Orientational anisotropy of bead-spring star chains during creep process

The Model and Outline of Calculation

We consider a monodisperse system of symmetric f-arm star chains obeying the Rouse–Ham dynamics9, 10 (Zimm–Kilb dynamics11 in the free-draining limit). Each arm is composed of N (≫1) segments having the friction coefficient ζ. Neighboring segments are connected with the Gaussian spring of equilibrium length a and spring constant κ = 3k_BT/a² (k_B = Boltzmann constant, T = absolute temperature); cf. Figure 1. For each arm, the segment index n runs from 0 to N, where n = 0 and N specify the branching point and free end of the arm, respectively.

For these star chains, we analyze growth of the orientational anisotropy of the bond vectors of the segments during the creep process under a constant stress in the linear viscoelastic regime. In the continuous treatment (valid for N ≫ 1), this anisotropy is quantified with the orientation function S(n,t):3, 12

(1)

Here, r (n, t) is the ξ component (ξ = x, y) of the spatial position r^[α] (n, t) of the n-th segment in α-th arm at time t, ∂r^[α] / ∂n denotes the bond vector for this segment, and 〈 … 〉 represents an ensemble average. S^[α] (n, t) is the orientation function of the α-th arm, and S(n,t) is defined as the average over all f arms.

In general, the shear stress of the monodisperse system of star chains at time t, σ(t), is related to S(n,t) as3, 12

(2)

Here, ν is the number density of the chains in the system. The time evolution of S(n,t) during the creep process is determined by the Rouse–Ham dynamics under the constraint that σ(t) is kept constant (=σ₀). This constraint results in the interplay/correlation of the Rouse–Ham eigenmodes, as discussed later in more details.

Equation of Motion

In the continuous treatment, the Rouse–Ham equation of motion for the segment position r^[α] (n, t) can be written as3, 8, 12

(3)

Here, κ is the spring constant, and F^[α] (n, t) is the Brownian force acting on n-th segment of α-th arm at time t: F^[α] (n, t) is modeled as a white noise characterized with the dyadic average (second-moment average),3

(4)

In eq 3, V(r^[α]) denotes the flow velocity of the frictional medium for the star chain at the position r^[α] (n, t). For the linear viscoelastic dynamics of our interest, the flow field is uniform in the system and V(r^[α]) can be written as

(5)

Here, represents the strain rate at time t (uniform in the system), and the x- and y-directions are chosen to be the velocity and velocity gradient directions, respectively.

Boundary Condition and Eigenmode Expansion

For the segment at the free end of the arm, the spring force is acting only from the inner-side segment (with n = N−1). The boundary condition representing this situation is identical to that for the linear Rouse chain and can be written, in the continuous treatment, as3, 10-12

(6)

The boundary condition representing the continuity of the star chain at the branching point is given by10, 11

(7)

In addition, we can regard the branching point segment (common to all arms) as a frictionless object and require the spring forces from all arms acting on this segment to be in balance at any time. This force balance requirement gives a boundary condition,10, 11

(8)

The segment position r^[α] (n, t) can be expanded with respect to the sine and cosine eigenfunctions being associated to eq 3 and satisfying the boundary condition, eq 6.11 This expansion is explicitly written as

(9)

where B (t; p) with η = o and e represent the amplitude vectors of the p-th order odd and even eigenmodes for α-th arm at time t. The amplitudes for the f arms are not totally independent but are correlated through the boundary conditions, eqs 7 and 8. The even modes appearing in eq 9 automatically satisfy eq 8 but their amplitudes B are subjected to a constraint due to eq 7,11

(10)

On the other hand, the odd modes automatically satisfy eq 7 but their amplitudes B are subjected to a constraint due to eq 8,11

(11)

Thus, among the p-th order eigenmodes for the f arms, only one even mode and f−1 odd modes behave as the independent eigenmodes.

Substituting eqs 9–11 in eq 1, we can express the orientation function in terms of the eigenmode amplitudes:

(12)

Here, X (t; p) and Y (t; p) represent the x and y components of B (t; p), respectively, and X_e (t; p) and Y_e (t; p) denote those of B_e (t; p). (In eq 12, the cross terms between the odd and even modes vanish because of eqs 10 and 11.) Solving the equation of motion (eq 3), we can calculate the averages 〈XY〉 appearing in eq 12 and obtain an explicit expression of S(n,t).

Eigenmode Analysis

From eqs 3 and 9–11, we can derive the time evolution equations for the amplitude vectors of respective odd and even modes, B (t; p) and B_e (t; p). Details of this derivation are described in Appendix A, and the equations for the x and y components of these vectors relevant to the calculation of S(n,t) (eq 12) are summarized as

(13)

(14)

Here, τ represents the longest viscoelastic relaxation time of the Rouse–Ham star chain:

(15)

F̂ (t; p) and F̂(t; p) indicate the ξ components (ξ = x, y) of the p-th order sine- and cosine-Fourier components of the Brownian force acting on β-th arm,

(16)

(17)

As can be noted from eqs 4, 16, and 17, the dyadic average of F̂ (t; p) (η = o, e and ξ = x, y) is given by

(18)

Note that eqs 13 and 14 are written for each arm but include the Fourier components of the Brownian forces acting on all arms. This feature, representing the coupled motion of the arms, results from the boundary conditions, eqs 10 and 11.

Equation 13 can be solved with a standard method to give the time evolution of Y (t; p) and X (t; p) for p-th odd mode:

(19)

and

(20)

Note that Y (t′; p) in eq 20 is written in the form of eq 19. Similarly, eq 14 gives

(21)

and

(22)

From eqs 19 and 20, the 〈X (t; p) Y (t; q)〉 term included in the orientation function (eq 12) can be straightforwardly calculated. Although eqs 19 and 20 show that this term is contributed from several types of averages such as 〈X (0; p) Y (0; q)〉 and 〈X (0; p) F̂ (t′; q)〉, most of these averages vanish: For example, 〈X (0; p) Y (0; q)〉 = 0 because the chains are at equilibrium at t = 0 (just before application of the stress) to have an isotropic conformation with no correlation between X (0; p) and Y (0; q), and 〈X (0; p) F̂ (t′; q)〉 = 0 because the chain conformation at t = 0 is not correlated with the Brownian force at time t′ > 0. From a careful inspection of eqs 19 and 20, we note that the 〈X (t; p) Y (t; q)〉 term has nonzero contributions only from two types of averages, 〈F̂ (t′; p) F̂ (t″; q)〉 and 〈Y (0; p) Y (0; q)〉, and is written, after some calculation, as

(23)

Similarly, from eqs 21 and 22, we find that the 〈X_e (t; p) Y_e (t; q)〉 term in eq 12 is contributed only from two types of averages, 〈F̂ (t′; p) F̂ (t″; q)〉 and 〈Y_e (0; p) Y_e (0; q)〉, and written as

(24)

In derivation of eqs 23 and 24, we have utilized eqs 15 and 18 to specify 〈F̂ (t′; p) F̂ (t″; q)〉 (with η = o, e) and τ, respectively.

Since the chains are at equilibrium at t = 0 (just before application of the stress), the averages remaining in eqs 23 and 24, 〈Y (0; p) Y (0; q)〉 and 〈Y_e (0; p) Y_e (0; q)〉, can be calculated from the equilibrium distribution function of the eigenmode amplitudes, as explained in Appendix B. The results are summarized as

(25)

(26)

Substituting eqs 23–26 into eq 12, we finally find a compact equation that determines the orientation function:

(27)

with

(28)

Note that this S(n,t) is contributed from no cross-average between the odd and even eigenmodes.

Calculation of S(n,t)

For the simplest strain-controlled process, the relaxation under the step strain , eqs 27 and 28 give

(29)

Utilizing this S(n,t) in eq 2, we recover the well-known expression of the relaxation modulus of the Rouse–Ham star chains10-12 (in the continuous limit):

(30)

For the other type of strain-controlled process, start-up of flow at a constant rate , eqs 27 and 28 give

(31)

In derivation of eq 31, we have replaced by a ratio of the steady state stress σ₀ to the zero-shear viscosity η₀ and further utilized the expression of η₀:12

(32)

For the stress-controlled creep process of our interest, the calculation of S(n,t) is less straightforward. For this case, the strain rate is not externally given but determined by the chains themselves under the requirement of constant stress σ₀. Equations 2, 27, and 28 give the equation that determines :

(33)

As can be noted from eq 30, eq 33 is equivalent to the linear viscoelastic constitutive equation, with for t′ < 0 (where the stress has not been applied).

Equation 33 can be solved with the Laplace transformation method described in Appendix C. The result can be conveniently summarized in the form of the creep compliance:

(34)

where η₀ is the zero-shear viscosity (eq 32) and J_R(t) is the recoverable compliance given by

(35)

with

(36)

The eigenvalues φ_q included in the retardation intensities j_q and retardation times λ_q are determined from

(37)

Some of the low order eigenvalues are summarized in Table 1. The retardation intensities j_q and eigenvalues φ_q satisfy the following summation rules, as explained in Appendixes C and D, respectively.

(38)

(39)

Utilizing the strain rate (with J(t) being given by eqs 34–36) in eq 27, we can find an expression of S(n,t) with the Laplace transformation method described in Appendix D. The results are summarized as

(40)

Here, λ_q is q-th retardation time specified by eq 36, and the n-dependent factors r₀(n) and r_q(n) are given by

(41)

(42)

For f = 2, all above results reduce to those for the linear Rouse chain obtained previously.4, 8 Namely, for f = 2, eq 37 reduces to the previously derived eigenvalue equation for the linear chain, tan θ_q = θ_q with θ_q = 2φ_q. Correspondingly, the expression of J_R(t) (eqs 35 and 36) reduces to the previous expression. The expression of S(n,t), eqs 40–42, were obtained after analytical summation of the eigenmodes of g(n,t) in the Laplace space (as described in Appendix D) and thus superficially different from the previous expression formulated before this summation (eqs 13 and 27 in ref.4). However, no real difference exists between the present and previous expressions of S(n,t), as explained later for the anisotropies of the eigenmode amplitudes.

Anisotropy Distribution along the Arm Contour

For the Rouse–Ham star chains with various arm numbers f, Figure 2 shows the normalized orientation function during the creep process, S° (n, t) = {Nfνk_BT/σ₀} S(n, t) (eq 40). This S° (n, t) is plotted against the normalized segment coordinate, n/N, at several reduced times, t/λ₁ = 0.01, 0.1, 0.3, 1, and ∞, with λ₁ being the longest retardation time (eq 36). For comparison, Figure 3 shows S° (n, t) after start-up of constant-rate flow (eq 31); the plots are shown at t/τ = 0.01, 0.1, 0.3, 1, and ∞, with τ being the longest relaxation time (eq 15). In both figures, the plots for f = 2 indicate the orientation function of the linear Rouse chain composed of 2N segments, and n = 0 corresponds to the segment at the chain center.

In Figure 2, the horizontal dotted line indicates the S° value expected for the affine deformation against the stress σ₀ (uniform deformation along the arm contour); S°_affine = 1/3. For all f values, S° (n, t) during the creep process at short t is only weakly dependent on n and its value in the major part of the range of 0 < n/N < 1 is close to this affine value; see the plots at t/λ₁ = 0.01 (triangles). In fact, in the limit of t → 0, eq 40 gives S° (n, t) = S°_affine in the entire range of 0 < n/N < 1 irrespective of the arm number f. This affine deformation enables the chain having finite retardation times to instantaneously respond to the applied stress, as fully discussed in the previous papers.5, 8 Note also that thenon-normalized orientation function has an infinitesimal value on the affine deformation, S(n,0) = σ₀/3Nfνk_B T → 0 for N → ∞, which corresponds to an infinitesimal value of the initial compliance, J(0) → 0 for N → ∞.

As seen in Figure 2, the flat, affine profile of S° changes with t in a way that the S° (n, t) value increases and decreases, respectively, for the segments near the branching point (n/N = 0) and the arm end (n/N = 1). Namely, S° (n, t) of the segments near the arm end approaches its steady state (S° (n, ∞) shown with the thick solid curves) after exhibiting the overshoot at short t. This overshoot is a natural consequence of the constant-stress requirement during the creep process: The total orientational anisotropy, being equivalent to the stress (cf. eq 2), is kept constant during this process, but the segments near the branching point cannot quickly attain their steady state anisotropy. (The anisotropy near the branching point grows considerably even at short t but the final growth to the steady state value occurs only slowly in a time scale of t ≥ λ₁; cf. Fig. 2.) Thus, the segments near the arm end, which can relax faster than the branching point segment, compensate for the delay in this final growth near the branching point by exhibiting the overshoot at short t. In other words, the overshoot reflects the interplay of the segments along the arm contour under the constant-stress requirement.

After start-up of the constant rate flow, S° (n, t) changes with t quite differently: As seen in Figure 3, S° (n, t) of all segments monotonically increases with t, and the approach to S° (n, ∞) is faster for the segments near the arm end. Thus, qualitatively speaking, the segments along the arm contour can determine their orientational anisotropy more independently under the constant-rate flow than in the creep process (because of the lack of the constant-stress requirement for the former). This difference between the creep and constant-rate flow processes can be noted also for the anisotropy of the eigenmode amplitude explained later.

Qualitatively, the above features are common to the star chains of any arm number f (including the linear chain with f = 2). However, we also note a quantitative difference for the star chains with various f. This difference can be most clearly demonstrated in Figure 4 where the S° (n, t) profiles of the stars with various f during the creep process are directly compared.

From eq 40, S° (n, ∞) in the steadily flowing state is given by

(43)

Of course, this S° (n, ∞) profile coincides with that under the rate-controlled flow given by eq 31 with t → ∞. (This coincidence can be easily noted by applying mathematical formulae,14, 15 Σ p⁻² cos px = { 3(x − π)² − π²} /12 for 0 ≤ x ≤ 2π and Σ (2p − 1)⁻² cos (2p − 1)x = π (π − |2x|) /8 for −π ≤ x ≤ π, to eq 31.)

As seen in Figure 4(b), the negative gradient of the steady state profile at n/N = 0, [dS° (n, ∞) / d }n / N}]_{n/N = 0}, is enhanced with increasing f > 2 (and approaches −2/3 for f → ∞; cf. eq 43). Thus, the orientational anisotropy in the steady state is more heterogeneously distributed along the arm contour for larger f. This behavior is a natural consequence of the fact that the branching point is subjected to the tensile forces from all arms and thus has a larger, steady state orientation for larger f. Since the total orientation is kept constant during the creep process, the enhancement of the orientation near the branching point reduces the orientation of the segments far from this point, thereby giving the larger heterogeneity for larger f.

The situation is essentially the same in the transient state at t/λ₁ = 0.3 [cf. Fig. 4(a)]. The orientation near the branching point is enhanced for larger f, as explained above. Under the constant-stress (constant total orientation) requirement, this enhancement allows S° (n, t) of the middle part of the arm (at around n/N = 0.5) to remain closer to the initial affine profile (S° (n, 0) = 1/3) shown with the horizontal dotted line.

Anisotropy of Eigenmode Amplitudes

Here, we focus on the time evolution of the anisotropy of the amplitude vector B of the Rouse–Ham eigenmode during the creep process. For this purpose, we utilize the xy component of the dyadic average 〈 BB〉 to define the normalized anisotropy of the eigenmode as

(44)

(45)

For f = 2, this definition reduces to the previous definition4 for the linear Rouse chain (composed of 2N segments).

The t dependencies A_odd and A_even represent the change of the orientational anisotropy of the star chain through the odd and even eigenmodes, the former involving no motion of the branching point while the latter activating the synchronized motion of all arms and being associated with the motion of the branching point. A_odd and A_even can be determined from a rheo-optical experiment for a series of star chains having the same architecture and differently located optical label and Fourier analysis of the resulting rheo-optical data with respect to the label location.4

These A(t;p) can be straightforwardly calculated from the orientation function S(n,t). From eq 12and eqs 23–26, we note that S(n,t) is written in terms of A(t;p) as

(46)

Thus, we can obtain A(t;p) as the Fourier components:

(47)

(48)

For S(n,t) explicitly given as the function of n and t (eqs 40–42), we conduct the integral shown in eqs 47 and 48 to find

(49)

and

(50)

For f = 2, eqs 49 and 50 reduce to the expression of A for the linear Rouse chain derived in the previous work (eq 27 in ref.4 with θ_q therein being equivalent to 2φ_q), confirming that eqs 40–42 derived in this study includes the previously obtained expression of S(n,t) for the linear Rouse chain as a special case.

For the star chains with various f in the creep process, Figure 5 shows the time evolution of low order eigenmode anisotropies, A_odd(t;p) and A_even(t;p) with p = 1 and 2. These anisotropies are plotted against the normalized time, t/λ₁ (λ₁ = longest retardation time). For comparison, Figure 6 shows the mode anisotropies after start-up of constant-rate flow plotted against the other type of the normalized time, t/τ (τ = longest relaxation time; cf. eq 31). This anisotropy is easily obtained from eq 31 as

(51)

(52)

In both Figures 5 and 6, thick solid and dotted curves indicate the odd mode anisotropies, A_odd(t;1) and A_odd(t;2), respectively, and the thin solid and dotted curves show the even mode anisotropies, A_even(t;1) and A_even(t;2).

As noted from eqs 51 and 52, the growth of the anisotropy for respective eigenmodes under the constant-rate flow has a single characteristic time, τ/(2p−1)2 for p-th odd mode and τ/4p² for p-th even mode. Correspondingly, the approach to the steadily flowing state (leveling-off of the plots shown in Fig. 6) is faster for higher order modes. These features indicate that respective eigenmodes behave as independent modes under the rate-controlled flow.

In contrast, during the creep process, the growth of the anisotropy of respective modes is associated with an infinite number of characteristic times λ_q; cf. eqs 49 and 50. Correspondingly, the anisotropy exhibits the overshoot except for the first mode (slowest odd mode; thick solid curve), and the steady state is attained simultaneously for all modes; see Figure 5. These features are a natural consequence of the interplay/correlation of the modes under the constant-stress requirement. The fast modes compensate for the slow growth of the anisotropy of the first odd mode at short t and thus all modes approach their steady state in a synchronous way. This compensation results in the overshoot of the anisotropy of the fast modes.

Finally, we turn our attention to a quantitative difference of the magnitude of interplay of the eigenmodes for the star chains with various f. This difference can be most clearly seen in Figure 7 where the anisotropies of the second odd mode and the lowest two even modes are normalized at their overshoot peaks and plotted against a reduced time, t/t_peak with t_peak being the time for the overshoot peak. The overshoot of these anisotropies is moderately enhanced with increasing f. This behavior is related to a relative decrease of the contribution of the even eigenmodes to S(n,t) with increasing f (compared to the odd mode contribution) noted in Figure 5 and eqs 49 and 50. The even modes of all orders p as well as the odd modes with p ≥ 2 exhibit the overshoot so as to compensate for the slow growth of the first odd mode, as explained above. Thus, the decrease of the even mode contribution requires the odd modes with p ≥ 2 and all even modes to make a larger compensation for the first odd mode, which results in the enhanced overshoot seen in Figure 7.