dissipative particle dynamics · mesoscale modelling techniques lattice gas cellular automata lgca...

Dissipative Particle Dynamics

Michael SeatonUKRI STFC Daresbury Laboratory

[email protected]

Mesoscale modelling• Fills simulation gap

between atomistic and macroscopic methods

• Both thermodynamics andhydrodynamics involved– Need atom-like and fluid-

like effects• Many complex systems

act at mesoscale, e.g.– Surfaces and interfaces– Microfluidics– Biological membranes– Phase behaviourSource: Nielsen et al., J Phys Cond Matter,

16, R481–R512 (2004)

Physical scales

Image courtesy: PV Coveney, UCL

Mesoscale modelling techniquesLattice Gas Cellular Automata LGCA Multiple Component LGCALattice Boltzmann Equation LBEMultiple Component LBEDissipative Particle Dynamics DPDSmooth particle hydrodynamics SPHLangevin Equation / Brownian Dynamics /Stokesian Dynamics SD

Hydrokinetic methods

Mesoscale simulation strategies• Bottom-up:

• Top-down:

Construct methodology for coarse-graining:(a) microscopic particles into sub-

thermodynamic populations of mesoscopic particles, and

(b) their force interactionsThese result in a commensurate reduction in number of degrees of freedom

Reverse engineer simulation rules to obtain desired behaviour (process may, in itself, provide insight)

Mesoscale simulation strategiesFij

Feff

Sub-thermodynamic

population

In a simple liquid, a single atom/molecule moves in the force-field of many neighbours and interacts or thermalizesstrongly: O(102) particles required to make a sub-thermodynamic population

Dissipative Particle Dynamics• A brief (incomplete!) history

– 1992: Hoogerbrugge and Koelman[1] devised DPD as off-lattice LGCA

– 1995: Español and Warren[2] provided enhanced theoretical basis to DPD

– 1997: Groot and Warren[3] devised systematic method to obtain DPD parameters for mesoscopic simulation

– 1999: Flekkøy and Coveney[4] demonstrated link between MD and DPD via coarse-graining and ensemble averaging

– 2003: Español and Revenga[5] combined DPD and SPH to give thermodynamically consistent model (SDPD)

[1] Hoogerbrugge and Koelman, EPL 19, 155–160 (1992)[2] Español and Warren, EPL 30, 191–196 (1995)[3] Groot and Warren, J Chem Phys 107, 4423–4435 (1997)[4] Flekkøy and Coveney, Phys Rev Lett 83, 1775–1778 (1999)[5] Español and Revenga, Phys Rev E 67, 026705 (2003)

DPD algorithm• Similar to classical MD• Condensed phase system modelled by particles

interacting via pairwise forces (usually soft)• System coupled to a heat bath using pairwise

stochastic and drag forces• At equilibrium (steady state), DPD system properties

calculated as ensemble averages

DPD algorithm• DPD computational ‘particles’ (‘beads’)

– Either populations of physical particles (coarse-graining) or ‘momentum carriers’

– If soft, are ‘transparent’ and can pass through each other

– Only discretising time, not position nor velocity– Newtonian dynamics: described by set of stochastic

differential equationsDPD equations of motion

Subscript ! identifies the !thDPD particle

"# =%&#%'

(# = )#%"#%'

DPD algorithm• Pairwise forces within short

distance (cut-off) calculated• Sum of forces determines

bead motion:

• Forces integrated numerically over time step Δ"

i

j #$%&#" = 1

)&*+,&

-&+. +*+,&

-&+0 +*+,&

-&+1

DPD algorithm• Dissipative force[1]

– Removes kinetic energy from centre-of-mass frame of particle pair (proportional to relative velocity)

• Random force

– Drives Brownian fluctuations

where

defining !"# as Gaussianfluctuations, such that:

$"#% = −()% *"# +,"# ⋅ ."# +,"# ."# = .# − .","# = ,# − ,"+,"# = ,"#/*"#

$"#0 = 1)0 *"# !"# +,"#

!"# 2 = 0, !"# 2 !"5#5 26 = 7""57##5 + 7"#57#"5 7 2 − 26

[1] Groot and Warren, J Chem Phys 107, 4423–4435 (1997)

DPD algorithm• Conservative force

– Can take any form, but usually defined as[1]:

CijF

ijA

ijrcr0

Finite force/potential at zero separation:soft repulsive force

!"#$ = &'"#($ )"# *+"#, )"# < ).

0, )"# ≥ ).where ). =

A"# =($ )"# =

Cut-off radiusMaximum repulsion1 − )"#/).


DPD algorithm• Groot-Warren potential:

– As !"# → 0, & → '()"#!*

– & ≥ 0 for all values of !"#• cf. Lennard-Jones potential:

– As !"# → 0, & → ∞

– &."/ = −2"# at !"# =3 25"#

– In practice, !* ≥ 2.55"#

&89 !"# = :;<)"#!* 1 −

!"#!*

<, !"# < !*

0, . !"# ≥ !*

&@A !"# = 42"#5"#!"#

;<

−5"#!"#

C

)"# = 6, !* = 12"# = 1, 5"# = 0.55

Note that E"#F = −∇& = − HIHJKL

MN"#

DPD algorithm• Forces are:

– Momentum conserving– Angular momentum conserving– Pairwise– Central

• Soft potentials appealing since:– Interactions are local and finite-ranged– Longer time steps possible than for classical MD

• Equilibria can be reached more quickly/directly

Theoretically most important

Fokker-Planck formulation• Determines relationship between dissipative and

random forces[1]

• Start with stochastic differential (Langevin) equations:

– where !"#$ = !"$# ≡ '#$!( is an increment of a Wiener (Brownian) process and

!)# = *#!( =+#,#

!(

!+# = -$.#

/#$0 1#$ −-$.#

345 1#$ 6)#$ ⋅ *#$ 6)#$ !( +-$.#

94: 1#$ 6)#$!"#$

!"#$!"#;$; = (=##; =$$; + =#$;=$#;)!(

[1] Español and Warren, EPL 30, 191–196 (1995)

Fokker-Planck formulation• Liouville’s equation:

• Can reformulate SDEs as a two-part Liouvilleoperator (Fokker-Planck formulation):

Conservative (Hamiltonian) term

Second derivative dissipative/ stochastic term

!"!# = − &

' ⋅ ∇* + , ⋅ ∇& " = −-."

. = ./ + .0

./ = − 12

&2'

!!*2

+ 12,452

,24/ 624!!&2

.0 =12

!!72

890 624 :*24 ⋅ ;24 + <=

2 9? 624=624

!!&2

− !!&4

Fokker-Planck formulation• At a steady state:

• For a conservative (Hamiltonian) system, solution is:

• This solution must hold when dissipation is included!• Therefore: !"#$% = 0

Under these conditions, dissipative and random forces act as system thermostat and

preserve hydrodynamics

(#$%() = !#$% = 0

#$% = 1+ exp − 1

k1234

546284

+ 123:; <4:

=" <4: = => <4:6

?6 = 2@12A

Fokker-Planck formulation• Possible to link dissipative/random forces with

macroscopic Navier-Stokes fluid equations[1]

– Ignoring conservative interactions, can derive an approximate Fokker-Planck-Boltzmann equation for evolution of particles (based on distribution functions)

– Can then apply Chapman-Enskog analysis to (eventually) attain Navier-Stokes equations

– Definitions of pressure tensors lead to relationships between dissipative force parameter, fluid viscosity and self-diffusivity

[1] Marsh et al., Phys Rev E 56, 1676–1691 (1997)

DPD fluid• Viscosity and self-diffusivity

– Normally assume screening function is

– Above Fokker-Planck analysis leads to following expressions based on kinetic and dissipative pressure tensors[1,2]:

!" #$% = '1 − #$%/#+, #$% < #+0, #$% ≥ #+

0 ≈ 454564789#+:

+ 2789#+=

1575? ≈ 45456

2789#+:[1] Marsh et al., Phys Rev E 56, 1676–1691 (1997)[2] Visser et al., J Chem Phys 214, 491–504 (2006)

DPD fluid• Viscosity and self-diffusivity

– Ratio of kinematic viscosity/diffusivity (Schmidt number) of order 1

• OK for gases, too low for liquids?

• Higher dissipative force parameters may make simulations unstable

– May need to consider alternative pairwise thermostats for higher viscosities/liquids[1]

[1] Lowe, EPL 47, 145–151 (1999)

DPD fluid• Equation of state

– Derived from virial theorem:

– For ‘standard DPD’ conservative force (single component), when ! > 2:

Scalar form of conservative force

Radial distribution function

$ = !&'( +13, -

./010. ⋅ 30.4

$ = !&'( +25!63 7

8

9:;< ; = ; ;6 >;

$ = !&'( + ?@!6. ? ≈ 0.101;DE

Applying DPD for simple fluids• Conservative force parameters

– Values for a single component can be determined from fluid compressibility and equation of state:

– e.g. for water (one molecule per bead), !"# ≈ 16 and:

!"# = ()*+,-./!0

!"# = 1-./

1213 0

!"# = 1 + 2673-./

7 ≈ 75-./3

Dimensionless form of compressibility: value

depends on real volume of a DPD bead


– Liquid-liquid interactions can be modelled by tuning !"#between different species

– Parameters can be mapped onto Flory-Huggins solution theory for polymer solutions (for !"" = !##)[1]:

Represents excess free

energy of mixing%"# =

2'( !"# − !""*+,

%"# ∝ Δ!"#



– Proportionality constant for given particle density can be determined from series of monomer simulations (finding local volume fraction !"):

– Proportionality constants hold for wide range of solubilities

plies that the first and second derivative of the chemical po-tential with respect to fA vanish, which leads to the criticalpoint

xcrit512S

1ANA

11

ANBD2

. ~19!

Since the present simulated system is fairly incompressible(k21516), and since the excess pressure is quadratic in thedensity, the soft sphere model is by nature very close to theFlory-Huggins lattice model. The free energy density thatcorresponds to the pressure of a single component, Eq. ~16!,is

f VkBT

5r ln r2r1aar2

kBT, ~20!

hence for a two component system of chains one expects

f VkBT

5rANAln rA1

rBNBln rB2

rANA

2rBNB

1a~aAArA

212aABrArB1aBBrB2 !

kBT. ~21!

If we choose aAA5aBB and assume that rA1rB is approxi-mately constant,

f V~rA1rB!kBT

'xNAln x1

~12x !

NBln~12x !1xx~12x !

1constants, ~22!

where we have set x5rA /(rA1rB) and made the tentativeidentification

x52a~aAB2aAA!~rA1rB!

kBT. ~23!

Apparently we have the correspondence ~soft spheres!f V /(rA1rB)5F ~Flory-Huggins!, with a x-parameter map-ping given by Eq. ~23!.

To test this relation simulations have been performed forbinary mixtures of both monomers and polymers, atr5rA1rB53 and r55, for repulsion parametersa5aAA5aBB525 and a515, respectively. When the excesspressure was measured as a function of the fraction x of Aparticles it was found that it is indeed proportional tox(12x). However, unlike the assumption in Eq. ~22! it wasfound that the prefactor of x(12x) is not simply linear inDa5aAB2a when Da is 2 to 5. In practice we are notinterested in small differences in the repulsion, but systemswill rather be chosen where segregation takes place, i.e.,x.xcrit. Now if x is much larger than the critical value,mean field theory is expected to be valid. This means that wecan use Eq. ~18! as the defining equation for the correspond-ing Flory-Huggins parameter.

Adopting this strategy, a system of size 838320 con-taining 3840 monomers was simulated. Half the particleswere of type A and in the initial configuration they wereplaced in the left half of the system, the other particles, oftype B , were placed in the right hand side. For these systems

the density profiles of A and B particles were sampled acrossthe interface. Averages of the density were taken over 105timesteps; the mean value of x over a slab where the densityis homogeneous was then taken to compute the correspond-ing Flory-Huggins parameter. An example of such a binarydensity profile is shown in Fig. 6. Note the small dip in thesum of the densities at the interface.

When the measured segregation parameter x is substi-tuted for fA in Eq. ~18!, the Flory-Huggins parameter formonomers is found. Now, when we are close to the criticalpoint (x52), we cannot expect this mean-field expression tohold, but when we calculate for x.3 this value should bereliable. The calculated x-parameter is shown in Fig. 7 fortwo densities as a function of the excess repulsion parameter.We find that for x.3 there is a very good linear relationbetween x and Da . Explicitly, we have

x5~0.28660.002!Da~r53 !,

x5~0.68960.002!Da~r55 !.~24!

These results partly confirm Eq. ~23! in that x is linear inDa , but the constant of proportionality is far from linear inthe density. Nevertheless, we can choose a fixed density, andhenceforth use Eqs. ~24! as an effective mapping on theFlory-Huggins theory.

FIG. 6. Density profile for r53 at repulsion parameters aAA5aBB525 andaAB537.

FIG. 7. Relation between excess repulsion and effective x-parameter.

4429R. D. Groot and P. B Warren: Dissipative particle dynamics

J. Chem. Phys., Vol. 107, No. 11, 15 September 1997

Downloaded 13 May 2009 to 148.79.162.144. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

plies that the first and second derivative of the chemical po-tential with respect to fA vanish, which leads to the criticalpoint

xcrit512S

1ANA

11

ANBD2

. ~19!

Since the present simulated system is fairly incompressible(k21516), and since the excess pressure is quadratic in thedensity, the soft sphere model is by nature very close to theFlory-Huggins lattice model. The free energy density thatcorresponds to the pressure of a single component, Eq. ~16!,is

f VkBT

5r ln r2r1aar2

kBT, ~20!

hence for a two component system of chains one expects

f VkBT

5rANAln rA1

rBNBln rB2

rANA

2rBNB

1a~aAArA

212aABrArB1aBBrB2 !

kBT. ~21!

If we choose aAA5aBB and assume that rA1rB is approxi-mately constant,

f V~rA1rB!kBT

'xNAln x1

~12x !

NBln~12x !1xx~12x !

1constants, ~22!

where we have set x5rA /(rA1rB) and made the tentativeidentification

x52a~aAB2aAA!~rA1rB!

kBT. ~23!

Apparently we have the correspondence ~soft spheres!f V /(rA1rB)5F ~Flory-Huggins!, with a x-parameter map-ping given by Eq. ~23!.

To test this relation simulations have been performed forbinary mixtures of both monomers and polymers, atr5rA1rB53 and r55, for repulsion parametersa5aAA5aBB525 and a515, respectively. When the excesspressure was measured as a function of the fraction x of Aparticles it was found that it is indeed proportional tox(12x). However, unlike the assumption in Eq. ~22! it wasfound that the prefactor of x(12x) is not simply linear inDa5aAB2a when Da is 2 to 5. In practice we are notinterested in small differences in the repulsion, but systemswill rather be chosen where segregation takes place, i.e.,x.xcrit. Now if x is much larger than the critical value,mean field theory is expected to be valid. This means that wecan use Eq. ~18! as the defining equation for the correspond-ing Flory-Huggins parameter.

Adopting this strategy, a system of size 838320 con-taining 3840 monomers was simulated. Half the particleswere of type A and in the initial configuration they wereplaced in the left half of the system, the other particles, oftype B , were placed in the right hand side. For these systems

the density profiles of A and B particles were sampled acrossthe interface. Averages of the density were taken over 105timesteps; the mean value of x over a slab where the densityis homogeneous was then taken to compute the correspond-ing Flory-Huggins parameter. An example of such a binarydensity profile is shown in Fig. 6. Note the small dip in thesum of the densities at the interface.

When the measured segregation parameter x is substi-tuted for fA in Eq. ~18!, the Flory-Huggins parameter formonomers is found. Now, when we are close to the criticalpoint (x52), we cannot expect this mean-field expression tohold, but when we calculate for x.3 this value should bereliable. The calculated x-parameter is shown in Fig. 7 fortwo densities as a function of the excess repulsion parameter.We find that for x.3 there is a very good linear relationbetween x and Da . Explicitly, we have

x5~0.28660.002!Da~r53 !,

x5~0.68960.002!Da~r55 !.~24!

These results partly confirm Eq. ~23! in that x is linear inDa , but the constant of proportionality is far from linear inthe density. Nevertheless, we can choose a fixed density, andhenceforth use Eqs. ~24! as an effective mapping on theFlory-Huggins theory.

FIG. 6. Density profile for r53 at repulsion parameters aAA5aBB525 andaAB537.

FIG. 7. Relation between excess repulsion and effective x-parameter.

4429R. D. Groot and P. B Warren: Dissipative particle dynamics

J. Chem. Phys., Vol. 107, No. 11, 15 September 1997

Downloaded 13 May 2009 to 148.79.162.144. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

#"$ =ln 1 − !"

!"1 − 2!"

Example: fluid/fluid separation• Carried out using

DL_MESO– Two species: 1500 beads

each, initially randomly distributed

– Bead masses and sizes identical

– Interaction parameter between species: !"" =!$$, !"$ = 4!""

– Separation complete after ~20 DPD time units

DPD Practical Exercise 1• Demonstrate how !"# relates to $"#• Carry out simulations of separating

beads with various !"# values

• Find volume fractions of one component

in bulk phases (%") and work out $"#:

$"# =ln 1 − %"

%"1 − 2%"

• Practical aspects:

– Compiling DL_MESO’s DPD code

– Python scripts to automate calculations

and plot $"# values

• Start from drfaustroll.gitlab.io/cs2019

(Day 4: Introduction to Dissipative

Particle Dynamics)

DPD calculations• Pairwise forces

– Only pairs within interaction cutoff need to be considered; reduces number of pairs to search

– Possible strategies to find particle pairs:

Linked-list cells

Verlet lists

Lists might need updating frequently for systems

with flow

DPD calculations• Random forces

– Fluctuations obtainable from Gaussian random numbers of zero mean, unity variance:

– Most random number generators give uniformly distributed numbers (0 ≤ # < 1), so either:

• Convert pairs using e.g. Box-Muller transforms• Use central limit theorem to use generated numbers

directly[1], i.e.

Gives ‘statistically similar’ results to true Gaussian

random numbers[2]

&'( = *'(Δ,-./

*'( ≈ 12 #'( −12

[1] Dünweg and Paul, Int J Mod Phys C 2, 817–827 (1991)[2] Groot and Warren, J Chem Phys 107, 4423–4435 (1997)

DPD calculations• Force integration (Velocity Verlet)

– Mid-step velocity:

– Positions:• Move beads

– Calculate forces

– Final velocity:

!" # + %&Δ# = !" # + Δ#2

*" #+"

," # + Δ# = ," # + !" # + %&Δ# Δ#

!" # + Δ# = !" # + %&Δ# + Δ#2

*" # + Δ#+"

*" # → *" # + Δ#

DPD calculationsSTART

Read simulation parameters; initialize

bead positions, velocities and forces

1st Velocity Verletstage: calculate

!" # + %&Δ# , )"(# + Δ#)

from)" # , !" # , ," #

Correct positions for periodic boundaries

Set # = 0Calculate ,"(# + Δ#) =

,"/ + ,"0 + ,"1

,"/ =234"

,"3/

,"0 =234"

,"30

,"1 =234"

,"31

Determine linked-list cells/Verlet lists

2nd Velocity Verletstage: calculate

!5 # + Δ#from

!" # + %&Δ# , ," # + Δ#

Calculate statistical properties; output

results, trajectories etc. at set intervals

Increment # by Δ#

YES

FINISH

NO

# < #7879:?

• Force integration problem

– Dissipative force needed at end of time step, but calculated using mid-step velocities:

• Error carried forward to next time step• Results in temperature offset that increases with time

step size

!"($ + Δ$)("($ + )

*Δ$)+"($ + Δ$)

DPD calculations

VV stage 1 VV stage 2Force

calculations!"($)("($)+"($)

!"($)("($ + )

*Δ$)+"($ + Δ$)

!"($ + Δ$)("($ + Δ$)+"($ + Δ$)

!",- $ + Δ$ = −01- 2", $ + Δ$ 3+", $ + Δ$ ⋅ (, $ + )*Δ$ − (" $ + )

*Δ$ 3+", $ + Δ$

DPD calculations• Force integration problem

– Solution 1: Estimate end-step velocity for dissipative force calculations[1]

• Requires empirical factor !; optimal value depends on system being modelled

– Solution 2: Recalculate dissipative forces at end of time step (DPD Velocity Verlet)[2]

• Can be made self-consistent by iterating dissipative forces/velocities

"#$ % + Δ% = #$ % + !Δ% )$ %*$

[1] Groot and Warren, J Chem Phys 107, 4423–4435 (1997)[2] Nikunen et al., Comp Phys Comms 153, 407–423 (2003)

DPD calculations• Force integration problem

– Solution 3: Apply exponential propagators for dissipative/random forces (e.g. Shardlow splitting[1])

• Uses velocities at start of time step• Conservative forces still integrated with Velocity Verlet

– Solution 4: Use an alternative pairwise thermostat (e.g. Lowe-Andersen[2], Peters[3])

• Applies pairwise velocity corrections at end of time step after integration of conservative forces

• Lowe-Andersen thermostat (and its variants) can increase Schmidt number

[1] Shardlow, SIAM J Sci Comput 24, 1267–1282 (2003)[2] Lowe, EPL 47, 145–151 (1999)[3] Peters, EPL 66, 311–317 (2004)

DPD calculationsSTART

Read simulation parameters; initialize

bead positions, velocities and forces

1st Velocity Verletstage: calculate

!" # + %&Δ# , )"(# + Δ#)

from)" # , !" # , ," #

Correct positions for periodic boundaries

Set # = 0Calculate ,"(# + Δ#) =

,"/ + ,"0 + ,"1

,"/ =234"

,"3/

,"0 =234"

,"30

,"1 =234"

,"31

Determine linked-list cells/Verlet lists

2nd Velocity Verletstage: calculate

!5 # + Δ#from

!" # + %&Δ# , ," # + Δ#

Calculate statistical properties; output

results, trajectories etc. at set intervals

Increment # by Δ#

YES

FINISH

NO

Recalculate,"0, ," # + Δ#and/or apply

thermostat/barostat

# < #7879:?

Applying DPD for complex fluids• DPD beads can be bonded together to form

molecular chains– Stretching, angles, dihedrals

– Used to give vestige (impression) of molecular structures– Vastly extends range of DPD simulations (e.g. polymers,

colloids, biological)

Example: vesicle formation[1]

• Four-bead amphiphilic molecule:

– Bonded by stiff harmonic springs with equilibrium length = cut-off radius

• Single equally-sized beads for water• Interactions: !"" = !$$ = !%% = !"% = 25, !"$ = !$% = 40• Simulation properties: +$, = 0.25, . = 6.7, Δ2 = 0.05

– Simulation carried out using DL_MESO in parallel on 96 cores of HECToR (former UK national supercomputer)

A = hydrophilic, B = hydrophobic

[1] Yamamoto et al., J Chem Phys 116, 5842–5849 (2002)

Example: vesicle formation• Spontaneous vesicle

formation– 41,472 beads in total

(9.7 vol% amphiphilic molecules)

• Molecules initially placed randomly in box

– 1,000,000 time steps (50,000 time units)

– Water omitted to show amphiphiles more clearly

Example: vesicle formation• Spontaneous vesicle

formation– Cross-section at last

time step– Vesicle formed from

large oblate micelle– Water incorporated

into vesicle– Applications include

e.g. drug delivery

Applying DPD for complex fluids• Charged particles

– Like classical MD, possible to apply electrostatic interactions between DPD beads

– Charged interactions act over large length scales• Need some method to deal with long-range interactions,

e.g. Ewald sum, Particle-Particle Particle-Mesh (P3M)– Cannot always use point charges with DPD beads!

• ‘Standard’ conservative interactions (soft, repulsive) are often insufficiently strong to prevent ion collapse[1]

• May need to smear out charges over finite volumes

[1] Terrón-Mejía et al., J Phys: Cond Mat 28, 425101 (2016)

Applying DPD for complex fluids• Charged particles: P3M approach[1]

– Construct grid for charge concentrations !∗, smearing out charges with function # $%

– Solve Poisson equation (iteratively) to give electrostatic field:

– Spatial differentiation of electrostatic field (summed over grid) gives force on ion:

∇ ⋅ ( ) ∇* = − -.!)/.0123435

!∗ = −Γ!∗

7%8 )% = −9%:;#; $% ∇* $;

[1] Groot, J Chem Phys 118, 11265–11277 (2003)

Applying DPD for complex fluids• Charged particles: Ewald sum approaches

– Modify Ewald sums to include charge smearing function in pairwise electrostatic potential, i.e.

– Smearing assumed to only affect real-space part:• Often needs sufficiently large real-space cutoff to converge with

Coulombic (point charge) distribution in reciprocal space• Reciprocal space parts carried out as usual

– Commonly used smearing functions:• Exponential (Slater-type) smearing[1]:• Gaussian charge smearing[2]:

!"#$ =Γ'"'#4)*"#

1 − - *"#

- * = 1 + /* 01234

- * = erfc *2:;

Can tune Ewald sum parameter <to coincide with :;: real-space terms then reduce to zero

[1] González-Melchor et al., J Chem Phys 125, 224107 (2006)[2] Warren et al., J Chem Phys 138, 204907 (2013)

Example: polyelectrolyte[1]

• Polyelectrolyte in salt solution– Modelled with Ewald

sum and Slater-type smearing

– Hydrophobic polymer of 50 charged beads

– Additional charged beads for salt, anions and counterions

– Can measure how far molecule stretches out using radius of gyration

[1] González-Melchor et al., J Chem Phys 125, 224107 (2006)

Boundary conditions• Most DPD simulations involve bulk systems at rest

(periodic BCs)• Can also subject them to flow fields, e.g. uniform

shear (Lees-Edwards BCs)[1]

If bead crosses upper boundary at time !, re-introduced at lower boundary with "-coordinate shifted by−$%! and "-component of velocity decreased by $%. (Vice versa if bead crosses lower boundary.)

Relative velocity between particle pairs across boundary adjusted to

remove wall speed difference[2]

&'()

*&'()[1] Lees and Edwards, J Phys C 5, 1921–1928 (1972)[2] Leimkuhler and Shang, J Comput Phys 324, 174–193 (2016)

Boundary conditions• Possible to represent solid

walls by planar reflections[1] or frozen DPD beads– Can produce artificial

density/temperature fluctuations at and near walls

– Often reduced by combination of both methods

• Used to model solid/liquid interactions– e.g. wetting, polymer brushes

[1] Warren et al., Phil Trans Roy Soc Lond A 361, 665–676 (2003)

Thermodynamics and DPD• Standard form of conservative interaction gives

quadratic equation of state:

– Possible to map fluid compressibility or pressure onto EOS, but not both!

– Not flexible enough to model thermodynamics of real systems in great detail

– Note that conservative force currently only depends on bead pair separation !"#

$ = &'() + +,&-

Thermodynamics and DPD• Many-body (density dependent) DPD[1,2]

– Suppose each DPD particle has a localised density !" and associated free energy (sum of bulk, mixing and excess components):

– Free energy density of system can be written:

Associated with ideal gas of force-free particles

# !" = #%&'( !" + #*"+",- !" + #.+/.00(!")

3 = 4! 5 # ! 5 65 →8 9"!"# !"

[1] Pagonabarraga and Frenkel, J Chem Phys 115, 5015–5026 (2001)[2] Trofimov et al., J Chem Phys 117, 9383–9394 (2002)

Thermodynamics and DPD• Many-body (density dependent) DPD

– Ideal gases have dissipation and mixing, thus excess free energy must be related to conservative interactions; tentatively given as

– Local density approximated as sum of weight functions between bead pairs:

This form of weight function can recover

‘standard’ DPD

!"# = −∇ '()"

*(+,-.,// *(

*" ='()"

0 1"( ='()"

1525 1 − 1"(16

7


– Many-body conservative interaction forces can be rewritten in pairwise additive form:

– Excess free energy form can be chosen to give required equation of state:

Spatial integral over system volume

!"# = −&'("

)*+,-+.. /" + )*+,-+.. /' 1# 2"' 34"'

5 ≈ /789 + /:;)+,-+..;/ −13>

?2"'1# 2"' @ 2"' AB AC AD


– Attractive interactions can be included, e.g. vapour-liquid systems[1]

Standard DPD force augmented by many-body (density-dependent) term:

which gives a van der Waals-type (cubic) equation of state when !"# < 0and & > 0:

(") =+#,"

!"#-) ."#, .0 + & 2" + 2# -) ."#, .3 45"#

[1] Warren, Phys Rev E 68, 066702 (2003)

6 = 2789 + :!"#2; + 2:&.3= 2> − @2; + A

Example: drop on surface• Formation of water drops

on hydrophobic surface– Two-parameter many-

body DPD: !"# > !##– Frozen bead walls with

bounce-back reflections– Interaction between water

and wall more repulsive– Constant gravitational

force on system

Molecular dynamics and DPD• DPD parameters can be fitted on bottom-up

basis, e.g. from MD simulations[1]

– Conservative parameters: Inverse Monte Carlo method using radial distribution function

– Dissipative/random parameters: Best fit to velocity autocorrelation function (VAF)

Second term negligible for low density gases

Related to diffusion:! = #

$ ∫&' ((*) ,*

- . = exp −3 .456

+ 8 9, .

; < = => 0 ⋅ =>(<)

[1] Lyubartsev et al., Soft Mat 1, 121–137 (2002)

Example: fitting DPD to MD• Obtaining dissipative

parameters– Initial MD simulation of

Lennard-Jones fluid (red particles)

– DPD beads (transparent spheres) applied afterwards: follow average dynamics of LJ fluid

– DPD random force parameter can be obtained by comparing VAFs of LJ particles and DPD beads

Video courtesy: Vlad Sokhan

Molecular dynamics and DPD• MD simulations can also be used to carry out top-

down fitting approach for DPD parameters– e.g. calculate mixing excess free energy between two

components from constant-pressure atomistic MD simulations of mixture and pure components [1]

– Mixing free energy related to Flory-Huggins !-parameter:

– Use determined relationship between !"# and $"#, e.g.

[1] Akkermans, J Chem Phys 128, 244904 (2008)

Δ&'"( = *"# − *" + *#

!"# =Δ&'"(-./0123"3#-

$"# ≈ $"" + 3.50!"#

Example: drug loading/release[1]

• Copolymer PAE-PEG with water and camptothecin

(CPT)

• Conservative force parameterisation

– Determined !-parameters for PAE, PEG and CPT from MD

simulations (see previous slide) with COMPASS force field

– Modelling at two pH levels (7.4 and 6.4): using hydrated

form of PAE with chloride ions in MD simulations for more

acidic system

[1] Luo and Jiang, J Control Release 162, 185–193 (2012)

namely dissipative particle dynamics (DPD) simulation was proposed[23]. DPD is a mesoscale method and can increase simulation scale byseveral orders of magnitude from atomistic simulation. This methodhas been used to investigate nanoparticlemicrostructures [24,25], poly-mer phase separation [26,27], membrane properties [28,29], etc. Drugloading/releasing in copolymers has also been examined by DPD simu-lation. Posocco et al. simulated the loading of nefidipine into poly(lacticacid) (PLA) hydrophobic core formed by PLA–PEG di-/tri-block copoly-mers [30]. Ahmad et al. quantitatively compared the loading efficienciesof prednisolone, paracetamol and isoniazid in PLA microspheres as de-termined from simulation and experiment [31]. Guo et al. investigatedthe drug loading of DOX in cholesterol conjugated polypeptideHis10Arg10andpH-sensitive swelling for drug releasing [32]. In the above-mentionedDPD simulation studies, the Flory–Huggins parameters χij usedwere esti-mated fromexperimentalmeasurements, solubility parameters, or Flory–Huggins lattice model. The estimation based on solubility parametersfollows “similarity and inter-miscibility principle” and works well onnon-polar systemswithout special interactions; for polar systems, how-ever, it is not reliable. On the other hand, the estimation based on Flory–Huggins latticemodel usually requires the comparable segment sizes oftwo components.

In this study, we integrate fully atomistic molecular dynamics (MD)and DPD simulations to investigate the loading/releasing of CPT in apH-sensitive diblock copolymer. The system chosen has been experi-mentally examined by Lee and coworkers [22]; therefore, the simula-tion results can be compared with measured data to validate themodels and methodology. The copolymer considered consists of PAEand methyl ether-capped PEG with a formula of PAE12580–PEG4850

[22]. Fig. 1 illustrates the chemical structures of PAE–PEG and hydro-phobic drug CPT. CPT has a planar pentacyclic ring structure that in-cludes a pyrrolo-quinoline moiety (rings A, B and C), conjugatedpyridone moiety (ring D) and one lactone ring (ring E). This large con-jugated ring structure forms an extended π bond (including rings A, B, Cand D). There is a hydroxyl group bonded with (S)-chiral carbon C20(also called (S)-CPT). The acid dissociation constant pKa of the quinolinegroup in CPT (rings A and B) is 1.18, decreased from 4.85 of a singlequinoline [33]. Here, the parameters χij between binary componentsare calculated from the energies ofmixing by atomistic MD simulations.This approach has been revealed to bemore accurate than the previous-ly usedmethods (solubility parameters or Flory–Huggins latticemodel)[34], [35]. DPD simulations are then conducted to examine themicellization/demicellization of PAE–PEG, the assembledmorphologiesat different conditions, and the pH-sensitive loading/releasing of CPT.

2. Simulation details

2.1. MD simulation

To calculate the Flory–Huggins parameters, MD simulation wasperformed for pure and binary components as listed in Table 1. All

the components were represented by the COMPASS force field[36–38], in which the total potential energy (Epot) is expressed

Epot ¼ Ebonded þ Enonbond þ Ecross¼ Eb þ Eθ þ Eϕ þ Eχ

! "þ EvdW þ ECoulombicð Þ þ Ecross

ð1Þ

where Eb is bond stretching energy, Eθ is bending energy, Eϕ is dihe-dral torsion energy, and Eχ is out-of-plane energy; the sum of thesefour items is bonded energy (Ebonded). EvdW is van der Waals energy,ECoulombic is electrostatic energy; and the sum is nonbonded energy(Enonbond). Ecross is the energy of cross terms between any two of bondeditems in the COMPASS, such as bond-angle and bond-bond cross terms.

Each system (either pure or binary mixture) was constructed bythe Amorphous Cell in Materials Studio 4.3 [39]. It should be notedthat in acidic environment, PAE is protonated at its tertiary aminegroups (PAEH). To electrically neutralize the charges on PAEH, chlo-ride ions were included in the system of PAEH. The packing tech-nique of Theodorou and Suter [40,41] and Meirovitch scanningmethod [42] were adopted to attempt achieving homogeneous sys-tem. To eliminate unfavorable contacts, the initial configurationswere subjected to 10,000 steps of energyminimization with an ener-gy convergence threshold of 1×10−4kcal∙mol−1 and a force conver-gence of 0.005kcal∙mol−1∙Å−1. The van der Waals interactions werecalculated with a cutoff of 12.5Å, a spline width of 1Å, and a bufferwidth of 0.5Å. Moreover, the Ewald summation with an accuracy of0.001kcal∙mol−1 was used to calculate the Coulombic interactions.After minimization, 3–6 configurations with the lowest energieswere chosen. For a pure component, thermal annealing from 1000to 300K was conducted, and followed by 5ns MD equilibrium simu-lation at isothermal and isobaric (NPT) conditions. The temperaturewas maintained at 298K by Nośe thermostat [43] and the pressurewas maintained at 1bar by Andersen method [44]. Trajectory wassaved every 1ps and the final 1.5ns were used to calculate the equi-librium density and potential energy. For binary components, theconfigurations were built with a density estimated from the volumefractions of two components, and 5ns MD equilibrium simulationwas performed at isothermal and isochoric (NVT) conditions. Thetemperature was also maintained at 298K and the final 1.5ns trajec-tory was used to calculate potential energy.

For binary components i and j, the Flory–Huggins parameter χij

can be estimated by

χij ¼ΔEmixV rRTϕiϕj V

ð2Þ

where R is gas constant and T is temperature; ϕi are ϕj are the volumefractions of components i and j, respectively; V is total volume and Vr

is reference volume; ΔEmix is the energy of mixing

ΔEmix ¼ Eij− Ei þ Ej! "

ð3Þ

Table 1Pure and binary components examined by MD simulations.

Components Number of molecules Density (g/cm3) δ (J/cm3)0.5 χ

H2O 900 0.958 46.40 −CPT 75 1.295 23.82 −PEG 50 1. 094 21.81 −PAE 50 0.986 18.08 −PAEH 50 0.968 22.87 −PAE/H2O 12/2700 0.953 − 28.67CPT/H2O 12/2700 0.980 − 6.64PAEH/H2O 12/2700 0.966 − −1.69PEG/CPT 50/15 1.125 − −0.78PAE/CPT 50/15 1.034 − −0.72PAEH/CPT 50/15 1.011 − 16.31PAE/PEG 25/25 1.040 − 6.24PAEH/PEG 25/25 1.025 − −0.39

CH3O

O

O

nN N O

OO

O m

N

N

O

O

OOH

Poly(β-amino ester)(PAE)PEG

Camptothecin (CPT)

12 3

456

78

9

10

1112

13

1415

1617

1819

2021

22

A B C

DE

Fig. 1. Chemical structures of PAE–PEG and CPT.

186 Z. Luo, J. Jiang / Journal of Controlled Release 162 (2012) 185–193

NANOMEDICIN

E

namely dissipative particle dynamics (DPD) simulation was proposed[23]. DPD is a mesoscale method and can increase simulation scale byseveral orders of magnitude from atomistic simulation. This methodhas been used to investigate nanoparticlemicrostructures [24,25], poly-mer phase separation [26,27], membrane properties [28,29], etc. Drugloading/releasing in copolymers has also been examined by DPD simu-lation. Posocco et al. simulated the loading of nefidipine into poly(lacticacid) (PLA) hydrophobic core formed by PLA–PEG di-/tri-block copoly-mers [30]. Ahmad et al. quantitatively compared the loading efficienciesof prednisolone, paracetamol and isoniazid in PLA microspheres as de-termined from simulation and experiment [31]. Guo et al. investigatedthe drug loading of DOX in cholesterol conjugated polypeptideHis10Arg10andpH-sensitive swelling for drug releasing [32]. In the above-mentionedDPD simulation studies, the Flory–Huggins parameters χij usedwere esti-mated fromexperimentalmeasurements, solubility parameters, or Flory–Huggins lattice model. The estimation based on solubility parametersfollows “similarity and inter-miscibility principle” and works well onnon-polar systemswithout special interactions; for polar systems, how-ever, it is not reliable. On the other hand, the estimation based on Flory–Huggins latticemodel usually requires the comparable segment sizes oftwo components.

In this study, we integrate fully atomistic molecular dynamics (MD)and DPD simulations to investigate the loading/releasing of CPT in apH-sensitive diblock copolymer. The system chosen has been experi-mentally examined by Lee and coworkers [22]; therefore, the simula-tion results can be compared with measured data to validate themodels and methodology. The copolymer considered consists of PAEand methyl ether-capped PEG with a formula of PAE12580–PEG4850

[22]. Fig. 1 illustrates the chemical structures of PAE–PEG and hydro-phobic drug CPT. CPT has a planar pentacyclic ring structure that in-cludes a pyrrolo-quinoline moiety (rings A, B and C), conjugatedpyridone moiety (ring D) and one lactone ring (ring E). This large con-jugated ring structure forms an extended π bond (including rings A, B, Cand D). There is a hydroxyl group bonded with (S)-chiral carbon C20(also called (S)-CPT). The acid dissociation constant pKa of the quinolinegroup in CPT (rings A and B) is 1.18, decreased from 4.85 of a singlequinoline [33]. Here, the parameters χij between binary componentsare calculated from the energies ofmixing by atomistic MD simulations.This approach has been revealed to bemore accurate than the previous-ly usedmethods (solubility parameters or Flory–Huggins latticemodel)[34], [35]. DPD simulations are then conducted to examine themicellization/demicellization of PAE–PEG, the assembledmorphologiesat different conditions, and the pH-sensitive loading/releasing of CPT.

2. Simulation details

2.1. MD simulation

To calculate the Flory–Huggins parameters, MD simulation wasperformed for pure and binary components as listed in Table 1. All

the components were represented by the COMPASS force field[36–38], in which the total potential energy (Epot) is expressed

Epot ¼ Ebonded þ Enonbond þ Ecross¼ Eb þ Eθ þ Eϕ þ Eχ

! "þ EvdW þ ECoulombicð Þ þ Ecross

ð1Þ

where Eb is bond stretching energy, Eθ is bending energy, Eϕ is dihe-dral torsion energy, and Eχ is out-of-plane energy; the sum of thesefour items is bonded energy (Ebonded). EvdW is van der Waals energy,ECoulombic is electrostatic energy; and the sum is nonbonded energy(Enonbond). Ecross is the energy of cross terms between any two of bondeditems in the COMPASS, such as bond-angle and bond-bond cross terms.

Each system (either pure or binary mixture) was constructed bythe Amorphous Cell in Materials Studio 4.3 [39]. It should be notedthat in acidic environment, PAE is protonated at its tertiary aminegroups (PAEH). To electrically neutralize the charges on PAEH, chlo-ride ions were included in the system of PAEH. The packing tech-nique of Theodorou and Suter [40,41] and Meirovitch scanningmethod [42] were adopted to attempt achieving homogeneous sys-tem. To eliminate unfavorable contacts, the initial configurationswere subjected to 10,000 steps of energyminimization with an ener-gy convergence threshold of 1×10−4kcal∙mol−1 and a force conver-gence of 0.005kcal∙mol−1∙Å−1. The van der Waals interactions werecalculated with a cutoff of 12.5Å, a spline width of 1Å, and a bufferwidth of 0.5Å. Moreover, the Ewald summation with an accuracy of0.001kcal∙mol−1 was used to calculate the Coulombic interactions.After minimization, 3–6 configurations with the lowest energieswere chosen. For a pure component, thermal annealing from 1000to 300K was conducted, and followed by 5ns MD equilibrium simu-lation at isothermal and isobaric (NPT) conditions. The temperaturewas maintained at 298K by Nośe thermostat [43] and the pressurewas maintained at 1bar by Andersen method [44]. Trajectory wassaved every 1ps and the final 1.5ns were used to calculate the equi-librium density and potential energy. For binary components, theconfigurations were built with a density estimated from the volumefractions of two components, and 5ns MD equilibrium simulationwas performed at isothermal and isochoric (NVT) conditions. Thetemperature was also maintained at 298K and the final 1.5ns trajec-tory was used to calculate potential energy.

For binary components i and j, the Flory–Huggins parameter χij

can be estimated by

χij ¼ΔEmixV rRTϕiϕj V

ð2Þ

where R is gas constant and T is temperature; ϕi are ϕj are the volumefractions of components i and j, respectively; V is total volume and Vr

is reference volume; ΔEmix is the energy of mixing

ΔEmix ¼ Eij− Ei þ Ej! "

ð3Þ

Table 1Pure and binary components examined by MD simulations.

Components Number of molecules Density (g/cm3) δ (J/cm3)0.5 χ

H2O 900 0.958 46.40 −CPT 75 1.295 23.82 −PEG 50 1. 094 21.81 −PAE 50 0.986 18.08 −PAEH 50 0.968 22.87 −PAE/H2O 12/2700 0.953 − 28.67CPT/H2O 12/2700 0.980 − 6.64PAEH/H2O 12/2700 0.966 − −1.69PEG/CPT 50/15 1.125 − −0.78PAE/CPT 50/15 1.034 − −0.72PAEH/CPT 50/15 1.011 − 16.31PAE/PEG 25/25 1.040 − 6.24PAEH/PEG 25/25 1.025 − −0.39

CH3O

O

O

nN N O

OO

O m

N

N

O

O

OOH

Poly(β-amino ester)(PAE)PEG

Camptothecin (CPT)

12 3

456

78

9

10

1112

13

1415

1617

1819

2021

22

A B C

DE

Fig. 1. Chemical structures of PAE–PEG and CPT.

186 Z. Luo, J. Jiang / Journal of Controlled Release 162 (2012) 185–193

NANOMEDICIN

E

Example: drug loading/release• Two sets of conservative force parameters !"#

H2O CPT PEG PAE

H2O 25.00 48.22 26.05 125.23

CPT 25.00 22.28 22.49

PEG 25.00 46.81

PAE 25.00

H2O CPT PEG PAE

H2O 25.00 48.22 26.05 19.10

CPT 25.00 22.28 82.04

PEG 25.00 23.63

PAE 25.00

Drug release: pH ≈ 6.4

PAE replaced with hydrated form: note changes to hydrophobicity between PAE and water/CPT/PEG

Drug loading: pH ≈ 7.4

Can bond PAE and PEG beads together to form copolymers (e.g. 28 PAE beads and 10 PEG beads represent PAE12580-PEG4850)

Example: drug loading/release• DPD simulations with DL_MESO

– Ran drug loading simulation from initial randomly distributed state over 900ns (1 million timesteps)

• PAE-PEG form vesicles that encapsulate CPT

– Last timeframe of drug loading simulation used as initial state for drug release

• Vesicles swell, break open and push out CPT due to drop in pH

– Drug release simulation ran for 90ns (100,000 timesteps)

– Loading efficiency closely matches experimental observation

Simulation courtesy: Nidhi Raj

Molecular dynamics and DPD• Coarse-graining in DPD

– Each DPD bead can represent:

• An individual atom?

• A functional group[1]

• An entire molecule

• A group of molecules

– Level of coarse-graining influences interactions

between DPD beads

– No single ‘best method’ to coarse-grain or determine

interactions

[1] Lopez et al., Proc Nat Acad Sci USA 101, 4431–4434 (2004)

Molecular dynamics and DPD• Coarse-graining in DPD

– Coarse-graining limit for fluids: artificial crystal formation occurs for large values of !"# > 250 ()*

– Conservative interaction strength scales with coarse-graining level:

• Linear relationship[1,2]:!"# ∝ ,

• Less linear (considers that -. ∝ ,/0)[3]:

!"# ∝ ,10

– Many-body DPD can offer greater coarse-graining, but greater risk of numerical instability

[1] Groot and Rabone, Biophys J 81, 725–736 (2001)[2] Trofimov, PhD Thesis, Technical University of Eindhoven (2003)[3] Füchslin et al., J Chem Phys 130, 214102 (2009)

Molecular dynamics and DPD• Relationship between MD and DPD[1-3]

– Can help prove DPD’s hydrodynamics– Assume DPD bead is collection of MD particles (e.g. atoms):

cell on Voronoi lattice– Define sampling function which can express macroscopic

observables, e.g. mass, momentum

!" # = % # − #"∑( % # − #(

#" = centre of DPD bead% # = localized function, e. g. exp − >

?

@?[1] Flekkøy and Coveney, Phys Rev Lett 83, 1775–1778 (1999)[2] Flekkøy et al., Phys Rev E 62, 2140–2157 (2000)[3] De Fabritiis et al., Phil Trans Roy Soc Lond A 360, 317–331 (2002)

Molecular dynamics and DPD• Relationship between MD and DPD

– Assuming MD particles are of equal mass:

• Mass of DPD bead:

• Momentum:

• Energy:

– If ∑" #" $ = 1:(")" =(

*+ ,(

")"-" =(

*+.*, (

"/"0 =(

*1*0

)" =(*#" 2* +

3" =(*#" 2* +.* ≡ )"-"

/"0 =(*#" 2* 5

6+7*6 + 56(9:*

; $*9

/"0 ≡ 56)"<"6 + /"=


– DPD equations of motion obtained from material derivatives of macroscopic quantities

Using quotient rule and substitutions

below

Overlap function

DPD centred parameters

"# $% = '"#'( =

))(

* $ − $#∑- * $ − $-

"# $% =.%"#- $% /%0 ⋅ $#- + $%0 ⋅ 3#-

"#- $% = 256 "# $% "- $%

$#- = $# − $-3#- = 3# − 3-$%0 = $% − 7

6 $# + $-/%0 = /% − 7

6 3# + 3-



• Mass:

• Momentum:

"# =%&

'# (& ) =%&'#* (& ) +&, ⋅ (#* + (&, ⋅ /#* =%

&"#*

0# =%&

12"#* /# + /* +%

*,&'#* (& 45, ⋅ (#* +%

*,&'#* (& )+&,(& ⋅ /#*

Momentum =lux tensors:4& = )+&+& + 1

2%CD&C (& − (C

4&, = )+&,+&, + 12%

CD&C (& − (C



• Energy:

Energy 'lux tensors:

/0,2 = 452 +789

:

;2:(52 + 5:) >2 − >:

/0,2@ = 452

@ + 789

:

;2:(52@ + 5:

@) >2 − >:

BCD =

EEF

7GHCIC

G +9J

7G HCJ

7GKCJ

G+9

J,2

LCJ >2 /0,2@ − 7

GM2@ ⋅ KCJ ⋅ >CJ

BCD = +9

J,2

LCJ >2 42@ − 7

GO52@ ⋅ KCJ >2

@ ⋅ KCJ


– Ensemble averaging of material derivatives• Need to average over states of MD particles to give

same properties (mass, velocity, energy) as DPD beads• Evolution can be represented (in terms of statistical

mechanics) by average and fluctuating parts• Assume that average velocity of MD particles between

DPD beads can be approximated by nearest neighbourinformation and determined from linear interpolation:

!" ≈ $" ⋅ $&'(&')

*&'


– Ensemble averaging of material derivatives

• Mass:

• Mass fluctuations visible in momentum and energy fluxes• Mass conservation otherwise observed

"# =%&

"#& + ("#&

"#& =%)*+#& ,) ,)- ⋅ /#&

("#& ≡ "#& − "#&


– Ensemble averaging of material derivatives• Momentum:

• Fluctuating force (similar to DPD random force):

"# =%&

'( )#* +# + +* +%

*-#* .& /& ⋅ .#*

"# +%&-#* .& 1 2&3.&3 ⋅ +#* +%

*45#*

46#* =%&-#* .7 /& − /& ⋅ .#* + 1 2&3.&3 − 2&3.&3 ⋅ +#*

46#* + '( 9)#* +# + +*


– Ensemble averaging of material derivatives• Assuming MD particles interact via (hard) Lennard-Jones

potential, following momentum flux equation applies:

• Using linear velocity gradients we (eventually) get:

• Momentum conservation, Galilean invarianceSimilar to: DPD conservative and dissipative forces

! "# = %!&& + (! − * ∇& + ∇& ,

./ =0#

12 3/4 5/ + 54 −0

46/4 1

2 7/ − 74 ⋅ 9:/4 +*;/4

5/4 + 5/4 ⋅ 9:/4 9:/4 +0#<=/4

"#$ = −'()#(*

+# − +(,#(

−'()#( -

. /# + /( ⋅ 23#( −4,#(

5#( + 5#( ⋅ 23#( 23#( ⋅ 5#(2

"#$ +'(

-. 7#(

5#(2

.+ )#(4,#(

9#(;#( ⋅ 5#("#<#+ "(<(

−'=>?#( ⋅

5#(2 + @A#(


– Ensemble averaging of material derivatives• Energy (carried out in similar way to momentum):

• Potential energy term similar to DPD’s• Terms similar to dissipative and random forces: DPD

thermostat!


– DPD can be derived from MD via coarse-graining and ensemble averaging

– Hydrodynamics emergent from both MD and DPD• Thermostats often used in MD can break local momentum

conservation (although global momentum is often correct)• DPD thermostat conserves both local and global momenta

‘automatically’– DPD parameters can be derived from MD simulations

DPD miscellany• Can couple barostats to DPD thermostat to give

constant pressure, surface area/tension ensembles[1]

– Applied during force integration (Velocity Verlet) stages– Similar approach to classical MD

• Constant-energy DPD (DPD-E)[2] also available:– Can associate temperature for each particle– Integrate internal energy as well as velocity– Possible to couple with barostat to give constant enthalpy

ensemble[3]

[1] Jakobsen, J Chem Phys 122, 124901 (2005)[2] Pastewka et al., Phys Rev E 73, 037701 (2006)[3] Lísal et al., J Chem Phys 135, 204105 (2011)

DPD Practical Exercises 2 and 3• Due to time, suggest that only one of these needs to

be completed today: you can each choose which one– Course materials available after today, so can

complete the other exercise later• Exercise 2: Mesophase behaviour

– Using DPD to find mesophase structures of two-bead amphiphilic molecules (dimers)

• Exercise 3: Transport properties– Finding viscosity of DPD fluid using Lees-Edwards

boundary conditions (and trying out another pairwise thermostat)

• Start from drfaustroll.gitlab.io/cs2019 (Day 5: Dissipative Particle Dynamics)

DPD Practical Exercise 2• Using DPD to find structures of two-bead amphiphilic

molecules (dimers) at various concentrations

• Run simulations at different dimer concentrations and work out mesophasic structures formed– Visualise shapes, use order parameters

• Try and find boundaries between mesophases

Isotropic (L1) Hexagonal (H1) Lamellar (Lα)

DPD Practical Exercise 3• Finding viscosity of particle-

based fluid with pairwise thermostats (DPD, Stoyanov-Groot)– Apply linear shear to box to give

constant shear rate (velocity gradient)

– Measure shear stresses– Plot stress vs. shear rate:

viscosity = gradient• Try and find relationships

between viscosity and thermostat parameter (! or Γ)

0 2 4 6 8 10Vertical position, y

-1

-0.5

0

0.5

1

Tim

e-av

erag

ed h

ori

zon

tal

vel

oci

ty,

<v x

>

dissipative particle dynamics · mesoscale modelling techniques lattice gas cellular automata lgca...

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