Depletion attraction between dendrimers

Depletion attraction between branched polymers

Summer Internship Project

Submitted By Prafull Kumar Sharma IIT Delhi

Supervised By

MSc. Martin Wengemeyer Prof. Dr. Jens-Uwe Sommer (Phd Student) (Group Leader)

Student: Prafull Kumar Sharma

Studies: Engineering Physics

Project Type: Summer Internship Project

Student ID: 2013PH10861 (IIT Delhi)

Address: SC-11,Satpura Hostel,IIT

Campus,New Delhi,110016

Phone-No.: +91-7530831251

E-Mail: [email protected]

New Delhi, 12/10/2016

Table of Contents

1MOTIVATION ............................................................................................................ 1

2INTRODUCTION ...................................................................................................... 12.1THEORY OF POLYMERS ........................................................................................................ 1

2.2DEPLETION ATTRACTION BETWEEN TWO COLLOIDS ........................................................... 2

2.3THEORY OF DENDRIMERS .................................................................................................... 3

2.4BOND-FLUCTUATION-METHOD ........................................................................................... 4

2.5 UMBRELLA SAMPLING ....................................................................................................... 6

3 RESULTS AND DISCUSSION ................................................................................ 7

4 REFERENCES ........................................................................................................ 11

List of Abbreviations and Symbols

MCS Monte Carlo Steps

MC Monte Carlo

LeMonADE Lattice Based extensible Monte-Carlo Algorithm and Development Environment

PMF Potential of mean force

BFM Bond Fluctuation Model

WHAM Weighted Histogram Analysis Method

c Density of solvent

g1 Generation of dendrimer 1

g2 Generation of dendrimer 2

Prafull Kumar Sharma (IPF,Dresden)

1 MOTIVATION

One of the major challenges for scientists is to design and modify molecules such as

polymers, to give them functionality suitable for purposes such as drug delivery, cata-

lysis, fluorescent sensors etc. For developing such kind of understanding, simulations

are one of the important way to understand and predict the effect of molecular design

on it's various physical properties.[1]

Dendrimers are basically synthetic polymers having treelike structure with branching.

These kind of polymers have their applications in bioengineering, material science and

chemical industry. Dendrimers are used as drug delivery agents due to large number of

sites having possibility to couple to an active species. These kind of molecules are

used in chemical reactions as catalysts. Dendrimers can also be used in sensor applica-

tions. However there is still a lot of need to increase our understanding about these

molecules in order to have more optimized designs and possibilities of new applica-

tions.[2] The main aim of this project is to understand “depletion attraction between

dendrimers” with or without solvent. In this project, we used Bond fluctuation model

(BFM) for simulation method along with umbrella sampling for free energy calcula-

tion. I would like to thank MSc Martin Wengemayer and MSc Ron Dockhorn for their

constant guidance throughout the project. I would like to thank IPF for their scholar-

ship. I would also like to acknowledge warm support of Prof. Sommer and the group

during the stay.

2 INTRODUCTION

2.1 Theory of Polymers

A polymer molecule consists of several identical repetitive units,called as

'monomers'.Every polymer can be thought of as linear chain of N connected segments

(first proposed by Werner Kuhn) ,now called Kuhn Segments. Each segment in a freely

jointed chain can randomly orient in any direction without the influence of any forces,

independent of the directions taken by other segments. While considering a real chain

consisting of bonds and with variable/fixed bond angles, dihedral angles, and bond

lengths, Polymers can be characterized in terms of bond length l, dihedral angle φ and

bond angle θ . In this simulation one bond vector represents one Kuhn segment, for

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this reason the distance between two connected monomers will be b. The dihedral

angle φ is defined as the rotation of the subsequent bond vector around the axis which

conforms to the direction of the previous bond vector.

The end to end vector Re-e for a polymer of N monomer can be written as,

Re-e = ∑i=1

i=N −1

r i+1−ri= ∑i=1

i=N −1

bi where ri is the position vector of ith monomer.

However if we calculate expectation value of above quantity for an ideal chain, it

would turn out to be zero. So for characterizing size/length/scaling of polymers, we

should calculate mean square end-to-end distance.

<Re-e 2`> = b (N-1)2γ where γ is flory exponent. For freely jointed real polymer chain,

theoretical value of γFlory is found to be equal to 3/5 which is indeed quite close to

value of 0.588 ,obtained in simulation. The radius of gyration Rg is used to character-

ize the span of polymeric chain i.e. span of largest dimension. It is defined similarly to

the moment of inertia by the average square distance between the monomers of the

chain to the chain’s centre of mass.[3]

2.2 Depletion Attraction between Two colloids

The Depletion Force is an attractive, short-range, entropic force which is a force res-

ulting from the system's thermodynamic tendency to increase its entropy rather than

from a particular underlying microscopic force.[4][5]

Suppose colloidal spheres are mixed with non-adsorbing depletants. Negative adsorp-

tion then results in an effective depletion layer near the surface. Presence of depletion

layer explains the presence of such attractive short range force. When the depletion

layer of two spheres overlap, free volume available for the depletants increases. Thus

free energy of the depletants is minimized by the states in which these colloidal

spheres are close together. Significance of this force can be realized in such way that

even though direct colloid-depletant interaction and colloid-colloid interactions are

both repulsive, there exists an attractive short range depletion force. For small de-

pletant concentration, the attraction equals the product of osmotic pressure and the

overlap volume.[6]Also due to overlapping of depletion layers, ΔS i.e. change in en-

tropy S, is positive which yields out loss of energy ΔF = -TΔS.

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(a) (b)

Figure 1: The depletion layers are indicated by short dashes (a).When there is no overlap of deple-

tion layers, osmotic pressure on the colloids due to depletants (in this case, polymer chains) is isotropic,

thus no entropic (depletion) force. (b) For overlapping depletion layers, the osmotic pressure on the

spheres is unbalanced, the excess pressure is indicated by arrows. Taken from “Phase Behaviour in Dis-

persions of Spheres and Stiff Chains”, R.Tuiner and Schmidit

2.3 Theory of Dendrimers

Dendrimers represent monodisperse macromolecules with regular and strongly

branched architecture. Dendrimers can be characterized by three parameters which are

g, s and f .The generation g, shows which cell number we are at. The functionality, f ,

is the number of chains branching out from each branching point (monomer).The

spacer length s indicates length of linear chain which composes dendrimer and acts a

basic component. Part of the interest in dendrimers is that one can vary g, s and f

along with the kinds of monomers used to produce a very rich variety of structures

with very different material properties.[7][8]

Figure 2: A schematic of a dendrimer

with functionality f = 3, showing the

generation number g, and the spacer

length s. The terminal groups are shown

darker than the other monomers. Taken

from Macromolecules, 37, page 3049,

2004.

Dendrimers with lower s and g

and high density can be modelled

as a soft sphere which imitates

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same as hard sphere and shows depletion attraction in presence of depletants. However

dendrimers show diffusion of each other and presence of hollow core makes it feasible

to work as a drug delivery agent, in addition to depletion. However these two effects

can only be effectively viewed only in higher s and g because for lower s and g ,these

soft spherical dendrimers would almost act like a hard sphere. Also one interesting

case is diffusion of a big generation dendrimer (say g=7) in to a small generation

dendrimer (say g=2) in a solvent because such diffusion creates a metastable state

(highly attractive force) which is of very huge significance in terms of dendrimer as a

drug delivery agent (with drug being small generation dendrimer in this case).

Figure 3: Dendrimers used in simulation for PMF calculation between these two, where spacer length

s=2,g1=g2=4.colour is marked for distinction between comformation of the two.

2.4 Bond-Fluctuation-Method

In 1988 I. Carmesin and K. Kremer introduced the Bond-Fluctuation-Model

(BFM) as a Monte Carlo algorithm for simulating conformation and dynamics of poly-

mer systems.[9][10] This method uses coarse-grained model which neglects chemical

properties of polymeric structures. In this method, monomers occupy 8 lattice posi-

tions to form a cube of edge length two in each positive direction, on a regular cubic

lattice. Each lattice position can be occupied by only one monomer in order to ensure

“excluded volume” effect (as per discussion in previous sections, 'excluded volume' is

one of the factor affecting depletion interaction between two spheres) . To ensure cut-

avoidance (no explicit test of local topology) on the single cubic lattice the bond

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length is required to be smaller than 4. Monomers are connected by bond vector which

are generally taken from a given set of 108 bond vectors. In our simulations, allowed

bond vectors B are made up from permutations and sign-variations of six base vectors

in 3-D:

Thus, bond lengths corresponding to such combination of bond vectors are

{2,√ 5 ,√ 6 , 3 ,√ 10 } .Above restrictions on monomers and bond vectors coupled

with shape and size of monomers fulfils criteria of self avoiding walk where one bond

unit is synonymous with Kuhn’s segment. The Monte Carlo step is performed in the

directions on the lattice’ axis, ΔB = P± (1, 0, 0) which means six possible directions

for monomers to move. Just like any other MC simulations, one direction in which a

monomer has to move, is selected randomly. Then firstly the new position is checked

for multiple lattice occupation and secondly whether the new bond vectors belong to

the allowed set B. If both condi-

tions are fulfilled, the move is

performed, otherwise rejected.

For a lattice with in total N

monomers one Monte Carlo Step

is defined as N attempts to move

a monomer as described above.

Thereby not every monomer

must be selected, but others can

be selected more than once.

Figure 4: Bond Fluctuation Model, as you can see one monomer occupies 8 lattice points on a regular

cube to form a cube and these monomers are connected by a bond vector from specific set of 108 bond

vectors. Taken from IPF theory polymer group applet

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For energetic interactions in MC simulations, we use metropolis algorithm. As in the

algorithm described above one monomer and one direction are selected randomly.

Then, the energetic contribution on the new position Enew is evaluated and compared to

the energy in the initial state Einit to get the energetic difference ΔE. If the new posi-

tion is more energetically favourable i.e. ΔE≤0, then the move is allowed. However if

new position is less energetically favourable i.e. ΔE>0, then the move is allowed with

probability, P =exp(- ΔE/kT) where k is Boltzmann constant and T is temperature.

In theoretical Polymer physics group at IPF, LeMonADE project has been developed

which implements BFM through a complex framework of C++ programs. In the pro-

gram, the boxsize, BS, is a magnitude of two (BSi= 2n with i = x, y, z and n ϵ N).In this

model, we impose periodic boundary conditions on monomers i.e. those who are leav-

ing the box will re-enter the box.The program saves the configuration of the

monomers in the box after a given MCS interval. For each save, the simulated struc-

ture has a specific conformation.

2.5 Umbrella Sampling

The quantification of a pair interaction in computer simulations can be obtained by re-

stricting dendrimers implementing a known potential. Well known spring potential

Vspring, and morse potential, Vmorse which is implemented in C++ framework to couple

with Simulation setup for dendrimers. We have used Morse potential in addition to

spring potential to see whether overlap of tails (important for better overlap of probab-

ility density thus sampling) can be improved.[11][12]

Also (r,rƿ 0)sim is probability distribution for system which is superposition of VMF and

Vspring i.e. simulation setup and (r,rƿ 0)spring is probability distribution arising due to

spring potential with equilibrium position r0 only, which can be written as:

ƿ(r,r0)spring = ps exp(-Vspring(r,r0))

ƿ(r,r0)sim = psim exp(-Vspring(r,r0)).exp(-VMF(r))

With the aid of Umbrella sampling ,we can exratct out potential of mean force VMF and

write the equation in this form by solving above two equations,

where ps and psim are normalization constants for spring potential and simulation setup

respectively. Similarly in case of Morse potential,we replace Vspring with Vmorse and per-

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form umbrella sampling in similar fashion. As you can see there is an arbitrary term in

formula for PMF which depends on normalization constants. Thus we need to shift

values of PMF in each window for overlapping.

3 Results and Discussion

In this project, Umbrella sampling was used for calculating Potential of mean force by

overlapping windows and eliminating biased potential effect by means of 'manually

shifting with the help of histogram analysis and Excel sheet', and WHAM algorithm as

well. In this graph you can see, we have generated results for PMF value between two

symmetric dendrimers of generation 3 and spacer length 1 ,as a function of distance

between their centre of mass. Also you can see PMF remains same regardless of

biased potential ,method of overlapping windows and adjustment, which is as expec-

ted due to fact in the end we eliminate effect of biased potential. In this graph, we have

used spring constant K=.50 for sampling over Rc-c >10 and K=1.50 for sampling over

Rc--c <10. Also optimized value of morse parameters a =.05 and D=100 ,where D is

well depth, is used. We used biased potential because the forces like depletion force

are very short range and to sample them we need to keep the dendrimers together and

very close to each other. In case of interaction between unsymmetric dendrimers , due

to bigger size of other dendrimer compared to symmetric dendrimer , PMF in case of

unsymmetric dendrimers case will be higher due to more repulsive force which can be

seen in the figure above as well. We also simulated the two dendrimers with biased po-

tential, in presence of solvent (in our case,linear chain polymers) with maximum dens-

ity, c=0.50 expecting a depletion force which could not be observed, which was expec-

ted.Reasons behind no observation of such force is most probably due to statistical er-

ror i.e. MCS were not sufficient enough to encompass most of phase space. Here force

of attraction involves diffusion component (in case of hard spheres, excluded volume

effect was only reason behind such short range force) as well. However one interesting

simulation would be to calculate PMF among one big dendrimer and one very small

dendrimer in the dense solvent (say, c=0.5).

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(a)

Figure 5: It should be noted that PMF is in unit of KBT and spacer length of both dendrimers

is set to 1.There is no solvent in this case.

*Here KBT is put as unity and PMF is calculated in unit of KBT.

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Figure 6: In the left side of the figure, you can see dendrimer solution with very high solvent density

c=0.5 (maximum possible) with red and green dendrimers inside. On the right hand side, dendrimers are

shown with red being generation 3 and green being generation 5 dendrimers. In this system we have

taken spacer length as 1.

There was a need of optimization due to fact that in case of smaller potential,

dendrimers might get away from each other due to weak force and roam around freely

thus making it impossible for us to sample PMF. In general we need a potential with

medium well depth and medium broadness for overlap. In case of spring potential, we

have only one parameter ,k (spring constant) which if increased will increase well

depth and decrease broadness of potential. However force of attraction needed to re-

strict them over a range of distance, depends on only one parameter k and with in-

creasing distance between dendrimers ,spring force will increase and pull back these

dendrimers from getting away. While morse potential allows us to have variable well

depth (change parameter D) and variable broadness (morse parameter ,a) quite inde-

pendently,it has issues regarding force of attraction because as you increase, distance

force will go to zero. And changing morse parameter and well depth won't always

change maximum magnitude of force of attraction that depends on product of these

two parameters, i.e. for a range of morse parameters and well depths,maximum mag-

nitude of force of attraction would be same (D*a= constant,yields an hyperbola) thus

while morse potential has independent control on depth and broadness, it has little and

sensitive control on force of attraction.

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(a)

(b)

Figure 7: It should be noted that PMF is in unit of KBT .There is Linear polymer chain solvent

in this case. (a) I have done overlapping of windows using histogram analysis method. this one

in paricular is for symmetric dendrimers of generation 4 and spacer length 2. (b) PMF for both

symmetric and unsymmetric dendrimers in solvent case.

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4 References

[1] E. Abbasi. Dendrimers: synthesis, applications, and properties. Nanoscale Research

Letters (2014) 9:247

[2] JK Kassube et al. Stereoselctive dendrimer catalysis. Topics in organometallic

chemistry. Volume 20,page 61-96 (june,2006), Springer,Hiedelberg

[3] Iwao Teraoka. Polymer solutions: An introduction to Physical properties.

Wiley,New York,2002.

[4] T Biben, P Bladon, and D Frenkel. Depletion effects in binary hard-sphere fluids.

Journal of Physics: Condensed Matter, 10799, 1996.

[5] Sho Asakura and Fumio Oosawa. Interaction between particles suspended in solu-

tions of macromolecules. Journal of Polymer Science, 33(126):183–192, December

1958

[6] HNW Lekkerkerker and Remco Tuinier. Colloids and the Depletion Interaction.

Springer,Heidelberg, London, New York, 2011.

[7] S.V. Lyulin et al. Effect of Solvent Quality and Electrostatic Interactions on Size

and Structure of Dendrimers. Brownian Dynamics Simulation and Mean-Field Theory.

Macromolecules 2004, 37, 3049-3063

[8] http://iitbmonash.org/Resources/Research_Opps_PDFs/IMURA_0017.pdf

[9] H. P. Deutsch and K. Binder. Interdiffusion and self-diffusion in polymer mixtures:

A Monte Carlo study. The Journal of Chemical Physics, 94(3):2294, 1991.

[10] I Carmesin and K Kremer. The bond fluctuation method: a new effective algo-

rithm for the dynamics of polymers in all spatial dimensions. Macromolecules, pages

2819–2823,1988.

[11] B. Roux. The calculation of potential of mean force using computer simulations

Computer physics communications 91 (1995),page 275-282

[12] Daan Frenkel and Brend Smit. Understanding Molecular Simulation: From algo-

rithms to applications, page 192-196.

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http://iitbmonash.org/Resources/Research_Opps_PDFs/IMURA_0017.pdf