Introducing time delay in the evolution of new technology: the case study of nanotechnology

1 Introduction

The aim of this article is to show the effectiveness of mathematical models previously used in the biology [1, 2] and material physics literature [3–5] in predicting the evolution of new technology and, in our case, nanotechnology. To achieve this, we apply the model of sigmoidal curves based on two different criteria. The first criterion regards expenditure on basic research in nanotechnology, while the second relates to scientific publications worldwide. When combined, the results of the two studies give us a clear growth picture of nanotechnology. More importantly, it helps estimate the value of the “time delay” or “maturity period” associated with its development and its effect on stability. The notion of nanotechnology was essentially introduced by Nobel laureate Richard Feynman in a celebrated lecture in Caltech [6]. Its implications were outlined, among others, by Eric Drexler in the mid-1980s [7]. However, it was not until 2000, with the initiative of then US President Bill Clinton, that the field witnessed a rapid development worldwide. Such terms as “nanomaterials” [8, 9], “nanomechanics” [5], “nanophysics” [10, 11], “nanobiology” [12, 13], and “nanomedicine” [14, 15] were introduced, followed by related research and corresponding industrial and commercial activities. Thus, assessing the growth of nanotechnology over the past years and predicting its evolution in the future is a desirable task. In fact, a question that naturally arises is whether or not current mathematical models (previously developed for biological populations and material defects as mentioned above) can be utilized for nanotechnology forecasts.

This issue is addressed in the present article. We focus on the well-known logistic equation, which is extensively used in biology, and also utilized to model the evolution of new technology/innovation and economy/financial markets [16–19]. The logistic equation results in the well-known sigmoid behavior (S-curve), which is also verified here for the case of nanotechnology. The new element that we introduce here, however, is the concept of time-delay [20, 21] and its effect on the stability of the corresponding S-curve. The time-delay parameter is identified with an “acquaintance” or “familiarization” period with the new technology, which occurs before development begins.

The logistic equation has been used to model the growth of various technologies and innovations, as already mentioned. The purpose of this contribution is to provide an initial exploration of its applicability in predicting the evolution of the emerging field of nanotechnology, as well as to introduce the time delay in that process.

We begin by listing the logistic equation [2] as follows:

and its solution

where N denotes the quantity of interest, N₀ is its initial value and the remaining of the coefficients are positive constants in appropriate units. The quantity N may denote biological species (e.g., bacteria, animals, humans) or material entities (e.g., molecules, voids, dislocations) growing at a rate r, which is limited by K. There are two equilibrium points of Eq. (1), one unstable (N=0) and one stable (N=K). It can be easily seen that if the initial value of N(0)=N₀ is such that N₀>K, then N(t) is decreasing monotonically approaching K; if K>N₀>K/2, then N(t) increases monotonically approaching K; and finally, if K/2>N₀, N(t) exhibits sigmoidal behavior [22, 23].

2 Introducing time delay in the evolution of nanotechnology

Basic research is not aimed at solving practical problems. However, some results of basic research are often used in applied research meant to solve specific problems. Either through basic or applied research, the developed innovations are usually promoted for industrial production by major manufacturers. However, for this to occur, manufacturers should be convinced to produce and sell the product. This procedure, which may be shorter or longer, is analogous to the time of pregnancy or incubation in biological systems [24]. It is the time that elapses from the moment of conception of an idea until its industrial application, i.e., the time delay that is necessary to ultimately promote new products to the market.

Considering this time delay or time lag (τ), which is needed for the maturation of an original idea (analogous to the “maturation” time in biological systems), the modified logistic equation can be expressed in mathematical form [21], depending on the way by which the time delay parameter is introduced. For example, one may simply assume that the per capita growth rate of change might depend on the time variable τ, as shown in Eq. (3) below. Of course, almost no innovation automatically passes into production. There is always a time lag required to complete the necessary procedures.

Let (1/r) be the characteristic return time. If (1/r) is large compared with the time delay τ, then the resulting behavior is asymptotically stable. However, if this time is reduced to a critical value, then oscillations occur around the equilibrium point [25, 26]. Roughly speaking, this may be interpreted as follows: This time delay in the adoption of the innovation involves risks that may arise from the time lag associated with the product’s acceptance. As the rate of acceptance grows, so does the probability to develop a better innovation that can replace it; otherwise, conditions of dissatisfaction with the product at hand may appear, forcing a company executive to shift investments to other sectors. Delay differential equations have extensively been studied in the field of mathematical biology [1, 21]. The purpose here is not to elaborate on a detailed analysis of the implications of delay equations of the logistic type in modeling technology/innovation aspects, but to illustrate the potential of such considerations in examining the stability of its saturation state.

To this end we re-write the logistic equation by considering its time delay extension in the form given by:

where r and K are, as before, positive constants, whereas the new time variable parameter τ denotes the time delay or time lag. In biological or physical systems, the time delay parameter denotes the period required for two individual species or entities to interact before contributing to the reduction of the slope of the growth curve, forcing it to eventually attain an asymptotic value in the case of τ=0. For this case, the solution for N(t) assumes the aforementioned exponential form given by Eq. (2). As already mentioned, the equilibrium solution N=0 is unstable, leading to exponential growth in small times, while the solution N=K is stable (for the case τ=0). Thus, we examine the stability of the solution N=K when small fluctuations occur for the case τ≠0. To this end, we rescale the variables entering in Eq. (3) by setting N(t)→N(t)/K, t→rt and τ→rτ, in order to deal with non-dimensional quantities. Thus, Eq. (3) now becomes:

Next, we consider small fluctuations (around the equilibrium state N=1), i.e., we search for solutions of Eq. (4) of the form given by:

where n represents a fluctuation of small magnitude. It follows [2] from Eq. (5) that

i.e., n(t) has an exponential dependence on t with the eigenvalue λ satisfying a transcendental equation with T>0. In general, λ is a complex number whose sign of the real part determines the stability of N=1 (for Re[λ]<0, N is stable; for Re[λ]>0, N is unstable). It is well-known [2] that instabilities with ascending oscillations of period

are possible. Thus, in dimensionless variables with rτ=(π/2)+ε=1.6 ε=0.0029, the period is given by:

This value compares well with numerical results giving a value 4.03τ for the period of the numerically computing limit cycle. When rτ=2.1, we have ε=0.5, and the corresponding period of the oscillations is 5.26τ, which compares well to the value of 4.54τ obtained numerically. Motivated by these analytical results, which are also verified by corresponding simulations, we return to Eq. (3) and choose various values of τ by first fixing the values of r and K from the fitting of experimental data for the case τ=0, r=0.479 and K=4.3×10⁹, as shown below.

3 First criterion: expenditures in basic nanotechnology research

In Figure 1, we present the expenditures of major investors on R&D in nanotechnology from 1997 to 2009. [More detailed data are included in the Appendix.] The fitting of these data by an S-curve is shown in Figure 2, which indicates a saturation value in the year 2015.

Figure 1

Government expenditure on R&D in nanotechnology.

Figure 2

Extraction of the S-curve for the evolution of spending in R&D in nanotechnology, based on the data extracted from Figure 1 and also given in the Appendix.

It should be pointed out that all of the above apply to basic research in nanotechnology. Based on the fact that a saturation of the field of nanotechnology as a whole (i.e., for basic research, applied research, and industrial production) is not anticipated in 2015, one may question the validity of the model beyond that year. This may suggest that a more elaborate evolutionary model must be sought. Such models have been developed for technology transfer by using multiple S-curves, i.e., using the saturation value for a first growth period segment as an initial value for a second (and third) growth period [17, 18]. Such transition may not be continuous if a mechanism or condition is developed, resulting in an unstable saturation value for the first period. This instability may be caused by another developing technology. Hence, a system of evolution equations should be used instead of Eq. (1) in order to model the competition of the two technologies under consideration. It may also be caused by a catastrophic event such as war or a collapse in the stock market. Such instability could be modeled by considering a time delay in Eq. (1); for example, using Eq. (3) or its dimensionless counterpart, Eq. (4).

The occurrence of a time delay in the development of new technology may result in abandoning it before going into industrial produce. The need to introduce time delays in new technology development is due to the following factors:

• time needed for understanding its importance by the academic and industrial communities;

• time needed for directing young scientists and engineers in this direction (including their training period);

• time needed for achieving initial promising results and establishing benchmark examples, indicating the necessity for basic R&D funding; and

• time needed for transition from the pilot scales to industrial production.

There is a fundamental difference in the use of time delay in the development of biological systems compared with the case of new technologies. In biological systems, the time lag is usually well-known: it is either deduced by the physics of the phenomenon at hand or by related experimental procedures. However, this is not the case for nanotechnology. Here, the time lag cannot be predicted in advance because there are no available measurements and experimental data to deduce it. Corresponding estimates may thus be deduced by resorting to data from other related indicators. Such estimates will be provided in the next section, where the second criterion for estimating the nanotechnology S-curve is considered. Before that, however, we give indicative values for the time delay to examine the behavior of the system as described by the model of the sigmoids.

Forτ=2.2 years (τ*=rτ=1.05)

The evolution of N (where N denotes the accumulated expenditures for basic R&D in nanotechnology) from the equilibrium state and the corresponding phase diagram are shown in Figure 3.

Figure 3

Time delay τ=2.2 years.

We observe that a time delay of just over 2 years generates oscillations for the evolution of nanotechnology, posing no threat to its course due to the fact that the oscillations are “damped” quickly. Thus, the risk of a catastrophic incident for its development is limited, and the probability of a system crash is small.

Forτ=3 years (τ*=rτ=1.45)

The evolution of N, as inferred by considering its equilibrium state and the corresponding phase diagram, are shown in Figure 4. In this case, we observe that the oscillations developed have larger amplitude and last significantly longer. This has the effect of increasing the probability that a catastrophic event will occur, which may be detrimental for the system.

Figure 4

Time delay τ=3 years and the corresponding phase diagram.

Forτ=3.75 years (τ*=rτ=1.8)

In this case we observe large amplitude oscillations of a rather long duration. The evolution of N and the corresponding phase diagram are shown in Figure 5.

Figure 5

Delay of 3.75 years causing very large amplitude oscillations.

4 Second criterion: scientific publications in nanotechnology

In order to obtain an estimate of the time delay associated with modeling nanotechnology evolution, we may utilize the performance indicator connected with the number of published scientific articles in the field worldwide. This is indicative of the academic interest in the field. This number is actually a measure of the technology’s advancement and the production of initial results that may eventually lead to its transition to industrial production. It may be argued that this is directly related to the amount of funding allocated for research in the area. The more funding spent in this direction, the larger is the number of researchers entering the field. For the case of nanotechnology, the number of published articles during the period 1996 to 2009 is shown schematically in Figure 6 [27, 28].

Figure 6

Schematic representation of the number of scientific publications in major countries.

The data shown in Figure 6 are then used for constructing an S-curve in order to predict, among other things, the saturation time associated with the evolution of nanotechnology. This is shown in Figure 7, from which it is deduced that the saturation time of research in the field of nanotechnology is estimated to be around 2021. It should be noted that this prediction concerns work done by scientists who are basically supported by government funding, since the transfer to applied research or to a production level has not yet been established.

Figure 7

Forecast for the evolution of nanotechnology using scientific publications as a measure.

It is noted from Figure 2 that the predicted saturation time is estimated to be around 2015. This may be expected, since in the previous case, only government expenditures on R&D in nanotechnology are taken into account, while in this case, the scientific publications are not financed exclusively by government funds.

The saturation time predicted for nanotechnology publications can be attributed to the following factors:

• the limited number of scientists with nanotechnology expertise;

• the limited funds supporting specialized research; and

• the exodus of a large portion of scientists who worked in R&D to the respective manufacturing industries once the new technology has shifted into the implementation stage. In their new industries, the scientists work on improvements of already available initial results, further elaborating on the design and optimization of production lines.

• a percentage of scientists establish their own spin-off companies, resulting to a scientific staff reduction, and their replacement may contribute to the observed saturation state.

5 Estimating time delays

A most significant information, extracted from Figure 7, is provided by the initial part of the sigmoidal curves, where a plateau corresponding to the first few years of the development of nanotechnology is seen. During this time, both investments on basic research (R&D) and the number of scientific publications remain approximately constant. However, by comparing the temporal development of the two curves, as shown in Figure 8, a time difference emerges. This time interval may be identified with the delay time necessary for the acceptance of the new technology. This is so because after this time, initial results have been produced, government and industrial funds are allocated, and more scientists (with more published articles) are encouraged to enter into the field. However, the transition to the new technology is not automatic. Initially, the organization of a new infrastructure is required, and to achieve this, we must convince the leaders of the academic community and policy makers. Then, young scientists must acquire relevant expertise and finally produce initial results. All these require time, which may result in a non-vanishing “time delay” for the system.

Figure 8

Use of the same system of axes for the two sigmoidal curves of Figures 2 and 7, in order to compare the time difference in their evolution. (The units on the vertical axis are total expenditures or total publications respectively.)

The comparison of the initial plateaus of the aforementioned two diagrams, gives a time delay of no more than 3 years. This time delay compares very well with the theoretically calculated time from the sigmoidal model discussed earlier. Further “delay” would increase the possibility for the new technology to be abandoned, thus forcing the industry to turn to technologies that are more readily exploited.

The value of the estimated delay period of about 3 years according to the diagram is very close to the time calculated in the case where the nanotechnology funding is taken as a measure. It also corresponds to the estimate obtained from the second criterion analyzed earlier.

6 Conclusions

In this article, we show that nanotechnology forecasts based on S-curves may yield interesting results. In particular, the introduction of the time delay provides a means for studying factors that may introduce instabilities to the system. We may obtain a reasonable estimate for the time interval, over which such instability may become catastrophic. We can also extract indicative information regarding trends for the evolution of nanotechnology, which can be used in developing government/private sector policies.

A first conclusion is that we are now in the process of transferring from basic research to the applied one. It is confirmed that nanotechnology progress is rapid, but it cannot be stated that this progress has reached the sufficient level to initiate mass production. This may happen when the applied research activity is transferred to industrial production. This is expected to occur after 2019, since the saturation of applied research is predicted to take place around 2023.

Another conclusion concerns the estimation of the time delay associated with the development of nanotechnology, which can be used for developing related policies in both the government and the private sector. It is determined that a time delay of the order of 3 years can be absorbed by the system without detrimental effects. Longer time intervals may lead to catastrophic instabilities, through large amplitude oscillations, which may cause risks and destruction. Such time delay is expected to appear in various nanotechnology sectors, e.g., nanomaterials, nanobiology, nanoelectronics, etc. This may be the next research task, and must be examined with analogous techniques such as by introducing time delays for each individual sector. Estimation of such “time delays” may enable potential users of an emerging technology to decide whether or not to participate in the new competitive environment.

It is finally mentioned that the potential of acquiring such knowledge is essential in revitalizing industrial sectors in countries that are no longer competitive due to strong international competition in other fields. For obvious reasons, a new industry may be relatively under-capitalized, but it should make an effort to devote sufficient resources to R&D and innovation. It should also be mentioned that once a new technology goes through the implementation stage, a large portion of scientists who worked for its development will contribute to the design and optimization of production lines. It is almost certain that some of these scientists will establish their own “spin-off” companies.