On recursive averaging processes and Hilbert space extensions of the contraction mapping principle

If T maps a convex domain D_T into itself, and if {ω_n} is a real sequence with range in (0, 1] then the recursive averaging process,

$\text{X}{}_{\mathrm{n+1}}\text{=(1?omega;}{}_{\mathrm{n}}\text{)}\text{X}{}_{\mathrm{n}}\text{+ω}{}_{\mathrm{n}}\text{+ω}{}_{\mathrm{n}}\text{T}\text{x}{}_{\mathrm{n}}\text{,}\text{x}{}_0\text{=ξ?D}{}_{\mathrm{T}}$

generates a sequence {x?_n}; with range in D_T. Under suitable conditions on D_T, T and {ω_n} the sequence {x?_n} will converge in some sense to a fixed point of T. We prove that if D_T is a closed convex subset of a complex Hilbert space H, if Tω = (1 ? ω) I + ωT is a strict contraction for some ω ? (0, 1], and if {ω_n} satisfies the conditions,

$\text{ω}{}_{\mathrm{n}}\text{→ 0}$

and

$\text{∑}\text{n=0}\text{∞}\text{ω}{}_{\mathrm{n}}\text{=∞}$

then, for arbitrary ξ ? D_T, {x?_n} converges strongly to (the unique) fixed point of T. We also prove that if D_T and {ω_n} satisfy the foregoing conditions, if T has at least one fixed point, and if Tω is non-expansive for some ω ? (0, 1], then for all ξ ? D_T, {x?_n} converges at least weakly to some fixed point of T. Finally, we apply these results to linear equations involving bounded normal operators and obtain an extension of the classical Neumann operator series.     

Eigenvalue localization for the tikhonova model of crystal growth

Matrix A with characteristic polynomial Q(z) is defined positive or negative Hurwitz according to whether Q(z) or Q(-z) is a Hurwitz polynomial. Leading principle sections of the Tikhonova growth matrix have associated characteristic polynomials P_n(-z) which satisfy the recursion

$\text{P}{}_{\mathrm{n+1}}\text{(z)=zP}{}_{\mathrm{n}}\text{(z)+}\text{1}\text{n(n+1)}\text{P}{}_{\mathrm{n-1}}\text{(z),P}{}_0\text{(z)=1,P}{}_1\text{(z)=1+z}$

That the Tikhonova growth matrix is negative Hurwitz is established through applying the Wall-Stieltjes theory of continued fraction expansions to show the P_n(-z) are Hurwitz polynomials. The Kayeya-Enestrom theorem and a procedure for refinement of the Gerschgorin estimate are used to obtain analytical bounds on spectral radii for the Tikhonova model, which provides estimates of maximal growth rates. The theory allows generalization to more complicated growth models.     

The reflection of hydrogen atoms from lithium fluoride

The paper describes the phenomena associated with the reflection of a sharply defined beam of hydrogen atoms from a crystal of LiF. Of primary interest is the fact that the atoms show interference effects in agreement with the wave mechanics theory and plane grating diffraction patterns are photographed. Evidence of the thermal agitation of the surface ions is obtained from the diffuse reflection with surrounds the specular beam.The Schrödinger wave equation for the motion of a free particle of mass m is

$\text{▽}{}^2\text{ψ ?}\text{4πmi}\text{h}\text{?ψ}\text{?t}\text{= 0}\text{(I)}$

. The solution of this equation corresponding to the kinetic energy $\text{mv}{}^2\text{2}$ is

where

$\text{v }\text{mv}{}^2\text{2}\text{and}\text{σ}\text{mv}\text{h}$

. The motion of such a particle should have the characteristics of a plane wave of frequency ν and wave-length $\text{λ =}\text{1}\text{σ}$. The experiments of various investigators¹ have shown the validity of the wave theory of the motion of the free electron and have given values of the wave-length in agreement with the theory.The free motion of atoms, ions and molecules should likewise have wave characteristics. In the case of the hydrogen atom, as the simplest example, the complete wave equation may be written in the form

$\text{I}\text{m}\text{▽}{}^2\text{x,y,zψ +}\text{I}\text{μ}\text{▽}{}^2\text{η,μζψ ?}\text{8π}{}^2\text{μ?ψ}\text{mh}{}^2\text{η}{}^2\text{+ μ}{}^2\text{+ ζ}{}^2$

$\text{?}\text{4πi}\text{h}\text{?ψ}\text{?t}\text{= 0,}\text{(3)}$

where x, y, z, are the coördinates of the center of mass of the atom and ξ, η, ζ the coördinates of the electron with respect to the center of mass. If m_? and m₊ are the masses of electron and proton, m and μ have the significance

$\text{m = m}{}_{\mathrm{?}}\text{+ m}{}_{\mathrm{+}}\text{and}\text{I}\text{μ}\text{=}\text{I}\text{m}{}_{\mathrm{?}}\text{+}\text{I}\text{m}{}_{\mathrm{+}}$

. Equation (3) is solved by

$\text{ψ = U}{}_1\text{(x,y,z) U}{}_2\text{(η, ν ζ) ?}{}^{\mathrm{<mtext>2\pi iEt</mtext><mtext>h</mtext>}}$

, where E may have a continuous set of values and represents the total energy. U₁ and U₂ must satisfy the equations

$\text{▽}{}_1{}^2\text{U}{}_1\text{+}\text{8π}{}^2\text{mβU}{}_1\text{h}{}^2\text{= 0,}\text{(4)}$

and

$\text{▽}{}_2{}^2\text{U}{}_2\text{+}\text{8π}{}^2\text{μ}\text{h}{}^2\text{(α ?}\text{μ?}\text{m}\text{η}{}^2\text{+ ν}{}^2\text{+ ζ}{}^2\text{)}\text{U}{}_2\text{= 0}\text{(5)}$

, where

$\text{α + β + E}$

.     

Mathematical models of information system use

This report presents the results from a study of mathematical models relating to the usage of information systems. For each of four models, the papers developed during the study provide three types of analyses: reviews of the literature relevant to the model, analytical studies, and tests of the models with data drawn from specific operational situations. (1) The Cobb-Douglas model: x₀ = ax₁^bx₂^(1?b).This classic production model, normally interpreted as applying to the relationship between production, labor, and capital, is applied to a number of information related contexts. These include specifically the performance of libraries, both public and academic, and the use of information resources by the nation's industry. The results confirm not only the utility of the Cobb-Douglas model in evaluation of the use of information resources, but demonstrate the extent to which those resources currently are being used at significantly less than optimum levels. (2) Mixture of Poissons:

$\text{χ}{}_0\text{=}\text{∑}\text{i=0}\text{n}\text{i}\text{∑}\text{j=0}\text{p}\text{n}{}_{\mathrm{j}}\text{e}{}^{\mathrm{mj}}\text{(m}{}_{\mathrm{j}}\text{)′/i!}$

where x₀ is the usage and (n_j,m_j),j = 0 to p, are the p + 1 components of the distribution. This model of heterogeneity is applied to the usage of library materials and of thesaurus terms. In each case, both the applicability and the analytical value of the model are demonstrated. (3) Inverse effects of distance: x = a e^?md if c(d) = rdx = ad^?m if c(d) = r log(d).These two models reflect different inverse effects of distance, the choice depending upon the cost of transportation. If the cost,c(d), is linear, the usage is inverse exponential; if logarithmic, the usage is inverse power. The literature that discusses the relationship between usage of facilities and the distance from them is reviewed. The models are tested with data from the usage of the Los Angeles Public Library, both Central Library and branches, based on a survey of 3662 users. (4) Weighted entropy:

$\text{S(x}{}_1\text{,x}{}_2\text{,...,x}{}_{\mathrm{n}}\text{)= -}\text{∑}\text{i=1}\text{n}\text{r(x}{}_{\mathrm{i}}\text{P(x}{}_{\mathrm{i}}\text{)}\text{log}\text{(p(x}{}_{\mathrm{i}}\text{)).}$

This generalization of the “entropy measure of information” is designed to accommodate the effects of “relevancy”, as measured by r(x), upon the performance of information retrieval systems. The relevant literature is reviewed and the application to retrieval systems is considered.     

Accurate approximate solutions to oscillatory problems with perturbing singular damping

In this paper we attempt to obtain approximate solutions of improved accuracy for a class of differential equations of the form

$\text{d}{}^2\text{y}\text{d}\text{x}{}^2\text{+εμ(x)}\text{d}\text{y}\text{d}\text{x}\text{+ω}{}^2{}_{\mathrm{c}}\text{y = 0}$

, where ε is a real parameter less than unity, ω_c is a positive real constant of order unity and μ(x) is a singular function of x in the region of interest. It does not appear to be possible to find a general analytic expression for the error estimate of the approximate solution. For the case μ(x) = x^?2, however, it is shown that the approximate solution is accurate to 0(ε²), as x → 0^? from negative values, by comparing it with the numerically integrated solution. For the same case, the approximate solution is orders of magnitude more accurate than Poincaré's first-order perturbation solution, which is accurate to $\text{0(}\text{ε}{}^2\text{ln}\text{|x|}\text{|x|}\text{)}$ as x → 0^?. This work arose in search of analytic solutions to a linearized form of the restricted three-body problem.     

An equation of state for a system with a single type of transformation

Based on theory of a previous paper, the writer has developed an equation of state for a system with a single type of transformation. This equation is of the form

$\text{h=A+Bv+Cp+Dpv?T(E+Fv+Gp+Hpv)}$

where h = ε + pv is the total heat, p the pressure, v the specific volume, T the temperature, and p, v, T are considered independent variables. A, B, C, etc., are constants for the system. The latent eat at constant (p, T) is given by

$\text{λp,T=(v}{}_2\text{?v}{}_1\text{)(}\text{?h}\text{?v}\text{)P,T= (v}{}_2\text{?v}{}_1\text{)[(B?TF)+p(D?TH)]}$

. These equations are checked with data on saturated and superheated ammonia, and the agreement is good to within a few tenths of a per cent. Also, checks with data on saturated and superheated steam show agreement within several per cent.     

The representation of radiation reaction in wave mechanics

It is well known that the wave mechanical ψ equation leads to the conclusion that the centroid of the wave mechanical electron should move according to the classical electrodynamic equation of motion in which, however, the terms representing what is commonly called radiation reaction are absent. If v is the velocity of the electron, the classical rate of change of momentum is $\text{md}\text{dt}\text{{}\text{v}\text{(}\text{I}\text{?}\text{v}{}^2\text{c}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{}}$. The equation of motion including radiation reaction terms may be regarded as obtainable by replacing this quantity by one obtained by operating upon it with the operator P^?1

$\text{P={I?}\text{α}{}_1\text{kd}\text{dt}\text{+}\text{α}{}_2\text{d}\text{dt}\text{(}\text{kd}\text{dt}\text{)?·}}{}^{\mathrm{?}}$

where α₁, α₂, etc., are constants and $\text{k = (}\text{I}\text{?}\text{v}{}^2\text{c}{}^2\text{)}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$. The main purpose of the paper is to show that if there be any relativistically invariant ψ equation which leads to the classical equation of motion without radiation reaction terms, then by replacing the vector and scalar potentials U and ? in that equation by P(U) and P(?), a relativistically invariant equation of motion will be obtained including the radiation reaction terms, provided that the $\text{d}\text{dt}$ in P be now regarded as $\text{?}\text{?t}$ + u · grad, where u is the velocity of the wave mechanical density distribution at a point. The purpose is to use the power to produce the equation of motion as a criterion for suggesting the proper modification of the ψ equation to apply in those cases where, on the classical theory, the electron would suffer great acceleration, as in ionization by rapidly moving corpuscles.     

On the compression of search trees

Let X=_x1,_x2,…,_xn$X=x_1\text{,}{x}_2\text{,}\dots \text{,}{x}_n$ be a sequence of non-decreasing integer values. Storing a compressed representation of X that supports access and search is a problem that occurs in many domains. The most common solution to this problem uses a linear list and encodes the differences between consecutive values with encodings that favor small numbers. This solution includes additional information (i.e. samples) to support efficient searching on the encoded values. We introduce a completely different alternative that achieves compression by encoding the differences in a search tree. Our proposal has many applications, such as the representation of posting lists, geographic data, sparse bitmaps, and compressed suffix arrays, to name just a few. The structure is practical and we provide an experimental evaluation to show that it is competitive with the existing techniques.     

The scattering of X-rays

Present Status of the Problem.—The scattering of X-rays is one of the outstanding problems of electromagnetic radiation which has not been solved satisfactorily. All theories (based on classical electrodynamics) presented thus far do not explain either the diminution in the scattering coefficient, or the observed asymmetry in the scattering, or both. Among such theories we may mention:J. J. Thompson's Theory.—Assuming that the scattering is done by a point electron, and making use of certain additional hypotheses, Thomson showed that the scattering coefficient of any substance is given by

$\text{σ=}\text{8πNpe}{}^4\text{3m}{}^2\text{c}{}^4$

and that the intensity of the scattered radiation is given by

$\text{I}{}_{\mathrm{\theta }}\text{=I}\text{e}{}^4\text{(I+cos}{}^2\text{θ)}\text{2r}{}^2\text{m}{}^2\text{c}{}^4$

where N is the number of atoms per c.c., p the number of electrons per atom, e the electronic charge, m the electronic mass, c the velocity of light, I_θ the intensity of the scattered radiation at an angle θ between the incident beam and the radius vector joining the centre of the electron and the point P distant r from the electron, and I is the intensity of the incident beam. This theory explains neither the asymmetry nor the decrease in the coefficient of scattering.Schott's Theory.—Among other things, the assumption is here made that the atom consists of coaxal rings of electron. The electrons in each ring are spaced at equal intervals and revolve with a uniform angular velocity, which, however, may be different for different rings. This theory fails to explain the observed diminution in the scattering coefficient.Debye's Theory.—In its essentials, Debye's theory has the same merits and demerits as that of Schott. Debye assumes that all the electrons in an atom are arranged in a single ring, and that they are spaced at equal intervals. This theory (and also Schott's theory) explains the asymmetry and the “excess scattering,” but is altogether unable to explain the diminution in the scattering coefficient.Modification of the Classical Theory.—The present paper presents a discussion of the possibility of modifying the classical theory (that of J. J. Thomson) so as to account for the decrease in the scattering coefficient as well as the dissymmetry. By assuming that the electron is made up of a number of parts—for simplicity, of two parts—it has been found possible to account for the diminution in the scattering coefficient without, at the same time, explaining the observed asymmetry. To accomplish both objects is what was aimed at in the combination of the present work with that of Debye. In this research the goal has not been perfection between predicted and observed results, but rather to discuss some possible modifications of the classical theory and their consequences.     

Stabilization of carrier-frequency servomechanisms. II. Design formulae for proportional-derivative networks

In an alternating current servomechanism, the error is proportional to the modulation envelope of a modulated-carrier error signal. It is shown in part I that for stability and fidelity of the servo, it is highly desirable that the effect of the controller includes a proportional-derivative action on the modulation envelope. This action may be obtained with various forms of RC networks, including the parallel “T,” bridge “T,” and Wien Bridge forms.This part contains detailed design procedures and tables of values for the various types of proportional-derivative networks. Several forms of parallel “T” networks arise from the fact that there are five independent time constants in the network, while in order to realize the desired transfer characteristic it is necessary to impose only four conditions. It is indicated how the remaining degree of freedom may be used to obtain the most suitable input and output impedances for the source and load impedances with which the parallel “T” is to be used. The derivations for the parallel “T” formulae are given in an Appendix.Tolerance requirements on the components of parallel “T” and bridge “T” networks are derived. If ±1 per cent components are used at 60 cycles, the resonant frequency will lie between 56.4 and 63.6 cycles, and the notch width (rejection band width) will be within ±0.99 cps. of the correct value. In order to guarantee that the phase shift at 60 cycles is within ±10°, the percentage deviation of each part must be less than ($\text{9.0}\text{T}{}_{\mathrm{d}}\text{ω}{}_0$), where ω₀ is the carrier angular frequency, T_d the derivative time constant.     

Residue matrices and related properties in two-variable reactance functions

In this paper, the characterization of a two-variable reactance polynomial φ(λ,μ) is given in terms of the residue matrices of a single variable reactance matrix, Y(λ). Specificially, if Y(λ) is expressed in terms of its partial fraction as Y(λ)=λH_∞+Σ$\text{H}{}_{\mathrm{i}}\text{+jβ}{}_{\mathrm{i}}\text{λ+jω}{}_{\mathrm{i}}$+G where the residue matrices in general are p.s.d. Hermitian, then the ranks of these residue matrices are fixed in relation to the construction of φ(λ,μ) as the determinant of the two-variable reactance matrix μ1+Y(λ). Three theorems concerning these ranks—one each corresponding to the finite poles, poles at ∞ and the behaviour at λ=0 of Y(λ) are stated and proved. Several properties following from these theorems are studied. Also, implications of these theorems from a network theoretic point of view, like the minimum number of gyrators required to synthesize Y(λ) to yield the specific type of φ(λ,μ) etc., are studied. In the sequel, the concept of “generalized compact pole conditions” is introduced. Finally, these results are applied for the generation of two-variable reactance functions and matrices.     

Report on the work of the Bartol Research Foundation, 1933–1934

Using the velocity analyzer of Zartman with improved technique the combined velocity spectrum of Bi atoms and Bi₂ molecules was obtained at 827°, 851°, 875°, 899°, 922°, 947° C. From the spectral distribution curves the relative abundance of Bi atoms and Bi₂ molecules in the beams at the above temperatures could be determined to 1 per cent. The vapor pressure curve of Bi was obtained experimentally by the method of effusion and the values so obtained were combined with the degree of dissociation of the vapor as computed from the beams to give the heat of dissociation. The heat of dissociation was computed from the data, assuming the pressure to be given by the temperature of the crucible T_c. In calculating the heat of dissociation, the equilibrium temperature was taken as that of the slit chamber T_s which was 24° above T_c. The results of these calculations plotted with log₁₀K_p as ordinates against $\text{1}\text{T}{}_{\mathrm{s}}$ give a straight line whose slope yields the value of the heat of dissociation as 77,100±1200 calories. The curves for the distribution of velocities observed and computed on the assumption of a given ratio of Bi atoms to Bi₂ molecules in the beam were compared in an attempt to test the law of distribution of velocities. On the high velocity side agreement in two curves was obtained within the limits of experimental accuracy. On the low velocity side important deviations were noted of such a sort that the observed curves below a velocity $\text{α}\text{2}$, (α is the most probable velocity) gave more molecules than the theory demanded. Other deviations were observed on some of the runs taken with a fourth slit in which a deficiency of molecules was observed between velocities of .75α and $\text{α}\text{2}$. This deviation was probably due to a warping of the fourth slit carriage due to heat. The nature of the variation at velocities less than $\text{α}\text{2}$ indicated the presence of molecules of greater mass than Bi₂ in the beam and at the lower temperatures a distinct peak corresponding to Bi₈ molecules was observed which were present to less than 2 per cent. The vapor pressure curve for Bi was determined by least square reduction of the observed points to be given by $\text{log}{}_{10}\text{P = ? 52.23 ×}\text{195.26}\text{T}\text{+ 8.56}$ between 1100° and 1220° abs. It lies very close to the extrapolated curve given in the International Critical Tables.     

Angular region: a measure of underdamped behavior

The natural modes of an underdamped dynamical system are given by the characteristic numbers of the quadratic operator pencil

$\text{P(s)=s}{}^2\text{I+sB+A,}$

where the operator A depends on the dissipative and reactive elements of the system, while B depends solely on the reactive elements. The operator P(s) for every applied stimulus vector signal x must satisfy:

$\text{(Bx,x)}{}^2\text{<4(Ax,x).}$

A measure of underdamped behaviour is suggested by predetermining an angular region |φ| containing all natural modes of the system,

$\text{|}\text{tan}\text{φ|?}\text{[4(Ax,x)?(Bx,x)}{}^2\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(Bx,x)}\text{.}$

When a comparison between positive operators A and B is available, say B²=KA, then

$\text{|}\text{tan}\text{φ|?}\text{√(4?K}{}^2\text{)}\text{K}\text{.}$

The paper is motivated by Duffin-Krein-Gohberg's earlier mathematical contributions.     

Adiabatic flow of hydrogen gas through a rocket nozzle with and without composition change

A procedure is described for determining the characteristics of adiabatic flow through a rocket nozzle with and without composition change. The method of calculation is illustrated for the expansion of pure hydrogen gas from a chamber temperature of 306° K. and a pressure of 20.42 atm. to atmospheric pressure.The study indicates that the exhaust velocity and temperature are highest for flow where complete equilibrium is reached at each temperature with respect to the reaction

$\text{H}{}_2\text{?2H}$

Flow with composition change requires a nozzle exit to nozzle throat area ratio somewhat greater than that determined for adiabatic flow without composition change for the same ratio of chamber pressure to exit pressure.The residence time in a given temperature range is computed as a function of gas temperature for the two types of flow. The results of this calculation may be used to determine the minimum required reaction rates which allow composition changes during flow through the nozzle.     

Initial conditioned solutions of a second-order nonlinear conservative differential equation with a periodically varying coefficient

The exact solution of the equation

$\text{d}{}^2\text{x}\text{dt}{}^2\text{+dx+d′f(wt)x}{}^3\text{=0,}$

where d, d' and w are positive constants, and ?(wt) is a rectangular periodic function of time is discussed. The equation describes approximately the transversal movement of a particle in an alternating gradient accelerator. The exact solution is obtained in the form of a composite recurrent relation containing five particular solutions. Each of these solutions corresponds to a specific well-defined area of the phase plane of the initial conditions. The dynamical behaviour and the stability of the movement are examined analytically.     

Chaotic behavior of discrete-time linear inclusion dynamical systems

Given any finite family of real d-by-d nonsingular matrices $\{S_1,\dots ,S_l\},$ by extending the well-known Li–Yorke chaos of a deterministic nonlinear dynamical system to a discrete-time linear inclusion or hybrid or switched system:

$\begin{array}{c}\hfill x_n\in \{S_k{x}_{n?1};1\le k\le l\},x_0\in {\mathbb{R}}^d\mathrm{and}n\ge 1,\end{array}$

we study the chaotic dynamics of the state trajectory (x_n(x₀, σ))_{n ≥ 1} with initial state $x_0\in {\mathbb{R}}^d,$ governed by a switching law $\sigma :\mathbb{N}\to \{1,\dots ,l\}$. Two sufficient conditions are given so that for a “large” set of switching laws σ, there exhibits the scrambled dynamics as follows: for all $x_0,y_0\in {\mathbb{R}}^d,x_0\ne y_0,$

$\begin{array}{c}\hfill \underset{n\to +\infty }{lim\; inf}\parallel x_n(x_0,\sigma )?x_n(y_0,\sigma )\parallel =0\mathrm{and}\underset{n\to +\infty }{lim\; sup}\parallel x_n(x_0,\sigma )?x_n(y_0,\sigma )\parallel =\infty .\end{array}$

This implies that there coexist positive, zero and negative Lyapunov exponents and that the trajectories (x_n(x₀, σ))_{n ≥ 1} are extremely sensitive to the initial states $x_0\in {\mathbb{R}}^d$. We also show that a periodically stable linear inclusion system, which may be product unbounded, does not exhibit any such chaotic behavior. An explicit simple example shows the discontinuity of Lyapunov exponents with respect to the switching laws.     

Separation of DNA by length in rotational flow: Lattice-Boltzmann-based simulations

We use a lattice-Boltzmann based Brownian dynamics simulation to investigate the separation of different lengths of DNA through the combination of a trapping force and the microflow created by counter-rotating vortices. We can separate most long DNA molecules from shorter chains that have lengths differing by as little as 30%. The sensitivity of this technique is determined by the flow rate, size of the trapping region, and the trapping strength. We expect that this technique can be used in microfluidic devices to separate long DNA fragments that result from techniques such as restriction enzyme digests of genomic DNA.The development of novel methods for manipulating biopolymers such as DNA is required for the continued advancement of microfluidic devices. Techniques such as restriction enzyme digests for genomic sequencing rely on the detection of DNA that differ in length by sometimes thousands of base pairs.¹ Methods that separate DNA strands with resolutions on the order of kilobase pairs are required to analyze the products of this technique. To gain an insight into possible techniques to separate polymers, it can be helpful to review the methods to separate particles in microfluidic devices. Experimental work has shown how hydrodynamic mechanisms can lead to separation of particles based on size and deformability.² Eddies, microvortices, and hydrodynamic tweezers have been used to trap and sort particles. The mechanism of the trapping and sorting arises from the differences between interactions of the particles with the fluid.^2–8 In particular, counter-rotating vortices have been used to sort particles and manipulate biopolymers. They have been used to deposit DNA precisely across electrodes⁹ and trap DNA.^10,11 Vortex flow may therefore be a good basis for a technique for sorting DNA by length.Streaming flow has been used in experiments to separate colloids of different size.^3,4 Particles are passed through a channel with a flow field driven by oscillating bubbles and pressure. The flow field becomes a combination of closed and open streamlines. The vortex flow is controlled by the accoustic driving of the bubbles while pressure controls the net flow of the fluid. Larger particles are trapped in the closed vortex flow created by the bubbles, while smaller particles can escape the neighborhood of a bubble in the open streamlines. This leads to efficient separation of particles with size differences as small as 1 μm.Previous work on DNA has shown that counter-rotating vortices can be used to trap DNA dynamically. Long strands of DNA have been observed to stretch between the centers of two counter-rotating vortices. The polymer stays trapped in this state for significant amounts of time.¹² In a different experiment, the vortices were used to thermally cycle the polymer and allow replication via the polymerase chain reaction (PCR). The DNA is also trapped against one wall by a thermophoretic force in these experiments.¹⁰ The strength of the trap is controlled by the gradient in temperature created by a focused infrared laser beam.Trapping DNA at one wall by counter-rotating vortices has also been explored in simulation and found to depend on the Peclet number, Pe = u_maxL/D_m, where u_max is the maximum speed of the vortex, L is the box size, and D_m is the diffusion coefficient of one bead in the polymer chain.¹¹ The trapping rate of the DNA was shown to depend on the competition between the flow compressing the DNA into the trap region and the diffusion of the DNA out of the trap. For the work presented here, Pe ≅ 2000, similar to the previous work done with the same simulation.We extend the previous work to investigate if counter-rotating vortices can be used to separate DNA of different lengths. We use the same type of simulation outlined in Refs. ¹¹ and ^13–17, based on the lattice-Boltzmann method. The simulation method has successfully modeled systems as diverse as thermophoresis of DNA,¹⁴ migration of DNA in a microchannel,¹⁶ and translocation of DNA through a micropore.^17,18 Using this method, the fluid is broken into a lattice with size, ΔL, chosen to be 0.5 μm, and is coupled to a worm-like chain model with Brownian dynamics for the polymer.^19,20 The fluid velocity distribution function, n_i(r, t), describes the fraction of fluid particles with a discretized velocity, c_i, at each lattice site.^21–24 A discrete velocity scheme with nineteen different velocities in three dimensions is used. The velocity distributions will evolve according to ni(r+ciΔτ,t+Δτ)=ni(r,t)+Lij[nj(r,t)−njeq(r,t)],(1)where L is a collision operator such that the fluid relaxes to the equilibrium distribution, nieq given by a second-order expansion of the Maxwell-Boltzmann distribution nieq=ρaci[1+(ci·u)/cs2+uu:(cici−cs2I)/(2cs4)],(2)where cs=1/3ΔLΔτ is the speed of sound. Δτ is the time step for the fluid in the simulation, Δτ = 8.8 × 10⁻⁵. The coefficients a^c_i are determined by satisfying a local isotropy condition ∑iaciciαciβciγciδ=cs4(δαβδγδ+δαγδbetaγ+δαδδβγ).(3)To simplify computation, the velocity distributions are transformed into moment space. The density ρ, momentum density j, and momentum flux density Π are some of the hydrodynamic moments of n_i(r, t). The equilibrium conditions for these three moments are given by ρ=∑nieq,(4) j=∑ci·nieq,(5) Π=∑nieq·cici.(6)L has eigenvalues τ0−1,τ1−1,…,τ18−1, which are the characteristic relaxation times of the moments. The Bhatanagar-Gross-Krook model is used to determine L:²⁵ the non-conserved moments have a single relaxation time, τ_s = 1.0. The conserved moments are density and momentum; for these, τ⁻¹= 0. Fluctuations are added to the fluid stress as in the method of Ladd.²⁴ We have also compared simulations with lattice sizes of 1 μm and 0.25 μm and found no significant differences in the results.The DNA used in the simulation is represented by a worm-like chain model parameterized to capture the dynamics of YOYO-stained λ DNA in bulk solution at room temperature.^15,16,26 Long, flexible DNA is modeled since techniques to separate long DNA molecules with kilobase pair resolution are necessary to complete techniques such as genomic level sequencing using restriction enzyme digests.¹ In addition, such DNA is often used in experiment. Its properties are similar to unstained DNA or DNA stained by other methods.²⁷ Each molecule is represented by N_b beads and N_b − 1 springs. A chain composed of N_b − 1 springs will have a contour length of (N_b − 1) × 2.1 μm. The forces acting on each monomer include: an excluded volume force, a non-linear spring force, the viscous drag force, a random force that produces Brownian motion, a repulsive force from the container walls, and an attractive trapping force only at one wall as shown in Fig. ​Fig.11.¹³ The excluded volume interaction between beads i and j located at r_i and r_j is modeled using the following potential: Uijev=12kBTνNks2(34πSs2)exp(−3|ri−rj|24Ss2),(7)where ν=σk3 is the excluded volume parameter with σ_k = 0.105 μm, the length of one Kuhn segment, N_ks = 19.8 is the number of Kuhn segments per spring, and Ss2=Nks/6)σk2 is the characteristic size of the bead. This excluded volume potential reproduces self avoiding walk statistics. The non-linear spring force is based on force-extension curves from experiments and is given by fijS=kBT2σk[(1−|rj−ri|Nksσk)−2+4|rj−ri|nKσk−1]rj−ri|rj−ri|,(8)which applies when N_ks ≫ 1.Open in a separate windowFIG. 1.Simulation set-up. Arrows indicate direction of fluid flow. The region where the trapping force is active is shaded, and its width (X_stick) is shown. The region used to determine the trapping rate is indicated by the area labeled trap region. Figure is not to scale, the trap region and X_stick are smaller than shown.The beads are modeled as freely draining but subject to a drag force given by F_f = ?6πηa(u_p ? u_f).(9)The beads are also subjected to a random forcing term that is drawn from a Gaussian distribution with zero mean and a variance σ_v = 2k_BTζΔt.(10)The random force reproduces Brownian motion. To conserve total momentum, the momentum change imparted to the beads through their interactions with the fluid is balanced by a momentum change in the fluid. The momentum change is distributed to the three closest fluid lattice sites using a linear interpolation scheme based on the proximity of the lattice site to the polymer beads. Through this momentum transfer, hydrodynamic interactions between the beads occur.The beads are repelled from the walls with a force of magnitude Fwall=250kBTσk3(xbead−xwall)2, xbead>(xwall−1),(11)where the repulsion range is 1ΔL. Each monomer will also be attracted to the top wall by a force with magnitude Fstick=KstickkBTσk3(xbead−xwall+10)2, xbead>(xwall−Xstick)(12)and range X_stick (see Fig. ​Fig.1).1). The sticking force is turned off every one out of one hundred time steps of the polymer (1% of the simulation time steps). We vary both X_stick and K_stick to achieve separation of the polymers.In previous experiments, DNA has been trapped against one wall by using thermophoresis,¹⁰ dielectrophoresis,²⁸ and nanoplasmonic tweezers.²⁹ In the case of thermophoresis, the trap strength (K_stick) can be controlled by tuning the intensity of the temperature gradient and the trap extension (X_stick) can be controlled through the area over which the gradient extends. Both of these are set through focusing of the laser used to produce local heating. Similarly, the trap parameters can be controlled when using plasmonic tweezers by controlling the laser beam exciting the nanoplasmonic structures. In dielectrophoresis, the DNA is trapped by an AC electric field and can be controlled by tuning the frequency and amplitude of the field.In this work, the number of polymers, N_p, is 10 unless otherwise noted, and the container size is 25 ΔL × 50 ΔL × 2 ΔL. The time step for the fluid is Δτ = 8.8 × 10⁻⁵ s, and for the polymer is Δt = 3.7 × 10⁻⁶ s. The total simulation time is over 100 chain relaxation times, allowing sufficient independent samples to perform statistical analysis.Two counter-rotating vortices, shown in 1, are produced by introducing external forces to the fluid bound by walls in the x-direction and periodic in the y and z. Two forces of equal magnitude push on the fluid in the upper x region (12ΔL < x < 25ΔL): one in the +y-direction along y = 10ΔL, and one in the –y-direction along y = 40ΔL. Such counter-rotating vortices can be produced in microfluidic channels using acoustically driven bubbles,^3,4,30 local heating,¹⁰ or plasmonic nanostructures.⁵ The flow speed is controlled by very different external mechanisms in each case. We therefore choose a simple model to produce fluid flow that is not specific to one mechanism.The simulations are started using random initial conditions, and therefore, both lengths of polymer are dispersed throughout the channel. Within a few minutes, the steady state configurations pictured in Figs. ​Figs.22 and ​and33 are reached. We define the steady state as when the number of polymer chains in the trap changes by less than one chain (10 beads) per 1000 polymer time steps. Intermittently, some polymers may still escape and re-enter the trap even in the steady state. Three final configurations are possible: Both the lengths of DNA have become trapped, both lengths continue to rotate freely, or the shorter strand has become trapped while the longer rotates freely. Two of these states leave the polymers mixed; in the third, the strands have separated by size.Open in a separate windowFIG. 2.Snapshots at t = 0Δt (left) and t = 2500Δt (right) showing the separation of 15-bead strands (grey) from 10-bead strands (black) of DNA. For these simulations, K_stick = 55 and Y_stick = 0.7ΔL.Open in a separate windowFIG. 3.Snapshots at t = 0Δt and t = 2500Δt showing the separation of 13-bead strands (grey) from 10-bead strands (black) of DNA. For these simulations, K_stick = 55 and Y_stick = 0.7ΔL as in Fig. ​Fig.2.2. Note that one long polymer is trapped, as well as all of the shorter polymers.By tuning the attractive wall force parameters and fluid flow, the separated steady state can be realized. We first set the flow parameters that allow the larger chains to rotate freely at the center of the vortices while the shorter chains rotate closer to the wall. The trap strength, K_stick, and extension, X_stick, are changed until the shorter polymers do not leave the trap. The same parameters were used to separate 10-bead chains from 15-bead and 13-bead chains.As shown in Fig. ​Fig.2,2, we have been able to separate shorter 10-bead chains from longer 15-bead chains. In the steady state, 97% of the rotating polymers were long polymers averaged over twenty simulations initialized with different random starting conditions. For three simulations, one small polymer would intermittently leave the trap region. In two of these simulations, one long polymer became stably trapped in the steady state. In another simulation, one 15-bead chain was intermittently trapped. On average, the trapped polymers were 5% 15-bead chains and 95% 10-bead chains. Again, 97% of the rotating polymers were 15-bead chains.Simulations conducted with 10-bead and 13-bead chains also showed significant separation of the two sizes as can be seen in Fig. ​Fig.3.3. In the steady state, 30% of the trapped polymers are 13-bead chains and 70% are 10-bead chains, averaged over twenty different random initial starting conditions and 1000 polymer time steps. Only 14.8% of the shorter polymers were not trapped, leading to 85.2% of the freely rotating chains being 13-bead chains. This is therefore a viable test to detect the presence of these longer chains.We have also separated 20-bead chains from 10-bead chains with all of the shorter chains trapped and all of the longer chains freely rotating in the steady-state. These results do not change for twenty different random initial starting conditions and 1000 polymer time steps. None of the longer polymers intermittently enter the trap region nor do any of the shorter polymers intermittently escape.The separation is achieved by tuning the trapping force and flow rate. Strong flows will push all the DNA molecules into the trap. The final state is mixed, with both short and long strands trapped. For flows that are too weak, the short molecules are not sufficiently compressed by the flow and therefore do not enter the trap region. The end state is mixed, with all polymers freely rotating. Separation is achieved when the flow rate is tuned so that the short strands are compressed against the channel wall while the long polymers rotate near the center of the vortices. The trap strength must then be set sufficiently high enough to prevent the short strands from being pulled by the hydrodynamic drag force out of the trap.The mechanism of the separation depends on the differences in the steady state configurations of the polymers and chances of a polymer escaping the trap. As shown in Fig. ​Fig.4,4, both longer and shorter chains are pulled into the trap region by the flow. However, the longer chains have a larger chance of a bead escaping into a region of the flow where the fluid velocity is sufficient to pull the entire strand out of the trap. As shown in Ref. ¹¹, the trapping rate depends on diffusion in a polymer depleted region near the trap, in agreement with classical theory which neglects bead-wall interactions. In addition, the theory depends on the single bead diffusion rate and does not take into account the elastic force holding the beads together. Diffusion becomes as significant as convection in the polymer depleted region leading to dependence on the Peclet number. Since longer polymers have more beads; they have more chances of a single bead diffusing through this layer into the region where convection is again more important. Thus, they are pulled out of the trap at a faster rate than the shorter chains.Open in a separate windowFIG. 4.N, number of beads in the trap region, versus time for 15-bead DNA strands (solid line) and 10-bead DNA strands (dashed line). Here, ΔT = 10000Δt. The simulation parameters are the same as in Fig. ​Fig.22.In addition, longer chains have a second trap resulting from the microflow. As shown in Ref. ¹², DNA in counter-rotating vortices can tumble at the center of one vortex or be stretched between the centers of the two vortices. We have observed both these conformations for the longer polymer strand. They are a stable trajectory for the longer polymer that remains outside of the trapping region. As seen in Fig. ​Fig.4,4, few monomers of the longer chains enter the trap region once the steady state has been reached. However, the shorter polymer rotates at a larger radius than the longer polymer as seen in Fig. ​Fig.5.5. The shorter polymers therefore are pushed back into the trap while the longer strands rotate stably outside the trapping region.Open in a separate windowFIG. 5.Trajectories of 15-bead DNA (grey) and 10-bead DNA (black). The position of each monomer is plotted for 100 consecutive time steps. Note that the longer polymers rotate in the center of the channel while the shorter polymers rotate at the edges. Simulation parameters are the same as in Fig. ​Fig.22.This mechanism is similar to the one proposed for the separation of colloids by size in Refs. ³ and ⁴. In that experimental work, the smaller colloidal particles rotated at larger radii. This allowed the smaller beads to be pushed out of the vicinity of the vortices by the streaming flow, while the larger beads continued to circle. However, in our simulations, we have the additional mechanism of separation based on the increased chance of a longer polymer escaping the trap region. This mechanism is important for maintaining the separation. Long polymers initially in the trap region or which diffuse into the trap would not be able to escape without it.We expect that this technique could be used to detect the sizes of DNA fragments on the order of thousands of base pairs. It relies on the flexibility of the molecule and its interaction with the flow. Common lab procedures such as restriction enzyme digests for DNA fingerprinting can produce these long fragments. Current techniques such as gel electrophoresis require significant time to separate the long strands that move more slowly through the matrix. This effect could therefore be a good candidate for developing a microfluidic analysis that is significantly faster than traditional procedures. Our separation occurs in minutes rather than in hours as for gel electrophoresis.As pointed out in Ref. ², hydrodynamic effects have been shown to be important for microfluidic devices for separation. We have demonstrated, in simulation, a novel hydrodynamic mechanism for separating polymers by length. We hope that these promising calculations will inspire experiments to verify these results.