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Second virial coefficient of bounded repulsive potentials
Accepted Manuscript Second virial coefficient of bounded repulsive potentials D.M. Heyes PII: DOI: Reference:
S0301-0104(18)30422-1 https://doi.org/10.1016/j.chemphys.2018.07.037 CHEMPH 10101
To appear in:
Chemical Physics
Received Date: Accepted Date:
25 April 2018 22 July 2018
Please cite this article as: D.M. Heyes, Second virial coefficient of bounded repulsive potentials, Chemical Physics (2018), doi: https://doi.org/10.1016/j.chemphys.2018.07.037
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Second virial coefficient of bounded repulsive potentials D.M. Heyes∗ The Physics Department, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom. †
(Dated: July 22, 2018)
Abstract Exact series expansion formulas are derived for the second virial coefficient, SVC, of a wide range of purely repulsive bounded potentials. The focus is on the consequences of the functional form, choice of disposable parameters, and temperature on the SVC characteristics. These include compressed and stretched exponential forms, and bounded inverse power (BIP) potentials, which are an extension of the inverse power system. The BIP have a quite different SVC temperature dependence to the exponential cases, characteristics which could be used to represent polymer based dispersed particles which are not accounted for by a gaussian. A new algebraic form of bounded potential is proposed whose SVC at high temperature tends to the Penetrable Sphere, but at low temperature to the BIP class.
1
I.
INTRODUCTION
There has been a rapid growth in multiphase liquid soft matter physics in the last few decades, driven in part by the diverse range of practical applications for these systems in the consumer products, and biomedical sensing and drug delivery areas.
In order to
gain an understanding of these systems, and for practical computational purposes, it has proved useful to represent the interaction of the dispersed particles by soft effective or coarse grained pair interactions1–3 . These typically have analytic forms which are different to atomic pair potentials, and are often (for polymeric particles) bounded, i .e., finite at the origin. The gaussian potential (or ‘Gaussian Core Model’ GCM as it is usually referred to), whose statistical mechanical properties were first explored by Stillinger in the 1970s4–6 is the archetypal example which has been used extensively to represent the effective potential between flexible linear polymer chains in solution7–11 . There is now a substantial literature on the phase behavior and other properties of the Gaussian and related potential functions12–18 .
In principle, polymeric particles in solution could be made whose effective interaction is still bounded but different from the gaussian by varying the chemical composition along the chain and the polymer molecule architecture. The polymer repeat unit number density may have a different radial dependence from the center of mass compared to an ideal random coil linear polymer, such as a star polymer, in which the radius of gyration is larger than that of the linear polymer containing the same number of statistical segment lengths19 . It is therefore of interest to compare and contrast the thermodynamic behavior of other effective potentials which are also bounded and could be used to represent different generic classes of polymer particle.
2
This work is concerned with comparing some alternative analytic potentials to the GCM which could potentially be used to represent effective interactions of other classes of soft dispersed particles on the mesoscale. The focus is on the second virial coefficient, b2 (or SVC) and its dependence on the temperature and potential parameter(s). This is the first configurational interaction term of a series expansion of the equation of state20 . The study of the SVC of simple molecular systems has a long history, dating back to at least Lennard-Jones,21–23 but its application to bounded potentials is more recent9,24 and much less well explored. One of the bounded potentials considered here is new in the context of soft matter. The SVC characteristics of bounded variants of the popular inverse power (IP) potential25 are also treated here. The interest is in purely repulsive bounded potentials, which would represent polymer particles in a good solvent,26,27 that is, one in which the polymer-solvent molecule attractions dominate.
Bounded potentials have also played an important role in coarse-grained fluid dynamics techniques such as Dissipitave Particle Dynamics (DPD),28 Smooth Particle Hydrodynamics (SPH)29 and various hybrids30 of them. These techniques occupy a place in the spectrum of methods between the Molecular Dynamics and Computational Fluid Dynamics extremes (i .e., atomistic to continuum) which are close or formally in the continuum limit (i .e., for DPD and SPH, respectively). The common feature of all of these schemes is that the (effective) interaction ‘potential’ (or ‘kernel’ in the SPH case) between the ‘particles’ mapping out the Lagrangian grid points is extremely soft and bounded. The local stress tensor used in SPH is a function of the local density, which at a grid site (particle position) is determined by summing the kernel contributions from neighboring particles. There is no unique functional form for the kernel. The local density and therefore local stress tensor at a particular grid point depends on the shape of the kernel. The results of this work could also therefore ultimately find 3
application in optimising SPH and related techniques.
II.
THEORY
Consider a radially symmetric interaction potential, φ(r), where r is the distance between the centers of the two particles. The potential can be expressed in terms of a characteristic energy and length, ǫ and σ, respectively. The temperature is in units of ǫ/kB , where kB is Boltzmann’s constant. To keep the notation as compact as possible, these constants are set to unity here. The second virial constant (SVC) in 3D is then given by the formula,23 b2 = −2π
Z
∞ 0
(e−φ(r)/T − 1)r 2 dr
(1)
Analytic formulas for the SVC from different potentials substituted in Eq. (1) are given below.
Penetrable Sphere The prototype example of a purely repulsive particle which has a bounded interaction is the so-called Penetrable Sphere (PS)31–33 φ(r) = 1,
r ≤ 1;
φ(r) = 0,
r > 1,
(2)
which could be considered to be a variant of the hard sphere potential, and gives from Eq. (1), b2 =
2π [1 − exp(−1/T )]. 3
(3)
The SVC ranges between the hard sphere value (i .e., 2π/3) for T << 1, to the ideal gas value (i .e., 0) when T >> 0. The phase behavior of this potential has been explored a number of times, in part because it can be used to represent the generic behavior of soft colloids and micelles. It also acts as the limiting form of a number of analytically continuous bounded 4
potentials in certain parameter limits, as will be demonstrated below.
Series expansion for the SVC For an arbitrary continuous potential, φ(r), by expanding the exponential in Eq. (1), b2 = −2π = −2π = −2π
Z
∞
r 2 dr 0
X (−1)k
k=1 X (−1)k Z ∞
k!T k
k=1
X (−1)k k!T k
k=1
k!T k
[φ(r)]k
r 2 dr[φ(r)]k
0
Ik ,
(4)
where, Ik =
Z
∞
r 2 dr[φ(r)]k .
(5)
0
For each potential the expression for SVC therefore reduces to deriving a formula for Ik to be inserted in Eq. (4). The series can be shown to converge for any T < 1 (as T ≥ 1 is not a problem as may be seen by inspection) by the ratio test rule because Ik is weakly dependent on k for large k, and convergence is dominated by the k! term The number of terms, k, in the series required for convergence to the exact answer (i .e., k!T k ∼ 1) follows k ∼ 1/T using Stirling’s approximation for k! in the large k limit.
Exponential potentials If m = 1, 2, 3, · · · , the exponential potential is φ(r) = exp(−r
2m
3 k −3/2m Γ( ), Ik = 2m 2m
),
(6)
where Γ(x) is the Gamma function of x. The term, Gaussian Core Model or ‘GCM’ is usually reserved for the m = 1 example, and the acronym ‘GCM-2m’ or the term, ‘Exponential’ is used here for the more general exponential potential form given in Eq. (6). The SVC expression for 5
m = 1 case is already in the literature9 1 π 3/2 X 1 (−1)k , b2 = − 3/2 2 k!k T k
(7)
k=1
and the more general expression for arbitrary m is, b2 = −
1 π 3 X 1 (−1)k Γ( ) . m 2m k=1 k!k 3/2m T k
(8)
The SVC of the Eq. (6) potential converges towards that of the PS potential in the m → ∞ limit. For any value of m the SVC eventually diverges from the PS limit at a low enough T because the effective hard sphere diameter of this type of particle tends to infinity as T → 0 (see the discussion at the end of this section).
As mentioned in the Introduction, the GCM has been used to represent polymers in solution. One unrealistic feature of the GCM potential when used in this context is that it is finite in the infinite r limit, which is inconsistent with the fact that any real polymer molecule has a maximum stretched out length. Therefore the actual effective potential should go to zero after several radii of gyration. This could be represented in a model potential by one that goes exactly to zero at a large distance, r0 , which prohibits a purely exponential form. Alternatively the GCM-2m potential can just be truncated at r = r0 , which is more straightforward to represent analytically. This truncated GCM-2m or ‘TGCM-2m’ potential is,34 φ(r) = exp(−r 2m ), Ik =
r ≤ r0 ;
k −3/2m 3 γ( , kr02m ) 2m 2m
φ(r) = 0,
r > r0 (9)
where γ(x, y) is the lower incomplete Gamma function, which tends to the Gamma function, Γ(x), in the y → ∞ limit (see Sect. 6.5 in Ref.35 ). The definition of Ik in Eq. (9) substituted in Eq. (4) gives the SVC of the TGCM-2m potential, bt . The contribution to the SVC from the 6
GCM-2m potential in the distance range, r0 → ∞ is therefore expressed just in the Ik term as34 k −3/2m δIk = 2m
3 3 Γ( ) − γ( , kr02m ) 2m 2m
k −3/2m 3 = Γ( , kr02m ) 2m 2m
(10)
where Γ(x, y) is the upper incomplete Gamma function, which tends to zero as y → 0. Let δb2 be the quantity when δIk of Eq. (10) is substituted in Eq. (4) instead of Ik . Therfore, b2 = bt + δb2 .
(11)
If we define T0 = exp(−r02m ) then for T << T0 , the SVC, b2 → 2πr03 /3, where r0 acts as an effective hard sphere diameter. Based on a Boltzmann factor criterion, an effective hard sphere diameter, σHS , of the GCM potential4,6,12 has been derived s T −1 , σHS = ln ln 2
(12)
which has been used to establish the low temperature part of the phase diagram of the GCM6 .
Algebraic Hat, AH A new potential in the context of soft matter, which is referred to here as the Algebraic Hat (AH), is 1 , φ(r) = a + rn
1 [3/n]−k Γ(k − n3 )Γ( n3 ) Ik = a , n Γ(k)
n>3
(13)
where a ≥ 034 . If a = 1 the SVC of this potential converges to that of the Penetrable Sphere potential as n → ∞. The generic form of Eq. (13) with n = 2 has been used as a single particle potential for metal clusters36 but not as an effective pair potential for mesoscopic particles in solution.
Bounded Inverse Power, BIP 7
The Inverse Power (IP) potential, φ(r) = 1/r n , where n is a positive constant (usually taken to be an integer), has found considerable use as a reference fluid in theories of the liquid state, second only to the hard sphere fluid, in part because of its simple density-temperature scaling behavior25 . This feature of the IP system has also been found in the condensed phase properties of other, more complex, model potential systems, a trend which is called ‘isomorphism’, and those liquids that obey this behavior are referred to as Roskilde Simple fluids37,38 . The bounded extension of this potential, called here the Bounded Inverse Power, BIP, potential is, φ(r) =
1 aq + r q
n/q
;
n > 3,
(14)
where a is a positive constant. The potential defined in Eq. (14) may also be referred to as the ‘Pq’ potential. The parameter, a, in Eq. (14) acts to soften the IP potential by diminishing the relative importance of the pair separation component when r is small compared to a. For small a the potential could be used to model polymer particles with strong excluded volume interactions, which discourage (but do not entirely eliminate) interpenetration. For larger a and relatively small n (but greater than 3) the potential is long ranged, being similar to the Kac potential (but without its normalisation factor)39 .
Two special cases of q in Eq. (14) are considered. The first is when q = 1, called P1, φ(r) =
1 a+r
n
;
n > 3.
(15)
The SVC of this potential is, b2 = 2π
2 X
q(p)b2,p ,
(16)
p=0
where q(0) = a2 , q(1) = −2a and q(2) = 1, and b2,p
1 T −(1+p)/n (1 + p) 1 a1+p [exp(− n ) − 1] + γ 1− , n . = (1 + p) Ta (1 + p) n Ta 8
(17)
The other case of Eq. (14) considered here is when q = 2, which is a potential referred to as, P2, φ(r) =
1 2 a + r2
n/2
;
n > 3,
(18)
which is X h(k) 1 (−3 + 2k) 1 3 2k (−3+2k)/n a [exp(− n ) − 1] + a T γ(1 + , n) . b2 = 2π 3 − 2k T a n Ta k=0
(19)
The summation in k involves h(k), h(k) =
(2k − 3) h(k − 1), 2k
(20)
for k ≥ 1, and h(0) = 1. The above formulas for the SVC of the P1 and P2 potentials are each the first term of a high temperature expansion of the corresponding bounded version of the Mie potential24 . The radial distribution functions generated by integral equation, mean field and molecular dynamics simulations of the P2 potential were explored in Ref.40 , but this is the first work where the SVCs of these two examples of a BIP potential have been investigated specifically in their own right.
Limiting forms of the SVC at high and low temperature are also of interest as they offer the prospect of simplifying analytic statistical mechanical treatments of mesoscopic systems. The high temperature (HT) limit SVC from Eq. (1) is 2π b2 = T
Z
∞
π r φ(r)dr − 2 T 2
0
= b2,1 − b2,2 · · ·
Z
0
∞ 2 2
r φ (r)dr + O
1 T3
, (21)
where b2,1 and b2,2 on the right hand side of Eq. (21), defined as positive quantities for these purely repulsive potentials, are the first two terms of a high temperature expansion of Eq. (1).
9
Although the analytic formula for the low temperature dependence of the SVC will depend on each potential form, following the previous literature, an effective hard sphere diameter, σHS can also be defined by solving α = exp(−φ(σHS )/T );
b2 =
2π 3 σ , 3 HS
(22)
for arbitrary potential. The value of α = 1/2 has been chosen in previous publications4,6,12 on the GCM, in which case Eq. (22) reduces to Eq. (12).
The SVC series expansions were evaluated using a code written in Fortran 90. The series were deemed to have converged when the difference between consecutive series terms was less than 10−10 . The converged values from the series expansions were confirmed using direct evaluation of the SVC employing the formal definition, given in Eq. (1).
Simpsons rule
numerical integration was used up to a distance where the potential energy at the truncation distance was 10−6 . The distance increment in the numerical integration was 0.001/(2n) for the exponential potentials and 0.001/n for the BIP and AH potentials. The calculations typically took less than a minute on a laptop to complete the temperature ranges covered in the figures.
10
III.
RESULTS AND DISCUSSION
Figure 1 compares examples of the GCM-2m potential of Eq. (6) for different values of m. With increasing exponent power, 2m, the potential converges towards the PS potential. Note the isosbestic point at r = 1, where φ(1) = e−1 in each case. In the general expression for Ik in Eq. (6), the variable m is not restricted to integer values. For m > 1/2 the potential is known as a ‘compressed’ exponential (CE), m = 1/2 is the exponential, and for m < 1/2 the potential is referred to as a ‘stretched’ exponential, an example of which is also given on Fig. 1. The stretched exponential (SE) does not appear to have been considered before in the context of bounded effective potentials for polymers in solution. For example, the stretched exponential could be used to represent a star polymer-like system12 which is still bounded at the origin. Figure 1 indicates that the large values of the SE potential are concentrated further from the origin and it has a slower rate of decay at large r than the CE cases. This leads to larger values of the SVC at a given temperature for smaller m value. The precise value of m could be adjusted to reproduce the effective interaction between polymer molecules of a specific architecture. The boundary of 2m = 1 between stretched and compressed exponentials marks also a transition between two different classes of SVC behavior. For m → 0 the potential tends to e−1 for a wider r range, and consequently the SVC tends to ∞ due the increasingly slow decay of φ(r), which is weighted by r 2 in the formula for the SVC given in Eq. (1). In contrast, as m → ∞ the SVC tends to the PS limit. Figure 2 provides a similar comparison for the AH potential with a = 1, which shows similar trends to the GCM-2m case, but has an isosbestic point at φ = 1/2. The significance is that GCM-2m type of behavior can be represented by an algebraic rather than exponential analytic form.
Figure 3 presents the P1 potential of Eq. (15), taking a = 1 for various values of n. It has a
11
cusped appearance near the origin (rather like the SE), which suggests that it might be useful in modelling star-like particles41 . The corresponding plots of the P2 potential of Eq. (18) are given in Fig. 4. The P2 potential has a point of inflection and therefore a maximum repulsive √ force at r = a/ 1 + n, unlike P1 which does not have a point of inflection. It also has a Maclaurin expansion which is even in r, unlike that of P1.
Figure 5 gives the second virial coefficients as a function of temperature for some examples of the GCM-2m type of potential for several m. The data are plotted in normalised form, b2 /b0 , where b0 = 2π/3 is the second virial coefficient of the hard sphere potential with the same unit of distance. The figure shows that at high temperature (T >> 1) the SVC all converge towards the PS model curve, and that for n = 100 the SVC is hardly distinguishable from the PS value even for T << 1. However, only the PS second virial coefficient is equal to b0 in the T → 0 limit. For each value of m there is always a low enough temperature where the SVC will diverge from the PS. This qualitative difference in behavior arises because the GCM-2m is a continuous potential, and the PS form is discontinuous. The effective hard sphere diameter of the GCM-2m potential diverges as T → 0.
Figure 6 shows b2 /b0 for the AH potential of Eq. (13) as a function of T for several n values, with a = 1. Again for high temperature (T > 1) the SVC curves converge to that of PS potential, and for T < 1 the function progressively converges towards the flat PS limit with increasingly n for not too low temperatures. There is the same qualitative difference with the PS potential as exhibited by the GCM-2m potentials in this limit.
Figure 7 presents b2 /b0 for the P1 potential of Eq. (15), mainly focusing on n = 12 . The SVC increases at all T as a decreases, and shows no evidence of approaching the PS limit at any 12
temperature, which is different to the AH case for the same n. The BIP potential tends to a−n , rather than a−1 (the AH case) and hence the greater SVC rate of divergence as a → 0 compared to the AH example. Figure 8 gives the corresponding quantities for the P2 potential of Eq. (18), which exhibits the same qualitative trends. The curves for the different a values using n = 12 are slightly closer together than those of P1, and they are shifted systematically to higher T .
Figure 9 explores the effectiveness of the high temperature limit expansion of Eq. (21) in representing the SVC, for the GCM-2m potential with 2m values ranging from 1/2 to 2 (the gaussian case). The figure, which presents the b2,1 and some b2,2 data, demonstrates that the first expansion term of the SVC or b2,1 represents well the SVC for temperatures larger than ca. 1, which extends to lower temperature as m decreases. In fact for the stretched exponential (2m = 1/2) the b2,1 term agrees well with b2 down to at least T ≃ 0.01, which is more than adequate to represent polymers in solution.
Figures 10 reveals in another way the extent to which the high temperature limit expansion of Eq. (21) i .e., b2,1 , can be used to replace the exact b2 value. The figure shows the m dependence of b2,1 /b2 for three temperatures, 0.1, 1 and 10. The quantity, b2,1 reproduces b2 well at a temperature of 10 for the whole m range shown, and also for the stretched exponential limit in the vicinity of m ≃ 1/4 (which is consistent with the trend seen in Fig. 9). For T = 1 for all m and T = 0.1 for m > 0.5 the first order high temperature expansion does not represent the SVC well.
Figures 11 and 12 show the corresponding SVC expansions for the AH, and P1 and P2 potentials, respectively. The trends are very similar to those of the GCM-2m potentials except for these cases there is no value of n where the first term in the high temperature expansion 13
represents the SVC in the vicinity of T ∼ 1 and lower temperature (note that n > 3 is a requirement for a finite SVC here). For high enough temperature, T >> 1, the AH, P1 and P2 potentials give b2,1 /b2 ratios of about 1, indicating that the first term in the High Temperature expansion is a good approximation of the SVC, as would be expected by inspection of Eq. (1).
Figure 13 compares the SVC of several cases of these potentials (symbols) with the prediction based on an effective hard sphere approximation from Eq. (22), which reveals that the HS approximation reproduces the exact results very well for T < 1. A value of α = 0.55 fits the GCM-4 data slightly better than the usual value chosen of 1/2 found in the literature, which in the GCM-4 case systematically slightly underestimates the exact SVC values.
A challenge is to distinguish the behavior of real polymers of various architectures in terms of the SVC of these different model potential systems. In polymer solution theory, the osmotic pressure, Π is expressed as a virial expansion42 MΠ = Z = 1 + MA2 c + MA3 c2 + · · · , RT c
(23)
where M is the molar mass, R is the Gas constant, c is the concentration in real units (e.g., in kg dm−3 ), and where A2 and A3 are the symbols usually used for the second and third virial coefficients, respectively, in the polymer solution literature. Equation (23) can be written in the form43 , Z = 1 + b2 ρ + · · · where A2 = Nav b2 /M 2 , Nav is Avogadro’s number and ρ is again the number of particles per unit volume. A so-called ‘interpenetration factor’, Ψ19 can usefully be defined which is sensitive to polymer architecture, Ψ=
2A2 M 2 , Nav (4π < Rg2 >)3/2
(24)
where < Rg2 > is the mean square radius of gyration (RoG). The factor Ψ ≈ 1 is typically several times larger for a star polymer than a linear chain with the same number of statistical 14
segment lengths, increasing with the number of arms19 . For high molar mass polymers, Ψ, in good solvents ranges from ca. 0.26 for linear polymers, and 0.6 to 1.1 for f = 4 to 18-arm star polymers,44,45 with a predicted upper limit of 2.13 for f → ∞,46 which is close to the hard-sphere limit.
One of the current problems in exploiting the above excluded volume theory developed in this work is that even in the GCM case when used as a model for linear polymers in a good solvent, there is a coefficient in the prefactor of the exponential term and in its argument, which are not known exactly and are functions of M 47 . Nevertheless, taking into account O(1) numerical constants, Ψ should be approximately equal to the ratio b2 /b0 in the region of T ≃ 1, a typical polymer solution reduced temperature,6,9 . It may be seen in Figs. 5-8 that b2 /b0 ∼ 1 at T ≃ 1. Figure 5, for example, for the GCM potential gives at T = 1 the value 1.5 for b2 /b0 , and this ratio increases dramatically as n decreases. For the AH potential the value is ca. 0.75 for the wide range of n values shown on Fig. 6. For the n = 12 P1 potential, the ratio b2 /b0 is seen in Fig. 7 to be, 1.0, 0.2 and 0.0 for a = 0.1, 0.5 and 1.0. Figure 8 shows that for the n = 12 P2 potential, the ratio is 1.2, 0.85 and 0.15 for a = 0.1, 0.5 and 1.0. By comparison with experimental Ψ values the ratio b2 /b0 could therefore in principle be used to determine the effective potential analytic form which matches well a particular polymeric system.
Although not relevant for polymer solutions (where T ∼ 1/2 to 1) some aspects of the low temperature fluid-solid aspects of the phase diagram of these systems (where T < 0.01,12 ) can be established from the Fourier Transform of the potential, which is discussed briefly here. The exponential system (m = 1 in Eq. (6)) undergoes reentrant melting (RM) at high density but the m ≥ 2 cases form multiple occupancy cystals (MOC, see ref.48 and the references quoted therein), and therefore no reentrant melting. This is determined on the basis of whether the 15
˜ is positive for all wavevectors (which gives RM) or crosses Fourier transform of the potential, φ, ˜ zero and goes negative at least once (which gives the MOC). For m equal to 1 the function, φ, ˜ has a minimum of −0.127 at k = 5.6, and is positive for all wavevectors k, and for m = 2, φ, for m = 3 the minimum is −0.200 at k = 5.7, both indicating MOC behavior at high density and low enough temperature.
The q = 2 example of the BIP potential has an analytic solution for φ˜ which is positive for all k at least up to n = 12 (see Eqs. (9-15e) in Ref.40 for the relevant theory). Although analytic ˜ solutions for φ(k) for the AH potential can be derived, they are rather complicated expressions in general, involving hypergeometric functions, and establishing whether they are negative for any k value is not trivial for the entire range of parameters. Some indication of the trends in the phase behavior, in regard to the RM vs. MOC issue can be explored by determining ˜ φ(k) by numerical integration. Applying this method to the BIP potential with n = 12, the ˜ q = 1 and 2 cases have φ(k) > 0 for all k which is consistent with RM behavior, whereas for q = 3 and 4 there are shallow negative regions for ca. k > 6.0, which suggest MOC. With increasing q in Eq. (14) the potentials exhibit a flatter top region in the vicinity of r ∼ 0 which intuitively one would associate with the conditions that would give rise to a MOC (i .e., a low energy penalty for a multiply occupied lattice site) rather than RM at a sufficiently high density.
For the AH potential and taking a = 1, increasing n produces a deeper negative region in ˜ φ(k). For example for n = 6 and 12 the minima are −0.134 and −0.272 at k = 5.3 and 5.5, respectively, which indicates MOC. With increasing n the AH potential for r < 1 progressively converges towards the PS potential, which takes on the appearance of the GCM when m is large. For n = 2 the behavior is inconclusive, and for n = 3 there is a weak negative region at
16
large k greater than about 5. Just as for the GCM and extensions, the preliminary evidence is that for the AH and Pq potentials there is a transition from RM to MOC behavior as the potential becomes flatter near the origin.
17
IV.
CONCLUSIONS
Series expansion formulas for the second virial coefficient, SVC, of various purely repulsive bounded potentials have been derived and compared here.
Several specific cases of an
exponential pair potential of the form, exp[−r 2m ], where m is a positive variable, were considered. This includes an example of a stretched exponential which occurs when m < 1/2. The stretched exponential is cusp shaped for small interparticle separations near the potential origin but decays slower than a gaussian at larger separations. This potential could be used to model a polymer in solution where the monomer density is concentrated further from the center of mass than a polymer exhibiting random coil or gaussian potential behavior. Examples include star polymers49 , dendrimers43,50,51 and comb polymers52 . In contrast, a more ‘compressed’ exponential than the gaussian might be expected to be a better match for a ring polymer53 . The first term in the high temperature expansion of the SVC of the stretched exponential is accurate for effective temperatures at and below that typical of polymers in solution (T ∼ 1 in the units used in this work). The SVC of a newly proposed, Algebraic Hat (AH) potential behaves like the penetrable sphere (PS) potential case at high temperature but like the bounded inverse power (BIP) potential at low temperature (i .e., T < 1). In addition to architectural dependencies, the effective interactions between two polymer molecules will also depend on the molar mass, as do the usual polymer molecular quantitive descriptors such as the radius of gyration, second virial coefficient and the intrinsic viscosity,49 .
One physically unrealistic feature of all the continuous bounded potentials considered here is that they are finite at infinite pair separation, which would require the molar mass of the polymer molecule to be infinite. This has a major effect on the T → 0 part of the phase diagram as the effective hard sphere diameter and hence the SVC (and presumably the higher
18
virial coefficients) diverges in that limit as a direct result of this feature. The physical state of all the model systems consided here (apart from the Penetrable Sphere case) in that limit is not well defined and physically unrealistic, which could have consequences for statistical mechanics treatments of these systems, e.g., in the context of thermodynamic integration. A truncated version of the gaussian (and more general exponential type) potential developed here is shown to remove this problem, and be a better representation of a polymer molecule in a good solvent. This in no way diminishes the significance of the GCM in polymer solution theory but points out that caution should be used when it is employed to represent these experimental systems at low temperature.
The BIP potential could represent different types of particle by varying the parameters, q, a and n. These range from small molecules forming a single component liquid (small a, and large n), then small solvent dispersed particles such as micelles,54 and microgels,55 through to large soft highly interpenetrable polymers (large a and variable n). The experimental virial coefficient of a polymer in solution scales approximately as the excluded volume of the molecule as determined by the radius of gyration, multiplied by what we call b2 10 The excluded volume of the molecule scales as the molar mass, M, as ∼ M 1.8 for a linear polymer8 whereas for a stiff rod it scales approximately as ∼ M 2 ,56 which is not too different. The Gaussian chain and stiff rod are at the two extremes of the topology of linear polymers in general. In defining the second virial coefficient this volume is multiplied by a second term, b2 , which depends on φ(r), representing implicitly the radial distribution of mass within the particle. It is this term that has been focused on in this work, as it is a key quantity in linking more precisely experimental and model data, but has not been explored systematically until the present work.
This study focuses on polymers in good solvents, where attractive interactions between the 19
chains can be neglected to a good approximation, which is why the effective center of mass potentials compared are purely repulsive. These potentials are also bounded (i .e., finite at the origin) which is appropriate for linear or branched chains where the chemical architecture varies the mass distribution with respect to the center of mass. For polymers in marginally poor solvents, a bounded version of the Lennard-Jones or more general Mie potentials could be used23,24 . Electrostatic interactions such as in polyelectrolytes and where ionic surfactants are present could also be introduced in the present treatment, although depending on the composition the effective interactions may cease to be of the bounded form.
To summarise, bounded potentials have been used in various aspects of modelling at a coarse grained level the properties and behavior of liquids on long length and time scales. The chemistry of the system and physical processes of interest are introduced by varying the analytic form of the potential. The present work makes a comparison of a range of bounded potentials using the fundamental thermodynamic quantity, the second virial coefficient, which can also be measured experimentally for polymers in solution. Incidently, the results and implications of this study of many different bounded potentials may go beyond polymeric systems, and could, for example, be used to fine tune the kernel (and therefore the modelled flow characteristics) produced by Smooth Particle Hydrodynamics and related simulation techniques.
Acknowledgments The author would like to thank Dr.
T. Crane (Department of Physics, Royal Holloway,
University of London, UK) for helpful software support.
20
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Figure Captions Figure 1: The exponential potential defined in Eq. (6). The 2m values are given on the figure. Note the figure includes data for the exponential case (‘1’) and an example of a stretched exponential, φ(r) = exp(−r 1/2 ) (‘1/2’).
Figure 2: The AH potential defined in Eq. (13), where a = 1. The potential exponent, n, values are given on the figure.
Figure 3:
The P1 potential defined in Eq. (15), with a = 1. The n values are given on the
figure.
Figure 4:
The P2 potential defined in Eq. (18), with a = 1. The n values are given on the
figure.
Figure 5: The second virial coefficients of the gaussian potentials defined in Eq. (6). The SVC of the hard sphere, b0 = 2π/3 is used as a normalising factor. The 2m values are given on the figure. The SVC of the PS potential is also given on the figure as ∗ symbols.
Figure 6: The second virial coefficients of the AH potential defined in Eq. (13), where a = 1. The n values are given on the figure. Otherwise the details are as for Fig. 5.
Figure 7:
The SVC of the P1 potential defined in Eq. (15). The key gives [a n] values for
each curve.
Figure 8: As for Fig. 7 except that the P2 potential defined in Eq. (18) is considered. 26
Figure 9: The second virial coefficients of the exponential potentials as a function temperature are shown as lines. The open square (2m = 1/2) and filled-in circles (2m = 2) are the b2,1 or first expansion term of Eq. (21). The triangles are (as indicated on the figure annotation) the second or b2,2 expansion term (which is a positive quantity as defined in Eq. (21)). The 2m values are given on the figure.
Figure 10: The ratio, b2,1 /b2 of the exponential or GCM-2m potential for three temperatures, 0.1, 1 and 10, which are indicated on the figure.
Figure 11: As for Fig. 10 except the AH potential defined in Eq. (13) is considered, with a = 1.
Figure 12: As for Fig. 10 except the SVC high temperature expansion of the P1 and P2 potentials are investigated, with a = 1. The open and filled in symbols are for P1 and P2, respectively.
Figure 13: The exact SVC of several potential cases given as symbols, compared with the effective hard sphere approximation based on Eq. (22) given as solid lines. The penetrable sphere solution (PS) is also given on the figure.
27
S0301-0104(18)30422-1 https://doi.org/10.1016/j.chemphys.2018.07.037 CHEMPH 10101
To appear in:
Chemical Physics
Received Date: Accepted Date:
25 April 2018 22 July 2018
Please cite this article as: D.M. Heyes, Second virial coefficient of bounded repulsive potentials, Chemical Physics (2018), doi: https://doi.org/10.1016/j.chemphys.2018.07.037
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Second virial coefficient of bounded repulsive potentials D.M. Heyes∗ The Physics Department, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom. †
(Dated: July 22, 2018)
Abstract Exact series expansion formulas are derived for the second virial coefficient, SVC, of a wide range of purely repulsive bounded potentials. The focus is on the consequences of the functional form, choice of disposable parameters, and temperature on the SVC characteristics. These include compressed and stretched exponential forms, and bounded inverse power (BIP) potentials, which are an extension of the inverse power system. The BIP have a quite different SVC temperature dependence to the exponential cases, characteristics which could be used to represent polymer based dispersed particles which are not accounted for by a gaussian. A new algebraic form of bounded potential is proposed whose SVC at high temperature tends to the Penetrable Sphere, but at low temperature to the BIP class.
1
I.
INTRODUCTION
There has been a rapid growth in multiphase liquid soft matter physics in the last few decades, driven in part by the diverse range of practical applications for these systems in the consumer products, and biomedical sensing and drug delivery areas.
In order to
gain an understanding of these systems, and for practical computational purposes, it has proved useful to represent the interaction of the dispersed particles by soft effective or coarse grained pair interactions1–3 . These typically have analytic forms which are different to atomic pair potentials, and are often (for polymeric particles) bounded, i .e., finite at the origin. The gaussian potential (or ‘Gaussian Core Model’ GCM as it is usually referred to), whose statistical mechanical properties were first explored by Stillinger in the 1970s4–6 is the archetypal example which has been used extensively to represent the effective potential between flexible linear polymer chains in solution7–11 . There is now a substantial literature on the phase behavior and other properties of the Gaussian and related potential functions12–18 .
In principle, polymeric particles in solution could be made whose effective interaction is still bounded but different from the gaussian by varying the chemical composition along the chain and the polymer molecule architecture. The polymer repeat unit number density may have a different radial dependence from the center of mass compared to an ideal random coil linear polymer, such as a star polymer, in which the radius of gyration is larger than that of the linear polymer containing the same number of statistical segment lengths19 . It is therefore of interest to compare and contrast the thermodynamic behavior of other effective potentials which are also bounded and could be used to represent different generic classes of polymer particle.
2
This work is concerned with comparing some alternative analytic potentials to the GCM which could potentially be used to represent effective interactions of other classes of soft dispersed particles on the mesoscale. The focus is on the second virial coefficient, b2 (or SVC) and its dependence on the temperature and potential parameter(s). This is the first configurational interaction term of a series expansion of the equation of state20 . The study of the SVC of simple molecular systems has a long history, dating back to at least Lennard-Jones,21–23 but its application to bounded potentials is more recent9,24 and much less well explored. One of the bounded potentials considered here is new in the context of soft matter. The SVC characteristics of bounded variants of the popular inverse power (IP) potential25 are also treated here. The interest is in purely repulsive bounded potentials, which would represent polymer particles in a good solvent,26,27 that is, one in which the polymer-solvent molecule attractions dominate.
Bounded potentials have also played an important role in coarse-grained fluid dynamics techniques such as Dissipitave Particle Dynamics (DPD),28 Smooth Particle Hydrodynamics (SPH)29 and various hybrids30 of them. These techniques occupy a place in the spectrum of methods between the Molecular Dynamics and Computational Fluid Dynamics extremes (i .e., atomistic to continuum) which are close or formally in the continuum limit (i .e., for DPD and SPH, respectively). The common feature of all of these schemes is that the (effective) interaction ‘potential’ (or ‘kernel’ in the SPH case) between the ‘particles’ mapping out the Lagrangian grid points is extremely soft and bounded. The local stress tensor used in SPH is a function of the local density, which at a grid site (particle position) is determined by summing the kernel contributions from neighboring particles. There is no unique functional form for the kernel. The local density and therefore local stress tensor at a particular grid point depends on the shape of the kernel. The results of this work could also therefore ultimately find 3
application in optimising SPH and related techniques.
II.
THEORY
Consider a radially symmetric interaction potential, φ(r), where r is the distance between the centers of the two particles. The potential can be expressed in terms of a characteristic energy and length, ǫ and σ, respectively. The temperature is in units of ǫ/kB , where kB is Boltzmann’s constant. To keep the notation as compact as possible, these constants are set to unity here. The second virial constant (SVC) in 3D is then given by the formula,23 b2 = −2π
Z
∞ 0
(e−φ(r)/T − 1)r 2 dr
(1)
Analytic formulas for the SVC from different potentials substituted in Eq. (1) are given below.
Penetrable Sphere The prototype example of a purely repulsive particle which has a bounded interaction is the so-called Penetrable Sphere (PS)31–33 φ(r) = 1,
r ≤ 1;
φ(r) = 0,
r > 1,
(2)
which could be considered to be a variant of the hard sphere potential, and gives from Eq. (1), b2 =
2π [1 − exp(−1/T )]. 3
(3)
The SVC ranges between the hard sphere value (i .e., 2π/3) for T << 1, to the ideal gas value (i .e., 0) when T >> 0. The phase behavior of this potential has been explored a number of times, in part because it can be used to represent the generic behavior of soft colloids and micelles. It also acts as the limiting form of a number of analytically continuous bounded 4
potentials in certain parameter limits, as will be demonstrated below.
Series expansion for the SVC For an arbitrary continuous potential, φ(r), by expanding the exponential in Eq. (1), b2 = −2π = −2π = −2π
Z
∞
r 2 dr 0
X (−1)k
k=1 X (−1)k Z ∞
k!T k
k=1
X (−1)k k!T k
k=1
k!T k
[φ(r)]k
r 2 dr[φ(r)]k
0
Ik ,
(4)
where, Ik =
Z
∞
r 2 dr[φ(r)]k .
(5)
0
For each potential the expression for SVC therefore reduces to deriving a formula for Ik to be inserted in Eq. (4). The series can be shown to converge for any T < 1 (as T ≥ 1 is not a problem as may be seen by inspection) by the ratio test rule because Ik is weakly dependent on k for large k, and convergence is dominated by the k! term The number of terms, k, in the series required for convergence to the exact answer (i .e., k!T k ∼ 1) follows k ∼ 1/T using Stirling’s approximation for k! in the large k limit.
Exponential potentials If m = 1, 2, 3, · · · , the exponential potential is φ(r) = exp(−r
2m
3 k −3/2m Γ( ), Ik = 2m 2m
),
(6)
where Γ(x) is the Gamma function of x. The term, Gaussian Core Model or ‘GCM’ is usually reserved for the m = 1 example, and the acronym ‘GCM-2m’ or the term, ‘Exponential’ is used here for the more general exponential potential form given in Eq. (6). The SVC expression for 5
m = 1 case is already in the literature9 1 π 3/2 X 1 (−1)k , b2 = − 3/2 2 k!k T k
(7)
k=1
and the more general expression for arbitrary m is, b2 = −
1 π 3 X 1 (−1)k Γ( ) . m 2m k=1 k!k 3/2m T k
(8)
The SVC of the Eq. (6) potential converges towards that of the PS potential in the m → ∞ limit. For any value of m the SVC eventually diverges from the PS limit at a low enough T because the effective hard sphere diameter of this type of particle tends to infinity as T → 0 (see the discussion at the end of this section).
As mentioned in the Introduction, the GCM has been used to represent polymers in solution. One unrealistic feature of the GCM potential when used in this context is that it is finite in the infinite r limit, which is inconsistent with the fact that any real polymer molecule has a maximum stretched out length. Therefore the actual effective potential should go to zero after several radii of gyration. This could be represented in a model potential by one that goes exactly to zero at a large distance, r0 , which prohibits a purely exponential form. Alternatively the GCM-2m potential can just be truncated at r = r0 , which is more straightforward to represent analytically. This truncated GCM-2m or ‘TGCM-2m’ potential is,34 φ(r) = exp(−r 2m ), Ik =
r ≤ r0 ;
k −3/2m 3 γ( , kr02m ) 2m 2m
φ(r) = 0,
r > r0 (9)
where γ(x, y) is the lower incomplete Gamma function, which tends to the Gamma function, Γ(x), in the y → ∞ limit (see Sect. 6.5 in Ref.35 ). The definition of Ik in Eq. (9) substituted in Eq. (4) gives the SVC of the TGCM-2m potential, bt . The contribution to the SVC from the 6
GCM-2m potential in the distance range, r0 → ∞ is therefore expressed just in the Ik term as34 k −3/2m δIk = 2m
3 3 Γ( ) − γ( , kr02m ) 2m 2m
k −3/2m 3 = Γ( , kr02m ) 2m 2m
(10)
where Γ(x, y) is the upper incomplete Gamma function, which tends to zero as y → 0. Let δb2 be the quantity when δIk of Eq. (10) is substituted in Eq. (4) instead of Ik . Therfore, b2 = bt + δb2 .
(11)
If we define T0 = exp(−r02m ) then for T << T0 , the SVC, b2 → 2πr03 /3, where r0 acts as an effective hard sphere diameter. Based on a Boltzmann factor criterion, an effective hard sphere diameter, σHS , of the GCM potential4,6,12 has been derived s T −1 , σHS = ln ln 2
(12)
which has been used to establish the low temperature part of the phase diagram of the GCM6 .
Algebraic Hat, AH A new potential in the context of soft matter, which is referred to here as the Algebraic Hat (AH), is 1 , φ(r) = a + rn
1 [3/n]−k Γ(k − n3 )Γ( n3 ) Ik = a , n Γ(k)
n>3
(13)
where a ≥ 034 . If a = 1 the SVC of this potential converges to that of the Penetrable Sphere potential as n → ∞. The generic form of Eq. (13) with n = 2 has been used as a single particle potential for metal clusters36 but not as an effective pair potential for mesoscopic particles in solution.
Bounded Inverse Power, BIP 7
The Inverse Power (IP) potential, φ(r) = 1/r n , where n is a positive constant (usually taken to be an integer), has found considerable use as a reference fluid in theories of the liquid state, second only to the hard sphere fluid, in part because of its simple density-temperature scaling behavior25 . This feature of the IP system has also been found in the condensed phase properties of other, more complex, model potential systems, a trend which is called ‘isomorphism’, and those liquids that obey this behavior are referred to as Roskilde Simple fluids37,38 . The bounded extension of this potential, called here the Bounded Inverse Power, BIP, potential is, φ(r) =
1 aq + r q
n/q
;
n > 3,
(14)
where a is a positive constant. The potential defined in Eq. (14) may also be referred to as the ‘Pq’ potential. The parameter, a, in Eq. (14) acts to soften the IP potential by diminishing the relative importance of the pair separation component when r is small compared to a. For small a the potential could be used to model polymer particles with strong excluded volume interactions, which discourage (but do not entirely eliminate) interpenetration. For larger a and relatively small n (but greater than 3) the potential is long ranged, being similar to the Kac potential (but without its normalisation factor)39 .
Two special cases of q in Eq. (14) are considered. The first is when q = 1, called P1, φ(r) =
1 a+r
n
;
n > 3.
(15)
The SVC of this potential is, b2 = 2π
2 X
q(p)b2,p ,
(16)
p=0
where q(0) = a2 , q(1) = −2a and q(2) = 1, and b2,p
1 T −(1+p)/n (1 + p) 1 a1+p [exp(− n ) − 1] + γ 1− , n . = (1 + p) Ta (1 + p) n Ta 8
(17)
The other case of Eq. (14) considered here is when q = 2, which is a potential referred to as, P2, φ(r) =
1 2 a + r2
n/2
;
n > 3,
(18)
which is X h(k) 1 (−3 + 2k) 1 3 2k (−3+2k)/n a [exp(− n ) − 1] + a T γ(1 + , n) . b2 = 2π 3 − 2k T a n Ta k=0
(19)
The summation in k involves h(k), h(k) =
(2k − 3) h(k − 1), 2k
(20)
for k ≥ 1, and h(0) = 1. The above formulas for the SVC of the P1 and P2 potentials are each the first term of a high temperature expansion of the corresponding bounded version of the Mie potential24 . The radial distribution functions generated by integral equation, mean field and molecular dynamics simulations of the P2 potential were explored in Ref.40 , but this is the first work where the SVCs of these two examples of a BIP potential have been investigated specifically in their own right.
Limiting forms of the SVC at high and low temperature are also of interest as they offer the prospect of simplifying analytic statistical mechanical treatments of mesoscopic systems. The high temperature (HT) limit SVC from Eq. (1) is 2π b2 = T
Z
∞
π r φ(r)dr − 2 T 2
0
= b2,1 − b2,2 · · ·
Z
0
∞ 2 2
r φ (r)dr + O
1 T3
, (21)
where b2,1 and b2,2 on the right hand side of Eq. (21), defined as positive quantities for these purely repulsive potentials, are the first two terms of a high temperature expansion of Eq. (1).
9
Although the analytic formula for the low temperature dependence of the SVC will depend on each potential form, following the previous literature, an effective hard sphere diameter, σHS can also be defined by solving α = exp(−φ(σHS )/T );
b2 =
2π 3 σ , 3 HS
(22)
for arbitrary potential. The value of α = 1/2 has been chosen in previous publications4,6,12 on the GCM, in which case Eq. (22) reduces to Eq. (12).
The SVC series expansions were evaluated using a code written in Fortran 90. The series were deemed to have converged when the difference between consecutive series terms was less than 10−10 . The converged values from the series expansions were confirmed using direct evaluation of the SVC employing the formal definition, given in Eq. (1).
Simpsons rule
numerical integration was used up to a distance where the potential energy at the truncation distance was 10−6 . The distance increment in the numerical integration was 0.001/(2n) for the exponential potentials and 0.001/n for the BIP and AH potentials. The calculations typically took less than a minute on a laptop to complete the temperature ranges covered in the figures.
10
III.
RESULTS AND DISCUSSION
Figure 1 compares examples of the GCM-2m potential of Eq. (6) for different values of m. With increasing exponent power, 2m, the potential converges towards the PS potential. Note the isosbestic point at r = 1, where φ(1) = e−1 in each case. In the general expression for Ik in Eq. (6), the variable m is not restricted to integer values. For m > 1/2 the potential is known as a ‘compressed’ exponential (CE), m = 1/2 is the exponential, and for m < 1/2 the potential is referred to as a ‘stretched’ exponential, an example of which is also given on Fig. 1. The stretched exponential (SE) does not appear to have been considered before in the context of bounded effective potentials for polymers in solution. For example, the stretched exponential could be used to represent a star polymer-like system12 which is still bounded at the origin. Figure 1 indicates that the large values of the SE potential are concentrated further from the origin and it has a slower rate of decay at large r than the CE cases. This leads to larger values of the SVC at a given temperature for smaller m value. The precise value of m could be adjusted to reproduce the effective interaction between polymer molecules of a specific architecture. The boundary of 2m = 1 between stretched and compressed exponentials marks also a transition between two different classes of SVC behavior. For m → 0 the potential tends to e−1 for a wider r range, and consequently the SVC tends to ∞ due the increasingly slow decay of φ(r), which is weighted by r 2 in the formula for the SVC given in Eq. (1). In contrast, as m → ∞ the SVC tends to the PS limit. Figure 2 provides a similar comparison for the AH potential with a = 1, which shows similar trends to the GCM-2m case, but has an isosbestic point at φ = 1/2. The significance is that GCM-2m type of behavior can be represented by an algebraic rather than exponential analytic form.
Figure 3 presents the P1 potential of Eq. (15), taking a = 1 for various values of n. It has a
11
cusped appearance near the origin (rather like the SE), which suggests that it might be useful in modelling star-like particles41 . The corresponding plots of the P2 potential of Eq. (18) are given in Fig. 4. The P2 potential has a point of inflection and therefore a maximum repulsive √ force at r = a/ 1 + n, unlike P1 which does not have a point of inflection. It also has a Maclaurin expansion which is even in r, unlike that of P1.
Figure 5 gives the second virial coefficients as a function of temperature for some examples of the GCM-2m type of potential for several m. The data are plotted in normalised form, b2 /b0 , where b0 = 2π/3 is the second virial coefficient of the hard sphere potential with the same unit of distance. The figure shows that at high temperature (T >> 1) the SVC all converge towards the PS model curve, and that for n = 100 the SVC is hardly distinguishable from the PS value even for T << 1. However, only the PS second virial coefficient is equal to b0 in the T → 0 limit. For each value of m there is always a low enough temperature where the SVC will diverge from the PS. This qualitative difference in behavior arises because the GCM-2m is a continuous potential, and the PS form is discontinuous. The effective hard sphere diameter of the GCM-2m potential diverges as T → 0.
Figure 6 shows b2 /b0 for the AH potential of Eq. (13) as a function of T for several n values, with a = 1. Again for high temperature (T > 1) the SVC curves converge to that of PS potential, and for T < 1 the function progressively converges towards the flat PS limit with increasingly n for not too low temperatures. There is the same qualitative difference with the PS potential as exhibited by the GCM-2m potentials in this limit.
Figure 7 presents b2 /b0 for the P1 potential of Eq. (15), mainly focusing on n = 12 . The SVC increases at all T as a decreases, and shows no evidence of approaching the PS limit at any 12
temperature, which is different to the AH case for the same n. The BIP potential tends to a−n , rather than a−1 (the AH case) and hence the greater SVC rate of divergence as a → 0 compared to the AH example. Figure 8 gives the corresponding quantities for the P2 potential of Eq. (18), which exhibits the same qualitative trends. The curves for the different a values using n = 12 are slightly closer together than those of P1, and they are shifted systematically to higher T .
Figure 9 explores the effectiveness of the high temperature limit expansion of Eq. (21) in representing the SVC, for the GCM-2m potential with 2m values ranging from 1/2 to 2 (the gaussian case). The figure, which presents the b2,1 and some b2,2 data, demonstrates that the first expansion term of the SVC or b2,1 represents well the SVC for temperatures larger than ca. 1, which extends to lower temperature as m decreases. In fact for the stretched exponential (2m = 1/2) the b2,1 term agrees well with b2 down to at least T ≃ 0.01, which is more than adequate to represent polymers in solution.
Figures 10 reveals in another way the extent to which the high temperature limit expansion of Eq. (21) i .e., b2,1 , can be used to replace the exact b2 value. The figure shows the m dependence of b2,1 /b2 for three temperatures, 0.1, 1 and 10. The quantity, b2,1 reproduces b2 well at a temperature of 10 for the whole m range shown, and also for the stretched exponential limit in the vicinity of m ≃ 1/4 (which is consistent with the trend seen in Fig. 9). For T = 1 for all m and T = 0.1 for m > 0.5 the first order high temperature expansion does not represent the SVC well.
Figures 11 and 12 show the corresponding SVC expansions for the AH, and P1 and P2 potentials, respectively. The trends are very similar to those of the GCM-2m potentials except for these cases there is no value of n where the first term in the high temperature expansion 13
represents the SVC in the vicinity of T ∼ 1 and lower temperature (note that n > 3 is a requirement for a finite SVC here). For high enough temperature, T >> 1, the AH, P1 and P2 potentials give b2,1 /b2 ratios of about 1, indicating that the first term in the High Temperature expansion is a good approximation of the SVC, as would be expected by inspection of Eq. (1).
Figure 13 compares the SVC of several cases of these potentials (symbols) with the prediction based on an effective hard sphere approximation from Eq. (22), which reveals that the HS approximation reproduces the exact results very well for T < 1. A value of α = 0.55 fits the GCM-4 data slightly better than the usual value chosen of 1/2 found in the literature, which in the GCM-4 case systematically slightly underestimates the exact SVC values.
A challenge is to distinguish the behavior of real polymers of various architectures in terms of the SVC of these different model potential systems. In polymer solution theory, the osmotic pressure, Π is expressed as a virial expansion42 MΠ = Z = 1 + MA2 c + MA3 c2 + · · · , RT c
(23)
where M is the molar mass, R is the Gas constant, c is the concentration in real units (e.g., in kg dm−3 ), and where A2 and A3 are the symbols usually used for the second and third virial coefficients, respectively, in the polymer solution literature. Equation (23) can be written in the form43 , Z = 1 + b2 ρ + · · · where A2 = Nav b2 /M 2 , Nav is Avogadro’s number and ρ is again the number of particles per unit volume. A so-called ‘interpenetration factor’, Ψ19 can usefully be defined which is sensitive to polymer architecture, Ψ=
2A2 M 2 , Nav (4π < Rg2 >)3/2
(24)
where < Rg2 > is the mean square radius of gyration (RoG). The factor Ψ ≈ 1 is typically several times larger for a star polymer than a linear chain with the same number of statistical 14
segment lengths, increasing with the number of arms19 . For high molar mass polymers, Ψ, in good solvents ranges from ca. 0.26 for linear polymers, and 0.6 to 1.1 for f = 4 to 18-arm star polymers,44,45 with a predicted upper limit of 2.13 for f → ∞,46 which is close to the hard-sphere limit.
One of the current problems in exploiting the above excluded volume theory developed in this work is that even in the GCM case when used as a model for linear polymers in a good solvent, there is a coefficient in the prefactor of the exponential term and in its argument, which are not known exactly and are functions of M 47 . Nevertheless, taking into account O(1) numerical constants, Ψ should be approximately equal to the ratio b2 /b0 in the region of T ≃ 1, a typical polymer solution reduced temperature,6,9 . It may be seen in Figs. 5-8 that b2 /b0 ∼ 1 at T ≃ 1. Figure 5, for example, for the GCM potential gives at T = 1 the value 1.5 for b2 /b0 , and this ratio increases dramatically as n decreases. For the AH potential the value is ca. 0.75 for the wide range of n values shown on Fig. 6. For the n = 12 P1 potential, the ratio b2 /b0 is seen in Fig. 7 to be, 1.0, 0.2 and 0.0 for a = 0.1, 0.5 and 1.0. Figure 8 shows that for the n = 12 P2 potential, the ratio is 1.2, 0.85 and 0.15 for a = 0.1, 0.5 and 1.0. By comparison with experimental Ψ values the ratio b2 /b0 could therefore in principle be used to determine the effective potential analytic form which matches well a particular polymeric system.
Although not relevant for polymer solutions (where T ∼ 1/2 to 1) some aspects of the low temperature fluid-solid aspects of the phase diagram of these systems (where T < 0.01,12 ) can be established from the Fourier Transform of the potential, which is discussed briefly here. The exponential system (m = 1 in Eq. (6)) undergoes reentrant melting (RM) at high density but the m ≥ 2 cases form multiple occupancy cystals (MOC, see ref.48 and the references quoted therein), and therefore no reentrant melting. This is determined on the basis of whether the 15
˜ is positive for all wavevectors (which gives RM) or crosses Fourier transform of the potential, φ, ˜ zero and goes negative at least once (which gives the MOC). For m equal to 1 the function, φ, ˜ has a minimum of −0.127 at k = 5.6, and is positive for all wavevectors k, and for m = 2, φ, for m = 3 the minimum is −0.200 at k = 5.7, both indicating MOC behavior at high density and low enough temperature.
The q = 2 example of the BIP potential has an analytic solution for φ˜ which is positive for all k at least up to n = 12 (see Eqs. (9-15e) in Ref.40 for the relevant theory). Although analytic ˜ solutions for φ(k) for the AH potential can be derived, they are rather complicated expressions in general, involving hypergeometric functions, and establishing whether they are negative for any k value is not trivial for the entire range of parameters. Some indication of the trends in the phase behavior, in regard to the RM vs. MOC issue can be explored by determining ˜ φ(k) by numerical integration. Applying this method to the BIP potential with n = 12, the ˜ q = 1 and 2 cases have φ(k) > 0 for all k which is consistent with RM behavior, whereas for q = 3 and 4 there are shallow negative regions for ca. k > 6.0, which suggest MOC. With increasing q in Eq. (14) the potentials exhibit a flatter top region in the vicinity of r ∼ 0 which intuitively one would associate with the conditions that would give rise to a MOC (i .e., a low energy penalty for a multiply occupied lattice site) rather than RM at a sufficiently high density.
For the AH potential and taking a = 1, increasing n produces a deeper negative region in ˜ φ(k). For example for n = 6 and 12 the minima are −0.134 and −0.272 at k = 5.3 and 5.5, respectively, which indicates MOC. With increasing n the AH potential for r < 1 progressively converges towards the PS potential, which takes on the appearance of the GCM when m is large. For n = 2 the behavior is inconclusive, and for n = 3 there is a weak negative region at
16
large k greater than about 5. Just as for the GCM and extensions, the preliminary evidence is that for the AH and Pq potentials there is a transition from RM to MOC behavior as the potential becomes flatter near the origin.
17
IV.
CONCLUSIONS
Series expansion formulas for the second virial coefficient, SVC, of various purely repulsive bounded potentials have been derived and compared here.
Several specific cases of an
exponential pair potential of the form, exp[−r 2m ], where m is a positive variable, were considered. This includes an example of a stretched exponential which occurs when m < 1/2. The stretched exponential is cusp shaped for small interparticle separations near the potential origin but decays slower than a gaussian at larger separations. This potential could be used to model a polymer in solution where the monomer density is concentrated further from the center of mass than a polymer exhibiting random coil or gaussian potential behavior. Examples include star polymers49 , dendrimers43,50,51 and comb polymers52 . In contrast, a more ‘compressed’ exponential than the gaussian might be expected to be a better match for a ring polymer53 . The first term in the high temperature expansion of the SVC of the stretched exponential is accurate for effective temperatures at and below that typical of polymers in solution (T ∼ 1 in the units used in this work). The SVC of a newly proposed, Algebraic Hat (AH) potential behaves like the penetrable sphere (PS) potential case at high temperature but like the bounded inverse power (BIP) potential at low temperature (i .e., T < 1). In addition to architectural dependencies, the effective interactions between two polymer molecules will also depend on the molar mass, as do the usual polymer molecular quantitive descriptors such as the radius of gyration, second virial coefficient and the intrinsic viscosity,49 .
One physically unrealistic feature of all the continuous bounded potentials considered here is that they are finite at infinite pair separation, which would require the molar mass of the polymer molecule to be infinite. This has a major effect on the T → 0 part of the phase diagram as the effective hard sphere diameter and hence the SVC (and presumably the higher
18
virial coefficients) diverges in that limit as a direct result of this feature. The physical state of all the model systems consided here (apart from the Penetrable Sphere case) in that limit is not well defined and physically unrealistic, which could have consequences for statistical mechanics treatments of these systems, e.g., in the context of thermodynamic integration. A truncated version of the gaussian (and more general exponential type) potential developed here is shown to remove this problem, and be a better representation of a polymer molecule in a good solvent. This in no way diminishes the significance of the GCM in polymer solution theory but points out that caution should be used when it is employed to represent these experimental systems at low temperature.
The BIP potential could represent different types of particle by varying the parameters, q, a and n. These range from small molecules forming a single component liquid (small a, and large n), then small solvent dispersed particles such as micelles,54 and microgels,55 through to large soft highly interpenetrable polymers (large a and variable n). The experimental virial coefficient of a polymer in solution scales approximately as the excluded volume of the molecule as determined by the radius of gyration, multiplied by what we call b2 10 The excluded volume of the molecule scales as the molar mass, M, as ∼ M 1.8 for a linear polymer8 whereas for a stiff rod it scales approximately as ∼ M 2 ,56 which is not too different. The Gaussian chain and stiff rod are at the two extremes of the topology of linear polymers in general. In defining the second virial coefficient this volume is multiplied by a second term, b2 , which depends on φ(r), representing implicitly the radial distribution of mass within the particle. It is this term that has been focused on in this work, as it is a key quantity in linking more precisely experimental and model data, but has not been explored systematically until the present work.
This study focuses on polymers in good solvents, where attractive interactions between the 19
chains can be neglected to a good approximation, which is why the effective center of mass potentials compared are purely repulsive. These potentials are also bounded (i .e., finite at the origin) which is appropriate for linear or branched chains where the chemical architecture varies the mass distribution with respect to the center of mass. For polymers in marginally poor solvents, a bounded version of the Lennard-Jones or more general Mie potentials could be used23,24 . Electrostatic interactions such as in polyelectrolytes and where ionic surfactants are present could also be introduced in the present treatment, although depending on the composition the effective interactions may cease to be of the bounded form.
To summarise, bounded potentials have been used in various aspects of modelling at a coarse grained level the properties and behavior of liquids on long length and time scales. The chemistry of the system and physical processes of interest are introduced by varying the analytic form of the potential. The present work makes a comparison of a range of bounded potentials using the fundamental thermodynamic quantity, the second virial coefficient, which can also be measured experimentally for polymers in solution. Incidently, the results and implications of this study of many different bounded potentials may go beyond polymeric systems, and could, for example, be used to fine tune the kernel (and therefore the modelled flow characteristics) produced by Smooth Particle Hydrodynamics and related simulation techniques.
Acknowledgments The author would like to thank Dr.
T. Crane (Department of Physics, Royal Holloway,
University of London, UK) for helpful software support.
20
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Figure Captions Figure 1: The exponential potential defined in Eq. (6). The 2m values are given on the figure. Note the figure includes data for the exponential case (‘1’) and an example of a stretched exponential, φ(r) = exp(−r 1/2 ) (‘1/2’).
Figure 2: The AH potential defined in Eq. (13), where a = 1. The potential exponent, n, values are given on the figure.
Figure 3:
The P1 potential defined in Eq. (15), with a = 1. The n values are given on the
figure.
Figure 4:
The P2 potential defined in Eq. (18), with a = 1. The n values are given on the
figure.
Figure 5: The second virial coefficients of the gaussian potentials defined in Eq. (6). The SVC of the hard sphere, b0 = 2π/3 is used as a normalising factor. The 2m values are given on the figure. The SVC of the PS potential is also given on the figure as ∗ symbols.
Figure 6: The second virial coefficients of the AH potential defined in Eq. (13), where a = 1. The n values are given on the figure. Otherwise the details are as for Fig. 5.
Figure 7:
The SVC of the P1 potential defined in Eq. (15). The key gives [a n] values for
each curve.
Figure 8: As for Fig. 7 except that the P2 potential defined in Eq. (18) is considered. 26
Figure 9: The second virial coefficients of the exponential potentials as a function temperature are shown as lines. The open square (2m = 1/2) and filled-in circles (2m = 2) are the b2,1 or first expansion term of Eq. (21). The triangles are (as indicated on the figure annotation) the second or b2,2 expansion term (which is a positive quantity as defined in Eq. (21)). The 2m values are given on the figure.
Figure 10: The ratio, b2,1 /b2 of the exponential or GCM-2m potential for three temperatures, 0.1, 1 and 10, which are indicated on the figure.
Figure 11: As for Fig. 10 except the AH potential defined in Eq. (13) is considered, with a = 1.
Figure 12: As for Fig. 10 except the SVC high temperature expansion of the P1 and P2 potentials are investigated, with a = 1. The open and filled in symbols are for P1 and P2, respectively.
Figure 13: The exact SVC of several potential cases given as symbols, compared with the effective hard sphere approximation based on Eq. (22) given as solid lines. The penetrable sphere solution (PS) is also given on the figure.
27