text:
# Time complexity of Hash table lookup

Suppose I have a hash table which stores the some strings. Let the index/key of this hash table be the length of the string. what is the time complexity of checking if the string of length K exists in your hashtable? is it O(1) or O(k) ?

• What do you think? How long does it take to read the string? Is it necessary? Apr 21 '20 at 8:13
• I don't have to read the string. All i need to know is that if a string of length k exists in my table. isn't this O(1) because I have my index as lengths and hence i'll just go to that index and see if a string is present. how can it be O(k)? Apr 21 '20 at 10:43

Here's how a hash table usually works:

• Given an input $$x$$, you compute its hash $$h$$.
• You look at cell $$h$$, and compare your input to the key $$k$$ stored there.
• If the key compares, great. Otherwise, what you do depends on the exact implementation.

As you can see, you have to compute the hash of your input, as well as compare it to a key stored at the table.

Typically we want the hash to depend on the entire input, has computing the hash takes at least linear time in the size of $$x$$. Similarly, comparing $$x$$ and $$k$$ takes non-constant time.

• Oh yes! I I forgot to account the time-complexity of generating the hash. This makes sense. Thanks ! Apr 21 '20 at 10:52

If the hash function takes the whole string into account, then the hash calculation takes $$\Omega(k)$$ time where $$k$$ is the length of the string. So the hash table lookup takes $$\Omega(k)$$ time.

The hash table lookup takes $$\Theta(k)$$ time if the hash calculation is done in linear time in the length of its input, which is typical for hash functions, and the lookup of the value takes $$O(k)$$ time. That part is usually not the case. With a typical hash table, looking up the bucket for the hash value in the table takes constant time $$O(1)$$ with respect to the length of the string $$k$$ and the size of the hash table $$n$$. But that's not the end of the lookup.

If the bucket is empty, the value is absent and the lookup ends there, having consumed $$\Theta(k)$$ time. If the bucket contains a value, you need to be compare it with the string that you're looking up, and that takes $$O(k)$$ time. If the value in the bucket is equal to the value that you're looking up, the lookup ends there, having consumed $$\Theta(k)$$ time. But what if the value is different? What happens next depends on the variant of the hash table data structure.

• In a variant where collisions are handled by growing the hash table until there are no more collisions, the lookup ends here and the cost of the lookup is indeed $$\Theta(k)$$. This variant has efficient lookup, but pays for it by making addition potentially very expensive (depending on the growing strategy when a collision happens).
• In a variant where collisions are handled by putting the value in the next bucket (for some traversal strategy), the search can potentially cover the whole table, with one string comparison for each bucket. The worst-case cost of each string comparison is $$\Theta(k)$$ (with a typical string comparison algorithm, the worst case is reached if the strings have the same length and differ towards the end). In the worst case, the total cost of the search is therefore $$O(k \cdot n)$$.
• In a variant where collisions are handled by storing values that land in the same bucket in a secondary lookup structure (such as a list, a hash table with a different hash function, or a search tree), the cost depends on what the secondary data structure is. It's unlikely to be $$O(1)$$ with respect to the number of values in the bucket, so the worst-case cost of the lookup is not $$O(k)$$.

Hash tables only have constant-time operations in the average case with some “reasonable” (or not) hypotheses over the data, not in the worst case. See also (When) is hash table lookup O(1)?


text:
## June 11, 2004

### Open string backgrounds from boundary state deformations

#### Posted by Urs Schreiber

Today Eric Forgy has asked me to summarize some aspects of results by Koji Hashimoto concerning the description of open string backgrounds by deformations of boundary states.

With the time and energy that I have I will certainly not succedd in giving a complete description starting from first principles, but I’ll try to convey the basic ideas and add some comments concerning my perspective on these matters. (The details are left to the comment section ;-)

So here’s the story: While thinking about how the deformations of closed string worldsheet theories of the form which we discussed a while ago in this entry generalize to open strings I began of course to think about boundary state formalism and some literature search then turned up the papers

Koji Hashimoto: Generalized supersymmetric boundary state (2000)

T. Maeda & T. Nakatsu & T. Oonishi: Non-linear Field Equations from Boundary State Formalism (2004)

Koji Hashimoto: The shape of nonabelian D-branes (2004)

Here is my perspective on what’s going on:

We had seen that all (massless NS, at least) backgrounds of the closed string come from deformations of the super-Virasoro constraints of the form ${d}_{K}\to {e}^{-W}{d}_{K}{e}^{W}$, ${d}_{K}^{†}\to {e}^{{W}^{†}}{d}_{K}^{†}{e}^{-{W}^{†}}$, where ${d}_{K}^{\left(†\right)}\sim {\mathrm{iT}}_{F}±{\overline{T}}_{F}$ are polar combinations of the left- and rightmoving supercurrents.

Some of these transformations are pure gauge. Namely if ${W}^{†}=-W$ then (and only then!) is the above transformation a global unitary tranformation of the entire super-Virasoro algebra. It has been shown how such unitary transformations encode gauge shifts and dualities of the background fields, precisely as expected (in retrospect, in my case) from string field theory considerations.

In particular, it can be shown that the gauge transformation $W=i\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}{X}^{\prime \mu }\left(\sigma \right){A}_{\mu }\left(X\left(\sigma \right)\right)$ corresponds to turning on a gauge field background, which couples trivially to closed strings, manifesting itself as a gauge-trivial shift $B↦B+dA$ of the 2-form NS-NS field. This can be understood by regarding a closed string as consisting of two glued open strings. The gauge field couples to the endpoints of these open strings and since they stick together (in this picture) one coupling cancels the other.

But this alrready suggests how the deformation generalizes to the open string: We have to rewrite the opens string theory as that of a closed string split in half.

This is precisely the moral content of the boundary state formalism. I have a brief description of this technique in what is currently appendix C.2 ‘Boundary states’ of my OSFT notes:

Imagine an open string propagating with both ends attached to some D-brane. The worldsheet is topologically the disk (with appropriate operator insertions at the boundary). This disk can equivalently be regarded as the half sphere glued to the brane. But from this point of view it represents the worldsheet of a closed string with a certain source at the brane. Therefore the open string disk correlator on the brane is physically the same as a closed string emission from the brane with a certain source term corresponding to the open string boundary condition. The source term at the boundary of the half sphere can be represented by an operator insertion in the full sphere. The state corresponding to this vertex insertion is the boundary state.

This is very similar in sprit to the method of mirror charges in elementary electrostatics. Simple but profound.

So what we have to do in fact consider just closed strings, and the deformations known of these, and then consider the modified inner product

(1)$⟨\psi ,\varphi {⟩}_{\mathrm{openstring}}:=⟨\psi ,\varphi \mid \alpha {⟩}_{\mathrm{closedstring}}\phantom{\rule{thinmathspace}{0ex}},$

where $\mid \alpha ⟩$ is the closed string state which encodes the desired boundary condition for the open string.

One curious observation is that the boundary state $\mid {\alpha }_{0}⟩$ which describes a flat space-filling D9 brane is nothing but the constant 0-form on loop space! For anyone knowing the respective formulas this is trivial, but to me this seems to be important, conceptually. For details of what I am talking about see what is currently section 3.6 ‘Boundary states and loop space formalism’ in my OSFT notes.

So let $U\left(A\right)=\mathrm{Tr}𝒫\mathrm{exp}\left(\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}{X}^{\prime \mu }{A}_{\mu }\right)$ be the deformation inducing a gauge field background for the closed string, as in equation (3.51) of my hep-th/0401175.

This suggests we study something like $\mid \alpha ⟩=U\left(A\right)\mid {\alpha }_{0}⟩$ if we want to describe open superstrings on $N$ flat D9 branes with a $U\left(N\right)$ gauge field turned on.

And, indeed, this is essentially what Hashimoto showed in JHEP 04 (2000) 023 to be the correct choice, at least with regard to the bosonic degrees of freedom. In addition one want the boundary state to be BRST-closed, i.e. annihilated by the BRST operator, because that sort of makes is an honest physical state of the closed string (a coherent state, in fact, describing the macroscopic excitation of a brane with that gauge field). So one has to add a fermionic component. This can be guessed and checked as in the above paper or derived with superfield formalism as in hep-th/0312260, but in any case the result is that we also need to multipliy by the unitary operator

(2)${U}^{f}\left(A\right)=\mathrm{exp}\left(\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}{ℰ}^{†}\cdot F\cdot {ℰ}^{†}\right)\mathrm{exp}\left(-\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}ℰ\cdot F\cdot ℰ\right)$

where ${ℰ}^{\left(†\right)}$ are the form creators/annihilators on loop space and $F={d}_{A}A$ is the field strength. This is natural, since according to the closed string formalism of section 3.3.2 these operators are asscociated with deformations turning on 2-form backgrounds of the form $\mathrm{dA}$.

So this is already the basic idea: A unitary and hence gauge-trivial deformation of the closed string becomes a non-trivial deformation of the open string formalism by ‘cutting it in half’, i.e. by inserting it only on one side of the inner product, instead of on two sides, so to say.

It should also not be a miracle how such deformations arise in the context of

A. Recknagel & V. Schomerus: Boundary deformation theory and Moduli spaces of D-branes,

because that was already clarified by J. Klusoň as reviewed (and referenced) in section 2.4 ‘CFTs from string field backgrounds’ in my notes (I’ll have to expand on that summary, though…).

The above method generalizes strightforwardly to tachyon backgrounds, finite number of gluon excitations (instead of coherent states) and what not. This and the derivation of the correct background equations of motion from the BRST-closedness condition is shown in great detail by Koji Hashimoto in hep-th/0312260 (thereby solving for the case of open striongs the little excercise that a kind referee suggested to me in the context of closed strings ;-)

It is nice to see how much physics this method captures. For instance on p. 5 of hep-th/0401043 Koji Hashimoto mentions that a set of D0 branes whose distribution is encoded in the matrices ${M}^{i}$ can be obtained from the simple boundary state $\mid {\alpha }_{D0}⟩$ of a single D0-brane simply by applying the unitary ‘translation’ operator with respect to the non-commutative coordinates ${M}^{i}$:

(3)$\mathrm{Tr}𝒫\mathrm{exp}\left(\oint {M}^{i}{\partial }_{i}\left(\sigma \right)\right)\mid {\alpha }_{D0}⟩\phantom{\rule{thinmathspace}{0ex}}.$

Note that if we tranlated these branes in this way and at the same time added a gauge field in the sense of the boundary state

(4)$\mathrm{Tr}𝒫\mathrm{exp}\left(\oint {M}^{i}\left({\mathrm{iX}}_{i}\left(\sigma \right)+{\partial }_{i}\left(\sigma \right)\right)\mid {\alpha }_{D0}⟩$

this is nothing but applying a Pohlmeyer operator to the original boundary state. Maybe I was not that far off with my outlook in hep-th/0403260 after all…

Posted at June 11, 2004 9:12 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/381

## 1 Comment & 0 Trackbacks

### Re: Open string backgrounds from boundary state deformations

[Due to ongoing problems with the sps newsserver I reproduce yet another sps post here:]

Urs Schreiber wrote in message news:

I have been looking at some papers by Koji Hashimoto, e.g. hep-th/0401043, hep-th/0312260 concerning deformations of boundary states in order to describe open string background field

Regarding the first paper, why does it not say anything about the relevant cohomology?

You mean the question whether the boundary state is in the BRST cohomology?

This has been studied for the abelian case in JHEP 04 (2000) 023 and for the general case in that hep-th/0312260, which, as you noticed, is not by Hashimoto but by Maeda,Nakatsu&Oonishi (sorry for confusing that). Of course the result is that the Wilson line over the closed string with which one acts on the boundary state of the bare brane has to come from a gauge field which satisfies the YM equations of motion.

But it is interesting to note that the above two papers have a completely complementary way of arriving at that conclusion:

In Hashimoto’s paper the boudnary state is not regularized. This makes it trivial to check its BRST-closedness, which is guaranteed. But since no regulation has been used this state is not well defined in general. Here the background equations of motion are precisely the condition that the state is well-defined, even without regularization.

In MN&O’s paper on the other hand the Wilson line is regulated. This makes it’s well-definedness as a state trivial, by construction. But not BRST-closedness has to be checked by hand, because in general it is broken by the regularization. Here the background YM equations of motion imply that even the regulated boundary state is still BRST closed.

from where does the associativity of the star product in OSFT originate?

I am not sure what this question has to do with the boudnary state issues that I mentioned, but it nevertheless has an easy answer:

The associativity is of course first of all demanded, since one wants to have an analogy between the star product and the ordinary wedge product of matrix-valued differential forms. But given the usual concerete definition of the star product associativity is easy to check:

(1)$A\star B$

is what you obtain after gluing the right half of the string (state) $A$ to the left half of $B$. Then

(2)$\left(A\star B\right)\star C$

is what you get by additioanlly gluing the right half of $B$ with the left half of $C$. Clearly this is the same as

(3)$A\star \left(B\star C\right)$

Is that what you were asking for?

Since this works so well for gluons and tachyons, the natural question is:

Is there anything known about boundary state deformations which describe for instance massice background fields of RR fields, for instance?

I don’t know, but perhaps if you can make your question more specific we might be able to think of something.

The motivation of this question is the following: As discussed above, it has been shown that turning on a gauge field on a brane corresponds to multiplying the respective boundary state with the unitary operator given by the (super) Wilson line of that gauge field along the closed string at the brane.

Now, this ‘deformation’ of the open string theory is precisely the one known from the closed string, where turning on a gauge field A (resulting in $B↦B+\mathrm{dA}$) is a pure gauge transformation and can be associated with a unitary transformation of the super-Virasoro generators. The only difference is that in the open string case the transformation is ‘cut in half’ (see here), which is the reson why it gives a genuinely new background, not just a gauge shift for the open string.

As discussed in hep-th/0401175 (for a summary see section 3.5 of my notes) this unitary $B↦B+\mathrm{dA}$ deformation of the closed string is just a special case of a general deformation of the background of the closed string. But it is not so easy to identify the deformations describing backgrounds that are not massless NS(-NS). That’s why I was asking if maybe this problem has equivalently be studied in terms of boundary state deformations.

For instance: What is the boundary state describing a macroscopic tachyonic excitation?

Posted by: Urs Schreiber on June 16, 2004 12:19 PM | Permalink | Reply to this

Post a New Comment

format: plain


text:
Abstract

Nozzle facilities, which can generate high Mach number flows, are the core portions of the supersonic wind tunnel. Different from traditional fixed nozzles, a flexible nozzle can deform to designed contours and supply steady core flows in several Mach numbers. Due to the high-quality demands from the thermo-aerodynamic testing, the deformation of the flexible nozzle plate should be carefully designed. This problem is usually converted into the large deformation problem of a cantilever with movable hinge boundary conditions. In this paper, a generalized variational method is established to analyze the deformation behavior of the flexible nozzle. By introducing axial deformation constraint and Lagrange multiplier, an analytical model is derived to predict the deformed morphology of the flexible plate. Finite element analyses (FEA) of a single-jack flexible nozzle model is performed to examine the predicted deformations and reaction forces. Furthermore, the large deformation experiments of an elastic cantilever with a movable hinge connection are carried out to simulate the scenarios in supersonic flexible nozzle facility. Both the FEA and experimental results show high accuracy of current theoretical model in deformation predictions. This method can also serve as a general approach in the design of flexible mechanisms with movable boundaries.

1 Introduction

Wind tunnels [1] are the facilities to supply aeronautical environments for thermo-aerodynamic testing. As a kernel portion in wind tunnels, the nozzle component [2] transforms the flow from reservoir condition to testing condition with a designed Mach number. In traditional fixed nozzle facilities, the contours of the nozzle plates are fixed, which means it can only generate a stream with a fixed Mach number. Therefore, the nozzles should be changed frequently according to the demands of various Mach number flows from aircraft testing. The fixed-flexible and full flexible nozzles [3,4] are developed to break these limits, where the flow Mach number is adjustable in one single nozzle. There are two common methods to achieve this purpose, i.e., flexible-wall [5,6] and asymmetric-sliding block nozzle [7,8] designs. The former adjusts the flow Mach numbers via the opposite nozzle plates symmetrically deforming to the designed profile curve and the latter only manipulates the plate at one side. The simplest flexible nozzle is the single-jack nozzle [4,6,9] as shown in Fig. 1(a). By geometry symmetry, this system can be simplified as an elastic cantilever hinged to a retractable rigid rod (jack), whose opposite end is hinged to a fixed point. During operations, the elongation or shortening of the jack leads to plate deformation which leads to jack rotation synchronously. At the design stage, it is important to establish the relationship between the rod elongation and the deformed plate contour. This mechanical model can be treated as a large deformation beam with a movable hinge boundary.

Fig. 1
Fig. 1
Close modal

At present, the common approach in designing flexible nozzle is finite element analyses (FEA) [1012], which cost a large amount of time for geometry and motion modeling. In the aspect of theories, most of the existing models are derived from empirical equations and engineering simplifications, which are not suitable for the scenarios of large deformation [5,13,14]. For large deformation beam problems, the optional approaches include small parameters perturbation method [15,16], elliptic integral method [1719], and shooting method [2023]. Actually, all these methods involve solving the deformation process during jack motions. In this paper, a generalized variational principle is established, which can calculate the beam deflection and boundary variation without complex iterations. In Sec. 2 the basic equations and their extensions for cantilever beam hinged to skew rod are derived. Then, the theoretical model is verified by FEA and experiments in Secs. 3 and 4, respectively. Finally, the work is concluded in the last section.

2 Analytical Model

For geometry symmetry, the single-jack flexible nozzle facility is simplified as an elastic cantilever hinged to a rigid jack at the right end as shown in Figs. 1(b)1(d). The jack mechanism contains translational and rotational degrees-of-freedom. In the simplest case, the jack is originally perpendicular to the beam axis in the Cartesian coordinate (x, y), as shown in Fig. 1(b). Here, we introduce an assumption that there is no axial deformation for the beam structure, which means its length remains unchanged during deformation. Hence, we can establish the potential energy functional with length restriction as
$Π=∫0x¯12EI[w″[1+(w′)2]32]2dx+λ(∫0x¯1+(w′)2dx−l)$
(1)
where the first term is the bending deformation energy with considerations of large deformation, and the second one represents the constraint condition for beam inextensibility. Since the hinge point is treated as a displacement boundary, according to the variational principle, the work done by the jack mechanism should not be included in the energy functional, which is also the key point in the current method. EI represents the bending stiffness, w denotes the deflection function of the beam, $x¯$ is the horizontal coordinate of the hinge point after deformation, the superscript “ʹ” is a derivative symbol standing for “d/dx”, and λ denotes the Lagrange multiplier. Based on the principle of minimization potential energy, the first-order variation of Π equals to zero, which leads to
$3w′(w″)2[1+(w′)2]4−w″′[1+(w′)2]3+λEIw′1+(w′)2=0$
(2)
$∫0x¯1+(w′)2dx−l=0$
(3)
From Eq. (2), the third-order derivative of w with respect to x can be expressed by its first and second orders as
$w′′′=3w′(w″)21+(w′)2+λEIw′[1+(w′)2]52$
(4)
Apparently, the fourth and higher-order derivatives of w can also be expressed in the similar way. Hereby, the general solution to Eq. (4) can be expressed in the form of the Taylor series as
$w=w(x0)+w′(x0)(x−x0)+w″(x0)2!(x−x0)2+w″′(x0)3!(x−x0)3+w(4)(x0)4!(x−x0)4+o(x5)$
(5)
By setting the expansion point x0 at the original point, the fourth-order derivative of w can be obtained by Eq. (4),
$w(4)=3(w″)3[1+5(w′)2][1+(w′)2]2+λw″[1+(w′)2]12EI(12(w′)4+13(w′)2+1)$
(6)
Considering the fixed boundary condition at the left end, the deflection function can be written as
$w=w″(0)2x2+124[3w″(0)3+λEIw″(0)]x4$
(7)
Set the coordinate of the right end as ($x¯$, $y¯$) after deformation, and we have
$w(x¯)=y¯$
(8)
The hinged boundary condition at the right end leads to
$w″(x¯)=0$
(9)
Since the beam is hinged to the rigid jack, the coordinates of its right end can also be expressed in the following terms
${x¯=l−(S0+ΔS)sinθy¯=(S0+ΔS)cosθ−S0$
(10)
where S0 and ΔS denote the original length and elongation of the jack, respectively and θ is the rotation angle of the jack. Simultaneous Eqs. (3) and (8)(10), the displacement function, the coordinates of the right endpoint, and the rotation angle of the jack are obtained.
For the case of non-vertical jack, only Eq. (10) needs to be rewritten. For the model in Figs. 1(c) and 1(d), Eq. (10) changes to
${x¯=l−[(S0+ΔS)sin(θ0+θ)−S0sinθ0]y¯=(S0+ΔS)cos(θ0+θ)−S0cosθ0$
(11)
where θ0 is the initial inclination angle of the jack.

3 Finite Element Analyses

3.1 The Cantilever Model With One Perpendicular Jack.

In this section, the commercial software abaqus is employed to simulate the models in Fig. 1. A four-node shell element with reduced integration is used to model the cantilever beam. The translator, a type of connector element in abaqus, is selected to model the jack, which can realize the translation and rotation movements of the rigid body.

The length ratio of the jack to beam is defined as
$α=S0l$
(12)
The applied loading on the jack is defined as the length variation versus its original value
$εapp=ΔSS0×100%$
(13)
Introducing an angle variable φ to denote the rotation angle at the right end of the beam
$φ=arctan(w′|x=x¯)$
(14)
The moment at the left end of the beam can be expressed as
$M0=−EIw″[1+(w′)2]32|x=0$
(15)
According to the equilibrium equation, M0 can also be expressed as
$M0=−P[x¯cos(θ0+θ)+y¯sin(θ0+θ)]$
(16)
The expression of reaction force P from jack can be given by Eqs. (15) and (16) as
$P=EIw″[1+(w′)2]32|x=0⋅1[x¯cos(θ0+θ)+y¯sin(θ0+θ)]$
(17)

The mechanical and geometric parameters used in the finite element model are listed in Table 1. E and ν represent Young’s modulus and Poisson's ratio, respectively. Two representative geometry configurations (α = 0.5 and 0.75) and three tensile loads (ɛapp = 30%, 40%, and 50%) are modeled, respectively. All the comparisons of deformations predicted by analytical analyses (ANA) and FEA are demonstrated in Figs. 2(a) and 2(b). Similarly, the deformations corresponding to compressive loads of −30%, −40%, and −50% are shown in Figs. 2(c) and 2(d). The results demonstrate that the current analytical model can predict the deformed configurations with high accuracy. And it has the ability to describe the boundary movements as well as the jack rotation at once.

Fig. 2
Fig. 2
Close modal
Table 1

Mechanical and geometric parameters of one-jack cantilever

E/GPaυl/mmb/mmh/mm
2.50.3150202
E/GPaυl/mmb/mmh/mm
2.50.3150202

In addition to the overall deformation, several important parameters are also compared in the case of α = 0.75. Figures 3(a)3(c) show the reaction force (P), the rotation angle (θ) of the jack, and the tangent angle (φ) at the right end of the beam in several tensile loads. It shows that the forces predicted by the analytical model are smaller than the results from FEA. Generally, the stiffness matrix derived in FEA is larger than that from elastic theory, which leads to a larger reaction force. Similarly, for compressive loads, the corresponding P, θ, and φ are also compared in Figs. 3(d)3(f), receptivity. The minus sign before ɛapp denotes the shortening (compressive) of the jack. It is apparently that the rotation angle of the jack is nonlinear with the applied loading, and its change rate accelerates with the growth of jack elongation (or shortening). But in contrast, the tangent angle at the right end of the beam varies linearly with the jack elongation (or shortening).

Fig. 3
Fig. 3
Close modal

3.2 The Model With One Slant Jack.

The cantilever hinged to a slant jack is also constructed by FEA to further verify the analytical model. The initial inclination angle θ0 and the length ratio α are set as 30 deg and 0.75, respectively. Other parameters remain the same as those in Sec. 3.1. The beam deflections with applied loads of 30%, 40%, and 50% are depicted in Fig. 4(a), respectively. The straight lines with triangle and circle signs represent the results from ANA and FEA, respectively. The results from FEA and theoretical models show good agreements with each other. With the increasing of applied loads, the force predicted by the analytic model and further deviates from that of FEA, as shown in Fig. 4(b). Different from the situation in perpendicular jack, the accumulation errors in rotation angle (Fig. 4(c)) dominate the calculation errors in reaction force, which beat the influence of stiffness matrix deviation. Due to the initial slope of the jack, it first rotates clockwise and then changed to counterclockwise with the increasing of applied load as shown in Fig. 4(c). Similarly, the tangent angle of the beam end increases approximately linear with the applied load as shown in Fig. 4(d). A similar case with the initial inclination angle of −30 deg is also modeled. The predictions of beam deflection, jack reaction force, jack rotation angle, and the tangent angle of the beam end are compared with those from ANA in Fig 5, respectively. It is notable that, the jack rotation angle is approximately linear with the increasing of the applied load, which is different from the previous two models. The selection of the jack initial angle not only affects the rotation direction, but also dominates the change rate of the angle-load curve.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal

4 Experiments

Several simple experiments are performed on a biaxial tensile test platform to further verify the analytical model as shown in Fig. 6(a). The beam sample was manufactured with thermoplastic polylactic acid by three-dimensional (3D) printing technology, whose length (l), width (b), and thickness (h) are 150 mm, 20 mm, and 1 mm respectively. An extra length of 50 mm was spared for clamped portion, and it was clamped by two permanent magnets at one end. A rigid rod was assembled with one of its ends hinged to the beam and the other one going through a shaft sleeve, which can rotate freely at a fixed point. By pushing or pulling the rod, it will rotate with the deformed beam, which can stimulate the mechanism of the single-jack nozzle. All experiment results were photographed by a digital camera.

Fig. 6
Fig. 6
Close modal

The deflections of the deformed beam were obtained by the tracing point method. Figures 6(b)6(e) show the deflection results from the analytical model and experiments in circle symbols and solid lines, respectively. Figures 6(b) and 6(c) demonstrate the case of θ0 set as 0 and a displacement load of 50% and −50%, respectively. The cases with initial inclination angles of 30 deg and −30 deg are shown in Figs. 6(d) and 6(e), respectively. After removal of the loads, the beam recovered to its initial state, which indicated the beam in elastic range during experiments.

As seen from the above results, the theoretical and experimental results are basically identical, which demonstrates that the proposed method has a high accuracy to predict the deformation of the single-jack cantilever beam structure.

5 Conclusions

In the current work, we proposed a generalized vibrational method to solve the deformation problem of the single-jack flexible nozzle structure, which can be extended to deal with the large deformation beam problem with a movable hinge boundary. The FEA and experimental validations reveal the high accuracy and feasibility of the current method. It also provides a basis for solving the problem of multi-jack flexible nozzle structure and can serve as guidelines for the design of wind tunnels.

Acknowledgment

This research was funded by the National Natural Science Foundation of China (NSFC) (Grant No. 12072150). This work is also supported by the Joint Fund of Advanced Aerospace Manufacturing Technology Research (Grant No. U1937601), the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics, Grant No. MCMS-I-0221Y01), and the National Natural Science Foundation of China for Creative Research Groups (Grant No. 51921003).

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Nomenclature

• l =

length of the beam

•
• w =

deflection function

•
• P =

reaction force

•
• M0 =

bending moment at the left end of the beam

•
• S0 =

initial length of the jack

•
• EI =

bending stiffness

•
• α =

the length ratio of jack to beam

•
• ΔS =

variation length of the jack

•
• ɛapp =

•
• θ =

rotation angle of the jack

•
• θ0 =

initial inclination angle of the jack

•
• λ =

Lagrange multiplier

•
• φ =

tangent angle at the right end of the beam

References

1.
Atieh
,
A.
,
Al Shariff
,
S.
, and
Ahmed
,
N.
,
2016
, “
Novel Wind Tunnel
,”
Sustainable Cities Soc.
,
25
, pp.
102
107
.
2.
Murugappan
,
S.
,
Gutmark
,
E. J.
,
Lakhamraju
,
R. R.
, and
Khosla
,
S.
,
2008
, “
Flow-Structure Interaction Effects on a Jet Emanating From a Flexible Nozzle
,”
Phys. Fluids
,
20
(
11
), p.
117105
.
3.
Erdmann
,
S. F.
,
1971
, “
A New Economic Flexible Nozzle for Supersonic Wind Tunnels
,”
J. Aircr.
,
8
(
1
), pp.
58
60
.
4.
Rom
,
J.
, and
Etsion
,
I.
,
1972
, “
Improved Flexible Supersonic Wind-Tunnel Nozzle Operated by a Single Jack
,”
AIAA J.
,
10
(
12
), pp.
1697
1699
.
5.
Rosen
,
J.
,
1955
, “
The Design and Calibration of a Variable Mach Number Nozzle
,”
Int. J. Aeronaut. Space Sci.
,
22
(
7
), pp.
484
490
.
6.
Lv
,
Z.
,
Xu
,
J.
,
Wu
,
F.
,
Chen
,
P.
, and
Wang
,
J.
,
2018
, “
Design of a Variable Mach Number Wind Tunnel Nozzle Operated by a Single Jack
,”
Aerosp. Sci. Technol.
,
77
, pp.
299
305
.
7.
Winarto
,
H.
, and
Stalker
,
R. J.
,
1984
, “
Design Parameters and Performance of Two-Dimensional, Asymmetric, ‘Sliding Block’, Variable Mach Number, Supersonic Nozzles
,”
Aeronaut. J.
,
88
(
876
), pp.
270
280
.
8.
Liepman
,
H. P.
,
1955
, “
An Analytic Method for the Design of Two-Dimensional Asymmetric Nozzles
,”
Int. J. Aeronaut. Space Sci.
,
22
(
10
), pp.
701
709
.
9.
Chen
,
P.
,
Wu
,
F.
,
Xu
,
J.
,
Feng
,
X.
, and
Yang
,
Q.
,
2016
, “
Design and Implementation of Rigid-Flexible Coupling for a Half-Flexible Single Jack Nozzle
,”
Chin. J. Aeronaut.
,
29
(
6
), pp.
1477
1483
.
10.
Sun
,
S.
,
Zhang
,
H.
,
Cheng
,
K.
, and
Wu
,
Y.
,
2007
, “
The Full Flowpath Analysis of a Hypersonic Vehicle
,”
Chin. J. Aeronaut.
,
20
(
5
), pp.
385
393
.
11.
Yu
,
C.
,
Chen
,
Z.
, and
Nie
,
X.
,
2012
, “
Multi-Jack Single-Drive Semi-Flexible Nozzle Mechanism Design and Simulation
,”
2nd International Conference on Frontiers of Manufacturing Science and Measuring Technology (ICFMM 2012)
,
Xi'an, China
,
June 12–13
, Trans Tech Publications Ltd., Vol. 503–504, pp.
892
895
.
12.
Jiao
,
X.
,
Chang
,
J.
,
Wang
,
Z.
, and
Yu
,
D.
,
2017
, “
Numerical Study on Hypersonic Nozzle-Inlet Starting Characteristics in a Shock Tunnel
,”
Acta Astronaut.
,
130
, pp.
167
179
.
13.
Guo
,
S. G.
,
Wang
,
Z. G.
, and
Zhao
,
Y. X.
,
2015
, “
Design of a Continuously Variable Mach-Number Nozzle
,”
J. Cent. South Univ.
,
22
(
2
), pp.
522
528
.
14.
Yang
,
Y.
,
Wen
,
C.
,
Wang
,
S. L.
,
Feng
,
Y. Q.
, and
Witt
,
P.
,
2014
, “
The Swirling Flow Structure in Supersonic Separators for Natural Gas Dehydration
,”
,
4
(
95
), pp.
52967
52972
.
15.
Su
,
Y.
,
Wu
,
J.
,
Fan
,
Z.
,
Hwang
,
K.-C.
,
Song
,
J.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2012
, “
Postbuckling Analysis and Its Application to Stretchable Electronics
,”
J. Mech. Phys. Solids
,
60
(
3
), pp.
487
508
.
16.
Chien
,
W. Z.
,
2002
, “
Second Order Approximation Solution of Nonlinear Large Deflection Problems of Yongjiang Railway Bridge in Ningbo
,”
Appl. Math. Mech. (Engl. Ed.)
,
23
(
5
), pp.
493
506
.
17.
Shoup
,
T. E.
, and
Mclarnan
,
C. W.
,
1971
, “
On the Use of the Undulating Elastica for the Analysis of Flexible Link Mechanisms
,”
J. Eng. Ind.
,
93
(
1
), pp.
263
267
.
18.
Liu
,
H. G.
,
Bian
,
K.
, and
Xiong
,
K.
,
2019
, “
Large Nonlinear Deflection Behavior of IPMC Actuators Analyzed With an Electromechanical Model
,”
Acta Mech. Sin.
,
35
(
5
), pp.
992
1000
.
19.
Xu
,
K.
,
Liu
,
H.
, and
Xiao
,
J.
,
2021
, “
Static Deflection Modeling of Combined Flexible Beams Using Elliptic Integral Solution
,”
Int. J. Non-Linear Mech.
,
129
, p.
103637
.
20.
Li
,
Z.
,
Yu
,
C.
,
Qi
,
L.
,
Xing
,
S.
,
Shi
,
Y.
, and
Gao
,
C.
,
2022
, “
Mechanical Behaviors of the Origami-Inspired Horseshoe-Shaped Solar Arrays
,”
Micromachines
,
13
(
5
), p.
732
.
21.
Wang
,
C. M.
, and
Kitipornchai
,
S.
,
1992
, “
Shooting Optimization Technique for Large Deflection Analysis of Structural Members
,”
Eng. Struct.
,
14
(
4
), pp.
231
240
.
22.
Nallathambi
,
A. K.
,
Rao
,
C. L.
, and
Srinivasan
,
S. M.
,
2010
, “
Large Deflection of Constant Curvature Cantilever Beam Under Follower Load
,”
Int. J. Mech. Sci.
,
52
(
3
), pp.
440
445
.
23.
Abdalla
,
H. M. A.
, and
Casagrande
,
D.
,
2020
, “
On the Longest Reach Problem in Large Deflection Elastic Rods
,”
Int. J. Non-Linear Mech.
,
119
(
1
), p.
103310
.


text:
# How do you find slope and intercepts to graph x = 5?

If $x = 5$ this means that the value of $x$ never changes. This means that the line is vertical so has no slope. It will intercept the $x$ axis at $\left(5 , 0\right)$ since at the $x$ axis $y = 0$ and x always equals 5.
There will be no $y$ intercept since all coordinates on the $y$ axis have an $x$ value of 0 and on this line $x$ is never equal to 0.


text:
OpenKattis
Reykjavik University

# Animal Classification

Alice and Bob are biologists. They like to group animals using classification trees. A classification tree is a binary tree where the leaves are labeled by the animals.

For example, Figure 1 below shows the classification of $5$ animals (labeled as $1, 2, 3, 4, 5$) by Alice and Bob. Alice’s classification has nine groups: $\{ 1\}$, $\{ 2\}$, $\{ 3\}$, $\{ 4\}$, $\{ 5\}$, $\{ 2, 5\}$, $\{ 1, 2, 5\}$, $\{ 1, 2, 3, 5\}$ and $\{ 1, 2, 3, 4, 5\}$. Bob’s classification also has nine groups: $\{ 1\}$, $\{ 2\}$, $\{ 3\}$, $\{ 4\}$, $\{ 5\}$, $\{ 1, 5\}$, $\{ 2, 3\}$, $\{ 1, 2, 3, 5\}$ and $\{ 1, 2, 3, 4, 5\}$.

As you can observe, Alice and Bob classify the animals differently. Carol is interested to know the number of common groups between the classifications of Alice and Bob. For the above example, there are $7$ common groups. They are: $\{ 1\}$, $\{ 2\}$, $\{ 3\}$, $\{ 4\}$, $\{ 5\}$, $\{ 1, 2, 3, 5\}$ and $\{ 1, 2, 3, 4, 5\}$.

Carol usually counts the number of common goups by hand. On one day, an outer space alien came to our earth. He asked Alice and Bob to classify aliens in outer space. The number $N$ of aliens can be as big as $100\, 000$. It is impossible to count the number of common goups by hand now. Can you make a program to help Carol?

## Input

The input consists of three lines:

• one line with one integer $N$ ($1 \leq N \leq 100\, 000$)

• the classification tree of Alice

• the classification tree of Bob

Note that both classification trees are leaf-labeled by $1$ to $N$. The tree is represented by the parathesis encoding. (Note that the encoding only have numbers and $3$ symbols, that is, ‘(’, ‘)’, ‘,’.) For the example in Figure 1, see Sample Input 1.

## Output

Output one line with a single integer that represents the number of common groups between Alice and Bob’s classification trees.

Sample Input 1 Sample Output 1
5
((3,(1,(5,2))),4)
(((5,1),(2,3)),4)

7

Sample Input 2 Sample Output 2
10
(1,(2,(3,(4,(5,(6,(7,(8,(9,10)))))))))
(((((((((1,2),3),4),5),6),7),8),9),10)

11

Sample Input 3 Sample Output 3
10
(10,(9,(8,(7,(6,(5,(4,(3,(2,1)))))))))
(((((((((1,2),3),4),5),6),7),8),9),10)

19


text:
Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Giant thermal expansion of a two-dimensional supramolecular network triggered by alkyl chain motion

Abstract

Thermal expansion, the response in shape, area or volume of a solid with heat, is usually large in molecular materials compared to their inorganic counterparts. Resulting from the intrinsic molecule flexibility, conformational changes or variable intermolecular interactions, the exact interplay between these mechanisms is however poorly understood down to the molecular level. Here, we investigate the structural variations of a two-dimensional supramolecular network on Au(111) consisting of shape persistent polyphenylene molecules equipped with peripheral dodecyl chains. By comparing high-resolution scanning probe microscopy and molecular dynamics simulations obtained at 5 and 300 K, we determine the thermal expansion coefficient of the assembly of 980 ± 110 × 10−6 K−1, twice larger than other molecular systems hitherto reported in the literature, and two orders of magnitude larger than conventional materials. This giant positive expansion originates from the increased mobility of the dodecyl chains with temperature that determine the intermolecular interactions and the network spacing.

Introduction

The expansion of a material upon heating, known as thermal expansion is a fundamental property of solid systems that governs many of their mechanical applications1,2,3. Typically, thermal expansion coefficients ($$\alpha =\frac{\Delta l}{{l}_{0}\Delta T}$$ with Δl is the length variation, l0 the initial length and ΔT the change of temperature) of inorganic materials are in the range of 0–20 × 10−6 K−1, whereas molecular counterparts vary from 50–60 × 10−6 K−1 (poly(styrene))2 up to  ~420 × 10−6 K−1 for porous coordination polymers1. Compared to inorganic materials, molecular architectures enable larger positive (or even negative4,5) expansions due to various mechanisms involving the intrinsic molecules flexibility, their propensity to conformational changes or the weak intermolecular interactions 2. To date, however, understanding of the underlying mechanisms governing such large expansion coefficients at the molecular scale remains elusive.

Supramolecular chemistry has opened these last years new avenues in controlling spontaneous bottom-up assemblies of molecules both in solution 6,7, in the solid state or at the solid interfaces8,9,10,11. Resulting from weak non-covalent intermolecular interactions, supramolecular assemblies enable the growth of well-extended and highly-ordered structures with tunable chemical, optochemical or mechanical properties9,10,11,12,13. In aqueous environments, extended polycyclic hydrocarbon molecule (PAH) assemblies have been also reported through the interdigitation of long alkyl chains14,15,16,17,18. While these organic architectures have already shown potential applications in photovoltaics and nanoelectronics19, the weak intermolecular interactions governing these assemblies and the intrinsic flexibility of the precursors also offer new opportunities for the design of responsive materials to external stimuli6,20. For instance, the thermal response of 2D supramolecular assemblies has never been investigated down to the molecular scale.

To address supramolecular systems at the nanoscale, atomic force microscopy (AFM) and scanning tunneling microscopy (STM) are suitable techniques since they allow the highest spatial resolutions of single molecules and their assemblies at surfaces21,22,23,24,25. The growth of two-dimensional supramolecular networks is nowadays successfully achieved at surfaces by the thermal evaporation of precursors in ultra high vacuum (UHV)8,9,10,23,25,26. This approach, however, limits the size of the precursors and prevents the use of fragile moieties such as long alkyl chains that might degrade during evaporation. In liquid, such size or composition limitations are not present11,13,14,15,16,17,18. In UHV conditions, the electrospray deposition (ESD) technique allows the introduction of large molecules. Therefore, it is possible to study in UHV conditions the adsorption properties of molecules and their assemblies that mimic established assemblies at the liquid–solid interface24,27,28,29,30,31,32,33. In this context, temperature-dependent STM/AFM investigations of supramolecular systems might enable the analysis of their thermoresponse down to the atomic scale.

Here, we investigate the thermal response of a 2D supramolecular assembly composed of shape persistent spoked wheel molecules equipped with peripheral dodecyl chains using non-contact AFM (nc-AFM), STM and molecular dynamic simulations (MD) at different temperatures. By comparing atomic-scale images of the assembly at 5 and 300 K we determined a giant positive thermal expansion of about 980 ± 110 × 10−6 K−1, which is supported by microsecond long temperature-dependent MD simulations. The alkyl chains attached to the molecular cores are found to undergo large thermal fluctuations (compared to the core) as a result of entropic effects and large anharmonic vibrations. This results in temperature-dependent intermolecular interactions that promote the giant expansion of the highly-ordered supramolecular network with temperature.

Results

Single spoked wheel molecules on Au(111)

The polyphenylene spoked wheel molecule (SW) 17 (Fig. 1a) is a purely sp2-hybridized molecule with a six-fold symmetry, composed of 37 benzene rings forming spokes (orange) and sides (green) of the molecular skeleton. Each spoke axis is functionalized with a peripheral dodecyl side chain.

To safely deposit the SW precursors onto Au(111), we used an electrospray deposition (ESD) device attached to the UHV setup24,27,28,29,30,31,32,33. The spray-deposition was always performed on the Au(111) substrates kept at room temperature and followed by an annealing step at 450 K. As shown in large-scale STM measurements (Supplementary Fig. 1), single SW molecules are observed intact on the surface indicating the low diffusion of the molecule on the gold surface and the absence of any degradation. Figure 1b shows a close-up STM image acquired at 5 K with a CO-terminated tip of such an isolated SW molecule. While both the core and the dodecyl chains are clearly visible, the arms (white arrow) appear fuzzier than the core due to their relative mobility under the influence of the scanning tip. Within the core, the spokes and sides regions are observed with slight differences in contrast suggesting a corrugation of the core. Contrarily to the chemical structure representation shown in Fig. 1a, the SW core adopts a three-fold symmetry.

The corresponding constant-height AFM image with a CO-terminated tip (Fig. 1c) further confirms this particular core symmetry. While each spoke parts of the core are resolved, only half of the side parts are visible in the image. The peripheral dodecyl chains are absent in the AFM image (but visible by STM). To rationalize this, we performed a 10 ns long molecular dynamic (MD) simulation of the SW molecule adsorption at 300 K (Supplementary Notes 1 and 2). A large corrugation of the final conformation is observed with the dodecyl chains stacked on the Au(111) compared to the protruding core. Experimentally, these local height variations prevent constant-height AFM imaging, more sensitive to height variations compared to STM (probing a convolution of local density of states and topography), resolving the entire molecule structure.

Thermal response of the spoked wheel assembly

Upon spray-deposition of the spoked-wheel molecules in the sub-monolayer range, we observed different aggregates on the surface varying from single molecules up to extended assemblies (see Supplementary Figs. 1b and 2). To investigate the effect of temperature on the SW assembly, extended close-packed networks (at least 200 nm in width and length) chosen to be of similar size to avoid influences of the domain size on the lattice parameter were investigated. Figure 2a, b show STM and AFM images of the networks acquired at 5 K and 300 K, respectively. Within the assembly, individual SW cores appear as bright protrusions at both temperatures forming a long-range hexagonal network.

A different molecule spacing of the networks is measured in Fig. 2a, b as shown by the line profiles (Fig. 2c) taken along the symmetry axis and the Fast Fourier Transform of the images (see inset Fig. 2a, b). We measured lattice parameters of 3.5 ± 0.1 nm of the assembly at 5 K increasing to 4.5 ± 0.2 nm at 300 K. This corresponds to lattice shrinking of  ~22% upon cooling. Note also that the preservation of the hexagonal nature of the assembly between these temperatures indicates the absence of any structural phase transitions. In conclusion, a thermal expansion coefficient of the SW on gold of α = 980 ± 110 × 10−6 K−1 is determined, which is two orders of magnitude larger than the expansion of the Au(111) of about α = 14 × 10−6 K−1.

Temperature-dependent core and chain mobility

Thermal evolution and mobility of a single SW molecule was numerically investigated using μs long all atom MD simulation on an Au(111) surface for different temperatures. Figure 3 provides a comparison of the mobility of the core and the chains of the SW molecule (see Supplementary Note 3). Snapshots of the final configuration are shown in Fig. 3a for each simulated temperature. Although displacements of the molecule core are marginal as a function of the temperature, the dodecyl chain fluctuations increase upon heating up the system (Supplementary Note 3 and Supplementary Movies 35). This is visualized in the snapshots (Fig. 3a) by the more straight positioning of the chains at 5 K in comparison to the increasingly tilted chains for 300 K, 420 K and finally 450 K.

To better quantify the fluctuations of the core and the chains, we compare in Fig. 3 the standard deviation of the mean square displacements (MSD) of the centre of mass of the molecule core $${\delta }_{{\rm{core}}}$$ and of the last atom of a dodecyl chain δchain at 5 K, 300 K, 420 K and 450 K, respectively. A larger δ corresponds to a larger mobility at a given temperature (see details in Supplementary Note 3). The core mobility is almost negligible at all considered temperatures (Fig. 3b). In contrast, the chain exhibits a much larger lateral displacement with increasing temperature. These chain fluctuations are characterized by rotations about the connection point to the core as well as in-plane bending of the chain (Supplementary Note 3 and Supplementary Movies 35). During simulations, the chains are continuously adsorbed on gold which is consistent with the experimental observations. The higher chain mobility compared to the core is also consistent with the experimental observation of accidental chain manipulations (Fig. 1b).

Influence of interdigitated dodecyl chains on thermoresponse

The thermal response of SW assemblies upon cooling (Fig. 4a) is numerically obtained by comparing MD simulations of a SW trimer at different temperatures. In the interest of the computational feasibility, given the sheer size and required simulation time of large assemblies, we considered a trimer whose core-core distance matched an equivalent assembly observed at 5 K (see Supplementary Fig. 4a). Further details on the simulation protocol are provided in Supplementary Note 1. Interestingly, small molecular assemblies displayed a core-core distance slightly bigger than larger assemblies, an effect possibly resulting from domain size. The simulations of this assembly (Fig. 4a, b, Supplementary Movies 12) confirm the significant variation of intermolecular spacing with temperature, namely from 3.9 nm at 5 K up to 5.1 nm at 450 K. The associated thermal expansion coefficient is thus 600 ± 120 × 10−6 K−1 in agreement with the experimental value.

Additionally, MD simulations show that the intermolecular spacing varies with the preservation of the hexagonal supramolecular structure, thereby excluding phase transition as a possible explanation of the thermal expansion. At 450 K, the chain mobility, initially constrained by the packing, increases as the network expands. This is shown by δchains as a function of time (Fig. 4c) that approaches an equilibrium value below the single molecule value on Au(111) at 450 K ($${\delta }_{{\rm{chains}}}^{1{\rm{SPW}}}=$$ 3.34 Å, see Supplementary Fig. 4) due to reduction by the intermolecular interaction in the assembly. While, at the single-molecule level (Fig. 3), the diffusion of the molecular core was found negligible for the considered temperatures, it appears relatively large within the assembly (Fig. 4a–c). The increased core mobility in the assembly is a consequence of the increased chain mobility at higher temperature. The mobility is enabled by the adjacent molecules forcing the network to expand.

Additionally, the expansion occurs in distinguishable events as marked by dotted lines in Fig. 4c, d. These events are separated by long periods where the core to core distance remains stable. A lively picture of the expansion mechanism is provided in the two insets in Fig. 4c. Prior to the expansion event 2 at t = 275 ns, one dodecyl arm (green) is close to the neighbouring molecule (pink), that restricts its lateral fluctuation but also increases the intermolecular interactions. After the expansion event (>300 ns), alkyl chains possess enough space to adopt a more extended configuration. The relationship between chain mobility and lattice expansion is particularly visible between 400 and 700 ns in Fig. 4c, d, where the chain mobility gradually decreases prior to the third and last expansion event.

Figure 4d shows the total energy of the SW trimer as a function of time during the MD simulation at 450 K. The network expansion leads to a decrease of the internal energy of the system which is accompanied by the decrease of the number of interdigitated chains. While at low temperature, this interdigitation governs the molecular assembly to a close-packed structure and increases the internal energy, the entropic contribution to the free energy at higher temperatures, manifested by the fluctuations of the dodecyl chains, compensates the loss of internal energy of the system thus favouring the lattice expansion.

The high lattice expansion may be understood as a conjunction of three properties of the dodecyl chains: 1- a weak interaction with the surface (facilitating a rapid increase of their oscillations with temperature), 2- the soft in-plane bending stiffness of the chains (favouring this oscillation mode over other rigid vibrations) and 3- the two-dimensional constraint of the chain mobility imposed by the surface and excluding chain fluctuations out of the surface plane.

Although the finite size of our simulations might affect the chain fluctuations and the intermolecular interaction, our simulations clearly support an abnormal thermal expansion coefficient for alkyl decorated molecules. Interestingly, small molecular assemblies obtained experimentally show a larger core-core distance (see Supplementary Fig. 4a). This could possibly explain the smaller thermal expansion coefficient obtained in our simulations. At last, the large thermal expansion coefficient obtained in the experiments seems to support the idea that although in smaller assemblies the cores possess an initial higher mobility, in the larger assemblies the long experimental time scale allows enough time for the chain fluctuations equilibrate giving rise to a giant thermal expansion the whole molecular network (an effect that is not disturbed even at step edges of the Au(111) surface).

Generalization of interdigitated 2D supramolecular networks to large thermal expansion

Besides unravelling the pivotal role of the alkyl chains into the giant thermal expansion of the SW assembly, we now discuss the generalization of this mechanism to interdigitated molecular systems at surfaces. Table 1 compares the lattice constants a measured at 5 and 300 K and the extracted thermal expansion α of the SW assemblies with hexabenzocoronene (HBC) assemblies with and without alkyl chains (see Supplementary Note 4). In a recent work24, we studied the assembly of HBC molecules equipped with six dodecyl side chains (HBC-6C12H25) on Au(111) at 5 and 300 K by combined STM/AFM measurements (see Supplementary Fig. 7). From the lattice measurements, we extract of a thermal expansion of this system of about α =  520 ± 60 × 10−6 K−1. While the same hexagonal ordering has been reported by Meissner et al. 34 from pristine HBC molecule (without alkyl chains), no lattice variation has been observed between 1.2 K and 300 K STM measurements, resulting in a thermal expansion of only α = 14 × 10−6 K−134,35.

Both the SW and HBC-6C12H25 molecular assemblies experience a similar giant thermal expansion, that we attribute to the increased fluctuations of their interdigitated long alkyl chains governing the intermolecular assemblies. The cores of these precursors are quite different accounting for the slight difference in observed thermal expansion since the HBC lies flat on the surface24 while the SW core tends to protrude upon adsorption. In contrast to these interdigitated systems, supramolecular networks formed by pristine HBC molecules do only show a negligible expansion34,35. This observation again points towards the central role of the alkyl chains in the thermal expansion mechanism of interdigitated systems. Based on these observations, we believe that future works might address further the influence of the molecule cores as well as the effect of the chain lengths. We also anticipate that the structural phase transition might be also observed in such alkyl-based expansion mechanism24. This might enable tailoring the thermoresponse of such supramolecular systems.

Discussion

We investigated the giant thermal expansion of a 2D supramolecular assembly composed of SW molecules on Au(111). Combining high-resolution STM/AFM images with MD calculations, we quantified a thermal expansion coefficient of α = 980 ± 110 × 10−6 K−1 value that can be explained by temperature-dependent fluctuations of alkyl side chains compared to the core of the molecules. The non-covalent intermolecular interaction governed by the alkyl chain interdigitation is thus the main factor responsible for the network expansion contrarily to the molecule core found to have a marginal role in the process. This mechanism is further generalised to two-dimensional interdigitated supramolecular systems at surfaces, in contrast to their pristine counterparts (without side chains) showing no thermal expansion. Such generalization might allow the rational design of new thermoresponsive 2D materials.

Methods

Sample preparation

Au(111) single crystals (Mateck GmbH) were prepared in UHV conditions by several cycles of Ar+ sputtering and annealing at 750 K. As result, atomically flat surfaces were obtained with large terraces separated by atomic steps.

Electrospray deposition

The electrospray deposition was performed on Au(111) samples kept at room temperature using a commercial system from MolecularSpray. The setup was connected to preparation chambers of both low and room temperature systems. The SW molecules were dissolved in a toluene and methanol mixture (ratio 5 to 1 in volume). During the spray deposition the pressure rose up to 1 × 10−6 mbar. Typical applied voltage were 1.5–2.0 kV with adjustment sometimes necessary during the spray to maintain stable conditions. Depositions were performed for around 30 min and were followed by a slight annealing at approx. 450 K in order to remove any solvent contaminants. We reproduced such depositions in both UHV apparatus with identical parameters that always lead to molecular coverage well-below the monolayer and non-uniform molecular densities at the macroscopic scale. This allowed us to investigate both single molecules and extended assemblies on the same sample (see Supplementary Figs. 1 and 2).

Room temperature AFM

Room temperature nc-AFM measurements were performed with a home-built non-contact atomic force microscope with Nanonis RC5 electronics. PPP-NCL cantilevers (Nanosensors) were used as sensors (typical resonance frequencies of f1st =  150 kHz and f2nd = 1 MHz, oscillation amplitude 2–5 nm and 200–800 pm, respectively). Their preparation consisted in an annealing for 1 h at 400 K followed by a tip Ar+ sputtering for 90 s at 680 eV at an Ar+ pressure of p  = 3 × 10−6 bar. Base pressure of UHV system was maintained to 2 × 10−11 mbar during the measurements. The calibration of the (XY) AFM scanner at 300 K was checked by imaging the atomic surface lattice of a KBr(001) crystal.

Low temperature AFM/STM

Low temperature STM/AFM measurements were realized with a low-temperature microscope (Omicron Nanotechnology GmbH) operated at 5 K with Nanonis RC5 electronics and based on a tuning fork sensor in the qPlus configuration (resonance frequency f = 25 KHz, at 5 K, oscillation amplitude of A = 50 pm). The STM experiments were conducted in the constant-current mode. The AFM experiments were performed in constant-height mode with a CO-terminated tips 21. The calibration of the (XY) scanner was checked by imaging the Au(111) surface with a CO-terminated tip.

SW synthesis

The SW molecules were prepared by multistep organic synthesis, through six-fold intramolecular Yamamoto coupling reaction of a carefully designed polyphenylene precursor bearing 12 bromo groups, which was obtained through a cobalt-catalysed cyclotrimerization of corresponding diarylacetylenes. The details of the synthesis and characterizations are reported in Ref. 17.

Molecular dynamic (MD) simulation details

MD simulations were performed using AMBER1836 with NVIDIA GPU acceleration37,38. To account for long-range electrostatic interactions, periodic boundary conditions and Particle Mesh Ewald39,40 were used with standard defaults and a real-space cutoff of 1.5 nm. Van der Waals interactions were truncated at the real space cutoff, and Lorentz-Berthelot mixing rules were used to determine the interaction parameters between different atoms. All simulations are preformed in vacuum conditions and the volume of the system was kept fixed and the temperature was adjusted by means of a Langevin thermostat 41 with a damping rate of 1 ps−1 which ensures fast thermalization with a minimal effect on the dynamics33 of the arms and slip dynamics governing the diffusion of the SW core. The SHAKE algorithm 42 was used to constrain bonds containing hydrogen, thus allowing the use of an integration time step of 1 fs. Coordinates were saved every 1000 steps.

Atomic level models and force fields

An Au(111) surface composed by three atomic layers, where the positions of the atoms in the lowest layer were fixed during the MD runs using a harmonic restrain of 5 kcal mol−1, was considered. Surfaces of two different sizes were considered, i.e. 10 × 10 nm2 (used in single molecule simulations shown in Fig. 3 and Supplementary Fig. 3) and 16 × 16 nm2 (used in three molecules simulations shown in Fig. 4 and Supplementary Fig. 4). SW initial structure was generated using the Avogadro software (see Supplementary Fig. 3a, b). Subsequently, the structure was relaxed via the energy minimization procedure described in the simulation protocol section. The interaction between the atoms composing the SW molecule was described using the General Amber Force Field (GAFF)43,44,45—including its latest refinements (version 1.8 from 2017) available in AMBER18 36 that better reproduce the high-level ab initio and high quality experimental data of a variety of molecular properties36. At the intramolecular level, GAFF is able to accurately reproduce the structure and vibrational properties of a wide range of organic molecules 43,44,45 (including biphenyl and aliphatic chains which are the basic building blocks of the SW). Moreover, at the intermolecular level, GAFF has been routinely and successfully applied to study supramolecular polymers45,46, mechanically interlocked molecules 47, molecular assembly processes48,49 as well as other thermophysical properties that strongly depend on intermolecular interactions45,50 (e.g. liquid density, heat of capacity, self-diffusivity among others). As for the gold atoms, the CHARMM-METAL force-field 51,52,53 was applied, which simultaneously describe the intrinsic properties of gold, while retaining thermodynamical consistency with AMBER force field 51,52,53. This force field has been extensively tested by studying the adsorption of small organic molecules (charged and uncharged and within the same chemical space as SW) against both density-functional-theory simulations as well as available experimental results51,52,53. Moreover, its suitability to describe the interaction with small organic molecules and their surface mobility has been recently validated through single molecule manipulation experiments33,54.

Simulations protocols

In total six different microsecond long MD simulations, that can be grouped in two stages, were performed. The simulation details, notably the preparation stages prior to production runs, are provided in Supplementary Note 1. Here only a brief outline of each stage is provided. Stage 1: Starting from the atomic configuration obtained after 10 ns long MD simulation of the SW adsorption at 300 K to Au(111), 4 different microsecond long MD simulations were performed. Each performed at a fixed constant temperature (i.e. at 5, 300, 420 and 450 K) allows the investigation of the influence of the temperature in the diffusion of the core as well as the chain mobility. Stage 2: In order to inspect the influence of the thermally enhanced arm mobility in the molecular assembly, two different microsecond long simulations were performed consisting in an assembly of three SW molecules (as shown in Fig. 4). In this case only the limiting temperatures, i.e. 5 and 450 K, were considered. At low temperatures the apparent lack of mobility of both the core and the chains simply confirms the stability of the assembly observed experimentally. At high temperatures (T = 450 K), and by performing very long MD simulations (microsecond long) one could observe the diffusion of the molecular bodies assisted by the chain mobility. Lower values of temperature were deliberately not considered, as in those cases the timescales of the diffusion dynamics increase significantly (as there is less thermal energy available) rendering the those MD simulation unfeasible, i.e. well beyond the current state of the art microsecond long regime on such large scale systems.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

References

1. 1.

Zhou, H.-L., Zhang, Y.-B., Zhang, J.-P. & Chen, X.-M. Supramolecular-jack-like guest in ultramicroporous crystal for exceptional thermal expansion behaviour. Nat. Commun. 6, 6917 (2015).

2. 2.

Liu, Z. et al. Negative thermal expansion in molecular materials. Chem. Commun. 54, 5164–5176 (2018).

3. 3.

Burg, J. A. & Dauskardt, R. H. Elastic and thermal expansion asymmetry in dense molecular materials. Nat. Mater. 15, 974–980 (2016).

4. 4.

Mary, T. A., Evans, J. S. O., Vogt, T. & Sleight, A. W. Negative thermal expansion from 0.3 to 1050 Kelvin in ZrW2O8. Science 272, 90–92 (1996).

5. 5.

Miller, W., Smith, C. W., Mackenzie, D. S. & Evans, K. E. Negative thermal expansion: a review. J. Mater. Sci. 44, 5441–5451 (2009).

6. 6.

Aida, T., Meijer, E. W. & Stupp, S. I. Functional supramolecular polymers. Science 335, 813–817 (2012).

7. 7.

Kolesnichenko, I. V. & Anslyn, E. V. Practical applications of supramolecular chemistry. Chem. Soc. Rev. 46, 2385–2390 (2017).

8. 8.

Theobald, J. A., Oxtoby, N. S., Phillips, M. A., Champness, N. R. & Beton, P. H. Controlling molecular deposition and layer structure with supramolecular surface assemblies. Nature 424, 1029–1031 (2003).

9. 9.

Barth, J. V., Costantini, G. & Kern, K. Engineering atomic and molecular nanostructures at surfaces. Nature 437, 671–679 (2005).

10. 10.

Barth, J. V. Molecular Architectonic on Metal Surfaces. Ann. Rev. Phys. Chem. 58, 375–407 (2007).

11. 11.

Mali, K. S., Pearce, N., De Feyter, S. & Champness, N. R. Frontiers of supramolecular chemistry at solid surfaces. Chem. Soc. Rev. 46, 2520–2542 (2017).

12. 12.

Liu, S.-X. et al. Structural studies of transition metal complexes with 4,5-bis(2-pyridylmethylsulfanyl)-4’,5’-ethylenedithiotetrathiafulvalene: probing their potential for the construction of multifunctional molecular assemblies. Inorg. Chem. 42, 4801–4803 (2003).

13. 13.

De Feyter, S. & De Schryver, F. C. Two-dimensional supramolecular self-assembly probed by scanning tunneling microscopy. Chem. Soc. Rev. 32, 139–150 (2003).

14. 14.

Tahara, K., Lei, S., Adisoejoso, J., De Feyter, S. & Tobe, Y. Supramolecular surface-confined architectures created by self-assembly of triangular phenyleneethynylene macrocycles via van der Waals interaction. Chem. Commun. 46, 8507–8525 (2010).

15. 15.

Mössinger, D. et al. Large all-hydrocarbon spoked wheels of high symmetry: modular synthesis, photophysical properties, and surface assembly. J. Am. Chem. Soc. 132, 1410–1423 (2010).

16. 16.

May, R., Jester, S.-S. & Höger, S. A giant molecular spoked wheel. J. Am. Chem. Soc. 136, 16732–16735 (2014).

17. 17.

Liu, Y. et al. A shape-persistent polyphenylene spoked wheel. J. Am. Chem. Soc. 138, 15539–15542 (2016).

18. 18.

Würsch, D. et al. Interactions between π -conjugated chromophores in a giant molecular spoked wheel. Chem. Commun. 53, 352–355 (2017).

19. 19.

Narita, A., Wang, X.-Y., Feng, X. & Müllen, K. New advances in nanographene chemistry. Chem. Soc. Rev. 44, 6616–6643 (2015).

20. 20.

Bocquet, f et al. Inhomogeneous relaxation of a molecular layer on an insulator due to compressive stress. Phys. Rev. Lett. 108, 206103 (2012).

21. 21.

Gross, L., Mohn, F., Moll, N., Liljeroth, P. & Meyer, G. The chemical structure of a molecule resolved by atomic force microscopy. Science 325, 1110–1114 (2009).

22. 22.

Iwata, K. et al. Chemical structure imaging of a single molecule by atomic force microscopy at room temperature. Nat. Commun. 6, 7766 (2015).

23. 23.

Gross, L. et al. Atomic force microscopy for molecular structure elucidation. Angew. Chem. Int. Edit. 57, 3888–3908 (2018).

24. 24.

Hinaut, A. et al. Electrospray deposition of structurally complex molecules revealed by atomic force microscopy. Nanoscale 10, 1337–1344 (2018).

25. 25.

Gießibl, F. J. The qPlus sensor, a powerful core for the atomic force microscope. Rev. Sci. Instrum. 90, 011101 (2019).

26. 26.

Elemans, J. A. A. W., Lei, S. & De Feyter, S. Molecular and supramolecular networks on surfaces: from two-dimensional crystal engineering to reactivity. Angew. Chem. Int. Edit. 48, 7298–7332 (2009).

27. 27.

Rauschenbach, S. et al. Electrospray ion beam deposition of clusters and biomolecules. Small 2, 540–547 (2006).

28. 28.

Satterley, C. J. et al. Electrospray deposition of fullerenes in ultra-high vacuum: in situ scanning tunneling microscopy and photoemission spectroscopy. Nanotechnology 18, 455304 (2007).

29. 29.

Hauptmann, N., Hamann, C., Tang, H. & Berndt, R. Switching and charging of a ruthenium dye on Ag(111). Phys. Chem. Chem. Phys. 15, 10326–10330 (2013).

30. 30.

Rinke, G. et al. Soft-landing electrospray ion beam deposition of sensitive oligoynes on surfaces in vacuum. Int. J. Mass Spectrom. 377, 228–234 (2015).

31. 31.

Gragnaniello, L. et al. Uniaxial 2D superlattice of Fe4 molecular magnets on graphene. Nano Lett. 17, 7177–7182 (2017).

32. 32.

Warr, D. A. et al. Sequencing conjugated polymers by eye. Science Advances 4, 9543 (2018).

33. 33.

Pawlak, R. et al. Conformations and cryo-force spectroscopy of spray-deposited single-strand DNA on gold. Nat. Commun. 10, 685 (2019).

34. 34.

Meissner, M. et al. Flexible 2D crystals of polycyclic aromatics stabilized by static distortion waves. ACS Nano 10, 6474–6483 (2016).

35. 35.

Wagner, C. et al. Repulsion between molecules on a metal: Monolayers and submonolayers of hexa-peri-hexabenzocoronene on Au(111). Phys. Rev. B 81, 035423 (2010).

36. 36.

Case, D. et al. AMBER 2018. (University of California, San Francisco, 2018).

37. 37.

Salomon-Ferrer, R., Götz, A. W., Poole, D., LeGrand, S. & Walker, R. C. Routine microsecond molecular dynamics simulations with AMBER on GPUs. 2. Explicit solvent particle mesh Ewald. J. Chem. Theory Comput. 9, 3878–3888 (2013).

38. 38.

Götz, A. W. et al. Routine microsecond molecular dynamics simulations with AMBER on GPUs. 1. Generalized born. J. Chem. Theory Comput. 8, 1542–1555 (2012).

39. 39.

Crowley, M., Darden, T., Cheatham, T. & Deerfield, D. Adventures in improving the scaling and accuracy of a parallel molecular dynamics program. J. Supercomput. 11, 255–278 (1997).

40. 40.

Essmann, U. et al. A smooth particle mesh Ewald method. J. Chem. Phys. 103, 8577–8593 (1995).

41. 41.

Loncharich, R. J., Brooks, B. R. & Pastor, R. W. Langevin dynamics of peptides: the frictional dependence of isomerization rates of N-acetylalanyl-N’-methylamide. Biopolymers 32, 523–535 (1992).

42. 42.

Miyamoto, S. & Kollman, P. A. Settle: an analytical version of the SHAKE and RATTLE algorithm for rigid water models. J. Comput. Chem. 13, 952–962 (1992).

43. 43.

Wang, J., Wolf, R. M., Caldwell, J. W., Kollman, P. A. & Case, D. A. Development and testing of a general amber force field. J. Comput. Chem. 25, 1157–1174 (2004).

44. 44.

Wang, J., Wang, W., Kollman, P. A. & Case, D. A. Automatic atom type and bond type perception in molecular mechanical calculations. J. Mol. Graph. Model. 25, 247–260 (2006).

45. 45.

Harrison, J. A. et al. Review of force fields and intermolecular potentials used in atomistic computational materials research. Appl. Phys. Rev. 5, 031104 (2018).

46. 46.

Baker, M. B. et al. Consequences of chirality on the dynamics of a water-soluble supramolecular polymer. Nat. Commun. 6, 6234 (2015).

47. 47.

Bazargan, G. & Sohlberg, K. Advances in modelling switchable mechanically interlocked molecular architectures. Int. Rev. Phys. Chem. 37, 1–82 (2018).

48. 48.

Chami, F. & Wilson, M. R. Molecular order in a chromonic liquid crystal: a molecular simulation study of the anionic azo dye sunset yellow. J. Am. Chem. Soc. 132, 7794–7802 (2010).

49. 49.

Bochicchio, D. & Pavan, G. M. Molecular modelling of supramolecular polymers. Advances in Physics: X 3, 1436408 (2018).

50. 50.

Sprenger, K. G., Jaeger, V. W. & Pfaendtner, J. The general AMBER force field (GAFF) can accurately predict thermodynamic and transport properties of many ionic liquids. J. Phys. Chem. B 119, 5882–5895 (2015).

51. 51.

Heinz, H. et al. Nature of molecular interactions of peptides with gold, palladium, and Pd-Au bimetal surfaces in aqueous solution. J. Am. Chem. Soc. 131, 9704–9714 (2009).

52. 52.

Heinz, H., Jha, K. C., Luettmer-Strathmann, J., Farmer, B. L. & Naik, R. R. Polarization at metal-biomolecular interfaces in solution. J. Royal Soc. Interface 8, 220–232 (2011).

53. 53.

Heinz, H. & Ramezani-Dakhel, H. Simulations of inorganic-bioorganic interfaces to discover new materials: insights, comparisons to experiment, challenges, and opportunities. Chem. Soc. Rev. 45, 412–448 (2016).

54. 54.

Ruiz, M. P. et al. Bioengineering a single-protein junction. J. Am. Chem. Soc. 139, 15337–15346 (2017).

Acknowledgements

Se.S., A.H., R.P., S.F., Z.L., T.G. and E.M. thank the Swiss National Science Foundation (SNF) and the Swiss Nanoscience Institute (SNI). J.G.V. acknowledges funding from a Marie Sklodowska-Curie Fellowship (DLV-795286) within the Horizons 2020 framework and also the computer resources, technical expertise and assistance provided by the Red Española de Supercomputación at the Power9 Supercomputer (BSC, Barcelona). E.M., R.P. and Se.S. thank the funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (ULTRADISS grant agreement No. [834402]).

Author information

Authors

Contributions

E.M. and A.H. conceived the experiments. Se.S. and A.H. performed spray deposition. Se.S. and A.H. performed the AFM measurements at room temperature, R.P. performed the experiments at low temperature and J.G.V conducted the simulations. A.N., Y.L. and X.F. synthesized the molecules. Se.S., A.H. and J.G.V. wrote the paper with the help of R.P. Se.S., A.H., R.P., J.G.V., Y.L., S.F., Z.L., X.F., K.M., T.G., A.N. and E.M. read and discussed the paper.

Corresponding authors

Correspondence to Antoine Hinaut or J. G. Vilhena or Ernst Meyer.

Ethics declarations

Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

Scherb, S., Hinaut, A., Pawlak, R. et al. Giant thermal expansion of a two-dimensional supramolecular network triggered by alkyl chain motion. Commun Mater 1, 8 (2020). https://doi.org/10.1038/s43246-020-0009-2

• Accepted:

• Published:

• Thermodynamic properties, mechanical properties and interatomic potential in solids: a Shou-Shi-Ling ( ) game

• Enrico Benassi
•  & Haiyan Fan

Continuum Mechanics and Thermodynamics (2021)