The liquid region

at^ν < R. In the other case we have to interprete the series as an asymptotic expansion of G(t). This will occur for series describing various long time asymptotes. Hence we call the sum P_∞

if for each N the following condition is satised [17]:

t→∞lim

We can use this condition for example to determine numerically α and a₁. In the following we will test our results numerically for the parameter setk= 2 listed in table 2.1 and^¯¯¯γ_Γ^˙¯¯¯= 10⁻¹⁰. With this we obtain ²_γ˙ = 8.41·10⁻⁶.

2.2.1 The liquid region

For the numerical calculations in this region we choose ²=−10⁻³.

The short time dynamics

On the left side of equation (2.18) we can neglect the eect of the shearing for suciently short times. We postulate the initial condition (2.21) which lead us to α = (t₀)^a, u =−aand Γ_0,0 = 0. The resulting equation leads to ν = 2a and ˜a = − |²|(t₀)^−2a. We observe that the sequence A_n converges to zero. We obtain numerically R₂₆₃ = 4.35, hence we expect that the series converges for log₁₀(Γt) < 5.25. The numerical results shown in gure 2.11 support our expectation. Increasing N also increases the accuracy. The two-parameter scaling law (2.25) allows us to introduce a natural time scale. By choosing Ω =|²|^2a1 we obtain |ˆ²|= 1 and ˆt= _t^t with ²

t_² ≡ t₀|²|^−2a1 .

For the numerical value we obtain Γt_²= 1.82·10⁴. The short time solution is represented by:

G(t) = |²|¹²

Figure 2.11: Numerical solution of the β-scaling equation and the short time power series (2.30) evaluated to the order N for ²=−10⁻³,^¯¯¯γ_Γ^˙¯¯¯= 10⁻¹⁰ and the parameter set k= 2. We observe a nite radius of convergence. Increasing N also increases the accuracy.

The intermediate time dynamics

As shown in gure 1.4, in the liquid region the nal decay ofφ(t)is not eected by the shearing.

Hence we rst consider the case of a vanishing shear rate. The nal decay requires an increasing

|G(t)|, hence we solve equation (2.18) by choosingα=−(τ₀)^−b andu=b. Then withΓ_0,0 = 0 we conclude ν =−2b, ˜a= − |²|(τ₀)^2b and ^¯¯¯t^−bG(t)^¯¯¯ → (τ₀)^−b for t → ∞. We use this fact to test our ansatz and for the numerical determination ofτ₀. Figure 2.12 shows that for a vanishing shear rate the exponentbdescribes the long time asymptote of theβ-correlator correctly and we

obtain Γτ₀ = 7.62·10⁶. We observe that the sequence A_n diverges and the numerical data are not sucient to decide weather R_n has a positive lower limit R or not. Hence we only expect that in the case of a vanishing shear rate the series represents an asymptotic expansion ofG(t). We dene

τ_² ≡ τ₀|²|^2b1

with the numerical value Γτ_² = 3.14·10⁴ and obtain:

G(t,γ˙ = 0) ∼ − |²|¹² µt

τ_²

¶_bà 1 +

X∞

n=1

(−1)ⁿA_n µ t

τ_²

¶_−2bn!

. (2.31)

Now we consider the case of a non-vanishing shear rate. For times where c^{( ˙γ)}( ˙γt)² ¿ |²| is satised, we can approximate:

G(t) ≈ G(t,γ˙ = 0).

Hence we expect a nite time window where we can approximate G(t) by evaluating equation (2.31) to some ordersN. Figure 2.13 shows that this window extends to times where theβ-scaling regime is already left and gure 2.14 shows that the window also extends to the convergence regime of the short time series. The fact that increasingN decreases the accuracy is an indication of a divergent series.

5 10 15 20 25

log₁₀(Γt) -4.34

-4.335 -4.33 -4.325 -4.32

log 10(|(Γt)-b G|)

|γ. /Γ|=0 -4.32836

Figure 2.12: The plot for ² = −10⁻³ and the parameter set k = 2 shows that for a vanishing shear rate the exponent b describes the long time asymptote of the β-correlator correctly. By tting a horizontal line we obtain the numerical value for −blog₁₀(Γτ₀).

-2 0 2 4 6 8 10 12 log₁₀(Γt)

-1 -0.5 0 0.5 1

G

numerical solution power series (N=1) power series (N=2) power series (N=5) power series (N=10) power series (N=20) power series (N=50)

Figure 2.13: Numerical solution of theβ-scaling equation and the intermediate time power series (2.31) evaluated to the order N for ²=−10⁻³, ^¯¯¯γ_Γ^˙¯¯¯= 10⁻¹⁰ and the parameter setk = 2. The fact that increasing N decreases the accuracy is an indication of a divergent series.

5.1 5.15 5.2 5.25 5.3

log₁₀(Γt) -0.1

-0.09 -0.08 -0.07 -0.06

G

numerical solution short time series (N=263) intermediate time series (N=1)

Figure 2.14: The plots show for ²=−10⁻³, ^¯¯¯_Γ^γ˙

¯¯

¯= 10⁻¹⁰ and the parameter set k= 2 that the range of validity of the intermediate time series (2.31) extend to the convergence regime of the short time series (2.30).

The long time dynamics

For suciently long times we can neglect ² on the left side of equation (2.14). We solve the resulting equation by choosing u= 1and Γ_0,1= 0. Then with Γ_0,0 = ¹₂ −λand Γ_0,1 = _2+ν¹ −λ We use this to test our ansatz and for the numerical determination of a₁. Figure 2.15 supports our ansatz and we obtain Γ^ca₁ = 2.03·10⁸. We observe that the sequence a_n diverges. The numerical data are not sucient to decide weather r_n has a positive lower limit r or not, hence we only expect that the series represents an asymptotic expansion of G(t). The fact that only low orders N lead to good approximations is an indication for a divergent series as shown in gure 2.16. We dene

τ_γ˙ ≡ 1

By complete induction we easily prove thatb_n≥2can also be determined by the following recursion formula:

Figure 2.17 shows the region where the intermediate time regime merges into the long time regime. We observe a gap but we also see that in our example the long time regime lies beyond theβ-scaling regime.

12.5 15 17.5 20 22.5 25 27.5 30 log₁₀(Γt)

8 8.1 8.2 8.3 8.4 8.5 8.6

log 10(-(G/(t/τ γ.)+1)(Γt)c )

numerical solution 8.30669

Figure 2.15: The plot for ²=−10⁻³,^¯¯¯_Γ^γ˙

¯¯

¯= 10⁻¹⁰ and the parameter set k= 2 shows that the exponents 1 and c describe the long time asymptote of the β-correlator correctly. By tting a horizontal line we obtain the numerical value for log₁₀(Γ^ca₁).

8 9 10 11 12 13 14 15 16

log₁₀(Γt) -50000

-40000 -30000 -20000 -10000 0 10000 20000

G

numerical solution power series (N=1) power series (N=2) power series (N=5) power series (N=10) power series (N=20)

Figure 2.16: Numerical solution of the β-scaling equation and the long time power series (2.32) evaluated to the order N for ² = −10⁻³, ^¯¯¯γ_Γ^˙¯¯¯ = 10⁻¹⁰ and the parameter set k = 2. The fact that only low orders N lead to good approximations is an indication of a divergent series.

12 12.5 13 13.5 14 log₁₀(Γt)

-15000 -10000 -5000 0

G

numerical solution

intermediate time series (N=1) long time series (N=3)

Figure 2.17: The region where the intermediate time regime merges into the long time regime for ²=−10⁻³,^¯¯¯γ_Γ^˙¯¯¯= 10⁻¹⁰and the parameter setk= 2. We observe a gap but we also see that in our example the long time regime lies beyond theβ-scaling regime.

The second intermediate time regime

To nd a solution for times between the range of validity of equation (2.31) and equation (2.32), we neglect ² on the left side of equation (2.18) and solve the resulting equation by choosing α = −(τ₀)^−b and u = b. This lead us to Γ_0,0 = 0, ν = 2−2b and ˜a = −c^{( ˙γ)}γ˙²(τ₀)^2b. We observe that the sequenceA_n diverges. But because of the behavior of the sequence R_n and the numerical result R₂₂₉ = 0.122 we expect that the series converges for log₁₀(Γt) < 14.4. The numerical results shown in gure 2.18 support our expectation. Increasing N also increases the accuracy. We dene

τ_b ≡ τ₀|γτ˙ ₀|^−1−b1

with the numerical valueΓτ_b = 1.93·10¹⁵. The second intermediate time solution is represented by:

G(t) = − |γτ˙ _b| µ t

τ_b

¶_bà 1 +

X∞

n=1

(−1)ⁿA_n^³c^{( ˙γ)´n} µ t

τ_b

¶_(2−2b)n!

. (2.33)

Figure 2.19 shows that equation (2.33) describes the regime between the range of validity of equation (2.31) and equation (2.32).

14 14.125 14.25 14.375 log₁₀(Γt)

-80000 -60000 -40000

G

numerical solution power series (N=1) power series (N=2) power series (N=5) power series (N=10) power series (N=20) power series (N=50)

Figure 2.18: Numerical solution of the β-scaling equation and the second intermediate time power series (2.33) evaluated to the order N for ²=−10⁻³, ^¯¯¯_Γ^γ˙

¯¯

¯= 10⁻¹⁰ and the parameter set k= 2. We observe a nite radius of convergence. IncreasingN also increases the accuracy.

12 12.5 13 13.5 14

log₁₀(Γt) -15000

-10000 -5000 0

G

numerical solution

intermediate time series (N=1) long time series (N=3)

second intermediate time series (N=229)

Figure 2.19: As shown for ²=−10⁻³,^¯¯¯γ_Γ^˙¯¯¯= 10⁻¹⁰ and the parameter setk= 2, equation (2.33) describes the regime between the range of validity of equation (2.31) and equation (2.32).

Im Dokument Scaling laws in the rheology of colloidal dispersions (Seite 37-45)