Systematic development of coarse-grained polymer models Patrick ...

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18 2.2. Brownian Dynamics Iterations j ≥ 3 proceed like iteration 2 except for one slight change Qi,j(t + δt)+ 2δt ζ FsprQi,j(t + δt) = Qi(t)+ 1 2 κ · Qi(t)+Qi,j−1(t + δt) δt + δt F ζ spr Qi+1,j−1(t + δt) + F spr Qi−1,j(t + δt) + Bi(t) . After iterations j ≥ 3 a residual error between iterations is calculated as ɛ = N−1 i=1 (2.20) Qi,j(t + δt) − Q i,j−1(t + δt) 2 . (2.21) The iterations are stopped when this ɛ is below a specified tolerance. We have chosen a tolerance of 10 −6 times the fully-extended length of a single spring. Note that it should be verified that the tolerance is small enough such that the results are independent of the size of the tolerance. Let us pause here to review the result of the semi-implicit method. The Brownian force contribution is treated in a explicit manner. Note that the Brownian force is evaluated once, during the first iteration, and the same value is used for all subsequent iterations. The flow force is treated in a type of midpoint algorithm. The flow force used is the average between the explicit force and the force at the next to last iteration. However, because the iterations have converged, the next to last iteration value and the final value are essentially the same. Therefore, the flow force is essentially an average between the initial force and final force. The spring force for the current connector vector i and the previous vector i − 1 in the current iteration are treated implicitly. The spring force in the next connector vector i+1 (not yet calculated in the current iteration) is used from the previous iteration. Since the iterations have converged, this is essentially like calculating all spring forces implicitly. Therefore, by construction, each spring will not be beyond the fully-extended length of a spring. The final issue to discuss in terms of the semi-implicit method is solving the nonlinear equation and the use of a look up table. Recall the equation to be solved is Q + 2δt ζ Fspr Q = R . (2.22) Because F spr Q is parallel to Q, we can deduce that R is also parallel and reduce the equation to a scalar equation by taking the dot product with a unit vector in the direction of Q. Wenowcan consider the scalar equation Q + 2δt ζ F spr Q = R, (2.23) where Q is the magnitude of the vector Q, R is the magnitude of the vector R, andF spr is the scalar spring force function defined by F spr Q = Q · F spr Q /Q . (2.24) Figure 2.1 shows a sketch of the solution to equation (2.23). The function is generally a smooth function which would lend itself to using a look up table with linear interpolation between points in the look up table. However, some issues must be noted and addressed. First, as the time step
2.2. Brownian Dynamics 19 Q/ℓ 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 1 2 3 4 5 R/ℓ Fig. 2.1: Sketch of the solution to the nonlinear equation for Q as a function of R used in the semi-implicit integration method, where ℓ is the fully-extended length of a spring. gets progressively smaller, the function has a sharp turn over from a slope of nearly 1 to a slope of nearly zero. This sharp turn over does not lend itself well to linear interpolation, and linearly interpolating in that region results in serious errors. Moreover, that is precisely the region of most importance because it is near full extension of the springs. The only times where the semi-implicit method is necessary is when the springs are near full extension. This can result in the strange result that as the time step is reduced, the error in the simulation actually increases because of the large error from the linear interpolation. Another word of caution with respect to a look up table concerns the maximum value of R within the table. Because the range of R extends to infinity but the look up table is necessarily finite, a method must be devised to choose the maximum R value in the look up table and to deal with situations in which the R value needed in the simulation is larger than the maximum value in the look up table. We can deal with these issues by expanding Q for large R. This is done by expanding the left hand side of equation (2.23) asQ approaches the fully-extended length. This series can then be inverted to obtain the series expansion of Q for large R. If the value of R in the simulation is larger than in the look up table, a truncated series expansion is used. The maximum value of R in the look up table and the number of terms used in the truncated expansion must be determined such that the error in Q because of the use of the truncated series is small enough. The last word of caution is concerning the accuracy required for Q, both in terms of the tolerance needed when generating the look up table, in terms of the linear interpolation and the
Page 1: Systematic development of coarse-gr
Page 4 and 5: a number of people I met at MIT tha
Page 6 and 7: 3.3 DimensionlessParameters........
Page 9 and 10: List of Figures 2.1 Sketch of the s
Page 11 and 12: 4.18 Relative error in the second m
Page 13: List of Tables 3.1 Summaryofdimensi
Page 16 and 17: 2 Abstract for other interactions b
Page 18 and 19: 4 1.2. General Polymer Physics prop
Page 20 and 21: 6 1.3. Progression of Coarse-graine
Page 22 and 23: 8 1.3. Progression of Coarse-graine
Page 24 and 25: 10 1.3. Progression of Coarse-grain
Page 26 and 27: 12 1.5. Importance of New Models So
Page 28 and 29: 14 2.2. Brownian Dynamics the effec
Page 30 and 31: 16 2.2. Brownian Dynamics where now
Page 34 and 35: 20 2.2. Brownian Dynamics use of a
Page 36 and 37: 22 3.2. Decoupled Springs fixed sol
Page 38 and 39: 24 3.3. Dimensionless Parameters pa
Page 40 and 41: 26 3.4. Force-extension Results 〈
Page 42 and 43: 28 3.5. Phase Space Visualization
Page 44 and 45: 30 3.6. Effective Persistence Lengt
Page 46 and 47: 32 3.6. Effective Persistence Lengt
Page 48 and 49: 34 3.7. Fluctuations 〈ˆztot〉m
Page 50 and 51: 36 3.8. Limiting Behavior α 1/2 δ
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Page 56 and 57: 42 3.9. FENE and Fraenkel Force Law
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4.2. Application to Freely Jointed
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4.3. Application to Worm-like Chain
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4.4. Summary and Outlook 85 freely
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CHAPTER 5 Low Weissenberg Number Re
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5.1. Retarded-motion Expansion Coef
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5.3. Influence of HI and EV 99 the
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CHAPTER 6 High Weissenberg Number R
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6.1. Longest Relaxation Time 103 ti
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6.1. Longest Relaxation Time 105 τ
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6.1. Longest Relaxation Time 107 τ
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6.2. Elongational Viscosity 109 ˆ
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6.2. Elongational Viscosity 111 ove
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6.2. Elongational Viscosity 113 whe
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6.2. Elongational Viscosity 115 6.2
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6.2. Elongational Viscosity 117 ˆ
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6.2. Elongational Viscosity 119 pol
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6.3. Influence of Hydrodynamic Inte
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CHAPTER 7 Nonhomogeneous Flow In th
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7.1. Dumbbell Model 125 consider (
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7.2. Long Chain Limit 127 〈Xf〉/
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7.3. Finite Extensibility 129 (L
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CHAPTER 8 Conclusions and Outlook I
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8.4. Nonhomogeneous Flow 133 presen
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Appendix A Appendices A.1 Fluctuati
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A.2. Retarded-motion Expansion Coef
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A.3. Example of the Behavior of the
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A.5. Calculation of the Response of
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A.5. Calculation of the Response of
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146 Bibliography [10] P. J. Flory,
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148 Bibliography [57] J. S. Hur, E.
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Systematic development of coarse-grained polymer models Patrick ...

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