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Chemotaxis of Active Janus Nanoparticles Mihail N Popescu, William E. Uspal, Clemens Bechinger, and Peer Fischer Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b02572 • Publication Date (Web): 26 Jul 2018 Downloaded from http://pubs.acs.org on July 26, 2018

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Chemotaxis of A tive Janus Nanoparti les Mihail N. Popes u,

∗ ,†

William E. Uspal,

†,‡

¶

Clemens Be hinger,

and Peer

∗,†,§

Fis her

†Max-Plan k-Institut

für Intelligente Systeme, Heisenbergstr. 3, D-70569, Stuttgart, Germany

‡IV.

Institut für Theoretis he Physik, Universität Stuttgart, Pfaenwaldring 57, D-70569 Stuttgart, Germany

¶Fa hberei h §Institut

Physik, Universität Konstanz, Universitätsstr. 10, 78464 Konstanz, Germany

für Physikalis he Chemie, Universität Stuttgart, Pfaenwaldring 55, D-70569 Stuttgart, Germany E-mail: popes uis.mpg.de; s heris.mpg.de

Abstra t While olloids and mole ules in solution exhibit passive Brownian motion, parti les that are partially overed with a atalyst, whi h promotes the transformation of a fuel dissolved in the solution, an a tively move. These a tive Janus parti les are known as hemi al nanomotors or self-propelling swimmers, and have been realized with a range of atalysts, sizes, and parti le geometries. Sin e their a tive translation depends on the fuel on entration, one expe ts that a tive olloidal parti les should also be able to swim towards a fuel sour e. Synthesizing and engineering nanoparti les with distin t

hemota ti properties may enable important developments, su h as parti les that an autonomously swim along a pH gradient towards a tumor. Chemotaxis requires that the parti les posses an a tive oupling of their orientation to a hemi al gradient. In this Perspe tive we provide a simple, intuitive des ription of the underlying me hanisms for

1

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hemotaxis, as well as the means to analyze and lassify a tive parti les that an show positive or negative hemotaxis. The lassi ation provides guidan e for engineering a spe i response, and is a useful organizing framework for the quantitative analysis and modeling of

hemota ti behaviors. Chemotaxis is emerging as an important fo us

area in the eld of a tive olloids, and promises a number of fas inating appli ations for nanoparti les and parti le-based delivery.

Keywords

hemotaxis, self-phoresis, hemi ally a tive parti les, diusion, nanoparti les, targeted delivery

∗∗∗ Nanoparti les that do not just randomly diuse around in solution, but that an sense a hemi al say, a an er marker and then a tively move towards it, promise to revolutionize drug delivery and medi al therapy. It may appear mira ulous that a parti le an a tively move by itself without any outside intervention, but there are now several established me hanisms, in luding hemi al rea tions, that an ause olloidal parti les to self-propel. It is therefore of interest to ask if a nanoparti le an utilize these hemi al rea tions to seek out a target, analogous to the In

positive hemotaxis

hemotaxis

an organism senses a hemi al attra tant and swims towards

higher on entrations of the hemi al (

taxis ).

of a motile organism.

A well-studied example of

vi e-versa for hemi al repellents in negative hemo-

hemotaxis

is found in the ba terium

Es heri hia oli (E.

oli ) and the way it adjusts its run-and-tumble behavior in response to a hemi al gradient. 1 The ba terium always swims in a random traje tory, known as a random walk, onsisting of relatively smooth runs whi h are interspersed by un ontrolled re-orientation tumbles (see Fig. 1a), when it briey reverses the rotation dire tion of its rotary motors that drive its heli al agella. While it swims,

E. oli an also dete t ertain mole ules and ount their number

2


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over a period of a few se onds, and thus it is able to sense if life gets better or worse. it

hemotaxes,

2

When

it an adjust its swimming behavior to respond to the mole ules it senses.

However, the ba terium annot simply steer towards the hemi al, and it an also not set the angle by whi h it tumbles, as it does not know where it is heading.

Figure 1: b) During

E. oli 's run- and tumble motion in the absen e of a hemotaxis.

hemotaxis E. oli extends the length of runs in the dire tion of in reasing

a) S hemati of

on entrations of the hemoattra tant (darker blue shading).

Instead of ae ting its overall orientation with respe t to the hemi al gradient,

E. oli

simply extends any runs that bring it to higher on entrations of the hemi al it likes, as shown s hemati ally in Fig.

1b.

During

hanges to its run-and-tumble motion.

2

hemotaxis, E. oli

Be ause

therefore makes temporal

E. oli 's timing hanges

involve a form of

memory and the swit hing of me hani al enzyme motors, this strategy is too omplex to be implemented in syntheti parti les at present. Implementation of

hemotaxis in parti les should therefore involve a pro ess that dire tly

ouples to the parti les' self-propulsion me hanism. It has been shown that nanoparti les with a radius of

∼

100 nm or more an swim by a me hanism known as self-phoresis .

3,4

For this the surfa e of the parti le is partially overed with a atalyst or with enzymes, whi h are able to hemi ally rea t with fuel mole ules that are present in solution (Fig. 3


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2).

Page 4 of 16

Due to the non-homogenous overage of the atalyst, a on entration gradient of the

rea tion produ ts builds up a ross the parti le and gives rise to a uid ow near the parti le's surfa e.

5,6

Be ause of momentum onservation the parti le moves in the opposite dire tion to

the uid ow, similar to a rowing boat, where the boat moves opposite to the stroke dire tion of the oars. Depending on the spe i surfa e rea tion and on the surfa e hemistry of the parti le, the parti le an move either with the rea tive side fa ing forward or ba kward (Fig. 2).

Figure 2:

The rea tion of fuel mole ules (blue) with the rea tive apped area of the parti le

leads to surfa e ows pointing either towards the ap or away from it (depending on the spe i s of the hemi al rea tion here indi ated by dierent olored rea tion produ ts s hemati ally shown as small dots). As des ribed in the text, the slip ows near the surfa e of the olloid with velo ity

u

lead to parti le motion with translational velo ity

U

in the

opposite dire tion to the respe tive slip ows, be ause the total for e in the system must be balan ed (vanish).

Typi ally, when the on entration of the fuel mole ules is in reased, the a tivity (speed) of the parti les will also in rease within some range (

hemokinesis ). 7 A self-propelled parti le

will thus speed up as it moves to higher fuel on entrations. One may wonder if this an

4


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ause a net migration of self-propelled parti les to fuel-ri h regions. However, the presen e of rotational Brownian motion will mean that it is equally probable for the parti le to point in the opposite dire tion, and then by the same me hanism it will speed away from the fuel. It an be shown that when there is no oupling between the dire tion of the fuel gradient and the orientation of the parti le there is no global parti le ux towards higher on entrations. A

ordingly,

hemokinesis

alone annot lead to

8

hemotaxis . 7,8

To avoid any onfusion, it seems bene ial to pinpoint the key features of

hemotaxis

in

the ase of hemi ally a tive Janus olloids. In analogy with other forms of taxis exhibited by mi roorganisms, su h as

rheotaxis or gravitaxis, a self-propelling parti le an hemotax

by

a tively hanging its orientation in response to a hemi al gradient and aligning its swimming dire tion with the dire tion of the gradient.

9

The re-orienting me hanism must arise as part

of its self-propulsion. The latter quali ation is ne essary, as a parti le that is passive (i.e., does not swim), but has some inherent asymmetry (e.g., a spheri al Janus parti le, or a rod-shaped parti le), an still re-orient in the external hemi al gradient while it is swept along be ause of diusiophoresis.

Figure 3:

10

a) A hemi ally a tive Janus parti le in a onstant fuel gradient orthogonal to

(b) Graphi al al ulation of the angular velo ity (rotation of

d.

d) indu ed by the orthogonally

oriented fuel gradient as in (a).

In order to gain some physi al intuition how

5

hemotaxis an be realized in a tive parti les,


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it is useful to analyze simple, well dened models; for example, the model of self-phoresis introdu ed by Golestanian.

6

Consider the s hemati in Fig. 3a. The spheri al Janus olloid

immersed in a gradient of its fuel onverts these fuel mole ules into produ t mole ules at the atalyti ap (shown as small dots). The lo al rate at whi h the produ t mole ules are generated is taken to be proportional to the lo al fuel on entration. orientation

d of the Janus parti le by the unit ve tor

6

One an dene the

pointing along the symmetry axis from

the inert side to the atalyti side of the olloid (see Fig. 3a). Examples of su h parti les are ina tive sili a or Au olloids where one half is overed with a atalyti ally a tive material su h as titania

11

or Pt.

12,13

The variations in the density of the produ t mole ules

c(r) along

the surfa e of the parti le gives rise to a ow, the phoreti slip with slip velo ity

us := −µ(rP )∇|| c(rP ) ,

where

µ(rP )

is the so- alled phoreti mobility, while

onto the tangent plane at the position

rP

∇||

(1)

is the proje tion of the gradient

on the surfa e of the parti le. The dire tion of

the slip ows, and hen e the sign of the phoreti mobility, is di ult to predi t

a priori

and

depends on the details of the hemi al rea tion and spe ies. The parti le will respond to the slip ows it has indu ed by translating with velo ity

U

(as s hemati ally shown in Fig.

s hemati ally shown in Fig. 3a);

2) and possibly rotating with angular velo ity

Ω

(as

U and Ω are given by averages of the hydrodynami slip ow

a ross the entire surfa e of the parti le

5,14

(see the Appendix). A straightforward al ulation

shows that for su h parti les to a tively rotate, i.e., experien e a rotational velo ity the phoreti mobility must vary a ross the surfa e

15

Ω 6= 0,

(see Eq. (3) below). This is naturally

the ase for a Janus olloid, whi h by onstru tion has an anisotropi surfa e (sin e it is made from two dierent materials) and thus dierent phoreti mobilities,

µcatal

and

µinert 6= µcatal ,

over the atalyti and inert sides, respe tively. If the axis of the parti le is aligned with the gradient, the parti le will not rotate, and

6


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its velo ity is to rst order given by

U = U0 (µinert + µcatal )d ,

where

U0

(2)

is a velo ity s ale proportional to the lo al density of fuel. The translational speed

is thus a fun tion of the average mobilities and there is no dependen e on the fuel gradient. Equation (2) thus des ribes

hemokinesis

(there is no

hemota ti

response).

Let's onsider the ase when the parti le's axis is not aligned with the gradient, e.g. due to rotational diusion, and points in dire tion perpendi ular to the gradient as shown in Fig. 3a. Now, the parti le an re-orient if there is ontrast in phoreti mobilities for the two halves of the Janus olloid:

3 (µcatal − µinert ) Ω = 8πa2

I

ω(rP ) ,

Γ

where ω(rP ) := c(rP )eφ ,

with

eφ

(3)

denoting the azimuthal unit ve tor in spheri al oordinates. This is a parti ularly

insightful result, be ause it provides a simple pi ture of

hemotaxis in a tive parti les without

any need for expli it algebra. Simple ve tor addition along the equatorial ir le

Γ (separating

a tive and ina tive regions, see Fig. 3a) is su ient to determine if and how the Janus olloid

an re-orient. Consider the strength and dire tion of the ve tors

Γ.

ω(rP ) := c(rP )eφ

Clearly, the density of produ t mole ules along the ir le

Γ

along the equator

is largest at the point where

the on entration of fuel mole ules (shown by the blue shading in Fig. 3a) is largest, i.e. for

φ = 90◦

in the dire tion of

+y

in Fig.

produ t mole ules will be smallest for

3a.

φ = 270◦

Similarly, the on entration of generated in dire tion of

−y .

Due to symmetry, the

variations in density of produ t mole ules are as indi ated by the olors along the ir le in Fig. 3b, and, a

ordingly,

|ω1| = |ω4|, |ω2 | = |ω3 |,

7

and


|ω1| > |ω2 |.

Γ

A

ording to Eq.

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(3), the addition of all ve tors

ω 's

along the ir le

ω

Γ

Page 8 of 16

will generate the overall rotation. The

on the right side of Fig. 3b with ve tor addition

ω1 + ω4

point towards

−x,

left side shows weaker generation of produ t mole ules and a smaller resultant along

+x.

ω

while the

that points

Therefore, the net result of the ontributions is a ve tor pointing in the negative

x-dire tion. To rst order, this response is dire tly proportional to the gradient in fuel on entration and an lead to

hemotaxis.

If the dieren e in the mobilities

µcatal − µinert > 0

rotates to align its a tive ap towards the high fuel region, while for

the parti le

µcatal − µinert < 0

the

response is reversed and the parti le rotates to align its a tive hemisphere away from the high fuel regions.

Figure 4:

The possible

hemota ti

gradient of fuel. The unit ve tor

d

behaviors of a hemi ally a tive Janus parti le in a

refers to the orientation of the parti le's axis (see the

main text).

One an thus on lude that to rst order the translation of the olloid and its

8


hemokineti

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behavior depends on the average mobility

µav := µcatal + µinert ,

whereas the rotation and its

hemota ti response depends on the dieren e ∆µ := µcatal −µinert This permits us to lassify the various

hemota ti responses exhibited by Janus olloids in the phase-diagram shown

in Fig. 4. Where do popular model atalyti Janus parti les t into our phase diagram? Con erning the verti al axis of the diagram, the dire tion of parti le motion is dire tly observable in most experimental setups. of a polystyrene

a

4,1618

Under typi al experimental onditions, spheri al parti les that onsist or sili a

12,19

ore with a hemispheri al oating of platinum are observed

to move with the inert-side forward in the presen e of hydrogen peroxide fuel. Au/Pt nanorods are generally observed to move in the dire tion of the platinum side in the presen e of H2 O2 . (Although both metals are involved in the redox rea tion, the nal rea tion produ t oxygen is produ ed on the platinum side, and hen e these parti les an be regarded as atalyst-forward .

3,20,21

) Light-a tivated TiO2 /SiO2 parti les in the presen e of H2 O2 an

(depending on the pH) move either inert-forward or atalyst-forward.

22

Con erning the horizontal axis of the diagram, the orientational response of a Janus parti le to a fuel gradient is dire tly related to the surfa e mobility ontrast, as dis ussed above (Eq.

(3)).

However, there are only few studies that have sought to measure this

rotation partially be ause of experimental di ulties in maintaining steady hemi al gradients, but also be ause early studies fo used on demonstrating parti le migration.

19,20

A re ent study onsidered Au/Pt nanorods near a hydrogel sponge loaded with H2 O2 fuel, and sought to distinguish hemota ti rotation of the nanorods from possible ee ts of adve tive ows.

A detailed quantitative analysis suggested that these parti les exhibit

weak rotation of their Pt end towards the fuel sour e,

21

whi h would pla e them in the top

right quandrant of Fig. 4 (a tive forward positive hemotaxis). Indire tly, surfa e mobility

ontrast an be inferred from the rotation of Janus parti les near solid surfa es or obsta les,

We note, though, that most of these observations have been made for parti les moving near a solid surfa e, and it is possible that the presen e of the surfa e ould ae t the dire tion of motion. However, studies of parti le motion in bulk liquid tend to onrm the inert-forward or atalyst-forward hara ter of motion inferred from the motion near a surfa e. a

11,16

9


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Page 10 of 16

where hemi al gradients develop from lo al onnement of the rea tion produ t.

These

studies suggest that SiO2 /Pt and polystyrene/Pt parti les would rotate their atalyti aps away from a fuel sour e, Fig.

12,17,23,24

whi h would pla e them in the bottom left quandrant of

4 (inert forward positive hemotaxis) while light-a tivated TiO2 /SiO2 parti les an

(depending on the pH) rotate their atalyti aps towards a fuel sour e

11

(inert forward

negative hemotaxis, bottom right quandrant of Fig. 4). Based on these onsiderations, we onsider that the quantitative hara terization and modeling of hemotaxis is emerging as an important fo us area in the a tive olloids ommunity. We anti ipate that Fig. 4 will provide a useful organizing framework for this work. Fig. 4 also highlights the importan e of the intrinsi ability of typi al Janus olloids, whi h by onstru tion have distin t phoreti mobilities asso iated with their two fa es, to rotate their axis in response to a fuel gradient, i.e., dire tly steer towards or away from a fuel sour e. For omparison, onsider another possible me hanism, statisti al in nature, for hemotaxis.

21

Suppose the parti le orientation is un onstrained and free to u tuate; the nite size of the parti le implies that the side fa ing the fuel sour e will see a slightly larger on entration than the opposite side. Hen e, the parti le moves slightly faster when the atalyti ap fa es the sour e of fuel then if it fa es in the opposite dire tion, whi h gives rise to biased diusion. The bias would be towards the fuel sour e for atalyst-forward parti les, and away from the fuel sour e for inert-forward parti les. Therefore, only two of the four quadrants in Fig. 4

an be a

essed through this me hanism. As a further remark on Fig. 4, we expe t that for parti les with less symmetri shapes, yet well dened polarity (e.g., Janus rods), the equivalent phase diagram will have the same four lasses of behavior, but the phase boundaries will be more omplex. Finally, the single parti le phase diagrams in Fig. 4 are parti ularly relevant for parti les and environments where thermal u tuations do not dominate. very small nanoparti les in open, low vis osity environments, the parti le level may be less important, but a e.g., enhan ed diusion.

hemota ti

at the single

response an then still emerge from,

25

10

hemotaxis

For


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Chemotaxis

oers a range of promising possibilities for parti les that swim with the help

of hemi al rea tions, and we expe t that there will be a range of fas inating appli ations. Engineering parti les for positive

hemotaxis

is a parti ularly important step towards ap-

pli ations in targeted drug delivery. The me hanisms outlined in this Perspe tive suggest how positive

hemotaxis

an be lassied and how it an be implemented in a tive parti les

( hemi al motors) that self-propel.

A knowledgement The authors are grateful for insightful dis ussions with H. Hess, R. Kapral, and A. Sen. M.N.P. and P.F. a knowledge funding from the Volkswagen Foundation.

Appendix At steady state, the rigid-body translational, of the sphere),

Ω,

U,

and rotational (with respe t to the enter

velo ities of a spheri al olloid moving in an unbounded, quies ent, in-

ompressible Newtonian uid, in the absen e of external for es or torques but subje t to a phoreti slip

us (rP ) := −µ(rP )∇|| c(rP ) , boundary ondition for the Stokes ow the hydrodynami slip:

u(r),

are given by the following surfa e averages of

5,14

U = −hus (rP )i , Ω=− Here

hXi :=

1 4π

R 2π R π 0

0

(4)

dφ dθ sin θ X,

3 her × us (rP )i . 2a

while

a

denotes the radius of the sphere.

• Ne essary ondition for a tive rotations

11


(5a)

(5b)

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In a oordinate system with the enter at the origin of the sphere, and dening

µ(θ, φ)c(r, θ, φ),

f (r, θ, φ) :=

one obtains after straightforward algebrai manipulations of Eq. (5b)

2a Ω = −her × [−µ(rP )∇|| c]i 3 er ||(∇c−∇|| c)

=

her × (µ(rP )∇c)i

=

her × [∇f (rP ) − c(rP )∇µ(rP )]i Z 1 dV ∇ × (∇f (r)) = {z } | 4π ≡0

|r|/R>1

− her × [c(rP )∇µ(rP )]i =

− her × [c(rP )∇µ(rP )]i ,

i.e.,

Ω=−

3 hc(rP ) [er × ∇|| µ(rP )]i . 2a

(6)

The onversion of the surfa e integral to a volume integral is a standard ve tor al ulus result (see, e.g., Eq. (20) in Appendix II of Ref. 26). For the parti le to rotate (i.e., gradient

∇|| µ(rP )

should not identi ally vanish, i.e., the phoreti mobility

Ω 6= 0),

must

the

vary over

the surfa e of the parti le. This is similar to the ondition for rotation in the ase of lassi phoresis.

5

Note that the equations governing the eld above. This means that Eq. (6) applies,

c(r)

have not been used in the derivation

inter alia, to su h parti les immersed in any kind of

fuel gradient. In this respe t, it generalizes the results in Ref. 9 and it orre ts the argument in Ref. 27.

• Rotation of a Janus parti le For a Janus parti le, the phoreti mobility may be approximated by a step fun tion, the step being lo ated at the equator ir le

Γ

separating the a tive and inert hemispheres (see

12


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Fig. 3a):

µ(rP ) = µcatal + (µinert − µcatal ) H(θ − π/2) , where

H(x)

gradient

denotes the Heaviside step fun tion (H(x

∇|| µ(rP )

then leads to a

δ(θ − π/2);

> 0) = 1, H(x < 0) = 0).

a

ordingly, the surfa e integral in Eq.

(7)

The (6)

redu es to the very simple line integral shown in Eq. (3) in the main text.

Referen es (1) Berg, H.; Brown, D.

(2) Berg, H.

Nature 1972, 239, 500505.

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