The complex viscosity of Möbius macromolecules

Using general rigid bead–rod theory, we explore the effect of twisting a macromolecule on its rheological properties in suspensions. We thus focus on macromolecules having the form of Möbius bands so that the number of twists can be incremented. We call these Möbius macromolecules . When represented in general rigid bead–rod theory, these macromolecules comprise beads whose centers all fall on a Möbius band. From first principles, we calculate the complex viscosity of twisted rings with zero to seven twists. We find that the zero-shear values of the viscosity and first normal stress coefficient increase with twisting. Furthermore, we find that the real part of the complex viscosity descends more rapidly, with frequency, with extent of twist. For the imaginary part of the complex viscosity, the more twisted, the higher the peak. For each part of the dimensionless complex viscosity and the first normal stress coefficient, the results fall on one of just three curves corresponding to zero, even, or odd numbers of twists. We also explore the effects of the length and the aspect ratio of twisted macromolecular suspensions. We close with a worked example for a suspension of triply twisted Möbius annulene.


I. INTRODUCTION
General rigid bead-rod theory affords the opportunity to explore how the macromolecular shape, even if complex, affects the rheology of polymeric liquids.We explore how twisting a macromolecule affects its complex viscosity.Previously, we studied the effect of twisting on linear chains, such as doubly helical macromolecules, including deoxyribonucleic acid (see Subsection V C of Ref. 1).However, for linear helical macromolecules, the chain length is proportional to the number of twists, so the effect of twisting is confounded by the chain length.We therefore explore the role of the number of twists in ring polymers on complex viscosity.Adding one or more twists to ring polymers produces Möbius macromolecules, which we define mathematically below.Although this work is mainly driven by curiosity, 2 its applications have not escaped our attention.For instance, to introduce chemical stability, organic chemists twist pi-bonded structures into ringed macromolecules. 3The more twisted, the more stable the macromolecule might be. 2,3For our symbols, dimensional and non-dimensional, listed in Tables I and II For linear viscoelastic behaviors, general rigid bead-rod theory has been evaluated for only a few very simple structures: rigid rings, the rigid tridumbbell, and three quadra-functional branched structures along a backbone (Table 16.7-1 of Ref. 5).More ambitiously, and still unpublished, recent work attacks other branch functionalities along a backbone, star branched architectures, be they planar or polyhedral, 6 and diblock copolymers. 7For oscillatory shear flow, the frequency dependencies of both parts of the complex viscosity are ARTICLE scitation.org/journal/phfFrom general rigid bead-rod theory, the viscous contribution to the dimensionless complex viscosity is given by [Eq.(40) of Ref. 9] and minus the elastic contribution by [Eq.(41) of Ref. 9] which are subject to the definition of small-amplitude oscillatory shear flow (SAOS) [Eq.(31) of Ref. 9], which is a restriction we will also use below.
and of the parts that are in-phase with cos 2ωt [Eq.(72) of Ref. 9], and out-of-phase with cos 2ωt [Eq.(73) of Ref. 9], which goes through a maximum, as does η ′′ /(η 0 − ηs) of Eq. ( 2). 9,13elow, we will capitalize on Eqs. ( 4)-( 6) subject to Eq. ( 3) to study the effect of architecture on Möbius macromolecules.Since general rigid bead-rod theory relies entirely on macromolecular orientation for explaining rheological responses of macromolecular suspensions, 17 the theory neglects interactions between macromolecules.We are attracted to general rigid beadrod theory for the accuracy of its predictions for its simplest special cases and to the rigid dumbbell, at least qualitatively, for most of the viscoelastic material functions measured in the laboratory (see Introduction of Refs.9, 10, 15, and 16).Perhaps the reason for the apparent success of the rigid dumbbell, even for flexible macromolecules, is that the overall orientation of the molecule is the most important factor in determining the rheological responses of elastic liquids.The chain expansion and contraction seem to be less important.Furthermore, the successes (Fig. 20

II. METHOD
We study macromolecules whose bead positions all lie on the surfaces of Möbius bands, with τ twists, given by where w is the strip width, s is the lateral position across the strip, θ is the cylindrical angular coordinate, and τ is the integer number of twists.The cylindrical angular coordinate, θ, thus defines the position along the strip.We call macromolecules whose bead positions satisfy Eq. ( 7) Möbius macromolecules.Otherwise put, Möbius macromolecules are objects whose representations in general rigid bead-rod theory comprise beads whose centers fall on Möbius bands.Equation ( 7) is parametric in θ, and all bands described by it intersect at the circle of support of radius R, Without twists, τ = 0, and Eq. ( 7) reduces to the parametric equation for a right circular cylinder, No twists (black), even number of twists (red) (Fig. 10), and odd number of twists (blue) (Fig. 11).Möbius bands from three rows of beads with aspect ratio ten.
Thus, we will compare twisted structures, exploring one to seven twists, with their untwisted counterpart, this right circular cylinder.When modeling Möbius macromolecules using general rigid bead-rod theory, we begin with a set of rigidly spaced beads with position vector ri, where the macromolecular center of mass R satisfies where the subscript i indicates the bead number, N is the total number of beads, and M ≡ ∑ N i=1 mi is the molecular weight.Since we construct our Möbius macromolecules with identical beads of diameter d and mass m, the molecular weight is given by M ≡ mN, and thus, the center of mass is which we will use below.
We next install molecular coordinates at the macromolecular center of mass, and we orient these Cartesian coordinates such that δ3 is along the polar axis of the moment of inertia ellipsoid (MIE).The position vector of the ith bead with respect to the center of mass is given by [Eq.( 16 Equation ( 7) refers to a coordinate system whose origin is arbitrarily set at the position [0,0,0].To get Ri, we insert Eq. ( 7) into Eq.( 12) to get R and then this into Eq.( 13).
We can define the principal moments of inertia We design each Möbius macromolecule by rigidly connecting each bead to its nearest neighbors, center to center, with massless dimensionless rods. 8Adjacent rows are indexed identically, By indexed identically, we mean that, for each row j, all beads have the same value of θ.From Eqs. ( 18) and ( 19), we learn that, if identically indexed beads are spaced evenly along and across the strip, We call such Möbius macromolecules axisymmetric.
Hassager derives the expression for the shear relaxation function from general rigid bead-rod theory for axisymmetric macromolecules, 17,18 G(s) ≡ (2η s + nζL The three quantities in Eq. ( 21), a, b, and λ, thus define completely the differences in linear viscoelastic behaviors arising between different axisymmetric Möbius macromolecules.Whereas we associate

Physics of Fluids
ARTICLE scitation.org/journal/phfa with the Dirac delta function contribution to the relaxation function given by Eq. ( 21), we associate b with the dying exponential.We call a, b, and ν the orientation constants of the macromolecule.
The relaxation time can be expressed as where For later convenience, we define which we will use below.
From Ref. 17 (Chap.16 of Refs.4, 6, or 9), we learn that the value of 2b/aν reflects lopsidedness, namely, the extent to which the macromolecule deviates from a spherically symmetric structure.
The minimum value of 2b/aν is 0, obtained for spherically symmetric structures such as rigid regular octahedra (Macromolecule 5 of Ref. 9), and the maximum is 3/2 for long slender bodies such as the rigid dumbbell (Macromolecule 1 in Table 4 of Ref. 9).
We classify macromolecules as prolate or oblate.By prolate, we mean that the macromolecule MIE is longer than it is wide.By oblate, we mean that the MIE is wider than it is long.Otherwise put, oblate and prolate macromolecules depart differently from spherical symmetry.
Dividing Eq. ( 25) by Eq. ( 27) normalizes the relaxation time of the general macromolecule to that of the simplest, which we will use below.Equations ( 22)-( 24) imply the quadratic  4)-( 6)] of one twist Möbius bands.Three rows of beads with aspect ratio ten (blue) (Fig. 11) and seven rows of beads with aspect ratio ten (red).
whose prolate lower branch is given by aν and whose oblate upper branch is given by From Eqs. ( 29)-(31), we learn that every axisymmetric macromolecule occupies one and only one position on either the upper or lower branch.The simple shapes given by parametric equation ( 7) are not to be confused with the more complicated shapes arising when thick flat elastic strips are curved into cylindrical or twisted bands.[21]

III. RESULTS
Tables III-VI compare the moments of inertia, orientation constants, lambda ratios, and zero-shear properties for every Möbius macromolecule studied in this paper.Table III includes the results for Möbius bands of three rows with aspect ratio R/w = 20.By R/w, we mean the ratio of the radius of the circle of support, R, to the band width, w, of the Möbius macromolecule.Table IV includes the results for Möbius bands of three rows with an aspect ratio of ten.Table V includes the results for Möbius bands of three rows with an aspect ratio of ten.Table VI includes the results for Möbius bands of three rows with an aspect ratio of seven.
From Columns 9 and 10 of Table III-VI, we learn that the zeroshear viscosities and first normal stress coefficients, made dimensionless with λ, for the untwisted Möbius macromolecule are much higher than any of its twisted counterparts.By contrast, we also learn that the zero-shear viscosities and first normal stress coefficients, made dimensionless with λ, for the twisted Möbius macromolecules do not differ from on another.From Table VII, on the other hand, we learn that the zero-shear viscosities and first normal stress coefficients, made dimensionless with λ 0 , do differ significantly and without pattern.

A. Twists
Figure 1 shows the effect of the number of twists on the viscous and elastic response of Möbius band macromolecules.We find that for the dimensionless real part of the complex viscosity [Eq.( 1)], the results fall on just three curves: one for zero twists [Eq.( 9)], another lower one for an odd number [Eq. ( 7) with τ odd], and an even lower one for an even number [Eq. ( 7) with τ even].This difference in η ′ (ω) increases with frequency [Fig.1(a)].We also find that for minus the dimensionless imaginary part of the complex viscosity [Eq.( 2)], the results also fall on just three curves: one for zero twists, another higher one for an odd number, and an even higher one for an even number.This difference in η ′′ (ω) does not vary with frequency [Fig.1(b)].We thus find that the effect of twisting the Möbius macromolecule on the parts of the complex viscosity is binary.This binary dependence on twist parity surprised us.
Figure 2 shows the effect of the number of twists on each of the three parts of the dimensionless complex first normal stress coefficient response of Möbius band macromolecules.We find that for the dimensionless displacement value of the first normal stress coefficient [Eq.( 4)], the results fall on just two curves: one for zero twists and the other higher one for twisted, be the number of twists (1) and ( 2)] of one twist Möbius macromolecules.Three rows of beads with aspect ratio ten (blue) (Fig. 11) and three rows of beads with aspect ratio twenty (red).4)-( 6)] of one twist Möbius bands.Three rows of beads with aspect ratio ten (blue) (Fig. 11) and three rows of beads with aspect ratio twenty (red).
even or odd.This difference decreases with frequency [Fig.2(a)].
Similarly, we find that for the part in-phase with cos 2ωt [Eq.( 5)], the results also fall on just two curves, and their difference also decreases with frequency [Fig.2(b)].We find likewise for minus the part outof-phase with cos 2ωt, Eq. ( 6) [Fig.2(c)].By contrast with the parts of the complex viscosity [Fig.1], we find that the effect of twisting the Möbius macromolecule on the in-phase and out-of-phase parts of the first normal stress coefficient is not binary.Our findings of binary twist parity for η * , and not for Ψ d 1 or Ψ * 1 , establish intuitive expectations for future work on twisted macromolecules.

B. Band width
To study the effect of band width, we compare a Möbius macromolecule with three rows of beads and an aspect ratio of ten to a Möbius macromolecule with seven rows of beads with the same aspect ratio of ten.Band width is the width of the Möbius macromolecule and is not to be confused with band width of signal processing.
Figure 3 shows the effect of Möbius macromolecule band width on the viscous and elastic response of Möbius band macromolecules.
We find that for the dimensionless real part of the complex viscosity [Eq.( 1)], the increase in band width lowers the curve.This difference in η ′ (ω) increases with frequency [Fig.3(a)].We also find that for minus the dimensionless imaginary part of the complex viscosity   2)], the increase in band width increases the peak value.This difference in η ′′ (ω) does not vary with frequency [Fig.3(b)].
Figure 4 shows the effect of the band width on the first normal stress of Möbius band macromolecules.We find that for the dimensionless displacement value of the first normal stress coefficient [Eq.( 4)], a decrease in band width lowers the curve.This difference decreases with frequency [Fig.4(a)].Similarly, we find that for the part in-phase with cos 2ωt [Eq.( 5)], a decrease also lowers the curve [Fig.4(b)].We find likewise for minus the part out-of-phase with cos 2ωt, Eq. ( 6) [Fig.4(c)].

C. Aspect ratio
To study the effect of aspect ratio, we compare a Möbius macromolecule with three rows of beads and an aspect ratio of ten to a Möbius macromolecule with seven rows of beads with the same aspect ratio of ten.
Figure 5 shows the effect of the chain aspect ratio on the viscous and elastic response of Möbius band macromolecules.We find that for the dimensionless real part of the complex viscosity [Eq.( 1)], the increase in aspect ratio lowers the curve.This difference in η ′ (ω) increases with frequency [Fig.5(a)].We also find that for minus the dimensionless imaginary part of the complex viscosity [Eq.( 2)], the increase in aspect ratio increases the peak value.This difference in η ′′ (ω) does not vary with frequency [Fig.5(b)].
Figure 6 shows the effect of the aspect ratio on the first normal stress of Möbius band macromolecules.We find that for the dimensionless displacement value of the first normal stress coefficient [Eq.( 4)], decreasing the aspect ratio lowers the curve.This difference decreases with frequency [Fig.6(a)].Similarly, we find that for the part in-phase with cos 2ωt [Eq.( 5)], a decrease also lowers the curve [Fig.6(b)].We find likewise for minus the part out-of-phase with cos 2ωt, Eq. ( 6) [Fig.6(c)].

IV. WORKED EXAMPLE: TRIPLY TWISTED MÖBIUS ANNULENE
We next turn our attention to a specific organic Möbius macromolecule, the triply twisted annulene, shown in Fig. 7, whose bead centers are given in the supplementary material of Ref. 3. Proceeding from these bead centers and following the method of Sec.II, we generate Table VIII from general rigid bead-rod theory.Then, using Eqs.( 1) and ( 2) with Table VIII, we predict the parts of the complex viscosity for a suspension of triply twisted annulene in small-amplitude oscillatory shear flow, after which we plot these in Fig. 8.To our knowledge and presumably due to the scarcity of the compound, the complex viscosity of such a suspension has yet to be measured.Using Eqs. ( 4)- (6) with Table VIII, we predict all three parts of the first normal stress coefficient response for a suspension of triply twisted annulene in small-amplitude oscillatory shear flow and plot these in Fig. 9.As far as we know, the first normal stress coefficient of such a suspension has yet to be measured.
From Table VIII, we learn that a triply twisted annulene ν ≅ 0.0085, which when inserted into Eq.( 3) yields which defines small-amplitude for triply twisted annulene in oscillatory shear flow.

V. CONCLUSION
In this paper, we calculate the complex viscosity [Eqs.We study macromolecules whose bead positions all lie on the surfaces of Möbius bands, described by Eq. ( 7), which includes the cylinder as a special case [Eq.( 9)].Equation ( 7) thus describes Möbius bands constructed by cutting and twisting a right circular cylinder.By contrast, Möbius bands can also be constructed by cutting and twisting a frustrated disk, and these surfaces are given by where all bands described by this intersect at the circle of support of radius, R [Eq. ( 8)].Without twists, τ = 0, and Eq.(33) reduces to which represents the frustrated disk.Generalizing Eqs. ( 7) and (33) yields for the surfaces of all Möbius bands constructed by cutting and twisting either cylinders (κ = 0) or frustrated disks(κ = π/2) − 1 2 w ≤ s ≤ 1 2 w, τ = 0, 1, 2, . . . .
We leave the study of this broader class of Möbius macromolecules for another day.
Since Möbius bands can evert, 22 in principle, so might Möbius macromolecules during flow, including oscillatory shear flow, the flow at issue in this paper.However, our general rigid bead-rod theory precludes eversion during flow.This being said, when everted, the principal moments of inertia of a Möbius macromolecule do not change.Our theory could be used to calculate the torque about the neutral axis of the Möbius macromolecules during flow.We could then see if this torque ever exceeds the critical torque for eversion.We leave this exploration for another day.
We have neglected interferences of solvent velocity profiles between adjacent beads.Called hydrodynamic interactions, 23 we leave this improvement for another day.Of course, general rigid bead-rod theory explains the elasticity of the liquid by means of macromolecular orientation alone.We also leave the role played by macromolecular compliance 24 in the rheology of Möbius macromolecules for another day.
In previous work, we managed to derive exact analytical expressions for the properties in Tables III-VIII of simpler architectures including shish-kebabs, rigid rings, and planar star-shaped polymers (see Table XV of Ref. 9).We have proceeded numerically from the bead positions.For the simplest general rigid bead-rod theory representation of Möbius macromolecules, one might proceed analytically from Eq. ( 14).Following the method of Sec.II, one might also attack interlocking Möbius macromolecules of like or unlike twist parities.We leave these interesting tasks for another day.
For rigid dumbbells, we know of reasonable predictions for the steady shear viscosity and first normal stress difference material functions of polymeric liquids (see Sec. 6 of Ref. 14).The rigiddumbbell suspension has also been shown to predict accurately the Cox-Merz rule. 25Of course, we can bridge general rigid bead-rod theory with continuum mechanics to arrive at approximate relations for the steady shear material functions.We would do so by inserting the bridging relations Eqs.(64)-(66) of Ref. 9 into the steady shear material functions for Oldroyd 8-constant fluids [Eqs.(74)-(76) of Ref. 9].The result could then be tested against the measured structure dependencies of the steady shear material functions.We leave this task for another day.When using the references cited herein, it is best to be mindful of corresponding ganged errata in Ref. 26.
, we follow those of the corresponding textbook treatments, Example 16.7-1 of Ref. 4 or Example 13.6-1 of Ref. 5.
of Ref. 1, Fig. 14 of Ref. 9, and Fig. 2 of Ref. 16) of general rigid bead-rod theory for concentrated systems suggests that interactions between macromolecules are also less important.

TABLE I .
Dimensional variables.Legend: M ≡ mass, L ≡ length, t ≡ time, and T ≡ temperature.
8redicted qualitatively.By qualitatively, we mean that the real part of the complex viscosity descends with frequency from its asymptotic zero-shear value and then inflects with frequency, while the imaginary part rises and then falls.8

TABLE II .
Dimensionless variables and groups.
Kanso andGiacomin bridged continuum theory to macromolecular, yielding expressions for the normal stress difference responses from general rigid bead-rod theory.By continuum theory, we mean any case of the Oldroyd 8-constant framework (see TableIIIof Ref. 10 and Table I of Refs.11 and 12), including the corotational Jeffreys model.The coefficients of the displacement term of By substituting Eqs.(64)-(66) of Ref. 9 into Eqs.(68)-(70) of Ref. 9,

TABLE III .
Möbius bands with three rows and an aspect ratio of twenty.

TABLE IV .
Möbius bands with three rows and an aspect ratio of ten.

TABLE V .
Möbius bands with seven rows and an aspect ratio of ten.

TABLE VI .
Möbius bands with seven rows and an aspect ratio of seven.

TABLE VII .
Zero-shear viscosities and first normal stress coefficients made dimensionless with λ 0 .