. The same as in Fig. 2 for IPC with ns = 2 at T∗ = 0.3 (red lines and circles) and T∗ = 0.25 (blue lines and triangles). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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All the rest thermodynamical properties can be obtained using the standard relations. For the pressure, chemical potential and internal energy we have:
∂ P = qkB T
A V
∂q
,
(32)
l = A + P /q,
U /N = −T
2
∂
(33) A N
∂T
.
(34)
5
4. Computer simulation To test the theory we performed Monte Carlo simulations in grand canonical ensemble (constant l, V, and T ). The colloids interact with the potential given by Eq. (1). We used periodic boundary conditions and the minimum image convention to mimic an infinite system of particles. Starting configurations were selected at random. In each move we randomly tried to translate or rotate random particle or insert new or remove random particle. In one cycle we on average tried to translate and rotate each particle and to make as many insertions or removals as were averaged number of particles in system. The simulations were allowed to equilibrate for 100,000 cycles and averages were taken for 20 series each consisted for another 100,000 cycles to obtain well converged results. In the system we had from 100 to 1000 particles depending on density of the system. Thermodynamic quantities such as energy were calculated as statistical averages over the course of the simulations [15]. Increasing the number of particles had no significant effect on the simulated quantities. 5. Results and discussion In this section we present and discuss numerical results for thermodynamical properties of three versions of the IPC model, calculated using our TPT2 approach and MC computer simulation method. We consider the models with one, two and three patches with the width of the patch-center square-well potential y = 0.1D and limiting angle h0 = 27◦ . The patches are symmetrically located
Fig. 5. The same as in Figs. 1 and 3 for IPC with ns = 3 at T∗ = 0.42 (red lines and circles) and T∗ = 0.3 (blue lines and triangles). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. The same as in Figs. 2 and 4 for IPC with ns = 3 at T∗ = 0.42 (red lines and circles) and T∗ = 0.3 (blue lines and triangles). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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on the hard-sphere surface of each particle. We calculate internal energy, pressure, chemical potentia and fractions of the free x0 and singly x1 bonded particles at different temperatures and densities. We also calculate the liquid-gas phase diagrams for two- and threepatch versions of the model. In what follows we are using reduce quantities, i.e. q∗ = qD3 , T∗ = kB T/|4|, U∗ = U/(|4|N), P∗ = PD3 /|4| and l ∗ = l /|4|. To determine the accuracy of the theoretical predictions we compare them against corresponding results from computer simulations. This comparison is presented in Figs. 1–7. In Figs. 1 and 2 we show our results for one-patch version of the model (ns = 1) at T∗ = 0.2 and T∗ = 0.3. Here agreement between theory and simulation for thermodynamical properties (Fig. 1) is very good for both values of the temperature and all values of the density. Predictions for the fraction of the free particles x0 are also very accurate (Fig. 2, upper panel); slightly less accurate are theoretical results for the fraction of singly bonded particles x1 (Fig. 2, lower panel). The model with two patches (ns = 2) was studied at T∗ = 0.25 and T∗ = 0.3 (Figs. 4 and 5). Here TPT2 results are accurate for thermodynamics at higher temperature; at lower temperature and higher density theoretical predictions for the energy and pressure (Fig. 3, upper and intermediate panels) U∗ and P∗ are less accurate. At the same time theoretical results for the chemical potential (Fig. 3, lower panel) at low temperature are in a good agreement with corresponding MC simulation results. Similar as in the case of the one-patch model predictions for the fraction of free particles for two-patch model are
accurate at all values of the temperature (and density) and prediction for the fraction of singly bonded particle becomes only qualitatively correct. The accuracy of the theoretical results for the model with three patches (Figs. 5 and 6), which was investigated at T∗ = 0.3 and T∗ = 0.42 is similar to that of the two-patch model, they are reasonably accurate at higher temperature and much less accurate at lower temperature. This decrease of the accuracy can be related to the increase of the number of particles with more than two bonds per center and to the ring formation. Both effects, which are neglected in the present version of the theory, become important with the temperature decrease and/or increase of the number of patches. Finally in Fig. 7 we show our results for the liquid-gas coexistence curve. Only results for two- and three-patch versions of the model are presented here, since for one-patch version of the model we have not been able to observe the liquid-gas phase transition. Comparison of the TPT2 and MC simulation results shows that while theoretical predictions for the critical temperature and density are relatively accurate, the overall shape of the corresponding phase diagram is much too narrow.
6. Conclusions We extended the second order thermodynamic perturbation theory (TPT2) of Wertheim [12] and applied it to study the equilibrium properties of the inverse patchy colloids (IPC) with up to three patches. We also generated a set of new computer simulation data for thermodynamical properties and liquid-gas phase diagram of the IPC model and compare against them corresponding theoretical predictions. The theory appears to be accurate for the one-patch version of the model at all values of the temperature and density studied and less accurate for two- and three-patch versions at lower temperature and higher density. TPT2 predictions for the critical temperature and density are relatively accurate, however the overall shape of the theoretical phase diagram appears to be too narrow.
Acknowledgments O. S. and Y. K. acknowledge financial support of the Ministry of Education and Science of Ukraine in the framework of joint Ukrainian-Austrian research project M/186-2016. T. U. is grateful for the support of the NIH (GM063592) and Slovenian Research Agency (P1 0103-0201, N1-0042) and the National Research, Development and Innovation Office of Hungary (SNN 116198). Appendix A. Equations for the density parameters s a Combining expressions for ca Eqs. (21)–(24) and Eqs. (27)–(30) we obtain the set of nonlinear equations for the density parameters s a . For ns = 1, 2 and 3 we have ⎧ 2 ˜ ⎪ ⎪ ⎨xp xc − x0 + x0 xp xc I2 = 0 2 ˜ x0 − xp + x0 xp I2 + x0 xp ˜I1 + x0 xp xc ˜J2 = 0 ⎪ ⎪ ⎩x − x + x2 x ˜I + x x ˜I + 1 x x2 ˜J = 0, c c 2 0 0 c 1 0 2 0
Fig. 7. The curve of liquid-gas coexistence for temperature T∗ vs density q∗ for IPC with ns = 2 (upper panel) and ns = 3 (lower panel). Blue lines denote results of the theory and red symbols stand for computer simulation results. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
2
(35)
c
⎧ 2 2 ⎪ ⎪ ⎪2x0 xp xcp − 2xp xc + x0 xpp xc − x0 = 0 ⎪ ⎪ ˜ ˜ ˜ ⎪ ⎪ ⎨x0 − xp + 2x0 xp xpp I2 + x0 xpp I1 + 2x0 xpp xcp J2 = 0 2 2 ˜ 2 xp − x0 xpp + x0 xpp I2 = 0 ⎪ ⎪ ⎪ ⎪ x0 − xc + 2x0 xpp xc ˜I2 + 4x0 xp xcp ˜I2 + 2x0 xcp ˜I1 + 2x0 x2cp ˜J2 = 0 ⎪ ⎪ ⎪ ⎩x x − x x + 2x2 x x ˜I = 0, p c
0 cp
0 pp cp 2
(36)
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and ⎧ ⎪ 3x20 xpp xcp − 2x3p xc − x30 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ x0 − xp + 3x0 xpp xppp ˜I2 + x0 xppp ˜I1 + 3x0 xppp xcpp ˜J2 = 0 ⎪ ⎪ ⎪ ⎪ 2 2 ˜ 2 ⎪ ⎪ ⎨xp − x0 xpp + x0 xppp I2 = 0
3x0 xp xpp − 2x3p − x20 xppp = 0 ⎪ ⎪ ⎪ ⎪ x − xc + 6x0 xppp xcp ˜I2 + 9x0 xpp xcpp ˜I2 + 3x0 xcpp ˜I1 + 4.5x0 x2cpp ˜J2 = 0 ⎪ ⎪ 0 ⎪ ⎪ 2 ˜ ⎪ ⎪ ⎪xp xc − x0 xcp + 3x0 xppp xcpp I2 = 0 ⎪ ⎩ 2 2x0 xp xcp − 2xp xc + x0 xpp xc − x20 xcpp = 0, (37) respectively. Here xa =
sa q
7
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, ˜I1 = qI1 , ˜J2 = q2 J2 , ˜I2 = q2 I2 .
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Please cite this article as: O. Stepanenko et al., Second-order thermodynamic perturbation theory for the inverse patchy colloids, Journal of Molecular Liquids (2016), http://dx.doi.org/10.1016/j.molliq.2016.09.081